TPTP Problem File: SLH0919^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Khovanskii_Theorem/0008_Khovanskii/prob_01132_043301__13704240_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1461 ( 439 unt; 277 typ; 0 def)
% Number of atoms : 4101 (1536 equ; 0 cnn)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 13159 ( 306 ~; 16 |; 393 &;10479 @)
% ( 0 <=>;1965 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 27 ( 26 usr)
% Number of type conns : 1408 (1408 >; 0 *; 0 +; 0 <<)
% Number of symbols : 254 ( 251 usr; 32 con; 0-3 aty)
% Number of variables : 4171 ( 586 ^;3503 !; 82 ?;4171 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:15:53.879
%------------------------------------------------------------------------------
% Could-be-implicit typings (26)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J_J,type,
set_se7082751667426497518st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J,type,
set_se5258582372428582328st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J_J_J,type,
set_se3806740948107030918omplex: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
set_set_set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
set_set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J_J,type,
set_set_set_set_int: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J_J,type,
set_set_set_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
set_set_set_int: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
set_set_complex: $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__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__Extended____Nonnegative____Real__Oennreal,type,
extend8495563244428889912nnreal: $tType ).
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $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__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (251)
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_Eo,type,
complete_Inf_Inf_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Int__Oint,type,
complete_Inf_Inf_int: set_int > int ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
complete_Inf_Inf_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Real__Oreal,type,
comple4887499456419720421f_real: set_real > real ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_I_Eo_J,type,
comple3063163877087187839_set_o: set_set_o > set_o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Complex__Ocomplex_J,type,
comple2956690151646016541omplex: set_set_complex > set_complex ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Int__Oint_J,type,
comple3628384868704368283et_int: set_set_int > set_int ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
comple184543376406953807st_nat: set_set_list_nat > set_list_nat ).
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_OInf__class_OInf_001t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
comple6723625652910419923omplex: set_set_set_complex > set_set_complex ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
comple7798297522565507409et_int: set_set_set_int > set_set_int ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
comple8462666950445340293st_nat: set_set_set_list_nat > set_set_list_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
comple1065008630642458357et_nat: set_set_set_nat > set_set_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J_J,type,
comple5875334597002396809omplex: set_se3806740948107030918omplex > set_set_set_complex ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
comple3254194022943978759et_int: set_set_set_set_int > set_set_set_int ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
comple5189992959352112827st_nat: set_se5258582372428582328st_nat > set_set_set_list_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
comple8067742441731897515et_nat: set_set_set_set_nat > set_set_set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
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_Finite__Set_Ocard_001_Eo,type,
finite_card_o: set_o > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
finite_card_complex: set_complex > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
finite_card_int: set_int > nat ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
finite_card_list_nat: set_list_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Real__Oreal,type,
finite_card_real: set_real > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Complex__Ocomplex_J,type,
finite903997441450111292omplex: set_set_complex > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Int__Oint_J,type,
finite_card_set_int: set_set_int > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
finite2364142230527598318st_nat: set_set_list_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
finite_card_set_nat: set_set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
finite6806414675520991474omplex: set_set_set_complex > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
finite7882580182802147440et_int: set_set_set_int > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
finite5070363488328301092st_nat: set_set_set_list_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
finite1149291290879098388et_nat: set_set_set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
finite3316566092449558106st_nat: set_se5258582372428582328st_nat > nat ).
thf(sy_c_Finite__Set_Ofinite_001_Eo,type,
finite_finite_o: set_o > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
finite3207457112153483333omplex: set_complex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
finite_finite_int: set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
finite8100373058378681591st_nat: set_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
finite_finite_real: set_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Complex__Ocomplex_J,type,
finite6551019134538273531omplex: set_set_complex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Int__Oint_J,type,
finite6197958912794628473et_int: set_set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
finite7047420756378620717st_nat: set_set_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
finite8937801997843863217omplex: set_set_set_complex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
finite4249678464180374575et_int: set_set_set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
finite1703049307278766691st_nat: set_set_set_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
finite6739761609112101331et_nat: set_set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
finite4405637976927726233st_nat: set_se5258582372428582328st_nat > $o ).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
one_one_complex: complex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nonnegative____Real__Oennreal,type,
one_on2969667320475766781nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
plus_plus_complex: complex > complex > complex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nonnegative____Real__Oennreal,type,
plus_p1859984266308609217nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
times_times_complex: complex > complex > complex ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
uminus1482373934393186551omplex: complex > complex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
zero_zero_complex: complex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nonnegative____Real__Oennreal,type,
zero_z7100319975126383169nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_Eo_001t__Complex__Ocomplex,type,
groups5328290441151304332omplex: ( $o > complex ) > set_o > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_Eo_001t__Extended____Nonnegative____Real__Oennreal,type,
groups7456689898616286486nnreal: ( $o > extend8495563244428889912nnreal ) > set_o > extend8495563244428889912nnreal ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_Eo_001t__Int__Oint,type,
groups8505340233167759370_o_int: ( $o > int ) > set_o > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_Eo_001t__Nat__Onat,type,
groups8507830703676809646_o_nat: ( $o > nat ) > set_o > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_Eo_001t__Real__Oreal,type,
groups8691415230153176458o_real: ( $o > real ) > set_o > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
groups7754918857620584856omplex: ( complex > complex ) > set_complex > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Extended____Nonnegative____Real__Oennreal,type,
groups6103019165529820194nnreal: ( complex > extend8495563244428889912nnreal ) > set_complex > extend8495563244428889912nnreal ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Int__Oint,type,
groups5690904116761175830ex_int: ( complex > int ) > set_complex > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Nat__Onat,type,
groups5693394587270226106ex_nat: ( complex > nat ) > set_complex > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Real__Oreal,type,
groups5808333547571424918x_real: ( complex > real ) > set_complex > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Complex__Ocomplex,type,
groups3049146728041665814omplex: ( int > complex ) > set_int > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Extended____Nonnegative____Real__Oennreal,type,
groups2558975329500312480nnreal: ( int > extend8495563244428889912nnreal ) > set_int > extend8495563244428889912nnreal ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Int__Oint,type,
groups4538972089207619220nt_int: ( int > int ) > set_int > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Nat__Onat,type,
groups4541462559716669496nt_nat: ( int > nat ) > set_int > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Real__Oreal,type,
groups8778361861064173332t_real: ( int > real ) > set_int > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__List__Olist_It__Nat__Onat_J_001t__Extended____Nonnegative____Real__Oennreal,type,
groups5253920722037313236nnreal: ( list_nat > extend8495563244428889912nnreal ) > set_list_nat > extend8495563244428889912nnreal ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__List__Olist_It__Nat__Onat_J_001t__Int__Oint,type,
groups4393565826250045896at_int: ( list_nat > int ) > set_list_nat > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
groups4396056296759096172at_nat: ( list_nat > nat ) > set_list_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__List__Olist_It__Nat__Onat_J_001t__Real__Oreal,type,
groups8399112307953289288t_real: ( list_nat > real ) > set_list_nat > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Complex__Ocomplex,type,
groups2073611262835488442omplex: ( nat > complex ) > set_nat > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Extended____Nonnegative____Real__Oennreal,type,
groups4868793261593263428nnreal: ( nat > extend8495563244428889912nnreal ) > set_nat > extend8495563244428889912nnreal ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Int__Oint,type,
groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Complex__Ocomplex,type,
groups5754745047067104278omplex: ( real > complex ) > set_real > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Complex__Ocomplex_J_001t__Int__Oint,type,
groups8756346999278610892ex_int: ( set_complex > int ) > set_set_complex > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Complex__Ocomplex_J_001t__Nat__Onat,type,
groups8758837469787661168ex_nat: ( set_complex > nat ) > set_set_complex > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Int__Oint_J_001t__Nat__Onat,type,
groups1258547046268367342nt_nat: ( set_int > nat ) > set_set_int > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__Int__Oint,type,
groups7312845317294741502at_int: ( set_list_nat > int ) > set_set_list_nat > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__Nat__Onat,type,
groups7315335787803791778at_nat: ( set_list_nat > nat ) > set_set_list_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__Real__Oreal,type,
groups6630292279250676606t_real: ( set_list_nat > real ) > set_set_list_nat > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
groups8294997508430121362at_nat: ( set_nat > nat ) > set_set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J_001t__Int__Oint,type,
groups9026133298672633218ex_int: ( set_set_complex > int ) > set_set_set_complex > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001t__Int__Oint,type,
groups1080061135233207040nt_int: ( set_set_int > int ) > set_set_set_int > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001t__Nat__Onat,type,
groups1082551605742257316nt_nat: ( set_set_int > nat ) > set_set_set_int > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_001t__Int__Oint,type,
groups7004213669654646580at_int: ( set_set_list_nat > int ) > set_set_set_list_nat > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_001t__Nat__Onat,type,
groups7006704140163696856at_nat: ( set_set_list_nat > nat ) > set_set_set_list_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Int__Oint,type,
groups7084729577923612836at_int: ( set_set_nat > int ) > set_set_set_nat > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Nat__Onat,type,
groups7087220048432663112at_nat: ( set_set_nat > nat ) > set_set_set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J_J_001t__Int__Oint,type,
groups988161218598451768ex_int: ( set_set_set_complex > int ) > set_se3806740948107030918omplex > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J_001t__Int__Oint,type,
groups256680394374805174nt_int: ( set_set_set_int > int ) > set_set_set_set_int > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_001t__Int__Oint,type,
groups8053974158537219946at_int: ( set_set_set_list_nat > int ) > set_se5258582372428582328st_nat > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001t__Int__Oint,type,
groups4601141224429267546at_int: ( set_set_set_nat > int ) > set_set_set_set_nat > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J_001t__Int__Oint,type,
groups2272353940797458592at_int: ( set_se5258582372428582328st_nat > int ) > set_se7082751667426497518st_nat > int ).
thf(sy_c_If_001t__Extended____Nonnegative____Real__Oennreal,type,
if_Ext9135588136721118450nnreal: $o > extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
semiri8010041392384452111omplex: nat > complex ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nonnegative____Real__Oennreal,type,
semiri6283507881447550617nnreal: nat > extend8495563244428889912nnreal ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nonnegative____Real__Oennreal,type,
bot_bo841427958541957580nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
bot_bot_set_o: set_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
bot_bot_set_int: set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
bot_bot_set_list_nat: set_list_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
bot_bo4474773400535771566omplex: set_set_complex ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
bot_bot_set_set_int: set_set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
bot_bo3886227569956363488st_nat: set_set_list_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J_J,type,
bot_bo92361985942245988omplex: set_set_set_complex ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
bot_bo2384636101374064866et_int: set_set_set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
bot_bo3499706412017099030st_nat: set_set_set_list_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
bot_bo7198184520161983622et_nat: set_set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J,type,
bot_bo1158166727579713100st_nat: set_se5258582372428582328st_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nonnegative____Real__Oennreal,type,
ord_le3935885782089961368nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
ord_le211207098394363844omplex: set_complex > set_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
ord_le4750530260501030778omplex: set_set_complex > set_set_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
ord_le4403425263959731960et_int: set_set_int > set_set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
ord_le1068707526560357548st_nat: set_set_list_nat > set_set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J_J,type,
ord_le314291461425487920omplex: set_set_set_complex > set_set_set_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
ord_le4317611570275147438et_int: set_set_set_int > set_set_set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
ord_le7100322305783427298st_nat: set_set_set_list_nat > set_set_set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le9131159989063066194et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J,type,
ord_le2499698639687704088st_nat: set_se5258582372428582328st_nat > set_se5258582372428582328st_nat > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
power_power_complex: complex > nat > complex ).
thf(sy_c_Power_Opower__class_Opower_001t__Extended____Nonnegative____Real__Oennreal,type,
power_6007165696250533058nnreal: extend8495563244428889912nnreal > nat > extend8495563244428889912nnreal ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
collect_set_list_nat: ( set_list_nat > $o ) > set_set_list_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
collec2434422415211999471omplex: ( set_set_complex > $o ) > set_set_set_complex ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
collect_set_set_int: ( set_set_int > $o ) > set_set_set_int ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
collec4691811733418234273st_nat: ( set_set_list_nat > $o ) > set_set_set_list_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
collect_set_set_nat: ( set_set_nat > $o ) > set_set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J_J,type,
collec7826410564325299621omplex: ( set_set_set_complex > $o ) > set_se3806740948107030918omplex ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
collec2387904720390651427et_int: ( set_set_set_int > $o ) > set_set_set_set_int ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
collec696328324557263319st_nat: ( set_set_set_list_nat > $o ) > set_se5258582372428582328st_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
collec7201453139178570183et_nat: ( set_set_set_nat > $o ) > set_set_set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J,type,
collec5980373920241216269st_nat: ( set_se5258582372428582328st_nat > $o ) > set_se7082751667426497518st_nat ).
thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Complex__Ocomplex,type,
image_o_complex: ( $o > complex ) > set_o > set_complex ).
thf(sy_c_Set_Oimage_001_Eo_001t__Int__Oint,type,
image_o_int: ( $o > int ) > set_o > set_int ).
thf(sy_c_Set_Oimage_001_Eo_001t__Nat__Onat,type,
image_o_nat: ( $o > nat ) > set_o > set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Real__Oreal,type,
image_o_real: ( $o > real ) > set_o > set_real ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_I_Eo_J,type,
image_o_set_o: ( $o > set_o ) > set_o > set_set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Complex__Ocomplex_J,type,
image_o_set_complex: ( $o > set_complex ) > set_o > set_set_complex ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Int__Oint_J,type,
image_o_set_int: ( $o > set_int ) > set_o > set_set_int ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
image_o_set_list_nat: ( $o > set_list_nat ) > set_o > set_set_list_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
image_o_set_nat: ( $o > set_nat ) > set_o > set_set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
image_1184936479116113517omplex: ( $o > set_set_complex ) > set_o > set_set_set_complex ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__Nat__Onat,type,
image_complex_nat: ( complex > nat ) > set_complex > set_nat ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Complex__Ocomplex,type,
image_int_complex: ( int > complex ) > set_int > set_complex ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
image_int_nat: ( int > nat ) > set_int > set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001_Eo,type,
image_list_nat_o: ( list_nat > $o ) > set_list_nat > set_o ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
image_8532145185254316925st_nat: ( list_nat > set_list_nat ) > set_list_nat > set_set_list_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
image_nat_o: ( nat > $o ) > set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Complex__Ocomplex,type,
image_nat_complex: ( nat > complex ) > set_nat > set_complex ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
image_nat_real: ( nat > real ) > set_nat > set_real ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_I_Eo_J,type,
image_nat_set_o: ( nat > set_o ) > set_nat > set_set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Complex__Ocomplex_J,type,
image_6594795319511438139omplex: ( nat > set_complex ) > set_nat > set_set_complex ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Int__Oint_J,type,
image_nat_set_int: ( nat > set_int ) > set_nat > set_set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
image_2883343038133793645st_nat: ( nat > set_list_nat ) > set_nat > set_set_list_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001_Eo,type,
image_set_o_o: ( set_o > $o ) > set_set_o > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Complex__Ocomplex_J_001_Eo,type,
image_set_complex_o: ( set_complex > $o ) > set_set_complex > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Complex__Ocomplex_J_001t__Set__Oset_It__Complex__Ocomplex_J,type,
image_7998606247489673935omplex: ( set_complex > set_complex ) > set_set_complex > set_set_complex ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001_Eo,type,
image_set_int_o: ( set_int > $o ) > set_set_int > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
image_524474410958335435et_int: ( set_int > set_int ) > set_set_int > set_set_int ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001_Eo,type,
image_set_list_nat_o: ( set_list_nat > $o ) > set_set_list_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
image_5143090206295581363st_nat: ( set_list_nat > set_list_nat ) > set_set_list_nat > set_set_list_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
image_set_nat_o: ( set_nat > $o ) > set_set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_001_Eo,type,
image_7778788071572860293_nat_o: ( set_set_list_nat > $o ) > set_set_set_list_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
image_799823415861195295st_nat: ( set_set_list_nat > set_set_list_nat ) > set_set_set_list_nat > set_set_set_list_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J_J_001_Eo,type,
image_3081631015407073793plex_o: ( set_set_set_complex > $o ) > set_se3806740948107030918omplex > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J_001_Eo,type,
image_2622566851371609091_int_o: ( set_set_set_int > $o ) > set_set_set_set_int > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_001_Eo,type,
image_3035193717992370127_nat_o: ( set_set_set_list_nat > $o ) > set_se5258582372428582328st_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001_Eo,type,
image_3488003393078953823_nat_o: ( set_set_set_nat > $o ) > set_set_set_set_nat > set_o ).
thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
arsinh_real: real > real ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $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__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__Complex__Ocomplex_J,type,
member_set_complex: set_complex > set_set_complex > $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__List__Olist_It__Nat__Onat_J_J,type,
member_set_list_nat: set_list_nat > set_set_list_nat > $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__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
member9015044028964487601omplex: set_set_complex > set_set_set_complex > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
member_set_set_int: set_set_int > set_set_set_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
member1029098694177496419st_nat: set_set_list_nat > set_set_set_list_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
member7304678173793621401st_nat: set_set_set_list_nat > set_se5258582372428582328st_nat > $o ).
thf(sy_v_C____,type,
c: nat > list_nat > set_list_nat ).
thf(sy_v_SUM1____,type,
sUM1: int ).
thf(sy_v_SUM2____,type,
sUM2: int ).
thf(sy_v_X____,type,
x: set_list_nat ).
thf(sy_v_n____,type,
n: nat ).
% Relevant facts (1174)
thf(fact_0__092_060open_062finite_AX_092_060close_062,axiom,
finite8100373058378681591st_nat @ x ).
% \<open>finite X\<close>
thf(fact_1_finC,axiom,
! [N: nat,X: list_nat] : ( finite8100373058378681591st_nat @ ( c @ N @ X ) ) ).
% finC
thf(fact_2_card__UNION__nonneg,axiom,
! [A: set_set_set_complex] :
( ( finite8937801997843863217omplex @ A )
=> ( ! [X2: set_set_complex] :
( ( member9015044028964487601omplex @ X2 @ A )
=> ( finite6551019134538273531omplex @ X2 ) )
=> ( ord_less_eq_int @ zero_zero_int
@ ( groups988161218598451768ex_int
@ ^ [I: set_set_set_complex] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ ( finite6806414675520991474omplex @ I ) @ one_one_nat ) ) @ ( semiri1314217659103216013at_int @ ( finite903997441450111292omplex @ ( comple6723625652910419923omplex @ I ) ) ) )
@ ( collec7826410564325299621omplex
@ ^ [I: set_set_set_complex] :
( ( ord_le314291461425487920omplex @ I @ A )
& ( I != bot_bo92361985942245988omplex ) ) ) ) ) ) ) ).
% card_UNION_nonneg
thf(fact_3_card__UNION__nonneg,axiom,
! [A: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A )
=> ( finite1152437895449049373et_nat @ X2 ) )
=> ( ord_less_eq_int @ zero_zero_int
@ ( groups4601141224429267546at_int
@ ^ [I: set_set_set_nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ ( finite1149291290879098388et_nat @ I ) @ one_one_nat ) ) @ ( semiri1314217659103216013at_int @ ( finite_card_set_nat @ ( comple1065008630642458357et_nat @ I ) ) ) )
@ ( collec7201453139178570183et_nat
@ ^ [I: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ I @ A )
& ( I != bot_bo7198184520161983622et_nat ) ) ) ) ) ) ) ).
% card_UNION_nonneg
thf(fact_4_card__UNION__nonneg,axiom,
! [A: set_set_set_int] :
( ( finite4249678464180374575et_int @ A )
=> ( ! [X2: set_set_int] :
( ( member_set_set_int @ X2 @ A )
=> ( finite6197958912794628473et_int @ X2 ) )
=> ( ord_less_eq_int @ zero_zero_int
@ ( groups256680394374805174nt_int
@ ^ [I: set_set_set_int] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ ( finite7882580182802147440et_int @ I ) @ one_one_nat ) ) @ ( semiri1314217659103216013at_int @ ( finite_card_set_int @ ( comple7798297522565507409et_int @ I ) ) ) )
@ ( collec2387904720390651427et_int
@ ^ [I: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ I @ A )
& ( I != bot_bo2384636101374064866et_int ) ) ) ) ) ) ) ).
% card_UNION_nonneg
thf(fact_5_card__UNION__nonneg,axiom,
! [A: set_se5258582372428582328st_nat] :
( ( finite4405637976927726233st_nat @ A )
=> ( ! [X2: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X2 @ A )
=> ( finite1703049307278766691st_nat @ X2 ) )
=> ( ord_less_eq_int @ zero_zero_int
@ ( groups2272353940797458592at_int
@ ^ [I: set_se5258582372428582328st_nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ ( finite3316566092449558106st_nat @ I ) @ one_one_nat ) ) @ ( semiri1314217659103216013at_int @ ( finite5070363488328301092st_nat @ ( comple5189992959352112827st_nat @ I ) ) ) )
@ ( collec5980373920241216269st_nat
@ ^ [I: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ I @ A )
& ( I != bot_bo1158166727579713100st_nat ) ) ) ) ) ) ) ).
% card_UNION_nonneg
thf(fact_6_card__UNION__nonneg,axiom,
! [A: set_set_set_list_nat] :
( ( finite1703049307278766691st_nat @ A )
=> ( ! [X2: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X2 @ A )
=> ( finite7047420756378620717st_nat @ X2 ) )
=> ( ord_less_eq_int @ zero_zero_int
@ ( groups8053974158537219946at_int
@ ^ [I: set_set_set_list_nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ ( finite5070363488328301092st_nat @ I ) @ one_one_nat ) ) @ ( semiri1314217659103216013at_int @ ( finite2364142230527598318st_nat @ ( comple8462666950445340293st_nat @ I ) ) ) )
@ ( collec696328324557263319st_nat
@ ^ [I: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ I @ A )
& ( I != bot_bo3499706412017099030st_nat ) ) ) ) ) ) ) ).
% card_UNION_nonneg
thf(fact_7_card__UNION__nonneg,axiom,
! [A: set_set_int] :
( ( finite6197958912794628473et_int @ A )
=> ( ! [X2: set_int] :
( ( member_set_int @ X2 @ A )
=> ( finite_finite_int @ X2 ) )
=> ( ord_less_eq_int @ zero_zero_int
@ ( groups1080061135233207040nt_int
@ ^ [I: set_set_int] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ ( finite_card_set_int @ I ) @ one_one_nat ) ) @ ( semiri1314217659103216013at_int @ ( finite_card_int @ ( comple3628384868704368283et_int @ I ) ) ) )
@ ( collect_set_set_int
@ ^ [I: set_set_int] :
( ( ord_le4403425263959731960et_int @ I @ A )
& ( I != bot_bot_set_set_int ) ) ) ) ) ) ) ).
% card_UNION_nonneg
thf(fact_8_card__UNION__nonneg,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( finite_finite_nat @ X2 ) )
=> ( ord_less_eq_int @ zero_zero_int
@ ( groups7084729577923612836at_int
@ ^ [I: set_set_nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ ( finite_card_set_nat @ I ) @ one_one_nat ) ) @ ( semiri1314217659103216013at_int @ ( finite_card_nat @ ( comple7806235888213564991et_nat @ I ) ) ) )
@ ( collect_set_set_nat
@ ^ [I: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ I @ A )
& ( I != bot_bot_set_set_nat ) ) ) ) ) ) ) ).
% card_UNION_nonneg
thf(fact_9_card__UNION__nonneg,axiom,
! [A: set_set_complex] :
( ( finite6551019134538273531omplex @ A )
=> ( ! [X2: set_complex] :
( ( member_set_complex @ X2 @ A )
=> ( finite3207457112153483333omplex @ X2 ) )
=> ( ord_less_eq_int @ zero_zero_int
@ ( groups9026133298672633218ex_int
@ ^ [I: set_set_complex] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ ( finite903997441450111292omplex @ I ) @ one_one_nat ) ) @ ( semiri1314217659103216013at_int @ ( finite_card_complex @ ( comple2956690151646016541omplex @ I ) ) ) )
@ ( collec2434422415211999471omplex
@ ^ [I: set_set_complex] :
( ( ord_le4750530260501030778omplex @ I @ A )
& ( I != bot_bo4474773400535771566omplex ) ) ) ) ) ) ) ).
% card_UNION_nonneg
thf(fact_10_card__UNION__nonneg,axiom,
! [A: set_set_list_nat] :
( ( finite7047420756378620717st_nat @ A )
=> ( ! [X2: set_list_nat] :
( ( member_set_list_nat @ X2 @ A )
=> ( finite8100373058378681591st_nat @ X2 ) )
=> ( ord_less_eq_int @ zero_zero_int
@ ( groups7004213669654646580at_int
@ ^ [I: set_set_list_nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ ( finite2364142230527598318st_nat @ I ) @ one_one_nat ) ) @ ( semiri1314217659103216013at_int @ ( finite_card_list_nat @ ( comple184543376406953807st_nat @ I ) ) ) )
@ ( collec4691811733418234273st_nat
@ ^ [I: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ I @ A )
& ( I != bot_bo3886227569956363488st_nat ) ) ) ) ) ) ) ).
% card_UNION_nonneg
thf(fact_11_SUM1__def,axiom,
( sUM1
= ( groups7004213669654646580at_int
@ ^ [I: set_set_list_nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ ( finite2364142230527598318st_nat @ I ) @ one_one_nat ) ) @ ( semiri1314217659103216013at_int @ ( finite_card_list_nat @ ( comple184543376406953807st_nat @ I ) ) ) )
@ ( collec4691811733418234273st_nat
@ ^ [I: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ I @ ( image_8532145185254316925st_nat @ ( c @ n ) @ x ) )
& ( I != bot_bo3886227569956363488st_nat ) ) ) ) ) ).
% SUM1_def
thf(fact_12_sum__constant,axiom,
! [Y: nat,A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [X3: nat] : Y
@ A )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_nat @ A ) ) @ Y ) ) ).
% sum_constant
thf(fact_13_sum__constant,axiom,
! [Y: int,A: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [X3: nat] : Y
@ A )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_nat @ A ) ) @ Y ) ) ).
% sum_constant
thf(fact_14_sum__constant,axiom,
! [Y: real,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X3: nat] : Y
@ A )
= ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_nat @ A ) ) @ Y ) ) ).
% sum_constant
thf(fact_15_sum__constant,axiom,
! [Y: nat,A: set_list_nat] :
( ( groups4396056296759096172at_nat
@ ^ [X3: list_nat] : Y
@ A )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_list_nat @ A ) ) @ Y ) ) ).
% sum_constant
thf(fact_16_sum__constant,axiom,
! [Y: int,A: set_list_nat] :
( ( groups4393565826250045896at_int
@ ^ [X3: list_nat] : Y
@ A )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_list_nat @ A ) ) @ Y ) ) ).
% sum_constant
thf(fact_17_sum__constant,axiom,
! [Y: real,A: set_list_nat] :
( ( groups8399112307953289288t_real
@ ^ [X3: list_nat] : Y
@ A )
= ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_list_nat @ A ) ) @ Y ) ) ).
% sum_constant
thf(fact_18_sum__constant,axiom,
! [Y: nat,A: set_set_list_nat] :
( ( groups7315335787803791778at_nat
@ ^ [X3: set_list_nat] : Y
@ A )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite2364142230527598318st_nat @ A ) ) @ Y ) ) ).
% sum_constant
thf(fact_19_sum__constant,axiom,
! [Y: real,A: set_set_list_nat] :
( ( groups6630292279250676606t_real
@ ^ [X3: set_list_nat] : Y
@ A )
= ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite2364142230527598318st_nat @ A ) ) @ Y ) ) ).
% sum_constant
thf(fact_20_sum__constant,axiom,
! [Y: int,A: set_set_list_nat] :
( ( groups7312845317294741502at_int
@ ^ [X3: set_list_nat] : Y
@ A )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite2364142230527598318st_nat @ A ) ) @ Y ) ) ).
% sum_constant
thf(fact_21_sum__constant,axiom,
! [Y: int,A: set_set_set_list_nat] :
( ( groups7004213669654646580at_int
@ ^ [X3: set_set_list_nat] : Y
@ A )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite5070363488328301092st_nat @ A ) ) @ Y ) ) ).
% sum_constant
thf(fact_22_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
= one_one_complex ) ).
% minus_one_mult_self
thf(fact_23_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
= one_one_int ) ).
% minus_one_mult_self
thf(fact_24_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
= one_one_real ) ).
% minus_one_mult_self
thf(fact_25_left__minus__one__mult__self,axiom,
! [N: nat,A2: complex] :
( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A2 ) )
= A2 ) ).
% left_minus_one_mult_self
thf(fact_26_left__minus__one__mult__self,axiom,
! [N: nat,A2: int] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A2 ) )
= A2 ) ).
% left_minus_one_mult_self
thf(fact_27_left__minus__one__mult__self,axiom,
! [N: nat,A2: real] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ A2 ) )
= A2 ) ).
% left_minus_one_mult_self
thf(fact_28_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_29_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_30_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_31_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_32_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_33_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_34_INF__const,axiom,
! [A: set_nat,F: $o] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [I2: nat] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_35_INF__const,axiom,
! [A: set_list_nat,F: $o] :
( ( A != bot_bot_set_list_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_list_nat_o
@ ^ [I2: list_nat] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_36_INF__const,axiom,
! [A: set_nat,F: set_list_nat] :
( ( A != bot_bot_set_nat )
=> ( ( comple184543376406953807st_nat
@ ( image_2883343038133793645st_nat
@ ^ [I2: nat] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_37_INF__const,axiom,
! [A: set_set_list_nat,F: $o] :
( ( A != bot_bo3886227569956363488st_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_set_list_nat_o
@ ^ [I2: set_list_nat] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_38_INF__const,axiom,
! [A: set_list_nat,F: set_list_nat] :
( ( A != bot_bot_set_list_nat )
=> ( ( comple184543376406953807st_nat
@ ( image_8532145185254316925st_nat
@ ^ [I2: list_nat] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_39_INF__const,axiom,
! [A: set_set_list_nat,F: set_list_nat] :
( ( A != bot_bo3886227569956363488st_nat )
=> ( ( comple184543376406953807st_nat
@ ( image_5143090206295581363st_nat
@ ^ [I2: set_list_nat] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_40_INF__const,axiom,
! [A: set_set_complex,F: $o] :
( ( A != bot_bo4474773400535771566omplex )
=> ( ( complete_Inf_Inf_o
@ ( image_set_complex_o
@ ^ [I2: set_complex] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_41_INF__const,axiom,
! [A: set_set_nat,F: $o] :
( ( A != bot_bot_set_set_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_set_nat_o
@ ^ [I2: set_nat] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_42_INF__const,axiom,
! [A: set_set_int,F: $o] :
( ( A != bot_bot_set_set_int )
=> ( ( complete_Inf_Inf_o
@ ( image_set_int_o
@ ^ [I2: set_int] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_43_INF__const,axiom,
! [A: set_nat,F: set_complex] :
( ( A != bot_bot_set_nat )
=> ( ( comple2956690151646016541omplex
@ ( image_6594795319511438139omplex
@ ^ [I2: nat] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_44_ccINF__const,axiom,
! [A: set_nat,F: $o] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [I2: nat] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_45_ccINF__const,axiom,
! [A: set_list_nat,F: $o] :
( ( A != bot_bot_set_list_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_list_nat_o
@ ^ [I2: list_nat] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_46_ccINF__const,axiom,
! [A: set_set_complex,F: $o] :
( ( A != bot_bo4474773400535771566omplex )
=> ( ( complete_Inf_Inf_o
@ ( image_set_complex_o
@ ^ [I2: set_complex] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_47_ccINF__const,axiom,
! [A: set_set_nat,F: $o] :
( ( A != bot_bot_set_set_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_set_nat_o
@ ^ [I2: set_nat] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_48_ccINF__const,axiom,
! [A: set_set_int,F: $o] :
( ( A != bot_bot_set_set_int )
=> ( ( complete_Inf_Inf_o
@ ( image_set_int_o
@ ^ [I2: set_int] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_49_ccINF__const,axiom,
! [A: set_nat,F: set_complex] :
( ( A != bot_bot_set_nat )
=> ( ( comple2956690151646016541omplex
@ ( image_6594795319511438139omplex
@ ^ [I2: nat] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_50_ccINF__const,axiom,
! [A: set_nat,F: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [I2: nat] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_51_ccINF__const,axiom,
! [A: set_nat,F: set_int] :
( ( A != bot_bot_set_nat )
=> ( ( comple3628384868704368283et_int
@ ( image_nat_set_int
@ ^ [I2: nat] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_52_ccINF__const,axiom,
! [A: set_nat,F: set_list_nat] :
( ( A != bot_bot_set_nat )
=> ( ( comple184543376406953807st_nat
@ ( image_2883343038133793645st_nat
@ ^ [I2: nat] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_53_ccINF__const,axiom,
! [A: set_set_list_nat,F: $o] :
( ( A != bot_bo3886227569956363488st_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_set_list_nat_o
@ ^ [I2: set_list_nat] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_54_cINF__const,axiom,
! [A: set_nat,C: $o] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [X3: nat] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_55_cINF__const,axiom,
! [A: set_int,C: int] :
( ( A != bot_bot_set_int )
=> ( ( complete_Inf_Inf_int
@ ( image_int_int
@ ^ [X3: int] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_56_cINF__const,axiom,
! [A: set_nat,C: int] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_int
@ ( image_nat_int
@ ^ [X3: nat] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_57_cINF__const,axiom,
! [A: set_nat,C: nat] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_nat
@ ( image_nat_nat
@ ^ [X3: nat] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_58_cINF__const,axiom,
! [A: set_real,C: real] :
( ( A != bot_bot_set_real )
=> ( ( comple4887499456419720421f_real
@ ( image_real_real
@ ^ [X3: real] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_59_cINF__const,axiom,
! [A: set_nat,C: real] :
( ( A != bot_bot_set_nat )
=> ( ( comple4887499456419720421f_real
@ ( image_nat_real
@ ^ [X3: nat] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_60_cINF__const,axiom,
! [A: set_list_nat,C: $o] :
( ( A != bot_bot_set_list_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_list_nat_o
@ ^ [X3: list_nat] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_61_cINF__const,axiom,
! [A: set_set_complex,C: $o] :
( ( A != bot_bo4474773400535771566omplex )
=> ( ( complete_Inf_Inf_o
@ ( image_set_complex_o
@ ^ [X3: set_complex] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_62_cINF__const,axiom,
! [A: set_set_nat,C: $o] :
( ( A != bot_bot_set_set_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_set_nat_o
@ ^ [X3: set_nat] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_63_cINF__const,axiom,
! [A: set_set_int,C: $o] :
( ( A != bot_bot_set_set_int )
=> ( ( complete_Inf_Inf_o
@ ( image_set_int_o
@ ^ [X3: set_int] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_64_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_65_of__nat__1,axiom,
( ( semiri8010041392384452111omplex @ one_one_nat )
= one_one_complex ) ).
% of_nat_1
thf(fact_66_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_67_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_68_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_69_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_complex
= ( semiri8010041392384452111omplex @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_70_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_71_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_72_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_73_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri8010041392384452111omplex @ N )
= one_one_complex )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_74_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_75_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_76_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_77_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_78_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_79_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= ( semiri1316708129612266289at_nat @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_80_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_81_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_82_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_83_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_84_Inter__iff,axiom,
! [A: $o,C2: set_set_o] :
( ( member_o @ A @ ( comple3063163877087187839_set_o @ C2 ) )
= ( ! [X3: set_o] :
( ( member_set_o @ X3 @ C2 )
=> ( member_o @ A @ X3 ) ) ) ) ).
% Inter_iff
thf(fact_85_Inter__iff,axiom,
! [A: list_nat,C2: set_set_list_nat] :
( ( member_list_nat @ A @ ( comple184543376406953807st_nat @ C2 ) )
= ( ! [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ C2 )
=> ( member_list_nat @ A @ X3 ) ) ) ) ).
% Inter_iff
thf(fact_86_Inter__iff,axiom,
! [A: complex,C2: set_set_complex] :
( ( member_complex @ A @ ( comple2956690151646016541omplex @ C2 ) )
= ( ! [X3: set_complex] :
( ( member_set_complex @ X3 @ C2 )
=> ( member_complex @ A @ X3 ) ) ) ) ).
% Inter_iff
thf(fact_87_Inter__iff,axiom,
! [A: nat,C2: set_set_nat] :
( ( member_nat @ A @ ( comple7806235888213564991et_nat @ C2 ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ C2 )
=> ( member_nat @ A @ X3 ) ) ) ) ).
% Inter_iff
thf(fact_88_Inter__iff,axiom,
! [A: int,C2: set_set_int] :
( ( member_int @ A @ ( comple3628384868704368283et_int @ C2 ) )
= ( ! [X3: set_int] :
( ( member_set_int @ X3 @ C2 )
=> ( member_int @ A @ X3 ) ) ) ) ).
% Inter_iff
thf(fact_89_Inter__iff,axiom,
! [A: set_list_nat,C2: set_set_set_list_nat] :
( ( member_set_list_nat @ A @ ( comple8462666950445340293st_nat @ C2 ) )
= ( ! [X3: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X3 @ C2 )
=> ( member_set_list_nat @ A @ X3 ) ) ) ) ).
% Inter_iff
thf(fact_90_InterI,axiom,
! [C2: set_set_o,A: $o] :
( ! [X4: set_o] :
( ( member_set_o @ X4 @ C2 )
=> ( member_o @ A @ X4 ) )
=> ( member_o @ A @ ( comple3063163877087187839_set_o @ C2 ) ) ) ).
% InterI
thf(fact_91_InterI,axiom,
! [C2: set_set_list_nat,A: list_nat] :
( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ C2 )
=> ( member_list_nat @ A @ X4 ) )
=> ( member_list_nat @ A @ ( comple184543376406953807st_nat @ C2 ) ) ) ).
% InterI
thf(fact_92_InterI,axiom,
! [C2: set_set_complex,A: complex] :
( ! [X4: set_complex] :
( ( member_set_complex @ X4 @ C2 )
=> ( member_complex @ A @ X4 ) )
=> ( member_complex @ A @ ( comple2956690151646016541omplex @ C2 ) ) ) ).
% InterI
thf(fact_93_InterI,axiom,
! [C2: set_set_nat,A: nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ C2 )
=> ( member_nat @ A @ X4 ) )
=> ( member_nat @ A @ ( comple7806235888213564991et_nat @ C2 ) ) ) ).
% InterI
thf(fact_94_InterI,axiom,
! [C2: set_set_int,A: int] :
( ! [X4: set_int] :
( ( member_set_int @ X4 @ C2 )
=> ( member_int @ A @ X4 ) )
=> ( member_int @ A @ ( comple3628384868704368283et_int @ C2 ) ) ) ).
% InterI
thf(fact_95_InterI,axiom,
! [C2: set_set_set_list_nat,A: set_list_nat] :
( ! [X4: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X4 @ C2 )
=> ( member_set_list_nat @ A @ X4 ) )
=> ( member_set_list_nat @ A @ ( comple8462666950445340293st_nat @ C2 ) ) ) ).
% InterI
thf(fact_96_finite__interval__int1,axiom,
! [A2: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_eq_int @ A2 @ I2 )
& ( ord_less_eq_int @ I2 @ B ) ) ) ) ).
% finite_interval_int1
thf(fact_97_double__eq__0__iff,axiom,
! [A2: int] :
( ( ( plus_plus_int @ A2 @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_98_double__eq__0__iff,axiom,
! [A2: real] :
( ( ( plus_plus_real @ A2 @ A2 )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_99_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_100_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_101_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_102_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri6283507881447550617nnreal @ M )
= zero_z7100319975126383169nnreal )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_103_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_104_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_105_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_106_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_z7100319975126383169nnreal
= ( semiri6283507881447550617nnreal @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_107_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_108_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_109_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_110_of__nat__0,axiom,
( ( semiri6283507881447550617nnreal @ zero_zero_nat )
= zero_z7100319975126383169nnreal ) ).
% of_nat_0
thf(fact_111_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_112_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_113_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_114_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_115_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_116_power__one,axiom,
! [N: nat] :
( ( power_power_real @ one_one_real @ N )
= one_one_real ) ).
% power_one
thf(fact_117_power__one,axiom,
! [N: nat] :
( ( power_power_complex @ one_one_complex @ N )
= one_one_complex ) ).
% power_one
thf(fact_118_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N ) )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% of_nat_mult
thf(fact_119_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_120_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_mult
thf(fact_121_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_122_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X5: nat,B: nat,W: nat] :
( ( ( semiri8010041392384452111omplex @ X5 )
= ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
= ( X5
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_123_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X5: nat,B: nat,W: nat] :
( ( ( semiri1314217659103216013at_int @ X5 )
= ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( X5
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_124_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X5: nat,B: nat,W: nat] :
( ( ( semiri5074537144036343181t_real @ X5 )
= ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( X5
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_125_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X5: nat,B: nat,W: nat] :
( ( ( semiri1316708129612266289at_nat @ X5 )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( X5
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_126_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X5: nat] :
( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
= ( semiri8010041392384452111omplex @ X5 ) )
= ( ( power_power_nat @ B @ W )
= X5 ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_127_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X5: nat] :
( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
= ( semiri1314217659103216013at_int @ X5 ) )
= ( ( power_power_nat @ B @ W )
= X5 ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_128_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X5: nat] :
( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
= ( semiri5074537144036343181t_real @ X5 ) )
= ( ( power_power_nat @ B @ W )
= X5 ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_129_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X5: nat] :
( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
= ( semiri1316708129612266289at_nat @ X5 ) )
= ( ( power_power_nat @ B @ W )
= X5 ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_130_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N ) )
= ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N ) ) ).
% of_nat_power
thf(fact_131_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
= ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% of_nat_power
thf(fact_132_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
= ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).
% of_nat_power
thf(fact_133_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% of_nat_power
thf(fact_134_power__one__right,axiom,
! [A2: int] :
( ( power_power_int @ A2 @ one_one_nat )
= A2 ) ).
% power_one_right
thf(fact_135_power__one__right,axiom,
! [A2: nat] :
( ( power_power_nat @ A2 @ one_one_nat )
= A2 ) ).
% power_one_right
thf(fact_136_power__one__right,axiom,
! [A2: real] :
( ( power_power_real @ A2 @ one_one_nat )
= A2 ) ).
% power_one_right
thf(fact_137_power__one__right,axiom,
! [A2: complex] :
( ( power_power_complex @ A2 @ one_one_nat )
= A2 ) ).
% power_one_right
thf(fact_138_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_139_sum_Oneutral__const,axiom,
! [A: set_set_set_list_nat] :
( ( groups7004213669654646580at_int
@ ^ [Uu: set_set_list_nat] : zero_zero_int
@ A )
= zero_zero_int ) ).
% sum.neutral_const
thf(fact_140_sum_Oneutral__const,axiom,
! [A: set_set_list_nat] :
( ( groups7312845317294741502at_int
@ ^ [Uu: set_list_nat] : zero_zero_int
@ A )
= zero_zero_int ) ).
% sum.neutral_const
thf(fact_141_sum_Oneutral__const,axiom,
! [A: set_set_set_nat] :
( ( groups7084729577923612836at_int
@ ^ [Uu: set_set_nat] : zero_zero_int
@ A )
= zero_zero_int ) ).
% sum.neutral_const
thf(fact_142_sum_Oneutral__const,axiom,
! [A: set_set_set_int] :
( ( groups1080061135233207040nt_int
@ ^ [Uu: set_set_int] : zero_zero_int
@ A )
= zero_zero_int ) ).
% sum.neutral_const
thf(fact_143_sum_Oneutral__const,axiom,
! [A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [Uu: nat] : zero_zero_real
@ A )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_144_sum_Oneutral__const,axiom,
! [A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [Uu: nat] : zero_zero_nat
@ A )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_145_sum_Oneutral__const,axiom,
! [A: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [Uu: nat] : zero_zero_int
@ A )
= zero_zero_int ) ).
% sum.neutral_const
thf(fact_146_INF__identity__eq,axiom,
! [A: set_set_list_nat] :
( ( comple184543376406953807st_nat
@ ( image_5143090206295581363st_nat
@ ^ [X3: set_list_nat] : X3
@ A ) )
= ( comple184543376406953807st_nat @ A ) ) ).
% INF_identity_eq
thf(fact_147_INF__identity__eq,axiom,
! [A: set_o] :
( ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [X3: $o] : X3
@ A ) )
= ( complete_Inf_Inf_o @ A ) ) ).
% INF_identity_eq
thf(fact_148_INF__identity__eq,axiom,
! [A: set_int] :
( ( complete_Inf_Inf_int
@ ( image_int_int
@ ^ [X3: int] : X3
@ A ) )
= ( complete_Inf_Inf_int @ A ) ) ).
% INF_identity_eq
thf(fact_149_INF__identity__eq,axiom,
! [A: set_nat] :
( ( complete_Inf_Inf_nat
@ ( image_nat_nat
@ ^ [X3: nat] : X3
@ A ) )
= ( complete_Inf_Inf_nat @ A ) ) ).
% INF_identity_eq
thf(fact_150_INF__identity__eq,axiom,
! [A: set_real] :
( ( comple4887499456419720421f_real
@ ( image_real_real
@ ^ [X3: real] : X3
@ A ) )
= ( comple4887499456419720421f_real @ A ) ) ).
% INF_identity_eq
thf(fact_151_INF__identity__eq,axiom,
! [A: set_set_complex] :
( ( comple2956690151646016541omplex
@ ( image_7998606247489673935omplex
@ ^ [X3: set_complex] : X3
@ A ) )
= ( comple2956690151646016541omplex @ A ) ) ).
% INF_identity_eq
thf(fact_152_INF__identity__eq,axiom,
! [A: set_set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_7916887816326733075et_nat
@ ^ [X3: set_nat] : X3
@ A ) )
= ( comple7806235888213564991et_nat @ A ) ) ).
% INF_identity_eq
thf(fact_153_INF__identity__eq,axiom,
! [A: set_set_int] :
( ( comple3628384868704368283et_int
@ ( image_524474410958335435et_int
@ ^ [X3: set_int] : X3
@ A ) )
= ( comple3628384868704368283et_int @ A ) ) ).
% INF_identity_eq
thf(fact_154_INF__identity__eq,axiom,
! [A: set_set_set_list_nat] :
( ( comple8462666950445340293st_nat
@ ( image_799823415861195295st_nat
@ ^ [X3: set_set_list_nat] : X3
@ A ) )
= ( comple8462666950445340293st_nat @ A ) ) ).
% INF_identity_eq
thf(fact_155_of__nat__sum,axiom,
! [F: set_set_list_nat > nat,A: set_set_set_list_nat] :
( ( semiri1314217659103216013at_int @ ( groups7006704140163696856at_nat @ F @ A ) )
= ( groups7004213669654646580at_int
@ ^ [X3: set_set_list_nat] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
@ A ) ) ).
% of_nat_sum
thf(fact_156_of__nat__sum,axiom,
! [F: set_list_nat > nat,A: set_set_list_nat] :
( ( semiri1314217659103216013at_int @ ( groups7315335787803791778at_nat @ F @ A ) )
= ( groups7312845317294741502at_int
@ ^ [X3: set_list_nat] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
@ A ) ) ).
% of_nat_sum
thf(fact_157_of__nat__sum,axiom,
! [F: set_set_nat > nat,A: set_set_set_nat] :
( ( semiri1314217659103216013at_int @ ( groups7087220048432663112at_nat @ F @ A ) )
= ( groups7084729577923612836at_int
@ ^ [X3: set_set_nat] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
@ A ) ) ).
% of_nat_sum
thf(fact_158_of__nat__sum,axiom,
! [F: set_set_int > nat,A: set_set_set_int] :
( ( semiri1314217659103216013at_int @ ( groups1082551605742257316nt_nat @ F @ A ) )
= ( groups1080061135233207040nt_int
@ ^ [X3: set_set_int] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
@ A ) ) ).
% of_nat_sum
thf(fact_159_of__nat__sum,axiom,
! [F: nat > nat,A: set_nat] :
( ( semiri5074537144036343181t_real @ ( groups3542108847815614940at_nat @ F @ A ) )
= ( groups6591440286371151544t_real
@ ^ [X3: nat] : ( semiri5074537144036343181t_real @ ( F @ X3 ) )
@ A ) ) ).
% of_nat_sum
thf(fact_160_of__nat__sum,axiom,
! [F: nat > nat,A: set_nat] :
( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A ) )
= ( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( semiri1316708129612266289at_nat @ ( F @ X3 ) )
@ A ) ) ).
% of_nat_sum
thf(fact_161_of__nat__sum,axiom,
! [F: nat > nat,A: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A ) )
= ( groups3539618377306564664at_int
@ ^ [X3: nat] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
@ A ) ) ).
% of_nat_sum
thf(fact_162_INT__iff,axiom,
! [B: list_nat,B2: list_nat > set_list_nat,A: set_list_nat] :
( ( member_list_nat @ B @ ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ B2 @ A ) ) )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( member_list_nat @ B @ ( B2 @ X3 ) ) ) ) ) ).
% INT_iff
thf(fact_163_INT__iff,axiom,
! [B: list_nat,B2: nat > set_list_nat,A: set_nat] :
( ( member_list_nat @ B @ ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ B2 @ A ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_list_nat @ B @ ( B2 @ X3 ) ) ) ) ) ).
% INT_iff
thf(fact_164_INT__iff,axiom,
! [B: list_nat,B2: set_list_nat > set_list_nat,A: set_set_list_nat] :
( ( member_list_nat @ B @ ( comple184543376406953807st_nat @ ( image_5143090206295581363st_nat @ B2 @ A ) ) )
= ( ! [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A )
=> ( member_list_nat @ B @ ( B2 @ X3 ) ) ) ) ) ).
% INT_iff
thf(fact_165_INT__I,axiom,
! [A: set_o,B: $o,B2: $o > set_o] :
( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( member_o @ B @ ( B2 @ X2 ) ) )
=> ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_166_INT__I,axiom,
! [A: set_nat,B: $o,B2: nat > set_o] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_o @ B @ ( B2 @ X2 ) ) )
=> ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_nat_set_o @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_167_INT__I,axiom,
! [A: set_o,B: complex,B2: $o > set_complex] :
( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( member_complex @ B @ ( B2 @ X2 ) ) )
=> ( member_complex @ B @ ( comple2956690151646016541omplex @ ( image_o_set_complex @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_168_INT__I,axiom,
! [A: set_nat,B: complex,B2: nat > set_complex] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_complex @ B @ ( B2 @ X2 ) ) )
=> ( member_complex @ B @ ( comple2956690151646016541omplex @ ( image_6594795319511438139omplex @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_169_INT__I,axiom,
! [A: set_o,B: nat,B2: $o > set_nat] :
( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( member_nat @ B @ ( B2 @ X2 ) ) )
=> ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_170_INT__I,axiom,
! [A: set_nat,B: nat,B2: nat > set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ B @ ( B2 @ X2 ) ) )
=> ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_171_INT__I,axiom,
! [A: set_o,B: int,B2: $o > set_int] :
( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( member_int @ B @ ( B2 @ X2 ) ) )
=> ( member_int @ B @ ( comple3628384868704368283et_int @ ( image_o_set_int @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_172_INT__I,axiom,
! [A: set_nat,B: int,B2: nat > set_int] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_int @ B @ ( B2 @ X2 ) ) )
=> ( member_int @ B @ ( comple3628384868704368283et_int @ ( image_nat_set_int @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_173_INT__I,axiom,
! [A: set_o,B: list_nat,B2: $o > set_list_nat] :
( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( member_list_nat @ B @ ( B2 @ X2 ) ) )
=> ( member_list_nat @ B @ ( comple184543376406953807st_nat @ ( image_o_set_list_nat @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_174_INT__I,axiom,
! [A: set_nat,B: list_nat,B2: nat > set_list_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_list_nat @ B @ ( B2 @ X2 ) ) )
=> ( member_list_nat @ B @ ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_175_sum__squares__eq__zero__iff,axiom,
! [X5: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X5 @ X5 ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X5 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_176_sum__squares__eq__zero__iff,axiom,
! [X5: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X5 @ X5 ) @ ( times_times_real @ Y @ Y ) )
= zero_zero_real )
= ( ( X5 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_177_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_le3935885782089961368nnreal @ ( semiri6283507881447550617nnreal @ M ) @ zero_z7100319975126383169nnreal )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_178_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_179_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_180_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_181_mem__Collect__eq,axiom,
! [A2: $o,P: $o > $o] :
( ( member_o @ A2 @ ( collect_o @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_182_mem__Collect__eq,axiom,
! [A2: set_set_list_nat,P: set_set_list_nat > $o] :
( ( member1029098694177496419st_nat @ A2 @ ( collec4691811733418234273st_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_183_mem__Collect__eq,axiom,
! [A2: int,P: int > $o] :
( ( member_int @ A2 @ ( collect_int @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_184_mem__Collect__eq,axiom,
! [A2: set_list_nat,P: set_list_nat > $o] :
( ( member_set_list_nat @ A2 @ ( collect_set_list_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_185_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_186_mem__Collect__eq,axiom,
! [A2: complex,P: complex > $o] :
( ( member_complex @ A2 @ ( collect_complex @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_187_mem__Collect__eq,axiom,
! [A2: set_set_complex,P: set_set_complex > $o] :
( ( member9015044028964487601omplex @ A2 @ ( collec2434422415211999471omplex @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_188_mem__Collect__eq,axiom,
! [A2: set_set_nat,P: set_set_nat > $o] :
( ( member_set_set_nat @ A2 @ ( collect_set_set_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_189_mem__Collect__eq,axiom,
! [A2: set_set_int,P: set_set_int > $o] :
( ( member_set_set_int @ A2 @ ( collect_set_set_int @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_190_mem__Collect__eq,axiom,
! [A2: set_set_set_list_nat,P: set_set_set_list_nat > $o] :
( ( member7304678173793621401st_nat @ A2 @ ( collec696328324557263319st_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_191_Collect__mem__eq,axiom,
! [A: set_o] :
( ( collect_o
@ ^ [X3: $o] : ( member_o @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_192_Collect__mem__eq,axiom,
! [A: set_set_set_list_nat] :
( ( collec4691811733418234273st_nat
@ ^ [X3: set_set_list_nat] : ( member1029098694177496419st_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_193_Collect__mem__eq,axiom,
! [A: set_int] :
( ( collect_int
@ ^ [X3: int] : ( member_int @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_194_Collect__mem__eq,axiom,
! [A: set_set_list_nat] :
( ( collect_set_list_nat
@ ^ [X3: set_list_nat] : ( member_set_list_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_195_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_196_Collect__mem__eq,axiom,
! [A: set_complex] :
( ( collect_complex
@ ^ [X3: complex] : ( member_complex @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_197_Collect__mem__eq,axiom,
! [A: set_set_set_complex] :
( ( collec2434422415211999471omplex
@ ^ [X3: set_set_complex] : ( member9015044028964487601omplex @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_198_Collect__mem__eq,axiom,
! [A: set_set_set_nat] :
( ( collect_set_set_nat
@ ^ [X3: set_set_nat] : ( member_set_set_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_199_Collect__mem__eq,axiom,
! [A: set_set_set_int] :
( ( collect_set_set_int
@ ^ [X3: set_set_int] : ( member_set_set_int @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_200_Collect__mem__eq,axiom,
! [A: set_se5258582372428582328st_nat] :
( ( collec696328324557263319st_nat
@ ^ [X3: set_set_set_list_nat] : ( member7304678173793621401st_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_201_Collect__cong,axiom,
! [P: set_set_list_nat > $o,Q: set_set_list_nat > $o] :
( ! [X2: set_set_list_nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collec4691811733418234273st_nat @ P )
= ( collec4691811733418234273st_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_202_Collect__cong,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X2: int] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_int @ P )
= ( collect_int @ Q ) ) ) ).
% Collect_cong
thf(fact_203_Collect__cong,axiom,
! [P: set_list_nat > $o,Q: set_list_nat > $o] :
( ! [X2: set_list_nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_set_list_nat @ P )
= ( collect_set_list_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_204_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_205_Collect__cong,axiom,
! [P: complex > $o,Q: complex > $o] :
( ! [X2: complex] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_complex @ P )
= ( collect_complex @ Q ) ) ) ).
% Collect_cong
thf(fact_206_Collect__cong,axiom,
! [P: set_set_complex > $o,Q: set_set_complex > $o] :
( ! [X2: set_set_complex] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collec2434422415211999471omplex @ P )
= ( collec2434422415211999471omplex @ Q ) ) ) ).
% Collect_cong
thf(fact_207_Collect__cong,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ! [X2: set_set_nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_set_set_nat @ P )
= ( collect_set_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_208_Collect__cong,axiom,
! [P: set_set_int > $o,Q: set_set_int > $o] :
( ! [X2: set_set_int] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_set_set_int @ P )
= ( collect_set_set_int @ Q ) ) ) ).
% Collect_cong
thf(fact_209_Collect__cong,axiom,
! [P: set_set_set_list_nat > $o,Q: set_set_set_list_nat > $o] :
( ! [X2: set_set_set_list_nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collec696328324557263319st_nat @ P )
= ( collec696328324557263319st_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_210_mult__minus1__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1_right
thf(fact_211_mult__minus1__right,axiom,
! [Z: real] :
( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1_right
thf(fact_212_mult__minus1__right,axiom,
! [Z: complex] :
( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ Z ) ) ).
% mult_minus1_right
thf(fact_213_mult__minus1,axiom,
! [Z: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1
thf(fact_214_mult__minus1,axiom,
! [Z: real] :
( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1
thf(fact_215_mult__minus1,axiom,
! [Z: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
= ( uminus1482373934393186551omplex @ Z ) ) ).
% mult_minus1
thf(fact_216_sum_Oempty,axiom,
! [G: nat > extend8495563244428889912nnreal] :
( ( groups4868793261593263428nnreal @ G @ bot_bot_set_nat )
= zero_z7100319975126383169nnreal ) ).
% sum.empty
thf(fact_217_sum_Oempty,axiom,
! [G: nat > real] :
( ( groups6591440286371151544t_real @ G @ bot_bot_set_nat )
= zero_zero_real ) ).
% sum.empty
thf(fact_218_sum_Oempty,axiom,
! [G: nat > nat] :
( ( groups3542108847815614940at_nat @ G @ bot_bot_set_nat )
= zero_zero_nat ) ).
% sum.empty
thf(fact_219_sum_Oempty,axiom,
! [G: nat > int] :
( ( groups3539618377306564664at_int @ G @ bot_bot_set_nat )
= zero_zero_int ) ).
% sum.empty
thf(fact_220_sum_Oempty,axiom,
! [G: list_nat > int] :
( ( groups4393565826250045896at_int @ G @ bot_bot_set_list_nat )
= zero_zero_int ) ).
% sum.empty
thf(fact_221_sum_Oempty,axiom,
! [G: list_nat > nat] :
( ( groups4396056296759096172at_nat @ G @ bot_bot_set_list_nat )
= zero_zero_nat ) ).
% sum.empty
thf(fact_222_sum_Oempty,axiom,
! [G: list_nat > real] :
( ( groups8399112307953289288t_real @ G @ bot_bot_set_list_nat )
= zero_zero_real ) ).
% sum.empty
thf(fact_223_sum_Oempty,axiom,
! [G: list_nat > extend8495563244428889912nnreal] :
( ( groups5253920722037313236nnreal @ G @ bot_bot_set_list_nat )
= zero_z7100319975126383169nnreal ) ).
% sum.empty
thf(fact_224_sum_Oempty,axiom,
! [G: set_complex > int] :
( ( groups8756346999278610892ex_int @ G @ bot_bo4474773400535771566omplex )
= zero_zero_int ) ).
% sum.empty
thf(fact_225_sum_Oempty,axiom,
! [G: set_complex > nat] :
( ( groups8758837469787661168ex_nat @ G @ bot_bo4474773400535771566omplex )
= zero_zero_nat ) ).
% sum.empty
thf(fact_226_sum__eq__0__iff,axiom,
! [F2: set_int,F: int > nat] :
( ( finite_finite_int @ F2 )
=> ( ( ( groups4541462559716669496nt_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X3: int] :
( ( member_int @ X3 @ F2 )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_227_sum__eq__0__iff,axiom,
! [F2: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ F2 )
=> ( ( ( groups5693394587270226106ex_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X3: complex] :
( ( member_complex @ X3 @ F2 )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_228_sum__eq__0__iff,axiom,
! [F2: set_int,F: int > extend8495563244428889912nnreal] :
( ( finite_finite_int @ F2 )
=> ( ( ( groups2558975329500312480nnreal @ F @ F2 )
= zero_z7100319975126383169nnreal )
= ( ! [X3: int] :
( ( member_int @ X3 @ F2 )
=> ( ( F @ X3 )
= zero_z7100319975126383169nnreal ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_229_sum__eq__0__iff,axiom,
! [F2: set_nat,F: nat > extend8495563244428889912nnreal] :
( ( finite_finite_nat @ F2 )
=> ( ( ( groups4868793261593263428nnreal @ F @ F2 )
= zero_z7100319975126383169nnreal )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ F2 )
=> ( ( F @ X3 )
= zero_z7100319975126383169nnreal ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_230_sum__eq__0__iff,axiom,
! [F2: set_complex,F: complex > extend8495563244428889912nnreal] :
( ( finite3207457112153483333omplex @ F2 )
=> ( ( ( groups6103019165529820194nnreal @ F @ F2 )
= zero_z7100319975126383169nnreal )
= ( ! [X3: complex] :
( ( member_complex @ X3 @ F2 )
=> ( ( F @ X3 )
= zero_z7100319975126383169nnreal ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_231_sum__eq__0__iff,axiom,
! [F2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ F2 )
=> ( ( ( groups3542108847815614940at_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ F2 )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_232_sum__eq__0__iff,axiom,
! [F2: set_list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ F2 )
=> ( ( ( groups4396056296759096172at_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ F2 )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_233_sum__eq__0__iff,axiom,
! [F2: set_set_complex,F: set_complex > nat] :
( ( finite6551019134538273531omplex @ F2 )
=> ( ( ( groups8758837469787661168ex_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X3: set_complex] :
( ( member_set_complex @ X3 @ F2 )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_234_sum__eq__0__iff,axiom,
! [F2: set_set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ F2 )
=> ( ( ( groups8294997508430121362at_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ F2 )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_235_sum__eq__0__iff,axiom,
! [F2: set_set_int,F: set_int > nat] :
( ( finite6197958912794628473et_int @ F2 )
=> ( ( ( groups1258547046268367342nt_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X3: set_int] :
( ( member_set_int @ X3 @ F2 )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_236_sum_Oinfinite,axiom,
! [A: set_int,G: int > int] :
( ~ ( finite_finite_int @ A )
=> ( ( groups4538972089207619220nt_int @ G @ A )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_237_sum_Oinfinite,axiom,
! [A: set_complex,G: complex > int] :
( ~ ( finite3207457112153483333omplex @ A )
=> ( ( groups5690904116761175830ex_int @ G @ A )
= zero_zero_int ) ) ).
% sum.infinite
thf(fact_238_sum_Oinfinite,axiom,
! [A: set_int,G: int > nat] :
( ~ ( finite_finite_int @ A )
=> ( ( groups4541462559716669496nt_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_239_sum_Oinfinite,axiom,
! [A: set_complex,G: complex > nat] :
( ~ ( finite3207457112153483333omplex @ A )
=> ( ( groups5693394587270226106ex_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_240_sum_Oinfinite,axiom,
! [A: set_int,G: int > real] :
( ~ ( finite_finite_int @ A )
=> ( ( groups8778361861064173332t_real @ G @ A )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_241_sum_Oinfinite,axiom,
! [A: set_complex,G: complex > real] :
( ~ ( finite3207457112153483333omplex @ A )
=> ( ( groups5808333547571424918x_real @ G @ A )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_242_sum_Oinfinite,axiom,
! [A: set_int,G: int > extend8495563244428889912nnreal] :
( ~ ( finite_finite_int @ A )
=> ( ( groups2558975329500312480nnreal @ G @ A )
= zero_z7100319975126383169nnreal ) ) ).
% sum.infinite
thf(fact_243_sum_Oinfinite,axiom,
! [A: set_nat,G: nat > extend8495563244428889912nnreal] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups4868793261593263428nnreal @ G @ A )
= zero_z7100319975126383169nnreal ) ) ).
% sum.infinite
thf(fact_244_sum_Oinfinite,axiom,
! [A: set_complex,G: complex > extend8495563244428889912nnreal] :
( ~ ( finite3207457112153483333omplex @ A )
=> ( ( groups6103019165529820194nnreal @ G @ A )
= zero_z7100319975126383169nnreal ) ) ).
% sum.infinite
thf(fact_245_sum_Oinfinite,axiom,
! [A: set_nat,G: nat > real] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups6591440286371151544t_real @ G @ A )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_246_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X5: nat,B: nat,W: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X5 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( ord_less_eq_nat @ X5 @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_247_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X5: nat,B: nat,W: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X5 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( ord_less_eq_nat @ X5 @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_248_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X5: nat,B: nat,W: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X5 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( ord_less_eq_nat @ X5 @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_249_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X5: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X5 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X5 ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_250_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X5: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X5 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X5 ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_251_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X5: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X5 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X5 ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_252_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri6283507881447550617nnreal @ ( plus_plus_nat @ M @ N ) )
= ( plus_p1859984266308609217nnreal @ ( semiri6283507881447550617nnreal @ M ) @ ( semiri6283507881447550617nnreal @ N ) ) ) ).
% of_nat_add
thf(fact_253_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_254_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_add
thf(fact_255_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_256_sum__zero__power,axiom,
! [A: set_nat,C: nat > complex] :
( ( ( ( finite_finite_nat @ A )
& ( member_nat @ zero_zero_nat @ A ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I2: nat] : ( times_times_complex @ ( C @ I2 ) @ ( power_power_complex @ zero_zero_complex @ I2 ) )
@ A )
= ( C @ zero_zero_nat ) ) )
& ( ~ ( ( finite_finite_nat @ A )
& ( member_nat @ zero_zero_nat @ A ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I2: nat] : ( times_times_complex @ ( C @ I2 ) @ ( power_power_complex @ zero_zero_complex @ I2 ) )
@ A )
= zero_zero_complex ) ) ) ).
% sum_zero_power
thf(fact_257_sum__zero__power,axiom,
! [A: set_nat,C: nat > real] :
( ( ( ( finite_finite_nat @ A )
& ( member_nat @ zero_zero_nat @ A ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I2: nat] : ( times_times_real @ ( C @ I2 ) @ ( power_power_real @ zero_zero_real @ I2 ) )
@ A )
= ( C @ zero_zero_nat ) ) )
& ( ~ ( ( finite_finite_nat @ A )
& ( member_nat @ zero_zero_nat @ A ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I2: nat] : ( times_times_real @ ( C @ I2 ) @ ( power_power_real @ zero_zero_real @ I2 ) )
@ A )
= zero_zero_real ) ) ) ).
% sum_zero_power
thf(fact_258_sum_Odelta_H,axiom,
! [S: set_o,A2: $o,B: $o > int] :
( ( finite_finite_o @ S )
=> ( ( ( member_o @ A2 @ S )
=> ( ( groups8505340233167759370_o_int
@ ^ [K: $o] : ( if_int @ ( A2 = K ) @ ( B @ K ) @ zero_zero_int )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_o @ A2 @ S )
=> ( ( groups8505340233167759370_o_int
@ ^ [K: $o] : ( if_int @ ( A2 = K ) @ ( B @ K ) @ zero_zero_int )
@ S )
= zero_zero_int ) ) ) ) ).
% sum.delta'
thf(fact_259_sum_Odelta_H,axiom,
! [S: set_int,A2: int,B: int > int] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A2 @ S )
=> ( ( groups4538972089207619220nt_int
@ ^ [K: int] : ( if_int @ ( A2 = K ) @ ( B @ K ) @ zero_zero_int )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_int @ A2 @ S )
=> ( ( groups4538972089207619220nt_int
@ ^ [K: int] : ( if_int @ ( A2 = K ) @ ( B @ K ) @ zero_zero_int )
@ S )
= zero_zero_int ) ) ) ) ).
% sum.delta'
thf(fact_260_sum_Odelta_H,axiom,
! [S: set_complex,A2: complex,B: complex > int] :
( ( finite3207457112153483333omplex @ S )
=> ( ( ( member_complex @ A2 @ S )
=> ( ( groups5690904116761175830ex_int
@ ^ [K: complex] : ( if_int @ ( A2 = K ) @ ( B @ K ) @ zero_zero_int )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_complex @ A2 @ S )
=> ( ( groups5690904116761175830ex_int
@ ^ [K: complex] : ( if_int @ ( A2 = K ) @ ( B @ K ) @ zero_zero_int )
@ S )
= zero_zero_int ) ) ) ) ).
% sum.delta'
thf(fact_261_sum_Odelta_H,axiom,
! [S: set_o,A2: $o,B: $o > nat] :
( ( finite_finite_o @ S )
=> ( ( ( member_o @ A2 @ S )
=> ( ( groups8507830703676809646_o_nat
@ ^ [K: $o] : ( if_nat @ ( A2 = K ) @ ( B @ K ) @ zero_zero_nat )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_o @ A2 @ S )
=> ( ( groups8507830703676809646_o_nat
@ ^ [K: $o] : ( if_nat @ ( A2 = K ) @ ( B @ K ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_262_sum_Odelta_H,axiom,
! [S: set_int,A2: int,B: int > nat] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A2 @ S )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K: int] : ( if_nat @ ( A2 = K ) @ ( B @ K ) @ zero_zero_nat )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_int @ A2 @ S )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K: int] : ( if_nat @ ( A2 = K ) @ ( B @ K ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_263_sum_Odelta_H,axiom,
! [S: set_complex,A2: complex,B: complex > nat] :
( ( finite3207457112153483333omplex @ S )
=> ( ( ( member_complex @ A2 @ S )
=> ( ( groups5693394587270226106ex_nat
@ ^ [K: complex] : ( if_nat @ ( A2 = K ) @ ( B @ K ) @ zero_zero_nat )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_complex @ A2 @ S )
=> ( ( groups5693394587270226106ex_nat
@ ^ [K: complex] : ( if_nat @ ( A2 = K ) @ ( B @ K ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_264_sum_Odelta_H,axiom,
! [S: set_o,A2: $o,B: $o > real] :
( ( finite_finite_o @ S )
=> ( ( ( member_o @ A2 @ S )
=> ( ( groups8691415230153176458o_real
@ ^ [K: $o] : ( if_real @ ( A2 = K ) @ ( B @ K ) @ zero_zero_real )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_o @ A2 @ S )
=> ( ( groups8691415230153176458o_real
@ ^ [K: $o] : ( if_real @ ( A2 = K ) @ ( B @ K ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_265_sum_Odelta_H,axiom,
! [S: set_int,A2: int,B: int > real] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A2 @ S )
=> ( ( groups8778361861064173332t_real
@ ^ [K: int] : ( if_real @ ( A2 = K ) @ ( B @ K ) @ zero_zero_real )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_int @ A2 @ S )
=> ( ( groups8778361861064173332t_real
@ ^ [K: int] : ( if_real @ ( A2 = K ) @ ( B @ K ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_266_sum_Odelta_H,axiom,
! [S: set_complex,A2: complex,B: complex > real] :
( ( finite3207457112153483333omplex @ S )
=> ( ( ( member_complex @ A2 @ S )
=> ( ( groups5808333547571424918x_real
@ ^ [K: complex] : ( if_real @ ( A2 = K ) @ ( B @ K ) @ zero_zero_real )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_complex @ A2 @ S )
=> ( ( groups5808333547571424918x_real
@ ^ [K: complex] : ( if_real @ ( A2 = K ) @ ( B @ K ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_267_sum_Odelta_H,axiom,
! [S: set_o,A2: $o,B: $o > extend8495563244428889912nnreal] :
( ( finite_finite_o @ S )
=> ( ( ( member_o @ A2 @ S )
=> ( ( groups7456689898616286486nnreal
@ ^ [K: $o] : ( if_Ext9135588136721118450nnreal @ ( A2 = K ) @ ( B @ K ) @ zero_z7100319975126383169nnreal )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_o @ A2 @ S )
=> ( ( groups7456689898616286486nnreal
@ ^ [K: $o] : ( if_Ext9135588136721118450nnreal @ ( A2 = K ) @ ( B @ K ) @ zero_z7100319975126383169nnreal )
@ S )
= zero_z7100319975126383169nnreal ) ) ) ) ).
% sum.delta'
thf(fact_268_sum_Odelta,axiom,
! [S: set_o,A2: $o,B: $o > int] :
( ( finite_finite_o @ S )
=> ( ( ( member_o @ A2 @ S )
=> ( ( groups8505340233167759370_o_int
@ ^ [K: $o] : ( if_int @ ( K = A2 ) @ ( B @ K ) @ zero_zero_int )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_o @ A2 @ S )
=> ( ( groups8505340233167759370_o_int
@ ^ [K: $o] : ( if_int @ ( K = A2 ) @ ( B @ K ) @ zero_zero_int )
@ S )
= zero_zero_int ) ) ) ) ).
% sum.delta
thf(fact_269_sum_Odelta,axiom,
! [S: set_int,A2: int,B: int > int] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A2 @ S )
=> ( ( groups4538972089207619220nt_int
@ ^ [K: int] : ( if_int @ ( K = A2 ) @ ( B @ K ) @ zero_zero_int )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_int @ A2 @ S )
=> ( ( groups4538972089207619220nt_int
@ ^ [K: int] : ( if_int @ ( K = A2 ) @ ( B @ K ) @ zero_zero_int )
@ S )
= zero_zero_int ) ) ) ) ).
% sum.delta
thf(fact_270_sum_Odelta,axiom,
! [S: set_complex,A2: complex,B: complex > int] :
( ( finite3207457112153483333omplex @ S )
=> ( ( ( member_complex @ A2 @ S )
=> ( ( groups5690904116761175830ex_int
@ ^ [K: complex] : ( if_int @ ( K = A2 ) @ ( B @ K ) @ zero_zero_int )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_complex @ A2 @ S )
=> ( ( groups5690904116761175830ex_int
@ ^ [K: complex] : ( if_int @ ( K = A2 ) @ ( B @ K ) @ zero_zero_int )
@ S )
= zero_zero_int ) ) ) ) ).
% sum.delta
thf(fact_271_sum_Odelta,axiom,
! [S: set_o,A2: $o,B: $o > nat] :
( ( finite_finite_o @ S )
=> ( ( ( member_o @ A2 @ S )
=> ( ( groups8507830703676809646_o_nat
@ ^ [K: $o] : ( if_nat @ ( K = A2 ) @ ( B @ K ) @ zero_zero_nat )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_o @ A2 @ S )
=> ( ( groups8507830703676809646_o_nat
@ ^ [K: $o] : ( if_nat @ ( K = A2 ) @ ( B @ K ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_272_sum_Odelta,axiom,
! [S: set_int,A2: int,B: int > nat] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A2 @ S )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K: int] : ( if_nat @ ( K = A2 ) @ ( B @ K ) @ zero_zero_nat )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_int @ A2 @ S )
=> ( ( groups4541462559716669496nt_nat
@ ^ [K: int] : ( if_nat @ ( K = A2 ) @ ( B @ K ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_273_sum_Odelta,axiom,
! [S: set_complex,A2: complex,B: complex > nat] :
( ( finite3207457112153483333omplex @ S )
=> ( ( ( member_complex @ A2 @ S )
=> ( ( groups5693394587270226106ex_nat
@ ^ [K: complex] : ( if_nat @ ( K = A2 ) @ ( B @ K ) @ zero_zero_nat )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_complex @ A2 @ S )
=> ( ( groups5693394587270226106ex_nat
@ ^ [K: complex] : ( if_nat @ ( K = A2 ) @ ( B @ K ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_274_sum_Odelta,axiom,
! [S: set_o,A2: $o,B: $o > real] :
( ( finite_finite_o @ S )
=> ( ( ( member_o @ A2 @ S )
=> ( ( groups8691415230153176458o_real
@ ^ [K: $o] : ( if_real @ ( K = A2 ) @ ( B @ K ) @ zero_zero_real )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_o @ A2 @ S )
=> ( ( groups8691415230153176458o_real
@ ^ [K: $o] : ( if_real @ ( K = A2 ) @ ( B @ K ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_275_sum_Odelta,axiom,
! [S: set_int,A2: int,B: int > real] :
( ( finite_finite_int @ S )
=> ( ( ( member_int @ A2 @ S )
=> ( ( groups8778361861064173332t_real
@ ^ [K: int] : ( if_real @ ( K = A2 ) @ ( B @ K ) @ zero_zero_real )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_int @ A2 @ S )
=> ( ( groups8778361861064173332t_real
@ ^ [K: int] : ( if_real @ ( K = A2 ) @ ( B @ K ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_276_sum_Odelta,axiom,
! [S: set_complex,A2: complex,B: complex > real] :
( ( finite3207457112153483333omplex @ S )
=> ( ( ( member_complex @ A2 @ S )
=> ( ( groups5808333547571424918x_real
@ ^ [K: complex] : ( if_real @ ( K = A2 ) @ ( B @ K ) @ zero_zero_real )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_complex @ A2 @ S )
=> ( ( groups5808333547571424918x_real
@ ^ [K: complex] : ( if_real @ ( K = A2 ) @ ( B @ K ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_277_sum_Odelta,axiom,
! [S: set_o,A2: $o,B: $o > extend8495563244428889912nnreal] :
( ( finite_finite_o @ S )
=> ( ( ( member_o @ A2 @ S )
=> ( ( groups7456689898616286486nnreal
@ ^ [K: $o] : ( if_Ext9135588136721118450nnreal @ ( K = A2 ) @ ( B @ K ) @ zero_z7100319975126383169nnreal )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_o @ A2 @ S )
=> ( ( groups7456689898616286486nnreal
@ ^ [K: $o] : ( if_Ext9135588136721118450nnreal @ ( K = A2 ) @ ( B @ K ) @ zero_z7100319975126383169nnreal )
@ S )
= zero_z7100319975126383169nnreal ) ) ) ) ).
% sum.delta
thf(fact_278_sum__eq__1__iff,axiom,
! [A: set_list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( ( groups4396056296759096172at_nat @ F @ A )
= one_one_nat )
= ( ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
& ( ( F @ X3 )
= one_one_nat )
& ! [Y2: list_nat] :
( ( member_list_nat @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( F @ Y2 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_279_sum__eq__1__iff,axiom,
! [A: set_int,F: int > nat] :
( ( finite_finite_int @ A )
=> ( ( ( groups4541462559716669496nt_nat @ F @ A )
= one_one_nat )
= ( ? [X3: int] :
( ( member_int @ X3 @ A )
& ( ( F @ X3 )
= one_one_nat )
& ! [Y2: int] :
( ( member_int @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( F @ Y2 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_280_sum__eq__1__iff,axiom,
! [A: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A )
=> ( ( ( groups5693394587270226106ex_nat @ F @ A )
= one_one_nat )
= ( ? [X3: complex] :
( ( member_complex @ X3 @ A )
& ( ( F @ X3 )
= one_one_nat )
& ! [Y2: complex] :
( ( member_complex @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( F @ Y2 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_281_sum__eq__1__iff,axiom,
! [A: set_set_list_nat,F: set_list_nat > nat] :
( ( finite7047420756378620717st_nat @ A )
=> ( ( ( groups7315335787803791778at_nat @ F @ A )
= one_one_nat )
= ( ? [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A )
& ( ( F @ X3 )
= one_one_nat )
& ! [Y2: set_list_nat] :
( ( member_set_list_nat @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( F @ Y2 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_282_sum__eq__1__iff,axiom,
! [A: set_set_complex,F: set_complex > nat] :
( ( finite6551019134538273531omplex @ A )
=> ( ( ( groups8758837469787661168ex_nat @ F @ A )
= one_one_nat )
= ( ? [X3: set_complex] :
( ( member_set_complex @ X3 @ A )
& ( ( F @ X3 )
= one_one_nat )
& ! [Y2: set_complex] :
( ( member_set_complex @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( F @ Y2 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_283_sum__eq__1__iff,axiom,
! [A: set_set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( ( groups8294997508430121362at_nat @ F @ A )
= one_one_nat )
= ( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ( ( F @ X3 )
= one_one_nat )
& ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( F @ Y2 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_284_sum__eq__1__iff,axiom,
! [A: set_set_int,F: set_int > nat] :
( ( finite6197958912794628473et_int @ A )
=> ( ( ( groups1258547046268367342nt_nat @ F @ A )
= one_one_nat )
= ( ? [X3: set_int] :
( ( member_set_int @ X3 @ A )
& ( ( F @ X3 )
= one_one_nat )
& ! [Y2: set_int] :
( ( member_set_int @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( F @ Y2 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_285_sum__eq__1__iff,axiom,
! [A: set_set_set_list_nat,F: set_set_list_nat > nat] :
( ( finite1703049307278766691st_nat @ A )
=> ( ( ( groups7006704140163696856at_nat @ F @ A )
= one_one_nat )
= ( ? [X3: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X3 @ A )
& ( ( F @ X3 )
= one_one_nat )
& ! [Y2: set_set_list_nat] :
( ( member1029098694177496419st_nat @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( F @ Y2 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_286_sum__eq__1__iff,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ( groups3542108847815614940at_nat @ F @ A )
= one_one_nat )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ( F @ X3 )
= one_one_nat )
& ! [Y2: nat] :
( ( member_nat @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( F @ Y2 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_287_sum__multicount,axiom,
! [S: set_o,T: set_o,R: $o > $o > $o,K2: nat] :
( ( finite_finite_o @ S )
=> ( ( finite_finite_o @ T )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ T )
=> ( ( finite_card_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ S )
& ( R @ I2 @ X2 ) ) ) )
= K2 ) )
=> ( ( groups8507830703676809646_o_nat
@ ^ [I2: $o] :
( finite_card_o
@ ( collect_o
@ ^ [J: $o] :
( ( member_o @ J @ T )
& ( R @ I2 @ J ) ) ) )
@ S )
= ( times_times_nat @ K2 @ ( finite_card_o @ T ) ) ) ) ) ) ).
% sum_multicount
thf(fact_288_sum__multicount,axiom,
! [S: set_o,T: set_int,R: $o > int > $o,K2: nat] :
( ( finite_finite_o @ S )
=> ( ( finite_finite_int @ T )
=> ( ! [X2: int] :
( ( member_int @ X2 @ T )
=> ( ( finite_card_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ S )
& ( R @ I2 @ X2 ) ) ) )
= K2 ) )
=> ( ( groups8507830703676809646_o_nat
@ ^ [I2: $o] :
( finite_card_int
@ ( collect_int
@ ^ [J: int] :
( ( member_int @ J @ T )
& ( R @ I2 @ J ) ) ) )
@ S )
= ( times_times_nat @ K2 @ ( finite_card_int @ T ) ) ) ) ) ) ).
% sum_multicount
thf(fact_289_sum__multicount,axiom,
! [S: set_o,T: set_nat,R: $o > nat > $o,K2: nat] :
( ( finite_finite_o @ S )
=> ( ( finite_finite_nat @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T )
=> ( ( finite_card_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ S )
& ( R @ I2 @ X2 ) ) ) )
= K2 ) )
=> ( ( groups8507830703676809646_o_nat
@ ^ [I2: $o] :
( finite_card_nat
@ ( collect_nat
@ ^ [J: nat] :
( ( member_nat @ J @ T )
& ( R @ I2 @ J ) ) ) )
@ S )
= ( times_times_nat @ K2 @ ( finite_card_nat @ T ) ) ) ) ) ) ).
% sum_multicount
thf(fact_290_sum__multicount,axiom,
! [S: set_o,T: set_complex,R: $o > complex > $o,K2: nat] :
( ( finite_finite_o @ S )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ T )
=> ( ( finite_card_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ S )
& ( R @ I2 @ X2 ) ) ) )
= K2 ) )
=> ( ( groups8507830703676809646_o_nat
@ ^ [I2: $o] :
( finite_card_complex
@ ( collect_complex
@ ^ [J: complex] :
( ( member_complex @ J @ T )
& ( R @ I2 @ J ) ) ) )
@ S )
= ( times_times_nat @ K2 @ ( finite_card_complex @ T ) ) ) ) ) ) ).
% sum_multicount
thf(fact_291_sum__multicount,axiom,
! [S: set_int,T: set_o,R: int > $o > $o,K2: nat] :
( ( finite_finite_int @ S )
=> ( ( finite_finite_o @ T )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ T )
=> ( ( finite_card_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ S )
& ( R @ I2 @ X2 ) ) ) )
= K2 ) )
=> ( ( groups4541462559716669496nt_nat
@ ^ [I2: int] :
( finite_card_o
@ ( collect_o
@ ^ [J: $o] :
( ( member_o @ J @ T )
& ( R @ I2 @ J ) ) ) )
@ S )
= ( times_times_nat @ K2 @ ( finite_card_o @ T ) ) ) ) ) ) ).
% sum_multicount
thf(fact_292_sum__multicount,axiom,
! [S: set_int,T: set_int,R: int > int > $o,K2: nat] :
( ( finite_finite_int @ S )
=> ( ( finite_finite_int @ T )
=> ( ! [X2: int] :
( ( member_int @ X2 @ T )
=> ( ( finite_card_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ S )
& ( R @ I2 @ X2 ) ) ) )
= K2 ) )
=> ( ( groups4541462559716669496nt_nat
@ ^ [I2: int] :
( finite_card_int
@ ( collect_int
@ ^ [J: int] :
( ( member_int @ J @ T )
& ( R @ I2 @ J ) ) ) )
@ S )
= ( times_times_nat @ K2 @ ( finite_card_int @ T ) ) ) ) ) ) ).
% sum_multicount
thf(fact_293_sum__multicount,axiom,
! [S: set_int,T: set_nat,R: int > nat > $o,K2: nat] :
( ( finite_finite_int @ S )
=> ( ( finite_finite_nat @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T )
=> ( ( finite_card_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ S )
& ( R @ I2 @ X2 ) ) ) )
= K2 ) )
=> ( ( groups4541462559716669496nt_nat
@ ^ [I2: int] :
( finite_card_nat
@ ( collect_nat
@ ^ [J: nat] :
( ( member_nat @ J @ T )
& ( R @ I2 @ J ) ) ) )
@ S )
= ( times_times_nat @ K2 @ ( finite_card_nat @ T ) ) ) ) ) ) ).
% sum_multicount
thf(fact_294_sum__multicount,axiom,
! [S: set_int,T: set_complex,R: int > complex > $o,K2: nat] :
( ( finite_finite_int @ S )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ T )
=> ( ( finite_card_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ S )
& ( R @ I2 @ X2 ) ) ) )
= K2 ) )
=> ( ( groups4541462559716669496nt_nat
@ ^ [I2: int] :
( finite_card_complex
@ ( collect_complex
@ ^ [J: complex] :
( ( member_complex @ J @ T )
& ( R @ I2 @ J ) ) ) )
@ S )
= ( times_times_nat @ K2 @ ( finite_card_complex @ T ) ) ) ) ) ) ).
% sum_multicount
thf(fact_295_sum__multicount,axiom,
! [S: set_complex,T: set_o,R: complex > $o > $o,K2: nat] :
( ( finite3207457112153483333omplex @ S )
=> ( ( finite_finite_o @ T )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ T )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [I2: complex] :
( ( member_complex @ I2 @ S )
& ( R @ I2 @ X2 ) ) ) )
= K2 ) )
=> ( ( groups5693394587270226106ex_nat
@ ^ [I2: complex] :
( finite_card_o
@ ( collect_o
@ ^ [J: $o] :
( ( member_o @ J @ T )
& ( R @ I2 @ J ) ) ) )
@ S )
= ( times_times_nat @ K2 @ ( finite_card_o @ T ) ) ) ) ) ) ).
% sum_multicount
thf(fact_296_sum__multicount,axiom,
! [S: set_complex,T: set_int,R: complex > int > $o,K2: nat] :
( ( finite3207457112153483333omplex @ S )
=> ( ( finite_finite_int @ T )
=> ( ! [X2: int] :
( ( member_int @ X2 @ T )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [I2: complex] :
( ( member_complex @ I2 @ S )
& ( R @ I2 @ X2 ) ) ) )
= K2 ) )
=> ( ( groups5693394587270226106ex_nat
@ ^ [I2: complex] :
( finite_card_int
@ ( collect_int
@ ^ [J: int] :
( ( member_int @ J @ T )
& ( R @ I2 @ J ) ) ) )
@ S )
= ( times_times_nat @ K2 @ ( finite_card_int @ T ) ) ) ) ) ) ).
% sum_multicount
thf(fact_297_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_298_Inf__set__def,axiom,
( comple3063163877087187839_set_o
= ( ^ [A3: set_set_o] :
( collect_o
@ ^ [X3: $o] : ( complete_Inf_Inf_o @ ( image_set_o_o @ ( member_o @ X3 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_299_Inf__set__def,axiom,
( comple2956690151646016541omplex
= ( ^ [A3: set_set_complex] :
( collect_complex
@ ^ [X3: complex] : ( complete_Inf_Inf_o @ ( image_set_complex_o @ ( member_complex @ X3 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_300_Inf__set__def,axiom,
( comple7806235888213564991et_nat
= ( ^ [A3: set_set_nat] :
( collect_nat
@ ^ [X3: nat] : ( complete_Inf_Inf_o @ ( image_set_nat_o @ ( member_nat @ X3 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_301_Inf__set__def,axiom,
( comple3628384868704368283et_int
= ( ^ [A3: set_set_int] :
( collect_int
@ ^ [X3: int] : ( complete_Inf_Inf_o @ ( image_set_int_o @ ( member_int @ X3 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_302_Inf__set__def,axiom,
( comple184543376406953807st_nat
= ( ^ [A3: set_set_list_nat] :
( collect_list_nat
@ ^ [X3: list_nat] : ( complete_Inf_Inf_o @ ( image_set_list_nat_o @ ( member_list_nat @ X3 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_303_Inf__set__def,axiom,
( comple5875334597002396809omplex
= ( ^ [A3: set_se3806740948107030918omplex] :
( collec2434422415211999471omplex
@ ^ [X3: set_set_complex] : ( complete_Inf_Inf_o @ ( image_3081631015407073793plex_o @ ( member9015044028964487601omplex @ X3 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_304_Inf__set__def,axiom,
( comple8067742441731897515et_nat
= ( ^ [A3: set_set_set_set_nat] :
( collect_set_set_nat
@ ^ [X3: set_set_nat] : ( complete_Inf_Inf_o @ ( image_3488003393078953823_nat_o @ ( member_set_set_nat @ X3 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_305_Inf__set__def,axiom,
( comple3254194022943978759et_int
= ( ^ [A3: set_set_set_set_int] :
( collect_set_set_int
@ ^ [X3: set_set_int] : ( complete_Inf_Inf_o @ ( image_2622566851371609091_int_o @ ( member_set_set_int @ X3 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_306_Inf__set__def,axiom,
( comple8462666950445340293st_nat
= ( ^ [A3: set_set_set_list_nat] :
( collect_set_list_nat
@ ^ [X3: set_list_nat] : ( complete_Inf_Inf_o @ ( image_7778788071572860293_nat_o @ ( member_set_list_nat @ X3 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_307_Inf__set__def,axiom,
( comple5189992959352112827st_nat
= ( ^ [A3: set_se5258582372428582328st_nat] :
( collec4691811733418234273st_nat
@ ^ [X3: set_set_list_nat] : ( complete_Inf_Inf_o @ ( image_3035193717992370127_nat_o @ ( member1029098694177496419st_nat @ X3 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_308_power__mult,axiom,
! [A2: int,M: nat,N: nat] :
( ( power_power_int @ A2 @ ( times_times_nat @ M @ N ) )
= ( power_power_int @ ( power_power_int @ A2 @ M ) @ N ) ) ).
% power_mult
thf(fact_309_power__mult,axiom,
! [A2: nat,M: nat,N: nat] :
( ( power_power_nat @ A2 @ ( times_times_nat @ M @ N ) )
= ( power_power_nat @ ( power_power_nat @ A2 @ M ) @ N ) ) ).
% power_mult
thf(fact_310_power__mult,axiom,
! [A2: real,M: nat,N: nat] :
( ( power_power_real @ A2 @ ( times_times_nat @ M @ N ) )
= ( power_power_real @ ( power_power_real @ A2 @ M ) @ N ) ) ).
% power_mult
thf(fact_311_power__mult,axiom,
! [A2: complex,M: nat,N: nat] :
( ( power_power_complex @ A2 @ ( times_times_nat @ M @ N ) )
= ( power_power_complex @ ( power_power_complex @ A2 @ M ) @ N ) ) ).
% power_mult
thf(fact_312_plus__int__code_I1_J,axiom,
! [K2: int] :
( ( plus_plus_int @ K2 @ zero_zero_int )
= K2 ) ).
% plus_int_code(1)
thf(fact_313_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_314_add__mult__distrib2,axiom,
! [K2: nat,M: nat,N: nat] :
( ( times_times_nat @ K2 @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) ) ) ).
% add_mult_distrib2
thf(fact_315_add__mult__distrib,axiom,
! [M: nat,N: nat,K2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K2 )
= ( plus_plus_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).
% add_mult_distrib
thf(fact_316_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_317_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_318_int__distrib_I1_J,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z2 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
% int_distrib(1)
thf(fact_319_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
% int_distrib(2)
thf(fact_320_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_321_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_322_sum__multicount__gen,axiom,
! [S2: set_o,T2: set_o,R: $o > $o > $o,K2: $o > nat] :
( ( finite_finite_o @ S2 )
=> ( ( finite_finite_o @ T2 )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ T2 )
=> ( ( finite_card_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ S2 )
& ( R @ I2 @ X2 ) ) ) )
= ( K2 @ X2 ) ) )
=> ( ( groups8507830703676809646_o_nat
@ ^ [I2: $o] :
( finite_card_o
@ ( collect_o
@ ^ [J: $o] :
( ( member_o @ J @ T2 )
& ( R @ I2 @ J ) ) ) )
@ S2 )
= ( groups8507830703676809646_o_nat @ K2 @ T2 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_323_sum__multicount__gen,axiom,
! [S2: set_o,T2: set_int,R: $o > int > $o,K2: int > nat] :
( ( finite_finite_o @ S2 )
=> ( ( finite_finite_int @ T2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ T2 )
=> ( ( finite_card_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ S2 )
& ( R @ I2 @ X2 ) ) ) )
= ( K2 @ X2 ) ) )
=> ( ( groups8507830703676809646_o_nat
@ ^ [I2: $o] :
( finite_card_int
@ ( collect_int
@ ^ [J: int] :
( ( member_int @ J @ T2 )
& ( R @ I2 @ J ) ) ) )
@ S2 )
= ( groups4541462559716669496nt_nat @ K2 @ T2 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_324_sum__multicount__gen,axiom,
! [S2: set_o,T2: set_complex,R: $o > complex > $o,K2: complex > nat] :
( ( finite_finite_o @ S2 )
=> ( ( finite3207457112153483333omplex @ T2 )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ T2 )
=> ( ( finite_card_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ S2 )
& ( R @ I2 @ X2 ) ) ) )
= ( K2 @ X2 ) ) )
=> ( ( groups8507830703676809646_o_nat
@ ^ [I2: $o] :
( finite_card_complex
@ ( collect_complex
@ ^ [J: complex] :
( ( member_complex @ J @ T2 )
& ( R @ I2 @ J ) ) ) )
@ S2 )
= ( groups5693394587270226106ex_nat @ K2 @ T2 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_325_sum__multicount__gen,axiom,
! [S2: set_int,T2: set_o,R: int > $o > $o,K2: $o > nat] :
( ( finite_finite_int @ S2 )
=> ( ( finite_finite_o @ T2 )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ T2 )
=> ( ( finite_card_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ S2 )
& ( R @ I2 @ X2 ) ) ) )
= ( K2 @ X2 ) ) )
=> ( ( groups4541462559716669496nt_nat
@ ^ [I2: int] :
( finite_card_o
@ ( collect_o
@ ^ [J: $o] :
( ( member_o @ J @ T2 )
& ( R @ I2 @ J ) ) ) )
@ S2 )
= ( groups8507830703676809646_o_nat @ K2 @ T2 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_326_sum__multicount__gen,axiom,
! [S2: set_int,T2: set_int,R: int > int > $o,K2: int > nat] :
( ( finite_finite_int @ S2 )
=> ( ( finite_finite_int @ T2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ T2 )
=> ( ( finite_card_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ S2 )
& ( R @ I2 @ X2 ) ) ) )
= ( K2 @ X2 ) ) )
=> ( ( groups4541462559716669496nt_nat
@ ^ [I2: int] :
( finite_card_int
@ ( collect_int
@ ^ [J: int] :
( ( member_int @ J @ T2 )
& ( R @ I2 @ J ) ) ) )
@ S2 )
= ( groups4541462559716669496nt_nat @ K2 @ T2 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_327_sum__multicount__gen,axiom,
! [S2: set_int,T2: set_complex,R: int > complex > $o,K2: complex > nat] :
( ( finite_finite_int @ S2 )
=> ( ( finite3207457112153483333omplex @ T2 )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ T2 )
=> ( ( finite_card_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ S2 )
& ( R @ I2 @ X2 ) ) ) )
= ( K2 @ X2 ) ) )
=> ( ( groups4541462559716669496nt_nat
@ ^ [I2: int] :
( finite_card_complex
@ ( collect_complex
@ ^ [J: complex] :
( ( member_complex @ J @ T2 )
& ( R @ I2 @ J ) ) ) )
@ S2 )
= ( groups5693394587270226106ex_nat @ K2 @ T2 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_328_sum__multicount__gen,axiom,
! [S2: set_complex,T2: set_o,R: complex > $o > $o,K2: $o > nat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( finite_finite_o @ T2 )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ T2 )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [I2: complex] :
( ( member_complex @ I2 @ S2 )
& ( R @ I2 @ X2 ) ) ) )
= ( K2 @ X2 ) ) )
=> ( ( groups5693394587270226106ex_nat
@ ^ [I2: complex] :
( finite_card_o
@ ( collect_o
@ ^ [J: $o] :
( ( member_o @ J @ T2 )
& ( R @ I2 @ J ) ) ) )
@ S2 )
= ( groups8507830703676809646_o_nat @ K2 @ T2 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_329_sum__multicount__gen,axiom,
! [S2: set_complex,T2: set_int,R: complex > int > $o,K2: int > nat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( finite_finite_int @ T2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ T2 )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [I2: complex] :
( ( member_complex @ I2 @ S2 )
& ( R @ I2 @ X2 ) ) ) )
= ( K2 @ X2 ) ) )
=> ( ( groups5693394587270226106ex_nat
@ ^ [I2: complex] :
( finite_card_int
@ ( collect_int
@ ^ [J: int] :
( ( member_int @ J @ T2 )
& ( R @ I2 @ J ) ) ) )
@ S2 )
= ( groups4541462559716669496nt_nat @ K2 @ T2 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_330_sum__multicount__gen,axiom,
! [S2: set_complex,T2: set_complex,R: complex > complex > $o,K2: complex > nat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( finite3207457112153483333omplex @ T2 )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ T2 )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [I2: complex] :
( ( member_complex @ I2 @ S2 )
& ( R @ I2 @ X2 ) ) ) )
= ( K2 @ X2 ) ) )
=> ( ( groups5693394587270226106ex_nat
@ ^ [I2: complex] :
( finite_card_complex
@ ( collect_complex
@ ^ [J: complex] :
( ( member_complex @ J @ T2 )
& ( R @ I2 @ J ) ) ) )
@ S2 )
= ( groups5693394587270226106ex_nat @ K2 @ T2 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_331_sum__multicount__gen,axiom,
! [S2: set_o,T2: set_nat,R: $o > nat > $o,K2: nat > nat] :
( ( finite_finite_o @ S2 )
=> ( ( finite_finite_nat @ T2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T2 )
=> ( ( finite_card_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ S2 )
& ( R @ I2 @ X2 ) ) ) )
= ( K2 @ X2 ) ) )
=> ( ( groups8507830703676809646_o_nat
@ ^ [I2: $o] :
( finite_card_nat
@ ( collect_nat
@ ^ [J: nat] :
( ( member_nat @ J @ T2 )
& ( R @ I2 @ J ) ) ) )
@ S2 )
= ( groups3542108847815614940at_nat @ K2 @ T2 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_332_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_333_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_334_power__0,axiom,
! [A2: int] :
( ( power_power_int @ A2 @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_335_power__0,axiom,
! [A2: nat] :
( ( power_power_nat @ A2 @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_336_power__0,axiom,
! [A2: real] :
( ( power_power_real @ A2 @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_337_power__0,axiom,
! [A2: complex] :
( ( power_power_complex @ A2 @ zero_zero_nat )
= one_one_complex ) ).
% power_0
thf(fact_338_int__ge__induct,axiom,
! [K2: int,I3: int,P: int > $o] :
( ( ord_less_eq_int @ K2 @ I3 )
=> ( ( P @ K2 )
=> ( ! [I4: int] :
( ( ord_less_eq_int @ K2 @ I4 )
=> ( ( P @ I4 )
=> ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_ge_induct
thf(fact_339_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z3: int] :
? [N2: nat] :
( Z3
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_340_card__eq__sum,axiom,
( finite2364142230527598318st_nat
= ( groups7315335787803791778at_nat
@ ^ [X3: set_list_nat] : one_one_nat ) ) ).
% card_eq_sum
thf(fact_341_card__eq__sum,axiom,
( finite_card_list_nat
= ( groups4396056296759096172at_nat
@ ^ [X3: list_nat] : one_one_nat ) ) ).
% card_eq_sum
thf(fact_342_card__eq__sum,axiom,
( finite903997441450111292omplex
= ( groups8758837469787661168ex_nat
@ ^ [X3: set_complex] : one_one_nat ) ) ).
% card_eq_sum
thf(fact_343_card__eq__sum,axiom,
( finite5070363488328301092st_nat
= ( groups7006704140163696856at_nat
@ ^ [X3: set_set_list_nat] : one_one_nat ) ) ).
% card_eq_sum
thf(fact_344_card__eq__sum,axiom,
( finite_card_set_nat
= ( groups8294997508430121362at_nat
@ ^ [X3: set_nat] : one_one_nat ) ) ).
% card_eq_sum
thf(fact_345_card__eq__sum,axiom,
( finite_card_set_int
= ( groups1258547046268367342nt_nat
@ ^ [X3: set_int] : one_one_nat ) ) ).
% card_eq_sum
thf(fact_346_card__eq__sum,axiom,
( finite_card_int
= ( groups4541462559716669496nt_nat
@ ^ [X3: int] : one_one_nat ) ) ).
% card_eq_sum
thf(fact_347_card__eq__sum,axiom,
( finite_card_nat
= ( groups3542108847815614940at_nat
@ ^ [X3: nat] : one_one_nat ) ) ).
% card_eq_sum
thf(fact_348_int__sum,axiom,
! [F: set_set_list_nat > nat,A: set_set_set_list_nat] :
( ( semiri1314217659103216013at_int @ ( groups7006704140163696856at_nat @ F @ A ) )
= ( groups7004213669654646580at_int
@ ^ [X3: set_set_list_nat] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
@ A ) ) ).
% int_sum
thf(fact_349_int__sum,axiom,
! [F: set_list_nat > nat,A: set_set_list_nat] :
( ( semiri1314217659103216013at_int @ ( groups7315335787803791778at_nat @ F @ A ) )
= ( groups7312845317294741502at_int
@ ^ [X3: set_list_nat] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
@ A ) ) ).
% int_sum
thf(fact_350_int__sum,axiom,
! [F: set_set_nat > nat,A: set_set_set_nat] :
( ( semiri1314217659103216013at_int @ ( groups7087220048432663112at_nat @ F @ A ) )
= ( groups7084729577923612836at_int
@ ^ [X3: set_set_nat] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
@ A ) ) ).
% int_sum
thf(fact_351_int__sum,axiom,
! [F: set_set_int > nat,A: set_set_set_int] :
( ( semiri1314217659103216013at_int @ ( groups1082551605742257316nt_nat @ F @ A ) )
= ( groups1080061135233207040nt_int
@ ^ [X3: set_set_int] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
@ A ) ) ).
% int_sum
thf(fact_352_int__sum,axiom,
! [F: nat > nat,A: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A ) )
= ( groups3539618377306564664at_int
@ ^ [X3: nat] : ( semiri1314217659103216013at_int @ ( F @ X3 ) )
@ A ) ) ).
% int_sum
thf(fact_353_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_354_sum_Oswap__restrict,axiom,
! [A: set_o,B2: set_nat,G: $o > nat > real,R: $o > nat > $o] :
( ( finite_finite_o @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups8691415230153176458o_real
@ ^ [X3: $o] :
( groups6591440286371151544t_real @ ( G @ X3 )
@ ( collect_nat
@ ^ [Y2: nat] :
( ( member_nat @ Y2 @ B2 )
& ( R @ X3 @ Y2 ) ) ) )
@ A )
= ( groups6591440286371151544t_real
@ ^ [Y2: nat] :
( groups8691415230153176458o_real
@ ^ [X3: $o] : ( G @ X3 @ Y2 )
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A )
& ( R @ X3 @ Y2 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_355_sum_Oswap__restrict,axiom,
! [A: set_int,B2: set_nat,G: int > nat > real,R: int > nat > $o] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups8778361861064173332t_real
@ ^ [X3: int] :
( groups6591440286371151544t_real @ ( G @ X3 )
@ ( collect_nat
@ ^ [Y2: nat] :
( ( member_nat @ Y2 @ B2 )
& ( R @ X3 @ Y2 ) ) ) )
@ A )
= ( groups6591440286371151544t_real
@ ^ [Y2: nat] :
( groups8778361861064173332t_real
@ ^ [X3: int] : ( G @ X3 @ Y2 )
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A )
& ( R @ X3 @ Y2 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_356_sum_Oswap__restrict,axiom,
! [A: set_complex,B2: set_nat,G: complex > nat > real,R: complex > nat > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups5808333547571424918x_real
@ ^ [X3: complex] :
( groups6591440286371151544t_real @ ( G @ X3 )
@ ( collect_nat
@ ^ [Y2: nat] :
( ( member_nat @ Y2 @ B2 )
& ( R @ X3 @ Y2 ) ) ) )
@ A )
= ( groups6591440286371151544t_real
@ ^ [Y2: nat] :
( groups5808333547571424918x_real
@ ^ [X3: complex] : ( G @ X3 @ Y2 )
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ A )
& ( R @ X3 @ Y2 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_357_sum_Oswap__restrict,axiom,
! [A: set_o,B2: set_nat,G: $o > nat > nat,R: $o > nat > $o] :
( ( finite_finite_o @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups8507830703676809646_o_nat
@ ^ [X3: $o] :
( groups3542108847815614940at_nat @ ( G @ X3 )
@ ( collect_nat
@ ^ [Y2: nat] :
( ( member_nat @ Y2 @ B2 )
& ( R @ X3 @ Y2 ) ) ) )
@ A )
= ( groups3542108847815614940at_nat
@ ^ [Y2: nat] :
( groups8507830703676809646_o_nat
@ ^ [X3: $o] : ( G @ X3 @ Y2 )
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A )
& ( R @ X3 @ Y2 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_358_sum_Oswap__restrict,axiom,
! [A: set_int,B2: set_nat,G: int > nat > nat,R: int > nat > $o] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups4541462559716669496nt_nat
@ ^ [X3: int] :
( groups3542108847815614940at_nat @ ( G @ X3 )
@ ( collect_nat
@ ^ [Y2: nat] :
( ( member_nat @ Y2 @ B2 )
& ( R @ X3 @ Y2 ) ) ) )
@ A )
= ( groups3542108847815614940at_nat
@ ^ [Y2: nat] :
( groups4541462559716669496nt_nat
@ ^ [X3: int] : ( G @ X3 @ Y2 )
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A )
& ( R @ X3 @ Y2 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_359_sum_Oswap__restrict,axiom,
! [A: set_complex,B2: set_nat,G: complex > nat > nat,R: complex > nat > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups5693394587270226106ex_nat
@ ^ [X3: complex] :
( groups3542108847815614940at_nat @ ( G @ X3 )
@ ( collect_nat
@ ^ [Y2: nat] :
( ( member_nat @ Y2 @ B2 )
& ( R @ X3 @ Y2 ) ) ) )
@ A )
= ( groups3542108847815614940at_nat
@ ^ [Y2: nat] :
( groups5693394587270226106ex_nat
@ ^ [X3: complex] : ( G @ X3 @ Y2 )
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ A )
& ( R @ X3 @ Y2 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_360_sum_Oswap__restrict,axiom,
! [A: set_o,B2: set_nat,G: $o > nat > int,R: $o > nat > $o] :
( ( finite_finite_o @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups8505340233167759370_o_int
@ ^ [X3: $o] :
( groups3539618377306564664at_int @ ( G @ X3 )
@ ( collect_nat
@ ^ [Y2: nat] :
( ( member_nat @ Y2 @ B2 )
& ( R @ X3 @ Y2 ) ) ) )
@ A )
= ( groups3539618377306564664at_int
@ ^ [Y2: nat] :
( groups8505340233167759370_o_int
@ ^ [X3: $o] : ( G @ X3 @ Y2 )
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A )
& ( R @ X3 @ Y2 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_361_sum_Oswap__restrict,axiom,
! [A: set_int,B2: set_nat,G: int > nat > int,R: int > nat > $o] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups4538972089207619220nt_int
@ ^ [X3: int] :
( groups3539618377306564664at_int @ ( G @ X3 )
@ ( collect_nat
@ ^ [Y2: nat] :
( ( member_nat @ Y2 @ B2 )
& ( R @ X3 @ Y2 ) ) ) )
@ A )
= ( groups3539618377306564664at_int
@ ^ [Y2: nat] :
( groups4538972089207619220nt_int
@ ^ [X3: int] : ( G @ X3 @ Y2 )
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A )
& ( R @ X3 @ Y2 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_362_sum_Oswap__restrict,axiom,
! [A: set_complex,B2: set_nat,G: complex > nat > int,R: complex > nat > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups5690904116761175830ex_int
@ ^ [X3: complex] :
( groups3539618377306564664at_int @ ( G @ X3 )
@ ( collect_nat
@ ^ [Y2: nat] :
( ( member_nat @ Y2 @ B2 )
& ( R @ X3 @ Y2 ) ) ) )
@ A )
= ( groups3539618377306564664at_int
@ ^ [Y2: nat] :
( groups5690904116761175830ex_int
@ ^ [X3: complex] : ( G @ X3 @ Y2 )
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ A )
& ( R @ X3 @ Y2 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_363_sum_Oswap__restrict,axiom,
! [A: set_nat,B2: set_o,G: nat > $o > real,R: nat > $o > $o] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_o @ B2 )
=> ( ( groups6591440286371151544t_real
@ ^ [X3: nat] :
( groups8691415230153176458o_real @ ( G @ X3 )
@ ( collect_o
@ ^ [Y2: $o] :
( ( member_o @ Y2 @ B2 )
& ( R @ X3 @ Y2 ) ) ) )
@ A )
= ( groups8691415230153176458o_real
@ ^ [Y2: $o] :
( groups6591440286371151544t_real
@ ^ [X3: nat] : ( G @ X3 @ Y2 )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( R @ X3 @ Y2 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_364_is__num__normalize_I1_J,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_365_is__num__normalize_I1_J,axiom,
! [A2: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A2 @ B ) @ C )
= ( plus_plus_real @ A2 @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_366_Sup_OSUP__cong,axiom,
! [A: set_list_nat,B2: set_list_nat,C2: list_nat > set_list_nat,D: list_nat > set_list_nat,Sup: set_set_list_nat > set_list_nat] :
( ( A = B2 )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_8532145185254316925st_nat @ C2 @ A ) )
= ( Sup @ ( image_8532145185254316925st_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_367_Sup_OSUP__cong,axiom,
! [A: set_int,B2: set_int,C2: int > int,D: int > int,Sup: set_int > int] :
( ( A = B2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_int_int @ C2 @ A ) )
= ( Sup @ ( image_int_int @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_368_Sup_OSUP__cong,axiom,
! [A: set_real,B2: set_real,C2: real > real,D: real > real,Sup: set_real > real] :
( ( A = B2 )
=> ( ! [X2: real] :
( ( member_real @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_real_real @ C2 @ A ) )
= ( Sup @ ( image_real_real @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_369_Sup_OSUP__cong,axiom,
! [A: set_list_nat,B2: set_list_nat,C2: list_nat > $o,D: list_nat > $o,Sup: set_o > $o] :
( ( A = B2 )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_list_nat_o @ C2 @ A ) )
= ( Sup @ ( image_list_nat_o @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_370_Sup_OSUP__cong,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat,C2: set_list_nat > $o,D: set_list_nat > $o,Sup: set_o > $o] :
( ( A = B2 )
=> ( ! [X2: set_list_nat] :
( ( member_set_list_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_set_list_nat_o @ C2 @ A ) )
= ( Sup @ ( image_set_list_nat_o @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_371_Sup_OSUP__cong,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat,C2: set_list_nat > set_list_nat,D: set_list_nat > set_list_nat,Sup: set_set_list_nat > set_list_nat] :
( ( A = B2 )
=> ( ! [X2: set_list_nat] :
( ( member_set_list_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_5143090206295581363st_nat @ C2 @ A ) )
= ( Sup @ ( image_5143090206295581363st_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_372_Sup_OSUP__cong,axiom,
! [A: set_nat,B2: set_nat,C2: nat > $o,D: nat > $o,Sup: set_o > $o] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_nat_o @ C2 @ A ) )
= ( Sup @ ( image_nat_o @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_373_Sup_OSUP__cong,axiom,
! [A: set_nat,B2: set_nat,C2: nat > set_list_nat,D: nat > set_list_nat,Sup: set_set_list_nat > set_list_nat] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_2883343038133793645st_nat @ C2 @ A ) )
= ( Sup @ ( image_2883343038133793645st_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_374_Inf_OINF__cong,axiom,
! [A: set_list_nat,B2: set_list_nat,C2: list_nat > set_list_nat,D: list_nat > set_list_nat,Inf: set_set_list_nat > set_list_nat] :
( ( A = B2 )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_8532145185254316925st_nat @ C2 @ A ) )
= ( Inf @ ( image_8532145185254316925st_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_375_Inf_OINF__cong,axiom,
! [A: set_int,B2: set_int,C2: int > int,D: int > int,Inf: set_int > int] :
( ( A = B2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_int_int @ C2 @ A ) )
= ( Inf @ ( image_int_int @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_376_Inf_OINF__cong,axiom,
! [A: set_real,B2: set_real,C2: real > real,D: real > real,Inf: set_real > real] :
( ( A = B2 )
=> ( ! [X2: real] :
( ( member_real @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_real_real @ C2 @ A ) )
= ( Inf @ ( image_real_real @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_377_Inf_OINF__cong,axiom,
! [A: set_list_nat,B2: set_list_nat,C2: list_nat > $o,D: list_nat > $o,Inf: set_o > $o] :
( ( A = B2 )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_list_nat_o @ C2 @ A ) )
= ( Inf @ ( image_list_nat_o @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_378_Inf_OINF__cong,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat,C2: set_list_nat > $o,D: set_list_nat > $o,Inf: set_o > $o] :
( ( A = B2 )
=> ( ! [X2: set_list_nat] :
( ( member_set_list_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_set_list_nat_o @ C2 @ A ) )
= ( Inf @ ( image_set_list_nat_o @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_379_Inf_OINF__cong,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat,C2: set_list_nat > set_list_nat,D: set_list_nat > set_list_nat,Inf: set_set_list_nat > set_list_nat] :
( ( A = B2 )
=> ( ! [X2: set_list_nat] :
( ( member_set_list_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_5143090206295581363st_nat @ C2 @ A ) )
= ( Inf @ ( image_5143090206295581363st_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_380_Inf_OINF__cong,axiom,
! [A: set_nat,B2: set_nat,C2: nat > $o,D: nat > $o,Inf: set_o > $o] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_nat_o @ C2 @ A ) )
= ( Inf @ ( image_nat_o @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_381_Inf_OINF__cong,axiom,
! [A: set_nat,B2: set_nat,C2: nat > set_list_nat,D: nat > set_list_nat,Inf: set_set_list_nat > set_list_nat] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_2883343038133793645st_nat @ C2 @ A ) )
= ( Inf @ ( image_2883343038133793645st_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_382_sum_Oreindex__bij__witness,axiom,
! [S: set_o,I3: nat > $o,J2: $o > nat,T: set_nat,H: nat > real,G: $o > real] :
( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( ( I3 @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( member_nat @ ( J2 @ A4 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J2 @ ( I3 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_o @ ( I3 @ B3 ) @ S ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8691415230153176458o_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_383_sum_Oreindex__bij__witness,axiom,
! [S: set_o,I3: nat > $o,J2: $o > nat,T: set_nat,H: nat > nat,G: $o > nat] :
( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( ( I3 @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( member_nat @ ( J2 @ A4 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J2 @ ( I3 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_o @ ( I3 @ B3 ) @ S ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8507830703676809646_o_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_384_sum_Oreindex__bij__witness,axiom,
! [S: set_o,I3: nat > $o,J2: $o > nat,T: set_nat,H: nat > int,G: $o > int] :
( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( ( I3 @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( member_nat @ ( J2 @ A4 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J2 @ ( I3 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_o @ ( I3 @ B3 ) @ S ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8505340233167759370_o_int @ G @ S )
= ( groups3539618377306564664at_int @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_385_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I3: $o > nat,J2: nat > $o,T: set_o,H: $o > real,G: nat > real] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( I3 @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( member_o @ ( J2 @ A4 ) @ T ) )
=> ( ! [B3: $o] :
( ( member_o @ B3 @ T )
=> ( ( J2 @ ( I3 @ B3 ) )
= B3 ) )
=> ( ! [B3: $o] :
( ( member_o @ B3 @ T )
=> ( member_nat @ ( I3 @ B3 ) @ S ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups8691415230153176458o_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_386_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I3: nat > nat,J2: nat > nat,T: set_nat,H: nat > real,G: nat > real] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( I3 @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( member_nat @ ( J2 @ A4 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J2 @ ( I3 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_nat @ ( I3 @ B3 ) @ S ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_387_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I3: $o > nat,J2: nat > $o,T: set_o,H: $o > nat,G: nat > nat] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( I3 @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( member_o @ ( J2 @ A4 ) @ T ) )
=> ( ! [B3: $o] :
( ( member_o @ B3 @ T )
=> ( ( J2 @ ( I3 @ B3 ) )
= B3 ) )
=> ( ! [B3: $o] :
( ( member_o @ B3 @ T )
=> ( member_nat @ ( I3 @ B3 ) @ S ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups8507830703676809646_o_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_388_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I3: nat > nat,J2: nat > nat,T: set_nat,H: nat > nat,G: nat > nat] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( I3 @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( member_nat @ ( J2 @ A4 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J2 @ ( I3 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_nat @ ( I3 @ B3 ) @ S ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_389_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I3: $o > nat,J2: nat > $o,T: set_o,H: $o > int,G: nat > int] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( I3 @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( member_o @ ( J2 @ A4 ) @ T ) )
=> ( ! [B3: $o] :
( ( member_o @ B3 @ T )
=> ( ( J2 @ ( I3 @ B3 ) )
= B3 ) )
=> ( ! [B3: $o] :
( ( member_o @ B3 @ T )
=> ( member_nat @ ( I3 @ B3 ) @ S ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ S )
= ( groups8505340233167759370_o_int @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_390_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I3: nat > nat,J2: nat > nat,T: set_nat,H: nat > int,G: nat > int] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( I3 @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( member_nat @ ( J2 @ A4 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J2 @ ( I3 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_nat @ ( I3 @ B3 ) @ S ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ S )
= ( groups3539618377306564664at_int @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_391_sum_Oreindex__bij__witness,axiom,
! [S: set_o,I3: set_list_nat > $o,J2: $o > set_list_nat,T: set_set_list_nat,H: set_list_nat > int,G: $o > int] :
( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( ( I3 @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( member_set_list_nat @ ( J2 @ A4 ) @ T ) )
=> ( ! [B3: set_list_nat] :
( ( member_set_list_nat @ B3 @ T )
=> ( ( J2 @ ( I3 @ B3 ) )
= B3 ) )
=> ( ! [B3: set_list_nat] :
( ( member_set_list_nat @ B3 @ T )
=> ( member_o @ ( I3 @ B3 ) @ S ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8505340233167759370_o_int @ G @ S )
= ( groups7312845317294741502at_int @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_392_sum_Oeq__general__inverses,axiom,
! [B2: set_nat,K2: nat > $o,A: set_o,H: $o > nat,Gamma: nat > real,Phi: $o > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ( ( member_o @ ( K2 @ Y3 ) @ A )
& ( ( H @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B2 )
& ( ( K2 @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8691415230153176458o_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_393_sum_Oeq__general__inverses,axiom,
! [B2: set_nat,K2: nat > $o,A: set_o,H: $o > nat,Gamma: nat > nat,Phi: $o > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ( ( member_o @ ( K2 @ Y3 ) @ A )
& ( ( H @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B2 )
& ( ( K2 @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8507830703676809646_o_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_394_sum_Oeq__general__inverses,axiom,
! [B2: set_nat,K2: nat > $o,A: set_o,H: $o > nat,Gamma: nat > int,Phi: $o > int] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ( ( member_o @ ( K2 @ Y3 ) @ A )
& ( ( H @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B2 )
& ( ( K2 @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8505340233167759370_o_int @ Phi @ A )
= ( groups3539618377306564664at_int @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_395_sum_Oeq__general__inverses,axiom,
! [B2: set_o,K2: $o > nat,A: set_nat,H: nat > $o,Gamma: $o > real,Phi: nat > real] :
( ! [Y3: $o] :
( ( member_o @ Y3 @ B2 )
=> ( ( member_nat @ ( K2 @ Y3 ) @ A )
& ( ( H @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_o @ ( H @ X2 ) @ B2 )
& ( ( K2 @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A )
= ( groups8691415230153176458o_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_396_sum_Oeq__general__inverses,axiom,
! [B2: set_nat,K2: nat > nat,A: set_nat,H: nat > nat,Gamma: nat > real,Phi: nat > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ( ( member_nat @ ( K2 @ Y3 ) @ A )
& ( ( H @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B2 )
& ( ( K2 @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_397_sum_Oeq__general__inverses,axiom,
! [B2: set_o,K2: $o > nat,A: set_nat,H: nat > $o,Gamma: $o > nat,Phi: nat > nat] :
( ! [Y3: $o] :
( ( member_o @ Y3 @ B2 )
=> ( ( member_nat @ ( K2 @ Y3 ) @ A )
& ( ( H @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_o @ ( H @ X2 ) @ B2 )
& ( ( K2 @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups8507830703676809646_o_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_398_sum_Oeq__general__inverses,axiom,
! [B2: set_nat,K2: nat > nat,A: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ( ( member_nat @ ( K2 @ Y3 ) @ A )
& ( ( H @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B2 )
& ( ( K2 @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_399_sum_Oeq__general__inverses,axiom,
! [B2: set_o,K2: $o > nat,A: set_nat,H: nat > $o,Gamma: $o > int,Phi: nat > int] :
( ! [Y3: $o] :
( ( member_o @ Y3 @ B2 )
=> ( ( member_nat @ ( K2 @ Y3 ) @ A )
& ( ( H @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_o @ ( H @ X2 ) @ B2 )
& ( ( K2 @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups3539618377306564664at_int @ Phi @ A )
= ( groups8505340233167759370_o_int @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_400_sum_Oeq__general__inverses,axiom,
! [B2: set_nat,K2: nat > nat,A: set_nat,H: nat > nat,Gamma: nat > int,Phi: nat > int] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ( ( member_nat @ ( K2 @ Y3 ) @ A )
& ( ( H @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B2 )
& ( ( K2 @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups3539618377306564664at_int @ Phi @ A )
= ( groups3539618377306564664at_int @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_401_sum_Oeq__general__inverses,axiom,
! [B2: set_set_list_nat,K2: set_list_nat > $o,A: set_o,H: $o > set_list_nat,Gamma: set_list_nat > int,Phi: $o > int] :
( ! [Y3: set_list_nat] :
( ( member_set_list_nat @ Y3 @ B2 )
=> ( ( member_o @ ( K2 @ Y3 ) @ A )
& ( ( H @ ( K2 @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ( member_set_list_nat @ ( H @ X2 ) @ B2 )
& ( ( K2 @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8505340233167759370_o_int @ Phi @ A )
= ( groups7312845317294741502at_int @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_402_sum_Oeq__general,axiom,
! [B2: set_nat,A: set_o,H: $o > nat,Gamma: nat > real,Phi: $o > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A )
& ( ( H @ X6 )
= Y3 )
& ! [Ya: $o] :
( ( ( member_o @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X6 ) ) ) )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8691415230153176458o_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_403_sum_Oeq__general,axiom,
! [B2: set_nat,A: set_o,H: $o > nat,Gamma: nat > nat,Phi: $o > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A )
& ( ( H @ X6 )
= Y3 )
& ! [Ya: $o] :
( ( ( member_o @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X6 ) ) ) )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8507830703676809646_o_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_404_sum_Oeq__general,axiom,
! [B2: set_nat,A: set_o,H: $o > nat,Gamma: nat > int,Phi: $o > int] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A )
& ( ( H @ X6 )
= Y3 )
& ! [Ya: $o] :
( ( ( member_o @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X6 ) ) ) )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8505340233167759370_o_int @ Phi @ A )
= ( groups3539618377306564664at_int @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_405_sum_Oeq__general,axiom,
! [B2: set_o,A: set_nat,H: nat > $o,Gamma: $o > real,Phi: nat > real] :
( ! [Y3: $o] :
( ( member_o @ Y3 @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ( H @ X6 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X6 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_o @ ( H @ X2 ) @ B2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A )
= ( groups8691415230153176458o_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_406_sum_Oeq__general,axiom,
! [B2: set_nat,A: set_nat,H: nat > nat,Gamma: nat > real,Phi: nat > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ( H @ X6 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X6 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_407_sum_Oeq__general,axiom,
! [B2: set_o,A: set_nat,H: nat > $o,Gamma: $o > nat,Phi: nat > nat] :
( ! [Y3: $o] :
( ( member_o @ Y3 @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ( H @ X6 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X6 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_o @ ( H @ X2 ) @ B2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups8507830703676809646_o_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_408_sum_Oeq__general,axiom,
! [B2: set_nat,A: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ( H @ X6 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X6 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_409_sum_Oeq__general,axiom,
! [B2: set_o,A: set_nat,H: nat > $o,Gamma: $o > int,Phi: nat > int] :
( ! [Y3: $o] :
( ( member_o @ Y3 @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ( H @ X6 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X6 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_o @ ( H @ X2 ) @ B2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups3539618377306564664at_int @ Phi @ A )
= ( groups8505340233167759370_o_int @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_410_sum_Oeq__general,axiom,
! [B2: set_nat,A: set_nat,H: nat > nat,Gamma: nat > int,Phi: nat > int] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ( H @ X6 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X6 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups3539618377306564664at_int @ Phi @ A )
= ( groups3539618377306564664at_int @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_411_sum_Oeq__general,axiom,
! [B2: set_set_list_nat,A: set_o,H: $o > set_list_nat,Gamma: set_list_nat > int,Phi: $o > int] :
( ! [Y3: set_list_nat] :
( ( member_set_list_nat @ Y3 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A )
& ( ( H @ X6 )
= Y3 )
& ! [Ya: $o] :
( ( ( member_o @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X6 ) ) ) )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ( member_set_list_nat @ ( H @ X2 ) @ B2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8505340233167759370_o_int @ Phi @ A )
= ( groups7312845317294741502at_int @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_412_sum_Ocong,axiom,
! [A: set_set_set_list_nat,B2: set_set_set_list_nat,G: set_set_list_nat > int,H: set_set_list_nat > int] :
( ( A = B2 )
=> ( ! [X2: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X2 @ B2 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups7004213669654646580at_int @ G @ A )
= ( groups7004213669654646580at_int @ H @ B2 ) ) ) ) ).
% sum.cong
thf(fact_413_sum_Ocong,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat,G: set_list_nat > int,H: set_list_nat > int] :
( ( A = B2 )
=> ( ! [X2: set_list_nat] :
( ( member_set_list_nat @ X2 @ B2 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups7312845317294741502at_int @ G @ A )
= ( groups7312845317294741502at_int @ H @ B2 ) ) ) ) ).
% sum.cong
thf(fact_414_sum_Ocong,axiom,
! [A: set_set_set_nat,B2: set_set_set_nat,G: set_set_nat > int,H: set_set_nat > int] :
( ( A = B2 )
=> ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ B2 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups7084729577923612836at_int @ G @ A )
= ( groups7084729577923612836at_int @ H @ B2 ) ) ) ) ).
% sum.cong
thf(fact_415_sum_Ocong,axiom,
! [A: set_set_set_int,B2: set_set_set_int,G: set_set_int > int,H: set_set_int > int] :
( ( A = B2 )
=> ( ! [X2: set_set_int] :
( ( member_set_set_int @ X2 @ B2 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups1080061135233207040nt_int @ G @ A )
= ( groups1080061135233207040nt_int @ H @ B2 ) ) ) ) ).
% sum.cong
thf(fact_416_sum_Ocong,axiom,
! [A: set_nat,B2: set_nat,G: nat > real,H: nat > real] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ A )
= ( groups6591440286371151544t_real @ H @ B2 ) ) ) ) ).
% sum.cong
thf(fact_417_sum_Ocong,axiom,
! [A: set_nat,B2: set_nat,G: nat > nat,H: nat > nat] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= ( groups3542108847815614940at_nat @ H @ B2 ) ) ) ) ).
% sum.cong
thf(fact_418_sum_Ocong,axiom,
! [A: set_nat,B2: set_nat,G: nat > int,H: nat > int] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ A )
= ( groups3539618377306564664at_int @ H @ B2 ) ) ) ) ).
% sum.cong
thf(fact_419_wellorder__InfI,axiom,
! [K2: nat,A: set_nat] :
( ( member_nat @ K2 @ A )
=> ( member_nat @ ( complete_Inf_Inf_nat @ A ) @ A ) ) ).
% wellorder_InfI
thf(fact_420_sum__mono__inv,axiom,
! [F: $o > int,I5: set_o,G: $o > int,I3: $o] :
( ( ( groups8505340233167759370_o_int @ F @ I5 )
= ( groups8505340233167759370_o_int @ G @ I5 ) )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I5 )
=> ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_o @ I3 @ I5 )
=> ( ( finite_finite_o @ I5 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_421_sum__mono__inv,axiom,
! [F: int > int,I5: set_int,G: int > int,I3: int] :
( ( ( groups4538972089207619220nt_int @ F @ I5 )
= ( groups4538972089207619220nt_int @ G @ I5 ) )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I5 )
=> ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_int @ I3 @ I5 )
=> ( ( finite_finite_int @ I5 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_422_sum__mono__inv,axiom,
! [F: complex > int,I5: set_complex,G: complex > int,I3: complex] :
( ( ( groups5690904116761175830ex_int @ F @ I5 )
= ( groups5690904116761175830ex_int @ G @ I5 ) )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ I5 )
=> ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_complex @ I3 @ I5 )
=> ( ( finite3207457112153483333omplex @ I5 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_423_sum__mono__inv,axiom,
! [F: $o > nat,I5: set_o,G: $o > nat,I3: $o] :
( ( ( groups8507830703676809646_o_nat @ F @ I5 )
= ( groups8507830703676809646_o_nat @ G @ I5 ) )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I5 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_o @ I3 @ I5 )
=> ( ( finite_finite_o @ I5 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_424_sum__mono__inv,axiom,
! [F: int > nat,I5: set_int,G: int > nat,I3: int] :
( ( ( groups4541462559716669496nt_nat @ F @ I5 )
= ( groups4541462559716669496nt_nat @ G @ I5 ) )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I5 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_int @ I3 @ I5 )
=> ( ( finite_finite_int @ I5 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_425_sum__mono__inv,axiom,
! [F: complex > nat,I5: set_complex,G: complex > nat,I3: complex] :
( ( ( groups5693394587270226106ex_nat @ F @ I5 )
= ( groups5693394587270226106ex_nat @ G @ I5 ) )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ I5 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_complex @ I3 @ I5 )
=> ( ( finite3207457112153483333omplex @ I5 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_426_sum__mono__inv,axiom,
! [F: $o > real,I5: set_o,G: $o > real,I3: $o] :
( ( ( groups8691415230153176458o_real @ F @ I5 )
= ( groups8691415230153176458o_real @ G @ I5 ) )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I5 )
=> ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_o @ I3 @ I5 )
=> ( ( finite_finite_o @ I5 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_427_sum__mono__inv,axiom,
! [F: int > real,I5: set_int,G: int > real,I3: int] :
( ( ( groups8778361861064173332t_real @ F @ I5 )
= ( groups8778361861064173332t_real @ G @ I5 ) )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I5 )
=> ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_int @ I3 @ I5 )
=> ( ( finite_finite_int @ I5 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_428_sum__mono__inv,axiom,
! [F: complex > real,I5: set_complex,G: complex > real,I3: complex] :
( ( ( groups5808333547571424918x_real @ F @ I5 )
= ( groups5808333547571424918x_real @ G @ I5 ) )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ I5 )
=> ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_complex @ I3 @ I5 )
=> ( ( finite3207457112153483333omplex @ I5 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_429_sum__mono__inv,axiom,
! [F: nat > real,I5: set_nat,G: nat > real,I3: nat] :
( ( ( groups6591440286371151544t_real @ F @ I5 )
= ( groups6591440286371151544t_real @ G @ I5 ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I5 )
=> ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_nat @ I3 @ I5 )
=> ( ( finite_finite_nat @ I5 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_430_cInf__le__finite,axiom,
! [X7: set_o,X5: $o] :
( ( finite_finite_o @ X7 )
=> ( ( member_o @ X5 @ X7 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ X7 ) @ X5 ) ) ) ).
% cInf_le_finite
thf(fact_431_cInf__le__finite,axiom,
! [X7: set_int,X5: int] :
( ( finite_finite_int @ X7 )
=> ( ( member_int @ X5 @ X7 )
=> ( ord_less_eq_int @ ( complete_Inf_Inf_int @ X7 ) @ X5 ) ) ) ).
% cInf_le_finite
thf(fact_432_cInf__le__finite,axiom,
! [X7: set_nat,X5: nat] :
( ( finite_finite_nat @ X7 )
=> ( ( member_nat @ X5 @ X7 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ X7 ) @ X5 ) ) ) ).
% cInf_le_finite
thf(fact_433_cInf__le__finite,axiom,
! [X7: set_real,X5: real] :
( ( finite_finite_real @ X7 )
=> ( ( member_real @ X5 @ X7 )
=> ( ord_less_eq_real @ ( comple4887499456419720421f_real @ X7 ) @ X5 ) ) ) ).
% cInf_le_finite
thf(fact_434_cInf__le__finite,axiom,
! [X7: set_set_complex,X5: set_complex] :
( ( finite6551019134538273531omplex @ X7 )
=> ( ( member_set_complex @ X5 @ X7 )
=> ( ord_le211207098394363844omplex @ ( comple2956690151646016541omplex @ X7 ) @ X5 ) ) ) ).
% cInf_le_finite
thf(fact_435_cInf__le__finite,axiom,
! [X7: set_set_nat,X5: set_nat] :
( ( finite1152437895449049373et_nat @ X7 )
=> ( ( member_set_nat @ X5 @ X7 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ X7 ) @ X5 ) ) ) ).
% cInf_le_finite
thf(fact_436_cInf__le__finite,axiom,
! [X7: set_set_int,X5: set_int] :
( ( finite6197958912794628473et_int @ X7 )
=> ( ( member_set_int @ X5 @ X7 )
=> ( ord_less_eq_set_int @ ( comple3628384868704368283et_int @ X7 ) @ X5 ) ) ) ).
% cInf_le_finite
thf(fact_437_cInf__le__finite,axiom,
! [X7: set_set_set_complex,X5: set_set_complex] :
( ( finite8937801997843863217omplex @ X7 )
=> ( ( member9015044028964487601omplex @ X5 @ X7 )
=> ( ord_le4750530260501030778omplex @ ( comple6723625652910419923omplex @ X7 ) @ X5 ) ) ) ).
% cInf_le_finite
thf(fact_438_cInf__le__finite,axiom,
! [X7: set_set_set_nat,X5: set_set_nat] :
( ( finite6739761609112101331et_nat @ X7 )
=> ( ( member_set_set_nat @ X5 @ X7 )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ X7 ) @ X5 ) ) ) ).
% cInf_le_finite
thf(fact_439_cInf__le__finite,axiom,
! [X7: set_set_set_int,X5: set_set_int] :
( ( finite4249678464180374575et_int @ X7 )
=> ( ( member_set_set_int @ X5 @ X7 )
=> ( ord_le4403425263959731960et_int @ ( comple7798297522565507409et_int @ X7 ) @ X5 ) ) ) ).
% cInf_le_finite
thf(fact_440_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_441_InterE,axiom,
! [A: $o,C2: set_set_o,X7: set_o] :
( ( member_o @ A @ ( comple3063163877087187839_set_o @ C2 ) )
=> ( ( member_set_o @ X7 @ C2 )
=> ( member_o @ A @ X7 ) ) ) ).
% InterE
thf(fact_442_InterE,axiom,
! [A: list_nat,C2: set_set_list_nat,X7: set_list_nat] :
( ( member_list_nat @ A @ ( comple184543376406953807st_nat @ C2 ) )
=> ( ( member_set_list_nat @ X7 @ C2 )
=> ( member_list_nat @ A @ X7 ) ) ) ).
% InterE
thf(fact_443_InterE,axiom,
! [A: complex,C2: set_set_complex,X7: set_complex] :
( ( member_complex @ A @ ( comple2956690151646016541omplex @ C2 ) )
=> ( ( member_set_complex @ X7 @ C2 )
=> ( member_complex @ A @ X7 ) ) ) ).
% InterE
thf(fact_444_InterE,axiom,
! [A: nat,C2: set_set_nat,X7: set_nat] :
( ( member_nat @ A @ ( comple7806235888213564991et_nat @ C2 ) )
=> ( ( member_set_nat @ X7 @ C2 )
=> ( member_nat @ A @ X7 ) ) ) ).
% InterE
thf(fact_445_InterE,axiom,
! [A: int,C2: set_set_int,X7: set_int] :
( ( member_int @ A @ ( comple3628384868704368283et_int @ C2 ) )
=> ( ( member_set_int @ X7 @ C2 )
=> ( member_int @ A @ X7 ) ) ) ).
% InterE
thf(fact_446_InterE,axiom,
! [A: set_list_nat,C2: set_set_set_list_nat,X7: set_set_list_nat] :
( ( member_set_list_nat @ A @ ( comple8462666950445340293st_nat @ C2 ) )
=> ( ( member1029098694177496419st_nat @ X7 @ C2 )
=> ( member_set_list_nat @ A @ X7 ) ) ) ).
% InterE
thf(fact_447_InterD,axiom,
! [A: $o,C2: set_set_o,X7: set_o] :
( ( member_o @ A @ ( comple3063163877087187839_set_o @ C2 ) )
=> ( ( member_set_o @ X7 @ C2 )
=> ( member_o @ A @ X7 ) ) ) ).
% InterD
thf(fact_448_InterD,axiom,
! [A: list_nat,C2: set_set_list_nat,X7: set_list_nat] :
( ( member_list_nat @ A @ ( comple184543376406953807st_nat @ C2 ) )
=> ( ( member_set_list_nat @ X7 @ C2 )
=> ( member_list_nat @ A @ X7 ) ) ) ).
% InterD
thf(fact_449_InterD,axiom,
! [A: complex,C2: set_set_complex,X7: set_complex] :
( ( member_complex @ A @ ( comple2956690151646016541omplex @ C2 ) )
=> ( ( member_set_complex @ X7 @ C2 )
=> ( member_complex @ A @ X7 ) ) ) ).
% InterD
thf(fact_450_InterD,axiom,
! [A: nat,C2: set_set_nat,X7: set_nat] :
( ( member_nat @ A @ ( comple7806235888213564991et_nat @ C2 ) )
=> ( ( member_set_nat @ X7 @ C2 )
=> ( member_nat @ A @ X7 ) ) ) ).
% InterD
thf(fact_451_InterD,axiom,
! [A: int,C2: set_set_int,X7: set_int] :
( ( member_int @ A @ ( comple3628384868704368283et_int @ C2 ) )
=> ( ( member_set_int @ X7 @ C2 )
=> ( member_int @ A @ X7 ) ) ) ).
% InterD
thf(fact_452_InterD,axiom,
! [A: set_list_nat,C2: set_set_set_list_nat,X7: set_set_list_nat] :
( ( member_set_list_nat @ A @ ( comple8462666950445340293st_nat @ C2 ) )
=> ( ( member1029098694177496419st_nat @ X7 @ C2 )
=> ( member_set_list_nat @ A @ X7 ) ) ) ).
% InterD
thf(fact_453_sum_Ofinite__Collect__op,axiom,
! [I5: set_o,X5: $o > int,Y: $o > int] :
( ( finite_finite_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ I5 )
& ( ( X5 @ I2 )
!= zero_zero_int ) ) ) )
=> ( ( finite_finite_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ I5 )
& ( ( Y @ I2 )
!= zero_zero_int ) ) ) )
=> ( finite_finite_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ I5 )
& ( ( plus_plus_int @ ( X5 @ I2 ) @ ( Y @ I2 ) )
!= zero_zero_int ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_454_sum_Ofinite__Collect__op,axiom,
! [I5: set_int,X5: int > int,Y: int > int] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( X5 @ I2 )
!= zero_zero_int ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( Y @ I2 )
!= zero_zero_int ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( plus_plus_int @ ( X5 @ I2 ) @ ( Y @ I2 ) )
!= zero_zero_int ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_455_sum_Ofinite__Collect__op,axiom,
! [I5: set_nat,X5: nat > int,Y: nat > int] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I5 )
& ( ( X5 @ I2 )
!= zero_zero_int ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I5 )
& ( ( Y @ I2 )
!= zero_zero_int ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I5 )
& ( ( plus_plus_int @ ( X5 @ I2 ) @ ( Y @ I2 ) )
!= zero_zero_int ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_456_sum_Ofinite__Collect__op,axiom,
! [I5: set_complex,X5: complex > int,Y: complex > int] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I2: complex] :
( ( member_complex @ I2 @ I5 )
& ( ( X5 @ I2 )
!= zero_zero_int ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I2: complex] :
( ( member_complex @ I2 @ I5 )
& ( ( Y @ I2 )
!= zero_zero_int ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I2: complex] :
( ( member_complex @ I2 @ I5 )
& ( ( plus_plus_int @ ( X5 @ I2 ) @ ( Y @ I2 ) )
!= zero_zero_int ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_457_sum_Ofinite__Collect__op,axiom,
! [I5: set_o,X5: $o > nat,Y: $o > nat] :
( ( finite_finite_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ I5 )
& ( ( X5 @ I2 )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ I5 )
& ( ( Y @ I2 )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ I5 )
& ( ( plus_plus_nat @ ( X5 @ I2 ) @ ( Y @ I2 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_458_sum_Ofinite__Collect__op,axiom,
! [I5: set_int,X5: int > nat,Y: int > nat] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( X5 @ I2 )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( Y @ I2 )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( plus_plus_nat @ ( X5 @ I2 ) @ ( Y @ I2 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_459_sum_Ofinite__Collect__op,axiom,
! [I5: set_nat,X5: nat > nat,Y: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I5 )
& ( ( X5 @ I2 )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I5 )
& ( ( Y @ I2 )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I5 )
& ( ( plus_plus_nat @ ( X5 @ I2 ) @ ( Y @ I2 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_460_sum_Ofinite__Collect__op,axiom,
! [I5: set_complex,X5: complex > nat,Y: complex > nat] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I2: complex] :
( ( member_complex @ I2 @ I5 )
& ( ( X5 @ I2 )
!= zero_zero_nat ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I2: complex] :
( ( member_complex @ I2 @ I5 )
& ( ( Y @ I2 )
!= zero_zero_nat ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I2: complex] :
( ( member_complex @ I2 @ I5 )
& ( ( plus_plus_nat @ ( X5 @ I2 ) @ ( Y @ I2 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_461_sum_Ofinite__Collect__op,axiom,
! [I5: set_o,X5: $o > real,Y: $o > real] :
( ( finite_finite_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ I5 )
& ( ( X5 @ I2 )
!= zero_zero_real ) ) ) )
=> ( ( finite_finite_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ I5 )
& ( ( Y @ I2 )
!= zero_zero_real ) ) ) )
=> ( finite_finite_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ I5 )
& ( ( plus_plus_real @ ( X5 @ I2 ) @ ( Y @ I2 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_462_sum_Ofinite__Collect__op,axiom,
! [I5: set_int,X5: int > real,Y: int > real] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( X5 @ I2 )
!= zero_zero_real ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( Y @ I2 )
!= zero_zero_real ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( plus_plus_real @ ( X5 @ I2 ) @ ( Y @ I2 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_463_prod_Ofinite__Collect__op,axiom,
! [I5: set_o,X5: $o > int,Y: $o > int] :
( ( finite_finite_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ I5 )
& ( ( X5 @ I2 )
!= one_one_int ) ) ) )
=> ( ( finite_finite_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ I5 )
& ( ( Y @ I2 )
!= one_one_int ) ) ) )
=> ( finite_finite_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ I5 )
& ( ( times_times_int @ ( X5 @ I2 ) @ ( Y @ I2 ) )
!= one_one_int ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_464_prod_Ofinite__Collect__op,axiom,
! [I5: set_int,X5: int > int,Y: int > int] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( X5 @ I2 )
!= one_one_int ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( Y @ I2 )
!= one_one_int ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( times_times_int @ ( X5 @ I2 ) @ ( Y @ I2 ) )
!= one_one_int ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_465_prod_Ofinite__Collect__op,axiom,
! [I5: set_nat,X5: nat > int,Y: nat > int] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I5 )
& ( ( X5 @ I2 )
!= one_one_int ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I5 )
& ( ( Y @ I2 )
!= one_one_int ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I5 )
& ( ( times_times_int @ ( X5 @ I2 ) @ ( Y @ I2 ) )
!= one_one_int ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_466_prod_Ofinite__Collect__op,axiom,
! [I5: set_complex,X5: complex > int,Y: complex > int] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I2: complex] :
( ( member_complex @ I2 @ I5 )
& ( ( X5 @ I2 )
!= one_one_int ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I2: complex] :
( ( member_complex @ I2 @ I5 )
& ( ( Y @ I2 )
!= one_one_int ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I2: complex] :
( ( member_complex @ I2 @ I5 )
& ( ( times_times_int @ ( X5 @ I2 ) @ ( Y @ I2 ) )
!= one_one_int ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_467_prod_Ofinite__Collect__op,axiom,
! [I5: set_o,X5: $o > nat,Y: $o > nat] :
( ( finite_finite_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ I5 )
& ( ( X5 @ I2 )
!= one_one_nat ) ) ) )
=> ( ( finite_finite_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ I5 )
& ( ( Y @ I2 )
!= one_one_nat ) ) ) )
=> ( finite_finite_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ I5 )
& ( ( times_times_nat @ ( X5 @ I2 ) @ ( Y @ I2 ) )
!= one_one_nat ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_468_prod_Ofinite__Collect__op,axiom,
! [I5: set_int,X5: int > nat,Y: int > nat] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( X5 @ I2 )
!= one_one_nat ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( Y @ I2 )
!= one_one_nat ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( times_times_nat @ ( X5 @ I2 ) @ ( Y @ I2 ) )
!= one_one_nat ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_469_prod_Ofinite__Collect__op,axiom,
! [I5: set_nat,X5: nat > nat,Y: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I5 )
& ( ( X5 @ I2 )
!= one_one_nat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I5 )
& ( ( Y @ I2 )
!= one_one_nat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I5 )
& ( ( times_times_nat @ ( X5 @ I2 ) @ ( Y @ I2 ) )
!= one_one_nat ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_470_prod_Ofinite__Collect__op,axiom,
! [I5: set_complex,X5: complex > nat,Y: complex > nat] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I2: complex] :
( ( member_complex @ I2 @ I5 )
& ( ( X5 @ I2 )
!= one_one_nat ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I2: complex] :
( ( member_complex @ I2 @ I5 )
& ( ( Y @ I2 )
!= one_one_nat ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I2: complex] :
( ( member_complex @ I2 @ I5 )
& ( ( times_times_nat @ ( X5 @ I2 ) @ ( Y @ I2 ) )
!= one_one_nat ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_471_prod_Ofinite__Collect__op,axiom,
! [I5: set_o,X5: $o > real,Y: $o > real] :
( ( finite_finite_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ I5 )
& ( ( X5 @ I2 )
!= one_one_real ) ) ) )
=> ( ( finite_finite_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ I5 )
& ( ( Y @ I2 )
!= one_one_real ) ) ) )
=> ( finite_finite_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ I5 )
& ( ( times_times_real @ ( X5 @ I2 ) @ ( Y @ I2 ) )
!= one_one_real ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_472_prod_Ofinite__Collect__op,axiom,
! [I5: set_int,X5: int > real,Y: int > real] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( X5 @ I2 )
!= one_one_real ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( Y @ I2 )
!= one_one_real ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( member_int @ I2 @ I5 )
& ( ( times_times_real @ ( X5 @ I2 ) @ ( Y @ I2 ) )
!= one_one_real ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_473_sum_Ointer__filter,axiom,
! [A: set_o,G: $o > int,P: $o > $o] :
( ( finite_finite_o @ A )
=> ( ( groups8505340233167759370_o_int @ G
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( groups8505340233167759370_o_int
@ ^ [X3: $o] : ( if_int @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_int )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_474_sum_Ointer__filter,axiom,
! [A: set_int,G: int > int,P: int > $o] :
( ( finite_finite_int @ A )
=> ( ( groups4538972089207619220nt_int @ G
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( groups4538972089207619220nt_int
@ ^ [X3: int] : ( if_int @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_int )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_475_sum_Ointer__filter,axiom,
! [A: set_complex,G: complex > int,P: complex > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ( groups5690904116761175830ex_int @ G
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( groups5690904116761175830ex_int
@ ^ [X3: complex] : ( if_int @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_int )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_476_sum_Ointer__filter,axiom,
! [A: set_o,G: $o > nat,P: $o > $o] :
( ( finite_finite_o @ A )
=> ( ( groups8507830703676809646_o_nat @ G
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( groups8507830703676809646_o_nat
@ ^ [X3: $o] : ( if_nat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_nat )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_477_sum_Ointer__filter,axiom,
! [A: set_int,G: int > nat,P: int > $o] :
( ( finite_finite_int @ A )
=> ( ( groups4541462559716669496nt_nat @ G
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( groups4541462559716669496nt_nat
@ ^ [X3: int] : ( if_nat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_nat )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_478_sum_Ointer__filter,axiom,
! [A: set_complex,G: complex > nat,P: complex > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ( groups5693394587270226106ex_nat @ G
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( groups5693394587270226106ex_nat
@ ^ [X3: complex] : ( if_nat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_nat )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_479_sum_Ointer__filter,axiom,
! [A: set_o,G: $o > real,P: $o > $o] :
( ( finite_finite_o @ A )
=> ( ( groups8691415230153176458o_real @ G
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( groups8691415230153176458o_real
@ ^ [X3: $o] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_real )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_480_sum_Ointer__filter,axiom,
! [A: set_int,G: int > real,P: int > $o] :
( ( finite_finite_int @ A )
=> ( ( groups8778361861064173332t_real @ G
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( groups8778361861064173332t_real
@ ^ [X3: int] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_real )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_481_sum_Ointer__filter,axiom,
! [A: set_complex,G: complex > real,P: complex > $o] :
( ( finite3207457112153483333omplex @ A )
=> ( ( groups5808333547571424918x_real @ G
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( groups5808333547571424918x_real
@ ^ [X3: complex] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_real )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_482_sum_Ointer__filter,axiom,
! [A: set_o,G: $o > extend8495563244428889912nnreal,P: $o > $o] :
( ( finite_finite_o @ A )
=> ( ( groups7456689898616286486nnreal @ G
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( groups7456689898616286486nnreal
@ ^ [X3: $o] : ( if_Ext9135588136721118450nnreal @ ( P @ X3 ) @ ( G @ X3 ) @ zero_z7100319975126383169nnreal )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_483_sum_Oimage__gen,axiom,
! [S: set_o,H: $o > real,G: $o > nat] :
( ( finite_finite_o @ S )
=> ( ( groups8691415230153176458o_real @ H @ S )
= ( groups6591440286371151544t_real
@ ^ [Y2: nat] :
( groups8691415230153176458o_real @ H
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) )
@ ( image_o_nat @ G @ S ) ) ) ) ).
% sum.image_gen
thf(fact_484_sum_Oimage__gen,axiom,
! [S: set_int,H: int > real,G: int > nat] :
( ( finite_finite_int @ S )
=> ( ( groups8778361861064173332t_real @ H @ S )
= ( groups6591440286371151544t_real
@ ^ [Y2: nat] :
( groups8778361861064173332t_real @ H
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) )
@ ( image_int_nat @ G @ S ) ) ) ) ).
% sum.image_gen
thf(fact_485_sum_Oimage__gen,axiom,
! [S: set_complex,H: complex > real,G: complex > nat] :
( ( finite3207457112153483333omplex @ S )
=> ( ( groups5808333547571424918x_real @ H @ S )
= ( groups6591440286371151544t_real
@ ^ [Y2: nat] :
( groups5808333547571424918x_real @ H
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) )
@ ( image_complex_nat @ G @ S ) ) ) ) ).
% sum.image_gen
thf(fact_486_sum_Oimage__gen,axiom,
! [S: set_o,H: $o > nat,G: $o > nat] :
( ( finite_finite_o @ S )
=> ( ( groups8507830703676809646_o_nat @ H @ S )
= ( groups3542108847815614940at_nat
@ ^ [Y2: nat] :
( groups8507830703676809646_o_nat @ H
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) )
@ ( image_o_nat @ G @ S ) ) ) ) ).
% sum.image_gen
thf(fact_487_sum_Oimage__gen,axiom,
! [S: set_int,H: int > nat,G: int > nat] :
( ( finite_finite_int @ S )
=> ( ( groups4541462559716669496nt_nat @ H @ S )
= ( groups3542108847815614940at_nat
@ ^ [Y2: nat] :
( groups4541462559716669496nt_nat @ H
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) )
@ ( image_int_nat @ G @ S ) ) ) ) ).
% sum.image_gen
thf(fact_488_sum_Oimage__gen,axiom,
! [S: set_complex,H: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ S )
=> ( ( groups5693394587270226106ex_nat @ H @ S )
= ( groups3542108847815614940at_nat
@ ^ [Y2: nat] :
( groups5693394587270226106ex_nat @ H
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) )
@ ( image_complex_nat @ G @ S ) ) ) ) ).
% sum.image_gen
thf(fact_489_sum_Oimage__gen,axiom,
! [S: set_o,H: $o > int,G: $o > nat] :
( ( finite_finite_o @ S )
=> ( ( groups8505340233167759370_o_int @ H @ S )
= ( groups3539618377306564664at_int
@ ^ [Y2: nat] :
( groups8505340233167759370_o_int @ H
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) )
@ ( image_o_nat @ G @ S ) ) ) ) ).
% sum.image_gen
thf(fact_490_sum_Oimage__gen,axiom,
! [S: set_int,H: int > int,G: int > nat] :
( ( finite_finite_int @ S )
=> ( ( groups4538972089207619220nt_int @ H @ S )
= ( groups3539618377306564664at_int
@ ^ [Y2: nat] :
( groups4538972089207619220nt_int @ H
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) )
@ ( image_int_nat @ G @ S ) ) ) ) ).
% sum.image_gen
thf(fact_491_sum_Oimage__gen,axiom,
! [S: set_complex,H: complex > int,G: complex > nat] :
( ( finite3207457112153483333omplex @ S )
=> ( ( groups5690904116761175830ex_int @ H @ S )
= ( groups3539618377306564664at_int
@ ^ [Y2: nat] :
( groups5690904116761175830ex_int @ H
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) )
@ ( image_complex_nat @ G @ S ) ) ) ) ).
% sum.image_gen
thf(fact_492_sum_Oimage__gen,axiom,
! [S: set_nat,H: nat > real,G: nat > $o] :
( ( finite_finite_nat @ S )
=> ( ( groups6591440286371151544t_real @ H @ S )
= ( groups8691415230153176458o_real
@ ^ [Y2: $o] :
( groups6591440286371151544t_real @ H
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) )
@ ( image_nat_o @ G @ S ) ) ) ) ).
% sum.image_gen
thf(fact_493_Sup_OSUP__identity__eq,axiom,
! [Sup: set_int > int,A: set_int] :
( ( Sup
@ ( image_int_int
@ ^ [X3: int] : X3
@ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_identity_eq
thf(fact_494_Sup_OSUP__identity__eq,axiom,
! [Sup: set_real > real,A: set_real] :
( ( Sup
@ ( image_real_real
@ ^ [X3: real] : X3
@ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_identity_eq
thf(fact_495_Sup_OSUP__identity__eq,axiom,
! [Sup: set_set_list_nat > set_list_nat,A: set_set_list_nat] :
( ( Sup
@ ( image_5143090206295581363st_nat
@ ^ [X3: set_list_nat] : X3
@ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_identity_eq
thf(fact_496_Inf_OINF__identity__eq,axiom,
! [Inf: set_int > int,A: set_int] :
( ( Inf
@ ( image_int_int
@ ^ [X3: int] : X3
@ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_identity_eq
thf(fact_497_Inf_OINF__identity__eq,axiom,
! [Inf: set_real > real,A: set_real] :
( ( Inf
@ ( image_real_real
@ ^ [X3: real] : X3
@ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_identity_eq
thf(fact_498_Inf_OINF__identity__eq,axiom,
! [Inf: set_set_list_nat > set_list_nat,A: set_set_list_nat] :
( ( Inf
@ ( image_5143090206295581363st_nat
@ ^ [X3: set_list_nat] : X3
@ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_identity_eq
thf(fact_499_sum_Oswap,axiom,
! [G: nat > nat > real,B2: set_nat,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I2: nat] : ( groups6591440286371151544t_real @ ( G @ I2 ) @ B2 )
@ A )
= ( groups6591440286371151544t_real
@ ^ [J: nat] :
( groups6591440286371151544t_real
@ ^ [I2: nat] : ( G @ I2 @ J )
@ A )
@ B2 ) ) ).
% sum.swap
thf(fact_500_sum_Oswap,axiom,
! [G: nat > nat > nat,B2: set_nat,A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( groups3542108847815614940at_nat @ ( G @ I2 ) @ B2 )
@ A )
= ( groups3542108847815614940at_nat
@ ^ [J: nat] :
( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( G @ I2 @ J )
@ A )
@ B2 ) ) ).
% sum.swap
thf(fact_501_sum_Oswap,axiom,
! [G: nat > nat > int,B2: set_nat,A: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [I2: nat] : ( groups3539618377306564664at_int @ ( G @ I2 ) @ B2 )
@ A )
= ( groups3539618377306564664at_int
@ ^ [J: nat] :
( groups3539618377306564664at_int
@ ^ [I2: nat] : ( G @ I2 @ J )
@ A )
@ B2 ) ) ).
% sum.swap
thf(fact_502_sum_Oswap,axiom,
! [G: set_list_nat > nat > int,B2: set_nat,A: set_set_list_nat] :
( ( groups7312845317294741502at_int
@ ^ [I2: set_list_nat] : ( groups3539618377306564664at_int @ ( G @ I2 ) @ B2 )
@ A )
= ( groups3539618377306564664at_int
@ ^ [J: nat] :
( groups7312845317294741502at_int
@ ^ [I2: set_list_nat] : ( G @ I2 @ J )
@ A )
@ B2 ) ) ).
% sum.swap
thf(fact_503_sum_Oswap,axiom,
! [G: set_set_nat > nat > int,B2: set_nat,A: set_set_set_nat] :
( ( groups7084729577923612836at_int
@ ^ [I2: set_set_nat] : ( groups3539618377306564664at_int @ ( G @ I2 ) @ B2 )
@ A )
= ( groups3539618377306564664at_int
@ ^ [J: nat] :
( groups7084729577923612836at_int
@ ^ [I2: set_set_nat] : ( G @ I2 @ J )
@ A )
@ B2 ) ) ).
% sum.swap
thf(fact_504_sum_Oswap,axiom,
! [G: set_set_int > nat > int,B2: set_nat,A: set_set_set_int] :
( ( groups1080061135233207040nt_int
@ ^ [I2: set_set_int] : ( groups3539618377306564664at_int @ ( G @ I2 ) @ B2 )
@ A )
= ( groups3539618377306564664at_int
@ ^ [J: nat] :
( groups1080061135233207040nt_int
@ ^ [I2: set_set_int] : ( G @ I2 @ J )
@ A )
@ B2 ) ) ).
% sum.swap
thf(fact_505_sum_Oswap,axiom,
! [G: nat > set_list_nat > int,B2: set_set_list_nat,A: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [I2: nat] : ( groups7312845317294741502at_int @ ( G @ I2 ) @ B2 )
@ A )
= ( groups7312845317294741502at_int
@ ^ [J: set_list_nat] :
( groups3539618377306564664at_int
@ ^ [I2: nat] : ( G @ I2 @ J )
@ A )
@ B2 ) ) ).
% sum.swap
thf(fact_506_sum_Oswap,axiom,
! [G: nat > set_set_nat > int,B2: set_set_set_nat,A: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [I2: nat] : ( groups7084729577923612836at_int @ ( G @ I2 ) @ B2 )
@ A )
= ( groups7084729577923612836at_int
@ ^ [J: set_set_nat] :
( groups3539618377306564664at_int
@ ^ [I2: nat] : ( G @ I2 @ J )
@ A )
@ B2 ) ) ).
% sum.swap
thf(fact_507_sum_Oswap,axiom,
! [G: nat > set_set_int > int,B2: set_set_set_int,A: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [I2: nat] : ( groups1080061135233207040nt_int @ ( G @ I2 ) @ B2 )
@ A )
= ( groups1080061135233207040nt_int
@ ^ [J: set_set_int] :
( groups3539618377306564664at_int
@ ^ [I2: nat] : ( G @ I2 @ J )
@ A )
@ B2 ) ) ).
% sum.swap
thf(fact_508_sum_Oswap,axiom,
! [G: set_set_list_nat > nat > int,B2: set_nat,A: set_set_set_list_nat] :
( ( groups7004213669654646580at_int
@ ^ [I2: set_set_list_nat] : ( groups3539618377306564664at_int @ ( G @ I2 ) @ B2 )
@ A )
= ( groups3539618377306564664at_int
@ ^ [J: nat] :
( groups7004213669654646580at_int
@ ^ [I2: set_set_list_nat] : ( G @ I2 @ J )
@ A )
@ B2 ) ) ).
% sum.swap
thf(fact_509_sum__nonneg__eq__0__iff,axiom,
! [A: set_o,F: $o > extend8495563244428889912nnreal] :
( ( finite_finite_o @ A )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ X2 ) ) )
=> ( ( ( groups7456689898616286486nnreal @ F @ A )
= zero_z7100319975126383169nnreal )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ( F @ X3 )
= zero_z7100319975126383169nnreal ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_510_sum__nonneg__eq__0__iff,axiom,
! [A: set_int,F: int > extend8495563244428889912nnreal] :
( ( finite_finite_int @ A )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ X2 ) ) )
=> ( ( ( groups2558975329500312480nnreal @ F @ A )
= zero_z7100319975126383169nnreal )
= ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ( F @ X3 )
= zero_z7100319975126383169nnreal ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_511_sum__nonneg__eq__0__iff,axiom,
! [A: set_nat,F: nat > extend8495563244428889912nnreal] :
( ( finite_finite_nat @ A )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ X2 ) ) )
=> ( ( ( groups4868793261593263428nnreal @ F @ A )
= zero_z7100319975126383169nnreal )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( F @ X3 )
= zero_z7100319975126383169nnreal ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_512_sum__nonneg__eq__0__iff,axiom,
! [A: set_complex,F: complex > extend8495563244428889912nnreal] :
( ( finite3207457112153483333omplex @ A )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ A )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ X2 ) ) )
=> ( ( ( groups6103019165529820194nnreal @ F @ A )
= zero_z7100319975126383169nnreal )
= ( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( ( F @ X3 )
= zero_z7100319975126383169nnreal ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_513_sum__nonneg__eq__0__iff,axiom,
! [A: set_o,F: $o > int] :
( ( finite_finite_o @ A )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X2 ) ) )
=> ( ( ( groups8505340233167759370_o_int @ F @ A )
= zero_zero_int )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ( F @ X3 )
= zero_zero_int ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_514_sum__nonneg__eq__0__iff,axiom,
! [A: set_int,F: int > int] :
( ( finite_finite_int @ A )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X2 ) ) )
=> ( ( ( groups4538972089207619220nt_int @ F @ A )
= zero_zero_int )
= ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ( F @ X3 )
= zero_zero_int ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_515_sum__nonneg__eq__0__iff,axiom,
! [A: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ A )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X2 ) ) )
=> ( ( ( groups5690904116761175830ex_int @ F @ A )
= zero_zero_int )
= ( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( ( F @ X3 )
= zero_zero_int ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_516_sum__nonneg__eq__0__iff,axiom,
! [A: set_o,F: $o > nat] :
( ( finite_finite_o @ A )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ( ( groups8507830703676809646_o_nat @ F @ A )
= zero_zero_nat )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_517_sum__nonneg__eq__0__iff,axiom,
! [A: set_int,F: int > nat] :
( ( finite_finite_int @ A )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F @ A )
= zero_zero_nat )
= ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_518_sum__nonneg__eq__0__iff,axiom,
! [A: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ( ( groups5693394587270226106ex_nat @ F @ A )
= zero_zero_nat )
= ( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( ( F @ X3 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_519_sum__le__included,axiom,
! [S2: set_int,T2: set_int,G: int > extend8495563244428889912nnreal,I3: int > int,F: int > extend8495563244428889912nnreal] :
( ( finite_finite_int @ S2 )
=> ( ( finite_finite_int @ T2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ T2 )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( G @ X2 ) ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ T2 )
& ( ( I3 @ Xa )
= X2 )
& ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_le3935885782089961368nnreal @ ( groups2558975329500312480nnreal @ F @ S2 ) @ ( groups2558975329500312480nnreal @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_520_sum__le__included,axiom,
! [S2: set_int,T2: set_nat,G: nat > extend8495563244428889912nnreal,I3: nat > int,F: int > extend8495563244428889912nnreal] :
( ( finite_finite_int @ S2 )
=> ( ( finite_finite_nat @ T2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T2 )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( G @ X2 ) ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T2 )
& ( ( I3 @ Xa )
= X2 )
& ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_le3935885782089961368nnreal @ ( groups2558975329500312480nnreal @ F @ S2 ) @ ( groups4868793261593263428nnreal @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_521_sum__le__included,axiom,
! [S2: set_int,T2: set_complex,G: complex > extend8495563244428889912nnreal,I3: complex > int,F: int > extend8495563244428889912nnreal] :
( ( finite_finite_int @ S2 )
=> ( ( finite3207457112153483333omplex @ T2 )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ T2 )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( G @ X2 ) ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S2 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T2 )
& ( ( I3 @ Xa )
= X2 )
& ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_le3935885782089961368nnreal @ ( groups2558975329500312480nnreal @ F @ S2 ) @ ( groups6103019165529820194nnreal @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_522_sum__le__included,axiom,
! [S2: set_nat,T2: set_int,G: int > extend8495563244428889912nnreal,I3: int > nat,F: nat > extend8495563244428889912nnreal] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_int @ T2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ T2 )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( G @ X2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ T2 )
& ( ( I3 @ Xa )
= X2 )
& ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_le3935885782089961368nnreal @ ( groups4868793261593263428nnreal @ F @ S2 ) @ ( groups2558975329500312480nnreal @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_523_sum__le__included,axiom,
! [S2: set_nat,T2: set_nat,G: nat > extend8495563244428889912nnreal,I3: nat > nat,F: nat > extend8495563244428889912nnreal] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_nat @ T2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T2 )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( G @ X2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T2 )
& ( ( I3 @ Xa )
= X2 )
& ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_le3935885782089961368nnreal @ ( groups4868793261593263428nnreal @ F @ S2 ) @ ( groups4868793261593263428nnreal @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_524_sum__le__included,axiom,
! [S2: set_nat,T2: set_complex,G: complex > extend8495563244428889912nnreal,I3: complex > nat,F: nat > extend8495563244428889912nnreal] :
( ( finite_finite_nat @ S2 )
=> ( ( finite3207457112153483333omplex @ T2 )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ T2 )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( G @ X2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S2 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T2 )
& ( ( I3 @ Xa )
= X2 )
& ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_le3935885782089961368nnreal @ ( groups4868793261593263428nnreal @ F @ S2 ) @ ( groups6103019165529820194nnreal @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_525_sum__le__included,axiom,
! [S2: set_complex,T2: set_int,G: int > extend8495563244428889912nnreal,I3: int > complex,F: complex > extend8495563244428889912nnreal] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( finite_finite_int @ T2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ T2 )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( G @ X2 ) ) )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ S2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ T2 )
& ( ( I3 @ Xa )
= X2 )
& ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_le3935885782089961368nnreal @ ( groups6103019165529820194nnreal @ F @ S2 ) @ ( groups2558975329500312480nnreal @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_526_sum__le__included,axiom,
! [S2: set_complex,T2: set_nat,G: nat > extend8495563244428889912nnreal,I3: nat > complex,F: complex > extend8495563244428889912nnreal] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( finite_finite_nat @ T2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T2 )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( G @ X2 ) ) )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ S2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T2 )
& ( ( I3 @ Xa )
= X2 )
& ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_le3935885782089961368nnreal @ ( groups6103019165529820194nnreal @ F @ S2 ) @ ( groups4868793261593263428nnreal @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_527_sum__le__included,axiom,
! [S2: set_complex,T2: set_complex,G: complex > extend8495563244428889912nnreal,I3: complex > complex,F: complex > extend8495563244428889912nnreal] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( finite3207457112153483333omplex @ T2 )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ T2 )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( G @ X2 ) ) )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ S2 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T2 )
& ( ( I3 @ Xa )
= X2 )
& ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_le3935885782089961368nnreal @ ( groups6103019165529820194nnreal @ F @ S2 ) @ ( groups6103019165529820194nnreal @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_528_sum__le__included,axiom,
! [S2: set_int,T2: set_int,G: int > int,I3: int > int,F: int > int] :
( ( finite_finite_int @ S2 )
=> ( ( finite_finite_int @ T2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ T2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( G @ X2 ) ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S2 )
=> ? [Xa: int] :
( ( member_int @ Xa @ T2 )
& ( ( I3 @ Xa )
= X2 )
& ( ord_less_eq_int @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F @ S2 ) @ ( groups4538972089207619220nt_int @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_529_sum_Orelated,axiom,
! [R: int > int > $o,S: set_int,H: int > int,G: int > int] :
( ( R @ zero_zero_int @ zero_zero_int )
=> ( ! [X1: int,Y1: int,X22: int,Y22: int] :
( ( ( R @ X1 @ X22 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_int @ X1 @ Y1 ) @ ( plus_plus_int @ X22 @ Y22 ) ) )
=> ( ( finite_finite_int @ S )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S )
=> ( R @ ( H @ X2 ) @ ( G @ X2 ) ) )
=> ( R @ ( groups4538972089207619220nt_int @ H @ S ) @ ( groups4538972089207619220nt_int @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_530_sum_Orelated,axiom,
! [R: int > int > $o,S: set_complex,H: complex > int,G: complex > int] :
( ( R @ zero_zero_int @ zero_zero_int )
=> ( ! [X1: int,Y1: int,X22: int,Y22: int] :
( ( ( R @ X1 @ X22 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_int @ X1 @ Y1 ) @ ( plus_plus_int @ X22 @ Y22 ) ) )
=> ( ( finite3207457112153483333omplex @ S )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ S )
=> ( R @ ( H @ X2 ) @ ( G @ X2 ) ) )
=> ( R @ ( groups5690904116761175830ex_int @ H @ S ) @ ( groups5690904116761175830ex_int @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_531_sum_Orelated,axiom,
! [R: nat > nat > $o,S: set_int,H: int > nat,G: int > nat] :
( ( R @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X1: nat,Y1: nat,X22: nat,Y22: nat] :
( ( ( R @ X1 @ X22 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X22 @ Y22 ) ) )
=> ( ( finite_finite_int @ S )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S )
=> ( R @ ( H @ X2 ) @ ( G @ X2 ) ) )
=> ( R @ ( groups4541462559716669496nt_nat @ H @ S ) @ ( groups4541462559716669496nt_nat @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_532_sum_Orelated,axiom,
! [R: nat > nat > $o,S: set_complex,H: complex > nat,G: complex > nat] :
( ( R @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X1: nat,Y1: nat,X22: nat,Y22: nat] :
( ( ( R @ X1 @ X22 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X22 @ Y22 ) ) )
=> ( ( finite3207457112153483333omplex @ S )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ S )
=> ( R @ ( H @ X2 ) @ ( G @ X2 ) ) )
=> ( R @ ( groups5693394587270226106ex_nat @ H @ S ) @ ( groups5693394587270226106ex_nat @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_533_sum_Orelated,axiom,
! [R: real > real > $o,S: set_int,H: int > real,G: int > real] :
( ( R @ zero_zero_real @ zero_zero_real )
=> ( ! [X1: real,Y1: real,X22: real,Y22: real] :
( ( ( R @ X1 @ X22 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X22 @ Y22 ) ) )
=> ( ( finite_finite_int @ S )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S )
=> ( R @ ( H @ X2 ) @ ( G @ X2 ) ) )
=> ( R @ ( groups8778361861064173332t_real @ H @ S ) @ ( groups8778361861064173332t_real @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_534_sum_Orelated,axiom,
! [R: real > real > $o,S: set_complex,H: complex > real,G: complex > real] :
( ( R @ zero_zero_real @ zero_zero_real )
=> ( ! [X1: real,Y1: real,X22: real,Y22: real] :
( ( ( R @ X1 @ X22 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X22 @ Y22 ) ) )
=> ( ( finite3207457112153483333omplex @ S )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ S )
=> ( R @ ( H @ X2 ) @ ( G @ X2 ) ) )
=> ( R @ ( groups5808333547571424918x_real @ H @ S ) @ ( groups5808333547571424918x_real @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_535_sum_Orelated,axiom,
! [R: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o,S: set_int,H: int > extend8495563244428889912nnreal,G: int > extend8495563244428889912nnreal] :
( ( R @ zero_z7100319975126383169nnreal @ zero_z7100319975126383169nnreal )
=> ( ! [X1: extend8495563244428889912nnreal,Y1: extend8495563244428889912nnreal,X22: extend8495563244428889912nnreal,Y22: extend8495563244428889912nnreal] :
( ( ( R @ X1 @ X22 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_p1859984266308609217nnreal @ X1 @ Y1 ) @ ( plus_p1859984266308609217nnreal @ X22 @ Y22 ) ) )
=> ( ( finite_finite_int @ S )
=> ( ! [X2: int] :
( ( member_int @ X2 @ S )
=> ( R @ ( H @ X2 ) @ ( G @ X2 ) ) )
=> ( R @ ( groups2558975329500312480nnreal @ H @ S ) @ ( groups2558975329500312480nnreal @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_536_sum_Orelated,axiom,
! [R: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o,S: set_nat,H: nat > extend8495563244428889912nnreal,G: nat > extend8495563244428889912nnreal] :
( ( R @ zero_z7100319975126383169nnreal @ zero_z7100319975126383169nnreal )
=> ( ! [X1: extend8495563244428889912nnreal,Y1: extend8495563244428889912nnreal,X22: extend8495563244428889912nnreal,Y22: extend8495563244428889912nnreal] :
( ( ( R @ X1 @ X22 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_p1859984266308609217nnreal @ X1 @ Y1 ) @ ( plus_p1859984266308609217nnreal @ X22 @ Y22 ) ) )
=> ( ( finite_finite_nat @ S )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( R @ ( H @ X2 ) @ ( G @ X2 ) ) )
=> ( R @ ( groups4868793261593263428nnreal @ H @ S ) @ ( groups4868793261593263428nnreal @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_537_sum_Orelated,axiom,
! [R: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o,S: set_complex,H: complex > extend8495563244428889912nnreal,G: complex > extend8495563244428889912nnreal] :
( ( R @ zero_z7100319975126383169nnreal @ zero_z7100319975126383169nnreal )
=> ( ! [X1: extend8495563244428889912nnreal,Y1: extend8495563244428889912nnreal,X22: extend8495563244428889912nnreal,Y22: extend8495563244428889912nnreal] :
( ( ( R @ X1 @ X22 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_p1859984266308609217nnreal @ X1 @ Y1 ) @ ( plus_p1859984266308609217nnreal @ X22 @ Y22 ) ) )
=> ( ( finite3207457112153483333omplex @ S )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ S )
=> ( R @ ( H @ X2 ) @ ( G @ X2 ) ) )
=> ( R @ ( groups6103019165529820194nnreal @ H @ S ) @ ( groups6103019165529820194nnreal @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_538_sum_Orelated,axiom,
! [R: real > real > $o,S: set_nat,H: nat > real,G: nat > real] :
( ( R @ zero_zero_real @ zero_zero_real )
=> ( ! [X1: real,Y1: real,X22: real,Y22: real] :
( ( ( R @ X1 @ X22 )
& ( R @ Y1 @ Y22 ) )
=> ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X22 @ Y22 ) ) )
=> ( ( finite_finite_nat @ S )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( R @ ( H @ X2 ) @ ( G @ X2 ) ) )
=> ( R @ ( groups6591440286371151544t_real @ H @ S ) @ ( groups6591440286371151544t_real @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_539_sum__nonneg__leq__bound,axiom,
! [S2: set_o,F: $o > extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,I3: $o] :
( ( finite_finite_o @ S2 )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ S2 )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ I4 ) ) )
=> ( ( ( groups7456689898616286486nnreal @ F @ S2 )
= B2 )
=> ( ( member_o @ I3 @ S2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ I3 ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_540_sum__nonneg__leq__bound,axiom,
! [S2: set_int,F: int > extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,I3: int] :
( ( finite_finite_int @ S2 )
=> ( ! [I4: int] :
( ( member_int @ I4 @ S2 )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ I4 ) ) )
=> ( ( ( groups2558975329500312480nnreal @ F @ S2 )
= B2 )
=> ( ( member_int @ I3 @ S2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ I3 ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_541_sum__nonneg__leq__bound,axiom,
! [S2: set_nat,F: nat > extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,I3: nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ S2 )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ I4 ) ) )
=> ( ( ( groups4868793261593263428nnreal @ F @ S2 )
= B2 )
=> ( ( member_nat @ I3 @ S2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ I3 ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_542_sum__nonneg__leq__bound,axiom,
! [S2: set_complex,F: complex > extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,I3: complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ S2 )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ I4 ) ) )
=> ( ( ( groups6103019165529820194nnreal @ F @ S2 )
= B2 )
=> ( ( member_complex @ I3 @ S2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ I3 ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_543_sum__nonneg__leq__bound,axiom,
! [S2: set_o,F: $o > int,B2: int,I3: $o] :
( ( finite_finite_o @ S2 )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ S2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) ) )
=> ( ( ( groups8505340233167759370_o_int @ F @ S2 )
= B2 )
=> ( ( member_o @ I3 @ S2 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_544_sum__nonneg__leq__bound,axiom,
! [S2: set_int,F: int > int,B2: int,I3: int] :
( ( finite_finite_int @ S2 )
=> ( ! [I4: int] :
( ( member_int @ I4 @ S2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) ) )
=> ( ( ( groups4538972089207619220nt_int @ F @ S2 )
= B2 )
=> ( ( member_int @ I3 @ S2 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_545_sum__nonneg__leq__bound,axiom,
! [S2: set_complex,F: complex > int,B2: int,I3: complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ S2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) ) )
=> ( ( ( groups5690904116761175830ex_int @ F @ S2 )
= B2 )
=> ( ( member_complex @ I3 @ S2 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_546_sum__nonneg__leq__bound,axiom,
! [S2: set_o,F: $o > nat,B2: nat,I3: $o] :
( ( finite_finite_o @ S2 )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups8507830703676809646_o_nat @ F @ S2 )
= B2 )
=> ( ( member_o @ I3 @ S2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_547_sum__nonneg__leq__bound,axiom,
! [S2: set_int,F: int > nat,B2: nat,I3: int] :
( ( finite_finite_int @ S2 )
=> ( ! [I4: int] :
( ( member_int @ I4 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F @ S2 )
= B2 )
=> ( ( member_int @ I3 @ S2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_548_sum__nonneg__leq__bound,axiom,
! [S2: set_complex,F: complex > nat,B2: nat,I3: complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups5693394587270226106ex_nat @ F @ S2 )
= B2 )
=> ( ( member_complex @ I3 @ S2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_549_sum__nonneg__0,axiom,
! [S2: set_o,F: $o > extend8495563244428889912nnreal,I3: $o] :
( ( finite_finite_o @ S2 )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ S2 )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ I4 ) ) )
=> ( ( ( groups7456689898616286486nnreal @ F @ S2 )
= zero_z7100319975126383169nnreal )
=> ( ( member_o @ I3 @ S2 )
=> ( ( F @ I3 )
= zero_z7100319975126383169nnreal ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_550_sum__nonneg__0,axiom,
! [S2: set_int,F: int > extend8495563244428889912nnreal,I3: int] :
( ( finite_finite_int @ S2 )
=> ( ! [I4: int] :
( ( member_int @ I4 @ S2 )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ I4 ) ) )
=> ( ( ( groups2558975329500312480nnreal @ F @ S2 )
= zero_z7100319975126383169nnreal )
=> ( ( member_int @ I3 @ S2 )
=> ( ( F @ I3 )
= zero_z7100319975126383169nnreal ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_551_sum__nonneg__0,axiom,
! [S2: set_nat,F: nat > extend8495563244428889912nnreal,I3: nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ S2 )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ I4 ) ) )
=> ( ( ( groups4868793261593263428nnreal @ F @ S2 )
= zero_z7100319975126383169nnreal )
=> ( ( member_nat @ I3 @ S2 )
=> ( ( F @ I3 )
= zero_z7100319975126383169nnreal ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_552_sum__nonneg__0,axiom,
! [S2: set_complex,F: complex > extend8495563244428889912nnreal,I3: complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ S2 )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ I4 ) ) )
=> ( ( ( groups6103019165529820194nnreal @ F @ S2 )
= zero_z7100319975126383169nnreal )
=> ( ( member_complex @ I3 @ S2 )
=> ( ( F @ I3 )
= zero_z7100319975126383169nnreal ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_553_sum__nonneg__0,axiom,
! [S2: set_o,F: $o > int,I3: $o] :
( ( finite_finite_o @ S2 )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ S2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) ) )
=> ( ( ( groups8505340233167759370_o_int @ F @ S2 )
= zero_zero_int )
=> ( ( member_o @ I3 @ S2 )
=> ( ( F @ I3 )
= zero_zero_int ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_554_sum__nonneg__0,axiom,
! [S2: set_int,F: int > int,I3: int] :
( ( finite_finite_int @ S2 )
=> ( ! [I4: int] :
( ( member_int @ I4 @ S2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) ) )
=> ( ( ( groups4538972089207619220nt_int @ F @ S2 )
= zero_zero_int )
=> ( ( member_int @ I3 @ S2 )
=> ( ( F @ I3 )
= zero_zero_int ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_555_sum__nonneg__0,axiom,
! [S2: set_complex,F: complex > int,I3: complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ S2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) ) )
=> ( ( ( groups5690904116761175830ex_int @ F @ S2 )
= zero_zero_int )
=> ( ( member_complex @ I3 @ S2 )
=> ( ( F @ I3 )
= zero_zero_int ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_556_sum__nonneg__0,axiom,
! [S2: set_o,F: $o > nat,I3: $o] :
( ( finite_finite_o @ S2 )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups8507830703676809646_o_nat @ F @ S2 )
= zero_zero_nat )
=> ( ( member_o @ I3 @ S2 )
=> ( ( F @ I3 )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_557_sum__nonneg__0,axiom,
! [S2: set_int,F: int > nat,I3: int] :
( ( finite_finite_int @ S2 )
=> ( ! [I4: int] :
( ( member_int @ I4 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F @ S2 )
= zero_zero_nat )
=> ( ( member_int @ I3 @ S2 )
=> ( ( F @ I3 )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_558_sum__nonneg__0,axiom,
! [S2: set_complex,F: complex > nat,I3: complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups5693394587270226106ex_nat @ F @ S2 )
= zero_zero_nat )
=> ( ( member_complex @ I3 @ S2 )
=> ( ( F @ I3 )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_559_sum_Ogroup,axiom,
! [S: set_nat,T: set_o,G: nat > $o,H: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_o @ T )
=> ( ( ord_less_eq_set_o @ ( image_nat_o @ G @ S ) @ T )
=> ( ( groups8691415230153176458o_real
@ ^ [Y2: $o] :
( groups6591440286371151544t_real @ H
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) )
@ T )
= ( groups6591440286371151544t_real @ H @ S ) ) ) ) ) ).
% sum.group
thf(fact_560_sum_Ogroup,axiom,
! [S: set_nat,T: set_int,G: nat > int,H: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ G @ S ) @ T )
=> ( ( groups8778361861064173332t_real
@ ^ [Y2: int] :
( groups6591440286371151544t_real @ H
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) )
@ T )
= ( groups6591440286371151544t_real @ H @ S ) ) ) ) ) ).
% sum.group
thf(fact_561_sum_Ogroup,axiom,
! [S: set_nat,T: set_complex,G: nat > complex,H: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ( ord_le211207098394363844omplex @ ( image_nat_complex @ G @ S ) @ T )
=> ( ( groups5808333547571424918x_real
@ ^ [Y2: complex] :
( groups6591440286371151544t_real @ H
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) )
@ T )
= ( groups6591440286371151544t_real @ H @ S ) ) ) ) ) ).
% sum.group
thf(fact_562_sum_Ogroup,axiom,
! [S: set_nat,T: set_o,G: nat > $o,H: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_o @ T )
=> ( ( ord_less_eq_set_o @ ( image_nat_o @ G @ S ) @ T )
=> ( ( groups8507830703676809646_o_nat
@ ^ [Y2: $o] :
( groups3542108847815614940at_nat @ H
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) )
@ T )
= ( groups3542108847815614940at_nat @ H @ S ) ) ) ) ) ).
% sum.group
thf(fact_563_sum_Ogroup,axiom,
! [S: set_nat,T: set_int,G: nat > int,H: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ G @ S ) @ T )
=> ( ( groups4541462559716669496nt_nat
@ ^ [Y2: int] :
( groups3542108847815614940at_nat @ H
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) )
@ T )
= ( groups3542108847815614940at_nat @ H @ S ) ) ) ) ) ).
% sum.group
thf(fact_564_sum_Ogroup,axiom,
! [S: set_nat,T: set_complex,G: nat > complex,H: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ( ord_le211207098394363844omplex @ ( image_nat_complex @ G @ S ) @ T )
=> ( ( groups5693394587270226106ex_nat
@ ^ [Y2: complex] :
( groups3542108847815614940at_nat @ H
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) )
@ T )
= ( groups3542108847815614940at_nat @ H @ S ) ) ) ) ) ).
% sum.group
thf(fact_565_sum_Ogroup,axiom,
! [S: set_nat,T: set_o,G: nat > $o,H: nat > int] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_o @ T )
=> ( ( ord_less_eq_set_o @ ( image_nat_o @ G @ S ) @ T )
=> ( ( groups8505340233167759370_o_int
@ ^ [Y2: $o] :
( groups3539618377306564664at_int @ H
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) )
@ T )
= ( groups3539618377306564664at_int @ H @ S ) ) ) ) ) ).
% sum.group
thf(fact_566_sum_Ogroup,axiom,
! [S: set_nat,T: set_int,G: nat > int,H: nat > int] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_int @ T )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ G @ S ) @ T )
=> ( ( groups4538972089207619220nt_int
@ ^ [Y2: int] :
( groups3539618377306564664at_int @ H
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) )
@ T )
= ( groups3539618377306564664at_int @ H @ S ) ) ) ) ) ).
% sum.group
thf(fact_567_sum_Ogroup,axiom,
! [S: set_nat,T: set_complex,G: nat > complex,H: nat > int] :
( ( finite_finite_nat @ S )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ( ord_le211207098394363844omplex @ ( image_nat_complex @ G @ S ) @ T )
=> ( ( groups5690904116761175830ex_int
@ ^ [Y2: complex] :
( groups3539618377306564664at_int @ H
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) )
@ T )
= ( groups3539618377306564664at_int @ H @ S ) ) ) ) ) ).
% sum.group
thf(fact_568_sum_Ogroup,axiom,
! [S: set_o,T: set_nat,G: $o > nat,H: $o > real] :
( ( finite_finite_o @ S )
=> ( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ ( image_o_nat @ G @ S ) @ T )
=> ( ( groups6591440286371151544t_real
@ ^ [Y2: nat] :
( groups8691415230153176458o_real @ H
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) )
@ T )
= ( groups8691415230153176458o_real @ H @ S ) ) ) ) ) ).
% sum.group
thf(fact_569_le__numeral__extra_I3_J,axiom,
ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ zero_z7100319975126383169nnreal ).
% le_numeral_extra(3)
thf(fact_570_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_571_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_572_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_573_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_574_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_575_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_576_power__not__zero,axiom,
! [A2: extend8495563244428889912nnreal,N: nat] :
( ( A2 != zero_z7100319975126383169nnreal )
=> ( ( power_6007165696250533058nnreal @ A2 @ N )
!= zero_z7100319975126383169nnreal ) ) ).
% power_not_zero
thf(fact_577_power__not__zero,axiom,
! [A2: int,N: nat] :
( ( A2 != zero_zero_int )
=> ( ( power_power_int @ A2 @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_578_power__not__zero,axiom,
! [A2: nat,N: nat] :
( ( A2 != zero_zero_nat )
=> ( ( power_power_nat @ A2 @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_579_power__not__zero,axiom,
! [A2: real,N: nat] :
( ( A2 != zero_zero_real )
=> ( ( power_power_real @ A2 @ N )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_580_power__not__zero,axiom,
! [A2: complex,N: nat] :
( ( A2 != zero_zero_complex )
=> ( ( power_power_complex @ A2 @ N )
!= zero_zero_complex ) ) ).
% power_not_zero
thf(fact_581_is__num__normalize_I8_J,axiom,
! [A2: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% is_num_normalize(8)
thf(fact_582_is__num__normalize_I8_J,axiom,
! [A2: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A2 @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A2 ) ) ) ).
% is_num_normalize(8)
thf(fact_583_is__num__normalize_I8_J,axiom,
! [A2: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A2 @ B ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A2 ) ) ) ).
% is_num_normalize(8)
thf(fact_584_of__nat__mono,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I3 ) @ ( semiri1314217659103216013at_int @ J2 ) ) ) ).
% of_nat_mono
thf(fact_585_of__nat__mono,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ ( semiri1316708129612266289at_nat @ J2 ) ) ) ).
% of_nat_mono
thf(fact_586_of__nat__mono,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I3 ) @ ( semiri5074537144036343181t_real @ J2 ) ) ) ).
% of_nat_mono
thf(fact_587_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_588_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_589_one__neq__neg__one,axiom,
( one_one_complex
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% one_neq_neg_one
thf(fact_590_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: $o > int,A: set_o] :
( ( ( groups8505340233167759370_o_int @ G @ A )
!= zero_zero_int )
=> ~ ! [A4: $o] :
( ( member_o @ A4 @ A )
=> ( ( G @ A4 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_591_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: $o > nat,A: set_o] :
( ( ( groups8507830703676809646_o_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A4: $o] :
( ( member_o @ A4 @ A )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_592_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: $o > real,A: set_o] :
( ( ( groups8691415230153176458o_real @ G @ A )
!= zero_zero_real )
=> ~ ! [A4: $o] :
( ( member_o @ A4 @ A )
=> ( ( G @ A4 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_593_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: $o > extend8495563244428889912nnreal,A: set_o] :
( ( ( groups7456689898616286486nnreal @ G @ A )
!= zero_z7100319975126383169nnreal )
=> ~ ! [A4: $o] :
( ( member_o @ A4 @ A )
=> ( ( G @ A4 )
= zero_z7100319975126383169nnreal ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_594_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > extend8495563244428889912nnreal,A: set_nat] :
( ( ( groups4868793261593263428nnreal @ G @ A )
!= zero_z7100319975126383169nnreal )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ( G @ A4 )
= zero_z7100319975126383169nnreal ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_595_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > real,A: set_nat] :
( ( ( groups6591440286371151544t_real @ G @ A )
!= zero_zero_real )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ( G @ A4 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_596_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > nat,A: set_nat] :
( ( ( groups3542108847815614940at_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_597_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > int,A: set_nat] :
( ( ( groups3539618377306564664at_int @ G @ A )
!= zero_zero_int )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ( G @ A4 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_598_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: set_list_nat > int,A: set_set_list_nat] :
( ( ( groups7312845317294741502at_int @ G @ A )
!= zero_zero_int )
=> ~ ! [A4: set_list_nat] :
( ( member_set_list_nat @ A4 @ A )
=> ( ( G @ A4 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_599_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: set_set_nat > int,A: set_set_set_nat] :
( ( ( groups7084729577923612836at_int @ G @ A )
!= zero_zero_int )
=> ~ ! [A4: set_set_nat] :
( ( member_set_set_nat @ A4 @ A )
=> ( ( G @ A4 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_600_sum_Oneutral,axiom,
! [A: set_set_set_list_nat,G: set_set_list_nat > int] :
( ! [X2: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_int ) )
=> ( ( groups7004213669654646580at_int @ G @ A )
= zero_zero_int ) ) ).
% sum.neutral
thf(fact_601_sum_Oneutral,axiom,
! [A: set_set_list_nat,G: set_list_nat > int] :
( ! [X2: set_list_nat] :
( ( member_set_list_nat @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_int ) )
=> ( ( groups7312845317294741502at_int @ G @ A )
= zero_zero_int ) ) ).
% sum.neutral
thf(fact_602_sum_Oneutral,axiom,
! [A: set_set_set_nat,G: set_set_nat > int] :
( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_int ) )
=> ( ( groups7084729577923612836at_int @ G @ A )
= zero_zero_int ) ) ).
% sum.neutral
thf(fact_603_sum_Oneutral,axiom,
! [A: set_set_set_int,G: set_set_int > int] :
( ! [X2: set_set_int] :
( ( member_set_set_int @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_int ) )
=> ( ( groups1080061135233207040nt_int @ G @ A )
= zero_zero_int ) ) ).
% sum.neutral
thf(fact_604_sum_Oneutral,axiom,
! [A: set_nat,G: nat > real] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_real ) )
=> ( ( groups6591440286371151544t_real @ G @ A )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_605_sum_Oneutral,axiom,
! [A: set_nat,G: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_606_sum_Oneutral,axiom,
! [A: set_nat,G: nat > int] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_int ) )
=> ( ( groups3539618377306564664at_int @ G @ A )
= zero_zero_int ) ) ).
% sum.neutral
thf(fact_607_power__commuting__commutes,axiom,
! [X5: int,Y: int,N: nat] :
( ( ( times_times_int @ X5 @ Y )
= ( times_times_int @ Y @ X5 ) )
=> ( ( times_times_int @ ( power_power_int @ X5 @ N ) @ Y )
= ( times_times_int @ Y @ ( power_power_int @ X5 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_608_power__commuting__commutes,axiom,
! [X5: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X5 @ Y )
= ( times_times_nat @ Y @ X5 ) )
=> ( ( times_times_nat @ ( power_power_nat @ X5 @ N ) @ Y )
= ( times_times_nat @ Y @ ( power_power_nat @ X5 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_609_power__commuting__commutes,axiom,
! [X5: real,Y: real,N: nat] :
( ( ( times_times_real @ X5 @ Y )
= ( times_times_real @ Y @ X5 ) )
=> ( ( times_times_real @ ( power_power_real @ X5 @ N ) @ Y )
= ( times_times_real @ Y @ ( power_power_real @ X5 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_610_power__commuting__commutes,axiom,
! [X5: complex,Y: complex,N: nat] :
( ( ( times_times_complex @ X5 @ Y )
= ( times_times_complex @ Y @ X5 ) )
=> ( ( times_times_complex @ ( power_power_complex @ X5 @ N ) @ Y )
= ( times_times_complex @ Y @ ( power_power_complex @ X5 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_611_power__mult__distrib,axiom,
! [A2: int,B: int,N: nat] :
( ( power_power_int @ ( times_times_int @ A2 @ B ) @ N )
= ( times_times_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_612_power__mult__distrib,axiom,
! [A2: nat,B: nat,N: nat] :
( ( power_power_nat @ ( times_times_nat @ A2 @ B ) @ N )
= ( times_times_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_613_power__mult__distrib,axiom,
! [A2: real,B: real,N: nat] :
( ( power_power_real @ ( times_times_real @ A2 @ B ) @ N )
= ( times_times_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_614_power__mult__distrib,axiom,
! [A2: complex,B: complex,N: nat] :
( ( power_power_complex @ ( times_times_complex @ A2 @ B ) @ N )
= ( times_times_complex @ ( power_power_complex @ A2 @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_615_power__commutes,axiom,
! [A2: int,N: nat] :
( ( times_times_int @ ( power_power_int @ A2 @ N ) @ A2 )
= ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ).
% power_commutes
thf(fact_616_power__commutes,axiom,
! [A2: nat,N: nat] :
( ( times_times_nat @ ( power_power_nat @ A2 @ N ) @ A2 )
= ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ).
% power_commutes
thf(fact_617_power__commutes,axiom,
! [A2: real,N: nat] :
( ( times_times_real @ ( power_power_real @ A2 @ N ) @ A2 )
= ( times_times_real @ A2 @ ( power_power_real @ A2 @ N ) ) ) ).
% power_commutes
thf(fact_618_power__commutes,axiom,
! [A2: complex,N: nat] :
( ( times_times_complex @ ( power_power_complex @ A2 @ N ) @ A2 )
= ( times_times_complex @ A2 @ ( power_power_complex @ A2 @ N ) ) ) ).
% power_commutes
thf(fact_619_wellorder__Inf__le1,axiom,
! [K2: nat,A: set_nat] :
( ( member_nat @ K2 @ A )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A ) @ K2 ) ) ).
% wellorder_Inf_le1
thf(fact_620_cInf__eq,axiom,
! [X7: set_int,A2: int] :
( ! [X2: int] :
( ( member_int @ X2 @ X7 )
=> ( ord_less_eq_int @ A2 @ X2 ) )
=> ( ! [Y3: int] :
( ! [X6: int] :
( ( member_int @ X6 @ X7 )
=> ( ord_less_eq_int @ Y3 @ X6 ) )
=> ( ord_less_eq_int @ Y3 @ A2 ) )
=> ( ( complete_Inf_Inf_int @ X7 )
= A2 ) ) ) ).
% cInf_eq
thf(fact_621_cInf__eq,axiom,
! [X7: set_nat,A2: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ X7 )
=> ( ord_less_eq_nat @ A2 @ X2 ) )
=> ( ! [Y3: nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ X7 )
=> ( ord_less_eq_nat @ Y3 @ X6 ) )
=> ( ord_less_eq_nat @ Y3 @ A2 ) )
=> ( ( complete_Inf_Inf_nat @ X7 )
= A2 ) ) ) ).
% cInf_eq
thf(fact_622_cInf__eq,axiom,
! [X7: set_real,A2: real] :
( ! [X2: real] :
( ( member_real @ X2 @ X7 )
=> ( ord_less_eq_real @ A2 @ X2 ) )
=> ( ! [Y3: real] :
( ! [X6: real] :
( ( member_real @ X6 @ X7 )
=> ( ord_less_eq_real @ Y3 @ X6 ) )
=> ( ord_less_eq_real @ Y3 @ A2 ) )
=> ( ( comple4887499456419720421f_real @ X7 )
= A2 ) ) ) ).
% cInf_eq
thf(fact_623_cInf__eq__minimum,axiom,
! [Z: $o,X7: set_o] :
( ( member_o @ Z @ X7 )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ X7 )
=> ( ord_less_eq_o @ Z @ X2 ) )
=> ( ( complete_Inf_Inf_o @ X7 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_624_cInf__eq__minimum,axiom,
! [Z: int,X7: set_int] :
( ( member_int @ Z @ X7 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ X7 )
=> ( ord_less_eq_int @ Z @ X2 ) )
=> ( ( complete_Inf_Inf_int @ X7 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_625_cInf__eq__minimum,axiom,
! [Z: nat,X7: set_nat] :
( ( member_nat @ Z @ X7 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X7 )
=> ( ord_less_eq_nat @ Z @ X2 ) )
=> ( ( complete_Inf_Inf_nat @ X7 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_626_cInf__eq__minimum,axiom,
! [Z: real,X7: set_real] :
( ( member_real @ Z @ X7 )
=> ( ! [X2: real] :
( ( member_real @ X2 @ X7 )
=> ( ord_less_eq_real @ Z @ X2 ) )
=> ( ( comple4887499456419720421f_real @ X7 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_627_cInf__eq__minimum,axiom,
! [Z: set_complex,X7: set_set_complex] :
( ( member_set_complex @ Z @ X7 )
=> ( ! [X2: set_complex] :
( ( member_set_complex @ X2 @ X7 )
=> ( ord_le211207098394363844omplex @ Z @ X2 ) )
=> ( ( comple2956690151646016541omplex @ X7 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_628_cInf__eq__minimum,axiom,
! [Z: set_nat,X7: set_set_nat] :
( ( member_set_nat @ Z @ X7 )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ X7 )
=> ( ord_less_eq_set_nat @ Z @ X2 ) )
=> ( ( comple7806235888213564991et_nat @ X7 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_629_cInf__eq__minimum,axiom,
! [Z: set_int,X7: set_set_int] :
( ( member_set_int @ Z @ X7 )
=> ( ! [X2: set_int] :
( ( member_set_int @ X2 @ X7 )
=> ( ord_less_eq_set_int @ Z @ X2 ) )
=> ( ( comple3628384868704368283et_int @ X7 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_630_cInf__eq__minimum,axiom,
! [Z: set_set_complex,X7: set_set_set_complex] :
( ( member9015044028964487601omplex @ Z @ X7 )
=> ( ! [X2: set_set_complex] :
( ( member9015044028964487601omplex @ X2 @ X7 )
=> ( ord_le4750530260501030778omplex @ Z @ X2 ) )
=> ( ( comple6723625652910419923omplex @ X7 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_631_cInf__eq__minimum,axiom,
! [Z: set_set_nat,X7: set_set_set_nat] :
( ( member_set_set_nat @ Z @ X7 )
=> ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ X7 )
=> ( ord_le6893508408891458716et_nat @ Z @ X2 ) )
=> ( ( comple1065008630642458357et_nat @ X7 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_632_cInf__eq__minimum,axiom,
! [Z: set_set_int,X7: set_set_set_int] :
( ( member_set_set_int @ Z @ X7 )
=> ( ! [X2: set_set_int] :
( ( member_set_set_int @ X2 @ X7 )
=> ( ord_le4403425263959731960et_int @ Z @ X2 ) )
=> ( ( comple7798297522565507409et_int @ X7 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_633_Inf__greatest,axiom,
! [A: set_set_set_complex,Z: set_set_complex] :
( ! [X2: set_set_complex] :
( ( member9015044028964487601omplex @ X2 @ A )
=> ( ord_le4750530260501030778omplex @ Z @ X2 ) )
=> ( ord_le4750530260501030778omplex @ Z @ ( comple6723625652910419923omplex @ A ) ) ) ).
% Inf_greatest
thf(fact_634_Inf__greatest,axiom,
! [A: set_set_set_nat,Z: set_set_nat] :
( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A )
=> ( ord_le6893508408891458716et_nat @ Z @ X2 ) )
=> ( ord_le6893508408891458716et_nat @ Z @ ( comple1065008630642458357et_nat @ A ) ) ) ).
% Inf_greatest
thf(fact_635_Inf__greatest,axiom,
! [A: set_set_set_int,Z: set_set_int] :
( ! [X2: set_set_int] :
( ( member_set_set_int @ X2 @ A )
=> ( ord_le4403425263959731960et_int @ Z @ X2 ) )
=> ( ord_le4403425263959731960et_int @ Z @ ( comple7798297522565507409et_int @ A ) ) ) ).
% Inf_greatest
thf(fact_636_Inf__greatest,axiom,
! [A: set_se5258582372428582328st_nat,Z: set_set_set_list_nat] :
( ! [X2: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X2 @ A )
=> ( ord_le7100322305783427298st_nat @ Z @ X2 ) )
=> ( ord_le7100322305783427298st_nat @ Z @ ( comple5189992959352112827st_nat @ A ) ) ) ).
% Inf_greatest
thf(fact_637_Inf__greatest,axiom,
! [A: set_set_list_nat,Z: set_list_nat] :
( ! [X2: set_list_nat] :
( ( member_set_list_nat @ X2 @ A )
=> ( ord_le6045566169113846134st_nat @ Z @ X2 ) )
=> ( ord_le6045566169113846134st_nat @ Z @ ( comple184543376406953807st_nat @ A ) ) ) ).
% Inf_greatest
thf(fact_638_Inf__greatest,axiom,
! [A: set_o,Z: $o] :
( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_o @ Z @ X2 ) )
=> ( ord_less_eq_o @ Z @ ( complete_Inf_Inf_o @ A ) ) ) ).
% Inf_greatest
thf(fact_639_Inf__greatest,axiom,
! [A: set_set_complex,Z: set_complex] :
( ! [X2: set_complex] :
( ( member_set_complex @ X2 @ A )
=> ( ord_le211207098394363844omplex @ Z @ X2 ) )
=> ( ord_le211207098394363844omplex @ Z @ ( comple2956690151646016541omplex @ A ) ) ) ).
% Inf_greatest
thf(fact_640_Inf__greatest,axiom,
! [A: set_set_nat,Z: set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ Z @ X2 ) )
=> ( ord_less_eq_set_nat @ Z @ ( comple7806235888213564991et_nat @ A ) ) ) ).
% Inf_greatest
thf(fact_641_Inf__greatest,axiom,
! [A: set_set_int,Z: set_int] :
( ! [X2: set_int] :
( ( member_set_int @ X2 @ A )
=> ( ord_less_eq_set_int @ Z @ X2 ) )
=> ( ord_less_eq_set_int @ Z @ ( comple3628384868704368283et_int @ A ) ) ) ).
% Inf_greatest
thf(fact_642_Inf__greatest,axiom,
! [A: set_set_set_list_nat,Z: set_set_list_nat] :
( ! [X2: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X2 @ A )
=> ( ord_le1068707526560357548st_nat @ Z @ X2 ) )
=> ( ord_le1068707526560357548st_nat @ Z @ ( comple8462666950445340293st_nat @ A ) ) ) ).
% Inf_greatest
thf(fact_643_le__Inf__iff,axiom,
! [B: set_set_complex,A: set_set_set_complex] :
( ( ord_le4750530260501030778omplex @ B @ ( comple6723625652910419923omplex @ A ) )
= ( ! [X3: set_set_complex] :
( ( member9015044028964487601omplex @ X3 @ A )
=> ( ord_le4750530260501030778omplex @ B @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_644_le__Inf__iff,axiom,
! [B: set_set_nat,A: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ ( comple1065008630642458357et_nat @ A ) )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
=> ( ord_le6893508408891458716et_nat @ B @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_645_le__Inf__iff,axiom,
! [B: set_set_int,A: set_set_set_int] :
( ( ord_le4403425263959731960et_int @ B @ ( comple7798297522565507409et_int @ A ) )
= ( ! [X3: set_set_int] :
( ( member_set_set_int @ X3 @ A )
=> ( ord_le4403425263959731960et_int @ B @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_646_le__Inf__iff,axiom,
! [B: set_set_set_list_nat,A: set_se5258582372428582328st_nat] :
( ( ord_le7100322305783427298st_nat @ B @ ( comple5189992959352112827st_nat @ A ) )
= ( ! [X3: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X3 @ A )
=> ( ord_le7100322305783427298st_nat @ B @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_647_le__Inf__iff,axiom,
! [B: set_list_nat,A: set_set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B @ ( comple184543376406953807st_nat @ A ) )
= ( ! [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A )
=> ( ord_le6045566169113846134st_nat @ B @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_648_le__Inf__iff,axiom,
! [B: $o,A: set_o] :
( ( ord_less_eq_o @ B @ ( complete_Inf_Inf_o @ A ) )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_o @ B @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_649_le__Inf__iff,axiom,
! [B: set_complex,A: set_set_complex] :
( ( ord_le211207098394363844omplex @ B @ ( comple2956690151646016541omplex @ A ) )
= ( ! [X3: set_complex] :
( ( member_set_complex @ X3 @ A )
=> ( ord_le211207098394363844omplex @ B @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_650_le__Inf__iff,axiom,
! [B: set_nat,A: set_set_nat] :
( ( ord_less_eq_set_nat @ B @ ( comple7806235888213564991et_nat @ A ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ B @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_651_le__Inf__iff,axiom,
! [B: set_int,A: set_set_int] :
( ( ord_less_eq_set_int @ B @ ( comple3628384868704368283et_int @ A ) )
= ( ! [X3: set_int] :
( ( member_set_int @ X3 @ A )
=> ( ord_less_eq_set_int @ B @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_652_le__Inf__iff,axiom,
! [B: set_set_list_nat,A: set_set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ B @ ( comple8462666950445340293st_nat @ A ) )
= ( ! [X3: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X3 @ A )
=> ( ord_le1068707526560357548st_nat @ B @ X3 ) ) ) ) ).
% le_Inf_iff
thf(fact_653_Inf__lower2,axiom,
! [U: set_set_complex,A: set_set_set_complex,V: set_set_complex] :
( ( member9015044028964487601omplex @ U @ A )
=> ( ( ord_le4750530260501030778omplex @ U @ V )
=> ( ord_le4750530260501030778omplex @ ( comple6723625652910419923omplex @ A ) @ V ) ) ) ).
% Inf_lower2
thf(fact_654_Inf__lower2,axiom,
! [U: set_set_nat,A: set_set_set_nat,V: set_set_nat] :
( ( member_set_set_nat @ U @ A )
=> ( ( ord_le6893508408891458716et_nat @ U @ V )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A ) @ V ) ) ) ).
% Inf_lower2
thf(fact_655_Inf__lower2,axiom,
! [U: set_set_int,A: set_set_set_int,V: set_set_int] :
( ( member_set_set_int @ U @ A )
=> ( ( ord_le4403425263959731960et_int @ U @ V )
=> ( ord_le4403425263959731960et_int @ ( comple7798297522565507409et_int @ A ) @ V ) ) ) ).
% Inf_lower2
thf(fact_656_Inf__lower2,axiom,
! [U: set_set_set_list_nat,A: set_se5258582372428582328st_nat,V: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ U @ A )
=> ( ( ord_le7100322305783427298st_nat @ U @ V )
=> ( ord_le7100322305783427298st_nat @ ( comple5189992959352112827st_nat @ A ) @ V ) ) ) ).
% Inf_lower2
thf(fact_657_Inf__lower2,axiom,
! [U: set_list_nat,A: set_set_list_nat,V: set_list_nat] :
( ( member_set_list_nat @ U @ A )
=> ( ( ord_le6045566169113846134st_nat @ U @ V )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ A ) @ V ) ) ) ).
% Inf_lower2
thf(fact_658_Inf__lower2,axiom,
! [U: $o,A: set_o,V: $o] :
( ( member_o @ U @ A )
=> ( ( ord_less_eq_o @ U @ V )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ V ) ) ) ).
% Inf_lower2
thf(fact_659_Inf__lower2,axiom,
! [U: set_complex,A: set_set_complex,V: set_complex] :
( ( member_set_complex @ U @ A )
=> ( ( ord_le211207098394363844omplex @ U @ V )
=> ( ord_le211207098394363844omplex @ ( comple2956690151646016541omplex @ A ) @ V ) ) ) ).
% Inf_lower2
thf(fact_660_Inf__lower2,axiom,
! [U: set_nat,A: set_set_nat,V: set_nat] :
( ( member_set_nat @ U @ A )
=> ( ( ord_less_eq_set_nat @ U @ V )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ V ) ) ) ).
% Inf_lower2
thf(fact_661_Inf__lower2,axiom,
! [U: set_int,A: set_set_int,V: set_int] :
( ( member_set_int @ U @ A )
=> ( ( ord_less_eq_set_int @ U @ V )
=> ( ord_less_eq_set_int @ ( comple3628384868704368283et_int @ A ) @ V ) ) ) ).
% Inf_lower2
thf(fact_662_Inf__lower2,axiom,
! [U: set_set_list_nat,A: set_set_set_list_nat,V: set_set_list_nat] :
( ( member1029098694177496419st_nat @ U @ A )
=> ( ( ord_le1068707526560357548st_nat @ U @ V )
=> ( ord_le1068707526560357548st_nat @ ( comple8462666950445340293st_nat @ A ) @ V ) ) ) ).
% Inf_lower2
thf(fact_663_Inf__lower,axiom,
! [X5: set_set_complex,A: set_set_set_complex] :
( ( member9015044028964487601omplex @ X5 @ A )
=> ( ord_le4750530260501030778omplex @ ( comple6723625652910419923omplex @ A ) @ X5 ) ) ).
% Inf_lower
thf(fact_664_Inf__lower,axiom,
! [X5: set_set_nat,A: set_set_set_nat] :
( ( member_set_set_nat @ X5 @ A )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A ) @ X5 ) ) ).
% Inf_lower
thf(fact_665_Inf__lower,axiom,
! [X5: set_set_int,A: set_set_set_int] :
( ( member_set_set_int @ X5 @ A )
=> ( ord_le4403425263959731960et_int @ ( comple7798297522565507409et_int @ A ) @ X5 ) ) ).
% Inf_lower
thf(fact_666_Inf__lower,axiom,
! [X5: set_set_set_list_nat,A: set_se5258582372428582328st_nat] :
( ( member7304678173793621401st_nat @ X5 @ A )
=> ( ord_le7100322305783427298st_nat @ ( comple5189992959352112827st_nat @ A ) @ X5 ) ) ).
% Inf_lower
thf(fact_667_Inf__lower,axiom,
! [X5: set_list_nat,A: set_set_list_nat] :
( ( member_set_list_nat @ X5 @ A )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ A ) @ X5 ) ) ).
% Inf_lower
thf(fact_668_Inf__lower,axiom,
! [X5: $o,A: set_o] :
( ( member_o @ X5 @ A )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ X5 ) ) ).
% Inf_lower
thf(fact_669_Inf__lower,axiom,
! [X5: set_complex,A: set_set_complex] :
( ( member_set_complex @ X5 @ A )
=> ( ord_le211207098394363844omplex @ ( comple2956690151646016541omplex @ A ) @ X5 ) ) ).
% Inf_lower
thf(fact_670_Inf__lower,axiom,
! [X5: set_nat,A: set_set_nat] :
( ( member_set_nat @ X5 @ A )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ X5 ) ) ).
% Inf_lower
thf(fact_671_Inf__lower,axiom,
! [X5: set_int,A: set_set_int] :
( ( member_set_int @ X5 @ A )
=> ( ord_less_eq_set_int @ ( comple3628384868704368283et_int @ A ) @ X5 ) ) ).
% Inf_lower
thf(fact_672_Inf__lower,axiom,
! [X5: set_set_list_nat,A: set_set_set_list_nat] :
( ( member1029098694177496419st_nat @ X5 @ A )
=> ( ord_le1068707526560357548st_nat @ ( comple8462666950445340293st_nat @ A ) @ X5 ) ) ).
% Inf_lower
thf(fact_673_Inf__mono,axiom,
! [B2: set_set_set_complex,A: set_set_set_complex] :
( ! [B3: set_set_complex] :
( ( member9015044028964487601omplex @ B3 @ B2 )
=> ? [X6: set_set_complex] :
( ( member9015044028964487601omplex @ X6 @ A )
& ( ord_le4750530260501030778omplex @ X6 @ B3 ) ) )
=> ( ord_le4750530260501030778omplex @ ( comple6723625652910419923omplex @ A ) @ ( comple6723625652910419923omplex @ B2 ) ) ) ).
% Inf_mono
thf(fact_674_Inf__mono,axiom,
! [B2: set_set_set_nat,A: set_set_set_nat] :
( ! [B3: set_set_nat] :
( ( member_set_set_nat @ B3 @ B2 )
=> ? [X6: set_set_nat] :
( ( member_set_set_nat @ X6 @ A )
& ( ord_le6893508408891458716et_nat @ X6 @ B3 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A ) @ ( comple1065008630642458357et_nat @ B2 ) ) ) ).
% Inf_mono
thf(fact_675_Inf__mono,axiom,
! [B2: set_set_set_int,A: set_set_set_int] :
( ! [B3: set_set_int] :
( ( member_set_set_int @ B3 @ B2 )
=> ? [X6: set_set_int] :
( ( member_set_set_int @ X6 @ A )
& ( ord_le4403425263959731960et_int @ X6 @ B3 ) ) )
=> ( ord_le4403425263959731960et_int @ ( comple7798297522565507409et_int @ A ) @ ( comple7798297522565507409et_int @ B2 ) ) ) ).
% Inf_mono
thf(fact_676_Inf__mono,axiom,
! [B2: set_se5258582372428582328st_nat,A: set_se5258582372428582328st_nat] :
( ! [B3: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ B3 @ B2 )
=> ? [X6: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X6 @ A )
& ( ord_le7100322305783427298st_nat @ X6 @ B3 ) ) )
=> ( ord_le7100322305783427298st_nat @ ( comple5189992959352112827st_nat @ A ) @ ( comple5189992959352112827st_nat @ B2 ) ) ) ).
% Inf_mono
thf(fact_677_Inf__mono,axiom,
! [B2: set_set_list_nat,A: set_set_list_nat] :
( ! [B3: set_list_nat] :
( ( member_set_list_nat @ B3 @ B2 )
=> ? [X6: set_list_nat] :
( ( member_set_list_nat @ X6 @ A )
& ( ord_le6045566169113846134st_nat @ X6 @ B3 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ A ) @ ( comple184543376406953807st_nat @ B2 ) ) ) ).
% Inf_mono
thf(fact_678_Inf__mono,axiom,
! [B2: set_o,A: set_o] :
( ! [B3: $o] :
( ( member_o @ B3 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A )
& ( ord_less_eq_o @ X6 @ B3 ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Inf_Inf_o @ B2 ) ) ) ).
% Inf_mono
thf(fact_679_Inf__mono,axiom,
! [B2: set_set_complex,A: set_set_complex] :
( ! [B3: set_complex] :
( ( member_set_complex @ B3 @ B2 )
=> ? [X6: set_complex] :
( ( member_set_complex @ X6 @ A )
& ( ord_le211207098394363844omplex @ X6 @ B3 ) ) )
=> ( ord_le211207098394363844omplex @ ( comple2956690151646016541omplex @ A ) @ ( comple2956690151646016541omplex @ B2 ) ) ) ).
% Inf_mono
thf(fact_680_Inf__mono,axiom,
! [B2: set_set_nat,A: set_set_nat] :
( ! [B3: set_nat] :
( ( member_set_nat @ B3 @ B2 )
=> ? [X6: set_nat] :
( ( member_set_nat @ X6 @ A )
& ( ord_less_eq_set_nat @ X6 @ B3 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ ( comple7806235888213564991et_nat @ B2 ) ) ) ).
% Inf_mono
thf(fact_681_Inf__mono,axiom,
! [B2: set_set_int,A: set_set_int] :
( ! [B3: set_int] :
( ( member_set_int @ B3 @ B2 )
=> ? [X6: set_int] :
( ( member_set_int @ X6 @ A )
& ( ord_less_eq_set_int @ X6 @ B3 ) ) )
=> ( ord_less_eq_set_int @ ( comple3628384868704368283et_int @ A ) @ ( comple3628384868704368283et_int @ B2 ) ) ) ).
% Inf_mono
thf(fact_682_Inf__mono,axiom,
! [B2: set_set_set_list_nat,A: set_set_set_list_nat] :
( ! [B3: set_set_list_nat] :
( ( member1029098694177496419st_nat @ B3 @ B2 )
=> ? [X6: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X6 @ A )
& ( ord_le1068707526560357548st_nat @ X6 @ B3 ) ) )
=> ( ord_le1068707526560357548st_nat @ ( comple8462666950445340293st_nat @ A ) @ ( comple8462666950445340293st_nat @ B2 ) ) ) ).
% Inf_mono
thf(fact_683_Inf__eqI,axiom,
! [A: set_set_set_complex,X5: set_set_complex] :
( ! [I4: set_set_complex] :
( ( member9015044028964487601omplex @ I4 @ A )
=> ( ord_le4750530260501030778omplex @ X5 @ I4 ) )
=> ( ! [Y3: set_set_complex] :
( ! [I6: set_set_complex] :
( ( member9015044028964487601omplex @ I6 @ A )
=> ( ord_le4750530260501030778omplex @ Y3 @ I6 ) )
=> ( ord_le4750530260501030778omplex @ Y3 @ X5 ) )
=> ( ( comple6723625652910419923omplex @ A )
= X5 ) ) ) ).
% Inf_eqI
thf(fact_684_Inf__eqI,axiom,
! [A: set_set_set_nat,X5: set_set_nat] :
( ! [I4: set_set_nat] :
( ( member_set_set_nat @ I4 @ A )
=> ( ord_le6893508408891458716et_nat @ X5 @ I4 ) )
=> ( ! [Y3: set_set_nat] :
( ! [I6: set_set_nat] :
( ( member_set_set_nat @ I6 @ A )
=> ( ord_le6893508408891458716et_nat @ Y3 @ I6 ) )
=> ( ord_le6893508408891458716et_nat @ Y3 @ X5 ) )
=> ( ( comple1065008630642458357et_nat @ A )
= X5 ) ) ) ).
% Inf_eqI
thf(fact_685_Inf__eqI,axiom,
! [A: set_set_set_int,X5: set_set_int] :
( ! [I4: set_set_int] :
( ( member_set_set_int @ I4 @ A )
=> ( ord_le4403425263959731960et_int @ X5 @ I4 ) )
=> ( ! [Y3: set_set_int] :
( ! [I6: set_set_int] :
( ( member_set_set_int @ I6 @ A )
=> ( ord_le4403425263959731960et_int @ Y3 @ I6 ) )
=> ( ord_le4403425263959731960et_int @ Y3 @ X5 ) )
=> ( ( comple7798297522565507409et_int @ A )
= X5 ) ) ) ).
% Inf_eqI
thf(fact_686_Inf__eqI,axiom,
! [A: set_se5258582372428582328st_nat,X5: set_set_set_list_nat] :
( ! [I4: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ I4 @ A )
=> ( ord_le7100322305783427298st_nat @ X5 @ I4 ) )
=> ( ! [Y3: set_set_set_list_nat] :
( ! [I6: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ I6 @ A )
=> ( ord_le7100322305783427298st_nat @ Y3 @ I6 ) )
=> ( ord_le7100322305783427298st_nat @ Y3 @ X5 ) )
=> ( ( comple5189992959352112827st_nat @ A )
= X5 ) ) ) ).
% Inf_eqI
thf(fact_687_Inf__eqI,axiom,
! [A: set_set_list_nat,X5: set_list_nat] :
( ! [I4: set_list_nat] :
( ( member_set_list_nat @ I4 @ A )
=> ( ord_le6045566169113846134st_nat @ X5 @ I4 ) )
=> ( ! [Y3: set_list_nat] :
( ! [I6: set_list_nat] :
( ( member_set_list_nat @ I6 @ A )
=> ( ord_le6045566169113846134st_nat @ Y3 @ I6 ) )
=> ( ord_le6045566169113846134st_nat @ Y3 @ X5 ) )
=> ( ( comple184543376406953807st_nat @ A )
= X5 ) ) ) ).
% Inf_eqI
thf(fact_688_Inf__eqI,axiom,
! [A: set_o,X5: $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_o @ X5 @ I4 ) )
=> ( ! [Y3: $o] :
( ! [I6: $o] :
( ( member_o @ I6 @ A )
=> ( ord_less_eq_o @ Y3 @ I6 ) )
=> ( ord_less_eq_o @ Y3 @ X5 ) )
=> ( ( complete_Inf_Inf_o @ A )
= X5 ) ) ) ).
% Inf_eqI
thf(fact_689_Inf__eqI,axiom,
! [A: set_set_complex,X5: set_complex] :
( ! [I4: set_complex] :
( ( member_set_complex @ I4 @ A )
=> ( ord_le211207098394363844omplex @ X5 @ I4 ) )
=> ( ! [Y3: set_complex] :
( ! [I6: set_complex] :
( ( member_set_complex @ I6 @ A )
=> ( ord_le211207098394363844omplex @ Y3 @ I6 ) )
=> ( ord_le211207098394363844omplex @ Y3 @ X5 ) )
=> ( ( comple2956690151646016541omplex @ A )
= X5 ) ) ) ).
% Inf_eqI
thf(fact_690_Inf__eqI,axiom,
! [A: set_set_nat,X5: set_nat] :
( ! [I4: set_nat] :
( ( member_set_nat @ I4 @ A )
=> ( ord_less_eq_set_nat @ X5 @ I4 ) )
=> ( ! [Y3: set_nat] :
( ! [I6: set_nat] :
( ( member_set_nat @ I6 @ A )
=> ( ord_less_eq_set_nat @ Y3 @ I6 ) )
=> ( ord_less_eq_set_nat @ Y3 @ X5 ) )
=> ( ( comple7806235888213564991et_nat @ A )
= X5 ) ) ) ).
% Inf_eqI
thf(fact_691_Inf__eqI,axiom,
! [A: set_set_int,X5: set_int] :
( ! [I4: set_int] :
( ( member_set_int @ I4 @ A )
=> ( ord_less_eq_set_int @ X5 @ I4 ) )
=> ( ! [Y3: set_int] :
( ! [I6: set_int] :
( ( member_set_int @ I6 @ A )
=> ( ord_less_eq_set_int @ Y3 @ I6 ) )
=> ( ord_less_eq_set_int @ Y3 @ X5 ) )
=> ( ( comple3628384868704368283et_int @ A )
= X5 ) ) ) ).
% Inf_eqI
thf(fact_692_Inf__eqI,axiom,
! [A: set_set_set_list_nat,X5: set_set_list_nat] :
( ! [I4: set_set_list_nat] :
( ( member1029098694177496419st_nat @ I4 @ A )
=> ( ord_le1068707526560357548st_nat @ X5 @ I4 ) )
=> ( ! [Y3: set_set_list_nat] :
( ! [I6: set_set_list_nat] :
( ( member1029098694177496419st_nat @ I6 @ A )
=> ( ord_le1068707526560357548st_nat @ Y3 @ I6 ) )
=> ( ord_le1068707526560357548st_nat @ Y3 @ X5 ) )
=> ( ( comple8462666950445340293st_nat @ A )
= X5 ) ) ) ).
% Inf_eqI
thf(fact_693_mult__of__nat__commute,axiom,
! [X5: nat,Y: complex] :
( ( times_times_complex @ ( semiri8010041392384452111omplex @ X5 ) @ Y )
= ( times_times_complex @ Y @ ( semiri8010041392384452111omplex @ X5 ) ) ) ).
% mult_of_nat_commute
thf(fact_694_mult__of__nat__commute,axiom,
! [X5: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X5 ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X5 ) ) ) ).
% mult_of_nat_commute
thf(fact_695_mult__of__nat__commute,axiom,
! [X5: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X5 ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X5 ) ) ) ).
% mult_of_nat_commute
thf(fact_696_mult__of__nat__commute,axiom,
! [X5: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X5 ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X5 ) ) ) ).
% mult_of_nat_commute
thf(fact_697_INF__cong,axiom,
! [A: set_o,B2: set_o,C2: $o > $o,D: $o > $o] :
( ( A = B2 )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ C2 @ A ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_698_INF__cong,axiom,
! [A: set_nat,B2: set_nat,C2: nat > $o,D: nat > $o] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ C2 @ A ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_699_INF__cong,axiom,
! [A: set_int,B2: set_int,C2: int > int,D: int > int] :
( ( A = B2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( complete_Inf_Inf_int @ ( image_int_int @ C2 @ A ) )
= ( complete_Inf_Inf_int @ ( image_int_int @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_700_INF__cong,axiom,
! [A: set_o,B2: set_o,C2: $o > int,D: $o > int] :
( ( A = B2 )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( complete_Inf_Inf_int @ ( image_o_int @ C2 @ A ) )
= ( complete_Inf_Inf_int @ ( image_o_int @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_701_INF__cong,axiom,
! [A: set_nat,B2: set_nat,C2: nat > int,D: nat > int] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( complete_Inf_Inf_int @ ( image_nat_int @ C2 @ A ) )
= ( complete_Inf_Inf_int @ ( image_nat_int @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_702_INF__cong,axiom,
! [A: set_o,B2: set_o,C2: $o > nat,D: $o > nat] :
( ( A = B2 )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_o_nat @ C2 @ A ) )
= ( complete_Inf_Inf_nat @ ( image_o_nat @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_703_INF__cong,axiom,
! [A: set_nat,B2: set_nat,C2: nat > nat,D: nat > nat] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_nat_nat @ C2 @ A ) )
= ( complete_Inf_Inf_nat @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_704_INF__cong,axiom,
! [A: set_real,B2: set_real,C2: real > real,D: real > real] :
( ( A = B2 )
=> ( ! [X2: real] :
( ( member_real @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( comple4887499456419720421f_real @ ( image_real_real @ C2 @ A ) )
= ( comple4887499456419720421f_real @ ( image_real_real @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_705_INF__cong,axiom,
! [A: set_o,B2: set_o,C2: $o > real,D: $o > real] :
( ( A = B2 )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( comple4887499456419720421f_real @ ( image_o_real @ C2 @ A ) )
= ( comple4887499456419720421f_real @ ( image_o_real @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_706_INF__cong,axiom,
! [A: set_nat,B2: set_nat,C2: nat > real,D: nat > real] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( comple4887499456419720421f_real @ ( image_nat_real @ C2 @ A ) )
= ( comple4887499456419720421f_real @ ( image_nat_real @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_707_Inter__greatest,axiom,
! [A: set_set_set_complex,C2: set_set_complex] :
( ! [X4: set_set_complex] :
( ( member9015044028964487601omplex @ X4 @ A )
=> ( ord_le4750530260501030778omplex @ C2 @ X4 ) )
=> ( ord_le4750530260501030778omplex @ C2 @ ( comple6723625652910419923omplex @ A ) ) ) ).
% Inter_greatest
thf(fact_708_Inter__greatest,axiom,
! [A: set_set_set_nat,C2: set_set_nat] :
( ! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A )
=> ( ord_le6893508408891458716et_nat @ C2 @ X4 ) )
=> ( ord_le6893508408891458716et_nat @ C2 @ ( comple1065008630642458357et_nat @ A ) ) ) ).
% Inter_greatest
thf(fact_709_Inter__greatest,axiom,
! [A: set_set_set_int,C2: set_set_int] :
( ! [X4: set_set_int] :
( ( member_set_set_int @ X4 @ A )
=> ( ord_le4403425263959731960et_int @ C2 @ X4 ) )
=> ( ord_le4403425263959731960et_int @ C2 @ ( comple7798297522565507409et_int @ A ) ) ) ).
% Inter_greatest
thf(fact_710_Inter__greatest,axiom,
! [A: set_se5258582372428582328st_nat,C2: set_set_set_list_nat] :
( ! [X4: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X4 @ A )
=> ( ord_le7100322305783427298st_nat @ C2 @ X4 ) )
=> ( ord_le7100322305783427298st_nat @ C2 @ ( comple5189992959352112827st_nat @ A ) ) ) ).
% Inter_greatest
thf(fact_711_Inter__greatest,axiom,
! [A: set_set_list_nat,C2: set_list_nat] :
( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ A )
=> ( ord_le6045566169113846134st_nat @ C2 @ X4 ) )
=> ( ord_le6045566169113846134st_nat @ C2 @ ( comple184543376406953807st_nat @ A ) ) ) ).
% Inter_greatest
thf(fact_712_Inter__greatest,axiom,
! [A: set_set_complex,C2: set_complex] :
( ! [X4: set_complex] :
( ( member_set_complex @ X4 @ A )
=> ( ord_le211207098394363844omplex @ C2 @ X4 ) )
=> ( ord_le211207098394363844omplex @ C2 @ ( comple2956690151646016541omplex @ A ) ) ) ).
% Inter_greatest
thf(fact_713_Inter__greatest,axiom,
! [A: set_set_nat,C2: set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ C2 @ X4 ) )
=> ( ord_less_eq_set_nat @ C2 @ ( comple7806235888213564991et_nat @ A ) ) ) ).
% Inter_greatest
thf(fact_714_Inter__greatest,axiom,
! [A: set_set_int,C2: set_int] :
( ! [X4: set_int] :
( ( member_set_int @ X4 @ A )
=> ( ord_less_eq_set_int @ C2 @ X4 ) )
=> ( ord_less_eq_set_int @ C2 @ ( comple3628384868704368283et_int @ A ) ) ) ).
% Inter_greatest
thf(fact_715_Inter__greatest,axiom,
! [A: set_set_set_list_nat,C2: set_set_list_nat] :
( ! [X4: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X4 @ A )
=> ( ord_le1068707526560357548st_nat @ C2 @ X4 ) )
=> ( ord_le1068707526560357548st_nat @ C2 @ ( comple8462666950445340293st_nat @ A ) ) ) ).
% Inter_greatest
thf(fact_716_Inter__lower,axiom,
! [B2: set_set_complex,A: set_set_set_complex] :
( ( member9015044028964487601omplex @ B2 @ A )
=> ( ord_le4750530260501030778omplex @ ( comple6723625652910419923omplex @ A ) @ B2 ) ) ).
% Inter_lower
thf(fact_717_Inter__lower,axiom,
! [B2: set_set_nat,A: set_set_set_nat] :
( ( member_set_set_nat @ B2 @ A )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A ) @ B2 ) ) ).
% Inter_lower
thf(fact_718_Inter__lower,axiom,
! [B2: set_set_int,A: set_set_set_int] :
( ( member_set_set_int @ B2 @ A )
=> ( ord_le4403425263959731960et_int @ ( comple7798297522565507409et_int @ A ) @ B2 ) ) ).
% Inter_lower
thf(fact_719_Inter__lower,axiom,
! [B2: set_set_set_list_nat,A: set_se5258582372428582328st_nat] :
( ( member7304678173793621401st_nat @ B2 @ A )
=> ( ord_le7100322305783427298st_nat @ ( comple5189992959352112827st_nat @ A ) @ B2 ) ) ).
% Inter_lower
thf(fact_720_Inter__lower,axiom,
! [B2: set_list_nat,A: set_set_list_nat] :
( ( member_set_list_nat @ B2 @ A )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ A ) @ B2 ) ) ).
% Inter_lower
thf(fact_721_Inter__lower,axiom,
! [B2: set_complex,A: set_set_complex] :
( ( member_set_complex @ B2 @ A )
=> ( ord_le211207098394363844omplex @ ( comple2956690151646016541omplex @ A ) @ B2 ) ) ).
% Inter_lower
thf(fact_722_Inter__lower,axiom,
! [B2: set_nat,A: set_set_nat] :
( ( member_set_nat @ B2 @ A )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ B2 ) ) ).
% Inter_lower
thf(fact_723_Inter__lower,axiom,
! [B2: set_int,A: set_set_int] :
( ( member_set_int @ B2 @ A )
=> ( ord_less_eq_set_int @ ( comple3628384868704368283et_int @ A ) @ B2 ) ) ).
% Inter_lower
thf(fact_724_Inter__lower,axiom,
! [B2: set_set_list_nat,A: set_set_set_list_nat] :
( ( member1029098694177496419st_nat @ B2 @ A )
=> ( ord_le1068707526560357548st_nat @ ( comple8462666950445340293st_nat @ A ) @ B2 ) ) ).
% Inter_lower
thf(fact_725_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_726_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_727_times__int__code_I1_J,axiom,
! [K2: int] :
( ( times_times_int @ K2 @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_728_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_729_int__cases2,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_730_sum__mono,axiom,
! [K3: set_o,F: $o > int,G: $o > int] :
( ! [I4: $o] :
( ( member_o @ I4 @ K3 )
=> ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_int @ ( groups8505340233167759370_o_int @ F @ K3 ) @ ( groups8505340233167759370_o_int @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_731_sum__mono,axiom,
! [K3: set_o,F: $o > nat,G: $o > nat] :
( ! [I4: $o] :
( ( member_o @ I4 @ K3 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ K3 ) @ ( groups8507830703676809646_o_nat @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_732_sum__mono,axiom,
! [K3: set_o,F: $o > real,G: $o > real] :
( ! [I4: $o] :
( ( member_o @ I4 @ K3 )
=> ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_real @ ( groups8691415230153176458o_real @ F @ K3 ) @ ( groups8691415230153176458o_real @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_733_sum__mono,axiom,
! [K3: set_set_set_list_nat,F: set_set_list_nat > int,G: set_set_list_nat > int] :
( ! [I4: set_set_list_nat] :
( ( member1029098694177496419st_nat @ I4 @ K3 )
=> ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_int @ ( groups7004213669654646580at_int @ F @ K3 ) @ ( groups7004213669654646580at_int @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_734_sum__mono,axiom,
! [K3: set_set_list_nat,F: set_list_nat > int,G: set_list_nat > int] :
( ! [I4: set_list_nat] :
( ( member_set_list_nat @ I4 @ K3 )
=> ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_int @ ( groups7312845317294741502at_int @ F @ K3 ) @ ( groups7312845317294741502at_int @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_735_sum__mono,axiom,
! [K3: set_set_set_nat,F: set_set_nat > int,G: set_set_nat > int] :
( ! [I4: set_set_nat] :
( ( member_set_set_nat @ I4 @ K3 )
=> ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_int @ ( groups7084729577923612836at_int @ F @ K3 ) @ ( groups7084729577923612836at_int @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_736_sum__mono,axiom,
! [K3: set_set_set_int,F: set_set_int > int,G: set_set_int > int] :
( ! [I4: set_set_int] :
( ( member_set_set_int @ I4 @ K3 )
=> ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_int @ ( groups1080061135233207040nt_int @ F @ K3 ) @ ( groups1080061135233207040nt_int @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_737_sum__mono,axiom,
! [K3: set_nat,F: nat > real,G: nat > real] :
( ! [I4: nat] :
( ( member_nat @ I4 @ K3 )
=> ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ K3 ) @ ( groups6591440286371151544t_real @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_738_sum__mono,axiom,
! [K3: set_nat,F: nat > nat,G: nat > nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ K3 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K3 ) @ ( groups3542108847815614940at_nat @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_739_sum__mono,axiom,
! [K3: set_nat,F: nat > int,G: nat > int] :
( ! [I4: nat] :
( ( member_nat @ I4 @ K3 )
=> ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K3 ) @ ( groups3539618377306564664at_int @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_740_sum__product,axiom,
! [F: nat > real,A: set_nat,G: nat > real,B2: set_nat] :
( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A ) @ ( groups6591440286371151544t_real @ G @ B2 ) )
= ( groups6591440286371151544t_real
@ ^ [I2: nat] :
( groups6591440286371151544t_real
@ ^ [J: nat] : ( times_times_real @ ( F @ I2 ) @ ( G @ J ) )
@ B2 )
@ A ) ) ).
% sum_product
thf(fact_741_sum__product,axiom,
! [F: nat > nat,A: set_nat,G: nat > nat,B2: set_nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ G @ B2 ) )
= ( groups3542108847815614940at_nat
@ ^ [I2: nat] :
( groups3542108847815614940at_nat
@ ^ [J: nat] : ( times_times_nat @ ( F @ I2 ) @ ( G @ J ) )
@ B2 )
@ A ) ) ).
% sum_product
thf(fact_742_sum__product,axiom,
! [F: nat > int,A: set_nat,G: nat > int,B2: set_nat] :
( ( times_times_int @ ( groups3539618377306564664at_int @ F @ A ) @ ( groups3539618377306564664at_int @ G @ B2 ) )
= ( groups3539618377306564664at_int
@ ^ [I2: nat] :
( groups3539618377306564664at_int
@ ^ [J: nat] : ( times_times_int @ ( F @ I2 ) @ ( G @ J ) )
@ B2 )
@ A ) ) ).
% sum_product
thf(fact_743_sum__product,axiom,
! [F: set_list_nat > int,A: set_set_list_nat,G: nat > int,B2: set_nat] :
( ( times_times_int @ ( groups7312845317294741502at_int @ F @ A ) @ ( groups3539618377306564664at_int @ G @ B2 ) )
= ( groups7312845317294741502at_int
@ ^ [I2: set_list_nat] :
( groups3539618377306564664at_int
@ ^ [J: nat] : ( times_times_int @ ( F @ I2 ) @ ( G @ J ) )
@ B2 )
@ A ) ) ).
% sum_product
thf(fact_744_sum__product,axiom,
! [F: set_set_nat > int,A: set_set_set_nat,G: nat > int,B2: set_nat] :
( ( times_times_int @ ( groups7084729577923612836at_int @ F @ A ) @ ( groups3539618377306564664at_int @ G @ B2 ) )
= ( groups7084729577923612836at_int
@ ^ [I2: set_set_nat] :
( groups3539618377306564664at_int
@ ^ [J: nat] : ( times_times_int @ ( F @ I2 ) @ ( G @ J ) )
@ B2 )
@ A ) ) ).
% sum_product
thf(fact_745_sum__product,axiom,
! [F: set_set_int > int,A: set_set_set_int,G: nat > int,B2: set_nat] :
( ( times_times_int @ ( groups1080061135233207040nt_int @ F @ A ) @ ( groups3539618377306564664at_int @ G @ B2 ) )
= ( groups1080061135233207040nt_int
@ ^ [I2: set_set_int] :
( groups3539618377306564664at_int
@ ^ [J: nat] : ( times_times_int @ ( F @ I2 ) @ ( G @ J ) )
@ B2 )
@ A ) ) ).
% sum_product
thf(fact_746_sum__product,axiom,
! [F: nat > int,A: set_nat,G: set_list_nat > int,B2: set_set_list_nat] :
( ( times_times_int @ ( groups3539618377306564664at_int @ F @ A ) @ ( groups7312845317294741502at_int @ G @ B2 ) )
= ( groups3539618377306564664at_int
@ ^ [I2: nat] :
( groups7312845317294741502at_int
@ ^ [J: set_list_nat] : ( times_times_int @ ( F @ I2 ) @ ( G @ J ) )
@ B2 )
@ A ) ) ).
% sum_product
thf(fact_747_sum__product,axiom,
! [F: nat > int,A: set_nat,G: set_set_nat > int,B2: set_set_set_nat] :
( ( times_times_int @ ( groups3539618377306564664at_int @ F @ A ) @ ( groups7084729577923612836at_int @ G @ B2 ) )
= ( groups3539618377306564664at_int
@ ^ [I2: nat] :
( groups7084729577923612836at_int
@ ^ [J: set_set_nat] : ( times_times_int @ ( F @ I2 ) @ ( G @ J ) )
@ B2 )
@ A ) ) ).
% sum_product
thf(fact_748_sum__product,axiom,
! [F: nat > int,A: set_nat,G: set_set_int > int,B2: set_set_set_int] :
( ( times_times_int @ ( groups3539618377306564664at_int @ F @ A ) @ ( groups1080061135233207040nt_int @ G @ B2 ) )
= ( groups3539618377306564664at_int
@ ^ [I2: nat] :
( groups1080061135233207040nt_int
@ ^ [J: set_set_int] : ( times_times_int @ ( F @ I2 ) @ ( G @ J ) )
@ B2 )
@ A ) ) ).
% sum_product
thf(fact_749_sum__product,axiom,
! [F: set_set_list_nat > int,A: set_set_set_list_nat,G: nat > int,B2: set_nat] :
( ( times_times_int @ ( groups7004213669654646580at_int @ F @ A ) @ ( groups3539618377306564664at_int @ G @ B2 ) )
= ( groups7004213669654646580at_int
@ ^ [I2: set_set_list_nat] :
( groups3539618377306564664at_int
@ ^ [J: nat] : ( times_times_int @ ( F @ I2 ) @ ( G @ J ) )
@ B2 )
@ A ) ) ).
% sum_product
thf(fact_750_sum__distrib__right,axiom,
! [F: set_set_list_nat > int,A: set_set_set_list_nat,R2: int] :
( ( times_times_int @ ( groups7004213669654646580at_int @ F @ A ) @ R2 )
= ( groups7004213669654646580at_int
@ ^ [N2: set_set_list_nat] : ( times_times_int @ ( F @ N2 ) @ R2 )
@ A ) ) ).
% sum_distrib_right
thf(fact_751_sum__distrib__right,axiom,
! [F: set_list_nat > int,A: set_set_list_nat,R2: int] :
( ( times_times_int @ ( groups7312845317294741502at_int @ F @ A ) @ R2 )
= ( groups7312845317294741502at_int
@ ^ [N2: set_list_nat] : ( times_times_int @ ( F @ N2 ) @ R2 )
@ A ) ) ).
% sum_distrib_right
thf(fact_752_sum__distrib__right,axiom,
! [F: set_set_nat > int,A: set_set_set_nat,R2: int] :
( ( times_times_int @ ( groups7084729577923612836at_int @ F @ A ) @ R2 )
= ( groups7084729577923612836at_int
@ ^ [N2: set_set_nat] : ( times_times_int @ ( F @ N2 ) @ R2 )
@ A ) ) ).
% sum_distrib_right
thf(fact_753_sum__distrib__right,axiom,
! [F: set_set_int > int,A: set_set_set_int,R2: int] :
( ( times_times_int @ ( groups1080061135233207040nt_int @ F @ A ) @ R2 )
= ( groups1080061135233207040nt_int
@ ^ [N2: set_set_int] : ( times_times_int @ ( F @ N2 ) @ R2 )
@ A ) ) ).
% sum_distrib_right
thf(fact_754_sum__distrib__right,axiom,
! [F: nat > real,A: set_nat,R2: real] :
( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A ) @ R2 )
= ( groups6591440286371151544t_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ R2 )
@ A ) ) ).
% sum_distrib_right
thf(fact_755_sum__distrib__right,axiom,
! [F: nat > nat,A: set_nat,R2: nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ R2 )
= ( groups3542108847815614940at_nat
@ ^ [N2: nat] : ( times_times_nat @ ( F @ N2 ) @ R2 )
@ A ) ) ).
% sum_distrib_right
thf(fact_756_sum__distrib__right,axiom,
! [F: nat > int,A: set_nat,R2: int] :
( ( times_times_int @ ( groups3539618377306564664at_int @ F @ A ) @ R2 )
= ( groups3539618377306564664at_int
@ ^ [N2: nat] : ( times_times_int @ ( F @ N2 ) @ R2 )
@ A ) ) ).
% sum_distrib_right
thf(fact_757_sum__distrib__left,axiom,
! [R2: int,F: set_set_list_nat > int,A: set_set_set_list_nat] :
( ( times_times_int @ R2 @ ( groups7004213669654646580at_int @ F @ A ) )
= ( groups7004213669654646580at_int
@ ^ [N2: set_set_list_nat] : ( times_times_int @ R2 @ ( F @ N2 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_758_sum__distrib__left,axiom,
! [R2: int,F: set_list_nat > int,A: set_set_list_nat] :
( ( times_times_int @ R2 @ ( groups7312845317294741502at_int @ F @ A ) )
= ( groups7312845317294741502at_int
@ ^ [N2: set_list_nat] : ( times_times_int @ R2 @ ( F @ N2 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_759_sum__distrib__left,axiom,
! [R2: int,F: set_set_nat > int,A: set_set_set_nat] :
( ( times_times_int @ R2 @ ( groups7084729577923612836at_int @ F @ A ) )
= ( groups7084729577923612836at_int
@ ^ [N2: set_set_nat] : ( times_times_int @ R2 @ ( F @ N2 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_760_sum__distrib__left,axiom,
! [R2: int,F: set_set_int > int,A: set_set_set_int] :
( ( times_times_int @ R2 @ ( groups1080061135233207040nt_int @ F @ A ) )
= ( groups1080061135233207040nt_int
@ ^ [N2: set_set_int] : ( times_times_int @ R2 @ ( F @ N2 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_761_sum__distrib__left,axiom,
! [R2: real,F: nat > real,A: set_nat] :
( ( times_times_real @ R2 @ ( groups6591440286371151544t_real @ F @ A ) )
= ( groups6591440286371151544t_real
@ ^ [N2: nat] : ( times_times_real @ R2 @ ( F @ N2 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_762_sum__distrib__left,axiom,
! [R2: nat,F: nat > nat,A: set_nat] :
( ( times_times_nat @ R2 @ ( groups3542108847815614940at_nat @ F @ A ) )
= ( groups3542108847815614940at_nat
@ ^ [N2: nat] : ( times_times_nat @ R2 @ ( F @ N2 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_763_sum__distrib__left,axiom,
! [R2: int,F: nat > int,A: set_nat] :
( ( times_times_int @ R2 @ ( groups3539618377306564664at_int @ F @ A ) )
= ( groups3539618377306564664at_int
@ ^ [N2: nat] : ( times_times_int @ R2 @ ( F @ N2 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_764_sum_Odistrib,axiom,
! [G: set_set_list_nat > int,H: set_set_list_nat > int,A: set_set_set_list_nat] :
( ( groups7004213669654646580at_int
@ ^ [X3: set_set_list_nat] : ( plus_plus_int @ ( G @ X3 ) @ ( H @ X3 ) )
@ A )
= ( plus_plus_int @ ( groups7004213669654646580at_int @ G @ A ) @ ( groups7004213669654646580at_int @ H @ A ) ) ) ).
% sum.distrib
thf(fact_765_sum_Odistrib,axiom,
! [G: set_list_nat > int,H: set_list_nat > int,A: set_set_list_nat] :
( ( groups7312845317294741502at_int
@ ^ [X3: set_list_nat] : ( plus_plus_int @ ( G @ X3 ) @ ( H @ X3 ) )
@ A )
= ( plus_plus_int @ ( groups7312845317294741502at_int @ G @ A ) @ ( groups7312845317294741502at_int @ H @ A ) ) ) ).
% sum.distrib
thf(fact_766_sum_Odistrib,axiom,
! [G: set_set_nat > int,H: set_set_nat > int,A: set_set_set_nat] :
( ( groups7084729577923612836at_int
@ ^ [X3: set_set_nat] : ( plus_plus_int @ ( G @ X3 ) @ ( H @ X3 ) )
@ A )
= ( plus_plus_int @ ( groups7084729577923612836at_int @ G @ A ) @ ( groups7084729577923612836at_int @ H @ A ) ) ) ).
% sum.distrib
thf(fact_767_sum_Odistrib,axiom,
! [G: set_set_int > int,H: set_set_int > int,A: set_set_set_int] :
( ( groups1080061135233207040nt_int
@ ^ [X3: set_set_int] : ( plus_plus_int @ ( G @ X3 ) @ ( H @ X3 ) )
@ A )
= ( plus_plus_int @ ( groups1080061135233207040nt_int @ G @ A ) @ ( groups1080061135233207040nt_int @ H @ A ) ) ) ).
% sum.distrib
thf(fact_768_sum_Odistrib,axiom,
! [G: nat > real,H: nat > real,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X3: nat] : ( plus_plus_real @ ( G @ X3 ) @ ( H @ X3 ) )
@ A )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A ) @ ( groups6591440286371151544t_real @ H @ A ) ) ) ).
% sum.distrib
thf(fact_769_sum_Odistrib,axiom,
! [G: nat > nat,H: nat > nat,A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [X3: nat] : ( plus_plus_nat @ ( G @ X3 ) @ ( H @ X3 ) )
@ A )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A ) @ ( groups3542108847815614940at_nat @ H @ A ) ) ) ).
% sum.distrib
thf(fact_770_sum_Odistrib,axiom,
! [G: nat > int,H: nat > int,A: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [X3: nat] : ( plus_plus_int @ ( G @ X3 ) @ ( H @ X3 ) )
@ A )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ A ) @ ( groups3539618377306564664at_int @ H @ A ) ) ) ).
% sum.distrib
thf(fact_771_sum__negf,axiom,
! [F: set_set_list_nat > int,A: set_set_set_list_nat] :
( ( groups7004213669654646580at_int
@ ^ [X3: set_set_list_nat] : ( uminus_uminus_int @ ( F @ X3 ) )
@ A )
= ( uminus_uminus_int @ ( groups7004213669654646580at_int @ F @ A ) ) ) ).
% sum_negf
thf(fact_772_sum__negf,axiom,
! [F: set_list_nat > int,A: set_set_list_nat] :
( ( groups7312845317294741502at_int
@ ^ [X3: set_list_nat] : ( uminus_uminus_int @ ( F @ X3 ) )
@ A )
= ( uminus_uminus_int @ ( groups7312845317294741502at_int @ F @ A ) ) ) ).
% sum_negf
thf(fact_773_sum__negf,axiom,
! [F: set_set_nat > int,A: set_set_set_nat] :
( ( groups7084729577923612836at_int
@ ^ [X3: set_set_nat] : ( uminus_uminus_int @ ( F @ X3 ) )
@ A )
= ( uminus_uminus_int @ ( groups7084729577923612836at_int @ F @ A ) ) ) ).
% sum_negf
thf(fact_774_sum__negf,axiom,
! [F: set_set_int > int,A: set_set_set_int] :
( ( groups1080061135233207040nt_int
@ ^ [X3: set_set_int] : ( uminus_uminus_int @ ( F @ X3 ) )
@ A )
= ( uminus_uminus_int @ ( groups1080061135233207040nt_int @ F @ A ) ) ) ).
% sum_negf
thf(fact_775_sum__negf,axiom,
! [F: nat > real,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X3: nat] : ( uminus_uminus_real @ ( F @ X3 ) )
@ A )
= ( uminus_uminus_real @ ( groups6591440286371151544t_real @ F @ A ) ) ) ).
% sum_negf
thf(fact_776_sum__negf,axiom,
! [F: nat > int,A: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [X3: nat] : ( uminus_uminus_int @ ( F @ X3 ) )
@ A )
= ( uminus_uminus_int @ ( groups3539618377306564664at_int @ F @ A ) ) ) ).
% sum_negf
thf(fact_777_INF__commute,axiom,
! [F: nat > nat > $o,B2: set_nat,A: set_nat] :
( ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [I2: nat] : ( complete_Inf_Inf_o @ ( image_nat_o @ ( F @ I2 ) @ B2 ) )
@ A ) )
= ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [J: nat] :
( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [I2: nat] : ( F @ I2 @ J )
@ A ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_778_INF__commute,axiom,
! [F: nat > list_nat > $o,B2: set_list_nat,A: set_nat] :
( ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [I2: nat] : ( complete_Inf_Inf_o @ ( image_list_nat_o @ ( F @ I2 ) @ B2 ) )
@ A ) )
= ( complete_Inf_Inf_o
@ ( image_list_nat_o
@ ^ [J: list_nat] :
( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [I2: nat] : ( F @ I2 @ J )
@ A ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_779_INF__commute,axiom,
! [F: list_nat > nat > $o,B2: set_nat,A: set_list_nat] :
( ( complete_Inf_Inf_o
@ ( image_list_nat_o
@ ^ [I2: list_nat] : ( complete_Inf_Inf_o @ ( image_nat_o @ ( F @ I2 ) @ B2 ) )
@ A ) )
= ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [J: nat] :
( complete_Inf_Inf_o
@ ( image_list_nat_o
@ ^ [I2: list_nat] : ( F @ I2 @ J )
@ A ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_780_INF__commute,axiom,
! [F: nat > nat > set_list_nat,B2: set_nat,A: set_nat] :
( ( comple184543376406953807st_nat
@ ( image_2883343038133793645st_nat
@ ^ [I2: nat] : ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ ( F @ I2 ) @ B2 ) )
@ A ) )
= ( comple184543376406953807st_nat
@ ( image_2883343038133793645st_nat
@ ^ [J: nat] :
( comple184543376406953807st_nat
@ ( image_2883343038133793645st_nat
@ ^ [I2: nat] : ( F @ I2 @ J )
@ A ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_781_INF__commute,axiom,
! [F: nat > set_list_nat > $o,B2: set_set_list_nat,A: set_nat] :
( ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [I2: nat] : ( complete_Inf_Inf_o @ ( image_set_list_nat_o @ ( F @ I2 ) @ B2 ) )
@ A ) )
= ( complete_Inf_Inf_o
@ ( image_set_list_nat_o
@ ^ [J: set_list_nat] :
( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [I2: nat] : ( F @ I2 @ J )
@ A ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_782_INF__commute,axiom,
! [F: list_nat > list_nat > $o,B2: set_list_nat,A: set_list_nat] :
( ( complete_Inf_Inf_o
@ ( image_list_nat_o
@ ^ [I2: list_nat] : ( complete_Inf_Inf_o @ ( image_list_nat_o @ ( F @ I2 ) @ B2 ) )
@ A ) )
= ( complete_Inf_Inf_o
@ ( image_list_nat_o
@ ^ [J: list_nat] :
( complete_Inf_Inf_o
@ ( image_list_nat_o
@ ^ [I2: list_nat] : ( F @ I2 @ J )
@ A ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_783_INF__commute,axiom,
! [F: set_list_nat > nat > $o,B2: set_nat,A: set_set_list_nat] :
( ( complete_Inf_Inf_o
@ ( image_set_list_nat_o
@ ^ [I2: set_list_nat] : ( complete_Inf_Inf_o @ ( image_nat_o @ ( F @ I2 ) @ B2 ) )
@ A ) )
= ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [J: nat] :
( complete_Inf_Inf_o
@ ( image_set_list_nat_o
@ ^ [I2: set_list_nat] : ( F @ I2 @ J )
@ A ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_784_INF__commute,axiom,
! [F: list_nat > nat > set_list_nat,B2: set_nat,A: set_list_nat] :
( ( comple184543376406953807st_nat
@ ( image_8532145185254316925st_nat
@ ^ [I2: list_nat] : ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ ( F @ I2 ) @ B2 ) )
@ A ) )
= ( comple184543376406953807st_nat
@ ( image_2883343038133793645st_nat
@ ^ [J: nat] :
( comple184543376406953807st_nat
@ ( image_8532145185254316925st_nat
@ ^ [I2: list_nat] : ( F @ I2 @ J )
@ A ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_785_INF__commute,axiom,
! [F: nat > list_nat > set_list_nat,B2: set_list_nat,A: set_nat] :
( ( comple184543376406953807st_nat
@ ( image_2883343038133793645st_nat
@ ^ [I2: nat] : ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ ( F @ I2 ) @ B2 ) )
@ A ) )
= ( comple184543376406953807st_nat
@ ( image_8532145185254316925st_nat
@ ^ [J: list_nat] :
( comple184543376406953807st_nat
@ ( image_2883343038133793645st_nat
@ ^ [I2: nat] : ( F @ I2 @ J )
@ A ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_786_INF__commute,axiom,
! [F: list_nat > set_list_nat > $o,B2: set_set_list_nat,A: set_list_nat] :
( ( complete_Inf_Inf_o
@ ( image_list_nat_o
@ ^ [I2: list_nat] : ( complete_Inf_Inf_o @ ( image_set_list_nat_o @ ( F @ I2 ) @ B2 ) )
@ A ) )
= ( complete_Inf_Inf_o
@ ( image_set_list_nat_o
@ ^ [J: set_list_nat] :
( complete_Inf_Inf_o
@ ( image_list_nat_o
@ ^ [I2: list_nat] : ( F @ I2 @ J )
@ A ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_787_INT__E,axiom,
! [B: $o,B2: $o > set_o,A: set_o,A2: $o] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A ) ) )
=> ( ~ ( member_o @ B @ ( B2 @ A2 ) )
=> ~ ( member_o @ A2 @ A ) ) ) ).
% INT_E
thf(fact_788_INT__E,axiom,
! [B: $o,B2: nat > set_o,A: set_nat,A2: nat] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_nat_set_o @ B2 @ A ) ) )
=> ( ~ ( member_o @ B @ ( B2 @ A2 ) )
=> ~ ( member_nat @ A2 @ A ) ) ) ).
% INT_E
thf(fact_789_INT__E,axiom,
! [B: complex,B2: $o > set_complex,A: set_o,A2: $o] :
( ( member_complex @ B @ ( comple2956690151646016541omplex @ ( image_o_set_complex @ B2 @ A ) ) )
=> ( ~ ( member_complex @ B @ ( B2 @ A2 ) )
=> ~ ( member_o @ A2 @ A ) ) ) ).
% INT_E
thf(fact_790_INT__E,axiom,
! [B: complex,B2: nat > set_complex,A: set_nat,A2: nat] :
( ( member_complex @ B @ ( comple2956690151646016541omplex @ ( image_6594795319511438139omplex @ B2 @ A ) ) )
=> ( ~ ( member_complex @ B @ ( B2 @ A2 ) )
=> ~ ( member_nat @ A2 @ A ) ) ) ).
% INT_E
thf(fact_791_INT__E,axiom,
! [B: nat,B2: $o > set_nat,A: set_o,A2: $o] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B2 @ A ) ) )
=> ( ~ ( member_nat @ B @ ( B2 @ A2 ) )
=> ~ ( member_o @ A2 @ A ) ) ) ).
% INT_E
thf(fact_792_INT__E,axiom,
! [B: nat,B2: nat > set_nat,A: set_nat,A2: nat] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B2 @ A ) ) )
=> ( ~ ( member_nat @ B @ ( B2 @ A2 ) )
=> ~ ( member_nat @ A2 @ A ) ) ) ).
% INT_E
thf(fact_793_INT__E,axiom,
! [B: int,B2: $o > set_int,A: set_o,A2: $o] :
( ( member_int @ B @ ( comple3628384868704368283et_int @ ( image_o_set_int @ B2 @ A ) ) )
=> ( ~ ( member_int @ B @ ( B2 @ A2 ) )
=> ~ ( member_o @ A2 @ A ) ) ) ).
% INT_E
thf(fact_794_INT__E,axiom,
! [B: int,B2: nat > set_int,A: set_nat,A2: nat] :
( ( member_int @ B @ ( comple3628384868704368283et_int @ ( image_nat_set_int @ B2 @ A ) ) )
=> ( ~ ( member_int @ B @ ( B2 @ A2 ) )
=> ~ ( member_nat @ A2 @ A ) ) ) ).
% INT_E
thf(fact_795_INT__E,axiom,
! [B: list_nat,B2: $o > set_list_nat,A: set_o,A2: $o] :
( ( member_list_nat @ B @ ( comple184543376406953807st_nat @ ( image_o_set_list_nat @ B2 @ A ) ) )
=> ( ~ ( member_list_nat @ B @ ( B2 @ A2 ) )
=> ~ ( member_o @ A2 @ A ) ) ) ).
% INT_E
thf(fact_796_INT__E,axiom,
! [B: list_nat,B2: nat > set_list_nat,A: set_nat,A2: nat] :
( ( member_list_nat @ B @ ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ B2 @ A ) ) )
=> ( ~ ( member_list_nat @ B @ ( B2 @ A2 ) )
=> ~ ( member_nat @ A2 @ A ) ) ) ).
% INT_E
thf(fact_797_INT__D,axiom,
! [B: $o,B2: $o > set_o,A: set_o,A2: $o] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A ) ) )
=> ( ( member_o @ A2 @ A )
=> ( member_o @ B @ ( B2 @ A2 ) ) ) ) ).
% INT_D
thf(fact_798_INT__D,axiom,
! [B: $o,B2: nat > set_o,A: set_nat,A2: nat] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_nat_set_o @ B2 @ A ) ) )
=> ( ( member_nat @ A2 @ A )
=> ( member_o @ B @ ( B2 @ A2 ) ) ) ) ).
% INT_D
thf(fact_799_INT__D,axiom,
! [B: complex,B2: $o > set_complex,A: set_o,A2: $o] :
( ( member_complex @ B @ ( comple2956690151646016541omplex @ ( image_o_set_complex @ B2 @ A ) ) )
=> ( ( member_o @ A2 @ A )
=> ( member_complex @ B @ ( B2 @ A2 ) ) ) ) ).
% INT_D
thf(fact_800_INT__D,axiom,
! [B: complex,B2: nat > set_complex,A: set_nat,A2: nat] :
( ( member_complex @ B @ ( comple2956690151646016541omplex @ ( image_6594795319511438139omplex @ B2 @ A ) ) )
=> ( ( member_nat @ A2 @ A )
=> ( member_complex @ B @ ( B2 @ A2 ) ) ) ) ).
% INT_D
thf(fact_801_INT__D,axiom,
! [B: nat,B2: $o > set_nat,A: set_o,A2: $o] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B2 @ A ) ) )
=> ( ( member_o @ A2 @ A )
=> ( member_nat @ B @ ( B2 @ A2 ) ) ) ) ).
% INT_D
thf(fact_802_INT__D,axiom,
! [B: nat,B2: nat > set_nat,A: set_nat,A2: nat] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B2 @ A ) ) )
=> ( ( member_nat @ A2 @ A )
=> ( member_nat @ B @ ( B2 @ A2 ) ) ) ) ).
% INT_D
thf(fact_803_INT__D,axiom,
! [B: int,B2: $o > set_int,A: set_o,A2: $o] :
( ( member_int @ B @ ( comple3628384868704368283et_int @ ( image_o_set_int @ B2 @ A ) ) )
=> ( ( member_o @ A2 @ A )
=> ( member_int @ B @ ( B2 @ A2 ) ) ) ) ).
% INT_D
thf(fact_804_INT__D,axiom,
! [B: int,B2: nat > set_int,A: set_nat,A2: nat] :
( ( member_int @ B @ ( comple3628384868704368283et_int @ ( image_nat_set_int @ B2 @ A ) ) )
=> ( ( member_nat @ A2 @ A )
=> ( member_int @ B @ ( B2 @ A2 ) ) ) ) ).
% INT_D
thf(fact_805_INT__D,axiom,
! [B: list_nat,B2: $o > set_list_nat,A: set_o,A2: $o] :
( ( member_list_nat @ B @ ( comple184543376406953807st_nat @ ( image_o_set_list_nat @ B2 @ A ) ) )
=> ( ( member_o @ A2 @ A )
=> ( member_list_nat @ B @ ( B2 @ A2 ) ) ) ) ).
% INT_D
thf(fact_806_INT__D,axiom,
! [B: list_nat,B2: nat > set_list_nat,A: set_nat,A2: nat] :
( ( member_list_nat @ B @ ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ B2 @ A ) ) )
=> ( ( member_nat @ A2 @ A )
=> ( member_list_nat @ B @ ( B2 @ A2 ) ) ) ) ).
% INT_D
thf(fact_807_zero__le__power,axiom,
! [A2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ N ) ) ) ).
% zero_le_power
thf(fact_808_zero__le__power,axiom,
! [A2: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A2 @ N ) ) ) ).
% zero_le_power
thf(fact_809_zero__le__power,axiom,
! [A2: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ N ) ) ) ).
% zero_le_power
thf(fact_810_power__mono,axiom,
! [A2: int,B: int,N: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono
thf(fact_811_power__mono,axiom,
! [A2: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_812_power__mono,axiom,
! [A2: real,B: real,N: nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ord_less_eq_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% power_mono
thf(fact_813_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_814_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_815_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_816_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_817_sum__nonpos,axiom,
! [A: set_o,F: $o > extend8495563244428889912nnreal] :
( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ zero_z7100319975126383169nnreal ) )
=> ( ord_le3935885782089961368nnreal @ ( groups7456689898616286486nnreal @ F @ A ) @ zero_z7100319975126383169nnreal ) ) ).
% sum_nonpos
thf(fact_818_sum__nonpos,axiom,
! [A: set_nat,F: nat > extend8495563244428889912nnreal] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ zero_z7100319975126383169nnreal ) )
=> ( ord_le3935885782089961368nnreal @ ( groups4868793261593263428nnreal @ F @ A ) @ zero_z7100319975126383169nnreal ) ) ).
% sum_nonpos
thf(fact_819_sum__nonpos,axiom,
! [A: set_o,F: $o > int] :
( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_int @ ( F @ X2 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups8505340233167759370_o_int @ F @ A ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_820_sum__nonpos,axiom,
! [A: set_o,F: $o > nat] :
( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_821_sum__nonpos,axiom,
! [A: set_o,F: $o > real] :
( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_real @ ( F @ X2 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8691415230153176458o_real @ F @ A ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_822_sum__nonpos,axiom,
! [A: set_nat,F: nat > real] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_real @ ( F @ X2 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ A ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_823_sum__nonpos,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_824_sum__nonpos,axiom,
! [A: set_nat,F: nat > int] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_int @ ( F @ X2 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_825_sum__nonpos,axiom,
! [A: set_set_list_nat,F: set_list_nat > int] :
( ! [X2: set_list_nat] :
( ( member_set_list_nat @ X2 @ A )
=> ( ord_less_eq_int @ ( F @ X2 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups7312845317294741502at_int @ F @ A ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_826_sum__nonpos,axiom,
! [A: set_set_set_nat,F: set_set_nat > int] :
( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A )
=> ( ord_less_eq_int @ ( F @ X2 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups7084729577923612836at_int @ F @ A ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_827_sum__nonneg,axiom,
! [A: set_o,F: $o > extend8495563244428889912nnreal] :
( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ X2 ) ) )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( groups7456689898616286486nnreal @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_828_sum__nonneg,axiom,
! [A: set_nat,F: nat > extend8495563244428889912nnreal] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ X2 ) ) )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( groups4868793261593263428nnreal @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_829_sum__nonneg,axiom,
! [A: set_o,F: $o > int] :
( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X2 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups8505340233167759370_o_int @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_830_sum__nonneg,axiom,
! [A: set_o,F: $o > nat] :
( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups8507830703676809646_o_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_831_sum__nonneg,axiom,
! [A: set_o,F: $o > real] :
( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8691415230153176458o_real @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_832_sum__nonneg,axiom,
! [A: set_nat,F: nat > real] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups6591440286371151544t_real @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_833_sum__nonneg,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_834_sum__nonneg,axiom,
! [A: set_nat,F: nat > int] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X2 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_835_sum__nonneg,axiom,
! [A: set_set_list_nat,F: set_list_nat > int] :
( ! [X2: set_list_nat] :
( ( member_set_list_nat @ X2 @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X2 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups7312845317294741502at_int @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_836_sum__nonneg,axiom,
! [A: set_set_set_nat,F: set_set_nat > int] :
( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X2 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups7084729577923612836at_int @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_837_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( semiri6283507881447550617nnreal @ N ) ) ).
% of_nat_0_le_iff
thf(fact_838_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_839_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_840_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_841_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_842_zero__neq__neg__one,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% zero_neq_neg_one
thf(fact_843_zero__neq__neg__one,axiom,
( zero_zero_complex
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% zero_neq_neg_one
thf(fact_844_power__increasing,axiom,
! [N: nat,N4: nat,A2: int] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_int @ one_one_int @ A2 )
=> ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ A2 @ N4 ) ) ) ) ).
% power_increasing
thf(fact_845_power__increasing,axiom,
! [N: nat,N4: nat,A2: nat] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A2 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ A2 @ N4 ) ) ) ) ).
% power_increasing
thf(fact_846_power__increasing,axiom,
! [N: nat,N4: nat,A2: real] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_real @ one_one_real @ A2 )
=> ( ord_less_eq_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ A2 @ N4 ) ) ) ) ).
% power_increasing
thf(fact_847_one__le__power,axiom,
! [A2: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A2 )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A2 @ N ) ) ) ).
% one_le_power
thf(fact_848_one__le__power,axiom,
! [A2: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A2 @ N ) ) ) ).
% one_le_power
thf(fact_849_one__le__power,axiom,
! [A2: real,N: nat] :
( ( ord_less_eq_real @ one_one_real @ A2 )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A2 @ N ) ) ) ).
% one_le_power
thf(fact_850_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_6007165696250533058nnreal @ zero_z7100319975126383169nnreal @ N )
= one_on2969667320475766781nnreal ) )
& ( ( N != zero_zero_nat )
=> ( ( power_6007165696250533058nnreal @ zero_z7100319975126383169nnreal @ N )
= zero_z7100319975126383169nnreal ) ) ) ).
% power_0_left
thf(fact_851_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_852_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_853_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_854_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_855_left__right__inverse__power,axiom,
! [X5: int,Y: int,N: nat] :
( ( ( times_times_int @ X5 @ Y )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X5 @ N ) @ ( power_power_int @ Y @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_856_left__right__inverse__power,axiom,
! [X5: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X5 @ Y )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X5 @ N ) @ ( power_power_nat @ Y @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_857_left__right__inverse__power,axiom,
! [X5: real,Y: real,N: nat] :
( ( ( times_times_real @ X5 @ Y )
= one_one_real )
=> ( ( times_times_real @ ( power_power_real @ X5 @ N ) @ ( power_power_real @ Y @ N ) )
= one_one_real ) ) ).
% left_right_inverse_power
thf(fact_858_left__right__inverse__power,axiom,
! [X5: complex,Y: complex,N: nat] :
( ( ( times_times_complex @ X5 @ Y )
= one_one_complex )
=> ( ( times_times_complex @ ( power_power_complex @ X5 @ N ) @ ( power_power_complex @ Y @ N ) )
= one_one_complex ) ) ).
% left_right_inverse_power
thf(fact_859_INF__eq,axiom,
! [A: set_o,B2: set_o,G: $o > $o,F: $o > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_o @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A )
& ( ord_less_eq_o @ ( F @ X6 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ A ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_860_INF__eq,axiom,
! [A: set_o,B2: set_nat,G: nat > $o,F: $o > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B2 )
& ( ord_less_eq_o @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A )
& ( ord_less_eq_o @ ( F @ X6 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ A ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_861_INF__eq,axiom,
! [A: set_nat,B2: set_o,G: $o > $o,F: nat > $o] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_o @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_less_eq_o @ ( F @ X6 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_862_INF__eq,axiom,
! [A: set_nat,B2: set_nat,G: nat > $o,F: nat > $o] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B2 )
& ( ord_less_eq_o @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_less_eq_o @ ( F @ X6 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_863_INF__eq,axiom,
! [A: set_list_nat,B2: set_o,G: $o > $o,F: list_nat > $o] :
( ! [I4: list_nat] :
( ( member_list_nat @ I4 @ A )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_o @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B2 )
=> ? [X6: list_nat] :
( ( member_list_nat @ X6 @ A )
& ( ord_less_eq_o @ ( F @ X6 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_list_nat_o @ F @ A ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_864_INF__eq,axiom,
! [A: set_list_nat,B2: set_nat,G: nat > $o,F: list_nat > $o] :
( ! [I4: list_nat] :
( ( member_list_nat @ I4 @ A )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B2 )
& ( ord_less_eq_o @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B2 )
=> ? [X6: list_nat] :
( ( member_list_nat @ X6 @ A )
& ( ord_less_eq_o @ ( F @ X6 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_list_nat_o @ F @ A ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_865_INF__eq,axiom,
! [A: set_o,B2: set_list_nat,G: list_nat > $o,F: $o > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ? [X6: list_nat] :
( ( member_list_nat @ X6 @ B2 )
& ( ord_less_eq_o @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J3: list_nat] :
( ( member_list_nat @ J3 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A )
& ( ord_less_eq_o @ ( F @ X6 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ A ) )
= ( complete_Inf_Inf_o @ ( image_list_nat_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_866_INF__eq,axiom,
! [A: set_nat,B2: set_list_nat,G: list_nat > $o,F: nat > $o] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ? [X6: list_nat] :
( ( member_list_nat @ X6 @ B2 )
& ( ord_less_eq_o @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J3: list_nat] :
( ( member_list_nat @ J3 @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_less_eq_o @ ( F @ X6 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A ) )
= ( complete_Inf_Inf_o @ ( image_list_nat_o @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_867_INF__eq,axiom,
! [A: set_o,B2: set_o,G: $o > set_complex,F: $o > set_complex] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_le211207098394363844omplex @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A )
& ( ord_le211207098394363844omplex @ ( F @ X6 ) @ ( G @ J3 ) ) ) )
=> ( ( comple2956690151646016541omplex @ ( image_o_set_complex @ F @ A ) )
= ( comple2956690151646016541omplex @ ( image_o_set_complex @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_868_INF__eq,axiom,
! [A: set_o,B2: set_nat,G: nat > set_complex,F: $o > set_complex] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B2 )
& ( ord_le211207098394363844omplex @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A )
& ( ord_le211207098394363844omplex @ ( F @ X6 ) @ ( G @ J3 ) ) ) )
=> ( ( comple2956690151646016541omplex @ ( image_o_set_complex @ F @ A ) )
= ( comple2956690151646016541omplex @ ( image_6594795319511438139omplex @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_869_cInf__eq__non__empty,axiom,
! [X7: set_o,A2: $o] :
( ( X7 != bot_bot_set_o )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ X7 )
=> ( ord_less_eq_o @ A2 @ X2 ) )
=> ( ! [Y3: $o] :
( ! [X6: $o] :
( ( member_o @ X6 @ X7 )
=> ( ord_less_eq_o @ Y3 @ X6 ) )
=> ( ord_less_eq_o @ Y3 @ A2 ) )
=> ( ( complete_Inf_Inf_o @ X7 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_870_cInf__eq__non__empty,axiom,
! [X7: set_int,A2: int] :
( ( X7 != bot_bot_set_int )
=> ( ! [X2: int] :
( ( member_int @ X2 @ X7 )
=> ( ord_less_eq_int @ A2 @ X2 ) )
=> ( ! [Y3: int] :
( ! [X6: int] :
( ( member_int @ X6 @ X7 )
=> ( ord_less_eq_int @ Y3 @ X6 ) )
=> ( ord_less_eq_int @ Y3 @ A2 ) )
=> ( ( complete_Inf_Inf_int @ X7 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_871_cInf__eq__non__empty,axiom,
! [X7: set_nat,A2: nat] :
( ( X7 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X7 )
=> ( ord_less_eq_nat @ A2 @ X2 ) )
=> ( ! [Y3: nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ X7 )
=> ( ord_less_eq_nat @ Y3 @ X6 ) )
=> ( ord_less_eq_nat @ Y3 @ A2 ) )
=> ( ( complete_Inf_Inf_nat @ X7 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_872_cInf__eq__non__empty,axiom,
! [X7: set_real,A2: real] :
( ( X7 != bot_bot_set_real )
=> ( ! [X2: real] :
( ( member_real @ X2 @ X7 )
=> ( ord_less_eq_real @ A2 @ X2 ) )
=> ( ! [Y3: real] :
( ! [X6: real] :
( ( member_real @ X6 @ X7 )
=> ( ord_less_eq_real @ Y3 @ X6 ) )
=> ( ord_less_eq_real @ Y3 @ A2 ) )
=> ( ( comple4887499456419720421f_real @ X7 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_873_cInf__eq__non__empty,axiom,
! [X7: set_set_complex,A2: set_complex] :
( ( X7 != bot_bo4474773400535771566omplex )
=> ( ! [X2: set_complex] :
( ( member_set_complex @ X2 @ X7 )
=> ( ord_le211207098394363844omplex @ A2 @ X2 ) )
=> ( ! [Y3: set_complex] :
( ! [X6: set_complex] :
( ( member_set_complex @ X6 @ X7 )
=> ( ord_le211207098394363844omplex @ Y3 @ X6 ) )
=> ( ord_le211207098394363844omplex @ Y3 @ A2 ) )
=> ( ( comple2956690151646016541omplex @ X7 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_874_cInf__eq__non__empty,axiom,
! [X7: set_set_nat,A2: set_nat] :
( ( X7 != bot_bot_set_set_nat )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ X7 )
=> ( ord_less_eq_set_nat @ A2 @ X2 ) )
=> ( ! [Y3: set_nat] :
( ! [X6: set_nat] :
( ( member_set_nat @ X6 @ X7 )
=> ( ord_less_eq_set_nat @ Y3 @ X6 ) )
=> ( ord_less_eq_set_nat @ Y3 @ A2 ) )
=> ( ( comple7806235888213564991et_nat @ X7 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_875_cInf__eq__non__empty,axiom,
! [X7: set_set_int,A2: set_int] :
( ( X7 != bot_bot_set_set_int )
=> ( ! [X2: set_int] :
( ( member_set_int @ X2 @ X7 )
=> ( ord_less_eq_set_int @ A2 @ X2 ) )
=> ( ! [Y3: set_int] :
( ! [X6: set_int] :
( ( member_set_int @ X6 @ X7 )
=> ( ord_less_eq_set_int @ Y3 @ X6 ) )
=> ( ord_less_eq_set_int @ Y3 @ A2 ) )
=> ( ( comple3628384868704368283et_int @ X7 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_876_cInf__eq__non__empty,axiom,
! [X7: set_set_set_complex,A2: set_set_complex] :
( ( X7 != bot_bo92361985942245988omplex )
=> ( ! [X2: set_set_complex] :
( ( member9015044028964487601omplex @ X2 @ X7 )
=> ( ord_le4750530260501030778omplex @ A2 @ X2 ) )
=> ( ! [Y3: set_set_complex] :
( ! [X6: set_set_complex] :
( ( member9015044028964487601omplex @ X6 @ X7 )
=> ( ord_le4750530260501030778omplex @ Y3 @ X6 ) )
=> ( ord_le4750530260501030778omplex @ Y3 @ A2 ) )
=> ( ( comple6723625652910419923omplex @ X7 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_877_cInf__eq__non__empty,axiom,
! [X7: set_set_set_nat,A2: set_set_nat] :
( ( X7 != bot_bo7198184520161983622et_nat )
=> ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ X7 )
=> ( ord_le6893508408891458716et_nat @ A2 @ X2 ) )
=> ( ! [Y3: set_set_nat] :
( ! [X6: set_set_nat] :
( ( member_set_set_nat @ X6 @ X7 )
=> ( ord_le6893508408891458716et_nat @ Y3 @ X6 ) )
=> ( ord_le6893508408891458716et_nat @ Y3 @ A2 ) )
=> ( ( comple1065008630642458357et_nat @ X7 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_878_cInf__eq__non__empty,axiom,
! [X7: set_set_set_int,A2: set_set_int] :
( ( X7 != bot_bo2384636101374064866et_int )
=> ( ! [X2: set_set_int] :
( ( member_set_set_int @ X2 @ X7 )
=> ( ord_le4403425263959731960et_int @ A2 @ X2 ) )
=> ( ! [Y3: set_set_int] :
( ! [X6: set_set_int] :
( ( member_set_set_int @ X6 @ X7 )
=> ( ord_le4403425263959731960et_int @ Y3 @ X6 ) )
=> ( ord_le4403425263959731960et_int @ Y3 @ A2 ) )
=> ( ( comple7798297522565507409et_int @ X7 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_879_cInf__greatest,axiom,
! [X7: set_o,Z: $o] :
( ( X7 != bot_bot_set_o )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ X7 )
=> ( ord_less_eq_o @ Z @ X2 ) )
=> ( ord_less_eq_o @ Z @ ( complete_Inf_Inf_o @ X7 ) ) ) ) ).
% cInf_greatest
thf(fact_880_cInf__greatest,axiom,
! [X7: set_int,Z: int] :
( ( X7 != bot_bot_set_int )
=> ( ! [X2: int] :
( ( member_int @ X2 @ X7 )
=> ( ord_less_eq_int @ Z @ X2 ) )
=> ( ord_less_eq_int @ Z @ ( complete_Inf_Inf_int @ X7 ) ) ) ) ).
% cInf_greatest
thf(fact_881_cInf__greatest,axiom,
! [X7: set_nat,Z: nat] :
( ( X7 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X7 )
=> ( ord_less_eq_nat @ Z @ X2 ) )
=> ( ord_less_eq_nat @ Z @ ( complete_Inf_Inf_nat @ X7 ) ) ) ) ).
% cInf_greatest
thf(fact_882_cInf__greatest,axiom,
! [X7: set_real,Z: real] :
( ( X7 != bot_bot_set_real )
=> ( ! [X2: real] :
( ( member_real @ X2 @ X7 )
=> ( ord_less_eq_real @ Z @ X2 ) )
=> ( ord_less_eq_real @ Z @ ( comple4887499456419720421f_real @ X7 ) ) ) ) ).
% cInf_greatest
thf(fact_883_cInf__greatest,axiom,
! [X7: set_set_complex,Z: set_complex] :
( ( X7 != bot_bo4474773400535771566omplex )
=> ( ! [X2: set_complex] :
( ( member_set_complex @ X2 @ X7 )
=> ( ord_le211207098394363844omplex @ Z @ X2 ) )
=> ( ord_le211207098394363844omplex @ Z @ ( comple2956690151646016541omplex @ X7 ) ) ) ) ).
% cInf_greatest
thf(fact_884_cInf__greatest,axiom,
! [X7: set_set_nat,Z: set_nat] :
( ( X7 != bot_bot_set_set_nat )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ X7 )
=> ( ord_less_eq_set_nat @ Z @ X2 ) )
=> ( ord_less_eq_set_nat @ Z @ ( comple7806235888213564991et_nat @ X7 ) ) ) ) ).
% cInf_greatest
thf(fact_885_cInf__greatest,axiom,
! [X7: set_set_int,Z: set_int] :
( ( X7 != bot_bot_set_set_int )
=> ( ! [X2: set_int] :
( ( member_set_int @ X2 @ X7 )
=> ( ord_less_eq_set_int @ Z @ X2 ) )
=> ( ord_less_eq_set_int @ Z @ ( comple3628384868704368283et_int @ X7 ) ) ) ) ).
% cInf_greatest
thf(fact_886_cInf__greatest,axiom,
! [X7: set_set_set_complex,Z: set_set_complex] :
( ( X7 != bot_bo92361985942245988omplex )
=> ( ! [X2: set_set_complex] :
( ( member9015044028964487601omplex @ X2 @ X7 )
=> ( ord_le4750530260501030778omplex @ Z @ X2 ) )
=> ( ord_le4750530260501030778omplex @ Z @ ( comple6723625652910419923omplex @ X7 ) ) ) ) ).
% cInf_greatest
thf(fact_887_cInf__greatest,axiom,
! [X7: set_set_set_nat,Z: set_set_nat] :
( ( X7 != bot_bo7198184520161983622et_nat )
=> ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ X7 )
=> ( ord_le6893508408891458716et_nat @ Z @ X2 ) )
=> ( ord_le6893508408891458716et_nat @ Z @ ( comple1065008630642458357et_nat @ X7 ) ) ) ) ).
% cInf_greatest
thf(fact_888_cInf__greatest,axiom,
! [X7: set_set_set_int,Z: set_set_int] :
( ( X7 != bot_bo2384636101374064866et_int )
=> ( ! [X2: set_set_int] :
( ( member_set_set_int @ X2 @ X7 )
=> ( ord_le4403425263959731960et_int @ Z @ X2 ) )
=> ( ord_le4403425263959731960et_int @ Z @ ( comple7798297522565507409et_int @ X7 ) ) ) ) ).
% cInf_greatest
thf(fact_889_Inf__less__eq,axiom,
! [A: set_set_set_complex,U: set_set_complex] :
( ! [V2: set_set_complex] :
( ( member9015044028964487601omplex @ V2 @ A )
=> ( ord_le4750530260501030778omplex @ V2 @ U ) )
=> ( ( A != bot_bo92361985942245988omplex )
=> ( ord_le4750530260501030778omplex @ ( comple6723625652910419923omplex @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_890_Inf__less__eq,axiom,
! [A: set_set_set_nat,U: set_set_nat] :
( ! [V2: set_set_nat] :
( ( member_set_set_nat @ V2 @ A )
=> ( ord_le6893508408891458716et_nat @ V2 @ U ) )
=> ( ( A != bot_bo7198184520161983622et_nat )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_891_Inf__less__eq,axiom,
! [A: set_set_set_int,U: set_set_int] :
( ! [V2: set_set_int] :
( ( member_set_set_int @ V2 @ A )
=> ( ord_le4403425263959731960et_int @ V2 @ U ) )
=> ( ( A != bot_bo2384636101374064866et_int )
=> ( ord_le4403425263959731960et_int @ ( comple7798297522565507409et_int @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_892_Inf__less__eq,axiom,
! [A: set_se5258582372428582328st_nat,U: set_set_set_list_nat] :
( ! [V2: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ V2 @ A )
=> ( ord_le7100322305783427298st_nat @ V2 @ U ) )
=> ( ( A != bot_bo1158166727579713100st_nat )
=> ( ord_le7100322305783427298st_nat @ ( comple5189992959352112827st_nat @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_893_Inf__less__eq,axiom,
! [A: set_set_list_nat,U: set_list_nat] :
( ! [V2: set_list_nat] :
( ( member_set_list_nat @ V2 @ A )
=> ( ord_le6045566169113846134st_nat @ V2 @ U ) )
=> ( ( A != bot_bo3886227569956363488st_nat )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_894_Inf__less__eq,axiom,
! [A: set_o,U: $o] :
( ! [V2: $o] :
( ( member_o @ V2 @ A )
=> ( ord_less_eq_o @ V2 @ U ) )
=> ( ( A != bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_895_Inf__less__eq,axiom,
! [A: set_set_complex,U: set_complex] :
( ! [V2: set_complex] :
( ( member_set_complex @ V2 @ A )
=> ( ord_le211207098394363844omplex @ V2 @ U ) )
=> ( ( A != bot_bo4474773400535771566omplex )
=> ( ord_le211207098394363844omplex @ ( comple2956690151646016541omplex @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_896_Inf__less__eq,axiom,
! [A: set_set_nat,U: set_nat] :
( ! [V2: set_nat] :
( ( member_set_nat @ V2 @ A )
=> ( ord_less_eq_set_nat @ V2 @ U ) )
=> ( ( A != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_897_Inf__less__eq,axiom,
! [A: set_set_int,U: set_int] :
( ! [V2: set_int] :
( ( member_set_int @ V2 @ A )
=> ( ord_less_eq_set_int @ V2 @ U ) )
=> ( ( A != bot_bot_set_set_int )
=> ( ord_less_eq_set_int @ ( comple3628384868704368283et_int @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_898_Inf__less__eq,axiom,
! [A: set_set_set_list_nat,U: set_set_list_nat] :
( ! [V2: set_set_list_nat] :
( ( member1029098694177496419st_nat @ V2 @ A )
=> ( ord_le1068707526560357548st_nat @ V2 @ U ) )
=> ( ( A != bot_bo3499706412017099030st_nat )
=> ( ord_le1068707526560357548st_nat @ ( comple8462666950445340293st_nat @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_899_Inf__superset__mono,axiom,
! [B2: set_set_set_complex,A: set_set_set_complex] :
( ( ord_le314291461425487920omplex @ B2 @ A )
=> ( ord_le4750530260501030778omplex @ ( comple6723625652910419923omplex @ A ) @ ( comple6723625652910419923omplex @ B2 ) ) ) ).
% Inf_superset_mono
thf(fact_900_Inf__superset__mono,axiom,
! [B2: set_set_set_nat,A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A ) @ ( comple1065008630642458357et_nat @ B2 ) ) ) ).
% Inf_superset_mono
thf(fact_901_Inf__superset__mono,axiom,
! [B2: set_set_set_int,A: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ B2 @ A )
=> ( ord_le4403425263959731960et_int @ ( comple7798297522565507409et_int @ A ) @ ( comple7798297522565507409et_int @ B2 ) ) ) ).
% Inf_superset_mono
thf(fact_902_Inf__superset__mono,axiom,
! [B2: set_se5258582372428582328st_nat,A: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ B2 @ A )
=> ( ord_le7100322305783427298st_nat @ ( comple5189992959352112827st_nat @ A ) @ ( comple5189992959352112827st_nat @ B2 ) ) ) ).
% Inf_superset_mono
thf(fact_903_Inf__superset__mono,axiom,
! [B2: set_set_list_nat,A: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ B2 @ A )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ A ) @ ( comple184543376406953807st_nat @ B2 ) ) ) ).
% Inf_superset_mono
thf(fact_904_Inf__superset__mono,axiom,
! [B2: set_o,A: set_o] :
( ( ord_less_eq_set_o @ B2 @ A )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Inf_Inf_o @ B2 ) ) ) ).
% Inf_superset_mono
thf(fact_905_Inf__superset__mono,axiom,
! [B2: set_set_complex,A: set_set_complex] :
( ( ord_le4750530260501030778omplex @ B2 @ A )
=> ( ord_le211207098394363844omplex @ ( comple2956690151646016541omplex @ A ) @ ( comple2956690151646016541omplex @ B2 ) ) ) ).
% Inf_superset_mono
thf(fact_906_Inf__superset__mono,axiom,
! [B2: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ ( comple7806235888213564991et_nat @ B2 ) ) ) ).
% Inf_superset_mono
thf(fact_907_Inf__superset__mono,axiom,
! [B2: set_set_int,A: set_set_int] :
( ( ord_le4403425263959731960et_int @ B2 @ A )
=> ( ord_less_eq_set_int @ ( comple3628384868704368283et_int @ A ) @ ( comple3628384868704368283et_int @ B2 ) ) ) ).
% Inf_superset_mono
thf(fact_908_Inf__superset__mono,axiom,
! [B2: set_set_set_list_nat,A: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ B2 @ A )
=> ( ord_le1068707526560357548st_nat @ ( comple8462666950445340293st_nat @ A ) @ ( comple8462666950445340293st_nat @ B2 ) ) ) ).
% Inf_superset_mono
thf(fact_909_INF__eq__const,axiom,
! [I5: set_o,F: $o > $o,X5: $o] :
( ( I5 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I5 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ I5 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_910_INF__eq__const,axiom,
! [I5: set_nat,F: nat > $o,X5: $o] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I5 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F @ I5 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_911_INF__eq__const,axiom,
! [I5: set_list_nat,F: list_nat > $o,X5: $o] :
( ( I5 != bot_bot_set_list_nat )
=> ( ! [I4: list_nat] :
( ( member_list_nat @ I4 @ I5 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( complete_Inf_Inf_o @ ( image_list_nat_o @ F @ I5 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_912_INF__eq__const,axiom,
! [I5: set_set_complex,F: set_complex > $o,X5: $o] :
( ( I5 != bot_bo4474773400535771566omplex )
=> ( ! [I4: set_complex] :
( ( member_set_complex @ I4 @ I5 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( complete_Inf_Inf_o @ ( image_set_complex_o @ F @ I5 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_913_INF__eq__const,axiom,
! [I5: set_set_nat,F: set_nat > $o,X5: $o] :
( ( I5 != bot_bot_set_set_nat )
=> ( ! [I4: set_nat] :
( ( member_set_nat @ I4 @ I5 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( complete_Inf_Inf_o @ ( image_set_nat_o @ F @ I5 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_914_INF__eq__const,axiom,
! [I5: set_set_int,F: set_int > $o,X5: $o] :
( ( I5 != bot_bot_set_set_int )
=> ( ! [I4: set_int] :
( ( member_set_int @ I4 @ I5 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( complete_Inf_Inf_o @ ( image_set_int_o @ F @ I5 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_915_INF__eq__const,axiom,
! [I5: set_o,F: $o > set_complex,X5: set_complex] :
( ( I5 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I5 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( comple2956690151646016541omplex @ ( image_o_set_complex @ F @ I5 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_916_INF__eq__const,axiom,
! [I5: set_nat,F: nat > set_complex,X5: set_complex] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I5 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( comple2956690151646016541omplex @ ( image_6594795319511438139omplex @ F @ I5 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_917_INF__eq__const,axiom,
! [I5: set_o,F: $o > set_nat,X5: set_nat] :
( ( I5 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I5 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ I5 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_918_INF__eq__const,axiom,
! [I5: set_nat,F: nat > set_nat,X5: set_nat] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I5 )
=> ( ( F @ I4 )
= X5 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ I5 ) )
= X5 ) ) ) ).
% INF_eq_const
thf(fact_919_power__add,axiom,
! [A2: int,M: nat,N: nat] :
( ( power_power_int @ A2 @ ( plus_plus_nat @ M @ N ) )
= ( times_times_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) ) ) ).
% power_add
thf(fact_920_power__add,axiom,
! [A2: nat,M: nat,N: nat] :
( ( power_power_nat @ A2 @ ( plus_plus_nat @ M @ N ) )
= ( times_times_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) ) ) ).
% power_add
thf(fact_921_power__add,axiom,
! [A2: real,M: nat,N: nat] :
( ( power_power_real @ A2 @ ( plus_plus_nat @ M @ N ) )
= ( times_times_real @ ( power_power_real @ A2 @ M ) @ ( power_power_real @ A2 @ N ) ) ) ).
% power_add
thf(fact_922_power__add,axiom,
! [A2: complex,M: nat,N: nat] :
( ( power_power_complex @ A2 @ ( plus_plus_nat @ M @ N ) )
= ( times_times_complex @ ( power_power_complex @ A2 @ M ) @ ( power_power_complex @ A2 @ N ) ) ) ).
% power_add
thf(fact_923_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_924_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_925_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_926_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( ( M = one_one_int )
& ( N = one_one_int ) )
| ( ( M
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_927_Inter__anti__mono,axiom,
! [B2: set_set_set_complex,A: set_set_set_complex] :
( ( ord_le314291461425487920omplex @ B2 @ A )
=> ( ord_le4750530260501030778omplex @ ( comple6723625652910419923omplex @ A ) @ ( comple6723625652910419923omplex @ B2 ) ) ) ).
% Inter_anti_mono
thf(fact_928_Inter__anti__mono,axiom,
! [B2: set_set_set_nat,A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A ) @ ( comple1065008630642458357et_nat @ B2 ) ) ) ).
% Inter_anti_mono
thf(fact_929_Inter__anti__mono,axiom,
! [B2: set_set_set_int,A: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ B2 @ A )
=> ( ord_le4403425263959731960et_int @ ( comple7798297522565507409et_int @ A ) @ ( comple7798297522565507409et_int @ B2 ) ) ) ).
% Inter_anti_mono
thf(fact_930_Inter__anti__mono,axiom,
! [B2: set_se5258582372428582328st_nat,A: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ B2 @ A )
=> ( ord_le7100322305783427298st_nat @ ( comple5189992959352112827st_nat @ A ) @ ( comple5189992959352112827st_nat @ B2 ) ) ) ).
% Inter_anti_mono
thf(fact_931_Inter__anti__mono,axiom,
! [B2: set_set_list_nat,A: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ B2 @ A )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ A ) @ ( comple184543376406953807st_nat @ B2 ) ) ) ).
% Inter_anti_mono
thf(fact_932_Inter__anti__mono,axiom,
! [B2: set_set_complex,A: set_set_complex] :
( ( ord_le4750530260501030778omplex @ B2 @ A )
=> ( ord_le211207098394363844omplex @ ( comple2956690151646016541omplex @ A ) @ ( comple2956690151646016541omplex @ B2 ) ) ) ).
% Inter_anti_mono
thf(fact_933_Inter__anti__mono,axiom,
! [B2: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ ( comple7806235888213564991et_nat @ B2 ) ) ) ).
% Inter_anti_mono
thf(fact_934_Inter__anti__mono,axiom,
! [B2: set_set_int,A: set_set_int] :
( ( ord_le4403425263959731960et_int @ B2 @ A )
=> ( ord_less_eq_set_int @ ( comple3628384868704368283et_int @ A ) @ ( comple3628384868704368283et_int @ B2 ) ) ) ).
% Inter_anti_mono
thf(fact_935_Inter__anti__mono,axiom,
! [B2: set_set_set_list_nat,A: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ B2 @ A )
=> ( ord_le1068707526560357548st_nat @ ( comple8462666950445340293st_nat @ A ) @ ( comple8462666950445340293st_nat @ B2 ) ) ) ).
% Inter_anti_mono
thf(fact_936_Inter__subset,axiom,
! [A: set_set_set_complex,B2: set_set_complex] :
( ! [X4: set_set_complex] :
( ( member9015044028964487601omplex @ X4 @ A )
=> ( ord_le4750530260501030778omplex @ X4 @ B2 ) )
=> ( ( A != bot_bo92361985942245988omplex )
=> ( ord_le4750530260501030778omplex @ ( comple6723625652910419923omplex @ A ) @ B2 ) ) ) ).
% Inter_subset
thf(fact_937_Inter__subset,axiom,
! [A: set_set_set_nat,B2: set_set_nat] :
( ! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A )
=> ( ord_le6893508408891458716et_nat @ X4 @ B2 ) )
=> ( ( A != bot_bo7198184520161983622et_nat )
=> ( ord_le6893508408891458716et_nat @ ( comple1065008630642458357et_nat @ A ) @ B2 ) ) ) ).
% Inter_subset
thf(fact_938_Inter__subset,axiom,
! [A: set_set_set_int,B2: set_set_int] :
( ! [X4: set_set_int] :
( ( member_set_set_int @ X4 @ A )
=> ( ord_le4403425263959731960et_int @ X4 @ B2 ) )
=> ( ( A != bot_bo2384636101374064866et_int )
=> ( ord_le4403425263959731960et_int @ ( comple7798297522565507409et_int @ A ) @ B2 ) ) ) ).
% Inter_subset
thf(fact_939_Inter__subset,axiom,
! [A: set_se5258582372428582328st_nat,B2: set_set_set_list_nat] :
( ! [X4: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X4 @ A )
=> ( ord_le7100322305783427298st_nat @ X4 @ B2 ) )
=> ( ( A != bot_bo1158166727579713100st_nat )
=> ( ord_le7100322305783427298st_nat @ ( comple5189992959352112827st_nat @ A ) @ B2 ) ) ) ).
% Inter_subset
thf(fact_940_Inter__subset,axiom,
! [A: set_set_list_nat,B2: set_list_nat] :
( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ A )
=> ( ord_le6045566169113846134st_nat @ X4 @ B2 ) )
=> ( ( A != bot_bo3886227569956363488st_nat )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ A ) @ B2 ) ) ) ).
% Inter_subset
thf(fact_941_Inter__subset,axiom,
! [A: set_set_complex,B2: set_complex] :
( ! [X4: set_complex] :
( ( member_set_complex @ X4 @ A )
=> ( ord_le211207098394363844omplex @ X4 @ B2 ) )
=> ( ( A != bot_bo4474773400535771566omplex )
=> ( ord_le211207098394363844omplex @ ( comple2956690151646016541omplex @ A ) @ B2 ) ) ) ).
% Inter_subset
thf(fact_942_Inter__subset,axiom,
! [A: set_set_nat,B2: set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ X4 @ B2 ) )
=> ( ( A != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ B2 ) ) ) ).
% Inter_subset
thf(fact_943_Inter__subset,axiom,
! [A: set_set_int,B2: set_int] :
( ! [X4: set_int] :
( ( member_set_int @ X4 @ A )
=> ( ord_less_eq_set_int @ X4 @ B2 ) )
=> ( ( A != bot_bot_set_set_int )
=> ( ord_less_eq_set_int @ ( comple3628384868704368283et_int @ A ) @ B2 ) ) ) ).
% Inter_subset
thf(fact_944_Inter__subset,axiom,
! [A: set_set_set_list_nat,B2: set_set_list_nat] :
( ! [X4: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X4 @ A )
=> ( ord_le1068707526560357548st_nat @ X4 @ B2 ) )
=> ( ( A != bot_bo3499706412017099030st_nat )
=> ( ord_le1068707526560357548st_nat @ ( comple8462666950445340293st_nat @ A ) @ B2 ) ) ) ).
% Inter_subset
thf(fact_945_INF__greatest,axiom,
! [A: set_o,U: $o,F: $o > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_o @ U @ ( F @ I4 ) ) )
=> ( ord_less_eq_o @ U @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_946_INF__greatest,axiom,
! [A: set_nat,U: $o,F: nat > $o] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ( ord_less_eq_o @ U @ ( F @ I4 ) ) )
=> ( ord_less_eq_o @ U @ ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_947_INF__greatest,axiom,
! [A: set_list_nat,U: $o,F: list_nat > $o] :
( ! [I4: list_nat] :
( ( member_list_nat @ I4 @ A )
=> ( ord_less_eq_o @ U @ ( F @ I4 ) ) )
=> ( ord_less_eq_o @ U @ ( complete_Inf_Inf_o @ ( image_list_nat_o @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_948_INF__greatest,axiom,
! [A: set_o,U: set_complex,F: $o > set_complex] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_le211207098394363844omplex @ U @ ( F @ I4 ) ) )
=> ( ord_le211207098394363844omplex @ U @ ( comple2956690151646016541omplex @ ( image_o_set_complex @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_949_INF__greatest,axiom,
! [A: set_nat,U: set_complex,F: nat > set_complex] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ( ord_le211207098394363844omplex @ U @ ( F @ I4 ) ) )
=> ( ord_le211207098394363844omplex @ U @ ( comple2956690151646016541omplex @ ( image_6594795319511438139omplex @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_950_INF__greatest,axiom,
! [A: set_o,U: set_nat,F: $o > set_nat] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_set_nat @ U @ ( F @ I4 ) ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_951_INF__greatest,axiom,
! [A: set_nat,U: set_nat,F: nat > set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ( ord_less_eq_set_nat @ U @ ( F @ I4 ) ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_952_INF__greatest,axiom,
! [A: set_o,U: set_int,F: $o > set_int] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_set_int @ U @ ( F @ I4 ) ) )
=> ( ord_less_eq_set_int @ U @ ( comple3628384868704368283et_int @ ( image_o_set_int @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_953_INF__greatest,axiom,
! [A: set_nat,U: set_int,F: nat > set_int] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ( ord_less_eq_set_int @ U @ ( F @ I4 ) ) )
=> ( ord_less_eq_set_int @ U @ ( comple3628384868704368283et_int @ ( image_nat_set_int @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_954_INF__greatest,axiom,
! [A: set_o,U: set_set_complex,F: $o > set_set_complex] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_le4750530260501030778omplex @ U @ ( F @ I4 ) ) )
=> ( ord_le4750530260501030778omplex @ U @ ( comple6723625652910419923omplex @ ( image_1184936479116113517omplex @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_955_le__INF__iff,axiom,
! [U: set_list_nat,F: list_nat > set_list_nat,A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ U @ ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ F @ A ) ) )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( ord_le6045566169113846134st_nat @ U @ ( F @ X3 ) ) ) ) ) ).
% le_INF_iff
thf(fact_956_le__INF__iff,axiom,
! [U: set_list_nat,F: nat > set_list_nat,A: set_nat] :
( ( ord_le6045566169113846134st_nat @ U @ ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ F @ A ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_le6045566169113846134st_nat @ U @ ( F @ X3 ) ) ) ) ) ).
% le_INF_iff
thf(fact_957_le__INF__iff,axiom,
! [U: set_list_nat,F: set_list_nat > set_list_nat,A: set_set_list_nat] :
( ( ord_le6045566169113846134st_nat @ U @ ( comple184543376406953807st_nat @ ( image_5143090206295581363st_nat @ F @ A ) ) )
= ( ! [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A )
=> ( ord_le6045566169113846134st_nat @ U @ ( F @ X3 ) ) ) ) ) ).
% le_INF_iff
thf(fact_958_le__INF__iff,axiom,
! [U: $o,F: nat > $o,A: set_nat] :
( ( ord_less_eq_o @ U @ ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_o @ U @ ( F @ X3 ) ) ) ) ) ).
% le_INF_iff
thf(fact_959_le__INF__iff,axiom,
! [U: $o,F: list_nat > $o,A: set_list_nat] :
( ( ord_less_eq_o @ U @ ( complete_Inf_Inf_o @ ( image_list_nat_o @ F @ A ) ) )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( ord_less_eq_o @ U @ ( F @ X3 ) ) ) ) ) ).
% le_INF_iff
thf(fact_960_le__INF__iff,axiom,
! [U: $o,F: set_list_nat > $o,A: set_set_list_nat] :
( ( ord_less_eq_o @ U @ ( complete_Inf_Inf_o @ ( image_set_list_nat_o @ F @ A ) ) )
= ( ! [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A )
=> ( ord_less_eq_o @ U @ ( F @ X3 ) ) ) ) ) ).
% le_INF_iff
thf(fact_961_INF__lower2,axiom,
! [I3: $o,A: set_o,F: $o > $o,U: $o] :
( ( member_o @ I3 @ A )
=> ( ( ord_less_eq_o @ ( F @ I3 ) @ U )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_962_INF__lower2,axiom,
! [I3: nat,A: set_nat,F: nat > $o,U: $o] :
( ( member_nat @ I3 @ A )
=> ( ( ord_less_eq_o @ ( F @ I3 ) @ U )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_963_INF__lower2,axiom,
! [I3: list_nat,A: set_list_nat,F: list_nat > $o,U: $o] :
( ( member_list_nat @ I3 @ A )
=> ( ( ord_less_eq_o @ ( F @ I3 ) @ U )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_list_nat_o @ F @ A ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_964_INF__lower2,axiom,
! [I3: $o,A: set_o,F: $o > set_complex,U: set_complex] :
( ( member_o @ I3 @ A )
=> ( ( ord_le211207098394363844omplex @ ( F @ I3 ) @ U )
=> ( ord_le211207098394363844omplex @ ( comple2956690151646016541omplex @ ( image_o_set_complex @ F @ A ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_965_INF__lower2,axiom,
! [I3: nat,A: set_nat,F: nat > set_complex,U: set_complex] :
( ( member_nat @ I3 @ A )
=> ( ( ord_le211207098394363844omplex @ ( F @ I3 ) @ U )
=> ( ord_le211207098394363844omplex @ ( comple2956690151646016541omplex @ ( image_6594795319511438139omplex @ F @ A ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_966_INF__lower2,axiom,
! [I3: $o,A: set_o,F: $o > set_nat,U: set_nat] :
( ( member_o @ I3 @ A )
=> ( ( ord_less_eq_set_nat @ ( F @ I3 ) @ U )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_967_INF__lower2,axiom,
! [I3: nat,A: set_nat,F: nat > set_nat,U: set_nat] :
( ( member_nat @ I3 @ A )
=> ( ( ord_less_eq_set_nat @ ( F @ I3 ) @ U )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_968_INF__lower2,axiom,
! [I3: $o,A: set_o,F: $o > set_int,U: set_int] :
( ( member_o @ I3 @ A )
=> ( ( ord_less_eq_set_int @ ( F @ I3 ) @ U )
=> ( ord_less_eq_set_int @ ( comple3628384868704368283et_int @ ( image_o_set_int @ F @ A ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_969_INF__lower2,axiom,
! [I3: nat,A: set_nat,F: nat > set_int,U: set_int] :
( ( member_nat @ I3 @ A )
=> ( ( ord_less_eq_set_int @ ( F @ I3 ) @ U )
=> ( ord_less_eq_set_int @ ( comple3628384868704368283et_int @ ( image_nat_set_int @ F @ A ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_970_INF__lower2,axiom,
! [I3: $o,A: set_o,F: $o > set_set_complex,U: set_set_complex] :
( ( member_o @ I3 @ A )
=> ( ( ord_le4750530260501030778omplex @ ( F @ I3 ) @ U )
=> ( ord_le4750530260501030778omplex @ ( comple6723625652910419923omplex @ ( image_1184936479116113517omplex @ F @ A ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_971_INF__mono_H,axiom,
! [F: list_nat > set_list_nat,G: list_nat > set_list_nat,A: set_list_nat] :
( ! [X2: list_nat] : ( ord_le6045566169113846134st_nat @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ F @ A ) ) @ ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ G @ A ) ) ) ) ).
% INF_mono'
thf(fact_972_INF__mono_H,axiom,
! [F: nat > set_list_nat,G: nat > set_list_nat,A: set_nat] :
( ! [X2: nat] : ( ord_le6045566169113846134st_nat @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ F @ A ) ) @ ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ G @ A ) ) ) ) ).
% INF_mono'
thf(fact_973_INF__mono_H,axiom,
! [F: set_list_nat > set_list_nat,G: set_list_nat > set_list_nat,A: set_set_list_nat] :
( ! [X2: set_list_nat] : ( ord_le6045566169113846134st_nat @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ ( image_5143090206295581363st_nat @ F @ A ) ) @ ( comple184543376406953807st_nat @ ( image_5143090206295581363st_nat @ G @ A ) ) ) ) ).
% INF_mono'
thf(fact_974_INF__mono_H,axiom,
! [F: nat > $o,G: nat > $o,A: set_nat] :
( ! [X2: nat] : ( ord_less_eq_o @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A ) ) @ ( complete_Inf_Inf_o @ ( image_nat_o @ G @ A ) ) ) ) ).
% INF_mono'
thf(fact_975_INF__mono_H,axiom,
! [F: list_nat > $o,G: list_nat > $o,A: set_list_nat] :
( ! [X2: list_nat] : ( ord_less_eq_o @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_list_nat_o @ F @ A ) ) @ ( complete_Inf_Inf_o @ ( image_list_nat_o @ G @ A ) ) ) ) ).
% INF_mono'
thf(fact_976_INF__mono_H,axiom,
! [F: set_list_nat > $o,G: set_list_nat > $o,A: set_set_list_nat] :
( ! [X2: set_list_nat] : ( ord_less_eq_o @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_set_list_nat_o @ F @ A ) ) @ ( complete_Inf_Inf_o @ ( image_set_list_nat_o @ G @ A ) ) ) ) ).
% INF_mono'
thf(fact_977_INF__lower,axiom,
! [I3: $o,A: set_o,F: $o > $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A ) ) @ ( F @ I3 ) ) ) ).
% INF_lower
thf(fact_978_INF__lower,axiom,
! [I3: nat,A: set_nat,F: nat > $o] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A ) ) @ ( F @ I3 ) ) ) ).
% INF_lower
thf(fact_979_INF__lower,axiom,
! [I3: list_nat,A: set_list_nat,F: list_nat > $o] :
( ( member_list_nat @ I3 @ A )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_list_nat_o @ F @ A ) ) @ ( F @ I3 ) ) ) ).
% INF_lower
thf(fact_980_INF__lower,axiom,
! [I3: $o,A: set_o,F: $o > set_complex] :
( ( member_o @ I3 @ A )
=> ( ord_le211207098394363844omplex @ ( comple2956690151646016541omplex @ ( image_o_set_complex @ F @ A ) ) @ ( F @ I3 ) ) ) ).
% INF_lower
thf(fact_981_INF__lower,axiom,
! [I3: nat,A: set_nat,F: nat > set_complex] :
( ( member_nat @ I3 @ A )
=> ( ord_le211207098394363844omplex @ ( comple2956690151646016541omplex @ ( image_6594795319511438139omplex @ F @ A ) ) @ ( F @ I3 ) ) ) ).
% INF_lower
thf(fact_982_INF__lower,axiom,
! [I3: $o,A: set_o,F: $o > set_nat] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A ) ) @ ( F @ I3 ) ) ) ).
% INF_lower
thf(fact_983_INF__lower,axiom,
! [I3: nat,A: set_nat,F: nat > set_nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( F @ I3 ) ) ) ).
% INF_lower
thf(fact_984_INF__lower,axiom,
! [I3: $o,A: set_o,F: $o > set_int] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_set_int @ ( comple3628384868704368283et_int @ ( image_o_set_int @ F @ A ) ) @ ( F @ I3 ) ) ) ).
% INF_lower
thf(fact_985_INF__lower,axiom,
! [I3: nat,A: set_nat,F: nat > set_int] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_set_int @ ( comple3628384868704368283et_int @ ( image_nat_set_int @ F @ A ) ) @ ( F @ I3 ) ) ) ).
% INF_lower
thf(fact_986_INF__lower,axiom,
! [I3: $o,A: set_o,F: $o > set_set_complex] :
( ( member_o @ I3 @ A )
=> ( ord_le4750530260501030778omplex @ ( comple6723625652910419923omplex @ ( image_1184936479116113517omplex @ F @ A ) ) @ ( F @ I3 ) ) ) ).
% INF_lower
thf(fact_987_INF__mono,axiom,
! [B2: set_o,A: set_nat,F: nat > $o,G: $o > $o] :
( ! [M2: $o] :
( ( member_o @ M2 @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_less_eq_o @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A ) ) @ ( complete_Inf_Inf_o @ ( image_o_o @ G @ B2 ) ) ) ) ).
% INF_mono
thf(fact_988_INF__mono,axiom,
! [B2: set_nat,A: set_nat,F: nat > $o,G: nat > $o] :
( ! [M2: nat] :
( ( member_nat @ M2 @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_less_eq_o @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A ) ) @ ( complete_Inf_Inf_o @ ( image_nat_o @ G @ B2 ) ) ) ) ).
% INF_mono
thf(fact_989_INF__mono,axiom,
! [B2: set_list_nat,A: set_nat,F: nat > $o,G: list_nat > $o] :
( ! [M2: list_nat] :
( ( member_list_nat @ M2 @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_less_eq_o @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A ) ) @ ( complete_Inf_Inf_o @ ( image_list_nat_o @ G @ B2 ) ) ) ) ).
% INF_mono
thf(fact_990_INF__mono,axiom,
! [B2: set_o,A: set_list_nat,F: list_nat > $o,G: $o > $o] :
( ! [M2: $o] :
( ( member_o @ M2 @ B2 )
=> ? [X6: list_nat] :
( ( member_list_nat @ X6 @ A )
& ( ord_less_eq_o @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_list_nat_o @ F @ A ) ) @ ( complete_Inf_Inf_o @ ( image_o_o @ G @ B2 ) ) ) ) ).
% INF_mono
thf(fact_991_INF__mono,axiom,
! [B2: set_nat,A: set_list_nat,F: list_nat > $o,G: nat > $o] :
( ! [M2: nat] :
( ( member_nat @ M2 @ B2 )
=> ? [X6: list_nat] :
( ( member_list_nat @ X6 @ A )
& ( ord_less_eq_o @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_list_nat_o @ F @ A ) ) @ ( complete_Inf_Inf_o @ ( image_nat_o @ G @ B2 ) ) ) ) ).
% INF_mono
thf(fact_992_INF__mono,axiom,
! [B2: set_o,A: set_nat,F: nat > set_list_nat,G: $o > set_list_nat] :
( ! [M2: $o] :
( ( member_o @ M2 @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_le6045566169113846134st_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ F @ A ) ) @ ( comple184543376406953807st_nat @ ( image_o_set_list_nat @ G @ B2 ) ) ) ) ).
% INF_mono
thf(fact_993_INF__mono,axiom,
! [B2: set_nat,A: set_nat,F: nat > set_list_nat,G: nat > set_list_nat] :
( ! [M2: nat] :
( ( member_nat @ M2 @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_le6045566169113846134st_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ F @ A ) ) @ ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ G @ B2 ) ) ) ) ).
% INF_mono
thf(fact_994_INF__mono,axiom,
! [B2: set_set_list_nat,A: set_nat,F: nat > $o,G: set_list_nat > $o] :
( ! [M2: set_list_nat] :
( ( member_set_list_nat @ M2 @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_less_eq_o @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A ) ) @ ( complete_Inf_Inf_o @ ( image_set_list_nat_o @ G @ B2 ) ) ) ) ).
% INF_mono
thf(fact_995_INF__mono,axiom,
! [B2: set_list_nat,A: set_list_nat,F: list_nat > $o,G: list_nat > $o] :
( ! [M2: list_nat] :
( ( member_list_nat @ M2 @ B2 )
=> ? [X6: list_nat] :
( ( member_list_nat @ X6 @ A )
& ( ord_less_eq_o @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_list_nat_o @ F @ A ) ) @ ( complete_Inf_Inf_o @ ( image_list_nat_o @ G @ B2 ) ) ) ) ).
% INF_mono
thf(fact_996_INF__mono,axiom,
! [B2: set_o,A: set_set_list_nat,F: set_list_nat > $o,G: $o > $o] :
( ! [M2: $o] :
( ( member_o @ M2 @ B2 )
=> ? [X6: set_list_nat] :
( ( member_set_list_nat @ X6 @ A )
& ( ord_less_eq_o @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_set_list_nat_o @ F @ A ) ) @ ( complete_Inf_Inf_o @ ( image_o_o @ G @ B2 ) ) ) ) ).
% INF_mono
thf(fact_997_INF__eqI,axiom,
! [A: set_o,X5: $o,F: $o > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_o @ X5 @ ( F @ I4 ) ) )
=> ( ! [Y3: $o] :
( ! [I6: $o] :
( ( member_o @ I6 @ A )
=> ( ord_less_eq_o @ Y3 @ ( F @ I6 ) ) )
=> ( ord_less_eq_o @ Y3 @ X5 ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ A ) )
= X5 ) ) ) ).
% INF_eqI
thf(fact_998_INF__eqI,axiom,
! [A: set_nat,X5: $o,F: nat > $o] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ( ord_less_eq_o @ X5 @ ( F @ I4 ) ) )
=> ( ! [Y3: $o] :
( ! [I6: nat] :
( ( member_nat @ I6 @ A )
=> ( ord_less_eq_o @ Y3 @ ( F @ I6 ) ) )
=> ( ord_less_eq_o @ Y3 @ X5 ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A ) )
= X5 ) ) ) ).
% INF_eqI
thf(fact_999_INF__eqI,axiom,
! [A: set_list_nat,X5: $o,F: list_nat > $o] :
( ! [I4: list_nat] :
( ( member_list_nat @ I4 @ A )
=> ( ord_less_eq_o @ X5 @ ( F @ I4 ) ) )
=> ( ! [Y3: $o] :
( ! [I6: list_nat] :
( ( member_list_nat @ I6 @ A )
=> ( ord_less_eq_o @ Y3 @ ( F @ I6 ) ) )
=> ( ord_less_eq_o @ Y3 @ X5 ) )
=> ( ( complete_Inf_Inf_o @ ( image_list_nat_o @ F @ A ) )
= X5 ) ) ) ).
% INF_eqI
thf(fact_1000_INF__eqI,axiom,
! [A: set_o,X5: set_complex,F: $o > set_complex] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_le211207098394363844omplex @ X5 @ ( F @ I4 ) ) )
=> ( ! [Y3: set_complex] :
( ! [I6: $o] :
( ( member_o @ I6 @ A )
=> ( ord_le211207098394363844omplex @ Y3 @ ( F @ I6 ) ) )
=> ( ord_le211207098394363844omplex @ Y3 @ X5 ) )
=> ( ( comple2956690151646016541omplex @ ( image_o_set_complex @ F @ A ) )
= X5 ) ) ) ).
% INF_eqI
thf(fact_1001_INF__eqI,axiom,
! [A: set_nat,X5: set_complex,F: nat > set_complex] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ( ord_le211207098394363844omplex @ X5 @ ( F @ I4 ) ) )
=> ( ! [Y3: set_complex] :
( ! [I6: nat] :
( ( member_nat @ I6 @ A )
=> ( ord_le211207098394363844omplex @ Y3 @ ( F @ I6 ) ) )
=> ( ord_le211207098394363844omplex @ Y3 @ X5 ) )
=> ( ( comple2956690151646016541omplex @ ( image_6594795319511438139omplex @ F @ A ) )
= X5 ) ) ) ).
% INF_eqI
thf(fact_1002_INF__eqI,axiom,
! [A: set_o,X5: set_nat,F: $o > set_nat] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_set_nat @ X5 @ ( F @ I4 ) ) )
=> ( ! [Y3: set_nat] :
( ! [I6: $o] :
( ( member_o @ I6 @ A )
=> ( ord_less_eq_set_nat @ Y3 @ ( F @ I6 ) ) )
=> ( ord_less_eq_set_nat @ Y3 @ X5 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A ) )
= X5 ) ) ) ).
% INF_eqI
thf(fact_1003_INF__eqI,axiom,
! [A: set_nat,X5: set_nat,F: nat > set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ( ord_less_eq_set_nat @ X5 @ ( F @ I4 ) ) )
=> ( ! [Y3: set_nat] :
( ! [I6: nat] :
( ( member_nat @ I6 @ A )
=> ( ord_less_eq_set_nat @ Y3 @ ( F @ I6 ) ) )
=> ( ord_less_eq_set_nat @ Y3 @ X5 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A ) )
= X5 ) ) ) ).
% INF_eqI
thf(fact_1004_INF__eqI,axiom,
! [A: set_o,X5: set_int,F: $o > set_int] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_set_int @ X5 @ ( F @ I4 ) ) )
=> ( ! [Y3: set_int] :
( ! [I6: $o] :
( ( member_o @ I6 @ A )
=> ( ord_less_eq_set_int @ Y3 @ ( F @ I6 ) ) )
=> ( ord_less_eq_set_int @ Y3 @ X5 ) )
=> ( ( comple3628384868704368283et_int @ ( image_o_set_int @ F @ A ) )
= X5 ) ) ) ).
% INF_eqI
thf(fact_1005_INF__eqI,axiom,
! [A: set_nat,X5: set_int,F: nat > set_int] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ( ord_less_eq_set_int @ X5 @ ( F @ I4 ) ) )
=> ( ! [Y3: set_int] :
( ! [I6: nat] :
( ( member_nat @ I6 @ A )
=> ( ord_less_eq_set_int @ Y3 @ ( F @ I6 ) ) )
=> ( ord_less_eq_set_int @ Y3 @ X5 ) )
=> ( ( comple3628384868704368283et_int @ ( image_nat_set_int @ F @ A ) )
= X5 ) ) ) ).
% INF_eqI
thf(fact_1006_INF__eqI,axiom,
! [A: set_o,X5: set_set_complex,F: $o > set_set_complex] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_le4750530260501030778omplex @ X5 @ ( F @ I4 ) ) )
=> ( ! [Y3: set_set_complex] :
( ! [I6: $o] :
( ( member_o @ I6 @ A )
=> ( ord_le4750530260501030778omplex @ Y3 @ ( F @ I6 ) ) )
=> ( ord_le4750530260501030778omplex @ Y3 @ X5 ) )
=> ( ( comple6723625652910419923omplex @ ( image_1184936479116113517omplex @ F @ A ) )
= X5 ) ) ) ).
% INF_eqI
thf(fact_1007_sum__fun__comp,axiom,
! [S: set_real,R: set_real,G: real > real,F: real > complex] :
( ( finite_finite_real @ S )
=> ( ( finite_finite_real @ R )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ G @ S ) @ R )
=> ( ( groups5754745047067104278omplex
@ ^ [X3: real] : ( F @ ( G @ X3 ) )
@ S )
= ( groups5754745047067104278omplex
@ ^ [Y2: real] :
( times_times_complex
@ ( semiri8010041392384452111omplex
@ ( finite_card_real
@ ( collect_real
@ ^ [X3: real] :
( ( member_real @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) ) )
@ ( F @ Y2 ) )
@ R ) ) ) ) ) ).
% sum_fun_comp
thf(fact_1008_sum__fun__comp,axiom,
! [S: set_o,R: set_int,G: $o > int,F: int > complex] :
( ( finite_finite_o @ S )
=> ( ( finite_finite_int @ R )
=> ( ( ord_less_eq_set_int @ ( image_o_int @ G @ S ) @ R )
=> ( ( groups5328290441151304332omplex
@ ^ [X3: $o] : ( F @ ( G @ X3 ) )
@ S )
= ( groups3049146728041665814omplex
@ ^ [Y2: int] :
( times_times_complex
@ ( semiri8010041392384452111omplex
@ ( finite_card_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) ) )
@ ( F @ Y2 ) )
@ R ) ) ) ) ) ).
% sum_fun_comp
thf(fact_1009_sum__fun__comp,axiom,
! [S: set_o,R: set_nat,G: $o > nat,F: nat > complex] :
( ( finite_finite_o @ S )
=> ( ( finite_finite_nat @ R )
=> ( ( ord_less_eq_set_nat @ ( image_o_nat @ G @ S ) @ R )
=> ( ( groups5328290441151304332omplex
@ ^ [X3: $o] : ( F @ ( G @ X3 ) )
@ S )
= ( groups2073611262835488442omplex
@ ^ [Y2: nat] :
( times_times_complex
@ ( semiri8010041392384452111omplex
@ ( finite_card_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) ) )
@ ( F @ Y2 ) )
@ R ) ) ) ) ) ).
% sum_fun_comp
thf(fact_1010_sum__fun__comp,axiom,
! [S: set_o,R: set_complex,G: $o > complex,F: complex > complex] :
( ( finite_finite_o @ S )
=> ( ( finite3207457112153483333omplex @ R )
=> ( ( ord_le211207098394363844omplex @ ( image_o_complex @ G @ S ) @ R )
=> ( ( groups5328290441151304332omplex
@ ^ [X3: $o] : ( F @ ( G @ X3 ) )
@ S )
= ( groups7754918857620584856omplex
@ ^ [Y2: complex] :
( times_times_complex
@ ( semiri8010041392384452111omplex
@ ( finite_card_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) ) )
@ ( F @ Y2 ) )
@ R ) ) ) ) ) ).
% sum_fun_comp
thf(fact_1011_sum__fun__comp,axiom,
! [S: set_int,R: set_int,G: int > int,F: int > complex] :
( ( finite_finite_int @ S )
=> ( ( finite_finite_int @ R )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ G @ S ) @ R )
=> ( ( groups3049146728041665814omplex
@ ^ [X3: int] : ( F @ ( G @ X3 ) )
@ S )
= ( groups3049146728041665814omplex
@ ^ [Y2: int] :
( times_times_complex
@ ( semiri8010041392384452111omplex
@ ( finite_card_int
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) ) )
@ ( F @ Y2 ) )
@ R ) ) ) ) ) ).
% sum_fun_comp
thf(fact_1012_sum__fun__comp,axiom,
! [S: set_int,R: set_nat,G: int > nat,F: nat > complex] :
( ( finite_finite_int @ S )
=> ( ( finite_finite_nat @ R )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ G @ S ) @ R )
=> ( ( groups3049146728041665814omplex
@ ^ [X3: int] : ( F @ ( G @ X3 ) )
@ S )
= ( groups2073611262835488442omplex
@ ^ [Y2: nat] :
( times_times_complex
@ ( semiri8010041392384452111omplex
@ ( finite_card_int
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) ) )
@ ( F @ Y2 ) )
@ R ) ) ) ) ) ).
% sum_fun_comp
thf(fact_1013_sum__fun__comp,axiom,
! [S: set_int,R: set_complex,G: int > complex,F: complex > complex] :
( ( finite_finite_int @ S )
=> ( ( finite3207457112153483333omplex @ R )
=> ( ( ord_le211207098394363844omplex @ ( image_int_complex @ G @ S ) @ R )
=> ( ( groups3049146728041665814omplex
@ ^ [X3: int] : ( F @ ( G @ X3 ) )
@ S )
= ( groups7754918857620584856omplex
@ ^ [Y2: complex] :
( times_times_complex
@ ( semiri8010041392384452111omplex
@ ( finite_card_int
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) ) )
@ ( F @ Y2 ) )
@ R ) ) ) ) ) ).
% sum_fun_comp
thf(fact_1014_sum__fun__comp,axiom,
! [S: set_nat,R: set_o,G: nat > $o,F: $o > complex] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_o @ R )
=> ( ( ord_less_eq_set_o @ ( image_nat_o @ G @ S ) @ R )
=> ( ( groups2073611262835488442omplex
@ ^ [X3: nat] : ( F @ ( G @ X3 ) )
@ S )
= ( groups5328290441151304332omplex
@ ^ [Y2: $o] :
( times_times_complex
@ ( semiri8010041392384452111omplex
@ ( finite_card_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) ) )
@ ( F @ Y2 ) )
@ R ) ) ) ) ) ).
% sum_fun_comp
thf(fact_1015_sum__fun__comp,axiom,
! [S: set_nat,R: set_int,G: nat > int,F: int > complex] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_int @ R )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ G @ S ) @ R )
=> ( ( groups2073611262835488442omplex
@ ^ [X3: nat] : ( F @ ( G @ X3 ) )
@ S )
= ( groups3049146728041665814omplex
@ ^ [Y2: int] :
( times_times_complex
@ ( semiri8010041392384452111omplex
@ ( finite_card_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) ) )
@ ( F @ Y2 ) )
@ R ) ) ) ) ) ).
% sum_fun_comp
thf(fact_1016_sum__fun__comp,axiom,
! [S: set_nat,R: set_nat,G: nat > nat,F: nat > complex] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_nat @ R )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ S ) @ R )
=> ( ( groups2073611262835488442omplex
@ ^ [X3: nat] : ( F @ ( G @ X3 ) )
@ S )
= ( groups2073611262835488442omplex
@ ^ [Y2: nat] :
( times_times_complex
@ ( semiri8010041392384452111omplex
@ ( finite_card_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ S )
& ( ( G @ X3 )
= Y2 ) ) ) ) )
@ ( F @ Y2 ) )
@ R ) ) ) ) ) ).
% sum_fun_comp
thf(fact_1017_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_1018_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1019_SUM2__def,axiom,
( sUM2
= ( groups7312845317294741502at_int
@ ^ [Y4: set_list_nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ ( finite_card_list_nat @ Y4 ) @ one_one_nat ) ) @ ( semiri1314217659103216013at_int @ ( finite_card_list_nat @ ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ ( c @ n ) @ Y4 ) ) ) ) )
@ ( collect_set_list_nat
@ ^ [Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ Y4 @ x )
& ( Y4 != bot_bot_set_list_nat ) ) ) ) ) ).
% SUM2_def
thf(fact_1020_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1021_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1022_nat__add__left__cancel__le,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1023_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1024_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1025_mult__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M )
= ( times_times_nat @ K2 @ N ) )
= ( ( M = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1026_mult__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ( times_times_nat @ M @ K2 )
= ( times_times_nat @ N @ K2 ) )
= ( ( M = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1027_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_1028_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1029_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1030_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1031_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1032_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1033_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1034_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1035_le__trans,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K2 )
=> ( ord_less_eq_nat @ I3 @ K2 ) ) ) ).
% le_trans
thf(fact_1036_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1037_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1038_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_1039_mult__le__mono,axiom,
! [I3: nat,J2: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K2 ) @ ( times_times_nat @ J2 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1040_mult__le__mono1,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K2 ) @ ( times_times_nat @ J2 @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_1041_mult__le__mono2,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I3 ) @ ( times_times_nat @ K2 @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_1042_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_1043_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1044_Inf__bool__def,axiom,
( complete_Inf_Inf_o
= ( ^ [A3: set_o] :
~ ( member_o @ $false @ A3 ) ) ) ).
% Inf_bool_def
thf(fact_1045_add__leE,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_1046_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1047_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1048_add__leD1,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1049_add__leD2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_1050_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_1051_add__le__mono,axiom,
! [I3: nat,J2: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_1052_add__le__mono1,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J2 @ K2 ) ) ) ).
% add_le_mono1
thf(fact_1053_trans__le__add1,axiom,
! [I3: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_le_add1
thf(fact_1054_trans__le__add2,axiom,
! [I3: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_le_add2
thf(fact_1055_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
? [K: nat] :
( N2
= ( plus_plus_nat @ M3 @ K ) ) ) ) ).
% nat_le_iff_add
thf(fact_1056_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_1057_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1058_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% nat_int
thf(fact_1059_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ Z )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_1060_nat__0__iff,axiom,
! [I3: int] :
( ( ( nat2 @ I3 )
= zero_zero_nat )
= ( ord_less_eq_int @ I3 @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_1061_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_1062_int__nat__eq,axiom,
! [Z: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_1063_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1064_Inf__int__def,axiom,
( complete_Inf_Inf_int
= ( ^ [X8: set_int] : ( uminus_uminus_int @ ( complete_Sup_Sup_int @ ( image_int_int @ uminus_uminus_int @ X8 ) ) ) ) ) ).
% Inf_int_def
thf(fact_1065_nat__mono,axiom,
! [X5: int,Y: int] :
( ( ord_less_eq_int @ X5 @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X5 ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_1066_ex__nat,axiom,
( ( ^ [P2: nat > $o] :
? [X9: nat] : ( P2 @ X9 ) )
= ( ^ [P3: nat > $o] :
? [X3: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
& ( P3 @ ( nat2 @ X3 ) ) ) ) ) ).
% ex_nat
thf(fact_1067_all__nat,axiom,
( ( ^ [P2: nat > $o] :
! [X9: nat] : ( P2 @ X9 ) )
= ( ^ [P3: nat > $o] :
! [X3: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( P3 @ ( nat2 @ X3 ) ) ) ) ) ).
% all_nat
thf(fact_1068_eq__nat__nat__iff,axiom,
! [Z: int,Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( ( nat2 @ Z )
= ( nat2 @ Z4 ) )
= ( Z = Z4 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_1069_nat__le__iff,axiom,
! [X5: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X5 ) @ N )
= ( ord_less_eq_int @ X5 @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_1070_nat__int__add,axiom,
! [A2: nat,B: nat] :
( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) )
= ( plus_plus_nat @ A2 @ B ) ) ).
% nat_int_add
thf(fact_1071_nat__0__le,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) ) ).
% nat_0_le
thf(fact_1072_int__eq__iff,axiom,
! [M: nat,Z: int] :
( ( ( semiri1314217659103216013at_int @ M )
= Z )
= ( ( M
= ( nat2 @ Z ) )
& ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% int_eq_iff
thf(fact_1073_bounded__Max__nat,axiom,
! [P: nat > $o,X5: nat,M4: nat] :
( ( P @ X5 )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M4 ) )
=> ~ ! [M2: nat] :
( ( P @ M2 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1074_nat__eq__iff,axiom,
! [W: int,M: nat] :
( ( ( nat2 @ W )
= M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1075_nat__eq__iff2,axiom,
! [M: nat,W: int] :
( ( M
= ( nat2 @ W ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1076_nat__add__distrib,axiom,
! [Z: int,Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( nat2 @ ( plus_plus_int @ Z @ Z4 ) )
= ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z4 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_1077_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_1078_nat__mult__distrib,axiom,
! [Z: int,Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( nat2 @ ( times_times_int @ Z @ Z4 ) )
= ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z4 ) ) ) ) ).
% nat_mult_distrib
thf(fact_1079_nat__power__eq,axiom,
! [Z: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( nat2 @ ( power_power_int @ Z @ N ) )
= ( power_power_nat @ ( nat2 @ Z ) @ N ) ) ) ).
% nat_power_eq
thf(fact_1080_nat__mult__distrib__neg,axiom,
! [Z: int,Z4: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ ( times_times_int @ Z @ Z4 ) )
= ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z4 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_1081_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M3: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_eq_nat @ X3 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1082_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1083_nat__times__as__int,axiom,
( times_times_nat
= ( ^ [A5: nat,B4: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% nat_times_as_int
thf(fact_1084_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_1085_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_1086_Inf__nat__def1,axiom,
! [K3: set_nat] :
( ( K3 != bot_bot_set_nat )
=> ( member_nat @ ( complete_Inf_Inf_nat @ K3 ) @ K3 ) ) ).
% Inf_nat_def1
thf(fact_1087_verit__la__generic,axiom,
! [A2: int,X5: int] :
( ( ord_less_eq_int @ A2 @ X5 )
| ( A2 = X5 )
| ( ord_less_eq_int @ X5 @ A2 ) ) ).
% verit_la_generic
thf(fact_1088_int__if,axiom,
! [P: $o,A2: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B ) )
= ( semiri1314217659103216013at_int @ A2 ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_1089_nat__int__comparison_I1_J,axiom,
( ( ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
= ( ^ [A5: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ A5 )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1090_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1091_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_1092_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_1093_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_1094_int__ops_I5_J,axiom,
! [A2: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A2 @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_1095_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1096_int__ops_I7_J,axiom,
! [A2: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A2 @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_1097_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_1098_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_1099_power__le__one__iff,axiom,
! [A2: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ ( power_power_real @ A2 @ N ) @ one_one_real )
= ( ( N = zero_zero_nat )
| ( ord_less_eq_real @ A2 @ one_one_real ) ) ) ) ).
% power_le_one_iff
thf(fact_1100_ennreal__add__bot,axiom,
! [X5: extend8495563244428889912nnreal] :
( ( plus_p1859984266308609217nnreal @ bot_bo841427958541957580nnreal @ X5 )
= X5 ) ).
% ennreal_add_bot
thf(fact_1101_real__of__nat__ge__one__iff,axiom,
! [N: nat] :
( ( ord_less_eq_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ one_one_nat @ N ) ) ).
% real_of_nat_ge_one_iff
thf(fact_1102_bot__ennreal,axiom,
bot_bo841427958541957580nnreal = zero_z7100319975126383169nnreal ).
% bot_ennreal
thf(fact_1103_Bernoulli__inequality,axiom,
! [X5: real,N: nat] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X5 )
=> ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X5 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X5 ) @ N ) ) ) ).
% Bernoulli_inequality
thf(fact_1104_arsinh__minus__real,axiom,
! [X5: real] :
( ( arsinh_real @ ( uminus_uminus_real @ X5 ) )
= ( uminus_uminus_real @ ( arsinh_real @ X5 ) ) ) ).
% arsinh_minus_real
thf(fact_1105_real__add__minus__iff,axiom,
! [X5: real,A2: real] :
( ( ( plus_plus_real @ X5 @ ( uminus_uminus_real @ A2 ) )
= zero_zero_real )
= ( X5 = A2 ) ) ).
% real_add_minus_iff
thf(fact_1106_Inf__real__def,axiom,
( comple4887499456419720421f_real
= ( ^ [X8: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X8 ) ) ) ) ) ).
% Inf_real_def
thf(fact_1107_real__minus__mult__self__le,axiom,
! [U: real,X5: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X5 @ X5 ) ) ).
% real_minus_mult_self_le
thf(fact_1108_real__add__le__0__iff,axiom,
! [X5: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X5 @ Y ) @ zero_zero_real )
= ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X5 ) ) ) ).
% real_add_le_0_iff
thf(fact_1109_real__0__le__add__iff,axiom,
! [X5: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X5 @ Y ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X5 ) @ Y ) ) ).
% real_0_le_add_iff
thf(fact_1110_real__eq__0__iff__le__ge__0,axiom,
! [X5: real] :
( ( X5 = zero_zero_real )
= ( ( ord_less_eq_real @ zero_zero_real @ X5 )
& ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ X5 ) ) ) ) ).
% real_eq_0_iff_le_ge_0
thf(fact_1111__092_060open_0620_A_060_An_092_060close_062,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% \<open>0 < n\<close>
thf(fact_1112_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_1113_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1114_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1115_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1116_nat__add__left__cancel__less,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1117_finite__Collect__less__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K2 ) ) ) ).
% finite_Collect_less_nat
thf(fact_1118_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1119_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1120_mult__less__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1121_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1122_nat__zero__less__power__iff,axiom,
! [X5: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X5 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X5 )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1123_mult__le__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1124_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ K4 )
=> ~ ( P @ I6 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1125_less__imp__add__positive,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ? [K4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K4 )
& ( ( plus_plus_nat @ I3 @ K4 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_1126_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
& ( M3 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1127_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1128_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1129_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1130_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1131_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J2: nat] :
( ! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1132_linorder__neqE__nat,axiom,
! [X5: nat,Y: nat] :
( ( X5 != Y )
=> ( ~ ( ord_less_nat @ X5 @ Y )
=> ( ord_less_nat @ Y @ X5 ) ) ) ).
% linorder_neqE_nat
thf(fact_1133_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1134_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1135_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1136_less__not__refl3,axiom,
! [S2: nat,T2: nat] :
( ( ord_less_nat @ S2 @ T2 )
=> ( S2 != T2 ) ) ).
% less_not_refl3
thf(fact_1137_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_1138_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1139_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_1140_bounded__nat__set__is__finite,axiom,
! [N4: set_nat,N: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ N4 )
=> ( ord_less_nat @ X2 @ N ) )
=> ( finite_finite_nat @ N4 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1141_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M3: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_nat @ X3 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1142_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I3: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( P @ K )
& ( ord_less_nat @ K @ I3 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_1143_add__lessD1,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I3 @ J2 ) @ K2 )
=> ( ord_less_nat @ I3 @ K2 ) ) ).
% add_lessD1
thf(fact_1144_add__less__mono,axiom,
! [I3: nat,J2: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_less_mono
thf(fact_1145_not__add__less1,axiom,
! [I3: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I3 @ J2 ) @ I3 ) ).
% not_add_less1
thf(fact_1146_not__add__less2,axiom,
! [J2: nat,I3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I3 ) @ I3 ) ).
% not_add_less2
thf(fact_1147_add__less__mono1,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J2 @ K2 ) ) ) ).
% add_less_mono1
thf(fact_1148_trans__less__add1,axiom,
! [I3: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_less_add1
thf(fact_1149_trans__less__add2,axiom,
! [I3: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_less_add2
thf(fact_1150_less__add__eq__less,axiom,
! [K2: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1151_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1152_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1153_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1154_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1155_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1156_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1157_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1158_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K2: nat] :
( ! [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( ord_less_nat @ ( F @ M2 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K2 ) @ ( F @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1159_mult__less__mono2,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ K2 @ I3 ) @ ( times_times_nat @ K2 @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_1160_mult__less__mono1,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ I3 @ K2 ) @ ( times_times_nat @ J2 @ K2 ) ) ) ) ).
% mult_less_mono1
thf(fact_1161_nat__power__less__imp__less,axiom,
! [I3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I3 )
=> ( ( ord_less_nat @ ( power_power_nat @ I3 @ M ) @ ( power_power_nat @ I3 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1162_int__cases4,axiom,
! [M: int] :
( ! [N3: nat] :
( M
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_1163_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_1164_real__archimedian__rdiv__eq__0,axiom,
! [X5: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X5 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ X5 ) @ C ) )
=> ( X5 = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1165_nat__mult__le__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1166_nat__mult__less__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1167_finite__nth__roots,axiom,
! [N: nat,C: complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z3: complex] :
( ( power_power_complex @ Z3 @ N )
= C ) ) ) ) ).
% finite_nth_roots
thf(fact_1168_finite__interval__int4,axiom,
! [A2: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_int @ A2 @ I2 )
& ( ord_less_int @ I2 @ B ) ) ) ) ).
% finite_interval_int4
thf(fact_1169_finite__interval__int2,axiom,
! [A2: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_eq_int @ A2 @ I2 )
& ( ord_less_int @ I2 @ B ) ) ) ) ).
% finite_interval_int2
thf(fact_1170_finite__interval__int3,axiom,
! [A2: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I2: int] :
( ( ord_less_int @ A2 @ I2 )
& ( ord_less_eq_int @ I2 @ B ) ) ) ) ).
% finite_interval_int3
thf(fact_1171_zless__nat__conj,axiom,
! [W: int,Z: int] :
( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ( ord_less_int @ zero_zero_int @ Z )
& ( ord_less_int @ W @ Z ) ) ) ).
% zless_nat_conj
thf(fact_1172_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_1173_zero__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% zero_less_nat_eq
% Helper facts (9)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X5: int,Y: int] :
( ( if_int @ $false @ X5 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X5: int,Y: int] :
( ( if_int @ $true @ X5 @ Y )
= X5 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X5: nat,Y: nat] :
( ( if_nat @ $false @ X5 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X5: nat,Y: nat] :
( ( if_nat @ $true @ X5 @ Y )
= X5 ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X5: real,Y: real] :
( ( if_real @ $false @ X5 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X5: real,Y: real] :
( ( if_real @ $true @ X5 @ Y )
= X5 ) ).
thf(help_If_3_1_If_001t__Extended____Nonnegative____Real__Oennreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Extended____Nonnegative____Real__Oennreal_T,axiom,
! [X5: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( if_Ext9135588136721118450nnreal @ $false @ X5 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Extended____Nonnegative____Real__Oennreal_T,axiom,
! [X5: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( if_Ext9135588136721118450nnreal @ $true @ X5 @ Y )
= X5 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ord_less_eq_int @ zero_zero_int
@ ( groups7004213669654646580at_int
@ ^ [I: set_set_list_nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ ( finite2364142230527598318st_nat @ I ) @ one_one_nat ) ) @ ( semiri1314217659103216013at_int @ ( finite_card_list_nat @ ( comple184543376406953807st_nat @ I ) ) ) )
@ ( collec4691811733418234273st_nat
@ ^ [I: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ I @ ( image_8532145185254316925st_nat @ ( c @ n ) @ x ) )
& ( I != bot_bo3886227569956363488st_nat ) ) ) ) ) ).
%------------------------------------------------------------------------------