TPTP Problem File: ITP228^3.p
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%------------------------------------------------------------------------------
% File : ITP228^3 : TPTP v8.2.0. Released v8.1.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_Insert 00400_024466
% Version : [Des22] axioms.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source : [Des22]
% Names : 0066_VEBT_Insert_00400_024466 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 10897 (4909 unt;1285 typ; 0 def)
% Number of atoms : 28331 (11619 equ; 0 cnn)
% Maximal formula atoms : 71 ( 2 avg)
% Number of connectives : 110018 (2816 ~; 486 |;1953 &;93278 @)
% ( 0 <=>;11485 =>; 0 <=; 0 <~>)
% Maximal formula depth : 39 ( 6 avg)
% Number of types : 160 ( 159 usr)
% Number of type conns : 5239 (5239 >; 0 *; 0 +; 0 <<)
% Number of symbols : 1129 (1126 usr; 85 con; 0-5 aty)
% Number of variables : 25689 (2354 ^;22657 !; 678 ?;25689 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% from the van Emde Boas Trees session in the Archive of Formal
% proofs -
% www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 2022-02-17 19:45:22.311
%------------------------------------------------------------------------------
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
list_P5707943133018811711nt_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
set_Pr958786334691620121nt_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
produc4411394909380815293omplex: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Complex__Ocomplex_J,type,
produc6979889472282505531omplex: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Real__Oreal_J,type,
produc8892588492097263291x_real: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Complex__Ocomplex_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Int__Oint_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
product_prod_nat_num: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Complex__Ocomplex_J_J,type,
set_list_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
set_set_complex: $tType ).
thf(ty_n_t__Option__Ooption_It__VEBT____Definitions__OVEBT_J,type,
option_VEBT_VEBT: $tType ).
thf(ty_n_t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
option_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
list_VEBT_VEBT: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
set_list_int: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
list_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
set_VEBT_VEBT: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
set_set_int: $tType ).
thf(ty_n_t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
set_Code_integer: $tType ).
thf(ty_n_t__Option__Ooption_It__Complex__Ocomplex_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
set_Product_unit: $tType ).
thf(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
list_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $tType ).
thf(ty_n_t__Option__Ooption_It__Real__Oreal_J,type,
option_real: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
filter_real: $tType ).
thf(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
option_num: $tType ).
thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
option_nat: $tType ).
thf(ty_n_t__Option__Ooption_It__Int__Oint_J,type,
option_int: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
filter_nat: $tType ).
thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_real: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Num__Onum_J,type,
list_num: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__VEBT____Definitions__OVEBT,type,
vEBT_VEBT: $tType ).
thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
set_rat: $tType ).
thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
set_num: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Code____Numeral__Ointeger,type,
code_integer: $tType ).
thf(ty_n_t__Extended____Nat__Oenat,type,
extended_enat: $tType ).
thf(ty_n_t__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Rat__Orat,type,
rat: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (1126)
thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
archim7802044766580827645g_real: real > int ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
archim3151403230148437115or_rat: rat > int ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
archim6058952711729229775r_real: real > int ).
thf(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
archim7778729529865785530nd_rat: rat > int ).
thf(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
archim8280529875227126926d_real: real > int ).
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial_001t__Code____Numeral__Ointeger,type,
gbinom8545251970709558553nteger: code_integer > nat > code_integer ).
thf(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
gbinomial_complex: complex > nat > complex ).
thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
gbinomial_int: int > nat > int ).
thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
gbinomial_nat: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
gbinomial_rat: rat > nat > rat ).
thf(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
gbinomial_real: real > nat > real ).
thf(sy_c_Bit__Operations_Oand__int__rel,type,
bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(sy_c_Bit__Operations_Oand__not__num,type,
bit_and_not_num: num > num > option_num ).
thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(sy_c_Bit__Operations_Oconcat__bit,type,
bit_concat_bit: nat > int > int > int ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
bit_ri7919022796975470100ot_int: int > int ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
bit_ri631733984087533419it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
bit_se725231765392027082nd_int: int > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
bit_se727722235901077358nd_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
bit_se8568078237143864401it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
bit_se8570568707652914677it_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Ointeger,type,
bit_se1345352211410354436nteger: nat > code_integer > code_integer ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
bit_se2159334234014336723it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
bit_se2161824704523386999it_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
bit_se2000444600071755411sk_int: nat > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
bit_se2002935070580805687sk_nat: nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Code____Numeral__Ointeger,type,
bit_se1080825931792720795nteger: code_integer > code_integer > code_integer ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
bit_se1409905431419307370or_int: int > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
bit_se1412395901928357646or_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Int__Oint,type,
bit_se545348938243370406it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
bit_se547839408752420682it_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Code____Numeral__Ointeger,type,
bit_se2793503036327961859nteger: nat > code_integer > code_integer ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
bit_se7879613467334960850it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
bit_se7882103937844011126it_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
bit_se2923211474154528505it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
bit_se2925701944663578781it_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Code____Numeral__Ointeger,type,
bit_se8260200283734997820nteger: nat > code_integer > code_integer ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
bit_se4203085406695923979it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
bit_se4205575877204974255it_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,type,
bit_se6526347334894502574or_int: int > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
bit_se6528837805403552850or_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
bit_se1146084159140164899it_int: int > nat > $o ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
bit_se1148574629649215175it_nat: nat > nat > $o ).
thf(sy_c_Bit__Operations_Otake__bit__num,type,
bit_take_bit_num: nat > num > option_num ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
bit_un7362597486090784418nd_num: num > num > option_num ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
bit_un4731106466462545111um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
bit_un2480387367778600638or_num: num > num > option_num ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(sy_c_Code__Numeral_Obit__cut__integer,type,
code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).
thf(sy_c_Code__Numeral_Odivmod__abs,type,
code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger ).
thf(sy_c_Code__Numeral_Odivmod__integer,type,
code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger ).
thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
code_int_of_integer: code_integer > int ).
thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
code_integer_of_int: int > code_integer ).
thf(sy_c_Code__Numeral_Ointeger__of__num,type,
code_integer_of_num: num > code_integer ).
thf(sy_c_Code__Numeral_Onat__of__integer,type,
code_nat_of_integer: code_integer > nat ).
thf(sy_c_Code__Numeral_Onum__of__integer,type,
code_num_of_integer: code_integer > num ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
comple7806235888213564991et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Int__Oint,type,
complete_Sup_Sup_int: set_int > int ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
comple1385675409528146559p_real: set_real > real ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complex_OArg,type,
arg: complex > real ).
thf(sy_c_Complex_Ocis,type,
cis: real > complex ).
thf(sy_c_Complex_Ocnj,type,
cnj: complex > complex ).
thf(sy_c_Complex_Ocomplex_OComplex,type,
complex2: real > real > complex ).
thf(sy_c_Complex_Ocomplex_OIm,type,
im: complex > real ).
thf(sy_c_Complex_Ocomplex_ORe,type,
re: complex > real ).
thf(sy_c_Complex_Ocsqrt,type,
csqrt: complex > complex ).
thf(sy_c_Complex_Oimaginary__unit,type,
imaginary_unit: complex ).
thf(sy_c_Deriv_Odifferentiable_001t__Real__Oreal_001t__Real__Oreal,type,
differ6690327859849518006l_real: ( real > real ) > filter_real > $o ).
thf(sy_c_Deriv_Ohas__derivative_001t__Real__Oreal_001t__Real__Oreal,type,
has_de1759254742604945161l_real: ( real > real ) > ( real > real ) > filter_real > $o ).
thf(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).
thf(sy_c_Divides_Oadjust__div,type,
adjust_div: product_prod_int_int > int ).
thf(sy_c_Divides_Odivmod__nat,type,
divmod_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Divides_Oeucl__rel__int,type,
eucl_rel_int: int > int > product_prod_int_int > $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Int__Oint,type,
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thf(sy_c_List_Olist__update_001t__Int__Oint,type,
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thf(sy_c_List_Olist__update_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_List_Oupt,type,
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thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel_001t__Nat__Onat,type,
set_fo3699595496184130361el_nat: produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
set_or1266510415728281911st_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
set_or7049704709247886629st_num: num > num > set_num ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
set_or633870826150836451st_rat: rat > rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
set_or1222579329274155063t_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Int__Oint_J,type,
set_or370866239135849197et_int: set_int > set_int > set_set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
set_ord_atLeast_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
set_ord_atLeast_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
set_ord_atMost_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Num__Onum,type,
set_ord_atMost_num: num > set_num ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Rat__Orat,type,
set_ord_atMost_rat: rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
set_ord_atMost_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Int__Oint_J,type,
set_or58775011639299419et_int: set_int > set_set_int ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
set_or6656581121297822940st_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
set_or5832277885323065728an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
set_or1633881224788618240n_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
set_or5849166863359141190n_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
set_ord_lessThan_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
set_ord_lessThan_num: num > set_num ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
set_ord_lessThan_rat: rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
set_or5984915006950818249n_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or890127255671739683et_nat: set_nat > set_set_nat ).
thf(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Int__Oint,type,
topolo4899668324122417113eq_int: ( nat > int ) > $o ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
topolo4902158794631467389eq_nat: ( nat > nat ) > $o ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Num__Onum,type,
topolo1459490580787246023eq_num: ( nat > num ) > $o ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Rat__Orat,type,
topolo4267028734544971653eq_rat: ( nat > rat ) > $o ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
topolo6980174941875973593q_real: ( nat > real ) > $o ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Int__Oint_J,type,
topolo3100542954746470799et_int: ( nat > set_int ) > $o ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
topolo2177554685111907308n_real: real > set_real > filter_real ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
topolo2815343760600316023s_real: real > filter_real ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
topolo4055970368930404560y_real: ( nat > real ) > $o ).
thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
topolo896644834953643431omplex: filter6041513312241820739omplex ).
thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
topolo1511823702728130853y_real: filter2146258269922977983l_real ).
thf(sy_c_Transcendental_Oarccos,type,
arccos: real > real ).
thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
arcosh_real: real > real ).
thf(sy_c_Transcendental_Oarcsin,type,
arcsin: real > real ).
thf(sy_c_Transcendental_Oarctan,type,
arctan: real > real ).
thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
arsinh_real: real > real ).
thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
artanh_real: real > real ).
thf(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
cos_complex: complex > complex ).
thf(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
cos_real: real > real ).
thf(sy_c_Transcendental_Ocos__coeff,type,
cos_coeff: nat > real ).
thf(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
cosh_real: real > real ).
thf(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
cot_real: real > real ).
thf(sy_c_Transcendental_Odiffs_001t__Code____Numeral__Ointeger,type,
diffs_Code_integer: ( nat > code_integer ) > nat > code_integer ).
thf(sy_c_Transcendental_Odiffs_001t__Complex__Ocomplex,type,
diffs_complex: ( nat > complex ) > nat > complex ).
thf(sy_c_Transcendental_Odiffs_001t__Int__Oint,type,
diffs_int: ( nat > int ) > nat > int ).
thf(sy_c_Transcendental_Odiffs_001t__Rat__Orat,type,
diffs_rat: ( nat > rat ) > nat > rat ).
thf(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
diffs_real: ( nat > real ) > nat > real ).
thf(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
exp_complex: complex > complex ).
thf(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
exp_real: real > real ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
thf(sy_c_Transcendental_Olog,type,
log: real > real > real ).
thf(sy_c_Transcendental_Opi,type,
pi: real ).
thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
powr_real: real > real > real ).
thf(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
sin_complex: complex > complex ).
thf(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
sin_real: real > real ).
thf(sy_c_Transcendental_Osin__coeff,type,
sin_coeff: nat > real ).
thf(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
sinh_real: real > real ).
thf(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
tan_complex: complex > complex ).
thf(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
tan_real: real > real ).
thf(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
tanh_complex: complex > complex ).
thf(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
tanh_real: real > real ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Nat__Onat,type,
transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
vEBT_Leaf: $o > $o > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
vEBT_size_VEBT: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
vEBT_VEBT_high: nat > nat > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
vEBT_VEBT_low: nat > nat > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_Oset__vebt,type,
vEBT_set_vebt: vEBT_VEBT > set_nat ).
thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
vEBT_vebt_buildup: nat > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
vEBT_v4011308405150292612up_rel: nat > nat > $o ).
thf(sy_c_VEBT__Insert_Ovebt__insert,type,
vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
thf(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
vEBT_VEBT_minNull: vEBT_VEBT > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).
thf(sy_c_VEBT__Member_Ovebt__member,type,
vEBT_vebt_member: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
accp_nat: ( nat > nat > $o ) > nat > $o ).
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,
accp_P6019419558468335806at_nat: ( produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o ) > produc4471711990508489141at_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).
thf(sy_c_Wellfounded_Ofinite__psubset_001t__Complex__Ocomplex,type,
finite8643634255014194347omplex: set_Pr6308028481084910985omplex ).
thf(sy_c_Wellfounded_Ofinite__psubset_001t__Int__Oint,type,
finite_psubset_int: set_Pr2522554150109002629et_int ).
thf(sy_c_Wellfounded_Ofinite__psubset_001t__Nat__Onat,type,
finite_psubset_nat: set_Pr5488025237498180813et_nat ).
thf(sy_c_Wellfounded_Ofinite__psubset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite469560695537375940at_nat: set_Pr4329608150637261639at_nat ).
thf(sy_c_Wellfounded_Omax__ext_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
max_ex8135407076693332796at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).
thf(sy_c_Wellfounded_Omeasure_001t__Int__Oint,type,
measure_int: ( int > nat ) > set_Pr958786334691620121nt_int ).
thf(sy_c_Wellfounded_Omin__ext_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
min_ex6901939911449802026at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).
thf(sy_c_Wellfounded_Opred__nat,type,
pred_nat: set_Pr1261947904930325089at_nat ).
thf(sy_c_Wellfounded_Owf_001t__Int__Oint,type,
wf_int: set_Pr958786334691620121nt_int > $o ).
thf(sy_c_fChoice_001t__Real__Oreal,type,
fChoice_real: ( real > $o ) > real ).
thf(sy_c_member_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
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thf(sy_c_member_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
member8845023287901829240e_term: ( int > option6357759511663192854e_term ) > set_in3461395444621081367e_term > $o ).
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member4242434998011752849e_term: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > set_Pr7604974323444597168e_term > $o ).
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member3222579708246209666e_term: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > set_Pr3642885161833720865e_term > $o ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
member_list_int: list_int > set_list_int > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Num__Onum,type,
member_num: num > set_num > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
member3068662437193594005nteger: produc8763457246119570046nteger > set_Pr8056137968301705908nteger > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
member7034335876925520548nt_int: produc7773217078559923341nt_int > set_Pr1872883991513573699nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
member4164122664394876845nteger: produc1908205239877642774nteger > set_Pr1281608226676607948nteger > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
member7618704894036264090nt_int: produc2285326912895808259nt_int > set_Pr9222295170931077689nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
member157494554546826820nteger: produc8923325533196201883nteger > set_Pr4811707699266497531nteger > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
member5793383173714906214omplex: produc4411394909380815293omplex > set_Pr5085853215250843933omplex > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Int__Oint_J,type,
member595073364599660772ex_int: produc6845221339535797307ex_int > set_Pr2254670189886740123ex_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Nat__Onat_J,type,
member4772924384108857480ex_nat: produc1799700322190218207ex_nat > set_Pr4744753334818466879ex_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Real__Oreal_J,type,
member47443559803733732x_real: produc8892588492097263291x_real > set_Pr1133549439701694107x_real > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__VEBT____Definitions__OVEBT_J,type,
member1978952105866562066T_VEBT: produc7151440242714718331T_VEBT > set_Pr4085867452638698417T_VEBT > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Complex__Ocomplex_J,type,
member8811922270175639012omplex: produc5838698208256999739omplex > set_Pr1846070511934368667omplex > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
member216504246829706758nt_nat: product_prod_int_nat > set_Pr3448869479623346877nt_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J,type,
member2744130022092475746t_real: produc679980390762269497t_real > set_Pr3538720872664544793t_real > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
member2056185340421749780T_VEBT: produc1531783533982839933T_VEBT > set_Pr8044002425091019955T_VEBT > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Complex__Ocomplex_J,type,
member7358116576843751780omplex: produc6979889472282505531omplex > set_Pr6591433984475009307omplex > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J,type,
member1627681773268152802al_int: produc8786904178792722361al_int > set_Pr1019928272762051225al_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
member5805532792777349510al_nat: produc3741383161447143261al_nat > set_Pr3510011417693777981al_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
member7849222048561428706l_real: produc2422161461964618553l_real > set_Pr6218003697084177305l_real > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
member7262085504369356948T_VEBT: produc3757001726724277373T_VEBT > set_Pr6019664923565264691T_VEBT > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Complex__Ocomplex_J_Mt__Set__Oset_It__Complex__Ocomplex_J_J,type,
member351165363924911826omplex: produc8064648209034914857omplex > set_Pr6308028481084910985omplex > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
member2572552093476627150et_int: produc2115011035271226405et_int > set_Pr2522554150109002629et_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member8757157785044589968at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Complex__Ocomplex_J,type,
member3207599676835851048omplex: produc8380087813684007313omplex > set_Pr216944050393469383omplex > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
member5419026705395827622BT_int: produc4894624898956917775BT_int > set_Pr5066593544530342725BT_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
member373505688050248522BT_nat: produc9072475918466114483BT_nat > set_Pr7556676689462069481BT_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
member8675245146396747942T_real: produc5170161368751668367T_real > set_Pr7765410600122031685T_real > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
member568628332442017744T_VEBT: produc8243902056947475879T_VEBT > set_Pr6192946355708809607T_VEBT > $o ).
thf(sy_c_member_001t__Rat__Orat,type,
member_rat: rat > set_rat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
member_set_int: set_int > set_set_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
thf(sy_v_deg____,type,
deg: nat ).
thf(sy_v_m____,type,
m: nat ).
thf(sy_v_ma____,type,
ma: nat ).
thf(sy_v_mi____,type,
mi: nat ).
thf(sy_v_na____,type,
na: nat ).
thf(sy_v_summary____,type,
summary: vEBT_VEBT ).
thf(sy_v_treeList____,type,
treeList: list_VEBT_VEBT ).
thf(sy_v_xa____,type,
xa: nat ).
thf(sy_v_ya____,type,
ya: nat ).
% Relevant facts (9575)
thf(fact_0_True,axiom,
xa = mi ).
% True
thf(fact_1_False,axiom,
~ ( ( ya = mi )
| ( ya = ma ) ) ).
% False
thf(fact_2__C0_C,axiom,
( ( ya != mi )
& ( ya != ma ) ) ).
% "0"
thf(fact_3__C5_Oprems_C_I3_J,axiom,
vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa ).
% "5.prems"(3)
thf(fact_4__C5_Ohyps_C_I7_J,axiom,
ord_less_eq_nat @ mi @ ma ).
% "5.hyps"(7)
thf(fact_5__C5_Ohyps_C_I6_J,axiom,
( ( mi = ma )
=> ! [X: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ treeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X @ X_1 ) ) ) ).
% "5.hyps"(6)
thf(fact_6__C001_C,axiom,
vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ deg ).
% "001"
thf(fact_7_not__min__Null__member,axiom,
! [T: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ T )
=> ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_12 ) ) ).
% not_min_Null_member
thf(fact_8_VEBT_Oinject_I1_J,axiom,
! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
= ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 )
& ( X13 = Y13 )
& ( X14 = Y14 ) ) ) ).
% VEBT.inject(1)
thf(fact_9_option_Oinject,axiom,
! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
( ( ( some_P7363390416028606310at_nat @ X2 )
= ( some_P7363390416028606310at_nat @ Y2 ) )
= ( X2 = Y2 ) ) ).
% option.inject
thf(fact_10_option_Oinject,axiom,
! [X2: num,Y2: num] :
( ( ( some_num @ X2 )
= ( some_num @ Y2 ) )
= ( X2 = Y2 ) ) ).
% option.inject
thf(fact_11_prod_Oinject,axiom,
! [X1: int,X2: int,Y1: int,Y2: int] :
( ( ( product_Pair_int_int @ X1 @ X2 )
= ( product_Pair_int_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_12_prod_Oinject,axiom,
! [X1: code_integer > option6357759511663192854e_term,X2: produc8923325533196201883nteger,Y1: code_integer > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
( ( ( produc6137756002093451184nteger @ X1 @ X2 )
= ( produc6137756002093451184nteger @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_13_prod_Oinject,axiom,
! [X1: produc6241069584506657477e_term > option6357759511663192854e_term,X2: produc8923325533196201883nteger,Y1: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
( ( ( produc8603105652947943368nteger @ X1 @ X2 )
= ( produc8603105652947943368nteger @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_14_prod_Oinject,axiom,
! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X2: product_prod_int_int,Y1: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] :
( ( ( produc5700946648718959541nt_int @ X1 @ X2 )
= ( produc5700946648718959541nt_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_15_prod_Oinject,axiom,
! [X1: int > option6357759511663192854e_term,X2: product_prod_int_int,Y1: int > option6357759511663192854e_term,Y2: product_prod_int_int] :
( ( ( produc4305682042979456191nt_int @ X1 @ X2 )
= ( produc4305682042979456191nt_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_16_old_Oprod_Oinject,axiom,
! [A: int,B: int,A2: int,B2: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_17_old_Oprod_Oinject,axiom,
! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,A2: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
( ( ( produc6137756002093451184nteger @ A @ B )
= ( produc6137756002093451184nteger @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_18_old_Oprod_Oinject,axiom,
! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A2: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
( ( ( produc8603105652947943368nteger @ A @ B )
= ( produc8603105652947943368nteger @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_19_old_Oprod_Oinject,axiom,
! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A2: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
( ( ( produc5700946648718959541nt_int @ A @ B )
= ( produc5700946648718959541nt_int @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_20_old_Oprod_Oinject,axiom,
! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A2: int > option6357759511663192854e_term,B2: product_prod_int_int] :
( ( ( produc4305682042979456191nt_int @ A @ B )
= ( produc4305682042979456191nt_int @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_21_prod__decode__aux_Ocases,axiom,
! [X3: product_prod_nat_nat] :
~ ! [K: nat,M: nat] :
( X3
!= ( product_Pair_nat_nat @ K @ M ) ) ).
% prod_decode_aux.cases
thf(fact_22__C5_Ohyps_C_I1_J,axiom,
vEBT_invar_vebt @ summary @ m ).
% "5.hyps"(1)
thf(fact_23_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_int_int] :
~ ! [A3: int,B3: int] :
( Y
!= ( product_Pair_int_int @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_24_old_Oprod_Oexhaust,axiom,
! [Y: produc8763457246119570046nteger] :
~ ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
( Y
!= ( produc6137756002093451184nteger @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_25_old_Oprod_Oexhaust,axiom,
! [Y: produc1908205239877642774nteger] :
~ ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
( Y
!= ( produc8603105652947943368nteger @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_26_old_Oprod_Oexhaust,axiom,
! [Y: produc2285326912895808259nt_int] :
~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
( Y
!= ( produc5700946648718959541nt_int @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_27_old_Oprod_Oexhaust,axiom,
! [Y: produc7773217078559923341nt_int] :
~ ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
( Y
!= ( produc4305682042979456191nt_int @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_28_surj__pair,axiom,
! [P: product_prod_int_int] :
? [X4: int,Y3: int] :
( P
= ( product_Pair_int_int @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_29_surj__pair,axiom,
! [P: produc8763457246119570046nteger] :
? [X4: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( P
= ( produc6137756002093451184nteger @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_30_surj__pair,axiom,
! [P: produc1908205239877642774nteger] :
? [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( P
= ( produc8603105652947943368nteger @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_31_surj__pair,axiom,
! [P: produc2285326912895808259nt_int] :
? [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
( P
= ( produc5700946648718959541nt_int @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_32_surj__pair,axiom,
! [P: produc7773217078559923341nt_int] :
? [X4: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
( P
= ( produc4305682042979456191nt_int @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_33_prod__cases,axiom,
! [P2: product_prod_int_int > $o,P: product_prod_int_int] :
( ! [A3: int,B3: int] : ( P2 @ ( product_Pair_int_int @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_34_prod__cases,axiom,
! [P2: produc8763457246119570046nteger > $o,P: produc8763457246119570046nteger] :
( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] : ( P2 @ ( produc6137756002093451184nteger @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_35_prod__cases,axiom,
! [P2: produc1908205239877642774nteger > $o,P: produc1908205239877642774nteger] :
( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] : ( P2 @ ( produc8603105652947943368nteger @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_36_prod__cases,axiom,
! [P2: produc2285326912895808259nt_int > $o,P: produc2285326912895808259nt_int] :
( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] : ( P2 @ ( produc5700946648718959541nt_int @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_37_prod__cases,axiom,
! [P2: produc7773217078559923341nt_int > $o,P: produc7773217078559923341nt_int] :
( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] : ( P2 @ ( produc4305682042979456191nt_int @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_38_deg__deg__n,axiom,
! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
=> ( Deg = N ) ) ).
% deg_deg_n
thf(fact_39_mi__eq__ma__no__ch,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
=> ( ( Mi = Ma )
=> ( ! [X: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X @ X_1 ) )
& ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 ) ) ) ) ).
% mi_eq_ma_no_ch
thf(fact_40_prod__induct3,axiom,
! [P2: produc8763457246119570046nteger > $o,X3: produc8763457246119570046nteger] :
( ! [A3: code_integer > option6357759511663192854e_term,B3: code_integer,C: code_integer] : ( P2 @ ( produc6137756002093451184nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct3
thf(fact_41_prod__induct3,axiom,
! [P2: produc1908205239877642774nteger > $o,X3: produc1908205239877642774nteger] :
( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: code_integer,C: code_integer] : ( P2 @ ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct3
thf(fact_42_prod__induct3,axiom,
! [P2: produc2285326912895808259nt_int > $o,X3: produc2285326912895808259nt_int] :
( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: int,C: int] : ( P2 @ ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct3
thf(fact_43_prod__induct3,axiom,
! [P2: produc7773217078559923341nt_int > $o,X3: produc7773217078559923341nt_int] :
( ! [A3: int > option6357759511663192854e_term,B3: int,C: int] : ( P2 @ ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct3
thf(fact_44_prod__cases3,axiom,
! [Y: produc8763457246119570046nteger] :
~ ! [A3: code_integer > option6357759511663192854e_term,B3: code_integer,C: code_integer] :
( Y
!= ( produc6137756002093451184nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_45_prod__cases3,axiom,
! [Y: produc1908205239877642774nteger] :
~ ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: code_integer,C: code_integer] :
( Y
!= ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_46_prod__cases3,axiom,
! [Y: produc2285326912895808259nt_int] :
~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: int,C: int] :
( Y
!= ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_47_prod__cases3,axiom,
! [Y: produc7773217078559923341nt_int] :
~ ! [A3: int > option6357759511663192854e_term,B3: int,C: int] :
( Y
!= ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_48_Pair__inject,axiom,
! [A: int,B: int,A2: int,B2: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_49_Pair__inject,axiom,
! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,A2: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
( ( ( produc6137756002093451184nteger @ A @ B )
= ( produc6137756002093451184nteger @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_50_Pair__inject,axiom,
! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A2: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
( ( ( produc8603105652947943368nteger @ A @ B )
= ( produc8603105652947943368nteger @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_51_Pair__inject,axiom,
! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A2: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
( ( ( produc5700946648718959541nt_int @ A @ B )
= ( produc5700946648718959541nt_int @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_52_Pair__inject,axiom,
! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A2: int > option6357759511663192854e_term,B2: product_prod_int_int] :
( ( ( produc4305682042979456191nt_int @ A @ B )
= ( produc4305682042979456191nt_int @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_53_valid__member__both__member__options,axiom,
! [T: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_V8194947554948674370ptions @ T @ X3 )
=> ( vEBT_vebt_member @ T @ X3 ) ) ) ).
% valid_member_both_member_options
thf(fact_54_both__member__options__equiv__member,axiom,
! [T: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_V8194947554948674370ptions @ T @ X3 )
= ( vEBT_vebt_member @ T @ X3 ) ) ) ).
% both_member_options_equiv_member
thf(fact_55_VEBT__internal_OminNull_Osimps_I5_J,axiom,
! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) ) ).
% VEBT_internal.minNull.simps(5)
thf(fact_56_set__vebt__set__vebt_H__valid,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_set_vebt @ T )
= ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% set_vebt_set_vebt'_valid
thf(fact_57_valid__eq,axiom,
vEBT_VEBT_valid = vEBT_invar_vebt ).
% valid_eq
thf(fact_58_valid__eq1,axiom,
! [T: vEBT_VEBT,D: nat] :
( ( vEBT_invar_vebt @ T @ D )
=> ( vEBT_VEBT_valid @ T @ D ) ) ).
% valid_eq1
thf(fact_59_valid__eq2,axiom,
! [T: vEBT_VEBT,D: nat] :
( ( vEBT_VEBT_valid @ T @ D )
=> ( vEBT_invar_vebt @ T @ D ) ) ).
% valid_eq2
thf(fact_60_min__Null__member,axiom,
! [T: vEBT_VEBT,X3: nat] :
( ( vEBT_VEBT_minNull @ T )
=> ~ ( vEBT_vebt_member @ T @ X3 ) ) ).
% min_Null_member
thf(fact_61_order__refl,axiom,
! [X3: set_int] : ( ord_less_eq_set_int @ X3 @ X3 ) ).
% order_refl
thf(fact_62_order__refl,axiom,
! [X3: rat] : ( ord_less_eq_rat @ X3 @ X3 ) ).
% order_refl
thf(fact_63_order__refl,axiom,
! [X3: num] : ( ord_less_eq_num @ X3 @ X3 ) ).
% order_refl
thf(fact_64_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_65_order__refl,axiom,
! [X3: int] : ( ord_less_eq_int @ X3 @ X3 ) ).
% order_refl
thf(fact_66_dual__order_Orefl,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% dual_order.refl
thf(fact_67_dual__order_Orefl,axiom,
! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% dual_order.refl
thf(fact_68_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_69_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_70_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_71_deg__SUcn__Node,axiom,
! [Tree: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
=> ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
( Tree
= ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList2 @ S ) ) ) ).
% deg_SUcn_Node
thf(fact_72_member__correct,axiom,
! [T: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_vebt_member @ T @ X3 )
= ( member_nat @ X3 @ ( vEBT_set_vebt @ T ) ) ) ) ).
% member_correct
thf(fact_73__C5_Ohyps_C_I3_J,axiom,
( m
= ( suc @ na ) ) ).
% "5.hyps"(3)
thf(fact_74_mem__Collect__eq,axiom,
! [A: complex,P2: complex > $o] :
( ( member_complex @ A @ ( collect_complex @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_75_mem__Collect__eq,axiom,
! [A: real,P2: real > $o] :
( ( member_real @ A @ ( collect_real @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_76_mem__Collect__eq,axiom,
! [A: list_nat,P2: list_nat > $o] :
( ( member_list_nat @ A @ ( collect_list_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_77_mem__Collect__eq,axiom,
! [A: set_nat,P2: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_78_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_79_mem__Collect__eq,axiom,
! [A: int,P2: int > $o] :
( ( member_int @ A @ ( collect_int @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_80_Collect__mem__eq,axiom,
! [A4: set_complex] :
( ( collect_complex
@ ^ [X5: complex] : ( member_complex @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_81_Collect__mem__eq,axiom,
! [A4: set_real] :
( ( collect_real
@ ^ [X5: real] : ( member_real @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_82_Collect__mem__eq,axiom,
! [A4: set_list_nat] :
( ( collect_list_nat
@ ^ [X5: list_nat] : ( member_list_nat @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_83_Collect__mem__eq,axiom,
! [A4: set_set_nat] :
( ( collect_set_nat
@ ^ [X5: set_nat] : ( member_set_nat @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_84_Collect__mem__eq,axiom,
! [A4: set_nat] :
( ( collect_nat
@ ^ [X5: nat] : ( member_nat @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_85_Collect__mem__eq,axiom,
! [A4: set_int] :
( ( collect_int
@ ^ [X5: int] : ( member_int @ X5 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_86_Collect__cong,axiom,
! [P2: real > $o,Q: real > $o] :
( ! [X4: real] :
( ( P2 @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_real @ P2 )
= ( collect_real @ Q ) ) ) ).
% Collect_cong
thf(fact_87_Collect__cong,axiom,
! [P2: list_nat > $o,Q: list_nat > $o] :
( ! [X4: list_nat] :
( ( P2 @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_list_nat @ P2 )
= ( collect_list_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_88_Collect__cong,axiom,
! [P2: set_nat > $o,Q: set_nat > $o] :
( ! [X4: set_nat] :
( ( P2 @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_set_nat @ P2 )
= ( collect_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_89_Collect__cong,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P2 @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_nat @ P2 )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_90_Collect__cong,axiom,
! [P2: int > $o,Q: int > $o] :
( ! [X4: int] :
( ( P2 @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_int @ P2 )
= ( collect_int @ Q ) ) ) ).
% Collect_cong
thf(fact_91_order__antisym__conv,axiom,
! [Y: set_int,X3: set_int] :
( ( ord_less_eq_set_int @ Y @ X3 )
=> ( ( ord_less_eq_set_int @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_92_order__antisym__conv,axiom,
! [Y: rat,X3: rat] :
( ( ord_less_eq_rat @ Y @ X3 )
=> ( ( ord_less_eq_rat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_93_order__antisym__conv,axiom,
! [Y: num,X3: num] :
( ( ord_less_eq_num @ Y @ X3 )
=> ( ( ord_less_eq_num @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_94_order__antisym__conv,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_95_order__antisym__conv,axiom,
! [Y: int,X3: int] :
( ( ord_less_eq_int @ Y @ X3 )
=> ( ( ord_less_eq_int @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_96_linorder__le__cases,axiom,
! [X3: rat,Y: rat] :
( ~ ( ord_less_eq_rat @ X3 @ Y )
=> ( ord_less_eq_rat @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_97_linorder__le__cases,axiom,
! [X3: num,Y: num] :
( ~ ( ord_less_eq_num @ X3 @ Y )
=> ( ord_less_eq_num @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_98_linorder__le__cases,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_99_linorder__le__cases,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_eq_int @ X3 @ Y )
=> ( ord_less_eq_int @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_100_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: rat > rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_101_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: rat > num,C2: num] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_102_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: rat > nat,C2: nat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_103_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: rat > int,C2: int] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_104_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > rat,C2: rat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_105_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > num,C2: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_106_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > nat,C2: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_107_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > int,C2: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_108_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > rat,C2: rat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_109_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > num,C2: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_110_ord__eq__le__subst,axiom,
! [A: rat,F: rat > rat,B: rat,C2: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_111_ord__eq__le__subst,axiom,
! [A: num,F: rat > num,B: rat,C2: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_112_ord__eq__le__subst,axiom,
! [A: nat,F: rat > nat,B: rat,C2: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_113_ord__eq__le__subst,axiom,
! [A: int,F: rat > int,B: rat,C2: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_114_ord__eq__le__subst,axiom,
! [A: rat,F: num > rat,B: num,C2: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_115_ord__eq__le__subst,axiom,
! [A: num,F: num > num,B: num,C2: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_116_ord__eq__le__subst,axiom,
! [A: nat,F: num > nat,B: num,C2: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_117_ord__eq__le__subst,axiom,
! [A: int,F: num > int,B: num,C2: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_118_ord__eq__le__subst,axiom,
! [A: rat,F: nat > rat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_119_ord__eq__le__subst,axiom,
! [A: num,F: nat > num,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_120_linorder__linear,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ X3 @ Y )
| ( ord_less_eq_rat @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_121_linorder__linear,axiom,
! [X3: num,Y: num] :
( ( ord_less_eq_num @ X3 @ Y )
| ( ord_less_eq_num @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_122_linorder__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_123_linorder__linear,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
| ( ord_less_eq_int @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_124_order__eq__refl,axiom,
! [X3: set_int,Y: set_int] :
( ( X3 = Y )
=> ( ord_less_eq_set_int @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_125_order__eq__refl,axiom,
! [X3: rat,Y: rat] :
( ( X3 = Y )
=> ( ord_less_eq_rat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_126_order__eq__refl,axiom,
! [X3: num,Y: num] :
( ( X3 = Y )
=> ( ord_less_eq_num @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_127_order__eq__refl,axiom,
! [X3: nat,Y: nat] :
( ( X3 = Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_128_order__eq__refl,axiom,
! [X3: int,Y: int] :
( ( X3 = Y )
=> ( ord_less_eq_int @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_129_order__subst2,axiom,
! [A: rat,B: rat,F: rat > rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_130_order__subst2,axiom,
! [A: rat,B: rat,F: rat > num,C2: num] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_131_order__subst2,axiom,
! [A: rat,B: rat,F: rat > nat,C2: nat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_132_order__subst2,axiom,
! [A: rat,B: rat,F: rat > int,C2: int] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_133_order__subst2,axiom,
! [A: num,B: num,F: num > rat,C2: rat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_134_order__subst2,axiom,
! [A: num,B: num,F: num > num,C2: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_135_order__subst2,axiom,
! [A: num,B: num,F: num > nat,C2: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_136_order__subst2,axiom,
! [A: num,B: num,F: num > int,C2: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_137_order__subst2,axiom,
! [A: nat,B: nat,F: nat > rat,C2: rat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_138_order__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C2: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_139_order__subst1,axiom,
! [A: rat,F: rat > rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_140_order__subst1,axiom,
! [A: rat,F: num > rat,B: num,C2: num] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_141_order__subst1,axiom,
! [A: rat,F: nat > rat,B: nat,C2: nat] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_142_order__subst1,axiom,
! [A: rat,F: int > rat,B: int,C2: int] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_143_order__subst1,axiom,
! [A: num,F: rat > num,B: rat,C2: rat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_144_order__subst1,axiom,
! [A: num,F: num > num,B: num,C2: num] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_145_order__subst1,axiom,
! [A: num,F: nat > num,B: nat,C2: nat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_146_order__subst1,axiom,
! [A: num,F: int > num,B: int,C2: int] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_147_order__subst1,axiom,
! [A: nat,F: rat > nat,B: rat,C2: rat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_148_order__subst1,axiom,
! [A: nat,F: num > nat,B: num,C2: num] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_149_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_int,Z: set_int] : Y4 = Z )
= ( ^ [A5: set_int,B4: set_int] :
( ( ord_less_eq_set_int @ A5 @ B4 )
& ( ord_less_eq_set_int @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_150_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: rat,Z: rat] : Y4 = Z )
= ( ^ [A5: rat,B4: rat] :
( ( ord_less_eq_rat @ A5 @ B4 )
& ( ord_less_eq_rat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_151_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: num,Z: num] : Y4 = Z )
= ( ^ [A5: num,B4: num] :
( ( ord_less_eq_num @ A5 @ B4 )
& ( ord_less_eq_num @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_152_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : Y4 = Z )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_153_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z: int] : Y4 = Z )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_154_antisym,axiom,
! [A: set_int,B: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( ord_less_eq_set_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_155_antisym,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_156_antisym,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_157_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_158_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_159_dual__order_Otrans,axiom,
! [B: set_int,A: set_int,C2: set_int] :
( ( ord_less_eq_set_int @ B @ A )
=> ( ( ord_less_eq_set_int @ C2 @ B )
=> ( ord_less_eq_set_int @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_160_dual__order_Otrans,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C2 @ B )
=> ( ord_less_eq_rat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_161_dual__order_Otrans,axiom,
! [B: num,A: num,C2: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ C2 @ B )
=> ( ord_less_eq_num @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_162_dual__order_Otrans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_163_dual__order_Otrans,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_eq_int @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_164_dual__order_Oantisym,axiom,
! [B: set_int,A: set_int] :
( ( ord_less_eq_set_int @ B @ A )
=> ( ( ord_less_eq_set_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_165_dual__order_Oantisym,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_166_dual__order_Oantisym,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_167_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_168_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_169_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_int,Z: set_int] : Y4 = Z )
= ( ^ [A5: set_int,B4: set_int] :
( ( ord_less_eq_set_int @ B4 @ A5 )
& ( ord_less_eq_set_int @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_170_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: rat,Z: rat] : Y4 = Z )
= ( ^ [A5: rat,B4: rat] :
( ( ord_less_eq_rat @ B4 @ A5 )
& ( ord_less_eq_rat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_171_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: num,Z: num] : Y4 = Z )
= ( ^ [A5: num,B4: num] :
( ( ord_less_eq_num @ B4 @ A5 )
& ( ord_less_eq_num @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_172_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : Y4 = Z )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_173_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: int,Z: int] : Y4 = Z )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_174_linorder__wlog,axiom,
! [P2: rat > rat > $o,A: rat,B: rat] :
( ! [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: rat,B3: rat] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_175_linorder__wlog,axiom,
! [P2: num > num > $o,A: num,B: num] :
( ! [A3: num,B3: num] :
( ( ord_less_eq_num @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: num,B3: num] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_176_linorder__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_177_linorder__wlog,axiom,
! [P2: int > int > $o,A: int,B: int] :
( ! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: int,B3: int] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_178_order__trans,axiom,
! [X3: set_int,Y: set_int,Z2: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y )
=> ( ( ord_less_eq_set_int @ Y @ Z2 )
=> ( ord_less_eq_set_int @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_179_order__trans,axiom,
! [X3: rat,Y: rat,Z2: rat] :
( ( ord_less_eq_rat @ X3 @ Y )
=> ( ( ord_less_eq_rat @ Y @ Z2 )
=> ( ord_less_eq_rat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_180_order__trans,axiom,
! [X3: num,Y: num,Z2: num] :
( ( ord_less_eq_num @ X3 @ Y )
=> ( ( ord_less_eq_num @ Y @ Z2 )
=> ( ord_less_eq_num @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_181_order__trans,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_182_order__trans,axiom,
! [X3: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_183_order_Otrans,axiom,
! [A: set_int,B: set_int,C2: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( ord_less_eq_set_int @ B @ C2 )
=> ( ord_less_eq_set_int @ A @ C2 ) ) ) ).
% order.trans
thf(fact_184_order_Otrans,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ord_less_eq_rat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_185_order_Otrans,axiom,
! [A: num,B: num,C2: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ord_less_eq_num @ A @ C2 ) ) ) ).
% order.trans
thf(fact_186_order_Otrans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_187_order_Otrans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% order.trans
thf(fact_188_order__antisym,axiom,
! [X3: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y )
=> ( ( ord_less_eq_set_int @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_189_order__antisym,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ X3 @ Y )
=> ( ( ord_less_eq_rat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_190_order__antisym,axiom,
! [X3: num,Y: num] :
( ( ord_less_eq_num @ X3 @ Y )
=> ( ( ord_less_eq_num @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_191_order__antisym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_192_order__antisym,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_193_ord__le__eq__trans,axiom,
! [A: set_int,B: set_int,C2: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_set_int @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_194_ord__le__eq__trans,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_rat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_195_ord__le__eq__trans,axiom,
! [A: num,B: num,C2: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_num @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_196_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_197_ord__le__eq__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_198_ord__eq__le__trans,axiom,
! [A: set_int,B: set_int,C2: set_int] :
( ( A = B )
=> ( ( ord_less_eq_set_int @ B @ C2 )
=> ( ord_less_eq_set_int @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_199_ord__eq__le__trans,axiom,
! [A: rat,B: rat,C2: rat] :
( ( A = B )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ord_less_eq_rat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_200_ord__eq__le__trans,axiom,
! [A: num,B: num,C2: num] :
( ( A = B )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ord_less_eq_num @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_201_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_202_ord__eq__le__trans,axiom,
! [A: int,B: int,C2: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_203_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_int,Z: set_int] : Y4 = Z )
= ( ^ [X5: set_int,Y5: set_int] :
( ( ord_less_eq_set_int @ X5 @ Y5 )
& ( ord_less_eq_set_int @ Y5 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_204_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: rat,Z: rat] : Y4 = Z )
= ( ^ [X5: rat,Y5: rat] :
( ( ord_less_eq_rat @ X5 @ Y5 )
& ( ord_less_eq_rat @ Y5 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_205_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: num,Z: num] : Y4 = Z )
= ( ^ [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
& ( ord_less_eq_num @ Y5 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_206_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : Y4 = Z )
= ( ^ [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_207_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z: int] : Y4 = Z )
= ( ^ [X5: int,Y5: int] :
( ( ord_less_eq_int @ X5 @ Y5 )
& ( ord_less_eq_int @ Y5 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_208_le__cases3,axiom,
! [X3: rat,Y: rat,Z2: rat] :
( ( ( ord_less_eq_rat @ X3 @ Y )
=> ~ ( ord_less_eq_rat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_rat @ Y @ X3 )
=> ~ ( ord_less_eq_rat @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq_rat @ X3 @ Z2 )
=> ~ ( ord_less_eq_rat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_rat @ Z2 @ Y )
=> ~ ( ord_less_eq_rat @ Y @ X3 ) )
=> ( ( ( ord_less_eq_rat @ Y @ Z2 )
=> ~ ( ord_less_eq_rat @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq_rat @ Z2 @ X3 )
=> ~ ( ord_less_eq_rat @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_209_le__cases3,axiom,
! [X3: num,Y: num,Z2: num] :
( ( ( ord_less_eq_num @ X3 @ Y )
=> ~ ( ord_less_eq_num @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_num @ Y @ X3 )
=> ~ ( ord_less_eq_num @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq_num @ X3 @ Z2 )
=> ~ ( ord_less_eq_num @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_num @ Z2 @ Y )
=> ~ ( ord_less_eq_num @ Y @ X3 ) )
=> ( ( ( ord_less_eq_num @ Y @ Z2 )
=> ~ ( ord_less_eq_num @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq_num @ Z2 @ X3 )
=> ~ ( ord_less_eq_num @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_210_le__cases3,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_211_le__cases3,axiom,
! [X3: int,Y: int,Z2: int] :
( ( ( ord_less_eq_int @ X3 @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X3 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X3 ) )
=> ( ( ( ord_less_eq_int @ Y @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_212_nle__le,axiom,
! [A: rat,B: rat] :
( ( ~ ( ord_less_eq_rat @ A @ B ) )
= ( ( ord_less_eq_rat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_213_nle__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_eq_num @ A @ B ) )
= ( ( ord_less_eq_num @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_214_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_215_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_216_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_217_lift__Suc__antimono__le,axiom,
! [F: nat > set_int,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_set_int @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_218_lift__Suc__antimono__le,axiom,
! [F: nat > rat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_219_lift__Suc__antimono__le,axiom,
! [F: nat > num,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_220_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_221_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_222_lift__Suc__mono__le,axiom,
! [F: nat > set_int,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_223_lift__Suc__mono__le,axiom,
! [F: nat > rat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_224_lift__Suc__mono__le,axiom,
! [F: nat > num,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_num @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_225_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_226_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_227_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_228_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_229_set__vebt__finite,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% set_vebt_finite
thf(fact_230_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X4: nat] : ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y3: nat,Z3: nat] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_231_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P2 @ M2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_232_full__nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P2 @ M3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% full_nat_induct
thf(fact_233_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_234_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_235__C5_Ohyps_C_I4_J,axiom,
( deg
= ( plus_plus_nat @ na @ m ) ) ).
% "5.hyps"(4)
thf(fact_236_Suc__inject,axiom,
! [X3: nat,Y: nat] :
( ( ( suc @ X3 )
= ( suc @ Y ) )
=> ( X3 = Y ) ) ).
% Suc_inject
thf(fact_237_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_238_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_239_le__trans,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_240_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_241_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_242_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_243_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K2: nat,B: nat] :
( ( P2 @ K2 )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X4: nat] :
( ( P2 @ X4 )
& ! [Y6: nat] :
( ( P2 @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_244_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_245_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_246_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_247_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
=> ? [M: nat] :
( M4
= ( suc @ M ) ) ) ).
% Suc_le_D
thf(fact_248_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_249_List_Ofinite__set,axiom,
! [Xs: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) ) ).
% List.finite_set
thf(fact_250_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_251_List_Ofinite__set,axiom,
! [Xs: list_int] : ( finite_finite_int @ ( set_int2 @ Xs ) ) ).
% List.finite_set
thf(fact_252_List_Ofinite__set,axiom,
! [Xs: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs ) ) ).
% List.finite_set
thf(fact_253_List_Ofinite__set,axiom,
! [Xs: list_P6011104703257516679at_nat] : ( finite6177210948735845034at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ).
% List.finite_set
thf(fact_254_infinite__nat__iff__unbounded__le,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M5: nat] :
? [N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
& ( member_nat @ N4 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_255_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M5: nat] :
! [X5: nat] :
( ( member_nat @ X5 @ N5 )
=> ( ord_less_eq_nat @ X5 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_256_finite__list,axiom,
! [A4: set_VEBT_VEBT] :
( ( finite5795047828879050333T_VEBT @ A4 )
=> ? [Xs2: list_VEBT_VEBT] :
( ( set_VEBT_VEBT2 @ Xs2 )
= A4 ) ) ).
% finite_list
thf(fact_257_finite__list,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ? [Xs2: list_nat] :
( ( set_nat2 @ Xs2 )
= A4 ) ) ).
% finite_list
thf(fact_258_finite__list,axiom,
! [A4: set_int] :
( ( finite_finite_int @ A4 )
=> ? [Xs2: list_int] :
( ( set_int2 @ Xs2 )
= A4 ) ) ).
% finite_list
thf(fact_259_finite__list,axiom,
! [A4: set_complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ? [Xs2: list_complex] :
( ( set_complex2 @ Xs2 )
= A4 ) ) ).
% finite_list
thf(fact_260_finite__list,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ? [Xs2: list_P6011104703257516679at_nat] :
( ( set_Pr5648618587558075414at_nat @ Xs2 )
= A4 ) ) ).
% finite_list
thf(fact_261_finite__has__maximal2,axiom,
! [A4: set_real,A: real] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A @ A4 )
=> ? [X4: real] :
( ( member_real @ X4 @ A4 )
& ( ord_less_eq_real @ A @ X4 )
& ! [Xa: real] :
( ( member_real @ Xa @ A4 )
=> ( ( ord_less_eq_real @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_262_finite__has__maximal2,axiom,
! [A4: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( member_set_nat @ A @ A4 )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A4 )
& ( ord_less_eq_set_nat @ A @ X4 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A4 )
=> ( ( ord_less_eq_set_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_263_finite__has__maximal2,axiom,
! [A4: set_set_int,A: set_int] :
( ( finite6197958912794628473et_int @ A4 )
=> ( ( member_set_int @ A @ A4 )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ A4 )
& ( ord_less_eq_set_int @ A @ X4 )
& ! [Xa: set_int] :
( ( member_set_int @ Xa @ A4 )
=> ( ( ord_less_eq_set_int @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_264_finite__has__maximal2,axiom,
! [A4: set_rat,A: rat] :
( ( finite_finite_rat @ A4 )
=> ( ( member_rat @ A @ A4 )
=> ? [X4: rat] :
( ( member_rat @ X4 @ A4 )
& ( ord_less_eq_rat @ A @ X4 )
& ! [Xa: rat] :
( ( member_rat @ Xa @ A4 )
=> ( ( ord_less_eq_rat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_265_finite__has__maximal2,axiom,
! [A4: set_num,A: num] :
( ( finite_finite_num @ A4 )
=> ( ( member_num @ A @ A4 )
=> ? [X4: num] :
( ( member_num @ X4 @ A4 )
& ( ord_less_eq_num @ A @ X4 )
& ! [Xa: num] :
( ( member_num @ Xa @ A4 )
=> ( ( ord_less_eq_num @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_266_finite__has__maximal2,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( ord_less_eq_nat @ A @ X4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_267_finite__has__maximal2,axiom,
! [A4: set_int,A: int] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ A @ A4 )
=> ? [X4: int] :
( ( member_int @ X4 @ A4 )
& ( ord_less_eq_int @ A @ X4 )
& ! [Xa: int] :
( ( member_int @ Xa @ A4 )
=> ( ( ord_less_eq_int @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_268_finite__has__minimal2,axiom,
! [A4: set_real,A: real] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A @ A4 )
=> ? [X4: real] :
( ( member_real @ X4 @ A4 )
& ( ord_less_eq_real @ X4 @ A )
& ! [Xa: real] :
( ( member_real @ Xa @ A4 )
=> ( ( ord_less_eq_real @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_269_finite__has__minimal2,axiom,
! [A4: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( member_set_nat @ A @ A4 )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A4 )
& ( ord_less_eq_set_nat @ X4 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A4 )
=> ( ( ord_less_eq_set_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_270_finite__has__minimal2,axiom,
! [A4: set_set_int,A: set_int] :
( ( finite6197958912794628473et_int @ A4 )
=> ( ( member_set_int @ A @ A4 )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ A4 )
& ( ord_less_eq_set_int @ X4 @ A )
& ! [Xa: set_int] :
( ( member_set_int @ Xa @ A4 )
=> ( ( ord_less_eq_set_int @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_271_finite__has__minimal2,axiom,
! [A4: set_rat,A: rat] :
( ( finite_finite_rat @ A4 )
=> ( ( member_rat @ A @ A4 )
=> ? [X4: rat] :
( ( member_rat @ X4 @ A4 )
& ( ord_less_eq_rat @ X4 @ A )
& ! [Xa: rat] :
( ( member_rat @ Xa @ A4 )
=> ( ( ord_less_eq_rat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_272_finite__has__minimal2,axiom,
! [A4: set_num,A: num] :
( ( finite_finite_num @ A4 )
=> ( ( member_num @ A @ A4 )
=> ? [X4: num] :
( ( member_num @ X4 @ A4 )
& ( ord_less_eq_num @ X4 @ A )
& ! [Xa: num] :
( ( member_num @ Xa @ A4 )
=> ( ( ord_less_eq_num @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_273_finite__has__minimal2,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( ord_less_eq_nat @ X4 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_274_finite__has__minimal2,axiom,
! [A4: set_int,A: int] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ A @ A4 )
=> ? [X4: int] :
( ( member_int @ X4 @ A4 )
& ( ord_less_eq_int @ X4 @ A )
& ! [Xa: int] :
( ( member_int @ Xa @ A4 )
=> ( ( ord_less_eq_int @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_275_vebt__insert_Osimps_I4_J,axiom,
! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X3 )
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) ) ).
% vebt_insert.simps(4)
thf(fact_276_vebt__member_Osimps_I4_J,axiom,
! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X3 ) ).
% vebt_member.simps(4)
thf(fact_277_valid__0__not,axiom,
! [T: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% valid_0_not
thf(fact_278_valid__tree__deg__neq__0,axiom,
! [T: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% valid_tree_deg_neq_0
thf(fact_279_even__odd__cases,axiom,
! [X3: nat] :
( ! [N3: nat] :
( X3
!= ( plus_plus_nat @ N3 @ N3 ) )
=> ~ ! [N3: nat] :
( X3
!= ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) ) ) ).
% even_odd_cases
thf(fact_280_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_281_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_282_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_283_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_284_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_285_nat__add__left__cancel__le,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_286_not__None__eq,axiom,
! [X3: option4927543243414619207at_nat] :
( ( X3 != none_P5556105721700978146at_nat )
= ( ? [Y5: product_prod_nat_nat] :
( X3
= ( some_P7363390416028606310at_nat @ Y5 ) ) ) ) ).
% not_None_eq
thf(fact_287_not__None__eq,axiom,
! [X3: option_num] :
( ( X3 != none_num )
= ( ? [Y5: num] :
( X3
= ( some_num @ Y5 ) ) ) ) ).
% not_None_eq
thf(fact_288_not__Some__eq,axiom,
! [X3: option4927543243414619207at_nat] :
( ( ! [Y5: product_prod_nat_nat] :
( X3
!= ( some_P7363390416028606310at_nat @ Y5 ) ) )
= ( X3 = none_P5556105721700978146at_nat ) ) ).
% not_Some_eq
thf(fact_289_not__Some__eq,axiom,
! [X3: option_num] :
( ( ! [Y5: num] :
( X3
!= ( some_num @ Y5 ) ) )
= ( X3 = none_num ) ) ).
% not_Some_eq
thf(fact_290_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_291_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_292_add__is__1,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_293_one__is__add,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_294_add__Suc__shift,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_295_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_296_nat__arith_Osuc1,axiom,
! [A4: nat,K2: nat,A: nat] :
( ( A4
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( suc @ A4 )
= ( plus_plus_nat @ K2 @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_297_option_Odistinct_I1_J,axiom,
! [X2: product_prod_nat_nat] :
( none_P5556105721700978146at_nat
!= ( some_P7363390416028606310at_nat @ X2 ) ) ).
% option.distinct(1)
thf(fact_298_option_Odistinct_I1_J,axiom,
! [X2: num] :
( none_num
!= ( some_num @ X2 ) ) ).
% option.distinct(1)
thf(fact_299_option_OdiscI,axiom,
! [Option: option4927543243414619207at_nat,X2: product_prod_nat_nat] :
( ( Option
= ( some_P7363390416028606310at_nat @ X2 ) )
=> ( Option != none_P5556105721700978146at_nat ) ) ).
% option.discI
thf(fact_300_option_OdiscI,axiom,
! [Option: option_num,X2: num] :
( ( Option
= ( some_num @ X2 ) )
=> ( Option != none_num ) ) ).
% option.discI
thf(fact_301_option_Oexhaust,axiom,
! [Y: option4927543243414619207at_nat] :
( ( Y != none_P5556105721700978146at_nat )
=> ~ ! [X22: product_prod_nat_nat] :
( Y
!= ( some_P7363390416028606310at_nat @ X22 ) ) ) ).
% option.exhaust
thf(fact_302_option_Oexhaust,axiom,
! [Y: option_num] :
( ( Y != none_num )
=> ~ ! [X22: num] :
( Y
!= ( some_num @ X22 ) ) ) ).
% option.exhaust
thf(fact_303_split__option__ex,axiom,
( ( ^ [P3: option4927543243414619207at_nat > $o] :
? [X6: option4927543243414619207at_nat] : ( P3 @ X6 ) )
= ( ^ [P4: option4927543243414619207at_nat > $o] :
( ( P4 @ none_P5556105721700978146at_nat )
| ? [X5: product_prod_nat_nat] : ( P4 @ ( some_P7363390416028606310at_nat @ X5 ) ) ) ) ) ).
% split_option_ex
thf(fact_304_split__option__ex,axiom,
( ( ^ [P3: option_num > $o] :
? [X6: option_num] : ( P3 @ X6 ) )
= ( ^ [P4: option_num > $o] :
( ( P4 @ none_num )
| ? [X5: num] : ( P4 @ ( some_num @ X5 ) ) ) ) ) ).
% split_option_ex
thf(fact_305_split__option__all,axiom,
( ( ^ [P3: option4927543243414619207at_nat > $o] :
! [X6: option4927543243414619207at_nat] : ( P3 @ X6 ) )
= ( ^ [P4: option4927543243414619207at_nat > $o] :
( ( P4 @ none_P5556105721700978146at_nat )
& ! [X5: product_prod_nat_nat] : ( P4 @ ( some_P7363390416028606310at_nat @ X5 ) ) ) ) ) ).
% split_option_all
thf(fact_306_split__option__all,axiom,
( ( ^ [P3: option_num > $o] :
! [X6: option_num] : ( P3 @ X6 ) )
= ( ^ [P4: option_num > $o] :
( ( P4 @ none_num )
& ! [X5: num] : ( P4 @ ( some_num @ X5 ) ) ) ) ) ).
% split_option_all
thf(fact_307_combine__options__cases,axiom,
! [X3: option4927543243414619207at_nat,P2: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
( ( ( X3 = none_P5556105721700978146at_nat )
=> ( P2 @ X3 @ Y ) )
=> ( ( ( Y = none_P5556105721700978146at_nat )
=> ( P2 @ X3 @ Y ) )
=> ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
( ( X3
= ( some_P7363390416028606310at_nat @ A3 ) )
=> ( ( Y
= ( some_P7363390416028606310at_nat @ B3 ) )
=> ( P2 @ X3 @ Y ) ) )
=> ( P2 @ X3 @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_308_combine__options__cases,axiom,
! [X3: option4927543243414619207at_nat,P2: option4927543243414619207at_nat > option_num > $o,Y: option_num] :
( ( ( X3 = none_P5556105721700978146at_nat )
=> ( P2 @ X3 @ Y ) )
=> ( ( ( Y = none_num )
=> ( P2 @ X3 @ Y ) )
=> ( ! [A3: product_prod_nat_nat,B3: num] :
( ( X3
= ( some_P7363390416028606310at_nat @ A3 ) )
=> ( ( Y
= ( some_num @ B3 ) )
=> ( P2 @ X3 @ Y ) ) )
=> ( P2 @ X3 @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_309_combine__options__cases,axiom,
! [X3: option_num,P2: option_num > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
( ( ( X3 = none_num )
=> ( P2 @ X3 @ Y ) )
=> ( ( ( Y = none_P5556105721700978146at_nat )
=> ( P2 @ X3 @ Y ) )
=> ( ! [A3: num,B3: product_prod_nat_nat] :
( ( X3
= ( some_num @ A3 ) )
=> ( ( Y
= ( some_P7363390416028606310at_nat @ B3 ) )
=> ( P2 @ X3 @ Y ) ) )
=> ( P2 @ X3 @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_310_combine__options__cases,axiom,
! [X3: option_num,P2: option_num > option_num > $o,Y: option_num] :
( ( ( X3 = none_num )
=> ( P2 @ X3 @ Y ) )
=> ( ( ( Y = none_num )
=> ( P2 @ X3 @ Y ) )
=> ( ! [A3: num,B3: num] :
( ( X3
= ( some_num @ A3 ) )
=> ( ( Y
= ( some_num @ B3 ) )
=> ( P2 @ X3 @ Y ) ) )
=> ( P2 @ X3 @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_311_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_312_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_313_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_314_add__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_315_add__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_316_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K2 @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_317_add__leD2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_318_add__leD1,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_319_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_320_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_321_add__leE,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_322_list__decode_Ocases,axiom,
! [X3: nat] :
( ( X3 != zero_zero_nat )
=> ~ ! [N3: nat] :
( X3
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_323_vebt__buildup_Ocases,axiom,
! [X3: nat] :
( ( X3 != zero_zero_nat )
=> ( ( X3
!= ( suc @ zero_zero_nat ) )
=> ~ ! [Va2: nat] :
( X3
!= ( suc @ ( suc @ Va2 ) ) ) ) ) ).
% vebt_buildup.cases
thf(fact_324_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M: nat] :
( N
= ( suc @ M ) ) ) ).
% not0_implies_Suc
thf(fact_325_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_326_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_327_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_328_zero__induct,axiom,
! [P2: nat > $o,K2: nat] :
( ( P2 @ K2 )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_329_diff__induct,axiom,
! [P2: nat > nat > $o,M2: nat,N: nat] :
( ! [X4: nat] : ( P2 @ X4 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X4: nat,Y3: nat] :
( ( P2 @ X4 @ Y3 )
=> ( P2 @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
=> ( P2 @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_330_nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) )
=> ( P2 @ N ) ) ) ).
% nat_induct
thf(fact_331_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_332_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_333_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_334_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_335_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_336_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_337_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_338_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_339_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_340_vebt__member_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X3: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X3 ) ).
% vebt_member.simps(2)
thf(fact_341_VEBT__internal_OminNull_Osimps_I4_J,axiom,
! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ).
% VEBT_internal.minNull.simps(4)
thf(fact_342_vebt__insert_Osimps_I2_J,axiom,
! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S3 ) @ X3 )
= ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S3 ) ) ).
% vebt_insert.simps(2)
thf(fact_343_rev__finite__subset,axiom,
! [B5: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( finite_finite_nat @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_344_rev__finite__subset,axiom,
! [B5: set_complex,A4: set_complex] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( finite3207457112153483333omplex @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_345_rev__finite__subset,axiom,
! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ B5 )
=> ( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( finite6177210948735845034at_nat @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_346_rev__finite__subset,axiom,
! [B5: set_int,A4: set_int] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( finite_finite_int @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_347_infinite__super,axiom,
! [S2: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_348_infinite__super,axiom,
! [S2: set_complex,T2: set_complex] :
( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ~ ( finite3207457112153483333omplex @ S2 )
=> ~ ( finite3207457112153483333omplex @ T2 ) ) ) ).
% infinite_super
thf(fact_349_infinite__super,axiom,
! [S2: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ S2 @ T2 )
=> ( ~ ( finite6177210948735845034at_nat @ S2 )
=> ~ ( finite6177210948735845034at_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_350_infinite__super,axiom,
! [S2: set_int,T2: set_int] :
( ( ord_less_eq_set_int @ S2 @ T2 )
=> ( ~ ( finite_finite_int @ S2 )
=> ~ ( finite_finite_int @ T2 ) ) ) ).
% infinite_super
thf(fact_351_finite__subset,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( finite_finite_nat @ B5 )
=> ( finite_finite_nat @ A4 ) ) ) ).
% finite_subset
thf(fact_352_finite__subset,axiom,
! [A4: set_complex,B5: set_complex] :
( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( finite3207457112153483333omplex @ A4 ) ) ) ).
% finite_subset
thf(fact_353_finite__subset,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( ( finite6177210948735845034at_nat @ B5 )
=> ( finite6177210948735845034at_nat @ A4 ) ) ) ).
% finite_subset
thf(fact_354_finite__subset,axiom,
! [A4: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( finite_finite_int @ B5 )
=> ( finite_finite_int @ A4 ) ) ) ).
% finite_subset
thf(fact_355_bounded__Max__nat,axiom,
! [P2: nat > $o,X3: nat,M6: nat] :
( ( P2 @ X3 )
=> ( ! [X4: nat] :
( ( P2 @ X4 )
=> ( ord_less_eq_nat @ X4 @ M6 ) )
=> ~ ! [M: nat] :
( ( P2 @ M )
=> ~ ! [X: nat] :
( ( P2 @ X )
=> ( ord_less_eq_nat @ X @ M ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_356_subset__code_I1_J,axiom,
! [Xs: list_complex,B5: set_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ B5 )
= ( ! [X5: complex] :
( ( member_complex @ X5 @ ( set_complex2 @ Xs ) )
=> ( member_complex @ X5 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_357_subset__code_I1_J,axiom,
! [Xs: list_real,B5: set_real] :
( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B5 )
= ( ! [X5: real] :
( ( member_real @ X5 @ ( set_real2 @ Xs ) )
=> ( member_real @ X5 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_358_subset__code_I1_J,axiom,
! [Xs: list_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B5 )
= ( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ ( set_set_nat2 @ Xs ) )
=> ( member_set_nat @ X5 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_359_subset__code_I1_J,axiom,
! [Xs: list_VEBT_VEBT,B5: set_VEBT_VEBT] :
( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ B5 )
= ( ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( member_VEBT_VEBT @ X5 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_360_subset__code_I1_J,axiom,
! [Xs: list_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B5 )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X5 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_361_subset__code_I1_J,axiom,
! [Xs: list_int,B5: set_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B5 )
= ( ! [X5: int] :
( ( member_int @ X5 @ ( set_int2 @ Xs ) )
=> ( member_int @ X5 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_362_vebt__insert_Osimps_I3_J,axiom,
! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) @ X3 )
= ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) ) ).
% vebt_insert.simps(3)
thf(fact_363_vebt__member_Osimps_I3_J,axiom,
! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X3: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X3 ) ).
% vebt_member.simps(3)
thf(fact_364_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_365_add__le__same__cancel1,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% add_le_same_cancel1
thf(fact_366_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_367_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_368_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_369_add__le__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% add_le_same_cancel2
thf(fact_370_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_371_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_372_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_373_le__add__same__cancel1,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
= ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% le_add_same_cancel1
thf(fact_374_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_375_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_376_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_377_le__add__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
= ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% le_add_same_cancel2
thf(fact_378_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_379_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_380_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_381_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_382_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_383_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_384_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_385_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_386_double__eq__0__iff,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_387_double__eq__0__iff,axiom,
! [A: rat] :
( ( ( plus_plus_rat @ A @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% double_eq_0_iff
thf(fact_388_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_389_add__0,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add_0
thf(fact_390_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_391_add__0,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% add_0
thf(fact_392_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_393_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_394_zero__eq__add__iff__both__eq__0,axiom,
! [X3: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X3 @ Y ) )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_395_add__eq__0__iff__both__eq__0,axiom,
! [X3: nat,Y: nat] :
( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_396_add__cancel__right__right,axiom,
! [A: complex,B: complex] :
( ( A
= ( plus_plus_complex @ A @ B ) )
= ( B = zero_zero_complex ) ) ).
% add_cancel_right_right
thf(fact_397_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_398_add__cancel__right__right,axiom,
! [A: rat,B: rat] :
( ( A
= ( plus_plus_rat @ A @ B ) )
= ( B = zero_zero_rat ) ) ).
% add_cancel_right_right
thf(fact_399_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_400_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_401_add__cancel__right__left,axiom,
! [A: complex,B: complex] :
( ( A
= ( plus_plus_complex @ B @ A ) )
= ( B = zero_zero_complex ) ) ).
% add_cancel_right_left
thf(fact_402_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_403_add__cancel__right__left,axiom,
! [A: rat,B: rat] :
( ( A
= ( plus_plus_rat @ B @ A ) )
= ( B = zero_zero_rat ) ) ).
% add_cancel_right_left
thf(fact_404_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_405_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_406_add__left__cancel,axiom,
! [A: real,B: real,C2: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_407_add__left__cancel,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ( plus_plus_rat @ A @ B )
= ( plus_plus_rat @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_408_add__left__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_409_add__left__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_410_add__right__cancel,axiom,
! [B: real,A: real,C2: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_411_add__right__cancel,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ( plus_plus_rat @ B @ A )
= ( plus_plus_rat @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_412_add__right__cancel,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_413_add__right__cancel,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_414_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_415_add__le__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_416_add__le__cancel__left,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
= ( ord_less_eq_rat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_417_add__le__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_418_add__le__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_419_add__le__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_420_add__le__cancel__right,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
= ( ord_less_eq_rat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_421_add__le__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_422_add__le__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_423_add_Oright__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.right_neutral
thf(fact_424_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_425_add_Oright__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% add.right_neutral
thf(fact_426_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_427_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_428_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_429_double__zero__sym,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( plus_plus_rat @ A @ A ) )
= ( A = zero_zero_rat ) ) ).
% double_zero_sym
thf(fact_430_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_431_add__cancel__left__left,axiom,
! [B: complex,A: complex] :
( ( ( plus_plus_complex @ B @ A )
= A )
= ( B = zero_zero_complex ) ) ).
% add_cancel_left_left
thf(fact_432_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_433_add__cancel__left__left,axiom,
! [B: rat,A: rat] :
( ( ( plus_plus_rat @ B @ A )
= A )
= ( B = zero_zero_rat ) ) ).
% add_cancel_left_left
thf(fact_434_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_435_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_436_add__cancel__left__right,axiom,
! [A: complex,B: complex] :
( ( ( plus_plus_complex @ A @ B )
= A )
= ( B = zero_zero_complex ) ) ).
% add_cancel_left_right
thf(fact_437_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_438_add__cancel__left__right,axiom,
! [A: rat,B: rat] :
( ( ( plus_plus_rat @ A @ B )
= A )
= ( B = zero_zero_rat ) ) ).
% add_cancel_left_right
thf(fact_439_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_440_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_441_zero__reorient,axiom,
! [X3: complex] :
( ( zero_zero_complex = X3 )
= ( X3 = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_442_zero__reorient,axiom,
! [X3: real] :
( ( zero_zero_real = X3 )
= ( X3 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_443_zero__reorient,axiom,
! [X3: rat] :
( ( zero_zero_rat = X3 )
= ( X3 = zero_zero_rat ) ) ).
% zero_reorient
thf(fact_444_zero__reorient,axiom,
! [X3: nat] :
( ( zero_zero_nat = X3 )
= ( X3 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_445_zero__reorient,axiom,
! [X3: int] :
( ( zero_zero_int = X3 )
= ( X3 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_446_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_447_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: rat,B: rat,C2: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_448_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_449_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_450_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: real,J: real,K2: real,L: real] :
( ( ( I = J )
& ( K2 = L ) )
=> ( ( plus_plus_real @ I @ K2 )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_451_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: rat,J: rat,K2: rat,L: rat] :
( ( ( I = J )
& ( K2 = L ) )
=> ( ( plus_plus_rat @ I @ K2 )
= ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_452_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( K2 = L ) )
=> ( ( plus_plus_nat @ I @ K2 )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_453_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( I = J )
& ( K2 = L ) )
=> ( ( plus_plus_int @ I @ K2 )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_454_group__cancel_Oadd1,axiom,
! [A4: real,K2: real,A: real,B: real] :
( ( A4
= ( plus_plus_real @ K2 @ A ) )
=> ( ( plus_plus_real @ A4 @ B )
= ( plus_plus_real @ K2 @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_455_group__cancel_Oadd1,axiom,
! [A4: rat,K2: rat,A: rat,B: rat] :
( ( A4
= ( plus_plus_rat @ K2 @ A ) )
=> ( ( plus_plus_rat @ A4 @ B )
= ( plus_plus_rat @ K2 @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_456_group__cancel_Oadd1,axiom,
! [A4: nat,K2: nat,A: nat,B: nat] :
( ( A4
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( plus_plus_nat @ A4 @ B )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_457_group__cancel_Oadd1,axiom,
! [A4: int,K2: int,A: int,B: int] :
( ( A4
= ( plus_plus_int @ K2 @ A ) )
=> ( ( plus_plus_int @ A4 @ B )
= ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_458_group__cancel_Oadd2,axiom,
! [B5: real,K2: real,B: real,A: real] :
( ( B5
= ( plus_plus_real @ K2 @ B ) )
=> ( ( plus_plus_real @ A @ B5 )
= ( plus_plus_real @ K2 @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_459_group__cancel_Oadd2,axiom,
! [B5: rat,K2: rat,B: rat,A: rat] :
( ( B5
= ( plus_plus_rat @ K2 @ B ) )
=> ( ( plus_plus_rat @ A @ B5 )
= ( plus_plus_rat @ K2 @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_460_group__cancel_Oadd2,axiom,
! [B5: nat,K2: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K2 @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_461_group__cancel_Oadd2,axiom,
! [B5: int,K2: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K2 @ B ) )
=> ( ( plus_plus_int @ A @ B5 )
= ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_462_add_Oassoc,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_463_add_Oassoc,axiom,
! [A: rat,B: rat,C2: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_464_add_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_465_add_Oassoc,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_466_add_Oleft__cancel,axiom,
! [A: real,B: real,C2: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C2 ) )
= ( B = C2 ) ) ).
% add.left_cancel
thf(fact_467_add_Oleft__cancel,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ( plus_plus_rat @ A @ B )
= ( plus_plus_rat @ A @ C2 ) )
= ( B = C2 ) ) ).
% add.left_cancel
thf(fact_468_add_Oleft__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
= ( B = C2 ) ) ).
% add.left_cancel
thf(fact_469_add_Oright__cancel,axiom,
! [B: real,A: real,C2: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C2 @ A ) )
= ( B = C2 ) ) ).
% add.right_cancel
thf(fact_470_add_Oright__cancel,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ( plus_plus_rat @ B @ A )
= ( plus_plus_rat @ C2 @ A ) )
= ( B = C2 ) ) ).
% add.right_cancel
thf(fact_471_add_Oright__cancel,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
= ( B = C2 ) ) ).
% add.right_cancel
thf(fact_472_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A5: real,B4: real] : ( plus_plus_real @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_473_add_Ocommute,axiom,
( plus_plus_rat
= ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_474_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_475_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A5: int,B4: int] : ( plus_plus_int @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_476_add_Oleft__commute,axiom,
! [B: real,A: real,C2: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C2 ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_477_add_Oleft__commute,axiom,
! [B: rat,A: rat,C2: rat] :
( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C2 ) )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_478_add_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C2 ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_479_add_Oleft__commute,axiom,
! [B: int,A: int,C2: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C2 ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_480_add__left__imp__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_481_add__left__imp__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ( plus_plus_rat @ A @ B )
= ( plus_plus_rat @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_482_add__left__imp__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_483_add__left__imp__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_484_add__right__imp__eq,axiom,
! [B: real,A: real,C2: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_485_add__right__imp__eq,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ( plus_plus_rat @ B @ A )
= ( plus_plus_rat @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_486_add__right__imp__eq,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_487_add__right__imp__eq,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_488_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_489_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K2: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K2 = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_490_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: rat,J: rat,K2: rat,L: rat] :
( ( ( ord_less_eq_rat @ I @ J )
& ( K2 = L ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_491_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K2 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_492_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K2 = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_493_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K2: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K2 @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_494_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: rat,J: rat,K2: rat,L: rat] :
( ( ( I = J )
& ( ord_less_eq_rat @ K2 @ L ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_495_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_496_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_497_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K2: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K2 @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_498_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: rat,J: rat,K2: rat,L: rat] :
( ( ( ord_less_eq_rat @ I @ J )
& ( ord_less_eq_rat @ K2 @ L ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_499_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_500_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_501_add__mono,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_502_add__mono,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C2 @ D )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_503_add__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_504_add__mono,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_505_add__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_506_add__left__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_507_add__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_508_add__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_509_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C: nat] :
( B
!= ( plus_plus_nat @ A @ C ) ) ) ).
% less_eqE
thf(fact_510_add__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_511_add__right__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_512_add__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_513_add__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_514_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
? [C3: nat] :
( B4
= ( plus_plus_nat @ A5 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_515_add__le__imp__le__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_516_add__le__imp__le__left,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
=> ( ord_less_eq_rat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_517_add__le__imp__le__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_518_add__le__imp__le__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_519_add__le__imp__le__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_520_add__le__imp__le__right,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
=> ( ord_less_eq_rat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_521_add__le__imp__le__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_522_add__le__imp__le__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_523_comm__monoid__add__class_Oadd__0,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_524_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_525_comm__monoid__add__class_Oadd__0,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_526_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_527_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_528_add_Ocomm__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.comm_neutral
thf(fact_529_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_530_add_Ocomm__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% add.comm_neutral
thf(fact_531_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_532_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_533_add_Ogroup__left__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add.group_left_neutral
thf(fact_534_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_535_add_Ogroup__left__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% add.group_left_neutral
thf(fact_536_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_537_add__nonpos__eq__0__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ( plus_plus_real @ X3 @ Y )
= zero_zero_real )
= ( ( X3 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_538_add__nonpos__eq__0__iff,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
=> ( ( ( plus_plus_rat @ X3 @ Y )
= zero_zero_rat )
= ( ( X3 = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_539_add__nonpos__eq__0__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_540_add__nonpos__eq__0__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X3 @ Y )
= zero_zero_int )
= ( ( X3 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_541_add__nonneg__eq__0__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ X3 @ Y )
= zero_zero_real )
= ( ( X3 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_542_add__nonneg__eq__0__iff,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ( plus_plus_rat @ X3 @ Y )
= zero_zero_rat )
= ( ( X3 = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_543_add__nonneg__eq__0__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_544_add__nonneg__eq__0__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X3 @ Y )
= zero_zero_int )
= ( ( X3 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_545_add__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_546_add__nonpos__nonpos,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% add_nonpos_nonpos
thf(fact_547_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_548_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_549_add__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_550_add__nonneg__nonneg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_551_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_552_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_553_add__increasing2,axiom,
! [C2: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_554_add__increasing2,axiom,
! [C2: rat,B: rat,A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ( ord_less_eq_rat @ B @ A )
=> ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_555_add__increasing2,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_556_add__increasing2,axiom,
! [C2: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_557_add__decreasing2,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_558_add__decreasing2,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_559_add__decreasing2,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_560_add__decreasing2,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_561_add__increasing,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_562_add__increasing,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_563_add__increasing,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_564_add__increasing,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_565_add__decreasing,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C2 @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_566_add__decreasing,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ C2 @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_567_add__decreasing,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_568_add__decreasing,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_569_option_Osize__gen_I2_J,axiom,
! [X3: product_prod_nat_nat > nat,X2: product_prod_nat_nat] :
( ( size_o8335143837870341156at_nat @ X3 @ ( some_P7363390416028606310at_nat @ X2 ) )
= ( plus_plus_nat @ ( X3 @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% option.size_gen(2)
thf(fact_570_option_Osize__gen_I2_J,axiom,
! [X3: num > nat,X2: num] :
( ( size_option_num @ X3 @ ( some_num @ X2 ) )
= ( plus_plus_nat @ ( X3 @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% option.size_gen(2)
thf(fact_571_subset__antisym,axiom,
! [A4: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( A4 = B5 ) ) ) ).
% subset_antisym
thf(fact_572_subsetI,axiom,
! [A4: set_complex,B5: set_complex] :
( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( member_complex @ X4 @ B5 ) )
=> ( ord_le211207098394363844omplex @ A4 @ B5 ) ) ).
% subsetI
thf(fact_573_subsetI,axiom,
! [A4: set_real,B5: set_real] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( member_real @ X4 @ B5 ) )
=> ( ord_less_eq_set_real @ A4 @ B5 ) ) ).
% subsetI
thf(fact_574_subsetI,axiom,
! [A4: set_set_nat,B5: set_set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A4 )
=> ( member_set_nat @ X4 @ B5 ) )
=> ( ord_le6893508408891458716et_nat @ A4 @ B5 ) ) ).
% subsetI
thf(fact_575_subsetI,axiom,
! [A4: set_nat,B5: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( member_nat @ X4 @ B5 ) )
=> ( ord_less_eq_set_nat @ A4 @ B5 ) ) ).
% subsetI
thf(fact_576_subsetI,axiom,
! [A4: set_int,B5: set_int] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( member_int @ X4 @ B5 ) )
=> ( ord_less_eq_set_int @ A4 @ B5 ) ) ).
% subsetI
thf(fact_577_option_Osize__gen_I1_J,axiom,
! [X3: product_prod_nat_nat > nat] :
( ( size_o8335143837870341156at_nat @ X3 @ none_P5556105721700978146at_nat )
= ( suc @ zero_zero_nat ) ) ).
% option.size_gen(1)
thf(fact_578_option_Osize__gen_I1_J,axiom,
! [X3: num > nat] :
( ( size_option_num @ X3 @ none_num )
= ( suc @ zero_zero_nat ) ) ).
% option.size_gen(1)
thf(fact_579_option_Osize_I3_J,axiom,
( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
= ( suc @ zero_zero_nat ) ) ).
% option.size(3)
thf(fact_580_option_Osize_I3_J,axiom,
( ( size_size_option_num @ none_num )
= ( suc @ zero_zero_nat ) ) ).
% option.size(3)
thf(fact_581_option_Osize_I4_J,axiom,
! [X2: product_prod_nat_nat] :
( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X2 ) )
= ( suc @ zero_zero_nat ) ) ).
% option.size(4)
thf(fact_582_option_Osize_I4_J,axiom,
! [X2: num] :
( ( size_size_option_num @ ( some_num @ X2 ) )
= ( suc @ zero_zero_nat ) ) ).
% option.size(4)
thf(fact_583_Euclid__induct,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( P2 @ A3 @ B3 )
= ( P2 @ B3 @ A3 ) )
=> ( ! [A3: nat] : ( P2 @ A3 @ zero_zero_nat )
=> ( ! [A3: nat,B3: nat] :
( ( P2 @ A3 @ B3 )
=> ( P2 @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
=> ( P2 @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_584_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_585_exists__least__lemma,axiom,
! [P2: nat > $o] :
( ~ ( P2 @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P2 @ X_1 )
=> ? [N3: nat] :
( ~ ( P2 @ N3 )
& ( P2 @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_586_add__0__iff,axiom,
! [B: complex,A: complex] :
( ( B
= ( plus_plus_complex @ B @ A ) )
= ( A = zero_zero_complex ) ) ).
% add_0_iff
thf(fact_587_add__0__iff,axiom,
! [B: real,A: real] :
( ( B
= ( plus_plus_real @ B @ A ) )
= ( A = zero_zero_real ) ) ).
% add_0_iff
thf(fact_588_add__0__iff,axiom,
! [B: rat,A: rat] :
( ( B
= ( plus_plus_rat @ B @ A ) )
= ( A = zero_zero_rat ) ) ).
% add_0_iff
thf(fact_589_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_590_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_591_triangle__0,axiom,
( ( nat_triangle @ zero_zero_nat )
= zero_zero_nat ) ).
% triangle_0
thf(fact_592_size__neq__size__imp__neq,axiom,
! [X3: vEBT_VEBT,Y: vEBT_VEBT] :
( ( ( size_size_VEBT_VEBT @ X3 )
!= ( size_size_VEBT_VEBT @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_593_size__neq__size__imp__neq,axiom,
! [X3: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ X3 )
!= ( size_s6755466524823107622T_VEBT @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_594_size__neq__size__imp__neq,axiom,
! [X3: num,Y: num] :
( ( ( size_size_num @ X3 )
!= ( size_size_num @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_595_size__neq__size__imp__neq,axiom,
! [X3: list_int,Y: list_int] :
( ( ( size_size_list_int @ X3 )
!= ( size_size_list_int @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_596_size__neq__size__imp__neq,axiom,
! [X3: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X3 )
!= ( size_size_list_nat @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_597_in__mono,axiom,
! [A4: set_complex,B5: set_complex,X3: complex] :
( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ( member_complex @ X3 @ A4 )
=> ( member_complex @ X3 @ B5 ) ) ) ).
% in_mono
thf(fact_598_in__mono,axiom,
! [A4: set_real,B5: set_real,X3: real] :
( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ( member_real @ X3 @ A4 )
=> ( member_real @ X3 @ B5 ) ) ) ).
% in_mono
thf(fact_599_in__mono,axiom,
! [A4: set_set_nat,B5: set_set_nat,X3: set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B5 )
=> ( ( member_set_nat @ X3 @ A4 )
=> ( member_set_nat @ X3 @ B5 ) ) ) ).
% in_mono
thf(fact_600_in__mono,axiom,
! [A4: set_nat,B5: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_nat @ X3 @ A4 )
=> ( member_nat @ X3 @ B5 ) ) ) ).
% in_mono
thf(fact_601_in__mono,axiom,
! [A4: set_int,B5: set_int,X3: int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( member_int @ X3 @ A4 )
=> ( member_int @ X3 @ B5 ) ) ) ).
% in_mono
thf(fact_602_subsetD,axiom,
! [A4: set_complex,B5: set_complex,C2: complex] :
( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ( member_complex @ C2 @ A4 )
=> ( member_complex @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_603_subsetD,axiom,
! [A4: set_real,B5: set_real,C2: real] :
( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ( member_real @ C2 @ A4 )
=> ( member_real @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_604_subsetD,axiom,
! [A4: set_set_nat,B5: set_set_nat,C2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B5 )
=> ( ( member_set_nat @ C2 @ A4 )
=> ( member_set_nat @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_605_subsetD,axiom,
! [A4: set_nat,B5: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_nat @ C2 @ A4 )
=> ( member_nat @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_606_subsetD,axiom,
! [A4: set_int,B5: set_int,C2: int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( member_int @ C2 @ A4 )
=> ( member_int @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_607_equalityE,axiom,
! [A4: set_int,B5: set_int] :
( ( A4 = B5 )
=> ~ ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ~ ( ord_less_eq_set_int @ B5 @ A4 ) ) ) ).
% equalityE
thf(fact_608_subset__eq,axiom,
( ord_le211207098394363844omplex
= ( ^ [A6: set_complex,B6: set_complex] :
! [X5: complex] :
( ( member_complex @ X5 @ A6 )
=> ( member_complex @ X5 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_609_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [X5: real] :
( ( member_real @ X5 @ A6 )
=> ( member_real @ X5 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_610_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A6: set_set_nat,B6: set_set_nat] :
! [X5: set_nat] :
( ( member_set_nat @ X5 @ A6 )
=> ( member_set_nat @ X5 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_611_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [X5: nat] :
( ( member_nat @ X5 @ A6 )
=> ( member_nat @ X5 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_612_subset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
! [X5: int] :
( ( member_int @ X5 @ A6 )
=> ( member_int @ X5 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_613_equalityD1,axiom,
! [A4: set_int,B5: set_int] :
( ( A4 = B5 )
=> ( ord_less_eq_set_int @ A4 @ B5 ) ) ).
% equalityD1
thf(fact_614_equalityD2,axiom,
! [A4: set_int,B5: set_int] :
( ( A4 = B5 )
=> ( ord_less_eq_set_int @ B5 @ A4 ) ) ).
% equalityD2
thf(fact_615_subset__iff,axiom,
( ord_le211207098394363844omplex
= ( ^ [A6: set_complex,B6: set_complex] :
! [T3: complex] :
( ( member_complex @ T3 @ A6 )
=> ( member_complex @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_616_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [T3: real] :
( ( member_real @ T3 @ A6 )
=> ( member_real @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_617_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A6: set_set_nat,B6: set_set_nat] :
! [T3: set_nat] :
( ( member_set_nat @ T3 @ A6 )
=> ( member_set_nat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_618_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A6 )
=> ( member_nat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_619_subset__iff,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
! [T3: int] :
( ( member_int @ T3 @ A6 )
=> ( member_int @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_620_subset__refl,axiom,
! [A4: set_int] : ( ord_less_eq_set_int @ A4 @ A4 ) ).
% subset_refl
thf(fact_621_Collect__mono,axiom,
! [P2: real > $o,Q: real > $o] :
( ! [X4: real] :
( ( P2 @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_real @ ( collect_real @ P2 ) @ ( collect_real @ Q ) ) ) ).
% Collect_mono
thf(fact_622_Collect__mono,axiom,
! [P2: list_nat > $o,Q: list_nat > $o] :
( ! [X4: list_nat] :
( ( P2 @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P2 ) @ ( collect_list_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_623_Collect__mono,axiom,
! [P2: set_nat > $o,Q: set_nat > $o] :
( ! [X4: set_nat] :
( ( P2 @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P2 ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_624_Collect__mono,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P2 @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_625_Collect__mono,axiom,
! [P2: int > $o,Q: int > $o] :
( ! [X4: int] :
( ( P2 @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_int @ ( collect_int @ P2 ) @ ( collect_int @ Q ) ) ) ).
% Collect_mono
thf(fact_626_subset__trans,axiom,
! [A4: set_int,B5: set_int,C4: set_int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( ord_less_eq_set_int @ B5 @ C4 )
=> ( ord_less_eq_set_int @ A4 @ C4 ) ) ) ).
% subset_trans
thf(fact_627_set__eq__subset,axiom,
( ( ^ [Y4: set_int,Z: set_int] : Y4 = Z )
= ( ^ [A6: set_int,B6: set_int] :
( ( ord_less_eq_set_int @ A6 @ B6 )
& ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_628_Collect__mono__iff,axiom,
! [P2: real > $o,Q: real > $o] :
( ( ord_less_eq_set_real @ ( collect_real @ P2 ) @ ( collect_real @ Q ) )
= ( ! [X5: real] :
( ( P2 @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_629_Collect__mono__iff,axiom,
! [P2: list_nat > $o,Q: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P2 ) @ ( collect_list_nat @ Q ) )
= ( ! [X5: list_nat] :
( ( P2 @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_630_Collect__mono__iff,axiom,
! [P2: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P2 ) @ ( collect_set_nat @ Q ) )
= ( ! [X5: set_nat] :
( ( P2 @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_631_Collect__mono__iff,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) )
= ( ! [X5: nat] :
( ( P2 @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_632_Collect__mono__iff,axiom,
! [P2: int > $o,Q: int > $o] :
( ( ord_less_eq_set_int @ ( collect_int @ P2 ) @ ( collect_int @ Q ) )
= ( ! [X5: int] :
( ( P2 @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_633_verit__sum__simplify,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% verit_sum_simplify
thf(fact_634_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_635_verit__sum__simplify,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% verit_sum_simplify
thf(fact_636_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_637_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_638_deg__not__0,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% deg_not_0
thf(fact_639_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_640_le__numeral__extra_I3_J,axiom,
ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% le_numeral_extra(3)
thf(fact_641_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_642_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_643_vebt__member_Ocases,axiom,
! [X3: produc9072475918466114483BT_nat] :
( ! [A3: $o,B3: $o,X4: nat] :
( X3
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X4 ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X4: nat] :
( X3
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X4 ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X4: nat] :
( X3
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X4 ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
( X3
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X4 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X3
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ X4 ) ) ) ) ) ) ).
% vebt_member.cases
thf(fact_644_vebt__insert_Ocases,axiom,
! [X3: produc9072475918466114483BT_nat] :
( ! [A3: $o,B3: $o,X4: nat] :
( X3
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X4 ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X4: nat] :
( X3
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S ) @ X4 ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X4: nat] :
( X3
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) @ X4 ) )
=> ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X3
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ X4 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X3
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ X4 ) ) ) ) ) ) ).
% vebt_insert.cases
thf(fact_645_VEBT__internal_Omembermima_Ocases,axiom,
! [X3: produc9072475918466114483BT_nat] :
( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
( X3
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
( X3
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X4: nat] :
( X3
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X4 ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
( X3
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ X4 ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X4: nat] :
( X3
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd ) @ X4 ) ) ) ) ) ) ).
% VEBT_internal.membermima.cases
thf(fact_646_count__notin,axiom,
! [X3: complex,Xs: list_complex] :
( ~ ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
=> ( ( count_list_complex @ Xs @ X3 )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_647_count__notin,axiom,
! [X3: real,Xs: list_real] :
( ~ ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( ( count_list_real @ Xs @ X3 )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_648_count__notin,axiom,
! [X3: set_nat,Xs: list_set_nat] :
( ~ ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
=> ( ( count_list_set_nat @ Xs @ X3 )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_649_count__notin,axiom,
! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
( ~ ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( ( count_list_VEBT_VEBT @ Xs @ X3 )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_650_count__notin,axiom,
! [X3: int,Xs: list_int] :
( ~ ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( ( count_list_int @ Xs @ X3 )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_651_count__notin,axiom,
! [X3: nat,Xs: list_nat] :
( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( count_list_nat @ Xs @ X3 )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_652_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X3: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb ) @ X3 )
= ( ( X3 = Mi )
| ( X3 = Ma ) ) ) ).
% VEBT_internal.membermima.simps(3)
thf(fact_653_Leaf__0__not,axiom,
! [A: $o,B: $o] :
~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).
% Leaf_0_not
thf(fact_654__092_060open_062x_A_061_Ami_A_092_060or_062_Ax_A_061_Ama_A_092_060or_062_Aboth__member__options_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_092_060close_062,axiom,
( ( xa = mi )
| ( xa = ma )
| ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) ).
% \<open>x = mi \<or> x = ma \<or> both_member_options (treeList ! high x n) (low x n)\<close>
thf(fact_655_VEBT__internal_OminNull_Oelims_I1_J,axiom,
! [X3: vEBT_VEBT,Y: $o] :
( ( ( vEBT_VEBT_minNull @ X3 )
= Y )
=> ( ( ( X3
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ Y )
=> ( ( ? [Uv2: $o] :
( X3
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> Y )
=> ( ( ? [Uu2: $o] :
( X3
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> Y )
=> ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ Y )
=> ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
=> Y ) ) ) ) ) ) ).
% VEBT_internal.minNull.elims(1)
thf(fact_656_inthall,axiom,
! [Xs: list_complex,P2: complex > $o,N: nat] :
( ! [X4: complex] :
( ( member_complex @ X4 @ ( set_complex2 @ Xs ) )
=> ( P2 @ X4 ) )
=> ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( P2 @ ( nth_complex @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_657_inthall,axiom,
! [Xs: list_real,P2: real > $o,N: nat] :
( ! [X4: real] :
( ( member_real @ X4 @ ( set_real2 @ Xs ) )
=> ( P2 @ X4 ) )
=> ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
=> ( P2 @ ( nth_real @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_658_inthall,axiom,
! [Xs: list_set_nat,P2: set_nat > $o,N: nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs ) )
=> ( P2 @ X4 ) )
=> ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
=> ( P2 @ ( nth_set_nat @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_659_inthall,axiom,
! [Xs: list_VEBT_VEBT,P2: vEBT_VEBT > $o,N: nat] :
( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( P2 @ X4 ) )
=> ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( P2 @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_660_inthall,axiom,
! [Xs: list_int,P2: int > $o,N: nat] :
( ! [X4: int] :
( ( member_int @ X4 @ ( set_int2 @ Xs ) )
=> ( P2 @ X4 ) )
=> ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( P2 @ ( nth_int @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_661_inthall,axiom,
! [Xs: list_nat,P2: nat > $o,N: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( P2 @ X4 ) )
=> ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( P2 @ ( nth_nat @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_662_bit__split__inv,axiom,
! [X3: nat,D: nat] :
( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X3 @ D ) @ ( vEBT_VEBT_low @ X3 @ D ) @ D )
= X3 ) ).
% bit_split_inv
thf(fact_663_VEBT_Oinject_I2_J,axiom,
! [X21: $o,X222: $o,Y21: $o,Y22: $o] :
( ( ( vEBT_Leaf @ X21 @ X222 )
= ( vEBT_Leaf @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% VEBT.inject(2)
thf(fact_664_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_665_add__less__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_666_add__less__cancel__left,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
= ( ord_less_rat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_667_add__less__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_668_add__less__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_669_add__less__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_670_add__less__cancel__right,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
= ( ord_less_rat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_671_add__less__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_672_add__less__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_673_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_674_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_675_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_676_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_677_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_678_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_679_nat__add__left__cancel__less,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_680_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_681_add__less__same__cancel1,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% add_less_same_cancel1
thf(fact_682_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_683_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_684_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_685_add__less__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% add_less_same_cancel2
thf(fact_686_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_687_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_688_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_689_less__add__same__cancel1,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
= ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% less_add_same_cancel1
thf(fact_690_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_691_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_692_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_693_less__add__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
= ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% less_add_same_cancel2
thf(fact_694_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_695_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_696_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_697_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_698_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_699_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_700_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_701_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_702_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_703_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_704_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_705_length__induct,axiom,
! [P2: list_VEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
( ! [Xs2: list_VEBT_VEBT] :
( ! [Ys: list_VEBT_VEBT] :
( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys ) @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( P2 @ Ys ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_706_length__induct,axiom,
! [P2: list_int > $o,Xs: list_int] :
( ! [Xs2: list_int] :
( ! [Ys: list_int] :
( ( ord_less_nat @ ( size_size_list_int @ Ys ) @ ( size_size_list_int @ Xs2 ) )
=> ( P2 @ Ys ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_707_length__induct,axiom,
! [P2: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P2 @ Ys ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_708_nth__equalityI,axiom,
! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs )
= ( size_s6755466524823107622T_VEBT @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( nth_VEBT_VEBT @ Xs @ I2 )
= ( nth_VEBT_VEBT @ Ys2 @ I2 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_709_nth__equalityI,axiom,
! [Xs: list_int,Ys2: list_int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ Xs @ I2 )
= ( nth_int @ Ys2 @ I2 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_710_nth__equalityI,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys2 @ I2 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_711_Skolem__list__nth,axiom,
! [K2: nat,P2: nat > vEBT_VEBT > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ? [X7: vEBT_VEBT] : ( P2 @ I3 @ X7 ) ) )
= ( ? [Xs3: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs3 )
= K2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ( P2 @ I3 @ ( nth_VEBT_VEBT @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_712_Skolem__list__nth,axiom,
! [K2: nat,P2: nat > int > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ? [X7: int] : ( P2 @ I3 @ X7 ) ) )
= ( ? [Xs3: list_int] :
( ( ( size_size_list_int @ Xs3 )
= K2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ( P2 @ I3 @ ( nth_int @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_713_Skolem__list__nth,axiom,
! [K2: nat,P2: nat > nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ? [X7: nat] : ( P2 @ I3 @ X7 ) ) )
= ( ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= K2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ( P2 @ I3 @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_714_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_VEBT_VEBT,Z: list_VEBT_VEBT] : Y4 = Z )
= ( ^ [Xs3: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs3 )
= ( size_s6755466524823107622T_VEBT @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
=> ( ( nth_VEBT_VEBT @ Xs3 @ I3 )
= ( nth_VEBT_VEBT @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_715_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_int,Z: list_int] : Y4 = Z )
= ( ^ [Xs3: list_int,Ys3: list_int] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_int @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs3 ) )
=> ( ( nth_int @ Xs3 @ I3 )
= ( nth_int @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_716_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_nat,Z: list_nat] : Y4 = Z )
= ( ^ [Xs3: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( nth_nat @ Xs3 @ I3 )
= ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_717_all__set__conv__all__nth,axiom,
! [Xs: list_VEBT_VEBT,P2: vEBT_VEBT > $o] :
( ( ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( P2 @ X5 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( P2 @ ( nth_VEBT_VEBT @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_718_all__set__conv__all__nth,axiom,
! [Xs: list_int,P2: int > $o] :
( ( ! [X5: int] :
( ( member_int @ X5 @ ( set_int2 @ Xs ) )
=> ( P2 @ X5 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
=> ( P2 @ ( nth_int @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_719_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P2: nat > $o] :
( ( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
=> ( P2 @ X5 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( P2 @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_720_all__nth__imp__all__set,axiom,
! [Xs: list_complex,P2: complex > $o,X3: complex] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( P2 @ ( nth_complex @ Xs @ I2 ) ) )
=> ( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
=> ( P2 @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_721_all__nth__imp__all__set,axiom,
! [Xs: list_real,P2: real > $o,X3: real] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs ) )
=> ( P2 @ ( nth_real @ Xs @ I2 ) ) )
=> ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( P2 @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_722_all__nth__imp__all__set,axiom,
! [Xs: list_set_nat,P2: set_nat > $o,X3: set_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Xs ) )
=> ( P2 @ ( nth_set_nat @ Xs @ I2 ) ) )
=> ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
=> ( P2 @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_723_all__nth__imp__all__set,axiom,
! [Xs: list_VEBT_VEBT,P2: vEBT_VEBT > $o,X3: vEBT_VEBT] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( P2 @ ( nth_VEBT_VEBT @ Xs @ I2 ) ) )
=> ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( P2 @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_724_all__nth__imp__all__set,axiom,
! [Xs: list_int,P2: int > $o,X3: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
=> ( P2 @ ( nth_int @ Xs @ I2 ) ) )
=> ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( P2 @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_725_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P2: nat > $o,X3: nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( P2 @ ( nth_nat @ Xs @ I2 ) ) )
=> ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P2 @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_726_in__set__conv__nth,axiom,
! [X3: complex,Xs: list_complex] :
( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs ) )
& ( ( nth_complex @ Xs @ I3 )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_727_in__set__conv__nth,axiom,
! [X3: real,Xs: list_real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs ) )
& ( ( nth_real @ Xs @ I3 )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_728_in__set__conv__nth,axiom,
! [X3: set_nat,Xs: list_set_nat] :
( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3254054031482475050et_nat @ Xs ) )
& ( ( nth_set_nat @ Xs @ I3 )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_729_in__set__conv__nth,axiom,
! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
& ( ( nth_VEBT_VEBT @ Xs @ I3 )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_730_in__set__conv__nth,axiom,
! [X3: int,Xs: list_int] :
( ( member_int @ X3 @ ( set_int2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
& ( ( nth_int @ Xs @ I3 )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_731_in__set__conv__nth,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I3 )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_732_list__ball__nth,axiom,
! [N: nat,Xs: list_VEBT_VEBT,P2: vEBT_VEBT > $o] :
( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( P2 @ X4 ) )
=> ( P2 @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_733_list__ball__nth,axiom,
! [N: nat,Xs: list_int,P2: int > $o] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( set_int2 @ Xs ) )
=> ( P2 @ X4 ) )
=> ( P2 @ ( nth_int @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_734_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P2: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( P2 @ X4 ) )
=> ( P2 @ ( nth_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_735_nth__mem,axiom,
! [N: nat,Xs: list_complex] :
( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( member_complex @ ( nth_complex @ Xs @ N ) @ ( set_complex2 @ Xs ) ) ) ).
% nth_mem
thf(fact_736_nth__mem,axiom,
! [N: nat,Xs: list_real] :
( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
=> ( member_real @ ( nth_real @ Xs @ N ) @ ( set_real2 @ Xs ) ) ) ).
% nth_mem
thf(fact_737_nth__mem,axiom,
! [N: nat,Xs: list_set_nat] :
( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
=> ( member_set_nat @ ( nth_set_nat @ Xs @ N ) @ ( set_set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_738_nth__mem,axiom,
! [N: nat,Xs: list_VEBT_VEBT] :
( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs @ N ) @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).
% nth_mem
thf(fact_739_nth__mem,axiom,
! [N: nat,Xs: list_int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( member_int @ ( nth_int @ Xs @ N ) @ ( set_int2 @ Xs ) ) ) ).
% nth_mem
thf(fact_740_nth__mem,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_741_finite__maxlen,axiom,
! [M6: set_list_VEBT_VEBT] :
( ( finite3004134309566078307T_VEBT @ M6 )
=> ? [N3: nat] :
! [X: list_VEBT_VEBT] :
( ( member2936631157270082147T_VEBT @ X @ M6 )
=> ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_742_finite__maxlen,axiom,
! [M6: set_list_int] :
( ( finite3922522038869484883st_int @ M6 )
=> ? [N3: nat] :
! [X: list_int] :
( ( member_list_int @ X @ M6 )
=> ( ord_less_nat @ ( size_size_list_int @ X ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_743_finite__maxlen,axiom,
! [M6: set_list_nat] :
( ( finite8100373058378681591st_nat @ M6 )
=> ? [N3: nat] :
! [X: list_nat] :
( ( member_list_nat @ X @ M6 )
=> ( ord_less_nat @ ( size_size_list_nat @ X ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_744_order__less__imp__not__less,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ~ ( ord_less_real @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_745_order__less__imp__not__less,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_rat @ X3 @ Y )
=> ~ ( ord_less_rat @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_746_order__less__imp__not__less,axiom,
! [X3: num,Y: num] :
( ( ord_less_num @ X3 @ Y )
=> ~ ( ord_less_num @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_747_order__less__imp__not__less,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_748_order__less__imp__not__less,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ~ ( ord_less_int @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_749_order__less__imp__not__eq2,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_750_order__less__imp__not__eq2,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_rat @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_751_order__less__imp__not__eq2,axiom,
! [X3: num,Y: num] :
( ( ord_less_num @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_752_order__less__imp__not__eq2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_753_order__less__imp__not__eq2,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_754_order__less__imp__not__eq,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_755_order__less__imp__not__eq,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_rat @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_756_order__less__imp__not__eq,axiom,
! [X3: num,Y: num] :
( ( ord_less_num @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_757_order__less__imp__not__eq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_758_order__less__imp__not__eq,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_759_linorder__less__linear,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
| ( X3 = Y )
| ( ord_less_real @ Y @ X3 ) ) ).
% linorder_less_linear
thf(fact_760_linorder__less__linear,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_rat @ X3 @ Y )
| ( X3 = Y )
| ( ord_less_rat @ Y @ X3 ) ) ).
% linorder_less_linear
thf(fact_761_linorder__less__linear,axiom,
! [X3: num,Y: num] :
( ( ord_less_num @ X3 @ Y )
| ( X3 = Y )
| ( ord_less_num @ Y @ X3 ) ) ).
% linorder_less_linear
thf(fact_762_linorder__less__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
| ( X3 = Y )
| ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_less_linear
thf(fact_763_linorder__less__linear,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
| ( X3 = Y )
| ( ord_less_int @ Y @ X3 ) ) ).
% linorder_less_linear
thf(fact_764_order__less__imp__triv,axiom,
! [X3: real,Y: real,P2: $o] :
( ( ord_less_real @ X3 @ Y )
=> ( ( ord_less_real @ Y @ X3 )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_765_order__less__imp__triv,axiom,
! [X3: rat,Y: rat,P2: $o] :
( ( ord_less_rat @ X3 @ Y )
=> ( ( ord_less_rat @ Y @ X3 )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_766_order__less__imp__triv,axiom,
! [X3: num,Y: num,P2: $o] :
( ( ord_less_num @ X3 @ Y )
=> ( ( ord_less_num @ Y @ X3 )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_767_order__less__imp__triv,axiom,
! [X3: nat,Y: nat,P2: $o] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ X3 )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_768_order__less__imp__triv,axiom,
! [X3: int,Y: int,P2: $o] :
( ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_int @ Y @ X3 )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_769_order__less__not__sym,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ~ ( ord_less_real @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_770_order__less__not__sym,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_rat @ X3 @ Y )
=> ~ ( ord_less_rat @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_771_order__less__not__sym,axiom,
! [X3: num,Y: num] :
( ( ord_less_num @ X3 @ Y )
=> ~ ( ord_less_num @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_772_order__less__not__sym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_773_order__less__not__sym,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ~ ( ord_less_int @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_774_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_775_order__less__subst2,axiom,
! [A: real,B: real,F: real > rat,C2: rat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_776_order__less__subst2,axiom,
! [A: real,B: real,F: real > num,C2: num] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_777_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_778_order__less__subst2,axiom,
! [A: real,B: real,F: real > int,C2: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_779_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > real,C2: real] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_780_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_781_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > num,C2: num] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_782_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > nat,C2: nat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_783_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > int,C2: int] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_784_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C2: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_785_order__less__subst1,axiom,
! [A: real,F: rat > real,B: rat,C2: rat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_786_order__less__subst1,axiom,
! [A: real,F: num > real,B: num,C2: num] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_787_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C2: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_788_order__less__subst1,axiom,
! [A: real,F: int > real,B: int,C2: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_789_order__less__subst1,axiom,
! [A: rat,F: real > rat,B: real,C2: real] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_790_order__less__subst1,axiom,
! [A: rat,F: rat > rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_791_order__less__subst1,axiom,
! [A: rat,F: num > rat,B: num,C2: num] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_792_order__less__subst1,axiom,
! [A: rat,F: nat > rat,B: nat,C2: nat] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_793_order__less__subst1,axiom,
! [A: rat,F: int > rat,B: int,C2: int] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_794_order__less__irrefl,axiom,
! [X3: real] :
~ ( ord_less_real @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_795_order__less__irrefl,axiom,
! [X3: rat] :
~ ( ord_less_rat @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_796_order__less__irrefl,axiom,
! [X3: num] :
~ ( ord_less_num @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_797_order__less__irrefl,axiom,
! [X3: nat] :
~ ( ord_less_nat @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_798_order__less__irrefl,axiom,
! [X3: int] :
~ ( ord_less_int @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_799_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_800_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > rat,C2: rat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_801_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > num,C2: num] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_802_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_803_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C2: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_804_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > real,C2: real] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_805_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_806_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > num,C2: num] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_807_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > nat,C2: nat] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_808_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > int,C2: int] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_809_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_810_ord__eq__less__subst,axiom,
! [A: rat,F: real > rat,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_811_ord__eq__less__subst,axiom,
! [A: num,F: real > num,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_812_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_813_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_814_ord__eq__less__subst,axiom,
! [A: real,F: rat > real,B: rat,C2: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_815_ord__eq__less__subst,axiom,
! [A: rat,F: rat > rat,B: rat,C2: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_816_ord__eq__less__subst,axiom,
! [A: num,F: rat > num,B: rat,C2: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_817_ord__eq__less__subst,axiom,
! [A: nat,F: rat > nat,B: rat,C2: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_818_ord__eq__less__subst,axiom,
! [A: int,F: rat > int,B: rat,C2: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_819_order__less__trans,axiom,
! [X3: real,Y: real,Z2: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X3 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_820_order__less__trans,axiom,
! [X3: rat,Y: rat,Z2: rat] :
( ( ord_less_rat @ X3 @ Y )
=> ( ( ord_less_rat @ Y @ Z2 )
=> ( ord_less_rat @ X3 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_821_order__less__trans,axiom,
! [X3: num,Y: num,Z2: num] :
( ( ord_less_num @ X3 @ Y )
=> ( ( ord_less_num @ Y @ Z2 )
=> ( ord_less_num @ X3 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_822_order__less__trans,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X3 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_823_order__less__trans,axiom,
! [X3: int,Y: int,Z2: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X3 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_824_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_825_order__less__asym_H,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ~ ( ord_less_rat @ B @ A ) ) ).
% order_less_asym'
thf(fact_826_order__less__asym_H,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order_less_asym'
thf(fact_827_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_828_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_829_linorder__neq__iff,axiom,
! [X3: real,Y: real] :
( ( X3 != Y )
= ( ( ord_less_real @ X3 @ Y )
| ( ord_less_real @ Y @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_830_linorder__neq__iff,axiom,
! [X3: rat,Y: rat] :
( ( X3 != Y )
= ( ( ord_less_rat @ X3 @ Y )
| ( ord_less_rat @ Y @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_831_linorder__neq__iff,axiom,
! [X3: num,Y: num] :
( ( X3 != Y )
= ( ( ord_less_num @ X3 @ Y )
| ( ord_less_num @ Y @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_832_linorder__neq__iff,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
= ( ( ord_less_nat @ X3 @ Y )
| ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_833_linorder__neq__iff,axiom,
! [X3: int,Y: int] :
( ( X3 != Y )
= ( ( ord_less_int @ X3 @ Y )
| ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_834_order__less__asym,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ~ ( ord_less_real @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_835_order__less__asym,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_rat @ X3 @ Y )
=> ~ ( ord_less_rat @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_836_order__less__asym,axiom,
! [X3: num,Y: num] :
( ( ord_less_num @ X3 @ Y )
=> ~ ( ord_less_num @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_837_order__less__asym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_838_order__less__asym,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ~ ( ord_less_int @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_839_linorder__neqE,axiom,
! [X3: real,Y: real] :
( ( X3 != Y )
=> ( ~ ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ Y @ X3 ) ) ) ).
% linorder_neqE
thf(fact_840_linorder__neqE,axiom,
! [X3: rat,Y: rat] :
( ( X3 != Y )
=> ( ~ ( ord_less_rat @ X3 @ Y )
=> ( ord_less_rat @ Y @ X3 ) ) ) ).
% linorder_neqE
thf(fact_841_linorder__neqE,axiom,
! [X3: num,Y: num] :
( ( X3 != Y )
=> ( ~ ( ord_less_num @ X3 @ Y )
=> ( ord_less_num @ Y @ X3 ) ) ) ).
% linorder_neqE
thf(fact_842_linorder__neqE,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neqE
thf(fact_843_linorder__neqE,axiom,
! [X3: int,Y: int] :
( ( X3 != Y )
=> ( ~ ( ord_less_int @ X3 @ Y )
=> ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_neqE
thf(fact_844_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_845_dual__order_Ostrict__implies__not__eq,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_846_dual__order_Ostrict__implies__not__eq,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_847_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_848_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_849_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_850_order_Ostrict__implies__not__eq,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_851_order_Ostrict__implies__not__eq,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_852_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_853_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_854_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C2 @ B )
=> ( ord_less_real @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_855_dual__order_Ostrict__trans,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ C2 @ B )
=> ( ord_less_rat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_856_dual__order_Ostrict__trans,axiom,
! [B: num,A: num,C2: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_num @ C2 @ B )
=> ( ord_less_num @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_857_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_858_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_859_not__less__iff__gr__or__eq,axiom,
! [X3: real,Y: real] :
( ( ~ ( ord_less_real @ X3 @ Y ) )
= ( ( ord_less_real @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_860_not__less__iff__gr__or__eq,axiom,
! [X3: rat,Y: rat] :
( ( ~ ( ord_less_rat @ X3 @ Y ) )
= ( ( ord_less_rat @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_861_not__less__iff__gr__or__eq,axiom,
! [X3: num,Y: num] :
( ( ~ ( ord_less_num @ X3 @ Y ) )
= ( ( ord_less_num @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_862_not__less__iff__gr__or__eq,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( ( ord_less_nat @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_863_not__less__iff__gr__or__eq,axiom,
! [X3: int,Y: int] :
( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( ( ord_less_int @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_864_order_Ostrict__trans,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_865_order_Ostrict__trans,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ord_less_rat @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_866_order_Ostrict__trans,axiom,
! [A: num,B: num,C2: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ B @ C2 )
=> ( ord_less_num @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_867_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_868_order_Ostrict__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_869_linorder__less__wlog,axiom,
! [P2: real > real > $o,A: real,B: real] :
( ! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: real] : ( P2 @ A3 @ A3 )
=> ( ! [A3: real,B3: real] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_870_linorder__less__wlog,axiom,
! [P2: rat > rat > $o,A: rat,B: rat] :
( ! [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: rat] : ( P2 @ A3 @ A3 )
=> ( ! [A3: rat,B3: rat] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_871_linorder__less__wlog,axiom,
! [P2: num > num > $o,A: num,B: num] :
( ! [A3: num,B3: num] :
( ( ord_less_num @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: num] : ( P2 @ A3 @ A3 )
=> ( ! [A3: num,B3: num] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_872_linorder__less__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P2 @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_873_linorder__less__wlog,axiom,
! [P2: int > int > $o,A: int,B: int] :
( ! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: int] : ( P2 @ A3 @ A3 )
=> ( ! [A3: int,B3: int] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_874_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
? [N4: nat] :
( ( P4 @ N4 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ~ ( P4 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_875_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_876_dual__order_Oirrefl,axiom,
! [A: rat] :
~ ( ord_less_rat @ A @ A ) ).
% dual_order.irrefl
thf(fact_877_dual__order_Oirrefl,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% dual_order.irrefl
thf(fact_878_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_879_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_880_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_881_dual__order_Oasym,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ~ ( ord_less_rat @ A @ B ) ) ).
% dual_order.asym
thf(fact_882_dual__order_Oasym,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ~ ( ord_less_num @ A @ B ) ) ).
% dual_order.asym
thf(fact_883_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_884_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_885_linorder__cases,axiom,
! [X3: real,Y: real] :
( ~ ( ord_less_real @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less_real @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_886_linorder__cases,axiom,
! [X3: rat,Y: rat] :
( ~ ( ord_less_rat @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less_rat @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_887_linorder__cases,axiom,
! [X3: num,Y: num] :
( ~ ( ord_less_num @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less_num @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_888_linorder__cases,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_889_linorder__cases,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_int @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_890_antisym__conv3,axiom,
! [Y: real,X3: real] :
( ~ ( ord_less_real @ Y @ X3 )
=> ( ( ~ ( ord_less_real @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_891_antisym__conv3,axiom,
! [Y: rat,X3: rat] :
( ~ ( ord_less_rat @ Y @ X3 )
=> ( ( ~ ( ord_less_rat @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_892_antisym__conv3,axiom,
! [Y: num,X3: num] :
( ~ ( ord_less_num @ Y @ X3 )
=> ( ( ~ ( ord_less_num @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_893_antisym__conv3,axiom,
! [Y: nat,X3: nat] :
( ~ ( ord_less_nat @ Y @ X3 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_894_antisym__conv3,axiom,
! [Y: int,X3: int] :
( ~ ( ord_less_int @ Y @ X3 )
=> ( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_895_less__induct,axiom,
! [P2: nat > $o,A: nat] :
( ! [X4: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X4 )
=> ( P2 @ Y6 ) )
=> ( P2 @ X4 ) )
=> ( P2 @ A ) ) ).
% less_induct
thf(fact_896_ord__less__eq__trans,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_897_ord__less__eq__trans,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_rat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_898_ord__less__eq__trans,axiom,
! [A: num,B: num,C2: num] :
( ( ord_less_num @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_num @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_899_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_900_ord__less__eq__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_901_ord__eq__less__trans,axiom,
! [A: real,B: real,C2: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_902_ord__eq__less__trans,axiom,
! [A: rat,B: rat,C2: rat] :
( ( A = B )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ord_less_rat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_903_ord__eq__less__trans,axiom,
! [A: num,B: num,C2: num] :
( ( A = B )
=> ( ( ord_less_num @ B @ C2 )
=> ( ord_less_num @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_904_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_905_ord__eq__less__trans,axiom,
! [A: int,B: int,C2: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_906_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_907_order_Oasym,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ~ ( ord_less_rat @ B @ A ) ) ).
% order.asym
thf(fact_908_order_Oasym,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order.asym
thf(fact_909_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_910_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_911_less__imp__neq,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_912_less__imp__neq,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_rat @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_913_less__imp__neq,axiom,
! [X3: num,Y: num] :
( ( ord_less_num @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_914_less__imp__neq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_915_less__imp__neq,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_916_dense,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ? [Z3: real] :
( ( ord_less_real @ X3 @ Z3 )
& ( ord_less_real @ Z3 @ Y ) ) ) ).
% dense
thf(fact_917_dense,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_rat @ X3 @ Y )
=> ? [Z3: rat] :
( ( ord_less_rat @ X3 @ Z3 )
& ( ord_less_rat @ Z3 @ Y ) ) ) ).
% dense
thf(fact_918_gt__ex,axiom,
! [X3: real] :
? [X_12: real] : ( ord_less_real @ X3 @ X_12 ) ).
% gt_ex
thf(fact_919_gt__ex,axiom,
! [X3: rat] :
? [X_12: rat] : ( ord_less_rat @ X3 @ X_12 ) ).
% gt_ex
thf(fact_920_gt__ex,axiom,
! [X3: nat] :
? [X_12: nat] : ( ord_less_nat @ X3 @ X_12 ) ).
% gt_ex
thf(fact_921_gt__ex,axiom,
! [X3: int] :
? [X_12: int] : ( ord_less_int @ X3 @ X_12 ) ).
% gt_ex
thf(fact_922_lt__ex,axiom,
! [X3: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X3 ) ).
% lt_ex
thf(fact_923_lt__ex,axiom,
! [X3: rat] :
? [Y3: rat] : ( ord_less_rat @ Y3 @ X3 ) ).
% lt_ex
thf(fact_924_lt__ex,axiom,
! [X3: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X3 ) ).
% lt_ex
thf(fact_925_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o,Uw: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).
% VEBT_internal.membermima.simps(1)
thf(fact_926_linorder__neqE__nat,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_927_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_928_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P2 @ M3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_929_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_930_less__not__refl3,axiom,
! [S3: nat,T: nat] :
( ( ord_less_nat @ S3 @ T )
=> ( S3 != T ) ) ).
% less_not_refl3
thf(fact_931_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_932_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_933_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_934_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_real @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_935_lift__Suc__mono__less__iff,axiom,
! [F: nat > rat,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_rat @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_936_lift__Suc__mono__less__iff,axiom,
! [F: nat > num,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_num @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_937_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_938_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_939_lift__Suc__mono__less,axiom,
! [F: nat > real,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_real @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_940_lift__Suc__mono__less,axiom,
! [F: nat > rat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_rat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_941_lift__Suc__mono__less,axiom,
! [F: nat > num,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_num @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_942_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_943_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_944_verit__comp__simplify1_I3_J,axiom,
! [B2: real,A2: real] :
( ( ~ ( ord_less_eq_real @ B2 @ A2 ) )
= ( ord_less_real @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_945_verit__comp__simplify1_I3_J,axiom,
! [B2: rat,A2: rat] :
( ( ~ ( ord_less_eq_rat @ B2 @ A2 ) )
= ( ord_less_rat @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_946_verit__comp__simplify1_I3_J,axiom,
! [B2: num,A2: num] :
( ( ~ ( ord_less_eq_num @ B2 @ A2 ) )
= ( ord_less_num @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_947_verit__comp__simplify1_I3_J,axiom,
! [B2: nat,A2: nat] :
( ( ~ ( ord_less_eq_nat @ B2 @ A2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_948_verit__comp__simplify1_I3_J,axiom,
! [B2: int,A2: int] :
( ( ~ ( ord_less_eq_int @ B2 @ A2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_949_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_950_less__numeral__extra_I3_J,axiom,
~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% less_numeral_extra(3)
thf(fact_951_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_952_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_953_VEBT_Osize_I4_J,axiom,
! [X21: $o,X222: $o] :
( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
= zero_zero_nat ) ).
% VEBT.size(4)
thf(fact_954_count__le__length,axiom,
! [Xs: list_VEBT_VEBT,X3: vEBT_VEBT] : ( ord_less_eq_nat @ ( count_list_VEBT_VEBT @ Xs @ X3 ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ).
% count_le_length
thf(fact_955_count__le__length,axiom,
! [Xs: list_int,X3: int] : ( ord_less_eq_nat @ ( count_list_int @ Xs @ X3 ) @ ( size_size_list_int @ Xs ) ) ).
% count_le_length
thf(fact_956_count__le__length,axiom,
! [Xs: list_nat,X3: nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs @ X3 ) @ ( size_size_list_nat @ Xs ) ) ).
% count_le_length
thf(fact_957_length__pos__if__in__set,axiom,
! [X3: complex,Xs: list_complex] :
( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_958_length__pos__if__in__set,axiom,
! [X3: real,Xs: list_real] :
( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_959_length__pos__if__in__set,axiom,
! [X3: set_nat,Xs: list_set_nat] :
( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_960_length__pos__if__in__set,axiom,
! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_961_length__pos__if__in__set,axiom,
! [X3: int,Xs: list_int] :
( ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_962_length__pos__if__in__set,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_963_VEBT__internal_Ovalid_H_Ocases,axiom,
! [X3: produc9072475918466114483BT_nat] :
( ! [Uu2: $o,Uv2: $o,D2: nat] :
( X3
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D2 ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
( X3
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Deg3 ) ) ) ).
% VEBT_internal.valid'.cases
thf(fact_964_VEBT_Oexhaust,axiom,
! [Y: vEBT_VEBT] :
( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
( Y
!= ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
=> ~ ! [X212: $o,X223: $o] :
( Y
!= ( vEBT_Leaf @ X212 @ X223 ) ) ) ).
% VEBT.exhaust
thf(fact_965_VEBT_Odistinct_I1_J,axiom,
! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
!= ( vEBT_Leaf @ X21 @ X222 ) ) ).
% VEBT.distinct(1)
thf(fact_966_leD,axiom,
! [Y: real,X3: real] :
( ( ord_less_eq_real @ Y @ X3 )
=> ~ ( ord_less_real @ X3 @ Y ) ) ).
% leD
thf(fact_967_leD,axiom,
! [Y: set_int,X3: set_int] :
( ( ord_less_eq_set_int @ Y @ X3 )
=> ~ ( ord_less_set_int @ X3 @ Y ) ) ).
% leD
thf(fact_968_leD,axiom,
! [Y: rat,X3: rat] :
( ( ord_less_eq_rat @ Y @ X3 )
=> ~ ( ord_less_rat @ X3 @ Y ) ) ).
% leD
thf(fact_969_leD,axiom,
! [Y: num,X3: num] :
( ( ord_less_eq_num @ Y @ X3 )
=> ~ ( ord_less_num @ X3 @ Y ) ) ).
% leD
thf(fact_970_leD,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_nat @ X3 @ Y ) ) ).
% leD
thf(fact_971_leD,axiom,
! [Y: int,X3: int] :
( ( ord_less_eq_int @ Y @ X3 )
=> ~ ( ord_less_int @ X3 @ Y ) ) ).
% leD
thf(fact_972_leI,axiom,
! [X3: real,Y: real] :
( ~ ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ Y @ X3 ) ) ).
% leI
thf(fact_973_leI,axiom,
! [X3: rat,Y: rat] :
( ~ ( ord_less_rat @ X3 @ Y )
=> ( ord_less_eq_rat @ Y @ X3 ) ) ).
% leI
thf(fact_974_leI,axiom,
! [X3: num,Y: num] :
( ~ ( ord_less_num @ X3 @ Y )
=> ( ord_less_eq_num @ Y @ X3 ) ) ).
% leI
thf(fact_975_leI,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ).
% leI
thf(fact_976_leI,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_int @ X3 @ Y )
=> ( ord_less_eq_int @ Y @ X3 ) ) ).
% leI
thf(fact_977_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_978_nless__le,axiom,
! [A: set_int,B: set_int] :
( ( ~ ( ord_less_set_int @ A @ B ) )
= ( ~ ( ord_less_eq_set_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_979_nless__le,axiom,
! [A: rat,B: rat] :
( ( ~ ( ord_less_rat @ A @ B ) )
= ( ~ ( ord_less_eq_rat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_980_nless__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_981_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_982_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_983_antisym__conv1,axiom,
! [X3: real,Y: real] :
( ~ ( ord_less_real @ X3 @ Y )
=> ( ( ord_less_eq_real @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_984_antisym__conv1,axiom,
! [X3: set_int,Y: set_int] :
( ~ ( ord_less_set_int @ X3 @ Y )
=> ( ( ord_less_eq_set_int @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_985_antisym__conv1,axiom,
! [X3: rat,Y: rat] :
( ~ ( ord_less_rat @ X3 @ Y )
=> ( ( ord_less_eq_rat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_986_antisym__conv1,axiom,
! [X3: num,Y: num] :
( ~ ( ord_less_num @ X3 @ Y )
=> ( ( ord_less_eq_num @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_987_antisym__conv1,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_988_antisym__conv1,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_989_antisym__conv2,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ~ ( ord_less_real @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_990_antisym__conv2,axiom,
! [X3: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y )
=> ( ( ~ ( ord_less_set_int @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_991_antisym__conv2,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ X3 @ Y )
=> ( ( ~ ( ord_less_rat @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_992_antisym__conv2,axiom,
! [X3: num,Y: num] :
( ( ord_less_eq_num @ X3 @ Y )
=> ( ( ~ ( ord_less_num @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_993_antisym__conv2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_994_antisym__conv2,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_995_dense__ge,axiom,
! [Z2: real,Y: real] :
( ! [X4: real] :
( ( ord_less_real @ Z2 @ X4 )
=> ( ord_less_eq_real @ Y @ X4 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_ge
thf(fact_996_dense__ge,axiom,
! [Z2: rat,Y: rat] :
( ! [X4: rat] :
( ( ord_less_rat @ Z2 @ X4 )
=> ( ord_less_eq_rat @ Y @ X4 ) )
=> ( ord_less_eq_rat @ Y @ Z2 ) ) ).
% dense_ge
thf(fact_997_dense__le,axiom,
! [Y: real,Z2: real] :
( ! [X4: real] :
( ( ord_less_real @ X4 @ Y )
=> ( ord_less_eq_real @ X4 @ Z2 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_le
thf(fact_998_dense__le,axiom,
! [Y: rat,Z2: rat] :
( ! [X4: rat] :
( ( ord_less_rat @ X4 @ Y )
=> ( ord_less_eq_rat @ X4 @ Z2 ) )
=> ( ord_less_eq_rat @ Y @ Z2 ) ) ).
% dense_le
thf(fact_999_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
& ~ ( ord_less_eq_real @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_1000_less__le__not__le,axiom,
( ord_less_set_int
= ( ^ [X5: set_int,Y5: set_int] :
( ( ord_less_eq_set_int @ X5 @ Y5 )
& ~ ( ord_less_eq_set_int @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_1001_less__le__not__le,axiom,
( ord_less_rat
= ( ^ [X5: rat,Y5: rat] :
( ( ord_less_eq_rat @ X5 @ Y5 )
& ~ ( ord_less_eq_rat @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_1002_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
& ~ ( ord_less_eq_num @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_1003_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_1004_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X5: int,Y5: int] :
( ( ord_less_eq_int @ X5 @ Y5 )
& ~ ( ord_less_eq_int @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_1005_not__le__imp__less,axiom,
! [Y: real,X3: real] :
( ~ ( ord_less_eq_real @ Y @ X3 )
=> ( ord_less_real @ X3 @ Y ) ) ).
% not_le_imp_less
thf(fact_1006_not__le__imp__less,axiom,
! [Y: rat,X3: rat] :
( ~ ( ord_less_eq_rat @ Y @ X3 )
=> ( ord_less_rat @ X3 @ Y ) ) ).
% not_le_imp_less
thf(fact_1007_not__le__imp__less,axiom,
! [Y: num,X3: num] :
( ~ ( ord_less_eq_num @ Y @ X3 )
=> ( ord_less_num @ X3 @ Y ) ) ).
% not_le_imp_less
thf(fact_1008_not__le__imp__less,axiom,
! [Y: nat,X3: nat] :
( ~ ( ord_less_eq_nat @ Y @ X3 )
=> ( ord_less_nat @ X3 @ Y ) ) ).
% not_le_imp_less
thf(fact_1009_not__le__imp__less,axiom,
! [Y: int,X3: int] :
( ~ ( ord_less_eq_int @ Y @ X3 )
=> ( ord_less_int @ X3 @ Y ) ) ).
% not_le_imp_less
thf(fact_1010_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B4: real] :
( ( ord_less_real @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1011_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_int
= ( ^ [A5: set_int,B4: set_int] :
( ( ord_less_set_int @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1012_order_Oorder__iff__strict,axiom,
( ord_less_eq_rat
= ( ^ [A5: rat,B4: rat] :
( ( ord_less_rat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1013_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A5: num,B4: num] :
( ( ord_less_num @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1014_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1015_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_int @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1016_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1017_order_Ostrict__iff__order,axiom,
( ord_less_set_int
= ( ^ [A5: set_int,B4: set_int] :
( ( ord_less_eq_set_int @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1018_order_Ostrict__iff__order,axiom,
( ord_less_rat
= ( ^ [A5: rat,B4: rat] :
( ( ord_less_eq_rat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1019_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A5: num,B4: num] :
( ( ord_less_eq_num @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1020_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1021_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1022_order_Ostrict__trans1,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1023_order_Ostrict__trans1,axiom,
! [A: set_int,B: set_int,C2: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( ord_less_set_int @ B @ C2 )
=> ( ord_less_set_int @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1024_order_Ostrict__trans1,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ord_less_rat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1025_order_Ostrict__trans1,axiom,
! [A: num,B: num,C2: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ B @ C2 )
=> ( ord_less_num @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1026_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1027_order_Ostrict__trans1,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1028_order_Ostrict__trans2,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1029_order_Ostrict__trans2,axiom,
! [A: set_int,B: set_int,C2: set_int] :
( ( ord_less_set_int @ A @ B )
=> ( ( ord_less_eq_set_int @ B @ C2 )
=> ( ord_less_set_int @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1030_order_Ostrict__trans2,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ord_less_rat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1031_order_Ostrict__trans2,axiom,
! [A: num,B: num,C2: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ord_less_num @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1032_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1033_order_Ostrict__trans2,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1034_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ A5 @ B4 )
& ~ ( ord_less_eq_real @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1035_order_Ostrict__iff__not,axiom,
( ord_less_set_int
= ( ^ [A5: set_int,B4: set_int] :
( ( ord_less_eq_set_int @ A5 @ B4 )
& ~ ( ord_less_eq_set_int @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1036_order_Ostrict__iff__not,axiom,
( ord_less_rat
= ( ^ [A5: rat,B4: rat] :
( ( ord_less_eq_rat @ A5 @ B4 )
& ~ ( ord_less_eq_rat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1037_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A5: num,B4: num] :
( ( ord_less_eq_num @ A5 @ B4 )
& ~ ( ord_less_eq_num @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1038_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1039_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ~ ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1040_dense__ge__bounded,axiom,
! [Z2: real,X3: real,Y: real] :
( ( ord_less_real @ Z2 @ X3 )
=> ( ! [W: real] :
( ( ord_less_real @ Z2 @ W )
=> ( ( ord_less_real @ W @ X3 )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_1041_dense__ge__bounded,axiom,
! [Z2: rat,X3: rat,Y: rat] :
( ( ord_less_rat @ Z2 @ X3 )
=> ( ! [W: rat] :
( ( ord_less_rat @ Z2 @ W )
=> ( ( ord_less_rat @ W @ X3 )
=> ( ord_less_eq_rat @ Y @ W ) ) )
=> ( ord_less_eq_rat @ Y @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_1042_dense__le__bounded,axiom,
! [X3: real,Y: real,Z2: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X3 @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z2 ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_1043_dense__le__bounded,axiom,
! [X3: rat,Y: rat,Z2: rat] :
( ( ord_less_rat @ X3 @ Y )
=> ( ! [W: rat] :
( ( ord_less_rat @ X3 @ W )
=> ( ( ord_less_rat @ W @ Y )
=> ( ord_less_eq_rat @ W @ Z2 ) ) )
=> ( ord_less_eq_rat @ Y @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_1044_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A5: real] :
( ( ord_less_real @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1045_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_int
= ( ^ [B4: set_int,A5: set_int] :
( ( ord_less_set_int @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1046_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_rat
= ( ^ [B4: rat,A5: rat] :
( ( ord_less_rat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1047_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B4: num,A5: num] :
( ( ord_less_num @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1048_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_nat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1049_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_int @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1050_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B4: real,A5: real] :
( ( ord_less_eq_real @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1051_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_int
= ( ^ [B4: set_int,A5: set_int] :
( ( ord_less_eq_set_int @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1052_dual__order_Ostrict__iff__order,axiom,
( ord_less_rat
= ( ^ [B4: rat,A5: rat] :
( ( ord_less_eq_rat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1053_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B4: num,A5: num] :
( ( ord_less_eq_num @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1054_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1055_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1056_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C2 @ B )
=> ( ord_less_real @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1057_dual__order_Ostrict__trans1,axiom,
! [B: set_int,A: set_int,C2: set_int] :
( ( ord_less_eq_set_int @ B @ A )
=> ( ( ord_less_set_int @ C2 @ B )
=> ( ord_less_set_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1058_dual__order_Ostrict__trans1,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_rat @ C2 @ B )
=> ( ord_less_rat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1059_dual__order_Ostrict__trans1,axiom,
! [B: num,A: num,C2: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_num @ C2 @ B )
=> ( ord_less_num @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1060_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1061_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1062_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C2 @ B )
=> ( ord_less_real @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1063_dual__order_Ostrict__trans2,axiom,
! [B: set_int,A: set_int,C2: set_int] :
( ( ord_less_set_int @ B @ A )
=> ( ( ord_less_eq_set_int @ C2 @ B )
=> ( ord_less_set_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1064_dual__order_Ostrict__trans2,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C2 @ B )
=> ( ord_less_rat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1065_dual__order_Ostrict__trans2,axiom,
! [B: num,A: num,C2: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_eq_num @ C2 @ B )
=> ( ord_less_num @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1066_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1067_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1068_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B4: real,A5: real] :
( ( ord_less_eq_real @ B4 @ A5 )
& ~ ( ord_less_eq_real @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1069_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_int
= ( ^ [B4: set_int,A5: set_int] :
( ( ord_less_eq_set_int @ B4 @ A5 )
& ~ ( ord_less_eq_set_int @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1070_dual__order_Ostrict__iff__not,axiom,
( ord_less_rat
= ( ^ [B4: rat,A5: rat] :
( ( ord_less_eq_rat @ B4 @ A5 )
& ~ ( ord_less_eq_rat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1071_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B4: num,A5: num] :
( ( ord_less_eq_num @ B4 @ A5 )
& ~ ( ord_less_eq_num @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1072_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1073_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ~ ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1074_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1075_order_Ostrict__implies__order,axiom,
! [A: set_int,B: set_int] :
( ( ord_less_set_int @ A @ B )
=> ( ord_less_eq_set_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1076_order_Ostrict__implies__order,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_eq_rat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1077_order_Ostrict__implies__order,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ord_less_eq_num @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1078_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1079_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1080_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1081_dual__order_Ostrict__implies__order,axiom,
! [B: set_int,A: set_int] :
( ( ord_less_set_int @ B @ A )
=> ( ord_less_eq_set_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1082_dual__order_Ostrict__implies__order,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ( ord_less_eq_rat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1083_dual__order_Ostrict__implies__order,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ord_less_eq_num @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1084_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1085_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1086_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_1087_order__le__less,axiom,
( ord_less_eq_set_int
= ( ^ [X5: set_int,Y5: set_int] :
( ( ord_less_set_int @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_1088_order__le__less,axiom,
( ord_less_eq_rat
= ( ^ [X5: rat,Y5: rat] :
( ( ord_less_rat @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_1089_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X5: num,Y5: num] :
( ( ord_less_num @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_1090_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_1091_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X5: int,Y5: int] :
( ( ord_less_int @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_1092_order__less__le,axiom,
( ord_less_real
= ( ^ [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_1093_order__less__le,axiom,
( ord_less_set_int
= ( ^ [X5: set_int,Y5: set_int] :
( ( ord_less_eq_set_int @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_1094_order__less__le,axiom,
( ord_less_rat
= ( ^ [X5: rat,Y5: rat] :
( ( ord_less_eq_rat @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_1095_order__less__le,axiom,
( ord_less_num
= ( ^ [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_1096_order__less__le,axiom,
( ord_less_nat
= ( ^ [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_1097_order__less__le,axiom,
( ord_less_int
= ( ^ [X5: int,Y5: int] :
( ( ord_less_eq_int @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_1098_linorder__not__le,axiom,
! [X3: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X3 @ Y ) )
= ( ord_less_real @ Y @ X3 ) ) ).
% linorder_not_le
thf(fact_1099_linorder__not__le,axiom,
! [X3: rat,Y: rat] :
( ( ~ ( ord_less_eq_rat @ X3 @ Y ) )
= ( ord_less_rat @ Y @ X3 ) ) ).
% linorder_not_le
thf(fact_1100_linorder__not__le,axiom,
! [X3: num,Y: num] :
( ( ~ ( ord_less_eq_num @ X3 @ Y ) )
= ( ord_less_num @ Y @ X3 ) ) ).
% linorder_not_le
thf(fact_1101_linorder__not__le,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X3 @ Y ) )
= ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_not_le
thf(fact_1102_linorder__not__le,axiom,
! [X3: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X3 @ Y ) )
= ( ord_less_int @ Y @ X3 ) ) ).
% linorder_not_le
thf(fact_1103_linorder__not__less,axiom,
! [X3: real,Y: real] :
( ( ~ ( ord_less_real @ X3 @ Y ) )
= ( ord_less_eq_real @ Y @ X3 ) ) ).
% linorder_not_less
thf(fact_1104_linorder__not__less,axiom,
! [X3: rat,Y: rat] :
( ( ~ ( ord_less_rat @ X3 @ Y ) )
= ( ord_less_eq_rat @ Y @ X3 ) ) ).
% linorder_not_less
thf(fact_1105_linorder__not__less,axiom,
! [X3: num,Y: num] :
( ( ~ ( ord_less_num @ X3 @ Y ) )
= ( ord_less_eq_num @ Y @ X3 ) ) ).
% linorder_not_less
thf(fact_1106_linorder__not__less,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_not_less
thf(fact_1107_linorder__not__less,axiom,
! [X3: int,Y: int] :
( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( ord_less_eq_int @ Y @ X3 ) ) ).
% linorder_not_less
thf(fact_1108_order__less__imp__le,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_1109_order__less__imp__le,axiom,
! [X3: set_int,Y: set_int] :
( ( ord_less_set_int @ X3 @ Y )
=> ( ord_less_eq_set_int @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_1110_order__less__imp__le,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_rat @ X3 @ Y )
=> ( ord_less_eq_rat @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_1111_order__less__imp__le,axiom,
! [X3: num,Y: num] :
( ( ord_less_num @ X3 @ Y )
=> ( ord_less_eq_num @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_1112_order__less__imp__le,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_1113_order__less__imp__le,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ord_less_eq_int @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_1114_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1115_order__le__neq__trans,axiom,
! [A: set_int,B: set_int] :
( ( ord_less_eq_set_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1116_order__le__neq__trans,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( A != B )
=> ( ord_less_rat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1117_order__le__neq__trans,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( A != B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1118_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1119_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1120_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1121_order__neq__le__trans,axiom,
! [A: set_int,B: set_int] :
( ( A != B )
=> ( ( ord_less_eq_set_int @ A @ B )
=> ( ord_less_set_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1122_order__neq__le__trans,axiom,
! [A: rat,B: rat] :
( ( A != B )
=> ( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_rat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1123_order__neq__le__trans,axiom,
! [A: num,B: num] :
( ( A != B )
=> ( ( ord_less_eq_num @ A @ B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1124_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1125_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1126_order__le__less__trans,axiom,
! [X3: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X3 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1127_order__le__less__trans,axiom,
! [X3: set_int,Y: set_int,Z2: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y )
=> ( ( ord_less_set_int @ Y @ Z2 )
=> ( ord_less_set_int @ X3 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1128_order__le__less__trans,axiom,
! [X3: rat,Y: rat,Z2: rat] :
( ( ord_less_eq_rat @ X3 @ Y )
=> ( ( ord_less_rat @ Y @ Z2 )
=> ( ord_less_rat @ X3 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1129_order__le__less__trans,axiom,
! [X3: num,Y: num,Z2: num] :
( ( ord_less_eq_num @ X3 @ Y )
=> ( ( ord_less_num @ Y @ Z2 )
=> ( ord_less_num @ X3 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1130_order__le__less__trans,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X3 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1131_order__le__less__trans,axiom,
! [X3: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X3 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1132_order__less__le__trans,axiom,
! [X3: real,Y: real,Z2: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_real @ X3 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1133_order__less__le__trans,axiom,
! [X3: set_int,Y: set_int,Z2: set_int] :
( ( ord_less_set_int @ X3 @ Y )
=> ( ( ord_less_eq_set_int @ Y @ Z2 )
=> ( ord_less_set_int @ X3 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1134_order__less__le__trans,axiom,
! [X3: rat,Y: rat,Z2: rat] :
( ( ord_less_rat @ X3 @ Y )
=> ( ( ord_less_eq_rat @ Y @ Z2 )
=> ( ord_less_rat @ X3 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1135_order__less__le__trans,axiom,
! [X3: num,Y: num,Z2: num] :
( ( ord_less_num @ X3 @ Y )
=> ( ( ord_less_eq_num @ Y @ Z2 )
=> ( ord_less_num @ X3 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1136_order__less__le__trans,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X3 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1137_order__less__le__trans,axiom,
! [X3: int,Y: int,Z2: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_int @ X3 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1138_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1139_order__le__less__subst1,axiom,
! [A: real,F: rat > real,B: rat,C2: rat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1140_order__le__less__subst1,axiom,
! [A: real,F: num > real,B: num,C2: num] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1141_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C2: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1142_order__le__less__subst1,axiom,
! [A: real,F: int > real,B: int,C2: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1143_order__le__less__subst1,axiom,
! [A: rat,F: real > rat,B: real,C2: real] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1144_order__le__less__subst1,axiom,
! [A: rat,F: rat > rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1145_order__le__less__subst1,axiom,
! [A: rat,F: num > rat,B: num,C2: num] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1146_order__le__less__subst1,axiom,
! [A: rat,F: nat > rat,B: nat,C2: nat] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1147_order__le__less__subst1,axiom,
! [A: rat,F: int > rat,B: int,C2: int] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1148_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > real,C2: real] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1149_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1150_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > num,C2: num] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1151_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > nat,C2: nat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1152_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > int,C2: int] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1153_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > real,C2: real] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1154_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > rat,C2: rat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1155_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > num,C2: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1156_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > nat,C2: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1157_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > int,C2: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1158_order__less__le__subst1,axiom,
! [A: real,F: rat > real,B: rat,C2: rat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1159_order__less__le__subst1,axiom,
! [A: rat,F: rat > rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1160_order__less__le__subst1,axiom,
! [A: num,F: rat > num,B: rat,C2: rat] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1161_order__less__le__subst1,axiom,
! [A: nat,F: rat > nat,B: rat,C2: rat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1162_order__less__le__subst1,axiom,
! [A: int,F: rat > int,B: rat,C2: rat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_eq_rat @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1163_order__less__le__subst1,axiom,
! [A: real,F: num > real,B: num,C2: num] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1164_order__less__le__subst1,axiom,
! [A: rat,F: num > rat,B: num,C2: num] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1165_order__less__le__subst1,axiom,
! [A: num,F: num > num,B: num,C2: num] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1166_order__less__le__subst1,axiom,
! [A: nat,F: num > nat,B: num,C2: num] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1167_order__less__le__subst1,axiom,
! [A: int,F: num > int,B: num,C2: num] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1168_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1169_order__less__le__subst2,axiom,
! [A: rat,B: rat,F: rat > real,C2: real] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1170_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > real,C2: real] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1171_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1172_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > real,C2: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1173_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > rat,C2: rat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1174_order__less__le__subst2,axiom,
! [A: rat,B: rat,F: rat > rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
=> ( ! [X4: rat,Y3: rat] :
( ( ord_less_rat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1175_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > rat,C2: rat] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
=> ( ! [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1176_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > rat,C2: rat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1177_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > rat,C2: rat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C2 )
=> ( ! [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1178_linorder__le__less__linear,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
| ( ord_less_real @ Y @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_1179_linorder__le__less__linear,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ X3 @ Y )
| ( ord_less_rat @ Y @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_1180_linorder__le__less__linear,axiom,
! [X3: num,Y: num] :
( ( ord_less_eq_num @ X3 @ Y )
| ( ord_less_num @ Y @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_1181_linorder__le__less__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_1182_linorder__le__less__linear,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
| ( ord_less_int @ Y @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_1183_order__le__imp__less__or__eq,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ord_less_real @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1184_order__le__imp__less__or__eq,axiom,
! [X3: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y )
=> ( ( ord_less_set_int @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1185_order__le__imp__less__or__eq,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ X3 @ Y )
=> ( ( ord_less_rat @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1186_order__le__imp__less__or__eq,axiom,
! [X3: num,Y: num] :
( ( ord_less_eq_num @ X3 @ Y )
=> ( ( ord_less_num @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1187_order__le__imp__less__or__eq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_nat @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1188_order__le__imp__less__or__eq,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_int @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1189_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_1190_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_1191_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_1192_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_1193_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K2: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_real @ K2 @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1194_add__mono__thms__linordered__field_I5_J,axiom,
! [I: rat,J: rat,K2: rat,L: rat] :
( ( ( ord_less_rat @ I @ J )
& ( ord_less_rat @ K2 @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1195_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1196_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1197_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K2: real,L: real] :
( ( ( I = J )
& ( ord_less_real @ K2 @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1198_add__mono__thms__linordered__field_I2_J,axiom,
! [I: rat,J: rat,K2: rat,L: rat] :
( ( ( I = J )
& ( ord_less_rat @ K2 @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1199_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1200_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1201_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K2: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( K2 = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1202_add__mono__thms__linordered__field_I1_J,axiom,
! [I: rat,J: rat,K2: rat,L: rat] :
( ( ( ord_less_rat @ I @ J )
& ( K2 = L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1203_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K2 = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1204_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K2 = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1205_add__strict__mono,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C2 @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1206_add__strict__mono,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ C2 @ D )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1207_add__strict__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1208_add__strict__mono,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1209_add__strict__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1210_add__strict__left__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1211_add__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1212_add__strict__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1213_add__strict__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_1214_add__strict__right__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_1215_add__strict__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_1216_add__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_1217_add__less__imp__less__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1218_add__less__imp__less__left,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
=> ( ord_less_rat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1219_add__less__imp__less__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1220_add__less__imp__less__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1221_add__less__imp__less__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1222_add__less__imp__less__right,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
=> ( ord_less_rat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1223_add__less__imp__less__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1224_add__less__imp__less__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1225_Nat_OlessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ I @ K2 )
=> ( ( K2
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1226_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_1227_Suc__lessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K2 )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1228_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1229_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_1230_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1231_Ex__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P2 @ I3 ) ) )
= ( ( P2 @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P2 @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1232_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_1233_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_1234_All__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P2 @ I3 ) ) )
= ( ( P2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P2 @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_1235_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M7: nat] :
( ( M2
= ( suc @ M7 ) )
& ( ord_less_nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1236_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_1237_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_1238_less__trans__Suc,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( suc @ I ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_1239_less__Suc__induct,axiom,
! [I: nat,J: nat,P2: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P2 @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ( P2 @ I2 @ J2 )
=> ( ( P2 @ J2 @ K )
=> ( P2 @ I2 @ K ) ) ) ) )
=> ( P2 @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1240_strict__inc__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P2 @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P2 @ ( suc @ I2 ) )
=> ( P2 @ I2 ) ) )
=> ( P2 @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1241_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1242_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_1243_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1244_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1245_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1246_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1247_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1248_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1249_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
& ( M5 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_1250_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_1251_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
| ( M5 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1252_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1253_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1254_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1255_less__add__eq__less,axiom,
! [K2: nat,L: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K2 @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_1256_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1257_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1258_add__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_1259_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1260_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1261_add__less__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1262_add__lessD1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
=> ( ord_less_nat @ I @ K2 ) ) ).
% add_lessD1
thf(fact_1263_VEBT__internal_OminNull_Osimps_I3_J,axiom,
! [Uu: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% VEBT_internal.minNull.simps(3)
thf(fact_1264_VEBT__internal_OminNull_Osimps_I2_J,axiom,
! [Uv: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% VEBT_internal.minNull.simps(2)
thf(fact_1265_VEBT__internal_OminNull_Osimps_I1_J,axiom,
vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% VEBT_internal.minNull.simps(1)
thf(fact_1266_unbounded__k__infinite,axiom,
! [K2: nat,S2: set_nat] :
( ! [M: nat] :
( ( ord_less_nat @ K2 @ M )
=> ? [N6: nat] :
( ( ord_less_nat @ M @ N6 )
& ( member_nat @ N6 @ S2 ) ) )
=> ~ ( finite_finite_nat @ S2 ) ) ).
% unbounded_k_infinite
thf(fact_1267_bounded__nat__set__is__finite,axiom,
! [N7: set_nat,N: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ N7 )
=> ( ord_less_nat @ X4 @ N ) )
=> ( finite_finite_nat @ N7 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1268_infinite__nat__iff__unbounded,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M5: nat] :
? [N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ( member_nat @ N4 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_1269_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M5: nat] :
! [X5: nat] :
( ( member_nat @ X5 @ N5 )
=> ( ord_less_nat @ X5 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1270_add__less__le__mono,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1271_add__less__le__mono,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C2 @ D )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1272_add__less__le__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1273_add__less__le__mono,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1274_add__le__less__mono,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C2 @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1275_add__le__less__mono,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ C2 @ D )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1276_add__le__less__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1277_add__le__less__mono,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1278_add__mono__thms__linordered__field_I3_J,axiom,
! [I: real,J: real,K2: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_eq_real @ K2 @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1279_add__mono__thms__linordered__field_I3_J,axiom,
! [I: rat,J: rat,K2: rat,L: rat] :
( ( ( ord_less_rat @ I @ J )
& ( ord_less_eq_rat @ K2 @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1280_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1281_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1282_add__mono__thms__linordered__field_I4_J,axiom,
! [I: real,J: real,K2: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_real @ K2 @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1283_add__mono__thms__linordered__field_I4_J,axiom,
! [I: rat,J: rat,K2: rat,L: rat] :
( ( ( ord_less_eq_rat @ I @ J )
& ( ord_less_rat @ K2 @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1284_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1285_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1286_add__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_1287_add__neg__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% add_neg_neg
thf(fact_1288_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_1289_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_1290_add__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1291_add__pos__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1292_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1293_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1294_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C: nat] :
( ( B
= ( plus_plus_nat @ A @ C ) )
=> ( C = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1295_pos__add__strict,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_1296_pos__add__strict,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_1297_pos__add__strict,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_1298_pos__add__strict,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_1299_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1300_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M: nat] :
( N
= ( suc @ M ) ) ) ).
% gr0_implies_Suc
thf(fact_1301_All__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P2 @ I3 ) ) )
= ( ( P2 @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P2 @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1302_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1303_Ex__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P2 @ I3 ) ) )
= ( ( P2 @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P2 @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1304_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1305_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1306_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_1307_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_1308_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_1309_inc__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P2 @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% inc_induct
thf(fact_1310_dec__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P2 @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) ) )
=> ( P2 @ J ) ) ) ) ).
% dec_induct
thf(fact_1311_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_1312_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_1313_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ~ ( P2 @ I4 ) )
& ( P2 @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_1314_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ? [K: nat] :
( N
= ( suc @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1315_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
? [K3: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M5 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1316_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_1317_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1318_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M2 @ Q2 ) ) ) ) ).
% less_natE
thf(fact_1319_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
& ( ( plus_plus_nat @ I @ K )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1320_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K2: nat] :
( ! [M: nat,N3: nat] :
( ( ord_less_nat @ M @ N3 )
=> ( ord_less_nat @ ( F @ M ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K2 ) @ ( F @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1321_VEBT__internal_Onaive__member_Ocases,axiom,
! [X3: produc9072475918466114483BT_nat] :
( ! [A3: $o,B3: $o,X4: nat] :
( X3
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X4 ) )
=> ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
( X3
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X4: nat] :
( X3
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) @ X4 ) ) ) ) ).
% VEBT_internal.naive_member.cases
thf(fact_1322_invar__vebt_Ointros_I1_J,axiom,
! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).
% invar_vebt.intros(1)
thf(fact_1323_add__strict__increasing2,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_1324_add__strict__increasing2,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_1325_add__strict__increasing2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_1326_add__strict__increasing2,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_1327_add__strict__increasing,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_1328_add__strict__increasing,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ B @ C2 )
=> ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_1329_add__strict__increasing,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_1330_add__strict__increasing,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_1331_add__pos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1332_add__pos__nonneg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1333_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1334_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1335_add__nonpos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_1336_add__nonpos__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% add_nonpos_neg
thf(fact_1337_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1338_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_1339_add__nonneg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1340_add__nonneg__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1341_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1342_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1343_add__neg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_1344_add__neg__nonpos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% add_neg_nonpos
thf(fact_1345_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1346_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_1347_VEBT__internal_OminNull_Ocases,axiom,
! [X3: vEBT_VEBT] :
( ( X3
!= ( vEBT_Leaf @ $false @ $false ) )
=> ( ! [Uv2: $o] :
( X3
!= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ! [Uu2: $o] :
( X3
!= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% VEBT_internal.minNull.cases
thf(fact_1348_verit__comp__simplify1_I2_J,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1349_verit__comp__simplify1_I2_J,axiom,
! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1350_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1351_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1352_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1353_verit__la__disequality,axiom,
! [A: rat,B: rat] :
( ( A = B )
| ~ ( ord_less_eq_rat @ A @ B )
| ~ ( ord_less_eq_rat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_1354_verit__la__disequality,axiom,
! [A: num,B: num] :
( ( A = B )
| ~ ( ord_less_eq_num @ A @ B )
| ~ ( ord_less_eq_num @ B @ A ) ) ).
% verit_la_disequality
thf(fact_1355_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_1356_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_1357_ex__least__nat__less,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_nat @ K @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K )
=> ~ ( P2 @ I4 ) )
& ( P2 @ ( suc @ K ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1358_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_1359_is__num__normalize_I1_J,axiom,
! [A: rat,B: rat,C2: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_1360_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_1361_VEBT__internal_OminNull_Oelims_I3_J,axiom,
! [X3: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X3 )
=> ( ! [Uv2: $o] :
( X3
!= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ! [Uu2: $o] :
( X3
!= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).
% VEBT_internal.minNull.elims(3)
thf(fact_1362_VEBT__internal_OminNull_Oelims_I2_J,axiom,
! [X3: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X3 )
=> ( ( X3
!= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).
% VEBT_internal.minNull.elims(2)
thf(fact_1363_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) ).
% VEBT_internal.membermima.simps(2)
thf(fact_1364_in__children__def,axiom,
( vEBT_V5917875025757280293ildren
= ( ^ [N4: nat,TreeList3: list_VEBT_VEBT,X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ X5 @ N4 ) ) @ ( vEBT_VEBT_low @ X5 @ N4 ) ) ) ) ).
% in_children_def
thf(fact_1365_member__valid__both__member__options,axiom,
! [Tree: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ Tree @ N )
=> ( ( vEBT_vebt_member @ Tree @ X3 )
=> ( ( vEBT_V5719532721284313246member @ Tree @ X3 )
| ( vEBT_VEBT_membermima @ Tree @ X3 ) ) ) ) ).
% member_valid_both_member_options
thf(fact_1366_buildup__gives__valid,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).
% buildup_gives_valid
thf(fact_1367_both__member__options__def,axiom,
( vEBT_V8194947554948674370ptions
= ( ^ [T3: vEBT_VEBT,X5: nat] :
( ( vEBT_V5719532721284313246member @ T3 @ X5 )
| ( vEBT_VEBT_membermima @ T3 @ X5 ) ) ) ) ).
% both_member_options_def
thf(fact_1368_field__le__epsilon,axiom,
! [X3: real,Y: real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_eq_real @ X3 @ ( plus_plus_real @ Y @ E ) ) )
=> ( ord_less_eq_real @ X3 @ Y ) ) ).
% field_le_epsilon
thf(fact_1369_field__le__epsilon,axiom,
! [X3: rat,Y: rat] :
( ! [E: rat] :
( ( ord_less_rat @ zero_zero_rat @ E )
=> ( ord_less_eq_rat @ X3 @ ( plus_plus_rat @ Y @ E ) ) )
=> ( ord_less_eq_rat @ X3 @ Y ) ) ).
% field_le_epsilon
thf(fact_1370_buildup__nothing__in__min__max,axiom,
! [N: nat,X3: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X3 ) ).
% buildup_nothing_in_min_max
thf(fact_1371_add__less__zeroD,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X3 @ Y ) @ zero_zero_real )
=> ( ( ord_less_real @ X3 @ zero_zero_real )
| ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_1372_add__less__zeroD,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ X3 @ Y ) @ zero_zero_rat )
=> ( ( ord_less_rat @ X3 @ zero_zero_rat )
| ( ord_less_rat @ Y @ zero_zero_rat ) ) ) ).
% add_less_zeroD
thf(fact_1373_add__less__zeroD,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X3 @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X3 @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_1374_deg1Leaf,axiom,
! [T: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T @ one_one_nat )
= ( ? [A5: $o,B4: $o] :
( T
= ( vEBT_Leaf @ A5 @ B4 ) ) ) ) ).
% deg1Leaf
thf(fact_1375_deg__1__Leaf,axiom,
! [T: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T @ one_one_nat )
=> ? [A3: $o,B3: $o] :
( T
= ( vEBT_Leaf @ A3 @ B3 ) ) ) ).
% deg_1_Leaf
thf(fact_1376_deg__1__Leafy,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( N = one_one_nat )
=> ? [A3: $o,B3: $o] :
( T
= ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ).
% deg_1_Leafy
thf(fact_1377_buildup__nothing__in__leaf,axiom,
! [N: nat,X3: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X3 ) ).
% buildup_nothing_in_leaf
thf(fact_1378_psubsetI,axiom,
! [A4: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( A4 != B5 )
=> ( ord_less_set_int @ A4 @ B5 ) ) ) ).
% psubsetI
thf(fact_1379_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1380_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_VEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_1381_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_int] :
( ( size_size_list_int @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_1382_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_1383_neq__if__length__neq,axiom,
! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs )
!= ( size_s6755466524823107622T_VEBT @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_1384_neq__if__length__neq,axiom,
! [Xs: list_int,Ys2: list_int] :
( ( ( size_size_list_int @ Xs )
!= ( size_size_list_int @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_1385_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_1386_one__reorient,axiom,
! [X3: complex] :
( ( one_one_complex = X3 )
= ( X3 = one_one_complex ) ) ).
% one_reorient
thf(fact_1387_one__reorient,axiom,
! [X3: real] :
( ( one_one_real = X3 )
= ( X3 = one_one_real ) ) ).
% one_reorient
thf(fact_1388_one__reorient,axiom,
! [X3: rat] :
( ( one_one_rat = X3 )
= ( X3 = one_one_rat ) ) ).
% one_reorient
thf(fact_1389_one__reorient,axiom,
! [X3: nat] :
( ( one_one_nat = X3 )
= ( X3 = one_one_nat ) ) ).
% one_reorient
thf(fact_1390_one__reorient,axiom,
! [X3: int] :
( ( one_one_int = X3 )
= ( X3 = one_one_int ) ) ).
% one_reorient
thf(fact_1391_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
% zero_neq_one
thf(fact_1392_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_1393_zero__neq__one,axiom,
zero_zero_rat != one_one_rat ).
% zero_neq_one
thf(fact_1394_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_1395_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_1396_finite__psubset__induct,axiom,
! [A4: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [B7: set_nat] :
( ( ord_less_set_nat @ B7 @ A7 )
=> ( P2 @ B7 ) )
=> ( P2 @ A7 ) ) )
=> ( P2 @ A4 ) ) ) ).
% finite_psubset_induct
thf(fact_1397_finite__psubset__induct,axiom,
! [A4: set_int,P2: set_int > $o] :
( ( finite_finite_int @ A4 )
=> ( ! [A7: set_int] :
( ( finite_finite_int @ A7 )
=> ( ! [B7: set_int] :
( ( ord_less_set_int @ B7 @ A7 )
=> ( P2 @ B7 ) )
=> ( P2 @ A7 ) ) )
=> ( P2 @ A4 ) ) ) ).
% finite_psubset_induct
thf(fact_1398_finite__psubset__induct,axiom,
! [A4: set_complex,P2: set_complex > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [A7: set_complex] :
( ( finite3207457112153483333omplex @ A7 )
=> ( ! [B7: set_complex] :
( ( ord_less_set_complex @ B7 @ A7 )
=> ( P2 @ B7 ) )
=> ( P2 @ A7 ) ) )
=> ( P2 @ A4 ) ) ) ).
% finite_psubset_induct
thf(fact_1399_finite__psubset__induct,axiom,
! [A4: set_Pr1261947904930325089at_nat,P2: set_Pr1261947904930325089at_nat > $o] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ! [A7: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A7 )
=> ( ! [B7: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ B7 @ A7 )
=> ( P2 @ B7 ) )
=> ( P2 @ A7 ) ) )
=> ( P2 @ A4 ) ) ) ).
% finite_psubset_induct
thf(fact_1400_psubsetE,axiom,
! [A4: set_int,B5: set_int] :
( ( ord_less_set_int @ A4 @ B5 )
=> ~ ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ord_less_eq_set_int @ B5 @ A4 ) ) ) ).
% psubsetE
thf(fact_1401_psubset__eq,axiom,
( ord_less_set_int
= ( ^ [A6: set_int,B6: set_int] :
( ( ord_less_eq_set_int @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_1402_psubset__imp__subset,axiom,
! [A4: set_int,B5: set_int] :
( ( ord_less_set_int @ A4 @ B5 )
=> ( ord_less_eq_set_int @ A4 @ B5 ) ) ).
% psubset_imp_subset
thf(fact_1403_psubset__subset__trans,axiom,
! [A4: set_int,B5: set_int,C4: set_int] :
( ( ord_less_set_int @ A4 @ B5 )
=> ( ( ord_less_eq_set_int @ B5 @ C4 )
=> ( ord_less_set_int @ A4 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_1404_subset__not__subset__eq,axiom,
( ord_less_set_int
= ( ^ [A6: set_int,B6: set_int] :
( ( ord_less_eq_set_int @ A6 @ B6 )
& ~ ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1405_subset__psubset__trans,axiom,
! [A4: set_int,B5: set_int,C4: set_int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( ord_less_set_int @ B5 @ C4 )
=> ( ord_less_set_int @ A4 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_1406_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
( ( ord_less_set_int @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1407_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_1408_le__numeral__extra_I4_J,axiom,
ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% le_numeral_extra(4)
thf(fact_1409_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1410_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_1411_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
! [A: $o,B: $o,X3: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X3 )
= ( ( ( X3 = zero_zero_nat )
=> A )
& ( ( X3 != zero_zero_nat )
=> ( ( ( X3 = one_one_nat )
=> B )
& ( X3 = one_one_nat ) ) ) ) ) ).
% VEBT_internal.naive_member.simps(1)
thf(fact_1412_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_1413_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% zero_less_one_class.zero_le_one
thf(fact_1414_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_1415_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_1416_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1417_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1418_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1419_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1420_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_1421_not__one__le__zero,axiom,
~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% not_one_le_zero
thf(fact_1422_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1423_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_1424_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_1425_not__one__less__zero,axiom,
~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% not_one_less_zero
thf(fact_1426_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1427_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_1428_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_1429_zero__less__one,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
% zero_less_one
thf(fact_1430_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1431_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_1432_add__mono1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% add_mono1
thf(fact_1433_add__mono1,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).
% add_mono1
thf(fact_1434_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_1435_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_1436_less__add__one,axiom,
! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% less_add_one
thf(fact_1437_less__add__one,axiom,
! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% less_add_one
thf(fact_1438_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_1439_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_1440_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_1441_zero__less__two,axiom,
ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% zero_less_two
thf(fact_1442_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1443_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_1444_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_1445_less__numeral__extra_I1_J,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
% less_numeral_extra(1)
thf(fact_1446_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_1447_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_1448_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1449_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1450_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1451_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1452_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o,D: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
= ( D = one_one_nat ) ) ).
% VEBT_internal.valid'.simps(1)
thf(fact_1453_vebt__buildup_Osimps_I1_J,axiom,
( ( vEBT_vebt_buildup @ zero_zero_nat )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(1)
thf(fact_1454_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).
% VEBT_internal.naive_member.simps(2)
thf(fact_1455_nat__induct__non__zero,axiom,
! [N: nat,P2: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P2 @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1456_vebt__member_Osimps_I1_J,axiom,
! [A: $o,B: $o,X3: nat] :
( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X3 )
= ( ( ( X3 = zero_zero_nat )
=> A )
& ( ( X3 != zero_zero_nat )
=> ( ( ( X3 = one_one_nat )
=> B )
& ( X3 = one_one_nat ) ) ) ) ) ).
% vebt_member.simps(1)
thf(fact_1457_vebt__insert_Osimps_I1_J,axiom,
! [X3: nat,A: $o,B: $o] :
( ( ( X3 = zero_zero_nat )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X3 )
= ( vEBT_Leaf @ $true @ B ) ) )
& ( ( X3 != zero_zero_nat )
=> ( ( ( X3 = one_one_nat )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X3 )
= ( vEBT_Leaf @ A @ $true ) ) )
& ( ( X3 != one_one_nat )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X3 )
= ( vEBT_Leaf @ A @ B ) ) ) ) ) ) ).
% vebt_insert.simps(1)
thf(fact_1458_vebt__buildup_Osimps_I2_J,axiom,
( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(2)
thf(fact_1459_discrete,axiom,
( ord_less_nat
= ( ^ [A5: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A5 @ one_one_nat ) ) ) ) ).
% discrete
thf(fact_1460_discrete,axiom,
( ord_less_int
= ( ^ [A5: int] : ( ord_less_eq_int @ ( plus_plus_int @ A5 @ one_one_int ) ) ) ) ).
% discrete
thf(fact_1461_buildup__gives__empty,axiom,
! [N: nat] :
( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
= bot_bot_set_nat ) ).
% buildup_gives_empty
thf(fact_1462_nth__enumerate__eq,axiom,
! [M2: nat,Xs: list_VEBT_VEBT,N: nat] :
( ( ord_less_nat @ M2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( nth_Pr744662078594809490T_VEBT @ ( enumerate_VEBT_VEBT @ N @ Xs ) @ M2 )
= ( produc599794634098209291T_VEBT @ ( plus_plus_nat @ N @ M2 ) @ ( nth_VEBT_VEBT @ Xs @ M2 ) ) ) ) ).
% nth_enumerate_eq
thf(fact_1463_nth__enumerate__eq,axiom,
! [M2: nat,Xs: list_int,N: nat] :
( ( ord_less_nat @ M2 @ ( size_size_list_int @ Xs ) )
=> ( ( nth_Pr3440142176431000676at_int @ ( enumerate_int @ N @ Xs ) @ M2 )
= ( product_Pair_nat_int @ ( plus_plus_nat @ N @ M2 ) @ ( nth_int @ Xs @ M2 ) ) ) ) ).
% nth_enumerate_eq
thf(fact_1464_nth__enumerate__eq,axiom,
! [M2: nat,Xs: list_nat,N: nat] :
( ( ord_less_nat @ M2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs ) @ M2 )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M2 ) @ ( nth_nat @ Xs @ M2 ) ) ) ) ).
% nth_enumerate_eq
thf(fact_1465_nat__descend__induct,axiom,
! [N: nat,P2: nat > $o,M2: nat] :
( ! [K: nat] :
( ( ord_less_nat @ N @ K )
=> ( P2 @ K ) )
=> ( ! [K: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K @ I4 )
=> ( P2 @ I4 ) )
=> ( P2 @ K ) ) )
=> ( P2 @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_1466_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_1467_field__lbound__gt__zero,axiom,
! [D1: rat,D22: rat] :
( ( ord_less_rat @ zero_zero_rat @ D1 )
=> ( ( ord_less_rat @ zero_zero_rat @ D22 )
=> ? [E: rat] :
( ( ord_less_rat @ zero_zero_rat @ E )
& ( ord_less_rat @ E @ D1 )
& ( ord_less_rat @ E @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_1468_complete__interval,axiom,
! [A: real,B: real,P2: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P2 @ A )
=> ( ~ ( P2 @ B )
=> ? [C: real] :
( ( ord_less_eq_real @ A @ C )
& ( ord_less_eq_real @ C @ B )
& ! [X: real] :
( ( ( ord_less_eq_real @ A @ X )
& ( ord_less_real @ X @ C ) )
=> ( P2 @ X ) )
& ! [D3: real] :
( ! [X4: real] :
( ( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_real @ X4 @ D3 ) )
=> ( P2 @ X4 ) )
=> ( ord_less_eq_real @ D3 @ C ) ) ) ) ) ) ).
% complete_interval
thf(fact_1469_complete__interval,axiom,
! [A: nat,B: nat,P2: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P2 @ A )
=> ( ~ ( P2 @ B )
=> ? [C: nat] :
( ( ord_less_eq_nat @ A @ C )
& ( ord_less_eq_nat @ C @ B )
& ! [X: nat] :
( ( ( ord_less_eq_nat @ A @ X )
& ( ord_less_nat @ X @ C ) )
=> ( P2 @ X ) )
& ! [D3: nat] :
( ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ D3 ) )
=> ( P2 @ X4 ) )
=> ( ord_less_eq_nat @ D3 @ C ) ) ) ) ) ) ).
% complete_interval
thf(fact_1470_complete__interval,axiom,
! [A: int,B: int,P2: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P2 @ A )
=> ( ~ ( P2 @ B )
=> ? [C: int] :
( ( ord_less_eq_int @ A @ C )
& ( ord_less_eq_int @ C @ B )
& ! [X: int] :
( ( ( ord_less_eq_int @ A @ X )
& ( ord_less_int @ X @ C ) )
=> ( P2 @ X ) )
& ! [D3: int] :
( ! [X4: int] :
( ( ( ord_less_eq_int @ A @ X4 )
& ( ord_less_int @ X4 @ D3 ) )
=> ( P2 @ X4 ) )
=> ( ord_less_eq_int @ D3 @ C ) ) ) ) ) ) ).
% complete_interval
thf(fact_1471_pinf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X: real] :
( ( ord_less_real @ Z3 @ X )
=> ~ ( ord_less_eq_real @ X @ T ) ) ).
% pinf(6)
thf(fact_1472_pinf_I6_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X: rat] :
( ( ord_less_rat @ Z3 @ X )
=> ~ ( ord_less_eq_rat @ X @ T ) ) ).
% pinf(6)
thf(fact_1473_pinf_I6_J,axiom,
! [T: num] :
? [Z3: num] :
! [X: num] :
( ( ord_less_num @ Z3 @ X )
=> ~ ( ord_less_eq_num @ X @ T ) ) ).
% pinf(6)
thf(fact_1474_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ T ) ) ).
% pinf(6)
thf(fact_1475_pinf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X: int] :
( ( ord_less_int @ Z3 @ X )
=> ~ ( ord_less_eq_int @ X @ T ) ) ).
% pinf(6)
thf(fact_1476_pinf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X: real] :
( ( ord_less_real @ Z3 @ X )
=> ( ord_less_eq_real @ T @ X ) ) ).
% pinf(8)
thf(fact_1477_pinf_I8_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X: rat] :
( ( ord_less_rat @ Z3 @ X )
=> ( ord_less_eq_rat @ T @ X ) ) ).
% pinf(8)
thf(fact_1478_pinf_I8_J,axiom,
! [T: num] :
? [Z3: num] :
! [X: num] :
( ( ord_less_num @ Z3 @ X )
=> ( ord_less_eq_num @ T @ X ) ) ).
% pinf(8)
thf(fact_1479_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ( ord_less_eq_nat @ T @ X ) ) ).
% pinf(8)
thf(fact_1480_pinf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X: int] :
( ( ord_less_int @ Z3 @ X )
=> ( ord_less_eq_int @ T @ X ) ) ).
% pinf(8)
thf(fact_1481_minf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X: real] :
( ( ord_less_real @ X @ Z3 )
=> ( ord_less_eq_real @ X @ T ) ) ).
% minf(6)
thf(fact_1482_minf_I6_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X: rat] :
( ( ord_less_rat @ X @ Z3 )
=> ( ord_less_eq_rat @ X @ T ) ) ).
% minf(6)
thf(fact_1483_minf_I6_J,axiom,
! [T: num] :
? [Z3: num] :
! [X: num] :
( ( ord_less_num @ X @ Z3 )
=> ( ord_less_eq_num @ X @ T ) ) ).
% minf(6)
thf(fact_1484_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ( ord_less_eq_nat @ X @ T ) ) ).
% minf(6)
thf(fact_1485_minf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X: int] :
( ( ord_less_int @ X @ Z3 )
=> ( ord_less_eq_int @ X @ T ) ) ).
% minf(6)
thf(fact_1486_minf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X: real] :
( ( ord_less_real @ X @ Z3 )
=> ~ ( ord_less_eq_real @ T @ X ) ) ).
% minf(8)
thf(fact_1487_minf_I8_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X: rat] :
( ( ord_less_rat @ X @ Z3 )
=> ~ ( ord_less_eq_rat @ T @ X ) ) ).
% minf(8)
thf(fact_1488_minf_I8_J,axiom,
! [T: num] :
? [Z3: num] :
! [X: num] :
( ( ord_less_num @ X @ Z3 )
=> ~ ( ord_less_eq_num @ T @ X ) ) ).
% minf(8)
thf(fact_1489_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X ) ) ).
% minf(8)
thf(fact_1490_minf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X: int] :
( ( ord_less_int @ X @ Z3 )
=> ~ ( ord_less_eq_int @ T @ X ) ) ).
% minf(8)
thf(fact_1491_empty__subsetI,axiom,
! [A4: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A4 ) ).
% empty_subsetI
thf(fact_1492_empty__subsetI,axiom,
! [A4: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A4 ) ).
% empty_subsetI
thf(fact_1493_empty__subsetI,axiom,
! [A4: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A4 ) ).
% empty_subsetI
thf(fact_1494_empty__subsetI,axiom,
! [A4: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A4 ) ).
% empty_subsetI
thf(fact_1495_subset__empty,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ bot_bo2099793752762293965at_nat )
= ( A4 = bot_bo2099793752762293965at_nat ) ) ).
% subset_empty
thf(fact_1496_subset__empty,axiom,
! [A4: set_real] :
( ( ord_less_eq_set_real @ A4 @ bot_bot_set_real )
= ( A4 = bot_bot_set_real ) ) ).
% subset_empty
thf(fact_1497_subset__empty,axiom,
! [A4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat )
= ( A4 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_1498_subset__empty,axiom,
! [A4: set_int] :
( ( ord_less_eq_set_int @ A4 @ bot_bot_set_int )
= ( A4 = bot_bot_set_int ) ) ).
% subset_empty
thf(fact_1499_length__enumerate,axiom,
! [N: nat,Xs: list_VEBT_VEBT] :
( ( size_s4762443039079500285T_VEBT @ ( enumerate_VEBT_VEBT @ N @ Xs ) )
= ( size_s6755466524823107622T_VEBT @ Xs ) ) ).
% length_enumerate
thf(fact_1500_length__enumerate,axiom,
! [N: nat,Xs: list_int] :
( ( size_s2970893825323803983at_int @ ( enumerate_int @ N @ Xs ) )
= ( size_size_list_int @ Xs ) ) ).
% length_enumerate
thf(fact_1501_length__enumerate,axiom,
! [N: nat,Xs: list_nat] :
( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_enumerate
thf(fact_1502_bot_Oextremum,axiom,
! [A: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A ) ).
% bot.extremum
thf(fact_1503_bot_Oextremum,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).
% bot.extremum
thf(fact_1504_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_1505_bot_Oextremum,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% bot.extremum
thf(fact_1506_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_1507_bot_Oextremum__unique,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
= ( A = bot_bo2099793752762293965at_nat ) ) ).
% bot.extremum_unique
thf(fact_1508_bot_Oextremum__unique,axiom,
! [A: set_real] :
( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
= ( A = bot_bot_set_real ) ) ).
% bot.extremum_unique
thf(fact_1509_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_1510_bot_Oextremum__unique,axiom,
! [A: set_int] :
( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
= ( A = bot_bot_set_int ) ) ).
% bot.extremum_unique
thf(fact_1511_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_1512_bot_Oextremum__uniqueI,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
=> ( A = bot_bo2099793752762293965at_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1513_bot_Oextremum__uniqueI,axiom,
! [A: set_real] :
( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
=> ( A = bot_bot_set_real ) ) ).
% bot.extremum_uniqueI
thf(fact_1514_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1515_bot_Oextremum__uniqueI,axiom,
! [A: set_int] :
( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
=> ( A = bot_bot_set_int ) ) ).
% bot.extremum_uniqueI
thf(fact_1516_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1517_bot_Onot__eq__extremum,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( A != bot_bo2099793752762293965at_nat )
= ( ord_le7866589430770878221at_nat @ bot_bo2099793752762293965at_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1518_bot_Onot__eq__extremum,axiom,
! [A: set_real] :
( ( A != bot_bot_set_real )
= ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1519_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1520_bot_Onot__eq__extremum,axiom,
! [A: set_int] :
( ( A != bot_bot_set_int )
= ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1521_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1522_bot_Oextremum__strict,axiom,
! [A: set_Pr1261947904930325089at_nat] :
~ ( ord_le7866589430770878221at_nat @ A @ bot_bo2099793752762293965at_nat ) ).
% bot.extremum_strict
thf(fact_1523_bot_Oextremum__strict,axiom,
! [A: set_real] :
~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).
% bot.extremum_strict
thf(fact_1524_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_1525_bot_Oextremum__strict,axiom,
! [A: set_int] :
~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).
% bot.extremum_strict
thf(fact_1526_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_1527_finite_OemptyI,axiom,
finite3207457112153483333omplex @ bot_bot_set_complex ).
% finite.emptyI
thf(fact_1528_finite_OemptyI,axiom,
finite6177210948735845034at_nat @ bot_bo2099793752762293965at_nat ).
% finite.emptyI
thf(fact_1529_finite_OemptyI,axiom,
finite_finite_real @ bot_bot_set_real ).
% finite.emptyI
thf(fact_1530_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_1531_finite_OemptyI,axiom,
finite_finite_int @ bot_bot_set_int ).
% finite.emptyI
thf(fact_1532_infinite__imp__nonempty,axiom,
! [S2: set_complex] :
( ~ ( finite3207457112153483333omplex @ S2 )
=> ( S2 != bot_bot_set_complex ) ) ).
% infinite_imp_nonempty
thf(fact_1533_infinite__imp__nonempty,axiom,
! [S2: set_Pr1261947904930325089at_nat] :
( ~ ( finite6177210948735845034at_nat @ S2 )
=> ( S2 != bot_bo2099793752762293965at_nat ) ) ).
% infinite_imp_nonempty
thf(fact_1534_infinite__imp__nonempty,axiom,
! [S2: set_real] :
( ~ ( finite_finite_real @ S2 )
=> ( S2 != bot_bot_set_real ) ) ).
% infinite_imp_nonempty
thf(fact_1535_infinite__imp__nonempty,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( S2 != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_1536_infinite__imp__nonempty,axiom,
! [S2: set_int] :
( ~ ( finite_finite_int @ S2 )
=> ( S2 != bot_bot_set_int ) ) ).
% infinite_imp_nonempty
thf(fact_1537_finite__transitivity__chain,axiom,
! [A4: set_set_nat,R: set_nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ! [X4: set_nat] :
~ ( R @ X4 @ X4 )
=> ( ! [X4: set_nat,Y3: set_nat,Z3: set_nat] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A4 )
=> ? [Y6: set_nat] :
( ( member_set_nat @ Y6 @ A4 )
& ( R @ X4 @ Y6 ) ) )
=> ( A4 = bot_bot_set_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1538_finite__transitivity__chain,axiom,
! [A4: set_complex,R: complex > complex > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [X4: complex] :
~ ( R @ X4 @ X4 )
=> ( ! [X4: complex,Y3: complex,Z3: complex] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ? [Y6: complex] :
( ( member_complex @ Y6 @ A4 )
& ( R @ X4 @ Y6 ) ) )
=> ( A4 = bot_bot_set_complex ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1539_finite__transitivity__chain,axiom,
! [A4: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ! [X4: product_prod_nat_nat] :
~ ( R @ X4 @ X4 )
=> ( ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat,Z3: product_prod_nat_nat] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A4 )
=> ? [Y6: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y6 @ A4 )
& ( R @ X4 @ Y6 ) ) )
=> ( A4 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1540_finite__transitivity__chain,axiom,
! [A4: set_real,R: real > real > $o] :
( ( finite_finite_real @ A4 )
=> ( ! [X4: real] :
~ ( R @ X4 @ X4 )
=> ( ! [X4: real,Y3: real,Z3: real] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ? [Y6: real] :
( ( member_real @ Y6 @ A4 )
& ( R @ X4 @ Y6 ) ) )
=> ( A4 = bot_bot_set_real ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1541_finite__transitivity__chain,axiom,
! [A4: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ! [X4: nat] :
~ ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y3: nat,Z3: nat] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ? [Y6: nat] :
( ( member_nat @ Y6 @ A4 )
& ( R @ X4 @ Y6 ) ) )
=> ( A4 = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1542_finite__transitivity__chain,axiom,
! [A4: set_int,R: int > int > $o] :
( ( finite_finite_int @ A4 )
=> ( ! [X4: int] :
~ ( R @ X4 @ X4 )
=> ( ! [X4: int,Y3: int,Z3: int] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ? [Y6: int] :
( ( member_int @ Y6 @ A4 )
& ( R @ X4 @ Y6 ) ) )
=> ( A4 = bot_bot_set_int ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1543_finite__has__maximal,axiom,
! [A4: set_real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ? [X4: real] :
( ( member_real @ X4 @ A4 )
& ! [Xa: real] :
( ( member_real @ Xa @ A4 )
=> ( ( ord_less_eq_real @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1544_finite__has__maximal,axiom,
! [A4: set_set_int] :
( ( finite6197958912794628473et_int @ A4 )
=> ( ( A4 != bot_bot_set_set_int )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ A4 )
& ! [Xa: set_int] :
( ( member_set_int @ Xa @ A4 )
=> ( ( ord_less_eq_set_int @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1545_finite__has__maximal,axiom,
! [A4: set_rat] :
( ( finite_finite_rat @ A4 )
=> ( ( A4 != bot_bot_set_rat )
=> ? [X4: rat] :
( ( member_rat @ X4 @ A4 )
& ! [Xa: rat] :
( ( member_rat @ Xa @ A4 )
=> ( ( ord_less_eq_rat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1546_finite__has__maximal,axiom,
! [A4: set_num] :
( ( finite_finite_num @ A4 )
=> ( ( A4 != bot_bot_set_num )
=> ? [X4: num] :
( ( member_num @ X4 @ A4 )
& ! [Xa: num] :
( ( member_num @ Xa @ A4 )
=> ( ( ord_less_eq_num @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1547_finite__has__maximal,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1548_finite__has__maximal,axiom,
! [A4: set_int] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ? [X4: int] :
( ( member_int @ X4 @ A4 )
& ! [Xa: int] :
( ( member_int @ Xa @ A4 )
=> ( ( ord_less_eq_int @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1549_finite__has__minimal,axiom,
! [A4: set_real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ? [X4: real] :
( ( member_real @ X4 @ A4 )
& ! [Xa: real] :
( ( member_real @ Xa @ A4 )
=> ( ( ord_less_eq_real @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1550_finite__has__minimal,axiom,
! [A4: set_set_int] :
( ( finite6197958912794628473et_int @ A4 )
=> ( ( A4 != bot_bot_set_set_int )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ A4 )
& ! [Xa: set_int] :
( ( member_set_int @ Xa @ A4 )
=> ( ( ord_less_eq_set_int @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1551_finite__has__minimal,axiom,
! [A4: set_rat] :
( ( finite_finite_rat @ A4 )
=> ( ( A4 != bot_bot_set_rat )
=> ? [X4: rat] :
( ( member_rat @ X4 @ A4 )
& ! [Xa: rat] :
( ( member_rat @ Xa @ A4 )
=> ( ( ord_less_eq_rat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1552_finite__has__minimal,axiom,
! [A4: set_num] :
( ( finite_finite_num @ A4 )
=> ( ( A4 != bot_bot_set_num )
=> ? [X4: num] :
( ( member_num @ X4 @ A4 )
& ! [Xa: num] :
( ( member_num @ Xa @ A4 )
=> ( ( ord_less_eq_num @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1553_finite__has__minimal,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1554_finite__has__minimal,axiom,
! [A4: set_int] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ? [X4: int] :
( ( member_int @ X4 @ A4 )
& ! [Xa: int] :
( ( member_int @ Xa @ A4 )
=> ( ( ord_less_eq_int @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1555_infinite__growing,axiom,
! [X8: set_real] :
( ( X8 != bot_bot_set_real )
=> ( ! [X4: real] :
( ( member_real @ X4 @ X8 )
=> ? [Xa: real] :
( ( member_real @ Xa @ X8 )
& ( ord_less_real @ X4 @ Xa ) ) )
=> ~ ( finite_finite_real @ X8 ) ) ) ).
% infinite_growing
thf(fact_1556_infinite__growing,axiom,
! [X8: set_rat] :
( ( X8 != bot_bot_set_rat )
=> ( ! [X4: rat] :
( ( member_rat @ X4 @ X8 )
=> ? [Xa: rat] :
( ( member_rat @ Xa @ X8 )
& ( ord_less_rat @ X4 @ Xa ) ) )
=> ~ ( finite_finite_rat @ X8 ) ) ) ).
% infinite_growing
thf(fact_1557_infinite__growing,axiom,
! [X8: set_num] :
( ( X8 != bot_bot_set_num )
=> ( ! [X4: num] :
( ( member_num @ X4 @ X8 )
=> ? [Xa: num] :
( ( member_num @ Xa @ X8 )
& ( ord_less_num @ X4 @ Xa ) ) )
=> ~ ( finite_finite_num @ X8 ) ) ) ).
% infinite_growing
thf(fact_1558_infinite__growing,axiom,
! [X8: set_nat] :
( ( X8 != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ X8 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X8 )
& ( ord_less_nat @ X4 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X8 ) ) ) ).
% infinite_growing
thf(fact_1559_infinite__growing,axiom,
! [X8: set_int] :
( ( X8 != bot_bot_set_int )
=> ( ! [X4: int] :
( ( member_int @ X4 @ X8 )
=> ? [Xa: int] :
( ( member_int @ Xa @ X8 )
& ( ord_less_int @ X4 @ Xa ) ) )
=> ~ ( finite_finite_int @ X8 ) ) ) ).
% infinite_growing
thf(fact_1560_ex__min__if__finite,axiom,
! [S2: set_real] :
( ( finite_finite_real @ S2 )
=> ( ( S2 != bot_bot_set_real )
=> ? [X4: real] :
( ( member_real @ X4 @ S2 )
& ~ ? [Xa: real] :
( ( member_real @ Xa @ S2 )
& ( ord_less_real @ Xa @ X4 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1561_ex__min__if__finite,axiom,
! [S2: set_rat] :
( ( finite_finite_rat @ S2 )
=> ( ( S2 != bot_bot_set_rat )
=> ? [X4: rat] :
( ( member_rat @ X4 @ S2 )
& ~ ? [Xa: rat] :
( ( member_rat @ Xa @ S2 )
& ( ord_less_rat @ Xa @ X4 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1562_ex__min__if__finite,axiom,
! [S2: set_num] :
( ( finite_finite_num @ S2 )
=> ( ( S2 != bot_bot_set_num )
=> ? [X4: num] :
( ( member_num @ X4 @ S2 )
& ~ ? [Xa: num] :
( ( member_num @ Xa @ S2 )
& ( ord_less_num @ Xa @ X4 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1563_ex__min__if__finite,axiom,
! [S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ S2 )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S2 )
& ( ord_less_nat @ Xa @ X4 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1564_ex__min__if__finite,axiom,
! [S2: set_int] :
( ( finite_finite_int @ S2 )
=> ( ( S2 != bot_bot_set_int )
=> ? [X4: int] :
( ( member_int @ X4 @ S2 )
& ~ ? [Xa: int] :
( ( member_int @ Xa @ S2 )
& ( ord_less_int @ Xa @ X4 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1565_divides__aux__eq,axiom,
! [Q3: nat,R2: nat] :
( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
= ( R2 = zero_zero_nat ) ) ).
% divides_aux_eq
thf(fact_1566_divides__aux__eq,axiom,
! [Q3: int,R2: int] :
( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q3 @ R2 ) )
= ( R2 = zero_zero_int ) ) ).
% divides_aux_eq
thf(fact_1567_arg__min__if__finite_I2_J,axiom,
! [S2: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( S2 != bot_bot_set_complex )
=> ~ ? [X: complex] :
( ( member_complex @ X @ S2 )
& ( ord_less_real @ ( F @ X ) @ ( F @ ( lattic8794016678065449205x_real @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1568_arg__min__if__finite_I2_J,axiom,
! [S2: set_real,F: real > real] :
( ( finite_finite_real @ S2 )
=> ( ( S2 != bot_bot_set_real )
=> ~ ? [X: real] :
( ( member_real @ X @ S2 )
& ( ord_less_real @ ( F @ X ) @ ( F @ ( lattic8440615504127631091l_real @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1569_arg__min__if__finite_I2_J,axiom,
! [S2: set_nat,F: nat > real] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ~ ? [X: nat] :
( ( member_nat @ X @ S2 )
& ( ord_less_real @ ( F @ X ) @ ( F @ ( lattic488527866317076247t_real @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1570_arg__min__if__finite_I2_J,axiom,
! [S2: set_int,F: int > real] :
( ( finite_finite_int @ S2 )
=> ( ( S2 != bot_bot_set_int )
=> ~ ? [X: int] :
( ( member_int @ X @ S2 )
& ( ord_less_real @ ( F @ X ) @ ( F @ ( lattic2675449441010098035t_real @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1571_arg__min__if__finite_I2_J,axiom,
! [S2: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( S2 != bot_bot_set_complex )
=> ~ ? [X: complex] :
( ( member_complex @ X @ S2 )
& ( ord_less_rat @ ( F @ X ) @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1572_arg__min__if__finite_I2_J,axiom,
! [S2: set_real,F: real > rat] :
( ( finite_finite_real @ S2 )
=> ( ( S2 != bot_bot_set_real )
=> ~ ? [X: real] :
( ( member_real @ X @ S2 )
& ( ord_less_rat @ ( F @ X ) @ ( F @ ( lattic4420706379359479199al_rat @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1573_arg__min__if__finite_I2_J,axiom,
! [S2: set_nat,F: nat > rat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ~ ? [X: nat] :
( ( member_nat @ X @ S2 )
& ( ord_less_rat @ ( F @ X ) @ ( F @ ( lattic6811802900495863747at_rat @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1574_arg__min__if__finite_I2_J,axiom,
! [S2: set_int,F: int > rat] :
( ( finite_finite_int @ S2 )
=> ( ( S2 != bot_bot_set_int )
=> ~ ? [X: int] :
( ( member_int @ X @ S2 )
& ( ord_less_rat @ ( F @ X ) @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1575_arg__min__if__finite_I2_J,axiom,
! [S2: set_complex,F: complex > num] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( S2 != bot_bot_set_complex )
=> ~ ? [X: complex] :
( ( member_complex @ X @ S2 )
& ( ord_less_num @ ( F @ X ) @ ( F @ ( lattic1922116423962787043ex_num @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1576_arg__min__if__finite_I2_J,axiom,
! [S2: set_real,F: real > num] :
( ( finite_finite_real @ S2 )
=> ( ( S2 != bot_bot_set_real )
=> ~ ? [X: real] :
( ( member_real @ X @ S2 )
& ( ord_less_num @ ( F @ X ) @ ( F @ ( lattic1613168225601753569al_num @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1577_arg__min__least,axiom,
! [S2: set_complex,Y: complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( S2 != bot_bot_set_complex )
=> ( ( member_complex @ Y @ S2 )
=> ( ord_less_eq_rat @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1578_arg__min__least,axiom,
! [S2: set_real,Y: real,F: real > rat] :
( ( finite_finite_real @ S2 )
=> ( ( S2 != bot_bot_set_real )
=> ( ( member_real @ Y @ S2 )
=> ( ord_less_eq_rat @ ( F @ ( lattic4420706379359479199al_rat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1579_arg__min__least,axiom,
! [S2: set_nat,Y: nat,F: nat > rat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S2 )
=> ( ord_less_eq_rat @ ( F @ ( lattic6811802900495863747at_rat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1580_arg__min__least,axiom,
! [S2: set_int,Y: int,F: int > rat] :
( ( finite_finite_int @ S2 )
=> ( ( S2 != bot_bot_set_int )
=> ( ( member_int @ Y @ S2 )
=> ( ord_less_eq_rat @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1581_arg__min__least,axiom,
! [S2: set_complex,Y: complex,F: complex > num] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( S2 != bot_bot_set_complex )
=> ( ( member_complex @ Y @ S2 )
=> ( ord_less_eq_num @ ( F @ ( lattic1922116423962787043ex_num @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1582_arg__min__least,axiom,
! [S2: set_real,Y: real,F: real > num] :
( ( finite_finite_real @ S2 )
=> ( ( S2 != bot_bot_set_real )
=> ( ( member_real @ Y @ S2 )
=> ( ord_less_eq_num @ ( F @ ( lattic1613168225601753569al_num @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1583_arg__min__least,axiom,
! [S2: set_nat,Y: nat,F: nat > num] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S2 )
=> ( ord_less_eq_num @ ( F @ ( lattic4004264746738138117at_num @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1584_arg__min__least,axiom,
! [S2: set_int,Y: int,F: int > num] :
( ( finite_finite_int @ S2 )
=> ( ( S2 != bot_bot_set_int )
=> ( ( member_int @ Y @ S2 )
=> ( ord_less_eq_num @ ( F @ ( lattic5003618458639192673nt_num @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1585_arg__min__least,axiom,
! [S2: set_complex,Y: complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( S2 != bot_bot_set_complex )
=> ( ( member_complex @ Y @ S2 )
=> ( ord_less_eq_nat @ ( F @ ( lattic5364784637807008409ex_nat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1586_arg__min__least,axiom,
! [S2: set_real,Y: real,F: real > nat] :
( ( finite_finite_real @ S2 )
=> ( ( S2 != bot_bot_set_real )
=> ( ( member_real @ Y @ S2 )
=> ( ord_less_eq_nat @ ( F @ ( lattic5055836439445974935al_nat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1587_subset__emptyI,axiom,
! [A4: set_complex] :
( ! [X4: complex] :
~ ( member_complex @ X4 @ A4 )
=> ( ord_le211207098394363844omplex @ A4 @ bot_bot_set_complex ) ) ).
% subset_emptyI
thf(fact_1588_subset__emptyI,axiom,
! [A4: set_set_nat] :
( ! [X4: set_nat] :
~ ( member_set_nat @ X4 @ A4 )
=> ( ord_le6893508408891458716et_nat @ A4 @ bot_bot_set_set_nat ) ) ).
% subset_emptyI
thf(fact_1589_subset__emptyI,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ! [X4: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ X4 @ A4 )
=> ( ord_le3146513528884898305at_nat @ A4 @ bot_bo2099793752762293965at_nat ) ) ).
% subset_emptyI
thf(fact_1590_subset__emptyI,axiom,
! [A4: set_real] :
( ! [X4: real] :
~ ( member_real @ X4 @ A4 )
=> ( ord_less_eq_set_real @ A4 @ bot_bot_set_real ) ) ).
% subset_emptyI
thf(fact_1591_subset__emptyI,axiom,
! [A4: set_nat] :
( ! [X4: nat] :
~ ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_1592_subset__emptyI,axiom,
! [A4: set_int] :
( ! [X4: int] :
~ ( member_int @ X4 @ A4 )
=> ( ord_less_eq_set_int @ A4 @ bot_bot_set_int ) ) ).
% subset_emptyI
thf(fact_1593_find__Some__iff2,axiom,
! [X3: product_prod_nat_nat,P2: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
( ( ( some_P7363390416028606310at_nat @ X3 )
= ( find_P8199882355184865565at_nat @ P2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
& ( P2 @ ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) )
& ( X3
= ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P2 @ ( nth_Pr7617993195940197384at_nat @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_1594_find__Some__iff2,axiom,
! [X3: num,P2: num > $o,Xs: list_num] :
( ( ( some_num @ X3 )
= ( find_num @ P2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_num @ Xs ) )
& ( P2 @ ( nth_num @ Xs @ I3 ) )
& ( X3
= ( nth_num @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P2 @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_1595_find__Some__iff2,axiom,
! [X3: vEBT_VEBT,P2: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
( ( ( some_VEBT_VEBT @ X3 )
= ( find_VEBT_VEBT @ P2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
& ( P2 @ ( nth_VEBT_VEBT @ Xs @ I3 ) )
& ( X3
= ( nth_VEBT_VEBT @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P2 @ ( nth_VEBT_VEBT @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_1596_find__Some__iff2,axiom,
! [X3: int,P2: int > $o,Xs: list_int] :
( ( ( some_int @ X3 )
= ( find_int @ P2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
& ( P2 @ ( nth_int @ Xs @ I3 ) )
& ( X3
= ( nth_int @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P2 @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_1597_find__Some__iff2,axiom,
! [X3: nat,P2: nat > $o,Xs: list_nat] :
( ( ( some_nat @ X3 )
= ( find_nat @ P2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
& ( P2 @ ( nth_nat @ Xs @ I3 ) )
& ( X3
= ( nth_nat @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P2 @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_1598_find__Some__iff,axiom,
! [P2: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat,X3: product_prod_nat_nat] :
( ( ( find_P8199882355184865565at_nat @ P2 @ Xs )
= ( some_P7363390416028606310at_nat @ X3 ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
& ( P2 @ ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) )
& ( X3
= ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P2 @ ( nth_Pr7617993195940197384at_nat @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_1599_find__Some__iff,axiom,
! [P2: num > $o,Xs: list_num,X3: num] :
( ( ( find_num @ P2 @ Xs )
= ( some_num @ X3 ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_num @ Xs ) )
& ( P2 @ ( nth_num @ Xs @ I3 ) )
& ( X3
= ( nth_num @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P2 @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_1600_find__Some__iff,axiom,
! [P2: vEBT_VEBT > $o,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
( ( ( find_VEBT_VEBT @ P2 @ Xs )
= ( some_VEBT_VEBT @ X3 ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
& ( P2 @ ( nth_VEBT_VEBT @ Xs @ I3 ) )
& ( X3
= ( nth_VEBT_VEBT @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P2 @ ( nth_VEBT_VEBT @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_1601_find__Some__iff,axiom,
! [P2: int > $o,Xs: list_int,X3: int] :
( ( ( find_int @ P2 @ Xs )
= ( some_int @ X3 ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
& ( P2 @ ( nth_int @ Xs @ I3 ) )
& ( X3
= ( nth_int @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P2 @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_1602_find__Some__iff,axiom,
! [P2: nat > $o,Xs: list_nat,X3: nat] :
( ( ( find_nat @ P2 @ Xs )
= ( some_nat @ X3 ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
& ( P2 @ ( nth_nat @ Xs @ I3 ) )
& ( X3
= ( nth_nat @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P2 @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_1603_nth__zip,axiom,
! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys2 ) )
=> ( ( nth_Pr4953567300277697838T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys2 ) @ I )
= ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_1604_nth__zip,axiom,
! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_int] :
( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys2 ) )
=> ( ( nth_Pr6837108013167703752BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys2 ) @ I )
= ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_int @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_1605_nth__zip,axiom,
! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_nat] :
( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
=> ( ( nth_Pr1791586995822124652BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys2 ) @ I )
= ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_nat @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_1606_nth__zip,axiom,
! [I: nat,Xs: list_int,Ys2: list_VEBT_VEBT] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys2 ) )
=> ( ( nth_Pr3474266648193625910T_VEBT @ ( zip_int_VEBT_VEBT @ Xs @ Ys2 ) @ I )
= ( produc3329399203697025711T_VEBT @ ( nth_int @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_1607_nth__zip,axiom,
! [I: nat,Xs: list_int,Ys2: list_int] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys2 ) )
=> ( ( nth_Pr4439495888332055232nt_int @ ( zip_int_int @ Xs @ Ys2 ) @ I )
= ( product_Pair_int_int @ ( nth_int @ Xs @ I ) @ ( nth_int @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_1608_nth__zip,axiom,
! [I: nat,Xs: list_int,Ys2: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
=> ( ( nth_Pr8617346907841251940nt_nat @ ( zip_int_nat @ Xs @ Ys2 ) @ I )
= ( product_Pair_int_nat @ ( nth_int @ Xs @ I ) @ ( nth_nat @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_1609_nth__zip,axiom,
! [I: nat,Xs: list_nat,Ys2: list_VEBT_VEBT] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys2 ) )
=> ( ( nth_Pr744662078594809490T_VEBT @ ( zip_nat_VEBT_VEBT @ Xs @ Ys2 ) @ I )
= ( produc599794634098209291T_VEBT @ ( nth_nat @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_1610_nth__zip,axiom,
! [I: nat,Xs: list_nat,Ys2: list_int] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys2 ) )
=> ( ( nth_Pr3440142176431000676at_int @ ( zip_nat_int @ Xs @ Ys2 ) @ I )
= ( product_Pair_nat_int @ ( nth_nat @ Xs @ I ) @ ( nth_int @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_1611_nth__zip,axiom,
! [I: nat,Xs: list_nat,Ys2: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( zip_nat_nat @ Xs @ Ys2 ) @ I )
= ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_1612_nth__zip,axiom,
! [I: nat,Xs: list_C878401137130745250e_term,Ys2: list_P5578671422887162913nteger] :
( ( ord_less_nat @ I @ ( size_s8902784113405215798e_term @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s6527587076508218509nteger @ Ys2 ) )
=> ( ( nth_Pr1312216959259138743nteger @ ( zip_Co8729459035503499408nteger @ Xs @ Ys2 ) @ I )
= ( produc6137756002093451184nteger @ ( nth_Co7464280458349004107e_term @ Xs @ I ) @ ( nth_Pr2304437835452373666nteger @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_1613_rotate1__length01,axiom,
! [Xs: list_VEBT_VEBT] :
( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ one_one_nat )
=> ( ( rotate1_VEBT_VEBT @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_1614_rotate1__length01,axiom,
! [Xs: list_int] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ one_one_nat )
=> ( ( rotate1_int @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_1615_rotate1__length01,axiom,
! [Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate1_nat @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_1616_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
= one_one_complex ) ).
% dbl_inc_simps(2)
thf(fact_1617_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_1618_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
= one_one_rat ) ).
% dbl_inc_simps(2)
thf(fact_1619_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_1620_VEBT_Osize__gen_I2_J,axiom,
! [X21: $o,X222: $o] :
( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
= zero_zero_nat ) ).
% VEBT.size_gen(2)
thf(fact_1621_set__rotate1,axiom,
! [Xs: list_VEBT_VEBT] :
( ( set_VEBT_VEBT2 @ ( rotate1_VEBT_VEBT @ Xs ) )
= ( set_VEBT_VEBT2 @ Xs ) ) ).
% set_rotate1
thf(fact_1622_set__rotate1,axiom,
! [Xs: list_int] :
( ( set_int2 @ ( rotate1_int @ Xs ) )
= ( set_int2 @ Xs ) ) ).
% set_rotate1
thf(fact_1623_set__rotate1,axiom,
! [Xs: list_nat] :
( ( set_nat2 @ ( rotate1_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% set_rotate1
thf(fact_1624_length__rotate1,axiom,
! [Xs: list_VEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT @ ( rotate1_VEBT_VEBT @ Xs ) )
= ( size_s6755466524823107622T_VEBT @ Xs ) ) ).
% length_rotate1
thf(fact_1625_length__rotate1,axiom,
! [Xs: list_int] :
( ( size_size_list_int @ ( rotate1_int @ Xs ) )
= ( size_size_list_int @ Xs ) ) ).
% length_rotate1
thf(fact_1626_length__rotate1,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate1
thf(fact_1627_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1628_find__cong,axiom,
! [Xs: list_complex,Ys2: list_complex,P2: complex > $o,Q: complex > $o] :
( ( Xs = Ys2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( set_complex2 @ Ys2 ) )
=> ( ( P2 @ X4 )
= ( Q @ X4 ) ) )
=> ( ( find_complex @ P2 @ Xs )
= ( find_complex @ Q @ Ys2 ) ) ) ) ).
% find_cong
thf(fact_1629_find__cong,axiom,
! [Xs: list_real,Ys2: list_real,P2: real > $o,Q: real > $o] :
( ( Xs = Ys2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( set_real2 @ Ys2 ) )
=> ( ( P2 @ X4 )
= ( Q @ X4 ) ) )
=> ( ( find_real @ P2 @ Xs )
= ( find_real @ Q @ Ys2 ) ) ) ) ).
% find_cong
thf(fact_1630_find__cong,axiom,
! [Xs: list_set_nat,Ys2: list_set_nat,P2: set_nat > $o,Q: set_nat > $o] :
( ( Xs = Ys2 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ ( set_set_nat2 @ Ys2 ) )
=> ( ( P2 @ X4 )
= ( Q @ X4 ) ) )
=> ( ( find_set_nat @ P2 @ Xs )
= ( find_set_nat @ Q @ Ys2 ) ) ) ) ).
% find_cong
thf(fact_1631_find__cong,axiom,
! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT,P2: vEBT_VEBT > $o,Q: vEBT_VEBT > $o] :
( ( Xs = Ys2 )
=> ( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Ys2 ) )
=> ( ( P2 @ X4 )
= ( Q @ X4 ) ) )
=> ( ( find_VEBT_VEBT @ P2 @ Xs )
= ( find_VEBT_VEBT @ Q @ Ys2 ) ) ) ) ).
% find_cong
thf(fact_1632_find__cong,axiom,
! [Xs: list_int,Ys2: list_int,P2: int > $o,Q: int > $o] :
( ( Xs = Ys2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( set_int2 @ Ys2 ) )
=> ( ( P2 @ X4 )
= ( Q @ X4 ) ) )
=> ( ( find_int @ P2 @ Xs )
= ( find_int @ Q @ Ys2 ) ) ) ) ).
% find_cong
thf(fact_1633_find__cong,axiom,
! [Xs: list_nat,Ys2: list_nat,P2: nat > $o,Q: nat > $o] :
( ( Xs = Ys2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Ys2 ) )
=> ( ( P2 @ X4 )
= ( Q @ X4 ) ) )
=> ( ( find_nat @ P2 @ Xs )
= ( find_nat @ Q @ Ys2 ) ) ) ) ).
% find_cong
thf(fact_1634_set__zip__rightD,axiom,
! [X3: int,Y: int,Xs: list_int,Ys2: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys2 ) ) )
=> ( member_int @ Y @ ( set_int2 @ Ys2 ) ) ) ).
% set_zip_rightD
thf(fact_1635_set__zip__rightD,axiom,
! [X3: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger,Xs: list_C878401137130745250e_term,Ys2: list_P5578671422887162913nteger] :
( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X3 @ Y ) @ ( set_Pr2999063419360598313nteger @ ( zip_Co8729459035503499408nteger @ Xs @ Ys2 ) ) )
=> ( member157494554546826820nteger @ Y @ ( set_Pr920681315882439344nteger @ Ys2 ) ) ) ).
% set_zip_rightD
thf(fact_1636_set__zip__rightD,axiom,
! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger,Xs: list_P1316552470764441098e_term,Ys2: list_P5578671422887162913nteger] :
( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y ) @ ( set_Pr2135590979564877377nteger @ ( zip_Pr8292346330294042792nteger @ Xs @ Ys2 ) ) )
=> ( member157494554546826820nteger @ Y @ ( set_Pr920681315882439344nteger @ Ys2 ) ) ) ).
% set_zip_rightD
thf(fact_1637_set__zip__rightD,axiom,
! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int,Xs: list_P1743416141875011707e_term,Ys2: list_P5707943133018811711nt_int] :
( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y ) @ ( set_Pr4943052134776177454nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys2 ) ) )
=> ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ Ys2 ) ) ) ).
% set_zip_rightD
thf(fact_1638_set__zip__rightD,axiom,
! [X3: int > option6357759511663192854e_term,Y: product_prod_int_int,Xs: list_i8448526496819171953e_term,Ys2: list_P5707943133018811711nt_int] :
( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y ) @ ( set_Pr1633835384712236856nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys2 ) ) )
=> ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ Ys2 ) ) ) ).
% set_zip_rightD
thf(fact_1639_set__zip__leftD,axiom,
! [X3: int,Y: int,Xs: list_int,Ys2: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys2 ) ) )
=> ( member_int @ X3 @ ( set_int2 @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_1640_set__zip__leftD,axiom,
! [X3: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger,Xs: list_C878401137130745250e_term,Ys2: list_P5578671422887162913nteger] :
( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X3 @ Y ) @ ( set_Pr2999063419360598313nteger @ ( zip_Co8729459035503499408nteger @ Xs @ Ys2 ) ) )
=> ( member1535805642427569193e_term @ X3 @ ( set_Co8062243466402858685e_term @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_1641_set__zip__leftD,axiom,
! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger,Xs: list_P1316552470764441098e_term,Ys2: list_P5578671422887162913nteger] :
( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y ) @ ( set_Pr2135590979564877377nteger @ ( zip_Pr8292346330294042792nteger @ Xs @ Ys2 ) ) )
=> ( member4242434998011752849e_term @ X3 @ ( set_Pr8342322266483756581e_term @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_1642_set__zip__leftD,axiom,
! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int,Xs: list_P1743416141875011707e_term,Ys2: list_P5707943133018811711nt_int] :
( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y ) @ ( set_Pr4943052134776177454nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys2 ) ) )
=> ( member3222579708246209666e_term @ X3 @ ( set_Pr16608062948090134e_term @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_1643_set__zip__leftD,axiom,
! [X3: int > option6357759511663192854e_term,Y: product_prod_int_int,Xs: list_i8448526496819171953e_term,Ys2: list_P5707943133018811711nt_int] :
( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y ) @ ( set_Pr1633835384712236856nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys2 ) ) )
=> ( member8845023287901829240e_term @ X3 @ ( set_in5217446777445088012e_term @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_1644_in__set__zipE,axiom,
! [X3: complex,Y: complex,Xs: list_complex,Ys2: list_complex] :
( ( member5793383173714906214omplex @ ( produc101793102246108661omplex @ X3 @ Y ) @ ( set_Pr8199049879907524818omplex @ ( zip_complex_complex @ Xs @ Ys2 ) ) )
=> ~ ( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
=> ~ ( member_complex @ Y @ ( set_complex2 @ Ys2 ) ) ) ) ).
% in_set_zipE
thf(fact_1645_in__set__zipE,axiom,
! [X3: complex,Y: real,Xs: list_complex,Ys2: list_real] :
( ( member47443559803733732x_real @ ( produc1746590499379883635x_real @ X3 @ Y ) @ ( set_Pr1225976482156248400x_real @ ( zip_complex_real @ Xs @ Ys2 ) ) )
=> ~ ( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
=> ~ ( member_real @ Y @ ( set_real2 @ Ys2 ) ) ) ) ).
% in_set_zipE
thf(fact_1646_in__set__zipE,axiom,
! [X3: real,Y: complex,Xs: list_real,Ys2: list_complex] :
( ( member7358116576843751780omplex @ ( produc1693001998875562995omplex @ X3 @ Y ) @ ( set_Pr8536649499196266448omplex @ ( zip_real_complex @ Xs @ Ys2 ) ) )
=> ~ ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ~ ( member_complex @ Y @ ( set_complex2 @ Ys2 ) ) ) ) ).
% in_set_zipE
thf(fact_1647_in__set__zipE,axiom,
! [X3: real,Y: real,Xs: list_real,Ys2: list_real] :
( ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ X3 @ Y ) @ ( set_Pr5999470521830281550l_real @ ( zip_real_real @ Xs @ Ys2 ) ) )
=> ~ ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ~ ( member_real @ Y @ ( set_real2 @ Ys2 ) ) ) ) ).
% in_set_zipE
thf(fact_1648_in__set__zipE,axiom,
! [X3: complex,Y: vEBT_VEBT,Xs: list_complex,Ys2: list_VEBT_VEBT] :
( ( member1978952105866562066T_VEBT @ ( produc2757191886755552429T_VEBT @ X3 @ Y ) @ ( set_Pr5158653123227461798T_VEBT @ ( zip_co9157518722488180109T_VEBT @ Xs @ Ys2 ) ) )
=> ~ ( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
=> ~ ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ Ys2 ) ) ) ) ).
% in_set_zipE
thf(fact_1649_in__set__zipE,axiom,
! [X3: real,Y: vEBT_VEBT,Xs: list_real,Ys2: list_VEBT_VEBT] :
( ( member7262085504369356948T_VEBT @ ( produc6931449550656315951T_VEBT @ X3 @ Y ) @ ( set_Pr8897343066327330088T_VEBT @ ( zip_real_VEBT_VEBT @ Xs @ Ys2 ) ) )
=> ~ ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ~ ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ Ys2 ) ) ) ) ).
% in_set_zipE
thf(fact_1650_in__set__zipE,axiom,
! [X3: complex,Y: int,Xs: list_complex,Ys2: list_int] :
( ( member595073364599660772ex_int @ ( produc1367138851071493491ex_int @ X3 @ Y ) @ ( set_Pr4995810437751016784ex_int @ ( zip_complex_int @ Xs @ Ys2 ) ) )
=> ~ ( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
=> ~ ( member_int @ Y @ ( set_int2 @ Ys2 ) ) ) ) ).
% in_set_zipE
thf(fact_1651_in__set__zipE,axiom,
! [X3: real,Y: int,Xs: list_real,Ys2: list_int] :
( ( member1627681773268152802al_int @ ( produc3179012173361985393al_int @ X3 @ Y ) @ ( set_Pr8219819362198175822al_int @ ( zip_real_int @ Xs @ Ys2 ) ) )
=> ~ ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ~ ( member_int @ Y @ ( set_int2 @ Ys2 ) ) ) ) ).
% in_set_zipE
thf(fact_1652_in__set__zipE,axiom,
! [X3: complex,Y: nat,Xs: list_complex,Ys2: list_nat] :
( ( member4772924384108857480ex_nat @ ( produc1369629321580543767ex_nat @ X3 @ Y ) @ ( set_Pr9173661457260213492ex_nat @ ( zip_complex_nat @ Xs @ Ys2 ) ) )
=> ~ ( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
=> ~ ( member_nat @ Y @ ( set_nat2 @ Ys2 ) ) ) ) ).
% in_set_zipE
thf(fact_1653_in__set__zipE,axiom,
! [X3: real,Y: nat,Xs: list_real,Ys2: list_nat] :
( ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X3 @ Y ) @ ( set_Pr3174298344852596722al_nat @ ( zip_real_nat @ Xs @ Ys2 ) ) )
=> ~ ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ~ ( member_nat @ Y @ ( set_nat2 @ Ys2 ) ) ) ) ).
% in_set_zipE
thf(fact_1654_zip__same,axiom,
! [A: complex,B: complex,Xs: list_complex] :
( ( member5793383173714906214omplex @ ( produc101793102246108661omplex @ A @ B ) @ ( set_Pr8199049879907524818omplex @ ( zip_complex_complex @ Xs @ Xs ) ) )
= ( ( member_complex @ A @ ( set_complex2 @ Xs ) )
& ( A = B ) ) ) ).
% zip_same
thf(fact_1655_zip__same,axiom,
! [A: real,B: real,Xs: list_real] :
( ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ A @ B ) @ ( set_Pr5999470521830281550l_real @ ( zip_real_real @ Xs @ Xs ) ) )
= ( ( member_real @ A @ ( set_real2 @ Xs ) )
& ( A = B ) ) ) ).
% zip_same
thf(fact_1656_zip__same,axiom,
! [A: set_nat,B: set_nat,Xs: list_set_nat] :
( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A @ B ) @ ( set_Pr9040384385603167362et_nat @ ( zip_set_nat_set_nat @ Xs @ Xs ) ) )
= ( ( member_set_nat @ A @ ( set_set_nat2 @ Xs ) )
& ( A = B ) ) ) ).
% zip_same
thf(fact_1657_zip__same,axiom,
! [A: vEBT_VEBT,B: vEBT_VEBT,Xs: list_VEBT_VEBT] :
( ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ A @ B ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Xs ) ) )
= ( ( member_VEBT_VEBT @ A @ ( set_VEBT_VEBT2 @ Xs ) )
& ( A = B ) ) ) ).
% zip_same
thf(fact_1658_zip__same,axiom,
! [A: nat,B: nat,Xs: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Xs ) ) )
= ( ( member_nat @ A @ ( set_nat2 @ Xs ) )
& ( A = B ) ) ) ).
% zip_same
thf(fact_1659_zip__same,axiom,
! [A: int,B: int,Xs: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Xs ) ) )
= ( ( member_int @ A @ ( set_int2 @ Xs ) )
& ( A = B ) ) ) ).
% zip_same
thf(fact_1660_in__set__impl__in__set__zip2,axiom,
! [Xs: list_VEBT_VEBT,Ys2: list_complex,Y: complex] :
( ( ( size_s6755466524823107622T_VEBT @ Xs )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( member_complex @ Y @ ( set_complex2 @ Ys2 ) )
=> ~ ! [X4: vEBT_VEBT] :
~ ( member3207599676835851048omplex @ ( produc5617778602380981643omplex @ X4 @ Y ) @ ( set_Pr6387300694196750780omplex @ ( zip_VE2794733401258833515omplex @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_1661_in__set__impl__in__set__zip2,axiom,
! [Xs: list_VEBT_VEBT,Ys2: list_real,Y: real] :
( ( ( size_s6755466524823107622T_VEBT @ Xs )
= ( size_size_list_real @ Ys2 ) )
=> ( ( member_real @ Y @ ( set_real2 @ Ys2 ) )
=> ~ ! [X4: vEBT_VEBT] :
~ ( member8675245146396747942T_real @ ( produc8117437818029410057T_real @ X4 @ Y ) @ ( set_Pr1087130671499945274T_real @ ( zip_VEBT_VEBT_real @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_1662_in__set__impl__in__set__zip2,axiom,
! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT,Y: vEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs )
= ( size_s6755466524823107622T_VEBT @ Ys2 ) )
=> ( ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ Ys2 ) )
=> ~ ! [X4: vEBT_VEBT] :
~ ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X4 @ Y ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_1663_in__set__impl__in__set__zip2,axiom,
! [Xs: list_VEBT_VEBT,Ys2: list_int,Y: int] :
( ( ( size_s6755466524823107622T_VEBT @ Xs )
= ( size_size_list_int @ Ys2 ) )
=> ( ( member_int @ Y @ ( set_int2 @ Ys2 ) )
=> ~ ! [X4: vEBT_VEBT] :
~ ( member5419026705395827622BT_int @ ( produc736041933913180425BT_int @ X4 @ Y ) @ ( set_Pr2853735649769556538BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_1664_in__set__impl__in__set__zip2,axiom,
! [Xs: list_VEBT_VEBT,Ys2: list_nat,Y: nat] :
( ( ( size_s6755466524823107622T_VEBT @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( member_nat @ Y @ ( set_nat2 @ Ys2 ) )
=> ~ ! [X4: vEBT_VEBT] :
~ ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X4 @ Y ) @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_1665_in__set__impl__in__set__zip2,axiom,
! [Xs: list_int,Ys2: list_complex,Y: complex] :
( ( ( size_size_list_int @ Xs )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( member_complex @ Y @ ( set_complex2 @ Ys2 ) )
=> ~ ! [X4: int] :
~ ( member8811922270175639012omplex @ ( produc7948753499206759283omplex @ X4 @ Y ) @ ( set_Pr3989287306472219216omplex @ ( zip_int_complex @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_1666_in__set__impl__in__set__zip2,axiom,
! [Xs: list_int,Ys2: list_real,Y: real] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_real @ Ys2 ) )
=> ( ( member_real @ Y @ ( set_real2 @ Ys2 ) )
=> ~ ! [X4: int] :
~ ( member2744130022092475746t_real @ ( produc801115645435158769t_real @ X4 @ Y ) @ ( set_Pr112895574167722958t_real @ ( zip_int_real @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_1667_in__set__impl__in__set__zip2,axiom,
! [Xs: list_int,Ys2: list_VEBT_VEBT,Y: vEBT_VEBT] :
( ( ( size_size_list_int @ Xs )
= ( size_s6755466524823107622T_VEBT @ Ys2 ) )
=> ( ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ Ys2 ) )
=> ~ ! [X4: int] :
~ ( member2056185340421749780T_VEBT @ ( produc3329399203697025711T_VEBT @ X4 @ Y ) @ ( set_Pr8714266321650254504T_VEBT @ ( zip_int_VEBT_VEBT @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_1668_in__set__impl__in__set__zip2,axiom,
! [Xs: list_int,Ys2: list_int,Y: int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys2 ) )
=> ( ( member_int @ Y @ ( set_int2 @ Ys2 ) )
=> ~ ! [X4: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_1669_in__set__impl__in__set__zip2,axiom,
! [Xs: list_int,Ys2: list_nat,Y: nat] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( member_nat @ Y @ ( set_nat2 @ Ys2 ) )
=> ~ ! [X4: int] :
~ ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ X4 @ Y ) @ ( set_Pr6647972299459129970nt_nat @ ( zip_int_nat @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_1670_in__set__impl__in__set__zip1,axiom,
! [Xs: list_complex,Ys2: list_VEBT_VEBT,X3: complex] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s6755466524823107622T_VEBT @ Ys2 ) )
=> ( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
=> ~ ! [Y3: vEBT_VEBT] :
~ ( member1978952105866562066T_VEBT @ ( produc2757191886755552429T_VEBT @ X3 @ Y3 ) @ ( set_Pr5158653123227461798T_VEBT @ ( zip_co9157518722488180109T_VEBT @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_1671_in__set__impl__in__set__zip1,axiom,
! [Xs: list_real,Ys2: list_VEBT_VEBT,X3: real] :
( ( ( size_size_list_real @ Xs )
= ( size_s6755466524823107622T_VEBT @ Ys2 ) )
=> ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ~ ! [Y3: vEBT_VEBT] :
~ ( member7262085504369356948T_VEBT @ ( produc6931449550656315951T_VEBT @ X3 @ Y3 ) @ ( set_Pr8897343066327330088T_VEBT @ ( zip_real_VEBT_VEBT @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_1672_in__set__impl__in__set__zip1,axiom,
! [Xs: list_complex,Ys2: list_int,X3: complex] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_int @ Ys2 ) )
=> ( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
=> ~ ! [Y3: int] :
~ ( member595073364599660772ex_int @ ( produc1367138851071493491ex_int @ X3 @ Y3 ) @ ( set_Pr4995810437751016784ex_int @ ( zip_complex_int @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_1673_in__set__impl__in__set__zip1,axiom,
! [Xs: list_real,Ys2: list_int,X3: real] :
( ( ( size_size_list_real @ Xs )
= ( size_size_list_int @ Ys2 ) )
=> ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ~ ! [Y3: int] :
~ ( member1627681773268152802al_int @ ( produc3179012173361985393al_int @ X3 @ Y3 ) @ ( set_Pr8219819362198175822al_int @ ( zip_real_int @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_1674_in__set__impl__in__set__zip1,axiom,
! [Xs: list_complex,Ys2: list_nat,X3: complex] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
=> ~ ! [Y3: nat] :
~ ( member4772924384108857480ex_nat @ ( produc1369629321580543767ex_nat @ X3 @ Y3 ) @ ( set_Pr9173661457260213492ex_nat @ ( zip_complex_nat @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_1675_in__set__impl__in__set__zip1,axiom,
! [Xs: list_real,Ys2: list_nat,X3: real] :
( ( ( size_size_list_real @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ~ ! [Y3: nat] :
~ ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X3 @ Y3 ) @ ( set_Pr3174298344852596722al_nat @ ( zip_real_nat @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_1676_in__set__impl__in__set__zip1,axiom,
! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT,X3: vEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs )
= ( size_s6755466524823107622T_VEBT @ Ys2 ) )
=> ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ~ ! [Y3: vEBT_VEBT] :
~ ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X3 @ Y3 ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_1677_in__set__impl__in__set__zip1,axiom,
! [Xs: list_VEBT_VEBT,Ys2: list_int,X3: vEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs )
= ( size_size_list_int @ Ys2 ) )
=> ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ~ ! [Y3: int] :
~ ( member5419026705395827622BT_int @ ( produc736041933913180425BT_int @ X3 @ Y3 ) @ ( set_Pr2853735649769556538BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_1678_in__set__impl__in__set__zip1,axiom,
! [Xs: list_VEBT_VEBT,Ys2: list_nat,X3: vEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ~ ! [Y3: nat] :
~ ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X3 @ Y3 ) @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_1679_in__set__impl__in__set__zip1,axiom,
! [Xs: list_int,Ys2: list_VEBT_VEBT,X3: int] :
( ( ( size_size_list_int @ Xs )
= ( size_s6755466524823107622T_VEBT @ Ys2 ) )
=> ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ~ ! [Y3: vEBT_VEBT] :
~ ( member2056185340421749780T_VEBT @ ( produc3329399203697025711T_VEBT @ X3 @ Y3 ) @ ( set_Pr8714266321650254504T_VEBT @ ( zip_int_VEBT_VEBT @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_1680_ssubst__Pair__rhs,axiom,
! [R2: int,S3: int,R: set_Pr958786334691620121nt_int,S4: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R2 @ S3 ) @ R )
=> ( ( S4 = S3 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R2 @ S4 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_1681_ssubst__Pair__rhs,axiom,
! [R2: code_integer > option6357759511663192854e_term,S3: produc8923325533196201883nteger,R: set_Pr8056137968301705908nteger,S4: produc8923325533196201883nteger] :
( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ R2 @ S3 ) @ R )
=> ( ( S4 = S3 )
=> ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ R2 @ S4 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_1682_ssubst__Pair__rhs,axiom,
! [R2: produc6241069584506657477e_term > option6357759511663192854e_term,S3: produc8923325533196201883nteger,R: set_Pr1281608226676607948nteger,S4: produc8923325533196201883nteger] :
( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ R2 @ S3 ) @ R )
=> ( ( S4 = S3 )
=> ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ R2 @ S4 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_1683_ssubst__Pair__rhs,axiom,
! [R2: produc8551481072490612790e_term > option6357759511663192854e_term,S3: product_prod_int_int,R: set_Pr9222295170931077689nt_int,S4: product_prod_int_int] :
( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R2 @ S3 ) @ R )
=> ( ( S4 = S3 )
=> ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R2 @ S4 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_1684_ssubst__Pair__rhs,axiom,
! [R2: int > option6357759511663192854e_term,S3: product_prod_int_int,R: set_Pr1872883991513573699nt_int,S4: product_prod_int_int] :
( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R2 @ S3 ) @ R )
=> ( ( S4 = S3 )
=> ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R2 @ S4 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_1685_find__None__iff,axiom,
! [P2: complex > $o,Xs: list_complex] :
( ( ( find_complex @ P2 @ Xs )
= none_complex )
= ( ~ ? [X5: complex] :
( ( member_complex @ X5 @ ( set_complex2 @ Xs ) )
& ( P2 @ X5 ) ) ) ) ).
% find_None_iff
thf(fact_1686_find__None__iff,axiom,
! [P2: real > $o,Xs: list_real] :
( ( ( find_real @ P2 @ Xs )
= none_real )
= ( ~ ? [X5: real] :
( ( member_real @ X5 @ ( set_real2 @ Xs ) )
& ( P2 @ X5 ) ) ) ) ).
% find_None_iff
thf(fact_1687_find__None__iff,axiom,
! [P2: set_nat > $o,Xs: list_set_nat] :
( ( ( find_set_nat @ P2 @ Xs )
= none_set_nat )
= ( ~ ? [X5: set_nat] :
( ( member_set_nat @ X5 @ ( set_set_nat2 @ Xs ) )
& ( P2 @ X5 ) ) ) ) ).
% find_None_iff
thf(fact_1688_find__None__iff,axiom,
! [P2: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
( ( ( find_VEBT_VEBT @ P2 @ Xs )
= none_VEBT_VEBT )
= ( ~ ? [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs ) )
& ( P2 @ X5 ) ) ) ) ).
% find_None_iff
thf(fact_1689_find__None__iff,axiom,
! [P2: int > $o,Xs: list_int] :
( ( ( find_int @ P2 @ Xs )
= none_int )
= ( ~ ? [X5: int] :
( ( member_int @ X5 @ ( set_int2 @ Xs ) )
& ( P2 @ X5 ) ) ) ) ).
% find_None_iff
thf(fact_1690_find__None__iff,axiom,
! [P2: nat > $o,Xs: list_nat] :
( ( ( find_nat @ P2 @ Xs )
= none_nat )
= ( ~ ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P2 @ X5 ) ) ) ) ).
% find_None_iff
thf(fact_1691_find__None__iff,axiom,
! [P2: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
( ( ( find_P8199882355184865565at_nat @ P2 @ Xs )
= none_P5556105721700978146at_nat )
= ( ~ ? [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
& ( P2 @ X5 ) ) ) ) ).
% find_None_iff
thf(fact_1692_find__None__iff,axiom,
! [P2: num > $o,Xs: list_num] :
( ( ( find_num @ P2 @ Xs )
= none_num )
= ( ~ ? [X5: num] :
( ( member_num @ X5 @ ( set_num2 @ Xs ) )
& ( P2 @ X5 ) ) ) ) ).
% find_None_iff
thf(fact_1693_find__None__iff2,axiom,
! [P2: complex > $o,Xs: list_complex] :
( ( none_complex
= ( find_complex @ P2 @ Xs ) )
= ( ~ ? [X5: complex] :
( ( member_complex @ X5 @ ( set_complex2 @ Xs ) )
& ( P2 @ X5 ) ) ) ) ).
% find_None_iff2
thf(fact_1694_find__None__iff2,axiom,
! [P2: real > $o,Xs: list_real] :
( ( none_real
= ( find_real @ P2 @ Xs ) )
= ( ~ ? [X5: real] :
( ( member_real @ X5 @ ( set_real2 @ Xs ) )
& ( P2 @ X5 ) ) ) ) ).
% find_None_iff2
thf(fact_1695_find__None__iff2,axiom,
! [P2: set_nat > $o,Xs: list_set_nat] :
( ( none_set_nat
= ( find_set_nat @ P2 @ Xs ) )
= ( ~ ? [X5: set_nat] :
( ( member_set_nat @ X5 @ ( set_set_nat2 @ Xs ) )
& ( P2 @ X5 ) ) ) ) ).
% find_None_iff2
thf(fact_1696_find__None__iff2,axiom,
! [P2: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
( ( none_VEBT_VEBT
= ( find_VEBT_VEBT @ P2 @ Xs ) )
= ( ~ ? [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs ) )
& ( P2 @ X5 ) ) ) ) ).
% find_None_iff2
thf(fact_1697_find__None__iff2,axiom,
! [P2: int > $o,Xs: list_int] :
( ( none_int
= ( find_int @ P2 @ Xs ) )
= ( ~ ? [X5: int] :
( ( member_int @ X5 @ ( set_int2 @ Xs ) )
& ( P2 @ X5 ) ) ) ) ).
% find_None_iff2
thf(fact_1698_find__None__iff2,axiom,
! [P2: nat > $o,Xs: list_nat] :
( ( none_nat
= ( find_nat @ P2 @ Xs ) )
= ( ~ ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P2 @ X5 ) ) ) ) ).
% find_None_iff2
thf(fact_1699_find__None__iff2,axiom,
! [P2: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
( ( none_P5556105721700978146at_nat
= ( find_P8199882355184865565at_nat @ P2 @ Xs ) )
= ( ~ ? [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
& ( P2 @ X5 ) ) ) ) ).
% find_None_iff2
thf(fact_1700_find__None__iff2,axiom,
! [P2: num > $o,Xs: list_num] :
( ( none_num
= ( find_num @ P2 @ Xs ) )
= ( ~ ? [X5: num] :
( ( member_num @ X5 @ ( set_num2 @ Xs ) )
& ( P2 @ X5 ) ) ) ) ).
% find_None_iff2
thf(fact_1701_dbl__inc__def,axiom,
( neg_nu8557863876264182079omplex
= ( ^ [X5: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X5 @ X5 ) @ one_one_complex ) ) ) ).
% dbl_inc_def
thf(fact_1702_dbl__inc__def,axiom,
( neg_nu8295874005876285629c_real
= ( ^ [X5: real] : ( plus_plus_real @ ( plus_plus_real @ X5 @ X5 ) @ one_one_real ) ) ) ).
% dbl_inc_def
thf(fact_1703_dbl__inc__def,axiom,
( neg_nu5219082963157363817nc_rat
= ( ^ [X5: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X5 @ X5 ) @ one_one_rat ) ) ) ).
% dbl_inc_def
thf(fact_1704_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X5: int] : ( plus_plus_int @ ( plus_plus_int @ X5 @ X5 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_1705__C5_Ohyps_C_I9_J,axiom,
( ( mi != ma )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
=> ( ( ( ( vEBT_VEBT_high @ ma @ na )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
& ! [X: nat] :
( ( ( ( vEBT_VEBT_high @ X @ na )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ ( vEBT_VEBT_low @ X @ na ) ) )
=> ( ( ord_less_nat @ mi @ X )
& ( ord_less_eq_nat @ X @ ma ) ) ) ) ) ) ).
% "5.hyps"(9)
thf(fact_1706_pair__lessI2,axiom,
! [A: nat,B: nat,S3: nat,T: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ S3 @ T )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).
% pair_lessI2
thf(fact_1707_intind,axiom,
! [I: nat,N: nat,P2: int > $o,X3: int] :
( ( ord_less_nat @ I @ N )
=> ( ( P2 @ X3 )
=> ( P2 @ ( nth_int @ ( replicate_int @ N @ X3 ) @ I ) ) ) ) ).
% intind
thf(fact_1708_intind,axiom,
! [I: nat,N: nat,P2: nat > $o,X3: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( P2 @ X3 )
=> ( P2 @ ( nth_nat @ ( replicate_nat @ N @ X3 ) @ I ) ) ) ) ).
% intind
thf(fact_1709_intind,axiom,
! [I: nat,N: nat,P2: vEBT_VEBT > $o,X3: vEBT_VEBT] :
( ( ord_less_nat @ I @ N )
=> ( ( P2 @ X3 )
=> ( P2 @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X3 ) @ I ) ) ) ) ).
% intind
thf(fact_1710_pair__less__iff1,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( product_Pair_nat_nat @ X3 @ Z2 ) ) @ fun_pair_less )
= ( ord_less_nat @ Y @ Z2 ) ) ).
% pair_less_iff1
thf(fact_1711_gen__length__def,axiom,
( gen_length_VEBT_VEBT
= ( ^ [N4: nat,Xs3: list_VEBT_VEBT] : ( plus_plus_nat @ N4 @ ( size_s6755466524823107622T_VEBT @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_1712_gen__length__def,axiom,
( gen_length_int
= ( ^ [N4: nat,Xs3: list_int] : ( plus_plus_nat @ N4 @ ( size_size_list_int @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_1713_gen__length__def,axiom,
( gen_length_nat
= ( ^ [N4: nat,Xs3: list_nat] : ( plus_plus_nat @ N4 @ ( size_size_list_nat @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_1714_set__encode__empty,axiom,
( ( nat_set_encode @ bot_bot_set_nat )
= zero_zero_nat ) ).
% set_encode_empty
thf(fact_1715_nth__rotate1,axiom,
! [N: nat,Xs: list_VEBT_VEBT] :
( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( nth_VEBT_VEBT @ ( rotate1_VEBT_VEBT @ Xs ) @ N )
= ( nth_VEBT_VEBT @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_1716_nth__rotate1,axiom,
! [N: nat,Xs: list_int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( rotate1_int @ Xs ) @ N )
= ( nth_int @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_1717_nth__rotate1,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rotate1_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_1718_length__code,axiom,
( size_s6755466524823107622T_VEBT
= ( gen_length_VEBT_VEBT @ zero_zero_nat ) ) ).
% length_code
thf(fact_1719_length__code,axiom,
( size_size_list_int
= ( gen_length_int @ zero_zero_nat ) ) ).
% length_code
thf(fact_1720_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_1721_nth__equal__first__eq,axiom,
! [X3: complex,Xs: list_complex,N: nat] :
( ~ ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( ( ( nth_complex @ ( cons_complex @ X3 @ Xs ) @ N )
= X3 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1722_nth__equal__first__eq,axiom,
! [X3: real,Xs: list_real,N: nat] :
( ~ ( member_real @ X3 @ ( set_real2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_real @ Xs ) )
=> ( ( ( nth_real @ ( cons_real @ X3 @ Xs ) @ N )
= X3 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1723_nth__equal__first__eq,axiom,
! [X3: set_nat,Xs: list_set_nat,N: nat] :
( ~ ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
=> ( ( ( nth_set_nat @ ( cons_set_nat @ X3 @ Xs ) @ N )
= X3 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1724_nth__equal__first__eq,axiom,
! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] :
( ~ ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X3 @ Xs ) @ N )
= X3 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1725_nth__equal__first__eq,axiom,
! [X3: int,Xs: list_int,N: nat] :
( ~ ( member_int @ X3 @ ( set_int2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( ( nth_int @ ( cons_int @ X3 @ Xs ) @ N )
= X3 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1726_nth__equal__first__eq,axiom,
! [X3: nat,Xs: list_nat,N: nat] :
( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ ( cons_nat @ X3 @ Xs ) @ N )
= X3 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1727__C5_Ohyps_C_I2_J,axiom,
( ( size_s6755466524823107622T_VEBT @ treeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).
% "5.hyps"(2)
thf(fact_1728__C5_Ohyps_C_I8_J,axiom,
ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% "5.hyps"(8)
thf(fact_1729__C5_Oprems_C_I1_J,axiom,
ord_less_nat @ xa @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% "5.prems"(1)
thf(fact_1730_list_Oinject,axiom,
! [X21: int,X222: list_int,Y21: int,Y22: list_int] :
( ( ( cons_int @ X21 @ X222 )
= ( cons_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_1731_list_Oinject,axiom,
! [X21: nat,X222: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X222 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_1732__C5_Oprems_C_I2_J,axiom,
ord_less_nat @ ya @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% "5.prems"(2)
thf(fact_1733_low__def,axiom,
( vEBT_VEBT_low
= ( ^ [X5: nat,N4: nat] : ( modulo_modulo_nat @ X5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% low_def
thf(fact_1734__C5_OIH_C_I1_J,axiom,
! [X: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ treeList ) )
=> ( ( vEBT_invar_vebt @ X @ na )
& ! [Xa: nat] :
( ( ord_less_nat @ Xa @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) )
=> ! [Xb: nat] :
( ( ord_less_nat @ Xb @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) )
=> ( ( vEBT_V8194947554948674370ptions @ X @ Xa )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ X @ Xb ) @ Xa ) ) ) ) ) ) ).
% "5.IH"(1)
thf(fact_1735__C5_Ohyps_C_I5_J,axiom,
! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ summary @ I4 ) ) ) ).
% "5.hyps"(5)
thf(fact_1736_high__bound__aux,axiom,
! [Ma: nat,N: nat,M2: nat] :
( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M2 ) ) )
=> ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).
% high_bound_aux
thf(fact_1737_member__bound,axiom,
! [Tree: vEBT_VEBT,X3: nat,N: nat] :
( ( vEBT_vebt_member @ Tree @ X3 )
=> ( ( vEBT_invar_vebt @ Tree @ N )
=> ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% member_bound
thf(fact_1738_numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% numeral_le_iff
thf(fact_1739_numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M2 ) @ ( numeral_numeral_rat @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% numeral_le_iff
thf(fact_1740_numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% numeral_le_iff
thf(fact_1741_numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% numeral_le_iff
thf(fact_1742_numeral__plus__numeral,axiom,
! [M2: num,N: num] :
( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M2 ) @ ( numera6690914467698888265omplex @ N ) )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ M2 @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_1743_numeral__plus__numeral,axiom,
! [M2: num,N: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( plus_plus_num @ M2 @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_1744_numeral__plus__numeral,axiom,
! [M2: num,N: num] :
( ( plus_plus_rat @ ( numeral_numeral_rat @ M2 ) @ ( numeral_numeral_rat @ N ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ M2 @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_1745_numeral__plus__numeral,axiom,
! [M2: num,N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_1746_numeral__plus__numeral,axiom,
! [M2: num,N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M2 @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_1747_add__numeral__left,axiom,
! [V: num,W2: num,Z2: complex] :
( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W2 ) @ Z2 ) )
= ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_1748_add__numeral__left,axiom,
! [V: num,W2: num,Z2: real] :
( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W2 ) @ Z2 ) )
= ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_1749_add__numeral__left,axiom,
! [V: num,W2: num,Z2: rat] :
( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W2 ) @ Z2 ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_1750_add__numeral__left,axiom,
! [V: num,W2: num,Z2: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W2 ) @ Z2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_1751_add__numeral__left,axiom,
! [V: num,W2: num,Z2: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W2 ) @ Z2 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_1752_mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mod_0
thf(fact_1753_mod__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mod_0
thf(fact_1754_mod__0,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
= zero_z3403309356797280102nteger ) ).
% mod_0
thf(fact_1755_mod__by__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ zero_zero_nat )
= A ) ).
% mod_by_0
thf(fact_1756_mod__by__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ zero_zero_int )
= A ) ).
% mod_by_0
thf(fact_1757_mod__by__0,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
= A ) ).
% mod_by_0
thf(fact_1758_mod__self,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ A )
= zero_zero_nat ) ).
% mod_self
thf(fact_1759_mod__self,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ A )
= zero_zero_int ) ).
% mod_self
thf(fact_1760_mod__self,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ A @ A )
= zero_z3403309356797280102nteger ) ).
% mod_self
thf(fact_1761_set__n__deg__not__0,axiom,
! [TreeList: list_VEBT_VEBT,N: nat,M2: nat] :
( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
=> ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).
% set_n_deg_not_0
thf(fact_1762_insert__simp__mima,axiom,
! [X3: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X3 = Mi )
| ( X3 = Ma ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% insert_simp_mima
thf(fact_1763_valid__insert__both__member__options__add,axiom,
! [T: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X3 ) @ X3 ) ) ) ).
% valid_insert_both_member_options_add
thf(fact_1764_replicate__eq__replicate,axiom,
! [M2: nat,X3: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
( ( ( replicate_VEBT_VEBT @ M2 @ X3 )
= ( replicate_VEBT_VEBT @ N @ Y ) )
= ( ( M2 = N )
& ( ( M2 != zero_zero_nat )
=> ( X3 = Y ) ) ) ) ).
% replicate_eq_replicate
thf(fact_1765_length__replicate,axiom,
! [N: nat,X3: vEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N @ X3 ) )
= N ) ).
% length_replicate
thf(fact_1766_length__replicate,axiom,
! [N: nat,X3: int] :
( ( size_size_list_int @ ( replicate_int @ N @ X3 ) )
= N ) ).
% length_replicate
thf(fact_1767_length__replicate,axiom,
! [N: nat,X3: nat] :
( ( size_size_list_nat @ ( replicate_nat @ N @ X3 ) )
= N ) ).
% length_replicate
thf(fact_1768_mi__ma__2__deg,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ( ord_less_eq_nat @ Mi @ Ma )
& ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).
% mi_ma_2_deg
thf(fact_1769__C5_OIH_C_I2_J,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
=> ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
=> ( ( vEBT_V8194947554948674370ptions @ summary @ X3 )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ summary @ Y ) @ X3 ) ) ) ) ).
% "5.IH"(2)
thf(fact_1770_mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% mod_by_1
thf(fact_1771_mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% mod_by_1
thf(fact_1772_mod__by__1,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
= zero_z3403309356797280102nteger ) ).
% mod_by_1
thf(fact_1773_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% Suc_numeral
thf(fact_1774_nth__Cons__Suc,axiom,
! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] :
( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X3 @ Xs ) @ ( suc @ N ) )
= ( nth_VEBT_VEBT @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1775_nth__Cons__Suc,axiom,
! [X3: int,Xs: list_int,N: nat] :
( ( nth_int @ ( cons_int @ X3 @ Xs ) @ ( suc @ N ) )
= ( nth_int @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1776_nth__Cons__Suc,axiom,
! [X3: nat,Xs: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X3 @ Xs ) @ ( suc @ N ) )
= ( nth_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1777_nth__Cons__0,axiom,
! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X3 @ Xs ) @ zero_zero_nat )
= X3 ) ).
% nth_Cons_0
thf(fact_1778_nth__Cons__0,axiom,
! [X3: int,Xs: list_int] :
( ( nth_int @ ( cons_int @ X3 @ Xs ) @ zero_zero_nat )
= X3 ) ).
% nth_Cons_0
thf(fact_1779_nth__Cons__0,axiom,
! [X3: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X3 @ Xs ) @ zero_zero_nat )
= X3 ) ).
% nth_Cons_0
thf(fact_1780_zip__Cons__Cons,axiom,
! [X3: int,Xs: list_int,Y: nat,Ys2: list_nat] :
( ( zip_int_nat @ ( cons_int @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) )
= ( cons_P7512249878480867347nt_nat @ ( product_Pair_int_nat @ X3 @ Y ) @ ( zip_int_nat @ Xs @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_1781_zip__Cons__Cons,axiom,
! [X3: nat,Xs: list_nat,Y: int,Ys2: list_int] :
( ( zip_nat_int @ ( cons_nat @ X3 @ Xs ) @ ( cons_int @ Y @ Ys2 ) )
= ( cons_P2335045147070616083at_int @ ( product_Pair_nat_int @ X3 @ Y ) @ ( zip_nat_int @ Xs @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_1782_zip__Cons__Cons,axiom,
! [X3: nat,Xs: list_nat,Y: nat,Ys2: list_nat] :
( ( zip_nat_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) )
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( zip_nat_nat @ Xs @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_1783_zip__Cons__Cons,axiom,
! [X3: int,Xs: list_int,Y: int,Ys2: list_int] :
( ( zip_int_int @ ( cons_int @ X3 @ Xs ) @ ( cons_int @ Y @ Ys2 ) )
= ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ X3 @ Y ) @ ( zip_int_int @ Xs @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_1784_zip__Cons__Cons,axiom,
! [X3: code_integer > option6357759511663192854e_term,Xs: list_C878401137130745250e_term,Y: produc8923325533196201883nteger,Ys2: list_P5578671422887162913nteger] :
( ( zip_Co8729459035503499408nteger @ ( cons_C6346100456420976220e_term @ X3 @ Xs ) @ ( cons_P9044669534377732177nteger @ Y @ Ys2 ) )
= ( cons_P7234078158855976520nteger @ ( produc6137756002093451184nteger @ X3 @ Y ) @ ( zip_Co8729459035503499408nteger @ Xs @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_1785_zip__Cons__Cons,axiom,
! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Xs: list_P1316552470764441098e_term,Y: produc8923325533196201883nteger,Ys2: list_P5578671422887162913nteger] :
( ( zip_Pr8292346330294042792nteger @ ( cons_P3591984656213271876e_term @ X3 @ Xs ) @ ( cons_P9044669534377732177nteger @ Y @ Ys2 ) )
= ( cons_P2100540245160168160nteger @ ( produc8603105652947943368nteger @ X3 @ Y ) @ ( zip_Pr8292346330294042792nteger @ Xs @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_1786_zip__Cons__Cons,axiom,
! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Xs: list_P1743416141875011707e_term,Y: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] :
( ( zip_Pr4168994715204986005nt_int @ ( cons_P2630085844062958645e_term @ X3 @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y @ Ys2 ) )
= ( cons_P6018425551955479501nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y ) @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_1787_zip__Cons__Cons,axiom,
! [X3: int > option6357759511663192854e_term,Xs: list_i8448526496819171953e_term,Y: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] :
( ( zip_in8766932505889695135nt_int @ ( cons_i7166360444231718571e_term @ X3 @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y @ Ys2 ) )
= ( cons_P2743708091642732631nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y ) @ ( zip_in8766932505889695135nt_int @ Xs @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_1788__092_060open_062low_Ax_An_A_060_A2_A_094_An_A_092_060and_062_Alow_Ay_An_A_060_A2_A_094_An_092_060close_062,axiom,
( ( ord_less_nat @ ( vEBT_VEBT_low @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) )
& ( ord_less_nat @ ( vEBT_VEBT_low @ ya @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ) ).
% \<open>low x n < 2 ^ n \<and> low y n < 2 ^ n\<close>
thf(fact_1789_Ball__set__replicate,axiom,
! [N: nat,A: int,P2: int > $o] :
( ( ! [X5: int] :
( ( member_int @ X5 @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
=> ( P2 @ X5 ) ) )
= ( ( P2 @ A )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_1790_Ball__set__replicate,axiom,
! [N: nat,A: nat,P2: nat > $o] :
( ( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
=> ( P2 @ X5 ) ) )
= ( ( P2 @ A )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_1791_Ball__set__replicate,axiom,
! [N: nat,A: vEBT_VEBT,P2: vEBT_VEBT > $o] :
( ( ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
=> ( P2 @ X5 ) ) )
= ( ( P2 @ A )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_1792_Bex__set__replicate,axiom,
! [N: nat,A: int,P2: int > $o] :
( ( ? [X5: int] :
( ( member_int @ X5 @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
& ( P2 @ X5 ) ) )
= ( ( P2 @ A )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_1793_Bex__set__replicate,axiom,
! [N: nat,A: nat,P2: nat > $o] :
( ( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
& ( P2 @ X5 ) ) )
= ( ( P2 @ A )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_1794_Bex__set__replicate,axiom,
! [N: nat,A: vEBT_VEBT,P2: vEBT_VEBT > $o] :
( ( ? [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
& ( P2 @ X5 ) ) )
= ( ( P2 @ A )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_1795_in__set__replicate,axiom,
! [X3: complex,N: nat,Y: complex] :
( ( member_complex @ X3 @ ( set_complex2 @ ( replicate_complex @ N @ Y ) ) )
= ( ( X3 = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_1796_in__set__replicate,axiom,
! [X3: real,N: nat,Y: real] :
( ( member_real @ X3 @ ( set_real2 @ ( replicate_real @ N @ Y ) ) )
= ( ( X3 = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_1797_in__set__replicate,axiom,
! [X3: set_nat,N: nat,Y: set_nat] :
( ( member_set_nat @ X3 @ ( set_set_nat2 @ ( replicate_set_nat @ N @ Y ) ) )
= ( ( X3 = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_1798_in__set__replicate,axiom,
! [X3: int,N: nat,Y: int] :
( ( member_int @ X3 @ ( set_int2 @ ( replicate_int @ N @ Y ) ) )
= ( ( X3 = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_1799_in__set__replicate,axiom,
! [X3: nat,N: nat,Y: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N @ Y ) ) )
= ( ( X3 = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_1800_in__set__replicate,axiom,
! [X3: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ Y ) ) )
= ( ( X3 = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_1801_nth__replicate,axiom,
! [I: nat,N: nat,X3: int] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_int @ ( replicate_int @ N @ X3 ) @ I )
= X3 ) ) ).
% nth_replicate
thf(fact_1802_nth__replicate,axiom,
! [I: nat,N: nat,X3: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_nat @ ( replicate_nat @ N @ X3 ) @ I )
= X3 ) ) ).
% nth_replicate
thf(fact_1803_nth__replicate,axiom,
! [I: nat,N: nat,X3: vEBT_VEBT] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X3 ) @ I )
= X3 ) ) ).
% nth_replicate
thf(fact_1804_xyprop,axiom,
( ( ord_less_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ ya @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ) ).
% xyprop
thf(fact_1805_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_1806_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_1807_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_1808_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_1809_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_1810_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_1811_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_1812_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_1813_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_1814_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ one_one_complex )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_1815_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
= ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_1816_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
= ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_1817_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_1818_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_1819_enumerate__simps_I2_J,axiom,
! [N: nat,X3: int,Xs: list_int] :
( ( enumerate_int @ N @ ( cons_int @ X3 @ Xs ) )
= ( cons_P2335045147070616083at_int @ ( product_Pair_nat_int @ N @ X3 ) @ ( enumerate_int @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_1820_enumerate__simps_I2_J,axiom,
! [N: nat,X3: nat,Xs: list_nat] :
( ( enumerate_nat @ N @ ( cons_nat @ X3 @ Xs ) )
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N @ X3 ) @ ( enumerate_nat @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_1821_one__add__one,axiom,
( ( plus_plus_complex @ one_one_complex @ one_one_complex )
= ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_1822_one__add__one,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_1823_one__add__one,axiom,
( ( plus_plus_rat @ one_one_rat @ one_one_rat )
= ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_1824_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_1825_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_1826_add__2__eq__Suc_H,axiom,
! [N: nat] :
( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc'
thf(fact_1827_add__2__eq__Suc,axiom,
! [N: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc
thf(fact_1828_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_1829_numeral__Bit0,axiom,
! [N: num] :
( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
= ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% numeral_Bit0
thf(fact_1830_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit0 @ N ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_Bit0
thf(fact_1831_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit0 @ N ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% numeral_Bit0
thf(fact_1832_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_Bit0
thf(fact_1833_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_Bit0
thf(fact_1834_replicate__Suc,axiom,
! [N: nat,X3: int] :
( ( replicate_int @ ( suc @ N ) @ X3 )
= ( cons_int @ X3 @ ( replicate_int @ N @ X3 ) ) ) ).
% replicate_Suc
thf(fact_1835_replicate__Suc,axiom,
! [N: nat,X3: nat] :
( ( replicate_nat @ ( suc @ N ) @ X3 )
= ( cons_nat @ X3 @ ( replicate_nat @ N @ X3 ) ) ) ).
% replicate_Suc
thf(fact_1836_replicate__Suc,axiom,
! [N: nat,X3: vEBT_VEBT] :
( ( replicate_VEBT_VEBT @ ( suc @ N ) @ X3 )
= ( cons_VEBT_VEBT @ X3 @ ( replicate_VEBT_VEBT @ N @ X3 ) ) ) ).
% replicate_Suc
thf(fact_1837_not__Cons__self2,axiom,
! [X3: int,Xs: list_int] :
( ( cons_int @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_1838_not__Cons__self2,axiom,
! [X3: nat,Xs: list_nat] :
( ( cons_nat @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_1839_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_1840_le__num__One__iff,axiom,
! [X3: num] :
( ( ord_less_eq_num @ X3 @ one )
= ( X3 = one ) ) ).
% le_num_One_iff
thf(fact_1841_cong__exp__iff__simps_I1_J,axiom,
! [N: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
= zero_zero_nat ) ).
% cong_exp_iff_simps(1)
thf(fact_1842_cong__exp__iff__simps_I1_J,axiom,
! [N: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
= zero_zero_int ) ).
% cong_exp_iff_simps(1)
thf(fact_1843_cong__exp__iff__simps_I1_J,axiom,
! [N: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
= zero_z3403309356797280102nteger ) ).
% cong_exp_iff_simps(1)
thf(fact_1844_cong__exp__iff__simps_I2_J,axiom,
! [N: num,Q3: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
= zero_zero_nat )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) )
= zero_zero_nat ) ) ).
% cong_exp_iff_simps(2)
thf(fact_1845_cong__exp__iff__simps_I2_J,axiom,
! [N: num,Q3: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
= zero_zero_int )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) )
= zero_zero_int ) ) ).
% cong_exp_iff_simps(2)
thf(fact_1846_cong__exp__iff__simps_I2_J,axiom,
! [N: num,Q3: num] :
( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
= zero_z3403309356797280102nteger )
= ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) )
= zero_z3403309356797280102nteger ) ) ).
% cong_exp_iff_simps(2)
thf(fact_1847_numeral__2__eq__2,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% numeral_2_eq_2
thf(fact_1848_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% nat_1_add_1
thf(fact_1849_gen__length__code_I2_J,axiom,
! [N: nat,X3: int,Xs: list_int] :
( ( gen_length_int @ N @ ( cons_int @ X3 @ Xs ) )
= ( gen_length_int @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_1850_gen__length__code_I2_J,axiom,
! [N: nat,X3: nat,Xs: list_nat] :
( ( gen_length_nat @ N @ ( cons_nat @ X3 @ Xs ) )
= ( gen_length_nat @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_1851_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
! [X3: nat,N: nat,M2: nat] :
( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M2 ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(1)
thf(fact_1852_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
! [X3: nat,N: nat,M2: nat] :
( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M2 ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( vEBT_VEBT_low @ X3 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(2)
thf(fact_1853_less__2__cases__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases_iff
thf(fact_1854_less__2__cases,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases
thf(fact_1855_zip__eq__ConsE,axiom,
! [Xs: list_int,Ys2: list_nat,Xy: product_prod_int_nat,Xys: list_P8198026277950538467nt_nat] :
( ( ( zip_int_nat @ Xs @ Ys2 )
= ( cons_P7512249878480867347nt_nat @ Xy @ Xys ) )
=> ~ ! [X4: int,Xs4: list_int] :
( ( Xs
= ( cons_int @ X4 @ Xs4 ) )
=> ! [Y3: nat,Ys4: list_nat] :
( ( Ys2
= ( cons_nat @ Y3 @ Ys4 ) )
=> ( ( Xy
= ( product_Pair_int_nat @ X4 @ Y3 ) )
=> ( Xys
!= ( zip_int_nat @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_1856_zip__eq__ConsE,axiom,
! [Xs: list_nat,Ys2: list_int,Xy: product_prod_nat_int,Xys: list_P3521021558325789923at_int] :
( ( ( zip_nat_int @ Xs @ Ys2 )
= ( cons_P2335045147070616083at_int @ Xy @ Xys ) )
=> ~ ! [X4: nat,Xs4: list_nat] :
( ( Xs
= ( cons_nat @ X4 @ Xs4 ) )
=> ! [Y3: int,Ys4: list_int] :
( ( Ys2
= ( cons_int @ Y3 @ Ys4 ) )
=> ( ( Xy
= ( product_Pair_nat_int @ X4 @ Y3 ) )
=> ( Xys
!= ( zip_nat_int @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_1857_zip__eq__ConsE,axiom,
! [Xs: list_nat,Ys2: list_nat,Xy: product_prod_nat_nat,Xys: list_P6011104703257516679at_nat] :
( ( ( zip_nat_nat @ Xs @ Ys2 )
= ( cons_P6512896166579812791at_nat @ Xy @ Xys ) )
=> ~ ! [X4: nat,Xs4: list_nat] :
( ( Xs
= ( cons_nat @ X4 @ Xs4 ) )
=> ! [Y3: nat,Ys4: list_nat] :
( ( Ys2
= ( cons_nat @ Y3 @ Ys4 ) )
=> ( ( Xy
= ( product_Pair_nat_nat @ X4 @ Y3 ) )
=> ( Xys
!= ( zip_nat_nat @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_1858_zip__eq__ConsE,axiom,
! [Xs: list_int,Ys2: list_int,Xy: product_prod_int_int,Xys: list_P5707943133018811711nt_int] :
( ( ( zip_int_int @ Xs @ Ys2 )
= ( cons_P3334398858971670639nt_int @ Xy @ Xys ) )
=> ~ ! [X4: int,Xs4: list_int] :
( ( Xs
= ( cons_int @ X4 @ Xs4 ) )
=> ! [Y3: int,Ys4: list_int] :
( ( Ys2
= ( cons_int @ Y3 @ Ys4 ) )
=> ( ( Xy
= ( product_Pair_int_int @ X4 @ Y3 ) )
=> ( Xys
!= ( zip_int_int @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_1859_zip__eq__ConsE,axiom,
! [Xs: list_C878401137130745250e_term,Ys2: list_P5578671422887162913nteger,Xy: produc8763457246119570046nteger,Xys: list_P5311841565141990158nteger] :
( ( ( zip_Co8729459035503499408nteger @ Xs @ Ys2 )
= ( cons_P7234078158855976520nteger @ Xy @ Xys ) )
=> ~ ! [X4: code_integer > option6357759511663192854e_term,Xs4: list_C878401137130745250e_term] :
( ( Xs
= ( cons_C6346100456420976220e_term @ X4 @ Xs4 ) )
=> ! [Y3: produc8923325533196201883nteger,Ys4: list_P5578671422887162913nteger] :
( ( Ys2
= ( cons_P9044669534377732177nteger @ Y3 @ Ys4 ) )
=> ( ( Xy
= ( produc6137756002093451184nteger @ X4 @ Y3 ) )
=> ( Xys
!= ( zip_Co8729459035503499408nteger @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_1860_zip__eq__ConsE,axiom,
! [Xs: list_P1316552470764441098e_term,Ys2: list_P5578671422887162913nteger,Xy: produc1908205239877642774nteger,Xys: list_P7828571989066258726nteger] :
( ( ( zip_Pr8292346330294042792nteger @ Xs @ Ys2 )
= ( cons_P2100540245160168160nteger @ Xy @ Xys ) )
=> ~ ! [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Xs4: list_P1316552470764441098e_term] :
( ( Xs
= ( cons_P3591984656213271876e_term @ X4 @ Xs4 ) )
=> ! [Y3: produc8923325533196201883nteger,Ys4: list_P5578671422887162913nteger] :
( ( Ys2
= ( cons_P9044669534377732177nteger @ Y3 @ Ys4 ) )
=> ( ( Xy
= ( produc8603105652947943368nteger @ X4 @ Y3 ) )
=> ( Xys
!= ( zip_Pr8292346330294042792nteger @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_1861_zip__eq__ConsE,axiom,
! [Xs: list_P1743416141875011707e_term,Ys2: list_P5707943133018811711nt_int,Xy: produc2285326912895808259nt_int,Xys: list_P651320350408439699nt_int] :
( ( ( zip_Pr4168994715204986005nt_int @ Xs @ Ys2 )
= ( cons_P6018425551955479501nt_int @ Xy @ Xys ) )
=> ~ ! [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Xs4: list_P1743416141875011707e_term] :
( ( Xs
= ( cons_P2630085844062958645e_term @ X4 @ Xs4 ) )
=> ! [Y3: product_prod_int_int,Ys4: list_P5707943133018811711nt_int] :
( ( Ys2
= ( cons_P3334398858971670639nt_int @ Y3 @ Ys4 ) )
=> ( ( Xy
= ( produc5700946648718959541nt_int @ X4 @ Y3 ) )
=> ( Xys
!= ( zip_Pr4168994715204986005nt_int @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_1862_zip__eq__ConsE,axiom,
! [Xs: list_i8448526496819171953e_term,Ys2: list_P5707943133018811711nt_int,Xy: produc7773217078559923341nt_int,Xys: list_P8915022641806594461nt_int] :
( ( ( zip_in8766932505889695135nt_int @ Xs @ Ys2 )
= ( cons_P2743708091642732631nt_int @ Xy @ Xys ) )
=> ~ ! [X4: int > option6357759511663192854e_term,Xs4: list_i8448526496819171953e_term] :
( ( Xs
= ( cons_i7166360444231718571e_term @ X4 @ Xs4 ) )
=> ! [Y3: product_prod_int_int,Ys4: list_P5707943133018811711nt_int] :
( ( Ys2
= ( cons_P3334398858971670639nt_int @ Y3 @ Ys4 ) )
=> ( ( Xy
= ( produc4305682042979456191nt_int @ X4 @ Y3 ) )
=> ( Xys
!= ( zip_in8766932505889695135nt_int @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_1863_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_1864_Suc__nat__number__of__add,axiom,
! [V: num,N: nat] :
( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_1865_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_complex
!= ( numera6690914467698888265omplex @ N ) ) ).
% zero_neq_numeral
thf(fact_1866_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( numeral_numeral_real @ N ) ) ).
% zero_neq_numeral
thf(fact_1867_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_rat
!= ( numeral_numeral_rat @ N ) ) ).
% zero_neq_numeral
thf(fact_1868_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_1869_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_1870_list_Oset__intros_I2_J,axiom,
! [Y: complex,X222: list_complex,X21: complex] :
( ( member_complex @ Y @ ( set_complex2 @ X222 ) )
=> ( member_complex @ Y @ ( set_complex2 @ ( cons_complex @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1871_list_Oset__intros_I2_J,axiom,
! [Y: real,X222: list_real,X21: real] :
( ( member_real @ Y @ ( set_real2 @ X222 ) )
=> ( member_real @ Y @ ( set_real2 @ ( cons_real @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1872_list_Oset__intros_I2_J,axiom,
! [Y: set_nat,X222: list_set_nat,X21: set_nat] :
( ( member_set_nat @ Y @ ( set_set_nat2 @ X222 ) )
=> ( member_set_nat @ Y @ ( set_set_nat2 @ ( cons_set_nat @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1873_list_Oset__intros_I2_J,axiom,
! [Y: vEBT_VEBT,X222: list_VEBT_VEBT,X21: vEBT_VEBT] :
( ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ X222 ) )
=> ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ ( cons_VEBT_VEBT @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1874_list_Oset__intros_I2_J,axiom,
! [Y: int,X222: list_int,X21: int] :
( ( member_int @ Y @ ( set_int2 @ X222 ) )
=> ( member_int @ Y @ ( set_int2 @ ( cons_int @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1875_list_Oset__intros_I2_J,axiom,
! [Y: nat,X222: list_nat,X21: nat] :
( ( member_nat @ Y @ ( set_nat2 @ X222 ) )
=> ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1876_list_Oset__intros_I1_J,axiom,
! [X21: complex,X222: list_complex] : ( member_complex @ X21 @ ( set_complex2 @ ( cons_complex @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_1877_list_Oset__intros_I1_J,axiom,
! [X21: real,X222: list_real] : ( member_real @ X21 @ ( set_real2 @ ( cons_real @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_1878_list_Oset__intros_I1_J,axiom,
! [X21: set_nat,X222: list_set_nat] : ( member_set_nat @ X21 @ ( set_set_nat2 @ ( cons_set_nat @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_1879_list_Oset__intros_I1_J,axiom,
! [X21: vEBT_VEBT,X222: list_VEBT_VEBT] : ( member_VEBT_VEBT @ X21 @ ( set_VEBT_VEBT2 @ ( cons_VEBT_VEBT @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_1880_list_Oset__intros_I1_J,axiom,
! [X21: int,X222: list_int] : ( member_int @ X21 @ ( set_int2 @ ( cons_int @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_1881_list_Oset__intros_I1_J,axiom,
! [X21: nat,X222: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_1882_list_Oset__cases,axiom,
! [E2: complex,A: list_complex] :
( ( member_complex @ E2 @ ( set_complex2 @ A ) )
=> ( ! [Z22: list_complex] :
( A
!= ( cons_complex @ E2 @ Z22 ) )
=> ~ ! [Z1: complex,Z22: list_complex] :
( ( A
= ( cons_complex @ Z1 @ Z22 ) )
=> ~ ( member_complex @ E2 @ ( set_complex2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1883_list_Oset__cases,axiom,
! [E2: real,A: list_real] :
( ( member_real @ E2 @ ( set_real2 @ A ) )
=> ( ! [Z22: list_real] :
( A
!= ( cons_real @ E2 @ Z22 ) )
=> ~ ! [Z1: real,Z22: list_real] :
( ( A
= ( cons_real @ Z1 @ Z22 ) )
=> ~ ( member_real @ E2 @ ( set_real2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1884_list_Oset__cases,axiom,
! [E2: set_nat,A: list_set_nat] :
( ( member_set_nat @ E2 @ ( set_set_nat2 @ A ) )
=> ( ! [Z22: list_set_nat] :
( A
!= ( cons_set_nat @ E2 @ Z22 ) )
=> ~ ! [Z1: set_nat,Z22: list_set_nat] :
( ( A
= ( cons_set_nat @ Z1 @ Z22 ) )
=> ~ ( member_set_nat @ E2 @ ( set_set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1885_list_Oset__cases,axiom,
! [E2: vEBT_VEBT,A: list_VEBT_VEBT] :
( ( member_VEBT_VEBT @ E2 @ ( set_VEBT_VEBT2 @ A ) )
=> ( ! [Z22: list_VEBT_VEBT] :
( A
!= ( cons_VEBT_VEBT @ E2 @ Z22 ) )
=> ~ ! [Z1: vEBT_VEBT,Z22: list_VEBT_VEBT] :
( ( A
= ( cons_VEBT_VEBT @ Z1 @ Z22 ) )
=> ~ ( member_VEBT_VEBT @ E2 @ ( set_VEBT_VEBT2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1886_list_Oset__cases,axiom,
! [E2: int,A: list_int] :
( ( member_int @ E2 @ ( set_int2 @ A ) )
=> ( ! [Z22: list_int] :
( A
!= ( cons_int @ E2 @ Z22 ) )
=> ~ ! [Z1: int,Z22: list_int] :
( ( A
= ( cons_int @ Z1 @ Z22 ) )
=> ~ ( member_int @ E2 @ ( set_int2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1887_list_Oset__cases,axiom,
! [E2: nat,A: list_nat] :
( ( member_nat @ E2 @ ( set_nat2 @ A ) )
=> ( ! [Z22: list_nat] :
( A
!= ( cons_nat @ E2 @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E2 @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1888_set__ConsD,axiom,
! [Y: complex,X3: complex,Xs: list_complex] :
( ( member_complex @ Y @ ( set_complex2 @ ( cons_complex @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_complex @ Y @ ( set_complex2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1889_set__ConsD,axiom,
! [Y: real,X3: real,Xs: list_real] :
( ( member_real @ Y @ ( set_real2 @ ( cons_real @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_real @ Y @ ( set_real2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1890_set__ConsD,axiom,
! [Y: set_nat,X3: set_nat,Xs: list_set_nat] :
( ( member_set_nat @ Y @ ( set_set_nat2 @ ( cons_set_nat @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_set_nat @ Y @ ( set_set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1891_set__ConsD,axiom,
! [Y: vEBT_VEBT,X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
( ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ ( cons_VEBT_VEBT @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1892_set__ConsD,axiom,
! [Y: int,X3: int,Xs: list_int] :
( ( member_int @ Y @ ( set_int2 @ ( cons_int @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_int @ Y @ ( set_int2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1893_set__ConsD,axiom,
! [Y: nat,X3: nat,Xs: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_nat @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1894_num_Osize_I4_J,axiom,
( ( size_size_num @ one )
= zero_zero_nat ) ).
% num.size(4)
thf(fact_1895_invar__vebt_Ointros_I2_J,axiom,
! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M2 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
=> ( ( M2 = N )
=> ( ( Deg
= ( plus_plus_nat @ N @ M2 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
=> ( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(2)
thf(fact_1896_invar__vebt_Ointros_I3_J,axiom,
! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M2 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
=> ( ( M2
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus_nat @ N @ M2 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
=> ( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(3)
thf(fact_1897_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_1898_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_1899_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_1900_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_1901_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_1902_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_1903_gcd__nat__induct,axiom,
! [P2: nat > nat > $o,M2: nat,N: nat] :
( ! [M: nat] : ( P2 @ M @ zero_zero_nat )
=> ( ! [M: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P2 @ N3 @ ( modulo_modulo_nat @ M @ N3 ) )
=> ( P2 @ M @ N3 ) ) )
=> ( P2 @ M2 @ N ) ) ) ).
% gcd_nat_induct
thf(fact_1904_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_le_zero
thf(fact_1905_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% not_numeral_le_zero
thf(fact_1906_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_1907_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_1908_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_le_numeral
thf(fact_1909_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% zero_le_numeral
thf(fact_1910_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_1911_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_le_numeral
thf(fact_1912_zero__less__numeral,axiom,
! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_less_numeral
thf(fact_1913_zero__less__numeral,axiom,
! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% zero_less_numeral
thf(fact_1914_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_1915_zero__less__numeral,axiom,
! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_less_numeral
thf(fact_1916_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_less_zero
thf(fact_1917_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% not_numeral_less_zero
thf(fact_1918_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_1919_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_1920_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% one_le_numeral
thf(fact_1921_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).
% one_le_numeral
thf(fact_1922_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_1923_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% one_le_numeral
thf(fact_1924_one__plus__numeral__commute,axiom,
! [X3: num] :
( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X3 ) )
= ( plus_plus_complex @ ( numera6690914467698888265omplex @ X3 ) @ one_one_complex ) ) ).
% one_plus_numeral_commute
thf(fact_1925_one__plus__numeral__commute,axiom,
! [X3: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X3 ) )
= ( plus_plus_real @ ( numeral_numeral_real @ X3 ) @ one_one_real ) ) ).
% one_plus_numeral_commute
thf(fact_1926_one__plus__numeral__commute,axiom,
! [X3: num] :
( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X3 ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ X3 ) @ one_one_rat ) ) ).
% one_plus_numeral_commute
thf(fact_1927_one__plus__numeral__commute,axiom,
! [X3: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X3 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X3 ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_1928_one__plus__numeral__commute,axiom,
! [X3: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X3 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X3 ) @ one_one_int ) ) ).
% one_plus_numeral_commute
thf(fact_1929_replicate__length__same,axiom,
! [Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( X4 = X3 ) )
=> ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs ) @ X3 )
= Xs ) ) ).
% replicate_length_same
thf(fact_1930_replicate__length__same,axiom,
! [Xs: list_int,X3: int] :
( ! [X4: int] :
( ( member_int @ X4 @ ( set_int2 @ Xs ) )
=> ( X4 = X3 ) )
=> ( ( replicate_int @ ( size_size_list_int @ Xs ) @ X3 )
= Xs ) ) ).
% replicate_length_same
thf(fact_1931_replicate__length__same,axiom,
! [Xs: list_nat,X3: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( X4 = X3 ) )
=> ( ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X3 )
= Xs ) ) ).
% replicate_length_same
thf(fact_1932_replicate__eqI,axiom,
! [Xs: list_complex,N: nat,X3: complex] :
( ( ( size_s3451745648224563538omplex @ Xs )
= N )
=> ( ! [Y3: complex] :
( ( member_complex @ Y3 @ ( set_complex2 @ Xs ) )
=> ( Y3 = X3 ) )
=> ( Xs
= ( replicate_complex @ N @ X3 ) ) ) ) ).
% replicate_eqI
thf(fact_1933_replicate__eqI,axiom,
! [Xs: list_real,N: nat,X3: real] :
( ( ( size_size_list_real @ Xs )
= N )
=> ( ! [Y3: real] :
( ( member_real @ Y3 @ ( set_real2 @ Xs ) )
=> ( Y3 = X3 ) )
=> ( Xs
= ( replicate_real @ N @ X3 ) ) ) ) ).
% replicate_eqI
thf(fact_1934_replicate__eqI,axiom,
! [Xs: list_set_nat,N: nat,X3: set_nat] :
( ( ( size_s3254054031482475050et_nat @ Xs )
= N )
=> ( ! [Y3: set_nat] :
( ( member_set_nat @ Y3 @ ( set_set_nat2 @ Xs ) )
=> ( Y3 = X3 ) )
=> ( Xs
= ( replicate_set_nat @ N @ X3 ) ) ) ) ).
% replicate_eqI
thf(fact_1935_replicate__eqI,axiom,
! [Xs: list_VEBT_VEBT,N: nat,X3: vEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs )
= N )
=> ( ! [Y3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Xs ) )
=> ( Y3 = X3 ) )
=> ( Xs
= ( replicate_VEBT_VEBT @ N @ X3 ) ) ) ) ).
% replicate_eqI
thf(fact_1936_replicate__eqI,axiom,
! [Xs: list_int,N: nat,X3: int] :
( ( ( size_size_list_int @ Xs )
= N )
=> ( ! [Y3: int] :
( ( member_int @ Y3 @ ( set_int2 @ Xs ) )
=> ( Y3 = X3 ) )
=> ( Xs
= ( replicate_int @ N @ X3 ) ) ) ) ).
% replicate_eqI
thf(fact_1937_replicate__eqI,axiom,
! [Xs: list_nat,N: nat,X3: nat] :
( ( ( size_size_list_nat @ Xs )
= N )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ Xs ) )
=> ( Y3 = X3 ) )
=> ( Xs
= ( replicate_nat @ N @ X3 ) ) ) ) ).
% replicate_eqI
thf(fact_1938_length__Suc__conv,axiom,
! [Xs: list_VEBT_VEBT,N: nat] :
( ( ( size_s6755466524823107622T_VEBT @ Xs )
= ( suc @ N ) )
= ( ? [Y5: vEBT_VEBT,Ys3: list_VEBT_VEBT] :
( ( Xs
= ( cons_VEBT_VEBT @ Y5 @ Ys3 ) )
& ( ( size_s6755466524823107622T_VEBT @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1939_length__Suc__conv,axiom,
! [Xs: list_int,N: nat] :
( ( ( size_size_list_int @ Xs )
= ( suc @ N ) )
= ( ? [Y5: int,Ys3: list_int] :
( ( Xs
= ( cons_int @ Y5 @ Ys3 ) )
& ( ( size_size_list_int @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1940_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y5: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y5 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1941_Suc__length__conv,axiom,
! [N: nat,Xs: list_VEBT_VEBT] :
( ( ( suc @ N )
= ( size_s6755466524823107622T_VEBT @ Xs ) )
= ( ? [Y5: vEBT_VEBT,Ys3: list_VEBT_VEBT] :
( ( Xs
= ( cons_VEBT_VEBT @ Y5 @ Ys3 ) )
& ( ( size_s6755466524823107622T_VEBT @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1942_Suc__length__conv,axiom,
! [N: nat,Xs: list_int] :
( ( ( suc @ N )
= ( size_size_list_int @ Xs ) )
= ( ? [Y5: int,Ys3: list_int] :
( ( Xs
= ( cons_int @ Y5 @ Ys3 ) )
& ( ( size_size_list_int @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1943_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y5: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y5 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1944_set__subset__Cons,axiom,
! [Xs: list_VEBT_VEBT,X3: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ ( set_VEBT_VEBT2 @ ( cons_VEBT_VEBT @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1945_set__subset__Cons,axiom,
! [Xs: list_nat,X3: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1946_set__subset__Cons,axiom,
! [Xs: list_int,X3: int] : ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ ( set_int2 @ ( cons_int @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1947_impossible__Cons,axiom,
! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT,X3: vEBT_VEBT] :
( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) )
=> ( Xs
!= ( cons_VEBT_VEBT @ X3 @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_1948_impossible__Cons,axiom,
! [Xs: list_int,Ys2: list_int,X3: int] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys2 ) )
=> ( Xs
!= ( cons_int @ X3 @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_1949_impossible__Cons,axiom,
! [Xs: list_nat,Ys2: list_nat,X3: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys2 ) )
=> ( Xs
!= ( cons_nat @ X3 @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_1950_set__encode__eq,axiom,
! [A4: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( ( nat_set_encode @ A4 )
= ( nat_set_encode @ B5 ) )
= ( A4 = B5 ) ) ) ) ).
% set_encode_eq
thf(fact_1951_find_Osimps_I2_J,axiom,
! [P2: int > $o,X3: int,Xs: list_int] :
( ( ( P2 @ X3 )
=> ( ( find_int @ P2 @ ( cons_int @ X3 @ Xs ) )
= ( some_int @ X3 ) ) )
& ( ~ ( P2 @ X3 )
=> ( ( find_int @ P2 @ ( cons_int @ X3 @ Xs ) )
= ( find_int @ P2 @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_1952_find_Osimps_I2_J,axiom,
! [P2: nat > $o,X3: nat,Xs: list_nat] :
( ( ( P2 @ X3 )
=> ( ( find_nat @ P2 @ ( cons_nat @ X3 @ Xs ) )
= ( some_nat @ X3 ) ) )
& ( ~ ( P2 @ X3 )
=> ( ( find_nat @ P2 @ ( cons_nat @ X3 @ Xs ) )
= ( find_nat @ P2 @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_1953_find_Osimps_I2_J,axiom,
! [P2: product_prod_nat_nat > $o,X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( ( P2 @ X3 )
=> ( ( find_P8199882355184865565at_nat @ P2 @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) )
= ( some_P7363390416028606310at_nat @ X3 ) ) )
& ( ~ ( P2 @ X3 )
=> ( ( find_P8199882355184865565at_nat @ P2 @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) )
= ( find_P8199882355184865565at_nat @ P2 @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_1954_find_Osimps_I2_J,axiom,
! [P2: num > $o,X3: num,Xs: list_num] :
( ( ( P2 @ X3 )
=> ( ( find_num @ P2 @ ( cons_num @ X3 @ Xs ) )
= ( some_num @ X3 ) ) )
& ( ~ ( P2 @ X3 )
=> ( ( find_num @ P2 @ ( cons_num @ X3 @ Xs ) )
= ( find_num @ P2 @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_1955_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_1956_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_1957_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_1958_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le6747313008572928689nteger @ A @ B )
=> ( ( modulo364778990260209775nteger @ A @ B )
= A ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_1959_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ B )
=> ( ( modulo_modulo_nat @ A @ B )
= A ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_1960_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ B )
=> ( ( modulo_modulo_int @ A @ B )
= A ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_1961_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_VEBT_VEBT] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s6755466524823107622T_VEBT @ Xs ) )
= ( ? [X5: vEBT_VEBT,Ys3: list_VEBT_VEBT] :
( ( Xs
= ( cons_VEBT_VEBT @ X5 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1962_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_int] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) )
= ( ? [X5: int,Ys3: list_int] :
( ( Xs
= ( cons_int @ X5 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1963_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X5: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ X5 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1964_invar__vebt_Ointros_I4_J,axiom,
! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M2 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
=> ( ( M2 = N )
=> ( ( Deg
= ( plus_plus_nat @ N @ M2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) )
=> ( ( ord_less_eq_nat @ Mi @ Ma )
=> ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
=> ( ( ord_less_nat @ Mi @ X4 )
& ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(4)
thf(fact_1965_num_Osize_I5_J,axiom,
! [X2: num] :
( ( size_size_num @ ( bit0 @ X2 ) )
= ( plus_plus_nat @ ( size_size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size(5)
thf(fact_1966_invar__vebt_Ointros_I5_J,axiom,
! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M2 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
=> ( ( M2
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus_nat @ N @ M2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) )
=> ( ( ord_less_eq_nat @ Mi @ Ma )
=> ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
=> ( ( ord_less_nat @ Mi @ X4 )
& ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(5)
thf(fact_1967_count__list_Osimps_I2_J,axiom,
! [X3: int,Y: int,Xs: list_int] :
( ( ( X3 = Y )
=> ( ( count_list_int @ ( cons_int @ X3 @ Xs ) @ Y )
= ( plus_plus_nat @ ( count_list_int @ Xs @ Y ) @ one_one_nat ) ) )
& ( ( X3 != Y )
=> ( ( count_list_int @ ( cons_int @ X3 @ Xs ) @ Y )
= ( count_list_int @ Xs @ Y ) ) ) ) ).
% count_list.simps(2)
thf(fact_1968_count__list_Osimps_I2_J,axiom,
! [X3: nat,Y: nat,Xs: list_nat] :
( ( ( X3 = Y )
=> ( ( count_list_nat @ ( cons_nat @ X3 @ Xs ) @ Y )
= ( plus_plus_nat @ ( count_list_nat @ Xs @ Y ) @ one_one_nat ) ) )
& ( ( X3 != Y )
=> ( ( count_list_nat @ ( cons_nat @ X3 @ Xs ) @ Y )
= ( count_list_nat @ Xs @ Y ) ) ) ) ).
% count_list.simps(2)
thf(fact_1969_set__encode__inf,axiom,
! [A4: set_nat] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( nat_set_encode @ A4 )
= zero_zero_nat ) ) ).
% set_encode_inf
thf(fact_1970_pair__lessI1,axiom,
! [A: nat,B: nat,S3: nat,T: nat] :
( ( ord_less_nat @ A @ B )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ).
% pair_lessI1
thf(fact_1971_list_Osize_I4_J,axiom,
! [X21: vEBT_VEBT,X222: list_VEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT @ ( cons_VEBT_VEBT @ X21 @ X222 ) )
= ( plus_plus_nat @ ( size_s6755466524823107622T_VEBT @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1972_list_Osize_I4_J,axiom,
! [X21: int,X222: list_int] :
( ( size_size_list_int @ ( cons_int @ X21 @ X222 ) )
= ( plus_plus_nat @ ( size_size_list_int @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1973_list_Osize_I4_J,axiom,
! [X21: nat,X222: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X21 @ X222 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1974_invar__vebt_Osimps,axiom,
( vEBT_invar_vebt
= ( ^ [A1: vEBT_VEBT,A22: nat] :
( ( ? [A5: $o,B4: $o] :
( A1
= ( vEBT_Leaf @ A5 @ B4 ) )
& ( A22
= ( suc @ zero_zero_nat ) ) )
| ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
( ( A1
= ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList3 @ Summary3 ) )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X5 @ N4 ) )
& ( vEBT_invar_vebt @ Summary3 @ N4 )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
& ( A22
= ( plus_plus_nat @ N4 @ N4 ) )
& ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X7 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X7 ) ) )
| ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
( ( A1
= ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList3 @ Summary3 ) )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X5 @ N4 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N4 ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
& ( A22
= ( plus_plus_nat @ N4 @ ( suc @ N4 ) ) )
& ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X7 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X7 ) ) )
| ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A1
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList3 @ Summary3 ) )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X5 @ N4 ) )
& ( vEBT_invar_vebt @ Summary3 @ N4 )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
& ( A22
= ( plus_plus_nat @ N4 @ N4 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X7 ) ) )
& ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
& ( ( Mi3 != Ma3 )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N4 )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N4 ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N4 )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N4 ) ) )
=> ( ( ord_less_nat @ Mi3 @ X5 )
& ( ord_less_eq_nat @ X5 @ Ma3 ) ) ) ) ) ) )
| ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A1
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList3 @ Summary3 ) )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X5 @ N4 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N4 ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
& ( A22
= ( plus_plus_nat @ N4 @ ( suc @ N4 ) ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X7 ) ) )
& ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
& ( ( Mi3 != Ma3 )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N4 )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N4 ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N4 )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N4 ) ) )
=> ( ( ord_less_nat @ Mi3 @ X5 )
& ( ord_less_eq_nat @ X5 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.simps
thf(fact_1975_invar__vebt_Ocases,axiom,
! [A12: vEBT_VEBT,A23: nat] :
( ( vEBT_invar_vebt @ A12 @ A23 )
=> ( ( ? [A3: $o,B3: $o] :
( A12
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ( A23
!= ( suc @ zero_zero_nat ) ) )
=> ( ! [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat] :
( ( A12
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( M = N3 )
=> ( ( Deg2
= ( plus_plus_nat @ N3 @ M ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
=> ~ ! [X: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X @ X_1 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat] :
( ( A12
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( M
= ( suc @ N3 ) )
=> ( ( Deg2
= ( plus_plus_nat @ N3 @ M ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
=> ~ ! [X: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X @ X_1 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A12
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( M = N3 )
=> ( ( Deg2
= ( plus_plus_nat @ N3 @ M ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X @ X_1 ) ) )
=> ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
=> ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
& ! [X: nat] :
( ( ( ( vEBT_VEBT_high @ X @ N3 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X @ N3 ) ) )
=> ( ( ord_less_nat @ Mi2 @ X )
& ( ord_less_eq_nat @ X @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ! [TreeList2: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A12
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( M
= ( suc @ N3 ) )
=> ( ( Deg2
= ( plus_plus_nat @ N3 @ M ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X @ X_1 ) ) )
=> ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
=> ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
& ! [X: nat] :
( ( ( ( vEBT_VEBT_high @ X @ N3 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X @ N3 ) ) )
=> ( ( ord_less_nat @ Mi2 @ X )
& ( ord_less_eq_nat @ X @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.cases
thf(fact_1976_mod2__gr__0,axiom,
! [M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ).
% mod2_gr_0
thf(fact_1977_add__self__mod__2,axiom,
! [M2: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ M2 @ M2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% add_self_mod_2
thf(fact_1978_not__mod2__eq__Suc__0__eq__0,axiom,
! [N: nat] :
( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= ( suc @ zero_zero_nat ) )
= ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ) ).
% not_mod2_eq_Suc_0_eq_0
thf(fact_1979_not__mod__2__eq__0__eq__1,axiom,
! [A: nat] :
( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= zero_zero_nat )
= ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_1980_not__mod__2__eq__0__eq__1,axiom,
! [A: int] :
( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= zero_zero_int )
= ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_1981_not__mod__2__eq__0__eq__1,axiom,
! [A: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
!= zero_z3403309356797280102nteger )
= ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_1982_not__mod__2__eq__1__eq__0,axiom,
! [A: nat] :
( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= one_one_nat )
= ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_1983_not__mod__2__eq__1__eq__0,axiom,
! [A: int] :
( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= one_one_int )
= ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_1984_not__mod__2__eq__1__eq__0,axiom,
! [A: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
!= one_one_Code_integer )
= ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= zero_z3403309356797280102nteger ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_1985_sum__power2__eq__zero__iff,axiom,
! [X3: rat,Y: rat] :
( ( ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_rat )
= ( ( X3 = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_1986_sum__power2__eq__zero__iff,axiom,
! [X3: real,Y: real] :
( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_real )
= ( ( X3 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_1987_sum__power2__eq__zero__iff,axiom,
! [X3: int,Y: int] :
( ( ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_int )
= ( ( X3 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_1988_zero__less__power2,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A != zero_zero_real ) ) ).
% zero_less_power2
thf(fact_1989_zero__less__power2,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A != zero_zero_rat ) ) ).
% zero_less_power2
thf(fact_1990_zero__less__power2,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A != zero_zero_int ) ) ).
% zero_less_power2
thf(fact_1991_power2__eq__iff__nonneg,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X3 = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_1992_power2__eq__iff__nonneg,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X3 = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_1993_power2__eq__iff__nonneg,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X3 = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_1994_power2__eq__iff__nonneg,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X3 = Y ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_1995_power2__less__eq__zero__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
= ( A = zero_zero_real ) ) ).
% power2_less_eq_zero_iff
thf(fact_1996_power2__less__eq__zero__iff,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% power2_less_eq_zero_iff
thf(fact_1997_power2__less__eq__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% power2_less_eq_zero_iff
thf(fact_1998_power__decreasing__iff,axiom,
! [B: real,M2: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ M2 ) @ ( power_power_real @ B @ N ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_1999_power__decreasing__iff,axiom,
! [B: rat,M2: nat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ( ord_less_rat @ B @ one_one_rat )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M2 ) @ ( power_power_rat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_2000_power__decreasing__iff,axiom,
! [B: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M2 ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_2001_power__decreasing__iff,axiom,
! [B: int,M2: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M2 ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_2002_mod__add__self2,axiom,
! [A: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_add_self2
thf(fact_2003_mod__add__self2,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_add_self2
thf(fact_2004_mod__add__self2,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_add_self2
thf(fact_2005_mod__add__self1,axiom,
! [B: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_add_self1
thf(fact_2006_mod__add__self1,axiom,
! [B: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_add_self1
thf(fact_2007_mod__add__self1,axiom,
! [B: code_integer,A: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_add_self1
thf(fact_2008_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
= zero_zero_rat ) ).
% power_0_Suc
thf(fact_2009_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_2010_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
= zero_zero_real ) ).
% power_0_Suc
thf(fact_2011_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_2012_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
= zero_zero_complex ) ).
% power_0_Suc
thf(fact_2013_power__zero__numeral,axiom,
! [K2: num] :
( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K2 ) )
= zero_zero_rat ) ).
% power_zero_numeral
thf(fact_2014_power__zero__numeral,axiom,
! [K2: num] :
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K2 ) )
= zero_zero_nat ) ).
% power_zero_numeral
thf(fact_2015_power__zero__numeral,axiom,
! [K2: num] :
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K2 ) )
= zero_zero_real ) ).
% power_zero_numeral
thf(fact_2016_power__zero__numeral,axiom,
! [K2: num] :
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K2 ) )
= zero_zero_int ) ).
% power_zero_numeral
thf(fact_2017_power__zero__numeral,axiom,
! [K2: num] :
( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K2 ) )
= zero_zero_complex ) ).
% power_zero_numeral
thf(fact_2018_power__Suc0__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_2019_power__Suc0__right,axiom,
! [A: real] :
( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_2020_power__Suc0__right,axiom,
! [A: int] :
( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_2021_power__Suc0__right,axiom,
! [A: complex] :
( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_2022_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_2023_nat__power__eq__Suc__0__iff,axiom,
! [X3: nat,M2: nat] :
( ( ( power_power_nat @ X3 @ M2 )
= ( suc @ zero_zero_nat ) )
= ( ( M2 = zero_zero_nat )
| ( X3
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_2024_nat__zero__less__power__iff,axiom,
! [X3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X3 )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_2025_mod__by__Suc__0,axiom,
! [M2: nat] :
( ( modulo_modulo_nat @ M2 @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% mod_by_Suc_0
thf(fact_2026_power__eq__0__iff,axiom,
! [A: rat,N: nat] :
( ( ( power_power_rat @ A @ N )
= zero_zero_rat )
= ( ( A = zero_zero_rat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_2027_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_2028_power__eq__0__iff,axiom,
! [A: real,N: nat] :
( ( ( power_power_real @ A @ N )
= zero_zero_real )
= ( ( A = zero_zero_real )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_2029_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( ( power_power_int @ A @ N )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_2030_power__eq__0__iff,axiom,
! [A: complex,N: nat] :
( ( ( power_power_complex @ A @ N )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_2031_power__strict__decreasing__iff,axiom,
! [B: real,M2: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_real @ ( power_power_real @ B @ M2 ) @ ( power_power_real @ B @ N ) )
= ( ord_less_nat @ N @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_2032_power__strict__decreasing__iff,axiom,
! [B: rat,M2: nat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ( ord_less_rat @ B @ one_one_rat )
=> ( ( ord_less_rat @ ( power_power_rat @ B @ M2 ) @ ( power_power_rat @ B @ N ) )
= ( ord_less_nat @ N @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_2033_power__strict__decreasing__iff,axiom,
! [B: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M2 ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_nat @ N @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_2034_power__strict__decreasing__iff,axiom,
! [B: int,M2: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M2 ) @ ( power_power_int @ B @ N ) )
= ( ord_less_nat @ N @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_2035_power__increasing__iff,axiom,
! [B: real,X3: nat,Y: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ X3 ) @ ( power_power_real @ B @ Y ) )
= ( ord_less_eq_nat @ X3 @ Y ) ) ) ).
% power_increasing_iff
thf(fact_2036_power__increasing__iff,axiom,
! [B: rat,X3: nat,Y: nat] :
( ( ord_less_rat @ one_one_rat @ B )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X3 ) @ ( power_power_rat @ B @ Y ) )
= ( ord_less_eq_nat @ X3 @ Y ) ) ) ).
% power_increasing_iff
thf(fact_2037_power__increasing__iff,axiom,
! [B: nat,X3: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X3 ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_eq_nat @ X3 @ Y ) ) ) ).
% power_increasing_iff
thf(fact_2038_power__increasing__iff,axiom,
! [B: int,X3: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X3 ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_eq_nat @ X3 @ Y ) ) ) ).
% power_increasing_iff
thf(fact_2039_power__mono__iff,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
= ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_2040_power__mono__iff,axiom,
! [A: rat,B: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
= ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_2041_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_2042_power__mono__iff,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_2043_zero__eq__power2,axiom,
! [A: rat] :
( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% zero_eq_power2
thf(fact_2044_zero__eq__power2,axiom,
! [A: nat] :
( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% zero_eq_power2
thf(fact_2045_zero__eq__power2,axiom,
! [A: real] :
( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% zero_eq_power2
thf(fact_2046_zero__eq__power2,axiom,
! [A: int] :
( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% zero_eq_power2
thf(fact_2047_zero__eq__power2,axiom,
! [A: complex] :
( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% zero_eq_power2
thf(fact_2048_mod2__Suc__Suc,axiom,
! [M2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% mod2_Suc_Suc
thf(fact_2049_two__realpow__ge__one,axiom,
! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% two_realpow_ge_one
thf(fact_2050_power__not__zero,axiom,
! [A: rat,N: nat] :
( ( A != zero_zero_rat )
=> ( ( power_power_rat @ A @ N )
!= zero_zero_rat ) ) ).
% power_not_zero
thf(fact_2051_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_2052_power__not__zero,axiom,
! [A: real,N: nat] :
( ( A != zero_zero_real )
=> ( ( power_power_real @ A @ N )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_2053_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_2054_power__not__zero,axiom,
! [A: complex,N: nat] :
( ( A != zero_zero_complex )
=> ( ( power_power_complex @ A @ N )
!= zero_zero_complex ) ) ).
% power_not_zero
thf(fact_2055_mod__add__right__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C2 ) ) @ C2 )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C2 ) ) ).
% mod_add_right_eq
thf(fact_2056_mod__add__right__eq,axiom,
! [A: int,B: int,C2: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C2 ) ) @ C2 )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% mod_add_right_eq
thf(fact_2057_mod__add__right__eq,axiom,
! [A: code_integer,B: code_integer,C2: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C2 ) ) @ C2 )
= ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C2 ) ) ).
% mod_add_right_eq
thf(fact_2058_mod__add__left__eq,axiom,
! [A: nat,C2: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C2 ) @ B ) @ C2 )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C2 ) ) ).
% mod_add_left_eq
thf(fact_2059_mod__add__left__eq,axiom,
! [A: int,C2: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C2 ) @ B ) @ C2 )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% mod_add_left_eq
thf(fact_2060_mod__add__left__eq,axiom,
! [A: code_integer,C2: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C2 ) @ B ) @ C2 )
= ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C2 ) ) ).
% mod_add_left_eq
thf(fact_2061_mod__add__cong,axiom,
! [A: nat,C2: nat,A2: nat,B: nat,B2: nat] :
( ( ( modulo_modulo_nat @ A @ C2 )
= ( modulo_modulo_nat @ A2 @ C2 ) )
=> ( ( ( modulo_modulo_nat @ B @ C2 )
= ( modulo_modulo_nat @ B2 @ C2 ) )
=> ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C2 ) ) ) ) ).
% mod_add_cong
thf(fact_2062_mod__add__cong,axiom,
! [A: int,C2: int,A2: int,B: int,B2: int] :
( ( ( modulo_modulo_int @ A @ C2 )
= ( modulo_modulo_int @ A2 @ C2 ) )
=> ( ( ( modulo_modulo_int @ B @ C2 )
= ( modulo_modulo_int @ B2 @ C2 ) )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( modulo_modulo_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 ) ) ) ) ).
% mod_add_cong
thf(fact_2063_mod__add__cong,axiom,
! [A: code_integer,C2: code_integer,A2: code_integer,B: code_integer,B2: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ C2 )
= ( modulo364778990260209775nteger @ A2 @ C2 ) )
=> ( ( ( modulo364778990260209775nteger @ B @ C2 )
= ( modulo364778990260209775nteger @ B2 @ C2 ) )
=> ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C2 )
= ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ C2 ) ) ) ) ).
% mod_add_cong
thf(fact_2064_mod__add__eq,axiom,
! [A: nat,C2: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C2 ) @ ( modulo_modulo_nat @ B @ C2 ) ) @ C2 )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C2 ) ) ).
% mod_add_eq
thf(fact_2065_mod__add__eq,axiom,
! [A: int,C2: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C2 ) @ ( modulo_modulo_int @ B @ C2 ) ) @ C2 )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% mod_add_eq
thf(fact_2066_mod__add__eq,axiom,
! [A: code_integer,C2: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C2 ) @ ( modulo364778990260209775nteger @ B @ C2 ) ) @ C2 )
= ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C2 ) ) ).
% mod_add_eq
thf(fact_2067_mod__Suc__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M2 @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ ( suc @ M2 ) ) @ N ) ) ).
% mod_Suc_Suc_eq
thf(fact_2068_mod__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M2 @ N ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M2 ) @ N ) ) ).
% mod_Suc_eq
thf(fact_2069_mod__less__eq__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ N ) @ M2 ) ).
% mod_less_eq_dividend
thf(fact_2070_power__mono,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% power_mono
thf(fact_2071_power__mono,axiom,
! [A: rat,B: rat,N: nat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_2072_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_2073_power__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono
thf(fact_2074_zero__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_le_power
thf(fact_2075_zero__le__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_2076_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_2077_zero__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_power
thf(fact_2078_zero__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_less_power
thf(fact_2079_zero__less__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_2080_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_2081_zero__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_less_power
thf(fact_2082_one__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% one_le_power
thf(fact_2083_one__le__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ one_one_rat @ A )
=> ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% one_le_power
thf(fact_2084_one__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% one_le_power
thf(fact_2085_one__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% one_le_power
thf(fact_2086_power__0,axiom,
! [A: rat] :
( ( power_power_rat @ A @ zero_zero_nat )
= one_one_rat ) ).
% power_0
thf(fact_2087_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_2088_power__0,axiom,
! [A: real] :
( ( power_power_real @ A @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_2089_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_2090_power__0,axiom,
! [A: complex] :
( ( power_power_complex @ A @ zero_zero_nat )
= one_one_complex ) ).
% power_0
thf(fact_2091_nat__power__less__imp__less,axiom,
! [I: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M2 ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_2092_mod__Suc,axiom,
! [M2: nat,N: nat] :
( ( ( ( suc @ ( modulo_modulo_nat @ M2 @ N ) )
= N )
=> ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N )
= zero_zero_nat ) )
& ( ( ( suc @ ( modulo_modulo_nat @ M2 @ N ) )
!= N )
=> ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( modulo_modulo_nat @ M2 @ N ) ) ) ) ) ).
% mod_Suc
thf(fact_2093_mod__induct,axiom,
! [P2: nat > $o,N: nat,P: nat,M2: nat] :
( ( P2 @ N )
=> ( ( ord_less_nat @ N @ P )
=> ( ( ord_less_nat @ M2 @ P )
=> ( ! [N3: nat] :
( ( ord_less_nat @ N3 @ P )
=> ( ( P2 @ N3 )
=> ( P2 @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P ) ) ) )
=> ( P2 @ M2 ) ) ) ) ) ).
% mod_induct
thf(fact_2094_mod__less__divisor,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( modulo_modulo_nat @ M2 @ N ) @ N ) ) ).
% mod_less_divisor
thf(fact_2095_mod__Suc__le__divisor,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ ( suc @ N ) ) @ N ) ).
% mod_Suc_le_divisor
thf(fact_2096_power__less__imp__less__base,axiom,
! [A: real,N: nat,B: real] :
( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_2097_power__less__imp__less__base,axiom,
! [A: rat,N: nat,B: rat] :
( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_2098_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_2099_power__less__imp__less__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_2100_power__le__one,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% power_le_one
thf(fact_2101_power__le__one,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).
% power_le_one
thf(fact_2102_power__le__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_2103_power__le__one,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_2104_power__inject__base,axiom,
! [A: real,N: nat,B: real] :
( ( ( power_power_real @ A @ ( suc @ N ) )
= ( power_power_real @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_2105_power__inject__base,axiom,
! [A: rat,N: nat,B: rat] :
( ( ( power_power_rat @ A @ ( suc @ N ) )
= ( power_power_rat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_2106_power__inject__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ ( suc @ N ) )
= ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_2107_power__inject__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ ( suc @ N ) )
= ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_2108_power__le__imp__le__base,axiom,
! [A: real,N: nat,B: real] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_2109_power__le__imp__le__base,axiom,
! [A: rat,N: nat,B: rat] :
( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_2110_power__le__imp__le__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_2111_power__le__imp__le__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_2112_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_rat @ zero_zero_rat @ N )
= one_one_rat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_rat @ zero_zero_rat @ N )
= zero_zero_rat ) ) ) ).
% power_0_left
thf(fact_2113_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_2114_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ) ).
% power_0_left
thf(fact_2115_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_2116_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= one_one_complex ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= zero_zero_complex ) ) ) ).
% power_0_left
thf(fact_2117_power__gt1,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_2118_power__gt1,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_2119_power__gt1,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_2120_power__gt1,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_2121_power__increasing,axiom,
! [N: nat,N7: nat,A: real] :
( ( ord_less_eq_nat @ N @ N7 )
=> ( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N7 ) ) ) ) ).
% power_increasing
thf(fact_2122_power__increasing,axiom,
! [N: nat,N7: nat,A: rat] :
( ( ord_less_eq_nat @ N @ N7 )
=> ( ( ord_less_eq_rat @ one_one_rat @ A )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N7 ) ) ) ) ).
% power_increasing
thf(fact_2123_power__increasing,axiom,
! [N: nat,N7: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N7 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N7 ) ) ) ) ).
% power_increasing
thf(fact_2124_power__increasing,axiom,
! [N: nat,N7: nat,A: int] :
( ( ord_less_eq_nat @ N @ N7 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N7 ) ) ) ) ).
% power_increasing
thf(fact_2125_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_rat @ zero_zero_rat @ N )
= zero_zero_rat ) ) ).
% zero_power
thf(fact_2126_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_2127_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ).
% zero_power
thf(fact_2128_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_2129_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= zero_zero_complex ) ) ).
% zero_power
thf(fact_2130_power__gt__expt,axiom,
! [N: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K2 @ ( power_power_nat @ N @ K2 ) ) ) ).
% power_gt_expt
thf(fact_2131_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% nat_one_le_power
thf(fact_2132_mod__le__divisor,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ N ) @ N ) ) ).
% mod_le_divisor
thf(fact_2133_power__Suc__le__self,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_2134_power__Suc__le__self,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_2135_power__Suc__le__self,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_2136_power__Suc__le__self,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_2137_power__Suc__less__one,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% power_Suc_less_one
thf(fact_2138_power__Suc__less__one,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ A @ one_one_rat )
=> ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).
% power_Suc_less_one
thf(fact_2139_power__Suc__less__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_2140_power__Suc__less__one,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% power_Suc_less_one
thf(fact_2141_power__strict__decreasing,axiom,
! [N: nat,N7: nat,A: real] :
( ( ord_less_nat @ N @ N7 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ N7 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_2142_power__strict__decreasing,axiom,
! [N: nat,N7: nat,A: rat] :
( ( ord_less_nat @ N @ N7 )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ A @ one_one_rat )
=> ( ord_less_rat @ ( power_power_rat @ A @ N7 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_2143_power__strict__decreasing,axiom,
! [N: nat,N7: nat,A: nat] :
( ( ord_less_nat @ N @ N7 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N7 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_2144_power__strict__decreasing,axiom,
! [N: nat,N7: nat,A: int] :
( ( ord_less_nat @ N @ N7 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N7 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_2145_power__decreasing,axiom,
! [N: nat,N7: nat,A: real] :
( ( ord_less_eq_nat @ N @ N7 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N7 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_2146_power__decreasing,axiom,
! [N: nat,N7: nat,A: rat] :
( ( ord_less_eq_nat @ N @ N7 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N7 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_2147_power__decreasing,axiom,
! [N: nat,N7: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N7 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N7 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_2148_power__decreasing,axiom,
! [N: nat,N7: nat,A: int] :
( ( ord_less_eq_nat @ N @ N7 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N7 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_2149_power__le__imp__le__exp,axiom,
! [A: real,M2: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_2150_power__le__imp__le__exp,axiom,
! [A: rat,M2: nat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M2 ) @ ( power_power_rat @ A @ N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_2151_power__le__imp__le__exp,axiom,
! [A: nat,M2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_2152_power__le__imp__le__exp,axiom,
! [A: int,M2: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_2153_power__eq__iff__eq__base,axiom,
! [N: nat,A: real,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ( power_power_real @ A @ N )
= ( power_power_real @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_2154_power__eq__iff__eq__base,axiom,
! [N: nat,A: rat,B: rat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ( power_power_rat @ A @ N )
= ( power_power_rat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_2155_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_2156_power__eq__iff__eq__base,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_2157_power__eq__imp__eq__base,axiom,
! [A: real,N: nat,B: real] :
( ( ( power_power_real @ A @ N )
= ( power_power_real @ B @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_2158_power__eq__imp__eq__base,axiom,
! [A: rat,N: nat,B: rat] :
( ( ( power_power_rat @ A @ N )
= ( power_power_rat @ B @ N ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_2159_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_2160_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_2161_zero__power2,axiom,
( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_rat ) ).
% zero_power2
thf(fact_2162_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% zero_power2
thf(fact_2163_zero__power2,axiom,
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real ) ).
% zero_power2
thf(fact_2164_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% zero_power2
thf(fact_2165_zero__power2,axiom,
( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_complex ) ).
% zero_power2
thf(fact_2166_self__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_2167_self__le__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ one_one_rat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_2168_self__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_2169_self__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_2170_one__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_2171_one__less__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_2172_one__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_2173_one__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_2174_power2__nat__le__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% power2_nat_le_imp_le
thf(fact_2175_power2__nat__le__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% power2_nat_le_eq_le
thf(fact_2176_self__le__ge2__pow,axiom,
! [K2: nat,M2: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
=> ( ord_less_eq_nat @ M2 @ ( power_power_nat @ K2 @ M2 ) ) ) ).
% self_le_ge2_pow
thf(fact_2177_zero__le__power2,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_2178_zero__le__power2,axiom,
! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_2179_zero__le__power2,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_2180_power2__eq__imp__eq,axiom,
! [X3: real,Y: real] :
( ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( X3 = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_2181_power2__eq__imp__eq,axiom,
! [X3: rat,Y: rat] :
( ( ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( X3 = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_2182_power2__eq__imp__eq,axiom,
! [X3: nat,Y: nat] :
( ( ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( X3 = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_2183_power2__eq__imp__eq,axiom,
! [X3: int,Y: int] :
( ( ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( X3 = Y ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_2184_power2__le__imp__le,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ X3 @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_2185_power2__le__imp__le,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ X3 @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_2186_power2__le__imp__le,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_2187_power2__le__imp__le,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ X3 @ Y ) ) ) ).
% power2_le_imp_le
thf(fact_2188_power__strict__mono,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_2189_power__strict__mono,axiom,
! [A: rat,B: rat,N: nat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_2190_power__strict__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_2191_power__strict__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_2192_power2__less__0,axiom,
! [A: real] :
~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% power2_less_0
thf(fact_2193_power2__less__0,axiom,
! [A: rat] :
~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% power2_less_0
thf(fact_2194_power2__less__0,axiom,
! [A: int] :
~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% power2_less_0
thf(fact_2195_nat__induct2,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ( P2 @ one_one_nat )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( P2 @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct2
thf(fact_2196_power2__less__imp__less,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_real @ X3 @ Y ) ) ) ).
% power2_less_imp_less
thf(fact_2197_power2__less__imp__less,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ord_less_rat @ X3 @ Y ) ) ) ).
% power2_less_imp_less
thf(fact_2198_power2__less__imp__less,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ord_less_nat @ X3 @ Y ) ) ) ).
% power2_less_imp_less
thf(fact_2199_power2__less__imp__less,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_int @ X3 @ Y ) ) ) ).
% power2_less_imp_less
thf(fact_2200_sum__power2__le__zero__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
= ( ( X3 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_2201_sum__power2__le__zero__iff,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
= ( ( X3 = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_2202_sum__power2__le__zero__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
= ( ( X3 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_2203_sum__power2__ge__zero,axiom,
! [X3: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_2204_sum__power2__ge__zero,axiom,
! [X3: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_2205_sum__power2__ge__zero,axiom,
! [X3: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_2206_not__sum__power2__lt__zero,axiom,
! [X3: real,Y: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% not_sum_power2_lt_zero
thf(fact_2207_not__sum__power2__lt__zero,axiom,
! [X3: rat,Y: rat] :
~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).
% not_sum_power2_lt_zero
thf(fact_2208_not__sum__power2__lt__zero,axiom,
! [X3: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% not_sum_power2_lt_zero
thf(fact_2209_sum__power2__gt__zero__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X3 != zero_zero_real )
| ( Y != zero_zero_real ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_2210_sum__power2__gt__zero__iff,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X3 != zero_zero_rat )
| ( Y != zero_zero_rat ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_2211_sum__power2__gt__zero__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X3 != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_2212_ex__power__ivl2,axiom,
! [B: nat,K2: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
=> ? [N3: nat] :
( ( ord_less_nat @ ( power_power_nat @ B @ N3 ) @ K2 )
& ( ord_less_eq_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_2213_ex__power__ivl1,axiom,
! [B: nat,K2: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_eq_nat @ one_one_nat @ K2 )
=> ? [N3: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ B @ N3 ) @ K2 )
& ( ord_less_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_2214__C00_C,axiom,
( ( deg
= ( plus_plus_nat @ na @ m ) )
& ( ( size_s6755466524823107622T_VEBT @ treeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
& ( ( suc @ na )
= m )
& ( ord_less_eq_nat @ one_one_nat @ na )
& ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg )
& ( na
= ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% "00"
thf(fact_2215_both__member__options__from__chilf__to__complete__tree,axiom,
! [X3: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( ( ord_less_eq_nat @ one_one_nat @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 ) ) ) ) ).
% both_member_options_from_chilf_to_complete_tree
thf(fact_2216_member__inv,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
& ( ( X3 = Mi )
| ( X3 = Ma )
| ( ( ord_less_nat @ X3 @ Ma )
& ( ord_less_nat @ Mi @ X3 )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
& ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% member_inv
thf(fact_2217_both__member__options__from__complete__tree__to__child,axiom,
! [Deg: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
( ( ord_less_eq_nat @ one_one_nat @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
| ( X3 = Mi )
| ( X3 = Ma ) ) ) ) ).
% both_member_options_from_complete_tree_to_child
thf(fact_2218_both__member__options__ding,axiom,
! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X3: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
=> ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X3 ) ) ) ) ).
% both_member_options_ding
thf(fact_2219_semiring__norm_I69_J,axiom,
! [M2: num] :
~ ( ord_less_eq_num @ ( bit0 @ M2 ) @ one ) ).
% semiring_norm(69)
thf(fact_2220_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_2221_bits__mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% bits_mod_by_1
thf(fact_2222_bits__mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% bits_mod_by_1
thf(fact_2223_bits__mod__by__1,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
= zero_z3403309356797280102nteger ) ).
% bits_mod_by_1
thf(fact_2224_exp__add__not__zero__imp__right,axiom,
! [M2: nat,N: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
!= zero_zero_nat ) ) ).
% exp_add_not_zero_imp_right
thf(fact_2225_exp__add__not__zero__imp__right,axiom,
! [M2: nat,N: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
!= zero_zero_int ) ) ).
% exp_add_not_zero_imp_right
thf(fact_2226_exp__add__not__zero__imp__left,axiom,
! [M2: nat,N: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
!= zero_zero_nat ) ) ).
% exp_add_not_zero_imp_left
thf(fact_2227_exp__add__not__zero__imp__left,axiom,
! [M2: nat,N: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 )
!= zero_zero_int ) ) ).
% exp_add_not_zero_imp_left
thf(fact_2228_pow__sum,axiom,
! [A: nat,B: nat] :
( ( 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 ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ).
% pow_sum
thf(fact_2229_high__def,axiom,
( vEBT_VEBT_high
= ( ^ [X5: nat,N4: nat] : ( divide_divide_nat @ X5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% high_def
thf(fact_2230_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_2231_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_2232_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_2233_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_2234_div__0,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
= zero_zero_complex ) ).
% div_0
thf(fact_2235_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_2236_div__0,axiom,
! [A: rat] :
( ( divide_divide_rat @ zero_zero_rat @ A )
= zero_zero_rat ) ).
% div_0
thf(fact_2237_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_2238_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_2239_divide__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ B )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% divide_eq_0_iff
thf(fact_2240_divide__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_2241_divide__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( ( divide_divide_rat @ A @ B )
= zero_zero_rat )
= ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% divide_eq_0_iff
thf(fact_2242_div__by__0,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% div_by_0
thf(fact_2243_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_2244_div__by__0,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% div_by_0
thf(fact_2245_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_2246_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_2247_divide__cancel__left,axiom,
! [C2: complex,A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ C2 @ A )
= ( divide1717551699836669952omplex @ C2 @ B ) )
= ( ( C2 = zero_zero_complex )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_2248_divide__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ( divide_divide_real @ C2 @ A )
= ( divide_divide_real @ C2 @ B ) )
= ( ( C2 = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_2249_divide__cancel__left,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ( divide_divide_rat @ C2 @ A )
= ( divide_divide_rat @ C2 @ B ) )
= ( ( C2 = zero_zero_rat )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_2250_divide__cancel__right,axiom,
! [A: complex,C2: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ C2 )
= ( divide1717551699836669952omplex @ B @ C2 ) )
= ( ( C2 = zero_zero_complex )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_2251_divide__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ( divide_divide_real @ A @ C2 )
= ( divide_divide_real @ B @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_2252_divide__cancel__right,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ( divide_divide_rat @ A @ C2 )
= ( divide_divide_rat @ B @ C2 ) )
= ( ( C2 = zero_zero_rat )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_2253_division__ring__divide__zero,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% division_ring_divide_zero
thf(fact_2254_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_2255_division__ring__divide__zero,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% division_ring_divide_zero
thf(fact_2256_bits__mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_mod_0
thf(fact_2257_bits__mod__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_mod_0
thf(fact_2258_bits__mod__0,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
= zero_z3403309356797280102nteger ) ).
% bits_mod_0
thf(fact_2259_semiring__norm_I6_J,axiom,
! [M2: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus_num @ M2 @ N ) ) ) ).
% semiring_norm(6)
thf(fact_2260_semiring__norm_I71_J,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% semiring_norm(71)
thf(fact_2261_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_2262_divide__eq__1__iff,axiom,
! [A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ B )
= one_one_complex )
= ( ( B != zero_zero_complex )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_2263_divide__eq__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_2264_divide__eq__1__iff,axiom,
! [A: rat,B: rat] :
( ( ( divide_divide_rat @ A @ B )
= one_one_rat )
= ( ( B != zero_zero_rat )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_2265_div__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ).
% div_self
thf(fact_2266_div__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% div_self
thf(fact_2267_div__self,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= one_one_rat ) ) ).
% div_self
thf(fact_2268_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_2269_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_2270_one__eq__divide__iff,axiom,
! [A: complex,B: complex] :
( ( one_one_complex
= ( divide1717551699836669952omplex @ A @ B ) )
= ( ( B != zero_zero_complex )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_2271_one__eq__divide__iff,axiom,
! [A: real,B: real] :
( ( one_one_real
= ( divide_divide_real @ A @ B ) )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_2272_one__eq__divide__iff,axiom,
! [A: rat,B: rat] :
( ( one_one_rat
= ( divide_divide_rat @ A @ B ) )
= ( ( B != zero_zero_rat )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_2273_divide__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ).
% divide_self
thf(fact_2274_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% divide_self
thf(fact_2275_divide__self,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= one_one_rat ) ) ).
% divide_self
thf(fact_2276_divide__self__if,axiom,
! [A: complex] :
( ( ( A = zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= zero_zero_complex ) )
& ( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ) ).
% divide_self_if
thf(fact_2277_divide__self__if,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= zero_zero_real ) )
& ( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ) ).
% divide_self_if
thf(fact_2278_divide__self__if,axiom,
! [A: rat] :
( ( ( A = zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= zero_zero_rat ) )
& ( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= one_one_rat ) ) ) ).
% divide_self_if
thf(fact_2279_divide__eq__eq__1,axiom,
! [B: real,A: real] :
( ( ( divide_divide_real @ B @ A )
= one_one_real )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_2280_divide__eq__eq__1,axiom,
! [B: rat,A: rat] :
( ( ( divide_divide_rat @ B @ A )
= one_one_rat )
= ( ( A != zero_zero_rat )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_2281_eq__divide__eq__1,axiom,
! [B: real,A: real] :
( ( one_one_real
= ( divide_divide_real @ B @ A ) )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_2282_eq__divide__eq__1,axiom,
! [B: rat,A: rat] :
( ( one_one_rat
= ( divide_divide_rat @ B @ A ) )
= ( ( A != zero_zero_rat )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_2283_one__divide__eq__0__iff,axiom,
! [A: real] :
( ( ( divide_divide_real @ one_one_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% one_divide_eq_0_iff
thf(fact_2284_one__divide__eq__0__iff,axiom,
! [A: rat] :
( ( ( divide_divide_rat @ one_one_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% one_divide_eq_0_iff
thf(fact_2285_zero__eq__1__divide__iff,axiom,
! [A: real] :
( ( zero_zero_real
= ( divide_divide_real @ one_one_real @ A ) )
= ( A = zero_zero_real ) ) ).
% zero_eq_1_divide_iff
thf(fact_2286_zero__eq__1__divide__iff,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( divide_divide_rat @ one_one_rat @ A ) )
= ( A = zero_zero_rat ) ) ).
% zero_eq_1_divide_iff
thf(fact_2287_bits__mod__div__trivial,axiom,
! [A: nat,B: nat] :
( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
= zero_zero_nat ) ).
% bits_mod_div_trivial
thf(fact_2288_bits__mod__div__trivial,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
= zero_zero_int ) ).
% bits_mod_div_trivial
thf(fact_2289_bits__mod__div__trivial,axiom,
! [A: code_integer,B: code_integer] :
( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
= zero_z3403309356797280102nteger ) ).
% bits_mod_div_trivial
thf(fact_2290_mod__div__trivial,axiom,
! [A: nat,B: nat] :
( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
= zero_zero_nat ) ).
% mod_div_trivial
thf(fact_2291_mod__div__trivial,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
= zero_zero_int ) ).
% mod_div_trivial
thf(fact_2292_mod__div__trivial,axiom,
! [A: code_integer,B: code_integer] :
( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
= zero_z3403309356797280102nteger ) ).
% mod_div_trivial
thf(fact_2293_div__by__Suc__0,axiom,
! [M2: nat] :
( ( divide_divide_nat @ M2 @ ( suc @ zero_zero_nat ) )
= M2 ) ).
% div_by_Suc_0
thf(fact_2294_div__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_2295_zero__le__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_divide_1_iff
thf(fact_2296_zero__le__divide__1__iff,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% zero_le_divide_1_iff
thf(fact_2297_divide__le__0__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% divide_le_0_1_iff
thf(fact_2298_divide__le__0__1__iff,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% divide_le_0_1_iff
thf(fact_2299_divide__less__0__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% divide_less_0_1_iff
thf(fact_2300_divide__less__0__1__iff,axiom,
! [A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% divide_less_0_1_iff
thf(fact_2301_divide__less__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ A @ B ) ) ) ).
% divide_less_eq_1_neg
thf(fact_2302_divide__less__eq__1__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ord_less_rat @ A @ B ) ) ) ).
% divide_less_eq_1_neg
thf(fact_2303_divide__less__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ B @ A ) ) ) ).
% divide_less_eq_1_pos
thf(fact_2304_divide__less__eq__1__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ord_less_rat @ B @ A ) ) ) ).
% divide_less_eq_1_pos
thf(fact_2305_less__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ B @ A ) ) ) ).
% less_divide_eq_1_neg
thf(fact_2306_less__divide__eq__1__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ord_less_rat @ B @ A ) ) ) ).
% less_divide_eq_1_neg
thf(fact_2307_less__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ A @ B ) ) ) ).
% less_divide_eq_1_pos
thf(fact_2308_less__divide__eq__1__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ord_less_rat @ A @ B ) ) ) ).
% less_divide_eq_1_pos
thf(fact_2309_zero__less__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_divide_1_iff
thf(fact_2310_zero__less__divide__1__iff,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% zero_less_divide_1_iff
thf(fact_2311_le__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% le_divide_eq_1_pos
thf(fact_2312_le__divide__eq__1__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ord_less_eq_rat @ A @ B ) ) ) ).
% le_divide_eq_1_pos
thf(fact_2313_le__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% le_divide_eq_1_neg
thf(fact_2314_le__divide__eq__1__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ord_less_eq_rat @ B @ A ) ) ) ).
% le_divide_eq_1_neg
thf(fact_2315_divide__le__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% divide_le_eq_1_pos
thf(fact_2316_divide__le__eq__1__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ord_less_eq_rat @ B @ A ) ) ) ).
% divide_le_eq_1_pos
thf(fact_2317_divide__le__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% divide_le_eq_1_neg
thf(fact_2318_divide__le__eq__1__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ord_less_eq_rat @ A @ B ) ) ) ).
% divide_le_eq_1_neg
thf(fact_2319_div2__Suc__Suc,axiom,
! [M2: nat] :
( ( divide_divide_nat @ ( suc @ ( suc @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% div2_Suc_Suc
thf(fact_2320_add__self__div__2,axiom,
! [M2: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ M2 @ M2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= M2 ) ).
% add_self_div_2
thf(fact_2321_bits__1__div__2,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% bits_1_div_2
thf(fact_2322_bits__1__div__2,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% bits_1_div_2
thf(fact_2323_one__div__two__eq__zero,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% one_div_two_eq_zero
thf(fact_2324_one__div__two__eq__zero,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% one_div_two_eq_zero
thf(fact_2325_real__arch__pow__inv,axiom,
! [Y: real,X3: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X3 @ one_one_real )
=> ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X3 @ N3 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_2326_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X5: real,Y5: real] :
( ( ord_less_real @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_2327_complete__real,axiom,
! [S2: set_real] :
( ? [X: real] : ( member_real @ X @ S2 )
=> ( ? [Z4: real] :
! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ X4 @ Z4 ) )
=> ? [Y3: real] :
( ! [X: real] :
( ( member_real @ X @ S2 )
=> ( ord_less_eq_real @ X @ Y3 ) )
& ! [Z4: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ X4 @ Z4 ) )
=> ( ord_less_eq_real @ Y3 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_2328_add__divide__distrib,axiom,
! [A: complex,B: complex,C2: complex] :
( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B ) @ C2 )
= ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C2 ) @ ( divide1717551699836669952omplex @ B @ C2 ) ) ) ).
% add_divide_distrib
thf(fact_2329_add__divide__distrib,axiom,
! [A: real,B: real,C2: real] :
( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B @ C2 ) ) ) ).
% add_divide_distrib
thf(fact_2330_add__divide__distrib,axiom,
! [A: rat,B: rat,C2: rat] :
( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
= ( plus_plus_rat @ ( divide_divide_rat @ A @ C2 ) @ ( divide_divide_rat @ B @ C2 ) ) ) ).
% add_divide_distrib
thf(fact_2331_div__le__mono,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ K2 ) @ ( divide_divide_nat @ N @ K2 ) ) ) ).
% div_le_mono
thf(fact_2332_div__le__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ).
% div_le_dividend
thf(fact_2333_divide__right__mono__neg,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B @ C2 ) @ ( divide_divide_real @ A @ C2 ) ) ) ) ).
% divide_right_mono_neg
thf(fact_2334_divide__right__mono__neg,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C2 ) @ ( divide_divide_rat @ A @ C2 ) ) ) ) ).
% divide_right_mono_neg
thf(fact_2335_divide__nonpos__nonpos,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_2336_divide__nonpos__nonpos,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_2337_divide__nonpos__nonneg,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_2338_divide__nonpos__nonneg,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y ) @ zero_zero_rat ) ) ) ).
% divide_nonpos_nonneg
thf(fact_2339_divide__nonneg__nonpos,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_2340_divide__nonneg__nonpos,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
=> ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y ) @ zero_zero_rat ) ) ) ).
% divide_nonneg_nonpos
thf(fact_2341_divide__nonneg__nonneg,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_2342_divide__nonneg__nonneg,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_2343_zero__le__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_divide_iff
thf(fact_2344_zero__le__divide__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% zero_le_divide_iff
thf(fact_2345_divide__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B @ C2 ) ) ) ) ).
% divide_right_mono
thf(fact_2346_divide__right__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C2 ) @ ( divide_divide_rat @ B @ C2 ) ) ) ) ).
% divide_right_mono
thf(fact_2347_divide__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% divide_le_0_iff
thf(fact_2348_divide__le__0__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% divide_le_0_iff
thf(fact_2349_divide__neg__neg,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ zero_zero_real )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y ) ) ) ) ).
% divide_neg_neg
thf(fact_2350_divide__neg__neg,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_rat @ X3 @ zero_zero_rat )
=> ( ( ord_less_rat @ Y @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y ) ) ) ) ).
% divide_neg_neg
thf(fact_2351_divide__neg__pos,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ ( divide_divide_real @ X3 @ Y ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_2352_divide__neg__pos,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_rat @ X3 @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y ) @ zero_zero_rat ) ) ) ).
% divide_neg_pos
thf(fact_2353_divide__pos__neg,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X3 @ Y ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_2354_divide__pos__neg,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ X3 )
=> ( ( ord_less_rat @ Y @ zero_zero_rat )
=> ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y ) @ zero_zero_rat ) ) ) ).
% divide_pos_neg
thf(fact_2355_divide__pos__pos,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y ) ) ) ) ).
% divide_pos_pos
thf(fact_2356_divide__pos__pos,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ X3 )
=> ( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y ) ) ) ) ).
% divide_pos_pos
thf(fact_2357_divide__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% divide_less_0_iff
thf(fact_2358_divide__less__0__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% divide_less_0_iff
thf(fact_2359_divide__less__cancel,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ A ) )
& ( C2 != zero_zero_real ) ) ) ).
% divide_less_cancel
thf(fact_2360_divide__less__cancel,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ A @ C2 ) @ ( divide_divide_rat @ B @ C2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ A @ B ) )
& ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ B @ A ) )
& ( C2 != zero_zero_rat ) ) ) ).
% divide_less_cancel
thf(fact_2361_zero__less__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_divide_iff
thf(fact_2362_zero__less__divide__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% zero_less_divide_iff
thf(fact_2363_divide__strict__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B @ C2 ) ) ) ) ).
% divide_strict_right_mono
thf(fact_2364_divide__strict__right__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ ( divide_divide_rat @ A @ C2 ) @ ( divide_divide_rat @ B @ C2 ) ) ) ) ).
% divide_strict_right_mono
thf(fact_2365_divide__strict__right__mono__neg,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B @ C2 ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_2366_divide__strict__right__mono__neg,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ ( divide_divide_rat @ A @ C2 ) @ ( divide_divide_rat @ B @ C2 ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_2367_right__inverse__eq,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ A @ B )
= one_one_complex )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_2368_right__inverse__eq,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_2369_right__inverse__eq,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( ( divide_divide_rat @ A @ B )
= one_one_rat )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_2370_mod__eq__self__iff__div__eq__0,axiom,
! [A: nat,B: nat] :
( ( ( modulo_modulo_nat @ A @ B )
= A )
= ( ( divide_divide_nat @ A @ B )
= zero_zero_nat ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_2371_mod__eq__self__iff__div__eq__0,axiom,
! [A: int,B: int] :
( ( ( modulo_modulo_int @ A @ B )
= A )
= ( ( divide_divide_int @ A @ B )
= zero_zero_int ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_2372_mod__eq__self__iff__div__eq__0,axiom,
! [A: code_integer,B: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ B )
= A )
= ( ( divide6298287555418463151nteger @ A @ B )
= zero_z3403309356797280102nteger ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_2373_div__add1__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C2 ) @ ( divide_divide_nat @ B @ C2 ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C2 ) @ ( modulo_modulo_nat @ B @ C2 ) ) @ C2 ) ) ) ).
% div_add1_eq
thf(fact_2374_div__add1__eq,axiom,
! [A: int,B: int,C2: int] :
( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C2 ) @ ( divide_divide_int @ B @ C2 ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C2 ) @ ( modulo_modulo_int @ B @ C2 ) ) @ C2 ) ) ) ).
% div_add1_eq
thf(fact_2375_div__add1__eq,axiom,
! [A: code_integer,B: code_integer,C2: code_integer] :
( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C2 )
= ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C2 ) @ ( divide6298287555418463151nteger @ B @ C2 ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C2 ) @ ( modulo364778990260209775nteger @ B @ C2 ) ) @ C2 ) ) ) ).
% div_add1_eq
thf(fact_2376_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N: nat] :
( ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M2 @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_2377_Suc__div__le__mono,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N ) @ ( divide_divide_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_2378_div__positive,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_positive
thf(fact_2379_div__positive,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_positive
thf(fact_2380_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ B )
=> ( ( divide_divide_nat @ A @ B )
= zero_zero_nat ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
thf(fact_2381_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ B )
=> ( ( divide_divide_int @ A @ B )
= zero_zero_int ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
thf(fact_2382_divide__nonpos__pos,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonpos_pos
thf(fact_2383_divide__nonpos__pos,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y ) @ zero_zero_rat ) ) ) ).
% divide_nonpos_pos
thf(fact_2384_divide__nonpos__neg,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y ) ) ) ) ).
% divide_nonpos_neg
thf(fact_2385_divide__nonpos__neg,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
=> ( ( ord_less_rat @ Y @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y ) ) ) ) ).
% divide_nonpos_neg
thf(fact_2386_divide__nonneg__pos,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y ) ) ) ) ).
% divide_nonneg_pos
thf(fact_2387_divide__nonneg__pos,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
=> ( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y ) ) ) ) ).
% divide_nonneg_pos
thf(fact_2388_divide__nonneg__neg,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonneg_neg
thf(fact_2389_divide__nonneg__neg,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
=> ( ( ord_less_rat @ Y @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y ) @ zero_zero_rat ) ) ) ).
% divide_nonneg_neg
thf(fact_2390_divide__le__cancel,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% divide_le_cancel
thf(fact_2391_divide__le__cancel,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C2 ) @ ( divide_divide_rat @ B @ C2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ A @ B ) )
& ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% divide_le_cancel
thf(fact_2392_frac__less2,axiom,
! [X3: real,Y: real,W2: real,Z2: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_real @ W2 @ Z2 )
=> ( ord_less_real @ ( divide_divide_real @ X3 @ Z2 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% frac_less2
thf(fact_2393_frac__less2,axiom,
! [X3: rat,Y: rat,W2: rat,Z2: rat] :
( ( ord_less_rat @ zero_zero_rat @ X3 )
=> ( ( ord_less_eq_rat @ X3 @ Y )
=> ( ( ord_less_rat @ zero_zero_rat @ W2 )
=> ( ( ord_less_rat @ W2 @ Z2 )
=> ( ord_less_rat @ ( divide_divide_rat @ X3 @ Z2 ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).
% frac_less2
thf(fact_2394_frac__less,axiom,
! [X3: real,Y: real,W2: real,Z2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_eq_real @ W2 @ Z2 )
=> ( ord_less_real @ ( divide_divide_real @ X3 @ Z2 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% frac_less
thf(fact_2395_frac__less,axiom,
! [X3: rat,Y: rat,W2: rat,Z2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
=> ( ( ord_less_rat @ X3 @ Y )
=> ( ( ord_less_rat @ zero_zero_rat @ W2 )
=> ( ( ord_less_eq_rat @ W2 @ Z2 )
=> ( ord_less_rat @ ( divide_divide_rat @ X3 @ Z2 ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).
% frac_less
thf(fact_2396_frac__le,axiom,
! [Y: real,X3: real,W2: real,Z2: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_eq_real @ W2 @ Z2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Z2 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% frac_le
thf(fact_2397_frac__le,axiom,
! [Y: rat,X3: rat,W2: rat,Z2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ X3 @ Y )
=> ( ( ord_less_rat @ zero_zero_rat @ W2 )
=> ( ( ord_less_eq_rat @ W2 @ Z2 )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Z2 ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).
% frac_le
thf(fact_2398_divide__less__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ A ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ A @ B ) )
| ( A = zero_zero_real ) ) ) ).
% divide_less_eq_1
thf(fact_2399_divide__less__eq__1,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ B @ A ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ A @ B ) )
| ( A = zero_zero_rat ) ) ) ).
% divide_less_eq_1
thf(fact_2400_less__divide__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% less_divide_eq_1
thf(fact_2401_less__divide__eq__1,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ A @ B ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ B @ A ) ) ) ) ).
% less_divide_eq_1
thf(fact_2402_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_2403_div__add__self1,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_2404_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_2405_div__add__self2,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_2406_gt__half__sum,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% gt_half_sum
thf(fact_2407_gt__half__sum,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).
% gt_half_sum
thf(fact_2408_less__half__sum,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% less_half_sum
thf(fact_2409_less__half__sum,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% less_half_sum
thf(fact_2410_div__greater__zero__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ N @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_2411_div__le__mono2,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K2 @ N ) @ ( divide_divide_nat @ K2 @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_2412_div__less__dividend,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_2413_div__eq__dividend__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ( divide_divide_nat @ M2 @ N )
= M2 )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_2414_div__less__mono,axiom,
! [A4: nat,B5: nat,N: nat] :
( ( ord_less_nat @ A4 @ B5 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( modulo_modulo_nat @ A4 @ N )
= zero_zero_nat )
=> ( ( ( modulo_modulo_nat @ B5 @ N )
= zero_zero_nat )
=> ( ord_less_nat @ ( divide_divide_nat @ A4 @ N ) @ ( divide_divide_nat @ B5 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_2415_div__exp__eq,axiom,
! [A: nat,M2: nat,N: nat] :
( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ).
% div_exp_eq
thf(fact_2416_div__exp__eq,axiom,
! [A: int,M2: nat,N: nat] :
( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ).
% div_exp_eq
thf(fact_2417_le__divide__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ A @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ A ) ) ) ) ).
% le_divide_eq_1
thf(fact_2418_le__divide__eq__1,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ A @ B ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% le_divide_eq_1
thf(fact_2419_divide__le__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ A ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ A @ B ) )
| ( A = zero_zero_real ) ) ) ).
% divide_le_eq_1
thf(fact_2420_divide__le__eq__1,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ B @ A ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ A @ B ) )
| ( A = zero_zero_rat ) ) ) ).
% divide_le_eq_1
thf(fact_2421_field__sum__of__halves,axiom,
! [X3: real] :
( ( plus_plus_real @ ( divide_divide_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= X3 ) ).
% field_sum_of_halves
thf(fact_2422_field__sum__of__halves,axiom,
! [X3: rat] :
( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X3 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
= X3 ) ).
% field_sum_of_halves
thf(fact_2423_bits__stable__imp__add__self,axiom,
! [A: nat] :
( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A )
=> ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_nat ) ) ).
% bits_stable_imp_add_self
thf(fact_2424_bits__stable__imp__add__self,axiom,
! [A: int] :
( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A )
=> ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= zero_zero_int ) ) ).
% bits_stable_imp_add_self
thf(fact_2425_bits__stable__imp__add__self,axiom,
! [A: code_integer] :
( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= A )
=> ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
= zero_z3403309356797280102nteger ) ) ).
% bits_stable_imp_add_self
thf(fact_2426_div__exp__mod__exp__eq,axiom,
! [A: nat,N: nat,M2: nat] :
( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
= ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M2 ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_2427_div__exp__mod__exp__eq,axiom,
! [A: int,N: nat,M2: nat] :
( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) )
= ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_2428_div__exp__mod__exp__eq,axiom,
! [A: code_integer,N: nat,M2: nat] :
( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) )
= ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M2 ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_2429_half__gt__zero__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% half_gt_zero_iff
thf(fact_2430_half__gt__zero__iff,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% half_gt_zero_iff
thf(fact_2431_half__gt__zero,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% half_gt_zero
thf(fact_2432_half__gt__zero,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% half_gt_zero
thf(fact_2433_field__less__half__sum,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ X3 @ ( divide_divide_real @ ( plus_plus_real @ X3 @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% field_less_half_sum
thf(fact_2434_field__less__half__sum,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_rat @ X3 @ Y )
=> ( ord_less_rat @ X3 @ ( divide_divide_rat @ ( plus_plus_rat @ X3 @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% field_less_half_sum
thf(fact_2435_Suc__n__div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% Suc_n_div_2_gt_zero
thf(fact_2436_div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% div_2_gt_zero
thf(fact_2437_verit__le__mono__div,axiom,
! [A4: nat,B5: nat,N: nat] :
( ( ord_less_nat @ A4 @ B5 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat
@ ( plus_plus_nat @ ( divide_divide_nat @ A4 @ N )
@ ( if_nat
@ ( ( modulo_modulo_nat @ B5 @ N )
= zero_zero_nat )
@ one_one_nat
@ zero_zero_nat ) )
@ ( divide_divide_nat @ B5 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_2438_not__exp__less__eq__0__int,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% not_exp_less_eq_0_int
thf(fact_2439_bit__concat__def,axiom,
( vEBT_VEBT_bit_concat
= ( ^ [H: nat,L2: nat,D4: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D4 ) ) @ L2 ) ) ) ).
% bit_concat_def
thf(fact_2440_low__inv,axiom,
! [X3: nat,N: nat,Y: nat] :
( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X3 ) @ N )
= X3 ) ) ).
% low_inv
thf(fact_2441_high__inv,axiom,
! [X3: nat,N: nat,Y: nat] :
( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X3 ) @ N )
= Y ) ) ).
% high_inv
thf(fact_2442_enat__ord__number_I1_J,axiom,
! [M2: num,N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(1)
thf(fact_2443_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_2444_fold__atLeastAtMost__nat_Opelims,axiom,
! [X3: nat > nat > nat,Xa2: nat,Xb2: nat,Xc: nat,Y: nat] :
( ( ( set_fo2584398358068434914at_nat @ X3 @ Xa2 @ Xb2 @ Xc )
= Y )
=> ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X3 @ ( produc487386426758144856at_nat @ Xa2 @ ( product_Pair_nat_nat @ Xb2 @ Xc ) ) ) )
=> ~ ( ( ( ( ord_less_nat @ Xb2 @ Xa2 )
=> ( Y = Xc ) )
& ( ~ ( ord_less_nat @ Xb2 @ Xa2 )
=> ( Y
= ( set_fo2584398358068434914at_nat @ X3 @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb2 @ ( X3 @ Xa2 @ Xc ) ) ) ) )
=> ~ ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X3 @ ( produc487386426758144856at_nat @ Xa2 @ ( product_Pair_nat_nat @ Xb2 @ Xc ) ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
thf(fact_2445_fold__atLeastAtMost__nat_Opsimps,axiom,
! [F: nat > nat > nat,A: nat,B: nat,Acc: nat] :
( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A @ ( product_Pair_nat_nat @ B @ Acc ) ) ) )
=> ( ( ( ord_less_nat @ B @ A )
=> ( ( set_fo2584398358068434914at_nat @ F @ A @ B @ Acc )
= Acc ) )
& ( ~ ( ord_less_nat @ B @ A )
=> ( ( set_fo2584398358068434914at_nat @ F @ A @ B @ Acc )
= ( set_fo2584398358068434914at_nat @ F @ ( plus_plus_nat @ A @ one_one_nat ) @ B @ ( F @ A @ Acc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.psimps
thf(fact_2446_insert__simp__excp,axiom,
! [Mi: nat,Deg: nat,TreeList: list_VEBT_VEBT,X3: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( ( ord_less_nat @ X3 @ Mi )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( X3 != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X3 @ ( 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 ) ) ) ) ) ) ) ).
% insert_simp_excp
thf(fact_2447_insert__simp__norm,axiom,
! [X3: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( ( ord_less_nat @ Mi @ X3 )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( X3 != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X3 @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% insert_simp_norm
thf(fact_2448_realpow__pos__nth,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ N )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_2449_zle__add1__eq__le,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_2450_list__update__overwrite,axiom,
! [Xs: list_VEBT_VEBT,I: nat,X3: vEBT_VEBT,Y: vEBT_VEBT] :
( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) @ I @ Y )
= ( list_u1324408373059187874T_VEBT @ Xs @ I @ Y ) ) ).
% list_update_overwrite
thf(fact_2451_mult__cancel__right,axiom,
! [A: complex,C2: complex,B: complex] :
( ( ( times_times_complex @ A @ C2 )
= ( times_times_complex @ B @ C2 ) )
= ( ( C2 = zero_zero_complex )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_2452_mult__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ( times_times_real @ A @ C2 )
= ( times_times_real @ B @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_2453_mult__cancel__right,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ( times_times_rat @ A @ C2 )
= ( times_times_rat @ B @ C2 ) )
= ( ( C2 = zero_zero_rat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_2454_mult__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_2455_mult__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ( times_times_int @ A @ C2 )
= ( times_times_int @ B @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_2456_mult__cancel__left,axiom,
! [C2: complex,A: complex,B: complex] :
( ( ( times_times_complex @ C2 @ A )
= ( times_times_complex @ C2 @ B ) )
= ( ( C2 = zero_zero_complex )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_2457_mult__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ( times_times_real @ C2 @ A )
= ( times_times_real @ C2 @ B ) )
= ( ( C2 = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_2458_mult__cancel__left,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ( times_times_rat @ C2 @ A )
= ( times_times_rat @ C2 @ B ) )
= ( ( C2 = zero_zero_rat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_2459_mult__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B ) )
= ( ( C2 = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_2460_mult__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ( times_times_int @ C2 @ A )
= ( times_times_int @ C2 @ B ) )
= ( ( C2 = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_2461_mult__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% mult_eq_0_iff
thf(fact_2462_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_2463_mult__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
= zero_zero_rat )
= ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% mult_eq_0_iff
thf(fact_2464_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_2465_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_2466_mult__zero__right,axiom,
! [A: complex] :
( ( times_times_complex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_zero_right
thf(fact_2467_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_2468_mult__zero__right,axiom,
! [A: rat] :
( ( times_times_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% mult_zero_right
thf(fact_2469_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_2470_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_2471_mult__zero__left,axiom,
! [A: complex] :
( ( times_times_complex @ zero_zero_complex @ A )
= zero_zero_complex ) ).
% mult_zero_left
thf(fact_2472_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_2473_mult__zero__left,axiom,
! [A: rat] :
( ( times_times_rat @ zero_zero_rat @ A )
= zero_zero_rat ) ).
% mult_zero_left
thf(fact_2474_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_2475_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_2476_mult_Oright__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.right_neutral
thf(fact_2477_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_2478_mult_Oright__neutral,axiom,
! [A: rat] :
( ( times_times_rat @ A @ one_one_rat )
= A ) ).
% mult.right_neutral
thf(fact_2479_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_2480_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_2481_mult__1,axiom,
! [A: complex] :
( ( times_times_complex @ one_one_complex @ A )
= A ) ).
% mult_1
thf(fact_2482_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_2483_mult__1,axiom,
! [A: rat] :
( ( times_times_rat @ one_one_rat @ A )
= A ) ).
% mult_1
thf(fact_2484_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_2485_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_2486_mult__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_2487_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_2488_mult__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M2 )
= ( times_times_nat @ K2 @ N ) )
= ( ( M2 = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_2489_mult__cancel2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ K2 )
= ( times_times_nat @ N @ K2 ) )
= ( ( M2 = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_2490_mod__neg__neg__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ K2 @ zero_zero_int )
=> ( ( ord_less_int @ L @ K2 )
=> ( ( modulo_modulo_int @ K2 @ L )
= K2 ) ) ) ).
% mod_neg_neg_trivial
thf(fact_2491_mod__pos__pos__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ( ( ord_less_int @ K2 @ L )
=> ( ( modulo_modulo_int @ K2 @ L )
= K2 ) ) ) ).
% mod_pos_pos_trivial
thf(fact_2492_div__pos__pos__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ( ( ord_less_int @ K2 @ L )
=> ( ( divide_divide_int @ K2 @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_2493_div__neg__neg__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ K2 @ zero_zero_int )
=> ( ( ord_less_int @ L @ K2 )
=> ( ( divide_divide_int @ K2 @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_2494_max__bot,axiom,
! [X3: set_Pr1261947904930325089at_nat] :
( ( ord_ma7524802468073614006at_nat @ bot_bo2099793752762293965at_nat @ X3 )
= X3 ) ).
% max_bot
thf(fact_2495_max__bot,axiom,
! [X3: set_real] :
( ( ord_max_set_real @ bot_bot_set_real @ X3 )
= X3 ) ).
% max_bot
thf(fact_2496_max__bot,axiom,
! [X3: set_nat] :
( ( ord_max_set_nat @ bot_bot_set_nat @ X3 )
= X3 ) ).
% max_bot
thf(fact_2497_max__bot,axiom,
! [X3: set_int] :
( ( ord_max_set_int @ bot_bot_set_int @ X3 )
= X3 ) ).
% max_bot
thf(fact_2498_max__bot,axiom,
! [X3: nat] :
( ( ord_max_nat @ bot_bot_nat @ X3 )
= X3 ) ).
% max_bot
thf(fact_2499_max__bot,axiom,
! [X3: extended_enat] :
( ( ord_ma741700101516333627d_enat @ bot_bo4199563552545308370d_enat @ X3 )
= X3 ) ).
% max_bot
thf(fact_2500_max__bot2,axiom,
! [X3: set_Pr1261947904930325089at_nat] :
( ( ord_ma7524802468073614006at_nat @ X3 @ bot_bo2099793752762293965at_nat )
= X3 ) ).
% max_bot2
thf(fact_2501_max__bot2,axiom,
! [X3: set_real] :
( ( ord_max_set_real @ X3 @ bot_bot_set_real )
= X3 ) ).
% max_bot2
thf(fact_2502_max__bot2,axiom,
! [X3: set_nat] :
( ( ord_max_set_nat @ X3 @ bot_bot_set_nat )
= X3 ) ).
% max_bot2
thf(fact_2503_max__bot2,axiom,
! [X3: set_int] :
( ( ord_max_set_int @ X3 @ bot_bot_set_int )
= X3 ) ).
% max_bot2
thf(fact_2504_max__bot2,axiom,
! [X3: nat] :
( ( ord_max_nat @ X3 @ bot_bot_nat )
= X3 ) ).
% max_bot2
thf(fact_2505_max__bot2,axiom,
! [X3: extended_enat] :
( ( ord_ma741700101516333627d_enat @ X3 @ bot_bo4199563552545308370d_enat )
= X3 ) ).
% max_bot2
thf(fact_2506_nat__1__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N ) )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_2507_nat__mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_2508_length__list__update,axiom,
! [Xs: list_VEBT_VEBT,I: nat,X3: vEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) )
= ( size_s6755466524823107622T_VEBT @ Xs ) ) ).
% length_list_update
thf(fact_2509_length__list__update,axiom,
! [Xs: list_int,I: nat,X3: int] :
( ( size_size_list_int @ ( list_update_int @ Xs @ I @ X3 ) )
= ( size_size_list_int @ Xs ) ) ).
% length_list_update
thf(fact_2510_length__list__update,axiom,
! [Xs: list_nat,I: nat,X3: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs @ I @ X3 ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_list_update
thf(fact_2511_max__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_max_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( suc @ ( ord_max_nat @ M2 @ N ) ) ) ).
% max_Suc_Suc
thf(fact_2512_max__nat_Oeq__neutr__iff,axiom,
! [A: nat,B: nat] :
( ( ( ord_max_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_2513_max__nat_Oleft__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ zero_zero_nat @ A )
= A ) ).
% max_nat.left_neutral
thf(fact_2514_max__nat_Oneutr__eq__iff,axiom,
! [A: nat,B: nat] :
( ( zero_zero_nat
= ( ord_max_nat @ A @ B ) )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_2515_max__nat_Oright__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ zero_zero_nat )
= A ) ).
% max_nat.right_neutral
thf(fact_2516_max__0L,axiom,
! [N: nat] :
( ( ord_max_nat @ zero_zero_nat @ N )
= N ) ).
% max_0L
thf(fact_2517_max__0R,axiom,
! [N: nat] :
( ( ord_max_nat @ N @ zero_zero_nat )
= N ) ).
% max_0R
thf(fact_2518_list__update__id,axiom,
! [Xs: list_int,I: nat] :
( ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_2519_list__update__id,axiom,
! [Xs: list_nat,I: nat] :
( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_2520_list__update__id,axiom,
! [Xs: list_VEBT_VEBT,I: nat] :
( ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( nth_VEBT_VEBT @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_2521_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs: list_int,X3: int] :
( ( I != J )
=> ( ( nth_int @ ( list_update_int @ Xs @ I @ X3 ) @ J )
= ( nth_int @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_2522_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs: list_nat,X3: nat] :
( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X3 ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_2523_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
( ( I != J )
=> ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) @ J )
= ( nth_VEBT_VEBT @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_2524_mult__cancel__right2,axiom,
! [A: complex,C2: complex] :
( ( ( times_times_complex @ A @ C2 )
= C2 )
= ( ( C2 = zero_zero_complex )
| ( A = one_one_complex ) ) ) ).
% mult_cancel_right2
thf(fact_2525_mult__cancel__right2,axiom,
! [A: real,C2: real] :
( ( ( times_times_real @ A @ C2 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_2526_mult__cancel__right2,axiom,
! [A: rat,C2: rat] :
( ( ( times_times_rat @ A @ C2 )
= C2 )
= ( ( C2 = zero_zero_rat )
| ( A = one_one_rat ) ) ) ).
% mult_cancel_right2
thf(fact_2527_mult__cancel__right2,axiom,
! [A: int,C2: int] :
( ( ( times_times_int @ A @ C2 )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_2528_mult__cancel__right1,axiom,
! [C2: complex,B: complex] :
( ( C2
= ( times_times_complex @ B @ C2 ) )
= ( ( C2 = zero_zero_complex )
| ( B = one_one_complex ) ) ) ).
% mult_cancel_right1
thf(fact_2529_mult__cancel__right1,axiom,
! [C2: real,B: real] :
( ( C2
= ( times_times_real @ B @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_2530_mult__cancel__right1,axiom,
! [C2: rat,B: rat] :
( ( C2
= ( times_times_rat @ B @ C2 ) )
= ( ( C2 = zero_zero_rat )
| ( B = one_one_rat ) ) ) ).
% mult_cancel_right1
thf(fact_2531_mult__cancel__right1,axiom,
! [C2: int,B: int] :
( ( C2
= ( times_times_int @ B @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_2532_mult__cancel__left2,axiom,
! [C2: complex,A: complex] :
( ( ( times_times_complex @ C2 @ A )
= C2 )
= ( ( C2 = zero_zero_complex )
| ( A = one_one_complex ) ) ) ).
% mult_cancel_left2
thf(fact_2533_mult__cancel__left2,axiom,
! [C2: real,A: real] :
( ( ( times_times_real @ C2 @ A )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_2534_mult__cancel__left2,axiom,
! [C2: rat,A: rat] :
( ( ( times_times_rat @ C2 @ A )
= C2 )
= ( ( C2 = zero_zero_rat )
| ( A = one_one_rat ) ) ) ).
% mult_cancel_left2
thf(fact_2535_mult__cancel__left2,axiom,
! [C2: int,A: int] :
( ( ( times_times_int @ C2 @ A )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_2536_mult__cancel__left1,axiom,
! [C2: complex,B: complex] :
( ( C2
= ( times_times_complex @ C2 @ B ) )
= ( ( C2 = zero_zero_complex )
| ( B = one_one_complex ) ) ) ).
% mult_cancel_left1
thf(fact_2537_mult__cancel__left1,axiom,
! [C2: real,B: real] :
( ( C2
= ( times_times_real @ C2 @ B ) )
= ( ( C2 = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_2538_mult__cancel__left1,axiom,
! [C2: rat,B: rat] :
( ( C2
= ( times_times_rat @ C2 @ B ) )
= ( ( C2 = zero_zero_rat )
| ( B = one_one_rat ) ) ) ).
% mult_cancel_left1
thf(fact_2539_mult__cancel__left1,axiom,
! [C2: int,B: int] :
( ( C2
= ( times_times_int @ C2 @ B ) )
= ( ( C2 = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_2540_sum__squares__eq__zero__iff,axiom,
! [X3: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y @ Y ) )
= zero_zero_real )
= ( ( X3 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_2541_sum__squares__eq__zero__iff,axiom,
! [X3: rat,Y: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y @ Y ) )
= zero_zero_rat )
= ( ( X3 = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_2542_sum__squares__eq__zero__iff,axiom,
! [X3: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X3 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_2543_div__mult__mult1__if,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ( C2 = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
= zero_zero_nat ) )
& ( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_2544_div__mult__mult1__if,axiom,
! [C2: int,A: int,B: int] :
( ( ( C2 = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= zero_zero_int ) )
& ( ( C2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_2545_div__mult__mult2,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_2546_div__mult__mult2,axiom,
! [C2: int,A: int,B: int] :
( ( C2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_2547_div__mult__mult1,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_2548_div__mult__mult1,axiom,
! [C2: int,A: int,B: int] :
( ( C2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_2549_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C2: complex,A: complex,B: complex] :
( ( C2 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C2 ) @ ( times_times_complex @ C2 @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_2550_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ C2 @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_2551_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C2: rat,A: rat,B: rat] :
( ( C2 != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ C2 @ B ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_2552_nonzero__mult__div__cancel__right,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_2553_nonzero__mult__div__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_2554_nonzero__mult__div__cancel__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_2555_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_2556_nonzero__mult__div__cancel__right,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_2557_nonzero__mult__divide__mult__cancel__right,axiom,
! [C2: complex,A: complex,B: complex] :
( ( C2 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C2 ) @ ( times_times_complex @ B @ C2 ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_2558_nonzero__mult__divide__mult__cancel__right,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_2559_nonzero__mult__divide__mult__cancel__right,axiom,
! [C2: rat,A: rat,B: rat] :
( ( C2 != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_2560_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C2: complex,A: complex,B: complex] :
( ( C2 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C2 @ A ) @ ( times_times_complex @ B @ C2 ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_2561_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ B @ C2 ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_2562_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C2: rat,A: rat,B: rat] :
( ( C2 != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ B @ C2 ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_2563_nonzero__mult__div__cancel__left,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_2564_nonzero__mult__div__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_2565_nonzero__mult__div__cancel__left,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_2566_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_2567_nonzero__mult__div__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_2568_nonzero__mult__divide__mult__cancel__left,axiom,
! [C2: complex,A: complex,B: complex] :
( ( C2 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C2 @ A ) @ ( times_times_complex @ C2 @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_2569_nonzero__mult__divide__mult__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_2570_nonzero__mult__divide__mult__cancel__left,axiom,
! [C2: rat,A: rat,B: rat] :
( ( C2 != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_2571_mult__divide__mult__cancel__left__if,axiom,
! [C2: complex,A: complex,B: complex] :
( ( ( C2 = zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C2 @ A ) @ ( times_times_complex @ C2 @ B ) )
= zero_zero_complex ) )
& ( ( C2 != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C2 @ A ) @ ( times_times_complex @ C2 @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_2572_mult__divide__mult__cancel__left__if,axiom,
! [C2: real,A: real,B: real] :
( ( ( C2 = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= zero_zero_real ) )
& ( ( C2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_2573_mult__divide__mult__cancel__left__if,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ( C2 = zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
= zero_zero_rat ) )
& ( ( C2 != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
= ( divide_divide_rat @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_2574_distrib__right__numeral,axiom,
! [A: complex,B: complex,V: num] :
( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
= ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_2575_distrib__right__numeral,axiom,
! [A: real,B: real,V: num] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
= ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_2576_distrib__right__numeral,axiom,
! [A: rat,B: rat,V: num] :
( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
= ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_2577_distrib__right__numeral,axiom,
! [A: nat,B: nat,V: num] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_2578_distrib__right__numeral,axiom,
! [A: int,B: int,V: num] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
= ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_2579_distrib__left__numeral,axiom,
! [V: num,B: complex,C2: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B @ C2 ) )
= ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C2 ) ) ) ).
% distrib_left_numeral
thf(fact_2580_distrib__left__numeral,axiom,
! [V: num,B: real,C2: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C2 ) ) ) ).
% distrib_left_numeral
thf(fact_2581_distrib__left__numeral,axiom,
! [V: num,B: rat,C2: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C2 ) )
= ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C2 ) ) ) ).
% distrib_left_numeral
thf(fact_2582_distrib__left__numeral,axiom,
! [V: num,B: nat,C2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C2 ) ) ) ).
% distrib_left_numeral
thf(fact_2583_distrib__left__numeral,axiom,
! [V: num,B: int,C2: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C2 ) ) ) ).
% distrib_left_numeral
thf(fact_2584_mod__mult__self2__is__0,axiom,
! [A: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
= zero_zero_nat ) ).
% mod_mult_self2_is_0
thf(fact_2585_mod__mult__self2__is__0,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
= zero_zero_int ) ).
% mod_mult_self2_is_0
thf(fact_2586_mod__mult__self2__is__0,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
= zero_z3403309356797280102nteger ) ).
% mod_mult_self2_is_0
thf(fact_2587_mod__mult__self1__is__0,axiom,
! [B: nat,A: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
= zero_zero_nat ) ).
% mod_mult_self1_is_0
thf(fact_2588_mod__mult__self1__is__0,axiom,
! [B: int,A: int] :
( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
= zero_zero_int ) ).
% mod_mult_self1_is_0
thf(fact_2589_mod__mult__self1__is__0,axiom,
! [B: code_integer,A: code_integer] :
( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
= zero_z3403309356797280102nteger ) ).
% mod_mult_self1_is_0
thf(fact_2590_mod__mult__self4,axiom,
! [B: nat,C2: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C2 ) @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self4
thf(fact_2591_mod__mult__self4,axiom,
! [B: int,C2: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C2 ) @ A ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self4
thf(fact_2592_mod__mult__self4,axiom,
! [B: code_integer,C2: code_integer,A: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C2 ) @ A ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_mult_self4
thf(fact_2593_mod__mult__self3,axiom,
! [C2: nat,B: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C2 @ B ) @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self3
thf(fact_2594_mod__mult__self3,axiom,
! [C2: int,B: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C2 @ B ) @ A ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self3
thf(fact_2595_mod__mult__self3,axiom,
! [C2: code_integer,B: code_integer,A: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C2 @ B ) @ A ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_mult_self3
thf(fact_2596_mod__mult__self2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C2 ) ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self2
thf(fact_2597_mod__mult__self2,axiom,
! [A: int,B: int,C2: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C2 ) ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self2
thf(fact_2598_mod__mult__self2,axiom,
! [A: code_integer,B: code_integer,C2: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C2 ) ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_mult_self2
thf(fact_2599_mod__mult__self1,axiom,
! [A: nat,C2: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C2 @ B ) ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self1
thf(fact_2600_mod__mult__self1,axiom,
! [A: int,C2: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C2 @ B ) ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self1
thf(fact_2601_mod__mult__self1,axiom,
! [A: code_integer,C2: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C2 @ B ) ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_mult_self1
thf(fact_2602_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
=> ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
= ( numera1916890842035813515d_enat @ V ) ) )
& ( ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
=> ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
= ( numera1916890842035813515d_enat @ U ) ) ) ) ).
% max_number_of(1)
thf(fact_2603_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
=> ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
= ( numera6620942414471956472nteger @ V ) ) )
& ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
=> ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
= ( numera6620942414471956472nteger @ U ) ) ) ) ).
% max_number_of(1)
thf(fact_2604_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
=> ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
= ( numeral_numeral_real @ V ) ) )
& ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
=> ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
= ( numeral_numeral_real @ U ) ) ) ) ).
% max_number_of(1)
thf(fact_2605_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
=> ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
= ( numeral_numeral_rat @ V ) ) )
& ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
=> ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
= ( numeral_numeral_rat @ U ) ) ) ) ).
% max_number_of(1)
thf(fact_2606_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
=> ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
= ( numeral_numeral_nat @ V ) ) )
& ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
=> ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
= ( numeral_numeral_nat @ U ) ) ) ) ).
% max_number_of(1)
thf(fact_2607_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
=> ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
= ( numeral_numeral_int @ V ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
=> ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
= ( numeral_numeral_int @ U ) ) ) ) ).
% max_number_of(1)
thf(fact_2608_max__0__1_I3_J,axiom,
! [X3: num] :
( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ X3 ) )
= ( numera1916890842035813515d_enat @ X3 ) ) ).
% max_0_1(3)
thf(fact_2609_max__0__1_I3_J,axiom,
! [X3: num] :
( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ X3 ) )
= ( numera6620942414471956472nteger @ X3 ) ) ).
% max_0_1(3)
thf(fact_2610_max__0__1_I3_J,axiom,
! [X3: num] :
( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X3 ) )
= ( numeral_numeral_real @ X3 ) ) ).
% max_0_1(3)
thf(fact_2611_max__0__1_I3_J,axiom,
! [X3: num] :
( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X3 ) )
= ( numeral_numeral_rat @ X3 ) ) ).
% max_0_1(3)
thf(fact_2612_max__0__1_I3_J,axiom,
! [X3: num] :
( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X3 ) )
= ( numeral_numeral_nat @ X3 ) ) ).
% max_0_1(3)
thf(fact_2613_max__0__1_I3_J,axiom,
! [X3: num] :
( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X3 ) )
= ( numeral_numeral_int @ X3 ) ) ).
% max_0_1(3)
thf(fact_2614_max__0__1_I4_J,axiom,
! [X3: num] :
( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X3 ) @ zero_z5237406670263579293d_enat )
= ( numera1916890842035813515d_enat @ X3 ) ) ).
% max_0_1(4)
thf(fact_2615_max__0__1_I4_J,axiom,
! [X3: num] :
( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ X3 ) @ zero_z3403309356797280102nteger )
= ( numera6620942414471956472nteger @ X3 ) ) ).
% max_0_1(4)
thf(fact_2616_max__0__1_I4_J,axiom,
! [X3: num] :
( ( ord_max_real @ ( numeral_numeral_real @ X3 ) @ zero_zero_real )
= ( numeral_numeral_real @ X3 ) ) ).
% max_0_1(4)
thf(fact_2617_max__0__1_I4_J,axiom,
! [X3: num] :
( ( ord_max_rat @ ( numeral_numeral_rat @ X3 ) @ zero_zero_rat )
= ( numeral_numeral_rat @ X3 ) ) ).
% max_0_1(4)
thf(fact_2618_max__0__1_I4_J,axiom,
! [X3: num] :
( ( ord_max_nat @ ( numeral_numeral_nat @ X3 ) @ zero_zero_nat )
= ( numeral_numeral_nat @ X3 ) ) ).
% max_0_1(4)
thf(fact_2619_max__0__1_I4_J,axiom,
! [X3: num] :
( ( ord_max_int @ ( numeral_numeral_int @ X3 ) @ zero_zero_int )
= ( numeral_numeral_int @ X3 ) ) ).
% max_0_1(4)
thf(fact_2620_max__0__1_I1_J,axiom,
( ( ord_max_real @ zero_zero_real @ one_one_real )
= one_one_real ) ).
% max_0_1(1)
thf(fact_2621_max__0__1_I1_J,axiom,
( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
= one_one_rat ) ).
% max_0_1(1)
thf(fact_2622_max__0__1_I1_J,axiom,
( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
= one_one_nat ) ).
% max_0_1(1)
thf(fact_2623_max__0__1_I1_J,axiom,
( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat )
= one_on7984719198319812577d_enat ) ).
% max_0_1(1)
thf(fact_2624_max__0__1_I1_J,axiom,
( ( ord_max_int @ zero_zero_int @ one_one_int )
= one_one_int ) ).
% max_0_1(1)
thf(fact_2625_max__0__1_I1_J,axiom,
( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer )
= one_one_Code_integer ) ).
% max_0_1(1)
thf(fact_2626_max__0__1_I2_J,axiom,
( ( ord_max_real @ one_one_real @ zero_zero_real )
= one_one_real ) ).
% max_0_1(2)
thf(fact_2627_max__0__1_I2_J,axiom,
( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
= one_one_rat ) ).
% max_0_1(2)
thf(fact_2628_max__0__1_I2_J,axiom,
( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
= one_one_nat ) ).
% max_0_1(2)
thf(fact_2629_max__0__1_I2_J,axiom,
( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat )
= one_on7984719198319812577d_enat ) ).
% max_0_1(2)
thf(fact_2630_max__0__1_I2_J,axiom,
( ( ord_max_int @ one_one_int @ zero_zero_int )
= one_one_int ) ).
% max_0_1(2)
thf(fact_2631_max__0__1_I2_J,axiom,
( ( ord_max_Code_integer @ one_one_Code_integer @ zero_z3403309356797280102nteger )
= one_one_Code_integer ) ).
% max_0_1(2)
thf(fact_2632_mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_2633_one__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M2 @ N ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_2634_mult__less__cancel2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_2635_nat__0__less__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_2636_nat__mult__less__cancel__disj,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_2637_mult__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( times_times_nat @ M2 @ ( suc @ N ) )
= ( plus_plus_nat @ M2 @ ( times_times_nat @ M2 @ N ) ) ) ).
% mult_Suc_right
thf(fact_2638_nat__mult__div__cancel__disj,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ( K2 = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= zero_zero_nat ) )
& ( ( K2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= ( divide_divide_nat @ M2 @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_2639_list__update__beyond,axiom,
! [Xs: list_VEBT_VEBT,I: nat,X3: vEBT_VEBT] :
( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ I )
=> ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 )
= Xs ) ) ).
% list_update_beyond
thf(fact_2640_list__update__beyond,axiom,
! [Xs: list_int,I: nat,X3: int] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ I )
=> ( ( list_update_int @ Xs @ I @ X3 )
= Xs ) ) ).
% list_update_beyond
thf(fact_2641_list__update__beyond,axiom,
! [Xs: list_nat,I: nat,X3: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
=> ( ( list_update_nat @ Xs @ I @ X3 )
= Xs ) ) ).
% list_update_beyond
thf(fact_2642_divide__le__eq__numeral1_I1_J,axiom,
! [B: real,W2: num,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) @ A )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_2643_divide__le__eq__numeral1_I1_J,axiom,
! [B: rat,W2: num,A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) @ A )
= ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_2644_le__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W2: num] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_2645_le__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W2: num] :
( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) @ B ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_2646_divide__eq__eq__numeral1_I1_J,axiom,
! [B: complex,W2: num,A: complex] :
( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W2 ) )
= A )
= ( ( ( ( numera6690914467698888265omplex @ W2 )
!= zero_zero_complex )
=> ( B
= ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W2 ) ) ) )
& ( ( ( numera6690914467698888265omplex @ W2 )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_2647_divide__eq__eq__numeral1_I1_J,axiom,
! [B: real,W2: num,A: real] :
( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) )
= A )
= ( ( ( ( numeral_numeral_real @ W2 )
!= zero_zero_real )
=> ( B
= ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) )
& ( ( ( numeral_numeral_real @ W2 )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_2648_divide__eq__eq__numeral1_I1_J,axiom,
! [B: rat,W2: num,A: rat] :
( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) )
= A )
= ( ( ( ( numeral_numeral_rat @ W2 )
!= zero_zero_rat )
=> ( B
= ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) )
& ( ( ( numeral_numeral_rat @ W2 )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_2649_eq__divide__eq__numeral1_I1_J,axiom,
! [A: complex,B: complex,W2: num] :
( ( A
= ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W2 ) ) )
= ( ( ( ( numera6690914467698888265omplex @ W2 )
!= zero_zero_complex )
=> ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W2 ) )
= B ) )
& ( ( ( numera6690914467698888265omplex @ W2 )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_2650_eq__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W2: num] :
( ( A
= ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
= ( ( ( ( numeral_numeral_real @ W2 )
!= zero_zero_real )
=> ( ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) )
= B ) )
& ( ( ( numeral_numeral_real @ W2 )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_2651_eq__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W2: num] :
( ( A
= ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
= ( ( ( ( numeral_numeral_rat @ W2 )
!= zero_zero_rat )
=> ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) )
= B ) )
& ( ( ( numeral_numeral_rat @ W2 )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_2652_nonzero__divide__mult__cancel__right,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
= ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_2653_nonzero__divide__mult__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_2654_nonzero__divide__mult__cancel__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
= ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_2655_nonzero__divide__mult__cancel__left,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
= ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_2656_nonzero__divide__mult__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_2657_nonzero__divide__mult__cancel__left,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
= ( divide_divide_rat @ one_one_rat @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_2658_div__mult__self4,axiom,
! [B: nat,C2: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C2 ) @ A ) @ B )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_2659_div__mult__self4,axiom,
! [B: int,C2: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C2 ) @ A ) @ B )
= ( plus_plus_int @ C2 @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_2660_div__mult__self3,axiom,
! [B: nat,C2: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C2 @ B ) @ A ) @ B )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_2661_div__mult__self3,axiom,
! [B: int,C2: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C2 @ B ) @ A ) @ B )
= ( plus_plus_int @ C2 @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_2662_div__mult__self2,axiom,
! [B: nat,A: nat,C2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C2 ) ) @ B )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_2663_div__mult__self2,axiom,
! [B: int,A: int,C2: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C2 ) ) @ B )
= ( plus_plus_int @ C2 @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_2664_div__mult__self1,axiom,
! [B: nat,A: nat,C2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C2 @ B ) ) @ B )
= ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_2665_div__mult__self1,axiom,
! [B: int,A: int,C2: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C2 @ B ) ) @ B )
= ( plus_plus_int @ C2 @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_2666_half__negative__int__iff,axiom,
! [K2: int] :
( ( ord_less_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( ord_less_int @ K2 @ zero_zero_int ) ) ).
% half_negative_int_iff
thf(fact_2667_one__le__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M2 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_2668_mult__le__cancel2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_2669_nat__mult__le__cancel__disj,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_2670_div__mult__self1__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M2 ) @ N )
= M2 ) ) ).
% div_mult_self1_is_m
thf(fact_2671_div__mult__self__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M2 @ N ) @ N )
= M2 ) ) ).
% div_mult_self_is_m
thf(fact_2672_power__add__numeral2,axiom,
! [A: complex,M2: num,N: num,B: complex] :
( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_2673_power__add__numeral2,axiom,
! [A: real,M2: num,N: num,B: real] :
( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_2674_power__add__numeral2,axiom,
! [A: rat,M2: num,N: num,B: rat] :
( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_2675_power__add__numeral2,axiom,
! [A: nat,M2: num,N: num,B: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_2676_power__add__numeral2,axiom,
! [A: int,M2: num,N: num,B: int] :
( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_2677_power__add__numeral,axiom,
! [A: complex,M2: num,N: num] :
( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).
% power_add_numeral
thf(fact_2678_power__add__numeral,axiom,
! [A: real,M2: num,N: num] :
( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).
% power_add_numeral
thf(fact_2679_power__add__numeral,axiom,
! [A: rat,M2: num,N: num] :
( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).
% power_add_numeral
thf(fact_2680_power__add__numeral,axiom,
! [A: nat,M2: num,N: num] :
( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).
% power_add_numeral
thf(fact_2681_power__add__numeral,axiom,
! [A: int,M2: num,N: num] :
( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).
% power_add_numeral
thf(fact_2682_Suc__mod__mult__self1,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M2 @ ( times_times_nat @ K2 @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self1
thf(fact_2683_Suc__mod__mult__self2,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M2 @ ( times_times_nat @ N @ K2 ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self2
thf(fact_2684_Suc__mod__mult__self3,axiom,
! [K2: nat,N: nat,M2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K2 @ N ) @ M2 ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self3
thf(fact_2685_Suc__mod__mult__self4,axiom,
! [N: nat,K2: nat,M2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K2 ) @ M2 ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self4
thf(fact_2686_nth__list__update__eq,axiom,
! [I: nat,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) @ I )
= X3 ) ) ).
% nth_list_update_eq
thf(fact_2687_nth__list__update__eq,axiom,
! [I: nat,Xs: list_int,X3: int] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( list_update_int @ Xs @ I @ X3 ) @ I )
= X3 ) ) ).
% nth_list_update_eq
thf(fact_2688_nth__list__update__eq,axiom,
! [I: nat,Xs: list_nat,X3: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X3 ) @ I )
= X3 ) ) ).
% nth_list_update_eq
thf(fact_2689_half__nonnegative__int__iff,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).
% half_nonnegative_int_iff
thf(fact_2690_Suc__times__numeral__mod__eq,axiom,
! [K2: num,N: nat] :
( ( ( numeral_numeral_nat @ K2 )
!= one_one_nat )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K2 ) @ N ) ) @ ( numeral_numeral_nat @ K2 ) )
= one_one_nat ) ) ).
% Suc_times_numeral_mod_eq
thf(fact_2691_set__swap,axiom,
! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( nth_VEBT_VEBT @ Xs @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs @ I ) ) )
= ( set_VEBT_VEBT2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_2692_set__swap,axiom,
! [I: nat,Xs: list_int,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
=> ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ J ) ) @ J @ ( nth_int @ Xs @ I ) ) )
= ( set_int2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_2693_set__swap,axiom,
! [I: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
= ( set_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_2694_zmod__le__nonneg__dividend,axiom,
! [M2: int,K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ M2 )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ M2 @ K2 ) @ M2 ) ) ).
% zmod_le_nonneg_dividend
thf(fact_2695_neg__mod__conj,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
& ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% neg_mod_conj
thf(fact_2696_pos__mod__conj,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
& ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).
% pos_mod_conj
thf(fact_2697_zmod__trivial__iff,axiom,
! [I: int,K2: int] :
( ( ( modulo_modulo_int @ I @ K2 )
= I )
= ( ( K2 = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K2 ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K2 @ I ) ) ) ) ).
% zmod_trivial_iff
thf(fact_2698_verit__le__mono__div__int,axiom,
! [A4: int,B5: int,N: int] :
( ( ord_less_int @ A4 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int
@ ( plus_plus_int @ ( divide_divide_int @ A4 @ N )
@ ( if_int
@ ( ( modulo_modulo_int @ B5 @ N )
= zero_zero_int )
@ one_one_int
@ zero_zero_int ) )
@ ( divide_divide_int @ B5 @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_2699_neg__mod__sign,axiom,
! [L: int,K2: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ K2 @ L ) @ zero_zero_int ) ) ).
% neg_mod_sign
thf(fact_2700_Euclidean__Division_Opos__mod__sign,axiom,
! [L: int,K2: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K2 @ L ) ) ) ).
% Euclidean_Division.pos_mod_sign
thf(fact_2701_mod__pos__neg__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K2 @ L ) @ zero_zero_int )
=> ( ( modulo_modulo_int @ K2 @ L )
= ( plus_plus_int @ K2 @ L ) ) ) ) ).
% mod_pos_neg_trivial
thf(fact_2702_zdiv__mono__strict,axiom,
! [A4: int,B5: int,N: int] :
( ( ord_less_int @ A4 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ( modulo_modulo_int @ A4 @ N )
= zero_zero_int )
=> ( ( ( modulo_modulo_int @ B5 @ N )
= zero_zero_int )
=> ( ord_less_int @ ( divide_divide_int @ A4 @ N ) @ ( divide_divide_int @ B5 @ N ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_2703_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_2704_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_2705_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_2706_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_2707_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_2708_pos__imp__zdiv__pos__iff,axiom,
! [K2: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K2 ) )
= ( ord_less_eq_int @ K2 @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_2709_div__nonpos__pos__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_2710_div__nonneg__neg__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_2711_div__positive__int,axiom,
! [L: int,K2: int] :
( ( ord_less_eq_int @ L @ K2 )
=> ( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K2 @ L ) ) ) ) ).
% div_positive_int
thf(fact_2712_div__int__pos__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K2 @ L ) )
= ( ( K2 = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K2 )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K2 @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_2713_zless__imp__add1__zle,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ Z2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_2714_zdiv__mono2__neg,axiom,
! [A: int,B2: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ B2 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_2715_zdiv__mono1__neg,axiom,
! [A: int,A2: int,B: int] :
( ( ord_less_eq_int @ A @ A2 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_2716_zdiv__eq__0__iff,axiom,
! [I: int,K2: int] :
( ( ( divide_divide_int @ I @ K2 )
= zero_zero_int )
= ( ( K2 = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K2 ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K2 @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_2717_zdiv__mono2,axiom,
! [A: int,B2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ B2 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_2718_zdiv__mono1,axiom,
! [A: int,A2: int,B: int] :
( ( ord_less_eq_int @ A @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A2 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_2719_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_2720_add1__zle__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% add1_zle_eq
thf(fact_2721_int__ge__induct,axiom,
! [K2: int,I: int,P2: int > $o] :
( ( ord_less_eq_int @ K2 @ I )
=> ( ( P2 @ K2 )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K2 @ I2 )
=> ( ( P2 @ I2 )
=> ( P2 @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_ge_induct
thf(fact_2722_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_2723_conj__le__cong,axiom,
! [X3: int,X9: int,P2: $o,P5: $o] :
( ( X3 = X9 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X9 )
=> ( P2 = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
& P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X9 )
& P5 ) ) ) ) ).
% conj_le_cong
thf(fact_2724_imp__le__cong,axiom,
! [X3: int,X9: int,P2: $o,P5: $o] :
( ( X3 = X9 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X9 )
=> ( P2 = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X9 )
=> P5 ) ) ) ) ).
% imp_le_cong
thf(fact_2725_verit__la__generic,axiom,
! [A: int,X3: int] :
( ( ord_less_eq_int @ A @ X3 )
| ( A = X3 )
| ( ord_less_eq_int @ X3 @ A ) ) ).
% verit_la_generic
thf(fact_2726_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_2727_i0__lb,axiom,
! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).
% i0_lb
thf(fact_2728_ile0__eq,axiom,
! [N: extended_enat] :
( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% ile0_eq
thf(fact_2729_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_2730_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_2731_int__div__less__self,axiom,
! [X3: int,K2: int] :
( ( ord_less_int @ zero_zero_int @ X3 )
=> ( ( ord_less_int @ one_one_int @ K2 )
=> ( ord_less_int @ ( divide_divide_int @ X3 @ K2 ) @ X3 ) ) ) ).
% int_div_less_self
thf(fact_2732_div__neg__pos__less0,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_2733_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_2734_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C2 )
= ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2735_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: rat,B: rat,C2: rat] :
( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C2 )
= ( times_times_rat @ A @ ( times_times_rat @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2736_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2737_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C2 )
= ( times_times_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2738_mult_Oassoc,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C2 )
= ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_2739_mult_Oassoc,axiom,
! [A: rat,B: rat,C2: rat] :
( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C2 )
= ( times_times_rat @ A @ ( times_times_rat @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_2740_mult_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_2741_mult_Oassoc,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C2 )
= ( times_times_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_2742_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A5: real,B4: real] : ( times_times_real @ B4 @ A5 ) ) ) ).
% mult.commute
thf(fact_2743_mult_Ocommute,axiom,
( times_times_rat
= ( ^ [A5: rat,B4: rat] : ( times_times_rat @ B4 @ A5 ) ) ) ).
% mult.commute
thf(fact_2744_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A5: nat,B4: nat] : ( times_times_nat @ B4 @ A5 ) ) ) ).
% mult.commute
thf(fact_2745_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A5: int,B4: int] : ( times_times_int @ B4 @ A5 ) ) ) ).
% mult.commute
thf(fact_2746_mult_Oleft__commute,axiom,
! [B: real,A: real,C2: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C2 ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_2747_mult_Oleft__commute,axiom,
! [B: rat,A: rat,C2: rat] :
( ( times_times_rat @ B @ ( times_times_rat @ A @ C2 ) )
= ( times_times_rat @ A @ ( times_times_rat @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_2748_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C2 ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_2749_mult_Oleft__commute,axiom,
! [B: int,A: int,C2: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C2 ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_2750_nat__mult__max__left,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( times_times_nat @ ( ord_max_nat @ M2 @ N ) @ Q3 )
= ( ord_max_nat @ ( times_times_nat @ M2 @ Q3 ) @ ( times_times_nat @ N @ Q3 ) ) ) ).
% nat_mult_max_left
thf(fact_2751_nat__mult__max__right,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( times_times_nat @ M2 @ ( ord_max_nat @ N @ Q3 ) )
= ( ord_max_nat @ ( times_times_nat @ M2 @ N ) @ ( times_times_nat @ M2 @ Q3 ) ) ) ).
% nat_mult_max_right
thf(fact_2752_list__update__swap,axiom,
! [I: nat,I5: nat,Xs: list_VEBT_VEBT,X3: vEBT_VEBT,X9: vEBT_VEBT] :
( ( I != I5 )
=> ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) @ I5 @ X9 )
= ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I5 @ X9 ) @ I @ X3 ) ) ) ).
% list_update_swap
thf(fact_2753_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_2754_plus__int__code_I1_J,axiom,
! [K2: int] :
( ( plus_plus_int @ K2 @ zero_zero_int )
= K2 ) ).
% plus_int_code(1)
thf(fact_2755_neg__mod__bound,axiom,
! [L: int,K2: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_int @ L @ ( modulo_modulo_int @ K2 @ L ) ) ) ).
% neg_mod_bound
thf(fact_2756_Euclidean__Division_Opos__mod__bound,axiom,
! [L: int,K2: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_int @ ( modulo_modulo_int @ K2 @ L ) @ L ) ) ).
% Euclidean_Division.pos_mod_bound
thf(fact_2757_zip__update,axiom,
! [Xs: list_VEBT_VEBT,I: nat,X3: vEBT_VEBT,Ys2: list_VEBT_VEBT,Y: vEBT_VEBT] :
( ( zip_VE537291747668921783T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) @ ( list_u1324408373059187874T_VEBT @ Ys2 @ I @ Y ) )
= ( list_u6961636818849549845T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys2 ) @ I @ ( produc537772716801021591T_VEBT @ X3 @ Y ) ) ) ).
% zip_update
thf(fact_2758_zip__update,axiom,
! [Xs: list_int,I: nat,X3: int,Ys2: list_int,Y: int] :
( ( zip_int_int @ ( list_update_int @ Xs @ I @ X3 ) @ ( list_update_int @ Ys2 @ I @ Y ) )
= ( list_u3002344382305578791nt_int @ ( zip_int_int @ Xs @ Ys2 ) @ I @ ( product_Pair_int_int @ X3 @ Y ) ) ) ).
% zip_update
thf(fact_2759_zip__update,axiom,
! [Xs: list_C878401137130745250e_term,I: nat,X3: code_integer > option6357759511663192854e_term,Ys2: list_P5578671422887162913nteger,Y: produc8923325533196201883nteger] :
( ( zip_Co8729459035503499408nteger @ ( list_u4743598893156345252e_term @ Xs @ I @ X3 ) @ ( list_u2254550707601501961nteger @ Ys2 @ I @ Y ) )
= ( list_u1133519416628930960nteger @ ( zip_Co8729459035503499408nteger @ Xs @ Ys2 ) @ I @ ( produc6137756002093451184nteger @ X3 @ Y ) ) ) ).
% zip_update
thf(fact_2760_zip__update,axiom,
! [Xs: list_P1316552470764441098e_term,I: nat,X3: produc6241069584506657477e_term > option6357759511663192854e_term,Ys2: list_P5578671422887162913nteger,Y: produc8923325533196201883nteger] :
( ( zip_Pr8292346330294042792nteger @ ( list_u877304756163299468e_term @ Xs @ I @ X3 ) @ ( list_u2254550707601501961nteger @ Ys2 @ I @ Y ) )
= ( list_u234853988314817064nteger @ ( zip_Pr8292346330294042792nteger @ Xs @ Ys2 ) @ I @ ( produc8603105652947943368nteger @ X3 @ Y ) ) ) ).
% zip_update
thf(fact_2761_zip__update,axiom,
! [Xs: list_P1743416141875011707e_term,I: nat,X3: produc8551481072490612790e_term > option6357759511663192854e_term,Ys2: list_P5707943133018811711nt_int,Y: product_prod_int_int] :
( ( zip_Pr4168994715204986005nt_int @ ( list_u3533491785856317309e_term @ Xs @ I @ X3 ) @ ( list_u3002344382305578791nt_int @ Ys2 @ I @ Y ) )
= ( list_u7736365598306452245nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys2 ) @ I @ ( produc5700946648718959541nt_int @ X3 @ Y ) ) ) ).
% zip_update
thf(fact_2762_zip__update,axiom,
! [Xs: list_i8448526496819171953e_term,I: nat,X3: int > option6357759511663192854e_term,Ys2: list_P5707943133018811711nt_int,Y: product_prod_int_int] :
( ( zip_in8766932505889695135nt_int @ ( list_u8946639151299769843e_term @ Xs @ I @ X3 ) @ ( list_u3002344382305578791nt_int @ Ys2 @ I @ Y ) )
= ( list_u4780935413889332127nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys2 ) @ I @ ( produc4305682042979456191nt_int @ X3 @ Y ) ) ) ).
% zip_update
thf(fact_2763_max__absorb2,axiom,
! [X3: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y )
=> ( ( ord_ma741700101516333627d_enat @ X3 @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_2764_max__absorb2,axiom,
! [X3: code_integer,Y: code_integer] :
( ( ord_le3102999989581377725nteger @ X3 @ Y )
=> ( ( ord_max_Code_integer @ X3 @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_2765_max__absorb2,axiom,
! [X3: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y )
=> ( ( ord_max_set_int @ X3 @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_2766_max__absorb2,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ X3 @ Y )
=> ( ( ord_max_rat @ X3 @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_2767_max__absorb2,axiom,
! [X3: num,Y: num] :
( ( ord_less_eq_num @ X3 @ Y )
=> ( ( ord_max_num @ X3 @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_2768_max__absorb2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_max_nat @ X3 @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_2769_max__absorb2,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_max_int @ X3 @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_2770_max__absorb1,axiom,
! [Y: extended_enat,X3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Y @ X3 )
=> ( ( ord_ma741700101516333627d_enat @ X3 @ Y )
= X3 ) ) ).
% max_absorb1
thf(fact_2771_max__absorb1,axiom,
! [Y: code_integer,X3: code_integer] :
( ( ord_le3102999989581377725nteger @ Y @ X3 )
=> ( ( ord_max_Code_integer @ X3 @ Y )
= X3 ) ) ).
% max_absorb1
thf(fact_2772_max__absorb1,axiom,
! [Y: set_int,X3: set_int] :
( ( ord_less_eq_set_int @ Y @ X3 )
=> ( ( ord_max_set_int @ X3 @ Y )
= X3 ) ) ).
% max_absorb1
thf(fact_2773_max__absorb1,axiom,
! [Y: rat,X3: rat] :
( ( ord_less_eq_rat @ Y @ X3 )
=> ( ( ord_max_rat @ X3 @ Y )
= X3 ) ) ).
% max_absorb1
thf(fact_2774_max__absorb1,axiom,
! [Y: num,X3: num] :
( ( ord_less_eq_num @ Y @ X3 )
=> ( ( ord_max_num @ X3 @ Y )
= X3 ) ) ).
% max_absorb1
thf(fact_2775_max__absorb1,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_max_nat @ X3 @ Y )
= X3 ) ) ).
% max_absorb1
thf(fact_2776_max__absorb1,axiom,
! [Y: int,X3: int] :
( ( ord_less_eq_int @ Y @ X3 )
=> ( ( ord_max_int @ X3 @ Y )
= X3 ) ) ).
% max_absorb1
thf(fact_2777_max__def,axiom,
( ord_ma741700101516333627d_enat
= ( ^ [A5: extended_enat,B4: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def
thf(fact_2778_max__def,axiom,
( ord_max_Code_integer
= ( ^ [A5: code_integer,B4: code_integer] : ( if_Code_integer @ ( ord_le3102999989581377725nteger @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def
thf(fact_2779_max__def,axiom,
( ord_max_set_int
= ( ^ [A5: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def
thf(fact_2780_max__def,axiom,
( ord_max_rat
= ( ^ [A5: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def
thf(fact_2781_max__def,axiom,
( ord_max_num
= ( ^ [A5: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def
thf(fact_2782_max__def,axiom,
( ord_max_nat
= ( ^ [A5: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def
thf(fact_2783_max__def,axiom,
( ord_max_int
= ( ^ [A5: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def
thf(fact_2784_max__add__distrib__right,axiom,
! [X3: real,Y: real,Z2: real] :
( ( plus_plus_real @ X3 @ ( ord_max_real @ Y @ Z2 ) )
= ( ord_max_real @ ( plus_plus_real @ X3 @ Y ) @ ( plus_plus_real @ X3 @ Z2 ) ) ) ).
% max_add_distrib_right
thf(fact_2785_max__add__distrib__right,axiom,
! [X3: rat,Y: rat,Z2: rat] :
( ( plus_plus_rat @ X3 @ ( ord_max_rat @ Y @ Z2 ) )
= ( ord_max_rat @ ( plus_plus_rat @ X3 @ Y ) @ ( plus_plus_rat @ X3 @ Z2 ) ) ) ).
% max_add_distrib_right
thf(fact_2786_max__add__distrib__right,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( plus_plus_nat @ X3 @ ( ord_max_nat @ Y @ Z2 ) )
= ( ord_max_nat @ ( plus_plus_nat @ X3 @ Y ) @ ( plus_plus_nat @ X3 @ Z2 ) ) ) ).
% max_add_distrib_right
thf(fact_2787_max__add__distrib__right,axiom,
! [X3: int,Y: int,Z2: int] :
( ( plus_plus_int @ X3 @ ( ord_max_int @ Y @ Z2 ) )
= ( ord_max_int @ ( plus_plus_int @ X3 @ Y ) @ ( plus_plus_int @ X3 @ Z2 ) ) ) ).
% max_add_distrib_right
thf(fact_2788_max__add__distrib__right,axiom,
! [X3: code_integer,Y: code_integer,Z2: code_integer] :
( ( plus_p5714425477246183910nteger @ X3 @ ( ord_max_Code_integer @ Y @ Z2 ) )
= ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X3 @ Y ) @ ( plus_p5714425477246183910nteger @ X3 @ Z2 ) ) ) ).
% max_add_distrib_right
thf(fact_2789_max__add__distrib__left,axiom,
! [X3: real,Y: real,Z2: real] :
( ( plus_plus_real @ ( ord_max_real @ X3 @ Y ) @ Z2 )
= ( ord_max_real @ ( plus_plus_real @ X3 @ Z2 ) @ ( plus_plus_real @ Y @ Z2 ) ) ) ).
% max_add_distrib_left
thf(fact_2790_max__add__distrib__left,axiom,
! [X3: rat,Y: rat,Z2: rat] :
( ( plus_plus_rat @ ( ord_max_rat @ X3 @ Y ) @ Z2 )
= ( ord_max_rat @ ( plus_plus_rat @ X3 @ Z2 ) @ ( plus_plus_rat @ Y @ Z2 ) ) ) ).
% max_add_distrib_left
thf(fact_2791_max__add__distrib__left,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ X3 @ Y ) @ Z2 )
= ( ord_max_nat @ ( plus_plus_nat @ X3 @ Z2 ) @ ( plus_plus_nat @ Y @ Z2 ) ) ) ).
% max_add_distrib_left
thf(fact_2792_max__add__distrib__left,axiom,
! [X3: int,Y: int,Z2: int] :
( ( plus_plus_int @ ( ord_max_int @ X3 @ Y ) @ Z2 )
= ( ord_max_int @ ( plus_plus_int @ X3 @ Z2 ) @ ( plus_plus_int @ Y @ Z2 ) ) ) ).
% max_add_distrib_left
thf(fact_2793_max__add__distrib__left,axiom,
! [X3: code_integer,Y: code_integer,Z2: code_integer] :
( ( plus_p5714425477246183910nteger @ ( ord_max_Code_integer @ X3 @ Y ) @ Z2 )
= ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X3 @ Z2 ) @ ( plus_p5714425477246183910nteger @ Y @ Z2 ) ) ) ).
% max_add_distrib_left
thf(fact_2794_nat__add__max__left,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ M2 @ N ) @ Q3 )
= ( ord_max_nat @ ( plus_plus_nat @ M2 @ Q3 ) @ ( plus_plus_nat @ N @ Q3 ) ) ) ).
% nat_add_max_left
thf(fact_2795_nat__add__max__right,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( plus_plus_nat @ M2 @ ( ord_max_nat @ N @ Q3 ) )
= ( ord_max_nat @ ( plus_plus_nat @ M2 @ N ) @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ).
% nat_add_max_right
thf(fact_2796_mult__right__cancel,axiom,
! [C2: complex,A: complex,B: complex] :
( ( C2 != zero_zero_complex )
=> ( ( ( times_times_complex @ A @ C2 )
= ( times_times_complex @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_2797_mult__right__cancel,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A @ C2 )
= ( times_times_real @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_2798_mult__right__cancel,axiom,
! [C2: rat,A: rat,B: rat] :
( ( C2 != zero_zero_rat )
=> ( ( ( times_times_rat @ A @ C2 )
= ( times_times_rat @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_2799_mult__right__cancel,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_2800_mult__right__cancel,axiom,
! [C2: int,A: int,B: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ A @ C2 )
= ( times_times_int @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_2801_mult__left__cancel,axiom,
! [C2: complex,A: complex,B: complex] :
( ( C2 != zero_zero_complex )
=> ( ( ( times_times_complex @ C2 @ A )
= ( times_times_complex @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_2802_mult__left__cancel,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ C2 @ A )
= ( times_times_real @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_2803_mult__left__cancel,axiom,
! [C2: rat,A: rat,B: rat] :
( ( C2 != zero_zero_rat )
=> ( ( ( times_times_rat @ C2 @ A )
= ( times_times_rat @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_2804_mult__left__cancel,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_2805_mult__left__cancel,axiom,
! [C2: int,A: int,B: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ C2 @ A )
= ( times_times_int @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_2806_no__zero__divisors,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( ( times_times_complex @ A @ B )
!= zero_zero_complex ) ) ) ).
% no_zero_divisors
thf(fact_2807_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_2808_no__zero__divisors,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( ( times_times_rat @ A @ B )
!= zero_zero_rat ) ) ) ).
% no_zero_divisors
thf(fact_2809_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_2810_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_2811_divisors__zero,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
= zero_zero_complex )
=> ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% divisors_zero
thf(fact_2812_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_2813_divisors__zero,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
= zero_zero_rat )
=> ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% divisors_zero
thf(fact_2814_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_2815_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_2816_mult__not__zero,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
!= zero_zero_complex )
=> ( ( A != zero_zero_complex )
& ( B != zero_zero_complex ) ) ) ).
% mult_not_zero
thf(fact_2817_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_2818_mult__not__zero,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
!= zero_zero_rat )
=> ( ( A != zero_zero_rat )
& ( B != zero_zero_rat ) ) ) ).
% mult_not_zero
thf(fact_2819_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_2820_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_2821_comm__monoid__mult__class_Omult__1,axiom,
! [A: complex] :
( ( times_times_complex @ one_one_complex @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_2822_comm__monoid__mult__class_Omult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_2823_comm__monoid__mult__class_Omult__1,axiom,
! [A: rat] :
( ( times_times_rat @ one_one_rat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_2824_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_2825_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_2826_mult_Ocomm__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.comm_neutral
thf(fact_2827_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_2828_mult_Ocomm__neutral,axiom,
! [A: rat] :
( ( times_times_rat @ A @ one_one_rat )
= A ) ).
% mult.comm_neutral
thf(fact_2829_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_2830_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_2831_combine__common__factor,axiom,
! [A: real,E2: real,B: real,C2: real] :
( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E2 ) @ C2 ) ) ).
% combine_common_factor
thf(fact_2832_combine__common__factor,axiom,
! [A: rat,E2: rat,B: rat,C2: rat] :
( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ C2 ) )
= ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E2 ) @ C2 ) ) ).
% combine_common_factor
thf(fact_2833_combine__common__factor,axiom,
! [A: nat,E2: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C2 ) ) ).
% combine_common_factor
thf(fact_2834_combine__common__factor,axiom,
! [A: int,E2: int,B: int,C2: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E2 ) @ C2 ) ) ).
% combine_common_factor
thf(fact_2835_distrib__right,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_2836_distrib__right,axiom,
! [A: rat,B: rat,C2: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
= ( plus_plus_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_2837_distrib__right,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_2838_distrib__right,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_2839_distrib__left,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) ) ) ).
% distrib_left
thf(fact_2840_distrib__left,axiom,
! [A: rat,B: rat,C2: rat] :
( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C2 ) )
= ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ).
% distrib_left
thf(fact_2841_distrib__left,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C2 ) ) ) ).
% distrib_left
thf(fact_2842_distrib__left,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ).
% distrib_left
thf(fact_2843_comm__semiring__class_Odistrib,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_2844_comm__semiring__class_Odistrib,axiom,
! [A: rat,B: rat,C2: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
= ( plus_plus_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_2845_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_2846_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_2847_ring__class_Oring__distribs_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_2848_ring__class_Oring__distribs_I1_J,axiom,
! [A: rat,B: rat,C2: rat] :
( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C2 ) )
= ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_2849_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_2850_ring__class_Oring__distribs_I2_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_2851_ring__class_Oring__distribs_I2_J,axiom,
! [A: rat,B: rat,C2: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
= ( plus_plus_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_2852_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_2853_crossproduct__noteq,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ( A != B )
& ( C2 != D ) )
= ( ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) )
!= ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_2854_crossproduct__noteq,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ( A != B )
& ( C2 != D ) )
= ( ( plus_plus_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) )
!= ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_2855_crossproduct__noteq,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ( A != B )
& ( C2 != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_2856_crossproduct__noteq,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ( A != B )
& ( C2 != D ) )
= ( ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) )
!= ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_2857_crossproduct__eq,axiom,
! [W2: real,Y: real,X3: real,Z2: real] :
( ( ( plus_plus_real @ ( times_times_real @ W2 @ Y ) @ ( times_times_real @ X3 @ Z2 ) )
= ( plus_plus_real @ ( times_times_real @ W2 @ Z2 ) @ ( times_times_real @ X3 @ Y ) ) )
= ( ( W2 = X3 )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_2858_crossproduct__eq,axiom,
! [W2: rat,Y: rat,X3: rat,Z2: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ W2 @ Y ) @ ( times_times_rat @ X3 @ Z2 ) )
= ( plus_plus_rat @ ( times_times_rat @ W2 @ Z2 ) @ ( times_times_rat @ X3 @ Y ) ) )
= ( ( W2 = X3 )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_2859_crossproduct__eq,axiom,
! [W2: nat,Y: nat,X3: nat,Z2: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W2 @ Y ) @ ( times_times_nat @ X3 @ Z2 ) )
= ( plus_plus_nat @ ( times_times_nat @ W2 @ Z2 ) @ ( times_times_nat @ X3 @ Y ) ) )
= ( ( W2 = X3 )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_2860_crossproduct__eq,axiom,
! [W2: int,Y: int,X3: int,Z2: int] :
( ( ( plus_plus_int @ ( times_times_int @ W2 @ Y ) @ ( times_times_int @ X3 @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W2 @ Z2 ) @ ( times_times_int @ X3 @ Y ) ) )
= ( ( W2 = X3 )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_2861_Suc__mult__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K2 ) @ M2 )
= ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( M2 = N ) ) ).
% Suc_mult_cancel1
thf(fact_2862_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_2863_nat__mult__eq__cancel__disj,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M2 )
= ( times_times_nat @ K2 @ N ) )
= ( ( K2 = zero_zero_nat )
| ( M2 = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_2864_mult__le__mono2,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J ) ) ) ).
% mult_le_mono2
thf(fact_2865_mult__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_2866_mult__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_2867_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_2868_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_2869_add__mult__distrib,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M2 @ N ) @ K2 )
= ( plus_plus_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).
% add_mult_distrib
thf(fact_2870_add__mult__distrib2,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( times_times_nat @ K2 @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) ) ) ).
% add_mult_distrib2
thf(fact_2871_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K2 ) ) ).
% left_add_mult_distrib
thf(fact_2872_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_2873_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_2874_mult__mono,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_2875_mult__mono,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C2 @ D )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_2876_mult__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_2877_mult__mono,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_2878_mult__mono_H,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_2879_mult__mono_H,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C2 @ D )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_2880_mult__mono_H,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_2881_mult__mono_H,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_2882_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_2883_zero__le__square,axiom,
! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% zero_le_square
thf(fact_2884_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_2885_split__mult__pos__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_2886_split__mult__pos__le,axiom,
! [A: rat,B: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_2887_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_2888_mult__left__mono__neg,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_2889_mult__left__mono__neg,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_2890_mult__left__mono__neg,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_2891_mult__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_2892_mult__nonpos__nonpos,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_2893_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_2894_mult__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% mult_left_mono
thf(fact_2895_mult__left__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).
% mult_left_mono
thf(fact_2896_mult__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% mult_left_mono
thf(fact_2897_mult__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% mult_left_mono
thf(fact_2898_mult__right__mono__neg,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_2899_mult__right__mono__neg,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_2900_mult__right__mono__neg,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_2901_mult__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_2902_mult__right__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_2903_mult__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_2904_mult__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_2905_mult__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_2906_mult__le__0__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_2907_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_2908_split__mult__neg__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_2909_split__mult__neg__le,axiom,
! [A: rat,B: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).
% split_mult_neg_le
thf(fact_2910_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_2911_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_2912_mult__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_2913_mult__nonneg__nonneg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_2914_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_2915_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_2916_mult__nonneg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_2917_mult__nonneg__nonpos,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_2918_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_2919_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_2920_mult__nonpos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_2921_mult__nonpos__nonneg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_2922_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_2923_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_2924_mult__nonneg__nonpos2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_2925_mult__nonneg__nonpos2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_2926_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_2927_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_2928_zero__le__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_2929_zero__le__mult__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% zero_le_mult_iff
thf(fact_2930_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_2931_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2932_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2933_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2934_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2935_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2936_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2937_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2938_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2939_mult__less__cancel__right__disj,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_2940_mult__less__cancel__right__disj,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
& ( ord_less_rat @ A @ B ) )
| ( ( ord_less_rat @ C2 @ zero_zero_rat )
& ( ord_less_rat @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_2941_mult__less__cancel__right__disj,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C2 @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_2942_mult__strict__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_2943_mult__strict__right__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_2944_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_2945_mult__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_2946_mult__strict__right__mono__neg,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_2947_mult__strict__right__mono__neg,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_2948_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_2949_mult__less__cancel__left__disj,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_2950_mult__less__cancel__left__disj,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
& ( ord_less_rat @ A @ B ) )
| ( ( ord_less_rat @ C2 @ zero_zero_rat )
& ( ord_less_rat @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_2951_mult__less__cancel__left__disj,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C2 @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_2952_mult__strict__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_2953_mult__strict__left__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_2954_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_2955_mult__strict__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_2956_mult__strict__left__mono__neg,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_2957_mult__strict__left__mono__neg,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_2958_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_2959_mult__less__cancel__left__pos,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ord_less_real @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_2960_mult__less__cancel__left__pos,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
= ( ord_less_rat @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_2961_mult__less__cancel__left__pos,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_2962_mult__less__cancel__left__neg,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ord_less_real @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_2963_mult__less__cancel__left__neg,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
= ( ord_less_rat @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_2964_mult__less__cancel__left__neg,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_2965_zero__less__mult__pos2,axiom,
! [B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_2966_zero__less__mult__pos2,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_2967_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_2968_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_2969_zero__less__mult__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_2970_zero__less__mult__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_2971_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_2972_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_2973_zero__less__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_2974_zero__less__mult__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% zero_less_mult_iff
thf(fact_2975_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_2976_mult__pos__neg2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_2977_mult__pos__neg2,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% mult_pos_neg2
thf(fact_2978_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_2979_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_2980_mult__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_2981_mult__pos__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_2982_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_2983_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_2984_mult__pos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_2985_mult__pos__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% mult_pos_neg
thf(fact_2986_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_2987_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_2988_mult__neg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_2989_mult__neg__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% mult_neg_pos
thf(fact_2990_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_2991_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_2992_mult__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_2993_mult__less__0__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_2994_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_2995_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_2996_not__square__less__zero,axiom,
! [A: rat] :
~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% not_square_less_zero
thf(fact_2997_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_2998_mult__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_2999_mult__neg__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_3000_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_3001_add__scale__eq__noteq,axiom,
! [R2: complex,A: complex,B: complex,C2: complex,D: complex] :
( ( R2 != zero_zero_complex )
=> ( ( ( A = B )
& ( C2 != D ) )
=> ( ( plus_plus_complex @ A @ ( times_times_complex @ R2 @ C2 ) )
!= ( plus_plus_complex @ B @ ( times_times_complex @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_3002_add__scale__eq__noteq,axiom,
! [R2: real,A: real,B: real,C2: real,D: real] :
( ( R2 != zero_zero_real )
=> ( ( ( A = B )
& ( C2 != D ) )
=> ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C2 ) )
!= ( plus_plus_real @ B @ ( times_times_real @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_3003_add__scale__eq__noteq,axiom,
! [R2: rat,A: rat,B: rat,C2: rat,D: rat] :
( ( R2 != zero_zero_rat )
=> ( ( ( A = B )
& ( C2 != D ) )
=> ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C2 ) )
!= ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_3004_add__scale__eq__noteq,axiom,
! [R2: nat,A: nat,B: nat,C2: nat,D: nat] :
( ( R2 != zero_zero_nat )
=> ( ( ( A = B )
& ( C2 != D ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C2 ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_3005_add__scale__eq__noteq,axiom,
! [R2: int,A: int,B: int,C2: int,D: int] :
( ( R2 != zero_zero_int )
=> ( ( ( A = B )
& ( C2 != D ) )
=> ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C2 ) )
!= ( plus_plus_int @ B @ ( times_times_int @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_3006_frac__eq__eq,axiom,
! [Y: complex,Z2: complex,X3: complex,W2: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z2 != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ X3 @ Y )
= ( divide1717551699836669952omplex @ W2 @ Z2 ) )
= ( ( times_times_complex @ X3 @ Z2 )
= ( times_times_complex @ W2 @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_3007_frac__eq__eq,axiom,
! [Y: real,Z2: real,X3: real,W2: real] :
( ( Y != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( ( divide_divide_real @ X3 @ Y )
= ( divide_divide_real @ W2 @ Z2 ) )
= ( ( times_times_real @ X3 @ Z2 )
= ( times_times_real @ W2 @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_3008_frac__eq__eq,axiom,
! [Y: rat,Z2: rat,X3: rat,W2: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z2 != zero_zero_rat )
=> ( ( ( divide_divide_rat @ X3 @ Y )
= ( divide_divide_rat @ W2 @ Z2 ) )
= ( ( times_times_rat @ X3 @ Z2 )
= ( times_times_rat @ W2 @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_3009_divide__eq__eq,axiom,
! [B: complex,C2: complex,A: complex] :
( ( ( divide1717551699836669952omplex @ B @ C2 )
= A )
= ( ( ( C2 != zero_zero_complex )
=> ( B
= ( times_times_complex @ A @ C2 ) ) )
& ( ( C2 = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% divide_eq_eq
thf(fact_3010_divide__eq__eq,axiom,
! [B: real,C2: real,A: real] :
( ( ( divide_divide_real @ B @ C2 )
= A )
= ( ( ( C2 != zero_zero_real )
=> ( B
= ( times_times_real @ A @ C2 ) ) )
& ( ( C2 = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_3011_divide__eq__eq,axiom,
! [B: rat,C2: rat,A: rat] :
( ( ( divide_divide_rat @ B @ C2 )
= A )
= ( ( ( C2 != zero_zero_rat )
=> ( B
= ( times_times_rat @ A @ C2 ) ) )
& ( ( C2 = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq
thf(fact_3012_eq__divide__eq,axiom,
! [A: complex,B: complex,C2: complex] :
( ( A
= ( divide1717551699836669952omplex @ B @ C2 ) )
= ( ( ( C2 != zero_zero_complex )
=> ( ( times_times_complex @ A @ C2 )
= B ) )
& ( ( C2 = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_divide_eq
thf(fact_3013_eq__divide__eq,axiom,
! [A: real,B: real,C2: real] :
( ( A
= ( divide_divide_real @ B @ C2 ) )
= ( ( ( C2 != zero_zero_real )
=> ( ( times_times_real @ A @ C2 )
= B ) )
& ( ( C2 = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_3014_eq__divide__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( A
= ( divide_divide_rat @ B @ C2 ) )
= ( ( ( C2 != zero_zero_rat )
=> ( ( times_times_rat @ A @ C2 )
= B ) )
& ( ( C2 = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq
thf(fact_3015_divide__eq__imp,axiom,
! [C2: complex,B: complex,A: complex] :
( ( C2 != zero_zero_complex )
=> ( ( B
= ( times_times_complex @ A @ C2 ) )
=> ( ( divide1717551699836669952omplex @ B @ C2 )
= A ) ) ) ).
% divide_eq_imp
thf(fact_3016_divide__eq__imp,axiom,
! [C2: real,B: real,A: real] :
( ( C2 != zero_zero_real )
=> ( ( B
= ( times_times_real @ A @ C2 ) )
=> ( ( divide_divide_real @ B @ C2 )
= A ) ) ) ).
% divide_eq_imp
thf(fact_3017_divide__eq__imp,axiom,
! [C2: rat,B: rat,A: rat] :
( ( C2 != zero_zero_rat )
=> ( ( B
= ( times_times_rat @ A @ C2 ) )
=> ( ( divide_divide_rat @ B @ C2 )
= A ) ) ) ).
% divide_eq_imp
thf(fact_3018_eq__divide__imp,axiom,
! [C2: complex,A: complex,B: complex] :
( ( C2 != zero_zero_complex )
=> ( ( ( times_times_complex @ A @ C2 )
= B )
=> ( A
= ( divide1717551699836669952omplex @ B @ C2 ) ) ) ) ).
% eq_divide_imp
thf(fact_3019_eq__divide__imp,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A @ C2 )
= B )
=> ( A
= ( divide_divide_real @ B @ C2 ) ) ) ) ).
% eq_divide_imp
thf(fact_3020_eq__divide__imp,axiom,
! [C2: rat,A: rat,B: rat] :
( ( C2 != zero_zero_rat )
=> ( ( ( times_times_rat @ A @ C2 )
= B )
=> ( A
= ( divide_divide_rat @ B @ C2 ) ) ) ) ).
% eq_divide_imp
thf(fact_3021_nonzero__divide__eq__eq,axiom,
! [C2: complex,B: complex,A: complex] :
( ( C2 != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ B @ C2 )
= A )
= ( B
= ( times_times_complex @ A @ C2 ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_3022_nonzero__divide__eq__eq,axiom,
! [C2: real,B: real,A: real] :
( ( C2 != zero_zero_real )
=> ( ( ( divide_divide_real @ B @ C2 )
= A )
= ( B
= ( times_times_real @ A @ C2 ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_3023_nonzero__divide__eq__eq,axiom,
! [C2: rat,B: rat,A: rat] :
( ( C2 != zero_zero_rat )
=> ( ( ( divide_divide_rat @ B @ C2 )
= A )
= ( B
= ( times_times_rat @ A @ C2 ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_3024_nonzero__eq__divide__eq,axiom,
! [C2: complex,A: complex,B: complex] :
( ( C2 != zero_zero_complex )
=> ( ( A
= ( divide1717551699836669952omplex @ B @ C2 ) )
= ( ( times_times_complex @ A @ C2 )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_3025_nonzero__eq__divide__eq,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( A
= ( divide_divide_real @ B @ C2 ) )
= ( ( times_times_real @ A @ C2 )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_3026_nonzero__eq__divide__eq,axiom,
! [C2: rat,A: rat,B: rat] :
( ( C2 != zero_zero_rat )
=> ( ( A
= ( divide_divide_rat @ B @ C2 ) )
= ( ( times_times_rat @ A @ C2 )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_3027_list__update__code_I3_J,axiom,
! [X3: int,Xs: list_int,I: nat,Y: int] :
( ( list_update_int @ ( cons_int @ X3 @ Xs ) @ ( suc @ I ) @ Y )
= ( cons_int @ X3 @ ( list_update_int @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_3028_list__update__code_I3_J,axiom,
! [X3: nat,Xs: list_nat,I: nat,Y: nat] :
( ( list_update_nat @ ( cons_nat @ X3 @ Xs ) @ ( suc @ I ) @ Y )
= ( cons_nat @ X3 @ ( list_update_nat @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_3029_list__update__code_I3_J,axiom,
! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT,I: nat,Y: vEBT_VEBT] :
( ( list_u1324408373059187874T_VEBT @ ( cons_VEBT_VEBT @ X3 @ Xs ) @ ( suc @ I ) @ Y )
= ( cons_VEBT_VEBT @ X3 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_3030_list__update__code_I2_J,axiom,
! [X3: int,Xs: list_int,Y: int] :
( ( list_update_int @ ( cons_int @ X3 @ Xs ) @ zero_zero_nat @ Y )
= ( cons_int @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_3031_list__update__code_I2_J,axiom,
! [X3: nat,Xs: list_nat,Y: nat] :
( ( list_update_nat @ ( cons_nat @ X3 @ Xs ) @ zero_zero_nat @ Y )
= ( cons_nat @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_3032_list__update__code_I2_J,axiom,
! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT,Y: vEBT_VEBT] :
( ( list_u1324408373059187874T_VEBT @ ( cons_VEBT_VEBT @ X3 @ Xs ) @ zero_zero_nat @ Y )
= ( cons_VEBT_VEBT @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_3033_power__Suc,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ A @ ( suc @ N ) )
= ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% power_Suc
thf(fact_3034_power__Suc,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( suc @ N ) )
= ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% power_Suc
thf(fact_3035_power__Suc,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ A @ ( suc @ N ) )
= ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% power_Suc
thf(fact_3036_power__Suc,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_Suc
thf(fact_3037_power__Suc,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ N ) )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_Suc
thf(fact_3038_power__Suc2,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ A @ ( suc @ N ) )
= ( times_times_complex @ ( power_power_complex @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_3039_power__Suc2,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( suc @ N ) )
= ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_3040_power__Suc2,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ A @ ( suc @ N ) )
= ( times_times_rat @ ( power_power_rat @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_3041_power__Suc2,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_3042_power__Suc2,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ N ) )
= ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_3043_set__update__subsetI,axiom,
! [Xs: list_complex,A4: set_complex,X3: complex,I: nat] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A4 )
=> ( ( member_complex @ X3 @ A4 )
=> ( ord_le211207098394363844omplex @ ( set_complex2 @ ( list_update_complex @ Xs @ I @ X3 ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_3044_set__update__subsetI,axiom,
! [Xs: list_real,A4: set_real,X3: real,I: nat] :
( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A4 )
=> ( ( member_real @ X3 @ A4 )
=> ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X3 ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_3045_set__update__subsetI,axiom,
! [Xs: list_set_nat,A4: set_set_nat,X3: set_nat,I: nat] :
( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ A4 )
=> ( ( member_set_nat @ X3 @ A4 )
=> ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ I @ X3 ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_3046_set__update__subsetI,axiom,
! [Xs: list_nat,A4: set_nat,X3: nat,I: nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A4 )
=> ( ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X3 ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_3047_set__update__subsetI,axiom,
! [Xs: list_VEBT_VEBT,A4: set_VEBT_VEBT,X3: vEBT_VEBT,I: nat] :
( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A4 )
=> ( ( member_VEBT_VEBT @ X3 @ A4 )
=> ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_3048_set__update__subsetI,axiom,
! [Xs: list_int,A4: set_int,X3: int,I: nat] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A4 )
=> ( ( member_int @ X3 @ A4 )
=> ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X3 ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_3049_mod__eqE,axiom,
! [A: int,C2: int,B: int] :
( ( ( modulo_modulo_int @ A @ C2 )
= ( modulo_modulo_int @ B @ C2 ) )
=> ~ ! [D2: int] :
( B
!= ( plus_plus_int @ A @ ( times_times_int @ C2 @ D2 ) ) ) ) ).
% mod_eqE
thf(fact_3050_mod__eqE,axiom,
! [A: code_integer,C2: code_integer,B: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ C2 )
= ( modulo364778990260209775nteger @ B @ C2 ) )
=> ~ ! [D2: code_integer] :
( B
!= ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C2 @ D2 ) ) ) ) ).
% mod_eqE
thf(fact_3051_Suc__mult__less__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K2 ) @ M2 ) @ ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_3052_power__add,axiom,
! [A: complex,M2: nat,N: nat] :
( ( power_power_complex @ A @ ( plus_plus_nat @ M2 @ N ) )
= ( times_times_complex @ ( power_power_complex @ A @ M2 ) @ ( power_power_complex @ A @ N ) ) ) ).
% power_add
thf(fact_3053_power__add,axiom,
! [A: real,M2: nat,N: nat] :
( ( power_power_real @ A @ ( plus_plus_nat @ M2 @ N ) )
= ( times_times_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N ) ) ) ).
% power_add
thf(fact_3054_power__add,axiom,
! [A: rat,M2: nat,N: nat] :
( ( power_power_rat @ A @ ( plus_plus_nat @ M2 @ N ) )
= ( times_times_rat @ ( power_power_rat @ A @ M2 ) @ ( power_power_rat @ A @ N ) ) ) ).
% power_add
thf(fact_3055_power__add,axiom,
! [A: nat,M2: nat,N: nat] :
( ( power_power_nat @ A @ ( plus_plus_nat @ M2 @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) ) ) ).
% power_add
thf(fact_3056_power__add,axiom,
! [A: int,M2: nat,N: nat] :
( ( power_power_int @ A @ ( plus_plus_nat @ M2 @ N ) )
= ( times_times_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) ) ) ).
% power_add
thf(fact_3057_mult__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ) ).
% mult_less_mono1
thf(fact_3058_mult__less__mono2,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_3059_nat__mult__less__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_3060_nat__mult__eq__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ( times_times_nat @ K2 @ M2 )
= ( times_times_nat @ K2 @ N ) )
= ( M2 = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_3061_Suc__mult__le__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K2 ) @ M2 ) @ ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_3062_mult__Suc,axiom,
! [M2: nat,N: nat] :
( ( times_times_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M2 @ N ) ) ) ).
% mult_Suc
thf(fact_3063_mult__eq__self__implies__10,axiom,
! [M2: nat,N: nat] :
( ( M2
= ( times_times_nat @ M2 @ N ) )
=> ( ( N = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_3064_div__times__less__eq__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N ) @ N ) @ M2 ) ).
% div_times_less_eq_dividend
thf(fact_3065_times__div__less__eq__dividend,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M2 @ N ) ) @ M2 ) ).
% times_div_less_eq_dividend
thf(fact_3066_mod__eq__0D,axiom,
! [M2: nat,D: nat] :
( ( ( modulo_modulo_nat @ M2 @ D )
= zero_zero_nat )
=> ? [Q2: nat] :
( M2
= ( times_times_nat @ D @ Q2 ) ) ) ).
% mod_eq_0D
thf(fact_3067_nat__mod__eq__iff,axiom,
! [X3: nat,N: nat,Y: nat] :
( ( ( modulo_modulo_nat @ X3 @ N )
= ( modulo_modulo_nat @ Y @ N ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus_nat @ X3 @ ( times_times_nat @ N @ Q1 ) )
= ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_3068_power__odd__eq,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_3069_power__odd__eq,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_3070_power__odd__eq,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_3071_power__odd__eq,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_3072_power__odd__eq,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_3073_mult__le__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_3074_mult__le__cancel__left,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ A @ B ) )
& ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_3075_mult__le__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_3076_mult__le__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_3077_mult__le__cancel__right,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ A @ B ) )
& ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_3078_mult__le__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_3079_mult__left__less__imp__less,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_3080_mult__left__less__imp__less,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_3081_mult__left__less__imp__less,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_3082_mult__left__less__imp__less,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_3083_mult__strict__mono,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C2 @ D )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_3084_mult__strict__mono,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ C2 @ D )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_3085_mult__strict__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_3086_mult__strict__mono,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_3087_mult__less__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_3088_mult__less__cancel__left,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ A @ B ) )
& ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_3089_mult__less__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_3090_mult__right__less__imp__less,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_3091_mult__right__less__imp__less,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_3092_mult__right__less__imp__less,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_3093_mult__right__less__imp__less,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_3094_mult__strict__mono_H,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_3095_mult__strict__mono_H,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ C2 @ D )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_3096_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_3097_mult__strict__mono_H,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_3098_mult__less__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_3099_mult__less__cancel__right,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ A @ B ) )
& ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_3100_mult__less__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_3101_mult__le__cancel__left__neg,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_3102_mult__le__cancel__left__neg,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
= ( ord_less_eq_rat @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_3103_mult__le__cancel__left__neg,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_3104_mult__le__cancel__left__pos,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_3105_mult__le__cancel__left__pos,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
= ( ord_less_eq_rat @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_3106_mult__le__cancel__left__pos,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_3107_mult__left__le__imp__le,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_3108_mult__left__le__imp__le,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
=> ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_3109_mult__left__le__imp__le,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_3110_mult__left__le__imp__le,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_3111_mult__right__le__imp__le,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_3112_mult__right__le__imp__le,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_3113_mult__right__le__imp__le,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_3114_mult__right__le__imp__le,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_3115_mult__le__less__imp__less,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C2 @ D )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_3116_mult__le__less__imp__less,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ C2 @ D )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_3117_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_3118_mult__le__less__imp__less,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_3119_mult__less__le__imp__less,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_3120_mult__less__le__imp__less,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C2 @ D )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_3121_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_3122_mult__less__le__imp__less,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_3123_mult__left__le__one__le,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y @ X3 ) @ X3 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_3124_mult__left__le__one__le,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ Y @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ Y @ X3 ) @ X3 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_3125_mult__left__le__one__le,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X3 ) @ X3 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_3126_mult__right__le__one__le,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X3 @ Y ) @ X3 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_3127_mult__right__le__one__le,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ Y @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ X3 @ Y ) @ X3 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_3128_mult__right__le__one__le,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X3 @ Y ) @ X3 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_3129_mult__le__one,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_3130_mult__le__one,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_eq_rat @ B @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).
% mult_le_one
thf(fact_3131_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_3132_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_3133_mult__left__le,axiom,
! [C2: real,A: real] :
( ( ord_less_eq_real @ C2 @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ A ) ) ) ).
% mult_left_le
thf(fact_3134_mult__left__le,axiom,
! [C2: rat,A: rat] :
( ( ord_less_eq_rat @ C2 @ one_one_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ A ) ) ) ).
% mult_left_le
thf(fact_3135_mult__left__le,axiom,
! [C2: nat,A: nat] :
( ( ord_less_eq_nat @ C2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ A ) ) ) ).
% mult_left_le
thf(fact_3136_mult__left__le,axiom,
! [C2: int,A: int] :
( ( ord_less_eq_int @ C2 @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ A ) ) ) ).
% mult_left_le
thf(fact_3137_sum__squares__le__zero__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
= ( ( X3 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_3138_sum__squares__le__zero__iff,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
= ( ( X3 = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_3139_sum__squares__le__zero__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X3 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_3140_sum__squares__ge__zero,axiom,
! [X3: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_3141_sum__squares__ge__zero,axiom,
! [X3: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_3142_sum__squares__ge__zero,axiom,
! [X3: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_3143_sum__squares__gt__zero__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y @ Y ) ) )
= ( ( X3 != zero_zero_real )
| ( Y != zero_zero_real ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_3144_sum__squares__gt__zero__iff,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y @ Y ) ) )
= ( ( X3 != zero_zero_rat )
| ( Y != zero_zero_rat ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_3145_sum__squares__gt__zero__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X3 != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_3146_not__sum__squares__lt__zero,axiom,
! [X3: real,Y: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).
% not_sum_squares_lt_zero
thf(fact_3147_not__sum__squares__lt__zero,axiom,
! [X3: rat,Y: rat] :
~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).
% not_sum_squares_lt_zero
thf(fact_3148_not__sum__squares__lt__zero,axiom,
! [X3: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_3149_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C2 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C2 ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3150_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C2 ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C2 ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3151_divide__strict__left__mono__neg,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C2 @ A ) @ ( divide_divide_real @ C2 @ B ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_3152_divide__strict__left__mono__neg,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ord_less_rat @ ( divide_divide_rat @ C2 @ A ) @ ( divide_divide_rat @ C2 @ B ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_3153_divide__strict__left__mono,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C2 @ A ) @ ( divide_divide_real @ C2 @ B ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_3154_divide__strict__left__mono,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ord_less_rat @ ( divide_divide_rat @ C2 @ A ) @ ( divide_divide_rat @ C2 @ B ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_3155_mult__imp__less__div__pos,axiom,
! [Y: real,Z2: real,X3: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( times_times_real @ Z2 @ Y ) @ X3 )
=> ( ord_less_real @ Z2 @ ( divide_divide_real @ X3 @ Y ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_3156_mult__imp__less__div__pos,axiom,
! [Y: rat,Z2: rat,X3: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_rat @ ( times_times_rat @ Z2 @ Y ) @ X3 )
=> ( ord_less_rat @ Z2 @ ( divide_divide_rat @ X3 @ Y ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_3157_mult__imp__div__pos__less,axiom,
! [Y: real,X3: real,Z2: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X3 @ ( times_times_real @ Z2 @ Y ) )
=> ( ord_less_real @ ( divide_divide_real @ X3 @ Y ) @ Z2 ) ) ) ).
% mult_imp_div_pos_less
thf(fact_3158_mult__imp__div__pos__less,axiom,
! [Y: rat,X3: rat,Z2: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_rat @ X3 @ ( times_times_rat @ Z2 @ Y ) )
=> ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y ) @ Z2 ) ) ) ).
% mult_imp_div_pos_less
thf(fact_3159_pos__less__divide__eq,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C2 ) )
= ( ord_less_real @ ( times_times_real @ A @ C2 ) @ B ) ) ) ).
% pos_less_divide_eq
thf(fact_3160_pos__less__divide__eq,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
= ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ B ) ) ) ).
% pos_less_divide_eq
thf(fact_3161_pos__divide__less__eq,axiom,
! [C2: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C2 ) @ A )
= ( ord_less_real @ B @ ( times_times_real @ A @ C2 ) ) ) ) ).
% pos_divide_less_eq
thf(fact_3162_pos__divide__less__eq,axiom,
! [C2: rat,B: rat,A: rat] :
( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B @ C2 ) @ A )
= ( ord_less_rat @ B @ ( times_times_rat @ A @ C2 ) ) ) ) ).
% pos_divide_less_eq
thf(fact_3163_neg__less__divide__eq,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C2 ) )
= ( ord_less_real @ B @ ( times_times_real @ A @ C2 ) ) ) ) ).
% neg_less_divide_eq
thf(fact_3164_neg__less__divide__eq,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
= ( ord_less_rat @ B @ ( times_times_rat @ A @ C2 ) ) ) ) ).
% neg_less_divide_eq
thf(fact_3165_neg__divide__less__eq,axiom,
! [C2: real,B: real,A: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C2 ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ C2 ) @ B ) ) ) ).
% neg_divide_less_eq
thf(fact_3166_neg__divide__less__eq,axiom,
! [C2: rat,B: rat,A: rat] :
( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B @ C2 ) @ A )
= ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ B ) ) ) ).
% neg_divide_less_eq
thf(fact_3167_less__divide__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C2 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_3168_less__divide__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ B @ ( times_times_rat @ A @ C2 ) ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_3169_divide__less__eq,axiom,
! [B: real,C2: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ C2 ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C2 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ B ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_3170_divide__less__eq,axiom,
! [B: rat,C2: rat,A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ C2 ) @ A )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ B @ ( times_times_rat @ A @ C2 ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ B ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_3171_divide__eq__eq__numeral_I1_J,axiom,
! [B: complex,C2: complex,W2: num] :
( ( ( divide1717551699836669952omplex @ B @ C2 )
= ( numera6690914467698888265omplex @ W2 ) )
= ( ( ( C2 != zero_zero_complex )
=> ( B
= ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C2 ) ) )
& ( ( C2 = zero_zero_complex )
=> ( ( numera6690914467698888265omplex @ W2 )
= zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_3172_divide__eq__eq__numeral_I1_J,axiom,
! [B: real,C2: real,W2: num] :
( ( ( divide_divide_real @ B @ C2 )
= ( numeral_numeral_real @ W2 ) )
= ( ( ( C2 != zero_zero_real )
=> ( B
= ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) ) )
& ( ( C2 = zero_zero_real )
=> ( ( numeral_numeral_real @ W2 )
= zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_3173_divide__eq__eq__numeral_I1_J,axiom,
! [B: rat,C2: rat,W2: num] :
( ( ( divide_divide_rat @ B @ C2 )
= ( numeral_numeral_rat @ W2 ) )
= ( ( ( C2 != zero_zero_rat )
=> ( B
= ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) ) )
& ( ( C2 = zero_zero_rat )
=> ( ( numeral_numeral_rat @ W2 )
= zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_3174_eq__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: complex,C2: complex] :
( ( ( numera6690914467698888265omplex @ W2 )
= ( divide1717551699836669952omplex @ B @ C2 ) )
= ( ( ( C2 != zero_zero_complex )
=> ( ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C2 )
= B ) )
& ( ( C2 = zero_zero_complex )
=> ( ( numera6690914467698888265omplex @ W2 )
= zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_3175_eq__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: real,C2: real] :
( ( ( numeral_numeral_real @ W2 )
= ( divide_divide_real @ B @ C2 ) )
= ( ( ( C2 != zero_zero_real )
=> ( ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 )
= B ) )
& ( ( C2 = zero_zero_real )
=> ( ( numeral_numeral_real @ W2 )
= zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_3176_eq__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: rat,C2: rat] :
( ( ( numeral_numeral_rat @ W2 )
= ( divide_divide_rat @ B @ C2 ) )
= ( ( ( C2 != zero_zero_rat )
=> ( ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 )
= B ) )
& ( ( C2 = zero_zero_rat )
=> ( ( numeral_numeral_rat @ W2 )
= zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_3177_divide__add__eq__iff,axiom,
! [Z2: complex,X3: complex,Y: complex] :
( ( Z2 != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Z2 ) @ Y )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% divide_add_eq_iff
thf(fact_3178_divide__add__eq__iff,axiom,
! [Z2: real,X3: real,Y: real] :
( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Z2 ) @ Y )
= ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% divide_add_eq_iff
thf(fact_3179_divide__add__eq__iff,axiom,
! [Z2: rat,X3: rat,Y: rat] :
( ( Z2 != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ Z2 ) @ Y )
= ( divide_divide_rat @ ( plus_plus_rat @ X3 @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% divide_add_eq_iff
thf(fact_3180_add__divide__eq__iff,axiom,
! [Z2: complex,X3: complex,Y: complex] :
( ( Z2 != zero_zero_complex )
=> ( ( plus_plus_complex @ X3 @ ( divide1717551699836669952omplex @ Y @ Z2 ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X3 @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% add_divide_eq_iff
thf(fact_3181_add__divide__eq__iff,axiom,
! [Z2: real,X3: real,Y: real] :
( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ X3 @ ( divide_divide_real @ Y @ Z2 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X3 @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% add_divide_eq_iff
thf(fact_3182_add__divide__eq__iff,axiom,
! [Z2: rat,X3: rat,Y: rat] :
( ( Z2 != zero_zero_rat )
=> ( ( plus_plus_rat @ X3 @ ( divide_divide_rat @ Y @ Z2 ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% add_divide_eq_iff
thf(fact_3183_add__num__frac,axiom,
! [Y: complex,Z2: complex,X3: complex] :
( ( Y != zero_zero_complex )
=> ( ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ X3 @ Y ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Z2 @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_3184_add__num__frac,axiom,
! [Y: real,Z2: real,X3: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ Z2 @ ( divide_divide_real @ X3 @ Y ) )
= ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Z2 @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_3185_add__num__frac,axiom,
! [Y: rat,Z2: rat,X3: rat] :
( ( Y != zero_zero_rat )
=> ( ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ X3 @ Y ) )
= ( divide_divide_rat @ ( plus_plus_rat @ X3 @ ( times_times_rat @ Z2 @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_3186_add__frac__num,axiom,
! [Y: complex,X3: complex,Z2: complex] :
( ( Y != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Y ) @ Z2 )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Z2 @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_3187_add__frac__num,axiom,
! [Y: real,X3: real,Z2: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Y ) @ Z2 )
= ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Z2 @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_3188_add__frac__num,axiom,
! [Y: rat,X3: rat,Z2: rat] :
( ( Y != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ Y ) @ Z2 )
= ( divide_divide_rat @ ( plus_plus_rat @ X3 @ ( times_times_rat @ Z2 @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_3189_add__frac__eq,axiom,
! [Y: complex,Z2: complex,X3: complex,W2: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z2 != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Y ) @ ( divide1717551699836669952omplex @ W2 @ Z2 ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X3 @ Z2 ) @ ( times_times_complex @ W2 @ Y ) ) @ ( times_times_complex @ Y @ Z2 ) ) ) ) ) ).
% add_frac_eq
thf(fact_3190_add__frac__eq,axiom,
! [Y: real,Z2: real,X3: real,W2: real] :
( ( Y != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Y ) @ ( divide_divide_real @ W2 @ Z2 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) ) ) ) ).
% add_frac_eq
thf(fact_3191_add__frac__eq,axiom,
! [Y: rat,Z2: rat,X3: rat,W2: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z2 != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ Y ) @ ( divide_divide_rat @ W2 @ Z2 ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) ) ) ) ).
% add_frac_eq
thf(fact_3192_add__divide__eq__if__simps_I1_J,axiom,
! [Z2: complex,A: complex,B: complex] :
( ( ( Z2 = zero_zero_complex )
=> ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
= A ) )
& ( ( Z2 != zero_zero_complex )
=> ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_3193_add__divide__eq__if__simps_I1_J,axiom,
! [Z2: real,A: real,B: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
= A ) )
& ( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_3194_add__divide__eq__if__simps_I1_J,axiom,
! [Z2: rat,A: rat,B: rat] :
( ( ( Z2 = zero_zero_rat )
=> ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
= A ) )
& ( ( Z2 != zero_zero_rat )
=> ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_3195_add__divide__eq__if__simps_I2_J,axiom,
! [Z2: complex,A: complex,B: complex] :
( ( ( Z2 = zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
= B ) )
& ( ( Z2 != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_3196_add__divide__eq__if__simps_I2_J,axiom,
! [Z2: real,A: real,B: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
= B ) )
& ( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_3197_add__divide__eq__if__simps_I2_J,axiom,
! [Z2: rat,A: rat,B: rat] :
( ( ( Z2 = zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
= B ) )
& ( ( Z2 != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
= ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_3198_div__mult1__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C2 ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C2 ) ) @ C2 ) ) ) ).
% div_mult1_eq
thf(fact_3199_div__mult1__eq,axiom,
! [A: int,B: int,C2: int] :
( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C2 ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C2 ) ) @ C2 ) ) ) ).
% div_mult1_eq
thf(fact_3200_div__mult1__eq,axiom,
! [A: code_integer,B: code_integer,C2: code_integer] :
( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C2 )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C2 ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C2 ) ) @ C2 ) ) ) ).
% div_mult1_eq
thf(fact_3201_cancel__div__mod__rules_I2_J,axiom,
! [B: nat,A: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C2 )
= ( plus_plus_nat @ A @ C2 ) ) ).
% cancel_div_mod_rules(2)
thf(fact_3202_cancel__div__mod__rules_I2_J,axiom,
! [B: int,A: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C2 )
= ( plus_plus_int @ A @ C2 ) ) ).
% cancel_div_mod_rules(2)
thf(fact_3203_cancel__div__mod__rules_I2_J,axiom,
! [B: code_integer,A: code_integer,C2: code_integer] :
( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C2 )
= ( plus_p5714425477246183910nteger @ A @ C2 ) ) ).
% cancel_div_mod_rules(2)
thf(fact_3204_cancel__div__mod__rules_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C2 )
= ( plus_plus_nat @ A @ C2 ) ) ).
% cancel_div_mod_rules(1)
thf(fact_3205_cancel__div__mod__rules_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C2 )
= ( plus_plus_int @ A @ C2 ) ) ).
% cancel_div_mod_rules(1)
thf(fact_3206_cancel__div__mod__rules_I1_J,axiom,
! [A: code_integer,B: code_integer,C2: code_integer] :
( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C2 )
= ( plus_p5714425477246183910nteger @ A @ C2 ) ) ).
% cancel_div_mod_rules(1)
thf(fact_3207_mod__div__decomp,axiom,
! [A: nat,B: nat] :
( A
= ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% mod_div_decomp
thf(fact_3208_mod__div__decomp,axiom,
! [A: int,B: int] :
( A
= ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).
% mod_div_decomp
thf(fact_3209_mod__div__decomp,axiom,
! [A: code_integer,B: code_integer] :
( A
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% mod_div_decomp
thf(fact_3210_div__mult__mod__eq,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
= A ) ).
% div_mult_mod_eq
thf(fact_3211_div__mult__mod__eq,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
= A ) ).
% div_mult_mod_eq
thf(fact_3212_div__mult__mod__eq,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
= A ) ).
% div_mult_mod_eq
thf(fact_3213_mod__div__mult__eq,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
= A ) ).
% mod_div_mult_eq
thf(fact_3214_mod__div__mult__eq,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
= A ) ).
% mod_div_mult_eq
thf(fact_3215_mod__div__mult__eq,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
= A ) ).
% mod_div_mult_eq
thf(fact_3216_mod__mult__div__eq,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
= A ) ).
% mod_mult_div_eq
thf(fact_3217_mod__mult__div__eq,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
= A ) ).
% mod_mult_div_eq
thf(fact_3218_mod__mult__div__eq,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
= A ) ).
% mod_mult_div_eq
thf(fact_3219_mult__div__mod__eq,axiom,
! [B: nat,A: nat] :
( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
= A ) ).
% mult_div_mod_eq
thf(fact_3220_mult__div__mod__eq,axiom,
! [B: int,A: int] :
( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
= A ) ).
% mult_div_mod_eq
thf(fact_3221_mult__div__mod__eq,axiom,
! [B: code_integer,A: code_integer] :
( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
= A ) ).
% mult_div_mod_eq
thf(fact_3222_set__update__memI,axiom,
! [N: nat,Xs: list_complex,X3: complex] :
( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs ) )
=> ( member_complex @ X3 @ ( set_complex2 @ ( list_update_complex @ Xs @ N @ X3 ) ) ) ) ).
% set_update_memI
thf(fact_3223_set__update__memI,axiom,
! [N: nat,Xs: list_real,X3: real] :
( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
=> ( member_real @ X3 @ ( set_real2 @ ( list_update_real @ Xs @ N @ X3 ) ) ) ) ).
% set_update_memI
thf(fact_3224_set__update__memI,axiom,
! [N: nat,Xs: list_set_nat,X3: set_nat] :
( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
=> ( member_set_nat @ X3 @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ N @ X3 ) ) ) ) ).
% set_update_memI
thf(fact_3225_set__update__memI,axiom,
! [N: nat,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ N @ X3 ) ) ) ) ).
% set_update_memI
thf(fact_3226_set__update__memI,axiom,
! [N: nat,Xs: list_int,X3: int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( member_int @ X3 @ ( set_int2 @ ( list_update_int @ Xs @ N @ X3 ) ) ) ) ).
% set_update_memI
thf(fact_3227_set__update__memI,axiom,
! [N: nat,Xs: list_nat,X3: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ X3 @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X3 ) ) ) ) ).
% set_update_memI
thf(fact_3228_n__less__n__mult__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M2 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_3229_n__less__m__mult__n,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_3230_one__less__mult,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% one_less_mult
thf(fact_3231_nat__mult__le__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_3232_list__update__same__conv,axiom,
! [I: nat,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 )
= Xs )
= ( ( nth_VEBT_VEBT @ Xs @ I )
= X3 ) ) ) ).
% list_update_same_conv
thf(fact_3233_list__update__same__conv,axiom,
! [I: nat,Xs: list_int,X3: int] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( ( ( list_update_int @ Xs @ I @ X3 )
= Xs )
= ( ( nth_int @ Xs @ I )
= X3 ) ) ) ).
% list_update_same_conv
thf(fact_3234_list__update__same__conv,axiom,
! [I: nat,Xs: list_nat,X3: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( list_update_nat @ Xs @ I @ X3 )
= Xs )
= ( ( nth_nat @ Xs @ I )
= X3 ) ) ) ).
% list_update_same_conv
thf(fact_3235_nth__list__update,axiom,
! [I: nat,Xs: list_VEBT_VEBT,J: nat,X3: vEBT_VEBT] :
( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) @ J )
= X3 ) )
& ( ( I != J )
=> ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) @ J )
= ( nth_VEBT_VEBT @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_3236_nth__list__update,axiom,
! [I: nat,Xs: list_int,J: nat,X3: int] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_int @ ( list_update_int @ Xs @ I @ X3 ) @ J )
= X3 ) )
& ( ( I != J )
=> ( ( nth_int @ ( list_update_int @ Xs @ I @ X3 ) @ J )
= ( nth_int @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_3237_nth__list__update,axiom,
! [I: nat,Xs: list_nat,J: nat,X3: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X3 ) @ J )
= X3 ) )
& ( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X3 ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_3238_div__less__iff__less__mult,axiom,
! [Q3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M2 @ Q3 ) @ N )
= ( ord_less_nat @ M2 @ ( times_times_nat @ N @ Q3 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_3239_nat__mult__div__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= ( divide_divide_nat @ M2 @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_3240_mod__eq__nat1E,axiom,
! [M2: nat,Q3: nat,N: nat] :
( ( ( modulo_modulo_nat @ M2 @ Q3 )
= ( modulo_modulo_nat @ N @ Q3 ) )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ~ ! [S: nat] :
( M2
!= ( plus_plus_nat @ N @ ( times_times_nat @ Q3 @ S ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_3241_mod__eq__nat2E,axiom,
! [M2: nat,Q3: nat,N: nat] :
( ( ( modulo_modulo_nat @ M2 @ Q3 )
= ( modulo_modulo_nat @ N @ Q3 ) )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ! [S: nat] :
( N
!= ( plus_plus_nat @ M2 @ ( times_times_nat @ Q3 @ S ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_3242_nat__mod__eq__lemma,axiom,
! [X3: nat,N: nat,Y: nat] :
( ( ( modulo_modulo_nat @ X3 @ N )
= ( modulo_modulo_nat @ Y @ N ) )
=> ( ( ord_less_eq_nat @ Y @ X3 )
=> ? [Q2: nat] :
( X3
= ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q2 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_3243_mod__mult2__eq,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( modulo_modulo_nat @ M2 @ ( times_times_nat @ N @ Q3 ) )
= ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M2 @ N ) @ Q3 ) ) @ ( modulo_modulo_nat @ M2 @ N ) ) ) ).
% mod_mult2_eq
thf(fact_3244_div__mod__decomp,axiom,
! [A4: nat,N: nat] :
( A4
= ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A4 @ N ) @ N ) @ ( modulo_modulo_nat @ A4 @ N ) ) ) ).
% div_mod_decomp
thf(fact_3245_field__le__mult__one__interval,axiom,
! [X3: real,Y: real] :
( ! [Z3: real] :
( ( ord_less_real @ zero_zero_real @ Z3 )
=> ( ( ord_less_real @ Z3 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Z3 @ X3 ) @ Y ) ) )
=> ( ord_less_eq_real @ X3 @ Y ) ) ).
% field_le_mult_one_interval
thf(fact_3246_field__le__mult__one__interval,axiom,
! [X3: rat,Y: rat] :
( ! [Z3: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z3 )
=> ( ( ord_less_rat @ Z3 @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ Z3 @ X3 ) @ Y ) ) )
=> ( ord_less_eq_rat @ X3 @ Y ) ) ).
% field_le_mult_one_interval
thf(fact_3247_mult__less__cancel__right2,axiom,
! [A: real,C2: real] :
( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ C2 )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_3248_mult__less__cancel__right2,axiom,
! [A: rat,C2: rat] :
( ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ C2 )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ A @ one_one_rat ) )
& ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_3249_mult__less__cancel__right2,axiom,
! [A: int,C2: int] :
( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ C2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_3250_mult__less__cancel__right1,axiom,
! [C2: real,B: real] :
( ( ord_less_real @ C2 @ ( times_times_real @ B @ C2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_3251_mult__less__cancel__right1,axiom,
! [C2: rat,B: rat] :
( ( ord_less_rat @ C2 @ ( times_times_rat @ B @ C2 ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ one_one_rat @ B ) )
& ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_3252_mult__less__cancel__right1,axiom,
! [C2: int,B: int] :
( ( ord_less_int @ C2 @ ( times_times_int @ B @ C2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_3253_mult__less__cancel__left2,axiom,
! [C2: real,A: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ C2 )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_3254_mult__less__cancel__left2,axiom,
! [C2: rat,A: rat] :
( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ C2 )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ A @ one_one_rat ) )
& ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_3255_mult__less__cancel__left2,axiom,
! [C2: int,A: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ C2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_3256_mult__less__cancel__left1,axiom,
! [C2: real,B: real] :
( ( ord_less_real @ C2 @ ( times_times_real @ C2 @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_3257_mult__less__cancel__left1,axiom,
! [C2: rat,B: rat] :
( ( ord_less_rat @ C2 @ ( times_times_rat @ C2 @ B ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ one_one_rat @ B ) )
& ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_3258_mult__less__cancel__left1,axiom,
! [C2: int,B: int] :
( ( ord_less_int @ C2 @ ( times_times_int @ C2 @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_3259_mult__le__cancel__right2,axiom,
! [A: real,C2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ C2 )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ one_one_real ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_3260_mult__le__cancel__right2,axiom,
! [A: rat,C2: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ C2 )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ A @ one_one_rat ) )
& ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_3261_mult__le__cancel__right2,axiom,
! [A: int,C2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ C2 )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_3262_mult__le__cancel__right1,axiom,
! [C2: real,B: real] :
( ( ord_less_eq_real @ C2 @ ( times_times_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ one_one_real @ B ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_3263_mult__le__cancel__right1,axiom,
! [C2: rat,B: rat] :
( ( ord_less_eq_rat @ C2 @ ( times_times_rat @ B @ C2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ one_one_rat @ B ) )
& ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_3264_mult__le__cancel__right1,axiom,
! [C2: int,B: int] :
( ( ord_less_eq_int @ C2 @ ( times_times_int @ B @ C2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_3265_mult__le__cancel__left2,axiom,
! [C2: real,A: real] :
( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ C2 )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ A @ one_one_real ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_3266_mult__le__cancel__left2,axiom,
! [C2: rat,A: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ C2 )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ A @ one_one_rat ) )
& ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_3267_mult__le__cancel__left2,axiom,
! [C2: int,A: int] :
( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ C2 )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_3268_mult__le__cancel__left1,axiom,
! [C2: real,B: real] :
( ( ord_less_eq_real @ C2 @ ( times_times_real @ C2 @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ one_one_real @ B ) )
& ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_3269_mult__le__cancel__left1,axiom,
! [C2: rat,B: rat] :
( ( ord_less_eq_rat @ C2 @ ( times_times_rat @ C2 @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ one_one_rat @ B ) )
& ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_3270_mult__le__cancel__left1,axiom,
! [C2: int,B: int] :
( ( ord_less_eq_int @ C2 @ ( times_times_int @ C2 @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_3271_divide__le__eq,axiom,
! [B: real,C2: real,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C2 ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ B @ ( times_times_real @ A @ C2 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ B ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_3272_divide__le__eq,axiom,
! [B: rat,C2: rat,A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C2 ) @ A )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C2 ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ B ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_3273_le__divide__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B @ ( times_times_real @ A @ C2 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_3274_le__divide__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C2 ) ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_3275_divide__left__mono,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C2 @ A ) @ ( divide_divide_real @ C2 @ B ) ) ) ) ) ).
% divide_left_mono
thf(fact_3276_divide__left__mono,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
=> ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ C2 @ A ) @ ( divide_divide_rat @ C2 @ B ) ) ) ) ) ).
% divide_left_mono
thf(fact_3277_neg__divide__le__eq,axiom,
! [C2: real,B: real,A: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C2 ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ B ) ) ) ).
% neg_divide_le_eq
thf(fact_3278_neg__divide__le__eq,axiom,
! [C2: rat,B: rat,A: rat] :
( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C2 ) @ A )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ B ) ) ) ).
% neg_divide_le_eq
thf(fact_3279_neg__le__divide__eq,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C2 ) )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ C2 ) ) ) ) ).
% neg_le_divide_eq
thf(fact_3280_neg__le__divide__eq,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
= ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C2 ) ) ) ) ).
% neg_le_divide_eq
thf(fact_3281_pos__divide__le__eq,axiom,
! [C2: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C2 ) @ A )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ C2 ) ) ) ) ).
% pos_divide_le_eq
thf(fact_3282_pos__divide__le__eq,axiom,
! [C2: rat,B: rat,A: rat] :
( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C2 ) @ A )
= ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C2 ) ) ) ) ).
% pos_divide_le_eq
thf(fact_3283_pos__le__divide__eq,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C2 ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ B ) ) ) ).
% pos_le_divide_eq
thf(fact_3284_pos__le__divide__eq,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ B ) ) ) ).
% pos_le_divide_eq
thf(fact_3285_mult__imp__div__pos__le,axiom,
! [Y: real,X3: real,Z2: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X3 @ ( times_times_real @ Z2 @ Y ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y ) @ Z2 ) ) ) ).
% mult_imp_div_pos_le
thf(fact_3286_mult__imp__div__pos__le,axiom,
! [Y: rat,X3: rat,Z2: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ X3 @ ( times_times_rat @ Z2 @ Y ) )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y ) @ Z2 ) ) ) ).
% mult_imp_div_pos_le
thf(fact_3287_mult__imp__le__div__pos,axiom,
! [Y: real,Z2: real,X3: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ Y ) @ X3 )
=> ( ord_less_eq_real @ Z2 @ ( divide_divide_real @ X3 @ Y ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_3288_mult__imp__le__div__pos,axiom,
! [Y: rat,Z2: rat,X3: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ Y ) @ X3 )
=> ( ord_less_eq_rat @ Z2 @ ( divide_divide_rat @ X3 @ Y ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_3289_divide__left__mono__neg,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C2 @ A ) @ ( divide_divide_real @ C2 @ B ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_3290_divide__left__mono__neg,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ C2 @ A ) @ ( divide_divide_rat @ C2 @ B ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_3291_convex__bound__le,axiom,
! [X3: real,A: real,Y: real,U: real,V: real] :
( ( ord_less_eq_real @ X3 @ A )
=> ( ( ord_less_eq_real @ Y @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ zero_zero_real @ V )
=> ( ( ( plus_plus_real @ U @ V )
= one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X3 ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_3292_convex__bound__le,axiom,
! [X3: rat,A: rat,Y: rat,U: rat,V: rat] :
( ( ord_less_eq_rat @ X3 @ A )
=> ( ( ord_less_eq_rat @ Y @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ U )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ V )
=> ( ( ( plus_plus_rat @ U @ V )
= one_one_rat )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X3 ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_3293_convex__bound__le,axiom,
! [X3: int,A: int,Y: int,U: int,V: int] :
( ( ord_less_eq_int @ X3 @ A )
=> ( ( ord_less_eq_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X3 ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_3294_less__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: real,C2: real] :
( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_3295_less__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: rat,C2: rat] :
( ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_3296_divide__less__eq__numeral_I1_J,axiom,
! [B: real,C2: real,W2: num] :
( ( ord_less_real @ ( divide_divide_real @ B @ C2 ) @ ( numeral_numeral_real @ W2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) @ B ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_3297_divide__less__eq__numeral_I1_J,axiom,
! [B: rat,C2: rat,W2: num] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ C2 ) @ ( numeral_numeral_rat @ W2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) @ B ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_3298_power__Suc__less,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_3299_power__Suc__less,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ A @ one_one_rat )
=> ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_3300_power__Suc__less,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_3301_power__Suc__less,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_3302_mult__2,axiom,
! [Z2: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_complex @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_3303_mult__2,axiom,
! [Z2: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_real @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_3304_mult__2,axiom,
! [Z2: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_rat @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_3305_mult__2,axiom,
! [Z2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_nat @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_3306_mult__2,axiom,
! [Z2: int] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_int @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_3307_mult__2__right,axiom,
! [Z2: complex] :
( ( times_times_complex @ Z2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
= ( plus_plus_complex @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_3308_mult__2__right,axiom,
! [Z2: real] :
( ( times_times_real @ Z2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
= ( plus_plus_real @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_3309_mult__2__right,axiom,
! [Z2: rat] :
( ( times_times_rat @ Z2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
= ( plus_plus_rat @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_3310_mult__2__right,axiom,
! [Z2: nat] :
( ( times_times_nat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_3311_mult__2__right,axiom,
! [Z2: int] :
( ( times_times_int @ Z2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_3312_left__add__twice,axiom,
! [A: complex,B: complex] :
( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B ) )
= ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_3313_left__add__twice,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_3314_left__add__twice,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
= ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_3315_left__add__twice,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_3316_left__add__twice,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_3317_Suc__double__not__eq__double,axiom,
! [M2: nat,N: nat] :
( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% Suc_double_not_eq_double
thf(fact_3318_double__not__eq__Suc__double,axiom,
! [M2: nat,N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
!= ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% double_not_eq_Suc_double
thf(fact_3319_div__nat__eqI,axiom,
! [N: nat,Q3: nat,M2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M2 )
=> ( ( ord_less_nat @ M2 @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
=> ( ( divide_divide_nat @ M2 @ N )
= Q3 ) ) ) ).
% div_nat_eqI
thf(fact_3320_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_eq_nat @ M2 @ ( divide_divide_nat @ N @ Q3 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M2 @ Q3 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_3321_dividend__less__times__div,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M2 @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M2 @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_3322_dividend__less__div__times,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M2 @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_3323_split__div,axiom,
! [P2: nat > $o,M2: nat,N: nat] :
( ( P2 @ ( divide_divide_nat @ M2 @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P2 @ zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ! [I3: nat,J3: nat] :
( ( ord_less_nat @ J3 @ N )
=> ( ( M2
= ( plus_plus_nat @ ( times_times_nat @ N @ I3 ) @ J3 ) )
=> ( P2 @ I3 ) ) ) ) ) ) ).
% split_div
thf(fact_3324_split__mod,axiom,
! [P2: nat > $o,M2: nat,N: nat] :
( ( P2 @ ( modulo_modulo_nat @ M2 @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P2 @ M2 ) )
& ( ( N != zero_zero_nat )
=> ! [I3: nat,J3: nat] :
( ( ord_less_nat @ J3 @ N )
=> ( ( M2
= ( plus_plus_nat @ ( times_times_nat @ N @ I3 ) @ J3 ) )
=> ( P2 @ J3 ) ) ) ) ) ) ).
% split_mod
thf(fact_3325_fold__atLeastAtMost__nat_Osimps,axiom,
( set_fo2584398358068434914at_nat
= ( ^ [F2: nat > nat > nat,A5: nat,B4: nat,Acc2: nat] : ( if_nat @ ( ord_less_nat @ B4 @ A5 ) @ Acc2 @ ( set_fo2584398358068434914at_nat @ F2 @ ( plus_plus_nat @ A5 @ one_one_nat ) @ B4 @ ( F2 @ A5 @ Acc2 ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_3326_fold__atLeastAtMost__nat_Oelims,axiom,
! [X3: nat > nat > nat,Xa2: nat,Xb2: nat,Xc: nat,Y: nat] :
( ( ( set_fo2584398358068434914at_nat @ X3 @ Xa2 @ Xb2 @ Xc )
= Y )
=> ( ( ( ord_less_nat @ Xb2 @ Xa2 )
=> ( Y = Xc ) )
& ( ~ ( ord_less_nat @ Xb2 @ Xa2 )
=> ( Y
= ( set_fo2584398358068434914at_nat @ X3 @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb2 @ ( X3 @ Xa2 @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_3327_convex__bound__lt,axiom,
! [X3: real,A: real,Y: real,U: real,V: real] :
( ( ord_less_real @ X3 @ A )
=> ( ( ord_less_real @ Y @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ zero_zero_real @ V )
=> ( ( ( plus_plus_real @ U @ V )
= one_one_real )
=> ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X3 ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_3328_convex__bound__lt,axiom,
! [X3: rat,A: rat,Y: rat,U: rat,V: rat] :
( ( ord_less_rat @ X3 @ A )
=> ( ( ord_less_rat @ Y @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ U )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ V )
=> ( ( ( plus_plus_rat @ U @ V )
= one_one_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X3 ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_3329_convex__bound__lt,axiom,
! [X3: int,A: int,Y: int,U: int,V: int] :
( ( ord_less_int @ X3 @ A )
=> ( ( ord_less_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X3 ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_3330_divide__le__eq__numeral_I1_J,axiom,
! [B: real,C2: real,W2: num] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C2 ) @ ( numeral_numeral_real @ W2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) @ B ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_3331_divide__le__eq__numeral_I1_J,axiom,
! [B: rat,C2: rat,W2: num] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C2 ) @ ( numeral_numeral_rat @ W2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) @ B ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_3332_le__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: real,C2: real] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_3333_le__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: rat,C2: rat] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_3334_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C2: code_integer,A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
=> ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C2 ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C2 ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_3335_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C2 ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_3336_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C2 ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_3337_sum__squares__bound,axiom,
! [X3: real,Y: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) @ Y ) @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_squares_bound
thf(fact_3338_sum__squares__bound,axiom,
! [X3: rat,Y: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X3 ) @ Y ) @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_squares_bound
thf(fact_3339_split__div_H,axiom,
! [P2: nat > $o,M2: nat,N: nat] :
( ( P2 @ ( divide_divide_nat @ M2 @ N ) )
= ( ( ( N = zero_zero_nat )
& ( P2 @ zero_zero_nat ) )
| ? [Q4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M2 )
& ( ord_less_nat @ M2 @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
& ( P2 @ Q4 ) ) ) ) ).
% split_div'
thf(fact_3340_Suc__times__mod__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M2 @ N ) ) @ M2 )
= one_one_nat ) ) ).
% Suc_times_mod_eq
thf(fact_3341_divmod__digit__0_I2_J,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
= ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_3342_divmod__digit__0_I2_J,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
= ( modulo_modulo_int @ A @ B ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_3343_divmod__digit__0_I2_J,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
= ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_3344_power2__sum,axiom,
! [X3: complex,Y: complex] :
( ( power_power_complex @ ( plus_plus_complex @ X3 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) @ Y ) ) ) ).
% power2_sum
thf(fact_3345_power2__sum,axiom,
! [X3: real,Y: real] :
( ( power_power_real @ ( plus_plus_real @ X3 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) @ Y ) ) ) ).
% power2_sum
thf(fact_3346_power2__sum,axiom,
! [X3: rat,Y: rat] :
( ( power_power_rat @ ( plus_plus_rat @ X3 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X3 ) @ Y ) ) ) ).
% power2_sum
thf(fact_3347_power2__sum,axiom,
! [X3: nat,Y: nat] :
( ( power_power_nat @ ( plus_plus_nat @ X3 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X3 ) @ Y ) ) ) ).
% power2_sum
thf(fact_3348_power2__sum,axiom,
! [X3: int,Y: int] :
( ( power_power_int @ ( plus_plus_int @ X3 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 ) @ Y ) ) ) ).
% power2_sum
thf(fact_3349_zero__le__even__power_H,axiom,
! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% zero_le_even_power'
thf(fact_3350_zero__le__even__power_H,axiom,
! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% zero_le_even_power'
thf(fact_3351_zero__le__even__power_H,axiom,
! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% zero_le_even_power'
thf(fact_3352_nat__bit__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( P2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( P2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_bit_induct
thf(fact_3353_arith__geo__mean,axiom,
! [U: real,X3: real,Y: real] :
( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_real @ X3 @ Y ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X3 @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_3354_arith__geo__mean,axiom,
! [U: rat,X3: rat,Y: rat] :
( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_rat @ X3 @ Y ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X3 @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_3355_triangle__def,axiom,
( nat_triangle
= ( ^ [N4: nat] : ( divide_divide_nat @ ( times_times_nat @ N4 @ ( suc @ N4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% triangle_def
thf(fact_3356_realpow__pos__nth2,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ ( suc @ N ) )
= A ) ) ) ).
% realpow_pos_nth2
thf(fact_3357_divmod__digit__0_I1_J,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_3358_divmod__digit__0_I1_J,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_3359_divmod__digit__0_I1_J,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
= ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_3360_odd__0__le__power__imp__0__le,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_3361_odd__0__le__power__imp__0__le,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_3362_odd__0__le__power__imp__0__le,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_3363_odd__power__less__zero,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).
% odd_power_less_zero
thf(fact_3364_odd__power__less__zero,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).
% odd_power_less_zero
thf(fact_3365_odd__power__less__zero,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).
% odd_power_less_zero
thf(fact_3366_mod__double__modulus,axiom,
! [M2: code_integer,X3: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M2 )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
=> ( ( ( modulo364778990260209775nteger @ X3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) )
= ( modulo364778990260209775nteger @ X3 @ M2 ) )
| ( ( modulo364778990260209775nteger @ X3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X3 @ M2 ) @ M2 ) ) ) ) ) ).
% mod_double_modulus
thf(fact_3367_mod__double__modulus,axiom,
! [M2: nat,X3: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
=> ( ( ( modulo_modulo_nat @ X3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
= ( modulo_modulo_nat @ X3 @ M2 ) )
| ( ( modulo_modulo_nat @ X3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
= ( plus_plus_nat @ ( modulo_modulo_nat @ X3 @ M2 ) @ M2 ) ) ) ) ) ).
% mod_double_modulus
thf(fact_3368_mod__double__modulus,axiom,
! [M2: int,X3: int] :
( ( ord_less_int @ zero_zero_int @ M2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ( modulo_modulo_int @ X3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) )
= ( modulo_modulo_int @ X3 @ M2 ) )
| ( ( modulo_modulo_int @ X3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) )
= ( plus_plus_int @ ( modulo_modulo_int @ X3 @ M2 ) @ M2 ) ) ) ) ) ).
% mod_double_modulus
thf(fact_3369_divmod__digit__1_I1_J,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
= ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_3370_divmod__digit__1_I1_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( 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 )
= ( divide_divide_nat @ A @ B ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_3371_divmod__digit__1_I1_J,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( 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 )
= ( divide_divide_int @ A @ B ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_3372_realpow__pos__nth__unique,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
& ( ( power_power_real @ X4 @ N )
= A )
& ! [Y6: real] :
( ( ( ord_less_real @ zero_zero_real @ Y6 )
& ( ( power_power_real @ Y6 @ N )
= A ) )
=> ( Y6 = X4 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_3373_set__bit__0,axiom,
! [A: nat] :
( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% set_bit_0
thf(fact_3374_set__bit__0,axiom,
! [A: int] :
( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% set_bit_0
thf(fact_3375_max_Obounded__iff,axiom,
! [B: extended_enat,C2: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C2 ) @ A )
= ( ( ord_le2932123472753598470d_enat @ B @ A )
& ( ord_le2932123472753598470d_enat @ C2 @ A ) ) ) ).
% max.bounded_iff
thf(fact_3376_max_Obounded__iff,axiom,
! [B: code_integer,C2: code_integer,A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ B @ C2 ) @ A )
= ( ( ord_le3102999989581377725nteger @ B @ A )
& ( ord_le3102999989581377725nteger @ C2 @ A ) ) ) ).
% max.bounded_iff
thf(fact_3377_max_Obounded__iff,axiom,
! [B: rat,C2: rat,A: rat] :
( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C2 ) @ A )
= ( ( ord_less_eq_rat @ B @ A )
& ( ord_less_eq_rat @ C2 @ A ) ) ) ).
% max.bounded_iff
thf(fact_3378_max_Obounded__iff,axiom,
! [B: num,C2: num,A: num] :
( ( ord_less_eq_num @ ( ord_max_num @ B @ C2 ) @ A )
= ( ( ord_less_eq_num @ B @ A )
& ( ord_less_eq_num @ C2 @ A ) ) ) ).
% max.bounded_iff
thf(fact_3379_max_Obounded__iff,axiom,
! [B: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C2 ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% max.bounded_iff
thf(fact_3380_max_Obounded__iff,axiom,
! [B: int,C2: int,A: int] :
( ( ord_less_eq_int @ ( ord_max_int @ B @ C2 ) @ A )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_eq_int @ C2 @ A ) ) ) ).
% max.bounded_iff
thf(fact_3381_max_Oabsorb2,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_ma741700101516333627d_enat @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_3382_max_Oabsorb2,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ B )
=> ( ( ord_max_Code_integer @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_3383_max_Oabsorb2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_max_rat @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_3384_max_Oabsorb2,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_max_num @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_3385_max_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_3386_max_Oabsorb2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_max_int @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_3387_max_Oabsorb1,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( ( ord_ma741700101516333627d_enat @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_3388_max_Oabsorb1,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le3102999989581377725nteger @ B @ A )
=> ( ( ord_max_Code_integer @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_3389_max_Oabsorb1,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_max_rat @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_3390_max_Oabsorb1,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_max_num @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_3391_max_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_3392_max_Oabsorb1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_max_int @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_3393_product__nth,axiom,
! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
=> ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys2 ) @ N )
= ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_3394_product__nth,axiom,
! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_int] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys2 ) ) )
=> ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys2 ) @ N )
= ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) @ ( nth_int @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_3395_product__nth,axiom,
! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) )
=> ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys2 ) @ N )
= ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) @ ( nth_nat @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_3396_product__nth,axiom,
! [N: nat,Xs: list_int,Ys2: list_VEBT_VEBT] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
=> ( ( nth_Pr3474266648193625910T_VEBT @ ( produc662631939642741121T_VEBT @ Xs @ Ys2 ) @ N )
= ( produc3329399203697025711T_VEBT @ ( nth_int @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_3397_product__nth,axiom,
! [N: nat,Xs: list_int,Ys2: list_int] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys2 ) ) )
=> ( ( nth_Pr4439495888332055232nt_int @ ( product_int_int @ Xs @ Ys2 ) @ N )
= ( product_Pair_int_int @ ( nth_int @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) @ ( nth_int @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_3398_product__nth,axiom,
! [N: nat,Xs: list_int,Ys2: list_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) )
=> ( ( nth_Pr8617346907841251940nt_nat @ ( product_int_nat @ Xs @ Ys2 ) @ N )
= ( product_Pair_int_nat @ ( nth_int @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) @ ( nth_nat @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_3399_product__nth,axiom,
! [N: nat,Xs: list_nat,Ys2: list_VEBT_VEBT] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
=> ( ( nth_Pr744662078594809490T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs @ Ys2 ) @ N )
= ( produc599794634098209291T_VEBT @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_3400_product__nth,axiom,
! [N: nat,Xs: list_nat,Ys2: list_int] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_int @ Ys2 ) ) )
=> ( ( nth_Pr3440142176431000676at_int @ ( product_nat_int @ Xs @ Ys2 ) @ N )
= ( product_Pair_nat_int @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) @ ( nth_int @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_3401_product__nth,axiom,
! [N: nat,Xs: list_nat,Ys2: list_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( product_nat_nat @ Xs @ Ys2 ) @ N )
= ( product_Pair_nat_nat @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) @ ( nth_nat @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_3402_product__nth,axiom,
! [N: nat,Xs: list_C878401137130745250e_term,Ys2: list_P5578671422887162913nteger] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s8902784113405215798e_term @ Xs ) @ ( size_s6527587076508218509nteger @ Ys2 ) ) )
=> ( ( nth_Pr1312216959259138743nteger @ ( produc4846348955484107138nteger @ Xs @ Ys2 ) @ N )
= ( produc6137756002093451184nteger @ ( nth_Co7464280458349004107e_term @ Xs @ ( divide_divide_nat @ N @ ( size_s6527587076508218509nteger @ Ys2 ) ) ) @ ( nth_Pr2304437835452373666nteger @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6527587076508218509nteger @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_3403_unset__bit__0,axiom,
! [A: nat] :
( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% unset_bit_0
thf(fact_3404_unset__bit__0,axiom,
! [A: int] :
( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% unset_bit_0
thf(fact_3405_set__bit__Suc,axiom,
! [N: nat,A: nat] :
( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A )
= ( 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 ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_3406_set__bit__Suc,axiom,
! [N: nat,A: code_integer] :
( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_3407_set__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A )
= ( 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 ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_3408_flip__bit__Suc,axiom,
! [N: nat,A: nat] :
( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A )
= ( 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 ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_3409_flip__bit__Suc,axiom,
! [N: nat,A: code_integer] :
( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_3410_flip__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A )
= ( 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 ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_3411_unset__bit__Suc,axiom,
! [N: nat,A: nat] :
( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A )
= ( 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 ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_3412_unset__bit__Suc,axiom,
! [N: nat,A: code_integer] :
( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_3413_unset__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A )
= ( 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 ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_3414_i0__less,axiom,
! [N: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
= ( N != zero_z5237406670263579293d_enat ) ) ).
% i0_less
thf(fact_3415_max__enat__simps_I2_J,axiom,
! [Q3: extended_enat] :
( ( ord_ma741700101516333627d_enat @ Q3 @ zero_z5237406670263579293d_enat )
= Q3 ) ).
% max_enat_simps(2)
thf(fact_3416_max__enat__simps_I3_J,axiom,
! [Q3: extended_enat] :
( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q3 )
= Q3 ) ).
% max_enat_simps(3)
thf(fact_3417_not__real__square__gt__zero,axiom,
! [X3: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X3 @ X3 ) ) )
= ( X3 = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_3418_unset__bit__nonnegative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K2 ) )
= ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).
% unset_bit_nonnegative_int_iff
thf(fact_3419_set__bit__nonnegative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K2 ) )
= ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).
% set_bit_nonnegative_int_iff
thf(fact_3420_flip__bit__nonnegative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K2 ) )
= ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).
% flip_bit_nonnegative_int_iff
thf(fact_3421_unset__bit__negative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K2 ) @ zero_zero_int )
= ( ord_less_int @ K2 @ zero_zero_int ) ) ).
% unset_bit_negative_int_iff
thf(fact_3422_set__bit__negative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K2 ) @ zero_zero_int )
= ( ord_less_int @ K2 @ zero_zero_int ) ) ).
% set_bit_negative_int_iff
thf(fact_3423_flip__bit__negative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K2 ) @ zero_zero_int )
= ( ord_less_int @ K2 @ zero_zero_int ) ) ).
% flip_bit_negative_int_iff
thf(fact_3424_length__product,axiom,
! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys2 ) )
= ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ).
% length_product
thf(fact_3425_length__product,axiom,
! [Xs: list_VEBT_VEBT,Ys2: list_int] :
( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys2 ) )
= ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys2 ) ) ) ).
% length_product
thf(fact_3426_length__product,axiom,
! [Xs: list_VEBT_VEBT,Ys2: list_nat] :
( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys2 ) )
= ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) ) ).
% length_product
thf(fact_3427_length__product,axiom,
! [Xs: list_int,Ys2: list_VEBT_VEBT] :
( ( size_s6639371672096860321T_VEBT @ ( produc662631939642741121T_VEBT @ Xs @ Ys2 ) )
= ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ).
% length_product
thf(fact_3428_length__product,axiom,
! [Xs: list_int,Ys2: list_int] :
( ( size_s5157815400016825771nt_int @ ( product_int_int @ Xs @ Ys2 ) )
= ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys2 ) ) ) ).
% length_product
thf(fact_3429_length__product,axiom,
! [Xs: list_int,Ys2: list_nat] :
( ( size_s7647898544948552527nt_nat @ ( product_int_nat @ Xs @ Ys2 ) )
= ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) ) ).
% length_product
thf(fact_3430_length__product,axiom,
! [Xs: list_nat,Ys2: list_VEBT_VEBT] :
( ( size_s4762443039079500285T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs @ Ys2 ) )
= ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ).
% length_product
thf(fact_3431_length__product,axiom,
! [Xs: list_nat,Ys2: list_int] :
( ( size_s2970893825323803983at_int @ ( product_nat_int @ Xs @ Ys2 ) )
= ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_int @ Ys2 ) ) ) ).
% length_product
thf(fact_3432_length__product,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( size_s5460976970255530739at_nat @ ( product_nat_nat @ Xs @ Ys2 ) )
= ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) ) ).
% length_product
thf(fact_3433_div__mod__decomp__int,axiom,
! [A4: int,N: int] :
( A4
= ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A4 @ N ) @ N ) @ ( modulo_modulo_int @ A4 @ N ) ) ) ).
% div_mod_decomp_int
thf(fact_3434_int__gr__induct,axiom,
! [K2: int,I: int,P2: int > $o] :
( ( ord_less_int @ K2 @ I )
=> ( ( P2 @ ( plus_plus_int @ K2 @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K2 @ I2 )
=> ( ( P2 @ I2 )
=> ( P2 @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_gr_induct
thf(fact_3435_zless__add1__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W2 @ Z2 )
| ( W2 = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_3436_zero__one__enat__neq_I1_J,axiom,
zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% zero_one_enat_neq(1)
thf(fact_3437_bot__enat__def,axiom,
bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% bot_enat_def
thf(fact_3438_not__iless0,axiom,
! [N: extended_enat] :
~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% not_iless0
thf(fact_3439_enat__0__less__mult__iff,axiom,
! [M2: extended_enat,N: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M2 @ N ) )
= ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M2 )
& ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).
% enat_0_less_mult_iff
thf(fact_3440_imult__is__0,axiom,
! [M2: extended_enat,N: extended_enat] :
( ( ( times_7803423173614009249d_enat @ M2 @ N )
= zero_z5237406670263579293d_enat )
= ( ( M2 = zero_z5237406670263579293d_enat )
| ( N = zero_z5237406670263579293d_enat ) ) ) ).
% imult_is_0
thf(fact_3441_int__distrib_I2_J,axiom,
! [W2: int,Z12: int,Z23: int] :
( ( times_times_int @ W2 @ ( plus_plus_int @ Z12 @ Z23 ) )
= ( plus_plus_int @ ( times_times_int @ W2 @ Z12 ) @ ( times_times_int @ W2 @ Z23 ) ) ) ).
% int_distrib(2)
thf(fact_3442_int__distrib_I1_J,axiom,
! [Z12: int,Z23: int,W2: int] :
( ( times_times_int @ ( plus_plus_int @ Z12 @ Z23 ) @ W2 )
= ( plus_plus_int @ ( times_times_int @ Z12 @ W2 ) @ ( times_times_int @ Z23 @ W2 ) ) ) ).
% int_distrib(1)
thf(fact_3443_iadd__is__0,axiom,
! [M2: extended_enat,N: extended_enat] :
( ( ( plus_p3455044024723400733d_enat @ M2 @ N )
= zero_z5237406670263579293d_enat )
= ( ( M2 = zero_z5237406670263579293d_enat )
& ( N = zero_z5237406670263579293d_enat ) ) ) ).
% iadd_is_0
thf(fact_3444_zmod__eq__0__iff,axiom,
! [M2: int,D: int] :
( ( ( modulo_modulo_int @ M2 @ D )
= zero_zero_int )
= ( ? [Q4: int] :
( M2
= ( times_times_int @ D @ Q4 ) ) ) ) ).
% zmod_eq_0_iff
thf(fact_3445_zmod__eq__0D,axiom,
! [M2: int,D: int] :
( ( ( modulo_modulo_int @ M2 @ D )
= zero_zero_int )
=> ? [Q2: int] :
( M2
= ( times_times_int @ D @ Q2 ) ) ) ).
% zmod_eq_0D
thf(fact_3446_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_3447_times__int__code_I1_J,axiom,
! [K2: int] :
( ( times_times_int @ K2 @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_3448_zmult__zless__mono2,axiom,
! [I: int,J: int,K2: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ord_less_int @ ( times_times_int @ K2 @ I ) @ ( times_times_int @ K2 @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_3449_pos__zmult__eq__1__iff,axiom,
! [M2: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M2 )
=> ( ( ( times_times_int @ M2 @ N )
= one_one_int )
= ( ( M2 = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_3450_unset__bit__less__eq,axiom,
! [N: nat,K2: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K2 ) @ K2 ) ).
% unset_bit_less_eq
thf(fact_3451_set__bit__greater__eq,axiom,
! [K2: int,N: nat] : ( ord_less_eq_int @ K2 @ ( bit_se7879613467334960850it_int @ N @ K2 ) ) ).
% set_bit_greater_eq
thf(fact_3452_zdiv__zmult2__eq,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C2 ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C2 ) ) ) ).
% zdiv_zmult2_eq
thf(fact_3453_split__zdiv,axiom,
! [P2: int > $o,N: int,K2: int] :
( ( P2 @ ( divide_divide_int @ N @ K2 ) )
= ( ( ( K2 = zero_zero_int )
=> ( P2 @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K2 )
=> ! [I3: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K2 )
& ( N
= ( plus_plus_int @ ( times_times_int @ K2 @ I3 ) @ J3 ) ) )
=> ( P2 @ I3 ) ) )
& ( ( ord_less_int @ K2 @ zero_zero_int )
=> ! [I3: int,J3: int] :
( ( ( ord_less_int @ K2 @ J3 )
& ( ord_less_eq_int @ J3 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K2 @ I3 ) @ J3 ) ) )
=> ( P2 @ I3 ) ) ) ) ) ).
% split_zdiv
thf(fact_3454_q__pos__lemma,axiom,
! [B2: int,Q5: int,R4: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q5 ) @ R4 ) )
=> ( ( ord_less_int @ R4 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).
% q_pos_lemma
thf(fact_3455_int__div__neg__eq,axiom,
! [A: int,B: int,Q3: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R2 )
=> ( ( divide_divide_int @ A @ B )
= Q3 ) ) ) ) ).
% int_div_neg_eq
thf(fact_3456_int__div__pos__eq,axiom,
! [A: int,B: int,Q3: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ R2 @ B )
=> ( ( divide_divide_int @ A @ B )
= Q3 ) ) ) ) ).
% int_div_pos_eq
thf(fact_3457_zdiv__mono2__lemma,axiom,
! [B: int,Q3: int,R2: int,B2: int,Q5: int,R4: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 )
= ( plus_plus_int @ ( times_times_int @ B2 @ Q5 ) @ R4 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q5 ) @ R4 ) )
=> ( ( ord_less_int @ R4 @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ B2 @ B )
=> ( ord_less_eq_int @ Q3 @ Q5 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_3458_incr__mult__lemma,axiom,
! [D: int,P2: int > $o,K2: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int] :
( ( P2 @ X4 )
=> ( P2 @ ( plus_plus_int @ X4 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ! [X: int] :
( ( P2 @ X )
=> ( P2 @ ( plus_plus_int @ X @ ( times_times_int @ K2 @ D ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_3459_zdiv__mono2__neg__lemma,axiom,
! [B: int,Q3: int,R2: int,B2: int,Q5: int,R4: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 )
= ( plus_plus_int @ ( times_times_int @ B2 @ Q5 ) @ R4 ) )
=> ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q5 ) @ R4 ) @ zero_zero_int )
=> ( ( ord_less_int @ R2 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ R4 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ B2 @ B )
=> ( ord_less_eq_int @ Q5 @ Q3 ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_3460_unique__quotient__lemma,axiom,
! [B: int,Q5: int,R4: int,Q3: int,R2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R4 )
=> ( ( ord_less_int @ R4 @ B )
=> ( ( ord_less_int @ R2 @ B )
=> ( ord_less_eq_int @ Q5 @ Q3 ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_3461_unique__quotient__lemma__neg,axiom,
! [B: int,Q5: int,R4: int,Q3: int,R2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R2 )
=> ( ( ord_less_int @ B @ R4 )
=> ( ord_less_eq_int @ Q3 @ Q5 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_3462_split__pos__lemma,axiom,
! [K2: int,P2: int > int > $o,N: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ( P2 @ ( divide_divide_int @ N @ K2 ) @ ( modulo_modulo_int @ N @ K2 ) )
= ( ! [I3: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K2 )
& ( N
= ( plus_plus_int @ ( times_times_int @ K2 @ I3 ) @ J3 ) ) )
=> ( P2 @ I3 @ J3 ) ) ) ) ) ).
% split_pos_lemma
thf(fact_3463_split__neg__lemma,axiom,
! [K2: int,P2: int > int > $o,N: int] :
( ( ord_less_int @ K2 @ zero_zero_int )
=> ( ( P2 @ ( divide_divide_int @ N @ K2 ) @ ( modulo_modulo_int @ N @ K2 ) )
= ( ! [I3: int,J3: int] :
( ( ( ord_less_int @ K2 @ J3 )
& ( ord_less_eq_int @ J3 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K2 @ I3 ) @ J3 ) ) )
=> ( P2 @ I3 @ J3 ) ) ) ) ) ).
% split_neg_lemma
thf(fact_3464_int__mod__pos__eq,axiom,
! [A: int,B: int,Q3: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ R2 @ B )
=> ( ( modulo_modulo_int @ A @ B )
= R2 ) ) ) ) ).
% int_mod_pos_eq
thf(fact_3465_int__mod__neg__eq,axiom,
! [A: int,B: int,Q3: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R2 )
=> ( ( modulo_modulo_int @ A @ B )
= R2 ) ) ) ) ).
% int_mod_neg_eq
thf(fact_3466_split__zmod,axiom,
! [P2: int > $o,N: int,K2: int] :
( ( P2 @ ( modulo_modulo_int @ N @ K2 ) )
= ( ( ( K2 = zero_zero_int )
=> ( P2 @ N ) )
& ( ( ord_less_int @ zero_zero_int @ K2 )
=> ! [I3: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K2 )
& ( N
= ( plus_plus_int @ ( times_times_int @ K2 @ I3 ) @ J3 ) ) )
=> ( P2 @ J3 ) ) )
& ( ( ord_less_int @ K2 @ zero_zero_int )
=> ! [I3: int,J3: int] :
( ( ( ord_less_int @ K2 @ J3 )
& ( ord_less_eq_int @ J3 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K2 @ I3 ) @ J3 ) ) )
=> ( P2 @ J3 ) ) ) ) ) ).
% split_zmod
thf(fact_3467_zmod__zmult2__eq,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C2 ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% zmod_zmult2_eq
thf(fact_3468_pos__zmod__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( 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 ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A ) ) ) ) ) ).
% pos_zmod_mult_2
thf(fact_3469_pos__zdiv__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( 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 ) )
= ( divide_divide_int @ B @ A ) ) ) ).
% pos_zdiv_mult_2
thf(fact_3470_neg__zdiv__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( 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 ) )
= ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).
% neg_zdiv_mult_2
thf(fact_3471_L2__set__mult__ineq__lemma,axiom,
! [A: real,C2: real,B: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C2 ) ) @ ( 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 @ C2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% L2_set_mult_ineq_lemma
thf(fact_3472_max_OcoboundedI2,axiom,
! [C2: extended_enat,B: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ C2 @ B )
=> ( ord_le2932123472753598470d_enat @ C2 @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_3473_max_OcoboundedI2,axiom,
! [C2: code_integer,B: code_integer,A: code_integer] :
( ( ord_le3102999989581377725nteger @ C2 @ B )
=> ( ord_le3102999989581377725nteger @ C2 @ ( ord_max_Code_integer @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_3474_max_OcoboundedI2,axiom,
! [C2: rat,B: rat,A: rat] :
( ( ord_less_eq_rat @ C2 @ B )
=> ( ord_less_eq_rat @ C2 @ ( ord_max_rat @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_3475_max_OcoboundedI2,axiom,
! [C2: num,B: num,A: num] :
( ( ord_less_eq_num @ C2 @ B )
=> ( ord_less_eq_num @ C2 @ ( ord_max_num @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_3476_max_OcoboundedI2,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ ( ord_max_nat @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_3477_max_OcoboundedI2,axiom,
! [C2: int,B: int,A: int] :
( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_eq_int @ C2 @ ( ord_max_int @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_3478_max_OcoboundedI1,axiom,
! [C2: extended_enat,A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ C2 @ A )
=> ( ord_le2932123472753598470d_enat @ C2 @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_3479_max_OcoboundedI1,axiom,
! [C2: code_integer,A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ C2 @ A )
=> ( ord_le3102999989581377725nteger @ C2 @ ( ord_max_Code_integer @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_3480_max_OcoboundedI1,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ C2 @ A )
=> ( ord_less_eq_rat @ C2 @ ( ord_max_rat @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_3481_max_OcoboundedI1,axiom,
! [C2: num,A: num,B: num] :
( ( ord_less_eq_num @ C2 @ A )
=> ( ord_less_eq_num @ C2 @ ( ord_max_num @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_3482_max_OcoboundedI1,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_eq_nat @ C2 @ ( ord_max_nat @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_3483_max_OcoboundedI1,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ C2 @ A )
=> ( ord_less_eq_int @ C2 @ ( ord_max_int @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_3484_max_Oabsorb__iff2,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [A5: extended_enat,B4: extended_enat] :
( ( ord_ma741700101516333627d_enat @ A5 @ B4 )
= B4 ) ) ) ).
% max.absorb_iff2
thf(fact_3485_max_Oabsorb__iff2,axiom,
( ord_le3102999989581377725nteger
= ( ^ [A5: code_integer,B4: code_integer] :
( ( ord_max_Code_integer @ A5 @ B4 )
= B4 ) ) ) ).
% max.absorb_iff2
thf(fact_3486_max_Oabsorb__iff2,axiom,
( ord_less_eq_rat
= ( ^ [A5: rat,B4: rat] :
( ( ord_max_rat @ A5 @ B4 )
= B4 ) ) ) ).
% max.absorb_iff2
thf(fact_3487_max_Oabsorb__iff2,axiom,
( ord_less_eq_num
= ( ^ [A5: num,B4: num] :
( ( ord_max_num @ A5 @ B4 )
= B4 ) ) ) ).
% max.absorb_iff2
thf(fact_3488_max_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_max_nat @ A5 @ B4 )
= B4 ) ) ) ).
% max.absorb_iff2
thf(fact_3489_max_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B4: int] :
( ( ord_max_int @ A5 @ B4 )
= B4 ) ) ) ).
% max.absorb_iff2
thf(fact_3490_max_Oabsorb__iff1,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [B4: extended_enat,A5: extended_enat] :
( ( ord_ma741700101516333627d_enat @ A5 @ B4 )
= A5 ) ) ) ).
% max.absorb_iff1
thf(fact_3491_max_Oabsorb__iff1,axiom,
( ord_le3102999989581377725nteger
= ( ^ [B4: code_integer,A5: code_integer] :
( ( ord_max_Code_integer @ A5 @ B4 )
= A5 ) ) ) ).
% max.absorb_iff1
thf(fact_3492_max_Oabsorb__iff1,axiom,
( ord_less_eq_rat
= ( ^ [B4: rat,A5: rat] :
( ( ord_max_rat @ A5 @ B4 )
= A5 ) ) ) ).
% max.absorb_iff1
thf(fact_3493_max_Oabsorb__iff1,axiom,
( ord_less_eq_num
= ( ^ [B4: num,A5: num] :
( ( ord_max_num @ A5 @ B4 )
= A5 ) ) ) ).
% max.absorb_iff1
thf(fact_3494_max_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_max_nat @ A5 @ B4 )
= A5 ) ) ) ).
% max.absorb_iff1
thf(fact_3495_max_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A5: int] :
( ( ord_max_int @ A5 @ B4 )
= A5 ) ) ) ).
% max.absorb_iff1
thf(fact_3496_le__max__iff__disj,axiom,
! [Z2: extended_enat,X3: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Z2 @ ( ord_ma741700101516333627d_enat @ X3 @ Y ) )
= ( ( ord_le2932123472753598470d_enat @ Z2 @ X3 )
| ( ord_le2932123472753598470d_enat @ Z2 @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_3497_le__max__iff__disj,axiom,
! [Z2: code_integer,X3: code_integer,Y: code_integer] :
( ( ord_le3102999989581377725nteger @ Z2 @ ( ord_max_Code_integer @ X3 @ Y ) )
= ( ( ord_le3102999989581377725nteger @ Z2 @ X3 )
| ( ord_le3102999989581377725nteger @ Z2 @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_3498_le__max__iff__disj,axiom,
! [Z2: rat,X3: rat,Y: rat] :
( ( ord_less_eq_rat @ Z2 @ ( ord_max_rat @ X3 @ Y ) )
= ( ( ord_less_eq_rat @ Z2 @ X3 )
| ( ord_less_eq_rat @ Z2 @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_3499_le__max__iff__disj,axiom,
! [Z2: num,X3: num,Y: num] :
( ( ord_less_eq_num @ Z2 @ ( ord_max_num @ X3 @ Y ) )
= ( ( ord_less_eq_num @ Z2 @ X3 )
| ( ord_less_eq_num @ Z2 @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_3500_le__max__iff__disj,axiom,
! [Z2: nat,X3: nat,Y: nat] :
( ( ord_less_eq_nat @ Z2 @ ( ord_max_nat @ X3 @ Y ) )
= ( ( ord_less_eq_nat @ Z2 @ X3 )
| ( ord_less_eq_nat @ Z2 @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_3501_le__max__iff__disj,axiom,
! [Z2: int,X3: int,Y: int] :
( ( ord_less_eq_int @ Z2 @ ( ord_max_int @ X3 @ Y ) )
= ( ( ord_less_eq_int @ Z2 @ X3 )
| ( ord_less_eq_int @ Z2 @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_3502_max_Ocobounded2,axiom,
! [B: extended_enat,A: extended_enat] : ( ord_le2932123472753598470d_enat @ B @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).
% max.cobounded2
thf(fact_3503_max_Ocobounded2,axiom,
! [B: code_integer,A: code_integer] : ( ord_le3102999989581377725nteger @ B @ ( ord_max_Code_integer @ A @ B ) ) ).
% max.cobounded2
thf(fact_3504_max_Ocobounded2,axiom,
! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).
% max.cobounded2
thf(fact_3505_max_Ocobounded2,axiom,
! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).
% max.cobounded2
thf(fact_3506_max_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded2
thf(fact_3507_max_Ocobounded2,axiom,
! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).
% max.cobounded2
thf(fact_3508_max_Ocobounded1,axiom,
! [A: extended_enat,B: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).
% max.cobounded1
thf(fact_3509_max_Ocobounded1,axiom,
! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( ord_max_Code_integer @ A @ B ) ) ).
% max.cobounded1
thf(fact_3510_max_Ocobounded1,axiom,
! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).
% max.cobounded1
thf(fact_3511_max_Ocobounded1,axiom,
! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).
% max.cobounded1
thf(fact_3512_max_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded1
thf(fact_3513_max_Ocobounded1,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).
% max.cobounded1
thf(fact_3514_max_Oorder__iff,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [B4: extended_enat,A5: extended_enat] :
( A5
= ( ord_ma741700101516333627d_enat @ A5 @ B4 ) ) ) ) ).
% max.order_iff
thf(fact_3515_max_Oorder__iff,axiom,
( ord_le3102999989581377725nteger
= ( ^ [B4: code_integer,A5: code_integer] :
( A5
= ( ord_max_Code_integer @ A5 @ B4 ) ) ) ) ).
% max.order_iff
thf(fact_3516_max_Oorder__iff,axiom,
( ord_less_eq_rat
= ( ^ [B4: rat,A5: rat] :
( A5
= ( ord_max_rat @ A5 @ B4 ) ) ) ) ).
% max.order_iff
thf(fact_3517_max_Oorder__iff,axiom,
( ord_less_eq_num
= ( ^ [B4: num,A5: num] :
( A5
= ( ord_max_num @ A5 @ B4 ) ) ) ) ).
% max.order_iff
thf(fact_3518_max_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( A5
= ( ord_max_nat @ A5 @ B4 ) ) ) ) ).
% max.order_iff
thf(fact_3519_max_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A5: int] :
( A5
= ( ord_max_int @ A5 @ B4 ) ) ) ) ).
% max.order_iff
thf(fact_3520_max_OboundedI,axiom,
! [B: extended_enat,A: extended_enat,C2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( ( ord_le2932123472753598470d_enat @ C2 @ A )
=> ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C2 ) @ A ) ) ) ).
% max.boundedI
thf(fact_3521_max_OboundedI,axiom,
! [B: code_integer,A: code_integer,C2: code_integer] :
( ( ord_le3102999989581377725nteger @ B @ A )
=> ( ( ord_le3102999989581377725nteger @ C2 @ A )
=> ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ B @ C2 ) @ A ) ) ) ).
% max.boundedI
thf(fact_3522_max_OboundedI,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C2 @ A )
=> ( ord_less_eq_rat @ ( ord_max_rat @ B @ C2 ) @ A ) ) ) ).
% max.boundedI
thf(fact_3523_max_OboundedI,axiom,
! [B: num,A: num,C2: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ C2 @ A )
=> ( ord_less_eq_num @ ( ord_max_num @ B @ C2 ) @ A ) ) ) ).
% max.boundedI
thf(fact_3524_max_OboundedI,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_eq_nat @ ( ord_max_nat @ B @ C2 ) @ A ) ) ) ).
% max.boundedI
thf(fact_3525_max_OboundedI,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C2 @ A )
=> ( ord_less_eq_int @ ( ord_max_int @ B @ C2 ) @ A ) ) ) ).
% max.boundedI
thf(fact_3526_max_OboundedE,axiom,
! [B: extended_enat,C2: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C2 ) @ A )
=> ~ ( ( ord_le2932123472753598470d_enat @ B @ A )
=> ~ ( ord_le2932123472753598470d_enat @ C2 @ A ) ) ) ).
% max.boundedE
thf(fact_3527_max_OboundedE,axiom,
! [B: code_integer,C2: code_integer,A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ B @ C2 ) @ A )
=> ~ ( ( ord_le3102999989581377725nteger @ B @ A )
=> ~ ( ord_le3102999989581377725nteger @ C2 @ A ) ) ) ).
% max.boundedE
thf(fact_3528_max_OboundedE,axiom,
! [B: rat,C2: rat,A: rat] :
( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C2 ) @ A )
=> ~ ( ( ord_less_eq_rat @ B @ A )
=> ~ ( ord_less_eq_rat @ C2 @ A ) ) ) ).
% max.boundedE
thf(fact_3529_max_OboundedE,axiom,
! [B: num,C2: num,A: num] :
( ( ord_less_eq_num @ ( ord_max_num @ B @ C2 ) @ A )
=> ~ ( ( ord_less_eq_num @ B @ A )
=> ~ ( ord_less_eq_num @ C2 @ A ) ) ) ).
% max.boundedE
thf(fact_3530_max_OboundedE,axiom,
! [B: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C2 ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% max.boundedE
thf(fact_3531_max_OboundedE,axiom,
! [B: int,C2: int,A: int] :
( ( ord_less_eq_int @ ( ord_max_int @ B @ C2 ) @ A )
=> ~ ( ( ord_less_eq_int @ B @ A )
=> ~ ( ord_less_eq_int @ C2 @ A ) ) ) ).
% max.boundedE
thf(fact_3532_max_OorderI,axiom,
! [A: extended_enat,B: extended_enat] :
( ( A
= ( ord_ma741700101516333627d_enat @ A @ B ) )
=> ( ord_le2932123472753598470d_enat @ B @ A ) ) ).
% max.orderI
thf(fact_3533_max_OorderI,axiom,
! [A: code_integer,B: code_integer] :
( ( A
= ( ord_max_Code_integer @ A @ B ) )
=> ( ord_le3102999989581377725nteger @ B @ A ) ) ).
% max.orderI
thf(fact_3534_max_OorderI,axiom,
! [A: rat,B: rat] :
( ( A
= ( ord_max_rat @ A @ B ) )
=> ( ord_less_eq_rat @ B @ A ) ) ).
% max.orderI
thf(fact_3535_max_OorderI,axiom,
! [A: num,B: num] :
( ( A
= ( ord_max_num @ A @ B ) )
=> ( ord_less_eq_num @ B @ A ) ) ).
% max.orderI
thf(fact_3536_max_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( ord_max_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% max.orderI
thf(fact_3537_max_OorderI,axiom,
! [A: int,B: int] :
( ( A
= ( ord_max_int @ A @ B ) )
=> ( ord_less_eq_int @ B @ A ) ) ).
% max.orderI
thf(fact_3538_max_OorderE,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( A
= ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% max.orderE
thf(fact_3539_max_OorderE,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le3102999989581377725nteger @ B @ A )
=> ( A
= ( ord_max_Code_integer @ A @ B ) ) ) ).
% max.orderE
thf(fact_3540_max_OorderE,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( A
= ( ord_max_rat @ A @ B ) ) ) ).
% max.orderE
thf(fact_3541_max_OorderE,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( A
= ( ord_max_num @ A @ B ) ) ) ).
% max.orderE
thf(fact_3542_max_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( ord_max_nat @ A @ B ) ) ) ).
% max.orderE
thf(fact_3543_max_OorderE,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( A
= ( ord_max_int @ A @ B ) ) ) ).
% max.orderE
thf(fact_3544_max_Omono,axiom,
! [C2: extended_enat,A: extended_enat,D: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ C2 @ A )
=> ( ( ord_le2932123472753598470d_enat @ D @ B )
=> ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ C2 @ D ) @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ) ).
% max.mono
thf(fact_3545_max_Omono,axiom,
! [C2: code_integer,A: code_integer,D: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ C2 @ A )
=> ( ( ord_le3102999989581377725nteger @ D @ B )
=> ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ C2 @ D ) @ ( ord_max_Code_integer @ A @ B ) ) ) ) ).
% max.mono
thf(fact_3546_max_Omono,axiom,
! [C2: rat,A: rat,D: rat,B: rat] :
( ( ord_less_eq_rat @ C2 @ A )
=> ( ( ord_less_eq_rat @ D @ B )
=> ( ord_less_eq_rat @ ( ord_max_rat @ C2 @ D ) @ ( ord_max_rat @ A @ B ) ) ) ) ).
% max.mono
thf(fact_3547_max_Omono,axiom,
! [C2: num,A: num,D: num,B: num] :
( ( ord_less_eq_num @ C2 @ A )
=> ( ( ord_less_eq_num @ D @ B )
=> ( ord_less_eq_num @ ( ord_max_num @ C2 @ D ) @ ( ord_max_num @ A @ B ) ) ) ) ).
% max.mono
thf(fact_3548_max_Omono,axiom,
! [C2: nat,A: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ A )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( ord_max_nat @ C2 @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).
% max.mono
thf(fact_3549_max_Omono,axiom,
! [C2: int,A: int,D: int,B: int] :
( ( ord_less_eq_int @ C2 @ A )
=> ( ( ord_less_eq_int @ D @ B )
=> ( ord_less_eq_int @ ( ord_max_int @ C2 @ D ) @ ( ord_max_int @ A @ B ) ) ) ) ).
% max.mono
thf(fact_3550_mult__le__cancel__iff2,axiom,
! [Z2: real,X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ X3 ) @ ( times_times_real @ Z2 @ Y ) )
= ( ord_less_eq_real @ X3 @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_3551_mult__le__cancel__iff2,axiom,
! [Z2: rat,X3: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z2 )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X3 ) @ ( times_times_rat @ Z2 @ Y ) )
= ( ord_less_eq_rat @ X3 @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_3552_mult__le__cancel__iff2,axiom,
! [Z2: int,X3: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z2 @ X3 ) @ ( times_times_int @ Z2 @ Y ) )
= ( ord_less_eq_int @ X3 @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_3553_mult__le__cancel__iff1,axiom,
! [Z2: real,X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ Y @ Z2 ) )
= ( ord_less_eq_real @ X3 @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_3554_mult__le__cancel__iff1,axiom,
! [Z2: rat,X3: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z2 )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ Y @ Z2 ) )
= ( ord_less_eq_rat @ X3 @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_3555_mult__le__cancel__iff1,axiom,
! [Z2: int,X3: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ ( times_times_int @ X3 @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
= ( ord_less_eq_int @ X3 @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_3556_even__succ__mod__exp,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_3557_even__succ__mod__exp,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_3558_even__succ__mod__exp,axiom,
! [A: code_integer,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_3559_length__Cons,axiom,
! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT @ ( cons_VEBT_VEBT @ X3 @ Xs ) )
= ( suc @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).
% length_Cons
thf(fact_3560_length__Cons,axiom,
! [X3: int,Xs: list_int] :
( ( size_size_list_int @ ( cons_int @ X3 @ Xs ) )
= ( suc @ ( size_size_list_int @ Xs ) ) ) ).
% length_Cons
thf(fact_3561_length__Cons,axiom,
! [X3: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X3 @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_3562_even__succ__div__exp,axiom,
! [A: code_integer,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_3563_even__succ__div__exp,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_3564_even__succ__div__exp,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_3565_signed__take__bit__Suc,axiom,
! [N: nat,A: code_integer] :
( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_3566_signed__take__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A )
= ( 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 ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_3567_nat__dvd__1__iff__1,axiom,
! [M2: nat] :
( ( dvd_dvd_nat @ M2 @ one_one_nat )
= ( M2 = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_3568_dvd__0__left__iff,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
= ( A = zero_z3403309356797280102nteger ) ) ).
% dvd_0_left_iff
thf(fact_3569_dvd__0__left__iff,axiom,
! [A: complex] :
( ( dvd_dvd_complex @ zero_zero_complex @ A )
= ( A = zero_zero_complex ) ) ).
% dvd_0_left_iff
thf(fact_3570_dvd__0__left__iff,axiom,
! [A: real] :
( ( dvd_dvd_real @ zero_zero_real @ A )
= ( A = zero_zero_real ) ) ).
% dvd_0_left_iff
thf(fact_3571_dvd__0__left__iff,axiom,
! [A: rat] :
( ( dvd_dvd_rat @ zero_zero_rat @ A )
= ( A = zero_zero_rat ) ) ).
% dvd_0_left_iff
thf(fact_3572_dvd__0__left__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_3573_dvd__0__left__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
= ( A = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_3574_dvd__0__right,axiom,
! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).
% dvd_0_right
thf(fact_3575_dvd__0__right,axiom,
! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).
% dvd_0_right
thf(fact_3576_dvd__0__right,axiom,
! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% dvd_0_right
thf(fact_3577_dvd__0__right,axiom,
! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).
% dvd_0_right
thf(fact_3578_dvd__0__right,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_3579_dvd__0__right,axiom,
! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% dvd_0_right
thf(fact_3580_dvd__add__triv__right__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
= ( dvd_dvd_Code_integer @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_3581_dvd__add__triv__right__iff,axiom,
! [A: real,B: real] :
( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ A ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_3582_dvd__add__triv__right__iff,axiom,
! [A: rat,B: rat] :
( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
= ( dvd_dvd_rat @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_3583_dvd__add__triv__right__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_3584_dvd__add__triv__right__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_3585_dvd__add__triv__left__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
= ( dvd_dvd_Code_integer @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_3586_dvd__add__triv__left__iff,axiom,
! [A: real,B: real] :
( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_3587_dvd__add__triv__left__iff,axiom,
! [A: rat,B: rat] :
( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
= ( dvd_dvd_rat @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_3588_dvd__add__triv__left__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_3589_dvd__add__triv__left__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_3590_dvd__1__left,axiom,
! [K2: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K2 ) ).
% dvd_1_left
thf(fact_3591_dvd__1__iff__1,axiom,
! [M2: nat] :
( ( dvd_dvd_nat @ M2 @ ( suc @ zero_zero_nat ) )
= ( M2
= ( suc @ zero_zero_nat ) ) ) ).
% dvd_1_iff_1
thf(fact_3592_nat__mult__dvd__cancel__disj,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= ( ( K2 = zero_zero_nat )
| ( dvd_dvd_nat @ M2 @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_3593_signed__take__bit__of__0,axiom,
! [N: nat] :
( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
= zero_zero_int ) ).
% signed_take_bit_of_0
thf(fact_3594_dvd__mult__cancel__left,axiom,
! [C2: code_integer,A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C2 @ A ) @ ( times_3573771949741848930nteger @ C2 @ B ) )
= ( ( C2 = zero_z3403309356797280102nteger )
| ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_3595_dvd__mult__cancel__left,axiom,
! [C2: complex,A: complex,B: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ C2 @ A ) @ ( times_times_complex @ C2 @ B ) )
= ( ( C2 = zero_zero_complex )
| ( dvd_dvd_complex @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_3596_dvd__mult__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( dvd_dvd_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ( C2 = zero_zero_real )
| ( dvd_dvd_real @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_3597_dvd__mult__cancel__left,axiom,
! [C2: rat,A: rat,B: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
= ( ( C2 = zero_zero_rat )
| ( dvd_dvd_rat @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_3598_dvd__mult__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ( C2 = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_3599_dvd__mult__cancel__right,axiom,
! [A: code_integer,C2: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C2 ) @ ( times_3573771949741848930nteger @ B @ C2 ) )
= ( ( C2 = zero_z3403309356797280102nteger )
| ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_3600_dvd__mult__cancel__right,axiom,
! [A: complex,C2: complex,B: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ A @ C2 ) @ ( times_times_complex @ B @ C2 ) )
= ( ( C2 = zero_zero_complex )
| ( dvd_dvd_complex @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_3601_dvd__mult__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( dvd_dvd_real @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_3602_dvd__mult__cancel__right,axiom,
! [A: rat,C2: rat,B: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
= ( ( C2 = zero_zero_rat )
| ( dvd_dvd_rat @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_3603_dvd__mult__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_3604_dvd__times__left__cancel__iff,axiom,
! [A: code_integer,B: code_integer,C2: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C2 ) )
= ( dvd_dvd_Code_integer @ B @ C2 ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_3605_dvd__times__left__cancel__iff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C2 ) )
= ( dvd_dvd_nat @ B @ C2 ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_3606_dvd__times__left__cancel__iff,axiom,
! [A: int,B: int,C2: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) )
= ( dvd_dvd_int @ B @ C2 ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_3607_dvd__times__right__cancel__iff,axiom,
! [A: code_integer,B: code_integer,C2: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C2 @ A ) )
= ( dvd_dvd_Code_integer @ B @ C2 ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_3608_dvd__times__right__cancel__iff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C2 @ A ) )
= ( dvd_dvd_nat @ B @ C2 ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_3609_dvd__times__right__cancel__iff,axiom,
! [A: int,B: int,C2: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C2 @ A ) )
= ( dvd_dvd_int @ B @ C2 ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_3610_dvd__add__times__triv__left__iff,axiom,
! [A: code_integer,C2: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C2 @ A ) @ B ) )
= ( dvd_dvd_Code_integer @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_3611_dvd__add__times__triv__left__iff,axiom,
! [A: real,C2: real,B: real] :
( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C2 @ A ) @ B ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_3612_dvd__add__times__triv__left__iff,axiom,
! [A: rat,C2: rat,B: rat] :
( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C2 @ A ) @ B ) )
= ( dvd_dvd_rat @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_3613_dvd__add__times__triv__left__iff,axiom,
! [A: nat,C2: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C2 @ A ) @ B ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_3614_dvd__add__times__triv__left__iff,axiom,
! [A: int,C2: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C2 @ A ) @ B ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_3615_dvd__add__times__triv__right__iff,axiom,
! [A: code_integer,B: code_integer,C2: code_integer] :
( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C2 @ A ) ) )
= ( dvd_dvd_Code_integer @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_3616_dvd__add__times__triv__right__iff,axiom,
! [A: real,B: real,C2: real] :
( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C2 @ A ) ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_3617_dvd__add__times__triv__right__iff,axiom,
! [A: rat,B: rat,C2: rat] :
( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C2 @ A ) ) )
= ( dvd_dvd_rat @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_3618_dvd__add__times__triv__right__iff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C2 @ A ) ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_3619_dvd__add__times__triv__right__iff,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C2 @ A ) ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_3620_div__add,axiom,
! [C2: code_integer,A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ C2 @ A )
=> ( ( dvd_dvd_Code_integer @ C2 @ B )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C2 )
= ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C2 ) @ ( divide6298287555418463151nteger @ B @ C2 ) ) ) ) ) ).
% div_add
thf(fact_3621_div__add,axiom,
! [C2: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C2 @ A )
=> ( ( dvd_dvd_nat @ C2 @ B )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C2 ) @ ( divide_divide_nat @ B @ C2 ) ) ) ) ) ).
% div_add
thf(fact_3622_div__add,axiom,
! [C2: int,A: int,B: int] :
( ( dvd_dvd_int @ C2 @ A )
=> ( ( dvd_dvd_int @ C2 @ B )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( divide_divide_int @ A @ C2 ) @ ( divide_divide_int @ B @ C2 ) ) ) ) ) ).
% div_add
thf(fact_3623_dvd__imp__mod__0,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( modulo_modulo_nat @ B @ A )
= zero_zero_nat ) ) ).
% dvd_imp_mod_0
thf(fact_3624_dvd__imp__mod__0,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( modulo_modulo_int @ B @ A )
= zero_zero_int ) ) ).
% dvd_imp_mod_0
thf(fact_3625_dvd__imp__mod__0,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( modulo364778990260209775nteger @ B @ A )
= zero_z3403309356797280102nteger ) ) ).
% dvd_imp_mod_0
thf(fact_3626_signed__take__bit__Suc__1,axiom,
! [N: nat] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
= one_one_int ) ).
% signed_take_bit_Suc_1
thf(fact_3627_even__Suc,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% even_Suc
thf(fact_3628_even__Suc__Suc__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% even_Suc_Suc_iff
thf(fact_3629_pow__divides__pow__iff,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( dvd_dvd_nat @ A @ B ) ) ) ).
% pow_divides_pow_iff
thf(fact_3630_pow__divides__pow__iff,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( dvd_dvd_int @ A @ B ) ) ) ).
% pow_divides_pow_iff
thf(fact_3631_even__add,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
= ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_add
thf(fact_3632_even__add,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_add
thf(fact_3633_even__add,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_add
thf(fact_3634_odd__add,axiom,
! [A: code_integer,B: code_integer] :
( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) )
= ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
!= ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% odd_add
thf(fact_3635_odd__add,axiom,
! [A: nat,B: nat] :
( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
= ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
!= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% odd_add
thf(fact_3636_odd__add,axiom,
! [A: int,B: int] :
( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
= ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
!= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% odd_add
thf(fact_3637_odd__Suc__div__two,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% odd_Suc_div_two
thf(fact_3638_even__Suc__div__two,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_Suc_div_two
thf(fact_3639_signed__take__bit__Suc__bit0,axiom,
! [N: nat,K2: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K2 ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_Suc_bit0
thf(fact_3640_zero__le__power__eq__numeral,axiom,
! [A: real,W2: num] :
( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_3641_zero__le__power__eq__numeral,axiom,
! [A: rat,W2: num] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_3642_zero__le__power__eq__numeral,axiom,
! [A: int,W2: num] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_3643_power__less__zero__eq,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% power_less_zero_eq
thf(fact_3644_power__less__zero__eq,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% power_less_zero_eq
thf(fact_3645_power__less__zero__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% power_less_zero_eq
thf(fact_3646_power__less__zero__eq__numeral,axiom,
! [A: real,W2: num] :
( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_real )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_3647_power__less__zero__eq__numeral,axiom,
! [A: rat,W2: num] :
( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_rat )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_3648_power__less__zero__eq__numeral,axiom,
! [A: int,W2: num] :
( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_3649_even__plus__one__iff,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
= ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_plus_one_iff
thf(fact_3650_even__plus__one__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_plus_one_iff
thf(fact_3651_even__plus__one__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_plus_one_iff
thf(fact_3652_zero__less__power__eq__numeral,axiom,
! [A: real,W2: num] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( ( numeral_numeral_nat @ W2 )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A != zero_zero_real ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_3653_zero__less__power__eq__numeral,axiom,
! [A: rat,W2: num] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( ( numeral_numeral_nat @ W2 )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A != zero_zero_rat ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_3654_zero__less__power__eq__numeral,axiom,
! [A: int,W2: num] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( ( numeral_numeral_nat @ W2 )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A != zero_zero_int ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_3655_even__succ__div__two,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_3656_even__succ__div__two,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_3657_even__succ__div__two,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_3658_odd__succ__div__two,axiom,
! [A: code_integer] :
( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% odd_succ_div_two
thf(fact_3659_odd__succ__div__two,axiom,
! [A: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% odd_succ_div_two
thf(fact_3660_odd__succ__div__two,axiom,
! [A: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% odd_succ_div_two
thf(fact_3661_even__succ__div__2,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_3662_even__succ__div__2,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_3663_even__succ__div__2,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_3664_even__power,axiom,
! [A: code_integer,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N ) )
= ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_3665_even__power,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_3666_even__power,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_3667_odd__two__times__div__two__succ,axiom,
! [A: code_integer] :
( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
= A ) ) ).
% odd_two_times_div_two_succ
thf(fact_3668_odd__two__times__div__two__succ,axiom,
! [A: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
= A ) ) ).
% odd_two_times_div_two_succ
thf(fact_3669_odd__two__times__div__two__succ,axiom,
! [A: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
= A ) ) ).
% odd_two_times_div_two_succ
thf(fact_3670_power__le__zero__eq__numeral,axiom,
! [A: real,W2: num] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_real )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_real @ A @ zero_zero_real ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A = zero_zero_real ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_3671_power__le__zero__eq__numeral,axiom,
! [A: rat,W2: num] :
( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_rat )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_rat @ A @ zero_zero_rat ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A = zero_zero_rat ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_3672_power__le__zero__eq__numeral,axiom,
! [A: int,W2: num] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_int @ A @ zero_zero_int ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A = zero_zero_int ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_3673_dvd__0__left,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
=> ( A = zero_z3403309356797280102nteger ) ) ).
% dvd_0_left
thf(fact_3674_dvd__0__left,axiom,
! [A: complex] :
( ( dvd_dvd_complex @ zero_zero_complex @ A )
=> ( A = zero_zero_complex ) ) ).
% dvd_0_left
thf(fact_3675_dvd__0__left,axiom,
! [A: real] :
( ( dvd_dvd_real @ zero_zero_real @ A )
=> ( A = zero_zero_real ) ) ).
% dvd_0_left
thf(fact_3676_dvd__0__left,axiom,
! [A: rat] :
( ( dvd_dvd_rat @ zero_zero_rat @ A )
=> ( A = zero_zero_rat ) ) ).
% dvd_0_left
thf(fact_3677_dvd__0__left,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_3678_dvd__0__left,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
=> ( A = zero_zero_int ) ) ).
% dvd_0_left
thf(fact_3679_dvd__field__iff,axiom,
( dvd_dvd_complex
= ( ^ [A5: complex,B4: complex] :
( ( A5 = zero_zero_complex )
=> ( B4 = zero_zero_complex ) ) ) ) ).
% dvd_field_iff
thf(fact_3680_dvd__field__iff,axiom,
( dvd_dvd_real
= ( ^ [A5: real,B4: real] :
( ( A5 = zero_zero_real )
=> ( B4 = zero_zero_real ) ) ) ) ).
% dvd_field_iff
thf(fact_3681_dvd__field__iff,axiom,
( dvd_dvd_rat
= ( ^ [A5: rat,B4: rat] :
( ( A5 = zero_zero_rat )
=> ( B4 = zero_zero_rat ) ) ) ) ).
% dvd_field_iff
thf(fact_3682_dvd__add__right__iff,axiom,
! [A: code_integer,B: code_integer,C2: code_integer] :
( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C2 ) )
= ( dvd_dvd_Code_integer @ A @ C2 ) ) ) ).
% dvd_add_right_iff
thf(fact_3683_dvd__add__right__iff,axiom,
! [A: real,B: real,C2: real] :
( ( dvd_dvd_real @ A @ B )
=> ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C2 ) )
= ( dvd_dvd_real @ A @ C2 ) ) ) ).
% dvd_add_right_iff
thf(fact_3684_dvd__add__right__iff,axiom,
! [A: rat,B: rat,C2: rat] :
( ( dvd_dvd_rat @ A @ B )
=> ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C2 ) )
= ( dvd_dvd_rat @ A @ C2 ) ) ) ).
% dvd_add_right_iff
thf(fact_3685_dvd__add__right__iff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C2 ) )
= ( dvd_dvd_nat @ A @ C2 ) ) ) ).
% dvd_add_right_iff
thf(fact_3686_dvd__add__right__iff,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( dvd_dvd_int @ A @ C2 ) ) ) ).
% dvd_add_right_iff
thf(fact_3687_dvd__add__left__iff,axiom,
! [A: code_integer,C2: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ C2 )
=> ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C2 ) )
= ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_3688_dvd__add__left__iff,axiom,
! [A: real,C2: real,B: real] :
( ( dvd_dvd_real @ A @ C2 )
=> ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C2 ) )
= ( dvd_dvd_real @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_3689_dvd__add__left__iff,axiom,
! [A: rat,C2: rat,B: rat] :
( ( dvd_dvd_rat @ A @ C2 )
=> ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C2 ) )
= ( dvd_dvd_rat @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_3690_dvd__add__left__iff,axiom,
! [A: nat,C2: nat,B: nat] :
( ( dvd_dvd_nat @ A @ C2 )
=> ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C2 ) )
= ( dvd_dvd_nat @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_3691_dvd__add__left__iff,axiom,
! [A: int,C2: int,B: int] :
( ( dvd_dvd_int @ A @ C2 )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_3692_dvd__add,axiom,
! [A: code_integer,B: code_integer,C2: code_integer] :
( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( dvd_dvd_Code_integer @ A @ C2 )
=> ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C2 ) ) ) ) ).
% dvd_add
thf(fact_3693_dvd__add,axiom,
! [A: real,B: real,C2: real] :
( ( dvd_dvd_real @ A @ B )
=> ( ( dvd_dvd_real @ A @ C2 )
=> ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ) ).
% dvd_add
thf(fact_3694_dvd__add,axiom,
! [A: rat,B: rat,C2: rat] :
( ( dvd_dvd_rat @ A @ B )
=> ( ( dvd_dvd_rat @ A @ C2 )
=> ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ) ).
% dvd_add
thf(fact_3695_dvd__add,axiom,
! [A: nat,B: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ C2 )
=> ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ) ).
% dvd_add
thf(fact_3696_dvd__add,axiom,
! [A: int,B: int,C2: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ C2 )
=> ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ) ).
% dvd_add
thf(fact_3697_gcd__nat_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_3698_gcd__nat_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ( dvd_dvd_nat @ A @ zero_zero_nat )
& ( A != zero_zero_nat ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_3699_gcd__nat_Oextremum__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_3700_gcd__nat_Oextremum__strict,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
& ( zero_zero_nat != A ) ) ).
% gcd_nat.extremum_strict
thf(fact_3701_gcd__nat_Oextremum,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_3702_zdvd__mult__cancel,axiom,
! [K2: int,M2: int,N: int] :
( ( dvd_dvd_int @ ( times_times_int @ K2 @ M2 ) @ ( times_times_int @ K2 @ N ) )
=> ( ( K2 != zero_zero_int )
=> ( dvd_dvd_int @ M2 @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_3703_zdvd__mono,axiom,
! [K2: int,M2: int,T: int] :
( ( K2 != zero_zero_int )
=> ( ( dvd_dvd_int @ M2 @ T )
= ( dvd_dvd_int @ ( times_times_int @ K2 @ M2 ) @ ( times_times_int @ K2 @ T ) ) ) ) ).
% zdvd_mono
thf(fact_3704_signed__take__bit__add,axiom,
! [N: nat,K2: int,L: int] :
( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
= ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K2 @ L ) ) ) ).
% signed_take_bit_add
thf(fact_3705_zdvd__reduce,axiom,
! [K2: int,N: int,M2: int] :
( ( dvd_dvd_int @ K2 @ ( plus_plus_int @ N @ ( times_times_int @ K2 @ M2 ) ) )
= ( dvd_dvd_int @ K2 @ N ) ) ).
% zdvd_reduce
thf(fact_3706_zdvd__period,axiom,
! [A: int,D: int,X3: int,T: int,C2: int] :
( ( dvd_dvd_int @ A @ D )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X3 @ T ) )
= ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X3 @ ( times_times_int @ C2 @ D ) ) @ T ) ) ) ) ).
% zdvd_period
thf(fact_3707_not__is__unit__0,axiom,
~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% not_is_unit_0
thf(fact_3708_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_3709_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_3710_pinf_I9_J,axiom,
! [D: code_integer,S3: code_integer] :
? [Z3: code_integer] :
! [X: code_integer] :
( ( ord_le6747313008572928689nteger @ Z3 @ X )
=> ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X @ S3 ) )
= ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X @ S3 ) ) ) ) ).
% pinf(9)
thf(fact_3711_pinf_I9_J,axiom,
! [D: real,S3: real] :
? [Z3: real] :
! [X: real] :
( ( ord_less_real @ Z3 @ X )
=> ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X @ S3 ) )
= ( dvd_dvd_real @ D @ ( plus_plus_real @ X @ S3 ) ) ) ) ).
% pinf(9)
thf(fact_3712_pinf_I9_J,axiom,
! [D: rat,S3: rat] :
? [Z3: rat] :
! [X: rat] :
( ( ord_less_rat @ Z3 @ X )
=> ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X @ S3 ) )
= ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X @ S3 ) ) ) ) ).
% pinf(9)
thf(fact_3713_pinf_I9_J,axiom,
! [D: nat,S3: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X @ S3 ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X @ S3 ) ) ) ) ).
% pinf(9)
thf(fact_3714_pinf_I9_J,axiom,
! [D: int,S3: int] :
? [Z3: int] :
! [X: int] :
( ( ord_less_int @ Z3 @ X )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X @ S3 ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X @ S3 ) ) ) ) ).
% pinf(9)
thf(fact_3715_pinf_I10_J,axiom,
! [D: code_integer,S3: code_integer] :
? [Z3: code_integer] :
! [X: code_integer] :
( ( ord_le6747313008572928689nteger @ Z3 @ X )
=> ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X @ S3 ) ) )
= ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X @ S3 ) ) ) ) ) ).
% pinf(10)
thf(fact_3716_pinf_I10_J,axiom,
! [D: real,S3: real] :
? [Z3: real] :
! [X: real] :
( ( ord_less_real @ Z3 @ X )
=> ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X @ S3 ) ) )
= ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X @ S3 ) ) ) ) ) ).
% pinf(10)
thf(fact_3717_pinf_I10_J,axiom,
! [D: rat,S3: rat] :
? [Z3: rat] :
! [X: rat] :
( ( ord_less_rat @ Z3 @ X )
=> ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X @ S3 ) ) )
= ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X @ S3 ) ) ) ) ) ).
% pinf(10)
thf(fact_3718_pinf_I10_J,axiom,
! [D: nat,S3: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ Z3 @ X )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X @ S3 ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X @ S3 ) ) ) ) ) ).
% pinf(10)
thf(fact_3719_pinf_I10_J,axiom,
! [D: int,S3: int] :
? [Z3: int] :
! [X: int] :
( ( ord_less_int @ Z3 @ X )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X @ S3 ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X @ S3 ) ) ) ) ) ).
% pinf(10)
thf(fact_3720_minf_I9_J,axiom,
! [D: code_integer,S3: code_integer] :
? [Z3: code_integer] :
! [X: code_integer] :
( ( ord_le6747313008572928689nteger @ X @ Z3 )
=> ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X @ S3 ) )
= ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X @ S3 ) ) ) ) ).
% minf(9)
thf(fact_3721_minf_I9_J,axiom,
! [D: real,S3: real] :
? [Z3: real] :
! [X: real] :
( ( ord_less_real @ X @ Z3 )
=> ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X @ S3 ) )
= ( dvd_dvd_real @ D @ ( plus_plus_real @ X @ S3 ) ) ) ) ).
% minf(9)
thf(fact_3722_minf_I9_J,axiom,
! [D: rat,S3: rat] :
? [Z3: rat] :
! [X: rat] :
( ( ord_less_rat @ X @ Z3 )
=> ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X @ S3 ) )
= ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X @ S3 ) ) ) ) ).
% minf(9)
thf(fact_3723_minf_I9_J,axiom,
! [D: nat,S3: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X @ S3 ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X @ S3 ) ) ) ) ).
% minf(9)
thf(fact_3724_minf_I9_J,axiom,
! [D: int,S3: int] :
? [Z3: int] :
! [X: int] :
( ( ord_less_int @ X @ Z3 )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X @ S3 ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X @ S3 ) ) ) ) ).
% minf(9)
thf(fact_3725_minf_I10_J,axiom,
! [D: code_integer,S3: code_integer] :
? [Z3: code_integer] :
! [X: code_integer] :
( ( ord_le6747313008572928689nteger @ X @ Z3 )
=> ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X @ S3 ) ) )
= ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X @ S3 ) ) ) ) ) ).
% minf(10)
thf(fact_3726_minf_I10_J,axiom,
! [D: real,S3: real] :
? [Z3: real] :
! [X: real] :
( ( ord_less_real @ X @ Z3 )
=> ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X @ S3 ) ) )
= ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X @ S3 ) ) ) ) ) ).
% minf(10)
thf(fact_3727_minf_I10_J,axiom,
! [D: rat,S3: rat] :
? [Z3: rat] :
! [X: rat] :
( ( ord_less_rat @ X @ Z3 )
=> ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X @ S3 ) ) )
= ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X @ S3 ) ) ) ) ) ).
% minf(10)
thf(fact_3728_minf_I10_J,axiom,
! [D: nat,S3: nat] :
? [Z3: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z3 )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X @ S3 ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X @ S3 ) ) ) ) ) ).
% minf(10)
thf(fact_3729_minf_I10_J,axiom,
! [D: int,S3: int] :
? [Z3: int] :
! [X: int] :
( ( ord_less_int @ X @ Z3 )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X @ S3 ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X @ S3 ) ) ) ) ) ).
% minf(10)
thf(fact_3730_dvd__div__eq__0__iff,axiom,
! [B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( ( divide6298287555418463151nteger @ A @ B )
= zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_3731_dvd__div__eq__0__iff,axiom,
! [B: complex,A: complex] :
( ( dvd_dvd_complex @ B @ A )
=> ( ( ( divide1717551699836669952omplex @ A @ B )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_3732_dvd__div__eq__0__iff,axiom,
! [B: real,A: real] :
( ( dvd_dvd_real @ B @ A )
=> ( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( A = zero_zero_real ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_3733_dvd__div__eq__0__iff,axiom,
! [B: rat,A: rat] :
( ( dvd_dvd_rat @ B @ A )
=> ( ( ( divide_divide_rat @ A @ B )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_3734_dvd__div__eq__0__iff,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ( ( ( divide_divide_nat @ A @ B )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_3735_dvd__div__eq__0__iff,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ( ( ( divide_divide_int @ A @ B )
= zero_zero_int )
= ( A = zero_zero_int ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_3736_div__plus__div__distrib__dvd__left,axiom,
! [C2: code_integer,A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ C2 @ A )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C2 )
= ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C2 ) @ ( divide6298287555418463151nteger @ B @ C2 ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_3737_div__plus__div__distrib__dvd__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C2 @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C2 ) @ ( divide_divide_nat @ B @ C2 ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_3738_div__plus__div__distrib__dvd__left,axiom,
! [C2: int,A: int,B: int] :
( ( dvd_dvd_int @ C2 @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( divide_divide_int @ A @ C2 ) @ ( divide_divide_int @ B @ C2 ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_3739_div__plus__div__distrib__dvd__right,axiom,
! [C2: code_integer,B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ C2 @ B )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C2 )
= ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C2 ) @ ( divide6298287555418463151nteger @ B @ C2 ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_3740_div__plus__div__distrib__dvd__right,axiom,
! [C2: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C2 @ B )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C2 ) @ ( divide_divide_nat @ B @ C2 ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_3741_div__plus__div__distrib__dvd__right,axiom,
! [C2: int,B: int,A: int] :
( ( dvd_dvd_int @ C2 @ B )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( divide_divide_int @ A @ C2 ) @ ( divide_divide_int @ B @ C2 ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_3742_mod__0__imp__dvd,axiom,
! [A: nat,B: nat] :
( ( ( modulo_modulo_nat @ A @ B )
= zero_zero_nat )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% mod_0_imp_dvd
thf(fact_3743_mod__0__imp__dvd,axiom,
! [A: int,B: int] :
( ( ( modulo_modulo_int @ A @ B )
= zero_zero_int )
=> ( dvd_dvd_int @ B @ A ) ) ).
% mod_0_imp_dvd
thf(fact_3744_mod__0__imp__dvd,axiom,
! [A: code_integer,B: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ B )
= zero_z3403309356797280102nteger )
=> ( dvd_dvd_Code_integer @ B @ A ) ) ).
% mod_0_imp_dvd
thf(fact_3745_dvd__eq__mod__eq__0,axiom,
( dvd_dvd_nat
= ( ^ [A5: nat,B4: nat] :
( ( modulo_modulo_nat @ B4 @ A5 )
= zero_zero_nat ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_3746_dvd__eq__mod__eq__0,axiom,
( dvd_dvd_int
= ( ^ [A5: int,B4: int] :
( ( modulo_modulo_int @ B4 @ A5 )
= zero_zero_int ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_3747_dvd__eq__mod__eq__0,axiom,
( dvd_dvd_Code_integer
= ( ^ [A5: code_integer,B4: code_integer] :
( ( modulo364778990260209775nteger @ B4 @ A5 )
= zero_z3403309356797280102nteger ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_3748_mod__eq__0__iff__dvd,axiom,
! [A: nat,B: nat] :
( ( ( modulo_modulo_nat @ A @ B )
= zero_zero_nat )
= ( dvd_dvd_nat @ B @ A ) ) ).
% mod_eq_0_iff_dvd
thf(fact_3749_mod__eq__0__iff__dvd,axiom,
! [A: int,B: int] :
( ( ( modulo_modulo_int @ A @ B )
= zero_zero_int )
= ( dvd_dvd_int @ B @ A ) ) ).
% mod_eq_0_iff_dvd
thf(fact_3750_mod__eq__0__iff__dvd,axiom,
! [A: code_integer,B: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ B )
= zero_z3403309356797280102nteger )
= ( dvd_dvd_Code_integer @ B @ A ) ) ).
% mod_eq_0_iff_dvd
thf(fact_3751_le__imp__power__dvd,axiom,
! [M2: nat,N: nat,A: code_integer] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M2 ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_3752_le__imp__power__dvd,axiom,
! [M2: nat,N: nat,A: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( dvd_dvd_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_3753_le__imp__power__dvd,axiom,
! [M2: nat,N: nat,A: real] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( dvd_dvd_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_3754_le__imp__power__dvd,axiom,
! [M2: nat,N: nat,A: int] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( dvd_dvd_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_3755_le__imp__power__dvd,axiom,
! [M2: nat,N: nat,A: complex] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( dvd_dvd_complex @ ( power_power_complex @ A @ M2 ) @ ( power_power_complex @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_3756_power__le__dvd,axiom,
! [A: code_integer,N: nat,B: code_integer,M2: nat] :
( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M2 ) @ B ) ) ) ).
% power_le_dvd
thf(fact_3757_power__le__dvd,axiom,
! [A: nat,N: nat,B: nat,M2: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( dvd_dvd_nat @ ( power_power_nat @ A @ M2 ) @ B ) ) ) ).
% power_le_dvd
thf(fact_3758_power__le__dvd,axiom,
! [A: real,N: nat,B: real,M2: nat] :
( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( dvd_dvd_real @ ( power_power_real @ A @ M2 ) @ B ) ) ) ).
% power_le_dvd
thf(fact_3759_power__le__dvd,axiom,
! [A: int,N: nat,B: int,M2: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( dvd_dvd_int @ ( power_power_int @ A @ M2 ) @ B ) ) ) ).
% power_le_dvd
thf(fact_3760_power__le__dvd,axiom,
! [A: complex,N: nat,B: complex,M2: nat] :
( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( dvd_dvd_complex @ ( power_power_complex @ A @ M2 ) @ B ) ) ) ).
% power_le_dvd
thf(fact_3761_dvd__power__le,axiom,
! [X3: code_integer,Y: code_integer,N: nat,M2: nat] :
( ( dvd_dvd_Code_integer @ X3 @ Y )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X3 @ N ) @ ( power_8256067586552552935nteger @ Y @ M2 ) ) ) ) ).
% dvd_power_le
thf(fact_3762_dvd__power__le,axiom,
! [X3: nat,Y: nat,N: nat,M2: nat] :
( ( dvd_dvd_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( dvd_dvd_nat @ ( power_power_nat @ X3 @ N ) @ ( power_power_nat @ Y @ M2 ) ) ) ) ).
% dvd_power_le
thf(fact_3763_dvd__power__le,axiom,
! [X3: real,Y: real,N: nat,M2: nat] :
( ( dvd_dvd_real @ X3 @ Y )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( dvd_dvd_real @ ( power_power_real @ X3 @ N ) @ ( power_power_real @ Y @ M2 ) ) ) ) ).
% dvd_power_le
thf(fact_3764_dvd__power__le,axiom,
! [X3: int,Y: int,N: nat,M2: nat] :
( ( dvd_dvd_int @ X3 @ Y )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( dvd_dvd_int @ ( power_power_int @ X3 @ N ) @ ( power_power_int @ Y @ M2 ) ) ) ) ).
% dvd_power_le
thf(fact_3765_dvd__power__le,axiom,
! [X3: complex,Y: complex,N: nat,M2: nat] :
( ( dvd_dvd_complex @ X3 @ Y )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( dvd_dvd_complex @ ( power_power_complex @ X3 @ N ) @ ( power_power_complex @ Y @ M2 ) ) ) ) ).
% dvd_power_le
thf(fact_3766_dvd__pos__nat,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M2 @ N )
=> ( ord_less_nat @ zero_zero_nat @ M2 ) ) ) ).
% dvd_pos_nat
thf(fact_3767_nat__dvd__not__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_nat @ M2 @ N )
=> ~ ( dvd_dvd_nat @ N @ M2 ) ) ) ).
% nat_dvd_not_less
thf(fact_3768_zdvd__antisym__nonneg,axiom,
! [M2: int,N: int] :
( ( ord_less_eq_int @ zero_zero_int @ M2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( dvd_dvd_int @ M2 @ N )
=> ( ( dvd_dvd_int @ N @ M2 )
=> ( M2 = N ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_3769_zdvd__not__zless,axiom,
! [M2: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M2 )
=> ( ( ord_less_int @ M2 @ N )
=> ~ ( dvd_dvd_int @ N @ M2 ) ) ) ).
% zdvd_not_zless
thf(fact_3770_bezout__add__nat,axiom,
! [A: nat,B: nat] :
? [D2: nat,X4: nat,Y3: nat] :
( ( dvd_dvd_nat @ D2 @ A )
& ( dvd_dvd_nat @ D2 @ B )
& ( ( ( times_times_nat @ A @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D2 ) )
| ( ( times_times_nat @ B @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D2 ) ) ) ) ).
% bezout_add_nat
thf(fact_3771_bezout__lemma__nat,axiom,
! [D: nat,A: nat,B: nat,X3: nat,Y: nat] :
( ( dvd_dvd_nat @ D @ A )
=> ( ( dvd_dvd_nat @ D @ B )
=> ( ( ( ( times_times_nat @ A @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D ) )
| ( ( times_times_nat @ B @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) )
=> ? [X4: nat,Y3: nat] :
( ( dvd_dvd_nat @ D @ A )
& ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
& ( ( ( times_times_nat @ A @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y3 ) @ D ) )
| ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_3772_unit__dvdE,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ~ ( ( A != zero_z3403309356797280102nteger )
=> ! [C: code_integer] :
( B
!= ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).
% unit_dvdE
thf(fact_3773_unit__dvdE,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ~ ( ( A != zero_zero_nat )
=> ! [C: nat] :
( B
!= ( times_times_nat @ A @ C ) ) ) ) ).
% unit_dvdE
thf(fact_3774_unit__dvdE,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ~ ( ( A != zero_zero_int )
=> ! [C: int] :
( B
!= ( times_times_int @ A @ C ) ) ) ) ).
% unit_dvdE
thf(fact_3775_unity__coeff__ex,axiom,
! [P2: code_integer > $o,L: code_integer] :
( ( ? [X5: code_integer] : ( P2 @ ( times_3573771949741848930nteger @ L @ X5 ) ) )
= ( ? [X5: code_integer] :
( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X5 @ zero_z3403309356797280102nteger ) )
& ( P2 @ X5 ) ) ) ) ).
% unity_coeff_ex
thf(fact_3776_unity__coeff__ex,axiom,
! [P2: complex > $o,L: complex] :
( ( ? [X5: complex] : ( P2 @ ( times_times_complex @ L @ X5 ) ) )
= ( ? [X5: complex] :
( ( dvd_dvd_complex @ L @ ( plus_plus_complex @ X5 @ zero_zero_complex ) )
& ( P2 @ X5 ) ) ) ) ).
% unity_coeff_ex
thf(fact_3777_unity__coeff__ex,axiom,
! [P2: real > $o,L: real] :
( ( ? [X5: real] : ( P2 @ ( times_times_real @ L @ X5 ) ) )
= ( ? [X5: real] :
( ( dvd_dvd_real @ L @ ( plus_plus_real @ X5 @ zero_zero_real ) )
& ( P2 @ X5 ) ) ) ) ).
% unity_coeff_ex
thf(fact_3778_unity__coeff__ex,axiom,
! [P2: rat > $o,L: rat] :
( ( ? [X5: rat] : ( P2 @ ( times_times_rat @ L @ X5 ) ) )
= ( ? [X5: rat] :
( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X5 @ zero_zero_rat ) )
& ( P2 @ X5 ) ) ) ) ).
% unity_coeff_ex
thf(fact_3779_unity__coeff__ex,axiom,
! [P2: nat > $o,L: nat] :
( ( ? [X5: nat] : ( P2 @ ( times_times_nat @ L @ X5 ) ) )
= ( ? [X5: nat] :
( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X5 @ zero_zero_nat ) )
& ( P2 @ X5 ) ) ) ) ).
% unity_coeff_ex
thf(fact_3780_unity__coeff__ex,axiom,
! [P2: int > $o,L: int] :
( ( ? [X5: int] : ( P2 @ ( times_times_int @ L @ X5 ) ) )
= ( ? [X5: int] :
( ( dvd_dvd_int @ L @ ( plus_plus_int @ X5 @ zero_zero_int ) )
& ( P2 @ X5 ) ) ) ) ).
% unity_coeff_ex
thf(fact_3781_dvd__div__eq__mult,axiom,
! [A: code_integer,B: code_integer,C2: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( ( divide6298287555418463151nteger @ B @ A )
= C2 )
= ( B
= ( times_3573771949741848930nteger @ C2 @ A ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_3782_dvd__div__eq__mult,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ A @ B )
=> ( ( ( divide_divide_nat @ B @ A )
= C2 )
= ( B
= ( times_times_nat @ C2 @ A ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_3783_dvd__div__eq__mult,axiom,
! [A: int,B: int,C2: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ A @ B )
=> ( ( ( divide_divide_int @ B @ A )
= C2 )
= ( B
= ( times_times_int @ C2 @ A ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_3784_div__dvd__iff__mult,axiom,
! [B: code_integer,A: code_integer,C2: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C2 )
= ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C2 @ B ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_3785_div__dvd__iff__mult,axiom,
! [B: nat,A: nat,C2: nat] :
( ( B != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C2 )
= ( dvd_dvd_nat @ A @ ( times_times_nat @ C2 @ B ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_3786_div__dvd__iff__mult,axiom,
! [B: int,A: int,C2: int] :
( ( B != zero_zero_int )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C2 )
= ( dvd_dvd_int @ A @ ( times_times_int @ C2 @ B ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_3787_dvd__div__iff__mult,axiom,
! [C2: code_integer,B: code_integer,A: code_integer] :
( ( C2 != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ C2 @ B )
=> ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C2 ) )
= ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C2 ) @ B ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_3788_dvd__div__iff__mult,axiom,
! [C2: nat,B: nat,A: nat] :
( ( C2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ C2 @ B )
=> ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C2 ) )
= ( dvd_dvd_nat @ ( times_times_nat @ A @ C2 ) @ B ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_3789_dvd__div__iff__mult,axiom,
! [C2: int,B: int,A: int] :
( ( C2 != zero_zero_int )
=> ( ( dvd_dvd_int @ C2 @ B )
=> ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C2 ) )
= ( dvd_dvd_int @ ( times_times_int @ A @ C2 ) @ B ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_3790_dvd__div__div__eq__mult,axiom,
! [A: code_integer,C2: code_integer,B: code_integer,D: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( C2 != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( dvd_dvd_Code_integer @ C2 @ D )
=> ( ( ( divide6298287555418463151nteger @ B @ A )
= ( divide6298287555418463151nteger @ D @ C2 ) )
= ( ( times_3573771949741848930nteger @ B @ C2 )
= ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_3791_dvd__div__div__eq__mult,axiom,
! [A: nat,C2: nat,B: nat,D: nat] :
( ( A != zero_zero_nat )
=> ( ( C2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ C2 @ D )
=> ( ( ( divide_divide_nat @ B @ A )
= ( divide_divide_nat @ D @ C2 ) )
= ( ( times_times_nat @ B @ C2 )
= ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_3792_dvd__div__div__eq__mult,axiom,
! [A: int,C2: int,B: int,D: int] :
( ( A != zero_zero_int )
=> ( ( C2 != zero_zero_int )
=> ( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ C2 @ D )
=> ( ( ( divide_divide_int @ B @ A )
= ( divide_divide_int @ D @ C2 ) )
= ( ( times_times_int @ B @ C2 )
= ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_3793_unit__div__eq__0__iff,axiom,
! [B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( ( divide6298287555418463151nteger @ A @ B )
= zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ) ).
% unit_div_eq_0_iff
thf(fact_3794_unit__div__eq__0__iff,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( ( divide_divide_nat @ A @ B )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ) ).
% unit_div_eq_0_iff
thf(fact_3795_unit__div__eq__0__iff,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( ( divide_divide_int @ A @ B )
= zero_zero_int )
= ( A = zero_zero_int ) ) ) ).
% unit_div_eq_0_iff
thf(fact_3796_unit__imp__mod__eq__0,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( modulo_modulo_nat @ A @ B )
= zero_zero_nat ) ) ).
% unit_imp_mod_eq_0
thf(fact_3797_unit__imp__mod__eq__0,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( modulo_modulo_int @ A @ B )
= zero_zero_int ) ) ).
% unit_imp_mod_eq_0
thf(fact_3798_unit__imp__mod__eq__0,axiom,
! [B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( modulo364778990260209775nteger @ A @ B )
= zero_z3403309356797280102nteger ) ) ).
% unit_imp_mod_eq_0
thf(fact_3799_is__unit__power__iff,axiom,
! [A: code_integer,N: nat] :
( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer )
= ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_3800_is__unit__power__iff,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_3801_is__unit__power__iff,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_3802_dvd__imp__le,axiom,
! [K2: nat,N: nat] :
( ( dvd_dvd_nat @ K2 @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% dvd_imp_le
thf(fact_3803_dvd__mult__cancel,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( dvd_dvd_nat @ M2 @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_3804_nat__mult__dvd__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= ( dvd_dvd_nat @ M2 @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_3805_bezout__add__strong__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [D2: nat,X4: nat,Y3: nat] :
( ( dvd_dvd_nat @ D2 @ A )
& ( dvd_dvd_nat @ D2 @ B )
& ( ( times_times_nat @ A @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D2 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_3806_zdvd__imp__le,axiom,
! [Z2: int,N: int] :
( ( dvd_dvd_int @ Z2 @ N )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int @ Z2 @ N ) ) ) ).
% zdvd_imp_le
thf(fact_3807_mod__greater__zero__iff__not__dvd,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M2 @ N ) )
= ( ~ ( dvd_dvd_nat @ N @ M2 ) ) ) ).
% mod_greater_zero_iff_not_dvd
thf(fact_3808_even__zero,axiom,
dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).
% even_zero
thf(fact_3809_even__zero,axiom,
dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% even_zero
thf(fact_3810_even__zero,axiom,
dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% even_zero
thf(fact_3811_is__unit__div__mult__cancel__right,axiom,
! [A: code_integer,B: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
= ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_3812_is__unit__div__mult__cancel__right,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
= ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_3813_is__unit__div__mult__cancel__right,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
= ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_3814_is__unit__div__mult__cancel__left,axiom,
! [A: code_integer,B: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
= ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_3815_is__unit__div__mult__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
= ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_3816_is__unit__div__mult__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
= ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_3817_is__unitE,axiom,
! [A: code_integer,C2: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ~ ( ( A != zero_z3403309356797280102nteger )
=> ! [B3: code_integer] :
( ( B3 != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
=> ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
= B3 )
=> ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B3 )
= A )
=> ( ( ( times_3573771949741848930nteger @ A @ B3 )
= one_one_Code_integer )
=> ( ( divide6298287555418463151nteger @ C2 @ A )
!= ( times_3573771949741848930nteger @ C2 @ B3 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_3818_is__unitE,axiom,
! [A: nat,C2: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ~ ( ( A != zero_zero_nat )
=> ! [B3: nat] :
( ( B3 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B3 @ one_one_nat )
=> ( ( ( divide_divide_nat @ one_one_nat @ A )
= B3 )
=> ( ( ( divide_divide_nat @ one_one_nat @ B3 )
= A )
=> ( ( ( times_times_nat @ A @ B3 )
= one_one_nat )
=> ( ( divide_divide_nat @ C2 @ A )
!= ( times_times_nat @ C2 @ B3 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_3819_is__unitE,axiom,
! [A: int,C2: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ~ ( ( A != zero_zero_int )
=> ! [B3: int] :
( ( B3 != zero_zero_int )
=> ( ( dvd_dvd_int @ B3 @ one_one_int )
=> ( ( ( divide_divide_int @ one_one_int @ A )
= B3 )
=> ( ( ( divide_divide_int @ one_one_int @ B3 )
= A )
=> ( ( ( times_times_int @ A @ B3 )
= one_one_int )
=> ( ( divide_divide_int @ C2 @ A )
!= ( times_times_int @ C2 @ B3 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_3820_odd__even__add,axiom,
! [A: code_integer,B: code_integer] :
( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).
% odd_even_add
thf(fact_3821_odd__even__add,axiom,
! [A: nat,B: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% odd_even_add
thf(fact_3822_odd__even__add,axiom,
! [A: int,B: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).
% odd_even_add
thf(fact_3823_dvd__power__iff,axiom,
! [X3: code_integer,M2: nat,N: nat] :
( ( X3 != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X3 @ M2 ) @ ( power_8256067586552552935nteger @ X3 @ N ) )
= ( ( dvd_dvd_Code_integer @ X3 @ one_one_Code_integer )
| ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_3824_dvd__power__iff,axiom,
! [X3: nat,M2: nat,N: nat] :
( ( X3 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ X3 @ M2 ) @ ( power_power_nat @ X3 @ N ) )
= ( ( dvd_dvd_nat @ X3 @ one_one_nat )
| ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_3825_dvd__power__iff,axiom,
! [X3: int,M2: nat,N: nat] :
( ( X3 != zero_zero_int )
=> ( ( dvd_dvd_int @ ( power_power_int @ X3 @ M2 ) @ ( power_power_int @ X3 @ N ) )
= ( ( dvd_dvd_int @ X3 @ one_one_int )
| ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_3826_dvd__power,axiom,
! [N: nat,X3: code_integer] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X3 = one_one_Code_integer ) )
=> ( dvd_dvd_Code_integer @ X3 @ ( power_8256067586552552935nteger @ X3 @ N ) ) ) ).
% dvd_power
thf(fact_3827_dvd__power,axiom,
! [N: nat,X3: rat] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X3 = one_one_rat ) )
=> ( dvd_dvd_rat @ X3 @ ( power_power_rat @ X3 @ N ) ) ) ).
% dvd_power
thf(fact_3828_dvd__power,axiom,
! [N: nat,X3: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X3 = one_one_nat ) )
=> ( dvd_dvd_nat @ X3 @ ( power_power_nat @ X3 @ N ) ) ) ).
% dvd_power
thf(fact_3829_dvd__power,axiom,
! [N: nat,X3: real] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X3 = one_one_real ) )
=> ( dvd_dvd_real @ X3 @ ( power_power_real @ X3 @ N ) ) ) ).
% dvd_power
thf(fact_3830_dvd__power,axiom,
! [N: nat,X3: int] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X3 = one_one_int ) )
=> ( dvd_dvd_int @ X3 @ ( power_power_int @ X3 @ N ) ) ) ).
% dvd_power
thf(fact_3831_dvd__power,axiom,
! [N: nat,X3: complex] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X3 = one_one_complex ) )
=> ( dvd_dvd_complex @ X3 @ ( power_power_complex @ X3 @ N ) ) ) ).
% dvd_power
thf(fact_3832_div2__even__ext__nat,axiom,
! [X3: nat,Y: nat] :
( ( ( divide_divide_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X3 )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y ) )
=> ( X3 = Y ) ) ) ).
% div2_even_ext_nat
thf(fact_3833_dvd__mult__cancel1,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ M2 @ N ) @ M2 )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel1
thf(fact_3834_dvd__mult__cancel2,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M2 ) @ M2 )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel2
thf(fact_3835_power__dvd__imp__le,axiom,
! [I: nat,M2: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ I @ M2 ) @ ( power_power_nat @ I @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ I )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_3836_mod__int__pos__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K2 @ L ) )
= ( ( dvd_dvd_int @ L @ K2 )
| ( ( L = zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ K2 ) )
| ( ord_less_int @ zero_zero_int @ L ) ) ) ).
% mod_int_pos_iff
thf(fact_3837_even__iff__mod__2__eq__zero,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
= ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_3838_even__iff__mod__2__eq__zero,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
= ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_3839_even__iff__mod__2__eq__zero,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
= ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= zero_z3403309356797280102nteger ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_3840_power__mono__odd,axiom,
! [N: nat,A: real,B: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% power_mono_odd
thf(fact_3841_power__mono__odd,axiom,
! [N: nat,A: rat,B: rat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% power_mono_odd
thf(fact_3842_power__mono__odd,axiom,
! [N: nat,A: int,B: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono_odd
thf(fact_3843_odd__pos,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% odd_pos
thf(fact_3844_dvd__power__iff__le,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ K2 @ M2 ) @ ( power_power_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% dvd_power_iff_le
thf(fact_3845_even__unset__bit__iff,axiom,
! [M2: nat,A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M2 @ A ) )
= ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
| ( M2 = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
thf(fact_3846_even__unset__bit__iff,axiom,
! [M2: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M2 @ A ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
| ( M2 = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
thf(fact_3847_even__unset__bit__iff,axiom,
! [M2: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M2 @ A ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
| ( M2 = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
thf(fact_3848_even__set__bit__iff,axiom,
! [M2: nat,A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M2 @ A ) )
= ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
& ( M2 != zero_zero_nat ) ) ) ).
% even_set_bit_iff
thf(fact_3849_even__set__bit__iff,axiom,
! [M2: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M2 @ A ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
& ( M2 != zero_zero_nat ) ) ) ).
% even_set_bit_iff
thf(fact_3850_even__set__bit__iff,axiom,
! [M2: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M2 @ A ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
& ( M2 != zero_zero_nat ) ) ) ).
% even_set_bit_iff
thf(fact_3851_even__flip__bit__iff,axiom,
! [M2: nat,A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M2 @ A ) )
= ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
!= ( M2 = zero_zero_nat ) ) ) ).
% even_flip_bit_iff
thf(fact_3852_even__flip__bit__iff,axiom,
! [M2: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M2 @ A ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
!= ( M2 = zero_zero_nat ) ) ) ).
% even_flip_bit_iff
thf(fact_3853_even__flip__bit__iff,axiom,
! [M2: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M2 @ A ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
!= ( M2 = zero_zero_nat ) ) ) ).
% even_flip_bit_iff
thf(fact_3854_oddE,axiom,
! [A: code_integer] :
( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ~ ! [B3: code_integer] :
( A
!= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) @ one_one_Code_integer ) ) ) ).
% oddE
thf(fact_3855_oddE,axiom,
! [A: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ~ ! [B3: nat] :
( A
!= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) @ one_one_nat ) ) ) ).
% oddE
thf(fact_3856_oddE,axiom,
! [A: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ~ ! [B3: int] :
( A
!= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ one_one_int ) ) ) ).
% oddE
thf(fact_3857_parity__cases,axiom,
! [A: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= zero_zero_nat ) )
=> ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= one_one_nat ) ) ) ).
% parity_cases
thf(fact_3858_parity__cases,axiom,
! [A: int] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= zero_zero_int ) )
=> ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= one_one_int ) ) ) ).
% parity_cases
thf(fact_3859_parity__cases,axiom,
! [A: code_integer] :
( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
!= zero_z3403309356797280102nteger ) )
=> ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
!= one_one_Code_integer ) ) ) ).
% parity_cases
thf(fact_3860_mod2__eq__if,axiom,
! [A: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ) ).
% mod2_eq_if
thf(fact_3861_mod2__eq__if,axiom,
! [A: int] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) )
& ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ) ) ).
% mod2_eq_if
thf(fact_3862_mod2__eq__if,axiom,
! [A: code_integer] :
( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= zero_z3403309356797280102nteger ) )
& ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ) ) ).
% mod2_eq_if
thf(fact_3863_zero__le__power__eq,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% zero_le_power_eq
thf(fact_3864_zero__le__power__eq,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% zero_le_power_eq
thf(fact_3865_zero__le__power__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% zero_le_power_eq
thf(fact_3866_zero__le__odd__power,axiom,
! [N: nat,A: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% zero_le_odd_power
thf(fact_3867_zero__le__odd__power,axiom,
! [N: nat,A: rat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).
% zero_le_odd_power
thf(fact_3868_zero__le__odd__power,axiom,
! [N: nat,A: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% zero_le_odd_power
thf(fact_3869_zero__le__even__power,axiom,
! [N: nat,A: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_le_even_power
thf(fact_3870_zero__le__even__power,axiom,
! [N: nat,A: rat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% zero_le_even_power
thf(fact_3871_zero__le__even__power,axiom,
! [N: nat,A: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_even_power
thf(fact_3872_signed__take__bit__int__greater__eq__self__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq_int @ K2 @ ( bit_ri631733984087533419it_int @ N @ K2 ) )
= ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% signed_take_bit_int_greater_eq_self_iff
thf(fact_3873_signed__take__bit__int__less__self__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) @ K2 )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K2 ) ) ).
% signed_take_bit_int_less_self_iff
thf(fact_3874_even__set__encode__iff,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A4 ) )
= ( ~ ( member_nat @ zero_zero_nat @ A4 ) ) ) ) ).
% even_set_encode_iff
thf(fact_3875_zero__less__power__eq,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
= ( ( N = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A != zero_zero_real ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% zero_less_power_eq
thf(fact_3876_zero__less__power__eq,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
= ( ( N = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A != zero_zero_rat ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% zero_less_power_eq
thf(fact_3877_zero__less__power__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
= ( ( N = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A != zero_zero_int ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% zero_less_power_eq
thf(fact_3878_power__le__zero__eq,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_real @ A @ zero_zero_real ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A = zero_zero_real ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_3879_power__le__zero__eq,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_rat @ A @ zero_zero_rat ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A = zero_zero_rat ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_3880_power__le__zero__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_int @ A @ zero_zero_int ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A = zero_zero_int ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_3881_mult__less__iff1,axiom,
! [Z2: real,X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ Y @ Z2 ) )
= ( ord_less_real @ X3 @ Y ) ) ) ).
% mult_less_iff1
thf(fact_3882_mult__less__iff1,axiom,
! [Z2: rat,X3: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z2 )
=> ( ( ord_less_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ Y @ Z2 ) )
= ( ord_less_rat @ X3 @ Y ) ) ) ).
% mult_less_iff1
thf(fact_3883_mult__less__iff1,axiom,
! [Z2: int,X3: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_int @ ( times_times_int @ X3 @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
= ( ord_less_int @ X3 @ Y ) ) ) ).
% mult_less_iff1
thf(fact_3884_pos__eucl__rel__int__mult__2,axiom,
! [B: int,A: int,Q3: int,R2: int] :
( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
=> ( 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 ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_3885_concat__bit__Suc,axiom,
! [N: nat,K2: int,L: int] :
( ( bit_concat_bit @ ( suc @ N ) @ K2 @ L )
= ( plus_plus_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).
% concat_bit_Suc
thf(fact_3886_flip__bit__0,axiom,
! [A: code_integer] :
( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
= ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_3887_flip__bit__0,axiom,
! [A: nat] :
( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
= ( 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 ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_3888_flip__bit__0,axiom,
! [A: int] :
( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
= ( 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 ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_3889_vebt__insert_Oelims,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X3 @ Xa2 )
= Y )
=> ( ! [A3: $o,B3: $o] :
( ( X3
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> ( Y
= ( vEBT_Leaf @ $true @ B3 ) ) )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> ( Y
= ( vEBT_Leaf @ A3 @ $true ) ) )
& ( ( Xa2 != one_one_nat )
=> ( Y
= ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S ) )
=> ( Y
!= ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S ) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) )
=> ( Y
!= ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) ) )
=> ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( if_VEBT_VEBT
@ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
& ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ).
% vebt_insert.elims
thf(fact_3890_set__decode__Suc,axiom,
! [N: nat,X3: nat] :
( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X3 ) )
= ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% set_decode_Suc
thf(fact_3891_even__mult__exp__div__exp__iff,axiom,
! [A: code_integer,M2: nat,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ord_less_nat @ N @ M2 )
| ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
= zero_z3403309356797280102nteger )
| ( ( ord_less_eq_nat @ M2 @ N )
& ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_3892_even__mult__exp__div__exp__iff,axiom,
! [A: nat,M2: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ord_less_nat @ N @ M2 )
| ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= zero_zero_nat )
| ( ( ord_less_eq_nat @ M2 @ N )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_3893_even__mult__exp__div__exp__iff,axiom,
! [A: int,M2: nat,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ord_less_nat @ N @ M2 )
| ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
= zero_zero_int )
| ( ( ord_less_eq_nat @ M2 @ N )
& ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_3894_num_Osize__gen_I2_J,axiom,
! [X2: num] :
( ( size_num @ ( bit0 @ X2 ) )
= ( plus_plus_nat @ ( size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size_gen(2)
thf(fact_3895_neg__zmod__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( 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 ) )
= ( 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 ) ) ) ).
% neg_zmod_mult_2
thf(fact_3896_set__vebt_H__def,axiom,
( vEBT_VEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T3 ) ) ) ) ).
% set_vebt'_def
thf(fact_3897_finite__Collect__conjI,axiom,
! [P2: real > $o,Q: real > $o] :
( ( ( finite_finite_real @ ( collect_real @ P2 ) )
| ( finite_finite_real @ ( collect_real @ Q ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [X5: real] :
( ( P2 @ X5 )
& ( Q @ X5 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_3898_finite__Collect__conjI,axiom,
! [P2: list_nat > $o,Q: list_nat > $o] :
( ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P2 ) )
| ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [X5: list_nat] :
( ( P2 @ X5 )
& ( Q @ X5 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_3899_finite__Collect__conjI,axiom,
! [P2: set_nat > $o,Q: set_nat > $o] :
( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P2 ) )
| ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X5: set_nat] :
( ( P2 @ X5 )
& ( Q @ X5 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_3900_finite__Collect__conjI,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X5: nat] :
( ( P2 @ X5 )
& ( Q @ X5 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_3901_finite__Collect__conjI,axiom,
! [P2: int > $o,Q: int > $o] :
( ( ( finite_finite_int @ ( collect_int @ P2 ) )
| ( finite_finite_int @ ( collect_int @ Q ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X5: int] :
( ( P2 @ X5 )
& ( Q @ X5 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_3902_finite__Collect__conjI,axiom,
! [P2: complex > $o,Q: complex > $o] :
( ( ( finite3207457112153483333omplex @ ( collect_complex @ P2 ) )
| ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X5: complex] :
( ( P2 @ X5 )
& ( Q @ X5 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_3903_finite__Collect__conjI,axiom,
! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P2 ) )
| ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) )
=> ( finite6177210948735845034at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X5: product_prod_nat_nat] :
( ( P2 @ X5 )
& ( Q @ X5 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_3904_finite__Collect__disjI,axiom,
! [P2: real > $o,Q: real > $o] :
( ( finite_finite_real
@ ( collect_real
@ ^ [X5: real] :
( ( P2 @ X5 )
| ( Q @ X5 ) ) ) )
= ( ( finite_finite_real @ ( collect_real @ P2 ) )
& ( finite_finite_real @ ( collect_real @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_3905_finite__Collect__disjI,axiom,
! [P2: list_nat > $o,Q: list_nat > $o] :
( ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [X5: list_nat] :
( ( P2 @ X5 )
| ( Q @ X5 ) ) ) )
= ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P2 ) )
& ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_3906_finite__Collect__disjI,axiom,
! [P2: set_nat > $o,Q: set_nat > $o] :
( ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [X5: set_nat] :
( ( P2 @ X5 )
| ( Q @ X5 ) ) ) )
= ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P2 ) )
& ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_3907_finite__Collect__disjI,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X5: nat] :
( ( P2 @ X5 )
| ( Q @ X5 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_3908_finite__Collect__disjI,axiom,
! [P2: int > $o,Q: int > $o] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X5: int] :
( ( P2 @ X5 )
| ( Q @ X5 ) ) ) )
= ( ( finite_finite_int @ ( collect_int @ P2 ) )
& ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_3909_finite__Collect__disjI,axiom,
! [P2: complex > $o,Q: complex > $o] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X5: complex] :
( ( P2 @ X5 )
| ( Q @ X5 ) ) ) )
= ( ( finite3207457112153483333omplex @ ( collect_complex @ P2 ) )
& ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_3910_finite__Collect__disjI,axiom,
! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ( finite6177210948735845034at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X5: product_prod_nat_nat] :
( ( P2 @ X5 )
| ( Q @ X5 ) ) ) )
= ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P2 ) )
& ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_3911_finite__interval__int1,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( ord_less_eq_int @ A @ I3 )
& ( ord_less_eq_int @ I3 @ B ) ) ) ) ).
% finite_interval_int1
thf(fact_3912_finite__interval__int4,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( ord_less_int @ A @ I3 )
& ( ord_less_int @ I3 @ B ) ) ) ) ).
% finite_interval_int4
thf(fact_3913_diff__self,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% diff_self
thf(fact_3914_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_3915_diff__self,axiom,
! [A: rat] :
( ( minus_minus_rat @ A @ A )
= zero_zero_rat ) ).
% diff_self
thf(fact_3916_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_3917_diff__0__right,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% diff_0_right
thf(fact_3918_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_3919_diff__0__right,axiom,
! [A: rat] :
( ( minus_minus_rat @ A @ zero_zero_rat )
= A ) ).
% diff_0_right
thf(fact_3920_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_3921_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_3922_diff__zero,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% diff_zero
thf(fact_3923_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_3924_diff__zero,axiom,
! [A: rat] :
( ( minus_minus_rat @ A @ zero_zero_rat )
= A ) ).
% diff_zero
thf(fact_3925_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_3926_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_3927_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_3928_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_3929_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: rat] :
( ( minus_minus_rat @ A @ A )
= zero_zero_rat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_3930_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_3931_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_3932_add__diff__cancel,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_3933_add__diff__cancel,axiom,
! [A: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_3934_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_3935_diff__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_3936_diff__add__cancel,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_3937_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_3938_add__diff__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_3939_add__diff__cancel__left,axiom,
! [C2: rat,A: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
= ( minus_minus_rat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_3940_add__diff__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_3941_add__diff__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_3942_add__diff__cancel__left_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_3943_add__diff__cancel__left_H,axiom,
! [A: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_3944_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_3945_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_3946_add__diff__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_3947_add__diff__cancel__right,axiom,
! [A: rat,C2: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
= ( minus_minus_rat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_3948_add__diff__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_3949_add__diff__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_3950_add__diff__cancel__right_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_3951_add__diff__cancel__right_H,axiom,
! [A: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_3952_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_3953_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_3954_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_3955_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_3956_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_3957_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_3958_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_3959_diff__diff__left,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% diff_diff_left
thf(fact_3960_of__bool__less__eq__iff,axiom,
! [P2: $o,Q: $o] :
( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
= ( P2
=> Q ) ) ).
% of_bool_less_eq_iff
thf(fact_3961_of__bool__less__eq__iff,axiom,
! [P2: $o,Q: $o] :
( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
= ( P2
=> Q ) ) ).
% of_bool_less_eq_iff
thf(fact_3962_of__bool__less__eq__iff,axiom,
! [P2: $o,Q: $o] :
( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ ( zero_n2684676970156552555ol_int @ Q ) )
= ( P2
=> Q ) ) ).
% of_bool_less_eq_iff
thf(fact_3963_of__bool__eq_I1_J,axiom,
( ( zero_n1201886186963655149omplex @ $false )
= zero_zero_complex ) ).
% of_bool_eq(1)
thf(fact_3964_of__bool__eq_I1_J,axiom,
( ( zero_n3304061248610475627l_real @ $false )
= zero_zero_real ) ).
% of_bool_eq(1)
thf(fact_3965_of__bool__eq_I1_J,axiom,
( ( zero_n2052037380579107095ol_rat @ $false )
= zero_zero_rat ) ).
% of_bool_eq(1)
thf(fact_3966_of__bool__eq_I1_J,axiom,
( ( zero_n2687167440665602831ol_nat @ $false )
= zero_zero_nat ) ).
% of_bool_eq(1)
thf(fact_3967_of__bool__eq_I1_J,axiom,
( ( zero_n2684676970156552555ol_int @ $false )
= zero_zero_int ) ).
% of_bool_eq(1)
thf(fact_3968_of__bool__eq__0__iff,axiom,
! [P2: $o] :
( ( ( zero_n1201886186963655149omplex @ P2 )
= zero_zero_complex )
= ~ P2 ) ).
% of_bool_eq_0_iff
thf(fact_3969_of__bool__eq__0__iff,axiom,
! [P2: $o] :
( ( ( zero_n3304061248610475627l_real @ P2 )
= zero_zero_real )
= ~ P2 ) ).
% of_bool_eq_0_iff
thf(fact_3970_of__bool__eq__0__iff,axiom,
! [P2: $o] :
( ( ( zero_n2052037380579107095ol_rat @ P2 )
= zero_zero_rat )
= ~ P2 ) ).
% of_bool_eq_0_iff
thf(fact_3971_of__bool__eq__0__iff,axiom,
! [P2: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P2 )
= zero_zero_nat )
= ~ P2 ) ).
% of_bool_eq_0_iff
thf(fact_3972_of__bool__eq__0__iff,axiom,
! [P2: $o] :
( ( ( zero_n2684676970156552555ol_int @ P2 )
= zero_zero_int )
= ~ P2 ) ).
% of_bool_eq_0_iff
thf(fact_3973_concat__bit__0,axiom,
! [K2: int,L: int] :
( ( bit_concat_bit @ zero_zero_nat @ K2 @ L )
= L ) ).
% concat_bit_0
thf(fact_3974_finite__Collect__subsets,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A4 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_3975_finite__Collect__subsets,axiom,
! [A4: set_complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( finite6551019134538273531omplex
@ ( collect_set_complex
@ ^ [B6: set_complex] : ( ord_le211207098394363844omplex @ B6 @ A4 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_3976_finite__Collect__subsets,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( finite9047747110432174090at_nat
@ ( collec5514110066124741708at_nat
@ ^ [B6: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ B6 @ A4 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_3977_finite__Collect__subsets,axiom,
! [A4: set_int] :
( ( finite_finite_int @ A4 )
=> ( finite6197958912794628473et_int
@ ( collect_set_int
@ ^ [B6: set_int] : ( ord_less_eq_set_int @ B6 @ A4 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_3978_finite__interval__int3,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( ord_less_int @ A @ I3 )
& ( ord_less_eq_int @ I3 @ B ) ) ) ) ).
% finite_interval_int3
thf(fact_3979_finite__interval__int2,axiom,
! [A: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( ord_less_eq_int @ A @ I3 )
& ( ord_less_int @ I3 @ B ) ) ) ) ).
% finite_interval_int2
thf(fact_3980_finite__Collect__less__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_nat @ N4 @ K2 ) ) ) ).
% finite_Collect_less_nat
thf(fact_3981_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_3982_set__decode__inverse,axiom,
! [N: nat] :
( ( nat_set_encode @ ( nat_set_decode @ N ) )
= N ) ).
% set_decode_inverse
thf(fact_3983_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_3984_diff__ge__0__iff__ge,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
= ( ord_less_eq_rat @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_3985_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_3986_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_3987_diff__gt__0__iff__gt,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
= ( ord_less_rat @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_3988_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_3989_le__add__diff__inverse2,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_3990_le__add__diff__inverse2,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_3991_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_3992_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_3993_le__add__diff__inverse,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_3994_le__add__diff__inverse,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_3995_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_3996_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_3997_diff__numeral__special_I9_J,axiom,
( ( minus_minus_complex @ one_one_complex @ one_one_complex )
= zero_zero_complex ) ).
% diff_numeral_special(9)
thf(fact_3998_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_3999_diff__numeral__special_I9_J,axiom,
( ( minus_minus_rat @ one_one_rat @ one_one_rat )
= zero_zero_rat ) ).
% diff_numeral_special(9)
thf(fact_4000_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_4001_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_4002_zero__less__of__bool__iff,axiom,
! [P2: $o] :
( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P2 ) )
= P2 ) ).
% zero_less_of_bool_iff
thf(fact_4003_zero__less__of__bool__iff,axiom,
! [P2: $o] :
( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) )
= P2 ) ).
% zero_less_of_bool_iff
thf(fact_4004_zero__less__of__bool__iff,axiom,
! [P2: $o] :
( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) )
= P2 ) ).
% zero_less_of_bool_iff
thf(fact_4005_zero__less__of__bool__iff,axiom,
! [P2: $o] :
( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P2 ) )
= P2 ) ).
% zero_less_of_bool_iff
thf(fact_4006_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_4007_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_4008_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_4009_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_4010_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_4011_Nat_Odiff__diff__right,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_4012_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_4013_Suc__0__mod__eq,axiom,
! [N: nat] :
( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
= ( zero_n2687167440665602831ol_nat
@ ( N
!= ( suc @ zero_zero_nat ) ) ) ) ).
% Suc_0_mod_eq
thf(fact_4014_concat__bit__nonnegative__iff,axiom,
! [N: nat,K2: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K2 @ L ) )
= ( ord_less_eq_int @ zero_zero_int @ L ) ) ).
% concat_bit_nonnegative_iff
thf(fact_4015_concat__bit__negative__iff,axiom,
! [N: nat,K2: int,L: int] :
( ( ord_less_int @ ( bit_concat_bit @ N @ K2 @ L ) @ zero_zero_int )
= ( ord_less_int @ L @ zero_zero_int ) ) ).
% concat_bit_negative_iff
thf(fact_4016_set__decode__zero,axiom,
( ( nat_set_decode @ zero_zero_nat )
= bot_bot_set_nat ) ).
% set_decode_zero
thf(fact_4017_set__encode__inverse,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( nat_set_decode @ ( nat_set_encode @ A4 ) )
= A4 ) ) ).
% set_encode_inverse
thf(fact_4018_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_4019_diff__Suc__diff__eq2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_4020_diff__Suc__diff__eq1,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_4021_zle__diff1__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_4022_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_4023_nth__Cons__numeral,axiom,
! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT,V: num] :
( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X3 @ Xs ) @ ( numeral_numeral_nat @ V ) )
= ( nth_VEBT_VEBT @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_4024_nth__Cons__numeral,axiom,
! [X3: int,Xs: list_int,V: num] :
( ( nth_int @ ( cons_int @ X3 @ Xs ) @ ( numeral_numeral_nat @ V ) )
= ( nth_int @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_4025_nth__Cons__numeral,axiom,
! [X3: nat,Xs: list_nat,V: num] :
( ( nth_nat @ ( cons_nat @ X3 @ Xs ) @ ( numeral_numeral_nat @ V ) )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_4026_even__diff,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B ) )
= ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ).
% even_diff
thf(fact_4027_even__diff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).
% even_diff
thf(fact_4028_of__bool__half__eq__0,axiom,
! [B: $o] :
( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% of_bool_half_eq_0
thf(fact_4029_of__bool__half__eq__0,axiom,
! [B: $o] :
( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% of_bool_half_eq_0
thf(fact_4030_nth__Cons__pos,axiom,
! [N: nat,X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X3 @ Xs ) @ N )
= ( nth_VEBT_VEBT @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_4031_nth__Cons__pos,axiom,
! [N: nat,X3: int,Xs: list_int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_int @ ( cons_int @ X3 @ Xs ) @ N )
= ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_4032_nth__Cons__pos,axiom,
! [N: nat,X3: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_nat @ ( cons_nat @ X3 @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_4033_set__decode__0,axiom,
! [X3: nat] :
( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X3 ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X3 ) ) ) ).
% set_decode_0
thf(fact_4034_odd__Suc__minus__one,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% odd_Suc_minus_one
thf(fact_4035_even__diff__nat,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ).
% even_diff_nat
thf(fact_4036_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_4037_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_4038_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_4039_bits__1__div__exp,axiom,
! [N: nat] :
( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% bits_1_div_exp
thf(fact_4040_bits__1__div__exp,axiom,
! [N: nat] :
( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% bits_1_div_exp
thf(fact_4041_one__div__2__pow__eq,axiom,
! [N: nat] :
( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% one_div_2_pow_eq
thf(fact_4042_one__div__2__pow__eq,axiom,
! [N: nat] :
( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% one_div_2_pow_eq
thf(fact_4043_one__mod__2__pow__eq,axiom,
! [N: nat] :
( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% one_mod_2_pow_eq
thf(fact_4044_one__mod__2__pow__eq,axiom,
! [N: nat] :
( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% one_mod_2_pow_eq
thf(fact_4045_one__mod__2__pow__eq,axiom,
! [N: nat] :
( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% one_mod_2_pow_eq
thf(fact_4046_finite__divisors__int,axiom,
! [I: int] :
( ( I != zero_zero_int )
=> ( finite_finite_int
@ ( collect_int
@ ^ [D4: int] : ( dvd_dvd_int @ D4 @ I ) ) ) ) ).
% finite_divisors_int
thf(fact_4047_small__lazy_H_Ocases,axiom,
! [X3: product_prod_int_int] :
~ ! [D2: int,I2: int] :
( X3
!= ( product_Pair_int_int @ D2 @ I2 ) ) ).
% small_lazy'.cases
thf(fact_4048_exhaustive__int_H_Ocases,axiom,
! [X3: produc7773217078559923341nt_int] :
~ ! [F3: int > option6357759511663192854e_term,D2: int,I2: int] :
( X3
!= ( produc4305682042979456191nt_int @ F3 @ ( product_Pair_int_int @ D2 @ I2 ) ) ) ).
% exhaustive_int'.cases
thf(fact_4049_full__exhaustive__int_H_Ocases,axiom,
! [X3: produc2285326912895808259nt_int] :
~ ! [F3: produc8551481072490612790e_term > option6357759511663192854e_term,D2: int,I2: int] :
( X3
!= ( produc5700946648718959541nt_int @ F3 @ ( product_Pair_int_int @ D2 @ I2 ) ) ) ).
% full_exhaustive_int'.cases
thf(fact_4050_zdvd__zdiffD,axiom,
! [K2: int,M2: int,N: int] :
( ( dvd_dvd_int @ K2 @ ( minus_minus_int @ M2 @ N ) )
=> ( ( dvd_dvd_int @ K2 @ N )
=> ( dvd_dvd_int @ K2 @ M2 ) ) ) ).
% zdvd_zdiffD
thf(fact_4051_dvd__antisym,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd_nat @ M2 @ N )
=> ( ( dvd_dvd_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% dvd_antisym
thf(fact_4052_dvd__diff__nat,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( dvd_dvd_nat @ K2 @ M2 )
=> ( ( dvd_dvd_nat @ K2 @ N )
=> ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_4053_subset__CollectI,axiom,
! [B5: set_complex,A4: set_complex,Q: complex > $o,P2: complex > $o] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ B5 )
=> ( ( Q @ X4 )
=> ( P2 @ X4 ) ) )
=> ( ord_le211207098394363844omplex
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ B5 )
& ( Q @ X5 ) ) )
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( P2 @ X5 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_4054_subset__CollectI,axiom,
! [B5: set_real,A4: set_real,Q: real > $o,P2: real > $o] :
( ( ord_less_eq_set_real @ B5 @ A4 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ B5 )
=> ( ( Q @ X4 )
=> ( P2 @ X4 ) ) )
=> ( ord_less_eq_set_real
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ B5 )
& ( Q @ X5 ) ) )
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( P2 @ X5 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_4055_subset__CollectI,axiom,
! [B5: set_list_nat,A4: set_list_nat,Q: list_nat > $o,P2: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B5 @ A4 )
=> ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ B5 )
=> ( ( Q @ X4 )
=> ( P2 @ X4 ) ) )
=> ( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X5: list_nat] :
( ( member_list_nat @ X5 @ B5 )
& ( Q @ X5 ) ) )
@ ( collect_list_nat
@ ^ [X5: list_nat] :
( ( member_list_nat @ X5 @ A4 )
& ( P2 @ X5 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_4056_subset__CollectI,axiom,
! [B5: set_set_nat,A4: set_set_nat,Q: set_nat > $o,P2: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ B5 @ A4 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ B5 )
=> ( ( Q @ X4 )
=> ( P2 @ X4 ) ) )
=> ( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X5: set_nat] :
( ( member_set_nat @ X5 @ B5 )
& ( Q @ X5 ) ) )
@ ( collect_set_nat
@ ^ [X5: set_nat] :
( ( member_set_nat @ X5 @ A4 )
& ( P2 @ X5 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_4057_subset__CollectI,axiom,
! [B5: set_nat,A4: set_nat,Q: nat > $o,P2: nat > $o] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B5 )
=> ( ( Q @ X4 )
=> ( P2 @ X4 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ B5 )
& ( Q @ X5 ) ) )
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( P2 @ X5 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_4058_subset__CollectI,axiom,
! [B5: set_int,A4: set_int,Q: int > $o,P2: int > $o] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ B5 )
=> ( ( Q @ X4 )
=> ( P2 @ X4 ) ) )
=> ( ord_less_eq_set_int
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ B5 )
& ( Q @ X5 ) ) )
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ A4 )
& ( P2 @ X5 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_4059_subset__Collect__iff,axiom,
! [B5: set_complex,A4: set_complex,P2: complex > $o] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( ord_le211207098394363844omplex @ B5
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( ! [X5: complex] :
( ( member_complex @ X5 @ B5 )
=> ( P2 @ X5 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_4060_subset__Collect__iff,axiom,
! [B5: set_real,A4: set_real,P2: real > $o] :
( ( ord_less_eq_set_real @ B5 @ A4 )
=> ( ( ord_less_eq_set_real @ B5
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( ! [X5: real] :
( ( member_real @ X5 @ B5 )
=> ( P2 @ X5 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_4061_subset__Collect__iff,axiom,
! [B5: set_list_nat,A4: set_list_nat,P2: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B5 @ A4 )
=> ( ( ord_le6045566169113846134st_nat @ B5
@ ( collect_list_nat
@ ^ [X5: list_nat] :
( ( member_list_nat @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( ! [X5: list_nat] :
( ( member_list_nat @ X5 @ B5 )
=> ( P2 @ X5 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_4062_subset__Collect__iff,axiom,
! [B5: set_set_nat,A4: set_set_nat,P2: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ B5 @ A4 )
=> ( ( ord_le6893508408891458716et_nat @ B5
@ ( collect_set_nat
@ ^ [X5: set_nat] :
( ( member_set_nat @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ B5 )
=> ( P2 @ X5 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_4063_subset__Collect__iff,axiom,
! [B5: set_nat,A4: set_nat,P2: nat > $o] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( ord_less_eq_set_nat @ B5
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ B5 )
=> ( P2 @ X5 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_4064_subset__Collect__iff,axiom,
! [B5: set_int,A4: set_int,P2: int > $o] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( ord_less_eq_set_int @ B5
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( ! [X5: int] :
( ( member_int @ X5 @ B5 )
=> ( P2 @ X5 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_4065_Collect__subset,axiom,
! [A4: set_complex,P2: complex > $o] :
( ord_le211207098394363844omplex
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( P2 @ X5 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_4066_Collect__subset,axiom,
! [A4: set_real,P2: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( P2 @ X5 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_4067_Collect__subset,axiom,
! [A4: set_list_nat,P2: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X5: list_nat] :
( ( member_list_nat @ X5 @ A4 )
& ( P2 @ X5 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_4068_Collect__subset,axiom,
! [A4: set_set_nat,P2: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X5: set_nat] :
( ( member_set_nat @ X5 @ A4 )
& ( P2 @ X5 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_4069_Collect__subset,axiom,
! [A4: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( P2 @ X5 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_4070_Collect__subset,axiom,
! [A4: set_int,P2: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ A4 )
& ( P2 @ X5 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_4071_less__eq__set__def,axiom,
( ord_le211207098394363844omplex
= ( ^ [A6: set_complex,B6: set_complex] :
( ord_le4573692005234683329plex_o
@ ^ [X5: complex] : ( member_complex @ X5 @ A6 )
@ ^ [X5: complex] : ( member_complex @ X5 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_4072_less__eq__set__def,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
( ord_less_eq_real_o
@ ^ [X5: real] : ( member_real @ X5 @ A6 )
@ ^ [X5: real] : ( member_real @ X5 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_4073_less__eq__set__def,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A6: set_set_nat,B6: set_set_nat] :
( ord_le3964352015994296041_nat_o
@ ^ [X5: set_nat] : ( member_set_nat @ X5 @ A6 )
@ ^ [X5: set_nat] : ( member_set_nat @ X5 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_4074_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ord_less_eq_nat_o
@ ^ [X5: nat] : ( member_nat @ X5 @ A6 )
@ ^ [X5: nat] : ( member_nat @ X5 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_4075_less__eq__set__def,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
( ord_less_eq_int_o
@ ^ [X5: int] : ( member_int @ X5 @ A6 )
@ ^ [X5: int] : ( member_int @ X5 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_4076_prop__restrict,axiom,
! [X3: complex,Z5: set_complex,X8: set_complex,P2: complex > $o] :
( ( member_complex @ X3 @ Z5 )
=> ( ( ord_le211207098394363844omplex @ Z5
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ X8 )
& ( P2 @ X5 ) ) ) )
=> ( P2 @ X3 ) ) ) ).
% prop_restrict
thf(fact_4077_prop__restrict,axiom,
! [X3: real,Z5: set_real,X8: set_real,P2: real > $o] :
( ( member_real @ X3 @ Z5 )
=> ( ( ord_less_eq_set_real @ Z5
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ X8 )
& ( P2 @ X5 ) ) ) )
=> ( P2 @ X3 ) ) ) ).
% prop_restrict
thf(fact_4078_prop__restrict,axiom,
! [X3: list_nat,Z5: set_list_nat,X8: set_list_nat,P2: list_nat > $o] :
( ( member_list_nat @ X3 @ Z5 )
=> ( ( ord_le6045566169113846134st_nat @ Z5
@ ( collect_list_nat
@ ^ [X5: list_nat] :
( ( member_list_nat @ X5 @ X8 )
& ( P2 @ X5 ) ) ) )
=> ( P2 @ X3 ) ) ) ).
% prop_restrict
thf(fact_4079_prop__restrict,axiom,
! [X3: set_nat,Z5: set_set_nat,X8: set_set_nat,P2: set_nat > $o] :
( ( member_set_nat @ X3 @ Z5 )
=> ( ( ord_le6893508408891458716et_nat @ Z5
@ ( collect_set_nat
@ ^ [X5: set_nat] :
( ( member_set_nat @ X5 @ X8 )
& ( P2 @ X5 ) ) ) )
=> ( P2 @ X3 ) ) ) ).
% prop_restrict
thf(fact_4080_prop__restrict,axiom,
! [X3: nat,Z5: set_nat,X8: set_nat,P2: nat > $o] :
( ( member_nat @ X3 @ Z5 )
=> ( ( ord_less_eq_set_nat @ Z5
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ X8 )
& ( P2 @ X5 ) ) ) )
=> ( P2 @ X3 ) ) ) ).
% prop_restrict
thf(fact_4081_prop__restrict,axiom,
! [X3: int,Z5: set_int,X8: set_int,P2: int > $o] :
( ( member_int @ X3 @ Z5 )
=> ( ( ord_less_eq_set_int @ Z5
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ X8 )
& ( P2 @ X5 ) ) ) )
=> ( P2 @ X3 ) ) ) ).
% prop_restrict
thf(fact_4082_Collect__restrict,axiom,
! [X8: set_complex,P2: complex > $o] :
( ord_le211207098394363844omplex
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ X8 )
& ( P2 @ X5 ) ) )
@ X8 ) ).
% Collect_restrict
thf(fact_4083_Collect__restrict,axiom,
! [X8: set_real,P2: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ X8 )
& ( P2 @ X5 ) ) )
@ X8 ) ).
% Collect_restrict
thf(fact_4084_Collect__restrict,axiom,
! [X8: set_list_nat,P2: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X5: list_nat] :
( ( member_list_nat @ X5 @ X8 )
& ( P2 @ X5 ) ) )
@ X8 ) ).
% Collect_restrict
thf(fact_4085_Collect__restrict,axiom,
! [X8: set_set_nat,P2: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X5: set_nat] :
( ( member_set_nat @ X5 @ X8 )
& ( P2 @ X5 ) ) )
@ X8 ) ).
% Collect_restrict
thf(fact_4086_Collect__restrict,axiom,
! [X8: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ X8 )
& ( P2 @ X5 ) ) )
@ X8 ) ).
% Collect_restrict
thf(fact_4087_Collect__restrict,axiom,
! [X8: set_int,P2: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ X8 )
& ( P2 @ X5 ) ) )
@ X8 ) ).
% Collect_restrict
thf(fact_4088_pigeonhole__infinite__rel,axiom,
! [A4: set_real,B5: set_nat,R: real > nat > $o] :
( ~ ( finite_finite_real @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B5 )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A5: real] :
( ( member_real @ A5 @ A4 )
& ( R @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_4089_pigeonhole__infinite__rel,axiom,
! [A4: set_real,B5: set_int,R: real > int > $o] :
( ~ ( finite_finite_real @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ B5 )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A5: real] :
( ( member_real @ A5 @ A4 )
& ( R @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_4090_pigeonhole__infinite__rel,axiom,
! [A4: set_real,B5: set_complex,R: real > complex > $o] :
( ~ ( finite_finite_real @ A4 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: complex] :
( ( member_complex @ X4 @ B5 )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A5: real] :
( ( member_real @ A5 @ A4 )
& ( R @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_4091_pigeonhole__infinite__rel,axiom,
! [A4: set_nat,B5: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B5 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A4 )
& ( R @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_4092_pigeonhole__infinite__rel,axiom,
! [A4: set_nat,B5: set_int,R: nat > int > $o] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ B5 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A4 )
& ( R @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_4093_pigeonhole__infinite__rel,axiom,
! [A4: set_nat,B5: set_complex,R: nat > complex > $o] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: complex] :
( ( member_complex @ X4 @ B5 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A4 )
& ( R @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_4094_pigeonhole__infinite__rel,axiom,
! [A4: set_int,B5: set_nat,R: int > nat > $o] :
( ~ ( finite_finite_int @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B5 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A5: int] :
( ( member_int @ A5 @ A4 )
& ( R @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_4095_pigeonhole__infinite__rel,axiom,
! [A4: set_int,B5: set_int,R: int > int > $o] :
( ~ ( finite_finite_int @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ? [Xa: int] :
( ( member_int @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ B5 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A5: int] :
( ( member_int @ A5 @ A4 )
& ( R @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_4096_pigeonhole__infinite__rel,axiom,
! [A4: set_int,B5: set_complex,R: int > complex > $o] :
( ~ ( finite_finite_int @ A4 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: complex] :
( ( member_complex @ X4 @ B5 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A5: int] :
( ( member_int @ A5 @ A4 )
& ( R @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_4097_pigeonhole__infinite__rel,axiom,
! [A4: set_complex,B5: set_nat,R: complex > nat > $o] :
( ~ ( finite3207457112153483333omplex @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B5 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B5 )
& ~ ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [A5: complex] :
( ( member_complex @ A5 @ A4 )
& ( R @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_4098_not__finite__existsD,axiom,
! [P2: real > $o] :
( ~ ( finite_finite_real @ ( collect_real @ P2 ) )
=> ? [X_12: real] : ( P2 @ X_12 ) ) ).
% not_finite_existsD
thf(fact_4099_not__finite__existsD,axiom,
! [P2: list_nat > $o] :
( ~ ( finite8100373058378681591st_nat @ ( collect_list_nat @ P2 ) )
=> ? [X_12: list_nat] : ( P2 @ X_12 ) ) ).
% not_finite_existsD
thf(fact_4100_not__finite__existsD,axiom,
! [P2: set_nat > $o] :
( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P2 ) )
=> ? [X_12: set_nat] : ( P2 @ X_12 ) ) ).
% not_finite_existsD
thf(fact_4101_not__finite__existsD,axiom,
! [P2: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P2 ) )
=> ? [X_12: nat] : ( P2 @ X_12 ) ) ).
% not_finite_existsD
thf(fact_4102_not__finite__existsD,axiom,
! [P2: int > $o] :
( ~ ( finite_finite_int @ ( collect_int @ P2 ) )
=> ? [X_12: int] : ( P2 @ X_12 ) ) ).
% not_finite_existsD
thf(fact_4103_not__finite__existsD,axiom,
! [P2: complex > $o] :
( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P2 ) )
=> ? [X_12: complex] : ( P2 @ X_12 ) ) ).
% not_finite_existsD
thf(fact_4104_not__finite__existsD,axiom,
! [P2: product_prod_nat_nat > $o] :
( ~ ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P2 ) )
=> ? [X_12: product_prod_nat_nat] : ( P2 @ X_12 ) ) ).
% not_finite_existsD
thf(fact_4105_diff__commute,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_4106_diff__right__commute,axiom,
! [A: real,C2: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C2 ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_4107_diff__right__commute,axiom,
! [A: rat,C2: rat,B: rat] :
( ( minus_minus_rat @ ( minus_minus_rat @ A @ C2 ) @ B )
= ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_4108_diff__right__commute,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_4109_diff__right__commute,axiom,
! [A: int,C2: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_4110_diff__eq__diff__eq,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C2 @ D ) )
=> ( ( A = B )
= ( C2 = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_4111_diff__eq__diff__eq,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ( minus_minus_rat @ A @ B )
= ( minus_minus_rat @ C2 @ D ) )
=> ( ( A = B )
= ( C2 = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_4112_diff__eq__diff__eq,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( A = B )
= ( C2 = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_4113_unique__quotient,axiom,
! [A: int,B: int,Q3: int,R2: int,Q5: int,R4: int] :
( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
=> ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q5 @ R4 ) )
=> ( Q3 = Q5 ) ) ) ).
% unique_quotient
thf(fact_4114_unique__remainder,axiom,
! [A: int,B: int,Q3: int,R2: int,Q5: int,R4: int] :
( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
=> ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q5 @ R4 ) )
=> ( R2 = R4 ) ) ) ).
% unique_remainder
thf(fact_4115_lambda__zero,axiom,
( ( ^ [H: complex] : zero_zero_complex )
= ( times_times_complex @ zero_zero_complex ) ) ).
% lambda_zero
thf(fact_4116_lambda__zero,axiom,
( ( ^ [H: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_4117_lambda__zero,axiom,
( ( ^ [H: rat] : zero_zero_rat )
= ( times_times_rat @ zero_zero_rat ) ) ).
% lambda_zero
thf(fact_4118_lambda__zero,axiom,
( ( ^ [H: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_4119_lambda__zero,axiom,
( ( ^ [H: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_4120_subset__divisors__dvd,axiom,
! [A: real,B: real] :
( ( ord_less_eq_set_real
@ ( collect_real
@ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
@ ( collect_real
@ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B ) ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% subset_divisors_dvd
thf(fact_4121_subset__divisors__dvd,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
@ ( collect_nat
@ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% subset_divisors_dvd
thf(fact_4122_subset__divisors__dvd,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le7084787975880047091nteger
@ ( collect_Code_integer
@ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A ) )
@ ( collect_Code_integer
@ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B ) ) )
= ( dvd_dvd_Code_integer @ A @ B ) ) ).
% subset_divisors_dvd
thf(fact_4123_subset__divisors__dvd,axiom,
! [A: int,B: int] :
( ( ord_less_eq_set_int
@ ( collect_int
@ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
@ ( collect_int
@ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% subset_divisors_dvd
thf(fact_4124_max__def__raw,axiom,
( ord_ma741700101516333627d_enat
= ( ^ [A5: extended_enat,B4: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def_raw
thf(fact_4125_max__def__raw,axiom,
( ord_max_Code_integer
= ( ^ [A5: code_integer,B4: code_integer] : ( if_Code_integer @ ( ord_le3102999989581377725nteger @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def_raw
thf(fact_4126_max__def__raw,axiom,
( ord_max_set_int
= ( ^ [A5: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def_raw
thf(fact_4127_max__def__raw,axiom,
( ord_max_rat
= ( ^ [A5: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def_raw
thf(fact_4128_max__def__raw,axiom,
( ord_max_num
= ( ^ [A5: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def_raw
thf(fact_4129_max__def__raw,axiom,
( ord_max_nat
= ( ^ [A5: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def_raw
thf(fact_4130_max__def__raw,axiom,
( ord_max_int
= ( ^ [A5: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% max_def_raw
thf(fact_4131_finite__M__bounded__by__nat,axiom,
! [P2: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( P2 @ K3 )
& ( ord_less_nat @ K3 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_4132_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_4133_diff__mono,axiom,
! [A: real,B: real,D: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D @ C2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_4134_diff__mono,axiom,
! [A: rat,B: rat,D: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ D @ C2 )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C2 ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_4135_diff__mono,axiom,
! [A: int,B: int,D: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_4136_diff__left__mono,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C2 @ A ) @ ( minus_minus_real @ C2 @ B ) ) ) ).
% diff_left_mono
thf(fact_4137_diff__left__mono,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ C2 @ A ) @ ( minus_minus_rat @ C2 @ B ) ) ) ).
% diff_left_mono
thf(fact_4138_diff__left__mono,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).
% diff_left_mono
thf(fact_4139_diff__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ C2 ) ) ) ).
% diff_right_mono
thf(fact_4140_diff__right__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C2 ) @ ( minus_minus_rat @ B @ C2 ) ) ) ).
% diff_right_mono
thf(fact_4141_diff__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).
% diff_right_mono
thf(fact_4142_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C2 @ D ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C2 @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_4143_diff__eq__diff__less__eq,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ( minus_minus_rat @ A @ B )
= ( minus_minus_rat @ C2 @ D ) )
=> ( ( ord_less_eq_rat @ A @ B )
= ( ord_less_eq_rat @ C2 @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_4144_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C2 @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_4145_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: complex,Z: complex] : Y4 = Z )
= ( ^ [A5: complex,B4: complex] :
( ( minus_minus_complex @ A5 @ B4 )
= zero_zero_complex ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_4146_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: real,Z: real] : Y4 = Z )
= ( ^ [A5: real,B4: real] :
( ( minus_minus_real @ A5 @ B4 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_4147_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: rat,Z: rat] : Y4 = Z )
= ( ^ [A5: rat,B4: rat] :
( ( minus_minus_rat @ A5 @ B4 )
= zero_zero_rat ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_4148_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z: int] : Y4 = Z )
= ( ^ [A5: int,B4: int] :
( ( minus_minus_int @ A5 @ B4 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_4149_diff__strict__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_4150_diff__strict__right__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( minus_minus_rat @ A @ C2 ) @ ( minus_minus_rat @ B @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_4151_diff__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_4152_diff__strict__left__mono,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C2 @ A ) @ ( minus_minus_real @ C2 @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_4153_diff__strict__left__mono,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_rat @ B @ A )
=> ( ord_less_rat @ ( minus_minus_rat @ C2 @ A ) @ ( minus_minus_rat @ C2 @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_4154_diff__strict__left__mono,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_4155_diff__eq__diff__less,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C2 @ D ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C2 @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_4156_diff__eq__diff__less,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ( minus_minus_rat @ A @ B )
= ( minus_minus_rat @ C2 @ D ) )
=> ( ( ord_less_rat @ A @ B )
= ( ord_less_rat @ C2 @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_4157_diff__eq__diff__less,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C2 @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_4158_diff__strict__mono,axiom,
! [A: real,B: real,D: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D @ C2 )
=> ( ord_less_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_4159_diff__strict__mono,axiom,
! [A: rat,B: rat,D: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ D @ C2 )
=> ( ord_less_rat @ ( minus_minus_rat @ A @ C2 ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_4160_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C2 )
=> ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_4161_add__diff__add,axiom,
! [A: real,C2: real,B: real,D: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D ) )
= ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C2 @ D ) ) ) ).
% add_diff_add
thf(fact_4162_add__diff__add,axiom,
! [A: rat,C2: rat,B: rat,D: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ D ) )
= ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ ( minus_minus_rat @ C2 @ D ) ) ) ).
% add_diff_add
thf(fact_4163_add__diff__add,axiom,
! [A: int,C2: int,B: int,D: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) )
= ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C2 @ D ) ) ) ).
% add_diff_add
thf(fact_4164_group__cancel_Osub1,axiom,
! [A4: real,K2: real,A: real,B: real] :
( ( A4
= ( plus_plus_real @ K2 @ A ) )
=> ( ( minus_minus_real @ A4 @ B )
= ( plus_plus_real @ K2 @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_4165_group__cancel_Osub1,axiom,
! [A4: rat,K2: rat,A: rat,B: rat] :
( ( A4
= ( plus_plus_rat @ K2 @ A ) )
=> ( ( minus_minus_rat @ A4 @ B )
= ( plus_plus_rat @ K2 @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_4166_group__cancel_Osub1,axiom,
! [A4: int,K2: int,A: int,B: int] :
( ( A4
= ( plus_plus_int @ K2 @ A ) )
=> ( ( minus_minus_int @ A4 @ B )
= ( plus_plus_int @ K2 @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_4167_diff__eq__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ( minus_minus_real @ A @ B )
= C2 )
= ( A
= ( plus_plus_real @ C2 @ B ) ) ) ).
% diff_eq_eq
thf(fact_4168_diff__eq__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ( minus_minus_rat @ A @ B )
= C2 )
= ( A
= ( plus_plus_rat @ C2 @ B ) ) ) ).
% diff_eq_eq
thf(fact_4169_diff__eq__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ( minus_minus_int @ A @ B )
= C2 )
= ( A
= ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_eq_eq
thf(fact_4170_eq__diff__eq,axiom,
! [A: real,C2: real,B: real] :
( ( A
= ( minus_minus_real @ C2 @ B ) )
= ( ( plus_plus_real @ A @ B )
= C2 ) ) ).
% eq_diff_eq
thf(fact_4171_eq__diff__eq,axiom,
! [A: rat,C2: rat,B: rat] :
( ( A
= ( minus_minus_rat @ C2 @ B ) )
= ( ( plus_plus_rat @ A @ B )
= C2 ) ) ).
% eq_diff_eq
thf(fact_4172_eq__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( A
= ( minus_minus_int @ C2 @ B ) )
= ( ( plus_plus_int @ A @ B )
= C2 ) ) ).
% eq_diff_eq
thf(fact_4173_add__diff__eq,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C2 ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C2 ) ) ).
% add_diff_eq
thf(fact_4174_add__diff__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C2 ) )
= ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C2 ) ) ).
% add_diff_eq
thf(fact_4175_add__diff__eq,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% add_diff_eq
thf(fact_4176_diff__diff__eq2,axiom,
! [A: real,B: real,C2: real] :
( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C2 ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ).
% diff_diff_eq2
thf(fact_4177_diff__diff__eq2,axiom,
! [A: rat,B: rat,C2: rat] :
( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C2 ) )
= ( minus_minus_rat @ ( plus_plus_rat @ A @ C2 ) @ B ) ) ).
% diff_diff_eq2
thf(fact_4178_diff__diff__eq2,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).
% diff_diff_eq2
thf(fact_4179_diff__add__eq,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ).
% diff_add_eq
thf(fact_4180_diff__add__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C2 )
= ( minus_minus_rat @ ( plus_plus_rat @ A @ C2 ) @ B ) ) ).
% diff_add_eq
thf(fact_4181_diff__add__eq,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).
% diff_add_eq
thf(fact_4182_diff__add__eq__diff__diff__swap,axiom,
! [A: real,B: real,C2: real] :
( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C2 ) )
= ( minus_minus_real @ ( minus_minus_real @ A @ C2 ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_4183_diff__add__eq__diff__diff__swap,axiom,
! [A: rat,B: rat,C2: rat] :
( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C2 ) )
= ( minus_minus_rat @ ( minus_minus_rat @ A @ C2 ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_4184_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_4185_add__implies__diff,axiom,
! [C2: real,B: real,A: real] :
( ( ( plus_plus_real @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_real @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_4186_add__implies__diff,axiom,
! [C2: rat,B: rat,A: rat] :
( ( ( plus_plus_rat @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_rat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_4187_add__implies__diff,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_4188_add__implies__diff,axiom,
! [C2: int,B: int,A: int] :
( ( ( plus_plus_int @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_4189_diff__diff__eq,axiom,
! [A: real,B: real,C2: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( minus_minus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_4190_diff__diff__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C2 )
= ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_4191_diff__diff__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_4192_diff__diff__eq,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_4193_zero__induct__lemma,axiom,
! [P2: nat > $o,K2: nat,I: nat] :
( ( P2 @ K2 )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ ( minus_minus_nat @ K2 @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_4194_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_4195_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_4196_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_4197_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_4198_dvd__minus__self,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N @ M2 ) )
= ( ( ord_less_nat @ N @ M2 )
| ( dvd_dvd_nat @ M2 @ N ) ) ) ).
% dvd_minus_self
thf(fact_4199_minus__int__code_I1_J,axiom,
! [K2: int] :
( ( minus_minus_int @ K2 @ zero_zero_int )
= K2 ) ).
% minus_int_code(1)
thf(fact_4200_eq__diff__iff,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ( minus_minus_nat @ M2 @ K2 )
= ( minus_minus_nat @ N @ K2 ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_4201_le__diff__iff,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_4202_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_4203_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_4204_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_4205_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_4206_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_4207_dvd__diffD,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M2 @ N ) )
=> ( ( dvd_dvd_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( dvd_dvd_nat @ K2 @ M2 ) ) ) ) ).
% dvd_diffD
thf(fact_4208_dvd__diffD1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M2 @ N ) )
=> ( ( dvd_dvd_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( dvd_dvd_nat @ K2 @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_4209_less__eq__dvd__minus,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( dvd_dvd_nat @ M2 @ N )
= ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N @ M2 ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_4210_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_4211_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_4212_diff__cancel2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_cancel2
thf(fact_4213_Nat_Odiff__cancel,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% Nat.diff_cancel
thf(fact_4214_diff__mult__distrib2,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( times_times_nat @ K2 @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_4215_diff__mult__distrib,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M2 @ N ) @ K2 )
= ( minus_minus_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).
% diff_mult_distrib
thf(fact_4216_int__distrib_I3_J,axiom,
! [Z12: int,Z23: int,W2: int] :
( ( times_times_int @ ( minus_minus_int @ Z12 @ Z23 ) @ W2 )
= ( minus_minus_int @ ( times_times_int @ Z12 @ W2 ) @ ( times_times_int @ Z23 @ W2 ) ) ) ).
% int_distrib(3)
thf(fact_4217_int__distrib_I4_J,axiom,
! [W2: int,Z12: int,Z23: int] :
( ( times_times_int @ W2 @ ( minus_minus_int @ Z12 @ Z23 ) )
= ( minus_minus_int @ ( times_times_int @ W2 @ Z12 ) @ ( times_times_int @ W2 @ Z23 ) ) ) ).
% int_distrib(4)
thf(fact_4218_max__diff__distrib__left,axiom,
! [X3: code_integer,Y: code_integer,Z2: code_integer] :
( ( minus_8373710615458151222nteger @ ( ord_max_Code_integer @ X3 @ Y ) @ Z2 )
= ( ord_max_Code_integer @ ( minus_8373710615458151222nteger @ X3 @ Z2 ) @ ( minus_8373710615458151222nteger @ Y @ Z2 ) ) ) ).
% max_diff_distrib_left
thf(fact_4219_max__diff__distrib__left,axiom,
! [X3: real,Y: real,Z2: real] :
( ( minus_minus_real @ ( ord_max_real @ X3 @ Y ) @ Z2 )
= ( ord_max_real @ ( minus_minus_real @ X3 @ Z2 ) @ ( minus_minus_real @ Y @ Z2 ) ) ) ).
% max_diff_distrib_left
thf(fact_4220_max__diff__distrib__left,axiom,
! [X3: rat,Y: rat,Z2: rat] :
( ( minus_minus_rat @ ( ord_max_rat @ X3 @ Y ) @ Z2 )
= ( ord_max_rat @ ( minus_minus_rat @ X3 @ Z2 ) @ ( minus_minus_rat @ Y @ Z2 ) ) ) ).
% max_diff_distrib_left
thf(fact_4221_max__diff__distrib__left,axiom,
! [X3: int,Y: int,Z2: int] :
( ( minus_minus_int @ ( ord_max_int @ X3 @ Y ) @ Z2 )
= ( ord_max_int @ ( minus_minus_int @ X3 @ Z2 ) @ ( minus_minus_int @ Y @ Z2 ) ) ) ).
% max_diff_distrib_left
thf(fact_4222_numeral__code_I2_J,axiom,
! [N: num] :
( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
= ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% numeral_code(2)
thf(fact_4223_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit0 @ N ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_code(2)
thf(fact_4224_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit0 @ N ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% numeral_code(2)
thf(fact_4225_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_code(2)
thf(fact_4226_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_code(2)
thf(fact_4227_set__vebt__def,axiom,
( vEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T3 ) ) ) ) ).
% set_vebt_def
thf(fact_4228_finite__divisors__nat,axiom,
! [M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M2 ) ) ) ) ).
% finite_divisors_nat
thf(fact_4229_concat__bit__assoc,axiom,
! [N: nat,K2: int,M2: nat,L: int,R2: int] :
( ( bit_concat_bit @ N @ K2 @ ( bit_concat_bit @ M2 @ L @ R2 ) )
= ( bit_concat_bit @ ( plus_plus_nat @ M2 @ N ) @ ( bit_concat_bit @ N @ K2 @ L ) @ R2 ) ) ).
% concat_bit_assoc
thf(fact_4230_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X5: nat] :
( collect_nat
@ ^ [N4: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ) ).
% set_decode_def
thf(fact_4231_eucl__rel__int__by0,axiom,
! [K2: int] : ( eucl_rel_int @ K2 @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K2 ) ) ).
% eucl_rel_int_by0
thf(fact_4232_div__int__unique,axiom,
! [K2: int,L: int,Q3: int,R2: int] :
( ( eucl_rel_int @ K2 @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
=> ( ( divide_divide_int @ K2 @ L )
= Q3 ) ) ).
% div_int_unique
thf(fact_4233_mod__int__unique,axiom,
! [K2: int,L: int,Q3: int,R2: int] :
( ( eucl_rel_int @ K2 @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
=> ( ( modulo_modulo_int @ K2 @ L )
= R2 ) ) ).
% mod_int_unique
thf(fact_4234_subset__decode__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( nat_set_decode @ M2 ) @ ( nat_set_decode @ N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% subset_decode_imp_le
thf(fact_4235_zero__less__eq__of__bool,axiom,
! [P2: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P2 ) ) ).
% zero_less_eq_of_bool
thf(fact_4236_zero__less__eq__of__bool,axiom,
! [P2: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) ) ).
% zero_less_eq_of_bool
thf(fact_4237_zero__less__eq__of__bool,axiom,
! [P2: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) ) ).
% zero_less_eq_of_bool
thf(fact_4238_zero__less__eq__of__bool,axiom,
! [P2: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P2 ) ) ).
% zero_less_eq_of_bool
thf(fact_4239_of__bool__less__eq__one,axiom,
! [P2: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P2 ) @ one_one_real ) ).
% of_bool_less_eq_one
thf(fact_4240_of__bool__less__eq__one,axiom,
! [P2: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) @ one_one_rat ) ).
% of_bool_less_eq_one
thf(fact_4241_of__bool__less__eq__one,axiom,
! [P2: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ one_one_nat ) ).
% of_bool_less_eq_one
thf(fact_4242_of__bool__less__eq__one,axiom,
! [P2: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ one_one_int ) ).
% of_bool_less_eq_one
thf(fact_4243_of__bool__def,axiom,
( zero_n1201886186963655149omplex
= ( ^ [P6: $o] : ( if_complex @ P6 @ one_one_complex @ zero_zero_complex ) ) ) ).
% of_bool_def
thf(fact_4244_of__bool__def,axiom,
( zero_n3304061248610475627l_real
= ( ^ [P6: $o] : ( if_real @ P6 @ one_one_real @ zero_zero_real ) ) ) ).
% of_bool_def
thf(fact_4245_of__bool__def,axiom,
( zero_n2052037380579107095ol_rat
= ( ^ [P6: $o] : ( if_rat @ P6 @ one_one_rat @ zero_zero_rat ) ) ) ).
% of_bool_def
thf(fact_4246_of__bool__def,axiom,
( zero_n2687167440665602831ol_nat
= ( ^ [P6: $o] : ( if_nat @ P6 @ one_one_nat @ zero_zero_nat ) ) ) ).
% of_bool_def
thf(fact_4247_of__bool__def,axiom,
( zero_n2684676970156552555ol_int
= ( ^ [P6: $o] : ( if_int @ P6 @ one_one_int @ zero_zero_int ) ) ) ).
% of_bool_def
thf(fact_4248_split__of__bool,axiom,
! [P2: complex > $o,P: $o] :
( ( P2 @ ( zero_n1201886186963655149omplex @ P ) )
= ( ( P
=> ( P2 @ one_one_complex ) )
& ( ~ P
=> ( P2 @ zero_zero_complex ) ) ) ) ).
% split_of_bool
thf(fact_4249_split__of__bool,axiom,
! [P2: real > $o,P: $o] :
( ( P2 @ ( zero_n3304061248610475627l_real @ P ) )
= ( ( P
=> ( P2 @ one_one_real ) )
& ( ~ P
=> ( P2 @ zero_zero_real ) ) ) ) ).
% split_of_bool
thf(fact_4250_split__of__bool,axiom,
! [P2: rat > $o,P: $o] :
( ( P2 @ ( zero_n2052037380579107095ol_rat @ P ) )
= ( ( P
=> ( P2 @ one_one_rat ) )
& ( ~ P
=> ( P2 @ zero_zero_rat ) ) ) ) ).
% split_of_bool
thf(fact_4251_split__of__bool,axiom,
! [P2: nat > $o,P: $o] :
( ( P2 @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( ( P
=> ( P2 @ one_one_nat ) )
& ( ~ P
=> ( P2 @ zero_zero_nat ) ) ) ) ).
% split_of_bool
thf(fact_4252_split__of__bool,axiom,
! [P2: int > $o,P: $o] :
( ( P2 @ ( zero_n2684676970156552555ol_int @ P ) )
= ( ( P
=> ( P2 @ one_one_int ) )
& ( ~ P
=> ( P2 @ zero_zero_int ) ) ) ) ).
% split_of_bool
thf(fact_4253_split__of__bool__asm,axiom,
! [P2: complex > $o,P: $o] :
( ( P2 @ ( zero_n1201886186963655149omplex @ P ) )
= ( ~ ( ( P
& ~ ( P2 @ one_one_complex ) )
| ( ~ P
& ~ ( P2 @ zero_zero_complex ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_4254_split__of__bool__asm,axiom,
! [P2: real > $o,P: $o] :
( ( P2 @ ( zero_n3304061248610475627l_real @ P ) )
= ( ~ ( ( P
& ~ ( P2 @ one_one_real ) )
| ( ~ P
& ~ ( P2 @ zero_zero_real ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_4255_split__of__bool__asm,axiom,
! [P2: rat > $o,P: $o] :
( ( P2 @ ( zero_n2052037380579107095ol_rat @ P ) )
= ( ~ ( ( P
& ~ ( P2 @ one_one_rat ) )
| ( ~ P
& ~ ( P2 @ zero_zero_rat ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_4256_split__of__bool__asm,axiom,
! [P2: nat > $o,P: $o] :
( ( P2 @ ( zero_n2687167440665602831ol_nat @ P ) )
= ( ~ ( ( P
& ~ ( P2 @ one_one_nat ) )
| ( ~ P
& ~ ( P2 @ zero_zero_nat ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_4257_split__of__bool__asm,axiom,
! [P2: int > $o,P: $o] :
( ( P2 @ ( zero_n2684676970156552555ol_int @ P ) )
= ( ~ ( ( P
& ~ ( P2 @ one_one_int ) )
| ( ~ P
& ~ ( P2 @ zero_zero_int ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_4258_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A5 @ B4 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_4259_le__iff__diff__le__0,axiom,
( ord_less_eq_rat
= ( ^ [A5: rat,B4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A5 @ B4 ) @ zero_zero_rat ) ) ) ).
% le_iff_diff_le_0
thf(fact_4260_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_4261_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A5: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A5 @ B4 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_4262_less__iff__diff__less__0,axiom,
( ord_less_rat
= ( ^ [A5: rat,B4: rat] : ( ord_less_rat @ ( minus_minus_rat @ A5 @ B4 ) @ zero_zero_rat ) ) ) ).
% less_iff_diff_less_0
thf(fact_4263_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_4264_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C2 )
= ( B
= ( plus_plus_nat @ C2 @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_4265_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_4266_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_4267_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_4268_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_4269_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A )
= ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_4270_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_4271_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_4272_le__add__diff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% le_add_diff
thf(fact_4273_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_4274_le__diff__eq,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ A @ ( minus_minus_real @ C2 @ B ) )
= ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C2 ) ) ).
% le_diff_eq
thf(fact_4275_le__diff__eq,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C2 @ B ) )
= ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C2 ) ) ).
% le_diff_eq
thf(fact_4276_le__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C2 @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% le_diff_eq
thf(fact_4277_diff__le__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( ord_less_eq_real @ A @ ( plus_plus_real @ C2 @ B ) ) ) ).
% diff_le_eq
thf(fact_4278_diff__le__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C2 )
= ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C2 @ B ) ) ) ).
% diff_le_eq
thf(fact_4279_diff__le__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_le_eq
thf(fact_4280_add__le__add__imp__diff__le,axiom,
! [I: real,K2: real,N: real,J: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I @ K2 ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K2 ) )
=> ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K2 ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K2 ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ N @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_4281_add__le__add__imp__diff__le,axiom,
! [I: rat,K2: rat,N: rat,J: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K2 ) @ N )
=> ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K2 ) )
=> ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K2 ) @ N )
=> ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K2 ) )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_4282_add__le__add__imp__diff__le,axiom,
! [I: nat,K2: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_4283_add__le__add__imp__diff__le,axiom,
! [I: int,K2: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K2 ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K2 ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_4284_add__le__imp__le__diff,axiom,
! [I: real,K2: real,N: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I @ K2 ) @ N )
=> ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_4285_add__le__imp__le__diff,axiom,
! [I: rat,K2: rat,N: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K2 ) @ N )
=> ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_4286_add__le__imp__le__diff,axiom,
! [I: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_4287_add__le__imp__le__diff,axiom,
! [I: int,K2: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_4288_finite__set__decode,axiom,
! [N: nat] : ( finite_finite_nat @ ( nat_set_decode @ N ) ) ).
% finite_set_decode
thf(fact_4289_less__diff__eq,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ A @ ( minus_minus_real @ C2 @ B ) )
= ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C2 ) ) ).
% less_diff_eq
thf(fact_4290_less__diff__eq,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_rat @ A @ ( minus_minus_rat @ C2 @ B ) )
= ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C2 ) ) ).
% less_diff_eq
thf(fact_4291_less__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C2 @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% less_diff_eq
thf(fact_4292_diff__less__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( ord_less_real @ A @ ( plus_plus_real @ C2 @ B ) ) ) ).
% diff_less_eq
thf(fact_4293_diff__less__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C2 )
= ( ord_less_rat @ A @ ( plus_plus_rat @ C2 @ B ) ) ) ).
% diff_less_eq
thf(fact_4294_diff__less__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( ord_less_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_less_eq
thf(fact_4295_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: real,B: real] :
( ~ ( ord_less_real @ A @ B )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_4296_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: rat,B: rat] :
( ~ ( ord_less_rat @ A @ B )
=> ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_4297_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_4298_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_4299_mult__diff__mult,axiom,
! [X3: real,Y: real,A: real,B: real] :
( ( minus_minus_real @ ( times_times_real @ X3 @ Y ) @ ( times_times_real @ A @ B ) )
= ( plus_plus_real @ ( times_times_real @ X3 @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X3 @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_4300_mult__diff__mult,axiom,
! [X3: rat,Y: rat,A: rat,B: rat] :
( ( minus_minus_rat @ ( times_times_rat @ X3 @ Y ) @ ( times_times_rat @ A @ B ) )
= ( plus_plus_rat @ ( times_times_rat @ X3 @ ( minus_minus_rat @ Y @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X3 @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_4301_mult__diff__mult,axiom,
! [X3: int,Y: int,A: int,B: int] :
( ( minus_minus_int @ ( times_times_int @ X3 @ Y ) @ ( times_times_int @ A @ B ) )
= ( plus_plus_int @ ( times_times_int @ X3 @ ( minus_minus_int @ Y @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X3 @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_4302_eq__add__iff1,axiom,
! [A: real,E2: real,C2: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C2 )
= D ) ) ).
% eq_add_iff1
thf(fact_4303_eq__add__iff1,axiom,
! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C2 )
= ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
= ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C2 )
= D ) ) ).
% eq_add_iff1
thf(fact_4304_eq__add__iff1,axiom,
! [A: int,E2: int,C2: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C2 )
= D ) ) ).
% eq_add_iff1
thf(fact_4305_eq__add__iff2,axiom,
! [A: real,E2: real,C2: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( C2
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_4306_eq__add__iff2,axiom,
! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C2 )
= ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
= ( C2
= ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_4307_eq__add__iff2,axiom,
! [A: int,E2: int,C2: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( C2
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_4308_square__diff__square__factored,axiom,
! [X3: real,Y: real] :
( ( minus_minus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y @ Y ) )
= ( times_times_real @ ( plus_plus_real @ X3 @ Y ) @ ( minus_minus_real @ X3 @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_4309_square__diff__square__factored,axiom,
! [X3: rat,Y: rat] :
( ( minus_minus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y @ Y ) )
= ( times_times_rat @ ( plus_plus_rat @ X3 @ Y ) @ ( minus_minus_rat @ X3 @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_4310_square__diff__square__factored,axiom,
! [X3: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X3 @ Y ) @ ( minus_minus_int @ X3 @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_4311_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_4312_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_4313_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_4314_Suc__diff__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_4315_less__diff__iff,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_4316_diff__less__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_4317_diff__add__0,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_4318_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less_nat @ M2 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_4319_less__diff__conv,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ).
% less_diff_conv
thf(fact_4320_le__diff__conv,axiom,
! [J: nat,K2: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ).
% le_diff_conv
thf(fact_4321_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_4322_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_4323_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_4324_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K2 )
= ( J
= ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_4325_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_4326_int__le__induct,axiom,
! [I: int,K2: int,P2: int > $o] :
( ( ord_less_eq_int @ I @ K2 )
=> ( ( P2 @ K2 )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K2 )
=> ( ( P2 @ I2 )
=> ( P2 @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_le_induct
thf(fact_4327_int__less__induct,axiom,
! [I: int,K2: int,P2: int > $o] :
( ( ord_less_int @ I @ K2 )
=> ( ( P2 @ ( minus_minus_int @ K2 @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K2 )
=> ( ( P2 @ I2 )
=> ( P2 @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_less_induct
thf(fact_4328_le__mod__geq,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( modulo_modulo_nat @ M2 @ N )
= ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ).
% le_mod_geq
thf(fact_4329_mod__eq__dvd__iff__nat,axiom,
! [N: nat,M2: nat,Q3: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( ( modulo_modulo_nat @ M2 @ Q3 )
= ( modulo_modulo_nat @ N @ Q3 ) )
= ( dvd_dvd_nat @ Q3 @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% mod_eq_dvd_iff_nat
thf(fact_4330_nat__minus__add__max,axiom,
! [N: nat,M2: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ N @ M2 ) @ M2 )
= ( ord_max_nat @ N @ M2 ) ) ).
% nat_minus_add_max
thf(fact_4331_finite__lists__length__eq,axiom,
! [A4: set_complex,N: nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( finite8712137658972009173omplex
@ ( collect_list_complex
@ ^ [Xs3: list_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs3 ) @ A4 )
& ( ( size_s3451745648224563538omplex @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_4332_finite__lists__length__eq,axiom,
! [A4: set_Pr1261947904930325089at_nat,N: nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( finite500796754983035824at_nat
@ ( collec3343600615725829874at_nat
@ ^ [Xs3: list_P6011104703257516679at_nat] :
( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs3 ) @ A4 )
& ( ( size_s5460976970255530739at_nat @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_4333_finite__lists__length__eq,axiom,
! [A4: set_VEBT_VEBT,N: nat] :
( ( finite5795047828879050333T_VEBT @ A4 )
=> ( finite3004134309566078307T_VEBT
@ ( collec5608196760682091941T_VEBT
@ ^ [Xs3: list_VEBT_VEBT] :
( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs3 ) @ A4 )
& ( ( size_s6755466524823107622T_VEBT @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_4334_finite__lists__length__eq,axiom,
! [A4: set_nat,N: nat] :
( ( finite_finite_nat @ A4 )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs3: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A4 )
& ( ( size_size_list_nat @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_4335_finite__lists__length__eq,axiom,
! [A4: set_int,N: nat] :
( ( finite_finite_int @ A4 )
=> ( finite3922522038869484883st_int
@ ( collect_list_int
@ ^ [Xs3: list_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs3 ) @ A4 )
& ( ( size_size_list_int @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_4336_finite__lists__length__le,axiom,
! [A4: set_complex,N: nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( finite8712137658972009173omplex
@ ( collect_list_complex
@ ^ [Xs3: list_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs3 ) @ A4 )
& ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_4337_finite__lists__length__le,axiom,
! [A4: set_Pr1261947904930325089at_nat,N: nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( finite500796754983035824at_nat
@ ( collec3343600615725829874at_nat
@ ^ [Xs3: list_P6011104703257516679at_nat] :
( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs3 ) @ A4 )
& ( ord_less_eq_nat @ ( size_s5460976970255530739at_nat @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_4338_finite__lists__length__le,axiom,
! [A4: set_VEBT_VEBT,N: nat] :
( ( finite5795047828879050333T_VEBT @ A4 )
=> ( finite3004134309566078307T_VEBT
@ ( collec5608196760682091941T_VEBT
@ ^ [Xs3: list_VEBT_VEBT] :
( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs3 ) @ A4 )
& ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_4339_finite__lists__length__le,axiom,
! [A4: set_nat,N: nat] :
( ( finite_finite_nat @ A4 )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs3: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A4 )
& ( ord_less_eq_nat @ ( size_size_list_nat @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_4340_finite__lists__length__le,axiom,
! [A4: set_int,N: nat] :
( ( finite_finite_int @ A4 )
=> ( finite3922522038869484883st_int
@ ( collect_list_int
@ ^ [Xs3: list_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs3 ) @ A4 )
& ( ord_less_eq_nat @ ( size_size_list_int @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_4341_ordered__ring__class_Ole__add__iff2,axiom,
! [A: real,E2: real,C2: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( ord_less_eq_real @ C2 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_4342_ordered__ring__class_Ole__add__iff2,axiom,
! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
= ( ord_less_eq_rat @ C2 @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_4343_ordered__ring__class_Ole__add__iff2,axiom,
! [A: int,E2: int,C2: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ord_less_eq_int @ C2 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_4344_ordered__ring__class_Ole__add__iff1,axiom,
! [A: real,E2: real,C2: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C2 ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_4345_ordered__ring__class_Ole__add__iff1,axiom,
! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
= ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C2 ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_4346_ordered__ring__class_Ole__add__iff1,axiom,
! [A: int,E2: int,C2: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C2 ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_4347_less__add__iff1,axiom,
! [A: real,E2: real,C2: real,B: real,D: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C2 ) @ D ) ) ).
% less_add_iff1
thf(fact_4348_less__add__iff1,axiom,
! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
= ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C2 ) @ D ) ) ).
% less_add_iff1
thf(fact_4349_less__add__iff1,axiom,
! [A: int,E2: int,C2: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C2 ) @ D ) ) ).
% less_add_iff1
thf(fact_4350_less__add__iff2,axiom,
! [A: real,E2: real,C2: real,B: real,D: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( ord_less_real @ C2 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% less_add_iff2
thf(fact_4351_less__add__iff2,axiom,
! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
= ( ord_less_rat @ C2 @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% less_add_iff2
thf(fact_4352_less__add__iff2,axiom,
! [A: int,E2: int,C2: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ord_less_int @ C2 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% less_add_iff2
thf(fact_4353_divide__diff__eq__iff,axiom,
! [Z2: complex,X3: complex,Y: complex] :
( ( Z2 != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X3 @ Z2 ) @ Y )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ X3 @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% divide_diff_eq_iff
thf(fact_4354_divide__diff__eq__iff,axiom,
! [Z2: real,X3: real,Y: real] :
( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X3 @ Z2 ) @ Y )
= ( divide_divide_real @ ( minus_minus_real @ X3 @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% divide_diff_eq_iff
thf(fact_4355_divide__diff__eq__iff,axiom,
! [Z2: rat,X3: rat,Y: rat] :
( ( Z2 != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ X3 @ Z2 ) @ Y )
= ( divide_divide_rat @ ( minus_minus_rat @ X3 @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% divide_diff_eq_iff
thf(fact_4356_diff__divide__eq__iff,axiom,
! [Z2: complex,X3: complex,Y: complex] :
( ( Z2 != zero_zero_complex )
=> ( ( minus_minus_complex @ X3 @ ( divide1717551699836669952omplex @ Y @ Z2 ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X3 @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% diff_divide_eq_iff
thf(fact_4357_diff__divide__eq__iff,axiom,
! [Z2: real,X3: real,Y: real] :
( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ X3 @ ( divide_divide_real @ Y @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% diff_divide_eq_iff
thf(fact_4358_diff__divide__eq__iff,axiom,
! [Z2: rat,X3: rat,Y: rat] :
( ( Z2 != zero_zero_rat )
=> ( ( minus_minus_rat @ X3 @ ( divide_divide_rat @ Y @ Z2 ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z2 ) @ Y ) @ Z2 ) ) ) ).
% diff_divide_eq_iff
thf(fact_4359_diff__frac__eq,axiom,
! [Y: complex,Z2: complex,X3: complex,W2: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z2 != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X3 @ Y ) @ ( divide1717551699836669952omplex @ W2 @ Z2 ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X3 @ Z2 ) @ ( times_times_complex @ W2 @ Y ) ) @ ( times_times_complex @ Y @ Z2 ) ) ) ) ) ).
% diff_frac_eq
thf(fact_4360_diff__frac__eq,axiom,
! [Y: real,Z2: real,X3: real,W2: real] :
( ( Y != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X3 @ Y ) @ ( divide_divide_real @ W2 @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) ) ) ) ).
% diff_frac_eq
thf(fact_4361_diff__frac__eq,axiom,
! [Y: rat,Z2: rat,X3: rat,W2: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z2 != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ X3 @ Y ) @ ( divide_divide_rat @ W2 @ Z2 ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) ) ) ) ).
% diff_frac_eq
thf(fact_4362_add__divide__eq__if__simps_I4_J,axiom,
! [Z2: complex,A: complex,B: complex] :
( ( ( Z2 = zero_zero_complex )
=> ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
= A ) )
& ( ( Z2 != zero_zero_complex )
=> ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_4363_add__divide__eq__if__simps_I4_J,axiom,
! [Z2: real,A: real,B: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
= A ) )
& ( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_4364_add__divide__eq__if__simps_I4_J,axiom,
! [Z2: rat,A: rat,B: rat] :
( ( ( Z2 = zero_zero_rat )
=> ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
= A ) )
& ( ( Z2 != zero_zero_rat )
=> ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_4365_square__diff__one__factored,axiom,
! [X3: complex] :
( ( minus_minus_complex @ ( times_times_complex @ X3 @ X3 ) @ one_one_complex )
= ( times_times_complex @ ( plus_plus_complex @ X3 @ one_one_complex ) @ ( minus_minus_complex @ X3 @ one_one_complex ) ) ) ).
% square_diff_one_factored
thf(fact_4366_square__diff__one__factored,axiom,
! [X3: real] :
( ( minus_minus_real @ ( times_times_real @ X3 @ X3 ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X3 @ one_one_real ) @ ( minus_minus_real @ X3 @ one_one_real ) ) ) ).
% square_diff_one_factored
thf(fact_4367_square__diff__one__factored,axiom,
! [X3: rat] :
( ( minus_minus_rat @ ( times_times_rat @ X3 @ X3 ) @ one_one_rat )
= ( times_times_rat @ ( plus_plus_rat @ X3 @ one_one_rat ) @ ( minus_minus_rat @ X3 @ one_one_rat ) ) ) ).
% square_diff_one_factored
thf(fact_4368_square__diff__one__factored,axiom,
! [X3: int] :
( ( minus_minus_int @ ( times_times_int @ X3 @ X3 ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X3 @ one_one_int ) @ ( minus_minus_int @ X3 @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_4369_inf__period_I4_J,axiom,
! [D: code_integer,D5: code_integer,T: code_integer] :
( ( dvd_dvd_Code_integer @ D @ D5 )
=> ! [X: code_integer,K4: code_integer] :
( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X @ T ) ) )
= ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X @ ( times_3573771949741848930nteger @ K4 @ D5 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_4370_inf__period_I4_J,axiom,
! [D: real,D5: real,T: real] :
( ( dvd_dvd_real @ D @ D5 )
=> ! [X: real,K4: real] :
( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X @ T ) ) )
= ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X @ ( times_times_real @ K4 @ D5 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_4371_inf__period_I4_J,axiom,
! [D: rat,D5: rat,T: rat] :
( ( dvd_dvd_rat @ D @ D5 )
=> ! [X: rat,K4: rat] :
( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X @ T ) ) )
= ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ K4 @ D5 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_4372_inf__period_I4_J,axiom,
! [D: int,D5: int,T: int] :
( ( dvd_dvd_int @ D @ D5 )
=> ! [X: int,K4: int] :
( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X @ T ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X @ ( times_times_int @ K4 @ D5 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_4373_inf__period_I3_J,axiom,
! [D: code_integer,D5: code_integer,T: code_integer] :
( ( dvd_dvd_Code_integer @ D @ D5 )
=> ! [X: code_integer,K4: code_integer] :
( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X @ T ) )
= ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X @ ( times_3573771949741848930nteger @ K4 @ D5 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_4374_inf__period_I3_J,axiom,
! [D: real,D5: real,T: real] :
( ( dvd_dvd_real @ D @ D5 )
=> ! [X: real,K4: real] :
( ( dvd_dvd_real @ D @ ( plus_plus_real @ X @ T ) )
= ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X @ ( times_times_real @ K4 @ D5 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_4375_inf__period_I3_J,axiom,
! [D: rat,D5: rat,T: rat] :
( ( dvd_dvd_rat @ D @ D5 )
=> ! [X: rat,K4: rat] :
( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X @ T ) )
= ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ K4 @ D5 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_4376_inf__period_I3_J,axiom,
! [D: int,D5: int,T: int] :
( ( dvd_dvd_int @ D @ D5 )
=> ! [X: int,K4: int] :
( ( dvd_dvd_int @ D @ ( plus_plus_int @ X @ T ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X @ ( times_times_int @ K4 @ D5 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_4377_eucl__rel__int__dividesI,axiom,
! [L: int,K2: int,Q3: int] :
( ( L != zero_zero_int )
=> ( ( K2
= ( times_times_int @ Q3 @ L ) )
=> ( eucl_rel_int @ K2 @ L @ ( product_Pair_int_int @ Q3 @ zero_zero_int ) ) ) ) ).
% eucl_rel_int_dividesI
thf(fact_4378_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_4379_nat__diff__split__asm,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P2 @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P2 @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_4380_nat__diff__split,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P2 @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
=> ( P2 @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_4381_less__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_4382_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_4383_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_4384_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_4385_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_4386_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M2
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_4387_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_4388_dvd__minus__add,axiom,
! [Q3: nat,N: nat,R2: nat,M2: nat] :
( ( ord_less_eq_nat @ Q3 @ N )
=> ( ( ord_less_eq_nat @ Q3 @ ( times_times_nat @ R2 @ M2 ) )
=> ( ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N @ Q3 ) )
= ( dvd_dvd_nat @ M2 @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M2 ) @ Q3 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_4389_plusinfinity,axiom,
! [D: int,P5: int > $o,P2: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int,K: int] :
( ( P5 @ X4 )
= ( P5 @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D ) ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [X_1: int] : ( P5 @ X_1 )
=> ? [X_12: int] : ( P2 @ X_12 ) ) ) ) ) ).
% plusinfinity
thf(fact_4390_minusinfinity,axiom,
! [D: int,P1: int > $o,P2: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int,K: int] :
( ( P1 @ X4 )
= ( P1 @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D ) ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( P2 @ X4 )
= ( P1 @ X4 ) ) )
=> ( ? [X_1: int] : ( P1 @ X_1 )
=> ? [X_12: int] : ( P2 @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_4391_int__induct,axiom,
! [P2: int > $o,K2: int,I: int] :
( ( P2 @ K2 )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K2 @ I2 )
=> ( ( P2 @ I2 )
=> ( P2 @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K2 )
=> ( ( P2 @ I2 )
=> ( P2 @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_induct
thf(fact_4392_mod__nat__eqI,axiom,
! [R2: nat,N: nat,M2: nat] :
( ( ord_less_nat @ R2 @ N )
=> ( ( ord_less_eq_nat @ R2 @ M2 )
=> ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M2 @ R2 ) )
=> ( ( modulo_modulo_nat @ M2 @ N )
= R2 ) ) ) ) ).
% mod_nat_eqI
thf(fact_4393_eucl__rel__int,axiom,
! [K2: int,L: int] : ( eucl_rel_int @ K2 @ L @ ( product_Pair_int_int @ ( divide_divide_int @ K2 @ L ) @ ( modulo_modulo_int @ K2 @ L ) ) ) ).
% eucl_rel_int
thf(fact_4394_exp__div__exp__eq,axiom,
! [M2: nat,N: nat] :
( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
!= zero_zero_nat )
& ( ord_less_eq_nat @ N @ M2 ) ) )
@ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% exp_div_exp_eq
thf(fact_4395_exp__div__exp__eq,axiom,
! [M2: nat,N: nat] :
( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( times_times_int
@ ( zero_n2684676970156552555ol_int
@ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 )
!= zero_zero_int )
& ( ord_less_eq_nat @ N @ M2 ) ) )
@ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% exp_div_exp_eq
thf(fact_4396_frac__le__eq,axiom,
! [Y: real,Z2: real,X3: real,W2: real] :
( ( Y != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y ) @ ( divide_divide_real @ W2 @ Z2 ) )
= ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) @ zero_zero_real ) ) ) ) ).
% frac_le_eq
thf(fact_4397_frac__le__eq,axiom,
! [Y: rat,Z2: rat,X3: rat,W2: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z2 != zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y ) @ ( divide_divide_rat @ W2 @ Z2 ) )
= ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).
% frac_le_eq
thf(fact_4398_frac__less__eq,axiom,
! [Y: real,Z2: real,X3: real,W2: real] :
( ( Y != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ X3 @ Y ) @ ( divide_divide_real @ W2 @ Z2 ) )
= ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) @ zero_zero_real ) ) ) ) ).
% frac_less_eq
thf(fact_4399_frac__less__eq,axiom,
! [Y: rat,Z2: rat,X3: rat,W2: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z2 != zero_zero_rat )
=> ( ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y ) @ ( divide_divide_rat @ W2 @ Z2 ) )
= ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).
% frac_less_eq
thf(fact_4400_power__diff,axiom,
! [A: complex,N: nat,M2: nat] :
( ( A != zero_zero_complex )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( power_power_complex @ A @ ( minus_minus_nat @ M2 @ N ) )
= ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M2 ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_4401_power__diff,axiom,
! [A: real,N: nat,M2: nat] :
( ( A != zero_zero_real )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( power_power_real @ A @ ( minus_minus_nat @ M2 @ N ) )
= ( divide_divide_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_4402_power__diff,axiom,
! [A: rat,N: nat,M2: nat] :
( ( A != zero_zero_rat )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( power_power_rat @ A @ ( minus_minus_nat @ M2 @ N ) )
= ( divide_divide_rat @ ( power_power_rat @ A @ M2 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_4403_power__diff,axiom,
! [A: nat,N: nat,M2: nat] :
( ( A != zero_zero_nat )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( power_power_nat @ A @ ( minus_minus_nat @ M2 @ N ) )
= ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_4404_power__diff,axiom,
! [A: int,N: nat,M2: nat] :
( ( A != zero_zero_int )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( power_power_int @ A @ ( minus_minus_nat @ M2 @ N ) )
= ( divide_divide_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_4405_even__diff__iff,axiom,
! [K2: int,L: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K2 @ L ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K2 @ L ) ) ) ).
% even_diff_iff
thf(fact_4406_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_4407_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_4408_div__geq,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( ( divide_divide_nat @ M2 @ N )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ) ).
% div_geq
thf(fact_4409_div__if,axiom,
( divide_divide_nat
= ( ^ [M5: nat,N4: nat] :
( if_nat
@ ( ( ord_less_nat @ M5 @ N4 )
| ( N4 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M5 @ N4 ) @ N4 ) ) ) ) ) ).
% div_if
thf(fact_4410_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_4411_vebt__buildup_Osimps_I3_J,axiom,
! [Va: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
= ( 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 ) ) ) ) ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
= ( 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 ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.simps(3)
thf(fact_4412_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_4413_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_4414_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% mult_eq_if
thf(fact_4415_nth__Cons_H,axiom,
! [N: nat,X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
( ( ( N = zero_zero_nat )
=> ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X3 @ Xs ) @ N )
= X3 ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X3 @ Xs ) @ N )
= ( nth_VEBT_VEBT @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_4416_nth__Cons_H,axiom,
! [N: nat,X3: int,Xs: list_int] :
( ( ( N = zero_zero_nat )
=> ( ( nth_int @ ( cons_int @ X3 @ Xs ) @ N )
= X3 ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_int @ ( cons_int @ X3 @ Xs ) @ N )
= ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_4417_nth__Cons_H,axiom,
! [N: nat,X3: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X3 @ Xs ) @ N )
= X3 ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X3 @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_4418_decr__mult__lemma,axiom,
! [D: int,P2: int > $o,K2: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int] :
( ( P2 @ X4 )
=> ( P2 @ ( minus_minus_int @ X4 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ! [X: int] :
( ( P2 @ X )
=> ( P2 @ ( minus_minus_int @ X @ ( times_times_int @ K2 @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_4419_mod__pos__geq,axiom,
! [L: int,K2: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ L @ K2 )
=> ( ( modulo_modulo_int @ K2 @ L )
= ( modulo_modulo_int @ ( minus_minus_int @ K2 @ L ) @ L ) ) ) ) ).
% mod_pos_geq
thf(fact_4420_neg__eucl__rel__int__mult__2,axiom,
! [B: int,A: int,Q3: int,R2: int] :
( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
=> ( 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 ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_4421_scaling__mono,axiom,
! [U: real,V: real,R2: real,S3: real] :
( ( ord_less_eq_real @ U @ V )
=> ( ( ord_less_eq_real @ zero_zero_real @ R2 )
=> ( ( ord_less_eq_real @ R2 @ S3 )
=> ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S3 ) ) @ V ) ) ) ) ).
% scaling_mono
thf(fact_4422_scaling__mono,axiom,
! [U: rat,V: rat,R2: rat,S3: rat] :
( ( ord_less_eq_rat @ U @ V )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
=> ( ( ord_less_eq_rat @ R2 @ S3 )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S3 ) ) @ V ) ) ) ) ).
% scaling_mono
thf(fact_4423_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [N: nat,M2: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) )
!= zero_zero_nat ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_4424_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [N: nat,M2: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) )
!= zero_zero_int ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_4425_power__diff__power__eq,axiom,
! [A: nat,N: nat,M2: nat] :
( ( A != zero_zero_nat )
=> ( ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
= ( power_power_nat @ A @ ( minus_minus_nat @ M2 @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M2 )
=> ( ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
= ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_4426_power__diff__power__eq,axiom,
! [A: int,N: nat,M2: nat] :
( ( A != zero_zero_int )
=> ( ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( divide_divide_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
= ( power_power_int @ A @ ( minus_minus_nat @ M2 @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M2 )
=> ( ( divide_divide_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
= ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_4427_power__eq__if,axiom,
( power_power_complex
= ( ^ [P6: complex,M5: nat] : ( if_complex @ ( M5 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P6 @ ( power_power_complex @ P6 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_4428_power__eq__if,axiom,
( power_power_real
= ( ^ [P6: real,M5: nat] : ( if_real @ ( M5 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P6 @ ( power_power_real @ P6 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_4429_power__eq__if,axiom,
( power_power_rat
= ( ^ [P6: rat,M5: nat] : ( if_rat @ ( M5 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P6 @ ( power_power_rat @ P6 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_4430_power__eq__if,axiom,
( power_power_nat
= ( ^ [P6: nat,M5: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P6 @ ( power_power_nat @ P6 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_4431_power__eq__if,axiom,
( power_power_int
= ( ^ [P6: int,M5: nat] : ( if_int @ ( M5 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P6 @ ( power_power_int @ P6 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_4432_vebt__buildup_Oelims,axiom,
! [X3: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X3 )
= Y )
=> ( ( ( X3 = zero_zero_nat )
=> ( Y
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ( ( ( X3
= ( suc @ zero_zero_nat ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Va2: nat] :
( ( X3
= ( suc @ ( suc @ Va2 ) ) )
=> ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( Y
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( Y
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.elims
thf(fact_4433_power__minus__mult,axiom,
! [N: nat,A: complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_complex @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_4434_power__minus__mult,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_real @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_4435_power__minus__mult,axiom,
! [N: nat,A: rat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_rat @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_4436_power__minus__mult,axiom,
! [N: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_nat @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_4437_power__minus__mult,axiom,
! [N: nat,A: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_int @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_4438_diff__le__diff__pow,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K2 @ M2 ) @ ( power_power_nat @ K2 @ N ) ) ) ) ).
% diff_le_diff_pow
thf(fact_4439_le__div__geq,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( divide_divide_nat @ M2 @ N )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_4440_num_Osize__gen_I1_J,axiom,
( ( size_num @ one )
= zero_zero_nat ) ).
% num.size_gen(1)
thf(fact_4441_nth__non__equal__first__eq,axiom,
! [X3: vEBT_VEBT,Y: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] :
( ( X3 != Y )
=> ( ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X3 @ Xs ) @ N )
= Y )
= ( ( ( nth_VEBT_VEBT @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_4442_nth__non__equal__first__eq,axiom,
! [X3: int,Y: int,Xs: list_int,N: nat] :
( ( X3 != Y )
=> ( ( ( nth_int @ ( cons_int @ X3 @ Xs ) @ N )
= Y )
= ( ( ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_4443_nth__non__equal__first__eq,axiom,
! [X3: nat,Y: nat,Xs: list_nat,N: nat] :
( ( X3 != Y )
=> ( ( ( nth_nat @ ( cons_nat @ X3 @ Xs ) @ N )
= Y )
= ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_4444_Cons__replicate__eq,axiom,
! [X3: int,Xs: list_int,N: nat,Y: int] :
( ( ( cons_int @ X3 @ Xs )
= ( replicate_int @ N @ Y ) )
= ( ( X3 = Y )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( Xs
= ( replicate_int @ ( minus_minus_nat @ N @ one_one_nat ) @ X3 ) ) ) ) ).
% Cons_replicate_eq
thf(fact_4445_Cons__replicate__eq,axiom,
! [X3: nat,Xs: list_nat,N: nat,Y: nat] :
( ( ( cons_nat @ X3 @ Xs )
= ( replicate_nat @ N @ Y ) )
= ( ( X3 = Y )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( Xs
= ( replicate_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ X3 ) ) ) ) ).
% Cons_replicate_eq
thf(fact_4446_Cons__replicate__eq,axiom,
! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat,Y: vEBT_VEBT] :
( ( ( cons_VEBT_VEBT @ X3 @ Xs )
= ( replicate_VEBT_VEBT @ N @ Y ) )
= ( ( X3 = Y )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( Xs
= ( replicate_VEBT_VEBT @ ( minus_minus_nat @ N @ one_one_nat ) @ X3 ) ) ) ) ).
% Cons_replicate_eq
thf(fact_4447_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S3 ) @ X3 )
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).
% VEBT_internal.naive_member.simps(3)
thf(fact_4448_bits__induct,axiom,
! [P2: nat > $o,A: nat] :
( ! [A3: nat] :
( ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A3 )
=> ( P2 @ A3 ) )
=> ( ! [A3: nat,B3: $o] :
( ( P2 @ A3 )
=> ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A3 )
=> ( P2 @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
=> ( P2 @ A ) ) ) ).
% bits_induct
thf(fact_4449_bits__induct,axiom,
! [P2: int > $o,A: int] :
( ! [A3: int] :
( ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A3 )
=> ( P2 @ A3 ) )
=> ( ! [A3: int,B3: $o] :
( ( P2 @ A3 )
=> ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A3 )
=> ( P2 @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
=> ( P2 @ A ) ) ) ).
% bits_induct
thf(fact_4450_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
! [V: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,X3: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd2 ) @ X3 )
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).
% VEBT_internal.membermima.simps(5)
thf(fact_4451_signed__take__bit__int__less__eq,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K2 )
=> ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) @ ( minus_minus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).
% signed_take_bit_int_less_eq
thf(fact_4452_div__pos__geq,axiom,
! [L: int,K2: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ L @ K2 )
=> ( ( divide_divide_int @ K2 @ L )
= ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K2 @ L ) @ L ) @ one_one_int ) ) ) ) ).
% div_pos_geq
thf(fact_4453_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList @ Vc ) @ X3 )
= ( ( X3 = Mi )
| ( X3 = Ma )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ).
% VEBT_internal.membermima.simps(4)
thf(fact_4454_vebt__member_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X3 )
= ( ( X3 != Mi )
=> ( ( X3 != Ma )
=> ( ~ ( ord_less_nat @ X3 @ Mi )
& ( ~ ( ord_less_nat @ X3 @ Mi )
=> ( ~ ( ord_less_nat @ Ma @ X3 )
& ( ~ ( ord_less_nat @ Ma @ X3 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.simps(5)
thf(fact_4455_power2__diff,axiom,
! [X3: complex,Y: complex] :
( ( power_power_complex @ ( minus_minus_complex @ X3 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) @ Y ) ) ) ).
% power2_diff
thf(fact_4456_power2__diff,axiom,
! [X3: real,Y: real] :
( ( power_power_real @ ( minus_minus_real @ X3 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) @ Y ) ) ) ).
% power2_diff
thf(fact_4457_power2__diff,axiom,
! [X3: rat,Y: rat] :
( ( power_power_rat @ ( minus_minus_rat @ X3 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X3 ) @ Y ) ) ) ).
% power2_diff
thf(fact_4458_power2__diff,axiom,
! [X3: int,Y: int] :
( ( power_power_int @ ( minus_minus_int @ X3 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 ) @ Y ) ) ) ).
% power2_diff
thf(fact_4459_mult__exp__mod__exp__eq,axiom,
! [M2: nat,N: nat,A: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_4460_mult__exp__mod__exp__eq,axiom,
! [M2: nat,N: nat,A: int] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_4461_mult__exp__mod__exp__eq,axiom,
! [M2: nat,N: nat,A: code_integer] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_4462_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
= Y )
=> ( ! [A3: $o,B3: $o] :
( ( X3
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ( Y
= ( ~ ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B3 )
& ( Xa2 = one_one_nat ) ) ) ) ) ) )
=> ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
=> Y )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [S: vEBT_VEBT] :
( X3
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
=> ( Y
= ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( 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 ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(1)
thf(fact_4463_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
=> ( ! [A3: $o,B3: $o] :
( ( X3
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B3 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [S: vEBT_VEBT] :
( X3
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( 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 ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(2)
thf(fact_4464_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
=> ( ! [A3: $o,B3: $o] :
( ( X3
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B3 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [S: vEBT_VEBT] :
( X3
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( 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 ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(3)
thf(fact_4465_even__mod__4__div__2,axiom,
! [N: nat] :
( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( suc @ zero_zero_nat ) )
=> ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_mod_4_div_2
thf(fact_4466_eucl__rel__int__iff,axiom,
! [K2: int,L: int,Q3: int,R2: int] :
( ( eucl_rel_int @ K2 @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
= ( ( K2
= ( plus_plus_int @ ( times_times_int @ L @ Q3 ) @ R2 ) )
& ( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
& ( ord_less_int @ R2 @ L ) ) )
& ( ~ ( ord_less_int @ zero_zero_int @ L )
=> ( ( ( ord_less_int @ L @ zero_zero_int )
=> ( ( ord_less_int @ L @ R2 )
& ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
& ( ~ ( ord_less_int @ L @ zero_zero_int )
=> ( Q3 = zero_zero_int ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_4467_int__power__div__base,axiom,
! [M2: nat,K2: int] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ( divide_divide_int @ ( power_power_int @ K2 @ M2 ) @ K2 )
= ( power_power_int @ K2 @ ( minus_minus_nat @ M2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_4468_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( 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 ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X3
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( 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 ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(2)
thf(fact_4469_divmod__digit__1_I2_J,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_4470_divmod__digit__1_I2_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_4471_divmod__digit__1_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_4472_even__mask__div__iff_H,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% even_mask_div_iff'
thf(fact_4473_even__mask__div__iff_H,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% even_mask_div_iff'
thf(fact_4474_even__mask__div__iff_H,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% even_mask_div_iff'
thf(fact_4475_vebt__member_Oelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ( vEBT_vebt_member @ X3 @ Xa2 )
=> ( ! [A3: $o,B3: $o] :
( ( X3
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B3 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(2)
thf(fact_4476_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X3
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> Y )
=> ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
=> Y )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
=> ( Y
= ( ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
=> ( Y
= ( ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( 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 ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X3
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd ) )
=> ( Y
= ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( 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 ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(1)
thf(fact_4477_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_membermima @ X3 @ Xa2 )
=> ( ! [Uu2: $o,Uv2: $o] :
( X3
!= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( 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 ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X3
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( 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 ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(3)
thf(fact_4478_vebt__insert_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X3 )
= ( if_VEBT_VEBT
@ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
& ~ ( ( X3 = Mi )
| ( X3 = Ma ) ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ X3 @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ Mi @ X3 ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( 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 @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( 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 @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) ) ).
% vebt_insert.simps(5)
thf(fact_4479_vebt__member_Oelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_vebt_member @ X3 @ Xa2 )
=> ( ! [A3: $o,B3: $o] :
( ( X3
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B3 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X3
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(3)
thf(fact_4480_vebt__member_Oelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X3 @ Xa2 )
= Y )
=> ( ! [A3: $o,B3: $o] :
( ( X3
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ( Y
= ( ~ ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B3 )
& ( Xa2 = one_one_nat ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
=> Y )
=> ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
=> Y )
=> ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
=> Y )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
= ( ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(1)
thf(fact_4481_even__mask__div__iff,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
= zero_z3403309356797280102nteger )
| ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% even_mask_div_iff
thf(fact_4482_even__mask__div__iff,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= zero_zero_nat )
| ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% even_mask_div_iff
thf(fact_4483_even__mask__div__iff,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
= zero_zero_int )
| ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% even_mask_div_iff
thf(fact_4484_divmod__step__eq,axiom,
! [L: num,R2: nat,Q3: nat] :
( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
=> ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
= ( 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 ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
=> ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
= ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).
% divmod_step_eq
thf(fact_4485_divmod__step__eq,axiom,
! [L: num,R2: int,Q3: int] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
=> ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
= ( 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 ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
=> ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
= ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).
% divmod_step_eq
thf(fact_4486_divmod__step__eq,axiom,
! [L: num,R2: code_integer,Q3: code_integer] :
( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
=> ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R2 ) )
= ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
& ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
=> ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R2 ) )
= ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).
% divmod_step_eq
thf(fact_4487_inrange,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( 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 ) ) ) ) ).
% inrange
thf(fact_4488_finite__nth__roots,axiom,
! [N: nat,C2: complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z6: complex] :
( ( power_power_complex @ Z6 @ N )
= C2 ) ) ) ) ).
% finite_nth_roots
thf(fact_4489_finite__roots__unity,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Z6: real] :
( ( power_power_real @ Z6 @ N )
= one_one_real ) ) ) ) ).
% finite_roots_unity
thf(fact_4490_finite__roots__unity,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z6: complex] :
( ( power_power_complex @ Z6 @ N )
= one_one_complex ) ) ) ) ).
% finite_roots_unity
thf(fact_4491_vebt__insert_Opelims,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X3 @ Xa2 )
= Y )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
=> ( ! [A3: $o,B3: $o] :
( ( X3
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ( ( ( ( Xa2 = zero_zero_nat )
=> ( Y
= ( vEBT_Leaf @ $true @ B3 ) ) )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> ( Y
= ( vEBT_Leaf @ A3 @ $true ) ) )
& ( ( Xa2 != one_one_nat )
=> ( Y
= ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S ) )
=> ( ( Y
= ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S ) @ Xa2 ) ) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) )
=> ( ( Y
= ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) @ Xa2 ) ) ) )
=> ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( if_VEBT_VEBT
@ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
& ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_insert.pelims
thf(fact_4492_prod_Ofinite__Collect__op,axiom,
! [I6: set_real,X3: real > complex,Y: real > complex] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I6 )
& ( ( X3 @ I3 )
!= one_one_complex ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I6 )
& ( ( Y @ I3 )
!= one_one_complex ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I6 )
& ( ( times_times_complex @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_4493_prod_Ofinite__Collect__op,axiom,
! [I6: set_nat,X3: nat > complex,Y: nat > complex] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I6 )
& ( ( X3 @ I3 )
!= one_one_complex ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I6 )
& ( ( Y @ I3 )
!= one_one_complex ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I6 )
& ( ( times_times_complex @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_4494_prod_Ofinite__Collect__op,axiom,
! [I6: set_int,X3: int > complex,Y: int > complex] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I6 )
& ( ( X3 @ I3 )
!= one_one_complex ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I6 )
& ( ( Y @ I3 )
!= one_one_complex ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I6 )
& ( ( times_times_complex @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_4495_prod_Ofinite__Collect__op,axiom,
! [I6: set_complex,X3: complex > complex,Y: complex > complex] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I6 )
& ( ( X3 @ I3 )
!= one_one_complex ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I6 )
& ( ( Y @ I3 )
!= one_one_complex ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I6 )
& ( ( times_times_complex @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_4496_prod_Ofinite__Collect__op,axiom,
! [I6: set_real,X3: real > real,Y: real > real] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I6 )
& ( ( X3 @ I3 )
!= one_one_real ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I6 )
& ( ( Y @ I3 )
!= one_one_real ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I6 )
& ( ( times_times_real @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= one_one_real ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_4497_prod_Ofinite__Collect__op,axiom,
! [I6: set_nat,X3: nat > real,Y: nat > real] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I6 )
& ( ( X3 @ I3 )
!= one_one_real ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I6 )
& ( ( Y @ I3 )
!= one_one_real ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I6 )
& ( ( times_times_real @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= one_one_real ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_4498_prod_Ofinite__Collect__op,axiom,
! [I6: set_int,X3: int > real,Y: int > real] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I6 )
& ( ( X3 @ I3 )
!= one_one_real ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I6 )
& ( ( Y @ I3 )
!= one_one_real ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I6 )
& ( ( times_times_real @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= one_one_real ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_4499_prod_Ofinite__Collect__op,axiom,
! [I6: set_complex,X3: complex > real,Y: complex > real] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I6 )
& ( ( X3 @ I3 )
!= one_one_real ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I6 )
& ( ( Y @ I3 )
!= one_one_real ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I6 )
& ( ( times_times_real @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= one_one_real ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_4500_prod_Ofinite__Collect__op,axiom,
! [I6: set_real,X3: real > rat,Y: real > rat] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I6 )
& ( ( X3 @ I3 )
!= one_one_rat ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I6 )
& ( ( Y @ I3 )
!= one_one_rat ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I6 )
& ( ( times_times_rat @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= one_one_rat ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_4501_prod_Ofinite__Collect__op,axiom,
! [I6: set_nat,X3: nat > rat,Y: nat > rat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I6 )
& ( ( X3 @ I3 )
!= one_one_rat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I6 )
& ( ( Y @ I3 )
!= one_one_rat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I6 )
& ( ( times_times_rat @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= one_one_rat ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_4502_finite__Diff2,axiom,
! [B5: set_int,A4: set_int] :
( ( finite_finite_int @ B5 )
=> ( ( finite_finite_int @ ( minus_minus_set_int @ A4 @ B5 ) )
= ( finite_finite_int @ A4 ) ) ) ).
% finite_Diff2
thf(fact_4503_finite__Diff2,axiom,
! [B5: set_complex,A4: set_complex] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
= ( finite3207457112153483333omplex @ A4 ) ) ) ).
% finite_Diff2
thf(fact_4504_finite__Diff2,axiom,
! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ B5 )
=> ( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) )
= ( finite6177210948735845034at_nat @ A4 ) ) ) ).
% finite_Diff2
thf(fact_4505_finite__Diff2,axiom,
! [B5: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( finite_finite_nat @ A4 ) ) ) ).
% finite_Diff2
thf(fact_4506_finite__Diff,axiom,
! [A4: set_int,B5: set_int] :
( ( finite_finite_int @ A4 )
=> ( finite_finite_int @ ( minus_minus_set_int @ A4 @ B5 ) ) ) ).
% finite_Diff
thf(fact_4507_finite__Diff,axiom,
! [A4: set_complex,B5: set_complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) ) ).
% finite_Diff
thf(fact_4508_finite__Diff,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) ) ) ).
% finite_Diff
thf(fact_4509_finite__Diff,axiom,
! [A4: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B5 ) ) ) ).
% finite_Diff
thf(fact_4510_idiff__0,axiom,
! [N: extended_enat] :
( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
= zero_z5237406670263579293d_enat ) ).
% idiff_0
thf(fact_4511_idiff__0__right,axiom,
! [N: extended_enat] :
( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
= N ) ).
% idiff_0_right
thf(fact_4512_Diff__eq__empty__iff,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ( minus_1356011639430497352at_nat @ A4 @ B5 )
= bot_bo2099793752762293965at_nat )
= ( ord_le3146513528884898305at_nat @ A4 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_4513_Diff__eq__empty__iff,axiom,
! [A4: set_real,B5: set_real] :
( ( ( minus_minus_set_real @ A4 @ B5 )
= bot_bot_set_real )
= ( ord_less_eq_set_real @ A4 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_4514_Diff__eq__empty__iff,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ( minus_minus_set_nat @ A4 @ B5 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A4 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_4515_Diff__eq__empty__iff,axiom,
! [A4: set_int,B5: set_int] :
( ( ( minus_minus_set_int @ A4 @ B5 )
= bot_bot_set_int )
= ( ord_less_eq_set_int @ A4 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_4516_atLeastAtMost__iff,axiom,
! [I: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L @ U ) )
= ( ( ord_less_eq_set_nat @ L @ I )
& ( ord_less_eq_set_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_4517_atLeastAtMost__iff,axiom,
! [I: set_int,L: set_int,U: set_int] :
( ( member_set_int @ I @ ( set_or370866239135849197et_int @ L @ U ) )
= ( ( ord_less_eq_set_int @ L @ I )
& ( ord_less_eq_set_int @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_4518_atLeastAtMost__iff,axiom,
! [I: rat,L: rat,U: rat] :
( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L @ U ) )
= ( ( ord_less_eq_rat @ L @ I )
& ( ord_less_eq_rat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_4519_atLeastAtMost__iff,axiom,
! [I: num,L: num,U: num] :
( ( member_num @ I @ ( set_or7049704709247886629st_num @ L @ U ) )
= ( ( ord_less_eq_num @ L @ I )
& ( ord_less_eq_num @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_4520_atLeastAtMost__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_eq_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_4521_atLeastAtMost__iff,axiom,
! [I: int,L: int,U: int] :
( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U ) )
= ( ( ord_less_eq_int @ L @ I )
& ( ord_less_eq_int @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_4522_atLeastAtMost__iff,axiom,
! [I: real,L: real,U: real] :
( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U ) )
= ( ( ord_less_eq_real @ L @ I )
& ( ord_less_eq_real @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_4523_Icc__eq__Icc,axiom,
! [L: set_int,H2: set_int,L3: set_int,H3: set_int] :
( ( ( set_or370866239135849197et_int @ L @ H2 )
= ( set_or370866239135849197et_int @ L3 @ H3 ) )
= ( ( ( L = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_set_int @ L @ H2 )
& ~ ( ord_less_eq_set_int @ L3 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_4524_Icc__eq__Icc,axiom,
! [L: rat,H2: rat,L3: rat,H3: rat] :
( ( ( set_or633870826150836451st_rat @ L @ H2 )
= ( set_or633870826150836451st_rat @ L3 @ H3 ) )
= ( ( ( L = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_rat @ L @ H2 )
& ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_4525_Icc__eq__Icc,axiom,
! [L: num,H2: num,L3: num,H3: num] :
( ( ( set_or7049704709247886629st_num @ L @ H2 )
= ( set_or7049704709247886629st_num @ L3 @ H3 ) )
= ( ( ( L = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_num @ L @ H2 )
& ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_4526_Icc__eq__Icc,axiom,
! [L: nat,H2: nat,L3: nat,H3: nat] :
( ( ( set_or1269000886237332187st_nat @ L @ H2 )
= ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
= ( ( ( L = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_nat @ L @ H2 )
& ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_4527_Icc__eq__Icc,axiom,
! [L: int,H2: int,L3: int,H3: int] :
( ( ( set_or1266510415728281911st_int @ L @ H2 )
= ( set_or1266510415728281911st_int @ L3 @ H3 ) )
= ( ( ( L = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_int @ L @ H2 )
& ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_4528_Icc__eq__Icc,axiom,
! [L: real,H2: real,L3: real,H3: real] :
( ( ( set_or1222579329274155063t_real @ L @ H2 )
= ( set_or1222579329274155063t_real @ L3 @ H3 ) )
= ( ( ( L = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_real @ L @ H2 )
& ~ ( ord_less_eq_real @ L3 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_4529_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% finite_atLeastAtMost
thf(fact_4530_atLeastatMost__empty__iff,axiom,
! [A: set_int,B: set_int] :
( ( ( set_or370866239135849197et_int @ A @ B )
= bot_bot_set_set_int )
= ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_4531_atLeastatMost__empty__iff,axiom,
! [A: rat,B: rat] :
( ( ( set_or633870826150836451st_rat @ A @ B )
= bot_bot_set_rat )
= ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_4532_atLeastatMost__empty__iff,axiom,
! [A: num,B: num] :
( ( ( set_or7049704709247886629st_num @ A @ B )
= bot_bot_set_num )
= ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_4533_atLeastatMost__empty__iff,axiom,
! [A: nat,B: nat] :
( ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat )
= ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_4534_atLeastatMost__empty__iff,axiom,
! [A: int,B: int] :
( ( ( set_or1266510415728281911st_int @ A @ B )
= bot_bot_set_int )
= ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_4535_atLeastatMost__empty__iff,axiom,
! [A: real,B: real] :
( ( ( set_or1222579329274155063t_real @ A @ B )
= bot_bot_set_real )
= ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_4536_atLeastatMost__empty__iff2,axiom,
! [A: set_int,B: set_int] :
( ( bot_bot_set_set_int
= ( set_or370866239135849197et_int @ A @ B ) )
= ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_4537_atLeastatMost__empty__iff2,axiom,
! [A: rat,B: rat] :
( ( bot_bot_set_rat
= ( set_or633870826150836451st_rat @ A @ B ) )
= ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_4538_atLeastatMost__empty__iff2,axiom,
! [A: num,B: num] :
( ( bot_bot_set_num
= ( set_or7049704709247886629st_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_4539_atLeastatMost__empty__iff2,axiom,
! [A: nat,B: nat] :
( ( bot_bot_set_nat
= ( set_or1269000886237332187st_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_4540_atLeastatMost__empty__iff2,axiom,
! [A: int,B: int] :
( ( bot_bot_set_int
= ( set_or1266510415728281911st_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_4541_atLeastatMost__empty__iff2,axiom,
! [A: real,B: real] :
( ( bot_bot_set_real
= ( set_or1222579329274155063t_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_4542_atLeastatMost__subset__iff,axiom,
! [A: set_int,B: set_int,C2: set_int,D: set_int] :
( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C2 @ D ) )
= ( ~ ( ord_less_eq_set_int @ A @ B )
| ( ( ord_less_eq_set_int @ C2 @ A )
& ( ord_less_eq_set_int @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_4543_atLeastatMost__subset__iff,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C2 @ D ) )
= ( ~ ( ord_less_eq_rat @ A @ B )
| ( ( ord_less_eq_rat @ C2 @ A )
& ( ord_less_eq_rat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_4544_atLeastatMost__subset__iff,axiom,
! [A: num,B: num,C2: num,D: num] :
( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C2 @ D ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( ( ord_less_eq_num @ C2 @ A )
& ( ord_less_eq_num @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_4545_atLeastatMost__subset__iff,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C2 @ D ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C2 @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_4546_atLeastatMost__subset__iff,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C2 @ D ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( ( ord_less_eq_int @ C2 @ A )
& ( ord_less_eq_int @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_4547_atLeastatMost__subset__iff,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C2 @ D ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C2 @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_4548_atLeastatMost__empty,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( set_or633870826150836451st_rat @ A @ B )
= bot_bot_set_rat ) ) ).
% atLeastatMost_empty
thf(fact_4549_atLeastatMost__empty,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ( set_or7049704709247886629st_num @ A @ B )
= bot_bot_set_num ) ) ).
% atLeastatMost_empty
thf(fact_4550_atLeastatMost__empty,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% atLeastatMost_empty
thf(fact_4551_atLeastatMost__empty,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( set_or1266510415728281911st_int @ A @ B )
= bot_bot_set_int ) ) ).
% atLeastatMost_empty
thf(fact_4552_atLeastatMost__empty,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ( set_or1222579329274155063t_real @ A @ B )
= bot_bot_set_real ) ) ).
% atLeastatMost_empty
thf(fact_4553_infinite__Icc__iff,axiom,
! [A: rat,B: rat] :
( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) )
= ( ord_less_rat @ A @ B ) ) ).
% infinite_Icc_iff
thf(fact_4554_infinite__Icc__iff,axiom,
! [A: real,B: real] :
( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) )
= ( ord_less_real @ A @ B ) ) ).
% infinite_Icc_iff
thf(fact_4555_Diff__infinite__finite,axiom,
! [T2: set_int,S2: set_int] :
( ( finite_finite_int @ T2 )
=> ( ~ ( finite_finite_int @ S2 )
=> ~ ( finite_finite_int @ ( minus_minus_set_int @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_4556_Diff__infinite__finite,axiom,
! [T2: set_complex,S2: set_complex] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ~ ( finite3207457112153483333omplex @ S2 )
=> ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_4557_Diff__infinite__finite,axiom,
! [T2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ T2 )
=> ( ~ ( finite6177210948735845034at_nat @ S2 )
=> ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_4558_Diff__infinite__finite,axiom,
! [T2: set_nat,S2: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_4559_Diff__mono,axiom,
! [A4: set_nat,C4: set_nat,D5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ C4 )
=> ( ( ord_less_eq_set_nat @ D5 @ B5 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ B5 ) @ ( minus_minus_set_nat @ C4 @ D5 ) ) ) ) ).
% Diff_mono
thf(fact_4560_Diff__mono,axiom,
! [A4: set_int,C4: set_int,D5: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ A4 @ C4 )
=> ( ( ord_less_eq_set_int @ D5 @ B5 )
=> ( ord_less_eq_set_int @ ( minus_minus_set_int @ A4 @ B5 ) @ ( minus_minus_set_int @ C4 @ D5 ) ) ) ) ).
% Diff_mono
thf(fact_4561_Diff__subset,axiom,
! [A4: set_nat,B5: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ B5 ) @ A4 ) ).
% Diff_subset
thf(fact_4562_Diff__subset,axiom,
! [A4: set_int,B5: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A4 @ B5 ) @ A4 ) ).
% Diff_subset
thf(fact_4563_double__diff,axiom,
! [A4: set_nat,B5: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ C4 )
=> ( ( minus_minus_set_nat @ B5 @ ( minus_minus_set_nat @ C4 @ A4 ) )
= A4 ) ) ) ).
% double_diff
thf(fact_4564_double__diff,axiom,
! [A4: set_int,B5: set_int,C4: set_int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( ord_less_eq_set_int @ B5 @ C4 )
=> ( ( minus_minus_set_int @ B5 @ ( minus_minus_set_int @ C4 @ A4 ) )
= A4 ) ) ) ).
% double_diff
thf(fact_4565_infinite__Icc,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).
% infinite_Icc
thf(fact_4566_infinite__Icc,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).
% infinite_Icc
thf(fact_4567_ex__nat__less,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [M5: nat] :
( ( ord_less_eq_nat @ M5 @ N )
& ( P2 @ M5 ) ) )
= ( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
& ( P2 @ X5 ) ) ) ) ).
% ex_nat_less
thf(fact_4568_all__nat__less,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [M5: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( P2 @ M5 ) ) )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( P2 @ X5 ) ) ) ) ).
% all_nat_less
thf(fact_4569_subset__eq__atLeast0__atMost__finite,axiom,
! [N7: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N7 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N7 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_4570_add__diff__assoc__enat,axiom,
! [Z2: extended_enat,Y: extended_enat,X3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Z2 @ Y )
=> ( ( plus_p3455044024723400733d_enat @ X3 @ ( minus_3235023915231533773d_enat @ Y @ Z2 ) )
= ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X3 @ Y ) @ Z2 ) ) ) ).
% add_diff_assoc_enat
thf(fact_4571_atLeastatMost__psubset__iff,axiom,
! [A: set_int,B: set_int,C2: set_int,D: set_int] :
( ( ord_less_set_set_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C2 @ D ) )
= ( ( ~ ( ord_less_eq_set_int @ A @ B )
| ( ( ord_less_eq_set_int @ C2 @ A )
& ( ord_less_eq_set_int @ B @ D )
& ( ( ord_less_set_int @ C2 @ A )
| ( ord_less_set_int @ B @ D ) ) ) )
& ( ord_less_eq_set_int @ C2 @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_4572_atLeastatMost__psubset__iff,axiom,
! [A: rat,B: rat,C2: rat,D: rat] :
( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C2 @ D ) )
= ( ( ~ ( ord_less_eq_rat @ A @ B )
| ( ( ord_less_eq_rat @ C2 @ A )
& ( ord_less_eq_rat @ B @ D )
& ( ( ord_less_rat @ C2 @ A )
| ( ord_less_rat @ B @ D ) ) ) )
& ( ord_less_eq_rat @ C2 @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_4573_atLeastatMost__psubset__iff,axiom,
! [A: num,B: num,C2: num,D: num] :
( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C2 @ D ) )
= ( ( ~ ( ord_less_eq_num @ A @ B )
| ( ( ord_less_eq_num @ C2 @ A )
& ( ord_less_eq_num @ B @ D )
& ( ( ord_less_num @ C2 @ A )
| ( ord_less_num @ B @ D ) ) ) )
& ( ord_less_eq_num @ C2 @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_4574_atLeastatMost__psubset__iff,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C2 @ D ) )
= ( ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C2 @ A )
& ( ord_less_eq_nat @ B @ D )
& ( ( ord_less_nat @ C2 @ A )
| ( ord_less_nat @ B @ D ) ) ) )
& ( ord_less_eq_nat @ C2 @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_4575_atLeastatMost__psubset__iff,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C2 @ D ) )
= ( ( ~ ( ord_less_eq_int @ A @ B )
| ( ( ord_less_eq_int @ C2 @ A )
& ( ord_less_eq_int @ B @ D )
& ( ( ord_less_int @ C2 @ A )
| ( ord_less_int @ B @ D ) ) ) )
& ( ord_less_eq_int @ C2 @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_4576_atLeastatMost__psubset__iff,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C2 @ D ) )
= ( ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C2 @ A )
& ( ord_less_eq_real @ B @ D )
& ( ( ord_less_real @ C2 @ A )
| ( ord_less_real @ B @ D ) ) ) )
& ( ord_less_eq_real @ C2 @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_4577_sum_Ofinite__Collect__op,axiom,
! [I6: set_real,X3: real > complex,Y: real > complex] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I6 )
& ( ( X3 @ I3 )
!= zero_zero_complex ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I6 )
& ( ( Y @ I3 )
!= zero_zero_complex ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I6 )
& ( ( plus_plus_complex @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_4578_sum_Ofinite__Collect__op,axiom,
! [I6: set_nat,X3: nat > complex,Y: nat > complex] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I6 )
& ( ( X3 @ I3 )
!= zero_zero_complex ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I6 )
& ( ( Y @ I3 )
!= zero_zero_complex ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I6 )
& ( ( plus_plus_complex @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_4579_sum_Ofinite__Collect__op,axiom,
! [I6: set_int,X3: int > complex,Y: int > complex] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I6 )
& ( ( X3 @ I3 )
!= zero_zero_complex ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I6 )
& ( ( Y @ I3 )
!= zero_zero_complex ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I6 )
& ( ( plus_plus_complex @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_4580_sum_Ofinite__Collect__op,axiom,
! [I6: set_complex,X3: complex > complex,Y: complex > complex] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I6 )
& ( ( X3 @ I3 )
!= zero_zero_complex ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I6 )
& ( ( Y @ I3 )
!= zero_zero_complex ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I6 )
& ( ( plus_plus_complex @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_4581_sum_Ofinite__Collect__op,axiom,
! [I6: set_real,X3: real > real,Y: real > real] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I6 )
& ( ( X3 @ I3 )
!= zero_zero_real ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I6 )
& ( ( Y @ I3 )
!= zero_zero_real ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I6 )
& ( ( plus_plus_real @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_4582_sum_Ofinite__Collect__op,axiom,
! [I6: set_nat,X3: nat > real,Y: nat > real] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I6 )
& ( ( X3 @ I3 )
!= zero_zero_real ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I6 )
& ( ( Y @ I3 )
!= zero_zero_real ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I6 )
& ( ( plus_plus_real @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_4583_sum_Ofinite__Collect__op,axiom,
! [I6: set_int,X3: int > real,Y: int > real] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I6 )
& ( ( X3 @ I3 )
!= zero_zero_real ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I6 )
& ( ( Y @ I3 )
!= zero_zero_real ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I3: int] :
( ( member_int @ I3 @ I6 )
& ( ( plus_plus_real @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_4584_sum_Ofinite__Collect__op,axiom,
! [I6: set_complex,X3: complex > real,Y: complex > real] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I6 )
& ( ( X3 @ I3 )
!= zero_zero_real ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I6 )
& ( ( Y @ I3 )
!= zero_zero_real ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I6 )
& ( ( plus_plus_real @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_4585_sum_Ofinite__Collect__op,axiom,
! [I6: set_real,X3: real > rat,Y: real > rat] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I6 )
& ( ( X3 @ I3 )
!= zero_zero_rat ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I6 )
& ( ( Y @ I3 )
!= zero_zero_rat ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I3: real] :
( ( member_real @ I3 @ I6 )
& ( ( plus_plus_rat @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_rat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_4586_sum_Ofinite__Collect__op,axiom,
! [I6: set_nat,X3: nat > rat,Y: nat > rat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I6 )
& ( ( X3 @ I3 )
!= zero_zero_rat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I6 )
& ( ( Y @ I3 )
!= zero_zero_rat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I6 )
& ( ( plus_plus_rat @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_rat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_4587_real__average__minus__first,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
= ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% real_average_minus_first
thf(fact_4588_real__average__minus__second,axiom,
! [B: real,A: real] :
( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
= ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% real_average_minus_second
thf(fact_4589_vebt__member_Opelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_vebt_member @ X3 @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
=> ( ! [A3: $o,B3: $o] :
( ( X3
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
=> ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B3 )
& ( Xa2 = one_one_nat ) ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) )
=> ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(3)
thf(fact_4590_vebt__member_Opelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X3 @ Xa2 )
= Y )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
=> ( ! [A3: $o,B3: $o] :
( ( X3
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ( ( Y
= ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B3 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(1)
thf(fact_4591_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
=> ( ! [A3: $o,B3: $o] :
( ( X3
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
=> ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B3 )
& ( Xa2 = one_one_nat ) ) ) ) ) )
=> ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) @ Xa2 ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( 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 ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(3)
thf(fact_4592_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
=> ( ! [A3: $o,B3: $o] :
( ( X3
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B3 )
& ( Xa2 = one_one_nat ) ) ) ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) @ Xa2 ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( 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 ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(2)
thf(fact_4593_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
= Y )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
=> ( ! [A3: $o,B3: $o] :
( ( X3
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ( ( Y
= ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B3 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
=> ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
=> ( ( Y
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( 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 ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) @ Xa2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(1)
thf(fact_4594_finite__atLeastAtMost__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( set_or1266510415728281911st_int @ L @ U ) ) ).
% finite_atLeastAtMost_int
thf(fact_4595_aset_I2_J,axiom,
! [D5: int,A4: set_int,P2: int > $o,Q: int > $o] :
( ! [X4: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A4 )
=> ( X4
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( P2 @ X4 )
=> ( P2 @ ( plus_plus_int @ X4 @ D5 ) ) ) )
=> ( ! [X4: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A4 )
=> ( X4
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( Q @ X4 )
=> ( Q @ ( plus_plus_int @ X4 @ D5 ) ) ) )
=> ! [X: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A4 )
=> ( X
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P2 @ X )
| ( Q @ X ) )
=> ( ( P2 @ ( plus_plus_int @ X @ D5 ) )
| ( Q @ ( plus_plus_int @ X @ D5 ) ) ) ) ) ) ) ).
% aset(2)
thf(fact_4596_aset_I1_J,axiom,
! [D5: int,A4: set_int,P2: int > $o,Q: int > $o] :
( ! [X4: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A4 )
=> ( X4
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( P2 @ X4 )
=> ( P2 @ ( plus_plus_int @ X4 @ D5 ) ) ) )
=> ( ! [X4: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A4 )
=> ( X4
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( Q @ X4 )
=> ( Q @ ( plus_plus_int @ X4 @ D5 ) ) ) )
=> ! [X: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A4 )
=> ( X
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P2 @ X )
& ( Q @ X ) )
=> ( ( P2 @ ( plus_plus_int @ X @ D5 ) )
& ( Q @ ( plus_plus_int @ X @ D5 ) ) ) ) ) ) ) ).
% aset(1)
thf(fact_4597_bset_I2_J,axiom,
! [D5: int,B5: set_int,P2: int > $o,Q: int > $o] :
( ! [X4: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B5 )
=> ( X4
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( P2 @ X4 )
=> ( P2 @ ( minus_minus_int @ X4 @ D5 ) ) ) )
=> ( ! [X4: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B5 )
=> ( X4
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( Q @ X4 )
=> ( Q @ ( minus_minus_int @ X4 @ D5 ) ) ) )
=> ! [X: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B5 )
=> ( X
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P2 @ X )
| ( Q @ X ) )
=> ( ( P2 @ ( minus_minus_int @ X @ D5 ) )
| ( Q @ ( minus_minus_int @ X @ D5 ) ) ) ) ) ) ) ).
% bset(2)
thf(fact_4598_bset_I1_J,axiom,
! [D5: int,B5: set_int,P2: int > $o,Q: int > $o] :
( ! [X4: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B5 )
=> ( X4
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( P2 @ X4 )
=> ( P2 @ ( minus_minus_int @ X4 @ D5 ) ) ) )
=> ( ! [X4: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B5 )
=> ( X4
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( Q @ X4 )
=> ( Q @ ( minus_minus_int @ X4 @ D5 ) ) ) )
=> ! [X: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B5 )
=> ( X
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P2 @ X )
& ( Q @ X ) )
=> ( ( P2 @ ( minus_minus_int @ X @ D5 ) )
& ( Q @ ( minus_minus_int @ X @ D5 ) ) ) ) ) ) ) ).
% bset(1)
thf(fact_4599_bset_I9_J,axiom,
! [D: int,D5: int,B5: set_int,T: int] :
( ( dvd_dvd_int @ D @ D5 )
=> ! [X: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B5 )
=> ( X
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X @ T ) )
=> ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X @ D5 ) @ T ) ) ) ) ) ).
% bset(9)
thf(fact_4600_bset_I10_J,axiom,
! [D: int,D5: int,B5: set_int,T: int] :
( ( dvd_dvd_int @ D @ D5 )
=> ! [X: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B5 )
=> ( X
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X @ T ) )
=> ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X @ D5 ) @ T ) ) ) ) ) ).
% bset(10)
thf(fact_4601_aset_I9_J,axiom,
! [D: int,D5: int,A4: set_int,T: int] :
( ( dvd_dvd_int @ D @ D5 )
=> ! [X: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A4 )
=> ( X
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X @ T ) )
=> ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X @ D5 ) @ T ) ) ) ) ) ).
% aset(9)
thf(fact_4602_aset_I10_J,axiom,
! [D: int,D5: int,A4: set_int,T: int] :
( ( dvd_dvd_int @ D @ D5 )
=> ! [X: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A4 )
=> ( X
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X @ T ) )
=> ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X @ D5 ) @ T ) ) ) ) ) ).
% aset(10)
thf(fact_4603_periodic__finite__ex,axiom,
! [D: int,P2: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int,K: int] :
( ( P2 @ X4 )
= ( P2 @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D ) ) ) )
=> ( ( ? [X7: int] : ( P2 @ X7 ) )
= ( ? [X5: int] :
( ( member_int @ X5 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
& ( P2 @ X5 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_4604_aset_I7_J,axiom,
! [D5: int,A4: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ! [X: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A4 )
=> ( X
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ T @ X )
=> ( ord_less_int @ T @ ( plus_plus_int @ X @ D5 ) ) ) ) ) ).
% aset(7)
thf(fact_4605_aset_I5_J,axiom,
! [D5: int,T: int,A4: set_int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ( ( member_int @ T @ A4 )
=> ! [X: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A4 )
=> ( X
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ X @ T )
=> ( ord_less_int @ ( plus_plus_int @ X @ D5 ) @ T ) ) ) ) ) ).
% aset(5)
thf(fact_4606_aset_I4_J,axiom,
! [D5: int,T: int,A4: set_int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ( ( member_int @ T @ A4 )
=> ! [X: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A4 )
=> ( X
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X != T )
=> ( ( plus_plus_int @ X @ D5 )
!= T ) ) ) ) ) ).
% aset(4)
thf(fact_4607_aset_I3_J,axiom,
! [D5: int,T: int,A4: set_int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A4 )
=> ! [X: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A4 )
=> ( X
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X = T )
=> ( ( plus_plus_int @ X @ D5 )
= T ) ) ) ) ) ).
% aset(3)
thf(fact_4608_bset_I7_J,axiom,
! [D5: int,T: int,B5: set_int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ( ( member_int @ T @ B5 )
=> ! [X: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B5 )
=> ( X
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ T @ X )
=> ( ord_less_int @ T @ ( minus_minus_int @ X @ D5 ) ) ) ) ) ) ).
% bset(7)
thf(fact_4609_bset_I5_J,axiom,
! [D5: int,B5: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ! [X: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B5 )
=> ( X
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ X @ T )
=> ( ord_less_int @ ( minus_minus_int @ X @ D5 ) @ T ) ) ) ) ).
% bset(5)
thf(fact_4610_bset_I4_J,axiom,
! [D5: int,T: int,B5: set_int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ( ( member_int @ T @ B5 )
=> ! [X: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B5 )
=> ( X
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X != T )
=> ( ( minus_minus_int @ X @ D5 )
!= T ) ) ) ) ) ).
% bset(4)
thf(fact_4611_bset_I3_J,axiom,
! [D5: int,T: int,B5: set_int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
=> ! [X: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B5 )
=> ( X
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X = T )
=> ( ( minus_minus_int @ X @ D5 )
= T ) ) ) ) ) ).
% bset(3)
thf(fact_4612_aset_I8_J,axiom,
! [D5: int,A4: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ! [X: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A4 )
=> ( X
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ T @ X )
=> ( ord_less_eq_int @ T @ ( plus_plus_int @ X @ D5 ) ) ) ) ) ).
% aset(8)
thf(fact_4613_aset_I6_J,axiom,
! [D5: int,T: int,A4: set_int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A4 )
=> ! [X: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A4 )
=> ( X
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ X @ T )
=> ( ord_less_eq_int @ ( plus_plus_int @ X @ D5 ) @ T ) ) ) ) ) ).
% aset(6)
thf(fact_4614_bset_I8_J,axiom,
! [D5: int,T: int,B5: set_int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
=> ! [X: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B5 )
=> ( X
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ T @ X )
=> ( ord_less_eq_int @ T @ ( minus_minus_int @ X @ D5 ) ) ) ) ) ) ).
% bset(8)
thf(fact_4615_bset_I6_J,axiom,
! [D5: int,B5: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ! [X: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B5 )
=> ( X
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ X @ T )
=> ( ord_less_eq_int @ ( minus_minus_int @ X @ D5 ) @ T ) ) ) ) ).
% bset(6)
thf(fact_4616_cpmi,axiom,
! [D5: int,P2: int > $o,P5: int > $o,B5: set_int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ! [X4: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B5 )
=> ( X4
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( P2 @ X4 )
=> ( P2 @ ( minus_minus_int @ X4 @ D5 ) ) ) )
=> ( ! [X4: int,K: int] :
( ( P5 @ X4 )
= ( P5 @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D5 ) ) ) )
=> ( ( ? [X7: int] : ( P2 @ X7 ) )
= ( ? [X5: int] :
( ( member_int @ X5 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
& ( P5 @ X5 ) )
| ? [X5: int] :
( ( member_int @ X5 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
& ? [Y5: int] :
( ( member_int @ Y5 @ B5 )
& ( P2 @ ( plus_plus_int @ Y5 @ X5 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_4617_cppi,axiom,
! [D5: int,P2: int > $o,P5: int > $o,A4: set_int] :
( ( ord_less_int @ zero_zero_int @ D5 )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ! [X4: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A4 )
=> ( X4
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( P2 @ X4 )
=> ( P2 @ ( plus_plus_int @ X4 @ D5 ) ) ) )
=> ( ! [X4: int,K: int] :
( ( P5 @ X4 )
= ( P5 @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D5 ) ) ) )
=> ( ( ? [X7: int] : ( P2 @ X7 ) )
= ( ? [X5: int] :
( ( member_int @ X5 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
& ( P5 @ X5 ) )
| ? [X5: int] :
( ( member_int @ X5 @ ( set_or1266510415728281911st_int @ one_one_int @ D5 ) )
& ? [Y5: int] :
( ( member_int @ Y5 @ A4 )
& ( P2 @ ( minus_minus_int @ Y5 @ X5 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_4618_vebt__member_Opelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ( vEBT_vebt_member @ X3 @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
=> ( ! [A3: $o,B3: $o] :
( ( X3
= ( vEBT_Leaf @ A3 @ B3 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B3 )
& ( Xa2 = one_one_nat ) ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) )
=> ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(2)
thf(fact_4619_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
= Y )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X3
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
=> ( ~ Y
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
=> ( ( Y
= ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
=> ( ( Y
= ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( 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 ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ Xa2 ) ) ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd ) )
=> ( ( Y
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( 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 ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(1)
thf(fact_4620_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_membermima @ X3 @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X3
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ Xa2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( 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 ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd ) @ Xa2 ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( 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 ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(3)
thf(fact_4621_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ Xa2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( 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 ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd ) @ Xa2 ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( 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 ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(2)
thf(fact_4622_arcosh__1,axiom,
( ( arcosh_real @ one_one_real )
= zero_zero_real ) ).
% arcosh_1
thf(fact_4623_Bolzano,axiom,
! [A: real,B: real,P2: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A3: real,B3: real,C: real] :
( ( P2 @ A3 @ B3 )
=> ( ( P2 @ B3 @ C )
=> ( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( P2 @ A3 @ C ) ) ) ) )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [A3: real,B3: real] :
( ( ( ord_less_eq_real @ A3 @ X4 )
& ( ord_less_eq_real @ X4 @ B3 )
& ( ord_less_real @ ( minus_minus_real @ B3 @ A3 ) @ D3 ) )
=> ( P2 @ A3 @ B3 ) ) ) ) )
=> ( P2 @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_4624_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_4625_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_4626_artanh__def,axiom,
( artanh_real
= ( ^ [X5: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X5 ) @ ( minus_minus_real @ one_one_real @ X5 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% artanh_def
thf(fact_4627_Divides_Oadjust__div__eq,axiom,
! [Q3: int,R2: int] :
( ( adjust_div @ ( product_Pair_int_int @ Q3 @ R2 ) )
= ( plus_plus_int @ Q3 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).
% Divides.adjust_div_eq
thf(fact_4628_signed__take__bit__rec,axiom,
( bit_ri6519982836138164636nteger
= ( ^ [N4: nat,A5: code_integer] : ( if_Code_integer @ ( N4 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_4629_signed__take__bit__rec,axiom,
( bit_ri631733984087533419it_int
= ( ^ [N4: nat,A5: int] : ( if_int @ ( N4 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_4630_vebt__buildup_Opelims,axiom,
! [X3: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X3 )
= Y )
=> ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X3 )
=> ( ( ( X3 = zero_zero_nat )
=> ( ( Y
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
=> ( ( ( X3
= ( suc @ zero_zero_nat ) )
=> ( ( Y
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
=> ~ ! [Va2: nat] :
( ( X3
= ( suc @ ( suc @ Va2 ) ) )
=> ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( Y
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( Y
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
=> ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.pelims
thf(fact_4631_diff__shunt__var,axiom,
! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( ( minus_1356011639430497352at_nat @ X3 @ Y )
= bot_bo2099793752762293965at_nat )
= ( ord_le3146513528884898305at_nat @ X3 @ Y ) ) ).
% diff_shunt_var
thf(fact_4632_diff__shunt__var,axiom,
! [X3: set_real,Y: set_real] :
( ( ( minus_minus_set_real @ X3 @ Y )
= bot_bot_set_real )
= ( ord_less_eq_set_real @ X3 @ Y ) ) ).
% diff_shunt_var
thf(fact_4633_diff__shunt__var,axiom,
! [X3: set_nat,Y: set_nat] :
( ( ( minus_minus_set_nat @ X3 @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X3 @ Y ) ) ).
% diff_shunt_var
thf(fact_4634_diff__shunt__var,axiom,
! [X3: set_int,Y: set_int] :
( ( ( minus_minus_set_int @ X3 @ Y )
= bot_bot_set_int )
= ( ord_less_eq_set_int @ X3 @ Y ) ) ).
% diff_shunt_var
thf(fact_4635_Sum__Icc__int,axiom,
! [M2: int,N: int] :
( ( ord_less_eq_int @ M2 @ N )
=> ( ( groups4538972089207619220nt_int
@ ^ [X5: int] : X5
@ ( set_or1266510415728281911st_int @ M2 @ N ) )
= ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N @ ( plus_plus_int @ N @ one_one_int ) ) @ ( times_times_int @ M2 @ ( minus_minus_int @ M2 @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% Sum_Icc_int
thf(fact_4636_divmod__step__def,axiom,
( unique5026877609467782581ep_nat
= ( ^ [L2: num] :
( produc2626176000494625587at_nat
@ ^ [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 ) ) ) ) ) ).
% divmod_step_def
thf(fact_4637_divmod__step__def,axiom,
( unique5024387138958732305ep_int
= ( ^ [L2: num] :
( produc4245557441103728435nt_int
@ ^ [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 ) ) ) ) ) ).
% divmod_step_def
thf(fact_4638_divmod__step__def,axiom,
( unique4921790084139445826nteger
= ( ^ [L2: num] :
( produc6916734918728496179nteger
@ ^ [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 ) ) ) ) ) ).
% divmod_step_def
thf(fact_4639_neg__equal__iff__equal,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_4640_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_4641_neg__equal__iff__equal,axiom,
! [A: complex,B: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= ( uminus1482373934393186551omplex @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_4642_neg__equal__iff__equal,axiom,
! [A: rat,B: rat] :
( ( ( uminus_uminus_rat @ A )
= ( uminus_uminus_rat @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_4643_neg__equal__iff__equal,axiom,
! [A: code_integer,B: code_integer] :
( ( ( uminus1351360451143612070nteger @ A )
= ( uminus1351360451143612070nteger @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_4644_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_4645_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_4646_add_Oinverse__inverse,axiom,
! [A: complex] :
( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_4647_add_Oinverse__inverse,axiom,
! [A: rat] :
( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_4648_add_Oinverse__inverse,axiom,
! [A: code_integer] :
( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_4649_Compl__subset__Compl__iff,axiom,
! [A4: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A4 ) @ ( uminus1532241313380277803et_int @ B5 ) )
= ( ord_less_eq_set_int @ B5 @ A4 ) ) ).
% Compl_subset_Compl_iff
thf(fact_4650_Compl__anti__mono,axiom,
! [A4: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B5 ) @ ( uminus1532241313380277803et_int @ A4 ) ) ) ).
% Compl_anti_mono
thf(fact_4651_compl__le__compl__iff,axiom,
! [X3: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X3 ) @ ( uminus1532241313380277803et_int @ Y ) )
= ( ord_less_eq_set_int @ Y @ X3 ) ) ).
% compl_le_compl_iff
thf(fact_4652_neg__le__iff__le,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_4653_neg__le__iff__le,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_4654_neg__le__iff__le,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
= ( ord_less_eq_rat @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_4655_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_4656_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_4657_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_4658_add_Oinverse__neutral,axiom,
( ( uminus1482373934393186551omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% add.inverse_neutral
thf(fact_4659_add_Oinverse__neutral,axiom,
( ( uminus_uminus_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% add.inverse_neutral
thf(fact_4660_add_Oinverse__neutral,axiom,
( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% add.inverse_neutral
thf(fact_4661_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A ) )
= ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_4662_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_4663_neg__0__equal__iff__equal,axiom,
! [A: complex] :
( ( zero_zero_complex
= ( uminus1482373934393186551omplex @ A ) )
= ( zero_zero_complex = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_4664_neg__0__equal__iff__equal,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( uminus_uminus_rat @ A ) )
= ( zero_zero_rat = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_4665_neg__0__equal__iff__equal,axiom,
! [A: code_integer] :
( ( zero_z3403309356797280102nteger
= ( uminus1351360451143612070nteger @ A ) )
= ( zero_z3403309356797280102nteger = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_4666_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_4667_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_4668_neg__equal__0__iff__equal,axiom,
! [A: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% neg_equal_0_iff_equal
thf(fact_4669_neg__equal__0__iff__equal,axiom,
! [A: rat] :
( ( ( uminus_uminus_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% neg_equal_0_iff_equal
thf(fact_4670_neg__equal__0__iff__equal,axiom,
! [A: code_integer] :
( ( ( uminus1351360451143612070nteger @ A )
= zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ).
% neg_equal_0_iff_equal
thf(fact_4671_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_4672_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_4673_equal__neg__zero,axiom,
! [A: rat] :
( ( A
= ( uminus_uminus_rat @ A ) )
= ( A = zero_zero_rat ) ) ).
% equal_neg_zero
thf(fact_4674_equal__neg__zero,axiom,
! [A: code_integer] :
( ( A
= ( uminus1351360451143612070nteger @ A ) )
= ( A = zero_z3403309356797280102nteger ) ) ).
% equal_neg_zero
thf(fact_4675_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_4676_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_4677_neg__equal__zero,axiom,
! [A: rat] :
( ( ( uminus_uminus_rat @ A )
= A )
= ( A = zero_zero_rat ) ) ).
% neg_equal_zero
thf(fact_4678_neg__equal__zero,axiom,
! [A: code_integer] :
( ( ( uminus1351360451143612070nteger @ A )
= A )
= ( A = zero_z3403309356797280102nteger ) ) ).
% neg_equal_zero
thf(fact_4679_neg__less__iff__less,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_4680_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_4681_neg__less__iff__less,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
= ( ord_less_rat @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_4682_neg__less__iff__less,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le6747313008572928689nteger @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_4683_add__minus__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_4684_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_4685_add__minus__cancel,axiom,
! [A: complex,B: complex] :
( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_4686_add__minus__cancel,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_4687_add__minus__cancel,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_4688_minus__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_4689_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_4690_minus__add__cancel,axiom,
! [A: complex,B: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_4691_minus__add__cancel,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_4692_minus__add__cancel,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_4693_minus__add__distrib,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% minus_add_distrib
thf(fact_4694_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_4695_minus__add__distrib,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% minus_add_distrib
thf(fact_4696_minus__add__distrib,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).
% minus_add_distrib
thf(fact_4697_minus__add__distrib,axiom,
! [A: code_integer,B: code_integer] :
( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% minus_add_distrib
thf(fact_4698_minus__diff__eq,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
= ( minus_minus_real @ B @ A ) ) ).
% minus_diff_eq
thf(fact_4699_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_4700_minus__diff__eq,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
= ( minus_minus_complex @ B @ A ) ) ).
% minus_diff_eq
thf(fact_4701_minus__diff__eq,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
= ( minus_minus_rat @ B @ A ) ) ).
% minus_diff_eq
thf(fact_4702_minus__diff__eq,axiom,
! [A: code_integer,B: code_integer] :
( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
= ( minus_8373710615458151222nteger @ B @ A ) ) ).
% minus_diff_eq
thf(fact_4703_ln__less__cancel__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y ) )
= ( ord_less_real @ X3 @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_4704_ln__inj__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ( ln_ln_real @ X3 )
= ( ln_ln_real @ Y ) )
= ( X3 = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_4705_real__add__minus__iff,axiom,
! [X3: real,A: real] :
( ( ( plus_plus_real @ X3 @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
= ( X3 = A ) ) ).
% real_add_minus_iff
thf(fact_4706_sum_Oneutral__const,axiom,
! [A4: set_int] :
( ( groups4538972089207619220nt_int
@ ^ [Uu3: int] : zero_zero_int
@ A4 )
= zero_zero_int ) ).
% sum.neutral_const
thf(fact_4707_sum_Oneutral__const,axiom,
! [A4: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [Uu3: complex] : zero_zero_complex
@ A4 )
= zero_zero_complex ) ).
% sum.neutral_const
thf(fact_4708_sum_Oneutral__const,axiom,
! [A4: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [Uu3: nat] : zero_zero_nat
@ A4 )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_4709_sum_Oneutral__const,axiom,
! [A4: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [Uu3: nat] : zero_zero_real
@ A4 )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_4710_case__prod__conv,axiom,
! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,A: nat,B: nat] :
( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_4711_case__prod__conv,axiom,
! [F: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat] :
( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_4712_case__prod__conv,axiom,
! [F: int > int > product_prod_int_int,A: int,B: int] :
( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_4713_case__prod__conv,axiom,
! [F: int > int > $o,A: int,B: int] :
( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_4714_case__prod__conv,axiom,
! [F: int > int > int,A: int,B: int] :
( ( produc8211389475949308722nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_4715_neg__less__eq__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_4716_neg__less__eq__nonneg,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
= ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_4717_neg__less__eq__nonneg,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_4718_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_4719_less__eq__neg__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_4720_less__eq__neg__nonpos,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% less_eq_neg_nonpos
thf(fact_4721_less__eq__neg__nonpos,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% less_eq_neg_nonpos
thf(fact_4722_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_4723_neg__le__0__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_le_0_iff_le
thf(fact_4724_neg__le__0__iff__le,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
= ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% neg_le_0_iff_le
thf(fact_4725_neg__le__0__iff__le,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% neg_le_0_iff_le
thf(fact_4726_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_4727_neg__0__le__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_4728_neg__0__le__iff__le,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% neg_0_le_iff_le
thf(fact_4729_neg__0__le__iff__le,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% neg_0_le_iff_le
thf(fact_4730_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_4731_neg__less__0__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_0_iff_less
thf(fact_4732_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_4733_neg__less__0__iff__less,axiom,
! [A: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% neg_less_0_iff_less
thf(fact_4734_neg__less__0__iff__less,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
= ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% neg_less_0_iff_less
thf(fact_4735_neg__0__less__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_4736_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_4737_neg__0__less__iff__less,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% neg_0_less_iff_less
thf(fact_4738_neg__0__less__iff__less,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% neg_0_less_iff_less
thf(fact_4739_neg__less__pos,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_pos
thf(fact_4740_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_4741_neg__less__pos,axiom,
! [A: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% neg_less_pos
thf(fact_4742_neg__less__pos,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
= ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% neg_less_pos
thf(fact_4743_less__neg__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_4744_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_4745_less__neg__neg,axiom,
! [A: rat] :
( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% less_neg_neg
thf(fact_4746_less__neg__neg,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% less_neg_neg
thf(fact_4747_add_Oright__inverse,axiom,
! [A: real] :
( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_4748_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_4749_add_Oright__inverse,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
= zero_zero_complex ) ).
% add.right_inverse
thf(fact_4750_add_Oright__inverse,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
= zero_zero_rat ) ).
% add.right_inverse
thf(fact_4751_add_Oright__inverse,axiom,
! [A: code_integer] :
( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
= zero_z3403309356797280102nteger ) ).
% add.right_inverse
thf(fact_4752_ab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_4753_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_4754_ab__left__minus,axiom,
! [A: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
= zero_zero_complex ) ).
% ab_left_minus
thf(fact_4755_ab__left__minus,axiom,
! [A: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
= zero_zero_rat ) ).
% ab_left_minus
thf(fact_4756_ab__left__minus,axiom,
! [A: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
= zero_z3403309356797280102nteger ) ).
% ab_left_minus
thf(fact_4757_diff__0,axiom,
! [A: real] :
( ( minus_minus_real @ zero_zero_real @ A )
= ( uminus_uminus_real @ A ) ) ).
% diff_0
thf(fact_4758_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_4759_diff__0,axiom,
! [A: complex] :
( ( minus_minus_complex @ zero_zero_complex @ A )
= ( uminus1482373934393186551omplex @ A ) ) ).
% diff_0
thf(fact_4760_diff__0,axiom,
! [A: rat] :
( ( minus_minus_rat @ zero_zero_rat @ A )
= ( uminus_uminus_rat @ A ) ) ).
% diff_0
thf(fact_4761_diff__0,axiom,
! [A: code_integer] :
( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
= ( uminus1351360451143612070nteger @ A ) ) ).
% diff_0
thf(fact_4762_verit__minus__simplify_I3_J,axiom,
! [B: real] :
( ( minus_minus_real @ zero_zero_real @ B )
= ( uminus_uminus_real @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_4763_verit__minus__simplify_I3_J,axiom,
! [B: int] :
( ( minus_minus_int @ zero_zero_int @ B )
= ( uminus_uminus_int @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_4764_verit__minus__simplify_I3_J,axiom,
! [B: complex] :
( ( minus_minus_complex @ zero_zero_complex @ B )
= ( uminus1482373934393186551omplex @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_4765_verit__minus__simplify_I3_J,axiom,
! [B: rat] :
( ( minus_minus_rat @ zero_zero_rat @ B )
= ( uminus_uminus_rat @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_4766_verit__minus__simplify_I3_J,axiom,
! [B: code_integer] :
( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
= ( uminus1351360451143612070nteger @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_4767_add__neg__numeral__simps_I3_J,axiom,
! [M2: num,N: num] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_4768_add__neg__numeral__simps_I3_J,axiom,
! [M2: num,N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_4769_add__neg__numeral__simps_I3_J,axiom,
! [M2: num,N: num] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M2 ) @ ( numera6690914467698888265omplex @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_4770_add__neg__numeral__simps_I3_J,axiom,
! [M2: num,N: num] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M2 ) @ ( numeral_numeral_rat @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_4771_add__neg__numeral__simps_I3_J,axiom,
! [M2: num,N: num] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_4772_diff__minus__eq__add,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
= ( plus_plus_real @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_4773_diff__minus__eq__add,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_4774_diff__minus__eq__add,axiom,
! [A: complex,B: complex] :
( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
= ( plus_plus_complex @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_4775_diff__minus__eq__add,axiom,
! [A: rat,B: rat] :
( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
= ( plus_plus_rat @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_4776_diff__minus__eq__add,axiom,
! [A: code_integer,B: code_integer] :
( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( plus_p5714425477246183910nteger @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_4777_uminus__add__conv__diff,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
= ( minus_minus_real @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_4778_uminus__add__conv__diff,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_4779_uminus__add__conv__diff,axiom,
! [A: complex,B: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
= ( minus_minus_complex @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_4780_uminus__add__conv__diff,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( minus_minus_rat @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_4781_uminus__add__conv__diff,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
= ( minus_8373710615458151222nteger @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_4782_sum_Oempty,axiom,
! [G: real > complex] :
( ( groups5754745047067104278omplex @ G @ bot_bot_set_real )
= zero_zero_complex ) ).
% sum.empty
thf(fact_4783_sum_Oempty,axiom,
! [G: real > real] :
( ( groups8097168146408367636l_real @ G @ bot_bot_set_real )
= zero_zero_real ) ).
% sum.empty
thf(fact_4784_sum_Oempty,axiom,
! [G: real > rat] :
( ( groups1300246762558778688al_rat @ G @ bot_bot_set_real )
= zero_zero_rat ) ).
% sum.empty
thf(fact_4785_sum_Oempty,axiom,
! [G: real > nat] :
( ( groups1935376822645274424al_nat @ G @ bot_bot_set_real )
= zero_zero_nat ) ).
% sum.empty
thf(fact_4786_sum_Oempty,axiom,
! [G: real > int] :
( ( groups1932886352136224148al_int @ G @ bot_bot_set_real )
= zero_zero_int ) ).
% sum.empty
thf(fact_4787_sum_Oempty,axiom,
! [G: nat > complex] :
( ( groups2073611262835488442omplex @ G @ bot_bot_set_nat )
= zero_zero_complex ) ).
% sum.empty
thf(fact_4788_sum_Oempty,axiom,
! [G: nat > rat] :
( ( groups2906978787729119204at_rat @ G @ bot_bot_set_nat )
= zero_zero_rat ) ).
% sum.empty
thf(fact_4789_sum_Oempty,axiom,
! [G: nat > int] :
( ( groups3539618377306564664at_int @ G @ bot_bot_set_nat )
= zero_zero_int ) ).
% sum.empty
thf(fact_4790_sum_Oempty,axiom,
! [G: int > complex] :
( ( groups3049146728041665814omplex @ G @ bot_bot_set_int )
= zero_zero_complex ) ).
% sum.empty
thf(fact_4791_sum_Oempty,axiom,
! [G: int > real] :
( ( groups8778361861064173332t_real @ G @ bot_bot_set_int )
= zero_zero_real ) ).
% sum.empty
thf(fact_4792_sum__eq__0__iff,axiom,
! [F4: set_int,F: int > nat] :
( ( finite_finite_int @ F4 )
=> ( ( ( groups4541462559716669496nt_nat @ F @ F4 )
= zero_zero_nat )
= ( ! [X5: int] :
( ( member_int @ X5 @ F4 )
=> ( ( F @ X5 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_4793_sum__eq__0__iff,axiom,
! [F4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ F4 )
=> ( ( ( groups5693394587270226106ex_nat @ F @ F4 )
= zero_zero_nat )
= ( ! [X5: complex] :
( ( member_complex @ X5 @ F4 )
=> ( ( F @ X5 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_4794_sum__eq__0__iff,axiom,
! [F4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ F4 )
=> ( ( ( groups977919841031483927at_nat @ F @ F4 )
= zero_zero_nat )
= ( ! [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ F4 )
=> ( ( F @ X5 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_4795_sum__eq__0__iff,axiom,
! [F4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ F4 )
=> ( ( ( groups3542108847815614940at_nat @ F @ F4 )
= zero_zero_nat )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ F4 )
=> ( ( F @ X5 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_4796_sum_Oinfinite,axiom,
! [A4: set_nat,G: nat > complex] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( groups2073611262835488442omplex @ G @ A4 )
= zero_zero_complex ) ) ).
% sum.infinite
thf(fact_4797_sum_Oinfinite,axiom,
! [A4: set_int,G: int > complex] :
( ~ ( finite_finite_int @ A4 )
=> ( ( groups3049146728041665814omplex @ G @ A4 )
= zero_zero_complex ) ) ).
% sum.infinite
thf(fact_4798_sum_Oinfinite,axiom,
! [A4: set_int,G: int > real] :
( ~ ( finite_finite_int @ A4 )
=> ( ( groups8778361861064173332t_real @ G @ A4 )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_4799_sum_Oinfinite,axiom,
! [A4: set_complex,G: complex > real] :
( ~ ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5808333547571424918x_real @ G @ A4 )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_4800_sum_Oinfinite,axiom,
! [A4: set_nat,G: nat > rat] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( groups2906978787729119204at_rat @ G @ A4 )
= zero_zero_rat ) ) ).
% sum.infinite
thf(fact_4801_sum_Oinfinite,axiom,
! [A4: set_int,G: int > rat] :
( ~ ( finite_finite_int @ A4 )
=> ( ( groups3906332499630173760nt_rat @ G @ A4 )
= zero_zero_rat ) ) ).
% sum.infinite
thf(fact_4802_sum_Oinfinite,axiom,
! [A4: set_complex,G: complex > rat] :
( ~ ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5058264527183730370ex_rat @ G @ A4 )
= zero_zero_rat ) ) ).
% sum.infinite
thf(fact_4803_sum_Oinfinite,axiom,
! [A4: set_int,G: int > nat] :
( ~ ( finite_finite_int @ A4 )
=> ( ( groups4541462559716669496nt_nat @ G @ A4 )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_4804_sum_Oinfinite,axiom,
! [A4: set_complex,G: complex > nat] :
( ~ ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5693394587270226106ex_nat @ G @ A4 )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_4805_sum_Oinfinite,axiom,
! [A4: set_nat,G: nat > int] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( groups3539618377306564664at_int @ G @ A4 )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_4806_ln__le__cancel__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y ) )
= ( ord_less_eq_real @ X3 @ Y ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_4807_ln__less__zero__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ ( ln_ln_real @ X3 ) @ zero_zero_real )
= ( ord_less_real @ X3 @ one_one_real ) ) ) ).
% ln_less_zero_iff
thf(fact_4808_ln__gt__zero__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
= ( ord_less_real @ one_one_real @ X3 ) ) ) ).
% ln_gt_zero_iff
thf(fact_4809_ln__eq__zero__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ( ln_ln_real @ X3 )
= zero_zero_real )
= ( X3 = one_one_real ) ) ) ).
% ln_eq_zero_iff
thf(fact_4810_ln__one,axiom,
( ( ln_ln_real @ one_one_real )
= zero_zero_real ) ).
% ln_one
thf(fact_4811_sum_Odelta,axiom,
! [S2: set_real,A: real,B: real > complex] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups5754745047067104278omplex
@ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups5754745047067104278omplex
@ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
@ S2 )
= zero_zero_complex ) ) ) ) ).
% sum.delta
thf(fact_4812_sum_Odelta,axiom,
! [S2: set_nat,A: nat,B: nat > complex] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
@ S2 )
= zero_zero_complex ) ) ) ) ).
% sum.delta
thf(fact_4813_sum_Odelta,axiom,
! [S2: set_int,A: int,B: int > complex] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups3049146728041665814omplex
@ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups3049146728041665814omplex
@ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
@ S2 )
= zero_zero_complex ) ) ) ) ).
% sum.delta
thf(fact_4814_sum_Odelta,axiom,
! [S2: set_real,A: real,B: real > real] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups8097168146408367636l_real
@ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups8097168146408367636l_real
@ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_4815_sum_Odelta,axiom,
! [S2: set_int,A: int,B: int > real] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups8778361861064173332t_real
@ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups8778361861064173332t_real
@ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_4816_sum_Odelta,axiom,
! [S2: set_complex,A: complex,B: complex > real] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups5808333547571424918x_real
@ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups5808333547571424918x_real
@ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_4817_sum_Odelta,axiom,
! [S2: set_real,A: real,B: real > rat] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups1300246762558778688al_rat
@ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups1300246762558778688al_rat
@ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= zero_zero_rat ) ) ) ) ).
% sum.delta
thf(fact_4818_sum_Odelta,axiom,
! [S2: set_nat,A: nat,B: nat > rat] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= zero_zero_rat ) ) ) ) ).
% sum.delta
thf(fact_4819_sum_Odelta,axiom,
! [S2: set_int,A: int,B: int > rat] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups3906332499630173760nt_rat
@ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups3906332499630173760nt_rat
@ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= zero_zero_rat ) ) ) ) ).
% sum.delta
thf(fact_4820_sum_Odelta,axiom,
! [S2: set_complex,A: complex,B: complex > rat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups5058264527183730370ex_rat
@ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups5058264527183730370ex_rat
@ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= zero_zero_rat ) ) ) ) ).
% sum.delta
thf(fact_4821_sum_Odelta_H,axiom,
! [S2: set_real,A: real,B: real > complex] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups5754745047067104278omplex
@ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups5754745047067104278omplex
@ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
@ S2 )
= zero_zero_complex ) ) ) ) ).
% sum.delta'
thf(fact_4822_sum_Odelta_H,axiom,
! [S2: set_nat,A: nat,B: nat > complex] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
@ S2 )
= zero_zero_complex ) ) ) ) ).
% sum.delta'
thf(fact_4823_sum_Odelta_H,axiom,
! [S2: set_int,A: int,B: int > complex] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups3049146728041665814omplex
@ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups3049146728041665814omplex
@ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
@ S2 )
= zero_zero_complex ) ) ) ) ).
% sum.delta'
thf(fact_4824_sum_Odelta_H,axiom,
! [S2: set_real,A: real,B: real > real] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups8097168146408367636l_real
@ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups8097168146408367636l_real
@ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_4825_sum_Odelta_H,axiom,
! [S2: set_int,A: int,B: int > real] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups8778361861064173332t_real
@ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups8778361861064173332t_real
@ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_4826_sum_Odelta_H,axiom,
! [S2: set_complex,A: complex,B: complex > real] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups5808333547571424918x_real
@ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups5808333547571424918x_real
@ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_4827_sum_Odelta_H,axiom,
! [S2: set_real,A: real,B: real > rat] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups1300246762558778688al_rat
@ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups1300246762558778688al_rat
@ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= zero_zero_rat ) ) ) ) ).
% sum.delta'
thf(fact_4828_sum_Odelta_H,axiom,
! [S2: set_nat,A: nat,B: nat > rat] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= zero_zero_rat ) ) ) ) ).
% sum.delta'
thf(fact_4829_sum_Odelta_H,axiom,
! [S2: set_int,A: int,B: int > rat] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups3906332499630173760nt_rat
@ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups3906332499630173760nt_rat
@ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= zero_zero_rat ) ) ) ) ).
% sum.delta'
thf(fact_4830_sum_Odelta_H,axiom,
! [S2: set_complex,A: complex,B: complex > rat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups5058264527183730370ex_rat
@ ^ [K3: complex] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups5058264527183730370ex_rat
@ ^ [K3: complex] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
@ S2 )
= zero_zero_rat ) ) ) ) ).
% sum.delta'
thf(fact_4831_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
= zero_zero_real ) ).
% add_neg_numeral_special(8)
thf(fact_4832_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_4833_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
= zero_zero_complex ) ).
% add_neg_numeral_special(8)
thf(fact_4834_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
= zero_zero_rat ) ).
% add_neg_numeral_special(8)
thf(fact_4835_add__neg__numeral__special_I8_J,axiom,
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
= zero_z3403309356797280102nteger ) ).
% add_neg_numeral_special(8)
thf(fact_4836_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% add_neg_numeral_special(7)
thf(fact_4837_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_4838_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= zero_zero_complex ) ).
% add_neg_numeral_special(7)
thf(fact_4839_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= zero_zero_rat ) ).
% add_neg_numeral_special(7)
thf(fact_4840_add__neg__numeral__special_I7_J,axiom,
( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= zero_z3403309356797280102nteger ) ).
% add_neg_numeral_special(7)
thf(fact_4841_diff__numeral__special_I12_J,axiom,
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% diff_numeral_special(12)
thf(fact_4842_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_4843_diff__numeral__special_I12_J,axiom,
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= zero_zero_complex ) ).
% diff_numeral_special(12)
thf(fact_4844_diff__numeral__special_I12_J,axiom,
( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
= zero_zero_rat ) ).
% diff_numeral_special(12)
thf(fact_4845_diff__numeral__special_I12_J,axiom,
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= zero_z3403309356797280102nteger ) ).
% diff_numeral_special(12)
thf(fact_4846_mod__minus1__right,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% mod_minus1_right
thf(fact_4847_mod__minus1__right,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= zero_z3403309356797280102nteger ) ).
% mod_minus1_right
thf(fact_4848_max__number__of_I2_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
=> ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
& ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
=> ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
= ( numeral_numeral_real @ U ) ) ) ) ).
% max_number_of(2)
thf(fact_4849_max__number__of_I2_J,axiom,
! [U: num,V: num] :
( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
=> ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
& ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
=> ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
= ( numera6620942414471956472nteger @ U ) ) ) ) ).
% max_number_of(2)
thf(fact_4850_max__number__of_I2_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
=> ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
& ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
=> ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
= ( numeral_numeral_rat @ U ) ) ) ) ).
% max_number_of(2)
thf(fact_4851_max__number__of_I2_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
=> ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
=> ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( numeral_numeral_int @ U ) ) ) ) ).
% max_number_of(2)
thf(fact_4852_max__number__of_I3_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
=> ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
= ( numeral_numeral_real @ V ) ) )
& ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
=> ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% max_number_of(3)
thf(fact_4853_max__number__of_I3_J,axiom,
! [U: num,V: num] :
( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
=> ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
= ( numera6620942414471956472nteger @ V ) ) )
& ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
=> ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% max_number_of(3)
thf(fact_4854_max__number__of_I3_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
=> ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
= ( numeral_numeral_rat @ V ) ) )
& ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
=> ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% max_number_of(3)
thf(fact_4855_max__number__of_I3_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
=> ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
= ( numeral_numeral_int @ V ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
=> ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% max_number_of(3)
thf(fact_4856_max__number__of_I4_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
=> ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
& ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
=> ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% max_number_of(4)
thf(fact_4857_max__number__of_I4_J,axiom,
! [U: num,V: num] :
( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
=> ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
& ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
=> ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% max_number_of(4)
thf(fact_4858_max__number__of_I4_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
=> ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
& ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
=> ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% max_number_of(4)
thf(fact_4859_max__number__of_I4_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
=> ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
=> ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% max_number_of(4)
thf(fact_4860_ln__le__zero__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ zero_zero_real )
= ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ).
% ln_le_zero_iff
thf(fact_4861_ln__ge__zero__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
= ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).
% ln_ge_zero_iff
thf(fact_4862_semiring__norm_I168_J,axiom,
! [V: num,W2: num,Y: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ Y ) )
= ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W2 ) ) ) @ Y ) ) ).
% semiring_norm(168)
thf(fact_4863_semiring__norm_I168_J,axiom,
! [V: num,W2: num,Y: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ Y ) )
= ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W2 ) ) ) @ Y ) ) ).
% semiring_norm(168)
thf(fact_4864_semiring__norm_I168_J,axiom,
! [V: num,W2: num,Y: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ Y ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W2 ) ) ) @ Y ) ) ).
% semiring_norm(168)
thf(fact_4865_semiring__norm_I168_J,axiom,
! [V: num,W2: num,Y: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ Y ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W2 ) ) ) @ Y ) ) ).
% semiring_norm(168)
thf(fact_4866_semiring__norm_I168_J,axiom,
! [V: num,W2: num,Y: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W2 ) ) @ Y ) )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W2 ) ) ) @ Y ) ) ).
% semiring_norm(168)
thf(fact_4867_diff__numeral__simps_I2_J,axiom,
! [M2: num,N: num] :
( ( minus_minus_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( numeral_numeral_real @ ( plus_plus_num @ M2 @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_4868_diff__numeral__simps_I2_J,axiom,
! [M2: num,N: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M2 @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_4869_diff__numeral__simps_I2_J,axiom,
! [M2: num,N: num] :
( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M2 ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ M2 @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_4870_diff__numeral__simps_I2_J,axiom,
! [M2: num,N: num] :
( ( minus_minus_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ M2 @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_4871_diff__numeral__simps_I2_J,axiom,
! [M2: num,N: num] :
( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( numera6620942414471956472nteger @ ( plus_plus_num @ M2 @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_4872_diff__numeral__simps_I3_J,axiom,
! [M2: num,N: num] :
( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( numeral_numeral_real @ N ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M2 @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_4873_diff__numeral__simps_I3_J,axiom,
! [M2: num,N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M2 @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_4874_diff__numeral__simps_I3_J,axiom,
! [M2: num,N: num] :
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) ) @ ( numera6690914467698888265omplex @ N ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M2 @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_4875_diff__numeral__simps_I3_J,axiom,
! [M2: num,N: num] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( numeral_numeral_rat @ N ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_4876_diff__numeral__simps_I3_J,axiom,
! [M2: num,N: num] :
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( numera6620942414471956472nteger @ N ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M2 @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_4877_neg__numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( ord_less_eq_num @ N @ M2 ) ) ).
% neg_numeral_le_iff
thf(fact_4878_neg__numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( ord_less_eq_num @ N @ M2 ) ) ).
% neg_numeral_le_iff
thf(fact_4879_neg__numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( ord_less_eq_num @ N @ M2 ) ) ).
% neg_numeral_le_iff
thf(fact_4880_neg__numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_eq_num @ N @ M2 ) ) ).
% neg_numeral_le_iff
thf(fact_4881_not__neg__one__le__neg__numeral__iff,axiom,
! [M2: num] :
( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) ) )
= ( M2 != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_4882_not__neg__one__le__neg__numeral__iff,axiom,
! [M2: num] :
( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) )
= ( M2 != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_4883_not__neg__one__le__neg__numeral__iff,axiom,
! [M2: num] :
( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) )
= ( M2 != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_4884_not__neg__one__le__neg__numeral__iff,axiom,
! [M2: num] :
( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) )
= ( M2 != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_4885_le__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W2: num] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_4886_le__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W2: num] :
( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
= ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_4887_divide__le__eq__numeral1_I2_J,axiom,
! [B: real,W2: num,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_4888_divide__le__eq__numeral1_I2_J,axiom,
! [B: rat,W2: num,A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_4889_eq__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W2: num] :
( ( A
= ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
= ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
!= zero_zero_real )
=> ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= B ) )
& ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_4890_eq__divide__eq__numeral1_I2_J,axiom,
! [A: complex,B: complex,W2: num] :
( ( A
= ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) )
= ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
!= zero_zero_complex )
=> ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
= B ) )
& ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_4891_eq__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W2: num] :
( ( A
= ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
= ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
!= zero_zero_rat )
=> ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= B ) )
& ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_4892_divide__eq__eq__numeral1_I2_J,axiom,
! [B: real,W2: num,A: real] :
( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= A )
= ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
!= zero_zero_real )
=> ( B
= ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) )
& ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_4893_divide__eq__eq__numeral1_I2_J,axiom,
! [B: complex,W2: num,A: complex] :
( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
= A )
= ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
!= zero_zero_complex )
=> ( B
= ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) ) )
& ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_4894_divide__eq__eq__numeral1_I2_J,axiom,
! [B: rat,W2: num,A: rat] :
( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= A )
= ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
!= zero_zero_rat )
=> ( B
= ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) )
& ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_4895_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_4896_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_4897_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_4898_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_4899_add__neg__numeral__special_I9_J,axiom,
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_4900_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_4901_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_4902_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_4903_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_4904_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_4905_diff__numeral__special_I4_J,axiom,
! [M2: num] :
( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ one_one_real )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M2 @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_4906_diff__numeral__special_I4_J,axiom,
! [M2: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M2 @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_4907_diff__numeral__special_I4_J,axiom,
! [M2: num] :
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) ) @ one_one_complex )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M2 @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_4908_diff__numeral__special_I4_J,axiom,
! [M2: num] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ one_one_rat )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M2 @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_4909_diff__numeral__special_I4_J,axiom,
! [M2: num] :
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ one_one_Code_integer )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M2 @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_4910_signed__take__bit__Suc__minus__bit0,axiom,
! [N: nat,K2: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K2 ) ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_Suc_minus_bit0
thf(fact_4911_signed__take__bit__0,axiom,
! [A: code_integer] :
( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
= ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% signed_take_bit_0
thf(fact_4912_signed__take__bit__0,axiom,
! [A: int] :
( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
= ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% signed_take_bit_0
thf(fact_4913_compl__mono,axiom,
! [X3: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y )
=> ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ ( uminus1532241313380277803et_int @ X3 ) ) ) ).
% compl_mono
thf(fact_4914_compl__le__swap1,axiom,
! [Y: set_int,X3: set_int] :
( ( ord_less_eq_set_int @ Y @ ( uminus1532241313380277803et_int @ X3 ) )
=> ( ord_less_eq_set_int @ X3 @ ( uminus1532241313380277803et_int @ Y ) ) ) ).
% compl_le_swap1
thf(fact_4915_compl__le__swap2,axiom,
! [Y: set_int,X3: set_int] :
( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ X3 )
=> ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X3 ) @ Y ) ) ).
% compl_le_swap2
thf(fact_4916_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( uminus_uminus_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_4917_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_4918_minus__equation__iff,axiom,
! [A: complex,B: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= B )
= ( ( uminus1482373934393186551omplex @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_4919_minus__equation__iff,axiom,
! [A: rat,B: rat] :
( ( ( uminus_uminus_rat @ A )
= B )
= ( ( uminus_uminus_rat @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_4920_minus__equation__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ( uminus1351360451143612070nteger @ A )
= B )
= ( ( uminus1351360451143612070nteger @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_4921_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_4922_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_4923_equation__minus__iff,axiom,
! [A: complex,B: complex] :
( ( A
= ( uminus1482373934393186551omplex @ B ) )
= ( B
= ( uminus1482373934393186551omplex @ A ) ) ) ).
% equation_minus_iff
thf(fact_4924_equation__minus__iff,axiom,
! [A: rat,B: rat] :
( ( A
= ( uminus_uminus_rat @ B ) )
= ( B
= ( uminus_uminus_rat @ A ) ) ) ).
% equation_minus_iff
thf(fact_4925_equation__minus__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( A
= ( uminus1351360451143612070nteger @ B ) )
= ( B
= ( uminus1351360451143612070nteger @ A ) ) ) ).
% equation_minus_iff
thf(fact_4926_prod_Ocase__distrib,axiom,
! [H2: $o > $o,F: int > int > $o,Prod: product_prod_int_int] :
( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
= ( produc4947309494688390418_int_o
@ ^ [X15: int,X23: int] : ( H2 @ ( F @ X15 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_4927_prod_Ocase__distrib,axiom,
! [H2: $o > int,F: int > int > $o,Prod: product_prod_int_int] :
( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
= ( produc8211389475949308722nt_int
@ ^ [X15: int,X23: int] : ( H2 @ ( F @ X15 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_4928_prod_Ocase__distrib,axiom,
! [H2: int > $o,F: int > int > int,Prod: product_prod_int_int] :
( ( H2 @ ( produc8211389475949308722nt_int @ F @ Prod ) )
= ( produc4947309494688390418_int_o
@ ^ [X15: int,X23: int] : ( H2 @ ( F @ X15 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_4929_prod_Ocase__distrib,axiom,
! [H2: int > int,F: int > int > int,Prod: product_prod_int_int] :
( ( H2 @ ( produc8211389475949308722nt_int @ F @ Prod ) )
= ( produc8211389475949308722nt_int
@ ^ [X15: int,X23: int] : ( H2 @ ( F @ X15 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_4930_prod_Ocase__distrib,axiom,
! [H2: product_prod_int_int > $o,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
( ( H2 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
= ( produc4947309494688390418_int_o
@ ^ [X15: int,X23: int] : ( H2 @ ( F @ X15 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_4931_prod_Ocase__distrib,axiom,
! [H2: product_prod_int_int > int,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
( ( H2 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
= ( produc8211389475949308722nt_int
@ ^ [X15: int,X23: int] : ( H2 @ ( F @ X15 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_4932_prod_Ocase__distrib,axiom,
! [H2: $o > product_prod_int_int,F: int > int > $o,Prod: product_prod_int_int] :
( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
= ( produc4245557441103728435nt_int
@ ^ [X15: int,X23: int] : ( H2 @ ( F @ X15 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_4933_prod_Ocase__distrib,axiom,
! [H2: int > product_prod_int_int,F: int > int > int,Prod: product_prod_int_int] :
( ( H2 @ ( produc8211389475949308722nt_int @ F @ Prod ) )
= ( produc4245557441103728435nt_int
@ ^ [X15: int,X23: int] : ( H2 @ ( F @ X15 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_4934_prod_Ocase__distrib,axiom,
! [H2: product_prod_int_int > product_prod_int_int,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
( ( H2 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
= ( produc4245557441103728435nt_int
@ ^ [X15: int,X23: int] : ( H2 @ ( F @ X15 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_4935_prod_Ocase__distrib,axiom,
! [H2: ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
( ( H2 @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
= ( produc8739625826339149834_nat_o
@ ^ [X15: nat,X23: nat] : ( H2 @ ( F @ X15 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_4936_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > complex,A4: set_real] :
( ( ( groups5754745047067104278omplex @ G @ A4 )
!= zero_zero_complex )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_complex ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_4937_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > complex,A4: set_nat] :
( ( ( groups2073611262835488442omplex @ G @ A4 )
!= zero_zero_complex )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_complex ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_4938_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > complex,A4: set_int] :
( ( ( groups3049146728041665814omplex @ G @ A4 )
!= zero_zero_complex )
=> ~ ! [A3: int] :
( ( member_int @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_complex ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_4939_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > real,A4: set_complex] :
( ( ( groups5808333547571424918x_real @ G @ A4 )
!= zero_zero_real )
=> ~ ! [A3: complex] :
( ( member_complex @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_4940_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > real,A4: set_real] :
( ( ( groups8097168146408367636l_real @ G @ A4 )
!= zero_zero_real )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_4941_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > real,A4: set_int] :
( ( ( groups8778361861064173332t_real @ G @ A4 )
!= zero_zero_real )
=> ~ ! [A3: int] :
( ( member_int @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_4942_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > rat,A4: set_complex] :
( ( ( groups5058264527183730370ex_rat @ G @ A4 )
!= zero_zero_rat )
=> ~ ! [A3: complex] :
( ( member_complex @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_rat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_4943_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > rat,A4: set_real] :
( ( ( groups1300246762558778688al_rat @ G @ A4 )
!= zero_zero_rat )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_rat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_4944_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > rat,A4: set_nat] :
( ( ( groups2906978787729119204at_rat @ G @ A4 )
!= zero_zero_rat )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_rat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_4945_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: int > rat,A4: set_int] :
( ( ( groups3906332499630173760nt_rat @ G @ A4 )
!= zero_zero_rat )
=> ~ ! [A3: int] :
( ( member_int @ A3 @ A4 )
=> ( ( G @ A3 )
= zero_zero_rat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_4946_sum_Oneutral,axiom,
! [A4: set_int,G: int > int] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ( G @ X4 )
= zero_zero_int ) )
=> ( ( groups4538972089207619220nt_int @ G @ A4 )
= zero_zero_int ) ) ).
% sum.neutral
thf(fact_4947_sum_Oneutral,axiom,
! [A4: set_complex,G: complex > complex] :
( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ( G @ X4 )
= zero_zero_complex ) )
=> ( ( groups7754918857620584856omplex @ G @ A4 )
= zero_zero_complex ) ) ).
% sum.neutral
thf(fact_4948_sum_Oneutral,axiom,
! [A4: set_nat,G: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ( G @ X4 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ A4 )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_4949_sum_Oneutral,axiom,
! [A4: set_nat,G: nat > real] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ( G @ X4 )
= zero_zero_real ) )
=> ( ( groups6591440286371151544t_real @ G @ A4 )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_4950_le__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% le_minus_iff
thf(fact_4951_le__minus__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% le_minus_iff
thf(fact_4952_le__minus__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
= ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% le_minus_iff
thf(fact_4953_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_4954_minus__le__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_4955_minus__le__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
= ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_4956_minus__le__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_4957_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_4958_le__imp__neg__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% le_imp_neg_le
thf(fact_4959_le__imp__neg__le,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ B )
=> ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% le_imp_neg_le
thf(fact_4960_le__imp__neg__le,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% le_imp_neg_le
thf(fact_4961_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_4962_minus__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_4963_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_4964_minus__less__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_4965_minus__less__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
= ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_4966_less__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% less_minus_iff
thf(fact_4967_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_4968_less__minus__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
= ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% less_minus_iff
thf(fact_4969_less__minus__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% less_minus_iff
thf(fact_4970_is__num__normalize_I8_J,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_4971_is__num__normalize_I8_J,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_4972_is__num__normalize_I8_J,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_4973_is__num__normalize_I8_J,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_4974_is__num__normalize_I8_J,axiom,
! [A: code_integer,B: code_integer] :
( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_4975_group__cancel_Oneg1,axiom,
! [A4: real,K2: real,A: real] :
( ( A4
= ( plus_plus_real @ K2 @ A ) )
=> ( ( uminus_uminus_real @ A4 )
= ( plus_plus_real @ ( uminus_uminus_real @ K2 ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_4976_group__cancel_Oneg1,axiom,
! [A4: int,K2: int,A: int] :
( ( A4
= ( plus_plus_int @ K2 @ A ) )
=> ( ( uminus_uminus_int @ A4 )
= ( plus_plus_int @ ( uminus_uminus_int @ K2 ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_4977_group__cancel_Oneg1,axiom,
! [A4: complex,K2: complex,A: complex] :
( ( A4
= ( plus_plus_complex @ K2 @ A ) )
=> ( ( uminus1482373934393186551omplex @ A4 )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K2 ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_4978_group__cancel_Oneg1,axiom,
! [A4: rat,K2: rat,A: rat] :
( ( A4
= ( plus_plus_rat @ K2 @ A ) )
=> ( ( uminus_uminus_rat @ A4 )
= ( plus_plus_rat @ ( uminus_uminus_rat @ K2 ) @ ( uminus_uminus_rat @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_4979_group__cancel_Oneg1,axiom,
! [A4: code_integer,K2: code_integer,A: code_integer] :
( ( A4
= ( plus_p5714425477246183910nteger @ K2 @ A ) )
=> ( ( uminus1351360451143612070nteger @ A4 )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_4980_add_Oinverse__distrib__swap,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_4981_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_4982_add_Oinverse__distrib__swap,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_4983_add_Oinverse__distrib__swap,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_4984_add_Oinverse__distrib__swap,axiom,
! [A: code_integer,B: code_integer] :
( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_4985_minus__diff__commute,axiom,
! [B: real,A: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
= ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_4986_minus__diff__commute,axiom,
! [B: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_4987_minus__diff__commute,axiom,
! [B: complex,A: complex] :
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B ) @ A )
= ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_4988_minus__diff__commute,axiom,
! [B: rat,A: rat] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
= ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_4989_minus__diff__commute,axiom,
! [B: code_integer,A: code_integer] :
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
= ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_4990_split__cong,axiom,
! [Q3: product_prod_nat_nat,F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: nat > nat > product_prod_nat_nat > product_prod_nat_nat,P: product_prod_nat_nat] :
( ! [X4: nat,Y3: nat] :
( ( ( product_Pair_nat_nat @ X4 @ Y3 )
= Q3 )
=> ( ( F @ X4 @ Y3 )
= ( G @ X4 @ Y3 ) ) )
=> ( ( P = Q3 )
=> ( ( produc27273713700761075at_nat @ F @ P )
= ( produc27273713700761075at_nat @ G @ Q3 ) ) ) ) ).
% split_cong
thf(fact_4991_split__cong,axiom,
! [Q3: product_prod_nat_nat,F: nat > nat > product_prod_nat_nat > $o,G: nat > nat > product_prod_nat_nat > $o,P: product_prod_nat_nat] :
( ! [X4: nat,Y3: nat] :
( ( ( product_Pair_nat_nat @ X4 @ Y3 )
= Q3 )
=> ( ( F @ X4 @ Y3 )
= ( G @ X4 @ Y3 ) ) )
=> ( ( P = Q3 )
=> ( ( produc8739625826339149834_nat_o @ F @ P )
= ( produc8739625826339149834_nat_o @ G @ Q3 ) ) ) ) ).
% split_cong
thf(fact_4992_split__cong,axiom,
! [Q3: product_prod_int_int,F: int > int > product_prod_int_int,G: int > int > product_prod_int_int,P: product_prod_int_int] :
( ! [X4: int,Y3: int] :
( ( ( product_Pair_int_int @ X4 @ Y3 )
= Q3 )
=> ( ( F @ X4 @ Y3 )
= ( G @ X4 @ Y3 ) ) )
=> ( ( P = Q3 )
=> ( ( produc4245557441103728435nt_int @ F @ P )
= ( produc4245557441103728435nt_int @ G @ Q3 ) ) ) ) ).
% split_cong
thf(fact_4993_split__cong,axiom,
! [Q3: product_prod_int_int,F: int > int > $o,G: int > int > $o,P: product_prod_int_int] :
( ! [X4: int,Y3: int] :
( ( ( product_Pair_int_int @ X4 @ Y3 )
= Q3 )
=> ( ( F @ X4 @ Y3 )
= ( G @ X4 @ Y3 ) ) )
=> ( ( P = Q3 )
=> ( ( produc4947309494688390418_int_o @ F @ P )
= ( produc4947309494688390418_int_o @ G @ Q3 ) ) ) ) ).
% split_cong
thf(fact_4994_split__cong,axiom,
! [Q3: product_prod_int_int,F: int > int > int,G: int > int > int,P: product_prod_int_int] :
( ! [X4: int,Y3: int] :
( ( ( product_Pair_int_int @ X4 @ Y3 )
= Q3 )
=> ( ( F @ X4 @ Y3 )
= ( G @ X4 @ Y3 ) ) )
=> ( ( P = Q3 )
=> ( ( produc8211389475949308722nt_int @ F @ P )
= ( produc8211389475949308722nt_int @ G @ Q3 ) ) ) ) ).
% split_cong
thf(fact_4995_old_Oprod_Ocase,axiom,
! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X1: nat,X2: nat] :
( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
= ( F @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_4996_old_Oprod_Ocase,axiom,
! [F: nat > nat > product_prod_nat_nat > $o,X1: nat,X2: nat] :
( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
= ( F @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_4997_old_Oprod_Ocase,axiom,
! [F: int > int > product_prod_int_int,X1: int,X2: int] :
( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ X1 @ X2 ) )
= ( F @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_4998_old_Oprod_Ocase,axiom,
! [F: int > int > $o,X1: int,X2: int] :
( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ X1 @ X2 ) )
= ( F @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_4999_old_Oprod_Ocase,axiom,
! [F: int > int > int,X1: int,X2: int] :
( ( produc8211389475949308722nt_int @ F @ ( product_Pair_int_int @ X1 @ X2 ) )
= ( F @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_5000_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_5001_sum__mono,axiom,
! [K5: set_complex,F: complex > rat,G: complex > rat] :
( ! [I2: complex] :
( ( member_complex @ I2 @ K5 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ K5 ) @ ( groups5058264527183730370ex_rat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5002_sum__mono,axiom,
! [K5: set_real,F: real > rat,G: real > rat] :
( ! [I2: real] :
( ( member_real @ I2 @ K5 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K5 ) @ ( groups1300246762558778688al_rat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5003_sum__mono,axiom,
! [K5: set_nat,F: nat > rat,G: nat > rat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ K5 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K5 ) @ ( groups2906978787729119204at_rat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5004_sum__mono,axiom,
! [K5: set_int,F: int > rat,G: int > rat] :
( ! [I2: int] :
( ( member_int @ I2 @ K5 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K5 ) @ ( groups3906332499630173760nt_rat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5005_sum__mono,axiom,
! [K5: set_complex,F: complex > nat,G: complex > nat] :
( ! [I2: complex] :
( ( member_complex @ I2 @ K5 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ K5 ) @ ( groups5693394587270226106ex_nat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5006_sum__mono,axiom,
! [K5: set_real,F: real > nat,G: real > nat] :
( ! [I2: real] :
( ( member_real @ I2 @ K5 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K5 ) @ ( groups1935376822645274424al_nat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5007_sum__mono,axiom,
! [K5: set_int,F: int > nat,G: int > nat] :
( ! [I2: int] :
( ( member_int @ I2 @ K5 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5008_sum__mono,axiom,
! [K5: set_complex,F: complex > int,G: complex > int] :
( ! [I2: complex] :
( ( member_complex @ I2 @ K5 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ K5 ) @ ( groups5690904116761175830ex_int @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5009_sum__mono,axiom,
! [K5: set_real,F: real > int,G: real > int] :
( ! [I2: real] :
( ( member_real @ I2 @ K5 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K5 ) @ ( groups1932886352136224148al_int @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5010_sum__mono,axiom,
! [K5: set_nat,F: nat > int,G: nat > int] :
( ! [I2: nat] :
( ( member_nat @ I2 @ K5 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K5 ) @ ( groups3539618377306564664at_int @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5011_sum_Odistrib,axiom,
! [G: int > int,H2: int > int,A4: set_int] :
( ( groups4538972089207619220nt_int
@ ^ [X5: int] : ( plus_plus_int @ ( G @ X5 ) @ ( H2 @ X5 ) )
@ A4 )
= ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A4 ) @ ( groups4538972089207619220nt_int @ H2 @ A4 ) ) ) ).
% sum.distrib
thf(fact_5012_sum_Odistrib,axiom,
! [G: complex > complex,H2: complex > complex,A4: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [X5: complex] : ( plus_plus_complex @ ( G @ X5 ) @ ( H2 @ X5 ) )
@ A4 )
= ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A4 ) @ ( groups7754918857620584856omplex @ H2 @ A4 ) ) ) ).
% sum.distrib
thf(fact_5013_sum_Odistrib,axiom,
! [G: nat > nat,H2: nat > nat,A4: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [X5: nat] : ( plus_plus_nat @ ( G @ X5 ) @ ( H2 @ X5 ) )
@ A4 )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A4 ) @ ( groups3542108847815614940at_nat @ H2 @ A4 ) ) ) ).
% sum.distrib
thf(fact_5014_sum_Odistrib,axiom,
! [G: nat > real,H2: nat > real,A4: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X5: nat] : ( plus_plus_real @ ( G @ X5 ) @ ( H2 @ X5 ) )
@ A4 )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A4 ) @ ( groups6591440286371151544t_real @ H2 @ A4 ) ) ) ).
% sum.distrib
thf(fact_5015_sum_Oswap__restrict,axiom,
! [A4: set_real,B5: set_int,G: real > int > int,R: real > int > $o] :
( ( finite_finite_real @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ( groups1932886352136224148al_int
@ ^ [X5: real] :
( groups4538972089207619220nt_int @ ( G @ X5 )
@ ( collect_int
@ ^ [Y5: int] :
( ( member_int @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups4538972089207619220nt_int
@ ^ [Y5: int] :
( groups1932886352136224148al_int
@ ^ [X5: real] : ( G @ X5 @ Y5 )
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_5016_sum_Oswap__restrict,axiom,
! [A4: set_nat,B5: set_int,G: nat > int > int,R: nat > int > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ( groups3539618377306564664at_int
@ ^ [X5: nat] :
( groups4538972089207619220nt_int @ ( G @ X5 )
@ ( collect_int
@ ^ [Y5: int] :
( ( member_int @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups4538972089207619220nt_int
@ ^ [Y5: int] :
( groups3539618377306564664at_int
@ ^ [X5: nat] : ( G @ X5 @ Y5 )
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_5017_sum_Oswap__restrict,axiom,
! [A4: set_complex,B5: set_int,G: complex > int > int,R: complex > int > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ( groups5690904116761175830ex_int
@ ^ [X5: complex] :
( groups4538972089207619220nt_int @ ( G @ X5 )
@ ( collect_int
@ ^ [Y5: int] :
( ( member_int @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups4538972089207619220nt_int
@ ^ [Y5: int] :
( groups5690904116761175830ex_int
@ ^ [X5: complex] : ( G @ X5 @ Y5 )
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_5018_sum_Oswap__restrict,axiom,
! [A4: set_real,B5: set_complex,G: real > complex > complex,R: real > complex > $o] :
( ( finite_finite_real @ A4 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( ( groups5754745047067104278omplex
@ ^ [X5: real] :
( groups7754918857620584856omplex @ ( G @ X5 )
@ ( collect_complex
@ ^ [Y5: complex] :
( ( member_complex @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups7754918857620584856omplex
@ ^ [Y5: complex] :
( groups5754745047067104278omplex
@ ^ [X5: real] : ( G @ X5 @ Y5 )
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_5019_sum_Oswap__restrict,axiom,
! [A4: set_nat,B5: set_complex,G: nat > complex > complex,R: nat > complex > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( ( groups2073611262835488442omplex
@ ^ [X5: nat] :
( groups7754918857620584856omplex @ ( G @ X5 )
@ ( collect_complex
@ ^ [Y5: complex] :
( ( member_complex @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups7754918857620584856omplex
@ ^ [Y5: complex] :
( groups2073611262835488442omplex
@ ^ [X5: nat] : ( G @ X5 @ Y5 )
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_5020_sum_Oswap__restrict,axiom,
! [A4: set_int,B5: set_complex,G: int > complex > complex,R: int > complex > $o] :
( ( finite_finite_int @ A4 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( ( groups3049146728041665814omplex
@ ^ [X5: int] :
( groups7754918857620584856omplex @ ( G @ X5 )
@ ( collect_complex
@ ^ [Y5: complex] :
( ( member_complex @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups7754918857620584856omplex
@ ^ [Y5: complex] :
( groups3049146728041665814omplex
@ ^ [X5: int] : ( G @ X5 @ Y5 )
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_5021_sum_Oswap__restrict,axiom,
! [A4: set_real,B5: set_nat,G: real > nat > nat,R: real > nat > $o] :
( ( finite_finite_real @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups1935376822645274424al_nat
@ ^ [X5: real] :
( groups3542108847815614940at_nat @ ( G @ X5 )
@ ( collect_nat
@ ^ [Y5: nat] :
( ( member_nat @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups3542108847815614940at_nat
@ ^ [Y5: nat] :
( groups1935376822645274424al_nat
@ ^ [X5: real] : ( G @ X5 @ Y5 )
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_5022_sum_Oswap__restrict,axiom,
! [A4: set_int,B5: set_nat,G: int > nat > nat,R: int > nat > $o] :
( ( finite_finite_int @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [X5: int] :
( groups3542108847815614940at_nat @ ( G @ X5 )
@ ( collect_nat
@ ^ [Y5: nat] :
( ( member_nat @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups3542108847815614940at_nat
@ ^ [Y5: nat] :
( groups4541462559716669496nt_nat
@ ^ [X5: int] : ( G @ X5 @ Y5 )
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_5023_sum_Oswap__restrict,axiom,
! [A4: set_complex,B5: set_nat,G: complex > nat > nat,R: complex > nat > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups5693394587270226106ex_nat
@ ^ [X5: complex] :
( groups3542108847815614940at_nat @ ( G @ X5 )
@ ( collect_nat
@ ^ [Y5: nat] :
( ( member_nat @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups3542108847815614940at_nat
@ ^ [Y5: nat] :
( groups5693394587270226106ex_nat
@ ^ [X5: complex] : ( G @ X5 @ Y5 )
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_5024_sum_Oswap__restrict,axiom,
! [A4: set_real,B5: set_nat,G: real > nat > real,R: real > nat > $o] :
( ( finite_finite_real @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups8097168146408367636l_real
@ ^ [X5: real] :
( groups6591440286371151544t_real @ ( G @ X5 )
@ ( collect_nat
@ ^ [Y5: nat] :
( ( member_nat @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups6591440286371151544t_real
@ ^ [Y5: nat] :
( groups8097168146408367636l_real
@ ^ [X5: real] : ( G @ X5 @ Y5 )
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% sum.swap_restrict
thf(fact_5025_cond__case__prod__eta,axiom,
! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
( ! [X4: nat,Y3: nat] :
( ( F @ X4 @ Y3 )
= ( G @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
=> ( ( produc27273713700761075at_nat @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_5026_cond__case__prod__eta,axiom,
! [F: nat > nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ! [X4: nat,Y3: nat] :
( ( F @ X4 @ Y3 )
= ( G @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
=> ( ( produc8739625826339149834_nat_o @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_5027_cond__case__prod__eta,axiom,
! [F: int > int > product_prod_int_int,G: product_prod_int_int > product_prod_int_int] :
( ! [X4: int,Y3: int] :
( ( F @ X4 @ Y3 )
= ( G @ ( product_Pair_int_int @ X4 @ Y3 ) ) )
=> ( ( produc4245557441103728435nt_int @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_5028_cond__case__prod__eta,axiom,
! [F: int > int > $o,G: product_prod_int_int > $o] :
( ! [X4: int,Y3: int] :
( ( F @ X4 @ Y3 )
= ( G @ ( product_Pair_int_int @ X4 @ Y3 ) ) )
=> ( ( produc4947309494688390418_int_o @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_5029_cond__case__prod__eta,axiom,
! [F: int > int > int,G: product_prod_int_int > int] :
( ! [X4: int,Y3: int] :
( ( F @ X4 @ Y3 )
= ( G @ ( product_Pair_int_int @ X4 @ Y3 ) ) )
=> ( ( produc8211389475949308722nt_int @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_5030_case__prod__eta,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
( ( produc27273713700761075at_nat
@ ^ [X5: nat,Y5: nat] : ( F @ ( product_Pair_nat_nat @ X5 @ Y5 ) ) )
= F ) ).
% case_prod_eta
thf(fact_5031_case__prod__eta,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( produc8739625826339149834_nat_o
@ ^ [X5: nat,Y5: nat] : ( F @ ( product_Pair_nat_nat @ X5 @ Y5 ) ) )
= F ) ).
% case_prod_eta
thf(fact_5032_case__prod__eta,axiom,
! [F: product_prod_int_int > product_prod_int_int] :
( ( produc4245557441103728435nt_int
@ ^ [X5: int,Y5: int] : ( F @ ( product_Pair_int_int @ X5 @ Y5 ) ) )
= F ) ).
% case_prod_eta
thf(fact_5033_case__prod__eta,axiom,
! [F: product_prod_int_int > $o] :
( ( produc4947309494688390418_int_o
@ ^ [X5: int,Y5: int] : ( F @ ( product_Pair_int_int @ X5 @ Y5 ) ) )
= F ) ).
% case_prod_eta
thf(fact_5034_case__prod__eta,axiom,
! [F: product_prod_int_int > int] :
( ( produc8211389475949308722nt_int
@ ^ [X5: int,Y5: int] : ( F @ ( product_Pair_int_int @ X5 @ Y5 ) ) )
= F ) ).
% case_prod_eta
thf(fact_5035_case__prodE2,axiom,
! [Q: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,P2: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Z2: product_prod_nat_nat] :
( ( Q @ ( produc27273713700761075at_nat @ P2 @ Z2 ) )
=> ~ ! [X4: nat,Y3: nat] :
( ( Z2
= ( product_Pair_nat_nat @ X4 @ Y3 ) )
=> ~ ( Q @ ( P2 @ X4 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_5036_case__prodE2,axiom,
! [Q: ( product_prod_nat_nat > $o ) > $o,P2: nat > nat > product_prod_nat_nat > $o,Z2: product_prod_nat_nat] :
( ( Q @ ( produc8739625826339149834_nat_o @ P2 @ Z2 ) )
=> ~ ! [X4: nat,Y3: nat] :
( ( Z2
= ( product_Pair_nat_nat @ X4 @ Y3 ) )
=> ~ ( Q @ ( P2 @ X4 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_5037_case__prodE2,axiom,
! [Q: product_prod_int_int > $o,P2: int > int > product_prod_int_int,Z2: product_prod_int_int] :
( ( Q @ ( produc4245557441103728435nt_int @ P2 @ Z2 ) )
=> ~ ! [X4: int,Y3: int] :
( ( Z2
= ( product_Pair_int_int @ X4 @ Y3 ) )
=> ~ ( Q @ ( P2 @ X4 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_5038_case__prodE2,axiom,
! [Q: $o > $o,P2: int > int > $o,Z2: product_prod_int_int] :
( ( Q @ ( produc4947309494688390418_int_o @ P2 @ Z2 ) )
=> ~ ! [X4: int,Y3: int] :
( ( Z2
= ( product_Pair_int_int @ X4 @ Y3 ) )
=> ~ ( Q @ ( P2 @ X4 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_5039_case__prodE2,axiom,
! [Q: int > $o,P2: int > int > int,Z2: product_prod_int_int] :
( ( Q @ ( produc8211389475949308722nt_int @ P2 @ Z2 ) )
=> ~ ! [X4: int,Y3: int] :
( ( Z2
= ( product_Pair_int_int @ X4 @ Y3 ) )
=> ~ ( Q @ ( P2 @ X4 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_5040_sum__nonpos,axiom,
! [A4: set_complex,F: complex > real] :
( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_5041_sum__nonpos,axiom,
! [A4: set_real,F: real > real] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_5042_sum__nonpos,axiom,
! [A4: set_int,F: int > real] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_5043_sum__nonpos,axiom,
! [A4: set_complex,F: complex > rat] :
( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_5044_sum__nonpos,axiom,
! [A4: set_real,F: real > rat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_5045_sum__nonpos,axiom,
! [A4: set_nat,F: nat > rat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_5046_sum__nonpos,axiom,
! [A4: set_int,F: int > rat] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_5047_sum__nonpos,axiom,
! [A4: set_complex,F: complex > nat] :
( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_5048_sum__nonpos,axiom,
! [A4: set_real,F: real > nat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_5049_sum__nonpos,axiom,
! [A4: set_int,F: int > nat] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_5050_sum__nonneg,axiom,
! [A4: set_complex,F: complex > real] :
( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_5051_sum__nonneg,axiom,
! [A4: set_real,F: real > real] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_5052_sum__nonneg,axiom,
! [A4: set_int,F: int > real] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_5053_sum__nonneg,axiom,
! [A4: set_complex,F: complex > rat] :
( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_5054_sum__nonneg,axiom,
! [A4: set_real,F: real > rat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_5055_sum__nonneg,axiom,
! [A4: set_nat,F: nat > rat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_5056_sum__nonneg,axiom,
! [A4: set_int,F: int > rat] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_5057_sum__nonneg,axiom,
! [A4: set_complex,F: complex > nat] :
( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_5058_sum__nonneg,axiom,
! [A4: set_real,F: real > nat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_5059_sum__nonneg,axiom,
! [A4: set_int,F: int > nat] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_5060_ln__add__one__self__le__self2,axiom,
! [X3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) @ X3 ) ) ).
% ln_add_one_self_le_self2
thf(fact_5061_sum__mono__inv,axiom,
! [F: real > rat,I6: set_real,G: real > rat,I: real] :
( ( ( groups1300246762558778688al_rat @ F @ I6 )
= ( groups1300246762558778688al_rat @ G @ I6 ) )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I6 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member_real @ I @ I6 )
=> ( ( finite_finite_real @ I6 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5062_sum__mono__inv,axiom,
! [F: nat > rat,I6: set_nat,G: nat > rat,I: nat] :
( ( ( groups2906978787729119204at_rat @ F @ I6 )
= ( groups2906978787729119204at_rat @ G @ I6 ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I6 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member_nat @ I @ I6 )
=> ( ( finite_finite_nat @ I6 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5063_sum__mono__inv,axiom,
! [F: int > rat,I6: set_int,G: int > rat,I: int] :
( ( ( groups3906332499630173760nt_rat @ F @ I6 )
= ( groups3906332499630173760nt_rat @ G @ I6 ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I6 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member_int @ I @ I6 )
=> ( ( finite_finite_int @ I6 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5064_sum__mono__inv,axiom,
! [F: complex > rat,I6: set_complex,G: complex > rat,I: complex] :
( ( ( groups5058264527183730370ex_rat @ F @ I6 )
= ( groups5058264527183730370ex_rat @ G @ I6 ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I6 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member_complex @ I @ I6 )
=> ( ( finite3207457112153483333omplex @ I6 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5065_sum__mono__inv,axiom,
! [F: real > nat,I6: set_real,G: real > nat,I: real] :
( ( ( groups1935376822645274424al_nat @ F @ I6 )
= ( groups1935376822645274424al_nat @ G @ I6 ) )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I6 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member_real @ I @ I6 )
=> ( ( finite_finite_real @ I6 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5066_sum__mono__inv,axiom,
! [F: int > nat,I6: set_int,G: int > nat,I: int] :
( ( ( groups4541462559716669496nt_nat @ F @ I6 )
= ( groups4541462559716669496nt_nat @ G @ I6 ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I6 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member_int @ I @ I6 )
=> ( ( finite_finite_int @ I6 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5067_sum__mono__inv,axiom,
! [F: complex > nat,I6: set_complex,G: complex > nat,I: complex] :
( ( ( groups5693394587270226106ex_nat @ F @ I6 )
= ( groups5693394587270226106ex_nat @ G @ I6 ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I6 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member_complex @ I @ I6 )
=> ( ( finite3207457112153483333omplex @ I6 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5068_sum__mono__inv,axiom,
! [F: real > int,I6: set_real,G: real > int,I: real] :
( ( ( groups1932886352136224148al_int @ F @ I6 )
= ( groups1932886352136224148al_int @ G @ I6 ) )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I6 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member_real @ I @ I6 )
=> ( ( finite_finite_real @ I6 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5069_sum__mono__inv,axiom,
! [F: nat > int,I6: set_nat,G: nat > int,I: nat] :
( ( ( groups3539618377306564664at_int @ F @ I6 )
= ( groups3539618377306564664at_int @ G @ I6 ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I6 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member_nat @ I @ I6 )
=> ( ( finite_finite_nat @ I6 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5070_sum__mono__inv,axiom,
! [F: complex > int,I6: set_complex,G: complex > int,I: complex] :
( ( ( groups5690904116761175830ex_int @ F @ I6 )
= ( groups5690904116761175830ex_int @ G @ I6 ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I6 )
=> ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member_complex @ I @ I6 )
=> ( ( finite3207457112153483333omplex @ I6 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5071_ln__less__self,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_real @ ( ln_ln_real @ X3 ) @ X3 ) ) ).
% ln_less_self
thf(fact_5072_not__numeral__le__neg__numeral,axiom,
! [M2: num,N: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_5073_not__numeral__le__neg__numeral,axiom,
! [M2: num,N: num] :
~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_5074_not__numeral__le__neg__numeral,axiom,
! [M2: num,N: num] :
~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_5075_not__numeral__le__neg__numeral,axiom,
! [M2: num,N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_5076_neg__numeral__le__numeral,axiom,
! [M2: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( numeral_numeral_real @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_5077_neg__numeral__le__numeral,axiom,
! [M2: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_5078_neg__numeral__le__numeral,axiom,
! [M2: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( numeral_numeral_rat @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_5079_neg__numeral__le__numeral,axiom,
! [M2: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_5080_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_5081_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_5082_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_complex
!= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_5083_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_rat
!= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_5084_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_z3403309356797280102nteger
!= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_5085_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(4)
thf(fact_5086_le__minus__one__simps_I4_J,axiom,
~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% le_minus_one_simps(4)
thf(fact_5087_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% le_minus_one_simps(4)
thf(fact_5088_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_5089_le__minus__one__simps_I2_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% le_minus_one_simps(2)
thf(fact_5090_le__minus__one__simps_I2_J,axiom,
ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% le_minus_one_simps(2)
thf(fact_5091_le__minus__one__simps_I2_J,axiom,
ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% le_minus_one_simps(2)
thf(fact_5092_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_5093_zero__neq__neg__one,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% zero_neq_neg_one
thf(fact_5094_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_5095_zero__neq__neg__one,axiom,
( zero_zero_complex
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% zero_neq_neg_one
thf(fact_5096_zero__neq__neg__one,axiom,
( zero_zero_rat
!= ( uminus_uminus_rat @ one_one_rat ) ) ).
% zero_neq_neg_one
thf(fact_5097_zero__neq__neg__one,axiom,
( zero_z3403309356797280102nteger
!= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% zero_neq_neg_one
thf(fact_5098_add__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= zero_zero_real )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% add_eq_0_iff
thf(fact_5099_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% add_eq_0_iff
thf(fact_5100_add__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( ( plus_plus_complex @ A @ B )
= zero_zero_complex )
= ( B
= ( uminus1482373934393186551omplex @ A ) ) ) ).
% add_eq_0_iff
thf(fact_5101_add__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( ( plus_plus_rat @ A @ B )
= zero_zero_rat )
= ( B
= ( uminus_uminus_rat @ A ) ) ) ).
% add_eq_0_iff
thf(fact_5102_add__eq__0__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ( plus_p5714425477246183910nteger @ A @ B )
= zero_z3403309356797280102nteger )
= ( B
= ( uminus1351360451143612070nteger @ A ) ) ) ).
% add_eq_0_iff
thf(fact_5103_ab__group__add__class_Oab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_group_add_class.ab_left_minus
thf(fact_5104_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_5105_ab__group__add__class_Oab__left__minus,axiom,
! [A: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
= zero_zero_complex ) ).
% ab_group_add_class.ab_left_minus
thf(fact_5106_ab__group__add__class_Oab__left__minus,axiom,
! [A: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
= zero_zero_rat ) ).
% ab_group_add_class.ab_left_minus
thf(fact_5107_ab__group__add__class_Oab__left__minus,axiom,
! [A: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
= zero_z3403309356797280102nteger ) ).
% ab_group_add_class.ab_left_minus
thf(fact_5108_add_Oinverse__unique,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= zero_zero_real )
=> ( ( uminus_uminus_real @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_5109_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_5110_add_Oinverse__unique,axiom,
! [A: complex,B: complex] :
( ( ( plus_plus_complex @ A @ B )
= zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_5111_add_Oinverse__unique,axiom,
! [A: rat,B: rat] :
( ( ( plus_plus_rat @ A @ B )
= zero_zero_rat )
=> ( ( uminus_uminus_rat @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_5112_add_Oinverse__unique,axiom,
! [A: code_integer,B: code_integer] :
( ( ( plus_p5714425477246183910nteger @ A @ B )
= zero_z3403309356797280102nteger )
=> ( ( uminus1351360451143612070nteger @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_5113_eq__neg__iff__add__eq__0,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( ( plus_plus_real @ A @ B )
= zero_zero_real ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_5114_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_5115_eq__neg__iff__add__eq__0,axiom,
! [A: complex,B: complex] :
( ( A
= ( uminus1482373934393186551omplex @ B ) )
= ( ( plus_plus_complex @ A @ B )
= zero_zero_complex ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_5116_eq__neg__iff__add__eq__0,axiom,
! [A: rat,B: rat] :
( ( A
= ( uminus_uminus_rat @ B ) )
= ( ( plus_plus_rat @ A @ B )
= zero_zero_rat ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_5117_eq__neg__iff__add__eq__0,axiom,
! [A: code_integer,B: code_integer] :
( ( A
= ( uminus1351360451143612070nteger @ B ) )
= ( ( plus_p5714425477246183910nteger @ A @ B )
= zero_z3403309356797280102nteger ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_5118_neg__eq__iff__add__eq__0,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( plus_plus_real @ A @ B )
= zero_zero_real ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_5119_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_5120_neg__eq__iff__add__eq__0,axiom,
! [A: complex,B: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= B )
= ( ( plus_plus_complex @ A @ B )
= zero_zero_complex ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_5121_neg__eq__iff__add__eq__0,axiom,
! [A: rat,B: rat] :
( ( ( uminus_uminus_rat @ A )
= B )
= ( ( plus_plus_rat @ A @ B )
= zero_zero_rat ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_5122_neg__eq__iff__add__eq__0,axiom,
! [A: code_integer,B: code_integer] :
( ( ( uminus1351360451143612070nteger @ A )
= B )
= ( ( plus_p5714425477246183910nteger @ A @ B )
= zero_z3403309356797280102nteger ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_5123_nonzero__minus__divide__divide,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_5124_nonzero__minus__divide__divide,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_5125_nonzero__minus__divide__divide,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_5126_nonzero__minus__divide__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_5127_nonzero__minus__divide__right,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_5128_nonzero__minus__divide__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
= ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_5129_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A5: real,B4: real] : ( plus_plus_real @ A5 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5130_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5131_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_complex
= ( ^ [A5: complex,B4: complex] : ( plus_plus_complex @ A5 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5132_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_rat
= ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ A5 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5133_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_8373710615458151222nteger
= ( ^ [A5: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A5 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5134_diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A5: real,B4: real] : ( plus_plus_real @ A5 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_5135_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_5136_diff__conv__add__uminus,axiom,
( minus_minus_complex
= ( ^ [A5: complex,B4: complex] : ( plus_plus_complex @ A5 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_5137_diff__conv__add__uminus,axiom,
( minus_minus_rat
= ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ A5 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_5138_diff__conv__add__uminus,axiom,
( minus_8373710615458151222nteger
= ( ^ [A5: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A5 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_5139_group__cancel_Osub2,axiom,
! [B5: real,K2: real,B: real,A: real] :
( ( B5
= ( plus_plus_real @ K2 @ B ) )
=> ( ( minus_minus_real @ A @ B5 )
= ( plus_plus_real @ ( uminus_uminus_real @ K2 ) @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_5140_group__cancel_Osub2,axiom,
! [B5: int,K2: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K2 @ B ) )
=> ( ( minus_minus_int @ A @ B5 )
= ( plus_plus_int @ ( uminus_uminus_int @ K2 ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_5141_group__cancel_Osub2,axiom,
! [B5: complex,K2: complex,B: complex,A: complex] :
( ( B5
= ( plus_plus_complex @ K2 @ B ) )
=> ( ( minus_minus_complex @ A @ B5 )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K2 ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_5142_group__cancel_Osub2,axiom,
! [B5: rat,K2: rat,B: rat,A: rat] :
( ( B5
= ( plus_plus_rat @ K2 @ B ) )
=> ( ( minus_minus_rat @ A @ B5 )
= ( plus_plus_rat @ ( uminus_uminus_rat @ K2 ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_5143_group__cancel_Osub2,axiom,
! [B5: code_integer,K2: code_integer,B: code_integer,A: code_integer] :
( ( B5
= ( plus_p5714425477246183910nteger @ K2 @ B ) )
=> ( ( minus_8373710615458151222nteger @ A @ B5 )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K2 ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_5144_subset__Compl__self__eq,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ ( uminus6524753893492686040at_nat @ A4 ) )
= ( A4 = bot_bo2099793752762293965at_nat ) ) ).
% subset_Compl_self_eq
thf(fact_5145_subset__Compl__self__eq,axiom,
! [A4: set_real] :
( ( ord_less_eq_set_real @ A4 @ ( uminus612125837232591019t_real @ A4 ) )
= ( A4 = bot_bot_set_real ) ) ).
% subset_Compl_self_eq
thf(fact_5146_subset__Compl__self__eq,axiom,
! [A4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ A4 ) )
= ( A4 = bot_bot_set_nat ) ) ).
% subset_Compl_self_eq
thf(fact_5147_subset__Compl__self__eq,axiom,
! [A4: set_int] :
( ( ord_less_eq_set_int @ A4 @ ( uminus1532241313380277803et_int @ A4 ) )
= ( A4 = bot_bot_set_int ) ) ).
% subset_Compl_self_eq
thf(fact_5148_real__minus__mult__self__le,axiom,
! [U: real,X3: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X3 @ X3 ) ) ).
% real_minus_mult_self_le
thf(fact_5149_pos__zmult__eq__1__iff__lemma,axiom,
! [M2: int,N: int] :
( ( ( times_times_int @ M2 @ N )
= one_one_int )
=> ( ( M2 = one_one_int )
| ( M2
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_5150_zmult__eq__1__iff,axiom,
! [M2: int,N: int] :
( ( ( times_times_int @ M2 @ N )
= one_one_int )
= ( ( ( M2 = one_one_int )
& ( N = one_one_int ) )
| ( ( M2
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_5151_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_5152_sum_Ointer__filter,axiom,
! [A4: set_real,G: real > complex,P2: real > $o] :
( ( finite_finite_real @ A4 )
=> ( ( groups5754745047067104278omplex @ G
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups5754745047067104278omplex
@ ^ [X5: real] : ( if_complex @ ( P2 @ X5 ) @ ( G @ X5 ) @ zero_zero_complex )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_5153_sum_Ointer__filter,axiom,
! [A4: set_nat,G: nat > complex,P2: nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( groups2073611262835488442omplex @ G
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups2073611262835488442omplex
@ ^ [X5: nat] : ( if_complex @ ( P2 @ X5 ) @ ( G @ X5 ) @ zero_zero_complex )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_5154_sum_Ointer__filter,axiom,
! [A4: set_int,G: int > complex,P2: int > $o] :
( ( finite_finite_int @ A4 )
=> ( ( groups3049146728041665814omplex @ G
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups3049146728041665814omplex
@ ^ [X5: int] : ( if_complex @ ( P2 @ X5 ) @ ( G @ X5 ) @ zero_zero_complex )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_5155_sum_Ointer__filter,axiom,
! [A4: set_real,G: real > real,P2: real > $o] :
( ( finite_finite_real @ A4 )
=> ( ( groups8097168146408367636l_real @ G
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups8097168146408367636l_real
@ ^ [X5: real] : ( if_real @ ( P2 @ X5 ) @ ( G @ X5 ) @ zero_zero_real )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_5156_sum_Ointer__filter,axiom,
! [A4: set_int,G: int > real,P2: int > $o] :
( ( finite_finite_int @ A4 )
=> ( ( groups8778361861064173332t_real @ G
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups8778361861064173332t_real
@ ^ [X5: int] : ( if_real @ ( P2 @ X5 ) @ ( G @ X5 ) @ zero_zero_real )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_5157_sum_Ointer__filter,axiom,
! [A4: set_complex,G: complex > real,P2: complex > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5808333547571424918x_real @ G
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups5808333547571424918x_real
@ ^ [X5: complex] : ( if_real @ ( P2 @ X5 ) @ ( G @ X5 ) @ zero_zero_real )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_5158_sum_Ointer__filter,axiom,
! [A4: set_real,G: real > rat,P2: real > $o] :
( ( finite_finite_real @ A4 )
=> ( ( groups1300246762558778688al_rat @ G
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups1300246762558778688al_rat
@ ^ [X5: real] : ( if_rat @ ( P2 @ X5 ) @ ( G @ X5 ) @ zero_zero_rat )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_5159_sum_Ointer__filter,axiom,
! [A4: set_nat,G: nat > rat,P2: nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( groups2906978787729119204at_rat @ G
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups2906978787729119204at_rat
@ ^ [X5: nat] : ( if_rat @ ( P2 @ X5 ) @ ( G @ X5 ) @ zero_zero_rat )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_5160_sum_Ointer__filter,axiom,
! [A4: set_int,G: int > rat,P2: int > $o] :
( ( finite_finite_int @ A4 )
=> ( ( groups3906332499630173760nt_rat @ G
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups3906332499630173760nt_rat
@ ^ [X5: int] : ( if_rat @ ( P2 @ X5 ) @ ( G @ X5 ) @ zero_zero_rat )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_5161_sum_Ointer__filter,axiom,
! [A4: set_complex,G: complex > rat,P2: complex > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5058264527183730370ex_rat @ G
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups5058264527183730370ex_rat
@ ^ [X5: complex] : ( if_rat @ ( P2 @ X5 ) @ ( G @ X5 ) @ zero_zero_rat )
@ A4 ) ) ) ).
% sum.inter_filter
thf(fact_5162_zmod__zminus2__not__zero,axiom,
! [K2: int,L: int] :
( ( ( modulo_modulo_int @ K2 @ ( uminus_uminus_int @ L ) )
!= zero_zero_int )
=> ( ( modulo_modulo_int @ K2 @ L )
!= zero_zero_int ) ) ).
% zmod_zminus2_not_zero
thf(fact_5163_zmod__zminus1__not__zero,axiom,
! [K2: int,L: int] :
( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K2 ) @ L )
!= zero_zero_int )
=> ( ( modulo_modulo_int @ K2 @ L )
!= zero_zero_int ) ) ).
% zmod_zminus1_not_zero
thf(fact_5164_minus__real__def,axiom,
( minus_minus_real
= ( ^ [X5: real,Y5: real] : ( plus_plus_real @ X5 @ ( uminus_uminus_real @ Y5 ) ) ) ) ).
% minus_real_def
thf(fact_5165_internal__case__prod__def,axiom,
produc1854806715440696265at_nat = produc27273713700761075at_nat ).
% internal_case_prod_def
thf(fact_5166_internal__case__prod__def,axiom,
produc4780622933104268256_nat_o = produc8739625826339149834_nat_o ).
% internal_case_prod_def
thf(fact_5167_internal__case__prod__def,axiom,
produc297006045350968285nt_int = produc4245557441103728435nt_int ).
% internal_case_prod_def
thf(fact_5168_internal__case__prod__def,axiom,
produc8005341501107743676_int_o = produc4947309494688390418_int_o ).
% internal_case_prod_def
thf(fact_5169_internal__case__prod__def,axiom,
produc7926200574084438792nt_int = produc8211389475949308722nt_int ).
% internal_case_prod_def
thf(fact_5170_ln__one__minus__pos__upper__bound,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ one_one_real )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X3 ) ) @ ( uminus_uminus_real @ X3 ) ) ) ) ).
% ln_one_minus_pos_upper_bound
thf(fact_5171_sum__le__included,axiom,
! [S3: set_int,T: set_int,G: int > real,I: int > int,F: int > real] :
( ( finite_finite_int @ S3 )
=> ( ( finite_finite_int @ T )
=> ( ! [X4: int] :
( ( member_int @ X4 @ T )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S3 )
=> ? [Xa: int] :
( ( member_int @ Xa @ T )
& ( ( I @ Xa )
= X4 )
& ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S3 ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5172_sum__le__included,axiom,
! [S3: set_int,T: set_complex,G: complex > real,I: complex > int,F: int > real] :
( ( finite_finite_int @ S3 )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ T )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S3 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T )
& ( ( I @ Xa )
= X4 )
& ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S3 ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5173_sum__le__included,axiom,
! [S3: set_complex,T: set_int,G: int > real,I: int > complex,F: complex > real] :
( ( finite3207457112153483333omplex @ S3 )
=> ( ( finite_finite_int @ T )
=> ( ! [X4: int] :
( ( member_int @ X4 @ T )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S3 )
=> ? [Xa: int] :
( ( member_int @ Xa @ T )
& ( ( I @ Xa )
= X4 )
& ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S3 ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5174_sum__le__included,axiom,
! [S3: set_complex,T: set_complex,G: complex > real,I: complex > complex,F: complex > real] :
( ( finite3207457112153483333omplex @ S3 )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ T )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S3 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T )
& ( ( I @ Xa )
= X4 )
& ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S3 ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5175_sum__le__included,axiom,
! [S3: set_nat,T: set_nat,G: nat > rat,I: nat > nat,F: nat > rat] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_nat @ T )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S3 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T )
& ( ( I @ Xa )
= X4 )
& ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S3 ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5176_sum__le__included,axiom,
! [S3: set_nat,T: set_int,G: int > rat,I: int > nat,F: nat > rat] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_int @ T )
=> ( ! [X4: int] :
( ( member_int @ X4 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S3 )
=> ? [Xa: int] :
( ( member_int @ Xa @ T )
& ( ( I @ Xa )
= X4 )
& ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S3 ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5177_sum__le__included,axiom,
! [S3: set_nat,T: set_complex,G: complex > rat,I: complex > nat,F: nat > rat] :
( ( finite_finite_nat @ S3 )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S3 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T )
& ( ( I @ Xa )
= X4 )
& ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S3 ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5178_sum__le__included,axiom,
! [S3: set_int,T: set_nat,G: nat > rat,I: nat > int,F: int > rat] :
( ( finite_finite_int @ S3 )
=> ( ( finite_finite_nat @ T )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S3 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T )
& ( ( I @ Xa )
= X4 )
& ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S3 ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5179_sum__le__included,axiom,
! [S3: set_int,T: set_int,G: int > rat,I: int > int,F: int > rat] :
( ( finite_finite_int @ S3 )
=> ( ( finite_finite_int @ T )
=> ( ! [X4: int] :
( ( member_int @ X4 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S3 )
=> ? [Xa: int] :
( ( member_int @ Xa @ T )
& ( ( I @ Xa )
= X4 )
& ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S3 ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5180_sum__le__included,axiom,
! [S3: set_int,T: set_complex,G: complex > rat,I: complex > int,F: int > rat] :
( ( finite_finite_int @ S3 )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S3 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T )
& ( ( I @ Xa )
= X4 )
& ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S3 ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5181_sum__nonneg__eq__0__iff,axiom,
! [A4: set_real,F: real > real] :
( ( finite_finite_real @ A4 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ( ( groups8097168146408367636l_real @ F @ A4 )
= zero_zero_real )
= ( ! [X5: real] :
( ( member_real @ X5 @ A4 )
=> ( ( F @ X5 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5182_sum__nonneg__eq__0__iff,axiom,
! [A4: set_int,F: int > real] :
( ( finite_finite_int @ A4 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ( ( groups8778361861064173332t_real @ F @ A4 )
= zero_zero_real )
= ( ! [X5: int] :
( ( member_int @ X5 @ A4 )
=> ( ( F @ X5 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5183_sum__nonneg__eq__0__iff,axiom,
! [A4: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ( ( groups5808333547571424918x_real @ F @ A4 )
= zero_zero_real )
= ( ! [X5: complex] :
( ( member_complex @ X5 @ A4 )
=> ( ( F @ X5 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5184_sum__nonneg__eq__0__iff,axiom,
! [A4: set_real,F: real > rat] :
( ( finite_finite_real @ A4 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ( ( groups1300246762558778688al_rat @ F @ A4 )
= zero_zero_rat )
= ( ! [X5: real] :
( ( member_real @ X5 @ A4 )
=> ( ( F @ X5 )
= zero_zero_rat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5185_sum__nonneg__eq__0__iff,axiom,
! [A4: set_nat,F: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ( ( groups2906978787729119204at_rat @ F @ A4 )
= zero_zero_rat )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ A4 )
=> ( ( F @ X5 )
= zero_zero_rat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5186_sum__nonneg__eq__0__iff,axiom,
! [A4: set_int,F: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ( ( groups3906332499630173760nt_rat @ F @ A4 )
= zero_zero_rat )
= ( ! [X5: int] :
( ( member_int @ X5 @ A4 )
=> ( ( F @ X5 )
= zero_zero_rat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5187_sum__nonneg__eq__0__iff,axiom,
! [A4: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ( ( groups5058264527183730370ex_rat @ F @ A4 )
= zero_zero_rat )
= ( ! [X5: complex] :
( ( member_complex @ X5 @ A4 )
=> ( ( F @ X5 )
= zero_zero_rat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5188_sum__nonneg__eq__0__iff,axiom,
! [A4: set_real,F: real > nat] :
( ( finite_finite_real @ A4 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ( ( groups1935376822645274424al_nat @ F @ A4 )
= zero_zero_nat )
= ( ! [X5: real] :
( ( member_real @ X5 @ A4 )
=> ( ( F @ X5 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5189_sum__nonneg__eq__0__iff,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F @ A4 )
= zero_zero_nat )
= ( ! [X5: int] :
( ( member_int @ X5 @ A4 )
=> ( ( F @ X5 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5190_sum__nonneg__eq__0__iff,axiom,
! [A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ( ( groups5693394587270226106ex_nat @ F @ A4 )
= zero_zero_nat )
= ( ! [X5: complex] :
( ( member_complex @ X5 @ A4 )
=> ( ( F @ X5 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5191_sum__strict__mono__ex1,axiom,
! [A4: set_int,F: int > real,G: int > real] :
( ( finite_finite_int @ A4 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X: int] :
( ( member_int @ X @ A4 )
& ( ord_less_real @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5192_sum__strict__mono__ex1,axiom,
! [A4: set_complex,F: complex > real,G: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X: complex] :
( ( member_complex @ X @ A4 )
& ( ord_less_real @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5193_sum__strict__mono__ex1,axiom,
! [A4: set_nat,F: nat > rat,G: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X: nat] :
( ( member_nat @ X @ A4 )
& ( ord_less_rat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5194_sum__strict__mono__ex1,axiom,
! [A4: set_int,F: int > rat,G: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X: int] :
( ( member_int @ X @ A4 )
& ( ord_less_rat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5195_sum__strict__mono__ex1,axiom,
! [A4: set_complex,F: complex > rat,G: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X: complex] :
( ( member_complex @ X @ A4 )
& ( ord_less_rat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5196_sum__strict__mono__ex1,axiom,
! [A4: set_int,F: int > nat,G: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X: int] :
( ( member_int @ X @ A4 )
& ( ord_less_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5197_sum__strict__mono__ex1,axiom,
! [A4: set_complex,F: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X: complex] :
( ( member_complex @ X @ A4 )
& ( ord_less_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5198_sum__strict__mono__ex1,axiom,
! [A4: set_nat,F: nat > int,G: nat > int] :
( ( finite_finite_nat @ A4 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X: nat] :
( ( member_nat @ X @ A4 )
& ( ord_less_int @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5199_sum__strict__mono__ex1,axiom,
! [A4: set_complex,F: complex > int,G: complex > int] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X: complex] :
( ( member_complex @ X @ A4 )
& ( ord_less_int @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5200_sum__strict__mono__ex1,axiom,
! [A4: set_int,F: int > int,G: int > int] :
( ( finite_finite_int @ A4 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ? [X: int] :
( ( member_int @ X @ A4 )
& ( ord_less_int @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5201_sum_Orelated,axiom,
! [R: complex > complex > $o,S2: set_nat,H2: nat > complex,G: nat > complex] :
( ( R @ zero_zero_complex @ zero_zero_complex )
=> ( ! [X16: complex,Y15: complex,X22: complex,Y23: complex] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_complex @ X16 @ Y15 ) @ ( plus_plus_complex @ X22 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups2073611262835488442omplex @ H2 @ S2 ) @ ( groups2073611262835488442omplex @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_5202_sum_Orelated,axiom,
! [R: complex > complex > $o,S2: set_int,H2: int > complex,G: int > complex] :
( ( R @ zero_zero_complex @ zero_zero_complex )
=> ( ! [X16: complex,Y15: complex,X22: complex,Y23: complex] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_complex @ X16 @ Y15 ) @ ( plus_plus_complex @ X22 @ Y23 ) ) )
=> ( ( finite_finite_int @ S2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups3049146728041665814omplex @ H2 @ S2 ) @ ( groups3049146728041665814omplex @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_5203_sum_Orelated,axiom,
! [R: real > real > $o,S2: set_int,H2: int > real,G: int > real] :
( ( R @ zero_zero_real @ zero_zero_real )
=> ( ! [X16: real,Y15: real,X22: real,Y23: real] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_real @ X16 @ Y15 ) @ ( plus_plus_real @ X22 @ Y23 ) ) )
=> ( ( finite_finite_int @ S2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups8778361861064173332t_real @ H2 @ S2 ) @ ( groups8778361861064173332t_real @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_5204_sum_Orelated,axiom,
! [R: real > real > $o,S2: set_complex,H2: complex > real,G: complex > real] :
( ( R @ zero_zero_real @ zero_zero_real )
=> ( ! [X16: real,Y15: real,X22: real,Y23: real] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_real @ X16 @ Y15 ) @ ( plus_plus_real @ X22 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups5808333547571424918x_real @ H2 @ S2 ) @ ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_5205_sum_Orelated,axiom,
! [R: rat > rat > $o,S2: set_nat,H2: nat > rat,G: nat > rat] :
( ( R @ zero_zero_rat @ zero_zero_rat )
=> ( ! [X16: rat,Y15: rat,X22: rat,Y23: rat] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X22 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups2906978787729119204at_rat @ H2 @ S2 ) @ ( groups2906978787729119204at_rat @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_5206_sum_Orelated,axiom,
! [R: rat > rat > $o,S2: set_int,H2: int > rat,G: int > rat] :
( ( R @ zero_zero_rat @ zero_zero_rat )
=> ( ! [X16: rat,Y15: rat,X22: rat,Y23: rat] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X22 @ Y23 ) ) )
=> ( ( finite_finite_int @ S2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups3906332499630173760nt_rat @ H2 @ S2 ) @ ( groups3906332499630173760nt_rat @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_5207_sum_Orelated,axiom,
! [R: rat > rat > $o,S2: set_complex,H2: complex > rat,G: complex > rat] :
( ( R @ zero_zero_rat @ zero_zero_rat )
=> ( ! [X16: rat,Y15: rat,X22: rat,Y23: rat] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X22 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups5058264527183730370ex_rat @ H2 @ S2 ) @ ( groups5058264527183730370ex_rat @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_5208_sum_Orelated,axiom,
! [R: nat > nat > $o,S2: set_int,H2: int > nat,G: int > nat] :
( ( R @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X16: nat,Y15: nat,X22: nat,Y23: nat] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_nat @ X16 @ Y15 ) @ ( plus_plus_nat @ X22 @ Y23 ) ) )
=> ( ( finite_finite_int @ S2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups4541462559716669496nt_nat @ H2 @ S2 ) @ ( groups4541462559716669496nt_nat @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_5209_sum_Orelated,axiom,
! [R: nat > nat > $o,S2: set_complex,H2: complex > nat,G: complex > nat] :
( ( R @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X16: nat,Y15: nat,X22: nat,Y23: nat] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_nat @ X16 @ Y15 ) @ ( plus_plus_nat @ X22 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups5693394587270226106ex_nat @ H2 @ S2 ) @ ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_5210_sum_Orelated,axiom,
! [R: int > int > $o,S2: set_nat,H2: nat > int,G: nat > int] :
( ( R @ zero_zero_int @ zero_zero_int )
=> ( ! [X16: int,Y15: int,X22: int,Y23: int] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_int @ X16 @ Y15 ) @ ( plus_plus_int @ X22 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups3539618377306564664at_int @ H2 @ S2 ) @ ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_5211_sum__strict__mono,axiom,
! [A4: set_complex,F: complex > real,G: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( A4 != bot_bot_set_complex )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5212_sum__strict__mono,axiom,
! [A4: set_real,F: real > real,G: real > real] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5213_sum__strict__mono,axiom,
! [A4: set_int,F: int > real,G: int > real] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5214_sum__strict__mono,axiom,
! [A4: set_complex,F: complex > rat,G: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( A4 != bot_bot_set_complex )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5215_sum__strict__mono,axiom,
! [A4: set_real,F: real > rat,G: real > rat] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5216_sum__strict__mono,axiom,
! [A4: set_nat,F: nat > rat,G: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5217_sum__strict__mono,axiom,
! [A4: set_int,F: int > rat,G: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5218_sum__strict__mono,axiom,
! [A4: set_complex,F: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( A4 != bot_bot_set_complex )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5219_sum__strict__mono,axiom,
! [A4: set_real,F: real > nat,G: real > nat] :
( ( finite_finite_real @ A4 )
=> ( ( A4 != bot_bot_set_real )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5220_sum__strict__mono,axiom,
! [A4: set_int,F: int > nat,G: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5221_ln__bound,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ X3 ) ) ).
% ln_bound
thf(fact_5222_ln__gt__zero,axiom,
! [X3: real] :
( ( ord_less_real @ one_one_real @ X3 )
=> ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) ) ) ).
% ln_gt_zero
thf(fact_5223_ln__less__zero,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ one_one_real )
=> ( ord_less_real @ ( ln_ln_real @ X3 ) @ zero_zero_real ) ) ) ).
% ln_less_zero
thf(fact_5224_ln__gt__zero__imp__gt__one,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_real @ one_one_real @ X3 ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_5225_ln__ge__zero,axiom,
! [X3: real] :
( ( ord_less_eq_real @ one_one_real @ X3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) ) ) ).
% ln_ge_zero
thf(fact_5226_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T4: set_real,S2: set_real,I: real > real,J: real > real,T2: set_real,G: real > complex,H2: real > complex] :
( ( finite_finite_real @ S5 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T2 @ T4 ) ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S5 )
=> ( ( G @ A3 )
= zero_zero_complex ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T4 )
=> ( ( H2 @ B3 )
= zero_zero_complex ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups5754745047067104278omplex @ G @ S2 )
= ( groups5754745047067104278omplex @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_5227_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T4: set_int,S2: set_real,I: int > real,J: real > int,T2: set_int,G: real > complex,H2: int > complex] :
( ( finite_finite_real @ S5 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T2 @ T4 ) ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T2 @ T4 ) )
=> ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S5 )
=> ( ( G @ A3 )
= zero_zero_complex ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ T4 )
=> ( ( H2 @ B3 )
= zero_zero_complex ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups5754745047067104278omplex @ G @ S2 )
= ( groups3049146728041665814omplex @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_5228_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_int,T4: set_real,S2: set_int,I: real > int,J: int > real,T2: set_real,G: int > complex,H2: real > complex] :
( ( finite_finite_int @ S5 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T2 @ T4 ) ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S5 )
=> ( ( G @ A3 )
= zero_zero_complex ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T4 )
=> ( ( H2 @ B3 )
= zero_zero_complex ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3049146728041665814omplex @ G @ S2 )
= ( groups5754745047067104278omplex @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_5229_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_int,T4: set_int,S2: set_int,I: int > int,J: int > int,T2: set_int,G: int > complex,H2: int > complex] :
( ( finite_finite_int @ S5 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T2 @ T4 ) ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T2 @ T4 ) )
=> ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S5 )
=> ( ( G @ A3 )
= zero_zero_complex ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ T4 )
=> ( ( H2 @ B3 )
= zero_zero_complex ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3049146728041665814omplex @ G @ S2 )
= ( groups3049146728041665814omplex @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_5230_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T4: set_real,S2: set_real,I: real > real,J: real > real,T2: set_real,G: real > real,H2: real > real] :
( ( finite_finite_real @ S5 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T2 @ T4 ) ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S5 )
=> ( ( G @ A3 )
= zero_zero_real ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T4 )
=> ( ( H2 @ B3 )
= zero_zero_real ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S2 )
= ( groups8097168146408367636l_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_5231_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T4: set_int,S2: set_real,I: int > real,J: real > int,T2: set_int,G: real > real,H2: int > real] :
( ( finite_finite_real @ S5 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T2 @ T4 ) ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T2 @ T4 ) )
=> ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S5 )
=> ( ( G @ A3 )
= zero_zero_real ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ T4 )
=> ( ( H2 @ B3 )
= zero_zero_real ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S2 )
= ( groups8778361861064173332t_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_5232_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T4: set_complex,S2: set_real,I: complex > real,J: real > complex,T2: set_complex,G: real > real,H2: complex > real] :
( ( finite_finite_real @ S5 )
=> ( ( finite3207457112153483333omplex @ T4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T2 @ T4 ) ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T2 @ T4 ) )
=> ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S5 )
=> ( ( G @ A3 )
= zero_zero_real ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T4 )
=> ( ( H2 @ B3 )
= zero_zero_real ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S2 )
= ( groups5808333547571424918x_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_5233_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_int,T4: set_real,S2: set_int,I: real > int,J: int > real,T2: set_real,G: int > real,H2: real > real] :
( ( finite_finite_int @ S5 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T2 @ T4 ) ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S5 )
=> ( ( G @ A3 )
= zero_zero_real ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T4 )
=> ( ( H2 @ B3 )
= zero_zero_real ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8778361861064173332t_real @ G @ S2 )
= ( groups8097168146408367636l_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_5234_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_int,T4: set_int,S2: set_int,I: int > int,J: int > int,T2: set_int,G: int > real,H2: int > real] :
( ( finite_finite_int @ S5 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T2 @ T4 ) ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T2 @ T4 ) )
=> ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S5 )
=> ( ( G @ A3 )
= zero_zero_real ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ T4 )
=> ( ( H2 @ B3 )
= zero_zero_real ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8778361861064173332t_real @ G @ S2 )
= ( groups8778361861064173332t_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_5235_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_int,T4: set_complex,S2: set_int,I: complex > int,J: int > complex,T2: set_complex,G: int > real,H2: complex > real] :
( ( finite_finite_int @ S5 )
=> ( ( finite3207457112153483333omplex @ T4 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T2 @ T4 ) ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T2 @ T4 ) )
=> ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S5 )
=> ( ( G @ A3 )
= zero_zero_real ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T4 )
=> ( ( H2 @ B3 )
= zero_zero_real ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8778361861064173332t_real @ G @ S2 )
= ( groups5808333547571424918x_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_5236_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_5237_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_5238_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_5239_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_5240_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% neg_numeral_le_zero
thf(fact_5241_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% neg_numeral_le_zero
thf(fact_5242_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% neg_numeral_le_zero
thf(fact_5243_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_le_zero
thf(fact_5244_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% neg_numeral_less_zero
thf(fact_5245_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_less_zero
thf(fact_5246_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% neg_numeral_less_zero
thf(fact_5247_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% neg_numeral_less_zero
thf(fact_5248_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_5249_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_5250_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_5251_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_5252_le__minus__one__simps_I1_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% le_minus_one_simps(1)
thf(fact_5253_le__minus__one__simps_I1_J,axiom,
ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% le_minus_one_simps(1)
thf(fact_5254_le__minus__one__simps_I1_J,axiom,
ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% le_minus_one_simps(1)
thf(fact_5255_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_5256_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(3)
thf(fact_5257_le__minus__one__simps_I3_J,axiom,
~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% le_minus_one_simps(3)
thf(fact_5258_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% le_minus_one_simps(3)
thf(fact_5259_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_5260_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(3)
thf(fact_5261_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_5262_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% less_minus_one_simps(3)
thf(fact_5263_less__minus__one__simps_I3_J,axiom,
~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% less_minus_one_simps(3)
thf(fact_5264_less__minus__one__simps_I1_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% less_minus_one_simps(1)
thf(fact_5265_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_5266_less__minus__one__simps_I1_J,axiom,
ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% less_minus_one_simps(1)
thf(fact_5267_less__minus__one__simps_I1_J,axiom,
ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% less_minus_one_simps(1)
thf(fact_5268_not__one__le__neg__numeral,axiom,
! [M2: num] :
~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) ) ).
% not_one_le_neg_numeral
thf(fact_5269_not__one__le__neg__numeral,axiom,
! [M2: num] :
~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) ).
% not_one_le_neg_numeral
thf(fact_5270_not__one__le__neg__numeral,axiom,
! [M2: num] :
~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) ).
% not_one_le_neg_numeral
thf(fact_5271_not__one__le__neg__numeral,axiom,
! [M2: num] :
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).
% not_one_le_neg_numeral
thf(fact_5272_not__numeral__le__neg__one,axiom,
! [M2: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% not_numeral_le_neg_one
thf(fact_5273_not__numeral__le__neg__one,axiom,
! [M2: num] :
~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% not_numeral_le_neg_one
thf(fact_5274_not__numeral__le__neg__one,axiom,
! [M2: num] :
~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% not_numeral_le_neg_one
thf(fact_5275_not__numeral__le__neg__one,axiom,
! [M2: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_le_neg_one
thf(fact_5276_neg__numeral__le__neg__one,axiom,
! [M2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% neg_numeral_le_neg_one
thf(fact_5277_neg__numeral__le__neg__one,axiom,
! [M2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% neg_numeral_le_neg_one
thf(fact_5278_neg__numeral__le__neg__one,axiom,
! [M2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% neg_numeral_le_neg_one
thf(fact_5279_neg__numeral__le__neg__one,axiom,
! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% neg_numeral_le_neg_one
thf(fact_5280_neg__one__le__numeral,axiom,
! [M2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M2 ) ) ).
% neg_one_le_numeral
thf(fact_5281_neg__one__le__numeral,axiom,
! [M2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M2 ) ) ).
% neg_one_le_numeral
thf(fact_5282_neg__one__le__numeral,axiom,
! [M2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M2 ) ) ).
% neg_one_le_numeral
thf(fact_5283_neg__one__le__numeral,axiom,
! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M2 ) ) ).
% neg_one_le_numeral
thf(fact_5284_neg__numeral__le__one,axiom,
! [M2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ one_one_real ) ).
% neg_numeral_le_one
thf(fact_5285_neg__numeral__le__one,axiom,
! [M2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ one_one_Code_integer ) ).
% neg_numeral_le_one
thf(fact_5286_neg__numeral__le__one,axiom,
! [M2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ one_one_rat ) ).
% neg_numeral_le_one
thf(fact_5287_neg__numeral__le__one,axiom,
! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) ).
% neg_numeral_le_one
thf(fact_5288_nonzero__neg__divide__eq__eq2,axiom,
! [B: real,C2: real,A: real] :
( ( B != zero_zero_real )
=> ( ( C2
= ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
= ( ( times_times_real @ C2 @ B )
= ( uminus_uminus_real @ A ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_5289_nonzero__neg__divide__eq__eq2,axiom,
! [B: complex,C2: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( C2
= ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
= ( ( times_times_complex @ C2 @ B )
= ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_5290_nonzero__neg__divide__eq__eq2,axiom,
! [B: rat,C2: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( C2
= ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
= ( ( times_times_rat @ C2 @ B )
= ( uminus_uminus_rat @ A ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_5291_nonzero__neg__divide__eq__eq,axiom,
! [B: real,A: real,C2: real] :
( ( B != zero_zero_real )
=> ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= C2 )
= ( ( uminus_uminus_real @ A )
= ( times_times_real @ C2 @ B ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_5292_nonzero__neg__divide__eq__eq,axiom,
! [B: complex,A: complex,C2: complex] :
( ( B != zero_zero_complex )
=> ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= C2 )
= ( ( uminus1482373934393186551omplex @ A )
= ( times_times_complex @ C2 @ B ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_5293_nonzero__neg__divide__eq__eq,axiom,
! [B: rat,A: rat,C2: rat] :
( ( B != zero_zero_rat )
=> ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
= C2 )
= ( ( uminus_uminus_rat @ A )
= ( times_times_rat @ C2 @ B ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_5294_minus__divide__eq__eq,axiom,
! [B: real,C2: real,A: real] :
( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) )
= A )
= ( ( ( C2 != zero_zero_real )
=> ( ( uminus_uminus_real @ B )
= ( times_times_real @ A @ C2 ) ) )
& ( ( C2 = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_5295_minus__divide__eq__eq,axiom,
! [B: complex,C2: complex,A: complex] :
( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C2 ) )
= A )
= ( ( ( C2 != zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ B )
= ( times_times_complex @ A @ C2 ) ) )
& ( ( C2 = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_5296_minus__divide__eq__eq,axiom,
! [B: rat,C2: rat,A: rat] :
( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) )
= A )
= ( ( ( C2 != zero_zero_rat )
=> ( ( uminus_uminus_rat @ B )
= ( times_times_rat @ A @ C2 ) ) )
& ( ( C2 = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_5297_eq__minus__divide__eq,axiom,
! [A: real,B: real,C2: real] :
( ( A
= ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) )
= ( ( ( C2 != zero_zero_real )
=> ( ( times_times_real @ A @ C2 )
= ( uminus_uminus_real @ B ) ) )
& ( ( C2 = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_5298_eq__minus__divide__eq,axiom,
! [A: complex,B: complex,C2: complex] :
( ( A
= ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C2 ) ) )
= ( ( ( C2 != zero_zero_complex )
=> ( ( times_times_complex @ A @ C2 )
= ( uminus1482373934393186551omplex @ B ) ) )
& ( ( C2 = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_5299_eq__minus__divide__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( A
= ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
= ( ( ( C2 != zero_zero_rat )
=> ( ( times_times_rat @ A @ C2 )
= ( uminus_uminus_rat @ B ) ) )
& ( ( C2 = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_5300_divide__eq__minus__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= ( uminus_uminus_real @ one_one_real ) )
= ( ( B != zero_zero_real )
& ( A
= ( uminus_uminus_real @ B ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_5301_divide__eq__minus__1__iff,axiom,
! [A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ B )
= ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( ( B != zero_zero_complex )
& ( A
= ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_5302_divide__eq__minus__1__iff,axiom,
! [A: rat,B: rat] :
( ( ( divide_divide_rat @ A @ B )
= ( uminus_uminus_rat @ one_one_rat ) )
= ( ( B != zero_zero_rat )
& ( A
= ( uminus_uminus_rat @ B ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_5303_sum__nonneg__0,axiom,
! [S3: set_real,F: real > real,I: real] :
( ( finite_finite_real @ S3 )
=> ( ! [I2: real] :
( ( member_real @ I2 @ S3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( ( groups8097168146408367636l_real @ F @ S3 )
= zero_zero_real )
=> ( ( member_real @ I @ S3 )
=> ( ( F @ I )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5304_sum__nonneg__0,axiom,
! [S3: set_int,F: int > real,I: int] :
( ( finite_finite_int @ S3 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ S3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( ( groups8778361861064173332t_real @ F @ S3 )
= zero_zero_real )
=> ( ( member_int @ I @ S3 )
=> ( ( F @ I )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5305_sum__nonneg__0,axiom,
! [S3: set_complex,F: complex > real,I: complex] :
( ( finite3207457112153483333omplex @ S3 )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ S3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( ( groups5808333547571424918x_real @ F @ S3 )
= zero_zero_real )
=> ( ( member_complex @ I @ S3 )
=> ( ( F @ I )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5306_sum__nonneg__0,axiom,
! [S3: set_real,F: real > rat,I: real] :
( ( finite_finite_real @ S3 )
=> ( ! [I2: real] :
( ( member_real @ I2 @ S3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ( ( groups1300246762558778688al_rat @ F @ S3 )
= zero_zero_rat )
=> ( ( member_real @ I @ S3 )
=> ( ( F @ I )
= zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5307_sum__nonneg__0,axiom,
! [S3: set_nat,F: nat > rat,I: nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ S3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ( ( groups2906978787729119204at_rat @ F @ S3 )
= zero_zero_rat )
=> ( ( member_nat @ I @ S3 )
=> ( ( F @ I )
= zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5308_sum__nonneg__0,axiom,
! [S3: set_int,F: int > rat,I: int] :
( ( finite_finite_int @ S3 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ S3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ( ( groups3906332499630173760nt_rat @ F @ S3 )
= zero_zero_rat )
=> ( ( member_int @ I @ S3 )
=> ( ( F @ I )
= zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5309_sum__nonneg__0,axiom,
! [S3: set_complex,F: complex > rat,I: complex] :
( ( finite3207457112153483333omplex @ S3 )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ S3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ( ( groups5058264527183730370ex_rat @ F @ S3 )
= zero_zero_rat )
=> ( ( member_complex @ I @ S3 )
=> ( ( F @ I )
= zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5310_sum__nonneg__0,axiom,
! [S3: set_real,F: real > nat,I: real] :
( ( finite_finite_real @ S3 )
=> ( ! [I2: real] :
( ( member_real @ I2 @ S3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ( ( groups1935376822645274424al_nat @ F @ S3 )
= zero_zero_nat )
=> ( ( member_real @ I @ S3 )
=> ( ( F @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5311_sum__nonneg__0,axiom,
! [S3: set_int,F: int > nat,I: int] :
( ( finite_finite_int @ S3 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ S3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F @ S3 )
= zero_zero_nat )
=> ( ( member_int @ I @ S3 )
=> ( ( F @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5312_sum__nonneg__0,axiom,
! [S3: set_complex,F: complex > nat,I: complex] :
( ( finite3207457112153483333omplex @ S3 )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ S3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ( ( groups5693394587270226106ex_nat @ F @ S3 )
= zero_zero_nat )
=> ( ( member_complex @ I @ S3 )
=> ( ( F @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5313_sum__nonneg__leq__bound,axiom,
! [S3: set_real,F: real > real,B5: real,I: real] :
( ( finite_finite_real @ S3 )
=> ( ! [I2: real] :
( ( member_real @ I2 @ S3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( ( groups8097168146408367636l_real @ F @ S3 )
= B5 )
=> ( ( member_real @ I @ S3 )
=> ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5314_sum__nonneg__leq__bound,axiom,
! [S3: set_int,F: int > real,B5: real,I: int] :
( ( finite_finite_int @ S3 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ S3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( ( groups8778361861064173332t_real @ F @ S3 )
= B5 )
=> ( ( member_int @ I @ S3 )
=> ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5315_sum__nonneg__leq__bound,axiom,
! [S3: set_complex,F: complex > real,B5: real,I: complex] :
( ( finite3207457112153483333omplex @ S3 )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ S3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( ( groups5808333547571424918x_real @ F @ S3 )
= B5 )
=> ( ( member_complex @ I @ S3 )
=> ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5316_sum__nonneg__leq__bound,axiom,
! [S3: set_real,F: real > rat,B5: rat,I: real] :
( ( finite_finite_real @ S3 )
=> ( ! [I2: real] :
( ( member_real @ I2 @ S3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ( ( groups1300246762558778688al_rat @ F @ S3 )
= B5 )
=> ( ( member_real @ I @ S3 )
=> ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5317_sum__nonneg__leq__bound,axiom,
! [S3: set_nat,F: nat > rat,B5: rat,I: nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ S3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ( ( groups2906978787729119204at_rat @ F @ S3 )
= B5 )
=> ( ( member_nat @ I @ S3 )
=> ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5318_sum__nonneg__leq__bound,axiom,
! [S3: set_int,F: int > rat,B5: rat,I: int] :
( ( finite_finite_int @ S3 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ S3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ( ( groups3906332499630173760nt_rat @ F @ S3 )
= B5 )
=> ( ( member_int @ I @ S3 )
=> ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5319_sum__nonneg__leq__bound,axiom,
! [S3: set_complex,F: complex > rat,B5: rat,I: complex] :
( ( finite3207457112153483333omplex @ S3 )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ S3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ( ( groups5058264527183730370ex_rat @ F @ S3 )
= B5 )
=> ( ( member_complex @ I @ S3 )
=> ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5320_sum__nonneg__leq__bound,axiom,
! [S3: set_real,F: real > nat,B5: nat,I: real] :
( ( finite_finite_real @ S3 )
=> ( ! [I2: real] :
( ( member_real @ I2 @ S3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ( ( groups1935376822645274424al_nat @ F @ S3 )
= B5 )
=> ( ( member_real @ I @ S3 )
=> ( ord_less_eq_nat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5321_sum__nonneg__leq__bound,axiom,
! [S3: set_int,F: int > nat,B5: nat,I: int] :
( ( finite_finite_int @ S3 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ S3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F @ S3 )
= B5 )
=> ( ( member_int @ I @ S3 )
=> ( ord_less_eq_nat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5322_sum__nonneg__leq__bound,axiom,
! [S3: set_complex,F: complex > nat,B5: nat,I: complex] :
( ( finite3207457112153483333omplex @ S3 )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ S3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ( ( groups5693394587270226106ex_nat @ F @ S3 )
= B5 )
=> ( ( member_complex @ I @ S3 )
=> ( ord_less_eq_nat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5323_sum_Osetdiff__irrelevant,axiom,
! [A4: set_real,G: real > complex] :
( ( finite_finite_real @ A4 )
=> ( ( groups5754745047067104278omplex @ G
@ ( minus_minus_set_real @ A4
@ ( collect_real
@ ^ [X5: real] :
( ( G @ X5 )
= zero_zero_complex ) ) ) )
= ( groups5754745047067104278omplex @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_5324_sum_Osetdiff__irrelevant,axiom,
! [A4: set_int,G: int > complex] :
( ( finite_finite_int @ A4 )
=> ( ( groups3049146728041665814omplex @ G
@ ( minus_minus_set_int @ A4
@ ( collect_int
@ ^ [X5: int] :
( ( G @ X5 )
= zero_zero_complex ) ) ) )
= ( groups3049146728041665814omplex @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_5325_sum_Osetdiff__irrelevant,axiom,
! [A4: set_real,G: real > real] :
( ( finite_finite_real @ A4 )
=> ( ( groups8097168146408367636l_real @ G
@ ( minus_minus_set_real @ A4
@ ( collect_real
@ ^ [X5: real] :
( ( G @ X5 )
= zero_zero_real ) ) ) )
= ( groups8097168146408367636l_real @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_5326_sum_Osetdiff__irrelevant,axiom,
! [A4: set_int,G: int > real] :
( ( finite_finite_int @ A4 )
=> ( ( groups8778361861064173332t_real @ G
@ ( minus_minus_set_int @ A4
@ ( collect_int
@ ^ [X5: int] :
( ( G @ X5 )
= zero_zero_real ) ) ) )
= ( groups8778361861064173332t_real @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_5327_sum_Osetdiff__irrelevant,axiom,
! [A4: set_complex,G: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5808333547571424918x_real @ G
@ ( minus_811609699411566653omplex @ A4
@ ( collect_complex
@ ^ [X5: complex] :
( ( G @ X5 )
= zero_zero_real ) ) ) )
= ( groups5808333547571424918x_real @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_5328_sum_Osetdiff__irrelevant,axiom,
! [A4: set_real,G: real > rat] :
( ( finite_finite_real @ A4 )
=> ( ( groups1300246762558778688al_rat @ G
@ ( minus_minus_set_real @ A4
@ ( collect_real
@ ^ [X5: real] :
( ( G @ X5 )
= zero_zero_rat ) ) ) )
= ( groups1300246762558778688al_rat @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_5329_sum_Osetdiff__irrelevant,axiom,
! [A4: set_int,G: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ( groups3906332499630173760nt_rat @ G
@ ( minus_minus_set_int @ A4
@ ( collect_int
@ ^ [X5: int] :
( ( G @ X5 )
= zero_zero_rat ) ) ) )
= ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_5330_sum_Osetdiff__irrelevant,axiom,
! [A4: set_complex,G: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5058264527183730370ex_rat @ G
@ ( minus_811609699411566653omplex @ A4
@ ( collect_complex
@ ^ [X5: complex] :
( ( G @ X5 )
= zero_zero_rat ) ) ) )
= ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_5331_sum_Osetdiff__irrelevant,axiom,
! [A4: set_real,G: real > nat] :
( ( finite_finite_real @ A4 )
=> ( ( groups1935376822645274424al_nat @ G
@ ( minus_minus_set_real @ A4
@ ( collect_real
@ ^ [X5: real] :
( ( G @ X5 )
= zero_zero_nat ) ) ) )
= ( groups1935376822645274424al_nat @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_5332_sum_Osetdiff__irrelevant,axiom,
! [A4: set_int,G: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( groups4541462559716669496nt_nat @ G
@ ( minus_minus_set_int @ A4
@ ( collect_int
@ ^ [X5: int] :
( ( G @ X5 )
= zero_zero_nat ) ) ) )
= ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_5333_real__0__less__add__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X3 @ Y ) )
= ( ord_less_real @ ( uminus_uminus_real @ X3 ) @ Y ) ) ).
% real_0_less_add_iff
thf(fact_5334_real__add__less__0__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X3 @ Y ) @ zero_zero_real )
= ( ord_less_real @ Y @ ( uminus_uminus_real @ X3 ) ) ) ).
% real_add_less_0_iff
thf(fact_5335_real__add__le__0__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X3 @ Y ) @ zero_zero_real )
= ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X3 ) ) ) ).
% real_add_le_0_iff
thf(fact_5336_real__0__le__add__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X3 @ Y ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X3 ) @ Y ) ) ).
% real_0_le_add_iff
thf(fact_5337_zmod__zminus1__eq__if,axiom,
! [A: int,B: int] :
( ( ( ( modulo_modulo_int @ A @ B )
= zero_zero_int )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
= zero_zero_int ) )
& ( ( ( modulo_modulo_int @ A @ B )
!= zero_zero_int )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% zmod_zminus1_eq_if
thf(fact_5338_zmod__zminus2__eq__if,axiom,
! [A: int,B: int] :
( ( ( ( modulo_modulo_int @ A @ B )
= zero_zero_int )
=> ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
= zero_zero_int ) )
& ( ( ( modulo_modulo_int @ A @ B )
!= zero_zero_int )
=> ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
= ( minus_minus_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ) ).
% zmod_zminus2_eq_if
thf(fact_5339_sum__pos2,axiom,
! [I6: set_real,I: real,F: real > real] :
( ( finite_finite_real @ I6 )
=> ( ( member_real @ I @ I6 )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I6 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I6 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5340_sum__pos2,axiom,
! [I6: set_int,I: int,F: int > real] :
( ( finite_finite_int @ I6 )
=> ( ( member_int @ I @ I6 )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I6 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I6 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5341_sum__pos2,axiom,
! [I6: set_complex,I: complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ( member_complex @ I @ I6 )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I6 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I6 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5342_sum__pos2,axiom,
! [I6: set_real,I: real,F: real > rat] :
( ( finite_finite_real @ I6 )
=> ( ( member_real @ I @ I6 )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I6 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I6 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5343_sum__pos2,axiom,
! [I6: set_nat,I: nat,F: nat > rat] :
( ( finite_finite_nat @ I6 )
=> ( ( member_nat @ I @ I6 )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I6 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I6 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5344_sum__pos2,axiom,
! [I6: set_int,I: int,F: int > rat] :
( ( finite_finite_int @ I6 )
=> ( ( member_int @ I @ I6 )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I6 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I6 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5345_sum__pos2,axiom,
! [I6: set_complex,I: complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ( member_complex @ I @ I6 )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I6 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I6 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5346_sum__pos2,axiom,
! [I6: set_real,I: real,F: real > nat] :
( ( finite_finite_real @ I6 )
=> ( ( member_real @ I @ I6 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I6 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I6 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5347_sum__pos2,axiom,
! [I6: set_int,I: int,F: int > nat] :
( ( finite_finite_int @ I6 )
=> ( ( member_int @ I @ I6 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I6 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I6 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5348_sum__pos2,axiom,
! [I6: set_complex,I: complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ( member_complex @ I @ I6 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I6 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I6 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5349_sum__pos,axiom,
! [I6: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ( I6 != bot_bot_set_complex )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I6 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I6 ) ) ) ) ) ).
% sum_pos
thf(fact_5350_sum__pos,axiom,
! [I6: set_real,F: real > real] :
( ( finite_finite_real @ I6 )
=> ( ( I6 != bot_bot_set_real )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I6 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I6 ) ) ) ) ) ).
% sum_pos
thf(fact_5351_sum__pos,axiom,
! [I6: set_int,F: int > real] :
( ( finite_finite_int @ I6 )
=> ( ( I6 != bot_bot_set_int )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I6 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I6 ) ) ) ) ) ).
% sum_pos
thf(fact_5352_sum__pos,axiom,
! [I6: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ( I6 != bot_bot_set_complex )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I6 )
=> ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I6 ) ) ) ) ) ).
% sum_pos
thf(fact_5353_sum__pos,axiom,
! [I6: set_real,F: real > rat] :
( ( finite_finite_real @ I6 )
=> ( ( I6 != bot_bot_set_real )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I6 )
=> ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I6 ) ) ) ) ) ).
% sum_pos
thf(fact_5354_sum__pos,axiom,
! [I6: set_nat,F: nat > rat] :
( ( finite_finite_nat @ I6 )
=> ( ( I6 != bot_bot_set_nat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I6 )
=> ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I6 ) ) ) ) ) ).
% sum_pos
thf(fact_5355_sum__pos,axiom,
! [I6: set_int,F: int > rat] :
( ( finite_finite_int @ I6 )
=> ( ( I6 != bot_bot_set_int )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I6 )
=> ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I6 ) ) ) ) ) ).
% sum_pos
thf(fact_5356_sum__pos,axiom,
! [I6: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ( I6 != bot_bot_set_complex )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I6 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I6 ) ) ) ) ) ).
% sum_pos
thf(fact_5357_sum__pos,axiom,
! [I6: set_real,F: real > nat] :
( ( finite_finite_real @ I6 )
=> ( ( I6 != bot_bot_set_real )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I6 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I6 ) ) ) ) ) ).
% sum_pos
thf(fact_5358_sum__pos,axiom,
! [I6: set_int,F: int > nat] :
( ( finite_finite_int @ I6 )
=> ( ( I6 != bot_bot_set_int )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I6 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I6 ) ) ) ) ) ).
% sum_pos
thf(fact_5359_ln__ge__zero__imp__ge__one,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_5360_sum_Omono__neutral__cong__right,axiom,
! [T2: set_real,S2: set_real,G: real > complex,H2: real > complex] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_complex ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups5754745047067104278omplex @ G @ T2 )
= ( groups5754745047067104278omplex @ H2 @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_5361_sum_Omono__neutral__cong__right,axiom,
! [T2: set_real,S2: set_real,G: real > real,H2: real > real] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_real ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ T2 )
= ( groups8097168146408367636l_real @ H2 @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_5362_sum_Omono__neutral__cong__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > real,H2: complex > real] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_real ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups5808333547571424918x_real @ G @ T2 )
= ( groups5808333547571424918x_real @ H2 @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_5363_sum_Omono__neutral__cong__right,axiom,
! [T2: set_real,S2: set_real,G: real > rat,H2: real > rat] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_rat ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups1300246762558778688al_rat @ G @ T2 )
= ( groups1300246762558778688al_rat @ H2 @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_5364_sum_Omono__neutral__cong__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > rat,H2: complex > rat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_rat ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups5058264527183730370ex_rat @ G @ T2 )
= ( groups5058264527183730370ex_rat @ H2 @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_5365_sum_Omono__neutral__cong__right,axiom,
! [T2: set_real,S2: set_real,G: real > nat,H2: real > nat] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_nat ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups1935376822645274424al_nat @ G @ T2 )
= ( groups1935376822645274424al_nat @ H2 @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_5366_sum_Omono__neutral__cong__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > nat,H2: complex > nat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_nat ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups5693394587270226106ex_nat @ G @ T2 )
= ( groups5693394587270226106ex_nat @ H2 @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_5367_sum_Omono__neutral__cong__right,axiom,
! [T2: set_real,S2: set_real,G: real > int,H2: real > int] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_int ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups1932886352136224148al_int @ G @ T2 )
= ( groups1932886352136224148al_int @ H2 @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_5368_sum_Omono__neutral__cong__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > int,H2: complex > int] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_int ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups5690904116761175830ex_int @ G @ T2 )
= ( groups5690904116761175830ex_int @ H2 @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_5369_sum_Omono__neutral__cong__right,axiom,
! [T2: set_nat,S2: set_nat,G: nat > complex,H2: nat > complex] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_complex ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups2073611262835488442omplex @ G @ T2 )
= ( groups2073611262835488442omplex @ H2 @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_5370_sum_Omono__neutral__cong__left,axiom,
! [T2: set_real,S2: set_real,H2: real > complex,G: real > complex] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( H2 @ X4 )
= zero_zero_complex ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups5754745047067104278omplex @ G @ S2 )
= ( groups5754745047067104278omplex @ H2 @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_5371_sum_Omono__neutral__cong__left,axiom,
! [T2: set_real,S2: set_real,H2: real > real,G: real > real] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( H2 @ X4 )
= zero_zero_real ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S2 )
= ( groups8097168146408367636l_real @ H2 @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_5372_sum_Omono__neutral__cong__left,axiom,
! [T2: set_complex,S2: set_complex,H2: complex > real,G: complex > real] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( H2 @ X4 )
= zero_zero_real ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups5808333547571424918x_real @ G @ S2 )
= ( groups5808333547571424918x_real @ H2 @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_5373_sum_Omono__neutral__cong__left,axiom,
! [T2: set_real,S2: set_real,H2: real > rat,G: real > rat] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( H2 @ X4 )
= zero_zero_rat ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups1300246762558778688al_rat @ G @ S2 )
= ( groups1300246762558778688al_rat @ H2 @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_5374_sum_Omono__neutral__cong__left,axiom,
! [T2: set_complex,S2: set_complex,H2: complex > rat,G: complex > rat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( H2 @ X4 )
= zero_zero_rat ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups5058264527183730370ex_rat @ G @ S2 )
= ( groups5058264527183730370ex_rat @ H2 @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_5375_sum_Omono__neutral__cong__left,axiom,
! [T2: set_real,S2: set_real,H2: real > nat,G: real > nat] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( H2 @ X4 )
= zero_zero_nat ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups1935376822645274424al_nat @ G @ S2 )
= ( groups1935376822645274424al_nat @ H2 @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_5376_sum_Omono__neutral__cong__left,axiom,
! [T2: set_complex,S2: set_complex,H2: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( H2 @ X4 )
= zero_zero_nat ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups5693394587270226106ex_nat @ G @ S2 )
= ( groups5693394587270226106ex_nat @ H2 @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_5377_sum_Omono__neutral__cong__left,axiom,
! [T2: set_real,S2: set_real,H2: real > int,G: real > int] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( H2 @ X4 )
= zero_zero_int ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups1932886352136224148al_int @ G @ S2 )
= ( groups1932886352136224148al_int @ H2 @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_5378_sum_Omono__neutral__cong__left,axiom,
! [T2: set_complex,S2: set_complex,H2: complex > int,G: complex > int] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( H2 @ X4 )
= zero_zero_int ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups5690904116761175830ex_int @ G @ S2 )
= ( groups5690904116761175830ex_int @ H2 @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_5379_sum_Omono__neutral__cong__left,axiom,
! [T2: set_nat,S2: set_nat,H2: nat > complex,G: nat > complex] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( H2 @ X4 )
= zero_zero_complex ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups2073611262835488442omplex @ G @ S2 )
= ( groups2073611262835488442omplex @ H2 @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_5380_sum_Omono__neutral__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > real] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_real ) )
=> ( ( groups5808333547571424918x_real @ G @ T2 )
= ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_5381_sum_Omono__neutral__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > rat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_rat ) )
=> ( ( groups5058264527183730370ex_rat @ G @ T2 )
= ( groups5058264527183730370ex_rat @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_5382_sum_Omono__neutral__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > nat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_nat ) )
=> ( ( groups5693394587270226106ex_nat @ G @ T2 )
= ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_5383_sum_Omono__neutral__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > int] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_int ) )
=> ( ( groups5690904116761175830ex_int @ G @ T2 )
= ( groups5690904116761175830ex_int @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_5384_sum_Omono__neutral__right,axiom,
! [T2: set_nat,S2: set_nat,G: nat > complex] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_complex ) )
=> ( ( groups2073611262835488442omplex @ G @ T2 )
= ( groups2073611262835488442omplex @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_5385_sum_Omono__neutral__right,axiom,
! [T2: set_nat,S2: set_nat,G: nat > rat] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_rat ) )
=> ( ( groups2906978787729119204at_rat @ G @ T2 )
= ( groups2906978787729119204at_rat @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_5386_sum_Omono__neutral__right,axiom,
! [T2: set_nat,S2: set_nat,G: nat > int] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_int ) )
=> ( ( groups3539618377306564664at_int @ G @ T2 )
= ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_5387_sum_Omono__neutral__right,axiom,
! [T2: set_int,S2: set_int,G: int > complex] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S2 @ T2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_complex ) )
=> ( ( groups3049146728041665814omplex @ G @ T2 )
= ( groups3049146728041665814omplex @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_5388_sum_Omono__neutral__right,axiom,
! [T2: set_int,S2: set_int,G: int > real] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S2 @ T2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_real ) )
=> ( ( groups8778361861064173332t_real @ G @ T2 )
= ( groups8778361861064173332t_real @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_5389_sum_Omono__neutral__right,axiom,
! [T2: set_int,S2: set_int,G: int > rat] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S2 @ T2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_rat ) )
=> ( ( groups3906332499630173760nt_rat @ G @ T2 )
= ( groups3906332499630173760nt_rat @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_5390_sum_Omono__neutral__left,axiom,
! [T2: set_complex,S2: set_complex,G: complex > real] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_real ) )
=> ( ( groups5808333547571424918x_real @ G @ S2 )
= ( groups5808333547571424918x_real @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_5391_sum_Omono__neutral__left,axiom,
! [T2: set_complex,S2: set_complex,G: complex > rat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_rat ) )
=> ( ( groups5058264527183730370ex_rat @ G @ S2 )
= ( groups5058264527183730370ex_rat @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_5392_sum_Omono__neutral__left,axiom,
! [T2: set_complex,S2: set_complex,G: complex > nat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_nat ) )
=> ( ( groups5693394587270226106ex_nat @ G @ S2 )
= ( groups5693394587270226106ex_nat @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_5393_sum_Omono__neutral__left,axiom,
! [T2: set_complex,S2: set_complex,G: complex > int] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_int ) )
=> ( ( groups5690904116761175830ex_int @ G @ S2 )
= ( groups5690904116761175830ex_int @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_5394_sum_Omono__neutral__left,axiom,
! [T2: set_nat,S2: set_nat,G: nat > complex] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_complex ) )
=> ( ( groups2073611262835488442omplex @ G @ S2 )
= ( groups2073611262835488442omplex @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_5395_sum_Omono__neutral__left,axiom,
! [T2: set_nat,S2: set_nat,G: nat > rat] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_rat ) )
=> ( ( groups2906978787729119204at_rat @ G @ S2 )
= ( groups2906978787729119204at_rat @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_5396_sum_Omono__neutral__left,axiom,
! [T2: set_nat,S2: set_nat,G: nat > int] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_int ) )
=> ( ( groups3539618377306564664at_int @ G @ S2 )
= ( groups3539618377306564664at_int @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_5397_sum_Omono__neutral__left,axiom,
! [T2: set_int,S2: set_int,G: int > complex] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S2 @ T2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_complex ) )
=> ( ( groups3049146728041665814omplex @ G @ S2 )
= ( groups3049146728041665814omplex @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_5398_sum_Omono__neutral__left,axiom,
! [T2: set_int,S2: set_int,G: int > real] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S2 @ T2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_real ) )
=> ( ( groups8778361861064173332t_real @ G @ S2 )
= ( groups8778361861064173332t_real @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_5399_sum_Omono__neutral__left,axiom,
! [T2: set_int,S2: set_int,G: int > rat] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S2 @ T2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T2 @ S2 ) )
=> ( ( G @ X4 )
= zero_zero_rat ) )
=> ( ( groups3906332499630173760nt_rat @ G @ S2 )
= ( groups3906332499630173760nt_rat @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_5400_sum_Osame__carrierI,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > complex,H2: real > complex] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_complex ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_complex ) )
=> ( ( ( groups5754745047067104278omplex @ G @ C4 )
= ( groups5754745047067104278omplex @ H2 @ C4 ) )
=> ( ( groups5754745047067104278omplex @ G @ A4 )
= ( groups5754745047067104278omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_5401_sum_Osame__carrierI,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > real,H2: real > real] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_real ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_real ) )
=> ( ( ( groups8097168146408367636l_real @ G @ C4 )
= ( groups8097168146408367636l_real @ H2 @ C4 ) )
=> ( ( groups8097168146408367636l_real @ G @ A4 )
= ( groups8097168146408367636l_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_5402_sum_Osame__carrierI,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_real ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_real ) )
=> ( ( ( groups5808333547571424918x_real @ G @ C4 )
= ( groups5808333547571424918x_real @ H2 @ C4 ) )
=> ( ( groups5808333547571424918x_real @ G @ A4 )
= ( groups5808333547571424918x_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_5403_sum_Osame__carrierI,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > rat,H2: real > rat] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_rat ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_rat ) )
=> ( ( ( groups1300246762558778688al_rat @ G @ C4 )
= ( groups1300246762558778688al_rat @ H2 @ C4 ) )
=> ( ( groups1300246762558778688al_rat @ G @ A4 )
= ( groups1300246762558778688al_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_5404_sum_Osame__carrierI,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > rat,H2: complex > rat] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_rat ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_rat ) )
=> ( ( ( groups5058264527183730370ex_rat @ G @ C4 )
= ( groups5058264527183730370ex_rat @ H2 @ C4 ) )
=> ( ( groups5058264527183730370ex_rat @ G @ A4 )
= ( groups5058264527183730370ex_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_5405_sum_Osame__carrierI,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > nat,H2: real > nat] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups1935376822645274424al_nat @ G @ C4 )
= ( groups1935376822645274424al_nat @ H2 @ C4 ) )
=> ( ( groups1935376822645274424al_nat @ G @ A4 )
= ( groups1935376822645274424al_nat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_5406_sum_Osame__carrierI,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > nat,H2: complex > nat] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups5693394587270226106ex_nat @ G @ C4 )
= ( groups5693394587270226106ex_nat @ H2 @ C4 ) )
=> ( ( groups5693394587270226106ex_nat @ G @ A4 )
= ( groups5693394587270226106ex_nat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_5407_sum_Osame__carrierI,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > int,H2: real > int] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_int ) )
=> ( ( ( groups1932886352136224148al_int @ G @ C4 )
= ( groups1932886352136224148al_int @ H2 @ C4 ) )
=> ( ( groups1932886352136224148al_int @ G @ A4 )
= ( groups1932886352136224148al_int @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_5408_sum_Osame__carrierI,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > int,H2: complex > int] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_int ) )
=> ( ( ( groups5690904116761175830ex_int @ G @ C4 )
= ( groups5690904116761175830ex_int @ H2 @ C4 ) )
=> ( ( groups5690904116761175830ex_int @ G @ A4 )
= ( groups5690904116761175830ex_int @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_5409_sum_Osame__carrierI,axiom,
! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > complex,H2: nat > complex] :
( ( finite_finite_nat @ C4 )
=> ( ( ord_less_eq_set_nat @ A4 @ C4 )
=> ( ( ord_less_eq_set_nat @ B5 @ C4 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_complex ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_complex ) )
=> ( ( ( groups2073611262835488442omplex @ G @ C4 )
= ( groups2073611262835488442omplex @ H2 @ C4 ) )
=> ( ( groups2073611262835488442omplex @ G @ A4 )
= ( groups2073611262835488442omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_5410_sum_Osame__carrier,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > complex,H2: real > complex] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_complex ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_complex ) )
=> ( ( ( groups5754745047067104278omplex @ G @ A4 )
= ( groups5754745047067104278omplex @ H2 @ B5 ) )
= ( ( groups5754745047067104278omplex @ G @ C4 )
= ( groups5754745047067104278omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_5411_sum_Osame__carrier,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > real,H2: real > real] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_real ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_real ) )
=> ( ( ( groups8097168146408367636l_real @ G @ A4 )
= ( groups8097168146408367636l_real @ H2 @ B5 ) )
= ( ( groups8097168146408367636l_real @ G @ C4 )
= ( groups8097168146408367636l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_5412_sum_Osame__carrier,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_real ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_real ) )
=> ( ( ( groups5808333547571424918x_real @ G @ A4 )
= ( groups5808333547571424918x_real @ H2 @ B5 ) )
= ( ( groups5808333547571424918x_real @ G @ C4 )
= ( groups5808333547571424918x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_5413_sum_Osame__carrier,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > rat,H2: real > rat] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_rat ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_rat ) )
=> ( ( ( groups1300246762558778688al_rat @ G @ A4 )
= ( groups1300246762558778688al_rat @ H2 @ B5 ) )
= ( ( groups1300246762558778688al_rat @ G @ C4 )
= ( groups1300246762558778688al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_5414_sum_Osame__carrier,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > rat,H2: complex > rat] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_rat ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_rat ) )
=> ( ( ( groups5058264527183730370ex_rat @ G @ A4 )
= ( groups5058264527183730370ex_rat @ H2 @ B5 ) )
= ( ( groups5058264527183730370ex_rat @ G @ C4 )
= ( groups5058264527183730370ex_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_5415_sum_Osame__carrier,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > nat,H2: real > nat] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups1935376822645274424al_nat @ G @ A4 )
= ( groups1935376822645274424al_nat @ H2 @ B5 ) )
= ( ( groups1935376822645274424al_nat @ G @ C4 )
= ( groups1935376822645274424al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_5416_sum_Osame__carrier,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > nat,H2: complex > nat] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups5693394587270226106ex_nat @ G @ A4 )
= ( groups5693394587270226106ex_nat @ H2 @ B5 ) )
= ( ( groups5693394587270226106ex_nat @ G @ C4 )
= ( groups5693394587270226106ex_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_5417_sum_Osame__carrier,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > int,H2: real > int] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_int ) )
=> ( ( ( groups1932886352136224148al_int @ G @ A4 )
= ( groups1932886352136224148al_int @ H2 @ B5 ) )
= ( ( groups1932886352136224148al_int @ G @ C4 )
= ( groups1932886352136224148al_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_5418_sum_Osame__carrier,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > int,H2: complex > int] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_int ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_int ) )
=> ( ( ( groups5690904116761175830ex_int @ G @ A4 )
= ( groups5690904116761175830ex_int @ H2 @ B5 ) )
= ( ( groups5690904116761175830ex_int @ G @ C4 )
= ( groups5690904116761175830ex_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_5419_sum_Osame__carrier,axiom,
! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > complex,H2: nat > complex] :
( ( finite_finite_nat @ C4 )
=> ( ( ord_less_eq_set_nat @ A4 @ C4 )
=> ( ( ord_less_eq_set_nat @ B5 @ C4 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
=> ( ( G @ A3 )
= zero_zero_complex ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= zero_zero_complex ) )
=> ( ( ( groups2073611262835488442omplex @ G @ A4 )
= ( groups2073611262835488442omplex @ H2 @ B5 ) )
= ( ( groups2073611262835488442omplex @ G @ C4 )
= ( groups2073611262835488442omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_5420_sum_Osubset__diff,axiom,
! [B5: set_complex,A4: set_complex,G: complex > real] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5808333547571424918x_real @ G @ A4 )
= ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups5808333547571424918x_real @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5421_sum_Osubset__diff,axiom,
! [B5: set_complex,A4: set_complex,G: complex > rat] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5058264527183730370ex_rat @ G @ A4 )
= ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups5058264527183730370ex_rat @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5422_sum_Osubset__diff,axiom,
! [B5: set_complex,A4: set_complex,G: complex > nat] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5693394587270226106ex_nat @ G @ A4 )
= ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups5693394587270226106ex_nat @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5423_sum_Osubset__diff,axiom,
! [B5: set_complex,A4: set_complex,G: complex > int] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups5690904116761175830ex_int @ G @ A4 )
= ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups5690904116761175830ex_int @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5424_sum_Osubset__diff,axiom,
! [B5: set_nat,A4: set_nat,G: nat > rat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( groups2906978787729119204at_rat @ G @ A4 )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups2906978787729119204at_rat @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5425_sum_Osubset__diff,axiom,
! [B5: set_nat,A4: set_nat,G: nat > int] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( groups3539618377306564664at_int @ G @ A4 )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups3539618377306564664at_int @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5426_sum_Osubset__diff,axiom,
! [B5: set_int,A4: set_int,G: int > real] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( finite_finite_int @ A4 )
=> ( ( groups8778361861064173332t_real @ G @ A4 )
= ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups8778361861064173332t_real @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5427_sum_Osubset__diff,axiom,
! [B5: set_int,A4: set_int,G: int > rat] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( finite_finite_int @ A4 )
=> ( ( groups3906332499630173760nt_rat @ G @ A4 )
= ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups3906332499630173760nt_rat @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5428_sum_Osubset__diff,axiom,
! [B5: set_int,A4: set_int,G: int > nat] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( finite_finite_int @ A4 )
=> ( ( groups4541462559716669496nt_nat @ G @ A4 )
= ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups4541462559716669496nt_nat @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5429_sum_Osubset__diff,axiom,
! [B5: set_int,A4: set_int,G: int > int] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( finite_finite_int @ A4 )
=> ( ( groups4538972089207619220nt_int @ G @ A4 )
= ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups4538972089207619220nt_int @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5430_sum__diff,axiom,
! [A4: set_complex,B5: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
= ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_5431_sum__diff,axiom,
! [A4: set_complex,B5: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
= ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_5432_sum__diff,axiom,
! [A4: set_complex,B5: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
= ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_5433_sum__diff,axiom,
! [A4: set_nat,B5: set_nat,F: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_5434_sum__diff,axiom,
! [A4: set_nat,B5: set_nat,F: nat > int] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_5435_sum__diff,axiom,
! [A4: set_int,B5: set_int,F: int > real] :
( ( finite_finite_int @ A4 )
=> ( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
= ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_5436_sum__diff,axiom,
! [A4: set_int,B5: set_int,F: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
= ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_5437_sum__diff,axiom,
! [A4: set_int,B5: set_int,F: int > int] :
( ( finite_finite_int @ A4 )
=> ( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
= ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_5438_sum__diff,axiom,
! [A4: set_complex,B5: set_complex,F: complex > complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( groups7754918857620584856omplex @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
= ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A4 ) @ ( groups7754918857620584856omplex @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_5439_sum__diff,axiom,
! [A4: set_nat,B5: set_nat,F: nat > real] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ ( groups6591440286371151544t_real @ F @ B5 ) ) ) ) ) ).
% sum_diff
thf(fact_5440_ln__add__one__self__le__self,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) @ X3 ) ) ).
% ln_add_one_self_le_self
thf(fact_5441_ln__mult,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ln_ln_real @ ( times_times_real @ X3 @ Y ) )
= ( plus_plus_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% ln_mult
thf(fact_5442_ln__eq__minus__one,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ( ln_ln_real @ X3 )
= ( minus_minus_real @ X3 @ one_one_real ) )
=> ( X3 = one_one_real ) ) ) ).
% ln_eq_minus_one
thf(fact_5443_ln__div,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ln_ln_real @ ( divide_divide_real @ X3 @ Y ) )
= ( minus_minus_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% ln_div
thf(fact_5444_pos__minus__divide__less__eq,axiom,
! [C2: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) @ A )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C2 ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_5445_pos__minus__divide__less__eq,axiom,
! [C2: rat,B: rat,A: rat] :
( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) @ A )
= ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_5446_pos__less__minus__divide__eq,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) )
= ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_5447_pos__less__minus__divide__eq,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
= ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_5448_neg__minus__divide__less__eq,axiom,
! [C2: real,B: real,A: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_5449_neg__minus__divide__less__eq,axiom,
! [C2: rat,B: rat,A: rat] :
( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) @ A )
= ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_5450_neg__less__minus__divide__eq,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C2 ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_5451_neg__less__minus__divide__eq,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
= ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_5452_minus__divide__less__eq,axiom,
! [B: real,C2: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C2 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( uminus_uminus_real @ B ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_5453_minus__divide__less__eq,axiom,
! [B: rat,C2: rat,A: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) @ A )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_5454_less__minus__divide__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( uminus_uminus_real @ B ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C2 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_5455_less__minus__divide__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_5456_eq__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: real,C2: real] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
= ( divide_divide_real @ B @ C2 ) )
= ( ( ( C2 != zero_zero_real )
=> ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 )
= B ) )
& ( ( C2 = zero_zero_real )
=> ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
= zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_5457_eq__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: complex,C2: complex] :
( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
= ( divide1717551699836669952omplex @ B @ C2 ) )
= ( ( ( C2 != zero_zero_complex )
=> ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C2 )
= B ) )
& ( ( C2 = zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
= zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_5458_eq__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: rat,C2: rat] :
( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
= ( divide_divide_rat @ B @ C2 ) )
= ( ( ( C2 != zero_zero_rat )
=> ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 )
= B ) )
& ( ( C2 = zero_zero_rat )
=> ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
= zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_5459_divide__eq__eq__numeral_I2_J,axiom,
! [B: real,C2: real,W2: num] :
( ( ( divide_divide_real @ B @ C2 )
= ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= ( ( ( C2 != zero_zero_real )
=> ( B
= ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) ) )
& ( ( C2 = zero_zero_real )
=> ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
= zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_5460_divide__eq__eq__numeral_I2_J,axiom,
! [B: complex,C2: complex,W2: num] :
( ( ( divide1717551699836669952omplex @ B @ C2 )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
= ( ( ( C2 != zero_zero_complex )
=> ( B
= ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C2 ) ) )
& ( ( C2 = zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
= zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_5461_divide__eq__eq__numeral_I2_J,axiom,
! [B: rat,C2: rat,W2: num] :
( ( ( divide_divide_rat @ B @ C2 )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= ( ( ( C2 != zero_zero_rat )
=> ( B
= ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) ) )
& ( ( C2 = zero_zero_rat )
=> ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
= zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_5462_add__divide__eq__if__simps_I3_J,axiom,
! [Z2: real,A: real,B: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
= B ) )
& ( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_5463_add__divide__eq__if__simps_I3_J,axiom,
! [Z2: complex,A: complex,B: complex] :
( ( ( Z2 = zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
= B ) )
& ( ( Z2 != zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_5464_add__divide__eq__if__simps_I3_J,axiom,
! [Z2: rat,A: rat,B: rat] :
( ( ( Z2 = zero_zero_rat )
=> ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
= B ) )
& ( ( Z2 != zero_zero_rat )
=> ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
= ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_5465_minus__divide__add__eq__iff,axiom,
! [Z2: real,X3: real,Y: real] :
( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X3 @ Z2 ) ) @ Y )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X3 ) @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_5466_minus__divide__add__eq__iff,axiom,
! [Z2: complex,X3: complex,Y: complex] :
( ( Z2 != zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X3 @ Z2 ) ) @ Y )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X3 ) @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_5467_minus__divide__add__eq__iff,axiom,
! [Z2: rat,X3: rat,Y: rat] :
( ( Z2 != zero_zero_rat )
=> ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X3 @ Z2 ) ) @ Y )
= ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X3 ) @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_5468_minus__divide__diff__eq__iff,axiom,
! [Z2: real,X3: real,Y: real] :
( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X3 @ Z2 ) ) @ Y )
= ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X3 ) @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_5469_minus__divide__diff__eq__iff,axiom,
! [Z2: complex,X3: complex,Y: complex] :
( ( Z2 != zero_zero_complex )
=> ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X3 @ Z2 ) ) @ Y )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X3 ) @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_5470_minus__divide__diff__eq__iff,axiom,
! [Z2: rat,X3: rat,Y: rat] :
( ( Z2 != zero_zero_rat )
=> ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X3 @ Z2 ) ) @ Y )
= ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X3 ) @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_5471_add__divide__eq__if__simps_I5_J,axiom,
! [Z2: real,A: real,B: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
= ( uminus_uminus_real @ B ) ) )
& ( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
= ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_5472_add__divide__eq__if__simps_I5_J,axiom,
! [Z2: complex,A: complex,B: complex] :
( ( ( Z2 = zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
= ( uminus1482373934393186551omplex @ B ) ) )
& ( ( Z2 != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_5473_add__divide__eq__if__simps_I5_J,axiom,
! [Z2: rat,A: rat,B: rat] :
( ( ( Z2 = zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
= ( uminus_uminus_rat @ B ) ) )
& ( ( Z2 != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
= ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_5474_add__divide__eq__if__simps_I6_J,axiom,
! [Z2: real,A: real,B: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
= ( uminus_uminus_real @ B ) ) )
& ( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
= ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_5475_add__divide__eq__if__simps_I6_J,axiom,
! [Z2: complex,A: complex,B: complex] :
( ( ( Z2 = zero_zero_complex )
=> ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
= ( uminus1482373934393186551omplex @ B ) ) )
& ( ( Z2 != zero_zero_complex )
=> ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_5476_add__divide__eq__if__simps_I6_J,axiom,
! [Z2: rat,A: rat,B: rat] :
( ( ( Z2 = zero_zero_rat )
=> ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
= ( uminus_uminus_rat @ B ) ) )
& ( ( Z2 != zero_zero_rat )
=> ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
= ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_5477_verit__less__mono__div__int2,axiom,
! [A4: int,B5: int,N: int] :
( ( ord_less_eq_int @ A4 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B5 @ N ) @ ( divide_divide_int @ A4 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_5478_div__eq__minus1,axiom,
! [B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% div_eq_minus1
thf(fact_5479_sum__mono2,axiom,
! [B5: set_real,A4: set_real,F: real > real] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5480_sum__mono2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
=> ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5481_sum__mono2,axiom,
! [B5: set_real,A4: set_real,F: real > rat] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
=> ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5482_sum__mono2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
=> ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5483_sum__mono2,axiom,
! [B5: set_nat,A4: set_nat,F: nat > rat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5484_sum__mono2,axiom,
! [B5: set_real,A4: set_real,F: real > nat] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5485_sum__mono2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
=> ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5486_sum__mono2,axiom,
! [B5: set_real,A4: set_real,F: real > int] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A4 ) @ ( groups1932886352136224148al_int @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5487_sum__mono2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
=> ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5488_sum__mono2,axiom,
! [B5: set_nat,A4: set_nat,F: nat > int] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5489_ln__le__minus__one,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( minus_minus_real @ X3 @ one_one_real ) ) ) ).
% ln_le_minus_one
thf(fact_5490_ln__diff__le,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y ) ) @ ( divide_divide_real @ ( minus_minus_real @ X3 @ Y ) @ Y ) ) ) ) ).
% ln_diff_le
thf(fact_5491_pos__minus__divide__le__eq,axiom,
! [C2: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) @ A )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C2 ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_5492_pos__minus__divide__le__eq,axiom,
! [C2: rat,B: rat,A: rat] :
( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) @ A )
= ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_5493_pos__le__minus__divide__eq,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_5494_pos__le__minus__divide__eq,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_5495_neg__minus__divide__le__eq,axiom,
! [C2: real,B: real,A: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_5496_neg__minus__divide__le__eq,axiom,
! [C2: rat,B: rat,A: rat] :
( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) @ A )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_5497_neg__le__minus__divide__eq,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C2 ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_5498_neg__le__minus__divide__eq,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
= ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_5499_minus__divide__le__eq,axiom,
! [B: real,C2: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C2 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( uminus_uminus_real @ B ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_5500_minus__divide__le__eq,axiom,
! [B: rat,C2: rat,A: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) @ A )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_5501_le__minus__divide__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( uminus_uminus_real @ B ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C2 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_5502_le__minus__divide__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_5503_less__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: real,C2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_5504_less__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: rat,C2: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_5505_divide__less__eq__numeral_I2_J,axiom,
! [B: real,C2: real,W2: num] :
( ( ord_less_real @ ( divide_divide_real @ B @ C2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) @ B ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_5506_divide__less__eq__numeral_I2_J,axiom,
! [B: rat,C2: rat,W2: num] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ C2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) @ B ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_5507_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K2 ) )
= ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5508_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K2 ) )
= ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5509_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K2 ) )
= ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5510_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K2 ) )
= ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5511_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K2 ) )
= ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5512_realpow__square__minus__le,axiom,
! [U: real,X3: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% realpow_square_minus_le
thf(fact_5513_ln__one__minus__pos__lower__bound,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X3 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ).
% ln_one_minus_pos_lower_bound
thf(fact_5514_minus__mod__int__eq,axiom,
! [L: int,K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ L )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ K2 ) @ L )
= ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K2 @ one_one_int ) @ L ) ) ) ) ).
% minus_mod_int_eq
thf(fact_5515_zmod__minus1,axiom,
! [B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
= ( minus_minus_int @ B @ one_one_int ) ) ) ).
% zmod_minus1
thf(fact_5516_sum__strict__mono2,axiom,
! [B5: set_real,A4: set_real,B: real,F: real > real] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ B5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5517_sum__strict__mono2,axiom,
! [B5: set_complex,A4: set_complex,B: complex,F: complex > real] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ B5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5518_sum__strict__mono2,axiom,
! [B5: set_real,A4: set_real,B: real,F: real > rat] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ B5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5519_sum__strict__mono2,axiom,
! [B5: set_complex,A4: set_complex,B: complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ B5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5520_sum__strict__mono2,axiom,
! [B5: set_nat,A4: set_nat,B: nat,F: nat > rat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_nat @ B @ ( minus_minus_set_nat @ B5 @ A4 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
=> ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5521_sum__strict__mono2,axiom,
! [B5: set_real,A4: set_real,B: real,F: real > nat] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ B5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5522_sum__strict__mono2,axiom,
! [B5: set_complex,A4: set_complex,B: complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ B5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5523_sum__strict__mono2,axiom,
! [B5: set_real,A4: set_real,B: real,F: real > int] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ B5 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A4 ) @ ( groups1932886352136224148al_int @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5524_sum__strict__mono2,axiom,
! [B5: set_complex,A4: set_complex,B: complex,F: complex > int] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ B5 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5525_sum__strict__mono2,axiom,
! [B5: set_nat,A4: set_nat,B: nat,F: nat > int] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_nat @ B @ ( minus_minus_set_nat @ B5 @ A4 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B5 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5526_zdiv__zminus1__eq__if,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( ( ( modulo_modulo_int @ A @ B )
= zero_zero_int )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
& ( ( ( modulo_modulo_int @ A @ B )
!= zero_zero_int )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% zdiv_zminus1_eq_if
thf(fact_5527_zdiv__zminus2__eq__if,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( ( ( modulo_modulo_int @ A @ B )
= zero_zero_int )
=> ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
& ( ( ( modulo_modulo_int @ A @ B )
!= zero_zero_int )
=> ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
= ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% zdiv_zminus2_eq_if
thf(fact_5528_zminus1__lemma,axiom,
! [A: int,B: int,Q3: int,R2: int] :
( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
=> ( ( B != zero_zero_int )
=> ( 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 ) ) ) ) ) ) ).
% zminus1_lemma
thf(fact_5529_le__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: real,C2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_5530_le__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: rat,C2: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_5531_divide__le__eq__numeral_I2_J,axiom,
! [B: real,C2: real,W2: num] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) @ B ) )
& ( ~ ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_5532_divide__le__eq__numeral_I2_J,axiom,
! [B: rat,C2: rat,W2: num] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
=> ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) @ B ) )
& ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_5533_square__le__1,axiom,
! [X3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
=> ( ( ord_less_eq_real @ X3 @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% square_le_1
thf(fact_5534_square__le__1,axiom,
! [X3: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X3 )
=> ( ( ord_le3102999989581377725nteger @ X3 @ one_one_Code_integer )
=> ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% square_le_1
thf(fact_5535_square__le__1,axiom,
! [X3: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X3 )
=> ( ( ord_less_eq_rat @ X3 @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% square_le_1
thf(fact_5536_square__le__1,axiom,
! [X3: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X3 )
=> ( ( ord_less_eq_int @ X3 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% square_le_1
thf(fact_5537_div__pos__neg__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K2 @ L ) @ zero_zero_int )
=> ( ( divide_divide_int @ K2 @ L )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_5538_signed__take__bit__int__greater__eq__minus__exp,axiom,
! [N: nat,K2: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_ri631733984087533419it_int @ N @ K2 ) ) ).
% signed_take_bit_int_greater_eq_minus_exp
thf(fact_5539_signed__take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) @ K2 )
= ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K2 ) ) ).
% signed_take_bit_int_less_eq_self_iff
thf(fact_5540_power__minus1__odd,axiom,
! [N: nat] :
( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% power_minus1_odd
thf(fact_5541_power__minus1__odd,axiom,
! [N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% power_minus1_odd
thf(fact_5542_power__minus1__odd,axiom,
! [N: nat] :
( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% power_minus1_odd
thf(fact_5543_power__minus1__odd,axiom,
! [N: nat] :
( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% power_minus1_odd
thf(fact_5544_power__minus1__odd,axiom,
! [N: nat] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% power_minus1_odd
thf(fact_5545_int__bit__induct,axiom,
! [P2: int > $o,K2: int] :
( ( P2 @ zero_zero_int )
=> ( ( P2 @ ( uminus_uminus_int @ one_one_int ) )
=> ( ! [K: int] :
( ( P2 @ K )
=> ( ( K != zero_zero_int )
=> ( P2 @ ( times_times_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
=> ( ! [K: int] :
( ( P2 @ K )
=> ( ( K
!= ( uminus_uminus_int @ one_one_int ) )
=> ( P2 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
=> ( P2 @ K2 ) ) ) ) ) ).
% int_bit_induct
thf(fact_5546_signed__take__bit__int__eq__self,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K2 )
=> ( ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( bit_ri631733984087533419it_int @ N @ K2 )
= K2 ) ) ) ).
% signed_take_bit_int_eq_self
thf(fact_5547_signed__take__bit__int__eq__self__iff,axiom,
! [N: nat,K2: int] :
( ( ( bit_ri631733984087533419it_int @ N @ K2 )
= K2 )
= ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K2 )
& ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% signed_take_bit_int_eq_self_iff
thf(fact_5548_divmod__step__nat__def,axiom,
( unique5026877609467782581ep_nat
= ( ^ [L2: num] :
( produc2626176000494625587at_nat
@ ^ [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 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_5549_ln__one__plus__pos__lower__bound,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ one_one_real )
=> ( ord_less_eq_real @ ( minus_minus_real @ X3 @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) ) ) ) ).
% ln_one_plus_pos_lower_bound
thf(fact_5550_divmod__step__int__def,axiom,
( unique5024387138958732305ep_int
= ( ^ [L2: num] :
( produc4245557441103728435nt_int
@ ^ [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 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_5551_signed__take__bit__int__greater__eq,axiom,
! [K2: int,N: nat] :
( ( ord_less_int @ K2 @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K2 ) ) ) ).
% signed_take_bit_int_greater_eq
thf(fact_5552_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
! [X3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
=> ( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) @ X3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_5553_tanh__ln__real,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( tanh_real @ ( ln_ln_real @ X3 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% tanh_ln_real
thf(fact_5554_divmod__algorithm__code_I5_J,axiom,
! [M2: num,N: num] :
( ( unique5052692396658037445od_int @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( produc4245557441103728435nt_int
@ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) )
@ ( unique5052692396658037445od_int @ M2 @ N ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_5555_divmod__algorithm__code_I5_J,axiom,
! [M2: num,N: num] :
( ( unique5055182867167087721od_nat @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( produc2626176000494625587at_nat
@ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) )
@ ( unique5055182867167087721od_nat @ M2 @ N ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_5556_divmod__algorithm__code_I5_J,axiom,
! [M2: num,N: num] :
( ( unique3479559517661332726nteger @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( produc6916734918728496179nteger
@ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) )
@ ( unique3479559517661332726nteger @ M2 @ N ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_5557_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M5: nat,N4: nat] :
( if_Pro6206227464963214023at_nat
@ ( ( N4 = zero_zero_nat )
| ( ord_less_nat @ M5 @ N4 ) )
@ ( product_Pair_nat_nat @ zero_zero_nat @ M5 )
@ ( produc2626176000494625587at_nat
@ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
@ ( divmod_nat @ ( minus_minus_nat @ M5 @ N4 ) @ N4 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_5558_signed__take__bit__Suc__minus__bit1,axiom,
! [N: nat,K2: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_Suc_minus_bit1
thf(fact_5559_abs__ln__one__plus__x__minus__x__bound,axiom,
! [X3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) @ X3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound
thf(fact_5560_signed__take__bit__Suc__bit1,axiom,
! [N: nat,K2: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_Suc_bit1
thf(fact_5561_abs__idempotent,axiom,
! [A: real] :
( ( abs_abs_real @ ( abs_abs_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_idempotent
thf(fact_5562_abs__idempotent,axiom,
! [A: int] :
( ( abs_abs_int @ ( abs_abs_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_idempotent
thf(fact_5563_abs__idempotent,axiom,
! [A: rat] :
( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
= ( abs_abs_rat @ A ) ) ).
% abs_idempotent
thf(fact_5564_abs__idempotent,axiom,
! [A: code_integer] :
( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
= ( abs_abs_Code_integer @ A ) ) ).
% abs_idempotent
thf(fact_5565_case__prodI,axiom,
! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( F @ A @ B )
=> ( produc127349428274296955eger_o @ F @ ( produc6137756002093451184nteger @ A @ B ) ) ) ).
% case_prodI
thf(fact_5566_case__prodI,axiom,
! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( F @ A @ B )
=> ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B ) ) ) ).
% case_prodI
thf(fact_5567_case__prodI,axiom,
! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
( ( F @ A @ B )
=> ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) ) ) ).
% case_prodI
thf(fact_5568_case__prodI,axiom,
! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
( ( F @ A @ B )
=> ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ).
% case_prodI
thf(fact_5569_case__prodI,axiom,
! [F: int > int > $o,A: int,B: int] :
( ( F @ A @ B )
=> ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) ) ) ).
% case_prodI
thf(fact_5570_case__prodI2,axiom,
! [P: produc8763457246119570046nteger,C2: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
( ( P
= ( produc6137756002093451184nteger @ A3 @ B3 ) )
=> ( C2 @ A3 @ B3 ) )
=> ( produc127349428274296955eger_o @ C2 @ P ) ) ).
% case_prodI2
thf(fact_5571_case__prodI2,axiom,
! [P: produc1908205239877642774nteger,C2: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
( ( P
= ( produc8603105652947943368nteger @ A3 @ B3 ) )
=> ( C2 @ A3 @ B3 ) )
=> ( produc6253627499356882019eger_o @ C2 @ P ) ) ).
% case_prodI2
thf(fact_5572_case__prodI2,axiom,
! [P: produc2285326912895808259nt_int,C2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
( ( P
= ( produc5700946648718959541nt_int @ A3 @ B3 ) )
=> ( C2 @ A3 @ B3 ) )
=> ( produc1573362020775583542_int_o @ C2 @ P ) ) ).
% case_prodI2
thf(fact_5573_case__prodI2,axiom,
! [P: produc7773217078559923341nt_int,C2: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
( ( P
= ( produc4305682042979456191nt_int @ A3 @ B3 ) )
=> ( C2 @ A3 @ B3 ) )
=> ( produc2558449545302689196_int_o @ C2 @ P ) ) ).
% case_prodI2
thf(fact_5574_case__prodI2,axiom,
! [P: product_prod_int_int,C2: int > int > $o] :
( ! [A3: int,B3: int] :
( ( P
= ( product_Pair_int_int @ A3 @ B3 ) )
=> ( C2 @ A3 @ B3 ) )
=> ( produc4947309494688390418_int_o @ C2 @ P ) ) ).
% case_prodI2
thf(fact_5575_mem__case__prodI,axiom,
! [Z2: complex,C2: int > int > set_complex,A: int,B: int] :
( ( member_complex @ Z2 @ ( C2 @ A @ B ) )
=> ( member_complex @ Z2 @ ( produc8580519160106071146omplex @ C2 @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5576_mem__case__prodI,axiom,
! [Z2: real,C2: int > int > set_real,A: int,B: int] :
( ( member_real @ Z2 @ ( C2 @ A @ B ) )
=> ( member_real @ Z2 @ ( produc6452406959799940328t_real @ C2 @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5577_mem__case__prodI,axiom,
! [Z2: nat,C2: int > int > set_nat,A: int,B: int] :
( ( member_nat @ Z2 @ ( C2 @ A @ B ) )
=> ( member_nat @ Z2 @ ( produc4251311855443802252et_nat @ C2 @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5578_mem__case__prodI,axiom,
! [Z2: int,C2: int > int > set_int,A: int,B: int] :
( ( member_int @ Z2 @ ( C2 @ A @ B ) )
=> ( member_int @ Z2 @ ( produc73460835934605544et_int @ C2 @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5579_mem__case__prodI,axiom,
! [Z2: set_nat,C2: int > int > set_set_nat,A: int,B: int] :
( ( member_set_nat @ Z2 @ ( C2 @ A @ B ) )
=> ( member_set_nat @ Z2 @ ( produc5233655623923918146et_nat @ C2 @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5580_mem__case__prodI,axiom,
! [Z2: complex,C2: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_complex,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( member_complex @ Z2 @ ( C2 @ A @ B ) )
=> ( member_complex @ Z2 @ ( produc2592262431452330817omplex @ C2 @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5581_mem__case__prodI,axiom,
! [Z2: real,C2: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( member_real @ Z2 @ ( C2 @ A @ B ) )
=> ( member_real @ Z2 @ ( produc815715089573277247t_real @ C2 @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5582_mem__case__prodI,axiom,
! [Z2: nat,C2: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( member_nat @ Z2 @ ( C2 @ A @ B ) )
=> ( member_nat @ Z2 @ ( produc3558942015123893603et_nat @ C2 @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5583_mem__case__prodI,axiom,
! [Z2: int,C2: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( member_int @ Z2 @ ( C2 @ A @ B ) )
=> ( member_int @ Z2 @ ( produc8604463032469472703et_int @ C2 @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5584_mem__case__prodI,axiom,
! [Z2: complex,C2: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_complex,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
( ( member_complex @ Z2 @ ( C2 @ A @ B ) )
=> ( member_complex @ Z2 @ ( produc7959293469001253456omplex @ C2 @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_5585_mem__case__prodI2,axiom,
! [P: product_prod_int_int,Z2: complex,C2: int > int > set_complex] :
( ! [A3: int,B3: int] :
( ( P
= ( product_Pair_int_int @ A3 @ B3 ) )
=> ( member_complex @ Z2 @ ( C2 @ A3 @ B3 ) ) )
=> ( member_complex @ Z2 @ ( produc8580519160106071146omplex @ C2 @ P ) ) ) ).
% mem_case_prodI2
thf(fact_5586_mem__case__prodI2,axiom,
! [P: product_prod_int_int,Z2: real,C2: int > int > set_real] :
( ! [A3: int,B3: int] :
( ( P
= ( product_Pair_int_int @ A3 @ B3 ) )
=> ( member_real @ Z2 @ ( C2 @ A3 @ B3 ) ) )
=> ( member_real @ Z2 @ ( produc6452406959799940328t_real @ C2 @ P ) ) ) ).
% mem_case_prodI2
thf(fact_5587_mem__case__prodI2,axiom,
! [P: product_prod_int_int,Z2: nat,C2: int > int > set_nat] :
( ! [A3: int,B3: int] :
( ( P
= ( product_Pair_int_int @ A3 @ B3 ) )
=> ( member_nat @ Z2 @ ( C2 @ A3 @ B3 ) ) )
=> ( member_nat @ Z2 @ ( produc4251311855443802252et_nat @ C2 @ P ) ) ) ).
% mem_case_prodI2
thf(fact_5588_mem__case__prodI2,axiom,
! [P: product_prod_int_int,Z2: int,C2: int > int > set_int] :
( ! [A3: int,B3: int] :
( ( P
= ( product_Pair_int_int @ A3 @ B3 ) )
=> ( member_int @ Z2 @ ( C2 @ A3 @ B3 ) ) )
=> ( member_int @ Z2 @ ( produc73460835934605544et_int @ C2 @ P ) ) ) ).
% mem_case_prodI2
thf(fact_5589_mem__case__prodI2,axiom,
! [P: product_prod_int_int,Z2: set_nat,C2: int > int > set_set_nat] :
( ! [A3: int,B3: int] :
( ( P
= ( product_Pair_int_int @ A3 @ B3 ) )
=> ( member_set_nat @ Z2 @ ( C2 @ A3 @ B3 ) ) )
=> ( member_set_nat @ Z2 @ ( produc5233655623923918146et_nat @ C2 @ P ) ) ) ).
% mem_case_prodI2
thf(fact_5590_mem__case__prodI2,axiom,
! [P: produc8763457246119570046nteger,Z2: complex,C2: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_complex] :
( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
( ( P
= ( produc6137756002093451184nteger @ A3 @ B3 ) )
=> ( member_complex @ Z2 @ ( C2 @ A3 @ B3 ) ) )
=> ( member_complex @ Z2 @ ( produc2592262431452330817omplex @ C2 @ P ) ) ) ).
% mem_case_prodI2
thf(fact_5591_mem__case__prodI2,axiom,
! [P: produc8763457246119570046nteger,Z2: real,C2: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real] :
( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
( ( P
= ( produc6137756002093451184nteger @ A3 @ B3 ) )
=> ( member_real @ Z2 @ ( C2 @ A3 @ B3 ) ) )
=> ( member_real @ Z2 @ ( produc815715089573277247t_real @ C2 @ P ) ) ) ).
% mem_case_prodI2
thf(fact_5592_mem__case__prodI2,axiom,
! [P: produc8763457246119570046nteger,Z2: nat,C2: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat] :
( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
( ( P
= ( produc6137756002093451184nteger @ A3 @ B3 ) )
=> ( member_nat @ Z2 @ ( C2 @ A3 @ B3 ) ) )
=> ( member_nat @ Z2 @ ( produc3558942015123893603et_nat @ C2 @ P ) ) ) ).
% mem_case_prodI2
thf(fact_5593_mem__case__prodI2,axiom,
! [P: produc8763457246119570046nteger,Z2: int,C2: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int] :
( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
( ( P
= ( produc6137756002093451184nteger @ A3 @ B3 ) )
=> ( member_int @ Z2 @ ( C2 @ A3 @ B3 ) ) )
=> ( member_int @ Z2 @ ( produc8604463032469472703et_int @ C2 @ P ) ) ) ).
% mem_case_prodI2
thf(fact_5594_mem__case__prodI2,axiom,
! [P: produc7773217078559923341nt_int,Z2: complex,C2: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_complex] :
( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
( ( P
= ( produc4305682042979456191nt_int @ A3 @ B3 ) )
=> ( member_complex @ Z2 @ ( C2 @ A3 @ B3 ) ) )
=> ( member_complex @ Z2 @ ( produc7959293469001253456omplex @ C2 @ P ) ) ) ).
% mem_case_prodI2
thf(fact_5595_case__prodI2_H,axiom,
! [P: product_prod_nat_nat,C2: nat > nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat] :
( ! [A3: nat,B3: nat] :
( ( ( product_Pair_nat_nat @ A3 @ B3 )
= P )
=> ( C2 @ A3 @ B3 @ X3 ) )
=> ( produc8739625826339149834_nat_o @ C2 @ P @ X3 ) ) ).
% case_prodI2'
thf(fact_5596_abs__0,axiom,
( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% abs_0
thf(fact_5597_abs__0,axiom,
( ( abs_abs_complex @ zero_zero_complex )
= zero_zero_complex ) ).
% abs_0
thf(fact_5598_abs__0,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_0
thf(fact_5599_abs__0,axiom,
( ( abs_abs_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% abs_0
thf(fact_5600_abs__0,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_0
thf(fact_5601_abs__zero,axiom,
( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% abs_zero
thf(fact_5602_abs__zero,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_zero
thf(fact_5603_abs__zero,axiom,
( ( abs_abs_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% abs_zero
thf(fact_5604_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_5605_abs__eq__0,axiom,
! [A: code_integer] :
( ( ( abs_abs_Code_integer @ A )
= zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ).
% abs_eq_0
thf(fact_5606_abs__eq__0,axiom,
! [A: real] :
( ( ( abs_abs_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_eq_0
thf(fact_5607_abs__eq__0,axiom,
! [A: rat] :
( ( ( abs_abs_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% abs_eq_0
thf(fact_5608_abs__eq__0,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_5609_abs__0__eq,axiom,
! [A: code_integer] :
( ( zero_z3403309356797280102nteger
= ( abs_abs_Code_integer @ A ) )
= ( A = zero_z3403309356797280102nteger ) ) ).
% abs_0_eq
thf(fact_5610_abs__0__eq,axiom,
! [A: real] :
( ( zero_zero_real
= ( abs_abs_real @ A ) )
= ( A = zero_zero_real ) ) ).
% abs_0_eq
thf(fact_5611_abs__0__eq,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( abs_abs_rat @ A ) )
= ( A = zero_zero_rat ) ) ).
% abs_0_eq
thf(fact_5612_abs__0__eq,axiom,
! [A: int] :
( ( zero_zero_int
= ( abs_abs_int @ A ) )
= ( A = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_5613_abs__add__abs,axiom,
! [A: code_integer,B: code_integer] :
( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
= ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% abs_add_abs
thf(fact_5614_abs__add__abs,axiom,
! [A: real,B: real] :
( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
= ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% abs_add_abs
thf(fact_5615_abs__add__abs,axiom,
! [A: rat,B: rat] :
( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
= ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% abs_add_abs
thf(fact_5616_abs__add__abs,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
= ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_add_abs
thf(fact_5617_abs__minus__cancel,axiom,
! [A: real] :
( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_minus_cancel
thf(fact_5618_abs__minus__cancel,axiom,
! [A: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_minus_cancel
thf(fact_5619_abs__minus__cancel,axiom,
! [A: rat] :
( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
= ( abs_abs_rat @ A ) ) ).
% abs_minus_cancel
thf(fact_5620_abs__minus__cancel,axiom,
! [A: code_integer] :
( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
= ( abs_abs_Code_integer @ A ) ) ).
% abs_minus_cancel
thf(fact_5621_tanh__0,axiom,
( ( tanh_complex @ zero_zero_complex )
= zero_zero_complex ) ).
% tanh_0
thf(fact_5622_tanh__0,axiom,
( ( tanh_real @ zero_zero_real )
= zero_zero_real ) ).
% tanh_0
thf(fact_5623_tanh__real__zero__iff,axiom,
! [X3: real] :
( ( ( tanh_real @ X3 )
= zero_zero_real )
= ( X3 = zero_zero_real ) ) ).
% tanh_real_zero_iff
thf(fact_5624_tanh__real__le__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( tanh_real @ X3 ) @ ( tanh_real @ Y ) )
= ( ord_less_eq_real @ X3 @ Y ) ) ).
% tanh_real_le_iff
thf(fact_5625_semiring__norm_I73_J,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% semiring_norm(73)
thf(fact_5626_abs__of__nonneg,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( abs_abs_Code_integer @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_5627_abs__of__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( abs_abs_real @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_5628_abs__of__nonneg,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( abs_abs_rat @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_5629_abs__of__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_5630_abs__le__self__iff,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
= ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% abs_le_self_iff
thf(fact_5631_abs__le__self__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% abs_le_self_iff
thf(fact_5632_abs__le__self__iff,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% abs_le_self_iff
thf(fact_5633_abs__le__self__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% abs_le_self_iff
thf(fact_5634_abs__le__zero__iff,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ).
% abs_le_zero_iff
thf(fact_5635_abs__le__zero__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_le_zero_iff
thf(fact_5636_abs__le__zero__iff,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% abs_le_zero_iff
thf(fact_5637_abs__le__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_le_zero_iff
thf(fact_5638_zero__less__abs__iff,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
= ( A != zero_z3403309356797280102nteger ) ) ).
% zero_less_abs_iff
thf(fact_5639_zero__less__abs__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
= ( A != zero_zero_real ) ) ).
% zero_less_abs_iff
thf(fact_5640_zero__less__abs__iff,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
= ( A != zero_zero_rat ) ) ).
% zero_less_abs_iff
thf(fact_5641_zero__less__abs__iff,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
= ( A != zero_zero_int ) ) ).
% zero_less_abs_iff
thf(fact_5642_semiring__norm_I7_J,axiom,
! [M2: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus_num @ M2 @ N ) ) ) ).
% semiring_norm(7)
thf(fact_5643_semiring__norm_I9_J,axiom,
! [M2: num,N: num] :
( ( plus_plus_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( bit1 @ ( plus_plus_num @ M2 @ N ) ) ) ).
% semiring_norm(9)
thf(fact_5644_semiring__norm_I16_J,axiom,
! [M2: num,N: num] :
( ( times_times_num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M2 @ N ) @ ( bit0 @ ( times_times_num @ M2 @ N ) ) ) ) ) ).
% semiring_norm(16)
thf(fact_5645_semiring__norm_I79_J,axiom,
! [M2: num,N: num] :
( ( ord_less_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% semiring_norm(79)
thf(fact_5646_semiring__norm_I74_J,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% semiring_norm(74)
thf(fact_5647_semiring__norm_I72_J,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% semiring_norm(72)
thf(fact_5648_semiring__norm_I70_J,axiom,
! [M2: num] :
~ ( ord_less_eq_num @ ( bit1 @ M2 ) @ one ) ).
% semiring_norm(70)
thf(fact_5649_tanh__real__neg__iff,axiom,
! [X3: real] :
( ( ord_less_real @ ( tanh_real @ X3 ) @ zero_zero_real )
= ( ord_less_real @ X3 @ zero_zero_real ) ) ).
% tanh_real_neg_iff
thf(fact_5650_tanh__real__pos__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X3 ) )
= ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% tanh_real_pos_iff
thf(fact_5651_tanh__real__nonpos__iff,axiom,
! [X3: real] :
( ( ord_less_eq_real @ ( tanh_real @ X3 ) @ zero_zero_real )
= ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).
% tanh_real_nonpos_iff
thf(fact_5652_tanh__real__nonneg__iff,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X3 ) )
= ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).
% tanh_real_nonneg_iff
thf(fact_5653_sum__abs,axiom,
! [F: int > int,A4: set_int] :
( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A4 ) )
@ ( groups4538972089207619220nt_int
@ ^ [I3: int] : ( abs_abs_int @ ( F @ I3 ) )
@ A4 ) ) ).
% sum_abs
thf(fact_5654_sum__abs,axiom,
! [F: nat > real,A4: set_nat] :
( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A4 ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( abs_abs_real @ ( F @ I3 ) )
@ A4 ) ) ).
% sum_abs
thf(fact_5655_divide__le__0__abs__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
= ( ( ord_less_eq_real @ A @ zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_le_0_abs_iff
thf(fact_5656_divide__le__0__abs__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) @ zero_zero_rat )
= ( ( ord_less_eq_rat @ A @ zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% divide_le_0_abs_iff
thf(fact_5657_zero__le__divide__abs__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
= ( ( ord_less_eq_real @ zero_zero_real @ A )
| ( B = zero_zero_real ) ) ) ).
% zero_le_divide_abs_iff
thf(fact_5658_zero__le__divide__abs__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) )
= ( ( ord_less_eq_rat @ zero_zero_rat @ A )
| ( B = zero_zero_rat ) ) ) ).
% zero_le_divide_abs_iff
thf(fact_5659_abs__of__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( abs_abs_real @ A )
= ( uminus_uminus_real @ A ) ) ) ).
% abs_of_nonpos
thf(fact_5660_abs__of__nonpos,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
=> ( ( abs_abs_Code_integer @ A )
= ( uminus1351360451143612070nteger @ A ) ) ) ).
% abs_of_nonpos
thf(fact_5661_abs__of__nonpos,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( abs_abs_rat @ A )
= ( uminus_uminus_rat @ A ) ) ) ).
% abs_of_nonpos
thf(fact_5662_abs__of__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( abs_abs_int @ A )
= ( uminus_uminus_int @ A ) ) ) ).
% abs_of_nonpos
thf(fact_5663_semiring__norm_I3_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% semiring_norm(3)
thf(fact_5664_semiring__norm_I4_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% semiring_norm(4)
thf(fact_5665_semiring__norm_I5_J,axiom,
! [M2: num] :
( ( plus_plus_num @ ( bit0 @ M2 ) @ one )
= ( bit1 @ M2 ) ) ).
% semiring_norm(5)
thf(fact_5666_semiring__norm_I8_J,axiom,
! [M2: num] :
( ( plus_plus_num @ ( bit1 @ M2 ) @ one )
= ( bit0 @ ( plus_plus_num @ M2 @ one ) ) ) ).
% semiring_norm(8)
thf(fact_5667_semiring__norm_I10_J,axiom,
! [M2: num,N: num] :
( ( plus_plus_num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M2 @ N ) @ one ) ) ) ).
% semiring_norm(10)
thf(fact_5668_sum__abs__ge__zero,axiom,
! [F: int > int,A4: set_int] :
( ord_less_eq_int @ zero_zero_int
@ ( groups4538972089207619220nt_int
@ ^ [I3: int] : ( abs_abs_int @ ( F @ I3 ) )
@ A4 ) ) ).
% sum_abs_ge_zero
thf(fact_5669_sum__abs__ge__zero,axiom,
! [F: nat > real,A4: set_nat] :
( ord_less_eq_real @ zero_zero_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( abs_abs_real @ ( F @ I3 ) )
@ A4 ) ) ).
% sum_abs_ge_zero
thf(fact_5670_zero__less__power__abs__iff,axiom,
! [A: code_integer,N: nat] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) )
= ( ( A != zero_z3403309356797280102nteger )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_5671_zero__less__power__abs__iff,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
= ( ( A != zero_zero_real )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_5672_zero__less__power__abs__iff,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) )
= ( ( A != zero_zero_rat )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_5673_zero__less__power__abs__iff,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
= ( ( A != zero_zero_int )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_5674_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > complex] :
( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= zero_zero_complex ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_5675_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > rat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= zero_zero_rat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_5676_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > int] :
( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= zero_zero_int ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_5677_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > nat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_5678_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > real] :
( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= zero_zero_real ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_5679_divmod__algorithm__code_I2_J,axiom,
! [M2: num] :
( ( unique5052692396658037445od_int @ M2 @ one )
= ( product_Pair_int_int @ ( numeral_numeral_int @ M2 ) @ zero_zero_int ) ) ).
% divmod_algorithm_code(2)
thf(fact_5680_divmod__algorithm__code_I2_J,axiom,
! [M2: num] :
( ( unique5055182867167087721od_nat @ M2 @ one )
= ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M2 ) @ zero_zero_nat ) ) ).
% divmod_algorithm_code(2)
thf(fact_5681_divmod__algorithm__code_I2_J,axiom,
! [M2: num] :
( ( unique3479559517661332726nteger @ M2 @ one )
= ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M2 ) @ zero_z3403309356797280102nteger ) ) ).
% divmod_algorithm_code(2)
thf(fact_5682_sum__zero__power,axiom,
! [A4: set_nat,C2: nat > complex] :
( ( ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C2 @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) )
@ A4 )
= ( C2 @ zero_zero_nat ) ) )
& ( ~ ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C2 @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) )
@ A4 )
= zero_zero_complex ) ) ) ).
% sum_zero_power
thf(fact_5683_sum__zero__power,axiom,
! [A4: set_nat,C2: nat > rat] :
( ( ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( C2 @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) )
@ A4 )
= ( C2 @ zero_zero_nat ) ) )
& ( ~ ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( C2 @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) )
@ A4 )
= zero_zero_rat ) ) ) ).
% sum_zero_power
thf(fact_5684_sum__zero__power,axiom,
! [A4: set_nat,C2: nat > real] :
( ( ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C2 @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) )
@ A4 )
= ( C2 @ zero_zero_nat ) ) )
& ( ~ ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C2 @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) )
@ A4 )
= zero_zero_real ) ) ) ).
% sum_zero_power
thf(fact_5685_div__Suc__eq__div__add3,axiom,
! [M2: nat,N: nat] :
( ( divide_divide_nat @ M2 @ ( suc @ ( suc @ ( suc @ N ) ) ) )
= ( divide_divide_nat @ M2 @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% div_Suc_eq_div_add3
thf(fact_5686_Suc__div__eq__add3__div__numeral,axiom,
! [M2: nat,V: num] :
( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral_nat @ V ) )
= ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ ( numeral_numeral_nat @ V ) ) ) ).
% Suc_div_eq_add3_div_numeral
thf(fact_5687_divmod__algorithm__code_I3_J,axiom,
! [N: num] :
( ( unique5052692396658037445od_int @ one @ ( bit0 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_5688_divmod__algorithm__code_I3_J,axiom,
! [N: num] :
( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_5689_divmod__algorithm__code_I3_J,axiom,
! [N: num] :
( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_5690_mod__Suc__eq__mod__add3,axiom,
! [M2: nat,N: nat] :
( ( modulo_modulo_nat @ M2 @ ( suc @ ( suc @ ( suc @ N ) ) ) )
= ( modulo_modulo_nat @ M2 @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% mod_Suc_eq_mod_add3
thf(fact_5691_Suc__mod__eq__add3__mod__numeral,axiom,
! [M2: nat,V: num] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral_nat @ V ) )
= ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ ( numeral_numeral_nat @ V ) ) ) ).
% Suc_mod_eq_add3_mod_numeral
thf(fact_5692_divmod__algorithm__code_I4_J,axiom,
! [N: num] :
( ( unique5052692396658037445od_int @ one @ ( bit1 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_5693_divmod__algorithm__code_I4_J,axiom,
! [N: num] :
( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_5694_divmod__algorithm__code_I4_J,axiom,
! [N: num] :
( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_5695_sum__zero__power_H,axiom,
! [A4: set_nat,C2: nat > complex,D: nat > complex] :
( ( ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C2 @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) ) @ ( D @ I3 ) )
@ A4 )
= ( divide1717551699836669952omplex @ ( C2 @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
& ( ~ ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C2 @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) ) @ ( D @ I3 ) )
@ A4 )
= zero_zero_complex ) ) ) ).
% sum_zero_power'
thf(fact_5696_sum__zero__power_H,axiom,
! [A4: set_nat,C2: nat > rat,D: nat > rat] :
( ( ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C2 @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) ) @ ( D @ I3 ) )
@ A4 )
= ( divide_divide_rat @ ( C2 @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
& ( ~ ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C2 @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) ) @ ( D @ I3 ) )
@ A4 )
= zero_zero_rat ) ) ) ).
% sum_zero_power'
thf(fact_5697_sum__zero__power_H,axiom,
! [A4: set_nat,C2: nat > real,D: nat > real] :
( ( ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( divide_divide_real @ ( times_times_real @ ( C2 @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) ) @ ( D @ I3 ) )
@ A4 )
= ( divide_divide_real @ ( C2 @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
& ( ~ ( ( finite_finite_nat @ A4 )
& ( member_nat @ zero_zero_nat @ A4 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( divide_divide_real @ ( times_times_real @ ( C2 @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) ) @ ( D @ I3 ) )
@ A4 )
= zero_zero_real ) ) ) ).
% sum_zero_power'
thf(fact_5698_divmod__algorithm__code_I8_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_num @ M2 @ N )
=> ( ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M2 ) ) ) ) )
& ( ~ ( ord_less_num @ M2 @ N )
=> ( ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_5699_divmod__algorithm__code_I8_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_num @ M2 @ N )
=> ( ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) ) ) )
& ( ~ ( ord_less_num @ M2 @ N )
=> ( ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_5700_divmod__algorithm__code_I8_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_num @ M2 @ N )
=> ( ( unique3479559517661332726nteger @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M2 ) ) ) ) )
& ( ~ ( ord_less_num @ M2 @ N )
=> ( ( unique3479559517661332726nteger @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_5701_zmod__numeral__Bit1,axiom,
! [V: num,W2: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W2 ) ) ) @ one_one_int ) ) ).
% zmod_numeral_Bit1
thf(fact_5702_divmod__algorithm__code_I7_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq_num @ M2 @ N )
=> ( ( unique5055182867167087721od_nat @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M2 ) ) ) ) )
& ( ~ ( ord_less_eq_num @ M2 @ N )
=> ( ( unique5055182867167087721od_nat @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_5703_divmod__algorithm__code_I7_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq_num @ M2 @ N )
=> ( ( unique5052692396658037445od_int @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) ) ) )
& ( ~ ( ord_less_eq_num @ M2 @ N )
=> ( ( unique5052692396658037445od_int @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_5704_divmod__algorithm__code_I7_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq_num @ M2 @ N )
=> ( ( unique3479559517661332726nteger @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M2 ) ) ) ) )
& ( ~ ( ord_less_eq_num @ M2 @ N )
=> ( ( unique3479559517661332726nteger @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_5705_divmod__algorithm__code_I6_J,axiom,
! [M2: num,N: num] :
( ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( produc4245557441103728435nt_int
@ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) @ one_one_int ) )
@ ( unique5052692396658037445od_int @ M2 @ N ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_5706_divmod__algorithm__code_I6_J,axiom,
! [M2: num,N: num] :
( ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( produc2626176000494625587at_nat
@ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) @ one_one_nat ) )
@ ( unique5055182867167087721od_nat @ M2 @ N ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_5707_divmod__algorithm__code_I6_J,axiom,
! [M2: num,N: num] :
( ( unique3479559517661332726nteger @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( produc6916734918728496179nteger
@ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) @ one_one_Code_integer ) )
@ ( unique3479559517661332726nteger @ M2 @ N ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_5708_abs__ge__self,axiom,
! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% abs_ge_self
thf(fact_5709_abs__ge__self,axiom,
! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).
% abs_ge_self
thf(fact_5710_abs__ge__self,axiom,
! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).
% abs_ge_self
thf(fact_5711_abs__ge__self,axiom,
! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% abs_ge_self
thf(fact_5712_abs__le__D1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% abs_le_D1
thf(fact_5713_abs__le__D1,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
=> ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% abs_le_D1
thf(fact_5714_abs__le__D1,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
=> ( ord_less_eq_rat @ A @ B ) ) ).
% abs_le_D1
thf(fact_5715_abs__le__D1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% abs_le_D1
thf(fact_5716_abs__eq__0__iff,axiom,
! [A: code_integer] :
( ( ( abs_abs_Code_integer @ A )
= zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ).
% abs_eq_0_iff
thf(fact_5717_abs__eq__0__iff,axiom,
! [A: complex] :
( ( ( abs_abs_complex @ A )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% abs_eq_0_iff
thf(fact_5718_abs__eq__0__iff,axiom,
! [A: real] :
( ( ( abs_abs_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_eq_0_iff
thf(fact_5719_abs__eq__0__iff,axiom,
! [A: rat] :
( ( ( abs_abs_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% abs_eq_0_iff
thf(fact_5720_abs__eq__0__iff,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0_iff
thf(fact_5721_abs__minus__commute,axiom,
! [A: code_integer,B: code_integer] :
( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
= ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_5722_abs__minus__commute,axiom,
! [A: real,B: real] :
( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
= ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_5723_abs__minus__commute,axiom,
! [A: rat,B: rat] :
( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
= ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_5724_abs__minus__commute,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
= ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_5725_mem__case__prodE,axiom,
! [Z2: complex,C2: int > int > set_complex,P: product_prod_int_int] :
( ( member_complex @ Z2 @ ( produc8580519160106071146omplex @ C2 @ P ) )
=> ~ ! [X4: int,Y3: int] :
( ( P
= ( product_Pair_int_int @ X4 @ Y3 ) )
=> ~ ( member_complex @ Z2 @ ( C2 @ X4 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_5726_mem__case__prodE,axiom,
! [Z2: real,C2: int > int > set_real,P: product_prod_int_int] :
( ( member_real @ Z2 @ ( produc6452406959799940328t_real @ C2 @ P ) )
=> ~ ! [X4: int,Y3: int] :
( ( P
= ( product_Pair_int_int @ X4 @ Y3 ) )
=> ~ ( member_real @ Z2 @ ( C2 @ X4 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_5727_mem__case__prodE,axiom,
! [Z2: nat,C2: int > int > set_nat,P: product_prod_int_int] :
( ( member_nat @ Z2 @ ( produc4251311855443802252et_nat @ C2 @ P ) )
=> ~ ! [X4: int,Y3: int] :
( ( P
= ( product_Pair_int_int @ X4 @ Y3 ) )
=> ~ ( member_nat @ Z2 @ ( C2 @ X4 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_5728_mem__case__prodE,axiom,
! [Z2: int,C2: int > int > set_int,P: product_prod_int_int] :
( ( member_int @ Z2 @ ( produc73460835934605544et_int @ C2 @ P ) )
=> ~ ! [X4: int,Y3: int] :
( ( P
= ( product_Pair_int_int @ X4 @ Y3 ) )
=> ~ ( member_int @ Z2 @ ( C2 @ X4 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_5729_mem__case__prodE,axiom,
! [Z2: set_nat,C2: int > int > set_set_nat,P: product_prod_int_int] :
( ( member_set_nat @ Z2 @ ( produc5233655623923918146et_nat @ C2 @ P ) )
=> ~ ! [X4: int,Y3: int] :
( ( P
= ( product_Pair_int_int @ X4 @ Y3 ) )
=> ~ ( member_set_nat @ Z2 @ ( C2 @ X4 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_5730_mem__case__prodE,axiom,
! [Z2: complex,C2: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_complex,P: produc8763457246119570046nteger] :
( ( member_complex @ Z2 @ ( produc2592262431452330817omplex @ C2 @ P ) )
=> ~ ! [X4: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( ( P
= ( produc6137756002093451184nteger @ X4 @ Y3 ) )
=> ~ ( member_complex @ Z2 @ ( C2 @ X4 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_5731_mem__case__prodE,axiom,
! [Z2: real,C2: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real,P: produc8763457246119570046nteger] :
( ( member_real @ Z2 @ ( produc815715089573277247t_real @ C2 @ P ) )
=> ~ ! [X4: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( ( P
= ( produc6137756002093451184nteger @ X4 @ Y3 ) )
=> ~ ( member_real @ Z2 @ ( C2 @ X4 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_5732_mem__case__prodE,axiom,
! [Z2: nat,C2: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,P: produc8763457246119570046nteger] :
( ( member_nat @ Z2 @ ( produc3558942015123893603et_nat @ C2 @ P ) )
=> ~ ! [X4: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( ( P
= ( produc6137756002093451184nteger @ X4 @ Y3 ) )
=> ~ ( member_nat @ Z2 @ ( C2 @ X4 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_5733_mem__case__prodE,axiom,
! [Z2: int,C2: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,P: produc8763457246119570046nteger] :
( ( member_int @ Z2 @ ( produc8604463032469472703et_int @ C2 @ P ) )
=> ~ ! [X4: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( ( P
= ( produc6137756002093451184nteger @ X4 @ Y3 ) )
=> ~ ( member_int @ Z2 @ ( C2 @ X4 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_5734_mem__case__prodE,axiom,
! [Z2: complex,C2: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_complex,P: produc7773217078559923341nt_int] :
( ( member_complex @ Z2 @ ( produc7959293469001253456omplex @ C2 @ P ) )
=> ~ ! [X4: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
( ( P
= ( produc4305682042979456191nt_int @ X4 @ Y3 ) )
=> ~ ( member_complex @ Z2 @ ( C2 @ X4 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_5735_case__prodD,axiom,
! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( produc127349428274296955eger_o @ F @ ( produc6137756002093451184nteger @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_5736_case__prodD,axiom,
! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_5737_case__prodD,axiom,
! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
( ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_5738_case__prodD,axiom,
! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
( ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_5739_case__prodD,axiom,
! [F: int > int > $o,A: int,B: int] :
( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_5740_case__prodE,axiom,
! [C2: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P: produc8763457246119570046nteger] :
( ( produc127349428274296955eger_o @ C2 @ P )
=> ~ ! [X4: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( ( P
= ( produc6137756002093451184nteger @ X4 @ Y3 ) )
=> ~ ( C2 @ X4 @ Y3 ) ) ) ).
% case_prodE
thf(fact_5741_case__prodE,axiom,
! [C2: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P: produc1908205239877642774nteger] :
( ( produc6253627499356882019eger_o @ C2 @ P )
=> ~ ! [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( ( P
= ( produc8603105652947943368nteger @ X4 @ Y3 ) )
=> ~ ( C2 @ X4 @ Y3 ) ) ) ).
% case_prodE
thf(fact_5742_case__prodE,axiom,
! [C2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,P: produc2285326912895808259nt_int] :
( ( produc1573362020775583542_int_o @ C2 @ P )
=> ~ ! [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
( ( P
= ( produc5700946648718959541nt_int @ X4 @ Y3 ) )
=> ~ ( C2 @ X4 @ Y3 ) ) ) ).
% case_prodE
thf(fact_5743_case__prodE,axiom,
! [C2: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,P: produc7773217078559923341nt_int] :
( ( produc2558449545302689196_int_o @ C2 @ P )
=> ~ ! [X4: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
( ( P
= ( produc4305682042979456191nt_int @ X4 @ Y3 ) )
=> ~ ( C2 @ X4 @ Y3 ) ) ) ).
% case_prodE
thf(fact_5744_case__prodE,axiom,
! [C2: int > int > $o,P: product_prod_int_int] :
( ( produc4947309494688390418_int_o @ C2 @ P )
=> ~ ! [X4: int,Y3: int] :
( ( P
= ( product_Pair_int_int @ X4 @ Y3 ) )
=> ~ ( C2 @ X4 @ Y3 ) ) ) ).
% case_prodE
thf(fact_5745_case__prodD_H,axiom,
! [R: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat,C2: product_prod_nat_nat] :
( ( produc8739625826339149834_nat_o @ R @ ( product_Pair_nat_nat @ A @ B ) @ C2 )
=> ( R @ A @ B @ C2 ) ) ).
% case_prodD'
thf(fact_5746_case__prodE_H,axiom,
! [C2: nat > nat > product_prod_nat_nat > $o,P: product_prod_nat_nat,Z2: product_prod_nat_nat] :
( ( produc8739625826339149834_nat_o @ C2 @ P @ Z2 )
=> ~ ! [X4: nat,Y3: nat] :
( ( P
= ( product_Pair_nat_nat @ X4 @ Y3 ) )
=> ~ ( C2 @ X4 @ Y3 @ Z2 ) ) ) ).
% case_prodE'
thf(fact_5747_abs__ge__zero,axiom,
! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).
% abs_ge_zero
thf(fact_5748_abs__ge__zero,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% abs_ge_zero
thf(fact_5749_abs__ge__zero,axiom,
! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).
% abs_ge_zero
thf(fact_5750_abs__ge__zero,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% abs_ge_zero
thf(fact_5751_abs__of__pos,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( abs_abs_Code_integer @ A )
= A ) ) ).
% abs_of_pos
thf(fact_5752_abs__of__pos,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( abs_abs_real @ A )
= A ) ) ).
% abs_of_pos
thf(fact_5753_abs__of__pos,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( abs_abs_rat @ A )
= A ) ) ).
% abs_of_pos
thf(fact_5754_abs__of__pos,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_pos
thf(fact_5755_abs__not__less__zero,axiom,
! [A: code_integer] :
~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).
% abs_not_less_zero
thf(fact_5756_abs__not__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% abs_not_less_zero
thf(fact_5757_abs__not__less__zero,axiom,
! [A: rat] :
~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).
% abs_not_less_zero
thf(fact_5758_abs__not__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_5759_abs__triangle__ineq,axiom,
! [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 ) ) ) ).
% abs_triangle_ineq
thf(fact_5760_abs__triangle__ineq,axiom,
! [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 ) ) ) ).
% abs_triangle_ineq
thf(fact_5761_abs__triangle__ineq,axiom,
! [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 ) ) ) ).
% abs_triangle_ineq
thf(fact_5762_abs__triangle__ineq,axiom,
! [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 ) ) ) ).
% abs_triangle_ineq
thf(fact_5763_abs__triangle__ineq2,axiom,
! [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 ) ) ) ).
% abs_triangle_ineq2
thf(fact_5764_abs__triangle__ineq2,axiom,
! [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 ) ) ) ).
% abs_triangle_ineq2
thf(fact_5765_abs__triangle__ineq2,axiom,
! [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 ) ) ) ).
% abs_triangle_ineq2
thf(fact_5766_abs__triangle__ineq2,axiom,
! [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 ) ) ) ).
% abs_triangle_ineq2
thf(fact_5767_abs__triangle__ineq3,axiom,
! [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 ) ) ) ).
% abs_triangle_ineq3
thf(fact_5768_abs__triangle__ineq3,axiom,
! [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 ) ) ) ).
% abs_triangle_ineq3
thf(fact_5769_abs__triangle__ineq3,axiom,
! [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 ) ) ) ).
% abs_triangle_ineq3
thf(fact_5770_abs__triangle__ineq3,axiom,
! [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 ) ) ) ).
% abs_triangle_ineq3
thf(fact_5771_abs__triangle__ineq2__sym,axiom,
! [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 ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_5772_abs__triangle__ineq2__sym,axiom,
! [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 ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_5773_abs__triangle__ineq2__sym,axiom,
! [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 ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_5774_abs__triangle__ineq2__sym,axiom,
! [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 ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_5775_nonzero__abs__divide,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% nonzero_abs_divide
thf(fact_5776_nonzero__abs__divide,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
= ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% nonzero_abs_divide
thf(fact_5777_abs__ge__minus__self,axiom,
! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).
% abs_ge_minus_self
thf(fact_5778_abs__ge__minus__self,axiom,
! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).
% abs_ge_minus_self
thf(fact_5779_abs__ge__minus__self,axiom,
! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).
% abs_ge_minus_self
thf(fact_5780_abs__ge__minus__self,axiom,
! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% abs_ge_minus_self
thf(fact_5781_abs__le__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
= ( ( ord_less_eq_real @ A @ B )
& ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% abs_le_iff
thf(fact_5782_abs__le__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
= ( ( ord_le3102999989581377725nteger @ A @ B )
& ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% abs_le_iff
thf(fact_5783_abs__le__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
= ( ( ord_less_eq_rat @ A @ B )
& ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% abs_le_iff
thf(fact_5784_abs__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
= ( ( ord_less_eq_int @ A @ B )
& ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% abs_le_iff
thf(fact_5785_abs__le__D2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% abs_le_D2
thf(fact_5786_abs__le__D2,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
=> ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% abs_le_D2
thf(fact_5787_abs__le__D2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% abs_le_D2
thf(fact_5788_abs__le__D2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% abs_le_D2
thf(fact_5789_abs__leI,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
=> ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).
% abs_leI
thf(fact_5790_abs__leI,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ B )
=> ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).
% abs_leI
thf(fact_5791_abs__leI,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).
% abs_leI
thf(fact_5792_abs__leI,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
=> ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% abs_leI
thf(fact_5793_sum__cong__Suc,axiom,
! [A4: set_nat,F: nat > nat,G: nat > nat] :
( ~ ( member_nat @ zero_zero_nat @ A4 )
=> ( ! [X4: nat] :
( ( member_nat @ ( suc @ X4 ) @ A4 )
=> ( ( F @ ( suc @ X4 ) )
= ( G @ ( suc @ X4 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ F @ A4 )
= ( groups3542108847815614940at_nat @ G @ A4 ) ) ) ) ).
% sum_cong_Suc
thf(fact_5794_sum__cong__Suc,axiom,
! [A4: set_nat,F: nat > real,G: nat > real] :
( ~ ( member_nat @ zero_zero_nat @ A4 )
=> ( ! [X4: nat] :
( ( member_nat @ ( suc @ X4 ) @ A4 )
=> ( ( F @ ( suc @ X4 ) )
= ( G @ ( suc @ X4 ) ) ) )
=> ( ( groups6591440286371151544t_real @ F @ A4 )
= ( groups6591440286371151544t_real @ G @ A4 ) ) ) ) ).
% sum_cong_Suc
thf(fact_5795_abs__real__def,axiom,
( abs_abs_real
= ( ^ [A5: real] : ( if_real @ ( ord_less_real @ A5 @ zero_zero_real ) @ ( uminus_uminus_real @ A5 ) @ A5 ) ) ) ).
% abs_real_def
thf(fact_5796_xor__num_Ocases,axiom,
! [X3: product_prod_num_num] :
( ( X3
!= ( product_Pair_num_num @ one @ one ) )
=> ( ! [N3: num] :
( X3
!= ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) )
=> ( ! [N3: num] :
( X3
!= ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) )
=> ( ! [M: num] :
( X3
!= ( product_Pair_num_num @ ( bit0 @ M ) @ one ) )
=> ( ! [M: num,N3: num] :
( X3
!= ( product_Pair_num_num @ ( bit0 @ M ) @ ( bit0 @ N3 ) ) )
=> ( ! [M: num,N3: num] :
( X3
!= ( product_Pair_num_num @ ( bit0 @ M ) @ ( bit1 @ N3 ) ) )
=> ( ! [M: num] :
( X3
!= ( product_Pair_num_num @ ( bit1 @ M ) @ one ) )
=> ( ! [M: num,N3: num] :
( X3
!= ( product_Pair_num_num @ ( bit1 @ M ) @ ( bit0 @ N3 ) ) )
=> ~ ! [M: num,N3: num] :
( X3
!= ( product_Pair_num_num @ ( bit1 @ M ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_5797_sum__subtractf__nat,axiom,
! [A4: set_complex,G: complex > nat,F: complex > nat] :
( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
=> ( ( groups5693394587270226106ex_nat
@ ^ [X5: complex] : ( minus_minus_nat @ ( F @ X5 ) @ ( G @ X5 ) )
@ A4 )
= ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_5798_sum__subtractf__nat,axiom,
! [A4: set_real,G: real > nat,F: real > nat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
=> ( ( groups1935376822645274424al_nat
@ ^ [X5: real] : ( minus_minus_nat @ ( F @ X5 ) @ ( G @ X5 ) )
@ A4 )
= ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ G @ A4 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_5799_sum__subtractf__nat,axiom,
! [A4: set_set_nat,G: set_nat > nat,F: set_nat > nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
=> ( ( groups8294997508430121362at_nat
@ ^ [X5: set_nat] : ( minus_minus_nat @ ( F @ X5 ) @ ( G @ X5 ) )
@ A4 )
= ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A4 ) @ ( groups8294997508430121362at_nat @ G @ A4 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_5800_sum__subtractf__nat,axiom,
! [A4: set_int,G: int > nat,F: int > nat] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
=> ( ( groups4541462559716669496nt_nat
@ ^ [X5: int] : ( minus_minus_nat @ ( F @ X5 ) @ ( G @ X5 ) )
@ A4 )
= ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_5801_sum__subtractf__nat,axiom,
! [A4: set_nat,G: nat > nat,F: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [X5: nat] : ( minus_minus_nat @ ( F @ X5 ) @ ( G @ X5 ) )
@ A4 )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ G @ A4 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_5802_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > nat,M2: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_5803_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > real,M2: nat,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_5804_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > nat,M2: nat,K2: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K2 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_5805_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > real,M2: nat,K2: nat,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) ) )
= ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K2 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_5806_dense__eq0__I,axiom,
! [X3: real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ E ) )
=> ( X3 = zero_zero_real ) ) ).
% dense_eq0_I
thf(fact_5807_dense__eq0__I,axiom,
! [X3: rat] :
( ! [E: rat] :
( ( ord_less_rat @ zero_zero_rat @ E )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ E ) )
=> ( X3 = zero_zero_rat ) ) ).
% dense_eq0_I
thf(fact_5808_abs__mult__pos,axiom,
! [X3: code_integer,Y: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
=> ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y ) @ X3 )
= ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y @ X3 ) ) ) ) ).
% abs_mult_pos
thf(fact_5809_abs__mult__pos,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( times_times_real @ ( abs_abs_real @ Y ) @ X3 )
= ( abs_abs_real @ ( times_times_real @ Y @ X3 ) ) ) ) ).
% abs_mult_pos
thf(fact_5810_abs__mult__pos,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
=> ( ( times_times_rat @ ( abs_abs_rat @ Y ) @ X3 )
= ( abs_abs_rat @ ( times_times_rat @ Y @ X3 ) ) ) ) ).
% abs_mult_pos
thf(fact_5811_abs__mult__pos,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( times_times_int @ ( abs_abs_int @ Y ) @ X3 )
= ( abs_abs_int @ ( times_times_int @ Y @ X3 ) ) ) ) ).
% abs_mult_pos
thf(fact_5812_abs__eq__mult,axiom,
! [A: code_integer,B: code_integer] :
( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
| ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
& ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
| ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
=> ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
= ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% abs_eq_mult
thf(fact_5813_abs__eq__mult,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
| ( ord_less_eq_real @ A @ zero_zero_real ) )
& ( ( ord_less_eq_real @ zero_zero_real @ B )
| ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
= ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% abs_eq_mult
thf(fact_5814_abs__eq__mult,axiom,
! [A: rat,B: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
| ( ord_less_eq_rat @ A @ zero_zero_rat ) )
& ( ( ord_less_eq_rat @ zero_zero_rat @ B )
| ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
=> ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
= ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% abs_eq_mult
thf(fact_5815_abs__eq__mult,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
| ( ord_less_eq_int @ A @ zero_zero_int ) )
& ( ( ord_less_eq_int @ zero_zero_int @ B )
| ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
= ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% abs_eq_mult
thf(fact_5816_abs__eq__iff_H,axiom,
! [A: real,B: real] :
( ( ( abs_abs_real @ A )
= B )
= ( ( ord_less_eq_real @ zero_zero_real @ B )
& ( ( A = B )
| ( A
= ( uminus_uminus_real @ B ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_5817_abs__eq__iff_H,axiom,
! [A: code_integer,B: code_integer] :
( ( ( abs_abs_Code_integer @ A )
= B )
= ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
& ( ( A = B )
| ( A
= ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_5818_abs__eq__iff_H,axiom,
! [A: rat,B: rat] :
( ( ( abs_abs_rat @ A )
= B )
= ( ( ord_less_eq_rat @ zero_zero_rat @ B )
& ( ( A = B )
| ( A
= ( uminus_uminus_rat @ B ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_5819_abs__eq__iff_H,axiom,
! [A: int,B: int] :
( ( ( abs_abs_int @ A )
= B )
= ( ( ord_less_eq_int @ zero_zero_int @ B )
& ( ( A = B )
| ( A
= ( uminus_uminus_int @ B ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_5820_eq__abs__iff_H,axiom,
! [A: real,B: real] :
( ( A
= ( abs_abs_real @ B ) )
= ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ( B = A )
| ( B
= ( uminus_uminus_real @ A ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_5821_eq__abs__iff_H,axiom,
! [A: code_integer,B: code_integer] :
( ( A
= ( abs_abs_Code_integer @ B ) )
= ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
& ( ( B = A )
| ( B
= ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_5822_eq__abs__iff_H,axiom,
! [A: rat,B: rat] :
( ( A
= ( abs_abs_rat @ B ) )
= ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ( B = A )
| ( B
= ( uminus_uminus_rat @ A ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_5823_eq__abs__iff_H,axiom,
! [A: int,B: int] :
( ( A
= ( abs_abs_int @ B ) )
= ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ( B = A )
| ( B
= ( uminus_uminus_int @ A ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_5824_abs__minus__le__zero,axiom,
! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).
% abs_minus_le_zero
thf(fact_5825_abs__minus__le__zero,axiom,
! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).
% abs_minus_le_zero
thf(fact_5826_abs__minus__le__zero,axiom,
! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).
% abs_minus_le_zero
thf(fact_5827_abs__minus__le__zero,axiom,
! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% abs_minus_le_zero
thf(fact_5828_abs__div__pos,axiom,
! [Y: real,X3: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( divide_divide_real @ ( abs_abs_real @ X3 ) @ Y )
= ( abs_abs_real @ ( divide_divide_real @ X3 @ Y ) ) ) ) ).
% abs_div_pos
thf(fact_5829_abs__div__pos,axiom,
! [Y: rat,X3: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ( divide_divide_rat @ ( abs_abs_rat @ X3 ) @ Y )
= ( abs_abs_rat @ ( divide_divide_rat @ X3 @ Y ) ) ) ) ).
% abs_div_pos
thf(fact_5830_zero__le__power__abs,axiom,
! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% zero_le_power_abs
thf(fact_5831_zero__le__power__abs,axiom,
! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% zero_le_power_abs
thf(fact_5832_zero__le__power__abs,axiom,
! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% zero_le_power_abs
thf(fact_5833_zero__le__power__abs,axiom,
! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% zero_le_power_abs
thf(fact_5834_abs__if,axiom,
( abs_abs_real
= ( ^ [A5: real] : ( if_real @ ( ord_less_real @ A5 @ zero_zero_real ) @ ( uminus_uminus_real @ A5 ) @ A5 ) ) ) ).
% abs_if
thf(fact_5835_abs__if,axiom,
( abs_abs_int
= ( ^ [A5: int] : ( if_int @ ( ord_less_int @ A5 @ zero_zero_int ) @ ( uminus_uminus_int @ A5 ) @ A5 ) ) ) ).
% abs_if
thf(fact_5836_abs__if,axiom,
( abs_abs_rat
= ( ^ [A5: rat] : ( if_rat @ ( ord_less_rat @ A5 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A5 ) @ A5 ) ) ) ).
% abs_if
thf(fact_5837_abs__if,axiom,
( abs_abs_Code_integer
= ( ^ [A5: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A5 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A5 ) @ A5 ) ) ) ).
% abs_if
thf(fact_5838_abs__if__raw,axiom,
( abs_abs_real
= ( ^ [A5: real] : ( if_real @ ( ord_less_real @ A5 @ zero_zero_real ) @ ( uminus_uminus_real @ A5 ) @ A5 ) ) ) ).
% abs_if_raw
thf(fact_5839_abs__if__raw,axiom,
( abs_abs_int
= ( ^ [A5: int] : ( if_int @ ( ord_less_int @ A5 @ zero_zero_int ) @ ( uminus_uminus_int @ A5 ) @ A5 ) ) ) ).
% abs_if_raw
thf(fact_5840_abs__if__raw,axiom,
( abs_abs_rat
= ( ^ [A5: rat] : ( if_rat @ ( ord_less_rat @ A5 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A5 ) @ A5 ) ) ) ).
% abs_if_raw
thf(fact_5841_abs__if__raw,axiom,
( abs_abs_Code_integer
= ( ^ [A5: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A5 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A5 ) @ A5 ) ) ) ).
% abs_if_raw
thf(fact_5842_abs__of__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( abs_abs_real @ A )
= ( uminus_uminus_real @ A ) ) ) ).
% abs_of_neg
thf(fact_5843_abs__of__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( abs_abs_int @ A )
= ( uminus_uminus_int @ A ) ) ) ).
% abs_of_neg
thf(fact_5844_abs__of__neg,axiom,
! [A: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( abs_abs_rat @ A )
= ( uminus_uminus_rat @ A ) ) ) ).
% abs_of_neg
thf(fact_5845_abs__of__neg,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
=> ( ( abs_abs_Code_integer @ A )
= ( uminus1351360451143612070nteger @ A ) ) ) ).
% abs_of_neg
thf(fact_5846_abs__diff__le__iff,axiom,
! [X3: code_integer,A: code_integer,R2: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X3 @ A ) ) @ R2 )
= ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X3 )
& ( ord_le3102999989581377725nteger @ X3 @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% abs_diff_le_iff
thf(fact_5847_abs__diff__le__iff,axiom,
! [X3: real,A: real,R2: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ A ) ) @ R2 )
= ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R2 ) @ X3 )
& ( ord_less_eq_real @ X3 @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% abs_diff_le_iff
thf(fact_5848_abs__diff__le__iff,axiom,
! [X3: rat,A: rat,R2: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X3 @ A ) ) @ R2 )
= ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R2 ) @ X3 )
& ( ord_less_eq_rat @ X3 @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% abs_diff_le_iff
thf(fact_5849_abs__diff__le__iff,axiom,
! [X3: int,A: int,R2: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ A ) ) @ R2 )
= ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X3 )
& ( ord_less_eq_int @ X3 @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% abs_diff_le_iff
thf(fact_5850_abs__triangle__ineq4,axiom,
! [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 ) ) ) ).
% abs_triangle_ineq4
thf(fact_5851_abs__triangle__ineq4,axiom,
! [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 ) ) ) ).
% abs_triangle_ineq4
thf(fact_5852_abs__triangle__ineq4,axiom,
! [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 ) ) ) ).
% abs_triangle_ineq4
thf(fact_5853_abs__triangle__ineq4,axiom,
! [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 ) ) ) ).
% abs_triangle_ineq4
thf(fact_5854_abs__diff__triangle__ineq,axiom,
! [A: code_integer,B: code_integer,C2: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( plus_p5714425477246183910nteger @ C2 @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C2 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ D ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_5855_abs__diff__triangle__ineq,axiom,
! [A: real,B: real,C2: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C2 @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C2 ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_5856_abs__diff__triangle__ineq,axiom,
! [A: rat,B: rat,C2: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ C2 @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C2 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_5857_abs__diff__triangle__ineq,axiom,
! [A: int,B: int,C2: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C2 @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_5858_abs__diff__less__iff,axiom,
! [X3: code_integer,A: code_integer,R2: code_integer] :
( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X3 @ A ) ) @ R2 )
= ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X3 )
& ( ord_le6747313008572928689nteger @ X3 @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_5859_abs__diff__less__iff,axiom,
! [X3: real,A: real,R2: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ A ) ) @ R2 )
= ( ( ord_less_real @ ( minus_minus_real @ A @ R2 ) @ X3 )
& ( ord_less_real @ X3 @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_5860_abs__diff__less__iff,axiom,
! [X3: rat,A: rat,R2: rat] :
( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X3 @ A ) ) @ R2 )
= ( ( ord_less_rat @ ( minus_minus_rat @ A @ R2 ) @ X3 )
& ( ord_less_rat @ X3 @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_5861_abs__diff__less__iff,axiom,
! [X3: int,A: int,R2: int] :
( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ A ) ) @ R2 )
= ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X3 )
& ( ord_less_int @ X3 @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_5862_sum__eq__Suc0__iff,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( ( groups4541462559716669496nt_nat @ F @ A4 )
= ( suc @ zero_zero_nat ) )
= ( ? [X5: int] :
( ( member_int @ X5 @ A4 )
& ( ( F @ X5 )
= ( suc @ zero_zero_nat ) )
& ! [Y5: int] :
( ( member_int @ Y5 @ A4 )
=> ( ( X5 != Y5 )
=> ( ( F @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_5863_sum__eq__Suc0__iff,axiom,
! [A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( groups5693394587270226106ex_nat @ F @ A4 )
= ( suc @ zero_zero_nat ) )
= ( ? [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( ( F @ X5 )
= ( suc @ zero_zero_nat ) )
& ! [Y5: complex] :
( ( member_complex @ Y5 @ A4 )
=> ( ( X5 != Y5 )
=> ( ( F @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_5864_sum__eq__Suc0__iff,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ( ( groups977919841031483927at_nat @ F @ A4 )
= ( suc @ zero_zero_nat ) )
= ( ? [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ A4 )
& ( ( F @ X5 )
= ( suc @ zero_zero_nat ) )
& ! [Y5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y5 @ A4 )
=> ( ( X5 != Y5 )
=> ( ( F @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_5865_sum__eq__Suc0__iff,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( groups3542108847815614940at_nat @ F @ A4 )
= ( suc @ zero_zero_nat ) )
= ( ? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( ( F @ X5 )
= ( suc @ zero_zero_nat ) )
& ! [Y5: nat] :
( ( member_nat @ Y5 @ A4 )
=> ( ( X5 != Y5 )
=> ( ( F @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_5866_sum__SucD,axiom,
! [F: nat > nat,A4: set_nat,N: nat] :
( ( ( groups3542108847815614940at_nat @ F @ A4 )
= ( suc @ N ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ).
% sum_SucD
thf(fact_5867_sum__eq__1__iff,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( ( groups4541462559716669496nt_nat @ F @ A4 )
= one_one_nat )
= ( ? [X5: int] :
( ( member_int @ X5 @ A4 )
& ( ( F @ X5 )
= one_one_nat )
& ! [Y5: int] :
( ( member_int @ Y5 @ A4 )
=> ( ( X5 != Y5 )
=> ( ( F @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_5868_sum__eq__1__iff,axiom,
! [A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( groups5693394587270226106ex_nat @ F @ A4 )
= one_one_nat )
= ( ? [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( ( F @ X5 )
= one_one_nat )
& ! [Y5: complex] :
( ( member_complex @ Y5 @ A4 )
=> ( ( X5 != Y5 )
=> ( ( F @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_5869_sum__eq__1__iff,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ( ( groups977919841031483927at_nat @ F @ A4 )
= one_one_nat )
= ( ? [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ A4 )
& ( ( F @ X5 )
= one_one_nat )
& ! [Y5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y5 @ A4 )
=> ( ( X5 != Y5 )
=> ( ( F @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_5870_sum__eq__1__iff,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( groups3542108847815614940at_nat @ F @ A4 )
= one_one_nat )
= ( ? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( ( F @ X5 )
= one_one_nat )
& ! [Y5: nat] :
( ( member_nat @ Y5 @ A4 )
=> ( ( X5 != Y5 )
=> ( ( F @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_5871_numeral__Bit1,axiom,
! [N: num] :
( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
= ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% numeral_Bit1
thf(fact_5872_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit1 @ N ) )
= ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% numeral_Bit1
thf(fact_5873_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit1 @ N ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% numeral_Bit1
thf(fact_5874_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% numeral_Bit1
thf(fact_5875_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit1 @ N ) )
= ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% numeral_Bit1
thf(fact_5876_eval__nat__numeral_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).
% eval_nat_numeral(3)
thf(fact_5877_lemma__interval__lt,axiom,
! [A: real,X3: real,B: real] :
( ( ord_less_real @ A @ X3 )
=> ( ( ord_less_real @ X3 @ B )
=> ? [D2: real] :
( ( ord_less_real @ zero_zero_real @ D2 )
& ! [Y6: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y6 ) ) @ D2 )
=> ( ( ord_less_real @ A @ Y6 )
& ( ord_less_real @ Y6 @ B ) ) ) ) ) ) ).
% lemma_interval_lt
thf(fact_5878_sum__power__add,axiom,
! [X3: complex,M2: nat,I6: set_nat] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( power_power_complex @ X3 @ ( plus_plus_nat @ M2 @ I3 ) )
@ I6 )
= ( times_times_complex @ ( power_power_complex @ X3 @ M2 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ I6 ) ) ) ).
% sum_power_add
thf(fact_5879_sum__power__add,axiom,
! [X3: rat,M2: nat,I6: set_nat] :
( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( power_power_rat @ X3 @ ( plus_plus_nat @ M2 @ I3 ) )
@ I6 )
= ( times_times_rat @ ( power_power_rat @ X3 @ M2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ I6 ) ) ) ).
% sum_power_add
thf(fact_5880_sum__power__add,axiom,
! [X3: int,M2: nat,I6: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( power_power_int @ X3 @ ( plus_plus_nat @ M2 @ I3 ) )
@ I6 )
= ( times_times_int @ ( power_power_int @ X3 @ M2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ I6 ) ) ) ).
% sum_power_add
thf(fact_5881_sum__power__add,axiom,
! [X3: real,M2: nat,I6: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( power_power_real @ X3 @ ( plus_plus_nat @ M2 @ I3 ) )
@ I6 )
= ( times_times_real @ ( power_power_real @ X3 @ M2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ I6 ) ) ) ).
% sum_power_add
thf(fact_5882_sum_OatLeastAtMost__rev,axiom,
! [G: nat > nat,N: nat,M2: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M2 ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ I3 ) )
@ ( set_or1269000886237332187st_nat @ N @ M2 ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_5883_sum_OatLeastAtMost__rev,axiom,
! [G: nat > real,N: nat,M2: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N @ M2 ) )
= ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ I3 ) )
@ ( set_or1269000886237332187st_nat @ N @ M2 ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_5884_numeral__code_I3_J,axiom,
! [N: num] :
( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
= ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% numeral_code(3)
thf(fact_5885_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit1 @ N ) )
= ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% numeral_code(3)
thf(fact_5886_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit1 @ N ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% numeral_code(3)
thf(fact_5887_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% numeral_code(3)
thf(fact_5888_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit1 @ N ) )
= ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% numeral_code(3)
thf(fact_5889_sum__nth__roots,axiom,
! [N: nat,C2: complex] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( groups7754918857620584856omplex
@ ^ [X5: complex] : X5
@ ( collect_complex
@ ^ [Z6: complex] :
( ( power_power_complex @ Z6 @ N )
= C2 ) ) )
= zero_zero_complex ) ) ).
% sum_nth_roots
thf(fact_5890_sum__roots__unity,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( groups7754918857620584856omplex
@ ^ [X5: complex] : X5
@ ( collect_complex
@ ^ [Z6: complex] :
( ( power_power_complex @ Z6 @ N )
= one_one_complex ) ) )
= zero_zero_complex ) ) ).
% sum_roots_unity
thf(fact_5891_abs__add__one__gt__zero,axiom,
! [X3: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X3 ) ) ) ).
% abs_add_one_gt_zero
thf(fact_5892_abs__add__one__gt__zero,axiom,
! [X3: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X3 ) ) ) ).
% abs_add_one_gt_zero
thf(fact_5893_abs__add__one__gt__zero,axiom,
! [X3: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X3 ) ) ) ).
% abs_add_one_gt_zero
thf(fact_5894_abs__add__one__gt__zero,axiom,
! [X3: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X3 ) ) ) ).
% abs_add_one_gt_zero
thf(fact_5895_sum__diff__nat,axiom,
! [B5: set_complex,A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
= ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_5896_sum__diff__nat,axiom,
! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ B5 )
=> ( ( ord_le3146513528884898305at_nat @ B5 @ A4 )
=> ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) )
= ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A4 ) @ ( groups977919841031483927at_nat @ F @ B5 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_5897_sum__diff__nat,axiom,
! [B5: set_int,A4: set_int,F: int > nat] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
= ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ F @ B5 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_5898_sum__diff__nat,axiom,
! [B5: set_nat,A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ F @ B5 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_5899_sum__shift__lb__Suc0__0,axiom,
! [F: nat > complex,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_complex )
=> ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_5900_sum__shift__lb__Suc0__0,axiom,
! [F: nat > rat,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_rat )
=> ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_5901_sum__shift__lb__Suc0__0,axiom,
! [F: nat > int,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_int )
=> ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_5902_sum__shift__lb__Suc0__0,axiom,
! [F: nat > nat,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_nat )
=> ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_5903_sum__shift__lb__Suc0__0,axiom,
! [F: nat > real,K2: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_real )
=> ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
= ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_5904_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_5905_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_5906_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_5907_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_5908_sum_Onat__ivl__Suc_H,axiom,
! [M2: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( G @ ( suc @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_5909_sum_Onat__ivl__Suc_H,axiom,
! [M2: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_int @ ( G @ ( suc @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_5910_sum_Onat__ivl__Suc_H,axiom,
! [M2: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ ( suc @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_5911_sum_Onat__ivl__Suc_H,axiom,
! [M2: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus_real @ ( G @ ( suc @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_5912_sum_OatLeast__Suc__atMost,axiom,
! [M2: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( plus_plus_rat @ ( G @ M2 ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_5913_sum_OatLeast__Suc__atMost,axiom,
! [M2: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( plus_plus_int @ ( G @ M2 ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_5914_sum_OatLeast__Suc__atMost,axiom,
! [M2: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( G @ M2 ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_5915_sum_OatLeast__Suc__atMost,axiom,
! [M2: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( plus_plus_real @ ( G @ M2 ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_5916_cong__exp__iff__simps_I3_J,axiom,
! [N: num,Q3: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
!= zero_zero_nat ) ).
% cong_exp_iff_simps(3)
thf(fact_5917_cong__exp__iff__simps_I3_J,axiom,
! [N: num,Q3: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
!= zero_zero_int ) ).
% cong_exp_iff_simps(3)
thf(fact_5918_cong__exp__iff__simps_I3_J,axiom,
! [N: num,Q3: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
!= zero_z3403309356797280102nteger ) ).
% cong_exp_iff_simps(3)
thf(fact_5919_numeral__3__eq__3,axiom,
( ( numeral_numeral_nat @ ( bit1 @ one ) )
= ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% numeral_3_eq_3
thf(fact_5920_lemma__interval,axiom,
! [A: real,X3: real,B: real] :
( ( ord_less_real @ A @ X3 )
=> ( ( ord_less_real @ X3 @ B )
=> ? [D2: real] :
( ( ord_less_real @ zero_zero_real @ D2 )
& ! [Y6: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y6 ) ) @ D2 )
=> ( ( ord_less_eq_real @ A @ Y6 )
& ( ord_less_eq_real @ Y6 @ B ) ) ) ) ) ) ).
% lemma_interval
thf(fact_5921_Suc3__eq__add__3,axiom,
! [N: nat] :
( ( suc @ ( suc @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% Suc3_eq_add_3
thf(fact_5922_sum_OSuc__reindex__ivl,axiom,
! [M2: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( G @ M2 )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_5923_sum_OSuc__reindex__ivl,axiom,
! [M2: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus_int @ ( G @ M2 )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_5924_sum_OSuc__reindex__ivl,axiom,
! [M2: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ M2 )
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_5925_sum_OSuc__reindex__ivl,axiom,
! [M2: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus_real @ ( G @ M2 )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_5926_sum__Suc__diff,axiom,
! [M2: nat,N: nat,F: nat > rat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( minus_minus_rat @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ M2 ) ) ) ) ).
% sum_Suc_diff
thf(fact_5927_sum__Suc__diff,axiom,
! [M2: nat,N: nat,F: nat > int] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ M2 ) ) ) ) ).
% sum_Suc_diff
thf(fact_5928_sum__Suc__diff,axiom,
! [M2: nat,N: nat,F: nat > real] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( minus_minus_real @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ M2 ) ) ) ) ).
% sum_Suc_diff
thf(fact_5929_sum__atLeastAtMost__code,axiom,
! [F: nat > complex,A: nat,B: nat] :
( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo1517530859248394432omplex
@ ^ [A5: nat] : ( plus_plus_complex @ ( F @ A5 ) )
@ A
@ B
@ zero_zero_complex ) ) ).
% sum_atLeastAtMost_code
thf(fact_5930_sum__atLeastAtMost__code,axiom,
! [F: nat > rat,A: nat,B: nat] :
( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo1949268297981939178at_rat
@ ^ [A5: nat] : ( plus_plus_rat @ ( F @ A5 ) )
@ A
@ B
@ zero_zero_rat ) ) ).
% sum_atLeastAtMost_code
thf(fact_5931_sum__atLeastAtMost__code,axiom,
! [F: nat > int,A: nat,B: nat] :
( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo2581907887559384638at_int
@ ^ [A5: nat] : ( plus_plus_int @ ( F @ A5 ) )
@ A
@ B
@ zero_zero_int ) ) ).
% sum_atLeastAtMost_code
thf(fact_5932_sum__atLeastAtMost__code,axiom,
! [F: nat > nat,A: nat,B: nat] :
( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo2584398358068434914at_nat
@ ^ [A5: nat] : ( plus_plus_nat @ ( F @ A5 ) )
@ A
@ B
@ zero_zero_nat ) ) ).
% sum_atLeastAtMost_code
thf(fact_5933_sum__atLeastAtMost__code,axiom,
! [F: nat > real,A: nat,B: nat] :
( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo3111899725591712190t_real
@ ^ [A5: nat] : ( plus_plus_real @ ( F @ A5 ) )
@ A
@ B
@ zero_zero_real ) ) ).
% sum_atLeastAtMost_code
thf(fact_5934_abs__le__square__iff,axiom,
! [X3: code_integer,Y: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X3 ) @ ( abs_abs_Code_integer @ Y ) )
= ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_5935_abs__le__square__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ ( abs_abs_real @ Y ) )
= ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_5936_abs__le__square__iff,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ ( abs_abs_rat @ Y ) )
= ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_5937_abs__le__square__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ X3 ) @ ( abs_abs_int @ Y ) )
= ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_5938_num_Osize_I6_J,axiom,
! [X32: num] :
( ( size_size_num @ ( bit1 @ X32 ) )
= ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size(6)
thf(fact_5939_num_Osize__gen_I3_J,axiom,
! [X32: num] :
( ( size_num @ ( bit1 @ X32 ) )
= ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size_gen(3)
thf(fact_5940_sum_Oub__add__nat,axiom,
! [M2: nat,N: nat,G: nat > rat,P: nat] :
( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_5941_sum_Oub__add__nat,axiom,
! [M2: nat,N: nat,G: nat > int,P: nat] :
( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_5942_sum_Oub__add__nat,axiom,
! [M2: nat,N: nat,G: nat > nat,P: nat] :
( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_5943_sum_Oub__add__nat,axiom,
! [M2: nat,N: nat,G: nat > real,P: nat] :
( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_5944_cong__exp__iff__simps_I11_J,axiom,
! [M2: num,Q3: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ Q3 ) )
= zero_zero_nat ) ) ).
% cong_exp_iff_simps(11)
thf(fact_5945_cong__exp__iff__simps_I11_J,axiom,
! [M2: num,Q3: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ Q3 ) )
= zero_zero_int ) ) ).
% cong_exp_iff_simps(11)
thf(fact_5946_cong__exp__iff__simps_I11_J,axiom,
! [M2: num,Q3: num] :
( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ Q3 ) )
= zero_z3403309356797280102nteger ) ) ).
% cong_exp_iff_simps(11)
thf(fact_5947_cong__exp__iff__simps_I7_J,axiom,
! [Q3: num,N: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) )
= zero_zero_nat ) ) ).
% cong_exp_iff_simps(7)
thf(fact_5948_cong__exp__iff__simps_I7_J,axiom,
! [Q3: num,N: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) )
= zero_zero_int ) ) ).
% cong_exp_iff_simps(7)
thf(fact_5949_cong__exp__iff__simps_I7_J,axiom,
! [Q3: num,N: num] :
( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) )
= zero_z3403309356797280102nteger ) ) ).
% cong_exp_iff_simps(7)
thf(fact_5950_Suc__div__eq__add3__div,axiom,
! [M2: nat,N: nat] :
( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ N )
= ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ N ) ) ).
% Suc_div_eq_add3_div
thf(fact_5951_sum__count__set,axiom,
! [Xs: list_complex,X8: set_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ X8 )
=> ( ( finite3207457112153483333omplex @ X8 )
=> ( ( groups5693394587270226106ex_nat @ ( count_list_complex @ Xs ) @ X8 )
= ( size_s3451745648224563538omplex @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_5952_sum__count__set,axiom,
! [Xs: list_P6011104703257516679at_nat,X8: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ X8 )
=> ( ( finite6177210948735845034at_nat @ X8 )
=> ( ( groups977919841031483927at_nat @ ( count_4203492906077236349at_nat @ Xs ) @ X8 )
= ( size_s5460976970255530739at_nat @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_5953_sum__count__set,axiom,
! [Xs: list_VEBT_VEBT,X8: set_VEBT_VEBT] :
( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ X8 )
=> ( ( finite5795047828879050333T_VEBT @ X8 )
=> ( ( groups771621172384141258BT_nat @ ( count_list_VEBT_VEBT @ Xs ) @ X8 )
= ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_5954_sum__count__set,axiom,
! [Xs: list_int,X8: set_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ X8 )
=> ( ( finite_finite_int @ X8 )
=> ( ( groups4541462559716669496nt_nat @ ( count_list_int @ Xs ) @ X8 )
= ( size_size_list_int @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_5955_sum__count__set,axiom,
! [Xs: list_nat,X8: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ X8 )
=> ( ( finite_finite_nat @ X8 )
=> ( ( groups3542108847815614940at_nat @ ( count_list_nat @ Xs ) @ X8 )
= ( size_size_list_nat @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_5956_Suc__mod__eq__add3__mod,axiom,
! [M2: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ N )
= ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ N ) ) ).
% Suc_mod_eq_add3_mod
thf(fact_5957_set__encode__def,axiom,
( nat_set_encode
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% set_encode_def
thf(fact_5958_divmod__int__def,axiom,
( unique5052692396658037445od_int
= ( ^ [M5: num,N4: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N4 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N4 ) ) ) ) ) ).
% divmod_int_def
thf(fact_5959_Divides_Oadjust__div__def,axiom,
( adjust_div
= ( produc8211389475949308722nt_int
@ ^ [Q4: int,R5: int] : ( plus_plus_int @ Q4 @ ( zero_n2684676970156552555ol_int @ ( R5 != zero_zero_int ) ) ) ) ) ).
% Divides.adjust_div_def
thf(fact_5960_abs__sqrt__wlog,axiom,
! [P2: code_integer > code_integer > $o,X3: code_integer] :
( ! [X4: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X4 )
=> ( P2 @ X4 @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P2 @ ( abs_abs_Code_integer @ X3 ) @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_5961_abs__sqrt__wlog,axiom,
! [P2: real > real > $o,X3: real] :
( ! [X4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
=> ( P2 @ X4 @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P2 @ ( abs_abs_real @ X3 ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_5962_abs__sqrt__wlog,axiom,
! [P2: rat > rat > $o,X3: rat] :
( ! [X4: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
=> ( P2 @ X4 @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P2 @ ( abs_abs_rat @ X3 ) @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_5963_abs__sqrt__wlog,axiom,
! [P2: int > int > $o,X3: int] :
( ! [X4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( P2 @ X4 @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P2 @ ( abs_abs_int @ X3 ) @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_5964_power2__le__iff__abs__le,axiom,
! [Y: code_integer,X3: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
=> ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X3 ) @ Y ) ) ) ).
% power2_le_iff_abs_le
thf(fact_5965_power2__le__iff__abs__le,axiom,
! [Y: real,X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ Y ) ) ) ).
% power2_le_iff_abs_le
thf(fact_5966_power2__le__iff__abs__le,axiom,
! [Y: rat,X3: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ Y ) ) ) ).
% power2_le_iff_abs_le
thf(fact_5967_power2__le__iff__abs__le,axiom,
! [Y: int,X3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_int @ ( abs_abs_int @ X3 ) @ Y ) ) ) ).
% power2_le_iff_abs_le
thf(fact_5968_abs__square__le__1,axiom,
! [X3: code_integer] :
( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
= ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X3 ) @ one_one_Code_integer ) ) ).
% abs_square_le_1
thf(fact_5969_abs__square__le__1,axiom,
! [X3: real] :
( ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
= ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real ) ) ).
% abs_square_le_1
thf(fact_5970_abs__square__le__1,axiom,
! [X3: rat] :
( ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
= ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ one_one_rat ) ) ).
% abs_square_le_1
thf(fact_5971_abs__square__le__1,axiom,
! [X3: int] :
( ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
= ( ord_less_eq_int @ ( abs_abs_int @ X3 ) @ one_one_int ) ) ).
% abs_square_le_1
thf(fact_5972_divmod__def,axiom,
( unique5052692396658037445od_int
= ( ^ [M5: num,N4: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N4 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N4 ) ) ) ) ) ).
% divmod_def
thf(fact_5973_divmod__def,axiom,
( unique5055182867167087721od_nat
= ( ^ [M5: num,N4: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N4 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N4 ) ) ) ) ) ).
% divmod_def
thf(fact_5974_divmod__def,axiom,
( unique3479559517661332726nteger
= ( ^ [M5: num,N4: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M5 ) @ ( numera6620942414471956472nteger @ N4 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M5 ) @ ( numera6620942414471956472nteger @ N4 ) ) ) ) ) ).
% divmod_def
thf(fact_5975_power__mono__even,axiom,
! [N: nat,A: code_integer,B: code_integer] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
=> ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% power_mono_even
thf(fact_5976_power__mono__even,axiom,
! [N: nat,A: real,B: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% power_mono_even
thf(fact_5977_power__mono__even,axiom,
! [N: nat,A: rat,B: rat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% power_mono_even
thf(fact_5978_power__mono__even,axiom,
! [N: nat,A: int,B: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono_even
thf(fact_5979_divmod_H__nat__def,axiom,
( unique5055182867167087721od_nat
= ( ^ [M5: num,N4: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N4 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N4 ) ) ) ) ) ).
% divmod'_nat_def
thf(fact_5980_convex__sum__bound__le,axiom,
! [I6: set_complex,X3: complex > code_integer,A: complex > code_integer,B: code_integer,Delta: code_integer] :
( ! [I2: complex] :
( ( member_complex @ I2 @ I6 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X3 @ I2 ) ) )
=> ( ( ( groups6621422865394947399nteger @ X3 @ I6 )
= one_one_Code_integer )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I6 )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_le3102999989581377725nteger
@ ( abs_abs_Code_integer
@ ( minus_8373710615458151222nteger
@ ( groups6621422865394947399nteger
@ ^ [I3: complex] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X3 @ I3 ) )
@ I6 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_5981_convex__sum__bound__le,axiom,
! [I6: set_real,X3: real > code_integer,A: real > code_integer,B: code_integer,Delta: code_integer] :
( ! [I2: real] :
( ( member_real @ I2 @ I6 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X3 @ I2 ) ) )
=> ( ( ( groups7713935264441627589nteger @ X3 @ I6 )
= one_one_Code_integer )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I6 )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_le3102999989581377725nteger
@ ( abs_abs_Code_integer
@ ( minus_8373710615458151222nteger
@ ( groups7713935264441627589nteger
@ ^ [I3: real] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X3 @ I3 ) )
@ I6 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_5982_convex__sum__bound__le,axiom,
! [I6: set_nat,X3: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
( ! [I2: nat] :
( ( member_nat @ I2 @ I6 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X3 @ I2 ) ) )
=> ( ( ( groups7501900531339628137nteger @ X3 @ I6 )
= one_one_Code_integer )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I6 )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_le3102999989581377725nteger
@ ( abs_abs_Code_integer
@ ( minus_8373710615458151222nteger
@ ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X3 @ I3 ) )
@ I6 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_5983_convex__sum__bound__le,axiom,
! [I6: set_int,X3: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
( ! [I2: int] :
( ( member_int @ I2 @ I6 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X3 @ I2 ) ) )
=> ( ( ( groups7873554091576472773nteger @ X3 @ I6 )
= one_one_Code_integer )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I6 )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_le3102999989581377725nteger
@ ( abs_abs_Code_integer
@ ( minus_8373710615458151222nteger
@ ( groups7873554091576472773nteger
@ ^ [I3: int] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X3 @ I3 ) )
@ I6 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_5984_convex__sum__bound__le,axiom,
! [I6: set_complex,X3: complex > real,A: complex > real,B: real,Delta: real] :
( ! [I2: complex] :
( ( member_complex @ I2 @ I6 )
=> ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I2 ) ) )
=> ( ( ( groups5808333547571424918x_real @ X3 @ I6 )
= one_one_real )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I6 )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_real
@ ( abs_abs_real
@ ( minus_minus_real
@ ( groups5808333547571424918x_real
@ ^ [I3: complex] : ( times_times_real @ ( A @ I3 ) @ ( X3 @ I3 ) )
@ I6 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_5985_convex__sum__bound__le,axiom,
! [I6: set_real,X3: real > real,A: real > real,B: real,Delta: real] :
( ! [I2: real] :
( ( member_real @ I2 @ I6 )
=> ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I2 ) ) )
=> ( ( ( groups8097168146408367636l_real @ X3 @ I6 )
= one_one_real )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I6 )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_real
@ ( abs_abs_real
@ ( minus_minus_real
@ ( groups8097168146408367636l_real
@ ^ [I3: real] : ( times_times_real @ ( A @ I3 ) @ ( X3 @ I3 ) )
@ I6 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_5986_convex__sum__bound__le,axiom,
! [I6: set_int,X3: int > real,A: int > real,B: real,Delta: real] :
( ! [I2: int] :
( ( member_int @ I2 @ I6 )
=> ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I2 ) ) )
=> ( ( ( groups8778361861064173332t_real @ X3 @ I6 )
= one_one_real )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I6 )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_real
@ ( abs_abs_real
@ ( minus_minus_real
@ ( groups8778361861064173332t_real
@ ^ [I3: int] : ( times_times_real @ ( A @ I3 ) @ ( X3 @ I3 ) )
@ I6 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_5987_convex__sum__bound__le,axiom,
! [I6: set_complex,X3: complex > rat,A: complex > rat,B: rat,Delta: rat] :
( ! [I2: complex] :
( ( member_complex @ I2 @ I6 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( X3 @ I2 ) ) )
=> ( ( ( groups5058264527183730370ex_rat @ X3 @ I6 )
= one_one_rat )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I6 )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_rat
@ ( abs_abs_rat
@ ( minus_minus_rat
@ ( groups5058264527183730370ex_rat
@ ^ [I3: complex] : ( times_times_rat @ ( A @ I3 ) @ ( X3 @ I3 ) )
@ I6 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_5988_convex__sum__bound__le,axiom,
! [I6: set_real,X3: real > rat,A: real > rat,B: rat,Delta: rat] :
( ! [I2: real] :
( ( member_real @ I2 @ I6 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( X3 @ I2 ) ) )
=> ( ( ( groups1300246762558778688al_rat @ X3 @ I6 )
= one_one_rat )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I6 )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_rat
@ ( abs_abs_rat
@ ( minus_minus_rat
@ ( groups1300246762558778688al_rat
@ ^ [I3: real] : ( times_times_rat @ ( A @ I3 ) @ ( X3 @ I3 ) )
@ I6 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_5989_convex__sum__bound__le,axiom,
! [I6: set_nat,X3: nat > rat,A: nat > rat,B: rat,Delta: rat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ I6 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( X3 @ I2 ) ) )
=> ( ( ( groups2906978787729119204at_rat @ X3 @ I6 )
= one_one_rat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I6 )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_rat
@ ( abs_abs_rat
@ ( minus_minus_rat
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( X3 @ I3 ) )
@ I6 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_5990_sum__natinterval__diff,axiom,
! [M2: nat,N: nat,F: nat > complex] :
( ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( minus_minus_complex @ ( F @ M2 ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= zero_zero_complex ) ) ) ).
% sum_natinterval_diff
thf(fact_5991_sum__natinterval__diff,axiom,
! [M2: nat,N: nat,F: nat > rat] :
( ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( minus_minus_rat @ ( F @ M2 ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= zero_zero_rat ) ) ) ).
% sum_natinterval_diff
thf(fact_5992_sum__natinterval__diff,axiom,
! [M2: nat,N: nat,F: nat > int] :
( ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( minus_minus_int @ ( F @ M2 ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= zero_zero_int ) ) ) ).
% sum_natinterval_diff
thf(fact_5993_sum__natinterval__diff,axiom,
! [M2: nat,N: nat,F: nat > real] :
( ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( minus_minus_real @ ( F @ M2 ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= zero_zero_real ) ) ) ).
% sum_natinterval_diff
thf(fact_5994_sum__telescope_H_H,axiom,
! [M2: nat,N: nat,F: nat > rat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) )
= ( minus_minus_rat @ ( F @ N ) @ ( F @ M2 ) ) ) ) ).
% sum_telescope''
thf(fact_5995_sum__telescope_H_H,axiom,
! [M2: nat,N: nat,F: nat > int] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) )
= ( minus_minus_int @ ( F @ N ) @ ( F @ M2 ) ) ) ) ).
% sum_telescope''
thf(fact_5996_sum__telescope_H_H,axiom,
! [M2: nat,N: nat,F: nat > real] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) )
= ( minus_minus_real @ ( F @ N ) @ ( F @ M2 ) ) ) ) ).
% sum_telescope''
thf(fact_5997_divmod__nat__def,axiom,
( divmod_nat
= ( ^ [M5: nat,N4: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M5 @ N4 ) @ ( modulo_modulo_nat @ M5 @ N4 ) ) ) ) ).
% divmod_nat_def
thf(fact_5998_sum__gp__multiplied,axiom,
! [M2: nat,N: nat,X3: complex] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X3 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) )
= ( minus_minus_complex @ ( power_power_complex @ X3 @ M2 ) @ ( power_power_complex @ X3 @ ( suc @ N ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_5999_sum__gp__multiplied,axiom,
! [M2: nat,N: nat,X3: rat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X3 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) )
= ( minus_minus_rat @ ( power_power_rat @ X3 @ M2 ) @ ( power_power_rat @ X3 @ ( suc @ N ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_6000_sum__gp__multiplied,axiom,
! [M2: nat,N: nat,X3: int] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X3 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) )
= ( minus_minus_int @ ( power_power_int @ X3 @ M2 ) @ ( power_power_int @ X3 @ ( suc @ N ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_6001_sum__gp__multiplied,axiom,
! [M2: nat,N: nat,X3: real] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X3 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) )
= ( minus_minus_real @ ( power_power_real @ X3 @ M2 ) @ ( power_power_real @ X3 @ ( suc @ N ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_6002_sum_Oin__pairs,axiom,
! [G: nat > rat,M2: nat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% sum.in_pairs
thf(fact_6003_sum_Oin__pairs,axiom,
! [G: nat > int,M2: nat,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% sum.in_pairs
thf(fact_6004_sum_Oin__pairs,axiom,
! [G: nat > nat,M2: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% sum.in_pairs
thf(fact_6005_sum_Oin__pairs,axiom,
! [G: nat > real,M2: nat,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% sum.in_pairs
thf(fact_6006_divmod__divmod__step,axiom,
( unique5055182867167087721od_nat
= ( ^ [M5: num,N4: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M5 @ N4 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M5 ) ) @ ( unique5026877609467782581ep_nat @ N4 @ ( unique5055182867167087721od_nat @ M5 @ ( bit0 @ N4 ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_6007_divmod__divmod__step,axiom,
( unique5052692396658037445od_int
= ( ^ [M5: num,N4: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M5 @ N4 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M5 ) ) @ ( unique5024387138958732305ep_int @ N4 @ ( unique5052692396658037445od_int @ M5 @ ( bit0 @ N4 ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_6008_divmod__divmod__step,axiom,
( unique3479559517661332726nteger
= ( ^ [M5: num,N4: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M5 @ N4 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M5 ) ) @ ( unique4921790084139445826nteger @ N4 @ ( unique3479559517661332726nteger @ M5 @ ( bit0 @ N4 ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_6009_mask__eq__sum__exp__nat,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
@ ( collect_nat
@ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_6010_gauss__sum__nat,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X5: nat] : X5
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_nat
thf(fact_6011_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ one_one_real )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) @ X3 ) ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_6012_arith__series__nat,axiom,
! [A: nat,D: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I3 @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( 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 ) ) ) ) ).
% arith_series_nat
thf(fact_6013_Sum__Icc__nat,axiom,
! [M2: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X5: nat] : X5
@ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M2 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% Sum_Icc_nat
thf(fact_6014_odd__mod__4__div__2,axiom,
! [N: nat] :
( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( numeral_numeral_nat @ ( bit1 @ one ) ) )
=> ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% odd_mod_4_div_2
thf(fact_6015_mod__exhaust__less__4,axiom,
! [M2: nat] :
( ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= zero_zero_nat )
| ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= one_one_nat )
| ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
| ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
% mod_exhaust_less_4
thf(fact_6016_sin__bound__lemma,axiom,
! [X3: real,Y: real,U: real,V: real] :
( ( X3 = Y )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X3 @ U ) @ Y ) ) @ V ) ) ) ).
% sin_bound_lemma
thf(fact_6017_signed__take__bit__numeral__minus__bit1,axiom,
! [L: num,K2: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_numeral_minus_bit1
thf(fact_6018_signed__take__bit__numeral__bit1,axiom,
! [L: num,K2: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_numeral_bit1
thf(fact_6019_sum__gp,axiom,
! [N: nat,M2: nat,X3: rat] :
( ( ( ord_less_nat @ N @ M2 )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= zero_zero_rat ) )
& ( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ( X3 = one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M2 ) ) ) )
& ( ( X3 != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X3 @ M2 ) @ ( power_power_rat @ X3 @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X3 ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_6020_sum__gp,axiom,
! [N: nat,M2: nat,X3: complex] :
( ( ( ord_less_nat @ N @ M2 )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= zero_zero_complex ) )
& ( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ( X3 = one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M2 ) ) ) )
& ( ( X3 != one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X3 @ M2 ) @ ( power_power_complex @ X3 @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X3 ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_6021_sum__gp,axiom,
! [N: nat,M2: nat,X3: real] :
( ( ( ord_less_nat @ N @ M2 )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= zero_zero_real ) )
& ( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ( X3 = one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M2 ) ) ) )
& ( ( X3 != one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X3 @ M2 ) @ ( power_power_real @ X3 @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_6022_of__int__code__if,axiom,
( ring_1_of_int_real
= ( ^ [K3: int] :
( if_real @ ( K3 = zero_zero_int ) @ zero_zero_real
@ ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_real
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_6023_of__int__code__if,axiom,
( ring_1_of_int_int
= ( ^ [K3: int] :
( if_int @ ( K3 = zero_zero_int ) @ zero_zero_int
@ ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_int
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_6024_of__int__code__if,axiom,
( ring_17405671764205052669omplex
= ( ^ [K3: int] :
( if_complex @ ( K3 = zero_zero_int ) @ zero_zero_complex
@ ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_complex
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_6025_of__int__code__if,axiom,
( ring_1_of_int_rat
= ( ^ [K3: int] :
( if_rat @ ( K3 = zero_zero_int ) @ zero_zero_rat
@ ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_rat
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_rat ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_6026_of__int__code__if,axiom,
( ring_18347121197199848620nteger
= ( ^ [K3: int] :
( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
@ ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_Code_integer
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_6027_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_6028_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= ( semiri5074537144036343181t_real @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_6029_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= ( semiri1316708129612266289at_nat @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_6030_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri8010041392384452111omplex @ M2 )
= ( semiri8010041392384452111omplex @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_6031_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri4939895301339042750nteger @ M2 )
= ( semiri4939895301339042750nteger @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_6032_of__int__eq__iff,axiom,
! [W2: int,Z2: int] :
( ( ( ring_1_of_int_real @ W2 )
= ( ring_1_of_int_real @ Z2 ) )
= ( W2 = Z2 ) ) ).
% of_int_eq_iff
thf(fact_6033_of__int__eq__iff,axiom,
! [W2: int,Z2: int] :
( ( ( ring_1_of_int_rat @ W2 )
= ( ring_1_of_int_rat @ Z2 ) )
= ( W2 = Z2 ) ) ).
% of_int_eq_iff
thf(fact_6034_of__int__eq__iff,axiom,
! [W2: int,Z2: int] :
( ( ( ring_17405671764205052669omplex @ W2 )
= ( ring_17405671764205052669omplex @ Z2 ) )
= ( W2 = Z2 ) ) ).
% of_int_eq_iff
thf(fact_6035_split__part,axiom,
! [P2: $o,Q: int > int > $o] :
( ( produc4947309494688390418_int_o
@ ^ [A5: int,B4: int] :
( P2
& ( Q @ A5 @ B4 ) ) )
= ( ^ [Ab: product_prod_int_int] :
( P2
& ( produc4947309494688390418_int_o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_6036_int__eq__iff__numeral,axiom,
! [M2: nat,V: num] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( numeral_numeral_int @ V ) )
= ( M2
= ( numeral_numeral_nat @ V ) ) ) ).
% int_eq_iff_numeral
thf(fact_6037_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
= ( semiri681578069525770553at_rat @ N ) ) ).
% abs_of_nat
thf(fact_6038_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% abs_of_nat
thf(fact_6039_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% abs_of_nat
thf(fact_6040_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
= ( semiri4939895301339042750nteger @ N ) ) ).
% abs_of_nat
thf(fact_6041_negative__eq__positive,axiom,
! [N: nat,M2: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M2 ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_6042_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_rat @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri681578069525770553at_rat @ N ) ) ).
% of_int_of_nat_eq
thf(fact_6043_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% of_int_of_nat_eq
thf(fact_6044_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_real @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% of_int_of_nat_eq
thf(fact_6045_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_17405671764205052669omplex @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri8010041392384452111omplex @ N ) ) ).
% of_int_of_nat_eq
thf(fact_6046_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_18347121197199848620nteger @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri4939895301339042750nteger @ N ) ) ).
% of_int_of_nat_eq
thf(fact_6047_of__int__abs,axiom,
! [X3: int] :
( ( ring_1_of_int_int @ ( abs_abs_int @ X3 ) )
= ( abs_abs_int @ ( ring_1_of_int_int @ X3 ) ) ) ).
% of_int_abs
thf(fact_6048_of__int__abs,axiom,
! [X3: int] :
( ( ring_18347121197199848620nteger @ ( abs_abs_int @ X3 ) )
= ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ X3 ) ) ) ).
% of_int_abs
thf(fact_6049_of__int__abs,axiom,
! [X3: int] :
( ( ring_1_of_int_real @ ( abs_abs_int @ X3 ) )
= ( abs_abs_real @ ( ring_1_of_int_real @ X3 ) ) ) ).
% of_int_abs
thf(fact_6050_of__int__abs,axiom,
! [X3: int] :
( ( ring_1_of_int_rat @ ( abs_abs_int @ X3 ) )
= ( abs_abs_rat @ ( ring_1_of_int_rat @ X3 ) ) ) ).
% of_int_abs
thf(fact_6051_negative__zle,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zle
thf(fact_6052_zdvd1__eq,axiom,
! [X3: int] :
( ( dvd_dvd_int @ X3 @ one_one_int )
= ( ( abs_abs_int @ X3 )
= one_one_int ) ) ).
% zdvd1_eq
thf(fact_6053_int__dvd__int__iff,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( dvd_dvd_nat @ M2 @ N ) ) ).
% int_dvd_int_iff
thf(fact_6054_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri681578069525770553at_rat @ M2 )
= zero_zero_rat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_6055_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_6056_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_6057_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_6058_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri8010041392384452111omplex @ M2 )
= zero_zero_complex )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_6059_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri4939895301339042750nteger @ M2 )
= zero_z3403309356797280102nteger )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_6060_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_rat
= ( semiri681578069525770553at_rat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_6061_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_6062_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_6063_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_6064_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_complex
= ( semiri8010041392384452111omplex @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_6065_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_z3403309356797280102nteger
= ( semiri4939895301339042750nteger @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_6066_of__nat__0,axiom,
( ( semiri681578069525770553at_rat @ zero_zero_nat )
= zero_zero_rat ) ).
% of_nat_0
thf(fact_6067_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_6068_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_6069_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_6070_of__nat__0,axiom,
( ( semiri8010041392384452111omplex @ zero_zero_nat )
= zero_zero_complex ) ).
% of_nat_0
thf(fact_6071_of__nat__0,axiom,
( ( semiri4939895301339042750nteger @ zero_zero_nat )
= zero_z3403309356797280102nteger ) ).
% of_nat_0
thf(fact_6072_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_6073_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_6074_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_6075_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_6076_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M2 ) @ ( semiri4939895301339042750nteger @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_6077_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_6078_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ M2 ) @ ( semiri4939895301339042750nteger @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_6079_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_6080_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_6081_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_6082_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% of_nat_add
thf(fact_6083_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_6084_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_add
thf(fact_6085_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_6086_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M2 ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% of_nat_add
thf(fact_6087_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri4939895301339042750nteger @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ M2 ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% of_nat_add
thf(fact_6088_of__nat__mult,axiom,
! [M2: nat,N: nat] :
( ( semiri681578069525770553at_rat @ ( times_times_nat @ M2 @ N ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% of_nat_mult
thf(fact_6089_of__nat__mult,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M2 @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_6090_of__nat__mult,axiom,
! [M2: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M2 @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_mult
thf(fact_6091_of__nat__mult,axiom,
! [M2: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M2 @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_6092_of__nat__mult,axiom,
! [M2: nat,N: nat] :
( ( semiri8010041392384452111omplex @ ( times_times_nat @ M2 @ N ) )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ M2 ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% of_nat_mult
thf(fact_6093_of__nat__mult,axiom,
! [M2: nat,N: nat] :
( ( semiri4939895301339042750nteger @ ( times_times_nat @ M2 @ N ) )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M2 ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% of_nat_mult
thf(fact_6094_of__int__0,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_0
thf(fact_6095_of__int__0,axiom,
( ( ring_1_of_int_real @ zero_zero_int )
= zero_zero_real ) ).
% of_int_0
thf(fact_6096_of__int__0,axiom,
( ( ring_1_of_int_rat @ zero_zero_int )
= zero_zero_rat ) ).
% of_int_0
thf(fact_6097_of__int__0,axiom,
( ( ring_17405671764205052669omplex @ zero_zero_int )
= zero_zero_complex ) ).
% of_int_0
thf(fact_6098_of__int__0__eq__iff,axiom,
! [Z2: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z2 ) )
= ( Z2 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_6099_of__int__0__eq__iff,axiom,
! [Z2: int] :
( ( zero_zero_real
= ( ring_1_of_int_real @ Z2 ) )
= ( Z2 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_6100_of__int__0__eq__iff,axiom,
! [Z2: int] :
( ( zero_zero_rat
= ( ring_1_of_int_rat @ Z2 ) )
= ( Z2 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_6101_of__int__0__eq__iff,axiom,
! [Z2: int] :
( ( zero_zero_complex
= ( ring_17405671764205052669omplex @ Z2 ) )
= ( Z2 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_6102_of__int__eq__0__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_int @ Z2 )
= zero_zero_int )
= ( Z2 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_6103_of__int__eq__0__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_real @ Z2 )
= zero_zero_real )
= ( Z2 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_6104_of__int__eq__0__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_rat @ Z2 )
= zero_zero_rat )
= ( Z2 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_6105_of__int__eq__0__iff,axiom,
! [Z2: int] :
( ( ( ring_17405671764205052669omplex @ Z2 )
= zero_zero_complex )
= ( Z2 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_6106_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri681578069525770553at_rat @ N )
= one_one_rat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_6107_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_6108_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri5074537144036343181t_real @ N )
= one_one_real )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_6109_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_6110_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri8010041392384452111omplex @ N )
= one_one_complex )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_6111_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri4939895301339042750nteger @ N )
= one_one_Code_integer )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_6112_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_rat
= ( semiri681578069525770553at_rat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_6113_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_6114_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_6115_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_6116_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_complex
= ( semiri8010041392384452111omplex @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_6117_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_Code_integer
= ( semiri4939895301339042750nteger @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_6118_of__nat__1,axiom,
( ( semiri681578069525770553at_rat @ one_one_nat )
= one_one_rat ) ).
% of_nat_1
thf(fact_6119_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_6120_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_6121_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_6122_of__nat__1,axiom,
( ( semiri8010041392384452111omplex @ one_one_nat )
= one_one_complex ) ).
% of_nat_1
thf(fact_6123_of__nat__1,axiom,
( ( semiri4939895301339042750nteger @ one_one_nat )
= one_one_Code_integer ) ).
% of_nat_1
thf(fact_6124_of__int__le__iff,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% of_int_le_iff
thf(fact_6125_of__int__le__iff,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% of_int_le_iff
thf(fact_6126_of__int__le__iff,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% of_int_le_iff
thf(fact_6127_of__int__eq__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ( ring_17405671764205052669omplex @ Z2 )
= ( numera6690914467698888265omplex @ N ) )
= ( Z2
= ( numeral_numeral_int @ N ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_6128_of__int__eq__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ( ring_1_of_int_real @ Z2 )
= ( numeral_numeral_real @ N ) )
= ( Z2
= ( numeral_numeral_int @ N ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_6129_of__int__eq__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ( ring_1_of_int_rat @ Z2 )
= ( numeral_numeral_rat @ N ) )
= ( Z2
= ( numeral_numeral_int @ N ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_6130_of__int__eq__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ( ring_1_of_int_int @ Z2 )
= ( numeral_numeral_int @ N ) )
= ( Z2
= ( numeral_numeral_int @ N ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_6131_of__int__numeral,axiom,
! [K2: num] :
( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K2 ) )
= ( numera6690914467698888265omplex @ K2 ) ) ).
% of_int_numeral
thf(fact_6132_of__int__numeral,axiom,
! [K2: num] :
( ( ring_1_of_int_real @ ( numeral_numeral_int @ K2 ) )
= ( numeral_numeral_real @ K2 ) ) ).
% of_int_numeral
thf(fact_6133_of__int__numeral,axiom,
! [K2: num] :
( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K2 ) )
= ( numeral_numeral_rat @ K2 ) ) ).
% of_int_numeral
thf(fact_6134_of__int__numeral,axiom,
! [K2: num] :
( ( ring_1_of_int_int @ ( numeral_numeral_int @ K2 ) )
= ( numeral_numeral_int @ K2 ) ) ).
% of_int_numeral
thf(fact_6135_of__int__less__iff,axiom,
! [W2: int,Z2: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% of_int_less_iff
thf(fact_6136_of__int__less__iff,axiom,
! [W2: int,Z2: int] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% of_int_less_iff
thf(fact_6137_of__int__less__iff,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% of_int_less_iff
thf(fact_6138_of__int__eq__1__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_int @ Z2 )
= one_one_int )
= ( Z2 = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_6139_of__int__eq__1__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_real @ Z2 )
= one_one_real )
= ( Z2 = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_6140_of__int__eq__1__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_rat @ Z2 )
= one_one_rat )
= ( Z2 = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_6141_of__int__eq__1__iff,axiom,
! [Z2: int] :
( ( ( ring_17405671764205052669omplex @ Z2 )
= one_one_complex )
= ( Z2 = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_6142_of__int__1,axiom,
( ( ring_1_of_int_int @ one_one_int )
= one_one_int ) ).
% of_int_1
thf(fact_6143_of__int__1,axiom,
( ( ring_1_of_int_real @ one_one_int )
= one_one_real ) ).
% of_int_1
thf(fact_6144_of__int__1,axiom,
( ( ring_1_of_int_rat @ one_one_int )
= one_one_rat ) ).
% of_int_1
thf(fact_6145_of__int__1,axiom,
( ( ring_17405671764205052669omplex @ one_one_int )
= one_one_complex ) ).
% of_int_1
thf(fact_6146_of__int__mult,axiom,
! [W2: int,Z2: int] :
( ( ring_17405671764205052669omplex @ ( times_times_int @ W2 @ Z2 ) )
= ( times_times_complex @ ( ring_17405671764205052669omplex @ W2 ) @ ( ring_17405671764205052669omplex @ Z2 ) ) ) ).
% of_int_mult
thf(fact_6147_of__int__mult,axiom,
! [W2: int,Z2: int] :
( ( ring_1_of_int_real @ ( times_times_int @ W2 @ Z2 ) )
= ( times_times_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% of_int_mult
thf(fact_6148_of__int__mult,axiom,
! [W2: int,Z2: int] :
( ( ring_1_of_int_rat @ ( times_times_int @ W2 @ Z2 ) )
= ( times_times_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% of_int_mult
thf(fact_6149_of__int__mult,axiom,
! [W2: int,Z2: int] :
( ( ring_1_of_int_int @ ( times_times_int @ W2 @ Z2 ) )
= ( times_times_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_mult
thf(fact_6150_of__int__add,axiom,
! [W2: int,Z2: int] :
( ( ring_1_of_int_int @ ( plus_plus_int @ W2 @ Z2 ) )
= ( plus_plus_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_add
thf(fact_6151_of__int__add,axiom,
! [W2: int,Z2: int] :
( ( ring_1_of_int_real @ ( plus_plus_int @ W2 @ Z2 ) )
= ( plus_plus_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% of_int_add
thf(fact_6152_of__int__add,axiom,
! [W2: int,Z2: int] :
( ( ring_1_of_int_rat @ ( plus_plus_int @ W2 @ Z2 ) )
= ( plus_plus_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% of_int_add
thf(fact_6153_of__int__add,axiom,
! [W2: int,Z2: int] :
( ( ring_17405671764205052669omplex @ ( plus_plus_int @ W2 @ Z2 ) )
= ( plus_plus_complex @ ( ring_17405671764205052669omplex @ W2 ) @ ( ring_17405671764205052669omplex @ Z2 ) ) ) ).
% of_int_add
thf(fact_6154_negative__zless,axiom,
! [N: nat,M2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zless
thf(fact_6155_of__int__minus,axiom,
! [Z2: int] :
( ( ring_1_of_int_real @ ( uminus_uminus_int @ Z2 ) )
= ( uminus_uminus_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% of_int_minus
thf(fact_6156_of__int__minus,axiom,
! [Z2: int] :
( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z2 ) )
= ( uminus_uminus_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_minus
thf(fact_6157_of__int__minus,axiom,
! [Z2: int] :
( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ Z2 ) )
= ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ Z2 ) ) ) ).
% of_int_minus
thf(fact_6158_of__int__minus,axiom,
! [Z2: int] :
( ( ring_1_of_int_rat @ ( uminus_uminus_int @ Z2 ) )
= ( uminus_uminus_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% of_int_minus
thf(fact_6159_of__int__minus,axiom,
! [Z2: int] :
( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ Z2 ) )
= ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ Z2 ) ) ) ).
% of_int_minus
thf(fact_6160_zabs__less__one__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
= ( Z2 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_6161_of__int__diff,axiom,
! [W2: int,Z2: int] :
( ( ring_17405671764205052669omplex @ ( minus_minus_int @ W2 @ Z2 ) )
= ( minus_minus_complex @ ( ring_17405671764205052669omplex @ W2 ) @ ( ring_17405671764205052669omplex @ Z2 ) ) ) ).
% of_int_diff
thf(fact_6162_of__int__diff,axiom,
! [W2: int,Z2: int] :
( ( ring_1_of_int_real @ ( minus_minus_int @ W2 @ Z2 ) )
= ( minus_minus_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% of_int_diff
thf(fact_6163_of__int__diff,axiom,
! [W2: int,Z2: int] :
( ( ring_1_of_int_rat @ ( minus_minus_int @ W2 @ Z2 ) )
= ( minus_minus_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% of_int_diff
thf(fact_6164_of__int__diff,axiom,
! [W2: int,Z2: int] :
( ( ring_1_of_int_int @ ( minus_minus_int @ W2 @ Z2 ) )
= ( minus_minus_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_diff
thf(fact_6165_pred__numeral__simps_I1_J,axiom,
( ( pred_numeral @ one )
= zero_zero_nat ) ).
% pred_numeral_simps(1)
thf(fact_6166_eq__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( ( numeral_numeral_nat @ K2 )
= ( suc @ N ) )
= ( ( pred_numeral @ K2 )
= N ) ) ).
% eq_numeral_Suc
thf(fact_6167_Suc__eq__numeral,axiom,
! [N: nat,K2: num] :
( ( ( suc @ N )
= ( numeral_numeral_nat @ K2 ) )
= ( N
= ( pred_numeral @ K2 ) ) ) ).
% Suc_eq_numeral
thf(fact_6168_of__int__power__eq__of__int__cancel__iff,axiom,
! [X3: int,B: int,W2: nat] :
( ( ( ring_1_of_int_rat @ X3 )
= ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
= ( X3
= ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_6169_of__int__power__eq__of__int__cancel__iff,axiom,
! [X3: int,B: int,W2: nat] :
( ( ( ring_1_of_int_real @ X3 )
= ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
= ( X3
= ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_6170_of__int__power__eq__of__int__cancel__iff,axiom,
! [X3: int,B: int,W2: nat] :
( ( ( ring_1_of_int_int @ X3 )
= ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
= ( X3
= ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_6171_of__int__power__eq__of__int__cancel__iff,axiom,
! [X3: int,B: int,W2: nat] :
( ( ( ring_17405671764205052669omplex @ X3 )
= ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W2 ) )
= ( X3
= ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_6172_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X3: int] :
( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 )
= ( ring_1_of_int_rat @ X3 ) )
= ( ( power_power_int @ B @ W2 )
= X3 ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_6173_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X3: int] :
( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 )
= ( ring_1_of_int_real @ X3 ) )
= ( ( power_power_int @ B @ W2 )
= X3 ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_6174_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X3: int] :
( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 )
= ( ring_1_of_int_int @ X3 ) )
= ( ( power_power_int @ B @ W2 )
= X3 ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_6175_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X3: int] :
( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W2 )
= ( ring_17405671764205052669omplex @ X3 ) )
= ( ( power_power_int @ B @ W2 )
= X3 ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_6176_of__int__power,axiom,
! [Z2: int,N: nat] :
( ( ring_1_of_int_rat @ ( power_power_int @ Z2 @ N ) )
= ( power_power_rat @ ( ring_1_of_int_rat @ Z2 ) @ N ) ) ).
% of_int_power
thf(fact_6177_of__int__power,axiom,
! [Z2: int,N: nat] :
( ( ring_1_of_int_real @ ( power_power_int @ Z2 @ N ) )
= ( power_power_real @ ( ring_1_of_int_real @ Z2 ) @ N ) ) ).
% of_int_power
thf(fact_6178_of__int__power,axiom,
! [Z2: int,N: nat] :
( ( ring_1_of_int_int @ ( power_power_int @ Z2 @ N ) )
= ( power_power_int @ ( ring_1_of_int_int @ Z2 ) @ N ) ) ).
% of_int_power
thf(fact_6179_of__int__power,axiom,
! [Z2: int,N: nat] :
( ( ring_17405671764205052669omplex @ ( power_power_int @ Z2 @ N ) )
= ( power_power_complex @ ( ring_17405671764205052669omplex @ Z2 ) @ N ) ) ).
% of_int_power
thf(fact_6180_of__int__of__bool,axiom,
! [P2: $o] :
( ( ring_1_of_int_real @ ( zero_n2684676970156552555ol_int @ P2 ) )
= ( zero_n3304061248610475627l_real @ P2 ) ) ).
% of_int_of_bool
thf(fact_6181_of__int__of__bool,axiom,
! [P2: $o] :
( ( ring_1_of_int_rat @ ( zero_n2684676970156552555ol_int @ P2 ) )
= ( zero_n2052037380579107095ol_rat @ P2 ) ) ).
% of_int_of_bool
thf(fact_6182_of__int__of__bool,axiom,
! [P2: $o] :
( ( ring_17405671764205052669omplex @ ( zero_n2684676970156552555ol_int @ P2 ) )
= ( zero_n1201886186963655149omplex @ P2 ) ) ).
% of_int_of_bool
thf(fact_6183_of__int__of__bool,axiom,
! [P2: $o] :
( ( ring_1_of_int_int @ ( zero_n2684676970156552555ol_int @ P2 ) )
= ( zero_n2684676970156552555ol_int @ P2 ) ) ).
% of_int_of_bool
thf(fact_6184_of__nat__of__bool,axiom,
! [P2: $o] :
( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P2 ) )
= ( zero_n3304061248610475627l_real @ P2 ) ) ).
% of_nat_of_bool
thf(fact_6185_of__nat__of__bool,axiom,
! [P2: $o] :
( ( semiri8010041392384452111omplex @ ( zero_n2687167440665602831ol_nat @ P2 ) )
= ( zero_n1201886186963655149omplex @ P2 ) ) ).
% of_nat_of_bool
thf(fact_6186_of__nat__of__bool,axiom,
! [P2: $o] :
( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P2 ) )
= ( zero_n356916108424825756nteger @ P2 ) ) ).
% of_nat_of_bool
thf(fact_6187_of__nat__of__bool,axiom,
! [P2: $o] :
( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) )
= ( zero_n2687167440665602831ol_nat @ P2 ) ) ).
% of_nat_of_bool
thf(fact_6188_of__nat__of__bool,axiom,
! [P2: $o] :
( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P2 ) )
= ( zero_n2684676970156552555ol_int @ P2 ) ) ).
% of_nat_of_bool
thf(fact_6189_of__nat__sum,axiom,
! [F: int > nat,A4: set_int] :
( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A4 ) )
= ( groups4538972089207619220nt_int
@ ^ [X5: int] : ( semiri1314217659103216013at_int @ ( F @ X5 ) )
@ A4 ) ) ).
% of_nat_sum
thf(fact_6190_of__nat__sum,axiom,
! [F: complex > nat,A4: set_complex] :
( ( semiri8010041392384452111omplex @ ( groups5693394587270226106ex_nat @ F @ A4 ) )
= ( groups7754918857620584856omplex
@ ^ [X5: complex] : ( semiri8010041392384452111omplex @ ( F @ X5 ) )
@ A4 ) ) ).
% of_nat_sum
thf(fact_6191_of__nat__sum,axiom,
! [F: nat > nat,A4: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A4 ) )
= ( groups3539618377306564664at_int
@ ^ [X5: nat] : ( semiri1314217659103216013at_int @ ( F @ X5 ) )
@ A4 ) ) ).
% of_nat_sum
thf(fact_6192_of__nat__sum,axiom,
! [F: nat > nat,A4: set_nat] :
( ( semiri8010041392384452111omplex @ ( groups3542108847815614940at_nat @ F @ A4 ) )
= ( groups2073611262835488442omplex
@ ^ [X5: nat] : ( semiri8010041392384452111omplex @ ( F @ X5 ) )
@ A4 ) ) ).
% of_nat_sum
thf(fact_6193_of__nat__sum,axiom,
! [F: nat > nat,A4: set_nat] :
( ( semiri4939895301339042750nteger @ ( groups3542108847815614940at_nat @ F @ A4 ) )
= ( groups7501900531339628137nteger
@ ^ [X5: nat] : ( semiri4939895301339042750nteger @ ( F @ X5 ) )
@ A4 ) ) ).
% of_nat_sum
thf(fact_6194_of__nat__sum,axiom,
! [F: nat > nat,A4: set_nat] :
( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) )
= ( groups3542108847815614940at_nat
@ ^ [X5: nat] : ( semiri1316708129612266289at_nat @ ( F @ X5 ) )
@ A4 ) ) ).
% of_nat_sum
thf(fact_6195_of__nat__sum,axiom,
! [F: nat > nat,A4: set_nat] :
( ( semiri5074537144036343181t_real @ ( groups3542108847815614940at_nat @ F @ A4 ) )
= ( groups6591440286371151544t_real
@ ^ [X5: nat] : ( semiri5074537144036343181t_real @ ( F @ X5 ) )
@ A4 ) ) ).
% of_nat_sum
thf(fact_6196_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_6197_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ M2 ) @ zero_z3403309356797280102nteger )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_6198_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M2 ) @ zero_zero_rat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_6199_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_6200_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_6201_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri681578069525770553at_rat @ ( suc @ M2 ) )
= ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_6202_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M2 ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_6203_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ M2 ) )
= ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_6204_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M2 ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_6205_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri8010041392384452111omplex @ ( suc @ M2 ) )
= ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_6206_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri4939895301339042750nteger @ ( suc @ M2 ) )
= ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( semiri4939895301339042750nteger @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_6207_numeral__le__real__of__nat__iff,axiom,
! [N: num,M2: nat] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M2 ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M2 ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_6208_less__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( ord_less_nat @ ( numeral_numeral_nat @ K2 ) @ ( suc @ N ) )
= ( ord_less_nat @ ( pred_numeral @ K2 ) @ N ) ) ).
% less_numeral_Suc
thf(fact_6209_less__Suc__numeral,axiom,
! [N: nat,K2: num] :
( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K2 ) )
= ( ord_less_nat @ N @ ( pred_numeral @ K2 ) ) ) ).
% less_Suc_numeral
thf(fact_6210_le__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K2 ) @ ( suc @ N ) )
= ( ord_less_eq_nat @ ( pred_numeral @ K2 ) @ N ) ) ).
% le_numeral_Suc
thf(fact_6211_le__Suc__numeral,axiom,
! [N: nat,K2: num] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K2 ) )
= ( ord_less_eq_nat @ N @ ( pred_numeral @ K2 ) ) ) ).
% le_Suc_numeral
thf(fact_6212_of__int__sum,axiom,
! [F: complex > int,A4: set_complex] :
( ( ring_17405671764205052669omplex @ ( groups5690904116761175830ex_int @ F @ A4 ) )
= ( groups7754918857620584856omplex
@ ^ [X5: complex] : ( ring_17405671764205052669omplex @ ( F @ X5 ) )
@ A4 ) ) ).
% of_int_sum
thf(fact_6213_of__int__sum,axiom,
! [F: nat > int,A4: set_nat] :
( ( ring_1_of_int_real @ ( groups3539618377306564664at_int @ F @ A4 ) )
= ( groups6591440286371151544t_real
@ ^ [X5: nat] : ( ring_1_of_int_real @ ( F @ X5 ) )
@ A4 ) ) ).
% of_int_sum
thf(fact_6214_of__int__sum,axiom,
! [F: int > int,A4: set_int] :
( ( ring_1_of_int_real @ ( groups4538972089207619220nt_int @ F @ A4 ) )
= ( groups8778361861064173332t_real
@ ^ [X5: int] : ( ring_1_of_int_real @ ( F @ X5 ) )
@ A4 ) ) ).
% of_int_sum
thf(fact_6215_of__int__sum,axiom,
! [F: int > int,A4: set_int] :
( ( ring_1_of_int_rat @ ( groups4538972089207619220nt_int @ F @ A4 ) )
= ( groups3906332499630173760nt_rat
@ ^ [X5: int] : ( ring_1_of_int_rat @ ( F @ X5 ) )
@ A4 ) ) ).
% of_int_sum
thf(fact_6216_of__int__sum,axiom,
! [F: int > int,A4: set_int] :
( ( ring_17405671764205052669omplex @ ( groups4538972089207619220nt_int @ F @ A4 ) )
= ( groups3049146728041665814omplex
@ ^ [X5: int] : ( ring_17405671764205052669omplex @ ( F @ X5 ) )
@ A4 ) ) ).
% of_int_sum
thf(fact_6217_of__int__sum,axiom,
! [F: int > int,A4: set_int] :
( ( ring_1_of_int_int @ ( groups4538972089207619220nt_int @ F @ A4 ) )
= ( groups4538972089207619220nt_int
@ ^ [X5: int] : ( ring_1_of_int_int @ ( F @ X5 ) )
@ A4 ) ) ).
% of_int_sum
thf(fact_6218_diff__Suc__numeral,axiom,
! [N: nat,K2: num] :
( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K2 ) )
= ( minus_minus_nat @ N @ ( pred_numeral @ K2 ) ) ) ).
% diff_Suc_numeral
thf(fact_6219_diff__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( minus_minus_nat @ ( numeral_numeral_nat @ K2 ) @ ( suc @ N ) )
= ( minus_minus_nat @ ( pred_numeral @ K2 ) @ N ) ) ).
% diff_numeral_Suc
thf(fact_6220_max__Suc__numeral,axiom,
! [N: nat,K2: num] :
( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K2 ) )
= ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K2 ) ) ) ) ).
% max_Suc_numeral
thf(fact_6221_max__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( ord_max_nat @ ( numeral_numeral_nat @ K2 ) @ ( suc @ N ) )
= ( suc @ ( ord_max_nat @ ( pred_numeral @ K2 ) @ N ) ) ) ).
% max_numeral_Suc
thf(fact_6222_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_6223_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_6224_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_6225_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_6226_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( semiri4939895301339042750nteger @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_6227_of__int__le__0__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
= ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_6228_of__int__le__0__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
= ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_6229_of__int__le__0__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
= ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_6230_of__int__0__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_le_iff
thf(fact_6231_of__int__0__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
= ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_le_iff
thf(fact_6232_of__int__0__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_le_iff
thf(fact_6233_of__int__0__less__iff,axiom,
! [Z2: int] :
( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_less_iff
thf(fact_6234_of__int__0__less__iff,axiom,
! [Z2: int] :
( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_less_iff
thf(fact_6235_of__int__0__less__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_less_iff
thf(fact_6236_of__int__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_6237_of__int__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_6238_of__int__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_6239_of__int__le__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_le_numeral_iff
thf(fact_6240_of__int__le__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N ) )
= ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_le_numeral_iff
thf(fact_6241_of__int__le__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_le_numeral_iff
thf(fact_6242_of__int__numeral__le__iff,axiom,
! [N: num,Z2: int] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% of_int_numeral_le_iff
thf(fact_6243_of__int__numeral__le__iff,axiom,
! [N: num,Z2: int] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z2 ) )
= ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% of_int_numeral_le_iff
thf(fact_6244_of__int__numeral__le__iff,axiom,
! [N: num,Z2: int] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% of_int_numeral_le_iff
thf(fact_6245_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X3: nat,B: nat,W2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
= ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_6246_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X3: nat,B: nat,W2: nat] :
( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ X3 ) @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W2 ) )
= ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_6247_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X3: nat,B: nat,W2: nat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
= ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_6248_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X3: nat,B: nat,W2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
= ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_6249_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X3: nat,B: nat,W2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
= ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_6250_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X3: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X3 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_6251_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X3: nat] :
( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W2 ) @ ( semiri4939895301339042750nteger @ X3 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_6252_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X3: nat] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) @ ( semiri681578069525770553at_rat @ X3 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_6253_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X3: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X3 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_6254_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X3: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X3 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_6255_of__int__less__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_less_numeral_iff
thf(fact_6256_of__int__less__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N ) )
= ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_less_numeral_iff
thf(fact_6257_of__int__less__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_less_numeral_iff
thf(fact_6258_of__int__numeral__less__iff,axiom,
! [N: num,Z2: int] :
( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% of_int_numeral_less_iff
thf(fact_6259_of__int__numeral__less__iff,axiom,
! [N: num,Z2: int] :
( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z2 ) )
= ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% of_int_numeral_less_iff
thf(fact_6260_of__int__numeral__less__iff,axiom,
! [N: num,Z2: int] :
( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% of_int_numeral_less_iff
thf(fact_6261_of__int__1__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% of_int_1_le_iff
thf(fact_6262_of__int__1__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z2 ) )
= ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% of_int_1_le_iff
thf(fact_6263_of__int__1__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% of_int_1_le_iff
thf(fact_6264_of__int__le__1__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real )
= ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_6265_of__int__le__1__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat )
= ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_6266_of__int__le__1__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
= ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_6267_of__int__1__less__iff,axiom,
! [Z2: int] :
( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% of_int_1_less_iff
thf(fact_6268_of__int__1__less__iff,axiom,
! [Z2: int] :
( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% of_int_1_less_iff
thf(fact_6269_of__int__1__less__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% of_int_1_less_iff
thf(fact_6270_of__int__less__1__iff,axiom,
! [Z2: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real )
= ( ord_less_int @ Z2 @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_6271_of__int__less__1__iff,axiom,
! [Z2: int] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat )
= ( ord_less_int @ Z2 @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_6272_of__int__less__1__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
= ( ord_less_int @ Z2 @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_6273_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X3: int] :
( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) @ ( ring_1_of_int_real @ X3 ) )
= ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X3 ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_6274_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X3: int] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) @ ( ring_1_of_int_rat @ X3 ) )
= ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X3 ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_6275_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X3: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) @ ( ring_1_of_int_int @ X3 ) )
= ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X3 ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_6276_of__int__power__le__of__int__cancel__iff,axiom,
! [X3: int,B: int,W2: nat] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ X3 ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
= ( ord_less_eq_int @ X3 @ ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_6277_of__int__power__le__of__int__cancel__iff,axiom,
! [X3: int,B: int,W2: nat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X3 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
= ( ord_less_eq_int @ X3 @ ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_6278_of__int__power__le__of__int__cancel__iff,axiom,
! [X3: int,B: int,W2: nat] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ X3 ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
= ( ord_less_eq_int @ X3 @ ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_6279_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X3: num,N: nat] :
( ( ( ring_17405671764205052669omplex @ Y )
= ( power_power_complex @ ( numera6690914467698888265omplex @ X3 ) @ N ) )
= ( Y
= ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_6280_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X3: num,N: nat] :
( ( ( ring_1_of_int_real @ Y )
= ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N ) )
= ( Y
= ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_6281_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X3: num,N: nat] :
( ( ( ring_1_of_int_rat @ Y )
= ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N ) )
= ( Y
= ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_6282_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X3: num,N: nat] :
( ( ( ring_1_of_int_int @ Y )
= ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) )
= ( Y
= ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_6283_numeral__power__eq__of__int__cancel__iff,axiom,
! [X3: num,N: nat,Y: int] :
( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X3 ) @ N )
= ( ring_17405671764205052669omplex @ Y ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N )
= Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_6284_numeral__power__eq__of__int__cancel__iff,axiom,
! [X3: num,N: nat,Y: int] :
( ( ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N )
= ( ring_1_of_int_real @ Y ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N )
= Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_6285_numeral__power__eq__of__int__cancel__iff,axiom,
! [X3: num,N: nat,Y: int] :
( ( ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N )
= ( ring_1_of_int_rat @ Y ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N )
= Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_6286_numeral__power__eq__of__int__cancel__iff,axiom,
! [X3: num,N: nat,Y: int] :
( ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N )
= ( ring_1_of_int_int @ Y ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N )
= Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_6287_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X3: int] :
( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) @ ( ring_1_of_int_real @ X3 ) )
= ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X3 ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_6288_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X3: int] :
( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) @ ( ring_1_of_int_rat @ X3 ) )
= ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X3 ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_6289_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X3: int] :
( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) @ ( ring_1_of_int_int @ X3 ) )
= ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X3 ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_6290_of__int__power__less__of__int__cancel__iff,axiom,
! [X3: int,B: int,W2: nat] :
( ( ord_less_real @ ( ring_1_of_int_real @ X3 ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
= ( ord_less_int @ X3 @ ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_6291_of__int__power__less__of__int__cancel__iff,axiom,
! [X3: int,B: int,W2: nat] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ X3 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
= ( ord_less_int @ X3 @ ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_6292_of__int__power__less__of__int__cancel__iff,axiom,
! [X3: int,B: int,W2: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ X3 ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
= ( ord_less_int @ X3 @ ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_6293_of__nat__zero__less__power__iff,axiom,
! [X3: nat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X3 ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X3 )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_6294_of__nat__zero__less__power__iff,axiom,
! [X3: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X3 ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X3 )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_6295_of__nat__zero__less__power__iff,axiom,
! [X3: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X3 ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X3 )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_6296_of__nat__zero__less__power__iff,axiom,
! [X3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X3 )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_6297_of__nat__zero__less__power__iff,axiom,
! [X3: nat,N: nat] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ X3 ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X3 )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_6298_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X3: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X3 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_6299_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X3: nat] :
( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) @ ( semiri4939895301339042750nteger @ X3 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_6300_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X3: nat] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X3 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_6301_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X3: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X3 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_6302_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X3: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X3 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_6303_of__nat__le__numeral__power__cancel__iff,axiom,
! [X3: nat,I: num,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
= ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_6304_of__nat__le__numeral__power__cancel__iff,axiom,
! [X3: nat,I: num,N: nat] :
( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ X3 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) )
= ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_6305_of__nat__le__numeral__power__cancel__iff,axiom,
! [X3: nat,I: num,N: nat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
= ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_6306_of__nat__le__numeral__power__cancel__iff,axiom,
! [X3: nat,I: num,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
= ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_6307_of__nat__le__numeral__power__cancel__iff,axiom,
! [X3: nat,I: num,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
= ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_6308_numeral__power__le__of__int__cancel__iff,axiom,
! [X3: num,N: nat,A: int] :
( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N ) @ ( ring_1_of_int_real @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) @ A ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_6309_numeral__power__le__of__int__cancel__iff,axiom,
! [X3: num,N: nat,A: int] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N ) @ ( ring_1_of_int_rat @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) @ A ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_6310_numeral__power__le__of__int__cancel__iff,axiom,
! [X3: num,N: nat,A: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) @ A ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_6311_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X3: num,N: nat] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_6312_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X3: num,N: nat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_6313_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X3: num,N: nat] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_6314_numeral__power__less__of__int__cancel__iff,axiom,
! [X3: num,N: nat,A: int] :
( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N ) @ ( ring_1_of_int_real @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) @ A ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_6315_numeral__power__less__of__int__cancel__iff,axiom,
! [X3: num,N: nat,A: int] :
( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N ) @ ( ring_1_of_int_rat @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) @ A ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_6316_numeral__power__less__of__int__cancel__iff,axiom,
! [X3: num,N: nat,A: int] :
( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) @ A ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_6317_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X3: num,N: nat] :
( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_6318_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X3: num,N: nat] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_6319_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X3: num,N: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_6320_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X3: num,N: nat] :
( ( ( ring_1_of_int_real @ Y )
= ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N ) )
= ( Y
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6321_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X3: num,N: nat] :
( ( ( ring_1_of_int_int @ Y )
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) )
= ( Y
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6322_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X3: num,N: nat] :
( ( ( ring_17405671764205052669omplex @ Y )
= ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X3 ) ) @ N ) )
= ( Y
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6323_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X3: num,N: nat] :
( ( ( ring_1_of_int_rat @ Y )
= ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N ) )
= ( Y
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6324_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X3: num,N: nat] :
( ( ( ring_18347121197199848620nteger @ Y )
= ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N ) )
= ( Y
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6325_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X3: num,N: nat,Y: int] :
( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N )
= ( ring_1_of_int_real @ Y ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N )
= Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6326_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X3: num,N: nat,Y: int] :
( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N )
= ( ring_1_of_int_int @ Y ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N )
= Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6327_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X3: num,N: nat,Y: int] :
( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X3 ) ) @ N )
= ( ring_17405671764205052669omplex @ Y ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N )
= Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6328_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X3: num,N: nat,Y: int] :
( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N )
= ( ring_1_of_int_rat @ Y ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N )
= Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6329_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X3: num,N: nat,Y: int] :
( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N )
= ( ring_18347121197199848620nteger @ Y ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N )
= Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6330_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X3: num,N: nat,A: int] :
( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) @ A ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6331_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X3: num,N: nat,A: int] :
( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) @ A ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6332_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X3: num,N: nat,A: int] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) @ A ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6333_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X3: num,N: nat,A: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) @ A ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6334_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X3: num,N: nat] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6335_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X3: num,N: nat] :
( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6336_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X3: num,N: nat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6337_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X3: num,N: nat] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6338_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X3: num,N: nat,A: int] :
( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) @ A ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6339_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X3: num,N: nat,A: int] :
( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) @ A ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6340_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X3: num,N: nat,A: int] :
( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) @ A ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6341_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X3: num,N: nat,A: int] :
( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) @ A ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6342_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X3: num,N: nat] :
( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6343_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X3: num,N: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6344_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X3: num,N: nat] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6345_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X3: num,N: nat] :
( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6346_Collect__case__prod__mono,axiom,
! [A4: int > int > $o,B5: int > int > $o] :
( ( ord_le6741204236512500942_int_o @ A4 @ B5 )
=> ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ A4 ) ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ B5 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_6347_prod_Odisc__eq__case,axiom,
! [Prod: product_prod_int_int] :
( produc4947309494688390418_int_o
@ ^ [Uu3: int,Uv3: int] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_6348_of__nat__less__of__int__iff,axiom,
! [N: nat,X3: int] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X3 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X3 ) ) ).
% of_nat_less_of_int_iff
thf(fact_6349_of__nat__less__of__int__iff,axiom,
! [N: nat,X3: int] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X3 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X3 ) ) ).
% of_nat_less_of_int_iff
thf(fact_6350_of__nat__less__of__int__iff,axiom,
! [N: nat,X3: int] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X3 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X3 ) ) ).
% of_nat_less_of_int_iff
thf(fact_6351_of__nat__less__of__int__iff,axiom,
! [N: nat,X3: int] :
( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ N ) @ ( ring_18347121197199848620nteger @ X3 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X3 ) ) ).
% of_nat_less_of_int_iff
thf(fact_6352_ex__le__of__int,axiom,
! [X3: real] :
? [Z3: int] : ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ Z3 ) ) ).
% ex_le_of_int
thf(fact_6353_ex__le__of__int,axiom,
! [X3: rat] :
? [Z3: int] : ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ Z3 ) ) ).
% ex_le_of_int
thf(fact_6354_mult__of__int__commute,axiom,
! [X3: int,Y: complex] :
( ( times_times_complex @ ( ring_17405671764205052669omplex @ X3 ) @ Y )
= ( times_times_complex @ Y @ ( ring_17405671764205052669omplex @ X3 ) ) ) ).
% mult_of_int_commute
thf(fact_6355_mult__of__int__commute,axiom,
! [X3: int,Y: real] :
( ( times_times_real @ ( ring_1_of_int_real @ X3 ) @ Y )
= ( times_times_real @ Y @ ( ring_1_of_int_real @ X3 ) ) ) ).
% mult_of_int_commute
thf(fact_6356_mult__of__int__commute,axiom,
! [X3: int,Y: rat] :
( ( times_times_rat @ ( ring_1_of_int_rat @ X3 ) @ Y )
= ( times_times_rat @ Y @ ( ring_1_of_int_rat @ X3 ) ) ) ).
% mult_of_int_commute
thf(fact_6357_mult__of__int__commute,axiom,
! [X3: int,Y: int] :
( ( times_times_int @ ( ring_1_of_int_int @ X3 ) @ Y )
= ( times_times_int @ Y @ ( ring_1_of_int_int @ X3 ) ) ) ).
% mult_of_int_commute
thf(fact_6358_real__arch__simple,axiom,
! [X3: real] :
? [N3: nat] : ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% real_arch_simple
thf(fact_6359_real__arch__simple,axiom,
! [X3: rat] :
? [N3: nat] : ( ord_less_eq_rat @ X3 @ ( semiri681578069525770553at_rat @ N3 ) ) ).
% real_arch_simple
thf(fact_6360_mult__of__nat__commute,axiom,
! [X3: nat,Y: rat] :
( ( times_times_rat @ ( semiri681578069525770553at_rat @ X3 ) @ Y )
= ( times_times_rat @ Y @ ( semiri681578069525770553at_rat @ X3 ) ) ) ).
% mult_of_nat_commute
thf(fact_6361_mult__of__nat__commute,axiom,
! [X3: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X3 ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X3 ) ) ) ).
% mult_of_nat_commute
thf(fact_6362_mult__of__nat__commute,axiom,
! [X3: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X3 ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X3 ) ) ) ).
% mult_of_nat_commute
thf(fact_6363_mult__of__nat__commute,axiom,
! [X3: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X3 ) ) ) ).
% mult_of_nat_commute
thf(fact_6364_mult__of__nat__commute,axiom,
! [X3: nat,Y: complex] :
( ( times_times_complex @ ( semiri8010041392384452111omplex @ X3 ) @ Y )
= ( times_times_complex @ Y @ ( semiri8010041392384452111omplex @ X3 ) ) ) ).
% mult_of_nat_commute
thf(fact_6365_mult__of__nat__commute,axiom,
! [X3: nat,Y: code_integer] :
( ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ X3 ) @ Y )
= ( times_3573771949741848930nteger @ Y @ ( semiri4939895301339042750nteger @ X3 ) ) ) ).
% mult_of_nat_commute
thf(fact_6366_of__int__max,axiom,
! [X3: int,Y: int] :
( ( ring_1_of_int_real @ ( ord_max_int @ X3 @ Y ) )
= ( ord_max_real @ ( ring_1_of_int_real @ X3 ) @ ( ring_1_of_int_real @ Y ) ) ) ).
% of_int_max
thf(fact_6367_of__int__max,axiom,
! [X3: int,Y: int] :
( ( ring_1_of_int_rat @ ( ord_max_int @ X3 @ Y ) )
= ( ord_max_rat @ ( ring_1_of_int_rat @ X3 ) @ ( ring_1_of_int_rat @ Y ) ) ) ).
% of_int_max
thf(fact_6368_of__int__max,axiom,
! [X3: int,Y: int] :
( ( ring_1_of_int_int @ ( ord_max_int @ X3 @ Y ) )
= ( ord_max_int @ ( ring_1_of_int_int @ X3 ) @ ( ring_1_of_int_int @ Y ) ) ) ).
% of_int_max
thf(fact_6369_of__int__max,axiom,
! [X3: int,Y: int] :
( ( ring_18347121197199848620nteger @ ( ord_max_int @ X3 @ Y ) )
= ( ord_max_Code_integer @ ( ring_18347121197199848620nteger @ X3 ) @ ( ring_18347121197199848620nteger @ Y ) ) ) ).
% of_int_max
thf(fact_6370_int__cases2,axiom,
! [Z2: int] :
( ! [N3: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_6371_zdvd__antisym__abs,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( abs_abs_int @ A )
= ( abs_abs_int @ B ) ) ) ) ).
% zdvd_antisym_abs
thf(fact_6372_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M: nat,N3: nat] :
( Z2
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_6373_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_6374_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( semiri4939895301339042750nteger @ N ) ) ).
% of_nat_0_le_iff
thf(fact_6375_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_6376_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_6377_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_6378_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M2 ) @ zero_zero_rat ) ).
% of_nat_less_0_iff
thf(fact_6379_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_6380_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_6381_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_6382_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M2 ) @ zero_z3403309356797280102nteger ) ).
% of_nat_less_0_iff
thf(fact_6383_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri681578069525770553at_rat @ ( suc @ N ) )
!= zero_zero_rat ) ).
% of_nat_neq_0
thf(fact_6384_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_6385_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ N ) )
!= zero_zero_real ) ).
% of_nat_neq_0
thf(fact_6386_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_6387_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri8010041392384452111omplex @ ( suc @ N ) )
!= zero_zero_complex ) ).
% of_nat_neq_0
thf(fact_6388_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri4939895301339042750nteger @ ( suc @ N ) )
!= zero_z3403309356797280102nteger ) ).
% of_nat_neq_0
thf(fact_6389_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_6390_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_6391_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_6392_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_6393_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M2 ) @ ( semiri4939895301339042750nteger @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_6394_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_6395_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_6396_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_6397_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_6398_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M2 ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_6399_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_6400_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ I ) @ ( semiri4939895301339042750nteger @ J ) ) ) ).
% of_nat_mono
thf(fact_6401_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% of_nat_mono
thf(fact_6402_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_6403_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_6404_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_6405_abs__zmult__eq__1,axiom,
! [M2: int,N: int] :
( ( ( abs_abs_int @ ( times_times_int @ M2 @ N ) )
= one_one_int )
=> ( ( abs_abs_int @ M2 )
= one_one_int ) ) ).
% abs_zmult_eq_1
thf(fact_6406_int__cases,axiom,
! [Z2: int] :
( ! [N3: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_6407_int__of__nat__induct,axiom,
! [P2: int > $o,Z2: int] :
( ! [N3: nat] : ( P2 @ ( semiri1314217659103216013at_int @ N3 ) )
=> ( ! [N3: nat] : ( P2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
=> ( P2 @ Z2 ) ) ) ).
% int_of_nat_induct
thf(fact_6408_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_6409_zle__int,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% zle_int
thf(fact_6410_zero__le__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ? [N3: nat] :
( K2
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_6411_nonneg__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ~ ! [N3: nat] :
( K2
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_6412_int__plus,axiom,
! [N: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% int_plus
thf(fact_6413_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_6414_zadd__int__left,axiom,
! [M2: nat,N: nat,Z2: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_6415_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W3: int,Z6: int] :
? [N4: nat] :
( Z6
= ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_6416_not__int__zless__negative,axiom,
! [N: nat,M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% not_int_zless_negative
thf(fact_6417_int__sum,axiom,
! [F: int > nat,A4: set_int] :
( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A4 ) )
= ( groups4538972089207619220nt_int
@ ^ [X5: int] : ( semiri1314217659103216013at_int @ ( F @ X5 ) )
@ A4 ) ) ).
% int_sum
thf(fact_6418_int__sum,axiom,
! [F: nat > nat,A4: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A4 ) )
= ( groups3539618377306564664at_int
@ ^ [X5: nat] : ( semiri1314217659103216013at_int @ ( F @ X5 ) )
@ A4 ) ) ).
% int_sum
thf(fact_6419_numeral__eq__Suc,axiom,
( numeral_numeral_nat
= ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% numeral_eq_Suc
thf(fact_6420_of__nat__max,axiom,
! [X3: nat,Y: nat] :
( ( semiri4216267220026989637d_enat @ ( ord_max_nat @ X3 @ Y ) )
= ( ord_ma741700101516333627d_enat @ ( semiri4216267220026989637d_enat @ X3 ) @ ( semiri4216267220026989637d_enat @ Y ) ) ) ).
% of_nat_max
thf(fact_6421_of__nat__max,axiom,
! [X3: nat,Y: nat] :
( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X3 @ Y ) )
= ( ord_max_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ).
% of_nat_max
thf(fact_6422_of__nat__max,axiom,
! [X3: nat,Y: nat] :
( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X3 @ Y ) )
= ( ord_max_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( semiri5074537144036343181t_real @ Y ) ) ) ).
% of_nat_max
thf(fact_6423_of__nat__max,axiom,
! [X3: nat,Y: nat] :
( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X3 @ Y ) )
= ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( semiri1316708129612266289at_nat @ Y ) ) ) ).
% of_nat_max
thf(fact_6424_of__nat__max,axiom,
! [X3: nat,Y: nat] :
( ( semiri4939895301339042750nteger @ ( ord_max_nat @ X3 @ Y ) )
= ( ord_max_Code_integer @ ( semiri4939895301339042750nteger @ X3 ) @ ( semiri4939895301339042750nteger @ Y ) ) ) ).
% of_nat_max
thf(fact_6425_infinite__int__iff__unbounded__le,axiom,
! [S2: set_int] :
( ( ~ ( finite_finite_int @ S2 ) )
= ( ! [M5: int] :
? [N4: int] :
( ( ord_less_eq_int @ M5 @ ( abs_abs_int @ N4 ) )
& ( member_int @ N4 @ S2 ) ) ) ) ).
% infinite_int_iff_unbounded_le
thf(fact_6426_infinite__int__iff__unbounded,axiom,
! [S2: set_int] :
( ( ~ ( finite_finite_int @ S2 ) )
= ( ! [M5: int] :
? [N4: int] :
( ( ord_less_int @ M5 @ ( abs_abs_int @ N4 ) )
& ( member_int @ N4 @ S2 ) ) ) ) ).
% infinite_int_iff_unbounded
thf(fact_6427_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_leq_as_int
thf(fact_6428_ex__less__of__nat__mult,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ X3 )
=> ? [N3: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ X3 ) ) ) ).
% ex_less_of_nat_mult
thf(fact_6429_ex__less__of__nat__mult,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ? [N3: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X3 ) ) ) ).
% ex_less_of_nat_mult
thf(fact_6430_of__nat__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_6431_of__nat__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_6432_of__nat__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% of_nat_diff
thf(fact_6433_of__nat__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_6434_of__nat__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri8010041392384452111omplex @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M2 ) @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).
% of_nat_diff
thf(fact_6435_of__nat__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri4939895301339042750nteger @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_8373710615458151222nteger @ ( semiri4939895301339042750nteger @ M2 ) @ ( semiri4939895301339042750nteger @ N ) ) ) ) ).
% of_nat_diff
thf(fact_6436_zabs__def,axiom,
( abs_abs_int
= ( ^ [I3: int] : ( if_int @ ( ord_less_int @ I3 @ zero_zero_int ) @ ( uminus_uminus_int @ I3 ) @ I3 ) ) ) ).
% zabs_def
thf(fact_6437_reals__Archimedean3,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ! [Y6: real] :
? [N3: nat] : ( ord_less_real @ Y6 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X3 ) ) ) ).
% reals_Archimedean3
thf(fact_6438_real__of__int__div4,axiom,
! [N: int,X3: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X3 ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X3 ) ) ) ).
% real_of_int_div4
thf(fact_6439_int__cases4,axiom,
! [M2: int] :
( ! [N3: nat] :
( M2
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_6440_dvd__imp__le__int,axiom,
! [I: int,D: int] :
( ( I != zero_zero_int )
=> ( ( dvd_dvd_int @ D @ I )
=> ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I ) ) ) ) ).
% dvd_imp_le_int
thf(fact_6441_real__of__nat__div4,axiom,
! [N: nat,X3: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X3 ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X3 ) ) ) ).
% real_of_nat_div4
thf(fact_6442_int__zle__neg,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_6443_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_6444_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_6445_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W3: int,Z6: int] :
? [N4: nat] :
( Z6
= ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_6446_abs__mod__less,axiom,
! [L: int,K2: int] :
( ( L != zero_zero_int )
=> ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K2 @ L ) ) @ ( abs_abs_int @ L ) ) ) ).
% abs_mod_less
thf(fact_6447_nonpos__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq_int @ K2 @ zero_zero_int )
=> ~ ! [N3: nat] :
( K2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_6448_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_6449_of__int__nonneg,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% of_int_nonneg
thf(fact_6450_of__int__nonneg,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% of_int_nonneg
thf(fact_6451_of__int__nonneg,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_nonneg
thf(fact_6452_of__int__leD,axiom,
! [N: int,X3: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X3 )
=> ( ( N = zero_zero_int )
| ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X3 ) ) ) ).
% of_int_leD
thf(fact_6453_of__int__leD,axiom,
! [N: int,X3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X3 )
=> ( ( N = zero_zero_int )
| ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).
% of_int_leD
thf(fact_6454_of__int__leD,axiom,
! [N: int,X3: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X3 )
=> ( ( N = zero_zero_int )
| ( ord_less_eq_rat @ one_one_rat @ X3 ) ) ) ).
% of_int_leD
thf(fact_6455_of__int__leD,axiom,
! [N: int,X3: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X3 )
=> ( ( N = zero_zero_int )
| ( ord_less_eq_int @ one_one_int @ X3 ) ) ) ).
% of_int_leD
thf(fact_6456_of__int__pos,axiom,
! [Z2: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% of_int_pos
thf(fact_6457_of__int__pos,axiom,
! [Z2: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% of_int_pos
thf(fact_6458_of__int__pos,axiom,
! [Z2: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_pos
thf(fact_6459_of__int__lessD,axiom,
! [N: int,X3: code_integer] :
( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X3 )
=> ( ( N = zero_zero_int )
| ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X3 ) ) ) ).
% of_int_lessD
thf(fact_6460_of__int__lessD,axiom,
! [N: int,X3: real] :
( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X3 )
=> ( ( N = zero_zero_int )
| ( ord_less_real @ one_one_real @ X3 ) ) ) ).
% of_int_lessD
thf(fact_6461_of__int__lessD,axiom,
! [N: int,X3: rat] :
( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X3 )
=> ( ( N = zero_zero_int )
| ( ord_less_rat @ one_one_rat @ X3 ) ) ) ).
% of_int_lessD
thf(fact_6462_of__int__lessD,axiom,
! [N: int,X3: int] :
( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X3 )
=> ( ( N = zero_zero_int )
| ( ord_less_int @ one_one_int @ X3 ) ) ) ).
% of_int_lessD
thf(fact_6463_floor__exists,axiom,
! [X3: real] :
? [Z3: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X3 )
& ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).
% floor_exists
thf(fact_6464_floor__exists,axiom,
! [X3: rat] :
? [Z3: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X3 )
& ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).
% floor_exists
thf(fact_6465_floor__exists1,axiom,
! [X3: real] :
? [X4: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ X4 ) @ X3 )
& ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ X4 @ one_one_int ) ) )
& ! [Y6: int] :
( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y6 ) @ X3 )
& ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ Y6 @ one_one_int ) ) ) )
=> ( Y6 = X4 ) ) ) ).
% floor_exists1
thf(fact_6466_floor__exists1,axiom,
! [X3: rat] :
? [X4: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X4 ) @ X3 )
& ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ X4 @ one_one_int ) ) )
& ! [Y6: int] :
( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y6 ) @ X3 )
& ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y6 @ one_one_int ) ) ) )
=> ( Y6 = X4 ) ) ) ).
% floor_exists1
thf(fact_6467_mod__mult2__eq_H,axiom,
! [A: int,M2: nat,N: nat] :
( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).
% mod_mult2_eq'
thf(fact_6468_mod__mult2__eq_H,axiom,
! [A: nat,M2: nat,N: nat] :
( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) )
= ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M2 ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M2 ) ) ) ) ).
% mod_mult2_eq'
thf(fact_6469_mod__mult2__eq_H,axiom,
! [A: code_integer,M2: nat,N: nat] :
( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M2 ) @ ( semiri4939895301339042750nteger @ N ) ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M2 ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M2 ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M2 ) ) ) ) ).
% mod_mult2_eq'
thf(fact_6470_of__int__neg__numeral,axiom,
! [K2: num] :
( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ K2 ) ) ) ).
% of_int_neg_numeral
thf(fact_6471_of__int__neg__numeral,axiom,
! [K2: num] :
( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) ) ).
% of_int_neg_numeral
thf(fact_6472_of__int__neg__numeral,axiom,
! [K2: num] :
( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K2 ) ) ) ).
% of_int_neg_numeral
thf(fact_6473_of__int__neg__numeral,axiom,
! [K2: num] :
( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ K2 ) ) ) ).
% of_int_neg_numeral
thf(fact_6474_of__int__neg__numeral,axiom,
! [K2: num] :
( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K2 ) ) ) ).
% of_int_neg_numeral
thf(fact_6475_int__le__real__less,axiom,
( ord_less_eq_int
= ( ^ [N4: int,M5: int] : ( ord_less_real @ ( ring_1_of_int_real @ N4 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M5 ) @ one_one_real ) ) ) ) ).
% int_le_real_less
thf(fact_6476_int__less__real__le,axiom,
( ord_less_int
= ( ^ [N4: int,M5: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N4 ) @ one_one_real ) @ ( ring_1_of_int_real @ M5 ) ) ) ) ).
% int_less_real_le
thf(fact_6477_pos__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ~ ! [N3: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_6478_zero__less__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K2
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_6479_int__cases3,axiom,
! [K2: int] :
( ( K2 != zero_zero_int )
=> ( ! [N3: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_6480_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N4: nat,M5: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N4 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M5 ) ) ) ) ).
% nat_less_real_le
thf(fact_6481_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N4: nat,M5: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M5 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_6482_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K2: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_6483_zdvd__mult__cancel1,axiom,
! [M2: int,N: int] :
( ( M2 != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ M2 @ N ) @ M2 )
= ( ( abs_abs_int @ N )
= one_one_int ) ) ) ).
% zdvd_mult_cancel1
thf(fact_6484_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_6485_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_6486_negD,axiom,
! [X3: int] :
( ( ord_less_int @ X3 @ zero_zero_int )
=> ? [N3: nat] :
( X3
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_6487_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_6488_real__of__int__div__aux,axiom,
! [X3: int,D: int] :
( ( divide_divide_real @ ( ring_1_of_int_real @ X3 ) @ ( ring_1_of_int_real @ D ) )
= ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X3 @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X3 @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% real_of_int_div_aux
thf(fact_6489_real__of__nat__div__aux,axiom,
! [X3: nat,D: nat] :
( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( semiri5074537144036343181t_real @ D ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X3 @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X3 @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% real_of_nat_div_aux
thf(fact_6490_nat__approx__posE,axiom,
! [E2: rat] :
( ( ord_less_rat @ zero_zero_rat @ E2 )
=> ~ ! [N3: nat] :
~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) ) @ E2 ) ) ).
% nat_approx_posE
thf(fact_6491_nat__approx__posE,axiom,
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ~ ! [N3: nat] :
~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ E2 ) ) ).
% nat_approx_posE
thf(fact_6492_inverse__of__nat__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( N != zero_zero_nat )
=> ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_6493_inverse__of__nat__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( N != zero_zero_nat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M2 ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_6494_even__add__abs__iff,axiom,
! [K2: int,L: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K2 @ ( abs_abs_int @ L ) ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K2 @ L ) ) ) ).
% even_add_abs_iff
thf(fact_6495_even__abs__add__iff,axiom,
! [K2: int,L: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K2 ) @ L ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K2 @ L ) ) ) ).
% even_abs_add_iff
thf(fact_6496_real__archimedian__rdiv__eq__0,axiom,
! [X3: real,C2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ! [M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ X3 ) @ C2 ) )
=> ( X3 = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_6497_real__of__int__div2,axiom,
! [N: int,X3: 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 @ X3 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X3 ) ) ) ) ).
% real_of_int_div2
thf(fact_6498_real__of__int__div3,axiom,
! [N: int,X3: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X3 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X3 ) ) ) @ one_one_real ) ).
% real_of_int_div3
thf(fact_6499_neg__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ K2 @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_6500_zdiff__int__split,axiom,
! [P2: int > $o,X3: nat,Y: nat] :
( ( P2 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X3 @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X3 )
=> ( P2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X3 @ Y )
=> ( P2 @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_6501_real__of__nat__div2,axiom,
! [N: nat,X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X3 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X3 ) ) ) ) ).
% real_of_nat_div2
thf(fact_6502_ln__realpow,axiom,
! [X3: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ln_ln_real @ ( power_power_real @ X3 @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X3 ) ) ) ) ).
% ln_realpow
thf(fact_6503_real__of__nat__div3,axiom,
! [N: nat,X3: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X3 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X3 ) ) ) @ one_one_real ) ).
% real_of_nat_div3
thf(fact_6504_nat__intermed__int__val,axiom,
! [M2: nat,N: nat,F: nat > int,K2: int] :
( ! [I2: nat] :
( ( ( ord_less_eq_nat @ M2 @ I2 )
& ( ord_less_nat @ I2 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_int @ ( F @ M2 ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ M2 @ I2 )
& ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K2 ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_6505_decr__lemma,axiom,
! [D: int,X3: int,Z2: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ ( minus_minus_int @ X3 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ Z2 ) ) @ one_one_int ) @ D ) ) @ Z2 ) ) ).
% decr_lemma
thf(fact_6506_incr__lemma,axiom,
! [D: int,Z2: int,X3: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ Z2 @ ( plus_plus_int @ X3 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ Z2 ) ) @ one_one_int ) @ D ) ) ) ) ).
% incr_lemma
thf(fact_6507_linear__plus__1__le__power,axiom,
! [X3: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X3 ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X3 @ one_one_real ) @ N ) ) ) ).
% linear_plus_1_le_power
thf(fact_6508_Bernoulli__inequality,axiom,
! [X3: real,N: nat] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
=> ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X3 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X3 ) @ N ) ) ) ).
% Bernoulli_inequality
thf(fact_6509_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K2: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K2 ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_6510_eq__diff__eq_H,axiom,
! [X3: real,Y: real,Z2: real] :
( ( X3
= ( minus_minus_real @ Y @ Z2 ) )
= ( Y
= ( plus_plus_real @ X3 @ Z2 ) ) ) ).
% eq_diff_eq'
thf(fact_6511_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K2: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K2 ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_6512_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% double_gauss_sum
thf(fact_6513_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% double_gauss_sum
thf(fact_6514_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% double_gauss_sum
thf(fact_6515_double__gauss__sum,axiom,
! [N: nat] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) ) ).
% double_gauss_sum
thf(fact_6516_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% double_gauss_sum
thf(fact_6517_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% double_gauss_sum
thf(fact_6518_double__arith__series,axiom,
! [A: rat,D: rat,N: nat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( 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 ) ) ) ) ).
% double_arith_series
thf(fact_6519_double__arith__series,axiom,
! [A: int,D: int,N: nat] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( 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 ) ) ) ) ).
% double_arith_series
thf(fact_6520_double__arith__series,axiom,
! [A: complex,D: complex,N: nat] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( 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 ) ) ) ) ).
% double_arith_series
thf(fact_6521_double__arith__series,axiom,
! [A: code_integer,D: code_integer,N: nat] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) )
@ ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_6522_double__arith__series,axiom,
! [A: nat,D: nat,N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( 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 ) ) ) ) ).
% double_arith_series
thf(fact_6523_double__arith__series,axiom,
! [A: real,D: real,N: nat] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( 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 ) ) ) ) ).
% double_arith_series
thf(fact_6524_gauss__sum,axiom,
! [N: nat] :
( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% gauss_sum
thf(fact_6525_gauss__sum,axiom,
! [N: nat] :
( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% gauss_sum
thf(fact_6526_gauss__sum,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% gauss_sum
thf(fact_6527_arith__series,axiom,
! [A: int,D: int,N: nat] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( 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 ) ) ) ) ).
% arith_series
thf(fact_6528_arith__series,axiom,
! [A: code_integer,D: code_integer,N: nat] :
( ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ D ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% arith_series
thf(fact_6529_arith__series,axiom,
! [A: nat,D: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( 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 ) ) ) ) ).
% arith_series
thf(fact_6530_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_6531_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_6532_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_6533_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_6534_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_6535_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_6536_Bernoulli__inequality__even,axiom,
! [N: nat,X3: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X3 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X3 ) @ N ) ) ) ).
% Bernoulli_inequality_even
thf(fact_6537_sum__gp__offset,axiom,
! [X3: rat,M2: nat,N: nat] :
( ( ( X3 = one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
= ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) )
& ( ( X3 != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
= ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X3 @ M2 ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ ( suc @ N ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X3 ) ) ) ) ) ).
% sum_gp_offset
thf(fact_6538_sum__gp__offset,axiom,
! [X3: complex,M2: nat,N: nat] :
( ( ( X3 = one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
= ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) )
& ( ( X3 != one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X3 @ M2 ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ ( suc @ N ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X3 ) ) ) ) ) ).
% sum_gp_offset
thf(fact_6539_sum__gp__offset,axiom,
! [X3: real,M2: nat,N: nat] :
( ( ( X3 = one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) )
& ( ( X3 != one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
= ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X3 @ M2 ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( suc @ N ) ) ) ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ).
% sum_gp_offset
thf(fact_6540_gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_6541_gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_6542_gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_6543_of__nat__code__if,axiom,
( semiri681578069525770553at_rat
= ( ^ [N4: nat] :
( if_rat @ ( N4 = zero_zero_nat ) @ zero_zero_rat
@ ( produc6207742614233964070at_rat
@ ^ [M5: nat,Q4: nat] : ( if_rat @ ( Q4 = zero_zero_nat ) @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M5 ) ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M5 ) ) @ one_one_rat ) )
@ ( divmod_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_6544_of__nat__code__if,axiom,
( semiri1314217659103216013at_int
= ( ^ [N4: nat] :
( if_int @ ( N4 = zero_zero_nat ) @ zero_zero_int
@ ( produc6840382203811409530at_int
@ ^ [M5: nat,Q4: nat] : ( if_int @ ( Q4 = zero_zero_nat ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M5 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M5 ) ) @ one_one_int ) )
@ ( divmod_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_6545_of__nat__code__if,axiom,
( semiri5074537144036343181t_real
= ( ^ [N4: nat] :
( if_real @ ( N4 = zero_zero_nat ) @ zero_zero_real
@ ( produc1703576794950452218t_real
@ ^ [M5: nat,Q4: nat] : ( if_real @ ( Q4 = zero_zero_nat ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M5 ) ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M5 ) ) @ one_one_real ) )
@ ( divmod_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_6546_of__nat__code__if,axiom,
( semiri1316708129612266289at_nat
= ( ^ [N4: nat] :
( if_nat @ ( N4 = zero_zero_nat ) @ zero_zero_nat
@ ( produc6842872674320459806at_nat
@ ^ [M5: nat,Q4: nat] : ( if_nat @ ( Q4 = zero_zero_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M5 ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M5 ) ) @ one_one_nat ) )
@ ( divmod_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_6547_of__nat__code__if,axiom,
( semiri8010041392384452111omplex
= ( ^ [N4: nat] :
( if_complex @ ( N4 = zero_zero_nat ) @ zero_zero_complex
@ ( produc1917071388513777916omplex
@ ^ [M5: nat,Q4: nat] : ( if_complex @ ( Q4 = zero_zero_nat ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M5 ) ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M5 ) ) @ one_one_complex ) )
@ ( divmod_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_6548_of__nat__code__if,axiom,
( semiri4939895301339042750nteger
= ( ^ [N4: nat] :
( if_Code_integer @ ( N4 = zero_zero_nat ) @ zero_z3403309356797280102nteger
@ ( produc1830744345554046123nteger
@ ^ [M5: nat,Q4: nat] : ( if_Code_integer @ ( Q4 = zero_zero_nat ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ M5 ) ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ M5 ) ) @ one_one_Code_integer ) )
@ ( divmod_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_6549_monoseq__arctan__series,axiom,
! [X3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
=> ( topolo6980174941875973593q_real
@ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% monoseq_arctan_series
thf(fact_6550_lemma__termdiff3,axiom,
! [H2: real,Z2: real,K5: real,N: nat] :
( ( H2 != zero_zero_real )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ K5 )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z2 @ H2 ) ) @ K5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ N ) @ ( power_power_real @ Z2 @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z2 @ ( 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 ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_6551_lemma__termdiff3,axiom,
! [H2: complex,Z2: complex,K5: real,N: nat] :
( ( H2 != zero_zero_complex )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ K5 )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z2 @ H2 ) ) @ K5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ N ) @ ( power_power_complex @ Z2 @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z2 @ ( 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 ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_6552_round__unique,axiom,
! [X3: real,Y: int] :
( ( ord_less_real @ ( minus_minus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y ) )
=> ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y ) @ ( plus_plus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( ( archim8280529875227126926d_real @ X3 )
= Y ) ) ) ).
% round_unique
thf(fact_6553_round__unique,axiom,
! [X3: rat,Y: int] :
( ( ord_less_rat @ ( minus_minus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y ) )
=> ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y ) @ ( plus_plus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
=> ( ( archim7778729529865785530nd_rat @ X3 )
= Y ) ) ) ).
% round_unique
thf(fact_6554_ln__series,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
=> ( ( ln_ln_real @ X3 )
= ( suminf_real
@ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X3 @ one_one_real ) @ ( suc @ N4 ) ) ) ) ) ) ) ).
% ln_series
thf(fact_6555_lemma__termdiff2,axiom,
! [H2: rat,Z2: rat,N: nat] :
( ( H2 != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ N ) @ ( power_power_rat @ Z2 @ N ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
= ( times_times_rat @ H2
@ ( groups2906978787729119204at_rat
@ ^ [P6: nat] :
( groups2906978787729119204at_rat
@ ^ [Q4: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ Q4 ) @ ( power_power_rat @ Z2 @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P6 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_6556_lemma__termdiff2,axiom,
! [H2: complex,Z2: complex,N: nat] :
( ( H2 != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ N ) @ ( power_power_complex @ Z2 @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
= ( times_times_complex @ H2
@ ( groups2073611262835488442omplex
@ ^ [P6: nat] :
( groups2073611262835488442omplex
@ ^ [Q4: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ Q4 ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P6 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_6557_lemma__termdiff2,axiom,
! [H2: real,Z2: real,N: nat] :
( ( H2 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ N ) @ ( power_power_real @ Z2 @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
= ( times_times_real @ H2
@ ( groups6591440286371151544t_real
@ ^ [P6: nat] :
( groups6591440286371151544t_real
@ ^ [Q4: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ Q4 ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P6 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_6558_lessThan__eq__iff,axiom,
! [X3: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X3 )
= ( set_ord_lessThan_nat @ Y ) )
= ( X3 = Y ) ) ).
% lessThan_eq_iff
thf(fact_6559_lessThan__eq__iff,axiom,
! [X3: int,Y: int] :
( ( ( set_ord_lessThan_int @ X3 )
= ( set_ord_lessThan_int @ Y ) )
= ( X3 = Y ) ) ).
% lessThan_eq_iff
thf(fact_6560_lessThan__eq__iff,axiom,
! [X3: real,Y: real] :
( ( ( set_or5984915006950818249n_real @ X3 )
= ( set_or5984915006950818249n_real @ Y ) )
= ( X3 = Y ) ) ).
% lessThan_eq_iff
thf(fact_6561_lessThan__iff,axiom,
! [I: set_nat,K2: set_nat] :
( ( member_set_nat @ I @ ( set_or890127255671739683et_nat @ K2 ) )
= ( ord_less_set_nat @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_6562_lessThan__iff,axiom,
! [I: rat,K2: rat] :
( ( member_rat @ I @ ( set_ord_lessThan_rat @ K2 ) )
= ( ord_less_rat @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_6563_lessThan__iff,axiom,
! [I: num,K2: num] :
( ( member_num @ I @ ( set_ord_lessThan_num @ K2 ) )
= ( ord_less_num @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_6564_lessThan__iff,axiom,
! [I: nat,K2: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K2 ) )
= ( ord_less_nat @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_6565_lessThan__iff,axiom,
! [I: int,K2: int] :
( ( member_int @ I @ ( set_ord_lessThan_int @ K2 ) )
= ( ord_less_int @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_6566_lessThan__iff,axiom,
! [I: real,K2: real] :
( ( member_real @ I @ ( set_or5984915006950818249n_real @ K2 ) )
= ( ord_less_real @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_6567_finite__lessThan,axiom,
! [K2: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K2 ) ) ).
% finite_lessThan
thf(fact_6568_lessThan__subset__iff,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X3 ) @ ( set_ord_lessThan_rat @ Y ) )
= ( ord_less_eq_rat @ X3 @ Y ) ) ).
% lessThan_subset_iff
thf(fact_6569_lessThan__subset__iff,axiom,
! [X3: num,Y: num] :
( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X3 ) @ ( set_ord_lessThan_num @ Y ) )
= ( ord_less_eq_num @ X3 @ Y ) ) ).
% lessThan_subset_iff
thf(fact_6570_lessThan__subset__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X3 ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X3 @ Y ) ) ).
% lessThan_subset_iff
thf(fact_6571_lessThan__subset__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X3 ) @ ( set_ord_lessThan_int @ Y ) )
= ( ord_less_eq_int @ X3 @ Y ) ) ).
% lessThan_subset_iff
thf(fact_6572_lessThan__subset__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X3 ) @ ( set_or5984915006950818249n_real @ Y ) )
= ( ord_less_eq_real @ X3 @ Y ) ) ).
% lessThan_subset_iff
thf(fact_6573_round__0,axiom,
( ( archim8280529875227126926d_real @ zero_zero_real )
= zero_zero_int ) ).
% round_0
thf(fact_6574_round__0,axiom,
( ( archim7778729529865785530nd_rat @ zero_zero_rat )
= zero_zero_int ) ).
% round_0
thf(fact_6575_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_6576_sum_OlessThan__Suc,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_6577_sum_OlessThan__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_6578_sum_OlessThan__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_6579_sum_OlessThan__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_6580_powser__zero,axiom,
! [F: nat > complex] :
( ( suminf_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ zero_zero_complex @ N4 ) ) )
= ( F @ zero_zero_nat ) ) ).
% powser_zero
thf(fact_6581_powser__zero,axiom,
! [F: nat > real] :
( ( suminf_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ zero_zero_real @ N4 ) ) )
= ( F @ zero_zero_nat ) ) ).
% powser_zero
thf(fact_6582_int__int__eq,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% int_int_eq
thf(fact_6583_less__by__empty,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( A4 = bot_bo2099793752762293965at_nat )
=> ( ord_le3146513528884898305at_nat @ A4 @ B5 ) ) ).
% less_by_empty
thf(fact_6584_lessThan__non__empty,axiom,
! [X3: int] :
( ( set_ord_lessThan_int @ X3 )
!= bot_bot_set_int ) ).
% lessThan_non_empty
thf(fact_6585_lessThan__non__empty,axiom,
! [X3: real] :
( ( set_or5984915006950818249n_real @ X3 )
!= bot_bot_set_real ) ).
% lessThan_non_empty
thf(fact_6586_infinite__Iio,axiom,
! [A: int] :
~ ( finite_finite_int @ ( set_ord_lessThan_int @ A ) ) ).
% infinite_Iio
thf(fact_6587_infinite__Iio,axiom,
! [A: real] :
~ ( finite_finite_real @ ( set_or5984915006950818249n_real @ A ) ) ).
% infinite_Iio
thf(fact_6588_lessThan__def,axiom,
( set_or890127255671739683et_nat
= ( ^ [U2: set_nat] :
( collect_set_nat
@ ^ [X5: set_nat] : ( ord_less_set_nat @ X5 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_6589_lessThan__def,axiom,
( set_ord_lessThan_rat
= ( ^ [U2: rat] :
( collect_rat
@ ^ [X5: rat] : ( ord_less_rat @ X5 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_6590_lessThan__def,axiom,
( set_ord_lessThan_num
= ( ^ [U2: num] :
( collect_num
@ ^ [X5: num] : ( ord_less_num @ X5 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_6591_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X5: nat] : ( ord_less_nat @ X5 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_6592_lessThan__def,axiom,
( set_ord_lessThan_int
= ( ^ [U2: int] :
( collect_int
@ ^ [X5: int] : ( ord_less_int @ X5 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_6593_lessThan__def,axiom,
( set_or5984915006950818249n_real
= ( ^ [U2: real] :
( collect_real
@ ^ [X5: real] : ( ord_less_real @ X5 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_6594_Iio__eq__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = bot_bot_nat ) ) ).
% Iio_eq_empty_iff
thf(fact_6595_lessThan__strict__subset__iff,axiom,
! [M2: rat,N: rat] :
( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M2 ) @ ( set_ord_lessThan_rat @ N ) )
= ( ord_less_rat @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_6596_lessThan__strict__subset__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_set_num @ ( set_ord_lessThan_num @ M2 ) @ ( set_ord_lessThan_num @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_6597_lessThan__strict__subset__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M2 ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_6598_lessThan__strict__subset__iff,axiom,
! [M2: int,N: int] :
( ( ord_less_set_int @ ( set_ord_lessThan_int @ M2 ) @ ( set_ord_lessThan_int @ N ) )
= ( ord_less_int @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_6599_lessThan__strict__subset__iff,axiom,
! [M2: real,N: real] :
( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M2 ) @ ( set_or5984915006950818249n_real @ N ) )
= ( ord_less_real @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_6600_complex__mod__minus__le__complex__mod,axiom,
! [X3: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X3 ) ) @ ( real_V1022390504157884413omplex @ X3 ) ) ).
% complex_mod_minus_le_complex_mod
thf(fact_6601_complex__mod__triangle__ineq2,axiom,
! [B: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B @ A ) ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).
% complex_mod_triangle_ineq2
thf(fact_6602_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_6603_finite__nat__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [S6: set_nat] :
? [K3: nat] : ( ord_less_eq_set_nat @ S6 @ ( set_ord_lessThan_nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded
thf(fact_6604_finite__nat__bounded,axiom,
! [S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ? [K: nat] : ( ord_less_eq_set_nat @ S2 @ ( set_ord_lessThan_nat @ K ) ) ) ).
% finite_nat_bounded
thf(fact_6605_round__mono,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_rat @ X3 @ Y )
=> ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X3 ) @ ( archim7778729529865785530nd_rat @ Y ) ) ) ).
% round_mono
thf(fact_6606_sum_Onat__diff__reindex,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nat_diff_reindex
thf(fact_6607_sum_Onat__diff__reindex,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nat_diff_reindex
thf(fact_6608_sum__diff__distrib,axiom,
! [Q: int > nat,P2: int > nat,N: int] :
( ! [X4: int] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P2 @ X4 ) )
=> ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P2 @ ( set_ord_lessThan_int @ N ) ) @ ( groups4541462559716669496nt_nat @ Q @ ( set_ord_lessThan_int @ N ) ) )
= ( groups4541462559716669496nt_nat
@ ^ [X5: int] : ( minus_minus_nat @ ( P2 @ X5 ) @ ( Q @ X5 ) )
@ ( set_ord_lessThan_int @ N ) ) ) ) ).
% sum_diff_distrib
thf(fact_6609_sum__diff__distrib,axiom,
! [Q: real > nat,P2: real > nat,N: real] :
( ! [X4: real] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P2 @ X4 ) )
=> ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P2 @ ( set_or5984915006950818249n_real @ N ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N ) ) )
= ( groups1935376822645274424al_nat
@ ^ [X5: real] : ( minus_minus_nat @ ( P2 @ X5 ) @ ( Q @ X5 ) )
@ ( set_or5984915006950818249n_real @ N ) ) ) ) ).
% sum_diff_distrib
thf(fact_6610_sum__diff__distrib,axiom,
! [Q: nat > nat,P2: nat > nat,N: nat] :
( ! [X4: nat] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P2 @ X4 ) )
=> ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P2 @ ( set_ord_lessThan_nat @ N ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [X5: nat] : ( minus_minus_nat @ ( P2 @ X5 ) @ ( Q @ X5 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum_diff_distrib
thf(fact_6611_sum_OlessThan__Suc__shift,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( G @ zero_zero_nat )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_6612_sum_OlessThan__Suc__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( G @ zero_zero_nat )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_6613_sum_OlessThan__Suc__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_6614_sum_OlessThan__Suc__shift,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( G @ zero_zero_nat )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_6615_sum__lessThan__telescope,axiom,
! [F: nat > rat,M2: nat] :
( ( groups2906978787729119204at_rat
@ ^ [N4: nat] : ( minus_minus_rat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
@ ( set_ord_lessThan_nat @ M2 ) )
= ( minus_minus_rat @ ( F @ M2 ) @ ( F @ zero_zero_nat ) ) ) ).
% sum_lessThan_telescope
thf(fact_6616_sum__lessThan__telescope,axiom,
! [F: nat > int,M2: nat] :
( ( groups3539618377306564664at_int
@ ^ [N4: nat] : ( minus_minus_int @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
@ ( set_ord_lessThan_nat @ M2 ) )
= ( minus_minus_int @ ( F @ M2 ) @ ( F @ zero_zero_nat ) ) ) ).
% sum_lessThan_telescope
thf(fact_6617_sum__lessThan__telescope,axiom,
! [F: nat > real,M2: nat] :
( ( groups6591440286371151544t_real
@ ^ [N4: nat] : ( minus_minus_real @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
@ ( set_ord_lessThan_nat @ M2 ) )
= ( minus_minus_real @ ( F @ M2 ) @ ( F @ zero_zero_nat ) ) ) ).
% sum_lessThan_telescope
thf(fact_6618_sum__lessThan__telescope_H,axiom,
! [F: nat > rat,M2: nat] :
( ( groups2906978787729119204at_rat
@ ^ [N4: nat] : ( minus_minus_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
@ ( set_ord_lessThan_nat @ M2 ) )
= ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M2 ) ) ) ).
% sum_lessThan_telescope'
thf(fact_6619_sum__lessThan__telescope_H,axiom,
! [F: nat > int,M2: nat] :
( ( groups3539618377306564664at_int
@ ^ [N4: nat] : ( minus_minus_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
@ ( set_ord_lessThan_nat @ M2 ) )
= ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M2 ) ) ) ).
% sum_lessThan_telescope'
thf(fact_6620_sum__lessThan__telescope_H,axiom,
! [F: nat > real,M2: nat] :
( ( groups6591440286371151544t_real
@ ^ [N4: nat] : ( minus_minus_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
@ ( set_ord_lessThan_nat @ M2 ) )
= ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M2 ) ) ) ).
% sum_lessThan_telescope'
thf(fact_6621_sum_OatLeast1__atMost__eq,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_6622_sum_OatLeast1__atMost__eq,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_6623_power__diff__1__eq,axiom,
! [X3: complex,N: nat] :
( ( minus_minus_complex @ ( power_power_complex @ X3 @ N ) @ one_one_complex )
= ( times_times_complex @ ( minus_minus_complex @ X3 @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_1_eq
thf(fact_6624_power__diff__1__eq,axiom,
! [X3: rat,N: nat] :
( ( minus_minus_rat @ ( power_power_rat @ X3 @ N ) @ one_one_rat )
= ( times_times_rat @ ( minus_minus_rat @ X3 @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_1_eq
thf(fact_6625_power__diff__1__eq,axiom,
! [X3: int,N: nat] :
( ( minus_minus_int @ ( power_power_int @ X3 @ N ) @ one_one_int )
= ( times_times_int @ ( minus_minus_int @ X3 @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_1_eq
thf(fact_6626_power__diff__1__eq,axiom,
! [X3: real,N: nat] :
( ( minus_minus_real @ ( power_power_real @ X3 @ N ) @ one_one_real )
= ( times_times_real @ ( minus_minus_real @ X3 @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_1_eq
thf(fact_6627_one__diff__power__eq,axiom,
! [X3: complex,N: nat] :
( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ N ) )
= ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X3 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq
thf(fact_6628_one__diff__power__eq,axiom,
! [X3: rat,N: nat] :
( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ N ) )
= ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X3 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq
thf(fact_6629_one__diff__power__eq,axiom,
! [X3: int,N: nat] :
( ( minus_minus_int @ one_one_int @ ( power_power_int @ X3 @ N ) )
= ( times_times_int @ ( minus_minus_int @ one_one_int @ X3 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq
thf(fact_6630_one__diff__power__eq,axiom,
! [X3: real,N: nat] :
( ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ N ) )
= ( times_times_real @ ( minus_minus_real @ one_one_real @ X3 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq
thf(fact_6631_geometric__sum,axiom,
! [X3: complex,N: nat] :
( ( X3 != one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X3 @ N ) @ one_one_complex ) @ ( minus_minus_complex @ X3 @ one_one_complex ) ) ) ) ).
% geometric_sum
thf(fact_6632_geometric__sum,axiom,
! [X3: rat,N: nat] :
( ( X3 != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X3 @ N ) @ one_one_rat ) @ ( minus_minus_rat @ X3 @ one_one_rat ) ) ) ) ).
% geometric_sum
thf(fact_6633_geometric__sum,axiom,
! [X3: real,N: nat] :
( ( X3 != one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
= ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X3 @ N ) @ one_one_real ) @ ( minus_minus_real @ X3 @ one_one_real ) ) ) ) ).
% geometric_sum
thf(fact_6634_monoseq__realpow,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ one_one_real )
=> ( topolo6980174941875973593q_real @ ( power_power_real @ X3 ) ) ) ) ).
% monoseq_realpow
thf(fact_6635_round__diff__minimal,axiom,
! [Z2: real,M2: int] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ Z2 ) ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ ( ring_1_of_int_real @ M2 ) ) ) ) ).
% round_diff_minimal
thf(fact_6636_round__diff__minimal,axiom,
! [Z2: rat,M2: int] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ Z2 @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ Z2 ) ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ Z2 @ ( ring_1_of_int_rat @ M2 ) ) ) ) ).
% round_diff_minimal
thf(fact_6637_sum__gp__strict,axiom,
! [X3: rat,N: nat] :
( ( ( X3 = one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
= ( semiri681578069525770553at_rat @ N ) ) )
& ( ( X3 != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
= ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ N ) ) @ ( minus_minus_rat @ one_one_rat @ X3 ) ) ) ) ) ).
% sum_gp_strict
thf(fact_6638_sum__gp__strict,axiom,
! [X3: complex,N: nat] :
( ( ( X3 = one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
= ( semiri8010041392384452111omplex @ N ) ) )
& ( ( X3 != one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ N ) ) @ ( minus_minus_complex @ one_one_complex @ X3 ) ) ) ) ) ).
% sum_gp_strict
thf(fact_6639_sum__gp__strict,axiom,
! [X3: real,N: nat] :
( ( ( X3 = one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) )
& ( ( X3 != one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
= ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ N ) ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ).
% sum_gp_strict
thf(fact_6640_lemma__termdiff1,axiom,
! [Z2: complex,H2: complex,M2: nat] :
( ( groups2073611262835488442omplex
@ ^ [P6: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ ( minus_minus_nat @ M2 @ P6 ) ) @ ( power_power_complex @ Z2 @ P6 ) ) @ ( power_power_complex @ Z2 @ M2 ) )
@ ( set_ord_lessThan_nat @ M2 ) )
= ( groups2073611262835488442omplex
@ ^ [P6: nat] : ( times_times_complex @ ( power_power_complex @ Z2 @ P6 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ ( minus_minus_nat @ M2 @ P6 ) ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ M2 @ P6 ) ) ) )
@ ( set_ord_lessThan_nat @ M2 ) ) ) ).
% lemma_termdiff1
thf(fact_6641_lemma__termdiff1,axiom,
! [Z2: rat,H2: rat,M2: nat] :
( ( groups2906978787729119204at_rat
@ ^ [P6: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ ( minus_minus_nat @ M2 @ P6 ) ) @ ( power_power_rat @ Z2 @ P6 ) ) @ ( power_power_rat @ Z2 @ M2 ) )
@ ( set_ord_lessThan_nat @ M2 ) )
= ( groups2906978787729119204at_rat
@ ^ [P6: nat] : ( times_times_rat @ ( power_power_rat @ Z2 @ P6 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ ( minus_minus_nat @ M2 @ P6 ) ) @ ( power_power_rat @ Z2 @ ( minus_minus_nat @ M2 @ P6 ) ) ) )
@ ( set_ord_lessThan_nat @ M2 ) ) ) ).
% lemma_termdiff1
thf(fact_6642_lemma__termdiff1,axiom,
! [Z2: int,H2: int,M2: nat] :
( ( groups3539618377306564664at_int
@ ^ [P6: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z2 @ H2 ) @ ( minus_minus_nat @ M2 @ P6 ) ) @ ( power_power_int @ Z2 @ P6 ) ) @ ( power_power_int @ Z2 @ M2 ) )
@ ( set_ord_lessThan_nat @ M2 ) )
= ( groups3539618377306564664at_int
@ ^ [P6: nat] : ( times_times_int @ ( power_power_int @ Z2 @ P6 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z2 @ H2 ) @ ( minus_minus_nat @ M2 @ P6 ) ) @ ( power_power_int @ Z2 @ ( minus_minus_nat @ M2 @ P6 ) ) ) )
@ ( set_ord_lessThan_nat @ M2 ) ) ) ).
% lemma_termdiff1
thf(fact_6643_lemma__termdiff1,axiom,
! [Z2: real,H2: real,M2: nat] :
( ( groups6591440286371151544t_real
@ ^ [P6: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ ( minus_minus_nat @ M2 @ P6 ) ) @ ( power_power_real @ Z2 @ P6 ) ) @ ( power_power_real @ Z2 @ M2 ) )
@ ( set_ord_lessThan_nat @ M2 ) )
= ( groups6591440286371151544t_real
@ ^ [P6: nat] : ( times_times_real @ ( power_power_real @ Z2 @ P6 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ ( minus_minus_nat @ M2 @ P6 ) ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ M2 @ P6 ) ) ) )
@ ( set_ord_lessThan_nat @ M2 ) ) ) ).
% lemma_termdiff1
thf(fact_6644_diff__power__eq__sum,axiom,
! [X3: complex,N: nat,Y: complex] :
( ( minus_minus_complex @ ( power_power_complex @ X3 @ ( suc @ N ) ) @ ( power_power_complex @ Y @ ( suc @ N ) ) )
= ( times_times_complex @ ( minus_minus_complex @ X3 @ Y )
@ ( groups2073611262835488442omplex
@ ^ [P6: nat] : ( times_times_complex @ ( power_power_complex @ X3 @ P6 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ N @ P6 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_6645_diff__power__eq__sum,axiom,
! [X3: rat,N: nat,Y: rat] :
( ( minus_minus_rat @ ( power_power_rat @ X3 @ ( suc @ N ) ) @ ( power_power_rat @ Y @ ( suc @ N ) ) )
= ( times_times_rat @ ( minus_minus_rat @ X3 @ Y )
@ ( groups2906978787729119204at_rat
@ ^ [P6: nat] : ( times_times_rat @ ( power_power_rat @ X3 @ P6 ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ N @ P6 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_6646_diff__power__eq__sum,axiom,
! [X3: int,N: nat,Y: int] :
( ( minus_minus_int @ ( power_power_int @ X3 @ ( suc @ N ) ) @ ( power_power_int @ Y @ ( suc @ N ) ) )
= ( times_times_int @ ( minus_minus_int @ X3 @ Y )
@ ( groups3539618377306564664at_int
@ ^ [P6: nat] : ( times_times_int @ ( power_power_int @ X3 @ P6 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ N @ P6 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_6647_diff__power__eq__sum,axiom,
! [X3: real,N: nat,Y: real] :
( ( minus_minus_real @ ( power_power_real @ X3 @ ( suc @ N ) ) @ ( power_power_real @ Y @ ( suc @ N ) ) )
= ( times_times_real @ ( minus_minus_real @ X3 @ Y )
@ ( groups6591440286371151544t_real
@ ^ [P6: nat] : ( times_times_real @ ( power_power_real @ X3 @ P6 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ N @ P6 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_6648_power__diff__sumr2,axiom,
! [X3: complex,N: nat,Y: complex] :
( ( minus_minus_complex @ ( power_power_complex @ X3 @ N ) @ ( power_power_complex @ Y @ N ) )
= ( times_times_complex @ ( minus_minus_complex @ X3 @ Y )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( power_power_complex @ Y @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) ) @ ( power_power_complex @ X3 @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_sumr2
thf(fact_6649_power__diff__sumr2,axiom,
! [X3: rat,N: nat,Y: rat] :
( ( minus_minus_rat @ ( power_power_rat @ X3 @ N ) @ ( power_power_rat @ Y @ N ) )
= ( times_times_rat @ ( minus_minus_rat @ X3 @ Y )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( power_power_rat @ Y @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) ) @ ( power_power_rat @ X3 @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_sumr2
thf(fact_6650_power__diff__sumr2,axiom,
! [X3: int,N: nat,Y: int] :
( ( minus_minus_int @ ( power_power_int @ X3 @ N ) @ ( power_power_int @ Y @ N ) )
= ( times_times_int @ ( minus_minus_int @ X3 @ Y )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ Y @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) ) @ ( power_power_int @ X3 @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_sumr2
thf(fact_6651_power__diff__sumr2,axiom,
! [X3: real,N: nat,Y: real] :
( ( minus_minus_real @ ( power_power_real @ X3 @ N ) @ ( power_power_real @ Y @ N ) )
= ( times_times_real @ ( minus_minus_real @ X3 @ Y )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ Y @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) ) @ ( power_power_real @ X3 @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_sumr2
thf(fact_6652_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: nat > code_integer,K5: code_integer,K2: nat] :
( ! [P7: nat] :
( ( ord_less_nat @ P7 @ N )
=> ( ord_le3102999989581377725nteger @ ( F @ P7 ) @ K5 ) )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ K5 )
=> ( ord_le3102999989581377725nteger @ ( groups7501900531339628137nteger @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ K5 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_6653_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: nat > rat,K5: rat,K2: nat] :
( ! [P7: nat] :
( ( ord_less_nat @ P7 @ N )
=> ( ord_less_eq_rat @ ( F @ P7 ) @ K5 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ K5 )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ K5 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_6654_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: nat > int,K5: int,K2: nat] :
( ! [P7: nat] :
( ( ord_less_nat @ P7 @ N )
=> ( ord_less_eq_int @ ( F @ P7 ) @ K5 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K5 )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ K5 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_6655_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: nat > nat,K5: nat,K2: nat] :
( ! [P7: nat] :
( ( ord_less_nat @ P7 @ N )
=> ( ord_less_eq_nat @ ( F @ P7 ) @ K5 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ K5 )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ K5 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_6656_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: nat > real,K5: real,K2: nat] :
( ! [P7: nat] :
( ( ord_less_nat @ P7 @ N )
=> ( ord_less_eq_real @ ( F @ P7 ) @ K5 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ K5 )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ K5 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_6657_one__diff__power__eq_H,axiom,
! [X3: complex,N: nat] :
( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ N ) )
= ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X3 )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( power_power_complex @ X3 @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq'
thf(fact_6658_one__diff__power__eq_H,axiom,
! [X3: rat,N: nat] :
( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ N ) )
= ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X3 )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( power_power_rat @ X3 @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq'
thf(fact_6659_one__diff__power__eq_H,axiom,
! [X3: int,N: nat] :
( ( minus_minus_int @ one_one_int @ ( power_power_int @ X3 @ N ) )
= ( times_times_int @ ( minus_minus_int @ one_one_int @ X3 )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( power_power_int @ X3 @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq'
thf(fact_6660_one__diff__power__eq_H,axiom,
! [X3: real,N: nat] :
( ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ N ) )
= ( times_times_real @ ( minus_minus_real @ one_one_real @ X3 )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( power_power_real @ X3 @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq'
thf(fact_6661_sum__split__even__odd,axiom,
! [F: nat > real,G: nat > real,N: nat] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( F @ I3 ) @ ( G @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ one_one_nat ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum_split_even_odd
thf(fact_6662_of__int__round__le,axiom,
! [X3: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X3 ) ) @ ( plus_plus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% of_int_round_le
thf(fact_6663_of__int__round__le,axiom,
! [X3: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X3 ) ) @ ( plus_plus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% of_int_round_le
thf(fact_6664_of__int__round__ge,axiom,
! [X3: real] : ( ord_less_eq_real @ ( minus_minus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X3 ) ) ) ).
% of_int_round_ge
thf(fact_6665_of__int__round__ge,axiom,
! [X3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X3 ) ) ) ).
% of_int_round_ge
thf(fact_6666_of__int__round__abs__le,axiom,
! [X3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X3 ) ) @ X3 ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% of_int_round_abs_le
thf(fact_6667_of__int__round__abs__le,axiom,
! [X3: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X3 ) ) @ X3 ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% of_int_round_abs_le
thf(fact_6668_norm__le__zero__iff,axiom,
! [X3: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X3 ) @ zero_zero_real )
= ( X3 = zero_zero_real ) ) ).
% norm_le_zero_iff
thf(fact_6669_norm__le__zero__iff,axiom,
! [X3: complex] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X3 ) @ zero_zero_real )
= ( X3 = zero_zero_complex ) ) ).
% norm_le_zero_iff
thf(fact_6670_zero__less__norm__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X3 ) )
= ( X3 != zero_zero_real ) ) ).
% zero_less_norm_iff
thf(fact_6671_zero__less__norm__iff,axiom,
! [X3: complex] :
( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X3 ) )
= ( X3 != zero_zero_complex ) ) ).
% zero_less_norm_iff
thf(fact_6672_norm__eq__zero,axiom,
! [X3: real] :
( ( ( real_V7735802525324610683m_real @ X3 )
= zero_zero_real )
= ( X3 = zero_zero_real ) ) ).
% norm_eq_zero
thf(fact_6673_norm__eq__zero,axiom,
! [X3: complex] :
( ( ( real_V1022390504157884413omplex @ X3 )
= zero_zero_real )
= ( X3 = zero_zero_complex ) ) ).
% norm_eq_zero
thf(fact_6674_norm__zero,axiom,
( ( real_V7735802525324610683m_real @ zero_zero_real )
= zero_zero_real ) ).
% norm_zero
thf(fact_6675_norm__zero,axiom,
( ( real_V1022390504157884413omplex @ zero_zero_complex )
= zero_zero_real ) ).
% norm_zero
thf(fact_6676_norm__power__diff,axiom,
! [Z2: real,W2: real,M2: nat] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W2 ) @ one_one_real )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( power_power_real @ Z2 @ M2 ) @ ( power_power_real @ W2 @ M2 ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z2 @ W2 ) ) ) ) ) ) ).
% norm_power_diff
thf(fact_6677_norm__power__diff,axiom,
! [Z2: complex,W2: complex,M2: nat] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W2 ) @ one_one_real )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( power_power_complex @ Z2 @ M2 ) @ ( power_power_complex @ W2 @ M2 ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z2 @ W2 ) ) ) ) ) ) ).
% norm_power_diff
thf(fact_6678_suminf__zero,axiom,
( ( suminf_complex
@ ^ [N4: nat] : zero_zero_complex )
= zero_zero_complex ) ).
% suminf_zero
thf(fact_6679_suminf__zero,axiom,
( ( suminf_real
@ ^ [N4: nat] : zero_zero_real )
= zero_zero_real ) ).
% suminf_zero
thf(fact_6680_suminf__zero,axiom,
( ( suminf_nat
@ ^ [N4: nat] : zero_zero_nat )
= zero_zero_nat ) ).
% suminf_zero
thf(fact_6681_suminf__zero,axiom,
( ( suminf_int
@ ^ [N4: nat] : zero_zero_int )
= zero_zero_int ) ).
% suminf_zero
thf(fact_6682_norm__not__less__zero,axiom,
! [X3: complex] :
~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ zero_zero_real ) ).
% norm_not_less_zero
thf(fact_6683_norm__ge__zero,axiom,
! [X3: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X3 ) ) ).
% norm_ge_zero
thf(fact_6684_sum__norm__le,axiom,
! [S2: set_real,F: real > complex,G: real > real] :
( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ S2 ) ) @ ( groups8097168146408367636l_real @ G @ S2 ) ) ) ).
% sum_norm_le
thf(fact_6685_sum__norm__le,axiom,
! [S2: set_set_nat,F: set_nat > complex,G: set_nat > real] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ S2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups8255218700646806128omplex @ F @ S2 ) ) @ ( groups5107569545109728110t_real @ G @ S2 ) ) ) ).
% sum_norm_le
thf(fact_6686_sum__norm__le,axiom,
! [S2: set_int,F: int > complex,G: int > real] :
( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups3049146728041665814omplex @ F @ S2 ) ) @ ( groups8778361861064173332t_real @ G @ S2 ) ) ) ).
% sum_norm_le
thf(fact_6687_sum__norm__le,axiom,
! [S2: set_nat,F: nat > complex,G: nat > real] :
( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S2 ) ) @ ( groups6591440286371151544t_real @ G @ S2 ) ) ) ).
% sum_norm_le
thf(fact_6688_sum__norm__le,axiom,
! [S2: set_complex,F: complex > complex,G: complex > real] :
( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ S2 ) ) @ ( groups5808333547571424918x_real @ G @ S2 ) ) ) ).
% sum_norm_le
thf(fact_6689_sum__norm__le,axiom,
! [S2: set_nat,F: nat > real,G: nat > real] :
( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S2 ) ) @ ( groups6591440286371151544t_real @ G @ S2 ) ) ) ).
% sum_norm_le
thf(fact_6690_norm__sum,axiom,
! [F: nat > complex,A4: set_nat] :
( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ A4 ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( real_V1022390504157884413omplex @ ( F @ I3 ) )
@ A4 ) ) ).
% norm_sum
thf(fact_6691_norm__sum,axiom,
! [F: complex > complex,A4: set_complex] :
( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ A4 ) )
@ ( groups5808333547571424918x_real
@ ^ [I3: complex] : ( real_V1022390504157884413omplex @ ( F @ I3 ) )
@ A4 ) ) ).
% norm_sum
thf(fact_6692_norm__sum,axiom,
! [F: nat > real,A4: set_nat] :
( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ A4 ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( real_V7735802525324610683m_real @ ( F @ I3 ) )
@ A4 ) ) ).
% norm_sum
thf(fact_6693_norm__uminus__minus,axiom,
! [X3: real,Y: real] :
( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X3 ) @ Y ) )
= ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y ) ) ) ).
% norm_uminus_minus
thf(fact_6694_norm__uminus__minus,axiom,
! [X3: complex,Y: complex] :
( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X3 ) @ Y ) )
= ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y ) ) ) ).
% norm_uminus_minus
thf(fact_6695_nonzero__norm__divide,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).
% nonzero_norm_divide
thf(fact_6696_nonzero__norm__divide,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).
% nonzero_norm_divide
thf(fact_6697_power__eq__imp__eq__norm,axiom,
! [W2: real,N: nat,Z2: real] :
( ( ( power_power_real @ W2 @ N )
= ( power_power_real @ Z2 @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( real_V7735802525324610683m_real @ W2 )
= ( real_V7735802525324610683m_real @ Z2 ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_6698_power__eq__imp__eq__norm,axiom,
! [W2: complex,N: nat,Z2: complex] :
( ( ( power_power_complex @ W2 @ N )
= ( power_power_complex @ Z2 @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( real_V1022390504157884413omplex @ W2 )
= ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_6699_norm__mult__ineq,axiom,
! [X3: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X3 @ Y ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).
% norm_mult_ineq
thf(fact_6700_norm__mult__ineq,axiom,
! [X3: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X3 @ Y ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).
% norm_mult_ineq
thf(fact_6701_norm__add__less,axiom,
! [X3: real,R2: real,Y: real,S3: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ R2 )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S3 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y ) ) @ ( plus_plus_real @ R2 @ S3 ) ) ) ) ).
% norm_add_less
thf(fact_6702_norm__add__less,axiom,
! [X3: complex,R2: real,Y: complex,S3: real] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ R2 )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S3 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y ) ) @ ( plus_plus_real @ R2 @ S3 ) ) ) ) ).
% norm_add_less
thf(fact_6703_norm__triangle__lt,axiom,
! [X3: real,Y: real,E2: real] :
( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y ) ) @ E2 ) ) ).
% norm_triangle_lt
thf(fact_6704_norm__triangle__lt,axiom,
! [X3: complex,Y: complex,E2: real] :
( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y ) ) @ E2 ) ) ).
% norm_triangle_lt
thf(fact_6705_norm__triangle__mono,axiom,
! [A: real,R2: real,B: real,S3: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R2 )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S3 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R2 @ S3 ) ) ) ) ).
% norm_triangle_mono
thf(fact_6706_norm__triangle__mono,axiom,
! [A: complex,R2: real,B: complex,S3: real] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R2 )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S3 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R2 @ S3 ) ) ) ) ).
% norm_triangle_mono
thf(fact_6707_norm__triangle__ineq,axiom,
! [X3: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).
% norm_triangle_ineq
thf(fact_6708_norm__triangle__ineq,axiom,
! [X3: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).
% norm_triangle_ineq
thf(fact_6709_norm__triangle__le,axiom,
! [X3: real,Y: real,E2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y ) ) @ E2 ) ) ).
% norm_triangle_le
thf(fact_6710_norm__triangle__le,axiom,
! [X3: complex,Y: complex,E2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y ) ) @ E2 ) ) ).
% norm_triangle_le
thf(fact_6711_norm__add__leD,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C2 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C2 ) ) ) ).
% norm_add_leD
thf(fact_6712_norm__add__leD,axiom,
! [A: complex,B: complex,C2: real] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C2 ) ) ) ).
% norm_add_leD
thf(fact_6713_norm__power__ineq,axiom,
! [X3: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X3 @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X3 ) @ N ) ) ).
% norm_power_ineq
thf(fact_6714_norm__power__ineq,axiom,
! [X3: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X3 @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X3 ) @ N ) ) ).
% norm_power_ineq
thf(fact_6715_norm__diff__triangle__less,axiom,
! [X3: real,Y: real,E1: real,Z2: real,E22: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Y ) ) @ E1 )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z2 ) ) @ E22 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% norm_diff_triangle_less
thf(fact_6716_norm__diff__triangle__less,axiom,
! [X3: complex,Y: complex,E1: real,Z2: complex,E22: real] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Y ) ) @ E1 )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z2 ) ) @ E22 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% norm_diff_triangle_less
thf(fact_6717_norm__triangle__le__diff,axiom,
! [X3: real,Y: real,E2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Y ) ) @ E2 ) ) ).
% norm_triangle_le_diff
thf(fact_6718_norm__triangle__le__diff,axiom,
! [X3: complex,Y: complex,E2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Y ) ) @ E2 ) ) ).
% norm_triangle_le_diff
thf(fact_6719_norm__diff__triangle__le,axiom,
! [X3: real,Y: real,E1: real,Z2: real,E22: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Y ) ) @ E1 )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z2 ) ) @ E22 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% norm_diff_triangle_le
thf(fact_6720_norm__diff__triangle__le,axiom,
! [X3: complex,Y: complex,E1: real,Z2: complex,E22: real] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Y ) ) @ E1 )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z2 ) ) @ E22 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% norm_diff_triangle_le
thf(fact_6721_norm__triangle__ineq4,axiom,
! [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 ) ) ) ).
% norm_triangle_ineq4
thf(fact_6722_norm__triangle__ineq4,axiom,
! [A: complex,B: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% norm_triangle_ineq4
thf(fact_6723_norm__triangle__sub,axiom,
! [X3: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Y ) ) ) ) ).
% norm_triangle_sub
thf(fact_6724_norm__triangle__sub,axiom,
! [X3: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Y ) ) ) ) ).
% norm_triangle_sub
thf(fact_6725_norm__diff__ineq,axiom,
! [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 ) ) ) ).
% norm_diff_ineq
thf(fact_6726_norm__diff__ineq,axiom,
! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) ) ).
% norm_diff_ineq
thf(fact_6727_norm__triangle__ineq2,axiom,
! [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 ) ) ) ).
% norm_triangle_ineq2
thf(fact_6728_norm__triangle__ineq2,axiom,
! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% norm_triangle_ineq2
thf(fact_6729_suminf__finite,axiom,
! [N7: set_nat,F: nat > complex] :
( ( finite_finite_nat @ N7 )
=> ( ! [N3: nat] :
( ~ ( member_nat @ N3 @ N7 )
=> ( ( F @ N3 )
= zero_zero_complex ) )
=> ( ( suminf_complex @ F )
= ( groups2073611262835488442omplex @ F @ N7 ) ) ) ) ).
% suminf_finite
thf(fact_6730_suminf__finite,axiom,
! [N7: set_nat,F: nat > int] :
( ( finite_finite_nat @ N7 )
=> ( ! [N3: nat] :
( ~ ( member_nat @ N3 @ N7 )
=> ( ( F @ N3 )
= zero_zero_int ) )
=> ( ( suminf_int @ F )
= ( groups3539618377306564664at_int @ F @ N7 ) ) ) ) ).
% suminf_finite
thf(fact_6731_suminf__finite,axiom,
! [N7: set_nat,F: nat > nat] :
( ( finite_finite_nat @ N7 )
=> ( ! [N3: nat] :
( ~ ( member_nat @ N3 @ N7 )
=> ( ( F @ N3 )
= zero_zero_nat ) )
=> ( ( suminf_nat @ F )
= ( groups3542108847815614940at_nat @ F @ N7 ) ) ) ) ).
% suminf_finite
thf(fact_6732_suminf__finite,axiom,
! [N7: set_nat,F: nat > real] :
( ( finite_finite_nat @ N7 )
=> ( ! [N3: nat] :
( ~ ( member_nat @ N3 @ N7 )
=> ( ( F @ N3 )
= zero_zero_real ) )
=> ( ( suminf_real @ F )
= ( groups6591440286371151544t_real @ F @ N7 ) ) ) ) ).
% suminf_finite
thf(fact_6733_power__eq__1__iff,axiom,
! [W2: real,N: nat] :
( ( ( power_power_real @ W2 @ N )
= one_one_real )
=> ( ( ( real_V7735802525324610683m_real @ W2 )
= one_one_real )
| ( N = zero_zero_nat ) ) ) ).
% power_eq_1_iff
thf(fact_6734_power__eq__1__iff,axiom,
! [W2: complex,N: nat] :
( ( ( power_power_complex @ W2 @ N )
= one_one_complex )
=> ( ( ( real_V1022390504157884413omplex @ W2 )
= one_one_real )
| ( N = zero_zero_nat ) ) ) ).
% power_eq_1_iff
thf(fact_6735_norm__diff__triangle__ineq,axiom,
! [A: real,B: real,C2: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C2 @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C2 ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% norm_diff_triangle_ineq
thf(fact_6736_norm__diff__triangle__ineq,axiom,
! [A: complex,B: complex,C2: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_plus_complex @ C2 @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C2 ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ D ) ) ) ) ).
% norm_diff_triangle_ineq
thf(fact_6737_norm__triangle__ineq3,axiom,
! [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 ) ) ) ).
% norm_triangle_ineq3
thf(fact_6738_norm__triangle__ineq3,axiom,
! [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 ) ) ) ).
% norm_triangle_ineq3
thf(fact_6739_sum__bounds__lt__plus1,axiom,
! [F: nat > nat,Mm: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
@ ( set_ord_lessThan_nat @ Mm ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_6740_sum__bounds__lt__plus1,axiom,
! [F: nat > real,Mm: nat] :
( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
@ ( set_ord_lessThan_nat @ Mm ) )
= ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_6741_arctan__series,axiom,
! [X3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
=> ( ( arctan @ X3 )
= ( suminf_real
@ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
% arctan_series
thf(fact_6742_sumr__cos__zero__one,axiom,
! [N: nat] :
( ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ zero_zero_real @ M5 ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= one_one_real ) ).
% sumr_cos_zero_one
thf(fact_6743_pi__series,axiom,
( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( suminf_real
@ ^ [K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% pi_series
thf(fact_6744_summable__arctan__series,axiom,
! [X3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
=> ( summable_real
@ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% summable_arctan_series
thf(fact_6745_pred__subset__eq2,axiom,
! [R: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
( ( ord_le6741204236512500942_int_o
@ ^ [X5: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y5 ) @ R )
@ ^ [X5: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y5 ) @ S2 ) )
= ( ord_le2843351958646193337nt_int @ R @ S2 ) ) ).
% pred_subset_eq2
thf(fact_6746_pred__subset__eq2,axiom,
! [R: set_Pr8056137968301705908nteger,S2: set_Pr8056137968301705908nteger] :
( ( ord_le3636971675376928563eger_o
@ ^ [X5: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X5 @ Y5 ) @ R )
@ ^ [X5: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X5 @ Y5 ) @ S2 ) )
= ( ord_le3216752416896350996nteger @ R @ S2 ) ) ).
% pred_subset_eq2
thf(fact_6747_pred__subset__eq2,axiom,
! [R: set_Pr1281608226676607948nteger,S2: set_Pr1281608226676607948nteger] :
( ( ord_le4340812435750786203eger_o
@ ^ [X5: produc6241069584506657477e_term > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X5 @ Y5 ) @ R )
@ ^ [X5: produc6241069584506657477e_term > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X5 @ Y5 ) @ S2 ) )
= ( ord_le653643898420964396nteger @ R @ S2 ) ) ).
% pred_subset_eq2
thf(fact_6748_pred__subset__eq2,axiom,
! [R: set_Pr9222295170931077689nt_int,S2: set_Pr9222295170931077689nt_int] :
( ( ord_le5643404153117327598_int_o
@ ^ [X5: produc8551481072490612790e_term > option6357759511663192854e_term,Y5: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X5 @ Y5 ) @ R )
@ ^ [X5: produc8551481072490612790e_term > option6357759511663192854e_term,Y5: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X5 @ Y5 ) @ S2 ) )
= ( ord_le8725513860283290265nt_int @ R @ S2 ) ) ).
% pred_subset_eq2
thf(fact_6749_pred__subset__eq2,axiom,
! [R: set_Pr1872883991513573699nt_int,S2: set_Pr1872883991513573699nt_int] :
( ( ord_le2124322318746777828_int_o
@ ^ [X5: int > option6357759511663192854e_term,Y5: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X5 @ Y5 ) @ R )
@ ^ [X5: int > option6357759511663192854e_term,Y5: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X5 @ Y5 ) @ S2 ) )
= ( ord_le135402666524580259nt_int @ R @ S2 ) ) ).
% pred_subset_eq2
thf(fact_6750_arctan__eq__zero__iff,axiom,
! [X3: real] :
( ( ( arctan @ X3 )
= zero_zero_real )
= ( X3 = zero_zero_real ) ) ).
% arctan_eq_zero_iff
thf(fact_6751_arctan__zero__zero,axiom,
( ( arctan @ zero_zero_real )
= zero_zero_real ) ).
% arctan_zero_zero
thf(fact_6752_summable__zero,axiom,
( summable_complex
@ ^ [N4: nat] : zero_zero_complex ) ).
% summable_zero
thf(fact_6753_summable__zero,axiom,
( summable_real
@ ^ [N4: nat] : zero_zero_real ) ).
% summable_zero
thf(fact_6754_summable__zero,axiom,
( summable_nat
@ ^ [N4: nat] : zero_zero_nat ) ).
% summable_zero
thf(fact_6755_summable__zero,axiom,
( summable_int
@ ^ [N4: nat] : zero_zero_int ) ).
% summable_zero
thf(fact_6756_summable__single,axiom,
! [I: nat,F: nat > complex] :
( summable_complex
@ ^ [R5: nat] : ( if_complex @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_complex ) ) ).
% summable_single
thf(fact_6757_summable__single,axiom,
! [I: nat,F: nat > real] :
( summable_real
@ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real ) ) ).
% summable_single
thf(fact_6758_summable__single,axiom,
! [I: nat,F: nat > nat] :
( summable_nat
@ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat ) ) ).
% summable_single
thf(fact_6759_summable__single,axiom,
! [I: nat,F: nat > int] :
( summable_int
@ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int ) ) ).
% summable_single
thf(fact_6760_summable__iff__shift,axiom,
! [F: nat > real,K2: nat] :
( ( summable_real
@ ^ [N4: nat] : ( F @ ( plus_plus_nat @ N4 @ K2 ) ) )
= ( summable_real @ F ) ) ).
% summable_iff_shift
thf(fact_6761_arctan__less__zero__iff,axiom,
! [X3: real] :
( ( ord_less_real @ ( arctan @ X3 ) @ zero_zero_real )
= ( ord_less_real @ X3 @ zero_zero_real ) ) ).
% arctan_less_zero_iff
thf(fact_6762_zero__less__arctan__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ ( arctan @ X3 ) )
= ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% zero_less_arctan_iff
thf(fact_6763_zero__le__arctan__iff,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X3 ) )
= ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).
% zero_le_arctan_iff
thf(fact_6764_arctan__le__zero__iff,axiom,
! [X3: real] :
( ( ord_less_eq_real @ ( arctan @ X3 ) @ zero_zero_real )
= ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).
% arctan_le_zero_iff
thf(fact_6765_cos__coeff__0,axiom,
( ( cos_coeff @ zero_zero_nat )
= one_one_real ) ).
% cos_coeff_0
thf(fact_6766_summable__cmult__iff,axiom,
! [C2: complex,F: nat > complex] :
( ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ C2 @ ( F @ N4 ) ) )
= ( ( C2 = zero_zero_complex )
| ( summable_complex @ F ) ) ) ).
% summable_cmult_iff
thf(fact_6767_summable__cmult__iff,axiom,
! [C2: real,F: nat > real] :
( ( summable_real
@ ^ [N4: nat] : ( times_times_real @ C2 @ ( F @ N4 ) ) )
= ( ( C2 = zero_zero_real )
| ( summable_real @ F ) ) ) ).
% summable_cmult_iff
thf(fact_6768_summable__divide__iff,axiom,
! [F: nat > complex,C2: complex] :
( ( summable_complex
@ ^ [N4: nat] : ( divide1717551699836669952omplex @ ( F @ N4 ) @ C2 ) )
= ( ( C2 = zero_zero_complex )
| ( summable_complex @ F ) ) ) ).
% summable_divide_iff
thf(fact_6769_summable__divide__iff,axiom,
! [F: nat > real,C2: real] :
( ( summable_real
@ ^ [N4: nat] : ( divide_divide_real @ ( F @ N4 ) @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( summable_real @ F ) ) ) ).
% summable_divide_iff
thf(fact_6770_summable__If__finite,axiom,
! [P2: nat > $o,F: nat > complex] :
( ( finite_finite_nat @ ( collect_nat @ P2 ) )
=> ( summable_complex
@ ^ [R5: nat] : ( if_complex @ ( P2 @ R5 ) @ ( F @ R5 ) @ zero_zero_complex ) ) ) ).
% summable_If_finite
thf(fact_6771_summable__If__finite,axiom,
! [P2: nat > $o,F: nat > real] :
( ( finite_finite_nat @ ( collect_nat @ P2 ) )
=> ( summable_real
@ ^ [R5: nat] : ( if_real @ ( P2 @ R5 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).
% summable_If_finite
thf(fact_6772_summable__If__finite,axiom,
! [P2: nat > $o,F: nat > nat] :
( ( finite_finite_nat @ ( collect_nat @ P2 ) )
=> ( summable_nat
@ ^ [R5: nat] : ( if_nat @ ( P2 @ R5 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).
% summable_If_finite
thf(fact_6773_summable__If__finite,axiom,
! [P2: nat > $o,F: nat > int] :
( ( finite_finite_nat @ ( collect_nat @ P2 ) )
=> ( summable_int
@ ^ [R5: nat] : ( if_int @ ( P2 @ R5 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).
% summable_If_finite
thf(fact_6774_summable__If__finite__set,axiom,
! [A4: set_nat,F: nat > complex] :
( ( finite_finite_nat @ A4 )
=> ( summable_complex
@ ^ [R5: nat] : ( if_complex @ ( member_nat @ R5 @ A4 ) @ ( F @ R5 ) @ zero_zero_complex ) ) ) ).
% summable_If_finite_set
thf(fact_6775_summable__If__finite__set,axiom,
! [A4: set_nat,F: nat > real] :
( ( finite_finite_nat @ A4 )
=> ( summable_real
@ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A4 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).
% summable_If_finite_set
thf(fact_6776_summable__If__finite__set,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( summable_nat
@ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A4 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).
% summable_If_finite_set
thf(fact_6777_summable__If__finite__set,axiom,
! [A4: set_nat,F: nat > int] :
( ( finite_finite_nat @ A4 )
=> ( summable_int
@ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A4 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).
% summable_If_finite_set
thf(fact_6778_pi__neq__zero,axiom,
pi != zero_zero_real ).
% pi_neq_zero
thf(fact_6779_arctan__monotone_H,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ ( arctan @ X3 ) @ ( arctan @ Y ) ) ) ).
% arctan_monotone'
thf(fact_6780_arctan__le__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( arctan @ X3 ) @ ( arctan @ Y ) )
= ( ord_less_eq_real @ X3 @ Y ) ) ).
% arctan_le_iff
thf(fact_6781_summable__const__iff,axiom,
! [C2: complex] :
( ( summable_complex
@ ^ [Uu3: nat] : C2 )
= ( C2 = zero_zero_complex ) ) ).
% summable_const_iff
thf(fact_6782_summable__const__iff,axiom,
! [C2: real] :
( ( summable_real
@ ^ [Uu3: nat] : C2 )
= ( C2 = zero_zero_real ) ) ).
% summable_const_iff
thf(fact_6783_summable__comparison__test_H,axiom,
! [G: nat > real,N7: nat,F: nat > real] :
( ( summable_real @ G )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ N7 @ N3 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
=> ( summable_real @ F ) ) ) ).
% summable_comparison_test'
thf(fact_6784_summable__comparison__test_H,axiom,
! [G: nat > real,N7: nat,F: nat > complex] :
( ( summable_real @ G )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ N7 @ N3 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
=> ( summable_complex @ F ) ) ) ).
% summable_comparison_test'
thf(fact_6785_summable__comparison__test,axiom,
! [F: nat > real,G: nat > real] :
( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ N8 @ N3 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
=> ( ( summable_real @ G )
=> ( summable_real @ F ) ) ) ).
% summable_comparison_test
thf(fact_6786_summable__comparison__test,axiom,
! [F: nat > complex,G: nat > real] :
( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ N8 @ N3 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
=> ( ( summable_real @ G )
=> ( summable_complex @ F ) ) ) ).
% summable_comparison_test
thf(fact_6787_summable__add,axiom,
! [F: nat > real,G: nat > real] :
( ( summable_real @ F )
=> ( ( summable_real @ G )
=> ( summable_real
@ ^ [N4: nat] : ( plus_plus_real @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ).
% summable_add
thf(fact_6788_summable__add,axiom,
! [F: nat > nat,G: nat > nat] :
( ( summable_nat @ F )
=> ( ( summable_nat @ G )
=> ( summable_nat
@ ^ [N4: nat] : ( plus_plus_nat @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ).
% summable_add
thf(fact_6789_summable__add,axiom,
! [F: nat > int,G: nat > int] :
( ( summable_int @ F )
=> ( ( summable_int @ G )
=> ( summable_int
@ ^ [N4: nat] : ( plus_plus_int @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ).
% summable_add
thf(fact_6790_summable__Suc__iff,axiom,
! [F: nat > real] :
( ( summable_real
@ ^ [N4: nat] : ( F @ ( suc @ N4 ) ) )
= ( summable_real @ F ) ) ).
% summable_Suc_iff
thf(fact_6791_summable__ignore__initial__segment,axiom,
! [F: nat > real,K2: nat] :
( ( summable_real @ F )
=> ( summable_real
@ ^ [N4: nat] : ( F @ ( plus_plus_nat @ N4 @ K2 ) ) ) ) ).
% summable_ignore_initial_segment
thf(fact_6792_suminf__le,axiom,
! [F: nat > real,G: nat > real] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
=> ( ( summable_real @ F )
=> ( ( summable_real @ G )
=> ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).
% suminf_le
thf(fact_6793_suminf__le,axiom,
! [F: nat > nat,G: nat > nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ N3 ) )
=> ( ( summable_nat @ F )
=> ( ( summable_nat @ G )
=> ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) ) ) ) ) ).
% suminf_le
thf(fact_6794_suminf__le,axiom,
! [F: nat > int,G: nat > int] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G @ N3 ) )
=> ( ( summable_int @ F )
=> ( ( summable_int @ G )
=> ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) ) ) ) ) ).
% suminf_le
thf(fact_6795_summable__finite,axiom,
! [N7: set_nat,F: nat > complex] :
( ( finite_finite_nat @ N7 )
=> ( ! [N3: nat] :
( ~ ( member_nat @ N3 @ N7 )
=> ( ( F @ N3 )
= zero_zero_complex ) )
=> ( summable_complex @ F ) ) ) ).
% summable_finite
thf(fact_6796_summable__finite,axiom,
! [N7: set_nat,F: nat > real] :
( ( finite_finite_nat @ N7 )
=> ( ! [N3: nat] :
( ~ ( member_nat @ N3 @ N7 )
=> ( ( F @ N3 )
= zero_zero_real ) )
=> ( summable_real @ F ) ) ) ).
% summable_finite
thf(fact_6797_summable__finite,axiom,
! [N7: set_nat,F: nat > nat] :
( ( finite_finite_nat @ N7 )
=> ( ! [N3: nat] :
( ~ ( member_nat @ N3 @ N7 )
=> ( ( F @ N3 )
= zero_zero_nat ) )
=> ( summable_nat @ F ) ) ) ).
% summable_finite
thf(fact_6798_summable__finite,axiom,
! [N7: set_nat,F: nat > int] :
( ( finite_finite_nat @ N7 )
=> ( ! [N3: nat] :
( ~ ( member_nat @ N3 @ N7 )
=> ( ( F @ N3 )
= zero_zero_int ) )
=> ( summable_int @ F ) ) ) ).
% summable_finite
thf(fact_6799_summable__mult__D,axiom,
! [C2: complex,F: nat > complex] :
( ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ C2 @ ( F @ N4 ) ) )
=> ( ( C2 != zero_zero_complex )
=> ( summable_complex @ F ) ) ) ).
% summable_mult_D
thf(fact_6800_summable__mult__D,axiom,
! [C2: real,F: nat > real] :
( ( summable_real
@ ^ [N4: nat] : ( times_times_real @ C2 @ ( F @ N4 ) ) )
=> ( ( C2 != zero_zero_real )
=> ( summable_real @ F ) ) ) ).
% summable_mult_D
thf(fact_6801_summable__zero__power,axiom,
summable_real @ ( power_power_real @ zero_zero_real ) ).
% summable_zero_power
thf(fact_6802_summable__zero__power,axiom,
summable_int @ ( power_power_int @ zero_zero_int ) ).
% summable_zero_power
thf(fact_6803_summable__zero__power,axiom,
summable_complex @ ( power_power_complex @ zero_zero_complex ) ).
% summable_zero_power
thf(fact_6804_pi__gt__zero,axiom,
ord_less_real @ zero_zero_real @ pi ).
% pi_gt_zero
thf(fact_6805_pi__not__less__zero,axiom,
~ ( ord_less_real @ pi @ zero_zero_real ) ).
% pi_not_less_zero
thf(fact_6806_pi__ge__zero,axiom,
ord_less_eq_real @ zero_zero_real @ pi ).
% pi_ge_zero
thf(fact_6807_suminf__add,axiom,
! [F: nat > real,G: nat > real] :
( ( summable_real @ F )
=> ( ( summable_real @ G )
=> ( ( plus_plus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
= ( suminf_real
@ ^ [N4: nat] : ( plus_plus_real @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ) ).
% suminf_add
thf(fact_6808_suminf__add,axiom,
! [F: nat > nat,G: nat > nat] :
( ( summable_nat @ F )
=> ( ( summable_nat @ G )
=> ( ( plus_plus_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) )
= ( suminf_nat
@ ^ [N4: nat] : ( plus_plus_nat @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ) ).
% suminf_add
thf(fact_6809_suminf__add,axiom,
! [F: nat > int,G: nat > int] :
( ( summable_int @ F )
=> ( ( summable_int @ G )
=> ( ( plus_plus_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) )
= ( suminf_int
@ ^ [N4: nat] : ( plus_plus_int @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ) ).
% suminf_add
thf(fact_6810_suminf__nonneg,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% suminf_nonneg
thf(fact_6811_suminf__nonneg,axiom,
! [F: nat > nat] :
( ( summable_nat @ F )
=> ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% suminf_nonneg
thf(fact_6812_suminf__nonneg,axiom,
! [F: nat > int] :
( ( summable_int @ F )
=> ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% suminf_nonneg
thf(fact_6813_suminf__eq__zero__iff,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
=> ( ( ( suminf_real @ F )
= zero_zero_real )
= ( ! [N4: nat] :
( ( F @ N4 )
= zero_zero_real ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_6814_suminf__eq__zero__iff,axiom,
! [F: nat > nat] :
( ( summable_nat @ F )
=> ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
=> ( ( ( suminf_nat @ F )
= zero_zero_nat )
= ( ! [N4: nat] :
( ( F @ N4 )
= zero_zero_nat ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_6815_suminf__eq__zero__iff,axiom,
! [F: nat > int] :
( ( summable_int @ F )
=> ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
=> ( ( ( suminf_int @ F )
= zero_zero_int )
= ( ! [N4: nat] :
( ( F @ N4 )
= zero_zero_int ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_6816_suminf__pos,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N3: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N3 ) )
=> ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% suminf_pos
thf(fact_6817_suminf__pos,axiom,
! [F: nat > nat] :
( ( summable_nat @ F )
=> ( ! [N3: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N3 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% suminf_pos
thf(fact_6818_suminf__pos,axiom,
! [F: nat > int] :
( ( summable_int @ F )
=> ( ! [N3: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N3 ) )
=> ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% suminf_pos
thf(fact_6819_summable__0__powser,axiom,
! [F: nat > complex] :
( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ zero_zero_complex @ N4 ) ) ) ).
% summable_0_powser
thf(fact_6820_summable__0__powser,axiom,
! [F: nat > real] :
( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ zero_zero_real @ N4 ) ) ) ).
% summable_0_powser
thf(fact_6821_summable__zero__power_H,axiom,
! [F: nat > complex] :
( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ zero_zero_complex @ N4 ) ) ) ).
% summable_zero_power'
thf(fact_6822_summable__zero__power_H,axiom,
! [F: nat > real] :
( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ zero_zero_real @ N4 ) ) ) ).
% summable_zero_power'
thf(fact_6823_summable__zero__power_H,axiom,
! [F: nat > int] :
( summable_int
@ ^ [N4: nat] : ( times_times_int @ ( F @ N4 ) @ ( power_power_int @ zero_zero_int @ N4 ) ) ) ).
% summable_zero_power'
thf(fact_6824_summable__powser__split__head,axiom,
! [F: nat > complex,Z2: complex] :
( ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ ( suc @ N4 ) ) @ ( power_power_complex @ Z2 @ N4 ) ) )
= ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z2 @ N4 ) ) ) ) ).
% summable_powser_split_head
thf(fact_6825_summable__powser__split__head,axiom,
! [F: nat > real,Z2: real] :
( ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ ( suc @ N4 ) ) @ ( power_power_real @ Z2 @ N4 ) ) )
= ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z2 @ N4 ) ) ) ) ).
% summable_powser_split_head
thf(fact_6826_powser__split__head_I3_J,axiom,
! [F: nat > complex,Z2: complex] :
( ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z2 @ N4 ) ) )
=> ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ ( suc @ N4 ) ) @ ( power_power_complex @ Z2 @ N4 ) ) ) ) ).
% powser_split_head(3)
thf(fact_6827_powser__split__head_I3_J,axiom,
! [F: nat > real,Z2: real] :
( ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z2 @ N4 ) ) )
=> ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ ( suc @ N4 ) ) @ ( power_power_real @ Z2 @ N4 ) ) ) ) ).
% powser_split_head(3)
thf(fact_6828_summable__powser__ignore__initial__segment,axiom,
! [F: nat > complex,M2: nat,Z2: complex] :
( ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N4 @ M2 ) ) @ ( power_power_complex @ Z2 @ N4 ) ) )
= ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z2 @ N4 ) ) ) ) ).
% summable_powser_ignore_initial_segment
thf(fact_6829_summable__powser__ignore__initial__segment,axiom,
! [F: nat > real,M2: nat,Z2: real] :
( ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N4 @ M2 ) ) @ ( power_power_real @ Z2 @ N4 ) ) )
= ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z2 @ N4 ) ) ) ) ).
% summable_powser_ignore_initial_segment
thf(fact_6830_summable__norm__comparison__test,axiom,
! [F: nat > complex,G: nat > real] :
( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ N8 @ N3 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
=> ( ( summable_real @ G )
=> ( summable_real
@ ^ [N4: nat] : ( real_V1022390504157884413omplex @ ( F @ N4 ) ) ) ) ) ).
% summable_norm_comparison_test
thf(fact_6831_summable__rabs__comparison__test,axiom,
! [F: nat > real,G: nat > real] :
( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ N8 @ N3 )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
=> ( ( summable_real @ G )
=> ( summable_real
@ ^ [N4: nat] : ( abs_abs_real @ ( F @ N4 ) ) ) ) ) ).
% summable_rabs_comparison_test
thf(fact_6832_summable__rabs,axiom,
! [F: nat > real] :
( ( summable_real
@ ^ [N4: nat] : ( abs_abs_real @ ( F @ N4 ) ) )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
@ ( suminf_real
@ ^ [N4: nat] : ( abs_abs_real @ ( F @ N4 ) ) ) ) ) ).
% summable_rabs
thf(fact_6833_suminf__pos2,axiom,
! [F: nat > real,I: nat] :
( ( summable_real @ F )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
=> ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).
% suminf_pos2
thf(fact_6834_suminf__pos2,axiom,
! [F: nat > nat,I: nat] :
( ( summable_nat @ F )
=> ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
=> ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).
% suminf_pos2
thf(fact_6835_suminf__pos2,axiom,
! [F: nat > int,I: nat] :
( ( summable_int @ F )
=> ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
=> ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).
% suminf_pos2
thf(fact_6836_suminf__pos__iff,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
=> ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
= ( ? [I3: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_6837_suminf__pos__iff,axiom,
! [F: nat > nat] :
( ( summable_nat @ F )
=> ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
= ( ? [I3: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_6838_suminf__pos__iff,axiom,
! [F: nat > int] :
( ( summable_int @ F )
=> ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
= ( ? [I3: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I3 ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_6839_suminf__le__const,axiom,
! [F: nat > int,X3: int] :
( ( summable_int @ F )
=> ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
=> ( ord_less_eq_int @ ( suminf_int @ F ) @ X3 ) ) ) ).
% suminf_le_const
thf(fact_6840_suminf__le__const,axiom,
! [F: nat > nat,X3: nat] :
( ( summable_nat @ F )
=> ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
=> ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X3 ) ) ) ).
% suminf_le_const
thf(fact_6841_suminf__le__const,axiom,
! [F: nat > real,X3: real] :
( ( summable_real @ F )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
=> ( ord_less_eq_real @ ( suminf_real @ F ) @ X3 ) ) ) ).
% suminf_le_const
thf(fact_6842_summableI__nonneg__bounded,axiom,
! [F: nat > int,X3: int] :
( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
=> ( summable_int @ F ) ) ) ).
% summableI_nonneg_bounded
thf(fact_6843_summableI__nonneg__bounded,axiom,
! [F: nat > nat,X3: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
=> ( summable_nat @ F ) ) ) ).
% summableI_nonneg_bounded
thf(fact_6844_summableI__nonneg__bounded,axiom,
! [F: nat > real,X3: real] :
( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
=> ( summable_real @ F ) ) ) ).
% summableI_nonneg_bounded
thf(fact_6845_suminf__split__head,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ( suminf_real
@ ^ [N4: nat] : ( F @ ( suc @ N4 ) ) )
= ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).
% suminf_split_head
thf(fact_6846_machin__Euler,axiom,
( ( 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 ) ) ) ) ) ) ) ) ) ) )
= ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% machin_Euler
thf(fact_6847_pi__ge__two,axiom,
ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).
% pi_ge_two
thf(fact_6848_summable__norm,axiom,
! [F: nat > real] :
( ( summable_real
@ ^ [N4: nat] : ( real_V7735802525324610683m_real @ ( F @ N4 ) ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( suminf_real @ F ) )
@ ( suminf_real
@ ^ [N4: nat] : ( real_V7735802525324610683m_real @ ( F @ N4 ) ) ) ) ) ).
% summable_norm
thf(fact_6849_summable__norm,axiom,
! [F: nat > complex] :
( ( summable_real
@ ^ [N4: nat] : ( real_V1022390504157884413omplex @ ( F @ N4 ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( suminf_complex @ F ) )
@ ( suminf_real
@ ^ [N4: nat] : ( real_V1022390504157884413omplex @ ( F @ N4 ) ) ) ) ) ).
% summable_norm
thf(fact_6850_sum__le__suminf,axiom,
! [F: nat > int,I6: set_nat] :
( ( summable_int @ F )
=> ( ( finite_finite_nat @ I6 )
=> ( ! [N3: nat] :
( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I6 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I6 ) @ ( suminf_int @ F ) ) ) ) ) ).
% sum_le_suminf
thf(fact_6851_sum__le__suminf,axiom,
! [F: nat > nat,I6: set_nat] :
( ( summable_nat @ F )
=> ( ( finite_finite_nat @ I6 )
=> ( ! [N3: nat] :
( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I6 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I6 ) @ ( suminf_nat @ F ) ) ) ) ) ).
% sum_le_suminf
thf(fact_6852_sum__le__suminf,axiom,
! [F: nat > real,I6: set_nat] :
( ( summable_real @ F )
=> ( ( finite_finite_nat @ I6 )
=> ( ! [N3: nat] :
( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I6 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I6 ) @ ( suminf_real @ F ) ) ) ) ) ).
% sum_le_suminf
thf(fact_6853_pred__equals__eq2,axiom,
! [R: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
( ( ( ^ [X5: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y5 ) @ R ) )
= ( ^ [X5: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y5 ) @ S2 ) ) )
= ( R = S2 ) ) ).
% pred_equals_eq2
thf(fact_6854_pred__equals__eq2,axiom,
! [R: set_Pr8056137968301705908nteger,S2: set_Pr8056137968301705908nteger] :
( ( ( ^ [X5: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X5 @ Y5 ) @ R ) )
= ( ^ [X5: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X5 @ Y5 ) @ S2 ) ) )
= ( R = S2 ) ) ).
% pred_equals_eq2
thf(fact_6855_pred__equals__eq2,axiom,
! [R: set_Pr1281608226676607948nteger,S2: set_Pr1281608226676607948nteger] :
( ( ( ^ [X5: produc6241069584506657477e_term > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X5 @ Y5 ) @ R ) )
= ( ^ [X5: produc6241069584506657477e_term > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X5 @ Y5 ) @ S2 ) ) )
= ( R = S2 ) ) ).
% pred_equals_eq2
thf(fact_6856_pred__equals__eq2,axiom,
! [R: set_Pr9222295170931077689nt_int,S2: set_Pr9222295170931077689nt_int] :
( ( ( ^ [X5: produc8551481072490612790e_term > option6357759511663192854e_term,Y5: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X5 @ Y5 ) @ R ) )
= ( ^ [X5: produc8551481072490612790e_term > option6357759511663192854e_term,Y5: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X5 @ Y5 ) @ S2 ) ) )
= ( R = S2 ) ) ).
% pred_equals_eq2
thf(fact_6857_pred__equals__eq2,axiom,
! [R: set_Pr1872883991513573699nt_int,S2: set_Pr1872883991513573699nt_int] :
( ( ( ^ [X5: int > option6357759511663192854e_term,Y5: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X5 @ Y5 ) @ R ) )
= ( ^ [X5: int > option6357759511663192854e_term,Y5: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X5 @ Y5 ) @ S2 ) ) )
= ( R = S2 ) ) ).
% pred_equals_eq2
thf(fact_6858_suminf__split__initial__segment,axiom,
! [F: nat > real,K2: nat] :
( ( summable_real @ F )
=> ( ( suminf_real @ F )
= ( plus_plus_real
@ ( suminf_real
@ ^ [N4: nat] : ( F @ ( plus_plus_nat @ N4 @ K2 ) ) )
@ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K2 ) ) ) ) ) ).
% suminf_split_initial_segment
thf(fact_6859_suminf__minus__initial__segment,axiom,
! [F: nat > real,K2: nat] :
( ( summable_real @ F )
=> ( ( suminf_real
@ ^ [N4: nat] : ( F @ ( plus_plus_nat @ N4 @ K2 ) ) )
= ( minus_minus_real @ ( suminf_real @ F ) @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K2 ) ) ) ) ) ).
% suminf_minus_initial_segment
thf(fact_6860_pi__half__neq__zero,axiom,
( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
!= zero_zero_real ) ).
% pi_half_neq_zero
thf(fact_6861_pi__half__le__two,axiom,
ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% pi_half_le_two
thf(fact_6862_sum__less__suminf,axiom,
! [F: nat > int,N: nat] :
( ( summable_int @ F )
=> ( ! [M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ord_less_int @ zero_zero_int @ ( F @ M ) ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ).
% sum_less_suminf
thf(fact_6863_sum__less__suminf,axiom,
! [F: nat > nat,N: nat] :
( ( summable_nat @ F )
=> ( ! [M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ M ) ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ).
% sum_less_suminf
thf(fact_6864_sum__less__suminf,axiom,
! [F: nat > real,N: nat] :
( ( summable_real @ F )
=> ( ! [M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ord_less_real @ zero_zero_real @ ( F @ M ) ) )
=> ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ).
% sum_less_suminf
thf(fact_6865_powser__split__head_I1_J,axiom,
! [F: nat > complex,Z2: complex] :
( ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z2 @ N4 ) ) )
=> ( ( suminf_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z2 @ N4 ) ) )
= ( plus_plus_complex @ ( F @ zero_zero_nat )
@ ( times_times_complex
@ ( suminf_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ ( suc @ N4 ) ) @ ( power_power_complex @ Z2 @ N4 ) ) )
@ Z2 ) ) ) ) ).
% powser_split_head(1)
thf(fact_6866_powser__split__head_I1_J,axiom,
! [F: nat > real,Z2: real] :
( ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z2 @ N4 ) ) )
=> ( ( suminf_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z2 @ N4 ) ) )
= ( plus_plus_real @ ( F @ zero_zero_nat )
@ ( times_times_real
@ ( suminf_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ ( suc @ N4 ) ) @ ( power_power_real @ Z2 @ N4 ) ) )
@ Z2 ) ) ) ) ).
% powser_split_head(1)
thf(fact_6867_powser__split__head_I2_J,axiom,
! [F: nat > complex,Z2: complex] :
( ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z2 @ N4 ) ) )
=> ( ( times_times_complex
@ ( suminf_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ ( suc @ N4 ) ) @ ( power_power_complex @ Z2 @ N4 ) ) )
@ Z2 )
= ( minus_minus_complex
@ ( suminf_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z2 @ N4 ) ) )
@ ( F @ zero_zero_nat ) ) ) ) ).
% powser_split_head(2)
thf(fact_6868_powser__split__head_I2_J,axiom,
! [F: nat > real,Z2: real] :
( ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z2 @ N4 ) ) )
=> ( ( times_times_real
@ ( suminf_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ ( suc @ N4 ) ) @ ( power_power_real @ Z2 @ N4 ) ) )
@ Z2 )
= ( minus_minus_real
@ ( suminf_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z2 @ N4 ) ) )
@ ( F @ zero_zero_nat ) ) ) ) ).
% powser_split_head(2)
thf(fact_6869_summable__partial__sum__bound,axiom,
! [F: nat > complex,E2: real] :
( ( summable_complex @ F )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ~ ! [N9: nat] :
~ ! [M3: nat] :
( ( ord_less_eq_nat @ N9 @ M3 )
=> ! [N6: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M3 @ N6 ) ) ) @ E2 ) ) ) ) ).
% summable_partial_sum_bound
thf(fact_6870_summable__partial__sum__bound,axiom,
! [F: nat > real,E2: real] :
( ( summable_real @ F )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ~ ! [N9: nat] :
~ ! [M3: nat] :
( ( ord_less_eq_nat @ N9 @ M3 )
=> ! [N6: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M3 @ N6 ) ) ) @ E2 ) ) ) ) ).
% summable_partial_sum_bound
thf(fact_6871_suminf__exist__split,axiom,
! [R2: real,F: nat > real] :
( ( ord_less_real @ zero_zero_real @ R2 )
=> ( ( summable_real @ F )
=> ? [N9: nat] :
! [N6: nat] :
( ( ord_less_eq_nat @ N9 @ N6 )
=> ( ord_less_real
@ ( real_V7735802525324610683m_real
@ ( suminf_real
@ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N6 ) ) ) )
@ R2 ) ) ) ) ).
% suminf_exist_split
thf(fact_6872_suminf__exist__split,axiom,
! [R2: real,F: nat > complex] :
( ( ord_less_real @ zero_zero_real @ R2 )
=> ( ( summable_complex @ F )
=> ? [N9: nat] :
! [N6: nat] :
( ( ord_less_eq_nat @ N9 @ N6 )
=> ( ord_less_real
@ ( real_V1022390504157884413omplex
@ ( suminf_complex
@ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N6 ) ) ) )
@ R2 ) ) ) ) ).
% suminf_exist_split
thf(fact_6873_summable__power__series,axiom,
! [F: nat > real,Z2: real] :
( ! [I2: nat] : ( ord_less_eq_real @ ( F @ I2 ) @ one_one_real )
=> ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_real @ Z2 @ one_one_real )
=> ( summable_real
@ ^ [I3: nat] : ( times_times_real @ ( F @ I3 ) @ ( power_power_real @ Z2 @ I3 ) ) ) ) ) ) ) ).
% summable_power_series
thf(fact_6874_Abel__lemma,axiom,
! [R2: real,R0: real,A: nat > complex,M6: real] :
( ( ord_less_eq_real @ zero_zero_real @ R2 )
=> ( ( ord_less_real @ R2 @ R0 )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N3 ) ) @ ( power_power_real @ R0 @ N3 ) ) @ M6 )
=> ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N4 ) ) @ ( power_power_real @ R2 @ N4 ) ) ) ) ) ) ).
% Abel_lemma
thf(fact_6875_pi__half__gt__zero,axiom,
ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% pi_half_gt_zero
thf(fact_6876_pi__half__ge__zero,axiom,
ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% pi_half_ge_zero
thf(fact_6877_summable__ratio__test,axiom,
! [C2: real,N7: nat,F: nat > real] :
( ( ord_less_real @ C2 @ one_one_real )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ N7 @ N3 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N3 ) ) ) @ ( times_times_real @ C2 @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) ) ) )
=> ( summable_real @ F ) ) ) ).
% summable_ratio_test
thf(fact_6878_summable__ratio__test,axiom,
! [C2: real,N7: nat,F: nat > complex] :
( ( ord_less_real @ C2 @ one_one_real )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ N7 @ N3 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N3 ) ) ) @ ( times_times_real @ C2 @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) ) ) )
=> ( summable_complex @ F ) ) ) ).
% summable_ratio_test
thf(fact_6879_sum__less__suminf2,axiom,
! [F: nat > int,N: nat,I: nat] :
( ( summable_int @ F )
=> ( ! [M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ M ) ) )
=> ( ( ord_less_eq_nat @ N @ I )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_6880_sum__less__suminf2,axiom,
! [F: nat > nat,N: nat,I: nat] :
( ( summable_nat @ F )
=> ( ! [M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M ) ) )
=> ( ( ord_less_eq_nat @ N @ I )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_6881_sum__less__suminf2,axiom,
! [F: nat > real,N: nat,I: nat] :
( ( summable_real @ F )
=> ( ! [M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ M ) ) )
=> ( ( ord_less_eq_nat @ N @ I )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
=> ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_6882_subrelI,axiom,
! [R2: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
( ! [X4: int,Y3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y3 ) @ R2 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y3 ) @ S3 ) )
=> ( ord_le2843351958646193337nt_int @ R2 @ S3 ) ) ).
% subrelI
thf(fact_6883_subrelI,axiom,
! [R2: set_Pr8056137968301705908nteger,S3: set_Pr8056137968301705908nteger] :
( ! [X4: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X4 @ Y3 ) @ R2 )
=> ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X4 @ Y3 ) @ S3 ) )
=> ( ord_le3216752416896350996nteger @ R2 @ S3 ) ) ).
% subrelI
thf(fact_6884_subrelI,axiom,
! [R2: set_Pr1281608226676607948nteger,S3: set_Pr1281608226676607948nteger] :
( ! [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X4 @ Y3 ) @ R2 )
=> ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X4 @ Y3 ) @ S3 ) )
=> ( ord_le653643898420964396nteger @ R2 @ S3 ) ) ).
% subrelI
thf(fact_6885_subrelI,axiom,
! [R2: set_Pr9222295170931077689nt_int,S3: set_Pr9222295170931077689nt_int] :
( ! [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y3 ) @ R2 )
=> ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y3 ) @ S3 ) )
=> ( ord_le8725513860283290265nt_int @ R2 @ S3 ) ) ).
% subrelI
thf(fact_6886_subrelI,axiom,
! [R2: set_Pr1872883991513573699nt_int,S3: set_Pr1872883991513573699nt_int] :
( ! [X4: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y3 ) @ R2 )
=> ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y3 ) @ S3 ) )
=> ( ord_le135402666524580259nt_int @ R2 @ S3 ) ) ).
% subrelI
thf(fact_6887_minus__pi__half__less__zero,axiom,
ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).
% minus_pi_half_less_zero
thf(fact_6888_arctan__add,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
=> ( ( ord_less_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( plus_plus_real @ ( arctan @ X3 ) @ ( arctan @ Y ) )
= ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X3 @ Y ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X3 @ Y ) ) ) ) ) ) ) ).
% arctan_add
thf(fact_6889_bot__empty__eq2,axiom,
( bot_bot_int_int_o
= ( ^ [X5: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y5 ) @ bot_bo1796632182523588997nt_int ) ) ) ).
% bot_empty_eq2
thf(fact_6890_bot__empty__eq2,axiom,
( bot_bo5358457235160185703eger_o
= ( ^ [X5: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X5 @ Y5 ) @ bot_bo3145834390647256904nteger ) ) ) ).
% bot_empty_eq2
thf(fact_6891_bot__empty__eq2,axiom,
( bot_bo3000040243691356879eger_o
= ( ^ [X5: produc6241069584506657477e_term > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X5 @ Y5 ) @ bot_bo5443222936135328352nteger ) ) ) ).
% bot_empty_eq2
thf(fact_6892_bot__empty__eq2,axiom,
( bot_bo8662317086119403298_int_o
= ( ^ [X5: produc8551481072490612790e_term > option6357759511663192854e_term,Y5: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X5 @ Y5 ) @ bot_bo572930865798478029nt_int ) ) ) ).
% bot_empty_eq2
thf(fact_6893_bot__empty__eq2,axiom,
( bot_bo1403522918969695512_int_o
= ( ^ [X5: int > option6357759511663192854e_term,Y5: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X5 @ Y5 ) @ bot_bo4508923176915781079nt_int ) ) ) ).
% bot_empty_eq2
thf(fact_6894_bot__empty__eq2,axiom,
( bot_bot_nat_nat_o
= ( ^ [X5: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y5 ) @ bot_bo2099793752762293965at_nat ) ) ) ).
% bot_empty_eq2
thf(fact_6895_pred__subset__eq,axiom,
! [R: set_complex,S2: set_complex] :
( ( ord_le4573692005234683329plex_o
@ ^ [X5: complex] : ( member_complex @ X5 @ R )
@ ^ [X5: complex] : ( member_complex @ X5 @ S2 ) )
= ( ord_le211207098394363844omplex @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_6896_pred__subset__eq,axiom,
! [R: set_real,S2: set_real] :
( ( ord_less_eq_real_o
@ ^ [X5: real] : ( member_real @ X5 @ R )
@ ^ [X5: real] : ( member_real @ X5 @ S2 ) )
= ( ord_less_eq_set_real @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_6897_pred__subset__eq,axiom,
! [R: set_set_nat,S2: set_set_nat] :
( ( ord_le3964352015994296041_nat_o
@ ^ [X5: set_nat] : ( member_set_nat @ X5 @ R )
@ ^ [X5: set_nat] : ( member_set_nat @ X5 @ S2 ) )
= ( ord_le6893508408891458716et_nat @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_6898_pred__subset__eq,axiom,
! [R: set_nat,S2: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X5: nat] : ( member_nat @ X5 @ R )
@ ^ [X5: nat] : ( member_nat @ X5 @ S2 ) )
= ( ord_less_eq_set_nat @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_6899_pred__subset__eq,axiom,
! [R: set_int,S2: set_int] :
( ( ord_less_eq_int_o
@ ^ [X5: int] : ( member_int @ X5 @ R )
@ ^ [X5: int] : ( member_int @ X5 @ S2 ) )
= ( ord_less_eq_set_int @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_6900_sum__pos__lt__pair,axiom,
! [F: nat > real,K2: nat] :
( ( summable_real @ F )
=> ( ! [D2: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K2 @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D2 ) ) ) @ ( F @ ( plus_plus_nat @ K2 @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D2 ) @ one_one_nat ) ) ) ) )
=> ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K2 ) ) @ ( suminf_real @ F ) ) ) ) ).
% sum_pos_lt_pair
thf(fact_6901_sin__cos__npi,axiom,
! [N: nat] :
( ( 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 ) ) ) )
= ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% sin_cos_npi
thf(fact_6902_cos__pi__eq__zero,axiom,
! [M2: nat] :
( ( cos_real @ ( divide_divide_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= zero_zero_real ) ).
% cos_pi_eq_zero
thf(fact_6903_accp__subset,axiom,
! [R1: product_prod_num_num > product_prod_num_num > $o,R22: product_prod_num_num > product_prod_num_num > $o] :
( ( ord_le2556027599737686990_num_o @ R1 @ R22 )
=> ( ord_le2239182809043710856_num_o @ ( accp_P3113834385874906142um_num @ R22 ) @ ( accp_P3113834385874906142um_num @ R1 ) ) ) ).
% accp_subset
thf(fact_6904_accp__subset,axiom,
! [R1: product_prod_nat_nat > product_prod_nat_nat > $o,R22: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( ord_le5604493270027003598_nat_o @ R1 @ R22 )
=> ( ord_le704812498762024988_nat_o @ ( accp_P4275260045618599050at_nat @ R22 ) @ ( accp_P4275260045618599050at_nat @ R1 ) ) ) ).
% accp_subset
thf(fact_6905_accp__subset,axiom,
! [R1: product_prod_int_int > product_prod_int_int > $o,R22: product_prod_int_int > product_prod_int_int > $o] :
( ( ord_le1598226405681992910_int_o @ R1 @ R22 )
=> ( ord_le8369615600986905444_int_o @ ( accp_P1096762738010456898nt_int @ R22 ) @ ( accp_P1096762738010456898nt_int @ R1 ) ) ) ).
% accp_subset
thf(fact_6906_accp__subset,axiom,
! [R1: list_nat > list_nat > $o,R22: list_nat > list_nat > $o] :
( ( ord_le6558929396352911974_nat_o @ R1 @ R22 )
=> ( ord_le1520216061033275535_nat_o @ ( accp_list_nat @ R22 ) @ ( accp_list_nat @ R1 ) ) ) ).
% accp_subset
thf(fact_6907_accp__subset,axiom,
! [R1: nat > nat > $o,R22: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ R1 @ R22 )
=> ( ord_less_eq_nat_o @ ( accp_nat @ R22 ) @ ( accp_nat @ R1 ) ) ) ).
% accp_subset
thf(fact_6908_geometric__deriv__sums,axiom,
! [Z2: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z2 ) @ one_one_real )
=> ( sums_real
@ ^ [N4: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) @ ( power_power_real @ Z2 @ N4 ) )
@ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% geometric_deriv_sums
thf(fact_6909_geometric__deriv__sums,axiom,
! [Z2: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ one_one_real )
=> ( sums_complex
@ ^ [N4: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N4 ) ) @ ( power_power_complex @ Z2 @ N4 ) )
@ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% geometric_deriv_sums
thf(fact_6910_monoseq__def,axiom,
( topolo6980174941875973593q_real
= ( ^ [X7: nat > real] :
( ! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_less_eq_real @ ( X7 @ M5 ) @ ( X7 @ N4 ) ) )
| ! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_less_eq_real @ ( X7 @ N4 ) @ ( X7 @ M5 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_6911_monoseq__def,axiom,
( topolo3100542954746470799et_int
= ( ^ [X7: nat > set_int] :
( ! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_less_eq_set_int @ ( X7 @ M5 ) @ ( X7 @ N4 ) ) )
| ! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_less_eq_set_int @ ( X7 @ N4 ) @ ( X7 @ M5 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_6912_monoseq__def,axiom,
( topolo4267028734544971653eq_rat
= ( ^ [X7: nat > rat] :
( ! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_less_eq_rat @ ( X7 @ M5 ) @ ( X7 @ N4 ) ) )
| ! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_less_eq_rat @ ( X7 @ N4 ) @ ( X7 @ M5 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_6913_monoseq__def,axiom,
( topolo1459490580787246023eq_num
= ( ^ [X7: nat > num] :
( ! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_less_eq_num @ ( X7 @ M5 ) @ ( X7 @ N4 ) ) )
| ! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_less_eq_num @ ( X7 @ N4 ) @ ( X7 @ M5 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_6914_monoseq__def,axiom,
( topolo4902158794631467389eq_nat
= ( ^ [X7: nat > nat] :
( ! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_less_eq_nat @ ( X7 @ M5 ) @ ( X7 @ N4 ) ) )
| ! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_less_eq_nat @ ( X7 @ N4 ) @ ( X7 @ M5 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_6915_monoseq__def,axiom,
( topolo4899668324122417113eq_int
= ( ^ [X7: nat > int] :
( ! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_less_eq_int @ ( X7 @ M5 ) @ ( X7 @ N4 ) ) )
| ! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_less_eq_int @ ( X7 @ N4 ) @ ( X7 @ M5 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_6916_monoI2,axiom,
! [X8: nat > real] :
( ! [M: nat,N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ord_less_eq_real @ ( X8 @ N3 ) @ ( X8 @ M ) ) )
=> ( topolo6980174941875973593q_real @ X8 ) ) ).
% monoI2
thf(fact_6917_monoI2,axiom,
! [X8: nat > set_int] :
( ! [M: nat,N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ord_less_eq_set_int @ ( X8 @ N3 ) @ ( X8 @ M ) ) )
=> ( topolo3100542954746470799et_int @ X8 ) ) ).
% monoI2
thf(fact_6918_monoI2,axiom,
! [X8: nat > rat] :
( ! [M: nat,N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ord_less_eq_rat @ ( X8 @ N3 ) @ ( X8 @ M ) ) )
=> ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% monoI2
thf(fact_6919_monoI2,axiom,
! [X8: nat > num] :
( ! [M: nat,N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ord_less_eq_num @ ( X8 @ N3 ) @ ( X8 @ M ) ) )
=> ( topolo1459490580787246023eq_num @ X8 ) ) ).
% monoI2
thf(fact_6920_monoI2,axiom,
! [X8: nat > nat] :
( ! [M: nat,N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ord_less_eq_nat @ ( X8 @ N3 ) @ ( X8 @ M ) ) )
=> ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% monoI2
thf(fact_6921_monoI2,axiom,
! [X8: nat > int] :
( ! [M: nat,N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ord_less_eq_int @ ( X8 @ N3 ) @ ( X8 @ M ) ) )
=> ( topolo4899668324122417113eq_int @ X8 ) ) ).
% monoI2
thf(fact_6922_sin__zero,axiom,
( ( sin_complex @ zero_zero_complex )
= zero_zero_complex ) ).
% sin_zero
thf(fact_6923_sin__zero,axiom,
( ( sin_real @ zero_zero_real )
= zero_zero_real ) ).
% sin_zero
thf(fact_6924_sums__zero,axiom,
( sums_complex
@ ^ [N4: nat] : zero_zero_complex
@ zero_zero_complex ) ).
% sums_zero
thf(fact_6925_sums__zero,axiom,
( sums_real
@ ^ [N4: nat] : zero_zero_real
@ zero_zero_real ) ).
% sums_zero
thf(fact_6926_sums__zero,axiom,
( sums_nat
@ ^ [N4: nat] : zero_zero_nat
@ zero_zero_nat ) ).
% sums_zero
thf(fact_6927_sums__zero,axiom,
( sums_int
@ ^ [N4: nat] : zero_zero_int
@ zero_zero_int ) ).
% sums_zero
thf(fact_6928_cos__zero,axiom,
( ( cos_complex @ zero_zero_complex )
= one_one_complex ) ).
% cos_zero
thf(fact_6929_cos__zero,axiom,
( ( cos_real @ zero_zero_real )
= one_one_real ) ).
% cos_zero
thf(fact_6930_sin__pi,axiom,
( ( sin_real @ pi )
= zero_zero_real ) ).
% sin_pi
thf(fact_6931_cos__periodic__pi2,axiom,
! [X3: real] :
( ( cos_real @ ( plus_plus_real @ pi @ X3 ) )
= ( uminus_uminus_real @ ( cos_real @ X3 ) ) ) ).
% cos_periodic_pi2
thf(fact_6932_cos__periodic__pi,axiom,
! [X3: real] :
( ( cos_real @ ( plus_plus_real @ X3 @ pi ) )
= ( uminus_uminus_real @ ( cos_real @ X3 ) ) ) ).
% cos_periodic_pi
thf(fact_6933_sin__periodic__pi2,axiom,
! [X3: real] :
( ( sin_real @ ( plus_plus_real @ pi @ X3 ) )
= ( uminus_uminus_real @ ( sin_real @ X3 ) ) ) ).
% sin_periodic_pi2
thf(fact_6934_sin__periodic__pi,axiom,
! [X3: real] :
( ( sin_real @ ( plus_plus_real @ X3 @ pi ) )
= ( uminus_uminus_real @ ( sin_real @ X3 ) ) ) ).
% sin_periodic_pi
thf(fact_6935_sin__cos__squared__add3,axiom,
! [X3: complex] :
( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ X3 ) ) @ ( times_times_complex @ ( sin_complex @ X3 ) @ ( sin_complex @ X3 ) ) )
= one_one_complex ) ).
% sin_cos_squared_add3
thf(fact_6936_sin__cos__squared__add3,axiom,
! [X3: real] :
( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ X3 ) ) @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ X3 ) ) )
= one_one_real ) ).
% sin_cos_squared_add3
thf(fact_6937_sin__npi,axiom,
! [N: nat] :
( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
= zero_zero_real ) ).
% sin_npi
thf(fact_6938_sin__npi2,axiom,
! [N: nat] :
( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
= zero_zero_real ) ).
% sin_npi2
thf(fact_6939_sin__npi__int,axiom,
! [N: int] :
( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
= zero_zero_real ) ).
% sin_npi_int
thf(fact_6940_powser__sums__zero__iff,axiom,
! [A: nat > complex,X3: complex] :
( ( sums_complex
@ ^ [N4: nat] : ( times_times_complex @ ( A @ N4 ) @ ( power_power_complex @ zero_zero_complex @ N4 ) )
@ X3 )
= ( ( A @ zero_zero_nat )
= X3 ) ) ).
% powser_sums_zero_iff
thf(fact_6941_powser__sums__zero__iff,axiom,
! [A: nat > real,X3: real] :
( ( sums_real
@ ^ [N4: nat] : ( times_times_real @ ( A @ N4 ) @ ( power_power_real @ zero_zero_real @ N4 ) )
@ X3 )
= ( ( A @ zero_zero_nat )
= X3 ) ) ).
% powser_sums_zero_iff
thf(fact_6942_cos__pi__half,axiom,
( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= zero_zero_real ) ).
% cos_pi_half
thf(fact_6943_sin__two__pi,axiom,
( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
= zero_zero_real ) ).
% sin_two_pi
thf(fact_6944_cos__periodic,axiom,
! [X3: real] :
( ( cos_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
= ( cos_real @ X3 ) ) ).
% cos_periodic
thf(fact_6945_sin__periodic,axiom,
! [X3: real] :
( ( sin_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
= ( sin_real @ X3 ) ) ).
% sin_periodic
thf(fact_6946_sin__cos__squared__add,axiom,
! [X3: real] :
( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real ) ).
% sin_cos_squared_add
thf(fact_6947_sin__cos__squared__add,axiom,
! [X3: complex] :
( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_complex ) ).
% sin_cos_squared_add
thf(fact_6948_sin__cos__squared__add2,axiom,
! [X3: real] :
( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real ) ).
% sin_cos_squared_add2
thf(fact_6949_sin__cos__squared__add2,axiom,
! [X3: complex] :
( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_complex ) ).
% sin_cos_squared_add2
thf(fact_6950_sin__2npi,axiom,
! [N: nat] :
( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
= zero_zero_real ) ).
% sin_2npi
thf(fact_6951_sin__int__2pin,axiom,
! [N: int] :
( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
= zero_zero_real ) ).
% sin_int_2pin
thf(fact_6952_cos__3over2__pi,axiom,
( ( cos_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
= zero_zero_real ) ).
% cos_3over2_pi
thf(fact_6953_cos__one__sin__zero,axiom,
! [X3: complex] :
( ( ( cos_complex @ X3 )
= one_one_complex )
=> ( ( sin_complex @ X3 )
= zero_zero_complex ) ) ).
% cos_one_sin_zero
thf(fact_6954_cos__one__sin__zero,axiom,
! [X3: real] :
( ( ( cos_real @ X3 )
= one_one_real )
=> ( ( sin_real @ X3 )
= zero_zero_real ) ) ).
% cos_one_sin_zero
thf(fact_6955_sums__0,axiom,
! [F: nat > complex] :
( ! [N3: nat] :
( ( F @ N3 )
= zero_zero_complex )
=> ( sums_complex @ F @ zero_zero_complex ) ) ).
% sums_0
thf(fact_6956_sums__0,axiom,
! [F: nat > real] :
( ! [N3: nat] :
( ( F @ N3 )
= zero_zero_real )
=> ( sums_real @ F @ zero_zero_real ) ) ).
% sums_0
thf(fact_6957_sums__0,axiom,
! [F: nat > nat] :
( ! [N3: nat] :
( ( F @ N3 )
= zero_zero_nat )
=> ( sums_nat @ F @ zero_zero_nat ) ) ).
% sums_0
thf(fact_6958_sums__0,axiom,
! [F: nat > int] :
( ! [N3: nat] :
( ( F @ N3 )
= zero_zero_int )
=> ( sums_int @ F @ zero_zero_int ) ) ).
% sums_0
thf(fact_6959_sin__add,axiom,
! [X3: real,Y: real] :
( ( sin_real @ ( plus_plus_real @ X3 @ Y ) )
= ( plus_plus_real @ ( times_times_real @ ( sin_real @ X3 ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( sin_real @ Y ) ) ) ) ).
% sin_add
thf(fact_6960_sums__le,axiom,
! [F: nat > real,G: nat > real,S3: real,T: real] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
=> ( ( sums_real @ F @ S3 )
=> ( ( sums_real @ G @ T )
=> ( ord_less_eq_real @ S3 @ T ) ) ) ) ).
% sums_le
thf(fact_6961_sums__le,axiom,
! [F: nat > nat,G: nat > nat,S3: nat,T: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ N3 ) )
=> ( ( sums_nat @ F @ S3 )
=> ( ( sums_nat @ G @ T )
=> ( ord_less_eq_nat @ S3 @ T ) ) ) ) ).
% sums_le
thf(fact_6962_sums__le,axiom,
! [F: nat > int,G: nat > int,S3: int,T: int] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G @ N3 ) )
=> ( ( sums_int @ F @ S3 )
=> ( ( sums_int @ G @ T )
=> ( ord_less_eq_int @ S3 @ T ) ) ) ) ).
% sums_le
thf(fact_6963_sums__single,axiom,
! [I: nat,F: nat > complex] :
( sums_complex
@ ^ [R5: nat] : ( if_complex @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_complex )
@ ( F @ I ) ) ).
% sums_single
thf(fact_6964_sums__single,axiom,
! [I: nat,F: nat > real] :
( sums_real
@ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real )
@ ( F @ I ) ) ).
% sums_single
thf(fact_6965_sums__single,axiom,
! [I: nat,F: nat > nat] :
( sums_nat
@ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat )
@ ( F @ I ) ) ).
% sums_single
thf(fact_6966_sums__single,axiom,
! [I: nat,F: nat > int] :
( sums_int
@ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int )
@ ( F @ I ) ) ).
% sums_single
thf(fact_6967_sums__add,axiom,
! [F: nat > real,A: real,G: nat > real,B: real] :
( ( sums_real @ F @ A )
=> ( ( sums_real @ G @ B )
=> ( sums_real
@ ^ [N4: nat] : ( plus_plus_real @ ( F @ N4 ) @ ( G @ N4 ) )
@ ( plus_plus_real @ A @ B ) ) ) ) ).
% sums_add
thf(fact_6968_sums__add,axiom,
! [F: nat > nat,A: nat,G: nat > nat,B: nat] :
( ( sums_nat @ F @ A )
=> ( ( sums_nat @ G @ B )
=> ( sums_nat
@ ^ [N4: nat] : ( plus_plus_nat @ ( F @ N4 ) @ ( G @ N4 ) )
@ ( plus_plus_nat @ A @ B ) ) ) ) ).
% sums_add
thf(fact_6969_sums__add,axiom,
! [F: nat > int,A: int,G: nat > int,B: int] :
( ( sums_int @ F @ A )
=> ( ( sums_int @ G @ B )
=> ( sums_int
@ ^ [N4: nat] : ( plus_plus_int @ ( F @ N4 ) @ ( G @ N4 ) )
@ ( plus_plus_int @ A @ B ) ) ) ) ).
% sums_add
thf(fact_6970_cos__add,axiom,
! [X3: real,Y: real] :
( ( cos_real @ ( plus_plus_real @ X3 @ Y ) )
= ( minus_minus_real @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ Y ) ) ) ) ).
% cos_add
thf(fact_6971_cos__diff,axiom,
! [X3: real,Y: real] :
( ( cos_real @ ( minus_minus_real @ X3 @ Y ) )
= ( plus_plus_real @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ Y ) ) ) ) ).
% cos_diff
thf(fact_6972_sin__zero__norm__cos__one,axiom,
! [X3: real] :
( ( ( sin_real @ X3 )
= zero_zero_real )
=> ( ( real_V7735802525324610683m_real @ ( cos_real @ X3 ) )
= one_one_real ) ) ).
% sin_zero_norm_cos_one
thf(fact_6973_sin__zero__norm__cos__one,axiom,
! [X3: complex] :
( ( ( sin_complex @ X3 )
= zero_zero_complex )
=> ( ( real_V1022390504157884413omplex @ ( cos_complex @ X3 ) )
= one_one_real ) ) ).
% sin_zero_norm_cos_one
thf(fact_6974_sin__zero__abs__cos__one,axiom,
! [X3: real] :
( ( ( sin_real @ X3 )
= zero_zero_real )
=> ( ( abs_abs_real @ ( cos_real @ X3 ) )
= one_one_real ) ) ).
% sin_zero_abs_cos_one
thf(fact_6975_sincos__principal__value,axiom,
! [X3: real] :
? [Y3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y3 )
& ( ord_less_eq_real @ Y3 @ pi )
& ( ( sin_real @ Y3 )
= ( sin_real @ X3 ) )
& ( ( cos_real @ Y3 )
= ( cos_real @ X3 ) ) ) ).
% sincos_principal_value
thf(fact_6976_sin__x__le__x,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ ( sin_real @ X3 ) @ X3 ) ) ).
% sin_x_le_x
thf(fact_6977_sin__le__one,axiom,
! [X3: real] : ( ord_less_eq_real @ ( sin_real @ X3 ) @ one_one_real ) ).
% sin_le_one
thf(fact_6978_cos__le__one,axiom,
! [X3: real] : ( ord_less_eq_real @ ( cos_real @ X3 ) @ one_one_real ) ).
% cos_le_one
thf(fact_6979_abs__sin__x__le__abs__x,axiom,
! [X3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X3 ) ) @ ( abs_abs_real @ X3 ) ) ).
% abs_sin_x_le_abs_x
thf(fact_6980_sums__mult2__iff,axiom,
! [C2: complex,F: nat > complex,D: complex] :
( ( C2 != zero_zero_complex )
=> ( ( sums_complex
@ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ C2 )
@ ( times_times_complex @ D @ C2 ) )
= ( sums_complex @ F @ D ) ) ) ).
% sums_mult2_iff
thf(fact_6981_sums__mult2__iff,axiom,
! [C2: real,F: nat > real,D: real] :
( ( C2 != zero_zero_real )
=> ( ( sums_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ C2 )
@ ( times_times_real @ D @ C2 ) )
= ( sums_real @ F @ D ) ) ) ).
% sums_mult2_iff
thf(fact_6982_sums__mult__iff,axiom,
! [C2: complex,F: nat > complex,D: complex] :
( ( C2 != zero_zero_complex )
=> ( ( sums_complex
@ ^ [N4: nat] : ( times_times_complex @ C2 @ ( F @ N4 ) )
@ ( times_times_complex @ C2 @ D ) )
= ( sums_complex @ F @ D ) ) ) ).
% sums_mult_iff
thf(fact_6983_sums__mult__iff,axiom,
! [C2: real,F: nat > real,D: real] :
( ( C2 != zero_zero_real )
=> ( ( sums_real
@ ^ [N4: nat] : ( times_times_real @ C2 @ ( F @ N4 ) )
@ ( times_times_real @ C2 @ D ) )
= ( sums_real @ F @ D ) ) ) ).
% sums_mult_iff
thf(fact_6984_cos__arctan__not__zero,axiom,
! [X3: real] :
( ( cos_real @ ( arctan @ X3 ) )
!= zero_zero_real ) ).
% cos_arctan_not_zero
thf(fact_6985_sin__cos__le1,axiom,
! [X3: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y ) ) ) ) @ one_one_real ) ).
% sin_cos_le1
thf(fact_6986_sums__mult__D,axiom,
! [C2: complex,F: nat > complex,A: complex] :
( ( sums_complex
@ ^ [N4: nat] : ( times_times_complex @ C2 @ ( F @ N4 ) )
@ A )
=> ( ( C2 != zero_zero_complex )
=> ( sums_complex @ F @ ( divide1717551699836669952omplex @ A @ C2 ) ) ) ) ).
% sums_mult_D
thf(fact_6987_sums__mult__D,axiom,
! [C2: real,F: nat > real,A: real] :
( ( sums_real
@ ^ [N4: nat] : ( times_times_real @ C2 @ ( F @ N4 ) )
@ A )
=> ( ( C2 != zero_zero_real )
=> ( sums_real @ F @ ( divide_divide_real @ A @ C2 ) ) ) ) ).
% sums_mult_D
thf(fact_6988_sums__Suc__imp,axiom,
! [F: nat > complex,S3: complex] :
( ( ( F @ zero_zero_nat )
= zero_zero_complex )
=> ( ( sums_complex
@ ^ [N4: nat] : ( F @ ( suc @ N4 ) )
@ S3 )
=> ( sums_complex @ F @ S3 ) ) ) ).
% sums_Suc_imp
thf(fact_6989_sums__Suc__imp,axiom,
! [F: nat > real,S3: real] :
( ( ( F @ zero_zero_nat )
= zero_zero_real )
=> ( ( sums_real
@ ^ [N4: nat] : ( F @ ( suc @ N4 ) )
@ S3 )
=> ( sums_real @ F @ S3 ) ) ) ).
% sums_Suc_imp
thf(fact_6990_sums__Suc__iff,axiom,
! [F: nat > real,S3: real] :
( ( sums_real
@ ^ [N4: nat] : ( F @ ( suc @ N4 ) )
@ S3 )
= ( sums_real @ F @ ( plus_plus_real @ S3 @ ( F @ zero_zero_nat ) ) ) ) ).
% sums_Suc_iff
thf(fact_6991_sums__Suc,axiom,
! [F: nat > real,L: real] :
( ( sums_real
@ ^ [N4: nat] : ( F @ ( suc @ N4 ) )
@ L )
=> ( sums_real @ F @ ( plus_plus_real @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% sums_Suc
thf(fact_6992_sums__Suc,axiom,
! [F: nat > nat,L: nat] :
( ( sums_nat
@ ^ [N4: nat] : ( F @ ( suc @ N4 ) )
@ L )
=> ( sums_nat @ F @ ( plus_plus_nat @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% sums_Suc
thf(fact_6993_sums__Suc,axiom,
! [F: nat > int,L: int] :
( ( sums_int
@ ^ [N4: nat] : ( F @ ( suc @ N4 ) )
@ L )
=> ( sums_int @ F @ ( plus_plus_int @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% sums_Suc
thf(fact_6994_sin__gt__zero,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ pi )
=> ( ord_less_real @ zero_zero_real @ ( sin_real @ X3 ) ) ) ) ).
% sin_gt_zero
thf(fact_6995_sin__x__ge__neg__x,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ X3 ) @ ( sin_real @ X3 ) ) ) ).
% sin_x_ge_neg_x
thf(fact_6996_sin__ge__zero,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ pi )
=> ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X3 ) ) ) ) ).
% sin_ge_zero
thf(fact_6997_sums__zero__iff__shift,axiom,
! [N: nat,F: nat > complex,S3: complex] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( F @ I2 )
= zero_zero_complex ) )
=> ( ( sums_complex
@ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N ) )
@ S3 )
= ( sums_complex @ F @ S3 ) ) ) ).
% sums_zero_iff_shift
thf(fact_6998_sums__zero__iff__shift,axiom,
! [N: nat,F: nat > real,S3: real] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( F @ I2 )
= zero_zero_real ) )
=> ( ( sums_real
@ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N ) )
@ S3 )
= ( sums_real @ F @ S3 ) ) ) ).
% sums_zero_iff_shift
thf(fact_6999_sin__ge__minus__one,axiom,
! [X3: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X3 ) ) ).
% sin_ge_minus_one
thf(fact_7000_cos__monotone__0__pi__le,axiom,
! [Y: real,X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ X3 )
=> ( ( ord_less_eq_real @ X3 @ pi )
=> ( ord_less_eq_real @ ( cos_real @ X3 ) @ ( cos_real @ Y ) ) ) ) ) ).
% cos_monotone_0_pi_le
thf(fact_7001_cos__mono__le__eq,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ pi )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ pi )
=> ( ( ord_less_eq_real @ ( cos_real @ X3 ) @ ( cos_real @ Y ) )
= ( ord_less_eq_real @ Y @ X3 ) ) ) ) ) ) ).
% cos_mono_le_eq
thf(fact_7002_cos__inj__pi,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ pi )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ pi )
=> ( ( ( cos_real @ X3 )
= ( cos_real @ Y ) )
=> ( X3 = Y ) ) ) ) ) ) ).
% cos_inj_pi
thf(fact_7003_cos__ge__minus__one,axiom,
! [X3: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X3 ) ) ).
% cos_ge_minus_one
thf(fact_7004_abs__sin__le__one,axiom,
! [X3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X3 ) ) @ one_one_real ) ).
% abs_sin_le_one
thf(fact_7005_abs__cos__le__one,axiom,
! [X3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X3 ) ) @ one_one_real ) ).
% abs_cos_le_one
thf(fact_7006_sin__times__sin,axiom,
! [W2: complex,Z2: complex] :
( ( times_times_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z2 ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) @ ( cos_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% sin_times_sin
thf(fact_7007_sin__times__sin,axiom,
! [W2: real,Z2: real] :
( ( times_times_real @ ( sin_real @ W2 ) @ ( sin_real @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( cos_real @ ( minus_minus_real @ W2 @ Z2 ) ) @ ( cos_real @ ( plus_plus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% sin_times_sin
thf(fact_7008_sin__times__cos,axiom,
! [W2: complex,Z2: complex] :
( ( times_times_complex @ ( sin_complex @ W2 ) @ ( cos_complex @ Z2 ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) @ ( sin_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% sin_times_cos
thf(fact_7009_sin__times__cos,axiom,
! [W2: real,Z2: real] :
( ( times_times_real @ ( sin_real @ W2 ) @ ( cos_real @ Z2 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( sin_real @ ( plus_plus_real @ W2 @ Z2 ) ) @ ( sin_real @ ( minus_minus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% sin_times_cos
thf(fact_7010_cos__times__sin,axiom,
! [W2: complex,Z2: complex] :
( ( times_times_complex @ ( cos_complex @ W2 ) @ ( sin_complex @ Z2 ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) @ ( sin_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% cos_times_sin
thf(fact_7011_cos__times__sin,axiom,
! [W2: real,Z2: real] :
( ( times_times_real @ ( cos_real @ W2 ) @ ( sin_real @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( sin_real @ ( plus_plus_real @ W2 @ Z2 ) ) @ ( sin_real @ ( minus_minus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% cos_times_sin
thf(fact_7012_sin__plus__sin,axiom,
! [W2: complex,Z2: complex] :
( ( plus_plus_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z2 ) )
= ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% sin_plus_sin
thf(fact_7013_sin__plus__sin,axiom,
! [W2: real,Z2: real] :
( ( plus_plus_real @ ( sin_real @ W2 ) @ ( sin_real @ Z2 ) )
= ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% sin_plus_sin
thf(fact_7014_sin__diff__sin,axiom,
! [W2: complex,Z2: complex] :
( ( minus_minus_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z2 ) )
= ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% sin_diff_sin
thf(fact_7015_sin__diff__sin,axiom,
! [W2: real,Z2: real] :
( ( minus_minus_real @ ( sin_real @ W2 ) @ ( sin_real @ Z2 ) )
= ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% sin_diff_sin
thf(fact_7016_cos__diff__cos,axiom,
! [W2: complex,Z2: complex] :
( ( minus_minus_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z2 ) )
= ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ Z2 @ W2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_diff_cos
thf(fact_7017_cos__diff__cos,axiom,
! [W2: real,Z2: real] :
( ( minus_minus_real @ ( cos_real @ W2 ) @ ( cos_real @ Z2 ) )
= ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ Z2 @ W2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_diff_cos
thf(fact_7018_sums__finite,axiom,
! [N7: set_nat,F: nat > complex] :
( ( finite_finite_nat @ N7 )
=> ( ! [N3: nat] :
( ~ ( member_nat @ N3 @ N7 )
=> ( ( F @ N3 )
= zero_zero_complex ) )
=> ( sums_complex @ F @ ( groups2073611262835488442omplex @ F @ N7 ) ) ) ) ).
% sums_finite
thf(fact_7019_sums__finite,axiom,
! [N7: set_nat,F: nat > int] :
( ( finite_finite_nat @ N7 )
=> ( ! [N3: nat] :
( ~ ( member_nat @ N3 @ N7 )
=> ( ( F @ N3 )
= zero_zero_int ) )
=> ( sums_int @ F @ ( groups3539618377306564664at_int @ F @ N7 ) ) ) ) ).
% sums_finite
thf(fact_7020_sums__finite,axiom,
! [N7: set_nat,F: nat > nat] :
( ( finite_finite_nat @ N7 )
=> ( ! [N3: nat] :
( ~ ( member_nat @ N3 @ N7 )
=> ( ( F @ N3 )
= zero_zero_nat ) )
=> ( sums_nat @ F @ ( groups3542108847815614940at_nat @ F @ N7 ) ) ) ) ).
% sums_finite
thf(fact_7021_sums__finite,axiom,
! [N7: set_nat,F: nat > real] :
( ( finite_finite_nat @ N7 )
=> ( ! [N3: nat] :
( ~ ( member_nat @ N3 @ N7 )
=> ( ( F @ N3 )
= zero_zero_real ) )
=> ( sums_real @ F @ ( groups6591440286371151544t_real @ F @ N7 ) ) ) ) ).
% sums_finite
thf(fact_7022_sums__If__finite,axiom,
! [P2: nat > $o,F: nat > complex] :
( ( finite_finite_nat @ ( collect_nat @ P2 ) )
=> ( sums_complex
@ ^ [R5: nat] : ( if_complex @ ( P2 @ R5 ) @ ( F @ R5 ) @ zero_zero_complex )
@ ( groups2073611262835488442omplex @ F @ ( collect_nat @ P2 ) ) ) ) ).
% sums_If_finite
thf(fact_7023_sums__If__finite,axiom,
! [P2: nat > $o,F: nat > int] :
( ( finite_finite_nat @ ( collect_nat @ P2 ) )
=> ( sums_int
@ ^ [R5: nat] : ( if_int @ ( P2 @ R5 ) @ ( F @ R5 ) @ zero_zero_int )
@ ( groups3539618377306564664at_int @ F @ ( collect_nat @ P2 ) ) ) ) ).
% sums_If_finite
thf(fact_7024_sums__If__finite,axiom,
! [P2: nat > $o,F: nat > nat] :
( ( finite_finite_nat @ ( collect_nat @ P2 ) )
=> ( sums_nat
@ ^ [R5: nat] : ( if_nat @ ( P2 @ R5 ) @ ( F @ R5 ) @ zero_zero_nat )
@ ( groups3542108847815614940at_nat @ F @ ( collect_nat @ P2 ) ) ) ) ).
% sums_If_finite
thf(fact_7025_sums__If__finite,axiom,
! [P2: nat > $o,F: nat > real] :
( ( finite_finite_nat @ ( collect_nat @ P2 ) )
=> ( sums_real
@ ^ [R5: nat] : ( if_real @ ( P2 @ R5 ) @ ( F @ R5 ) @ zero_zero_real )
@ ( groups6591440286371151544t_real @ F @ ( collect_nat @ P2 ) ) ) ) ).
% sums_If_finite
thf(fact_7026_sums__If__finite__set,axiom,
! [A4: set_nat,F: nat > complex] :
( ( finite_finite_nat @ A4 )
=> ( sums_complex
@ ^ [R5: nat] : ( if_complex @ ( member_nat @ R5 @ A4 ) @ ( F @ R5 ) @ zero_zero_complex )
@ ( groups2073611262835488442omplex @ F @ A4 ) ) ) ).
% sums_If_finite_set
thf(fact_7027_sums__If__finite__set,axiom,
! [A4: set_nat,F: nat > int] :
( ( finite_finite_nat @ A4 )
=> ( sums_int
@ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A4 ) @ ( F @ R5 ) @ zero_zero_int )
@ ( groups3539618377306564664at_int @ F @ A4 ) ) ) ).
% sums_If_finite_set
thf(fact_7028_sums__If__finite__set,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( sums_nat
@ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A4 ) @ ( F @ R5 ) @ zero_zero_nat )
@ ( groups3542108847815614940at_nat @ F @ A4 ) ) ) ).
% sums_If_finite_set
thf(fact_7029_sums__If__finite__set,axiom,
! [A4: set_nat,F: nat > real] :
( ( finite_finite_nat @ A4 )
=> ( sums_real
@ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A4 ) @ ( F @ R5 ) @ zero_zero_real )
@ ( groups6591440286371151544t_real @ F @ A4 ) ) ) ).
% sums_If_finite_set
thf(fact_7030_powser__sums__if,axiom,
! [M2: nat,Z2: complex] :
( sums_complex
@ ^ [N4: nat] : ( times_times_complex @ ( if_complex @ ( N4 = M2 ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z2 @ N4 ) )
@ ( power_power_complex @ Z2 @ M2 ) ) ).
% powser_sums_if
thf(fact_7031_powser__sums__if,axiom,
! [M2: nat,Z2: real] :
( sums_real
@ ^ [N4: nat] : ( times_times_real @ ( if_real @ ( N4 = M2 ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z2 @ N4 ) )
@ ( power_power_real @ Z2 @ M2 ) ) ).
% powser_sums_if
thf(fact_7032_powser__sums__if,axiom,
! [M2: nat,Z2: int] :
( sums_int
@ ^ [N4: nat] : ( times_times_int @ ( if_int @ ( N4 = M2 ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z2 @ N4 ) )
@ ( power_power_int @ Z2 @ M2 ) ) ).
% powser_sums_if
thf(fact_7033_powser__sums__zero,axiom,
! [A: nat > complex] :
( sums_complex
@ ^ [N4: nat] : ( times_times_complex @ ( A @ N4 ) @ ( power_power_complex @ zero_zero_complex @ N4 ) )
@ ( A @ zero_zero_nat ) ) ).
% powser_sums_zero
thf(fact_7034_powser__sums__zero,axiom,
! [A: nat > real] :
( sums_real
@ ^ [N4: nat] : ( times_times_real @ ( A @ N4 ) @ ( power_power_real @ zero_zero_real @ N4 ) )
@ ( A @ zero_zero_nat ) ) ).
% powser_sums_zero
thf(fact_7035_cos__two__neq__zero,axiom,
( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
!= zero_zero_real ) ).
% cos_two_neq_zero
thf(fact_7036_cos__mono__less__eq,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ pi )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ pi )
=> ( ( ord_less_real @ ( cos_real @ X3 ) @ ( cos_real @ Y ) )
= ( ord_less_real @ Y @ X3 ) ) ) ) ) ) ).
% cos_mono_less_eq
thf(fact_7037_cos__monotone__0__pi,axiom,
! [Y: real,X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ Y @ X3 )
=> ( ( ord_less_eq_real @ X3 @ pi )
=> ( ord_less_real @ ( cos_real @ X3 ) @ ( cos_real @ Y ) ) ) ) ) ).
% cos_monotone_0_pi
thf(fact_7038_sin__eq__0__pi,axiom,
! [X3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X3 )
=> ( ( ord_less_real @ X3 @ pi )
=> ( ( ( sin_real @ X3 )
= zero_zero_real )
=> ( X3 = zero_zero_real ) ) ) ) ).
% sin_eq_0_pi
thf(fact_7039_sums__iff__shift,axiom,
! [F: nat > real,N: nat,S3: real] :
( ( sums_real
@ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N ) )
@ S3 )
= ( sums_real @ F @ ( plus_plus_real @ S3 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% sums_iff_shift
thf(fact_7040_sums__iff__shift_H,axiom,
! [F: nat > real,N: nat,S3: real] :
( ( sums_real
@ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N ) )
@ ( minus_minus_real @ S3 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) )
= ( sums_real @ F @ S3 ) ) ).
% sums_iff_shift'
thf(fact_7041_sums__split__initial__segment,axiom,
! [F: nat > real,S3: real,N: nat] :
( ( sums_real @ F @ S3 )
=> ( sums_real
@ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N ) )
@ ( minus_minus_real @ S3 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% sums_split_initial_segment
thf(fact_7042_sin__zero__pi__iff,axiom,
! [X3: real] :
( ( ord_less_real @ ( abs_abs_real @ X3 ) @ pi )
=> ( ( ( sin_real @ X3 )
= zero_zero_real )
= ( X3 = zero_zero_real ) ) ) ).
% sin_zero_pi_iff
thf(fact_7043_cos__monotone__minus__pi__0_H,axiom,
! [Y: real,X3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
=> ( ( ord_less_eq_real @ Y @ X3 )
=> ( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ord_less_eq_real @ ( cos_real @ Y ) @ ( cos_real @ X3 ) ) ) ) ) ).
% cos_monotone_minus_pi_0'
thf(fact_7044_sums__If__finite__set_H,axiom,
! [G: nat > real,S2: real,A4: set_nat,S5: real,F: nat > real] :
( ( sums_real @ G @ S2 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( S5
= ( plus_plus_real @ S2
@ ( groups6591440286371151544t_real
@ ^ [N4: nat] : ( minus_minus_real @ ( F @ N4 ) @ ( G @ N4 ) )
@ A4 ) ) )
=> ( sums_real
@ ^ [N4: nat] : ( if_real @ ( member_nat @ N4 @ A4 ) @ ( F @ N4 ) @ ( G @ N4 ) )
@ S5 ) ) ) ) ).
% sums_If_finite_set'
thf(fact_7045_sin__zero__iff__int2,axiom,
! [X3: real] :
( ( ( sin_real @ X3 )
= zero_zero_real )
= ( ? [I3: int] :
( X3
= ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ pi ) ) ) ) ).
% sin_zero_iff_int2
thf(fact_7046_sincos__total__pi,axiom,
! [Y: real,X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real )
=> ? [T5: real] :
( ( ord_less_eq_real @ zero_zero_real @ T5 )
& ( ord_less_eq_real @ T5 @ pi )
& ( X3
= ( cos_real @ T5 ) )
& ( Y
= ( sin_real @ T5 ) ) ) ) ) ).
% sincos_total_pi
thf(fact_7047_sin__expansion__lemma,axiom,
! [X3: real,M2: nat] :
( ( sin_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M2 ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
= ( cos_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% sin_expansion_lemma
thf(fact_7048_cos__expansion__lemma,axiom,
! [X3: real,M2: nat] :
( ( cos_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M2 ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
= ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% cos_expansion_lemma
thf(fact_7049_sin__gt__zero__02,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
=> ( ord_less_real @ zero_zero_real @ ( sin_real @ X3 ) ) ) ) ).
% sin_gt_zero_02
thf(fact_7050_cos__two__less__zero,axiom,
ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% cos_two_less_zero
thf(fact_7051_cos__two__le__zero,axiom,
ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% cos_two_le_zero
thf(fact_7052_cos__is__zero,axiom,
? [X4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
& ( ord_less_eq_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
& ( ( cos_real @ X4 )
= zero_zero_real )
& ! [Y6: real] :
( ( ( ord_less_eq_real @ zero_zero_real @ Y6 )
& ( ord_less_eq_real @ Y6 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
& ( ( cos_real @ Y6 )
= zero_zero_real ) )
=> ( Y6 = X4 ) ) ) ).
% cos_is_zero
thf(fact_7053_cos__monotone__minus__pi__0,axiom,
! [Y: real,X3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
=> ( ( ord_less_real @ Y @ X3 )
=> ( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ord_less_real @ ( cos_real @ Y ) @ ( cos_real @ X3 ) ) ) ) ) ).
% cos_monotone_minus_pi_0
thf(fact_7054_cos__total,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ? [X4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
& ( ord_less_eq_real @ X4 @ pi )
& ( ( cos_real @ X4 )
= Y )
& ! [Y6: real] :
( ( ( ord_less_eq_real @ zero_zero_real @ Y6 )
& ( ord_less_eq_real @ Y6 @ pi )
& ( ( cos_real @ Y6 )
= Y ) )
=> ( Y6 = X4 ) ) ) ) ) ).
% cos_total
thf(fact_7055_sincos__total__pi__half,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real )
=> ? [T5: real] :
( ( ord_less_eq_real @ zero_zero_real @ T5 )
& ( ord_less_eq_real @ T5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( X3
= ( cos_real @ T5 ) )
& ( Y
= ( sin_real @ T5 ) ) ) ) ) ) ).
% sincos_total_pi_half
thf(fact_7056_sincos__total__2pi__le,axiom,
! [X3: real,Y: real] :
( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real )
=> ? [T5: real] :
( ( ord_less_eq_real @ zero_zero_real @ T5 )
& ( ord_less_eq_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
& ( X3
= ( cos_real @ T5 ) )
& ( Y
= ( sin_real @ T5 ) ) ) ) ).
% sincos_total_2pi_le
thf(fact_7057_sincos__total__2pi,axiom,
! [X3: real,Y: real] :
( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real )
=> ~ ! [T5: real] :
( ( ord_less_eq_real @ zero_zero_real @ T5 )
=> ( ( ord_less_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
=> ( ( X3
= ( cos_real @ T5 ) )
=> ( Y
!= ( sin_real @ T5 ) ) ) ) ) ) ).
% sincos_total_2pi
thf(fact_7058_sin__pi__divide__n__ge__0,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% sin_pi_divide_n_ge_0
thf(fact_7059_cos__plus__cos,axiom,
! [W2: complex,Z2: complex] :
( ( plus_plus_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z2 ) )
= ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_plus_cos
thf(fact_7060_cos__plus__cos,axiom,
! [W2: real,Z2: real] :
( ( plus_plus_real @ ( cos_real @ W2 ) @ ( cos_real @ Z2 ) )
= ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_plus_cos
thf(fact_7061_cos__times__cos,axiom,
! [W2: complex,Z2: complex] :
( ( times_times_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z2 ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) @ ( cos_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% cos_times_cos
thf(fact_7062_cos__times__cos,axiom,
! [W2: real,Z2: real] :
( ( times_times_real @ ( cos_real @ W2 ) @ ( cos_real @ Z2 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( cos_real @ ( minus_minus_real @ W2 @ Z2 ) ) @ ( cos_real @ ( plus_plus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% cos_times_cos
thf(fact_7063_power__half__series,axiom,
( sums_real
@ ^ [N4: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N4 ) )
@ one_one_real ) ).
% power_half_series
thf(fact_7064_sin__gt__zero2,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ zero_zero_real @ ( sin_real @ X3 ) ) ) ) ).
% sin_gt_zero2
thf(fact_7065_sin__lt__zero,axiom,
! [X3: real] :
( ( ord_less_real @ pi @ X3 )
=> ( ( ord_less_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
=> ( ord_less_real @ ( sin_real @ X3 ) @ zero_zero_real ) ) ) ).
% sin_lt_zero
thf(fact_7066_cos__double__less__one,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
=> ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) @ one_one_real ) ) ) ).
% cos_double_less_one
thf(fact_7067_cos__gt__zero,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ zero_zero_real @ ( cos_real @ X3 ) ) ) ) ).
% cos_gt_zero
thf(fact_7068_sin__inj__pi,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
=> ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
=> ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ( sin_real @ X3 )
= ( sin_real @ Y ) )
=> ( X3 = Y ) ) ) ) ) ) ).
% sin_inj_pi
thf(fact_7069_sin__mono__le__eq,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
=> ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
=> ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( sin_real @ X3 ) @ ( sin_real @ Y ) )
= ( ord_less_eq_real @ X3 @ Y ) ) ) ) ) ) ).
% sin_mono_le_eq
thf(fact_7070_sin__monotone__2pi__le,axiom,
! [Y: real,X3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
=> ( ( ord_less_eq_real @ Y @ X3 )
=> ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( sin_real @ Y ) @ ( sin_real @ X3 ) ) ) ) ) ).
% sin_monotone_2pi_le
thf(fact_7071_sums__if_H,axiom,
! [G: nat > real,X3: real] :
( ( sums_real @ G @ X3 )
=> ( sums_real
@ ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
@ X3 ) ) ).
% sums_if'
thf(fact_7072_sums__if,axiom,
! [G: nat > real,X3: real,F: nat > real,Y: real] :
( ( sums_real @ G @ X3 )
=> ( ( sums_real @ F @ Y )
=> ( sums_real
@ ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ ( F @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_real @ X3 @ Y ) ) ) ) ).
% sums_if
thf(fact_7073_sin__le__zero,axiom,
! [X3: real] :
( ( ord_less_eq_real @ pi @ X3 )
=> ( ( ord_less_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
=> ( ord_less_eq_real @ ( sin_real @ X3 ) @ zero_zero_real ) ) ) ).
% sin_le_zero
thf(fact_7074_sin__less__zero,axiom,
! [X3: real] :
( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 )
=> ( ( ord_less_real @ X3 @ zero_zero_real )
=> ( ord_less_real @ ( sin_real @ X3 ) @ zero_zero_real ) ) ) ).
% sin_less_zero
thf(fact_7075_sin__monotone__2pi,axiom,
! [Y: real,X3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
=> ( ( ord_less_real @ Y @ X3 )
=> ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ ( sin_real @ Y ) @ ( sin_real @ X3 ) ) ) ) ) ).
% sin_monotone_2pi
thf(fact_7076_sin__mono__less__eq,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
=> ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
=> ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ ( sin_real @ X3 ) @ ( sin_real @ Y ) )
= ( ord_less_real @ X3 @ Y ) ) ) ) ) ) ).
% sin_mono_less_eq
thf(fact_7077_sin__total,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ? [X4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
& ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( sin_real @ X4 )
= Y )
& ! [Y6: real] :
( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y6 )
& ( ord_less_eq_real @ Y6 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( sin_real @ Y6 )
= Y ) )
=> ( Y6 = X4 ) ) ) ) ) ).
% sin_total
thf(fact_7078_cos__gt__zero__pi,axiom,
! [X3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
=> ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ zero_zero_real @ ( cos_real @ X3 ) ) ) ) ).
% cos_gt_zero_pi
thf(fact_7079_cos__ge__zero,axiom,
! [X3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
=> ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X3 ) ) ) ) ).
% cos_ge_zero
thf(fact_7080_accp__subset__induct,axiom,
! [D5: product_prod_num_num > $o,R: product_prod_num_num > product_prod_num_num > $o,X3: product_prod_num_num,P2: product_prod_num_num > $o] :
( ( ord_le2239182809043710856_num_o @ D5 @ ( accp_P3113834385874906142um_num @ R ) )
=> ( ! [X4: product_prod_num_num,Z3: product_prod_num_num] :
( ( D5 @ X4 )
=> ( ( R @ Z3 @ X4 )
=> ( D5 @ Z3 ) ) )
=> ( ( D5 @ X3 )
=> ( ! [X4: product_prod_num_num] :
( ( D5 @ X4 )
=> ( ! [Z4: product_prod_num_num] :
( ( R @ Z4 @ X4 )
=> ( P2 @ Z4 ) )
=> ( P2 @ X4 ) ) )
=> ( P2 @ X3 ) ) ) ) ) ).
% accp_subset_induct
thf(fact_7081_accp__subset__induct,axiom,
! [D5: product_prod_nat_nat > $o,R: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
( ( ord_le704812498762024988_nat_o @ D5 @ ( accp_P4275260045618599050at_nat @ R ) )
=> ( ! [X4: product_prod_nat_nat,Z3: product_prod_nat_nat] :
( ( D5 @ X4 )
=> ( ( R @ Z3 @ X4 )
=> ( D5 @ Z3 ) ) )
=> ( ( D5 @ X3 )
=> ( ! [X4: product_prod_nat_nat] :
( ( D5 @ X4 )
=> ( ! [Z4: product_prod_nat_nat] :
( ( R @ Z4 @ X4 )
=> ( P2 @ Z4 ) )
=> ( P2 @ X4 ) ) )
=> ( P2 @ X3 ) ) ) ) ) ).
% accp_subset_induct
thf(fact_7082_accp__subset__induct,axiom,
! [D5: product_prod_int_int > $o,R: product_prod_int_int > product_prod_int_int > $o,X3: product_prod_int_int,P2: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ D5 @ ( accp_P1096762738010456898nt_int @ R ) )
=> ( ! [X4: product_prod_int_int,Z3: product_prod_int_int] :
( ( D5 @ X4 )
=> ( ( R @ Z3 @ X4 )
=> ( D5 @ Z3 ) ) )
=> ( ( D5 @ X3 )
=> ( ! [X4: product_prod_int_int] :
( ( D5 @ X4 )
=> ( ! [Z4: product_prod_int_int] :
( ( R @ Z4 @ X4 )
=> ( P2 @ Z4 ) )
=> ( P2 @ X4 ) ) )
=> ( P2 @ X3 ) ) ) ) ) ).
% accp_subset_induct
thf(fact_7083_accp__subset__induct,axiom,
! [D5: list_nat > $o,R: list_nat > list_nat > $o,X3: list_nat,P2: list_nat > $o] :
( ( ord_le1520216061033275535_nat_o @ D5 @ ( accp_list_nat @ R ) )
=> ( ! [X4: list_nat,Z3: list_nat] :
( ( D5 @ X4 )
=> ( ( R @ Z3 @ X4 )
=> ( D5 @ Z3 ) ) )
=> ( ( D5 @ X3 )
=> ( ! [X4: list_nat] :
( ( D5 @ X4 )
=> ( ! [Z4: list_nat] :
( ( R @ Z4 @ X4 )
=> ( P2 @ Z4 ) )
=> ( P2 @ X4 ) ) )
=> ( P2 @ X3 ) ) ) ) ) ).
% accp_subset_induct
thf(fact_7084_accp__subset__induct,axiom,
! [D5: nat > $o,R: nat > nat > $o,X3: nat,P2: nat > $o] :
( ( ord_less_eq_nat_o @ D5 @ ( accp_nat @ R ) )
=> ( ! [X4: nat,Z3: nat] :
( ( D5 @ X4 )
=> ( ( R @ Z3 @ X4 )
=> ( D5 @ Z3 ) ) )
=> ( ( D5 @ X3 )
=> ( ! [X4: nat] :
( ( D5 @ X4 )
=> ( ! [Z4: nat] :
( ( R @ Z4 @ X4 )
=> ( P2 @ Z4 ) )
=> ( P2 @ X4 ) ) )
=> ( P2 @ X3 ) ) ) ) ) ).
% accp_subset_induct
thf(fact_7085_sin__pi__divide__n__gt__0,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% sin_pi_divide_n_gt_0
thf(fact_7086_sin__zero__iff__int,axiom,
! [X3: real] :
( ( ( sin_real @ X3 )
= zero_zero_real )
= ( ? [I3: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
& ( X3
= ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% sin_zero_iff_int
thf(fact_7087_cos__zero__iff__int,axiom,
! [X3: real] :
( ( ( cos_real @ X3 )
= zero_zero_real )
= ( ? [I3: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
& ( X3
= ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% cos_zero_iff_int
thf(fact_7088_sin__zero__lemma,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ( sin_real @ X3 )
= zero_zero_real )
=> ? [N3: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
& ( X3
= ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% sin_zero_lemma
thf(fact_7089_sin__zero__iff,axiom,
! [X3: real] :
( ( ( sin_real @ X3 )
= zero_zero_real )
= ( ? [N4: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
& ( X3
= ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
| ? [N4: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
& ( X3
= ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% sin_zero_iff
thf(fact_7090_cos__zero__lemma,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ( cos_real @ X3 )
= zero_zero_real )
=> ? [N3: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
& ( X3
= ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% cos_zero_lemma
thf(fact_7091_cos__zero__iff,axiom,
! [X3: real] :
( ( ( cos_real @ X3 )
= zero_zero_real )
= ( ? [N4: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
& ( X3
= ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
| ? [N4: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
& ( X3
= ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% cos_zero_iff
thf(fact_7092_monoseq__Suc,axiom,
( topolo6980174941875973593q_real
= ( ^ [X7: nat > real] :
( ! [N4: nat] : ( ord_less_eq_real @ ( X7 @ N4 ) @ ( X7 @ ( suc @ N4 ) ) )
| ! [N4: nat] : ( ord_less_eq_real @ ( X7 @ ( suc @ N4 ) ) @ ( X7 @ N4 ) ) ) ) ) ).
% monoseq_Suc
thf(fact_7093_monoseq__Suc,axiom,
( topolo3100542954746470799et_int
= ( ^ [X7: nat > set_int] :
( ! [N4: nat] : ( ord_less_eq_set_int @ ( X7 @ N4 ) @ ( X7 @ ( suc @ N4 ) ) )
| ! [N4: nat] : ( ord_less_eq_set_int @ ( X7 @ ( suc @ N4 ) ) @ ( X7 @ N4 ) ) ) ) ) ).
% monoseq_Suc
thf(fact_7094_monoseq__Suc,axiom,
( topolo4267028734544971653eq_rat
= ( ^ [X7: nat > rat] :
( ! [N4: nat] : ( ord_less_eq_rat @ ( X7 @ N4 ) @ ( X7 @ ( suc @ N4 ) ) )
| ! [N4: nat] : ( ord_less_eq_rat @ ( X7 @ ( suc @ N4 ) ) @ ( X7 @ N4 ) ) ) ) ) ).
% monoseq_Suc
thf(fact_7095_monoseq__Suc,axiom,
( topolo1459490580787246023eq_num
= ( ^ [X7: nat > num] :
( ! [N4: nat] : ( ord_less_eq_num @ ( X7 @ N4 ) @ ( X7 @ ( suc @ N4 ) ) )
| ! [N4: nat] : ( ord_less_eq_num @ ( X7 @ ( suc @ N4 ) ) @ ( X7 @ N4 ) ) ) ) ) ).
% monoseq_Suc
thf(fact_7096_monoseq__Suc,axiom,
( topolo4902158794631467389eq_nat
= ( ^ [X7: nat > nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( X7 @ N4 ) @ ( X7 @ ( suc @ N4 ) ) )
| ! [N4: nat] : ( ord_less_eq_nat @ ( X7 @ ( suc @ N4 ) ) @ ( X7 @ N4 ) ) ) ) ) ).
% monoseq_Suc
thf(fact_7097_monoseq__Suc,axiom,
( topolo4899668324122417113eq_int
= ( ^ [X7: nat > int] :
( ! [N4: nat] : ( ord_less_eq_int @ ( X7 @ N4 ) @ ( X7 @ ( suc @ N4 ) ) )
| ! [N4: nat] : ( ord_less_eq_int @ ( X7 @ ( suc @ N4 ) ) @ ( X7 @ N4 ) ) ) ) ) ).
% monoseq_Suc
thf(fact_7098_mono__SucI2,axiom,
! [X8: nat > real] :
( ! [N3: nat] : ( ord_less_eq_real @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) )
=> ( topolo6980174941875973593q_real @ X8 ) ) ).
% mono_SucI2
thf(fact_7099_mono__SucI2,axiom,
! [X8: nat > set_int] :
( ! [N3: nat] : ( ord_less_eq_set_int @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) )
=> ( topolo3100542954746470799et_int @ X8 ) ) ).
% mono_SucI2
thf(fact_7100_mono__SucI2,axiom,
! [X8: nat > rat] :
( ! [N3: nat] : ( ord_less_eq_rat @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) )
=> ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% mono_SucI2
thf(fact_7101_mono__SucI2,axiom,
! [X8: nat > num] :
( ! [N3: nat] : ( ord_less_eq_num @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) )
=> ( topolo1459490580787246023eq_num @ X8 ) ) ).
% mono_SucI2
thf(fact_7102_mono__SucI2,axiom,
! [X8: nat > nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) )
=> ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% mono_SucI2
thf(fact_7103_mono__SucI2,axiom,
! [X8: nat > int] :
( ! [N3: nat] : ( ord_less_eq_int @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) )
=> ( topolo4899668324122417113eq_int @ X8 ) ) ).
% mono_SucI2
thf(fact_7104_mono__SucI1,axiom,
! [X8: nat > real] :
( ! [N3: nat] : ( ord_less_eq_real @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
=> ( topolo6980174941875973593q_real @ X8 ) ) ).
% mono_SucI1
thf(fact_7105_mono__SucI1,axiom,
! [X8: nat > set_int] :
( ! [N3: nat] : ( ord_less_eq_set_int @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
=> ( topolo3100542954746470799et_int @ X8 ) ) ).
% mono_SucI1
thf(fact_7106_mono__SucI1,axiom,
! [X8: nat > rat] :
( ! [N3: nat] : ( ord_less_eq_rat @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
=> ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% mono_SucI1
thf(fact_7107_mono__SucI1,axiom,
! [X8: nat > num] :
( ! [N3: nat] : ( ord_less_eq_num @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
=> ( topolo1459490580787246023eq_num @ X8 ) ) ).
% mono_SucI1
thf(fact_7108_mono__SucI1,axiom,
! [X8: nat > nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
=> ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% mono_SucI1
thf(fact_7109_mono__SucI1,axiom,
! [X8: nat > int] :
( ! [N3: nat] : ( ord_less_eq_int @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
=> ( topolo4899668324122417113eq_int @ X8 ) ) ).
% mono_SucI1
thf(fact_7110_monoI1,axiom,
! [X8: nat > real] :
( ! [M: nat,N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ord_less_eq_real @ ( X8 @ M ) @ ( X8 @ N3 ) ) )
=> ( topolo6980174941875973593q_real @ X8 ) ) ).
% monoI1
thf(fact_7111_monoI1,axiom,
! [X8: nat > set_int] :
( ! [M: nat,N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ord_less_eq_set_int @ ( X8 @ M ) @ ( X8 @ N3 ) ) )
=> ( topolo3100542954746470799et_int @ X8 ) ) ).
% monoI1
thf(fact_7112_monoI1,axiom,
! [X8: nat > rat] :
( ! [M: nat,N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ord_less_eq_rat @ ( X8 @ M ) @ ( X8 @ N3 ) ) )
=> ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% monoI1
thf(fact_7113_monoI1,axiom,
! [X8: nat > num] :
( ! [M: nat,N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ord_less_eq_num @ ( X8 @ M ) @ ( X8 @ N3 ) ) )
=> ( topolo1459490580787246023eq_num @ X8 ) ) ).
% monoI1
thf(fact_7114_monoI1,axiom,
! [X8: nat > nat] :
( ! [M: nat,N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ord_less_eq_nat @ ( X8 @ M ) @ ( X8 @ N3 ) ) )
=> ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% monoI1
thf(fact_7115_monoI1,axiom,
! [X8: nat > int] :
( ! [M: nat,N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ord_less_eq_int @ ( X8 @ M ) @ ( X8 @ N3 ) ) )
=> ( topolo4899668324122417113eq_int @ X8 ) ) ).
% monoI1
thf(fact_7116_Maclaurin__cos__expansion2,axiom,
! [X3: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [T5: real] :
( ( ord_less_real @ zero_zero_real @ T5 )
& ( ord_less_real @ T5 @ X3 )
& ( ( cos_real @ X3 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X3 @ M5 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( 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 @ X3 @ N ) ) ) ) ) ) ) ).
% Maclaurin_cos_expansion2
thf(fact_7117_Maclaurin__minus__cos__expansion,axiom,
! [N: nat,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ X3 @ zero_zero_real )
=> ? [T5: real] :
( ( ord_less_real @ X3 @ T5 )
& ( ord_less_real @ T5 @ zero_zero_real )
& ( ( cos_real @ X3 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X3 @ M5 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( 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 @ X3 @ N ) ) ) ) ) ) ) ).
% Maclaurin_minus_cos_expansion
thf(fact_7118_Maclaurin__cos__expansion,axiom,
! [X3: real,N: nat] :
? [T5: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X3 ) )
& ( ( cos_real @ X3 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X3 @ M5 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( 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 @ X3 @ N ) ) ) ) ) ).
% Maclaurin_cos_expansion
thf(fact_7119_diffs__equiv,axiom,
! [C2: nat > real,X3: real] :
( ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( diffs_real @ C2 @ N4 ) @ ( power_power_real @ X3 @ N4 ) ) )
=> ( sums_real
@ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( C2 @ N4 ) ) @ ( power_power_real @ X3 @ ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) ) ) )
@ ( suminf_real
@ ^ [N4: nat] : ( times_times_real @ ( diffs_real @ C2 @ N4 ) @ ( power_power_real @ X3 @ N4 ) ) ) ) ) ).
% diffs_equiv
thf(fact_7120_diffs__equiv,axiom,
! [C2: nat > complex,X3: complex] :
( ( summable_complex
@ ^ [N4: nat] : ( times_times_complex @ ( diffs_complex @ C2 @ N4 ) @ ( power_power_complex @ X3 @ N4 ) ) )
=> ( sums_complex
@ ^ [N4: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N4 ) @ ( C2 @ N4 ) ) @ ( power_power_complex @ X3 @ ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) ) ) )
@ ( suminf_complex
@ ^ [N4: nat] : ( times_times_complex @ ( diffs_complex @ C2 @ N4 ) @ ( power_power_complex @ X3 @ N4 ) ) ) ) ) ).
% diffs_equiv
thf(fact_7121_tan__double,axiom,
! [X3: complex] :
( ( ( cos_complex @ X3 )
!= zero_zero_complex )
=> ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) )
!= zero_zero_complex )
=> ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X3 ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% tan_double
thf(fact_7122_tan__double,axiom,
! [X3: real] :
( ( ( cos_real @ X3 )
!= zero_zero_real )
=> ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) )
!= zero_zero_real )
=> ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) )
= ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X3 ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% tan_double
thf(fact_7123_in__measure,axiom,
! [X3: int,Y: int,F: int > nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y ) @ ( measure_int @ F ) )
= ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) ) ).
% in_measure
thf(fact_7124_tan__pi,axiom,
( ( tan_real @ pi )
= zero_zero_real ) ).
% tan_pi
thf(fact_7125_tan__zero,axiom,
( ( tan_complex @ zero_zero_complex )
= zero_zero_complex ) ).
% tan_zero
thf(fact_7126_tan__zero,axiom,
( ( tan_real @ zero_zero_real )
= zero_zero_real ) ).
% tan_zero
thf(fact_7127_tan__periodic__pi,axiom,
! [X3: real] :
( ( tan_real @ ( plus_plus_real @ X3 @ pi ) )
= ( tan_real @ X3 ) ) ).
% tan_periodic_pi
thf(fact_7128_fact__0,axiom,
( ( semiri773545260158071498ct_rat @ zero_zero_nat )
= one_one_rat ) ).
% fact_0
thf(fact_7129_fact__0,axiom,
( ( semiri1406184849735516958ct_int @ zero_zero_nat )
= one_one_int ) ).
% fact_0
thf(fact_7130_fact__0,axiom,
( ( semiri2265585572941072030t_real @ zero_zero_nat )
= one_one_real ) ).
% fact_0
thf(fact_7131_fact__0,axiom,
( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
= one_one_nat ) ).
% fact_0
thf(fact_7132_fact__0,axiom,
( ( semiri5044797733671781792omplex @ zero_zero_nat )
= one_one_complex ) ).
% fact_0
thf(fact_7133_fact__Suc__0,axiom,
( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
= one_one_rat ) ).
% fact_Suc_0
thf(fact_7134_fact__Suc__0,axiom,
( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
= one_one_int ) ).
% fact_Suc_0
thf(fact_7135_fact__Suc__0,axiom,
( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
= one_one_real ) ).
% fact_Suc_0
thf(fact_7136_fact__Suc__0,axiom,
( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% fact_Suc_0
thf(fact_7137_fact__Suc__0,axiom,
( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
= one_one_complex ) ).
% fact_Suc_0
thf(fact_7138_fact__Suc,axiom,
! [N: nat] :
( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% fact_Suc
thf(fact_7139_fact__Suc,axiom,
! [N: nat] :
( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% fact_Suc
thf(fact_7140_fact__Suc,axiom,
! [N: nat] :
( ( semiri3624122377584611663nteger @ ( suc @ N ) )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( suc @ N ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% fact_Suc
thf(fact_7141_fact__Suc,axiom,
! [N: nat] :
( ( semiri2265585572941072030t_real @ ( suc @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% fact_Suc
thf(fact_7142_fact__Suc,axiom,
! [N: nat] :
( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% fact_Suc
thf(fact_7143_fact__Suc,axiom,
! [N: nat] :
( ( semiri5044797733671781792omplex @ ( suc @ N ) )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% fact_Suc
thf(fact_7144_tan__npi,axiom,
! [N: nat] :
( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
= zero_zero_real ) ).
% tan_npi
thf(fact_7145_tan__periodic__n,axiom,
! [X3: real,N: num] :
( ( tan_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ N ) @ pi ) ) )
= ( tan_real @ X3 ) ) ).
% tan_periodic_n
thf(fact_7146_tan__periodic__nat,axiom,
! [X3: real,N: nat] :
( ( tan_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) ) )
= ( tan_real @ X3 ) ) ).
% tan_periodic_nat
thf(fact_7147_tan__periodic__int,axiom,
! [X3: real,I: int] :
( ( tan_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( ring_1_of_int_real @ I ) @ pi ) ) )
= ( tan_real @ X3 ) ) ).
% tan_periodic_int
thf(fact_7148_tan__periodic,axiom,
! [X3: real] :
( ( tan_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
= ( tan_real @ X3 ) ) ).
% tan_periodic
thf(fact_7149_fact__nonzero,axiom,
! [N: nat] :
( ( semiri773545260158071498ct_rat @ N )
!= zero_zero_rat ) ).
% fact_nonzero
thf(fact_7150_fact__nonzero,axiom,
! [N: nat] :
( ( semiri1406184849735516958ct_int @ N )
!= zero_zero_int ) ).
% fact_nonzero
thf(fact_7151_fact__nonzero,axiom,
! [N: nat] :
( ( semiri2265585572941072030t_real @ N )
!= zero_zero_real ) ).
% fact_nonzero
thf(fact_7152_fact__nonzero,axiom,
! [N: nat] :
( ( semiri1408675320244567234ct_nat @ N )
!= zero_zero_nat ) ).
% fact_nonzero
thf(fact_7153_fact__nonzero,axiom,
! [N: nat] :
( ( semiri5044797733671781792omplex @ N )
!= zero_zero_complex ) ).
% fact_nonzero
thf(fact_7154_fact__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% fact_ge_zero
thf(fact_7155_fact__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% fact_ge_zero
thf(fact_7156_fact__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% fact_ge_zero
thf(fact_7157_fact__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_ge_zero
thf(fact_7158_fact__not__neg,axiom,
! [N: nat] :
~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).
% fact_not_neg
thf(fact_7159_fact__not__neg,axiom,
! [N: nat] :
~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).
% fact_not_neg
thf(fact_7160_fact__not__neg,axiom,
! [N: nat] :
~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).
% fact_not_neg
thf(fact_7161_fact__not__neg,axiom,
! [N: nat] :
~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).
% fact_not_neg
thf(fact_7162_fact__gt__zero,axiom,
! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% fact_gt_zero
thf(fact_7163_fact__gt__zero,axiom,
! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% fact_gt_zero
thf(fact_7164_fact__gt__zero,axiom,
! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% fact_gt_zero
thf(fact_7165_fact__gt__zero,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_gt_zero
thf(fact_7166_fact__ge__1,axiom,
! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% fact_ge_1
thf(fact_7167_fact__ge__1,axiom,
! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% fact_ge_1
thf(fact_7168_fact__ge__1,axiom,
! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% fact_ge_1
thf(fact_7169_fact__ge__1,axiom,
! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_ge_1
thf(fact_7170_fact__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M2 ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% fact_mono
thf(fact_7171_fact__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M2 ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% fact_mono
thf(fact_7172_fact__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M2 ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% fact_mono
thf(fact_7173_fact__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% fact_mono
thf(fact_7174_fact__dvd,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M2 ) ) ) ).
% fact_dvd
thf(fact_7175_fact__dvd,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M2 ) ) ) ).
% fact_dvd
thf(fact_7176_fact__dvd,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ M2 ) ) ) ).
% fact_dvd
thf(fact_7177_fact__dvd,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M2 ) ) ) ).
% fact_dvd
thf(fact_7178_fact__less__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_nat @ M2 @ N )
=> ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M2 ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ) ).
% fact_less_mono
thf(fact_7179_fact__less__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_nat @ M2 @ N )
=> ( ord_less_int @ ( semiri1406184849735516958ct_int @ M2 ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).
% fact_less_mono
thf(fact_7180_fact__less__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_nat @ M2 @ N )
=> ( ord_less_real @ ( semiri2265585572941072030t_real @ M2 ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ).
% fact_less_mono
thf(fact_7181_fact__less__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% fact_less_mono
thf(fact_7182_fact__fact__dvd__fact,axiom,
! [K2: nat,N: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K2 ) @ ( semiri3624122377584611663nteger @ N ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K2 @ N ) ) ) ).
% fact_fact_dvd_fact
thf(fact_7183_fact__fact__dvd__fact,axiom,
! [K2: nat,N: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K2 ) @ ( semiri773545260158071498ct_rat @ N ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K2 @ N ) ) ) ).
% fact_fact_dvd_fact
thf(fact_7184_fact__fact__dvd__fact,axiom,
! [K2: nat,N: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K2 ) @ ( semiri1406184849735516958ct_int @ N ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K2 @ N ) ) ) ).
% fact_fact_dvd_fact
thf(fact_7185_fact__fact__dvd__fact,axiom,
! [K2: nat,N: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K2 @ N ) ) ) ).
% fact_fact_dvd_fact
thf(fact_7186_fact__fact__dvd__fact,axiom,
! [K2: nat,N: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K2 ) @ ( semiri1408675320244567234ct_nat @ N ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K2 @ N ) ) ) ).
% fact_fact_dvd_fact
thf(fact_7187_fact__mod,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M2 ) )
= zero_zero_int ) ) ).
% fact_mod
thf(fact_7188_fact__mod,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M2 ) )
= zero_z3403309356797280102nteger ) ) ).
% fact_mod
thf(fact_7189_fact__mod,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M2 ) )
= zero_zero_nat ) ) ).
% fact_mod
thf(fact_7190_fact__le__power,axiom,
! [N: nat] : ( ord_le3102999989581377725nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri4939895301339042750nteger @ ( power_power_nat @ N @ N ) ) ) ).
% fact_le_power
thf(fact_7191_fact__le__power,axiom,
! [N: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N @ N ) ) ) ).
% fact_le_power
thf(fact_7192_fact__le__power,axiom,
! [N: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N @ N ) ) ) ).
% fact_le_power
thf(fact_7193_fact__le__power,axiom,
! [N: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N @ N ) ) ) ).
% fact_le_power
thf(fact_7194_fact__le__power,axiom,
! [N: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N @ N ) ) ) ).
% fact_le_power
thf(fact_7195_choose__dvd,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K2 ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% choose_dvd
thf(fact_7196_choose__dvd,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% choose_dvd
thf(fact_7197_choose__dvd,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K2 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% choose_dvd
thf(fact_7198_choose__dvd,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% choose_dvd
thf(fact_7199_choose__dvd,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% choose_dvd
thf(fact_7200_diffs__def,axiom,
( diffs_rat
= ( ^ [C3: nat > rat,N4: nat] : ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N4 ) ) @ ( C3 @ ( suc @ N4 ) ) ) ) ) ).
% diffs_def
thf(fact_7201_diffs__def,axiom,
( diffs_int
= ( ^ [C3: nat > int,N4: nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) @ ( C3 @ ( suc @ N4 ) ) ) ) ) ).
% diffs_def
thf(fact_7202_diffs__def,axiom,
( diffs_real
= ( ^ [C3: nat > real,N4: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) @ ( C3 @ ( suc @ N4 ) ) ) ) ) ).
% diffs_def
thf(fact_7203_diffs__def,axiom,
( diffs_complex
= ( ^ [C3: nat > complex,N4: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N4 ) ) @ ( C3 @ ( suc @ N4 ) ) ) ) ) ).
% diffs_def
thf(fact_7204_diffs__def,axiom,
( diffs_Code_integer
= ( ^ [C3: nat > code_integer,N4: nat] : ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( suc @ N4 ) ) @ ( C3 @ ( suc @ N4 ) ) ) ) ) ).
% diffs_def
thf(fact_7205_square__fact__le__2__fact,axiom,
! [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 ) ) ) ).
% square_fact_le_2_fact
thf(fact_7206_fact__num__eq__if,axiom,
( semiri773545260158071498ct_rat
= ( ^ [M5: nat] : ( if_rat @ ( M5 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M5 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_7207_fact__num__eq__if,axiom,
( semiri1406184849735516958ct_int
= ( ^ [M5: nat] : ( if_int @ ( M5 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_7208_fact__num__eq__if,axiom,
( semiri3624122377584611663nteger
= ( ^ [M5: nat] : ( if_Code_integer @ ( M5 = zero_zero_nat ) @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M5 ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_7209_fact__num__eq__if,axiom,
( semiri2265585572941072030t_real
= ( ^ [M5: nat] : ( if_real @ ( M5 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M5 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_7210_fact__num__eq__if,axiom,
( semiri1408675320244567234ct_nat
= ( ^ [M5: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M5 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_7211_fact__num__eq__if,axiom,
( semiri5044797733671781792omplex
= ( ^ [M5: nat] : ( if_complex @ ( M5 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M5 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_7212_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri773545260158071498ct_rat @ N )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_7213_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri1406184849735516958ct_int @ N )
= ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_7214_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri3624122377584611663nteger @ N )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_7215_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri2265585572941072030t_real @ N )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_7216_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri1408675320244567234ct_nat @ N )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_7217_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri5044797733671781792omplex @ N )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_7218_tan__gt__zero,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ zero_zero_real @ ( tan_real @ X3 ) ) ) ) ).
% tan_gt_zero
thf(fact_7219_lemma__tan__total,axiom,
! [Y: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ? [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
& ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ord_less_real @ Y @ ( tan_real @ X4 ) ) ) ) ).
% lemma_tan_total
thf(fact_7220_add__tan__eq,axiom,
! [X3: complex,Y: complex] :
( ( ( cos_complex @ X3 )
!= zero_zero_complex )
=> ( ( ( cos_complex @ Y )
!= zero_zero_complex )
=> ( ( plus_plus_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y ) )
= ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X3 @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ Y ) ) ) ) ) ) ).
% add_tan_eq
thf(fact_7221_add__tan__eq,axiom,
! [X3: real,Y: real] :
( ( ( cos_real @ X3 )
!= zero_zero_real )
=> ( ( ( cos_real @ Y )
!= zero_zero_real )
=> ( ( plus_plus_real @ ( tan_real @ X3 ) @ ( tan_real @ Y ) )
= ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X3 @ Y ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y ) ) ) ) ) ) ).
% add_tan_eq
thf(fact_7222_tan__pos__pi2__le,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X3 ) ) ) ) ).
% tan_pos_pi2_le
thf(fact_7223_tan__total__pos,axiom,
! [Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ? [X4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
& ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ X4 )
= Y ) ) ) ).
% tan_total_pos
thf(fact_7224_tan__less__zero,axiom,
! [X3: real] :
( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 )
=> ( ( ord_less_real @ X3 @ zero_zero_real )
=> ( ord_less_real @ ( tan_real @ X3 ) @ zero_zero_real ) ) ) ).
% tan_less_zero
thf(fact_7225_tan__mono__le,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( tan_real @ X3 ) @ ( tan_real @ Y ) ) ) ) ) ).
% tan_mono_le
thf(fact_7226_tan__mono__le__eq,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
=> ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
=> ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( tan_real @ X3 ) @ ( tan_real @ Y ) )
= ( ord_less_eq_real @ X3 @ Y ) ) ) ) ) ) ).
% tan_mono_le_eq
thf(fact_7227_Maclaurin__zero,axiom,
! [X3: real,N: nat,Diff: nat > complex > real] :
( ( X3 = zero_zero_real )
=> ( ( N != zero_zero_nat )
=> ( ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_complex ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X3 @ M5 ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( Diff @ zero_zero_nat @ zero_zero_complex ) ) ) ) ).
% Maclaurin_zero
thf(fact_7228_Maclaurin__zero,axiom,
! [X3: real,N: nat,Diff: nat > real > real] :
( ( X3 = zero_zero_real )
=> ( ( N != zero_zero_nat )
=> ( ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X3 @ M5 ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).
% Maclaurin_zero
thf(fact_7229_Maclaurin__zero,axiom,
! [X3: real,N: nat,Diff: nat > rat > real] :
( ( X3 = zero_zero_real )
=> ( ( N != zero_zero_nat )
=> ( ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_rat ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X3 @ M5 ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( Diff @ zero_zero_nat @ zero_zero_rat ) ) ) ) ).
% Maclaurin_zero
thf(fact_7230_Maclaurin__zero,axiom,
! [X3: real,N: nat,Diff: nat > nat > real] :
( ( X3 = zero_zero_real )
=> ( ( N != zero_zero_nat )
=> ( ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_nat ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X3 @ M5 ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( Diff @ zero_zero_nat @ zero_zero_nat ) ) ) ) ).
% Maclaurin_zero
thf(fact_7231_Maclaurin__zero,axiom,
! [X3: real,N: nat,Diff: nat > int > real] :
( ( X3 = zero_zero_real )
=> ( ( N != zero_zero_nat )
=> ( ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_int ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X3 @ M5 ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( Diff @ zero_zero_nat @ zero_zero_int ) ) ) ) ).
% Maclaurin_zero
thf(fact_7232_Maclaurin__lemma,axiom,
! [H2: real,F: real > real,J: nat > real,N: nat] :
( ( ord_less_real @ zero_zero_real @ H2 )
=> ? [B8: real] :
( ( F @ H2 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H2 @ M5 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ B8 @ ( divide_divide_real @ ( power_power_real @ H2 @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).
% Maclaurin_lemma
thf(fact_7233_tan__add,axiom,
! [X3: complex,Y: complex] :
( ( ( cos_complex @ X3 )
!= zero_zero_complex )
=> ( ( ( cos_complex @ Y )
!= zero_zero_complex )
=> ( ( ( cos_complex @ ( plus_plus_complex @ X3 @ Y ) )
!= zero_zero_complex )
=> ( ( tan_complex @ ( plus_plus_complex @ X3 @ Y ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).
% tan_add
thf(fact_7234_tan__add,axiom,
! [X3: real,Y: real] :
( ( ( cos_real @ X3 )
!= zero_zero_real )
=> ( ( ( cos_real @ Y )
!= zero_zero_real )
=> ( ( ( cos_real @ ( plus_plus_real @ X3 @ Y ) )
!= zero_zero_real )
=> ( ( tan_real @ ( plus_plus_real @ X3 @ Y ) )
= ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X3 ) @ ( tan_real @ Y ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X3 ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).
% tan_add
thf(fact_7235_tan__diff,axiom,
! [X3: complex,Y: complex] :
( ( ( cos_complex @ X3 )
!= zero_zero_complex )
=> ( ( ( cos_complex @ Y )
!= zero_zero_complex )
=> ( ( ( cos_complex @ ( minus_minus_complex @ X3 @ Y ) )
!= zero_zero_complex )
=> ( ( tan_complex @ ( minus_minus_complex @ X3 @ Y ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).
% tan_diff
thf(fact_7236_tan__diff,axiom,
! [X3: real,Y: real] :
( ( ( cos_real @ X3 )
!= zero_zero_real )
=> ( ( ( cos_real @ Y )
!= zero_zero_real )
=> ( ( ( cos_real @ ( minus_minus_real @ X3 @ Y ) )
!= zero_zero_real )
=> ( ( tan_real @ ( minus_minus_real @ X3 @ Y ) )
= ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X3 ) @ ( tan_real @ Y ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X3 ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).
% tan_diff
thf(fact_7237_lemma__tan__add1,axiom,
! [X3: complex,Y: complex] :
( ( ( cos_complex @ X3 )
!= zero_zero_complex )
=> ( ( ( cos_complex @ Y )
!= zero_zero_complex )
=> ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y ) ) )
= ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X3 @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ Y ) ) ) ) ) ) ).
% lemma_tan_add1
thf(fact_7238_lemma__tan__add1,axiom,
! [X3: real,Y: real] :
( ( ( cos_real @ X3 )
!= zero_zero_real )
=> ( ( ( cos_real @ Y )
!= zero_zero_real )
=> ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X3 ) @ ( tan_real @ Y ) ) )
= ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X3 @ Y ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y ) ) ) ) ) ) ).
% lemma_tan_add1
thf(fact_7239_tan__half,axiom,
( tan_complex
= ( ^ [X5: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X5 ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X5 ) ) @ one_one_complex ) ) ) ) ).
% tan_half
thf(fact_7240_tan__half,axiom,
( tan_real
= ( ^ [X5: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X5 ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X5 ) ) @ one_one_real ) ) ) ) ).
% tan_half
thf(fact_7241_cos__coeff__def,axiom,
( cos_coeff
= ( ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N4 ) ) @ zero_zero_real ) ) ) ).
% cos_coeff_def
thf(fact_7242_sin__paired,axiom,
! [X3: real] :
( sums_real
@ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
@ ( sin_real @ X3 ) ) ).
% sin_paired
thf(fact_7243_Maclaurin__sin__expansion3,axiom,
! [N: nat,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ? [T5: real] :
( ( ord_less_real @ zero_zero_real @ T5 )
& ( ord_less_real @ T5 @ X3 )
& ( ( sin_real @ X3 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X3 @ M5 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( 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 @ X3 @ N ) ) ) ) ) ) ) ).
% Maclaurin_sin_expansion3
thf(fact_7244_Maclaurin__sin__expansion4,axiom,
! [X3: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ? [T5: real] :
( ( ord_less_real @ zero_zero_real @ T5 )
& ( ord_less_eq_real @ T5 @ X3 )
& ( ( sin_real @ X3 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X3 @ M5 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( 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 @ X3 @ N ) ) ) ) ) ) ).
% Maclaurin_sin_expansion4
thf(fact_7245_Maclaurin__sin__expansion2,axiom,
! [X3: real,N: nat] :
? [T5: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X3 ) )
& ( ( sin_real @ X3 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X3 @ M5 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( 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 @ X3 @ N ) ) ) ) ) ).
% Maclaurin_sin_expansion2
thf(fact_7246_Maclaurin__sin__expansion,axiom,
! [X3: real,N: nat] :
? [T5: real] :
( ( sin_real @ X3 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X3 @ M5 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( 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 @ X3 @ N ) ) ) ) ).
% Maclaurin_sin_expansion
thf(fact_7247_sin__coeff__def,axiom,
( sin_coeff
= ( ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N4 ) ) ) ) ) ).
% sin_coeff_def
thf(fact_7248_complex__unimodular__polar,axiom,
! [Z2: complex] :
( ( ( real_V1022390504157884413omplex @ Z2 )
= one_one_real )
=> ~ ! [T5: real] :
( ( ord_less_eq_real @ zero_zero_real @ T5 )
=> ( ( ord_less_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
=> ( Z2
!= ( complex2 @ ( cos_real @ T5 ) @ ( sin_real @ T5 ) ) ) ) ) ) ).
% complex_unimodular_polar
thf(fact_7249_sin__coeff__0,axiom,
( ( sin_coeff @ zero_zero_nat )
= zero_zero_real ) ).
% sin_coeff_0
thf(fact_7250_fact__ge__self,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_ge_self
thf(fact_7251_fact__mono__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% fact_mono_nat
thf(fact_7252_fact__less__mono__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% fact_less_mono_nat
thf(fact_7253_Complex__eq__numeral,axiom,
! [A: real,B: real,W2: num] :
( ( ( complex2 @ A @ B )
= ( numera6690914467698888265omplex @ W2 ) )
= ( ( A
= ( numeral_numeral_real @ W2 ) )
& ( B = zero_zero_real ) ) ) ).
% Complex_eq_numeral
thf(fact_7254_zero__complex_Ocode,axiom,
( zero_zero_complex
= ( complex2 @ zero_zero_real @ zero_zero_real ) ) ).
% zero_complex.code
thf(fact_7255_Complex__eq__0,axiom,
! [A: real,B: real] :
( ( ( complex2 @ A @ B )
= zero_zero_complex )
= ( ( A = zero_zero_real )
& ( B = zero_zero_real ) ) ) ).
% Complex_eq_0
thf(fact_7256_complex__add,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( plus_plus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C2 @ D ) )
= ( complex2 @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ).
% complex_add
thf(fact_7257_fact__ge__Suc__0__nat,axiom,
! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_ge_Suc_0_nat
thf(fact_7258_dvd__fact,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( dvd_dvd_nat @ M2 @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% dvd_fact
thf(fact_7259_Complex__eq__neg__numeral,axiom,
! [A: real,B: real,W2: num] :
( ( ( complex2 @ A @ B )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
= ( ( A
= ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
& ( B = zero_zero_real ) ) ) ).
% Complex_eq_neg_numeral
thf(fact_7260_complex__mult,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( times_times_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C2 @ D ) )
= ( complex2 @ ( minus_minus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% complex_mult
thf(fact_7261_one__complex_Ocode,axiom,
( one_one_complex
= ( complex2 @ one_one_real @ zero_zero_real ) ) ).
% one_complex.code
thf(fact_7262_Complex__eq__1,axiom,
! [A: real,B: real] :
( ( ( complex2 @ A @ B )
= one_one_complex )
= ( ( A = one_one_real )
& ( B = zero_zero_real ) ) ) ).
% Complex_eq_1
thf(fact_7263_fact__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) )
= ( times_times_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M2 @ N ) ) ) ) ) ).
% fact_diff_Suc
thf(fact_7264_fact__div__fact__le__pow,axiom,
! [R2: nat,N: nat] :
( ( ord_less_eq_nat @ R2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R2 ) ) ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% fact_div_fact_le_pow
thf(fact_7265_Complex__sum_H,axiom,
! [F: nat > real,S3: set_nat] :
( ( groups2073611262835488442omplex
@ ^ [X5: nat] : ( complex2 @ ( F @ X5 ) @ zero_zero_real )
@ S3 )
= ( complex2 @ ( groups6591440286371151544t_real @ F @ S3 ) @ zero_zero_real ) ) ).
% Complex_sum'
thf(fact_7266_Complex__sum_H,axiom,
! [F: complex > real,S3: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [X5: complex] : ( complex2 @ ( F @ X5 ) @ zero_zero_real )
@ S3 )
= ( complex2 @ ( groups5808333547571424918x_real @ F @ S3 ) @ zero_zero_real ) ) ).
% Complex_sum'
thf(fact_7267_Complex__eq__neg__1,axiom,
! [A: real,B: real] :
( ( ( complex2 @ A @ B )
= ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( ( A
= ( uminus_uminus_real @ one_one_real ) )
& ( B = zero_zero_real ) ) ) ).
% Complex_eq_neg_1
thf(fact_7268_sin__coeff__Suc,axiom,
! [N: nat] :
( ( sin_coeff @ ( suc @ N ) )
= ( divide_divide_real @ ( cos_coeff @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% sin_coeff_Suc
thf(fact_7269_cos__coeff__Suc,axiom,
! [N: nat] :
( ( cos_coeff @ ( suc @ N ) )
= ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% cos_coeff_Suc
thf(fact_7270_sin__tan,axiom,
! [X3: real] :
( ( ord_less_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( sin_real @ X3 )
= ( divide_divide_real @ ( tan_real @ X3 ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% sin_tan
thf(fact_7271_cos__tan,axiom,
! [X3: real] :
( ( ord_less_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( cos_real @ X3 )
= ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% cos_tan
thf(fact_7272_Maclaurin__exp__lt,axiom,
! [X3: real,N: nat] :
( ( X3 != zero_zero_real )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [T5: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T5 ) )
& ( ord_less_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X3 ) )
& ( ( exp_real @ X3 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( divide_divide_real @ ( power_power_real @ X3 @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ) ).
% Maclaurin_exp_lt
thf(fact_7273_in__finite__psubset,axiom,
! [A4: set_nat,B5: set_nat] :
( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A4 @ B5 ) @ finite_psubset_nat )
= ( ( ord_less_set_nat @ A4 @ B5 )
& ( finite_finite_nat @ B5 ) ) ) ).
% in_finite_psubset
thf(fact_7274_in__finite__psubset,axiom,
! [A4: set_int,B5: set_int] :
( ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ A4 @ B5 ) @ finite_psubset_int )
= ( ( ord_less_set_int @ A4 @ B5 )
& ( finite_finite_int @ B5 ) ) ) ).
% in_finite_psubset
thf(fact_7275_in__finite__psubset,axiom,
! [A4: set_complex,B5: set_complex] :
( ( member351165363924911826omplex @ ( produc3790773574474814305omplex @ A4 @ B5 ) @ finite8643634255014194347omplex )
= ( ( ord_less_set_complex @ A4 @ B5 )
& ( finite3207457112153483333omplex @ B5 ) ) ) ).
% in_finite_psubset
thf(fact_7276_in__finite__psubset,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A4 @ B5 ) @ finite469560695537375940at_nat )
= ( ( ord_le7866589430770878221at_nat @ A4 @ B5 )
& ( finite6177210948735845034at_nat @ B5 ) ) ) ).
% in_finite_psubset
thf(fact_7277_pochhammer__double,axiom,
! [Z2: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( 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 @ Z2 @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% pochhammer_double
thf(fact_7278_pochhammer__double,axiom,
! [Z2: real,N: nat] :
( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( 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 @ Z2 @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% pochhammer_double
thf(fact_7279_pochhammer__double,axiom,
! [Z2: complex,N: nat] :
( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( 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 @ Z2 @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% pochhammer_double
thf(fact_7280_of__nat__code,axiom,
( semiri681578069525770553at_rat
= ( ^ [N4: nat] :
( semiri7787848453975740701ux_rat
@ ^ [I3: rat] : ( plus_plus_rat @ I3 @ one_one_rat )
@ N4
@ zero_zero_rat ) ) ) ).
% of_nat_code
thf(fact_7281_of__nat__code,axiom,
( semiri1314217659103216013at_int
= ( ^ [N4: nat] :
( semiri8420488043553186161ux_int
@ ^ [I3: int] : ( plus_plus_int @ I3 @ one_one_int )
@ N4
@ zero_zero_int ) ) ) ).
% of_nat_code
thf(fact_7282_of__nat__code,axiom,
( semiri5074537144036343181t_real
= ( ^ [N4: nat] :
( semiri7260567687927622513x_real
@ ^ [I3: real] : ( plus_plus_real @ I3 @ one_one_real )
@ N4
@ zero_zero_real ) ) ) ).
% of_nat_code
thf(fact_7283_of__nat__code,axiom,
( semiri1316708129612266289at_nat
= ( ^ [N4: nat] :
( semiri8422978514062236437ux_nat
@ ^ [I3: nat] : ( plus_plus_nat @ I3 @ one_one_nat )
@ N4
@ zero_zero_nat ) ) ) ).
% of_nat_code
thf(fact_7284_of__nat__code,axiom,
( semiri8010041392384452111omplex
= ( ^ [N4: nat] :
( semiri2816024913162550771omplex
@ ^ [I3: complex] : ( plus_plus_complex @ I3 @ one_one_complex )
@ N4
@ zero_zero_complex ) ) ) ).
% of_nat_code
thf(fact_7285_of__nat__code,axiom,
( semiri4939895301339042750nteger
= ( ^ [N4: nat] :
( semiri4055485073559036834nteger
@ ^ [I3: code_integer] : ( plus_p5714425477246183910nteger @ I3 @ one_one_Code_integer )
@ N4
@ zero_z3403309356797280102nteger ) ) ) ).
% of_nat_code
thf(fact_7286_real__sqrt__eq__zero__cancel__iff,axiom,
! [X3: real] :
( ( ( sqrt @ X3 )
= zero_zero_real )
= ( X3 = zero_zero_real ) ) ).
% real_sqrt_eq_zero_cancel_iff
thf(fact_7287_real__sqrt__zero,axiom,
( ( sqrt @ zero_zero_real )
= zero_zero_real ) ).
% real_sqrt_zero
thf(fact_7288_real__sqrt__le__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( sqrt @ X3 ) @ ( sqrt @ Y ) )
= ( ord_less_eq_real @ X3 @ Y ) ) ).
% real_sqrt_le_iff
thf(fact_7289_exp__le__cancel__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( exp_real @ X3 ) @ ( exp_real @ Y ) )
= ( ord_less_eq_real @ X3 @ Y ) ) ).
% exp_le_cancel_iff
thf(fact_7290_exp__zero,axiom,
( ( exp_complex @ zero_zero_complex )
= one_one_complex ) ).
% exp_zero
thf(fact_7291_exp__zero,axiom,
( ( exp_real @ zero_zero_real )
= one_one_real ) ).
% exp_zero
thf(fact_7292_real__sqrt__gt__0__iff,axiom,
! [Y: real] :
( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y ) )
= ( ord_less_real @ zero_zero_real @ Y ) ) ).
% real_sqrt_gt_0_iff
thf(fact_7293_real__sqrt__lt__0__iff,axiom,
! [X3: real] :
( ( ord_less_real @ ( sqrt @ X3 ) @ zero_zero_real )
= ( ord_less_real @ X3 @ zero_zero_real ) ) ).
% real_sqrt_lt_0_iff
thf(fact_7294_real__sqrt__le__0__iff,axiom,
! [X3: real] :
( ( ord_less_eq_real @ ( sqrt @ X3 ) @ zero_zero_real )
= ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).
% real_sqrt_le_0_iff
thf(fact_7295_real__sqrt__ge__0__iff,axiom,
! [Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y ) )
= ( ord_less_eq_real @ zero_zero_real @ Y ) ) ).
% real_sqrt_ge_0_iff
thf(fact_7296_real__sqrt__ge__1__iff,axiom,
! [Y: real] :
( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y ) )
= ( ord_less_eq_real @ one_one_real @ Y ) ) ).
% real_sqrt_ge_1_iff
thf(fact_7297_real__sqrt__le__1__iff,axiom,
! [X3: real] :
( ( ord_less_eq_real @ ( sqrt @ X3 ) @ one_one_real )
= ( ord_less_eq_real @ X3 @ one_one_real ) ) ).
% real_sqrt_le_1_iff
thf(fact_7298_pochhammer__0,axiom,
! [A: complex] :
( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
= one_one_complex ) ).
% pochhammer_0
thf(fact_7299_pochhammer__0,axiom,
! [A: real] :
( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
= one_one_real ) ).
% pochhammer_0
thf(fact_7300_pochhammer__0,axiom,
! [A: rat] :
( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
= one_one_rat ) ).
% pochhammer_0
thf(fact_7301_pochhammer__0,axiom,
! [A: nat] :
( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% pochhammer_0
thf(fact_7302_pochhammer__0,axiom,
! [A: int] :
( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
= one_one_int ) ).
% pochhammer_0
thf(fact_7303_exp__eq__one__iff,axiom,
! [X3: real] :
( ( ( exp_real @ X3 )
= one_one_real )
= ( X3 = zero_zero_real ) ) ).
% exp_eq_one_iff
thf(fact_7304_exp__less__one__iff,axiom,
! [X3: real] :
( ( ord_less_real @ ( exp_real @ X3 ) @ one_one_real )
= ( ord_less_real @ X3 @ zero_zero_real ) ) ).
% exp_less_one_iff
thf(fact_7305_one__less__exp__iff,axiom,
! [X3: real] :
( ( ord_less_real @ one_one_real @ ( exp_real @ X3 ) )
= ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% one_less_exp_iff
thf(fact_7306_one__le__exp__iff,axiom,
! [X3: real] :
( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X3 ) )
= ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).
% one_le_exp_iff
thf(fact_7307_exp__le__one__iff,axiom,
! [X3: real] :
( ( ord_less_eq_real @ ( exp_real @ X3 ) @ one_one_real )
= ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).
% exp_le_one_iff
thf(fact_7308_exp__ln__iff,axiom,
! [X3: real] :
( ( ( exp_real @ ( ln_ln_real @ X3 ) )
= X3 )
= ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% exp_ln_iff
thf(fact_7309_exp__ln,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( exp_real @ ( ln_ln_real @ X3 ) )
= X3 ) ) ).
% exp_ln
thf(fact_7310_real__sqrt__pow2,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( power_power_real @ ( sqrt @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= X3 ) ) ).
% real_sqrt_pow2
thf(fact_7311_real__sqrt__pow2__iff,axiom,
! [X3: real] :
( ( ( power_power_real @ ( sqrt @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= X3 )
= ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).
% real_sqrt_pow2_iff
thf(fact_7312_real__sqrt__sum__squares__mult__squared__eq,axiom,
! [X3: real,Y: real,Xa2: real,Ya: real] :
( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% real_sqrt_sum_squares_mult_squared_eq
thf(fact_7313_norm__exp,axiom,
! [X3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ X3 ) ) @ ( exp_real @ ( real_V7735802525324610683m_real @ X3 ) ) ) ).
% norm_exp
thf(fact_7314_norm__exp,axiom,
! [X3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ X3 ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ X3 ) ) ) ).
% norm_exp
thf(fact_7315_real__sqrt__le__mono,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ ( sqrt @ X3 ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_le_mono
thf(fact_7316_exp__not__eq__zero,axiom,
! [X3: complex] :
( ( exp_complex @ X3 )
!= zero_zero_complex ) ).
% exp_not_eq_zero
thf(fact_7317_exp__not__eq__zero,axiom,
! [X3: real] :
( ( exp_real @ X3 )
!= zero_zero_real ) ).
% exp_not_eq_zero
thf(fact_7318_real__sqrt__gt__zero,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_real @ zero_zero_real @ ( sqrt @ X3 ) ) ) ).
% real_sqrt_gt_zero
thf(fact_7319_exp__total,axiom,
! [Y: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ? [X4: real] :
( ( exp_real @ X4 )
= Y ) ) ).
% exp_total
thf(fact_7320_exp__gt__zero,axiom,
! [X3: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X3 ) ) ).
% exp_gt_zero
thf(fact_7321_not__exp__less__zero,axiom,
! [X3: real] :
~ ( ord_less_real @ ( exp_real @ X3 ) @ zero_zero_real ) ).
% not_exp_less_zero
thf(fact_7322_real__sqrt__ge__zero,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X3 ) ) ) ).
% real_sqrt_ge_zero
thf(fact_7323_real__sqrt__eq__zero__cancel,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ( sqrt @ X3 )
= zero_zero_real )
=> ( X3 = zero_zero_real ) ) ) ).
% real_sqrt_eq_zero_cancel
thf(fact_7324_exp__ge__zero,axiom,
! [X3: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X3 ) ) ).
% exp_ge_zero
thf(fact_7325_not__exp__le__zero,axiom,
! [X3: real] :
~ ( ord_less_eq_real @ ( exp_real @ X3 ) @ zero_zero_real ) ).
% not_exp_le_zero
thf(fact_7326_real__sqrt__ge__one,axiom,
! [X3: real] :
( ( ord_less_eq_real @ one_one_real @ X3 )
=> ( ord_less_eq_real @ one_one_real @ ( sqrt @ X3 ) ) ) ).
% real_sqrt_ge_one
thf(fact_7327_mult__exp__exp,axiom,
! [X3: real,Y: real] :
( ( times_times_real @ ( exp_real @ X3 ) @ ( exp_real @ Y ) )
= ( exp_real @ ( plus_plus_real @ X3 @ Y ) ) ) ).
% mult_exp_exp
thf(fact_7328_exp__add__commuting,axiom,
! [X3: real,Y: real] :
( ( ( times_times_real @ X3 @ Y )
= ( times_times_real @ Y @ X3 ) )
=> ( ( exp_real @ ( plus_plus_real @ X3 @ Y ) )
= ( times_times_real @ ( exp_real @ X3 ) @ ( exp_real @ Y ) ) ) ) ).
% exp_add_commuting
thf(fact_7329_pochhammer__pos,axiom,
! [X3: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X3 @ N ) ) ) ).
% pochhammer_pos
thf(fact_7330_pochhammer__pos,axiom,
! [X3: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ X3 )
=> ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X3 @ N ) ) ) ).
% pochhammer_pos
thf(fact_7331_pochhammer__pos,axiom,
! [X3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ X3 )
=> ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X3 @ N ) ) ) ).
% pochhammer_pos
thf(fact_7332_pochhammer__pos,axiom,
! [X3: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ X3 )
=> ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X3 @ N ) ) ) ).
% pochhammer_pos
thf(fact_7333_pochhammer__eq__0__mono,axiom,
! [A: complex,N: nat,M2: nat] :
( ( ( comm_s2602460028002588243omplex @ A @ N )
= zero_zero_complex )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( comm_s2602460028002588243omplex @ A @ M2 )
= zero_zero_complex ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_7334_pochhammer__eq__0__mono,axiom,
! [A: real,N: nat,M2: nat] :
( ( ( comm_s7457072308508201937r_real @ A @ N )
= zero_zero_real )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( comm_s7457072308508201937r_real @ A @ M2 )
= zero_zero_real ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_7335_pochhammer__eq__0__mono,axiom,
! [A: rat,N: nat,M2: nat] :
( ( ( comm_s4028243227959126397er_rat @ A @ N )
= zero_zero_rat )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( comm_s4028243227959126397er_rat @ A @ M2 )
= zero_zero_rat ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_7336_pochhammer__neq__0__mono,axiom,
! [A: complex,M2: nat,N: nat] :
( ( ( comm_s2602460028002588243omplex @ A @ M2 )
!= zero_zero_complex )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( comm_s2602460028002588243omplex @ A @ N )
!= zero_zero_complex ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_7337_pochhammer__neq__0__mono,axiom,
! [A: real,M2: nat,N: nat] :
( ( ( comm_s7457072308508201937r_real @ A @ M2 )
!= zero_zero_real )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( comm_s7457072308508201937r_real @ A @ N )
!= zero_zero_real ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_7338_pochhammer__neq__0__mono,axiom,
! [A: rat,M2: nat,N: nat] :
( ( ( comm_s4028243227959126397er_rat @ A @ M2 )
!= zero_zero_rat )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( comm_s4028243227959126397er_rat @ A @ N )
!= zero_zero_rat ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_7339_exp__gt__one,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_real @ one_one_real @ ( exp_real @ X3 ) ) ) ).
% exp_gt_one
thf(fact_7340_real__div__sqrt,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( divide_divide_real @ X3 @ ( sqrt @ X3 ) )
= ( sqrt @ X3 ) ) ) ).
% real_div_sqrt
thf(fact_7341_sqrt__add__le__add__sqrt,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X3 @ Y ) ) @ ( plus_plus_real @ ( sqrt @ X3 ) @ ( sqrt @ Y ) ) ) ) ) ).
% sqrt_add_le_add_sqrt
thf(fact_7342_exp__ge__add__one__self,axiom,
! [X3: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( exp_real @ X3 ) ) ).
% exp_ge_add_one_self
thf(fact_7343_le__real__sqrt__sumsq,axiom,
! [X3: real,Y: real] : ( ord_less_eq_real @ X3 @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y @ Y ) ) ) ) ).
% le_real_sqrt_sumsq
thf(fact_7344_pochhammer__nonneg,axiom,
! [X3: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X3 @ N ) ) ) ).
% pochhammer_nonneg
thf(fact_7345_pochhammer__nonneg,axiom,
! [X3: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ X3 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X3 @ N ) ) ) ).
% pochhammer_nonneg
thf(fact_7346_pochhammer__nonneg,axiom,
! [X3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ X3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X3 @ N ) ) ) ).
% pochhammer_nonneg
thf(fact_7347_pochhammer__nonneg,axiom,
! [X3: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ X3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X3 @ N ) ) ) ).
% pochhammer_nonneg
thf(fact_7348_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
= one_one_complex ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
= zero_zero_complex ) ) ) ).
% pochhammer_0_left
thf(fact_7349_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
= zero_zero_real ) ) ) ).
% pochhammer_0_left
thf(fact_7350_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
= one_one_rat ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
= zero_zero_rat ) ) ) ).
% pochhammer_0_left
thf(fact_7351_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% pochhammer_0_left
thf(fact_7352_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% pochhammer_0_left
thf(fact_7353_exp__ge__add__one__self__aux,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( exp_real @ X3 ) ) ) ).
% exp_ge_add_one_self_aux
thf(fact_7354_lemma__exp__total,axiom,
! [Y: real] :
( ( ord_less_eq_real @ one_one_real @ Y )
=> ? [X4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
& ( ord_less_eq_real @ X4 @ ( minus_minus_real @ Y @ one_one_real ) )
& ( ( exp_real @ X4 )
= Y ) ) ) ).
% lemma_exp_total
thf(fact_7355_ln__ge__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ Y @ ( ln_ln_real @ X3 ) )
= ( ord_less_eq_real @ ( exp_real @ Y ) @ X3 ) ) ) ).
% ln_ge_iff
thf(fact_7356_ln__x__over__x__mono,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y ) @ Y ) @ ( divide_divide_real @ ( ln_ln_real @ X3 ) @ X3 ) ) ) ) ).
% ln_x_over_x_mono
thf(fact_7357_pochhammer__rec,axiom,
! [A: complex,N: nat] :
( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
= ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_7358_pochhammer__rec,axiom,
! [A: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
= ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_7359_pochhammer__rec,axiom,
! [A: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
= ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_7360_pochhammer__rec,axiom,
! [A: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_7361_pochhammer__rec,axiom,
! [A: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
= ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_7362_pochhammer__Suc,axiom,
! [A: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
= ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_7363_pochhammer__Suc,axiom,
! [A: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
= ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_7364_pochhammer__Suc,axiom,
! [A: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
= ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_7365_pochhammer__Suc,axiom,
! [A: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_7366_pochhammer__Suc,axiom,
! [A: complex,N: nat] :
( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
= ( times_times_complex @ ( comm_s2602460028002588243omplex @ A @ N ) @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_7367_pochhammer__Suc,axiom,
! [A: code_integer,N: nat] :
( ( comm_s8582702949713902594nteger @ A @ ( suc @ N ) )
= ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ A @ N ) @ ( plus_p5714425477246183910nteger @ A @ ( semiri4939895301339042750nteger @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_7368_pochhammer__rec_H,axiom,
! [Z2: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ Z2 @ ( suc @ N ) )
= ( times_times_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z2 @ N ) ) ) ).
% pochhammer_rec'
thf(fact_7369_pochhammer__rec_H,axiom,
! [Z2: int,N: nat] :
( ( comm_s4660882817536571857er_int @ Z2 @ ( suc @ N ) )
= ( times_times_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z2 @ N ) ) ) ).
% pochhammer_rec'
thf(fact_7370_pochhammer__rec_H,axiom,
! [Z2: real,N: nat] :
( ( comm_s7457072308508201937r_real @ Z2 @ ( suc @ N ) )
= ( times_times_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ N ) ) @ ( comm_s7457072308508201937r_real @ Z2 @ N ) ) ) ).
% pochhammer_rec'
thf(fact_7371_pochhammer__rec_H,axiom,
! [Z2: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ Z2 @ ( suc @ N ) )
= ( times_times_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z2 @ N ) ) ) ).
% pochhammer_rec'
thf(fact_7372_pochhammer__rec_H,axiom,
! [Z2: complex,N: nat] :
( ( comm_s2602460028002588243omplex @ Z2 @ ( suc @ N ) )
= ( times_times_complex @ ( plus_plus_complex @ Z2 @ ( semiri8010041392384452111omplex @ N ) ) @ ( comm_s2602460028002588243omplex @ Z2 @ N ) ) ) ).
% pochhammer_rec'
thf(fact_7373_pochhammer__rec_H,axiom,
! [Z2: code_integer,N: nat] :
( ( comm_s8582702949713902594nteger @ Z2 @ ( suc @ N ) )
= ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ Z2 @ ( semiri4939895301339042750nteger @ N ) ) @ ( comm_s8582702949713902594nteger @ Z2 @ N ) ) ) ).
% pochhammer_rec'
thf(fact_7374_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K2 )
= zero_zero_rat ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_7375_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K2 )
= zero_zero_int ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_7376_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K2 )
= zero_zero_real ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_7377_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K2 )
= zero_zero_complex ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_7378_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K2 )
= zero_z3403309356797280102nteger ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_7379_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K2: nat] :
( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K2 )
= zero_zero_rat )
= ( ord_less_nat @ N @ K2 ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_7380_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K2: nat] :
( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K2 )
= zero_zero_int )
= ( ord_less_nat @ N @ K2 ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_7381_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K2: nat] :
( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K2 )
= zero_zero_real )
= ( ord_less_nat @ N @ K2 ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_7382_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K2: nat] :
( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K2 )
= zero_zero_complex )
= ( ord_less_nat @ N @ K2 ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_7383_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K2: nat] :
( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K2 )
= zero_z3403309356797280102nteger )
= ( ord_less_nat @ N @ K2 ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_7384_pochhammer__eq__0__iff,axiom,
! [A: rat,N: nat] :
( ( ( comm_s4028243227959126397er_rat @ A @ N )
= zero_zero_rat )
= ( ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ( A
= ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_7385_pochhammer__eq__0__iff,axiom,
! [A: real,N: nat] :
( ( ( comm_s7457072308508201937r_real @ A @ N )
= zero_zero_real )
= ( ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ( A
= ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K3 ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_7386_pochhammer__eq__0__iff,axiom,
! [A: complex,N: nat] :
( ( ( comm_s2602460028002588243omplex @ A @ N )
= zero_zero_complex )
= ( ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ( A
= ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K3 ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_7387_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K2 )
!= zero_zero_rat ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_7388_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K2 )
!= zero_zero_int ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_7389_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K2 )
!= zero_zero_real ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_7390_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K2 )
!= zero_zero_complex ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_7391_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K2 )
!= zero_z3403309356797280102nteger ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_7392_pochhammer__product_H,axiom,
! [Z2: rat,N: nat,M2: nat] :
( ( comm_s4028243227959126397er_rat @ Z2 @ ( plus_plus_nat @ N @ M2 ) )
= ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ N ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ N ) ) @ M2 ) ) ) ).
% pochhammer_product'
thf(fact_7393_pochhammer__product_H,axiom,
! [Z2: int,N: nat,M2: nat] :
( ( comm_s4660882817536571857er_int @ Z2 @ ( plus_plus_nat @ N @ M2 ) )
= ( times_times_int @ ( comm_s4660882817536571857er_int @ Z2 @ N ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ N ) ) @ M2 ) ) ) ).
% pochhammer_product'
thf(fact_7394_pochhammer__product_H,axiom,
! [Z2: real,N: nat,M2: nat] :
( ( comm_s7457072308508201937r_real @ Z2 @ ( plus_plus_nat @ N @ M2 ) )
= ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ N ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ N ) ) @ M2 ) ) ) ).
% pochhammer_product'
thf(fact_7395_pochhammer__product_H,axiom,
! [Z2: nat,N: nat,M2: nat] :
( ( comm_s4663373288045622133er_nat @ Z2 @ ( plus_plus_nat @ N @ M2 ) )
= ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z2 @ N ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ N ) ) @ M2 ) ) ) ).
% pochhammer_product'
thf(fact_7396_pochhammer__product_H,axiom,
! [Z2: complex,N: nat,M2: nat] :
( ( comm_s2602460028002588243omplex @ Z2 @ ( plus_plus_nat @ N @ M2 ) )
= ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z2 @ N ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( semiri8010041392384452111omplex @ N ) ) @ M2 ) ) ) ).
% pochhammer_product'
thf(fact_7397_pochhammer__product_H,axiom,
! [Z2: code_integer,N: nat,M2: nat] :
( ( comm_s8582702949713902594nteger @ Z2 @ ( plus_plus_nat @ N @ M2 ) )
= ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ Z2 @ N ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ Z2 @ ( semiri4939895301339042750nteger @ N ) ) @ M2 ) ) ) ).
% pochhammer_product'
thf(fact_7398_real__le__rsqrt,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
=> ( ord_less_eq_real @ X3 @ ( sqrt @ Y ) ) ) ).
% real_le_rsqrt
thf(fact_7399_sqrt__le__D,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( sqrt @ X3 ) @ Y )
=> ( ord_less_eq_real @ X3 @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sqrt_le_D
thf(fact_7400_exp__le,axiom,
ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).
% exp_le
thf(fact_7401_exp__divide__power__eq,axiom,
! [N: nat,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X3 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
= ( exp_real @ X3 ) ) ) ).
% exp_divide_power_eq
thf(fact_7402_exp__divide__power__eq,axiom,
! [N: nat,X3: complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X3 @ ( semiri8010041392384452111omplex @ N ) ) ) @ N )
= ( exp_complex @ X3 ) ) ) ).
% exp_divide_power_eq
thf(fact_7403_tanh__altdef,axiom,
( tanh_real
= ( ^ [X5: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X5 ) @ ( exp_real @ ( uminus_uminus_real @ X5 ) ) ) @ ( plus_plus_real @ ( exp_real @ X5 ) @ ( exp_real @ ( uminus_uminus_real @ X5 ) ) ) ) ) ) ).
% tanh_altdef
thf(fact_7404_tanh__altdef,axiom,
( tanh_complex
= ( ^ [X5: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X5 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X5 ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X5 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X5 ) ) ) ) ) ) ).
% tanh_altdef
thf(fact_7405_pochhammer__product,axiom,
! [M2: nat,N: nat,Z2: rat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( comm_s4028243227959126397er_rat @ Z2 @ N )
= ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ M2 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).
% pochhammer_product
thf(fact_7406_pochhammer__product,axiom,
! [M2: nat,N: nat,Z2: int] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( comm_s4660882817536571857er_int @ Z2 @ N )
= ( times_times_int @ ( comm_s4660882817536571857er_int @ Z2 @ M2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).
% pochhammer_product
thf(fact_7407_pochhammer__product,axiom,
! [M2: nat,N: nat,Z2: real] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( comm_s7457072308508201937r_real @ Z2 @ N )
= ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ M2 ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).
% pochhammer_product
thf(fact_7408_pochhammer__product,axiom,
! [M2: nat,N: nat,Z2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( comm_s4663373288045622133er_nat @ Z2 @ N )
= ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z2 @ M2 ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).
% pochhammer_product
thf(fact_7409_pochhammer__product,axiom,
! [M2: nat,N: nat,Z2: complex] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( comm_s2602460028002588243omplex @ Z2 @ N )
= ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z2 @ M2 ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( semiri8010041392384452111omplex @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).
% pochhammer_product
thf(fact_7410_pochhammer__product,axiom,
! [M2: nat,N: nat,Z2: code_integer] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( comm_s8582702949713902594nteger @ Z2 @ N )
= ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ Z2 @ M2 ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ Z2 @ ( semiri4939895301339042750nteger @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).
% pochhammer_product
thf(fact_7411_real__le__lsqrt,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X3 @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( sqrt @ X3 ) @ Y ) ) ) ) ).
% real_le_lsqrt
thf(fact_7412_real__sqrt__unique,axiom,
! [Y: real,X3: real] :
( ( ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( sqrt @ X3 )
= Y ) ) ) ).
% real_sqrt_unique
thf(fact_7413_lemma__real__divide__sqrt__less,axiom,
! [U: real] :
( ( ord_less_real @ zero_zero_real @ U )
=> ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).
% lemma_real_divide_sqrt_less
thf(fact_7414_real__sqrt__sum__squares__eq__cancel2,axiom,
! [X3: real,Y: real] :
( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= Y )
=> ( X3 = zero_zero_real ) ) ).
% real_sqrt_sum_squares_eq_cancel2
thf(fact_7415_real__sqrt__sum__squares__eq__cancel,axiom,
! [X3: real,Y: real] :
( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= X3 )
=> ( Y = zero_zero_real ) ) ).
% real_sqrt_sum_squares_eq_cancel
thf(fact_7416_exp__half__le2,axiom,
ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% exp_half_le2
thf(fact_7417_real__sqrt__sum__squares__ge1,axiom,
! [X3: real,Y: real] : ( ord_less_eq_real @ X3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge1
thf(fact_7418_real__sqrt__sum__squares__ge2,axiom,
! [Y: real,X3: real] : ( ord_less_eq_real @ Y @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge2
thf(fact_7419_real__sqrt__sum__squares__triangle__ineq,axiom,
! [A: real,C2: real,B: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A @ C2 ) @ ( 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 @ C2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% real_sqrt_sum_squares_triangle_ineq
thf(fact_7420_sqrt__ge__absD,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ ( sqrt @ Y ) )
=> ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) ).
% sqrt_ge_absD
thf(fact_7421_pochhammer__absorb__comp,axiom,
! [R2: rat,K2: nat] :
( ( times_times_rat @ ( minus_minus_rat @ R2 @ ( semiri681578069525770553at_rat @ K2 ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R2 ) @ K2 ) )
= ( times_times_rat @ R2 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R2 ) @ one_one_rat ) @ K2 ) ) ) ).
% pochhammer_absorb_comp
thf(fact_7422_pochhammer__absorb__comp,axiom,
! [R2: int,K2: nat] :
( ( times_times_int @ ( minus_minus_int @ R2 @ ( semiri1314217659103216013at_int @ K2 ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R2 ) @ K2 ) )
= ( times_times_int @ R2 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R2 ) @ one_one_int ) @ K2 ) ) ) ).
% pochhammer_absorb_comp
thf(fact_7423_pochhammer__absorb__comp,axiom,
! [R2: real,K2: nat] :
( ( times_times_real @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R2 ) @ K2 ) )
= ( times_times_real @ R2 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R2 ) @ one_one_real ) @ K2 ) ) ) ).
% pochhammer_absorb_comp
thf(fact_7424_pochhammer__absorb__comp,axiom,
! [R2: complex,K2: nat] :
( ( times_times_complex @ ( minus_minus_complex @ R2 @ ( semiri8010041392384452111omplex @ K2 ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R2 ) @ K2 ) )
= ( times_times_complex @ R2 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R2 ) @ one_one_complex ) @ K2 ) ) ) ).
% pochhammer_absorb_comp
thf(fact_7425_pochhammer__absorb__comp,axiom,
! [R2: code_integer,K2: nat] :
( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R2 @ ( semiri4939895301339042750nteger @ K2 ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R2 ) @ K2 ) )
= ( times_3573771949741848930nteger @ R2 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R2 ) @ one_one_Code_integer ) @ K2 ) ) ) ).
% pochhammer_absorb_comp
thf(fact_7426_real__less__lsqrt,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X3 @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ ( sqrt @ X3 ) @ Y ) ) ) ) ).
% real_less_lsqrt
thf(fact_7427_sqrt__sum__squares__le__sum,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X3 @ Y ) ) ) ) ).
% sqrt_sum_squares_le_sum
thf(fact_7428_real__sqrt__ge__abs1,axiom,
! [X3: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% real_sqrt_ge_abs1
thf(fact_7429_real__sqrt__ge__abs2,axiom,
! [Y: real,X3: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% real_sqrt_ge_abs2
thf(fact_7430_sqrt__sum__squares__le__sum__abs,axiom,
! [X3: real,Y: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X3 ) @ ( abs_abs_real @ Y ) ) ) ).
% sqrt_sum_squares_le_sum_abs
thf(fact_7431_ln__sqrt,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ln_ln_real @ ( sqrt @ X3 ) )
= ( divide_divide_real @ ( ln_ln_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% ln_sqrt
thf(fact_7432_arsinh__real__def,axiom,
( arsinh_real
= ( ^ [X5: real] : ( ln_ln_real @ ( plus_plus_real @ X5 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% arsinh_real_def
thf(fact_7433_complex__norm,axiom,
! [X3: real,Y: real] :
( ( real_V1022390504157884413omplex @ ( complex2 @ X3 @ Y ) )
= ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% complex_norm
thf(fact_7434_exp__bound__half,axiom,
! [Z2: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% exp_bound_half
thf(fact_7435_exp__bound__half,axiom,
! [Z2: complex] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% exp_bound_half
thf(fact_7436_pochhammer__minus,axiom,
! [B: rat,K2: nat] :
( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K2 )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K2 ) ) @ one_one_rat ) @ K2 ) ) ) ).
% pochhammer_minus
thf(fact_7437_pochhammer__minus,axiom,
! [B: int,K2: nat] :
( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K2 )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K2 ) ) @ one_one_int ) @ K2 ) ) ) ).
% pochhammer_minus
thf(fact_7438_pochhammer__minus,axiom,
! [B: real,K2: nat] :
( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K2 )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K2 ) ) @ one_one_real ) @ K2 ) ) ) ).
% pochhammer_minus
thf(fact_7439_pochhammer__minus,axiom,
! [B: complex,K2: nat] :
( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K2 )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K2 ) ) @ one_one_complex ) @ K2 ) ) ) ).
% pochhammer_minus
thf(fact_7440_pochhammer__minus,axiom,
! [B: code_integer,K2: nat] :
( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K2 )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K2 ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K2 ) ) @ one_one_Code_integer ) @ K2 ) ) ) ).
% pochhammer_minus
thf(fact_7441_pochhammer__minus_H,axiom,
! [B: rat,K2: nat] :
( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K2 ) ) @ one_one_rat ) @ K2 )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K2 ) ) ) ).
% pochhammer_minus'
thf(fact_7442_pochhammer__minus_H,axiom,
! [B: int,K2: nat] :
( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K2 ) ) @ one_one_int ) @ K2 )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K2 ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K2 ) ) ) ).
% pochhammer_minus'
thf(fact_7443_pochhammer__minus_H,axiom,
! [B: real,K2: nat] :
( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K2 ) ) @ one_one_real ) @ K2 )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K2 ) ) ) ).
% pochhammer_minus'
thf(fact_7444_pochhammer__minus_H,axiom,
! [B: complex,K2: nat] :
( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K2 ) ) @ one_one_complex ) @ K2 )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K2 ) ) ) ).
% pochhammer_minus'
thf(fact_7445_pochhammer__minus_H,axiom,
! [B: code_integer,K2: nat] :
( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K2 ) ) @ one_one_Code_integer ) @ K2 )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K2 ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K2 ) ) ) ).
% pochhammer_minus'
thf(fact_7446_finite__psubset__def,axiom,
( finite_psubset_nat
= ( collec6662362479098859352et_nat
@ ( produc6247414631856714078_nat_o
@ ^ [A6: set_nat,B6: set_nat] :
( ( ord_less_set_nat @ A6 @ B6 )
& ( finite_finite_nat @ B6 ) ) ) ) ) ).
% finite_psubset_def
thf(fact_7447_finite__psubset__def,axiom,
( finite_psubset_int
= ( collec957716948307931664et_int
@ ( produc4109468873575309990_int_o
@ ^ [A6: set_int,B6: set_int] :
( ( ord_less_set_int @ A6 @ B6 )
& ( finite_finite_int @ B6 ) ) ) ) ) ).
% finite_psubset_def
thf(fact_7448_finite__psubset__def,axiom,
( finite8643634255014194347omplex
= ( collec5108298041176329748omplex
@ ( produc3914248068834153634plex_o
@ ^ [A6: set_complex,B6: set_complex] :
( ( ord_less_set_complex @ A6 @ B6 )
& ( finite3207457112153483333omplex @ B6 ) ) ) ) ) ).
% finite_psubset_def
thf(fact_7449_finite__psubset__def,axiom,
( finite469560695537375940at_nat
= ( collec6321179662152712658at_nat
@ ( produc410239310623530412_nat_o
@ ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ A6 @ B6 )
& ( finite6177210948735845034at_nat @ B6 ) ) ) ) ) ).
% finite_psubset_def
thf(fact_7450_arsinh__real__aux,axiom,
! [X3: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% arsinh_real_aux
thf(fact_7451_exp__bound,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ one_one_real )
=> ( ord_less_eq_real @ ( exp_real @ X3 ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% exp_bound
thf(fact_7452_real__sqrt__sum__squares__mult__ge__zero,axiom,
! [X3: real,Y: real,Xa2: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% real_sqrt_sum_squares_mult_ge_zero
thf(fact_7453_real__sqrt__power__even,axiom,
! [N: nat,X3: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( power_power_real @ ( sqrt @ X3 ) @ N )
= ( power_power_real @ X3 @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% real_sqrt_power_even
thf(fact_7454_arith__geo__mean__sqrt,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X3 @ Y ) ) @ ( divide_divide_real @ ( plus_plus_real @ X3 @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% arith_geo_mean_sqrt
thf(fact_7455_real__exp__bound__lemma,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( exp_real @ X3 ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) ) ) ) ).
% real_exp_bound_lemma
thf(fact_7456_cos__x__y__le__one,axiom,
! [X3: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).
% cos_x_y_le_one
thf(fact_7457_real__sqrt__sum__squares__less,axiom,
! [X3: real,U: real,Y: real] :
( ( ord_less_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( ( ord_less_real @ ( abs_abs_real @ Y ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).
% real_sqrt_sum_squares_less
thf(fact_7458_arcosh__real__def,axiom,
! [X3: real] :
( ( ord_less_eq_real @ one_one_real @ X3 )
=> ( ( arcosh_real @ X3 )
= ( ln_ln_real @ ( plus_plus_real @ X3 @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% arcosh_real_def
thf(fact_7459_exp__ge__one__plus__x__over__n__power__n,axiom,
! [N: nat,X3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X3 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X3 ) ) ) ) ).
% exp_ge_one_plus_x_over_n_power_n
thf(fact_7460_exp__ge__one__minus__x__over__n__power__n,axiom,
! [X3: real,N: nat] :
( ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X3 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) ) ) ).
% exp_ge_one_minus_x_over_n_power_n
thf(fact_7461_cos__arctan,axiom,
! [X3: real] :
( ( cos_real @ ( arctan @ X3 ) )
= ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% cos_arctan
thf(fact_7462_sin__arctan,axiom,
! [X3: real] :
( ( sin_real @ ( arctan @ X3 ) )
= ( divide_divide_real @ X3 @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% sin_arctan
thf(fact_7463_pochhammer__code,axiom,
( comm_s4028243227959126397er_rat
= ( ^ [A5: rat,N4: nat] :
( if_rat @ ( N4 = zero_zero_nat ) @ one_one_rat
@ ( set_fo1949268297981939178at_rat
@ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A5 @ ( semiri681578069525770553at_rat @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N4 @ one_one_nat )
@ one_one_rat ) ) ) ) ).
% pochhammer_code
thf(fact_7464_pochhammer__code,axiom,
( comm_s4660882817536571857er_int
= ( ^ [A5: int,N4: nat] :
( if_int @ ( N4 = zero_zero_nat ) @ one_one_int
@ ( set_fo2581907887559384638at_int
@ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A5 @ ( semiri1314217659103216013at_int @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N4 @ one_one_nat )
@ one_one_int ) ) ) ) ).
% pochhammer_code
thf(fact_7465_pochhammer__code,axiom,
( comm_s7457072308508201937r_real
= ( ^ [A5: real,N4: nat] :
( if_real @ ( N4 = zero_zero_nat ) @ one_one_real
@ ( set_fo3111899725591712190t_real
@ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A5 @ ( semiri5074537144036343181t_real @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N4 @ one_one_nat )
@ one_one_real ) ) ) ) ).
% pochhammer_code
thf(fact_7466_pochhammer__code,axiom,
( comm_s2602460028002588243omplex
= ( ^ [A5: complex,N4: nat] :
( if_complex @ ( N4 = zero_zero_nat ) @ one_one_complex
@ ( set_fo1517530859248394432omplex
@ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A5 @ ( semiri8010041392384452111omplex @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N4 @ one_one_nat )
@ one_one_complex ) ) ) ) ).
% pochhammer_code
thf(fact_7467_pochhammer__code,axiom,
( comm_s8582702949713902594nteger
= ( ^ [A5: code_integer,N4: nat] :
( if_Code_integer @ ( N4 = zero_zero_nat ) @ one_one_Code_integer
@ ( set_fo1084959871951514735nteger
@ ^ [O: nat] : ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A5 @ ( semiri4939895301339042750nteger @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N4 @ one_one_nat )
@ one_one_Code_integer ) ) ) ) ).
% pochhammer_code
thf(fact_7468_pochhammer__code,axiom,
( comm_s4663373288045622133er_nat
= ( ^ [A5: nat,N4: nat] :
( if_nat @ ( N4 = zero_zero_nat ) @ one_one_nat
@ ( set_fo2584398358068434914at_nat
@ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A5 @ ( semiri1316708129612266289at_nat @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N4 @ one_one_nat )
@ one_one_nat ) ) ) ) ).
% pochhammer_code
thf(fact_7469_exp__bound__lemma,axiom,
! [Z2: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z2 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V7735802525324610683m_real @ Z2 ) ) ) ) ) ).
% exp_bound_lemma
thf(fact_7470_exp__bound__lemma,axiom,
! [Z2: complex] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z2 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ) ).
% exp_bound_lemma
thf(fact_7471_Maclaurin__exp__le,axiom,
! [X3: real,N: nat] :
? [T5: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X3 ) )
& ( ( exp_real @ X3 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( divide_divide_real @ ( power_power_real @ X3 @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ).
% Maclaurin_exp_le
thf(fact_7472_sqrt__sum__squares__half__less,axiom,
! [X3: real,U: real,Y: real] :
( ( ord_less_real @ X3 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ Y @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).
% sqrt_sum_squares_half_less
thf(fact_7473_exp__lower__Taylor__quadratic,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( divide_divide_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X3 ) ) ) ).
% exp_lower_Taylor_quadratic
thf(fact_7474_sin__cos__sqrt,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X3 ) )
=> ( ( sin_real @ X3 )
= ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% sin_cos_sqrt
thf(fact_7475_arctan__half,axiom,
( arctan
= ( ^ [X5: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X5 @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% arctan_half
thf(fact_7476_tanh__real__altdef,axiom,
( tanh_real
= ( ^ [X5: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X5 ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X5 ) ) ) ) ) ) ).
% tanh_real_altdef
thf(fact_7477_pochhammer__times__pochhammer__half,axiom,
! [Z2: rat,N: nat] :
( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ ( suc @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
= ( groups73079841787564623at_rat
@ ^ [K3: nat] : ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_7478_pochhammer__times__pochhammer__half,axiom,
! [Z2: real,N: nat] :
( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ ( suc @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
= ( groups129246275422532515t_real
@ ^ [K3: nat] : ( plus_plus_real @ Z2 @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_7479_pochhammer__times__pochhammer__half,axiom,
! [Z2: complex,N: nat] :
( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z2 @ ( suc @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
= ( groups6464643781859351333omplex
@ ^ [K3: nat] : ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_7480_cos__arcsin,axiom,
! [X3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
=> ( ( ord_less_eq_real @ X3 @ one_one_real )
=> ( ( cos_real @ ( arcsin @ X3 ) )
= ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% cos_arcsin
thf(fact_7481_sin__arccos__abs,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( sin_real @ ( arccos @ Y ) )
= ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% sin_arccos_abs
thf(fact_7482_sin__arccos,axiom,
! [X3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
=> ( ( ord_less_eq_real @ X3 @ one_one_real )
=> ( ( sin_real @ ( arccos @ X3 ) )
= ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% sin_arccos
thf(fact_7483_or__int__unfold,axiom,
( bit_se1409905431419307370or_int
= ( ^ [K3: int,L2: int] :
( if_int
@ ( ( K3
= ( uminus_uminus_int @ one_one_int ) )
| ( L2
= ( uminus_uminus_int @ one_one_int ) ) )
@ ( uminus_uminus_int @ one_one_int )
@ ( if_int @ ( K3 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K3 @ ( 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 @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% or_int_unfold
thf(fact_7484_gchoose__row__sum__weighted,axiom,
! [R2: rat,M2: nat] :
( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ R2 @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ R2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M2 ) )
= ( times_times_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ ( suc @ M2 ) ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ R2 @ ( suc @ M2 ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_7485_gchoose__row__sum__weighted,axiom,
! [R2: complex,M2: nat] :
( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ R2 @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ R2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M2 ) )
= ( times_times_complex @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ ( suc @ M2 ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ R2 @ ( suc @ M2 ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_7486_gchoose__row__sum__weighted,axiom,
! [R2: real,M2: nat] :
( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ R2 @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ R2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M2 ) )
= ( times_times_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( suc @ M2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ R2 @ ( suc @ M2 ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_7487_or_Oleft__neutral,axiom,
! [A: int] :
( ( bit_se1409905431419307370or_int @ zero_zero_int @ A )
= A ) ).
% or.left_neutral
thf(fact_7488_or_Oleft__neutral,axiom,
! [A: nat] :
( ( bit_se1412395901928357646or_nat @ zero_zero_nat @ A )
= A ) ).
% or.left_neutral
thf(fact_7489_or_Oright__neutral,axiom,
! [A: int] :
( ( bit_se1409905431419307370or_int @ A @ zero_zero_int )
= A ) ).
% or.right_neutral
thf(fact_7490_or_Oright__neutral,axiom,
! [A: nat] :
( ( bit_se1412395901928357646or_nat @ A @ zero_zero_nat )
= A ) ).
% or.right_neutral
thf(fact_7491_arcsin__0,axiom,
( ( arcsin @ zero_zero_real )
= zero_zero_real ) ).
% arcsin_0
thf(fact_7492_of__nat__prod,axiom,
! [F: int > nat,A4: set_int] :
( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F @ A4 ) )
= ( groups1705073143266064639nt_int
@ ^ [X5: int] : ( semiri1314217659103216013at_int @ ( F @ X5 ) )
@ A4 ) ) ).
% of_nat_prod
thf(fact_7493_of__nat__prod,axiom,
! [F: nat > nat,A4: set_nat] :
( ( semiri5074537144036343181t_real @ ( groups708209901874060359at_nat @ F @ A4 ) )
= ( groups129246275422532515t_real
@ ^ [X5: nat] : ( semiri5074537144036343181t_real @ ( F @ X5 ) )
@ A4 ) ) ).
% of_nat_prod
thf(fact_7494_of__nat__prod,axiom,
! [F: nat > nat,A4: set_nat] :
( ( semiri8010041392384452111omplex @ ( groups708209901874060359at_nat @ F @ A4 ) )
= ( groups6464643781859351333omplex
@ ^ [X5: nat] : ( semiri8010041392384452111omplex @ ( F @ X5 ) )
@ A4 ) ) ).
% of_nat_prod
thf(fact_7495_of__nat__prod,axiom,
! [F: nat > nat,A4: set_nat] :
( ( semiri4939895301339042750nteger @ ( groups708209901874060359at_nat @ F @ A4 ) )
= ( groups3455450783089532116nteger
@ ^ [X5: nat] : ( semiri4939895301339042750nteger @ ( F @ X5 ) )
@ A4 ) ) ).
% of_nat_prod
thf(fact_7496_of__nat__prod,axiom,
! [F: nat > nat,A4: set_nat] :
( ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ F @ A4 ) )
= ( groups708209901874060359at_nat
@ ^ [X5: nat] : ( semiri1316708129612266289at_nat @ ( F @ X5 ) )
@ A4 ) ) ).
% of_nat_prod
thf(fact_7497_of__nat__prod,axiom,
! [F: nat > nat,A4: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F @ A4 ) )
= ( groups705719431365010083at_int
@ ^ [X5: nat] : ( semiri1314217659103216013at_int @ ( F @ X5 ) )
@ A4 ) ) ).
% of_nat_prod
thf(fact_7498_of__int__prod,axiom,
! [F: nat > int,A4: set_nat] :
( ( ring_1_of_int_real @ ( groups705719431365010083at_int @ F @ A4 ) )
= ( groups129246275422532515t_real
@ ^ [X5: nat] : ( ring_1_of_int_real @ ( F @ X5 ) )
@ A4 ) ) ).
% of_int_prod
thf(fact_7499_of__int__prod,axiom,
! [F: nat > int,A4: set_nat] :
( ( ring_1_of_int_rat @ ( groups705719431365010083at_int @ F @ A4 ) )
= ( groups73079841787564623at_rat
@ ^ [X5: nat] : ( ring_1_of_int_rat @ ( F @ X5 ) )
@ A4 ) ) ).
% of_int_prod
thf(fact_7500_of__int__prod,axiom,
! [F: nat > int,A4: set_nat] :
( ( ring_17405671764205052669omplex @ ( groups705719431365010083at_int @ F @ A4 ) )
= ( groups6464643781859351333omplex
@ ^ [X5: nat] : ( ring_17405671764205052669omplex @ ( F @ X5 ) )
@ A4 ) ) ).
% of_int_prod
thf(fact_7501_of__int__prod,axiom,
! [F: nat > int,A4: set_nat] :
( ( ring_1_of_int_int @ ( groups705719431365010083at_int @ F @ A4 ) )
= ( groups705719431365010083at_int
@ ^ [X5: nat] : ( ring_1_of_int_int @ ( F @ X5 ) )
@ A4 ) ) ).
% of_int_prod
thf(fact_7502_of__int__prod,axiom,
! [F: int > int,A4: set_int] :
( ( ring_1_of_int_real @ ( groups1705073143266064639nt_int @ F @ A4 ) )
= ( groups2316167850115554303t_real
@ ^ [X5: int] : ( ring_1_of_int_real @ ( F @ X5 ) )
@ A4 ) ) ).
% of_int_prod
thf(fact_7503_of__int__prod,axiom,
! [F: int > int,A4: set_int] :
( ( ring_1_of_int_rat @ ( groups1705073143266064639nt_int @ F @ A4 ) )
= ( groups1072433553688619179nt_rat
@ ^ [X5: int] : ( ring_1_of_int_rat @ ( F @ X5 ) )
@ A4 ) ) ).
% of_int_prod
thf(fact_7504_of__int__prod,axiom,
! [F: int > int,A4: set_int] :
( ( ring_17405671764205052669omplex @ ( groups1705073143266064639nt_int @ F @ A4 ) )
= ( groups7440179247065528705omplex
@ ^ [X5: int] : ( ring_17405671764205052669omplex @ ( F @ X5 ) )
@ A4 ) ) ).
% of_int_prod
thf(fact_7505_of__int__prod,axiom,
! [F: int > int,A4: set_int] :
( ( ring_1_of_int_int @ ( groups1705073143266064639nt_int @ F @ A4 ) )
= ( groups1705073143266064639nt_int
@ ^ [X5: int] : ( ring_1_of_int_int @ ( F @ X5 ) )
@ A4 ) ) ).
% of_int_prod
thf(fact_7506_prod__zero__iff,axiom,
! [A4: set_nat,F: nat > complex] :
( ( finite_finite_nat @ A4 )
=> ( ( ( groups6464643781859351333omplex @ F @ A4 )
= zero_zero_complex )
= ( ? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_complex ) ) ) ) ) ).
% prod_zero_iff
thf(fact_7507_prod__zero__iff,axiom,
! [A4: set_int,F: int > complex] :
( ( finite_finite_int @ A4 )
=> ( ( ( groups7440179247065528705omplex @ F @ A4 )
= zero_zero_complex )
= ( ? [X5: int] :
( ( member_int @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_complex ) ) ) ) ) ).
% prod_zero_iff
thf(fact_7508_prod__zero__iff,axiom,
! [A4: set_complex,F: complex > complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( groups3708469109370488835omplex @ F @ A4 )
= zero_zero_complex )
= ( ? [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_complex ) ) ) ) ) ).
% prod_zero_iff
thf(fact_7509_prod__zero__iff,axiom,
! [A4: set_nat,F: nat > real] :
( ( finite_finite_nat @ A4 )
=> ( ( ( groups129246275422532515t_real @ F @ A4 )
= zero_zero_real )
= ( ? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_real ) ) ) ) ) ).
% prod_zero_iff
thf(fact_7510_prod__zero__iff,axiom,
! [A4: set_int,F: int > real] :
( ( finite_finite_int @ A4 )
=> ( ( ( groups2316167850115554303t_real @ F @ A4 )
= zero_zero_real )
= ( ? [X5: int] :
( ( member_int @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_real ) ) ) ) ) ).
% prod_zero_iff
thf(fact_7511_prod__zero__iff,axiom,
! [A4: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( groups766887009212190081x_real @ F @ A4 )
= zero_zero_real )
= ( ? [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_real ) ) ) ) ) ).
% prod_zero_iff
thf(fact_7512_prod__zero__iff,axiom,
! [A4: set_nat,F: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( groups73079841787564623at_rat @ F @ A4 )
= zero_zero_rat )
= ( ? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_rat ) ) ) ) ) ).
% prod_zero_iff
thf(fact_7513_prod__zero__iff,axiom,
! [A4: set_int,F: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ( ( groups1072433553688619179nt_rat @ F @ A4 )
= zero_zero_rat )
= ( ? [X5: int] :
( ( member_int @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_rat ) ) ) ) ) ).
% prod_zero_iff
thf(fact_7514_prod__zero__iff,axiom,
! [A4: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( groups225925009352817453ex_rat @ F @ A4 )
= zero_zero_rat )
= ( ? [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_rat ) ) ) ) ) ).
% prod_zero_iff
thf(fact_7515_prod__zero__iff,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( ( groups1707563613775114915nt_nat @ F @ A4 )
= zero_zero_nat )
= ( ? [X5: int] :
( ( member_int @ X5 @ A4 )
& ( ( F @ X5 )
= zero_zero_nat ) ) ) ) ) ).
% prod_zero_iff
thf(fact_7516_prod_Oinfinite,axiom,
! [A4: set_nat,G: nat > complex] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( groups6464643781859351333omplex @ G @ A4 )
= one_one_complex ) ) ).
% prod.infinite
thf(fact_7517_prod_Oinfinite,axiom,
! [A4: set_int,G: int > complex] :
( ~ ( finite_finite_int @ A4 )
=> ( ( groups7440179247065528705omplex @ G @ A4 )
= one_one_complex ) ) ).
% prod.infinite
thf(fact_7518_prod_Oinfinite,axiom,
! [A4: set_complex,G: complex > complex] :
( ~ ( finite3207457112153483333omplex @ A4 )
=> ( ( groups3708469109370488835omplex @ G @ A4 )
= one_one_complex ) ) ).
% prod.infinite
thf(fact_7519_prod_Oinfinite,axiom,
! [A4: set_nat,G: nat > real] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( groups129246275422532515t_real @ G @ A4 )
= one_one_real ) ) ).
% prod.infinite
thf(fact_7520_prod_Oinfinite,axiom,
! [A4: set_int,G: int > real] :
( ~ ( finite_finite_int @ A4 )
=> ( ( groups2316167850115554303t_real @ G @ A4 )
= one_one_real ) ) ).
% prod.infinite
thf(fact_7521_prod_Oinfinite,axiom,
! [A4: set_complex,G: complex > real] :
( ~ ( finite3207457112153483333omplex @ A4 )
=> ( ( groups766887009212190081x_real @ G @ A4 )
= one_one_real ) ) ).
% prod.infinite
thf(fact_7522_prod_Oinfinite,axiom,
! [A4: set_nat,G: nat > rat] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( groups73079841787564623at_rat @ G @ A4 )
= one_one_rat ) ) ).
% prod.infinite
thf(fact_7523_prod_Oinfinite,axiom,
! [A4: set_int,G: int > rat] :
( ~ ( finite_finite_int @ A4 )
=> ( ( groups1072433553688619179nt_rat @ G @ A4 )
= one_one_rat ) ) ).
% prod.infinite
thf(fact_7524_prod_Oinfinite,axiom,
! [A4: set_complex,G: complex > rat] :
( ~ ( finite3207457112153483333omplex @ A4 )
=> ( ( groups225925009352817453ex_rat @ G @ A4 )
= one_one_rat ) ) ).
% prod.infinite
thf(fact_7525_prod_Oinfinite,axiom,
! [A4: set_int,G: int > nat] :
( ~ ( finite_finite_int @ A4 )
=> ( ( groups1707563613775114915nt_nat @ G @ A4 )
= one_one_nat ) ) ).
% prod.infinite
thf(fact_7526_dvd__prod__eqI,axiom,
! [A4: set_real,A: real,B: nat,F: real > nat] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A @ A4 )
=> ( ( B
= ( F @ A ) )
=> ( dvd_dvd_nat @ B @ ( groups4696554848551431203al_nat @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7527_dvd__prod__eqI,axiom,
! [A4: set_int,A: int,B: nat,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ A @ A4 )
=> ( ( B
= ( F @ A ) )
=> ( dvd_dvd_nat @ B @ ( groups1707563613775114915nt_nat @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7528_dvd__prod__eqI,axiom,
! [A4: set_complex,A: complex,B: nat,F: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( member_complex @ A @ A4 )
=> ( ( B
= ( F @ A ) )
=> ( dvd_dvd_nat @ B @ ( groups861055069439313189ex_nat @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7529_dvd__prod__eqI,axiom,
! [A4: set_real,A: real,B: int,F: real > int] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A @ A4 )
=> ( ( B
= ( F @ A ) )
=> ( dvd_dvd_int @ B @ ( groups4694064378042380927al_int @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7530_dvd__prod__eqI,axiom,
! [A4: set_complex,A: complex,B: int,F: complex > int] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( member_complex @ A @ A4 )
=> ( ( B
= ( F @ A ) )
=> ( dvd_dvd_int @ B @ ( groups858564598930262913ex_int @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7531_dvd__prod__eqI,axiom,
! [A4: set_real,A: real,B: code_integer,F: real > code_integer] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A @ A4 )
=> ( ( B
= ( F @ A ) )
=> ( dvd_dvd_Code_integer @ B @ ( groups6225526099057966256nteger @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7532_dvd__prod__eqI,axiom,
! [A4: set_nat,A: nat,B: code_integer,F: nat > code_integer] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ( ( B
= ( F @ A ) )
=> ( dvd_dvd_Code_integer @ B @ ( groups3455450783089532116nteger @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7533_dvd__prod__eqI,axiom,
! [A4: set_int,A: int,B: code_integer,F: int > code_integer] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ A @ A4 )
=> ( ( B
= ( F @ A ) )
=> ( dvd_dvd_Code_integer @ B @ ( groups3827104343326376752nteger @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7534_dvd__prod__eqI,axiom,
! [A4: set_complex,A: complex,B: code_integer,F: complex > code_integer] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( member_complex @ A @ A4 )
=> ( ( B
= ( F @ A ) )
=> ( dvd_dvd_Code_integer @ B @ ( groups8682486955453173170nteger @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7535_dvd__prod__eqI,axiom,
! [A4: set_nat,A: nat,B: nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ( ( B
= ( F @ A ) )
=> ( dvd_dvd_nat @ B @ ( groups708209901874060359at_nat @ F @ A4 ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_7536_dvd__prodI,axiom,
! [A4: set_real,A: real,F: real > nat] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A @ A4 )
=> ( dvd_dvd_nat @ ( F @ A ) @ ( groups4696554848551431203al_nat @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7537_dvd__prodI,axiom,
! [A4: set_int,A: int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ A @ A4 )
=> ( dvd_dvd_nat @ ( F @ A ) @ ( groups1707563613775114915nt_nat @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7538_dvd__prodI,axiom,
! [A4: set_complex,A: complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( member_complex @ A @ A4 )
=> ( dvd_dvd_nat @ ( F @ A ) @ ( groups861055069439313189ex_nat @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7539_dvd__prodI,axiom,
! [A4: set_real,A: real,F: real > int] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A @ A4 )
=> ( dvd_dvd_int @ ( F @ A ) @ ( groups4694064378042380927al_int @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7540_dvd__prodI,axiom,
! [A4: set_complex,A: complex,F: complex > int] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( member_complex @ A @ A4 )
=> ( dvd_dvd_int @ ( F @ A ) @ ( groups858564598930262913ex_int @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7541_dvd__prodI,axiom,
! [A4: set_real,A: real,F: real > code_integer] :
( ( finite_finite_real @ A4 )
=> ( ( member_real @ A @ A4 )
=> ( dvd_dvd_Code_integer @ ( F @ A ) @ ( groups6225526099057966256nteger @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7542_dvd__prodI,axiom,
! [A4: set_nat,A: nat,F: nat > code_integer] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ( dvd_dvd_Code_integer @ ( F @ A ) @ ( groups3455450783089532116nteger @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7543_dvd__prodI,axiom,
! [A4: set_int,A: int,F: int > code_integer] :
( ( finite_finite_int @ A4 )
=> ( ( member_int @ A @ A4 )
=> ( dvd_dvd_Code_integer @ ( F @ A ) @ ( groups3827104343326376752nteger @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7544_dvd__prodI,axiom,
! [A4: set_complex,A: complex,F: complex > code_integer] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( member_complex @ A @ A4 )
=> ( dvd_dvd_Code_integer @ ( F @ A ) @ ( groups8682486955453173170nteger @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7545_dvd__prodI,axiom,
! [A4: set_nat,A: nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ( dvd_dvd_nat @ ( F @ A ) @ ( groups708209901874060359at_nat @ F @ A4 ) ) ) ) ).
% dvd_prodI
thf(fact_7546_gbinomial__0_I2_J,axiom,
! [K2: nat] :
( ( gbinomial_complex @ zero_zero_complex @ ( suc @ K2 ) )
= zero_zero_complex ) ).
% gbinomial_0(2)
thf(fact_7547_gbinomial__0_I2_J,axiom,
! [K2: nat] :
( ( gbinomial_real @ zero_zero_real @ ( suc @ K2 ) )
= zero_zero_real ) ).
% gbinomial_0(2)
thf(fact_7548_gbinomial__0_I2_J,axiom,
! [K2: nat] :
( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K2 ) )
= zero_zero_rat ) ).
% gbinomial_0(2)
thf(fact_7549_gbinomial__0_I2_J,axiom,
! [K2: nat] :
( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K2 ) )
= zero_zero_nat ) ).
% gbinomial_0(2)
thf(fact_7550_gbinomial__0_I2_J,axiom,
! [K2: nat] :
( ( gbinomial_int @ zero_zero_int @ ( suc @ K2 ) )
= zero_zero_int ) ).
% gbinomial_0(2)
thf(fact_7551_gbinomial__0_I1_J,axiom,
! [A: complex] :
( ( gbinomial_complex @ A @ zero_zero_nat )
= one_one_complex ) ).
% gbinomial_0(1)
thf(fact_7552_gbinomial__0_I1_J,axiom,
! [A: real] :
( ( gbinomial_real @ A @ zero_zero_nat )
= one_one_real ) ).
% gbinomial_0(1)
thf(fact_7553_gbinomial__0_I1_J,axiom,
! [A: rat] :
( ( gbinomial_rat @ A @ zero_zero_nat )
= one_one_rat ) ).
% gbinomial_0(1)
thf(fact_7554_gbinomial__0_I1_J,axiom,
! [A: nat] :
( ( gbinomial_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% gbinomial_0(1)
thf(fact_7555_gbinomial__0_I1_J,axiom,
! [A: int] :
( ( gbinomial_int @ A @ zero_zero_nat )
= one_one_int ) ).
% gbinomial_0(1)
thf(fact_7556_arccos__1,axiom,
( ( arccos @ one_one_real )
= zero_zero_real ) ).
% arccos_1
thf(fact_7557_or__nonnegative__int__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K2 @ L ) )
= ( ( ord_less_eq_int @ zero_zero_int @ K2 )
& ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% or_nonnegative_int_iff
thf(fact_7558_or__negative__int__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K2 @ L ) @ zero_zero_int )
= ( ( ord_less_int @ K2 @ zero_zero_int )
| ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% or_negative_int_iff
thf(fact_7559_prod_Odelta_H,axiom,
! [S2: set_real,A: real,B: real > complex] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups713298508707869441omplex
@ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups713298508707869441omplex
@ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
@ S2 )
= one_one_complex ) ) ) ) ).
% prod.delta'
thf(fact_7560_prod_Odelta_H,axiom,
! [S2: set_nat,A: nat,B: nat > complex] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups6464643781859351333omplex
@ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups6464643781859351333omplex
@ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
@ S2 )
= one_one_complex ) ) ) ) ).
% prod.delta'
thf(fact_7561_prod_Odelta_H,axiom,
! [S2: set_int,A: int,B: int > complex] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups7440179247065528705omplex
@ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups7440179247065528705omplex
@ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
@ S2 )
= one_one_complex ) ) ) ) ).
% prod.delta'
thf(fact_7562_prod_Odelta_H,axiom,
! [S2: set_complex,A: complex,B: complex > complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups3708469109370488835omplex
@ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups3708469109370488835omplex
@ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
@ S2 )
= one_one_complex ) ) ) ) ).
% prod.delta'
thf(fact_7563_prod_Odelta_H,axiom,
! [S2: set_real,A: real,B: real > real] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups1681761925125756287l_real
@ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups1681761925125756287l_real
@ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= one_one_real ) ) ) ) ).
% prod.delta'
thf(fact_7564_prod_Odelta_H,axiom,
! [S2: set_nat,A: nat,B: nat > real] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups129246275422532515t_real
@ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups129246275422532515t_real
@ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= one_one_real ) ) ) ) ).
% prod.delta'
thf(fact_7565_prod_Odelta_H,axiom,
! [S2: set_int,A: int,B: int > real] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups2316167850115554303t_real
@ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups2316167850115554303t_real
@ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= one_one_real ) ) ) ) ).
% prod.delta'
thf(fact_7566_prod_Odelta_H,axiom,
! [S2: set_complex,A: complex,B: complex > real] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups766887009212190081x_real
@ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups766887009212190081x_real
@ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= one_one_real ) ) ) ) ).
% prod.delta'
thf(fact_7567_prod_Odelta_H,axiom,
! [S2: set_real,A: real,B: real > rat] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups4061424788464935467al_rat
@ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups4061424788464935467al_rat
@ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
@ S2 )
= one_one_rat ) ) ) ) ).
% prod.delta'
thf(fact_7568_prod_Odelta_H,axiom,
! [S2: set_nat,A: nat,B: nat > rat] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups73079841787564623at_rat
@ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups73079841787564623at_rat
@ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
@ S2 )
= one_one_rat ) ) ) ) ).
% prod.delta'
thf(fact_7569_prod_Odelta,axiom,
! [S2: set_real,A: real,B: real > complex] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups713298508707869441omplex
@ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups713298508707869441omplex
@ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
@ S2 )
= one_one_complex ) ) ) ) ).
% prod.delta
thf(fact_7570_prod_Odelta,axiom,
! [S2: set_nat,A: nat,B: nat > complex] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups6464643781859351333omplex
@ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups6464643781859351333omplex
@ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
@ S2 )
= one_one_complex ) ) ) ) ).
% prod.delta
thf(fact_7571_prod_Odelta,axiom,
! [S2: set_int,A: int,B: int > complex] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups7440179247065528705omplex
@ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups7440179247065528705omplex
@ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
@ S2 )
= one_one_complex ) ) ) ) ).
% prod.delta
thf(fact_7572_prod_Odelta,axiom,
! [S2: set_complex,A: complex,B: complex > complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups3708469109370488835omplex
@ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups3708469109370488835omplex
@ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
@ S2 )
= one_one_complex ) ) ) ) ).
% prod.delta
thf(fact_7573_prod_Odelta,axiom,
! [S2: set_real,A: real,B: real > real] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups1681761925125756287l_real
@ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups1681761925125756287l_real
@ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= one_one_real ) ) ) ) ).
% prod.delta
thf(fact_7574_prod_Odelta,axiom,
! [S2: set_nat,A: nat,B: nat > real] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups129246275422532515t_real
@ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups129246275422532515t_real
@ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= one_one_real ) ) ) ) ).
% prod.delta
thf(fact_7575_prod_Odelta,axiom,
! [S2: set_int,A: int,B: int > real] :
( ( finite_finite_int @ S2 )
=> ( ( ( member_int @ A @ S2 )
=> ( ( groups2316167850115554303t_real
@ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S2 )
=> ( ( groups2316167850115554303t_real
@ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= one_one_real ) ) ) ) ).
% prod.delta
thf(fact_7576_prod_Odelta,axiom,
! [S2: set_complex,A: complex,B: complex > real] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( ( member_complex @ A @ S2 )
=> ( ( groups766887009212190081x_real
@ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S2 )
=> ( ( groups766887009212190081x_real
@ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
@ S2 )
= one_one_real ) ) ) ) ).
% prod.delta
thf(fact_7577_prod_Odelta,axiom,
! [S2: set_real,A: real,B: real > rat] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups4061424788464935467al_rat
@ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups4061424788464935467al_rat
@ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
@ S2 )
= one_one_rat ) ) ) ) ).
% prod.delta
thf(fact_7578_prod_Odelta,axiom,
! [S2: set_nat,A: nat,B: nat > rat] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups73079841787564623at_rat
@ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups73079841787564623at_rat
@ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
@ S2 )
= one_one_rat ) ) ) ) ).
% prod.delta
thf(fact_7579_prod_OlessThan__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% prod.lessThan_Suc
thf(fact_7580_prod_OlessThan__Suc,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% prod.lessThan_Suc
thf(fact_7581_prod_OlessThan__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% prod.lessThan_Suc
thf(fact_7582_prod_OlessThan__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% prod.lessThan_Suc
thf(fact_7583_cos__arccos,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( cos_real @ ( arccos @ Y ) )
= Y ) ) ) ).
% cos_arccos
thf(fact_7584_sin__arcsin,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( sin_real @ ( arcsin @ Y ) )
= Y ) ) ) ).
% sin_arcsin
thf(fact_7585_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > complex] :
( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= one_one_complex ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_7586_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > real] :
( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= one_one_real ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_7587_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > rat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= one_one_rat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_7588_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > nat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= one_one_nat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_7589_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M2: nat,G: nat > int] :
( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= one_one_int ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_7590_arccos__0,axiom,
( ( arccos @ zero_zero_real )
= ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% arccos_0
thf(fact_7591_or__numerals_I7_J,axiom,
! [X3: num,Y: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X3 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X3 ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% or_numerals(7)
thf(fact_7592_or__numerals_I7_J,axiom,
! [X3: num,Y: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X3 ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% or_numerals(7)
thf(fact_7593_or__numerals_I6_J,axiom,
! [X3: num,Y: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X3 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X3 ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% or_numerals(6)
thf(fact_7594_or__numerals_I6_J,axiom,
! [X3: num,Y: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X3 ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% or_numerals(6)
thf(fact_7595_or__numerals_I4_J,axiom,
! [X3: num,Y: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X3 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X3 ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).
% or_numerals(4)
thf(fact_7596_or__numerals_I4_J,axiom,
! [X3: num,Y: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X3 ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% or_numerals(4)
thf(fact_7597_or__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( bit_se1409905431419307370or_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( B = zero_zero_int ) ) ) ).
% or_eq_0_iff
thf(fact_7598_or__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( bit_se1412395901928357646or_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% or_eq_0_iff
thf(fact_7599_bit_Odisj__zero__right,axiom,
! [X3: int] :
( ( bit_se1409905431419307370or_int @ X3 @ zero_zero_int )
= X3 ) ).
% bit.disj_zero_right
thf(fact_7600_prod_Oswap__restrict,axiom,
! [A4: set_real,B5: set_nat,G: real > nat > nat,R: real > nat > $o] :
( ( finite_finite_real @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups4696554848551431203al_nat
@ ^ [X5: real] :
( groups708209901874060359at_nat @ ( G @ X5 )
@ ( collect_nat
@ ^ [Y5: nat] :
( ( member_nat @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups708209901874060359at_nat
@ ^ [Y5: nat] :
( groups4696554848551431203al_nat
@ ^ [X5: real] : ( G @ X5 @ Y5 )
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7601_prod_Oswap__restrict,axiom,
! [A4: set_int,B5: set_nat,G: int > nat > nat,R: int > nat > $o] :
( ( finite_finite_int @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups1707563613775114915nt_nat
@ ^ [X5: int] :
( groups708209901874060359at_nat @ ( G @ X5 )
@ ( collect_nat
@ ^ [Y5: nat] :
( ( member_nat @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups708209901874060359at_nat
@ ^ [Y5: nat] :
( groups1707563613775114915nt_nat
@ ^ [X5: int] : ( G @ X5 @ Y5 )
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7602_prod_Oswap__restrict,axiom,
! [A4: set_complex,B5: set_nat,G: complex > nat > nat,R: complex > nat > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups861055069439313189ex_nat
@ ^ [X5: complex] :
( groups708209901874060359at_nat @ ( G @ X5 )
@ ( collect_nat
@ ^ [Y5: nat] :
( ( member_nat @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups708209901874060359at_nat
@ ^ [Y5: nat] :
( groups861055069439313189ex_nat
@ ^ [X5: complex] : ( G @ X5 @ Y5 )
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7603_prod_Oswap__restrict,axiom,
! [A4: set_real,B5: set_nat,G: real > nat > int,R: real > nat > $o] :
( ( finite_finite_real @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups4694064378042380927al_int
@ ^ [X5: real] :
( groups705719431365010083at_int @ ( G @ X5 )
@ ( collect_nat
@ ^ [Y5: nat] :
( ( member_nat @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups705719431365010083at_int
@ ^ [Y5: nat] :
( groups4694064378042380927al_int
@ ^ [X5: real] : ( G @ X5 @ Y5 )
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7604_prod_Oswap__restrict,axiom,
! [A4: set_complex,B5: set_nat,G: complex > nat > int,R: complex > nat > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( groups858564598930262913ex_int
@ ^ [X5: complex] :
( groups705719431365010083at_int @ ( G @ X5 )
@ ( collect_nat
@ ^ [Y5: nat] :
( ( member_nat @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups705719431365010083at_int
@ ^ [Y5: nat] :
( groups858564598930262913ex_int
@ ^ [X5: complex] : ( G @ X5 @ Y5 )
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7605_prod_Oswap__restrict,axiom,
! [A4: set_real,B5: set_int,G: real > int > int,R: real > int > $o] :
( ( finite_finite_real @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ( groups4694064378042380927al_int
@ ^ [X5: real] :
( groups1705073143266064639nt_int @ ( G @ X5 )
@ ( collect_int
@ ^ [Y5: int] :
( ( member_int @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups1705073143266064639nt_int
@ ^ [Y5: int] :
( groups4694064378042380927al_int
@ ^ [X5: real] : ( G @ X5 @ Y5 )
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7606_prod_Oswap__restrict,axiom,
! [A4: set_complex,B5: set_int,G: complex > int > int,R: complex > int > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ( groups858564598930262913ex_int
@ ^ [X5: complex] :
( groups1705073143266064639nt_int @ ( G @ X5 )
@ ( collect_int
@ ^ [Y5: int] :
( ( member_int @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups1705073143266064639nt_int
@ ^ [Y5: int] :
( groups858564598930262913ex_int
@ ^ [X5: complex] : ( G @ X5 @ Y5 )
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7607_prod_Oswap__restrict,axiom,
! [A4: set_nat,B5: set_real,G: nat > real > nat,R: nat > real > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_real @ B5 )
=> ( ( groups708209901874060359at_nat
@ ^ [X5: nat] :
( groups4696554848551431203al_nat @ ( G @ X5 )
@ ( collect_real
@ ^ [Y5: real] :
( ( member_real @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups4696554848551431203al_nat
@ ^ [Y5: real] :
( groups708209901874060359at_nat
@ ^ [X5: nat] : ( G @ X5 @ Y5 )
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7608_prod_Oswap__restrict,axiom,
! [A4: set_nat,B5: set_int,G: nat > int > nat,R: nat > int > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ( groups708209901874060359at_nat
@ ^ [X5: nat] :
( groups1707563613775114915nt_nat @ ( G @ X5 )
@ ( collect_int
@ ^ [Y5: int] :
( ( member_int @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups1707563613775114915nt_nat
@ ^ [Y5: int] :
( groups708209901874060359at_nat
@ ^ [X5: nat] : ( G @ X5 @ Y5 )
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7609_prod_Oswap__restrict,axiom,
! [A4: set_nat,B5: set_complex,G: nat > complex > nat,R: nat > complex > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( ( groups708209901874060359at_nat
@ ^ [X5: nat] :
( groups861055069439313189ex_nat @ ( G @ X5 )
@ ( collect_complex
@ ^ [Y5: complex] :
( ( member_complex @ Y5 @ B5 )
& ( R @ X5 @ Y5 ) ) ) )
@ A4 )
= ( groups861055069439313189ex_nat
@ ^ [Y5: complex] :
( groups708209901874060359at_nat
@ ^ [X5: nat] : ( G @ X5 @ Y5 )
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( R @ X5 @ Y5 ) ) ) )
@ B5 ) ) ) ) ).
% prod.swap_restrict
thf(fact_7610_prod__nonneg,axiom,
! [A4: set_nat,F: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A4 ) ) ) ).
% prod_nonneg
thf(fact_7611_prod__nonneg,axiom,
! [A4: set_nat,F: nat > int] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A4 ) ) ) ).
% prod_nonneg
thf(fact_7612_prod__nonneg,axiom,
! [A4: set_int,F: int > int] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A4 ) ) ) ).
% prod_nonneg
thf(fact_7613_prod__mono,axiom,
! [A4: set_complex,F: complex > real,G: complex > real] :
( ! [I2: complex] :
( ( member_complex @ I2 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
& ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A4 ) @ ( groups766887009212190081x_real @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_7614_prod__mono,axiom,
! [A4: set_real,F: real > real,G: real > real] :
( ! [I2: real] :
( ( member_real @ I2 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
& ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ ( groups1681761925125756287l_real @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_7615_prod__mono,axiom,
! [A4: set_nat,F: nat > real,G: nat > real] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
& ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A4 ) @ ( groups129246275422532515t_real @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_7616_prod__mono,axiom,
! [A4: set_int,F: int > real,G: int > real] :
( ! [I2: int] :
( ( member_int @ I2 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
& ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ ( groups2316167850115554303t_real @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_7617_prod__mono,axiom,
! [A4: set_complex,F: complex > rat,G: complex > rat] :
( ! [I2: complex] :
( ( member_complex @ I2 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
& ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) @ ( groups225925009352817453ex_rat @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_7618_prod__mono,axiom,
! [A4: set_real,F: real > rat,G: real > rat] :
( ! [I2: real] :
( ( member_real @ I2 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
& ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ ( groups4061424788464935467al_rat @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_7619_prod__mono,axiom,
! [A4: set_nat,F: nat > rat,G: nat > rat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
& ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ ( groups73079841787564623at_rat @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_7620_prod__mono,axiom,
! [A4: set_int,F: int > rat,G: int > rat] :
( ! [I2: int] :
( ( member_int @ I2 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
& ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ ( groups1072433553688619179nt_rat @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_7621_prod__mono,axiom,
! [A4: set_complex,F: complex > nat,G: complex > nat] :
( ! [I2: complex] :
( ( member_complex @ I2 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
& ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) @ ( groups861055069439313189ex_nat @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_7622_prod__mono,axiom,
! [A4: set_real,F: real > nat,G: real > nat] :
( ! [I2: real] :
( ( member_real @ I2 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
& ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) @ ( groups4696554848551431203al_nat @ G @ A4 ) ) ) ).
% prod_mono
thf(fact_7623_prod__pos,axiom,
! [A4: set_nat,F: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A4 ) ) ) ).
% prod_pos
thf(fact_7624_prod__pos,axiom,
! [A4: set_nat,F: nat > int] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A4 ) ) ) ).
% prod_pos
thf(fact_7625_prod__pos,axiom,
! [A4: set_int,F: int > int] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_int @ zero_zero_int @ ( F @ X4 ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A4 ) ) ) ).
% prod_pos
thf(fact_7626_prod__ge__1,axiom,
! [A4: set_complex,F: complex > real] :
( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups766887009212190081x_real @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_7627_prod__ge__1,axiom,
! [A4: set_real,F: real > real] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_7628_prod__ge__1,axiom,
! [A4: set_nat,F: nat > real] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_7629_prod__ge__1,axiom,
! [A4: set_int,F: int > real] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_7630_prod__ge__1,axiom,
! [A4: set_complex,F: complex > rat] :
( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_7631_prod__ge__1,axiom,
! [A4: set_real,F: real > rat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_7632_prod__ge__1,axiom,
! [A4: set_nat,F: nat > rat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_7633_prod__ge__1,axiom,
! [A4: set_int,F: int > rat] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_7634_prod__ge__1,axiom,
! [A4: set_complex,F: complex > nat] :
( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ord_less_eq_nat @ one_one_nat @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_7635_prod__ge__1,axiom,
! [A4: set_real,F: real > nat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_nat @ one_one_nat @ ( F @ X4 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_7636_prod__zero,axiom,
! [A4: set_nat,F: nat > complex] :
( ( finite_finite_nat @ A4 )
=> ( ? [X: nat] :
( ( member_nat @ X @ A4 )
& ( ( F @ X )
= zero_zero_complex ) )
=> ( ( groups6464643781859351333omplex @ F @ A4 )
= zero_zero_complex ) ) ) ).
% prod_zero
thf(fact_7637_prod__zero,axiom,
! [A4: set_int,F: int > complex] :
( ( finite_finite_int @ A4 )
=> ( ? [X: int] :
( ( member_int @ X @ A4 )
& ( ( F @ X )
= zero_zero_complex ) )
=> ( ( groups7440179247065528705omplex @ F @ A4 )
= zero_zero_complex ) ) ) ).
% prod_zero
thf(fact_7638_prod__zero,axiom,
! [A4: set_complex,F: complex > complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ? [X: complex] :
( ( member_complex @ X @ A4 )
& ( ( F @ X )
= zero_zero_complex ) )
=> ( ( groups3708469109370488835omplex @ F @ A4 )
= zero_zero_complex ) ) ) ).
% prod_zero
thf(fact_7639_prod__zero,axiom,
! [A4: set_nat,F: nat > real] :
( ( finite_finite_nat @ A4 )
=> ( ? [X: nat] :
( ( member_nat @ X @ A4 )
& ( ( F @ X )
= zero_zero_real ) )
=> ( ( groups129246275422532515t_real @ F @ A4 )
= zero_zero_real ) ) ) ).
% prod_zero
thf(fact_7640_prod__zero,axiom,
! [A4: set_int,F: int > real] :
( ( finite_finite_int @ A4 )
=> ( ? [X: int] :
( ( member_int @ X @ A4 )
& ( ( F @ X )
= zero_zero_real ) )
=> ( ( groups2316167850115554303t_real @ F @ A4 )
= zero_zero_real ) ) ) ).
% prod_zero
thf(fact_7641_prod__zero,axiom,
! [A4: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ? [X: complex] :
( ( member_complex @ X @ A4 )
& ( ( F @ X )
= zero_zero_real ) )
=> ( ( groups766887009212190081x_real @ F @ A4 )
= zero_zero_real ) ) ) ).
% prod_zero
thf(fact_7642_prod__zero,axiom,
! [A4: set_nat,F: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ? [X: nat] :
( ( member_nat @ X @ A4 )
& ( ( F @ X )
= zero_zero_rat ) )
=> ( ( groups73079841787564623at_rat @ F @ A4 )
= zero_zero_rat ) ) ) ).
% prod_zero
thf(fact_7643_prod__zero,axiom,
! [A4: set_int,F: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ? [X: int] :
( ( member_int @ X @ A4 )
& ( ( F @ X )
= zero_zero_rat ) )
=> ( ( groups1072433553688619179nt_rat @ F @ A4 )
= zero_zero_rat ) ) ) ).
% prod_zero
thf(fact_7644_prod__zero,axiom,
! [A4: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ? [X: complex] :
( ( member_complex @ X @ A4 )
& ( ( F @ X )
= zero_zero_rat ) )
=> ( ( groups225925009352817453ex_rat @ F @ A4 )
= zero_zero_rat ) ) ) ).
% prod_zero
thf(fact_7645_prod__zero,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ? [X: int] :
( ( member_int @ X @ A4 )
& ( ( F @ X )
= zero_zero_nat ) )
=> ( ( groups1707563613775114915nt_nat @ F @ A4 )
= zero_zero_nat ) ) ) ).
% prod_zero
thf(fact_7646_or__greater__eq,axiom,
! [L: int,K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ L )
=> ( ord_less_eq_int @ K2 @ ( bit_se1409905431419307370or_int @ K2 @ L ) ) ) ).
% or_greater_eq
thf(fact_7647_OR__lower,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X3 @ Y ) ) ) ) ).
% OR_lower
thf(fact_7648_prod_Ointer__filter,axiom,
! [A4: set_real,G: real > complex,P2: real > $o] :
( ( finite_finite_real @ A4 )
=> ( ( groups713298508707869441omplex @ G
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups713298508707869441omplex
@ ^ [X5: real] : ( if_complex @ ( P2 @ X5 ) @ ( G @ X5 ) @ one_one_complex )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_7649_prod_Ointer__filter,axiom,
! [A4: set_nat,G: nat > complex,P2: nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( groups6464643781859351333omplex @ G
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups6464643781859351333omplex
@ ^ [X5: nat] : ( if_complex @ ( P2 @ X5 ) @ ( G @ X5 ) @ one_one_complex )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_7650_prod_Ointer__filter,axiom,
! [A4: set_int,G: int > complex,P2: int > $o] :
( ( finite_finite_int @ A4 )
=> ( ( groups7440179247065528705omplex @ G
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups7440179247065528705omplex
@ ^ [X5: int] : ( if_complex @ ( P2 @ X5 ) @ ( G @ X5 ) @ one_one_complex )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_7651_prod_Ointer__filter,axiom,
! [A4: set_complex,G: complex > complex,P2: complex > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups3708469109370488835omplex @ G
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups3708469109370488835omplex
@ ^ [X5: complex] : ( if_complex @ ( P2 @ X5 ) @ ( G @ X5 ) @ one_one_complex )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_7652_prod_Ointer__filter,axiom,
! [A4: set_real,G: real > real,P2: real > $o] :
( ( finite_finite_real @ A4 )
=> ( ( groups1681761925125756287l_real @ G
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups1681761925125756287l_real
@ ^ [X5: real] : ( if_real @ ( P2 @ X5 ) @ ( G @ X5 ) @ one_one_real )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_7653_prod_Ointer__filter,axiom,
! [A4: set_nat,G: nat > real,P2: nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( groups129246275422532515t_real @ G
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups129246275422532515t_real
@ ^ [X5: nat] : ( if_real @ ( P2 @ X5 ) @ ( G @ X5 ) @ one_one_real )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_7654_prod_Ointer__filter,axiom,
! [A4: set_int,G: int > real,P2: int > $o] :
( ( finite_finite_int @ A4 )
=> ( ( groups2316167850115554303t_real @ G
@ ( collect_int
@ ^ [X5: int] :
( ( member_int @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups2316167850115554303t_real
@ ^ [X5: int] : ( if_real @ ( P2 @ X5 ) @ ( G @ X5 ) @ one_one_real )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_7655_prod_Ointer__filter,axiom,
! [A4: set_complex,G: complex > real,P2: complex > $o] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups766887009212190081x_real @ G
@ ( collect_complex
@ ^ [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups766887009212190081x_real
@ ^ [X5: complex] : ( if_real @ ( P2 @ X5 ) @ ( G @ X5 ) @ one_one_real )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_7656_prod_Ointer__filter,axiom,
! [A4: set_real,G: real > rat,P2: real > $o] :
( ( finite_finite_real @ A4 )
=> ( ( groups4061424788464935467al_rat @ G
@ ( collect_real
@ ^ [X5: real] :
( ( member_real @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups4061424788464935467al_rat
@ ^ [X5: real] : ( if_rat @ ( P2 @ X5 ) @ ( G @ X5 ) @ one_one_rat )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_7657_prod_Ointer__filter,axiom,
! [A4: set_nat,G: nat > rat,P2: nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( groups73079841787564623at_rat @ G
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( P2 @ X5 ) ) ) )
= ( groups73079841787564623at_rat
@ ^ [X5: nat] : ( if_rat @ ( P2 @ X5 ) @ ( G @ X5 ) @ one_one_rat )
@ A4 ) ) ) ).
% prod.inter_filter
thf(fact_7658_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > nat,M2: nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_7659_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > int,M2: nat,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_7660_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > nat,M2: nat,K2: nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K2 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_7661_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > int,M2: nat,K2: nat,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K2 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_7662_prod__le__1,axiom,
! [A4: set_complex,F: complex > real] :
( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
& ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A4 ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_7663_prod__le__1,axiom,
! [A4: set_real,F: real > real] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
& ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_7664_prod__le__1,axiom,
! [A4: set_nat,F: nat > real] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
& ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A4 ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_7665_prod__le__1,axiom,
! [A4: set_int,F: int > real] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
& ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_7666_prod__le__1,axiom,
! [A4: set_complex,F: complex > rat] :
( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
& ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_7667_prod__le__1,axiom,
! [A4: set_real,F: real > rat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
& ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_7668_prod__le__1,axiom,
! [A4: set_nat,F: nat > rat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
& ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_7669_prod__le__1,axiom,
! [A4: set_int,F: int > rat] :
( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
& ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_7670_prod__le__1,axiom,
! [A4: set_complex,F: complex > nat] :
( ! [X4: complex] :
( ( member_complex @ X4 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) )
& ( ord_less_eq_nat @ ( F @ X4 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_7671_prod__le__1,axiom,
! [A4: set_real,F: real > nat] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) )
& ( ord_less_eq_nat @ ( F @ X4 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_7672_prod_Orelated,axiom,
! [R: complex > complex > $o,S2: set_nat,H2: nat > complex,G: nat > complex] :
( ( R @ one_one_complex @ one_one_complex )
=> ( ! [X16: complex,Y15: complex,X22: complex,Y23: complex] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_complex @ X16 @ Y15 ) @ ( times_times_complex @ X22 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups6464643781859351333omplex @ H2 @ S2 ) @ ( groups6464643781859351333omplex @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_7673_prod_Orelated,axiom,
! [R: complex > complex > $o,S2: set_int,H2: int > complex,G: int > complex] :
( ( R @ one_one_complex @ one_one_complex )
=> ( ! [X16: complex,Y15: complex,X22: complex,Y23: complex] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_complex @ X16 @ Y15 ) @ ( times_times_complex @ X22 @ Y23 ) ) )
=> ( ( finite_finite_int @ S2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups7440179247065528705omplex @ H2 @ S2 ) @ ( groups7440179247065528705omplex @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_7674_prod_Orelated,axiom,
! [R: complex > complex > $o,S2: set_complex,H2: complex > complex,G: complex > complex] :
( ( R @ one_one_complex @ one_one_complex )
=> ( ! [X16: complex,Y15: complex,X22: complex,Y23: complex] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_complex @ X16 @ Y15 ) @ ( times_times_complex @ X22 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups3708469109370488835omplex @ H2 @ S2 ) @ ( groups3708469109370488835omplex @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_7675_prod_Orelated,axiom,
! [R: real > real > $o,S2: set_nat,H2: nat > real,G: nat > real] :
( ( R @ one_one_real @ one_one_real )
=> ( ! [X16: real,Y15: real,X22: real,Y23: real] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_real @ X16 @ Y15 ) @ ( times_times_real @ X22 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups129246275422532515t_real @ H2 @ S2 ) @ ( groups129246275422532515t_real @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_7676_prod_Orelated,axiom,
! [R: real > real > $o,S2: set_int,H2: int > real,G: int > real] :
( ( R @ one_one_real @ one_one_real )
=> ( ! [X16: real,Y15: real,X22: real,Y23: real] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_real @ X16 @ Y15 ) @ ( times_times_real @ X22 @ Y23 ) ) )
=> ( ( finite_finite_int @ S2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups2316167850115554303t_real @ H2 @ S2 ) @ ( groups2316167850115554303t_real @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_7677_prod_Orelated,axiom,
! [R: real > real > $o,S2: set_complex,H2: complex > real,G: complex > real] :
( ( R @ one_one_real @ one_one_real )
=> ( ! [X16: real,Y15: real,X22: real,Y23: real] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_real @ X16 @ Y15 ) @ ( times_times_real @ X22 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups766887009212190081x_real @ H2 @ S2 ) @ ( groups766887009212190081x_real @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_7678_prod_Orelated,axiom,
! [R: rat > rat > $o,S2: set_nat,H2: nat > rat,G: nat > rat] :
( ( R @ one_one_rat @ one_one_rat )
=> ( ! [X16: rat,Y15: rat,X22: rat,Y23: rat] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_rat @ X16 @ Y15 ) @ ( times_times_rat @ X22 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups73079841787564623at_rat @ H2 @ S2 ) @ ( groups73079841787564623at_rat @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_7679_prod_Orelated,axiom,
! [R: rat > rat > $o,S2: set_int,H2: int > rat,G: int > rat] :
( ( R @ one_one_rat @ one_one_rat )
=> ( ! [X16: rat,Y15: rat,X22: rat,Y23: rat] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_rat @ X16 @ Y15 ) @ ( times_times_rat @ X22 @ Y23 ) ) )
=> ( ( finite_finite_int @ S2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups1072433553688619179nt_rat @ H2 @ S2 ) @ ( groups1072433553688619179nt_rat @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_7680_prod_Orelated,axiom,
! [R: rat > rat > $o,S2: set_complex,H2: complex > rat,G: complex > rat] :
( ( R @ one_one_rat @ one_one_rat )
=> ( ! [X16: rat,Y15: rat,X22: rat,Y23: rat] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_rat @ X16 @ Y15 ) @ ( times_times_rat @ X22 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups225925009352817453ex_rat @ H2 @ S2 ) @ ( groups225925009352817453ex_rat @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_7681_prod_Orelated,axiom,
! [R: nat > nat > $o,S2: set_int,H2: int > nat,G: int > nat] :
( ( R @ one_one_nat @ one_one_nat )
=> ( ! [X16: nat,Y15: nat,X22: nat,Y23: nat] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_nat @ X16 @ Y15 ) @ ( times_times_nat @ X22 @ Y23 ) ) )
=> ( ( finite_finite_int @ S2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
=> ( R @ ( groups1707563613775114915nt_nat @ H2 @ S2 ) @ ( groups1707563613775114915nt_nat @ G @ S2 ) ) ) ) ) ) ).
% prod.related
thf(fact_7682_prod__dvd__prod__subset2,axiom,
! [B5: set_real,A4: set_real,F: real > nat,G: real > nat] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ A4 )
=> ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) @ ( groups4696554848551431203al_nat @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_7683_prod__dvd__prod__subset2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ A4 )
=> ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) @ ( groups861055069439313189ex_nat @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_7684_prod__dvd__prod__subset2,axiom,
! [B5: set_real,A4: set_real,F: real > int,G: real > int] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ A4 )
=> ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_int @ ( groups4694064378042380927al_int @ F @ A4 ) @ ( groups4694064378042380927al_int @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_7685_prod__dvd__prod__subset2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > int,G: complex > int] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ A4 )
=> ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A4 ) @ ( groups858564598930262913ex_int @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_7686_prod__dvd__prod__subset2,axiom,
! [B5: set_real,A4: set_real,F: real > code_integer,G: real > code_integer] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ A4 )
=> ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_Code_integer @ ( groups6225526099057966256nteger @ F @ A4 ) @ ( groups6225526099057966256nteger @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_7687_prod__dvd__prod__subset2,axiom,
! [B5: set_nat,A4: set_nat,F: nat > code_integer,G: nat > code_integer] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A4 ) @ ( groups3455450783089532116nteger @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_7688_prod__dvd__prod__subset2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > code_integer,G: complex > code_integer] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ A4 )
=> ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A4 ) @ ( groups8682486955453173170nteger @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_7689_prod__dvd__prod__subset2,axiom,
! [B5: set_int,A4: set_int,F: int > nat,G: int > nat] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ A4 )
=> ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ ( groups1707563613775114915nt_nat @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_7690_prod__dvd__prod__subset2,axiom,
! [B5: set_int,A4: set_int,F: int > code_integer,G: int > code_integer] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ A4 )
=> ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_Code_integer @ ( groups3827104343326376752nteger @ F @ A4 ) @ ( groups3827104343326376752nteger @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_7691_prod__dvd__prod__subset2,axiom,
! [B5: set_nat,A4: set_nat,F: nat > nat,G: nat > nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
=> ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ ( groups708209901874060359at_nat @ G @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_7692_prod__dvd__prod__subset,axiom,
! [B5: set_complex,A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) @ ( groups861055069439313189ex_nat @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_7693_prod__dvd__prod__subset,axiom,
! [B5: set_complex,A4: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A4 ) @ ( groups858564598930262913ex_int @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_7694_prod__dvd__prod__subset,axiom,
! [B5: set_nat,A4: set_nat,F: nat > code_integer] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A4 ) @ ( groups3455450783089532116nteger @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_7695_prod__dvd__prod__subset,axiom,
! [B5: set_complex,A4: set_complex,F: complex > code_integer] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A4 ) @ ( groups8682486955453173170nteger @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_7696_prod__dvd__prod__subset,axiom,
! [B5: set_int,A4: set_int,F: int > nat] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ ( groups1707563613775114915nt_nat @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_7697_prod__dvd__prod__subset,axiom,
! [B5: set_int,A4: set_int,F: int > code_integer] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( dvd_dvd_Code_integer @ ( groups3827104343326376752nteger @ F @ A4 ) @ ( groups3827104343326376752nteger @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_7698_prod__dvd__prod__subset,axiom,
! [B5: set_nat,A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ ( groups708209901874060359at_nat @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_7699_prod__dvd__prod__subset,axiom,
! [B5: set_nat,A4: set_nat,F: nat > int] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A4 ) @ ( groups705719431365010083at_int @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_7700_prod__dvd__prod__subset,axiom,
! [B5: set_int,A4: set_int,F: int > int] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( dvd_dvd_int @ ( groups1705073143266064639nt_int @ F @ A4 ) @ ( groups1705073143266064639nt_int @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_7701_prod__dvd__prod__subset,axiom,
! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ B5 )
=> ( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( dvd_dvd_nat @ ( groups4077766827762148844at_nat @ F @ A4 ) @ ( groups4077766827762148844at_nat @ F @ B5 ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_7702_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T4: set_real,S2: set_real,I: real > real,J: real > real,T2: set_real,G: real > complex,H2: real > complex] :
( ( finite_finite_real @ S5 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T2 @ T4 ) ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S5 )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T4 )
=> ( ( H2 @ B3 )
= one_one_complex ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups713298508707869441omplex @ G @ S2 )
= ( groups713298508707869441omplex @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7703_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T4: set_int,S2: set_real,I: int > real,J: real > int,T2: set_int,G: real > complex,H2: int > complex] :
( ( finite_finite_real @ S5 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T2 @ T4 ) ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T2 @ T4 ) )
=> ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S5 )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ T4 )
=> ( ( H2 @ B3 )
= one_one_complex ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups713298508707869441omplex @ G @ S2 )
= ( groups7440179247065528705omplex @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7704_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T4: set_complex,S2: set_real,I: complex > real,J: real > complex,T2: set_complex,G: real > complex,H2: complex > complex] :
( ( finite_finite_real @ S5 )
=> ( ( finite3207457112153483333omplex @ T4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T2 @ T4 ) ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T2 @ T4 ) )
=> ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S5 )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T4 )
=> ( ( H2 @ B3 )
= one_one_complex ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups713298508707869441omplex @ G @ S2 )
= ( groups3708469109370488835omplex @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7705_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_int,T4: set_real,S2: set_int,I: real > int,J: int > real,T2: set_real,G: int > complex,H2: real > complex] :
( ( finite_finite_int @ S5 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T2 @ T4 ) ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S5 )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T4 )
=> ( ( H2 @ B3 )
= one_one_complex ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups7440179247065528705omplex @ G @ S2 )
= ( groups713298508707869441omplex @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7706_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_int,T4: set_int,S2: set_int,I: int > int,J: int > int,T2: set_int,G: int > complex,H2: int > complex] :
( ( finite_finite_int @ S5 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T2 @ T4 ) ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T2 @ T4 ) )
=> ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S5 )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ T4 )
=> ( ( H2 @ B3 )
= one_one_complex ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups7440179247065528705omplex @ G @ S2 )
= ( groups7440179247065528705omplex @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7707_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_int,T4: set_complex,S2: set_int,I: complex > int,J: int > complex,T2: set_complex,G: int > complex,H2: complex > complex] :
( ( finite_finite_int @ S5 )
=> ( ( finite3207457112153483333omplex @ T4 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S5 ) )
=> ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T2 @ T4 ) ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T2 @ T4 ) )
=> ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S5 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S5 )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T4 )
=> ( ( H2 @ B3 )
= one_one_complex ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups7440179247065528705omplex @ G @ S2 )
= ( groups3708469109370488835omplex @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7708_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_complex,T4: set_real,S2: set_complex,I: real > complex,J: complex > real,T2: set_real,G: complex > complex,H2: real > complex] :
( ( finite3207457112153483333omplex @ S5 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
=> ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T2 @ T4 ) ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( member_complex @ ( I @ B3 ) @ ( minus_811609699411566653omplex @ S2 @ S5 ) ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ S5 )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T4 )
=> ( ( H2 @ B3 )
= one_one_complex ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3708469109370488835omplex @ G @ S2 )
= ( groups713298508707869441omplex @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7709_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_complex,T4: set_int,S2: set_complex,I: int > complex,J: complex > int,T2: set_int,G: complex > complex,H2: int > complex] :
( ( finite3207457112153483333omplex @ S5 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
=> ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T2 @ T4 ) ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ T2 @ T4 ) )
=> ( member_complex @ ( I @ B3 ) @ ( minus_811609699411566653omplex @ S2 @ S5 ) ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ S5 )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B3: int] :
( ( member_int @ B3 @ T4 )
=> ( ( H2 @ B3 )
= one_one_complex ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3708469109370488835omplex @ G @ S2 )
= ( groups7440179247065528705omplex @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7710_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_complex,T4: set_complex,S2: set_complex,I: complex > complex,J: complex > complex,T2: set_complex,G: complex > complex,H2: complex > complex] :
( ( finite3207457112153483333omplex @ S5 )
=> ( ( finite3207457112153483333omplex @ T4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S2 @ S5 ) )
=> ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T2 @ T4 ) ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T2 @ T4 ) )
=> ( member_complex @ ( I @ B3 ) @ ( minus_811609699411566653omplex @ S2 @ S5 ) ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ S5 )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T4 )
=> ( ( H2 @ B3 )
= one_one_complex ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3708469109370488835omplex @ G @ S2 )
= ( groups3708469109370488835omplex @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7711_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_real,T4: set_real,S2: set_real,I: real > real,J: real > real,T2: set_real,G: real > real,H2: real > real] :
( ( finite_finite_real @ S5 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( ( I @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S2 @ S5 ) )
=> ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T2 @ T4 ) ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S5 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S5 )
=> ( ( G @ A3 )
= one_one_real ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T4 )
=> ( ( H2 @ B3 )
= one_one_real ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S2 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups1681761925125756287l_real @ G @ S2 )
= ( groups1681761925125756287l_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7712_gbinomial__Suc__Suc,axiom,
! [A: complex,K2: nat] :
( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K2 ) )
= ( plus_plus_complex @ ( gbinomial_complex @ A @ K2 ) @ ( gbinomial_complex @ A @ ( suc @ K2 ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_7713_gbinomial__Suc__Suc,axiom,
! [A: real,K2: nat] :
( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K2 ) )
= ( plus_plus_real @ ( gbinomial_real @ A @ K2 ) @ ( gbinomial_real @ A @ ( suc @ K2 ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_7714_gbinomial__Suc__Suc,axiom,
! [A: rat,K2: nat] :
( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K2 ) )
= ( plus_plus_rat @ ( gbinomial_rat @ A @ K2 ) @ ( gbinomial_rat @ A @ ( suc @ K2 ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_7715_gbinomial__of__nat__symmetric,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K2 )
= ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).
% gbinomial_of_nat_symmetric
thf(fact_7716_gbinomial__of__nat__symmetric,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ K2 )
= ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N ) @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).
% gbinomial_of_nat_symmetric
thf(fact_7717_prod_Osetdiff__irrelevant,axiom,
! [A4: set_real,G: real > complex] :
( ( finite_finite_real @ A4 )
=> ( ( groups713298508707869441omplex @ G
@ ( minus_minus_set_real @ A4
@ ( collect_real
@ ^ [X5: real] :
( ( G @ X5 )
= one_one_complex ) ) ) )
= ( groups713298508707869441omplex @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_7718_prod_Osetdiff__irrelevant,axiom,
! [A4: set_int,G: int > complex] :
( ( finite_finite_int @ A4 )
=> ( ( groups7440179247065528705omplex @ G
@ ( minus_minus_set_int @ A4
@ ( collect_int
@ ^ [X5: int] :
( ( G @ X5 )
= one_one_complex ) ) ) )
= ( groups7440179247065528705omplex @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_7719_prod_Osetdiff__irrelevant,axiom,
! [A4: set_complex,G: complex > complex] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups3708469109370488835omplex @ G
@ ( minus_811609699411566653omplex @ A4
@ ( collect_complex
@ ^ [X5: complex] :
( ( G @ X5 )
= one_one_complex ) ) ) )
= ( groups3708469109370488835omplex @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_7720_prod_Osetdiff__irrelevant,axiom,
! [A4: set_real,G: real > real] :
( ( finite_finite_real @ A4 )
=> ( ( groups1681761925125756287l_real @ G
@ ( minus_minus_set_real @ A4
@ ( collect_real
@ ^ [X5: real] :
( ( G @ X5 )
= one_one_real ) ) ) )
= ( groups1681761925125756287l_real @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_7721_prod_Osetdiff__irrelevant,axiom,
! [A4: set_int,G: int > real] :
( ( finite_finite_int @ A4 )
=> ( ( groups2316167850115554303t_real @ G
@ ( minus_minus_set_int @ A4
@ ( collect_int
@ ^ [X5: int] :
( ( G @ X5 )
= one_one_real ) ) ) )
= ( groups2316167850115554303t_real @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_7722_prod_Osetdiff__irrelevant,axiom,
! [A4: set_complex,G: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups766887009212190081x_real @ G
@ ( minus_811609699411566653omplex @ A4
@ ( collect_complex
@ ^ [X5: complex] :
( ( G @ X5 )
= one_one_real ) ) ) )
= ( groups766887009212190081x_real @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_7723_prod_Osetdiff__irrelevant,axiom,
! [A4: set_real,G: real > rat] :
( ( finite_finite_real @ A4 )
=> ( ( groups4061424788464935467al_rat @ G
@ ( minus_minus_set_real @ A4
@ ( collect_real
@ ^ [X5: real] :
( ( G @ X5 )
= one_one_rat ) ) ) )
= ( groups4061424788464935467al_rat @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_7724_prod_Osetdiff__irrelevant,axiom,
! [A4: set_int,G: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ( groups1072433553688619179nt_rat @ G
@ ( minus_minus_set_int @ A4
@ ( collect_int
@ ^ [X5: int] :
( ( G @ X5 )
= one_one_rat ) ) ) )
= ( groups1072433553688619179nt_rat @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_7725_prod_Osetdiff__irrelevant,axiom,
! [A4: set_complex,G: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups225925009352817453ex_rat @ G
@ ( minus_811609699411566653omplex @ A4
@ ( collect_complex
@ ^ [X5: complex] :
( ( G @ X5 )
= one_one_rat ) ) ) )
= ( groups225925009352817453ex_rat @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_7726_prod_Osetdiff__irrelevant,axiom,
! [A4: set_real,G: real > nat] :
( ( finite_finite_real @ A4 )
=> ( ( groups4696554848551431203al_nat @ G
@ ( minus_minus_set_real @ A4
@ ( collect_real
@ ^ [X5: real] :
( ( G @ X5 )
= one_one_nat ) ) ) )
= ( groups4696554848551431203al_nat @ G @ A4 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_7727_exp__sum,axiom,
! [I6: set_int,F: int > real] :
( ( finite_finite_int @ I6 )
=> ( ( exp_real @ ( groups8778361861064173332t_real @ F @ I6 ) )
= ( groups2316167850115554303t_real
@ ^ [X5: int] : ( exp_real @ ( F @ X5 ) )
@ I6 ) ) ) ).
% exp_sum
thf(fact_7728_exp__sum,axiom,
! [I6: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ( exp_real @ ( groups5808333547571424918x_real @ F @ I6 ) )
= ( groups766887009212190081x_real
@ ^ [X5: complex] : ( exp_real @ ( F @ X5 ) )
@ I6 ) ) ) ).
% exp_sum
thf(fact_7729_exp__sum,axiom,
! [I6: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > real] :
( ( finite6177210948735845034at_nat @ I6 )
=> ( ( exp_real @ ( groups4567486121110086003t_real @ F @ I6 ) )
= ( groups6036352826371341000t_real
@ ^ [X5: product_prod_nat_nat] : ( exp_real @ ( F @ X5 ) )
@ I6 ) ) ) ).
% exp_sum
thf(fact_7730_exp__sum,axiom,
! [I6: set_complex,F: complex > complex] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ( exp_complex @ ( groups7754918857620584856omplex @ F @ I6 ) )
= ( groups3708469109370488835omplex
@ ^ [X5: complex] : ( exp_complex @ ( F @ X5 ) )
@ I6 ) ) ) ).
% exp_sum
thf(fact_7731_exp__sum,axiom,
! [I6: set_nat,F: nat > real] :
( ( finite_finite_nat @ I6 )
=> ( ( exp_real @ ( groups6591440286371151544t_real @ F @ I6 ) )
= ( groups129246275422532515t_real
@ ^ [X5: nat] : ( exp_real @ ( F @ X5 ) )
@ I6 ) ) ) ).
% exp_sum
thf(fact_7732_prod_Onat__diff__reindex,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.nat_diff_reindex
thf(fact_7733_prod_Onat__diff__reindex,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.nat_diff_reindex
thf(fact_7734_prod_OatLeastAtMost__rev,axiom,
! [G: nat > nat,N: nat,M2: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M2 ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ I3 ) )
@ ( set_or1269000886237332187st_nat @ N @ M2 ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_7735_prod_OatLeastAtMost__rev,axiom,
! [G: nat > int,N: nat,M2: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ N @ M2 ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ I3 ) )
@ ( set_or1269000886237332187st_nat @ N @ M2 ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_7736_gbinomial__Suc,axiom,
! [A: rat,K2: nat] :
( ( gbinomial_rat @ A @ ( suc @ K2 ) )
= ( divide_divide_rat
@ ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) )
@ ( semiri773545260158071498ct_rat @ ( suc @ K2 ) ) ) ) ).
% gbinomial_Suc
thf(fact_7737_gbinomial__Suc,axiom,
! [A: code_integer,K2: nat] :
( ( gbinom8545251970709558553nteger @ A @ ( suc @ K2 ) )
= ( divide6298287555418463151nteger
@ ( groups3455450783089532116nteger
@ ^ [I3: nat] : ( minus_8373710615458151222nteger @ A @ ( semiri4939895301339042750nteger @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) )
@ ( semiri3624122377584611663nteger @ ( suc @ K2 ) ) ) ) ).
% gbinomial_Suc
thf(fact_7738_gbinomial__Suc,axiom,
! [A: real,K2: nat] :
( ( gbinomial_real @ A @ ( suc @ K2 ) )
= ( divide_divide_real
@ ( groups129246275422532515t_real
@ ^ [I3: nat] : ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) )
@ ( semiri2265585572941072030t_real @ ( suc @ K2 ) ) ) ) ).
% gbinomial_Suc
thf(fact_7739_gbinomial__Suc,axiom,
! [A: complex,K2: nat] :
( ( gbinomial_complex @ A @ ( suc @ K2 ) )
= ( divide1717551699836669952omplex
@ ( groups6464643781859351333omplex
@ ^ [I3: nat] : ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) )
@ ( semiri5044797733671781792omplex @ ( suc @ K2 ) ) ) ) ).
% gbinomial_Suc
thf(fact_7740_gbinomial__Suc,axiom,
! [A: nat,K2: nat] :
( ( gbinomial_nat @ A @ ( suc @ K2 ) )
= ( divide_divide_nat
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( minus_minus_nat @ A @ ( semiri1316708129612266289at_nat @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) )
@ ( semiri1408675320244567234ct_nat @ ( suc @ K2 ) ) ) ) ).
% gbinomial_Suc
thf(fact_7741_gbinomial__Suc,axiom,
! [A: int,K2: nat] :
( ( gbinomial_int @ A @ ( suc @ K2 ) )
= ( divide_divide_int
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( minus_minus_int @ A @ ( semiri1314217659103216013at_int @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) )
@ ( semiri1406184849735516958ct_int @ ( suc @ K2 ) ) ) ) ).
% gbinomial_Suc
thf(fact_7742_less__1__prod2,axiom,
! [I6: set_real,I: real,F: real > real] :
( ( finite_finite_real @ I6 )
=> ( ( member_real @ I @ I6 )
=> ( ( ord_less_real @ one_one_real @ ( F @ I ) )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I6 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I6 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7743_less__1__prod2,axiom,
! [I6: set_nat,I: nat,F: nat > real] :
( ( finite_finite_nat @ I6 )
=> ( ( member_nat @ I @ I6 )
=> ( ( ord_less_real @ one_one_real @ ( F @ I ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I6 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I6 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7744_less__1__prod2,axiom,
! [I6: set_int,I: int,F: int > real] :
( ( finite_finite_int @ I6 )
=> ( ( member_int @ I @ I6 )
=> ( ( ord_less_real @ one_one_real @ ( F @ I ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I6 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I6 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7745_less__1__prod2,axiom,
! [I6: set_complex,I: complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ( member_complex @ I @ I6 )
=> ( ( ord_less_real @ one_one_real @ ( F @ I ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I6 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I6 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7746_less__1__prod2,axiom,
! [I6: set_real,I: real,F: real > rat] :
( ( finite_finite_real @ I6 )
=> ( ( member_real @ I @ I6 )
=> ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I6 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I6 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7747_less__1__prod2,axiom,
! [I6: set_nat,I: nat,F: nat > rat] :
( ( finite_finite_nat @ I6 )
=> ( ( member_nat @ I @ I6 )
=> ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I6 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I6 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7748_less__1__prod2,axiom,
! [I6: set_int,I: int,F: int > rat] :
( ( finite_finite_int @ I6 )
=> ( ( member_int @ I @ I6 )
=> ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I6 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I6 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7749_less__1__prod2,axiom,
! [I6: set_complex,I: complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ( member_complex @ I @ I6 )
=> ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I6 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I6 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7750_less__1__prod2,axiom,
! [I6: set_real,I: real,F: real > int] :
( ( finite_finite_real @ I6 )
=> ( ( member_real @ I @ I6 )
=> ( ( ord_less_int @ one_one_int @ ( F @ I ) )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I6 )
=> ( ord_less_eq_int @ one_one_int @ ( F @ I2 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I6 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7751_less__1__prod2,axiom,
! [I6: set_complex,I: complex,F: complex > int] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ( member_complex @ I @ I6 )
=> ( ( ord_less_int @ one_one_int @ ( F @ I ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I6 )
=> ( ord_less_eq_int @ one_one_int @ ( F @ I2 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I6 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7752_arccos__le__arccos,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( arccos @ Y ) @ ( arccos @ X3 ) ) ) ) ) ).
% arccos_le_arccos
thf(fact_7753_less__1__prod,axiom,
! [I6: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ( I6 != bot_bot_set_complex )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I6 )
=> ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I6 ) ) ) ) ) ).
% less_1_prod
thf(fact_7754_less__1__prod,axiom,
! [I6: set_real,F: real > real] :
( ( finite_finite_real @ I6 )
=> ( ( I6 != bot_bot_set_real )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I6 )
=> ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I6 ) ) ) ) ) ).
% less_1_prod
thf(fact_7755_less__1__prod,axiom,
! [I6: set_nat,F: nat > real] :
( ( finite_finite_nat @ I6 )
=> ( ( I6 != bot_bot_set_nat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I6 )
=> ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I6 ) ) ) ) ) ).
% less_1_prod
thf(fact_7756_less__1__prod,axiom,
! [I6: set_int,F: int > real] :
( ( finite_finite_int @ I6 )
=> ( ( I6 != bot_bot_set_int )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I6 )
=> ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I6 ) ) ) ) ) ).
% less_1_prod
thf(fact_7757_less__1__prod,axiom,
! [I6: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ( I6 != bot_bot_set_complex )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I6 )
=> ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I6 ) ) ) ) ) ).
% less_1_prod
thf(fact_7758_less__1__prod,axiom,
! [I6: set_real,F: real > rat] :
( ( finite_finite_real @ I6 )
=> ( ( I6 != bot_bot_set_real )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I6 )
=> ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I6 ) ) ) ) ) ).
% less_1_prod
thf(fact_7759_less__1__prod,axiom,
! [I6: set_nat,F: nat > rat] :
( ( finite_finite_nat @ I6 )
=> ( ( I6 != bot_bot_set_nat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I6 )
=> ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I6 ) ) ) ) ) ).
% less_1_prod
thf(fact_7760_less__1__prod,axiom,
! [I6: set_int,F: int > rat] :
( ( finite_finite_int @ I6 )
=> ( ( I6 != bot_bot_set_int )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I6 )
=> ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I6 ) ) ) ) ) ).
% less_1_prod
thf(fact_7761_less__1__prod,axiom,
! [I6: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ( I6 != bot_bot_set_complex )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I6 )
=> ( ord_less_int @ one_one_int @ ( F @ I2 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I6 ) ) ) ) ) ).
% less_1_prod
thf(fact_7762_less__1__prod,axiom,
! [I6: set_real,F: real > int] :
( ( finite_finite_real @ I6 )
=> ( ( I6 != bot_bot_set_real )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I6 )
=> ( ord_less_int @ one_one_int @ ( F @ I2 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I6 ) ) ) ) ) ).
% less_1_prod
thf(fact_7763_arccos__le__mono,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( arccos @ X3 ) @ ( arccos @ Y ) )
= ( ord_less_eq_real @ Y @ X3 ) ) ) ) ).
% arccos_le_mono
thf(fact_7764_arccos__eq__iff,axiom,
! [X3: real,Y: real] :
( ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
& ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real ) )
=> ( ( ( arccos @ X3 )
= ( arccos @ Y ) )
= ( X3 = Y ) ) ) ).
% arccos_eq_iff
thf(fact_7765_arcsin__le__arcsin,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( arcsin @ X3 ) @ ( arcsin @ Y ) ) ) ) ) ).
% arcsin_le_arcsin
thf(fact_7766_arcsin__minus,axiom,
! [X3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
=> ( ( ord_less_eq_real @ X3 @ one_one_real )
=> ( ( arcsin @ ( uminus_uminus_real @ X3 ) )
= ( uminus_uminus_real @ ( arcsin @ X3 ) ) ) ) ) ).
% arcsin_minus
thf(fact_7767_prod_Osubset__diff,axiom,
! [B5: set_complex,A4: set_complex,G: complex > real] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups766887009212190081x_real @ G @ A4 )
= ( times_times_real @ ( groups766887009212190081x_real @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups766887009212190081x_real @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7768_prod_Osubset__diff,axiom,
! [B5: set_nat,A4: set_nat,G: nat > real] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( groups129246275422532515t_real @ G @ A4 )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups129246275422532515t_real @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7769_prod_Osubset__diff,axiom,
! [B5: set_complex,A4: set_complex,G: complex > rat] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups225925009352817453ex_rat @ G @ A4 )
= ( times_times_rat @ ( groups225925009352817453ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups225925009352817453ex_rat @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7770_prod_Osubset__diff,axiom,
! [B5: set_nat,A4: set_nat,G: nat > rat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( groups73079841787564623at_rat @ G @ A4 )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups73079841787564623at_rat @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7771_prod_Osubset__diff,axiom,
! [B5: set_complex,A4: set_complex,G: complex > nat] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups861055069439313189ex_nat @ G @ A4 )
= ( times_times_nat @ ( groups861055069439313189ex_nat @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups861055069439313189ex_nat @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7772_prod_Osubset__diff,axiom,
! [B5: set_complex,A4: set_complex,G: complex > int] :
( ( ord_le211207098394363844omplex @ B5 @ A4 )
=> ( ( finite3207457112153483333omplex @ A4 )
=> ( ( groups858564598930262913ex_int @ G @ A4 )
= ( times_times_int @ ( groups858564598930262913ex_int @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups858564598930262913ex_int @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7773_prod_Osubset__diff,axiom,
! [B5: set_int,A4: set_int,G: int > real] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( finite_finite_int @ A4 )
=> ( ( groups2316167850115554303t_real @ G @ A4 )
= ( times_times_real @ ( groups2316167850115554303t_real @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups2316167850115554303t_real @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7774_prod_Osubset__diff,axiom,
! [B5: set_int,A4: set_int,G: int > rat] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( finite_finite_int @ A4 )
=> ( ( groups1072433553688619179nt_rat @ G @ A4 )
= ( times_times_rat @ ( groups1072433553688619179nt_rat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups1072433553688619179nt_rat @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7775_prod_Osubset__diff,axiom,
! [B5: set_int,A4: set_int,G: int > nat] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( finite_finite_int @ A4 )
=> ( ( groups1707563613775114915nt_nat @ G @ A4 )
= ( times_times_nat @ ( groups1707563613775114915nt_nat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups1707563613775114915nt_nat @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7776_prod_Osubset__diff,axiom,
! [B5: set_nat,A4: set_nat,G: nat > nat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( groups708209901874060359at_nat @ G @ A4 )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups708209901874060359at_nat @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7777_prod_Omono__neutral__cong__right,axiom,
! [T2: set_real,S2: set_real,G: real > complex,H2: real > complex] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_complex ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups713298508707869441omplex @ G @ T2 )
= ( groups713298508707869441omplex @ H2 @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7778_prod_Omono__neutral__cong__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > complex,H2: complex > complex] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_complex ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups3708469109370488835omplex @ G @ T2 )
= ( groups3708469109370488835omplex @ H2 @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7779_prod_Omono__neutral__cong__right,axiom,
! [T2: set_real,S2: set_real,G: real > real,H2: real > real] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_real ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups1681761925125756287l_real @ G @ T2 )
= ( groups1681761925125756287l_real @ H2 @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7780_prod_Omono__neutral__cong__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > real,H2: complex > real] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_real ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups766887009212190081x_real @ G @ T2 )
= ( groups766887009212190081x_real @ H2 @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7781_prod_Omono__neutral__cong__right,axiom,
! [T2: set_real,S2: set_real,G: real > rat,H2: real > rat] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_rat ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups4061424788464935467al_rat @ G @ T2 )
= ( groups4061424788464935467al_rat @ H2 @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7782_prod_Omono__neutral__cong__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > rat,H2: complex > rat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_rat ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups225925009352817453ex_rat @ G @ T2 )
= ( groups225925009352817453ex_rat @ H2 @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7783_prod_Omono__neutral__cong__right,axiom,
! [T2: set_real,S2: set_real,G: real > nat,H2: real > nat] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_nat ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups4696554848551431203al_nat @ G @ T2 )
= ( groups4696554848551431203al_nat @ H2 @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7784_prod_Omono__neutral__cong__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > nat,H2: complex > nat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_nat ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups861055069439313189ex_nat @ G @ T2 )
= ( groups861055069439313189ex_nat @ H2 @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7785_prod_Omono__neutral__cong__right,axiom,
! [T2: set_real,S2: set_real,G: real > int,H2: real > int] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_int ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups4694064378042380927al_int @ G @ T2 )
= ( groups4694064378042380927al_int @ H2 @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7786_prod_Omono__neutral__cong__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > int,H2: complex > int] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_int ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups858564598930262913ex_int @ G @ T2 )
= ( groups858564598930262913ex_int @ H2 @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7787_prod_Omono__neutral__cong__left,axiom,
! [T2: set_real,S2: set_real,H2: real > complex,G: real > complex] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( H2 @ X4 )
= one_one_complex ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups713298508707869441omplex @ G @ S2 )
= ( groups713298508707869441omplex @ H2 @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7788_prod_Omono__neutral__cong__left,axiom,
! [T2: set_complex,S2: set_complex,H2: complex > complex,G: complex > complex] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( H2 @ X4 )
= one_one_complex ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups3708469109370488835omplex @ G @ S2 )
= ( groups3708469109370488835omplex @ H2 @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7789_prod_Omono__neutral__cong__left,axiom,
! [T2: set_real,S2: set_real,H2: real > real,G: real > real] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( H2 @ X4 )
= one_one_real ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups1681761925125756287l_real @ G @ S2 )
= ( groups1681761925125756287l_real @ H2 @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7790_prod_Omono__neutral__cong__left,axiom,
! [T2: set_complex,S2: set_complex,H2: complex > real,G: complex > real] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( H2 @ X4 )
= one_one_real ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups766887009212190081x_real @ G @ S2 )
= ( groups766887009212190081x_real @ H2 @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7791_prod_Omono__neutral__cong__left,axiom,
! [T2: set_real,S2: set_real,H2: real > rat,G: real > rat] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( H2 @ X4 )
= one_one_rat ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups4061424788464935467al_rat @ G @ S2 )
= ( groups4061424788464935467al_rat @ H2 @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7792_prod_Omono__neutral__cong__left,axiom,
! [T2: set_complex,S2: set_complex,H2: complex > rat,G: complex > rat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( H2 @ X4 )
= one_one_rat ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups225925009352817453ex_rat @ G @ S2 )
= ( groups225925009352817453ex_rat @ H2 @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7793_prod_Omono__neutral__cong__left,axiom,
! [T2: set_real,S2: set_real,H2: real > nat,G: real > nat] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( H2 @ X4 )
= one_one_nat ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups4696554848551431203al_nat @ G @ S2 )
= ( groups4696554848551431203al_nat @ H2 @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7794_prod_Omono__neutral__cong__left,axiom,
! [T2: set_complex,S2: set_complex,H2: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( H2 @ X4 )
= one_one_nat ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups861055069439313189ex_nat @ G @ S2 )
= ( groups861055069439313189ex_nat @ H2 @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7795_prod_Omono__neutral__cong__left,axiom,
! [T2: set_real,S2: set_real,H2: real > int,G: real > int] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( H2 @ X4 )
= one_one_int ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups4694064378042380927al_int @ G @ S2 )
= ( groups4694064378042380927al_int @ H2 @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7796_prod_Omono__neutral__cong__left,axiom,
! [T2: set_complex,S2: set_complex,H2: complex > int,G: complex > int] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( H2 @ X4 )
= one_one_int ) )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( G @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups858564598930262913ex_int @ G @ S2 )
= ( groups858564598930262913ex_int @ H2 @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7797_prod_Omono__neutral__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > complex] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_complex ) )
=> ( ( groups3708469109370488835omplex @ G @ T2 )
= ( groups3708469109370488835omplex @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7798_prod_Omono__neutral__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > real] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_real ) )
=> ( ( groups766887009212190081x_real @ G @ T2 )
= ( groups766887009212190081x_real @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7799_prod_Omono__neutral__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > rat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_rat ) )
=> ( ( groups225925009352817453ex_rat @ G @ T2 )
= ( groups225925009352817453ex_rat @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7800_prod_Omono__neutral__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > nat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_nat ) )
=> ( ( groups861055069439313189ex_nat @ G @ T2 )
= ( groups861055069439313189ex_nat @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7801_prod_Omono__neutral__right,axiom,
! [T2: set_complex,S2: set_complex,G: complex > int] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_int ) )
=> ( ( groups858564598930262913ex_int @ G @ T2 )
= ( groups858564598930262913ex_int @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7802_prod_Omono__neutral__right,axiom,
! [T2: set_nat,S2: set_nat,G: nat > complex] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_complex ) )
=> ( ( groups6464643781859351333omplex @ G @ T2 )
= ( groups6464643781859351333omplex @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7803_prod_Omono__neutral__right,axiom,
! [T2: set_nat,S2: set_nat,G: nat > real] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_real ) )
=> ( ( groups129246275422532515t_real @ G @ T2 )
= ( groups129246275422532515t_real @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7804_prod_Omono__neutral__right,axiom,
! [T2: set_nat,S2: set_nat,G: nat > rat] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_rat ) )
=> ( ( groups73079841787564623at_rat @ G @ T2 )
= ( groups73079841787564623at_rat @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7805_prod_Omono__neutral__right,axiom,
! [T2: set_int,S2: set_int,G: int > complex] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S2 @ T2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_complex ) )
=> ( ( groups7440179247065528705omplex @ G @ T2 )
= ( groups7440179247065528705omplex @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7806_prod_Omono__neutral__right,axiom,
! [T2: set_int,S2: set_int,G: int > real] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S2 @ T2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_real ) )
=> ( ( groups2316167850115554303t_real @ G @ T2 )
= ( groups2316167850115554303t_real @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7807_prod_Omono__neutral__left,axiom,
! [T2: set_complex,S2: set_complex,G: complex > complex] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_complex ) )
=> ( ( groups3708469109370488835omplex @ G @ S2 )
= ( groups3708469109370488835omplex @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7808_prod_Omono__neutral__left,axiom,
! [T2: set_complex,S2: set_complex,G: complex > real] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_real ) )
=> ( ( groups766887009212190081x_real @ G @ S2 )
= ( groups766887009212190081x_real @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7809_prod_Omono__neutral__left,axiom,
! [T2: set_complex,S2: set_complex,G: complex > rat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_rat ) )
=> ( ( groups225925009352817453ex_rat @ G @ S2 )
= ( groups225925009352817453ex_rat @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7810_prod_Omono__neutral__left,axiom,
! [T2: set_complex,S2: set_complex,G: complex > nat] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_nat ) )
=> ( ( groups861055069439313189ex_nat @ G @ S2 )
= ( groups861055069439313189ex_nat @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7811_prod_Omono__neutral__left,axiom,
! [T2: set_complex,S2: set_complex,G: complex > int] :
( ( finite3207457112153483333omplex @ T2 )
=> ( ( ord_le211207098394363844omplex @ S2 @ T2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_int ) )
=> ( ( groups858564598930262913ex_int @ G @ S2 )
= ( groups858564598930262913ex_int @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7812_prod_Omono__neutral__left,axiom,
! [T2: set_nat,S2: set_nat,G: nat > complex] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_complex ) )
=> ( ( groups6464643781859351333omplex @ G @ S2 )
= ( groups6464643781859351333omplex @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7813_prod_Omono__neutral__left,axiom,
! [T2: set_nat,S2: set_nat,G: nat > real] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_real ) )
=> ( ( groups129246275422532515t_real @ G @ S2 )
= ( groups129246275422532515t_real @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7814_prod_Omono__neutral__left,axiom,
! [T2: set_nat,S2: set_nat,G: nat > rat] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_rat ) )
=> ( ( groups73079841787564623at_rat @ G @ S2 )
= ( groups73079841787564623at_rat @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7815_prod_Omono__neutral__left,axiom,
! [T2: set_int,S2: set_int,G: int > complex] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S2 @ T2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_complex ) )
=> ( ( groups7440179247065528705omplex @ G @ S2 )
= ( groups7440179247065528705omplex @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7816_prod_Omono__neutral__left,axiom,
! [T2: set_int,S2: set_int,G: int > real] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S2 @ T2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ T2 @ S2 ) )
=> ( ( G @ X4 )
= one_one_real ) )
=> ( ( groups2316167850115554303t_real @ G @ S2 )
= ( groups2316167850115554303t_real @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7817_prod_Osame__carrierI,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > complex,H2: real > complex] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_complex ) )
=> ( ( ( groups713298508707869441omplex @ G @ C4 )
= ( groups713298508707869441omplex @ H2 @ C4 ) )
=> ( ( groups713298508707869441omplex @ G @ A4 )
= ( groups713298508707869441omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7818_prod_Osame__carrierI,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > complex,H2: complex > complex] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_complex ) )
=> ( ( ( groups3708469109370488835omplex @ G @ C4 )
= ( groups3708469109370488835omplex @ H2 @ C4 ) )
=> ( ( groups3708469109370488835omplex @ G @ A4 )
= ( groups3708469109370488835omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7819_prod_Osame__carrierI,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > real,H2: real > real] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_real ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_real ) )
=> ( ( ( groups1681761925125756287l_real @ G @ C4 )
= ( groups1681761925125756287l_real @ H2 @ C4 ) )
=> ( ( groups1681761925125756287l_real @ G @ A4 )
= ( groups1681761925125756287l_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7820_prod_Osame__carrierI,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_real ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_real ) )
=> ( ( ( groups766887009212190081x_real @ G @ C4 )
= ( groups766887009212190081x_real @ H2 @ C4 ) )
=> ( ( groups766887009212190081x_real @ G @ A4 )
= ( groups766887009212190081x_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7821_prod_Osame__carrierI,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > rat,H2: real > rat] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_rat ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_rat ) )
=> ( ( ( groups4061424788464935467al_rat @ G @ C4 )
= ( groups4061424788464935467al_rat @ H2 @ C4 ) )
=> ( ( groups4061424788464935467al_rat @ G @ A4 )
= ( groups4061424788464935467al_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7822_prod_Osame__carrierI,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > rat,H2: complex > rat] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_rat ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_rat ) )
=> ( ( ( groups225925009352817453ex_rat @ G @ C4 )
= ( groups225925009352817453ex_rat @ H2 @ C4 ) )
=> ( ( groups225925009352817453ex_rat @ G @ A4 )
= ( groups225925009352817453ex_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7823_prod_Osame__carrierI,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > nat,H2: real > nat] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_nat ) )
=> ( ( ( groups4696554848551431203al_nat @ G @ C4 )
= ( groups4696554848551431203al_nat @ H2 @ C4 ) )
=> ( ( groups4696554848551431203al_nat @ G @ A4 )
= ( groups4696554848551431203al_nat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7824_prod_Osame__carrierI,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > nat,H2: complex > nat] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_nat ) )
=> ( ( ( groups861055069439313189ex_nat @ G @ C4 )
= ( groups861055069439313189ex_nat @ H2 @ C4 ) )
=> ( ( groups861055069439313189ex_nat @ G @ A4 )
= ( groups861055069439313189ex_nat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7825_prod_Osame__carrierI,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > int,H2: real > int] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_int ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_int ) )
=> ( ( ( groups4694064378042380927al_int @ G @ C4 )
= ( groups4694064378042380927al_int @ H2 @ C4 ) )
=> ( ( groups4694064378042380927al_int @ G @ A4 )
= ( groups4694064378042380927al_int @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7826_prod_Osame__carrierI,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > int,H2: complex > int] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_int ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_int ) )
=> ( ( ( groups858564598930262913ex_int @ G @ C4 )
= ( groups858564598930262913ex_int @ H2 @ C4 ) )
=> ( ( groups858564598930262913ex_int @ G @ A4 )
= ( groups858564598930262913ex_int @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7827_prod_Osame__carrier,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > complex,H2: real > complex] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_complex ) )
=> ( ( ( groups713298508707869441omplex @ G @ A4 )
= ( groups713298508707869441omplex @ H2 @ B5 ) )
= ( ( groups713298508707869441omplex @ G @ C4 )
= ( groups713298508707869441omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7828_prod_Osame__carrier,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > complex,H2: complex > complex] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_complex ) )
=> ( ( ( groups3708469109370488835omplex @ G @ A4 )
= ( groups3708469109370488835omplex @ H2 @ B5 ) )
= ( ( groups3708469109370488835omplex @ G @ C4 )
= ( groups3708469109370488835omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7829_prod_Osame__carrier,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > real,H2: real > real] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_real ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_real ) )
=> ( ( ( groups1681761925125756287l_real @ G @ A4 )
= ( groups1681761925125756287l_real @ H2 @ B5 ) )
= ( ( groups1681761925125756287l_real @ G @ C4 )
= ( groups1681761925125756287l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7830_prod_Osame__carrier,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_real ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_real ) )
=> ( ( ( groups766887009212190081x_real @ G @ A4 )
= ( groups766887009212190081x_real @ H2 @ B5 ) )
= ( ( groups766887009212190081x_real @ G @ C4 )
= ( groups766887009212190081x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7831_prod_Osame__carrier,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > rat,H2: real > rat] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_rat ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_rat ) )
=> ( ( ( groups4061424788464935467al_rat @ G @ A4 )
= ( groups4061424788464935467al_rat @ H2 @ B5 ) )
= ( ( groups4061424788464935467al_rat @ G @ C4 )
= ( groups4061424788464935467al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7832_prod_Osame__carrier,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > rat,H2: complex > rat] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_rat ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_rat ) )
=> ( ( ( groups225925009352817453ex_rat @ G @ A4 )
= ( groups225925009352817453ex_rat @ H2 @ B5 ) )
= ( ( groups225925009352817453ex_rat @ G @ C4 )
= ( groups225925009352817453ex_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7833_prod_Osame__carrier,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > nat,H2: real > nat] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_nat ) )
=> ( ( ( groups4696554848551431203al_nat @ G @ A4 )
= ( groups4696554848551431203al_nat @ H2 @ B5 ) )
= ( ( groups4696554848551431203al_nat @ G @ C4 )
= ( groups4696554848551431203al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7834_prod_Osame__carrier,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > nat,H2: complex > nat] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_nat ) )
=> ( ( ( groups861055069439313189ex_nat @ G @ A4 )
= ( groups861055069439313189ex_nat @ H2 @ B5 ) )
= ( ( groups861055069439313189ex_nat @ G @ C4 )
= ( groups861055069439313189ex_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7835_prod_Osame__carrier,axiom,
! [C4: set_real,A4: set_real,B5: set_real,G: real > int,H2: real > int] :
( ( finite_finite_real @ C4 )
=> ( ( ord_less_eq_set_real @ A4 @ C4 )
=> ( ( ord_less_eq_set_real @ B5 @ C4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_int ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_int ) )
=> ( ( ( groups4694064378042380927al_int @ G @ A4 )
= ( groups4694064378042380927al_int @ H2 @ B5 ) )
= ( ( groups4694064378042380927al_int @ G @ C4 )
= ( groups4694064378042380927al_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7836_prod_Osame__carrier,axiom,
! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > int,H2: complex > int] :
( ( finite3207457112153483333omplex @ C4 )
=> ( ( ord_le211207098394363844omplex @ A4 @ C4 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
=> ( ( G @ A3 )
= one_one_int ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
=> ( ( H2 @ B3 )
= one_one_int ) )
=> ( ( ( groups858564598930262913ex_int @ G @ A4 )
= ( groups858564598930262913ex_int @ H2 @ B5 ) )
= ( ( groups858564598930262913ex_int @ G @ C4 )
= ( groups858564598930262913ex_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7837_arcsin__le__mono,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( arcsin @ X3 ) @ ( arcsin @ Y ) )
= ( ord_less_eq_real @ X3 @ Y ) ) ) ) ).
% arcsin_le_mono
thf(fact_7838_arcsin__eq__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( ( arcsin @ X3 )
= ( arcsin @ Y ) )
= ( X3 = Y ) ) ) ) ).
% arcsin_eq_iff
thf(fact_7839_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_7840_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_7841_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_7842_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_7843_prod_Onat__ivl__Suc_H,axiom,
! [M2: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( times_times_real @ ( G @ ( suc @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_7844_prod_Onat__ivl__Suc_H,axiom,
! [M2: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( times_times_rat @ ( G @ ( suc @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_7845_prod_Onat__ivl__Suc_H,axiom,
! [M2: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( times_times_nat @ ( G @ ( suc @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_7846_prod_Onat__ivl__Suc_H,axiom,
! [M2: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
= ( times_times_int @ ( G @ ( suc @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_7847_prod_OatLeast__Suc__atMost,axiom,
! [M2: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( times_times_real @ ( G @ M2 ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_7848_prod_OatLeast__Suc__atMost,axiom,
! [M2: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( times_times_rat @ ( G @ M2 ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_7849_prod_OatLeast__Suc__atMost,axiom,
! [M2: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( times_times_nat @ ( G @ M2 ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_7850_prod_OatLeast__Suc__atMost,axiom,
! [M2: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( times_times_int @ ( G @ M2 ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_7851_gbinomial__addition__formula,axiom,
! [A: complex,K2: nat] :
( ( gbinomial_complex @ A @ ( suc @ K2 ) )
= ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( suc @ K2 ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K2 ) ) ) ).
% gbinomial_addition_formula
thf(fact_7852_gbinomial__addition__formula,axiom,
! [A: real,K2: nat] :
( ( gbinomial_real @ A @ ( suc @ K2 ) )
= ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( suc @ K2 ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K2 ) ) ) ).
% gbinomial_addition_formula
thf(fact_7853_gbinomial__addition__formula,axiom,
! [A: rat,K2: nat] :
( ( gbinomial_rat @ A @ ( suc @ K2 ) )
= ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( suc @ K2 ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K2 ) ) ) ).
% gbinomial_addition_formula
thf(fact_7854_gbinomial__ge__n__over__k__pow__k,axiom,
! [K2: nat,A: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K2 ) @ A )
=> ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K2 ) ) @ K2 ) @ ( gbinomial_real @ A @ K2 ) ) ) ).
% gbinomial_ge_n_over_k_pow_k
thf(fact_7855_gbinomial__ge__n__over__k__pow__k,axiom,
! [K2: nat,A: rat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K2 ) @ A )
=> ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K2 ) ) @ K2 ) @ ( gbinomial_rat @ A @ K2 ) ) ) ).
% gbinomial_ge_n_over_k_pow_k
thf(fact_7856_gbinomial__mult__1,axiom,
! [A: rat,K2: nat] :
( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K2 ) )
= ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K2 ) @ ( gbinomial_rat @ A @ K2 ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K2 ) ) @ ( gbinomial_rat @ A @ ( suc @ K2 ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_7857_gbinomial__mult__1,axiom,
! [A: real,K2: nat] :
( ( times_times_real @ A @ ( gbinomial_real @ A @ K2 ) )
= ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K2 ) @ ( gbinomial_real @ A @ K2 ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K2 ) ) @ ( gbinomial_real @ A @ ( suc @ K2 ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_7858_gbinomial__mult__1,axiom,
! [A: complex,K2: nat] :
( ( times_times_complex @ A @ ( gbinomial_complex @ A @ K2 ) )
= ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K2 ) @ ( gbinomial_complex @ A @ K2 ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K2 ) ) @ ( gbinomial_complex @ A @ ( suc @ K2 ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_7859_gbinomial__mult__1_H,axiom,
! [A: rat,K2: nat] :
( ( times_times_rat @ ( gbinomial_rat @ A @ K2 ) @ A )
= ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K2 ) @ ( gbinomial_rat @ A @ K2 ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K2 ) ) @ ( gbinomial_rat @ A @ ( suc @ K2 ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_7860_gbinomial__mult__1_H,axiom,
! [A: real,K2: nat] :
( ( times_times_real @ ( gbinomial_real @ A @ K2 ) @ A )
= ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K2 ) @ ( gbinomial_real @ A @ K2 ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K2 ) ) @ ( gbinomial_real @ A @ ( suc @ K2 ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_7861_gbinomial__mult__1_H,axiom,
! [A: complex,K2: nat] :
( ( times_times_complex @ ( gbinomial_complex @ A @ K2 ) @ A )
= ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K2 ) @ ( gbinomial_complex @ A @ K2 ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K2 ) ) @ ( gbinomial_complex @ A @ ( suc @ K2 ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_7862_prod_OlessThan__Suc__shift,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_real @ ( G @ zero_zero_nat )
@ ( groups129246275422532515t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_7863_prod_OlessThan__Suc__shift,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( G @ zero_zero_nat )
@ ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_7864_prod_OlessThan__Suc__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( G @ zero_zero_nat )
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_7865_prod_OlessThan__Suc__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( G @ zero_zero_nat )
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_7866_prod_OSuc__reindex__ivl,axiom,
! [M2: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times_real @ ( G @ M2 )
@ ( groups129246275422532515t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_7867_prod_OSuc__reindex__ivl,axiom,
! [M2: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times_rat @ ( G @ M2 )
@ ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_7868_prod_OSuc__reindex__ivl,axiom,
! [M2: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times_nat @ ( G @ M2 )
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_7869_prod_OSuc__reindex__ivl,axiom,
! [M2: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times_int @ ( G @ M2 )
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_7870_prod_OatLeast1__atMost__eq,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups708209901874060359at_nat
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_7871_prod_OatLeast1__atMost__eq,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups705719431365010083at_int
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_7872_prod__atLeastAtMost__code,axiom,
! [F: nat > complex,A: nat,B: nat] :
( ( groups6464643781859351333omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo1517530859248394432omplex
@ ^ [A5: nat] : ( times_times_complex @ ( F @ A5 ) )
@ A
@ B
@ one_one_complex ) ) ).
% prod_atLeastAtMost_code
thf(fact_7873_prod__atLeastAtMost__code,axiom,
! [F: nat > real,A: nat,B: nat] :
( ( groups129246275422532515t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo3111899725591712190t_real
@ ^ [A5: nat] : ( times_times_real @ ( F @ A5 ) )
@ A
@ B
@ one_one_real ) ) ).
% prod_atLeastAtMost_code
thf(fact_7874_prod__atLeastAtMost__code,axiom,
! [F: nat > rat,A: nat,B: nat] :
( ( groups73079841787564623at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo1949268297981939178at_rat
@ ^ [A5: nat] : ( times_times_rat @ ( F @ A5 ) )
@ A
@ B
@ one_one_rat ) ) ).
% prod_atLeastAtMost_code
thf(fact_7875_prod__atLeastAtMost__code,axiom,
! [F: nat > nat,A: nat,B: nat] :
( ( groups708209901874060359at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo2584398358068434914at_nat
@ ^ [A5: nat] : ( times_times_nat @ ( F @ A5 ) )
@ A
@ B
@ one_one_nat ) ) ).
% prod_atLeastAtMost_code
thf(fact_7876_prod__atLeastAtMost__code,axiom,
! [F: nat > int,A: nat,B: nat] :
( ( groups705719431365010083at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo2581907887559384638at_int
@ ^ [A5: nat] : ( times_times_int @ ( F @ A5 ) )
@ A
@ B
@ one_one_int ) ) ).
% prod_atLeastAtMost_code
thf(fact_7877_prod__mono__strict,axiom,
! [A4: set_complex,F: complex > real,G: complex > real] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
& ( ord_less_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ( A4 != bot_bot_set_complex )
=> ( ord_less_real @ ( groups766887009212190081x_real @ F @ A4 ) @ ( groups766887009212190081x_real @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7878_prod__mono__strict,axiom,
! [A4: set_real,F: real > real,G: real > real] :
( ( finite_finite_real @ A4 )
=> ( ! [I2: real] :
( ( member_real @ I2 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
& ( ord_less_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ( A4 != bot_bot_set_real )
=> ( ord_less_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ ( groups1681761925125756287l_real @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7879_prod__mono__strict,axiom,
! [A4: set_nat,F: nat > real,G: nat > real] :
( ( finite_finite_nat @ A4 )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
& ( ord_less_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ( A4 != bot_bot_set_nat )
=> ( ord_less_real @ ( groups129246275422532515t_real @ F @ A4 ) @ ( groups129246275422532515t_real @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7880_prod__mono__strict,axiom,
! [A4: set_int,F: int > real,G: int > real] :
( ( finite_finite_int @ A4 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
& ( ord_less_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ( A4 != bot_bot_set_int )
=> ( ord_less_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ ( groups2316167850115554303t_real @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7881_prod__mono__strict,axiom,
! [A4: set_complex,F: complex > rat,G: complex > rat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
& ( ord_less_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ( A4 != bot_bot_set_complex )
=> ( ord_less_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) @ ( groups225925009352817453ex_rat @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7882_prod__mono__strict,axiom,
! [A4: set_real,F: real > rat,G: real > rat] :
( ( finite_finite_real @ A4 )
=> ( ! [I2: real] :
( ( member_real @ I2 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
& ( ord_less_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ( A4 != bot_bot_set_real )
=> ( ord_less_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ ( groups4061424788464935467al_rat @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7883_prod__mono__strict,axiom,
! [A4: set_nat,F: nat > rat,G: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
& ( ord_less_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ( A4 != bot_bot_set_nat )
=> ( ord_less_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ ( groups73079841787564623at_rat @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7884_prod__mono__strict,axiom,
! [A4: set_int,F: int > rat,G: int > rat] :
( ( finite_finite_int @ A4 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ A4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
& ( ord_less_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ( A4 != bot_bot_set_int )
=> ( ord_less_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ ( groups1072433553688619179nt_rat @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7885_prod__mono__strict,axiom,
! [A4: set_complex,F: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
& ( ord_less_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ( A4 != bot_bot_set_complex )
=> ( ord_less_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) @ ( groups861055069439313189ex_nat @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7886_prod__mono__strict,axiom,
! [A4: set_real,F: real > nat,G: real > nat] :
( ( finite_finite_real @ A4 )
=> ( ! [I2: real] :
( ( member_real @ I2 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
& ( ord_less_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ( A4 != bot_bot_set_real )
=> ( ord_less_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) @ ( groups4696554848551431203al_nat @ G @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7887_even__prod__iff,axiom,
! [A4: set_nat,F: nat > code_integer] :
( ( finite_finite_nat @ A4 )
=> ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups3455450783089532116nteger @ F @ A4 ) )
= ( ? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X5 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_7888_even__prod__iff,axiom,
! [A4: set_int,F: int > code_integer] :
( ( finite_finite_int @ A4 )
=> ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups3827104343326376752nteger @ F @ A4 ) )
= ( ? [X5: int] :
( ( member_int @ X5 @ A4 )
& ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X5 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_7889_even__prod__iff,axiom,
! [A4: set_complex,F: complex > code_integer] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups8682486955453173170nteger @ F @ A4 ) )
= ( ? [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X5 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_7890_even__prod__iff,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups1707563613775114915nt_nat @ F @ A4 ) )
= ( ? [X5: int] :
( ( member_int @ X5 @ A4 )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X5 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_7891_even__prod__iff,axiom,
! [A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups861055069439313189ex_nat @ F @ A4 ) )
= ( ? [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X5 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_7892_even__prod__iff,axiom,
! [A4: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups858564598930262913ex_int @ F @ A4 ) )
= ( ? [X5: complex] :
( ( member_complex @ X5 @ A4 )
& ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X5 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_7893_even__prod__iff,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups708209901874060359at_nat @ F @ A4 ) )
= ( ? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X5 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_7894_even__prod__iff,axiom,
! [A4: set_nat,F: nat > int] :
( ( finite_finite_nat @ A4 )
=> ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups705719431365010083at_int @ F @ A4 ) )
= ( ? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X5 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_7895_even__prod__iff,axiom,
! [A4: set_int,F: int > int] :
( ( finite_finite_int @ A4 )
=> ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups1705073143266064639nt_int @ F @ A4 ) )
= ( ? [X5: int] :
( ( member_int @ X5 @ A4 )
& ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X5 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_7896_even__prod__iff,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > code_integer] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups1230400874837758585nteger @ F @ A4 ) )
= ( ? [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ A4 )
& ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X5 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_7897_arccos__lbound,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) ) ) ) ).
% arccos_lbound
thf(fact_7898_arccos__less__arccos,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
=> ( ( ord_less_real @ X3 @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_real @ ( arccos @ Y ) @ ( arccos @ X3 ) ) ) ) ) ).
% arccos_less_arccos
thf(fact_7899_arccos__less__mono,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( ord_less_real @ ( arccos @ X3 ) @ ( arccos @ Y ) )
= ( ord_less_real @ Y @ X3 ) ) ) ) ).
% arccos_less_mono
thf(fact_7900_arccos__ubound,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ).
% arccos_ubound
thf(fact_7901_arccos__cos,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ pi )
=> ( ( arccos @ ( cos_real @ X3 ) )
= X3 ) ) ) ).
% arccos_cos
thf(fact_7902_arcsin__less__arcsin,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
=> ( ( ord_less_real @ X3 @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_real @ ( arcsin @ X3 ) @ ( arcsin @ Y ) ) ) ) ) ).
% arcsin_less_arcsin
thf(fact_7903_prod_Oub__add__nat,axiom,
! [M2: nat,N: nat,G: nat > real,P: nat] :
( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_7904_prod_Oub__add__nat,axiom,
! [M2: nat,N: nat,G: nat > rat,P: nat] :
( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_7905_prod_Oub__add__nat,axiom,
! [M2: nat,N: nat,G: nat > nat,P: nat] :
( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_7906_prod_Oub__add__nat,axiom,
! [M2: nat,N: nat,G: nat > int,P: nat] :
( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_7907_arcsin__less__mono,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( ord_less_real @ ( arcsin @ X3 ) @ ( arcsin @ Y ) )
= ( ord_less_real @ X3 @ Y ) ) ) ) ).
% arcsin_less_mono
thf(fact_7908_cos__arccos__abs,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( cos_real @ ( arccos @ Y ) )
= Y ) ) ).
% cos_arccos_abs
thf(fact_7909_arccos__cos__eq__abs,axiom,
! [Theta: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
=> ( ( arccos @ ( cos_real @ Theta ) )
= ( abs_abs_real @ Theta ) ) ) ).
% arccos_cos_eq_abs
thf(fact_7910_Suc__times__gbinomial,axiom,
! [K2: nat,A: rat] :
( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K2 ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K2 ) ) )
= ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K2 ) ) ) ).
% Suc_times_gbinomial
thf(fact_7911_Suc__times__gbinomial,axiom,
! [K2: nat,A: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K2 ) ) @ ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K2 ) ) )
= ( times_times_real @ ( plus_plus_real @ A @ one_one_real ) @ ( gbinomial_real @ A @ K2 ) ) ) ).
% Suc_times_gbinomial
thf(fact_7912_Suc__times__gbinomial,axiom,
! [K2: nat,A: complex] :
( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K2 ) ) @ ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K2 ) ) )
= ( times_times_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( gbinomial_complex @ A @ K2 ) ) ) ).
% Suc_times_gbinomial
thf(fact_7913_gbinomial__absorption,axiom,
! [K2: nat,A: rat] :
( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K2 ) ) @ ( gbinomial_rat @ A @ ( suc @ K2 ) ) )
= ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K2 ) ) ) ).
% gbinomial_absorption
thf(fact_7914_gbinomial__absorption,axiom,
! [K2: nat,A: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K2 ) ) @ ( gbinomial_real @ A @ ( suc @ K2 ) ) )
= ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K2 ) ) ) ).
% gbinomial_absorption
thf(fact_7915_gbinomial__absorption,axiom,
! [K2: nat,A: complex] :
( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K2 ) ) @ ( gbinomial_complex @ A @ ( suc @ K2 ) ) )
= ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K2 ) ) ) ).
% gbinomial_absorption
thf(fact_7916_gbinomial__trinomial__revision,axiom,
! [K2: nat,M2: nat,A: rat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( times_times_rat @ ( gbinomial_rat @ A @ M2 ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M2 ) @ K2 ) )
= ( times_times_rat @ ( gbinomial_rat @ A @ K2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K2 ) ) @ ( minus_minus_nat @ M2 @ K2 ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_7917_gbinomial__trinomial__revision,axiom,
! [K2: nat,M2: nat,A: real] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( times_times_real @ ( gbinomial_real @ A @ M2 ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M2 ) @ K2 ) )
= ( times_times_real @ ( gbinomial_real @ A @ K2 ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( minus_minus_nat @ M2 @ K2 ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_7918_gbinomial__trinomial__revision,axiom,
! [K2: nat,M2: nat,A: complex] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( times_times_complex @ ( gbinomial_complex @ A @ M2 ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M2 ) @ K2 ) )
= ( times_times_complex @ ( gbinomial_complex @ A @ K2 ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K2 ) ) @ ( minus_minus_nat @ M2 @ K2 ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_7919_norm__prod__diff,axiom,
! [I6: set_complex,Z2: complex > real,W2: complex > real] :
( ! [I2: complex] :
( ( member_complex @ I2 @ I6 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z2 @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I6 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W2 @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups766887009212190081x_real @ Z2 @ I6 ) @ ( groups766887009212190081x_real @ W2 @ I6 ) ) )
@ ( groups5808333547571424918x_real
@ ^ [I3: complex] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z2 @ I3 ) @ ( W2 @ I3 ) ) )
@ I6 ) ) ) ) ).
% norm_prod_diff
thf(fact_7920_norm__prod__diff,axiom,
! [I6: set_real,Z2: real > real,W2: real > real] :
( ! [I2: real] :
( ( member_real @ I2 @ I6 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z2 @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I6 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W2 @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups1681761925125756287l_real @ Z2 @ I6 ) @ ( groups1681761925125756287l_real @ W2 @ I6 ) ) )
@ ( groups8097168146408367636l_real
@ ^ [I3: real] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z2 @ I3 ) @ ( W2 @ I3 ) ) )
@ I6 ) ) ) ) ).
% norm_prod_diff
thf(fact_7921_norm__prod__diff,axiom,
! [I6: set_set_nat,Z2: set_nat > real,W2: set_nat > real] :
( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ I6 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z2 @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ I6 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W2 @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups3619160379726066777t_real @ Z2 @ I6 ) @ ( groups3619160379726066777t_real @ W2 @ I6 ) ) )
@ ( groups5107569545109728110t_real
@ ^ [I3: set_nat] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z2 @ I3 ) @ ( W2 @ I3 ) ) )
@ I6 ) ) ) ) ).
% norm_prod_diff
thf(fact_7922_norm__prod__diff,axiom,
! [I6: set_int,Z2: int > real,W2: int > real] :
( ! [I2: int] :
( ( member_int @ I2 @ I6 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z2 @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I6 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W2 @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups2316167850115554303t_real @ Z2 @ I6 ) @ ( groups2316167850115554303t_real @ W2 @ I6 ) ) )
@ ( groups8778361861064173332t_real
@ ^ [I3: int] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z2 @ I3 ) @ ( W2 @ I3 ) ) )
@ I6 ) ) ) ) ).
% norm_prod_diff
thf(fact_7923_norm__prod__diff,axiom,
! [I6: set_complex,Z2: complex > complex,W2: complex > complex] :
( ! [I2: complex] :
( ( member_complex @ I2 @ I6 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z2 @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I6 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W2 @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups3708469109370488835omplex @ Z2 @ I6 ) @ ( groups3708469109370488835omplex @ W2 @ I6 ) ) )
@ ( groups5808333547571424918x_real
@ ^ [I3: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z2 @ I3 ) @ ( W2 @ I3 ) ) )
@ I6 ) ) ) ) ).
% norm_prod_diff
thf(fact_7924_norm__prod__diff,axiom,
! [I6: set_real,Z2: real > complex,W2: real > complex] :
( ! [I2: real] :
( ( member_real @ I2 @ I6 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z2 @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I6 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W2 @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups713298508707869441omplex @ Z2 @ I6 ) @ ( groups713298508707869441omplex @ W2 @ I6 ) ) )
@ ( groups8097168146408367636l_real
@ ^ [I3: real] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z2 @ I3 ) @ ( W2 @ I3 ) ) )
@ I6 ) ) ) ) ).
% norm_prod_diff
thf(fact_7925_norm__prod__diff,axiom,
! [I6: set_set_nat,Z2: set_nat > complex,W2: set_nat > complex] :
( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ I6 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z2 @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ I6 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W2 @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups1092910753850256091omplex @ Z2 @ I6 ) @ ( groups1092910753850256091omplex @ W2 @ I6 ) ) )
@ ( groups5107569545109728110t_real
@ ^ [I3: set_nat] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z2 @ I3 ) @ ( W2 @ I3 ) ) )
@ I6 ) ) ) ) ).
% norm_prod_diff
thf(fact_7926_norm__prod__diff,axiom,
! [I6: set_int,Z2: int > complex,W2: int > complex] :
( ! [I2: int] :
( ( member_int @ I2 @ I6 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z2 @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I6 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W2 @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups7440179247065528705omplex @ Z2 @ I6 ) @ ( groups7440179247065528705omplex @ W2 @ I6 ) ) )
@ ( groups8778361861064173332t_real
@ ^ [I3: int] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z2 @ I3 ) @ ( W2 @ I3 ) ) )
@ I6 ) ) ) ) ).
% norm_prod_diff
thf(fact_7927_norm__prod__diff,axiom,
! [I6: set_nat,Z2: nat > real,W2: nat > real] :
( ! [I2: nat] :
( ( member_nat @ I2 @ I6 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z2 @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I6 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W2 @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups129246275422532515t_real @ Z2 @ I6 ) @ ( groups129246275422532515t_real @ W2 @ I6 ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z2 @ I3 ) @ ( W2 @ I3 ) ) )
@ I6 ) ) ) ) ).
% norm_prod_diff
thf(fact_7928_norm__prod__diff,axiom,
! [I6: set_nat,Z2: nat > complex,W2: nat > complex] :
( ! [I2: nat] :
( ( member_nat @ I2 @ I6 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z2 @ I2 ) ) @ one_one_real ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I6 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W2 @ I2 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups6464643781859351333omplex @ Z2 @ I6 ) @ ( groups6464643781859351333omplex @ W2 @ I6 ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z2 @ I3 ) @ ( W2 @ I3 ) ) )
@ I6 ) ) ) ) ).
% norm_prod_diff
thf(fact_7929_fact__eq__fact__times,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri1408675320244567234ct_nat @ M2 )
= ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
@ ( groups708209901874060359at_nat
@ ^ [X5: nat] : X5
@ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M2 ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_7930_prod__mono2,axiom,
! [B5: set_real,A4: set_real,F: real > real] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
=> ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ ( groups1681761925125756287l_real @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_7931_prod__mono2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
=> ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A4 ) @ ( groups766887009212190081x_real @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_7932_prod__mono2,axiom,
! [B5: set_nat,A4: set_nat,F: nat > real] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
=> ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
=> ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A4 ) @ ( groups129246275422532515t_real @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_7933_prod__mono2,axiom,
! [B5: set_real,A4: set_real,F: real > rat] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
=> ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ ( groups4061424788464935467al_rat @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_7934_prod__mono2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
=> ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) @ ( groups225925009352817453ex_rat @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_7935_prod__mono2,axiom,
! [B5: set_nat,A4: set_nat,F: nat > rat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
=> ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ ( groups73079841787564623at_rat @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_7936_prod__mono2,axiom,
! [B5: set_real,A4: set_real,F: real > int] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A4 @ B5 )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
=> ( ord_less_eq_int @ one_one_int @ ( F @ B3 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ A4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ A3 ) ) )
=> ( ord_less_eq_int @ ( groups4694064378042380927al_int @ F @ A4 ) @ ( groups4694064378042380927al_int @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_7937_prod__mono2,axiom,
! [B5: set_complex,A4: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A4 @ B5 )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
=> ( ord_less_eq_int @ one_one_int @ ( F @ B3 ) ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ A4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ A3 ) ) )
=> ( ord_less_eq_int @ ( groups858564598930262913ex_int @ F @ A4 ) @ ( groups858564598930262913ex_int @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_7938_prod__mono2,axiom,
! [B5: set_int,A4: set_int,F: int > real] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A4 ) )
=> ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ A4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
=> ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ ( groups2316167850115554303t_real @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_7939_prod__mono2,axiom,
! [B5: set_int,A4: set_int,F: int > rat] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ! [B3: int] :
( ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A4 ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ A4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
=> ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ ( groups1072433553688619179nt_rat @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_7940_arccos__lt__bounded,axiom,
! [Y: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_real @ Y @ one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y ) )
& ( ord_less_real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% arccos_lt_bounded
thf(fact_7941_arccos__bounded,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
& ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% arccos_bounded
thf(fact_7942_gbinomial__rec,axiom,
! [A: rat,K2: nat] :
( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K2 ) )
= ( times_times_rat @ ( gbinomial_rat @ A @ K2 ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K2 ) ) ) ) ) ).
% gbinomial_rec
thf(fact_7943_gbinomial__rec,axiom,
! [A: real,K2: nat] :
( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K2 ) )
= ( times_times_real @ ( gbinomial_real @ A @ K2 ) @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K2 ) ) ) ) ) ).
% gbinomial_rec
thf(fact_7944_gbinomial__rec,axiom,
! [A: complex,K2: nat] :
( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K2 ) )
= ( times_times_complex @ ( gbinomial_complex @ A @ K2 ) @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K2 ) ) ) ) ) ).
% gbinomial_rec
thf(fact_7945_gbinomial__factors,axiom,
! [A: rat,K2: nat] :
( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K2 ) )
= ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K2 ) ) ) @ ( gbinomial_rat @ A @ K2 ) ) ) ).
% gbinomial_factors
thf(fact_7946_gbinomial__factors,axiom,
! [A: real,K2: nat] :
( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K2 ) )
= ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K2 ) ) ) @ ( gbinomial_real @ A @ K2 ) ) ) ).
% gbinomial_factors
thf(fact_7947_gbinomial__factors,axiom,
! [A: complex,K2: nat] :
( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K2 ) )
= ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K2 ) ) ) @ ( gbinomial_complex @ A @ K2 ) ) ) ).
% gbinomial_factors
thf(fact_7948_sin__arccos__nonzero,axiom,
! [X3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
=> ( ( ord_less_real @ X3 @ one_one_real )
=> ( ( sin_real @ ( arccos @ X3 ) )
!= zero_zero_real ) ) ) ).
% sin_arccos_nonzero
thf(fact_7949_arccos__cos2,axiom,
! [X3: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X3 )
=> ( ( arccos @ ( cos_real @ X3 ) )
= ( uminus_uminus_real @ X3 ) ) ) ) ).
% arccos_cos2
thf(fact_7950_arccos__minus,axiom,
! [X3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
=> ( ( ord_less_eq_real @ X3 @ one_one_real )
=> ( ( arccos @ ( uminus_uminus_real @ X3 ) )
= ( minus_minus_real @ pi @ ( arccos @ X3 ) ) ) ) ) ).
% arccos_minus
thf(fact_7951_cos__arcsin__nonzero,axiom,
! [X3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
=> ( ( ord_less_real @ X3 @ one_one_real )
=> ( ( cos_real @ ( arcsin @ X3 ) )
!= zero_zero_real ) ) ) ).
% cos_arcsin_nonzero
thf(fact_7952_pochhammer__Suc__prod,axiom,
! [A: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
= ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_7953_pochhammer__Suc__prod,axiom,
! [A: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
= ( groups129246275422532515t_real
@ ^ [I3: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_7954_pochhammer__Suc__prod,axiom,
! [A: complex,N: nat] :
( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
= ( groups6464643781859351333omplex
@ ^ [I3: nat] : ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_7955_pochhammer__Suc__prod,axiom,
! [A: code_integer,N: nat] :
( ( comm_s8582702949713902594nteger @ A @ ( suc @ N ) )
= ( groups3455450783089532116nteger
@ ^ [I3: nat] : ( plus_p5714425477246183910nteger @ A @ ( semiri4939895301339042750nteger @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_7956_pochhammer__Suc__prod,axiom,
! [A: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_7957_pochhammer__Suc__prod,axiom,
! [A: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ I3 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_7958_pochhammer__prod__rev,axiom,
( comm_s4028243227959126397er_rat
= ( ^ [A5: rat,N4: nat] :
( groups73079841787564623at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ A5 @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N4 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_7959_pochhammer__prod__rev,axiom,
( comm_s7457072308508201937r_real
= ( ^ [A5: real,N4: nat] :
( groups129246275422532515t_real
@ ^ [I3: nat] : ( plus_plus_real @ A5 @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N4 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_7960_pochhammer__prod__rev,axiom,
( comm_s2602460028002588243omplex
= ( ^ [A5: complex,N4: nat] :
( groups6464643781859351333omplex
@ ^ [I3: nat] : ( plus_plus_complex @ A5 @ ( semiri8010041392384452111omplex @ ( minus_minus_nat @ N4 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_7961_pochhammer__prod__rev,axiom,
( comm_s8582702949713902594nteger
= ( ^ [A5: code_integer,N4: nat] :
( groups3455450783089532116nteger
@ ^ [I3: nat] : ( plus_p5714425477246183910nteger @ A5 @ ( semiri4939895301339042750nteger @ ( minus_minus_nat @ N4 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_7962_pochhammer__prod__rev,axiom,
( comm_s4663373288045622133er_nat
= ( ^ [A5: nat,N4: nat] :
( groups708209901874060359at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A5 @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N4 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_7963_pochhammer__prod__rev,axiom,
( comm_s4660882817536571857er_int
= ( ^ [A5: int,N4: nat] :
( groups705719431365010083at_int
@ ^ [I3: nat] : ( plus_plus_int @ A5 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N4 @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_7964_fact__div__fact,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) )
= ( groups708209901874060359at_nat
@ ^ [X5: nat] : X5
@ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M2 ) ) ) ) ).
% fact_div_fact
thf(fact_7965_one__or__eq,axiom,
! [A: code_integer] :
( ( bit_se1080825931792720795nteger @ one_one_Code_integer @ A )
= ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% one_or_eq
thf(fact_7966_one__or__eq,axiom,
! [A: int] :
( ( bit_se1409905431419307370or_int @ one_one_int @ A )
= ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% one_or_eq
thf(fact_7967_one__or__eq,axiom,
! [A: nat] :
( ( bit_se1412395901928357646or_nat @ one_one_nat @ A )
= ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% one_or_eq
thf(fact_7968_or__one__eq,axiom,
! [A: code_integer] :
( ( bit_se1080825931792720795nteger @ A @ one_one_Code_integer )
= ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% or_one_eq
thf(fact_7969_or__one__eq,axiom,
! [A: int] :
( ( bit_se1409905431419307370or_int @ A @ one_one_int )
= ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% or_one_eq
thf(fact_7970_or__one__eq,axiom,
! [A: nat] :
( ( bit_se1412395901928357646or_nat @ A @ one_one_nat )
= ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% or_one_eq
thf(fact_7971_OR__upper,axiom,
! [X3: int,N: nat,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_int @ X3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ord_less_int @ ( bit_se1409905431419307370or_int @ X3 @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% OR_upper
thf(fact_7972_gbinomial__minus,axiom,
! [A: rat,K2: nat] :
( ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K2 )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K2 ) ) @ one_one_rat ) @ K2 ) ) ) ).
% gbinomial_minus
thf(fact_7973_gbinomial__minus,axiom,
! [A: real,K2: nat] :
( ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K2 )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K2 ) ) @ one_one_real ) @ K2 ) ) ) ).
% gbinomial_minus
thf(fact_7974_gbinomial__minus,axiom,
! [A: complex,K2: nat] :
( ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K2 )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K2 ) ) @ one_one_complex ) @ K2 ) ) ) ).
% gbinomial_minus
thf(fact_7975_prod_Oin__pairs,axiom,
! [G: nat > real,M2: nat,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups129246275422532515t_real
@ ^ [I3: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% prod.in_pairs
thf(fact_7976_prod_Oin__pairs,axiom,
! [G: nat > rat,M2: nat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% prod.in_pairs
thf(fact_7977_prod_Oin__pairs,axiom,
! [G: nat > nat,M2: nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% prod.in_pairs
thf(fact_7978_prod_Oin__pairs,axiom,
! [G: nat > int,M2: nat,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% prod.in_pairs
thf(fact_7979_gbinomial__reduce__nat,axiom,
! [K2: nat,A: complex] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( gbinomial_complex @ A @ K2 )
= ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K2 ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_7980_gbinomial__reduce__nat,axiom,
! [K2: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( gbinomial_real @ A @ K2 )
= ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K2 ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_7981_gbinomial__reduce__nat,axiom,
! [K2: nat,A: rat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( gbinomial_rat @ A @ K2 )
= ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K2 ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_7982_pochhammer__Suc__prod__rev,axiom,
! [A: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
= ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_7983_pochhammer__Suc__prod__rev,axiom,
! [A: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
= ( groups129246275422532515t_real
@ ^ [I3: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_7984_pochhammer__Suc__prod__rev,axiom,
! [A: complex,N: nat] :
( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
= ( groups6464643781859351333omplex
@ ^ [I3: nat] : ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ ( minus_minus_nat @ N @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_7985_pochhammer__Suc__prod__rev,axiom,
! [A: code_integer,N: nat] :
( ( comm_s8582702949713902594nteger @ A @ ( suc @ N ) )
= ( groups3455450783089532116nteger
@ ^ [I3: nat] : ( plus_p5714425477246183910nteger @ A @ ( semiri4939895301339042750nteger @ ( minus_minus_nat @ N @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_7986_pochhammer__Suc__prod__rev,axiom,
! [A: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_7987_pochhammer__Suc__prod__rev,axiom,
! [A: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ I3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_7988_arccos,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
& ( ord_less_eq_real @ ( arccos @ Y ) @ pi )
& ( ( cos_real @ ( arccos @ Y ) )
= Y ) ) ) ) ).
% arccos
thf(fact_7989_gbinomial__pochhammer_H,axiom,
( gbinomial_rat
= ( ^ [A5: rat,K3: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A5 @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_7990_gbinomial__pochhammer_H,axiom,
( gbinomial_real
= ( ^ [A5: real,K3: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A5 @ ( semiri5074537144036343181t_real @ K3 ) ) @ one_one_real ) @ K3 ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_7991_gbinomial__pochhammer_H,axiom,
( gbinomial_complex
= ( ^ [A5: complex,K3: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A5 @ ( semiri8010041392384452111omplex @ K3 ) ) @ one_one_complex ) @ K3 ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_7992_arccos__minus__abs,axiom,
! [X3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
=> ( ( arccos @ ( uminus_uminus_real @ X3 ) )
= ( minus_minus_real @ pi @ ( arccos @ X3 ) ) ) ) ).
% arccos_minus_abs
thf(fact_7993_or__int__rec,axiom,
( bit_se1409905431419307370or_int
= ( ^ [K3: int,L2: int] :
( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
| ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% or_int_rec
thf(fact_7994_gbinomial__sum__up__index,axiom,
! [K2: nat,N: nat] :
( ( groups2906978787729119204at_rat
@ ^ [J3: nat] : ( gbinomial_rat @ ( semiri681578069525770553at_rat @ J3 ) @ K2 )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ).
% gbinomial_sum_up_index
thf(fact_7995_gbinomial__sum__up__index,axiom,
! [K2: nat,N: nat] :
( ( groups2073611262835488442omplex
@ ^ [J3: nat] : ( gbinomial_complex @ ( semiri8010041392384452111omplex @ J3 ) @ K2 )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ).
% gbinomial_sum_up_index
thf(fact_7996_gbinomial__sum__up__index,axiom,
! [K2: nat,N: nat] :
( ( groups6591440286371151544t_real
@ ^ [J3: nat] : ( gbinomial_real @ ( semiri5074537144036343181t_real @ J3 ) @ K2 )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ).
% gbinomial_sum_up_index
thf(fact_7997_gbinomial__absorption_H,axiom,
! [K2: nat,A: rat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( gbinomial_rat @ A @ K2 )
= ( times_times_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K2 ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_7998_gbinomial__absorption_H,axiom,
! [K2: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( gbinomial_real @ A @ K2 )
= ( times_times_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_7999_gbinomial__absorption_H,axiom,
! [K2: nat,A: complex] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( gbinomial_complex @ A @ K2 )
= ( times_times_complex @ ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ K2 ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_8000_arccos__le__pi2,axiom,
! [Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( arccos @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% arccos_le_pi2
thf(fact_8001_arcsin__lbound,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) ) ) ) ).
% arcsin_lbound
thf(fact_8002_arcsin__ubound,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% arcsin_ubound
thf(fact_8003_arcsin__bounded,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% arcsin_bounded
thf(fact_8004_arcsin__sin,axiom,
! [X3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
=> ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( arcsin @ ( sin_real @ X3 ) )
= X3 ) ) ) ).
% arcsin_sin
thf(fact_8005_arcsin,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( sin_real @ ( arcsin @ Y ) )
= Y ) ) ) ) ).
% arcsin
thf(fact_8006_arcsin__pi,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq_real @ ( arcsin @ Y ) @ pi )
& ( ( sin_real @ ( arcsin @ Y ) )
= Y ) ) ) ) ).
% arcsin_pi
thf(fact_8007_arcsin__le__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
=> ( ( ord_less_eq_real @ X3 @ one_one_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
=> ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( arcsin @ X3 ) @ Y )
= ( ord_less_eq_real @ X3 @ ( sin_real @ Y ) ) ) ) ) ) ) ).
% arcsin_le_iff
thf(fact_8008_le__arcsin__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
=> ( ( ord_less_eq_real @ X3 @ one_one_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
=> ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ Y @ ( arcsin @ X3 ) )
= ( ord_less_eq_real @ ( sin_real @ Y ) @ X3 ) ) ) ) ) ) ).
% le_arcsin_iff
thf(fact_8009_gbinomial__code,axiom,
( gbinomial_rat
= ( ^ [A5: rat,K3: nat] :
( if_rat @ ( K3 = zero_zero_nat ) @ one_one_rat
@ ( divide_divide_rat
@ ( set_fo1949268297981939178at_rat
@ ^ [L2: nat] : ( times_times_rat @ ( minus_minus_rat @ A5 @ ( semiri681578069525770553at_rat @ L2 ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ K3 @ one_one_nat )
@ one_one_rat )
@ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ) ).
% gbinomial_code
thf(fact_8010_gbinomial__code,axiom,
( gbinomial_real
= ( ^ [A5: real,K3: nat] :
( if_real @ ( K3 = zero_zero_nat ) @ one_one_real
@ ( divide_divide_real
@ ( set_fo3111899725591712190t_real
@ ^ [L2: nat] : ( times_times_real @ ( minus_minus_real @ A5 @ ( semiri5074537144036343181t_real @ L2 ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ K3 @ one_one_nat )
@ one_one_real )
@ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ) ).
% gbinomial_code
thf(fact_8011_gbinomial__code,axiom,
( gbinomial_complex
= ( ^ [A5: complex,K3: nat] :
( if_complex @ ( K3 = zero_zero_nat ) @ one_one_complex
@ ( divide1717551699836669952omplex
@ ( set_fo1517530859248394432omplex
@ ^ [L2: nat] : ( times_times_complex @ ( minus_minus_complex @ A5 @ ( semiri8010041392384452111omplex @ L2 ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ K3 @ one_one_nat )
@ one_one_complex )
@ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ) ).
% gbinomial_code
thf(fact_8012_gbinomial__partial__row__sum,axiom,
! [A: rat,M2: nat] :
( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
@ ( set_ord_atMost_nat @ M2 ) )
= ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M2 ) @ one_one_rat ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ A @ ( plus_plus_nat @ M2 @ one_one_nat ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_8013_gbinomial__partial__row__sum,axiom,
! [A: complex,M2: nat] :
( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
@ ( set_ord_atMost_nat @ M2 ) )
= ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M2 ) @ one_one_complex ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ A @ ( plus_plus_nat @ M2 @ one_one_nat ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_8014_gbinomial__partial__row__sum,axiom,
! [A: real,M2: nat] :
( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ A @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K3 ) ) )
@ ( set_ord_atMost_nat @ M2 ) )
= ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ A @ ( plus_plus_nat @ M2 @ one_one_nat ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_8015_gbinomial__r__part__sum,axiom,
! [M2: nat] :
( ( groups2906978787729119204at_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M2 ) ) @ one_one_rat ) ) @ ( set_ord_atMost_nat @ M2 ) )
= ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).
% gbinomial_r_part_sum
thf(fact_8016_gbinomial__r__part__sum,axiom,
! [M2: nat] :
( ( groups2073611262835488442omplex @ ( gbinomial_complex @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M2 ) ) @ one_one_complex ) ) @ ( set_ord_atMost_nat @ M2 ) )
= ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).
% gbinomial_r_part_sum
thf(fact_8017_gbinomial__r__part__sum,axiom,
! [M2: nat] :
( ( groups6591440286371151544t_real @ ( gbinomial_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ one_one_real ) ) @ ( set_ord_atMost_nat @ M2 ) )
= ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).
% gbinomial_r_part_sum
thf(fact_8018_central__binomial__lower__bound,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( 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 ) ) ) ) ).
% central_binomial_lower_bound
thf(fact_8019_Maclaurin__sin__bound,axiom,
! [X3: real,N: nat] :
( ord_less_eq_real
@ ( abs_abs_real
@ ( minus_minus_real @ ( sin_real @ X3 )
@ ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X3 @ M5 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) )
@ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X3 ) @ N ) ) ) ).
% Maclaurin_sin_bound
thf(fact_8020_divmod__BitM__2__eq,axiom,
! [M2: num] :
( ( unique5052692396658037445od_int @ ( bitM @ M2 ) @ ( bit0 @ one ) )
= ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ one_one_int ) ) ).
% divmod_BitM_2_eq
thf(fact_8021_atMost__eq__iff,axiom,
! [X3: nat,Y: nat] :
( ( ( set_ord_atMost_nat @ X3 )
= ( set_ord_atMost_nat @ Y ) )
= ( X3 = Y ) ) ).
% atMost_eq_iff
thf(fact_8022_atMost__eq__iff,axiom,
! [X3: int,Y: int] :
( ( ( set_ord_atMost_int @ X3 )
= ( set_ord_atMost_int @ Y ) )
= ( X3 = Y ) ) ).
% atMost_eq_iff
thf(fact_8023_inverse__nonzero__iff__nonzero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_8024_inverse__nonzero__iff__nonzero,axiom,
! [A: complex] :
( ( ( invers8013647133539491842omplex @ A )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_8025_inverse__nonzero__iff__nonzero,axiom,
! [A: rat] :
( ( ( inverse_inverse_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_8026_inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% inverse_zero
thf(fact_8027_inverse__zero,axiom,
( ( invers8013647133539491842omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% inverse_zero
thf(fact_8028_inverse__zero,axiom,
( ( inverse_inverse_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% inverse_zero
thf(fact_8029_atMost__iff,axiom,
! [I: real,K2: real] :
( ( member_real @ I @ ( set_ord_atMost_real @ K2 ) )
= ( ord_less_eq_real @ I @ K2 ) ) ).
% atMost_iff
thf(fact_8030_atMost__iff,axiom,
! [I: set_nat,K2: set_nat] :
( ( member_set_nat @ I @ ( set_or4236626031148496127et_nat @ K2 ) )
= ( ord_less_eq_set_nat @ I @ K2 ) ) ).
% atMost_iff
thf(fact_8031_atMost__iff,axiom,
! [I: set_int,K2: set_int] :
( ( member_set_int @ I @ ( set_or58775011639299419et_int @ K2 ) )
= ( ord_less_eq_set_int @ I @ K2 ) ) ).
% atMost_iff
thf(fact_8032_atMost__iff,axiom,
! [I: rat,K2: rat] :
( ( member_rat @ I @ ( set_ord_atMost_rat @ K2 ) )
= ( ord_less_eq_rat @ I @ K2 ) ) ).
% atMost_iff
thf(fact_8033_atMost__iff,axiom,
! [I: num,K2: num] :
( ( member_num @ I @ ( set_ord_atMost_num @ K2 ) )
= ( ord_less_eq_num @ I @ K2 ) ) ).
% atMost_iff
thf(fact_8034_atMost__iff,axiom,
! [I: nat,K2: nat] :
( ( member_nat @ I @ ( set_ord_atMost_nat @ K2 ) )
= ( ord_less_eq_nat @ I @ K2 ) ) ).
% atMost_iff
thf(fact_8035_atMost__iff,axiom,
! [I: int,K2: int] :
( ( member_int @ I @ ( set_ord_atMost_int @ K2 ) )
= ( ord_less_eq_int @ I @ K2 ) ) ).
% atMost_iff
thf(fact_8036_binomial__Suc__n,axiom,
! [N: nat] :
( ( binomial @ ( suc @ N ) @ N )
= ( suc @ N ) ) ).
% binomial_Suc_n
thf(fact_8037_finite__atMost,axiom,
! [K2: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K2 ) ) ).
% finite_atMost
thf(fact_8038_inverse__nonnegative__iff__nonnegative,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_8039_inverse__nonnegative__iff__nonnegative,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_8040_inverse__nonpositive__iff__nonpositive,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_8041_inverse__nonpositive__iff__nonpositive,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_8042_inverse__positive__iff__positive,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% inverse_positive_iff_positive
thf(fact_8043_inverse__positive__iff__positive,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% inverse_positive_iff_positive
thf(fact_8044_inverse__negative__iff__negative,axiom,
! [A: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% inverse_negative_iff_negative
thf(fact_8045_inverse__negative__iff__negative,axiom,
! [A: rat] :
( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% inverse_negative_iff_negative
thf(fact_8046_inverse__less__iff__less__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_real @ B @ A ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_8047_inverse__less__iff__less__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
= ( ord_less_rat @ B @ A ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_8048_inverse__less__iff__less,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_real @ B @ A ) ) ) ) ).
% inverse_less_iff_less
thf(fact_8049_inverse__less__iff__less,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
= ( ord_less_rat @ B @ A ) ) ) ) ).
% inverse_less_iff_less
thf(fact_8050_atMost__subset__iff,axiom,
! [X3: set_int,Y: set_int] :
( ( ord_le4403425263959731960et_int @ ( set_or58775011639299419et_int @ X3 ) @ ( set_or58775011639299419et_int @ Y ) )
= ( ord_less_eq_set_int @ X3 @ Y ) ) ).
% atMost_subset_iff
thf(fact_8051_atMost__subset__iff,axiom,
! [X3: rat,Y: rat] :
( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ X3 ) @ ( set_ord_atMost_rat @ Y ) )
= ( ord_less_eq_rat @ X3 @ Y ) ) ).
% atMost_subset_iff
thf(fact_8052_atMost__subset__iff,axiom,
! [X3: num,Y: num] :
( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ X3 ) @ ( set_ord_atMost_num @ Y ) )
= ( ord_less_eq_num @ X3 @ Y ) ) ).
% atMost_subset_iff
thf(fact_8053_atMost__subset__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X3 ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X3 @ Y ) ) ).
% atMost_subset_iff
thf(fact_8054_atMost__subset__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X3 ) @ ( set_ord_atMost_int @ Y ) )
= ( ord_less_eq_int @ X3 @ Y ) ) ).
% atMost_subset_iff
thf(fact_8055_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ zero_zero_nat ) )
= N ) ).
% binomial_1
thf(fact_8056_binomial__0__Suc,axiom,
! [K2: nat] :
( ( binomial @ zero_zero_nat @ ( suc @ K2 ) )
= zero_zero_nat ) ).
% binomial_0_Suc
thf(fact_8057_binomial__eq__0__iff,axiom,
! [N: nat,K2: nat] :
( ( ( binomial @ N @ K2 )
= zero_zero_nat )
= ( ord_less_nat @ N @ K2 ) ) ).
% binomial_eq_0_iff
thf(fact_8058_binomial__Suc__Suc,axiom,
! [N: nat,K2: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K2 ) )
= ( plus_plus_nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ ( suc @ K2 ) ) ) ) ).
% binomial_Suc_Suc
thf(fact_8059_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ zero_zero_nat )
= one_one_nat ) ).
% binomial_n_0
thf(fact_8060_prod__eq__1__iff,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( ( groups1707563613775114915nt_nat @ F @ A4 )
= one_one_nat )
= ( ! [X5: int] :
( ( member_int @ X5 @ A4 )
=> ( ( F @ X5 )
= one_one_nat ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_8061_prod__eq__1__iff,axiom,
! [A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ( groups861055069439313189ex_nat @ F @ A4 )
= one_one_nat )
= ( ! [X5: complex] :
( ( member_complex @ X5 @ A4 )
=> ( ( F @ X5 )
= one_one_nat ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_8062_prod__eq__1__iff,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ( ( groups4077766827762148844at_nat @ F @ A4 )
= one_one_nat )
= ( ! [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ A4 )
=> ( ( F @ X5 )
= one_one_nat ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_8063_prod__eq__1__iff,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( groups708209901874060359at_nat @ F @ A4 )
= one_one_nat )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ A4 )
=> ( ( F @ X5 )
= one_one_nat ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_8064_inverse__le__iff__le,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ) ).
% inverse_le_iff_le
thf(fact_8065_inverse__le__iff__le,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
= ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% inverse_le_iff_le
thf(fact_8066_inverse__le__iff__le__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_8067_inverse__le__iff__le__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
= ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_8068_right__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
= one_one_real ) ) ).
% right_inverse
thf(fact_8069_right__inverse,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( times_times_complex @ A @ ( invers8013647133539491842omplex @ A ) )
= one_one_complex ) ) ).
% right_inverse
thf(fact_8070_right__inverse,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( times_times_rat @ A @ ( inverse_inverse_rat @ A ) )
= one_one_rat ) ) ).
% right_inverse
thf(fact_8071_left__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
= one_one_real ) ) ).
% left_inverse
thf(fact_8072_left__inverse,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
= one_one_complex ) ) ).
% left_inverse
thf(fact_8073_left__inverse,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
= one_one_rat ) ) ).
% left_inverse
thf(fact_8074_or__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% or_nat_numerals(2)
thf(fact_8075_or__nat__numerals_I4_J,axiom,
! [X3: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
= ( numeral_numeral_nat @ ( bit1 @ X3 ) ) ) ).
% or_nat_numerals(4)
thf(fact_8076_Icc__subset__Iic__iff,axiom,
! [L: set_int,H2: set_int,H3: set_int] :
( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ L @ H2 ) @ ( set_or58775011639299419et_int @ H3 ) )
= ( ~ ( ord_less_eq_set_int @ L @ H2 )
| ( ord_less_eq_set_int @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8077_Icc__subset__Iic__iff,axiom,
! [L: rat,H2: rat,H3: rat] :
( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ L @ H2 ) @ ( set_ord_atMost_rat @ H3 ) )
= ( ~ ( ord_less_eq_rat @ L @ H2 )
| ( ord_less_eq_rat @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8078_Icc__subset__Iic__iff,axiom,
! [L: num,H2: num,H3: num] :
( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ L @ H2 ) @ ( set_ord_atMost_num @ H3 ) )
= ( ~ ( ord_less_eq_num @ L @ H2 )
| ( ord_less_eq_num @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8079_Icc__subset__Iic__iff,axiom,
! [L: nat,H2: nat,H3: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H2 ) @ ( set_ord_atMost_nat @ H3 ) )
= ( ~ ( ord_less_eq_nat @ L @ H2 )
| ( ord_less_eq_nat @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8080_Icc__subset__Iic__iff,axiom,
! [L: int,H2: int,H3: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L @ H2 ) @ ( set_ord_atMost_int @ H3 ) )
= ( ~ ( ord_less_eq_int @ L @ H2 )
| ( ord_less_eq_int @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8081_Icc__subset__Iic__iff,axiom,
! [L: real,H2: real,H3: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L @ H2 ) @ ( set_ord_atMost_real @ H3 ) )
= ( ~ ( ord_less_eq_real @ L @ H2 )
| ( ord_less_eq_real @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8082_sum_OatMost__Suc,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atMost_Suc
thf(fact_8083_sum_OatMost__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atMost_Suc
thf(fact_8084_sum_OatMost__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atMost_Suc
thf(fact_8085_sum_OatMost__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atMost_Suc
thf(fact_8086_zero__less__binomial__iff,axiom,
! [N: nat,K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K2 ) )
= ( ord_less_eq_nat @ K2 @ N ) ) ).
% zero_less_binomial_iff
thf(fact_8087_prod_OatMost__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atMost_Suc
thf(fact_8088_prod_OatMost__Suc,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atMost_Suc
thf(fact_8089_prod_OatMost__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atMost_Suc
thf(fact_8090_prod_OatMost__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atMost_Suc
thf(fact_8091_prod__pos__nat__iff,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) )
= ( ! [X5: int] :
( ( member_int @ X5 @ A4 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_8092_prod__pos__nat__iff,axiom,
! [A4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A4 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) )
= ( ! [X5: complex] :
( ( member_complex @ X5 @ A4 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_8093_prod__pos__nat__iff,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( groups4077766827762148844at_nat @ F @ A4 ) )
= ( ! [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ A4 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_8094_prod__pos__nat__iff,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A4 ) )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ A4 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_8095_or__nat__numerals_I3_J,axiom,
! [X3: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
= ( numeral_numeral_nat @ ( bit1 @ X3 ) ) ) ).
% or_nat_numerals(3)
thf(fact_8096_or__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% or_nat_numerals(1)
thf(fact_8097_int__prod,axiom,
! [F: int > nat,A4: set_int] :
( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F @ A4 ) )
= ( groups1705073143266064639nt_int
@ ^ [X5: int] : ( semiri1314217659103216013at_int @ ( F @ X5 ) )
@ A4 ) ) ).
% int_prod
thf(fact_8098_int__prod,axiom,
! [F: nat > nat,A4: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F @ A4 ) )
= ( groups705719431365010083at_int
@ ^ [X5: nat] : ( semiri1314217659103216013at_int @ ( F @ X5 ) )
@ A4 ) ) ).
% int_prod
thf(fact_8099_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% field_class.field_inverse_zero
thf(fact_8100_field__class_Ofield__inverse__zero,axiom,
( ( invers8013647133539491842omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% field_class.field_inverse_zero
thf(fact_8101_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% field_class.field_inverse_zero
thf(fact_8102_inverse__zero__imp__zero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ).
% inverse_zero_imp_zero
thf(fact_8103_inverse__zero__imp__zero,axiom,
! [A: complex] :
( ( ( invers8013647133539491842omplex @ A )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ).
% inverse_zero_imp_zero
thf(fact_8104_inverse__zero__imp__zero,axiom,
! [A: rat] :
( ( ( inverse_inverse_rat @ A )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ).
% inverse_zero_imp_zero
thf(fact_8105_nonzero__inverse__eq__imp__eq,axiom,
! [A: real,B: real] :
( ( ( inverse_inverse_real @ A )
= ( inverse_inverse_real @ B ) )
=> ( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( A = B ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_8106_nonzero__inverse__eq__imp__eq,axiom,
! [A: complex,B: complex] :
( ( ( invers8013647133539491842omplex @ A )
= ( invers8013647133539491842omplex @ B ) )
=> ( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( A = B ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_8107_nonzero__inverse__eq__imp__eq,axiom,
! [A: rat,B: rat] :
( ( ( inverse_inverse_rat @ A )
= ( inverse_inverse_rat @ B ) )
=> ( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( A = B ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_8108_nonzero__inverse__inverse__eq,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
= A ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_8109_nonzero__inverse__inverse__eq,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
= A ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_8110_nonzero__inverse__inverse__eq,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
= A ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_8111_nonzero__imp__inverse__nonzero,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ A )
!= zero_zero_real ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_8112_nonzero__imp__inverse__nonzero,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ A )
!= zero_zero_complex ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_8113_nonzero__imp__inverse__nonzero,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( inverse_inverse_rat @ A )
!= zero_zero_rat ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_8114_nonzero__norm__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ A ) )
= ( inverse_inverse_real @ ( real_V7735802525324610683m_real @ A ) ) ) ) ).
% nonzero_norm_inverse
thf(fact_8115_nonzero__norm__inverse,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ A ) )
= ( inverse_inverse_real @ ( real_V1022390504157884413omplex @ A ) ) ) ) ).
% nonzero_norm_inverse
thf(fact_8116_sum__choose__upper,axiom,
! [M2: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( binomial @ K3 @ M2 )
@ ( set_ord_atMost_nat @ N ) )
= ( binomial @ ( suc @ N ) @ ( suc @ M2 ) ) ) ).
% sum_choose_upper
thf(fact_8117_not__empty__eq__Iic__eq__empty,axiom,
! [H2: real] :
( bot_bot_set_real
!= ( set_ord_atMost_real @ H2 ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_8118_not__empty__eq__Iic__eq__empty,axiom,
! [H2: nat] :
( bot_bot_set_nat
!= ( set_ord_atMost_nat @ H2 ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_8119_not__empty__eq__Iic__eq__empty,axiom,
! [H2: int] :
( bot_bot_set_int
!= ( set_ord_atMost_int @ H2 ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_8120_infinite__Iic,axiom,
! [A: int] :
~ ( finite_finite_int @ ( set_ord_atMost_int @ A ) ) ).
% infinite_Iic
thf(fact_8121_not__Iic__eq__Icc,axiom,
! [H3: int,L: int,H2: int] :
( ( set_ord_atMost_int @ H3 )
!= ( set_or1266510415728281911st_int @ L @ H2 ) ) ).
% not_Iic_eq_Icc
thf(fact_8122_not__Iic__eq__Icc,axiom,
! [H3: real,L: real,H2: real] :
( ( set_ord_atMost_real @ H3 )
!= ( set_or1222579329274155063t_real @ L @ H2 ) ) ).
% not_Iic_eq_Icc
thf(fact_8123_atMost__def,axiom,
( set_ord_atMost_real
= ( ^ [U2: real] :
( collect_real
@ ^ [X5: real] : ( ord_less_eq_real @ X5 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_8124_atMost__def,axiom,
( set_or4236626031148496127et_nat
= ( ^ [U2: set_nat] :
( collect_set_nat
@ ^ [X5: set_nat] : ( ord_less_eq_set_nat @ X5 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_8125_atMost__def,axiom,
( set_or58775011639299419et_int
= ( ^ [U2: set_int] :
( collect_set_int
@ ^ [X5: set_int] : ( ord_less_eq_set_int @ X5 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_8126_atMost__def,axiom,
( set_ord_atMost_rat
= ( ^ [U2: rat] :
( collect_rat
@ ^ [X5: rat] : ( ord_less_eq_rat @ X5 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_8127_atMost__def,axiom,
( set_ord_atMost_num
= ( ^ [U2: num] :
( collect_num
@ ^ [X5: num] : ( ord_less_eq_num @ X5 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_8128_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X5: nat] : ( ord_less_eq_nat @ X5 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_8129_atMost__def,axiom,
( set_ord_atMost_int
= ( ^ [U2: int] :
( collect_int
@ ^ [X5: int] : ( ord_less_eq_int @ X5 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_8130_sum__choose__lower,axiom,
! [R2: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K3 ) @ K3 )
@ ( set_ord_atMost_nat @ N ) )
= ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N ) ) @ N ) ) ).
% sum_choose_lower
thf(fact_8131_choose__rising__sum_I2_J,axiom,
! [N: nat,M2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
@ ( set_ord_atMost_nat @ M2 ) )
= ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M2 ) @ one_one_nat ) @ M2 ) ) ).
% choose_rising_sum(2)
thf(fact_8132_choose__rising__sum_I1_J,axiom,
! [N: nat,M2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
@ ( set_ord_atMost_nat @ M2 ) )
= ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M2 ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% choose_rising_sum(1)
thf(fact_8133_norm__inverse__le__norm,axiom,
! [R2: real,X3: real] :
( ( ord_less_eq_real @ R2 @ ( real_V7735802525324610683m_real @ X3 ) )
=> ( ( ord_less_real @ zero_zero_real @ R2 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ X3 ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).
% norm_inverse_le_norm
thf(fact_8134_norm__inverse__le__norm,axiom,
! [R2: real,X3: complex] :
( ( ord_less_eq_real @ R2 @ ( real_V1022390504157884413omplex @ X3 ) )
=> ( ( ord_less_real @ zero_zero_real @ R2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ X3 ) ) @ ( inverse_inverse_real @ R2 ) ) ) ) ).
% norm_inverse_le_norm
thf(fact_8135_binomial__eq__0,axiom,
! [N: nat,K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( ( binomial @ N @ K2 )
= zero_zero_nat ) ) ).
% binomial_eq_0
thf(fact_8136_positive__imp__inverse__positive,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) ) ) ).
% positive_imp_inverse_positive
thf(fact_8137_positive__imp__inverse__positive,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) ) ) ).
% positive_imp_inverse_positive
thf(fact_8138_negative__imp__inverse__negative,axiom,
! [A: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real ) ) ).
% negative_imp_inverse_negative
thf(fact_8139_negative__imp__inverse__negative,axiom,
! [A: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat ) ) ).
% negative_imp_inverse_negative
thf(fact_8140_inverse__positive__imp__positive,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
=> ( ( A != zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ).
% inverse_positive_imp_positive
thf(fact_8141_inverse__positive__imp__positive,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
=> ( ( A != zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ A ) ) ) ).
% inverse_positive_imp_positive
thf(fact_8142_inverse__negative__imp__negative,axiom,
! [A: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
=> ( ( A != zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% inverse_negative_imp_negative
thf(fact_8143_inverse__negative__imp__negative,axiom,
! [A: rat] :
( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
=> ( ( A != zero_zero_rat )
=> ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% inverse_negative_imp_negative
thf(fact_8144_less__imp__inverse__less__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_8145_less__imp__inverse__less__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_8146_inverse__less__imp__less__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_8147_inverse__less__imp__less__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ B @ A ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_8148_less__imp__inverse__less,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% less_imp_inverse_less
thf(fact_8149_less__imp__inverse__less,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% less_imp_inverse_less
thf(fact_8150_inverse__less__imp__less,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ B @ A ) ) ) ).
% inverse_less_imp_less
thf(fact_8151_inverse__less__imp__less,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_rat @ B @ A ) ) ) ).
% inverse_less_imp_less
thf(fact_8152_nonzero__inverse__mult__distrib,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
= ( times_times_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_8153_nonzero__inverse__mult__distrib,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B ) )
= ( times_times_complex @ ( invers8013647133539491842omplex @ B ) @ ( invers8013647133539491842omplex @ A ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_8154_nonzero__inverse__mult__distrib,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B ) )
= ( times_times_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_8155_nonzero__inverse__minus__eq,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
= ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_8156_nonzero__inverse__minus__eq,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
= ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_8157_nonzero__inverse__minus__eq,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ A ) )
= ( uminus_uminus_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_8158_Suc__times__binomial,axiom,
! [K2: nat,N: nat] :
( ( times_times_nat @ ( suc @ K2 ) @ ( binomial @ ( suc @ N ) @ ( suc @ K2 ) ) )
= ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K2 ) ) ) ).
% Suc_times_binomial
thf(fact_8159_Suc__times__binomial__eq,axiom,
! [N: nat,K2: nat] :
( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K2 ) )
= ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K2 ) ) @ ( suc @ K2 ) ) ) ).
% Suc_times_binomial_eq
thf(fact_8160_binomial__symmetric,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( binomial @ N @ K2 )
= ( binomial @ N @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).
% binomial_symmetric
thf(fact_8161_power__mult__power__inverse__commute,axiom,
! [X3: real,M2: nat,N: nat] :
( ( times_times_real @ ( power_power_real @ X3 @ M2 ) @ ( power_power_real @ ( inverse_inverse_real @ X3 ) @ N ) )
= ( times_times_real @ ( power_power_real @ ( inverse_inverse_real @ X3 ) @ N ) @ ( power_power_real @ X3 @ M2 ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_8162_power__mult__power__inverse__commute,axiom,
! [X3: complex,M2: nat,N: nat] :
( ( times_times_complex @ ( power_power_complex @ X3 @ M2 ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X3 ) @ N ) )
= ( times_times_complex @ ( power_power_complex @ ( invers8013647133539491842omplex @ X3 ) @ N ) @ ( power_power_complex @ X3 @ M2 ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_8163_power__mult__power__inverse__commute,axiom,
! [X3: rat,M2: nat,N: nat] :
( ( times_times_rat @ ( power_power_rat @ X3 @ M2 ) @ ( power_power_rat @ ( inverse_inverse_rat @ X3 ) @ N ) )
= ( times_times_rat @ ( power_power_rat @ ( inverse_inverse_rat @ X3 ) @ N ) @ ( power_power_rat @ X3 @ M2 ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_8164_power__mult__inverse__distrib,axiom,
! [X3: real,M2: nat] :
( ( times_times_real @ ( power_power_real @ X3 @ M2 ) @ ( inverse_inverse_real @ X3 ) )
= ( times_times_real @ ( inverse_inverse_real @ X3 ) @ ( power_power_real @ X3 @ M2 ) ) ) ).
% power_mult_inverse_distrib
thf(fact_8165_power__mult__inverse__distrib,axiom,
! [X3: complex,M2: nat] :
( ( times_times_complex @ ( power_power_complex @ X3 @ M2 ) @ ( invers8013647133539491842omplex @ X3 ) )
= ( times_times_complex @ ( invers8013647133539491842omplex @ X3 ) @ ( power_power_complex @ X3 @ M2 ) ) ) ).
% power_mult_inverse_distrib
thf(fact_8166_power__mult__inverse__distrib,axiom,
! [X3: rat,M2: nat] :
( ( times_times_rat @ ( power_power_rat @ X3 @ M2 ) @ ( inverse_inverse_rat @ X3 ) )
= ( times_times_rat @ ( inverse_inverse_rat @ X3 ) @ ( power_power_rat @ X3 @ M2 ) ) ) ).
% power_mult_inverse_distrib
thf(fact_8167_choose__mult__lemma,axiom,
! [M2: nat,R2: nat,K2: nat] :
( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M2 @ R2 ) @ K2 ) @ ( plus_plus_nat @ M2 @ K2 ) ) @ ( binomial @ ( plus_plus_nat @ M2 @ K2 ) @ K2 ) )
= ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M2 @ R2 ) @ K2 ) @ K2 ) @ ( binomial @ ( plus_plus_nat @ M2 @ R2 ) @ M2 ) ) ) ).
% choose_mult_lemma
thf(fact_8168_nonzero__abs__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
= ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ) ).
% nonzero_abs_inverse
thf(fact_8169_nonzero__abs__inverse,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( abs_abs_rat @ ( inverse_inverse_rat @ A ) )
= ( inverse_inverse_rat @ ( abs_abs_rat @ A ) ) ) ) ).
% nonzero_abs_inverse
thf(fact_8170_binomial__le__pow,axiom,
! [R2: nat,N: nat] :
( ( ord_less_eq_nat @ R2 @ N )
=> ( ord_less_eq_nat @ ( binomial @ N @ R2 ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% binomial_le_pow
thf(fact_8171_mult__inverse__of__int__commute,axiom,
! [Xa2: int,X3: real] :
( ( times_times_real @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa2 ) ) @ X3 )
= ( times_times_real @ X3 @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa2 ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_8172_mult__inverse__of__int__commute,axiom,
! [Xa2: int,X3: complex] :
( ( times_times_complex @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa2 ) ) @ X3 )
= ( times_times_complex @ X3 @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa2 ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_8173_mult__inverse__of__int__commute,axiom,
! [Xa2: int,X3: rat] :
( ( times_times_rat @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa2 ) ) @ X3 )
= ( times_times_rat @ X3 @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa2 ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_8174_atMost__atLeast0,axiom,
( set_ord_atMost_nat
= ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% atMost_atLeast0
thf(fact_8175_lessThan__Suc__atMost,axiom,
! [K2: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K2 ) )
= ( set_ord_atMost_nat @ K2 ) ) ).
% lessThan_Suc_atMost
thf(fact_8176_sum__choose__diagonal,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N @ K3 ) @ ( minus_minus_nat @ M2 @ K3 ) )
@ ( set_ord_atMost_nat @ M2 ) )
= ( binomial @ ( suc @ N ) @ M2 ) ) ) ).
% sum_choose_diagonal
thf(fact_8177_vandermonde,axiom,
! [M2: nat,N: nat,R2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M2 @ K3 ) @ ( binomial @ N @ ( minus_minus_nat @ R2 @ K3 ) ) )
@ ( set_ord_atMost_nat @ R2 ) )
= ( binomial @ ( plus_plus_nat @ M2 @ N ) @ R2 ) ) ).
% vandermonde
thf(fact_8178_not__Iic__le__Icc,axiom,
! [H2: int,L3: int,H3: int] :
~ ( ord_less_eq_set_int @ ( set_ord_atMost_int @ H2 ) @ ( set_or1266510415728281911st_int @ L3 @ H3 ) ) ).
% not_Iic_le_Icc
thf(fact_8179_not__Iic__le__Icc,axiom,
! [H2: real,L3: real,H3: real] :
~ ( ord_less_eq_set_real @ ( set_ord_atMost_real @ H2 ) @ ( set_or1222579329274155063t_real @ L3 @ H3 ) ) ).
% not_Iic_le_Icc
thf(fact_8180_prod_Otriangle__reindex__eq,axiom,
! [G: nat > nat > nat,N: nat] :
( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I3: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [K3: nat] :
( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_8181_prod_Otriangle__reindex__eq,axiom,
! [G: nat > nat > int,N: nat] :
( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I3: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups705719431365010083at_int
@ ^ [K3: nat] :
( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_8182_finite__nat__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [S6: set_nat] :
? [K3: nat] : ( ord_less_eq_set_nat @ S6 @ ( set_ord_atMost_nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded_le
thf(fact_8183_binomial,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
= ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial
thf(fact_8184_inverse__le__imp__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ B @ A ) ) ) ).
% inverse_le_imp_le
thf(fact_8185_inverse__le__imp__le,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_eq_rat @ B @ A ) ) ) ).
% inverse_le_imp_le
thf(fact_8186_le__imp__inverse__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% le_imp_inverse_le
thf(fact_8187_le__imp__inverse__le,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_eq_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% le_imp_inverse_le
thf(fact_8188_inverse__le__imp__le__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_8189_inverse__le__imp__le__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ A ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_8190_le__imp__inverse__le__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_8191_le__imp__inverse__le__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_8192_inverse__le__1__iff,axiom,
! [X3: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ X3 ) @ one_one_real )
= ( ( ord_less_eq_real @ X3 @ zero_zero_real )
| ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).
% inverse_le_1_iff
thf(fact_8193_inverse__le__1__iff,axiom,
! [X3: rat] :
( ( ord_less_eq_rat @ ( inverse_inverse_rat @ X3 ) @ one_one_rat )
= ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
| ( ord_less_eq_rat @ one_one_rat @ X3 ) ) ) ).
% inverse_le_1_iff
thf(fact_8194_one__less__inverse__iff,axiom,
! [X3: real] :
( ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ X3 ) )
= ( ( ord_less_real @ zero_zero_real @ X3 )
& ( ord_less_real @ X3 @ one_one_real ) ) ) ).
% one_less_inverse_iff
thf(fact_8195_one__less__inverse__iff,axiom,
! [X3: rat] :
( ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ X3 ) )
= ( ( ord_less_rat @ zero_zero_rat @ X3 )
& ( ord_less_rat @ X3 @ one_one_rat ) ) ) ).
% one_less_inverse_iff
thf(fact_8196_one__less__inverse,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% one_less_inverse
thf(fact_8197_one__less__inverse,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ A @ one_one_rat )
=> ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% one_less_inverse
thf(fact_8198_zero__less__binomial,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K2 ) ) ) ).
% zero_less_binomial
thf(fact_8199_field__class_Ofield__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
= one_one_real ) ) ).
% field_class.field_inverse
thf(fact_8200_field__class_Ofield__inverse,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
= one_one_complex ) ) ).
% field_class.field_inverse
thf(fact_8201_field__class_Ofield__inverse,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
= one_one_rat ) ) ).
% field_class.field_inverse
thf(fact_8202_division__ring__inverse__add,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( plus_plus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_8203_division__ring__inverse__add,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
= ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( plus_plus_complex @ A @ B ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_8204_division__ring__inverse__add,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
= ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( plus_plus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_8205_inverse__add,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( times_times_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( inverse_inverse_real @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% inverse_add
thf(fact_8206_inverse__add,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
= ( times_times_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( invers8013647133539491842omplex @ A ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% inverse_add
thf(fact_8207_inverse__add,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
= ( times_times_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( inverse_inverse_rat @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% inverse_add
thf(fact_8208_division__ring__inverse__diff,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ B @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_8209_division__ring__inverse__diff,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
= ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ B @ A ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_8210_division__ring__inverse__diff,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
= ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ B @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_8211_nonzero__inverse__eq__divide,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ A )
= ( divide_divide_real @ one_one_real @ A ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_8212_nonzero__inverse__eq__divide,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ A )
= ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_8213_nonzero__inverse__eq__divide,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( inverse_inverse_rat @ A )
= ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_8214_Suc__times__binomial__add,axiom,
! [A: nat,B: nat] :
( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
= ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).
% Suc_times_binomial_add
thf(fact_8215_binomial__ring,axiom,
! [A: rat,B: rat,N: nat] :
( ( power_power_rat @ ( plus_plus_rat @ A @ B ) @ N )
= ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K3 ) ) @ ( power_power_rat @ A @ K3 ) ) @ ( power_power_rat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial_ring
thf(fact_8216_binomial__ring,axiom,
! [A: int,B: int,N: nat] :
( ( power_power_int @ ( plus_plus_int @ A @ B ) @ N )
= ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N @ K3 ) ) @ ( power_power_int @ A @ K3 ) ) @ ( power_power_int @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial_ring
thf(fact_8217_binomial__ring,axiom,
! [A: complex,B: complex,N: nat] :
( ( power_power_complex @ ( plus_plus_complex @ A @ B ) @ N )
= ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K3 ) ) @ ( power_power_complex @ A @ K3 ) ) @ ( power_power_complex @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial_ring
thf(fact_8218_binomial__ring,axiom,
! [A: code_integer,B: code_integer,N: nat] :
( ( power_8256067586552552935nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ N )
= ( groups7501900531339628137nteger
@ ^ [K3: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( binomial @ N @ K3 ) ) @ ( power_8256067586552552935nteger @ A @ K3 ) ) @ ( power_8256067586552552935nteger @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial_ring
thf(fact_8219_binomial__ring,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
= ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial_ring
thf(fact_8220_binomial__ring,axiom,
! [A: real,B: real,N: nat] :
( ( power_power_real @ ( plus_plus_real @ A @ B ) @ N )
= ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K3 ) ) @ ( power_power_real @ A @ K3 ) ) @ ( power_power_real @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial_ring
thf(fact_8221_choose__mult,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( times_times_nat @ ( binomial @ N @ M2 ) @ ( binomial @ M2 @ K2 ) )
= ( times_times_nat @ ( binomial @ N @ K2 ) @ ( binomial @ ( minus_minus_nat @ N @ K2 ) @ ( minus_minus_nat @ M2 @ K2 ) ) ) ) ) ) ).
% choose_mult
thf(fact_8222_binomial__Suc__Suc__eq__times,axiom,
! [N: nat,K2: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K2 ) )
= ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K2 ) ) @ ( suc @ K2 ) ) ) ).
% binomial_Suc_Suc_eq_times
thf(fact_8223_pochhammer__binomial__sum,axiom,
! [A: rat,B: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ B ) @ N )
= ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K3 ) ) @ ( comm_s4028243227959126397er_rat @ A @ K3 ) ) @ ( comm_s4028243227959126397er_rat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% pochhammer_binomial_sum
thf(fact_8224_pochhammer__binomial__sum,axiom,
! [A: int,B: int,N: nat] :
( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ B ) @ N )
= ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N @ K3 ) ) @ ( comm_s4660882817536571857er_int @ A @ K3 ) ) @ ( comm_s4660882817536571857er_int @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% pochhammer_binomial_sum
thf(fact_8225_pochhammer__binomial__sum,axiom,
! [A: complex,B: complex,N: nat] :
( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ B ) @ N )
= ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K3 ) ) @ ( comm_s2602460028002588243omplex @ A @ K3 ) ) @ ( comm_s2602460028002588243omplex @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% pochhammer_binomial_sum
thf(fact_8226_pochhammer__binomial__sum,axiom,
! [A: code_integer,B: code_integer,N: nat] :
( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ N )
= ( groups7501900531339628137nteger
@ ^ [K3: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( binomial @ N @ K3 ) ) @ ( comm_s8582702949713902594nteger @ A @ K3 ) ) @ ( comm_s8582702949713902594nteger @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% pochhammer_binomial_sum
thf(fact_8227_pochhammer__binomial__sum,axiom,
! [A: real,B: real,N: nat] :
( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ B ) @ N )
= ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K3 ) ) @ ( comm_s7457072308508201937r_real @ A @ K3 ) ) @ ( comm_s7457072308508201937r_real @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% pochhammer_binomial_sum
thf(fact_8228_prod_Otriangle__reindex,axiom,
! [G: nat > nat > nat,N: nat] :
( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I3: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [K3: nat] :
( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.triangle_reindex
thf(fact_8229_prod_Otriangle__reindex,axiom,
! [G: nat > nat > int,N: nat] :
( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I3: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups705719431365010083at_int
@ ^ [K3: nat] :
( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.triangle_reindex
thf(fact_8230_Iic__subset__Iio__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ A ) @ ( set_ord_lessThan_rat @ B ) )
= ( ord_less_rat @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_8231_Iic__subset__Iio__iff,axiom,
! [A: num,B: num] :
( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ A ) @ ( set_ord_lessThan_num @ B ) )
= ( ord_less_num @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_8232_Iic__subset__Iio__iff,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_8233_Iic__subset__Iio__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ A ) @ ( set_ord_lessThan_int @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_8234_Iic__subset__Iio__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ A ) @ ( set_or5984915006950818249n_real @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_8235_prod__int__eq,axiom,
! [I: nat,J: nat] :
( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ J ) )
= ( groups1705073143266064639nt_int
@ ^ [X5: int] : X5
@ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).
% prod_int_eq
thf(fact_8236_eval__nat__numeral_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).
% eval_nat_numeral(2)
thf(fact_8237_inverse__le__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_eq_real @ B @ A ) )
& ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
=> ( ord_less_eq_real @ A @ B ) ) ) ) ).
% inverse_le_iff
thf(fact_8238_inverse__le__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ord_less_eq_rat @ B @ A ) )
& ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
=> ( ord_less_eq_rat @ A @ B ) ) ) ) ).
% inverse_le_iff
thf(fact_8239_inverse__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ B @ A ) )
& ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
=> ( ord_less_real @ A @ B ) ) ) ) ).
% inverse_less_iff
thf(fact_8240_inverse__less__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ord_less_rat @ B @ A ) )
& ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
=> ( ord_less_rat @ A @ B ) ) ) ) ).
% inverse_less_iff
thf(fact_8241_one__le__inverse,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% one_le_inverse
thf(fact_8242_one__le__inverse,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% one_le_inverse
thf(fact_8243_inverse__less__1__iff,axiom,
! [X3: real] :
( ( ord_less_real @ ( inverse_inverse_real @ X3 ) @ one_one_real )
= ( ( ord_less_eq_real @ X3 @ zero_zero_real )
| ( ord_less_real @ one_one_real @ X3 ) ) ) ).
% inverse_less_1_iff
thf(fact_8244_inverse__less__1__iff,axiom,
! [X3: rat] :
( ( ord_less_rat @ ( inverse_inverse_rat @ X3 ) @ one_one_rat )
= ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
| ( ord_less_rat @ one_one_rat @ X3 ) ) ) ).
% inverse_less_1_iff
thf(fact_8245_one__le__inverse__iff,axiom,
! [X3: real] :
( ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ X3 ) )
= ( ( ord_less_real @ zero_zero_real @ X3 )
& ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ).
% one_le_inverse_iff
thf(fact_8246_one__le__inverse__iff,axiom,
! [X3: rat] :
( ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ X3 ) )
= ( ( ord_less_rat @ zero_zero_rat @ X3 )
& ( ord_less_eq_rat @ X3 @ one_one_rat ) ) ) ).
% one_le_inverse_iff
thf(fact_8247_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus_num @ ( bitM @ N ) @ one )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_8248_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_8249_inverse__diff__inverse,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ) ).
% inverse_diff_inverse
thf(fact_8250_inverse__diff__inverse,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
= ( uminus1482373934393186551omplex @ ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ A @ B ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ) ).
% inverse_diff_inverse
thf(fact_8251_inverse__diff__inverse,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
= ( uminus_uminus_rat @ ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ) ).
% inverse_diff_inverse
thf(fact_8252_reals__Archimedean,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ? [N3: nat] : ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ X3 ) ) ).
% reals_Archimedean
thf(fact_8253_reals__Archimedean,axiom,
! [X3: rat] :
( ( ord_less_rat @ zero_zero_rat @ X3 )
=> ? [N3: nat] : ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) ) @ X3 ) ) ).
% reals_Archimedean
thf(fact_8254_choose__alternating__linear__sum,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I3 ) @ ( semiri681578069525770553at_rat @ I3 ) ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_rat ) ) ).
% choose_alternating_linear_sum
thf(fact_8255_choose__alternating__linear__sum,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I3 ) @ ( semiri1314217659103216013at_int @ I3 ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_int ) ) ).
% choose_alternating_linear_sum
thf(fact_8256_choose__alternating__linear__sum,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I3 ) @ ( semiri8010041392384452111omplex @ I3 ) ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) ) ).
% choose_alternating_linear_sum
thf(fact_8257_choose__alternating__linear__sum,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ( ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I3 ) @ ( semiri4939895301339042750nteger @ I3 ) ) @ ( semiri4939895301339042750nteger @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_z3403309356797280102nteger ) ) ).
% choose_alternating_linear_sum
thf(fact_8258_choose__alternating__linear__sum,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( semiri5074537144036343181t_real @ I3 ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) ) ).
% choose_alternating_linear_sum
thf(fact_8259_binomial__absorption,axiom,
! [K2: nat,N: nat] :
( ( times_times_nat @ ( suc @ K2 ) @ ( binomial @ N @ ( suc @ K2 ) ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K2 ) ) ) ).
% binomial_absorption
thf(fact_8260_binomial__r__part__sum,axiom,
! [M2: nat] :
( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M2 ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).
% binomial_r_part_sum
thf(fact_8261_forall__pos__mono__1,axiom,
! [P2: real > $o,E2: real] :
( ! [D2: real,E: real] :
( ( ord_less_real @ D2 @ E )
=> ( ( P2 @ D2 )
=> ( P2 @ E ) ) )
=> ( ! [N3: nat] : ( P2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( P2 @ E2 ) ) ) ) ).
% forall_pos_mono_1
thf(fact_8262_binomial__fact__lemma,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( binomial @ N @ K2 ) )
= ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% binomial_fact_lemma
thf(fact_8263_real__arch__inverse,axiom,
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
= ( ? [N4: nat] :
( ( N4 != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) @ E2 ) ) ) ) ).
% real_arch_inverse
thf(fact_8264_forall__pos__mono,axiom,
! [P2: real > $o,E2: real] :
( ! [D2: real,E: real] :
( ( ord_less_real @ D2 @ E )
=> ( ( P2 @ D2 )
=> ( P2 @ E ) ) )
=> ( ! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( P2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( P2 @ E2 ) ) ) ) ).
% forall_pos_mono
thf(fact_8265_sqrt__divide__self__eq,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( divide_divide_real @ ( sqrt @ X3 ) @ X3 )
= ( inverse_inverse_real @ ( sqrt @ X3 ) ) ) ) ).
% sqrt_divide_self_eq
thf(fact_8266_ln__inverse,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ln_ln_real @ ( inverse_inverse_real @ X3 ) )
= ( uminus_uminus_real @ ( ln_ln_real @ X3 ) ) ) ) ).
% ln_inverse
thf(fact_8267_choose__alternating__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I3 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_rat ) ) ).
% choose_alternating_sum
thf(fact_8268_choose__alternating__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I3 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_int ) ) ).
% choose_alternating_sum
thf(fact_8269_choose__alternating__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I3 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) ) ).
% choose_alternating_sum
thf(fact_8270_choose__alternating__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I3 ) @ ( semiri4939895301339042750nteger @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_z3403309356797280102nteger ) ) ).
% choose_alternating_sum
thf(fact_8271_choose__alternating__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) ) ).
% choose_alternating_sum
thf(fact_8272_sum_OatMost__Suc__shift,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( G @ zero_zero_nat )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_8273_sum_OatMost__Suc__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( G @ zero_zero_nat )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_8274_sum_OatMost__Suc__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_8275_sum_OatMost__Suc__shift,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( G @ zero_zero_nat )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_8276_sum__telescope,axiom,
! [F: nat > rat,I: nat] :
( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( minus_minus_rat @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
@ ( set_ord_atMost_nat @ I ) )
= ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).
% sum_telescope
thf(fact_8277_sum__telescope,axiom,
! [F: nat > int,I: nat] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( minus_minus_int @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
@ ( set_ord_atMost_nat @ I ) )
= ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).
% sum_telescope
thf(fact_8278_sum__telescope,axiom,
! [F: nat > real,I: nat] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( minus_minus_real @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
@ ( set_ord_atMost_nat @ I ) )
= ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).
% sum_telescope
thf(fact_8279_polyfun__eq__coeffs,axiom,
! [C2: nat > complex,N: nat,D: nat > complex] :
( ( ! [X5: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C2 @ I3 ) @ ( power_power_complex @ X5 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( D @ I3 ) @ ( power_power_complex @ X5 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) )
= ( ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( C2 @ I3 )
= ( D @ I3 ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_8280_polyfun__eq__coeffs,axiom,
! [C2: nat > real,N: nat,D: nat > real] :
( ( ! [X5: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C2 @ I3 ) @ ( power_power_real @ X5 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( D @ I3 ) @ ( power_power_real @ X5 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) )
= ( ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( C2 @ I3 )
= ( D @ I3 ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_8281_bounded__imp__summable,axiom,
! [A: nat > int,B5: int] :
( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B5 )
=> ( summable_int @ A ) ) ) ).
% bounded_imp_summable
thf(fact_8282_bounded__imp__summable,axiom,
! [A: nat > nat,B5: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B5 )
=> ( summable_nat @ A ) ) ) ).
% bounded_imp_summable
thf(fact_8283_bounded__imp__summable,axiom,
! [A: nat > real,B5: real] :
( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B5 )
=> ( summable_real @ A ) ) ) ).
% bounded_imp_summable
thf(fact_8284_prod_OatMost__Suc__shift,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_real @ ( G @ zero_zero_nat )
@ ( groups129246275422532515t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_8285_prod_OatMost__Suc__shift,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( G @ zero_zero_nat )
@ ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_8286_prod_OatMost__Suc__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( G @ zero_zero_nat )
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_8287_prod_OatMost__Suc__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( G @ zero_zero_nat )
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_8288_prod__int__plus__eq,axiom,
! [I: nat,J: nat] :
( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
= ( groups1705073143266064639nt_int
@ ^ [X5: int] : X5
@ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).
% prod_int_plus_eq
thf(fact_8289_sum_Onested__swap_H,axiom,
! [A: nat > nat > nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( groups3542108847815614940at_nat @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [J3: nat] :
( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( A @ I3 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nested_swap'
thf(fact_8290_sum_Onested__swap_H,axiom,
! [A: nat > nat > real,N: nat] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( groups6591440286371151544t_real @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups6591440286371151544t_real
@ ^ [J3: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( A @ I3 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nested_swap'
thf(fact_8291_prod_Onested__swap_H,axiom,
! [A: nat > nat > nat,N: nat] :
( ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( groups708209901874060359at_nat @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups708209901874060359at_nat
@ ^ [J3: nat] :
( groups708209901874060359at_nat
@ ^ [I3: nat] : ( A @ I3 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.nested_swap'
thf(fact_8292_prod_Onested__swap_H,axiom,
! [A: nat > nat > int,N: nat] :
( ( groups705719431365010083at_int
@ ^ [I3: nat] : ( groups705719431365010083at_int @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups705719431365010083at_int
@ ^ [J3: nat] :
( groups705719431365010083at_int
@ ^ [I3: nat] : ( A @ I3 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.nested_swap'
thf(fact_8293_binomial__ge__n__over__k__pow__k,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ K2 ) ) @ K2 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K2 ) ) ) ) ).
% binomial_ge_n_over_k_pow_k
thf(fact_8294_binomial__ge__n__over__k__pow__k,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri681578069525770553at_rat @ K2 ) ) @ K2 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K2 ) ) ) ) ).
% binomial_ge_n_over_k_pow_k
thf(fact_8295_binomial__maximum_H,axiom,
! [N: nat,K2: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K2 ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% binomial_maximum'
thf(fact_8296_binomial__mono,axiom,
! [K2: nat,K6: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ K6 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
=> ( ord_less_eq_nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ K6 ) ) ) ) ).
% binomial_mono
thf(fact_8297_binomial__antimono,axiom,
! [K2: nat,K6: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ K6 )
=> ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K2 )
=> ( ( ord_less_eq_nat @ K6 @ N )
=> ( ord_less_eq_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K2 ) ) ) ) ) ).
% binomial_antimono
thf(fact_8298_binomial__maximum,axiom,
! [N: nat,K2: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% binomial_maximum
thf(fact_8299_binomial__le__pow2,axiom,
! [N: nat,K2: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% binomial_le_pow2
thf(fact_8300_choose__reduce__nat,axiom,
! [N: nat,K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( binomial @ N @ K2 )
= ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K2 ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_8301_ex__inverse__of__nat__less,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ X3 ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_8302_ex__inverse__of__nat__less,axiom,
! [X3: rat] :
( ( ord_less_rat @ zero_zero_rat @ X3 )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ N3 ) ) @ X3 ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_8303_times__binomial__minus1__eq,axiom,
! [K2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( times_times_nat @ K2 @ ( binomial @ N @ K2 ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_8304_power__diff__conv__inverse,axiom,
! [X3: real,M2: nat,N: nat] :
( ( X3 != zero_zero_real )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( power_power_real @ X3 @ ( minus_minus_nat @ N @ M2 ) )
= ( times_times_real @ ( power_power_real @ X3 @ N ) @ ( power_power_real @ ( inverse_inverse_real @ X3 ) @ M2 ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_8305_power__diff__conv__inverse,axiom,
! [X3: complex,M2: nat,N: nat] :
( ( X3 != zero_zero_complex )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( power_power_complex @ X3 @ ( minus_minus_nat @ N @ M2 ) )
= ( times_times_complex @ ( power_power_complex @ X3 @ N ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X3 ) @ M2 ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_8306_power__diff__conv__inverse,axiom,
! [X3: rat,M2: nat,N: nat] :
( ( X3 != zero_zero_rat )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( power_power_rat @ X3 @ ( minus_minus_nat @ N @ M2 ) )
= ( times_times_rat @ ( power_power_rat @ X3 @ N ) @ ( power_power_rat @ ( inverse_inverse_rat @ X3 ) @ M2 ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_8307_binomial__altdef__nat,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( binomial @ N @ K2 )
= ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K2 ) ) ) ) ) ) ).
% binomial_altdef_nat
thf(fact_8308_zero__polynom__imp__zero__coeffs,axiom,
! [C2: nat > complex,N: nat,K2: nat] :
( ! [W: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C2 @ I3 ) @ ( power_power_complex @ W @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( C2 @ K2 )
= zero_zero_complex ) ) ) ).
% zero_polynom_imp_zero_coeffs
thf(fact_8309_zero__polynom__imp__zero__coeffs,axiom,
! [C2: nat > real,N: nat,K2: nat] :
( ! [W: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C2 @ I3 ) @ ( power_power_real @ W @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( C2 @ K2 )
= zero_zero_real ) ) ) ).
% zero_polynom_imp_zero_coeffs
thf(fact_8310_polyfun__eq__0,axiom,
! [C2: nat > complex,N: nat] :
( ( ! [X5: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C2 @ I3 ) @ ( power_power_complex @ X5 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) )
= ( ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( C2 @ I3 )
= zero_zero_complex ) ) ) ) ).
% polyfun_eq_0
thf(fact_8311_polyfun__eq__0,axiom,
! [C2: nat > real,N: nat] :
( ( ! [X5: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C2 @ I3 ) @ ( power_power_real @ X5 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) )
= ( ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( C2 @ I3 )
= zero_zero_real ) ) ) ) ).
% polyfun_eq_0
thf(fact_8312_ln__prod,axiom,
! [I6: set_real,F: real > real] :
( ( finite_finite_real @ I6 )
=> ( ! [I2: real] :
( ( member_real @ I2 @ I6 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( ln_ln_real @ ( groups1681761925125756287l_real @ F @ I6 ) )
= ( groups8097168146408367636l_real
@ ^ [X5: real] : ( ln_ln_real @ ( F @ X5 ) )
@ I6 ) ) ) ) ).
% ln_prod
thf(fact_8313_ln__prod,axiom,
! [I6: set_set_nat,F: set_nat > real] :
( ( finite1152437895449049373et_nat @ I6 )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ I6 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( ln_ln_real @ ( groups3619160379726066777t_real @ F @ I6 ) )
= ( groups5107569545109728110t_real
@ ^ [X5: set_nat] : ( ln_ln_real @ ( F @ X5 ) )
@ I6 ) ) ) ) ).
% ln_prod
thf(fact_8314_ln__prod,axiom,
! [I6: set_int,F: int > real] :
( ( finite_finite_int @ I6 )
=> ( ! [I2: int] :
( ( member_int @ I2 @ I6 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( ln_ln_real @ ( groups2316167850115554303t_real @ F @ I6 ) )
= ( groups8778361861064173332t_real
@ ^ [X5: int] : ( ln_ln_real @ ( F @ X5 ) )
@ I6 ) ) ) ) ).
% ln_prod
thf(fact_8315_ln__prod,axiom,
! [I6: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I6 )
=> ( ! [I2: complex] :
( ( member_complex @ I2 @ I6 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( ln_ln_real @ ( groups766887009212190081x_real @ F @ I6 ) )
= ( groups5808333547571424918x_real
@ ^ [X5: complex] : ( ln_ln_real @ ( F @ X5 ) )
@ I6 ) ) ) ) ).
% ln_prod
thf(fact_8316_ln__prod,axiom,
! [I6: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > real] :
( ( finite6177210948735845034at_nat @ I6 )
=> ( ! [I2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I2 @ I6 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( ln_ln_real @ ( groups6036352826371341000t_real @ F @ I6 ) )
= ( groups4567486121110086003t_real
@ ^ [X5: product_prod_nat_nat] : ( ln_ln_real @ ( F @ X5 ) )
@ I6 ) ) ) ) ).
% ln_prod
thf(fact_8317_ln__prod,axiom,
! [I6: set_nat,F: nat > real] :
( ( finite_finite_nat @ I6 )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I6 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
=> ( ( ln_ln_real @ ( groups129246275422532515t_real @ F @ I6 ) )
= ( groups6591440286371151544t_real
@ ^ [X5: nat] : ( ln_ln_real @ ( F @ X5 ) )
@ I6 ) ) ) ) ).
% ln_prod
thf(fact_8318_sum_OatMost__shift,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_rat @ ( G @ zero_zero_nat )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.atMost_shift
thf(fact_8319_sum_OatMost__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_int @ ( G @ zero_zero_nat )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.atMost_shift
thf(fact_8320_sum_OatMost__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_nat @ ( G @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.atMost_shift
thf(fact_8321_sum_OatMost__shift,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_real @ ( G @ zero_zero_nat )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.atMost_shift
thf(fact_8322_sum__up__index__split,axiom,
! [F: nat > rat,M2: nat,N: nat] :
( ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ).
% sum_up_index_split
thf(fact_8323_sum__up__index__split,axiom,
! [F: nat > int,M2: nat,N: nat] :
( ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ).
% sum_up_index_split
thf(fact_8324_sum__up__index__split,axiom,
! [F: nat > nat,M2: nat,N: nat] :
( ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ).
% sum_up_index_split
thf(fact_8325_sum__up__index__split,axiom,
! [F: nat > real,M2: nat,N: nat] :
( ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ).
% sum_up_index_split
thf(fact_8326_prod_OatMost__shift,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N ) )
= ( times_times_real @ ( G @ zero_zero_nat )
@ ( groups129246275422532515t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.atMost_shift
thf(fact_8327_prod_OatMost__shift,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N ) )
= ( times_times_rat @ ( G @ zero_zero_nat )
@ ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.atMost_shift
thf(fact_8328_prod_OatMost__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N ) )
= ( times_times_nat @ ( G @ zero_zero_nat )
@ ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.atMost_shift
thf(fact_8329_prod_OatMost__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N ) )
= ( times_times_int @ ( G @ zero_zero_nat )
@ ( groups705719431365010083at_int
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.atMost_shift
thf(fact_8330_gbinomial__parallel__sum,axiom,
! [A: rat,N: nat] :
( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( gbinomial_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K3 ) ) @ K3 )
@ ( set_ord_atMost_nat @ N ) )
= ( gbinomial_rat @ ( plus_plus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) @ one_one_rat ) @ N ) ) ).
% gbinomial_parallel_sum
thf(fact_8331_gbinomial__parallel__sum,axiom,
! [A: complex,N: nat] :
( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( gbinomial_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K3 ) ) @ K3 )
@ ( set_ord_atMost_nat @ N ) )
= ( gbinomial_complex @ ( plus_plus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ N ) ) ).
% gbinomial_parallel_sum
thf(fact_8332_gbinomial__parallel__sum,axiom,
! [A: real,N: nat] :
( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( gbinomial_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K3 ) ) @ K3 )
@ ( set_ord_atMost_nat @ N ) )
= ( gbinomial_real @ ( plus_plus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) @ one_one_real ) @ N ) ) ).
% gbinomial_parallel_sum
thf(fact_8333_sum_Otriangle__reindex__eq,axiom,
! [G: nat > nat > nat,N: nat] :
( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I3: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [K3: nat] :
( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_8334_sum_Otriangle__reindex__eq,axiom,
! [G: nat > nat > real,N: nat] :
( ( groups4567486121110086003t_real @ ( produc1703576794950452218t_real @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I3: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups6591440286371151544t_real
@ ^ [K3: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_8335_binomial__less__binomial__Suc,axiom,
! [K2: nat,N: nat] :
( ( ord_less_nat @ K2 @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ord_less_nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ ( suc @ K2 ) ) ) ) ).
% binomial_less_binomial_Suc
thf(fact_8336_binomial__strict__mono,axiom,
! [K2: nat,K6: nat,N: nat] :
( ( ord_less_nat @ K2 @ K6 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
=> ( ord_less_nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ K6 ) ) ) ) ).
% binomial_strict_mono
thf(fact_8337_binomial__strict__antimono,axiom,
! [K2: nat,K6: nat,N: nat] :
( ( ord_less_nat @ K2 @ K6 )
=> ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) )
=> ( ( ord_less_eq_nat @ K6 @ N )
=> ( ord_less_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K2 ) ) ) ) ) ).
% binomial_strict_antimono
thf(fact_8338_central__binomial__odd,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( binomial @ N @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% central_binomial_odd
thf(fact_8339_binomial__addition__formula,axiom,
! [N: nat,K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( binomial @ N @ ( suc @ K2 ) )
= ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K2 ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K2 ) ) ) ) ).
% binomial_addition_formula
thf(fact_8340_choose__odd__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] :
( if_rat
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
@ ( semiri681578069525770553at_rat @ ( binomial @ N @ I3 ) )
@ zero_zero_rat )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_odd_sum
thf(fact_8341_choose__odd__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] :
( if_int
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
@ ( semiri1314217659103216013at_int @ ( binomial @ N @ I3 ) )
@ zero_zero_int )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_odd_sum
thf(fact_8342_choose__odd__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] :
( if_complex
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
@ ( semiri8010041392384452111omplex @ ( binomial @ N @ I3 ) )
@ zero_zero_complex )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_odd_sum
thf(fact_8343_choose__odd__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) )
@ ( groups7501900531339628137nteger
@ ^ [I3: nat] :
( if_Code_integer
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
@ ( semiri4939895301339042750nteger @ ( binomial @ N @ I3 ) )
@ zero_z3403309356797280102nteger )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_odd_sum
thf(fact_8344_choose__odd__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] :
( if_real
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
@ ( semiri5074537144036343181t_real @ ( binomial @ N @ I3 ) )
@ zero_zero_real )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_odd_sum
thf(fact_8345_choose__even__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( if_rat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I3 ) ) @ zero_zero_rat )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_even_sum
thf(fact_8346_choose__even__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( if_int @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I3 ) ) @ zero_zero_int )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_even_sum
thf(fact_8347_choose__even__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I3 ) ) @ zero_zero_complex )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_even_sum
thf(fact_8348_choose__even__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) )
@ ( groups7501900531339628137nteger
@ ^ [I3: nat] : ( if_Code_integer @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri4939895301339042750nteger @ ( binomial @ N @ I3 ) ) @ zero_z3403309356797280102nteger )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_even_sum
thf(fact_8349_choose__even__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I3 ) ) @ zero_zero_real )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_even_sum
thf(fact_8350_binomial__fact,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K2 ) )
= ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K2 ) ) ) ) ) ) ).
% binomial_fact
thf(fact_8351_binomial__fact,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K2 ) )
= ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K2 ) ) ) ) ) ) ).
% binomial_fact
thf(fact_8352_binomial__fact,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K2 ) )
= ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K2 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K2 ) ) ) ) ) ) ).
% binomial_fact
thf(fact_8353_fact__binomial,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K2 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K2 ) ) )
= ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K2 ) ) ) ) ) ).
% fact_binomial
thf(fact_8354_fact__binomial,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( times_times_real @ ( semiri2265585572941072030t_real @ K2 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K2 ) ) )
= ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K2 ) ) ) ) ) ).
% fact_binomial
thf(fact_8355_fact__binomial,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K2 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K2 ) ) )
= ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K2 ) ) ) ) ) ).
% fact_binomial
thf(fact_8356_sum__gp__basic,axiom,
! [X3: complex,N: nat] :
( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X3 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_atMost_nat @ N ) ) )
= ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ ( suc @ N ) ) ) ) ).
% sum_gp_basic
thf(fact_8357_sum__gp__basic,axiom,
! [X3: rat,N: nat] :
( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X3 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_atMost_nat @ N ) ) )
= ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ ( suc @ N ) ) ) ) ).
% sum_gp_basic
thf(fact_8358_sum__gp__basic,axiom,
! [X3: int,N: nat] :
( ( times_times_int @ ( minus_minus_int @ one_one_int @ X3 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_ord_atMost_nat @ N ) ) )
= ( minus_minus_int @ one_one_int @ ( power_power_int @ X3 @ ( suc @ N ) ) ) ) ).
% sum_gp_basic
thf(fact_8359_sum__gp__basic,axiom,
! [X3: real,N: nat] :
( ( times_times_real @ ( minus_minus_real @ one_one_real @ X3 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_atMost_nat @ N ) ) )
= ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( suc @ N ) ) ) ) ).
% sum_gp_basic
thf(fact_8360_exp__plus__inverse__exp,axiom,
! [X3: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X3 ) @ ( inverse_inverse_real @ ( exp_real @ X3 ) ) ) ) ).
% exp_plus_inverse_exp
thf(fact_8361_polyfun__roots__finite,axiom,
! [C2: nat > complex,K2: nat,N: nat] :
( ( ( C2 @ K2 )
!= zero_zero_complex )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z6: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C2 @ I3 ) @ ( power_power_complex @ Z6 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) ) ) ) ) ).
% polyfun_roots_finite
thf(fact_8362_polyfun__roots__finite,axiom,
! [C2: nat > real,K2: nat,N: nat] :
( ( ( C2 @ K2 )
!= zero_zero_real )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Z6: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C2 @ I3 ) @ ( power_power_real @ Z6 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) ) ) ) ) ).
% polyfun_roots_finite
thf(fact_8363_polyfun__finite__roots,axiom,
! [C2: nat > complex,N: nat] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X5: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C2 @ I3 ) @ ( power_power_complex @ X5 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) ) )
= ( ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( C2 @ I3 )
!= zero_zero_complex ) ) ) ) ).
% polyfun_finite_roots
thf(fact_8364_polyfun__finite__roots,axiom,
! [C2: nat > real,N: nat] :
( ( finite_finite_real
@ ( collect_real
@ ^ [X5: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C2 @ I3 ) @ ( power_power_real @ X5 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) ) )
= ( ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( C2 @ I3 )
!= zero_zero_real ) ) ) ) ).
% polyfun_finite_roots
thf(fact_8365_polyfun__linear__factor__root,axiom,
! [C2: nat > complex,A: complex,N: nat] :
( ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C2 @ I3 ) @ ( power_power_complex @ A @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex )
=> ~ ! [B3: nat > complex] :
~ ! [Z4: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C2 @ I3 ) @ ( power_power_complex @ Z4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_complex @ ( minus_minus_complex @ Z4 @ A )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( B3 @ I3 ) @ ( power_power_complex @ Z4 @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_8366_polyfun__linear__factor__root,axiom,
! [C2: nat > rat,A: rat,N: nat] :
( ( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( C2 @ I3 ) @ ( power_power_rat @ A @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_rat )
=> ~ ! [B3: nat > rat] :
~ ! [Z4: rat] :
( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( C2 @ I3 ) @ ( power_power_rat @ Z4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_rat @ ( minus_minus_rat @ Z4 @ A )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( B3 @ I3 ) @ ( power_power_rat @ Z4 @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_8367_polyfun__linear__factor__root,axiom,
! [C2: nat > int,A: int,N: nat] :
( ( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( C2 @ I3 ) @ ( power_power_int @ A @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_int )
=> ~ ! [B3: nat > int] :
~ ! [Z4: int] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( C2 @ I3 ) @ ( power_power_int @ Z4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_int @ ( minus_minus_int @ Z4 @ A )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( B3 @ I3 ) @ ( power_power_int @ Z4 @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_8368_polyfun__linear__factor__root,axiom,
! [C2: nat > real,A: real,N: nat] :
( ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C2 @ I3 ) @ ( power_power_real @ A @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real )
=> ~ ! [B3: nat > real] :
~ ! [Z4: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C2 @ I3 ) @ ( power_power_real @ Z4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_real @ ( minus_minus_real @ Z4 @ A )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( B3 @ I3 ) @ ( power_power_real @ Z4 @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_8369_polyfun__linear__factor,axiom,
! [C2: nat > complex,N: nat,A: complex] :
? [B3: nat > complex] :
! [Z4: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C2 @ I3 ) @ ( power_power_complex @ Z4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_complex
@ ( times_times_complex @ ( minus_minus_complex @ Z4 @ A )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( B3 @ I3 ) @ ( power_power_complex @ Z4 @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) )
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C2 @ I3 ) @ ( power_power_complex @ A @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% polyfun_linear_factor
thf(fact_8370_polyfun__linear__factor,axiom,
! [C2: nat > rat,N: nat,A: rat] :
? [B3: nat > rat] :
! [Z4: rat] :
( ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( C2 @ I3 ) @ ( power_power_rat @ Z4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_rat
@ ( times_times_rat @ ( minus_minus_rat @ Z4 @ A )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( B3 @ I3 ) @ ( power_power_rat @ Z4 @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) )
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( C2 @ I3 ) @ ( power_power_rat @ A @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% polyfun_linear_factor
thf(fact_8371_polyfun__linear__factor,axiom,
! [C2: nat > int,N: nat,A: int] :
? [B3: nat > int] :
! [Z4: int] :
( ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( C2 @ I3 ) @ ( power_power_int @ Z4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_int
@ ( times_times_int @ ( minus_minus_int @ Z4 @ A )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( B3 @ I3 ) @ ( power_power_int @ Z4 @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) )
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( C2 @ I3 ) @ ( power_power_int @ A @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% polyfun_linear_factor
thf(fact_8372_polyfun__linear__factor,axiom,
! [C2: nat > real,N: nat,A: real] :
? [B3: nat > real] :
! [Z4: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C2 @ I3 ) @ ( power_power_real @ Z4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_real
@ ( times_times_real @ ( minus_minus_real @ Z4 @ A )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( B3 @ I3 ) @ ( power_power_real @ Z4 @ I3 ) )
@ ( set_ord_lessThan_nat @ N ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C2 @ I3 ) @ ( power_power_real @ A @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% polyfun_linear_factor
thf(fact_8373_sum__power__shift,axiom,
! [M2: nat,N: nat,X3: complex] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( times_times_complex @ ( power_power_complex @ X3 @ M2 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_8374_sum__power__shift,axiom,
! [M2: nat,N: nat,X3: rat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( times_times_rat @ ( power_power_rat @ X3 @ M2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_8375_sum__power__shift,axiom,
! [M2: nat,N: nat,X3: int] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( times_times_int @ ( power_power_int @ X3 @ M2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_8376_sum__power__shift,axiom,
! [M2: nat,N: nat,X3: real] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
= ( times_times_real @ ( power_power_real @ X3 @ M2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_8377_sum_Otriangle__reindex,axiom,
! [G: nat > nat > nat,N: nat] :
( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I3: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [K3: nat] :
( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.triangle_reindex
thf(fact_8378_sum_Otriangle__reindex,axiom,
! [G: nat > nat > real,N: nat] :
( ( groups4567486121110086003t_real @ ( produc1703576794950452218t_real @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I3: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
= ( groups6591440286371151544t_real
@ ^ [K3: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.triangle_reindex
thf(fact_8379_plus__inverse__ge__2,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X3 @ ( inverse_inverse_real @ X3 ) ) ) ) ).
% plus_inverse_ge_2
thf(fact_8380_real__inv__sqrt__pow2,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( inverse_inverse_real @ X3 ) ) ) ).
% real_inv_sqrt_pow2
thf(fact_8381_sum_Oin__pairs__0,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.in_pairs_0
thf(fact_8382_sum_Oin__pairs__0,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.in_pairs_0
thf(fact_8383_sum_Oin__pairs__0,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.in_pairs_0
thf(fact_8384_sum_Oin__pairs__0,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.in_pairs_0
thf(fact_8385_polynomial__product,axiom,
! [M2: nat,A: nat > complex,N: nat,B: nat > complex,X3: complex] :
( ! [I2: nat] :
( ( ord_less_nat @ M2 @ I2 )
=> ( ( A @ I2 )
= zero_zero_complex ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N @ J2 )
=> ( ( B @ J2 )
= zero_zero_complex ) )
=> ( ( times_times_complex
@ ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ X3 @ I3 ) )
@ ( set_ord_atMost_nat @ M2 ) )
@ ( groups2073611262835488442omplex
@ ^ [J3: nat] : ( times_times_complex @ ( B @ J3 ) @ ( power_power_complex @ X3 @ J3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( groups2073611262835488442omplex
@ ^ [R5: nat] :
( times_times_complex
@ ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost_nat @ R5 ) )
@ ( power_power_complex @ X3 @ R5 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).
% polynomial_product
thf(fact_8386_polynomial__product,axiom,
! [M2: nat,A: nat > rat,N: nat,B: nat > rat,X3: rat] :
( ! [I2: nat] :
( ( ord_less_nat @ M2 @ I2 )
=> ( ( A @ I2 )
= zero_zero_rat ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N @ J2 )
=> ( ( B @ J2 )
= zero_zero_rat ) )
=> ( ( times_times_rat
@ ( groups2906978787729119204at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ X3 @ I3 ) )
@ ( set_ord_atMost_nat @ M2 ) )
@ ( groups2906978787729119204at_rat
@ ^ [J3: nat] : ( times_times_rat @ ( B @ J3 ) @ ( power_power_rat @ X3 @ J3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( groups2906978787729119204at_rat
@ ^ [R5: nat] :
( times_times_rat
@ ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost_nat @ R5 ) )
@ ( power_power_rat @ X3 @ R5 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).
% polynomial_product
thf(fact_8387_polynomial__product,axiom,
! [M2: nat,A: nat > int,N: nat,B: nat > int,X3: int] :
( ! [I2: nat] :
( ( ord_less_nat @ M2 @ I2 )
=> ( ( A @ I2 )
= zero_zero_int ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N @ J2 )
=> ( ( B @ J2 )
= zero_zero_int ) )
=> ( ( times_times_int
@ ( groups3539618377306564664at_int
@ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ X3 @ I3 ) )
@ ( set_ord_atMost_nat @ M2 ) )
@ ( groups3539618377306564664at_int
@ ^ [J3: nat] : ( times_times_int @ ( B @ J3 ) @ ( power_power_int @ X3 @ J3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( groups3539618377306564664at_int
@ ^ [R5: nat] :
( times_times_int
@ ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( times_times_int @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost_nat @ R5 ) )
@ ( power_power_int @ X3 @ R5 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).
% polynomial_product
thf(fact_8388_polynomial__product,axiom,
! [M2: nat,A: nat > real,N: nat,B: nat > real,X3: real] :
( ! [I2: nat] :
( ( ord_less_nat @ M2 @ I2 )
=> ( ( A @ I2 )
= zero_zero_real ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N @ J2 )
=> ( ( B @ J2 )
= zero_zero_real ) )
=> ( ( times_times_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ X3 @ I3 ) )
@ ( set_ord_atMost_nat @ M2 ) )
@ ( groups6591440286371151544t_real
@ ^ [J3: nat] : ( times_times_real @ ( B @ J3 ) @ ( power_power_real @ X3 @ J3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( groups6591440286371151544t_real
@ ^ [R5: nat] :
( times_times_real
@ ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost_nat @ R5 ) )
@ ( power_power_real @ X3 @ R5 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).
% polynomial_product
thf(fact_8389_prod_Oin__pairs__0,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups129246275422532515t_real
@ ^ [I3: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.in_pairs_0
thf(fact_8390_prod_Oin__pairs__0,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups73079841787564623at_rat
@ ^ [I3: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.in_pairs_0
thf(fact_8391_prod_Oin__pairs__0,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [I3: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.in_pairs_0
thf(fact_8392_prod_Oin__pairs__0,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups705719431365010083at_int
@ ^ [I3: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.in_pairs_0
thf(fact_8393_polyfun__eq__const,axiom,
! [C2: nat > complex,N: nat,K2: complex] :
( ( ! [X5: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C2 @ I3 ) @ ( power_power_complex @ X5 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= K2 ) )
= ( ( ( C2 @ zero_zero_nat )
= K2 )
& ! [X5: nat] :
( ( member_nat @ X5 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) )
=> ( ( C2 @ X5 )
= zero_zero_complex ) ) ) ) ).
% polyfun_eq_const
thf(fact_8394_polyfun__eq__const,axiom,
! [C2: nat > real,N: nat,K2: real] :
( ( ! [X5: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C2 @ I3 ) @ ( power_power_real @ X5 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= K2 ) )
= ( ( ( C2 @ zero_zero_nat )
= K2 )
& ! [X5: nat] :
( ( member_nat @ X5 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) )
=> ( ( C2 @ X5 )
= zero_zero_real ) ) ) ) ).
% polyfun_eq_const
thf(fact_8395_polynomial__product__nat,axiom,
! [M2: nat,A: nat > nat,N: nat,B: nat > nat,X3: nat] :
( ! [I2: nat] :
( ( ord_less_nat @ M2 @ I2 )
=> ( ( A @ I2 )
= zero_zero_nat ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N @ J2 )
=> ( ( B @ J2 )
= zero_zero_nat ) )
=> ( ( times_times_nat
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( power_power_nat @ X3 @ I3 ) )
@ ( set_ord_atMost_nat @ M2 ) )
@ ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X3 @ J3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [R5: nat] :
( times_times_nat
@ ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( times_times_nat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost_nat @ R5 ) )
@ ( power_power_nat @ X3 @ R5 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).
% polynomial_product_nat
thf(fact_8396_Suc__0__or__eq,axiom,
! [N: nat] :
( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
= ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% Suc_0_or_eq
thf(fact_8397_or__Suc__0__eq,axiom,
! [N: nat] :
( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
= ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% or_Suc_0_eq
thf(fact_8398_or__nat__rec,axiom,
( bit_se1412395901928357646or_nat
= ( ^ [M5: nat,N4: nat] :
( plus_plus_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 )
| ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
@ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% or_nat_rec
thf(fact_8399_real__le__x__sinh,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ X3 @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X3 ) @ ( inverse_inverse_real @ ( exp_real @ X3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% real_le_x_sinh
thf(fact_8400_sum_Ozero__middle,axiom,
! [P: nat,K2: nat,G: nat > complex,H2: nat > complex] :
( ( ord_less_eq_nat @ one_one_nat @ P )
=> ( ( ord_less_eq_nat @ K2 @ P )
=> ( ( groups2073611262835488442omplex
@ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_complex @ ( J3 = K2 ) @ zero_zero_complex @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P ) )
= ( groups2073611262835488442omplex
@ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_8401_sum_Ozero__middle,axiom,
! [P: nat,K2: nat,G: nat > rat,H2: nat > rat] :
( ( ord_less_eq_nat @ one_one_nat @ P )
=> ( ( ord_less_eq_nat @ K2 @ P )
=> ( ( groups2906978787729119204at_rat
@ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K2 ) @ zero_zero_rat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P ) )
= ( groups2906978787729119204at_rat
@ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_8402_sum_Ozero__middle,axiom,
! [P: nat,K2: nat,G: nat > int,H2: nat > int] :
( ( ord_less_eq_nat @ one_one_nat @ P )
=> ( ( ord_less_eq_nat @ K2 @ P )
=> ( ( groups3539618377306564664at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K2 ) @ zero_zero_int @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P ) )
= ( groups3539618377306564664at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_8403_sum_Ozero__middle,axiom,
! [P: nat,K2: nat,G: nat > nat,H2: nat > nat] :
( ( ord_less_eq_nat @ one_one_nat @ P )
=> ( ( ord_less_eq_nat @ K2 @ P )
=> ( ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K2 ) @ zero_zero_nat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P ) )
= ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_8404_sum_Ozero__middle,axiom,
! [P: nat,K2: nat,G: nat > real,H2: nat > real] :
( ( ord_less_eq_nat @ one_one_nat @ P )
=> ( ( ord_less_eq_nat @ K2 @ P )
=> ( ( groups6591440286371151544t_real
@ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K2 ) @ zero_zero_real @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P ) )
= ( groups6591440286371151544t_real
@ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_8405_prod_Ozero__middle,axiom,
! [P: nat,K2: nat,G: nat > complex,H2: nat > complex] :
( ( ord_less_eq_nat @ one_one_nat @ P )
=> ( ( ord_less_eq_nat @ K2 @ P )
=> ( ( groups6464643781859351333omplex
@ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_complex @ ( J3 = K2 ) @ one_one_complex @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P ) )
= ( groups6464643781859351333omplex
@ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_8406_prod_Ozero__middle,axiom,
! [P: nat,K2: nat,G: nat > real,H2: nat > real] :
( ( ord_less_eq_nat @ one_one_nat @ P )
=> ( ( ord_less_eq_nat @ K2 @ P )
=> ( ( groups129246275422532515t_real
@ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K2 ) @ one_one_real @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P ) )
= ( groups129246275422532515t_real
@ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_8407_prod_Ozero__middle,axiom,
! [P: nat,K2: nat,G: nat > rat,H2: nat > rat] :
( ( ord_less_eq_nat @ one_one_nat @ P )
=> ( ( ord_less_eq_nat @ K2 @ P )
=> ( ( groups73079841787564623at_rat
@ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K2 ) @ one_one_rat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P ) )
= ( groups73079841787564623at_rat
@ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_8408_prod_Ozero__middle,axiom,
! [P: nat,K2: nat,G: nat > nat,H2: nat > nat] :
( ( ord_less_eq_nat @ one_one_nat @ P )
=> ( ( ord_less_eq_nat @ K2 @ P )
=> ( ( groups708209901874060359at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K2 ) @ one_one_nat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P ) )
= ( groups708209901874060359at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_8409_prod_Ozero__middle,axiom,
! [P: nat,K2: nat,G: nat > int,H2: nat > int] :
( ( ord_less_eq_nat @ one_one_nat @ P )
=> ( ( ord_less_eq_nat @ K2 @ P )
=> ( ( groups705719431365010083at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K2 ) @ one_one_int @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P ) )
= ( groups705719431365010083at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_8410_real__le__abs__sinh,axiom,
! [X3: real] : ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X3 ) @ ( inverse_inverse_real @ ( exp_real @ X3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% real_le_abs_sinh
thf(fact_8411_gbinomial__partial__sum__poly,axiom,
! [M2: nat,A: rat,X3: rat,Y: rat] :
( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M2 ) @ A ) @ K3 ) @ ( power_power_rat @ X3 @ K3 ) ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ M2 @ K3 ) ) )
@ ( set_ord_atMost_nat @ M2 ) )
= ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K3 ) @ ( power_power_rat @ ( uminus_uminus_rat @ X3 ) @ K3 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X3 @ Y ) @ ( minus_minus_nat @ M2 @ K3 ) ) )
@ ( set_ord_atMost_nat @ M2 ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_8412_gbinomial__partial__sum__poly,axiom,
! [M2: nat,A: complex,X3: complex,Y: complex] :
( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M2 ) @ A ) @ K3 ) @ ( power_power_complex @ X3 @ K3 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ M2 @ K3 ) ) )
@ ( set_ord_atMost_nat @ M2 ) )
= ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K3 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ X3 ) @ K3 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X3 @ Y ) @ ( minus_minus_nat @ M2 @ K3 ) ) )
@ ( set_ord_atMost_nat @ M2 ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_8413_gbinomial__partial__sum__poly,axiom,
! [M2: nat,A: real,X3: real,Y: real] :
( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ A ) @ K3 ) @ ( power_power_real @ X3 @ K3 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ M2 @ K3 ) ) )
@ ( set_ord_atMost_nat @ M2 ) )
= ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K3 ) @ ( power_power_real @ ( uminus_uminus_real @ X3 ) @ K3 ) ) @ ( power_power_real @ ( plus_plus_real @ X3 @ Y ) @ ( minus_minus_nat @ M2 @ K3 ) ) )
@ ( set_ord_atMost_nat @ M2 ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_8414_or__nat__unfold,axiom,
( bit_se1412395901928357646or_nat
= ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = zero_zero_nat ) @ M5 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_8415_binomial__code,axiom,
( binomial
= ( ^ [N4: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N4 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N4 @ ( minus_minus_nat @ N4 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N4 @ K3 ) @ one_one_nat ) @ N4 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).
% binomial_code
thf(fact_8416_cot__less__zero,axiom,
! [X3: real] :
( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 )
=> ( ( ord_less_real @ X3 @ zero_zero_real )
=> ( ord_less_real @ ( cot_real @ X3 ) @ zero_zero_real ) ) ) ).
% cot_less_zero
thf(fact_8417_of__nat__id,axiom,
( semiri1316708129612266289at_nat
= ( ^ [N4: nat] : N4 ) ) ).
% of_nat_id
thf(fact_8418_cot__pi,axiom,
( ( cot_real @ pi )
= zero_zero_real ) ).
% cot_pi
thf(fact_8419_cot__npi,axiom,
! [N: nat] :
( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
= zero_zero_real ) ).
% cot_npi
thf(fact_8420_cot__periodic,axiom,
! [X3: real] :
( ( cot_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
= ( cot_real @ X3 ) ) ).
% cot_periodic
thf(fact_8421_complex__inverse,axiom,
! [A: real,B: real] :
( ( invers8013647133539491842omplex @ ( complex2 @ A @ B ) )
= ( 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 ) ) ) ) ) ) ) ).
% complex_inverse
thf(fact_8422_cot__gt__zero,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ zero_zero_real @ ( cot_real @ X3 ) ) ) ) ).
% cot_gt_zero
thf(fact_8423_log__base__10__eq1,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X3 )
= ( 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 @ X3 ) ) ) ) ).
% log_base_10_eq1
thf(fact_8424_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D4: int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [Z7: int,Z6: int] :
( ( ord_less_eq_int @ D4 @ Z7 )
& ( ord_less_int @ Z7 @ Z6 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_8425_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D4: int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [Z7: int,Z6: int] :
( ( ord_less_eq_int @ D4 @ Z6 )
& ( ord_less_int @ Z7 @ Z6 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_8426_sinh__real__zero__iff,axiom,
! [X3: real] :
( ( ( sinh_real @ X3 )
= zero_zero_real )
= ( X3 = zero_zero_real ) ) ).
% sinh_real_zero_iff
thf(fact_8427_sinh__real__le__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( sinh_real @ X3 ) @ ( sinh_real @ Y ) )
= ( ord_less_eq_real @ X3 @ Y ) ) ).
% sinh_real_le_iff
thf(fact_8428_sinh__real__neg__iff,axiom,
! [X3: real] :
( ( ord_less_real @ ( sinh_real @ X3 ) @ zero_zero_real )
= ( ord_less_real @ X3 @ zero_zero_real ) ) ).
% sinh_real_neg_iff
thf(fact_8429_sinh__real__pos__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X3 ) )
= ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% sinh_real_pos_iff
thf(fact_8430_sinh__real__nonpos__iff,axiom,
! [X3: real] :
( ( ord_less_eq_real @ ( sinh_real @ X3 ) @ zero_zero_real )
= ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).
% sinh_real_nonpos_iff
thf(fact_8431_sinh__real__nonneg__iff,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X3 ) )
= ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).
% sinh_real_nonneg_iff
thf(fact_8432_log__one,axiom,
! [A: real] :
( ( log @ A @ one_one_real )
= zero_zero_real ) ).
% log_one
thf(fact_8433_log__eq__one,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log @ A @ A )
= one_one_real ) ) ) ).
% log_eq_one
thf(fact_8434_log__less__cancel__iff,axiom,
! [A: real,X3: real,Y: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( log @ A @ X3 ) @ ( log @ A @ Y ) )
= ( ord_less_real @ X3 @ Y ) ) ) ) ) ).
% log_less_cancel_iff
thf(fact_8435_log__less__one__cancel__iff,axiom,
! [A: real,X3: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ ( log @ A @ X3 ) @ one_one_real )
= ( ord_less_real @ X3 @ A ) ) ) ) ).
% log_less_one_cancel_iff
thf(fact_8436_one__less__log__cancel__iff,axiom,
! [A: real,X3: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ one_one_real @ ( log @ A @ X3 ) )
= ( ord_less_real @ A @ X3 ) ) ) ) ).
% one_less_log_cancel_iff
thf(fact_8437_log__less__zero__cancel__iff,axiom,
! [A: real,X3: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ ( log @ A @ X3 ) @ zero_zero_real )
= ( ord_less_real @ X3 @ one_one_real ) ) ) ) ).
% log_less_zero_cancel_iff
thf(fact_8438_zero__less__log__cancel__iff,axiom,
! [A: real,X3: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X3 ) )
= ( ord_less_real @ one_one_real @ X3 ) ) ) ) ).
% zero_less_log_cancel_iff
thf(fact_8439_log__le__cancel__iff,axiom,
! [A: real,X3: real,Y: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( log @ A @ X3 ) @ ( log @ A @ Y ) )
= ( ord_less_eq_real @ X3 @ Y ) ) ) ) ) ).
% log_le_cancel_iff
thf(fact_8440_log__le__one__cancel__iff,axiom,
! [A: real,X3: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ ( log @ A @ X3 ) @ one_one_real )
= ( ord_less_eq_real @ X3 @ A ) ) ) ) ).
% log_le_one_cancel_iff
thf(fact_8441_one__le__log__cancel__iff,axiom,
! [A: real,X3: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X3 ) )
= ( ord_less_eq_real @ A @ X3 ) ) ) ) ).
% one_le_log_cancel_iff
thf(fact_8442_log__le__zero__cancel__iff,axiom,
! [A: real,X3: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ ( log @ A @ X3 ) @ zero_zero_real )
= ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ) ).
% log_le_zero_cancel_iff
thf(fact_8443_zero__le__log__cancel__iff,axiom,
! [A: real,X3: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X3 ) )
= ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ) ).
% zero_le_log_cancel_iff
thf(fact_8444_log__pow__cancel,axiom,
! [A: real,B: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log @ A @ ( power_power_real @ A @ B ) )
= ( semiri5074537144036343181t_real @ B ) ) ) ) ).
% log_pow_cancel
thf(fact_8445_cosh__real__nonzero,axiom,
! [X3: real] :
( ( cosh_real @ X3 )
!= zero_zero_real ) ).
% cosh_real_nonzero
thf(fact_8446_sinh__le__cosh__real,axiom,
! [X3: real] : ( ord_less_eq_real @ ( sinh_real @ X3 ) @ ( cosh_real @ X3 ) ) ).
% sinh_le_cosh_real
thf(fact_8447_cosh__real__pos,axiom,
! [X3: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X3 ) ) ).
% cosh_real_pos
thf(fact_8448_cosh__real__nonpos__le__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y ) )
= ( ord_less_eq_real @ Y @ X3 ) ) ) ) ).
% cosh_real_nonpos_le_iff
thf(fact_8449_cosh__real__nonneg__le__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y ) )
= ( ord_less_eq_real @ X3 @ Y ) ) ) ) ).
% cosh_real_nonneg_le_iff
thf(fact_8450_cosh__real__nonneg,axiom,
! [X3: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X3 ) ) ).
% cosh_real_nonneg
thf(fact_8451_cosh__real__ge__1,axiom,
! [X3: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X3 ) ) ).
% cosh_real_ge_1
thf(fact_8452_cosh__real__strict__mono,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y ) ) ) ) ).
% cosh_real_strict_mono
thf(fact_8453_cosh__real__nonneg__less__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y ) )
= ( ord_less_real @ X3 @ Y ) ) ) ) ).
% cosh_real_nonneg_less_iff
thf(fact_8454_cosh__real__nonpos__less__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ord_less_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y ) )
= ( ord_less_real @ Y @ X3 ) ) ) ) ).
% cosh_real_nonpos_less_iff
thf(fact_8455_arcosh__cosh__real,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( arcosh_real @ ( cosh_real @ X3 ) )
= X3 ) ) ).
% arcosh_cosh_real
thf(fact_8456_log__base__change,axiom,
! [A: real,B: real,X3: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log @ B @ X3 )
= ( divide_divide_real @ ( log @ A @ X3 ) @ ( log @ A @ B ) ) ) ) ) ).
% log_base_change
thf(fact_8457_log__mult,axiom,
! [A: real,X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( log @ A @ ( times_times_real @ X3 @ Y ) )
= ( plus_plus_real @ ( log @ A @ X3 ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).
% log_mult
thf(fact_8458_log__divide,axiom,
! [A: real,X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( log @ A @ ( divide_divide_real @ X3 @ Y ) )
= ( minus_minus_real @ ( log @ A @ X3 ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).
% log_divide
thf(fact_8459_le__log__of__power,axiom,
! [B: real,N: nat,M2: real] :
( ( ord_less_eq_real @ ( power_power_real @ B @ N ) @ M2 )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M2 ) ) ) ) ).
% le_log_of_power
thf(fact_8460_log__base__pow,axiom,
! [A: real,N: nat,X3: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( log @ ( power_power_real @ A @ N ) @ X3 )
= ( divide_divide_real @ ( log @ A @ X3 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% log_base_pow
thf(fact_8461_log__nat__power,axiom,
! [X3: real,B: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( log @ B @ ( power_power_real @ X3 @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X3 ) ) ) ) ).
% log_nat_power
thf(fact_8462_log__inverse,axiom,
! [A: real,X3: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( log @ A @ ( inverse_inverse_real @ X3 ) )
= ( uminus_uminus_real @ ( log @ A @ X3 ) ) ) ) ) ) ).
% log_inverse
thf(fact_8463_log__of__power__less,axiom,
! [M2: nat,B: real,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( power_power_real @ B @ N ) )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% log_of_power_less
thf(fact_8464_log__eq__div__ln__mult__log,axiom,
! [A: real,B: real,X3: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( log @ A @ X3 )
= ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X3 ) ) ) ) ) ) ) ) ).
% log_eq_div_ln_mult_log
thf(fact_8465_log__of__power__le,axiom,
! [M2: nat,B: real,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( power_power_real @ B @ N ) )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% log_of_power_le
thf(fact_8466_le__log2__of__power,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M2 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).
% le_log2_of_power
thf(fact_8467_log2__of__power__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% log2_of_power_less
thf(fact_8468_cosh__ln__real,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( cosh_real @ ( ln_ln_real @ X3 ) )
= ( divide_divide_real @ ( plus_plus_real @ X3 @ ( inverse_inverse_real @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% cosh_ln_real
thf(fact_8469_log2__of__power__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% log2_of_power_le
thf(fact_8470_log__base__10__eq2,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X3 )
= ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X3 ) ) ) ) ).
% log_base_10_eq2
thf(fact_8471_sinh__ln__real,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( sinh_real @ ( ln_ln_real @ X3 ) )
= ( divide_divide_real @ ( minus_minus_real @ X3 @ ( inverse_inverse_real @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% sinh_ln_real
thf(fact_8472_ceiling__log__nat__eq__powr__iff,axiom,
! [B: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
= ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K2 )
& ( ord_less_eq_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% ceiling_log_nat_eq_powr_iff
thf(fact_8473_ceiling__log__nat__eq__if,axiom,
! [B: nat,N: nat,K2: nat] :
( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K2 )
=> ( ( ord_less_eq_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).
% ceiling_log_nat_eq_if
thf(fact_8474_ceiling__log2__div2,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
= ( 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 ) ) ) ).
% ceiling_log2_div2
thf(fact_8475_floor__log__nat__eq__powr__iff,axiom,
! [B: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
= ( semiri1314217659103216013at_int @ N ) )
= ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K2 )
& ( ord_less_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% floor_log_nat_eq_powr_iff
thf(fact_8476_floor__log__nat__eq__if,axiom,
! [B: nat,N: nat,K2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K2 )
=> ( ( ord_less_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
= ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% floor_log_nat_eq_if
thf(fact_8477_real__of__int__floor__add__one__gt,axiom,
! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% real_of_int_floor_add_one_gt
thf(fact_8478_floor__eq,axiom,
! [N: int,X3: real] :
( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X3 )
=> ( ( ord_less_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
=> ( ( archim6058952711729229775r_real @ X3 )
= N ) ) ) ).
% floor_eq
thf(fact_8479_real__of__int__floor__add__one__ge,axiom,
! [R2: real] : ( ord_less_eq_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% real_of_int_floor_add_one_ge
thf(fact_8480_real__of__int__floor__ge__diff__one,axiom,
! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% real_of_int_floor_ge_diff_one
thf(fact_8481_floor__eq2,axiom,
! [N: int,X3: real] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X3 )
=> ( ( ord_less_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
=> ( ( archim6058952711729229775r_real @ X3 )
= N ) ) ) ).
% floor_eq2
thf(fact_8482_floor__divide__real__eq__div,axiom,
! [B: int,A: real] :
( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B ) ) )
= ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B ) ) ) ).
% floor_divide_real_eq_div
thf(fact_8483_floor__log2__div2,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
= ( 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 ) ) ) ).
% floor_log2_div2
thf(fact_8484_ceiling__log__eq__powr__iff,axiom,
! [X3: real,B: real,K2: nat] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( ( archim7802044766580827645g_real @ ( log @ B @ X3 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ K2 ) @ one_one_int ) )
= ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K2 ) ) @ X3 )
& ( ord_less_eq_real @ X3 @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ) ) ) ) ) ).
% ceiling_log_eq_powr_iff
thf(fact_8485_upto_Opinduct,axiom,
! [A0: int,A12: int,P2: int > int > $o] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
=> ( ! [I2: int,J2: int] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J2 ) )
=> ( ( ( ord_less_eq_int @ I2 @ J2 )
=> ( P2 @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) )
=> ( P2 @ I2 @ J2 ) ) )
=> ( P2 @ A0 @ A12 ) ) ) ).
% upto.pinduct
thf(fact_8486_arcsin__def,axiom,
( arcsin
= ( ^ [Y5: real] :
( the_real
@ ^ [X5: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
& ( ord_less_eq_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( sin_real @ X5 )
= Y5 ) ) ) ) ) ).
% arcsin_def
thf(fact_8487_modulo__int__unfold,axiom,
! [L: int,K2: int,N: nat,M2: nat] :
( ( ( ( ( sgn_sgn_int @ L )
= zero_zero_int )
| ( ( sgn_sgn_int @ K2 )
= zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) )
& ( ~ ( ( ( sgn_sgn_int @ L )
= zero_zero_int )
| ( ( sgn_sgn_int @ K2 )
= zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( ( ( ( sgn_sgn_int @ K2 )
= ( sgn_sgn_int @ L ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M2 @ N ) ) ) ) )
& ( ( ( sgn_sgn_int @ K2 )
!= ( sgn_sgn_int @ L ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( times_times_int @ ( sgn_sgn_int @ L )
@ ( minus_minus_int
@ ( semiri1314217659103216013at_int
@ ( times_times_nat @ N
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ N @ M2 ) ) ) )
@ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M2 @ N ) ) ) ) ) ) ) ) ) ).
% modulo_int_unfold
thf(fact_8488_powr__gt__zero,axiom,
! [X3: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( powr_real @ X3 @ A ) )
= ( X3 != zero_zero_real ) ) ).
% powr_gt_zero
thf(fact_8489_powr__nonneg__iff,axiom,
! [A: real,X3: real] :
( ( ord_less_eq_real @ ( powr_real @ A @ X3 ) @ zero_zero_real )
= ( A = zero_zero_real ) ) ).
% powr_nonneg_iff
thf(fact_8490_powr__eq__one__iff,axiom,
! [A: real,X3: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( powr_real @ A @ X3 )
= one_one_real )
= ( X3 = zero_zero_real ) ) ) ).
% powr_eq_one_iff
thf(fact_8491_powr__one__gt__zero__iff,axiom,
! [X3: real] :
( ( ( powr_real @ X3 @ one_one_real )
= X3 )
= ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).
% powr_one_gt_zero_iff
thf(fact_8492_powr__one,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( powr_real @ X3 @ one_one_real )
= X3 ) ) ).
% powr_one
thf(fact_8493_powr__le__cancel__iff,axiom,
! [X3: real,A: real,B: real] :
( ( ord_less_real @ one_one_real @ X3 )
=> ( ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% powr_le_cancel_iff
thf(fact_8494_sgn__mult__dvd__iff,axiom,
! [R2: int,L: int,K2: int] :
( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L ) @ K2 )
= ( ( dvd_dvd_int @ L @ K2 )
& ( ( R2 = zero_zero_int )
=> ( K2 = zero_zero_int ) ) ) ) ).
% sgn_mult_dvd_iff
thf(fact_8495_mult__sgn__dvd__iff,axiom,
! [L: int,R2: int,K2: int] :
( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R2 ) ) @ K2 )
= ( ( dvd_dvd_int @ L @ K2 )
& ( ( R2 = zero_zero_int )
=> ( K2 = zero_zero_int ) ) ) ) ).
% mult_sgn_dvd_iff
thf(fact_8496_dvd__sgn__mult__iff,axiom,
! [L: int,R2: int,K2: int] :
( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K2 ) )
= ( ( dvd_dvd_int @ L @ K2 )
| ( R2 = zero_zero_int ) ) ) ).
% dvd_sgn_mult_iff
thf(fact_8497_dvd__mult__sgn__iff,axiom,
! [L: int,K2: int,R2: int] :
( ( dvd_dvd_int @ L @ ( times_times_int @ K2 @ ( sgn_sgn_int @ R2 ) ) )
= ( ( dvd_dvd_int @ L @ K2 )
| ( R2 = zero_zero_int ) ) ) ).
% dvd_mult_sgn_iff
thf(fact_8498_powr__log__cancel,axiom,
! [A: real,X3: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( powr_real @ A @ ( log @ A @ X3 ) )
= X3 ) ) ) ) ).
% powr_log_cancel
thf(fact_8499_log__powr__cancel,axiom,
! [A: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log @ A @ ( powr_real @ A @ Y ) )
= Y ) ) ) ).
% log_powr_cancel
thf(fact_8500_powr__numeral,axiom,
! [X3: real,N: num] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( powr_real @ X3 @ ( numeral_numeral_real @ N ) )
= ( power_power_real @ X3 @ ( numeral_numeral_nat @ N ) ) ) ) ).
% powr_numeral
thf(fact_8501_powr__non__neg,axiom,
! [A: real,X3: real] :
~ ( ord_less_real @ ( powr_real @ A @ X3 ) @ zero_zero_real ) ).
% powr_non_neg
thf(fact_8502_powr__less__mono2__neg,axiom,
! [A: real,X3: real,Y: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X3 @ A ) ) ) ) ) ).
% powr_less_mono2_neg
thf(fact_8503_powr__mono2,axiom,
! [A: real,X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% powr_mono2
thf(fact_8504_powr__ge__pzero,axiom,
! [X3: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X3 @ Y ) ) ).
% powr_ge_pzero
thf(fact_8505_powr__mono,axiom,
! [A: real,B: real,X3: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ one_one_real @ X3 )
=> ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B ) ) ) ) ).
% powr_mono
thf(fact_8506_int__sgnE,axiom,
! [K2: int] :
~ ! [N3: nat,L4: int] :
( K2
!= ( times_times_int @ ( sgn_sgn_int @ L4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_sgnE
thf(fact_8507_powr__less__mono2,axiom,
! [A: real,X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% powr_less_mono2
thf(fact_8508_powr__mono2_H,axiom,
! [A: real,X3: real,Y: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X3 @ A ) ) ) ) ) ).
% powr_mono2'
thf(fact_8509_powr__inj,axiom,
! [A: real,X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ( powr_real @ A @ X3 )
= ( powr_real @ A @ Y ) )
= ( X3 = Y ) ) ) ) ).
% powr_inj
thf(fact_8510_gr__one__powr,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ one_one_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ one_one_real @ ( powr_real @ X3 @ Y ) ) ) ) ).
% gr_one_powr
thf(fact_8511_ge__one__powr__ge__zero,axiom,
! [X3: real,A: real] :
( ( ord_less_eq_real @ one_one_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ one_one_real @ ( powr_real @ X3 @ A ) ) ) ) ).
% ge_one_powr_ge_zero
thf(fact_8512_powr__mono__both,axiom,
! [A: real,B: real,X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ one_one_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y @ B ) ) ) ) ) ) ).
% powr_mono_both
thf(fact_8513_powr__le1,axiom,
! [A: real,X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ one_one_real )
=> ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ one_one_real ) ) ) ) ).
% powr_le1
thf(fact_8514_powr__divide,axiom,
! [X3: real,Y: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( powr_real @ ( divide_divide_real @ X3 @ Y ) @ A )
= ( divide_divide_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% powr_divide
thf(fact_8515_powr__mult,axiom,
! [X3: real,Y: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( powr_real @ ( times_times_real @ X3 @ Y ) @ A )
= ( times_times_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% powr_mult
thf(fact_8516_inverse__powr,axiom,
! [Y: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( powr_real @ ( inverse_inverse_real @ Y ) @ A )
= ( inverse_inverse_real @ ( powr_real @ Y @ A ) ) ) ) ).
% inverse_powr
thf(fact_8517_log__base__powr,axiom,
! [A: real,B: real,X3: real] :
( ( A != zero_zero_real )
=> ( ( log @ ( powr_real @ A @ B ) @ X3 )
= ( divide_divide_real @ ( log @ A @ X3 ) @ B ) ) ) ).
% log_base_powr
thf(fact_8518_ln__powr,axiom,
! [X3: real,Y: real] :
( ( X3 != zero_zero_real )
=> ( ( ln_ln_real @ ( powr_real @ X3 @ Y ) )
= ( times_times_real @ Y @ ( ln_ln_real @ X3 ) ) ) ) ).
% ln_powr
thf(fact_8519_log__powr,axiom,
! [X3: real,B: real,Y: real] :
( ( X3 != zero_zero_real )
=> ( ( log @ B @ ( powr_real @ X3 @ Y ) )
= ( times_times_real @ Y @ ( log @ B @ X3 ) ) ) ) ).
% log_powr
thf(fact_8520_sgn__mod,axiom,
! [L: int,K2: int] :
( ( L != zero_zero_int )
=> ( ~ ( dvd_dvd_int @ L @ K2 )
=> ( ( sgn_sgn_int @ ( modulo_modulo_int @ K2 @ L ) )
= ( sgn_sgn_int @ L ) ) ) ) ).
% sgn_mod
thf(fact_8521_ln__neg__is__const,axiom,
! [X3: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ln_ln_real @ X3 )
= ( the_real
@ ^ [X5: real] : $false ) ) ) ).
% ln_neg_is_const
thf(fact_8522_powr__realpow,axiom,
! [X3: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( powr_real @ X3 @ ( semiri5074537144036343181t_real @ N ) )
= ( power_power_real @ X3 @ N ) ) ) ).
% powr_realpow
thf(fact_8523_powr__less__iff,axiom,
! [B: real,X3: real,Y: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ ( powr_real @ B @ Y ) @ X3 )
= ( ord_less_real @ Y @ ( log @ B @ X3 ) ) ) ) ) ).
% powr_less_iff
thf(fact_8524_less__powr__iff,axiom,
! [B: real,X3: real,Y: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ ( powr_real @ B @ Y ) )
= ( ord_less_real @ ( log @ B @ X3 ) @ Y ) ) ) ) ).
% less_powr_iff
thf(fact_8525_log__less__iff,axiom,
! [B: real,X3: real,Y: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ ( log @ B @ X3 ) @ Y )
= ( ord_less_real @ X3 @ ( powr_real @ B @ Y ) ) ) ) ) ).
% log_less_iff
thf(fact_8526_less__log__iff,axiom,
! [B: real,X3: real,Y: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ Y @ ( log @ B @ X3 ) )
= ( ord_less_real @ ( powr_real @ B @ Y ) @ X3 ) ) ) ) ).
% less_log_iff
thf(fact_8527_zsgn__def,axiom,
( sgn_sgn_int
= ( ^ [I3: int] : ( if_int @ ( I3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zsgn_def
thf(fact_8528_div__sgn__abs__cancel,axiom,
! [V: int,K2: int,L: int] :
( ( V != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L ) ) )
= ( divide_divide_int @ ( abs_abs_int @ K2 ) @ ( abs_abs_int @ L ) ) ) ) ).
% div_sgn_abs_cancel
thf(fact_8529_powr__neg__one,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( powr_real @ X3 @ ( uminus_uminus_real @ one_one_real ) )
= ( divide_divide_real @ one_one_real @ X3 ) ) ) ).
% powr_neg_one
thf(fact_8530_powr__mult__base,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( times_times_real @ X3 @ ( powr_real @ X3 @ Y ) )
= ( powr_real @ X3 @ ( plus_plus_real @ one_one_real @ Y ) ) ) ) ).
% powr_mult_base
thf(fact_8531_le__log__iff,axiom,
! [B: real,X3: real,Y: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ Y @ ( log @ B @ X3 ) )
= ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X3 ) ) ) ) ).
% le_log_iff
thf(fact_8532_log__le__iff,axiom,
! [B: real,X3: real,Y: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ ( log @ B @ X3 ) @ Y )
= ( ord_less_eq_real @ X3 @ ( powr_real @ B @ Y ) ) ) ) ) ).
% log_le_iff
thf(fact_8533_le__powr__iff,axiom,
! [B: real,X3: real,Y: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ ( powr_real @ B @ Y ) )
= ( ord_less_eq_real @ ( log @ B @ X3 ) @ Y ) ) ) ) ).
% le_powr_iff
thf(fact_8534_powr__le__iff,axiom,
! [B: real,X3: real,Y: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X3 )
= ( ord_less_eq_real @ Y @ ( log @ B @ X3 ) ) ) ) ) ).
% powr_le_iff
thf(fact_8535_ln__powr__bound,axiom,
! [X3: real,A: real] :
( ( ord_less_eq_real @ one_one_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( divide_divide_real @ ( powr_real @ X3 @ A ) @ A ) ) ) ) ).
% ln_powr_bound
thf(fact_8536_ln__powr__bound2,axiom,
! [X3: real,A: real] :
( ( ord_less_real @ one_one_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X3 ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X3 ) ) ) ) ).
% ln_powr_bound2
thf(fact_8537_add__log__eq__powr,axiom,
! [B: real,X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( plus_plus_real @ Y @ ( log @ B @ X3 ) )
= ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y ) @ X3 ) ) ) ) ) ) ).
% add_log_eq_powr
thf(fact_8538_log__add__eq__powr,axiom,
! [B: real,X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( plus_plus_real @ ( log @ B @ X3 ) @ Y )
= ( log @ B @ ( times_times_real @ X3 @ ( powr_real @ B @ Y ) ) ) ) ) ) ) ).
% log_add_eq_powr
thf(fact_8539_minus__log__eq__powr,axiom,
! [B: real,X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( minus_minus_real @ Y @ ( log @ B @ X3 ) )
= ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y ) @ X3 ) ) ) ) ) ) ).
% minus_log_eq_powr
thf(fact_8540_arccos__def,axiom,
( arccos
= ( ^ [Y5: real] :
( the_real
@ ^ [X5: real] :
( ( ord_less_eq_real @ zero_zero_real @ X5 )
& ( ord_less_eq_real @ X5 @ pi )
& ( ( cos_real @ X5 )
= Y5 ) ) ) ) ) ).
% arccos_def
thf(fact_8541_eucl__rel__int__remainderI,axiom,
! [R2: int,L: int,K2: int,Q3: int] :
( ( ( sgn_sgn_int @ R2 )
= ( sgn_sgn_int @ L ) )
=> ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
=> ( ( K2
= ( plus_plus_int @ ( times_times_int @ Q3 @ L ) @ R2 ) )
=> ( eucl_rel_int @ K2 @ L @ ( product_Pair_int_int @ Q3 @ R2 ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_8542_log__minus__eq__powr,axiom,
! [B: real,X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( minus_minus_real @ ( log @ B @ X3 ) @ Y )
= ( log @ B @ ( times_times_real @ X3 @ ( powr_real @ B @ ( uminus_uminus_real @ Y ) ) ) ) ) ) ) ) ).
% log_minus_eq_powr
thf(fact_8543_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A1: int,A22: int,A32: product_prod_int_int] :
( ? [K3: int] :
( ( A1 = K3 )
& ( A22 = zero_zero_int )
& ( A32
= ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
| ? [L2: int,K3: int,Q4: int] :
( ( A1 = K3 )
& ( A22 = L2 )
& ( A32
= ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
& ( L2 != zero_zero_int )
& ( K3
= ( times_times_int @ Q4 @ L2 ) ) )
| ? [R5: int,L2: int,K3: int,Q4: int] :
( ( A1 = K3 )
& ( A22 = L2 )
& ( A32
= ( product_Pair_int_int @ Q4 @ R5 ) )
& ( ( sgn_sgn_int @ R5 )
= ( sgn_sgn_int @ L2 ) )
& ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L2 ) )
& ( K3
= ( plus_plus_int @ ( times_times_int @ Q4 @ L2 ) @ R5 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_8544_eucl__rel__int_Ocases,axiom,
! [A12: int,A23: int,A33: product_prod_int_int] :
( ( eucl_rel_int @ A12 @ A23 @ A33 )
=> ( ( ( A23 = zero_zero_int )
=> ( A33
!= ( product_Pair_int_int @ zero_zero_int @ A12 ) ) )
=> ( ! [Q2: int] :
( ( A33
= ( product_Pair_int_int @ Q2 @ zero_zero_int ) )
=> ( ( A23 != zero_zero_int )
=> ( A12
!= ( times_times_int @ Q2 @ A23 ) ) ) )
=> ~ ! [R3: int,Q2: int] :
( ( A33
= ( product_Pair_int_int @ Q2 @ R3 ) )
=> ( ( ( sgn_sgn_int @ R3 )
= ( sgn_sgn_int @ A23 ) )
=> ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A23 ) )
=> ( A12
!= ( plus_plus_int @ ( times_times_int @ Q2 @ A23 ) @ R3 ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_8545_div__noneq__sgn__abs,axiom,
! [L: int,K2: int] :
( ( L != zero_zero_int )
=> ( ( ( sgn_sgn_int @ K2 )
!= ( sgn_sgn_int @ L ) )
=> ( ( divide_divide_int @ K2 @ L )
= ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K2 ) @ ( abs_abs_int @ L ) ) )
@ ( zero_n2684676970156552555ol_int
@ ~ ( dvd_dvd_int @ L @ K2 ) ) ) ) ) ) ).
% div_noneq_sgn_abs
thf(fact_8546_powr__half__sqrt,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( powr_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= ( sqrt @ X3 ) ) ) ).
% powr_half_sqrt
thf(fact_8547_powr__neg__numeral,axiom,
! [X3: real,N: num] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( powr_real @ X3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% powr_neg_numeral
thf(fact_8548_pi__half,axiom,
( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
= ( the_real
@ ^ [X5: real] :
( ( ord_less_eq_real @ zero_zero_real @ X5 )
& ( ord_less_eq_real @ X5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
& ( ( cos_real @ X5 )
= zero_zero_real ) ) ) ) ).
% pi_half
thf(fact_8549_pi__def,axiom,
( pi
= ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
@ ( the_real
@ ^ [X5: real] :
( ( ord_less_eq_real @ zero_zero_real @ X5 )
& ( ord_less_eq_real @ X5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
& ( ( cos_real @ X5 )
= zero_zero_real ) ) ) ) ) ).
% pi_def
thf(fact_8550_floor__log__eq__powr__iff,axiom,
! [X3: real,B: real,K2: int] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( ( archim6058952711729229775r_real @ ( log @ B @ X3 ) )
= K2 )
= ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K2 ) ) @ X3 )
& ( ord_less_real @ X3 @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K2 @ one_one_int ) ) ) ) ) ) ) ) ).
% floor_log_eq_powr_iff
thf(fact_8551_divide__int__unfold,axiom,
! [L: int,K2: int,N: nat,M2: nat] :
( ( ( ( ( sgn_sgn_int @ L )
= zero_zero_int )
| ( ( sgn_sgn_int @ K2 )
= zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_int ) )
& ( ~ ( ( ( sgn_sgn_int @ L )
= zero_zero_int )
| ( ( sgn_sgn_int @ K2 )
= zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( ( ( ( sgn_sgn_int @ K2 )
= ( sgn_sgn_int @ L ) )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M2 @ N ) ) ) )
& ( ( ( sgn_sgn_int @ K2 )
!= ( sgn_sgn_int @ L ) )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( uminus_uminus_int
@ ( semiri1314217659103216013at_int
@ ( plus_plus_nat @ ( divide_divide_nat @ M2 @ N )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ N @ M2 ) ) ) ) ) ) ) ) ) ) ).
% divide_int_unfold
thf(fact_8552_modulo__int__def,axiom,
( modulo_modulo_int
= ( ^ [K3: int,L2: int] :
( if_int @ ( L2 = zero_zero_int ) @ K3
@ ( if_int
@ ( ( sgn_sgn_int @ K3 )
= ( sgn_sgn_int @ L2 ) )
@ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) )
@ ( times_times_int @ ( sgn_sgn_int @ L2 )
@ ( minus_minus_int
@ ( times_times_int @ ( abs_abs_int @ L2 )
@ ( zero_n2684676970156552555ol_int
@ ~ ( dvd_dvd_int @ L2 @ K3 ) ) )
@ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ) ) ) ).
% modulo_int_def
thf(fact_8553_divide__int__def,axiom,
( divide_divide_int
= ( ^ [K3: int,L2: int] :
( if_int @ ( L2 = zero_zero_int ) @ zero_zero_int
@ ( if_int
@ ( ( sgn_sgn_int @ K3 )
= ( sgn_sgn_int @ L2 ) )
@ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) )
@ ( uminus_uminus_int
@ ( semiri1314217659103216013at_int
@ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_int @ L2 @ K3 ) ) ) ) ) ) ) ) ) ).
% divide_int_def
thf(fact_8554_sgn__le__0__iff,axiom,
! [X3: real] :
( ( ord_less_eq_real @ ( sgn_sgn_real @ X3 ) @ zero_zero_real )
= ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).
% sgn_le_0_iff
thf(fact_8555_zero__le__sgn__iff,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X3 ) )
= ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).
% zero_le_sgn_iff
thf(fact_8556_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% nat_int
thf(fact_8557_nat__numeral,axiom,
! [K2: num] :
( ( nat2 @ ( numeral_numeral_int @ K2 ) )
= ( numeral_numeral_nat @ K2 ) ) ).
% nat_numeral
thf(fact_8558_nat__of__bool,axiom,
! [P2: $o] :
( ( nat2 @ ( zero_n2684676970156552555ol_int @ P2 ) )
= ( zero_n2687167440665602831ol_nat @ P2 ) ) ).
% nat_of_bool
thf(fact_8559_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_8560_nat__le__0,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ Z2 @ zero_zero_int )
=> ( ( nat2 @ Z2 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_8561_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_8562_zless__nat__conj,axiom,
! [W2: int,Z2: int] :
( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ( ord_less_int @ zero_zero_int @ Z2 )
& ( ord_less_int @ W2 @ Z2 ) ) ) ).
% zless_nat_conj
thf(fact_8563_nat__neg__numeral,axiom,
! [K2: num] :
( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
= zero_zero_nat ) ).
% nat_neg_numeral
thf(fact_8564_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_8565_int__nat__eq,axiom,
! [Z2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= Z2 ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_8566_zero__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% zero_less_nat_eq
thf(fact_8567_diff__nat__numeral,axiom,
! [V: num,V3: num] :
( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V3 ) )
= ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V3 ) ) ) ) ).
% diff_nat_numeral
thf(fact_8568_numeral__power__eq__nat__cancel__iff,axiom,
! [X3: num,N: nat,Y: int] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
= ( nat2 @ Y ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N )
= Y ) ) ).
% numeral_power_eq_nat_cancel_iff
thf(fact_8569_nat__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X3: num,N: nat] :
( ( ( nat2 @ Y )
= ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) )
= ( Y
= ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).
% nat_eq_numeral_power_cancel_iff
thf(fact_8570_dvd__nat__abs__iff,axiom,
! [N: nat,K2: int] :
( ( dvd_dvd_nat @ N @ ( nat2 @ ( abs_abs_int @ K2 ) ) )
= ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ N ) @ K2 ) ) ).
% dvd_nat_abs_iff
thf(fact_8571_nat__abs__dvd__iff,axiom,
! [K2: int,N: nat] :
( ( dvd_dvd_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ N )
= ( dvd_dvd_int @ K2 @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_abs_dvd_iff
thf(fact_8572_nat__ceiling__le__eq,axiom,
! [X3: real,A: nat] :
( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X3 ) ) @ A )
= ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% nat_ceiling_le_eq
thf(fact_8573_one__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% one_less_nat_eq
thf(fact_8574_nat__numeral__diff__1,axiom,
! [V: num] :
( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
= ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).
% nat_numeral_diff_1
thf(fact_8575_nat__less__numeral__power__cancel__iff,axiom,
! [A: int,X3: num,N: nat] :
( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).
% nat_less_numeral_power_cancel_iff
thf(fact_8576_numeral__power__less__nat__cancel__iff,axiom,
! [X3: num,N: nat,A: int] :
( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) @ ( nat2 @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) @ A ) ) ).
% numeral_power_less_nat_cancel_iff
thf(fact_8577_nat__le__numeral__power__cancel__iff,axiom,
! [A: int,X3: num,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).
% nat_le_numeral_power_cancel_iff
thf(fact_8578_numeral__power__le__nat__cancel__iff,axiom,
! [X3: num,N: nat,A: int] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) @ ( nat2 @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) @ A ) ) ).
% numeral_power_le_nat_cancel_iff
thf(fact_8579_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_8580_nat__mono,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_8581_eq__nat__nat__iff,axiom,
! [Z2: int,Z8: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z8 )
=> ( ( ( nat2 @ Z2 )
= ( nat2 @ Z8 ) )
= ( Z2 = Z8 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_8582_all__nat,axiom,
( ( ^ [P3: nat > $o] :
! [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
! [X5: int] :
( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( P4 @ ( nat2 @ X5 ) ) ) ) ) ).
% all_nat
thf(fact_8583_ex__nat,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
? [X5: int] :
( ( ord_less_eq_int @ zero_zero_int @ X5 )
& ( P4 @ ( nat2 @ X5 ) ) ) ) ) ).
% ex_nat
thf(fact_8584_complex__of__real__def,axiom,
( real_V4546457046886955230omplex
= ( ^ [R5: real] : ( complex2 @ R5 @ zero_zero_real ) ) ) ).
% complex_of_real_def
thf(fact_8585_complex__of__real__code,axiom,
( real_V4546457046886955230omplex
= ( ^ [X5: real] : ( complex2 @ X5 @ zero_zero_real ) ) ) ).
% complex_of_real_code
thf(fact_8586_complex__eq__cancel__iff2,axiom,
! [X3: real,Y: real,Xa2: real] :
( ( ( complex2 @ X3 @ Y )
= ( real_V4546457046886955230omplex @ Xa2 ) )
= ( ( X3 = Xa2 )
& ( Y = zero_zero_real ) ) ) ).
% complex_eq_cancel_iff2
thf(fact_8587_nat__mono__iff,axiom,
! [Z2: int,W2: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W2 @ Z2 ) ) ) ).
% nat_mono_iff
thf(fact_8588_zless__nat__eq__int__zless,axiom,
! [M2: nat,Z2: int] :
( ( ord_less_nat @ M2 @ ( nat2 @ Z2 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ Z2 ) ) ).
% zless_nat_eq_int_zless
thf(fact_8589_nat__le__iff,axiom,
! [X3: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X3 ) @ N )
= ( ord_less_eq_int @ X3 @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_8590_int__eq__iff,axiom,
! [M2: nat,Z2: int] :
( ( ( semiri1314217659103216013at_int @ M2 )
= Z2 )
= ( ( M2
= ( nat2 @ Z2 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ) ).
% int_eq_iff
thf(fact_8591_nat__0__le,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= Z2 ) ) ).
% nat_0_le
thf(fact_8592_nat__int__add,axiom,
! [A: nat,B: nat] :
( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
= ( plus_plus_nat @ A @ B ) ) ).
% nat_int_add
thf(fact_8593_nat__abs__mult__distrib,axiom,
! [W2: int,Z2: int] :
( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W2 @ Z2 ) ) )
= ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W2 ) ) @ ( nat2 @ ( abs_abs_int @ Z2 ) ) ) ) ).
% nat_abs_mult_distrib
thf(fact_8594_Complex__add__complex__of__real,axiom,
! [X3: real,Y: real,R2: real] :
( ( plus_plus_complex @ ( complex2 @ X3 @ Y ) @ ( real_V4546457046886955230omplex @ R2 ) )
= ( complex2 @ ( plus_plus_real @ X3 @ R2 ) @ Y ) ) ).
% Complex_add_complex_of_real
thf(fact_8595_complex__of__real__add__Complex,axiom,
! [R2: real,X3: real,Y: real] :
( ( plus_plus_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( complex2 @ X3 @ Y ) )
= ( complex2 @ ( plus_plus_real @ R2 @ X3 ) @ Y ) ) ).
% complex_of_real_add_Complex
thf(fact_8596_nat__plus__as__int,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_8597_real__nat__ceiling__ge,axiom,
! [X3: real] : ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X3 ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_8598_sgn__real__def,axiom,
( sgn_sgn_real
= ( ^ [A5: real] : ( if_real @ ( A5 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A5 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% sgn_real_def
thf(fact_8599_nat__less__eq__zless,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W2 @ Z2 ) ) ) ).
% nat_less_eq_zless
thf(fact_8600_nat__le__eq__zle,axiom,
! [W2: int,Z2: int] :
( ( ( ord_less_int @ zero_zero_int @ W2 )
| ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ) ).
% nat_le_eq_zle
thf(fact_8601_nat__eq__iff2,axiom,
! [M2: nat,W2: int] :
( ( M2
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_8602_nat__eq__iff,axiom,
! [W2: int,M2: nat] :
( ( ( nat2 @ W2 )
= M2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_8603_split__nat,axiom,
! [P2: nat > $o,I: int] :
( ( P2 @ ( nat2 @ I ) )
= ( ! [N4: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N4 ) )
=> ( P2 @ N4 ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P2 @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_8604_le__nat__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K2 ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K2 ) ) ) ).
% le_nat_iff
thf(fact_8605_nat__add__distrib,axiom,
! [Z2: int,Z8: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z8 )
=> ( ( nat2 @ ( plus_plus_int @ Z2 @ Z8 ) )
= ( plus_plus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_8606_nat__mult__distrib,axiom,
! [Z2: int,Z8: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( nat2 @ ( times_times_int @ Z2 @ Z8 ) )
= ( times_times_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) ) ) ).
% nat_mult_distrib
thf(fact_8607_Suc__as__int,axiom,
( suc
= ( ^ [A5: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ one_one_int ) ) ) ) ).
% Suc_as_int
thf(fact_8608_nat__diff__distrib,axiom,
! [Z8: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z8 )
=> ( ( ord_less_eq_int @ Z8 @ Z2 )
=> ( ( nat2 @ ( minus_minus_int @ Z2 @ Z8 ) )
= ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_8609_nat__diff__distrib_H,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( minus_minus_int @ X3 @ Y ) )
= ( minus_minus_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_8610_nat__abs__triangle__ineq,axiom,
! [K2: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K2 @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_8611_nat__div__distrib_H,axiom,
! [Y: int,X3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( divide_divide_int @ X3 @ Y ) )
= ( divide_divide_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib'
thf(fact_8612_nat__div__distrib,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( nat2 @ ( divide_divide_int @ X3 @ Y ) )
= ( divide_divide_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib
thf(fact_8613_nat__floor__neg,axiom,
! [X3: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X3 ) )
= zero_zero_nat ) ) ).
% nat_floor_neg
thf(fact_8614_nat__power__eq,axiom,
! [Z2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( nat2 @ ( power_power_int @ Z2 @ N ) )
= ( power_power_nat @ ( nat2 @ Z2 ) @ N ) ) ) ).
% nat_power_eq
thf(fact_8615_nat__mod__distrib,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( modulo_modulo_int @ X3 @ Y ) )
= ( modulo_modulo_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_mod_distrib
thf(fact_8616_floor__eq3,axiom,
! [N: nat,X3: real] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X3 )
=> ( ( ord_less_real @ X3 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X3 ) )
= N ) ) ) ).
% floor_eq3
thf(fact_8617_le__nat__floor,axiom,
! [X3: nat,A: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X3 ) @ A )
=> ( ord_less_eq_nat @ X3 @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% le_nat_floor
thf(fact_8618_nat__2,axiom,
( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% nat_2
thf(fact_8619_sgn__power__injE,axiom,
! [A: real,N: nat,X3: real,B: real] :
( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
= X3 )
=> ( ( X3
= ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ).
% sgn_power_injE
thf(fact_8620_Suc__nat__eq__nat__zadd1,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( suc @ ( nat2 @ Z2 ) )
= ( nat2 @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_8621_nat__less__iff,axiom,
! [W2: int,M2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ M2 )
= ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).
% nat_less_iff
thf(fact_8622_nat__mult__distrib__neg,axiom,
! [Z2: int,Z8: int] :
( ( ord_less_eq_int @ Z2 @ zero_zero_int )
=> ( ( nat2 @ ( times_times_int @ Z2 @ Z8 ) )
= ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z2 ) ) @ ( nat2 @ ( uminus_uminus_int @ Z8 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_8623_nat__abs__int__diff,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_eq_nat @ A @ B )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
= ( minus_minus_nat @ B @ A ) ) )
& ( ~ ( ord_less_eq_nat @ A @ B )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
= ( minus_minus_nat @ A @ B ) ) ) ) ).
% nat_abs_int_diff
thf(fact_8624_floor__eq4,axiom,
! [N: nat,X3: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X3 )
=> ( ( ord_less_real @ X3 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X3 ) )
= N ) ) ) ).
% floor_eq4
thf(fact_8625_diff__nat__eq__if,axiom,
! [Z8: int,Z2: int] :
( ( ( ord_less_int @ Z8 @ zero_zero_int )
=> ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) )
= ( nat2 @ Z2 ) ) )
& ( ~ ( ord_less_int @ Z8 @ zero_zero_int )
=> ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) )
= ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z2 @ Z8 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z2 @ Z8 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_8626_nat__dvd__iff,axiom,
! [Z2: int,M2: nat] :
( ( dvd_dvd_nat @ ( nat2 @ Z2 ) @ M2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( dvd_dvd_int @ Z2 @ ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_dvd_iff
thf(fact_8627_floor__real__def,axiom,
( archim6058952711729229775r_real
= ( ^ [X5: real] :
( the_int
@ ^ [Z6: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z6 ) @ X5 )
& ( ord_less_real @ X5 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z6 @ one_one_int ) ) ) ) ) ) ) ).
% floor_real_def
thf(fact_8628_even__nat__iff,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K2 ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) ) ) ).
% even_nat_iff
thf(fact_8629_powr__real__of__int,axiom,
! [X3: real,N: int] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( powr_real @ X3 @ ( ring_1_of_int_real @ N ) )
= ( power_power_real @ X3 @ ( nat2 @ N ) ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( powr_real @ X3 @ ( ring_1_of_int_real @ N ) )
= ( inverse_inverse_real @ ( power_power_real @ X3 @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).
% powr_real_of_int
thf(fact_8630_arctan__inverse,axiom,
! [X3: real] :
( ( X3 != zero_zero_real )
=> ( ( arctan @ ( divide_divide_real @ one_one_real @ X3 ) )
= ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X3 ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X3 ) ) ) ) ).
% arctan_inverse
thf(fact_8631_powr__int,axiom,
! [X3: real,I: int] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( powr_real @ X3 @ ( ring_1_of_int_real @ I ) )
= ( power_power_real @ X3 @ ( nat2 @ I ) ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( powr_real @ X3 @ ( ring_1_of_int_real @ I ) )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ X3 @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).
% powr_int
thf(fact_8632_and__int__unfold,axiom,
( bit_se725231765392027082nd_int
= ( ^ [K3: int,L2: int] :
( if_int
@ ( ( K3 = zero_zero_int )
| ( L2 = zero_zero_int ) )
@ zero_zero_int
@ ( if_int
@ ( K3
= ( uminus_uminus_int @ one_one_int ) )
@ L2
@ ( if_int
@ ( L2
= ( uminus_uminus_int @ one_one_int ) )
@ K3
@ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K3 @ ( 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 @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% and_int_unfold
thf(fact_8633_VEBT__internal_OminNull_Opelims_I1_J,axiom,
! [X3: vEBT_VEBT,Y: $o] :
( ( ( vEBT_VEBT_minNull @ X3 )
= Y )
=> ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
=> ( ( ( X3
= ( vEBT_Leaf @ $false @ $false ) )
=> ( Y
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
=> ( ! [Uv2: $o] :
( ( X3
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ~ Y
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
=> ( ! [Uu2: $o] :
( ( X3
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( ~ Y
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
=> ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
=> ( Y
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
=> ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
=> ( ~ Y
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(1)
thf(fact_8634_set__encode__insert,axiom,
! [A4: set_nat,N: nat] :
( ( finite_finite_nat @ A4 )
=> ( ~ ( member_nat @ N @ A4 )
=> ( ( nat_set_encode @ ( insert_nat @ N @ A4 ) )
= ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A4 ) ) ) ) ) ).
% set_encode_insert
thf(fact_8635_mask__nat__positive__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% mask_nat_positive_iff
thf(fact_8636_concat__bit__of__zero__2,axiom,
! [N: nat,K2: int] :
( ( bit_concat_bit @ N @ K2 @ zero_zero_int )
= ( bit_se2923211474154528505it_int @ N @ K2 ) ) ).
% concat_bit_of_zero_2
thf(fact_8637_and__nonnegative__int__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K2 @ L ) )
= ( ( ord_less_eq_int @ zero_zero_int @ K2 )
| ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% and_nonnegative_int_iff
thf(fact_8638_and__negative__int__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K2 @ L ) @ zero_zero_int )
= ( ( ord_less_int @ K2 @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% and_negative_int_iff
thf(fact_8639_take__bit__of__Suc__0,axiom,
! [N: nat] :
( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
= ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% take_bit_of_Suc_0
thf(fact_8640_atMost__0,axiom,
( ( set_ord_atMost_nat @ zero_zero_nat )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% atMost_0
thf(fact_8641_and__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
= zero_zero_int ) ).
% and_minus_numerals(5)
thf(fact_8642_and__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= zero_zero_int ) ).
% and_minus_numerals(1)
thf(fact_8643_take__bit__nat__less__eq__self,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M2 ) @ M2 ) ).
% take_bit_nat_less_eq_self
thf(fact_8644_take__bit__tightened__less__eq__nat,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M2 @ Q3 ) @ ( bit_se2925701944663578781it_nat @ N @ Q3 ) ) ) ).
% take_bit_tightened_less_eq_nat
thf(fact_8645_less__eq__mask,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% less_eq_mask
thf(fact_8646_take__bit__nat__eq,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K2 ) )
= ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K2 ) ) ) ) ).
% take_bit_nat_eq
thf(fact_8647_nat__take__bit__eq,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K2 ) )
= ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K2 ) ) ) ) ).
% nat_take_bit_eq
thf(fact_8648_take__bit__eq__mask__iff,axiom,
! [N: nat,K2: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K2 )
= ( bit_se2000444600071755411sk_int @ N ) )
= ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K2 @ one_one_int ) )
= zero_zero_int ) ) ).
% take_bit_eq_mask_iff
thf(fact_8649_AND__lower,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X3 @ Y ) ) ) ).
% AND_lower
thf(fact_8650_AND__upper1,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X3 @ Y ) @ X3 ) ) ).
% AND_upper1
thf(fact_8651_AND__upper2,axiom,
! [Y: int,X3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X3 @ Y ) @ Y ) ) ).
% AND_upper2
thf(fact_8652_AND__upper1_H,axiom,
! [Y: int,Z2: int,Ya: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z2 ) ) ) ).
% AND_upper1'
thf(fact_8653_AND__upper2_H,axiom,
! [Y: int,Z2: int,X3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X3 @ Y ) @ Z2 ) ) ) ).
% AND_upper2'
thf(fact_8654_take__bit__tightened__less__eq__int,axiom,
! [M2: nat,N: nat,K2: int] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M2 @ K2 ) @ ( bit_se2923211474154528505it_int @ N @ K2 ) ) ) ).
% take_bit_tightened_less_eq_int
thf(fact_8655_take__bit__nonnegative,axiom,
! [N: nat,K2: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) ) ).
% take_bit_nonnegative
thf(fact_8656_take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) @ K2 )
= ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).
% take_bit_int_less_eq_self_iff
thf(fact_8657_take__bit__int__greater__self__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less_int @ K2 @ ( bit_se2923211474154528505it_int @ N @ K2 ) )
= ( ord_less_int @ K2 @ zero_zero_int ) ) ).
% take_bit_int_greater_self_iff
thf(fact_8658_not__take__bit__negative,axiom,
! [N: nat,K2: int] :
~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) @ zero_zero_int ) ).
% not_take_bit_negative
thf(fact_8659_plus__and__or,axiom,
! [X3: int,Y: int] :
( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X3 @ Y ) @ ( bit_se1409905431419307370or_int @ X3 @ Y ) )
= ( plus_plus_int @ X3 @ Y ) ) ).
% plus_and_or
thf(fact_8660_lessThan__Suc,axiom,
! [K2: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K2 ) )
= ( insert_nat @ K2 @ ( set_ord_lessThan_nat @ K2 ) ) ) ).
% lessThan_Suc
thf(fact_8661_atMost__Suc,axiom,
! [K2: nat] :
( ( set_ord_atMost_nat @ ( suc @ K2 ) )
= ( insert_nat @ ( suc @ K2 ) @ ( set_ord_atMost_nat @ K2 ) ) ) ).
% atMost_Suc
thf(fact_8662_mask__nonnegative__int,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).
% mask_nonnegative_int
thf(fact_8663_not__mask__negative__int,axiom,
! [N: nat] :
~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% not_mask_negative_int
thf(fact_8664_AND__upper2_H_H,axiom,
! [Y: int,Z2: int,X3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ ( bit_se725231765392027082nd_int @ X3 @ Y ) @ Z2 ) ) ) ).
% AND_upper2''
thf(fact_8665_AND__upper1_H_H,axiom,
! [Y: int,Z2: int,Ya: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z2 ) ) ) ).
% AND_upper1''
thf(fact_8666_and__less__eq,axiom,
! [L: int,K2: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K2 @ L ) @ K2 ) ) ).
% and_less_eq
thf(fact_8667_take__bit__decr__eq,axiom,
! [N: nat,K2: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K2 )
!= zero_zero_int )
=> ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K2 @ one_one_int ) )
= ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) @ one_one_int ) ) ) ).
% take_bit_decr_eq
thf(fact_8668_atLeast0__atMost__Suc,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% atLeast0_atMost_Suc
thf(fact_8669_Icc__eq__insert__lb__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( set_or1269000886237332187st_nat @ M2 @ N )
= ( insert_nat @ M2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_8670_atLeastAtMostSuc__conv,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) )
= ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_8671_atLeastAtMost__insertL,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( insert_nat @ M2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) )
= ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).
% atLeastAtMost_insertL
thf(fact_8672_lessThan__nat__numeral,axiom,
! [K2: num] :
( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K2 ) )
= ( insert_nat @ ( pred_numeral @ K2 ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K2 ) ) ) ) ).
% lessThan_nat_numeral
thf(fact_8673_atMost__nat__numeral,axiom,
! [K2: num] :
( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K2 ) )
= ( insert_nat @ ( numeral_numeral_nat @ K2 ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K2 ) ) ) ) ).
% atMost_nat_numeral
thf(fact_8674_less__mask,axiom,
! [N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% less_mask
thf(fact_8675_take__bit__eq__mask__iff__exp__dvd,axiom,
! [N: nat,K2: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K2 )
= ( bit_se2000444600071755411sk_int @ N ) )
= ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K2 @ one_one_int ) ) ) ).
% take_bit_eq_mask_iff_exp_dvd
thf(fact_8676_atLeast1__atMost__eq__remove0,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
= ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_8677_take__bit__nat__less__self__iff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M2 ) @ M2 )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M2 ) ) ).
% take_bit_nat_less_self_iff
thf(fact_8678_Suc__mask__eq__exp,axiom,
! [N: nat] :
( ( suc @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% Suc_mask_eq_exp
thf(fact_8679_take__bit__Suc__minus__bit0,axiom,
! [N: nat,K2: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K2 ) ) ) )
= ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_minus_bit0
thf(fact_8680_take__bit__int__greater__eq__self__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq_int @ K2 @ ( bit_se2923211474154528505it_int @ N @ K2 ) )
= ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% take_bit_int_greater_eq_self_iff
thf(fact_8681_take__bit__int__less__self__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) @ K2 )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K2 ) ) ).
% take_bit_int_less_self_iff
thf(fact_8682_set__decode__plus__power__2,axiom,
! [N: nat,Z2: nat] :
( ~ ( member_nat @ N @ ( nat_set_decode @ Z2 ) )
=> ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z2 ) )
= ( insert_nat @ N @ ( nat_set_decode @ Z2 ) ) ) ) ).
% set_decode_plus_power_2
thf(fact_8683_take__bit__int__eq__self,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ( ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( bit_se2923211474154528505it_int @ N @ K2 )
= K2 ) ) ) ).
% take_bit_int_eq_self
thf(fact_8684_take__bit__int__eq__self__iff,axiom,
! [N: nat,K2: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K2 )
= K2 )
= ( ( ord_less_eq_int @ zero_zero_int @ K2 )
& ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% take_bit_int_eq_self_iff
thf(fact_8685_take__bit__incr__eq,axiom,
! [N: nat,K2: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K2 )
!= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
=> ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K2 @ one_one_int ) )
= ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) ) ) ) ).
% take_bit_incr_eq
thf(fact_8686_VEBT__internal_OminNull_Opelims_I3_J,axiom,
! [X3: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X3 )
=> ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
=> ( ! [Uv2: $o] :
( ( X3
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
=> ( ! [Uu2: $o] :
( ( X3
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
=> ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(3)
thf(fact_8687_VEBT__internal_OminNull_Opelims_I2_J,axiom,
! [X3: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X3 )
=> ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
=> ( ( ( X3
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(2)
thf(fact_8688_take__bit__int__less__eq,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) @ ( minus_minus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% take_bit_int_less_eq
thf(fact_8689_take__bit__int__greater__eq,axiom,
! [K2: int,N: nat] :
( ( ord_less_int @ K2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K2 ) ) ) ).
% take_bit_int_greater_eq
thf(fact_8690_signed__take__bit__eq__take__bit__shift,axiom,
( bit_ri631733984087533419it_int
= ( ^ [N4: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% signed_take_bit_eq_take_bit_shift
thf(fact_8691_and__int__rec,axiom,
( bit_se725231765392027082nd_int
= ( ^ [K3: int,L2: int] :
( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% and_int_rec
thf(fact_8692_take__bit__minus__small__eq,axiom,
! [K2: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K2 ) )
= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K2 ) ) ) ) ).
% take_bit_minus_small_eq
thf(fact_8693_floor__rat__def,axiom,
( archim3151403230148437115or_rat
= ( ^ [X5: rat] :
( the_int
@ ^ [Z6: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z6 ) @ X5 )
& ( ord_less_rat @ X5 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z6 @ one_one_int ) ) ) ) ) ) ) ).
% floor_rat_def
thf(fact_8694_take__bit__numeral__minus__bit1,axiom,
! [L: num,K2: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K2 ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% take_bit_numeral_minus_bit1
thf(fact_8695_and__int_Oelims,axiom,
! [X3: int,Xa2: int,Y: int] :
( ( ( bit_se725231765392027082nd_int @ X3 @ Xa2 )
= Y )
=> ( ( ( ( member_int @ X3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y
= ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
& ( ~ ( ( member_int @ X3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y
= ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% and_int.elims
thf(fact_8696_and__int_Osimps,axiom,
( bit_se725231765392027082nd_int
= ( ^ [K3: int,L2: int] :
( if_int
@ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
@ ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
@ ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% and_int.simps
thf(fact_8697_and__nat__numerals_I3_J,axiom,
! [X3: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% and_nat_numerals(3)
thf(fact_8698_and__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= zero_zero_nat ) ).
% and_nat_numerals(1)
thf(fact_8699_and__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= one_one_nat ) ).
% and_nat_numerals(2)
thf(fact_8700_and__nat__numerals_I4_J,axiom,
! [X3: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% and_nat_numerals(4)
thf(fact_8701_and__Suc__0__eq,axiom,
! [N: nat] :
( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
= ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% and_Suc_0_eq
thf(fact_8702_Suc__0__and__eq,axiom,
! [N: nat] :
( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
= ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% Suc_0_and_eq
thf(fact_8703_sgn__rat__def,axiom,
( sgn_sgn_rat
= ( ^ [A5: rat] : ( if_rat @ ( A5 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A5 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% sgn_rat_def
thf(fact_8704_abs__rat__def,axiom,
( abs_abs_rat
= ( ^ [A5: rat] : ( if_rat @ ( ord_less_rat @ A5 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A5 ) @ A5 ) ) ) ).
% abs_rat_def
thf(fact_8705_less__eq__rat__def,axiom,
( ord_less_eq_rat
= ( ^ [X5: rat,Y5: rat] :
( ( ord_less_rat @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% less_eq_rat_def
thf(fact_8706_obtain__pos__sum,axiom,
! [R2: rat] :
( ( ord_less_rat @ zero_zero_rat @ R2 )
=> ~ ! [S: rat] :
( ( ord_less_rat @ zero_zero_rat @ S )
=> ! [T5: rat] :
( ( ord_less_rat @ zero_zero_rat @ T5 )
=> ( R2
!= ( plus_plus_rat @ S @ T5 ) ) ) ) ) ).
% obtain_pos_sum
thf(fact_8707_add__inc,axiom,
! [X3: num,Y: num] :
( ( plus_plus_num @ X3 @ ( inc @ Y ) )
= ( inc @ ( plus_plus_num @ X3 @ Y ) ) ) ).
% add_inc
thf(fact_8708_add__One,axiom,
! [X3: num] :
( ( plus_plus_num @ X3 @ one )
= ( inc @ X3 ) ) ).
% add_One
thf(fact_8709_mult__inc,axiom,
! [X3: num,Y: num] :
( ( times_times_num @ X3 @ ( inc @ Y ) )
= ( plus_plus_num @ ( times_times_num @ X3 @ Y ) @ X3 ) ) ).
% mult_inc
thf(fact_8710_atLeastAtMostPlus1__int__conv,axiom,
! [M2: int,N: int] :
( ( ord_less_eq_int @ M2 @ ( plus_plus_int @ one_one_int @ N ) )
=> ( ( set_or1266510415728281911st_int @ M2 @ ( plus_plus_int @ one_one_int @ N ) )
= ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M2 @ N ) ) ) ) ).
% atLeastAtMostPlus1_int_conv
thf(fact_8711_simp__from__to,axiom,
( set_or1266510415728281911st_int
= ( ^ [I3: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I3 ) @ bot_bot_set_int @ ( insert_int @ I3 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) ) ) ) ).
% simp_from_to
thf(fact_8712_and__nat__unfold,axiom,
( bit_se727722235901077358nd_nat
= ( ^ [M5: nat,N4: nat] :
( if_nat
@ ( ( M5 = zero_zero_nat )
| ( N4 = zero_zero_nat ) )
@ zero_zero_nat
@ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_8713_and__nat__rec,axiom,
( bit_se727722235901077358nd_nat
= ( ^ [M5: nat,N4: nat] :
( plus_plus_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 )
& ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
@ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% and_nat_rec
thf(fact_8714_take__bit__Suc__minus__bit1,axiom,
! [N: nat,K2: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K2 ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% take_bit_Suc_minus_bit1
thf(fact_8715_and__int_Opelims,axiom,
! [X3: int,Xa2: int,Y: int] :
( ( ( bit_se725231765392027082nd_int @ X3 @ Xa2 )
= Y )
=> ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) )
=> ~ ( ( ( ( ( member_int @ X3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y
= ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
& ( ~ ( ( member_int @ X3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y
= ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
=> ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) ) ) ) ) ).
% and_int.pelims
thf(fact_8716_and__int_Opsimps,axiom,
! [K2: int,L: int] :
( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L ) )
=> ( ( ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( ( bit_se725231765392027082nd_int @ K2 @ L )
= ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
& ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( ( bit_se725231765392027082nd_int @ K2 @ L )
= ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% and_int.psimps
thf(fact_8717_and__int_Opinduct,axiom,
! [A0: int,A12: int,P2: int > int > $o] :
( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
=> ( ! [K: int,L4: int] :
( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L4 ) )
=> ( ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( P2 @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
=> ( P2 @ K @ L4 ) ) )
=> ( P2 @ A0 @ A12 ) ) ) ).
% and_int.pinduct
thf(fact_8718_signed__take__bit__eq__take__bit__minus,axiom,
( bit_ri631733984087533419it_int
= ( ^ [N4: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N4 ) ) ) ) ) ) ).
% signed_take_bit_eq_take_bit_minus
thf(fact_8719_signed__take__bit__nonnegative__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) )
= ( ~ ( bit_se1146084159140164899it_int @ K2 @ N ) ) ) ).
% signed_take_bit_nonnegative_iff
thf(fact_8720_signed__take__bit__negative__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) @ zero_zero_int )
= ( bit_se1146084159140164899it_int @ K2 @ N ) ) ).
% signed_take_bit_negative_iff
thf(fact_8721_bit__minus__numeral__Bit0__Suc__iff,axiom,
! [W2: num,N: nat] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) ) @ ( suc @ N ) )
= ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ N ) ) ).
% bit_minus_numeral_Bit0_Suc_iff
thf(fact_8722_bit__minus__numeral__Bit1__Suc__iff,axiom,
! [W2: num,N: nat] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W2 ) ) ) @ ( suc @ N ) )
= ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W2 ) @ N ) ) ) ).
% bit_minus_numeral_Bit1_Suc_iff
thf(fact_8723_diff__rat__def,axiom,
( minus_minus_rat
= ( ^ [Q4: rat,R5: rat] : ( plus_plus_rat @ Q4 @ ( uminus_uminus_rat @ R5 ) ) ) ) ).
% diff_rat_def
thf(fact_8724_bit__imp__take__bit__positive,axiom,
! [N: nat,M2: nat,K2: int] :
( ( ord_less_nat @ N @ M2 )
=> ( ( bit_se1146084159140164899it_int @ K2 @ N )
=> ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M2 @ K2 ) ) ) ) ).
% bit_imp_take_bit_positive
thf(fact_8725_bit__concat__bit__iff,axiom,
! [M2: nat,K2: int,L: int,N: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M2 @ K2 @ L ) @ N )
= ( ( ( ord_less_nat @ N @ M2 )
& ( bit_se1146084159140164899it_int @ K2 @ N ) )
| ( ( ord_less_eq_nat @ M2 @ N )
& ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).
% bit_concat_bit_iff
thf(fact_8726_int__bit__bound,axiom,
! [K2: int] :
~ ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ N3 @ M3 )
=> ( ( bit_se1146084159140164899it_int @ K2 @ M3 )
= ( bit_se1146084159140164899it_int @ K2 @ N3 ) ) )
=> ~ ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( bit_se1146084159140164899it_int @ K2 @ ( minus_minus_nat @ N3 @ one_one_nat ) )
= ( ~ ( bit_se1146084159140164899it_int @ K2 @ N3 ) ) ) ) ) ).
% int_bit_bound
thf(fact_8727_set__bit__eq,axiom,
( bit_se7879613467334960850it_int
= ( ^ [N4: nat,K3: int] :
( plus_plus_int @ K3
@ ( times_times_int
@ ( zero_n2684676970156552555ol_int
@ ~ ( bit_se1146084159140164899it_int @ K3 @ N4 ) )
@ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% set_bit_eq
thf(fact_8728_take__bit__Suc__from__most,axiom,
! [N: nat,K2: int] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K2 )
= ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N ) ) ) @ ( bit_se2923211474154528505it_int @ N @ K2 ) ) ) ).
% take_bit_Suc_from_most
thf(fact_8729_rat__inverse__code,axiom,
! [P: rat] :
( ( quotient_of @ ( inverse_inverse_rat @ P ) )
= ( produc4245557441103728435nt_int
@ ^ [A5: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( A5 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A5 ) @ B4 ) @ ( abs_abs_int @ A5 ) ) )
@ ( quotient_of @ P ) ) ) ).
% rat_inverse_code
thf(fact_8730_cmod__complex__polar,axiom,
! [R2: real,A: real] :
( ( 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 ) ) ) ) ) )
= ( abs_abs_real @ R2 ) ) ).
% cmod_complex_polar
thf(fact_8731_cmod__unit__one,axiom,
! [A: real] :
( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) )
= one_one_real ) ).
% cmod_unit_one
thf(fact_8732_Suc__0__xor__eq,axiom,
! [N: nat] :
( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
= ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% Suc_0_xor_eq
thf(fact_8733_quotient__of__number_I3_J,axiom,
! [K2: num] :
( ( quotient_of @ ( numeral_numeral_rat @ K2 ) )
= ( product_Pair_int_int @ ( numeral_numeral_int @ K2 ) @ one_one_int ) ) ).
% quotient_of_number(3)
thf(fact_8734_rat__one__code,axiom,
( ( quotient_of @ one_one_rat )
= ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).
% rat_one_code
thf(fact_8735_rat__zero__code,axiom,
( ( quotient_of @ zero_zero_rat )
= ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% rat_zero_code
thf(fact_8736_xor__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% xor_nat_numerals(1)
thf(fact_8737_xor__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).
% xor_nat_numerals(2)
thf(fact_8738_xor__nat__numerals_I3_J,axiom,
! [X3: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
= ( numeral_numeral_nat @ ( bit1 @ X3 ) ) ) ).
% xor_nat_numerals(3)
thf(fact_8739_xor__nat__numerals_I4_J,axiom,
! [X3: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
= ( numeral_numeral_nat @ ( bit0 @ X3 ) ) ) ).
% xor_nat_numerals(4)
thf(fact_8740_quotient__of__number_I5_J,axiom,
! [K2: num] :
( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K2 ) ) )
= ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) @ one_one_int ) ) ).
% quotient_of_number(5)
thf(fact_8741_quotient__of__number_I4_J,axiom,
( ( quotient_of @ ( uminus_uminus_rat @ one_one_rat ) )
= ( product_Pair_int_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ) ) ).
% quotient_of_number(4)
thf(fact_8742_complex__i__not__zero,axiom,
imaginary_unit != zero_zero_complex ).
% complex_i_not_zero
thf(fact_8743_not__bit__Suc__0__Suc,axiom,
! [N: nat] :
~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).
% not_bit_Suc_0_Suc
thf(fact_8744_bit__Suc__0__iff,axiom,
! [N: nat] :
( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
= ( N = zero_zero_nat ) ) ).
% bit_Suc_0_iff
thf(fact_8745_not__bit__Suc__0__numeral,axiom,
! [N: num] :
~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).
% not_bit_Suc_0_numeral
thf(fact_8746_quotient__of__div,axiom,
! [R2: rat,N: int,D: int] :
( ( ( quotient_of @ R2 )
= ( product_Pair_int_int @ N @ D ) )
=> ( R2
= ( divide_divide_rat @ ( ring_1_of_int_rat @ N ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).
% quotient_of_div
thf(fact_8747_Complex__eq__i,axiom,
! [X3: real,Y: real] :
( ( ( complex2 @ X3 @ Y )
= imaginary_unit )
= ( ( X3 = zero_zero_real )
& ( Y = one_one_real ) ) ) ).
% Complex_eq_i
thf(fact_8748_imaginary__unit_Ocode,axiom,
( imaginary_unit
= ( complex2 @ zero_zero_real @ one_one_real ) ) ).
% imaginary_unit.code
thf(fact_8749_quotient__of__denom__pos,axiom,
! [R2: rat,P: int,Q3: int] :
( ( ( quotient_of @ R2 )
= ( product_Pair_int_int @ P @ Q3 ) )
=> ( ord_less_int @ zero_zero_int @ Q3 ) ) ).
% quotient_of_denom_pos
thf(fact_8750_bit__nat__iff,axiom,
! [K2: int,N: nat] :
( ( bit_se1148574629649215175it_nat @ ( nat2 @ K2 ) @ N )
= ( ( ord_less_eq_int @ zero_zero_int @ K2 )
& ( bit_se1146084159140164899it_int @ K2 @ N ) ) ) ).
% bit_nat_iff
thf(fact_8751_i__complex__of__real,axiom,
! [R2: real] :
( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R2 ) )
= ( complex2 @ zero_zero_real @ R2 ) ) ).
% i_complex_of_real
thf(fact_8752_complex__of__real__i,axiom,
! [R2: real] :
( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ imaginary_unit )
= ( complex2 @ zero_zero_real @ R2 ) ) ).
% complex_of_real_i
thf(fact_8753_Complex__eq,axiom,
( complex2
= ( ^ [A5: real,B4: real] : ( plus_plus_complex @ ( real_V4546457046886955230omplex @ A5 ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B4 ) ) ) ) ) ).
% Complex_eq
thf(fact_8754_rat__uminus__code,axiom,
! [P: rat] :
( ( quotient_of @ ( uminus_uminus_rat @ P ) )
= ( produc4245557441103728435nt_int
@ ^ [A5: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A5 ) )
@ ( quotient_of @ P ) ) ) ).
% rat_uminus_code
thf(fact_8755_rat__abs__code,axiom,
! [P: rat] :
( ( quotient_of @ ( abs_abs_rat @ P ) )
= ( produc4245557441103728435nt_int
@ ^ [A5: int] : ( product_Pair_int_int @ ( abs_abs_int @ A5 ) )
@ ( quotient_of @ P ) ) ) ).
% rat_abs_code
thf(fact_8756_rat__less__eq__code,axiom,
( ord_less_eq_rat
= ( ^ [P6: rat,Q4: rat] :
( produc4947309494688390418_int_o
@ ^ [A5: int,C3: int] :
( produc4947309494688390418_int_o
@ ^ [B4: int,D4: int] : ( ord_less_eq_int @ ( times_times_int @ A5 @ D4 ) @ ( times_times_int @ C3 @ B4 ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P6 ) ) ) ) ).
% rat_less_eq_code
thf(fact_8757_complex__split__polar,axiom,
! [Z2: complex] :
? [R3: real,A3: real] :
( Z2
= ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A3 ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A3 ) ) ) ) ) ) ).
% complex_split_polar
thf(fact_8758_xor__nat__unfold,axiom,
( bit_se6528837805403552850or_nat
= ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = zero_zero_nat ) @ M5 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_8759_xor__nat__rec,axiom,
( bit_se6528837805403552850or_nat
= ( ^ [M5: nat,N4: nat] :
( plus_plus_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
!= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
@ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% xor_nat_rec
thf(fact_8760_xor__Suc__0__eq,axiom,
! [N: nat] :
( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
= ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% xor_Suc_0_eq
thf(fact_8761_csqrt__ii,axiom,
( ( csqrt @ imaginary_unit )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ one_one_complex @ imaginary_unit ) @ ( real_V4546457046886955230omplex @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% csqrt_ii
thf(fact_8762_quotient__of__int,axiom,
! [A: int] :
( ( quotient_of @ ( of_int @ A ) )
= ( product_Pair_int_int @ A @ one_one_int ) ) ).
% quotient_of_int
thf(fact_8763_rat__minus__code,axiom,
! [P: rat,Q3: rat] :
( ( quotient_of @ ( minus_minus_rat @ P @ Q3 ) )
= ( produc4245557441103728435nt_int
@ ^ [A5: int,C3: int] :
( produc4245557441103728435nt_int
@ ^ [B4: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A5 @ D4 ) @ ( times_times_int @ B4 @ C3 ) ) @ ( times_times_int @ C3 @ D4 ) ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P ) ) ) ).
% rat_minus_code
thf(fact_8764_rat__plus__code,axiom,
! [P: rat,Q3: rat] :
( ( quotient_of @ ( plus_plus_rat @ P @ Q3 ) )
= ( produc4245557441103728435nt_int
@ ^ [A5: int,C3: int] :
( produc4245557441103728435nt_int
@ ^ [B4: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A5 @ D4 ) @ ( times_times_int @ B4 @ C3 ) ) @ ( times_times_int @ C3 @ D4 ) ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P ) ) ) ).
% rat_plus_code
thf(fact_8765_xor__nonnegative__int__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K2 @ L ) )
= ( ( ord_less_eq_int @ zero_zero_int @ K2 )
= ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% xor_nonnegative_int_iff
thf(fact_8766_xor__negative__int__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K2 @ L ) @ zero_zero_int )
= ( ( ord_less_int @ K2 @ zero_zero_int )
!= ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% xor_negative_int_iff
thf(fact_8767_csqrt__0,axiom,
( ( csqrt @ zero_zero_complex )
= zero_zero_complex ) ).
% csqrt_0
thf(fact_8768_csqrt__eq__0,axiom,
! [Z2: complex] :
( ( ( csqrt @ Z2 )
= zero_zero_complex )
= ( Z2 = zero_zero_complex ) ) ).
% csqrt_eq_0
thf(fact_8769_normalize__denom__zero,axiom,
! [P: int] :
( ( normalize @ ( product_Pair_int_int @ P @ zero_zero_int ) )
= ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% normalize_denom_zero
thf(fact_8770_normalize__negative,axiom,
! [Q3: int,P: int] :
( ( ord_less_int @ Q3 @ zero_zero_int )
=> ( ( normalize @ ( product_Pair_int_int @ P @ Q3 ) )
= ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P ) @ ( uminus_uminus_int @ Q3 ) ) ) ) ) ).
% normalize_negative
thf(fact_8771_XOR__lower,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X3 @ Y ) ) ) ) ).
% XOR_lower
thf(fact_8772_normalize__denom__pos,axiom,
! [R2: product_prod_int_int,P: int,Q3: int] :
( ( ( normalize @ R2 )
= ( product_Pair_int_int @ P @ Q3 ) )
=> ( ord_less_int @ zero_zero_int @ Q3 ) ) ).
% normalize_denom_pos
thf(fact_8773_normalize__crossproduct,axiom,
! [Q3: int,S3: int,P: int,R2: int] :
( ( Q3 != zero_zero_int )
=> ( ( S3 != zero_zero_int )
=> ( ( ( normalize @ ( product_Pair_int_int @ P @ Q3 ) )
= ( normalize @ ( product_Pair_int_int @ R2 @ S3 ) ) )
=> ( ( times_times_int @ P @ S3 )
= ( times_times_int @ R2 @ Q3 ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_8774_of__real__sqrt,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( real_V4546457046886955230omplex @ ( sqrt @ X3 ) )
= ( csqrt @ ( real_V4546457046886955230omplex @ X3 ) ) ) ) ).
% of_real_sqrt
thf(fact_8775_XOR__upper,axiom,
! [X3: int,N: nat,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_int @ X3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ord_less_int @ ( bit_se6526347334894502574or_int @ X3 @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% XOR_upper
thf(fact_8776_rat__divide__code,axiom,
! [P: rat,Q3: rat] :
( ( quotient_of @ ( divide_divide_rat @ P @ Q3 ) )
= ( produc4245557441103728435nt_int
@ ^ [A5: int,C3: int] :
( produc4245557441103728435nt_int
@ ^ [B4: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A5 @ D4 ) @ ( times_times_int @ C3 @ B4 ) ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P ) ) ) ).
% rat_divide_code
thf(fact_8777_rat__times__code,axiom,
! [P: rat,Q3: rat] :
( ( quotient_of @ ( times_times_rat @ P @ Q3 ) )
= ( produc4245557441103728435nt_int
@ ^ [A5: int,C3: int] :
( produc4245557441103728435nt_int
@ ^ [B4: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A5 @ B4 ) @ ( times_times_int @ C3 @ D4 ) ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P ) ) ) ).
% rat_times_code
thf(fact_8778_xor__int__rec,axiom,
( bit_se6526347334894502574or_int
= ( ^ [K3: int,L2: int] :
( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) )
!= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% xor_int_rec
thf(fact_8779_Frct__code__post_I5_J,axiom,
! [K2: num] :
( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K2 ) ) )
= ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K2 ) ) ) ).
% Frct_code_post(5)
thf(fact_8780_xor__int__unfold,axiom,
( bit_se6526347334894502574or_int
= ( ^ [K3: int,L2: int] :
( if_int
@ ( K3
= ( uminus_uminus_int @ one_one_int ) )
@ ( bit_ri7919022796975470100ot_int @ L2 )
@ ( if_int
@ ( L2
= ( uminus_uminus_int @ one_one_int ) )
@ ( bit_ri7919022796975470100ot_int @ K3 )
@ ( if_int @ ( K3 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_int_unfold
thf(fact_8781_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I3: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_8782_not__negative__int__iff,axiom,
! [K2: int] :
( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).
% not_negative_int_iff
thf(fact_8783_not__nonnegative__int__iff,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K2 ) )
= ( ord_less_int @ K2 @ zero_zero_int ) ) ).
% not_nonnegative_int_iff
thf(fact_8784_and__not__numerals_I1_J,axiom,
( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= zero_zero_int ) ).
% and_not_numerals(1)
thf(fact_8785_or__not__numerals_I1_J,axiom,
( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% or_not_numerals(1)
thf(fact_8786_and__not__numerals_I3_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= zero_zero_int ) ).
% and_not_numerals(3)
thf(fact_8787_or__not__numerals_I7_J,axiom,
! [M2: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% or_not_numerals(7)
thf(fact_8788_Frct__code__post_I2_J,axiom,
! [A: int] :
( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
= zero_zero_rat ) ).
% Frct_code_post(2)
thf(fact_8789_Frct__code__post_I1_J,axiom,
! [A: int] :
( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
= zero_zero_rat ) ).
% Frct_code_post(1)
thf(fact_8790_Frct__code__post_I7_J,axiom,
! [A: int,B: int] :
( ( frct @ ( product_Pair_int_int @ ( uminus_uminus_int @ A ) @ B ) )
= ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% Frct_code_post(7)
thf(fact_8791_Frct__code__post_I8_J,axiom,
! [A: int,B: int] :
( ( frct @ ( product_Pair_int_int @ A @ ( uminus_uminus_int @ B ) ) )
= ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% Frct_code_post(8)
thf(fact_8792_Frct__code__post_I3_J,axiom,
( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
= one_one_rat ) ).
% Frct_code_post(3)
thf(fact_8793_or__not__numerals_I5_J,axiom,
! [M2: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% or_not_numerals(5)
thf(fact_8794_Frct__code__post_I4_J,axiom,
! [K2: num] :
( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K2 ) @ one_one_int ) )
= ( numeral_numeral_rat @ K2 ) ) ).
% Frct_code_post(4)
thf(fact_8795_and__not__numerals_I8_J,axiom,
! [M2: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% and_not_numerals(8)
thf(fact_8796_or__not__numerals_I8_J,axiom,
! [M2: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% or_not_numerals(8)
thf(fact_8797_or__not__numerals_I9_J,axiom,
! [M2: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% or_not_numerals(9)
thf(fact_8798_not__int__rec,axiom,
( bit_ri7919022796975470100ot_int
= ( ^ [K3: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% not_int_rec
thf(fact_8799_Frct__code__post_I6_J,axiom,
! [K2: num,L: num] :
( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K2 ) @ ( numeral_numeral_int @ L ) ) )
= ( divide_divide_rat @ ( numeral_numeral_rat @ K2 ) @ ( numeral_numeral_rat @ L ) ) ) ).
% Frct_code_post(6)
thf(fact_8800_Sum__Ico__nat,axiom,
! [M2: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X5: nat] : X5
@ ( set_or4665077453230672383an_nat @ M2 @ N ) )
= ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M2 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% Sum_Ico_nat
thf(fact_8801_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).
% finite_atLeastLessThan
thf(fact_8802_atLeastLessThan__singleton,axiom,
! [M2: nat] :
( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ M2 ) )
= ( insert_nat @ M2 @ bot_bot_set_nat ) ) ).
% atLeastLessThan_singleton
thf(fact_8803_all__nat__less__eq,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N )
=> ( P2 @ M5 ) ) )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P2 @ X5 ) ) ) ) ).
% all_nat_less_eq
thf(fact_8804_ex__nat__less__eq,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [M5: nat] :
( ( ord_less_nat @ M5 @ N )
& ( P2 @ M5 ) ) )
= ( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P2 @ X5 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_8805_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_8806_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_8807_atLeastLessThan0,axiom,
! [M2: nat] :
( ( set_or4665077453230672383an_nat @ M2 @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_8808_atLeast0__lessThan__Suc,axiom,
! [N: nat] :
( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% atLeast0_lessThan_Suc
thf(fact_8809_subset__eq__atLeast0__lessThan__finite,axiom,
! [N7: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N7 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_8810_atLeastLessThanSuc,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) )
= ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) )
= bot_bot_set_nat ) ) ) ).
% atLeastLessThanSuc
thf(fact_8811_prod__Suc__fact,axiom,
! [N: nat] :
( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
= ( semiri1408675320244567234ct_nat @ N ) ) ).
% prod_Suc_fact
thf(fact_8812_prod__Suc__Suc__fact,axiom,
! [N: nat] :
( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( semiri1408675320244567234ct_nat @ N ) ) ).
% prod_Suc_Suc_fact
thf(fact_8813_atLeastLessThan__nat__numeral,axiom,
! [M2: nat,K2: num] :
( ( ( ord_less_eq_nat @ M2 @ ( pred_numeral @ K2 ) )
=> ( ( set_or4665077453230672383an_nat @ M2 @ ( numeral_numeral_nat @ K2 ) )
= ( insert_nat @ ( pred_numeral @ K2 ) @ ( set_or4665077453230672383an_nat @ M2 @ ( pred_numeral @ K2 ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M2 @ ( pred_numeral @ K2 ) )
=> ( ( set_or4665077453230672383an_nat @ M2 @ ( numeral_numeral_nat @ K2 ) )
= bot_bot_set_nat ) ) ) ).
% atLeastLessThan_nat_numeral
thf(fact_8814_atLeast1__lessThan__eq__remove0,axiom,
! [N: nat] :
( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
= ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_8815_sum__power2,axiom,
! [K2: nat] :
( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K2 ) )
= ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) @ one_one_nat ) ) ).
% sum_power2
thf(fact_8816_Chebyshev__sum__upper__nat,axiom,
! [N: nat,A: nat > nat,B: nat > nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ N )
=> ( ord_less_eq_nat @ ( A @ I2 ) @ ( A @ J2 ) ) ) )
=> ( ! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ N )
=> ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I2 ) ) ) )
=> ( ord_less_eq_nat
@ ( times_times_nat @ N
@ ( groups3542108847815614940at_nat
@ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( B @ I3 ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
@ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% Chebyshev_sum_upper_nat
thf(fact_8817_VEBT_Osize__gen_I1_J,axiom,
! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% VEBT.size_gen(1)
thf(fact_8818_VEBT_Osize_I3_J,axiom,
! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% VEBT.size(3)
thf(fact_8819_finite__atLeastLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L @ U ) ) ).
% finite_atLeastLessThan_int
thf(fact_8820_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).
% finite_atLeastZeroLessThan_int
thf(fact_8821_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or4662586982721622107an_int @ L @ ( plus_plus_int @ U @ one_one_int ) )
= ( set_or1266510415728281911st_int @ L @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_8822_Cauchy__iff2,axiom,
( topolo4055970368930404560y_real
= ( ^ [X7: nat > real] :
! [J3: nat] :
? [M8: nat] :
! [M5: nat] :
( ( ord_less_eq_nat @ M8 @ M5 )
=> ! [N4: nat] :
( ( ord_less_eq_nat @ M8 @ N4 )
=> ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X7 @ M5 ) @ ( X7 @ N4 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% Cauchy_iff2
thf(fact_8823_divmod__step__integer__def,axiom,
( unique4921790084139445826nteger
= ( ^ [L2: num] :
( produc6916734918728496179nteger
@ ^ [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 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_8824_csqrt_Osimps_I1_J,axiom,
! [Z2: complex] :
( ( re @ ( csqrt @ Z2 ) )
= ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% csqrt.simps(1)
thf(fact_8825_times__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L )
= zero_z3403309356797280102nteger ) ).
% times_integer_code(2)
thf(fact_8826_times__integer__code_I1_J,axiom,
! [K2: code_integer] :
( ( times_3573771949741848930nteger @ K2 @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% times_integer_code(1)
thf(fact_8827_minus__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L )
= ( uminus1351360451143612070nteger @ L ) ) ).
% minus_integer_code(2)
thf(fact_8828_minus__integer__code_I1_J,axiom,
! [K2: code_integer] :
( ( minus_8373710615458151222nteger @ K2 @ zero_z3403309356797280102nteger )
= K2 ) ).
% minus_integer_code(1)
thf(fact_8829_full__exhaustive__integer_H_Ocases,axiom,
! [X3: produc1908205239877642774nteger] :
~ ! [F3: produc6241069584506657477e_term > option6357759511663192854e_term,D2: code_integer,I2: code_integer] :
( X3
!= ( produc8603105652947943368nteger @ F3 @ ( produc1086072967326762835nteger @ D2 @ I2 ) ) ) ).
% full_exhaustive_integer'.cases
thf(fact_8830_exhaustive__integer_H_Ocases,axiom,
! [X3: produc8763457246119570046nteger] :
~ ! [F3: code_integer > option6357759511663192854e_term,D2: code_integer,I2: code_integer] :
( X3
!= ( produc6137756002093451184nteger @ F3 @ ( produc1086072967326762835nteger @ D2 @ I2 ) ) ) ).
% exhaustive_integer'.cases
thf(fact_8831_plus__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L )
= L ) ).
% plus_integer_code(2)
thf(fact_8832_plus__integer__code_I1_J,axiom,
! [K2: code_integer] :
( ( plus_p5714425477246183910nteger @ K2 @ zero_z3403309356797280102nteger )
= K2 ) ).
% plus_integer_code(1)
thf(fact_8833_less__eq__integer__code_I1_J,axiom,
ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).
% less_eq_integer_code(1)
thf(fact_8834_sgn__integer__code,axiom,
( sgn_sgn_Code_integer
= ( ^ [K3: code_integer] : ( if_Code_integer @ ( K3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).
% sgn_integer_code
thf(fact_8835_divmod__integer_H__def,axiom,
( unique3479559517661332726nteger
= ( ^ [M5: num,N4: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M5 ) @ ( numera6620942414471956472nteger @ N4 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M5 ) @ ( numera6620942414471956472nteger @ N4 ) ) ) ) ) ).
% divmod_integer'_def
thf(fact_8836_imaginary__unit_Osimps_I1_J,axiom,
( ( re @ imaginary_unit )
= zero_zero_real ) ).
% imaginary_unit.simps(1)
thf(fact_8837_complex__Re__le__cmod,axiom,
! [X3: complex] : ( ord_less_eq_real @ ( re @ X3 ) @ ( real_V1022390504157884413omplex @ X3 ) ) ).
% complex_Re_le_cmod
thf(fact_8838_zero__complex_Osimps_I1_J,axiom,
( ( re @ zero_zero_complex )
= zero_zero_real ) ).
% zero_complex.simps(1)
thf(fact_8839_plus__complex_Osimps_I1_J,axiom,
! [X3: complex,Y: complex] :
( ( re @ ( plus_plus_complex @ X3 @ Y ) )
= ( plus_plus_real @ ( re @ X3 ) @ ( re @ Y ) ) ) ).
% plus_complex.simps(1)
thf(fact_8840_abs__Re__le__cmod,axiom,
! [X3: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X3 ) ) @ ( real_V1022390504157884413omplex @ X3 ) ) ).
% abs_Re_le_cmod
thf(fact_8841_Re__csqrt,axiom,
! [Z2: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z2 ) ) ) ).
% Re_csqrt
thf(fact_8842_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_8843_zero__integer_Orsp,axiom,
zero_zero_int = zero_zero_int ).
% zero_integer.rsp
thf(fact_8844_cmod__plus__Re__le__0__iff,axiom,
! [Z2: complex] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ zero_zero_real )
= ( ( re @ Z2 )
= ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ).
% cmod_plus_Re_le_0_iff
thf(fact_8845_csqrt_Ocode,axiom,
( csqrt
= ( ^ [Z6: complex] :
( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z6 ) @ ( re @ Z6 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
@ ( times_times_real
@ ( if_real
@ ( ( im @ Z6 )
= zero_zero_real )
@ one_one_real
@ ( sgn_sgn_real @ ( im @ Z6 ) ) )
@ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z6 ) @ ( re @ Z6 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% csqrt.code
thf(fact_8846_csqrt_Osimps_I2_J,axiom,
! [Z2: complex] :
( ( im @ ( csqrt @ Z2 ) )
= ( times_times_real
@ ( if_real
@ ( ( im @ Z2 )
= zero_zero_real )
@ one_one_real
@ ( sgn_sgn_real @ ( im @ Z2 ) ) )
@ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% csqrt.simps(2)
thf(fact_8847_integer__of__int__code,axiom,
( code_integer_of_int
= ( ^ [K3: int] :
( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
@ ( if_Code_integer
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% integer_of_int_code
thf(fact_8848_csqrt__of__real__nonpos,axiom,
! [X3: complex] :
( ( ( im @ X3 )
= zero_zero_real )
=> ( ( ord_less_eq_real @ ( re @ X3 ) @ zero_zero_real )
=> ( ( csqrt @ X3 )
= ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X3 ) ) ) ) ) ) ) ) ).
% csqrt_of_real_nonpos
thf(fact_8849_complex__Im__of__int,axiom,
! [Z2: int] :
( ( im @ ( ring_17405671764205052669omplex @ Z2 ) )
= zero_zero_real ) ).
% complex_Im_of_int
thf(fact_8850_complex__Im__fact,axiom,
! [N: nat] :
( ( im @ ( semiri5044797733671781792omplex @ N ) )
= zero_zero_real ) ).
% complex_Im_fact
thf(fact_8851_complex__Im__of__nat,axiom,
! [N: nat] :
( ( im @ ( semiri8010041392384452111omplex @ N ) )
= zero_zero_real ) ).
% complex_Im_of_nat
thf(fact_8852_Im__complex__of__real,axiom,
! [Z2: real] :
( ( im @ ( real_V4546457046886955230omplex @ Z2 ) )
= zero_zero_real ) ).
% Im_complex_of_real
thf(fact_8853_Im__power__real,axiom,
! [X3: complex,N: nat] :
( ( ( im @ X3 )
= zero_zero_real )
=> ( ( im @ ( power_power_complex @ X3 @ N ) )
= zero_zero_real ) ) ).
% Im_power_real
thf(fact_8854_complex__Im__numeral,axiom,
! [V: num] :
( ( im @ ( numera6690914467698888265omplex @ V ) )
= zero_zero_real ) ).
% complex_Im_numeral
thf(fact_8855_Re__power__real,axiom,
! [X3: complex,N: nat] :
( ( ( im @ X3 )
= zero_zero_real )
=> ( ( re @ ( power_power_complex @ X3 @ N ) )
= ( power_power_real @ ( re @ X3 ) @ N ) ) ) ).
% Re_power_real
thf(fact_8856_csqrt__of__real__nonneg,axiom,
! [X3: complex] :
( ( ( im @ X3 )
= zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X3 ) )
=> ( ( csqrt @ X3 )
= ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X3 ) ) ) ) ) ) ).
% csqrt_of_real_nonneg
thf(fact_8857_csqrt__minus,axiom,
! [X3: complex] :
( ( ( ord_less_real @ ( im @ X3 ) @ zero_zero_real )
| ( ( ( im @ X3 )
= zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ ( re @ X3 ) ) ) )
=> ( ( csqrt @ ( uminus1482373934393186551omplex @ X3 ) )
= ( times_times_complex @ imaginary_unit @ ( csqrt @ X3 ) ) ) ) ).
% csqrt_minus
thf(fact_8858_uminus__integer__code_I1_J,axiom,
( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% uminus_integer_code(1)
thf(fact_8859_zero__integer__def,axiom,
( zero_z3403309356797280102nteger
= ( code_integer_of_int @ zero_zero_int ) ) ).
% zero_integer_def
thf(fact_8860_less__integer__code_I1_J,axiom,
~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).
% less_integer_code(1)
thf(fact_8861_abs__integer__code,axiom,
( abs_abs_Code_integer
= ( ^ [K3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K3 ) @ K3 ) ) ) ).
% abs_integer_code
thf(fact_8862_zero__complex_Osimps_I2_J,axiom,
( ( im @ zero_zero_complex )
= zero_zero_real ) ).
% zero_complex.simps(2)
thf(fact_8863_one__complex_Osimps_I2_J,axiom,
( ( im @ one_one_complex )
= zero_zero_real ) ).
% one_complex.simps(2)
thf(fact_8864_plus__integer_Oabs__eq,axiom,
! [Xa2: int,X3: int] :
( ( plus_p5714425477246183910nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X3 ) )
= ( code_integer_of_int @ ( plus_plus_int @ Xa2 @ X3 ) ) ) ).
% plus_integer.abs_eq
thf(fact_8865_less__eq__integer_Oabs__eq,axiom,
! [Xa2: int,X3: int] :
( ( ord_le3102999989581377725nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X3 ) )
= ( ord_less_eq_int @ Xa2 @ X3 ) ) ).
% less_eq_integer.abs_eq
thf(fact_8866_plus__complex_Osimps_I2_J,axiom,
! [X3: complex,Y: complex] :
( ( im @ ( plus_plus_complex @ X3 @ Y ) )
= ( plus_plus_real @ ( im @ X3 ) @ ( im @ Y ) ) ) ).
% plus_complex.simps(2)
thf(fact_8867_abs__Im__le__cmod,axiom,
! [X3: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X3 ) ) @ ( real_V1022390504157884413omplex @ X3 ) ) ).
% abs_Im_le_cmod
thf(fact_8868_times__complex_Osimps_I2_J,axiom,
! [X3: complex,Y: complex] :
( ( im @ ( times_times_complex @ X3 @ Y ) )
= ( plus_plus_real @ ( times_times_real @ ( re @ X3 ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X3 ) @ ( re @ Y ) ) ) ) ).
% times_complex.simps(2)
thf(fact_8869_cmod__eq__Re,axiom,
! [Z2: complex] :
( ( ( im @ Z2 )
= zero_zero_real )
=> ( ( real_V1022390504157884413omplex @ Z2 )
= ( abs_abs_real @ ( re @ Z2 ) ) ) ) ).
% cmod_eq_Re
thf(fact_8870_cmod__eq__Im,axiom,
! [Z2: complex] :
( ( ( re @ Z2 )
= zero_zero_real )
=> ( ( real_V1022390504157884413omplex @ Z2 )
= ( abs_abs_real @ ( im @ Z2 ) ) ) ) ).
% cmod_eq_Im
thf(fact_8871_Im__eq__0,axiom,
! [Z2: complex] :
( ( ( abs_abs_real @ ( re @ Z2 ) )
= ( real_V1022390504157884413omplex @ Z2 ) )
=> ( ( im @ Z2 )
= zero_zero_real ) ) ).
% Im_eq_0
thf(fact_8872_cmod__Im__le__iff,axiom,
! [X3: complex,Y: complex] :
( ( ( re @ X3 )
= ( re @ Y ) )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y ) )
= ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X3 ) ) @ ( abs_abs_real @ ( im @ Y ) ) ) ) ) ).
% cmod_Im_le_iff
thf(fact_8873_cmod__Re__le__iff,axiom,
! [X3: complex,Y: complex] :
( ( ( im @ X3 )
= ( im @ Y ) )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y ) )
= ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X3 ) ) @ ( abs_abs_real @ ( re @ Y ) ) ) ) ) ).
% cmod_Re_le_iff
thf(fact_8874_plus__complex_Ocode,axiom,
( plus_plus_complex
= ( ^ [X5: complex,Y5: complex] : ( complex2 @ ( plus_plus_real @ ( re @ X5 ) @ ( re @ Y5 ) ) @ ( plus_plus_real @ ( im @ X5 ) @ ( im @ Y5 ) ) ) ) ) ).
% plus_complex.code
thf(fact_8875_csqrt__principal,axiom,
! [Z2: complex] :
( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z2 ) ) )
| ( ( ( re @ ( csqrt @ Z2 ) )
= zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z2 ) ) ) ) ) ).
% csqrt_principal
thf(fact_8876_cmod__le,axiom,
! [Z2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z2 ) ) @ ( abs_abs_real @ ( im @ Z2 ) ) ) ) ).
% cmod_le
thf(fact_8877_complex__eq,axiom,
! [A: complex] :
( A
= ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( re @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( im @ A ) ) ) ) ) ).
% complex_eq
thf(fact_8878_times__complex_Ocode,axiom,
( times_times_complex
= ( ^ [X5: complex,Y5: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X5 ) @ ( re @ Y5 ) ) @ ( times_times_real @ ( im @ X5 ) @ ( im @ Y5 ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X5 ) @ ( im @ Y5 ) ) @ ( times_times_real @ ( im @ X5 ) @ ( re @ Y5 ) ) ) ) ) ) ).
% times_complex.code
thf(fact_8879_cmod__power2,axiom,
! [Z2: complex] :
( ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% cmod_power2
thf(fact_8880_complex__eq__0,axiom,
! [Z2: complex] :
( ( Z2 = zero_zero_complex )
= ( ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_real ) ) ).
% complex_eq_0
thf(fact_8881_norm__complex__def,axiom,
( real_V1022390504157884413omplex
= ( ^ [Z6: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% norm_complex_def
thf(fact_8882_inverse__complex_Osimps_I1_J,axiom,
! [X3: complex] :
( ( re @ ( invers8013647133539491842omplex @ X3 ) )
= ( divide_divide_real @ ( re @ X3 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% inverse_complex.simps(1)
thf(fact_8883_complex__neq__0,axiom,
! [Z2: complex] :
( ( Z2 != zero_zero_complex )
= ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% complex_neq_0
thf(fact_8884_Re__divide,axiom,
! [X3: complex,Y: complex] :
( ( re @ ( divide1717551699836669952omplex @ X3 @ Y ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X3 ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X3 ) @ ( 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 ) ) ) ) ) ) ).
% Re_divide
thf(fact_8885_csqrt__square,axiom,
! [B: complex] :
( ( ( ord_less_real @ zero_zero_real @ ( re @ B ) )
| ( ( ( re @ B )
= zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ ( im @ B ) ) ) )
=> ( ( csqrt @ ( power_power_complex @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= B ) ) ).
% csqrt_square
thf(fact_8886_csqrt__unique,axiom,
! [W2: complex,Z2: complex] :
( ( ( power_power_complex @ W2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= Z2 )
=> ( ( ( ord_less_real @ zero_zero_real @ ( re @ W2 ) )
| ( ( ( re @ W2 )
= zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ ( im @ W2 ) ) ) )
=> ( ( csqrt @ Z2 )
= W2 ) ) ) ).
% csqrt_unique
thf(fact_8887_inverse__complex_Osimps_I2_J,axiom,
! [X3: complex] :
( ( im @ ( invers8013647133539491842omplex @ X3 ) )
= ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X3 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% inverse_complex.simps(2)
thf(fact_8888_Im__divide,axiom,
! [X3: complex,Y: complex] :
( ( im @ ( divide1717551699836669952omplex @ X3 @ Y ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X3 ) @ ( re @ Y ) ) @ ( times_times_real @ ( re @ X3 ) @ ( 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 ) ) ) ) ) ) ).
% Im_divide
thf(fact_8889_complex__abs__le__norm,axiom,
! [Z2: complex] : ( ord_less_eq_real @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z2 ) ) @ ( abs_abs_real @ ( im @ Z2 ) ) ) @ ( times_times_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( real_V1022390504157884413omplex @ Z2 ) ) ) ).
% complex_abs_le_norm
thf(fact_8890_complex__unit__circle,axiom,
! [Z2: complex] :
( ( Z2 != zero_zero_complex )
=> ( ( plus_plus_real @ ( power_power_real @ ( divide_divide_real @ ( re @ Z2 ) @ ( real_V1022390504157884413omplex @ Z2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( divide_divide_real @ ( im @ Z2 ) @ ( real_V1022390504157884413omplex @ Z2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real ) ) ).
% complex_unit_circle
thf(fact_8891_inverse__complex_Ocode,axiom,
( invers8013647133539491842omplex
= ( ^ [X5: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X5 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X5 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% inverse_complex.code
thf(fact_8892_Complex__divide,axiom,
( divide1717551699836669952omplex
= ( ^ [X5: complex,Y5: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X5 ) @ ( re @ Y5 ) ) @ ( times_times_real @ ( im @ X5 ) @ ( im @ Y5 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X5 ) @ ( re @ Y5 ) ) @ ( times_times_real @ ( re @ X5 ) @ ( im @ Y5 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% Complex_divide
thf(fact_8893_cos__Arg__i__mult__zero,axiom,
! [Y: complex] :
( ( Y != zero_zero_complex )
=> ( ( ( re @ Y )
= zero_zero_real )
=> ( ( cos_real @ ( arg @ Y ) )
= zero_zero_real ) ) ) ).
% cos_Arg_i_mult_zero
thf(fact_8894_complex__mult__cnj,axiom,
! [Z2: complex] :
( ( times_times_complex @ Z2 @ ( cnj @ Z2 ) )
= ( real_V4546457046886955230omplex @ ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% complex_mult_cnj
thf(fact_8895_complex__cnj__zero,axiom,
( ( cnj @ zero_zero_complex )
= zero_zero_complex ) ).
% complex_cnj_zero
thf(fact_8896_complex__cnj__zero__iff,axiom,
! [Z2: complex] :
( ( ( cnj @ Z2 )
= zero_zero_complex )
= ( Z2 = zero_zero_complex ) ) ).
% complex_cnj_zero_iff
thf(fact_8897_complex__cnj__add,axiom,
! [X3: complex,Y: complex] :
( ( cnj @ ( plus_plus_complex @ X3 @ Y ) )
= ( plus_plus_complex @ ( cnj @ X3 ) @ ( cnj @ Y ) ) ) ).
% complex_cnj_add
thf(fact_8898_complex__In__mult__cnj__zero,axiom,
! [Z2: complex] :
( ( im @ ( times_times_complex @ Z2 @ ( cnj @ Z2 ) ) )
= zero_zero_real ) ).
% complex_In_mult_cnj_zero
thf(fact_8899_Arg__zero,axiom,
( ( arg @ zero_zero_complex )
= zero_zero_real ) ).
% Arg_zero
thf(fact_8900_Re__complex__div__eq__0,axiom,
! [A: complex,B: complex] :
( ( ( re @ ( divide1717551699836669952omplex @ A @ B ) )
= zero_zero_real )
= ( ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) )
= zero_zero_real ) ) ).
% Re_complex_div_eq_0
thf(fact_8901_Im__complex__div__eq__0,axiom,
! [A: complex,B: complex] :
( ( ( im @ ( divide1717551699836669952omplex @ A @ B ) )
= zero_zero_real )
= ( ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) )
= zero_zero_real ) ) ).
% Im_complex_div_eq_0
thf(fact_8902_Re__complex__div__lt__0,axiom,
! [A: complex,B: complex] :
( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
= ( ord_less_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% Re_complex_div_lt_0
thf(fact_8903_Re__complex__div__gt__0,axiom,
! [A: complex,B: complex] :
( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
= ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% Re_complex_div_gt_0
thf(fact_8904_Re__complex__div__ge__0,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
= ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% Re_complex_div_ge_0
thf(fact_8905_Re__complex__div__le__0,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
= ( ord_less_eq_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% Re_complex_div_le_0
thf(fact_8906_Im__complex__div__gt__0,axiom,
! [A: complex,B: complex] :
( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
= ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% Im_complex_div_gt_0
thf(fact_8907_Im__complex__div__lt__0,axiom,
! [A: complex,B: complex] :
( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
= ( ord_less_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% Im_complex_div_lt_0
thf(fact_8908_Im__complex__div__le__0,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
= ( ord_less_eq_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% Im_complex_div_le_0
thf(fact_8909_Im__complex__div__ge__0,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
= ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% Im_complex_div_ge_0
thf(fact_8910_complex__div__gt__0,axiom,
! [A: complex,B: complex] :
( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
= ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) )
& ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
= ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ) ).
% complex_div_gt_0
thf(fact_8911_Arg__bounded,axiom,
! [Z2: complex] :
( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z2 ) )
& ( ord_less_eq_real @ ( arg @ Z2 ) @ pi ) ) ).
% Arg_bounded
thf(fact_8912_complex__add__cnj,axiom,
! [Z2: complex] :
( ( plus_plus_complex @ Z2 @ ( cnj @ Z2 ) )
= ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z2 ) ) ) ) ).
% complex_add_cnj
thf(fact_8913_cnj__add__mult__eq__Re,axiom,
! [Z2: complex,W2: complex] :
( ( plus_plus_complex @ ( times_times_complex @ Z2 @ ( cnj @ W2 ) ) @ ( times_times_complex @ ( cnj @ Z2 ) @ W2 ) )
= ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z2 @ ( cnj @ W2 ) ) ) ) ) ) ).
% cnj_add_mult_eq_Re
thf(fact_8914_integer__of__num_I3_J,axiom,
! [N: num] :
( ( code_integer_of_num @ ( bit1 @ N ) )
= ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ one_one_Code_integer ) ) ).
% integer_of_num(3)
thf(fact_8915_int__of__integer__code,axiom,
( code_int_of_integer
= ( ^ [K3: code_integer] :
( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
@ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
@ ( produc1553301316500091796er_int
@ ^ [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 ) )
@ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% int_of_integer_code
thf(fact_8916_bit__cut__integer__def,axiom,
( code_bit_cut_integer
= ( ^ [K3: code_integer] :
( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
@ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K3 ) ) ) ) ).
% bit_cut_integer_def
thf(fact_8917_num__of__integer__code,axiom,
( code_num_of_integer
= ( ^ [K3: code_integer] :
( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
@ ( produc7336495610019696514er_num
@ ^ [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 ) )
@ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% num_of_integer_code
thf(fact_8918_zero__integer_Orep__eq,axiom,
( ( code_int_of_integer @ zero_z3403309356797280102nteger )
= zero_zero_int ) ).
% zero_integer.rep_eq
thf(fact_8919_plus__integer_Orep__eq,axiom,
! [X3: code_integer,Xa2: code_integer] :
( ( code_int_of_integer @ ( plus_p5714425477246183910nteger @ X3 @ Xa2 ) )
= ( plus_plus_int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% plus_integer.rep_eq
thf(fact_8920_less__eq__integer_Orep__eq,axiom,
( ord_le3102999989581377725nteger
= ( ^ [X5: code_integer,Xa4: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ X5 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% less_eq_integer.rep_eq
thf(fact_8921_integer__less__eq__iff,axiom,
( ord_le3102999989581377725nteger
= ( ^ [K3: code_integer,L2: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% integer_less_eq_iff
thf(fact_8922_integer__of__num_I2_J,axiom,
! [N: num] :
( ( code_integer_of_num @ ( bit0 @ N ) )
= ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% integer_of_num(2)
thf(fact_8923_divmod__integer__def,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K3 @ L2 ) @ ( modulo364778990260209775nteger @ K3 @ L2 ) ) ) ) ).
% divmod_integer_def
thf(fact_8924_bit__cut__integer__code,axiom,
( code_bit_cut_integer
= ( ^ [K3: code_integer] :
( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
@ ( produc9125791028180074456eger_o
@ ^ [R5: code_integer,S7: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S7 ) ) @ ( S7 = one_one_Code_integer ) )
@ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_cut_integer_code
thf(fact_8925_nat__of__integer__code,axiom,
( code_nat_of_integer
= ( ^ [K3: code_integer] :
( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
@ ( produc1555791787009142072er_nat
@ ^ [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 ) )
@ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% nat_of_integer_code
thf(fact_8926_card__lessThan,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
= U ) ).
% card_lessThan
thf(fact_8927_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_8928_card__atMost,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
= ( suc @ U ) ) ).
% card_atMost
thf(fact_8929_card__atLeastLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( minus_minus_nat @ U @ L ) ) ).
% card_atLeastLessThan
thf(fact_8930_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I3: nat] : ( ord_less_eq_nat @ I3 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_8931_of__nat__of__integer,axiom,
! [K2: code_integer] :
( ( semiri4939895301339042750nteger @ ( code_nat_of_integer @ K2 ) )
= ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ K2 ) ) ).
% of_nat_of_integer
thf(fact_8932_card__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).
% card_atLeastAtMost
thf(fact_8933_card__atLeastLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card_int @ ( set_or4662586982721622107an_int @ L @ U ) )
= ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).
% card_atLeastLessThan_int
thf(fact_8934_nat__of__integer__non__positive,axiom,
! [K2: code_integer] :
( ( ord_le3102999989581377725nteger @ K2 @ zero_z3403309356797280102nteger )
=> ( ( code_nat_of_integer @ K2 )
= zero_zero_nat ) ) ).
% nat_of_integer_non_positive
thf(fact_8935_card__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card_int @ ( set_or1266510415728281911st_int @ L @ U ) )
= ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L ) @ one_one_int ) ) ) ).
% card_atLeastAtMost_int
thf(fact_8936_card__less,axiom,
! [M6: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M6 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M6 )
& ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_8937_card__less__Suc,axiom,
! [M6: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M6 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ ( suc @ K3 ) @ M6 )
& ( ord_less_nat @ K3 @ I ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M6 )
& ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_8938_card__less__Suc2,axiom,
! [M6: set_nat,I: nat] :
( ~ ( member_nat @ zero_zero_nat @ M6 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ ( suc @ K3 ) @ M6 )
& ( ord_less_nat @ K3 @ I ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M6 )
& ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_8939_card__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
= ( nat2 @ U ) ) ).
% card_atLeastZeroLessThan_int
thf(fact_8940_nat__of__integer__code__post_I1_J,axiom,
( ( code_nat_of_integer @ zero_z3403309356797280102nteger )
= zero_zero_nat ) ).
% nat_of_integer_code_post(1)
thf(fact_8941_subset__card__intvl__is__intvl,axiom,
! [A4: set_nat,K2: nat] :
( ( ord_less_eq_set_nat @ A4 @ ( set_or4665077453230672383an_nat @ K2 @ ( plus_plus_nat @ K2 @ ( finite_card_nat @ A4 ) ) ) )
=> ( A4
= ( set_or4665077453230672383an_nat @ K2 @ ( plus_plus_nat @ K2 @ ( finite_card_nat @ A4 ) ) ) ) ) ).
% subset_card_intvl_is_intvl
thf(fact_8942_divmod__abs__code_I6_J,axiom,
! [J: code_integer] :
( ( code_divmod_abs @ zero_z3403309356797280102nteger @ J )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ) ).
% divmod_abs_code(6)
thf(fact_8943_subset__eq__atLeast0__lessThan__card,axiom,
! [N7: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ N7 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_8944_card__sum__le__nat__sum,axiom,
! [S2: set_nat] :
( ord_less_eq_nat
@ ( groups3542108847815614940at_nat
@ ^ [X5: nat] : X5
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
@ ( groups3542108847815614940at_nat
@ ^ [X5: nat] : X5
@ S2 ) ) ).
% card_sum_le_nat_sum
thf(fact_8945_card__nth__roots,axiom,
! [C2: complex,N: nat] :
( ( C2 != zero_zero_complex )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z6: complex] :
( ( power_power_complex @ Z6 @ N )
= C2 ) ) )
= N ) ) ) ).
% card_nth_roots
thf(fact_8946_card__roots__unity__eq,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z6: complex] :
( ( power_power_complex @ Z6 @ N )
= one_one_complex ) ) )
= N ) ) ).
% card_roots_unity_eq
thf(fact_8947_divmod__abs__code_I5_J,axiom,
! [J: code_integer] :
( ( code_divmod_abs @ J @ zero_z3403309356797280102nteger )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ J ) ) ) ).
% divmod_abs_code(5)
thf(fact_8948_divmod__abs__def,axiom,
( code_divmod_abs
= ( ^ [K3: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L2 ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L2 ) ) ) ) ) ).
% divmod_abs_def
thf(fact_8949_divmod__integer__code,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L2: code_integer] :
( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
@ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L2 )
@ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L2 )
@ ( produc6916734918728496179nteger
@ ^ [R5: code_integer,S7: code_integer] : ( if_Pro6119634080678213985nteger @ ( S7 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L2 @ S7 ) ) )
@ ( code_divmod_abs @ K3 @ L2 ) ) )
@ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
@ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
@ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L2 )
@ ( produc6916734918728496179nteger
@ ^ [R5: code_integer,S7: code_integer] : ( if_Pro6119634080678213985nteger @ ( S7 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L2 ) @ S7 ) ) )
@ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ) ).
% divmod_integer_code
thf(fact_8950_pair__leqI2,axiom,
! [A: nat,B: nat,S3: nat,T: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ S3 @ T )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ) ).
% pair_leqI2
thf(fact_8951_pair__leqI1,axiom,
! [A: nat,B: nat,S3: nat,T: nat] :
( ( ord_less_nat @ A @ B )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ).
% pair_leqI1
thf(fact_8952_nat_Odisc__eq__case_I2_J,axiom,
! [Nat: nat] :
( ( Nat != zero_zero_nat )
= ( case_nat_o @ $false
@ ^ [Uu3: nat] : $true
@ Nat ) ) ).
% nat.disc_eq_case(2)
thf(fact_8953_nat_Odisc__eq__case_I1_J,axiom,
! [Nat: nat] :
( ( Nat = zero_zero_nat )
= ( case_nat_o @ $true
@ ^ [Uu3: nat] : $false
@ Nat ) ) ).
% nat.disc_eq_case(1)
thf(fact_8954_less__eq__nat_Osimps_I2_J,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( case_nat_o @ $false @ ( ord_less_eq_nat @ M2 ) @ N ) ) ).
% less_eq_nat.simps(2)
thf(fact_8955_max__Suc1,axiom,
! [N: nat,M2: nat] :
( ( ord_max_nat @ ( suc @ N ) @ M2 )
= ( case_nat_nat @ ( suc @ N )
@ ^ [M7: nat] : ( suc @ ( ord_max_nat @ N @ M7 ) )
@ M2 ) ) ).
% max_Suc1
thf(fact_8956_max__Suc2,axiom,
! [M2: nat,N: nat] :
( ( ord_max_nat @ M2 @ ( suc @ N ) )
= ( case_nat_nat @ ( suc @ N )
@ ^ [M7: nat] : ( suc @ ( ord_max_nat @ M7 @ N ) )
@ M2 ) ) ).
% max_Suc2
thf(fact_8957_diff__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N ) )
= ( case_nat_nat @ zero_zero_nat
@ ^ [K3: nat] : K3
@ ( minus_minus_nat @ M2 @ N ) ) ) ).
% diff_Suc
thf(fact_8958_binomial__def,axiom,
( binomial
= ( ^ [N4: nat,K3: nat] :
( finite_card_set_nat
@ ( collect_set_nat
@ ^ [K7: set_nat] :
( ( member_set_nat @ K7 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N4 ) ) )
& ( ( finite_card_nat @ K7 )
= K3 ) ) ) ) ) ) ).
% binomial_def
thf(fact_8959_push__bit__nonnegative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K2 ) )
= ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).
% push_bit_nonnegative_int_iff
thf(fact_8960_push__bit__negative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K2 ) @ zero_zero_int )
= ( ord_less_int @ K2 @ zero_zero_int ) ) ).
% push_bit_negative_int_iff
thf(fact_8961_concat__bit__of__zero__1,axiom,
! [N: nat,L: int] :
( ( bit_concat_bit @ N @ zero_zero_int @ L )
= ( bit_se545348938243370406it_int @ N @ L ) ) ).
% concat_bit_of_zero_1
thf(fact_8962_push__bit__of__Suc__0,axiom,
! [N: nat] :
( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% push_bit_of_Suc_0
thf(fact_8963_bit__push__bit__iff__int,axiom,
! [M2: nat,K2: int,N: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M2 @ K2 ) @ N )
= ( ( ord_less_eq_nat @ M2 @ N )
& ( bit_se1146084159140164899it_int @ K2 @ ( minus_minus_nat @ N @ M2 ) ) ) ) ).
% bit_push_bit_iff_int
thf(fact_8964_bit__push__bit__iff__nat,axiom,
! [M2: nat,Q3: nat,N: nat] :
( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M2 @ Q3 ) @ N )
= ( ( ord_less_eq_nat @ M2 @ N )
& ( bit_se1148574629649215175it_nat @ Q3 @ ( minus_minus_nat @ N @ M2 ) ) ) ) ).
% bit_push_bit_iff_nat
thf(fact_8965_concat__bit__eq,axiom,
( bit_concat_bit
= ( ^ [N4: nat,K3: int,L2: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N4 @ K3 ) @ ( bit_se545348938243370406it_int @ N4 @ L2 ) ) ) ) ).
% concat_bit_eq
thf(fact_8966_pred__def,axiom,
( pred
= ( case_nat_nat @ zero_zero_nat
@ ^ [X23: nat] : X23 ) ) ).
% pred_def
thf(fact_8967_wmin__insertI,axiom,
! [X3: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ X3 @ XS )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ fun_pair_leq )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_weak )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_weak ) ) ) ) ).
% wmin_insertI
thf(fact_8968_wmax__insertI,axiom,
! [Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ Y @ YS )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ fun_pair_leq )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_max_weak )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X3 @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).
% wmax_insertI
thf(fact_8969_bezw__0,axiom,
! [X3: nat] :
( ( bezw @ X3 @ zero_zero_nat )
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% bezw_0
thf(fact_8970_prod__decode__aux_Oelims,axiom,
! [X3: nat,Xa2: nat,Y: product_prod_nat_nat] :
( ( ( nat_prod_decode_aux @ X3 @ Xa2 )
= Y )
=> ( ( ( ord_less_eq_nat @ Xa2 @ X3 )
=> ( Y
= ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X3 @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ Xa2 @ X3 )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X3 ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_8971_wmax__emptyI,axiom,
! [X8: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ X8 )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ X8 ) @ fun_max_weak ) ) ).
% wmax_emptyI
thf(fact_8972_wmin__emptyI,axiom,
! [X8: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X8 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).
% wmin_emptyI
thf(fact_8973_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K3: nat,M5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M5 @ K3 ) @ ( product_Pair_nat_nat @ M5 @ ( minus_minus_nat @ K3 @ M5 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M5 @ ( suc @ K3 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_8974_prod__decode__aux_Opelims,axiom,
! [X3: nat,Xa2: nat,Y: product_prod_nat_nat] :
( ( ( nat_prod_decode_aux @ X3 @ Xa2 )
= Y )
=> ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) )
=> ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X3 )
=> ( Y
= ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X3 @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ Xa2 @ X3 )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X3 ) ) ) ) ) )
=> ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) ) ) ) ) ).
% prod_decode_aux.pelims
thf(fact_8975_smax__insertI,axiom,
! [Y: product_prod_nat_nat,Y7: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,X8: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ Y @ Y7 )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ fun_pair_less )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X8 @ Y7 ) @ fun_max_strict )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X3 @ X8 ) @ Y7 ) @ fun_max_strict ) ) ) ) ).
% smax_insertI
thf(fact_8976_smin__insertI,axiom,
! [X3: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ X3 @ XS )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ fun_pair_less )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_strict )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_strict ) ) ) ) ).
% smin_insertI
thf(fact_8977_Suc__0__div__numeral,axiom,
! [K2: num] :
( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K2 ) )
= ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K2 ) ) ) ).
% Suc_0_div_numeral
thf(fact_8978_smin__emptyI,axiom,
! [X8: set_Pr1261947904930325089at_nat] :
( ( X8 != bot_bo2099793752762293965at_nat )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X8 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).
% smin_emptyI
thf(fact_8979_smax__emptyI,axiom,
! [Y7: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ Y7 )
=> ( ( Y7 != bot_bo2099793752762293965at_nat )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ Y7 ) @ fun_max_strict ) ) ) ).
% smax_emptyI
thf(fact_8980_Suc__0__mod__numeral,axiom,
! [K2: num] :
( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K2 ) )
= ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K2 ) ) ) ).
% Suc_0_mod_numeral
thf(fact_8981_finite__enumerate,axiom,
! [S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ? [R3: nat > nat] :
( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S2 ) ) )
& ! [N6: nat] :
( ( ord_less_nat @ N6 @ ( finite_card_nat @ S2 ) )
=> ( member_nat @ ( R3 @ N6 ) @ S2 ) ) ) ) ).
% finite_enumerate
thf(fact_8982_divmod__integer__eq__cases,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L2: code_integer] :
( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
@ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
@ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
@ ( if_Pro6119634080678213985nteger
@ ( ( sgn_sgn_Code_integer @ K3 )
= ( sgn_sgn_Code_integer @ L2 ) )
@ ( code_divmod_abs @ K3 @ L2 )
@ ( produc6916734918728496179nteger
@ ^ [R5: code_integer,S7: code_integer] : ( if_Pro6119634080678213985nteger @ ( S7 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L2 ) @ S7 ) ) )
@ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_8983_rat__sgn__code,axiom,
! [P: rat] :
( ( quotient_of @ ( sgn_sgn_rat @ P ) )
= ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P ) ) ) @ one_one_int ) ) ).
% rat_sgn_code
thf(fact_8984_bezw_Oelims,axiom,
! [X3: nat,Xa2: nat,Y: product_prod_int_int] :
( ( ( bezw @ X3 @ Xa2 )
= Y )
=> ( ( ( Xa2 = zero_zero_nat )
=> ( Y
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
& ( ( Xa2 != zero_zero_nat )
=> ( Y
= ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Xa2 ) ) ) ) ) ) ) ) ) ).
% bezw.elims
thf(fact_8985_card_Ocomp__fun__commute__on,axiom,
( ( comp_nat_nat_nat @ suc @ suc )
= ( comp_nat_nat_nat @ suc @ suc ) ) ).
% card.comp_fun_commute_on
thf(fact_8986_quotient__of__denom__pos_H,axiom,
! [R2: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R2 ) ) ) ).
% quotient_of_denom_pos'
thf(fact_8987_bezw__non__0,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_nat @ zero_zero_nat @ Y )
=> ( ( bezw @ X3 @ Y )
= ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X3 @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X3 @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X3 @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Y ) ) ) ) ) ) ) ).
% bezw_non_0
thf(fact_8988_bezw_Osimps,axiom,
( bezw
= ( ^ [X5: nat,Y5: nat] : ( if_Pro3027730157355071871nt_int @ ( Y5 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y5 @ ( modulo_modulo_nat @ X5 @ Y5 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y5 @ ( modulo_modulo_nat @ X5 @ Y5 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y5 @ ( modulo_modulo_nat @ X5 @ Y5 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X5 @ Y5 ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_8989_bezw_Opelims,axiom,
! [X3: nat,Xa2: nat,Y: product_prod_int_int] :
( ( ( bezw @ X3 @ Xa2 )
= Y )
=> ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) )
=> ~ ( ( ( ( Xa2 = zero_zero_nat )
=> ( Y
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
& ( ( Xa2 != zero_zero_nat )
=> ( Y
= ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Xa2 ) ) ) ) ) ) ) )
=> ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) ) ) ) ) ).
% bezw.pelims
thf(fact_8990_normalize__def,axiom,
( normalize
= ( ^ [P6: product_prod_int_int] :
( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P6 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P6 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P6 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) )
@ ( if_Pro3027730157355071871nt_int
@ ( ( product_snd_int_int @ P6 )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P6 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P6 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) ) ) ) ) ) ) ).
% normalize_def
thf(fact_8991_gcd__pos__int,axiom,
! [M2: int,N: int] :
( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M2 @ N ) )
= ( ( M2 != zero_zero_int )
| ( N != zero_zero_int ) ) ) ).
% gcd_pos_int
thf(fact_8992_gcd__0__left__int,axiom,
! [X3: int] :
( ( gcd_gcd_int @ zero_zero_int @ X3 )
= ( abs_abs_int @ X3 ) ) ).
% gcd_0_left_int
thf(fact_8993_gcd__0__int,axiom,
! [X3: int] :
( ( gcd_gcd_int @ X3 @ zero_zero_int )
= ( abs_abs_int @ X3 ) ) ).
% gcd_0_int
thf(fact_8994_gcd__ge__0__int,axiom,
! [X3: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X3 @ Y ) ) ).
% gcd_ge_0_int
thf(fact_8995_bezout__int,axiom,
! [X3: int,Y: int] :
? [U3: int,V2: int] :
( ( plus_plus_int @ ( times_times_int @ U3 @ X3 ) @ ( times_times_int @ V2 @ Y ) )
= ( gcd_gcd_int @ X3 @ Y ) ) ).
% bezout_int
thf(fact_8996_gcd__le1__int,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).
% gcd_le1_int
thf(fact_8997_gcd__le2__int,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).
% gcd_le2_int
thf(fact_8998_gcd__cases__int,axiom,
! [X3: int,Y: int,P2: int > $o] :
( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( P2 @ ( gcd_gcd_int @ X3 @ Y ) ) ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( P2 @ ( gcd_gcd_int @ X3 @ ( uminus_uminus_int @ Y ) ) ) ) )
=> ( ( ( ord_less_eq_int @ X3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( P2 @ ( gcd_gcd_int @ ( uminus_uminus_int @ X3 ) @ Y ) ) ) )
=> ( ( ( ord_less_eq_int @ X3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( P2 @ ( gcd_gcd_int @ ( uminus_uminus_int @ X3 ) @ ( uminus_uminus_int @ Y ) ) ) ) )
=> ( P2 @ ( gcd_gcd_int @ X3 @ Y ) ) ) ) ) ) ).
% gcd_cases_int
thf(fact_8999_gcd__unique__int,axiom,
! [D: int,A: int,B: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ D )
& ( dvd_dvd_int @ D @ A )
& ( dvd_dvd_int @ D @ B )
& ! [E3: int] :
( ( ( dvd_dvd_int @ E3 @ A )
& ( dvd_dvd_int @ E3 @ B ) )
=> ( dvd_dvd_int @ E3 @ D ) ) )
= ( D
= ( gcd_gcd_int @ A @ B ) ) ) ).
% gcd_unique_int
thf(fact_9000_gcd__non__0__int,axiom,
! [Y: int,X3: int] :
( ( ord_less_int @ zero_zero_int @ Y )
=> ( ( gcd_gcd_int @ X3 @ Y )
= ( gcd_gcd_int @ Y @ ( modulo_modulo_int @ X3 @ Y ) ) ) ) ).
% gcd_non_0_int
thf(fact_9001_gcd__code__int,axiom,
( gcd_gcd_int
= ( ^ [K3: int,L2: int] : ( abs_abs_int @ ( if_int @ ( L2 = zero_zero_int ) @ K3 @ ( gcd_gcd_int @ L2 @ ( modulo_modulo_int @ ( abs_abs_int @ K3 ) @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ).
% gcd_code_int
thf(fact_9002_Arg__correct,axiom,
! [Z2: complex] :
( ( Z2 != zero_zero_complex )
=> ( ( ( sgn_sgn_complex @ Z2 )
= ( cis @ ( arg @ Z2 ) ) )
& ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z2 ) )
& ( ord_less_eq_real @ ( arg @ Z2 ) @ pi ) ) ) ).
% Arg_correct
thf(fact_9003_gcd__nat_Oeq__neutr__iff,axiom,
! [A: nat,B: nat] :
( ( ( gcd_gcd_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% gcd_nat.eq_neutr_iff
thf(fact_9004_gcd__nat_Oleft__neutral,axiom,
! [A: nat] :
( ( gcd_gcd_nat @ zero_zero_nat @ A )
= A ) ).
% gcd_nat.left_neutral
thf(fact_9005_gcd__nat_Oneutr__eq__iff,axiom,
! [A: nat,B: nat] :
( ( zero_zero_nat
= ( gcd_gcd_nat @ A @ B ) )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% gcd_nat.neutr_eq_iff
thf(fact_9006_gcd__nat_Oright__neutral,axiom,
! [A: nat] :
( ( gcd_gcd_nat @ A @ zero_zero_nat )
= A ) ).
% gcd_nat.right_neutral
thf(fact_9007_gcd__0__nat,axiom,
! [X3: nat] :
( ( gcd_gcd_nat @ X3 @ zero_zero_nat )
= X3 ) ).
% gcd_0_nat
thf(fact_9008_gcd__0__left__nat,axiom,
! [X3: nat] :
( ( gcd_gcd_nat @ zero_zero_nat @ X3 )
= X3 ) ).
% gcd_0_left_nat
thf(fact_9009_gcd__Suc__0,axiom,
! [M2: nat] :
( ( gcd_gcd_nat @ M2 @ ( suc @ zero_zero_nat ) )
= ( suc @ zero_zero_nat ) ) ).
% gcd_Suc_0
thf(fact_9010_gcd__pos__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M2 @ N ) )
= ( ( M2 != zero_zero_nat )
| ( N != zero_zero_nat ) ) ) ).
% gcd_pos_nat
thf(fact_9011_cis__zero,axiom,
( ( cis @ zero_zero_real )
= one_one_complex ) ).
% cis_zero
thf(fact_9012_real__eq__imaginary__iff,axiom,
! [Y: complex,X3: complex] :
( ( member_complex @ Y @ real_V2521375963428798218omplex )
=> ( ( member_complex @ X3 @ real_V2521375963428798218omplex )
=> ( ( X3
= ( times_times_complex @ imaginary_unit @ Y ) )
= ( ( X3 = zero_zero_complex )
& ( Y = zero_zero_complex ) ) ) ) ) ).
% real_eq_imaginary_iff
thf(fact_9013_imaginary__eq__real__iff,axiom,
! [Y: complex,X3: complex] :
( ( member_complex @ Y @ real_V2521375963428798218omplex )
=> ( ( member_complex @ X3 @ real_V2521375963428798218omplex )
=> ( ( ( times_times_complex @ imaginary_unit @ Y )
= X3 )
= ( ( X3 = zero_zero_complex )
& ( Y = zero_zero_complex ) ) ) ) ) ).
% imaginary_eq_real_iff
thf(fact_9014_cis__neq__zero,axiom,
! [A: real] :
( ( cis @ A )
!= zero_zero_complex ) ).
% cis_neq_zero
thf(fact_9015_gcd__le2__nat,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).
% gcd_le2_nat
thf(fact_9016_gcd__le1__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).
% gcd_le1_nat
thf(fact_9017_gcd__diff1__nat,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( gcd_gcd_nat @ ( minus_minus_nat @ M2 @ N ) @ N )
= ( gcd_gcd_nat @ M2 @ N ) ) ) ).
% gcd_diff1_nat
thf(fact_9018_gcd__diff2__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M2 ) @ N )
= ( gcd_gcd_nat @ M2 @ N ) ) ) ).
% gcd_diff2_nat
thf(fact_9019_gcd__non__0__nat,axiom,
! [Y: nat,X3: nat] :
( ( Y != zero_zero_nat )
=> ( ( gcd_gcd_nat @ X3 @ Y )
= ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X3 @ Y ) ) ) ) ).
% gcd_non_0_nat
thf(fact_9020_gcd__nat_Osimps,axiom,
( gcd_gcd_nat
= ( ^ [X5: nat,Y5: nat] : ( if_nat @ ( Y5 = zero_zero_nat ) @ X5 @ ( gcd_gcd_nat @ Y5 @ ( modulo_modulo_nat @ X5 @ Y5 ) ) ) ) ) ).
% gcd_nat.simps
thf(fact_9021_gcd__nat_Oelims,axiom,
! [X3: nat,Xa2: nat,Y: nat] :
( ( ( gcd_gcd_nat @ X3 @ Xa2 )
= Y )
=> ( ( ( Xa2 = zero_zero_nat )
=> ( Y = X3 ) )
& ( ( Xa2 != zero_zero_nat )
=> ( Y
= ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) ) ) ) ).
% gcd_nat.elims
thf(fact_9022_complex__is__Real__iff,axiom,
! [Z2: complex] :
( ( member_complex @ Z2 @ real_V2521375963428798218omplex )
= ( ( im @ Z2 )
= zero_zero_real ) ) ).
% complex_is_Real_iff
thf(fact_9023_Complex__in__Reals,axiom,
! [X3: real] : ( member_complex @ ( complex2 @ X3 @ zero_zero_real ) @ real_V2521375963428798218omplex ) ).
% Complex_in_Reals
thf(fact_9024_cis__mult,axiom,
! [A: real,B: real] :
( ( times_times_complex @ ( cis @ A ) @ ( cis @ B ) )
= ( cis @ ( plus_plus_real @ A @ B ) ) ) ).
% cis_mult
thf(fact_9025_bezout__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [X4: nat,Y3: nat] :
( ( times_times_nat @ A @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% bezout_nat
thf(fact_9026_bezout__gcd__nat_H,axiom,
! [B: nat,A: nat] :
? [X4: nat,Y3: nat] :
( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y3 ) @ ( times_times_nat @ A @ X4 ) )
& ( ( minus_minus_nat @ ( times_times_nat @ A @ X4 ) @ ( times_times_nat @ B @ Y3 ) )
= ( gcd_gcd_nat @ A @ B ) ) )
| ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y3 ) @ ( times_times_nat @ B @ X4 ) )
& ( ( minus_minus_nat @ ( times_times_nat @ B @ X4 ) @ ( times_times_nat @ A @ Y3 ) )
= ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% bezout_gcd_nat'
thf(fact_9027_cis__Arg,axiom,
! [Z2: complex] :
( ( Z2 != zero_zero_complex )
=> ( ( cis @ ( arg @ Z2 ) )
= ( sgn_sgn_complex @ Z2 ) ) ) ).
% cis_Arg
thf(fact_9028_gcd__code__integer,axiom,
( gcd_gcd_Code_integer
= ( ^ [K3: code_integer,L2: code_integer] : ( abs_abs_Code_integer @ ( if_Code_integer @ ( L2 = zero_z3403309356797280102nteger ) @ K3 @ ( gcd_gcd_Code_integer @ L2 @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L2 ) ) ) ) ) ) ) ).
% gcd_code_integer
thf(fact_9029_cis__Arg__unique,axiom,
! [Z2: complex,X3: real] :
( ( ( sgn_sgn_complex @ Z2 )
= ( cis @ X3 ) )
=> ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X3 )
=> ( ( ord_less_eq_real @ X3 @ pi )
=> ( ( arg @ Z2 )
= X3 ) ) ) ) ).
% cis_Arg_unique
thf(fact_9030_bezw__aux,axiom,
! [X3: nat,Y: nat] :
( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X3 @ Y ) )
= ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X3 @ Y ) ) @ ( semiri1314217659103216013at_int @ X3 ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X3 @ Y ) ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ) ).
% bezw_aux
thf(fact_9031_gcd__nat_Opelims,axiom,
! [X3: nat,Xa2: nat,Y: nat] :
( ( ( gcd_gcd_nat @ X3 @ Xa2 )
= Y )
=> ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) )
=> ~ ( ( ( ( Xa2 = zero_zero_nat )
=> ( Y = X3 ) )
& ( ( Xa2 != zero_zero_nat )
=> ( Y
= ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) ) )
=> ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) ) ) ) ) ).
% gcd_nat.pelims
thf(fact_9032_bij__betw__roots__unity,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( bij_betw_nat_complex
@ ^ [K3: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K3 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
@ ( set_ord_lessThan_nat @ N )
@ ( collect_complex
@ ^ [Z6: complex] :
( ( power_power_complex @ Z6 @ N )
= one_one_complex ) ) ) ) ).
% bij_betw_roots_unity
thf(fact_9033_Arg__def,axiom,
( arg
= ( ^ [Z6: complex] :
( if_real @ ( Z6 = zero_zero_complex ) @ zero_zero_real
@ ( fChoice_real
@ ^ [A5: real] :
( ( ( sgn_sgn_complex @ Z6 )
= ( cis @ A5 ) )
& ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A5 )
& ( ord_less_eq_real @ A5 @ pi ) ) ) ) ) ) ).
% Arg_def
thf(fact_9034_drop__bit__nonnegative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K2 ) )
= ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).
% drop_bit_nonnegative_int_iff
thf(fact_9035_drop__bit__negative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K2 ) @ zero_zero_int )
= ( ord_less_int @ K2 @ zero_zero_int ) ) ).
% drop_bit_negative_int_iff
thf(fact_9036_drop__bit__of__Suc__0,axiom,
! [N: nat] :
( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
= ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% drop_bit_of_Suc_0
thf(fact_9037_drop__bit__Suc__minus__bit0,axiom,
! [N: nat,K2: num] :
( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K2 ) ) ) )
= ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) ) ) ).
% drop_bit_Suc_minus_bit0
thf(fact_9038_drop__bit__Suc__minus__bit1,axiom,
! [N: nat,K2: num] :
( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) ) )
= ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K2 ) ) ) ) ) ).
% drop_bit_Suc_minus_bit1
thf(fact_9039_root__powr__inverse,axiom,
! [N: nat,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( root @ N @ X3 )
= ( powr_real @ X3 @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% root_powr_inverse
thf(fact_9040_card__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card_int @ ( set_or5832277885323065728an_int @ L @ U ) )
= ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L @ one_one_int ) ) ) ) ).
% card_greaterThanLessThan_int
thf(fact_9041_real__root__zero,axiom,
! [N: nat] :
( ( root @ N @ zero_zero_real )
= zero_zero_real ) ).
% real_root_zero
thf(fact_9042_finite__greaterThanLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( set_or5832277885323065728an_int @ L @ U ) ) ).
% finite_greaterThanLessThan_int
thf(fact_9043_real__root__Suc__0,axiom,
! [X3: real] :
( ( root @ ( suc @ zero_zero_nat ) @ X3 )
= X3 ) ).
% real_root_Suc_0
thf(fact_9044_root__0,axiom,
! [X3: real] :
( ( root @ zero_zero_nat @ X3 )
= zero_zero_real ) ).
% root_0
thf(fact_9045_real__root__eq__iff,axiom,
! [N: nat,X3: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X3 )
= ( root @ N @ Y ) )
= ( X3 = Y ) ) ) ).
% real_root_eq_iff
thf(fact_9046_real__root__eq__0__iff,axiom,
! [N: nat,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X3 )
= zero_zero_real )
= ( X3 = zero_zero_real ) ) ) ).
% real_root_eq_0_iff
thf(fact_9047_real__root__less__iff,axiom,
! [N: nat,X3: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X3 ) @ ( root @ N @ Y ) )
= ( ord_less_real @ X3 @ Y ) ) ) ).
% real_root_less_iff
thf(fact_9048_real__root__le__iff,axiom,
! [N: nat,X3: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X3 ) @ ( root @ N @ Y ) )
= ( ord_less_eq_real @ X3 @ Y ) ) ) ).
% real_root_le_iff
thf(fact_9049_real__root__one,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ one_one_real )
= one_one_real ) ) ).
% real_root_one
thf(fact_9050_real__root__eq__1__iff,axiom,
! [N: nat,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X3 )
= one_one_real )
= ( X3 = one_one_real ) ) ) ).
% real_root_eq_1_iff
thf(fact_9051_real__root__gt__0__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y ) )
= ( ord_less_real @ zero_zero_real @ Y ) ) ) ).
% real_root_gt_0_iff
thf(fact_9052_real__root__lt__0__iff,axiom,
! [N: nat,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X3 ) @ zero_zero_real )
= ( ord_less_real @ X3 @ zero_zero_real ) ) ) ).
% real_root_lt_0_iff
thf(fact_9053_real__root__le__0__iff,axiom,
! [N: nat,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X3 ) @ zero_zero_real )
= ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ) ).
% real_root_le_0_iff
thf(fact_9054_real__root__ge__0__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y ) )
= ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).
% real_root_ge_0_iff
thf(fact_9055_real__root__gt__1__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ one_one_real @ ( root @ N @ Y ) )
= ( ord_less_real @ one_one_real @ Y ) ) ) ).
% real_root_gt_1_iff
thf(fact_9056_real__root__lt__1__iff,axiom,
! [N: nat,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X3 ) @ one_one_real )
= ( ord_less_real @ X3 @ one_one_real ) ) ) ).
% real_root_lt_1_iff
thf(fact_9057_real__root__le__1__iff,axiom,
! [N: nat,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X3 ) @ one_one_real )
= ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ).
% real_root_le_1_iff
thf(fact_9058_real__root__ge__1__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y ) )
= ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).
% real_root_ge_1_iff
thf(fact_9059_real__root__pow__pos2,axiom,
! [N: nat,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( power_power_real @ ( root @ N @ X3 ) @ N )
= X3 ) ) ) ).
% real_root_pow_pos2
thf(fact_9060_real__root__pos__pos__le,axiom,
! [X3: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X3 ) ) ) ).
% real_root_pos_pos_le
thf(fact_9061_real__root__less__mono,axiom,
! [N: nat,X3: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ ( root @ N @ X3 ) @ ( root @ N @ Y ) ) ) ) ).
% real_root_less_mono
thf(fact_9062_real__root__le__mono,axiom,
! [N: nat,X3: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ ( root @ N @ X3 ) @ ( root @ N @ Y ) ) ) ) ).
% real_root_le_mono
thf(fact_9063_real__root__power,axiom,
! [N: nat,X3: real,K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( power_power_real @ X3 @ K2 ) )
= ( power_power_real @ ( root @ N @ X3 ) @ K2 ) ) ) ).
% real_root_power
thf(fact_9064_real__root__abs,axiom,
! [N: nat,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( abs_abs_real @ X3 ) )
= ( abs_abs_real @ ( root @ N @ X3 ) ) ) ) ).
% real_root_abs
thf(fact_9065_sgn__root,axiom,
! [N: nat,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( sgn_sgn_real @ ( root @ N @ X3 ) )
= ( sgn_sgn_real @ X3 ) ) ) ).
% sgn_root
thf(fact_9066_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
= ( set_or5832277885323065728an_int @ L @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_9067_real__root__gt__zero,axiom,
! [N: nat,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_real @ zero_zero_real @ ( root @ N @ X3 ) ) ) ) ).
% real_root_gt_zero
thf(fact_9068_real__root__strict__decreasing,axiom,
! [N: nat,N7: nat,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ N @ N7 )
=> ( ( ord_less_real @ one_one_real @ X3 )
=> ( ord_less_real @ ( root @ N7 @ X3 ) @ ( root @ N @ X3 ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_9069_root__abs__power,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y @ N ) ) )
= ( abs_abs_real @ Y ) ) ) ).
% root_abs_power
thf(fact_9070_real__root__pos__pos,axiom,
! [N: nat,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X3 ) ) ) ) ).
% real_root_pos_pos
thf(fact_9071_real__root__strict__increasing,axiom,
! [N: nat,N7: nat,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ N @ N7 )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ one_one_real )
=> ( ord_less_real @ ( root @ N @ X3 ) @ ( root @ N7 @ X3 ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_9072_real__root__decreasing,axiom,
! [N: nat,N7: nat,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ N7 )
=> ( ( ord_less_eq_real @ one_one_real @ X3 )
=> ( ord_less_eq_real @ ( root @ N7 @ X3 ) @ ( root @ N @ X3 ) ) ) ) ) ).
% real_root_decreasing
thf(fact_9073_real__root__pow__pos,axiom,
! [N: nat,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( power_power_real @ ( root @ N @ X3 ) @ N )
= X3 ) ) ) ).
% real_root_pow_pos
thf(fact_9074_real__root__power__cancel,axiom,
! [N: nat,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( root @ N @ ( power_power_real @ X3 @ N ) )
= X3 ) ) ) ).
% real_root_power_cancel
thf(fact_9075_real__root__pos__unique,axiom,
! [N: nat,Y: real,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( power_power_real @ Y @ N )
= X3 )
=> ( ( root @ N @ X3 )
= Y ) ) ) ) ).
% real_root_pos_unique
thf(fact_9076_real__root__increasing,axiom,
! [N: nat,N7: nat,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ N7 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ one_one_real )
=> ( ord_less_eq_real @ ( root @ N @ X3 ) @ ( root @ N7 @ X3 ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_9077_root__sgn__power,axiom,
! [N: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) ) )
= Y ) ) ).
% root_sgn_power
thf(fact_9078_sgn__power__root,axiom,
! [N: nat,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X3 ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X3 ) ) @ N ) )
= X3 ) ) ).
% sgn_power_root
thf(fact_9079_bij__betw__nth__root__unity,axiom,
! [C2: complex,N: nat] :
( ( C2 != zero_zero_complex )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C2 ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
@ ( collect_complex
@ ^ [Z6: complex] :
( ( power_power_complex @ Z6 @ N )
= one_one_complex ) )
@ ( collect_complex
@ ^ [Z6: complex] :
( ( power_power_complex @ Z6 @ N )
= C2 ) ) ) ) ) ).
% bij_betw_nth_root_unity
thf(fact_9080_ln__root,axiom,
! [N: nat,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ln_ln_real @ ( root @ N @ B ) )
= ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% ln_root
thf(fact_9081_log__root,axiom,
! [N: nat,A: real,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( log @ B @ ( root @ N @ A ) )
= ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% log_root
thf(fact_9082_log__base__root,axiom,
! [N: nat,B: real,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( log @ ( root @ N @ B ) @ X3 )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X3 ) ) ) ) ) ).
% log_base_root
thf(fact_9083_split__root,axiom,
! [P2: real > $o,N: nat,X3: real] :
( ( P2 @ ( root @ N @ X3 ) )
= ( ( ( N = zero_zero_nat )
=> ( P2 @ zero_zero_real ) )
& ( ( ord_less_nat @ zero_zero_nat @ N )
=> ! [Y5: real] :
( ( ( times_times_real @ ( sgn_sgn_real @ Y5 ) @ ( power_power_real @ ( abs_abs_real @ Y5 ) @ N ) )
= X3 )
=> ( P2 @ Y5 ) ) ) ) ) ).
% split_root
thf(fact_9084_sin__integer__2pi,axiom,
! [N: real] :
( ( member_real @ N @ ring_1_Ints_real )
=> ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
= zero_zero_real ) ) ).
% sin_integer_2pi
thf(fact_9085_finite__greaterThanLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% finite_greaterThanLessThan
thf(fact_9086_Suc__funpow,axiom,
! [N: nat] :
( ( compow_nat_nat @ N @ suc )
= ( plus_plus_nat @ N ) ) ).
% Suc_funpow
thf(fact_9087_card__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L @ U ) )
= ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).
% card_greaterThanLessThan
thf(fact_9088_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
= ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% atLeastSucLessThan_greaterThanLessThan
thf(fact_9089_sin__times__pi__eq__0,axiom,
! [X3: real] :
( ( ( sin_real @ ( times_times_real @ X3 @ pi ) )
= zero_zero_real )
= ( member_real @ X3 @ ring_1_Ints_real ) ) ).
% sin_times_pi_eq_0
thf(fact_9090_complex__is__Int__iff,axiom,
! [Z2: complex] :
( ( member_complex @ Z2 @ ring_1_Ints_complex )
= ( ( ( im @ Z2 )
= zero_zero_real )
& ? [I3: int] :
( ( re @ Z2 )
= ( ring_1_of_int_real @ I3 ) ) ) ) ).
% complex_is_Int_iff
thf(fact_9091_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
@ ^ [X5: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ X5 )
@ ^ [X5: nat,Y5: nat] : ( ord_less_nat @ Y5 @ X5 ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_9092_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
@ ^ [M5: nat,N4: nat] :
( ( dvd_dvd_nat @ M5 @ N4 )
& ( M5 != N4 ) ) ) ).
% gcd_nat.semilattice_neutr_order_axioms
thf(fact_9093_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_9094_upto_Opelims,axiom,
! [X3: int,Xa2: int,Y: list_int] :
( ( ( upto @ X3 @ Xa2 )
= Y )
=> ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) )
=> ~ ( ( ( ( ord_less_eq_int @ X3 @ Xa2 )
=> ( Y
= ( cons_int @ X3 @ ( upto @ ( plus_plus_int @ X3 @ one_one_int ) @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq_int @ X3 @ Xa2 )
=> ( Y = nil_int ) ) )
=> ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) ) ) ) ) ).
% upto.pelims
thf(fact_9095_upto_Opsimps,axiom,
! [I: int,J: int] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
=> ( ( ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
& ( ~ ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= nil_int ) ) ) ) ).
% upto.psimps
thf(fact_9096_upto__Nil,axiom,
! [I: int,J: int] :
( ( ( upto @ I @ J )
= nil_int )
= ( ord_less_int @ J @ I ) ) ).
% upto_Nil
thf(fact_9097_upto__Nil2,axiom,
! [I: int,J: int] :
( ( nil_int
= ( upto @ I @ J ) )
= ( ord_less_int @ J @ I ) ) ).
% upto_Nil2
thf(fact_9098_upto__empty,axiom,
! [J: int,I: int] :
( ( ord_less_int @ J @ I )
=> ( ( upto @ I @ J )
= nil_int ) ) ).
% upto_empty
thf(fact_9099_upto__single,axiom,
! [I: int] :
( ( upto @ I @ I )
= ( cons_int @ I @ nil_int ) ) ).
% upto_single
thf(fact_9100_nth__upto,axiom,
! [I: int,K2: nat,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K2 ) ) @ J )
=> ( ( nth_int @ ( upto @ I @ J ) @ K2 )
= ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K2 ) ) ) ) ).
% nth_upto
thf(fact_9101_length__upto,axiom,
! [I: int,J: int] :
( ( size_size_list_int @ ( upto @ I @ J ) )
= ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I ) @ one_one_int ) ) ) ).
% length_upto
thf(fact_9102_upto__rec__numeral_I1_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
= ( cons_int @ ( numeral_numeral_int @ M2 ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
= nil_int ) ) ) ).
% upto_rec_numeral(1)
thf(fact_9103_upto__rec__numeral_I2_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( cons_int @ ( numeral_numeral_int @ M2 ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= nil_int ) ) ) ).
% upto_rec_numeral(2)
thf(fact_9104_upto__rec__numeral_I3_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
= ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
= nil_int ) ) ) ).
% upto_rec_numeral(3)
thf(fact_9105_upto__rec__numeral_I4_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= nil_int ) ) ) ).
% upto_rec_numeral(4)
thf(fact_9106_upto__split1,axiom,
! [I: int,J: int,K2: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K2 )
=> ( ( upto @ I @ K2 )
= ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K2 ) ) ) ) ) ).
% upto_split1
thf(fact_9107_upto__split2,axiom,
! [I: int,J: int,K2: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K2 )
=> ( ( upto @ I @ K2 )
= ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K2 ) ) ) ) ) ).
% upto_split2
thf(fact_9108_upto__aux__def,axiom,
( upto_aux
= ( ^ [I3: int,J3: int] : ( append_int @ ( upto @ I3 @ J3 ) ) ) ) ).
% upto_aux_def
thf(fact_9109_atLeastAtMost__upto,axiom,
( set_or1266510415728281911st_int
= ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ I3 @ J3 ) ) ) ) ).
% atLeastAtMost_upto
thf(fact_9110_distinct__upto,axiom,
! [I: int,J: int] : ( distinct_int @ ( upto @ I @ J ) ) ).
% distinct_upto
thf(fact_9111_upto__code,axiom,
( upto
= ( ^ [I3: int,J3: int] : ( upto_aux @ I3 @ J3 @ nil_int ) ) ) ).
% upto_code
thf(fact_9112_upto__rec2,axiom,
! [I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% upto_rec2
thf(fact_9113_upto__split3,axiom,
! [I: int,J: int,K2: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K2 )
=> ( ( upto @ I @ K2 )
= ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K2 ) ) ) ) ) ) ).
% upto_split3
thf(fact_9114_atLeastLessThan__upto,axiom,
( set_or4662586982721622107an_int
= ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% atLeastLessThan_upto
thf(fact_9115_upto__rec1,axiom,
! [I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).
% upto_rec1
thf(fact_9116_upto_Oelims,axiom,
! [X3: int,Xa2: int,Y: list_int] :
( ( ( upto @ X3 @ Xa2 )
= Y )
=> ( ( ( ord_less_eq_int @ X3 @ Xa2 )
=> ( Y
= ( cons_int @ X3 @ ( upto @ ( plus_plus_int @ X3 @ one_one_int ) @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq_int @ X3 @ Xa2 )
=> ( Y = nil_int ) ) ) ) ).
% upto.elims
thf(fact_9117_upto_Osimps,axiom,
( upto
= ( ^ [I3: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I3 @ J3 ) @ ( cons_int @ I3 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).
% upto.simps
thf(fact_9118_greaterThanLessThan__upto,axiom,
( set_or5832277885323065728an_int
= ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% greaterThanLessThan_upto
thf(fact_9119_take__bit__numeral__minus__numeral__int,axiom,
! [M2: num,N: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( case_option_int_num @ zero_zero_int
@ ^ [Q4: num] : ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M2 ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_int @ Q4 ) ) )
@ ( bit_take_bit_num @ ( numeral_numeral_nat @ M2 ) @ N ) ) ) ).
% take_bit_numeral_minus_numeral_int
thf(fact_9120_and__minus__numerals_I3_J,axiom,
! [M2: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M2 @ ( bitM @ N ) ) ) ) ).
% and_minus_numerals(3)
thf(fact_9121_and__minus__numerals_I7_J,axiom,
! [N: num,M2: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M2 ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M2 @ ( bitM @ N ) ) ) ) ).
% and_minus_numerals(7)
thf(fact_9122_take__bit__num__simps_I1_J,axiom,
! [M2: num] :
( ( bit_take_bit_num @ zero_zero_nat @ M2 )
= none_num ) ).
% take_bit_num_simps(1)
thf(fact_9123_take__bit__num__simps_I2_J,axiom,
! [N: nat] :
( ( bit_take_bit_num @ ( suc @ N ) @ one )
= ( some_num @ one ) ) ).
% take_bit_num_simps(2)
thf(fact_9124_take__bit__num__simps_I3_J,axiom,
! [N: nat,M2: num] :
( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M2 ) )
= ( case_o6005452278849405969um_num @ none_num
@ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ N @ M2 ) ) ) ).
% take_bit_num_simps(3)
thf(fact_9125_take__bit__num__simps_I4_J,axiom,
! [N: nat,M2: num] :
( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M2 ) )
= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M2 ) ) ) ) ).
% take_bit_num_simps(4)
thf(fact_9126_take__bit__num__simps_I6_J,axiom,
! [R2: num,M2: num] :
( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit0 @ M2 ) )
= ( case_o6005452278849405969um_num @ none_num
@ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M2 ) ) ) ).
% take_bit_num_simps(6)
thf(fact_9127_and__minus__numerals_I8_J,axiom,
! [N: num,M2: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M2 ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M2 @ ( bit0 @ N ) ) ) ) ).
% and_minus_numerals(8)
thf(fact_9128_and__minus__numerals_I4_J,axiom,
! [M2: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M2 @ ( bit0 @ N ) ) ) ) ).
% and_minus_numerals(4)
thf(fact_9129_and__not__num_Osimps_I1_J,axiom,
( ( bit_and_not_num @ one @ one )
= none_num ) ).
% and_not_num.simps(1)
thf(fact_9130_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
! [N: nat,M2: num] :
( ( bit_take_bit_num @ N @ ( bit0 @ M2 ) )
= ( case_nat_option_num @ none_num
@ ^ [N4: nat] :
( case_o6005452278849405969um_num @ none_num
@ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ N4 @ M2 ) )
@ N ) ) ).
% Code_Abstract_Nat.take_bit_num_code(2)
thf(fact_9131_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
! [N: nat] :
( ( bit_take_bit_num @ N @ one )
= ( case_nat_option_num @ none_num
@ ^ [N4: nat] : ( some_num @ one )
@ N ) ) ).
% Code_Abstract_Nat.take_bit_num_code(1)
thf(fact_9132_and__not__num_Osimps_I3_J,axiom,
! [N: num] :
( ( bit_and_not_num @ one @ ( bit1 @ N ) )
= none_num ) ).
% and_not_num.simps(3)
thf(fact_9133_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
! [N: nat,M2: num] :
( ( bit_take_bit_num @ N @ ( bit1 @ M2 ) )
= ( case_nat_option_num @ none_num
@ ^ [N4: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N4 @ M2 ) ) )
@ N ) ) ).
% Code_Abstract_Nat.take_bit_num_code(3)
thf(fact_9134_and__not__num__eq__None__iff,axiom,
! [M2: num,N: num] :
( ( ( bit_and_not_num @ M2 @ N )
= none_num )
= ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
= zero_zero_int ) ) ).
% and_not_num_eq_None_iff
thf(fact_9135_int__numeral__and__not__num,axiom,
! [M2: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M2 @ N ) ) ) ).
% int_numeral_and_not_num
thf(fact_9136_int__numeral__not__and__num,axiom,
! [M2: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N @ M2 ) ) ) ).
% int_numeral_not_and_num
thf(fact_9137_Bit__Operations_Otake__bit__num__code,axiom,
( bit_take_bit_num
= ( ^ [N4: nat,M5: num] :
( produc478579273971653890on_num
@ ^ [A5: nat,X5: num] :
( case_nat_option_num @ none_num
@ ^ [O: nat] :
( case_num_option_num @ ( some_num @ one )
@ ^ [P6: num] :
( case_o6005452278849405969um_num @ none_num
@ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ O @ P6 ) )
@ ^ [P6: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P6 ) ) )
@ X5 )
@ A5 )
@ ( product_Pair_nat_num @ N4 @ M5 ) ) ) ) ).
% Bit_Operations.take_bit_num_code
thf(fact_9138_times__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
( ( times_times_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
= ( abs_Integ
@ ( produc27273713700761075at_nat
@ ^ [X5: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X5 @ U2 ) @ ( times_times_nat @ Y5 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X5 @ V4 ) @ ( times_times_nat @ Y5 @ U2 ) ) ) )
@ Xa2
@ X3 ) ) ) ).
% times_int.abs_eq
thf(fact_9139_int_Oabs__induct,axiom,
! [P2: int > $o,X3: int] :
( ! [Y3: product_prod_nat_nat] : ( P2 @ ( abs_Integ @ Y3 ) )
=> ( P2 @ X3 ) ) ).
% int.abs_induct
thf(fact_9140_eq__Abs__Integ,axiom,
! [Z2: int] :
~ ! [X4: nat,Y3: nat] :
( Z2
!= ( abs_Integ @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) ) ).
% eq_Abs_Integ
thf(fact_9141_nat_Oabs__eq,axiom,
! [X3: product_prod_nat_nat] :
( ( nat2 @ ( abs_Integ @ X3 ) )
= ( produc6842872674320459806at_nat @ minus_minus_nat @ X3 ) ) ).
% nat.abs_eq
thf(fact_9142_zero__int__def,axiom,
( zero_zero_int
= ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% zero_int_def
thf(fact_9143_int__def,axiom,
( semiri1314217659103216013at_int
= ( ^ [N4: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N4 @ zero_zero_nat ) ) ) ) ).
% int_def
thf(fact_9144_uminus__int_Oabs__eq,axiom,
! [X3: product_prod_nat_nat] :
( ( uminus_uminus_int @ ( abs_Integ @ X3 ) )
= ( abs_Integ
@ ( produc2626176000494625587at_nat
@ ^ [X5: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X5 )
@ X3 ) ) ) ).
% uminus_int.abs_eq
thf(fact_9145_one__int__def,axiom,
( one_one_int
= ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% one_int_def
thf(fact_9146_less__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
( ( ord_less_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
= ( produc8739625826339149834_nat_o
@ ^ [X5: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X5 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) )
@ Xa2
@ X3 ) ) ).
% less_int.abs_eq
thf(fact_9147_less__eq__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
= ( produc8739625826339149834_nat_o
@ ^ [X5: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X5 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) )
@ Xa2
@ X3 ) ) ).
% less_eq_int.abs_eq
thf(fact_9148_plus__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
( ( plus_plus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
= ( abs_Integ
@ ( produc27273713700761075at_nat
@ ^ [X5: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X5 @ U2 ) @ ( plus_plus_nat @ Y5 @ V4 ) ) )
@ Xa2
@ X3 ) ) ) ).
% plus_int.abs_eq
thf(fact_9149_minus__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
( ( minus_minus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
= ( abs_Integ
@ ( produc27273713700761075at_nat
@ ^ [X5: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X5 @ V4 ) @ ( plus_plus_nat @ Y5 @ U2 ) ) )
@ Xa2
@ X3 ) ) ) ).
% minus_int.abs_eq
thf(fact_9150_take__bit__num__def,axiom,
( bit_take_bit_num
= ( ^ [N4: nat,M5: num] :
( if_option_num
@ ( ( bit_se2925701944663578781it_nat @ N4 @ ( numeral_numeral_nat @ M5 ) )
= zero_zero_nat )
@ none_num
@ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N4 @ ( numeral_numeral_nat @ M5 ) ) ) ) ) ) ) ).
% take_bit_num_def
thf(fact_9151_and__not__num_Oelims,axiom,
! [X3: num,Xa2: num,Y: option_num] :
( ( ( bit_and_not_num @ X3 @ Xa2 )
= Y )
=> ( ( ( X3 = one )
=> ( ( Xa2 = one )
=> ( Y != none_num ) ) )
=> ( ( ( X3 = one )
=> ( ? [N3: num] :
( Xa2
= ( bit0 @ N3 ) )
=> ( Y
!= ( some_num @ one ) ) ) )
=> ( ( ( X3 = one )
=> ( ? [N3: num] :
( Xa2
= ( bit1 @ N3 ) )
=> ( Y != none_num ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ( ( Xa2 = one )
=> ( Y
!= ( some_num @ ( bit0 @ M ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y
!= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N3 ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y
!= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N3 ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ( ( Xa2 = one )
=> ( Y
!= ( some_num @ ( bit0 @ M ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y
!= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
@ ( bit_and_not_num @ M @ N3 ) ) ) ) )
=> ~ ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y
!= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.elims
thf(fact_9152_max__rpair__set,axiom,
fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).
% max_rpair_set
thf(fact_9153_num__of__nat_Osimps_I1_J,axiom,
( ( num_of_nat @ zero_zero_nat )
= one ) ).
% num_of_nat.simps(1)
thf(fact_9154_numeral__num__of__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
= N ) ) ).
% numeral_num_of_nat
thf(fact_9155_num__of__nat__One,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ one_one_nat )
=> ( ( num_of_nat @ N )
= one ) ) ).
% num_of_nat_One
thf(fact_9156_num__of__nat__double,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
= ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% num_of_nat_double
thf(fact_9157_num__of__nat__plus__distrib,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_num @ ( num_of_nat @ M2 ) @ ( num_of_nat @ N ) ) ) ) ) ).
% num_of_nat_plus_distrib
thf(fact_9158_num__of__nat_Osimps_I2_J,axiom,
! [N: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( suc @ N ) )
= ( inc @ ( num_of_nat @ N ) ) ) )
& ( ~ ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( suc @ N ) )
= one ) ) ) ).
% num_of_nat.simps(2)
thf(fact_9159_min__rpair__set,axiom,
fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).
% min_rpair_set
thf(fact_9160_and__not__num_Opelims,axiom,
! [X3: num,Xa2: num,Y: option_num] :
( ( ( bit_and_not_num @ X3 @ Xa2 )
= Y )
=> ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X3 @ Xa2 ) )
=> ( ( ( X3 = one )
=> ( ( Xa2 = one )
=> ( ( Y = none_num )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
=> ( ( ( X3 = one )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y
= ( some_num @ one ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ( ( X3 = one )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y = none_num )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ( ( Xa2 = one )
=> ( ( Y
= ( some_num @ ( bit0 @ M ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M ) @ one ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y
= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y
= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M ) @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ( ( Xa2 = one )
=> ( ( Y
= ( some_num @ ( bit0 @ M ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M ) @ one ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y
= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
@ ( bit_and_not_num @ M @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ~ ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y
= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.pelims
thf(fact_9161_and__num_Oelims,axiom,
! [X3: num,Xa2: num,Y: option_num] :
( ( ( bit_un7362597486090784418nd_num @ X3 @ Xa2 )
= Y )
=> ( ( ( X3 = one )
=> ( ( Xa2 = one )
=> ( Y
!= ( some_num @ one ) ) ) )
=> ( ( ( X3 = one )
=> ( ? [N3: num] :
( Xa2
= ( bit0 @ N3 ) )
=> ( Y != none_num ) ) )
=> ( ( ( X3 = one )
=> ( ? [N3: num] :
( Xa2
= ( bit1 @ N3 ) )
=> ( Y
!= ( some_num @ one ) ) ) )
=> ( ( ? [M: num] :
( X3
= ( bit0 @ M ) )
=> ( ( Xa2 = one )
=> ( Y != none_num ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y
!= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N3 ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y
!= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N3 ) ) ) ) )
=> ( ( ? [M: num] :
( X3
= ( bit1 @ M ) )
=> ( ( Xa2 = one )
=> ( Y
!= ( some_num @ one ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y
!= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N3 ) ) ) ) )
=> ~ ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y
!= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
@ ( bit_un7362597486090784418nd_num @ M @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.elims
thf(fact_9162_and__num_Osimps_I2_J,axiom,
! [N: num] :
( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N ) )
= none_num ) ).
% and_num.simps(2)
thf(fact_9163_and__num_Osimps_I4_J,axiom,
! [M2: num] :
( ( bit_un7362597486090784418nd_num @ ( bit0 @ M2 ) @ one )
= none_num ) ).
% and_num.simps(4)
thf(fact_9164_and__num_Opelims,axiom,
! [X3: num,Xa2: num,Y: option_num] :
( ( ( bit_un7362597486090784418nd_num @ X3 @ Xa2 )
= Y )
=> ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X3 @ Xa2 ) )
=> ( ( ( X3 = one )
=> ( ( Xa2 = one )
=> ( ( Y
= ( some_num @ one ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
=> ( ( ( X3 = one )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y = none_num )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ( ( X3 = one )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y
= ( some_num @ one ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ( ( Xa2 = one )
=> ( ( Y = none_num )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M ) @ one ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y
= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y
= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M ) @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ( ( Xa2 = one )
=> ( ( Y
= ( some_num @ one ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M ) @ one ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y
= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ~ ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y
= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
@ ( bit_un7362597486090784418nd_num @ M @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.pelims
thf(fact_9165_xor__num_Oelims,axiom,
! [X3: num,Xa2: num,Y: option_num] :
( ( ( bit_un2480387367778600638or_num @ X3 @ Xa2 )
= Y )
=> ( ( ( X3 = one )
=> ( ( Xa2 = one )
=> ( Y != none_num ) ) )
=> ( ( ( X3 = one )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y
!= ( some_num @ ( bit1 @ N3 ) ) ) ) )
=> ( ( ( X3 = one )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y
!= ( some_num @ ( bit0 @ N3 ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ( ( Xa2 = one )
=> ( Y
!= ( some_num @ ( bit1 @ M ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y
!= ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N3 ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y
!= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N3 ) ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ( ( Xa2 = one )
=> ( Y
!= ( some_num @ ( bit0 @ M ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y
!= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N3 ) ) ) ) ) )
=> ~ ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y
!= ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.elims
thf(fact_9166_less__eq__int_Orep__eq,axiom,
( ord_less_eq_int
= ( ^ [X5: int,Xa4: int] :
( produc8739625826339149834_nat_o
@ ^ [Y5: nat,Z6: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y5 @ V4 ) @ ( plus_plus_nat @ U2 @ Z6 ) ) )
@ ( rep_Integ @ X5 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_eq_int.rep_eq
thf(fact_9167_xor__num_Osimps_I1_J,axiom,
( ( bit_un2480387367778600638or_num @ one @ one )
= none_num ) ).
% xor_num.simps(1)
thf(fact_9168_nat_Orep__eq,axiom,
( nat2
= ( ^ [X5: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X5 ) ) ) ) ).
% nat.rep_eq
thf(fact_9169_less__int_Orep__eq,axiom,
( ord_less_int
= ( ^ [X5: int,Xa4: int] :
( produc8739625826339149834_nat_o
@ ^ [Y5: nat,Z6: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y5 @ V4 ) @ ( plus_plus_nat @ U2 @ Z6 ) ) )
@ ( rep_Integ @ X5 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_int.rep_eq
thf(fact_9170_xor__num_Opelims,axiom,
! [X3: num,Xa2: num,Y: option_num] :
( ( ( bit_un2480387367778600638or_num @ X3 @ Xa2 )
= Y )
=> ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X3 @ Xa2 ) )
=> ( ( ( X3 = one )
=> ( ( Xa2 = one )
=> ( ( Y = none_num )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
=> ( ( ( X3 = one )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y
= ( some_num @ ( bit1 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ( ( X3 = one )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y
= ( some_num @ ( bit0 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ( ( Xa2 = one )
=> ( ( Y
= ( some_num @ ( bit1 @ M ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M ) @ one ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y
= ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit0 @ M ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y
= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N3 ) ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M ) @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ( ( Xa2 = one )
=> ( ( Y
= ( some_num @ ( bit0 @ M ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M ) @ one ) ) ) ) )
=> ( ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y
= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N3 ) ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ~ ! [M: num] :
( ( X3
= ( bit1 @ M ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y
= ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.pelims
thf(fact_9171_uminus__int__def,axiom,
( uminus_uminus_int
= ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
@ ( produc2626176000494625587at_nat
@ ^ [X5: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X5 ) ) ) ) ).
% uminus_int_def
thf(fact_9172_times__int__def,axiom,
( times_times_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X5: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X5 @ U2 ) @ ( times_times_nat @ Y5 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X5 @ V4 ) @ ( times_times_nat @ Y5 @ U2 ) ) ) ) ) ) ) ).
% times_int_def
thf(fact_9173_minus__int__def,axiom,
( minus_minus_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X5: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X5 @ V4 ) @ ( plus_plus_nat @ Y5 @ U2 ) ) ) ) ) ) ).
% minus_int_def
thf(fact_9174_plus__int__def,axiom,
( plus_plus_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X5: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X5 @ U2 ) @ ( plus_plus_nat @ Y5 @ V4 ) ) ) ) ) ) ).
% plus_int_def
thf(fact_9175_prod__encode__def,axiom,
( nat_prod_encode
= ( produc6842872674320459806at_nat
@ ^ [M5: nat,N4: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M5 @ N4 ) ) @ M5 ) ) ) ).
% prod_encode_def
thf(fact_9176_Gcd__remove0__nat,axiom,
! [M6: set_nat] :
( ( finite_finite_nat @ M6 )
=> ( ( gcd_Gcd_nat @ M6 )
= ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M6 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% Gcd_remove0_nat
thf(fact_9177_prod__encode__eq,axiom,
! [X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( ( nat_prod_encode @ X3 )
= ( nat_prod_encode @ Y ) )
= ( X3 = Y ) ) ).
% prod_encode_eq
thf(fact_9178_le__prod__encode__1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% le_prod_encode_1
thf(fact_9179_le__prod__encode__2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% le_prod_encode_2
thf(fact_9180_prod__encode__prod__decode__aux,axiom,
! [K2: nat,M2: nat] :
( ( nat_prod_encode @ ( nat_prod_decode_aux @ K2 @ M2 ) )
= ( plus_plus_nat @ ( nat_triangle @ K2 ) @ M2 ) ) ).
% prod_encode_prod_decode_aux
thf(fact_9181_list__encode_Oelims,axiom,
! [X3: list_nat,Y: nat] :
( ( ( nat_list_encode @ X3 )
= Y )
=> ( ( ( X3 = nil_nat )
=> ( Y != zero_zero_nat ) )
=> ~ ! [X4: nat,Xs2: list_nat] :
( ( X3
= ( cons_nat @ X4 @ Xs2 ) )
=> ( Y
!= ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).
% list_encode.elims
thf(fact_9182_list__encode__eq,axiom,
! [X3: list_nat,Y: list_nat] :
( ( ( nat_list_encode @ X3 )
= ( nat_list_encode @ Y ) )
= ( X3 = Y ) ) ).
% list_encode_eq
thf(fact_9183_list__encode_Ocases,axiom,
! [X3: list_nat] :
( ( X3 != nil_nat )
=> ~ ! [X4: nat,Xs2: list_nat] :
( X3
!= ( cons_nat @ X4 @ Xs2 ) ) ) ).
% list_encode.cases
thf(fact_9184_list__encode_Osimps_I1_J,axiom,
( ( nat_list_encode @ nil_nat )
= zero_zero_nat ) ).
% list_encode.simps(1)
thf(fact_9185_Gcd__int__greater__eq__0,axiom,
! [K5: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K5 ) ) ).
% Gcd_int_greater_eq_0
thf(fact_9186_list__encode_Osimps_I2_J,axiom,
! [X3: nat,Xs: list_nat] :
( ( nat_list_encode @ ( cons_nat @ X3 @ Xs ) )
= ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs ) ) ) ) ) ).
% list_encode.simps(2)
thf(fact_9187_sub__BitM__One__eq,axiom,
! [N: num] :
( ( neg_numeral_sub_int @ ( bitM @ N ) @ one )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N @ one ) ) ) ).
% sub_BitM_One_eq
thf(fact_9188_list__encode_Opelims,axiom,
! [X3: list_nat,Y: nat] :
( ( ( nat_list_encode @ X3 )
= Y )
=> ( ( accp_list_nat @ nat_list_encode_rel @ X3 )
=> ( ( ( X3 = nil_nat )
=> ( ( Y = zero_zero_nat )
=> ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
=> ~ ! [X4: nat,Xs2: list_nat] :
( ( X3
= ( cons_nat @ X4 @ Xs2 ) )
=> ( ( Y
= ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs2 ) ) ) ) )
=> ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X4 @ Xs2 ) ) ) ) ) ) ) ).
% list_encode.pelims
thf(fact_9189_pred__nat__def,axiom,
( pred_nat
= ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [M5: nat,N4: nat] :
( N4
= ( suc @ M5 ) ) ) ) ) ).
% pred_nat_def
thf(fact_9190_sorted__list__of__set__atMost__Suc,axiom,
! [K2: nat] :
( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K2 ) ) )
= ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K2 ) ) @ ( cons_nat @ ( suc @ K2 ) @ nil_nat ) ) ) ).
% sorted_list_of_set_atMost_Suc
thf(fact_9191_sorted__list__of__set__lessThan__Suc,axiom,
! [K2: nat] :
( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K2 ) ) )
= ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K2 ) ) @ ( cons_nat @ K2 @ nil_nat ) ) ) ).
% sorted_list_of_set_lessThan_Suc
thf(fact_9192_sorted__list__of__set__greaterThanLessThan,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ ( suc @ I ) @ J )
=> ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
= ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanLessThan
thf(fact_9193_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J: nat,I: nat] :
( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
=> ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
= ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_9194_image__minus__const__atLeastLessThan__nat,axiom,
! [C2: nat,Y: nat,X3: nat] :
( ( ( ord_less_nat @ C2 @ Y )
=> ( ( image_nat_nat
@ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C2 )
@ ( set_or4665077453230672383an_nat @ X3 @ Y ) )
= ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X3 @ C2 ) @ ( minus_minus_nat @ Y @ C2 ) ) ) )
& ( ~ ( ord_less_nat @ C2 @ Y )
=> ( ( ( ord_less_nat @ X3 @ Y )
=> ( ( image_nat_nat
@ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C2 )
@ ( set_or4665077453230672383an_nat @ X3 @ Y ) )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
& ( ~ ( ord_less_nat @ X3 @ Y )
=> ( ( image_nat_nat
@ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C2 )
@ ( set_or4665077453230672383an_nat @ X3 @ Y ) )
= bot_bot_set_nat ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_9195_bij__betw__Suc,axiom,
! [M6: set_nat,N7: set_nat] :
( ( bij_betw_nat_nat @ suc @ M6 @ N7 )
= ( ( image_nat_nat @ suc @ M6 )
= N7 ) ) ).
% bij_betw_Suc
thf(fact_9196_image__Suc__atLeastAtMost,axiom,
! [I: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
= ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_9197_image__Suc__atLeastLessThan,axiom,
! [I: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
= ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_9198_zero__notin__Suc__image,axiom,
! [A4: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A4 ) ) ).
% zero_notin_Suc_image
thf(fact_9199_image__Suc__lessThan,axiom,
! [N: nat] :
( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
= ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).
% image_Suc_lessThan
thf(fact_9200_image__Suc__atMost,axiom,
! [N: nat] :
( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
= ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).
% image_Suc_atMost
thf(fact_9201_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).
% atLeast0_atMost_Suc_eq_insert_0
thf(fact_9202_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).
% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_9203_lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_lessThan_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_9204_atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_atMost_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_9205_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J: nat,I: nat] :
( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
=> ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
= ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_9206_rat__floor__lemma,axiom,
! [A: int,B: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B ) ) @ ( fract @ A @ B ) )
& ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ) ).
% rat_floor_lemma
thf(fact_9207_finite__greaterThanAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% finite_greaterThanAtMost
thf(fact_9208_card__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L @ U ) )
= ( minus_minus_nat @ U @ L ) ) ).
% card_greaterThanAtMost
thf(fact_9209_less__rat,axiom,
! [B: int,D: int,A: int,C2: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C2 @ D ) )
= ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C2 @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% less_rat
thf(fact_9210_add__rat,axiom,
! [B: int,D: int,A: int,C2: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( plus_plus_rat @ ( fract @ A @ B ) @ ( fract @ C2 @ D ) )
= ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C2 @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% add_rat
thf(fact_9211_le__rat,axiom,
! [B: int,D: int,A: int,C2: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ ( fract @ C2 @ D ) )
= ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C2 @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% le_rat
thf(fact_9212_diff__rat,axiom,
! [B: int,D: int,A: int,C2: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( minus_minus_rat @ ( fract @ A @ B ) @ ( fract @ C2 @ D ) )
= ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C2 @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% diff_rat
thf(fact_9213_eq__rat_I3_J,axiom,
! [A: int,C2: int] :
( ( fract @ zero_zero_int @ A )
= ( fract @ zero_zero_int @ C2 ) ) ).
% eq_rat(3)
thf(fact_9214_eq__rat_I2_J,axiom,
! [A: int] :
( ( fract @ A @ zero_zero_int )
= ( fract @ zero_zero_int @ one_one_int ) ) ).
% eq_rat(2)
thf(fact_9215_Rat__induct__pos,axiom,
! [P2: rat > $o,Q3: rat] :
( ! [A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( P2 @ ( fract @ A3 @ B3 ) ) )
=> ( P2 @ Q3 ) ) ).
% Rat_induct_pos
thf(fact_9216_eq__rat_I1_J,axiom,
! [B: int,D: int,A: int,C2: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( ( fract @ A @ B )
= ( fract @ C2 @ D ) )
= ( ( times_times_int @ A @ D )
= ( times_times_int @ C2 @ B ) ) ) ) ) ).
% eq_rat(1)
thf(fact_9217_mult__rat__cancel,axiom,
! [C2: int,A: int,B: int] :
( ( C2 != zero_zero_int )
=> ( ( fract @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( fract @ A @ B ) ) ) ).
% mult_rat_cancel
thf(fact_9218_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% atLeastSucAtMost_greaterThanAtMost
thf(fact_9219_rat__number__collapse_I1_J,axiom,
! [K2: int] :
( ( fract @ zero_zero_int @ K2 )
= zero_zero_rat ) ).
% rat_number_collapse(1)
thf(fact_9220_rat__number__collapse_I6_J,axiom,
! [K2: int] :
( ( fract @ K2 @ zero_zero_int )
= zero_zero_rat ) ).
% rat_number_collapse(6)
thf(fact_9221_quotient__of__eq,axiom,
! [A: int,B: int,P: int,Q3: int] :
( ( ( quotient_of @ ( fract @ A @ B ) )
= ( product_Pair_int_int @ P @ Q3 ) )
=> ( ( fract @ P @ Q3 )
= ( fract @ A @ B ) ) ) ).
% quotient_of_eq
thf(fact_9222_normalize__eq,axiom,
! [A: int,B: int,P: int,Q3: int] :
( ( ( normalize @ ( product_Pair_int_int @ A @ B ) )
= ( product_Pair_int_int @ P @ Q3 ) )
=> ( ( fract @ P @ Q3 )
= ( fract @ A @ B ) ) ) ).
% normalize_eq
thf(fact_9223_finite__int__iff__bounded,axiom,
( finite_finite_int
= ( ^ [S6: set_int] :
? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S6 ) @ ( set_ord_lessThan_int @ K3 ) ) ) ) ).
% finite_int_iff_bounded
thf(fact_9224_finite__int__iff__bounded__le,axiom,
( finite_finite_int
= ( ^ [S6: set_int] :
? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S6 ) @ ( set_ord_atMost_int @ K3 ) ) ) ) ).
% finite_int_iff_bounded_le
thf(fact_9225_Zero__rat__def,axiom,
( zero_zero_rat
= ( fract @ zero_zero_int @ one_one_int ) ) ).
% Zero_rat_def
thf(fact_9226_image__int__atLeastAtMost,axiom,
! [A: nat,B: nat] :
( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% image_int_atLeastAtMost
thf(fact_9227_image__int__atLeastLessThan,axiom,
! [A: nat,B: nat] :
( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A @ B ) )
= ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% image_int_atLeastLessThan
thf(fact_9228_quotient__of__Fract,axiom,
! [A: int,B: int] :
( ( quotient_of @ ( fract @ A @ B ) )
= ( normalize @ ( product_Pair_int_int @ A @ B ) ) ) ).
% quotient_of_Fract
thf(fact_9229_Fract__less__zero__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_rat @ ( fract @ A @ B ) @ zero_zero_rat )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% Fract_less_zero_iff
thf(fact_9230_zero__less__Fract__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% zero_less_Fract_iff
thf(fact_9231_one__less__Fract__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% one_less_Fract_iff
thf(fact_9232_Fract__less__one__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_rat @ ( fract @ A @ B ) @ one_one_rat )
= ( ord_less_int @ A @ B ) ) ) ).
% Fract_less_one_iff
thf(fact_9233_Fract__add__one,axiom,
! [N: int,M2: int] :
( ( N != zero_zero_int )
=> ( ( fract @ ( plus_plus_int @ M2 @ N ) @ N )
= ( plus_plus_rat @ ( fract @ M2 @ N ) @ one_one_rat ) ) ) ).
% Fract_add_one
thf(fact_9234_infinite__int__iff__infinite__nat__abs,axiom,
! [S2: set_int] :
( ( ~ ( finite_finite_int @ S2 ) )
= ( ~ ( finite_finite_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ S2 ) ) ) ) ).
% infinite_int_iff_infinite_nat_abs
thf(fact_9235_sorted__list__of__set__greaterThanAtMost,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ ( suc @ I ) @ J )
=> ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
= ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanAtMost
thf(fact_9236_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] :
( ( image_int_int
@ ^ [X5: int] : ( plus_plus_int @ X5 @ L )
@ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
= ( set_or4662586982721622107an_int @ L @ U ) ) ).
% image_add_int_atLeastLessThan
thf(fact_9237_zero__le__Fract__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% zero_le_Fract_iff
thf(fact_9238_Fract__le__zero__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ zero_zero_rat )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% Fract_le_zero_iff
thf(fact_9239_one__le__Fract__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% one_le_Fract_iff
thf(fact_9240_Fract__le__one__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% Fract_le_one_iff
thf(fact_9241_image__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
= ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).
% image_atLeastZeroLessThan_int
thf(fact_9242_finite__greaterThanAtMost__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( set_or6656581121297822940st_int @ L @ U ) ) ).
% finite_greaterThanAtMost_int
thf(fact_9243_card__greaterThanAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card_int @ ( set_or6656581121297822940st_int @ L @ U ) )
= ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).
% card_greaterThanAtMost_int
thf(fact_9244_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
= ( set_or6656581121297822940st_int @ L @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_9245_greaterThanAtMost__upto,axiom,
( set_or6656581121297822940st_int
= ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) ) ) ).
% greaterThanAtMost_upto
thf(fact_9246_UN__lessThan__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_lessThan_UNIV
thf(fact_9247_UN__atMost__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atMost_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_atMost_UNIV
thf(fact_9248_sup__int__def,axiom,
sup_sup_int = ord_max_int ).
% sup_int_def
thf(fact_9249_sup__nat__def,axiom,
sup_sup_nat = ord_max_nat ).
% sup_nat_def
thf(fact_9250_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_9251_nat__not__finite,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% nat_not_finite
thf(fact_9252_atLeastLessThan__add__Un,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K2 ) )
= ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K2 ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_9253_range__mod,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( image_nat_nat
@ ^ [M5: nat] : ( modulo_modulo_nat @ M5 @ N )
@ top_top_set_nat )
= ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% range_mod
thf(fact_9254_suminf__eq__SUP__real,axiom,
! [X8: nat > real] :
( ( summable_real @ X8 )
=> ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X8 @ I2 ) )
=> ( ( suminf_real @ X8 )
= ( comple1385675409528146559p_real
@ ( image_nat_real
@ ^ [I3: nat] : ( groups6591440286371151544t_real @ X8 @ ( set_ord_lessThan_nat @ I3 ) )
@ top_top_set_nat ) ) ) ) ) ).
% suminf_eq_SUP_real
thf(fact_9255_UNIV__nat__eq,axiom,
( top_top_set_nat
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).
% UNIV_nat_eq
thf(fact_9256_min__weak__def,axiom,
( fun_min_weak
= ( sup_su5525570899277871387at_nat @ ( min_ex6901939911449802026at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).
% min_weak_def
thf(fact_9257_card__UNIV__unit,axiom,
( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
= one_one_nat ) ).
% card_UNIV_unit
thf(fact_9258_range__mult,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
= ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
& ( ( A != zero_zero_real )
=> ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
= top_top_set_real ) ) ) ).
% range_mult
thf(fact_9259_infinite__UNIV__int,axiom,
~ ( finite_finite_int @ top_top_set_int ) ).
% infinite_UNIV_int
thf(fact_9260_bij__list__encode,axiom,
bij_be8532844293280997160at_nat @ nat_list_encode @ top_top_set_list_nat @ top_top_set_nat ).
% bij_list_encode
thf(fact_9261_surj__list__encode,axiom,
( ( image_list_nat_nat @ nat_list_encode @ top_top_set_list_nat )
= top_top_set_nat ) ).
% surj_list_encode
thf(fact_9262_bij__prod__encode,axiom,
bij_be5333170631980326235at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat @ top_top_set_nat ).
% bij_prod_encode
thf(fact_9263_surj__prod__encode,axiom,
( ( image_2486076414777270412at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat )
= top_top_set_nat ) ).
% surj_prod_encode
thf(fact_9264_int__in__range__abs,axiom,
! [N: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N ) @ ( image_int_int @ abs_abs_int @ top_top_set_int ) ) ).
% int_in_range_abs
thf(fact_9265_max__weak__def,axiom,
( fun_max_weak
= ( sup_su5525570899277871387at_nat @ ( max_ex8135407076693332796at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).
% max_weak_def
thf(fact_9266_root__def,axiom,
( root
= ( ^ [N4: nat,X5: real] :
( if_real @ ( N4 = zero_zero_nat ) @ zero_zero_real
@ ( the_in5290026491893676941l_real @ top_top_set_real
@ ^ [Y5: real] : ( times_times_real @ ( sgn_sgn_real @ Y5 ) @ ( power_power_real @ ( abs_abs_real @ Y5 ) @ N4 ) )
@ X5 ) ) ) ) ).
% root_def
thf(fact_9267_DERIV__real__root__generic,axiom,
! [N: nat,X3: real,D5: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( X3 != zero_zero_real )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( D5
= ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X3 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_real @ X3 @ zero_zero_real )
=> ( D5
= ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X3 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
=> ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( D5
= ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X3 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
=> ( has_fi5821293074295781190e_real @ ( root @ N ) @ D5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ) ) ).
% DERIV_real_root_generic
thf(fact_9268_DERIV__even__real__root,axiom,
! [N: nat,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_real @ X3 @ zero_zero_real )
=> ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X3 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ).
% DERIV_even_real_root
thf(fact_9269_has__real__derivative__neg__dec__right,axiom,
! [F: real > real,L: real,X3: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S2 ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [D2: real] :
( ( ord_less_real @ zero_zero_real @ D2 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( member_real @ ( plus_plus_real @ X3 @ H4 ) @ S2 )
=> ( ( ord_less_real @ H4 @ D2 )
=> ( ord_less_real @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_right
thf(fact_9270_has__real__derivative__pos__inc__right,axiom,
! [F: real > real,L: real,X3: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S2 ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [D2: real] :
( ( ord_less_real @ zero_zero_real @ D2 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( member_real @ ( plus_plus_real @ X3 @ H4 ) @ S2 )
=> ( ( ord_less_real @ H4 @ D2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_right
thf(fact_9271_has__real__derivative__pos__inc__left,axiom,
! [F: real > real,L: real,X3: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S2 ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [D2: real] :
( ( ord_less_real @ zero_zero_real @ D2 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( member_real @ ( minus_minus_real @ X3 @ H4 ) @ S2 )
=> ( ( ord_less_real @ H4 @ D2 )
=> ( ord_less_real @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_left
thf(fact_9272_has__real__derivative__neg__dec__left,axiom,
! [F: real > real,L: real,X3: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S2 ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [D2: real] :
( ( ord_less_real @ zero_zero_real @ D2 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( member_real @ ( minus_minus_real @ X3 @ H4 ) @ S2 )
=> ( ( ord_less_real @ H4 @ D2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_left
thf(fact_9273_DERIV__isconst__all,axiom,
! [F: real > real,X3: real,Y: real] :
( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
=> ( ( F @ X3 )
= ( F @ Y ) ) ) ).
% DERIV_isconst_all
thf(fact_9274_DERIV__pos__inc__right,axiom,
! [F: real > real,L: real,X3: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [D2: real] :
( ( ord_less_real @ zero_zero_real @ D2 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( ord_less_real @ H4 @ D2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ).
% DERIV_pos_inc_right
thf(fact_9275_DERIV__neg__dec__right,axiom,
! [F: real > real,L: real,X3: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [D2: real] :
( ( ord_less_real @ zero_zero_real @ D2 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( ord_less_real @ H4 @ D2 )
=> ( ord_less_real @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ).
% DERIV_neg_dec_right
thf(fact_9276_DERIV__neg__dec__left,axiom,
! [F: real > real,L: real,X3: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [D2: real] :
( ( ord_less_real @ zero_zero_real @ D2 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( ord_less_real @ H4 @ D2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ).
% DERIV_neg_dec_left
thf(fact_9277_DERIV__pos__inc__left,axiom,
! [F: real > real,L: real,X3: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [D2: real] :
( ( ord_less_real @ zero_zero_real @ D2 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( ord_less_real @ H4 @ D2 )
=> ( ord_less_real @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ).
% DERIV_pos_inc_left
thf(fact_9278_DERIV__isconst3,axiom,
! [A: real,B: real,X3: real,Y: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
=> ( ( member_real @ Y @ ( set_or1633881224788618240n_real @ A @ B ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
=> ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
=> ( ( F @ X3 )
= ( F @ Y ) ) ) ) ) ) ).
% DERIV_isconst3
thf(fact_9279_DERIV__neg__imp__decreasing,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B )
=> ? [Y6: real] :
( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_real @ Y6 @ zero_zero_real ) ) ) )
=> ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% DERIV_neg_imp_decreasing
thf(fact_9280_DERIV__pos__imp__increasing,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B )
=> ? [Y6: real] :
( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_real @ zero_zero_real @ Y6 ) ) ) )
=> ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% DERIV_pos_imp_increasing
thf(fact_9281_DERIV__nonneg__imp__nondecreasing,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B )
=> ? [Y6: real] :
( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_eq_real @ zero_zero_real @ Y6 ) ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% DERIV_nonneg_imp_nondecreasing
thf(fact_9282_DERIV__nonpos__imp__nonincreasing,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B )
=> ? [Y6: real] :
( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_eq_real @ Y6 @ zero_zero_real ) ) ) )
=> ( ord_less_eq_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% DERIV_nonpos_imp_nonincreasing
thf(fact_9283_deriv__nonneg__imp__mono,axiom,
! [A: real,B: real,G: real > real,G2: real > real] :
( ! [X4: real] :
( ( member_real @ X4 @ ( set_or1222579329274155063t_real @ A @ B ) )
=> ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( set_or1222579329274155063t_real @ A @ B ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X4 ) ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( G @ A ) @ ( G @ B ) ) ) ) ) ).
% deriv_nonneg_imp_mono
thf(fact_9284_MVT2,axiom,
! [A: real,B: real,F: real > real,F5: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B )
=> ( has_fi5821293074295781190e_real @ F @ ( F5 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
=> ? [Z3: real] :
( ( ord_less_real @ A @ Z3 )
& ( ord_less_real @ Z3 @ B )
& ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
= ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F5 @ Z3 ) ) ) ) ) ) ).
% MVT2
thf(fact_9285_DERIV__local__const,axiom,
! [F: real > real,L: real,X3: real,D: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Y3: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y3 ) ) @ D )
=> ( ( F @ X3 )
= ( F @ Y3 ) ) )
=> ( L = zero_zero_real ) ) ) ) ).
% DERIV_local_const
thf(fact_9286_DERIV__ln,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% DERIV_ln
thf(fact_9287_DERIV__const__average,axiom,
! [A: real,B: real,V: real > real,K2: real] :
( ( A != B )
=> ( ! [X4: real] : ( has_fi5821293074295781190e_real @ V @ K2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
=> ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= ( divide_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% DERIV_const_average
thf(fact_9288_DERIV__local__max,axiom,
! [F: real > real,L: real,X3: real,D: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Y3: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y3 ) ) @ D )
=> ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X3 ) ) )
=> ( L = zero_zero_real ) ) ) ) ).
% DERIV_local_max
thf(fact_9289_DERIV__local__min,axiom,
! [F: real > real,L: real,X3: real,D: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Y3: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y3 ) ) @ D )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( L = zero_zero_real ) ) ) ) ).
% DERIV_local_min
thf(fact_9290_DERIV__ln__divide,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% DERIV_ln_divide
thf(fact_9291_DERIV__pow,axiom,
! [N: nat,X3: real,S3: set_real] :
( has_fi5821293074295781190e_real
@ ^ [X5: real] : ( power_power_real @ X5 @ N )
@ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X3 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
@ ( topolo2177554685111907308n_real @ X3 @ S3 ) ) ).
% DERIV_pow
thf(fact_9292_has__real__derivative__powr,axiom,
! [Z2: real,R2: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( has_fi5821293074295781190e_real
@ ^ [Z6: real] : ( powr_real @ Z6 @ R2 )
@ ( times_times_real @ R2 @ ( powr_real @ Z2 @ ( minus_minus_real @ R2 @ one_one_real ) ) )
@ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) ).
% has_real_derivative_powr
thf(fact_9293_DERIV__series_H,axiom,
! [F: real > nat > real,F5: real > nat > real,X0: real,A: real,B: real,L5: nat > real] :
( ! [N3: nat] :
( has_fi5821293074295781190e_real
@ ^ [X5: real] : ( F @ X5 @ N3 )
@ ( F5 @ X0 @ N3 )
@ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
=> ( summable_real @ ( F @ X4 ) ) )
=> ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B ) )
=> ( ( summable_real @ ( F5 @ X0 ) )
=> ( ( summable_real @ L5 )
=> ( ! [N3: nat,X4: real,Y3: real] :
( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
=> ( ( member_real @ Y3 @ ( set_or1633881224788618240n_real @ A @ B ) )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X4 @ N3 ) @ ( F @ Y3 @ N3 ) ) ) @ ( times_times_real @ ( L5 @ N3 ) @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y3 ) ) ) ) ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X5: real] : ( suminf_real @ ( F @ X5 ) )
@ ( suminf_real @ ( F5 @ X0 ) )
@ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% DERIV_series'
thf(fact_9294_DERIV__log,axiom,
! [X3: real,B: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( has_fi5821293074295781190e_real @ ( log @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X3 ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% DERIV_log
thf(fact_9295_DERIV__fun__powr,axiom,
! [G: real > real,M2: real,X3: real,R2: real] :
( ( has_fi5821293074295781190e_real @ G @ M2 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ ( G @ X3 ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X5: real] : ( powr_real @ ( G @ X5 ) @ R2 )
@ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G @ X3 ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M2 )
@ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).
% DERIV_fun_powr
thf(fact_9296_DERIV__powr,axiom,
! [G: real > real,M2: real,X3: real,F: real > real,R2: real] :
( ( has_fi5821293074295781190e_real @ G @ M2 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ ( G @ X3 ) )
=> ( ( has_fi5821293074295781190e_real @ F @ R2 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X5: real] : ( powr_real @ ( G @ X5 ) @ ( F @ X5 ) )
@ ( times_times_real @ ( powr_real @ ( G @ X3 ) @ ( F @ X3 ) ) @ ( plus_plus_real @ ( times_times_real @ R2 @ ( ln_ln_real @ ( G @ X3 ) ) ) @ ( divide_divide_real @ ( times_times_real @ M2 @ ( F @ X3 ) ) @ ( G @ X3 ) ) ) )
@ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ).
% DERIV_powr
thf(fact_9297_DERIV__real__sqrt,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% DERIV_real_sqrt
thf(fact_9298_DERIV__arctan,axiom,
! [X3: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ).
% DERIV_arctan
thf(fact_9299_arsinh__real__has__field__derivative,axiom,
! [X3: real,A4: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ A4 ) ) ).
% arsinh_real_has_field_derivative
thf(fact_9300_DERIV__real__sqrt__generic,axiom,
! [X3: real,D5: real] :
( ( X3 != zero_zero_real )
=> ( ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( D5
= ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( ( ( ord_less_real @ X3 @ zero_zero_real )
=> ( D5
= ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( has_fi5821293074295781190e_real @ sqrt @ D5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ).
% DERIV_real_sqrt_generic
thf(fact_9301_DERIV__power__series_H,axiom,
! [R: real,F: nat > real,X0: real] :
( ! [X4: real] :
( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
=> ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( F @ N4 ) @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ ( power_power_real @ X4 @ N4 ) ) ) )
=> ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
=> ( ( ord_less_real @ zero_zero_real @ R )
=> ( has_fi5821293074295781190e_real
@ ^ [X5: real] :
( suminf_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ X5 @ ( suc @ N4 ) ) ) )
@ ( suminf_real
@ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( F @ N4 ) @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ ( power_power_real @ X0 @ N4 ) ) )
@ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% DERIV_power_series'
thf(fact_9302_DERIV__real__root,axiom,
! [N: nat,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X3 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).
% DERIV_real_root
thf(fact_9303_Maclaurin__all__le,axiom,
! [Diff: nat > real > real,F: real > real,X3: real,N: nat] :
( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
=> ? [T5: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X3 ) )
& ( ( F @ X3 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X3 @ M5 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ) ).
% Maclaurin_all_le
thf(fact_9304_Maclaurin__all__le__objl,axiom,
! [Diff: nat > real > real,F: real > real,X3: real,N: nat] :
( ( ( ( Diff @ zero_zero_nat )
= F )
& ! [M: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
=> ? [T5: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X3 ) )
& ( ( F @ X3 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X3 @ M5 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ).
% Maclaurin_all_le_objl
thf(fact_9305_DERIV__odd__real__root,axiom,
! [N: nat,X3: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( X3 != zero_zero_real )
=> ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X3 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).
% DERIV_odd_real_root
thf(fact_9306_Maclaurin,axiom,
! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
( ( ord_less_real @ zero_zero_real @ H2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M: nat,T5: real] :
( ( ( ord_less_nat @ M @ N )
& ( ord_less_eq_real @ zero_zero_real @ T5 )
& ( ord_less_eq_real @ T5 @ H2 ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
=> ? [T5: real] :
( ( ord_less_real @ zero_zero_real @ T5 )
& ( ord_less_real @ T5 @ H2 )
& ( ( F @ H2 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H2 @ M5 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin
thf(fact_9307_Maclaurin2,axiom,
! [H2: real,Diff: nat > real > real,F: real > real,N: nat] :
( ( ord_less_real @ zero_zero_real @ H2 )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M: nat,T5: real] :
( ( ( ord_less_nat @ M @ N )
& ( ord_less_eq_real @ zero_zero_real @ T5 )
& ( ord_less_eq_real @ T5 @ H2 ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
=> ? [T5: real] :
( ( ord_less_real @ zero_zero_real @ T5 )
& ( ord_less_eq_real @ T5 @ H2 )
& ( ( F @ H2 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H2 @ M5 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ).
% Maclaurin2
thf(fact_9308_Maclaurin__minus,axiom,
! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
( ( ord_less_real @ H2 @ zero_zero_real )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M: nat,T5: real] :
( ( ( ord_less_nat @ M @ N )
& ( ord_less_eq_real @ H2 @ T5 )
& ( ord_less_eq_real @ T5 @ zero_zero_real ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
=> ? [T5: real] :
( ( ord_less_real @ H2 @ T5 )
& ( ord_less_real @ T5 @ zero_zero_real )
& ( ( F @ H2 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H2 @ M5 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_minus
thf(fact_9309_Maclaurin__all__lt,axiom,
! [Diff: nat > real > real,F: real > real,N: nat,X3: real] :
( ( ( Diff @ zero_zero_nat )
= F )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( X3 != zero_zero_real )
=> ( ! [M: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
=> ? [T5: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T5 ) )
& ( ord_less_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X3 ) )
& ( ( F @ X3 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X3 @ M5 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_all_lt
thf(fact_9310_Maclaurin__bi__le,axiom,
! [Diff: nat > real > real,F: real > real,N: nat,X3: real] :
( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M: nat,T5: real] :
( ( ( ord_less_nat @ M @ N )
& ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X3 ) ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
=> ? [T5: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X3 ) )
& ( ( F @ X3 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X3 @ M5 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ) ).
% Maclaurin_bi_le
thf(fact_9311_Taylor__down,axiom,
! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M: nat,T5: real] :
( ( ( ord_less_nat @ M @ N )
& ( ord_less_eq_real @ A @ T5 )
& ( ord_less_eq_real @ T5 @ B ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
=> ( ( ord_less_real @ A @ C2 )
=> ( ( ord_less_eq_real @ C2 @ B )
=> ? [T5: real] :
( ( ord_less_real @ A @ T5 )
& ( ord_less_real @ T5 @ C2 )
& ( ( F @ A )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C2 ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C2 ) @ M5 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_down
thf(fact_9312_Taylor__up,axiom,
! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C2: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M: nat,T5: real] :
( ( ( ord_less_nat @ M @ N )
& ( ord_less_eq_real @ A @ T5 )
& ( ord_less_eq_real @ T5 @ B ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
=> ( ( ord_less_eq_real @ A @ C2 )
=> ( ( ord_less_real @ C2 @ B )
=> ? [T5: real] :
( ( ord_less_real @ C2 @ T5 )
& ( ord_less_real @ T5 @ B )
& ( ( F @ B )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C2 ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C2 ) @ M5 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_up
thf(fact_9313_Taylor,axiom,
! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C2: real,X3: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M: nat,T5: real] :
( ( ( ord_less_nat @ M @ N )
& ( ord_less_eq_real @ A @ T5 )
& ( ord_less_eq_real @ T5 @ B ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
=> ( ( ord_less_eq_real @ A @ C2 )
=> ( ( ord_less_eq_real @ C2 @ B )
=> ( ( ord_less_eq_real @ A @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B )
=> ( ( X3 != C2 )
=> ? [T5: real] :
( ( ( ord_less_real @ X3 @ C2 )
=> ( ( ord_less_real @ X3 @ T5 )
& ( ord_less_real @ T5 @ C2 ) ) )
& ( ~ ( ord_less_real @ X3 @ C2 )
=> ( ( ord_less_real @ C2 @ T5 )
& ( ord_less_real @ T5 @ X3 ) ) )
& ( ( F @ X3 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C2 ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ X3 @ C2 ) @ M5 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X3 @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Taylor
thf(fact_9314_Maclaurin__lemma2,axiom,
! [N: nat,H2: real,Diff: nat > real > real,K2: nat,B5: real] :
( ! [M: nat,T5: real] :
( ( ( ord_less_nat @ M @ N )
& ( ord_less_eq_real @ zero_zero_real @ T5 )
& ( ord_less_eq_real @ T5 @ H2 ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
=> ( ( N
= ( suc @ K2 ) )
=> ! [M3: nat,T6: real] :
( ( ( ord_less_nat @ M3 @ N )
& ( ord_less_eq_real @ zero_zero_real @ T6 )
& ( ord_less_eq_real @ T6 @ H2 ) )
=> ( has_fi5821293074295781190e_real
@ ^ [U2: real] :
( minus_minus_real @ ( Diff @ M3 @ U2 )
@ ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [P6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M3 @ P6 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ U2 @ P6 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M3 ) ) )
@ ( times_times_real @ B5 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N @ M3 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M3 ) ) ) ) ) )
@ ( minus_minus_real @ ( Diff @ ( suc @ M3 ) @ T6 )
@ ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [P6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M3 ) @ P6 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ T6 @ P6 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) )
@ ( times_times_real @ B5 @ ( divide_divide_real @ ( power_power_real @ T6 @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) ) ) ) )
@ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) ) ) ) ).
% Maclaurin_lemma2
thf(fact_9315_DERIV__arctan__series,axiom,
! [X3: real] :
( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
=> ( has_fi5821293074295781190e_real
@ ^ [X10: real] :
( suminf_real
@ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X10 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
@ ( suminf_real
@ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X3 @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
@ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% DERIV_arctan_series
thf(fact_9316_surj__int__encode,axiom,
( ( image_int_nat @ nat_int_encode @ top_top_set_int )
= top_top_set_nat ) ).
% surj_int_encode
thf(fact_9317_int__encode__eq,axiom,
! [X3: int,Y: int] :
( ( ( nat_int_encode @ X3 )
= ( nat_int_encode @ Y ) )
= ( X3 = Y ) ) ).
% int_encode_eq
thf(fact_9318_isCont__Lb__Ub,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [X4: real] :
( ( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_eq_real @ X4 @ B ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F ) )
=> ? [L6: real,M9: real] :
( ! [X: real] :
( ( ( ord_less_eq_real @ A @ X )
& ( ord_less_eq_real @ X @ B ) )
=> ( ( ord_less_eq_real @ L6 @ ( F @ X ) )
& ( ord_less_eq_real @ ( F @ X ) @ M9 ) ) )
& ! [Y6: real] :
( ( ( ord_less_eq_real @ L6 @ Y6 )
& ( ord_less_eq_real @ Y6 @ M9 ) )
=> ? [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_eq_real @ X4 @ B )
& ( ( F @ X4 )
= Y6 ) ) ) ) ) ) ).
% isCont_Lb_Ub
thf(fact_9319_LIM__fun__less__zero,axiom,
! [F: real > real,L: real,C2: real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ! [X: real] :
( ( ( X != C2 )
& ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C2 @ X ) ) @ R3 ) )
=> ( ord_less_real @ ( F @ X ) @ zero_zero_real ) ) ) ) ) ).
% LIM_fun_less_zero
thf(fact_9320_LIM__fun__not__zero,axiom,
! [F: real > real,L: real,C2: real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
=> ( ( L != zero_zero_real )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ! [X: real] :
( ( ( X != C2 )
& ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C2 @ X ) ) @ R3 ) )
=> ( ( F @ X )
!= zero_zero_real ) ) ) ) ) ).
% LIM_fun_not_zero
thf(fact_9321_LIM__fun__gt__zero,axiom,
! [F: real > real,L: real,C2: real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ! [X: real] :
( ( ( X != C2 )
& ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C2 @ X ) ) @ R3 ) )
=> ( ord_less_real @ zero_zero_real @ ( F @ X ) ) ) ) ) ) ).
% LIM_fun_gt_zero
thf(fact_9322_isCont__inverse__function2,axiom,
! [A: real,X3: real,B: real,G: real > real,F: real > real] :
( ( ord_less_real @ A @ X3 )
=> ( ( ord_less_real @ X3 @ B )
=> ( ! [Z3: real] :
( ( ord_less_eq_real @ A @ Z3 )
=> ( ( ord_less_eq_real @ Z3 @ B )
=> ( ( G @ ( F @ Z3 ) )
= Z3 ) ) )
=> ( ! [Z3: real] :
( ( ord_less_eq_real @ A @ Z3 )
=> ( ( ord_less_eq_real @ Z3 @ B )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X3 ) @ top_top_set_real ) @ G ) ) ) ) ) ).
% isCont_inverse_function2
thf(fact_9323_isCont__ln,axiom,
! [X3: real] :
( ( X3 != zero_zero_real )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ ln_ln_real ) ) ).
% isCont_ln
thf(fact_9324_bij__int__encode,axiom,
bij_betw_int_nat @ nat_int_encode @ top_top_set_int @ top_top_set_nat ).
% bij_int_encode
thf(fact_9325_LIM__cos__div__sin,axiom,
( filterlim_real_real
@ ^ [X5: real] : ( divide_divide_real @ ( cos_real @ X5 ) @ ( sin_real @ X5 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).
% LIM_cos_div_sin
thf(fact_9326_DERIV__inverse__function,axiom,
! [F: real > real,D5: real,G: real > real,X3: real,A: real,B: real] :
( ( has_fi5821293074295781190e_real @ F @ D5 @ ( topolo2177554685111907308n_real @ ( G @ X3 ) @ top_top_set_real ) )
=> ( ( D5 != zero_zero_real )
=> ( ( ord_less_real @ A @ X3 )
=> ( ( ord_less_real @ X3 @ B )
=> ( ! [Y3: real] :
( ( ord_less_real @ A @ Y3 )
=> ( ( ord_less_real @ Y3 @ B )
=> ( ( F @ ( G @ Y3 ) )
= Y3 ) ) )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ G )
=> ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D5 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% DERIV_inverse_function
thf(fact_9327_LIM__less__bound,axiom,
! [B: real,X3: real,F: real > real] :
( ( ord_less_real @ B @ X3 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ B @ X3 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) ) ) ) ).
% LIM_less_bound
thf(fact_9328_isCont__inverse__function,axiom,
! [D: real,X3: real,G: real > real,F: real > real] :
( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Z3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z3 @ X3 ) ) @ D )
=> ( ( G @ ( F @ Z3 ) )
= Z3 ) )
=> ( ! [Z3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z3 @ X3 ) ) @ D )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X3 ) @ top_top_set_real ) @ G ) ) ) ) ).
% isCont_inverse_function
thf(fact_9329_GMVT_H,axiom,
! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F5: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [Z3: real] :
( ( ord_less_eq_real @ A @ Z3 )
=> ( ( ord_less_eq_real @ Z3 @ B )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) ) )
=> ( ! [Z3: real] :
( ( ord_less_eq_real @ A @ Z3 )
=> ( ( ord_less_eq_real @ Z3 @ B )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ G ) ) )
=> ( ! [Z3: real] :
( ( ord_less_real @ A @ Z3 )
=> ( ( ord_less_real @ Z3 @ B )
=> ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z3 ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) )
=> ( ! [Z3: real] :
( ( ord_less_real @ A @ Z3 )
=> ( ( ord_less_real @ Z3 @ B )
=> ( has_fi5821293074295781190e_real @ F @ ( F5 @ Z3 ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) )
=> ? [C: real] :
( ( ord_less_real @ A @ C )
& ( ord_less_real @ C @ B )
& ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C ) )
= ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F5 @ C ) ) ) ) ) ) ) ) ) ).
% GMVT'
thf(fact_9330_summable__Leibniz_I2_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ( topolo6980174941875973593q_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
=> ! [N6: nat] :
( member_real
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
@ ( set_or1222579329274155063t_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% summable_Leibniz(2)
thf(fact_9331_summable__Leibniz_I3_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ( topolo6980174941875973593q_real @ A )
=> ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
=> ! [N6: nat] :
( member_real
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
@ ( set_or1222579329274155063t_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) ) ) ) ) ) ) ).
% summable_Leibniz(3)
thf(fact_9332_summable__Leibniz_H_I5_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
=> ( filterlim_nat_real
@ ^ [N4: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
@ ( topolo2815343760600316023s_real
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
@ at_top_nat ) ) ) ) ).
% summable_Leibniz'(5)
thf(fact_9333_filterlim__Suc,axiom,
filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).
% filterlim_Suc
thf(fact_9334_mult__nat__left__at__top,axiom,
! [C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( filterlim_nat_nat @ ( times_times_nat @ C2 ) @ at_top_nat @ at_top_nat ) ) ).
% mult_nat_left_at_top
thf(fact_9335_mult__nat__right__at__top,axiom,
! [C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( filterlim_nat_nat
@ ^ [X5: nat] : ( times_times_nat @ X5 @ C2 )
@ at_top_nat
@ at_top_nat ) ) ).
% mult_nat_right_at_top
thf(fact_9336_monoseq__convergent,axiom,
! [X8: nat > real,B5: real] :
( ( topolo6980174941875973593q_real @ X8 )
=> ( ! [I2: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X8 @ I2 ) ) @ B5 )
=> ~ ! [L6: real] :
~ ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat ) ) ) ).
% monoseq_convergent
thf(fact_9337_nested__sequence__unique,axiom,
! [F: nat > real,G: nat > real] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N3 ) ) @ ( G @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
=> ( ( filterlim_nat_real
@ ^ [N4: nat] : ( minus_minus_real @ ( F @ N4 ) @ ( G @ N4 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat )
=> ? [L4: real] :
( ! [N6: nat] : ( ord_less_eq_real @ ( F @ N6 ) @ L4 )
& ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
& ! [N6: nat] : ( ord_less_eq_real @ L4 @ ( G @ N6 ) )
& ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).
% nested_sequence_unique
thf(fact_9338_LIMSEQ__inverse__zero,axiom,
! [X8: nat > real] :
( ! [R3: real] :
? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ N8 @ N3 )
=> ( ord_less_real @ R3 @ ( X8 @ N3 ) ) )
=> ( filterlim_nat_real
@ ^ [N4: nat] : ( inverse_inverse_real @ ( X8 @ N4 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% LIMSEQ_inverse_zero
thf(fact_9339_lim__inverse__n_H,axiom,
( filterlim_nat_real
@ ^ [N4: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N4 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ).
% lim_inverse_n'
thf(fact_9340_LIMSEQ__root__const,axiom,
! [C2: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( filterlim_nat_real
@ ^ [N4: nat] : ( root @ N4 @ C2 )
@ ( topolo2815343760600316023s_real @ one_one_real )
@ at_top_nat ) ) ).
% LIMSEQ_root_const
thf(fact_9341_LIMSEQ__inverse__real__of__nat,axiom,
( filterlim_nat_real
@ ^ [N4: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat
thf(fact_9342_LIMSEQ__inverse__real__of__nat__add,axiom,
! [R2: real] :
( filterlim_nat_real
@ ^ [N4: nat] : ( plus_plus_real @ R2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) )
@ ( topolo2815343760600316023s_real @ R2 )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat_add
thf(fact_9343_increasing__LIMSEQ,axiom,
! [F: nat > real,L: real] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ L )
=> ( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [N6: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N6 ) @ E ) ) )
=> ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).
% increasing_LIMSEQ
thf(fact_9344_LIMSEQ__realpow__zero,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ one_one_real )
=> ( filterlim_nat_real @ ( power_power_real @ X3 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% LIMSEQ_realpow_zero
thf(fact_9345_LIMSEQ__divide__realpow__zero,axiom,
! [X3: real,A: real] :
( ( ord_less_real @ one_one_real @ X3 )
=> ( filterlim_nat_real
@ ^ [N4: nat] : ( divide_divide_real @ A @ ( power_power_real @ X3 @ N4 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% LIMSEQ_divide_realpow_zero
thf(fact_9346_LIMSEQ__abs__realpow__zero2,axiom,
! [C2: real] :
( ( ord_less_real @ ( abs_abs_real @ C2 ) @ one_one_real )
=> ( filterlim_nat_real @ ( power_power_real @ C2 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% LIMSEQ_abs_realpow_zero2
thf(fact_9347_LIMSEQ__abs__realpow__zero,axiom,
! [C2: real] :
( ( ord_less_real @ ( abs_abs_real @ C2 ) @ one_one_real )
=> ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C2 ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% LIMSEQ_abs_realpow_zero
thf(fact_9348_LIMSEQ__inverse__realpow__zero,axiom,
! [X3: real] :
( ( ord_less_real @ one_one_real @ X3 )
=> ( filterlim_nat_real
@ ^ [N4: nat] : ( inverse_inverse_real @ ( power_power_real @ X3 @ N4 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% LIMSEQ_inverse_realpow_zero
thf(fact_9349_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
! [R2: real] :
( filterlim_nat_real
@ ^ [N4: nat] : ( plus_plus_real @ R2 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) ) )
@ ( topolo2815343760600316023s_real @ R2 )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_9350_tendsto__exp__limit__sequentially,axiom,
! [X3: real] :
( filterlim_nat_real
@ ^ [N4: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X3 @ ( semiri5074537144036343181t_real @ N4 ) ) ) @ N4 )
@ ( topolo2815343760600316023s_real @ ( exp_real @ X3 ) )
@ at_top_nat ) ).
% tendsto_exp_limit_sequentially
thf(fact_9351_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [R2: real] :
( filterlim_nat_real
@ ^ [N4: nat] : ( times_times_real @ R2 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) ) ) )
@ ( topolo2815343760600316023s_real @ R2 )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_9352_summable__Leibniz_I1_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ( topolo6980174941875973593q_real @ A )
=> ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( A @ N4 ) ) ) ) ) ).
% summable_Leibniz(1)
thf(fact_9353_summable,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
=> ( summable_real
@ ^ [N4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( A @ N4 ) ) ) ) ) ) ).
% summable
thf(fact_9354_cos__limit__1,axiom,
! [Theta: nat > real] :
( ( filterlim_nat_real
@ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
@ ( topolo2815343760600316023s_real @ one_one_real )
@ at_top_nat )
=> ? [K: nat > int] :
( filterlim_nat_real
@ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% cos_limit_1
thf(fact_9355_summable__Leibniz_I4_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ( topolo6980174941875973593q_real @ A )
=> ( filterlim_nat_real
@ ^ [N4: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
@ ( topolo2815343760600316023s_real
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
@ at_top_nat ) ) ) ).
% summable_Leibniz(4)
thf(fact_9356_zeroseq__arctan__series,axiom,
! [X3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
=> ( filterlim_nat_real
@ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% zeroseq_arctan_series
thf(fact_9357_summable__Leibniz_H_I3_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
=> ( filterlim_nat_real
@ ^ [N4: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
@ ( topolo2815343760600316023s_real
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
@ at_top_nat ) ) ) ) ).
% summable_Leibniz'(3)
thf(fact_9358_summable__Leibniz_H_I2_J,axiom,
! [A: nat > real,N: nat] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
=> ( ord_less_eq_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) ) ) ) ) ).
% summable_Leibniz'(2)
thf(fact_9359_sums__alternating__upper__lower,axiom,
! [A: nat > real] :
( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
=> ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ? [L4: real] :
( ! [N6: nat] :
( ord_less_eq_real
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
@ L4 )
& ( filterlim_nat_real
@ ^ [N4: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
@ ( topolo2815343760600316023s_real @ L4 )
@ at_top_nat )
& ! [N6: nat] :
( ord_less_eq_real @ L4
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) )
& ( filterlim_nat_real
@ ^ [N4: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
@ ( topolo2815343760600316023s_real @ L4 )
@ at_top_nat ) ) ) ) ) ).
% sums_alternating_upper_lower
thf(fact_9360_summable__Leibniz_I5_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ( topolo6980174941875973593q_real @ A )
=> ( filterlim_nat_real
@ ^ [N4: nat] :
( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
@ ( topolo2815343760600316023s_real
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
@ at_top_nat ) ) ) ).
% summable_Leibniz(5)
thf(fact_9361_summable__Leibniz_H_I4_J,axiom,
! [A: nat > real,N: nat] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
=> ( ord_less_eq_real
@ ( suminf_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).
% summable_Leibniz'(4)
thf(fact_9362_tendsto__exp__limit__at__right,axiom,
! [X3: real] :
( filterlim_real_real
@ ^ [Y5: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X3 @ Y5 ) ) @ ( divide_divide_real @ one_one_real @ Y5 ) )
@ ( topolo2815343760600316023s_real @ ( exp_real @ X3 ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% tendsto_exp_limit_at_right
thf(fact_9363_tendsto__arcosh__at__left__1,axiom,
filterlim_real_real @ arcosh_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5849166863359141190n_real @ one_one_real ) ) ).
% tendsto_arcosh_at_left_1
thf(fact_9364_atLeast__0,axiom,
( ( set_ord_atLeast_nat @ zero_zero_nat )
= top_top_set_nat ) ).
% atLeast_0
thf(fact_9365_atLeast__Suc__greaterThan,axiom,
! [K2: nat] :
( ( set_ord_atLeast_nat @ ( suc @ K2 ) )
= ( set_or1210151606488870762an_nat @ K2 ) ) ).
% atLeast_Suc_greaterThan
thf(fact_9366_INT__greaterThan__UNIV,axiom,
( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
= bot_bot_set_nat ) ).
% INT_greaterThan_UNIV
thf(fact_9367_greaterThan__0,axiom,
( ( set_or1210151606488870762an_nat @ zero_zero_nat )
= ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% greaterThan_0
thf(fact_9368_greaterThan__Suc,axiom,
! [K2: nat] :
( ( set_or1210151606488870762an_nat @ ( suc @ K2 ) )
= ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K2 ) @ ( insert_nat @ ( suc @ K2 ) @ bot_bot_set_nat ) ) ) ).
% greaterThan_Suc
thf(fact_9369_UN__atLeast__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atLeast_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_atLeast_UNIV
thf(fact_9370_atLeast__Suc,axiom,
! [K2: nat] :
( ( set_ord_atLeast_nat @ ( suc @ K2 ) )
= ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K2 ) @ ( insert_nat @ K2 @ bot_bot_set_nat ) ) ) ).
% atLeast_Suc
thf(fact_9371_filterlim__pow__at__bot__odd,axiom,
! [N: nat,F: real > real,F4: filter_real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
=> ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( filterlim_real_real
@ ^ [X5: real] : ( power_power_real @ ( F @ X5 ) @ N )
@ at_bot_real
@ F4 ) ) ) ) ).
% filterlim_pow_at_bot_odd
thf(fact_9372_at__bot__le__at__infinity,axiom,
ord_le4104064031414453916r_real @ at_bot_real @ at_infinity_real ).
% at_bot_le_at_infinity
thf(fact_9373_exp__at__bot,axiom,
filterlim_real_real @ exp_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_bot_real ).
% exp_at_bot
thf(fact_9374_filterlim__inverse__at__bot__neg,axiom,
filterlim_real_real @ inverse_inverse_real @ at_bot_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5984915006950818249n_real @ zero_zero_real ) ) ).
% filterlim_inverse_at_bot_neg
thf(fact_9375_ln__at__0,axiom,
filterlim_real_real @ ln_ln_real @ at_bot_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ).
% ln_at_0
thf(fact_9376_DERIV__pos__imp__increasing__at__bot,axiom,
! [B: real,F: real > real,Flim: real] :
( ! [X4: real] :
( ( ord_less_eq_real @ X4 @ B )
=> ? [Y6: real] :
( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_real @ zero_zero_real @ Y6 ) ) )
=> ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
=> ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% DERIV_pos_imp_increasing_at_bot
thf(fact_9377_Gcd__eq__Max,axiom,
! [M6: set_nat] :
( ( finite_finite_nat @ M6 )
=> ( ( M6 != bot_bot_set_nat )
=> ( ~ ( member_nat @ zero_zero_nat @ M6 )
=> ( ( gcd_Gcd_nat @ M6 )
= ( lattic8265883725875713057ax_nat
@ ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [M5: nat] :
( collect_nat
@ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M5 ) )
@ M6 ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_9378_eventually__sequentially__Suc,axiom,
! [P2: nat > $o] :
( ( eventually_nat
@ ^ [I3: nat] : ( P2 @ ( suc @ I3 ) )
@ at_top_nat )
= ( eventually_nat @ P2 @ at_top_nat ) ) ).
% eventually_sequentially_Suc
thf(fact_9379_eventually__sequentially__seg,axiom,
! [P2: nat > $o,K2: nat] :
( ( eventually_nat
@ ^ [N4: nat] : ( P2 @ ( plus_plus_nat @ N4 @ K2 ) )
@ at_top_nat )
= ( eventually_nat @ P2 @ at_top_nat ) ) ).
% eventually_sequentially_seg
thf(fact_9380_Max__divisors__self__nat,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ N ) ) )
= N ) ) ).
% Max_divisors_self_nat
thf(fact_9381_eventually__sequentiallyI,axiom,
! [C2: nat,P2: nat > $o] :
( ! [X4: nat] :
( ( ord_less_eq_nat @ C2 @ X4 )
=> ( P2 @ X4 ) )
=> ( eventually_nat @ P2 @ at_top_nat ) ) ).
% eventually_sequentiallyI
thf(fact_9382_eventually__sequentially,axiom,
! [P2: nat > $o] :
( ( eventually_nat @ P2 @ at_top_nat )
= ( ? [N5: nat] :
! [N4: nat] :
( ( ord_less_eq_nat @ N5 @ N4 )
=> ( P2 @ N4 ) ) ) ) ).
% eventually_sequentially
thf(fact_9383_le__sequentially,axiom,
! [F4: filter_nat] :
( ( ord_le2510731241096832064er_nat @ F4 @ at_top_nat )
= ( ! [N5: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N5 ) @ F4 ) ) ) ).
% le_sequentially
thf(fact_9384_sequentially__offset,axiom,
! [P2: nat > $o,K2: nat] :
( ( eventually_nat @ P2 @ at_top_nat )
=> ( eventually_nat
@ ^ [I3: nat] : ( P2 @ ( plus_plus_nat @ I3 @ K2 ) )
@ at_top_nat ) ) ).
% sequentially_offset
thf(fact_9385_eventually__at__right__to__0,axiom,
! [P2: real > $o,A: real] :
( ( eventually_real @ P2 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
= ( eventually_real
@ ^ [X5: real] : ( P2 @ ( plus_plus_real @ X5 @ A ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% eventually_at_right_to_0
thf(fact_9386_card__le__Suc__Max,axiom,
! [S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ S2 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S2 ) ) ) ) ).
% card_le_Suc_Max
thf(fact_9387_Sup__nat__def,axiom,
( complete_Sup_Sup_nat
= ( ^ [X7: set_nat] : ( if_nat @ ( X7 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X7 ) ) ) ) ).
% Sup_nat_def
thf(fact_9388_divide__nat__def,axiom,
( divide_divide_nat
= ( ^ [M5: nat,N4: nat] :
( if_nat @ ( N4 = zero_zero_nat ) @ zero_zero_nat
@ ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N4 ) @ M5 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_9389_gcd__is__Max__divisors__nat,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( gcd_gcd_nat @ M2 @ N )
= ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [D4: nat] :
( ( dvd_dvd_nat @ D4 @ M2 )
& ( dvd_dvd_nat @ D4 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_nat
thf(fact_9390_lhopital__left,axiom,
! [F: real > real,X3: real,G: real > real,G2: real > real,F5: real > real,F4: filter_real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X3 @ ( set_or5984915006950818249n_real @ X3 ) ) )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X3 @ ( set_or5984915006950818249n_real @ X3 ) ) )
=> ( ( eventually_real
@ ^ [X5: real] :
( ( G @ X5 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X3 @ ( set_or5984915006950818249n_real @ X3 ) ) )
=> ( ( eventually_real
@ ^ [X5: real] :
( ( G2 @ X5 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X3 @ ( set_or5984915006950818249n_real @ X3 ) ) )
=> ( ( eventually_real
@ ^ [X5: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X3 @ ( set_or5984915006950818249n_real @ X3 ) ) )
=> ( ( eventually_real
@ ^ [X5: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X3 @ ( set_or5984915006950818249n_real @ X3 ) ) )
=> ( ( filterlim_real_real
@ ^ [X5: real] : ( divide_divide_real @ ( F5 @ X5 ) @ ( G2 @ X5 ) )
@ F4
@ ( topolo2177554685111907308n_real @ X3 @ ( set_or5984915006950818249n_real @ X3 ) ) )
=> ( filterlim_real_real
@ ^ [X5: real] : ( divide_divide_real @ ( F @ X5 ) @ ( G @ X5 ) )
@ F4
@ ( topolo2177554685111907308n_real @ X3 @ ( set_or5984915006950818249n_real @ X3 ) ) ) ) ) ) ) ) ) ) ).
% lhopital_left
thf(fact_9391_lhopital,axiom,
! [F: real > real,X3: real,G: real > real,G2: real > real,F5: real > real,F4: filter_real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X5: real] :
( ( G @ X5 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X5: real] :
( ( G2 @ X5 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X5: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X5: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( filterlim_real_real
@ ^ [X5: real] : ( divide_divide_real @ ( F5 @ X5 ) @ ( G2 @ X5 ) )
@ F4
@ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( filterlim_real_real
@ ^ [X5: real] : ( divide_divide_real @ ( F @ X5 ) @ ( G @ X5 ) )
@ F4
@ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ) ) ) ) ).
% lhopital
thf(fact_9392_lhopital__right__0,axiom,
! [F0: real > real,G0: real > real,G2: real > real,F5: real > real,F4: filter_real] :
( ( filterlim_real_real @ F0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( filterlim_real_real @ G0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( eventually_real
@ ^ [X5: real] :
( ( G0 @ X5 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( eventually_real
@ ^ [X5: real] :
( ( G2 @ X5 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( eventually_real
@ ^ [X5: real] : ( has_fi5821293074295781190e_real @ F0 @ ( F5 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( eventually_real
@ ^ [X5: real] : ( has_fi5821293074295781190e_real @ G0 @ ( G2 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( filterlim_real_real
@ ^ [X5: real] : ( divide_divide_real @ ( F5 @ X5 ) @ ( G2 @ X5 ) )
@ F4
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( filterlim_real_real
@ ^ [X5: real] : ( divide_divide_real @ ( F0 @ X5 ) @ ( G0 @ X5 ) )
@ F4
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ) ) ).
% lhopital_right_0
thf(fact_9393_lhopital__right,axiom,
! [F: real > real,X3: real,G: real > real,G2: real > real,F5: real > real,F4: filter_real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X3 @ ( set_or5849166863359141190n_real @ X3 ) ) )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X3 @ ( set_or5849166863359141190n_real @ X3 ) ) )
=> ( ( eventually_real
@ ^ [X5: real] :
( ( G @ X5 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X3 @ ( set_or5849166863359141190n_real @ X3 ) ) )
=> ( ( eventually_real
@ ^ [X5: real] :
( ( G2 @ X5 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X3 @ ( set_or5849166863359141190n_real @ X3 ) ) )
=> ( ( eventually_real
@ ^ [X5: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X3 @ ( set_or5849166863359141190n_real @ X3 ) ) )
=> ( ( eventually_real
@ ^ [X5: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X3 @ ( set_or5849166863359141190n_real @ X3 ) ) )
=> ( ( filterlim_real_real
@ ^ [X5: real] : ( divide_divide_real @ ( F5 @ X5 ) @ ( G2 @ X5 ) )
@ F4
@ ( topolo2177554685111907308n_real @ X3 @ ( set_or5849166863359141190n_real @ X3 ) ) )
=> ( filterlim_real_real
@ ^ [X5: real] : ( divide_divide_real @ ( F @ X5 ) @ ( G @ X5 ) )
@ F4
@ ( topolo2177554685111907308n_real @ X3 @ ( set_or5849166863359141190n_real @ X3 ) ) ) ) ) ) ) ) ) ) ).
% lhopital_right
thf(fact_9394_Max__divisors__self__int,axiom,
! [N: int] :
( ( N != zero_zero_int )
=> ( ( lattic8263393255366662781ax_int
@ ( collect_int
@ ^ [D4: int] : ( dvd_dvd_int @ D4 @ N ) ) )
= ( abs_abs_int @ N ) ) ) ).
% Max_divisors_self_int
thf(fact_9395_at__top__le__at__infinity,axiom,
ord_le4104064031414453916r_real @ at_top_real @ at_infinity_real ).
% at_top_le_at_infinity
thf(fact_9396_gcd__is__Max__divisors__int,axiom,
! [N: int,M2: int] :
( ( N != zero_zero_int )
=> ( ( gcd_gcd_int @ M2 @ N )
= ( lattic8263393255366662781ax_int
@ ( collect_int
@ ^ [D4: int] :
( ( dvd_dvd_int @ D4 @ M2 )
& ( dvd_dvd_int @ D4 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_int
thf(fact_9397_eventually__at__top__to__right,axiom,
! [P2: real > $o] :
( ( eventually_real @ P2 @ at_top_real )
= ( eventually_real
@ ^ [X5: real] : ( P2 @ ( inverse_inverse_real @ X5 ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% eventually_at_top_to_right
thf(fact_9398_eventually__at__right__to__top,axiom,
! [P2: real > $o] :
( ( eventually_real @ P2 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
= ( eventually_real
@ ^ [X5: real] : ( P2 @ ( inverse_inverse_real @ X5 ) )
@ at_top_real ) ) ).
% eventually_at_right_to_top
thf(fact_9399_ln__x__over__x__tendsto__0,axiom,
( filterlim_real_real
@ ^ [X5: real] : ( divide_divide_real @ ( ln_ln_real @ X5 ) @ X5 )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_real ) ).
% ln_x_over_x_tendsto_0
thf(fact_9400_filterlim__inverse__at__right__top,axiom,
filterlim_real_real @ inverse_inverse_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) @ at_top_real ).
% filterlim_inverse_at_right_top
thf(fact_9401_filterlim__inverse__at__top__right,axiom,
filterlim_real_real @ inverse_inverse_real @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ).
% filterlim_inverse_at_top_right
thf(fact_9402_tendsto__power__div__exp__0,axiom,
! [K2: nat] :
( filterlim_real_real
@ ^ [X5: real] : ( divide_divide_real @ ( power_power_real @ X5 @ K2 ) @ ( exp_real @ X5 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_real ) ).
% tendsto_power_div_exp_0
thf(fact_9403_lhopital__at__top,axiom,
! [G: real > real,X3: real,G2: real > real,F: real > real,F5: real > real,Y: real] :
( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X5: real] :
( ( G2 @ X5 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X5: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( eventually_real
@ ^ [X5: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( filterlim_real_real
@ ^ [X5: real] : ( divide_divide_real @ ( F5 @ X5 ) @ ( G2 @ X5 ) )
@ ( topolo2815343760600316023s_real @ Y )
@ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( filterlim_real_real
@ ^ [X5: real] : ( divide_divide_real @ ( F @ X5 ) @ ( G @ X5 ) )
@ ( topolo2815343760600316023s_real @ Y )
@ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ) ) ).
% lhopital_at_top
thf(fact_9404_lhospital__at__top__at__top,axiom,
! [G: real > real,G2: real > real,F: real > real,F5: real > real,X3: real] :
( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
=> ( ( eventually_real
@ ^ [X5: real] :
( ( G2 @ X5 )
!= zero_zero_real )
@ at_top_real )
=> ( ( eventually_real
@ ^ [X5: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
@ at_top_real )
=> ( ( eventually_real
@ ^ [X5: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
@ at_top_real )
=> ( ( filterlim_real_real
@ ^ [X5: real] : ( divide_divide_real @ ( F5 @ X5 ) @ ( G2 @ X5 ) )
@ ( topolo2815343760600316023s_real @ X3 )
@ at_top_real )
=> ( filterlim_real_real
@ ^ [X5: real] : ( divide_divide_real @ ( F @ X5 ) @ ( G @ X5 ) )
@ ( topolo2815343760600316023s_real @ X3 )
@ at_top_real ) ) ) ) ) ) ).
% lhospital_at_top_at_top
thf(fact_9405_tendsto__exp__limit__at__top,axiom,
! [X3: real] :
( filterlim_real_real
@ ^ [Y5: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X3 @ Y5 ) ) @ Y5 )
@ ( topolo2815343760600316023s_real @ ( exp_real @ X3 ) )
@ at_top_real ) ).
% tendsto_exp_limit_at_top
thf(fact_9406_DERIV__neg__imp__decreasing__at__top,axiom,
! [B: real,F: real > real,Flim: real] :
( ! [X4: real] :
( ( ord_less_eq_real @ B @ X4 )
=> ? [Y6: real] :
( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_real @ Y6 @ zero_zero_real ) ) )
=> ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
=> ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% DERIV_neg_imp_decreasing_at_top
thf(fact_9407_lhopital__right__0__at__top,axiom,
! [G: real > real,G2: real > real,F: real > real,F5: real > real,X3: real] :
( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( eventually_real
@ ^ [X5: real] :
( ( G2 @ X5 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( eventually_real
@ ^ [X5: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( eventually_real
@ ^ [X5: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( ( filterlim_real_real
@ ^ [X5: real] : ( divide_divide_real @ ( F5 @ X5 ) @ ( G2 @ X5 ) )
@ ( topolo2815343760600316023s_real @ X3 )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
=> ( filterlim_real_real
@ ^ [X5: real] : ( divide_divide_real @ ( F @ X5 ) @ ( G @ X5 ) )
@ ( topolo2815343760600316023s_real @ X3 )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ).
% lhopital_right_0_at_top
thf(fact_9408_lhopital__right__at__top,axiom,
! [G: real > real,X3: real,G2: real > real,F: real > real,F5: real > real,Y: real] :
( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X3 @ ( set_or5849166863359141190n_real @ X3 ) ) )
=> ( ( eventually_real
@ ^ [X5: real] :
( ( G2 @ X5 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X3 @ ( set_or5849166863359141190n_real @ X3 ) ) )
=> ( ( eventually_real
@ ^ [X5: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X3 @ ( set_or5849166863359141190n_real @ X3 ) ) )
=> ( ( eventually_real
@ ^ [X5: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X3 @ ( set_or5849166863359141190n_real @ X3 ) ) )
=> ( ( filterlim_real_real
@ ^ [X5: real] : ( divide_divide_real @ ( F5 @ X5 ) @ ( G2 @ X5 ) )
@ ( topolo2815343760600316023s_real @ Y )
@ ( topolo2177554685111907308n_real @ X3 @ ( set_or5849166863359141190n_real @ X3 ) ) )
=> ( filterlim_real_real
@ ^ [X5: real] : ( divide_divide_real @ ( F @ X5 ) @ ( G @ X5 ) )
@ ( topolo2815343760600316023s_real @ Y )
@ ( topolo2177554685111907308n_real @ X3 @ ( set_or5849166863359141190n_real @ X3 ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top
thf(fact_9409_lhopital__left__at__top,axiom,
! [G: real > real,X3: real,G2: real > real,F: real > real,F5: real > real,Y: real] :
( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X3 @ ( set_or5984915006950818249n_real @ X3 ) ) )
=> ( ( eventually_real
@ ^ [X5: real] :
( ( G2 @ X5 )
!= zero_zero_real )
@ ( topolo2177554685111907308n_real @ X3 @ ( set_or5984915006950818249n_real @ X3 ) ) )
=> ( ( eventually_real
@ ^ [X5: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X3 @ ( set_or5984915006950818249n_real @ X3 ) ) )
=> ( ( eventually_real
@ ^ [X5: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
@ ( topolo2177554685111907308n_real @ X3 @ ( set_or5984915006950818249n_real @ X3 ) ) )
=> ( ( filterlim_real_real
@ ^ [X5: real] : ( divide_divide_real @ ( F5 @ X5 ) @ ( G2 @ X5 ) )
@ ( topolo2815343760600316023s_real @ Y )
@ ( topolo2177554685111907308n_real @ X3 @ ( set_or5984915006950818249n_real @ X3 ) ) )
=> ( filterlim_real_real
@ ^ [X5: real] : ( divide_divide_real @ ( F @ X5 ) @ ( G @ X5 ) )
@ ( topolo2815343760600316023s_real @ Y )
@ ( topolo2177554685111907308n_real @ X3 @ ( set_or5984915006950818249n_real @ X3 ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top
thf(fact_9410_filterlim__pow__at__bot__even,axiom,
! [N: nat,F: real > real,F4: filter_real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( filterlim_real_real
@ ^ [X5: real] : ( power_power_real @ ( F @ X5 ) @ N )
@ at_top_real
@ F4 ) ) ) ) ).
% filterlim_pow_at_bot_even
thf(fact_9411_GreatestI__nat,axiom,
! [P2: nat > $o,K2: nat,B: nat] :
( ( P2 @ K2 )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( P2 @ ( order_Greatest_nat @ P2 ) ) ) ) ).
% GreatestI_nat
thf(fact_9412_Greatest__le__nat,axiom,
! [P2: nat > $o,K2: nat,B: nat] :
( ( P2 @ K2 )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( ord_less_eq_nat @ K2 @ ( order_Greatest_nat @ P2 ) ) ) ) ).
% Greatest_le_nat
thf(fact_9413_GreatestI__ex__nat,axiom,
! [P2: nat > $o,B: nat] :
( ? [X_1: nat] : ( P2 @ X_1 )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( P2 @ ( order_Greatest_nat @ P2 ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_9414_Bseq__eq__bounded,axiom,
! [F: nat > real,A: real,B: real] :
( ( ord_less_eq_set_real @ ( image_nat_real @ F @ top_top_set_nat ) @ ( set_or1222579329274155063t_real @ A @ B ) )
=> ( bfun_nat_real @ F @ at_top_nat ) ) ).
% Bseq_eq_bounded
thf(fact_9415_Bseq__realpow,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ one_one_real )
=> ( bfun_nat_real @ ( power_power_real @ X3 ) @ at_top_nat ) ) ) ).
% Bseq_realpow
thf(fact_9416_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X3
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Y
= ( Xa2 != one_one_nat ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
= ( ~ ( ( Deg2 = Xa2 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X7 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
& ! [X5: nat] :
( ( ord_less_nat @ X5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X5 )
=> ( ( ord_less_nat @ Mi3 @ X5 )
& ( ord_less_eq_nat @ X5 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.elims(1)
thf(fact_9417_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_valid @ X3 @ Xa2 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X3
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Xa2 != one_one_nat ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ~ ( ( Deg2 = Xa2 )
& ! [X: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X7 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
& ! [X5: nat] :
( ( ord_less_nat @ X5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X5 )
=> ( ( ord_less_nat @ Mi3 @ X5 )
& ( ord_less_eq_nat @ X5 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(2)
thf(fact_9418_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList @ Summary ) @ Deg4 )
= ( ( Deg = Deg4 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X7 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X7 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
& ! [X5: nat] :
( ( ord_less_nat @ X5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X5 )
=> ( ( ord_less_nat @ Mi3 @ X5 )
& ( ord_less_eq_nat @ X5 @ Ma3 ) ) ) ) ) ) ) )
@ Mima2 ) ) ) ).
% VEBT_internal.valid'.simps(2)
thf(fact_9419_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_valid @ X3 @ Xa2 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X3
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Xa2 = one_one_nat ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Deg2 = Xa2 )
& ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X7 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
& ! [X5: nat] :
( ( ord_less_nat @ X5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X5 )
=> ( ( ord_less_nat @ Mi3 @ X5 )
& ( ord_less_eq_nat @ X5 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(3)
thf(fact_9420_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_valid @ X3 @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X3
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
=> ( Xa2 = one_one_nat ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) )
=> ( ( Deg2 = Xa2 )
& ! [X4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X7 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
& ! [X5: nat] :
( ( ord_less_nat @ X5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X5 )
=> ( ( ord_less_nat @ Mi3 @ X5 )
& ( ord_less_eq_nat @ X5 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(3)
thf(fact_9421_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_valid @ X3 @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X3
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
=> ( Xa2 != one_one_nat ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) )
=> ~ ( ( Deg2 = Xa2 )
& ! [X: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X7 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
& ! [X5: nat] :
( ( ord_less_nat @ X5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X5 )
=> ( ( ord_less_nat @ Mi3 @ X5 )
& ( ord_less_eq_nat @ X5 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(2)
thf(fact_9422_Sup__int__def,axiom,
( complete_Sup_Sup_int
= ( ^ [X7: set_int] :
( the_int
@ ^ [X5: int] :
( ( member_int @ X5 @ X7 )
& ! [Y5: int] :
( ( member_int @ Y5 @ X7 )
=> ( ord_less_eq_int @ Y5 @ X5 ) ) ) ) ) ) ).
% Sup_int_def
thf(fact_9423_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
= Y )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X3
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Y
= ( Xa2 = one_one_nat ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X3
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( ( Deg2 = Xa2 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X7 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
& ! [X5: nat] :
( ( ord_less_nat @ X5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X5 )
=> ( ( ord_less_nat @ Mi3 @ X5 )
& ( ord_less_eq_nat @ X5 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(1)
thf(fact_9424_GMVT,axiom,
! [A: real,B: real,F: real > real,G: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [X4: real] :
( ( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_eq_real @ X4 @ B ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F ) )
=> ( ! [X4: real] :
( ( ( ord_less_real @ A @ X4 )
& ( ord_less_real @ X4 @ B ) )
=> ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
=> ( ! [X4: real] :
( ( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_eq_real @ X4 @ B ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ G ) )
=> ( ! [X4: real] :
( ( ( ord_less_real @ A @ X4 )
& ( ord_less_real @ X4 @ B ) )
=> ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
=> ? [G_c: real,F_c: real,C: real] :
( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
& ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
& ( ord_less_real @ A @ C )
& ( ord_less_real @ C @ B )
& ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
= ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).
% GMVT
thf(fact_9425_less__eq,axiom,
! [M2: nat,N: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M2 @ N ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% less_eq
thf(fact_9426_remdups__upt,axiom,
! [M2: nat,N: nat] :
( ( remdups_nat @ ( upt @ M2 @ N ) )
= ( upt @ M2 @ N ) ) ).
% remdups_upt
thf(fact_9427_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size_list_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ I ) ) ).
% length_upt
thf(fact_9428_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( upt @ I @ J )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_9429_sorted__list__of__set__range,axiom,
! [M2: nat,N: nat] :
( ( linord2614967742042102400et_nat @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
= ( upt @ M2 @ N ) ) ).
% sorted_list_of_set_range
thf(fact_9430_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( ( upt @ I @ J )
= nil_nat )
= ( ( J = zero_zero_nat )
| ( ord_less_eq_nat @ J @ I ) ) ) ).
% upt_eq_Nil_conv
thf(fact_9431_nth__upt,axiom,
! [I: nat,K2: nat,J: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ J )
=> ( ( nth_nat @ ( upt @ I @ J ) @ K2 )
= ( plus_plus_nat @ I @ K2 ) ) ) ).
% nth_upt
thf(fact_9432_upt__rec__numeral,axiom,
! [M2: num,N: num] :
( ( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( cons_nat @ ( numeral_numeral_nat @ M2 ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
& ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= nil_nat ) ) ) ).
% upt_rec_numeral
thf(fact_9433_greaterThanLessThan__upt,axiom,
( set_or5834768355832116004an_nat
= ( ^ [N4: nat,M5: nat] : ( set_nat2 @ ( upt @ ( suc @ N4 ) @ M5 ) ) ) ) ).
% greaterThanLessThan_upt
thf(fact_9434_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_9435_upt__conv__Cons__Cons,axiom,
! [M2: nat,N: nat,Ns: list_nat,Q3: nat] :
( ( ( cons_nat @ M2 @ ( cons_nat @ N @ Ns ) )
= ( upt @ M2 @ Q3 ) )
= ( ( cons_nat @ N @ Ns )
= ( upt @ ( suc @ M2 ) @ Q3 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_9436_greaterThanAtMost__upt,axiom,
( set_or6659071591806873216st_nat
= ( ^ [N4: nat,M5: nat] : ( set_nat2 @ ( upt @ ( suc @ N4 ) @ ( suc @ M5 ) ) ) ) ) ).
% greaterThanAtMost_upt
thf(fact_9437_atLeastLessThan__upt,axiom,
( set_or4665077453230672383an_nat
= ( ^ [I3: nat,J3: nat] : ( set_nat2 @ ( upt @ I3 @ J3 ) ) ) ) ).
% atLeastLessThan_upt
thf(fact_9438_atLeastAtMost__upt,axiom,
( set_or1269000886237332187st_nat
= ( ^ [N4: nat,M5: nat] : ( set_nat2 @ ( upt @ N4 @ ( suc @ M5 ) ) ) ) ) ).
% atLeastAtMost_upt
thf(fact_9439_atLeast__upt,axiom,
( set_ord_lessThan_nat
= ( ^ [N4: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N4 ) ) ) ) ).
% atLeast_upt
thf(fact_9440_distinct__upt,axiom,
! [I: nat,J: nat] : ( distinct_nat @ ( upt @ I @ J ) ) ).
% distinct_upt
thf(fact_9441_map__Suc__upt,axiom,
! [M2: nat,N: nat] :
( ( map_nat_nat @ suc @ ( upt @ M2 @ N ) )
= ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_9442_map__add__upt,axiom,
! [N: nat,M2: nat] :
( ( map_nat_nat
@ ^ [I3: nat] : ( plus_plus_nat @ I3 @ N )
@ ( upt @ zero_zero_nat @ M2 ) )
= ( upt @ N @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% map_add_upt
thf(fact_9443_atMost__upto,axiom,
( set_ord_atMost_nat
= ( ^ [N4: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N4 ) ) ) ) ) ).
% atMost_upto
thf(fact_9444_upt__conv__Cons,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( upt @ I @ J )
= ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% upt_conv_Cons
thf(fact_9445_map__decr__upt,axiom,
! [M2: nat,N: nat] :
( ( map_nat_nat
@ ^ [N4: nat] : ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) )
@ ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( upt @ M2 @ N ) ) ).
% map_decr_upt
thf(fact_9446_upt__add__eq__append,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( plus_plus_nat @ J @ K2 ) )
= ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K2 ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_9447_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X3: nat,Xs: list_nat] :
( ( ( upt @ I @ J )
= ( cons_nat @ X3 @ Xs ) )
= ( ( ord_less_nat @ I @ J )
& ( I = X3 )
& ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_9448_upt__rec,axiom,
( upt
= ( ^ [I3: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I3 @ J3 ) @ ( cons_nat @ I3 @ ( upt @ ( suc @ I3 ) @ J3 ) ) @ nil_nat ) ) ) ).
% upt_rec
thf(fact_9449_upt__Suc,axiom,
! [I: nat,J: nat] :
( ( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
& ( ~ ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= nil_nat ) ) ) ).
% upt_Suc
thf(fact_9450_upt__Suc__append,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% upt_Suc_append
thf(fact_9451_tl__upt,axiom,
! [M2: nat,N: nat] :
( ( tl_nat @ ( upt @ M2 @ N ) )
= ( upt @ ( suc @ M2 ) @ N ) ) ).
% tl_upt
thf(fact_9452_hd__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( hd_nat @ ( upt @ I @ J ) )
= I ) ) ).
% hd_upt
thf(fact_9453_sum__list__upt,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( groups4561878855575611511st_nat @ ( upt @ M2 @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [X5: nat] : X5
@ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) ) ).
% sum_list_upt
thf(fact_9454_card__length__sum__list__rec,axiom,
! [M2: nat,N7: nat] :
( ( ord_less_eq_nat @ one_one_nat @ M2 )
=> ( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L2: list_nat] :
( ( ( size_size_list_nat @ L2 )
= M2 )
& ( ( groups4561878855575611511st_nat @ L2 )
= N7 ) ) ) )
= ( plus_plus_nat
@ ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L2: list_nat] :
( ( ( size_size_list_nat @ L2 )
= ( minus_minus_nat @ M2 @ one_one_nat ) )
& ( ( groups4561878855575611511st_nat @ L2 )
= N7 ) ) ) )
@ ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L2: list_nat] :
( ( ( size_size_list_nat @ L2 )
= M2 )
& ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L2 ) @ one_one_nat )
= N7 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_9455_card__length__sum__list,axiom,
! [M2: nat,N7: nat] :
( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L2: list_nat] :
( ( ( size_size_list_nat @ L2 )
= M2 )
& ( ( groups4561878855575611511st_nat @ L2 )
= N7 ) ) ) )
= ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N7 @ M2 ) @ one_one_nat ) @ N7 ) ) ).
% card_length_sum_list
thf(fact_9456_sorted__upt,axiom,
! [M2: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M2 @ N ) ) ).
% sorted_upt
thf(fact_9457_sorted__wrt__upt,axiom,
! [M2: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M2 @ N ) ) ).
% sorted_wrt_upt
thf(fact_9458_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I: nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_9459_sorted__wrt__upto,axiom,
! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).
% sorted_wrt_upto
thf(fact_9460_sorted__upto,axiom,
! [M2: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M2 @ N ) ) ).
% sorted_upto
thf(fact_9461_range__abs__Nats,axiom,
( ( image_int_int @ abs_abs_int @ top_top_set_int )
= semiring_1_Nats_int ) ).
% range_abs_Nats
thf(fact_9462_min__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_min_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( suc @ ( ord_min_nat @ M2 @ N ) ) ) ).
% min_Suc_Suc
thf(fact_9463_min__0L,axiom,
! [N: nat] :
( ( ord_min_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% min_0L
thf(fact_9464_min__0R,axiom,
! [N: nat] :
( ( ord_min_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0R
thf(fact_9465_min__Suc__numeral,axiom,
! [N: nat,K2: num] :
( ( ord_min_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K2 ) )
= ( suc @ ( ord_min_nat @ N @ ( pred_numeral @ K2 ) ) ) ) ).
% min_Suc_numeral
thf(fact_9466_min__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( ord_min_nat @ ( numeral_numeral_nat @ K2 ) @ ( suc @ N ) )
= ( suc @ ( ord_min_nat @ ( pred_numeral @ K2 ) @ N ) ) ) ).
% min_numeral_Suc
thf(fact_9467_nat__mult__min__left,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( times_times_nat @ ( ord_min_nat @ M2 @ N ) @ Q3 )
= ( ord_min_nat @ ( times_times_nat @ M2 @ Q3 ) @ ( times_times_nat @ N @ Q3 ) ) ) ).
% nat_mult_min_left
thf(fact_9468_nat__mult__min__right,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( times_times_nat @ M2 @ ( ord_min_nat @ N @ Q3 ) )
= ( ord_min_nat @ ( times_times_nat @ M2 @ N ) @ ( times_times_nat @ M2 @ Q3 ) ) ) ).
% nat_mult_min_right
thf(fact_9469_min__diff,axiom,
! [M2: nat,I: nat,N: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M2 @ I ) @ ( minus_minus_nat @ N @ I ) )
= ( minus_minus_nat @ ( ord_min_nat @ M2 @ N ) @ I ) ) ).
% min_diff
thf(fact_9470_inf__nat__def,axiom,
inf_inf_nat = ord_min_nat ).
% inf_nat_def
thf(fact_9471_min__Suc2,axiom,
! [M2: nat,N: nat] :
( ( ord_min_nat @ M2 @ ( suc @ N ) )
= ( case_nat_nat @ zero_zero_nat
@ ^ [M7: nat] : ( suc @ ( ord_min_nat @ M7 @ N ) )
@ M2 ) ) ).
% min_Suc2
thf(fact_9472_min__Suc1,axiom,
! [N: nat,M2: nat] :
( ( ord_min_nat @ ( suc @ N ) @ M2 )
= ( case_nat_nat @ zero_zero_nat
@ ^ [M7: nat] : ( suc @ ( ord_min_nat @ N @ M7 ) )
@ M2 ) ) ).
% min_Suc1
thf(fact_9473_complex__is__Nat__iff,axiom,
! [Z2: complex] :
( ( member_complex @ Z2 @ semiri3842193898606819883omplex )
= ( ( ( im @ Z2 )
= zero_zero_real )
& ? [I3: nat] :
( ( re @ Z2 )
= ( semiri5074537144036343181t_real @ I3 ) ) ) ) ).
% complex_is_Nat_iff
thf(fact_9474_Rats__eq__int__div__nat,axiom,
( field_5140801741446780682s_real
= ( collect_real
@ ^ [Uu3: real] :
? [I3: int,N4: nat] :
( ( Uu3
= ( divide_divide_real @ ( ring_1_of_int_real @ I3 ) @ ( semiri5074537144036343181t_real @ N4 ) ) )
& ( N4 != zero_zero_nat ) ) ) ) ).
% Rats_eq_int_div_nat
thf(fact_9475_min__enat__simps_I2_J,axiom,
! [Q3: extended_enat] :
( ( ord_mi8085742599997312461d_enat @ Q3 @ zero_z5237406670263579293d_enat )
= zero_z5237406670263579293d_enat ) ).
% min_enat_simps(2)
thf(fact_9476_min__enat__simps_I3_J,axiom,
! [Q3: extended_enat] :
( ( ord_mi8085742599997312461d_enat @ zero_z5237406670263579293d_enat @ Q3 )
= zero_z5237406670263579293d_enat ) ).
% min_enat_simps(3)
thf(fact_9477_inf__int__def,axiom,
inf_inf_int = ord_min_int ).
% inf_int_def
thf(fact_9478_Rats__no__top__le,axiom,
! [X3: real] :
? [X4: real] :
( ( member_real @ X4 @ field_5140801741446780682s_real )
& ( ord_less_eq_real @ X3 @ X4 ) ) ).
% Rats_no_top_le
thf(fact_9479_Rats__eq__int__div__int,axiom,
( field_5140801741446780682s_real
= ( collect_real
@ ^ [Uu3: real] :
? [I3: int,J3: int] :
( ( Uu3
= ( divide_divide_real @ ( ring_1_of_int_real @ I3 ) @ ( ring_1_of_int_real @ J3 ) ) )
& ( J3 != zero_zero_int ) ) ) ) ).
% Rats_eq_int_div_int
thf(fact_9480_Rolle,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ A )
= ( F @ B ) )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ! [X4: real] :
( ( ord_less_real @ A @ X4 )
=> ( ( ord_less_real @ X4 @ B )
=> ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
=> ? [Z3: real] :
( ( ord_less_real @ A @ Z3 )
& ( ord_less_real @ Z3 @ B )
& ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) ) ) ) ) ).
% Rolle
thf(fact_9481_continuous__on__arcosh_H,axiom,
! [A4: set_real,F: real > real] :
( ( topolo5044208981011980120l_real @ A4 @ F )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
=> ( topolo5044208981011980120l_real @ A4
@ ^ [X5: real] : ( arcosh_real @ ( F @ X5 ) ) ) ) ) ).
% continuous_on_arcosh'
thf(fact_9482_continuous__image__closed__interval,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ? [C: real,D2: real] :
( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
= ( set_or1222579329274155063t_real @ C @ D2 ) )
& ( ord_less_eq_real @ C @ D2 ) ) ) ) ).
% continuous_image_closed_interval
thf(fact_9483_continuous__on__arcosh,axiom,
! [A4: set_real] :
( ( ord_less_eq_set_real @ A4 @ ( set_ord_atLeast_real @ one_one_real ) )
=> ( topolo5044208981011980120l_real @ A4 @ arcosh_real ) ) ).
% continuous_on_arcosh
thf(fact_9484_continuous__on__artanh,axiom,
! [A4: set_real] :
( ( ord_less_eq_set_real @ A4 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
=> ( topolo5044208981011980120l_real @ A4 @ artanh_real ) ) ).
% continuous_on_artanh
thf(fact_9485_Rolle__deriv,axiom,
! [A: real,B: real,F: real > real,F5: real > real > real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ A )
= ( F @ B ) )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ! [X4: real] :
( ( ord_less_real @ A @ X4 )
=> ( ( ord_less_real @ X4 @ B )
=> ( has_de1759254742604945161l_real @ F @ ( F5 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
=> ? [Z3: real] :
( ( ord_less_real @ A @ Z3 )
& ( ord_less_real @ Z3 @ B )
& ( ( F5 @ Z3 )
= ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).
% Rolle_deriv
thf(fact_9486_DERIV__isconst__end,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ! [X4: real] :
( ( ord_less_real @ A @ X4 )
=> ( ( ord_less_real @ X4 @ B )
=> ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
=> ( ( F @ B )
= ( F @ A ) ) ) ) ) ).
% DERIV_isconst_end
thf(fact_9487_DERIV__neg__imp__decreasing__open,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [X4: real] :
( ( ord_less_real @ A @ X4 )
=> ( ( ord_less_real @ X4 @ B )
=> ? [Y6: real] :
( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_real @ Y6 @ zero_zero_real ) ) ) )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ) ).
% DERIV_neg_imp_decreasing_open
thf(fact_9488_DERIV__pos__imp__increasing__open,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [X4: real] :
( ( ord_less_real @ A @ X4 )
=> ( ( ord_less_real @ X4 @ B )
=> ? [Y6: real] :
( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
& ( ord_less_real @ zero_zero_real @ Y6 ) ) ) )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).
% DERIV_pos_imp_increasing_open
thf(fact_9489_DERIV__isconst2,axiom,
! [A: real,B: real,F: real > real,X3: real] :
( ( ord_less_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ! [X4: real] :
( ( ord_less_real @ A @ X4 )
=> ( ( ord_less_real @ X4 @ B )
=> ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
=> ( ( ord_less_eq_real @ A @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B )
=> ( ( F @ X3 )
= ( F @ A ) ) ) ) ) ) ) ).
% DERIV_isconst2
thf(fact_9490_inj__sgn__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( inj_on_real_real
@ ^ [Y5: real] : ( times_times_real @ ( sgn_sgn_real @ Y5 ) @ ( power_power_real @ ( abs_abs_real @ Y5 ) @ N ) )
@ top_top_set_real ) ) ).
% inj_sgn_power
thf(fact_9491_pred__nat__trancl__eq__le,axiom,
! [M2: nat,N: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M2 @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% pred_nat_trancl_eq_le
thf(fact_9492_log__inj,axiom,
! [B: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( inj_on_real_real @ ( log @ B ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% log_inj
thf(fact_9493_inj__on__diff__nat,axiom,
! [N7: set_nat,K2: nat] :
( ! [N3: nat] :
( ( member_nat @ N3 @ N7 )
=> ( ord_less_eq_nat @ K2 @ N3 ) )
=> ( inj_on_nat_nat
@ ^ [N4: nat] : ( minus_minus_nat @ N4 @ K2 )
@ N7 ) ) ).
% inj_on_diff_nat
thf(fact_9494_inj__on__set__encode,axiom,
inj_on_set_nat_nat @ nat_set_encode @ ( collect_set_nat @ finite_finite_nat ) ).
% inj_on_set_encode
thf(fact_9495_inj__int__encode,axiom,
! [A4: set_int] : ( inj_on_int_nat @ nat_int_encode @ A4 ) ).
% inj_int_encode
thf(fact_9496_inj__prod__encode,axiom,
! [A4: set_Pr1261947904930325089at_nat] : ( inj_on2178005380612969504at_nat @ nat_prod_encode @ A4 ) ).
% inj_prod_encode
thf(fact_9497_inj__Suc,axiom,
! [N7: set_nat] : ( inj_on_nat_nat @ suc @ N7 ) ).
% inj_Suc
thf(fact_9498_inj__list__encode,axiom,
! [A4: set_list_nat] : ( inj_on_list_nat_nat @ nat_list_encode @ A4 ) ).
% inj_list_encode
thf(fact_9499_summable__reindex,axiom,
! [F: nat > real,G: nat > nat] :
( ( summable_real @ F )
=> ( ( inj_on_nat_nat @ G @ top_top_set_nat )
=> ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
=> ( summable_real @ ( comp_nat_real_nat @ F @ G ) ) ) ) ) ).
% summable_reindex
thf(fact_9500_suminf__reindex__mono,axiom,
! [F: nat > real,G: nat > nat] :
( ( summable_real @ F )
=> ( ( inj_on_nat_nat @ G @ top_top_set_nat )
=> ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
=> ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G ) ) @ ( suminf_real @ F ) ) ) ) ) ).
% suminf_reindex_mono
thf(fact_9501_suminf__reindex,axiom,
! [F: nat > real,G: nat > nat] :
( ( summable_real @ F )
=> ( ( inj_on_nat_nat @ G @ top_top_set_nat )
=> ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
=> ( ! [X4: nat] :
( ~ ( member_nat @ X4 @ ( image_nat_nat @ G @ top_top_set_nat ) )
=> ( ( F @ X4 )
= zero_zero_real ) )
=> ( ( suminf_real @ ( comp_nat_real_nat @ F @ G ) )
= ( suminf_real @ F ) ) ) ) ) ) ).
% suminf_reindex
thf(fact_9502_measure__function__int,axiom,
fun_is_measure_int @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) ).
% measure_function_int
thf(fact_9503_surj__int__decode,axiom,
( ( image_nat_int @ nat_int_decode @ top_top_set_nat )
= top_top_set_int ) ).
% surj_int_decode
thf(fact_9504_int__encode__inverse,axiom,
! [X3: int] :
( ( nat_int_decode @ ( nat_int_encode @ X3 ) )
= X3 ) ).
% int_encode_inverse
thf(fact_9505_int__decode__inverse,axiom,
! [N: nat] :
( ( nat_int_encode @ ( nat_int_decode @ N ) )
= N ) ).
% int_decode_inverse
thf(fact_9506_int__decode__eq,axiom,
! [X3: nat,Y: nat] :
( ( ( nat_int_decode @ X3 )
= ( nat_int_decode @ Y ) )
= ( X3 = Y ) ) ).
% int_decode_eq
thf(fact_9507_inj__int__decode,axiom,
! [A4: set_nat] : ( inj_on_nat_int @ nat_int_decode @ A4 ) ).
% inj_int_decode
thf(fact_9508_bij__int__decode,axiom,
bij_betw_nat_int @ nat_int_decode @ top_top_set_nat @ top_top_set_int ).
% bij_int_decode
thf(fact_9509_powr__real__of__int_H,axiom,
! [X3: real,N: int] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ( X3 != zero_zero_real )
| ( ord_less_int @ zero_zero_int @ N ) )
=> ( ( powr_real @ X3 @ ( ring_1_of_int_real @ N ) )
= ( power_int_real @ X3 @ N ) ) ) ) ).
% powr_real_of_int'
thf(fact_9510_pairs__le__eq__Sigma,axiom,
! [M2: nat] :
( ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I3: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J3 ) @ M2 ) ) )
= ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M2 )
@ ^ [R5: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M2 @ R5 ) ) ) ) ).
% pairs_le_eq_Sigma
thf(fact_9511_uniformity__complex__def,axiom,
( topolo896644834953643431omplex
= ( comple8358262395181532106omplex
@ ( image_5971271580939081552omplex
@ ^ [E3: real] :
( princi3496590319149328850omplex
@ ( collec8663557070575231912omplex
@ ( produc6771430404735790350plex_o
@ ^ [X5: complex,Y5: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X5 @ Y5 ) @ E3 ) ) ) )
@ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% uniformity_complex_def
thf(fact_9512_uniformity__real__def,axiom,
( topolo1511823702728130853y_real
= ( comple2936214249959783750l_real
@ ( image_2178119161166701260l_real
@ ^ [E3: real] :
( princi6114159922880469582l_real
@ ( collec3799799289383736868l_real
@ ( produc5414030515140494994real_o
@ ^ [X5: real,Y5: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X5 @ Y5 ) @ E3 ) ) ) )
@ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% uniformity_real_def
thf(fact_9513_take__upt,axiom,
! [I: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M2 ) @ N )
=> ( ( take_nat @ M2 @ ( upt @ I @ N ) )
= ( upt @ I @ ( plus_plus_nat @ I @ M2 ) ) ) ) ).
% take_upt
thf(fact_9514_eventually__prod__sequentially,axiom,
! [P2: product_prod_nat_nat > $o] :
( ( eventu1038000079068216329at_nat @ P2 @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
= ( ? [N5: nat] :
! [M5: nat] :
( ( ord_less_eq_nat @ N5 @ M5 )
=> ! [N4: nat] :
( ( ord_less_eq_nat @ N5 @ N4 )
=> ( P2 @ ( product_Pair_nat_nat @ N4 @ M5 ) ) ) ) ) ) ).
% eventually_prod_sequentially
thf(fact_9515_drop__upt,axiom,
! [M2: nat,I: nat,J: nat] :
( ( drop_nat @ M2 @ ( upt @ I @ J ) )
= ( upt @ ( plus_plus_nat @ I @ M2 ) @ J ) ) ).
% drop_upt
thf(fact_9516_at__right__to__0,axiom,
! [A: real] :
( ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) )
= ( filtermap_real_real
@ ^ [X5: real] : ( plus_plus_real @ X5 @ A )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% at_right_to_0
thf(fact_9517_at__right__to__top,axiom,
( ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) )
= ( filtermap_real_real @ inverse_inverse_real @ at_top_real ) ) ).
% at_right_to_top
thf(fact_9518_at__top__to__right,axiom,
( at_top_real
= ( filtermap_real_real @ inverse_inverse_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% at_top_to_right
thf(fact_9519_filtermap__ln__at__right,axiom,
( ( filtermap_real_real @ ln_ln_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
= at_bot_real ) ).
% filtermap_ln_at_right
thf(fact_9520_pos__deriv__imp__strict__mono,axiom,
! [F: real > real,F5: real > real] :
( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
=> ( ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( F5 @ X4 ) )
=> ( order_7092887310737990675l_real @ F ) ) ) ).
% pos_deriv_imp_strict_mono
thf(fact_9521_of__nat__eq__id,axiom,
semiri1316708129612266289at_nat = id_nat ).
% of_nat_eq_id
thf(fact_9522_strict__mono__imp__increasing,axiom,
! [F: nat > nat,N: nat] :
( ( order_5726023648592871131at_nat @ F )
=> ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% strict_mono_imp_increasing
thf(fact_9523_infinite__enumerate,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ? [R3: nat > nat] :
( ( order_5726023648592871131at_nat @ R3 )
& ! [N6: nat] : ( member_nat @ ( R3 @ N6 ) @ S2 ) ) ) ).
% infinite_enumerate
thf(fact_9524_less__eq__int__def,axiom,
( ord_less_eq_int
= ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
@ ( produc8739625826339149834_nat_o
@ ^ [X5: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X5 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) ) ) ) ) ).
% less_eq_int_def
thf(fact_9525_less__int__def,axiom,
( ord_less_int
= ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
@ ( produc8739625826339149834_nat_o
@ ^ [X5: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X5 @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) ) ) ) ) ).
% less_int_def
thf(fact_9526_nat__def,axiom,
( nat2
= ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).
% nat_def
thf(fact_9527_vimage__Suc__insert__Suc,axiom,
! [N: nat,A4: set_nat] :
( ( vimage_nat_nat @ suc @ ( insert_nat @ ( suc @ N ) @ A4 ) )
= ( insert_nat @ N @ ( vimage_nat_nat @ suc @ A4 ) ) ) ).
% vimage_Suc_insert_Suc
thf(fact_9528_vimage__Suc__insert__0,axiom,
! [A4: set_nat] :
( ( vimage_nat_nat @ suc @ ( insert_nat @ zero_zero_nat @ A4 ) )
= ( vimage_nat_nat @ suc @ A4 ) ) ).
% vimage_Suc_insert_0
thf(fact_9529_finite__vimage__Suc__iff,axiom,
! [F4: set_nat] :
( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F4 ) )
= ( finite_finite_nat @ F4 ) ) ).
% finite_vimage_Suc_iff
thf(fact_9530_set__decode__div__2,axiom,
! [X3: nat] :
( ( nat_set_decode @ ( divide_divide_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( vimage_nat_nat @ suc @ ( nat_set_decode @ X3 ) ) ) ).
% set_decode_div_2
thf(fact_9531_set__encode__vimage__Suc,axiom,
! [A4: set_nat] :
( ( nat_set_encode @ ( vimage_nat_nat @ suc @ A4 ) )
= ( divide_divide_nat @ ( nat_set_encode @ A4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% set_encode_vimage_Suc
thf(fact_9532_positive__rat,axiom,
! [A: int,B: int] :
( ( positive @ ( fract @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% positive_rat
thf(fact_9533_Rat_Opositive__add,axiom,
! [X3: rat,Y: rat] :
( ( positive @ X3 )
=> ( ( positive @ Y )
=> ( positive @ ( plus_plus_rat @ X3 @ Y ) ) ) ) ).
% Rat.positive_add
thf(fact_9534_Rat_Opositive__zero,axiom,
~ ( positive @ zero_zero_rat ) ).
% Rat.positive_zero
thf(fact_9535_Rat_Opositive__minus,axiom,
! [X3: rat] :
( ~ ( positive @ X3 )
=> ( ( X3 != zero_zero_rat )
=> ( positive @ ( uminus_uminus_rat @ X3 ) ) ) ) ).
% Rat.positive_minus
thf(fact_9536_Rat_Opositive_Orep__eq,axiom,
( positive
= ( ^ [X5: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X5 ) ) @ ( product_snd_int_int @ ( rep_Rat @ X5 ) ) ) ) ) ) ).
% Rat.positive.rep_eq
thf(fact_9537_Rat_Opositive__def,axiom,
( positive
= ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
@ ^ [X5: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X5 ) @ ( product_snd_int_int @ X5 ) ) ) ) ) ).
% Rat.positive_def
thf(fact_9538_Restr__natLeq,axiom,
! [N: nat] :
( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
@ ( produc457027306803732586at_nat
@ ( collect_nat
@ ^ [X5: nat] : ( ord_less_nat @ X5 @ N ) )
@ ^ [Uu3: nat] :
( collect_nat
@ ^ [X5: nat] : ( ord_less_nat @ X5 @ N ) ) ) )
= ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ N )
& ( ord_less_nat @ Y5 @ N )
& ( ord_less_eq_nat @ X5 @ Y5 ) ) ) ) ) ).
% Restr_natLeq
thf(fact_9539_natLeq__def,axiom,
( bNF_Ca8665028551170535155natLeq
= ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_eq_nat ) ) ) ).
% natLeq_def
thf(fact_9540_Restr__natLeq2,axiom,
! [N: nat] :
( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
@ ( produc457027306803732586at_nat @ ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
@ ^ [Uu3: nat] : ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N ) ) )
= ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ N )
& ( ord_less_nat @ Y5 @ N )
& ( ord_less_eq_nat @ X5 @ Y5 ) ) ) ) ) ).
% Restr_natLeq2
thf(fact_9541_Field__natLeq__on,axiom,
! [N: nat] :
( ( field_nat
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ N )
& ( ord_less_nat @ Y5 @ N )
& ( ord_less_eq_nat @ X5 @ Y5 ) ) ) ) )
= ( collect_nat
@ ^ [X5: nat] : ( ord_less_nat @ X5 @ N ) ) ) ).
% Field_natLeq_on
thf(fact_9542_wf__int__ge__less__than2,axiom,
! [D: int] : ( wf_int @ ( int_ge_less_than2 @ D ) ) ).
% wf_int_ge_less_than2
thf(fact_9543_wf__int__ge__less__than,axiom,
! [D: int] : ( wf_int @ ( int_ge_less_than @ D ) ) ).
% wf_int_ge_less_than
thf(fact_9544_mono__Suc,axiom,
order_mono_nat_nat @ suc ).
% mono_Suc
thf(fact_9545_mono__times__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).
% mono_times_nat
thf(fact_9546_mono__ge2__power__minus__self,axiom,
! [K2: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
=> ( order_mono_nat_nat
@ ^ [M5: nat] : ( minus_minus_nat @ ( power_power_nat @ K2 @ M5 ) @ M5 ) ) ) ).
% mono_ge2_power_minus_self
thf(fact_9547_incseq__bounded,axiom,
! [X8: nat > real,B5: real] :
( ( order_mono_nat_real @ X8 )
=> ( ! [I2: nat] : ( ord_less_eq_real @ ( X8 @ I2 ) @ B5 )
=> ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).
% incseq_bounded
thf(fact_9548_incseq__convergent,axiom,
! [X8: nat > real,B5: real] :
( ( order_mono_nat_real @ X8 )
=> ( ! [I2: nat] : ( ord_less_eq_real @ ( X8 @ I2 ) @ B5 )
=> ~ ! [L6: real] :
( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
=> ~ ! [I4: nat] : ( ord_less_eq_real @ ( X8 @ I4 ) @ L6 ) ) ) ) ).
% incseq_convergent
thf(fact_9549_nonneg__incseq__Bseq__subseq__iff,axiom,
! [F: nat > real,G: nat > nat] :
( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
=> ( ( order_mono_nat_real @ F )
=> ( ( order_5726023648592871131at_nat @ G )
=> ( ( bfun_nat_real
@ ^ [X5: nat] : ( F @ ( G @ X5 ) )
@ at_top_nat )
= ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).
% nonneg_incseq_Bseq_subseq_iff
thf(fact_9550_quotient__of__def,axiom,
( quotient_of
= ( ^ [X5: rat] :
( the_Pr4378521158711661632nt_int
@ ^ [Pair: product_prod_int_int] :
( ( X5
= ( fract @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) )
& ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Pair ) )
& ( algebr932160517623751201me_int @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) ) ) ) ) ).
% quotient_of_def
thf(fact_9551_normalize__stable,axiom,
! [Q3: int,P: int] :
( ( ord_less_int @ zero_zero_int @ Q3 )
=> ( ( algebr932160517623751201me_int @ P @ Q3 )
=> ( ( normalize @ ( product_Pair_int_int @ P @ Q3 ) )
= ( product_Pair_int_int @ P @ Q3 ) ) ) ) ).
% normalize_stable
thf(fact_9552_quotient__of__coprime,axiom,
! [R2: rat,P: int,Q3: int] :
( ( ( quotient_of @ R2 )
= ( product_Pair_int_int @ P @ Q3 ) )
=> ( algebr932160517623751201me_int @ P @ Q3 ) ) ).
% quotient_of_coprime
thf(fact_9553_normalize__coprime,axiom,
! [R2: product_prod_int_int,P: int,Q3: int] :
( ( ( normalize @ R2 )
= ( product_Pair_int_int @ P @ Q3 ) )
=> ( algebr932160517623751201me_int @ P @ Q3 ) ) ).
% normalize_coprime
thf(fact_9554_Rat__cases,axiom,
! [Q3: rat] :
~ ! [A3: int,B3: int] :
( ( Q3
= ( fract @ A3 @ B3 ) )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ~ ( algebr932160517623751201me_int @ A3 @ B3 ) ) ) ).
% Rat_cases
thf(fact_9555_Rat__induct,axiom,
! [P2: rat > $o,Q3: rat] :
( ! [A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( algebr932160517623751201me_int @ A3 @ B3 )
=> ( P2 @ ( fract @ A3 @ B3 ) ) ) )
=> ( P2 @ Q3 ) ) ).
% Rat_induct
thf(fact_9556_Rat__cases__nonzero,axiom,
! [Q3: rat] :
( ! [A3: int,B3: int] :
( ( Q3
= ( fract @ A3 @ B3 ) )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( A3 != zero_zero_int )
=> ~ ( algebr932160517623751201me_int @ A3 @ B3 ) ) ) )
=> ( Q3 = zero_zero_rat ) ) ).
% Rat_cases_nonzero
thf(fact_9557_quotient__of__unique,axiom,
! [R2: rat] :
? [X4: product_prod_int_int] :
( ( R2
= ( fract @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) )
& ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ X4 ) )
& ( algebr932160517623751201me_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) )
& ! [Y6: product_prod_int_int] :
( ( ( R2
= ( fract @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ Y6 ) ) )
& ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Y6 ) )
& ( algebr932160517623751201me_int @ ( product_fst_int_int @ Y6 ) @ ( product_snd_int_int @ Y6 ) ) )
=> ( Y6 = X4 ) ) ) ).
% quotient_of_unique
thf(fact_9558_coprime__Suc__left__nat,axiom,
! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ N ) @ N ) ).
% coprime_Suc_left_nat
thf(fact_9559_coprime__Suc__right__nat,axiom,
! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ N ) ) ).
% coprime_Suc_right_nat
thf(fact_9560_coprime__Suc__0__right,axiom,
! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ zero_zero_nat ) ) ).
% coprime_Suc_0_right
thf(fact_9561_coprime__Suc__0__left,axiom,
! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N ) ).
% coprime_Suc_0_left
thf(fact_9562_coprime__diff__one__right__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( algebr934650988132801477me_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% coprime_diff_one_right_nat
thf(fact_9563_coprime__diff__one__left__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).
% coprime_diff_one_left_nat
thf(fact_9564_Gcd__nat__set__eq__fold,axiom,
! [Xs: list_nat] :
( ( gcd_Gcd_nat @ ( set_nat2 @ Xs ) )
= ( fold_nat_nat @ gcd_gcd_nat @ Xs @ zero_zero_nat ) ) ).
% Gcd_nat_set_eq_fold
thf(fact_9565_Rats__abs__nat__div__natE,axiom,
! [X3: real] :
( ( member_real @ X3 @ field_5140801741446780682s_real )
=> ~ ! [M: nat,N3: nat] :
( ( N3 != zero_zero_nat )
=> ( ( ( abs_abs_real @ X3 )
= ( divide_divide_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
=> ~ ( algebr934650988132801477me_nat @ M @ N3 ) ) ) ) ).
% Rats_abs_nat_div_natE
thf(fact_9566_Gcd__int__set__eq__fold,axiom,
! [Xs: list_int] :
( ( gcd_Gcd_int @ ( set_int2 @ Xs ) )
= ( fold_int_int @ gcd_gcd_int @ Xs @ zero_zero_int ) ) ).
% Gcd_int_set_eq_fold
thf(fact_9567_sort__upt,axiom,
! [M2: nat,N: nat] :
( ( linord738340561235409698at_nat
@ ^ [X5: nat] : X5
@ ( upt @ M2 @ N ) )
= ( upt @ M2 @ N ) ) ).
% sort_upt
thf(fact_9568_sort__upto,axiom,
! [I: int,J: int] :
( ( linord1735203802627413978nt_int
@ ^ [X5: int] : X5
@ ( upto @ I @ J ) )
= ( upto @ I @ J ) ) ).
% sort_upto
thf(fact_9569_vanishes__mult__bounded,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ? [A8: rat] :
( ( ord_less_rat @ zero_zero_rat @ A8 )
& ! [N3: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N3 ) ) @ A8 ) )
=> ( ( vanishes @ Y7 )
=> ( vanishes
@ ^ [N4: nat] : ( times_times_rat @ ( X8 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ).
% vanishes_mult_bounded
thf(fact_9570_vanishes__const,axiom,
! [C2: rat] :
( ( vanishes
@ ^ [N4: nat] : C2 )
= ( C2 = zero_zero_rat ) ) ).
% vanishes_const
thf(fact_9571_vanishes__add,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( vanishes @ X8 )
=> ( ( vanishes @ Y7 )
=> ( vanishes
@ ^ [N4: nat] : ( plus_plus_rat @ ( X8 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ).
% vanishes_add
thf(fact_9572_vanishesD,axiom,
! [X8: nat > rat,R2: rat] :
( ( vanishes @ X8 )
=> ( ( ord_less_rat @ zero_zero_rat @ R2 )
=> ? [K: nat] :
! [N6: nat] :
( ( ord_less_eq_nat @ K @ N6 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N6 ) ) @ R2 ) ) ) ) ).
% vanishesD
thf(fact_9573_vanishesI,axiom,
! [X8: nat > rat] :
( ! [R3: rat] :
( ( ord_less_rat @ zero_zero_rat @ R3 )
=> ? [K4: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ K4 @ N3 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N3 ) ) @ R3 ) ) )
=> ( vanishes @ X8 ) ) ).
% vanishesI
thf(fact_9574_vanishes__def,axiom,
( vanishes
= ( ^ [X7: nat > rat] :
! [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
=> ? [K3: nat] :
! [N4: nat] :
( ( ord_less_eq_nat @ K3 @ N4 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X7 @ N4 ) ) @ R5 ) ) ) ) ) ).
% vanishes_def
% Helper facts (36)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X3: int,Y: int] :
( ( if_int @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X3: int,Y: int] :
( ( if_int @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
! [X3: num,Y: num] :
( ( if_num @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
! [X3: num,Y: num] :
( ( if_num @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
! [X3: rat,Y: rat] :
( ( if_rat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
! [X3: rat,Y: rat] :
( ( if_rat @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X3: real,Y: real] :
( ( if_real @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X3: real,Y: real] :
( ( if_real @ $true @ X3 @ Y )
= X3 ) ).
thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
! [P2: real > $o] :
( ( P2 @ ( fChoice_real @ P2 ) )
= ( ? [X7: real] : ( P2 @ X7 ) ) ) ).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
! [X3: complex,Y: complex] :
( ( if_complex @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
! [X3: complex,Y: complex] :
( ( if_complex @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
! [X3: extended_enat,Y: extended_enat] :
( ( if_Extended_enat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
! [X3: extended_enat,Y: extended_enat] :
( ( if_Extended_enat @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
! [X3: code_integer,Y: code_integer] :
( ( if_Code_integer @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
! [X3: code_integer,Y: code_integer] :
( ( if_Code_integer @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
! [X3: set_int,Y: set_int] :
( ( if_set_int @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
! [X3: set_int,Y: set_int] :
( ( if_set_int @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
! [X3: vEBT_VEBT,Y: vEBT_VEBT] :
( ( if_VEBT_VEBT @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
! [X3: vEBT_VEBT,Y: vEBT_VEBT] :
( ( if_VEBT_VEBT @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X3: list_int,Y: list_int] :
( ( if_list_int @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X3: list_int,Y: list_int] :
( ( if_list_int @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X3: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X3: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
! [X3: option_num,Y: option_num] :
( ( if_option_num @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
! [X3: option_num,Y: option_num] :
( ( if_option_num @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X3: product_prod_int_int,Y: product_prod_int_int] :
( ( if_Pro3027730157355071871nt_int @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X3: product_prod_int_int,Y: product_prod_int_int] :
( ( if_Pro3027730157355071871nt_int @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
! [X3: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
( ( if_Pro5737122678794959658eger_o @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
! [X3: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
( ( if_Pro5737122678794959658eger_o @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
! [X3: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
( ( if_Pro6119634080678213985nteger @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
! [X3: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
( ( if_Pro6119634080678213985nteger @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ ya ) @ xa ).
%------------------------------------------------------------------------------