TPTP Problem File: SLH0577^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Universal_Hash_Families/0033_Preliminary_Results/prob_00013_000298__18383540_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1538 ( 727 unt; 262 typ; 0 def)
% Number of atoms : 3407 (1847 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 9708 ( 418 ~; 10 |; 207 &;8024 @)
% ( 0 <=>;1049 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Number of types : 34 ( 33 usr)
% Number of type conns : 1357 (1357 >; 0 *; 0 +; 0 <<)
% Number of symbols : 232 ( 229 usr; 39 con; 0-4 aty)
% Number of variables : 3668 ( 713 ^;2844 !; 111 ?;3668 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:43:16.267
%------------------------------------------------------------------------------
% Could-be-implicit typings (33)
thf(ty_n_t__Set__Oset_I_062_It__Extended____Real__Oereal_Mt__Extended____Real__Oereal_J_J,type,
set_Ex2354994561656779803_ereal: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_J,type,
set_se4580700918925141924nnreal: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Extended____Nonnegative____Real__Oennreal_M_Eo_J_J,type,
set_Ex70502500924464887real_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Extended____Real__Oereal_J_J,type,
set_na7152043825411390613_ereal: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Extended____Real__Oereal_Mt__Nat__Onat_J_J,type,
set_Ex8414784459666926319al_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
set_Ex3793607809372303086nnreal: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Extended____Real__Oereal_J_J,type,
set_se6634062954251873166_ereal: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__b_Mt__Extended____Real__Oereal_J_J,type,
set_b_Extended_ereal: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Extended____Real__Oereal_Mtf__b_J_J,type,
set_Extended_ereal_b: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Extended____Real__Oereal_M_Eo_J_J,type,
set_Extended_ereal_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_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__Extended____Nonnegative____Real__Oennreal,type,
extend8495563244428889912nnreal: $tType ).
thf(ty_n_t__Set__Oset_It__Extended____Real__Oereal_J,type,
set_Extended_ereal: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__b_Mt__Nat__Onat_J_J,type,
set_b_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mtf__b_J_J,type,
set_nat_b: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__c_J_J,type,
set_set_c: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
set_set_b: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__b_Mtf__b_J_J,type,
set_b_b: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__c_M_Eo_J_J,type,
set_c_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__b_M_Eo_J_J,type,
set_b_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
set_o_o: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Extended____Real__Oereal,type,
extended_ereal: $tType ).
thf(ty_n_t__Set__Oset_Itf__c_J,type,
set_c: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $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_tf__c,type,
c: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (229)
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_I_Eo_M_Eo_J,type,
complete_Inf_Inf_o_o: set_o_o > $o > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Extended____Nonnegative____Real__Oennreal_M_Eo_J,type,
comple2110304272711893406real_o: set_Ex70502500924464887real_o > extend8495563244428889912nnreal > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Extended____Real__Oereal_M_Eo_J,type,
comple5376498136088350824real_o: set_Extended_ereal_o > extended_ereal > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Nat__Onat_M_Eo_J,type,
comple6214475593288795910_nat_o: set_nat_o > nat > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_Itf__b_M_Eo_J,type,
complete_Inf_Inf_b_o: set_b_o > b > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_Itf__c_M_Eo_J,type,
complete_Inf_Inf_c_o: set_c_o > c > $o ).
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__Extended____Nonnegative____Real__Oennreal,type,
comple7330758040695736817nnreal: set_Ex3793607809372303086nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Extended____Real__Oereal,type,
comple3556804143462414037_ereal: set_Extended_ereal > extended_ereal ).
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__Extended____Nonnegative____Real__Oennreal_J,type,
comple5724520875574609319nnreal: set_se4580700918925141924nnreal > set_Ex3793607809372303086nnreal ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
comple4418415374894819509_ereal: set_se6634062954251873166_ereal > set_Extended_ereal ).
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_Itf__b_J,type,
comple6135023382983342438_set_b: set_set_b > set_b ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_Itf__c_J,type,
comple6135023387286571239_set_c: set_set_c > set_c ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_I_Eo_M_Eo_J,type,
complete_Sup_Sup_o_o: set_o_o > $o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Extended____Nonnegative____Real__Oennreal_M_Eo_J,type,
comple5476927491321936772real_o: set_Ex70502500924464887real_o > extend8495563244428889912nnreal > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Extended____Real__Oereal_M_Eo_J,type,
comple8551942733113566466real_o: set_Extended_ereal_o > extended_ereal > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_Eo_J,type,
comple8317665133742190828_nat_o: set_nat_o > nat > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_Itf__b_M_Eo_J,type,
complete_Sup_Sup_b_o: set_b_o > b > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_Itf__c_M_Eo_J,type,
complete_Sup_Sup_c_o: set_c_o > c > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Extended____Nonnegative____Real__Oennreal,type,
comple6814414086264997003nnreal: set_Ex3793607809372303086nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Extended____Real__Oereal,type,
comple8415311339701865915_ereal: set_Extended_ereal > extended_ereal ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_Eo_J,type,
comple90263536869209701_set_o: set_set_o > set_o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
comple4226387801268262977nnreal: set_se4580700918925141924nnreal > set_Ex3793607809372303086nnreal ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
comple4319282863272126363_ereal: set_se6634062954251873166_ereal > set_Extended_ereal ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__b_J,type,
comple2307003614231284044_set_b: set_set_b > set_b ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__c_J,type,
comple2307003618534512845_set_c: set_set_c > set_c ).
thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity_001t__Extended____Real__Oereal,type,
extend1530274965995635425_ereal: extended_ereal ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Extended____Nonnegative____Real__Oennreal,type,
monoto2291723841412853873nnreal: set_nat > ( nat > nat > $o ) > ( extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ) > ( nat > extend8495563244428889912nnreal ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Extended____Real__Oereal,type,
monoto8452838292781035605_ereal: set_nat > ( nat > nat > $o ) > ( extended_ereal > extended_ereal > $o ) > ( nat > extended_ereal ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nonnegative____Real__Oennreal,type,
minus_8429688780609304081nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Real__Oereal,type,
minus_2816186181549245109_ereal: extended_ereal > extended_ereal > extended_ereal ).
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__Extended____Nonnegative____Real__Oennreal_J,type,
minus_104578273773384135nnreal: set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
minus_1264018925008434325_ereal: set_Extended_ereal > set_Extended_ereal > set_Extended_ereal ).
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_Itf__b_J,type,
minus_minus_set_b: set_b > set_b > set_b ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__c_J,type,
minus_minus_set_c: set_c > set_c > set_c ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nonnegative____Real__Oennreal,type,
plus_p1859984266308609217nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Real__Oereal,type,
plus_p7876563987511257093_ereal: extended_ereal > extended_ereal > extended_ereal ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Extended____Real__Oereal,type,
uminus27091377158695749_ereal: extended_ereal > extended_ereal ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
uminus5895154729394068773_ereal: set_Extended_ereal > set_Extended_ereal ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nonnegative____Real__Oennreal,type,
zero_z7100319975126383169nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Real__Oereal,type,
zero_z2744965634713055877_ereal: extended_ereal ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_Eo,type,
inf_inf_o: $o > $o > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Extended____Nonnegative____Real__Oennreal,type,
inf_in7439215052339218890nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Extended____Real__Oereal,type,
inf_in2794916579150040252_ereal: extended_ereal > extended_ereal > extended_ereal ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_Eo_J,type,
inf_inf_set_o: set_o > set_o > set_o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
inf_in3368558534146122112nnreal: set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
inf_in2779415704524776092_ereal: set_Extended_ereal > set_Extended_ereal > set_Extended_ereal ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__b_J,type,
inf_inf_set_b: set_b > set_b > set_b ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__c_J,type,
inf_inf_set_c: set_c > set_c > set_c ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_Eo_M_Eo_J,type,
bot_bot_o_o: $o > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Extended____Nonnegative____Real__Oennreal_M_Eo_J,type,
bot_bo412624608084785539real_o: extend8495563244428889912nnreal > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Extended____Real__Oereal_M_Eo_J,type,
bot_bo5519581617326455619real_o: extended_ereal > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__b_M_Eo_J,type,
bot_bot_b_o: b > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__c_M_Eo_J,type,
bot_bot_c_o: c > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nonnegative____Real__Oennreal,type,
bot_bo841427958541957580nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Real__Oereal,type,
bot_bo2710585358178759738_ereal: extended_ereal ).
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__Extended____Nonnegative____Real__Oennreal_J,type,
bot_bo4854962954004695426nnreal: set_Ex3793607809372303086nnreal ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
bot_bo8367695208629047834_ereal: set_Extended_ereal ).
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__Extended____Nonnegative____Real__Oennreal_J_J,type,
bot_bo2988155216863113784nnreal: set_se4580700918925141924nnreal ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Extended____Real__Oereal_J_J,type,
bot_bo7400643019497942010_ereal: set_se6634062954251873166_ereal ).
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_Itf__b_J,type,
bot_bot_set_b: set_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__c_J,type,
bot_bot_set_c: set_c ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nonnegative____Real__Oennreal,type,
ord_le7381754540660121996nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nonnegative____Real__Oennreal,type,
ord_le3935885782089961368nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Real__Oereal,type,
ord_le1083603963089353582_ereal: extended_ereal > extended_ereal > $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_It__Extended____Nonnegative____Real__Oennreal_J,type,
ord_le6787938422905777998nnreal: set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
ord_le1644982726543182158_ereal: set_Extended_ereal > set_Extended_ereal > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_Eo_M_Eo_J,type,
top_top_o_o: $o > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Extended____Nonnegative____Real__Oennreal_M_Eo_J,type,
top_to5118619752887738471real_o: extend8495563244428889912nnreal > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Extended____Real__Oereal_M_Eo_J,type,
top_to6999531812125281119real_o: extended_ereal > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_Itf__b_M_Eo_J,type,
top_top_b_o: b > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_Itf__c_M_Eo_J,type,
top_top_c_o: c > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_Eo,type,
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thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__c_J,type,
image_set_b_set_c: ( set_b > set_c ) > set_set_b > set_set_c ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__c_J_001_062_Itf__c_M_Eo_J,type,
image_set_c_c_o: ( set_c > c > $o ) > set_set_c > set_c_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__c_J_001_Eo,type,
image_set_c_o: ( set_c > $o ) > set_set_c > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__c_J_001t__Set__Oset_Itf__b_J,type,
image_set_c_set_b: ( set_c > set_b ) > set_set_c > set_set_b ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__c_J_001t__Set__Oset_Itf__c_J,type,
image_set_c_set_c: ( set_c > set_c ) > set_set_c > set_set_c ).
thf(sy_c_Set_Oimage_001tf__b_001_Eo,type,
image_b_o: ( b > $o ) > set_b > set_o ).
thf(sy_c_Set_Oimage_001tf__b_001t__Extended____Real__Oereal,type,
image_5319725110001000852_ereal: ( b > extended_ereal ) > set_b > set_Extended_ereal ).
thf(sy_c_Set_Oimage_001tf__b_001t__Nat__Onat,type,
image_b_nat: ( b > nat ) > set_b > set_nat ).
thf(sy_c_Set_Oimage_001tf__b_001t__Set__Oset_I_Eo_J,type,
image_b_set_o: ( b > set_o ) > set_b > set_set_o ).
thf(sy_c_Set_Oimage_001tf__b_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
image_8773349707370420084_ereal: ( b > set_Extended_ereal ) > set_b > set_se6634062954251873166_ereal ).
thf(sy_c_Set_Oimage_001tf__b_001t__Set__Oset_It__Nat__Onat_J,type,
image_b_set_nat: ( b > set_nat ) > set_b > set_set_nat ).
thf(sy_c_Set_Oimage_001tf__b_001t__Set__Oset_Itf__b_J,type,
image_b_set_b: ( b > set_b ) > set_b > set_set_b ).
thf(sy_c_Set_Oimage_001tf__b_001tf__b,type,
image_b_b: ( b > b ) > set_b > set_b ).
thf(sy_c_Set_Oimage_001tf__b_001tf__c,type,
image_b_c: ( b > c ) > set_b > set_c ).
thf(sy_c_Set_Oimage_001tf__c_001_Eo,type,
image_c_o: ( c > $o ) > set_c > set_o ).
thf(sy_c_Set_Oimage_001tf__c_001t__Extended____Real__Oereal,type,
image_2233968868011006291_ereal: ( c > extended_ereal ) > set_c > set_Extended_ereal ).
thf(sy_c_Set_Oimage_001tf__c_001t__Set__Oset_I_Eo_J,type,
image_c_set_o: ( c > set_o ) > set_c > set_set_o ).
thf(sy_c_Set_Oimage_001tf__c_001tf__b,type,
image_c_b: ( c > b ) > set_c > set_b ).
thf(sy_c_Set_Oimage_001tf__c_001tf__c,type,
image_c_c: ( c > c ) > set_c > set_c ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Extended____Nonnegative____Real__Oennreal,type,
member7908768830364227535nnreal: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal > $o ).
thf(sy_c_member_001t__Extended____Real__Oereal,type,
member2350847679896131959_ereal: extended_ereal > set_Extended_ereal > $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__Extended____Nonnegative____Real__Oennreal_J,type,
member603777416030116741nnreal: set_Ex3793607809372303086nnreal > set_se4580700918925141924nnreal > $o ).
thf(sy_c_member_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
member5519481007471526743_ereal: set_Extended_ereal > set_se6634062954251873166_ereal > $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_Itf__b_J,type,
member_set_b: set_b > set_set_b > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__c_J,type,
member_set_c: set_c > set_set_c > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_c_member_001tf__c,type,
member_c: c > set_c > $o ).
thf(sy_v_P,type,
p: a > $o ).
thf(sy_v_f,type,
f: a > b ).
thf(sy_v_g,type,
g: a > c ).
thf(sy_v_h,type,
h: c > b ).
% Relevant facts (1275)
thf(fact_0_assms,axiom,
! [X: a] :
( ( p @ X )
=> ( ( f @ X )
= ( h @ ( g @ X ) ) ) ) ).
% assms
thf(fact_1_image__ident,axiom,
! [Y: set_c] :
( ( image_c_c
@ ^ [X2: c] : X2
@ Y )
= Y ) ).
% image_ident
thf(fact_2_image__ident,axiom,
! [Y: set_b] :
( ( image_b_b
@ ^ [X2: b] : X2
@ Y )
= Y ) ).
% image_ident
thf(fact_3_image__ident,axiom,
! [Y: set_Extended_ereal] :
( ( image_6042159593519690757_ereal
@ ^ [X2: extended_ereal] : X2
@ Y )
= Y ) ).
% image_ident
thf(fact_4_image__eqI,axiom,
! [B: b,F: c > b,X: c,A: set_c] :
( ( B
= ( F @ X ) )
=> ( ( member_c @ X @ A )
=> ( member_b @ B @ ( image_c_b @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_5_image__eqI,axiom,
! [B: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member7908768830364227535nnreal @ B @ ( image_8459861568512453903nnreal @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_6_image__eqI,axiom,
! [B: extended_ereal,F: nat > extended_ereal,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member2350847679896131959_ereal @ B @ ( image_4309273772856505399_ereal @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_7_image__eqI,axiom,
! [B: $o,F: $o > $o,X: $o,A: set_o] :
( ( B
= ( F @ X ) )
=> ( ( member_o @ X @ A )
=> ( member_o @ B @ ( image_o_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_8_image__eqI,axiom,
! [B: extended_ereal,F: $o > extended_ereal,X: $o,A: set_o] :
( ( B
= ( F @ X ) )
=> ( ( member_o @ X @ A )
=> ( member2350847679896131959_ereal @ B @ ( image_7729549296133164475_ereal @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_9_image__eqI,axiom,
! [B: $o,F: extended_ereal > $o,X: extended_ereal,A: set_Extended_ereal] :
( ( B
= ( F @ X ) )
=> ( ( member2350847679896131959_ereal @ X @ A )
=> ( member_o @ B @ ( image_951975095941678543real_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_10_image__eqI,axiom,
! [B: extended_ereal,F: extended_ereal > extended_ereal,X: extended_ereal,A: set_Extended_ereal] :
( ( B
= ( F @ X ) )
=> ( ( member2350847679896131959_ereal @ X @ A )
=> ( member2350847679896131959_ereal @ B @ ( image_6042159593519690757_ereal @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_11_image__eqI,axiom,
! [B: nat,F: $o > nat,X: $o,A: set_o] :
( ( B
= ( F @ X ) )
=> ( ( member_o @ X @ A )
=> ( member_nat @ B @ ( image_o_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_12_image__eqI,axiom,
! [B: extend8495563244428889912nnreal,F: $o > extend8495563244428889912nnreal,X: $o,A: set_o] :
( ( B
= ( F @ X ) )
=> ( ( member_o @ X @ A )
=> ( member7908768830364227535nnreal @ B @ ( image_3342735880743421067nnreal @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_13_image__eqI,axiom,
! [B: c,F: $o > c,X: $o,A: set_o] :
( ( B
= ( F @ X ) )
=> ( ( member_o @ X @ A )
=> ( member_c @ B @ ( image_o_c @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_14_Setcompr__eq__image,axiom,
! [F: nat > extended_ereal,A: set_nat] :
( ( collec5835592288176408249_ereal
@ ^ [Uu: extended_ereal] :
? [X2: nat] :
( ( Uu
= ( F @ X2 ) )
& ( member_nat @ X2 @ A ) ) )
= ( image_4309273772856505399_ereal @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_15_Setcompr__eq__image,axiom,
! [F: nat > extend8495563244428889912nnreal,A: set_nat] :
( ( collec6648975593938027277nnreal
@ ^ [Uu: extend8495563244428889912nnreal] :
? [X2: nat] :
( ( Uu
= ( F @ X2 ) )
& ( member_nat @ X2 @ A ) ) )
= ( image_8459861568512453903nnreal @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_16_Setcompr__eq__image,axiom,
! [F: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( collec5835592288176408249_ereal
@ ^ [Uu: extended_ereal] :
? [X2: extended_ereal] :
( ( Uu
= ( F @ X2 ) )
& ( member2350847679896131959_ereal @ X2 @ A ) ) )
= ( image_6042159593519690757_ereal @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_17_Setcompr__eq__image,axiom,
! [F: c > b,A: set_c] :
( ( collect_b
@ ^ [Uu: b] :
? [X2: c] :
( ( Uu
= ( F @ X2 ) )
& ( member_c @ X2 @ A ) ) )
= ( image_c_b @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_18_Setcompr__eq__image,axiom,
! [F: $o > b,A: set_o] :
( ( collect_b
@ ^ [Uu: b] :
? [X2: $o] :
( ( Uu
= ( F @ X2 ) )
& ( member_o @ X2 @ A ) ) )
= ( image_o_b @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_19_Setcompr__eq__image,axiom,
! [F: extended_ereal > b,A: set_Extended_ereal] :
( ( collect_b
@ ^ [Uu: b] :
? [X2: extended_ereal] :
( ( Uu
= ( F @ X2 ) )
& ( member2350847679896131959_ereal @ X2 @ A ) ) )
= ( image_3724615099042636214real_b @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_20_Setcompr__eq__image,axiom,
! [F: $o > c,A: set_o] :
( ( collect_c
@ ^ [Uu: c] :
? [X2: $o] :
( ( Uu
= ( F @ X2 ) )
& ( member_o @ X2 @ A ) ) )
= ( image_o_c @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_21_Setcompr__eq__image,axiom,
! [F: extended_ereal > c,A: set_Extended_ereal] :
( ( collect_c
@ ^ [Uu: c] :
? [X2: extended_ereal] :
( ( Uu
= ( F @ X2 ) )
& ( member2350847679896131959_ereal @ X2 @ A ) ) )
= ( image_3724615099042636215real_c @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_22_Setcompr__eq__image,axiom,
! [F: nat > b,A: set_nat] :
( ( collect_b
@ ^ [Uu: b] :
? [X2: nat] :
( ( Uu
= ( F @ X2 ) )
& ( member_nat @ X2 @ A ) ) )
= ( image_nat_b @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_23_Setcompr__eq__image,axiom,
! [F: extend8495563244428889912nnreal > b,A: set_Ex3793607809372303086nnreal] :
( ( collect_b
@ ^ [Uu: b] :
? [X2: extend8495563244428889912nnreal] :
( ( Uu
= ( F @ X2 ) )
& ( member7908768830364227535nnreal @ X2 @ A ) ) )
= ( image_7862617044475835264real_b @ F @ A ) ) ).
% Setcompr_eq_image
thf(fact_24_setcompr__eq__image,axiom,
! [F: nat > extended_ereal,P: nat > $o] :
( ( collec5835592288176408249_ereal
@ ^ [Uu: extended_ereal] :
? [X2: nat] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) )
= ( image_4309273772856505399_ereal @ F @ ( collect_nat @ P ) ) ) ).
% setcompr_eq_image
thf(fact_25_setcompr__eq__image,axiom,
! [F: extended_ereal > extended_ereal,P: extended_ereal > $o] :
( ( collec5835592288176408249_ereal
@ ^ [Uu: extended_ereal] :
? [X2: extended_ereal] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) )
= ( image_6042159593519690757_ereal @ F @ ( collec5835592288176408249_ereal @ P ) ) ) ).
% setcompr_eq_image
thf(fact_26_setcompr__eq__image,axiom,
! [F: nat > extend8495563244428889912nnreal,P: nat > $o] :
( ( collec6648975593938027277nnreal
@ ^ [Uu: extend8495563244428889912nnreal] :
? [X2: nat] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) )
= ( image_8459861568512453903nnreal @ F @ ( collect_nat @ P ) ) ) ).
% setcompr_eq_image
thf(fact_27_setcompr__eq__image,axiom,
! [F: b > b,P: b > $o] :
( ( collect_b
@ ^ [Uu: b] :
? [X2: b] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) )
= ( image_b_b @ F @ ( collect_b @ P ) ) ) ).
% setcompr_eq_image
thf(fact_28_setcompr__eq__image,axiom,
! [F: c > b,P: c > $o] :
( ( collect_b
@ ^ [Uu: b] :
? [X2: c] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) )
= ( image_c_b @ F @ ( collect_c @ P ) ) ) ).
% setcompr_eq_image
thf(fact_29_setcompr__eq__image,axiom,
! [F: b > c,P: b > $o] :
( ( collect_c
@ ^ [Uu: c] :
? [X2: b] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) )
= ( image_b_c @ F @ ( collect_b @ P ) ) ) ).
% setcompr_eq_image
thf(fact_30_setcompr__eq__image,axiom,
! [F: c > c,P: c > $o] :
( ( collect_c
@ ^ [Uu: c] :
? [X2: c] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) )
= ( image_c_c @ F @ ( collect_c @ P ) ) ) ).
% setcompr_eq_image
thf(fact_31_setcompr__eq__image,axiom,
! [F: extended_ereal > b,P: extended_ereal > $o] :
( ( collect_b
@ ^ [Uu: b] :
? [X2: extended_ereal] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) )
= ( image_3724615099042636214real_b @ F @ ( collec5835592288176408249_ereal @ P ) ) ) ).
% setcompr_eq_image
thf(fact_32_setcompr__eq__image,axiom,
! [F: extend8495563244428889912nnreal > b,P: extend8495563244428889912nnreal > $o] :
( ( collect_b
@ ^ [Uu: b] :
? [X2: extend8495563244428889912nnreal] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) )
= ( image_7862617044475835264real_b @ F @ ( collec6648975593938027277nnreal @ P ) ) ) ).
% setcompr_eq_image
thf(fact_33_setcompr__eq__image,axiom,
! [F: nat > b,P: nat > $o] :
( ( collect_b
@ ^ [Uu: b] :
? [X2: nat] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) )
= ( image_nat_b @ F @ ( collect_nat @ P ) ) ) ).
% setcompr_eq_image
thf(fact_34_Inf_OINF__identity__eq,axiom,
! [Inf: set_c > c,A: set_c] :
( ( Inf
@ ( image_c_c
@ ^ [X2: c] : X2
@ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_identity_eq
thf(fact_35_Inf_OINF__identity__eq,axiom,
! [Inf: set_b > b,A: set_b] :
( ( Inf
@ ( image_b_b
@ ^ [X2: b] : X2
@ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_identity_eq
thf(fact_36_Inf_OINF__identity__eq,axiom,
! [Inf: set_Extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( Inf
@ ( image_6042159593519690757_ereal
@ ^ [X2: extended_ereal] : X2
@ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_identity_eq
thf(fact_37_Sup_OSUP__identity__eq,axiom,
! [Sup: set_c > c,A: set_c] :
( ( Sup
@ ( image_c_c
@ ^ [X2: c] : X2
@ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_identity_eq
thf(fact_38_Sup_OSUP__identity__eq,axiom,
! [Sup: set_b > b,A: set_b] :
( ( Sup
@ ( image_b_b
@ ^ [X2: b] : X2
@ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_identity_eq
thf(fact_39_Sup_OSUP__identity__eq,axiom,
! [Sup: set_Extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( Sup
@ ( image_6042159593519690757_ereal
@ ^ [X2: extended_ereal] : X2
@ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_identity_eq
thf(fact_40_imageE,axiom,
! [B: b,F: c > b,A: set_c] :
( ( member_b @ B @ ( image_c_b @ F @ A ) )
=> ~ ! [X3: c] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_c @ X3 @ A ) ) ) ).
% imageE
thf(fact_41_imageE,axiom,
! [B: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,A: set_nat] :
( ( member7908768830364227535nnreal @ B @ ( image_8459861568512453903nnreal @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_42_imageE,axiom,
! [B: $o,F: $o > $o,A: set_o] :
( ( member_o @ B @ ( image_o_o @ F @ A ) )
=> ~ ! [X3: $o] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% imageE
thf(fact_43_imageE,axiom,
! [B: $o,F: extended_ereal > $o,A: set_Extended_ereal] :
( ( member_o @ B @ ( image_951975095941678543real_o @ F @ A ) )
=> ~ ! [X3: extended_ereal] :
( ( B
= ( F @ X3 ) )
=> ~ ( member2350847679896131959_ereal @ X3 @ A ) ) ) ).
% imageE
thf(fact_44_imageE,axiom,
! [B: extended_ereal,F: nat > extended_ereal,A: set_nat] :
( ( member2350847679896131959_ereal @ B @ ( image_4309273772856505399_ereal @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_45_imageE,axiom,
! [B: extended_ereal,F: $o > extended_ereal,A: set_o] :
( ( member2350847679896131959_ereal @ B @ ( image_7729549296133164475_ereal @ F @ A ) )
=> ~ ! [X3: $o] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% imageE
thf(fact_46_imageE,axiom,
! [B: extended_ereal,F: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ B @ ( image_6042159593519690757_ereal @ F @ A ) )
=> ~ ! [X3: extended_ereal] :
( ( B
= ( F @ X3 ) )
=> ~ ( member2350847679896131959_ereal @ X3 @ A ) ) ) ).
% imageE
thf(fact_47_imageE,axiom,
! [B: $o,F: nat > $o,A: set_nat] :
( ( member_o @ B @ ( image_nat_o @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_48_imageE,axiom,
! [B: $o,F: extend8495563244428889912nnreal > $o,A: set_Ex3793607809372303086nnreal] :
( ( member_o @ B @ ( image_3162942742313426073real_o @ F @ A ) )
=> ~ ! [X3: extend8495563244428889912nnreal] :
( ( B
= ( F @ X3 ) )
=> ~ ( member7908768830364227535nnreal @ X3 @ A ) ) ) ).
% imageE
thf(fact_49_imageE,axiom,
! [B: $o,F: c > $o,A: set_c] :
( ( member_o @ B @ ( image_c_o @ F @ A ) )
=> ~ ! [X3: c] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_c @ X3 @ A ) ) ) ).
% imageE
thf(fact_50_image__image,axiom,
! [F: b > b,G: c > b,A: set_c] :
( ( image_b_b @ F @ ( image_c_b @ G @ A ) )
= ( image_c_b
@ ^ [X2: c] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_51_image__image,axiom,
! [F: extended_ereal > extend8495563244428889912nnreal,G: nat > extended_ereal,A: set_nat] :
( ( image_8614087454967683265nnreal @ F @ ( image_4309273772856505399_ereal @ G @ A ) )
= ( image_8459861568512453903nnreal
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_52_image__image,axiom,
! [F: extend8495563244428889912nnreal > extended_ereal,G: nat > extend8495563244428889912nnreal,A: set_nat] :
( ( image_6393943237584228047_ereal @ F @ ( image_8459861568512453903nnreal @ G @ A ) )
= ( image_4309273772856505399_ereal
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_53_image__image,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,G: nat > extend8495563244428889912nnreal,A: set_nat] :
( ( image_8394674774369097847nnreal @ F @ ( image_8459861568512453903nnreal @ G @ A ) )
= ( image_8459861568512453903nnreal
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_54_image__image,axiom,
! [F: c > b,G: c > c,A: set_c] :
( ( image_c_b @ F @ ( image_c_c @ G @ A ) )
= ( image_c_b
@ ^ [X2: c] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_55_image__image,axiom,
! [F: nat > extended_ereal,G: nat > nat,A: set_nat] :
( ( image_4309273772856505399_ereal @ F @ ( image_nat_nat @ G @ A ) )
= ( image_4309273772856505399_ereal
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_56_image__image,axiom,
! [F: nat > extended_ereal,G: extended_ereal > nat,A: set_Extended_ereal] :
( ( image_4309273772856505399_ereal @ F @ ( image_7659842161140344153al_nat @ G @ A ) )
= ( image_6042159593519690757_ereal
@ ^ [X2: extended_ereal] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_57_image__image,axiom,
! [F: extended_ereal > extended_ereal,G: nat > extended_ereal,A: set_nat] :
( ( image_6042159593519690757_ereal @ F @ ( image_4309273772856505399_ereal @ G @ A ) )
= ( image_4309273772856505399_ereal
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_58_image__image,axiom,
! [F: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( image_6042159593519690757_ereal @ F @ ( image_6042159593519690757_ereal @ G @ A ) )
= ( image_6042159593519690757_ereal
@ ^ [X2: extended_ereal] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_59_image__image,axiom,
! [F: nat > extend8495563244428889912nnreal,G: nat > nat,A: set_nat] :
( ( image_8459861568512453903nnreal @ F @ ( image_nat_nat @ G @ A ) )
= ( image_8459861568512453903nnreal
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_60_Compr__image__eq,axiom,
! [F: nat > extend8495563244428889912nnreal,A: set_nat,P: extend8495563244428889912nnreal > $o] :
( ( collec6648975593938027277nnreal
@ ^ [X2: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X2 @ ( image_8459861568512453903nnreal @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_8459861568512453903nnreal @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_61_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_62_Compr__image__eq,axiom,
! [F: extended_ereal > $o,A: set_Extended_ereal,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_951975095941678543real_o @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_951975095941678543real_o @ F
@ ( collec5835592288176408249_ereal
@ ^ [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_63_Compr__image__eq,axiom,
! [F: nat > extended_ereal,A: set_nat,P: extended_ereal > $o] :
( ( collec5835592288176408249_ereal
@ ^ [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ ( image_4309273772856505399_ereal @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_4309273772856505399_ereal @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_64_Compr__image__eq,axiom,
! [F: $o > extended_ereal,A: set_o,P: extended_ereal > $o] :
( ( collec5835592288176408249_ereal
@ ^ [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ ( image_7729549296133164475_ereal @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_7729549296133164475_ereal @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_65_Compr__image__eq,axiom,
! [F: extended_ereal > extended_ereal,A: set_Extended_ereal,P: extended_ereal > $o] :
( ( collec5835592288176408249_ereal
@ ^ [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ ( image_6042159593519690757_ereal @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_6042159593519690757_ereal @ F
@ ( collec5835592288176408249_ereal
@ ^ [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_66_Compr__image__eq,axiom,
! [F: b > $o,A: set_b,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_b_o @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_b_o @ F
@ ( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_67_Compr__image__eq,axiom,
! [F: b > extended_ereal,A: set_b,P: extended_ereal > $o] :
( ( collec5835592288176408249_ereal
@ ^ [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ ( image_5319725110001000852_ereal @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_5319725110001000852_ereal @ F
@ ( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_68_Compr__image__eq,axiom,
! [F: c > $o,A: set_c,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_c_o @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_c_o @ F
@ ( collect_c
@ ^ [X2: c] :
( ( member_c @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_69_Compr__image__eq,axiom,
! [F: c > extended_ereal,A: set_c,P: extended_ereal > $o] :
( ( collec5835592288176408249_ereal
@ ^ [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ ( image_2233968868011006291_ereal @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_2233968868011006291_ereal @ F
@ ( collect_c
@ ^ [X2: c] :
( ( member_c @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_70_Inf_OINF__cong,axiom,
! [A: set_Extended_ereal,B2: set_Extended_ereal,C: extended_ereal > extended_ereal,D: extended_ereal > extended_ereal,Inf: set_Extended_ereal > extended_ereal] :
( ( A = B2 )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_6042159593519690757_ereal @ C @ A ) )
= ( Inf @ ( image_6042159593519690757_ereal @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_71_Inf_OINF__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > extended_ereal,D: nat > extended_ereal,Inf: set_Extended_ereal > extended_ereal] :
( ( A = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_4309273772856505399_ereal @ C @ A ) )
= ( Inf @ ( image_4309273772856505399_ereal @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_72_Inf_OINF__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > extend8495563244428889912nnreal,D: nat > extend8495563244428889912nnreal,Inf: set_Ex3793607809372303086nnreal > extend8495563244428889912nnreal] :
( ( A = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_8459861568512453903nnreal @ C @ A ) )
= ( Inf @ ( image_8459861568512453903nnreal @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_73_Inf_OINF__cong,axiom,
! [A: set_c,B2: set_c,C: c > b,D: c > b,Inf: set_b > b] :
( ( A = B2 )
=> ( ! [X3: c] :
( ( member_c @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_c_b @ C @ A ) )
= ( Inf @ ( image_c_b @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_74_Inf_OINF__cong,axiom,
! [A: set_c,B2: set_c,C: c > c,D: c > c,Inf: set_c > c] :
( ( A = B2 )
=> ( ! [X3: c] :
( ( member_c @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_c_c @ C @ A ) )
= ( Inf @ ( image_c_c @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_75_Inf_OINF__cong,axiom,
! [A: set_b,B2: set_b,C: b > extended_ereal,D: b > extended_ereal,Inf: set_Extended_ereal > extended_ereal] :
( ( A = B2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_5319725110001000852_ereal @ C @ A ) )
= ( Inf @ ( image_5319725110001000852_ereal @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_76_Inf_OINF__cong,axiom,
! [A: set_b,B2: set_b,C: b > $o,D: b > $o,Inf: set_o > $o] :
( ( A = B2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_b_o @ C @ A ) )
= ( Inf @ ( image_b_o @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_77_Inf_OINF__cong,axiom,
! [A: set_b,B2: set_b,C: b > c,D: b > c,Inf: set_c > c] :
( ( A = B2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_b_c @ C @ A ) )
= ( Inf @ ( image_b_c @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_78_Inf_OINF__cong,axiom,
! [A: set_b,B2: set_b,C: b > b,D: b > b,Inf: set_b > b] :
( ( A = B2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_b_b @ C @ A ) )
= ( Inf @ ( image_b_b @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_79_Sup_OSUP__cong,axiom,
! [A: set_Extended_ereal,B2: set_Extended_ereal,C: extended_ereal > extended_ereal,D: extended_ereal > extended_ereal,Sup: set_Extended_ereal > extended_ereal] :
( ( A = B2 )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_6042159593519690757_ereal @ C @ A ) )
= ( Sup @ ( image_6042159593519690757_ereal @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_80_Sup_OSUP__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > extended_ereal,D: nat > extended_ereal,Sup: set_Extended_ereal > extended_ereal] :
( ( A = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_4309273772856505399_ereal @ C @ A ) )
= ( Sup @ ( image_4309273772856505399_ereal @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_81_Sup_OSUP__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > extend8495563244428889912nnreal,D: nat > extend8495563244428889912nnreal,Sup: set_Ex3793607809372303086nnreal > extend8495563244428889912nnreal] :
( ( A = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_8459861568512453903nnreal @ C @ A ) )
= ( Sup @ ( image_8459861568512453903nnreal @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_82_Sup_OSUP__cong,axiom,
! [A: set_c,B2: set_c,C: c > b,D: c > b,Sup: set_b > b] :
( ( A = B2 )
=> ( ! [X3: c] :
( ( member_c @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_c_b @ C @ A ) )
= ( Sup @ ( image_c_b @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_83_Sup_OSUP__cong,axiom,
! [A: set_c,B2: set_c,C: c > c,D: c > c,Sup: set_c > c] :
( ( A = B2 )
=> ( ! [X3: c] :
( ( member_c @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_c_c @ C @ A ) )
= ( Sup @ ( image_c_c @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_84_Sup_OSUP__cong,axiom,
! [A: set_b,B2: set_b,C: b > extended_ereal,D: b > extended_ereal,Sup: set_Extended_ereal > extended_ereal] :
( ( A = B2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_5319725110001000852_ereal @ C @ A ) )
= ( Sup @ ( image_5319725110001000852_ereal @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_85_Sup_OSUP__cong,axiom,
! [A: set_b,B2: set_b,C: b > $o,D: b > $o,Sup: set_o > $o] :
( ( A = B2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_b_o @ C @ A ) )
= ( Sup @ ( image_b_o @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_86_Sup_OSUP__cong,axiom,
! [A: set_b,B2: set_b,C: b > c,D: b > c,Sup: set_c > c] :
( ( A = B2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_b_c @ C @ A ) )
= ( Sup @ ( image_b_c @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_87_Sup_OSUP__cong,axiom,
! [A: set_b,B2: set_b,C: b > b,D: b > b,Sup: set_b > b] :
( ( A = B2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_b_b @ C @ A ) )
= ( Sup @ ( image_b_b @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_88_imageI,axiom,
! [X: $o,A: set_o,F: $o > $o] :
( ( member_o @ X @ A )
=> ( member_o @ ( F @ X ) @ ( image_o_o @ F @ A ) ) ) ).
% imageI
thf(fact_89_imageI,axiom,
! [X: $o,A: set_o,F: $o > extended_ereal] :
( ( member_o @ X @ A )
=> ( member2350847679896131959_ereal @ ( F @ X ) @ ( image_7729549296133164475_ereal @ F @ A ) ) ) ).
% imageI
thf(fact_90_imageI,axiom,
! [X: $o,A: set_o,F: $o > nat] :
( ( member_o @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_o_nat @ F @ A ) ) ) ).
% imageI
thf(fact_91_imageI,axiom,
! [X: $o,A: set_o,F: $o > extend8495563244428889912nnreal] :
( ( member_o @ X @ A )
=> ( member7908768830364227535nnreal @ ( F @ X ) @ ( image_3342735880743421067nnreal @ F @ A ) ) ) ).
% imageI
thf(fact_92_imageI,axiom,
! [X: $o,A: set_o,F: $o > c] :
( ( member_o @ X @ A )
=> ( member_c @ ( F @ X ) @ ( image_o_c @ F @ A ) ) ) ).
% imageI
thf(fact_93_imageI,axiom,
! [X: $o,A: set_o,F: $o > b] :
( ( member_o @ X @ A )
=> ( member_b @ ( F @ X ) @ ( image_o_b @ F @ A ) ) ) ).
% imageI
thf(fact_94_imageI,axiom,
! [X: extended_ereal,A: set_Extended_ereal,F: extended_ereal > $o] :
( ( member2350847679896131959_ereal @ X @ A )
=> ( member_o @ ( F @ X ) @ ( image_951975095941678543real_o @ F @ A ) ) ) ).
% imageI
thf(fact_95_imageI,axiom,
! [X: extended_ereal,A: set_Extended_ereal,F: extended_ereal > extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A )
=> ( member2350847679896131959_ereal @ ( F @ X ) @ ( image_6042159593519690757_ereal @ F @ A ) ) ) ).
% imageI
thf(fact_96_imageI,axiom,
! [X: extended_ereal,A: set_Extended_ereal,F: extended_ereal > nat] :
( ( member2350847679896131959_ereal @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_7659842161140344153al_nat @ F @ A ) ) ) ).
% imageI
thf(fact_97_imageI,axiom,
! [X: extended_ereal,A: set_Extended_ereal,F: extended_ereal > extend8495563244428889912nnreal] :
( ( member2350847679896131959_ereal @ X @ A )
=> ( member7908768830364227535nnreal @ ( F @ X ) @ ( image_8614087454967683265nnreal @ F @ A ) ) ) ).
% imageI
thf(fact_98_rev__image__eqI,axiom,
! [X: $o,A: set_o,B: $o,F: $o > $o] :
( ( member_o @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_o_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_99_rev__image__eqI,axiom,
! [X: $o,A: set_o,B: extended_ereal,F: $o > extended_ereal] :
( ( member_o @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member2350847679896131959_ereal @ B @ ( image_7729549296133164475_ereal @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_100_rev__image__eqI,axiom,
! [X: $o,A: set_o,B: nat,F: $o > nat] :
( ( member_o @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_o_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_101_rev__image__eqI,axiom,
! [X: $o,A: set_o,B: extend8495563244428889912nnreal,F: $o > extend8495563244428889912nnreal] :
( ( member_o @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member7908768830364227535nnreal @ B @ ( image_3342735880743421067nnreal @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_102_rev__image__eqI,axiom,
! [X: $o,A: set_o,B: c,F: $o > c] :
( ( member_o @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_c @ B @ ( image_o_c @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_103_rev__image__eqI,axiom,
! [X: $o,A: set_o,B: b,F: $o > b] :
( ( member_o @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_b @ B @ ( image_o_b @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_104_rev__image__eqI,axiom,
! [X: extended_ereal,A: set_Extended_ereal,B: $o,F: extended_ereal > $o] :
( ( member2350847679896131959_ereal @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_951975095941678543real_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_105_rev__image__eqI,axiom,
! [X: extended_ereal,A: set_Extended_ereal,B: extended_ereal,F: extended_ereal > extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member2350847679896131959_ereal @ B @ ( image_6042159593519690757_ereal @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_106_rev__image__eqI,axiom,
! [X: extended_ereal,A: set_Extended_ereal,B: nat,F: extended_ereal > nat] :
( ( member2350847679896131959_ereal @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_7659842161140344153al_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_107_rev__image__eqI,axiom,
! [X: extended_ereal,A: set_Extended_ereal,B: extend8495563244428889912nnreal,F: extended_ereal > extend8495563244428889912nnreal] :
( ( member2350847679896131959_ereal @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member7908768830364227535nnreal @ B @ ( image_8614087454967683265nnreal @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_108_ball__imageD,axiom,
! [F: c > b,A: set_c,P: b > $o] :
( ! [X3: b] :
( ( member_b @ X3 @ ( image_c_b @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: c] :
( ( member_c @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_109_ball__imageD,axiom,
! [F: nat > extended_ereal,A: set_nat,P: extended_ereal > $o] :
( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ ( image_4309273772856505399_ereal @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_110_ball__imageD,axiom,
! [F: extended_ereal > extended_ereal,A: set_Extended_ereal,P: extended_ereal > $o] :
( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ ( image_6042159593519690757_ereal @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_111_ball__imageD,axiom,
! [F: nat > extend8495563244428889912nnreal,A: set_nat,P: extend8495563244428889912nnreal > $o] :
( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ ( image_8459861568512453903nnreal @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_112_ball__imageD,axiom,
! [F: c > c,A: set_c,P: c > $o] :
( ! [X3: c] :
( ( member_c @ X3 @ ( image_c_c @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: c] :
( ( member_c @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_113_ball__imageD,axiom,
! [F: b > extended_ereal,A: set_b,P: extended_ereal > $o] :
( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ ( image_5319725110001000852_ereal @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: b] :
( ( member_b @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_114_ball__imageD,axiom,
! [F: b > $o,A: set_b,P: $o > $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ ( image_b_o @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: b] :
( ( member_b @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_115_ball__imageD,axiom,
! [F: b > c,A: set_b,P: c > $o] :
( ! [X3: c] :
( ( member_c @ X3 @ ( image_b_c @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: b] :
( ( member_b @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_116_ball__imageD,axiom,
! [F: b > b,A: set_b,P: b > $o] :
( ! [X3: b] :
( ( member_b @ X3 @ ( image_b_b @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: b] :
( ( member_b @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_117_image__cong,axiom,
! [M: set_Extended_ereal,N: set_Extended_ereal,F: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal] :
( ( M = N )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_6042159593519690757_ereal @ F @ M )
= ( image_6042159593519690757_ereal @ G @ N ) ) ) ) ).
% image_cong
thf(fact_118_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > extended_ereal,G: nat > extended_ereal] :
( ( M = N )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_4309273772856505399_ereal @ F @ M )
= ( image_4309273772856505399_ereal @ G @ N ) ) ) ) ).
% image_cong
thf(fact_119_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > extend8495563244428889912nnreal,G: nat > extend8495563244428889912nnreal] :
( ( M = N )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_8459861568512453903nnreal @ F @ M )
= ( image_8459861568512453903nnreal @ G @ N ) ) ) ) ).
% image_cong
thf(fact_120_image__cong,axiom,
! [M: set_c,N: set_c,F: c > b,G: c > b] :
( ( M = N )
=> ( ! [X3: c] :
( ( member_c @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_c_b @ F @ M )
= ( image_c_b @ G @ N ) ) ) ) ).
% image_cong
thf(fact_121_image__cong,axiom,
! [M: set_c,N: set_c,F: c > c,G: c > c] :
( ( M = N )
=> ( ! [X3: c] :
( ( member_c @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_c_c @ F @ M )
= ( image_c_c @ G @ N ) ) ) ) ).
% image_cong
thf(fact_122_image__cong,axiom,
! [M: set_b,N: set_b,F: b > extended_ereal,G: b > extended_ereal] :
( ( M = N )
=> ( ! [X3: b] :
( ( member_b @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_5319725110001000852_ereal @ F @ M )
= ( image_5319725110001000852_ereal @ G @ N ) ) ) ) ).
% image_cong
thf(fact_123_image__cong,axiom,
! [M: set_b,N: set_b,F: b > $o,G: b > $o] :
( ( M = N )
=> ( ! [X3: b] :
( ( member_b @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_b_o @ F @ M )
= ( image_b_o @ G @ N ) ) ) ) ).
% image_cong
thf(fact_124_image__cong,axiom,
! [M: set_b,N: set_b,F: b > c,G: b > c] :
( ( M = N )
=> ( ! [X3: b] :
( ( member_b @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_b_c @ F @ M )
= ( image_b_c @ G @ N ) ) ) ) ).
% image_cong
thf(fact_125_image__cong,axiom,
! [M: set_b,N: set_b,F: b > b,G: b > b] :
( ( M = N )
=> ( ! [X3: b] :
( ( member_b @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_b_b @ F @ M )
= ( image_b_b @ G @ N ) ) ) ) ).
% image_cong
thf(fact_126_bex__imageD,axiom,
! [F: c > b,A: set_c,P: b > $o] :
( ? [X4: b] :
( ( member_b @ X4 @ ( image_c_b @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: c] :
( ( member_c @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_127_bex__imageD,axiom,
! [F: nat > extended_ereal,A: set_nat,P: extended_ereal > $o] :
( ? [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ ( image_4309273772856505399_ereal @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_128_bex__imageD,axiom,
! [F: extended_ereal > extended_ereal,A: set_Extended_ereal,P: extended_ereal > $o] :
( ? [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ ( image_6042159593519690757_ereal @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_129_bex__imageD,axiom,
! [F: nat > extend8495563244428889912nnreal,A: set_nat,P: extend8495563244428889912nnreal > $o] :
( ? [X4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X4 @ ( image_8459861568512453903nnreal @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_130_bex__imageD,axiom,
! [F: c > c,A: set_c,P: c > $o] :
( ? [X4: c] :
( ( member_c @ X4 @ ( image_c_c @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: c] :
( ( member_c @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_131_bex__imageD,axiom,
! [F: b > extended_ereal,A: set_b,P: extended_ereal > $o] :
( ? [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ ( image_5319725110001000852_ereal @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: b] :
( ( member_b @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_132_bex__imageD,axiom,
! [F: b > $o,A: set_b,P: $o > $o] :
( ? [X4: $o] :
( ( member_o @ X4 @ ( image_b_o @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: b] :
( ( member_b @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_133_bex__imageD,axiom,
! [F: b > c,A: set_b,P: c > $o] :
( ? [X4: c] :
( ( member_c @ X4 @ ( image_b_c @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: b] :
( ( member_b @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_134_bex__imageD,axiom,
! [F: b > b,A: set_b,P: b > $o] :
( ? [X4: b] :
( ( member_b @ X4 @ ( image_b_b @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: b] :
( ( member_b @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_135_image__iff,axiom,
! [Z: $o,F: b > $o,A: set_b] :
( ( member_o @ Z @ ( image_b_o @ F @ A ) )
= ( ? [X2: b] :
( ( member_b @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_136_image__iff,axiom,
! [Z: extended_ereal,F: nat > extended_ereal,A: set_nat] :
( ( member2350847679896131959_ereal @ Z @ ( image_4309273772856505399_ereal @ F @ A ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_137_image__iff,axiom,
! [Z: extended_ereal,F: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ Z @ ( image_6042159593519690757_ereal @ F @ A ) )
= ( ? [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_138_image__iff,axiom,
! [Z: extended_ereal,F: b > extended_ereal,A: set_b] :
( ( member2350847679896131959_ereal @ Z @ ( image_5319725110001000852_ereal @ F @ A ) )
= ( ? [X2: b] :
( ( member_b @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_139_image__iff,axiom,
! [Z: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,A: set_nat] :
( ( member7908768830364227535nnreal @ Z @ ( image_8459861568512453903nnreal @ F @ A ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_140_image__iff,axiom,
! [Z: c,F: c > c,A: set_c] :
( ( member_c @ Z @ ( image_c_c @ F @ A ) )
= ( ? [X2: c] :
( ( member_c @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_141_image__iff,axiom,
! [Z: c,F: b > c,A: set_b] :
( ( member_c @ Z @ ( image_b_c @ F @ A ) )
= ( ? [X2: b] :
( ( member_b @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_142_image__iff,axiom,
! [Z: b,F: c > b,A: set_c] :
( ( member_b @ Z @ ( image_c_b @ F @ A ) )
= ( ? [X2: c] :
( ( member_c @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_143_image__iff,axiom,
! [Z: b,F: b > b,A: set_b] :
( ( member_b @ Z @ ( image_b_b @ F @ A ) )
= ( ? [X2: b] :
( ( member_b @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_144_full__SetCompr__eq,axiom,
! [F: c > b] :
( ( collect_b
@ ^ [U: b] :
? [X2: c] :
( U
= ( F @ X2 ) ) )
= ( image_c_b @ F @ top_top_set_c ) ) ).
% full_SetCompr_eq
thf(fact_145_full__SetCompr__eq,axiom,
! [F: b > b] :
( ( collect_b
@ ^ [U: b] :
? [X2: b] :
( U
= ( F @ X2 ) ) )
= ( image_b_b @ F @ top_top_set_b ) ) ).
% full_SetCompr_eq
thf(fact_146_full__SetCompr__eq,axiom,
! [F: c > c] :
( ( collect_c
@ ^ [U: c] :
? [X2: c] :
( U
= ( F @ X2 ) ) )
= ( image_c_c @ F @ top_top_set_c ) ) ).
% full_SetCompr_eq
thf(fact_147_full__SetCompr__eq,axiom,
! [F: b > c] :
( ( collect_c
@ ^ [U: c] :
? [X2: b] :
( U
= ( F @ X2 ) ) )
= ( image_b_c @ F @ top_top_set_b ) ) ).
% full_SetCompr_eq
thf(fact_148_full__SetCompr__eq,axiom,
! [F: b > extended_ereal] :
( ( collec5835592288176408249_ereal
@ ^ [U: extended_ereal] :
? [X2: b] :
( U
= ( F @ X2 ) ) )
= ( image_5319725110001000852_ereal @ F @ top_top_set_b ) ) ).
% full_SetCompr_eq
thf(fact_149_full__SetCompr__eq,axiom,
! [F: b > $o] :
( ( collect_o
@ ^ [U: $o] :
? [X2: b] :
( U
= ( F @ X2 ) ) )
= ( image_b_o @ F @ top_top_set_b ) ) ).
% full_SetCompr_eq
thf(fact_150_full__SetCompr__eq,axiom,
! [F: nat > b] :
( ( collect_b
@ ^ [U: b] :
? [X2: nat] :
( U
= ( F @ X2 ) ) )
= ( image_nat_b @ F @ top_top_set_nat ) ) ).
% full_SetCompr_eq
thf(fact_151_full__SetCompr__eq,axiom,
! [F: nat > c] :
( ( collect_c
@ ^ [U: c] :
? [X2: nat] :
( U
= ( F @ X2 ) ) )
= ( image_nat_c @ F @ top_top_set_nat ) ) ).
% full_SetCompr_eq
thf(fact_152_full__SetCompr__eq,axiom,
! [F: nat > extended_ereal] :
( ( collec5835592288176408249_ereal
@ ^ [U: extended_ereal] :
? [X2: nat] :
( U
= ( F @ X2 ) ) )
= ( image_4309273772856505399_ereal @ F @ top_top_set_nat ) ) ).
% full_SetCompr_eq
thf(fact_153_full__SetCompr__eq,axiom,
! [F: nat > extend8495563244428889912nnreal] :
( ( collec6648975593938027277nnreal
@ ^ [U: extend8495563244428889912nnreal] :
? [X2: nat] :
( U
= ( F @ X2 ) ) )
= ( image_8459861568512453903nnreal @ F @ top_top_set_nat ) ) ).
% full_SetCompr_eq
thf(fact_154_SUP__identity__eq,axiom,
! [A: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X2: $o] : X2
@ A ) )
= ( complete_Sup_Sup_o @ A ) ) ).
% SUP_identity_eq
thf(fact_155_SUP__identity__eq,axiom,
! [A: set_Extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X2: extended_ereal] : X2
@ A ) )
= ( comple8415311339701865915_ereal @ A ) ) ).
% SUP_identity_eq
thf(fact_156_SUP__identity__eq,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( comple6814414086264997003nnreal
@ ( image_8394674774369097847nnreal
@ ^ [X2: extend8495563244428889912nnreal] : X2
@ A ) )
= ( comple6814414086264997003nnreal @ A ) ) ).
% SUP_identity_eq
thf(fact_157_SUP__identity__eq,axiom,
! [A: set_nat] :
( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A ) )
= ( complete_Sup_Sup_nat @ A ) ) ).
% SUP_identity_eq
thf(fact_158_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_159_INF__identity__eq,axiom,
! [A: set_Extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X2: extended_ereal] : X2
@ A ) )
= ( comple3556804143462414037_ereal @ A ) ) ).
% INF_identity_eq
thf(fact_160_INF__identity__eq,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( comple7330758040695736817nnreal
@ ( image_8394674774369097847nnreal
@ ^ [X2: extend8495563244428889912nnreal] : X2
@ A ) )
= ( comple7330758040695736817nnreal @ A ) ) ).
% INF_identity_eq
thf(fact_161_range__composition,axiom,
! [F: b > extended_ereal,G: c > b] :
( ( image_2233968868011006291_ereal
@ ^ [X2: c] : ( F @ ( G @ X2 ) )
@ top_top_set_c )
= ( image_5319725110001000852_ereal @ F @ ( image_c_b @ G @ top_top_set_c ) ) ) ).
% range_composition
thf(fact_162_range__composition,axiom,
! [F: b > $o,G: c > b] :
( ( image_c_o
@ ^ [X2: c] : ( F @ ( G @ X2 ) )
@ top_top_set_c )
= ( image_b_o @ F @ ( image_c_b @ G @ top_top_set_c ) ) ) ).
% range_composition
thf(fact_163_range__composition,axiom,
! [F: c > b,G: c > c] :
( ( image_c_b
@ ^ [X2: c] : ( F @ ( G @ X2 ) )
@ top_top_set_c )
= ( image_c_b @ F @ ( image_c_c @ G @ top_top_set_c ) ) ) ).
% range_composition
thf(fact_164_range__composition,axiom,
! [F: b > b,G: c > b] :
( ( image_c_b
@ ^ [X2: c] : ( F @ ( G @ X2 ) )
@ top_top_set_c )
= ( image_b_b @ F @ ( image_c_b @ G @ top_top_set_c ) ) ) ).
% range_composition
thf(fact_165_range__composition,axiom,
! [F: c > c,G: c > c] :
( ( image_c_c
@ ^ [X2: c] : ( F @ ( G @ X2 ) )
@ top_top_set_c )
= ( image_c_c @ F @ ( image_c_c @ G @ top_top_set_c ) ) ) ).
% range_composition
thf(fact_166_range__composition,axiom,
! [F: b > c,G: c > b] :
( ( image_c_c
@ ^ [X2: c] : ( F @ ( G @ X2 ) )
@ top_top_set_c )
= ( image_b_c @ F @ ( image_c_b @ G @ top_top_set_c ) ) ) ).
% range_composition
thf(fact_167_range__composition,axiom,
! [F: $o > extended_ereal,G: b > $o] :
( ( image_5319725110001000852_ereal
@ ^ [X2: b] : ( F @ ( G @ X2 ) )
@ top_top_set_b )
= ( image_7729549296133164475_ereal @ F @ ( image_b_o @ G @ top_top_set_b ) ) ) ).
% range_composition
thf(fact_168_range__composition,axiom,
! [F: c > extended_ereal,G: b > c] :
( ( image_5319725110001000852_ereal
@ ^ [X2: b] : ( F @ ( G @ X2 ) )
@ top_top_set_b )
= ( image_2233968868011006291_ereal @ F @ ( image_b_c @ G @ top_top_set_b ) ) ) ).
% range_composition
thf(fact_169_range__composition,axiom,
! [F: nat > extended_ereal,G: b > nat] :
( ( image_5319725110001000852_ereal
@ ^ [X2: b] : ( F @ ( G @ X2 ) )
@ top_top_set_b )
= ( image_4309273772856505399_ereal @ F @ ( image_b_nat @ G @ top_top_set_b ) ) ) ).
% range_composition
thf(fact_170_range__composition,axiom,
! [F: extended_ereal > extended_ereal,G: b > extended_ereal] :
( ( image_5319725110001000852_ereal
@ ^ [X2: b] : ( F @ ( G @ X2 ) )
@ top_top_set_b )
= ( image_6042159593519690757_ereal @ F @ ( image_5319725110001000852_ereal @ G @ top_top_set_b ) ) ) ).
% range_composition
thf(fact_171_rangeE,axiom,
! [B: $o,F: b > $o] :
( ( member_o @ B @ ( image_b_o @ F @ top_top_set_b ) )
=> ~ ! [X3: b] :
( B
= ( ~ ( F @ X3 ) ) ) ) ).
% rangeE
thf(fact_172_rangeE,axiom,
! [B: extended_ereal,F: b > extended_ereal] :
( ( member2350847679896131959_ereal @ B @ ( image_5319725110001000852_ereal @ F @ top_top_set_b ) )
=> ~ ! [X3: b] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_173_rangeE,axiom,
! [B: c,F: c > c] :
( ( member_c @ B @ ( image_c_c @ F @ top_top_set_c ) )
=> ~ ! [X3: c] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_174_rangeE,axiom,
! [B: c,F: b > c] :
( ( member_c @ B @ ( image_b_c @ F @ top_top_set_b ) )
=> ~ ! [X3: b] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_175_rangeE,axiom,
! [B: b,F: c > b] :
( ( member_b @ B @ ( image_c_b @ F @ top_top_set_c ) )
=> ~ ! [X3: c] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_176_rangeE,axiom,
! [B: b,F: b > b] :
( ( member_b @ B @ ( image_b_b @ F @ top_top_set_b ) )
=> ~ ! [X3: b] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_177_rangeE,axiom,
! [B: $o,F: nat > $o] :
( ( member_o @ B @ ( image_nat_o @ F @ top_top_set_nat ) )
=> ~ ! [X3: nat] :
( B
= ( ~ ( F @ X3 ) ) ) ) ).
% rangeE
thf(fact_178_rangeE,axiom,
! [B: extended_ereal,F: nat > extended_ereal] :
( ( member2350847679896131959_ereal @ B @ ( image_4309273772856505399_ereal @ F @ top_top_set_nat ) )
=> ~ ! [X3: nat] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_179_rangeE,axiom,
! [B: nat,F: nat > nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ~ ! [X3: nat] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_180_rangeE,axiom,
! [B: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ B @ ( image_8459861568512453903nnreal @ F @ top_top_set_nat ) )
=> ~ ! [X3: nat] :
( B
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_181_image__is__empty,axiom,
! [F: c > b,A: set_c] :
( ( ( image_c_b @ F @ A )
= bot_bot_set_b )
= ( A = bot_bot_set_c ) ) ).
% image_is_empty
thf(fact_182_image__is__empty,axiom,
! [F: c > c,A: set_c] :
( ( ( image_c_c @ F @ A )
= bot_bot_set_c )
= ( A = bot_bot_set_c ) ) ).
% image_is_empty
thf(fact_183_image__is__empty,axiom,
! [F: b > $o,A: set_b] :
( ( ( image_b_o @ F @ A )
= bot_bot_set_o )
= ( A = bot_bot_set_b ) ) ).
% image_is_empty
thf(fact_184_image__is__empty,axiom,
! [F: b > c,A: set_b] :
( ( ( image_b_c @ F @ A )
= bot_bot_set_c )
= ( A = bot_bot_set_b ) ) ).
% image_is_empty
thf(fact_185_image__is__empty,axiom,
! [F: b > b,A: set_b] :
( ( ( image_b_b @ F @ A )
= bot_bot_set_b )
= ( A = bot_bot_set_b ) ) ).
% image_is_empty
thf(fact_186_image__is__empty,axiom,
! [F: b > extended_ereal,A: set_b] :
( ( ( image_5319725110001000852_ereal @ F @ A )
= bot_bo8367695208629047834_ereal )
= ( A = bot_bot_set_b ) ) ).
% image_is_empty
thf(fact_187_image__is__empty,axiom,
! [F: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F @ A )
= bot_bo8367695208629047834_ereal )
= ( A = bot_bo8367695208629047834_ereal ) ) ).
% image_is_empty
thf(fact_188_image__is__empty,axiom,
! [F: extend8495563244428889912nnreal > extended_ereal,A: set_Ex3793607809372303086nnreal] :
( ( ( image_6393943237584228047_ereal @ F @ A )
= bot_bo8367695208629047834_ereal )
= ( A = bot_bo4854962954004695426nnreal ) ) ).
% image_is_empty
thf(fact_189_image__is__empty,axiom,
! [F: nat > extended_ereal,A: set_nat] :
( ( ( image_4309273772856505399_ereal @ F @ A )
= bot_bo8367695208629047834_ereal )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_190_image__is__empty,axiom,
! [F: extended_ereal > extend8495563244428889912nnreal,A: set_Extended_ereal] :
( ( ( image_8614087454967683265nnreal @ F @ A )
= bot_bo4854962954004695426nnreal )
= ( A = bot_bo8367695208629047834_ereal ) ) ).
% image_is_empty
thf(fact_191_empty__is__image,axiom,
! [F: c > b,A: set_c] :
( ( bot_bot_set_b
= ( image_c_b @ F @ A ) )
= ( A = bot_bot_set_c ) ) ).
% empty_is_image
thf(fact_192_empty__is__image,axiom,
! [F: c > c,A: set_c] :
( ( bot_bot_set_c
= ( image_c_c @ F @ A ) )
= ( A = bot_bot_set_c ) ) ).
% empty_is_image
thf(fact_193_empty__is__image,axiom,
! [F: b > $o,A: set_b] :
( ( bot_bot_set_o
= ( image_b_o @ F @ A ) )
= ( A = bot_bot_set_b ) ) ).
% empty_is_image
thf(fact_194_empty__is__image,axiom,
! [F: b > c,A: set_b] :
( ( bot_bot_set_c
= ( image_b_c @ F @ A ) )
= ( A = bot_bot_set_b ) ) ).
% empty_is_image
thf(fact_195_empty__is__image,axiom,
! [F: b > b,A: set_b] :
( ( bot_bot_set_b
= ( image_b_b @ F @ A ) )
= ( A = bot_bot_set_b ) ) ).
% empty_is_image
thf(fact_196_empty__is__image,axiom,
! [F: b > extended_ereal,A: set_b] :
( ( bot_bo8367695208629047834_ereal
= ( image_5319725110001000852_ereal @ F @ A ) )
= ( A = bot_bot_set_b ) ) ).
% empty_is_image
thf(fact_197_empty__is__image,axiom,
! [F: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( bot_bo8367695208629047834_ereal
= ( image_6042159593519690757_ereal @ F @ A ) )
= ( A = bot_bo8367695208629047834_ereal ) ) ).
% empty_is_image
thf(fact_198_empty__is__image,axiom,
! [F: extend8495563244428889912nnreal > extended_ereal,A: set_Ex3793607809372303086nnreal] :
( ( bot_bo8367695208629047834_ereal
= ( image_6393943237584228047_ereal @ F @ A ) )
= ( A = bot_bo4854962954004695426nnreal ) ) ).
% empty_is_image
thf(fact_199_empty__is__image,axiom,
! [F: nat > extended_ereal,A: set_nat] :
( ( bot_bo8367695208629047834_ereal
= ( image_4309273772856505399_ereal @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_200_empty__is__image,axiom,
! [F: extended_ereal > extend8495563244428889912nnreal,A: set_Extended_ereal] :
( ( bot_bo4854962954004695426nnreal
= ( image_8614087454967683265nnreal @ F @ A ) )
= ( A = bot_bo8367695208629047834_ereal ) ) ).
% empty_is_image
thf(fact_201_image__empty,axiom,
! [F: c > b] :
( ( image_c_b @ F @ bot_bot_set_c )
= bot_bot_set_b ) ).
% image_empty
thf(fact_202_image__empty,axiom,
! [F: c > c] :
( ( image_c_c @ F @ bot_bot_set_c )
= bot_bot_set_c ) ).
% image_empty
thf(fact_203_image__empty,axiom,
! [F: b > $o] :
( ( image_b_o @ F @ bot_bot_set_b )
= bot_bot_set_o ) ).
% image_empty
thf(fact_204_image__empty,axiom,
! [F: b > c] :
( ( image_b_c @ F @ bot_bot_set_b )
= bot_bot_set_c ) ).
% image_empty
thf(fact_205_image__empty,axiom,
! [F: b > b] :
( ( image_b_b @ F @ bot_bot_set_b )
= bot_bot_set_b ) ).
% image_empty
thf(fact_206_image__empty,axiom,
! [F: b > extended_ereal] :
( ( image_5319725110001000852_ereal @ F @ bot_bot_set_b )
= bot_bo8367695208629047834_ereal ) ).
% image_empty
thf(fact_207_image__empty,axiom,
! [F: extended_ereal > extended_ereal] :
( ( image_6042159593519690757_ereal @ F @ bot_bo8367695208629047834_ereal )
= bot_bo8367695208629047834_ereal ) ).
% image_empty
thf(fact_208_image__empty,axiom,
! [F: extended_ereal > extend8495563244428889912nnreal] :
( ( image_8614087454967683265nnreal @ F @ bot_bo8367695208629047834_ereal )
= bot_bo4854962954004695426nnreal ) ).
% image_empty
thf(fact_209_image__empty,axiom,
! [F: extended_ereal > nat] :
( ( image_7659842161140344153al_nat @ F @ bot_bo8367695208629047834_ereal )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_210_image__empty,axiom,
! [F: extend8495563244428889912nnreal > extended_ereal] :
( ( image_6393943237584228047_ereal @ F @ bot_bo4854962954004695426nnreal )
= bot_bo8367695208629047834_ereal ) ).
% image_empty
thf(fact_211_image__Union,axiom,
! [F: c > b,S: set_set_c] :
( ( image_c_b @ F @ ( comple2307003618534512845_set_c @ S ) )
= ( comple2307003614231284044_set_b @ ( image_set_c_set_b @ ( image_c_b @ F ) @ S ) ) ) ).
% image_Union
thf(fact_212_image__Union,axiom,
! [F: nat > extended_ereal,S: set_set_nat] :
( ( image_4309273772856505399_ereal @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple4319282863272126363_ereal @ ( image_8825259783980156129_ereal @ ( image_4309273772856505399_ereal @ F ) @ S ) ) ) ).
% image_Union
thf(fact_213_image__Union,axiom,
! [F: extended_ereal > extended_ereal,S: set_se6634062954251873166_ereal] :
( ( image_6042159593519690757_ereal @ F @ ( comple4319282863272126363_ereal @ S ) )
= ( comple4319282863272126363_ereal @ ( image_6293272304431515653_ereal @ ( image_6042159593519690757_ereal @ F ) @ S ) ) ) ).
% image_Union
thf(fact_214_image__Union,axiom,
! [F: nat > extend8495563244428889912nnreal,S: set_set_nat] :
( ( image_8459861568512453903nnreal @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple4226387801268262977nnreal @ ( image_2240520088648803451nnreal @ ( image_8459861568512453903nnreal @ F ) @ S ) ) ) ).
% image_Union
thf(fact_215_image__Union,axiom,
! [F: c > c,S: set_set_c] :
( ( image_c_c @ F @ ( comple2307003618534512845_set_c @ S ) )
= ( comple2307003618534512845_set_c @ ( image_set_c_set_c @ ( image_c_c @ F ) @ S ) ) ) ).
% image_Union
thf(fact_216_image__Union,axiom,
! [F: b > extended_ereal,S: set_set_b] :
( ( image_5319725110001000852_ereal @ F @ ( comple2307003614231284044_set_b @ S ) )
= ( comple4319282863272126363_ereal @ ( image_1305302568177716884_ereal @ ( image_5319725110001000852_ereal @ F ) @ S ) ) ) ).
% image_Union
thf(fact_217_image__Union,axiom,
! [F: b > $o,S: set_set_b] :
( ( image_b_o @ F @ ( comple2307003614231284044_set_b @ S ) )
= ( comple90263536869209701_set_o @ ( image_set_b_set_o @ ( image_b_o @ F ) @ S ) ) ) ).
% image_Union
thf(fact_218_image__Union,axiom,
! [F: b > c,S: set_set_b] :
( ( image_b_c @ F @ ( comple2307003614231284044_set_b @ S ) )
= ( comple2307003618534512845_set_c @ ( image_set_b_set_c @ ( image_b_c @ F ) @ S ) ) ) ).
% image_Union
thf(fact_219_image__Union,axiom,
! [F: b > b,S: set_set_b] :
( ( image_b_b @ F @ ( comple2307003614231284044_set_b @ S ) )
= ( comple2307003614231284044_set_b @ ( image_set_b_set_b @ ( image_b_b @ F ) @ S ) ) ) ).
% image_Union
thf(fact_220_Inter__UNIV__conv_I2_J,axiom,
! [A: set_set_nat] :
( ( top_top_set_nat
= ( comple7806235888213564991et_nat @ A ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( X2 = top_top_set_nat ) ) ) ) ).
% Inter_UNIV_conv(2)
thf(fact_221_Inter__UNIV__conv_I2_J,axiom,
! [A: set_se6634062954251873166_ereal] :
( ( top_to5683747375963461374_ereal
= ( comple4418415374894819509_ereal @ A ) )
= ( ! [X2: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ X2 @ A )
=> ( X2 = top_to5683747375963461374_ereal ) ) ) ) ).
% Inter_UNIV_conv(2)
thf(fact_222_Inter__UNIV__conv_I1_J,axiom,
! [A: set_set_nat] :
( ( ( comple7806235888213564991et_nat @ A )
= top_top_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( X2 = top_top_set_nat ) ) ) ) ).
% Inter_UNIV_conv(1)
thf(fact_223_Inter__UNIV__conv_I1_J,axiom,
! [A: set_se6634062954251873166_ereal] :
( ( ( comple4418415374894819509_ereal @ A )
= top_to5683747375963461374_ereal )
= ( ! [X2: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ X2 @ A )
=> ( X2 = top_to5683747375963461374_ereal ) ) ) ) ).
% Inter_UNIV_conv(1)
thf(fact_224_UNIV__I,axiom,
! [X: $o] : ( member_o @ X @ top_top_set_o ) ).
% UNIV_I
thf(fact_225_UNIV__I,axiom,
! [X: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X @ top_to7994903218803871134nnreal ) ).
% UNIV_I
thf(fact_226_UNIV__I,axiom,
! [X: c] : ( member_c @ X @ top_top_set_c ) ).
% UNIV_I
thf(fact_227_UNIV__I,axiom,
! [X: b] : ( member_b @ X @ top_top_set_b ) ).
% UNIV_I
thf(fact_228_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_229_UNIV__I,axiom,
! [X: extended_ereal] : ( member2350847679896131959_ereal @ X @ top_to5683747375963461374_ereal ) ).
% UNIV_I
thf(fact_230_empty__iff,axiom,
! [C2: $o] :
~ ( member_o @ C2 @ bot_bot_set_o ) ).
% empty_iff
thf(fact_231_empty__iff,axiom,
! [C2: c] :
~ ( member_c @ C2 @ bot_bot_set_c ) ).
% empty_iff
thf(fact_232_empty__iff,axiom,
! [C2: b] :
~ ( member_b @ C2 @ bot_bot_set_b ) ).
% empty_iff
thf(fact_233_empty__iff,axiom,
! [C2: extended_ereal] :
~ ( member2350847679896131959_ereal @ C2 @ bot_bo8367695208629047834_ereal ) ).
% empty_iff
thf(fact_234_empty__iff,axiom,
! [C2: extend8495563244428889912nnreal] :
~ ( member7908768830364227535nnreal @ C2 @ bot_bo4854962954004695426nnreal ) ).
% empty_iff
thf(fact_235_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_236_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_237_all__not__in__conv,axiom,
! [A: set_c] :
( ( ! [X2: c] :
~ ( member_c @ X2 @ A ) )
= ( A = bot_bot_set_c ) ) ).
% all_not_in_conv
thf(fact_238_all__not__in__conv,axiom,
! [A: set_b] :
( ( ! [X2: b] :
~ ( member_b @ X2 @ A ) )
= ( A = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_239_all__not__in__conv,axiom,
! [A: set_Extended_ereal] :
( ( ! [X2: extended_ereal] :
~ ( member2350847679896131959_ereal @ X2 @ A ) )
= ( A = bot_bo8367695208629047834_ereal ) ) ).
% all_not_in_conv
thf(fact_240_all__not__in__conv,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( ! [X2: extend8495563244428889912nnreal] :
~ ( member7908768830364227535nnreal @ X2 @ A ) )
= ( A = bot_bo4854962954004695426nnreal ) ) ).
% all_not_in_conv
thf(fact_241_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_242_Collect__empty__eq,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( ! [X2: b] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_243_Collect__empty__eq,axiom,
! [P: c > $o] :
( ( ( collect_c @ P )
= bot_bot_set_c )
= ( ! [X2: c] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_244_Collect__empty__eq,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( ! [X2: $o] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_245_Collect__empty__eq,axiom,
! [P: extended_ereal > $o] :
( ( ( collec5835592288176408249_ereal @ P )
= bot_bo8367695208629047834_ereal )
= ( ! [X2: extended_ereal] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_246_Collect__empty__eq,axiom,
! [P: extend8495563244428889912nnreal > $o] :
( ( ( collec6648975593938027277nnreal @ P )
= bot_bo4854962954004695426nnreal )
= ( ! [X2: extend8495563244428889912nnreal] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_247_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_248_empty__Collect__eq,axiom,
! [P: b > $o] :
( ( bot_bot_set_b
= ( collect_b @ P ) )
= ( ! [X2: b] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_249_empty__Collect__eq,axiom,
! [P: c > $o] :
( ( bot_bot_set_c
= ( collect_c @ P ) )
= ( ! [X2: c] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_250_empty__Collect__eq,axiom,
! [P: $o > $o] :
( ( bot_bot_set_o
= ( collect_o @ P ) )
= ( ! [X2: $o] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_251_empty__Collect__eq,axiom,
! [P: extended_ereal > $o] :
( ( bot_bo8367695208629047834_ereal
= ( collec5835592288176408249_ereal @ P ) )
= ( ! [X2: extended_ereal] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_252_empty__Collect__eq,axiom,
! [P: extend8495563244428889912nnreal > $o] :
( ( bot_bo4854962954004695426nnreal
= ( collec6648975593938027277nnreal @ P ) )
= ( ! [X2: extend8495563244428889912nnreal] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_253_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_254_UnionI,axiom,
! [X5: set_o,C: set_set_o,A: $o] :
( ( member_set_o @ X5 @ C )
=> ( ( member_o @ A @ X5 )
=> ( member_o @ A @ ( comple90263536869209701_set_o @ C ) ) ) ) ).
% UnionI
thf(fact_255_UnionI,axiom,
! [X5: set_Extended_ereal,C: set_se6634062954251873166_ereal,A: extended_ereal] :
( ( member5519481007471526743_ereal @ X5 @ C )
=> ( ( member2350847679896131959_ereal @ A @ X5 )
=> ( member2350847679896131959_ereal @ A @ ( comple4319282863272126363_ereal @ C ) ) ) ) ).
% UnionI
thf(fact_256_UnionI,axiom,
! [X5: set_nat,C: set_set_nat,A: nat] :
( ( member_set_nat @ X5 @ C )
=> ( ( member_nat @ A @ X5 )
=> ( member_nat @ A @ ( comple7399068483239264473et_nat @ C ) ) ) ) ).
% UnionI
thf(fact_257_UnionI,axiom,
! [X5: set_Ex3793607809372303086nnreal,C: set_se4580700918925141924nnreal,A: extend8495563244428889912nnreal] :
( ( member603777416030116741nnreal @ X5 @ C )
=> ( ( member7908768830364227535nnreal @ A @ X5 )
=> ( member7908768830364227535nnreal @ A @ ( comple4226387801268262977nnreal @ C ) ) ) ) ).
% UnionI
thf(fact_258_UnionI,axiom,
! [X5: set_c,C: set_set_c,A: c] :
( ( member_set_c @ X5 @ C )
=> ( ( member_c @ A @ X5 )
=> ( member_c @ A @ ( comple2307003618534512845_set_c @ C ) ) ) ) ).
% UnionI
thf(fact_259_UnionI,axiom,
! [X5: set_b,C: set_set_b,A: b] :
( ( member_set_b @ X5 @ C )
=> ( ( member_b @ A @ X5 )
=> ( member_b @ A @ ( comple2307003614231284044_set_b @ C ) ) ) ) ).
% UnionI
thf(fact_260_Union__iff,axiom,
! [A: $o,C: set_set_o] :
( ( member_o @ A @ ( comple90263536869209701_set_o @ C ) )
= ( ? [X2: set_o] :
( ( member_set_o @ X2 @ C )
& ( member_o @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_261_Union__iff,axiom,
! [A: extended_ereal,C: set_se6634062954251873166_ereal] :
( ( member2350847679896131959_ereal @ A @ ( comple4319282863272126363_ereal @ C ) )
= ( ? [X2: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ X2 @ C )
& ( member2350847679896131959_ereal @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_262_Union__iff,axiom,
! [A: nat,C: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C ) )
= ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ C )
& ( member_nat @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_263_Union__iff,axiom,
! [A: extend8495563244428889912nnreal,C: set_se4580700918925141924nnreal] :
( ( member7908768830364227535nnreal @ A @ ( comple4226387801268262977nnreal @ C ) )
= ( ? [X2: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X2 @ C )
& ( member7908768830364227535nnreal @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_264_Union__iff,axiom,
! [A: c,C: set_set_c] :
( ( member_c @ A @ ( comple2307003618534512845_set_c @ C ) )
= ( ? [X2: set_c] :
( ( member_set_c @ X2 @ C )
& ( member_c @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_265_Union__iff,axiom,
! [A: b,C: set_set_b] :
( ( member_b @ A @ ( comple2307003614231284044_set_b @ C ) )
= ( ? [X2: set_b] :
( ( member_set_b @ X2 @ C )
& ( member_b @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_266_INT__I,axiom,
! [A: set_o,B: $o,B2: $o > set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_o @ B @ ( B2 @ X3 ) ) )
=> ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_267_INT__I,axiom,
! [A: set_o,B: extended_ereal,B2: $o > set_Extended_ereal] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member2350847679896131959_ereal @ B @ ( B2 @ X3 ) ) )
=> ( member2350847679896131959_ereal @ B @ ( comple4418415374894819509_ereal @ ( image_6375117163256653723_ereal @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_268_INT__I,axiom,
! [A: set_o,B: nat,B2: $o > set_nat] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_nat @ B @ ( B2 @ X3 ) ) )
=> ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_269_INT__I,axiom,
! [A: set_o,B: extend8495563244428889912nnreal,B2: $o > set_Ex3793607809372303086nnreal] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member7908768830364227535nnreal @ B @ ( B2 @ X3 ) ) )
=> ( member7908768830364227535nnreal @ B @ ( comple5724520875574609319nnreal @ ( image_1679811975146592321nnreal @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_270_INT__I,axiom,
! [A: set_o,B: c,B2: $o > set_c] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_c @ B @ ( B2 @ X3 ) ) )
=> ( member_c @ B @ ( comple6135023387286571239_set_c @ ( image_o_set_c @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_271_INT__I,axiom,
! [A: set_o,B: b,B2: $o > set_b] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_b @ B @ ( B2 @ X3 ) ) )
=> ( member_b @ B @ ( comple6135023382983342438_set_b @ ( image_o_set_b @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_272_INT__I,axiom,
! [A: set_Extended_ereal,B: $o,B2: extended_ereal > set_o] :
( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
=> ( member_o @ B @ ( B2 @ X3 ) ) )
=> ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_2973972673614065839_set_o @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_273_INT__I,axiom,
! [A: set_Extended_ereal,B: extended_ereal,B2: extended_ereal > set_Extended_ereal] :
( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
=> ( member2350847679896131959_ereal @ B @ ( B2 @ X3 ) ) )
=> ( member2350847679896131959_ereal @ B @ ( comple4418415374894819509_ereal @ ( image_5562094264469218789_ereal @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_274_INT__I,axiom,
! [A: set_Extended_ereal,B: nat,B2: extended_ereal > set_nat] :
( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
=> ( member_nat @ B @ ( B2 @ X3 ) ) )
=> ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_3090908713637162255et_nat @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_275_INT__I,axiom,
! [A: set_Extended_ereal,B: extend8495563244428889912nnreal,B2: extended_ereal > set_Ex3793607809372303086nnreal] :
( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
=> ( member7908768830364227535nnreal @ B @ ( B2 @ X3 ) ) )
=> ( member7908768830364227535nnreal @ B @ ( comple5724520875574609319nnreal @ ( image_6588766411312125047nnreal @ B2 @ A ) ) ) ) ).
% INT_I
thf(fact_276_mem__Collect__eq,axiom,
! [A2: b,P: b > $o] :
( ( member_b @ A2 @ ( collect_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_277_mem__Collect__eq,axiom,
! [A2: c,P: c > $o] :
( ( member_c @ A2 @ ( collect_c @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_278_mem__Collect__eq,axiom,
! [A2: extended_ereal,P: extended_ereal > $o] :
( ( member2350847679896131959_ereal @ A2 @ ( collec5835592288176408249_ereal @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_279_mem__Collect__eq,axiom,
! [A2: extend8495563244428889912nnreal,P: extend8495563244428889912nnreal > $o] :
( ( member7908768830364227535nnreal @ A2 @ ( collec6648975593938027277nnreal @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_280_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_281_mem__Collect__eq,axiom,
! [A2: $o,P: $o > $o] :
( ( member_o @ A2 @ ( collect_o @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_282_Collect__mem__eq,axiom,
! [A: set_b] :
( ( collect_b
@ ^ [X2: b] : ( member_b @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_283_Collect__mem__eq,axiom,
! [A: set_c] :
( ( collect_c
@ ^ [X2: c] : ( member_c @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_284_Collect__mem__eq,axiom,
! [A: set_Extended_ereal] :
( ( collec5835592288176408249_ereal
@ ^ [X2: extended_ereal] : ( member2350847679896131959_ereal @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_285_Collect__mem__eq,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( collec6648975593938027277nnreal
@ ^ [X2: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_286_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_287_Collect__mem__eq,axiom,
! [A: set_o] :
( ( collect_o
@ ^ [X2: $o] : ( member_o @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_288_Collect__cong,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X3: b] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_b @ P )
= ( collect_b @ Q ) ) ) ).
% Collect_cong
thf(fact_289_Collect__cong,axiom,
! [P: c > $o,Q: c > $o] :
( ! [X3: c] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_c @ P )
= ( collect_c @ Q ) ) ) ).
% Collect_cong
thf(fact_290_Collect__cong,axiom,
! [P: extended_ereal > $o,Q: extended_ereal > $o] :
( ! [X3: extended_ereal] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec5835592288176408249_ereal @ P )
= ( collec5835592288176408249_ereal @ Q ) ) ) ).
% Collect_cong
thf(fact_291_Collect__cong,axiom,
! [P: extend8495563244428889912nnreal > $o,Q: extend8495563244428889912nnreal > $o] :
( ! [X3: extend8495563244428889912nnreal] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec6648975593938027277nnreal @ P )
= ( collec6648975593938027277nnreal @ Q ) ) ) ).
% Collect_cong
thf(fact_292_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_293_Collect__cong,axiom,
! [P: $o > $o,Q: $o > $o] :
( ! [X3: $o] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_o @ P )
= ( collect_o @ Q ) ) ) ).
% Collect_cong
thf(fact_294_Sup__bot__conv_I2_J,axiom,
! [A: set_se6634062954251873166_ereal] :
( ( bot_bo8367695208629047834_ereal
= ( comple4319282863272126363_ereal @ A ) )
= ( ! [X2: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ X2 @ A )
=> ( X2 = bot_bo8367695208629047834_ereal ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_295_Sup__bot__conv_I2_J,axiom,
! [A: set_se4580700918925141924nnreal] :
( ( bot_bo4854962954004695426nnreal
= ( comple4226387801268262977nnreal @ A ) )
= ( ! [X2: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X2 @ A )
=> ( X2 = bot_bo4854962954004695426nnreal ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_296_Sup__bot__conv_I2_J,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_297_Sup__bot__conv_I2_J,axiom,
! [A: set_o] :
( ( bot_bot_o
= ( complete_Sup_Sup_o @ A ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( X2 = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_298_Sup__bot__conv_I2_J,axiom,
! [A: set_Extended_ereal] :
( ( bot_bo2710585358178759738_ereal
= ( comple8415311339701865915_ereal @ A ) )
= ( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A )
=> ( X2 = bot_bo2710585358178759738_ereal ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_299_Sup__bot__conv_I2_J,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( bot_bo841427958541957580nnreal
= ( comple6814414086264997003nnreal @ A ) )
= ( ! [X2: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X2 @ A )
=> ( X2 = bot_bo841427958541957580nnreal ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_300_Sup__bot__conv_I1_J,axiom,
! [A: set_se6634062954251873166_ereal] :
( ( ( comple4319282863272126363_ereal @ A )
= bot_bo8367695208629047834_ereal )
= ( ! [X2: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ X2 @ A )
=> ( X2 = bot_bo8367695208629047834_ereal ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_301_Sup__bot__conv_I1_J,axiom,
! [A: set_se4580700918925141924nnreal] :
( ( ( comple4226387801268262977nnreal @ A )
= bot_bo4854962954004695426nnreal )
= ( ! [X2: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X2 @ A )
=> ( X2 = bot_bo4854962954004695426nnreal ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_302_Sup__bot__conv_I1_J,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_303_Sup__bot__conv_I1_J,axiom,
! [A: set_o] :
( ( ( complete_Sup_Sup_o @ A )
= bot_bot_o )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( X2 = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_304_Sup__bot__conv_I1_J,axiom,
! [A: set_Extended_ereal] :
( ( ( comple8415311339701865915_ereal @ A )
= bot_bo2710585358178759738_ereal )
= ( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A )
=> ( X2 = bot_bo2710585358178759738_ereal ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_305_Sup__bot__conv_I1_J,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( ( comple6814414086264997003nnreal @ A )
= bot_bo841427958541957580nnreal )
= ( ! [X2: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X2 @ A )
=> ( X2 = bot_bo841427958541957580nnreal ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_306_Inf__top__conv_I2_J,axiom,
! [A: set_set_nat] :
( ( top_top_set_nat
= ( comple7806235888213564991et_nat @ A ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( X2 = top_top_set_nat ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_307_Inf__top__conv_I2_J,axiom,
! [A: set_se6634062954251873166_ereal] :
( ( top_to5683747375963461374_ereal
= ( comple4418415374894819509_ereal @ A ) )
= ( ! [X2: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ X2 @ A )
=> ( X2 = top_to5683747375963461374_ereal ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_308_Inf__top__conv_I2_J,axiom,
! [A: set_o] :
( ( top_top_o
= ( complete_Inf_Inf_o @ A ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( X2 = top_top_o ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_309_Inf__top__conv_I2_J,axiom,
! [A: set_Extended_ereal] :
( ( top_to6662034908053899550_ereal
= ( comple3556804143462414037_ereal @ A ) )
= ( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A )
=> ( X2 = top_to6662034908053899550_ereal ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_310_Inf__top__conv_I2_J,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( top_to1496364449551166952nnreal
= ( comple7330758040695736817nnreal @ A ) )
= ( ! [X2: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X2 @ A )
=> ( X2 = top_to1496364449551166952nnreal ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_311_Inf__top__conv_I1_J,axiom,
! [A: set_set_nat] :
( ( ( comple7806235888213564991et_nat @ A )
= top_top_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( X2 = top_top_set_nat ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_312_Inf__top__conv_I1_J,axiom,
! [A: set_se6634062954251873166_ereal] :
( ( ( comple4418415374894819509_ereal @ A )
= top_to5683747375963461374_ereal )
= ( ! [X2: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ X2 @ A )
=> ( X2 = top_to5683747375963461374_ereal ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_313_Inf__top__conv_I1_J,axiom,
! [A: set_o] :
( ( ( complete_Inf_Inf_o @ A )
= top_top_o )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( X2 = top_top_o ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_314_Inf__top__conv_I1_J,axiom,
! [A: set_Extended_ereal] :
( ( ( comple3556804143462414037_ereal @ A )
= top_to6662034908053899550_ereal )
= ( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A )
=> ( X2 = top_to6662034908053899550_ereal ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_315_Inf__top__conv_I1_J,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( ( comple7330758040695736817nnreal @ A )
= top_to1496364449551166952nnreal )
= ( ! [X2: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X2 @ A )
=> ( X2 = top_to1496364449551166952nnreal ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_316_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_b
@ ^ [S2: b] : P )
= top_top_set_b ) )
& ( ~ P
=> ( ( collect_b
@ ^ [S2: b] : P )
= bot_bot_set_b ) ) ) ).
% Collect_const
thf(fact_317_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_c
@ ^ [S2: c] : P )
= top_top_set_c ) )
& ( ~ P
=> ( ( collect_c
@ ^ [S2: c] : P )
= bot_bot_set_c ) ) ) ).
% Collect_const
thf(fact_318_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_o
@ ^ [S2: $o] : P )
= top_top_set_o ) )
& ( ~ P
=> ( ( collect_o
@ ^ [S2: $o] : P )
= bot_bot_set_o ) ) ) ).
% Collect_const
thf(fact_319_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collec6648975593938027277nnreal
@ ^ [S2: extend8495563244428889912nnreal] : P )
= top_to7994903218803871134nnreal ) )
& ( ~ P
=> ( ( collec6648975593938027277nnreal
@ ^ [S2: extend8495563244428889912nnreal] : P )
= bot_bo4854962954004695426nnreal ) ) ) ).
% Collect_const
thf(fact_320_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_nat
@ ^ [S2: nat] : P )
= top_top_set_nat ) )
& ( ~ P
=> ( ( collect_nat
@ ^ [S2: nat] : P )
= bot_bot_set_nat ) ) ) ).
% Collect_const
thf(fact_321_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collec5835592288176408249_ereal
@ ^ [S2: extended_ereal] : P )
= top_to5683747375963461374_ereal ) )
& ( ~ P
=> ( ( collec5835592288176408249_ereal
@ ^ [S2: extended_ereal] : P )
= bot_bo8367695208629047834_ereal ) ) ) ).
% Collect_const
thf(fact_322_UN__I,axiom,
! [A2: $o,A: set_o,B: $o,B2: $o > set_o] :
( ( member_o @ A2 @ A )
=> ( ( member_o @ B @ ( B2 @ A2 ) )
=> ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A ) ) ) ) ) ).
% UN_I
thf(fact_323_UN__I,axiom,
! [A2: $o,A: set_o,B: extended_ereal,B2: $o > set_Extended_ereal] :
( ( member_o @ A2 @ A )
=> ( ( member2350847679896131959_ereal @ B @ ( B2 @ A2 ) )
=> ( member2350847679896131959_ereal @ B @ ( comple4319282863272126363_ereal @ ( image_6375117163256653723_ereal @ B2 @ A ) ) ) ) ) ).
% UN_I
thf(fact_324_UN__I,axiom,
! [A2: $o,A: set_o,B: nat,B2: $o > set_nat] :
( ( member_o @ A2 @ A )
=> ( ( member_nat @ B @ ( B2 @ A2 ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A ) ) ) ) ) ).
% UN_I
thf(fact_325_UN__I,axiom,
! [A2: $o,A: set_o,B: extend8495563244428889912nnreal,B2: $o > set_Ex3793607809372303086nnreal] :
( ( member_o @ A2 @ A )
=> ( ( member7908768830364227535nnreal @ B @ ( B2 @ A2 ) )
=> ( member7908768830364227535nnreal @ B @ ( comple4226387801268262977nnreal @ ( image_1679811975146592321nnreal @ B2 @ A ) ) ) ) ) ).
% UN_I
thf(fact_326_UN__I,axiom,
! [A2: $o,A: set_o,B: c,B2: $o > set_c] :
( ( member_o @ A2 @ A )
=> ( ( member_c @ B @ ( B2 @ A2 ) )
=> ( member_c @ B @ ( comple2307003618534512845_set_c @ ( image_o_set_c @ B2 @ A ) ) ) ) ) ).
% UN_I
thf(fact_327_UN__I,axiom,
! [A2: $o,A: set_o,B: b,B2: $o > set_b] :
( ( member_o @ A2 @ A )
=> ( ( member_b @ B @ ( B2 @ A2 ) )
=> ( member_b @ B @ ( comple2307003614231284044_set_b @ ( image_o_set_b @ B2 @ A ) ) ) ) ) ).
% UN_I
thf(fact_328_UN__I,axiom,
! [A2: extended_ereal,A: set_Extended_ereal,B: $o,B2: extended_ereal > set_o] :
( ( member2350847679896131959_ereal @ A2 @ A )
=> ( ( member_o @ B @ ( B2 @ A2 ) )
=> ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_2973972673614065839_set_o @ B2 @ A ) ) ) ) ) ).
% UN_I
thf(fact_329_UN__I,axiom,
! [A2: extended_ereal,A: set_Extended_ereal,B: extended_ereal,B2: extended_ereal > set_Extended_ereal] :
( ( member2350847679896131959_ereal @ A2 @ A )
=> ( ( member2350847679896131959_ereal @ B @ ( B2 @ A2 ) )
=> ( member2350847679896131959_ereal @ B @ ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ B2 @ A ) ) ) ) ) ).
% UN_I
thf(fact_330_UN__I,axiom,
! [A2: extended_ereal,A: set_Extended_ereal,B: nat,B2: extended_ereal > set_nat] :
( ( member2350847679896131959_ereal @ A2 @ A )
=> ( ( member_nat @ B @ ( B2 @ A2 ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_3090908713637162255et_nat @ B2 @ A ) ) ) ) ) ).
% UN_I
thf(fact_331_UN__I,axiom,
! [A2: extended_ereal,A: set_Extended_ereal,B: extend8495563244428889912nnreal,B2: extended_ereal > set_Ex3793607809372303086nnreal] :
( ( member2350847679896131959_ereal @ A2 @ A )
=> ( ( member7908768830364227535nnreal @ B @ ( B2 @ A2 ) )
=> ( member7908768830364227535nnreal @ B @ ( comple4226387801268262977nnreal @ ( image_6588766411312125047nnreal @ B2 @ A ) ) ) ) ) ).
% UN_I
thf(fact_332_Sup__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Sup_UNIV
thf(fact_333_Sup__UNIV,axiom,
( ( comple4319282863272126363_ereal @ top_to4757929550322229470_ereal )
= top_to5683747375963461374_ereal ) ).
% Sup_UNIV
thf(fact_334_Sup__UNIV,axiom,
( ( complete_Sup_Sup_o @ top_top_set_o )
= top_top_o ) ).
% Sup_UNIV
thf(fact_335_Sup__UNIV,axiom,
( ( comple8415311339701865915_ereal @ top_to5683747375963461374_ereal )
= top_to6662034908053899550_ereal ) ).
% Sup_UNIV
thf(fact_336_Sup__UNIV,axiom,
( ( comple6814414086264997003nnreal @ top_to7994903218803871134nnreal )
= top_to1496364449551166952nnreal ) ).
% Sup_UNIV
thf(fact_337_Sup__empty,axiom,
( ( comple4319282863272126363_ereal @ bot_bo7400643019497942010_ereal )
= bot_bo8367695208629047834_ereal ) ).
% Sup_empty
thf(fact_338_Sup__empty,axiom,
( ( comple4226387801268262977nnreal @ bot_bo2988155216863113784nnreal )
= bot_bo4854962954004695426nnreal ) ).
% Sup_empty
thf(fact_339_Sup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Sup_empty
thf(fact_340_Sup__empty,axiom,
( ( complete_Sup_Sup_o @ bot_bot_set_o )
= bot_bot_o ) ).
% Sup_empty
thf(fact_341_Sup__empty,axiom,
( ( comple8415311339701865915_ereal @ bot_bo8367695208629047834_ereal )
= bot_bo2710585358178759738_ereal ) ).
% Sup_empty
thf(fact_342_Sup__empty,axiom,
( ( comple6814414086264997003nnreal @ bot_bo4854962954004695426nnreal )
= bot_bo841427958541957580nnreal ) ).
% Sup_empty
thf(fact_343_Inf__UNIV,axiom,
( ( comple4418415374894819509_ereal @ top_to4757929550322229470_ereal )
= bot_bo8367695208629047834_ereal ) ).
% Inf_UNIV
thf(fact_344_Inf__UNIV,axiom,
( ( comple5724520875574609319nnreal @ top_to3356475028079756884nnreal )
= bot_bo4854962954004695426nnreal ) ).
% Inf_UNIV
thf(fact_345_Inf__UNIV,axiom,
( ( comple7806235888213564991et_nat @ top_top_set_set_nat )
= bot_bot_set_nat ) ).
% Inf_UNIV
thf(fact_346_Inf__UNIV,axiom,
( ( complete_Inf_Inf_o @ top_top_set_o )
= bot_bot_o ) ).
% Inf_UNIV
thf(fact_347_Inf__UNIV,axiom,
( ( comple3556804143462414037_ereal @ top_to5683747375963461374_ereal )
= bot_bo2710585358178759738_ereal ) ).
% Inf_UNIV
thf(fact_348_Inf__UNIV,axiom,
( ( comple7330758040695736817nnreal @ top_to7994903218803871134nnreal )
= bot_bo841427958541957580nnreal ) ).
% Inf_UNIV
thf(fact_349_Inf__empty,axiom,
( ( comple7806235888213564991et_nat @ bot_bot_set_set_nat )
= top_top_set_nat ) ).
% Inf_empty
thf(fact_350_Inf__empty,axiom,
( ( comple4418415374894819509_ereal @ bot_bo7400643019497942010_ereal )
= top_to5683747375963461374_ereal ) ).
% Inf_empty
thf(fact_351_Inf__empty,axiom,
( ( complete_Inf_Inf_o @ bot_bot_set_o )
= top_top_o ) ).
% Inf_empty
thf(fact_352_Inf__empty,axiom,
( ( comple3556804143462414037_ereal @ bot_bo8367695208629047834_ereal )
= top_to6662034908053899550_ereal ) ).
% Inf_empty
thf(fact_353_Inf__empty,axiom,
( ( comple7330758040695736817nnreal @ bot_bo4854962954004695426nnreal )
= top_to1496364449551166952nnreal ) ).
% Inf_empty
thf(fact_354_SUP__bot__conv_I2_J,axiom,
! [B2: b > $o,A: set_b] :
( ( bot_bot_o
= ( complete_Sup_Sup_o @ ( image_b_o @ B2 @ A ) ) )
= ( ! [X2: b] :
( ( member_b @ X2 @ A )
=> ( ( B2 @ X2 )
= bot_bot_o ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_355_SUP__bot__conv_I2_J,axiom,
! [B2: nat > extended_ereal,A: set_nat] :
( ( bot_bo2710585358178759738_ereal
= ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ B2 @ A ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( B2 @ X2 )
= bot_bo2710585358178759738_ereal ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_356_SUP__bot__conv_I2_J,axiom,
! [B2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( bot_bo2710585358178759738_ereal
= ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ B2 @ A ) ) )
= ( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A )
=> ( ( B2 @ X2 )
= bot_bo2710585358178759738_ereal ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_357_SUP__bot__conv_I2_J,axiom,
! [B2: b > extended_ereal,A: set_b] :
( ( bot_bo2710585358178759738_ereal
= ( comple8415311339701865915_ereal @ ( image_5319725110001000852_ereal @ B2 @ A ) ) )
= ( ! [X2: b] :
( ( member_b @ X2 @ A )
=> ( ( B2 @ X2 )
= bot_bo2710585358178759738_ereal ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_358_SUP__bot__conv_I2_J,axiom,
! [B2: nat > extend8495563244428889912nnreal,A: set_nat] :
( ( bot_bo841427958541957580nnreal
= ( comple6814414086264997003nnreal @ ( image_8459861568512453903nnreal @ B2 @ A ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( B2 @ X2 )
= bot_bo841427958541957580nnreal ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_359_SUP__bot__conv_I1_J,axiom,
! [B2: b > $o,A: set_b] :
( ( ( complete_Sup_Sup_o @ ( image_b_o @ B2 @ A ) )
= bot_bot_o )
= ( ! [X2: b] :
( ( member_b @ X2 @ A )
=> ( ( B2 @ X2 )
= bot_bot_o ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_360_SUP__bot__conv_I1_J,axiom,
! [B2: nat > extended_ereal,A: set_nat] :
( ( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ B2 @ A ) )
= bot_bo2710585358178759738_ereal )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( B2 @ X2 )
= bot_bo2710585358178759738_ereal ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_361_SUP__bot__conv_I1_J,axiom,
! [B2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ B2 @ A ) )
= bot_bo2710585358178759738_ereal )
= ( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A )
=> ( ( B2 @ X2 )
= bot_bo2710585358178759738_ereal ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_362_SUP__bot__conv_I1_J,axiom,
! [B2: b > extended_ereal,A: set_b] :
( ( ( comple8415311339701865915_ereal @ ( image_5319725110001000852_ereal @ B2 @ A ) )
= bot_bo2710585358178759738_ereal )
= ( ! [X2: b] :
( ( member_b @ X2 @ A )
=> ( ( B2 @ X2 )
= bot_bo2710585358178759738_ereal ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_363_SUP__bot__conv_I1_J,axiom,
! [B2: nat > extend8495563244428889912nnreal,A: set_nat] :
( ( ( comple6814414086264997003nnreal @ ( image_8459861568512453903nnreal @ B2 @ A ) )
= bot_bo841427958541957580nnreal )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( B2 @ X2 )
= bot_bo841427958541957580nnreal ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_364_SUP__bot,axiom,
! [A: set_b] :
( ( complete_Sup_Sup_o
@ ( image_b_o
@ ^ [X2: b] : bot_bot_o
@ A ) )
= bot_bot_o ) ).
% SUP_bot
thf(fact_365_SUP__bot,axiom,
! [A: set_nat] :
( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X2: nat] : bot_bo2710585358178759738_ereal
@ A ) )
= bot_bo2710585358178759738_ereal ) ).
% SUP_bot
thf(fact_366_SUP__bot,axiom,
! [A: set_Extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X2: extended_ereal] : bot_bo2710585358178759738_ereal
@ A ) )
= bot_bo2710585358178759738_ereal ) ).
% SUP_bot
thf(fact_367_SUP__bot,axiom,
! [A: set_b] :
( ( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [X2: b] : bot_bo2710585358178759738_ereal
@ A ) )
= bot_bo2710585358178759738_ereal ) ).
% SUP_bot
thf(fact_368_SUP__bot,axiom,
! [A: set_nat] :
( ( comple6814414086264997003nnreal
@ ( image_8459861568512453903nnreal
@ ^ [X2: nat] : bot_bo841427958541957580nnreal
@ A ) )
= bot_bo841427958541957580nnreal ) ).
% SUP_bot
thf(fact_369_SUP__const,axiom,
! [A: set_b,F: $o] :
( ( A != bot_bot_set_b )
=> ( ( complete_Sup_Sup_o
@ ( image_b_o
@ ^ [I: b] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_370_SUP__const,axiom,
! [A: set_Extended_ereal,F: $o] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( complete_Sup_Sup_o
@ ( image_951975095941678543real_o
@ ^ [I: extended_ereal] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_371_SUP__const,axiom,
! [A: set_Ex3793607809372303086nnreal,F: $o] :
( ( A != bot_bo4854962954004695426nnreal )
=> ( ( complete_Sup_Sup_o
@ ( image_3162942742313426073real_o
@ ^ [I: extend8495563244428889912nnreal] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_372_SUP__const,axiom,
! [A: set_nat,F: $o] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [I: nat] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_373_SUP__const,axiom,
! [A: set_b,F: extended_ereal] :
( ( A != bot_bot_set_b )
=> ( ( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [I: b] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_374_SUP__const,axiom,
! [A: set_Extended_ereal,F: extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I: extended_ereal] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_375_SUP__const,axiom,
! [A: set_Ex3793607809372303086nnreal,F: extended_ereal] :
( ( A != bot_bo4854962954004695426nnreal )
=> ( ( comple8415311339701865915_ereal
@ ( image_6393943237584228047_ereal
@ ^ [I: extend8495563244428889912nnreal] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_376_SUP__const,axiom,
! [A: set_nat,F: extended_ereal] :
( ( A != bot_bot_set_nat )
=> ( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I: nat] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_377_SUP__const,axiom,
! [A: set_Extended_ereal,F: extend8495563244428889912nnreal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple6814414086264997003nnreal
@ ( image_8614087454967683265nnreal
@ ^ [I: extended_ereal] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_378_SUP__const,axiom,
! [A: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal] :
( ( A != bot_bo4854962954004695426nnreal )
=> ( ( comple6814414086264997003nnreal
@ ( image_8394674774369097847nnreal
@ ^ [I: extend8495563244428889912nnreal] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_379_INT__constant,axiom,
! [A: set_Extended_ereal,C2: set_nat] :
( ( ( A = bot_bo8367695208629047834_ereal )
=> ( ( comple7806235888213564991et_nat
@ ( image_3090908713637162255et_nat
@ ^ [Y2: extended_ereal] : C2
@ A ) )
= top_top_set_nat ) )
& ( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple7806235888213564991et_nat
@ ( image_3090908713637162255et_nat
@ ^ [Y2: extended_ereal] : C2
@ A ) )
= C2 ) ) ) ).
% INT_constant
thf(fact_380_INT__constant,axiom,
! [A: set_Ex3793607809372303086nnreal,C2: set_nat] :
( ( ( A = bot_bo4854962954004695426nnreal )
=> ( ( comple7806235888213564991et_nat
@ ( image_2869339492569777349et_nat
@ ^ [Y2: extend8495563244428889912nnreal] : C2
@ A ) )
= top_top_set_nat ) )
& ( ( A != bot_bo4854962954004695426nnreal )
=> ( ( comple7806235888213564991et_nat
@ ( image_2869339492569777349et_nat
@ ^ [Y2: extend8495563244428889912nnreal] : C2
@ A ) )
= C2 ) ) ) ).
% INT_constant
thf(fact_381_INT__constant,axiom,
! [A: set_nat,C2: set_nat] :
( ( ( A = bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [Y2: nat] : C2
@ A ) )
= top_top_set_nat ) )
& ( ( A != bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [Y2: nat] : C2
@ A ) )
= C2 ) ) ) ).
% INT_constant
thf(fact_382_INT__constant,axiom,
! [A: set_Extended_ereal,C2: set_Extended_ereal] :
( ( ( A = bot_bo8367695208629047834_ereal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5562094264469218789_ereal
@ ^ [Y2: extended_ereal] : C2
@ A ) )
= top_to5683747375963461374_ereal ) )
& ( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5562094264469218789_ereal
@ ^ [Y2: extended_ereal] : C2
@ A ) )
= C2 ) ) ) ).
% INT_constant
thf(fact_383_INT__constant,axiom,
! [A: set_Ex3793607809372303086nnreal,C2: set_Extended_ereal] :
( ( ( A = bot_bo4854962954004695426nnreal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5929344197358196911_ereal
@ ^ [Y2: extend8495563244428889912nnreal] : C2
@ A ) )
= top_to5683747375963461374_ereal ) )
& ( ( A != bot_bo4854962954004695426nnreal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5929344197358196911_ereal
@ ^ [Y2: extend8495563244428889912nnreal] : C2
@ A ) )
= C2 ) ) ) ).
% INT_constant
thf(fact_384_INT__constant,axiom,
! [A: set_nat,C2: set_Extended_ereal] :
( ( ( A = bot_bot_set_nat )
=> ( ( comple4418415374894819509_ereal
@ ( image_305533323056406039_ereal
@ ^ [Y2: nat] : C2
@ A ) )
= top_to5683747375963461374_ereal ) )
& ( ( A != bot_bot_set_nat )
=> ( ( comple4418415374894819509_ereal
@ ( image_305533323056406039_ereal
@ ^ [Y2: nat] : C2
@ A ) )
= C2 ) ) ) ).
% INT_constant
thf(fact_385_INF__top__conv_I2_J,axiom,
! [B2: b > $o,A: set_b] :
( ( top_top_o
= ( complete_Inf_Inf_o @ ( image_b_o @ B2 @ A ) ) )
= ( ! [X2: b] :
( ( member_b @ X2 @ A )
=> ( ( B2 @ X2 )
= top_top_o ) ) ) ) ).
% INF_top_conv(2)
thf(fact_386_INF__top__conv_I2_J,axiom,
! [B2: nat > extended_ereal,A: set_nat] :
( ( top_to6662034908053899550_ereal
= ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ B2 @ A ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( B2 @ X2 )
= top_to6662034908053899550_ereal ) ) ) ) ).
% INF_top_conv(2)
thf(fact_387_INF__top__conv_I2_J,axiom,
! [B2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( top_to6662034908053899550_ereal
= ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ B2 @ A ) ) )
= ( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A )
=> ( ( B2 @ X2 )
= top_to6662034908053899550_ereal ) ) ) ) ).
% INF_top_conv(2)
thf(fact_388_INF__top__conv_I2_J,axiom,
! [B2: b > extended_ereal,A: set_b] :
( ( top_to6662034908053899550_ereal
= ( comple3556804143462414037_ereal @ ( image_5319725110001000852_ereal @ B2 @ A ) ) )
= ( ! [X2: b] :
( ( member_b @ X2 @ A )
=> ( ( B2 @ X2 )
= top_to6662034908053899550_ereal ) ) ) ) ).
% INF_top_conv(2)
thf(fact_389_INF__top__conv_I2_J,axiom,
! [B2: nat > extend8495563244428889912nnreal,A: set_nat] :
( ( top_to1496364449551166952nnreal
= ( comple7330758040695736817nnreal @ ( image_8459861568512453903nnreal @ B2 @ A ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( B2 @ X2 )
= top_to1496364449551166952nnreal ) ) ) ) ).
% INF_top_conv(2)
thf(fact_390_INF__top__conv_I1_J,axiom,
! [B2: b > $o,A: set_b] :
( ( ( complete_Inf_Inf_o @ ( image_b_o @ B2 @ A ) )
= top_top_o )
= ( ! [X2: b] :
( ( member_b @ X2 @ A )
=> ( ( B2 @ X2 )
= top_top_o ) ) ) ) ).
% INF_top_conv(1)
thf(fact_391_INF__top__conv_I1_J,axiom,
! [B2: nat > extended_ereal,A: set_nat] :
( ( ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ B2 @ A ) )
= top_to6662034908053899550_ereal )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( B2 @ X2 )
= top_to6662034908053899550_ereal ) ) ) ) ).
% INF_top_conv(1)
thf(fact_392_INF__top__conv_I1_J,axiom,
! [B2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ B2 @ A ) )
= top_to6662034908053899550_ereal )
= ( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A )
=> ( ( B2 @ X2 )
= top_to6662034908053899550_ereal ) ) ) ) ).
% INF_top_conv(1)
thf(fact_393_INF__top__conv_I1_J,axiom,
! [B2: b > extended_ereal,A: set_b] :
( ( ( comple3556804143462414037_ereal @ ( image_5319725110001000852_ereal @ B2 @ A ) )
= top_to6662034908053899550_ereal )
= ( ! [X2: b] :
( ( member_b @ X2 @ A )
=> ( ( B2 @ X2 )
= top_to6662034908053899550_ereal ) ) ) ) ).
% INF_top_conv(1)
thf(fact_394_INF__top__conv_I1_J,axiom,
! [B2: nat > extend8495563244428889912nnreal,A: set_nat] :
( ( ( comple7330758040695736817nnreal @ ( image_8459861568512453903nnreal @ B2 @ A ) )
= top_to1496364449551166952nnreal )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( B2 @ X2 )
= top_to1496364449551166952nnreal ) ) ) ) ).
% INF_top_conv(1)
thf(fact_395_INF__top,axiom,
! [A: set_b] :
( ( complete_Inf_Inf_o
@ ( image_b_o
@ ^ [X2: b] : top_top_o
@ A ) )
= top_top_o ) ).
% INF_top
thf(fact_396_INF__top,axiom,
! [A: set_nat] :
( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X2: nat] : top_to6662034908053899550_ereal
@ A ) )
= top_to6662034908053899550_ereal ) ).
% INF_top
thf(fact_397_INF__top,axiom,
! [A: set_Extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X2: extended_ereal] : top_to6662034908053899550_ereal
@ A ) )
= top_to6662034908053899550_ereal ) ).
% INF_top
thf(fact_398_INF__top,axiom,
! [A: set_b] :
( ( comple3556804143462414037_ereal
@ ( image_5319725110001000852_ereal
@ ^ [X2: b] : top_to6662034908053899550_ereal
@ A ) )
= top_to6662034908053899550_ereal ) ).
% INF_top
thf(fact_399_INF__top,axiom,
! [A: set_nat] :
( ( comple7330758040695736817nnreal
@ ( image_8459861568512453903nnreal
@ ^ [X2: nat] : top_to1496364449551166952nnreal
@ A ) )
= top_to1496364449551166952nnreal ) ).
% INF_top
thf(fact_400_INF__const,axiom,
! [A: set_b,F: $o] :
( ( A != bot_bot_set_b )
=> ( ( complete_Inf_Inf_o
@ ( image_b_o
@ ^ [I: b] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_401_INF__const,axiom,
! [A: set_Extended_ereal,F: $o] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( complete_Inf_Inf_o
@ ( image_951975095941678543real_o
@ ^ [I: extended_ereal] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_402_INF__const,axiom,
! [A: set_Ex3793607809372303086nnreal,F: $o] :
( ( A != bot_bo4854962954004695426nnreal )
=> ( ( complete_Inf_Inf_o
@ ( image_3162942742313426073real_o
@ ^ [I: extend8495563244428889912nnreal] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_403_INF__const,axiom,
! [A: set_nat,F: $o] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [I: nat] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_404_INF__const,axiom,
! [A: set_b,F: extended_ereal] :
( ( A != bot_bot_set_b )
=> ( ( comple3556804143462414037_ereal
@ ( image_5319725110001000852_ereal
@ ^ [I: b] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_405_INF__const,axiom,
! [A: set_Extended_ereal,F: extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I: extended_ereal] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_406_INF__const,axiom,
! [A: set_Ex3793607809372303086nnreal,F: extended_ereal] :
( ( A != bot_bo4854962954004695426nnreal )
=> ( ( comple3556804143462414037_ereal
@ ( image_6393943237584228047_ereal
@ ^ [I: extend8495563244428889912nnreal] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_407_INF__const,axiom,
! [A: set_nat,F: extended_ereal] :
( ( A != bot_bot_set_nat )
=> ( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I: nat] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_408_INF__const,axiom,
! [A: set_Extended_ereal,F: extend8495563244428889912nnreal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple7330758040695736817nnreal
@ ( image_8614087454967683265nnreal
@ ^ [I: extended_ereal] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_409_INF__const,axiom,
! [A: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal] :
( ( A != bot_bo4854962954004695426nnreal )
=> ( ( comple7330758040695736817nnreal
@ ( image_8394674774369097847nnreal
@ ^ [I: extend8495563244428889912nnreal] : F
@ A ) )
= F ) ) ).
% INF_const
thf(fact_410_UN__constant,axiom,
! [A: set_Extended_ereal,C2: set_Extended_ereal] :
( ( ( A = bot_bo8367695208629047834_ereal )
=> ( ( comple4319282863272126363_ereal
@ ( image_5562094264469218789_ereal
@ ^ [Y2: extended_ereal] : C2
@ A ) )
= bot_bo8367695208629047834_ereal ) )
& ( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple4319282863272126363_ereal
@ ( image_5562094264469218789_ereal
@ ^ [Y2: extended_ereal] : C2
@ A ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_411_UN__constant,axiom,
! [A: set_Extended_ereal,C2: set_Ex3793607809372303086nnreal] :
( ( ( A = bot_bo8367695208629047834_ereal )
=> ( ( comple4226387801268262977nnreal
@ ( image_6588766411312125047nnreal
@ ^ [Y2: extended_ereal] : C2
@ A ) )
= bot_bo4854962954004695426nnreal ) )
& ( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple4226387801268262977nnreal
@ ( image_6588766411312125047nnreal
@ ^ [Y2: extended_ereal] : C2
@ A ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_412_UN__constant,axiom,
! [A: set_Extended_ereal,C2: set_nat] :
( ( ( A = bot_bo8367695208629047834_ereal )
=> ( ( comple7399068483239264473et_nat
@ ( image_3090908713637162255et_nat
@ ^ [Y2: extended_ereal] : C2
@ A ) )
= bot_bot_set_nat ) )
& ( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple7399068483239264473et_nat
@ ( image_3090908713637162255et_nat
@ ^ [Y2: extended_ereal] : C2
@ A ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_413_UN__constant,axiom,
! [A: set_Ex3793607809372303086nnreal,C2: set_Extended_ereal] :
( ( ( A = bot_bo4854962954004695426nnreal )
=> ( ( comple4319282863272126363_ereal
@ ( image_5929344197358196911_ereal
@ ^ [Y2: extend8495563244428889912nnreal] : C2
@ A ) )
= bot_bo8367695208629047834_ereal ) )
& ( ( A != bot_bo4854962954004695426nnreal )
=> ( ( comple4319282863272126363_ereal
@ ( image_5929344197358196911_ereal
@ ^ [Y2: extend8495563244428889912nnreal] : C2
@ A ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_414_UN__constant,axiom,
! [A: set_Ex3793607809372303086nnreal,C2: set_Ex3793607809372303086nnreal] :
( ( ( A = bot_bo4854962954004695426nnreal )
=> ( ( comple4226387801268262977nnreal
@ ( image_205196257943321645nnreal
@ ^ [Y2: extend8495563244428889912nnreal] : C2
@ A ) )
= bot_bo4854962954004695426nnreal ) )
& ( ( A != bot_bo4854962954004695426nnreal )
=> ( ( comple4226387801268262977nnreal
@ ( image_205196257943321645nnreal
@ ^ [Y2: extend8495563244428889912nnreal] : C2
@ A ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_415_UN__constant,axiom,
! [A: set_Ex3793607809372303086nnreal,C2: set_nat] :
( ( ( A = bot_bo4854962954004695426nnreal )
=> ( ( comple7399068483239264473et_nat
@ ( image_2869339492569777349et_nat
@ ^ [Y2: extend8495563244428889912nnreal] : C2
@ A ) )
= bot_bot_set_nat ) )
& ( ( A != bot_bo4854962954004695426nnreal )
=> ( ( comple7399068483239264473et_nat
@ ( image_2869339492569777349et_nat
@ ^ [Y2: extend8495563244428889912nnreal] : C2
@ A ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_416_UN__constant,axiom,
! [A: set_nat,C2: set_Extended_ereal] :
( ( ( A = bot_bot_set_nat )
=> ( ( comple4319282863272126363_ereal
@ ( image_305533323056406039_ereal
@ ^ [Y2: nat] : C2
@ A ) )
= bot_bo8367695208629047834_ereal ) )
& ( ( A != bot_bot_set_nat )
=> ( ( comple4319282863272126363_ereal
@ ( image_305533323056406039_ereal
@ ^ [Y2: nat] : C2
@ A ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_417_UN__constant,axiom,
! [A: set_nat,C2: set_Ex3793607809372303086nnreal] :
( ( ( A = bot_bot_set_nat )
=> ( ( comple4226387801268262977nnreal
@ ( image_3394822847079329989nnreal
@ ^ [Y2: nat] : C2
@ A ) )
= bot_bo4854962954004695426nnreal ) )
& ( ( A != bot_bot_set_nat )
=> ( ( comple4226387801268262977nnreal
@ ( image_3394822847079329989nnreal
@ ^ [Y2: nat] : C2
@ A ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_418_UN__constant,axiom,
! [A: set_nat,C2: set_nat] :
( ( ( A = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y2: nat] : C2
@ A ) )
= bot_bot_set_nat ) )
& ( ( A != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y2: nat] : C2
@ A ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_419_INF__empty,axiom,
! [F: b > $o] :
( ( complete_Inf_Inf_o @ ( image_b_o @ F @ bot_bot_set_b ) )
= top_top_o ) ).
% INF_empty
thf(fact_420_INF__empty,axiom,
! [F: extended_ereal > $o] :
( ( complete_Inf_Inf_o @ ( image_951975095941678543real_o @ F @ bot_bo8367695208629047834_ereal ) )
= top_top_o ) ).
% INF_empty
thf(fact_421_INF__empty,axiom,
! [F: extend8495563244428889912nnreal > $o] :
( ( complete_Inf_Inf_o @ ( image_3162942742313426073real_o @ F @ bot_bo4854962954004695426nnreal ) )
= top_top_o ) ).
% INF_empty
thf(fact_422_INF__empty,axiom,
! [F: nat > $o] :
( ( complete_Inf_Inf_o @ ( image_nat_o @ F @ bot_bot_set_nat ) )
= top_top_o ) ).
% INF_empty
thf(fact_423_INF__empty,axiom,
! [F: b > extended_ereal] :
( ( comple3556804143462414037_ereal @ ( image_5319725110001000852_ereal @ F @ bot_bot_set_b ) )
= top_to6662034908053899550_ereal ) ).
% INF_empty
thf(fact_424_INF__empty,axiom,
! [F: extended_ereal > extended_ereal] :
( ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F @ bot_bo8367695208629047834_ereal ) )
= top_to6662034908053899550_ereal ) ).
% INF_empty
thf(fact_425_INF__empty,axiom,
! [F: extend8495563244428889912nnreal > extended_ereal] :
( ( comple3556804143462414037_ereal @ ( image_6393943237584228047_ereal @ F @ bot_bo4854962954004695426nnreal ) )
= top_to6662034908053899550_ereal ) ).
% INF_empty
thf(fact_426_INF__empty,axiom,
! [F: nat > extended_ereal] :
( ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F @ bot_bot_set_nat ) )
= top_to6662034908053899550_ereal ) ).
% INF_empty
thf(fact_427_INF__empty,axiom,
! [F: extended_ereal > extend8495563244428889912nnreal] :
( ( comple7330758040695736817nnreal @ ( image_8614087454967683265nnreal @ F @ bot_bo8367695208629047834_ereal ) )
= top_to1496364449551166952nnreal ) ).
% INF_empty
thf(fact_428_INF__empty,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( comple7330758040695736817nnreal @ ( image_8394674774369097847nnreal @ F @ bot_bo4854962954004695426nnreal ) )
= top_to1496364449551166952nnreal ) ).
% INF_empty
thf(fact_429_SUP__empty,axiom,
! [F: b > $o] :
( ( complete_Sup_Sup_o @ ( image_b_o @ F @ bot_bot_set_b ) )
= bot_bot_o ) ).
% SUP_empty
thf(fact_430_SUP__empty,axiom,
! [F: extended_ereal > $o] :
( ( complete_Sup_Sup_o @ ( image_951975095941678543real_o @ F @ bot_bo8367695208629047834_ereal ) )
= bot_bot_o ) ).
% SUP_empty
thf(fact_431_SUP__empty,axiom,
! [F: extend8495563244428889912nnreal > $o] :
( ( complete_Sup_Sup_o @ ( image_3162942742313426073real_o @ F @ bot_bo4854962954004695426nnreal ) )
= bot_bot_o ) ).
% SUP_empty
thf(fact_432_SUP__empty,axiom,
! [F: nat > $o] :
( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ bot_bot_set_nat ) )
= bot_bot_o ) ).
% SUP_empty
thf(fact_433_SUP__empty,axiom,
! [F: b > extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_5319725110001000852_ereal @ F @ bot_bot_set_b ) )
= bot_bo2710585358178759738_ereal ) ).
% SUP_empty
thf(fact_434_SUP__empty,axiom,
! [F: extended_ereal > extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F @ bot_bo8367695208629047834_ereal ) )
= bot_bo2710585358178759738_ereal ) ).
% SUP_empty
thf(fact_435_SUP__empty,axiom,
! [F: extend8495563244428889912nnreal > extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_6393943237584228047_ereal @ F @ bot_bo4854962954004695426nnreal ) )
= bot_bo2710585358178759738_ereal ) ).
% SUP_empty
thf(fact_436_SUP__empty,axiom,
! [F: nat > extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F @ bot_bot_set_nat ) )
= bot_bo2710585358178759738_ereal ) ).
% SUP_empty
thf(fact_437_SUP__empty,axiom,
! [F: extended_ereal > extend8495563244428889912nnreal] :
( ( comple6814414086264997003nnreal @ ( image_8614087454967683265nnreal @ F @ bot_bo8367695208629047834_ereal ) )
= bot_bo841427958541957580nnreal ) ).
% SUP_empty
thf(fact_438_SUP__empty,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( comple6814414086264997003nnreal @ ( image_8394674774369097847nnreal @ F @ bot_bo4854962954004695426nnreal ) )
= bot_bo841427958541957580nnreal ) ).
% SUP_empty
thf(fact_439_INF__constant,axiom,
! [C2: $o,A: set_b] :
( ( complete_Inf_Inf_o
@ ( image_b_o
@ ^ [Y2: b] : C2
@ A ) )
= ( ( ( A = bot_bot_set_b )
=> top_top_o )
& ( ( A != bot_bot_set_b )
=> C2 ) ) ) ).
% INF_constant
thf(fact_440_INF__constant,axiom,
! [C2: $o,A: set_Extended_ereal] :
( ( complete_Inf_Inf_o
@ ( image_951975095941678543real_o
@ ^ [Y2: extended_ereal] : C2
@ A ) )
= ( ( ( A = bot_bo8367695208629047834_ereal )
=> top_top_o )
& ( ( A != bot_bo8367695208629047834_ereal )
=> C2 ) ) ) ).
% INF_constant
thf(fact_441_INF__constant,axiom,
! [C2: $o,A: set_Ex3793607809372303086nnreal] :
( ( complete_Inf_Inf_o
@ ( image_3162942742313426073real_o
@ ^ [Y2: extend8495563244428889912nnreal] : C2
@ A ) )
= ( ( ( A = bot_bo4854962954004695426nnreal )
=> top_top_o )
& ( ( A != bot_bo4854962954004695426nnreal )
=> C2 ) ) ) ).
% INF_constant
thf(fact_442_INF__constant,axiom,
! [C2: $o,A: set_nat] :
( ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [Y2: nat] : C2
@ A ) )
= ( ( ( A = bot_bot_set_nat )
=> top_top_o )
& ( ( A != bot_bot_set_nat )
=> C2 ) ) ) ).
% INF_constant
thf(fact_443_INF__constant,axiom,
! [A: set_b,C2: extended_ereal] :
( ( ( A = bot_bot_set_b )
=> ( ( comple3556804143462414037_ereal
@ ( image_5319725110001000852_ereal
@ ^ [Y2: b] : C2
@ A ) )
= top_to6662034908053899550_ereal ) )
& ( ( A != bot_bot_set_b )
=> ( ( comple3556804143462414037_ereal
@ ( image_5319725110001000852_ereal
@ ^ [Y2: b] : C2
@ A ) )
= C2 ) ) ) ).
% INF_constant
thf(fact_444_INF__constant,axiom,
! [A: set_Extended_ereal,C2: extended_ereal] :
( ( ( A = bot_bo8367695208629047834_ereal )
=> ( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [Y2: extended_ereal] : C2
@ A ) )
= top_to6662034908053899550_ereal ) )
& ( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [Y2: extended_ereal] : C2
@ A ) )
= C2 ) ) ) ).
% INF_constant
thf(fact_445_INF__constant,axiom,
! [A: set_Ex3793607809372303086nnreal,C2: extended_ereal] :
( ( ( A = bot_bo4854962954004695426nnreal )
=> ( ( comple3556804143462414037_ereal
@ ( image_6393943237584228047_ereal
@ ^ [Y2: extend8495563244428889912nnreal] : C2
@ A ) )
= top_to6662034908053899550_ereal ) )
& ( ( A != bot_bo4854962954004695426nnreal )
=> ( ( comple3556804143462414037_ereal
@ ( image_6393943237584228047_ereal
@ ^ [Y2: extend8495563244428889912nnreal] : C2
@ A ) )
= C2 ) ) ) ).
% INF_constant
thf(fact_446_INF__constant,axiom,
! [A: set_nat,C2: extended_ereal] :
( ( ( A = bot_bot_set_nat )
=> ( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [Y2: nat] : C2
@ A ) )
= top_to6662034908053899550_ereal ) )
& ( ( A != bot_bot_set_nat )
=> ( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [Y2: nat] : C2
@ A ) )
= C2 ) ) ) ).
% INF_constant
thf(fact_447_INF__constant,axiom,
! [A: set_Extended_ereal,C2: extend8495563244428889912nnreal] :
( ( ( A = bot_bo8367695208629047834_ereal )
=> ( ( comple7330758040695736817nnreal
@ ( image_8614087454967683265nnreal
@ ^ [Y2: extended_ereal] : C2
@ A ) )
= top_to1496364449551166952nnreal ) )
& ( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple7330758040695736817nnreal
@ ( image_8614087454967683265nnreal
@ ^ [Y2: extended_ereal] : C2
@ A ) )
= C2 ) ) ) ).
% INF_constant
thf(fact_448_INF__constant,axiom,
! [A: set_Ex3793607809372303086nnreal,C2: extend8495563244428889912nnreal] :
( ( ( A = bot_bo4854962954004695426nnreal )
=> ( ( comple7330758040695736817nnreal
@ ( image_8394674774369097847nnreal
@ ^ [Y2: extend8495563244428889912nnreal] : C2
@ A ) )
= top_to1496364449551166952nnreal ) )
& ( ( A != bot_bo4854962954004695426nnreal )
=> ( ( comple7330758040695736817nnreal
@ ( image_8394674774369097847nnreal
@ ^ [Y2: extend8495563244428889912nnreal] : C2
@ A ) )
= C2 ) ) ) ).
% INF_constant
thf(fact_449_SUP__constant,axiom,
! [C2: $o,A: set_b] :
( ( complete_Sup_Sup_o
@ ( image_b_o
@ ^ [Y2: b] : C2
@ A ) )
= ( ( ( A = bot_bot_set_b )
=> bot_bot_o )
& ( ( A != bot_bot_set_b )
=> C2 ) ) ) ).
% SUP_constant
thf(fact_450_SUP__constant,axiom,
! [C2: $o,A: set_Extended_ereal] :
( ( complete_Sup_Sup_o
@ ( image_951975095941678543real_o
@ ^ [Y2: extended_ereal] : C2
@ A ) )
= ( ( ( A = bot_bo8367695208629047834_ereal )
=> bot_bot_o )
& ( ( A != bot_bo8367695208629047834_ereal )
=> C2 ) ) ) ).
% SUP_constant
thf(fact_451_SUP__constant,axiom,
! [C2: $o,A: set_Ex3793607809372303086nnreal] :
( ( complete_Sup_Sup_o
@ ( image_3162942742313426073real_o
@ ^ [Y2: extend8495563244428889912nnreal] : C2
@ A ) )
= ( ( ( A = bot_bo4854962954004695426nnreal )
=> bot_bot_o )
& ( ( A != bot_bo4854962954004695426nnreal )
=> C2 ) ) ) ).
% SUP_constant
thf(fact_452_SUP__constant,axiom,
! [C2: $o,A: set_nat] :
( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y2: nat] : C2
@ A ) )
= ( ( ( A = bot_bot_set_nat )
=> bot_bot_o )
& ( ( A != bot_bot_set_nat )
=> C2 ) ) ) ).
% SUP_constant
thf(fact_453_SUP__constant,axiom,
! [A: set_b,C2: extended_ereal] :
( ( ( A = bot_bot_set_b )
=> ( ( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [Y2: b] : C2
@ A ) )
= bot_bo2710585358178759738_ereal ) )
& ( ( A != bot_bot_set_b )
=> ( ( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [Y2: b] : C2
@ A ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_454_SUP__constant,axiom,
! [A: set_Extended_ereal,C2: extended_ereal] :
( ( ( A = bot_bo8367695208629047834_ereal )
=> ( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [Y2: extended_ereal] : C2
@ A ) )
= bot_bo2710585358178759738_ereal ) )
& ( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [Y2: extended_ereal] : C2
@ A ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_455_SUP__constant,axiom,
! [A: set_Ex3793607809372303086nnreal,C2: extended_ereal] :
( ( ( A = bot_bo4854962954004695426nnreal )
=> ( ( comple8415311339701865915_ereal
@ ( image_6393943237584228047_ereal
@ ^ [Y2: extend8495563244428889912nnreal] : C2
@ A ) )
= bot_bo2710585358178759738_ereal ) )
& ( ( A != bot_bo4854962954004695426nnreal )
=> ( ( comple8415311339701865915_ereal
@ ( image_6393943237584228047_ereal
@ ^ [Y2: extend8495563244428889912nnreal] : C2
@ A ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_456_SUP__constant,axiom,
! [A: set_nat,C2: extended_ereal] :
( ( ( A = bot_bot_set_nat )
=> ( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [Y2: nat] : C2
@ A ) )
= bot_bo2710585358178759738_ereal ) )
& ( ( A != bot_bot_set_nat )
=> ( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [Y2: nat] : C2
@ A ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_457_SUP__constant,axiom,
! [A: set_Extended_ereal,C2: extend8495563244428889912nnreal] :
( ( ( A = bot_bo8367695208629047834_ereal )
=> ( ( comple6814414086264997003nnreal
@ ( image_8614087454967683265nnreal
@ ^ [Y2: extended_ereal] : C2
@ A ) )
= bot_bo841427958541957580nnreal ) )
& ( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple6814414086264997003nnreal
@ ( image_8614087454967683265nnreal
@ ^ [Y2: extended_ereal] : C2
@ A ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_458_SUP__constant,axiom,
! [A: set_Ex3793607809372303086nnreal,C2: extend8495563244428889912nnreal] :
( ( ( A = bot_bo4854962954004695426nnreal )
=> ( ( comple6814414086264997003nnreal
@ ( image_8394674774369097847nnreal
@ ^ [Y2: extend8495563244428889912nnreal] : C2
@ A ) )
= bot_bo841427958541957580nnreal ) )
& ( ( A != bot_bo4854962954004695426nnreal )
=> ( ( comple6814414086264997003nnreal
@ ( image_8394674774369097847nnreal
@ ^ [Y2: extend8495563244428889912nnreal] : C2
@ A ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_459_emptyE,axiom,
! [A2: $o] :
~ ( member_o @ A2 @ bot_bot_set_o ) ).
% emptyE
thf(fact_460_emptyE,axiom,
! [A2: c] :
~ ( member_c @ A2 @ bot_bot_set_c ) ).
% emptyE
thf(fact_461_emptyE,axiom,
! [A2: b] :
~ ( member_b @ A2 @ bot_bot_set_b ) ).
% emptyE
thf(fact_462_emptyE,axiom,
! [A2: extended_ereal] :
~ ( member2350847679896131959_ereal @ A2 @ bot_bo8367695208629047834_ereal ) ).
% emptyE
thf(fact_463_emptyE,axiom,
! [A2: extend8495563244428889912nnreal] :
~ ( member7908768830364227535nnreal @ A2 @ bot_bo4854962954004695426nnreal ) ).
% emptyE
thf(fact_464_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_465_equals0D,axiom,
! [A: set_o,A2: $o] :
( ( A = bot_bot_set_o )
=> ~ ( member_o @ A2 @ A ) ) ).
% equals0D
thf(fact_466_equals0D,axiom,
! [A: set_c,A2: c] :
( ( A = bot_bot_set_c )
=> ~ ( member_c @ A2 @ A ) ) ).
% equals0D
thf(fact_467_equals0D,axiom,
! [A: set_b,A2: b] :
( ( A = bot_bot_set_b )
=> ~ ( member_b @ A2 @ A ) ) ).
% equals0D
thf(fact_468_equals0D,axiom,
! [A: set_Extended_ereal,A2: extended_ereal] :
( ( A = bot_bo8367695208629047834_ereal )
=> ~ ( member2350847679896131959_ereal @ A2 @ A ) ) ).
% equals0D
thf(fact_469_equals0D,axiom,
! [A: set_Ex3793607809372303086nnreal,A2: extend8495563244428889912nnreal] :
( ( A = bot_bo4854962954004695426nnreal )
=> ~ ( member7908768830364227535nnreal @ A2 @ A ) ) ).
% equals0D
thf(fact_470_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_471_equals0I,axiom,
! [A: set_o] :
( ! [Y3: $o] :
~ ( member_o @ Y3 @ A )
=> ( A = bot_bot_set_o ) ) ).
% equals0I
thf(fact_472_equals0I,axiom,
! [A: set_c] :
( ! [Y3: c] :
~ ( member_c @ Y3 @ A )
=> ( A = bot_bot_set_c ) ) ).
% equals0I
thf(fact_473_equals0I,axiom,
! [A: set_b] :
( ! [Y3: b] :
~ ( member_b @ Y3 @ A )
=> ( A = bot_bot_set_b ) ) ).
% equals0I
thf(fact_474_equals0I,axiom,
! [A: set_Extended_ereal] :
( ! [Y3: extended_ereal] :
~ ( member2350847679896131959_ereal @ Y3 @ A )
=> ( A = bot_bo8367695208629047834_ereal ) ) ).
% equals0I
thf(fact_475_equals0I,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ! [Y3: extend8495563244428889912nnreal] :
~ ( member7908768830364227535nnreal @ Y3 @ A )
=> ( A = bot_bo4854962954004695426nnreal ) ) ).
% equals0I
thf(fact_476_equals0I,axiom,
! [A: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_477_UNIV__eq__I,axiom,
! [A: set_o] :
( ! [X3: $o] : ( member_o @ X3 @ A )
=> ( top_top_set_o = A ) ) ).
% UNIV_eq_I
thf(fact_478_UNIV__eq__I,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ! [X3: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X3 @ A )
=> ( top_to7994903218803871134nnreal = A ) ) ).
% UNIV_eq_I
thf(fact_479_UNIV__eq__I,axiom,
! [A: set_c] :
( ! [X3: c] : ( member_c @ X3 @ A )
=> ( top_top_set_c = A ) ) ).
% UNIV_eq_I
thf(fact_480_UNIV__eq__I,axiom,
! [A: set_b] :
( ! [X3: b] : ( member_b @ X3 @ A )
=> ( top_top_set_b = A ) ) ).
% UNIV_eq_I
thf(fact_481_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X3: nat] : ( member_nat @ X3 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_482_UNIV__eq__I,axiom,
! [A: set_Extended_ereal] :
( ! [X3: extended_ereal] : ( member2350847679896131959_ereal @ X3 @ A )
=> ( top_to5683747375963461374_ereal = A ) ) ).
% UNIV_eq_I
thf(fact_483_ex__in__conv,axiom,
! [A: set_o] :
( ( ? [X2: $o] : ( member_o @ X2 @ A ) )
= ( A != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_484_ex__in__conv,axiom,
! [A: set_c] :
( ( ? [X2: c] : ( member_c @ X2 @ A ) )
= ( A != bot_bot_set_c ) ) ).
% ex_in_conv
thf(fact_485_ex__in__conv,axiom,
! [A: set_b] :
( ( ? [X2: b] : ( member_b @ X2 @ A ) )
= ( A != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_486_ex__in__conv,axiom,
! [A: set_Extended_ereal] :
( ( ? [X2: extended_ereal] : ( member2350847679896131959_ereal @ X2 @ A ) )
= ( A != bot_bo8367695208629047834_ereal ) ) ).
% ex_in_conv
thf(fact_487_ex__in__conv,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( ? [X2: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X2 @ A ) )
= ( A != bot_bo4854962954004695426nnreal ) ) ).
% ex_in_conv
thf(fact_488_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_489_UNIV__witness,axiom,
? [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% UNIV_witness
thf(fact_490_UNIV__witness,axiom,
? [X3: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X3 @ top_to7994903218803871134nnreal ) ).
% UNIV_witness
thf(fact_491_UNIV__witness,axiom,
? [X3: c] : ( member_c @ X3 @ top_top_set_c ) ).
% UNIV_witness
thf(fact_492_UNIV__witness,axiom,
? [X3: b] : ( member_b @ X3 @ top_top_set_b ) ).
% UNIV_witness
thf(fact_493_UNIV__witness,axiom,
? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_494_UNIV__witness,axiom,
? [X3: extended_ereal] : ( member2350847679896131959_ereal @ X3 @ top_to5683747375963461374_ereal ) ).
% UNIV_witness
thf(fact_495_empty__not__UNIV,axiom,
bot_bo4854962954004695426nnreal != top_to7994903218803871134nnreal ).
% empty_not_UNIV
thf(fact_496_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_497_empty__not__UNIV,axiom,
bot_bo8367695208629047834_ereal != top_to5683747375963461374_ereal ).
% empty_not_UNIV
thf(fact_498_UnionE,axiom,
! [A: $o,C: set_set_o] :
( ( member_o @ A @ ( comple90263536869209701_set_o @ C ) )
=> ~ ! [X6: set_o] :
( ( member_o @ A @ X6 )
=> ~ ( member_set_o @ X6 @ C ) ) ) ).
% UnionE
thf(fact_499_UnionE,axiom,
! [A: extended_ereal,C: set_se6634062954251873166_ereal] :
( ( member2350847679896131959_ereal @ A @ ( comple4319282863272126363_ereal @ C ) )
=> ~ ! [X6: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ A @ X6 )
=> ~ ( member5519481007471526743_ereal @ X6 @ C ) ) ) ).
% UnionE
thf(fact_500_UnionE,axiom,
! [A: nat,C: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C ) )
=> ~ ! [X6: set_nat] :
( ( member_nat @ A @ X6 )
=> ~ ( member_set_nat @ X6 @ C ) ) ) ).
% UnionE
thf(fact_501_UnionE,axiom,
! [A: extend8495563244428889912nnreal,C: set_se4580700918925141924nnreal] :
( ( member7908768830364227535nnreal @ A @ ( comple4226387801268262977nnreal @ C ) )
=> ~ ! [X6: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ A @ X6 )
=> ~ ( member603777416030116741nnreal @ X6 @ C ) ) ) ).
% UnionE
thf(fact_502_UnionE,axiom,
! [A: c,C: set_set_c] :
( ( member_c @ A @ ( comple2307003618534512845_set_c @ C ) )
=> ~ ! [X6: set_c] :
( ( member_c @ A @ X6 )
=> ~ ( member_set_c @ X6 @ C ) ) ) ).
% UnionE
thf(fact_503_UnionE,axiom,
! [A: b,C: set_set_b] :
( ( member_b @ A @ ( comple2307003614231284044_set_b @ C ) )
=> ~ ! [X6: set_b] :
( ( member_b @ A @ X6 )
=> ~ ( member_set_b @ X6 @ C ) ) ) ).
% UnionE
thf(fact_504_UN__empty,axiom,
! [B2: extended_ereal > set_Extended_ereal] :
( ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ B2 @ bot_bo8367695208629047834_ereal ) )
= bot_bo8367695208629047834_ereal ) ).
% UN_empty
thf(fact_505_UN__empty,axiom,
! [B2: extended_ereal > set_Ex3793607809372303086nnreal] :
( ( comple4226387801268262977nnreal @ ( image_6588766411312125047nnreal @ B2 @ bot_bo8367695208629047834_ereal ) )
= bot_bo4854962954004695426nnreal ) ).
% UN_empty
thf(fact_506_UN__empty,axiom,
! [B2: extended_ereal > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_3090908713637162255et_nat @ B2 @ bot_bo8367695208629047834_ereal ) )
= bot_bot_set_nat ) ).
% UN_empty
thf(fact_507_UN__empty,axiom,
! [B2: extend8495563244428889912nnreal > set_Extended_ereal] :
( ( comple4319282863272126363_ereal @ ( image_5929344197358196911_ereal @ B2 @ bot_bo4854962954004695426nnreal ) )
= bot_bo8367695208629047834_ereal ) ).
% UN_empty
thf(fact_508_UN__empty,axiom,
! [B2: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal] :
( ( comple4226387801268262977nnreal @ ( image_205196257943321645nnreal @ B2 @ bot_bo4854962954004695426nnreal ) )
= bot_bo4854962954004695426nnreal ) ).
% UN_empty
thf(fact_509_UN__empty,axiom,
! [B2: extend8495563244428889912nnreal > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_2869339492569777349et_nat @ B2 @ bot_bo4854962954004695426nnreal ) )
= bot_bot_set_nat ) ).
% UN_empty
thf(fact_510_UN__empty,axiom,
! [B2: nat > set_Extended_ereal] :
( ( comple4319282863272126363_ereal @ ( image_305533323056406039_ereal @ B2 @ bot_bot_set_nat ) )
= bot_bo8367695208629047834_ereal ) ).
% UN_empty
thf(fact_511_UN__empty,axiom,
! [B2: nat > set_Ex3793607809372303086nnreal] :
( ( comple4226387801268262977nnreal @ ( image_3394822847079329989nnreal @ B2 @ bot_bot_set_nat ) )
= bot_bo4854962954004695426nnreal ) ).
% UN_empty
thf(fact_512_UN__empty,axiom,
! [B2: nat > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% UN_empty
thf(fact_513_INT__empty,axiom,
! [B2: extended_ereal > set_nat] :
( ( comple7806235888213564991et_nat @ ( image_3090908713637162255et_nat @ B2 @ bot_bo8367695208629047834_ereal ) )
= top_top_set_nat ) ).
% INT_empty
thf(fact_514_INT__empty,axiom,
! [B2: extend8495563244428889912nnreal > set_nat] :
( ( comple7806235888213564991et_nat @ ( image_2869339492569777349et_nat @ B2 @ bot_bo4854962954004695426nnreal ) )
= top_top_set_nat ) ).
% INT_empty
thf(fact_515_INT__empty,axiom,
! [B2: nat > set_nat] :
( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B2 @ bot_bot_set_nat ) )
= top_top_set_nat ) ).
% INT_empty
thf(fact_516_INT__empty,axiom,
! [B2: extended_ereal > set_Extended_ereal] :
( ( comple4418415374894819509_ereal @ ( image_5562094264469218789_ereal @ B2 @ bot_bo8367695208629047834_ereal ) )
= top_to5683747375963461374_ereal ) ).
% INT_empty
thf(fact_517_INT__empty,axiom,
! [B2: extend8495563244428889912nnreal > set_Extended_ereal] :
( ( comple4418415374894819509_ereal @ ( image_5929344197358196911_ereal @ B2 @ bot_bo4854962954004695426nnreal ) )
= top_to5683747375963461374_ereal ) ).
% INT_empty
thf(fact_518_INT__empty,axiom,
! [B2: nat > set_Extended_ereal] :
( ( comple4418415374894819509_ereal @ ( image_305533323056406039_ereal @ B2 @ bot_bot_set_nat ) )
= top_to5683747375963461374_ereal ) ).
% INT_empty
thf(fact_519_Inter__UNIV,axiom,
( ( comple4418415374894819509_ereal @ top_to4757929550322229470_ereal )
= bot_bo8367695208629047834_ereal ) ).
% Inter_UNIV
thf(fact_520_Inter__UNIV,axiom,
( ( comple5724520875574609319nnreal @ top_to3356475028079756884nnreal )
= bot_bo4854962954004695426nnreal ) ).
% Inter_UNIV
thf(fact_521_Inter__UNIV,axiom,
( ( comple7806235888213564991et_nat @ top_top_set_set_nat )
= bot_bot_set_nat ) ).
% Inter_UNIV
thf(fact_522_Union__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Union_UNIV
thf(fact_523_Union__UNIV,axiom,
( ( comple4319282863272126363_ereal @ top_to4757929550322229470_ereal )
= top_to5683747375963461374_ereal ) ).
% Union_UNIV
thf(fact_524_Inter__empty,axiom,
( ( comple7806235888213564991et_nat @ bot_bot_set_set_nat )
= top_top_set_nat ) ).
% Inter_empty
thf(fact_525_Inter__empty,axiom,
( ( comple4418415374894819509_ereal @ bot_bo7400643019497942010_ereal )
= top_to5683747375963461374_ereal ) ).
% Inter_empty
thf(fact_526_Sup__set__def,axiom,
( comple2307003614231284044_set_b
= ( ^ [A3: set_set_b] :
( collect_b
@ ^ [X2: b] : ( complete_Sup_Sup_o @ ( image_set_b_o @ ( member_b @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_527_Sup__set__def,axiom,
( comple2307003618534512845_set_c
= ( ^ [A3: set_set_c] :
( collect_c
@ ^ [X2: c] : ( complete_Sup_Sup_o @ ( image_set_c_o @ ( member_c @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_528_Sup__set__def,axiom,
( comple4319282863272126363_ereal
= ( ^ [A3: set_se6634062954251873166_ereal] :
( collec5835592288176408249_ereal
@ ^ [X2: extended_ereal] : ( complete_Sup_Sup_o @ ( image_1946622920212178927real_o @ ( member2350847679896131959_ereal @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_529_Sup__set__def,axiom,
( comple4226387801268262977nnreal
= ( ^ [A3: set_se4580700918925141924nnreal] :
( collec6648975593938027277nnreal
@ ^ [X2: extend8495563244428889912nnreal] : ( complete_Sup_Sup_o @ ( image_2954085599833420643real_o @ ( member7908768830364227535nnreal @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_530_Sup__set__def,axiom,
( comple7399068483239264473et_nat
= ( ^ [A3: set_set_nat] :
( collect_nat
@ ^ [X2: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_531_Sup__set__def,axiom,
( comple90263536869209701_set_o
= ( ^ [A3: set_set_o] :
( collect_o
@ ^ [X2: $o] : ( complete_Sup_Sup_o @ ( image_set_o_o @ ( member_o @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_532_Union__empty,axiom,
( ( comple4319282863272126363_ereal @ bot_bo7400643019497942010_ereal )
= bot_bo8367695208629047834_ereal ) ).
% Union_empty
thf(fact_533_Union__empty,axiom,
( ( comple4226387801268262977nnreal @ bot_bo2988155216863113784nnreal )
= bot_bo4854962954004695426nnreal ) ).
% Union_empty
thf(fact_534_Union__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Union_empty
thf(fact_535_Union__empty__conv,axiom,
! [A: set_se6634062954251873166_ereal] :
( ( ( comple4319282863272126363_ereal @ A )
= bot_bo8367695208629047834_ereal )
= ( ! [X2: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ X2 @ A )
=> ( X2 = bot_bo8367695208629047834_ereal ) ) ) ) ).
% Union_empty_conv
thf(fact_536_Union__empty__conv,axiom,
! [A: set_se4580700918925141924nnreal] :
( ( ( comple4226387801268262977nnreal @ A )
= bot_bo4854962954004695426nnreal )
= ( ! [X2: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X2 @ A )
=> ( X2 = bot_bo4854962954004695426nnreal ) ) ) ) ).
% Union_empty_conv
thf(fact_537_Union__empty__conv,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_538_empty__Union__conv,axiom,
! [A: set_se6634062954251873166_ereal] :
( ( bot_bo8367695208629047834_ereal
= ( comple4319282863272126363_ereal @ A ) )
= ( ! [X2: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ X2 @ A )
=> ( X2 = bot_bo8367695208629047834_ereal ) ) ) ) ).
% empty_Union_conv
thf(fact_539_empty__Union__conv,axiom,
! [A: set_se4580700918925141924nnreal] :
( ( bot_bo4854962954004695426nnreal
= ( comple4226387801268262977nnreal @ A ) )
= ( ! [X2: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X2 @ A )
=> ( X2 = bot_bo4854962954004695426nnreal ) ) ) ) ).
% empty_Union_conv
thf(fact_540_empty__Union__conv,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_541_INF__eq__const,axiom,
! [I2: set_o,F: $o > $o,X: $o] :
( ( I2 != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ I2 ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_542_INF__eq__const,axiom,
! [I2: set_c,F: c > $o,X: $o] :
( ( I2 != bot_bot_set_c )
=> ( ! [I3: c] :
( ( member_c @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Inf_Inf_o @ ( image_c_o @ F @ I2 ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_543_INF__eq__const,axiom,
! [I2: set_b,F: b > $o,X: $o] :
( ( I2 != bot_bot_set_b )
=> ( ! [I3: b] :
( ( member_b @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Inf_Inf_o @ ( image_b_o @ F @ I2 ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_544_INF__eq__const,axiom,
! [I2: set_Extended_ereal,F: extended_ereal > $o,X: $o] :
( ( I2 != bot_bo8367695208629047834_ereal )
=> ( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Inf_Inf_o @ ( image_951975095941678543real_o @ F @ I2 ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_545_INF__eq__const,axiom,
! [I2: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > $o,X: $o] :
( ( I2 != bot_bo4854962954004695426nnreal )
=> ( ! [I3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Inf_Inf_o @ ( image_3162942742313426073real_o @ F @ I2 ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_546_INF__eq__const,axiom,
! [I2: set_nat,F: nat > $o,X: $o] :
( ( I2 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ F @ I2 ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_547_INF__eq__const,axiom,
! [I2: set_o,F: $o > extended_ereal,X: extended_ereal] :
( ( I2 != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple3556804143462414037_ereal @ ( image_7729549296133164475_ereal @ F @ I2 ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_548_INF__eq__const,axiom,
! [I2: set_c,F: c > extended_ereal,X: extended_ereal] :
( ( I2 != bot_bot_set_c )
=> ( ! [I3: c] :
( ( member_c @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple3556804143462414037_ereal @ ( image_2233968868011006291_ereal @ F @ I2 ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_549_INF__eq__const,axiom,
! [I2: set_b,F: b > extended_ereal,X: extended_ereal] :
( ( I2 != bot_bot_set_b )
=> ( ! [I3: b] :
( ( member_b @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple3556804143462414037_ereal @ ( image_5319725110001000852_ereal @ F @ I2 ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_550_INF__eq__const,axiom,
! [I2: set_Extended_ereal,F: extended_ereal > extended_ereal,X: extended_ereal] :
( ( I2 != bot_bo8367695208629047834_ereal )
=> ( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F @ I2 ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_551_SUP__eq__const,axiom,
! [I2: set_o,F: $o > $o,X: $o] :
( ( I2 != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ I2 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_552_SUP__eq__const,axiom,
! [I2: set_c,F: c > $o,X: $o] :
( ( I2 != bot_bot_set_c )
=> ( ! [I3: c] :
( ( member_c @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_c_o @ F @ I2 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_553_SUP__eq__const,axiom,
! [I2: set_b,F: b > $o,X: $o] :
( ( I2 != bot_bot_set_b )
=> ( ! [I3: b] :
( ( member_b @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_b_o @ F @ I2 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_554_SUP__eq__const,axiom,
! [I2: set_Extended_ereal,F: extended_ereal > $o,X: $o] :
( ( I2 != bot_bo8367695208629047834_ereal )
=> ( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_951975095941678543real_o @ F @ I2 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_555_SUP__eq__const,axiom,
! [I2: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > $o,X: $o] :
( ( I2 != bot_bo4854962954004695426nnreal )
=> ( ! [I3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_3162942742313426073real_o @ F @ I2 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_556_SUP__eq__const,axiom,
! [I2: set_nat,F: nat > $o,X: $o] :
( ( I2 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ I2 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_557_SUP__eq__const,axiom,
! [I2: set_o,F: $o > extended_ereal,X: extended_ereal] :
( ( I2 != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple8415311339701865915_ereal @ ( image_7729549296133164475_ereal @ F @ I2 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_558_SUP__eq__const,axiom,
! [I2: set_c,F: c > extended_ereal,X: extended_ereal] :
( ( I2 != bot_bot_set_c )
=> ( ! [I3: c] :
( ( member_c @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple8415311339701865915_ereal @ ( image_2233968868011006291_ereal @ F @ I2 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_559_SUP__eq__const,axiom,
! [I2: set_b,F: b > extended_ereal,X: extended_ereal] :
( ( I2 != bot_bot_set_b )
=> ( ! [I3: b] :
( ( member_b @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple8415311339701865915_ereal @ ( image_5319725110001000852_ereal @ F @ I2 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_560_SUP__eq__const,axiom,
! [I2: set_Extended_ereal,F: extended_ereal > extended_ereal,X: extended_ereal] :
( ( I2 != bot_bo8367695208629047834_ereal )
=> ( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F @ I2 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_561_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_562_INT__E,axiom,
! [B: $o,B2: extended_ereal > set_o,A: set_Extended_ereal,A2: extended_ereal] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_2973972673614065839_set_o @ B2 @ A ) ) )
=> ( ~ ( member_o @ B @ ( B2 @ A2 ) )
=> ~ ( member2350847679896131959_ereal @ A2 @ A ) ) ) ).
% INT_E
thf(fact_563_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_564_INT__E,axiom,
! [B: $o,B2: extend8495563244428889912nnreal > set_o,A: set_Ex3793607809372303086nnreal,A2: extend8495563244428889912nnreal] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_4514577626304257913_set_o @ B2 @ A ) ) )
=> ( ~ ( member_o @ B @ ( B2 @ A2 ) )
=> ~ ( member7908768830364227535nnreal @ A2 @ A ) ) ) ).
% INT_E
thf(fact_565_INT__E,axiom,
! [B: $o,B2: c > set_o,A: set_c,A2: c] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_c_set_o @ B2 @ A ) ) )
=> ( ~ ( member_o @ B @ ( B2 @ A2 ) )
=> ~ ( member_c @ A2 @ A ) ) ) ).
% INT_E
thf(fact_566_INT__E,axiom,
! [B: $o,B2: b > set_o,A: set_b,A2: b] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_b_set_o @ B2 @ A ) ) )
=> ( ~ ( member_o @ B @ ( B2 @ A2 ) )
=> ~ ( member_b @ A2 @ A ) ) ) ).
% INT_E
thf(fact_567_INT__E,axiom,
! [B: extended_ereal,B2: $o > set_Extended_ereal,A: set_o,A2: $o] :
( ( member2350847679896131959_ereal @ B @ ( comple4418415374894819509_ereal @ ( image_6375117163256653723_ereal @ B2 @ A ) ) )
=> ( ~ ( member2350847679896131959_ereal @ B @ ( B2 @ A2 ) )
=> ~ ( member_o @ A2 @ A ) ) ) ).
% INT_E
thf(fact_568_INT__E,axiom,
! [B: extended_ereal,B2: extended_ereal > set_Extended_ereal,A: set_Extended_ereal,A2: extended_ereal] :
( ( member2350847679896131959_ereal @ B @ ( comple4418415374894819509_ereal @ ( image_5562094264469218789_ereal @ B2 @ A ) ) )
=> ( ~ ( member2350847679896131959_ereal @ B @ ( B2 @ A2 ) )
=> ~ ( member2350847679896131959_ereal @ A2 @ A ) ) ) ).
% INT_E
thf(fact_569_INT__E,axiom,
! [B: extended_ereal,B2: nat > set_Extended_ereal,A: set_nat,A2: nat] :
( ( member2350847679896131959_ereal @ B @ ( comple4418415374894819509_ereal @ ( image_305533323056406039_ereal @ B2 @ A ) ) )
=> ( ~ ( member2350847679896131959_ereal @ B @ ( B2 @ A2 ) )
=> ~ ( member_nat @ A2 @ A ) ) ) ).
% INT_E
thf(fact_570_INT__E,axiom,
! [B: extended_ereal,B2: extend8495563244428889912nnreal > set_Extended_ereal,A: set_Ex3793607809372303086nnreal,A2: extend8495563244428889912nnreal] :
( ( member2350847679896131959_ereal @ B @ ( comple4418415374894819509_ereal @ ( image_5929344197358196911_ereal @ B2 @ A ) ) )
=> ( ~ ( member2350847679896131959_ereal @ B @ ( B2 @ A2 ) )
=> ~ ( member7908768830364227535nnreal @ A2 @ A ) ) ) ).
% INT_E
thf(fact_571_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_572_INT__D,axiom,
! [B: $o,B2: extended_ereal > set_o,A: set_Extended_ereal,A2: extended_ereal] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_2973972673614065839_set_o @ B2 @ A ) ) )
=> ( ( member2350847679896131959_ereal @ A2 @ A )
=> ( member_o @ B @ ( B2 @ A2 ) ) ) ) ).
% INT_D
thf(fact_573_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_574_INT__D,axiom,
! [B: $o,B2: extend8495563244428889912nnreal > set_o,A: set_Ex3793607809372303086nnreal,A2: extend8495563244428889912nnreal] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_4514577626304257913_set_o @ B2 @ A ) ) )
=> ( ( member7908768830364227535nnreal @ A2 @ A )
=> ( member_o @ B @ ( B2 @ A2 ) ) ) ) ).
% INT_D
thf(fact_575_INT__D,axiom,
! [B: $o,B2: c > set_o,A: set_c,A2: c] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_c_set_o @ B2 @ A ) ) )
=> ( ( member_c @ A2 @ A )
=> ( member_o @ B @ ( B2 @ A2 ) ) ) ) ).
% INT_D
thf(fact_576_INT__D,axiom,
! [B: $o,B2: b > set_o,A: set_b,A2: b] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_b_set_o @ B2 @ A ) ) )
=> ( ( member_b @ A2 @ A )
=> ( member_o @ B @ ( B2 @ A2 ) ) ) ) ).
% INT_D
thf(fact_577_INT__D,axiom,
! [B: extended_ereal,B2: $o > set_Extended_ereal,A: set_o,A2: $o] :
( ( member2350847679896131959_ereal @ B @ ( comple4418415374894819509_ereal @ ( image_6375117163256653723_ereal @ B2 @ A ) ) )
=> ( ( member_o @ A2 @ A )
=> ( member2350847679896131959_ereal @ B @ ( B2 @ A2 ) ) ) ) ).
% INT_D
thf(fact_578_INT__D,axiom,
! [B: extended_ereal,B2: extended_ereal > set_Extended_ereal,A: set_Extended_ereal,A2: extended_ereal] :
( ( member2350847679896131959_ereal @ B @ ( comple4418415374894819509_ereal @ ( image_5562094264469218789_ereal @ B2 @ A ) ) )
=> ( ( member2350847679896131959_ereal @ A2 @ A )
=> ( member2350847679896131959_ereal @ B @ ( B2 @ A2 ) ) ) ) ).
% INT_D
thf(fact_579_INT__D,axiom,
! [B: extended_ereal,B2: nat > set_Extended_ereal,A: set_nat,A2: nat] :
( ( member2350847679896131959_ereal @ B @ ( comple4418415374894819509_ereal @ ( image_305533323056406039_ereal @ B2 @ A ) ) )
=> ( ( member_nat @ A2 @ A )
=> ( member2350847679896131959_ereal @ B @ ( B2 @ A2 ) ) ) ) ).
% INT_D
thf(fact_580_INT__D,axiom,
! [B: extended_ereal,B2: extend8495563244428889912nnreal > set_Extended_ereal,A: set_Ex3793607809372303086nnreal,A2: extend8495563244428889912nnreal] :
( ( member2350847679896131959_ereal @ B @ ( comple4418415374894819509_ereal @ ( image_5929344197358196911_ereal @ B2 @ A ) ) )
=> ( ( member7908768830364227535nnreal @ A2 @ A )
=> ( member2350847679896131959_ereal @ B @ ( B2 @ A2 ) ) ) ) ).
% INT_D
thf(fact_581_UN__E,axiom,
! [B: $o,B2: $o > set_o,A: set_o] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A ) ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ~ ( member_o @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_582_UN__E,axiom,
! [B: $o,B2: extended_ereal > set_o,A: set_Extended_ereal] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_2973972673614065839_set_o @ B2 @ A ) ) )
=> ~ ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
=> ~ ( member_o @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_583_UN__E,axiom,
! [B: $o,B2: nat > set_o,A: set_nat] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ A ) ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ~ ( member_o @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_584_UN__E,axiom,
! [B: $o,B2: extend8495563244428889912nnreal > set_o,A: set_Ex3793607809372303086nnreal] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_4514577626304257913_set_o @ B2 @ A ) ) )
=> ~ ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A )
=> ~ ( member_o @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_585_UN__E,axiom,
! [B: $o,B2: c > set_o,A: set_c] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_c_set_o @ B2 @ A ) ) )
=> ~ ! [X3: c] :
( ( member_c @ X3 @ A )
=> ~ ( member_o @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_586_UN__E,axiom,
! [B: $o,B2: b > set_o,A: set_b] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_b_set_o @ B2 @ A ) ) )
=> ~ ! [X3: b] :
( ( member_b @ X3 @ A )
=> ~ ( member_o @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_587_UN__E,axiom,
! [B: extended_ereal,B2: $o > set_Extended_ereal,A: set_o] :
( ( member2350847679896131959_ereal @ B @ ( comple4319282863272126363_ereal @ ( image_6375117163256653723_ereal @ B2 @ A ) ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ~ ( member2350847679896131959_ereal @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_588_UN__E,axiom,
! [B: extended_ereal,B2: extended_ereal > set_Extended_ereal,A: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ B @ ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ B2 @ A ) ) )
=> ~ ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
=> ~ ( member2350847679896131959_ereal @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_589_UN__E,axiom,
! [B: extended_ereal,B2: nat > set_Extended_ereal,A: set_nat] :
( ( member2350847679896131959_ereal @ B @ ( comple4319282863272126363_ereal @ ( image_305533323056406039_ereal @ B2 @ A ) ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ~ ( member2350847679896131959_ereal @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_590_UN__E,axiom,
! [B: extended_ereal,B2: extend8495563244428889912nnreal > set_Extended_ereal,A: set_Ex3793607809372303086nnreal] :
( ( member2350847679896131959_ereal @ B @ ( comple4319282863272126363_ereal @ ( image_5929344197358196911_ereal @ B2 @ A ) ) )
=> ~ ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A )
=> ~ ( member2350847679896131959_ereal @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_591_UNIV__def,axiom,
( top_top_set_b
= ( collect_b
@ ^ [X2: b] : $true ) ) ).
% UNIV_def
thf(fact_592_UNIV__def,axiom,
( top_top_set_c
= ( collect_c
@ ^ [X2: c] : $true ) ) ).
% UNIV_def
thf(fact_593_UNIV__def,axiom,
( top_to7994903218803871134nnreal
= ( collec6648975593938027277nnreal
@ ^ [X2: extend8495563244428889912nnreal] : $true ) ) ).
% UNIV_def
thf(fact_594_UNIV__def,axiom,
( top_top_set_o
= ( collect_o
@ ^ [X2: $o] : $true ) ) ).
% UNIV_def
thf(fact_595_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X2: nat] : $true ) ) ).
% UNIV_def
thf(fact_596_UNIV__def,axiom,
( top_to5683747375963461374_ereal
= ( collec5835592288176408249_ereal
@ ^ [X2: extended_ereal] : $true ) ) ).
% UNIV_def
thf(fact_597_empty__def,axiom,
( bot_bot_set_b
= ( collect_b
@ ^ [X2: b] : $false ) ) ).
% empty_def
thf(fact_598_empty__def,axiom,
( bot_bot_set_c
= ( collect_c
@ ^ [X2: c] : $false ) ) ).
% empty_def
thf(fact_599_empty__def,axiom,
( bot_bot_set_o
= ( collect_o
@ ^ [X2: $o] : $false ) ) ).
% empty_def
thf(fact_600_empty__def,axiom,
( bot_bo8367695208629047834_ereal
= ( collec5835592288176408249_ereal
@ ^ [X2: extended_ereal] : $false ) ) ).
% empty_def
thf(fact_601_empty__def,axiom,
( bot_bo4854962954004695426nnreal
= ( collec6648975593938027277nnreal
@ ^ [X2: extend8495563244428889912nnreal] : $false ) ) ).
% empty_def
thf(fact_602_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X2: nat] : $false ) ) ).
% empty_def
thf(fact_603_SUP__UNION,axiom,
! [F: b > $o,G: b > set_b,A: set_b] :
( ( complete_Sup_Sup_o @ ( image_b_o @ F @ ( comple2307003614231284044_set_b @ ( image_b_set_b @ G @ A ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_b_o
@ ^ [Y2: b] : ( complete_Sup_Sup_o @ ( image_b_o @ F @ ( G @ Y2 ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_604_SUP__UNION,axiom,
! [F: nat > extended_ereal,G: nat > set_nat,A: set_nat] :
( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A ) ) ) )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [Y2: nat] : ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F @ ( G @ Y2 ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_605_SUP__UNION,axiom,
! [F: nat > extended_ereal,G: extended_ereal > set_nat,A: set_Extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F @ ( comple7399068483239264473et_nat @ ( image_3090908713637162255et_nat @ G @ A ) ) ) )
= ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [Y2: extended_ereal] : ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F @ ( G @ Y2 ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_606_SUP__UNION,axiom,
! [F: nat > extended_ereal,G: b > set_nat,A: set_b] :
( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F @ ( comple7399068483239264473et_nat @ ( image_b_set_nat @ G @ A ) ) ) )
= ( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [Y2: b] : ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F @ ( G @ Y2 ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_607_SUP__UNION,axiom,
! [F: extended_ereal > extended_ereal,G: nat > set_Extended_ereal,A: set_nat] :
( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F @ ( comple4319282863272126363_ereal @ ( image_305533323056406039_ereal @ G @ A ) ) ) )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [Y2: nat] : ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F @ ( G @ Y2 ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_608_SUP__UNION,axiom,
! [F: extended_ereal > extended_ereal,G: extended_ereal > set_Extended_ereal,A: set_Extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F @ ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ G @ A ) ) ) )
= ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [Y2: extended_ereal] : ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F @ ( G @ Y2 ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_609_SUP__UNION,axiom,
! [F: extended_ereal > extended_ereal,G: b > set_Extended_ereal,A: set_b] :
( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F @ ( comple4319282863272126363_ereal @ ( image_8773349707370420084_ereal @ G @ A ) ) ) )
= ( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [Y2: b] : ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F @ ( G @ Y2 ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_610_SUP__UNION,axiom,
! [F: b > extended_ereal,G: nat > set_b,A: set_nat] :
( ( comple8415311339701865915_ereal @ ( image_5319725110001000852_ereal @ F @ ( comple2307003614231284044_set_b @ ( image_nat_set_b @ G @ A ) ) ) )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [Y2: nat] : ( comple8415311339701865915_ereal @ ( image_5319725110001000852_ereal @ F @ ( G @ Y2 ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_611_SUP__UNION,axiom,
! [F: b > extended_ereal,G: extended_ereal > set_b,A: set_Extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_5319725110001000852_ereal @ F @ ( comple2307003614231284044_set_b @ ( image_1981423240844246678_set_b @ G @ A ) ) ) )
= ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [Y2: extended_ereal] : ( comple8415311339701865915_ereal @ ( image_5319725110001000852_ereal @ F @ ( G @ Y2 ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_612_SUP__UNION,axiom,
! [F: b > extended_ereal,G: b > set_b,A: set_b] :
( ( comple8415311339701865915_ereal @ ( image_5319725110001000852_ereal @ F @ ( comple2307003614231284044_set_b @ ( image_b_set_b @ G @ A ) ) ) )
= ( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [Y2: b] : ( comple8415311339701865915_ereal @ ( image_5319725110001000852_ereal @ F @ ( G @ Y2 ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_613_Collect__ex__eq,axiom,
! [P: b > nat > $o] :
( ( collect_b
@ ^ [X2: b] :
? [X7: nat] : ( P @ X2 @ X7 ) )
= ( comple2307003614231284044_set_b
@ ( image_nat_set_b
@ ^ [Y2: nat] :
( collect_b
@ ^ [X2: b] : ( P @ X2 @ Y2 ) )
@ top_top_set_nat ) ) ) ).
% Collect_ex_eq
thf(fact_614_Collect__ex__eq,axiom,
! [P: c > nat > $o] :
( ( collect_c
@ ^ [X2: c] :
? [X7: nat] : ( P @ X2 @ X7 ) )
= ( comple2307003618534512845_set_c
@ ( image_nat_set_c
@ ^ [Y2: nat] :
( collect_c
@ ^ [X2: c] : ( P @ X2 @ Y2 ) )
@ top_top_set_nat ) ) ) ).
% Collect_ex_eq
thf(fact_615_Collect__ex__eq,axiom,
! [P: extended_ereal > nat > $o] :
( ( collec5835592288176408249_ereal
@ ^ [X2: extended_ereal] :
? [X7: nat] : ( P @ X2 @ X7 ) )
= ( comple4319282863272126363_ereal
@ ( image_305533323056406039_ereal
@ ^ [Y2: nat] :
( collec5835592288176408249_ereal
@ ^ [X2: extended_ereal] : ( P @ X2 @ Y2 ) )
@ top_top_set_nat ) ) ) ).
% Collect_ex_eq
thf(fact_616_Collect__ex__eq,axiom,
! [P: extend8495563244428889912nnreal > nat > $o] :
( ( collec6648975593938027277nnreal
@ ^ [X2: extend8495563244428889912nnreal] :
? [X7: nat] : ( P @ X2 @ X7 ) )
= ( comple4226387801268262977nnreal
@ ( image_3394822847079329989nnreal
@ ^ [Y2: nat] :
( collec6648975593938027277nnreal
@ ^ [X2: extend8495563244428889912nnreal] : ( P @ X2 @ Y2 ) )
@ top_top_set_nat ) ) ) ).
% Collect_ex_eq
thf(fact_617_Collect__ex__eq,axiom,
! [P: nat > nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
? [X7: nat] : ( P @ X2 @ X7 ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y2: nat] :
( collect_nat
@ ^ [X2: nat] : ( P @ X2 @ Y2 ) )
@ top_top_set_nat ) ) ) ).
% Collect_ex_eq
thf(fact_618_Collect__ex__eq,axiom,
! [P: $o > nat > $o] :
( ( collect_o
@ ^ [X2: $o] :
? [X7: nat] : ( P @ X2 @ X7 ) )
= ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [Y2: nat] :
( collect_o
@ ^ [X2: $o] : ( P @ X2 @ Y2 ) )
@ top_top_set_nat ) ) ) ).
% Collect_ex_eq
thf(fact_619_Collect__ex__eq,axiom,
! [P: b > extended_ereal > $o] :
( ( collect_b
@ ^ [X2: b] :
? [X7: extended_ereal] : ( P @ X2 @ X7 ) )
= ( comple2307003614231284044_set_b
@ ( image_1981423240844246678_set_b
@ ^ [Y2: extended_ereal] :
( collect_b
@ ^ [X2: b] : ( P @ X2 @ Y2 ) )
@ top_to5683747375963461374_ereal ) ) ) ).
% Collect_ex_eq
thf(fact_620_Collect__ex__eq,axiom,
! [P: c > extended_ereal > $o] :
( ( collect_c
@ ^ [X2: c] :
? [X7: extended_ereal] : ( P @ X2 @ X7 ) )
= ( comple2307003618534512845_set_c
@ ( image_1981423245147475479_set_c
@ ^ [Y2: extended_ereal] :
( collect_c
@ ^ [X2: c] : ( P @ X2 @ Y2 ) )
@ top_to5683747375963461374_ereal ) ) ) ).
% Collect_ex_eq
thf(fact_621_Collect__ex__eq,axiom,
! [P: extended_ereal > extended_ereal > $o] :
( ( collec5835592288176408249_ereal
@ ^ [X2: extended_ereal] :
? [X7: extended_ereal] : ( P @ X2 @ X7 ) )
= ( comple4319282863272126363_ereal
@ ( image_5562094264469218789_ereal
@ ^ [Y2: extended_ereal] :
( collec5835592288176408249_ereal
@ ^ [X2: extended_ereal] : ( P @ X2 @ Y2 ) )
@ top_to5683747375963461374_ereal ) ) ) ).
% Collect_ex_eq
thf(fact_622_Collect__ex__eq,axiom,
! [P: extend8495563244428889912nnreal > extended_ereal > $o] :
( ( collec6648975593938027277nnreal
@ ^ [X2: extend8495563244428889912nnreal] :
? [X7: extended_ereal] : ( P @ X2 @ X7 ) )
= ( comple4226387801268262977nnreal
@ ( image_6588766411312125047nnreal
@ ^ [Y2: extended_ereal] :
( collec6648975593938027277nnreal
@ ^ [X2: extend8495563244428889912nnreal] : ( P @ X2 @ Y2 ) )
@ top_to5683747375963461374_ereal ) ) ) ).
% Collect_ex_eq
thf(fact_623_INF__cong,axiom,
! [A: set_o,B2: set_o,C: $o > $o,D: $o > $o] :
( ( A = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ C @ A ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_624_INF__cong,axiom,
! [A: set_Extended_ereal,B2: set_Extended_ereal,C: extended_ereal > $o,D: extended_ereal > $o] :
( ( A = B2 )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_951975095941678543real_o @ C @ A ) )
= ( complete_Inf_Inf_o @ ( image_951975095941678543real_o @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_625_INF__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > $o,D: nat > $o] :
( ( A = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_nat_o @ C @ A ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_626_INF__cong,axiom,
! [A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,C: extend8495563244428889912nnreal > $o,D: extend8495563244428889912nnreal > $o] :
( ( A = B2 )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_3162942742313426073real_o @ C @ A ) )
= ( complete_Inf_Inf_o @ ( image_3162942742313426073real_o @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_627_INF__cong,axiom,
! [A: set_c,B2: set_c,C: c > $o,D: c > $o] :
( ( A = B2 )
=> ( ! [X3: c] :
( ( member_c @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_c_o @ C @ A ) )
= ( complete_Inf_Inf_o @ ( image_c_o @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_628_INF__cong,axiom,
! [A: set_b,B2: set_b,C: b > $o,D: b > $o] :
( ( A = B2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_b_o @ C @ A ) )
= ( complete_Inf_Inf_o @ ( image_b_o @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_629_INF__cong,axiom,
! [A: set_o,B2: set_o,C: $o > extended_ereal,D: $o > extended_ereal] :
( ( A = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple3556804143462414037_ereal @ ( image_7729549296133164475_ereal @ C @ A ) )
= ( comple3556804143462414037_ereal @ ( image_7729549296133164475_ereal @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_630_INF__cong,axiom,
! [A: set_Extended_ereal,B2: set_Extended_ereal,C: extended_ereal > extended_ereal,D: extended_ereal > extended_ereal] :
( ( A = B2 )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ C @ A ) )
= ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_631_INF__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > extended_ereal,D: nat > extended_ereal] :
( ( A = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ C @ A ) )
= ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_632_INF__cong,axiom,
! [A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,C: extend8495563244428889912nnreal > extended_ereal,D: extend8495563244428889912nnreal > extended_ereal] :
( ( A = B2 )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple3556804143462414037_ereal @ ( image_6393943237584228047_ereal @ C @ A ) )
= ( comple3556804143462414037_ereal @ ( image_6393943237584228047_ereal @ D @ B2 ) ) ) ) ) ).
% INF_cong
thf(fact_633_SUP__cong,axiom,
! [A: set_o,B2: set_o,C: $o > $o,D: $o > $o] :
( ( A = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ C @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_634_SUP__cong,axiom,
! [A: set_Extended_ereal,B2: set_Extended_ereal,C: extended_ereal > $o,D: extended_ereal > $o] :
( ( A = B2 )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_951975095941678543real_o @ C @ A ) )
= ( complete_Sup_Sup_o @ ( image_951975095941678543real_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_635_SUP__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > $o,D: nat > $o] :
( ( A = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ C @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_636_SUP__cong,axiom,
! [A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,C: extend8495563244428889912nnreal > $o,D: extend8495563244428889912nnreal > $o] :
( ( A = B2 )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_3162942742313426073real_o @ C @ A ) )
= ( complete_Sup_Sup_o @ ( image_3162942742313426073real_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_637_SUP__cong,axiom,
! [A: set_c,B2: set_c,C: c > $o,D: c > $o] :
( ( A = B2 )
=> ( ! [X3: c] :
( ( member_c @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_c_o @ C @ A ) )
= ( complete_Sup_Sup_o @ ( image_c_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_638_SUP__cong,axiom,
! [A: set_b,B2: set_b,C: b > $o,D: b > $o] :
( ( A = B2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_b_o @ C @ A ) )
= ( complete_Sup_Sup_o @ ( image_b_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_639_SUP__cong,axiom,
! [A: set_o,B2: set_o,C: $o > extended_ereal,D: $o > extended_ereal] :
( ( A = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_7729549296133164475_ereal @ C @ A ) )
= ( comple8415311339701865915_ereal @ ( image_7729549296133164475_ereal @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_640_SUP__cong,axiom,
! [A: set_Extended_ereal,B2: set_Extended_ereal,C: extended_ereal > extended_ereal,D: extended_ereal > extended_ereal] :
( ( A = B2 )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ C @ A ) )
= ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_641_SUP__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > extended_ereal,D: nat > extended_ereal] :
( ( A = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ C @ A ) )
= ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_642_SUP__cong,axiom,
! [A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,C: extend8495563244428889912nnreal > extended_ereal,D: extend8495563244428889912nnreal > extended_ereal] :
( ( A = B2 )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_6393943237584228047_ereal @ C @ A ) )
= ( comple8415311339701865915_ereal @ ( image_6393943237584228047_ereal @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_643_rangeI,axiom,
! [F: b > $o,X: b] : ( member_o @ ( F @ X ) @ ( image_b_o @ F @ top_top_set_b ) ) ).
% rangeI
thf(fact_644_rangeI,axiom,
! [F: b > extended_ereal,X: b] : ( member2350847679896131959_ereal @ ( F @ X ) @ ( image_5319725110001000852_ereal @ F @ top_top_set_b ) ) ).
% rangeI
thf(fact_645_rangeI,axiom,
! [F: c > c,X: c] : ( member_c @ ( F @ X ) @ ( image_c_c @ F @ top_top_set_c ) ) ).
% rangeI
thf(fact_646_rangeI,axiom,
! [F: b > c,X: b] : ( member_c @ ( F @ X ) @ ( image_b_c @ F @ top_top_set_b ) ) ).
% rangeI
thf(fact_647_rangeI,axiom,
! [F: c > b,X: c] : ( member_b @ ( F @ X ) @ ( image_c_b @ F @ top_top_set_c ) ) ).
% rangeI
thf(fact_648_rangeI,axiom,
! [F: b > b,X: b] : ( member_b @ ( F @ X ) @ ( image_b_b @ F @ top_top_set_b ) ) ).
% rangeI
thf(fact_649_rangeI,axiom,
! [F: nat > $o,X: nat] : ( member_o @ ( F @ X ) @ ( image_nat_o @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_650_rangeI,axiom,
! [F: nat > extended_ereal,X: nat] : ( member2350847679896131959_ereal @ ( F @ X ) @ ( image_4309273772856505399_ereal @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_651_rangeI,axiom,
! [F: nat > nat,X: nat] : ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_652_rangeI,axiom,
! [F: nat > extend8495563244428889912nnreal,X: nat] : ( member7908768830364227535nnreal @ ( F @ X ) @ ( image_8459861568512453903nnreal @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_653_range__eqI,axiom,
! [B: $o,F: b > $o,X: b] :
( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_b_o @ F @ top_top_set_b ) ) ) ).
% range_eqI
thf(fact_654_range__eqI,axiom,
! [B: extended_ereal,F: b > extended_ereal,X: b] :
( ( B
= ( F @ X ) )
=> ( member2350847679896131959_ereal @ B @ ( image_5319725110001000852_ereal @ F @ top_top_set_b ) ) ) ).
% range_eqI
thf(fact_655_range__eqI,axiom,
! [B: c,F: c > c,X: c] :
( ( B
= ( F @ X ) )
=> ( member_c @ B @ ( image_c_c @ F @ top_top_set_c ) ) ) ).
% range_eqI
thf(fact_656_range__eqI,axiom,
! [B: c,F: b > c,X: b] :
( ( B
= ( F @ X ) )
=> ( member_c @ B @ ( image_b_c @ F @ top_top_set_b ) ) ) ).
% range_eqI
thf(fact_657_range__eqI,axiom,
! [B: b,F: c > b,X: c] :
( ( B
= ( F @ X ) )
=> ( member_b @ B @ ( image_c_b @ F @ top_top_set_c ) ) ) ).
% range_eqI
thf(fact_658_range__eqI,axiom,
! [B: b,F: b > b,X: b] :
( ( B
= ( F @ X ) )
=> ( member_b @ B @ ( image_b_b @ F @ top_top_set_b ) ) ) ).
% range_eqI
thf(fact_659_range__eqI,axiom,
! [B: $o,F: nat > $o,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_nat_o @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_660_range__eqI,axiom,
! [B: extended_ereal,F: nat > extended_ereal,X: nat] :
( ( B
= ( F @ X ) )
=> ( member2350847679896131959_ereal @ B @ ( image_4309273772856505399_ereal @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_661_range__eqI,axiom,
! [B: nat,F: nat > nat,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_662_range__eqI,axiom,
! [B: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,X: nat] :
( ( B
= ( F @ X ) )
=> ( member7908768830364227535nnreal @ B @ ( image_8459861568512453903nnreal @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_663_INF__commute,axiom,
! [F: b > b > $o,B2: set_b,A: set_b] :
( ( complete_Inf_Inf_o
@ ( image_b_o
@ ^ [I: b] : ( complete_Inf_Inf_o @ ( image_b_o @ ( F @ I ) @ B2 ) )
@ A ) )
= ( complete_Inf_Inf_o
@ ( image_b_o
@ ^ [J: b] :
( complete_Inf_Inf_o
@ ( image_b_o
@ ^ [I: b] : ( F @ I @ J )
@ A ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_664_INF__commute,axiom,
! [F: nat > nat > extended_ereal,B2: set_nat,A: set_nat] :
( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I: nat] : ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ ( F @ I ) @ B2 ) )
@ A ) )
= ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [J: nat] :
( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I: nat] : ( F @ I @ J )
@ A ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_665_INF__commute,axiom,
! [F: nat > extended_ereal > extended_ereal,B2: set_Extended_ereal,A: set_nat] :
( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I: nat] : ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ ( F @ I ) @ B2 ) )
@ A ) )
= ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [J: extended_ereal] :
( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I: nat] : ( F @ I @ J )
@ A ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_666_INF__commute,axiom,
! [F: nat > b > extended_ereal,B2: set_b,A: set_nat] :
( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I: nat] : ( comple3556804143462414037_ereal @ ( image_5319725110001000852_ereal @ ( F @ I ) @ B2 ) )
@ A ) )
= ( comple3556804143462414037_ereal
@ ( image_5319725110001000852_ereal
@ ^ [J: b] :
( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I: nat] : ( F @ I @ J )
@ A ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_667_INF__commute,axiom,
! [F: extended_ereal > nat > extended_ereal,B2: set_nat,A: set_Extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I: extended_ereal] : ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ ( F @ I ) @ B2 ) )
@ A ) )
= ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [J: nat] :
( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I: extended_ereal] : ( F @ I @ J )
@ A ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_668_INF__commute,axiom,
! [F: extended_ereal > extended_ereal > extended_ereal,B2: set_Extended_ereal,A: set_Extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I: extended_ereal] : ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ ( F @ I ) @ B2 ) )
@ A ) )
= ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [J: extended_ereal] :
( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I: extended_ereal] : ( F @ I @ J )
@ A ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_669_INF__commute,axiom,
! [F: extended_ereal > b > extended_ereal,B2: set_b,A: set_Extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I: extended_ereal] : ( comple3556804143462414037_ereal @ ( image_5319725110001000852_ereal @ ( F @ I ) @ B2 ) )
@ A ) )
= ( comple3556804143462414037_ereal
@ ( image_5319725110001000852_ereal
@ ^ [J: b] :
( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I: extended_ereal] : ( F @ I @ J )
@ A ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_670_INF__commute,axiom,
! [F: b > nat > extended_ereal,B2: set_nat,A: set_b] :
( ( comple3556804143462414037_ereal
@ ( image_5319725110001000852_ereal
@ ^ [I: b] : ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ ( F @ I ) @ B2 ) )
@ A ) )
= ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [J: nat] :
( comple3556804143462414037_ereal
@ ( image_5319725110001000852_ereal
@ ^ [I: b] : ( F @ I @ J )
@ A ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_671_INF__commute,axiom,
! [F: b > extended_ereal > extended_ereal,B2: set_Extended_ereal,A: set_b] :
( ( comple3556804143462414037_ereal
@ ( image_5319725110001000852_ereal
@ ^ [I: b] : ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ ( F @ I ) @ B2 ) )
@ A ) )
= ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [J: extended_ereal] :
( comple3556804143462414037_ereal
@ ( image_5319725110001000852_ereal
@ ^ [I: b] : ( F @ I @ J )
@ A ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_672_INF__commute,axiom,
! [F: b > b > extended_ereal,B2: set_b,A: set_b] :
( ( comple3556804143462414037_ereal
@ ( image_5319725110001000852_ereal
@ ^ [I: b] : ( comple3556804143462414037_ereal @ ( image_5319725110001000852_ereal @ ( F @ I ) @ B2 ) )
@ A ) )
= ( comple3556804143462414037_ereal
@ ( image_5319725110001000852_ereal
@ ^ [J: b] :
( comple3556804143462414037_ereal
@ ( image_5319725110001000852_ereal
@ ^ [I: b] : ( F @ I @ J )
@ A ) )
@ B2 ) ) ) ).
% INF_commute
thf(fact_673_SUP__commute,axiom,
! [F: b > b > $o,B2: set_b,A: set_b] :
( ( complete_Sup_Sup_o
@ ( image_b_o
@ ^ [I: b] : ( complete_Sup_Sup_o @ ( image_b_o @ ( F @ I ) @ B2 ) )
@ A ) )
= ( complete_Sup_Sup_o
@ ( image_b_o
@ ^ [J: b] :
( complete_Sup_Sup_o
@ ( image_b_o
@ ^ [I: b] : ( F @ I @ J )
@ A ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_674_SUP__commute,axiom,
! [F: nat > nat > extended_ereal,B2: set_nat,A: set_nat] :
( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I: nat] : ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ ( F @ I ) @ B2 ) )
@ A ) )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [J: nat] :
( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I: nat] : ( F @ I @ J )
@ A ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_675_SUP__commute,axiom,
! [F: nat > extended_ereal > extended_ereal,B2: set_Extended_ereal,A: set_nat] :
( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I: nat] : ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ ( F @ I ) @ B2 ) )
@ A ) )
= ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [J: extended_ereal] :
( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I: nat] : ( F @ I @ J )
@ A ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_676_SUP__commute,axiom,
! [F: nat > b > extended_ereal,B2: set_b,A: set_nat] :
( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I: nat] : ( comple8415311339701865915_ereal @ ( image_5319725110001000852_ereal @ ( F @ I ) @ B2 ) )
@ A ) )
= ( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [J: b] :
( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I: nat] : ( F @ I @ J )
@ A ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_677_SUP__commute,axiom,
! [F: extended_ereal > nat > extended_ereal,B2: set_nat,A: set_Extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I: extended_ereal] : ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ ( F @ I ) @ B2 ) )
@ A ) )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [J: nat] :
( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I: extended_ereal] : ( F @ I @ J )
@ A ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_678_SUP__commute,axiom,
! [F: extended_ereal > extended_ereal > extended_ereal,B2: set_Extended_ereal,A: set_Extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I: extended_ereal] : ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ ( F @ I ) @ B2 ) )
@ A ) )
= ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [J: extended_ereal] :
( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I: extended_ereal] : ( F @ I @ J )
@ A ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_679_SUP__commute,axiom,
! [F: extended_ereal > b > extended_ereal,B2: set_b,A: set_Extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I: extended_ereal] : ( comple8415311339701865915_ereal @ ( image_5319725110001000852_ereal @ ( F @ I ) @ B2 ) )
@ A ) )
= ( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [J: b] :
( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I: extended_ereal] : ( F @ I @ J )
@ A ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_680_SUP__commute,axiom,
! [F: b > nat > extended_ereal,B2: set_nat,A: set_b] :
( ( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [I: b] : ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ ( F @ I ) @ B2 ) )
@ A ) )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [J: nat] :
( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [I: b] : ( F @ I @ J )
@ A ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_681_SUP__commute,axiom,
! [F: b > extended_ereal > extended_ereal,B2: set_Extended_ereal,A: set_b] :
( ( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [I: b] : ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ ( F @ I ) @ B2 ) )
@ A ) )
= ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [J: extended_ereal] :
( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [I: b] : ( F @ I @ J )
@ A ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_682_SUP__commute,axiom,
! [F: b > b > extended_ereal,B2: set_b,A: set_b] :
( ( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [I: b] : ( comple8415311339701865915_ereal @ ( image_5319725110001000852_ereal @ ( F @ I ) @ B2 ) )
@ A ) )
= ( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [J: b] :
( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [I: b] : ( F @ I @ J )
@ A ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_683_ccINF__empty,axiom,
! [F: b > $o] :
( ( complete_Inf_Inf_o @ ( image_b_o @ F @ bot_bot_set_b ) )
= top_top_o ) ).
% ccINF_empty
thf(fact_684_ccINF__empty,axiom,
! [F: extended_ereal > $o] :
( ( complete_Inf_Inf_o @ ( image_951975095941678543real_o @ F @ bot_bo8367695208629047834_ereal ) )
= top_top_o ) ).
% ccINF_empty
thf(fact_685_ccINF__empty,axiom,
! [F: extend8495563244428889912nnreal > $o] :
( ( complete_Inf_Inf_o @ ( image_3162942742313426073real_o @ F @ bot_bo4854962954004695426nnreal ) )
= top_top_o ) ).
% ccINF_empty
thf(fact_686_ccINF__empty,axiom,
! [F: nat > $o] :
( ( complete_Inf_Inf_o @ ( image_nat_o @ F @ bot_bot_set_nat ) )
= top_top_o ) ).
% ccINF_empty
thf(fact_687_ccINF__empty,axiom,
! [F: b > extended_ereal] :
( ( comple3556804143462414037_ereal @ ( image_5319725110001000852_ereal @ F @ bot_bot_set_b ) )
= top_to6662034908053899550_ereal ) ).
% ccINF_empty
thf(fact_688_ccINF__empty,axiom,
! [F: extended_ereal > extended_ereal] :
( ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F @ bot_bo8367695208629047834_ereal ) )
= top_to6662034908053899550_ereal ) ).
% ccINF_empty
thf(fact_689_ccINF__empty,axiom,
! [F: extend8495563244428889912nnreal > extended_ereal] :
( ( comple3556804143462414037_ereal @ ( image_6393943237584228047_ereal @ F @ bot_bo4854962954004695426nnreal ) )
= top_to6662034908053899550_ereal ) ).
% ccINF_empty
thf(fact_690_ccINF__empty,axiom,
! [F: nat > extended_ereal] :
( ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F @ bot_bot_set_nat ) )
= top_to6662034908053899550_ereal ) ).
% ccINF_empty
thf(fact_691_ccINF__empty,axiom,
! [F: extended_ereal > extend8495563244428889912nnreal] :
( ( comple7330758040695736817nnreal @ ( image_8614087454967683265nnreal @ F @ bot_bo8367695208629047834_ereal ) )
= top_to1496364449551166952nnreal ) ).
% ccINF_empty
thf(fact_692_ccINF__empty,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( comple7330758040695736817nnreal @ ( image_8394674774369097847nnreal @ F @ bot_bo4854962954004695426nnreal ) )
= top_to1496364449551166952nnreal ) ).
% ccINF_empty
thf(fact_693_ccSUP__empty,axiom,
! [F: b > $o] :
( ( complete_Sup_Sup_o @ ( image_b_o @ F @ bot_bot_set_b ) )
= bot_bot_o ) ).
% ccSUP_empty
thf(fact_694_ccSUP__empty,axiom,
! [F: extended_ereal > $o] :
( ( complete_Sup_Sup_o @ ( image_951975095941678543real_o @ F @ bot_bo8367695208629047834_ereal ) )
= bot_bot_o ) ).
% ccSUP_empty
thf(fact_695_ccSUP__empty,axiom,
! [F: extend8495563244428889912nnreal > $o] :
( ( complete_Sup_Sup_o @ ( image_3162942742313426073real_o @ F @ bot_bo4854962954004695426nnreal ) )
= bot_bot_o ) ).
% ccSUP_empty
thf(fact_696_ccSUP__empty,axiom,
! [F: nat > $o] :
( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ bot_bot_set_nat ) )
= bot_bot_o ) ).
% ccSUP_empty
thf(fact_697_ccSUP__empty,axiom,
! [F: b > extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_5319725110001000852_ereal @ F @ bot_bot_set_b ) )
= bot_bo2710585358178759738_ereal ) ).
% ccSUP_empty
thf(fact_698_ccSUP__empty,axiom,
! [F: extended_ereal > extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F @ bot_bo8367695208629047834_ereal ) )
= bot_bo2710585358178759738_ereal ) ).
% ccSUP_empty
thf(fact_699_ccSUP__empty,axiom,
! [F: extend8495563244428889912nnreal > extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_6393943237584228047_ereal @ F @ bot_bo4854962954004695426nnreal ) )
= bot_bo2710585358178759738_ereal ) ).
% ccSUP_empty
thf(fact_700_ccSUP__empty,axiom,
! [F: nat > extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F @ bot_bot_set_nat ) )
= bot_bo2710585358178759738_ereal ) ).
% ccSUP_empty
thf(fact_701_ccSUP__empty,axiom,
! [F: extended_ereal > extend8495563244428889912nnreal] :
( ( comple6814414086264997003nnreal @ ( image_8614087454967683265nnreal @ F @ bot_bo8367695208629047834_ereal ) )
= bot_bo841427958541957580nnreal ) ).
% ccSUP_empty
thf(fact_702_ccSUP__empty,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( comple6814414086264997003nnreal @ ( image_8394674774369097847nnreal @ F @ bot_bo4854962954004695426nnreal ) )
= bot_bo841427958541957580nnreal ) ).
% ccSUP_empty
thf(fact_703_cINF__const,axiom,
! [A: set_b,C2: $o] :
( ( A != bot_bot_set_b )
=> ( ( complete_Inf_Inf_o
@ ( image_b_o
@ ^ [X2: b] : C2
@ A ) )
= C2 ) ) ).
% cINF_const
thf(fact_704_cINF__const,axiom,
! [A: set_Extended_ereal,C2: $o] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( complete_Inf_Inf_o
@ ( image_951975095941678543real_o
@ ^ [X2: extended_ereal] : C2
@ A ) )
= C2 ) ) ).
% cINF_const
thf(fact_705_cINF__const,axiom,
! [A: set_Ex3793607809372303086nnreal,C2: $o] :
( ( A != bot_bo4854962954004695426nnreal )
=> ( ( complete_Inf_Inf_o
@ ( image_3162942742313426073real_o
@ ^ [X2: extend8495563244428889912nnreal] : C2
@ A ) )
= C2 ) ) ).
% cINF_const
thf(fact_706_cINF__const,axiom,
! [A: set_nat,C2: $o] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [X2: nat] : C2
@ A ) )
= C2 ) ) ).
% cINF_const
thf(fact_707_cINF__const,axiom,
! [A: set_b,C2: extended_ereal] :
( ( A != bot_bot_set_b )
=> ( ( comple3556804143462414037_ereal
@ ( image_5319725110001000852_ereal
@ ^ [X2: b] : C2
@ A ) )
= C2 ) ) ).
% cINF_const
thf(fact_708_cINF__const,axiom,
! [A: set_Extended_ereal,C2: extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X2: extended_ereal] : C2
@ A ) )
= C2 ) ) ).
% cINF_const
thf(fact_709_cINF__const,axiom,
! [A: set_Ex3793607809372303086nnreal,C2: extended_ereal] :
( ( A != bot_bo4854962954004695426nnreal )
=> ( ( comple3556804143462414037_ereal
@ ( image_6393943237584228047_ereal
@ ^ [X2: extend8495563244428889912nnreal] : C2
@ A ) )
= C2 ) ) ).
% cINF_const
thf(fact_710_cINF__const,axiom,
! [A: set_nat,C2: extended_ereal] :
( ( A != bot_bot_set_nat )
=> ( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X2: nat] : C2
@ A ) )
= C2 ) ) ).
% cINF_const
thf(fact_711_cINF__const,axiom,
! [A: set_Extended_ereal,C2: extend8495563244428889912nnreal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple7330758040695736817nnreal
@ ( image_8614087454967683265nnreal
@ ^ [X2: extended_ereal] : C2
@ A ) )
= C2 ) ) ).
% cINF_const
thf(fact_712_cINF__const,axiom,
! [A: set_Ex3793607809372303086nnreal,C2: extend8495563244428889912nnreal] :
( ( A != bot_bo4854962954004695426nnreal )
=> ( ( comple7330758040695736817nnreal
@ ( image_8394674774369097847nnreal
@ ^ [X2: extend8495563244428889912nnreal] : C2
@ A ) )
= C2 ) ) ).
% cINF_const
thf(fact_713_ccINF__const,axiom,
! [A: set_b,F: $o] :
( ( A != bot_bot_set_b )
=> ( ( complete_Inf_Inf_o
@ ( image_b_o
@ ^ [I: b] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_714_ccINF__const,axiom,
! [A: set_Extended_ereal,F: $o] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( complete_Inf_Inf_o
@ ( image_951975095941678543real_o
@ ^ [I: extended_ereal] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_715_ccINF__const,axiom,
! [A: set_Ex3793607809372303086nnreal,F: $o] :
( ( A != bot_bo4854962954004695426nnreal )
=> ( ( complete_Inf_Inf_o
@ ( image_3162942742313426073real_o
@ ^ [I: extend8495563244428889912nnreal] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_716_ccINF__const,axiom,
! [A: set_nat,F: $o] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [I: nat] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_717_ccINF__const,axiom,
! [A: set_b,F: extended_ereal] :
( ( A != bot_bot_set_b )
=> ( ( comple3556804143462414037_ereal
@ ( image_5319725110001000852_ereal
@ ^ [I: b] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_718_ccINF__const,axiom,
! [A: set_Extended_ereal,F: extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I: extended_ereal] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_719_ccINF__const,axiom,
! [A: set_Ex3793607809372303086nnreal,F: extended_ereal] :
( ( A != bot_bo4854962954004695426nnreal )
=> ( ( comple3556804143462414037_ereal
@ ( image_6393943237584228047_ereal
@ ^ [I: extend8495563244428889912nnreal] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_720_ccINF__const,axiom,
! [A: set_nat,F: extended_ereal] :
( ( A != bot_bot_set_nat )
=> ( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I: nat] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_721_ccINF__const,axiom,
! [A: set_Extended_ereal,F: extend8495563244428889912nnreal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple7330758040695736817nnreal
@ ( image_8614087454967683265nnreal
@ ^ [I: extended_ereal] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_722_ccINF__const,axiom,
! [A: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal] :
( ( A != bot_bo4854962954004695426nnreal )
=> ( ( comple7330758040695736817nnreal
@ ( image_8394674774369097847nnreal
@ ^ [I: extend8495563244428889912nnreal] : F
@ A ) )
= F ) ) ).
% ccINF_const
thf(fact_723_ccINF__top,axiom,
! [A: set_b] :
( ( complete_Inf_Inf_o
@ ( image_b_o
@ ^ [X2: b] : top_top_o
@ A ) )
= top_top_o ) ).
% ccINF_top
thf(fact_724_ccINF__top,axiom,
! [A: set_nat] :
( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X2: nat] : top_to6662034908053899550_ereal
@ A ) )
= top_to6662034908053899550_ereal ) ).
% ccINF_top
thf(fact_725_ccINF__top,axiom,
! [A: set_Extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X2: extended_ereal] : top_to6662034908053899550_ereal
@ A ) )
= top_to6662034908053899550_ereal ) ).
% ccINF_top
thf(fact_726_ccINF__top,axiom,
! [A: set_b] :
( ( comple3556804143462414037_ereal
@ ( image_5319725110001000852_ereal
@ ^ [X2: b] : top_to6662034908053899550_ereal
@ A ) )
= top_to6662034908053899550_ereal ) ).
% ccINF_top
thf(fact_727_ccINF__top,axiom,
! [A: set_nat] :
( ( comple7330758040695736817nnreal
@ ( image_8459861568512453903nnreal
@ ^ [X2: nat] : top_to1496364449551166952nnreal
@ A ) )
= top_to1496364449551166952nnreal ) ).
% ccINF_top
thf(fact_728_ccSUP__const,axiom,
! [A: set_b,F: $o] :
( ( A != bot_bot_set_b )
=> ( ( complete_Sup_Sup_o
@ ( image_b_o
@ ^ [I: b] : F
@ A ) )
= F ) ) ).
% ccSUP_const
thf(fact_729_ccSUP__const,axiom,
! [A: set_Extended_ereal,F: $o] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( complete_Sup_Sup_o
@ ( image_951975095941678543real_o
@ ^ [I: extended_ereal] : F
@ A ) )
= F ) ) ).
% ccSUP_const
thf(fact_730_ccSUP__const,axiom,
! [A: set_Ex3793607809372303086nnreal,F: $o] :
( ( A != bot_bo4854962954004695426nnreal )
=> ( ( complete_Sup_Sup_o
@ ( image_3162942742313426073real_o
@ ^ [I: extend8495563244428889912nnreal] : F
@ A ) )
= F ) ) ).
% ccSUP_const
thf(fact_731_ccSUP__const,axiom,
! [A: set_nat,F: $o] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [I: nat] : F
@ A ) )
= F ) ) ).
% ccSUP_const
thf(fact_732_ccSUP__const,axiom,
! [A: set_b,F: extended_ereal] :
( ( A != bot_bot_set_b )
=> ( ( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [I: b] : F
@ A ) )
= F ) ) ).
% ccSUP_const
thf(fact_733_ccSUP__const,axiom,
! [A: set_Extended_ereal,F: extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I: extended_ereal] : F
@ A ) )
= F ) ) ).
% ccSUP_const
thf(fact_734_ccSUP__const,axiom,
! [A: set_Ex3793607809372303086nnreal,F: extended_ereal] :
( ( A != bot_bo4854962954004695426nnreal )
=> ( ( comple8415311339701865915_ereal
@ ( image_6393943237584228047_ereal
@ ^ [I: extend8495563244428889912nnreal] : F
@ A ) )
= F ) ) ).
% ccSUP_const
thf(fact_735_ccSUP__const,axiom,
! [A: set_nat,F: extended_ereal] :
( ( A != bot_bot_set_nat )
=> ( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I: nat] : F
@ A ) )
= F ) ) ).
% ccSUP_const
thf(fact_736_ccSUP__const,axiom,
! [A: set_Extended_ereal,F: extend8495563244428889912nnreal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple6814414086264997003nnreal
@ ( image_8614087454967683265nnreal
@ ^ [I: extended_ereal] : F
@ A ) )
= F ) ) ).
% ccSUP_const
thf(fact_737_ccSUP__const,axiom,
! [A: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal] :
( ( A != bot_bo4854962954004695426nnreal )
=> ( ( comple6814414086264997003nnreal
@ ( image_8394674774369097847nnreal
@ ^ [I: extend8495563244428889912nnreal] : F
@ A ) )
= F ) ) ).
% ccSUP_const
thf(fact_738_cSUP__const,axiom,
! [A: set_b,C2: $o] :
( ( A != bot_bot_set_b )
=> ( ( complete_Sup_Sup_o
@ ( image_b_o
@ ^ [X2: b] : C2
@ A ) )
= C2 ) ) ).
% cSUP_const
thf(fact_739_cSUP__const,axiom,
! [A: set_Extended_ereal,C2: $o] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( complete_Sup_Sup_o
@ ( image_951975095941678543real_o
@ ^ [X2: extended_ereal] : C2
@ A ) )
= C2 ) ) ).
% cSUP_const
thf(fact_740_cSUP__const,axiom,
! [A: set_Ex3793607809372303086nnreal,C2: $o] :
( ( A != bot_bo4854962954004695426nnreal )
=> ( ( complete_Sup_Sup_o
@ ( image_3162942742313426073real_o
@ ^ [X2: extend8495563244428889912nnreal] : C2
@ A ) )
= C2 ) ) ).
% cSUP_const
thf(fact_741_cSUP__const,axiom,
! [A: set_nat,C2: $o] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [X2: nat] : C2
@ A ) )
= C2 ) ) ).
% cSUP_const
thf(fact_742_cSUP__const,axiom,
! [A: set_b,C2: extended_ereal] :
( ( A != bot_bot_set_b )
=> ( ( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [X2: b] : C2
@ A ) )
= C2 ) ) ).
% cSUP_const
thf(fact_743_cSUP__const,axiom,
! [A: set_Extended_ereal,C2: extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X2: extended_ereal] : C2
@ A ) )
= C2 ) ) ).
% cSUP_const
thf(fact_744_cSUP__const,axiom,
! [A: set_Ex3793607809372303086nnreal,C2: extended_ereal] :
( ( A != bot_bo4854962954004695426nnreal )
=> ( ( comple8415311339701865915_ereal
@ ( image_6393943237584228047_ereal
@ ^ [X2: extend8495563244428889912nnreal] : C2
@ A ) )
= C2 ) ) ).
% cSUP_const
thf(fact_745_cSUP__const,axiom,
! [A: set_nat,C2: extended_ereal] :
( ( A != bot_bot_set_nat )
=> ( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X2: nat] : C2
@ A ) )
= C2 ) ) ).
% cSUP_const
thf(fact_746_cSUP__const,axiom,
! [A: set_Extended_ereal,C2: extend8495563244428889912nnreal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple6814414086264997003nnreal
@ ( image_8614087454967683265nnreal
@ ^ [X2: extended_ereal] : C2
@ A ) )
= C2 ) ) ).
% cSUP_const
thf(fact_747_cSUP__const,axiom,
! [A: set_Ex3793607809372303086nnreal,C2: extend8495563244428889912nnreal] :
( ( A != bot_bo4854962954004695426nnreal )
=> ( ( comple6814414086264997003nnreal
@ ( image_8394674774369097847nnreal
@ ^ [X2: extend8495563244428889912nnreal] : C2
@ A ) )
= C2 ) ) ).
% cSUP_const
thf(fact_748_ccSUP__bot,axiom,
! [A: set_b] :
( ( complete_Sup_Sup_o
@ ( image_b_o
@ ^ [X2: b] : bot_bot_o
@ A ) )
= bot_bot_o ) ).
% ccSUP_bot
thf(fact_749_ccSUP__bot,axiom,
! [A: set_nat] :
( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X2: nat] : bot_bo2710585358178759738_ereal
@ A ) )
= bot_bo2710585358178759738_ereal ) ).
% ccSUP_bot
thf(fact_750_ccSUP__bot,axiom,
! [A: set_Extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X2: extended_ereal] : bot_bo2710585358178759738_ereal
@ A ) )
= bot_bo2710585358178759738_ereal ) ).
% ccSUP_bot
thf(fact_751_ccSUP__bot,axiom,
! [A: set_b] :
( ( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [X2: b] : bot_bo2710585358178759738_ereal
@ A ) )
= bot_bo2710585358178759738_ereal ) ).
% ccSUP_bot
thf(fact_752_ccSUP__bot,axiom,
! [A: set_nat] :
( ( comple6814414086264997003nnreal
@ ( image_8459861568512453903nnreal
@ ^ [X2: nat] : bot_bo841427958541957580nnreal
@ A ) )
= bot_bo841427958541957580nnreal ) ).
% ccSUP_bot
thf(fact_753_ccInf__empty,axiom,
( ( comple7806235888213564991et_nat @ bot_bot_set_set_nat )
= top_top_set_nat ) ).
% ccInf_empty
thf(fact_754_ccInf__empty,axiom,
( ( comple4418415374894819509_ereal @ bot_bo7400643019497942010_ereal )
= top_to5683747375963461374_ereal ) ).
% ccInf_empty
thf(fact_755_ccInf__empty,axiom,
( ( complete_Inf_Inf_o @ bot_bot_set_o )
= top_top_o ) ).
% ccInf_empty
thf(fact_756_ccInf__empty,axiom,
( ( comple3556804143462414037_ereal @ bot_bo8367695208629047834_ereal )
= top_to6662034908053899550_ereal ) ).
% ccInf_empty
thf(fact_757_ccInf__empty,axiom,
( ( comple7330758040695736817nnreal @ bot_bo4854962954004695426nnreal )
= top_to1496364449551166952nnreal ) ).
% ccInf_empty
thf(fact_758_ccSup__empty,axiom,
( ( comple4319282863272126363_ereal @ bot_bo7400643019497942010_ereal )
= bot_bo8367695208629047834_ereal ) ).
% ccSup_empty
thf(fact_759_ccSup__empty,axiom,
( ( comple4226387801268262977nnreal @ bot_bo2988155216863113784nnreal )
= bot_bo4854962954004695426nnreal ) ).
% ccSup_empty
thf(fact_760_ccSup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% ccSup_empty
thf(fact_761_ccSup__empty,axiom,
( ( complete_Sup_Sup_o @ bot_bot_set_o )
= bot_bot_o ) ).
% ccSup_empty
thf(fact_762_ccSup__empty,axiom,
( ( comple8415311339701865915_ereal @ bot_bo8367695208629047834_ereal )
= bot_bo2710585358178759738_ereal ) ).
% ccSup_empty
thf(fact_763_ccSup__empty,axiom,
( ( comple6814414086264997003nnreal @ bot_bo4854962954004695426nnreal )
= bot_bo841427958541957580nnreal ) ).
% ccSup_empty
thf(fact_764_INF__SUP,axiom,
! [P: b > b > $o] :
( ( complete_Inf_Inf_o
@ ( image_b_o
@ ^ [Y2: b] :
( complete_Sup_Sup_o
@ ( image_b_o
@ ^ [X2: b] : ( P @ X2 @ Y2 )
@ top_top_set_b ) )
@ top_top_set_b ) )
= ( complete_Sup_Sup_o
@ ( image_b_b_o
@ ^ [F2: b > b] :
( complete_Inf_Inf_o
@ ( image_b_o
@ ^ [X2: b] : ( P @ ( F2 @ X2 ) @ X2 )
@ top_top_set_b ) )
@ top_top_set_b_b ) ) ) ).
% INF_SUP
thf(fact_765_INF__SUP,axiom,
! [P: b > nat > $o] :
( ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [Y2: nat] :
( complete_Sup_Sup_o
@ ( image_b_o
@ ^ [X2: b] : ( P @ X2 @ Y2 )
@ top_top_set_b ) )
@ top_top_set_nat ) )
= ( complete_Sup_Sup_o
@ ( image_nat_b_o
@ ^ [F2: nat > b] :
( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [X2: nat] : ( P @ ( F2 @ X2 ) @ X2 )
@ top_top_set_nat ) )
@ top_top_set_nat_b ) ) ) ).
% INF_SUP
thf(fact_766_INF__SUP,axiom,
! [P: b > extended_ereal > $o] :
( ( complete_Inf_Inf_o
@ ( image_951975095941678543real_o
@ ^ [Y2: extended_ereal] :
( complete_Sup_Sup_o
@ ( image_b_o
@ ^ [X2: b] : ( P @ X2 @ Y2 )
@ top_top_set_b ) )
@ top_to5683747375963461374_ereal ) )
= ( complete_Sup_Sup_o
@ ( image_438845630295331515al_b_o
@ ^ [F2: extended_ereal > b] :
( complete_Inf_Inf_o
@ ( image_951975095941678543real_o
@ ^ [X2: extended_ereal] : ( P @ ( F2 @ X2 ) @ X2 )
@ top_to5683747375963461374_ereal ) )
@ top_to5435482696333698940real_b ) ) ) ).
% INF_SUP
thf(fact_767_INF__SUP,axiom,
! [P: nat > b > $o] :
( ( complete_Inf_Inf_o
@ ( image_b_o
@ ^ [Y2: b] :
( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [X2: nat] : ( P @ X2 @ Y2 )
@ top_top_set_nat ) )
@ top_top_set_b ) )
= ( complete_Sup_Sup_o
@ ( image_b_nat_o
@ ^ [F2: b > nat] :
( complete_Inf_Inf_o
@ ( image_b_o
@ ^ [X2: b] : ( P @ ( F2 @ X2 ) @ X2 )
@ top_top_set_b ) )
@ top_top_set_b_nat ) ) ) ).
% INF_SUP
thf(fact_768_INF__SUP,axiom,
! [P: nat > nat > $o] :
( ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [Y2: nat] :
( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [X2: nat] : ( P @ X2 @ Y2 )
@ top_top_set_nat ) )
@ top_top_set_nat ) )
= ( complete_Sup_Sup_o
@ ( image_nat_nat_o
@ ^ [F2: nat > nat] :
( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [X2: nat] : ( P @ ( F2 @ X2 ) @ X2 )
@ top_top_set_nat ) )
@ top_top_set_nat_nat ) ) ) ).
% INF_SUP
thf(fact_769_INF__SUP,axiom,
! [P: nat > extended_ereal > $o] :
( ( complete_Inf_Inf_o
@ ( image_951975095941678543real_o
@ ^ [Y2: extended_ereal] :
( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [X2: nat] : ( P @ X2 @ Y2 )
@ top_top_set_nat ) )
@ top_to5683747375963461374_ereal ) )
= ( complete_Sup_Sup_o
@ ( image_1537912518788876622_nat_o
@ ^ [F2: extended_ereal > nat] :
( complete_Inf_Inf_o
@ ( image_951975095941678543real_o
@ ^ [X2: extended_ereal] : ( P @ ( F2 @ X2 ) @ X2 )
@ top_to5683747375963461374_ereal ) )
@ top_to2398772004365740095al_nat ) ) ) ).
% INF_SUP
thf(fact_770_INF__SUP,axiom,
! [P: extended_ereal > b > $o] :
( ( complete_Inf_Inf_o
@ ( image_b_o
@ ^ [Y2: b] :
( complete_Sup_Sup_o
@ ( image_951975095941678543real_o
@ ^ [X2: extended_ereal] : ( P @ X2 @ Y2 )
@ top_to5683747375963461374_ereal ) )
@ top_top_set_b ) )
= ( complete_Sup_Sup_o
@ ( image_8104747410577683677real_o
@ ^ [F2: b > extended_ereal] :
( complete_Inf_Inf_o
@ ( image_b_o
@ ^ [X2: b] : ( P @ ( F2 @ X2 ) @ X2 )
@ top_top_set_b ) )
@ top_to685074653192944346_ereal ) ) ) ).
% INF_SUP
thf(fact_771_INF__SUP,axiom,
! [P: extended_ereal > nat > $o] :
( ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [Y2: nat] :
( complete_Sup_Sup_o
@ ( image_951975095941678543real_o
@ ^ [X2: extended_ereal] : ( P @ X2 @ Y2 )
@ top_to5683747375963461374_ereal ) )
@ top_top_set_nat ) )
= ( complete_Sup_Sup_o
@ ( image_8272980733007922984real_o
@ ^ [F2: nat > extended_ereal] :
( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [X2: nat] : ( P @ ( F2 @ X2 ) @ X2 )
@ top_top_set_nat ) )
@ top_to1136031370110204389_ereal ) ) ) ).
% INF_SUP
thf(fact_772_INF__SUP,axiom,
! [P: extended_ereal > extended_ereal > $o] :
( ( complete_Inf_Inf_o
@ ( image_951975095941678543real_o
@ ^ [Y2: extended_ereal] :
( complete_Sup_Sup_o
@ ( image_951975095941678543real_o
@ ^ [X2: extended_ereal] : ( P @ X2 @ Y2 )
@ top_to5683747375963461374_ereal ) )
@ top_to5683747375963461374_ereal ) )
= ( complete_Sup_Sup_o
@ ( image_341501477734635180real_o
@ ^ [F2: extended_ereal > extended_ereal] :
( complete_Inf_Inf_o
@ ( image_951975095941678543real_o
@ ^ [X2: extended_ereal] : ( P @ ( F2 @ X2 ) @ X2 )
@ top_to5683747375963461374_ereal ) )
@ top_to908700840774984395_ereal ) ) ) ).
% INF_SUP
thf(fact_773_INF__SUP,axiom,
! [P: b > b > extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_5319725110001000852_ereal
@ ^ [Y2: b] :
( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [X2: b] : ( P @ X2 @ Y2 )
@ top_top_set_b ) )
@ top_top_set_b ) )
= ( comple8415311339701865915_ereal
@ ( image_2440594391108767394_ereal
@ ^ [F2: b > b] :
( comple3556804143462414037_ereal
@ ( image_5319725110001000852_ereal
@ ^ [X2: b] : ( P @ ( F2 @ X2 ) @ X2 )
@ top_top_set_b ) )
@ top_top_set_b_b ) ) ) ).
% INF_SUP
thf(fact_774_InterI,axiom,
! [C: set_set_o,A: $o] :
( ! [X6: set_o] :
( ( member_set_o @ X6 @ C )
=> ( member_o @ A @ X6 ) )
=> ( member_o @ A @ ( comple3063163877087187839_set_o @ C ) ) ) ).
% InterI
thf(fact_775_InterI,axiom,
! [C: set_se6634062954251873166_ereal,A: extended_ereal] :
( ! [X6: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ X6 @ C )
=> ( member2350847679896131959_ereal @ A @ X6 ) )
=> ( member2350847679896131959_ereal @ A @ ( comple4418415374894819509_ereal @ C ) ) ) ).
% InterI
thf(fact_776_InterI,axiom,
! [C: set_set_nat,A: nat] :
( ! [X6: set_nat] :
( ( member_set_nat @ X6 @ C )
=> ( member_nat @ A @ X6 ) )
=> ( member_nat @ A @ ( comple7806235888213564991et_nat @ C ) ) ) ).
% InterI
thf(fact_777_InterI,axiom,
! [C: set_se4580700918925141924nnreal,A: extend8495563244428889912nnreal] :
( ! [X6: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X6 @ C )
=> ( member7908768830364227535nnreal @ A @ X6 ) )
=> ( member7908768830364227535nnreal @ A @ ( comple5724520875574609319nnreal @ C ) ) ) ).
% InterI
thf(fact_778_InterI,axiom,
! [C: set_set_c,A: c] :
( ! [X6: set_c] :
( ( member_set_c @ X6 @ C )
=> ( member_c @ A @ X6 ) )
=> ( member_c @ A @ ( comple6135023387286571239_set_c @ C ) ) ) ).
% InterI
thf(fact_779_InterI,axiom,
! [C: set_set_b,A: b] :
( ! [X6: set_b] :
( ( member_set_b @ X6 @ C )
=> ( member_b @ A @ X6 ) )
=> ( member_b @ A @ ( comple6135023382983342438_set_b @ C ) ) ) ).
% InterI
thf(fact_780_Inter__iff,axiom,
! [A: $o,C: set_set_o] :
( ( member_o @ A @ ( comple3063163877087187839_set_o @ C ) )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ C )
=> ( member_o @ A @ X2 ) ) ) ) ).
% Inter_iff
thf(fact_781_Inter__iff,axiom,
! [A: extended_ereal,C: set_se6634062954251873166_ereal] :
( ( member2350847679896131959_ereal @ A @ ( comple4418415374894819509_ereal @ C ) )
= ( ! [X2: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ X2 @ C )
=> ( member2350847679896131959_ereal @ A @ X2 ) ) ) ) ).
% Inter_iff
thf(fact_782_Inter__iff,axiom,
! [A: nat,C: set_set_nat] :
( ( member_nat @ A @ ( comple7806235888213564991et_nat @ C ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ C )
=> ( member_nat @ A @ X2 ) ) ) ) ).
% Inter_iff
thf(fact_783_Inter__iff,axiom,
! [A: extend8495563244428889912nnreal,C: set_se4580700918925141924nnreal] :
( ( member7908768830364227535nnreal @ A @ ( comple5724520875574609319nnreal @ C ) )
= ( ! [X2: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X2 @ C )
=> ( member7908768830364227535nnreal @ A @ X2 ) ) ) ) ).
% Inter_iff
thf(fact_784_Inter__iff,axiom,
! [A: c,C: set_set_c] :
( ( member_c @ A @ ( comple6135023387286571239_set_c @ C ) )
= ( ! [X2: set_c] :
( ( member_set_c @ X2 @ C )
=> ( member_c @ A @ X2 ) ) ) ) ).
% Inter_iff
thf(fact_785_Inter__iff,axiom,
! [A: b,C: set_set_b] :
( ( member_b @ A @ ( comple6135023382983342438_set_b @ C ) )
= ( ! [X2: set_b] :
( ( member_set_b @ X2 @ C )
=> ( member_b @ A @ X2 ) ) ) ) ).
% Inter_iff
thf(fact_786_InterD,axiom,
! [A: $o,C: set_set_o,X5: set_o] :
( ( member_o @ A @ ( comple3063163877087187839_set_o @ C ) )
=> ( ( member_set_o @ X5 @ C )
=> ( member_o @ A @ X5 ) ) ) ).
% InterD
thf(fact_787_InterD,axiom,
! [A: extended_ereal,C: set_se6634062954251873166_ereal,X5: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ A @ ( comple4418415374894819509_ereal @ C ) )
=> ( ( member5519481007471526743_ereal @ X5 @ C )
=> ( member2350847679896131959_ereal @ A @ X5 ) ) ) ).
% InterD
thf(fact_788_InterD,axiom,
! [A: nat,C: set_set_nat,X5: set_nat] :
( ( member_nat @ A @ ( comple7806235888213564991et_nat @ C ) )
=> ( ( member_set_nat @ X5 @ C )
=> ( member_nat @ A @ X5 ) ) ) ).
% InterD
thf(fact_789_InterD,axiom,
! [A: extend8495563244428889912nnreal,C: set_se4580700918925141924nnreal,X5: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ A @ ( comple5724520875574609319nnreal @ C ) )
=> ( ( member603777416030116741nnreal @ X5 @ C )
=> ( member7908768830364227535nnreal @ A @ X5 ) ) ) ).
% InterD
thf(fact_790_InterD,axiom,
! [A: c,C: set_set_c,X5: set_c] :
( ( member_c @ A @ ( comple6135023387286571239_set_c @ C ) )
=> ( ( member_set_c @ X5 @ C )
=> ( member_c @ A @ X5 ) ) ) ).
% InterD
thf(fact_791_InterD,axiom,
! [A: b,C: set_set_b,X5: set_b] :
( ( member_b @ A @ ( comple6135023382983342438_set_b @ C ) )
=> ( ( member_set_b @ X5 @ C )
=> ( member_b @ A @ X5 ) ) ) ).
% InterD
thf(fact_792_InterE,axiom,
! [A: $o,C: set_set_o,X5: set_o] :
( ( member_o @ A @ ( comple3063163877087187839_set_o @ C ) )
=> ( ( member_set_o @ X5 @ C )
=> ( member_o @ A @ X5 ) ) ) ).
% InterE
thf(fact_793_InterE,axiom,
! [A: extended_ereal,C: set_se6634062954251873166_ereal,X5: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ A @ ( comple4418415374894819509_ereal @ C ) )
=> ( ( member5519481007471526743_ereal @ X5 @ C )
=> ( member2350847679896131959_ereal @ A @ X5 ) ) ) ).
% InterE
thf(fact_794_InterE,axiom,
! [A: nat,C: set_set_nat,X5: set_nat] :
( ( member_nat @ A @ ( comple7806235888213564991et_nat @ C ) )
=> ( ( member_set_nat @ X5 @ C )
=> ( member_nat @ A @ X5 ) ) ) ).
% InterE
thf(fact_795_InterE,axiom,
! [A: extend8495563244428889912nnreal,C: set_se4580700918925141924nnreal,X5: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ A @ ( comple5724520875574609319nnreal @ C ) )
=> ( ( member603777416030116741nnreal @ X5 @ C )
=> ( member7908768830364227535nnreal @ A @ X5 ) ) ) ).
% InterE
thf(fact_796_InterE,axiom,
! [A: c,C: set_set_c,X5: set_c] :
( ( member_c @ A @ ( comple6135023387286571239_set_c @ C ) )
=> ( ( member_set_c @ X5 @ C )
=> ( member_c @ A @ X5 ) ) ) ).
% InterE
thf(fact_797_InterE,axiom,
! [A: b,C: set_set_b,X5: set_b] :
( ( member_b @ A @ ( comple6135023382983342438_set_b @ C ) )
=> ( ( member_set_b @ X5 @ C )
=> ( member_b @ A @ X5 ) ) ) ).
% InterE
thf(fact_798_Inf__set__def,axiom,
( comple6135023382983342438_set_b
= ( ^ [A3: set_set_b] :
( collect_b
@ ^ [X2: b] : ( complete_Inf_Inf_o @ ( image_set_b_o @ ( member_b @ X2 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_799_Inf__set__def,axiom,
( comple6135023387286571239_set_c
= ( ^ [A3: set_set_c] :
( collect_c
@ ^ [X2: c] : ( complete_Inf_Inf_o @ ( image_set_c_o @ ( member_c @ X2 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_800_Inf__set__def,axiom,
( comple4418415374894819509_ereal
= ( ^ [A3: set_se6634062954251873166_ereal] :
( collec5835592288176408249_ereal
@ ^ [X2: extended_ereal] : ( complete_Inf_Inf_o @ ( image_1946622920212178927real_o @ ( member2350847679896131959_ereal @ X2 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_801_Inf__set__def,axiom,
( comple5724520875574609319nnreal
= ( ^ [A3: set_se4580700918925141924nnreal] :
( collec6648975593938027277nnreal
@ ^ [X2: extend8495563244428889912nnreal] : ( complete_Inf_Inf_o @ ( image_2954085599833420643real_o @ ( member7908768830364227535nnreal @ X2 ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_802_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_803_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_804_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_805_top__set__def,axiom,
( top_top_set_b
= ( collect_b @ top_top_b_o ) ) ).
% top_set_def
thf(fact_806_top__set__def,axiom,
( top_top_set_c
= ( collect_c @ top_top_c_o ) ) ).
% top_set_def
thf(fact_807_top__set__def,axiom,
( top_to7994903218803871134nnreal
= ( collec6648975593938027277nnreal @ top_to5118619752887738471real_o ) ) ).
% top_set_def
thf(fact_808_top__set__def,axiom,
( top_top_set_o
= ( collect_o @ top_top_o_o ) ) ).
% top_set_def
thf(fact_809_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_810_top__set__def,axiom,
( top_to5683747375963461374_ereal
= ( collec5835592288176408249_ereal @ top_to6999531812125281119real_o ) ) ).
% top_set_def
thf(fact_811_bot__set__def,axiom,
( bot_bot_set_b
= ( collect_b @ bot_bot_b_o ) ) ).
% bot_set_def
thf(fact_812_bot__set__def,axiom,
( bot_bot_set_c
= ( collect_c @ bot_bot_c_o ) ) ).
% bot_set_def
thf(fact_813_bot__set__def,axiom,
( bot_bot_set_o
= ( collect_o @ bot_bot_o_o ) ) ).
% bot_set_def
thf(fact_814_bot__set__def,axiom,
( bot_bo8367695208629047834_ereal
= ( collec5835592288176408249_ereal @ bot_bo5519581617326455619real_o ) ) ).
% bot_set_def
thf(fact_815_bot__set__def,axiom,
( bot_bo4854962954004695426nnreal
= ( collec6648975593938027277nnreal @ bot_bo412624608084785539real_o ) ) ).
% bot_set_def
thf(fact_816_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_817_wellorder__InfI,axiom,
! [K: nat,A: set_nat] :
( ( member_nat @ K @ A )
=> ( member_nat @ ( complete_Inf_Inf_nat @ A ) @ A ) ) ).
% wellorder_InfI
thf(fact_818_SUP__INF,axiom,
! [P: b > b > $o] :
( ( complete_Sup_Sup_o
@ ( image_b_o
@ ^ [Y2: b] :
( complete_Inf_Inf_o
@ ( image_b_o
@ ^ [X2: b] : ( P @ X2 @ Y2 )
@ top_top_set_b ) )
@ top_top_set_b ) )
= ( complete_Inf_Inf_o
@ ( image_b_b_o
@ ^ [X2: b > b] :
( complete_Sup_Sup_o
@ ( image_b_o
@ ^ [Y2: b] : ( P @ ( X2 @ Y2 ) @ Y2 )
@ top_top_set_b ) )
@ top_top_set_b_b ) ) ) ).
% SUP_INF
thf(fact_819_SUP__INF,axiom,
! [P: b > nat > $o] :
( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y2: nat] :
( complete_Inf_Inf_o
@ ( image_b_o
@ ^ [X2: b] : ( P @ X2 @ Y2 )
@ top_top_set_b ) )
@ top_top_set_nat ) )
= ( complete_Inf_Inf_o
@ ( image_nat_b_o
@ ^ [X2: nat > b] :
( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y2: nat] : ( P @ ( X2 @ Y2 ) @ Y2 )
@ top_top_set_nat ) )
@ top_top_set_nat_b ) ) ) ).
% SUP_INF
thf(fact_820_SUP__INF,axiom,
! [P: b > extended_ereal > $o] :
( ( complete_Sup_Sup_o
@ ( image_951975095941678543real_o
@ ^ [Y2: extended_ereal] :
( complete_Inf_Inf_o
@ ( image_b_o
@ ^ [X2: b] : ( P @ X2 @ Y2 )
@ top_top_set_b ) )
@ top_to5683747375963461374_ereal ) )
= ( complete_Inf_Inf_o
@ ( image_438845630295331515al_b_o
@ ^ [X2: extended_ereal > b] :
( complete_Sup_Sup_o
@ ( image_951975095941678543real_o
@ ^ [Y2: extended_ereal] : ( P @ ( X2 @ Y2 ) @ Y2 )
@ top_to5683747375963461374_ereal ) )
@ top_to5435482696333698940real_b ) ) ) ).
% SUP_INF
thf(fact_821_SUP__INF,axiom,
! [P: nat > b > $o] :
( ( complete_Sup_Sup_o
@ ( image_b_o
@ ^ [Y2: b] :
( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [X2: nat] : ( P @ X2 @ Y2 )
@ top_top_set_nat ) )
@ top_top_set_b ) )
= ( complete_Inf_Inf_o
@ ( image_b_nat_o
@ ^ [X2: b > nat] :
( complete_Sup_Sup_o
@ ( image_b_o
@ ^ [Y2: b] : ( P @ ( X2 @ Y2 ) @ Y2 )
@ top_top_set_b ) )
@ top_top_set_b_nat ) ) ) ).
% SUP_INF
thf(fact_822_SUP__INF,axiom,
! [P: nat > nat > $o] :
( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y2: nat] :
( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [X2: nat] : ( P @ X2 @ Y2 )
@ top_top_set_nat ) )
@ top_top_set_nat ) )
= ( complete_Inf_Inf_o
@ ( image_nat_nat_o
@ ^ [X2: nat > nat] :
( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y2: nat] : ( P @ ( X2 @ Y2 ) @ Y2 )
@ top_top_set_nat ) )
@ top_top_set_nat_nat ) ) ) ).
% SUP_INF
thf(fact_823_SUP__INF,axiom,
! [P: nat > extended_ereal > $o] :
( ( complete_Sup_Sup_o
@ ( image_951975095941678543real_o
@ ^ [Y2: extended_ereal] :
( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [X2: nat] : ( P @ X2 @ Y2 )
@ top_top_set_nat ) )
@ top_to5683747375963461374_ereal ) )
= ( complete_Inf_Inf_o
@ ( image_1537912518788876622_nat_o
@ ^ [X2: extended_ereal > nat] :
( complete_Sup_Sup_o
@ ( image_951975095941678543real_o
@ ^ [Y2: extended_ereal] : ( P @ ( X2 @ Y2 ) @ Y2 )
@ top_to5683747375963461374_ereal ) )
@ top_to2398772004365740095al_nat ) ) ) ).
% SUP_INF
thf(fact_824_SUP__INF,axiom,
! [P: extended_ereal > b > $o] :
( ( complete_Sup_Sup_o
@ ( image_b_o
@ ^ [Y2: b] :
( complete_Inf_Inf_o
@ ( image_951975095941678543real_o
@ ^ [X2: extended_ereal] : ( P @ X2 @ Y2 )
@ top_to5683747375963461374_ereal ) )
@ top_top_set_b ) )
= ( complete_Inf_Inf_o
@ ( image_8104747410577683677real_o
@ ^ [X2: b > extended_ereal] :
( complete_Sup_Sup_o
@ ( image_b_o
@ ^ [Y2: b] : ( P @ ( X2 @ Y2 ) @ Y2 )
@ top_top_set_b ) )
@ top_to685074653192944346_ereal ) ) ) ).
% SUP_INF
thf(fact_825_SUP__INF,axiom,
! [P: extended_ereal > nat > $o] :
( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y2: nat] :
( complete_Inf_Inf_o
@ ( image_951975095941678543real_o
@ ^ [X2: extended_ereal] : ( P @ X2 @ Y2 )
@ top_to5683747375963461374_ereal ) )
@ top_top_set_nat ) )
= ( complete_Inf_Inf_o
@ ( image_8272980733007922984real_o
@ ^ [X2: nat > extended_ereal] :
( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y2: nat] : ( P @ ( X2 @ Y2 ) @ Y2 )
@ top_top_set_nat ) )
@ top_to1136031370110204389_ereal ) ) ) ).
% SUP_INF
thf(fact_826_SUP__INF,axiom,
! [P: extended_ereal > extended_ereal > $o] :
( ( complete_Sup_Sup_o
@ ( image_951975095941678543real_o
@ ^ [Y2: extended_ereal] :
( complete_Inf_Inf_o
@ ( image_951975095941678543real_o
@ ^ [X2: extended_ereal] : ( P @ X2 @ Y2 )
@ top_to5683747375963461374_ereal ) )
@ top_to5683747375963461374_ereal ) )
= ( complete_Inf_Inf_o
@ ( image_341501477734635180real_o
@ ^ [X2: extended_ereal > extended_ereal] :
( complete_Sup_Sup_o
@ ( image_951975095941678543real_o
@ ^ [Y2: extended_ereal] : ( P @ ( X2 @ Y2 ) @ Y2 )
@ top_to5683747375963461374_ereal ) )
@ top_to908700840774984395_ereal ) ) ) ).
% SUP_INF
thf(fact_827_SUP__INF,axiom,
! [P: b > b > extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [Y2: b] :
( comple3556804143462414037_ereal
@ ( image_5319725110001000852_ereal
@ ^ [X2: b] : ( P @ X2 @ Y2 )
@ top_top_set_b ) )
@ top_top_set_b ) )
= ( comple3556804143462414037_ereal
@ ( image_2440594391108767394_ereal
@ ^ [X2: b > b] :
( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [Y2: b] : ( P @ ( X2 @ Y2 ) @ Y2 )
@ top_top_set_b ) )
@ top_top_set_b_b ) ) ) ).
% SUP_INF
thf(fact_828_iso__tuple__UNIV__I,axiom,
! [X: $o] : ( member_o @ X @ top_top_set_o ) ).
% iso_tuple_UNIV_I
thf(fact_829_iso__tuple__UNIV__I,axiom,
! [X: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X @ top_to7994903218803871134nnreal ) ).
% iso_tuple_UNIV_I
thf(fact_830_iso__tuple__UNIV__I,axiom,
! [X: c] : ( member_c @ X @ top_top_set_c ) ).
% iso_tuple_UNIV_I
thf(fact_831_iso__tuple__UNIV__I,axiom,
! [X: b] : ( member_b @ X @ top_top_set_b ) ).
% iso_tuple_UNIV_I
thf(fact_832_iso__tuple__UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_833_iso__tuple__UNIV__I,axiom,
! [X: extended_ereal] : ( member2350847679896131959_ereal @ X @ top_to5683747375963461374_ereal ) ).
% iso_tuple_UNIV_I
thf(fact_834_Inf__bool__def,axiom,
( complete_Inf_Inf_o
= ( ^ [A3: set_o] :
~ ( member_o @ $false @ A3 ) ) ) ).
% Inf_bool_def
thf(fact_835_top__empty__eq,axiom,
( top_top_o_o
= ( ^ [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ) ) ).
% top_empty_eq
thf(fact_836_top__empty__eq,axiom,
( top_to5118619752887738471real_o
= ( ^ [X2: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X2 @ top_to7994903218803871134nnreal ) ) ) ).
% top_empty_eq
thf(fact_837_top__empty__eq,axiom,
( top_top_c_o
= ( ^ [X2: c] : ( member_c @ X2 @ top_top_set_c ) ) ) ).
% top_empty_eq
thf(fact_838_top__empty__eq,axiom,
( top_top_b_o
= ( ^ [X2: b] : ( member_b @ X2 @ top_top_set_b ) ) ) ).
% top_empty_eq
thf(fact_839_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_840_top__empty__eq,axiom,
( top_to6999531812125281119real_o
= ( ^ [X2: extended_ereal] : ( member2350847679896131959_ereal @ X2 @ top_to5683747375963461374_ereal ) ) ) ).
% top_empty_eq
thf(fact_841_bot__empty__eq,axiom,
( bot_bot_o_o
= ( ^ [X2: $o] : ( member_o @ X2 @ bot_bot_set_o ) ) ) ).
% bot_empty_eq
thf(fact_842_bot__empty__eq,axiom,
( bot_bot_c_o
= ( ^ [X2: c] : ( member_c @ X2 @ bot_bot_set_c ) ) ) ).
% bot_empty_eq
thf(fact_843_bot__empty__eq,axiom,
( bot_bot_b_o
= ( ^ [X2: b] : ( member_b @ X2 @ bot_bot_set_b ) ) ) ).
% bot_empty_eq
thf(fact_844_bot__empty__eq,axiom,
( bot_bo5519581617326455619real_o
= ( ^ [X2: extended_ereal] : ( member2350847679896131959_ereal @ X2 @ bot_bo8367695208629047834_ereal ) ) ) ).
% bot_empty_eq
thf(fact_845_bot__empty__eq,axiom,
( bot_bo412624608084785539real_o
= ( ^ [X2: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X2 @ bot_bo4854962954004695426nnreal ) ) ) ).
% bot_empty_eq
thf(fact_846_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X2: nat] : ( member_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_847_Sup__SUP__eq,axiom,
( complete_Sup_Sup_b_o
= ( ^ [S3: set_b_o,X2: b] : ( member_b @ X2 @ ( comple2307003614231284044_set_b @ ( image_b_o_set_b @ collect_b @ S3 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_848_Sup__SUP__eq,axiom,
( complete_Sup_Sup_c_o
= ( ^ [S3: set_c_o,X2: c] : ( member_c @ X2 @ ( comple2307003618534512845_set_c @ ( image_c_o_set_c @ collect_c @ S3 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_849_Sup__SUP__eq,axiom,
( comple8551942733113566466real_o
= ( ^ [S3: set_Extended_ereal_o,X2: extended_ereal] : ( member2350847679896131959_ereal @ X2 @ ( comple4319282863272126363_ereal @ ( image_169545030887771000_ereal @ collec5835592288176408249_ereal @ S3 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_850_Sup__SUP__eq,axiom,
( comple5476927491321936772real_o
= ( ^ [S3: set_Ex70502500924464887real_o,X2: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X2 @ ( comple4226387801268262977nnreal @ ( image_7529257491699830976nnreal @ collec6648975593938027277nnreal @ S3 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_851_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S3: set_nat_o,X2: nat] : ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S3 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_852_Sup__SUP__eq,axiom,
( complete_Sup_Sup_o_o
= ( ^ [S3: set_o_o,X2: $o] : ( member_o @ X2 @ ( comple90263536869209701_set_o @ ( image_o_o_set_o @ collect_o @ S3 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_853_Inf__INT__eq,axiom,
( complete_Inf_Inf_b_o
= ( ^ [S3: set_b_o,X2: b] : ( member_b @ X2 @ ( comple6135023382983342438_set_b @ ( image_b_o_set_b @ collect_b @ S3 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_854_Inf__INT__eq,axiom,
( complete_Inf_Inf_c_o
= ( ^ [S3: set_c_o,X2: c] : ( member_c @ X2 @ ( comple6135023387286571239_set_c @ ( image_c_o_set_c @ collect_c @ S3 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_855_Inf__INT__eq,axiom,
( comple5376498136088350824real_o
= ( ^ [S3: set_Extended_ereal_o,X2: extended_ereal] : ( member2350847679896131959_ereal @ X2 @ ( comple4418415374894819509_ereal @ ( image_169545030887771000_ereal @ collec5835592288176408249_ereal @ S3 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_856_Inf__INT__eq,axiom,
( comple2110304272711893406real_o
= ( ^ [S3: set_Ex70502500924464887real_o,X2: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X2 @ ( comple5724520875574609319nnreal @ ( image_7529257491699830976nnreal @ collec6648975593938027277nnreal @ S3 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_857_Inf__INT__eq,axiom,
( comple6214475593288795910_nat_o
= ( ^ [S3: set_nat_o,X2: nat] : ( member_nat @ X2 @ ( comple7806235888213564991et_nat @ ( image_nat_o_set_nat @ collect_nat @ S3 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_858_Inf__INT__eq,axiom,
( complete_Inf_Inf_o_o
= ( ^ [S3: set_o_o,X2: $o] : ( member_o @ X2 @ ( comple3063163877087187839_set_o @ ( image_o_o_set_o @ collect_o @ S3 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_859_SUP__Sup__eq,axiom,
! [S: set_set_o] :
( ( complete_Sup_Sup_o_o
@ ( image_set_o_o_o
@ ^ [I: set_o,X2: $o] : ( member_o @ X2 @ I )
@ S ) )
= ( ^ [X2: $o] : ( member_o @ X2 @ ( comple90263536869209701_set_o @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_860_SUP__Sup__eq,axiom,
! [S: set_se6634062954251873166_ereal] :
( ( comple8551942733113566466real_o
@ ( image_6529656333506721048real_o
@ ^ [I: set_Extended_ereal,X2: extended_ereal] : ( member2350847679896131959_ereal @ X2 @ I )
@ S ) )
= ( ^ [X2: extended_ereal] : ( member2350847679896131959_ereal @ X2 @ ( comple4319282863272126363_ereal @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_861_SUP__Sup__eq,axiom,
! [S: set_set_nat] :
( ( comple8317665133742190828_nat_o
@ ( image_set_nat_nat_o
@ ^ [I: set_nat,X2: nat] : ( member_nat @ X2 @ I )
@ S ) )
= ( ^ [X2: nat] : ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_862_SUP__Sup__eq,axiom,
! [S: set_se4580700918925141924nnreal] :
( ( comple5476927491321936772real_o
@ ( image_8646621328128423522real_o
@ ^ [I: set_Ex3793607809372303086nnreal,X2: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X2 @ I )
@ S ) )
= ( ^ [X2: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X2 @ ( comple4226387801268262977nnreal @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_863_SUP__Sup__eq,axiom,
! [S: set_set_c] :
( ( complete_Sup_Sup_c_o
@ ( image_set_c_c_o
@ ^ [I: set_c,X2: c] : ( member_c @ X2 @ I )
@ S ) )
= ( ^ [X2: c] : ( member_c @ X2 @ ( comple2307003618534512845_set_c @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_864_SUP__Sup__eq,axiom,
! [S: set_set_b] :
( ( complete_Sup_Sup_b_o
@ ( image_set_b_b_o
@ ^ [I: set_b,X2: b] : ( member_b @ X2 @ I )
@ S ) )
= ( ^ [X2: b] : ( member_b @ X2 @ ( comple2307003614231284044_set_b @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_865_INF__Int__eq,axiom,
! [S: set_set_o] :
( ( complete_Inf_Inf_o_o
@ ( image_set_o_o_o
@ ^ [I: set_o,X2: $o] : ( member_o @ X2 @ I )
@ S ) )
= ( ^ [X2: $o] : ( member_o @ X2 @ ( comple3063163877087187839_set_o @ S ) ) ) ) ).
% INF_Int_eq
thf(fact_866_INF__Int__eq,axiom,
! [S: set_se6634062954251873166_ereal] :
( ( comple5376498136088350824real_o
@ ( image_6529656333506721048real_o
@ ^ [I: set_Extended_ereal,X2: extended_ereal] : ( member2350847679896131959_ereal @ X2 @ I )
@ S ) )
= ( ^ [X2: extended_ereal] : ( member2350847679896131959_ereal @ X2 @ ( comple4418415374894819509_ereal @ S ) ) ) ) ).
% INF_Int_eq
thf(fact_867_INF__Int__eq,axiom,
! [S: set_set_nat] :
( ( comple6214475593288795910_nat_o
@ ( image_set_nat_nat_o
@ ^ [I: set_nat,X2: nat] : ( member_nat @ X2 @ I )
@ S ) )
= ( ^ [X2: nat] : ( member_nat @ X2 @ ( comple7806235888213564991et_nat @ S ) ) ) ) ).
% INF_Int_eq
thf(fact_868_INF__Int__eq,axiom,
! [S: set_se4580700918925141924nnreal] :
( ( comple2110304272711893406real_o
@ ( image_8646621328128423522real_o
@ ^ [I: set_Ex3793607809372303086nnreal,X2: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X2 @ I )
@ S ) )
= ( ^ [X2: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X2 @ ( comple5724520875574609319nnreal @ S ) ) ) ) ).
% INF_Int_eq
thf(fact_869_INF__Int__eq,axiom,
! [S: set_set_c] :
( ( complete_Inf_Inf_c_o
@ ( image_set_c_c_o
@ ^ [I: set_c,X2: c] : ( member_c @ X2 @ I )
@ S ) )
= ( ^ [X2: c] : ( member_c @ X2 @ ( comple6135023387286571239_set_c @ S ) ) ) ) ).
% INF_Int_eq
thf(fact_870_INF__Int__eq,axiom,
! [S: set_set_b] :
( ( complete_Inf_Inf_b_o
@ ( image_set_b_b_o
@ ^ [I: set_b,X2: b] : ( member_b @ X2 @ I )
@ S ) )
= ( ^ [X2: b] : ( member_b @ X2 @ ( comple6135023382983342438_set_b @ S ) ) ) ) ).
% INF_Int_eq
thf(fact_871_Collect__empty__eq__bot,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( P = bot_bot_b_o ) ) ).
% Collect_empty_eq_bot
thf(fact_872_Collect__empty__eq__bot,axiom,
! [P: c > $o] :
( ( ( collect_c @ P )
= bot_bot_set_c )
= ( P = bot_bot_c_o ) ) ).
% Collect_empty_eq_bot
thf(fact_873_Collect__empty__eq__bot,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( P = bot_bot_o_o ) ) ).
% Collect_empty_eq_bot
thf(fact_874_Collect__empty__eq__bot,axiom,
! [P: extended_ereal > $o] :
( ( ( collec5835592288176408249_ereal @ P )
= bot_bo8367695208629047834_ereal )
= ( P = bot_bo5519581617326455619real_o ) ) ).
% Collect_empty_eq_bot
thf(fact_875_Collect__empty__eq__bot,axiom,
! [P: extend8495563244428889912nnreal > $o] :
( ( ( collec6648975593938027277nnreal @ P )
= bot_bo4854962954004695426nnreal )
= ( P = bot_bo412624608084785539real_o ) ) ).
% Collect_empty_eq_bot
thf(fact_876_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_877_INT__simps_I4_J,axiom,
! [C: set_Extended_ereal,A: set_nat,B2: extended_ereal > set_nat] :
( ( ( C = bot_bo8367695208629047834_ereal )
=> ( ( comple7806235888213564991et_nat
@ ( image_3090908713637162255et_nat
@ ^ [X2: extended_ereal] : ( minus_minus_set_nat @ A @ ( B2 @ X2 ) )
@ C ) )
= top_top_set_nat ) )
& ( ( C != bot_bo8367695208629047834_ereal )
=> ( ( comple7806235888213564991et_nat
@ ( image_3090908713637162255et_nat
@ ^ [X2: extended_ereal] : ( minus_minus_set_nat @ A @ ( B2 @ X2 ) )
@ C ) )
= ( minus_minus_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_3090908713637162255et_nat @ B2 @ C ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_878_INT__simps_I4_J,axiom,
! [C: set_Ex3793607809372303086nnreal,A: set_nat,B2: extend8495563244428889912nnreal > set_nat] :
( ( ( C = bot_bo4854962954004695426nnreal )
=> ( ( comple7806235888213564991et_nat
@ ( image_2869339492569777349et_nat
@ ^ [X2: extend8495563244428889912nnreal] : ( minus_minus_set_nat @ A @ ( B2 @ X2 ) )
@ C ) )
= top_top_set_nat ) )
& ( ( C != bot_bo4854962954004695426nnreal )
=> ( ( comple7806235888213564991et_nat
@ ( image_2869339492569777349et_nat
@ ^ [X2: extend8495563244428889912nnreal] : ( minus_minus_set_nat @ A @ ( B2 @ X2 ) )
@ C ) )
= ( minus_minus_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_2869339492569777349et_nat @ B2 @ C ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_879_INT__simps_I4_J,axiom,
! [C: set_nat,A: set_nat,B2: nat > set_nat] :
( ( ( C = bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( minus_minus_set_nat @ A @ ( B2 @ X2 ) )
@ C ) )
= top_top_set_nat ) )
& ( ( C != bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( minus_minus_set_nat @ A @ ( B2 @ X2 ) )
@ C ) )
= ( minus_minus_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ C ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_880_INT__simps_I4_J,axiom,
! [C: set_Extended_ereal,A: set_Extended_ereal,B2: extended_ereal > set_Extended_ereal] :
( ( ( C = bot_bo8367695208629047834_ereal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5562094264469218789_ereal
@ ^ [X2: extended_ereal] : ( minus_1264018925008434325_ereal @ A @ ( B2 @ X2 ) )
@ C ) )
= top_to5683747375963461374_ereal ) )
& ( ( C != bot_bo8367695208629047834_ereal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5562094264469218789_ereal
@ ^ [X2: extended_ereal] : ( minus_1264018925008434325_ereal @ A @ ( B2 @ X2 ) )
@ C ) )
= ( minus_1264018925008434325_ereal @ A @ ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ B2 @ C ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_881_INT__simps_I4_J,axiom,
! [C: set_Ex3793607809372303086nnreal,A: set_Extended_ereal,B2: extend8495563244428889912nnreal > set_Extended_ereal] :
( ( ( C = bot_bo4854962954004695426nnreal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5929344197358196911_ereal
@ ^ [X2: extend8495563244428889912nnreal] : ( minus_1264018925008434325_ereal @ A @ ( B2 @ X2 ) )
@ C ) )
= top_to5683747375963461374_ereal ) )
& ( ( C != bot_bo4854962954004695426nnreal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5929344197358196911_ereal
@ ^ [X2: extend8495563244428889912nnreal] : ( minus_1264018925008434325_ereal @ A @ ( B2 @ X2 ) )
@ C ) )
= ( minus_1264018925008434325_ereal @ A @ ( comple4319282863272126363_ereal @ ( image_5929344197358196911_ereal @ B2 @ C ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_882_INT__simps_I4_J,axiom,
! [C: set_nat,A: set_Extended_ereal,B2: nat > set_Extended_ereal] :
( ( ( C = bot_bot_set_nat )
=> ( ( comple4418415374894819509_ereal
@ ( image_305533323056406039_ereal
@ ^ [X2: nat] : ( minus_1264018925008434325_ereal @ A @ ( B2 @ X2 ) )
@ C ) )
= top_to5683747375963461374_ereal ) )
& ( ( C != bot_bot_set_nat )
=> ( ( comple4418415374894819509_ereal
@ ( image_305533323056406039_ereal
@ ^ [X2: nat] : ( minus_1264018925008434325_ereal @ A @ ( B2 @ X2 ) )
@ C ) )
= ( minus_1264018925008434325_ereal @ A @ ( comple4319282863272126363_ereal @ ( image_305533323056406039_ereal @ B2 @ C ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_883_surjD,axiom,
! [F: c > b,Y4: b] :
( ( ( image_c_b @ F @ top_top_set_c )
= top_top_set_b )
=> ? [X3: c] :
( Y4
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_884_surjD,axiom,
! [F: c > c,Y4: c] :
( ( ( image_c_c @ F @ top_top_set_c )
= top_top_set_c )
=> ? [X3: c] :
( Y4
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_885_surjD,axiom,
! [F: b > $o,Y4: $o] :
( ( ( image_b_o @ F @ top_top_set_b )
= top_top_set_o )
=> ? [X3: b] :
( Y4
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_886_surjD,axiom,
! [F: b > c,Y4: c] :
( ( ( image_b_c @ F @ top_top_set_b )
= top_top_set_c )
=> ? [X3: b] :
( Y4
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_887_surjD,axiom,
! [F: b > b,Y4: b] :
( ( ( image_b_b @ F @ top_top_set_b )
= top_top_set_b )
=> ? [X3: b] :
( Y4
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_888_surjD,axiom,
! [F: b > extended_ereal,Y4: extended_ereal] :
( ( ( image_5319725110001000852_ereal @ F @ top_top_set_b )
= top_to5683747375963461374_ereal )
=> ? [X3: b] :
( Y4
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_889_surjD,axiom,
! [F: nat > extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ( image_8459861568512453903nnreal @ F @ top_top_set_nat )
= top_to7994903218803871134nnreal )
=> ? [X3: nat] :
( Y4
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_890_surjD,axiom,
! [F: nat > nat,Y4: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X3: nat] :
( Y4
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_891_surjD,axiom,
! [F: nat > extended_ereal,Y4: extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F @ top_top_set_nat )
= top_to5683747375963461374_ereal )
=> ? [X3: nat] :
( Y4
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_892_surjD,axiom,
! [F: extended_ereal > nat,Y4: nat] :
( ( ( image_7659842161140344153al_nat @ F @ top_to5683747375963461374_ereal )
= top_top_set_nat )
=> ? [X3: extended_ereal] :
( Y4
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_893_surjE,axiom,
! [F: c > b,Y4: b] :
( ( ( image_c_b @ F @ top_top_set_c )
= top_top_set_b )
=> ~ ! [X3: c] :
( Y4
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_894_surjE,axiom,
! [F: c > c,Y4: c] :
( ( ( image_c_c @ F @ top_top_set_c )
= top_top_set_c )
=> ~ ! [X3: c] :
( Y4
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_895_surjE,axiom,
! [F: b > $o,Y4: $o] :
( ( ( image_b_o @ F @ top_top_set_b )
= top_top_set_o )
=> ~ ! [X3: b] :
( Y4
= ( ~ ( F @ X3 ) ) ) ) ).
% surjE
thf(fact_896_surjE,axiom,
! [F: b > c,Y4: c] :
( ( ( image_b_c @ F @ top_top_set_b )
= top_top_set_c )
=> ~ ! [X3: b] :
( Y4
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_897_surjE,axiom,
! [F: b > b,Y4: b] :
( ( ( image_b_b @ F @ top_top_set_b )
= top_top_set_b )
=> ~ ! [X3: b] :
( Y4
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_898_surjE,axiom,
! [F: b > extended_ereal,Y4: extended_ereal] :
( ( ( image_5319725110001000852_ereal @ F @ top_top_set_b )
= top_to5683747375963461374_ereal )
=> ~ ! [X3: b] :
( Y4
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_899_surjE,axiom,
! [F: nat > extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ( image_8459861568512453903nnreal @ F @ top_top_set_nat )
= top_to7994903218803871134nnreal )
=> ~ ! [X3: nat] :
( Y4
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_900_surjE,axiom,
! [F: nat > nat,Y4: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X3: nat] :
( Y4
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_901_surjE,axiom,
! [F: nat > extended_ereal,Y4: extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F @ top_top_set_nat )
= top_to5683747375963461374_ereal )
=> ~ ! [X3: nat] :
( Y4
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_902_surjE,axiom,
! [F: extended_ereal > nat,Y4: nat] :
( ( ( image_7659842161140344153al_nat @ F @ top_to5683747375963461374_ereal )
= top_top_set_nat )
=> ~ ! [X3: extended_ereal] :
( Y4
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_903_surjI,axiom,
! [G: c > b,F: b > c] :
( ! [X3: b] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_c_b @ G @ top_top_set_c )
= top_top_set_b ) ) ).
% surjI
thf(fact_904_surjI,axiom,
! [G: c > c,F: c > c] :
( ! [X3: c] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_c_c @ G @ top_top_set_c )
= top_top_set_c ) ) ).
% surjI
thf(fact_905_surjI,axiom,
! [G: b > $o,F: $o > b] :
( ! [X3: $o] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_b_o @ G @ top_top_set_b )
= top_top_set_o ) ) ).
% surjI
thf(fact_906_surjI,axiom,
! [G: b > c,F: c > b] :
( ! [X3: c] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_b_c @ G @ top_top_set_b )
= top_top_set_c ) ) ).
% surjI
thf(fact_907_surjI,axiom,
! [G: b > b,F: b > b] :
( ! [X3: b] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_b_b @ G @ top_top_set_b )
= top_top_set_b ) ) ).
% surjI
thf(fact_908_surjI,axiom,
! [G: b > extended_ereal,F: extended_ereal > b] :
( ! [X3: extended_ereal] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_5319725110001000852_ereal @ G @ top_top_set_b )
= top_to5683747375963461374_ereal ) ) ).
% surjI
thf(fact_909_surjI,axiom,
! [G: nat > extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > nat] :
( ! [X3: extend8495563244428889912nnreal] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_8459861568512453903nnreal @ G @ top_top_set_nat )
= top_to7994903218803871134nnreal ) ) ).
% surjI
thf(fact_910_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_911_surjI,axiom,
! [G: nat > extended_ereal,F: extended_ereal > nat] :
( ! [X3: extended_ereal] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_4309273772856505399_ereal @ G @ top_top_set_nat )
= top_to5683747375963461374_ereal ) ) ).
% surjI
thf(fact_912_surjI,axiom,
! [G: extended_ereal > nat,F: nat > extended_ereal] :
( ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_7659842161140344153al_nat @ G @ top_to5683747375963461374_ereal )
= top_top_set_nat ) ) ).
% surjI
thf(fact_913_surj__def,axiom,
! [F: c > b] :
( ( ( image_c_b @ F @ top_top_set_c )
= top_top_set_b )
= ( ! [Y2: b] :
? [X2: c] :
( Y2
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_914_surj__def,axiom,
! [F: c > c] :
( ( ( image_c_c @ F @ top_top_set_c )
= top_top_set_c )
= ( ! [Y2: c] :
? [X2: c] :
( Y2
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_915_surj__def,axiom,
! [F: b > $o] :
( ( ( image_b_o @ F @ top_top_set_b )
= top_top_set_o )
= ( ! [Y2: $o] :
? [X2: b] :
( Y2
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_916_surj__def,axiom,
! [F: b > c] :
( ( ( image_b_c @ F @ top_top_set_b )
= top_top_set_c )
= ( ! [Y2: c] :
? [X2: b] :
( Y2
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_917_surj__def,axiom,
! [F: b > b] :
( ( ( image_b_b @ F @ top_top_set_b )
= top_top_set_b )
= ( ! [Y2: b] :
? [X2: b] :
( Y2
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_918_surj__def,axiom,
! [F: b > extended_ereal] :
( ( ( image_5319725110001000852_ereal @ F @ top_top_set_b )
= top_to5683747375963461374_ereal )
= ( ! [Y2: extended_ereal] :
? [X2: b] :
( Y2
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_919_surj__def,axiom,
! [F: nat > extend8495563244428889912nnreal] :
( ( ( image_8459861568512453903nnreal @ F @ top_top_set_nat )
= top_to7994903218803871134nnreal )
= ( ! [Y2: extend8495563244428889912nnreal] :
? [X2: nat] :
( Y2
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_920_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y2: nat] :
? [X2: nat] :
( Y2
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_921_surj__def,axiom,
! [F: nat > extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F @ top_top_set_nat )
= top_to5683747375963461374_ereal )
= ( ! [Y2: extended_ereal] :
? [X2: nat] :
( Y2
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_922_surj__def,axiom,
! [F: extended_ereal > nat] :
( ( ( image_7659842161140344153al_nat @ F @ top_to5683747375963461374_ereal )
= top_top_set_nat )
= ( ! [Y2: nat] :
? [X2: extended_ereal] :
( Y2
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_923_INT__simps_I3_J,axiom,
! [C: set_Extended_ereal,A: extended_ereal > set_nat,B2: set_nat] :
( ( ( C = bot_bo8367695208629047834_ereal )
=> ( ( comple7806235888213564991et_nat
@ ( image_3090908713637162255et_nat
@ ^ [X2: extended_ereal] : ( minus_minus_set_nat @ ( A @ X2 ) @ B2 )
@ C ) )
= top_top_set_nat ) )
& ( ( C != bot_bo8367695208629047834_ereal )
=> ( ( comple7806235888213564991et_nat
@ ( image_3090908713637162255et_nat
@ ^ [X2: extended_ereal] : ( minus_minus_set_nat @ ( A @ X2 ) @ B2 )
@ C ) )
= ( minus_minus_set_nat @ ( comple7806235888213564991et_nat @ ( image_3090908713637162255et_nat @ A @ C ) ) @ B2 ) ) ) ) ).
% INT_simps(3)
thf(fact_924_INT__simps_I3_J,axiom,
! [C: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal > set_nat,B2: set_nat] :
( ( ( C = bot_bo4854962954004695426nnreal )
=> ( ( comple7806235888213564991et_nat
@ ( image_2869339492569777349et_nat
@ ^ [X2: extend8495563244428889912nnreal] : ( minus_minus_set_nat @ ( A @ X2 ) @ B2 )
@ C ) )
= top_top_set_nat ) )
& ( ( C != bot_bo4854962954004695426nnreal )
=> ( ( comple7806235888213564991et_nat
@ ( image_2869339492569777349et_nat
@ ^ [X2: extend8495563244428889912nnreal] : ( minus_minus_set_nat @ ( A @ X2 ) @ B2 )
@ C ) )
= ( minus_minus_set_nat @ ( comple7806235888213564991et_nat @ ( image_2869339492569777349et_nat @ A @ C ) ) @ B2 ) ) ) ) ).
% INT_simps(3)
thf(fact_925_INT__simps_I3_J,axiom,
! [C: set_nat,A: nat > set_nat,B2: set_nat] :
( ( ( C = bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( minus_minus_set_nat @ ( A @ X2 ) @ B2 )
@ C ) )
= top_top_set_nat ) )
& ( ( C != bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( minus_minus_set_nat @ ( A @ X2 ) @ B2 )
@ C ) )
= ( minus_minus_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A @ C ) ) @ B2 ) ) ) ) ).
% INT_simps(3)
thf(fact_926_INT__simps_I3_J,axiom,
! [C: set_Extended_ereal,A: extended_ereal > set_Extended_ereal,B2: set_Extended_ereal] :
( ( ( C = bot_bo8367695208629047834_ereal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5562094264469218789_ereal
@ ^ [X2: extended_ereal] : ( minus_1264018925008434325_ereal @ ( A @ X2 ) @ B2 )
@ C ) )
= top_to5683747375963461374_ereal ) )
& ( ( C != bot_bo8367695208629047834_ereal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5562094264469218789_ereal
@ ^ [X2: extended_ereal] : ( minus_1264018925008434325_ereal @ ( A @ X2 ) @ B2 )
@ C ) )
= ( minus_1264018925008434325_ereal @ ( comple4418415374894819509_ereal @ ( image_5562094264469218789_ereal @ A @ C ) ) @ B2 ) ) ) ) ).
% INT_simps(3)
thf(fact_927_INT__simps_I3_J,axiom,
! [C: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal > set_Extended_ereal,B2: set_Extended_ereal] :
( ( ( C = bot_bo4854962954004695426nnreal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5929344197358196911_ereal
@ ^ [X2: extend8495563244428889912nnreal] : ( minus_1264018925008434325_ereal @ ( A @ X2 ) @ B2 )
@ C ) )
= top_to5683747375963461374_ereal ) )
& ( ( C != bot_bo4854962954004695426nnreal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5929344197358196911_ereal
@ ^ [X2: extend8495563244428889912nnreal] : ( minus_1264018925008434325_ereal @ ( A @ X2 ) @ B2 )
@ C ) )
= ( minus_1264018925008434325_ereal @ ( comple4418415374894819509_ereal @ ( image_5929344197358196911_ereal @ A @ C ) ) @ B2 ) ) ) ) ).
% INT_simps(3)
thf(fact_928_INT__simps_I3_J,axiom,
! [C: set_nat,A: nat > set_Extended_ereal,B2: set_Extended_ereal] :
( ( ( C = bot_bot_set_nat )
=> ( ( comple4418415374894819509_ereal
@ ( image_305533323056406039_ereal
@ ^ [X2: nat] : ( minus_1264018925008434325_ereal @ ( A @ X2 ) @ B2 )
@ C ) )
= top_to5683747375963461374_ereal ) )
& ( ( C != bot_bot_set_nat )
=> ( ( comple4418415374894819509_ereal
@ ( image_305533323056406039_ereal
@ ^ [X2: nat] : ( minus_1264018925008434325_ereal @ ( A @ X2 ) @ B2 )
@ C ) )
= ( minus_1264018925008434325_ereal @ ( comple4418415374894819509_ereal @ ( image_305533323056406039_ereal @ A @ C ) ) @ B2 ) ) ) ) ).
% INT_simps(3)
thf(fact_929_INT__simps_I2_J,axiom,
! [C: set_Extended_ereal,A: set_nat,B2: extended_ereal > set_nat] :
( ( ( C = bot_bo8367695208629047834_ereal )
=> ( ( comple7806235888213564991et_nat
@ ( image_3090908713637162255et_nat
@ ^ [X2: extended_ereal] : ( inf_inf_set_nat @ A @ ( B2 @ X2 ) )
@ C ) )
= top_top_set_nat ) )
& ( ( C != bot_bo8367695208629047834_ereal )
=> ( ( comple7806235888213564991et_nat
@ ( image_3090908713637162255et_nat
@ ^ [X2: extended_ereal] : ( inf_inf_set_nat @ A @ ( B2 @ X2 ) )
@ C ) )
= ( inf_inf_set_nat @ A @ ( comple7806235888213564991et_nat @ ( image_3090908713637162255et_nat @ B2 @ C ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_930_INT__simps_I2_J,axiom,
! [C: set_Ex3793607809372303086nnreal,A: set_nat,B2: extend8495563244428889912nnreal > set_nat] :
( ( ( C = bot_bo4854962954004695426nnreal )
=> ( ( comple7806235888213564991et_nat
@ ( image_2869339492569777349et_nat
@ ^ [X2: extend8495563244428889912nnreal] : ( inf_inf_set_nat @ A @ ( B2 @ X2 ) )
@ C ) )
= top_top_set_nat ) )
& ( ( C != bot_bo4854962954004695426nnreal )
=> ( ( comple7806235888213564991et_nat
@ ( image_2869339492569777349et_nat
@ ^ [X2: extend8495563244428889912nnreal] : ( inf_inf_set_nat @ A @ ( B2 @ X2 ) )
@ C ) )
= ( inf_inf_set_nat @ A @ ( comple7806235888213564991et_nat @ ( image_2869339492569777349et_nat @ B2 @ C ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_931_INT__simps_I2_J,axiom,
! [C: set_nat,A: set_nat,B2: nat > set_nat] :
( ( ( C = bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( inf_inf_set_nat @ A @ ( B2 @ X2 ) )
@ C ) )
= top_top_set_nat ) )
& ( ( C != bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( inf_inf_set_nat @ A @ ( B2 @ X2 ) )
@ C ) )
= ( inf_inf_set_nat @ A @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B2 @ C ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_932_INT__simps_I2_J,axiom,
! [C: set_Extended_ereal,A: set_Extended_ereal,B2: extended_ereal > set_Extended_ereal] :
( ( ( C = bot_bo8367695208629047834_ereal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5562094264469218789_ereal
@ ^ [X2: extended_ereal] : ( inf_in2779415704524776092_ereal @ A @ ( B2 @ X2 ) )
@ C ) )
= top_to5683747375963461374_ereal ) )
& ( ( C != bot_bo8367695208629047834_ereal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5562094264469218789_ereal
@ ^ [X2: extended_ereal] : ( inf_in2779415704524776092_ereal @ A @ ( B2 @ X2 ) )
@ C ) )
= ( inf_in2779415704524776092_ereal @ A @ ( comple4418415374894819509_ereal @ ( image_5562094264469218789_ereal @ B2 @ C ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_933_INT__simps_I2_J,axiom,
! [C: set_Ex3793607809372303086nnreal,A: set_Extended_ereal,B2: extend8495563244428889912nnreal > set_Extended_ereal] :
( ( ( C = bot_bo4854962954004695426nnreal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5929344197358196911_ereal
@ ^ [X2: extend8495563244428889912nnreal] : ( inf_in2779415704524776092_ereal @ A @ ( B2 @ X2 ) )
@ C ) )
= top_to5683747375963461374_ereal ) )
& ( ( C != bot_bo4854962954004695426nnreal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5929344197358196911_ereal
@ ^ [X2: extend8495563244428889912nnreal] : ( inf_in2779415704524776092_ereal @ A @ ( B2 @ X2 ) )
@ C ) )
= ( inf_in2779415704524776092_ereal @ A @ ( comple4418415374894819509_ereal @ ( image_5929344197358196911_ereal @ B2 @ C ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_934_INT__simps_I2_J,axiom,
! [C: set_nat,A: set_Extended_ereal,B2: nat > set_Extended_ereal] :
( ( ( C = bot_bot_set_nat )
=> ( ( comple4418415374894819509_ereal
@ ( image_305533323056406039_ereal
@ ^ [X2: nat] : ( inf_in2779415704524776092_ereal @ A @ ( B2 @ X2 ) )
@ C ) )
= top_to5683747375963461374_ereal ) )
& ( ( C != bot_bot_set_nat )
=> ( ( comple4418415374894819509_ereal
@ ( image_305533323056406039_ereal
@ ^ [X2: nat] : ( inf_in2779415704524776092_ereal @ A @ ( B2 @ X2 ) )
@ C ) )
= ( inf_in2779415704524776092_ereal @ A @ ( comple4418415374894819509_ereal @ ( image_305533323056406039_ereal @ B2 @ C ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_935_Int__iff,axiom,
! [C2: $o,A: set_o,B2: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A @ B2 ) )
= ( ( member_o @ C2 @ A )
& ( member_o @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_936_Int__iff,axiom,
! [C2: extended_ereal,A: set_Extended_ereal,B2: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ C2 @ ( inf_in2779415704524776092_ereal @ A @ B2 ) )
= ( ( member2350847679896131959_ereal @ C2 @ A )
& ( member2350847679896131959_ereal @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_937_Int__iff,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B2 ) )
= ( ( member_nat @ C2 @ A )
& ( member_nat @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_938_Int__iff,axiom,
! [C2: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ C2 @ ( inf_in3368558534146122112nnreal @ A @ B2 ) )
= ( ( member7908768830364227535nnreal @ C2 @ A )
& ( member7908768830364227535nnreal @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_939_Int__iff,axiom,
! [C2: c,A: set_c,B2: set_c] :
( ( member_c @ C2 @ ( inf_inf_set_c @ A @ B2 ) )
= ( ( member_c @ C2 @ A )
& ( member_c @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_940_Int__iff,axiom,
! [C2: b,A: set_b,B2: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A @ B2 ) )
= ( ( member_b @ C2 @ A )
& ( member_b @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_941_IntI,axiom,
! [C2: $o,A: set_o,B2: set_o] :
( ( member_o @ C2 @ A )
=> ( ( member_o @ C2 @ B2 )
=> ( member_o @ C2 @ ( inf_inf_set_o @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_942_IntI,axiom,
! [C2: extended_ereal,A: set_Extended_ereal,B2: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ C2 @ A )
=> ( ( member2350847679896131959_ereal @ C2 @ B2 )
=> ( member2350847679896131959_ereal @ C2 @ ( inf_in2779415704524776092_ereal @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_943_IntI,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ A )
=> ( ( member_nat @ C2 @ B2 )
=> ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_944_IntI,axiom,
! [C2: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ C2 @ A )
=> ( ( member7908768830364227535nnreal @ C2 @ B2 )
=> ( member7908768830364227535nnreal @ C2 @ ( inf_in3368558534146122112nnreal @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_945_IntI,axiom,
! [C2: c,A: set_c,B2: set_c] :
( ( member_c @ C2 @ A )
=> ( ( member_c @ C2 @ B2 )
=> ( member_c @ C2 @ ( inf_inf_set_c @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_946_IntI,axiom,
! [C2: b,A: set_b,B2: set_b] :
( ( member_b @ C2 @ A )
=> ( ( member_b @ C2 @ B2 )
=> ( member_b @ C2 @ ( inf_inf_set_b @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_947_Diff__iff,axiom,
! [C2: $o,A: set_o,B2: set_o] :
( ( member_o @ C2 @ ( minus_minus_set_o @ A @ B2 ) )
= ( ( member_o @ C2 @ A )
& ~ ( member_o @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_948_Diff__iff,axiom,
! [C2: extended_ereal,A: set_Extended_ereal,B2: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ C2 @ ( minus_1264018925008434325_ereal @ A @ B2 ) )
= ( ( member2350847679896131959_ereal @ C2 @ A )
& ~ ( member2350847679896131959_ereal @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_949_Diff__iff,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B2 ) )
= ( ( member_nat @ C2 @ A )
& ~ ( member_nat @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_950_Diff__iff,axiom,
! [C2: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ C2 @ ( minus_104578273773384135nnreal @ A @ B2 ) )
= ( ( member7908768830364227535nnreal @ C2 @ A )
& ~ ( member7908768830364227535nnreal @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_951_Diff__iff,axiom,
! [C2: c,A: set_c,B2: set_c] :
( ( member_c @ C2 @ ( minus_minus_set_c @ A @ B2 ) )
= ( ( member_c @ C2 @ A )
& ~ ( member_c @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_952_Diff__iff,axiom,
! [C2: b,A: set_b,B2: set_b] :
( ( member_b @ C2 @ ( minus_minus_set_b @ A @ B2 ) )
= ( ( member_b @ C2 @ A )
& ~ ( member_b @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_953_DiffI,axiom,
! [C2: $o,A: set_o,B2: set_o] :
( ( member_o @ C2 @ A )
=> ( ~ ( member_o @ C2 @ B2 )
=> ( member_o @ C2 @ ( minus_minus_set_o @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_954_DiffI,axiom,
! [C2: extended_ereal,A: set_Extended_ereal,B2: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ C2 @ A )
=> ( ~ ( member2350847679896131959_ereal @ C2 @ B2 )
=> ( member2350847679896131959_ereal @ C2 @ ( minus_1264018925008434325_ereal @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_955_DiffI,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ A )
=> ( ~ ( member_nat @ C2 @ B2 )
=> ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_956_DiffI,axiom,
! [C2: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ C2 @ A )
=> ( ~ ( member7908768830364227535nnreal @ C2 @ B2 )
=> ( member7908768830364227535nnreal @ C2 @ ( minus_104578273773384135nnreal @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_957_DiffI,axiom,
! [C2: c,A: set_c,B2: set_c] :
( ( member_c @ C2 @ A )
=> ( ~ ( member_c @ C2 @ B2 )
=> ( member_c @ C2 @ ( minus_minus_set_c @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_958_DiffI,axiom,
! [C2: b,A: set_b,B2: set_b] :
( ( member_b @ C2 @ A )
=> ( ~ ( member_b @ C2 @ B2 )
=> ( member_b @ C2 @ ( minus_minus_set_b @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_959_Int__UNIV,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A @ B2 )
= top_top_set_nat )
= ( ( A = top_top_set_nat )
& ( B2 = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_960_Int__UNIV,axiom,
! [A: set_Extended_ereal,B2: set_Extended_ereal] :
( ( ( inf_in2779415704524776092_ereal @ A @ B2 )
= top_to5683747375963461374_ereal )
= ( ( A = top_to5683747375963461374_ereal )
& ( B2 = top_to5683747375963461374_ereal ) ) ) ).
% Int_UNIV
thf(fact_961_Diff__cancel,axiom,
! [A: set_Extended_ereal] :
( ( minus_1264018925008434325_ereal @ A @ A )
= bot_bo8367695208629047834_ereal ) ).
% Diff_cancel
thf(fact_962_Diff__cancel,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( minus_104578273773384135nnreal @ A @ A )
= bot_bo4854962954004695426nnreal ) ).
% Diff_cancel
thf(fact_963_Diff__cancel,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ A )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_964_empty__Diff,axiom,
! [A: set_Extended_ereal] :
( ( minus_1264018925008434325_ereal @ bot_bo8367695208629047834_ereal @ A )
= bot_bo8367695208629047834_ereal ) ).
% empty_Diff
thf(fact_965_empty__Diff,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( minus_104578273773384135nnreal @ bot_bo4854962954004695426nnreal @ A )
= bot_bo4854962954004695426nnreal ) ).
% empty_Diff
thf(fact_966_empty__Diff,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_967_Diff__empty,axiom,
! [A: set_Extended_ereal] :
( ( minus_1264018925008434325_ereal @ A @ bot_bo8367695208629047834_ereal )
= A ) ).
% Diff_empty
thf(fact_968_Diff__empty,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( minus_104578273773384135nnreal @ A @ bot_bo4854962954004695426nnreal )
= A ) ).
% Diff_empty
thf(fact_969_Diff__empty,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Diff_empty
thf(fact_970_Diff__UNIV,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( minus_104578273773384135nnreal @ A @ top_to7994903218803871134nnreal )
= bot_bo4854962954004695426nnreal ) ).
% Diff_UNIV
thf(fact_971_Diff__UNIV,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Diff_UNIV
thf(fact_972_Diff__UNIV,axiom,
! [A: set_Extended_ereal] :
( ( minus_1264018925008434325_ereal @ A @ top_to5683747375963461374_ereal )
= bot_bo8367695208629047834_ereal ) ).
% Diff_UNIV
thf(fact_973_Diff__disjoint,axiom,
! [A: set_Extended_ereal,B2: set_Extended_ereal] :
( ( inf_in2779415704524776092_ereal @ A @ ( minus_1264018925008434325_ereal @ B2 @ A ) )
= bot_bo8367695208629047834_ereal ) ).
% Diff_disjoint
thf(fact_974_Diff__disjoint,axiom,
! [A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( inf_in3368558534146122112nnreal @ A @ ( minus_104578273773384135nnreal @ B2 @ A ) )
= bot_bo4854962954004695426nnreal ) ).
% Diff_disjoint
thf(fact_975_Diff__disjoint,axiom,
! [A: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B2 @ A ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_976_INT__simps_I1_J,axiom,
! [C: set_Extended_ereal,A: extended_ereal > set_nat,B2: set_nat] :
( ( ( C = bot_bo8367695208629047834_ereal )
=> ( ( comple7806235888213564991et_nat
@ ( image_3090908713637162255et_nat
@ ^ [X2: extended_ereal] : ( inf_inf_set_nat @ ( A @ X2 ) @ B2 )
@ C ) )
= top_top_set_nat ) )
& ( ( C != bot_bo8367695208629047834_ereal )
=> ( ( comple7806235888213564991et_nat
@ ( image_3090908713637162255et_nat
@ ^ [X2: extended_ereal] : ( inf_inf_set_nat @ ( A @ X2 ) @ B2 )
@ C ) )
= ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_3090908713637162255et_nat @ A @ C ) ) @ B2 ) ) ) ) ).
% INT_simps(1)
thf(fact_977_INT__simps_I1_J,axiom,
! [C: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal > set_nat,B2: set_nat] :
( ( ( C = bot_bo4854962954004695426nnreal )
=> ( ( comple7806235888213564991et_nat
@ ( image_2869339492569777349et_nat
@ ^ [X2: extend8495563244428889912nnreal] : ( inf_inf_set_nat @ ( A @ X2 ) @ B2 )
@ C ) )
= top_top_set_nat ) )
& ( ( C != bot_bo4854962954004695426nnreal )
=> ( ( comple7806235888213564991et_nat
@ ( image_2869339492569777349et_nat
@ ^ [X2: extend8495563244428889912nnreal] : ( inf_inf_set_nat @ ( A @ X2 ) @ B2 )
@ C ) )
= ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_2869339492569777349et_nat @ A @ C ) ) @ B2 ) ) ) ) ).
% INT_simps(1)
thf(fact_978_INT__simps_I1_J,axiom,
! [C: set_nat,A: nat > set_nat,B2: set_nat] :
( ( ( C = bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( inf_inf_set_nat @ ( A @ X2 ) @ B2 )
@ C ) )
= top_top_set_nat ) )
& ( ( C != bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( inf_inf_set_nat @ ( A @ X2 ) @ B2 )
@ C ) )
= ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A @ C ) ) @ B2 ) ) ) ) ).
% INT_simps(1)
thf(fact_979_INT__simps_I1_J,axiom,
! [C: set_Extended_ereal,A: extended_ereal > set_Extended_ereal,B2: set_Extended_ereal] :
( ( ( C = bot_bo8367695208629047834_ereal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5562094264469218789_ereal
@ ^ [X2: extended_ereal] : ( inf_in2779415704524776092_ereal @ ( A @ X2 ) @ B2 )
@ C ) )
= top_to5683747375963461374_ereal ) )
& ( ( C != bot_bo8367695208629047834_ereal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5562094264469218789_ereal
@ ^ [X2: extended_ereal] : ( inf_in2779415704524776092_ereal @ ( A @ X2 ) @ B2 )
@ C ) )
= ( inf_in2779415704524776092_ereal @ ( comple4418415374894819509_ereal @ ( image_5562094264469218789_ereal @ A @ C ) ) @ B2 ) ) ) ) ).
% INT_simps(1)
thf(fact_980_INT__simps_I1_J,axiom,
! [C: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal > set_Extended_ereal,B2: set_Extended_ereal] :
( ( ( C = bot_bo4854962954004695426nnreal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5929344197358196911_ereal
@ ^ [X2: extend8495563244428889912nnreal] : ( inf_in2779415704524776092_ereal @ ( A @ X2 ) @ B2 )
@ C ) )
= top_to5683747375963461374_ereal ) )
& ( ( C != bot_bo4854962954004695426nnreal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5929344197358196911_ereal
@ ^ [X2: extend8495563244428889912nnreal] : ( inf_in2779415704524776092_ereal @ ( A @ X2 ) @ B2 )
@ C ) )
= ( inf_in2779415704524776092_ereal @ ( comple4418415374894819509_ereal @ ( image_5929344197358196911_ereal @ A @ C ) ) @ B2 ) ) ) ) ).
% INT_simps(1)
thf(fact_981_INT__simps_I1_J,axiom,
! [C: set_nat,A: nat > set_Extended_ereal,B2: set_Extended_ereal] :
( ( ( C = bot_bot_set_nat )
=> ( ( comple4418415374894819509_ereal
@ ( image_305533323056406039_ereal
@ ^ [X2: nat] : ( inf_in2779415704524776092_ereal @ ( A @ X2 ) @ B2 )
@ C ) )
= top_to5683747375963461374_ereal ) )
& ( ( C != bot_bot_set_nat )
=> ( ( comple4418415374894819509_ereal
@ ( image_305533323056406039_ereal
@ ^ [X2: nat] : ( inf_in2779415704524776092_ereal @ ( A @ X2 ) @ B2 )
@ C ) )
= ( inf_in2779415704524776092_ereal @ ( comple4418415374894819509_ereal @ ( image_305533323056406039_ereal @ A @ C ) ) @ B2 ) ) ) ) ).
% INT_simps(1)
thf(fact_982_Int__Diff__disjoint,axiom,
! [A: set_Extended_ereal,B2: set_Extended_ereal] :
( ( inf_in2779415704524776092_ereal @ ( inf_in2779415704524776092_ereal @ A @ B2 ) @ ( minus_1264018925008434325_ereal @ A @ B2 ) )
= bot_bo8367695208629047834_ereal ) ).
% Int_Diff_disjoint
thf(fact_983_Int__Diff__disjoint,axiom,
! [A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( inf_in3368558534146122112nnreal @ ( inf_in3368558534146122112nnreal @ A @ B2 ) @ ( minus_104578273773384135nnreal @ A @ B2 ) )
= bot_bo4854962954004695426nnreal ) ).
% Int_Diff_disjoint
thf(fact_984_Int__Diff__disjoint,axiom,
! [A: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ ( minus_minus_set_nat @ A @ B2 ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_985_Diff__triv,axiom,
! [A: set_Extended_ereal,B2: set_Extended_ereal] :
( ( ( inf_in2779415704524776092_ereal @ A @ B2 )
= bot_bo8367695208629047834_ereal )
=> ( ( minus_1264018925008434325_ereal @ A @ B2 )
= A ) ) ).
% Diff_triv
thf(fact_986_Diff__triv,axiom,
! [A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( ( inf_in3368558534146122112nnreal @ A @ B2 )
= bot_bo4854962954004695426nnreal )
=> ( ( minus_104578273773384135nnreal @ A @ B2 )
= A ) ) ).
% Diff_triv
thf(fact_987_Diff__triv,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A @ B2 )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A @ B2 )
= A ) ) ).
% Diff_triv
thf(fact_988_DiffD2,axiom,
! [C2: $o,A: set_o,B2: set_o] :
( ( member_o @ C2 @ ( minus_minus_set_o @ A @ B2 ) )
=> ~ ( member_o @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_989_DiffD2,axiom,
! [C2: extended_ereal,A: set_Extended_ereal,B2: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ C2 @ ( minus_1264018925008434325_ereal @ A @ B2 ) )
=> ~ ( member2350847679896131959_ereal @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_990_DiffD2,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B2 ) )
=> ~ ( member_nat @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_991_DiffD2,axiom,
! [C2: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ C2 @ ( minus_104578273773384135nnreal @ A @ B2 ) )
=> ~ ( member7908768830364227535nnreal @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_992_DiffD2,axiom,
! [C2: c,A: set_c,B2: set_c] :
( ( member_c @ C2 @ ( minus_minus_set_c @ A @ B2 ) )
=> ~ ( member_c @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_993_DiffD2,axiom,
! [C2: b,A: set_b,B2: set_b] :
( ( member_b @ C2 @ ( minus_minus_set_b @ A @ B2 ) )
=> ~ ( member_b @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_994_DiffD1,axiom,
! [C2: $o,A: set_o,B2: set_o] :
( ( member_o @ C2 @ ( minus_minus_set_o @ A @ B2 ) )
=> ( member_o @ C2 @ A ) ) ).
% DiffD1
thf(fact_995_DiffD1,axiom,
! [C2: extended_ereal,A: set_Extended_ereal,B2: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ C2 @ ( minus_1264018925008434325_ereal @ A @ B2 ) )
=> ( member2350847679896131959_ereal @ C2 @ A ) ) ).
% DiffD1
thf(fact_996_DiffD1,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B2 ) )
=> ( member_nat @ C2 @ A ) ) ).
% DiffD1
thf(fact_997_DiffD1,axiom,
! [C2: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ C2 @ ( minus_104578273773384135nnreal @ A @ B2 ) )
=> ( member7908768830364227535nnreal @ C2 @ A ) ) ).
% DiffD1
thf(fact_998_DiffD1,axiom,
! [C2: c,A: set_c,B2: set_c] :
( ( member_c @ C2 @ ( minus_minus_set_c @ A @ B2 ) )
=> ( member_c @ C2 @ A ) ) ).
% DiffD1
thf(fact_999_DiffD1,axiom,
! [C2: b,A: set_b,B2: set_b] :
( ( member_b @ C2 @ ( minus_minus_set_b @ A @ B2 ) )
=> ( member_b @ C2 @ A ) ) ).
% DiffD1
thf(fact_1000_IntD2,axiom,
! [C2: $o,A: set_o,B2: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A @ B2 ) )
=> ( member_o @ C2 @ B2 ) ) ).
% IntD2
thf(fact_1001_IntD2,axiom,
! [C2: extended_ereal,A: set_Extended_ereal,B2: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ C2 @ ( inf_in2779415704524776092_ereal @ A @ B2 ) )
=> ( member2350847679896131959_ereal @ C2 @ B2 ) ) ).
% IntD2
thf(fact_1002_IntD2,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B2 ) )
=> ( member_nat @ C2 @ B2 ) ) ).
% IntD2
thf(fact_1003_IntD2,axiom,
! [C2: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ C2 @ ( inf_in3368558534146122112nnreal @ A @ B2 ) )
=> ( member7908768830364227535nnreal @ C2 @ B2 ) ) ).
% IntD2
thf(fact_1004_IntD2,axiom,
! [C2: c,A: set_c,B2: set_c] :
( ( member_c @ C2 @ ( inf_inf_set_c @ A @ B2 ) )
=> ( member_c @ C2 @ B2 ) ) ).
% IntD2
thf(fact_1005_IntD2,axiom,
! [C2: b,A: set_b,B2: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A @ B2 ) )
=> ( member_b @ C2 @ B2 ) ) ).
% IntD2
thf(fact_1006_IntD1,axiom,
! [C2: $o,A: set_o,B2: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A @ B2 ) )
=> ( member_o @ C2 @ A ) ) ).
% IntD1
thf(fact_1007_IntD1,axiom,
! [C2: extended_ereal,A: set_Extended_ereal,B2: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ C2 @ ( inf_in2779415704524776092_ereal @ A @ B2 ) )
=> ( member2350847679896131959_ereal @ C2 @ A ) ) ).
% IntD1
thf(fact_1008_IntD1,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B2 ) )
=> ( member_nat @ C2 @ A ) ) ).
% IntD1
thf(fact_1009_IntD1,axiom,
! [C2: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ C2 @ ( inf_in3368558534146122112nnreal @ A @ B2 ) )
=> ( member7908768830364227535nnreal @ C2 @ A ) ) ).
% IntD1
thf(fact_1010_IntD1,axiom,
! [C2: c,A: set_c,B2: set_c] :
( ( member_c @ C2 @ ( inf_inf_set_c @ A @ B2 ) )
=> ( member_c @ C2 @ A ) ) ).
% IntD1
thf(fact_1011_IntD1,axiom,
! [C2: b,A: set_b,B2: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A @ B2 ) )
=> ( member_b @ C2 @ A ) ) ).
% IntD1
thf(fact_1012_DiffE,axiom,
! [C2: $o,A: set_o,B2: set_o] :
( ( member_o @ C2 @ ( minus_minus_set_o @ A @ B2 ) )
=> ~ ( ( member_o @ C2 @ A )
=> ( member_o @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_1013_DiffE,axiom,
! [C2: extended_ereal,A: set_Extended_ereal,B2: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ C2 @ ( minus_1264018925008434325_ereal @ A @ B2 ) )
=> ~ ( ( member2350847679896131959_ereal @ C2 @ A )
=> ( member2350847679896131959_ereal @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_1014_DiffE,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B2 ) )
=> ~ ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_1015_DiffE,axiom,
! [C2: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ C2 @ ( minus_104578273773384135nnreal @ A @ B2 ) )
=> ~ ( ( member7908768830364227535nnreal @ C2 @ A )
=> ( member7908768830364227535nnreal @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_1016_DiffE,axiom,
! [C2: c,A: set_c,B2: set_c] :
( ( member_c @ C2 @ ( minus_minus_set_c @ A @ B2 ) )
=> ~ ( ( member_c @ C2 @ A )
=> ( member_c @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_1017_DiffE,axiom,
! [C2: b,A: set_b,B2: set_b] :
( ( member_b @ C2 @ ( minus_minus_set_b @ A @ B2 ) )
=> ~ ( ( member_b @ C2 @ A )
=> ( member_b @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_1018_IntE,axiom,
! [C2: $o,A: set_o,B2: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A @ B2 ) )
=> ~ ( ( member_o @ C2 @ A )
=> ~ ( member_o @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_1019_IntE,axiom,
! [C2: extended_ereal,A: set_Extended_ereal,B2: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ C2 @ ( inf_in2779415704524776092_ereal @ A @ B2 ) )
=> ~ ( ( member2350847679896131959_ereal @ C2 @ A )
=> ~ ( member2350847679896131959_ereal @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_1020_IntE,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B2 ) )
=> ~ ( ( member_nat @ C2 @ A )
=> ~ ( member_nat @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_1021_IntE,axiom,
! [C2: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ C2 @ ( inf_in3368558534146122112nnreal @ A @ B2 ) )
=> ~ ( ( member7908768830364227535nnreal @ C2 @ A )
=> ~ ( member7908768830364227535nnreal @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_1022_IntE,axiom,
! [C2: c,A: set_c,B2: set_c] :
( ( member_c @ C2 @ ( inf_inf_set_c @ A @ B2 ) )
=> ~ ( ( member_c @ C2 @ A )
=> ~ ( member_c @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_1023_IntE,axiom,
! [C2: b,A: set_b,B2: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A @ B2 ) )
=> ~ ( ( member_b @ C2 @ A )
=> ~ ( member_b @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_1024_set__diff__eq,axiom,
( minus_minus_set_b
= ( ^ [A3: set_b,B3: set_b] :
( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ A3 )
& ~ ( member_b @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1025_set__diff__eq,axiom,
( minus_minus_set_c
= ( ^ [A3: set_c,B3: set_c] :
( collect_c
@ ^ [X2: c] :
( ( member_c @ X2 @ A3 )
& ~ ( member_c @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1026_set__diff__eq,axiom,
( minus_1264018925008434325_ereal
= ( ^ [A3: set_Extended_ereal,B3: set_Extended_ereal] :
( collec5835592288176408249_ereal
@ ^ [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A3 )
& ~ ( member2350847679896131959_ereal @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1027_set__diff__eq,axiom,
( minus_104578273773384135nnreal
= ( ^ [A3: set_Ex3793607809372303086nnreal,B3: set_Ex3793607809372303086nnreal] :
( collec6648975593938027277nnreal
@ ^ [X2: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X2 @ A3 )
& ~ ( member7908768830364227535nnreal @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1028_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_1029_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_1030_Collect__conj__eq,axiom,
! [P: b > $o,Q: b > $o] :
( ( collect_b
@ ^ [X2: b] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_1031_Collect__conj__eq,axiom,
! [P: c > $o,Q: c > $o] :
( ( collect_c
@ ^ [X2: c] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_c @ ( collect_c @ P ) @ ( collect_c @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_1032_Collect__conj__eq,axiom,
! [P: extended_ereal > $o,Q: extended_ereal > $o] :
( ( collec5835592288176408249_ereal
@ ^ [X2: extended_ereal] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_in2779415704524776092_ereal @ ( collec5835592288176408249_ereal @ P ) @ ( collec5835592288176408249_ereal @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_1033_Collect__conj__eq,axiom,
! [P: extend8495563244428889912nnreal > $o,Q: extend8495563244428889912nnreal > $o] :
( ( collec6648975593938027277nnreal
@ ^ [X2: extend8495563244428889912nnreal] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_in3368558534146122112nnreal @ ( collec6648975593938027277nnreal @ P ) @ ( collec6648975593938027277nnreal @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_1034_Collect__conj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_1035_Collect__conj__eq,axiom,
! [P: $o > $o,Q: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_o @ ( collect_o @ P ) @ ( collect_o @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_1036_Int__Collect,axiom,
! [X: b,A: set_b,P: b > $o] :
( ( member_b @ X @ ( inf_inf_set_b @ A @ ( collect_b @ P ) ) )
= ( ( member_b @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_1037_Int__Collect,axiom,
! [X: c,A: set_c,P: c > $o] :
( ( member_c @ X @ ( inf_inf_set_c @ A @ ( collect_c @ P ) ) )
= ( ( member_c @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_1038_Int__Collect,axiom,
! [X: extended_ereal,A: set_Extended_ereal,P: extended_ereal > $o] :
( ( member2350847679896131959_ereal @ X @ ( inf_in2779415704524776092_ereal @ A @ ( collec5835592288176408249_ereal @ P ) ) )
= ( ( member2350847679896131959_ereal @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_1039_Int__Collect,axiom,
! [X: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal,P: extend8495563244428889912nnreal > $o] :
( ( member7908768830364227535nnreal @ X @ ( inf_in3368558534146122112nnreal @ A @ ( collec6648975593938027277nnreal @ P ) ) )
= ( ( member7908768830364227535nnreal @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_1040_Int__Collect,axiom,
! [X: nat,A: set_nat,P: nat > $o] :
( ( member_nat @ X @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_1041_Int__Collect,axiom,
! [X: $o,A: set_o,P: $o > $o] :
( ( member_o @ X @ ( inf_inf_set_o @ A @ ( collect_o @ P ) ) )
= ( ( member_o @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_1042_Int__def,axiom,
( inf_inf_set_b
= ( ^ [A3: set_b,B3: set_b] :
( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ A3 )
& ( member_b @ X2 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_1043_Int__def,axiom,
( inf_inf_set_c
= ( ^ [A3: set_c,B3: set_c] :
( collect_c
@ ^ [X2: c] :
( ( member_c @ X2 @ A3 )
& ( member_c @ X2 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_1044_Int__def,axiom,
( inf_in2779415704524776092_ereal
= ( ^ [A3: set_Extended_ereal,B3: set_Extended_ereal] :
( collec5835592288176408249_ereal
@ ^ [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A3 )
& ( member2350847679896131959_ereal @ X2 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_1045_Int__def,axiom,
( inf_in3368558534146122112nnreal
= ( ^ [A3: set_Ex3793607809372303086nnreal,B3: set_Ex3793607809372303086nnreal] :
( collec6648975593938027277nnreal
@ ^ [X2: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X2 @ A3 )
& ( member7908768830364227535nnreal @ X2 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_1046_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_1047_Int__def,axiom,
( inf_inf_set_o
= ( ^ [A3: set_o,B3: set_o] :
( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A3 )
& ( member_o @ X2 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_1048_Int__UNIV__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ top_top_set_nat )
= A ) ).
% Int_UNIV_right
thf(fact_1049_Int__UNIV__right,axiom,
! [A: set_Extended_ereal] :
( ( inf_in2779415704524776092_ereal @ A @ top_to5683747375963461374_ereal )
= A ) ).
% Int_UNIV_right
thf(fact_1050_Int__UNIV__left,axiom,
! [B2: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ B2 )
= B2 ) ).
% Int_UNIV_left
thf(fact_1051_Int__UNIV__left,axiom,
! [B2: set_Extended_ereal] :
( ( inf_in2779415704524776092_ereal @ top_to5683747375963461374_ereal @ B2 )
= B2 ) ).
% Int_UNIV_left
thf(fact_1052_disjoint__iff__not__equal,axiom,
! [A: set_Extended_ereal,B2: set_Extended_ereal] :
( ( ( inf_in2779415704524776092_ereal @ A @ B2 )
= bot_bo8367695208629047834_ereal )
= ( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A )
=> ! [Y2: extended_ereal] :
( ( member2350847679896131959_ereal @ Y2 @ B2 )
=> ( X2 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1053_disjoint__iff__not__equal,axiom,
! [A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( ( inf_in3368558534146122112nnreal @ A @ B2 )
= bot_bo4854962954004695426nnreal )
= ( ! [X2: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X2 @ A )
=> ! [Y2: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ Y2 @ B2 )
=> ( X2 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1054_disjoint__iff__not__equal,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A @ B2 )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ B2 )
=> ( X2 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1055_Int__empty__right,axiom,
! [A: set_Extended_ereal] :
( ( inf_in2779415704524776092_ereal @ A @ bot_bo8367695208629047834_ereal )
= bot_bo8367695208629047834_ereal ) ).
% Int_empty_right
thf(fact_1056_Int__empty__right,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( inf_in3368558534146122112nnreal @ A @ bot_bo4854962954004695426nnreal )
= bot_bo4854962954004695426nnreal ) ).
% Int_empty_right
thf(fact_1057_Int__empty__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_1058_Int__empty__left,axiom,
! [B2: set_Extended_ereal] :
( ( inf_in2779415704524776092_ereal @ bot_bo8367695208629047834_ereal @ B2 )
= bot_bo8367695208629047834_ereal ) ).
% Int_empty_left
thf(fact_1059_Int__empty__left,axiom,
! [B2: set_Ex3793607809372303086nnreal] :
( ( inf_in3368558534146122112nnreal @ bot_bo4854962954004695426nnreal @ B2 )
= bot_bo4854962954004695426nnreal ) ).
% Int_empty_left
thf(fact_1060_Int__empty__left,axiom,
! [B2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B2 )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_1061_disjoint__iff,axiom,
! [A: set_o,B2: set_o] :
( ( ( inf_inf_set_o @ A @ B2 )
= bot_bot_set_o )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ~ ( member_o @ X2 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1062_disjoint__iff,axiom,
! [A: set_c,B2: set_c] :
( ( ( inf_inf_set_c @ A @ B2 )
= bot_bot_set_c )
= ( ! [X2: c] :
( ( member_c @ X2 @ A )
=> ~ ( member_c @ X2 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1063_disjoint__iff,axiom,
! [A: set_b,B2: set_b] :
( ( ( inf_inf_set_b @ A @ B2 )
= bot_bot_set_b )
= ( ! [X2: b] :
( ( member_b @ X2 @ A )
=> ~ ( member_b @ X2 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1064_disjoint__iff,axiom,
! [A: set_Extended_ereal,B2: set_Extended_ereal] :
( ( ( inf_in2779415704524776092_ereal @ A @ B2 )
= bot_bo8367695208629047834_ereal )
= ( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A )
=> ~ ( member2350847679896131959_ereal @ X2 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1065_disjoint__iff,axiom,
! [A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( ( inf_in3368558534146122112nnreal @ A @ B2 )
= bot_bo4854962954004695426nnreal )
= ( ! [X2: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X2 @ A )
=> ~ ( member7908768830364227535nnreal @ X2 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1066_disjoint__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A @ B2 )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ~ ( member_nat @ X2 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1067_Int__emptyI,axiom,
! [A: set_o,B2: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ~ ( member_o @ X3 @ B2 ) )
=> ( ( inf_inf_set_o @ A @ B2 )
= bot_bot_set_o ) ) ).
% Int_emptyI
thf(fact_1068_Int__emptyI,axiom,
! [A: set_c,B2: set_c] :
( ! [X3: c] :
( ( member_c @ X3 @ A )
=> ~ ( member_c @ X3 @ B2 ) )
=> ( ( inf_inf_set_c @ A @ B2 )
= bot_bot_set_c ) ) ).
% Int_emptyI
thf(fact_1069_Int__emptyI,axiom,
! [A: set_b,B2: set_b] :
( ! [X3: b] :
( ( member_b @ X3 @ A )
=> ~ ( member_b @ X3 @ B2 ) )
=> ( ( inf_inf_set_b @ A @ B2 )
= bot_bot_set_b ) ) ).
% Int_emptyI
thf(fact_1070_Int__emptyI,axiom,
! [A: set_Extended_ereal,B2: set_Extended_ereal] :
( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
=> ~ ( member2350847679896131959_ereal @ X3 @ B2 ) )
=> ( ( inf_in2779415704524776092_ereal @ A @ B2 )
= bot_bo8367695208629047834_ereal ) ) ).
% Int_emptyI
thf(fact_1071_Int__emptyI,axiom,
! [A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A )
=> ~ ( member7908768830364227535nnreal @ X3 @ B2 ) )
=> ( ( inf_in3368558534146122112nnreal @ A @ B2 )
= bot_bo4854962954004695426nnreal ) ) ).
% Int_emptyI
thf(fact_1072_Int__emptyI,axiom,
! [A: set_nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ~ ( member_nat @ X3 @ B2 ) )
=> ( ( inf_inf_set_nat @ A @ B2 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_1073_INF__UNIV__bool__expand,axiom,
! [A: $o > $o] :
( ( complete_Inf_Inf_o @ ( image_o_o @ A @ top_top_set_o ) )
= ( inf_inf_o @ ( A @ $true ) @ ( A @ $false ) ) ) ).
% INF_UNIV_bool_expand
thf(fact_1074_INF__UNIV__bool__expand,axiom,
! [A: $o > extended_ereal] :
( ( comple3556804143462414037_ereal @ ( image_7729549296133164475_ereal @ A @ top_top_set_o ) )
= ( inf_in2794916579150040252_ereal @ ( A @ $true ) @ ( A @ $false ) ) ) ).
% INF_UNIV_bool_expand
thf(fact_1075_INF__UNIV__bool__expand,axiom,
! [A: $o > extend8495563244428889912nnreal] :
( ( comple7330758040695736817nnreal @ ( image_3342735880743421067nnreal @ A @ top_top_set_o ) )
= ( inf_in7439215052339218890nnreal @ ( A @ $true ) @ ( A @ $false ) ) ) ).
% INF_UNIV_bool_expand
thf(fact_1076_inf__Sup,axiom,
! [A2: $o,B2: set_o] :
( ( inf_inf_o @ A2 @ ( complete_Sup_Sup_o @ B2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ ( inf_inf_o @ A2 ) @ B2 ) ) ) ).
% inf_Sup
thf(fact_1077_inf__Sup,axiom,
! [A2: extended_ereal,B2: set_Extended_ereal] :
( ( inf_in2794916579150040252_ereal @ A2 @ ( comple8415311339701865915_ereal @ B2 ) )
= ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ ( inf_in2794916579150040252_ereal @ A2 ) @ B2 ) ) ) ).
% inf_Sup
thf(fact_1078_inf__Sup,axiom,
! [A2: extend8495563244428889912nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( inf_in7439215052339218890nnreal @ A2 @ ( comple6814414086264997003nnreal @ B2 ) )
= ( comple6814414086264997003nnreal @ ( image_8394674774369097847nnreal @ ( inf_in7439215052339218890nnreal @ A2 ) @ B2 ) ) ) ).
% inf_Sup
thf(fact_1079_Sup__inf__eq__bot__iff,axiom,
! [B2: set_se6634062954251873166_ereal,A2: set_Extended_ereal] :
( ( ( inf_in2779415704524776092_ereal @ ( comple4319282863272126363_ereal @ B2 ) @ A2 )
= bot_bo8367695208629047834_ereal )
= ( ! [X2: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ X2 @ B2 )
=> ( ( inf_in2779415704524776092_ereal @ X2 @ A2 )
= bot_bo8367695208629047834_ereal ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_1080_Sup__inf__eq__bot__iff,axiom,
! [B2: set_se4580700918925141924nnreal,A2: set_Ex3793607809372303086nnreal] :
( ( ( inf_in3368558534146122112nnreal @ ( comple4226387801268262977nnreal @ B2 ) @ A2 )
= bot_bo4854962954004695426nnreal )
= ( ! [X2: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X2 @ B2 )
=> ( ( inf_in3368558534146122112nnreal @ X2 @ A2 )
= bot_bo4854962954004695426nnreal ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_1081_Sup__inf__eq__bot__iff,axiom,
! [B2: set_set_nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ B2 ) @ A2 )
= bot_bot_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ B2 )
=> ( ( inf_inf_set_nat @ X2 @ A2 )
= bot_bot_set_nat ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_1082_Sup__inf__eq__bot__iff,axiom,
! [B2: set_o,A2: $o] :
( ( ( inf_inf_o @ ( complete_Sup_Sup_o @ B2 ) @ A2 )
= bot_bot_o )
= ( ! [X2: $o] :
( ( member_o @ X2 @ B2 )
=> ( ( inf_inf_o @ X2 @ A2 )
= bot_bot_o ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_1083_Sup__inf__eq__bot__iff,axiom,
! [B2: set_Extended_ereal,A2: extended_ereal] :
( ( ( inf_in2794916579150040252_ereal @ ( comple8415311339701865915_ereal @ B2 ) @ A2 )
= bot_bo2710585358178759738_ereal )
= ( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ B2 )
=> ( ( inf_in2794916579150040252_ereal @ X2 @ A2 )
= bot_bo2710585358178759738_ereal ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_1084_Sup__inf__eq__bot__iff,axiom,
! [B2: set_Ex3793607809372303086nnreal,A2: extend8495563244428889912nnreal] :
( ( ( inf_in7439215052339218890nnreal @ ( comple6814414086264997003nnreal @ B2 ) @ A2 )
= bot_bo841427958541957580nnreal )
= ( ! [X2: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X2 @ B2 )
=> ( ( inf_in7439215052339218890nnreal @ X2 @ A2 )
= bot_bo841427958541957580nnreal ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_1085_Union__disjoint,axiom,
! [C: set_se6634062954251873166_ereal,A: set_Extended_ereal] :
( ( ( inf_in2779415704524776092_ereal @ ( comple4319282863272126363_ereal @ C ) @ A )
= bot_bo8367695208629047834_ereal )
= ( ! [X2: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ X2 @ C )
=> ( ( inf_in2779415704524776092_ereal @ X2 @ A )
= bot_bo8367695208629047834_ereal ) ) ) ) ).
% Union_disjoint
thf(fact_1086_Union__disjoint,axiom,
! [C: set_se4580700918925141924nnreal,A: set_Ex3793607809372303086nnreal] :
( ( ( inf_in3368558534146122112nnreal @ ( comple4226387801268262977nnreal @ C ) @ A )
= bot_bo4854962954004695426nnreal )
= ( ! [X2: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X2 @ C )
=> ( ( inf_in3368558534146122112nnreal @ X2 @ A )
= bot_bo4854962954004695426nnreal ) ) ) ) ).
% Union_disjoint
thf(fact_1087_Union__disjoint,axiom,
! [C: set_set_nat,A: set_nat] :
( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ C ) @ A )
= bot_bot_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ C )
=> ( ( inf_inf_set_nat @ X2 @ A )
= bot_bot_set_nat ) ) ) ) ).
% Union_disjoint
thf(fact_1088_SUP__inf__distrib2,axiom,
! [F: b > $o,A: set_b,G: b > $o,B2: set_b] :
( ( inf_inf_o @ ( complete_Sup_Sup_o @ ( image_b_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_b_o @ G @ B2 ) ) )
= ( complete_Sup_Sup_o
@ ( image_b_o
@ ^ [A4: b] :
( complete_Sup_Sup_o
@ ( image_b_o
@ ^ [B4: b] : ( inf_inf_o @ ( F @ A4 ) @ ( G @ B4 ) )
@ B2 ) )
@ A ) ) ) ).
% SUP_inf_distrib2
thf(fact_1089_SUP__inf__distrib2,axiom,
! [F: nat > extended_ereal,A: set_nat,G: nat > extended_ereal,B2: set_nat] :
( ( inf_in2794916579150040252_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ B2 ) ) )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [A4: nat] :
( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [B4: nat] : ( inf_in2794916579150040252_ereal @ ( F @ A4 ) @ ( G @ B4 ) )
@ B2 ) )
@ A ) ) ) ).
% SUP_inf_distrib2
thf(fact_1090_SUP__inf__distrib2,axiom,
! [F: nat > extended_ereal,A: set_nat,G: extended_ereal > extended_ereal,B2: set_Extended_ereal] :
( ( inf_in2794916579150040252_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ G @ B2 ) ) )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [A4: nat] :
( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [B4: extended_ereal] : ( inf_in2794916579150040252_ereal @ ( F @ A4 ) @ ( G @ B4 ) )
@ B2 ) )
@ A ) ) ) ).
% SUP_inf_distrib2
thf(fact_1091_SUP__inf__distrib2,axiom,
! [F: nat > extended_ereal,A: set_nat,G: b > extended_ereal,B2: set_b] :
( ( inf_in2794916579150040252_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_5319725110001000852_ereal @ G @ B2 ) ) )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [A4: nat] :
( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [B4: b] : ( inf_in2794916579150040252_ereal @ ( F @ A4 ) @ ( G @ B4 ) )
@ B2 ) )
@ A ) ) ) ).
% SUP_inf_distrib2
thf(fact_1092_SUP__inf__distrib2,axiom,
! [F: extended_ereal > extended_ereal,A: set_Extended_ereal,G: nat > extended_ereal,B2: set_nat] :
( ( inf_in2794916579150040252_ereal @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ B2 ) ) )
= ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [A4: extended_ereal] :
( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [B4: nat] : ( inf_in2794916579150040252_ereal @ ( F @ A4 ) @ ( G @ B4 ) )
@ B2 ) )
@ A ) ) ) ).
% SUP_inf_distrib2
thf(fact_1093_SUP__inf__distrib2,axiom,
! [F: extended_ereal > extended_ereal,A: set_Extended_ereal,G: extended_ereal > extended_ereal,B2: set_Extended_ereal] :
( ( inf_in2794916579150040252_ereal @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ G @ B2 ) ) )
= ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [A4: extended_ereal] :
( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [B4: extended_ereal] : ( inf_in2794916579150040252_ereal @ ( F @ A4 ) @ ( G @ B4 ) )
@ B2 ) )
@ A ) ) ) ).
% SUP_inf_distrib2
thf(fact_1094_SUP__inf__distrib2,axiom,
! [F: extended_ereal > extended_ereal,A: set_Extended_ereal,G: b > extended_ereal,B2: set_b] :
( ( inf_in2794916579150040252_ereal @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_5319725110001000852_ereal @ G @ B2 ) ) )
= ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [A4: extended_ereal] :
( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [B4: b] : ( inf_in2794916579150040252_ereal @ ( F @ A4 ) @ ( G @ B4 ) )
@ B2 ) )
@ A ) ) ) ).
% SUP_inf_distrib2
thf(fact_1095_SUP__inf__distrib2,axiom,
! [F: b > extended_ereal,A: set_b,G: nat > extended_ereal,B2: set_nat] :
( ( inf_in2794916579150040252_ereal @ ( comple8415311339701865915_ereal @ ( image_5319725110001000852_ereal @ F @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ B2 ) ) )
= ( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [A4: b] :
( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [B4: nat] : ( inf_in2794916579150040252_ereal @ ( F @ A4 ) @ ( G @ B4 ) )
@ B2 ) )
@ A ) ) ) ).
% SUP_inf_distrib2
thf(fact_1096_SUP__inf__distrib2,axiom,
! [F: b > extended_ereal,A: set_b,G: extended_ereal > extended_ereal,B2: set_Extended_ereal] :
( ( inf_in2794916579150040252_ereal @ ( comple8415311339701865915_ereal @ ( image_5319725110001000852_ereal @ F @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ G @ B2 ) ) )
= ( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [A4: b] :
( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [B4: extended_ereal] : ( inf_in2794916579150040252_ereal @ ( F @ A4 ) @ ( G @ B4 ) )
@ B2 ) )
@ A ) ) ) ).
% SUP_inf_distrib2
thf(fact_1097_SUP__inf__distrib2,axiom,
! [F: b > extended_ereal,A: set_b,G: b > extended_ereal,B2: set_b] :
( ( inf_in2794916579150040252_ereal @ ( comple8415311339701865915_ereal @ ( image_5319725110001000852_ereal @ F @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_5319725110001000852_ereal @ G @ B2 ) ) )
= ( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [A4: b] :
( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [B4: b] : ( inf_in2794916579150040252_ereal @ ( F @ A4 ) @ ( G @ B4 ) )
@ B2 ) )
@ A ) ) ) ).
% SUP_inf_distrib2
thf(fact_1098_inf__SUP,axiom,
! [A2: $o,F: b > $o,B2: set_b] :
( ( inf_inf_o @ A2 @ ( complete_Sup_Sup_o @ ( image_b_o @ F @ B2 ) ) )
= ( complete_Sup_Sup_o
@ ( image_b_o
@ ^ [B4: b] : ( inf_inf_o @ A2 @ ( F @ B4 ) )
@ B2 ) ) ) ).
% inf_SUP
thf(fact_1099_inf__SUP,axiom,
! [A2: extended_ereal,F: nat > extended_ereal,B2: set_nat] :
( ( inf_in2794916579150040252_ereal @ A2 @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F @ B2 ) ) )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [B4: nat] : ( inf_in2794916579150040252_ereal @ A2 @ ( F @ B4 ) )
@ B2 ) ) ) ).
% inf_SUP
thf(fact_1100_inf__SUP,axiom,
! [A2: extended_ereal,F: extended_ereal > extended_ereal,B2: set_Extended_ereal] :
( ( inf_in2794916579150040252_ereal @ A2 @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F @ B2 ) ) )
= ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [B4: extended_ereal] : ( inf_in2794916579150040252_ereal @ A2 @ ( F @ B4 ) )
@ B2 ) ) ) ).
% inf_SUP
thf(fact_1101_inf__SUP,axiom,
! [A2: extended_ereal,F: b > extended_ereal,B2: set_b] :
( ( inf_in2794916579150040252_ereal @ A2 @ ( comple8415311339701865915_ereal @ ( image_5319725110001000852_ereal @ F @ B2 ) ) )
= ( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [B4: b] : ( inf_in2794916579150040252_ereal @ A2 @ ( F @ B4 ) )
@ B2 ) ) ) ).
% inf_SUP
thf(fact_1102_inf__SUP,axiom,
! [A2: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,B2: set_nat] :
( ( inf_in7439215052339218890nnreal @ A2 @ ( comple6814414086264997003nnreal @ ( image_8459861568512453903nnreal @ F @ B2 ) ) )
= ( comple6814414086264997003nnreal
@ ( image_8459861568512453903nnreal
@ ^ [B4: nat] : ( inf_in7439215052339218890nnreal @ A2 @ ( F @ B4 ) )
@ B2 ) ) ) ).
% inf_SUP
thf(fact_1103_Sup__inf,axiom,
! [B2: set_o,A2: $o] :
( ( inf_inf_o @ ( complete_Sup_Sup_o @ B2 ) @ A2 )
= ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [B4: $o] : ( inf_inf_o @ B4 @ A2 )
@ B2 ) ) ) ).
% Sup_inf
thf(fact_1104_Sup__inf,axiom,
! [B2: set_Extended_ereal,A2: extended_ereal] :
( ( inf_in2794916579150040252_ereal @ ( comple8415311339701865915_ereal @ B2 ) @ A2 )
= ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [B4: extended_ereal] : ( inf_in2794916579150040252_ereal @ B4 @ A2 )
@ B2 ) ) ) ).
% Sup_inf
thf(fact_1105_Sup__inf,axiom,
! [B2: set_Ex3793607809372303086nnreal,A2: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ ( comple6814414086264997003nnreal @ B2 ) @ A2 )
= ( comple6814414086264997003nnreal
@ ( image_8394674774369097847nnreal
@ ^ [B4: extend8495563244428889912nnreal] : ( inf_in7439215052339218890nnreal @ B4 @ A2 )
@ B2 ) ) ) ).
% Sup_inf
thf(fact_1106_SUP__inf,axiom,
! [F: b > $o,B2: set_b,A2: $o] :
( ( inf_inf_o @ ( complete_Sup_Sup_o @ ( image_b_o @ F @ B2 ) ) @ A2 )
= ( complete_Sup_Sup_o
@ ( image_b_o
@ ^ [B4: b] : ( inf_inf_o @ ( F @ B4 ) @ A2 )
@ B2 ) ) ) ).
% SUP_inf
thf(fact_1107_SUP__inf,axiom,
! [F: nat > extended_ereal,B2: set_nat,A2: extended_ereal] :
( ( inf_in2794916579150040252_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F @ B2 ) ) @ A2 )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [B4: nat] : ( inf_in2794916579150040252_ereal @ ( F @ B4 ) @ A2 )
@ B2 ) ) ) ).
% SUP_inf
thf(fact_1108_SUP__inf,axiom,
! [F: extended_ereal > extended_ereal,B2: set_Extended_ereal,A2: extended_ereal] :
( ( inf_in2794916579150040252_ereal @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F @ B2 ) ) @ A2 )
= ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [B4: extended_ereal] : ( inf_in2794916579150040252_ereal @ ( F @ B4 ) @ A2 )
@ B2 ) ) ) ).
% SUP_inf
thf(fact_1109_SUP__inf,axiom,
! [F: b > extended_ereal,B2: set_b,A2: extended_ereal] :
( ( inf_in2794916579150040252_ereal @ ( comple8415311339701865915_ereal @ ( image_5319725110001000852_ereal @ F @ B2 ) ) @ A2 )
= ( comple8415311339701865915_ereal
@ ( image_5319725110001000852_ereal
@ ^ [B4: b] : ( inf_in2794916579150040252_ereal @ ( F @ B4 ) @ A2 )
@ B2 ) ) ) ).
% SUP_inf
thf(fact_1110_SUP__inf,axiom,
! [F: nat > extend8495563244428889912nnreal,B2: set_nat,A2: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ ( comple6814414086264997003nnreal @ ( image_8459861568512453903nnreal @ F @ B2 ) ) @ A2 )
= ( comple6814414086264997003nnreal
@ ( image_8459861568512453903nnreal
@ ^ [B4: nat] : ( inf_in7439215052339218890nnreal @ ( F @ B4 ) @ A2 )
@ B2 ) ) ) ).
% SUP_inf
thf(fact_1111_INF__absorb,axiom,
! [K: $o,I2: set_o,A: $o > $o] :
( ( member_o @ K @ I2 )
=> ( ( inf_inf_o @ ( A @ K ) @ ( complete_Inf_Inf_o @ ( image_o_o @ A @ I2 ) ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ A @ I2 ) ) ) ) ).
% INF_absorb
thf(fact_1112_INF__absorb,axiom,
! [K: extended_ereal,I2: set_Extended_ereal,A: extended_ereal > $o] :
( ( member2350847679896131959_ereal @ K @ I2 )
=> ( ( inf_inf_o @ ( A @ K ) @ ( complete_Inf_Inf_o @ ( image_951975095941678543real_o @ A @ I2 ) ) )
= ( complete_Inf_Inf_o @ ( image_951975095941678543real_o @ A @ I2 ) ) ) ) ).
% INF_absorb
thf(fact_1113_INF__absorb,axiom,
! [K: nat,I2: set_nat,A: nat > $o] :
( ( member_nat @ K @ I2 )
=> ( ( inf_inf_o @ ( A @ K ) @ ( complete_Inf_Inf_o @ ( image_nat_o @ A @ I2 ) ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ A @ I2 ) ) ) ) ).
% INF_absorb
thf(fact_1114_INF__absorb,axiom,
! [K: extend8495563244428889912nnreal,I2: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal > $o] :
( ( member7908768830364227535nnreal @ K @ I2 )
=> ( ( inf_inf_o @ ( A @ K ) @ ( complete_Inf_Inf_o @ ( image_3162942742313426073real_o @ A @ I2 ) ) )
= ( complete_Inf_Inf_o @ ( image_3162942742313426073real_o @ A @ I2 ) ) ) ) ).
% INF_absorb
thf(fact_1115_INF__absorb,axiom,
! [K: c,I2: set_c,A: c > $o] :
( ( member_c @ K @ I2 )
=> ( ( inf_inf_o @ ( A @ K ) @ ( complete_Inf_Inf_o @ ( image_c_o @ A @ I2 ) ) )
= ( complete_Inf_Inf_o @ ( image_c_o @ A @ I2 ) ) ) ) ).
% INF_absorb
thf(fact_1116_INF__absorb,axiom,
! [K: b,I2: set_b,A: b > $o] :
( ( member_b @ K @ I2 )
=> ( ( inf_inf_o @ ( A @ K ) @ ( complete_Inf_Inf_o @ ( image_b_o @ A @ I2 ) ) )
= ( complete_Inf_Inf_o @ ( image_b_o @ A @ I2 ) ) ) ) ).
% INF_absorb
thf(fact_1117_INF__absorb,axiom,
! [K: $o,I2: set_o,A: $o > extended_ereal] :
( ( member_o @ K @ I2 )
=> ( ( inf_in2794916579150040252_ereal @ ( A @ K ) @ ( comple3556804143462414037_ereal @ ( image_7729549296133164475_ereal @ A @ I2 ) ) )
= ( comple3556804143462414037_ereal @ ( image_7729549296133164475_ereal @ A @ I2 ) ) ) ) ).
% INF_absorb
thf(fact_1118_INF__absorb,axiom,
! [K: extended_ereal,I2: set_Extended_ereal,A: extended_ereal > extended_ereal] :
( ( member2350847679896131959_ereal @ K @ I2 )
=> ( ( inf_in2794916579150040252_ereal @ ( A @ K ) @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ A @ I2 ) ) )
= ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ A @ I2 ) ) ) ) ).
% INF_absorb
thf(fact_1119_INF__absorb,axiom,
! [K: nat,I2: set_nat,A: nat > extended_ereal] :
( ( member_nat @ K @ I2 )
=> ( ( inf_in2794916579150040252_ereal @ ( A @ K ) @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ A @ I2 ) ) )
= ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ A @ I2 ) ) ) ) ).
% INF_absorb
thf(fact_1120_INF__absorb,axiom,
! [K: extend8495563244428889912nnreal,I2: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal > extended_ereal] :
( ( member7908768830364227535nnreal @ K @ I2 )
=> ( ( inf_in2794916579150040252_ereal @ ( A @ K ) @ ( comple3556804143462414037_ereal @ ( image_6393943237584228047_ereal @ A @ I2 ) ) )
= ( comple3556804143462414037_ereal @ ( image_6393943237584228047_ereal @ A @ I2 ) ) ) ) ).
% INF_absorb
thf(fact_1121_INF__inf__distrib,axiom,
! [F: b > $o,A: set_b,G: b > $o] :
( ( inf_inf_o @ ( complete_Inf_Inf_o @ ( image_b_o @ F @ A ) ) @ ( complete_Inf_Inf_o @ ( image_b_o @ G @ A ) ) )
= ( complete_Inf_Inf_o
@ ( image_b_o
@ ^ [A4: b] : ( inf_inf_o @ ( F @ A4 ) @ ( G @ A4 ) )
@ A ) ) ) ).
% INF_inf_distrib
thf(fact_1122_INF__inf__distrib,axiom,
! [F: nat > extended_ereal,A: set_nat,G: nat > extended_ereal] :
( ( inf_in2794916579150040252_ereal @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F @ A ) ) @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ G @ A ) ) )
= ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [A4: nat] : ( inf_in2794916579150040252_ereal @ ( F @ A4 ) @ ( G @ A4 ) )
@ A ) ) ) ).
% INF_inf_distrib
thf(fact_1123_INF__inf__distrib,axiom,
! [F: extended_ereal > extended_ereal,A: set_Extended_ereal,G: extended_ereal > extended_ereal] :
( ( inf_in2794916579150040252_ereal @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F @ A ) ) @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ G @ A ) ) )
= ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [A4: extended_ereal] : ( inf_in2794916579150040252_ereal @ ( F @ A4 ) @ ( G @ A4 ) )
@ A ) ) ) ).
% INF_inf_distrib
thf(fact_1124_INF__inf__distrib,axiom,
! [F: b > extended_ereal,A: set_b,G: b > extended_ereal] :
( ( inf_in2794916579150040252_ereal @ ( comple3556804143462414037_ereal @ ( image_5319725110001000852_ereal @ F @ A ) ) @ ( comple3556804143462414037_ereal @ ( image_5319725110001000852_ereal @ G @ A ) ) )
= ( comple3556804143462414037_ereal
@ ( image_5319725110001000852_ereal
@ ^ [A4: b] : ( inf_in2794916579150040252_ereal @ ( F @ A4 ) @ ( G @ A4 ) )
@ A ) ) ) ).
% INF_inf_distrib
thf(fact_1125_INF__inf__distrib,axiom,
! [F: nat > extend8495563244428889912nnreal,A: set_nat,G: nat > extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ ( comple7330758040695736817nnreal @ ( image_8459861568512453903nnreal @ F @ A ) ) @ ( comple7330758040695736817nnreal @ ( image_8459861568512453903nnreal @ G @ A ) ) )
= ( comple7330758040695736817nnreal
@ ( image_8459861568512453903nnreal
@ ^ [A4: nat] : ( inf_in7439215052339218890nnreal @ ( F @ A4 ) @ ( G @ A4 ) )
@ A ) ) ) ).
% INF_inf_distrib
thf(fact_1126_INF__inf__const2,axiom,
! [I2: set_b,F: b > $o,X: $o] :
( ( I2 != bot_bot_set_b )
=> ( ( complete_Inf_Inf_o
@ ( image_b_o
@ ^ [I: b] : ( inf_inf_o @ ( F @ I ) @ X )
@ I2 ) )
= ( inf_inf_o @ ( complete_Inf_Inf_o @ ( image_b_o @ F @ I2 ) ) @ X ) ) ) ).
% INF_inf_const2
thf(fact_1127_INF__inf__const2,axiom,
! [I2: set_Extended_ereal,F: extended_ereal > $o,X: $o] :
( ( I2 != bot_bo8367695208629047834_ereal )
=> ( ( complete_Inf_Inf_o
@ ( image_951975095941678543real_o
@ ^ [I: extended_ereal] : ( inf_inf_o @ ( F @ I ) @ X )
@ I2 ) )
= ( inf_inf_o @ ( complete_Inf_Inf_o @ ( image_951975095941678543real_o @ F @ I2 ) ) @ X ) ) ) ).
% INF_inf_const2
thf(fact_1128_INF__inf__const2,axiom,
! [I2: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > $o,X: $o] :
( ( I2 != bot_bo4854962954004695426nnreal )
=> ( ( complete_Inf_Inf_o
@ ( image_3162942742313426073real_o
@ ^ [I: extend8495563244428889912nnreal] : ( inf_inf_o @ ( F @ I ) @ X )
@ I2 ) )
= ( inf_inf_o @ ( complete_Inf_Inf_o @ ( image_3162942742313426073real_o @ F @ I2 ) ) @ X ) ) ) ).
% INF_inf_const2
thf(fact_1129_INF__inf__const2,axiom,
! [I2: set_nat,F: nat > $o,X: $o] :
( ( I2 != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [I: nat] : ( inf_inf_o @ ( F @ I ) @ X )
@ I2 ) )
= ( inf_inf_o @ ( complete_Inf_Inf_o @ ( image_nat_o @ F @ I2 ) ) @ X ) ) ) ).
% INF_inf_const2
thf(fact_1130_INF__inf__const2,axiom,
! [I2: set_b,F: b > extended_ereal,X: extended_ereal] :
( ( I2 != bot_bot_set_b )
=> ( ( comple3556804143462414037_ereal
@ ( image_5319725110001000852_ereal
@ ^ [I: b] : ( inf_in2794916579150040252_ereal @ ( F @ I ) @ X )
@ I2 ) )
= ( inf_in2794916579150040252_ereal @ ( comple3556804143462414037_ereal @ ( image_5319725110001000852_ereal @ F @ I2 ) ) @ X ) ) ) ).
% INF_inf_const2
thf(fact_1131_INF__inf__const2,axiom,
! [I2: set_Extended_ereal,F: extended_ereal > extended_ereal,X: extended_ereal] :
( ( I2 != bot_bo8367695208629047834_ereal )
=> ( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I: extended_ereal] : ( inf_in2794916579150040252_ereal @ ( F @ I ) @ X )
@ I2 ) )
= ( inf_in2794916579150040252_ereal @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F @ I2 ) ) @ X ) ) ) ).
% INF_inf_const2
thf(fact_1132_INF__inf__const2,axiom,
! [I2: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > extended_ereal,X: extended_ereal] :
( ( I2 != bot_bo4854962954004695426nnreal )
=> ( ( comple3556804143462414037_ereal
@ ( image_6393943237584228047_ereal
@ ^ [I: extend8495563244428889912nnreal] : ( inf_in2794916579150040252_ereal @ ( F @ I ) @ X )
@ I2 ) )
= ( inf_in2794916579150040252_ereal @ ( comple3556804143462414037_ereal @ ( image_6393943237584228047_ereal @ F @ I2 ) ) @ X ) ) ) ).
% INF_inf_const2
thf(fact_1133_INF__inf__const2,axiom,
! [I2: set_nat,F: nat > extended_ereal,X: extended_ereal] :
( ( I2 != bot_bot_set_nat )
=> ( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I: nat] : ( inf_in2794916579150040252_ereal @ ( F @ I ) @ X )
@ I2 ) )
= ( inf_in2794916579150040252_ereal @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F @ I2 ) ) @ X ) ) ) ).
% INF_inf_const2
thf(fact_1134_INF__inf__const2,axiom,
! [I2: set_Extended_ereal,F: extended_ereal > extend8495563244428889912nnreal,X: extend8495563244428889912nnreal] :
( ( I2 != bot_bo8367695208629047834_ereal )
=> ( ( comple7330758040695736817nnreal
@ ( image_8614087454967683265nnreal
@ ^ [I: extended_ereal] : ( inf_in7439215052339218890nnreal @ ( F @ I ) @ X )
@ I2 ) )
= ( inf_in7439215052339218890nnreal @ ( comple7330758040695736817nnreal @ ( image_8614087454967683265nnreal @ F @ I2 ) ) @ X ) ) ) ).
% INF_inf_const2
thf(fact_1135_INF__inf__const2,axiom,
! [I2: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,X: extend8495563244428889912nnreal] :
( ( I2 != bot_bo4854962954004695426nnreal )
=> ( ( comple7330758040695736817nnreal
@ ( image_8394674774369097847nnreal
@ ^ [I: extend8495563244428889912nnreal] : ( inf_in7439215052339218890nnreal @ ( F @ I ) @ X )
@ I2 ) )
= ( inf_in7439215052339218890nnreal @ ( comple7330758040695736817nnreal @ ( image_8394674774369097847nnreal @ F @ I2 ) ) @ X ) ) ) ).
% INF_inf_const2
thf(fact_1136_INF__inf__const1,axiom,
! [I2: set_b,X: $o,F: b > $o] :
( ( I2 != bot_bot_set_b )
=> ( ( complete_Inf_Inf_o
@ ( image_b_o
@ ^ [I: b] : ( inf_inf_o @ X @ ( F @ I ) )
@ I2 ) )
= ( inf_inf_o @ X @ ( complete_Inf_Inf_o @ ( image_b_o @ F @ I2 ) ) ) ) ) ).
% INF_inf_const1
thf(fact_1137_INF__inf__const1,axiom,
! [I2: set_Extended_ereal,X: $o,F: extended_ereal > $o] :
( ( I2 != bot_bo8367695208629047834_ereal )
=> ( ( complete_Inf_Inf_o
@ ( image_951975095941678543real_o
@ ^ [I: extended_ereal] : ( inf_inf_o @ X @ ( F @ I ) )
@ I2 ) )
= ( inf_inf_o @ X @ ( complete_Inf_Inf_o @ ( image_951975095941678543real_o @ F @ I2 ) ) ) ) ) ).
% INF_inf_const1
thf(fact_1138_INF__inf__const1,axiom,
! [I2: set_Ex3793607809372303086nnreal,X: $o,F: extend8495563244428889912nnreal > $o] :
( ( I2 != bot_bo4854962954004695426nnreal )
=> ( ( complete_Inf_Inf_o
@ ( image_3162942742313426073real_o
@ ^ [I: extend8495563244428889912nnreal] : ( inf_inf_o @ X @ ( F @ I ) )
@ I2 ) )
= ( inf_inf_o @ X @ ( complete_Inf_Inf_o @ ( image_3162942742313426073real_o @ F @ I2 ) ) ) ) ) ).
% INF_inf_const1
thf(fact_1139_INF__inf__const1,axiom,
! [I2: set_nat,X: $o,F: nat > $o] :
( ( I2 != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [I: nat] : ( inf_inf_o @ X @ ( F @ I ) )
@ I2 ) )
= ( inf_inf_o @ X @ ( complete_Inf_Inf_o @ ( image_nat_o @ F @ I2 ) ) ) ) ) ).
% INF_inf_const1
thf(fact_1140_INF__inf__const1,axiom,
! [I2: set_b,X: extended_ereal,F: b > extended_ereal] :
( ( I2 != bot_bot_set_b )
=> ( ( comple3556804143462414037_ereal
@ ( image_5319725110001000852_ereal
@ ^ [I: b] : ( inf_in2794916579150040252_ereal @ X @ ( F @ I ) )
@ I2 ) )
= ( inf_in2794916579150040252_ereal @ X @ ( comple3556804143462414037_ereal @ ( image_5319725110001000852_ereal @ F @ I2 ) ) ) ) ) ).
% INF_inf_const1
thf(fact_1141_INF__inf__const1,axiom,
! [I2: set_Extended_ereal,X: extended_ereal,F: extended_ereal > extended_ereal] :
( ( I2 != bot_bo8367695208629047834_ereal )
=> ( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I: extended_ereal] : ( inf_in2794916579150040252_ereal @ X @ ( F @ I ) )
@ I2 ) )
= ( inf_in2794916579150040252_ereal @ X @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F @ I2 ) ) ) ) ) ).
% INF_inf_const1
thf(fact_1142_INF__inf__const1,axiom,
! [I2: set_Ex3793607809372303086nnreal,X: extended_ereal,F: extend8495563244428889912nnreal > extended_ereal] :
( ( I2 != bot_bo4854962954004695426nnreal )
=> ( ( comple3556804143462414037_ereal
@ ( image_6393943237584228047_ereal
@ ^ [I: extend8495563244428889912nnreal] : ( inf_in2794916579150040252_ereal @ X @ ( F @ I ) )
@ I2 ) )
= ( inf_in2794916579150040252_ereal @ X @ ( comple3556804143462414037_ereal @ ( image_6393943237584228047_ereal @ F @ I2 ) ) ) ) ) ).
% INF_inf_const1
thf(fact_1143_INF__inf__const1,axiom,
! [I2: set_nat,X: extended_ereal,F: nat > extended_ereal] :
( ( I2 != bot_bot_set_nat )
=> ( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I: nat] : ( inf_in2794916579150040252_ereal @ X @ ( F @ I ) )
@ I2 ) )
= ( inf_in2794916579150040252_ereal @ X @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F @ I2 ) ) ) ) ) ).
% INF_inf_const1
thf(fact_1144_INF__inf__const1,axiom,
! [I2: set_Extended_ereal,X: extend8495563244428889912nnreal,F: extended_ereal > extend8495563244428889912nnreal] :
( ( I2 != bot_bo8367695208629047834_ereal )
=> ( ( comple7330758040695736817nnreal
@ ( image_8614087454967683265nnreal
@ ^ [I: extended_ereal] : ( inf_in7439215052339218890nnreal @ X @ ( F @ I ) )
@ I2 ) )
= ( inf_in7439215052339218890nnreal @ X @ ( comple7330758040695736817nnreal @ ( image_8614087454967683265nnreal @ F @ I2 ) ) ) ) ) ).
% INF_inf_const1
thf(fact_1145_INF__inf__const1,axiom,
! [I2: set_Ex3793607809372303086nnreal,X: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( I2 != bot_bo4854962954004695426nnreal )
=> ( ( comple7330758040695736817nnreal
@ ( image_8394674774369097847nnreal
@ ^ [I: extend8495563244428889912nnreal] : ( inf_in7439215052339218890nnreal @ X @ ( F @ I ) )
@ I2 ) )
= ( inf_in7439215052339218890nnreal @ X @ ( comple7330758040695736817nnreal @ ( image_8394674774369097847nnreal @ F @ I2 ) ) ) ) ) ).
% INF_inf_const1
thf(fact_1146_INT__extend__simps_I3_J,axiom,
! [C: set_Extended_ereal,A: extended_ereal > set_nat,B2: set_nat] :
( ( ( C = bot_bo8367695208629047834_ereal )
=> ( ( minus_minus_set_nat @ ( comple7806235888213564991et_nat @ ( image_3090908713637162255et_nat @ A @ C ) ) @ B2 )
= ( minus_minus_set_nat @ top_top_set_nat @ B2 ) ) )
& ( ( C != bot_bo8367695208629047834_ereal )
=> ( ( minus_minus_set_nat @ ( comple7806235888213564991et_nat @ ( image_3090908713637162255et_nat @ A @ C ) ) @ B2 )
= ( comple7806235888213564991et_nat
@ ( image_3090908713637162255et_nat
@ ^ [X2: extended_ereal] : ( minus_minus_set_nat @ ( A @ X2 ) @ B2 )
@ C ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_1147_INT__extend__simps_I3_J,axiom,
! [C: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal > set_nat,B2: set_nat] :
( ( ( C = bot_bo4854962954004695426nnreal )
=> ( ( minus_minus_set_nat @ ( comple7806235888213564991et_nat @ ( image_2869339492569777349et_nat @ A @ C ) ) @ B2 )
= ( minus_minus_set_nat @ top_top_set_nat @ B2 ) ) )
& ( ( C != bot_bo4854962954004695426nnreal )
=> ( ( minus_minus_set_nat @ ( comple7806235888213564991et_nat @ ( image_2869339492569777349et_nat @ A @ C ) ) @ B2 )
= ( comple7806235888213564991et_nat
@ ( image_2869339492569777349et_nat
@ ^ [X2: extend8495563244428889912nnreal] : ( minus_minus_set_nat @ ( A @ X2 ) @ B2 )
@ C ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_1148_INT__extend__simps_I3_J,axiom,
! [C: set_nat,A: nat > set_nat,B2: set_nat] :
( ( ( C = bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A @ C ) ) @ B2 )
= ( minus_minus_set_nat @ top_top_set_nat @ B2 ) ) )
& ( ( C != bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A @ C ) ) @ B2 )
= ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( minus_minus_set_nat @ ( A @ X2 ) @ B2 )
@ C ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_1149_INT__extend__simps_I3_J,axiom,
! [C: set_Extended_ereal,A: extended_ereal > set_Extended_ereal,B2: set_Extended_ereal] :
( ( ( C = bot_bo8367695208629047834_ereal )
=> ( ( minus_1264018925008434325_ereal @ ( comple4418415374894819509_ereal @ ( image_5562094264469218789_ereal @ A @ C ) ) @ B2 )
= ( minus_1264018925008434325_ereal @ top_to5683747375963461374_ereal @ B2 ) ) )
& ( ( C != bot_bo8367695208629047834_ereal )
=> ( ( minus_1264018925008434325_ereal @ ( comple4418415374894819509_ereal @ ( image_5562094264469218789_ereal @ A @ C ) ) @ B2 )
= ( comple4418415374894819509_ereal
@ ( image_5562094264469218789_ereal
@ ^ [X2: extended_ereal] : ( minus_1264018925008434325_ereal @ ( A @ X2 ) @ B2 )
@ C ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_1150_INT__extend__simps_I3_J,axiom,
! [C: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal > set_Extended_ereal,B2: set_Extended_ereal] :
( ( ( C = bot_bo4854962954004695426nnreal )
=> ( ( minus_1264018925008434325_ereal @ ( comple4418415374894819509_ereal @ ( image_5929344197358196911_ereal @ A @ C ) ) @ B2 )
= ( minus_1264018925008434325_ereal @ top_to5683747375963461374_ereal @ B2 ) ) )
& ( ( C != bot_bo4854962954004695426nnreal )
=> ( ( minus_1264018925008434325_ereal @ ( comple4418415374894819509_ereal @ ( image_5929344197358196911_ereal @ A @ C ) ) @ B2 )
= ( comple4418415374894819509_ereal
@ ( image_5929344197358196911_ereal
@ ^ [X2: extend8495563244428889912nnreal] : ( minus_1264018925008434325_ereal @ ( A @ X2 ) @ B2 )
@ C ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_1151_INT__extend__simps_I3_J,axiom,
! [C: set_nat,A: nat > set_Extended_ereal,B2: set_Extended_ereal] :
( ( ( C = bot_bot_set_nat )
=> ( ( minus_1264018925008434325_ereal @ ( comple4418415374894819509_ereal @ ( image_305533323056406039_ereal @ A @ C ) ) @ B2 )
= ( minus_1264018925008434325_ereal @ top_to5683747375963461374_ereal @ B2 ) ) )
& ( ( C != bot_bot_set_nat )
=> ( ( minus_1264018925008434325_ereal @ ( comple4418415374894819509_ereal @ ( image_305533323056406039_ereal @ A @ C ) ) @ B2 )
= ( comple4418415374894819509_ereal
@ ( image_305533323056406039_ereal
@ ^ [X2: nat] : ( minus_1264018925008434325_ereal @ ( A @ X2 ) @ B2 )
@ C ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_1152_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_Extended_ereal] :
( ( inf_in2779415704524776092_ereal @ X @ bot_bo8367695208629047834_ereal )
= bot_bo8367695208629047834_ereal ) ).
% boolean_algebra.conj_zero_right
thf(fact_1153_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_Ex3793607809372303086nnreal] :
( ( inf_in3368558534146122112nnreal @ X @ bot_bo4854962954004695426nnreal )
= bot_bo4854962954004695426nnreal ) ).
% boolean_algebra.conj_zero_right
thf(fact_1154_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_1155_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_Ex3793607809372303086nnreal] :
( ( inf_in3368558534146122112nnreal @ bot_bo4854962954004695426nnreal @ X )
= bot_bo4854962954004695426nnreal ) ).
% boolean_algebra.conj_zero_left
thf(fact_1156_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_1157_ereal__decseq__uminus,axiom,
! [F: nat > extended_ereal] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X2: extended_ereal,Y2: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y2 @ X2 )
@ ^ [X2: nat] : ( uminus27091377158695749_ereal @ ( F @ X2 ) ) )
= ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ F ) ) ).
% ereal_decseq_uminus
thf(fact_1158_ereal__incseq__uminus,axiom,
! [F: nat > extended_ereal] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal
@ ^ [X2: nat] : ( uminus27091377158695749_ereal @ ( F @ X2 ) ) )
= ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X2: extended_ereal,Y2: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y2 @ X2 )
@ F ) ) ).
% ereal_incseq_uminus
thf(fact_1159_Inf__countable__INF,axiom,
! [A: set_Extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ? [F3: nat > extended_ereal] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X2: extended_ereal,Y2: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y2 @ X2 )
@ F3 )
& ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F3 @ top_top_set_nat ) @ A )
& ( ( comple3556804143462414037_ereal @ A )
= ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F3 @ top_top_set_nat ) ) ) ) ) ).
% Inf_countable_INF
thf(fact_1160_Sup__countable__SUP,axiom,
! [A: set_Extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ? [F3: nat > extended_ereal] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ F3 )
& ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F3 @ top_top_set_nat ) @ A )
& ( ( comple8415311339701865915_ereal @ A )
= ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F3 @ top_top_set_nat ) ) ) ) ) ).
% Sup_countable_SUP
thf(fact_1161_ereal__Inf__uminus__image__eq,axiom,
! [S: set_Extended_ereal] :
( ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S ) )
= ( uminus27091377158695749_ereal @ ( comple8415311339701865915_ereal @ S ) ) ) ).
% ereal_Inf_uminus_image_eq
thf(fact_1162_ereal__Sup__uminus__image__eq,axiom,
! [S: set_Extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S ) )
= ( uminus27091377158695749_ereal @ ( comple3556804143462414037_ereal @ S ) ) ) ).
% ereal_Sup_uminus_image_eq
thf(fact_1163_ereal__range__uminus,axiom,
( ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal ) ).
% ereal_range_uminus
thf(fact_1164_ereal__minus__minus__image,axiom,
! [S: set_Extended_ereal] :
( ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S ) )
= S ) ).
% ereal_minus_minus_image
thf(fact_1165_ereal__uminus__complement,axiom,
! [S: set_Extended_ereal] :
( ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ ( uminus5895154729394068773_ereal @ S ) )
= ( uminus5895154729394068773_ereal @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S ) ) ) ).
% ereal_uminus_complement
thf(fact_1166_ereal__complete__uminus__eq,axiom,
! [S: set_Extended_ereal,X: extended_ereal] :
( ( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S ) )
=> ( ord_le1083603963089353582_ereal @ X2 @ X ) )
& ! [Z2: extended_ereal] :
( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S ) )
=> ( ord_le1083603963089353582_ereal @ X2 @ Z2 ) )
=> ( ord_le1083603963089353582_ereal @ X @ Z2 ) ) )
= ( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ S )
=> ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ X ) @ X2 ) )
& ! [Z2: extended_ereal] :
( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ S )
=> ( ord_le1083603963089353582_ereal @ Z2 @ X2 ) )
=> ( ord_le1083603963089353582_ereal @ Z2 @ ( uminus27091377158695749_ereal @ X ) ) ) ) ) ).
% ereal_complete_uminus_eq
thf(fact_1167_ereal__image__uminus__shift,axiom,
! [X5: set_Extended_ereal,Y: set_Extended_ereal] :
( ( ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ X5 )
= Y )
= ( X5
= ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ Y ) ) ) ).
% ereal_image_uminus_shift
thf(fact_1168_ereal__minus__mono,axiom,
! [A: extended_ereal,B2: extended_ereal,D: extended_ereal,C: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B2 )
=> ( ( ord_le1083603963089353582_ereal @ D @ C )
=> ( ord_le1083603963089353582_ereal @ ( minus_2816186181549245109_ereal @ A @ C ) @ ( minus_2816186181549245109_ereal @ B2 @ D ) ) ) ) ).
% ereal_minus_mono
thf(fact_1169_ennreal__Sup__countable__SUP,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( A != bot_bo4854962954004695426nnreal )
=> ? [F3: nat > extend8495563244428889912nnreal] :
( ( monoto2291723841412853873nnreal @ top_top_set_nat @ ord_less_eq_nat @ ord_le3935885782089961368nnreal @ F3 )
& ( ord_le6787938422905777998nnreal @ ( image_8459861568512453903nnreal @ F3 @ top_top_set_nat ) @ A )
& ( ( comple6814414086264997003nnreal @ A )
= ( comple6814414086264997003nnreal @ ( image_8459861568512453903nnreal @ F3 @ top_top_set_nat ) ) ) ) ) ).
% ennreal_Sup_countable_SUP
thf(fact_1170_ennreal__Inf__countable__INF,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( A != bot_bo4854962954004695426nnreal )
=> ? [F3: nat > extend8495563244428889912nnreal] :
( ( monoto2291723841412853873nnreal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X2: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ Y2 @ X2 )
@ F3 )
& ( ord_le6787938422905777998nnreal @ ( image_8459861568512453903nnreal @ F3 @ top_top_set_nat ) @ A )
& ( ( comple7330758040695736817nnreal @ A )
= ( comple7330758040695736817nnreal @ ( image_8459861568512453903nnreal @ F3 @ top_top_set_nat ) ) ) ) ) ).
% ennreal_Inf_countable_INF
thf(fact_1171_diff__diff__cancel,axiom,
! [I4: nat,N2: nat] :
( ( ord_less_eq_nat @ I4 @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I4 ) )
= I4 ) ) ).
% diff_diff_cancel
thf(fact_1172_ennreal__minus__eq__top,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A2 @ B )
= top_to1496364449551166952nnreal )
= ( A2 = top_to1496364449551166952nnreal ) ) ).
% ennreal_minus_eq_top
thf(fact_1173_ennreal__top__minus,axiom,
! [X: extend8495563244428889912nnreal] :
( ( minus_8429688780609304081nnreal @ top_to1496364449551166952nnreal @ X )
= top_to1496364449551166952nnreal ) ).
% ennreal_top_minus
thf(fact_1174_ennreal__minus__cancel__iff,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A2 @ B )
= ( minus_8429688780609304081nnreal @ A2 @ C2 ) )
= ( ( B = C2 )
| ( ( ord_le3935885782089961368nnreal @ A2 @ B )
& ( ord_le3935885782089961368nnreal @ A2 @ C2 ) )
| ( A2 = top_to1496364449551166952nnreal ) ) ) ).
% ennreal_minus_cancel_iff
thf(fact_1175_ennreal__minus__cancel,axiom,
! [C2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( C2 != top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ A2 @ C2 )
=> ( ( ord_le3935885782089961368nnreal @ B @ C2 )
=> ( ( ( minus_8429688780609304081nnreal @ C2 @ A2 )
= ( minus_8429688780609304081nnreal @ C2 @ B ) )
=> ( A2 = B ) ) ) ) ) ).
% ennreal_minus_cancel
thf(fact_1176_ennreal__minus__mono,axiom,
! [A2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal,D2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ C2 )
=> ( ( ord_le3935885782089961368nnreal @ D2 @ B )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A2 @ B ) @ ( minus_8429688780609304081nnreal @ C2 @ D2 ) ) ) ) ).
% ennreal_minus_mono
thf(fact_1177_ennreal__mono__minus,axiom,
! [C2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ C2 @ B )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A2 @ B ) @ ( minus_8429688780609304081nnreal @ A2 @ C2 ) ) ) ).
% ennreal_mono_minus
thf(fact_1178_diff__le__self__ennreal,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A2 @ B ) @ A2 ) ).
% diff_le_self_ennreal
thf(fact_1179_ennreal__diff__le__mono__left,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A2 @ C2 ) @ B ) ) ).
% ennreal_diff_le_mono_left
thf(fact_1180_neq__top__trans,axiom,
! [Y4: extend8495563244428889912nnreal,X: extend8495563244428889912nnreal] :
( ( Y4 != top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ X @ Y4 )
=> ( X != top_to1496364449551166952nnreal ) ) ) ).
% neq_top_trans
thf(fact_1181_diff__diff__commute__ennreal,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( minus_8429688780609304081nnreal @ ( minus_8429688780609304081nnreal @ A2 @ B ) @ C2 )
= ( minus_8429688780609304081nnreal @ ( minus_8429688780609304081nnreal @ A2 @ C2 ) @ B ) ) ).
% diff_diff_commute_ennreal
thf(fact_1182_diff__diff__commute__ereal,axiom,
! [X: extended_ereal,Y4: extended_ereal,Z: extended_ereal] :
( ( minus_2816186181549245109_ereal @ ( minus_2816186181549245109_ereal @ X @ Y4 ) @ Z )
= ( minus_2816186181549245109_ereal @ ( minus_2816186181549245109_ereal @ X @ Z ) @ Y4 ) ) ).
% diff_diff_commute_ereal
thf(fact_1183_diff__commute,axiom,
! [I4: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J2 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I4 @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_1184_diff__le__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1185_le__diff__iff_H,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A2 ) @ ( minus_minus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1186_diff__le__self,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).
% diff_le_self
thf(fact_1187_diff__le__mono,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_1188_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1189_le__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1190_eq__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M2 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1191_ennreal__SUP__add,axiom,
! [F: nat > extend8495563244428889912nnreal,G: nat > extend8495563244428889912nnreal] :
( ( monoto2291723841412853873nnreal @ top_top_set_nat @ ord_less_eq_nat @ ord_le3935885782089961368nnreal @ F )
=> ( ( monoto2291723841412853873nnreal @ top_top_set_nat @ ord_less_eq_nat @ ord_le3935885782089961368nnreal @ G )
=> ( ( comple6814414086264997003nnreal
@ ( image_8459861568512453903nnreal
@ ^ [I: nat] : ( plus_p1859984266308609217nnreal @ ( F @ I ) @ ( G @ I ) )
@ top_top_set_nat ) )
= ( plus_p1859984266308609217nnreal @ ( comple6814414086264997003nnreal @ ( image_8459861568512453903nnreal @ F @ top_top_set_nat ) ) @ ( comple6814414086264997003nnreal @ ( image_8459861568512453903nnreal @ G @ top_top_set_nat ) ) ) ) ) ) ).
% ennreal_SUP_add
thf(fact_1192_ereal__minus_I4_J,axiom,
! [X: extended_ereal] :
( ( minus_2816186181549245109_ereal @ extend1530274965995635425_ereal @ X )
= extend1530274965995635425_ereal ) ).
% ereal_minus(4)
thf(fact_1193_ennreal__add__eq__top,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( plus_p1859984266308609217nnreal @ A2 @ B )
= top_to1496364449551166952nnreal )
= ( ( A2 = top_to1496364449551166952nnreal )
| ( B = top_to1496364449551166952nnreal ) ) ) ).
% ennreal_add_eq_top
thf(fact_1194_add__top__left__ennreal,axiom,
! [X: extend8495563244428889912nnreal] :
( ( plus_p1859984266308609217nnreal @ top_to1496364449551166952nnreal @ X )
= top_to1496364449551166952nnreal ) ).
% add_top_left_ennreal
thf(fact_1195_add__top__right__ennreal,axiom,
! [X: extend8495563244428889912nnreal] :
( ( plus_p1859984266308609217nnreal @ X @ top_to1496364449551166952nnreal )
= top_to1496364449551166952nnreal ) ).
% add_top_right_ennreal
thf(fact_1196_ereal__minus_I5_J,axiom,
( ( minus_2816186181549245109_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ extend1530274965995635425_ereal )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% ereal_minus(5)
thf(fact_1197_add__diff__eq__iff__ennreal,axiom,
! [X: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ( plus_p1859984266308609217nnreal @ X @ ( minus_8429688780609304081nnreal @ Y4 @ X ) )
= Y4 )
= ( ord_le3935885782089961368nnreal @ X @ Y4 ) ) ).
% add_diff_eq_iff_ennreal
thf(fact_1198_ennreal__add__diff__cancel__right,axiom,
! [Y4: extend8495563244428889912nnreal,X: extend8495563244428889912nnreal] :
( ( Y4 != top_to1496364449551166952nnreal )
=> ( ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ X @ Y4 ) @ Y4 )
= X ) ) ).
% ennreal_add_diff_cancel_right
thf(fact_1199_ennreal__add__diff__cancel__left,axiom,
! [Y4: extend8495563244428889912nnreal,X: extend8495563244428889912nnreal] :
( ( Y4 != top_to1496364449551166952nnreal )
=> ( ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ Y4 @ X ) @ Y4 )
= X ) ) ).
% ennreal_add_diff_cancel_left
thf(fact_1200_diff__add__eq__diff__diff__swap__ennreal,axiom,
! [X: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] :
( ( minus_8429688780609304081nnreal @ X @ ( plus_p1859984266308609217nnreal @ Y4 @ Z ) )
= ( minus_8429688780609304081nnreal @ ( minus_8429688780609304081nnreal @ X @ Y4 ) @ Z ) ) ).
% diff_add_eq_diff_diff_swap_ennreal
thf(fact_1201_Sup__eq__PInfty,axiom,
! [S: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ extend1530274965995635425_ereal @ S )
=> ( ( comple8415311339701865915_ereal @ S )
= extend1530274965995635425_ereal ) ) ).
% Sup_eq_PInfty
thf(fact_1202_top__ereal__def,axiom,
top_to6662034908053899550_ereal = extend1530274965995635425_ereal ).
% top_ereal_def
thf(fact_1203_add__diff__inverse__ennreal,axiom,
! [X: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X @ Y4 )
=> ( ( plus_p1859984266308609217nnreal @ X @ ( minus_8429688780609304081nnreal @ Y4 @ X ) )
= Y4 ) ) ).
% add_diff_inverse_ennreal
thf(fact_1204_diff__add__cancel__ennreal,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B )
=> ( ( plus_p1859984266308609217nnreal @ ( minus_8429688780609304081nnreal @ B @ A2 ) @ A2 )
= B ) ) ).
% diff_add_cancel_ennreal
thf(fact_1205_diff__add__assoc2__ennreal,axiom,
! [B: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B @ A2 )
=> ( ( plus_p1859984266308609217nnreal @ ( minus_8429688780609304081nnreal @ A2 @ B ) @ C2 )
= ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ A2 @ C2 ) @ B ) ) ) ).
% diff_add_assoc2_ennreal
thf(fact_1206_ennreal__diff__add__assoc,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B )
=> ( ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ C2 @ B ) @ A2 )
= ( plus_p1859984266308609217nnreal @ C2 @ ( minus_8429688780609304081nnreal @ B @ A2 ) ) ) ) ).
% ennreal_diff_add_assoc
thf(fact_1207_ennreal__ineq__diff__add,axiom,
! [B: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B @ A2 )
=> ( A2
= ( plus_p1859984266308609217nnreal @ B @ ( minus_8429688780609304081nnreal @ A2 @ B ) ) ) ) ).
% ennreal_ineq_diff_add
thf(fact_1208_diff__add__self__ennreal,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( ord_le3935885782089961368nnreal @ A2 @ B )
=> ( ( plus_p1859984266308609217nnreal @ ( minus_8429688780609304081nnreal @ B @ A2 ) @ A2 )
= B ) )
& ( ~ ( ord_le3935885782089961368nnreal @ A2 @ B )
=> ( ( plus_p1859984266308609217nnreal @ ( minus_8429688780609304081nnreal @ B @ A2 ) @ A2 )
= A2 ) ) ) ).
% diff_add_self_ennreal
thf(fact_1209_add__diff__self__ennreal,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( ord_le3935885782089961368nnreal @ A2 @ B )
=> ( ( plus_p1859984266308609217nnreal @ A2 @ ( minus_8429688780609304081nnreal @ B @ A2 ) )
= B ) )
& ( ~ ( ord_le3935885782089961368nnreal @ A2 @ B )
=> ( ( plus_p1859984266308609217nnreal @ A2 @ ( minus_8429688780609304081nnreal @ B @ A2 ) )
= A2 ) ) ) ).
% add_diff_self_ennreal
thf(fact_1210_add__diff__le__ennreal,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ A2 @ B ) @ C2 ) @ ( plus_p1859984266308609217nnreal @ A2 @ ( minus_8429688780609304081nnreal @ B @ C2 ) ) ) ).
% add_diff_le_ennreal
thf(fact_1211_add__diff__eq__ennreal,axiom,
! [Z: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal,X: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Z @ Y4 )
=> ( ( plus_p1859984266308609217nnreal @ X @ ( minus_8429688780609304081nnreal @ Y4 @ Z ) )
= ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ X @ Y4 ) @ Z ) ) ) ).
% add_diff_eq_ennreal
thf(fact_1212_diff__diff__ennreal_H,axiom,
! [Z: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal,X: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Z @ Y4 )
=> ( ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ Y4 @ Z ) @ X )
=> ( ( minus_8429688780609304081nnreal @ X @ ( minus_8429688780609304081nnreal @ Y4 @ Z ) )
= ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ X @ Z ) @ Y4 ) ) ) ) ).
% diff_diff_ennreal'
thf(fact_1213_ennreal__minus__le__iff,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A2 @ B ) @ C2 )
= ( ( ord_le3935885782089961368nnreal @ A2 @ ( plus_p1859984266308609217nnreal @ B @ C2 ) )
& ( ( ( A2 = top_to1496364449551166952nnreal )
& ( B = top_to1496364449551166952nnreal ) )
=> ( C2 = top_to1496364449551166952nnreal ) ) ) ) ).
% ennreal_minus_le_iff
thf(fact_1214_ereal__minus__eq__PInfty__iff,axiom,
! [X: extended_ereal,Y4: extended_ereal] :
( ( ( minus_2816186181549245109_ereal @ X @ Y4 )
= extend1530274965995635425_ereal )
= ( ( Y4
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
| ( X = extend1530274965995635425_ereal ) ) ) ).
% ereal_minus_eq_PInfty_iff
thf(fact_1215_ereal__minus__eq__minus__iff,axiom,
! [A2: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( ( minus_2816186181549245109_ereal @ A2 @ B )
= ( minus_2816186181549245109_ereal @ A2 @ C2 ) )
= ( ( B = C2 )
| ( A2 = extend1530274965995635425_ereal )
| ( ( A2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
& ( B
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
& ( C2
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ).
% ereal_minus_eq_minus_iff
thf(fact_1216_ereal__minus__diff__eq,axiom,
! [X: extended_ereal,Y4: extended_ereal] :
( ( ( X = extend1530274965995635425_ereal )
=> ( Y4 != extend1530274965995635425_ereal ) )
=> ( ( ( X
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Y4
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) )
=> ( ( uminus27091377158695749_ereal @ ( minus_2816186181549245109_ereal @ X @ Y4 ) )
= ( minus_2816186181549245109_ereal @ Y4 @ X ) ) ) ) ).
% ereal_minus_diff_eq
thf(fact_1217_INF__ennreal__add__const,axiom,
! [F: nat > extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( comple7330758040695736817nnreal
@ ( image_8459861568512453903nnreal
@ ^ [I: nat] : ( plus_p1859984266308609217nnreal @ ( F @ I ) @ C2 )
@ top_top_set_nat ) )
= ( plus_p1859984266308609217nnreal @ ( comple7330758040695736817nnreal @ ( image_8459861568512453903nnreal @ F @ top_top_set_nat ) ) @ C2 ) ) ).
% INF_ennreal_add_const
thf(fact_1218_INF__ennreal__const__add,axiom,
! [C2: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal] :
( ( comple7330758040695736817nnreal
@ ( image_8459861568512453903nnreal
@ ^ [I: nat] : ( plus_p1859984266308609217nnreal @ C2 @ ( F @ I ) )
@ top_top_set_nat ) )
= ( plus_p1859984266308609217nnreal @ C2 @ ( comple7330758040695736817nnreal @ ( image_8459861568512453903nnreal @ F @ top_top_set_nat ) ) ) ) ).
% INF_ennreal_const_add
thf(fact_1219_diff__diff__left,axiom,
! [I4: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J2 ) @ K )
= ( minus_minus_nat @ I4 @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% diff_diff_left
thf(fact_1220_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ I4 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I4 @ J2 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1221_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I4 )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I4 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1222_Nat_Odiff__diff__right,axiom,
! [K: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I4 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I4 @ K ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_1223_ereal__minus_I6_J,axiom,
! [X: extended_ereal,Y4: extended_ereal] :
( ( minus_2816186181549245109_ereal @ X @ ( uminus27091377158695749_ereal @ Y4 ) )
= ( plus_p7876563987511257093_ereal @ X @ Y4 ) ) ).
% ereal_minus(6)
thf(fact_1224_ereal__diff__add__assoc2,axiom,
! [X: extended_ereal,Y4: extended_ereal,Z: extended_ereal] :
( ( minus_2816186181549245109_ereal @ ( plus_p7876563987511257093_ereal @ X @ Y4 ) @ Z )
= ( plus_p7876563987511257093_ereal @ ( minus_2816186181549245109_ereal @ X @ Z ) @ Y4 ) ) ).
% ereal_diff_add_assoc2
thf(fact_1225_diff__add__eq__ereal,axiom,
! [A2: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ ( minus_2816186181549245109_ereal @ A2 @ B ) @ C2 )
= ( minus_2816186181549245109_ereal @ ( plus_p7876563987511257093_ereal @ A2 @ C2 ) @ B ) ) ).
% diff_add_eq_ereal
thf(fact_1226_add__diff__eq__ereal,axiom,
! [X: extended_ereal,Y4: extended_ereal,Z: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ X @ ( minus_2816186181549245109_ereal @ Y4 @ Z ) )
= ( minus_2816186181549245109_ereal @ ( plus_p7876563987511257093_ereal @ X @ Y4 ) @ Z ) ) ).
% add_diff_eq_ereal
thf(fact_1227_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_1228_diff__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_cancel2
thf(fact_1229_diff__add__inverse,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ N2 )
= M2 ) ).
% diff_add_inverse
thf(fact_1230_diff__add__inverse2,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ N2 )
= M2 ) ).
% diff_add_inverse2
thf(fact_1231_le__diff__conv,axiom,
! [J2: nat,K: nat,I4: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K ) @ I4 )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I4 @ K ) ) ) ).
% le_diff_conv
thf(fact_1232_Nat_Ole__diff__conv2,axiom,
! [K: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_eq_nat @ I4 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1233_Nat_Odiff__add__assoc,axiom,
! [K: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I4 @ J2 ) @ K )
= ( plus_plus_nat @ I4 @ ( minus_minus_nat @ J2 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1234_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J2: nat,I4: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I4 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I4 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1235_Nat_Ole__imp__diff__is__add,axiom,
! [I4: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I4 )
= K )
= ( J2
= ( plus_plus_nat @ K @ I4 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1236_minus__ereal__def,axiom,
( minus_2816186181549245109_ereal
= ( ^ [X2: extended_ereal,Y2: extended_ereal] : ( plus_p7876563987511257093_ereal @ X2 @ ( uminus27091377158695749_ereal @ Y2 ) ) ) ) ).
% minus_ereal_def
thf(fact_1237_ereal__add__uminus__conv__diff,axiom,
! [X: extended_ereal,Y4: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ ( uminus27091377158695749_ereal @ X ) @ Y4 )
= ( minus_2816186181549245109_ereal @ Y4 @ X ) ) ).
% ereal_add_uminus_conv_diff
thf(fact_1238_ereal__ineq__diff__add,axiom,
! [B: extended_ereal,A2: extended_ereal] :
( ( B
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( ord_le1083603963089353582_ereal @ B @ A2 )
=> ( A2
= ( plus_p7876563987511257093_ereal @ B @ ( minus_2816186181549245109_ereal @ A2 @ B ) ) ) ) ) ).
% ereal_ineq_diff_add
thf(fact_1239_INF__ereal__add,axiom,
! [F: nat > extended_ereal,G: nat > extended_ereal] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X2: extended_ereal,Y2: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y2 @ X2 )
@ F )
=> ( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X2: extended_ereal,Y2: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y2 @ X2 )
@ G )
=> ( ! [I3: nat] :
( ( F @ I3 )
!= extend1530274965995635425_ereal )
=> ( ! [I3: nat] :
( ( G @ I3 )
!= extend1530274965995635425_ereal )
=> ( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I: nat] : ( plus_p7876563987511257093_ereal @ ( F @ I ) @ ( G @ I ) )
@ top_top_set_nat ) )
= ( plus_p7876563987511257093_ereal @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F @ top_top_set_nat ) ) @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ G @ top_top_set_nat ) ) ) ) ) ) ) ) ).
% INF_ereal_add
thf(fact_1240_SUP__ereal__add,axiom,
! [F: nat > extended_ereal,G: nat > extended_ereal] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ F )
=> ( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ G )
=> ( ! [I3: nat] :
( ( F @ I3 )
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ! [I3: nat] :
( ( G @ I3 )
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I: nat] : ( plus_p7876563987511257093_ereal @ ( F @ I ) @ ( G @ I ) )
@ top_top_set_nat ) )
= ( plus_p7876563987511257093_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F @ top_top_set_nat ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ top_top_set_nat ) ) ) ) ) ) ) ) ).
% SUP_ereal_add
thf(fact_1241_SUP__ereal__add__pos,axiom,
! [F: nat > extended_ereal,G: nat > extended_ereal] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ F )
=> ( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ G )
=> ( ! [I3: nat] : ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F @ I3 ) )
=> ( ! [I3: nat] : ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( G @ I3 ) )
=> ( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I: nat] : ( plus_p7876563987511257093_ereal @ ( F @ I ) @ ( G @ I ) )
@ top_top_set_nat ) )
= ( plus_p7876563987511257093_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F @ top_top_set_nat ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ top_top_set_nat ) ) ) ) ) ) ) ) ).
% SUP_ereal_add_pos
thf(fact_1242_ereal__minus_I7_J,axiom,
! [X: extended_ereal] :
( ( minus_2816186181549245109_ereal @ X @ zero_z2744965634713055877_ereal )
= X ) ).
% ereal_minus(7)
thf(fact_1243_ereal__minus_I8_J,axiom,
! [X: extended_ereal] :
( ( minus_2816186181549245109_ereal @ zero_z2744965634713055877_ereal @ X )
= ( uminus27091377158695749_ereal @ X ) ) ).
% ereal_minus(8)
thf(fact_1244_ereal__diff__le__mono__left,axiom,
! [X: extended_ereal,Z: extended_ereal,Y4: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ Z )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ Y4 )
=> ( ord_le1083603963089353582_ereal @ ( minus_2816186181549245109_ereal @ X @ Y4 ) @ Z ) ) ) ).
% ereal_diff_le_mono_left
thf(fact_1245_ereal__diff__positive,axiom,
! [A2: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A2 @ B )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( minus_2816186181549245109_ereal @ B @ A2 ) ) ) ).
% ereal_diff_positive
thf(fact_1246_ereal__diff__le__self,axiom,
! [Y4: extended_ereal,X: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ Y4 )
=> ( ord_le1083603963089353582_ereal @ ( minus_2816186181549245109_ereal @ X @ Y4 ) @ X ) ) ).
% ereal_diff_le_self
thf(fact_1247_ereal__diff__nonpos,axiom,
! [A2: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A2 @ B )
=> ( ( ( A2 = extend1530274965995635425_ereal )
=> ( B != extend1530274965995635425_ereal ) )
=> ( ( ( A2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( B
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) )
=> ( ord_le1083603963089353582_ereal @ ( minus_2816186181549245109_ereal @ A2 @ B ) @ zero_z2744965634713055877_ereal ) ) ) ) ).
% ereal_diff_nonpos
thf(fact_1248_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1249_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1250_ennreal__minus__zero,axiom,
! [A2: extend8495563244428889912nnreal] :
( ( minus_8429688780609304081nnreal @ A2 @ zero_z7100319975126383169nnreal )
= A2 ) ).
% ennreal_minus_zero
thf(fact_1251_zero__minus__ennreal,axiom,
! [A2: extend8495563244428889912nnreal] :
( ( minus_8429688780609304081nnreal @ zero_z7100319975126383169nnreal @ A2 )
= zero_z7100319975126383169nnreal ) ).
% zero_minus_ennreal
thf(fact_1252_diff__is__0__eq_H,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1253_diff__is__0__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_1254_ennreal__diff__self,axiom,
! [A2: extend8495563244428889912nnreal] :
( ( A2 != top_to1496364449551166952nnreal )
=> ( ( minus_8429688780609304081nnreal @ A2 @ A2 )
= zero_z7100319975126383169nnreal ) ) ).
% ennreal_diff_self
thf(fact_1255_diff__add__0,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1256_diffs0__imp__equal,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M2 )
= zero_zero_nat )
=> ( M2 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_1257_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_1258_ennreal__zero__neq__top,axiom,
zero_z7100319975126383169nnreal != top_to1496364449551166952nnreal ).
% ennreal_zero_neq_top
thf(fact_1259_ennreal__minus__eq__0,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A2 @ B )
= zero_z7100319975126383169nnreal )
=> ( ord_le3935885782089961368nnreal @ A2 @ B ) ) ).
% ennreal_minus_eq_0
thf(fact_1260_minus__top__ennreal,axiom,
! [X: extend8495563244428889912nnreal] :
( ( ( X = top_to1496364449551166952nnreal )
=> ( ( minus_8429688780609304081nnreal @ X @ top_to1496364449551166952nnreal )
= top_to1496364449551166952nnreal ) )
& ( ( X != top_to1496364449551166952nnreal )
=> ( ( minus_8429688780609304081nnreal @ X @ top_to1496364449551166952nnreal )
= zero_z7100319975126383169nnreal ) ) ) ).
% minus_top_ennreal
thf(fact_1261_diff__diff__ennreal_H_H,axiom,
! [Z: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal,X: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Z @ Y4 )
=> ( ( ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ Y4 @ Z ) @ X )
=> ( ( minus_8429688780609304081nnreal @ X @ ( minus_8429688780609304081nnreal @ Y4 @ Z ) )
= ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ X @ Z ) @ Y4 ) ) )
& ( ~ ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ Y4 @ Z ) @ X )
=> ( ( minus_8429688780609304081nnreal @ X @ ( minus_8429688780609304081nnreal @ Y4 @ Z ) )
= zero_z7100319975126383169nnreal ) ) ) ) ).
% diff_diff_ennreal''
thf(fact_1262_ennreal__le__minus__iff,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ ( minus_8429688780609304081nnreal @ B @ C2 ) )
= ( ( ord_le3935885782089961368nnreal @ ( plus_p1859984266308609217nnreal @ A2 @ C2 ) @ B )
| ( ( A2 = zero_z7100319975126383169nnreal )
& ( ord_le3935885782089961368nnreal @ B @ C2 ) ) ) ) ).
% ennreal_le_minus_iff
thf(fact_1263_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1264_ennreal__add__less__top,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ ( plus_p1859984266308609217nnreal @ A2 @ B ) @ top_to1496364449551166952nnreal )
= ( ( ord_le7381754540660121996nnreal @ A2 @ top_to1496364449551166952nnreal )
& ( ord_le7381754540660121996nnreal @ B @ top_to1496364449551166952nnreal ) ) ) ).
% ennreal_add_less_top
thf(fact_1265_diff__gr0__ennreal,axiom,
! [B: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B @ A2 )
=> ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( minus_8429688780609304081nnreal @ A2 @ B ) ) ) ).
% diff_gr0_ennreal
thf(fact_1266_ennreal__zero__less__top,axiom,
ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ top_to1496364449551166952nnreal ).
% ennreal_zero_less_top
thf(fact_1267_ennreal__between,axiom,
! [E: extend8495563244428889912nnreal,X: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ E )
=> ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ X )
=> ( ( ord_le7381754540660121996nnreal @ X @ top_to1496364449551166952nnreal )
=> ( ord_le7381754540660121996nnreal @ ( minus_8429688780609304081nnreal @ X @ E ) @ X ) ) ) ) ).
% ennreal_between
thf(fact_1268_ennreal__minus__pos__iff,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( ord_le7381754540660121996nnreal @ A2 @ top_to1496364449551166952nnreal )
| ( ord_le7381754540660121996nnreal @ B @ top_to1496364449551166952nnreal ) )
=> ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( minus_8429688780609304081nnreal @ A2 @ B ) )
=> ( ord_le7381754540660121996nnreal @ B @ A2 ) ) ) ).
% ennreal_minus_pos_iff
thf(fact_1269_diff__gt__0__iff__gt__ennreal,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( minus_8429688780609304081nnreal @ A2 @ B ) )
= ( ( ( A2 = top_to1496364449551166952nnreal )
& ( B = top_to1496364449551166952nnreal ) )
| ( ord_le7381754540660121996nnreal @ B @ A2 ) ) ) ).
% diff_gt_0_iff_gt_ennreal
thf(fact_1270_diff__less__top__ennreal,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ ( minus_8429688780609304081nnreal @ A2 @ B ) @ top_to1496364449551166952nnreal )
= ( ord_le7381754540660121996nnreal @ A2 @ top_to1496364449551166952nnreal ) ) ).
% diff_less_top_ennreal
thf(fact_1271_ennreal__mono__minus__cancel,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A2 @ B ) @ ( minus_8429688780609304081nnreal @ A2 @ C2 ) )
=> ( ( ord_le7381754540660121996nnreal @ A2 @ top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ B @ A2 )
=> ( ( ord_le3935885782089961368nnreal @ C2 @ A2 )
=> ( ord_le3935885782089961368nnreal @ C2 @ B ) ) ) ) ) ).
% ennreal_mono_minus_cancel
thf(fact_1272_diff__eq__0__iff__ennreal,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A2 @ B )
= zero_z7100319975126383169nnreal )
= ( ( ord_le7381754540660121996nnreal @ A2 @ top_to1496364449551166952nnreal )
& ( ord_le3935885782089961368nnreal @ A2 @ B ) ) ) ).
% diff_eq_0_iff_ennreal
thf(fact_1273_diff__eq__0__ennreal,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ A2 @ B )
=> ( ( minus_8429688780609304081nnreal @ A2 @ B )
= zero_z7100319975126383169nnreal ) ) ) ).
% diff_eq_0_ennreal
thf(fact_1274_less__diff__eq__ennreal,axiom,
! [B: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ( ord_le7381754540660121996nnreal @ B @ top_to1496364449551166952nnreal )
| ( ord_le7381754540660121996nnreal @ C2 @ top_to1496364449551166952nnreal ) )
=> ( ( ord_le7381754540660121996nnreal @ A2 @ ( minus_8429688780609304081nnreal @ B @ C2 ) )
= ( ord_le7381754540660121996nnreal @ ( plus_p1859984266308609217nnreal @ A2 @ C2 ) @ B ) ) ) ).
% less_diff_eq_ennreal
% Conjectures (1)
thf(conj_0,conjecture,
( ( collect_b
@ ^ [Uu: b] :
? [X2: a] :
( ( Uu
= ( f @ X2 ) )
& ( p @ X2 ) ) )
= ( image_c_b @ h
@ ( collect_c
@ ^ [Uu: c] :
? [X2: a] :
( ( Uu
= ( g @ X2 ) )
& ( p @ X2 ) ) ) ) ) ).
%------------------------------------------------------------------------------