TPTP Problem File: SLH0918^1.p
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%------------------------------------------------------------------------------
% 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_01178_045801__13727842_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1486 ( 590 unt; 207 typ; 0 def)
% Number of atoms : 3585 (1541 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 11734 ( 376 ~; 39 |; 275 &;9596 @)
% ( 0 <=>;1448 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 14 ( 13 usr)
% Number of type conns : 1227 (1227 >; 0 *; 0 +; 0 <<)
% Number of symbols : 197 ( 194 usr; 20 con; 0-3 aty)
% Number of variables : 3978 ( 342 ^;3558 !; 78 ?;3978 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:15:58.925
%------------------------------------------------------------------------------
% Could-be-implicit typings (13)
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__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__List__Olist_It__Nat__Onat_J_J_J,type,
set_set_list_nat: $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_I_Eo_J_J,type,
set_set_o: $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__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (194)
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__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__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__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__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_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__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__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
finite5070363488328301092st_nat: set_set_set_list_nat > nat ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__Nat__Onat_J_001t__Int__Oint_001t__List__Olist_It__Nat__Onat_J,type,
comp_l2071074322182594513st_nat: ( list_nat > int ) > ( list_nat > list_nat ) > list_nat > int ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__Nat__Onat_J_001t__Int__Oint_001t__Nat__Onat,type,
comp_l4538574248832168257nt_nat: ( list_nat > int ) > ( nat > list_nat ) > nat > int ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
comp_l3539220536931113701at_nat: ( list_nat > nat ) > ( nat > list_nat ) > nat > nat ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__Nat__Onat_J,type,
comp_l6547402907347976123st_nat: ( list_nat > set_list_nat ) > ( list_nat > list_nat ) > list_nat > set_list_nat ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Int__Oint_001t__List__Olist_It__Nat__Onat_J,type,
comp_n779016382214257217st_nat: ( nat > int ) > ( list_nat > nat ) > list_nat > int ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Int__Oint_001t__Nat__Onat,type,
comp_nat_int_nat: ( nat > int ) > ( nat > nat ) > nat > int ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Int__Oint_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
comp_n6688923842565920887st_nat: ( nat > int ) > ( set_list_nat > nat ) > set_list_nat > int ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
comp_n3993578768073458149st_nat: ( nat > nat ) > ( list_nat > nat ) > list_nat > nat ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001_Eo_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__Int__Oint_001t__Nat__Onat,type,
comp_s4814495803593316215nt_nat: ( set_list_nat > int ) > ( nat > set_list_nat ) > nat > int ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__Nat__Onat_J,type,
comp_s803412145318520817st_nat: ( set_list_nat > set_list_nat ) > ( list_nat > set_list_nat ) > list_nat > set_list_nat ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_001t__List__Olist_It__Nat__Onat_J,type,
comp_s2760078235918225447st_nat: ( set_list_nat > set_set_list_nat ) > ( list_nat > set_list_nat ) > list_nat > set_set_list_nat ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
comp_s319613270319683165st_nat: ( set_list_nat > set_set_list_nat ) > ( set_list_nat > set_list_nat ) > set_list_nat > set_set_list_nat ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_001_Eo_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
comp_s1552796267380350143st_nat: ( set_set_list_nat > $o ) > ( set_list_nat > set_set_list_nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_001t__Int__Oint_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
comp_s3399862273134644595st_nat: ( set_set_list_nat > int ) > ( set_list_nat > set_set_list_nat ) > set_list_nat > int ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
comp_s4316523840537751389st_nat: ( set_set_list_nat > set_list_nat ) > ( set_list_nat > set_set_list_nat ) > set_list_nat > set_list_nat ).
thf(sy_c_Fun_Ocomp_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_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
comp_s2115431755894073235st_nat: ( set_set_list_nat > set_set_list_nat ) > ( set_list_nat > set_set_list_nat ) > set_list_nat > set_set_list_nat ).
thf(sy_c_Fun_Oinj__on_001_Eo_001t__List__Olist_It__Nat__Onat_J,type,
inj_on_o_list_nat: ( $o > list_nat ) > set_o > $o ).
thf(sy_c_Fun_Oinj__on_001_Eo_001t__Nat__Onat,type,
inj_on_o_nat: ( $o > nat ) > set_o > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__Int__Oint,type,
inj_on_list_nat_int: ( list_nat > int ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
inj_on3049792774292151987st_nat: ( list_nat > list_nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
inj_on_list_nat_nat: ( list_nat > nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
inj_on8624761805129053417st_nat: ( list_nat > set_list_nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Int__Oint,type,
inj_on_nat_int: ( nat > int ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
inj_on_nat_list_nat: ( nat > list_nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
inj_on2924389301302751961st_nat: ( nat > set_list_nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__Int__Oint,type,
inj_on1513901456888566197at_int: ( set_list_nat > int ) > set_set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__Nat__Onat_J,type,
inj_on6502772119031385577st_nat: ( set_list_nat > list_nat ) > set_set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__Nat__Onat,type,
inj_on1516391927397616473at_nat: ( set_list_nat > nat ) > set_set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
inj_on211238342484403029st_nat: ( set_list_nat > set_set_list_nat ) > set_set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_001t__Int__Oint,type,
inj_on2362802109445287147at_int: ( set_set_list_nat > int ) > set_set_set_list_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__List__Olist_It__Nat__Onat_J,type,
minus_minus_list_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_Eo_J,type,
minus_minus_set_o: set_o > set_o > set_o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
minus_1490147023502258853st_nat: set_set_list_nat > set_set_list_nat > set_set_list_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
minus_7520675481749758683st_nat: set_set_set_list_nat > set_set_set_list_nat > set_set_set_list_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J,type,
minus_1885081636186270737st_nat: set_se5258582372428582328st_nat > set_se5258582372428582328st_nat > set_se5258582372428582328st_nat ).
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_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__List__Olist_It__Nat__Onat_J,type,
plus_plus_list_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
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_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_I_Eo_J,type,
uminus_uminus_set_o: set_o > set_o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
uminus3195874150345416415st_nat: set_list_nat > set_list_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
uminus8693322751144048149st_nat: set_set_list_nat > set_set_list_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
uminus8338265383663789643st_nat: set_set_set_list_nat > set_set_set_list_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J,type,
uminus3509280191954366337st_nat: set_se5258582372428582328st_nat > set_se5258582372428582328st_nat ).
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__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_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__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__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__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__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__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
groups4561878855575611511st_nat: list_nat > nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Khovanskii_Opointwise__le,type,
pointwise_le: list_nat > list_nat > $o ).
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_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
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__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__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__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J,type,
bot_bo1158166727579713100st_nat: set_se5258582372428582328st_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__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,
bot_bo3510429811944912002st_nat: set_se7082751667426497518st_nat ).
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__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__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__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__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__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__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__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_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
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__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__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__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_OPow_001t__List__Olist_It__Nat__Onat_J,type,
pow_list_nat: set_list_nat > set_set_list_nat ).
thf(sy_c_Set_OPow_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
pow_set_list_nat: set_set_list_nat > set_set_set_list_nat ).
thf(sy_c_Set_OPow_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
pow_set_set_list_nat: set_set_set_list_nat > set_se5258582372428582328st_nat ).
thf(sy_c_Set_OPow_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
pow_se7925038334826946341st_nat: set_se5258582372428582328st_nat > 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__List__Olist_It__Nat__Onat_J,type,
image_o_list_nat: ( $o > list_nat ) > set_o > set_list_nat ).
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__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__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__List__Olist_It__Nat__Onat_J_J_J,type,
image_4575192727348083743st_nat: ( $o > set_set_list_nat ) > set_o > set_set_set_list_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
image_839146393129944149st_nat: ( $o > set_set_set_list_nat ) > set_o > set_se5258582372428582328st_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__Int__Oint,type,
image_list_nat_int: ( list_nat > int ) > set_list_nat > set_int ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
image_7976474329151083847st_nat: ( list_nat > list_nat ) > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
image_list_nat_nat: ( list_nat > nat ) > set_list_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_I_Eo_J,type,
image_list_nat_set_o: ( list_nat > set_o ) > set_list_nat > set_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__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_1775855109352712557et_nat: ( list_nat > set_nat ) > set_list_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
image_1427552562080725683st_nat: ( list_nat > set_set_list_nat ) > set_list_nat > set_set_set_list_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
image_4549367993744225001st_nat: ( list_nat > set_set_set_list_nat ) > set_list_nat > set_se5258582372428582328st_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__Int__Oint,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
image_nat_list_nat: ( nat > list_nat ) > set_nat > set_list_nat ).
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__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__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__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
image_3702150970216043171st_nat: ( nat > set_set_list_nat ) > set_nat > set_set_set_list_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
image_3522889415257180889st_nat: ( nat > set_set_set_list_nat ) > set_nat > set_se5258582372428582328st_nat ).
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_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
image_set_o_set_o: ( set_o > set_o ) > set_set_o > set_set_o ).
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__Int__Oint,type,
image_1472855193719607881at_int: ( set_list_nat > int ) > set_set_list_nat > set_int ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__Nat__Onat_J,type,
image_6410155499156649085st_nat: ( set_list_nat > list_nat ) > set_set_list_nat > set_list_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__Nat__Onat,type,
image_1475345664228658157at_nat: ( set_list_nat > nat ) > set_set_list_nat > set_nat ).
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__List__Olist_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7015791558925140899et_nat: ( set_list_nat > set_nat ) > set_set_list_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
image_920372530894776809st_nat: ( set_list_nat > set_set_list_nat ) > set_set_list_nat > set_set_set_list_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
image_463335885318752799st_nat: ( set_list_nat > set_set_set_list_nat ) > set_set_list_nat > set_se5258582372428582328st_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__Int__Oint,type,
image_8275292939657174399at_int: ( set_set_list_nat > int ) > set_set_set_list_nat > set_int ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_001t__List__Olist_It__Nat__Onat_J,type,
image_8319865106727365811st_nat: ( set_set_list_nat > list_nat ) > set_set_set_list_nat > set_list_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
image_5970735819588453097st_nat: ( set_set_list_nat > set_list_nat ) > set_set_set_list_nat > set_set_list_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_8873905749617817305et_nat: ( set_set_list_nat > set_nat ) > set_set_set_list_nat > set_set_nat ).
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__List__Olist_It__Nat__Onat_J_J_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
image_2453375416348523093st_nat: ( set_set_list_nat > set_set_set_list_nat ) > set_set_set_list_nat > set_se5258582372428582328st_nat ).
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__List__Olist_It__Nat__Onat_J_J_J_J_001t__List__Olist_It__Nat__Onat_J,type,
image_27328595625754089st_nat: ( set_set_set_list_nat > list_nat ) > set_se5258582372428582328st_nat > set_list_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
image_7505702526219074591st_nat: ( set_set_set_list_nat > set_list_nat ) > set_se5258582372428582328st_nat > set_set_list_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_5197920692104315663et_nat: ( set_set_set_list_nat > set_nat ) > set_se5258582372428582328st_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
image_904579742278570837st_nat: ( set_set_set_list_nat > set_set_list_nat ) > set_se5258582372428582328st_nat > set_set_set_list_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
image_5561069522181849995st_nat: ( set_set_set_list_nat > set_set_set_list_nat ) > set_se5258582372428582328st_nat > set_se5258582372428582328st_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J,type,
image_7413923974343677687st_nat: ( set_se5258582372428582328st_nat > set_se5258582372428582328st_nat ) > set_se7082751667426497518st_nat > set_se7082751667426497518st_nat ).
thf(sy_c_Set_Oinsert_001_Eo,type,
insert_o: $o > set_o > set_o ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
insert_list_nat: list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
insert_set_list_nat: set_list_nat > set_set_list_nat > set_set_list_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
insert721039030076969596st_nat: set_set_list_nat > set_set_set_list_nat > set_set_set_list_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
insert8008074249615255218st_nat: set_set_set_list_nat > set_se5258582372428582328st_nat > set_se5258582372428582328st_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J,type,
insert4845049477442163688st_nat: set_se5258582372428582328st_nat > set_se7082751667426497518st_nat > set_se7082751667426497518st_nat ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $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__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $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__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__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_c_member_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J_J_J,type,
member4920245652161177295st_nat: set_se5258582372428582328st_nat > set_se7082751667426497518st_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 (1275)
thf(fact_0__092_060open_062_092_060And_062A_O_AA_A_092_060subseteq_062_AX_A_092_060Longrightarrow_062_Ainj__on_A_IC_An_J_AA_092_060close_062,axiom,
! [A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ x )
=> ( inj_on8624761805129053417st_nat @ ( c @ n ) @ A ) ) ).
% \<open>\<And>A. A \<subseteq> X \<Longrightarrow> inj_on (C n) A\<close>
thf(fact_1_inj__C,axiom,
inj_on8624761805129053417st_nat @ ( c @ n ) @ x ).
% inj_C
thf(fact_2_SUM2__def,axiom,
( sUM2
= ( groups7312845317294741502at_int
@ ^ [Y: set_list_nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ ( finite_card_list_nat @ Y ) @ one_one_nat ) ) @ ( semiri1314217659103216013at_int @ ( finite_card_list_nat @ ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ ( c @ n ) @ Y ) ) ) ) )
@ ( collect_set_list_nat
@ ^ [Y: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ Y @ x )
& ( Y != bot_bot_set_list_nat ) ) ) ) ) ).
% SUM2_def
thf(fact_3_calculation,axiom,
( sUM1
= ( groups7312845317294741502at_int
@ ( comp_s3399862273134644595st_nat
@ ^ [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 ) ) ) )
@ ( image_8532145185254316925st_nat @ ( c @ n ) ) )
@ ( minus_1490147023502258853st_nat @ ( pow_list_nat @ x ) @ ( insert_set_list_nat @ bot_bot_set_list_nat @ bot_bo3886227569956363488st_nat ) ) ) ) ).
% calculation
thf(fact_4_Pow__empty,axiom,
( ( pow_se7925038334826946341st_nat @ bot_bo1158166727579713100st_nat )
= ( insert4845049477442163688st_nat @ bot_bo1158166727579713100st_nat @ bot_bo3510429811944912002st_nat ) ) ).
% Pow_empty
thf(fact_5_Pow__empty,axiom,
( ( pow_list_nat @ bot_bot_set_list_nat )
= ( insert_set_list_nat @ bot_bot_set_list_nat @ bot_bo3886227569956363488st_nat ) ) ).
% Pow_empty
thf(fact_6_Pow__empty,axiom,
( ( pow_set_list_nat @ bot_bo3886227569956363488st_nat )
= ( insert721039030076969596st_nat @ bot_bo3886227569956363488st_nat @ bot_bo3499706412017099030st_nat ) ) ).
% Pow_empty
thf(fact_7_Pow__empty,axiom,
( ( pow_set_set_list_nat @ bot_bo3499706412017099030st_nat )
= ( insert8008074249615255218st_nat @ bot_bo3499706412017099030st_nat @ bot_bo1158166727579713100st_nat ) ) ).
% Pow_empty
thf(fact_8_Pow__singleton__iff,axiom,
! [X: set_se5258582372428582328st_nat,Y2: set_se5258582372428582328st_nat] :
( ( ( pow_se7925038334826946341st_nat @ X )
= ( insert4845049477442163688st_nat @ Y2 @ bot_bo3510429811944912002st_nat ) )
= ( ( X = bot_bo1158166727579713100st_nat )
& ( Y2 = bot_bo1158166727579713100st_nat ) ) ) ).
% Pow_singleton_iff
thf(fact_9_Pow__singleton__iff,axiom,
! [X: set_set_set_list_nat,Y2: set_set_set_list_nat] :
( ( ( pow_set_set_list_nat @ X )
= ( insert8008074249615255218st_nat @ Y2 @ bot_bo1158166727579713100st_nat ) )
= ( ( X = bot_bo3499706412017099030st_nat )
& ( Y2 = bot_bo3499706412017099030st_nat ) ) ) ).
% Pow_singleton_iff
thf(fact_10_Pow__singleton__iff,axiom,
! [X: set_list_nat,Y2: set_list_nat] :
( ( ( pow_list_nat @ X )
= ( insert_set_list_nat @ Y2 @ bot_bo3886227569956363488st_nat ) )
= ( ( X = bot_bot_set_list_nat )
& ( Y2 = bot_bot_set_list_nat ) ) ) ).
% Pow_singleton_iff
thf(fact_11_Pow__singleton__iff,axiom,
! [X: set_set_list_nat,Y2: set_set_list_nat] :
( ( ( pow_set_list_nat @ X )
= ( insert721039030076969596st_nat @ Y2 @ bot_bo3499706412017099030st_nat ) )
= ( ( X = bot_bo3886227569956363488st_nat )
& ( Y2 = bot_bo3886227569956363488st_nat ) ) ) ).
% Pow_singleton_iff
thf(fact_12_sum__constant,axiom,
! [Y3: nat,A: set_set_set_list_nat] :
( ( groups7006704140163696856at_nat
@ ^ [X2: set_set_list_nat] : Y3
@ A )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite5070363488328301092st_nat @ A ) ) @ Y3 ) ) ).
% sum_constant
thf(fact_13_sum__constant,axiom,
! [Y3: nat,A: set_set_list_nat] :
( ( groups7315335787803791778at_nat
@ ^ [X2: set_list_nat] : Y3
@ A )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite2364142230527598318st_nat @ A ) ) @ Y3 ) ) ).
% sum_constant
thf(fact_14_sum__constant,axiom,
! [Y3: nat,A: set_list_nat] :
( ( groups4396056296759096172at_nat
@ ^ [X2: list_nat] : Y3
@ A )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_list_nat @ A ) ) @ Y3 ) ) ).
% sum_constant
thf(fact_15_sum__constant,axiom,
! [Y3: nat,A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : Y3
@ A )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_nat @ A ) ) @ Y3 ) ) ).
% sum_constant
thf(fact_16_sum__constant,axiom,
! [Y3: int,A: set_list_nat] :
( ( groups4393565826250045896at_int
@ ^ [X2: list_nat] : Y3
@ A )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_list_nat @ A ) ) @ Y3 ) ) ).
% sum_constant
thf(fact_17_sum__constant,axiom,
! [Y3: int,A: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [X2: nat] : Y3
@ A )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_nat @ A ) ) @ Y3 ) ) ).
% sum_constant
thf(fact_18_sum__constant,axiom,
! [Y3: int,A: set_set_list_nat] :
( ( groups7312845317294741502at_int
@ ^ [X2: set_list_nat] : Y3
@ A )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite2364142230527598318st_nat @ A ) ) @ Y3 ) ) ).
% sum_constant
thf(fact_19_sum__constant,axiom,
! [Y3: int,A: set_set_set_list_nat] :
( ( groups7004213669654646580at_int
@ ^ [X2: set_set_list_nat] : Y3
@ A )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite5070363488328301092st_nat @ A ) ) @ Y3 ) ) ).
% sum_constant
thf(fact_20_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_21_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_22_inj__on__imageC,axiom,
inj_on211238342484403029st_nat @ ( image_8532145185254316925st_nat @ ( c @ n ) ) @ ( minus_1490147023502258853st_nat @ ( pow_list_nat @ x ) @ ( insert_set_list_nat @ bot_bot_set_list_nat @ bot_bo3886227569956363488st_nat ) ) ).
% inj_on_imageC
thf(fact_23_INF__const,axiom,
! [A: set_set_list_nat,F: set_set_list_nat] :
( ( A != bot_bo3886227569956363488st_nat )
=> ( ( comple8462666950445340293st_nat
@ ( image_920372530894776809st_nat
@ ^ [I2: set_list_nat] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_24_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_25_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_26_INF__const,axiom,
! [A: set_set_set_list_nat,F: set_list_nat] :
( ( A != bot_bo3499706412017099030st_nat )
=> ( ( comple184543376406953807st_nat
@ ( image_5970735819588453097st_nat
@ ^ [I2: set_set_list_nat] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_27_INF__const,axiom,
! [A: set_se5258582372428582328st_nat,F: set_list_nat] :
( ( A != bot_bo1158166727579713100st_nat )
=> ( ( comple184543376406953807st_nat
@ ( image_7505702526219074591st_nat
@ ^ [I2: set_set_set_list_nat] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_28_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_29_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_30_INF__const,axiom,
! [A: set_set_set_list_nat,F: $o] :
( ( A != bot_bo3499706412017099030st_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_7778788071572860293_nat_o
@ ^ [I2: set_set_list_nat] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_31_INF__const,axiom,
! [A: set_se5258582372428582328st_nat,F: $o] :
( ( A != bot_bo1158166727579713100st_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_3035193717992370127_nat_o
@ ^ [I2: set_set_set_list_nat] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_32_ccINF__const,axiom,
! [A: set_set_list_nat,F: set_set_list_nat] :
( ( A != bot_bo3886227569956363488st_nat )
=> ( ( comple8462666950445340293st_nat
@ ( image_920372530894776809st_nat
@ ^ [I2: set_list_nat] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_33_ccINF__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 ) ) ).
% ccINF_const
thf(fact_34_ccINF__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 ) ) ).
% ccINF_const
thf(fact_35_ccINF__const,axiom,
! [A: set_set_set_list_nat,F: set_list_nat] :
( ( A != bot_bo3499706412017099030st_nat )
=> ( ( comple184543376406953807st_nat
@ ( image_5970735819588453097st_nat
@ ^ [I2: set_set_list_nat] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_36_ccINF__const,axiom,
! [A: set_se5258582372428582328st_nat,F: set_list_nat] :
( ( A != bot_bo1158166727579713100st_nat )
=> ( ( comple184543376406953807st_nat
@ ( image_7505702526219074591st_nat
@ ^ [I2: set_set_set_list_nat] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_37_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_38_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_39_ccINF__const,axiom,
! [A: set_set_set_list_nat,F: $o] :
( ( A != bot_bo3499706412017099030st_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_7778788071572860293_nat_o
@ ^ [I2: set_set_list_nat] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_40_ccINF__const,axiom,
! [A: set_se5258582372428582328st_nat,F: $o] :
( ( A != bot_bo1158166727579713100st_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_3035193717992370127_nat_o
@ ^ [I2: set_set_set_list_nat] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_41_cINF__const,axiom,
! [A: set_set_list_nat,C: set_set_list_nat] :
( ( A != bot_bo3886227569956363488st_nat )
=> ( ( comple8462666950445340293st_nat
@ ( image_920372530894776809st_nat
@ ^ [X2: set_list_nat] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_42_cINF__const,axiom,
! [A: set_list_nat,C: set_list_nat] :
( ( A != bot_bot_set_list_nat )
=> ( ( comple184543376406953807st_nat
@ ( image_8532145185254316925st_nat
@ ^ [X2: list_nat] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_43_cINF__const,axiom,
! [A: set_set_list_nat,C: set_list_nat] :
( ( A != bot_bo3886227569956363488st_nat )
=> ( ( comple184543376406953807st_nat
@ ( image_5143090206295581363st_nat
@ ^ [X2: set_list_nat] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_44_cINF__const,axiom,
! [A: set_set_set_list_nat,C: set_list_nat] :
( ( A != bot_bo3499706412017099030st_nat )
=> ( ( comple184543376406953807st_nat
@ ( image_5970735819588453097st_nat
@ ^ [X2: set_set_list_nat] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_45_cINF__const,axiom,
! [A: set_se5258582372428582328st_nat,C: set_list_nat] :
( ( A != bot_bo1158166727579713100st_nat )
=> ( ( comple184543376406953807st_nat
@ ( image_7505702526219074591st_nat
@ ^ [X2: set_set_set_list_nat] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_46_cINF__const,axiom,
! [A: set_list_nat,C: $o] :
( ( A != bot_bot_set_list_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_list_nat_o
@ ^ [X2: list_nat] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_47_cINF__const,axiom,
! [A: set_set_list_nat,C: $o] :
( ( A != bot_bo3886227569956363488st_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_set_list_nat_o
@ ^ [X2: set_list_nat] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_48_cINF__const,axiom,
! [A: set_set_set_list_nat,C: $o] :
( ( A != bot_bo3499706412017099030st_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_7778788071572860293_nat_o
@ ^ [X2: set_set_list_nat] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_49_cINF__const,axiom,
! [A: set_se5258582372428582328st_nat,C: $o] :
( ( A != bot_bo1158166727579713100st_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_3035193717992370127_nat_o
@ ^ [X2: set_set_set_list_nat] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_50_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_51_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_52_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_53_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_54_image__eqI,axiom,
! [B: nat,F: nat > nat,X3: nat,A: set_nat] :
( ( B
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_55_image__eqI,axiom,
! [B: $o,F: nat > $o,X3: nat,A: set_nat] :
( ( B
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member_o @ B @ ( image_nat_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_56_image__eqI,axiom,
! [B: nat,F: $o > nat,X3: $o,A: set_o] :
( ( B
= ( F @ X3 ) )
=> ( ( member_o @ X3 @ A )
=> ( member_nat @ B @ ( image_o_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_57_image__eqI,axiom,
! [B: $o,F: $o > $o,X3: $o,A: set_o] :
( ( B
= ( F @ X3 ) )
=> ( ( member_o @ X3 @ A )
=> ( member_o @ B @ ( image_o_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_58_image__eqI,axiom,
! [B: list_nat,F: nat > list_nat,X3: nat,A: set_nat] :
( ( B
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member_list_nat @ B @ ( image_nat_list_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_59_image__eqI,axiom,
! [B: list_nat,F: $o > list_nat,X3: $o,A: set_o] :
( ( B
= ( F @ X3 ) )
=> ( ( member_o @ X3 @ A )
=> ( member_list_nat @ B @ ( image_o_list_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_60_image__eqI,axiom,
! [B: nat,F: list_nat > nat,X3: list_nat,A: set_list_nat] :
( ( B
= ( F @ X3 ) )
=> ( ( member_list_nat @ X3 @ A )
=> ( member_nat @ B @ ( image_list_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_61_image__eqI,axiom,
! [B: $o,F: list_nat > $o,X3: list_nat,A: set_list_nat] :
( ( B
= ( F @ X3 ) )
=> ( ( member_list_nat @ X3 @ A )
=> ( member_o @ B @ ( image_list_nat_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_62_image__eqI,axiom,
! [B: list_nat,F: list_nat > list_nat,X3: list_nat,A: set_list_nat] :
( ( B
= ( F @ X3 ) )
=> ( ( member_list_nat @ X3 @ A )
=> ( member_list_nat @ B @ ( image_7976474329151083847st_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_63_image__eqI,axiom,
! [B: set_list_nat,F: list_nat > set_list_nat,X3: list_nat,A: set_list_nat] :
( ( B
= ( F @ X3 ) )
=> ( ( member_list_nat @ X3 @ A )
=> ( member_set_list_nat @ B @ ( image_8532145185254316925st_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_64_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_65_empty__Collect__eq,axiom,
! [P: list_nat > $o] :
( ( bot_bot_set_list_nat
= ( collect_list_nat @ P ) )
= ( ! [X2: list_nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_66_empty__Collect__eq,axiom,
! [P: set_list_nat > $o] :
( ( bot_bo3886227569956363488st_nat
= ( collect_set_list_nat @ P ) )
= ( ! [X2: set_list_nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_67_empty__Collect__eq,axiom,
! [P: set_set_list_nat > $o] :
( ( bot_bo3499706412017099030st_nat
= ( collec4691811733418234273st_nat @ P ) )
= ( ! [X2: set_set_list_nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_68_empty__Collect__eq,axiom,
! [P: set_set_set_list_nat > $o] :
( ( bot_bo1158166727579713100st_nat
= ( collec696328324557263319st_nat @ P ) )
= ( ! [X2: set_set_set_list_nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_69_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_70_Collect__empty__eq,axiom,
! [P: list_nat > $o] :
( ( ( collect_list_nat @ P )
= bot_bot_set_list_nat )
= ( ! [X2: list_nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_71_Collect__empty__eq,axiom,
! [P: set_list_nat > $o] :
( ( ( collect_set_list_nat @ P )
= bot_bo3886227569956363488st_nat )
= ( ! [X2: set_list_nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_72_Collect__empty__eq,axiom,
! [P: set_set_list_nat > $o] :
( ( ( collec4691811733418234273st_nat @ P )
= bot_bo3499706412017099030st_nat )
= ( ! [X2: set_set_list_nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_73_Collect__empty__eq,axiom,
! [P: set_set_set_list_nat > $o] :
( ( ( collec696328324557263319st_nat @ P )
= bot_bo1158166727579713100st_nat )
= ( ! [X2: set_set_set_list_nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_74_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_75_all__not__in__conv,axiom,
! [A: set_o] :
( ( ! [X2: $o] :
~ ( member_o @ X2 @ A ) )
= ( A = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_76_all__not__in__conv,axiom,
! [A: set_list_nat] :
( ( ! [X2: list_nat] :
~ ( member_list_nat @ X2 @ A ) )
= ( A = bot_bot_set_list_nat ) ) ).
% all_not_in_conv
thf(fact_77_all__not__in__conv,axiom,
! [A: set_set_list_nat] :
( ( ! [X2: set_list_nat] :
~ ( member_set_list_nat @ X2 @ A ) )
= ( A = bot_bo3886227569956363488st_nat ) ) ).
% all_not_in_conv
thf(fact_78_all__not__in__conv,axiom,
! [A: set_set_set_list_nat] :
( ( ! [X2: set_set_list_nat] :
~ ( member1029098694177496419st_nat @ X2 @ A ) )
= ( A = bot_bo3499706412017099030st_nat ) ) ).
% all_not_in_conv
thf(fact_79_all__not__in__conv,axiom,
! [A: set_se5258582372428582328st_nat] :
( ( ! [X2: set_set_set_list_nat] :
~ ( member7304678173793621401st_nat @ X2 @ A ) )
= ( A = bot_bo1158166727579713100st_nat ) ) ).
% all_not_in_conv
thf(fact_80_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_81_empty__iff,axiom,
! [C: $o] :
~ ( member_o @ C @ bot_bot_set_o ) ).
% empty_iff
thf(fact_82_empty__iff,axiom,
! [C: list_nat] :
~ ( member_list_nat @ C @ bot_bot_set_list_nat ) ).
% empty_iff
thf(fact_83_empty__iff,axiom,
! [C: set_list_nat] :
~ ( member_set_list_nat @ C @ bot_bo3886227569956363488st_nat ) ).
% empty_iff
thf(fact_84_empty__iff,axiom,
! [C: set_set_list_nat] :
~ ( member1029098694177496419st_nat @ C @ bot_bo3499706412017099030st_nat ) ).
% empty_iff
thf(fact_85_empty__iff,axiom,
! [C: set_set_set_list_nat] :
~ ( member7304678173793621401st_nat @ C @ bot_bo1158166727579713100st_nat ) ).
% empty_iff
thf(fact_86_Compl__subset__Compl__iff,axiom,
! [A: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( uminus3195874150345416415st_nat @ A ) @ ( uminus3195874150345416415st_nat @ B2 ) )
= ( ord_le6045566169113846134st_nat @ B2 @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_87_Compl__subset__Compl__iff,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ ( uminus8693322751144048149st_nat @ A ) @ ( uminus8693322751144048149st_nat @ B2 ) )
= ( ord_le1068707526560357548st_nat @ B2 @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_88_Compl__subset__Compl__iff,axiom,
! [A: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ ( uminus8338265383663789643st_nat @ A ) @ ( uminus8338265383663789643st_nat @ B2 ) )
= ( ord_le7100322305783427298st_nat @ B2 @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_89_Compl__anti__mono,axiom,
! [A: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B2 )
=> ( ord_le6045566169113846134st_nat @ ( uminus3195874150345416415st_nat @ B2 ) @ ( uminus3195874150345416415st_nat @ A ) ) ) ).
% Compl_anti_mono
thf(fact_90_Compl__anti__mono,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A @ B2 )
=> ( ord_le1068707526560357548st_nat @ ( uminus8693322751144048149st_nat @ B2 ) @ ( uminus8693322751144048149st_nat @ A ) ) ) ).
% Compl_anti_mono
thf(fact_91_Compl__anti__mono,axiom,
! [A: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A @ B2 )
=> ( ord_le7100322305783427298st_nat @ ( uminus8338265383663789643st_nat @ B2 ) @ ( uminus8338265383663789643st_nat @ A ) ) ) ).
% Compl_anti_mono
thf(fact_92_subset__antisym,axiom,
! [A: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B2 )
=> ( ( ord_le6045566169113846134st_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_93_subset__antisym,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A @ B2 )
=> ( ( ord_le1068707526560357548st_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_94_subset__antisym,axiom,
! [A: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A @ B2 )
=> ( ( ord_le7100322305783427298st_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_95_subsetI,axiom,
! [A: set_nat,B2: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ X4 @ B2 ) )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_96_subsetI,axiom,
! [A: set_o,B2: set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( member_o @ X4 @ B2 ) )
=> ( ord_less_eq_set_o @ A @ B2 ) ) ).
% subsetI
thf(fact_97_subsetI,axiom,
! [A: set_list_nat,B2: set_list_nat] :
( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
=> ( member_list_nat @ X4 @ B2 ) )
=> ( ord_le6045566169113846134st_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_98_subsetI,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat] :
( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ A )
=> ( member_set_list_nat @ X4 @ B2 ) )
=> ( ord_le1068707526560357548st_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_99_subsetI,axiom,
! [A: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ! [X4: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X4 @ A )
=> ( member1029098694177496419st_nat @ X4 @ B2 ) )
=> ( ord_le7100322305783427298st_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_100_insert__absorb2,axiom,
! [X3: set_list_nat,A: set_set_list_nat] :
( ( insert_set_list_nat @ X3 @ ( insert_set_list_nat @ X3 @ A ) )
= ( insert_set_list_nat @ X3 @ A ) ) ).
% insert_absorb2
thf(fact_101_insert__absorb2,axiom,
! [X3: set_set_list_nat,A: set_set_set_list_nat] :
( ( insert721039030076969596st_nat @ X3 @ ( insert721039030076969596st_nat @ X3 @ A ) )
= ( insert721039030076969596st_nat @ X3 @ A ) ) ).
% insert_absorb2
thf(fact_102_insert__absorb2,axiom,
! [X3: set_set_set_list_nat,A: set_se5258582372428582328st_nat] :
( ( insert8008074249615255218st_nat @ X3 @ ( insert8008074249615255218st_nat @ X3 @ A ) )
= ( insert8008074249615255218st_nat @ X3 @ A ) ) ).
% insert_absorb2
thf(fact_103_insert__iff,axiom,
! [A2: set_list_nat,B: set_list_nat,A: set_set_list_nat] :
( ( member_set_list_nat @ A2 @ ( insert_set_list_nat @ B @ A ) )
= ( ( A2 = B )
| ( member_set_list_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_104_insert__iff,axiom,
! [A2: set_set_list_nat,B: set_set_list_nat,A: set_set_set_list_nat] :
( ( member1029098694177496419st_nat @ A2 @ ( insert721039030076969596st_nat @ B @ A ) )
= ( ( A2 = B )
| ( member1029098694177496419st_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_105_insert__iff,axiom,
! [A2: set_set_set_list_nat,B: set_set_set_list_nat,A: set_se5258582372428582328st_nat] :
( ( member7304678173793621401st_nat @ A2 @ ( insert8008074249615255218st_nat @ B @ A ) )
= ( ( A2 = B )
| ( member7304678173793621401st_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_106_insert__iff,axiom,
! [A2: nat,B: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
= ( ( A2 = B )
| ( member_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_107_insert__iff,axiom,
! [A2: $o,B: $o,A: set_o] :
( ( member_o @ A2 @ ( insert_o @ B @ A ) )
= ( ( A2 = B )
| ( member_o @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_108_insert__iff,axiom,
! [A2: list_nat,B: list_nat,A: set_list_nat] :
( ( member_list_nat @ A2 @ ( insert_list_nat @ B @ A ) )
= ( ( A2 = B )
| ( member_list_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_109_insertCI,axiom,
! [A2: set_list_nat,B2: set_set_list_nat,B: set_list_nat] :
( ( ~ ( member_set_list_nat @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_set_list_nat @ A2 @ ( insert_set_list_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_110_insertCI,axiom,
! [A2: set_set_list_nat,B2: set_set_set_list_nat,B: set_set_list_nat] :
( ( ~ ( member1029098694177496419st_nat @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member1029098694177496419st_nat @ A2 @ ( insert721039030076969596st_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_111_insertCI,axiom,
! [A2: set_set_set_list_nat,B2: set_se5258582372428582328st_nat,B: set_set_set_list_nat] :
( ( ~ ( member7304678173793621401st_nat @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member7304678173793621401st_nat @ A2 @ ( insert8008074249615255218st_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_112_insertCI,axiom,
! [A2: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_113_insertCI,axiom,
! [A2: $o,B2: set_o,B: $o] :
( ( ~ ( member_o @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_o @ A2 @ ( insert_o @ B @ B2 ) ) ) ).
% insertCI
thf(fact_114_insertCI,axiom,
! [A2: list_nat,B2: set_list_nat,B: list_nat] :
( ( ~ ( member_list_nat @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_list_nat @ A2 @ ( insert_list_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_115_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_116_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= ( semiri1316708129612266289at_nat @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_117_Diff__idemp,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat] :
( ( minus_1490147023502258853st_nat @ ( minus_1490147023502258853st_nat @ A @ B2 ) @ B2 )
= ( minus_1490147023502258853st_nat @ A @ B2 ) ) ).
% Diff_idemp
thf(fact_118_Diff__idemp,axiom,
! [A: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ( minus_7520675481749758683st_nat @ ( minus_7520675481749758683st_nat @ A @ B2 ) @ B2 )
= ( minus_7520675481749758683st_nat @ A @ B2 ) ) ).
% Diff_idemp
thf(fact_119_Diff__iff,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) )
= ( ( member_nat @ C @ A )
& ~ ( member_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_120_Diff__iff,axiom,
! [C: $o,A: set_o,B2: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A @ B2 ) )
= ( ( member_o @ C @ A )
& ~ ( member_o @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_121_Diff__iff,axiom,
! [C: list_nat,A: set_list_nat,B2: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A @ B2 ) )
= ( ( member_list_nat @ C @ A )
& ~ ( member_list_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_122_Diff__iff,axiom,
! [C: set_list_nat,A: set_set_list_nat,B2: set_set_list_nat] :
( ( member_set_list_nat @ C @ ( minus_1490147023502258853st_nat @ A @ B2 ) )
= ( ( member_set_list_nat @ C @ A )
& ~ ( member_set_list_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_123_Diff__iff,axiom,
! [C: set_set_list_nat,A: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ( member1029098694177496419st_nat @ C @ ( minus_7520675481749758683st_nat @ A @ B2 ) )
= ( ( member1029098694177496419st_nat @ C @ A )
& ~ ( member1029098694177496419st_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_124_DiffI,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ A )
=> ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_125_DiffI,axiom,
! [C: $o,A: set_o,B2: set_o] :
( ( member_o @ C @ A )
=> ( ~ ( member_o @ C @ B2 )
=> ( member_o @ C @ ( minus_minus_set_o @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_126_DiffI,axiom,
! [C: list_nat,A: set_list_nat,B2: set_list_nat] :
( ( member_list_nat @ C @ A )
=> ( ~ ( member_list_nat @ C @ B2 )
=> ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_127_DiffI,axiom,
! [C: set_list_nat,A: set_set_list_nat,B2: set_set_list_nat] :
( ( member_set_list_nat @ C @ A )
=> ( ~ ( member_set_list_nat @ C @ B2 )
=> ( member_set_list_nat @ C @ ( minus_1490147023502258853st_nat @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_128_DiffI,axiom,
! [C: set_set_list_nat,A: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ( member1029098694177496419st_nat @ C @ A )
=> ( ~ ( member1029098694177496419st_nat @ C @ B2 )
=> ( member1029098694177496419st_nat @ C @ ( minus_7520675481749758683st_nat @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_129_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_130_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_131_Inter__iff,axiom,
! [A: nat,C2: set_set_nat] :
( ( member_nat @ A @ ( comple7806235888213564991et_nat @ C2 ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ C2 )
=> ( member_nat @ A @ X2 ) ) ) ) ).
% Inter_iff
thf(fact_132_Inter__iff,axiom,
! [A: $o,C2: set_set_o] :
( ( member_o @ A @ ( comple3063163877087187839_set_o @ C2 ) )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ C2 )
=> ( member_o @ A @ X2 ) ) ) ) ).
% Inter_iff
thf(fact_133_Inter__iff,axiom,
! [A: list_nat,C2: set_set_list_nat] :
( ( member_list_nat @ A @ ( comple184543376406953807st_nat @ C2 ) )
= ( ! [X2: set_list_nat] :
( ( member_set_list_nat @ X2 @ C2 )
=> ( member_list_nat @ A @ X2 ) ) ) ) ).
% Inter_iff
thf(fact_134_InterI,axiom,
! [C2: set_set_nat,A: nat] :
( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ C2 )
=> ( member_nat @ A @ X5 ) )
=> ( member_nat @ A @ ( comple7806235888213564991et_nat @ C2 ) ) ) ).
% InterI
thf(fact_135_InterI,axiom,
! [C2: set_set_o,A: $o] :
( ! [X5: set_o] :
( ( member_set_o @ X5 @ C2 )
=> ( member_o @ A @ X5 ) )
=> ( member_o @ A @ ( comple3063163877087187839_set_o @ C2 ) ) ) ).
% InterI
thf(fact_136_InterI,axiom,
! [C2: set_set_list_nat,A: list_nat] :
( ! [X5: set_list_nat] :
( ( member_set_list_nat @ X5 @ C2 )
=> ( member_list_nat @ A @ X5 ) )
=> ( member_list_nat @ A @ ( comple184543376406953807st_nat @ C2 ) ) ) ).
% InterI
thf(fact_137_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_138_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_139_image__is__empty,axiom,
! [F: list_nat > list_nat,A: set_list_nat] :
( ( ( image_7976474329151083847st_nat @ F @ A )
= bot_bot_set_list_nat )
= ( A = bot_bot_set_list_nat ) ) ).
% image_is_empty
thf(fact_140_image__is__empty,axiom,
! [F: set_list_nat > list_nat,A: set_set_list_nat] :
( ( ( image_6410155499156649085st_nat @ F @ A )
= bot_bot_set_list_nat )
= ( A = bot_bo3886227569956363488st_nat ) ) ).
% image_is_empty
thf(fact_141_image__is__empty,axiom,
! [F: list_nat > set_list_nat,A: set_list_nat] :
( ( ( image_8532145185254316925st_nat @ F @ A )
= bot_bo3886227569956363488st_nat )
= ( A = bot_bot_set_list_nat ) ) ).
% image_is_empty
thf(fact_142_image__is__empty,axiom,
! [F: set_set_list_nat > list_nat,A: set_set_set_list_nat] :
( ( ( image_8319865106727365811st_nat @ F @ A )
= bot_bot_set_list_nat )
= ( A = bot_bo3499706412017099030st_nat ) ) ).
% image_is_empty
thf(fact_143_image__is__empty,axiom,
! [F: set_list_nat > set_list_nat,A: set_set_list_nat] :
( ( ( image_5143090206295581363st_nat @ F @ A )
= bot_bo3886227569956363488st_nat )
= ( A = bot_bo3886227569956363488st_nat ) ) ).
% image_is_empty
thf(fact_144_image__is__empty,axiom,
! [F: list_nat > set_set_list_nat,A: set_list_nat] :
( ( ( image_1427552562080725683st_nat @ F @ A )
= bot_bo3499706412017099030st_nat )
= ( A = bot_bot_set_list_nat ) ) ).
% image_is_empty
thf(fact_145_image__is__empty,axiom,
! [F: set_set_set_list_nat > list_nat,A: set_se5258582372428582328st_nat] :
( ( ( image_27328595625754089st_nat @ F @ A )
= bot_bot_set_list_nat )
= ( A = bot_bo1158166727579713100st_nat ) ) ).
% image_is_empty
thf(fact_146_image__is__empty,axiom,
! [F: set_set_list_nat > set_list_nat,A: set_set_set_list_nat] :
( ( ( image_5970735819588453097st_nat @ F @ A )
= bot_bo3886227569956363488st_nat )
= ( A = bot_bo3499706412017099030st_nat ) ) ).
% image_is_empty
thf(fact_147_image__is__empty,axiom,
! [F: set_list_nat > set_set_list_nat,A: set_set_list_nat] :
( ( ( image_920372530894776809st_nat @ F @ A )
= bot_bo3499706412017099030st_nat )
= ( A = bot_bo3886227569956363488st_nat ) ) ).
% image_is_empty
thf(fact_148_image__is__empty,axiom,
! [F: list_nat > set_set_set_list_nat,A: set_list_nat] :
( ( ( image_4549367993744225001st_nat @ F @ A )
= bot_bo1158166727579713100st_nat )
= ( A = bot_bot_set_list_nat ) ) ).
% image_is_empty
thf(fact_149_empty__is__image,axiom,
! [F: list_nat > list_nat,A: set_list_nat] :
( ( bot_bot_set_list_nat
= ( image_7976474329151083847st_nat @ F @ A ) )
= ( A = bot_bot_set_list_nat ) ) ).
% empty_is_image
thf(fact_150_empty__is__image,axiom,
! [F: set_list_nat > list_nat,A: set_set_list_nat] :
( ( bot_bot_set_list_nat
= ( image_6410155499156649085st_nat @ F @ A ) )
= ( A = bot_bo3886227569956363488st_nat ) ) ).
% empty_is_image
thf(fact_151_empty__is__image,axiom,
! [F: list_nat > set_list_nat,A: set_list_nat] :
( ( bot_bo3886227569956363488st_nat
= ( image_8532145185254316925st_nat @ F @ A ) )
= ( A = bot_bot_set_list_nat ) ) ).
% empty_is_image
thf(fact_152_empty__is__image,axiom,
! [F: set_set_list_nat > list_nat,A: set_set_set_list_nat] :
( ( bot_bot_set_list_nat
= ( image_8319865106727365811st_nat @ F @ A ) )
= ( A = bot_bo3499706412017099030st_nat ) ) ).
% empty_is_image
thf(fact_153_empty__is__image,axiom,
! [F: set_list_nat > set_list_nat,A: set_set_list_nat] :
( ( bot_bo3886227569956363488st_nat
= ( image_5143090206295581363st_nat @ F @ A ) )
= ( A = bot_bo3886227569956363488st_nat ) ) ).
% empty_is_image
thf(fact_154_empty__is__image,axiom,
! [F: list_nat > set_set_list_nat,A: set_list_nat] :
( ( bot_bo3499706412017099030st_nat
= ( image_1427552562080725683st_nat @ F @ A ) )
= ( A = bot_bot_set_list_nat ) ) ).
% empty_is_image
thf(fact_155_empty__is__image,axiom,
! [F: set_set_set_list_nat > list_nat,A: set_se5258582372428582328st_nat] :
( ( bot_bot_set_list_nat
= ( image_27328595625754089st_nat @ F @ A ) )
= ( A = bot_bo1158166727579713100st_nat ) ) ).
% empty_is_image
thf(fact_156_empty__is__image,axiom,
! [F: set_set_list_nat > set_list_nat,A: set_set_set_list_nat] :
( ( bot_bo3886227569956363488st_nat
= ( image_5970735819588453097st_nat @ F @ A ) )
= ( A = bot_bo3499706412017099030st_nat ) ) ).
% empty_is_image
thf(fact_157_empty__is__image,axiom,
! [F: set_list_nat > set_set_list_nat,A: set_set_list_nat] :
( ( bot_bo3499706412017099030st_nat
= ( image_920372530894776809st_nat @ F @ A ) )
= ( A = bot_bo3886227569956363488st_nat ) ) ).
% empty_is_image
thf(fact_158_empty__is__image,axiom,
! [F: list_nat > set_set_set_list_nat,A: set_list_nat] :
( ( bot_bo1158166727579713100st_nat
= ( image_4549367993744225001st_nat @ F @ A ) )
= ( A = bot_bot_set_list_nat ) ) ).
% empty_is_image
thf(fact_159_image__empty,axiom,
! [F: list_nat > list_nat] :
( ( image_7976474329151083847st_nat @ F @ bot_bot_set_list_nat )
= bot_bot_set_list_nat ) ).
% image_empty
thf(fact_160_image__empty,axiom,
! [F: list_nat > set_list_nat] :
( ( image_8532145185254316925st_nat @ F @ bot_bot_set_list_nat )
= bot_bo3886227569956363488st_nat ) ).
% image_empty
thf(fact_161_image__empty,axiom,
! [F: set_list_nat > list_nat] :
( ( image_6410155499156649085st_nat @ F @ bot_bo3886227569956363488st_nat )
= bot_bot_set_list_nat ) ).
% image_empty
thf(fact_162_image__empty,axiom,
! [F: list_nat > set_set_list_nat] :
( ( image_1427552562080725683st_nat @ F @ bot_bot_set_list_nat )
= bot_bo3499706412017099030st_nat ) ).
% image_empty
thf(fact_163_image__empty,axiom,
! [F: set_list_nat > set_list_nat] :
( ( image_5143090206295581363st_nat @ F @ bot_bo3886227569956363488st_nat )
= bot_bo3886227569956363488st_nat ) ).
% image_empty
thf(fact_164_image__empty,axiom,
! [F: set_set_list_nat > list_nat] :
( ( image_8319865106727365811st_nat @ F @ bot_bo3499706412017099030st_nat )
= bot_bot_set_list_nat ) ).
% image_empty
thf(fact_165_image__empty,axiom,
! [F: list_nat > set_set_set_list_nat] :
( ( image_4549367993744225001st_nat @ F @ bot_bot_set_list_nat )
= bot_bo1158166727579713100st_nat ) ).
% image_empty
thf(fact_166_image__empty,axiom,
! [F: set_list_nat > set_set_list_nat] :
( ( image_920372530894776809st_nat @ F @ bot_bo3886227569956363488st_nat )
= bot_bo3499706412017099030st_nat ) ).
% image_empty
thf(fact_167_image__empty,axiom,
! [F: set_set_list_nat > set_list_nat] :
( ( image_5970735819588453097st_nat @ F @ bot_bo3499706412017099030st_nat )
= bot_bo3886227569956363488st_nat ) ).
% image_empty
thf(fact_168_image__empty,axiom,
! [F: set_set_set_list_nat > list_nat] :
( ( image_27328595625754089st_nat @ F @ bot_bo1158166727579713100st_nat )
= bot_bot_set_list_nat ) ).
% image_empty
thf(fact_169_empty__subsetI,axiom,
! [A: set_se5258582372428582328st_nat] : ( ord_le2499698639687704088st_nat @ bot_bo1158166727579713100st_nat @ A ) ).
% empty_subsetI
thf(fact_170_empty__subsetI,axiom,
! [A: set_list_nat] : ( ord_le6045566169113846134st_nat @ bot_bot_set_list_nat @ A ) ).
% empty_subsetI
thf(fact_171_empty__subsetI,axiom,
! [A: set_set_list_nat] : ( ord_le1068707526560357548st_nat @ bot_bo3886227569956363488st_nat @ A ) ).
% empty_subsetI
thf(fact_172_empty__subsetI,axiom,
! [A: set_set_set_list_nat] : ( ord_le7100322305783427298st_nat @ bot_bo3499706412017099030st_nat @ A ) ).
% empty_subsetI
thf(fact_173_subset__empty,axiom,
! [A: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ A @ bot_bo1158166727579713100st_nat )
= ( A = bot_bo1158166727579713100st_nat ) ) ).
% subset_empty
thf(fact_174_subset__empty,axiom,
! [A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ bot_bot_set_list_nat )
= ( A = bot_bot_set_list_nat ) ) ).
% subset_empty
thf(fact_175_subset__empty,axiom,
! [A: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A @ bot_bo3886227569956363488st_nat )
= ( A = bot_bo3886227569956363488st_nat ) ) ).
% subset_empty
thf(fact_176_subset__empty,axiom,
! [A: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A @ bot_bo3499706412017099030st_nat )
= ( A = bot_bo3499706412017099030st_nat ) ) ).
% subset_empty
thf(fact_177_insert__image,axiom,
! [X3: set_list_nat,A: set_set_list_nat,F: set_list_nat > set_set_list_nat] :
( ( member_set_list_nat @ X3 @ A )
=> ( ( insert721039030076969596st_nat @ ( F @ X3 ) @ ( image_920372530894776809st_nat @ F @ A ) )
= ( image_920372530894776809st_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_178_insert__image,axiom,
! [X3: nat,A: set_nat,F: nat > set_list_nat] :
( ( member_nat @ X3 @ A )
=> ( ( insert_set_list_nat @ ( F @ X3 ) @ ( image_2883343038133793645st_nat @ F @ A ) )
= ( image_2883343038133793645st_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_179_insert__image,axiom,
! [X3: nat,A: set_nat,F: nat > set_set_list_nat] :
( ( member_nat @ X3 @ A )
=> ( ( insert721039030076969596st_nat @ ( F @ X3 ) @ ( image_3702150970216043171st_nat @ F @ A ) )
= ( image_3702150970216043171st_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_180_insert__image,axiom,
! [X3: nat,A: set_nat,F: nat > set_set_set_list_nat] :
( ( member_nat @ X3 @ A )
=> ( ( insert8008074249615255218st_nat @ ( F @ X3 ) @ ( image_3522889415257180889st_nat @ F @ A ) )
= ( image_3522889415257180889st_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_181_insert__image,axiom,
! [X3: $o,A: set_o,F: $o > set_list_nat] :
( ( member_o @ X3 @ A )
=> ( ( insert_set_list_nat @ ( F @ X3 ) @ ( image_o_set_list_nat @ F @ A ) )
= ( image_o_set_list_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_182_insert__image,axiom,
! [X3: $o,A: set_o,F: $o > set_set_list_nat] :
( ( member_o @ X3 @ A )
=> ( ( insert721039030076969596st_nat @ ( F @ X3 ) @ ( image_4575192727348083743st_nat @ F @ A ) )
= ( image_4575192727348083743st_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_183_insert__image,axiom,
! [X3: $o,A: set_o,F: $o > set_set_set_list_nat] :
( ( member_o @ X3 @ A )
=> ( ( insert8008074249615255218st_nat @ ( F @ X3 ) @ ( image_839146393129944149st_nat @ F @ A ) )
= ( image_839146393129944149st_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_184_insert__image,axiom,
! [X3: list_nat,A: set_list_nat,F: list_nat > set_list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( ( insert_set_list_nat @ ( F @ X3 ) @ ( image_8532145185254316925st_nat @ F @ A ) )
= ( image_8532145185254316925st_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_185_insert__image,axiom,
! [X3: list_nat,A: set_list_nat,F: list_nat > set_set_list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( ( insert721039030076969596st_nat @ ( F @ X3 ) @ ( image_1427552562080725683st_nat @ F @ A ) )
= ( image_1427552562080725683st_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_186_insert__image,axiom,
! [X3: list_nat,A: set_list_nat,F: list_nat > set_set_set_list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( ( insert8008074249615255218st_nat @ ( F @ X3 ) @ ( image_4549367993744225001st_nat @ F @ A ) )
= ( image_4549367993744225001st_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_187_image__insert,axiom,
! [F: list_nat > set_list_nat,A2: list_nat,B2: set_list_nat] :
( ( image_8532145185254316925st_nat @ F @ ( insert_list_nat @ A2 @ B2 ) )
= ( insert_set_list_nat @ ( F @ A2 ) @ ( image_8532145185254316925st_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_188_image__insert,axiom,
! [F: set_list_nat > set_list_nat,A2: set_list_nat,B2: set_set_list_nat] :
( ( image_5143090206295581363st_nat @ F @ ( insert_set_list_nat @ A2 @ B2 ) )
= ( insert_set_list_nat @ ( F @ A2 ) @ ( image_5143090206295581363st_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_189_image__insert,axiom,
! [F: set_list_nat > set_set_list_nat,A2: set_list_nat,B2: set_set_list_nat] :
( ( image_920372530894776809st_nat @ F @ ( insert_set_list_nat @ A2 @ B2 ) )
= ( insert721039030076969596st_nat @ ( F @ A2 ) @ ( image_920372530894776809st_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_190_image__insert,axiom,
! [F: set_list_nat > set_set_set_list_nat,A2: set_list_nat,B2: set_set_list_nat] :
( ( image_463335885318752799st_nat @ F @ ( insert_set_list_nat @ A2 @ B2 ) )
= ( insert8008074249615255218st_nat @ ( F @ A2 ) @ ( image_463335885318752799st_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_191_image__insert,axiom,
! [F: set_set_list_nat > set_list_nat,A2: set_set_list_nat,B2: set_set_set_list_nat] :
( ( image_5970735819588453097st_nat @ F @ ( insert721039030076969596st_nat @ A2 @ B2 ) )
= ( insert_set_list_nat @ ( F @ A2 ) @ ( image_5970735819588453097st_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_192_image__insert,axiom,
! [F: set_set_list_nat > set_set_list_nat,A2: set_set_list_nat,B2: set_set_set_list_nat] :
( ( image_799823415861195295st_nat @ F @ ( insert721039030076969596st_nat @ A2 @ B2 ) )
= ( insert721039030076969596st_nat @ ( F @ A2 ) @ ( image_799823415861195295st_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_193_image__insert,axiom,
! [F: set_set_list_nat > set_set_set_list_nat,A2: set_set_list_nat,B2: set_set_set_list_nat] :
( ( image_2453375416348523093st_nat @ F @ ( insert721039030076969596st_nat @ A2 @ B2 ) )
= ( insert8008074249615255218st_nat @ ( F @ A2 ) @ ( image_2453375416348523093st_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_194_image__insert,axiom,
! [F: set_set_set_list_nat > set_list_nat,A2: set_set_set_list_nat,B2: set_se5258582372428582328st_nat] :
( ( image_7505702526219074591st_nat @ F @ ( insert8008074249615255218st_nat @ A2 @ B2 ) )
= ( insert_set_list_nat @ ( F @ A2 ) @ ( image_7505702526219074591st_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_195_image__insert,axiom,
! [F: set_set_set_list_nat > set_set_list_nat,A2: set_set_set_list_nat,B2: set_se5258582372428582328st_nat] :
( ( image_904579742278570837st_nat @ F @ ( insert8008074249615255218st_nat @ A2 @ B2 ) )
= ( insert721039030076969596st_nat @ ( F @ A2 ) @ ( image_904579742278570837st_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_196_image__insert,axiom,
! [F: set_set_set_list_nat > set_set_set_list_nat,A2: set_set_set_list_nat,B2: set_se5258582372428582328st_nat] :
( ( image_5561069522181849995st_nat @ F @ ( insert8008074249615255218st_nat @ A2 @ B2 ) )
= ( insert8008074249615255218st_nat @ ( F @ A2 ) @ ( image_5561069522181849995st_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_197_mem__Collect__eq,axiom,
! [A2: $o,P: $o > $o] :
( ( member_o @ A2 @ ( collect_o @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_198_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_199_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_200_mem__Collect__eq,axiom,
! [A2: list_nat,P: list_nat > $o] :
( ( member_list_nat @ A2 @ ( collect_list_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_201_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_202_Collect__mem__eq,axiom,
! [A: set_o] :
( ( collect_o
@ ^ [X2: $o] : ( member_o @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_203_Collect__mem__eq,axiom,
! [A: set_set_list_nat] :
( ( collect_set_list_nat
@ ^ [X2: set_list_nat] : ( member_set_list_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_204_Collect__mem__eq,axiom,
! [A: set_set_set_list_nat] :
( ( collec4691811733418234273st_nat
@ ^ [X2: set_set_list_nat] : ( member1029098694177496419st_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_205_Collect__mem__eq,axiom,
! [A: set_list_nat] :
( ( collect_list_nat
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_206_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_207_Collect__cong,axiom,
! [P: set_list_nat > $o,Q: set_list_nat > $o] :
( ! [X4: set_list_nat] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_set_list_nat @ P )
= ( collect_set_list_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_208_Collect__cong,axiom,
! [P: set_set_list_nat > $o,Q: set_set_list_nat > $o] :
( ! [X4: set_set_list_nat] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collec4691811733418234273st_nat @ P )
= ( collec4691811733418234273st_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_209_Collect__cong,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X4: list_nat] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_list_nat @ P )
= ( collect_list_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_210_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_211_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_212_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_213_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_214_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_215_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_216_singletonI,axiom,
! [A2: $o] : ( member_o @ A2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) ).
% singletonI
thf(fact_217_singletonI,axiom,
! [A2: list_nat] : ( member_list_nat @ A2 @ ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) ).
% singletonI
thf(fact_218_singletonI,axiom,
! [A2: set_list_nat] : ( member_set_list_nat @ A2 @ ( insert_set_list_nat @ A2 @ bot_bo3886227569956363488st_nat ) ) ).
% singletonI
thf(fact_219_singletonI,axiom,
! [A2: set_set_list_nat] : ( member1029098694177496419st_nat @ A2 @ ( insert721039030076969596st_nat @ A2 @ bot_bo3499706412017099030st_nat ) ) ).
% singletonI
thf(fact_220_singletonI,axiom,
! [A2: set_set_set_list_nat] : ( member7304678173793621401st_nat @ A2 @ ( insert8008074249615255218st_nat @ A2 @ bot_bo1158166727579713100st_nat ) ) ).
% singletonI
thf(fact_221_insert__subset,axiom,
! [X3: set_set_set_list_nat,A: set_se5258582372428582328st_nat,B2: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ ( insert8008074249615255218st_nat @ X3 @ A ) @ B2 )
= ( ( member7304678173793621401st_nat @ X3 @ B2 )
& ( ord_le2499698639687704088st_nat @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_222_insert__subset,axiom,
! [X3: nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ A ) @ B2 )
= ( ( member_nat @ X3 @ B2 )
& ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_223_insert__subset,axiom,
! [X3: $o,A: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ ( insert_o @ X3 @ A ) @ B2 )
= ( ( member_o @ X3 @ B2 )
& ( ord_less_eq_set_o @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_224_insert__subset,axiom,
! [X3: list_nat,A: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( insert_list_nat @ X3 @ A ) @ B2 )
= ( ( member_list_nat @ X3 @ B2 )
& ( ord_le6045566169113846134st_nat @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_225_insert__subset,axiom,
! [X3: set_list_nat,A: set_set_list_nat,B2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ ( insert_set_list_nat @ X3 @ A ) @ B2 )
= ( ( member_set_list_nat @ X3 @ B2 )
& ( ord_le1068707526560357548st_nat @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_226_insert__subset,axiom,
! [X3: set_set_list_nat,A: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ ( insert721039030076969596st_nat @ X3 @ A ) @ B2 )
= ( ( member1029098694177496419st_nat @ X3 @ B2 )
& ( ord_le7100322305783427298st_nat @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_227_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X3: nat,B: nat,W: nat] :
( ( ( semiri1314217659103216013at_int @ X3 )
= ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( X3
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_228_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X3: nat,B: nat,W: nat] :
( ( ( semiri1316708129612266289at_nat @ X3 )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( X3
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_229_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X3: nat] :
( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
= ( semiri1314217659103216013at_int @ X3 ) )
= ( ( power_power_nat @ B @ W )
= X3 ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_230_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X3: nat] :
( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
= ( semiri1316708129612266289at_nat @ X3 ) )
= ( ( power_power_nat @ B @ W )
= X3 ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_231_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_232_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_233_Diff__cancel,axiom,
! [A: set_list_nat] :
( ( minus_7954133019191499631st_nat @ A @ A )
= bot_bot_set_list_nat ) ).
% Diff_cancel
thf(fact_234_Diff__cancel,axiom,
! [A: set_se5258582372428582328st_nat] :
( ( minus_1885081636186270737st_nat @ A @ A )
= bot_bo1158166727579713100st_nat ) ).
% Diff_cancel
thf(fact_235_Diff__cancel,axiom,
! [A: set_set_list_nat] :
( ( minus_1490147023502258853st_nat @ A @ A )
= bot_bo3886227569956363488st_nat ) ).
% Diff_cancel
thf(fact_236_Diff__cancel,axiom,
! [A: set_set_set_list_nat] :
( ( minus_7520675481749758683st_nat @ A @ A )
= bot_bo3499706412017099030st_nat ) ).
% Diff_cancel
thf(fact_237_empty__Diff,axiom,
! [A: set_list_nat] :
( ( minus_7954133019191499631st_nat @ bot_bot_set_list_nat @ A )
= bot_bot_set_list_nat ) ).
% empty_Diff
thf(fact_238_empty__Diff,axiom,
! [A: set_se5258582372428582328st_nat] :
( ( minus_1885081636186270737st_nat @ bot_bo1158166727579713100st_nat @ A )
= bot_bo1158166727579713100st_nat ) ).
% empty_Diff
thf(fact_239_empty__Diff,axiom,
! [A: set_set_list_nat] :
( ( minus_1490147023502258853st_nat @ bot_bo3886227569956363488st_nat @ A )
= bot_bo3886227569956363488st_nat ) ).
% empty_Diff
thf(fact_240_empty__Diff,axiom,
! [A: set_set_set_list_nat] :
( ( minus_7520675481749758683st_nat @ bot_bo3499706412017099030st_nat @ A )
= bot_bo3499706412017099030st_nat ) ).
% empty_Diff
thf(fact_241_Diff__empty,axiom,
! [A: set_list_nat] :
( ( minus_7954133019191499631st_nat @ A @ bot_bot_set_list_nat )
= A ) ).
% Diff_empty
thf(fact_242_Diff__empty,axiom,
! [A: set_se5258582372428582328st_nat] :
( ( minus_1885081636186270737st_nat @ A @ bot_bo1158166727579713100st_nat )
= A ) ).
% Diff_empty
thf(fact_243_Diff__empty,axiom,
! [A: set_set_list_nat] :
( ( minus_1490147023502258853st_nat @ A @ bot_bo3886227569956363488st_nat )
= A ) ).
% Diff_empty
thf(fact_244_Diff__empty,axiom,
! [A: set_set_set_list_nat] :
( ( minus_7520675481749758683st_nat @ A @ bot_bo3499706412017099030st_nat )
= A ) ).
% Diff_empty
thf(fact_245_insert__Diff1,axiom,
! [X3: set_set_set_list_nat,B2: set_se5258582372428582328st_nat,A: set_se5258582372428582328st_nat] :
( ( member7304678173793621401st_nat @ X3 @ B2 )
=> ( ( minus_1885081636186270737st_nat @ ( insert8008074249615255218st_nat @ X3 @ A ) @ B2 )
= ( minus_1885081636186270737st_nat @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_246_insert__Diff1,axiom,
! [X3: nat,B2: set_nat,A: set_nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A ) @ B2 )
= ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_247_insert__Diff1,axiom,
! [X3: $o,B2: set_o,A: set_o] :
( ( member_o @ X3 @ B2 )
=> ( ( minus_minus_set_o @ ( insert_o @ X3 @ A ) @ B2 )
= ( minus_minus_set_o @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_248_insert__Diff1,axiom,
! [X3: list_nat,B2: set_list_nat,A: set_list_nat] :
( ( member_list_nat @ X3 @ B2 )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat @ X3 @ A ) @ B2 )
= ( minus_7954133019191499631st_nat @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_249_insert__Diff1,axiom,
! [X3: set_list_nat,B2: set_set_list_nat,A: set_set_list_nat] :
( ( member_set_list_nat @ X3 @ B2 )
=> ( ( minus_1490147023502258853st_nat @ ( insert_set_list_nat @ X3 @ A ) @ B2 )
= ( minus_1490147023502258853st_nat @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_250_insert__Diff1,axiom,
! [X3: set_set_list_nat,B2: set_set_set_list_nat,A: set_set_set_list_nat] :
( ( member1029098694177496419st_nat @ X3 @ B2 )
=> ( ( minus_7520675481749758683st_nat @ ( insert721039030076969596st_nat @ X3 @ A ) @ B2 )
= ( minus_7520675481749758683st_nat @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_251_Diff__insert0,axiom,
! [X3: set_set_set_list_nat,A: set_se5258582372428582328st_nat,B2: set_se5258582372428582328st_nat] :
( ~ ( member7304678173793621401st_nat @ X3 @ A )
=> ( ( minus_1885081636186270737st_nat @ A @ ( insert8008074249615255218st_nat @ X3 @ B2 ) )
= ( minus_1885081636186270737st_nat @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_252_Diff__insert0,axiom,
! [X3: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat @ X3 @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ B2 ) )
= ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_253_Diff__insert0,axiom,
! [X3: $o,A: set_o,B2: set_o] :
( ~ ( member_o @ X3 @ A )
=> ( ( minus_minus_set_o @ A @ ( insert_o @ X3 @ B2 ) )
= ( minus_minus_set_o @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_254_Diff__insert0,axiom,
! [X3: list_nat,A: set_list_nat,B2: set_list_nat] :
( ~ ( member_list_nat @ X3 @ A )
=> ( ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat @ X3 @ B2 ) )
= ( minus_7954133019191499631st_nat @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_255_Diff__insert0,axiom,
! [X3: set_list_nat,A: set_set_list_nat,B2: set_set_list_nat] :
( ~ ( member_set_list_nat @ X3 @ A )
=> ( ( minus_1490147023502258853st_nat @ A @ ( insert_set_list_nat @ X3 @ B2 ) )
= ( minus_1490147023502258853st_nat @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_256_Diff__insert0,axiom,
! [X3: set_set_list_nat,A: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ~ ( member1029098694177496419st_nat @ X3 @ A )
=> ( ( minus_7520675481749758683st_nat @ A @ ( insert721039030076969596st_nat @ X3 @ B2 ) )
= ( minus_7520675481749758683st_nat @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_257_power__one__right,axiom,
! [A2: int] :
( ( power_power_int @ A2 @ one_one_nat )
= A2 ) ).
% power_one_right
thf(fact_258_power__one__right,axiom,
! [A2: nat] :
( ( power_power_nat @ A2 @ one_one_nat )
= A2 ) ).
% power_one_right
thf(fact_259_Pow__iff,axiom,
! [A: set_list_nat,B2: set_list_nat] :
( ( member_set_list_nat @ A @ ( pow_list_nat @ B2 ) )
= ( ord_le6045566169113846134st_nat @ A @ B2 ) ) ).
% Pow_iff
thf(fact_260_Pow__iff,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat] :
( ( member1029098694177496419st_nat @ A @ ( pow_set_list_nat @ B2 ) )
= ( ord_le1068707526560357548st_nat @ A @ B2 ) ) ).
% Pow_iff
thf(fact_261_Pow__iff,axiom,
! [A: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ A @ ( pow_set_set_list_nat @ B2 ) )
= ( ord_le7100322305783427298st_nat @ A @ B2 ) ) ).
% Pow_iff
thf(fact_262_PowI,axiom,
! [A: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B2 )
=> ( member_set_list_nat @ A @ ( pow_list_nat @ B2 ) ) ) ).
% PowI
thf(fact_263_PowI,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A @ B2 )
=> ( member1029098694177496419st_nat @ A @ ( pow_set_list_nat @ B2 ) ) ) ).
% PowI
thf(fact_264_PowI,axiom,
! [A: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A @ B2 )
=> ( member7304678173793621401st_nat @ A @ ( pow_set_set_list_nat @ B2 ) ) ) ).
% PowI
thf(fact_265_singleton__conv2,axiom,
! [A2: nat] :
( ( collect_nat
@ ( ^ [Y4: nat,Z: nat] : ( Y4 = Z )
@ A2 ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_266_singleton__conv2,axiom,
! [A2: list_nat] :
( ( collect_list_nat
@ ( ^ [Y4: list_nat,Z: list_nat] : ( Y4 = Z )
@ A2 ) )
= ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) ).
% singleton_conv2
thf(fact_267_singleton__conv2,axiom,
! [A2: set_list_nat] :
( ( collect_set_list_nat
@ ( ^ [Y4: set_list_nat,Z: set_list_nat] : ( Y4 = Z )
@ A2 ) )
= ( insert_set_list_nat @ A2 @ bot_bo3886227569956363488st_nat ) ) ).
% singleton_conv2
thf(fact_268_singleton__conv2,axiom,
! [A2: set_set_list_nat] :
( ( collec4691811733418234273st_nat
@ ( ^ [Y4: set_set_list_nat,Z: set_set_list_nat] : ( Y4 = Z )
@ A2 ) )
= ( insert721039030076969596st_nat @ A2 @ bot_bo3499706412017099030st_nat ) ) ).
% singleton_conv2
thf(fact_269_singleton__conv2,axiom,
! [A2: set_set_set_list_nat] :
( ( collec696328324557263319st_nat
@ ( ^ [Y4: set_set_set_list_nat,Z: set_set_set_list_nat] : ( Y4 = Z )
@ A2 ) )
= ( insert8008074249615255218st_nat @ A2 @ bot_bo1158166727579713100st_nat ) ) ).
% singleton_conv2
thf(fact_270_singleton__conv,axiom,
! [A2: nat] :
( ( collect_nat
@ ^ [X2: nat] : ( X2 = A2 ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_271_singleton__conv,axiom,
! [A2: list_nat] :
( ( collect_list_nat
@ ^ [X2: list_nat] : ( X2 = A2 ) )
= ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) ).
% singleton_conv
thf(fact_272_singleton__conv,axiom,
! [A2: set_list_nat] :
( ( collect_set_list_nat
@ ^ [X2: set_list_nat] : ( X2 = A2 ) )
= ( insert_set_list_nat @ A2 @ bot_bo3886227569956363488st_nat ) ) ).
% singleton_conv
thf(fact_273_singleton__conv,axiom,
! [A2: set_set_list_nat] :
( ( collec4691811733418234273st_nat
@ ^ [X2: set_set_list_nat] : ( X2 = A2 ) )
= ( insert721039030076969596st_nat @ A2 @ bot_bo3499706412017099030st_nat ) ) ).
% singleton_conv
thf(fact_274_singleton__conv,axiom,
! [A2: set_set_set_list_nat] :
( ( collec696328324557263319st_nat
@ ^ [X2: set_set_set_list_nat] : ( X2 = A2 ) )
= ( insert8008074249615255218st_nat @ A2 @ bot_bo1158166727579713100st_nat ) ) ).
% singleton_conv
thf(fact_275_INF__identity__eq,axiom,
! [A: set_set_list_nat] :
( ( comple184543376406953807st_nat
@ ( image_5143090206295581363st_nat
@ ^ [X2: set_list_nat] : X2
@ A ) )
= ( comple184543376406953807st_nat @ A ) ) ).
% INF_identity_eq
thf(fact_276_INF__identity__eq,axiom,
! [A: set_o] :
( ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [X2: $o] : X2
@ A ) )
= ( complete_Inf_Inf_o @ A ) ) ).
% INF_identity_eq
thf(fact_277_INT__iff,axiom,
! [B: set_list_nat,B2: set_list_nat > set_set_list_nat,A: set_set_list_nat] :
( ( member_set_list_nat @ B @ ( comple8462666950445340293st_nat @ ( image_920372530894776809st_nat @ B2 @ A ) ) )
= ( ! [X2: set_list_nat] :
( ( member_set_list_nat @ X2 @ A )
=> ( member_set_list_nat @ B @ ( B2 @ X2 ) ) ) ) ) ).
% INT_iff
thf(fact_278_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 ) ) )
= ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
=> ( member_list_nat @ B @ ( B2 @ X2 ) ) ) ) ) ).
% INT_iff
thf(fact_279_INT__I,axiom,
! [A: set_set_list_nat,B: set_list_nat,B2: set_list_nat > set_set_list_nat] :
( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ A )
=> ( member_set_list_nat @ B @ ( B2 @ X4 ) ) )
=> ( member_set_list_nat @ B @ ( comple8462666950445340293st_nat @ ( image_920372530894776809st_nat @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_280_INT__I,axiom,
! [A: set_nat,B: nat,B2: nat > set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ B @ ( B2 @ X4 ) ) )
=> ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_281_INT__I,axiom,
! [A: set_nat,B: $o,B2: nat > set_o] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_o @ B @ ( B2 @ X4 ) ) )
=> ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_nat_set_o @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_282_INT__I,axiom,
! [A: set_o,B: nat,B2: $o > set_nat] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( member_nat @ B @ ( B2 @ X4 ) ) )
=> ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_283_INT__I,axiom,
! [A: set_o,B: $o,B2: $o > set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( member_o @ B @ ( B2 @ X4 ) ) )
=> ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_284_INT__I,axiom,
! [A: set_list_nat,B: nat,B2: list_nat > set_nat] :
( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
=> ( member_nat @ B @ ( B2 @ X4 ) ) )
=> ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_1775855109352712557et_nat @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_285_INT__I,axiom,
! [A: set_list_nat,B: $o,B2: list_nat > set_o] :
( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
=> ( member_o @ B @ ( B2 @ X4 ) ) )
=> ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_list_nat_set_o @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_286_INT__I,axiom,
! [A: set_nat,B: list_nat,B2: nat > set_list_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_list_nat @ B @ ( B2 @ X4 ) ) )
=> ( member_list_nat @ B @ ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_287_INT__I,axiom,
! [A: set_o,B: list_nat,B2: $o > set_list_nat] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( member_list_nat @ B @ ( B2 @ X4 ) ) )
=> ( member_list_nat @ B @ ( comple184543376406953807st_nat @ ( image_o_set_list_nat @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_288_INT__I,axiom,
! [A: set_list_nat,B: list_nat,B2: list_nat > set_list_nat] :
( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
=> ( member_list_nat @ B @ ( B2 @ X4 ) ) )
=> ( member_list_nat @ B @ ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_289_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X3: nat,B: nat,W: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_290_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X3: nat,B: nat,W: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_291_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X3: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X3 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X3 ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_292_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X3: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X3 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X3 ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_293_subset__Compl__singleton,axiom,
! [A: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ~ ( member_nat @ B @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_294_subset__Compl__singleton,axiom,
! [A: set_o,B: $o] :
( ( ord_less_eq_set_o @ A @ ( uminus_uminus_set_o @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( ~ ( member_o @ B @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_295_subset__Compl__singleton,axiom,
! [A: set_se5258582372428582328st_nat,B: set_set_set_list_nat] :
( ( ord_le2499698639687704088st_nat @ A @ ( uminus3509280191954366337st_nat @ ( insert8008074249615255218st_nat @ B @ bot_bo1158166727579713100st_nat ) ) )
= ( ~ ( member7304678173793621401st_nat @ B @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_296_subset__Compl__singleton,axiom,
! [A: set_list_nat,B: list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ ( uminus3195874150345416415st_nat @ ( insert_list_nat @ B @ bot_bot_set_list_nat ) ) )
= ( ~ ( member_list_nat @ B @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_297_subset__Compl__singleton,axiom,
! [A: set_set_list_nat,B: set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A @ ( uminus8693322751144048149st_nat @ ( insert_set_list_nat @ B @ bot_bo3886227569956363488st_nat ) ) )
= ( ~ ( member_set_list_nat @ B @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_298_subset__Compl__singleton,axiom,
! [A: set_set_set_list_nat,B: set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A @ ( uminus8338265383663789643st_nat @ ( insert721039030076969596st_nat @ B @ bot_bo3499706412017099030st_nat ) ) )
= ( ~ ( member1029098694177496419st_nat @ B @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_299_singleton__insert__inj__eq_H,axiom,
! [A2: set_set_set_list_nat,A: set_se5258582372428582328st_nat,B: set_set_set_list_nat] :
( ( ( insert8008074249615255218st_nat @ A2 @ A )
= ( insert8008074249615255218st_nat @ B @ bot_bo1158166727579713100st_nat ) )
= ( ( A2 = B )
& ( ord_le2499698639687704088st_nat @ A @ ( insert8008074249615255218st_nat @ B @ bot_bo1158166727579713100st_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_300_singleton__insert__inj__eq_H,axiom,
! [A2: list_nat,A: set_list_nat,B: list_nat] :
( ( ( insert_list_nat @ A2 @ A )
= ( insert_list_nat @ B @ bot_bot_set_list_nat ) )
= ( ( A2 = B )
& ( ord_le6045566169113846134st_nat @ A @ ( insert_list_nat @ B @ bot_bot_set_list_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_301_singleton__insert__inj__eq_H,axiom,
! [A2: set_list_nat,A: set_set_list_nat,B: set_list_nat] :
( ( ( insert_set_list_nat @ A2 @ A )
= ( insert_set_list_nat @ B @ bot_bo3886227569956363488st_nat ) )
= ( ( A2 = B )
& ( ord_le1068707526560357548st_nat @ A @ ( insert_set_list_nat @ B @ bot_bo3886227569956363488st_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_302_singleton__insert__inj__eq_H,axiom,
! [A2: set_set_list_nat,A: set_set_set_list_nat,B: set_set_list_nat] :
( ( ( insert721039030076969596st_nat @ A2 @ A )
= ( insert721039030076969596st_nat @ B @ bot_bo3499706412017099030st_nat ) )
= ( ( A2 = B )
& ( ord_le7100322305783427298st_nat @ A @ ( insert721039030076969596st_nat @ B @ bot_bo3499706412017099030st_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_303_singleton__insert__inj__eq,axiom,
! [B: set_set_set_list_nat,A2: set_set_set_list_nat,A: set_se5258582372428582328st_nat] :
( ( ( insert8008074249615255218st_nat @ B @ bot_bo1158166727579713100st_nat )
= ( insert8008074249615255218st_nat @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_le2499698639687704088st_nat @ A @ ( insert8008074249615255218st_nat @ B @ bot_bo1158166727579713100st_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_304_singleton__insert__inj__eq,axiom,
! [B: list_nat,A2: list_nat,A: set_list_nat] :
( ( ( insert_list_nat @ B @ bot_bot_set_list_nat )
= ( insert_list_nat @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_le6045566169113846134st_nat @ A @ ( insert_list_nat @ B @ bot_bot_set_list_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_305_singleton__insert__inj__eq,axiom,
! [B: set_list_nat,A2: set_list_nat,A: set_set_list_nat] :
( ( ( insert_set_list_nat @ B @ bot_bo3886227569956363488st_nat )
= ( insert_set_list_nat @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_le1068707526560357548st_nat @ A @ ( insert_set_list_nat @ B @ bot_bo3886227569956363488st_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_306_singleton__insert__inj__eq,axiom,
! [B: set_set_list_nat,A2: set_set_list_nat,A: set_set_set_list_nat] :
( ( ( insert721039030076969596st_nat @ B @ bot_bo3499706412017099030st_nat )
= ( insert721039030076969596st_nat @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_le7100322305783427298st_nat @ A @ ( insert721039030076969596st_nat @ B @ bot_bo3499706412017099030st_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_307_Diff__eq__empty__iff,axiom,
! [A: set_se5258582372428582328st_nat,B2: set_se5258582372428582328st_nat] :
( ( ( minus_1885081636186270737st_nat @ A @ B2 )
= bot_bo1158166727579713100st_nat )
= ( ord_le2499698639687704088st_nat @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_308_Diff__eq__empty__iff,axiom,
! [A: set_list_nat,B2: set_list_nat] :
( ( ( minus_7954133019191499631st_nat @ A @ B2 )
= bot_bot_set_list_nat )
= ( ord_le6045566169113846134st_nat @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_309_Diff__eq__empty__iff,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat] :
( ( ( minus_1490147023502258853st_nat @ A @ B2 )
= bot_bo3886227569956363488st_nat )
= ( ord_le1068707526560357548st_nat @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_310_Diff__eq__empty__iff,axiom,
! [A: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ( ( minus_7520675481749758683st_nat @ A @ B2 )
= bot_bo3499706412017099030st_nat )
= ( ord_le7100322305783427298st_nat @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_311_cInf__singleton,axiom,
! [X3: set_set_list_nat] :
( ( comple8462666950445340293st_nat @ ( insert721039030076969596st_nat @ X3 @ bot_bo3499706412017099030st_nat ) )
= X3 ) ).
% cInf_singleton
thf(fact_312_cInf__singleton,axiom,
! [X3: set_set_set_list_nat] :
( ( comple5189992959352112827st_nat @ ( insert8008074249615255218st_nat @ X3 @ bot_bo1158166727579713100st_nat ) )
= X3 ) ).
% cInf_singleton
thf(fact_313_cInf__singleton,axiom,
! [X3: set_list_nat] :
( ( comple184543376406953807st_nat @ ( insert_set_list_nat @ X3 @ bot_bo3886227569956363488st_nat ) )
= X3 ) ).
% cInf_singleton
thf(fact_314_cInf__singleton,axiom,
! [X3: $o] :
( ( complete_Inf_Inf_o @ ( insert_o @ X3 @ bot_bot_set_o ) )
= X3 ) ).
% cInf_singleton
thf(fact_315_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_316_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_317_insert__Diff__single,axiom,
! [A2: list_nat,A: set_list_nat] :
( ( insert_list_nat @ A2 @ ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) )
= ( insert_list_nat @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_318_insert__Diff__single,axiom,
! [A2: set_set_set_list_nat,A: set_se5258582372428582328st_nat] :
( ( insert8008074249615255218st_nat @ A2 @ ( minus_1885081636186270737st_nat @ A @ ( insert8008074249615255218st_nat @ A2 @ bot_bo1158166727579713100st_nat ) ) )
= ( insert8008074249615255218st_nat @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_319_insert__Diff__single,axiom,
! [A2: set_list_nat,A: set_set_list_nat] :
( ( insert_set_list_nat @ A2 @ ( minus_1490147023502258853st_nat @ A @ ( insert_set_list_nat @ A2 @ bot_bo3886227569956363488st_nat ) ) )
= ( insert_set_list_nat @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_320_insert__Diff__single,axiom,
! [A2: set_set_list_nat,A: set_set_set_list_nat] :
( ( insert721039030076969596st_nat @ A2 @ ( minus_7520675481749758683st_nat @ A @ ( insert721039030076969596st_nat @ A2 @ bot_bo3499706412017099030st_nat ) ) )
= ( insert721039030076969596st_nat @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_321_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_322_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_323__092_060open_062SUM1_A_061_A_I_092_060Sum_062I_092_060in_062_I_096_J_A_IC_An_J_A_096_A_IPow_AX_A_N_A_123_123_125_125_J_O_A_I_N_A1_J_A_094_A_Icard_AI_A_L_A1_J_A_K_Aint_A_Icard_A_I_092_060Inter_062_AI_J_J_J_092_060close_062,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 ) ) ) )
@ ( image_920372530894776809st_nat @ ( image_8532145185254316925st_nat @ ( c @ n ) ) @ ( minus_1490147023502258853st_nat @ ( pow_list_nat @ x ) @ ( insert_set_list_nat @ bot_bot_set_list_nat @ bot_bo3886227569956363488st_nat ) ) ) ) ) ).
% \<open>SUM1 = (\<Sum>I\<in>(`) (C n) ` (Pow X - {{}}). (- 1) ^ (card I + 1) * int (card (\<Inter> I)))\<close>
thf(fact_324__092_060open_062_I_092_060Sum_062I_092_060in_062_I_096_J_A_IC_An_J_A_096_A_IPow_AX_A_N_A_123_123_125_125_J_O_A_I_N_A1_J_A_094_A_Icard_AI_A_L_A1_J_A_K_Aint_A_Icard_A_I_092_060Inter_062_AI_J_J_J_A_061_Asum_A_I_I_092_060lambda_062I_O_A_I_N_A1_J_A_094_A_Icard_AI_A_L_A1_J_A_K_Aint_A_Icard_A_I_092_060Inter_062_AI_J_J_J_A_092_060circ_062_A_I_096_J_A_IC_An_J_J_A_IPow_AX_A_N_A_123_123_125_125_J_092_060close_062,axiom,
( ( 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 ) ) ) )
@ ( image_920372530894776809st_nat @ ( image_8532145185254316925st_nat @ ( c @ n ) ) @ ( minus_1490147023502258853st_nat @ ( pow_list_nat @ x ) @ ( insert_set_list_nat @ bot_bot_set_list_nat @ bot_bo3886227569956363488st_nat ) ) ) )
= ( groups7312845317294741502at_int
@ ( comp_s3399862273134644595st_nat
@ ^ [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 ) ) ) )
@ ( image_8532145185254316925st_nat @ ( c @ n ) ) )
@ ( minus_1490147023502258853st_nat @ ( pow_list_nat @ x ) @ ( insert_set_list_nat @ bot_bot_set_list_nat @ bot_bo3886227569956363488st_nat ) ) ) ) ).
% \<open>(\<Sum>I\<in>(`) (C n) ` (Pow X - {{}}). (- 1) ^ (card I + 1) * int (card (\<Inter> I))) = sum ((\<lambda>I. (- 1) ^ (card I + 1) * int (card (\<Inter> I))) \<circ> (`) (C n)) (Pow X - {{}})\<close>
thf(fact_325_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_326_Inf__set__def,axiom,
( comple3063163877087187839_set_o
= ( ^ [A3: set_set_o] :
( collect_o
@ ^ [X2: $o] : ( complete_Inf_Inf_o @ ( image_set_o_o @ ( member_o @ X2 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_327_Inf__set__def,axiom,
( comple8462666950445340293st_nat
= ( ^ [A3: set_set_set_list_nat] :
( collect_set_list_nat
@ ^ [X2: set_list_nat] : ( complete_Inf_Inf_o @ ( image_7778788071572860293_nat_o @ ( member_set_list_nat @ X2 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_328_Inf__set__def,axiom,
( comple5189992959352112827st_nat
= ( ^ [A3: set_se5258582372428582328st_nat] :
( collec4691811733418234273st_nat
@ ^ [X2: set_set_list_nat] : ( complete_Inf_Inf_o @ ( image_3035193717992370127_nat_o @ ( member1029098694177496419st_nat @ X2 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_329_Inf__set__def,axiom,
( comple7806235888213564991et_nat
= ( ^ [A3: set_set_nat] :
( collect_nat
@ ^ [X2: nat] : ( complete_Inf_Inf_o @ ( image_set_nat_o @ ( member_nat @ X2 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_330_Inf__set__def,axiom,
( comple184543376406953807st_nat
= ( ^ [A3: set_set_list_nat] :
( collect_list_nat
@ ^ [X2: list_nat] : ( complete_Inf_Inf_o @ ( image_set_list_nat_o @ ( member_list_nat @ X2 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_331_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_332_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_333_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_334_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_335_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_336_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_337_subset__Compl__self__eq,axiom,
! [A: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ A @ ( uminus3509280191954366337st_nat @ A ) )
= ( A = bot_bo1158166727579713100st_nat ) ) ).
% subset_Compl_self_eq
thf(fact_338_subset__Compl__self__eq,axiom,
! [A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ ( uminus3195874150345416415st_nat @ A ) )
= ( A = bot_bot_set_list_nat ) ) ).
% subset_Compl_self_eq
thf(fact_339_subset__Compl__self__eq,axiom,
! [A: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A @ ( uminus8693322751144048149st_nat @ A ) )
= ( A = bot_bo3886227569956363488st_nat ) ) ).
% subset_Compl_self_eq
thf(fact_340_subset__Compl__self__eq,axiom,
! [A: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A @ ( uminus8338265383663789643st_nat @ A ) )
= ( A = bot_bo3499706412017099030st_nat ) ) ).
% subset_Compl_self_eq
thf(fact_341_card__eq__sum,axiom,
( finite5070363488328301092st_nat
= ( groups7006704140163696856at_nat
@ ^ [X2: set_set_list_nat] : one_one_nat ) ) ).
% card_eq_sum
thf(fact_342_card__eq__sum,axiom,
( finite_card_nat
= ( groups3542108847815614940at_nat
@ ^ [X2: nat] : one_one_nat ) ) ).
% card_eq_sum
thf(fact_343_card__eq__sum,axiom,
( finite_card_list_nat
= ( groups4396056296759096172at_nat
@ ^ [X2: list_nat] : one_one_nat ) ) ).
% card_eq_sum
thf(fact_344_card__eq__sum,axiom,
( finite2364142230527598318st_nat
= ( groups7315335787803791778at_nat
@ ^ [X2: set_list_nat] : one_one_nat ) ) ).
% card_eq_sum
thf(fact_345_in__image__insert__iff,axiom,
! [B2: set_set_nat,X3: nat,A: set_nat] :
( ! [C3: set_nat] :
( ( member_set_nat @ C3 @ B2 )
=> ~ ( member_nat @ X3 @ C3 ) )
=> ( ( member_set_nat @ A @ ( image_7916887816326733075et_nat @ ( insert_nat @ X3 ) @ B2 ) )
= ( ( member_nat @ X3 @ A )
& ( member_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_346_in__image__insert__iff,axiom,
! [B2: set_set_o,X3: $o,A: set_o] :
( ! [C3: set_o] :
( ( member_set_o @ C3 @ B2 )
=> ~ ( member_o @ X3 @ C3 ) )
=> ( ( member_set_o @ A @ ( image_set_o_set_o @ ( insert_o @ X3 ) @ B2 ) )
= ( ( member_o @ X3 @ A )
& ( member_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X3 @ bot_bot_set_o ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_347_in__image__insert__iff,axiom,
! [B2: set_set_list_nat,X3: list_nat,A: set_list_nat] :
( ! [C3: set_list_nat] :
( ( member_set_list_nat @ C3 @ B2 )
=> ~ ( member_list_nat @ X3 @ C3 ) )
=> ( ( member_set_list_nat @ A @ ( image_5143090206295581363st_nat @ ( insert_list_nat @ X3 ) @ B2 ) )
= ( ( member_list_nat @ X3 @ A )
& ( member_set_list_nat @ ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat @ X3 @ bot_bot_set_list_nat ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_348_in__image__insert__iff,axiom,
! [B2: set_se7082751667426497518st_nat,X3: set_set_set_list_nat,A: set_se5258582372428582328st_nat] :
( ! [C3: set_se5258582372428582328st_nat] :
( ( member4920245652161177295st_nat @ C3 @ B2 )
=> ~ ( member7304678173793621401st_nat @ X3 @ C3 ) )
=> ( ( member4920245652161177295st_nat @ A @ ( image_7413923974343677687st_nat @ ( insert8008074249615255218st_nat @ X3 ) @ B2 ) )
= ( ( member7304678173793621401st_nat @ X3 @ A )
& ( member4920245652161177295st_nat @ ( minus_1885081636186270737st_nat @ A @ ( insert8008074249615255218st_nat @ X3 @ bot_bo1158166727579713100st_nat ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_349_in__image__insert__iff,axiom,
! [B2: set_set_set_list_nat,X3: set_list_nat,A: set_set_list_nat] :
( ! [C3: set_set_list_nat] :
( ( member1029098694177496419st_nat @ C3 @ B2 )
=> ~ ( member_set_list_nat @ X3 @ C3 ) )
=> ( ( member1029098694177496419st_nat @ A @ ( image_799823415861195295st_nat @ ( insert_set_list_nat @ X3 ) @ B2 ) )
= ( ( member_set_list_nat @ X3 @ A )
& ( member1029098694177496419st_nat @ ( minus_1490147023502258853st_nat @ A @ ( insert_set_list_nat @ X3 @ bot_bo3886227569956363488st_nat ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_350_in__image__insert__iff,axiom,
! [B2: set_se5258582372428582328st_nat,X3: set_set_list_nat,A: set_set_set_list_nat] :
( ! [C3: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ C3 @ B2 )
=> ~ ( member1029098694177496419st_nat @ X3 @ C3 ) )
=> ( ( member7304678173793621401st_nat @ A @ ( image_5561069522181849995st_nat @ ( insert721039030076969596st_nat @ X3 ) @ B2 ) )
= ( ( member1029098694177496419st_nat @ X3 @ A )
& ( member7304678173793621401st_nat @ ( minus_7520675481749758683st_nat @ A @ ( insert721039030076969596st_nat @ X3 @ bot_bo3499706412017099030st_nat ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_351_Compl__insert,axiom,
! [X3: list_nat,A: set_list_nat] :
( ( uminus3195874150345416415st_nat @ ( insert_list_nat @ X3 @ A ) )
= ( minus_7954133019191499631st_nat @ ( uminus3195874150345416415st_nat @ A ) @ ( insert_list_nat @ X3 @ bot_bot_set_list_nat ) ) ) ).
% Compl_insert
thf(fact_352_Compl__insert,axiom,
! [X3: set_set_set_list_nat,A: set_se5258582372428582328st_nat] :
( ( uminus3509280191954366337st_nat @ ( insert8008074249615255218st_nat @ X3 @ A ) )
= ( minus_1885081636186270737st_nat @ ( uminus3509280191954366337st_nat @ A ) @ ( insert8008074249615255218st_nat @ X3 @ bot_bo1158166727579713100st_nat ) ) ) ).
% Compl_insert
thf(fact_353_Compl__insert,axiom,
! [X3: set_list_nat,A: set_set_list_nat] :
( ( uminus8693322751144048149st_nat @ ( insert_set_list_nat @ X3 @ A ) )
= ( minus_1490147023502258853st_nat @ ( uminus8693322751144048149st_nat @ A ) @ ( insert_set_list_nat @ X3 @ bot_bo3886227569956363488st_nat ) ) ) ).
% Compl_insert
thf(fact_354_Compl__insert,axiom,
! [X3: set_set_list_nat,A: set_set_set_list_nat] :
( ( uminus8338265383663789643st_nat @ ( insert721039030076969596st_nat @ X3 @ A ) )
= ( minus_7520675481749758683st_nat @ ( uminus8338265383663789643st_nat @ A ) @ ( insert721039030076969596st_nat @ X3 @ bot_bo3499706412017099030st_nat ) ) ) ).
% Compl_insert
thf(fact_355_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X3 @ A )
=> ( ( B
= ( F @ X3 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_356_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B: $o,F: nat > $o] :
( ( member_nat @ X3 @ A )
=> ( ( B
= ( F @ X3 ) )
=> ( member_o @ B @ ( image_nat_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_357_rev__image__eqI,axiom,
! [X3: $o,A: set_o,B: nat,F: $o > nat] :
( ( member_o @ X3 @ A )
=> ( ( B
= ( F @ X3 ) )
=> ( member_nat @ B @ ( image_o_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_358_rev__image__eqI,axiom,
! [X3: $o,A: set_o,B: $o,F: $o > $o] :
( ( member_o @ X3 @ A )
=> ( ( B
= ( F @ X3 ) )
=> ( member_o @ B @ ( image_o_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_359_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B: list_nat,F: nat > list_nat] :
( ( member_nat @ X3 @ A )
=> ( ( B
= ( F @ X3 ) )
=> ( member_list_nat @ B @ ( image_nat_list_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_360_rev__image__eqI,axiom,
! [X3: $o,A: set_o,B: list_nat,F: $o > list_nat] :
( ( member_o @ X3 @ A )
=> ( ( B
= ( F @ X3 ) )
=> ( member_list_nat @ B @ ( image_o_list_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_361_rev__image__eqI,axiom,
! [X3: list_nat,A: set_list_nat,B: nat,F: list_nat > nat] :
( ( member_list_nat @ X3 @ A )
=> ( ( B
= ( F @ X3 ) )
=> ( member_nat @ B @ ( image_list_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_362_rev__image__eqI,axiom,
! [X3: list_nat,A: set_list_nat,B: $o,F: list_nat > $o] :
( ( member_list_nat @ X3 @ A )
=> ( ( B
= ( F @ X3 ) )
=> ( member_o @ B @ ( image_list_nat_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_363_rev__image__eqI,axiom,
! [X3: list_nat,A: set_list_nat,B: list_nat,F: list_nat > list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( ( B
= ( F @ X3 ) )
=> ( member_list_nat @ B @ ( image_7976474329151083847st_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_364_rev__image__eqI,axiom,
! [X3: list_nat,A: set_list_nat,B: set_list_nat,F: list_nat > set_list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( ( B
= ( F @ X3 ) )
=> ( member_set_list_nat @ B @ ( image_8532145185254316925st_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_365_ball__imageD,axiom,
! [F: list_nat > set_list_nat,A: set_list_nat,P: set_list_nat > $o] :
( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ ( image_8532145185254316925st_nat @ F @ A ) )
=> ( P @ X4 ) )
=> ! [X6: list_nat] :
( ( member_list_nat @ X6 @ A )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_366_ball__imageD,axiom,
! [F: set_list_nat > set_set_list_nat,A: set_set_list_nat,P: set_set_list_nat > $o] :
( ! [X4: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X4 @ ( image_920372530894776809st_nat @ F @ A ) )
=> ( P @ X4 ) )
=> ! [X6: set_list_nat] :
( ( member_set_list_nat @ X6 @ A )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_367_image__cong,axiom,
! [M2: set_set_list_nat,N2: set_set_list_nat,F: set_list_nat > set_set_list_nat,G: set_list_nat > set_set_list_nat] :
( ( M2 = N2 )
=> ( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ N2 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_920372530894776809st_nat @ F @ M2 )
= ( image_920372530894776809st_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_368_image__cong,axiom,
! [M2: set_list_nat,N2: set_list_nat,F: list_nat > set_list_nat,G: list_nat > set_list_nat] :
( ( M2 = N2 )
=> ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ N2 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_8532145185254316925st_nat @ F @ M2 )
= ( image_8532145185254316925st_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_369_bex__imageD,axiom,
! [F: list_nat > set_list_nat,A: set_list_nat,P: set_list_nat > $o] :
( ? [X6: set_list_nat] :
( ( member_set_list_nat @ X6 @ ( image_8532145185254316925st_nat @ F @ A ) )
& ( P @ X6 ) )
=> ? [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_370_bex__imageD,axiom,
! [F: set_list_nat > set_set_list_nat,A: set_set_list_nat,P: set_set_list_nat > $o] :
( ? [X6: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X6 @ ( image_920372530894776809st_nat @ F @ A ) )
& ( P @ X6 ) )
=> ? [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_371_image__iff,axiom,
! [Z2: set_list_nat,F: list_nat > set_list_nat,A: set_list_nat] :
( ( member_set_list_nat @ Z2 @ ( image_8532145185254316925st_nat @ F @ A ) )
= ( ? [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_372_image__iff,axiom,
! [Z2: set_set_list_nat,F: set_list_nat > set_set_list_nat,A: set_set_list_nat] :
( ( member1029098694177496419st_nat @ Z2 @ ( image_920372530894776809st_nat @ F @ A ) )
= ( ? [X2: set_list_nat] :
( ( member_set_list_nat @ X2 @ A )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_373_imageI,axiom,
! [X3: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_374_imageI,axiom,
! [X3: nat,A: set_nat,F: nat > $o] :
( ( member_nat @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ ( image_nat_o @ F @ A ) ) ) ).
% imageI
thf(fact_375_imageI,axiom,
! [X3: $o,A: set_o,F: $o > nat] :
( ( member_o @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ ( image_o_nat @ F @ A ) ) ) ).
% imageI
thf(fact_376_imageI,axiom,
! [X3: $o,A: set_o,F: $o > $o] :
( ( member_o @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ ( image_o_o @ F @ A ) ) ) ).
% imageI
thf(fact_377_imageI,axiom,
! [X3: nat,A: set_nat,F: nat > list_nat] :
( ( member_nat @ X3 @ A )
=> ( member_list_nat @ ( F @ X3 ) @ ( image_nat_list_nat @ F @ A ) ) ) ).
% imageI
thf(fact_378_imageI,axiom,
! [X3: $o,A: set_o,F: $o > list_nat] :
( ( member_o @ X3 @ A )
=> ( member_list_nat @ ( F @ X3 ) @ ( image_o_list_nat @ F @ A ) ) ) ).
% imageI
thf(fact_379_imageI,axiom,
! [X3: list_nat,A: set_list_nat,F: list_nat > nat] :
( ( member_list_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ ( image_list_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_380_imageI,axiom,
! [X3: list_nat,A: set_list_nat,F: list_nat > $o] :
( ( member_list_nat @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ ( image_list_nat_o @ F @ A ) ) ) ).
% imageI
thf(fact_381_imageI,axiom,
! [X3: list_nat,A: set_list_nat,F: list_nat > list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( member_list_nat @ ( F @ X3 ) @ ( image_7976474329151083847st_nat @ F @ A ) ) ) ).
% imageI
thf(fact_382_imageI,axiom,
! [X3: list_nat,A: set_list_nat,F: list_nat > set_list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( member_set_list_nat @ ( F @ X3 ) @ ( image_8532145185254316925st_nat @ F @ A ) ) ) ).
% imageI
thf(fact_383_Sup_OSUP__cong,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat,C2: set_list_nat > set_set_list_nat,D: set_list_nat > set_set_list_nat,Sup: set_set_set_list_nat > set_set_list_nat] :
( ( A = B2 )
=> ( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_920372530894776809st_nat @ C2 @ A ) )
= ( Sup @ ( image_920372530894776809st_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_384_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 )
=> ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_8532145185254316925st_nat @ C2 @ A ) )
= ( Sup @ ( image_8532145185254316925st_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_385_Inf_OINF__cong,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat,C2: set_list_nat > set_set_list_nat,D: set_list_nat > set_set_list_nat,Inf: set_set_set_list_nat > set_set_list_nat] :
( ( A = B2 )
=> ( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_920372530894776809st_nat @ C2 @ A ) )
= ( Inf @ ( image_920372530894776809st_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_386_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 )
=> ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_8532145185254316925st_nat @ C2 @ A ) )
= ( Inf @ ( image_8532145185254316925st_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_387_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_388_ex__in__conv,axiom,
! [A: set_o] :
( ( ? [X2: $o] : ( member_o @ X2 @ A ) )
= ( A != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_389_ex__in__conv,axiom,
! [A: set_list_nat] :
( ( ? [X2: list_nat] : ( member_list_nat @ X2 @ A ) )
= ( A != bot_bot_set_list_nat ) ) ).
% ex_in_conv
thf(fact_390_ex__in__conv,axiom,
! [A: set_set_list_nat] :
( ( ? [X2: set_list_nat] : ( member_set_list_nat @ X2 @ A ) )
= ( A != bot_bo3886227569956363488st_nat ) ) ).
% ex_in_conv
thf(fact_391_ex__in__conv,axiom,
! [A: set_set_set_list_nat] :
( ( ? [X2: set_set_list_nat] : ( member1029098694177496419st_nat @ X2 @ A ) )
= ( A != bot_bo3499706412017099030st_nat ) ) ).
% ex_in_conv
thf(fact_392_ex__in__conv,axiom,
! [A: set_se5258582372428582328st_nat] :
( ( ? [X2: set_set_set_list_nat] : ( member7304678173793621401st_nat @ X2 @ A ) )
= ( A != bot_bo1158166727579713100st_nat ) ) ).
% ex_in_conv
thf(fact_393_equals0I,axiom,
! [A: set_nat] :
( ! [Y5: nat] :
~ ( member_nat @ Y5 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_394_equals0I,axiom,
! [A: set_o] :
( ! [Y5: $o] :
~ ( member_o @ Y5 @ A )
=> ( A = bot_bot_set_o ) ) ).
% equals0I
thf(fact_395_equals0I,axiom,
! [A: set_list_nat] :
( ! [Y5: list_nat] :
~ ( member_list_nat @ Y5 @ A )
=> ( A = bot_bot_set_list_nat ) ) ).
% equals0I
thf(fact_396_equals0I,axiom,
! [A: set_set_list_nat] :
( ! [Y5: set_list_nat] :
~ ( member_set_list_nat @ Y5 @ A )
=> ( A = bot_bo3886227569956363488st_nat ) ) ).
% equals0I
thf(fact_397_equals0I,axiom,
! [A: set_set_set_list_nat] :
( ! [Y5: set_set_list_nat] :
~ ( member1029098694177496419st_nat @ Y5 @ A )
=> ( A = bot_bo3499706412017099030st_nat ) ) ).
% equals0I
thf(fact_398_equals0I,axiom,
! [A: set_se5258582372428582328st_nat] :
( ! [Y5: set_set_set_list_nat] :
~ ( member7304678173793621401st_nat @ Y5 @ A )
=> ( A = bot_bo1158166727579713100st_nat ) ) ).
% equals0I
thf(fact_399_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_400_equals0D,axiom,
! [A: set_o,A2: $o] :
( ( A = bot_bot_set_o )
=> ~ ( member_o @ A2 @ A ) ) ).
% equals0D
thf(fact_401_equals0D,axiom,
! [A: set_list_nat,A2: list_nat] :
( ( A = bot_bot_set_list_nat )
=> ~ ( member_list_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_402_equals0D,axiom,
! [A: set_set_list_nat,A2: set_list_nat] :
( ( A = bot_bo3886227569956363488st_nat )
=> ~ ( member_set_list_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_403_equals0D,axiom,
! [A: set_set_set_list_nat,A2: set_set_list_nat] :
( ( A = bot_bo3499706412017099030st_nat )
=> ~ ( member1029098694177496419st_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_404_equals0D,axiom,
! [A: set_se5258582372428582328st_nat,A2: set_set_set_list_nat] :
( ( A = bot_bo1158166727579713100st_nat )
=> ~ ( member7304678173793621401st_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_405_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_406_emptyE,axiom,
! [A2: $o] :
~ ( member_o @ A2 @ bot_bot_set_o ) ).
% emptyE
thf(fact_407_emptyE,axiom,
! [A2: list_nat] :
~ ( member_list_nat @ A2 @ bot_bot_set_list_nat ) ).
% emptyE
thf(fact_408_emptyE,axiom,
! [A2: set_list_nat] :
~ ( member_set_list_nat @ A2 @ bot_bo3886227569956363488st_nat ) ).
% emptyE
thf(fact_409_emptyE,axiom,
! [A2: set_set_list_nat] :
~ ( member1029098694177496419st_nat @ A2 @ bot_bo3499706412017099030st_nat ) ).
% emptyE
thf(fact_410_emptyE,axiom,
! [A2: set_set_set_list_nat] :
~ ( member7304678173793621401st_nat @ A2 @ bot_bo1158166727579713100st_nat ) ).
% emptyE
thf(fact_411_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_412_Collect__mono__iff,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
= ( ! [X2: list_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_413_Collect__mono__iff,axiom,
! [P: set_list_nat > $o,Q: set_list_nat > $o] :
( ( ord_le1068707526560357548st_nat @ ( collect_set_list_nat @ P ) @ ( collect_set_list_nat @ Q ) )
= ( ! [X2: set_list_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_414_Collect__mono__iff,axiom,
! [P: set_set_list_nat > $o,Q: set_set_list_nat > $o] :
( ( ord_le7100322305783427298st_nat @ ( collec4691811733418234273st_nat @ P ) @ ( collec4691811733418234273st_nat @ Q ) )
= ( ! [X2: set_set_list_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_415_set__eq__subset,axiom,
( ( ^ [Y4: set_list_nat,Z: set_list_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_list_nat,B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B3 )
& ( ord_le6045566169113846134st_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_416_set__eq__subset,axiom,
( ( ^ [Y4: set_set_list_nat,Z: set_set_list_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_set_list_nat,B3: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A3 @ B3 )
& ( ord_le1068707526560357548st_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_417_set__eq__subset,axiom,
( ( ^ [Y4: set_set_set_list_nat,Z: set_set_set_list_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_set_set_list_nat,B3: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A3 @ B3 )
& ( ord_le7100322305783427298st_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_418_subset__trans,axiom,
! [A: set_list_nat,B2: set_list_nat,C2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B2 )
=> ( ( ord_le6045566169113846134st_nat @ B2 @ C2 )
=> ( ord_le6045566169113846134st_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_419_subset__trans,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat,C2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A @ B2 )
=> ( ( ord_le1068707526560357548st_nat @ B2 @ C2 )
=> ( ord_le1068707526560357548st_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_420_subset__trans,axiom,
! [A: set_set_set_list_nat,B2: set_set_set_list_nat,C2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A @ B2 )
=> ( ( ord_le7100322305783427298st_nat @ B2 @ C2 )
=> ( ord_le7100322305783427298st_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_421_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_422_Collect__mono,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X4: list_nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_423_Collect__mono,axiom,
! [P: set_list_nat > $o,Q: set_list_nat > $o] :
( ! [X4: set_list_nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le1068707526560357548st_nat @ ( collect_set_list_nat @ P ) @ ( collect_set_list_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_424_Collect__mono,axiom,
! [P: set_set_list_nat > $o,Q: set_set_list_nat > $o] :
( ! [X4: set_set_list_nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le7100322305783427298st_nat @ ( collec4691811733418234273st_nat @ P ) @ ( collec4691811733418234273st_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_425_subset__refl,axiom,
! [A: set_list_nat] : ( ord_le6045566169113846134st_nat @ A @ A ) ).
% subset_refl
thf(fact_426_subset__refl,axiom,
! [A: set_set_list_nat] : ( ord_le1068707526560357548st_nat @ A @ A ) ).
% subset_refl
thf(fact_427_subset__refl,axiom,
! [A: set_set_set_list_nat] : ( ord_le7100322305783427298st_nat @ A @ A ) ).
% subset_refl
thf(fact_428_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_429_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
! [T: $o] :
( ( member_o @ T @ A3 )
=> ( member_o @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_430_subset__iff,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A3: set_list_nat,B3: set_list_nat] :
! [T: list_nat] :
( ( member_list_nat @ T @ A3 )
=> ( member_list_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_431_subset__iff,axiom,
( ord_le1068707526560357548st_nat
= ( ^ [A3: set_set_list_nat,B3: set_set_list_nat] :
! [T: set_list_nat] :
( ( member_set_list_nat @ T @ A3 )
=> ( member_set_list_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_432_subset__iff,axiom,
( ord_le7100322305783427298st_nat
= ( ^ [A3: set_set_set_list_nat,B3: set_set_set_list_nat] :
! [T: set_set_list_nat] :
( ( member1029098694177496419st_nat @ T @ A3 )
=> ( member1029098694177496419st_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_433_equalityD2,axiom,
! [A: set_list_nat,B2: set_list_nat] :
( ( A = B2 )
=> ( ord_le6045566169113846134st_nat @ B2 @ A ) ) ).
% equalityD2
thf(fact_434_equalityD2,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat] :
( ( A = B2 )
=> ( ord_le1068707526560357548st_nat @ B2 @ A ) ) ).
% equalityD2
thf(fact_435_equalityD2,axiom,
! [A: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ( A = B2 )
=> ( ord_le7100322305783427298st_nat @ B2 @ A ) ) ).
% equalityD2
thf(fact_436_equalityD1,axiom,
! [A: set_list_nat,B2: set_list_nat] :
( ( A = B2 )
=> ( ord_le6045566169113846134st_nat @ A @ B2 ) ) ).
% equalityD1
thf(fact_437_equalityD1,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat] :
( ( A = B2 )
=> ( ord_le1068707526560357548st_nat @ A @ B2 ) ) ).
% equalityD1
thf(fact_438_equalityD1,axiom,
! [A: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ( A = B2 )
=> ( ord_le7100322305783427298st_nat @ A @ B2 ) ) ).
% equalityD1
thf(fact_439_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( member_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_440_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
! [X2: $o] :
( ( member_o @ X2 @ A3 )
=> ( member_o @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_441_subset__eq,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A3: set_list_nat,B3: set_list_nat] :
! [X2: list_nat] :
( ( member_list_nat @ X2 @ A3 )
=> ( member_list_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_442_subset__eq,axiom,
( ord_le1068707526560357548st_nat
= ( ^ [A3: set_set_list_nat,B3: set_set_list_nat] :
! [X2: set_list_nat] :
( ( member_set_list_nat @ X2 @ A3 )
=> ( member_set_list_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_443_subset__eq,axiom,
( ord_le7100322305783427298st_nat
= ( ^ [A3: set_set_set_list_nat,B3: set_set_set_list_nat] :
! [X2: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X2 @ A3 )
=> ( member1029098694177496419st_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_444_equalityE,axiom,
! [A: set_list_nat,B2: set_list_nat] :
( ( A = B2 )
=> ~ ( ( ord_le6045566169113846134st_nat @ A @ B2 )
=> ~ ( ord_le6045566169113846134st_nat @ B2 @ A ) ) ) ).
% equalityE
thf(fact_445_equalityE,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat] :
( ( A = B2 )
=> ~ ( ( ord_le1068707526560357548st_nat @ A @ B2 )
=> ~ ( ord_le1068707526560357548st_nat @ B2 @ A ) ) ) ).
% equalityE
thf(fact_446_equalityE,axiom,
! [A: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ( A = B2 )
=> ~ ( ( ord_le7100322305783427298st_nat @ A @ B2 )
=> ~ ( ord_le7100322305783427298st_nat @ B2 @ A ) ) ) ).
% equalityE
thf(fact_447_subsetD,axiom,
! [A: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_448_subsetD,axiom,
! [A: set_o,B2: set_o,C: $o] :
( ( ord_less_eq_set_o @ A @ B2 )
=> ( ( member_o @ C @ A )
=> ( member_o @ C @ B2 ) ) ) ).
% subsetD
thf(fact_449_subsetD,axiom,
! [A: set_list_nat,B2: set_list_nat,C: list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B2 )
=> ( ( member_list_nat @ C @ A )
=> ( member_list_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_450_subsetD,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat,C: set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A @ B2 )
=> ( ( member_set_list_nat @ C @ A )
=> ( member_set_list_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_451_subsetD,axiom,
! [A: set_set_set_list_nat,B2: set_set_set_list_nat,C: set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A @ B2 )
=> ( ( member1029098694177496419st_nat @ C @ A )
=> ( member1029098694177496419st_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_452_in__mono,axiom,
! [A: set_nat,B2: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_453_in__mono,axiom,
! [A: set_o,B2: set_o,X3: $o] :
( ( ord_less_eq_set_o @ A @ B2 )
=> ( ( member_o @ X3 @ A )
=> ( member_o @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_454_in__mono,axiom,
! [A: set_list_nat,B2: set_list_nat,X3: list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B2 )
=> ( ( member_list_nat @ X3 @ A )
=> ( member_list_nat @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_455_in__mono,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat,X3: set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A @ B2 )
=> ( ( member_set_list_nat @ X3 @ A )
=> ( member_set_list_nat @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_456_in__mono,axiom,
! [A: set_set_set_list_nat,B2: set_set_set_list_nat,X3: set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A @ B2 )
=> ( ( member1029098694177496419st_nat @ X3 @ A )
=> ( member1029098694177496419st_nat @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_457_mk__disjoint__insert,axiom,
! [A2: set_list_nat,A: set_set_list_nat] :
( ( member_set_list_nat @ A2 @ A )
=> ? [B4: set_set_list_nat] :
( ( A
= ( insert_set_list_nat @ A2 @ B4 ) )
& ~ ( member_set_list_nat @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_458_mk__disjoint__insert,axiom,
! [A2: set_set_list_nat,A: set_set_set_list_nat] :
( ( member1029098694177496419st_nat @ A2 @ A )
=> ? [B4: set_set_set_list_nat] :
( ( A
= ( insert721039030076969596st_nat @ A2 @ B4 ) )
& ~ ( member1029098694177496419st_nat @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_459_mk__disjoint__insert,axiom,
! [A2: set_set_set_list_nat,A: set_se5258582372428582328st_nat] :
( ( member7304678173793621401st_nat @ A2 @ A )
=> ? [B4: set_se5258582372428582328st_nat] :
( ( A
= ( insert8008074249615255218st_nat @ A2 @ B4 ) )
& ~ ( member7304678173793621401st_nat @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_460_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ? [B4: set_nat] :
( ( A
= ( insert_nat @ A2 @ B4 ) )
& ~ ( member_nat @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_461_mk__disjoint__insert,axiom,
! [A2: $o,A: set_o] :
( ( member_o @ A2 @ A )
=> ? [B4: set_o] :
( ( A
= ( insert_o @ A2 @ B4 ) )
& ~ ( member_o @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_462_mk__disjoint__insert,axiom,
! [A2: list_nat,A: set_list_nat] :
( ( member_list_nat @ A2 @ A )
=> ? [B4: set_list_nat] :
( ( A
= ( insert_list_nat @ A2 @ B4 ) )
& ~ ( member_list_nat @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_463_insert__commute,axiom,
! [X3: set_list_nat,Y3: set_list_nat,A: set_set_list_nat] :
( ( insert_set_list_nat @ X3 @ ( insert_set_list_nat @ Y3 @ A ) )
= ( insert_set_list_nat @ Y3 @ ( insert_set_list_nat @ X3 @ A ) ) ) ).
% insert_commute
thf(fact_464_insert__commute,axiom,
! [X3: set_set_list_nat,Y3: set_set_list_nat,A: set_set_set_list_nat] :
( ( insert721039030076969596st_nat @ X3 @ ( insert721039030076969596st_nat @ Y3 @ A ) )
= ( insert721039030076969596st_nat @ Y3 @ ( insert721039030076969596st_nat @ X3 @ A ) ) ) ).
% insert_commute
thf(fact_465_insert__commute,axiom,
! [X3: set_set_set_list_nat,Y3: set_set_set_list_nat,A: set_se5258582372428582328st_nat] :
( ( insert8008074249615255218st_nat @ X3 @ ( insert8008074249615255218st_nat @ Y3 @ A ) )
= ( insert8008074249615255218st_nat @ Y3 @ ( insert8008074249615255218st_nat @ X3 @ A ) ) ) ).
% insert_commute
thf(fact_466_insert__eq__iff,axiom,
! [A2: set_list_nat,A: set_set_list_nat,B: set_list_nat,B2: set_set_list_nat] :
( ~ ( member_set_list_nat @ A2 @ A )
=> ( ~ ( member_set_list_nat @ B @ B2 )
=> ( ( ( insert_set_list_nat @ A2 @ A )
= ( insert_set_list_nat @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C4: set_set_list_nat] :
( ( A
= ( insert_set_list_nat @ B @ C4 ) )
& ~ ( member_set_list_nat @ B @ C4 )
& ( B2
= ( insert_set_list_nat @ A2 @ C4 ) )
& ~ ( member_set_list_nat @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_467_insert__eq__iff,axiom,
! [A2: set_set_list_nat,A: set_set_set_list_nat,B: set_set_list_nat,B2: set_set_set_list_nat] :
( ~ ( member1029098694177496419st_nat @ A2 @ A )
=> ( ~ ( member1029098694177496419st_nat @ B @ B2 )
=> ( ( ( insert721039030076969596st_nat @ A2 @ A )
= ( insert721039030076969596st_nat @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C4: set_set_set_list_nat] :
( ( A
= ( insert721039030076969596st_nat @ B @ C4 ) )
& ~ ( member1029098694177496419st_nat @ B @ C4 )
& ( B2
= ( insert721039030076969596st_nat @ A2 @ C4 ) )
& ~ ( member1029098694177496419st_nat @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_468_insert__eq__iff,axiom,
! [A2: set_set_set_list_nat,A: set_se5258582372428582328st_nat,B: set_set_set_list_nat,B2: set_se5258582372428582328st_nat] :
( ~ ( member7304678173793621401st_nat @ A2 @ A )
=> ( ~ ( member7304678173793621401st_nat @ B @ B2 )
=> ( ( ( insert8008074249615255218st_nat @ A2 @ A )
= ( insert8008074249615255218st_nat @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C4: set_se5258582372428582328st_nat] :
( ( A
= ( insert8008074249615255218st_nat @ B @ C4 ) )
& ~ ( member7304678173793621401st_nat @ B @ C4 )
& ( B2
= ( insert8008074249615255218st_nat @ A2 @ C4 ) )
& ~ ( member7304678173793621401st_nat @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_469_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B: nat,B2: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ B @ B2 )
=> ( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C4: set_nat] :
( ( A
= ( insert_nat @ B @ C4 ) )
& ~ ( member_nat @ B @ C4 )
& ( B2
= ( insert_nat @ A2 @ C4 ) )
& ~ ( member_nat @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_470_insert__eq__iff,axiom,
! [A2: $o,A: set_o,B: $o,B2: set_o] :
( ~ ( member_o @ A2 @ A )
=> ( ~ ( member_o @ B @ B2 )
=> ( ( ( insert_o @ A2 @ A )
= ( insert_o @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 = ~ B )
=> ? [C4: set_o] :
( ( A
= ( insert_o @ B @ C4 ) )
& ~ ( member_o @ B @ C4 )
& ( B2
= ( insert_o @ A2 @ C4 ) )
& ~ ( member_o @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_471_insert__eq__iff,axiom,
! [A2: list_nat,A: set_list_nat,B: list_nat,B2: set_list_nat] :
( ~ ( member_list_nat @ A2 @ A )
=> ( ~ ( member_list_nat @ B @ B2 )
=> ( ( ( insert_list_nat @ A2 @ A )
= ( insert_list_nat @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C4: set_list_nat] :
( ( A
= ( insert_list_nat @ B @ C4 ) )
& ~ ( member_list_nat @ B @ C4 )
& ( B2
= ( insert_list_nat @ A2 @ C4 ) )
& ~ ( member_list_nat @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_472_insert__absorb,axiom,
! [A2: set_list_nat,A: set_set_list_nat] :
( ( member_set_list_nat @ A2 @ A )
=> ( ( insert_set_list_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_473_insert__absorb,axiom,
! [A2: set_set_list_nat,A: set_set_set_list_nat] :
( ( member1029098694177496419st_nat @ A2 @ A )
=> ( ( insert721039030076969596st_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_474_insert__absorb,axiom,
! [A2: set_set_set_list_nat,A: set_se5258582372428582328st_nat] :
( ( member7304678173793621401st_nat @ A2 @ A )
=> ( ( insert8008074249615255218st_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_475_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_476_insert__absorb,axiom,
! [A2: $o,A: set_o] :
( ( member_o @ A2 @ A )
=> ( ( insert_o @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_477_insert__absorb,axiom,
! [A2: list_nat,A: set_list_nat] :
( ( member_list_nat @ A2 @ A )
=> ( ( insert_list_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_478_insert__ident,axiom,
! [X3: set_list_nat,A: set_set_list_nat,B2: set_set_list_nat] :
( ~ ( member_set_list_nat @ X3 @ A )
=> ( ~ ( member_set_list_nat @ X3 @ B2 )
=> ( ( ( insert_set_list_nat @ X3 @ A )
= ( insert_set_list_nat @ X3 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_479_insert__ident,axiom,
! [X3: set_set_list_nat,A: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ~ ( member1029098694177496419st_nat @ X3 @ A )
=> ( ~ ( member1029098694177496419st_nat @ X3 @ B2 )
=> ( ( ( insert721039030076969596st_nat @ X3 @ A )
= ( insert721039030076969596st_nat @ X3 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_480_insert__ident,axiom,
! [X3: set_set_set_list_nat,A: set_se5258582372428582328st_nat,B2: set_se5258582372428582328st_nat] :
( ~ ( member7304678173793621401st_nat @ X3 @ A )
=> ( ~ ( member7304678173793621401st_nat @ X3 @ B2 )
=> ( ( ( insert8008074249615255218st_nat @ X3 @ A )
= ( insert8008074249615255218st_nat @ X3 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_481_insert__ident,axiom,
! [X3: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat @ X3 @ A )
=> ( ~ ( member_nat @ X3 @ B2 )
=> ( ( ( insert_nat @ X3 @ A )
= ( insert_nat @ X3 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_482_insert__ident,axiom,
! [X3: $o,A: set_o,B2: set_o] :
( ~ ( member_o @ X3 @ A )
=> ( ~ ( member_o @ X3 @ B2 )
=> ( ( ( insert_o @ X3 @ A )
= ( insert_o @ X3 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_483_insert__ident,axiom,
! [X3: list_nat,A: set_list_nat,B2: set_list_nat] :
( ~ ( member_list_nat @ X3 @ A )
=> ( ~ ( member_list_nat @ X3 @ B2 )
=> ( ( ( insert_list_nat @ X3 @ A )
= ( insert_list_nat @ X3 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_484_Set_Oset__insert,axiom,
! [X3: set_list_nat,A: set_set_list_nat] :
( ( member_set_list_nat @ X3 @ A )
=> ~ ! [B4: set_set_list_nat] :
( ( A
= ( insert_set_list_nat @ X3 @ B4 ) )
=> ( member_set_list_nat @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_485_Set_Oset__insert,axiom,
! [X3: set_set_list_nat,A: set_set_set_list_nat] :
( ( member1029098694177496419st_nat @ X3 @ A )
=> ~ ! [B4: set_set_set_list_nat] :
( ( A
= ( insert721039030076969596st_nat @ X3 @ B4 ) )
=> ( member1029098694177496419st_nat @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_486_Set_Oset__insert,axiom,
! [X3: set_set_set_list_nat,A: set_se5258582372428582328st_nat] :
( ( member7304678173793621401st_nat @ X3 @ A )
=> ~ ! [B4: set_se5258582372428582328st_nat] :
( ( A
= ( insert8008074249615255218st_nat @ X3 @ B4 ) )
=> ( member7304678173793621401st_nat @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_487_Set_Oset__insert,axiom,
! [X3: nat,A: set_nat] :
( ( member_nat @ X3 @ A )
=> ~ ! [B4: set_nat] :
( ( A
= ( insert_nat @ X3 @ B4 ) )
=> ( member_nat @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_488_Set_Oset__insert,axiom,
! [X3: $o,A: set_o] :
( ( member_o @ X3 @ A )
=> ~ ! [B4: set_o] :
( ( A
= ( insert_o @ X3 @ B4 ) )
=> ( member_o @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_489_Set_Oset__insert,axiom,
! [X3: list_nat,A: set_list_nat] :
( ( member_list_nat @ X3 @ A )
=> ~ ! [B4: set_list_nat] :
( ( A
= ( insert_list_nat @ X3 @ B4 ) )
=> ( member_list_nat @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_490_insertI2,axiom,
! [A2: set_list_nat,B2: set_set_list_nat,B: set_list_nat] :
( ( member_set_list_nat @ A2 @ B2 )
=> ( member_set_list_nat @ A2 @ ( insert_set_list_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_491_insertI2,axiom,
! [A2: set_set_list_nat,B2: set_set_set_list_nat,B: set_set_list_nat] :
( ( member1029098694177496419st_nat @ A2 @ B2 )
=> ( member1029098694177496419st_nat @ A2 @ ( insert721039030076969596st_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_492_insertI2,axiom,
! [A2: set_set_set_list_nat,B2: set_se5258582372428582328st_nat,B: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ A2 @ B2 )
=> ( member7304678173793621401st_nat @ A2 @ ( insert8008074249615255218st_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_493_insertI2,axiom,
! [A2: nat,B2: set_nat,B: nat] :
( ( member_nat @ A2 @ B2 )
=> ( member_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_494_insertI2,axiom,
! [A2: $o,B2: set_o,B: $o] :
( ( member_o @ A2 @ B2 )
=> ( member_o @ A2 @ ( insert_o @ B @ B2 ) ) ) ).
% insertI2
thf(fact_495_insertI2,axiom,
! [A2: list_nat,B2: set_list_nat,B: list_nat] :
( ( member_list_nat @ A2 @ B2 )
=> ( member_list_nat @ A2 @ ( insert_list_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_496_insertI1,axiom,
! [A2: set_list_nat,B2: set_set_list_nat] : ( member_set_list_nat @ A2 @ ( insert_set_list_nat @ A2 @ B2 ) ) ).
% insertI1
thf(fact_497_insertI1,axiom,
! [A2: set_set_list_nat,B2: set_set_set_list_nat] : ( member1029098694177496419st_nat @ A2 @ ( insert721039030076969596st_nat @ A2 @ B2 ) ) ).
% insertI1
thf(fact_498_insertI1,axiom,
! [A2: set_set_set_list_nat,B2: set_se5258582372428582328st_nat] : ( member7304678173793621401st_nat @ A2 @ ( insert8008074249615255218st_nat @ A2 @ B2 ) ) ).
% insertI1
thf(fact_499_insertI1,axiom,
! [A2: nat,B2: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B2 ) ) ).
% insertI1
thf(fact_500_insertI1,axiom,
! [A2: $o,B2: set_o] : ( member_o @ A2 @ ( insert_o @ A2 @ B2 ) ) ).
% insertI1
thf(fact_501_insertI1,axiom,
! [A2: list_nat,B2: set_list_nat] : ( member_list_nat @ A2 @ ( insert_list_nat @ A2 @ B2 ) ) ).
% insertI1
thf(fact_502_insertE,axiom,
! [A2: set_list_nat,B: set_list_nat,A: set_set_list_nat] :
( ( member_set_list_nat @ A2 @ ( insert_set_list_nat @ B @ A ) )
=> ( ( A2 != B )
=> ( member_set_list_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_503_insertE,axiom,
! [A2: set_set_list_nat,B: set_set_list_nat,A: set_set_set_list_nat] :
( ( member1029098694177496419st_nat @ A2 @ ( insert721039030076969596st_nat @ B @ A ) )
=> ( ( A2 != B )
=> ( member1029098694177496419st_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_504_insertE,axiom,
! [A2: set_set_set_list_nat,B: set_set_set_list_nat,A: set_se5258582372428582328st_nat] :
( ( member7304678173793621401st_nat @ A2 @ ( insert8008074249615255218st_nat @ B @ A ) )
=> ( ( A2 != B )
=> ( member7304678173793621401st_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_505_insertE,axiom,
! [A2: nat,B: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
=> ( ( A2 != B )
=> ( member_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_506_insertE,axiom,
! [A2: $o,B: $o,A: set_o] :
( ( member_o @ A2 @ ( insert_o @ B @ A ) )
=> ( ( A2 = ~ B )
=> ( member_o @ A2 @ A ) ) ) ).
% insertE
thf(fact_507_insertE,axiom,
! [A2: list_nat,B: list_nat,A: set_list_nat] :
( ( member_list_nat @ A2 @ ( insert_list_nat @ B @ A ) )
=> ( ( A2 != B )
=> ( member_list_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_508_sum_Oreindex__bij__witness,axiom,
! [S: set_o,I3: nat > $o,J: $o > nat,T2: set_nat,H: nat > int,G: $o > int] :
( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( ( I3 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( member_nat @ ( J @ A4 ) @ T2 ) )
=> ( ! [B5: nat] :
( ( member_nat @ B5 @ T2 )
=> ( ( J @ ( I3 @ B5 ) )
= B5 ) )
=> ( ! [B5: nat] :
( ( member_nat @ B5 @ T2 )
=> ( member_o @ ( I3 @ B5 ) @ S ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8505340233167759370_o_int @ G @ S )
= ( groups3539618377306564664at_int @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_509_sum_Oreindex__bij__witness,axiom,
! [S: set_o,I3: nat > $o,J: $o > nat,T2: set_nat,H: nat > nat,G: $o > nat] :
( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( ( I3 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( member_nat @ ( J @ A4 ) @ T2 ) )
=> ( ! [B5: nat] :
( ( member_nat @ B5 @ T2 )
=> ( ( J @ ( I3 @ B5 ) )
= B5 ) )
=> ( ! [B5: nat] :
( ( member_nat @ B5 @ T2 )
=> ( member_o @ ( I3 @ B5 ) @ S ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8507830703676809646_o_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_510_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I3: $o > nat,J: nat > $o,T2: set_o,H: $o > int,G: nat > int] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( I3 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( member_o @ ( J @ A4 ) @ T2 ) )
=> ( ! [B5: $o] :
( ( member_o @ B5 @ T2 )
=> ( ( J @ ( I3 @ B5 ) )
= B5 ) )
=> ( ! [B5: $o] :
( ( member_o @ B5 @ T2 )
=> ( member_nat @ ( I3 @ B5 ) @ S ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ S )
= ( groups8505340233167759370_o_int @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_511_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I3: nat > nat,J: nat > nat,T2: set_nat,H: nat > int,G: nat > int] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( I3 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( member_nat @ ( J @ A4 ) @ T2 ) )
=> ( ! [B5: nat] :
( ( member_nat @ B5 @ T2 )
=> ( ( J @ ( I3 @ B5 ) )
= B5 ) )
=> ( ! [B5: nat] :
( ( member_nat @ B5 @ T2 )
=> ( member_nat @ ( I3 @ B5 ) @ S ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ S )
= ( groups3539618377306564664at_int @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_512_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I3: $o > nat,J: nat > $o,T2: set_o,H: $o > nat,G: nat > nat] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( I3 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( member_o @ ( J @ A4 ) @ T2 ) )
=> ( ! [B5: $o] :
( ( member_o @ B5 @ T2 )
=> ( ( J @ ( I3 @ B5 ) )
= B5 ) )
=> ( ! [B5: $o] :
( ( member_o @ B5 @ T2 )
=> ( member_nat @ ( I3 @ B5 ) @ S ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups8507830703676809646_o_nat @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_513_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I3: nat > nat,J: nat > nat,T2: set_nat,H: nat > nat,G: nat > nat] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( I3 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( member_nat @ ( J @ A4 ) @ T2 ) )
=> ( ! [B5: nat] :
( ( member_nat @ B5 @ T2 )
=> ( ( J @ ( I3 @ B5 ) )
= B5 ) )
=> ( ! [B5: nat] :
( ( member_nat @ B5 @ T2 )
=> ( member_nat @ ( I3 @ B5 ) @ S ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_514_sum_Oreindex__bij__witness,axiom,
! [S: set_o,I3: list_nat > $o,J: $o > list_nat,T2: set_list_nat,H: list_nat > int,G: $o > int] :
( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( ( I3 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( member_list_nat @ ( J @ A4 ) @ T2 ) )
=> ( ! [B5: list_nat] :
( ( member_list_nat @ B5 @ T2 )
=> ( ( J @ ( I3 @ B5 ) )
= B5 ) )
=> ( ! [B5: list_nat] :
( ( member_list_nat @ B5 @ T2 )
=> ( member_o @ ( I3 @ B5 ) @ S ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8505340233167759370_o_int @ G @ S )
= ( groups4393565826250045896at_int @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_515_sum_Oreindex__bij__witness,axiom,
! [S: set_o,I3: list_nat > $o,J: $o > list_nat,T2: set_list_nat,H: list_nat > nat,G: $o > nat] :
( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( ( I3 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( member_list_nat @ ( J @ A4 ) @ T2 ) )
=> ( ! [B5: list_nat] :
( ( member_list_nat @ B5 @ T2 )
=> ( ( J @ ( I3 @ B5 ) )
= B5 ) )
=> ( ! [B5: list_nat] :
( ( member_list_nat @ B5 @ T2 )
=> ( member_o @ ( I3 @ B5 ) @ S ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8507830703676809646_o_nat @ G @ S )
= ( groups4396056296759096172at_nat @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_516_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I3: list_nat > nat,J: nat > list_nat,T2: set_list_nat,H: list_nat > int,G: nat > int] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( I3 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( member_list_nat @ ( J @ A4 ) @ T2 ) )
=> ( ! [B5: list_nat] :
( ( member_list_nat @ B5 @ T2 )
=> ( ( J @ ( I3 @ B5 ) )
= B5 ) )
=> ( ! [B5: list_nat] :
( ( member_list_nat @ B5 @ T2 )
=> ( member_nat @ ( I3 @ B5 ) @ S ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ S )
= ( groups4393565826250045896at_int @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_517_sum_Oreindex__bij__witness,axiom,
! [S: set_list_nat,I3: $o > list_nat,J: list_nat > $o,T2: set_o,H: $o > int,G: list_nat > int] :
( ! [A4: list_nat] :
( ( member_list_nat @ A4 @ S )
=> ( ( I3 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: list_nat] :
( ( member_list_nat @ A4 @ S )
=> ( member_o @ ( J @ A4 ) @ T2 ) )
=> ( ! [B5: $o] :
( ( member_o @ B5 @ T2 )
=> ( ( J @ ( I3 @ B5 ) )
= B5 ) )
=> ( ! [B5: $o] :
( ( member_o @ B5 @ T2 )
=> ( member_list_nat @ ( I3 @ B5 ) @ S ) )
=> ( ! [A4: list_nat] :
( ( member_list_nat @ A4 @ S )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups4393565826250045896at_int @ G @ S )
= ( groups8505340233167759370_o_int @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_518_sum_Oeq__general__inverses,axiom,
! [B2: set_nat,K: nat > $o,A: set_o,H: $o > nat,Gamma: nat > int,Phi: $o > int] :
( ! [Y5: nat] :
( ( member_nat @ Y5 @ B2 )
=> ( ( member_o @ ( K @ Y5 ) @ A )
& ( ( H @ ( K @ Y5 ) )
= Y5 ) ) )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ( member_nat @ ( H @ X4 ) @ B2 )
& ( ( K @ ( H @ X4 ) )
= X4 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups8505340233167759370_o_int @ Phi @ A )
= ( groups3539618377306564664at_int @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_519_sum_Oeq__general__inverses,axiom,
! [B2: set_nat,K: nat > $o,A: set_o,H: $o > nat,Gamma: nat > nat,Phi: $o > nat] :
( ! [Y5: nat] :
( ( member_nat @ Y5 @ B2 )
=> ( ( member_o @ ( K @ Y5 ) @ A )
& ( ( H @ ( K @ Y5 ) )
= Y5 ) ) )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ( member_nat @ ( H @ X4 ) @ B2 )
& ( ( K @ ( H @ X4 ) )
= X4 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups8507830703676809646_o_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_520_sum_Oeq__general__inverses,axiom,
! [B2: set_o,K: $o > nat,A: set_nat,H: nat > $o,Gamma: $o > int,Phi: nat > int] :
( ! [Y5: $o] :
( ( member_o @ Y5 @ B2 )
=> ( ( member_nat @ ( K @ Y5 ) @ A )
& ( ( H @ ( K @ Y5 ) )
= Y5 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ( member_o @ ( H @ X4 ) @ B2 )
& ( ( K @ ( H @ X4 ) )
= X4 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3539618377306564664at_int @ Phi @ A )
= ( groups8505340233167759370_o_int @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_521_sum_Oeq__general__inverses,axiom,
! [B2: set_nat,K: nat > nat,A: set_nat,H: nat > nat,Gamma: nat > int,Phi: nat > int] :
( ! [Y5: nat] :
( ( member_nat @ Y5 @ B2 )
=> ( ( member_nat @ ( K @ Y5 ) @ A )
& ( ( H @ ( K @ Y5 ) )
= Y5 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ( member_nat @ ( H @ X4 ) @ B2 )
& ( ( K @ ( H @ X4 ) )
= X4 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3539618377306564664at_int @ Phi @ A )
= ( groups3539618377306564664at_int @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_522_sum_Oeq__general__inverses,axiom,
! [B2: set_o,K: $o > nat,A: set_nat,H: nat > $o,Gamma: $o > nat,Phi: nat > nat] :
( ! [Y5: $o] :
( ( member_o @ Y5 @ B2 )
=> ( ( member_nat @ ( K @ Y5 ) @ A )
& ( ( H @ ( K @ Y5 ) )
= Y5 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ( member_o @ ( H @ X4 ) @ B2 )
& ( ( K @ ( H @ X4 ) )
= X4 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups8507830703676809646_o_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_523_sum_Oeq__general__inverses,axiom,
! [B2: set_nat,K: nat > nat,A: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y5: nat] :
( ( member_nat @ Y5 @ B2 )
=> ( ( member_nat @ ( K @ Y5 ) @ A )
& ( ( H @ ( K @ Y5 ) )
= Y5 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ( member_nat @ ( H @ X4 ) @ B2 )
& ( ( K @ ( H @ X4 ) )
= X4 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_524_sum_Oeq__general__inverses,axiom,
! [B2: set_list_nat,K: list_nat > $o,A: set_o,H: $o > list_nat,Gamma: list_nat > int,Phi: $o > int] :
( ! [Y5: list_nat] :
( ( member_list_nat @ Y5 @ B2 )
=> ( ( member_o @ ( K @ Y5 ) @ A )
& ( ( H @ ( K @ Y5 ) )
= Y5 ) ) )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ( member_list_nat @ ( H @ X4 ) @ B2 )
& ( ( K @ ( H @ X4 ) )
= X4 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups8505340233167759370_o_int @ Phi @ A )
= ( groups4393565826250045896at_int @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_525_sum_Oeq__general__inverses,axiom,
! [B2: set_list_nat,K: list_nat > $o,A: set_o,H: $o > list_nat,Gamma: list_nat > nat,Phi: $o > nat] :
( ! [Y5: list_nat] :
( ( member_list_nat @ Y5 @ B2 )
=> ( ( member_o @ ( K @ Y5 ) @ A )
& ( ( H @ ( K @ Y5 ) )
= Y5 ) ) )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ( member_list_nat @ ( H @ X4 ) @ B2 )
& ( ( K @ ( H @ X4 ) )
= X4 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups8507830703676809646_o_nat @ Phi @ A )
= ( groups4396056296759096172at_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_526_sum_Oeq__general__inverses,axiom,
! [B2: set_list_nat,K: list_nat > nat,A: set_nat,H: nat > list_nat,Gamma: list_nat > int,Phi: nat > int] :
( ! [Y5: list_nat] :
( ( member_list_nat @ Y5 @ B2 )
=> ( ( member_nat @ ( K @ Y5 ) @ A )
& ( ( H @ ( K @ Y5 ) )
= Y5 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ( member_list_nat @ ( H @ X4 ) @ B2 )
& ( ( K @ ( H @ X4 ) )
= X4 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3539618377306564664at_int @ Phi @ A )
= ( groups4393565826250045896at_int @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_527_sum_Oeq__general__inverses,axiom,
! [B2: set_o,K: $o > list_nat,A: set_list_nat,H: list_nat > $o,Gamma: $o > int,Phi: list_nat > int] :
( ! [Y5: $o] :
( ( member_o @ Y5 @ B2 )
=> ( ( member_list_nat @ ( K @ Y5 ) @ A )
& ( ( H @ ( K @ Y5 ) )
= Y5 ) ) )
=> ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
=> ( ( member_o @ ( H @ X4 ) @ B2 )
& ( ( K @ ( H @ X4 ) )
= X4 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups4393565826250045896at_int @ Phi @ A )
= ( groups8505340233167759370_o_int @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_528_sum_Oeq__general,axiom,
! [B2: set_nat,A: set_o,H: $o > nat,Gamma: nat > int,Phi: $o > int] :
( ! [Y5: nat] :
( ( member_nat @ Y5 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A )
& ( ( H @ X6 )
= Y5 )
& ! [Ya: $o] :
( ( ( member_o @ Ya @ A )
& ( ( H @ Ya )
= Y5 ) )
=> ( Ya = X6 ) ) ) )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ( member_nat @ ( H @ X4 ) @ B2 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups8505340233167759370_o_int @ Phi @ A )
= ( groups3539618377306564664at_int @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_529_sum_Oeq__general,axiom,
! [B2: set_nat,A: set_o,H: $o > nat,Gamma: nat > nat,Phi: $o > nat] :
( ! [Y5: nat] :
( ( member_nat @ Y5 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A )
& ( ( H @ X6 )
= Y5 )
& ! [Ya: $o] :
( ( ( member_o @ Ya @ A )
& ( ( H @ Ya )
= Y5 ) )
=> ( Ya = X6 ) ) ) )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ( member_nat @ ( H @ X4 ) @ B2 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups8507830703676809646_o_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_530_sum_Oeq__general,axiom,
! [B2: set_o,A: set_nat,H: nat > $o,Gamma: $o > int,Phi: nat > int] :
( ! [Y5: $o] :
( ( member_o @ Y5 @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ( H @ X6 )
= Y5 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y5 ) )
=> ( Ya = X6 ) ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ( member_o @ ( H @ X4 ) @ B2 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3539618377306564664at_int @ Phi @ A )
= ( groups8505340233167759370_o_int @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_531_sum_Oeq__general,axiom,
! [B2: set_nat,A: set_nat,H: nat > nat,Gamma: nat > int,Phi: nat > int] :
( ! [Y5: nat] :
( ( member_nat @ Y5 @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ( H @ X6 )
= Y5 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y5 ) )
=> ( Ya = X6 ) ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ( member_nat @ ( H @ X4 ) @ B2 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3539618377306564664at_int @ Phi @ A )
= ( groups3539618377306564664at_int @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_532_sum_Oeq__general,axiom,
! [B2: set_o,A: set_nat,H: nat > $o,Gamma: $o > nat,Phi: nat > nat] :
( ! [Y5: $o] :
( ( member_o @ Y5 @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ( H @ X6 )
= Y5 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y5 ) )
=> ( Ya = X6 ) ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ( member_o @ ( H @ X4 ) @ B2 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups8507830703676809646_o_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_533_sum_Oeq__general,axiom,
! [B2: set_nat,A: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y5: nat] :
( ( member_nat @ Y5 @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ( H @ X6 )
= Y5 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y5 ) )
=> ( Ya = X6 ) ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ( member_nat @ ( H @ X4 ) @ B2 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_534_sum_Oeq__general,axiom,
! [B2: set_list_nat,A: set_o,H: $o > list_nat,Gamma: list_nat > int,Phi: $o > int] :
( ! [Y5: list_nat] :
( ( member_list_nat @ Y5 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A )
& ( ( H @ X6 )
= Y5 )
& ! [Ya: $o] :
( ( ( member_o @ Ya @ A )
& ( ( H @ Ya )
= Y5 ) )
=> ( Ya = X6 ) ) ) )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ( member_list_nat @ ( H @ X4 ) @ B2 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups8505340233167759370_o_int @ Phi @ A )
= ( groups4393565826250045896at_int @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_535_sum_Oeq__general,axiom,
! [B2: set_list_nat,A: set_o,H: $o > list_nat,Gamma: list_nat > nat,Phi: $o > nat] :
( ! [Y5: list_nat] :
( ( member_list_nat @ Y5 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A )
& ( ( H @ X6 )
= Y5 )
& ! [Ya: $o] :
( ( ( member_o @ Ya @ A )
& ( ( H @ Ya )
= Y5 ) )
=> ( Ya = X6 ) ) ) )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ( member_list_nat @ ( H @ X4 ) @ B2 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups8507830703676809646_o_nat @ Phi @ A )
= ( groups4396056296759096172at_nat @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_536_sum_Oeq__general,axiom,
! [B2: set_list_nat,A: set_nat,H: nat > list_nat,Gamma: list_nat > int,Phi: nat > int] :
( ! [Y5: list_nat] :
( ( member_list_nat @ Y5 @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ( H @ X6 )
= Y5 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y5 ) )
=> ( Ya = X6 ) ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ( member_list_nat @ ( H @ X4 ) @ B2 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups3539618377306564664at_int @ Phi @ A )
= ( groups4393565826250045896at_int @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_537_sum_Oeq__general,axiom,
! [B2: set_o,A: set_list_nat,H: list_nat > $o,Gamma: $o > int,Phi: list_nat > int] :
( ! [Y5: $o] :
( ( member_o @ Y5 @ B2 )
=> ? [X6: list_nat] :
( ( member_list_nat @ X6 @ A )
& ( ( H @ X6 )
= Y5 )
& ! [Ya: list_nat] :
( ( ( member_list_nat @ Ya @ A )
& ( ( H @ Ya )
= Y5 ) )
=> ( Ya = X6 ) ) ) )
=> ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
=> ( ( member_o @ ( H @ X4 ) @ B2 )
& ( ( Gamma @ ( H @ X4 ) )
= ( Phi @ X4 ) ) ) )
=> ( ( groups4393565826250045896at_int @ Phi @ A )
= ( groups8505340233167759370_o_int @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_538_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 )
=> ( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ B2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups7312845317294741502at_int @ G @ A )
= ( groups7312845317294741502at_int @ H @ B2 ) ) ) ) ).
% sum.cong
thf(fact_539_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 )
=> ( ! [X4: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X4 @ B2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups7004213669654646580at_int @ G @ A )
= ( groups7004213669654646580at_int @ H @ B2 ) ) ) ) ).
% sum.cong
thf(fact_540_sum_Ocong,axiom,
! [A: set_nat,B2: set_nat,G: nat > int,H: nat > int] :
( ( A = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ A )
= ( groups3539618377306564664at_int @ H @ B2 ) ) ) ) ).
% sum.cong
thf(fact_541_sum_Ocong,axiom,
! [A: set_list_nat,B2: set_list_nat,G: list_nat > int,H: list_nat > int] :
( ( A = B2 )
=> ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ B2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups4393565826250045896at_int @ G @ A )
= ( groups4393565826250045896at_int @ H @ B2 ) ) ) ) ).
% sum.cong
thf(fact_542_sum_Ocong,axiom,
! [A: set_nat,B2: set_nat,G: nat > nat,H: nat > nat] :
( ( A = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= ( groups3542108847815614940at_nat @ H @ B2 ) ) ) ) ).
% sum.cong
thf(fact_543_sum_Ocong,axiom,
! [A: set_list_nat,B2: set_list_nat,G: list_nat > nat,H: list_nat > nat] :
( ( A = B2 )
=> ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ B2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups4396056296759096172at_nat @ G @ A )
= ( groups4396056296759096172at_nat @ H @ B2 ) ) ) ) ).
% sum.cong
thf(fact_544_sum_Ocong,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat,G: set_list_nat > nat,H: set_list_nat > nat] :
( ( A = B2 )
=> ( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ B2 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups7315335787803791778at_nat @ G @ A )
= ( groups7315335787803791778at_nat @ H @ B2 ) ) ) ) ).
% sum.cong
thf(fact_545_sum_Oreindex__cong,axiom,
! [L: $o > nat,B2: set_o,A: set_nat,G: nat > int,H: $o > int] :
( ( inj_on_o_nat @ L @ B2 )
=> ( ( A
= ( image_o_nat @ L @ B2 ) )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( G @ ( L @ X4 ) )
= ( H @ X4 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ A )
= ( groups8505340233167759370_o_int @ H @ B2 ) ) ) ) ) ).
% sum.reindex_cong
thf(fact_546_sum_Oreindex__cong,axiom,
! [L: nat > nat,B2: set_nat,A: set_nat,G: nat > int,H: nat > int] :
( ( inj_on_nat_nat @ L @ B2 )
=> ( ( A
= ( image_nat_nat @ L @ B2 ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( G @ ( L @ X4 ) )
= ( H @ X4 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ A )
= ( groups3539618377306564664at_int @ H @ B2 ) ) ) ) ) ).
% sum.reindex_cong
thf(fact_547_sum_Oreindex__cong,axiom,
! [L: $o > nat,B2: set_o,A: set_nat,G: nat > nat,H: $o > nat] :
( ( inj_on_o_nat @ L @ B2 )
=> ( ( A
= ( image_o_nat @ L @ B2 ) )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( G @ ( L @ X4 ) )
= ( H @ X4 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= ( groups8507830703676809646_o_nat @ H @ B2 ) ) ) ) ) ).
% sum.reindex_cong
thf(fact_548_sum_Oreindex__cong,axiom,
! [L: nat > nat,B2: set_nat,A: set_nat,G: nat > nat,H: nat > nat] :
( ( inj_on_nat_nat @ L @ B2 )
=> ( ( A
= ( image_nat_nat @ L @ B2 ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( G @ ( L @ X4 ) )
= ( H @ X4 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= ( groups3542108847815614940at_nat @ H @ B2 ) ) ) ) ) ).
% sum.reindex_cong
thf(fact_549_sum_Oreindex__cong,axiom,
! [L: list_nat > nat,B2: set_list_nat,A: set_nat,G: nat > int,H: list_nat > int] :
( ( inj_on_list_nat_nat @ L @ B2 )
=> ( ( A
= ( image_list_nat_nat @ L @ B2 ) )
=> ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ B2 )
=> ( ( G @ ( L @ X4 ) )
= ( H @ X4 ) ) )
=> ( ( groups3539618377306564664at_int @ G @ A )
= ( groups4393565826250045896at_int @ H @ B2 ) ) ) ) ) ).
% sum.reindex_cong
thf(fact_550_sum_Oreindex__cong,axiom,
! [L: $o > list_nat,B2: set_o,A: set_list_nat,G: list_nat > int,H: $o > int] :
( ( inj_on_o_list_nat @ L @ B2 )
=> ( ( A
= ( image_o_list_nat @ L @ B2 ) )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( G @ ( L @ X4 ) )
= ( H @ X4 ) ) )
=> ( ( groups4393565826250045896at_int @ G @ A )
= ( groups8505340233167759370_o_int @ H @ B2 ) ) ) ) ) ).
% sum.reindex_cong
thf(fact_551_sum_Oreindex__cong,axiom,
! [L: nat > list_nat,B2: set_nat,A: set_list_nat,G: list_nat > int,H: nat > int] :
( ( inj_on_nat_list_nat @ L @ B2 )
=> ( ( A
= ( image_nat_list_nat @ L @ B2 ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( G @ ( L @ X4 ) )
= ( H @ X4 ) ) )
=> ( ( groups4393565826250045896at_int @ G @ A )
= ( groups3539618377306564664at_int @ H @ B2 ) ) ) ) ) ).
% sum.reindex_cong
thf(fact_552_sum_Oreindex__cong,axiom,
! [L: list_nat > nat,B2: set_list_nat,A: set_nat,G: nat > nat,H: list_nat > nat] :
( ( inj_on_list_nat_nat @ L @ B2 )
=> ( ( A
= ( image_list_nat_nat @ L @ B2 ) )
=> ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ B2 )
=> ( ( G @ ( L @ X4 ) )
= ( H @ X4 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= ( groups4396056296759096172at_nat @ H @ B2 ) ) ) ) ) ).
% sum.reindex_cong
thf(fact_553_sum_Oreindex__cong,axiom,
! [L: $o > list_nat,B2: set_o,A: set_list_nat,G: list_nat > nat,H: $o > nat] :
( ( inj_on_o_list_nat @ L @ B2 )
=> ( ( A
= ( image_o_list_nat @ L @ B2 ) )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( G @ ( L @ X4 ) )
= ( H @ X4 ) ) )
=> ( ( groups4396056296759096172at_nat @ G @ A )
= ( groups8507830703676809646_o_nat @ H @ B2 ) ) ) ) ) ).
% sum.reindex_cong
thf(fact_554_sum_Oreindex__cong,axiom,
! [L: nat > list_nat,B2: set_nat,A: set_list_nat,G: list_nat > nat,H: nat > nat] :
( ( inj_on_nat_list_nat @ L @ B2 )
=> ( ( A
= ( image_nat_list_nat @ L @ B2 ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( G @ ( L @ X4 ) )
= ( H @ X4 ) ) )
=> ( ( groups4396056296759096172at_nat @ G @ A )
= ( groups3542108847815614940at_nat @ H @ B2 ) ) ) ) ) ).
% sum.reindex_cong
thf(fact_555_wellorder__InfI,axiom,
! [K: nat,A: set_nat] :
( ( member_nat @ K @ A )
=> ( member_nat @ ( complete_Inf_Inf_nat @ A ) @ A ) ) ).
% wellorder_InfI
thf(fact_556_DiffD2,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) )
=> ~ ( member_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_557_DiffD2,axiom,
! [C: $o,A: set_o,B2: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A @ B2 ) )
=> ~ ( member_o @ C @ B2 ) ) ).
% DiffD2
thf(fact_558_DiffD2,axiom,
! [C: list_nat,A: set_list_nat,B2: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A @ B2 ) )
=> ~ ( member_list_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_559_DiffD2,axiom,
! [C: set_list_nat,A: set_set_list_nat,B2: set_set_list_nat] :
( ( member_set_list_nat @ C @ ( minus_1490147023502258853st_nat @ A @ B2 ) )
=> ~ ( member_set_list_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_560_DiffD2,axiom,
! [C: set_set_list_nat,A: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ( member1029098694177496419st_nat @ C @ ( minus_7520675481749758683st_nat @ A @ B2 ) )
=> ~ ( member1029098694177496419st_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_561_DiffD1,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) )
=> ( member_nat @ C @ A ) ) ).
% DiffD1
thf(fact_562_DiffD1,axiom,
! [C: $o,A: set_o,B2: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A @ B2 ) )
=> ( member_o @ C @ A ) ) ).
% DiffD1
thf(fact_563_DiffD1,axiom,
! [C: list_nat,A: set_list_nat,B2: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A @ B2 ) )
=> ( member_list_nat @ C @ A ) ) ).
% DiffD1
thf(fact_564_DiffD1,axiom,
! [C: set_list_nat,A: set_set_list_nat,B2: set_set_list_nat] :
( ( member_set_list_nat @ C @ ( minus_1490147023502258853st_nat @ A @ B2 ) )
=> ( member_set_list_nat @ C @ A ) ) ).
% DiffD1
thf(fact_565_DiffD1,axiom,
! [C: set_set_list_nat,A: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ( member1029098694177496419st_nat @ C @ ( minus_7520675481749758683st_nat @ A @ B2 ) )
=> ( member1029098694177496419st_nat @ C @ A ) ) ).
% DiffD1
thf(fact_566_DiffE,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) )
=> ~ ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_567_DiffE,axiom,
! [C: $o,A: set_o,B2: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A @ B2 ) )
=> ~ ( ( member_o @ C @ A )
=> ( member_o @ C @ B2 ) ) ) ).
% DiffE
thf(fact_568_DiffE,axiom,
! [C: list_nat,A: set_list_nat,B2: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A @ B2 ) )
=> ~ ( ( member_list_nat @ C @ A )
=> ( member_list_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_569_DiffE,axiom,
! [C: set_list_nat,A: set_set_list_nat,B2: set_set_list_nat] :
( ( member_set_list_nat @ C @ ( minus_1490147023502258853st_nat @ A @ B2 ) )
=> ~ ( ( member_set_list_nat @ C @ A )
=> ( member_set_list_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_570_DiffE,axiom,
! [C: set_set_list_nat,A: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ( member1029098694177496419st_nat @ C @ ( minus_7520675481749758683st_nat @ A @ B2 ) )
=> ~ ( ( member1029098694177496419st_nat @ C @ A )
=> ( member1029098694177496419st_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_571_InterE,axiom,
! [A: nat,C2: set_set_nat,X: set_nat] :
( ( member_nat @ A @ ( comple7806235888213564991et_nat @ C2 ) )
=> ( ( member_set_nat @ X @ C2 )
=> ( member_nat @ A @ X ) ) ) ).
% InterE
thf(fact_572_InterE,axiom,
! [A: $o,C2: set_set_o,X: set_o] :
( ( member_o @ A @ ( comple3063163877087187839_set_o @ C2 ) )
=> ( ( member_set_o @ X @ C2 )
=> ( member_o @ A @ X ) ) ) ).
% InterE
thf(fact_573_InterE,axiom,
! [A: list_nat,C2: set_set_list_nat,X: set_list_nat] :
( ( member_list_nat @ A @ ( comple184543376406953807st_nat @ C2 ) )
=> ( ( member_set_list_nat @ X @ C2 )
=> ( member_list_nat @ A @ X ) ) ) ).
% InterE
thf(fact_574_InterD,axiom,
! [A: nat,C2: set_set_nat,X: set_nat] :
( ( member_nat @ A @ ( comple7806235888213564991et_nat @ C2 ) )
=> ( ( member_set_nat @ X @ C2 )
=> ( member_nat @ A @ X ) ) ) ).
% InterD
thf(fact_575_InterD,axiom,
! [A: $o,C2: set_set_o,X: set_o] :
( ( member_o @ A @ ( comple3063163877087187839_set_o @ C2 ) )
=> ( ( member_set_o @ X @ C2 )
=> ( member_o @ A @ X ) ) ) ).
% InterD
thf(fact_576_InterD,axiom,
! [A: list_nat,C2: set_set_list_nat,X: set_list_nat] :
( ( member_list_nat @ A @ ( comple184543376406953807st_nat @ C2 ) )
=> ( ( member_set_list_nat @ X @ C2 )
=> ( member_list_nat @ A @ X ) ) ) ).
% InterD
thf(fact_577_inj__on__Inter,axiom,
! [S: set_set_nat,F: nat > nat] :
( ( S != bot_bot_set_set_nat )
=> ( ! [A5: set_nat] :
( ( member_set_nat @ A5 @ S )
=> ( inj_on_nat_nat @ F @ A5 ) )
=> ( inj_on_nat_nat @ F @ ( comple7806235888213564991et_nat @ S ) ) ) ) ).
% inj_on_Inter
thf(fact_578_inj__on__Inter,axiom,
! [S: set_set_set_list_nat,F: set_list_nat > set_set_list_nat] :
( ( S != bot_bo3499706412017099030st_nat )
=> ( ! [A5: set_set_list_nat] :
( ( member1029098694177496419st_nat @ A5 @ S )
=> ( inj_on211238342484403029st_nat @ F @ A5 ) )
=> ( inj_on211238342484403029st_nat @ F @ ( comple8462666950445340293st_nat @ S ) ) ) ) ).
% inj_on_Inter
thf(fact_579_inj__on__Inter,axiom,
! [S: set_set_list_nat,F: list_nat > set_list_nat] :
( ( S != bot_bo3886227569956363488st_nat )
=> ( ! [A5: set_list_nat] :
( ( member_set_list_nat @ A5 @ S )
=> ( inj_on8624761805129053417st_nat @ F @ A5 ) )
=> ( inj_on8624761805129053417st_nat @ F @ ( comple184543376406953807st_nat @ S ) ) ) ) ).
% inj_on_Inter
thf(fact_580_Pow__top,axiom,
! [A: set_list_nat] : ( member_set_list_nat @ A @ ( pow_list_nat @ A ) ) ).
% Pow_top
thf(fact_581_Pow__top,axiom,
! [A: set_set_list_nat] : ( member1029098694177496419st_nat @ A @ ( pow_set_list_nat @ A ) ) ).
% Pow_top
thf(fact_582_Pow__top,axiom,
! [A: set_set_set_list_nat] : ( member7304678173793621401st_nat @ A @ ( pow_set_set_list_nat @ A ) ) ).
% Pow_top
thf(fact_583_sum_Oimage__eq,axiom,
! [G: set_list_nat > int,A: set_set_list_nat] :
( ( inj_on1513901456888566197at_int @ G @ A )
=> ( ( groups4538972089207619220nt_int
@ ^ [X2: int] : X2
@ ( image_1472855193719607881at_int @ G @ A ) )
= ( groups7312845317294741502at_int @ G @ A ) ) ) ).
% sum.image_eq
thf(fact_584_sum_Oimage__eq,axiom,
! [G: set_set_list_nat > int,A: set_set_set_list_nat] :
( ( inj_on2362802109445287147at_int @ G @ A )
=> ( ( groups4538972089207619220nt_int
@ ^ [X2: int] : X2
@ ( image_8275292939657174399at_int @ G @ A ) )
= ( groups7004213669654646580at_int @ G @ A ) ) ) ).
% sum.image_eq
thf(fact_585_sum_Oimage__eq,axiom,
! [G: nat > int,A: set_nat] :
( ( inj_on_nat_int @ G @ A )
=> ( ( groups4538972089207619220nt_int
@ ^ [X2: int] : X2
@ ( image_nat_int @ G @ A ) )
= ( groups3539618377306564664at_int @ G @ A ) ) ) ).
% sum.image_eq
thf(fact_586_sum_Oimage__eq,axiom,
! [G: list_nat > int,A: set_list_nat] :
( ( inj_on_list_nat_int @ G @ A )
=> ( ( groups4538972089207619220nt_int
@ ^ [X2: int] : X2
@ ( image_list_nat_int @ G @ A ) )
= ( groups4393565826250045896at_int @ G @ A ) ) ) ).
% sum.image_eq
thf(fact_587_sum_Oimage__eq,axiom,
! [G: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ G @ A )
=> ( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : X2
@ ( image_nat_nat @ G @ A ) )
= ( groups3542108847815614940at_nat @ G @ A ) ) ) ).
% sum.image_eq
thf(fact_588_sum_Oimage__eq,axiom,
! [G: list_nat > nat,A: set_list_nat] :
( ( inj_on_list_nat_nat @ G @ A )
=> ( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : X2
@ ( image_list_nat_nat @ G @ A ) )
= ( groups4396056296759096172at_nat @ G @ A ) ) ) ).
% sum.image_eq
thf(fact_589_sum_Oimage__eq,axiom,
! [G: set_list_nat > nat,A: set_set_list_nat] :
( ( inj_on1516391927397616473at_nat @ G @ A )
=> ( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : X2
@ ( image_1475345664228658157at_nat @ G @ A ) )
= ( groups7315335787803791778at_nat @ G @ A ) ) ) ).
% sum.image_eq
thf(fact_590_Compr__image__eq,axiom,
! [F: $o > $o,A: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_o_o @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_o_o @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_591_Compr__image__eq,axiom,
! [F: nat > $o,A: set_nat,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_nat_o @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_nat_o @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_592_Compr__image__eq,axiom,
! [F: $o > nat,A: set_o,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_o_nat @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_o_nat @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_593_Compr__image__eq,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_594_Compr__image__eq,axiom,
! [F: list_nat > $o,A: set_list_nat,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_list_nat_o @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_list_nat_o @ F
@ ( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_595_Compr__image__eq,axiom,
! [F: $o > list_nat,A: set_o,P: list_nat > $o] :
( ( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ ( image_o_list_nat @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_o_list_nat @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_596_Compr__image__eq,axiom,
! [F: nat > list_nat,A: set_nat,P: list_nat > $o] :
( ( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ ( image_nat_list_nat @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_nat_list_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_597_Compr__image__eq,axiom,
! [F: list_nat > nat,A: set_list_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_list_nat_nat @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_list_nat_nat @ F
@ ( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_598_Compr__image__eq,axiom,
! [F: set_list_nat > $o,A: set_set_list_nat,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_set_list_nat_o @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_set_list_nat_o @ F
@ ( collect_set_list_nat
@ ^ [X2: set_list_nat] :
( ( member_set_list_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_599_Compr__image__eq,axiom,
! [F: $o > set_list_nat,A: set_o,P: set_list_nat > $o] :
( ( collect_set_list_nat
@ ^ [X2: set_list_nat] :
( ( member_set_list_nat @ X2 @ ( image_o_set_list_nat @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_o_set_list_nat @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_600_image__image,axiom,
! [F: set_list_nat > set_list_nat,G: list_nat > set_list_nat,A: set_list_nat] :
( ( image_5143090206295581363st_nat @ F @ ( image_8532145185254316925st_nat @ G @ A ) )
= ( image_8532145185254316925st_nat
@ ^ [X2: list_nat] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_601_image__image,axiom,
! [F: set_set_list_nat > set_set_list_nat,G: set_list_nat > set_set_list_nat,A: set_set_list_nat] :
( ( image_799823415861195295st_nat @ F @ ( image_920372530894776809st_nat @ G @ A ) )
= ( image_920372530894776809st_nat
@ ^ [X2: set_list_nat] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_602_image__image,axiom,
! [F: list_nat > set_list_nat,G: list_nat > list_nat,A: set_list_nat] :
( ( image_8532145185254316925st_nat @ F @ ( image_7976474329151083847st_nat @ G @ A ) )
= ( image_8532145185254316925st_nat
@ ^ [X2: list_nat] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_603_image__image,axiom,
! [F: set_list_nat > set_set_list_nat,G: set_list_nat > set_list_nat,A: set_set_list_nat] :
( ( image_920372530894776809st_nat @ F @ ( image_5143090206295581363st_nat @ G @ A ) )
= ( image_920372530894776809st_nat
@ ^ [X2: set_list_nat] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_604_image__image,axiom,
! [F: set_list_nat > set_set_list_nat,G: list_nat > set_list_nat,A: set_list_nat] :
( ( image_920372530894776809st_nat @ F @ ( image_8532145185254316925st_nat @ G @ A ) )
= ( image_1427552562080725683st_nat
@ ^ [X2: list_nat] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_605_imageE,axiom,
! [B: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ~ ! [X4: nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_nat @ X4 @ A ) ) ) ).
% imageE
thf(fact_606_imageE,axiom,
! [B: nat,F: $o > nat,A: set_o] :
( ( member_nat @ B @ ( image_o_nat @ F @ A ) )
=> ~ ! [X4: $o] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_o @ X4 @ A ) ) ) ).
% imageE
thf(fact_607_imageE,axiom,
! [B: $o,F: nat > $o,A: set_nat] :
( ( member_o @ B @ ( image_nat_o @ F @ A ) )
=> ~ ! [X4: nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_nat @ X4 @ A ) ) ) ).
% imageE
thf(fact_608_imageE,axiom,
! [B: $o,F: $o > $o,A: set_o] :
( ( member_o @ B @ ( image_o_o @ F @ A ) )
=> ~ ! [X4: $o] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_o @ X4 @ A ) ) ) ).
% imageE
thf(fact_609_imageE,axiom,
! [B: nat,F: list_nat > nat,A: set_list_nat] :
( ( member_nat @ B @ ( image_list_nat_nat @ F @ A ) )
=> ~ ! [X4: list_nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_list_nat @ X4 @ A ) ) ) ).
% imageE
thf(fact_610_imageE,axiom,
! [B: $o,F: list_nat > $o,A: set_list_nat] :
( ( member_o @ B @ ( image_list_nat_o @ F @ A ) )
=> ~ ! [X4: list_nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_list_nat @ X4 @ A ) ) ) ).
% imageE
thf(fact_611_imageE,axiom,
! [B: list_nat,F: nat > list_nat,A: set_nat] :
( ( member_list_nat @ B @ ( image_nat_list_nat @ F @ A ) )
=> ~ ! [X4: nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_nat @ X4 @ A ) ) ) ).
% imageE
thf(fact_612_imageE,axiom,
! [B: list_nat,F: $o > list_nat,A: set_o] :
( ( member_list_nat @ B @ ( image_o_list_nat @ F @ A ) )
=> ~ ! [X4: $o] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_o @ X4 @ A ) ) ) ).
% imageE
thf(fact_613_imageE,axiom,
! [B: list_nat,F: list_nat > list_nat,A: set_list_nat] :
( ( member_list_nat @ B @ ( image_7976474329151083847st_nat @ F @ A ) )
=> ~ ! [X4: list_nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_list_nat @ X4 @ A ) ) ) ).
% imageE
thf(fact_614_imageE,axiom,
! [B: set_list_nat,F: list_nat > set_list_nat,A: set_list_nat] :
( ( member_set_list_nat @ B @ ( image_8532145185254316925st_nat @ F @ A ) )
=> ~ ! [X4: list_nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_list_nat @ X4 @ A ) ) ) ).
% imageE
thf(fact_615_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X2: nat] : $false ) ) ).
% empty_def
thf(fact_616_empty__def,axiom,
( bot_bot_set_list_nat
= ( collect_list_nat
@ ^ [X2: list_nat] : $false ) ) ).
% empty_def
thf(fact_617_empty__def,axiom,
( bot_bo3886227569956363488st_nat
= ( collect_set_list_nat
@ ^ [X2: set_list_nat] : $false ) ) ).
% empty_def
thf(fact_618_empty__def,axiom,
( bot_bo3499706412017099030st_nat
= ( collec4691811733418234273st_nat
@ ^ [X2: set_set_list_nat] : $false ) ) ).
% empty_def
thf(fact_619_empty__def,axiom,
( bot_bo1158166727579713100st_nat
= ( collec696328324557263319st_nat
@ ^ [X2: set_set_set_list_nat] : $false ) ) ).
% empty_def
thf(fact_620_Collect__subset,axiom,
! [A: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_621_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_622_Collect__subset,axiom,
! [A: set_list_nat,P: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_623_Collect__subset,axiom,
! [A: set_set_list_nat,P: set_list_nat > $o] :
( ord_le1068707526560357548st_nat
@ ( collect_set_list_nat
@ ^ [X2: set_list_nat] :
( ( member_set_list_nat @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_624_Collect__subset,axiom,
! [A: set_set_set_list_nat,P: set_set_list_nat > $o] :
( ord_le7100322305783427298st_nat
@ ( collec4691811733418234273st_nat
@ ^ [X2: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_625_insert__Collect,axiom,
! [A2: set_set_set_list_nat,P: set_set_set_list_nat > $o] :
( ( insert8008074249615255218st_nat @ A2 @ ( collec696328324557263319st_nat @ P ) )
= ( collec696328324557263319st_nat
@ ^ [U: set_set_set_list_nat] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_626_insert__Collect,axiom,
! [A2: set_list_nat,P: set_list_nat > $o] :
( ( insert_set_list_nat @ A2 @ ( collect_set_list_nat @ P ) )
= ( collect_set_list_nat
@ ^ [U: set_list_nat] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_627_insert__Collect,axiom,
! [A2: set_set_list_nat,P: set_set_list_nat > $o] :
( ( insert721039030076969596st_nat @ A2 @ ( collec4691811733418234273st_nat @ P ) )
= ( collec4691811733418234273st_nat
@ ^ [U: set_set_list_nat] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_628_insert__Collect,axiom,
! [A2: list_nat,P: list_nat > $o] :
( ( insert_list_nat @ A2 @ ( collect_list_nat @ P ) )
= ( collect_list_nat
@ ^ [U: list_nat] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_629_insert__Collect,axiom,
! [A2: nat,P: nat > $o] :
( ( insert_nat @ A2 @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [U: nat] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_630_insert__compr,axiom,
( insert8008074249615255218st_nat
= ( ^ [A6: set_set_set_list_nat,B3: set_se5258582372428582328st_nat] :
( collec696328324557263319st_nat
@ ^ [X2: set_set_set_list_nat] :
( ( X2 = A6 )
| ( member7304678173793621401st_nat @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_631_insert__compr,axiom,
( insert_o
= ( ^ [A6: $o,B3: set_o] :
( collect_o
@ ^ [X2: $o] :
( ( X2 = A6 )
| ( member_o @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_632_insert__compr,axiom,
( insert_set_list_nat
= ( ^ [A6: set_list_nat,B3: set_set_list_nat] :
( collect_set_list_nat
@ ^ [X2: set_list_nat] :
( ( X2 = A6 )
| ( member_set_list_nat @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_633_insert__compr,axiom,
( insert721039030076969596st_nat
= ( ^ [A6: set_set_list_nat,B3: set_set_set_list_nat] :
( collec4691811733418234273st_nat
@ ^ [X2: set_set_list_nat] :
( ( X2 = A6 )
| ( member1029098694177496419st_nat @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_634_insert__compr,axiom,
( insert_list_nat
= ( ^ [A6: list_nat,B3: set_list_nat] :
( collect_list_nat
@ ^ [X2: list_nat] :
( ( X2 = A6 )
| ( member_list_nat @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_635_insert__compr,axiom,
( insert_nat
= ( ^ [A6: nat,B3: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( X2 = A6 )
| ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_636_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
@ ^ [J2: nat] :
( groups3539618377306564664at_int
@ ^ [I2: nat] : ( G @ I2 @ J2 )
@ A )
@ B2 ) ) ).
% sum.swap
thf(fact_637_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
@ ^ [J2: nat] :
( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( G @ I2 @ J2 )
@ A )
@ B2 ) ) ).
% sum.swap
thf(fact_638_sum_Oswap,axiom,
! [G: nat > list_nat > int,B2: set_list_nat,A: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [I2: nat] : ( groups4393565826250045896at_int @ ( G @ I2 ) @ B2 )
@ A )
= ( groups4393565826250045896at_int
@ ^ [J2: list_nat] :
( groups3539618377306564664at_int
@ ^ [I2: nat] : ( G @ I2 @ J2 )
@ A )
@ B2 ) ) ).
% sum.swap
thf(fact_639_sum_Oswap,axiom,
! [G: list_nat > nat > int,B2: set_nat,A: set_list_nat] :
( ( groups4393565826250045896at_int
@ ^ [I2: list_nat] : ( groups3539618377306564664at_int @ ( G @ I2 ) @ B2 )
@ A )
= ( groups3539618377306564664at_int
@ ^ [J2: nat] :
( groups4393565826250045896at_int
@ ^ [I2: list_nat] : ( G @ I2 @ J2 )
@ A )
@ B2 ) ) ).
% sum.swap
thf(fact_640_sum_Oswap,axiom,
! [G: nat > list_nat > nat,B2: set_list_nat,A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( groups4396056296759096172at_nat @ ( G @ I2 ) @ B2 )
@ A )
= ( groups4396056296759096172at_nat
@ ^ [J2: list_nat] :
( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( G @ I2 @ J2 )
@ A )
@ B2 ) ) ).
% sum.swap
thf(fact_641_sum_Oswap,axiom,
! [G: list_nat > nat > nat,B2: set_nat,A: set_list_nat] :
( ( groups4396056296759096172at_nat
@ ^ [I2: list_nat] : ( groups3542108847815614940at_nat @ ( G @ I2 ) @ B2 )
@ A )
= ( groups3542108847815614940at_nat
@ ^ [J2: nat] :
( groups4396056296759096172at_nat
@ ^ [I2: list_nat] : ( G @ I2 @ J2 )
@ A )
@ B2 ) ) ).
% sum.swap
thf(fact_642_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
@ ^ [J2: nat] :
( groups7312845317294741502at_int
@ ^ [I2: set_list_nat] : ( G @ I2 @ J2 )
@ A )
@ B2 ) ) ).
% sum.swap
thf(fact_643_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
@ ^ [J2: set_list_nat] :
( groups3539618377306564664at_int
@ ^ [I2: nat] : ( G @ I2 @ J2 )
@ A )
@ B2 ) ) ).
% sum.swap
thf(fact_644_sum_Oswap,axiom,
! [G: list_nat > list_nat > int,B2: set_list_nat,A: set_list_nat] :
( ( groups4393565826250045896at_int
@ ^ [I2: list_nat] : ( groups4393565826250045896at_int @ ( G @ I2 ) @ B2 )
@ A )
= ( groups4393565826250045896at_int
@ ^ [J2: list_nat] :
( groups4393565826250045896at_int
@ ^ [I2: list_nat] : ( G @ I2 @ J2 )
@ A )
@ B2 ) ) ).
% sum.swap
thf(fact_645_sum_Oswap,axiom,
! [G: nat > set_list_nat > nat,B2: set_set_list_nat,A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( groups7315335787803791778at_nat @ ( G @ I2 ) @ B2 )
@ A )
= ( groups7315335787803791778at_nat
@ ^ [J2: set_list_nat] :
( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( G @ I2 @ J2 )
@ A )
@ B2 ) ) ).
% sum.swap
thf(fact_646_set__diff__eq,axiom,
( minus_minus_set_o
= ( ^ [A3: set_o,B3: set_o] :
( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A3 )
& ~ ( member_o @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_647_set__diff__eq,axiom,
( minus_7954133019191499631st_nat
= ( ^ [A3: set_list_nat,B3: set_list_nat] :
( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ A3 )
& ~ ( member_list_nat @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_648_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ~ ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_649_set__diff__eq,axiom,
( minus_1490147023502258853st_nat
= ( ^ [A3: set_set_list_nat,B3: set_set_list_nat] :
( collect_set_list_nat
@ ^ [X2: set_list_nat] :
( ( member_set_list_nat @ X2 @ A3 )
& ~ ( member_set_list_nat @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_650_set__diff__eq,axiom,
( minus_7520675481749758683st_nat
= ( ^ [A3: set_set_set_list_nat,B3: set_set_set_list_nat] :
( collec4691811733418234273st_nat
@ ^ [X2: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X2 @ A3 )
& ~ ( member1029098694177496419st_nat @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_651_sum_Oreindex,axiom,
! [H: nat > nat,A: set_nat,G: nat > int] :
( ( inj_on_nat_nat @ H @ A )
=> ( ( groups3539618377306564664at_int @ G @ ( image_nat_nat @ H @ A ) )
= ( groups3539618377306564664at_int @ ( comp_nat_int_nat @ G @ H ) @ A ) ) ) ).
% sum.reindex
thf(fact_652_sum_Oreindex,axiom,
! [H: nat > nat,A: set_nat,G: nat > nat] :
( ( inj_on_nat_nat @ H @ A )
=> ( ( groups3542108847815614940at_nat @ G @ ( image_nat_nat @ H @ A ) )
= ( groups3542108847815614940at_nat @ ( comp_nat_nat_nat @ G @ H ) @ A ) ) ) ).
% sum.reindex
thf(fact_653_sum_Oreindex,axiom,
! [H: list_nat > nat,A: set_list_nat,G: nat > int] :
( ( inj_on_list_nat_nat @ H @ A )
=> ( ( groups3539618377306564664at_int @ G @ ( image_list_nat_nat @ H @ A ) )
= ( groups4393565826250045896at_int @ ( comp_n779016382214257217st_nat @ G @ H ) @ A ) ) ) ).
% sum.reindex
thf(fact_654_sum_Oreindex,axiom,
! [H: nat > list_nat,A: set_nat,G: list_nat > int] :
( ( inj_on_nat_list_nat @ H @ A )
=> ( ( groups4393565826250045896at_int @ G @ ( image_nat_list_nat @ H @ A ) )
= ( groups3539618377306564664at_int @ ( comp_l4538574248832168257nt_nat @ G @ H ) @ A ) ) ) ).
% sum.reindex
thf(fact_655_sum_Oreindex,axiom,
! [H: list_nat > nat,A: set_list_nat,G: nat > nat] :
( ( inj_on_list_nat_nat @ H @ A )
=> ( ( groups3542108847815614940at_nat @ G @ ( image_list_nat_nat @ H @ A ) )
= ( groups4396056296759096172at_nat @ ( comp_n3993578768073458149st_nat @ G @ H ) @ A ) ) ) ).
% sum.reindex
thf(fact_656_sum_Oreindex,axiom,
! [H: nat > list_nat,A: set_nat,G: list_nat > nat] :
( ( inj_on_nat_list_nat @ H @ A )
=> ( ( groups4396056296759096172at_nat @ G @ ( image_nat_list_nat @ H @ A ) )
= ( groups3542108847815614940at_nat @ ( comp_l3539220536931113701at_nat @ G @ H ) @ A ) ) ) ).
% sum.reindex
thf(fact_657_sum_Oreindex,axiom,
! [H: nat > set_list_nat,A: set_nat,G: set_list_nat > int] :
( ( inj_on2924389301302751961st_nat @ H @ A )
=> ( ( groups7312845317294741502at_int @ G @ ( image_2883343038133793645st_nat @ H @ A ) )
= ( groups3539618377306564664at_int @ ( comp_s4814495803593316215nt_nat @ G @ H ) @ A ) ) ) ).
% sum.reindex
thf(fact_658_sum_Oreindex,axiom,
! [H: set_list_nat > nat,A: set_set_list_nat,G: nat > int] :
( ( inj_on1516391927397616473at_nat @ H @ A )
=> ( ( groups3539618377306564664at_int @ G @ ( image_1475345664228658157at_nat @ H @ A ) )
= ( groups7312845317294741502at_int @ ( comp_n6688923842565920887st_nat @ G @ H ) @ A ) ) ) ).
% sum.reindex
thf(fact_659_sum_Oreindex,axiom,
! [H: list_nat > list_nat,A: set_list_nat,G: list_nat > int] :
( ( inj_on3049792774292151987st_nat @ H @ A )
=> ( ( groups4393565826250045896at_int @ G @ ( image_7976474329151083847st_nat @ H @ A ) )
= ( groups4393565826250045896at_int @ ( comp_l2071074322182594513st_nat @ G @ H ) @ A ) ) ) ).
% sum.reindex
thf(fact_660_sum_Oreindex,axiom,
! [H: set_list_nat > nat,A: set_set_list_nat,G: nat > nat] :
( ( inj_on1516391927397616473at_nat @ H @ A )
=> ( ( groups3542108847815614940at_nat @ G @ ( image_1475345664228658157at_nat @ H @ A ) )
= ( groups7315335787803791778at_nat @ ( comp_n690403151923122203st_nat @ G @ H ) @ A ) ) ) ).
% sum.reindex
thf(fact_661_inj__on__INTER,axiom,
! [I4: set_nat,F: nat > nat,A: nat > set_nat] :
( ( I4 != bot_bot_set_nat )
=> ( ! [I5: nat] :
( ( member_nat @ I5 @ I4 )
=> ( inj_on_nat_nat @ F @ ( A @ I5 ) ) )
=> ( inj_on_nat_nat @ F @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A @ I4 ) ) ) ) ) ).
% inj_on_INTER
thf(fact_662_inj__on__INTER,axiom,
! [I4: set_o,F: nat > nat,A: $o > set_nat] :
( ( I4 != bot_bot_set_o )
=> ( ! [I5: $o] :
( ( member_o @ I5 @ I4 )
=> ( inj_on_nat_nat @ F @ ( A @ I5 ) ) )
=> ( inj_on_nat_nat @ F @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ A @ I4 ) ) ) ) ) ).
% inj_on_INTER
thf(fact_663_inj__on__INTER,axiom,
! [I4: set_list_nat,F: nat > nat,A: list_nat > set_nat] :
( ( I4 != bot_bot_set_list_nat )
=> ( ! [I5: list_nat] :
( ( member_list_nat @ I5 @ I4 )
=> ( inj_on_nat_nat @ F @ ( A @ I5 ) ) )
=> ( inj_on_nat_nat @ F @ ( comple7806235888213564991et_nat @ ( image_1775855109352712557et_nat @ A @ I4 ) ) ) ) ) ).
% inj_on_INTER
thf(fact_664_inj__on__INTER,axiom,
! [I4: set_set_list_nat,F: nat > nat,A: set_list_nat > set_nat] :
( ( I4 != bot_bo3886227569956363488st_nat )
=> ( ! [I5: set_list_nat] :
( ( member_set_list_nat @ I5 @ I4 )
=> ( inj_on_nat_nat @ F @ ( A @ I5 ) ) )
=> ( inj_on_nat_nat @ F @ ( comple7806235888213564991et_nat @ ( image_7015791558925140899et_nat @ A @ I4 ) ) ) ) ) ).
% inj_on_INTER
thf(fact_665_inj__on__INTER,axiom,
! [I4: set_set_set_list_nat,F: nat > nat,A: set_set_list_nat > set_nat] :
( ( I4 != bot_bo3499706412017099030st_nat )
=> ( ! [I5: set_set_list_nat] :
( ( member1029098694177496419st_nat @ I5 @ I4 )
=> ( inj_on_nat_nat @ F @ ( A @ I5 ) ) )
=> ( inj_on_nat_nat @ F @ ( comple7806235888213564991et_nat @ ( image_8873905749617817305et_nat @ A @ I4 ) ) ) ) ) ).
% inj_on_INTER
thf(fact_666_inj__on__INTER,axiom,
! [I4: set_nat,F: list_nat > set_list_nat,A: nat > set_list_nat] :
( ( I4 != bot_bot_set_nat )
=> ( ! [I5: nat] :
( ( member_nat @ I5 @ I4 )
=> ( inj_on8624761805129053417st_nat @ F @ ( A @ I5 ) ) )
=> ( inj_on8624761805129053417st_nat @ F @ ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ A @ I4 ) ) ) ) ) ).
% inj_on_INTER
thf(fact_667_inj__on__INTER,axiom,
! [I4: set_o,F: list_nat > set_list_nat,A: $o > set_list_nat] :
( ( I4 != bot_bot_set_o )
=> ( ! [I5: $o] :
( ( member_o @ I5 @ I4 )
=> ( inj_on8624761805129053417st_nat @ F @ ( A @ I5 ) ) )
=> ( inj_on8624761805129053417st_nat @ F @ ( comple184543376406953807st_nat @ ( image_o_set_list_nat @ A @ I4 ) ) ) ) ) ).
% inj_on_INTER
thf(fact_668_inj__on__INTER,axiom,
! [I4: set_se5258582372428582328st_nat,F: nat > nat,A: set_set_set_list_nat > set_nat] :
( ( I4 != bot_bo1158166727579713100st_nat )
=> ( ! [I5: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ I5 @ I4 )
=> ( inj_on_nat_nat @ F @ ( A @ I5 ) ) )
=> ( inj_on_nat_nat @ F @ ( comple7806235888213564991et_nat @ ( image_5197920692104315663et_nat @ A @ I4 ) ) ) ) ) ).
% inj_on_INTER
thf(fact_669_inj__on__INTER,axiom,
! [I4: set_list_nat,F: list_nat > set_list_nat,A: list_nat > set_list_nat] :
( ( I4 != bot_bot_set_list_nat )
=> ( ! [I5: list_nat] :
( ( member_list_nat @ I5 @ I4 )
=> ( inj_on8624761805129053417st_nat @ F @ ( A @ I5 ) ) )
=> ( inj_on8624761805129053417st_nat @ F @ ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ A @ I4 ) ) ) ) ) ).
% inj_on_INTER
thf(fact_670_inj__on__INTER,axiom,
! [I4: set_nat,F: set_list_nat > set_set_list_nat,A: nat > set_set_list_nat] :
( ( I4 != bot_bot_set_nat )
=> ( ! [I5: nat] :
( ( member_nat @ I5 @ I4 )
=> ( inj_on211238342484403029st_nat @ F @ ( A @ I5 ) ) )
=> ( inj_on211238342484403029st_nat @ F @ ( comple8462666950445340293st_nat @ ( image_3702150970216043171st_nat @ A @ I4 ) ) ) ) ) ).
% inj_on_INTER
thf(fact_671_of__nat__mono,axiom,
! [I3: nat,J: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_672_of__nat__mono,axiom,
! [I3: nat,J: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I3 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_673_subset__image__iff,axiom,
! [B2: set_list_nat,F: list_nat > list_nat,A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B2 @ ( image_7976474329151083847st_nat @ F @ A ) )
= ( ? [AA: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ AA @ A )
& ( B2
= ( image_7976474329151083847st_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_674_subset__image__iff,axiom,
! [B2: set_list_nat,F: set_list_nat > list_nat,A: set_set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B2 @ ( image_6410155499156649085st_nat @ F @ A ) )
= ( ? [AA: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ AA @ A )
& ( B2
= ( image_6410155499156649085st_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_675_subset__image__iff,axiom,
! [B2: set_list_nat,F: set_set_list_nat > list_nat,A: set_set_set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B2 @ ( image_8319865106727365811st_nat @ F @ A ) )
= ( ? [AA: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ AA @ A )
& ( B2
= ( image_8319865106727365811st_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_676_subset__image__iff,axiom,
! [B2: set_set_list_nat,F: list_nat > set_list_nat,A: set_list_nat] :
( ( ord_le1068707526560357548st_nat @ B2 @ ( image_8532145185254316925st_nat @ F @ A ) )
= ( ? [AA: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ AA @ A )
& ( B2
= ( image_8532145185254316925st_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_677_subset__image__iff,axiom,
! [B2: set_set_list_nat,F: set_list_nat > set_list_nat,A: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ B2 @ ( image_5143090206295581363st_nat @ F @ A ) )
= ( ? [AA: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ AA @ A )
& ( B2
= ( image_5143090206295581363st_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_678_subset__image__iff,axiom,
! [B2: set_set_list_nat,F: set_set_list_nat > set_list_nat,A: set_set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ B2 @ ( image_5970735819588453097st_nat @ F @ A ) )
= ( ? [AA: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ AA @ A )
& ( B2
= ( image_5970735819588453097st_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_679_subset__image__iff,axiom,
! [B2: set_set_set_list_nat,F: list_nat > set_set_list_nat,A: set_list_nat] :
( ( ord_le7100322305783427298st_nat @ B2 @ ( image_1427552562080725683st_nat @ F @ A ) )
= ( ? [AA: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ AA @ A )
& ( B2
= ( image_1427552562080725683st_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_680_subset__image__iff,axiom,
! [B2: set_set_set_list_nat,F: set_list_nat > set_set_list_nat,A: set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ B2 @ ( image_920372530894776809st_nat @ F @ A ) )
= ( ? [AA: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ AA @ A )
& ( B2
= ( image_920372530894776809st_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_681_subset__image__iff,axiom,
! [B2: set_set_set_list_nat,F: set_set_list_nat > set_set_list_nat,A: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ B2 @ ( image_799823415861195295st_nat @ F @ A ) )
= ( ? [AA: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ AA @ A )
& ( B2
= ( image_799823415861195295st_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_682_image__subset__iff,axiom,
! [F: list_nat > set_list_nat,A: set_list_nat,B2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ ( image_8532145185254316925st_nat @ F @ A ) @ B2 )
= ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
=> ( member_set_list_nat @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_683_image__subset__iff,axiom,
! [F: set_list_nat > set_set_list_nat,A: set_set_list_nat,B2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ ( image_920372530894776809st_nat @ F @ A ) @ B2 )
= ( ! [X2: set_list_nat] :
( ( member_set_list_nat @ X2 @ A )
=> ( member1029098694177496419st_nat @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_684_subset__imageE,axiom,
! [B2: set_list_nat,F: list_nat > list_nat,A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B2 @ ( image_7976474329151083847st_nat @ F @ A ) )
=> ~ ! [C3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ C3 @ A )
=> ( B2
!= ( image_7976474329151083847st_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_685_subset__imageE,axiom,
! [B2: set_list_nat,F: set_list_nat > list_nat,A: set_set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B2 @ ( image_6410155499156649085st_nat @ F @ A ) )
=> ~ ! [C3: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ C3 @ A )
=> ( B2
!= ( image_6410155499156649085st_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_686_subset__imageE,axiom,
! [B2: set_list_nat,F: set_set_list_nat > list_nat,A: set_set_set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B2 @ ( image_8319865106727365811st_nat @ F @ A ) )
=> ~ ! [C3: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ C3 @ A )
=> ( B2
!= ( image_8319865106727365811st_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_687_subset__imageE,axiom,
! [B2: set_set_list_nat,F: list_nat > set_list_nat,A: set_list_nat] :
( ( ord_le1068707526560357548st_nat @ B2 @ ( image_8532145185254316925st_nat @ F @ A ) )
=> ~ ! [C3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ C3 @ A )
=> ( B2
!= ( image_8532145185254316925st_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_688_subset__imageE,axiom,
! [B2: set_set_list_nat,F: set_list_nat > set_list_nat,A: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ B2 @ ( image_5143090206295581363st_nat @ F @ A ) )
=> ~ ! [C3: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ C3 @ A )
=> ( B2
!= ( image_5143090206295581363st_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_689_subset__imageE,axiom,
! [B2: set_set_list_nat,F: set_set_list_nat > set_list_nat,A: set_set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ B2 @ ( image_5970735819588453097st_nat @ F @ A ) )
=> ~ ! [C3: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ C3 @ A )
=> ( B2
!= ( image_5970735819588453097st_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_690_subset__imageE,axiom,
! [B2: set_set_set_list_nat,F: list_nat > set_set_list_nat,A: set_list_nat] :
( ( ord_le7100322305783427298st_nat @ B2 @ ( image_1427552562080725683st_nat @ F @ A ) )
=> ~ ! [C3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ C3 @ A )
=> ( B2
!= ( image_1427552562080725683st_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_691_subset__imageE,axiom,
! [B2: set_set_set_list_nat,F: set_list_nat > set_set_list_nat,A: set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ B2 @ ( image_920372530894776809st_nat @ F @ A ) )
=> ~ ! [C3: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ C3 @ A )
=> ( B2
!= ( image_920372530894776809st_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_692_subset__imageE,axiom,
! [B2: set_set_set_list_nat,F: set_set_list_nat > set_set_list_nat,A: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ B2 @ ( image_799823415861195295st_nat @ F @ A ) )
=> ~ ! [C3: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ C3 @ A )
=> ( B2
!= ( image_799823415861195295st_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_693_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B2: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_694_image__subsetI,axiom,
! [A: set_nat,F: nat > $o,B2: set_o] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_o @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_695_image__subsetI,axiom,
! [A: set_o,F: $o > nat,B2: set_nat] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_o_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_696_image__subsetI,axiom,
! [A: set_o,F: $o > $o,B2: set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( member_o @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_697_image__subsetI,axiom,
! [A: set_list_nat,F: list_nat > nat,B2: set_nat] :
( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_list_nat_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_698_image__subsetI,axiom,
! [A: set_list_nat,F: list_nat > $o,B2: set_o] :
( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
=> ( member_o @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_list_nat_o @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_699_image__subsetI,axiom,
! [A: set_nat,F: nat > list_nat,B2: set_list_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_list_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_le6045566169113846134st_nat @ ( image_nat_list_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_700_image__subsetI,axiom,
! [A: set_o,F: $o > list_nat,B2: set_list_nat] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( member_list_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_le6045566169113846134st_nat @ ( image_o_list_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_701_image__subsetI,axiom,
! [A: set_list_nat,F: list_nat > list_nat,B2: set_list_nat] :
( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
=> ( member_list_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_le6045566169113846134st_nat @ ( image_7976474329151083847st_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_702_image__subsetI,axiom,
! [A: set_nat,F: nat > set_list_nat,B2: set_set_list_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_set_list_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_le1068707526560357548st_nat @ ( image_2883343038133793645st_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_703_image__mono,axiom,
! [A: set_list_nat,B2: set_list_nat,F: list_nat > list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B2 )
=> ( ord_le6045566169113846134st_nat @ ( image_7976474329151083847st_nat @ F @ A ) @ ( image_7976474329151083847st_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_704_image__mono,axiom,
! [A: set_list_nat,B2: set_list_nat,F: list_nat > set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B2 )
=> ( ord_le1068707526560357548st_nat @ ( image_8532145185254316925st_nat @ F @ A ) @ ( image_8532145185254316925st_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_705_image__mono,axiom,
! [A: set_list_nat,B2: set_list_nat,F: list_nat > set_set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B2 )
=> ( ord_le7100322305783427298st_nat @ ( image_1427552562080725683st_nat @ F @ A ) @ ( image_1427552562080725683st_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_706_image__mono,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat,F: set_list_nat > list_nat] :
( ( ord_le1068707526560357548st_nat @ A @ B2 )
=> ( ord_le6045566169113846134st_nat @ ( image_6410155499156649085st_nat @ F @ A ) @ ( image_6410155499156649085st_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_707_image__mono,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat,F: set_list_nat > set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A @ B2 )
=> ( ord_le1068707526560357548st_nat @ ( image_5143090206295581363st_nat @ F @ A ) @ ( image_5143090206295581363st_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_708_image__mono,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat,F: set_list_nat > set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A @ B2 )
=> ( ord_le7100322305783427298st_nat @ ( image_920372530894776809st_nat @ F @ A ) @ ( image_920372530894776809st_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_709_image__mono,axiom,
! [A: set_set_set_list_nat,B2: set_set_set_list_nat,F: set_set_list_nat > list_nat] :
( ( ord_le7100322305783427298st_nat @ A @ B2 )
=> ( ord_le6045566169113846134st_nat @ ( image_8319865106727365811st_nat @ F @ A ) @ ( image_8319865106727365811st_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_710_image__mono,axiom,
! [A: set_set_set_list_nat,B2: set_set_set_list_nat,F: set_set_list_nat > set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A @ B2 )
=> ( ord_le1068707526560357548st_nat @ ( image_5970735819588453097st_nat @ F @ A ) @ ( image_5970735819588453097st_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_711_image__mono,axiom,
! [A: set_set_set_list_nat,B2: set_set_set_list_nat,F: set_set_list_nat > set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A @ B2 )
=> ( ord_le7100322305783427298st_nat @ ( image_799823415861195295st_nat @ F @ A ) @ ( image_799823415861195295st_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_712_power__commuting__commutes,axiom,
! [X3: int,Y3: int,N: nat] :
( ( ( times_times_int @ X3 @ Y3 )
= ( times_times_int @ Y3 @ X3 ) )
=> ( ( times_times_int @ ( power_power_int @ X3 @ N ) @ Y3 )
= ( times_times_int @ Y3 @ ( power_power_int @ X3 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_713_power__commuting__commutes,axiom,
! [X3: nat,Y3: nat,N: nat] :
( ( ( times_times_nat @ X3 @ Y3 )
= ( times_times_nat @ Y3 @ X3 ) )
=> ( ( times_times_nat @ ( power_power_nat @ X3 @ N ) @ Y3 )
= ( times_times_nat @ Y3 @ ( power_power_nat @ X3 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_714_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_715_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_716_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_717_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_718_wellorder__Inf__le1,axiom,
! [K: nat,A: set_nat] :
( ( member_nat @ K @ A )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A ) @ K ) ) ).
% wellorder_Inf_le1
thf(fact_719_cInf__eq,axiom,
! [X: set_nat,A2: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ X )
=> ( ord_less_eq_nat @ A2 @ X4 ) )
=> ( ! [Y5: nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ X )
=> ( ord_less_eq_nat @ Y5 @ X6 ) )
=> ( ord_less_eq_nat @ Y5 @ A2 ) )
=> ( ( complete_Inf_Inf_nat @ X )
= A2 ) ) ) ).
% cInf_eq
thf(fact_720_cInf__eq,axiom,
! [X: set_int,A2: int] :
( ! [X4: int] :
( ( member_int @ X4 @ X )
=> ( ord_less_eq_int @ A2 @ X4 ) )
=> ( ! [Y5: int] :
( ! [X6: int] :
( ( member_int @ X6 @ X )
=> ( ord_less_eq_int @ Y5 @ X6 ) )
=> ( ord_less_eq_int @ Y5 @ A2 ) )
=> ( ( complete_Inf_Inf_int @ X )
= A2 ) ) ) ).
% cInf_eq
thf(fact_721_cInf__eq__minimum,axiom,
! [Z2: set_set_list_nat,X: set_set_set_list_nat] :
( ( member1029098694177496419st_nat @ Z2 @ X )
=> ( ! [X4: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X4 @ X )
=> ( ord_le1068707526560357548st_nat @ Z2 @ X4 ) )
=> ( ( comple8462666950445340293st_nat @ X )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_722_cInf__eq__minimum,axiom,
! [Z2: nat,X: set_nat] :
( ( member_nat @ Z2 @ X )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ X )
=> ( ord_less_eq_nat @ Z2 @ X4 ) )
=> ( ( complete_Inf_Inf_nat @ X )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_723_cInf__eq__minimum,axiom,
! [Z2: int,X: set_int] :
( ( member_int @ Z2 @ X )
=> ( ! [X4: int] :
( ( member_int @ X4 @ X )
=> ( ord_less_eq_int @ Z2 @ X4 ) )
=> ( ( complete_Inf_Inf_int @ X )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_724_cInf__eq__minimum,axiom,
! [Z2: set_set_set_list_nat,X: set_se5258582372428582328st_nat] :
( ( member7304678173793621401st_nat @ Z2 @ X )
=> ( ! [X4: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X4 @ X )
=> ( ord_le7100322305783427298st_nat @ Z2 @ X4 ) )
=> ( ( comple5189992959352112827st_nat @ X )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_725_cInf__eq__minimum,axiom,
! [Z2: set_list_nat,X: set_set_list_nat] :
( ( member_set_list_nat @ Z2 @ X )
=> ( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ X )
=> ( ord_le6045566169113846134st_nat @ Z2 @ X4 ) )
=> ( ( comple184543376406953807st_nat @ X )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_726_cInf__eq__minimum,axiom,
! [Z2: $o,X: set_o] :
( ( member_o @ Z2 @ X )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ X )
=> ( ord_less_eq_o @ Z2 @ X4 ) )
=> ( ( complete_Inf_Inf_o @ X )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_727_Inf__greatest,axiom,
! [A: set_set_set_list_nat,Z2: set_set_list_nat] :
( ! [X4: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X4 @ A )
=> ( ord_le1068707526560357548st_nat @ Z2 @ X4 ) )
=> ( ord_le1068707526560357548st_nat @ Z2 @ ( comple8462666950445340293st_nat @ A ) ) ) ).
% Inf_greatest
thf(fact_728_Inf__greatest,axiom,
! [A: set_se5258582372428582328st_nat,Z2: set_set_set_list_nat] :
( ! [X4: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X4 @ A )
=> ( ord_le7100322305783427298st_nat @ Z2 @ X4 ) )
=> ( ord_le7100322305783427298st_nat @ Z2 @ ( comple5189992959352112827st_nat @ A ) ) ) ).
% Inf_greatest
thf(fact_729_Inf__greatest,axiom,
! [A: set_set_list_nat,Z2: set_list_nat] :
( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ A )
=> ( ord_le6045566169113846134st_nat @ Z2 @ X4 ) )
=> ( ord_le6045566169113846134st_nat @ Z2 @ ( comple184543376406953807st_nat @ A ) ) ) ).
% Inf_greatest
thf(fact_730_Inf__greatest,axiom,
! [A: set_o,Z2: $o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_o @ Z2 @ X4 ) )
=> ( ord_less_eq_o @ Z2 @ ( complete_Inf_Inf_o @ A ) ) ) ).
% Inf_greatest
thf(fact_731_le__Inf__iff,axiom,
! [B: set_set_list_nat,A: set_set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ B @ ( comple8462666950445340293st_nat @ A ) )
= ( ! [X2: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X2 @ A )
=> ( ord_le1068707526560357548st_nat @ B @ X2 ) ) ) ) ).
% le_Inf_iff
thf(fact_732_le__Inf__iff,axiom,
! [B: set_set_set_list_nat,A: set_se5258582372428582328st_nat] :
( ( ord_le7100322305783427298st_nat @ B @ ( comple5189992959352112827st_nat @ A ) )
= ( ! [X2: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X2 @ A )
=> ( ord_le7100322305783427298st_nat @ B @ X2 ) ) ) ) ).
% le_Inf_iff
thf(fact_733_le__Inf__iff,axiom,
! [B: set_list_nat,A: set_set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B @ ( comple184543376406953807st_nat @ A ) )
= ( ! [X2: set_list_nat] :
( ( member_set_list_nat @ X2 @ A )
=> ( ord_le6045566169113846134st_nat @ B @ X2 ) ) ) ) ).
% le_Inf_iff
thf(fact_734_le__Inf__iff,axiom,
! [B: $o,A: set_o] :
( ( ord_less_eq_o @ B @ ( complete_Inf_Inf_o @ A ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_o @ B @ X2 ) ) ) ) ).
% le_Inf_iff
thf(fact_735_Inf__lower2,axiom,
! [U2: set_set_list_nat,A: set_set_set_list_nat,V: set_set_list_nat] :
( ( member1029098694177496419st_nat @ U2 @ A )
=> ( ( ord_le1068707526560357548st_nat @ U2 @ V )
=> ( ord_le1068707526560357548st_nat @ ( comple8462666950445340293st_nat @ A ) @ V ) ) ) ).
% Inf_lower2
thf(fact_736_Inf__lower2,axiom,
! [U2: set_set_set_list_nat,A: set_se5258582372428582328st_nat,V: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ U2 @ A )
=> ( ( ord_le7100322305783427298st_nat @ U2 @ V )
=> ( ord_le7100322305783427298st_nat @ ( comple5189992959352112827st_nat @ A ) @ V ) ) ) ).
% Inf_lower2
thf(fact_737_Inf__lower2,axiom,
! [U2: set_list_nat,A: set_set_list_nat,V: set_list_nat] :
( ( member_set_list_nat @ U2 @ A )
=> ( ( ord_le6045566169113846134st_nat @ U2 @ V )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ A ) @ V ) ) ) ).
% Inf_lower2
thf(fact_738_Inf__lower2,axiom,
! [U2: $o,A: set_o,V: $o] :
( ( member_o @ U2 @ A )
=> ( ( ord_less_eq_o @ U2 @ V )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ V ) ) ) ).
% Inf_lower2
thf(fact_739_Inf__lower,axiom,
! [X3: set_set_list_nat,A: set_set_set_list_nat] :
( ( member1029098694177496419st_nat @ X3 @ A )
=> ( ord_le1068707526560357548st_nat @ ( comple8462666950445340293st_nat @ A ) @ X3 ) ) ).
% Inf_lower
thf(fact_740_Inf__lower,axiom,
! [X3: set_set_set_list_nat,A: set_se5258582372428582328st_nat] :
( ( member7304678173793621401st_nat @ X3 @ A )
=> ( ord_le7100322305783427298st_nat @ ( comple5189992959352112827st_nat @ A ) @ X3 ) ) ).
% Inf_lower
thf(fact_741_Inf__lower,axiom,
! [X3: set_list_nat,A: set_set_list_nat] :
( ( member_set_list_nat @ X3 @ A )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ A ) @ X3 ) ) ).
% Inf_lower
thf(fact_742_Inf__lower,axiom,
! [X3: $o,A: set_o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ X3 ) ) ).
% Inf_lower
thf(fact_743_Inf__mono,axiom,
! [B2: set_set_set_list_nat,A: set_set_set_list_nat] :
( ! [B5: set_set_list_nat] :
( ( member1029098694177496419st_nat @ B5 @ B2 )
=> ? [X6: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X6 @ A )
& ( ord_le1068707526560357548st_nat @ X6 @ B5 ) ) )
=> ( ord_le1068707526560357548st_nat @ ( comple8462666950445340293st_nat @ A ) @ ( comple8462666950445340293st_nat @ B2 ) ) ) ).
% Inf_mono
thf(fact_744_Inf__mono,axiom,
! [B2: set_se5258582372428582328st_nat,A: set_se5258582372428582328st_nat] :
( ! [B5: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ B5 @ B2 )
=> ? [X6: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X6 @ A )
& ( ord_le7100322305783427298st_nat @ X6 @ B5 ) ) )
=> ( ord_le7100322305783427298st_nat @ ( comple5189992959352112827st_nat @ A ) @ ( comple5189992959352112827st_nat @ B2 ) ) ) ).
% Inf_mono
thf(fact_745_Inf__mono,axiom,
! [B2: set_set_list_nat,A: set_set_list_nat] :
( ! [B5: set_list_nat] :
( ( member_set_list_nat @ B5 @ B2 )
=> ? [X6: set_list_nat] :
( ( member_set_list_nat @ X6 @ A )
& ( ord_le6045566169113846134st_nat @ X6 @ B5 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ A ) @ ( comple184543376406953807st_nat @ B2 ) ) ) ).
% Inf_mono
thf(fact_746_Inf__mono,axiom,
! [B2: set_o,A: set_o] :
( ! [B5: $o] :
( ( member_o @ B5 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A )
& ( ord_less_eq_o @ X6 @ B5 ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Inf_Inf_o @ B2 ) ) ) ).
% Inf_mono
thf(fact_747_Inf__eqI,axiom,
! [A: set_set_set_list_nat,X3: set_set_list_nat] :
( ! [I5: set_set_list_nat] :
( ( member1029098694177496419st_nat @ I5 @ A )
=> ( ord_le1068707526560357548st_nat @ X3 @ I5 ) )
=> ( ! [Y5: set_set_list_nat] :
( ! [I6: set_set_list_nat] :
( ( member1029098694177496419st_nat @ I6 @ A )
=> ( ord_le1068707526560357548st_nat @ Y5 @ I6 ) )
=> ( ord_le1068707526560357548st_nat @ Y5 @ X3 ) )
=> ( ( comple8462666950445340293st_nat @ A )
= X3 ) ) ) ).
% Inf_eqI
thf(fact_748_Inf__eqI,axiom,
! [A: set_se5258582372428582328st_nat,X3: set_set_set_list_nat] :
( ! [I5: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ I5 @ A )
=> ( ord_le7100322305783427298st_nat @ X3 @ I5 ) )
=> ( ! [Y5: set_set_set_list_nat] :
( ! [I6: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ I6 @ A )
=> ( ord_le7100322305783427298st_nat @ Y5 @ I6 ) )
=> ( ord_le7100322305783427298st_nat @ Y5 @ X3 ) )
=> ( ( comple5189992959352112827st_nat @ A )
= X3 ) ) ) ).
% Inf_eqI
thf(fact_749_Inf__eqI,axiom,
! [A: set_set_list_nat,X3: set_list_nat] :
( ! [I5: set_list_nat] :
( ( member_set_list_nat @ I5 @ A )
=> ( ord_le6045566169113846134st_nat @ X3 @ I5 ) )
=> ( ! [Y5: set_list_nat] :
( ! [I6: set_list_nat] :
( ( member_set_list_nat @ I6 @ A )
=> ( ord_le6045566169113846134st_nat @ Y5 @ I6 ) )
=> ( ord_le6045566169113846134st_nat @ Y5 @ X3 ) )
=> ( ( comple184543376406953807st_nat @ A )
= X3 ) ) ) ).
% Inf_eqI
thf(fact_750_Inf__eqI,axiom,
! [A: set_o,X3: $o] :
( ! [I5: $o] :
( ( member_o @ I5 @ A )
=> ( ord_less_eq_o @ X3 @ I5 ) )
=> ( ! [Y5: $o] :
( ! [I6: $o] :
( ( member_o @ I6 @ A )
=> ( ord_less_eq_o @ Y5 @ I6 ) )
=> ( ord_less_eq_o @ Y5 @ X3 ) )
=> ( ( complete_Inf_Inf_o @ A )
= X3 ) ) ) ).
% Inf_eqI
thf(fact_751_mult__of__nat__commute,axiom,
! [X3: nat,Y3: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X3 ) @ Y3 )
= ( times_times_int @ Y3 @ ( semiri1314217659103216013at_int @ X3 ) ) ) ).
% mult_of_nat_commute
thf(fact_752_mult__of__nat__commute,axiom,
! [X3: nat,Y3: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ Y3 )
= ( times_times_nat @ Y3 @ ( semiri1316708129612266289at_nat @ X3 ) ) ) ).
% mult_of_nat_commute
thf(fact_753_singleton__inject,axiom,
! [A2: list_nat,B: list_nat] :
( ( ( insert_list_nat @ A2 @ bot_bot_set_list_nat )
= ( insert_list_nat @ B @ bot_bot_set_list_nat ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_754_singleton__inject,axiom,
! [A2: set_list_nat,B: set_list_nat] :
( ( ( insert_set_list_nat @ A2 @ bot_bo3886227569956363488st_nat )
= ( insert_set_list_nat @ B @ bot_bo3886227569956363488st_nat ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_755_singleton__inject,axiom,
! [A2: set_set_list_nat,B: set_set_list_nat] :
( ( ( insert721039030076969596st_nat @ A2 @ bot_bo3499706412017099030st_nat )
= ( insert721039030076969596st_nat @ B @ bot_bo3499706412017099030st_nat ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_756_singleton__inject,axiom,
! [A2: set_set_set_list_nat,B: set_set_set_list_nat] :
( ( ( insert8008074249615255218st_nat @ A2 @ bot_bo1158166727579713100st_nat )
= ( insert8008074249615255218st_nat @ B @ bot_bo1158166727579713100st_nat ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_757_insert__not__empty,axiom,
! [A2: list_nat,A: set_list_nat] :
( ( insert_list_nat @ A2 @ A )
!= bot_bot_set_list_nat ) ).
% insert_not_empty
thf(fact_758_insert__not__empty,axiom,
! [A2: set_list_nat,A: set_set_list_nat] :
( ( insert_set_list_nat @ A2 @ A )
!= bot_bo3886227569956363488st_nat ) ).
% insert_not_empty
thf(fact_759_insert__not__empty,axiom,
! [A2: set_set_list_nat,A: set_set_set_list_nat] :
( ( insert721039030076969596st_nat @ A2 @ A )
!= bot_bo3499706412017099030st_nat ) ).
% insert_not_empty
thf(fact_760_insert__not__empty,axiom,
! [A2: set_set_set_list_nat,A: set_se5258582372428582328st_nat] :
( ( insert8008074249615255218st_nat @ A2 @ A )
!= bot_bo1158166727579713100st_nat ) ).
% insert_not_empty
thf(fact_761_doubleton__eq__iff,axiom,
! [A2: list_nat,B: list_nat,C: list_nat,D2: list_nat] :
( ( ( insert_list_nat @ A2 @ ( insert_list_nat @ B @ bot_bot_set_list_nat ) )
= ( insert_list_nat @ C @ ( insert_list_nat @ D2 @ bot_bot_set_list_nat ) ) )
= ( ( ( A2 = C )
& ( B = D2 ) )
| ( ( A2 = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_762_doubleton__eq__iff,axiom,
! [A2: set_list_nat,B: set_list_nat,C: set_list_nat,D2: set_list_nat] :
( ( ( insert_set_list_nat @ A2 @ ( insert_set_list_nat @ B @ bot_bo3886227569956363488st_nat ) )
= ( insert_set_list_nat @ C @ ( insert_set_list_nat @ D2 @ bot_bo3886227569956363488st_nat ) ) )
= ( ( ( A2 = C )
& ( B = D2 ) )
| ( ( A2 = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_763_doubleton__eq__iff,axiom,
! [A2: set_set_list_nat,B: set_set_list_nat,C: set_set_list_nat,D2: set_set_list_nat] :
( ( ( insert721039030076969596st_nat @ A2 @ ( insert721039030076969596st_nat @ B @ bot_bo3499706412017099030st_nat ) )
= ( insert721039030076969596st_nat @ C @ ( insert721039030076969596st_nat @ D2 @ bot_bo3499706412017099030st_nat ) ) )
= ( ( ( A2 = C )
& ( B = D2 ) )
| ( ( A2 = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_764_doubleton__eq__iff,axiom,
! [A2: set_set_set_list_nat,B: set_set_set_list_nat,C: set_set_set_list_nat,D2: set_set_set_list_nat] :
( ( ( insert8008074249615255218st_nat @ A2 @ ( insert8008074249615255218st_nat @ B @ bot_bo1158166727579713100st_nat ) )
= ( insert8008074249615255218st_nat @ C @ ( insert8008074249615255218st_nat @ D2 @ bot_bo1158166727579713100st_nat ) ) )
= ( ( ( A2 = C )
& ( B = D2 ) )
| ( ( A2 = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_765_singleton__iff,axiom,
! [B: nat,A2: nat] :
( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_766_singleton__iff,axiom,
! [B: $o,A2: $o] :
( ( member_o @ B @ ( insert_o @ A2 @ bot_bot_set_o ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_767_singleton__iff,axiom,
! [B: list_nat,A2: list_nat] :
( ( member_list_nat @ B @ ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_768_singleton__iff,axiom,
! [B: set_list_nat,A2: set_list_nat] :
( ( member_set_list_nat @ B @ ( insert_set_list_nat @ A2 @ bot_bo3886227569956363488st_nat ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_769_singleton__iff,axiom,
! [B: set_set_list_nat,A2: set_set_list_nat] :
( ( member1029098694177496419st_nat @ B @ ( insert721039030076969596st_nat @ A2 @ bot_bo3499706412017099030st_nat ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_770_singleton__iff,axiom,
! [B: set_set_set_list_nat,A2: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ B @ ( insert8008074249615255218st_nat @ A2 @ bot_bo1158166727579713100st_nat ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_771_singletonD,axiom,
! [B: nat,A2: nat] :
( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_772_singletonD,axiom,
! [B: $o,A2: $o] :
( ( member_o @ B @ ( insert_o @ A2 @ bot_bot_set_o ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_773_singletonD,axiom,
! [B: list_nat,A2: list_nat] :
( ( member_list_nat @ B @ ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_774_singletonD,axiom,
! [B: set_list_nat,A2: set_list_nat] :
( ( member_set_list_nat @ B @ ( insert_set_list_nat @ A2 @ bot_bo3886227569956363488st_nat ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_775_singletonD,axiom,
! [B: set_set_list_nat,A2: set_set_list_nat] :
( ( member1029098694177496419st_nat @ B @ ( insert721039030076969596st_nat @ A2 @ bot_bo3499706412017099030st_nat ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_776_singletonD,axiom,
! [B: set_set_set_list_nat,A2: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ B @ ( insert8008074249615255218st_nat @ A2 @ bot_bo1158166727579713100st_nat ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_777_subset__insertI2,axiom,
! [A: set_se5258582372428582328st_nat,B2: set_se5258582372428582328st_nat,B: set_set_set_list_nat] :
( ( ord_le2499698639687704088st_nat @ A @ B2 )
=> ( ord_le2499698639687704088st_nat @ A @ ( insert8008074249615255218st_nat @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_778_subset__insertI2,axiom,
! [A: set_list_nat,B2: set_list_nat,B: list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B2 )
=> ( ord_le6045566169113846134st_nat @ A @ ( insert_list_nat @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_779_subset__insertI2,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat,B: set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A @ B2 )
=> ( ord_le1068707526560357548st_nat @ A @ ( insert_set_list_nat @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_780_subset__insertI2,axiom,
! [A: set_set_set_list_nat,B2: set_set_set_list_nat,B: set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A @ B2 )
=> ( ord_le7100322305783427298st_nat @ A @ ( insert721039030076969596st_nat @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_781_subset__insertI,axiom,
! [B2: set_se5258582372428582328st_nat,A2: set_set_set_list_nat] : ( ord_le2499698639687704088st_nat @ B2 @ ( insert8008074249615255218st_nat @ A2 @ B2 ) ) ).
% subset_insertI
thf(fact_782_subset__insertI,axiom,
! [B2: set_list_nat,A2: list_nat] : ( ord_le6045566169113846134st_nat @ B2 @ ( insert_list_nat @ A2 @ B2 ) ) ).
% subset_insertI
thf(fact_783_subset__insertI,axiom,
! [B2: set_set_list_nat,A2: set_list_nat] : ( ord_le1068707526560357548st_nat @ B2 @ ( insert_set_list_nat @ A2 @ B2 ) ) ).
% subset_insertI
thf(fact_784_subset__insertI,axiom,
! [B2: set_set_set_list_nat,A2: set_set_list_nat] : ( ord_le7100322305783427298st_nat @ B2 @ ( insert721039030076969596st_nat @ A2 @ B2 ) ) ).
% subset_insertI
thf(fact_785_subset__insert,axiom,
! [X3: set_set_set_list_nat,A: set_se5258582372428582328st_nat,B2: set_se5258582372428582328st_nat] :
( ~ ( member7304678173793621401st_nat @ X3 @ A )
=> ( ( ord_le2499698639687704088st_nat @ A @ ( insert8008074249615255218st_nat @ X3 @ B2 ) )
= ( ord_le2499698639687704088st_nat @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_786_subset__insert,axiom,
! [X3: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat @ X3 @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X3 @ B2 ) )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_787_subset__insert,axiom,
! [X3: $o,A: set_o,B2: set_o] :
( ~ ( member_o @ X3 @ A )
=> ( ( ord_less_eq_set_o @ A @ ( insert_o @ X3 @ B2 ) )
= ( ord_less_eq_set_o @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_788_subset__insert,axiom,
! [X3: list_nat,A: set_list_nat,B2: set_list_nat] :
( ~ ( member_list_nat @ X3 @ A )
=> ( ( ord_le6045566169113846134st_nat @ A @ ( insert_list_nat @ X3 @ B2 ) )
= ( ord_le6045566169113846134st_nat @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_789_subset__insert,axiom,
! [X3: set_list_nat,A: set_set_list_nat,B2: set_set_list_nat] :
( ~ ( member_set_list_nat @ X3 @ A )
=> ( ( ord_le1068707526560357548st_nat @ A @ ( insert_set_list_nat @ X3 @ B2 ) )
= ( ord_le1068707526560357548st_nat @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_790_subset__insert,axiom,
! [X3: set_set_list_nat,A: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ~ ( member1029098694177496419st_nat @ X3 @ A )
=> ( ( ord_le7100322305783427298st_nat @ A @ ( insert721039030076969596st_nat @ X3 @ B2 ) )
= ( ord_le7100322305783427298st_nat @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_791_insert__mono,axiom,
! [C2: set_se5258582372428582328st_nat,D: set_se5258582372428582328st_nat,A2: set_set_set_list_nat] :
( ( ord_le2499698639687704088st_nat @ C2 @ D )
=> ( ord_le2499698639687704088st_nat @ ( insert8008074249615255218st_nat @ A2 @ C2 ) @ ( insert8008074249615255218st_nat @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_792_insert__mono,axiom,
! [C2: set_list_nat,D: set_list_nat,A2: list_nat] :
( ( ord_le6045566169113846134st_nat @ C2 @ D )
=> ( ord_le6045566169113846134st_nat @ ( insert_list_nat @ A2 @ C2 ) @ ( insert_list_nat @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_793_insert__mono,axiom,
! [C2: set_set_list_nat,D: set_set_list_nat,A2: set_list_nat] :
( ( ord_le1068707526560357548st_nat @ C2 @ D )
=> ( ord_le1068707526560357548st_nat @ ( insert_set_list_nat @ A2 @ C2 ) @ ( insert_set_list_nat @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_794_insert__mono,axiom,
! [C2: set_set_set_list_nat,D: set_set_set_list_nat,A2: set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ C2 @ D )
=> ( ord_le7100322305783427298st_nat @ ( insert721039030076969596st_nat @ A2 @ C2 ) @ ( insert721039030076969596st_nat @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_795_Sup_OSUP__image,axiom,
! [Sup: set_set_list_nat > set_list_nat,G: set_list_nat > set_list_nat,F: list_nat > set_list_nat,A: set_list_nat] :
( ( Sup @ ( image_5143090206295581363st_nat @ G @ ( image_8532145185254316925st_nat @ F @ A ) ) )
= ( Sup @ ( image_8532145185254316925st_nat @ ( comp_s803412145318520817st_nat @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_796_Sup_OSUP__image,axiom,
! [Sup: set_int > int,G: set_set_list_nat > int,F: set_list_nat > set_set_list_nat,A: set_set_list_nat] :
( ( Sup @ ( image_8275292939657174399at_int @ G @ ( image_920372530894776809st_nat @ F @ A ) ) )
= ( Sup @ ( image_1472855193719607881at_int @ ( comp_s3399862273134644595st_nat @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_797_Sup_OSUP__image,axiom,
! [Sup: set_set_set_list_nat > set_set_list_nat,G: set_set_list_nat > set_set_list_nat,F: set_list_nat > set_set_list_nat,A: set_set_list_nat] :
( ( Sup @ ( image_799823415861195295st_nat @ G @ ( image_920372530894776809st_nat @ F @ A ) ) )
= ( Sup @ ( image_920372530894776809st_nat @ ( comp_s2115431755894073235st_nat @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_798_Sup_OSUP__image,axiom,
! [Sup: set_set_list_nat > set_list_nat,G: list_nat > set_list_nat,F: list_nat > list_nat,A: set_list_nat] :
( ( Sup @ ( image_8532145185254316925st_nat @ G @ ( image_7976474329151083847st_nat @ F @ A ) ) )
= ( Sup @ ( image_8532145185254316925st_nat @ ( comp_l6547402907347976123st_nat @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_799_Sup_OSUP__image,axiom,
! [Sup: set_set_set_list_nat > set_set_list_nat,G: set_list_nat > set_set_list_nat,F: set_list_nat > set_list_nat,A: set_set_list_nat] :
( ( Sup @ ( image_920372530894776809st_nat @ G @ ( image_5143090206295581363st_nat @ F @ A ) ) )
= ( Sup @ ( image_920372530894776809st_nat @ ( comp_s319613270319683165st_nat @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_800_Sup_OSUP__image,axiom,
! [Sup: set_set_set_list_nat > set_set_list_nat,G: set_list_nat > set_set_list_nat,F: list_nat > set_list_nat,A: set_list_nat] :
( ( Sup @ ( image_920372530894776809st_nat @ G @ ( image_8532145185254316925st_nat @ F @ A ) ) )
= ( Sup @ ( image_1427552562080725683st_nat @ ( comp_s2760078235918225447st_nat @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_801_Inf_OINF__image,axiom,
! [Inf: set_set_list_nat > set_list_nat,G: set_list_nat > set_list_nat,F: list_nat > set_list_nat,A: set_list_nat] :
( ( Inf @ ( image_5143090206295581363st_nat @ G @ ( image_8532145185254316925st_nat @ F @ A ) ) )
= ( Inf @ ( image_8532145185254316925st_nat @ ( comp_s803412145318520817st_nat @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_802_Inf_OINF__image,axiom,
! [Inf: set_int > int,G: set_set_list_nat > int,F: set_list_nat > set_set_list_nat,A: set_set_list_nat] :
( ( Inf @ ( image_8275292939657174399at_int @ G @ ( image_920372530894776809st_nat @ F @ A ) ) )
= ( Inf @ ( image_1472855193719607881at_int @ ( comp_s3399862273134644595st_nat @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_803_Inf_OINF__image,axiom,
! [Inf: set_set_set_list_nat > set_set_list_nat,G: set_set_list_nat > set_set_list_nat,F: set_list_nat > set_set_list_nat,A: set_set_list_nat] :
( ( Inf @ ( image_799823415861195295st_nat @ G @ ( image_920372530894776809st_nat @ F @ A ) ) )
= ( Inf @ ( image_920372530894776809st_nat @ ( comp_s2115431755894073235st_nat @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_804_Inf_OINF__image,axiom,
! [Inf: set_set_list_nat > set_list_nat,G: list_nat > set_list_nat,F: list_nat > list_nat,A: set_list_nat] :
( ( Inf @ ( image_8532145185254316925st_nat @ G @ ( image_7976474329151083847st_nat @ F @ A ) ) )
= ( Inf @ ( image_8532145185254316925st_nat @ ( comp_l6547402907347976123st_nat @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_805_Inf_OINF__image,axiom,
! [Inf: set_set_set_list_nat > set_set_list_nat,G: set_list_nat > set_set_list_nat,F: set_list_nat > set_list_nat,A: set_set_list_nat] :
( ( Inf @ ( image_920372530894776809st_nat @ G @ ( image_5143090206295581363st_nat @ F @ A ) ) )
= ( Inf @ ( image_920372530894776809st_nat @ ( comp_s319613270319683165st_nat @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_806_Inf_OINF__image,axiom,
! [Inf: set_set_set_list_nat > set_set_list_nat,G: set_list_nat > set_set_list_nat,F: list_nat > set_list_nat,A: set_list_nat] :
( ( Inf @ ( image_920372530894776809st_nat @ G @ ( image_8532145185254316925st_nat @ F @ A ) ) )
= ( Inf @ ( image_1427552562080725683st_nat @ ( comp_s2760078235918225447st_nat @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_807_INF__cong,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat,C2: set_list_nat > set_set_list_nat,D: set_list_nat > set_set_list_nat] :
( ( A = B2 )
=> ( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple8462666950445340293st_nat @ ( image_920372530894776809st_nat @ C2 @ A ) )
= ( comple8462666950445340293st_nat @ ( image_920372530894776809st_nat @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_808_INF__cong,axiom,
! [A: set_nat,B2: set_nat,C2: nat > set_list_nat,D: nat > set_list_nat] :
( ( A = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ C2 @ A ) )
= ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_809_INF__cong,axiom,
! [A: set_o,B2: set_o,C2: $o > set_list_nat,D: $o > set_list_nat] :
( ( A = B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple184543376406953807st_nat @ ( image_o_set_list_nat @ C2 @ A ) )
= ( comple184543376406953807st_nat @ ( image_o_set_list_nat @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_810_INF__cong,axiom,
! [A: set_list_nat,B2: set_list_nat,C2: list_nat > set_list_nat,D: list_nat > set_list_nat] :
( ( A = B2 )
=> ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ C2 @ A ) )
= ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_811_INF__cong,axiom,
! [A: set_nat,B2: set_nat,C2: nat > $o,D: nat > $o] :
( ( A = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ C2 @ A ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_812_INF__cong,axiom,
! [A: set_o,B2: set_o,C2: $o > $o,D: $o > $o] :
( ( A = B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ C2 @ A ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_813_INF__cong,axiom,
! [A: set_list_nat,B2: set_list_nat,C2: list_nat > $o,D: list_nat > $o] :
( ( A = B2 )
=> ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_list_nat_o @ C2 @ A ) )
= ( complete_Inf_Inf_o @ ( image_list_nat_o @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_814_double__diff,axiom,
! [A: set_list_nat,B2: set_list_nat,C2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B2 )
=> ( ( ord_le6045566169113846134st_nat @ B2 @ C2 )
=> ( ( minus_7954133019191499631st_nat @ B2 @ ( minus_7954133019191499631st_nat @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_815_double__diff,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat,C2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A @ B2 )
=> ( ( ord_le1068707526560357548st_nat @ B2 @ C2 )
=> ( ( minus_1490147023502258853st_nat @ B2 @ ( minus_1490147023502258853st_nat @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_816_double__diff,axiom,
! [A: set_set_set_list_nat,B2: set_set_set_list_nat,C2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A @ B2 )
=> ( ( ord_le7100322305783427298st_nat @ B2 @ C2 )
=> ( ( minus_7520675481749758683st_nat @ B2 @ ( minus_7520675481749758683st_nat @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_817_Diff__subset,axiom,
! [A: set_list_nat,B2: set_list_nat] : ( ord_le6045566169113846134st_nat @ ( minus_7954133019191499631st_nat @ A @ B2 ) @ A ) ).
% Diff_subset
thf(fact_818_Diff__subset,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat] : ( ord_le1068707526560357548st_nat @ ( minus_1490147023502258853st_nat @ A @ B2 ) @ A ) ).
% Diff_subset
thf(fact_819_Diff__subset,axiom,
! [A: set_set_set_list_nat,B2: set_set_set_list_nat] : ( ord_le7100322305783427298st_nat @ ( minus_7520675481749758683st_nat @ A @ B2 ) @ A ) ).
% Diff_subset
thf(fact_820_Diff__mono,axiom,
! [A: set_list_nat,C2: set_list_nat,D: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ C2 )
=> ( ( ord_le6045566169113846134st_nat @ D @ B2 )
=> ( ord_le6045566169113846134st_nat @ ( minus_7954133019191499631st_nat @ A @ B2 ) @ ( minus_7954133019191499631st_nat @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_821_Diff__mono,axiom,
! [A: set_set_list_nat,C2: set_set_list_nat,D: set_set_list_nat,B2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A @ C2 )
=> ( ( ord_le1068707526560357548st_nat @ D @ B2 )
=> ( ord_le1068707526560357548st_nat @ ( minus_1490147023502258853st_nat @ A @ B2 ) @ ( minus_1490147023502258853st_nat @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_822_Diff__mono,axiom,
! [A: set_set_set_list_nat,C2: set_set_set_list_nat,D: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A @ C2 )
=> ( ( ord_le7100322305783427298st_nat @ D @ B2 )
=> ( ord_le7100322305783427298st_nat @ ( minus_7520675481749758683st_nat @ A @ B2 ) @ ( minus_7520675481749758683st_nat @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_823_insert__Diff__if,axiom,
! [X3: set_set_set_list_nat,B2: set_se5258582372428582328st_nat,A: set_se5258582372428582328st_nat] :
( ( ( member7304678173793621401st_nat @ X3 @ B2 )
=> ( ( minus_1885081636186270737st_nat @ ( insert8008074249615255218st_nat @ X3 @ A ) @ B2 )
= ( minus_1885081636186270737st_nat @ A @ B2 ) ) )
& ( ~ ( member7304678173793621401st_nat @ X3 @ B2 )
=> ( ( minus_1885081636186270737st_nat @ ( insert8008074249615255218st_nat @ X3 @ A ) @ B2 )
= ( insert8008074249615255218st_nat @ X3 @ ( minus_1885081636186270737st_nat @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_824_insert__Diff__if,axiom,
! [X3: nat,B2: set_nat,A: set_nat] :
( ( ( member_nat @ X3 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A ) @ B2 )
= ( minus_minus_set_nat @ A @ B2 ) ) )
& ( ~ ( member_nat @ X3 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A ) @ B2 )
= ( insert_nat @ X3 @ ( minus_minus_set_nat @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_825_insert__Diff__if,axiom,
! [X3: $o,B2: set_o,A: set_o] :
( ( ( member_o @ X3 @ B2 )
=> ( ( minus_minus_set_o @ ( insert_o @ X3 @ A ) @ B2 )
= ( minus_minus_set_o @ A @ B2 ) ) )
& ( ~ ( member_o @ X3 @ B2 )
=> ( ( minus_minus_set_o @ ( insert_o @ X3 @ A ) @ B2 )
= ( insert_o @ X3 @ ( minus_minus_set_o @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_826_insert__Diff__if,axiom,
! [X3: list_nat,B2: set_list_nat,A: set_list_nat] :
( ( ( member_list_nat @ X3 @ B2 )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat @ X3 @ A ) @ B2 )
= ( minus_7954133019191499631st_nat @ A @ B2 ) ) )
& ( ~ ( member_list_nat @ X3 @ B2 )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat @ X3 @ A ) @ B2 )
= ( insert_list_nat @ X3 @ ( minus_7954133019191499631st_nat @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_827_insert__Diff__if,axiom,
! [X3: set_list_nat,B2: set_set_list_nat,A: set_set_list_nat] :
( ( ( member_set_list_nat @ X3 @ B2 )
=> ( ( minus_1490147023502258853st_nat @ ( insert_set_list_nat @ X3 @ A ) @ B2 )
= ( minus_1490147023502258853st_nat @ A @ B2 ) ) )
& ( ~ ( member_set_list_nat @ X3 @ B2 )
=> ( ( minus_1490147023502258853st_nat @ ( insert_set_list_nat @ X3 @ A ) @ B2 )
= ( insert_set_list_nat @ X3 @ ( minus_1490147023502258853st_nat @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_828_insert__Diff__if,axiom,
! [X3: set_set_list_nat,B2: set_set_set_list_nat,A: set_set_set_list_nat] :
( ( ( member1029098694177496419st_nat @ X3 @ B2 )
=> ( ( minus_7520675481749758683st_nat @ ( insert721039030076969596st_nat @ X3 @ A ) @ B2 )
= ( minus_7520675481749758683st_nat @ A @ B2 ) ) )
& ( ~ ( member1029098694177496419st_nat @ X3 @ B2 )
=> ( ( minus_7520675481749758683st_nat @ ( insert721039030076969596st_nat @ X3 @ A ) @ B2 )
= ( insert721039030076969596st_nat @ X3 @ ( minus_7520675481749758683st_nat @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_829_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_830_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_831_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_832_Inter__greatest,axiom,
! [A: set_set_set_list_nat,C2: set_set_list_nat] :
( ! [X5: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X5 @ A )
=> ( ord_le1068707526560357548st_nat @ C2 @ X5 ) )
=> ( ord_le1068707526560357548st_nat @ C2 @ ( comple8462666950445340293st_nat @ A ) ) ) ).
% Inter_greatest
thf(fact_833_Inter__greatest,axiom,
! [A: set_se5258582372428582328st_nat,C2: set_set_set_list_nat] :
( ! [X5: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X5 @ A )
=> ( ord_le7100322305783427298st_nat @ C2 @ X5 ) )
=> ( ord_le7100322305783427298st_nat @ C2 @ ( comple5189992959352112827st_nat @ A ) ) ) ).
% Inter_greatest
thf(fact_834_Inter__greatest,axiom,
! [A: set_set_list_nat,C2: set_list_nat] :
( ! [X5: set_list_nat] :
( ( member_set_list_nat @ X5 @ A )
=> ( ord_le6045566169113846134st_nat @ C2 @ X5 ) )
=> ( ord_le6045566169113846134st_nat @ C2 @ ( comple184543376406953807st_nat @ A ) ) ) ).
% Inter_greatest
thf(fact_835_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_836_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_837_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_838_image__Pow__surj,axiom,
! [F: list_nat > list_nat,A: set_list_nat,B2: set_list_nat] :
( ( ( image_7976474329151083847st_nat @ F @ A )
= B2 )
=> ( ( image_5143090206295581363st_nat @ ( image_7976474329151083847st_nat @ F ) @ ( pow_list_nat @ A ) )
= ( pow_list_nat @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_839_image__Pow__surj,axiom,
! [F: list_nat > set_set_list_nat,A: set_list_nat,B2: set_set_set_list_nat] :
( ( ( image_1427552562080725683st_nat @ F @ A )
= B2 )
=> ( ( image_463335885318752799st_nat @ ( image_1427552562080725683st_nat @ F ) @ ( pow_list_nat @ A ) )
= ( pow_set_set_list_nat @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_840_image__Pow__surj,axiom,
! [F: set_list_nat > list_nat,A: set_set_list_nat,B2: set_list_nat] :
( ( ( image_6410155499156649085st_nat @ F @ A )
= B2 )
=> ( ( image_5970735819588453097st_nat @ ( image_6410155499156649085st_nat @ F ) @ ( pow_set_list_nat @ A ) )
= ( pow_list_nat @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_841_image__Pow__surj,axiom,
! [F: set_list_nat > set_list_nat,A: set_set_list_nat,B2: set_set_list_nat] :
( ( ( image_5143090206295581363st_nat @ F @ A )
= B2 )
=> ( ( image_799823415861195295st_nat @ ( image_5143090206295581363st_nat @ F ) @ ( pow_set_list_nat @ A ) )
= ( pow_set_list_nat @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_842_image__Pow__surj,axiom,
! [F: set_set_list_nat > list_nat,A: set_set_set_list_nat,B2: set_list_nat] :
( ( ( image_8319865106727365811st_nat @ F @ A )
= B2 )
=> ( ( image_7505702526219074591st_nat @ ( image_8319865106727365811st_nat @ F ) @ ( pow_set_set_list_nat @ A ) )
= ( pow_list_nat @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_843_image__Pow__surj,axiom,
! [F: set_set_list_nat > set_list_nat,A: set_set_set_list_nat,B2: set_set_list_nat] :
( ( ( image_5970735819588453097st_nat @ F @ A )
= B2 )
=> ( ( image_904579742278570837st_nat @ ( image_5970735819588453097st_nat @ F ) @ ( pow_set_set_list_nat @ A ) )
= ( pow_set_list_nat @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_844_image__Pow__surj,axiom,
! [F: set_set_list_nat > set_set_list_nat,A: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ( ( image_799823415861195295st_nat @ F @ A )
= B2 )
=> ( ( image_5561069522181849995st_nat @ ( image_799823415861195295st_nat @ F ) @ ( pow_set_set_list_nat @ A ) )
= ( pow_set_set_list_nat @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_845_image__Pow__surj,axiom,
! [F: list_nat > set_list_nat,A: set_list_nat,B2: set_set_list_nat] :
( ( ( image_8532145185254316925st_nat @ F @ A )
= B2 )
=> ( ( image_920372530894776809st_nat @ ( image_8532145185254316925st_nat @ F ) @ ( pow_list_nat @ A ) )
= ( pow_set_list_nat @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_846_image__Pow__surj,axiom,
! [F: set_list_nat > set_set_list_nat,A: set_set_list_nat,B2: set_set_set_list_nat] :
( ( ( image_920372530894776809st_nat @ F @ A )
= B2 )
=> ( ( image_2453375416348523093st_nat @ ( image_920372530894776809st_nat @ F ) @ ( pow_set_list_nat @ A ) )
= ( pow_set_set_list_nat @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_847_Pow__bottom,axiom,
! [B2: set_list_nat] : ( member_set_list_nat @ bot_bot_set_list_nat @ ( pow_list_nat @ B2 ) ) ).
% Pow_bottom
thf(fact_848_Pow__bottom,axiom,
! [B2: set_set_list_nat] : ( member1029098694177496419st_nat @ bot_bo3886227569956363488st_nat @ ( pow_set_list_nat @ B2 ) ) ).
% Pow_bottom
thf(fact_849_Pow__bottom,axiom,
! [B2: set_set_set_list_nat] : ( member7304678173793621401st_nat @ bot_bo3499706412017099030st_nat @ ( pow_set_set_list_nat @ B2 ) ) ).
% Pow_bottom
thf(fact_850_Pow__bottom,axiom,
! [B2: set_se5258582372428582328st_nat] : ( member4920245652161177295st_nat @ bot_bo1158166727579713100st_nat @ ( pow_se7925038334826946341st_nat @ B2 ) ) ).
% Pow_bottom
thf(fact_851_Pow__mono,axiom,
! [A: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B2 )
=> ( ord_le1068707526560357548st_nat @ ( pow_list_nat @ A ) @ ( pow_list_nat @ B2 ) ) ) ).
% Pow_mono
thf(fact_852_Pow__mono,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A @ B2 )
=> ( ord_le7100322305783427298st_nat @ ( pow_set_list_nat @ A ) @ ( pow_set_list_nat @ B2 ) ) ) ).
% Pow_mono
thf(fact_853_Pow__mono,axiom,
! [A: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A @ B2 )
=> ( ord_le2499698639687704088st_nat @ ( pow_set_set_list_nat @ A ) @ ( pow_set_set_list_nat @ B2 ) ) ) ).
% Pow_mono
thf(fact_854_PowD,axiom,
! [A: set_list_nat,B2: set_list_nat] :
( ( member_set_list_nat @ A @ ( pow_list_nat @ B2 ) )
=> ( ord_le6045566169113846134st_nat @ A @ B2 ) ) ).
% PowD
thf(fact_855_PowD,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat] :
( ( member1029098694177496419st_nat @ A @ ( pow_set_list_nat @ B2 ) )
=> ( ord_le1068707526560357548st_nat @ A @ B2 ) ) ).
% PowD
thf(fact_856_PowD,axiom,
! [A: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ A @ ( pow_set_set_list_nat @ B2 ) )
=> ( ord_le7100322305783427298st_nat @ A @ B2 ) ) ).
% PowD
thf(fact_857_Cantors__paradox,axiom,
! [A: set_set_set_list_nat] :
~ ? [F2: set_set_list_nat > set_set_set_list_nat] :
( ( image_2453375416348523093st_nat @ F2 @ A )
= ( pow_set_set_list_nat @ A ) ) ).
% Cantors_paradox
thf(fact_858_Cantors__paradox,axiom,
! [A: set_list_nat] :
~ ? [F2: list_nat > set_list_nat] :
( ( image_8532145185254316925st_nat @ F2 @ A )
= ( pow_list_nat @ A ) ) ).
% Cantors_paradox
thf(fact_859_Cantors__paradox,axiom,
! [A: set_set_list_nat] :
~ ? [F2: set_list_nat > set_set_list_nat] :
( ( image_920372530894776809st_nat @ F2 @ A )
= ( pow_set_list_nat @ A ) ) ).
% Cantors_paradox
thf(fact_860_image__INT,axiom,
! [F: nat > nat,C2: set_nat,A: set_nat,B2: nat > set_nat,J: nat] :
( ( inj_on_nat_nat @ F @ C2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( B2 @ X4 ) @ C2 ) )
=> ( ( member_nat @ J @ A )
=> ( ( image_nat_nat @ F @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B2 @ A ) ) )
= ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( image_nat_nat @ F @ ( B2 @ X2 ) )
@ A ) ) ) ) ) ) ).
% image_INT
thf(fact_861_image__INT,axiom,
! [F: nat > nat,C2: set_nat,A: set_o,B2: $o > set_nat,J: $o] :
( ( inj_on_nat_nat @ F @ C2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( B2 @ X4 ) @ C2 ) )
=> ( ( member_o @ J @ A )
=> ( ( image_nat_nat @ F @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B2 @ A ) ) )
= ( comple7806235888213564991et_nat
@ ( image_o_set_nat
@ ^ [X2: $o] : ( image_nat_nat @ F @ ( B2 @ X2 ) )
@ A ) ) ) ) ) ) ).
% image_INT
thf(fact_862_image__INT,axiom,
! [F: nat > nat,C2: set_nat,A: set_list_nat,B2: list_nat > set_nat,J: list_nat] :
( ( inj_on_nat_nat @ F @ C2 )
=> ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( B2 @ X4 ) @ C2 ) )
=> ( ( member_list_nat @ J @ A )
=> ( ( image_nat_nat @ F @ ( comple7806235888213564991et_nat @ ( image_1775855109352712557et_nat @ B2 @ A ) ) )
= ( comple7806235888213564991et_nat
@ ( image_1775855109352712557et_nat
@ ^ [X2: list_nat] : ( image_nat_nat @ F @ ( B2 @ X2 ) )
@ A ) ) ) ) ) ) ).
% image_INT
thf(fact_863_image__INT,axiom,
! [F: list_nat > list_nat,C2: set_list_nat,A: set_nat,B2: nat > set_list_nat,J: nat] :
( ( inj_on3049792774292151987st_nat @ F @ C2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_le6045566169113846134st_nat @ ( B2 @ X4 ) @ C2 ) )
=> ( ( member_nat @ J @ A )
=> ( ( image_7976474329151083847st_nat @ F @ ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ B2 @ A ) ) )
= ( comple184543376406953807st_nat
@ ( image_2883343038133793645st_nat
@ ^ [X2: nat] : ( image_7976474329151083847st_nat @ F @ ( B2 @ X2 ) )
@ A ) ) ) ) ) ) ).
% image_INT
thf(fact_864_image__INT,axiom,
! [F: list_nat > list_nat,C2: set_list_nat,A: set_o,B2: $o > set_list_nat,J: $o] :
( ( inj_on3049792774292151987st_nat @ F @ C2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_le6045566169113846134st_nat @ ( B2 @ X4 ) @ C2 ) )
=> ( ( member_o @ J @ A )
=> ( ( image_7976474329151083847st_nat @ F @ ( comple184543376406953807st_nat @ ( image_o_set_list_nat @ B2 @ A ) ) )
= ( comple184543376406953807st_nat
@ ( image_o_set_list_nat
@ ^ [X2: $o] : ( image_7976474329151083847st_nat @ F @ ( B2 @ X2 ) )
@ A ) ) ) ) ) ) ).
% image_INT
thf(fact_865_image__INT,axiom,
! [F: set_list_nat > list_nat,C2: set_set_list_nat,A: set_nat,B2: nat > set_set_list_nat,J: nat] :
( ( inj_on6502772119031385577st_nat @ F @ C2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_le1068707526560357548st_nat @ ( B2 @ X4 ) @ C2 ) )
=> ( ( member_nat @ J @ A )
=> ( ( image_6410155499156649085st_nat @ F @ ( comple8462666950445340293st_nat @ ( image_3702150970216043171st_nat @ B2 @ A ) ) )
= ( comple184543376406953807st_nat
@ ( image_2883343038133793645st_nat
@ ^ [X2: nat] : ( image_6410155499156649085st_nat @ F @ ( B2 @ X2 ) )
@ A ) ) ) ) ) ) ).
% image_INT
thf(fact_866_image__INT,axiom,
! [F: set_list_nat > list_nat,C2: set_set_list_nat,A: set_o,B2: $o > set_set_list_nat,J: $o] :
( ( inj_on6502772119031385577st_nat @ F @ C2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_le1068707526560357548st_nat @ ( B2 @ X4 ) @ C2 ) )
=> ( ( member_o @ J @ A )
=> ( ( image_6410155499156649085st_nat @ F @ ( comple8462666950445340293st_nat @ ( image_4575192727348083743st_nat @ B2 @ A ) ) )
= ( comple184543376406953807st_nat
@ ( image_o_set_list_nat
@ ^ [X2: $o] : ( image_6410155499156649085st_nat @ F @ ( B2 @ X2 ) )
@ A ) ) ) ) ) ) ).
% image_INT
thf(fact_867_image__INT,axiom,
! [F: list_nat > set_list_nat,C2: set_list_nat,A: set_nat,B2: nat > set_list_nat,J: nat] :
( ( inj_on8624761805129053417st_nat @ F @ C2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_le6045566169113846134st_nat @ ( B2 @ X4 ) @ C2 ) )
=> ( ( member_nat @ J @ A )
=> ( ( image_8532145185254316925st_nat @ F @ ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ B2 @ A ) ) )
= ( comple8462666950445340293st_nat
@ ( image_3702150970216043171st_nat
@ ^ [X2: nat] : ( image_8532145185254316925st_nat @ F @ ( B2 @ X2 ) )
@ A ) ) ) ) ) ) ).
% image_INT
thf(fact_868_image__INT,axiom,
! [F: list_nat > set_list_nat,C2: set_list_nat,A: set_o,B2: $o > set_list_nat,J: $o] :
( ( inj_on8624761805129053417st_nat @ F @ C2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_le6045566169113846134st_nat @ ( B2 @ X4 ) @ C2 ) )
=> ( ( member_o @ J @ A )
=> ( ( image_8532145185254316925st_nat @ F @ ( comple184543376406953807st_nat @ ( image_o_set_list_nat @ B2 @ A ) ) )
= ( comple8462666950445340293st_nat
@ ( image_4575192727348083743st_nat
@ ^ [X2: $o] : ( image_8532145185254316925st_nat @ F @ ( B2 @ X2 ) )
@ A ) ) ) ) ) ) ).
% image_INT
thf(fact_869_image__INT,axiom,
! [F: list_nat > list_nat,C2: set_list_nat,A: set_list_nat,B2: list_nat > set_list_nat,J: list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ C2 )
=> ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
=> ( ord_le6045566169113846134st_nat @ ( B2 @ X4 ) @ C2 ) )
=> ( ( member_list_nat @ J @ A )
=> ( ( image_7976474329151083847st_nat @ F @ ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ B2 @ A ) ) )
= ( comple184543376406953807st_nat
@ ( image_8532145185254316925st_nat
@ ^ [X2: list_nat] : ( image_7976474329151083847st_nat @ F @ ( B2 @ X2 ) )
@ A ) ) ) ) ) ) ).
% image_INT
thf(fact_870_Pow__not__empty,axiom,
! [A: set_list_nat] :
( ( pow_list_nat @ A )
!= bot_bo3886227569956363488st_nat ) ).
% Pow_not_empty
thf(fact_871_Pow__not__empty,axiom,
! [A: set_set_list_nat] :
( ( pow_set_list_nat @ A )
!= bot_bo3499706412017099030st_nat ) ).
% Pow_not_empty
thf(fact_872_Pow__not__empty,axiom,
! [A: set_set_set_list_nat] :
( ( pow_set_set_list_nat @ A )
!= bot_bo1158166727579713100st_nat ) ).
% Pow_not_empty
thf(fact_873_sum__mono,axiom,
! [K2: set_o,F: $o > nat,G: $o > nat] :
( ! [I5: $o] :
( ( member_o @ I5 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I5 ) @ ( G @ I5 ) ) )
=> ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ K2 ) @ ( groups8507830703676809646_o_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_874_sum__mono,axiom,
! [K2: set_o,F: $o > int,G: $o > int] :
( ! [I5: $o] :
( ( member_o @ I5 @ K2 )
=> ( ord_less_eq_int @ ( F @ I5 ) @ ( G @ I5 ) ) )
=> ( ord_less_eq_int @ ( groups8505340233167759370_o_int @ F @ K2 ) @ ( groups8505340233167759370_o_int @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_875_sum__mono,axiom,
! [K2: set_set_list_nat,F: set_list_nat > int,G: set_list_nat > int] :
( ! [I5: set_list_nat] :
( ( member_set_list_nat @ I5 @ K2 )
=> ( ord_less_eq_int @ ( F @ I5 ) @ ( G @ I5 ) ) )
=> ( ord_less_eq_int @ ( groups7312845317294741502at_int @ F @ K2 ) @ ( groups7312845317294741502at_int @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_876_sum__mono,axiom,
! [K2: set_set_set_list_nat,F: set_set_list_nat > int,G: set_set_list_nat > int] :
( ! [I5: set_set_list_nat] :
( ( member1029098694177496419st_nat @ I5 @ K2 )
=> ( ord_less_eq_int @ ( F @ I5 ) @ ( G @ I5 ) ) )
=> ( ord_less_eq_int @ ( groups7004213669654646580at_int @ F @ K2 ) @ ( groups7004213669654646580at_int @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_877_sum__mono,axiom,
! [K2: set_nat,F: nat > int,G: nat > int] :
( ! [I5: nat] :
( ( member_nat @ I5 @ K2 )
=> ( ord_less_eq_int @ ( F @ I5 ) @ ( G @ I5 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K2 ) @ ( groups3539618377306564664at_int @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_878_sum__mono,axiom,
! [K2: set_list_nat,F: list_nat > int,G: list_nat > int] :
( ! [I5: list_nat] :
( ( member_list_nat @ I5 @ K2 )
=> ( ord_less_eq_int @ ( F @ I5 ) @ ( G @ I5 ) ) )
=> ( ord_less_eq_int @ ( groups4393565826250045896at_int @ F @ K2 ) @ ( groups4393565826250045896at_int @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_879_sum__mono,axiom,
! [K2: set_nat,F: nat > nat,G: nat > nat] :
( ! [I5: nat] :
( ( member_nat @ I5 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I5 ) @ ( G @ I5 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K2 ) @ ( groups3542108847815614940at_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_880_sum__mono,axiom,
! [K2: set_list_nat,F: list_nat > nat,G: list_nat > nat] :
( ! [I5: list_nat] :
( ( member_list_nat @ I5 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I5 ) @ ( G @ I5 ) ) )
=> ( ord_less_eq_nat @ ( groups4396056296759096172at_nat @ F @ K2 ) @ ( groups4396056296759096172at_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_881_sum__mono,axiom,
! [K2: set_set_list_nat,F: set_list_nat > nat,G: set_list_nat > nat] :
( ! [I5: set_list_nat] :
( ( member_set_list_nat @ I5 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I5 ) @ ( G @ I5 ) ) )
=> ( ord_less_eq_nat @ ( groups7315335787803791778at_nat @ F @ K2 ) @ ( groups7315335787803791778at_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_882_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
@ ^ [J2: nat] : ( times_times_int @ ( F @ I2 ) @ ( G @ J2 ) )
@ B2 )
@ A ) ) ).
% sum_product
thf(fact_883_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
@ ^ [J2: nat] : ( times_times_nat @ ( F @ I2 ) @ ( G @ J2 ) )
@ B2 )
@ A ) ) ).
% sum_product
thf(fact_884_sum__product,axiom,
! [F: nat > int,A: set_nat,G: list_nat > int,B2: set_list_nat] :
( ( times_times_int @ ( groups3539618377306564664at_int @ F @ A ) @ ( groups4393565826250045896at_int @ G @ B2 ) )
= ( groups3539618377306564664at_int
@ ^ [I2: nat] :
( groups4393565826250045896at_int
@ ^ [J2: list_nat] : ( times_times_int @ ( F @ I2 ) @ ( G @ J2 ) )
@ B2 )
@ A ) ) ).
% sum_product
thf(fact_885_sum__product,axiom,
! [F: list_nat > int,A: set_list_nat,G: nat > int,B2: set_nat] :
( ( times_times_int @ ( groups4393565826250045896at_int @ F @ A ) @ ( groups3539618377306564664at_int @ G @ B2 ) )
= ( groups4393565826250045896at_int
@ ^ [I2: list_nat] :
( groups3539618377306564664at_int
@ ^ [J2: nat] : ( times_times_int @ ( F @ I2 ) @ ( G @ J2 ) )
@ B2 )
@ A ) ) ).
% sum_product
thf(fact_886_sum__product,axiom,
! [F: nat > nat,A: set_nat,G: list_nat > nat,B2: set_list_nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups4396056296759096172at_nat @ G @ B2 ) )
= ( groups3542108847815614940at_nat
@ ^ [I2: nat] :
( groups4396056296759096172at_nat
@ ^ [J2: list_nat] : ( times_times_nat @ ( F @ I2 ) @ ( G @ J2 ) )
@ B2 )
@ A ) ) ).
% sum_product
thf(fact_887_sum__product,axiom,
! [F: list_nat > nat,A: set_list_nat,G: nat > nat,B2: set_nat] :
( ( times_times_nat @ ( groups4396056296759096172at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ G @ B2 ) )
= ( groups4396056296759096172at_nat
@ ^ [I2: list_nat] :
( groups3542108847815614940at_nat
@ ^ [J2: nat] : ( times_times_nat @ ( F @ I2 ) @ ( G @ J2 ) )
@ B2 )
@ A ) ) ).
% sum_product
thf(fact_888_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
@ ^ [J2: nat] : ( times_times_int @ ( F @ I2 ) @ ( G @ J2 ) )
@ B2 )
@ A ) ) ).
% sum_product
thf(fact_889_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
@ ^ [J2: set_list_nat] : ( times_times_int @ ( F @ I2 ) @ ( G @ J2 ) )
@ B2 )
@ A ) ) ).
% sum_product
thf(fact_890_sum__product,axiom,
! [F: list_nat > int,A: set_list_nat,G: list_nat > int,B2: set_list_nat] :
( ( times_times_int @ ( groups4393565826250045896at_int @ F @ A ) @ ( groups4393565826250045896at_int @ G @ B2 ) )
= ( groups4393565826250045896at_int
@ ^ [I2: list_nat] :
( groups4393565826250045896at_int
@ ^ [J2: list_nat] : ( times_times_int @ ( F @ I2 ) @ ( G @ J2 ) )
@ B2 )
@ A ) ) ).
% sum_product
thf(fact_891_sum__product,axiom,
! [F: nat > nat,A: set_nat,G: set_list_nat > nat,B2: set_set_list_nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups7315335787803791778at_nat @ G @ B2 ) )
= ( groups3542108847815614940at_nat
@ ^ [I2: nat] :
( groups7315335787803791778at_nat
@ ^ [J2: set_list_nat] : ( times_times_nat @ ( F @ I2 ) @ ( G @ J2 ) )
@ B2 )
@ A ) ) ).
% sum_product
thf(fact_892_sum__distrib__right,axiom,
! [F: set_list_nat > int,A: set_set_list_nat,R: int] :
( ( times_times_int @ ( groups7312845317294741502at_int @ F @ A ) @ R )
= ( groups7312845317294741502at_int
@ ^ [N3: set_list_nat] : ( times_times_int @ ( F @ N3 ) @ R )
@ A ) ) ).
% sum_distrib_right
thf(fact_893_sum__distrib__right,axiom,
! [F: set_set_list_nat > int,A: set_set_set_list_nat,R: int] :
( ( times_times_int @ ( groups7004213669654646580at_int @ F @ A ) @ R )
= ( groups7004213669654646580at_int
@ ^ [N3: set_set_list_nat] : ( times_times_int @ ( F @ N3 ) @ R )
@ A ) ) ).
% sum_distrib_right
thf(fact_894_sum__distrib__right,axiom,
! [F: nat > int,A: set_nat,R: int] :
( ( times_times_int @ ( groups3539618377306564664at_int @ F @ A ) @ R )
= ( groups3539618377306564664at_int
@ ^ [N3: nat] : ( times_times_int @ ( F @ N3 ) @ R )
@ A ) ) ).
% sum_distrib_right
thf(fact_895_sum__distrib__right,axiom,
! [F: list_nat > int,A: set_list_nat,R: int] :
( ( times_times_int @ ( groups4393565826250045896at_int @ F @ A ) @ R )
= ( groups4393565826250045896at_int
@ ^ [N3: list_nat] : ( times_times_int @ ( F @ N3 ) @ R )
@ A ) ) ).
% sum_distrib_right
thf(fact_896_sum__distrib__right,axiom,
! [F: nat > nat,A: set_nat,R: nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ R )
= ( groups3542108847815614940at_nat
@ ^ [N3: nat] : ( times_times_nat @ ( F @ N3 ) @ R )
@ A ) ) ).
% sum_distrib_right
thf(fact_897_sum__distrib__right,axiom,
! [F: list_nat > nat,A: set_list_nat,R: nat] :
( ( times_times_nat @ ( groups4396056296759096172at_nat @ F @ A ) @ R )
= ( groups4396056296759096172at_nat
@ ^ [N3: list_nat] : ( times_times_nat @ ( F @ N3 ) @ R )
@ A ) ) ).
% sum_distrib_right
thf(fact_898_sum__distrib__right,axiom,
! [F: set_list_nat > nat,A: set_set_list_nat,R: nat] :
( ( times_times_nat @ ( groups7315335787803791778at_nat @ F @ A ) @ R )
= ( groups7315335787803791778at_nat
@ ^ [N3: set_list_nat] : ( times_times_nat @ ( F @ N3 ) @ R )
@ A ) ) ).
% sum_distrib_right
thf(fact_899_sum__distrib__left,axiom,
! [R: int,F: set_list_nat > int,A: set_set_list_nat] :
( ( times_times_int @ R @ ( groups7312845317294741502at_int @ F @ A ) )
= ( groups7312845317294741502at_int
@ ^ [N3: set_list_nat] : ( times_times_int @ R @ ( F @ N3 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_900_sum__distrib__left,axiom,
! [R: int,F: set_set_list_nat > int,A: set_set_set_list_nat] :
( ( times_times_int @ R @ ( groups7004213669654646580at_int @ F @ A ) )
= ( groups7004213669654646580at_int
@ ^ [N3: set_set_list_nat] : ( times_times_int @ R @ ( F @ N3 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_901_sum__distrib__left,axiom,
! [R: int,F: nat > int,A: set_nat] :
( ( times_times_int @ R @ ( groups3539618377306564664at_int @ F @ A ) )
= ( groups3539618377306564664at_int
@ ^ [N3: nat] : ( times_times_int @ R @ ( F @ N3 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_902_sum__distrib__left,axiom,
! [R: int,F: list_nat > int,A: set_list_nat] :
( ( times_times_int @ R @ ( groups4393565826250045896at_int @ F @ A ) )
= ( groups4393565826250045896at_int
@ ^ [N3: list_nat] : ( times_times_int @ R @ ( F @ N3 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_903_sum__distrib__left,axiom,
! [R: nat,F: nat > nat,A: set_nat] :
( ( times_times_nat @ R @ ( groups3542108847815614940at_nat @ F @ A ) )
= ( groups3542108847815614940at_nat
@ ^ [N3: nat] : ( times_times_nat @ R @ ( F @ N3 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_904_sum__distrib__left,axiom,
! [R: nat,F: list_nat > nat,A: set_list_nat] :
( ( times_times_nat @ R @ ( groups4396056296759096172at_nat @ F @ A ) )
= ( groups4396056296759096172at_nat
@ ^ [N3: list_nat] : ( times_times_nat @ R @ ( F @ N3 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_905_sum__distrib__left,axiom,
! [R: nat,F: set_list_nat > nat,A: set_set_list_nat] :
( ( times_times_nat @ R @ ( groups7315335787803791778at_nat @ F @ A ) )
= ( groups7315335787803791778at_nat
@ ^ [N3: set_list_nat] : ( times_times_nat @ R @ ( F @ N3 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_906_Collect__conv__if2,axiom,
! [P: nat > $o,A2: nat] :
( ( ( P @ A2 )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( A2 = X2 )
& ( P @ X2 ) ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( A2 = X2 )
& ( P @ X2 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_907_Collect__conv__if2,axiom,
! [P: list_nat > $o,A2: list_nat] :
( ( ( P @ A2 )
=> ( ( collect_list_nat
@ ^ [X2: list_nat] :
( ( A2 = X2 )
& ( P @ X2 ) ) )
= ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_list_nat
@ ^ [X2: list_nat] :
( ( A2 = X2 )
& ( P @ X2 ) ) )
= bot_bot_set_list_nat ) ) ) ).
% Collect_conv_if2
thf(fact_908_Collect__conv__if2,axiom,
! [P: set_list_nat > $o,A2: set_list_nat] :
( ( ( P @ A2 )
=> ( ( collect_set_list_nat
@ ^ [X2: set_list_nat] :
( ( A2 = X2 )
& ( P @ X2 ) ) )
= ( insert_set_list_nat @ A2 @ bot_bo3886227569956363488st_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_set_list_nat
@ ^ [X2: set_list_nat] :
( ( A2 = X2 )
& ( P @ X2 ) ) )
= bot_bo3886227569956363488st_nat ) ) ) ).
% Collect_conv_if2
thf(fact_909_Collect__conv__if2,axiom,
! [P: set_set_list_nat > $o,A2: set_set_list_nat] :
( ( ( P @ A2 )
=> ( ( collec4691811733418234273st_nat
@ ^ [X2: set_set_list_nat] :
( ( A2 = X2 )
& ( P @ X2 ) ) )
= ( insert721039030076969596st_nat @ A2 @ bot_bo3499706412017099030st_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collec4691811733418234273st_nat
@ ^ [X2: set_set_list_nat] :
( ( A2 = X2 )
& ( P @ X2 ) ) )
= bot_bo3499706412017099030st_nat ) ) ) ).
% Collect_conv_if2
thf(fact_910_Collect__conv__if2,axiom,
! [P: set_set_set_list_nat > $o,A2: set_set_set_list_nat] :
( ( ( P @ A2 )
=> ( ( collec696328324557263319st_nat
@ ^ [X2: set_set_set_list_nat] :
( ( A2 = X2 )
& ( P @ X2 ) ) )
= ( insert8008074249615255218st_nat @ A2 @ bot_bo1158166727579713100st_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collec696328324557263319st_nat
@ ^ [X2: set_set_set_list_nat] :
( ( A2 = X2 )
& ( P @ X2 ) ) )
= bot_bo1158166727579713100st_nat ) ) ) ).
% Collect_conv_if2
thf(fact_911_Collect__conv__if,axiom,
! [P: nat > $o,A2: nat] :
( ( ( P @ A2 )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( X2 = A2 )
& ( P @ X2 ) ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( X2 = A2 )
& ( P @ X2 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_912_Collect__conv__if,axiom,
! [P: list_nat > $o,A2: list_nat] :
( ( ( P @ A2 )
=> ( ( collect_list_nat
@ ^ [X2: list_nat] :
( ( X2 = A2 )
& ( P @ X2 ) ) )
= ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_list_nat
@ ^ [X2: list_nat] :
( ( X2 = A2 )
& ( P @ X2 ) ) )
= bot_bot_set_list_nat ) ) ) ).
% Collect_conv_if
thf(fact_913_Collect__conv__if,axiom,
! [P: set_list_nat > $o,A2: set_list_nat] :
( ( ( P @ A2 )
=> ( ( collect_set_list_nat
@ ^ [X2: set_list_nat] :
( ( X2 = A2 )
& ( P @ X2 ) ) )
= ( insert_set_list_nat @ A2 @ bot_bo3886227569956363488st_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_set_list_nat
@ ^ [X2: set_list_nat] :
( ( X2 = A2 )
& ( P @ X2 ) ) )
= bot_bo3886227569956363488st_nat ) ) ) ).
% Collect_conv_if
thf(fact_914_Collect__conv__if,axiom,
! [P: set_set_list_nat > $o,A2: set_set_list_nat] :
( ( ( P @ A2 )
=> ( ( collec4691811733418234273st_nat
@ ^ [X2: set_set_list_nat] :
( ( X2 = A2 )
& ( P @ X2 ) ) )
= ( insert721039030076969596st_nat @ A2 @ bot_bo3499706412017099030st_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collec4691811733418234273st_nat
@ ^ [X2: set_set_list_nat] :
( ( X2 = A2 )
& ( P @ X2 ) ) )
= bot_bo3499706412017099030st_nat ) ) ) ).
% Collect_conv_if
thf(fact_915_Collect__conv__if,axiom,
! [P: set_set_set_list_nat > $o,A2: set_set_set_list_nat] :
( ( ( P @ A2 )
=> ( ( collec696328324557263319st_nat
@ ^ [X2: set_set_set_list_nat] :
( ( X2 = A2 )
& ( P @ X2 ) ) )
= ( insert8008074249615255218st_nat @ A2 @ bot_bo1158166727579713100st_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collec696328324557263319st_nat
@ ^ [X2: set_set_set_list_nat] :
( ( X2 = A2 )
& ( P @ X2 ) ) )
= bot_bo1158166727579713100st_nat ) ) ) ).
% Collect_conv_if
thf(fact_916_sum_Odistrib,axiom,
! [G: set_list_nat > int,H: set_list_nat > int,A: set_set_list_nat] :
( ( groups7312845317294741502at_int
@ ^ [X2: set_list_nat] : ( plus_plus_int @ ( G @ X2 ) @ ( H @ X2 ) )
@ A )
= ( plus_plus_int @ ( groups7312845317294741502at_int @ G @ A ) @ ( groups7312845317294741502at_int @ H @ A ) ) ) ).
% sum.distrib
thf(fact_917_sum_Odistrib,axiom,
! [G: set_set_list_nat > int,H: set_set_list_nat > int,A: set_set_set_list_nat] :
( ( groups7004213669654646580at_int
@ ^ [X2: set_set_list_nat] : ( plus_plus_int @ ( G @ X2 ) @ ( H @ X2 ) )
@ A )
= ( plus_plus_int @ ( groups7004213669654646580at_int @ G @ A ) @ ( groups7004213669654646580at_int @ H @ A ) ) ) ).
% sum.distrib
thf(fact_918_sum_Odistrib,axiom,
! [G: nat > int,H: nat > int,A: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [X2: nat] : ( plus_plus_int @ ( G @ X2 ) @ ( H @ X2 ) )
@ A )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ A ) @ ( groups3539618377306564664at_int @ H @ A ) ) ) ).
% sum.distrib
thf(fact_919_sum_Odistrib,axiom,
! [G: list_nat > int,H: list_nat > int,A: set_list_nat] :
( ( groups4393565826250045896at_int
@ ^ [X2: list_nat] : ( plus_plus_int @ ( G @ X2 ) @ ( H @ X2 ) )
@ A )
= ( plus_plus_int @ ( groups4393565826250045896at_int @ G @ A ) @ ( groups4393565826250045896at_int @ H @ A ) ) ) ).
% sum.distrib
thf(fact_920_sum_Odistrib,axiom,
! [G: nat > nat,H: nat > nat,A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( plus_plus_nat @ ( G @ X2 ) @ ( H @ X2 ) )
@ A )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A ) @ ( groups3542108847815614940at_nat @ H @ A ) ) ) ).
% sum.distrib
thf(fact_921_sum_Odistrib,axiom,
! [G: list_nat > nat,H: list_nat > nat,A: set_list_nat] :
( ( groups4396056296759096172at_nat
@ ^ [X2: list_nat] : ( plus_plus_nat @ ( G @ X2 ) @ ( H @ X2 ) )
@ A )
= ( plus_plus_nat @ ( groups4396056296759096172at_nat @ G @ A ) @ ( groups4396056296759096172at_nat @ H @ A ) ) ) ).
% sum.distrib
thf(fact_922_sum_Odistrib,axiom,
! [G: set_list_nat > nat,H: set_list_nat > nat,A: set_set_list_nat] :
( ( groups7315335787803791778at_nat
@ ^ [X2: set_list_nat] : ( plus_plus_nat @ ( G @ X2 ) @ ( H @ X2 ) )
@ A )
= ( plus_plus_nat @ ( groups7315335787803791778at_nat @ G @ A ) @ ( groups7315335787803791778at_nat @ H @ A ) ) ) ).
% sum.distrib
thf(fact_923_sum__subtractf,axiom,
! [F: set_list_nat > int,G: set_list_nat > int,A: set_set_list_nat] :
( ( groups7312845317294741502at_int
@ ^ [X2: set_list_nat] : ( minus_minus_int @ ( F @ X2 ) @ ( G @ X2 ) )
@ A )
= ( minus_minus_int @ ( groups7312845317294741502at_int @ F @ A ) @ ( groups7312845317294741502at_int @ G @ A ) ) ) ).
% sum_subtractf
thf(fact_924_sum__subtractf,axiom,
! [F: set_set_list_nat > int,G: set_set_list_nat > int,A: set_set_set_list_nat] :
( ( groups7004213669654646580at_int
@ ^ [X2: set_set_list_nat] : ( minus_minus_int @ ( F @ X2 ) @ ( G @ X2 ) )
@ A )
= ( minus_minus_int @ ( groups7004213669654646580at_int @ F @ A ) @ ( groups7004213669654646580at_int @ G @ A ) ) ) ).
% sum_subtractf
thf(fact_925_sum__subtractf,axiom,
! [F: nat > int,G: nat > int,A: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [X2: nat] : ( minus_minus_int @ ( F @ X2 ) @ ( G @ X2 ) )
@ A )
= ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A ) @ ( groups3539618377306564664at_int @ G @ A ) ) ) ).
% sum_subtractf
thf(fact_926_sum__subtractf,axiom,
! [F: list_nat > int,G: list_nat > int,A: set_list_nat] :
( ( groups4393565826250045896at_int
@ ^ [X2: list_nat] : ( minus_minus_int @ ( F @ X2 ) @ ( G @ X2 ) )
@ A )
= ( minus_minus_int @ ( groups4393565826250045896at_int @ F @ A ) @ ( groups4393565826250045896at_int @ G @ A ) ) ) ).
% sum_subtractf
thf(fact_927_sum__negf,axiom,
! [F: set_list_nat > int,A: set_set_list_nat] :
( ( groups7312845317294741502at_int
@ ^ [X2: set_list_nat] : ( uminus_uminus_int @ ( F @ X2 ) )
@ A )
= ( uminus_uminus_int @ ( groups7312845317294741502at_int @ F @ A ) ) ) ).
% sum_negf
thf(fact_928_sum__negf,axiom,
! [F: set_set_list_nat > int,A: set_set_set_list_nat] :
( ( groups7004213669654646580at_int
@ ^ [X2: set_set_list_nat] : ( uminus_uminus_int @ ( F @ X2 ) )
@ A )
= ( uminus_uminus_int @ ( groups7004213669654646580at_int @ F @ A ) ) ) ).
% sum_negf
thf(fact_929_sum__negf,axiom,
! [F: nat > int,A: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [X2: nat] : ( uminus_uminus_int @ ( F @ X2 ) )
@ A )
= ( uminus_uminus_int @ ( groups3539618377306564664at_int @ F @ A ) ) ) ).
% sum_negf
thf(fact_930_sum__negf,axiom,
! [F: list_nat > int,A: set_list_nat] :
( ( groups4393565826250045896at_int
@ ^ [X2: list_nat] : ( uminus_uminus_int @ ( F @ X2 ) )
@ A )
= ( uminus_uminus_int @ ( groups4393565826250045896at_int @ F @ A ) ) ) ).
% sum_negf
thf(fact_931_INF__commute,axiom,
! [F: set_list_nat > set_list_nat > set_set_list_nat,B2: set_set_list_nat,A: set_set_list_nat] :
( ( comple8462666950445340293st_nat
@ ( image_920372530894776809st_nat
@ ^ [I2: set_list_nat] : ( comple8462666950445340293st_nat @ ( image_920372530894776809st_nat @ ( F @ I2 ) @ B2 ) )
@ A ) )
= ( comple8462666950445340293st_nat
@ ( image_920372530894776809st_nat
@ ^ [J2: set_list_nat] :
( comple8462666950445340293st_nat
@ ( image_920372530894776809st_nat
@ ^ [I2: set_list_nat] : ( F @ I2 @ J2 )
@ A ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_932_INF__commute,axiom,
! [F: list_nat > list_nat > set_list_nat,B2: set_list_nat,A: set_list_nat] :
( ( comple184543376406953807st_nat
@ ( image_8532145185254316925st_nat
@ ^ [I2: list_nat] : ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ ( F @ I2 ) @ B2 ) )
@ A ) )
= ( comple184543376406953807st_nat
@ ( image_8532145185254316925st_nat
@ ^ [J2: list_nat] :
( comple184543376406953807st_nat
@ ( image_8532145185254316925st_nat
@ ^ [I2: list_nat] : ( F @ I2 @ J2 )
@ A ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_933_INT__E,axiom,
! [B: set_list_nat,B2: set_list_nat > set_set_list_nat,A: set_set_list_nat,A2: set_list_nat] :
( ( member_set_list_nat @ B @ ( comple8462666950445340293st_nat @ ( image_920372530894776809st_nat @ B2 @ A ) ) )
=> ( ~ ( member_set_list_nat @ B @ ( B2 @ A2 ) )
=> ~ ( member_set_list_nat @ A2 @ A ) ) ) ).
% INT_E
thf(fact_934_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_935_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_936_INT__E,axiom,
! [B: nat,B2: list_nat > set_nat,A: set_list_nat,A2: list_nat] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_1775855109352712557et_nat @ B2 @ A ) ) )
=> ( ~ ( member_nat @ B @ ( B2 @ A2 ) )
=> ~ ( member_list_nat @ A2 @ A ) ) ) ).
% INT_E
thf(fact_937_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_938_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_939_INT__E,axiom,
! [B: $o,B2: list_nat > set_o,A: set_list_nat,A2: list_nat] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_list_nat_set_o @ B2 @ A ) ) )
=> ( ~ ( member_o @ B @ ( B2 @ A2 ) )
=> ~ ( member_list_nat @ A2 @ A ) ) ) ).
% INT_E
thf(fact_940_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_941_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_942_INT__E,axiom,
! [B: list_nat,B2: list_nat > set_list_nat,A: set_list_nat,A2: list_nat] :
( ( member_list_nat @ B @ ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ B2 @ A ) ) )
=> ( ~ ( member_list_nat @ B @ ( B2 @ A2 ) )
=> ~ ( member_list_nat @ A2 @ A ) ) ) ).
% INT_E
thf(fact_943_INT__D,axiom,
! [B: set_list_nat,B2: set_list_nat > set_set_list_nat,A: set_set_list_nat,A2: set_list_nat] :
( ( member_set_list_nat @ B @ ( comple8462666950445340293st_nat @ ( image_920372530894776809st_nat @ B2 @ A ) ) )
=> ( ( member_set_list_nat @ A2 @ A )
=> ( member_set_list_nat @ B @ ( B2 @ A2 ) ) ) ) ).
% INT_D
thf(fact_944_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_945_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_946_INT__D,axiom,
! [B: nat,B2: list_nat > set_nat,A: set_list_nat,A2: list_nat] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_1775855109352712557et_nat @ B2 @ A ) ) )
=> ( ( member_list_nat @ A2 @ A )
=> ( member_nat @ B @ ( B2 @ A2 ) ) ) ) ).
% INT_D
thf(fact_947_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_948_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_949_INT__D,axiom,
! [B: $o,B2: list_nat > set_o,A: set_list_nat,A2: list_nat] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_list_nat_set_o @ B2 @ A ) ) )
=> ( ( member_list_nat @ A2 @ A )
=> ( member_o @ B @ ( B2 @ A2 ) ) ) ) ).
% INT_D
thf(fact_950_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_951_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_952_INT__D,axiom,
! [B: list_nat,B2: list_nat > set_list_nat,A: set_list_nat,A2: list_nat] :
( ( member_list_nat @ B @ ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ B2 @ A ) ) )
=> ( ( member_list_nat @ A2 @ A )
=> ( member_list_nat @ B @ ( B2 @ A2 ) ) ) ) ).
% INT_D
thf(fact_953_Pow__def,axiom,
( pow_list_nat
= ( ^ [A3: set_list_nat] :
( collect_set_list_nat
@ ^ [B3: set_list_nat] : ( ord_le6045566169113846134st_nat @ B3 @ A3 ) ) ) ) ).
% Pow_def
thf(fact_954_Pow__def,axiom,
( pow_set_list_nat
= ( ^ [A3: set_set_list_nat] :
( collec4691811733418234273st_nat
@ ^ [B3: set_set_list_nat] : ( ord_le1068707526560357548st_nat @ B3 @ A3 ) ) ) ) ).
% Pow_def
thf(fact_955_Pow__def,axiom,
( pow_set_set_list_nat
= ( ^ [A3: set_set_set_list_nat] :
( collec696328324557263319st_nat
@ ^ [B3: set_set_set_list_nat] : ( ord_le7100322305783427298st_nat @ B3 @ A3 ) ) ) ) ).
% Pow_def
thf(fact_956_power__increasing,axiom,
! [N: nat,N2: nat,A2: nat] :
( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A2 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ A2 @ N2 ) ) ) ) ).
% power_increasing
thf(fact_957_power__increasing,axiom,
! [N: nat,N2: nat,A2: int] :
( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_int @ one_one_int @ A2 )
=> ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ A2 @ N2 ) ) ) ) ).
% power_increasing
thf(fact_958_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_959_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_960_left__right__inverse__power,axiom,
! [X3: int,Y3: int,N: nat] :
( ( ( times_times_int @ X3 @ Y3 )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X3 @ N ) @ ( power_power_int @ Y3 @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_961_left__right__inverse__power,axiom,
! [X3: nat,Y3: nat,N: nat] :
( ( ( times_times_nat @ X3 @ Y3 )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X3 @ N ) @ ( power_power_nat @ Y3 @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_962_INF__eq,axiom,
! [A: set_nat,B2: set_nat,G: nat > $o,F: nat > $o] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B2 )
& ( ord_less_eq_o @ ( G @ X6 ) @ ( F @ I5 ) ) ) )
=> ( ! [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_963_INF__eq,axiom,
! [A: set_nat,B2: set_o,G: $o > $o,F: nat > $o] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_o @ ( G @ X6 ) @ ( F @ I5 ) ) ) )
=> ( ! [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_964_INF__eq,axiom,
! [A: set_o,B2: set_nat,G: nat > $o,F: $o > $o] :
( ! [I5: $o] :
( ( member_o @ I5 @ A )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B2 )
& ( ord_less_eq_o @ ( G @ X6 ) @ ( F @ I5 ) ) ) )
=> ( ! [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_965_INF__eq,axiom,
! [A: set_o,B2: set_o,G: $o > $o,F: $o > $o] :
( ! [I5: $o] :
( ( member_o @ I5 @ A )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_o @ ( G @ X6 ) @ ( F @ I5 ) ) ) )
=> ( ! [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_966_INF__eq,axiom,
! [A: set_nat,B2: set_list_nat,G: list_nat > $o,F: nat > $o] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A )
=> ? [X6: list_nat] :
( ( member_list_nat @ X6 @ B2 )
& ( ord_less_eq_o @ ( G @ X6 ) @ ( F @ I5 ) ) ) )
=> ( ! [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_967_INF__eq,axiom,
! [A: set_o,B2: set_list_nat,G: list_nat > $o,F: $o > $o] :
( ! [I5: $o] :
( ( member_o @ I5 @ A )
=> ? [X6: list_nat] :
( ( member_list_nat @ X6 @ B2 )
& ( ord_less_eq_o @ ( G @ X6 ) @ ( F @ I5 ) ) ) )
=> ( ! [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_968_INF__eq,axiom,
! [A: set_list_nat,B2: set_nat,G: nat > $o,F: list_nat > $o] :
( ! [I5: list_nat] :
( ( member_list_nat @ I5 @ A )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B2 )
& ( ord_less_eq_o @ ( G @ X6 ) @ ( F @ I5 ) ) ) )
=> ( ! [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_969_INF__eq,axiom,
! [A: set_list_nat,B2: set_o,G: $o > $o,F: list_nat > $o] :
( ! [I5: list_nat] :
( ( member_list_nat @ I5 @ A )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_o @ ( G @ X6 ) @ ( F @ I5 ) ) ) )
=> ( ! [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_970_INF__eq,axiom,
! [A: set_nat,B2: set_nat,G: nat > set_list_nat,F: nat > set_list_nat] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B2 )
& ( ord_le6045566169113846134st_nat @ ( G @ X6 ) @ ( F @ I5 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_le6045566169113846134st_nat @ ( F @ X6 ) @ ( G @ J3 ) ) ) )
=> ( ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ F @ A ) )
= ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_971_INF__eq,axiom,
! [A: set_nat,B2: set_o,G: $o > set_list_nat,F: nat > set_list_nat] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_le6045566169113846134st_nat @ ( G @ X6 ) @ ( F @ I5 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_le6045566169113846134st_nat @ ( F @ X6 ) @ ( G @ J3 ) ) ) )
=> ( ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ F @ A ) )
= ( comple184543376406953807st_nat @ ( image_o_set_list_nat @ G @ B2 ) ) ) ) ) ).
% INF_eq
thf(fact_972_cInf__eq__non__empty,axiom,
! [X: set_set_set_list_nat,A2: set_set_list_nat] :
( ( X != bot_bo3499706412017099030st_nat )
=> ( ! [X4: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X4 @ X )
=> ( ord_le1068707526560357548st_nat @ A2 @ X4 ) )
=> ( ! [Y5: set_set_list_nat] :
( ! [X6: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X6 @ X )
=> ( ord_le1068707526560357548st_nat @ Y5 @ X6 ) )
=> ( ord_le1068707526560357548st_nat @ Y5 @ A2 ) )
=> ( ( comple8462666950445340293st_nat @ X )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_973_cInf__eq__non__empty,axiom,
! [X: set_nat,A2: nat] :
( ( X != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ X )
=> ( ord_less_eq_nat @ A2 @ X4 ) )
=> ( ! [Y5: nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ X )
=> ( ord_less_eq_nat @ Y5 @ X6 ) )
=> ( ord_less_eq_nat @ Y5 @ A2 ) )
=> ( ( complete_Inf_Inf_nat @ X )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_974_cInf__eq__non__empty,axiom,
! [X: set_int,A2: int] :
( ( X != bot_bot_set_int )
=> ( ! [X4: int] :
( ( member_int @ X4 @ X )
=> ( ord_less_eq_int @ A2 @ X4 ) )
=> ( ! [Y5: int] :
( ! [X6: int] :
( ( member_int @ X6 @ X )
=> ( ord_less_eq_int @ Y5 @ X6 ) )
=> ( ord_less_eq_int @ Y5 @ A2 ) )
=> ( ( complete_Inf_Inf_int @ X )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_975_cInf__eq__non__empty,axiom,
! [X: set_se5258582372428582328st_nat,A2: set_set_set_list_nat] :
( ( X != bot_bo1158166727579713100st_nat )
=> ( ! [X4: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X4 @ X )
=> ( ord_le7100322305783427298st_nat @ A2 @ X4 ) )
=> ( ! [Y5: set_set_set_list_nat] :
( ! [X6: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X6 @ X )
=> ( ord_le7100322305783427298st_nat @ Y5 @ X6 ) )
=> ( ord_le7100322305783427298st_nat @ Y5 @ A2 ) )
=> ( ( comple5189992959352112827st_nat @ X )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_976_cInf__eq__non__empty,axiom,
! [X: set_set_list_nat,A2: set_list_nat] :
( ( X != bot_bo3886227569956363488st_nat )
=> ( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ X )
=> ( ord_le6045566169113846134st_nat @ A2 @ X4 ) )
=> ( ! [Y5: set_list_nat] :
( ! [X6: set_list_nat] :
( ( member_set_list_nat @ X6 @ X )
=> ( ord_le6045566169113846134st_nat @ Y5 @ X6 ) )
=> ( ord_le6045566169113846134st_nat @ Y5 @ A2 ) )
=> ( ( comple184543376406953807st_nat @ X )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_977_cInf__eq__non__empty,axiom,
! [X: set_o,A2: $o] :
( ( X != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ X )
=> ( ord_less_eq_o @ A2 @ X4 ) )
=> ( ! [Y5: $o] :
( ! [X6: $o] :
( ( member_o @ X6 @ X )
=> ( ord_less_eq_o @ Y5 @ X6 ) )
=> ( ord_less_eq_o @ Y5 @ A2 ) )
=> ( ( complete_Inf_Inf_o @ X )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_978_cInf__greatest,axiom,
! [X: set_set_set_list_nat,Z2: set_set_list_nat] :
( ( X != bot_bo3499706412017099030st_nat )
=> ( ! [X4: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X4 @ X )
=> ( ord_le1068707526560357548st_nat @ Z2 @ X4 ) )
=> ( ord_le1068707526560357548st_nat @ Z2 @ ( comple8462666950445340293st_nat @ X ) ) ) ) ).
% cInf_greatest
thf(fact_979_cInf__greatest,axiom,
! [X: set_nat,Z2: nat] :
( ( X != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ X )
=> ( ord_less_eq_nat @ Z2 @ X4 ) )
=> ( ord_less_eq_nat @ Z2 @ ( complete_Inf_Inf_nat @ X ) ) ) ) ).
% cInf_greatest
thf(fact_980_cInf__greatest,axiom,
! [X: set_int,Z2: int] :
( ( X != bot_bot_set_int )
=> ( ! [X4: int] :
( ( member_int @ X4 @ X )
=> ( ord_less_eq_int @ Z2 @ X4 ) )
=> ( ord_less_eq_int @ Z2 @ ( complete_Inf_Inf_int @ X ) ) ) ) ).
% cInf_greatest
thf(fact_981_cInf__greatest,axiom,
! [X: set_se5258582372428582328st_nat,Z2: set_set_set_list_nat] :
( ( X != bot_bo1158166727579713100st_nat )
=> ( ! [X4: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X4 @ X )
=> ( ord_le7100322305783427298st_nat @ Z2 @ X4 ) )
=> ( ord_le7100322305783427298st_nat @ Z2 @ ( comple5189992959352112827st_nat @ X ) ) ) ) ).
% cInf_greatest
thf(fact_982_cInf__greatest,axiom,
! [X: set_set_list_nat,Z2: set_list_nat] :
( ( X != bot_bo3886227569956363488st_nat )
=> ( ! [X4: set_list_nat] :
( ( member_set_list_nat @ X4 @ X )
=> ( ord_le6045566169113846134st_nat @ Z2 @ X4 ) )
=> ( ord_le6045566169113846134st_nat @ Z2 @ ( comple184543376406953807st_nat @ X ) ) ) ) ).
% cInf_greatest
thf(fact_983_cInf__greatest,axiom,
! [X: set_o,Z2: $o] :
( ( X != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ X )
=> ( ord_less_eq_o @ Z2 @ X4 ) )
=> ( ord_less_eq_o @ Z2 @ ( complete_Inf_Inf_o @ X ) ) ) ) ).
% cInf_greatest
thf(fact_984_Inf__less__eq,axiom,
! [A: set_set_set_list_nat,U2: set_set_list_nat] :
( ! [V2: set_set_list_nat] :
( ( member1029098694177496419st_nat @ V2 @ A )
=> ( ord_le1068707526560357548st_nat @ V2 @ U2 ) )
=> ( ( A != bot_bo3499706412017099030st_nat )
=> ( ord_le1068707526560357548st_nat @ ( comple8462666950445340293st_nat @ A ) @ U2 ) ) ) ).
% Inf_less_eq
thf(fact_985_Inf__less__eq,axiom,
! [A: set_se5258582372428582328st_nat,U2: set_set_set_list_nat] :
( ! [V2: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ V2 @ A )
=> ( ord_le7100322305783427298st_nat @ V2 @ U2 ) )
=> ( ( A != bot_bo1158166727579713100st_nat )
=> ( ord_le7100322305783427298st_nat @ ( comple5189992959352112827st_nat @ A ) @ U2 ) ) ) ).
% Inf_less_eq
thf(fact_986_Inf__less__eq,axiom,
! [A: set_set_list_nat,U2: set_list_nat] :
( ! [V2: set_list_nat] :
( ( member_set_list_nat @ V2 @ A )
=> ( ord_le6045566169113846134st_nat @ V2 @ U2 ) )
=> ( ( A != bot_bo3886227569956363488st_nat )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ A ) @ U2 ) ) ) ).
% Inf_less_eq
thf(fact_987_Inf__less__eq,axiom,
! [A: set_o,U2: $o] :
( ! [V2: $o] :
( ( member_o @ V2 @ A )
=> ( ord_less_eq_o @ V2 @ U2 ) )
=> ( ( A != bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ U2 ) ) ) ).
% Inf_less_eq
thf(fact_988_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_989_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_990_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_991_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_992_subset__singleton__iff,axiom,
! [X: set_se5258582372428582328st_nat,A2: set_set_set_list_nat] :
( ( ord_le2499698639687704088st_nat @ X @ ( insert8008074249615255218st_nat @ A2 @ bot_bo1158166727579713100st_nat ) )
= ( ( X = bot_bo1158166727579713100st_nat )
| ( X
= ( insert8008074249615255218st_nat @ A2 @ bot_bo1158166727579713100st_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_993_subset__singleton__iff,axiom,
! [X: set_list_nat,A2: list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) )
= ( ( X = bot_bot_set_list_nat )
| ( X
= ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_994_subset__singleton__iff,axiom,
! [X: set_set_list_nat,A2: set_list_nat] :
( ( ord_le1068707526560357548st_nat @ X @ ( insert_set_list_nat @ A2 @ bot_bo3886227569956363488st_nat ) )
= ( ( X = bot_bo3886227569956363488st_nat )
| ( X
= ( insert_set_list_nat @ A2 @ bot_bo3886227569956363488st_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_995_subset__singleton__iff,axiom,
! [X: set_set_set_list_nat,A2: set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ X @ ( insert721039030076969596st_nat @ A2 @ bot_bo3499706412017099030st_nat ) )
= ( ( X = bot_bo3499706412017099030st_nat )
| ( X
= ( insert721039030076969596st_nat @ A2 @ bot_bo3499706412017099030st_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_996_subset__singletonD,axiom,
! [A: set_se5258582372428582328st_nat,X3: set_set_set_list_nat] :
( ( ord_le2499698639687704088st_nat @ A @ ( insert8008074249615255218st_nat @ X3 @ bot_bo1158166727579713100st_nat ) )
=> ( ( A = bot_bo1158166727579713100st_nat )
| ( A
= ( insert8008074249615255218st_nat @ X3 @ bot_bo1158166727579713100st_nat ) ) ) ) ).
% subset_singletonD
thf(fact_997_subset__singletonD,axiom,
! [A: set_list_nat,X3: list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ ( insert_list_nat @ X3 @ bot_bot_set_list_nat ) )
=> ( ( A = bot_bot_set_list_nat )
| ( A
= ( insert_list_nat @ X3 @ bot_bot_set_list_nat ) ) ) ) ).
% subset_singletonD
thf(fact_998_subset__singletonD,axiom,
! [A: set_set_list_nat,X3: set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A @ ( insert_set_list_nat @ X3 @ bot_bo3886227569956363488st_nat ) )
=> ( ( A = bot_bo3886227569956363488st_nat )
| ( A
= ( insert_set_list_nat @ X3 @ bot_bo3886227569956363488st_nat ) ) ) ) ).
% subset_singletonD
thf(fact_999_subset__singletonD,axiom,
! [A: set_set_set_list_nat,X3: set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A @ ( insert721039030076969596st_nat @ X3 @ bot_bo3499706412017099030st_nat ) )
=> ( ( A = bot_bo3499706412017099030st_nat )
| ( A
= ( insert721039030076969596st_nat @ X3 @ bot_bo3499706412017099030st_nat ) ) ) ) ).
% subset_singletonD
thf(fact_1000_INF__eq__const,axiom,
! [I4: set_nat,F: nat > $o,X3: $o] :
( ( I4 != bot_bot_set_nat )
=> ( ! [I5: nat] :
( ( member_nat @ I5 @ I4 )
=> ( ( F @ I5 )
= X3 ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F @ I4 ) )
= X3 ) ) ) ).
% INF_eq_const
thf(fact_1001_INF__eq__const,axiom,
! [I4: set_o,F: $o > $o,X3: $o] :
( ( I4 != bot_bot_set_o )
=> ( ! [I5: $o] :
( ( member_o @ I5 @ I4 )
=> ( ( F @ I5 )
= X3 ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ I4 ) )
= X3 ) ) ) ).
% INF_eq_const
thf(fact_1002_INF__eq__const,axiom,
! [I4: set_list_nat,F: list_nat > $o,X3: $o] :
( ( I4 != bot_bot_set_list_nat )
=> ( ! [I5: list_nat] :
( ( member_list_nat @ I5 @ I4 )
=> ( ( F @ I5 )
= X3 ) )
=> ( ( complete_Inf_Inf_o @ ( image_list_nat_o @ F @ I4 ) )
= X3 ) ) ) ).
% INF_eq_const
thf(fact_1003_INF__eq__const,axiom,
! [I4: set_nat,F: nat > set_list_nat,X3: set_list_nat] :
( ( I4 != bot_bot_set_nat )
=> ( ! [I5: nat] :
( ( member_nat @ I5 @ I4 )
=> ( ( F @ I5 )
= X3 ) )
=> ( ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ F @ I4 ) )
= X3 ) ) ) ).
% INF_eq_const
thf(fact_1004_INF__eq__const,axiom,
! [I4: set_o,F: $o > set_list_nat,X3: set_list_nat] :
( ( I4 != bot_bot_set_o )
=> ( ! [I5: $o] :
( ( member_o @ I5 @ I4 )
=> ( ( F @ I5 )
= X3 ) )
=> ( ( comple184543376406953807st_nat @ ( image_o_set_list_nat @ F @ I4 ) )
= X3 ) ) ) ).
% INF_eq_const
thf(fact_1005_INF__eq__const,axiom,
! [I4: set_set_list_nat,F: set_list_nat > $o,X3: $o] :
( ( I4 != bot_bo3886227569956363488st_nat )
=> ( ! [I5: set_list_nat] :
( ( member_set_list_nat @ I5 @ I4 )
=> ( ( F @ I5 )
= X3 ) )
=> ( ( complete_Inf_Inf_o @ ( image_set_list_nat_o @ F @ I4 ) )
= X3 ) ) ) ).
% INF_eq_const
thf(fact_1006_INF__eq__const,axiom,
! [I4: set_list_nat,F: list_nat > set_list_nat,X3: set_list_nat] :
( ( I4 != bot_bot_set_list_nat )
=> ( ! [I5: list_nat] :
( ( member_list_nat @ I5 @ I4 )
=> ( ( F @ I5 )
= X3 ) )
=> ( ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ F @ I4 ) )
= X3 ) ) ) ).
% INF_eq_const
thf(fact_1007_INF__eq__const,axiom,
! [I4: set_set_set_list_nat,F: set_set_list_nat > $o,X3: $o] :
( ( I4 != bot_bo3499706412017099030st_nat )
=> ( ! [I5: set_set_list_nat] :
( ( member1029098694177496419st_nat @ I5 @ I4 )
=> ( ( F @ I5 )
= X3 ) )
=> ( ( complete_Inf_Inf_o @ ( image_7778788071572860293_nat_o @ F @ I4 ) )
= X3 ) ) ) ).
% INF_eq_const
thf(fact_1008_INF__eq__const,axiom,
! [I4: set_set_list_nat,F: set_list_nat > set_list_nat,X3: set_list_nat] :
( ( I4 != bot_bo3886227569956363488st_nat )
=> ( ! [I5: set_list_nat] :
( ( member_set_list_nat @ I5 @ I4 )
=> ( ( F @ I5 )
= X3 ) )
=> ( ( comple184543376406953807st_nat @ ( image_5143090206295581363st_nat @ F @ I4 ) )
= X3 ) ) ) ).
% INF_eq_const
thf(fact_1009_INF__eq__const,axiom,
! [I4: set_se5258582372428582328st_nat,F: set_set_set_list_nat > $o,X3: $o] :
( ( I4 != bot_bo1158166727579713100st_nat )
=> ( ! [I5: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ I5 @ I4 )
=> ( ( F @ I5 )
= X3 ) )
=> ( ( complete_Inf_Inf_o @ ( image_3035193717992370127_nat_o @ F @ I4 ) )
= X3 ) ) ) ).
% INF_eq_const
thf(fact_1010_image__diff__subset,axiom,
! [F: set_list_nat > list_nat,A: set_set_list_nat,B2: set_set_list_nat] : ( ord_le6045566169113846134st_nat @ ( minus_7954133019191499631st_nat @ ( image_6410155499156649085st_nat @ F @ A ) @ ( image_6410155499156649085st_nat @ F @ B2 ) ) @ ( image_6410155499156649085st_nat @ F @ ( minus_1490147023502258853st_nat @ A @ B2 ) ) ) ).
% image_diff_subset
thf(fact_1011_image__diff__subset,axiom,
! [F: set_set_list_nat > list_nat,A: set_set_set_list_nat,B2: set_set_set_list_nat] : ( ord_le6045566169113846134st_nat @ ( minus_7954133019191499631st_nat @ ( image_8319865106727365811st_nat @ F @ A ) @ ( image_8319865106727365811st_nat @ F @ B2 ) ) @ ( image_8319865106727365811st_nat @ F @ ( minus_7520675481749758683st_nat @ A @ B2 ) ) ) ).
% image_diff_subset
thf(fact_1012_image__diff__subset,axiom,
! [F: list_nat > set_list_nat,A: set_list_nat,B2: set_list_nat] : ( ord_le1068707526560357548st_nat @ ( minus_1490147023502258853st_nat @ ( image_8532145185254316925st_nat @ F @ A ) @ ( image_8532145185254316925st_nat @ F @ B2 ) ) @ ( image_8532145185254316925st_nat @ F @ ( minus_7954133019191499631st_nat @ A @ B2 ) ) ) ).
% image_diff_subset
thf(fact_1013_image__diff__subset,axiom,
! [F: set_list_nat > set_list_nat,A: set_set_list_nat,B2: set_set_list_nat] : ( ord_le1068707526560357548st_nat @ ( minus_1490147023502258853st_nat @ ( image_5143090206295581363st_nat @ F @ A ) @ ( image_5143090206295581363st_nat @ F @ B2 ) ) @ ( image_5143090206295581363st_nat @ F @ ( minus_1490147023502258853st_nat @ A @ B2 ) ) ) ).
% image_diff_subset
thf(fact_1014_image__diff__subset,axiom,
! [F: set_set_list_nat > set_list_nat,A: set_set_set_list_nat,B2: set_set_set_list_nat] : ( ord_le1068707526560357548st_nat @ ( minus_1490147023502258853st_nat @ ( image_5970735819588453097st_nat @ F @ A ) @ ( image_5970735819588453097st_nat @ F @ B2 ) ) @ ( image_5970735819588453097st_nat @ F @ ( minus_7520675481749758683st_nat @ A @ B2 ) ) ) ).
% image_diff_subset
thf(fact_1015_image__diff__subset,axiom,
! [F: set_list_nat > set_set_list_nat,A: set_set_list_nat,B2: set_set_list_nat] : ( ord_le7100322305783427298st_nat @ ( minus_7520675481749758683st_nat @ ( image_920372530894776809st_nat @ F @ A ) @ ( image_920372530894776809st_nat @ F @ B2 ) ) @ ( image_920372530894776809st_nat @ F @ ( minus_1490147023502258853st_nat @ A @ B2 ) ) ) ).
% image_diff_subset
thf(fact_1016_image__diff__subset,axiom,
! [F: set_set_list_nat > set_set_list_nat,A: set_set_set_list_nat,B2: set_set_set_list_nat] : ( ord_le7100322305783427298st_nat @ ( minus_7520675481749758683st_nat @ ( image_799823415861195295st_nat @ F @ A ) @ ( image_799823415861195295st_nat @ F @ B2 ) ) @ ( image_799823415861195295st_nat @ F @ ( minus_7520675481749758683st_nat @ A @ B2 ) ) ) ).
% image_diff_subset
thf(fact_1017_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_1018_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_1019_Diff__insert__absorb,axiom,
! [X3: nat,A: set_nat] :
( ~ ( member_nat @ X3 @ A )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1020_Diff__insert__absorb,axiom,
! [X3: $o,A: set_o] :
( ~ ( member_o @ X3 @ A )
=> ( ( minus_minus_set_o @ ( insert_o @ X3 @ A ) @ ( insert_o @ X3 @ bot_bot_set_o ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1021_Diff__insert__absorb,axiom,
! [X3: list_nat,A: set_list_nat] :
( ~ ( member_list_nat @ X3 @ A )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat @ X3 @ A ) @ ( insert_list_nat @ X3 @ bot_bot_set_list_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1022_Diff__insert__absorb,axiom,
! [X3: set_set_set_list_nat,A: set_se5258582372428582328st_nat] :
( ~ ( member7304678173793621401st_nat @ X3 @ A )
=> ( ( minus_1885081636186270737st_nat @ ( insert8008074249615255218st_nat @ X3 @ A ) @ ( insert8008074249615255218st_nat @ X3 @ bot_bo1158166727579713100st_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1023_Diff__insert__absorb,axiom,
! [X3: set_list_nat,A: set_set_list_nat] :
( ~ ( member_set_list_nat @ X3 @ A )
=> ( ( minus_1490147023502258853st_nat @ ( insert_set_list_nat @ X3 @ A ) @ ( insert_set_list_nat @ X3 @ bot_bo3886227569956363488st_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1024_Diff__insert__absorb,axiom,
! [X3: set_set_list_nat,A: set_set_set_list_nat] :
( ~ ( member1029098694177496419st_nat @ X3 @ A )
=> ( ( minus_7520675481749758683st_nat @ ( insert721039030076969596st_nat @ X3 @ A ) @ ( insert721039030076969596st_nat @ X3 @ bot_bo3499706412017099030st_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1025_Diff__insert2,axiom,
! [A: set_list_nat,A2: list_nat,B2: set_list_nat] :
( ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat @ A2 @ B2 ) )
= ( minus_7954133019191499631st_nat @ ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_1026_Diff__insert2,axiom,
! [A: set_se5258582372428582328st_nat,A2: set_set_set_list_nat,B2: set_se5258582372428582328st_nat] :
( ( minus_1885081636186270737st_nat @ A @ ( insert8008074249615255218st_nat @ A2 @ B2 ) )
= ( minus_1885081636186270737st_nat @ ( minus_1885081636186270737st_nat @ A @ ( insert8008074249615255218st_nat @ A2 @ bot_bo1158166727579713100st_nat ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_1027_Diff__insert2,axiom,
! [A: set_set_list_nat,A2: set_list_nat,B2: set_set_list_nat] :
( ( minus_1490147023502258853st_nat @ A @ ( insert_set_list_nat @ A2 @ B2 ) )
= ( minus_1490147023502258853st_nat @ ( minus_1490147023502258853st_nat @ A @ ( insert_set_list_nat @ A2 @ bot_bo3886227569956363488st_nat ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_1028_Diff__insert2,axiom,
! [A: set_set_set_list_nat,A2: set_set_list_nat,B2: set_set_set_list_nat] :
( ( minus_7520675481749758683st_nat @ A @ ( insert721039030076969596st_nat @ A2 @ B2 ) )
= ( minus_7520675481749758683st_nat @ ( minus_7520675481749758683st_nat @ A @ ( insert721039030076969596st_nat @ A2 @ bot_bo3499706412017099030st_nat ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_1029_insert__Diff,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1030_insert__Diff,axiom,
! [A2: $o,A: set_o] :
( ( member_o @ A2 @ A )
=> ( ( insert_o @ A2 @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1031_insert__Diff,axiom,
! [A2: list_nat,A: set_list_nat] :
( ( member_list_nat @ A2 @ A )
=> ( ( insert_list_nat @ A2 @ ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1032_insert__Diff,axiom,
! [A2: set_set_set_list_nat,A: set_se5258582372428582328st_nat] :
( ( member7304678173793621401st_nat @ A2 @ A )
=> ( ( insert8008074249615255218st_nat @ A2 @ ( minus_1885081636186270737st_nat @ A @ ( insert8008074249615255218st_nat @ A2 @ bot_bo1158166727579713100st_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1033_insert__Diff,axiom,
! [A2: set_list_nat,A: set_set_list_nat] :
( ( member_set_list_nat @ A2 @ A )
=> ( ( insert_set_list_nat @ A2 @ ( minus_1490147023502258853st_nat @ A @ ( insert_set_list_nat @ A2 @ bot_bo3886227569956363488st_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1034_insert__Diff,axiom,
! [A2: set_set_list_nat,A: set_set_set_list_nat] :
( ( member1029098694177496419st_nat @ A2 @ A )
=> ( ( insert721039030076969596st_nat @ A2 @ ( minus_7520675481749758683st_nat @ A @ ( insert721039030076969596st_nat @ A2 @ bot_bo3499706412017099030st_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1035_Diff__insert,axiom,
! [A: set_list_nat,A2: list_nat,B2: set_list_nat] :
( ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat @ A2 @ B2 ) )
= ( minus_7954133019191499631st_nat @ ( minus_7954133019191499631st_nat @ A @ B2 ) @ ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) ) ).
% Diff_insert
thf(fact_1036_Diff__insert,axiom,
! [A: set_se5258582372428582328st_nat,A2: set_set_set_list_nat,B2: set_se5258582372428582328st_nat] :
( ( minus_1885081636186270737st_nat @ A @ ( insert8008074249615255218st_nat @ A2 @ B2 ) )
= ( minus_1885081636186270737st_nat @ ( minus_1885081636186270737st_nat @ A @ B2 ) @ ( insert8008074249615255218st_nat @ A2 @ bot_bo1158166727579713100st_nat ) ) ) ).
% Diff_insert
thf(fact_1037_Diff__insert,axiom,
! [A: set_set_list_nat,A2: set_list_nat,B2: set_set_list_nat] :
( ( minus_1490147023502258853st_nat @ A @ ( insert_set_list_nat @ A2 @ B2 ) )
= ( minus_1490147023502258853st_nat @ ( minus_1490147023502258853st_nat @ A @ B2 ) @ ( insert_set_list_nat @ A2 @ bot_bo3886227569956363488st_nat ) ) ) ).
% Diff_insert
thf(fact_1038_Diff__insert,axiom,
! [A: set_set_set_list_nat,A2: set_set_list_nat,B2: set_set_set_list_nat] :
( ( minus_7520675481749758683st_nat @ A @ ( insert721039030076969596st_nat @ A2 @ B2 ) )
= ( minus_7520675481749758683st_nat @ ( minus_7520675481749758683st_nat @ A @ B2 ) @ ( insert721039030076969596st_nat @ A2 @ bot_bo3499706412017099030st_nat ) ) ) ).
% Diff_insert
thf(fact_1039_subset__Diff__insert,axiom,
! [A: set_se5258582372428582328st_nat,B2: set_se5258582372428582328st_nat,X3: set_set_set_list_nat,C2: set_se5258582372428582328st_nat] :
( ( ord_le2499698639687704088st_nat @ A @ ( minus_1885081636186270737st_nat @ B2 @ ( insert8008074249615255218st_nat @ X3 @ C2 ) ) )
= ( ( ord_le2499698639687704088st_nat @ A @ ( minus_1885081636186270737st_nat @ B2 @ C2 ) )
& ~ ( member7304678173793621401st_nat @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1040_subset__Diff__insert,axiom,
! [A: set_nat,B2: set_nat,X3: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B2 @ ( insert_nat @ X3 @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B2 @ C2 ) )
& ~ ( member_nat @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1041_subset__Diff__insert,axiom,
! [A: set_o,B2: set_o,X3: $o,C2: set_o] :
( ( ord_less_eq_set_o @ A @ ( minus_minus_set_o @ B2 @ ( insert_o @ X3 @ C2 ) ) )
= ( ( ord_less_eq_set_o @ A @ ( minus_minus_set_o @ B2 @ C2 ) )
& ~ ( member_o @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1042_subset__Diff__insert,axiom,
! [A: set_list_nat,B2: set_list_nat,X3: list_nat,C2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ ( minus_7954133019191499631st_nat @ B2 @ ( insert_list_nat @ X3 @ C2 ) ) )
= ( ( ord_le6045566169113846134st_nat @ A @ ( minus_7954133019191499631st_nat @ B2 @ C2 ) )
& ~ ( member_list_nat @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1043_subset__Diff__insert,axiom,
! [A: set_set_list_nat,B2: set_set_list_nat,X3: set_list_nat,C2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A @ ( minus_1490147023502258853st_nat @ B2 @ ( insert_set_list_nat @ X3 @ C2 ) ) )
= ( ( ord_le1068707526560357548st_nat @ A @ ( minus_1490147023502258853st_nat @ B2 @ C2 ) )
& ~ ( member_set_list_nat @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1044_subset__Diff__insert,axiom,
! [A: set_set_set_list_nat,B2: set_set_set_list_nat,X3: set_set_list_nat,C2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ A @ ( minus_7520675481749758683st_nat @ B2 @ ( insert721039030076969596st_nat @ X3 @ C2 ) ) )
= ( ( ord_le7100322305783427298st_nat @ A @ ( minus_7520675481749758683st_nat @ B2 @ C2 ) )
& ~ ( member1029098694177496419st_nat @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1045_INF__image,axiom,
! [G: set_set_list_nat > int,F: set_list_nat > set_set_list_nat,A: set_set_list_nat] :
( ( complete_Inf_Inf_int @ ( image_8275292939657174399at_int @ G @ ( image_920372530894776809st_nat @ F @ A ) ) )
= ( complete_Inf_Inf_int @ ( image_1472855193719607881at_int @ ( comp_s3399862273134644595st_nat @ G @ F ) @ A ) ) ) ).
% INF_image
thf(fact_1046_INF__image,axiom,
! [G: set_set_list_nat > set_set_list_nat,F: set_list_nat > set_set_list_nat,A: set_set_list_nat] :
( ( comple8462666950445340293st_nat @ ( image_799823415861195295st_nat @ G @ ( image_920372530894776809st_nat @ F @ A ) ) )
= ( comple8462666950445340293st_nat @ ( image_920372530894776809st_nat @ ( comp_s2115431755894073235st_nat @ G @ F ) @ A ) ) ) ).
% INF_image
thf(fact_1047_INF__image,axiom,
! [G: set_list_nat > set_set_list_nat,F: set_list_nat > set_list_nat,A: set_set_list_nat] :
( ( comple8462666950445340293st_nat @ ( image_920372530894776809st_nat @ G @ ( image_5143090206295581363st_nat @ F @ A ) ) )
= ( comple8462666950445340293st_nat @ ( image_920372530894776809st_nat @ ( comp_s319613270319683165st_nat @ G @ F ) @ A ) ) ) ).
% INF_image
thf(fact_1048_INF__image,axiom,
! [G: set_list_nat > set_set_list_nat,F: list_nat > set_list_nat,A: set_list_nat] :
( ( comple8462666950445340293st_nat @ ( image_920372530894776809st_nat @ G @ ( image_8532145185254316925st_nat @ F @ A ) ) )
= ( comple8462666950445340293st_nat @ ( image_1427552562080725683st_nat @ ( comp_s2760078235918225447st_nat @ G @ F ) @ A ) ) ) ).
% INF_image
thf(fact_1049_INF__image,axiom,
! [G: set_list_nat > set_list_nat,F: list_nat > set_list_nat,A: set_list_nat] :
( ( comple184543376406953807st_nat @ ( image_5143090206295581363st_nat @ G @ ( image_8532145185254316925st_nat @ F @ A ) ) )
= ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ ( comp_s803412145318520817st_nat @ G @ F ) @ A ) ) ) ).
% INF_image
thf(fact_1050_INF__image,axiom,
! [G: set_set_list_nat > set_list_nat,F: set_list_nat > set_set_list_nat,A: set_set_list_nat] :
( ( comple184543376406953807st_nat @ ( image_5970735819588453097st_nat @ G @ ( image_920372530894776809st_nat @ F @ A ) ) )
= ( comple184543376406953807st_nat @ ( image_5143090206295581363st_nat @ ( comp_s4316523840537751389st_nat @ G @ F ) @ A ) ) ) ).
% INF_image
thf(fact_1051_INF__image,axiom,
! [G: list_nat > set_list_nat,F: list_nat > list_nat,A: set_list_nat] :
( ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ G @ ( image_7976474329151083847st_nat @ F @ A ) ) )
= ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ ( comp_l6547402907347976123st_nat @ G @ F ) @ A ) ) ) ).
% INF_image
thf(fact_1052_INF__image,axiom,
! [G: set_list_nat > $o,F: list_nat > set_list_nat,A: set_list_nat] :
( ( complete_Inf_Inf_o @ ( image_set_list_nat_o @ G @ ( image_8532145185254316925st_nat @ F @ A ) ) )
= ( complete_Inf_Inf_o @ ( image_list_nat_o @ ( comp_s2218972281003795519st_nat @ G @ F ) @ A ) ) ) ).
% INF_image
thf(fact_1053_INF__image,axiom,
! [G: set_set_list_nat > $o,F: set_list_nat > set_set_list_nat,A: set_set_list_nat] :
( ( complete_Inf_Inf_o @ ( image_7778788071572860293_nat_o @ G @ ( image_920372530894776809st_nat @ F @ A ) ) )
= ( complete_Inf_Inf_o @ ( image_set_list_nat_o @ ( comp_s1552796267380350143st_nat @ G @ F ) @ A ) ) ) ).
% INF_image
thf(fact_1054_image__Pow__mono,axiom,
! [F: list_nat > list_nat,A: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( image_7976474329151083847st_nat @ F @ A ) @ B2 )
=> ( ord_le1068707526560357548st_nat @ ( image_5143090206295581363st_nat @ ( image_7976474329151083847st_nat @ F ) @ ( pow_list_nat @ A ) ) @ ( pow_list_nat @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_1055_image__Pow__mono,axiom,
! [F: set_list_nat > list_nat,A: set_set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( image_6410155499156649085st_nat @ F @ A ) @ B2 )
=> ( ord_le1068707526560357548st_nat @ ( image_5970735819588453097st_nat @ ( image_6410155499156649085st_nat @ F ) @ ( pow_set_list_nat @ A ) ) @ ( pow_list_nat @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_1056_image__Pow__mono,axiom,
! [F: set_set_list_nat > list_nat,A: set_set_set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( image_8319865106727365811st_nat @ F @ A ) @ B2 )
=> ( ord_le1068707526560357548st_nat @ ( image_7505702526219074591st_nat @ ( image_8319865106727365811st_nat @ F ) @ ( pow_set_set_list_nat @ A ) ) @ ( pow_list_nat @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_1057_image__Pow__mono,axiom,
! [F: set_list_nat > set_list_nat,A: set_set_list_nat,B2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ ( image_5143090206295581363st_nat @ F @ A ) @ B2 )
=> ( ord_le7100322305783427298st_nat @ ( image_799823415861195295st_nat @ ( image_5143090206295581363st_nat @ F ) @ ( pow_set_list_nat @ A ) ) @ ( pow_set_list_nat @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_1058_image__Pow__mono,axiom,
! [F: set_set_list_nat > set_list_nat,A: set_set_set_list_nat,B2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ ( image_5970735819588453097st_nat @ F @ A ) @ B2 )
=> ( ord_le7100322305783427298st_nat @ ( image_904579742278570837st_nat @ ( image_5970735819588453097st_nat @ F ) @ ( pow_set_set_list_nat @ A ) ) @ ( pow_set_list_nat @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_1059_image__Pow__mono,axiom,
! [F: list_nat > set_list_nat,A: set_list_nat,B2: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ ( image_8532145185254316925st_nat @ F @ A ) @ B2 )
=> ( ord_le7100322305783427298st_nat @ ( image_920372530894776809st_nat @ ( image_8532145185254316925st_nat @ F ) @ ( pow_list_nat @ A ) ) @ ( pow_set_list_nat @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_1060_image__Pow__mono,axiom,
! [F: list_nat > set_set_list_nat,A: set_list_nat,B2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ ( image_1427552562080725683st_nat @ F @ A ) @ B2 )
=> ( ord_le2499698639687704088st_nat @ ( image_463335885318752799st_nat @ ( image_1427552562080725683st_nat @ F ) @ ( pow_list_nat @ A ) ) @ ( pow_set_set_list_nat @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_1061_image__Pow__mono,axiom,
! [F: set_set_list_nat > set_set_list_nat,A: set_set_set_list_nat,B2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ ( image_799823415861195295st_nat @ F @ A ) @ B2 )
=> ( ord_le2499698639687704088st_nat @ ( image_5561069522181849995st_nat @ ( image_799823415861195295st_nat @ F ) @ ( pow_set_set_list_nat @ A ) ) @ ( pow_set_set_list_nat @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_1062_image__Pow__mono,axiom,
! [F: set_list_nat > set_set_list_nat,A: set_set_list_nat,B2: set_set_set_list_nat] :
( ( ord_le7100322305783427298st_nat @ ( image_920372530894776809st_nat @ F @ A ) @ B2 )
=> ( ord_le2499698639687704088st_nat @ ( image_2453375416348523093st_nat @ ( image_920372530894776809st_nat @ F ) @ ( pow_set_list_nat @ A ) ) @ ( pow_set_set_list_nat @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_1063_Inter__subset,axiom,
! [A: set_set_set_list_nat,B2: set_set_list_nat] :
( ! [X5: set_set_list_nat] :
( ( member1029098694177496419st_nat @ X5 @ A )
=> ( ord_le1068707526560357548st_nat @ X5 @ B2 ) )
=> ( ( A != bot_bo3499706412017099030st_nat )
=> ( ord_le1068707526560357548st_nat @ ( comple8462666950445340293st_nat @ A ) @ B2 ) ) ) ).
% Inter_subset
thf(fact_1064_Inter__subset,axiom,
! [A: set_se5258582372428582328st_nat,B2: set_set_set_list_nat] :
( ! [X5: set_set_set_list_nat] :
( ( member7304678173793621401st_nat @ X5 @ A )
=> ( ord_le7100322305783427298st_nat @ X5 @ B2 ) )
=> ( ( A != bot_bo1158166727579713100st_nat )
=> ( ord_le7100322305783427298st_nat @ ( comple5189992959352112827st_nat @ A ) @ B2 ) ) ) ).
% Inter_subset
thf(fact_1065_Inter__subset,axiom,
! [A: set_set_list_nat,B2: set_list_nat] :
( ! [X5: set_list_nat] :
( ( member_set_list_nat @ X5 @ A )
=> ( ord_le6045566169113846134st_nat @ X5 @ B2 ) )
=> ( ( A != bot_bo3886227569956363488st_nat )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ A ) @ B2 ) ) ) ).
% Inter_subset
thf(fact_1066_INF__greatest,axiom,
! [A: set_nat,U2: $o,F: nat > $o] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A )
=> ( ord_less_eq_o @ U2 @ ( F @ I5 ) ) )
=> ( ord_less_eq_o @ U2 @ ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_1067_INF__greatest,axiom,
! [A: set_o,U2: $o,F: $o > $o] :
( ! [I5: $o] :
( ( member_o @ I5 @ A )
=> ( ord_less_eq_o @ U2 @ ( F @ I5 ) ) )
=> ( ord_less_eq_o @ U2 @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_1068_INF__greatest,axiom,
! [A: set_list_nat,U2: $o,F: list_nat > $o] :
( ! [I5: list_nat] :
( ( member_list_nat @ I5 @ A )
=> ( ord_less_eq_o @ U2 @ ( F @ I5 ) ) )
=> ( ord_less_eq_o @ U2 @ ( complete_Inf_Inf_o @ ( image_list_nat_o @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_1069_INF__greatest,axiom,
! [A: set_nat,U2: set_list_nat,F: nat > set_list_nat] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A )
=> ( ord_le6045566169113846134st_nat @ U2 @ ( F @ I5 ) ) )
=> ( ord_le6045566169113846134st_nat @ U2 @ ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_1070_INF__greatest,axiom,
! [A: set_o,U2: set_list_nat,F: $o > set_list_nat] :
( ! [I5: $o] :
( ( member_o @ I5 @ A )
=> ( ord_le6045566169113846134st_nat @ U2 @ ( F @ I5 ) ) )
=> ( ord_le6045566169113846134st_nat @ U2 @ ( comple184543376406953807st_nat @ ( image_o_set_list_nat @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_1071_INF__greatest,axiom,
! [A: set_nat,U2: set_set_list_nat,F: nat > set_set_list_nat] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A )
=> ( ord_le1068707526560357548st_nat @ U2 @ ( F @ I5 ) ) )
=> ( ord_le1068707526560357548st_nat @ U2 @ ( comple8462666950445340293st_nat @ ( image_3702150970216043171st_nat @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_1072_INF__greatest,axiom,
! [A: set_o,U2: set_set_list_nat,F: $o > set_set_list_nat] :
( ! [I5: $o] :
( ( member_o @ I5 @ A )
=> ( ord_le1068707526560357548st_nat @ U2 @ ( F @ I5 ) ) )
=> ( ord_le1068707526560357548st_nat @ U2 @ ( comple8462666950445340293st_nat @ ( image_4575192727348083743st_nat @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_1073_INF__greatest,axiom,
! [A: set_list_nat,U2: set_list_nat,F: list_nat > set_list_nat] :
( ! [I5: list_nat] :
( ( member_list_nat @ I5 @ A )
=> ( ord_le6045566169113846134st_nat @ U2 @ ( F @ I5 ) ) )
=> ( ord_le6045566169113846134st_nat @ U2 @ ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_1074_INF__greatest,axiom,
! [A: set_list_nat,U2: set_set_list_nat,F: list_nat > set_set_list_nat] :
( ! [I5: list_nat] :
( ( member_list_nat @ I5 @ A )
=> ( ord_le1068707526560357548st_nat @ U2 @ ( F @ I5 ) ) )
=> ( ord_le1068707526560357548st_nat @ U2 @ ( comple8462666950445340293st_nat @ ( image_1427552562080725683st_nat @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_1075_INF__greatest,axiom,
! [A: set_nat,U2: set_set_set_list_nat,F: nat > set_set_set_list_nat] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A )
=> ( ord_le7100322305783427298st_nat @ U2 @ ( F @ I5 ) ) )
=> ( ord_le7100322305783427298st_nat @ U2 @ ( comple5189992959352112827st_nat @ ( image_3522889415257180889st_nat @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_1076_le__INF__iff,axiom,
! [U2: set_set_list_nat,F: set_list_nat > set_set_list_nat,A: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ U2 @ ( comple8462666950445340293st_nat @ ( image_920372530894776809st_nat @ F @ A ) ) )
= ( ! [X2: set_list_nat] :
( ( member_set_list_nat @ X2 @ A )
=> ( ord_le1068707526560357548st_nat @ U2 @ ( F @ X2 ) ) ) ) ) ).
% le_INF_iff
thf(fact_1077_le__INF__iff,axiom,
! [U2: set_list_nat,F: list_nat > set_list_nat,A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ U2 @ ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ F @ A ) ) )
= ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
=> ( ord_le6045566169113846134st_nat @ U2 @ ( F @ X2 ) ) ) ) ) ).
% le_INF_iff
thf(fact_1078_INF__lower2,axiom,
! [I3: nat,A: set_nat,F: nat > $o,U2: $o] :
( ( member_nat @ I3 @ A )
=> ( ( ord_less_eq_o @ ( F @ I3 ) @ U2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_nat_o @ F @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_1079_INF__lower2,axiom,
! [I3: $o,A: set_o,F: $o > $o,U2: $o] :
( ( member_o @ I3 @ A )
=> ( ( ord_less_eq_o @ ( F @ I3 ) @ U2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_1080_INF__lower2,axiom,
! [I3: list_nat,A: set_list_nat,F: list_nat > $o,U2: $o] :
( ( member_list_nat @ I3 @ A )
=> ( ( ord_less_eq_o @ ( F @ I3 ) @ U2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_list_nat_o @ F @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_1081_INF__lower2,axiom,
! [I3: nat,A: set_nat,F: nat > set_list_nat,U2: set_list_nat] :
( ( member_nat @ I3 @ A )
=> ( ( ord_le6045566169113846134st_nat @ ( F @ I3 ) @ U2 )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ ( image_2883343038133793645st_nat @ F @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_1082_INF__lower2,axiom,
! [I3: $o,A: set_o,F: $o > set_list_nat,U2: set_list_nat] :
( ( member_o @ I3 @ A )
=> ( ( ord_le6045566169113846134st_nat @ ( F @ I3 ) @ U2 )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ ( image_o_set_list_nat @ F @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_1083_INF__lower2,axiom,
! [I3: nat,A: set_nat,F: nat > set_set_list_nat,U2: set_set_list_nat] :
( ( member_nat @ I3 @ A )
=> ( ( ord_le1068707526560357548st_nat @ ( F @ I3 ) @ U2 )
=> ( ord_le1068707526560357548st_nat @ ( comple8462666950445340293st_nat @ ( image_3702150970216043171st_nat @ F @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_1084_INF__lower2,axiom,
! [I3: $o,A: set_o,F: $o > set_set_list_nat,U2: set_set_list_nat] :
( ( member_o @ I3 @ A )
=> ( ( ord_le1068707526560357548st_nat @ ( F @ I3 ) @ U2 )
=> ( ord_le1068707526560357548st_nat @ ( comple8462666950445340293st_nat @ ( image_4575192727348083743st_nat @ F @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_1085_INF__lower2,axiom,
! [I3: list_nat,A: set_list_nat,F: list_nat > set_list_nat,U2: set_list_nat] :
( ( member_list_nat @ I3 @ A )
=> ( ( ord_le6045566169113846134st_nat @ ( F @ I3 ) @ U2 )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ F @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_1086_INF__lower2,axiom,
! [I3: list_nat,A: set_list_nat,F: list_nat > set_set_list_nat,U2: set_set_list_nat] :
( ( member_list_nat @ I3 @ A )
=> ( ( ord_le1068707526560357548st_nat @ ( F @ I3 ) @ U2 )
=> ( ord_le1068707526560357548st_nat @ ( comple8462666950445340293st_nat @ ( image_1427552562080725683st_nat @ F @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_1087_INF__lower2,axiom,
! [I3: nat,A: set_nat,F: nat > set_set_set_list_nat,U2: set_set_set_list_nat] :
( ( member_nat @ I3 @ A )
=> ( ( ord_le7100322305783427298st_nat @ ( F @ I3 ) @ U2 )
=> ( ord_le7100322305783427298st_nat @ ( comple5189992959352112827st_nat @ ( image_3522889415257180889st_nat @ F @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_1088_INF__mono_H,axiom,
! [F: set_list_nat > set_set_list_nat,G: set_list_nat > set_set_list_nat,A: set_set_list_nat] :
( ! [X4: set_list_nat] : ( ord_le1068707526560357548st_nat @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_le1068707526560357548st_nat @ ( comple8462666950445340293st_nat @ ( image_920372530894776809st_nat @ F @ A ) ) @ ( comple8462666950445340293st_nat @ ( image_920372530894776809st_nat @ G @ A ) ) ) ) ).
% INF_mono'
thf(fact_1089_INF__mono_H,axiom,
! [F: list_nat > set_list_nat,G: list_nat > set_list_nat,A: set_list_nat] :
( ! [X4: list_nat] : ( ord_le6045566169113846134st_nat @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ F @ A ) ) @ ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ G @ A ) ) ) ) ).
% INF_mono'
thf(fact_1090_INF__lower,axiom,
! [I3: list_nat,A: set_list_nat,F: list_nat > set_list_nat] :
( ( member_list_nat @ I3 @ A )
=> ( ord_le6045566169113846134st_nat @ ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ F @ A ) ) @ ( F @ I3 ) ) ) ).
% INF_lower
thf(fact_1091_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_1092_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_1093_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_1094__092_060open_062_I_096_J_A_IC_An_J_A_096_A_IPow_AX_A_N_A_123_123_125_125_J_A_061_APow_A_IC_An_A_096_AX_J_A_N_A_123_123_125_125_092_060close_062,axiom,
( ( image_920372530894776809st_nat @ ( image_8532145185254316925st_nat @ ( c @ n ) ) @ ( minus_1490147023502258853st_nat @ ( pow_list_nat @ x ) @ ( insert_set_list_nat @ bot_bot_set_list_nat @ bot_bo3886227569956363488st_nat ) ) )
= ( minus_7520675481749758683st_nat @ ( pow_set_list_nat @ ( image_8532145185254316925st_nat @ ( c @ n ) @ x ) ) @ ( insert721039030076969596st_nat @ bot_bo3886227569956363488st_nat @ bot_bo3499706412017099030st_nat ) ) ) ).
% \<open>(`) (C n) ` (Pow X - {{}}) = Pow (C n ` X) - {{}}\<close>
thf(fact_1095_diff__diff__cancel,axiom,
! [I3: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_1096_diff__diff__left,axiom,
! [I3: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J ) @ K )
= ( minus_minus_nat @ I3 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1097_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1098_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I3 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I3 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1099_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1100_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1101_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_1102__092_060open_062Pow_A_IC_An_A_096_AX_J_A_N_A_123_123_125_125_A_092_060subseteq_062_A_I_096_J_A_IC_An_J_A_096_A_IPow_AX_A_N_A_123_123_125_125_J_092_060close_062,axiom,
ord_le7100322305783427298st_nat @ ( minus_7520675481749758683st_nat @ ( pow_set_list_nat @ ( image_8532145185254316925st_nat @ ( c @ n ) @ x ) ) @ ( insert721039030076969596st_nat @ bot_bo3886227569956363488st_nat @ bot_bo3499706412017099030st_nat ) ) @ ( image_920372530894776809st_nat @ ( image_8532145185254316925st_nat @ ( c @ n ) ) @ ( minus_1490147023502258853st_nat @ ( pow_list_nat @ x ) @ ( insert_set_list_nat @ bot_bot_set_list_nat @ bot_bo3886227569956363488st_nat ) ) ) ).
% \<open>Pow (C n ` X) - {{}} \<subseteq> (`) (C n) ` (Pow X - {{}})\<close>
thf(fact_1103_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_1104_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1105_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1106_le__trans,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I3 @ K ) ) ) ).
% le_trans
thf(fact_1107_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1108_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1109_int__induct,axiom,
! [P: int > $o,K: int,I3: int] :
( ( P @ K )
=> ( ! [I5: int] :
( ( ord_less_eq_int @ K @ I5 )
=> ( ( P @ I5 )
=> ( P @ ( plus_plus_int @ I5 @ one_one_int ) ) ) )
=> ( ! [I5: int] :
( ( ord_less_eq_int @ I5 @ K )
=> ( ( P @ I5 )
=> ( P @ ( minus_minus_int @ I5 @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_induct
thf(fact_1110_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_1111_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1112_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1113_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z3: int] :
? [N3: nat] :
( Z3
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1114_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1115_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1116_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1117_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1118_mult__le__mono,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1119_int__ge__induct,axiom,
! [K: int,I3: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ K )
=> ( ! [I5: int] :
( ( ord_less_eq_int @ K @ I5 )
=> ( ( P @ I5 )
=> ( P @ ( plus_plus_int @ I5 @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_ge_induct
thf(fact_1120_int__le__induct,axiom,
! [I3: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ K )
=> ( ! [I5: int] :
( ( ord_less_eq_int @ I5 @ K )
=> ( ( P @ I5 )
=> ( P @ ( minus_minus_int @ I5 @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_le_induct
thf(fact_1121_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1122_mult__le__mono1,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1123_mult__le__mono2,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I3 ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1124_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_1125_inj__on__diff__nat,axiom,
! [N2: set_nat,K: nat] :
( ! [N4: nat] :
( ( member_nat @ N4 @ N2 )
=> ( ord_less_eq_nat @ K @ N4 ) )
=> ( inj_on_nat_nat
@ ^ [N3: nat] : ( minus_minus_nat @ N3 @ K )
@ N2 ) ) ).
% inj_on_diff_nat
thf(fact_1126_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1127_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1128_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1129_Inf__bool__def,axiom,
( complete_Inf_Inf_o
= ( ^ [A3: set_o] :
~ ( member_o @ $false @ A3 ) ) ) ).
% Inf_bool_def
thf(fact_1130_Nat_Ole__imp__diff__is__add,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ( minus_minus_nat @ J @ I3 )
= K )
= ( J
= ( plus_plus_nat @ K @ I3 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1131_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I3 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I3 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1132_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J ) @ K )
= ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1133_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1134_le__diff__conv,axiom,
! [J: nat,K: nat,I3: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I3 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I3 @ K ) ) ) ).
% le_diff_conv
thf(fact_1135_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M3: nat,N4: nat] :
( Z2
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% int_diff_cases
thf(fact_1136_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1137_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_1138_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1139_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_1140_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1141_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1142_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1143_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1144_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1145_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1146_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1147_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1148_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1149_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_1150_add__le__mono,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1151_add__le__mono1,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1152_trans__le__add1,axiom,
! [I3: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1153_trans__le__add2,axiom,
! [I3: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1154_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus_nat @ M4 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1155_zadd__int__left,axiom,
! [M: nat,N: nat,Z2: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_1156_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_1157_int__cases2,axiom,
! [Z2: int] :
( ! [N4: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ! [N4: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% int_cases2
thf(fact_1158_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_1159_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_1160_diff__commute,axiom,
! [I3: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K ) @ J ) ) ).
% diff_commute
thf(fact_1161_nat__diff__add__eq2,axiom,
! [I3: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I3 ) @ U2 ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1162_nat__diff__add__eq1,axiom,
! [J: nat,I3: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J ) @ U2 ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1163_nat__le__add__iff2,axiom,
! [I3: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I3 ) @ U2 ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1164_nat__le__add__iff1,axiom,
! [J: nat,I3: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I3 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J ) @ U2 ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1165_left__add__mult__distrib,axiom,
! [I3: nat,U2: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I3 @ U2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I3 @ J ) @ U2 ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1166_nat__eq__add__iff1,axiom,
! [J: nat,I3: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I3 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I3 @ U2 ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J ) @ U2 ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1167_nat__eq__add__iff2,axiom,
! [I3: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I3 @ U2 ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I3 ) @ U2 ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1168__092_060open_0620_A_092_060le_062_ASUM1_092_060close_062,axiom,
ord_less_eq_int @ zero_zero_int @ sUM1 ).
% \<open>0 \<le> SUM1\<close>
thf(fact_1169_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1170_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1171_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1172_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1173_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1174_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1175_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1176_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1177_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N4: nat] :
( K
= ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1178_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N4: nat] :
( K
!= ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% nonneg_int_cases
thf(fact_1179_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_1180_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N4: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1181_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1182_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1183_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1184_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_1185_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1186_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1187_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1188_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_1189_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1190_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1191_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1192_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1193_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1194_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_1195_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1196_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1197_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1198_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1199_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1200_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_1201_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1202_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1203_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1204_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_1205_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_1206_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1207_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1208_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_1209_verit__la__generic,axiom,
! [A2: int,X3: int] :
( ( ord_less_eq_int @ A2 @ X3 )
| ( A2 = X3 )
| ( ord_less_eq_int @ X3 @ A2 ) ) ).
% verit_la_generic
thf(fact_1210_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A6: nat,B6: nat] :
( ( semiri1314217659103216013at_int @ A6 )
= ( semiri1314217659103216013at_int @ B6 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1211_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_1212_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_1213_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M4: nat,N3: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1214_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B6: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A6 ) @ ( semiri1314217659103216013at_int @ B6 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1215_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_1216_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_1217_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1218_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_1219_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B6: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A6 ) @ ( semiri1314217659103216013at_int @ B6 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1220_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M2: nat] :
( ( P @ X3 )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M2 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1221_n__ge__X,axiom,
! [X3: list_nat] :
( ( member_list_nat @ X3 @ x )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ X3 ) @ n ) ) ).
% n_ge_X
thf(fact_1222_C__def,axiom,
( c
= ( ^ [N3: nat,X7: list_nat] :
( collect_list_nat
@ ^ [X2: list_nat] :
( ( pointwise_le @ X7 @ X2 )
& ( ( groups4561878855575611511st_nat @ X2 )
= N3 ) ) ) ) ) ).
% C_def
thf(fact_1223_pointwise__le__refl,axiom,
! [X3: list_nat] : ( pointwise_le @ X3 @ X3 ) ).
% pointwise_le_refl
thf(fact_1224_sum__list__minus,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( pointwise_le @ Xs @ Ys )
=> ( ( groups4561878855575611511st_nat @ ( minus_minus_list_nat @ Ys @ Xs ) )
= ( minus_minus_nat @ ( groups4561878855575611511st_nat @ Ys ) @ ( groups4561878855575611511st_nat @ Xs ) ) ) ) ).
% sum_list_minus
thf(fact_1225_pointwise__le__imp___092_060sigma_062,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( pointwise_le @ Xs @ Ys )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ).
% pointwise_le_imp_\<sigma>
thf(fact_1226_pairwise__minus__cancel,axiom,
! [Z2: list_nat,X3: list_nat,Y3: list_nat] :
( ( pointwise_le @ Z2 @ X3 )
=> ( ( pointwise_le @ Z2 @ Y3 )
=> ( ( ( minus_minus_list_nat @ X3 @ Z2 )
= ( minus_minus_list_nat @ Y3 @ Z2 ) )
=> ( X3 = Y3 ) ) ) ) ).
% pairwise_minus_cancel
thf(fact_1227_pointwise__le__trans,axiom,
! [X3: list_nat,Y3: list_nat,Z2: list_nat] :
( ( pointwise_le @ X3 @ Y3 )
=> ( ( pointwise_le @ Y3 @ Z2 )
=> ( pointwise_le @ X3 @ Z2 ) ) ) ).
% pointwise_le_trans
thf(fact_1228_pointwise__le__antisym,axiom,
! [X3: list_nat,Y3: list_nat] :
( ( pointwise_le @ X3 @ Y3 )
=> ( ( pointwise_le @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% pointwise_le_antisym
thf(fact_1229_extend__tuple,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs ) @ N )
=> ( ( ( size_size_list_nat @ Xs )
!= zero_zero_nat )
=> ~ ! [Ys2: list_nat] :
( ( ( groups4561878855575611511st_nat @ Ys2 )
= N )
=> ~ ( pointwise_le @ Xs @ Ys2 ) ) ) ) ).
% extend_tuple
thf(fact_1230_pointwise__le__plus,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( pointwise_le @ Xs @ Ys )
=> ( ( ord_less_eq_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Zs ) )
=> ( pointwise_le @ Xs @ ( plus_plus_list_nat @ Ys @ Zs ) ) ) ) ).
% pointwise_le_plus
thf(fact_1231_card__length__sum__list__rec,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ one_one_nat @ M )
=> ( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L2: list_nat] :
( ( ( size_size_list_nat @ L2 )
= M )
& ( ( groups4561878855575611511st_nat @ L2 )
= N2 ) ) ) )
= ( plus_plus_nat
@ ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L2: list_nat] :
( ( ( size_size_list_nat @ L2 )
= ( minus_minus_nat @ M @ one_one_nat ) )
& ( ( groups4561878855575611511st_nat @ L2 )
= N2 ) ) ) )
@ ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L2: list_nat] :
( ( ( size_size_list_nat @ L2 )
= M )
& ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L2 ) @ one_one_nat )
= N2 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_1232_C__empty__iff,axiom,
! [X8: list_nat] :
( ( ( size_size_list_nat @ X8 )
!= zero_zero_nat )
=> ( ( ( c @ n @ X8 )
= bot_bot_set_list_nat )
= ( ord_less_nat @ n @ ( groups4561878855575611511st_nat @ X8 ) ) ) ) ).
% C_empty_iff
thf(fact_1233__092_060open_0620_A_060_An_092_060close_062,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% \<open>0 < n\<close>
thf(fact_1234_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_1235_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1236_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1237_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_1238_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1239_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_1240_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1241_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1242_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_1243_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1244_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1245_nat__zero__less__power__iff,axiom,
! [X3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X3 )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1246_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1247_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1248_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1249_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1250_less__diff__conv,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1251_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1252_diff__less__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1253_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1254_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1255_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1256_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1257_mult__less__mono2,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I3 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1258_mult__less__mono1,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1259_less__imp__add__positive,axiom,
! [I3: nat,J: nat] :
( ( ord_less_nat @ I3 @ J )
=> ? [K4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K4 )
& ( ( plus_plus_nat @ I3 @ K4 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1260_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_1261_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1262_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1263_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1264_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1265_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1266_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1267_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1268_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_1269_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J: nat] :
( ! [I5: nat,J3: nat] :
( ( ord_less_nat @ I5 @ J3 )
=> ( ord_less_nat @ ( F @ I5 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1270_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_1271_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_1272_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
| ( M4 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1273_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_1274_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
& ( M4 != N3 ) ) ) ) ).
% nat_less_le
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y3: nat] :
( ( if_nat @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y3: nat] :
( ( if_nat @ $true @ X3 @ Y3 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( groups7312845317294741502at_int
@ ( comp_s3399862273134644595st_nat
@ ^ [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 ) ) ) )
@ ( image_8532145185254316925st_nat @ ( c @ n ) ) )
@ ( minus_1490147023502258853st_nat @ ( pow_list_nat @ x ) @ ( insert_set_list_nat @ bot_bot_set_list_nat @ bot_bo3886227569956363488st_nat ) ) )
= ( groups7312845317294741502at_int
@ ^ [Y: set_list_nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ ( finite_card_list_nat @ Y ) @ one_one_nat ) ) @ ( semiri1314217659103216013at_int @ ( finite_card_list_nat @ ( comple184543376406953807st_nat @ ( image_8532145185254316925st_nat @ ( c @ n ) @ Y ) ) ) ) )
@ ( collect_set_list_nat
@ ^ [Y: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ Y @ x )
& ( Y != bot_bot_set_list_nat ) ) ) ) ) ).
%------------------------------------------------------------------------------