TPTP Problem File: SLH0031^1.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 : Prefix_Free_Code_Combinators/0000_Prefix_Free_Code_Combinators/prob_00158_005760__11821182_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1844 ( 528 unt; 556 typ; 0 def)
% Number of atoms : 3814 (1577 equ; 0 cnn)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 13769 ( 278 ~; 15 |; 202 &;11624 @)
% ( 0 <=>;1650 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 59 ( 58 usr)
% Number of type conns : 3035 (3035 >; 0 *; 0 +; 0 <<)
% Number of symbols : 501 ( 498 usr; 42 con; 0-5 aty)
% Number of variables : 4436 ( 691 ^;3639 !; 106 ?;4436 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:57:22.188
%------------------------------------------------------------------------------
% Could-be-implicit typings (58)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_It__List__Olist_I_Eo_J_J_Mt__Option__Ooption_It__List__Olist_I_Eo_J_J_J_J_J,type,
set_se8915165130606451687list_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Product____Type__Oprod_It__Option__Ooption_It__List__Olist_I_Eo_J_J_Mt__Option__Ooption_It__List__Olist_I_Eo_J_J_J_M_Eo_J_J,type,
set_Pr5416514491829664972st_o_o: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_It__List__Olist_I_Eo_J_J_Mt__Option__Ooption_It__List__Olist_I_Eo_J_J_J_J,type,
set_Pr7497825696620840711list_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Option__Ooption_It__List__Olist_I_Eo_J_J_M_062_It__Option__Ooption_It__List__Olist_I_Eo_J_J_M_Eo_J_J_J,type,
set_op7420493560124892494st_o_o: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__List__Olist_I_Eo_J_J_Mt__Option__Ooption_It__List__Olist_I_Eo_J_J_J,type,
produc4882884732533091879list_o: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Extended____Real__Oereal_Mt__Extended____Real__Oereal_J_J,type,
set_Pr2129990008675586951_ereal: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Extended____Real__Oereal_Mt__Product____Type__Oprod_Itf__a_Mtf__b_J_J_J,type,
set_Ex8012342718228902517od_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Extended____Real__Oereal_J_J,type,
set_Pr8411329518592215081_ereal: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Extended____Real__Oereal_Mt__Nat__Onat_J_J,type,
set_Pr450698115992974979al_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Extended____Real__Oereal_Mt__Extended____Real__Oereal_J,type,
produc5501587555545223847_ereal: $tType ).
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__Product____Type__Oprod_Itf__a_Mt__Extended____Real__Oereal_J_J,type,
set_Pr937448535721291095_ereal: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Extended____Real__Oereal_Mtf__a_J_J,type,
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set_na3619395881858068463od_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Option__Ooption_It__List__Olist_I_Eo_J_J_J,type,
produc2335322370636471565list_o: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__List__Olist_I_Eo_J_J_Mtf__b_J,type,
produc7323895778178278594st_o_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__List__Olist_I_Eo_J_J_J,type,
set_Pr3077528766752018087list_o: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Extended____Real__Oereal_J,type,
produc7896515489273245043_ereal: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Extended____Real__Oereal_Mt__Nat__Onat_J,type,
produc5547956294148499661al_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Extended____Real__Oereal_J_J,type,
set_op5201211589336098036_ereal: $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__Option__Ooption_It__Option__Ooption_It__List__Olist_I_Eo_J_J_J,type,
option_option_list_o: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Extended____Real__Oereal_J,type,
produc3654056799218114807_ereal: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Extended____Real__Oereal_Mtf__a_J,type,
produc1703666962261337815real_a: $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_It__Option__Ooption_It__List__Olist_I_Eo_J_J_J,type,
set_option_list_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Extended____Real__Oereal_J_J,type,
set_a_Extended_ereal: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J,type,
set_Product_prod_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
set_Product_prod_a_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__List__Olist_I_Eo_J_J,type,
produc5884233991663340231list_o: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
set_option_nat: $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__Option__Ooption_It__Extended____Real__Oereal_J,type,
option2379537754664488340_ereal: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Extended____Real__Oereal_J,type,
set_Extended_ereal: $tType ).
thf(ty_n_t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
option_list_o: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
product_prod_b_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__b_Mtf__a_J,type,
product_prod_b_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
product_prod_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
product_prod_a_a: $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_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
set_list_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
set_a_o: $tType ).
thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
option_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Option__Ooption_Itf__b_J,type,
option_b: $tType ).
thf(ty_n_t__Extended____Real__Oereal,type,
extended_ereal: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__List__Olist_I_Eo_J,type,
list_o: $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__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (498)
thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
bNF_Ca7786835802997198645list_o: set_option_list_o > set_Pr7497825696620840711list_o > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_001t__Extended____Real__Oereal,type,
bNF_Ca4784764557604061198_ereal: set_Pr7497825696620840711list_o > ( option_list_o > extended_ereal ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_001t__Nat__Onat,type,
bNF_Ca9081405844473788304_o_nat: set_Pr7497825696620840711list_o > ( option_list_o > nat ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
bNF_Ca6937576274950426094_ereal: set_Pr7497825696620840711list_o > ( option_list_o > set_Extended_ereal ) > $o ).
thf(sy_c_BNF__Def_OGr_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
bNF_Gr945911885070196386_ereal: set_Extended_ereal > ( extended_ereal > extended_ereal ) > set_Pr2129990008675586951_ereal ).
thf(sy_c_BNF__Def_OGr_001t__Nat__Onat_001t__Extended____Real__Oereal,type,
bNF_Gr8352086143671499994_ereal: set_nat > ( nat > extended_ereal ) > set_Pr8411329518592215081_ereal ).
thf(sy_c_BNF__Def_OGr_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
bNF_Gr4294414969041315106list_o: set_option_list_o > ( option_list_o > option_list_o ) > set_Pr7497825696620840711list_o ).
thf(sy_c_BNF__Def_OGrp_001_062_It__Extended____Real__Oereal_Mt__Product____Type__Oprod_Itf__a_Mtf__b_J_J_001_062_It__Extended____Real__Oereal_Mtf__a_J,type,
bNF_Gr4066342199408912420real_a: set_Ex8012342718228902517od_a_b > ( ( extended_ereal > product_prod_a_b ) > extended_ereal > a ) > ( extended_ereal > product_prod_a_b ) > ( extended_ereal > a ) > $o ).
thf(sy_c_BNF__Def_OGrp_001_062_It__Extended____Real__Oereal_Mt__Product____Type__Oprod_Itf__a_Mtf__b_J_J_001_062_It__Extended____Real__Oereal_Mtf__b_J,type,
bNF_Gr4066342203712141221real_b: set_Ex8012342718228902517od_a_b > ( ( extended_ereal > product_prod_a_b ) > extended_ereal > b ) > ( extended_ereal > product_prod_a_b ) > ( extended_ereal > b ) > $o ).
thf(sy_c_BNF__Def_OGrp_001_062_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__a_Mtf__b_J_J_001_062_It__Nat__Onat_Mtf__a_J,type,
bNF_Gr2261899126669558756_nat_a: set_na3619395881858068463od_a_b > ( ( nat > product_prod_a_b ) > nat > a ) > ( nat > product_prod_a_b ) > ( nat > a ) > $o ).
thf(sy_c_BNF__Def_OGrp_001_062_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__a_Mtf__b_J_J_001_062_It__Nat__Onat_Mtf__b_J,type,
bNF_Gr2261899130972787557_nat_b: set_na3619395881858068463od_a_b > ( ( nat > product_prod_a_b ) > nat > b ) > ( nat > product_prod_a_b ) > ( nat > b ) > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
bNF_Gr4257726243395767246_ereal: set_Extended_ereal > ( extended_ereal > extended_ereal ) > extended_ereal > extended_ereal > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
bNF_Gr3846222354243918800al_nat: set_Extended_ereal > ( extended_ereal > nat ) > extended_ereal > nat > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Nat__Onat_001t__Extended____Real__Oereal,type,
bNF_Gr495653965960080046_ereal: set_nat > ( nat > extended_ereal ) > nat > extended_ereal > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_Grp_nat_nat: set_nat > ( nat > nat ) > nat > nat > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001tf__a,type,
bNF_Gr7842136747927947300_a_b_a: set_Product_prod_a_b > ( product_prod_a_b > a ) > product_prod_a_b > a > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001tf__b,type,
bNF_Gr7842136747927947301_a_b_b: set_Product_prod_a_b > ( product_prod_a_b > b ) > product_prod_a_b > b > $o ).
thf(sy_c_BNF__Def_OGrp_001tf__a_001tf__b,type,
bNF_Grp_a_b: set_a > ( a > b ) > a > b > $o ).
thf(sy_c_BNF__Def_OfstOp_001tf__a_001tf__b_001tf__b,type,
bNF_fstOp_a_b_b: ( a > b > $o ) > ( b > b > $o ) > product_prod_a_b > product_prod_a_b ).
thf(sy_c_BNF__Def_Orel__fun_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
bNF_re3416630401399921757_ereal: ( extended_ereal > extended_ereal > $o ) > ( extended_ereal > extended_ereal > $o ) > ( extended_ereal > extended_ereal ) > ( extended_ereal > extended_ereal ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal_001tf__a,type,
bNF_re5691446141301026317real_a: ( extended_ereal > extended_ereal > $o ) > ( extended_ereal > a > $o ) > ( extended_ereal > extended_ereal ) > ( extended_ereal > a ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal_001tf__a_001t__Extended____Real__Oereal,type,
bNF_re1148940357394609709_ereal: ( extended_ereal > extended_ereal > $o ) > ( a > extended_ereal > $o ) > ( extended_ereal > a ) > ( extended_ereal > extended_ereal ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal_001tf__a_001tf__a,type,
bNF_re4205385778126815197al_a_a: ( extended_ereal > extended_ereal > $o ) > ( a > a > $o ) > ( extended_ereal > a ) > ( extended_ereal > a ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal_001tf__a_001tf__b,type,
bNF_re4205385778126815198al_a_b: ( extended_ereal > extended_ereal > $o ) > ( a > b > $o ) > ( extended_ereal > a ) > ( extended_ereal > b ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
bNF_re5337380594787866367_ereal: ( nat > nat > $o ) > ( extended_ereal > extended_ereal > $o ) > ( nat > extended_ereal ) > ( nat > extended_ereal ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Extended____Real__Oereal_001tf__a,type,
bNF_re6402089080707531695real_a: ( nat > nat > $o ) > ( extended_ereal > a > $o ) > ( nat > extended_ereal ) > ( nat > a ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001tf__a_001t__Extended____Real__Oereal,type,
bNF_re1859583296801115087_ereal: ( nat > nat > $o ) > ( a > extended_ereal > $o ) > ( nat > a ) > ( nat > extended_ereal ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001tf__a_001tf__a,type,
bNF_re4153754443986628735at_a_a: ( nat > nat > $o ) > ( a > a > $o ) > ( nat > a ) > ( nat > a ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001tf__a_001tf__b,type,
bNF_re4153754443986628736at_a_b: ( nat > nat > $o ) > ( a > b > $o ) > ( nat > a ) > ( nat > b ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001tf__a_001tf__b_001_062_Itf__a_M_Eo_J_001_062_Itf__b_M_Eo_J,type,
bNF_re5830743871565202077_o_b_o: ( a > b > $o ) > ( ( a > $o ) > ( b > $o ) > $o ) > ( a > a > $o ) > ( b > b > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001tf__a_001tf__b_001_Eo_001_Eo,type,
bNF_rel_fun_a_b_o_o: ( a > b > $o ) > ( $o > $o > $o ) > ( a > $o ) > ( b > $o ) > $o ).
thf(sy_c_BNF__Def_OsndOp_001tf__a_001tf__a_001tf__b,type,
bNF_sndOp_a_a_b: ( a > a > $o ) > ( a > b > $o ) > product_prod_a_b > product_prod_a_b ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001tf__a_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
bNF_Gr7877113298573333156list_o: set_a > ( a > option_list_o ) > ( a > option_list_o ) > set_Pr7497825696620840711list_o ).
thf(sy_c_BNF__Wellorder__Constructions_OFunc_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
bNF_We7694149557721242900_ereal: set_Extended_ereal > set_Extended_ereal > set_Ex2354994561656779803_ereal ).
thf(sy_c_BNF__Wellorder__Constructions_OFunc_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
bNF_We489741039315895306al_nat: set_Extended_ereal > set_nat > set_Ex8414784459666926319al_nat ).
thf(sy_c_BNF__Wellorder__Constructions_OFunc_001t__Nat__Onat_001t__Extended____Real__Oereal,type,
bNF_We6362544687886832360_ereal: set_nat > set_Extended_ereal > set_na7152043825411390613_ereal ).
thf(sy_c_BNF__Wellorder__Constructions_OFunc_001tf__a_001t__Extended____Real__Oereal,type,
bNF_We381379198075229540_ereal: set_a > set_Extended_ereal > set_a_Extended_ereal ).
thf(sy_c_BNF__Wellorder__Constructions_OFunc__map_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
bNF_We4175038070499737256_ereal: set_Extended_ereal > ( extended_ereal > extended_ereal ) > ( extended_ereal > extended_ereal ) > ( extended_ereal > extended_ereal ) > extended_ereal > extended_ereal ).
thf(sy_c_BNF__Wellorder__Constructions_OFunc__map_001t__Extended____Real__Oereal_001t__Nat__Onat_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
bNF_We3429895033966940150_ereal: set_Extended_ereal > ( nat > extended_ereal ) > ( extended_ereal > extended_ereal ) > ( extended_ereal > nat ) > extended_ereal > extended_ereal ).
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thf(sy_c_member_001_062_Itf__a_Mt__Extended____Real__Oereal_J,type,
member5529087853344971084_ereal: ( a > extended_ereal ) > set_a_Extended_ereal > $o ).
thf(sy_c_member_001t__Extended____Real__Oereal,type,
member2350847679896131959_ereal: extended_ereal > set_Extended_ereal > $o ).
thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
member_list_o: list_o > set_list_o > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Extended____Real__Oereal_J,type,
member4803645955555566013_ereal: option2379537754664488340_ereal > set_op5201211589336098036_ereal > $o ).
thf(sy_c_member_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
member_option_list_o: option_list_o > set_option_list_o > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Nat__Onat_J,type,
member_option_nat: option_nat > set_option_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Option__Ooption_It__List__Olist_I_Eo_J_J_Mt__Option__Ooption_It__List__Olist_I_Eo_J_J_J,type,
member1589324699396745552list_o: produc4882884732533091879list_o > set_Pr7497825696620840711list_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_I_Eo_J_J,type,
member7948383622993546480list_o: produc5884233991663340231list_o > set_Pr3077528766752018087list_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
member1426531481828664017od_a_b: product_prod_a_b > set_Product_prod_a_b > $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__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_e1,type,
e1: a > option_list_o ).
thf(sy_v_e2,type,
e2: a > b > option_list_o ).
thf(sy_v_x____,type,
x: product_prod_a_b ).
thf(sy_v_y____,type,
y: product_prod_a_b ).
% Relevant facts (1276)
thf(fact_0_b,axiom,
( ( product_fst_a_b @ x )
= ( product_fst_a_b @ y ) ) ).
% b
thf(fact_1_c,axiom,
( ( product_snd_a_b @ x )
= ( product_snd_a_b @ y ) ) ).
% c
thf(fact_2_d,axiom,
prefix454693708527911765comp_o @ ( e1 @ ( product_fst_a_b @ x ) ) @ ( e1 @ ( product_fst_a_b @ y ) ) ).
% d
thf(fact_3__092_060open_062opt__comp_A_Ie2_A_Ifst_Ax_J_A_Isnd_Ax_J_J_A_Ie2_A_Ifst_Ax_J_A_Isnd_Ay_J_J_092_060close_062,axiom,
prefix454693708527911765comp_o @ ( e2 @ ( product_fst_a_b @ x ) @ ( product_snd_a_b @ x ) ) @ ( e2 @ ( product_fst_a_b @ x ) @ ( product_snd_a_b @ y ) ) ).
% \<open>opt_comp (e2 (fst x) (snd x)) (e2 (fst x) (snd y))\<close>
thf(fact_4_a,axiom,
prefix454693708527911765comp_o @ ( prefix6990428588352057089od_a_b @ e1 @ e2 @ x ) @ ( prefix6990428588352057089od_a_b @ e1 @ e2 @ y ) ).
% a
thf(fact_5_assms_I1_J,axiom,
prefix7485107378405021920ding_a @ e1 ).
% assms(1)
thf(fact_6__092_060open_062fst_Ax_A_092_060in_062_Adom_Ae1_092_060close_062,axiom,
member_a @ ( product_fst_a_b @ x ) @ ( dom_a_list_o @ e1 ) ).
% \<open>fst x \<in> dom e1\<close>
thf(fact_7_opt__comp__sym,axiom,
( prefix454693708527911765comp_o
= ( ^ [X: option_list_o,Y: option_list_o] : ( prefix454693708527911765comp_o @ Y @ X ) ) ) ).
% opt_comp_sym
thf(fact_8_assms_I2_J,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( dom_a_list_o @ e1 ) )
=> ( prefix7485107378405021921ding_b @ ( e2 @ X2 ) ) ) ).
% assms(2)
thf(fact_9_is__encodingD,axiom,
! [E: a > option_list_o,X2: a,Y2: a] :
( ( prefix7485107378405021920ding_a @ E )
=> ( ( prefix454693708527911765comp_o @ ( E @ X2 ) @ ( E @ Y2 ) )
=> ( X2 = Y2 ) ) ) ).
% is_encodingD
thf(fact_10_is__encodingD,axiom,
! [E: b > option_list_o,X2: b,Y2: b] :
( ( prefix7485107378405021921ding_b @ E )
=> ( ( prefix454693708527911765comp_o @ ( E @ X2 ) @ ( E @ Y2 ) )
=> ( X2 = Y2 ) ) ) ).
% is_encodingD
thf(fact_11_is__encodingI__2,axiom,
! [E: a > option_list_o] :
( ! [X3: a,Y3: a] :
( ( prefix454693708527911765comp_o @ ( E @ X3 ) @ ( E @ Y3 ) )
=> ( X3 = Y3 ) )
=> ( prefix7485107378405021920ding_a @ E ) ) ).
% is_encodingI_2
thf(fact_12_is__encodingI__2,axiom,
! [E: b > option_list_o] :
( ! [X3: b,Y3: b] :
( ( prefix454693708527911765comp_o @ ( E @ X3 ) @ ( E @ Y3 ) )
=> ( X3 = Y3 ) )
=> ( prefix7485107378405021921ding_b @ E ) ) ).
% is_encodingI_2
thf(fact_13_prod_Oexpand,axiom,
! [Prod: product_prod_a_b,Prod2: product_prod_a_b] :
( ( ( ( product_fst_a_b @ Prod )
= ( product_fst_a_b @ Prod2 ) )
& ( ( product_snd_a_b @ Prod )
= ( product_snd_a_b @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_14_prod__eqI,axiom,
! [P: product_prod_a_b,Q: product_prod_a_b] :
( ( ( product_fst_a_b @ P )
= ( product_fst_a_b @ Q ) )
=> ( ( ( product_snd_a_b @ P )
= ( product_snd_a_b @ Q ) )
=> ( P = Q ) ) ) ).
% prod_eqI
thf(fact_15_exE__realizer_H,axiom,
! [P2: b > a > $o,P: product_prod_a_b] :
( ( P2 @ ( product_snd_a_b @ P ) @ ( product_fst_a_b @ P ) )
=> ~ ! [X3: a,Y3: b] :
~ ( P2 @ Y3 @ X3 ) ) ).
% exE_realizer'
thf(fact_16_prod__eq__iff,axiom,
( ( ^ [Y4: product_prod_a_b,Z: product_prod_a_b] : ( Y4 = Z ) )
= ( ^ [S: product_prod_a_b,T: product_prod_a_b] :
( ( ( product_fst_a_b @ S )
= ( product_fst_a_b @ T ) )
& ( ( product_snd_a_b @ S )
= ( product_snd_a_b @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_17_encode__dependent__prod__def,axiom,
( prefix6990428588352057089od_a_b
= ( ^ [E2: a > option_list_o,F: a > b > option_list_o,X: product_prod_a_b] : ( prefix5314359684614007693append @ ( E2 @ ( product_fst_a_b @ X ) ) @ ( F @ ( product_fst_a_b @ X ) @ ( product_snd_a_b @ X ) ) ) ) ) ).
% encode_dependent_prod_def
thf(fact_18_encoding__imp__inj,axiom,
! [F2: a > option_list_o] :
( ( prefix7485107378405021920ding_a @ F2 )
=> ( inj_on374126998980950615list_o @ F2 @ ( dom_a_list_o @ F2 ) ) ) ).
% encoding_imp_inj
thf(fact_19_encoding__imp__inj,axiom,
! [F2: b > option_list_o] :
( ( prefix7485107378405021921ding_b @ F2 )
=> ( inj_on8261448415883032086list_o @ F2 @ ( dom_b_list_o @ F2 ) ) ) ).
% encoding_imp_inj
thf(fact_20_is__encoding__def,axiom,
( prefix7485107378405021920ding_a
= ( ^ [F: a > option_list_o] :
! [X: a,Y: a] :
( ( prefix8824957607401505554efix_o @ ( F @ X ) @ ( F @ Y ) )
=> ( X = Y ) ) ) ) ).
% is_encoding_def
thf(fact_21_is__encoding__def,axiom,
( prefix7485107378405021921ding_b
= ( ^ [F: b > option_list_o] :
! [X: b,Y: b] :
( ( prefix8824957607401505554efix_o @ ( F @ X ) @ ( F @ Y ) )
=> ( X = Y ) ) ) ) ).
% is_encoding_def
thf(fact_22_opt__comp__append,axiom,
! [X2: option_list_o,Y2: option_list_o,Z2: option_list_o] :
( ( prefix454693708527911765comp_o @ ( prefix5314359684614007693append @ X2 @ Y2 ) @ Z2 )
=> ( prefix454693708527911765comp_o @ X2 @ Z2 ) ) ).
% opt_comp_append
thf(fact_23_opt__comp__append__2,axiom,
! [X2: option_list_o,Y2: option_list_o,Z2: option_list_o] :
( ( prefix454693708527911765comp_o @ X2 @ ( prefix5314359684614007693append @ Y2 @ Z2 ) )
=> ( prefix454693708527911765comp_o @ X2 @ Y2 ) ) ).
% opt_comp_append_2
thf(fact_24_opt__comp__append__3,axiom,
! [X2: option_list_o,Y2: option_list_o,Z2: option_list_o] :
( ( prefix454693708527911765comp_o @ ( prefix5314359684614007693append @ X2 @ Y2 ) @ ( prefix5314359684614007693append @ X2 @ Z2 ) )
=> ( prefix454693708527911765comp_o @ Y2 @ Z2 ) ) ).
% opt_comp_append_3
thf(fact_25_opt__comp__def,axiom,
( prefix454693708527911765comp_o
= ( ^ [X: option_list_o,Y: option_list_o] :
( ( prefix8824957607401505554efix_o @ X @ Y )
| ( prefix8824957607401505554efix_o @ Y @ X ) ) ) ) ).
% opt_comp_def
thf(fact_26_fst__swap,axiom,
! [X2: product_prod_b_a] :
( ( product_fst_a_b @ ( product_swap_b_a @ X2 ) )
= ( product_snd_b_a @ X2 ) ) ).
% fst_swap
thf(fact_27_fst__swap,axiom,
! [X2: product_prod_a_b] :
( ( product_fst_b_a @ ( product_swap_a_b @ X2 ) )
= ( product_snd_a_b @ X2 ) ) ).
% fst_swap
thf(fact_28_snd__swap,axiom,
! [X2: product_prod_a_b] :
( ( product_snd_b_a @ ( product_swap_a_b @ X2 ) )
= ( product_fst_a_b @ X2 ) ) ).
% snd_swap
thf(fact_29_snd__swap,axiom,
! [X2: product_prod_b_a] :
( ( product_snd_a_b @ ( product_swap_b_a @ X2 ) )
= ( product_fst_b_a @ X2 ) ) ).
% snd_swap
thf(fact_30_prod_Oswap__def,axiom,
( produc4566859058638526903list_o
= ( ^ [P3: produc4882884732533091879list_o] : ( produc5745850523778858007list_o @ ( produc3339157721029416773list_o @ P3 ) @ ( produc8474152268468583939list_o @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_31_prod_Oswap__def,axiom,
( product_swap_a_b
= ( ^ [P3: product_prod_a_b] : ( product_Pair_b_a @ ( product_snd_a_b @ P3 ) @ ( product_fst_a_b @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_32_inj__on__inverseI,axiom,
! [A: set_Extended_ereal,G: extended_ereal > extended_ereal,F2: extended_ereal > extended_ereal] :
( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
=> ( ( G @ ( F2 @ X3 ) )
= X3 ) )
=> ( inj_on7162434037990268785_ereal @ F2 @ A ) ) ).
% inj_on_inverseI
thf(fact_33_inj__on__contraD,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,X2: extended_ereal,Y2: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A )
=> ( ( X2 != Y2 )
=> ( ( member2350847679896131959_ereal @ X2 @ A )
=> ( ( member2350847679896131959_ereal @ Y2 @ A )
=> ( ( F2 @ X2 )
!= ( F2 @ Y2 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_34_inj__on__eq__iff,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,X2: extended_ereal,Y2: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A )
=> ( ( member2350847679896131959_ereal @ X2 @ A )
=> ( ( member2350847679896131959_ereal @ Y2 @ A )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_35_inj__on__cong,axiom,
! [A: set_Extended_ereal,F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal] :
( ! [A2: extended_ereal] :
( ( member2350847679896131959_ereal @ A2 @ A )
=> ( ( F2 @ A2 )
= ( G @ A2 ) ) )
=> ( ( inj_on7162434037990268785_ereal @ F2 @ A )
= ( inj_on7162434037990268785_ereal @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_36_inj__on__def,axiom,
( inj_on7162434037990268785_ereal
= ( ^ [F: extended_ereal > extended_ereal,A3: set_Extended_ereal] :
! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A3 )
=> ! [Y: extended_ereal] :
( ( member2350847679896131959_ereal @ Y @ A3 )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ) ) ) ).
% inj_on_def
thf(fact_37_inj__onI,axiom,
! [A: set_Extended_ereal,F2: extended_ereal > extended_ereal] :
( ! [X3: extended_ereal,Y3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
=> ( ( member2350847679896131959_ereal @ Y3 @ A )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y3 ) )
=> ( X3 = Y3 ) ) ) )
=> ( inj_on7162434037990268785_ereal @ F2 @ A ) ) ).
% inj_onI
thf(fact_38_inj__onD,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,X2: extended_ereal,Y2: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y2 ) )
=> ( ( member2350847679896131959_ereal @ X2 @ A )
=> ( ( member2350847679896131959_ereal @ Y2 @ A )
=> ( X2 = Y2 ) ) ) ) ) ).
% inj_onD
thf(fact_39_prod_Ocollapse,axiom,
! [Prod: produc4882884732533091879list_o] :
( ( produc5745850523778858007list_o @ ( produc8474152268468583939list_o @ Prod ) @ ( produc3339157721029416773list_o @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_40_prod_Ocollapse,axiom,
! [Prod: product_prod_a_b] :
( ( product_Pair_a_b @ ( product_fst_a_b @ Prod ) @ ( product_snd_a_b @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_41_inj__on__map__add__dom,axiom,
! [M: a > option_list_o,M2: a > option_list_o] :
( ( inj_on374126998980950615list_o @ ( map_add_a_list_o @ M @ M2 ) @ ( dom_a_list_o @ M2 ) )
= ( inj_on374126998980950615list_o @ M2 @ ( dom_a_list_o @ M2 ) ) ) ).
% inj_on_map_add_dom
thf(fact_42_old_Oprod_Oinject,axiom,
! [A4: option_list_o,B: option_list_o,A5: option_list_o,B2: option_list_o] :
( ( ( produc5745850523778858007list_o @ A4 @ B )
= ( produc5745850523778858007list_o @ A5 @ B2 ) )
= ( ( A4 = A5 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_43_prod_Oinject,axiom,
! [X1: option_list_o,X22: option_list_o,Y1: option_list_o,Y22: option_list_o] :
( ( ( produc5745850523778858007list_o @ X1 @ X22 )
= ( produc5745850523778858007list_o @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_44_swap__simp,axiom,
! [X2: option_list_o,Y2: option_list_o] :
( ( produc4566859058638526903list_o @ ( produc5745850523778858007list_o @ X2 @ Y2 ) )
= ( produc5745850523778858007list_o @ Y2 @ X2 ) ) ).
% swap_simp
thf(fact_45_mem__case__prodE,axiom,
! [Z2: a,C: option_list_o > option_list_o > set_a,P: produc4882884732533091879list_o] :
( ( member_a @ Z2 @ ( produc8989019892752111410_set_a @ C @ P ) )
=> ~ ! [X3: option_list_o,Y3: option_list_o] :
( ( P
= ( produc5745850523778858007list_o @ X3 @ Y3 ) )
=> ~ ( member_a @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_46_mem__Collect__eq,axiom,
! [A4: a,P2: a > $o] :
( ( member_a @ A4 @ ( collect_a @ P2 ) )
= ( P2 @ A4 ) ) ).
% mem_Collect_eq
thf(fact_47_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_48_Pair__inject,axiom,
! [A4: option_list_o,B: option_list_o,A5: option_list_o,B2: option_list_o] :
( ( ( produc5745850523778858007list_o @ A4 @ B )
= ( produc5745850523778858007list_o @ A5 @ B2 ) )
=> ~ ( ( A4 = A5 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_49_prod__cases,axiom,
! [P2: produc4882884732533091879list_o > $o,P: produc4882884732533091879list_o] :
( ! [A2: option_list_o,B3: option_list_o] : ( P2 @ ( produc5745850523778858007list_o @ A2 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_50_surj__pair,axiom,
! [P: produc4882884732533091879list_o] :
? [X3: option_list_o,Y3: option_list_o] :
( P
= ( produc5745850523778858007list_o @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_51_old_Oprod_Oexhaust,axiom,
! [Y2: produc4882884732533091879list_o] :
~ ! [A2: option_list_o,B3: option_list_o] :
( Y2
!= ( produc5745850523778858007list_o @ A2 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_52_map__add__dom__app__simps_I3_J,axiom,
! [M: a,L2: a > option_list_o,L1: a > option_list_o] :
( ~ ( member_a @ M @ ( dom_a_list_o @ L2 ) )
=> ( ( map_add_a_list_o @ L1 @ L2 @ M )
= ( L1 @ M ) ) ) ).
% map_add_dom_app_simps(3)
thf(fact_53_map__add__dom__app__simps_I2_J,axiom,
! [M: a,L1: a > option_list_o,L2: a > option_list_o] :
( ~ ( member_a @ M @ ( dom_a_list_o @ L1 ) )
=> ( ( map_add_a_list_o @ L1 @ L2 @ M )
= ( L2 @ M ) ) ) ).
% map_add_dom_app_simps(2)
thf(fact_54_map__add__dom__app__simps_I1_J,axiom,
! [M: a,L2: a > option_list_o,L1: a > option_list_o] :
( ( member_a @ M @ ( dom_a_list_o @ L2 ) )
=> ( ( map_add_a_list_o @ L1 @ L2 @ M )
= ( L2 @ M ) ) ) ).
% map_add_dom_app_simps(1)
thf(fact_55_fst__eqD,axiom,
! [X2: option_list_o,Y2: option_list_o,A4: option_list_o] :
( ( ( produc8474152268468583939list_o @ ( produc5745850523778858007list_o @ X2 @ Y2 ) )
= A4 )
=> ( X2 = A4 ) ) ).
% fst_eqD
thf(fact_56_fst__eqD,axiom,
! [X2: a,Y2: b,A4: a] :
( ( ( product_fst_a_b @ ( product_Pair_a_b @ X2 @ Y2 ) )
= A4 )
=> ( X2 = A4 ) ) ).
% fst_eqD
thf(fact_57_fst__conv,axiom,
! [X1: option_list_o,X22: option_list_o] :
( ( produc8474152268468583939list_o @ ( produc5745850523778858007list_o @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_58_fst__conv,axiom,
! [X1: a,X22: b] :
( ( product_fst_a_b @ ( product_Pair_a_b @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_59_snd__eqD,axiom,
! [X2: option_list_o,Y2: option_list_o,A4: option_list_o] :
( ( ( produc3339157721029416773list_o @ ( produc5745850523778858007list_o @ X2 @ Y2 ) )
= A4 )
=> ( Y2 = A4 ) ) ).
% snd_eqD
thf(fact_60_snd__eqD,axiom,
! [X2: a,Y2: b,A4: b] :
( ( ( product_snd_a_b @ ( product_Pair_a_b @ X2 @ Y2 ) )
= A4 )
=> ( Y2 = A4 ) ) ).
% snd_eqD
thf(fact_61_snd__conv,axiom,
! [X1: option_list_o,X22: option_list_o] :
( ( produc3339157721029416773list_o @ ( produc5745850523778858007list_o @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_62_snd__conv,axiom,
! [X1: a,X22: b] :
( ( product_snd_a_b @ ( product_Pair_a_b @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_63_Product__Type_OCollect__case__prodD,axiom,
! [X2: product_prod_a_b,A: a > b > $o] :
( ( member1426531481828664017od_a_b @ X2 @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ A ) ) )
=> ( A @ ( product_fst_a_b @ X2 ) @ ( product_snd_a_b @ X2 ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_64_prod_Oexhaust__sel,axiom,
! [Prod: produc4882884732533091879list_o] :
( Prod
= ( produc5745850523778858007list_o @ ( produc8474152268468583939list_o @ Prod ) @ ( produc3339157721029416773list_o @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_65_prod_Oexhaust__sel,axiom,
! [Prod: product_prod_a_b] :
( Prod
= ( product_Pair_a_b @ ( product_fst_a_b @ Prod ) @ ( product_snd_a_b @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_66_exI__realizer,axiom,
! [P2: option_list_o > option_list_o > $o,Y2: option_list_o,X2: option_list_o] :
( ( P2 @ Y2 @ X2 )
=> ( P2 @ ( produc3339157721029416773list_o @ ( produc5745850523778858007list_o @ X2 @ Y2 ) ) @ ( produc8474152268468583939list_o @ ( produc5745850523778858007list_o @ X2 @ Y2 ) ) ) ) ).
% exI_realizer
thf(fact_67_exI__realizer,axiom,
! [P2: b > a > $o,Y2: b,X2: a] :
( ( P2 @ Y2 @ X2 )
=> ( P2 @ ( product_snd_a_b @ ( product_Pair_a_b @ X2 @ Y2 ) ) @ ( product_fst_a_b @ ( product_Pair_a_b @ X2 @ Y2 ) ) ) ) ).
% exI_realizer
thf(fact_68_conjI__realizer,axiom,
! [P2: option_list_o > $o,P: option_list_o,Q2: option_list_o > $o,Q: option_list_o] :
( ( P2 @ P )
=> ( ( Q2 @ Q )
=> ( ( P2 @ ( produc8474152268468583939list_o @ ( produc5745850523778858007list_o @ P @ Q ) ) )
& ( Q2 @ ( produc3339157721029416773list_o @ ( produc5745850523778858007list_o @ P @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_69_conjI__realizer,axiom,
! [P2: a > $o,P: a,Q2: b > $o,Q: b] :
( ( P2 @ P )
=> ( ( Q2 @ Q )
=> ( ( P2 @ ( product_fst_a_b @ ( product_Pair_a_b @ P @ Q ) ) )
& ( Q2 @ ( product_snd_a_b @ ( product_Pair_a_b @ P @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_70_surjective__pairing,axiom,
! [T2: produc4882884732533091879list_o] :
( T2
= ( produc5745850523778858007list_o @ ( produc8474152268468583939list_o @ T2 ) @ ( produc3339157721029416773list_o @ T2 ) ) ) ).
% surjective_pairing
thf(fact_71_surjective__pairing,axiom,
! [T2: product_prod_a_b] :
( T2
= ( product_Pair_a_b @ ( product_fst_a_b @ T2 ) @ ( product_snd_a_b @ T2 ) ) ) ).
% surjective_pairing
thf(fact_72_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P2: option_list_o > option_list_o > $o,X2: option_list_o,Y2: option_list_o,A4: produc4882884732533091879list_o] :
( ( P2 @ X2 @ Y2 )
=> ( ( A4
= ( produc5745850523778858007list_o @ X2 @ Y2 ) )
=> ( P2 @ ( produc8474152268468583939list_o @ A4 ) @ ( produc3339157721029416773list_o @ A4 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_73_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P2: a > b > $o,X2: a,Y2: b,A4: product_prod_a_b] :
( ( P2 @ X2 @ Y2 )
=> ( ( A4
= ( product_Pair_a_b @ X2 @ Y2 ) )
=> ( P2 @ ( product_fst_a_b @ A4 ) @ ( product_snd_a_b @ A4 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_74_sndI,axiom,
! [X2: produc4882884732533091879list_o,Y2: option_list_o,Z2: option_list_o] :
( ( X2
= ( produc5745850523778858007list_o @ Y2 @ Z2 ) )
=> ( ( produc3339157721029416773list_o @ X2 )
= Z2 ) ) ).
% sndI
thf(fact_75_sndI,axiom,
! [X2: product_prod_a_b,Y2: a,Z2: b] :
( ( X2
= ( product_Pair_a_b @ Y2 @ Z2 ) )
=> ( ( product_snd_a_b @ X2 )
= Z2 ) ) ).
% sndI
thf(fact_76_eq__snd__iff,axiom,
! [B: option_list_o,P: produc4882884732533091879list_o] :
( ( B
= ( produc3339157721029416773list_o @ P ) )
= ( ? [A6: option_list_o] :
( P
= ( produc5745850523778858007list_o @ A6 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_77_eq__snd__iff,axiom,
! [B: b,P: product_prod_a_b] :
( ( B
= ( product_snd_a_b @ P ) )
= ( ? [A6: a] :
( P
= ( product_Pair_a_b @ A6 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_78_fstI,axiom,
! [X2: produc4882884732533091879list_o,Y2: option_list_o,Z2: option_list_o] :
( ( X2
= ( produc5745850523778858007list_o @ Y2 @ Z2 ) )
=> ( ( produc8474152268468583939list_o @ X2 )
= Y2 ) ) ).
% fstI
thf(fact_79_fstI,axiom,
! [X2: product_prod_a_b,Y2: a,Z2: b] :
( ( X2
= ( product_Pair_a_b @ Y2 @ Z2 ) )
=> ( ( product_fst_a_b @ X2 )
= Y2 ) ) ).
% fstI
thf(fact_80_eq__fst__iff,axiom,
! [A4: option_list_o,P: produc4882884732533091879list_o] :
( ( A4
= ( produc8474152268468583939list_o @ P ) )
= ( ? [B4: option_list_o] :
( P
= ( produc5745850523778858007list_o @ A4 @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_81_eq__fst__iff,axiom,
! [A4: a,P: product_prod_a_b] :
( ( A4
= ( product_fst_a_b @ P ) )
= ( ? [B4: b] :
( P
= ( product_Pair_a_b @ A4 @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_82_curryI,axiom,
! [F2: produc4882884732533091879list_o > $o,A4: option_list_o,B: option_list_o] :
( ( F2 @ ( produc5745850523778858007list_o @ A4 @ B ) )
=> ( produc7606196288476282253st_o_o @ F2 @ A4 @ B ) ) ).
% curryI
thf(fact_83_curryD,axiom,
! [F2: produc4882884732533091879list_o > $o,A4: option_list_o,B: option_list_o] :
( ( produc7606196288476282253st_o_o @ F2 @ A4 @ B )
=> ( F2 @ ( produc5745850523778858007list_o @ A4 @ B ) ) ) ).
% curryD
thf(fact_84_curryE,axiom,
! [F2: produc4882884732533091879list_o > $o,A4: option_list_o,B: option_list_o] :
( ( produc7606196288476282253st_o_o @ F2 @ A4 @ B )
=> ( F2 @ ( produc5745850523778858007list_o @ A4 @ B ) ) ) ).
% curryE
thf(fact_85_image2__eqI,axiom,
! [B: option_list_o,F2: a > option_list_o,X2: a,C: option_list_o,G: a > option_list_o,A: set_a] :
( ( B
= ( F2 @ X2 ) )
=> ( ( C
= ( G @ X2 ) )
=> ( ( member_a @ X2 @ A )
=> ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ B @ C ) @ ( bNF_Gr7877113298573333156list_o @ A @ F2 @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_86_ssubst__Pair__rhs,axiom,
! [R: option_list_o,S2: option_list_o,R2: set_Pr7497825696620840711list_o,S3: option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ R @ S2 ) @ R2 )
=> ( ( S3 = S2 )
=> ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ R @ S3 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_87_inj__on__fst__graph,axiom,
! [M: a > option_b] : ( inj_on4343330552946057671_a_b_a @ product_fst_a_b @ ( graph_a_b @ M ) ) ).
% inj_on_fst_graph
thf(fact_88_dom__map__add,axiom,
! [M: a > option_list_o,N: a > option_list_o] :
( ( dom_a_list_o @ ( map_add_a_list_o @ M @ N ) )
= ( sup_sup_set_a @ ( dom_a_list_o @ N ) @ ( dom_a_list_o @ M ) ) ) ).
% dom_map_add
thf(fact_89_graph__domD,axiom,
! [X2: produc5884233991663340231list_o,M: a > option_list_o] :
( ( member7948383622993546480list_o @ X2 @ ( graph_a_list_o @ M ) )
=> ( member_a @ ( product_fst_a_list_o @ X2 ) @ ( dom_a_list_o @ M ) ) ) ).
% graph_domD
thf(fact_90_graph__domD,axiom,
! [X2: product_prod_a_b,M: a > option_b] :
( ( member1426531481828664017od_a_b @ X2 @ ( graph_a_b @ M ) )
=> ( member_a @ ( product_fst_a_b @ X2 ) @ ( dom_a_b @ M ) ) ) ).
% graph_domD
thf(fact_91_pair__in__swap__image,axiom,
! [Y2: option_list_o,X2: option_list_o,A: set_Pr7497825696620840711list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ Y2 @ X2 ) @ ( image_8752170789471566789list_o @ produc4566859058638526903list_o @ A ) )
= ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X2 @ Y2 ) @ A ) ) ).
% pair_in_swap_image
thf(fact_92_Collect__case__prod__Grp__in,axiom,
! [Z2: product_prod_a_b,A: set_a,F2: a > b] :
( ( member1426531481828664017od_a_b @ Z2 @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ ( bNF_Grp_a_b @ A @ F2 ) ) ) )
=> ( member_a @ ( product_fst_a_b @ Z2 ) @ A ) ) ).
% Collect_case_prod_Grp_in
thf(fact_93_inj__on__Un__image__eq__iff,axiom,
! [F2: nat > extended_ereal,A: set_nat,B5: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ ( sup_sup_set_nat @ A @ B5 ) )
=> ( ( ( image_4309273772856505399_ereal @ F2 @ A )
= ( image_4309273772856505399_ereal @ F2 @ B5 ) )
= ( A = B5 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_94_inj__on__Un__image__eq__iff,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,B5: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ ( sup_su2680283192902082946_ereal @ A @ B5 ) )
=> ( ( ( image_6042159593519690757_ereal @ F2 @ A )
= ( image_6042159593519690757_ereal @ F2 @ B5 ) )
= ( A = B5 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_95_inj__on__image__iff,axiom,
! [A: set_Extended_ereal,G: extended_ereal > extended_ereal,F2: extended_ereal > extended_ereal] :
( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
=> ! [Xa: extended_ereal] :
( ( member2350847679896131959_ereal @ Xa @ A )
=> ( ( ( G @ ( F2 @ X3 ) )
= ( G @ ( F2 @ Xa ) ) )
= ( ( G @ X3 )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on7162434037990268785_ereal @ F2 @ A )
=> ( ( inj_on7162434037990268785_ereal @ G @ ( image_6042159593519690757_ereal @ F2 @ A ) )
= ( inj_on7162434037990268785_ereal @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_96_fst__graph__eq__dom,axiom,
! [M: a > option_list_o] :
( ( image_8953000392185216508st_o_a @ product_fst_a_list_o @ ( graph_a_list_o @ M ) )
= ( dom_a_list_o @ M ) ) ).
% fst_graph_eq_dom
thf(fact_97_fst__graph__eq__dom,axiom,
! [M: a > option_b] :
( ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( graph_a_b @ M ) )
= ( dom_a_b @ M ) ) ).
% fst_graph_eq_dom
thf(fact_98_UnCI,axiom,
! [C: a,B5: set_a,A: set_a] :
( ( ~ ( member_a @ C @ B5 )
=> ( member_a @ C @ A ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B5 ) ) ) ).
% UnCI
thf(fact_99_Un__iff,axiom,
! [C: a,A: set_a,B5: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B5 ) )
= ( ( member_a @ C @ A )
| ( member_a @ C @ B5 ) ) ) ).
% Un_iff
thf(fact_100_image__eqI,axiom,
! [B: extended_ereal,F2: extended_ereal > extended_ereal,X2: extended_ereal,A: set_Extended_ereal] :
( ( B
= ( F2 @ X2 ) )
=> ( ( member2350847679896131959_ereal @ X2 @ A )
=> ( member2350847679896131959_ereal @ B @ ( image_6042159593519690757_ereal @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_101_image__eqI,axiom,
! [B: extended_ereal,F2: nat > extended_ereal,X2: nat,A: set_nat] :
( ( B
= ( F2 @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member2350847679896131959_ereal @ B @ ( image_4309273772856505399_ereal @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_102_image__eqI,axiom,
! [B: a,F2: a > a,X2: a,A: set_a] :
( ( B
= ( F2 @ X2 ) )
=> ( ( member_a @ X2 @ A )
=> ( member_a @ B @ ( image_a_a @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_103_image__Un,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,B5: set_Extended_ereal] :
( ( image_6042159593519690757_ereal @ F2 @ ( sup_su2680283192902082946_ereal @ A @ B5 ) )
= ( sup_su2680283192902082946_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) @ ( image_6042159593519690757_ereal @ F2 @ B5 ) ) ) ).
% image_Un
thf(fact_104_image__Un,axiom,
! [F2: nat > extended_ereal,A: set_nat,B5: set_nat] :
( ( image_4309273772856505399_ereal @ F2 @ ( sup_sup_set_nat @ A @ B5 ) )
= ( sup_su2680283192902082946_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) @ ( image_4309273772856505399_ereal @ F2 @ B5 ) ) ) ).
% image_Un
thf(fact_105_rev__image__eqI,axiom,
! [X2: extended_ereal,A: set_Extended_ereal,B: extended_ereal,F2: extended_ereal > extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A )
=> ( ( B
= ( F2 @ X2 ) )
=> ( member2350847679896131959_ereal @ B @ ( image_6042159593519690757_ereal @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_106_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B: extended_ereal,F2: nat > extended_ereal] :
( ( member_nat @ X2 @ A )
=> ( ( B
= ( F2 @ X2 ) )
=> ( member2350847679896131959_ereal @ B @ ( image_4309273772856505399_ereal @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_107_rev__image__eqI,axiom,
! [X2: a,A: set_a,B: a,F2: a > a] :
( ( member_a @ X2 @ A )
=> ( ( B
= ( F2 @ X2 ) )
=> ( member_a @ B @ ( image_a_a @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_108_ball__imageD,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,P2: extended_ereal > $o] :
( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ ( image_6042159593519690757_ereal @ F2 @ A ) )
=> ( P2 @ X3 ) )
=> ! [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ A )
=> ( P2 @ ( F2 @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_109_ball__imageD,axiom,
! [F2: nat > extended_ereal,A: set_nat,P2: extended_ereal > $o] :
( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ ( image_4309273772856505399_ereal @ F2 @ A ) )
=> ( P2 @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P2 @ ( F2 @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_110_image__cong,axiom,
! [M3: set_Extended_ereal,N2: set_Extended_ereal,F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal] :
( ( M3 = N2 )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ N2 )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_6042159593519690757_ereal @ F2 @ M3 )
= ( image_6042159593519690757_ereal @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_111_image__cong,axiom,
! [M3: set_nat,N2: set_nat,F2: nat > extended_ereal,G: nat > extended_ereal] :
( ( M3 = N2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N2 )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_4309273772856505399_ereal @ F2 @ M3 )
= ( image_4309273772856505399_ereal @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_112_bex__imageD,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,P2: extended_ereal > $o] :
( ? [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ ( image_6042159593519690757_ereal @ F2 @ A ) )
& ( P2 @ X4 ) )
=> ? [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
& ( P2 @ ( F2 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_113_bex__imageD,axiom,
! [F2: nat > extended_ereal,A: set_nat,P2: extended_ereal > $o] :
( ? [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ ( image_4309273772856505399_ereal @ F2 @ A ) )
& ( P2 @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P2 @ ( F2 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_114_image__iff,axiom,
! [Z2: extended_ereal,F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ Z2 @ ( image_6042159593519690757_ereal @ F2 @ A ) )
= ( ? [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A )
& ( Z2
= ( F2 @ X ) ) ) ) ) ).
% image_iff
thf(fact_115_image__iff,axiom,
! [Z2: extended_ereal,F2: nat > extended_ereal,A: set_nat] :
( ( member2350847679896131959_ereal @ Z2 @ ( image_4309273772856505399_ereal @ F2 @ A ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A )
& ( Z2
= ( F2 @ X ) ) ) ) ) ).
% image_iff
thf(fact_116_imageI,axiom,
! [X2: extended_ereal,A: set_Extended_ereal,F2: extended_ereal > extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A )
=> ( member2350847679896131959_ereal @ ( F2 @ X2 ) @ ( image_6042159593519690757_ereal @ F2 @ A ) ) ) ).
% imageI
thf(fact_117_imageI,axiom,
! [X2: nat,A: set_nat,F2: nat > extended_ereal] :
( ( member_nat @ X2 @ A )
=> ( member2350847679896131959_ereal @ ( F2 @ X2 ) @ ( image_4309273772856505399_ereal @ F2 @ A ) ) ) ).
% imageI
thf(fact_118_imageI,axiom,
! [X2: a,A: set_a,F2: a > a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F2 @ X2 ) @ ( image_a_a @ F2 @ A ) ) ) ).
% imageI
thf(fact_119_UnI2,axiom,
! [C: a,B5: set_a,A: set_a] :
( ( member_a @ C @ B5 )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B5 ) ) ) ).
% UnI2
thf(fact_120_UnI1,axiom,
! [C: a,A: set_a,B5: set_a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B5 ) ) ) ).
% UnI1
thf(fact_121_UnE,axiom,
! [C: a,A: set_a,B5: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B5 ) )
=> ( ~ ( member_a @ C @ A )
=> ( member_a @ C @ B5 ) ) ) ).
% UnE
thf(fact_122_snd__graph__ran,axiom,
! [M: a > option_b] :
( ( image_2802296252294471260_a_b_b @ product_snd_a_b @ ( graph_a_b @ M ) )
= ( ran_a_b @ M ) ) ).
% snd_graph_ran
thf(fact_123_Collect__split__mono__strong,axiom,
! [X5: set_a,A: set_Product_prod_a_b,Y5: set_b,P2: a > b > $o,Q2: a > b > $o] :
( ( X5
= ( image_2802296252294471259_a_b_a @ product_fst_a_b @ A ) )
=> ( ( Y5
= ( image_2802296252294471260_a_b_b @ product_snd_a_b @ A ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ X5 )
=> ! [Xa: b] :
( ( member_b @ Xa @ Y5 )
=> ( ( P2 @ X3 @ Xa )
=> ( Q2 @ X3 @ Xa ) ) ) )
=> ( ( ord_le817736998455962536od_a_b @ A @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ P2 ) ) )
=> ( ord_le817736998455962536od_a_b @ A @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ Q2 ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_124_Collect__case__prod__Grp__eqD,axiom,
! [Z2: product_prod_a_b,A: set_a,F2: a > b] :
( ( member1426531481828664017od_a_b @ Z2 @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ ( bNF_Grp_a_b @ A @ F2 ) ) ) )
=> ( ( comp_a9170378079104387268od_a_b @ F2 @ product_fst_a_b @ Z2 )
= ( product_snd_a_b @ Z2 ) ) ) ).
% Collect_case_prod_Grp_eqD
thf(fact_125_the__inv__into__onto,axiom,
! [F2: nat > extended_ereal,A: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ A )
=> ( ( image_7659842161140344153al_nat @ ( the_in5959796611709155849_ereal @ A @ F2 ) @ ( image_4309273772856505399_ereal @ F2 @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_126_the__inv__into__onto,axiom,
! [F2: extended_ereal > nat,A: set_Extended_ereal] :
( ( inj_on318729178700965101al_nat @ F2 @ A )
=> ( ( image_4309273772856505399_ereal @ ( the_in86992963138218795al_nat @ A @ F2 ) @ ( image_7659842161140344153al_nat @ F2 @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_127_the__inv__into__onto,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A )
=> ( ( image_6042159593519690757_ereal @ ( the_in1141389326992810419_ereal @ A @ F2 ) @ ( image_6042159593519690757_ereal @ F2 @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_128_Sup_OSUP__cong,axiom,
! [A: set_Extended_ereal,B5: set_Extended_ereal,C2: extended_ereal > extended_ereal,D: extended_ereal > extended_ereal,Sup: set_Extended_ereal > extended_ereal] :
( ( A = B5 )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ B5 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_6042159593519690757_ereal @ C2 @ A ) )
= ( Sup @ ( image_6042159593519690757_ereal @ D @ B5 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_129_Sup_OSUP__cong,axiom,
! [A: set_nat,B5: set_nat,C2: nat > extended_ereal,D: nat > extended_ereal,Sup: set_Extended_ereal > extended_ereal] :
( ( A = B5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B5 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_4309273772856505399_ereal @ C2 @ A ) )
= ( Sup @ ( image_4309273772856505399_ereal @ D @ B5 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_130_le__sup__iff,axiom,
! [X2: extended_ereal,Y2: extended_ereal,Z2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( sup_su7653423775389492130_ereal @ X2 @ Y2 ) @ Z2 )
= ( ( ord_le1083603963089353582_ereal @ X2 @ Z2 )
& ( ord_le1083603963089353582_ereal @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_131_le__sup__iff,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ Y2 ) @ Z2 )
= ( ( ord_less_eq_nat @ X2 @ Z2 )
& ( ord_less_eq_nat @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_132_le__sup__iff,axiom,
! [X2: set_Extended_ereal,Y2: set_Extended_ereal,Z2: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ ( sup_su2680283192902082946_ereal @ X2 @ Y2 ) @ Z2 )
= ( ( ord_le1644982726543182158_ereal @ X2 @ Z2 )
& ( ord_le1644982726543182158_ereal @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_133_sup_Obounded__iff,axiom,
! [B: extended_ereal,C: extended_ereal,A4: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( sup_su7653423775389492130_ereal @ B @ C ) @ A4 )
= ( ( ord_le1083603963089353582_ereal @ B @ A4 )
& ( ord_le1083603963089353582_ereal @ C @ A4 ) ) ) ).
% sup.bounded_iff
thf(fact_134_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A4: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A4 )
= ( ( ord_less_eq_nat @ B @ A4 )
& ( ord_less_eq_nat @ C @ A4 ) ) ) ).
% sup.bounded_iff
thf(fact_135_sup_Obounded__iff,axiom,
! [B: set_Extended_ereal,C: set_Extended_ereal,A4: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ ( sup_su2680283192902082946_ereal @ B @ C ) @ A4 )
= ( ( ord_le1644982726543182158_ereal @ B @ A4 )
& ( ord_le1644982726543182158_ereal @ C @ A4 ) ) ) ).
% sup.bounded_iff
thf(fact_136_boolean__algebra_Odisj__one__left,axiom,
! [X2: set_Extended_ereal] :
( ( sup_su2680283192902082946_ereal @ top_to5683747375963461374_ereal @ X2 )
= top_to5683747375963461374_ereal ) ).
% boolean_algebra.disj_one_left
thf(fact_137_boolean__algebra_Odisj__one__left,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X2 )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_left
thf(fact_138_boolean__algebra_Odisj__one__right,axiom,
! [X2: set_Extended_ereal] :
( ( sup_su2680283192902082946_ereal @ X2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal ) ).
% boolean_algebra.disj_one_right
thf(fact_139_boolean__algebra_Odisj__one__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ top_top_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_right
thf(fact_140_sup__top__left,axiom,
! [X2: set_Extended_ereal] :
( ( sup_su2680283192902082946_ereal @ top_to5683747375963461374_ereal @ X2 )
= top_to5683747375963461374_ereal ) ).
% sup_top_left
thf(fact_141_sup__top__left,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X2 )
= top_top_set_nat ) ).
% sup_top_left
thf(fact_142_sup__top__right,axiom,
! [X2: set_Extended_ereal] :
( ( sup_su2680283192902082946_ereal @ X2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal ) ).
% sup_top_right
thf(fact_143_sup__top__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ top_top_set_nat )
= top_top_set_nat ) ).
% sup_top_right
thf(fact_144_Un__subset__iff,axiom,
! [A: set_Extended_ereal,B5: set_Extended_ereal,C2: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ ( sup_su2680283192902082946_ereal @ A @ B5 ) @ C2 )
= ( ( ord_le1644982726543182158_ereal @ A @ C2 )
& ( ord_le1644982726543182158_ereal @ B5 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_145_Inf_OINF__image,axiom,
! [Inf: set_Extended_ereal > extended_ereal,G: extended_ereal > extended_ereal,F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( Inf @ ( image_6042159593519690757_ereal @ G @ ( image_6042159593519690757_ereal @ F2 @ A ) ) )
= ( Inf @ ( image_6042159593519690757_ereal @ ( comp_E9177254828515427499_ereal @ G @ F2 ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_146_Inf_OINF__image,axiom,
! [Inf: set_Extended_ereal > extended_ereal,G: extended_ereal > extended_ereal,F2: nat > extended_ereal,A: set_nat] :
( ( Inf @ ( image_6042159593519690757_ereal @ G @ ( image_4309273772856505399_ereal @ F2 @ A ) ) )
= ( Inf @ ( image_4309273772856505399_ereal @ ( comp_E3726099860353345075al_nat @ G @ F2 ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_147_Inf_OINF__image,axiom,
! [Inf: set_Extended_ereal > extended_ereal,G: nat > extended_ereal,F2: extended_ereal > nat,A: set_Extended_ereal] :
( ( Inf @ ( image_4309273772856505399_ereal @ G @ ( image_7659842161140344153al_nat @ F2 @ A ) ) )
= ( Inf @ ( image_6042159593519690757_ereal @ ( comp_n261702227720650419_ereal @ G @ F2 ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_148_Inf_OINF__image,axiom,
! [Inf: set_Extended_ereal > extended_ereal,G: nat > extended_ereal,F2: nat > nat,A: set_nat] :
( ( Inf @ ( image_4309273772856505399_ereal @ G @ ( image_nat_nat @ F2 @ A ) ) )
= ( Inf @ ( image_4309273772856505399_ereal @ ( comp_n13370146242399787al_nat @ G @ F2 ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_149_Sup_OSUP__image,axiom,
! [Sup: set_Extended_ereal > extended_ereal,G: extended_ereal > extended_ereal,F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( Sup @ ( image_6042159593519690757_ereal @ G @ ( image_6042159593519690757_ereal @ F2 @ A ) ) )
= ( Sup @ ( image_6042159593519690757_ereal @ ( comp_E9177254828515427499_ereal @ G @ F2 ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_150_Sup_OSUP__image,axiom,
! [Sup: set_Extended_ereal > extended_ereal,G: extended_ereal > extended_ereal,F2: nat > extended_ereal,A: set_nat] :
( ( Sup @ ( image_6042159593519690757_ereal @ G @ ( image_4309273772856505399_ereal @ F2 @ A ) ) )
= ( Sup @ ( image_4309273772856505399_ereal @ ( comp_E3726099860353345075al_nat @ G @ F2 ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_151_Sup_OSUP__image,axiom,
! [Sup: set_Extended_ereal > extended_ereal,G: nat > extended_ereal,F2: extended_ereal > nat,A: set_Extended_ereal] :
( ( Sup @ ( image_4309273772856505399_ereal @ G @ ( image_7659842161140344153al_nat @ F2 @ A ) ) )
= ( Sup @ ( image_6042159593519690757_ereal @ ( comp_n261702227720650419_ereal @ G @ F2 ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_152_Sup_OSUP__image,axiom,
! [Sup: set_Extended_ereal > extended_ereal,G: nat > extended_ereal,F2: nat > nat,A: set_nat] :
( ( Sup @ ( image_4309273772856505399_ereal @ G @ ( image_nat_nat @ F2 @ A ) ) )
= ( Sup @ ( image_4309273772856505399_ereal @ ( comp_n13370146242399787al_nat @ G @ F2 ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_153_range__subsetD,axiom,
! [F2: extended_ereal > a,B5: set_a,I: extended_ereal] :
( ( ord_less_eq_set_a @ ( image_3724615099042636213real_a @ F2 @ top_to5683747375963461374_ereal ) @ B5 )
=> ( member_a @ ( F2 @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_154_range__subsetD,axiom,
! [F2: nat > a,B5: set_a,I: nat] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F2 @ top_top_set_nat ) @ B5 )
=> ( member_a @ ( F2 @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_155_range__subsetD,axiom,
! [F2: extended_ereal > extended_ereal,B5: set_Extended_ereal,I: extended_ereal] :
( ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal ) @ B5 )
=> ( member2350847679896131959_ereal @ ( F2 @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_156_range__subsetD,axiom,
! [F2: nat > extended_ereal,B5: set_Extended_ereal,I: nat] :
( ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat ) @ B5 )
=> ( member2350847679896131959_ereal @ ( F2 @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_157_comp__surj,axiom,
! [F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ( ( ( image_6042159593519690757_ereal @ G @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ( ( image_6042159593519690757_ereal @ ( comp_E9177254828515427499_ereal @ G @ F2 ) @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal ) ) ) ).
% comp_surj
thf(fact_158_comp__surj,axiom,
! [F2: extended_ereal > extended_ereal,G: extended_ereal > nat] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ( ( ( image_7659842161140344153al_nat @ G @ top_to5683747375963461374_ereal )
= top_top_set_nat )
=> ( ( image_7659842161140344153al_nat @ ( comp_E375531472069506321_ereal @ G @ F2 ) @ top_to5683747375963461374_ereal )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_159_comp__surj,axiom,
! [F2: extended_ereal > nat,G: nat > extended_ereal] :
( ( ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_nat )
=> ( ( ( image_4309273772856505399_ereal @ G @ top_top_set_nat )
= top_to5683747375963461374_ereal )
=> ( ( image_6042159593519690757_ereal @ ( comp_n261702227720650419_ereal @ G @ F2 ) @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal ) ) ) ).
% comp_surj
thf(fact_160_comp__surj,axiom,
! [F2: extended_ereal > nat,G: nat > nat] :
( ( ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_nat )
=> ( ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_7659842161140344153al_nat @ ( comp_n5886173794813336841_ereal @ G @ F2 ) @ top_to5683747375963461374_ereal )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_161_comp__surj,axiom,
! [F2: nat > extended_ereal,G: extended_ereal > extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat )
= top_to5683747375963461374_ereal )
=> ( ( ( image_6042159593519690757_ereal @ G @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ( ( image_4309273772856505399_ereal @ ( comp_E3726099860353345075al_nat @ G @ F2 ) @ top_top_set_nat )
= top_to5683747375963461374_ereal ) ) ) ).
% comp_surj
thf(fact_162_comp__surj,axiom,
! [F2: nat > extended_ereal,G: extended_ereal > nat] :
( ( ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat )
= top_to5683747375963461374_ereal )
=> ( ( ( image_7659842161140344153al_nat @ G @ top_to5683747375963461374_ereal )
= top_top_set_nat )
=> ( ( image_nat_nat @ ( comp_E7502005551946643277at_nat @ G @ F2 ) @ top_top_set_nat )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_163_comp__surj,axiom,
! [F2: nat > nat,G: nat > extended_ereal] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ( image_4309273772856505399_ereal @ G @ top_top_set_nat )
= top_to5683747375963461374_ereal )
=> ( ( image_4309273772856505399_ereal @ ( comp_n13370146242399787al_nat @ G @ F2 ) @ top_top_set_nat )
= top_to5683747375963461374_ereal ) ) ) ).
% comp_surj
thf(fact_164_comp__surj,axiom,
! [F2: nat > nat,G: nat > nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F2 ) @ top_top_set_nat )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_165_inj__compose,axiom,
! [F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ( inj_on7162434037990268785_ereal @ G @ top_to5683747375963461374_ereal )
=> ( inj_on7162434037990268785_ereal @ ( comp_E9177254828515427499_ereal @ F2 @ G ) @ top_to5683747375963461374_ereal ) ) ) ).
% inj_compose
thf(fact_166_inj__compose,axiom,
! [F2: extended_ereal > extended_ereal,G: nat > extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ( inj_on6191532827271902155_ereal @ G @ top_top_set_nat )
=> ( inj_on6191532827271902155_ereal @ ( comp_E3726099860353345075al_nat @ F2 @ G ) @ top_top_set_nat ) ) ) ).
% inj_compose
thf(fact_167_inj__compose,axiom,
! [F2: nat > extended_ereal,G: extended_ereal > nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ top_top_set_nat )
=> ( ( inj_on318729178700965101al_nat @ G @ top_to5683747375963461374_ereal )
=> ( inj_on7162434037990268785_ereal @ ( comp_n261702227720650419_ereal @ F2 @ G ) @ top_to5683747375963461374_ereal ) ) ) ).
% inj_compose
thf(fact_168_the__inv__f__f,axiom,
! [F2: extended_ereal > extended_ereal,X2: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ( the_in1141389326992810419_ereal @ top_to5683747375963461374_ereal @ F2 @ ( F2 @ X2 ) )
= X2 ) ) ).
% the_inv_f_f
thf(fact_169_image__eq__imp__comp,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,G: extended_ereal > extended_ereal,B5: set_Extended_ereal,H: extended_ereal > extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ A )
= ( image_6042159593519690757_ereal @ G @ B5 ) )
=> ( ( image_6042159593519690757_ereal @ ( comp_E9177254828515427499_ereal @ H @ F2 ) @ A )
= ( image_6042159593519690757_ereal @ ( comp_E9177254828515427499_ereal @ H @ G ) @ B5 ) ) ) ).
% image_eq_imp_comp
thf(fact_170_image__eq__imp__comp,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,G: nat > extended_ereal,B5: set_nat,H: extended_ereal > extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ A )
= ( image_4309273772856505399_ereal @ G @ B5 ) )
=> ( ( image_6042159593519690757_ereal @ ( comp_E9177254828515427499_ereal @ H @ F2 ) @ A )
= ( image_4309273772856505399_ereal @ ( comp_E3726099860353345075al_nat @ H @ G ) @ B5 ) ) ) ).
% image_eq_imp_comp
thf(fact_171_image__eq__imp__comp,axiom,
! [F2: nat > extended_ereal,A: set_nat,G: extended_ereal > extended_ereal,B5: set_Extended_ereal,H: extended_ereal > extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F2 @ A )
= ( image_6042159593519690757_ereal @ G @ B5 ) )
=> ( ( image_4309273772856505399_ereal @ ( comp_E3726099860353345075al_nat @ H @ F2 ) @ A )
= ( image_6042159593519690757_ereal @ ( comp_E9177254828515427499_ereal @ H @ G ) @ B5 ) ) ) ).
% image_eq_imp_comp
thf(fact_172_image__eq__imp__comp,axiom,
! [F2: nat > extended_ereal,A: set_nat,G: nat > extended_ereal,B5: set_nat,H: extended_ereal > extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F2 @ A )
= ( image_4309273772856505399_ereal @ G @ B5 ) )
=> ( ( image_4309273772856505399_ereal @ ( comp_E3726099860353345075al_nat @ H @ F2 ) @ A )
= ( image_4309273772856505399_ereal @ ( comp_E3726099860353345075al_nat @ H @ G ) @ B5 ) ) ) ).
% image_eq_imp_comp
thf(fact_173_image__comp,axiom,
! [F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal,R: set_Extended_ereal] :
( ( image_6042159593519690757_ereal @ F2 @ ( image_6042159593519690757_ereal @ G @ R ) )
= ( image_6042159593519690757_ereal @ ( comp_E9177254828515427499_ereal @ F2 @ G ) @ R ) ) ).
% image_comp
thf(fact_174_image__comp,axiom,
! [F2: extended_ereal > extended_ereal,G: nat > extended_ereal,R: set_nat] :
( ( image_6042159593519690757_ereal @ F2 @ ( image_4309273772856505399_ereal @ G @ R ) )
= ( image_4309273772856505399_ereal @ ( comp_E3726099860353345075al_nat @ F2 @ G ) @ R ) ) ).
% image_comp
thf(fact_175_image__comp,axiom,
! [F2: nat > extended_ereal,G: extended_ereal > nat,R: set_Extended_ereal] :
( ( image_4309273772856505399_ereal @ F2 @ ( image_7659842161140344153al_nat @ G @ R ) )
= ( image_6042159593519690757_ereal @ ( comp_n261702227720650419_ereal @ F2 @ G ) @ R ) ) ).
% image_comp
thf(fact_176_image__comp,axiom,
! [F2: nat > extended_ereal,G: nat > nat,R: set_nat] :
( ( image_4309273772856505399_ereal @ F2 @ ( image_nat_nat @ G @ R ) )
= ( image_4309273772856505399_ereal @ ( comp_n13370146242399787al_nat @ F2 @ G ) @ R ) ) ).
% image_comp
thf(fact_177_inj__image__subset__iff,axiom,
! [F2: nat > extended_ereal,A: set_nat,B5: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ top_top_set_nat )
=> ( ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) @ ( image_4309273772856505399_ereal @ F2 @ B5 ) )
= ( ord_less_eq_set_nat @ A @ B5 ) ) ) ).
% inj_image_subset_iff
thf(fact_178_inj__image__subset__iff,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,B5: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) @ ( image_6042159593519690757_ereal @ F2 @ B5 ) )
= ( ord_le1644982726543182158_ereal @ A @ B5 ) ) ) ).
% inj_image_subset_iff
thf(fact_179_inj__on__imageI2,axiom,
! [F3: extended_ereal > extended_ereal,F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ ( comp_E9177254828515427499_ereal @ F3 @ F2 ) @ A )
=> ( inj_on7162434037990268785_ereal @ F2 @ A ) ) ).
% inj_on_imageI2
thf(fact_180_the__inv__into__comp,axiom,
! [F2: extended_ereal > extended_ereal,G: nat > extended_ereal,A: set_nat,X2: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ ( image_4309273772856505399_ereal @ G @ A ) )
=> ( ( inj_on6191532827271902155_ereal @ G @ A )
=> ( ( member2350847679896131959_ereal @ X2 @ ( image_6042159593519690757_ereal @ F2 @ ( image_4309273772856505399_ereal @ G @ A ) ) )
=> ( ( the_in5959796611709155849_ereal @ A @ ( comp_E3726099860353345075al_nat @ F2 @ G ) @ X2 )
= ( comp_E375531472069506321_ereal @ ( the_in5959796611709155849_ereal @ A @ G ) @ ( the_in1141389326992810419_ereal @ ( image_4309273772856505399_ereal @ G @ A ) @ F2 ) @ X2 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_181_the__inv__into__comp,axiom,
! [F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal,A: set_Extended_ereal,X2: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ ( image_6042159593519690757_ereal @ G @ A ) )
=> ( ( inj_on7162434037990268785_ereal @ G @ A )
=> ( ( member2350847679896131959_ereal @ X2 @ ( image_6042159593519690757_ereal @ F2 @ ( image_6042159593519690757_ereal @ G @ A ) ) )
=> ( ( the_in1141389326992810419_ereal @ A @ ( comp_E9177254828515427499_ereal @ F2 @ G ) @ X2 )
= ( comp_E9177254828515427499_ereal @ ( the_in1141389326992810419_ereal @ A @ G ) @ ( the_in1141389326992810419_ereal @ ( image_6042159593519690757_ereal @ G @ A ) @ F2 ) @ X2 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_182_the__inv__into__comp,axiom,
! [F2: extended_ereal > a,G: nat > extended_ereal,A: set_nat,X2: a] :
( ( inj_on8242634198667403041real_a @ F2 @ ( image_4309273772856505399_ereal @ G @ A ) )
=> ( ( inj_on6191532827271902155_ereal @ G @ A )
=> ( ( member_a @ X2 @ ( image_3724615099042636213real_a @ F2 @ ( image_4309273772856505399_ereal @ G @ A ) ) )
=> ( ( the_inv_into_nat_a @ A @ ( comp_E5637448798707004259_a_nat @ F2 @ G ) @ X2 )
= ( comp_E446008263030514881_nat_a @ ( the_in5959796611709155849_ereal @ A @ G ) @ ( the_in377810665034427491real_a @ ( image_4309273772856505399_ereal @ G @ A ) @ F2 ) @ X2 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_183_the__inv__into__comp,axiom,
! [F2: extended_ereal > a,G: extended_ereal > extended_ereal,A: set_Extended_ereal,X2: a] :
( ( inj_on8242634198667403041real_a @ F2 @ ( image_6042159593519690757_ereal @ G @ A ) )
=> ( ( inj_on7162434037990268785_ereal @ G @ A )
=> ( ( member_a @ X2 @ ( image_3724615099042636213real_a @ F2 @ ( image_6042159593519690757_ereal @ G @ A ) ) )
=> ( ( the_in377810665034427491real_a @ A @ ( comp_E6551704282591734651_ereal @ F2 @ G ) @ X2 )
= ( comp_E1870838029643375451real_a @ ( the_in1141389326992810419_ereal @ A @ G ) @ ( the_in377810665034427491real_a @ ( image_6042159593519690757_ereal @ G @ A ) @ F2 ) @ X2 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_184_the__inv__into__into,axiom,
! [F2: nat > extended_ereal,A: set_nat,X2: extended_ereal,B5: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ A )
=> ( ( member2350847679896131959_ereal @ X2 @ ( image_4309273772856505399_ereal @ F2 @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B5 )
=> ( member_nat @ ( the_in5959796611709155849_ereal @ A @ F2 @ X2 ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_185_the__inv__into__into,axiom,
! [F2: a > a,A: set_a,X2: a,B5: set_a] :
( ( inj_on_a_a @ F2 @ A )
=> ( ( member_a @ X2 @ ( image_a_a @ F2 @ A ) )
=> ( ( ord_less_eq_set_a @ A @ B5 )
=> ( member_a @ ( the_inv_into_a_a @ A @ F2 @ X2 ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_186_the__inv__into__into,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,X2: extended_ereal,B5: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A )
=> ( ( member2350847679896131959_ereal @ X2 @ ( image_6042159593519690757_ereal @ F2 @ A ) )
=> ( ( ord_le1644982726543182158_ereal @ A @ B5 )
=> ( member2350847679896131959_ereal @ ( the_in1141389326992810419_ereal @ A @ F2 @ X2 ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_187_the__inv__into__into,axiom,
! [F2: extended_ereal > a,A: set_Extended_ereal,X2: a,B5: set_Extended_ereal] :
( ( inj_on8242634198667403041real_a @ F2 @ A )
=> ( ( member_a @ X2 @ ( image_3724615099042636213real_a @ F2 @ A ) )
=> ( ( ord_le1644982726543182158_ereal @ A @ B5 )
=> ( member2350847679896131959_ereal @ ( the_in377810665034427491real_a @ A @ F2 @ X2 ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_188_image__mono,axiom,
! [A: set_nat,B5: set_nat,F2: nat > extended_ereal] :
( ( ord_less_eq_set_nat @ A @ B5 )
=> ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) @ ( image_4309273772856505399_ereal @ F2 @ B5 ) ) ) ).
% image_mono
thf(fact_189_image__mono,axiom,
! [A: set_Extended_ereal,B5: set_Extended_ereal,F2: extended_ereal > extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A @ B5 )
=> ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) @ ( image_6042159593519690757_ereal @ F2 @ B5 ) ) ) ).
% image_mono
thf(fact_190_image__subsetI,axiom,
! [A: set_a,F2: a > a,B5: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ ( F2 @ X3 ) @ B5 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A ) @ B5 ) ) ).
% image_subsetI
thf(fact_191_image__subsetI,axiom,
! [A: set_Extended_ereal,F2: extended_ereal > extended_ereal,B5: set_Extended_ereal] :
( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
=> ( member2350847679896131959_ereal @ ( F2 @ X3 ) @ B5 ) )
=> ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) @ B5 ) ) ).
% image_subsetI
thf(fact_192_image__subsetI,axiom,
! [A: set_nat,F2: nat > extended_ereal,B5: set_Extended_ereal] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member2350847679896131959_ereal @ ( F2 @ X3 ) @ B5 ) )
=> ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) @ B5 ) ) ).
% image_subsetI
thf(fact_193_image__subsetI,axiom,
! [A: set_a,F2: a > extended_ereal,B5: set_Extended_ereal] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member2350847679896131959_ereal @ ( F2 @ X3 ) @ B5 ) )
=> ( ord_le1644982726543182158_ereal @ ( image_8405481351990995413_ereal @ F2 @ A ) @ B5 ) ) ).
% image_subsetI
thf(fact_194_subset__imageE,axiom,
! [B5: set_Extended_ereal,F2: nat > extended_ereal,A: set_nat] :
( ( ord_le1644982726543182158_ereal @ B5 @ ( image_4309273772856505399_ereal @ F2 @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B5
!= ( image_4309273772856505399_ereal @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_195_subset__imageE,axiom,
! [B5: set_Extended_ereal,F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B5 @ ( image_6042159593519690757_ereal @ F2 @ A ) )
=> ~ ! [C3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ C3 @ A )
=> ( B5
!= ( image_6042159593519690757_ereal @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_196_image__subset__iff,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,B5: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) @ B5 )
= ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A )
=> ( member2350847679896131959_ereal @ ( F2 @ X ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_197_image__subset__iff,axiom,
! [F2: nat > extended_ereal,A: set_nat,B5: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) @ B5 )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member2350847679896131959_ereal @ ( F2 @ X ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_198_subset__image__iff,axiom,
! [B5: set_Extended_ereal,F2: nat > extended_ereal,A: set_nat] :
( ( ord_le1644982726543182158_ereal @ B5 @ ( image_4309273772856505399_ereal @ F2 @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B5
= ( image_4309273772856505399_ereal @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_199_subset__image__iff,axiom,
! [B5: set_Extended_ereal,F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B5 @ ( image_6042159593519690757_ereal @ F2 @ A ) )
= ( ? [AA: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ AA @ A )
& ( B5
= ( image_6042159593519690757_ereal @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_200_inf__sup__ord_I4_J,axiom,
! [Y2: extended_ereal,X2: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y2 @ ( sup_su7653423775389492130_ereal @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_201_inf__sup__ord_I4_J,axiom,
! [Y2: nat,X2: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_202_inf__sup__ord_I4_J,axiom,
! [Y2: set_Extended_ereal,X2: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ Y2 @ ( sup_su2680283192902082946_ereal @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_203_inf__sup__ord_I3_J,axiom,
! [X2: extended_ereal,Y2: extended_ereal] : ( ord_le1083603963089353582_ereal @ X2 @ ( sup_su7653423775389492130_ereal @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_204_inf__sup__ord_I3_J,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_205_inf__sup__ord_I3_J,axiom,
! [X2: set_Extended_ereal,Y2: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ X2 @ ( sup_su2680283192902082946_ereal @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_206_le__supE,axiom,
! [A4: extended_ereal,B: extended_ereal,X2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( sup_su7653423775389492130_ereal @ A4 @ B ) @ X2 )
=> ~ ( ( ord_le1083603963089353582_ereal @ A4 @ X2 )
=> ~ ( ord_le1083603963089353582_ereal @ B @ X2 ) ) ) ).
% le_supE
thf(fact_207_le__supE,axiom,
! [A4: nat,B: nat,X2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A4 @ B ) @ X2 )
=> ~ ( ( ord_less_eq_nat @ A4 @ X2 )
=> ~ ( ord_less_eq_nat @ B @ X2 ) ) ) ).
% le_supE
thf(fact_208_le__supE,axiom,
! [A4: set_Extended_ereal,B: set_Extended_ereal,X2: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ ( sup_su2680283192902082946_ereal @ A4 @ B ) @ X2 )
=> ~ ( ( ord_le1644982726543182158_ereal @ A4 @ X2 )
=> ~ ( ord_le1644982726543182158_ereal @ B @ X2 ) ) ) ).
% le_supE
thf(fact_209_le__supI,axiom,
! [A4: extended_ereal,X2: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A4 @ X2 )
=> ( ( ord_le1083603963089353582_ereal @ B @ X2 )
=> ( ord_le1083603963089353582_ereal @ ( sup_su7653423775389492130_ereal @ A4 @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_210_le__supI,axiom,
! [A4: nat,X2: nat,B: nat] :
( ( ord_less_eq_nat @ A4 @ X2 )
=> ( ( ord_less_eq_nat @ B @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A4 @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_211_le__supI,axiom,
! [A4: set_Extended_ereal,X2: set_Extended_ereal,B: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A4 @ X2 )
=> ( ( ord_le1644982726543182158_ereal @ B @ X2 )
=> ( ord_le1644982726543182158_ereal @ ( sup_su2680283192902082946_ereal @ A4 @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_212_sup__ge1,axiom,
! [X2: extended_ereal,Y2: extended_ereal] : ( ord_le1083603963089353582_ereal @ X2 @ ( sup_su7653423775389492130_ereal @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_213_sup__ge1,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_214_sup__ge1,axiom,
! [X2: set_Extended_ereal,Y2: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ X2 @ ( sup_su2680283192902082946_ereal @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_215_sup__ge2,axiom,
! [Y2: extended_ereal,X2: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y2 @ ( sup_su7653423775389492130_ereal @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_216_sup__ge2,axiom,
! [Y2: nat,X2: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_217_sup__ge2,axiom,
! [Y2: set_Extended_ereal,X2: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ Y2 @ ( sup_su2680283192902082946_ereal @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_218_le__supI1,axiom,
! [X2: extended_ereal,A4: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X2 @ A4 )
=> ( ord_le1083603963089353582_ereal @ X2 @ ( sup_su7653423775389492130_ereal @ A4 @ B ) ) ) ).
% le_supI1
thf(fact_219_le__supI1,axiom,
! [X2: nat,A4: nat,B: nat] :
( ( ord_less_eq_nat @ X2 @ A4 )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A4 @ B ) ) ) ).
% le_supI1
thf(fact_220_le__supI1,axiom,
! [X2: set_Extended_ereal,A4: set_Extended_ereal,B: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ X2 @ A4 )
=> ( ord_le1644982726543182158_ereal @ X2 @ ( sup_su2680283192902082946_ereal @ A4 @ B ) ) ) ).
% le_supI1
thf(fact_221_le__supI2,axiom,
! [X2: extended_ereal,B: extended_ereal,A4: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X2 @ B )
=> ( ord_le1083603963089353582_ereal @ X2 @ ( sup_su7653423775389492130_ereal @ A4 @ B ) ) ) ).
% le_supI2
thf(fact_222_le__supI2,axiom,
! [X2: nat,B: nat,A4: nat] :
( ( ord_less_eq_nat @ X2 @ B )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A4 @ B ) ) ) ).
% le_supI2
thf(fact_223_le__supI2,axiom,
! [X2: set_Extended_ereal,B: set_Extended_ereal,A4: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ X2 @ B )
=> ( ord_le1644982726543182158_ereal @ X2 @ ( sup_su2680283192902082946_ereal @ A4 @ B ) ) ) ).
% le_supI2
thf(fact_224_sup_Omono,axiom,
! [C: extended_ereal,A4: extended_ereal,D2: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ C @ A4 )
=> ( ( ord_le1083603963089353582_ereal @ D2 @ B )
=> ( ord_le1083603963089353582_ereal @ ( sup_su7653423775389492130_ereal @ C @ D2 ) @ ( sup_su7653423775389492130_ereal @ A4 @ B ) ) ) ) ).
% sup.mono
thf(fact_225_sup_Omono,axiom,
! [C: nat,A4: nat,D2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A4 )
=> ( ( ord_less_eq_nat @ D2 @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A4 @ B ) ) ) ) ).
% sup.mono
thf(fact_226_sup_Omono,axiom,
! [C: set_Extended_ereal,A4: set_Extended_ereal,D2: set_Extended_ereal,B: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ C @ A4 )
=> ( ( ord_le1644982726543182158_ereal @ D2 @ B )
=> ( ord_le1644982726543182158_ereal @ ( sup_su2680283192902082946_ereal @ C @ D2 ) @ ( sup_su2680283192902082946_ereal @ A4 @ B ) ) ) ) ).
% sup.mono
thf(fact_227_sup__mono,axiom,
! [A4: extended_ereal,C: extended_ereal,B: extended_ereal,D2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A4 @ C )
=> ( ( ord_le1083603963089353582_ereal @ B @ D2 )
=> ( ord_le1083603963089353582_ereal @ ( sup_su7653423775389492130_ereal @ A4 @ B ) @ ( sup_su7653423775389492130_ereal @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_228_sup__mono,axiom,
! [A4: nat,C: nat,B: nat,D2: nat] :
( ( ord_less_eq_nat @ A4 @ C )
=> ( ( ord_less_eq_nat @ B @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A4 @ B ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_229_sup__mono,axiom,
! [A4: set_Extended_ereal,C: set_Extended_ereal,B: set_Extended_ereal,D2: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A4 @ C )
=> ( ( ord_le1644982726543182158_ereal @ B @ D2 )
=> ( ord_le1644982726543182158_ereal @ ( sup_su2680283192902082946_ereal @ A4 @ B ) @ ( sup_su2680283192902082946_ereal @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_230_sup__least,axiom,
! [Y2: extended_ereal,X2: extended_ereal,Z2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ Y2 @ X2 )
=> ( ( ord_le1083603963089353582_ereal @ Z2 @ X2 )
=> ( ord_le1083603963089353582_ereal @ ( sup_su7653423775389492130_ereal @ Y2 @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_231_sup__least,axiom,
! [Y2: nat,X2: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y2 @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_232_sup__least,axiom,
! [Y2: set_Extended_ereal,X2: set_Extended_ereal,Z2: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ Y2 @ X2 )
=> ( ( ord_le1644982726543182158_ereal @ Z2 @ X2 )
=> ( ord_le1644982726543182158_ereal @ ( sup_su2680283192902082946_ereal @ Y2 @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_233_le__iff__sup,axiom,
( ord_le1083603963089353582_ereal
= ( ^ [X: extended_ereal,Y: extended_ereal] :
( ( sup_su7653423775389492130_ereal @ X @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_234_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_235_le__iff__sup,axiom,
( ord_le1644982726543182158_ereal
= ( ^ [X: set_Extended_ereal,Y: set_Extended_ereal] :
( ( sup_su2680283192902082946_ereal @ X @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_236_sup_OorderE,axiom,
! [B: extended_ereal,A4: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ B @ A4 )
=> ( A4
= ( sup_su7653423775389492130_ereal @ A4 @ B ) ) ) ).
% sup.orderE
thf(fact_237_sup_OorderE,axiom,
! [B: nat,A4: nat] :
( ( ord_less_eq_nat @ B @ A4 )
=> ( A4
= ( sup_sup_nat @ A4 @ B ) ) ) ).
% sup.orderE
thf(fact_238_sup_OorderE,axiom,
! [B: set_Extended_ereal,A4: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B @ A4 )
=> ( A4
= ( sup_su2680283192902082946_ereal @ A4 @ B ) ) ) ).
% sup.orderE
thf(fact_239_sup_OorderI,axiom,
! [A4: extended_ereal,B: extended_ereal] :
( ( A4
= ( sup_su7653423775389492130_ereal @ A4 @ B ) )
=> ( ord_le1083603963089353582_ereal @ B @ A4 ) ) ).
% sup.orderI
thf(fact_240_sup_OorderI,axiom,
! [A4: nat,B: nat] :
( ( A4
= ( sup_sup_nat @ A4 @ B ) )
=> ( ord_less_eq_nat @ B @ A4 ) ) ).
% sup.orderI
thf(fact_241_sup_OorderI,axiom,
! [A4: set_Extended_ereal,B: set_Extended_ereal] :
( ( A4
= ( sup_su2680283192902082946_ereal @ A4 @ B ) )
=> ( ord_le1644982726543182158_ereal @ B @ A4 ) ) ).
% sup.orderI
thf(fact_242_sup__unique,axiom,
! [F2: extended_ereal > extended_ereal > extended_ereal,X2: extended_ereal,Y2: extended_ereal] :
( ! [X3: extended_ereal,Y3: extended_ereal] : ( ord_le1083603963089353582_ereal @ X3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: extended_ereal,Y3: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: extended_ereal,Y3: extended_ereal,Z3: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ Y3 @ X3 )
=> ( ( ord_le1083603963089353582_ereal @ Z3 @ X3 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ Y3 @ Z3 ) @ X3 ) ) )
=> ( ( sup_su7653423775389492130_ereal @ X2 @ Y2 )
= ( F2 @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_243_sup__unique,axiom,
! [F2: nat > nat > nat,X2: nat,Y2: nat] :
( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ X3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ( ord_less_eq_nat @ ( F2 @ Y3 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X2 @ Y2 )
= ( F2 @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_244_sup__unique,axiom,
! [F2: set_Extended_ereal > set_Extended_ereal > set_Extended_ereal,X2: set_Extended_ereal,Y2: set_Extended_ereal] :
( ! [X3: set_Extended_ereal,Y3: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ X3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: set_Extended_ereal,Y3: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ Y3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: set_Extended_ereal,Y3: set_Extended_ereal,Z3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ Y3 @ X3 )
=> ( ( ord_le1644982726543182158_ereal @ Z3 @ X3 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ Y3 @ Z3 ) @ X3 ) ) )
=> ( ( sup_su2680283192902082946_ereal @ X2 @ Y2 )
= ( F2 @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_245_sup_Oabsorb1,axiom,
! [B: extended_ereal,A4: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ B @ A4 )
=> ( ( sup_su7653423775389492130_ereal @ A4 @ B )
= A4 ) ) ).
% sup.absorb1
thf(fact_246_sup_Oabsorb1,axiom,
! [B: nat,A4: nat] :
( ( ord_less_eq_nat @ B @ A4 )
=> ( ( sup_sup_nat @ A4 @ B )
= A4 ) ) ).
% sup.absorb1
thf(fact_247_sup_Oabsorb1,axiom,
! [B: set_Extended_ereal,A4: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B @ A4 )
=> ( ( sup_su2680283192902082946_ereal @ A4 @ B )
= A4 ) ) ).
% sup.absorb1
thf(fact_248_sup_Oabsorb2,axiom,
! [A4: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A4 @ B )
=> ( ( sup_su7653423775389492130_ereal @ A4 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_249_sup_Oabsorb2,axiom,
! [A4: nat,B: nat] :
( ( ord_less_eq_nat @ A4 @ B )
=> ( ( sup_sup_nat @ A4 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_250_sup_Oabsorb2,axiom,
! [A4: set_Extended_ereal,B: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A4 @ B )
=> ( ( sup_su2680283192902082946_ereal @ A4 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_251_sup__absorb1,axiom,
! [Y2: extended_ereal,X2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ Y2 @ X2 )
=> ( ( sup_su7653423775389492130_ereal @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_252_sup__absorb1,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( sup_sup_nat @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_253_sup__absorb1,axiom,
! [Y2: set_Extended_ereal,X2: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ Y2 @ X2 )
=> ( ( sup_su2680283192902082946_ereal @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_254_sup__absorb2,axiom,
! [X2: extended_ereal,Y2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
=> ( ( sup_su7653423775389492130_ereal @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_255_sup__absorb2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( sup_sup_nat @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_256_sup__absorb2,axiom,
! [X2: set_Extended_ereal,Y2: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ X2 @ Y2 )
=> ( ( sup_su2680283192902082946_ereal @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_257_sup_OboundedE,axiom,
! [B: extended_ereal,C: extended_ereal,A4: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( sup_su7653423775389492130_ereal @ B @ C ) @ A4 )
=> ~ ( ( ord_le1083603963089353582_ereal @ B @ A4 )
=> ~ ( ord_le1083603963089353582_ereal @ C @ A4 ) ) ) ).
% sup.boundedE
thf(fact_258_sup_OboundedE,axiom,
! [B: nat,C: nat,A4: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A4 )
=> ~ ( ( ord_less_eq_nat @ B @ A4 )
=> ~ ( ord_less_eq_nat @ C @ A4 ) ) ) ).
% sup.boundedE
thf(fact_259_sup_OboundedE,axiom,
! [B: set_Extended_ereal,C: set_Extended_ereal,A4: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ ( sup_su2680283192902082946_ereal @ B @ C ) @ A4 )
=> ~ ( ( ord_le1644982726543182158_ereal @ B @ A4 )
=> ~ ( ord_le1644982726543182158_ereal @ C @ A4 ) ) ) ).
% sup.boundedE
thf(fact_260_sup_OboundedI,axiom,
! [B: extended_ereal,A4: extended_ereal,C: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ B @ A4 )
=> ( ( ord_le1083603963089353582_ereal @ C @ A4 )
=> ( ord_le1083603963089353582_ereal @ ( sup_su7653423775389492130_ereal @ B @ C ) @ A4 ) ) ) ).
% sup.boundedI
thf(fact_261_sup_OboundedI,axiom,
! [B: nat,A4: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A4 )
=> ( ( ord_less_eq_nat @ C @ A4 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A4 ) ) ) ).
% sup.boundedI
thf(fact_262_sup_OboundedI,axiom,
! [B: set_Extended_ereal,A4: set_Extended_ereal,C: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B @ A4 )
=> ( ( ord_le1644982726543182158_ereal @ C @ A4 )
=> ( ord_le1644982726543182158_ereal @ ( sup_su2680283192902082946_ereal @ B @ C ) @ A4 ) ) ) ).
% sup.boundedI
thf(fact_263_sup_Oorder__iff,axiom,
( ord_le1083603963089353582_ereal
= ( ^ [B4: extended_ereal,A6: extended_ereal] :
( A6
= ( sup_su7653423775389492130_ereal @ A6 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_264_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A6: nat] :
( A6
= ( sup_sup_nat @ A6 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_265_sup_Oorder__iff,axiom,
( ord_le1644982726543182158_ereal
= ( ^ [B4: set_Extended_ereal,A6: set_Extended_ereal] :
( A6
= ( sup_su2680283192902082946_ereal @ A6 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_266_sup_Ocobounded1,axiom,
! [A4: extended_ereal,B: extended_ereal] : ( ord_le1083603963089353582_ereal @ A4 @ ( sup_su7653423775389492130_ereal @ A4 @ B ) ) ).
% sup.cobounded1
thf(fact_267_sup_Ocobounded1,axiom,
! [A4: nat,B: nat] : ( ord_less_eq_nat @ A4 @ ( sup_sup_nat @ A4 @ B ) ) ).
% sup.cobounded1
thf(fact_268_sup_Ocobounded1,axiom,
! [A4: set_Extended_ereal,B: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ A4 @ ( sup_su2680283192902082946_ereal @ A4 @ B ) ) ).
% sup.cobounded1
thf(fact_269_sup_Ocobounded2,axiom,
! [B: extended_ereal,A4: extended_ereal] : ( ord_le1083603963089353582_ereal @ B @ ( sup_su7653423775389492130_ereal @ A4 @ B ) ) ).
% sup.cobounded2
thf(fact_270_sup_Ocobounded2,axiom,
! [B: nat,A4: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A4 @ B ) ) ).
% sup.cobounded2
thf(fact_271_sup_Ocobounded2,axiom,
! [B: set_Extended_ereal,A4: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ B @ ( sup_su2680283192902082946_ereal @ A4 @ B ) ) ).
% sup.cobounded2
thf(fact_272_sup_Oabsorb__iff1,axiom,
( ord_le1083603963089353582_ereal
= ( ^ [B4: extended_ereal,A6: extended_ereal] :
( ( sup_su7653423775389492130_ereal @ A6 @ B4 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_273_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A6: nat] :
( ( sup_sup_nat @ A6 @ B4 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_274_sup_Oabsorb__iff1,axiom,
( ord_le1644982726543182158_ereal
= ( ^ [B4: set_Extended_ereal,A6: set_Extended_ereal] :
( ( sup_su2680283192902082946_ereal @ A6 @ B4 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_275_sup_Oabsorb__iff2,axiom,
( ord_le1083603963089353582_ereal
= ( ^ [A6: extended_ereal,B4: extended_ereal] :
( ( sup_su7653423775389492130_ereal @ A6 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_276_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B4: nat] :
( ( sup_sup_nat @ A6 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_277_sup_Oabsorb__iff2,axiom,
( ord_le1644982726543182158_ereal
= ( ^ [A6: set_Extended_ereal,B4: set_Extended_ereal] :
( ( sup_su2680283192902082946_ereal @ A6 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_278_sup_OcoboundedI1,axiom,
! [C: extended_ereal,A4: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ C @ A4 )
=> ( ord_le1083603963089353582_ereal @ C @ ( sup_su7653423775389492130_ereal @ A4 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_279_sup_OcoboundedI1,axiom,
! [C: nat,A4: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A4 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A4 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_280_sup_OcoboundedI1,axiom,
! [C: set_Extended_ereal,A4: set_Extended_ereal,B: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ C @ A4 )
=> ( ord_le1644982726543182158_ereal @ C @ ( sup_su2680283192902082946_ereal @ A4 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_281_sup_OcoboundedI2,axiom,
! [C: extended_ereal,B: extended_ereal,A4: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ C @ B )
=> ( ord_le1083603963089353582_ereal @ C @ ( sup_su7653423775389492130_ereal @ A4 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_282_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A4: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A4 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_283_sup_OcoboundedI2,axiom,
! [C: set_Extended_ereal,B: set_Extended_ereal,A4: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ C @ B )
=> ( ord_le1644982726543182158_ereal @ C @ ( sup_su2680283192902082946_ereal @ A4 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_284_surj__def,axiom,
! [F2: extended_ereal > extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
= ( ! [Y: extended_ereal] :
? [X: extended_ereal] :
( Y
= ( F2 @ X ) ) ) ) ).
% surj_def
thf(fact_285_surj__def,axiom,
! [F2: extended_ereal > nat] :
( ( ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_nat )
= ( ! [Y: nat] :
? [X: extended_ereal] :
( Y
= ( F2 @ X ) ) ) ) ).
% surj_def
thf(fact_286_surj__def,axiom,
! [F2: nat > extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat )
= top_to5683747375963461374_ereal )
= ( ! [Y: extended_ereal] :
? [X: nat] :
( Y
= ( F2 @ X ) ) ) ) ).
% surj_def
thf(fact_287_surj__def,axiom,
! [F2: nat > nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y: nat] :
? [X: nat] :
( Y
= ( F2 @ X ) ) ) ) ).
% surj_def
thf(fact_288_surjI,axiom,
! [G: extended_ereal > extended_ereal,F2: extended_ereal > extended_ereal] :
( ! [X3: extended_ereal] :
( ( G @ ( F2 @ X3 ) )
= X3 )
=> ( ( image_6042159593519690757_ereal @ G @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal ) ) ).
% surjI
thf(fact_289_surjI,axiom,
! [G: extended_ereal > nat,F2: nat > extended_ereal] :
( ! [X3: nat] :
( ( G @ ( F2 @ X3 ) )
= X3 )
=> ( ( image_7659842161140344153al_nat @ G @ top_to5683747375963461374_ereal )
= top_top_set_nat ) ) ).
% surjI
thf(fact_290_surjI,axiom,
! [G: nat > extended_ereal,F2: extended_ereal > nat] :
( ! [X3: extended_ereal] :
( ( G @ ( F2 @ X3 ) )
= X3 )
=> ( ( image_4309273772856505399_ereal @ G @ top_top_set_nat )
= top_to5683747375963461374_ereal ) ) ).
% surjI
thf(fact_291_surjI,axiom,
! [G: nat > nat,F2: nat > nat] :
( ! [X3: nat] :
( ( G @ ( F2 @ X3 ) )
= X3 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_292_surjE,axiom,
! [F2: extended_ereal > extended_ereal,Y2: extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ~ ! [X3: extended_ereal] :
( Y2
!= ( F2 @ X3 ) ) ) ).
% surjE
thf(fact_293_surjE,axiom,
! [F2: extended_ereal > nat,Y2: nat] :
( ( ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_nat )
=> ~ ! [X3: extended_ereal] :
( Y2
!= ( F2 @ X3 ) ) ) ).
% surjE
thf(fact_294_surjE,axiom,
! [F2: nat > extended_ereal,Y2: extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat )
= top_to5683747375963461374_ereal )
=> ~ ! [X3: nat] :
( Y2
!= ( F2 @ X3 ) ) ) ).
% surjE
thf(fact_295_surjE,axiom,
! [F2: nat > nat,Y2: nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X3: nat] :
( Y2
!= ( F2 @ X3 ) ) ) ).
% surjE
thf(fact_296_surjD,axiom,
! [F2: extended_ereal > extended_ereal,Y2: extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ? [X3: extended_ereal] :
( Y2
= ( F2 @ X3 ) ) ) ).
% surjD
thf(fact_297_surjD,axiom,
! [F2: extended_ereal > nat,Y2: nat] :
( ( ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_nat )
=> ? [X3: extended_ereal] :
( Y2
= ( F2 @ X3 ) ) ) ).
% surjD
thf(fact_298_surjD,axiom,
! [F2: nat > extended_ereal,Y2: extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat )
= top_to5683747375963461374_ereal )
=> ? [X3: nat] :
( Y2
= ( F2 @ X3 ) ) ) ).
% surjD
thf(fact_299_surjD,axiom,
! [F2: nat > nat,Y2: nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ? [X3: nat] :
( Y2
= ( F2 @ X3 ) ) ) ).
% surjD
thf(fact_300_rangeI,axiom,
! [F2: extended_ereal > extended_ereal,X2: extended_ereal] : ( member2350847679896131959_ereal @ ( F2 @ X2 ) @ ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal ) ) ).
% rangeI
thf(fact_301_rangeI,axiom,
! [F2: extended_ereal > a,X2: extended_ereal] : ( member_a @ ( F2 @ X2 ) @ ( image_3724615099042636213real_a @ F2 @ top_to5683747375963461374_ereal ) ) ).
% rangeI
thf(fact_302_rangeI,axiom,
! [F2: nat > extended_ereal,X2: nat] : ( member2350847679896131959_ereal @ ( F2 @ X2 ) @ ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat ) ) ).
% rangeI
thf(fact_303_rangeI,axiom,
! [F2: nat > a,X2: nat] : ( member_a @ ( F2 @ X2 ) @ ( image_nat_a @ F2 @ top_top_set_nat ) ) ).
% rangeI
thf(fact_304_range__eqI,axiom,
! [B: extended_ereal,F2: extended_ereal > extended_ereal,X2: extended_ereal] :
( ( B
= ( F2 @ X2 ) )
=> ( member2350847679896131959_ereal @ B @ ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal ) ) ) ).
% range_eqI
thf(fact_305_range__eqI,axiom,
! [B: a,F2: extended_ereal > a,X2: extended_ereal] :
( ( B
= ( F2 @ X2 ) )
=> ( member_a @ B @ ( image_3724615099042636213real_a @ F2 @ top_to5683747375963461374_ereal ) ) ) ).
% range_eqI
thf(fact_306_range__eqI,axiom,
! [B: extended_ereal,F2: nat > extended_ereal,X2: nat] :
( ( B
= ( F2 @ X2 ) )
=> ( member2350847679896131959_ereal @ B @ ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_307_range__eqI,axiom,
! [B: a,F2: nat > a,X2: nat] :
( ( B
= ( F2 @ X2 ) )
=> ( member_a @ B @ ( image_nat_a @ F2 @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_308_inj__on__subset,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,B5: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A )
=> ( ( ord_le1644982726543182158_ereal @ B5 @ A )
=> ( inj_on7162434037990268785_ereal @ F2 @ B5 ) ) ) ).
% inj_on_subset
thf(fact_309_subset__inj__on,axiom,
! [F2: extended_ereal > extended_ereal,B5: set_Extended_ereal,A: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ B5 )
=> ( ( ord_le1644982726543182158_ereal @ A @ B5 )
=> ( inj_on7162434037990268785_ereal @ F2 @ A ) ) ) ).
% subset_inj_on
thf(fact_310_Un__mono,axiom,
! [A: set_Extended_ereal,C2: set_Extended_ereal,B5: set_Extended_ereal,D: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A @ C2 )
=> ( ( ord_le1644982726543182158_ereal @ B5 @ D )
=> ( ord_le1644982726543182158_ereal @ ( sup_su2680283192902082946_ereal @ A @ B5 ) @ ( sup_su2680283192902082946_ereal @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_311_Un__least,axiom,
! [A: set_Extended_ereal,C2: set_Extended_ereal,B5: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A @ C2 )
=> ( ( ord_le1644982726543182158_ereal @ B5 @ C2 )
=> ( ord_le1644982726543182158_ereal @ ( sup_su2680283192902082946_ereal @ A @ B5 ) @ C2 ) ) ) ).
% Un_least
thf(fact_312_Un__upper1,axiom,
! [A: set_Extended_ereal,B5: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ A @ ( sup_su2680283192902082946_ereal @ A @ B5 ) ) ).
% Un_upper1
thf(fact_313_Un__upper2,axiom,
! [B5: set_Extended_ereal,A: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ B5 @ ( sup_su2680283192902082946_ereal @ A @ B5 ) ) ).
% Un_upper2
thf(fact_314_Un__absorb1,axiom,
! [A: set_Extended_ereal,B5: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A @ B5 )
=> ( ( sup_su2680283192902082946_ereal @ A @ B5 )
= B5 ) ) ).
% Un_absorb1
thf(fact_315_Un__absorb2,axiom,
! [B5: set_Extended_ereal,A: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B5 @ A )
=> ( ( sup_su2680283192902082946_ereal @ A @ B5 )
= A ) ) ).
% Un_absorb2
thf(fact_316_subset__UnE,axiom,
! [C2: set_Extended_ereal,A: set_Extended_ereal,B5: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ C2 @ ( sup_su2680283192902082946_ereal @ A @ B5 ) )
=> ~ ! [A7: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A7 @ A )
=> ! [B6: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B6 @ B5 )
=> ( C2
!= ( sup_su2680283192902082946_ereal @ A7 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_317_subset__Un__eq,axiom,
( ord_le1644982726543182158_ereal
= ( ^ [A3: set_Extended_ereal,B7: set_Extended_ereal] :
( ( sup_su2680283192902082946_ereal @ A3 @ B7 )
= B7 ) ) ) ).
% subset_Un_eq
thf(fact_318_injD,axiom,
! [F2: extended_ereal > extended_ereal,X2: extended_ereal,Y2: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y2 ) )
=> ( X2 = Y2 ) ) ) ).
% injD
thf(fact_319_injI,axiom,
! [F2: extended_ereal > extended_ereal] :
( ! [X3: extended_ereal,Y3: extended_ereal] :
( ( ( F2 @ X3 )
= ( F2 @ Y3 ) )
=> ( X3 = Y3 ) )
=> ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal ) ) ).
% injI
thf(fact_320_inj__eq,axiom,
! [F2: extended_ereal > extended_ereal,X2: extended_ereal,Y2: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% inj_eq
thf(fact_321_inj__def,axiom,
! [F2: extended_ereal > extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
= ( ! [X: extended_ereal,Y: extended_ereal] :
( ( ( F2 @ X )
= ( F2 @ Y ) )
=> ( X = Y ) ) ) ) ).
% inj_def
thf(fact_322_Un__UNIV__left,axiom,
! [B5: set_Extended_ereal] :
( ( sup_su2680283192902082946_ereal @ top_to5683747375963461374_ereal @ B5 )
= top_to5683747375963461374_ereal ) ).
% Un_UNIV_left
thf(fact_323_Un__UNIV__left,axiom,
! [B5: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ B5 )
= top_top_set_nat ) ).
% Un_UNIV_left
thf(fact_324_Un__UNIV__right,axiom,
! [A: set_Extended_ereal] :
( ( sup_su2680283192902082946_ereal @ A @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal ) ).
% Un_UNIV_right
thf(fact_325_Un__UNIV__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ top_top_set_nat )
= top_top_set_nat ) ).
% Un_UNIV_right
thf(fact_326_the__inv__into__f__f,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,X2: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A )
=> ( ( member2350847679896131959_ereal @ X2 @ A )
=> ( ( the_in1141389326992810419_ereal @ A @ F2 @ ( F2 @ X2 ) )
= X2 ) ) ) ).
% the_inv_into_f_f
thf(fact_327_the__inv__into__f__eq,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,X2: extended_ereal,Y2: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A )
=> ( ( ( F2 @ X2 )
= Y2 )
=> ( ( member2350847679896131959_ereal @ X2 @ A )
=> ( ( the_in1141389326992810419_ereal @ A @ F2 @ Y2 )
= X2 ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_328_comp__inj__on,axiom,
! [F2: nat > extended_ereal,A: set_nat,G: extended_ereal > extended_ereal] :
( ( inj_on6191532827271902155_ereal @ F2 @ A )
=> ( ( inj_on7162434037990268785_ereal @ G @ ( image_4309273772856505399_ereal @ F2 @ A ) )
=> ( inj_on6191532827271902155_ereal @ ( comp_E3726099860353345075al_nat @ G @ F2 ) @ A ) ) ) ).
% comp_inj_on
thf(fact_329_comp__inj__on,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,G: extended_ereal > extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A )
=> ( ( inj_on7162434037990268785_ereal @ G @ ( image_6042159593519690757_ereal @ F2 @ A ) )
=> ( inj_on7162434037990268785_ereal @ ( comp_E9177254828515427499_ereal @ G @ F2 ) @ A ) ) ) ).
% comp_inj_on
thf(fact_330_inj__on__imageI,axiom,
! [G: extended_ereal > extended_ereal,F2: nat > extended_ereal,A: set_nat] :
( ( inj_on6191532827271902155_ereal @ ( comp_E3726099860353345075al_nat @ G @ F2 ) @ A )
=> ( inj_on7162434037990268785_ereal @ G @ ( image_4309273772856505399_ereal @ F2 @ A ) ) ) ).
% inj_on_imageI
thf(fact_331_inj__on__imageI,axiom,
! [G: extended_ereal > extended_ereal,F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ ( comp_E9177254828515427499_ereal @ G @ F2 ) @ A )
=> ( inj_on7162434037990268785_ereal @ G @ ( image_6042159593519690757_ereal @ F2 @ A ) ) ) ).
% inj_on_imageI
thf(fact_332_comp__inj__on__iff,axiom,
! [F2: nat > extended_ereal,A: set_nat,F3: extended_ereal > extended_ereal] :
( ( inj_on6191532827271902155_ereal @ F2 @ A )
=> ( ( inj_on7162434037990268785_ereal @ F3 @ ( image_4309273772856505399_ereal @ F2 @ A ) )
= ( inj_on6191532827271902155_ereal @ ( comp_E3726099860353345075al_nat @ F3 @ F2 ) @ A ) ) ) ).
% comp_inj_on_iff
thf(fact_333_comp__inj__on__iff,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,F3: extended_ereal > extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A )
=> ( ( inj_on7162434037990268785_ereal @ F3 @ ( image_6042159593519690757_ereal @ F2 @ A ) )
= ( inj_on7162434037990268785_ereal @ ( comp_E9177254828515427499_ereal @ F3 @ F2 ) @ A ) ) ) ).
% comp_inj_on_iff
thf(fact_334_inj__on__image__eq__iff,axiom,
! [F2: nat > extended_ereal,C2: set_nat,A: set_nat,B5: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ C2 )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B5 @ C2 )
=> ( ( ( image_4309273772856505399_ereal @ F2 @ A )
= ( image_4309273772856505399_ereal @ F2 @ B5 ) )
= ( A = B5 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_335_inj__on__image__eq__iff,axiom,
! [F2: extended_ereal > extended_ereal,C2: set_Extended_ereal,A: set_Extended_ereal,B5: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ C2 )
=> ( ( ord_le1644982726543182158_ereal @ A @ C2 )
=> ( ( ord_le1644982726543182158_ereal @ B5 @ C2 )
=> ( ( ( image_6042159593519690757_ereal @ F2 @ A )
= ( image_6042159593519690757_ereal @ F2 @ B5 ) )
= ( A = B5 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_336_inj__on__image__mem__iff,axiom,
! [F2: nat > extended_ereal,B5: set_nat,A4: nat,A: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ B5 )
=> ( ( member_nat @ A4 @ B5 )
=> ( ( ord_less_eq_set_nat @ A @ B5 )
=> ( ( member2350847679896131959_ereal @ ( F2 @ A4 ) @ ( image_4309273772856505399_ereal @ F2 @ A ) )
= ( member_nat @ A4 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_337_inj__on__image__mem__iff,axiom,
! [F2: a > a,B5: set_a,A4: a,A: set_a] :
( ( inj_on_a_a @ F2 @ B5 )
=> ( ( member_a @ A4 @ B5 )
=> ( ( ord_less_eq_set_a @ A @ B5 )
=> ( ( member_a @ ( F2 @ A4 ) @ ( image_a_a @ F2 @ A ) )
= ( member_a @ A4 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_338_inj__on__image__mem__iff,axiom,
! [F2: extended_ereal > extended_ereal,B5: set_Extended_ereal,A4: extended_ereal,A: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ B5 )
=> ( ( member2350847679896131959_ereal @ A4 @ B5 )
=> ( ( ord_le1644982726543182158_ereal @ A @ B5 )
=> ( ( member2350847679896131959_ereal @ ( F2 @ A4 ) @ ( image_6042159593519690757_ereal @ F2 @ A ) )
= ( member2350847679896131959_ereal @ A4 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_339_inj__on__image__mem__iff,axiom,
! [F2: extended_ereal > a,B5: set_Extended_ereal,A4: extended_ereal,A: set_Extended_ereal] :
( ( inj_on8242634198667403041real_a @ F2 @ B5 )
=> ( ( member2350847679896131959_ereal @ A4 @ B5 )
=> ( ( ord_le1644982726543182158_ereal @ A @ B5 )
=> ( ( member_a @ ( F2 @ A4 ) @ ( image_3724615099042636213real_a @ F2 @ A ) )
= ( member2350847679896131959_ereal @ A4 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_340_inj__image__mem__iff,axiom,
! [F2: a > a,A4: a,A: set_a] :
( ( inj_on_a_a @ F2 @ top_top_set_a )
=> ( ( member_a @ ( F2 @ A4 ) @ ( image_a_a @ F2 @ A ) )
= ( member_a @ A4 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_341_inj__image__mem__iff,axiom,
! [F2: extended_ereal > extended_ereal,A4: extended_ereal,A: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ( member2350847679896131959_ereal @ ( F2 @ A4 ) @ ( image_6042159593519690757_ereal @ F2 @ A ) )
= ( member2350847679896131959_ereal @ A4 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_342_inj__image__mem__iff,axiom,
! [F2: extended_ereal > a,A4: extended_ereal,A: set_Extended_ereal] :
( ( inj_on8242634198667403041real_a @ F2 @ top_to5683747375963461374_ereal )
=> ( ( member_a @ ( F2 @ A4 ) @ ( image_3724615099042636213real_a @ F2 @ A ) )
= ( member2350847679896131959_ereal @ A4 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_343_inj__image__mem__iff,axiom,
! [F2: nat > extended_ereal,A4: nat,A: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ top_top_set_nat )
=> ( ( member2350847679896131959_ereal @ ( F2 @ A4 ) @ ( image_4309273772856505399_ereal @ F2 @ A ) )
= ( member_nat @ A4 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_344_inj__image__mem__iff,axiom,
! [F2: nat > a,A4: nat,A: set_nat] :
( ( inj_on_nat_a @ F2 @ top_top_set_nat )
=> ( ( member_a @ ( F2 @ A4 ) @ ( image_nat_a @ F2 @ A ) )
= ( member_nat @ A4 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_345_inj__image__eq__iff,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,B5: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ( ( image_6042159593519690757_ereal @ F2 @ A )
= ( image_6042159593519690757_ereal @ F2 @ B5 ) )
= ( A = B5 ) ) ) ).
% inj_image_eq_iff
thf(fact_346_inj__image__eq__iff,axiom,
! [F2: nat > extended_ereal,A: set_nat,B5: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ top_top_set_nat )
=> ( ( ( image_4309273772856505399_ereal @ F2 @ A )
= ( image_4309273772856505399_ereal @ F2 @ B5 ) )
= ( A = B5 ) ) ) ).
% inj_image_eq_iff
thf(fact_347_range__ex1__eq,axiom,
! [F2: extended_ereal > extended_ereal,B: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ( member2350847679896131959_ereal @ B @ ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal ) )
= ( ? [X: extended_ereal] :
( ( B
= ( F2 @ X ) )
& ! [Y: extended_ereal] :
( ( B
= ( F2 @ Y ) )
=> ( Y = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_348_range__ex1__eq,axiom,
! [F2: extended_ereal > a,B: a] :
( ( inj_on8242634198667403041real_a @ F2 @ top_to5683747375963461374_ereal )
=> ( ( member_a @ B @ ( image_3724615099042636213real_a @ F2 @ top_to5683747375963461374_ereal ) )
= ( ? [X: extended_ereal] :
( ( B
= ( F2 @ X ) )
& ! [Y: extended_ereal] :
( ( B
= ( F2 @ Y ) )
=> ( Y = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_349_range__ex1__eq,axiom,
! [F2: nat > extended_ereal,B: extended_ereal] :
( ( inj_on6191532827271902155_ereal @ F2 @ top_top_set_nat )
=> ( ( member2350847679896131959_ereal @ B @ ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat ) )
= ( ? [X: nat] :
( ( B
= ( F2 @ X ) )
& ! [Y: nat] :
( ( B
= ( F2 @ Y ) )
=> ( Y = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_350_range__ex1__eq,axiom,
! [F2: nat > a,B: a] :
( ( inj_on_nat_a @ F2 @ top_top_set_nat )
=> ( ( member_a @ B @ ( image_nat_a @ F2 @ top_top_set_nat ) )
= ( ? [X: nat] :
( ( B
= ( F2 @ X ) )
& ! [Y: nat] :
( ( B
= ( F2 @ Y ) )
=> ( Y = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_351_fst__fstOp,axiom,
! [P2: a > b > $o,Q2: b > b > $o] :
( product_fst_a_b
= ( comp_P2648956292766268207od_a_b @ product_fst_a_b @ ( bNF_fstOp_a_b_b @ P2 @ Q2 ) ) ) ).
% fst_fstOp
thf(fact_352_snd__sndOp,axiom,
! [P2: a > a > $o,Q2: a > b > $o] :
( product_snd_a_b
= ( comp_P2009515992434452078od_a_b @ product_snd_a_b @ ( bNF_sndOp_a_a_b @ P2 @ Q2 ) ) ) ).
% snd_sndOp
thf(fact_353_inj__on__the__inv__into,axiom,
! [F2: nat > extended_ereal,A: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ A )
=> ( inj_on318729178700965101al_nat @ ( the_in5959796611709155849_ereal @ A @ F2 ) @ ( image_4309273772856505399_ereal @ F2 @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_354_inj__on__the__inv__into,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A )
=> ( inj_on7162434037990268785_ereal @ ( the_in1141389326992810419_ereal @ A @ F2 ) @ ( image_6042159593519690757_ereal @ F2 @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_355_f__the__inv__into__f,axiom,
! [F2: nat > extended_ereal,A: set_nat,Y2: extended_ereal] :
( ( inj_on6191532827271902155_ereal @ F2 @ A )
=> ( ( member2350847679896131959_ereal @ Y2 @ ( image_4309273772856505399_ereal @ F2 @ A ) )
=> ( ( F2 @ ( the_in5959796611709155849_ereal @ A @ F2 @ Y2 ) )
= Y2 ) ) ) ).
% f_the_inv_into_f
thf(fact_356_f__the__inv__into__f,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,Y2: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A )
=> ( ( member2350847679896131959_ereal @ Y2 @ ( image_6042159593519690757_ereal @ F2 @ A ) )
=> ( ( F2 @ ( the_in1141389326992810419_ereal @ A @ F2 @ Y2 ) )
= Y2 ) ) ) ).
% f_the_inv_into_f
thf(fact_357_graph__ranD,axiom,
! [X2: product_prod_a_b,M: a > option_b] :
( ( member1426531481828664017od_a_b @ X2 @ ( graph_a_b @ M ) )
=> ( member_b @ ( product_snd_a_b @ X2 ) @ ( ran_a_b @ M ) ) ) ).
% graph_ranD
thf(fact_358_Inf_OINF__cong,axiom,
! [A: set_Extended_ereal,B5: set_Extended_ereal,C2: extended_ereal > extended_ereal,D: extended_ereal > extended_ereal,Inf: set_Extended_ereal > extended_ereal] :
( ( A = B5 )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ B5 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_6042159593519690757_ereal @ C2 @ A ) )
= ( Inf @ ( image_6042159593519690757_ereal @ D @ B5 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_359_Inf_OINF__cong,axiom,
! [A: set_nat,B5: set_nat,C2: nat > extended_ereal,D: nat > extended_ereal,Inf: set_Extended_ereal > extended_ereal] :
( ( A = B5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B5 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_4309273772856505399_ereal @ C2 @ A ) )
= ( Inf @ ( image_4309273772856505399_ereal @ D @ B5 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_360_range__snd,axiom,
( ( image_2802296252294471260_a_b_b @ product_snd_a_b @ top_to8134405472303993176od_a_b )
= top_top_set_b ) ).
% range_snd
thf(fact_361_range__fst,axiom,
( ( image_2802296252294471259_a_b_a @ product_fst_a_b @ top_to8134405472303993176od_a_b )
= top_top_set_a ) ).
% range_fst
thf(fact_362_all__subset__image__inj,axiom,
! [F2: nat > extended_ereal,S4: set_nat,P2: set_Extended_ereal > $o] :
( ( ! [T3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ T3 @ ( image_4309273772856505399_ereal @ F2 @ S4 ) )
=> ( P2 @ T3 ) ) )
= ( ! [T3: set_nat] :
( ( ( ord_less_eq_set_nat @ T3 @ S4 )
& ( inj_on6191532827271902155_ereal @ F2 @ T3 ) )
=> ( P2 @ ( image_4309273772856505399_ereal @ F2 @ T3 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_363_all__subset__image__inj,axiom,
! [F2: extended_ereal > extended_ereal,S4: set_Extended_ereal,P2: set_Extended_ereal > $o] :
( ( ! [T3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ T3 @ ( image_6042159593519690757_ereal @ F2 @ S4 ) )
=> ( P2 @ T3 ) ) )
= ( ! [T3: set_Extended_ereal] :
( ( ( ord_le1644982726543182158_ereal @ T3 @ S4 )
& ( inj_on7162434037990268785_ereal @ F2 @ T3 ) )
=> ( P2 @ ( image_6042159593519690757_ereal @ F2 @ T3 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_364_ex__subset__image__inj,axiom,
! [F2: nat > extended_ereal,S4: set_nat,P2: set_Extended_ereal > $o] :
( ( ? [T3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ T3 @ ( image_4309273772856505399_ereal @ F2 @ S4 ) )
& ( P2 @ T3 ) ) )
= ( ? [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S4 )
& ( inj_on6191532827271902155_ereal @ F2 @ T3 )
& ( P2 @ ( image_4309273772856505399_ereal @ F2 @ T3 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_365_ex__subset__image__inj,axiom,
! [F2: extended_ereal > extended_ereal,S4: set_Extended_ereal,P2: set_Extended_ereal > $o] :
( ( ? [T3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ T3 @ ( image_6042159593519690757_ereal @ F2 @ S4 ) )
& ( P2 @ T3 ) ) )
= ( ? [T3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ T3 @ S4 )
& ( inj_on7162434037990268785_ereal @ F2 @ T3 )
& ( P2 @ ( image_6042159593519690757_ereal @ F2 @ T3 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_366_subset__image__inj,axiom,
! [S4: set_Extended_ereal,F2: nat > extended_ereal,T4: set_nat] :
( ( ord_le1644982726543182158_ereal @ S4 @ ( image_4309273772856505399_ereal @ F2 @ T4 ) )
= ( ? [U: set_nat] :
( ( ord_less_eq_set_nat @ U @ T4 )
& ( inj_on6191532827271902155_ereal @ F2 @ U )
& ( S4
= ( image_4309273772856505399_ereal @ F2 @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_367_subset__image__inj,axiom,
! [S4: set_Extended_ereal,F2: extended_ereal > extended_ereal,T4: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ S4 @ ( image_6042159593519690757_ereal @ F2 @ T4 ) )
= ( ? [U: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ U @ T4 )
& ( inj_on7162434037990268785_ereal @ F2 @ U )
& ( S4
= ( image_6042159593519690757_ereal @ F2 @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_368_fun_Oset__map,axiom,
! [F2: extended_ereal > extended_ereal,V: extended_ereal > extended_ereal] :
( ( image_6042159593519690757_ereal @ ( comp_E9177254828515427499_ereal @ F2 @ V ) @ top_to5683747375963461374_ereal )
= ( image_6042159593519690757_ereal @ F2 @ ( image_6042159593519690757_ereal @ V @ top_to5683747375963461374_ereal ) ) ) ).
% fun.set_map
thf(fact_369_fun_Oset__map,axiom,
! [F2: nat > extended_ereal,V: extended_ereal > nat] :
( ( image_6042159593519690757_ereal @ ( comp_n261702227720650419_ereal @ F2 @ V ) @ top_to5683747375963461374_ereal )
= ( image_4309273772856505399_ereal @ F2 @ ( image_7659842161140344153al_nat @ V @ top_to5683747375963461374_ereal ) ) ) ).
% fun.set_map
thf(fact_370_fun_Oset__map,axiom,
! [F2: extended_ereal > extended_ereal,V: nat > extended_ereal] :
( ( image_4309273772856505399_ereal @ ( comp_E3726099860353345075al_nat @ F2 @ V ) @ top_top_set_nat )
= ( image_6042159593519690757_ereal @ F2 @ ( image_4309273772856505399_ereal @ V @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_371_fun_Oset__map,axiom,
! [F2: nat > extended_ereal,V: nat > nat] :
( ( image_4309273772856505399_ereal @ ( comp_n13370146242399787al_nat @ F2 @ V ) @ top_top_set_nat )
= ( image_4309273772856505399_ereal @ F2 @ ( image_nat_nat @ V @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_372_subrelI,axiom,
! [R: set_Pr7497825696620840711list_o,S2: set_Pr7497825696620840711list_o] :
( ! [X3: option_list_o,Y3: option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X3 @ Y3 ) @ R )
=> ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X3 @ Y3 ) @ S2 ) )
=> ( ord_le7655427003059986087list_o @ R @ S2 ) ) ).
% subrelI
thf(fact_373_fun_Omap__ident__strong,axiom,
! [T2: extended_ereal > extended_ereal,F2: extended_ereal > extended_ereal] :
( ! [Z3: extended_ereal] :
( ( member2350847679896131959_ereal @ Z3 @ ( image_6042159593519690757_ereal @ T2 @ top_to5683747375963461374_ereal ) )
=> ( ( F2 @ Z3 )
= Z3 ) )
=> ( ( comp_E9177254828515427499_ereal @ F2 @ T2 )
= T2 ) ) ).
% fun.map_ident_strong
thf(fact_374_fun_Omap__ident__strong,axiom,
! [T2: extended_ereal > a,F2: a > a] :
( ! [Z3: a] :
( ( member_a @ Z3 @ ( image_3724615099042636213real_a @ T2 @ top_to5683747375963461374_ereal ) )
=> ( ( F2 @ Z3 )
= Z3 ) )
=> ( ( comp_a2309826877912173099_ereal @ F2 @ T2 )
= T2 ) ) ).
% fun.map_ident_strong
thf(fact_375_fun_Omap__ident__strong,axiom,
! [T2: nat > extended_ereal,F2: extended_ereal > extended_ereal] :
( ! [Z3: extended_ereal] :
( ( member2350847679896131959_ereal @ Z3 @ ( image_4309273772856505399_ereal @ T2 @ top_top_set_nat ) )
=> ( ( F2 @ Z3 )
= Z3 ) )
=> ( ( comp_E3726099860353345075al_nat @ F2 @ T2 )
= T2 ) ) ).
% fun.map_ident_strong
thf(fact_376_fun_Omap__ident__strong,axiom,
! [T2: nat > a,F2: a > a] :
( ! [Z3: a] :
( ( member_a @ Z3 @ ( image_nat_a @ T2 @ top_top_set_nat ) )
=> ( ( F2 @ Z3 )
= Z3 ) )
=> ( ( comp_a_a_nat @ F2 @ T2 )
= T2 ) ) ).
% fun.map_ident_strong
thf(fact_377_all__subset__image,axiom,
! [F2: nat > extended_ereal,A: set_nat,P2: set_Extended_ereal > $o] :
( ( ! [B7: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B7 @ ( image_4309273772856505399_ereal @ F2 @ A ) )
=> ( P2 @ B7 ) ) )
= ( ! [B7: set_nat] :
( ( ord_less_eq_set_nat @ B7 @ A )
=> ( P2 @ ( image_4309273772856505399_ereal @ F2 @ B7 ) ) ) ) ) ).
% all_subset_image
thf(fact_378_all__subset__image,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,P2: set_Extended_ereal > $o] :
( ( ! [B7: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B7 @ ( image_6042159593519690757_ereal @ F2 @ A ) )
=> ( P2 @ B7 ) ) )
= ( ! [B7: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B7 @ A )
=> ( P2 @ ( image_6042159593519690757_ereal @ F2 @ B7 ) ) ) ) ) ).
% all_subset_image
thf(fact_379_snd__comp__apsnd,axiom,
! [F2: b > b] :
( ( comp_P2009515992434452078od_a_b @ product_snd_a_b @ ( product_apsnd_b_b_a @ F2 ) )
= ( comp_b3886954628874447685od_a_b @ F2 @ product_snd_a_b ) ) ).
% snd_comp_apsnd
thf(fact_380_fst__comp__apfst,axiom,
! [F2: a > a] :
( ( comp_P2648956292766268207od_a_b @ product_fst_a_b @ ( product_apfst_a_a_b @ F2 ) )
= ( comp_a586446342581427589od_a_b @ F2 @ product_fst_a_b ) ) ).
% fst_comp_apfst
thf(fact_381_prod_Oinj__map,axiom,
! [F1: extended_ereal > extended_ereal,F22: extended_ereal > extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F1 @ top_to5683747375963461374_ereal )
=> ( ( inj_on7162434037990268785_ereal @ F22 @ top_to5683747375963461374_ereal )
=> ( inj_on7007621943451069233_ereal @ ( produc7788783332699689718_ereal @ F1 @ F22 ) @ top_to3798671025730093271_ereal ) ) ) ).
% prod.inj_map
thf(fact_382_map__prod__simp,axiom,
! [F2: option_list_o > option_list_o,G: option_list_o > option_list_o,A4: option_list_o,B: option_list_o] :
( ( produc2083626781540638710list_o @ F2 @ G @ ( produc5745850523778858007list_o @ A4 @ B ) )
= ( produc5745850523778858007list_o @ ( F2 @ A4 ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_383_fst__map__prod,axiom,
! [F2: a > a,G: b > b,X2: product_prod_a_b] :
( ( product_fst_a_b @ ( produc1231242867592151606_a_b_b @ F2 @ G @ X2 ) )
= ( F2 @ ( product_fst_a_b @ X2 ) ) ) ).
% fst_map_prod
thf(fact_384_snd__map__prod,axiom,
! [F2: a > a,G: b > b,X2: product_prod_a_b] :
( ( product_snd_a_b @ ( produc1231242867592151606_a_b_b @ F2 @ G @ X2 ) )
= ( G @ ( product_snd_a_b @ X2 ) ) ) ).
% snd_map_prod
thf(fact_385_apfst__conv,axiom,
! [F2: option_list_o > option_list_o,X2: option_list_o,Y2: option_list_o] :
( ( produc6343591382087220648list_o @ F2 @ ( produc5745850523778858007list_o @ X2 @ Y2 ) )
= ( produc5745850523778858007list_o @ ( F2 @ X2 ) @ Y2 ) ) ).
% apfst_conv
thf(fact_386_apsnd__conv,axiom,
! [F2: option_list_o > option_list_o,X2: option_list_o,Y2: option_list_o] :
( ( produc1422399317716192230list_o @ F2 @ ( produc5745850523778858007list_o @ X2 @ Y2 ) )
= ( produc5745850523778858007list_o @ X2 @ ( F2 @ Y2 ) ) ) ).
% apsnd_conv
thf(fact_387_fst__apfst,axiom,
! [F2: a > a,X2: product_prod_a_b] :
( ( product_fst_a_b @ ( product_apfst_a_a_b @ F2 @ X2 ) )
= ( F2 @ ( product_fst_a_b @ X2 ) ) ) ).
% fst_apfst
thf(fact_388_snd__apfst,axiom,
! [F2: a > a,X2: product_prod_a_b] :
( ( product_snd_a_b @ ( product_apfst_a_a_b @ F2 @ X2 ) )
= ( product_snd_a_b @ X2 ) ) ).
% snd_apfst
thf(fact_389_fst__apsnd,axiom,
! [F2: b > b,X2: product_prod_a_b] :
( ( product_fst_a_b @ ( product_apsnd_b_b_a @ F2 @ X2 ) )
= ( product_fst_a_b @ X2 ) ) ).
% fst_apsnd
thf(fact_390_snd__apsnd,axiom,
! [F2: b > b,X2: product_prod_a_b] :
( ( product_snd_a_b @ ( product_apsnd_b_b_a @ F2 @ X2 ) )
= ( F2 @ ( product_snd_a_b @ X2 ) ) ) ).
% snd_apsnd
thf(fact_391_map__prod__imageI,axiom,
! [A4: option_list_o,B: option_list_o,R2: set_Pr7497825696620840711list_o,F2: option_list_o > option_list_o,G: option_list_o > option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ A4 @ B ) @ R2 )
=> ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ ( F2 @ A4 ) @ ( G @ B ) ) @ ( image_8752170789471566789list_o @ ( produc2083626781540638710list_o @ F2 @ G ) @ R2 ) ) ) ).
% map_prod_imageI
thf(fact_392_snd__comp__apfst,axiom,
! [F2: a > a] :
( ( comp_P2009515992434452078od_a_b @ product_snd_a_b @ ( product_apfst_a_a_b @ F2 ) )
= product_snd_a_b ) ).
% snd_comp_apfst
thf(fact_393_fst__comp__apsnd,axiom,
! [F2: b > b] :
( ( comp_P2648956292766268207od_a_b @ product_fst_a_b @ ( product_apsnd_b_b_a @ F2 ) )
= product_fst_a_b ) ).
% fst_comp_apsnd
thf(fact_394_fst__comp__map__prod,axiom,
! [F2: a > a,G: b > b] :
( ( comp_P2648956292766268207od_a_b @ product_fst_a_b @ ( produc1231242867592151606_a_b_b @ F2 @ G ) )
= ( comp_a586446342581427589od_a_b @ F2 @ product_fst_a_b ) ) ).
% fst_comp_map_prod
thf(fact_395_snd__comp__map__prod,axiom,
! [F2: a > a,G: b > b] :
( ( comp_P2009515992434452078od_a_b @ product_snd_a_b @ ( produc1231242867592151606_a_b_b @ F2 @ G ) )
= ( comp_b3886954628874447685od_a_b @ G @ product_snd_a_b ) ) ).
% snd_comp_map_prod
thf(fact_396_apfst__apsnd,axiom,
! [F2: a > option_list_o,G: b > option_list_o,X2: product_prod_a_b] :
( ( produc4092373055763648706list_o @ F2 @ ( produc1993804381082613787st_o_a @ G @ X2 ) )
= ( produc5745850523778858007list_o @ ( F2 @ ( product_fst_a_b @ X2 ) ) @ ( G @ ( product_snd_a_b @ X2 ) ) ) ) ).
% apfst_apsnd
thf(fact_397_apsnd__apfst,axiom,
! [F2: b > option_list_o,G: a > option_list_o,X2: product_prod_a_b] :
( ( produc761593816161674369list_o @ F2 @ ( produc7556171008775904733st_o_b @ G @ X2 ) )
= ( produc5745850523778858007list_o @ ( G @ ( product_fst_a_b @ X2 ) ) @ ( F2 @ ( product_snd_a_b @ X2 ) ) ) ) ).
% apsnd_apfst
thf(fact_398_fun_Orel__cong,axiom,
! [X2: extended_ereal > extended_ereal,Ya: extended_ereal > extended_ereal,Y2: extended_ereal > extended_ereal,Xa2: extended_ereal > extended_ereal,R2: extended_ereal > extended_ereal > $o,Ra: extended_ereal > extended_ereal > $o] :
( ( X2 = Ya )
=> ( ( Y2 = Xa2 )
=> ( ! [Z3: extended_ereal,Yb: extended_ereal] :
( ( member2350847679896131959_ereal @ Z3 @ ( image_6042159593519690757_ereal @ Ya @ top_to5683747375963461374_ereal ) )
=> ( ( member2350847679896131959_ereal @ Yb @ ( image_6042159593519690757_ereal @ Xa2 @ top_to5683747375963461374_ereal ) )
=> ( ( R2 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( bNF_re3416630401399921757_ereal
@ ^ [Y4: extended_ereal,Z: extended_ereal] : ( Y4 = Z )
@ R2
@ X2
@ Y2 )
= ( bNF_re3416630401399921757_ereal
@ ^ [Y4: extended_ereal,Z: extended_ereal] : ( Y4 = Z )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_399_fun_Orel__cong,axiom,
! [X2: extended_ereal > extended_ereal,Ya: extended_ereal > extended_ereal,Y2: extended_ereal > a,Xa2: extended_ereal > a,R2: extended_ereal > a > $o,Ra: extended_ereal > a > $o] :
( ( X2 = Ya )
=> ( ( Y2 = Xa2 )
=> ( ! [Z3: extended_ereal,Yb: a] :
( ( member2350847679896131959_ereal @ Z3 @ ( image_6042159593519690757_ereal @ Ya @ top_to5683747375963461374_ereal ) )
=> ( ( member_a @ Yb @ ( image_3724615099042636213real_a @ Xa2 @ top_to5683747375963461374_ereal ) )
=> ( ( R2 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( bNF_re5691446141301026317real_a
@ ^ [Y4: extended_ereal,Z: extended_ereal] : ( Y4 = Z )
@ R2
@ X2
@ Y2 )
= ( bNF_re5691446141301026317real_a
@ ^ [Y4: extended_ereal,Z: extended_ereal] : ( Y4 = Z )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_400_fun_Orel__cong,axiom,
! [X2: extended_ereal > a,Ya: extended_ereal > a,Y2: extended_ereal > extended_ereal,Xa2: extended_ereal > extended_ereal,R2: a > extended_ereal > $o,Ra: a > extended_ereal > $o] :
( ( X2 = Ya )
=> ( ( Y2 = Xa2 )
=> ( ! [Z3: a,Yb: extended_ereal] :
( ( member_a @ Z3 @ ( image_3724615099042636213real_a @ Ya @ top_to5683747375963461374_ereal ) )
=> ( ( member2350847679896131959_ereal @ Yb @ ( image_6042159593519690757_ereal @ Xa2 @ top_to5683747375963461374_ereal ) )
=> ( ( R2 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( bNF_re1148940357394609709_ereal
@ ^ [Y4: extended_ereal,Z: extended_ereal] : ( Y4 = Z )
@ R2
@ X2
@ Y2 )
= ( bNF_re1148940357394609709_ereal
@ ^ [Y4: extended_ereal,Z: extended_ereal] : ( Y4 = Z )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_401_fun_Orel__cong,axiom,
! [X2: extended_ereal > a,Ya: extended_ereal > a,Y2: extended_ereal > a,Xa2: extended_ereal > a,R2: a > a > $o,Ra: a > a > $o] :
( ( X2 = Ya )
=> ( ( Y2 = Xa2 )
=> ( ! [Z3: a,Yb: a] :
( ( member_a @ Z3 @ ( image_3724615099042636213real_a @ Ya @ top_to5683747375963461374_ereal ) )
=> ( ( member_a @ Yb @ ( image_3724615099042636213real_a @ Xa2 @ top_to5683747375963461374_ereal ) )
=> ( ( R2 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( bNF_re4205385778126815197al_a_a
@ ^ [Y4: extended_ereal,Z: extended_ereal] : ( Y4 = Z )
@ R2
@ X2
@ Y2 )
= ( bNF_re4205385778126815197al_a_a
@ ^ [Y4: extended_ereal,Z: extended_ereal] : ( Y4 = Z )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_402_fun_Orel__cong,axiom,
! [X2: nat > extended_ereal,Ya: nat > extended_ereal,Y2: nat > extended_ereal,Xa2: nat > extended_ereal,R2: extended_ereal > extended_ereal > $o,Ra: extended_ereal > extended_ereal > $o] :
( ( X2 = Ya )
=> ( ( Y2 = Xa2 )
=> ( ! [Z3: extended_ereal,Yb: extended_ereal] :
( ( member2350847679896131959_ereal @ Z3 @ ( image_4309273772856505399_ereal @ Ya @ top_top_set_nat ) )
=> ( ( member2350847679896131959_ereal @ Yb @ ( image_4309273772856505399_ereal @ Xa2 @ top_top_set_nat ) )
=> ( ( R2 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( bNF_re5337380594787866367_ereal
@ ^ [Y4: nat,Z: nat] : ( Y4 = Z )
@ R2
@ X2
@ Y2 )
= ( bNF_re5337380594787866367_ereal
@ ^ [Y4: nat,Z: nat] : ( Y4 = Z )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_403_fun_Orel__cong,axiom,
! [X2: nat > extended_ereal,Ya: nat > extended_ereal,Y2: nat > a,Xa2: nat > a,R2: extended_ereal > a > $o,Ra: extended_ereal > a > $o] :
( ( X2 = Ya )
=> ( ( Y2 = Xa2 )
=> ( ! [Z3: extended_ereal,Yb: a] :
( ( member2350847679896131959_ereal @ Z3 @ ( image_4309273772856505399_ereal @ Ya @ top_top_set_nat ) )
=> ( ( member_a @ Yb @ ( image_nat_a @ Xa2 @ top_top_set_nat ) )
=> ( ( R2 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( bNF_re6402089080707531695real_a
@ ^ [Y4: nat,Z: nat] : ( Y4 = Z )
@ R2
@ X2
@ Y2 )
= ( bNF_re6402089080707531695real_a
@ ^ [Y4: nat,Z: nat] : ( Y4 = Z )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_404_fun_Orel__cong,axiom,
! [X2: nat > a,Ya: nat > a,Y2: nat > extended_ereal,Xa2: nat > extended_ereal,R2: a > extended_ereal > $o,Ra: a > extended_ereal > $o] :
( ( X2 = Ya )
=> ( ( Y2 = Xa2 )
=> ( ! [Z3: a,Yb: extended_ereal] :
( ( member_a @ Z3 @ ( image_nat_a @ Ya @ top_top_set_nat ) )
=> ( ( member2350847679896131959_ereal @ Yb @ ( image_4309273772856505399_ereal @ Xa2 @ top_top_set_nat ) )
=> ( ( R2 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( bNF_re1859583296801115087_ereal
@ ^ [Y4: nat,Z: nat] : ( Y4 = Z )
@ R2
@ X2
@ Y2 )
= ( bNF_re1859583296801115087_ereal
@ ^ [Y4: nat,Z: nat] : ( Y4 = Z )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_405_fun_Orel__cong,axiom,
! [X2: nat > a,Ya: nat > a,Y2: nat > a,Xa2: nat > a,R2: a > a > $o,Ra: a > a > $o] :
( ( X2 = Ya )
=> ( ( Y2 = Xa2 )
=> ( ! [Z3: a,Yb: a] :
( ( member_a @ Z3 @ ( image_nat_a @ Ya @ top_top_set_nat ) )
=> ( ( member_a @ Yb @ ( image_nat_a @ Xa2 @ top_top_set_nat ) )
=> ( ( R2 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( bNF_re4153754443986628735at_a_a
@ ^ [Y4: nat,Z: nat] : ( Y4 = Z )
@ R2
@ X2
@ Y2 )
= ( bNF_re4153754443986628735at_a_a
@ ^ [Y4: nat,Z: nat] : ( Y4 = Z )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_406_fun_Orel__mono__strong,axiom,
! [R2: extended_ereal > extended_ereal > $o,X2: extended_ereal > extended_ereal,Y2: extended_ereal > extended_ereal,Ra: extended_ereal > extended_ereal > $o] :
( ( bNF_re3416630401399921757_ereal
@ ^ [Y4: extended_ereal,Z: extended_ereal] : ( Y4 = Z )
@ R2
@ X2
@ Y2 )
=> ( ! [Z3: extended_ereal,Yb: extended_ereal] :
( ( member2350847679896131959_ereal @ Z3 @ ( image_6042159593519690757_ereal @ X2 @ top_to5683747375963461374_ereal ) )
=> ( ( member2350847679896131959_ereal @ Yb @ ( image_6042159593519690757_ereal @ Y2 @ top_to5683747375963461374_ereal ) )
=> ( ( R2 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re3416630401399921757_ereal
@ ^ [Y4: extended_ereal,Z: extended_ereal] : ( Y4 = Z )
@ Ra
@ X2
@ Y2 ) ) ) ).
% fun.rel_mono_strong
thf(fact_407_fun_Orel__mono__strong,axiom,
! [R2: extended_ereal > a > $o,X2: extended_ereal > extended_ereal,Y2: extended_ereal > a,Ra: extended_ereal > a > $o] :
( ( bNF_re5691446141301026317real_a
@ ^ [Y4: extended_ereal,Z: extended_ereal] : ( Y4 = Z )
@ R2
@ X2
@ Y2 )
=> ( ! [Z3: extended_ereal,Yb: a] :
( ( member2350847679896131959_ereal @ Z3 @ ( image_6042159593519690757_ereal @ X2 @ top_to5683747375963461374_ereal ) )
=> ( ( member_a @ Yb @ ( image_3724615099042636213real_a @ Y2 @ top_to5683747375963461374_ereal ) )
=> ( ( R2 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re5691446141301026317real_a
@ ^ [Y4: extended_ereal,Z: extended_ereal] : ( Y4 = Z )
@ Ra
@ X2
@ Y2 ) ) ) ).
% fun.rel_mono_strong
thf(fact_408_fun_Orel__mono__strong,axiom,
! [R2: a > extended_ereal > $o,X2: extended_ereal > a,Y2: extended_ereal > extended_ereal,Ra: a > extended_ereal > $o] :
( ( bNF_re1148940357394609709_ereal
@ ^ [Y4: extended_ereal,Z: extended_ereal] : ( Y4 = Z )
@ R2
@ X2
@ Y2 )
=> ( ! [Z3: a,Yb: extended_ereal] :
( ( member_a @ Z3 @ ( image_3724615099042636213real_a @ X2 @ top_to5683747375963461374_ereal ) )
=> ( ( member2350847679896131959_ereal @ Yb @ ( image_6042159593519690757_ereal @ Y2 @ top_to5683747375963461374_ereal ) )
=> ( ( R2 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re1148940357394609709_ereal
@ ^ [Y4: extended_ereal,Z: extended_ereal] : ( Y4 = Z )
@ Ra
@ X2
@ Y2 ) ) ) ).
% fun.rel_mono_strong
thf(fact_409_fun_Orel__mono__strong,axiom,
! [R2: a > a > $o,X2: extended_ereal > a,Y2: extended_ereal > a,Ra: a > a > $o] :
( ( bNF_re4205385778126815197al_a_a
@ ^ [Y4: extended_ereal,Z: extended_ereal] : ( Y4 = Z )
@ R2
@ X2
@ Y2 )
=> ( ! [Z3: a,Yb: a] :
( ( member_a @ Z3 @ ( image_3724615099042636213real_a @ X2 @ top_to5683747375963461374_ereal ) )
=> ( ( member_a @ Yb @ ( image_3724615099042636213real_a @ Y2 @ top_to5683747375963461374_ereal ) )
=> ( ( R2 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re4205385778126815197al_a_a
@ ^ [Y4: extended_ereal,Z: extended_ereal] : ( Y4 = Z )
@ Ra
@ X2
@ Y2 ) ) ) ).
% fun.rel_mono_strong
thf(fact_410_fun_Orel__mono__strong,axiom,
! [R2: extended_ereal > extended_ereal > $o,X2: nat > extended_ereal,Y2: nat > extended_ereal,Ra: extended_ereal > extended_ereal > $o] :
( ( bNF_re5337380594787866367_ereal
@ ^ [Y4: nat,Z: nat] : ( Y4 = Z )
@ R2
@ X2
@ Y2 )
=> ( ! [Z3: extended_ereal,Yb: extended_ereal] :
( ( member2350847679896131959_ereal @ Z3 @ ( image_4309273772856505399_ereal @ X2 @ top_top_set_nat ) )
=> ( ( member2350847679896131959_ereal @ Yb @ ( image_4309273772856505399_ereal @ Y2 @ top_top_set_nat ) )
=> ( ( R2 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re5337380594787866367_ereal
@ ^ [Y4: nat,Z: nat] : ( Y4 = Z )
@ Ra
@ X2
@ Y2 ) ) ) ).
% fun.rel_mono_strong
thf(fact_411_fun_Orel__mono__strong,axiom,
! [R2: extended_ereal > a > $o,X2: nat > extended_ereal,Y2: nat > a,Ra: extended_ereal > a > $o] :
( ( bNF_re6402089080707531695real_a
@ ^ [Y4: nat,Z: nat] : ( Y4 = Z )
@ R2
@ X2
@ Y2 )
=> ( ! [Z3: extended_ereal,Yb: a] :
( ( member2350847679896131959_ereal @ Z3 @ ( image_4309273772856505399_ereal @ X2 @ top_top_set_nat ) )
=> ( ( member_a @ Yb @ ( image_nat_a @ Y2 @ top_top_set_nat ) )
=> ( ( R2 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re6402089080707531695real_a
@ ^ [Y4: nat,Z: nat] : ( Y4 = Z )
@ Ra
@ X2
@ Y2 ) ) ) ).
% fun.rel_mono_strong
thf(fact_412_fun_Orel__mono__strong,axiom,
! [R2: a > extended_ereal > $o,X2: nat > a,Y2: nat > extended_ereal,Ra: a > extended_ereal > $o] :
( ( bNF_re1859583296801115087_ereal
@ ^ [Y4: nat,Z: nat] : ( Y4 = Z )
@ R2
@ X2
@ Y2 )
=> ( ! [Z3: a,Yb: extended_ereal] :
( ( member_a @ Z3 @ ( image_nat_a @ X2 @ top_top_set_nat ) )
=> ( ( member2350847679896131959_ereal @ Yb @ ( image_4309273772856505399_ereal @ Y2 @ top_top_set_nat ) )
=> ( ( R2 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re1859583296801115087_ereal
@ ^ [Y4: nat,Z: nat] : ( Y4 = Z )
@ Ra
@ X2
@ Y2 ) ) ) ).
% fun.rel_mono_strong
thf(fact_413_fun_Orel__mono__strong,axiom,
! [R2: a > a > $o,X2: nat > a,Y2: nat > a,Ra: a > a > $o] :
( ( bNF_re4153754443986628735at_a_a
@ ^ [Y4: nat,Z: nat] : ( Y4 = Z )
@ R2
@ X2
@ Y2 )
=> ( ! [Z3: a,Yb: a] :
( ( member_a @ Z3 @ ( image_nat_a @ X2 @ top_top_set_nat ) )
=> ( ( member_a @ Yb @ ( image_nat_a @ Y2 @ top_top_set_nat ) )
=> ( ( R2 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re4153754443986628735at_a_a
@ ^ [Y4: nat,Z: nat] : ( Y4 = Z )
@ Ra
@ X2
@ Y2 ) ) ) ).
% fun.rel_mono_strong
thf(fact_414_fun_Orel__refl__strong,axiom,
! [X2: extended_ereal > extended_ereal,Ra: extended_ereal > extended_ereal > $o] :
( ! [Z3: extended_ereal] :
( ( member2350847679896131959_ereal @ Z3 @ ( image_6042159593519690757_ereal @ X2 @ top_to5683747375963461374_ereal ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_re3416630401399921757_ereal
@ ^ [Y4: extended_ereal,Z: extended_ereal] : ( Y4 = Z )
@ Ra
@ X2
@ X2 ) ) ).
% fun.rel_refl_strong
thf(fact_415_fun_Orel__refl__strong,axiom,
! [X2: extended_ereal > a,Ra: a > a > $o] :
( ! [Z3: a] :
( ( member_a @ Z3 @ ( image_3724615099042636213real_a @ X2 @ top_to5683747375963461374_ereal ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_re4205385778126815197al_a_a
@ ^ [Y4: extended_ereal,Z: extended_ereal] : ( Y4 = Z )
@ Ra
@ X2
@ X2 ) ) ).
% fun.rel_refl_strong
thf(fact_416_fun_Orel__refl__strong,axiom,
! [X2: nat > extended_ereal,Ra: extended_ereal > extended_ereal > $o] :
( ! [Z3: extended_ereal] :
( ( member2350847679896131959_ereal @ Z3 @ ( image_4309273772856505399_ereal @ X2 @ top_top_set_nat ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_re5337380594787866367_ereal
@ ^ [Y4: nat,Z: nat] : ( Y4 = Z )
@ Ra
@ X2
@ X2 ) ) ).
% fun.rel_refl_strong
thf(fact_417_fun_Orel__refl__strong,axiom,
! [X2: nat > a,Ra: a > a > $o] :
( ! [Z3: a] :
( ( member_a @ Z3 @ ( image_nat_a @ X2 @ top_top_set_nat ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_re4153754443986628735at_a_a
@ ^ [Y4: nat,Z: nat] : ( Y4 = Z )
@ Ra
@ X2
@ X2 ) ) ).
% fun.rel_refl_strong
thf(fact_418_prod__fun__imageE,axiom,
! [C: produc4882884732533091879list_o,F2: option_list_o > option_list_o,G: option_list_o > option_list_o,R2: set_Pr7497825696620840711list_o] :
( ( member1589324699396745552list_o @ C @ ( image_8752170789471566789list_o @ ( produc2083626781540638710list_o @ F2 @ G ) @ R2 ) )
=> ~ ! [X3: option_list_o,Y3: option_list_o] :
( ( C
= ( produc5745850523778858007list_o @ ( F2 @ X3 ) @ ( G @ Y3 ) ) )
=> ~ ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X3 @ Y3 ) @ R2 ) ) ) ).
% prod_fun_imageE
thf(fact_419_map__prod__surj,axiom,
! [F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ( ( ( image_6042159593519690757_ereal @ G @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ( ( image_959328165755419589_ereal @ ( produc7788783332699689718_ereal @ F2 @ G ) @ top_to3798671025730093271_ereal )
= top_to3798671025730093271_ereal ) ) ) ).
% map_prod_surj
thf(fact_420_map__prod__surj,axiom,
! [F2: extended_ereal > extended_ereal,G: extended_ereal > nat] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ( ( ( image_7659842161140344153al_nat @ G @ top_to5683747375963461374_ereal )
= top_top_set_nat )
=> ( ( image_5023592649136443183al_nat @ ( produc8611138313008152232al_nat @ F2 @ G ) @ top_to3798671025730093271_ereal )
= top_to7896853287916821811al_nat ) ) ) ).
% map_prod_surj
thf(fact_421_map__prod__surj,axiom,
! [F2: extended_ereal > extended_ereal,G: nat > extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ( ( ( image_4309273772856505399_ereal @ G @ top_top_set_nat )
= top_to5683747375963461374_ereal )
=> ( ( image_6926512632986509267_ereal @ ( produc5260569924724313478_ereal @ F2 @ G ) @ top_to7896853287916821811al_nat )
= top_to3798671025730093271_ereal ) ) ) ).
% map_prod_surj
thf(fact_422_map__prod__surj,axiom,
! [F2: extended_ereal > extended_ereal,G: nat > nat] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ( ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_2725866490533164705al_nat @ ( produc8678206924122515480at_nat @ F2 @ G ) @ top_to7896853287916821811al_nat )
= top_to7896853287916821811al_nat ) ) ) ).
% map_prod_surj
thf(fact_423_map__prod__surj,axiom,
! [F2: extended_ereal > nat,G: extended_ereal > extended_ereal] :
( ( ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_nat )
=> ( ( ( image_6042159593519690757_ereal @ G @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ( ( image_7372151844261188565_ereal @ ( produc5146740680375457576_ereal @ F2 @ G ) @ top_to3798671025730093271_ereal )
= top_to6634112653661286105_ereal ) ) ) ).
% map_prod_surj
thf(fact_424_map__prod__surj,axiom,
! [F2: extended_ereal > nat,G: extended_ereal > nat] :
( ( ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_nat )
=> ( ( ( image_7659842161140344153al_nat @ G @ top_to5683747375963461374_ereal )
= top_top_set_nat )
=> ( ( image_8705176418032253727at_nat @ ( produc1189571518418271990al_nat @ F2 @ G ) @ top_to3798671025730093271_ereal )
= top_to4669805908274784177at_nat ) ) ) ).
% map_prod_surj
thf(fact_425_map__prod__surj,axiom,
! [F2: extended_ereal > nat,G: nat > extended_ereal] :
( ( ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_nat )
=> ( ( ( image_4309273772856505399_ereal @ G @ top_top_set_nat )
= top_to5683747375963461374_ereal )
=> ( ( image_5074425685657910087_ereal @ ( produc7062375166989209044_ereal @ F2 @ G ) @ top_to7896853287916821811al_nat )
= top_to6634112653661286105_ereal ) ) ) ).
% map_prod_surj
thf(fact_426_map__prod__surj,axiom,
! [F2: extended_ereal > nat,G: nat > nat] :
( ( ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_nat )
=> ( ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_8765427385416706093at_nat @ ( produc348661627770995530at_nat @ F2 @ G ) @ top_to7896853287916821811al_nat )
= top_to4669805908274784177at_nat ) ) ) ).
% map_prod_surj
thf(fact_427_map__prod__surj,axiom,
! [F2: nat > extended_ereal,G: extended_ereal > extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat )
= top_to5683747375963461374_ereal )
=> ( ( ( image_6042159593519690757_ereal @ G @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ( ( image_3845515196057536685_ereal @ ( produc7290494158911329542_ereal @ F2 @ G ) @ top_to6634112653661286105_ereal )
= top_to3798671025730093271_ereal ) ) ) ).
% map_prod_surj
thf(fact_428_map__prod__surj,axiom,
! [F2: nat > extended_ereal,G: extended_ereal > nat] :
( ( ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat )
= top_to5683747375963461374_ereal )
=> ( ( ( image_7659842161140344153al_nat @ G @ top_to5683747375963461374_ereal )
= top_top_set_nat )
=> ( ( image_5859397937574370375al_nat @ ( produc3928991975285440664al_nat @ F2 @ G ) @ top_to6634112653661286105_ereal )
= top_to7896853287916821811al_nat ) ) ) ).
% map_prod_surj
thf(fact_429_apfst__convE,axiom,
! [Q: produc4882884732533091879list_o,F2: option_list_o > option_list_o,P: produc4882884732533091879list_o] :
( ( Q
= ( produc6343591382087220648list_o @ F2 @ P ) )
=> ~ ! [X3: option_list_o,Y3: option_list_o] :
( ( P
= ( produc5745850523778858007list_o @ X3 @ Y3 ) )
=> ( Q
!= ( produc5745850523778858007list_o @ ( F2 @ X3 ) @ Y3 ) ) ) ) ).
% apfst_convE
thf(fact_430_inj__image__Compl__subset,axiom,
! [F2: nat > extended_ereal,A: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ top_top_set_nat )
=> ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F2 @ ( uminus5710092332889474511et_nat @ A ) ) @ ( uminus5895154729394068773_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_431_inj__image__Compl__subset,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F2 @ ( uminus5895154729394068773_ereal @ A ) ) @ ( uminus5895154729394068773_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_432_sup__compl__top__left1,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ ( sup_sup_set_nat @ X2 @ Y2 ) )
= top_top_set_nat ) ).
% sup_compl_top_left1
thf(fact_433_sup__compl__top__left1,axiom,
! [X2: set_Extended_ereal,Y2: set_Extended_ereal] :
( ( sup_su2680283192902082946_ereal @ ( uminus5895154729394068773_ereal @ X2 ) @ ( sup_su2680283192902082946_ereal @ X2 @ Y2 ) )
= top_to5683747375963461374_ereal ) ).
% sup_compl_top_left1
thf(fact_434_sup__compl__top__left2,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ Y2 ) )
= top_top_set_nat ) ).
% sup_compl_top_left2
thf(fact_435_sup__compl__top__left2,axiom,
! [X2: set_Extended_ereal,Y2: set_Extended_ereal] :
( ( sup_su2680283192902082946_ereal @ X2 @ ( sup_su2680283192902082946_ereal @ ( uminus5895154729394068773_ereal @ X2 ) @ Y2 ) )
= top_to5683747375963461374_ereal ) ).
% sup_compl_top_left2
thf(fact_436_boolean__algebra_Odisj__cancel__left,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ X2 )
= top_top_set_nat ) ).
% boolean_algebra.disj_cancel_left
thf(fact_437_boolean__algebra_Odisj__cancel__left,axiom,
! [X2: set_Extended_ereal] :
( ( sup_su2680283192902082946_ereal @ ( uminus5895154729394068773_ereal @ X2 ) @ X2 )
= top_to5683747375963461374_ereal ) ).
% boolean_algebra.disj_cancel_left
thf(fact_438_boolean__algebra_Odisj__cancel__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ X2 ) )
= top_top_set_nat ) ).
% boolean_algebra.disj_cancel_right
thf(fact_439_boolean__algebra_Odisj__cancel__right,axiom,
! [X2: set_Extended_ereal] :
( ( sup_su2680283192902082946_ereal @ X2 @ ( uminus5895154729394068773_ereal @ X2 ) )
= top_to5683747375963461374_ereal ) ).
% boolean_algebra.disj_cancel_right
thf(fact_440_sup__cancel__left1,axiom,
! [X2: set_nat,A4: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X2 @ A4 ) @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ B ) )
= top_top_set_nat ) ).
% sup_cancel_left1
thf(fact_441_sup__cancel__left1,axiom,
! [X2: set_Extended_ereal,A4: set_Extended_ereal,B: set_Extended_ereal] :
( ( sup_su2680283192902082946_ereal @ ( sup_su2680283192902082946_ereal @ X2 @ A4 ) @ ( sup_su2680283192902082946_ereal @ ( uminus5895154729394068773_ereal @ X2 ) @ B ) )
= top_to5683747375963461374_ereal ) ).
% sup_cancel_left1
thf(fact_442_sup__cancel__left2,axiom,
! [X2: set_nat,A4: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ A4 ) @ ( sup_sup_set_nat @ X2 @ B ) )
= top_top_set_nat ) ).
% sup_cancel_left2
thf(fact_443_sup__cancel__left2,axiom,
! [X2: set_Extended_ereal,A4: set_Extended_ereal,B: set_Extended_ereal] :
( ( sup_su2680283192902082946_ereal @ ( sup_su2680283192902082946_ereal @ ( uminus5895154729394068773_ereal @ X2 ) @ A4 ) @ ( sup_su2680283192902082946_ereal @ X2 @ B ) )
= top_to5683747375963461374_ereal ) ).
% sup_cancel_left2
thf(fact_444_Compl__partition,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ ( uminus5710092332889474511et_nat @ A ) )
= top_top_set_nat ) ).
% Compl_partition
thf(fact_445_Compl__partition,axiom,
! [A: set_Extended_ereal] :
( ( sup_su2680283192902082946_ereal @ A @ ( uminus5895154729394068773_ereal @ A ) )
= top_to5683747375963461374_ereal ) ).
% Compl_partition
thf(fact_446_Compl__partition2,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ A )
= top_top_set_nat ) ).
% Compl_partition2
thf(fact_447_Compl__partition2,axiom,
! [A: set_Extended_ereal] :
( ( sup_su2680283192902082946_ereal @ ( uminus5895154729394068773_ereal @ A ) @ A )
= top_to5683747375963461374_ereal ) ).
% Compl_partition2
thf(fact_448_sup__shunt,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ( sup_sup_set_nat @ X2 @ Y2 )
= top_top_set_nat )
= ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ Y2 ) ) ).
% sup_shunt
thf(fact_449_sup__shunt,axiom,
! [X2: set_Extended_ereal,Y2: set_Extended_ereal] :
( ( ( sup_su2680283192902082946_ereal @ X2 @ Y2 )
= top_to5683747375963461374_ereal )
= ( ord_le1644982726543182158_ereal @ ( uminus5895154729394068773_ereal @ X2 ) @ Y2 ) ) ).
% sup_shunt
thf(fact_450_surj__Compl__image__subset,axiom,
! [F2: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_nat_nat @ F2 @ A ) ) @ ( image_nat_nat @ F2 @ ( uminus5710092332889474511et_nat @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_451_surj__Compl__image__subset,axiom,
! [F2: extended_ereal > nat,A: set_Extended_ereal] :
( ( ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_7659842161140344153al_nat @ F2 @ A ) ) @ ( image_7659842161140344153al_nat @ F2 @ ( uminus5895154729394068773_ereal @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_452_surj__Compl__image__subset,axiom,
! [F2: nat > extended_ereal,A: set_nat] :
( ( ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat )
= top_to5683747375963461374_ereal )
=> ( ord_le1644982726543182158_ereal @ ( uminus5895154729394068773_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) @ ( image_4309273772856505399_ereal @ F2 @ ( uminus5710092332889474511et_nat @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_453_surj__Compl__image__subset,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ( ord_le1644982726543182158_ereal @ ( uminus5895154729394068773_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) @ ( image_6042159593519690757_ereal @ F2 @ ( uminus5895154729394068773_ereal @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_454_the__inv__f__o__f__id,axiom,
! [F2: extended_ereal > extended_ereal,Z2: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ( comp_E9177254828515427499_ereal @ ( the_in1141389326992810419_ereal @ top_to5683747375963461374_ereal @ F2 ) @ F2 @ Z2 )
= ( id_Extended_ereal @ Z2 ) ) ) ).
% the_inv_f_o_f_id
thf(fact_455_relChain__def,axiom,
( bNF_Ca4784764557604061198_ereal
= ( ^ [R3: set_Pr7497825696620840711list_o,As: option_list_o > extended_ereal] :
! [I2: option_list_o,J: option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ I2 @ J ) @ R3 )
=> ( ord_le1083603963089353582_ereal @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_456_relChain__def,axiom,
( bNF_Ca9081405844473788304_o_nat
= ( ^ [R3: set_Pr7497825696620840711list_o,As: option_list_o > nat] :
! [I2: option_list_o,J: option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ I2 @ J ) @ R3 )
=> ( ord_less_eq_nat @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_457_relChain__def,axiom,
( bNF_Ca6937576274950426094_ereal
= ( ^ [R3: set_Pr7497825696620840711list_o,As: option_list_o > set_Extended_ereal] :
! [I2: option_list_o,J: option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ I2 @ J ) @ R3 )
=> ( ord_le1644982726543182158_ereal @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_458_case__prod__Pair,axiom,
( ( produc1290809171805591803list_o @ produc5745850523778858007list_o )
= id_Pro5747325552096439704list_o ) ).
% case_prod_Pair
thf(fact_459_image__id,axiom,
( ( image_6042159593519690757_ereal @ id_Extended_ereal )
= id_set1423601066293951391_ereal ) ).
% image_id
thf(fact_460_boolean__algebra__class_Ominus__comp__minus,axiom,
( ( comp_s2342961461332800139_ereal @ uminus5895154729394068773_ereal @ uminus5895154729394068773_ereal )
= id_set1423601066293951391_ereal ) ).
% boolean_algebra_class.minus_comp_minus
thf(fact_461_Sup_OSUP__id__eq,axiom,
! [Sup: set_Extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( Sup @ ( image_6042159593519690757_ereal @ id_Extended_ereal @ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_id_eq
thf(fact_462_Inf_OINF__id__eq,axiom,
! [Inf: set_Extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( Inf @ ( image_6042159593519690757_ereal @ id_Extended_ereal @ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_id_eq
thf(fact_463_inj__on__id,axiom,
! [A: set_Extended_ereal] : ( inj_on7162434037990268785_ereal @ id_Extended_ereal @ A ) ).
% inj_on_id
thf(fact_464_Grp__UNIV__id,axiom,
! [F2: extended_ereal > extended_ereal] :
( ( F2 = id_Extended_ereal )
=> ( ( relcom7112103985748706525_ereal @ ( conver695035103646831605_ereal @ ( bNF_Gr4257726243395767246_ereal @ top_to5683747375963461374_ereal @ F2 ) ) @ ( bNF_Gr4257726243395767246_ereal @ top_to5683747375963461374_ereal @ F2 ) )
= ( bNF_Gr4257726243395767246_ereal @ top_to5683747375963461374_ereal @ F2 ) ) ) ).
% Grp_UNIV_id
thf(fact_465_Grp__UNIV__id,axiom,
! [F2: nat > nat] :
( ( F2 = id_nat )
=> ( ( relcompp_nat_nat_nat @ ( conversep_nat_nat @ ( bNF_Grp_nat_nat @ top_top_set_nat @ F2 ) ) @ ( bNF_Grp_nat_nat @ top_top_set_nat @ F2 ) )
= ( bNF_Grp_nat_nat @ top_top_set_nat @ F2 ) ) ) ).
% Grp_UNIV_id
thf(fact_466_surj__id,axiom,
( ( image_6042159593519690757_ereal @ id_Extended_ereal @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal ) ).
% surj_id
thf(fact_467_surj__id,axiom,
( ( image_nat_nat @ id_nat @ top_top_set_nat )
= top_top_set_nat ) ).
% surj_id
thf(fact_468_in__rel__def,axiom,
( fun_in5754580152604529986list_o
= ( ^ [R4: set_Pr7497825696620840711list_o,X: option_list_o,Y: option_list_o] : ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X @ Y ) @ R4 ) ) ) ).
% in_rel_def
thf(fact_469_Grp__UNIV__idI,axiom,
! [X2: extended_ereal,Y2: extended_ereal] :
( ( X2 = Y2 )
=> ( bNF_Gr4257726243395767246_ereal @ top_to5683747375963461374_ereal @ id_Extended_ereal @ X2 @ Y2 ) ) ).
% Grp_UNIV_idI
thf(fact_470_Grp__UNIV__idI,axiom,
! [X2: nat,Y2: nat] :
( ( X2 = Y2 )
=> ( bNF_Grp_nat_nat @ top_top_set_nat @ id_nat @ X2 @ Y2 ) ) ).
% Grp_UNIV_idI
thf(fact_471_eq__alt,axiom,
( ( ^ [Y4: extended_ereal,Z: extended_ereal] : ( Y4 = Z ) )
= ( bNF_Gr4257726243395767246_ereal @ top_to5683747375963461374_ereal @ id_Extended_ereal ) ) ).
% eq_alt
thf(fact_472_eq__alt,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( bNF_Grp_nat_nat @ top_top_set_nat @ id_nat ) ) ).
% eq_alt
thf(fact_473_Grp__fst__snd,axiom,
! [R2: a > b > $o] :
( ( relcom9179970356352878820_a_b_b @ ( conver9146098269634272843_a_b_a @ ( bNF_Gr7842136747927947300_a_b_a @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R2 ) ) @ product_fst_a_b ) ) @ ( bNF_Gr7842136747927947301_a_b_b @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R2 ) ) @ product_snd_a_b ) )
= R2 ) ).
% Grp_fst_snd
thf(fact_474_inv__o__cancel,axiom,
! [F2: extended_ereal > extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ( comp_E9177254828515427499_ereal @ ( hilber7422611030134141132_ereal @ top_to5683747375963461374_ereal @ F2 ) @ F2 )
= id_Extended_ereal ) ) ).
% inv_o_cancel
thf(fact_475_Quotient__alt__def5,axiom,
( quotie5409145007053189782_ereal
= ( ^ [R4: extended_ereal > extended_ereal > $o,Abs: extended_ereal > extended_ereal,Rep: extended_ereal > extended_ereal,T3: extended_ereal > extended_ereal > $o] :
( ( ord_le6654028770825229838real_o @ T3 @ ( bNF_Gr4257726243395767246_ereal @ top_to5683747375963461374_ereal @ Abs ) )
& ( ord_le6654028770825229838real_o @ ( bNF_Gr4257726243395767246_ereal @ top_to5683747375963461374_ereal @ Rep ) @ ( conver695035103646831605_ereal @ T3 ) )
& ( R4
= ( relcom7112103985748706525_ereal @ T3 @ ( conver695035103646831605_ereal @ T3 ) ) ) ) ) ) ).
% Quotient_alt_def5
thf(fact_476_Quotient__alt__def5,axiom,
( quotie1598486591011391240al_nat
= ( ^ [R4: extended_ereal > extended_ereal > $o,Abs: extended_ereal > nat,Rep: nat > extended_ereal,T3: extended_ereal > nat > $o] :
( ( ord_le3354276052480486518_nat_o @ T3 @ ( bNF_Gr3846222354243918800al_nat @ top_to5683747375963461374_ereal @ Abs ) )
& ( ord_le3392000996217782014real_o @ ( bNF_Gr495653965960080046_ereal @ top_top_set_nat @ Rep ) @ ( conver918952640114508137al_nat @ T3 ) )
& ( R4
= ( relcom4844919796881324383_ereal @ T3 @ ( conver918952640114508137al_nat @ T3 ) ) ) ) ) ) ).
% Quotient_alt_def5
thf(fact_477_Quotient__alt__def5,axiom,
( quotie7471290239582328294_ereal
= ( ^ [R4: nat > nat > $o,Abs: nat > extended_ereal,Rep: extended_ereal > nat,T3: nat > extended_ereal > $o] :
( ( ord_le3392000996217782014real_o @ T3 @ ( bNF_Gr495653965960080046_ereal @ top_top_set_nat @ Abs ) )
& ( ord_le3354276052480486518_nat_o @ ( bNF_Gr3846222354243918800al_nat @ top_to5683747375963461374_ereal @ Rep ) @ ( conver6791756288685445191_ereal @ T3 ) )
& ( R4
= ( relcom3981851657757828445al_nat @ T3 @ ( conver6791756288685445191_ereal @ T3 ) ) ) ) ) ) ).
% Quotient_alt_def5
thf(fact_478_Quotient__alt__def5,axiom,
( quotient_nat_nat
= ( ^ [R4: nat > nat > $o,Abs: nat > nat,Rep: nat > nat,T3: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ T3 @ ( bNF_Grp_nat_nat @ top_top_set_nat @ Abs ) )
& ( ord_le2646555220125990790_nat_o @ ( bNF_Grp_nat_nat @ top_top_set_nat @ Rep ) @ ( conversep_nat_nat @ T3 ) )
& ( R4
= ( relcompp_nat_nat_nat @ T3 @ ( conversep_nat_nat @ T3 ) ) ) ) ) ) ).
% Quotient_alt_def5
thf(fact_479_inv__into__f__f,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,X2: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A )
=> ( ( member2350847679896131959_ereal @ X2 @ A )
=> ( ( hilber7422611030134141132_ereal @ A @ F2 @ ( F2 @ X2 ) )
= X2 ) ) ) ).
% inv_into_f_f
thf(fact_480_inv__into__image__cancel,axiom,
! [F2: nat > extended_ereal,A: set_nat,S4: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ A )
=> ( ( ord_less_eq_set_nat @ S4 @ A )
=> ( ( image_7659842161140344153al_nat @ ( hilber6822286039850061104_ereal @ A @ F2 ) @ ( image_4309273772856505399_ereal @ F2 @ S4 ) )
= S4 ) ) ) ).
% inv_into_image_cancel
thf(fact_481_inv__into__image__cancel,axiom,
! [F2: extended_ereal > nat,A: set_Extended_ereal,S4: set_Extended_ereal] :
( ( inj_on318729178700965101al_nat @ F2 @ A )
=> ( ( ord_le1644982726543182158_ereal @ S4 @ A )
=> ( ( image_4309273772856505399_ereal @ ( hilber949482391279124050al_nat @ A @ F2 ) @ ( image_7659842161140344153al_nat @ F2 @ S4 ) )
= S4 ) ) ) ).
% inv_into_image_cancel
thf(fact_482_inv__into__image__cancel,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,S4: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A )
=> ( ( ord_le1644982726543182158_ereal @ S4 @ A )
=> ( ( image_6042159593519690757_ereal @ ( hilber7422611030134141132_ereal @ A @ F2 ) @ ( image_6042159593519690757_ereal @ F2 @ S4 ) )
= S4 ) ) ) ).
% inv_into_image_cancel
thf(fact_483_inv__into__f__eq,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,X2: extended_ereal,Y2: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A )
=> ( ( member2350847679896131959_ereal @ X2 @ A )
=> ( ( ( F2 @ X2 )
= Y2 )
=> ( ( hilber7422611030134141132_ereal @ A @ F2 @ Y2 )
= X2 ) ) ) ) ).
% inv_into_f_eq
thf(fact_484_inv__into__injective,axiom,
! [A: set_Extended_ereal,F2: extended_ereal > extended_ereal,X2: extended_ereal,Y2: extended_ereal] :
( ( ( hilber7422611030134141132_ereal @ A @ F2 @ X2 )
= ( hilber7422611030134141132_ereal @ A @ F2 @ Y2 ) )
=> ( ( member2350847679896131959_ereal @ X2 @ ( image_6042159593519690757_ereal @ F2 @ A ) )
=> ( ( member2350847679896131959_ereal @ Y2 @ ( image_6042159593519690757_ereal @ F2 @ A ) )
=> ( X2 = Y2 ) ) ) ) ).
% inv_into_injective
thf(fact_485_inv__into__injective,axiom,
! [A: set_nat,F2: nat > extended_ereal,X2: extended_ereal,Y2: extended_ereal] :
( ( ( hilber6822286039850061104_ereal @ A @ F2 @ X2 )
= ( hilber6822286039850061104_ereal @ A @ F2 @ Y2 ) )
=> ( ( member2350847679896131959_ereal @ X2 @ ( image_4309273772856505399_ereal @ F2 @ A ) )
=> ( ( member2350847679896131959_ereal @ Y2 @ ( image_4309273772856505399_ereal @ F2 @ A ) )
=> ( X2 = Y2 ) ) ) ) ).
% inv_into_injective
thf(fact_486_inv__into__into,axiom,
! [X2: extended_ereal,F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ ( image_6042159593519690757_ereal @ F2 @ A ) )
=> ( member2350847679896131959_ereal @ ( hilber7422611030134141132_ereal @ A @ F2 @ X2 ) @ A ) ) ).
% inv_into_into
thf(fact_487_inv__into__into,axiom,
! [X2: extended_ereal,F2: nat > extended_ereal,A: set_nat] :
( ( member2350847679896131959_ereal @ X2 @ ( image_4309273772856505399_ereal @ F2 @ A ) )
=> ( member_nat @ ( hilber6822286039850061104_ereal @ A @ F2 @ X2 ) @ A ) ) ).
% inv_into_into
thf(fact_488_inv__into__into,axiom,
! [X2: a,F2: a > a,A: set_a] :
( ( member_a @ X2 @ ( image_a_a @ F2 @ A ) )
=> ( member_a @ ( hilbert_inv_into_a_a @ A @ F2 @ X2 ) @ A ) ) ).
% inv_into_into
thf(fact_489_f__inv__into__f,axiom,
! [Y2: extended_ereal,F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ Y2 @ ( image_6042159593519690757_ereal @ F2 @ A ) )
=> ( ( F2 @ ( hilber7422611030134141132_ereal @ A @ F2 @ Y2 ) )
= Y2 ) ) ).
% f_inv_into_f
thf(fact_490_f__inv__into__f,axiom,
! [Y2: extended_ereal,F2: nat > extended_ereal,A: set_nat] :
( ( member2350847679896131959_ereal @ Y2 @ ( image_4309273772856505399_ereal @ F2 @ A ) )
=> ( ( F2 @ ( hilber6822286039850061104_ereal @ A @ F2 @ Y2 ) )
= Y2 ) ) ).
% f_inv_into_f
thf(fact_491_surj__f__inv__f,axiom,
! [F2: extended_ereal > extended_ereal,Y2: extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ( ( F2 @ ( hilber7422611030134141132_ereal @ top_to5683747375963461374_ereal @ F2 @ Y2 ) )
= Y2 ) ) ).
% surj_f_inv_f
thf(fact_492_surj__f__inv__f,axiom,
! [F2: extended_ereal > nat,Y2: nat] :
( ( ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_nat )
=> ( ( F2 @ ( hilber949482391279124050al_nat @ top_to5683747375963461374_ereal @ F2 @ Y2 ) )
= Y2 ) ) ).
% surj_f_inv_f
thf(fact_493_surj__f__inv__f,axiom,
! [F2: nat > extended_ereal,Y2: extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat )
= top_to5683747375963461374_ereal )
=> ( ( F2 @ ( hilber6822286039850061104_ereal @ top_top_set_nat @ F2 @ Y2 ) )
= Y2 ) ) ).
% surj_f_inv_f
thf(fact_494_surj__f__inv__f,axiom,
! [F2: nat > nat,Y2: nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( F2 @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 @ Y2 ) )
= Y2 ) ) ).
% surj_f_inv_f
thf(fact_495_surj__iff__all,axiom,
! [F2: extended_ereal > extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
= ( ! [X: extended_ereal] :
( ( F2 @ ( hilber7422611030134141132_ereal @ top_to5683747375963461374_ereal @ F2 @ X ) )
= X ) ) ) ).
% surj_iff_all
thf(fact_496_surj__iff__all,axiom,
! [F2: extended_ereal > nat] :
( ( ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_nat )
= ( ! [X: nat] :
( ( F2 @ ( hilber949482391279124050al_nat @ top_to5683747375963461374_ereal @ F2 @ X ) )
= X ) ) ) ).
% surj_iff_all
thf(fact_497_surj__iff__all,axiom,
! [F2: nat > extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat )
= top_to5683747375963461374_ereal )
= ( ! [X: extended_ereal] :
( ( F2 @ ( hilber6822286039850061104_ereal @ top_top_set_nat @ F2 @ X ) )
= X ) ) ) ).
% surj_iff_all
thf(fact_498_surj__iff__all,axiom,
! [F2: nat > nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
= ( ! [X: nat] :
( ( F2 @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 @ X ) )
= X ) ) ) ).
% surj_iff_all
thf(fact_499_image__f__inv__f,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ( ( image_6042159593519690757_ereal @ F2 @ ( image_6042159593519690757_ereal @ ( hilber7422611030134141132_ereal @ top_to5683747375963461374_ereal @ F2 ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_500_image__f__inv__f,axiom,
! [F2: extended_ereal > nat,A: set_nat] :
( ( ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_nat )
=> ( ( image_7659842161140344153al_nat @ F2 @ ( image_4309273772856505399_ereal @ ( hilber949482391279124050al_nat @ top_to5683747375963461374_ereal @ F2 ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_501_image__f__inv__f,axiom,
! [F2: nat > extended_ereal,A: set_Extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat )
= top_to5683747375963461374_ereal )
=> ( ( image_4309273772856505399_ereal @ F2 @ ( image_7659842161140344153al_nat @ ( hilber6822286039850061104_ereal @ top_top_set_nat @ F2 ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_502_image__f__inv__f,axiom,
! [F2: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_nat_nat @ F2 @ ( image_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 ) @ A ) )
= A ) ) ).
% image_f_inv_f
thf(fact_503_surj__imp__inv__eq,axiom,
! [F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ( ! [X3: extended_ereal] :
( ( G @ ( F2 @ X3 ) )
= X3 )
=> ( ( hilber7422611030134141132_ereal @ top_to5683747375963461374_ereal @ F2 )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_504_surj__imp__inv__eq,axiom,
! [F2: extended_ereal > nat,G: nat > extended_ereal] :
( ( ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_nat )
=> ( ! [X3: extended_ereal] :
( ( G @ ( F2 @ X3 ) )
= X3 )
=> ( ( hilber949482391279124050al_nat @ top_to5683747375963461374_ereal @ F2 )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_505_surj__imp__inv__eq,axiom,
! [F2: nat > extended_ereal,G: extended_ereal > nat] :
( ( ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat )
= top_to5683747375963461374_ereal )
=> ( ! [X3: nat] :
( ( G @ ( F2 @ X3 ) )
= X3 )
=> ( ( hilber6822286039850061104_ereal @ top_top_set_nat @ F2 )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_506_surj__imp__inv__eq,axiom,
! [F2: nat > nat,G: nat > nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ! [X3: nat] :
( ( G @ ( F2 @ X3 ) )
= X3 )
=> ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_507_GrD1,axiom,
! [X2: option_list_o,Fx: option_list_o,A: set_option_list_o,F2: option_list_o > option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X2 @ Fx ) @ ( bNF_Gr4294414969041315106list_o @ A @ F2 ) )
=> ( member_option_list_o @ X2 @ A ) ) ).
% GrD1
thf(fact_508_GrD2,axiom,
! [X2: option_list_o,Fx: option_list_o,A: set_option_list_o,F2: option_list_o > option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X2 @ Fx ) @ ( bNF_Gr4294414969041315106list_o @ A @ F2 ) )
=> ( ( F2 @ X2 )
= Fx ) ) ).
% GrD2
thf(fact_509_image__inv__into__cancel,axiom,
! [F2: extended_ereal > nat,A: set_Extended_ereal,A8: set_nat,B8: set_nat] :
( ( ( image_7659842161140344153al_nat @ F2 @ A )
= A8 )
=> ( ( ord_less_eq_set_nat @ B8 @ A8 )
=> ( ( image_7659842161140344153al_nat @ F2 @ ( image_4309273772856505399_ereal @ ( hilber949482391279124050al_nat @ A @ F2 ) @ B8 ) )
= B8 ) ) ) ).
% image_inv_into_cancel
thf(fact_510_image__inv__into__cancel,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,A8: set_Extended_ereal,B8: set_Extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ A )
= A8 )
=> ( ( ord_le1644982726543182158_ereal @ B8 @ A8 )
=> ( ( image_6042159593519690757_ereal @ F2 @ ( image_6042159593519690757_ereal @ ( hilber7422611030134141132_ereal @ A @ F2 ) @ B8 ) )
= B8 ) ) ) ).
% image_inv_into_cancel
thf(fact_511_image__inv__into__cancel,axiom,
! [F2: nat > extended_ereal,A: set_nat,A8: set_Extended_ereal,B8: set_Extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F2 @ A )
= A8 )
=> ( ( ord_le1644982726543182158_ereal @ B8 @ A8 )
=> ( ( image_4309273772856505399_ereal @ F2 @ ( image_7659842161140344153al_nat @ ( hilber6822286039850061104_ereal @ A @ F2 ) @ B8 ) )
= B8 ) ) ) ).
% image_inv_into_cancel
thf(fact_512_inv__f__f,axiom,
! [F2: extended_ereal > extended_ereal,X2: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ( hilber7422611030134141132_ereal @ top_to5683747375963461374_ereal @ F2 @ ( F2 @ X2 ) )
= X2 ) ) ).
% inv_f_f
thf(fact_513_inv__f__eq,axiom,
! [F2: extended_ereal > extended_ereal,X2: extended_ereal,Y2: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ( ( F2 @ X2 )
= Y2 )
=> ( ( hilber7422611030134141132_ereal @ top_to5683747375963461374_ereal @ F2 @ Y2 )
= X2 ) ) ) ).
% inv_f_eq
thf(fact_514_inj__imp__inv__eq,axiom,
! [F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ! [X3: extended_ereal] :
( ( F2 @ ( G @ X3 ) )
= X3 )
=> ( ( hilber7422611030134141132_ereal @ top_to5683747375963461374_ereal @ F2 )
= G ) ) ) ).
% inj_imp_inv_eq
thf(fact_515_surj__imp__inj__inv,axiom,
! [F2: extended_ereal > extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ( inj_on7162434037990268785_ereal @ ( hilber7422611030134141132_ereal @ top_to5683747375963461374_ereal @ F2 ) @ top_to5683747375963461374_ereal ) ) ).
% surj_imp_inj_inv
thf(fact_516_surj__imp__inj__inv,axiom,
! [F2: extended_ereal > nat] :
( ( ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_nat )
=> ( inj_on6191532827271902155_ereal @ ( hilber949482391279124050al_nat @ top_to5683747375963461374_ereal @ F2 ) @ top_top_set_nat ) ) ).
% surj_imp_inj_inv
thf(fact_517_surj__imp__inj__inv,axiom,
! [F2: nat > extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat )
= top_to5683747375963461374_ereal )
=> ( inj_on318729178700965101al_nat @ ( hilber6822286039850061104_ereal @ top_top_set_nat @ F2 ) @ top_to5683747375963461374_ereal ) ) ).
% surj_imp_inj_inv
thf(fact_518_surj__imp__inj__inv,axiom,
! [F2: nat > nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( inj_on_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 ) @ top_top_set_nat ) ) ).
% surj_imp_inj_inv
thf(fact_519_inj__imp__surj__inv,axiom,
! [F2: extended_ereal > extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ( image_6042159593519690757_ereal @ ( hilber7422611030134141132_ereal @ top_to5683747375963461374_ereal @ F2 ) @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal ) ) ).
% inj_imp_surj_inv
thf(fact_520_inj__imp__surj__inv,axiom,
! [F2: extended_ereal > nat] :
( ( inj_on318729178700965101al_nat @ F2 @ top_to5683747375963461374_ereal )
=> ( ( image_4309273772856505399_ereal @ ( hilber949482391279124050al_nat @ top_to5683747375963461374_ereal @ F2 ) @ top_top_set_nat )
= top_to5683747375963461374_ereal ) ) ).
% inj_imp_surj_inv
thf(fact_521_inj__imp__surj__inv,axiom,
! [F2: nat > extended_ereal] :
( ( inj_on6191532827271902155_ereal @ F2 @ top_top_set_nat )
=> ( ( image_7659842161140344153al_nat @ ( hilber6822286039850061104_ereal @ top_top_set_nat @ F2 ) @ top_to5683747375963461374_ereal )
= top_top_set_nat ) ) ).
% inj_imp_surj_inv
thf(fact_522_inj__imp__surj__inv,axiom,
! [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( image_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 ) @ top_top_set_nat )
= top_top_set_nat ) ) ).
% inj_imp_surj_inv
thf(fact_523_image__inv__f__f,axiom,
! [F2: extended_ereal > nat,A: set_Extended_ereal] :
( ( inj_on318729178700965101al_nat @ F2 @ top_to5683747375963461374_ereal )
=> ( ( image_4309273772856505399_ereal @ ( hilber949482391279124050al_nat @ top_to5683747375963461374_ereal @ F2 ) @ ( image_7659842161140344153al_nat @ F2 @ A ) )
= A ) ) ).
% image_inv_f_f
thf(fact_524_image__inv__f__f,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ( image_6042159593519690757_ereal @ ( hilber7422611030134141132_ereal @ top_to5683747375963461374_ereal @ F2 ) @ ( image_6042159593519690757_ereal @ F2 @ A ) )
= A ) ) ).
% image_inv_f_f
thf(fact_525_image__inv__f__f,axiom,
! [F2: nat > extended_ereal,A: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ top_top_set_nat )
=> ( ( image_7659842161140344153al_nat @ ( hilber6822286039850061104_ereal @ top_top_set_nat @ F2 ) @ ( image_4309273772856505399_ereal @ F2 @ A ) )
= A ) ) ).
% image_inv_f_f
thf(fact_526_inj__transfer,axiom,
! [F2: extended_ereal > extended_ereal,P2: extended_ereal > $o,X2: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ! [Y3: extended_ereal] :
( ( member2350847679896131959_ereal @ Y3 @ ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal ) )
=> ( P2 @ ( hilber7422611030134141132_ereal @ top_to5683747375963461374_ereal @ F2 @ Y3 ) ) )
=> ( P2 @ X2 ) ) ) ).
% inj_transfer
thf(fact_527_inj__transfer,axiom,
! [F2: extended_ereal > a,P2: extended_ereal > $o,X2: extended_ereal] :
( ( inj_on8242634198667403041real_a @ F2 @ top_to5683747375963461374_ereal )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ ( image_3724615099042636213real_a @ F2 @ top_to5683747375963461374_ereal ) )
=> ( P2 @ ( hilber3276319488169350396real_a @ top_to5683747375963461374_ereal @ F2 @ Y3 ) ) )
=> ( P2 @ X2 ) ) ) ).
% inj_transfer
thf(fact_528_inj__transfer,axiom,
! [F2: nat > extended_ereal,P2: nat > $o,X2: nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ top_top_set_nat )
=> ( ! [Y3: extended_ereal] :
( ( member2350847679896131959_ereal @ Y3 @ ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat ) )
=> ( P2 @ ( hilber6822286039850061104_ereal @ top_top_set_nat @ F2 @ Y3 ) ) )
=> ( P2 @ X2 ) ) ) ).
% inj_transfer
thf(fact_529_inj__transfer,axiom,
! [F2: nat > a,P2: nat > $o,X2: nat] :
( ( inj_on_nat_a @ F2 @ top_top_set_nat )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ ( image_nat_a @ F2 @ top_top_set_nat ) )
=> ( P2 @ ( hilber2795491120104822624_nat_a @ top_top_set_nat @ F2 @ Y3 ) ) )
=> ( P2 @ X2 ) ) ) ).
% inj_transfer
thf(fact_530_inj__on__inv__into,axiom,
! [B5: set_Extended_ereal,F2: nat > extended_ereal,A: set_nat] :
( ( ord_le1644982726543182158_ereal @ B5 @ ( image_4309273772856505399_ereal @ F2 @ A ) )
=> ( inj_on318729178700965101al_nat @ ( hilber6822286039850061104_ereal @ A @ F2 ) @ B5 ) ) ).
% inj_on_inv_into
thf(fact_531_inj__on__inv__into,axiom,
! [B5: set_Extended_ereal,F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B5 @ ( image_6042159593519690757_ereal @ F2 @ A ) )
=> ( inj_on7162434037990268785_ereal @ ( hilber7422611030134141132_ereal @ A @ F2 ) @ B5 ) ) ).
% inj_on_inv_into
thf(fact_532_inv__into__comp,axiom,
! [F2: extended_ereal > extended_ereal,G: nat > extended_ereal,A: set_nat,X2: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ ( image_4309273772856505399_ereal @ G @ A ) )
=> ( ( inj_on6191532827271902155_ereal @ G @ A )
=> ( ( member2350847679896131959_ereal @ X2 @ ( image_6042159593519690757_ereal @ F2 @ ( image_4309273772856505399_ereal @ G @ A ) ) )
=> ( ( hilber6822286039850061104_ereal @ A @ ( comp_E3726099860353345075al_nat @ F2 @ G ) @ X2 )
= ( comp_E375531472069506321_ereal @ ( hilber6822286039850061104_ereal @ A @ G ) @ ( hilber7422611030134141132_ereal @ ( image_4309273772856505399_ereal @ G @ A ) @ F2 ) @ X2 ) ) ) ) ) ).
% inv_into_comp
thf(fact_533_inv__into__comp,axiom,
! [F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal,A: set_Extended_ereal,X2: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ ( image_6042159593519690757_ereal @ G @ A ) )
=> ( ( inj_on7162434037990268785_ereal @ G @ A )
=> ( ( member2350847679896131959_ereal @ X2 @ ( image_6042159593519690757_ereal @ F2 @ ( image_6042159593519690757_ereal @ G @ A ) ) )
=> ( ( hilber7422611030134141132_ereal @ A @ ( comp_E9177254828515427499_ereal @ F2 @ G ) @ X2 )
= ( comp_E9177254828515427499_ereal @ ( hilber7422611030134141132_ereal @ A @ G ) @ ( hilber7422611030134141132_ereal @ ( image_6042159593519690757_ereal @ G @ A ) @ F2 ) @ X2 ) ) ) ) ) ).
% inv_into_comp
thf(fact_534_inv__into__comp,axiom,
! [F2: extended_ereal > a,G: nat > extended_ereal,A: set_nat,X2: a] :
( ( inj_on8242634198667403041real_a @ F2 @ ( image_4309273772856505399_ereal @ G @ A ) )
=> ( ( inj_on6191532827271902155_ereal @ G @ A )
=> ( ( member_a @ X2 @ ( image_3724615099042636213real_a @ F2 @ ( image_4309273772856505399_ereal @ G @ A ) ) )
=> ( ( hilber2795491120104822624_nat_a @ A @ ( comp_E5637448798707004259_a_nat @ F2 @ G ) @ X2 )
= ( comp_E446008263030514881_nat_a @ ( hilber6822286039850061104_ereal @ A @ G ) @ ( hilber3276319488169350396real_a @ ( image_4309273772856505399_ereal @ G @ A ) @ F2 ) @ X2 ) ) ) ) ) ).
% inv_into_comp
thf(fact_535_inv__into__comp,axiom,
! [F2: extended_ereal > a,G: extended_ereal > extended_ereal,A: set_Extended_ereal,X2: a] :
( ( inj_on8242634198667403041real_a @ F2 @ ( image_6042159593519690757_ereal @ G @ A ) )
=> ( ( inj_on7162434037990268785_ereal @ G @ A )
=> ( ( member_a @ X2 @ ( image_3724615099042636213real_a @ F2 @ ( image_6042159593519690757_ereal @ G @ A ) ) )
=> ( ( hilber3276319488169350396real_a @ A @ ( comp_E6551704282591734651_ereal @ F2 @ G ) @ X2 )
= ( comp_E1870838029643375451real_a @ ( hilber7422611030134141132_ereal @ A @ G ) @ ( hilber3276319488169350396real_a @ ( image_6042159593519690757_ereal @ G @ A ) @ F2 ) @ X2 ) ) ) ) ) ).
% inv_into_comp
thf(fact_536_surj__iff,axiom,
! [F2: extended_ereal > extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
= ( ( comp_E9177254828515427499_ereal @ F2 @ ( hilber7422611030134141132_ereal @ top_to5683747375963461374_ereal @ F2 ) )
= id_Extended_ereal ) ) ).
% surj_iff
thf(fact_537_surj__iff,axiom,
! [F2: extended_ereal > nat] :
( ( ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_nat )
= ( ( comp_E7502005551946643277at_nat @ F2 @ ( hilber949482391279124050al_nat @ top_to5683747375963461374_ereal @ F2 ) )
= id_nat ) ) ).
% surj_iff
thf(fact_538_surj__iff,axiom,
! [F2: nat > extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat )
= top_to5683747375963461374_ereal )
= ( ( comp_n261702227720650419_ereal @ F2 @ ( hilber6822286039850061104_ereal @ top_top_set_nat @ F2 ) )
= id_Extended_ereal ) ) ).
% surj_iff
thf(fact_539_surj__iff,axiom,
! [F2: nat > nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
= ( ( comp_nat_nat_nat @ F2 @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 ) )
= id_nat ) ) ).
% surj_iff
thf(fact_540_inj__iff,axiom,
! [F2: extended_ereal > extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
= ( ( comp_E9177254828515427499_ereal @ ( hilber7422611030134141132_ereal @ top_to5683747375963461374_ereal @ F2 ) @ F2 )
= id_Extended_ereal ) ) ).
% inj_iff
thf(fact_541_bijection_Oinj__inv,axiom,
! [F2: extended_ereal > extended_ereal] :
( ( hilber6088754731438466237_ereal @ F2 )
=> ( inj_on7162434037990268785_ereal @ ( hilber7422611030134141132_ereal @ top_to5683747375963461374_ereal @ F2 ) @ top_to5683747375963461374_ereal ) ) ).
% bijection.inj_inv
thf(fact_542_bijection_Oinj__inv,axiom,
! [F2: nat > nat] :
( ( hilber5277034221543178913on_nat @ F2 )
=> ( inj_on_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 ) @ top_top_set_nat ) ) ).
% bijection.inj_inv
thf(fact_543_bijection_Osurj__inv,axiom,
! [F2: extended_ereal > extended_ereal] :
( ( hilber6088754731438466237_ereal @ F2 )
=> ( ( image_6042159593519690757_ereal @ ( hilber7422611030134141132_ereal @ top_to5683747375963461374_ereal @ F2 ) @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal ) ) ).
% bijection.surj_inv
thf(fact_544_bijection_Osurj__inv,axiom,
! [F2: nat > nat] :
( ( hilber5277034221543178913on_nat @ F2 )
=> ( ( image_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 ) @ top_top_set_nat )
= top_top_set_nat ) ) ).
% bijection.surj_inv
thf(fact_545_fun_Oin__rel,axiom,
! [R2: a > b > $o,A4: extended_ereal > a,B: extended_ereal > b] :
( ( bNF_re4205385778126815198al_a_b
@ ^ [Y4: extended_ereal,Z: extended_ereal] : ( Y4 = Z )
@ R2
@ A4
@ B )
= ( ? [Z4: extended_ereal > product_prod_a_b] :
( ( member2022240716180195542od_a_b @ Z4
@ ( collec4637262903798613268od_a_b
@ ^ [X: extended_ereal > product_prod_a_b] : ( ord_le817736998455962536od_a_b @ ( image_2061744219110304863od_a_b @ X @ top_to5683747375963461374_ereal ) @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R2 ) ) ) ) )
& ( ( comp_P5204766043101469397_ereal @ product_fst_a_b @ Z4 )
= A4 )
& ( ( comp_P2119009801111474836_ereal @ product_snd_a_b @ Z4 )
= B ) ) ) ) ).
% fun.in_rel
thf(fact_546_fun_Oin__rel,axiom,
! [R2: a > b > $o,A4: nat > a,B: nat > b] :
( ( bNF_re4153754443986628736at_a_b
@ ^ [Y4: nat,Z: nat] : ( Y4 = Z )
@ R2
@ A4
@ B )
= ( ? [Z4: nat > product_prod_a_b] :
( ( member980272314738517048od_a_b @ Z4
@ ( collec2451373400991203194od_a_b
@ ^ [X: nat > product_prod_a_b] : ( ord_le817736998455962536od_a_b @ ( image_372941892535967121od_a_b @ X @ top_top_set_nat ) @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R2 ) ) ) ) )
& ( ( comp_P2362932533289333385_a_nat @ product_fst_a_b @ Z4 )
= A4 )
& ( ( comp_P3598376862245727882_b_nat @ product_snd_a_b @ Z4 )
= B ) ) ) ) ).
% fun.in_rel
thf(fact_547_fun_Orel__compp__Grp,axiom,
! [R2: a > b > $o] :
( ( bNF_re4205385778126815198al_a_b
@ ^ [Y4: extended_ereal,Z: extended_ereal] : ( Y4 = Z )
@ R2 )
= ( relcom5425792838349012253real_b
@ ( conver2344420769666178251real_a
@ ( bNF_Gr4066342199408912420real_a
@ ( collec4637262903798613268od_a_b
@ ^ [X: extended_ereal > product_prod_a_b] : ( ord_le817736998455962536od_a_b @ ( image_2061744219110304863od_a_b @ X @ top_to5683747375963461374_ereal ) @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R2 ) ) ) )
@ ( comp_P5204766043101469397_ereal @ product_fst_a_b ) ) )
@ ( bNF_Gr4066342203712141221real_b
@ ( collec4637262903798613268od_a_b
@ ^ [X: extended_ereal > product_prod_a_b] : ( ord_le817736998455962536od_a_b @ ( image_2061744219110304863od_a_b @ X @ top_to5683747375963461374_ereal ) @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R2 ) ) ) )
@ ( comp_P2119009801111474836_ereal @ product_snd_a_b ) ) ) ) ).
% fun.rel_compp_Grp
thf(fact_548_fun_Orel__compp__Grp,axiom,
! [R2: a > b > $o] :
( ( bNF_re4153754443986628736at_a_b
@ ^ [Y4: nat,Z: nat] : ( Y4 = Z )
@ R2 )
= ( relcom8441821119780287755_nat_b
@ ( conver7239854674026122507_nat_a
@ ( bNF_Gr2261899126669558756_nat_a
@ ( collec2451373400991203194od_a_b
@ ^ [X: nat > product_prod_a_b] : ( ord_le817736998455962536od_a_b @ ( image_372941892535967121od_a_b @ X @ top_top_set_nat ) @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R2 ) ) ) )
@ ( comp_P2362932533289333385_a_nat @ product_fst_a_b ) ) )
@ ( bNF_Gr2261899130972787557_nat_b
@ ( collec2451373400991203194od_a_b
@ ^ [X: nat > product_prod_a_b] : ( ord_le817736998455962536od_a_b @ ( image_372941892535967121od_a_b @ X @ top_top_set_nat ) @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R2 ) ) ) )
@ ( comp_P3598376862245727882_b_nat @ product_snd_a_b ) ) ) ) ).
% fun.rel_compp_Grp
thf(fact_549_inj__setminus,axiom,
! [A: set_se6634062954251873166_ereal] : ( inj_on5406440306785145713_ereal @ uminus5895154729394068773_ereal @ A ) ).
% inj_setminus
thf(fact_550_image__ident,axiom,
! [Y5: set_Extended_ereal] :
( ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : X
@ Y5 )
= Y5 ) ).
% image_ident
thf(fact_551_case__prodI,axiom,
! [F2: option_list_o > option_list_o > $o,A4: option_list_o,B: option_list_o] :
( ( F2 @ A4 @ B )
=> ( produc2885628514414960748st_o_o @ F2 @ ( produc5745850523778858007list_o @ A4 @ B ) ) ) ).
% case_prodI
thf(fact_552_case__prodI2,axiom,
! [P: produc4882884732533091879list_o,C: option_list_o > option_list_o > $o] :
( ! [A2: option_list_o,B3: option_list_o] :
( ( P
= ( produc5745850523778858007list_o @ A2 @ B3 ) )
=> ( C @ A2 @ B3 ) )
=> ( produc2885628514414960748st_o_o @ C @ P ) ) ).
% case_prodI2
thf(fact_553_mem__case__prodI2,axiom,
! [P: produc4882884732533091879list_o,Z2: a,C: option_list_o > option_list_o > set_a] :
( ! [A2: option_list_o,B3: option_list_o] :
( ( P
= ( produc5745850523778858007list_o @ A2 @ B3 ) )
=> ( member_a @ Z2 @ ( C @ A2 @ B3 ) ) )
=> ( member_a @ Z2 @ ( produc8989019892752111410_set_a @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_554_mem__case__prodI,axiom,
! [Z2: a,C: option_list_o > option_list_o > set_a,A4: option_list_o,B: option_list_o] :
( ( member_a @ Z2 @ ( C @ A4 @ B ) )
=> ( member_a @ Z2 @ ( produc8989019892752111410_set_a @ C @ ( produc5745850523778858007list_o @ A4 @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_555_pair__imageI,axiom,
! [A4: option_list_o,B: option_list_o,A: set_Pr7497825696620840711list_o,F2: option_list_o > option_list_o > a] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ A4 @ B ) @ A )
=> ( member_a @ ( F2 @ A4 @ B ) @ ( image_8562475727554508956st_o_a @ ( produc2009808905996238290st_o_a @ F2 ) @ A ) ) ) ).
% pair_imageI
thf(fact_556_fst__def,axiom,
( product_fst_a_b
= ( produc6028431345588019473_a_b_a
@ ^ [X12: a,X23: b] : X12 ) ) ).
% fst_def
thf(fact_557_snd__def,axiom,
( product_snd_a_b
= ( produc6028431345588019474_a_b_b
@ ^ [X12: a,X23: b] : X23 ) ) ).
% snd_def
thf(fact_558_case__prodE,axiom,
! [C: option_list_o > option_list_o > $o,P: produc4882884732533091879list_o] :
( ( produc2885628514414960748st_o_o @ C @ P )
=> ~ ! [X3: option_list_o,Y3: option_list_o] :
( ( P
= ( produc5745850523778858007list_o @ X3 @ Y3 ) )
=> ~ ( C @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_559_case__prodD,axiom,
! [F2: option_list_o > option_list_o > $o,A4: option_list_o,B: option_list_o] :
( ( produc2885628514414960748st_o_o @ F2 @ ( produc5745850523778858007list_o @ A4 @ B ) )
=> ( F2 @ A4 @ B ) ) ).
% case_prodD
thf(fact_560_case__prod__Pair__iden,axiom,
! [P: produc4882884732533091879list_o] :
( ( produc1290809171805591803list_o @ produc5745850523778858007list_o @ P )
= P ) ).
% case_prod_Pair_iden
thf(fact_561_subset__Collect__iff,axiom,
! [B5: set_a,A: set_a,P2: a > $o] :
( ( ord_less_eq_set_a @ B5 @ A )
=> ( ( ord_less_eq_set_a @ B5
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A )
& ( P2 @ X ) ) ) )
= ( ! [X: a] :
( ( member_a @ X @ B5 )
=> ( P2 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_562_subset__Collect__iff,axiom,
! [B5: set_Extended_ereal,A: set_Extended_ereal,P2: extended_ereal > $o] :
( ( ord_le1644982726543182158_ereal @ B5 @ A )
=> ( ( ord_le1644982726543182158_ereal @ B5
@ ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A )
& ( P2 @ X ) ) ) )
= ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ B5 )
=> ( P2 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_563_subset__CollectI,axiom,
! [B5: set_a,A: set_a,Q2: a > $o,P2: a > $o] :
( ( ord_less_eq_set_a @ B5 @ A )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B5 )
=> ( ( Q2 @ X3 )
=> ( P2 @ X3 ) ) )
=> ( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ B5 )
& ( Q2 @ X ) ) )
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A )
& ( P2 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_564_subset__CollectI,axiom,
! [B5: set_Extended_ereal,A: set_Extended_ereal,Q2: extended_ereal > $o,P2: extended_ereal > $o] :
( ( ord_le1644982726543182158_ereal @ B5 @ A )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ B5 )
=> ( ( Q2 @ X3 )
=> ( P2 @ X3 ) ) )
=> ( ord_le1644982726543182158_ereal
@ ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ B5 )
& ( Q2 @ X ) ) )
@ ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A )
& ( P2 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_565_Collect__restrict,axiom,
! [X5: set_a,P2: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ X5 )
& ( P2 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_566_Collect__restrict,axiom,
! [X5: set_Extended_ereal,P2: extended_ereal > $o] :
( ord_le1644982726543182158_ereal
@ ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ X5 )
& ( P2 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_567_prop__restrict,axiom,
! [X2: a,Z5: set_a,X5: set_a,P2: a > $o] :
( ( member_a @ X2 @ Z5 )
=> ( ( ord_less_eq_set_a @ Z5
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ X5 )
& ( P2 @ X ) ) ) )
=> ( P2 @ X2 ) ) ) ).
% prop_restrict
thf(fact_568_prop__restrict,axiom,
! [X2: extended_ereal,Z5: set_Extended_ereal,X5: set_Extended_ereal,P2: extended_ereal > $o] :
( ( member2350847679896131959_ereal @ X2 @ Z5 )
=> ( ( ord_le1644982726543182158_ereal @ Z5
@ ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ X5 )
& ( P2 @ X ) ) ) )
=> ( P2 @ X2 ) ) ) ).
% prop_restrict
thf(fact_569_sup__Un__eq,axiom,
! [R2: set_a,S4: set_a] :
( ( sup_sup_a_o
@ ^ [X: a] : ( member_a @ X @ R2 )
@ ^ [X: a] : ( member_a @ X @ S4 ) )
= ( ^ [X: a] : ( member_a @ X @ ( sup_sup_set_a @ R2 @ S4 ) ) ) ) ).
% sup_Un_eq
thf(fact_570_image__Collect__subsetI,axiom,
! [P2: extended_ereal > $o,F2: extended_ereal > extended_ereal,B5: set_Extended_ereal] :
( ! [X3: extended_ereal] :
( ( P2 @ X3 )
=> ( member2350847679896131959_ereal @ ( F2 @ X3 ) @ B5 ) )
=> ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F2 @ ( collec5835592288176408249_ereal @ P2 ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_571_image__Collect__subsetI,axiom,
! [P2: nat > $o,F2: nat > extended_ereal,B5: set_Extended_ereal] :
( ! [X3: nat] :
( ( P2 @ X3 )
=> ( member2350847679896131959_ereal @ ( F2 @ X3 ) @ B5 ) )
=> ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F2 @ ( collect_nat @ P2 ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_572_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B7: set_a] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A3 )
| ( member_a @ X @ B7 ) ) ) ) ) ).
% Un_def
thf(fact_573_sup__set__def,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B7: set_a] :
( collect_a
@ ( sup_sup_a_o
@ ^ [X: a] : ( member_a @ X @ A3 )
@ ^ [X: a] : ( member_a @ X @ B7 ) ) ) ) ) ).
% sup_set_def
thf(fact_574_Inf_OINF__identity__eq,axiom,
! [Inf: set_Extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( Inf
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : X
@ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_identity_eq
thf(fact_575_Sup_OSUP__identity__eq,axiom,
! [Sup: set_Extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( Sup
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : X
@ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_identity_eq
thf(fact_576_Compr__image__eq,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,P2: extended_ereal > $o] :
( ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ ( image_6042159593519690757_ereal @ F2 @ A ) )
& ( P2 @ X ) ) )
= ( image_6042159593519690757_ereal @ F2
@ ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A )
& ( P2 @ ( F2 @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_577_Compr__image__eq,axiom,
! [F2: nat > extended_ereal,A: set_nat,P2: extended_ereal > $o] :
( ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ ( image_4309273772856505399_ereal @ F2 @ A ) )
& ( P2 @ X ) ) )
= ( image_4309273772856505399_ereal @ F2
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P2 @ ( F2 @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_578_Compr__image__eq,axiom,
! [F2: a > a,A: set_a,P2: a > $o] :
( ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( image_a_a @ F2 @ A ) )
& ( P2 @ X ) ) )
= ( image_a_a @ F2
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A )
& ( P2 @ ( F2 @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_579_image__image,axiom,
! [F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( image_6042159593519690757_ereal @ F2 @ ( image_6042159593519690757_ereal @ G @ A ) )
= ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : ( F2 @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_580_image__image,axiom,
! [F2: extended_ereal > extended_ereal,G: nat > extended_ereal,A: set_nat] :
( ( image_6042159593519690757_ereal @ F2 @ ( image_4309273772856505399_ereal @ G @ A ) )
= ( image_4309273772856505399_ereal
@ ^ [X: nat] : ( F2 @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_581_image__image,axiom,
! [F2: nat > extended_ereal,G: extended_ereal > nat,A: set_Extended_ereal] :
( ( image_4309273772856505399_ereal @ F2 @ ( image_7659842161140344153al_nat @ G @ A ) )
= ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : ( F2 @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_582_image__image,axiom,
! [F2: nat > extended_ereal,G: nat > nat,A: set_nat] :
( ( image_4309273772856505399_ereal @ F2 @ ( image_nat_nat @ G @ A ) )
= ( image_4309273772856505399_ereal
@ ^ [X: nat] : ( F2 @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_583_imageE,axiom,
! [B: extended_ereal,F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ B @ ( image_6042159593519690757_ereal @ F2 @ A ) )
=> ~ ! [X3: extended_ereal] :
( ( B
= ( F2 @ X3 ) )
=> ~ ( member2350847679896131959_ereal @ X3 @ A ) ) ) ).
% imageE
thf(fact_584_imageE,axiom,
! [B: extended_ereal,F2: nat > extended_ereal,A: set_nat] :
( ( member2350847679896131959_ereal @ B @ ( image_4309273772856505399_ereal @ F2 @ A ) )
=> ~ ! [X3: nat] :
( ( B
= ( F2 @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_585_imageE,axiom,
! [B: a,F2: a > a,A: set_a] :
( ( member_a @ B @ ( image_a_a @ F2 @ A ) )
=> ~ ! [X3: a] :
( ( B
= ( F2 @ X3 ) )
=> ~ ( member_a @ X3 @ A ) ) ) ).
% imageE
thf(fact_586_rangeE,axiom,
! [B: extended_ereal,F2: extended_ereal > extended_ereal] :
( ( member2350847679896131959_ereal @ B @ ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal ) )
=> ~ ! [X3: extended_ereal] :
( B
!= ( F2 @ X3 ) ) ) ).
% rangeE
thf(fact_587_rangeE,axiom,
! [B: a,F2: extended_ereal > a] :
( ( member_a @ B @ ( image_3724615099042636213real_a @ F2 @ top_to5683747375963461374_ereal ) )
=> ~ ! [X3: extended_ereal] :
( B
!= ( F2 @ X3 ) ) ) ).
% rangeE
thf(fact_588_rangeE,axiom,
! [B: extended_ereal,F2: nat > extended_ereal] :
( ( member2350847679896131959_ereal @ B @ ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat ) )
=> ~ ! [X3: nat] :
( B
!= ( F2 @ X3 ) ) ) ).
% rangeE
thf(fact_589_rangeE,axiom,
! [B: a,F2: nat > a] :
( ( member_a @ B @ ( image_nat_a @ F2 @ top_top_set_nat ) )
=> ~ ! [X3: nat] :
( B
!= ( F2 @ X3 ) ) ) ).
% rangeE
thf(fact_590_range__composition,axiom,
! [F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal] :
( ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : ( F2 @ ( G @ X ) )
@ top_to5683747375963461374_ereal )
= ( image_6042159593519690757_ereal @ F2 @ ( image_6042159593519690757_ereal @ G @ top_to5683747375963461374_ereal ) ) ) ).
% range_composition
thf(fact_591_range__composition,axiom,
! [F2: nat > extended_ereal,G: extended_ereal > nat] :
( ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : ( F2 @ ( G @ X ) )
@ top_to5683747375963461374_ereal )
= ( image_4309273772856505399_ereal @ F2 @ ( image_7659842161140344153al_nat @ G @ top_to5683747375963461374_ereal ) ) ) ).
% range_composition
thf(fact_592_range__composition,axiom,
! [F2: extended_ereal > extended_ereal,G: nat > extended_ereal] :
( ( image_4309273772856505399_ereal
@ ^ [X: nat] : ( F2 @ ( G @ X ) )
@ top_top_set_nat )
= ( image_6042159593519690757_ereal @ F2 @ ( image_4309273772856505399_ereal @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_593_range__composition,axiom,
! [F2: nat > extended_ereal,G: nat > nat] :
( ( image_4309273772856505399_ereal
@ ^ [X: nat] : ( F2 @ ( G @ X ) )
@ top_top_set_nat )
= ( image_4309273772856505399_ereal @ F2 @ ( image_nat_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_594_pred__equals__eq2,axiom,
! [R2: set_Pr7497825696620840711list_o,S4: set_Pr7497825696620840711list_o] :
( ( ( ^ [X: option_list_o,Y: option_list_o] : ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X @ Y ) @ R2 ) )
= ( ^ [X: option_list_o,Y: option_list_o] : ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X @ Y ) @ S4 ) ) )
= ( R2 = S4 ) ) ).
% pred_equals_eq2
thf(fact_595_inj__on__id2,axiom,
! [A: set_Extended_ereal] :
( inj_on7162434037990268785_ereal
@ ^ [X: extended_ereal] : X
@ A ) ).
% inj_on_id2
thf(fact_596_top__empty__eq2,axiom,
( top_to2083282274478383934st_o_o
= ( ^ [X: option_list_o,Y: option_list_o] : ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X @ Y ) @ top_to4710185308193272919list_o ) ) ) ).
% top_empty_eq2
thf(fact_597_sup__Un__eq2,axiom,
! [R2: set_Pr7497825696620840711list_o,S4: set_Pr7497825696620840711list_o] :
( ( sup_su7401349752647199682st_o_o
@ ^ [X: option_list_o,Y: option_list_o] : ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X @ Y ) @ R2 )
@ ^ [X: option_list_o,Y: option_list_o] : ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X @ Y ) @ S4 ) )
= ( ^ [X: option_list_o,Y: option_list_o] : ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X @ Y ) @ ( sup_su2413444806983345371list_o @ R2 @ S4 ) ) ) ) ).
% sup_Un_eq2
thf(fact_598_inj__on__convol__ident,axiom,
! [F2: option_list_o > option_list_o,X5: set_option_list_o] :
( inj_on408740241426942132list_o
@ ^ [X: option_list_o] : ( produc5745850523778858007list_o @ X @ ( F2 @ X ) )
@ X5 ) ).
% inj_on_convol_ident
thf(fact_599_Collect__imp__eq,axiom,
! [P2: extended_ereal > $o,Q2: extended_ereal > $o] :
( ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( P2 @ X )
=> ( Q2 @ X ) ) )
= ( sup_su2680283192902082946_ereal @ ( uminus5895154729394068773_ereal @ ( collec5835592288176408249_ereal @ P2 ) ) @ ( collec5835592288176408249_ereal @ Q2 ) ) ) ).
% Collect_imp_eq
thf(fact_600_pred__subset__eq2,axiom,
! [R2: set_Pr7497825696620840711list_o,S4: set_Pr7497825696620840711list_o] :
( ( ord_le2124072808290184590st_o_o
@ ^ [X: option_list_o,Y: option_list_o] : ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X @ Y ) @ R2 )
@ ^ [X: option_list_o,Y: option_list_o] : ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X @ Y ) @ S4 ) )
= ( ord_le7655427003059986087list_o @ R2 @ S4 ) ) ).
% pred_subset_eq2
thf(fact_601_swap__inj__on,axiom,
! [A: set_Pr7497825696620840711list_o] :
( inj_on2117460296820009777list_o
@ ( produc1290809171805591803list_o
@ ^ [I2: option_list_o,J: option_list_o] : ( produc5745850523778858007list_o @ J @ I2 ) )
@ A ) ).
% swap_inj_on
thf(fact_602_predicate2__transferD,axiom,
! [R1: a > b > $o,R22: a > b > $o,P2: a > a > $o,Q2: b > b > $o,A4: product_prod_a_b,A: set_Product_prod_a_b,B: product_prod_a_b,B5: set_Product_prod_a_b] :
( ( bNF_re5830743871565202077_o_b_o @ R1
@ ( bNF_rel_fun_a_b_o_o @ R22
@ ^ [Y4: $o,Z: $o] : ( Y4 = Z ) )
@ P2
@ Q2 )
=> ( ( member1426531481828664017od_a_b @ A4 @ A )
=> ( ( member1426531481828664017od_a_b @ B @ B5 )
=> ( ( ord_le817736998455962536od_a_b @ A @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R1 ) ) )
=> ( ( ord_le817736998455962536od_a_b @ B5 @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R22 ) ) )
=> ( ( P2 @ ( product_fst_a_b @ A4 ) @ ( product_fst_a_b @ B ) )
= ( Q2 @ ( product_snd_a_b @ A4 ) @ ( product_snd_a_b @ B ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_603_fst__diag__id,axiom,
! [Z2: option_list_o] :
( ( comp_P3073798813330040196list_o @ produc8474152268468583939list_o
@ ^ [X: option_list_o] : ( produc5745850523778858007list_o @ X @ X )
@ Z2 )
= ( id_option_list_o @ Z2 ) ) ).
% fst_diag_id
thf(fact_604_snd__diag__id,axiom,
! [Z2: option_list_o] :
( ( comp_P3073798813330040196list_o @ produc3339157721029416773list_o
@ ^ [X: option_list_o] : ( produc5745850523778858007list_o @ X @ X )
@ Z2 )
= ( id_option_list_o @ Z2 ) ) ).
% snd_diag_id
thf(fact_605_fst__diag__fst,axiom,
( ( comp_P6892385181472540974od_a_b @ product_fst_a_a
@ ( comp_a1036870397537576092od_a_b
@ ^ [X: a] : ( product_Pair_a_a @ X @ X )
@ product_fst_a_b ) )
= product_fst_a_b ) ).
% fst_diag_fst
thf(fact_606_snd__diag__snd,axiom,
( ( comp_P5715990135234274861od_a_b @ product_snd_b_b
@ ( comp_b6573281471806798941od_a_b
@ ^ [X: b] : ( product_Pair_b_b @ X @ X )
@ product_snd_a_b ) )
= product_snd_a_b ) ).
% snd_diag_snd
thf(fact_607_If__the__inv__into__f__f,axiom,
! [I: nat,C2: set_nat,G: nat > extended_ereal,X2: nat] :
( ( member_nat @ I @ C2 )
=> ( ( inj_on6191532827271902155_ereal @ G @ C2 )
=> ( ( comp_E7502005551946643277at_nat
@ ^ [I2: extended_ereal] : ( if_nat @ ( member2350847679896131959_ereal @ I2 @ ( image_4309273772856505399_ereal @ G @ C2 ) ) @ ( the_in5959796611709155849_ereal @ C2 @ G @ I2 ) @ X2 )
@ G
@ I )
= ( id_nat @ I ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_608_If__the__inv__into__f__f,axiom,
! [I: extended_ereal,C2: set_Extended_ereal,G: extended_ereal > extended_ereal,X2: extended_ereal] :
( ( member2350847679896131959_ereal @ I @ C2 )
=> ( ( inj_on7162434037990268785_ereal @ G @ C2 )
=> ( ( comp_E9177254828515427499_ereal
@ ^ [I2: extended_ereal] : ( if_Extended_ereal @ ( member2350847679896131959_ereal @ I2 @ ( image_6042159593519690757_ereal @ G @ C2 ) ) @ ( the_in1141389326992810419_ereal @ C2 @ G @ I2 ) @ X2 )
@ G
@ I )
= ( id_Extended_ereal @ I ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_609_If__the__inv__into__f__f,axiom,
! [I: a,C2: set_a,G: a > a,X2: a] :
( ( member_a @ I @ C2 )
=> ( ( inj_on_a_a @ G @ C2 )
=> ( ( comp_a_a_a
@ ^ [I2: a] : ( if_a @ ( member_a @ I2 @ ( image_a_a @ G @ C2 ) ) @ ( the_inv_into_a_a @ C2 @ G @ I2 ) @ X2 )
@ G
@ I )
= ( id_a @ I ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_610_snd__fst__flip,axiom,
( produc3339157721029416773list_o
= ( comp_P3188503004689993927list_o @ produc8474152268468583939list_o
@ ( produc1290809171805591803list_o
@ ^ [X: option_list_o,Y: option_list_o] : ( produc5745850523778858007list_o @ Y @ X ) ) ) ) ).
% snd_fst_flip
thf(fact_611_snd__fst__flip,axiom,
( product_snd_b_a
= ( comp_P9084872743345139181od_b_a @ product_fst_a_b
@ ( produc4348216232050026237od_a_b
@ ^ [X: b,Y: a] : ( product_Pair_a_b @ Y @ X ) ) ) ) ).
% snd_fst_flip
thf(fact_612_snd__fst__flip,axiom,
( product_snd_a_b
= ( comp_P736046987085771820od_a_b @ product_fst_b_a
@ ( produc6204743795672244857od_b_a
@ ^ [X: a,Y: b] : ( product_Pair_b_a @ Y @ X ) ) ) ) ).
% snd_fst_flip
thf(fact_613_fst__snd__flip,axiom,
( produc8474152268468583939list_o
= ( comp_P3188503004689993927list_o @ produc3339157721029416773list_o
@ ( produc1290809171805591803list_o
@ ^ [X: option_list_o,Y: option_list_o] : ( produc5745850523778858007list_o @ Y @ X ) ) ) ) ).
% fst_snd_flip
thf(fact_614_fst__snd__flip,axiom,
( product_fst_a_b
= ( comp_P1375487287417587949od_a_b @ product_snd_b_a
@ ( produc6204743795672244857od_b_a
@ ^ [X: a,Y: b] : ( product_Pair_b_a @ Y @ X ) ) ) ) ).
% fst_snd_flip
thf(fact_615_fst__snd__flip,axiom,
( product_fst_b_a
= ( comp_P8445432443013323052od_b_a @ product_snd_a_b
@ ( produc4348216232050026237od_a_b
@ ^ [X: b,Y: a] : ( product_Pair_a_b @ Y @ X ) ) ) ) ).
% fst_snd_flip
thf(fact_616_snd__diag__fst,axiom,
( ( comp_P6892385181472540974od_a_b @ product_snd_a_a
@ ( comp_a1036870397537576092od_a_b
@ ^ [X: a] : ( product_Pair_a_a @ X @ X )
@ product_fst_a_b ) )
= product_fst_a_b ) ).
% snd_diag_fst
thf(fact_617_fst__diag__snd,axiom,
( ( comp_P5715990135234274861od_a_b @ product_fst_b_b
@ ( comp_b6573281471806798941od_a_b
@ ^ [X: b] : ( product_Pair_b_b @ X @ X )
@ product_snd_a_b ) )
= product_snd_a_b ) ).
% fst_diag_snd
thf(fact_618_bijection_Osurj,axiom,
! [F2: extended_ereal > extended_ereal] :
( ( hilber6088754731438466237_ereal @ F2 )
=> ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal ) ) ).
% bijection.surj
thf(fact_619_bijection_Osurj,axiom,
! [F2: nat > nat] :
( ( hilber5277034221543178913on_nat @ F2 )
=> ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat ) ) ).
% bijection.surj
thf(fact_620_flip__pred,axiom,
! [A: set_Pr7497825696620840711list_o,R2: option_list_o > option_list_o > $o] :
( ( ord_le7655427003059986087list_o @ A @ ( collec8383569672281745810list_o @ ( produc2885628514414960748st_o_o @ ( conver6027910057834070645list_o @ R2 ) ) ) )
=> ( ord_le7655427003059986087list_o
@ ( image_8752170789471566789list_o
@ ( produc1290809171805591803list_o
@ ^ [X: option_list_o,Y: option_list_o] : ( produc5745850523778858007list_o @ Y @ X ) )
@ A )
@ ( collec8383569672281745810list_o @ ( produc2885628514414960748st_o_o @ R2 ) ) ) ) ).
% flip_pred
thf(fact_621_bijection_Oinj,axiom,
! [F2: extended_ereal > extended_ereal] :
( ( hilber6088754731438466237_ereal @ F2 )
=> ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal ) ) ).
% bijection.inj
thf(fact_622_bijection_Oinj,axiom,
! [F2: nat > nat] :
( ( hilber5277034221543178913on_nat @ F2 )
=> ( inj_on_nat_nat @ F2 @ top_top_set_nat ) ) ).
% bijection.inj
thf(fact_623_sorted__list__of__set_Oinj__on,axiom,
( inj_on7162434037990268785_ereal
@ ^ [X: extended_ereal] : X
@ top_to5683747375963461374_ereal ) ).
% sorted_list_of_set.inj_on
thf(fact_624_sorted__list__of__set_Oinj__on,axiom,
( inj_on_nat_nat
@ ^ [X: nat] : X
@ top_top_set_nat ) ).
% sorted_list_of_set.inj_on
thf(fact_625_inj__on__image__Fpow,axiom,
! [F2: nat > extended_ereal,A: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ A )
=> ( inj_on1463964778812548213_ereal @ ( image_4309273772856505399_ereal @ F2 ) @ ( finite_Fpow_nat @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_626_inj__on__image__Fpow,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A )
=> ( inj_on5406440306785145713_ereal @ ( image_6042159593519690757_ereal @ F2 ) @ ( finite2137394461708460340_ereal @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_627_graph__eq__to__snd__dom,axiom,
( graph_a_list_o
= ( ^ [M4: a > option_list_o] :
( image_1329905666413968462list_o
@ ^ [X: a] : ( produc6899623729582506167list_o @ X @ ( the_list_o @ ( M4 @ X ) ) )
@ ( dom_a_list_o @ M4 ) ) ) ) ).
% graph_eq_to_snd_dom
thf(fact_628_graph__eq__to__snd__dom,axiom,
( graph_1138927715465675390list_o
= ( ^ [M4: option_list_o > option_option_list_o] :
( image_8264744964252371784list_o
@ ^ [X: option_list_o] : ( produc5745850523778858007list_o @ X @ ( the_option_list_o @ ( M4 @ X ) ) )
@ ( dom_op7091944507009107026list_o @ M4 ) ) ) ) ).
% graph_eq_to_snd_dom
thf(fact_629_inj__on__apsnd,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( inj_on7007621943451069233_ereal @ ( produc5246002025674687312_ereal @ F2 )
@ ( produc8095709571603465288_ereal @ top_to5683747375963461374_ereal
@ ^ [Uu: extended_ereal] : A ) )
= ( inj_on7162434037990268785_ereal @ F2 @ A ) ) ).
% inj_on_apsnd
thf(fact_630_inj__on__apsnd,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( inj_on113668994391194201_ereal @ ( produc6756617771423349966al_nat @ F2 )
@ ( produc870331913724930228_ereal @ top_top_set_nat
@ ^ [Uu: nat] : A ) )
= ( inj_on7162434037990268785_ereal @ F2 @ A ) ) ).
% inj_on_apsnd
thf(fact_631_inj__on__apfst,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( inj_on7007621943451069233_ereal @ ( produc7707886444459085074_ereal @ F2 )
@ ( produc8095709571603465288_ereal @ A
@ ^ [Uu: extended_ereal] : top_to5683747375963461374_ereal ) )
= ( inj_on7162434037990268785_ereal @ F2 @ A ) ) ).
% inj_on_apfst
thf(fact_632_inj__on__apfst,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( inj_on3854950389080018957al_nat @ ( produc6531249619332995980al_nat @ F2 )
@ ( produc4220900302008768982al_nat @ A
@ ^ [Uu: extended_ereal] : top_top_set_nat ) )
= ( inj_on7162434037990268785_ereal @ F2 @ A ) ) ).
% inj_on_apfst
thf(fact_633_mem__Sigma__iff,axiom,
! [A4: a,B: a,A: set_a,B5: a > set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ B ) @ ( product_Sigma_a_a @ A @ B5 ) )
= ( ( member_a @ A4 @ A )
& ( member_a @ B @ ( B5 @ A4 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_634_mem__Sigma__iff,axiom,
! [A4: option_list_o,B: option_list_o,A: set_option_list_o,B5: option_list_o > set_option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ A4 @ B ) @ ( produc7548132086704134856list_o @ A @ B5 ) )
= ( ( member_option_list_o @ A4 @ A )
& ( member_option_list_o @ B @ ( B5 @ A4 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_635_SigmaI,axiom,
! [A4: a,A: set_a,B: a,B5: a > set_a] :
( ( member_a @ A4 @ A )
=> ( ( member_a @ B @ ( B5 @ A4 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ B ) @ ( product_Sigma_a_a @ A @ B5 ) ) ) ) ).
% SigmaI
thf(fact_636_SigmaI,axiom,
! [A4: option_list_o,A: set_option_list_o,B: option_list_o,B5: option_list_o > set_option_list_o] :
( ( member_option_list_o @ A4 @ A )
=> ( ( member_option_list_o @ B @ ( B5 @ A4 ) )
=> ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ A4 @ B ) @ ( produc7548132086704134856list_o @ A @ B5 ) ) ) ) ).
% SigmaI
thf(fact_637_Compl__Times__UNIV2,axiom,
! [A: set_Extended_ereal] :
( ( uminus6085167613793993086_ereal
@ ( produc8095709571603465288_ereal @ A
@ ^ [Uu: extended_ereal] : top_to5683747375963461374_ereal ) )
= ( produc8095709571603465288_ereal @ ( uminus5895154729394068773_ereal @ A )
@ ^ [Uu: extended_ereal] : top_to5683747375963461374_ereal ) ) ).
% Compl_Times_UNIV2
thf(fact_638_Compl__Times__UNIV2,axiom,
! [A: set_Extended_ereal] :
( ( uminus6262216577377593932al_nat
@ ( produc4220900302008768982al_nat @ A
@ ^ [Uu: extended_ereal] : top_top_set_nat ) )
= ( produc4220900302008768982al_nat @ ( uminus5895154729394068773_ereal @ A )
@ ^ [Uu: extended_ereal] : top_top_set_nat ) ) ).
% Compl_Times_UNIV2
thf(fact_639_Compl__Times__UNIV1,axiom,
! [A: set_Extended_ereal] :
( ( uminus6085167613793993086_ereal
@ ( produc8095709571603465288_ereal @ top_to5683747375963461374_ereal
@ ^ [Uu: extended_ereal] : A ) )
= ( produc8095709571603465288_ereal @ top_to5683747375963461374_ereal
@ ^ [Uu: extended_ereal] : ( uminus5895154729394068773_ereal @ A ) ) ) ).
% Compl_Times_UNIV1
thf(fact_640_Compl__Times__UNIV1,axiom,
! [A: set_Extended_ereal] :
( ( uminus4999475943122058226_ereal
@ ( produc870331913724930228_ereal @ top_top_set_nat
@ ^ [Uu: nat] : A ) )
= ( produc870331913724930228_ereal @ top_top_set_nat
@ ^ [Uu: nat] : ( uminus5895154729394068773_ereal @ A ) ) ) ).
% Compl_Times_UNIV1
thf(fact_641_UNIV__Times__UNIV,axiom,
( ( produc8095709571603465288_ereal @ top_to5683747375963461374_ereal
@ ^ [Uu: extended_ereal] : top_to5683747375963461374_ereal )
= top_to3798671025730093271_ereal ) ).
% UNIV_Times_UNIV
thf(fact_642_UNIV__Times__UNIV,axiom,
( ( produc4220900302008768982al_nat @ top_to5683747375963461374_ereal
@ ^ [Uu: extended_ereal] : top_top_set_nat )
= top_to7896853287916821811al_nat ) ).
% UNIV_Times_UNIV
thf(fact_643_UNIV__Times__UNIV,axiom,
( ( produc870331913724930228_ereal @ top_top_set_nat
@ ^ [Uu: nat] : top_to5683747375963461374_ereal )
= top_to6634112653661286105_ereal ) ).
% UNIV_Times_UNIV
thf(fact_644_UNIV__Times__UNIV,axiom,
( ( produc457027306803732586at_nat @ top_top_set_nat
@ ^ [Uu: nat] : top_top_set_nat )
= top_to4669805908274784177at_nat ) ).
% UNIV_Times_UNIV
thf(fact_645_SigmaE,axiom,
! [C: product_prod_a_a,A: set_a,B5: a > set_a] :
( ( member1426531477525435216od_a_a @ C @ ( product_Sigma_a_a @ A @ B5 ) )
=> ~ ! [X3: a] :
( ( member_a @ X3 @ A )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( B5 @ X3 ) )
=> ( C
!= ( product_Pair_a_a @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_646_SigmaE,axiom,
! [C: produc4882884732533091879list_o,A: set_option_list_o,B5: option_list_o > set_option_list_o] :
( ( member1589324699396745552list_o @ C @ ( produc7548132086704134856list_o @ A @ B5 ) )
=> ~ ! [X3: option_list_o] :
( ( member_option_list_o @ X3 @ A )
=> ! [Y3: option_list_o] :
( ( member_option_list_o @ Y3 @ ( B5 @ X3 ) )
=> ( C
!= ( produc5745850523778858007list_o @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_647_SigmaD1,axiom,
! [A4: option_list_o,B: option_list_o,A: set_option_list_o,B5: option_list_o > set_option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ A4 @ B ) @ ( produc7548132086704134856list_o @ A @ B5 ) )
=> ( member_option_list_o @ A4 @ A ) ) ).
% SigmaD1
thf(fact_648_SigmaD2,axiom,
! [A4: option_list_o,B: option_list_o,A: set_option_list_o,B5: option_list_o > set_option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ A4 @ B ) @ ( produc7548132086704134856list_o @ A @ B5 ) )
=> ( member_option_list_o @ B @ ( B5 @ A4 ) ) ) ).
% SigmaD2
thf(fact_649_SigmaE2,axiom,
! [A4: a,B: a,A: set_a,B5: a > set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ B ) @ ( product_Sigma_a_a @ A @ B5 ) )
=> ~ ( ( member_a @ A4 @ A )
=> ~ ( member_a @ B @ ( B5 @ A4 ) ) ) ) ).
% SigmaE2
thf(fact_650_SigmaE2,axiom,
! [A4: option_list_o,B: option_list_o,A: set_option_list_o,B5: option_list_o > set_option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ A4 @ B ) @ ( produc7548132086704134856list_o @ A @ B5 ) )
=> ~ ( ( member_option_list_o @ A4 @ A )
=> ~ ( member_option_list_o @ B @ ( B5 @ A4 ) ) ) ) ).
% SigmaE2
thf(fact_651_Sigma__mono,axiom,
! [A: set_a,C2: set_a,B5: a > set_Extended_ereal,D: a > set_Extended_ereal] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ord_le1644982726543182158_ereal @ ( B5 @ X3 ) @ ( D @ X3 ) ) )
=> ( ord_le8132973195874727159_ereal @ ( produc7264369102385879704_ereal @ A @ B5 ) @ ( produc7264369102385879704_ereal @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_652_Sigma__mono,axiom,
! [A: set_Extended_ereal,C2: set_Extended_ereal,B5: extended_ereal > set_Extended_ereal,D: extended_ereal > set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A @ C2 )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
=> ( ord_le1644982726543182158_ereal @ ( B5 @ X3 ) @ ( D @ X3 ) ) )
=> ( ord_le8239133294219471655_ereal @ ( produc8095709571603465288_ereal @ A @ B5 ) @ ( produc8095709571603465288_ereal @ C2 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_653_Times__subset__cancel2,axiom,
! [X2: a,C2: set_a,A: set_Extended_ereal,B5: set_Extended_ereal] :
( ( member_a @ X2 @ C2 )
=> ( ( ord_le7790192137282702039real_a
@ ( produc2583502849437520504real_a @ A
@ ^ [Uu: extended_ereal] : C2 )
@ ( produc2583502849437520504real_a @ B5
@ ^ [Uu: extended_ereal] : C2 ) )
= ( ord_le1644982726543182158_ereal @ A @ B5 ) ) ) ).
% Times_subset_cancel2
thf(fact_654_mem__Times__iff,axiom,
! [X2: product_prod_a_a,A: set_a,B5: set_a] :
( ( member1426531477525435216od_a_a @ X2
@ ( product_Sigma_a_a @ A
@ ^ [Uu: a] : B5 ) )
= ( ( member_a @ ( product_fst_a_a @ X2 ) @ A )
& ( member_a @ ( product_snd_a_a @ X2 ) @ B5 ) ) ) ).
% mem_Times_iff
thf(fact_655_mem__Times__iff,axiom,
! [X2: product_prod_a_b,A: set_a,B5: set_b] :
( ( member1426531481828664017od_a_b @ X2
@ ( product_Sigma_a_b @ A
@ ^ [Uu: a] : B5 ) )
= ( ( member_a @ ( product_fst_a_b @ X2 ) @ A )
& ( member_b @ ( product_snd_a_b @ X2 ) @ B5 ) ) ) ).
% mem_Times_iff
thf(fact_656_swap__product,axiom,
! [A: set_option_list_o,B5: set_option_list_o] :
( ( image_8752170789471566789list_o
@ ( produc1290809171805591803list_o
@ ^ [I2: option_list_o,J: option_list_o] : ( produc5745850523778858007list_o @ J @ I2 ) )
@ ( produc7548132086704134856list_o @ A
@ ^ [Uu: option_list_o] : B5 ) )
= ( produc7548132086704134856list_o @ B5
@ ^ [Uu: option_list_o] : A ) ) ).
% swap_product
thf(fact_657_map__prod__surj__on,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,A8: set_Extended_ereal,G: extended_ereal > extended_ereal,B5: set_Extended_ereal,B8: set_Extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ A )
= A8 )
=> ( ( ( image_6042159593519690757_ereal @ G @ B5 )
= B8 )
=> ( ( image_959328165755419589_ereal @ ( produc7788783332699689718_ereal @ F2 @ G )
@ ( produc8095709571603465288_ereal @ A
@ ^ [Uu: extended_ereal] : B5 ) )
= ( produc8095709571603465288_ereal @ A8
@ ^ [Uu: extended_ereal] : B8 ) ) ) ) ).
% map_prod_surj_on
thf(fact_658_map__prod__surj__on,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,A8: set_Extended_ereal,G: nat > extended_ereal,B5: set_nat,B8: set_Extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ A )
= A8 )
=> ( ( ( image_4309273772856505399_ereal @ G @ B5 )
= B8 )
=> ( ( image_6926512632986509267_ereal @ ( produc5260569924724313478_ereal @ F2 @ G )
@ ( produc4220900302008768982al_nat @ A
@ ^ [Uu: extended_ereal] : B5 ) )
= ( produc8095709571603465288_ereal @ A8
@ ^ [Uu: extended_ereal] : B8 ) ) ) ) ).
% map_prod_surj_on
thf(fact_659_map__prod__surj__on,axiom,
! [F2: nat > extended_ereal,A: set_nat,A8: set_Extended_ereal,G: extended_ereal > extended_ereal,B5: set_Extended_ereal,B8: set_Extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F2 @ A )
= A8 )
=> ( ( ( image_6042159593519690757_ereal @ G @ B5 )
= B8 )
=> ( ( image_3845515196057536685_ereal @ ( produc7290494158911329542_ereal @ F2 @ G )
@ ( produc870331913724930228_ereal @ A
@ ^ [Uu: nat] : B5 ) )
= ( produc8095709571603465288_ereal @ A8
@ ^ [Uu: extended_ereal] : B8 ) ) ) ) ).
% map_prod_surj_on
thf(fact_660_map__prod__surj__on,axiom,
! [F2: nat > extended_ereal,A: set_nat,A8: set_Extended_ereal,G: nat > extended_ereal,B5: set_nat,B8: set_Extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F2 @ A )
= A8 )
=> ( ( ( image_4309273772856505399_ereal @ G @ B5 )
= B8 )
=> ( ( image_7762317921424436459_ereal @ ( produc578423587001601910_ereal @ F2 @ G )
@ ( produc457027306803732586at_nat @ A
@ ^ [Uu: nat] : B5 ) )
= ( produc8095709571603465288_ereal @ A8
@ ^ [Uu: extended_ereal] : B8 ) ) ) ) ).
% map_prod_surj_on
thf(fact_661_image__Fpow__mono,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,B5: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) @ B5 )
=> ( ord_le5287700718633833262_ereal @ ( image_6293272304431515653_ereal @ ( image_6042159593519690757_ereal @ F2 ) @ ( finite2137394461708460340_ereal @ A ) ) @ ( finite2137394461708460340_ereal @ B5 ) ) ) ).
% image_Fpow_mono
thf(fact_662_image__Fpow__mono,axiom,
! [F2: nat > extended_ereal,A: set_nat,B5: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) @ B5 )
=> ( ord_le5287700718633833262_ereal @ ( image_8825259783980156129_ereal @ ( image_4309273772856505399_ereal @ F2 ) @ ( finite_Fpow_nat @ A ) ) @ ( finite2137394461708460340_ereal @ B5 ) ) ) ).
% image_Fpow_mono
thf(fact_663_map__prod__inj__on,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,G: extended_ereal > extended_ereal,B5: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A )
=> ( ( inj_on7162434037990268785_ereal @ G @ B5 )
=> ( inj_on7007621943451069233_ereal @ ( produc7788783332699689718_ereal @ F2 @ G )
@ ( produc8095709571603465288_ereal @ A
@ ^ [Uu: extended_ereal] : B5 ) ) ) ) ).
% map_prod_inj_on
thf(fact_664_subset__fst__imageI,axiom,
! [A: set_a,B5: set_b,S4: set_Product_prod_a_b,Y2: b] :
( ( ord_le817736998455962536od_a_b
@ ( product_Sigma_a_b @ A
@ ^ [Uu: a] : B5 )
@ S4 )
=> ( ( member_b @ Y2 @ B5 )
=> ( ord_less_eq_set_a @ A @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ S4 ) ) ) ) ).
% subset_fst_imageI
thf(fact_665_subset__fst__imageI,axiom,
! [A: set_Extended_ereal,B5: set_a,S4: set_Pr594667477129265975real_a,Y2: a] :
( ( ord_le7790192137282702039real_a
@ ( produc2583502849437520504real_a @ A
@ ^ [Uu: extended_ereal] : B5 )
@ S4 )
=> ( ( member_a @ Y2 @ B5 )
=> ( ord_le1644982726543182158_ereal @ A @ ( image_5786011877969985724_ereal @ produc7562294591749751347real_a @ S4 ) ) ) ) ).
% subset_fst_imageI
thf(fact_666_subset__snd__imageI,axiom,
! [A: set_a,B5: set_b,S4: set_Product_prod_a_b,X2: a] :
( ( ord_le817736998455962536od_a_b
@ ( product_Sigma_a_b @ A
@ ^ [Uu: a] : B5 )
@ S4 )
=> ( ( member_a @ X2 @ A )
=> ( ord_less_eq_set_b @ B5 @ ( image_2802296252294471260_a_b_b @ product_snd_a_b @ S4 ) ) ) ) ).
% subset_snd_imageI
thf(fact_667_subset__snd__imageI,axiom,
! [A: set_a,B5: set_Extended_ereal,S4: set_Pr937448535721291095_ereal,X2: a] :
( ( ord_le8132973195874727159_ereal
@ ( produc7264369102385879704_ereal @ A
@ ^ [Uu: a] : B5 )
@ S4 )
=> ( ( member_a @ X2 @ A )
=> ( ord_le1644982726543182158_ereal @ B5 @ ( image_5059165124122689180_ereal @ produc2480566465808596629_ereal @ S4 ) ) ) ) ).
% subset_snd_imageI
thf(fact_668_subset__fst__snd,axiom,
! [A: set_Product_prod_a_b] :
( ord_le817736998455962536od_a_b @ A
@ ( product_Sigma_a_b @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ A )
@ ^ [Uu: a] : ( image_2802296252294471260_a_b_b @ product_snd_a_b @ A ) ) ) ).
% subset_fst_snd
thf(fact_669_image__paired__Times,axiom,
! [F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal,A: set_Extended_ereal,B5: set_Extended_ereal] :
( ( image_959328165755419589_ereal
@ ( produc1215357860076780539_ereal
@ ^ [X: extended_ereal,Y: extended_ereal] : ( produc7614594614994623895_ereal @ ( F2 @ X ) @ ( G @ Y ) ) )
@ ( produc8095709571603465288_ereal @ A
@ ^ [Uu: extended_ereal] : B5 ) )
= ( produc8095709571603465288_ereal @ ( image_6042159593519690757_ereal @ F2 @ A )
@ ^ [Uu: extended_ereal] : ( image_6042159593519690757_ereal @ G @ B5 ) ) ) ).
% image_paired_Times
thf(fact_670_image__paired__Times,axiom,
! [F2: extended_ereal > extended_ereal,G: nat > extended_ereal,A: set_Extended_ereal,B5: set_nat] :
( ( image_6926512632986509267_ereal
@ ( produc976730521393319219_ereal
@ ^ [X: extended_ereal,Y: nat] : ( produc7614594614994623895_ereal @ ( F2 @ X ) @ ( G @ Y ) ) )
@ ( produc4220900302008768982al_nat @ A
@ ^ [Uu: extended_ereal] : B5 ) )
= ( produc8095709571603465288_ereal @ ( image_6042159593519690757_ereal @ F2 @ A )
@ ^ [Uu: extended_ereal] : ( image_4309273772856505399_ereal @ G @ B5 ) ) ) ).
% image_paired_Times
thf(fact_671_image__paired__Times,axiom,
! [F2: nat > extended_ereal,G: extended_ereal > extended_ereal,A: set_nat,B5: set_Extended_ereal] :
( ( image_3845515196057536685_ereal
@ ( produc7030432313249384917_ereal
@ ^ [X: nat,Y: extended_ereal] : ( produc7614594614994623895_ereal @ ( F2 @ X ) @ ( G @ Y ) ) )
@ ( produc870331913724930228_ereal @ A
@ ^ [Uu: nat] : B5 ) )
= ( produc8095709571603465288_ereal @ ( image_4309273772856505399_ereal @ F2 @ A )
@ ^ [Uu: extended_ereal] : ( image_6042159593519690757_ereal @ G @ B5 ) ) ) ).
% image_paired_Times
thf(fact_672_image__paired__Times,axiom,
! [F2: nat > extended_ereal,G: nat > extended_ereal,A: set_nat,B5: set_nat] :
( ( image_7762317921424436459_ereal
@ ( produc3135378909069315545_ereal
@ ^ [X: nat,Y: nat] : ( produc7614594614994623895_ereal @ ( F2 @ X ) @ ( G @ Y ) ) )
@ ( produc457027306803732586at_nat @ A
@ ^ [Uu: nat] : B5 ) )
= ( produc8095709571603465288_ereal @ ( image_4309273772856505399_ereal @ F2 @ A )
@ ^ [Uu: extended_ereal] : ( image_4309273772856505399_ereal @ G @ B5 ) ) ) ).
% image_paired_Times
thf(fact_673_Gr__incl,axiom,
! [A: set_Extended_ereal,F2: extended_ereal > extended_ereal,B5: set_Extended_ereal] :
( ( ord_le8239133294219471655_ereal @ ( bNF_Gr945911885070196386_ereal @ A @ F2 )
@ ( produc8095709571603465288_ereal @ A
@ ^ [Uu: extended_ereal] : B5 ) )
= ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) @ B5 ) ) ).
% Gr_incl
thf(fact_674_Gr__incl,axiom,
! [A: set_nat,F2: nat > extended_ereal,B5: set_Extended_ereal] :
( ( ord_le3920121919471048841_ereal @ ( bNF_Gr8352086143671499994_ereal @ A @ F2 )
@ ( produc870331913724930228_ereal @ A
@ ^ [Uu: nat] : B5 ) )
= ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) @ B5 ) ) ).
% Gr_incl
thf(fact_675_SUP__identity__eq,axiom,
! [A: set_Extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : X
@ A ) )
= ( comple8415311339701865915_ereal @ A ) ) ).
% SUP_identity_eq
thf(fact_676_UN__I,axiom,
! [A4: a,A: set_a,B: a,B5: a > set_a] :
( ( member_a @ A4 @ A )
=> ( ( member_a @ B @ ( B5 @ A4 ) )
=> ( member_a @ B @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B5 @ A ) ) ) ) ) ).
% UN_I
thf(fact_677_SUP__id__eq,axiom,
! [A: set_Extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ id_Extended_ereal @ A ) )
= ( comple8415311339701865915_ereal @ A ) ) ).
% SUP_id_eq
thf(fact_678_UN__E,axiom,
! [B: a,B5: a > set_a,A: set_a] :
( ( member_a @ B @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B5 @ A ) ) )
=> ~ ! [X3: a] :
( ( member_a @ X3 @ A )
=> ~ ( member_a @ B @ ( B5 @ X3 ) ) ) ) ).
% UN_E
thf(fact_679_SUP__UNION,axiom,
! [F2: extended_ereal > extended_ereal,G: extended_ereal > set_Extended_ereal,A: set_Extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ G @ A ) ) ) )
= ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [Y: extended_ereal] : ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ ( G @ Y ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_680_SUP__UNION,axiom,
! [F2: extended_ereal > extended_ereal,G: nat > set_Extended_ereal,A: set_nat] :
( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ ( comple4319282863272126363_ereal @ ( image_305533323056406039_ereal @ G @ A ) ) ) )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [Y: nat] : ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ ( G @ Y ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_681_SUP__UNION,axiom,
! [F2: nat > extended_ereal,G: extended_ereal > set_nat,A: set_Extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ ( comple7399068483239264473et_nat @ ( image_3090908713637162255et_nat @ G @ A ) ) ) )
= ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [Y: extended_ereal] : ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ ( G @ Y ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_682_SUP__UNION,axiom,
! [F2: nat > extended_ereal,G: nat > set_nat,A: set_nat] :
( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A ) ) ) )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [Y: nat] : ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ ( G @ Y ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_683_SUP__cong,axiom,
! [A: set_Extended_ereal,B5: set_Extended_ereal,C2: extended_ereal > extended_ereal,D: extended_ereal > extended_ereal] :
( ( A = B5 )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ B5 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ C2 @ A ) )
= ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ D @ B5 ) ) ) ) ) ).
% SUP_cong
thf(fact_684_SUP__cong,axiom,
! [A: set_nat,B5: set_nat,C2: nat > extended_ereal,D: nat > extended_ereal] :
( ( A = B5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B5 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ C2 @ A ) )
= ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ D @ B5 ) ) ) ) ) ).
% SUP_cong
thf(fact_685_SUP__cong,axiom,
! [A: set_a,B5: set_a,C2: a > extended_ereal,D: a > extended_ereal] :
( ( A = B5 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B5 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ C2 @ A ) )
= ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ D @ B5 ) ) ) ) ) ).
% SUP_cong
thf(fact_686_SUP__commute,axiom,
! [F2: extended_ereal > extended_ereal > extended_ereal,B5: set_Extended_ereal,A: set_Extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ ( F2 @ I2 ) @ B5 ) )
@ A ) )
= ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [J: extended_ereal] :
( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( F2 @ I2 @ J )
@ A ) )
@ B5 ) ) ) ).
% SUP_commute
thf(fact_687_SUP__commute,axiom,
! [F2: extended_ereal > nat > extended_ereal,B5: set_nat,A: set_Extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ ( F2 @ I2 ) @ B5 ) )
@ A ) )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [J: nat] :
( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( F2 @ I2 @ J )
@ A ) )
@ B5 ) ) ) ).
% SUP_commute
thf(fact_688_SUP__commute,axiom,
! [F2: nat > extended_ereal > extended_ereal,B5: set_Extended_ereal,A: set_nat] :
( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ ( F2 @ I2 ) @ B5 ) )
@ A ) )
= ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [J: extended_ereal] :
( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( F2 @ I2 @ J )
@ A ) )
@ B5 ) ) ) ).
% SUP_commute
thf(fact_689_SUP__commute,axiom,
! [F2: nat > nat > extended_ereal,B5: set_nat,A: set_nat] :
( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ ( F2 @ I2 ) @ B5 ) )
@ A ) )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [J: nat] :
( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( F2 @ I2 @ J )
@ A ) )
@ B5 ) ) ) ).
% SUP_commute
thf(fact_690_SUP__UNIV__bool__expand,axiom,
! [A: $o > extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_7729549296133164475_ereal @ A @ top_top_set_o ) )
= ( sup_su7653423775389492130_ereal @ ( A @ $true ) @ ( A @ $false ) ) ) ).
% SUP_UNIV_bool_expand
thf(fact_691_Union__natural,axiom,
! [F2: extended_ereal > extended_ereal] :
( ( comp_s3049225012970524235_ereal @ comple4319282863272126363_ereal @ ( image_6293272304431515653_ereal @ ( image_6042159593519690757_ereal @ F2 ) ) )
= ( comp_s23437526320367723_ereal @ ( image_6042159593519690757_ereal @ F2 ) @ comple4319282863272126363_ereal ) ) ).
% Union_natural
thf(fact_692_Union__natural,axiom,
! [F2: nat > extended_ereal] :
( ( comp_s7111974376712559807et_nat @ comple4319282863272126363_ereal @ ( image_8825259783980156129_ereal @ ( image_4309273772856505399_ereal @ F2 ) ) )
= ( comp_s7602107544143488833et_nat @ ( image_4309273772856505399_ereal @ F2 ) @ comple7399068483239264473et_nat ) ) ).
% Union_natural
thf(fact_693_image__Union,axiom,
! [F2: extended_ereal > extended_ereal,S4: set_se6634062954251873166_ereal] :
( ( image_6042159593519690757_ereal @ F2 @ ( comple4319282863272126363_ereal @ S4 ) )
= ( comple4319282863272126363_ereal @ ( image_6293272304431515653_ereal @ ( image_6042159593519690757_ereal @ F2 ) @ S4 ) ) ) ).
% image_Union
thf(fact_694_image__Union,axiom,
! [F2: nat > extended_ereal,S4: set_set_nat] :
( ( image_4309273772856505399_ereal @ F2 @ ( comple7399068483239264473et_nat @ S4 ) )
= ( comple4319282863272126363_ereal @ ( image_8825259783980156129_ereal @ ( image_4309273772856505399_ereal @ F2 ) @ S4 ) ) ) ).
% image_Union
thf(fact_695_UN__mono,axiom,
! [A: set_a,B5: set_a,F2: a > set_Extended_ereal,G: a > set_Extended_ereal] :
( ( ord_less_eq_set_a @ A @ B5 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ F2 @ A ) ) @ ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ G @ B5 ) ) ) ) ) ).
% UN_mono
thf(fact_696_UN__mono,axiom,
! [A: set_Extended_ereal,B5: set_Extended_ereal,F2: extended_ereal > set_Extended_ereal,G: extended_ereal > set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A @ B5 )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ F2 @ A ) ) @ ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ G @ B5 ) ) ) ) ) ).
% UN_mono
thf(fact_697_UN__least,axiom,
! [A: set_a,B5: a > set_Extended_ereal,C2: set_Extended_ereal] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ord_le1644982726543182158_ereal @ ( B5 @ X3 ) @ C2 ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ B5 @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_698_UN__upper,axiom,
! [A4: a,A: set_a,B5: a > set_Extended_ereal] :
( ( member_a @ A4 @ A )
=> ( ord_le1644982726543182158_ereal @ ( B5 @ A4 ) @ ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ B5 @ A ) ) ) ) ).
% UN_upper
thf(fact_699_SUP__eq,axiom,
! [A: set_a,B5: set_a,F2: a > set_Extended_ereal,G: a > set_Extended_ereal] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X4: a] :
( ( member_a @ X4 @ B5 )
& ( ord_le1644982726543182158_ereal @ ( F2 @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B5 )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ord_le1644982726543182158_ereal @ ( G @ J2 ) @ ( F2 @ X4 ) ) ) )
=> ( ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ F2 @ A ) )
= ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ G @ B5 ) ) ) ) ) ).
% SUP_eq
thf(fact_700_SUP__eq,axiom,
! [A: set_Extended_ereal,B5: set_Extended_ereal,F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal] :
( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ A )
=> ? [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ B5 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: extended_ereal] :
( ( member2350847679896131959_ereal @ J2 @ B5 )
=> ? [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ A )
& ( ord_le1083603963089353582_ereal @ ( G @ J2 ) @ ( F2 @ X4 ) ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) )
= ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ G @ B5 ) ) ) ) ) ).
% SUP_eq
thf(fact_701_SUP__eq,axiom,
! [A: set_Extended_ereal,B5: set_nat,F2: extended_ereal > extended_ereal,G: nat > extended_ereal] :
( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B5 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B5 )
=> ? [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ A )
& ( ord_le1083603963089353582_ereal @ ( G @ J2 ) @ ( F2 @ X4 ) ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) )
= ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ B5 ) ) ) ) ) ).
% SUP_eq
thf(fact_702_SUP__eq,axiom,
! [A: set_nat,B5: set_Extended_ereal,F2: nat > extended_ereal,G: extended_ereal > extended_ereal] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ B5 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: extended_ereal] :
( ( member2350847679896131959_ereal @ J2 @ B5 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_le1083603963089353582_ereal @ ( G @ J2 ) @ ( F2 @ X4 ) ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) )
= ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ G @ B5 ) ) ) ) ) ).
% SUP_eq
thf(fact_703_SUP__eq,axiom,
! [A: set_nat,B5: set_nat,F2: nat > extended_ereal,G: nat > extended_ereal] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B5 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B5 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_le1083603963089353582_ereal @ ( G @ J2 ) @ ( F2 @ X4 ) ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) )
= ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ B5 ) ) ) ) ) ).
% SUP_eq
thf(fact_704_SUP__eq,axiom,
! [A: set_Extended_ereal,B5: set_a,F2: extended_ereal > extended_ereal,G: a > extended_ereal] :
( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ A )
=> ? [X4: a] :
( ( member_a @ X4 @ B5 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B5 )
=> ? [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ A )
& ( ord_le1083603963089353582_ereal @ ( G @ J2 ) @ ( F2 @ X4 ) ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) )
= ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ G @ B5 ) ) ) ) ) ).
% SUP_eq
thf(fact_705_SUP__eq,axiom,
! [A: set_nat,B5: set_a,F2: nat > extended_ereal,G: a > extended_ereal] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X4: a] :
( ( member_a @ X4 @ B5 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B5 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_le1083603963089353582_ereal @ ( G @ J2 ) @ ( F2 @ X4 ) ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) )
= ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ G @ B5 ) ) ) ) ) ).
% SUP_eq
thf(fact_706_SUP__eq,axiom,
! [A: set_a,B5: set_Extended_ereal,F2: a > extended_ereal,G: extended_ereal > extended_ereal] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ B5 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: extended_ereal] :
( ( member2350847679896131959_ereal @ J2 @ B5 )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ord_le1083603963089353582_ereal @ ( G @ J2 ) @ ( F2 @ X4 ) ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ F2 @ A ) )
= ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ G @ B5 ) ) ) ) ) ).
% SUP_eq
thf(fact_707_SUP__eq,axiom,
! [A: set_a,B5: set_nat,F2: a > extended_ereal,G: nat > extended_ereal] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B5 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B5 )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ord_le1083603963089353582_ereal @ ( G @ J2 ) @ ( F2 @ X4 ) ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ F2 @ A ) )
= ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ B5 ) ) ) ) ) ).
% SUP_eq
thf(fact_708_SUP__eq,axiom,
! [A: set_a,B5: set_a,F2: a > extended_ereal,G: a > extended_ereal] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X4: a] :
( ( member_a @ X4 @ B5 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B5 )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ord_le1083603963089353582_ereal @ ( G @ J2 ) @ ( F2 @ X4 ) ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ F2 @ A ) )
= ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ G @ B5 ) ) ) ) ) ).
% SUP_eq
thf(fact_709_SUP__image,axiom,
! [G: extended_ereal > extended_ereal,F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ G @ ( image_6042159593519690757_ereal @ F2 @ A ) ) )
= ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ ( comp_E9177254828515427499_ereal @ G @ F2 ) @ A ) ) ) ).
% SUP_image
thf(fact_710_SUP__image,axiom,
! [G: extended_ereal > extended_ereal,F2: nat > extended_ereal,A: set_nat] :
( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ G @ ( image_4309273772856505399_ereal @ F2 @ A ) ) )
= ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ ( comp_E3726099860353345075al_nat @ G @ F2 ) @ A ) ) ) ).
% SUP_image
thf(fact_711_SUP__image,axiom,
! [G: nat > extended_ereal,F2: extended_ereal > nat,A: set_Extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ ( image_7659842161140344153al_nat @ F2 @ A ) ) )
= ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ ( comp_n261702227720650419_ereal @ G @ F2 ) @ A ) ) ) ).
% SUP_image
thf(fact_712_SUP__image,axiom,
! [G: nat > extended_ereal,F2: nat > nat,A: set_nat] :
( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ ( image_nat_nat @ F2 @ A ) ) )
= ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ ( comp_n13370146242399787al_nat @ G @ F2 ) @ A ) ) ) ).
% SUP_image
thf(fact_713_Sup__union__distrib,axiom,
! [A: set_Extended_ereal,B5: set_Extended_ereal] :
( ( comple8415311339701865915_ereal @ ( sup_su2680283192902082946_ereal @ A @ B5 ) )
= ( sup_su7653423775389492130_ereal @ ( comple8415311339701865915_ereal @ A ) @ ( comple8415311339701865915_ereal @ B5 ) ) ) ).
% Sup_union_distrib
thf(fact_714_SUP__eqI,axiom,
! [A: set_a,F2: a > set_Extended_ereal,X2: set_Extended_ereal] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ I3 ) @ X2 ) )
=> ( ! [Y3: set_Extended_ereal] :
( ! [I4: a] :
( ( member_a @ I4 @ A )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ I4 ) @ Y3 ) )
=> ( ord_le1644982726543182158_ereal @ X2 @ Y3 ) )
=> ( ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ F2 @ A ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_715_SUP__eqI,axiom,
! [A: set_Extended_ereal,F2: extended_ereal > extended_ereal,X2: extended_ereal] :
( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ A )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I3 ) @ X2 ) )
=> ( ! [Y3: extended_ereal] :
( ! [I4: extended_ereal] :
( ( member2350847679896131959_ereal @ I4 @ A )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I4 ) @ Y3 ) )
=> ( ord_le1083603963089353582_ereal @ X2 @ Y3 ) )
=> ( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_716_SUP__eqI,axiom,
! [A: set_nat,F2: nat > extended_ereal,X2: extended_ereal] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I3 ) @ X2 ) )
=> ( ! [Y3: extended_ereal] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I4 ) @ Y3 ) )
=> ( ord_le1083603963089353582_ereal @ X2 @ Y3 ) )
=> ( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_717_SUP__eqI,axiom,
! [A: set_a,F2: a > extended_ereal,X2: extended_ereal] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I3 ) @ X2 ) )
=> ( ! [Y3: extended_ereal] :
( ! [I4: a] :
( ( member_a @ I4 @ A )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I4 ) @ Y3 ) )
=> ( ord_le1083603963089353582_ereal @ X2 @ Y3 ) )
=> ( ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ F2 @ A ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_718_SUP__mono,axiom,
! [A: set_Extended_ereal,B5: set_Extended_ereal,F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal] :
( ! [N3: extended_ereal] :
( ( member2350847679896131959_ereal @ N3 @ A )
=> ? [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ B5 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ N3 ) @ ( G @ X4 ) ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ G @ B5 ) ) ) ) ).
% SUP_mono
thf(fact_719_SUP__mono,axiom,
! [A: set_Extended_ereal,B5: set_nat,F2: extended_ereal > extended_ereal,G: nat > extended_ereal] :
( ! [N3: extended_ereal] :
( ( member2350847679896131959_ereal @ N3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B5 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ N3 ) @ ( G @ X4 ) ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ B5 ) ) ) ) ).
% SUP_mono
thf(fact_720_SUP__mono,axiom,
! [A: set_nat,B5: set_Extended_ereal,F2: nat > extended_ereal,G: extended_ereal > extended_ereal] :
( ! [N3: nat] :
( ( member_nat @ N3 @ A )
=> ? [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ B5 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ N3 ) @ ( G @ X4 ) ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ G @ B5 ) ) ) ) ).
% SUP_mono
thf(fact_721_SUP__mono,axiom,
! [A: set_nat,B5: set_nat,F2: nat > extended_ereal,G: nat > extended_ereal] :
( ! [N3: nat] :
( ( member_nat @ N3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B5 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ N3 ) @ ( G @ X4 ) ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ B5 ) ) ) ) ).
% SUP_mono
thf(fact_722_SUP__mono,axiom,
! [A: set_a,B5: set_Extended_ereal,F2: a > extended_ereal,G: extended_ereal > extended_ereal] :
( ! [N3: a] :
( ( member_a @ N3 @ A )
=> ? [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ B5 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ N3 ) @ ( G @ X4 ) ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ F2 @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ G @ B5 ) ) ) ) ).
% SUP_mono
thf(fact_723_SUP__mono,axiom,
! [A: set_a,B5: set_nat,F2: a > extended_ereal,G: nat > extended_ereal] :
( ! [N3: a] :
( ( member_a @ N3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B5 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ N3 ) @ ( G @ X4 ) ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ F2 @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ B5 ) ) ) ) ).
% SUP_mono
thf(fact_724_SUP__least,axiom,
! [A: set_a,F2: a > set_Extended_ereal,U2: set_Extended_ereal] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ I3 ) @ U2 ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ F2 @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_725_SUP__least,axiom,
! [A: set_Extended_ereal,F2: extended_ereal > extended_ereal,U2: extended_ereal] :
( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ A )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I3 ) @ U2 ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_726_SUP__least,axiom,
! [A: set_nat,F2: nat > extended_ereal,U2: extended_ereal] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I3 ) @ U2 ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_727_SUP__least,axiom,
! [A: set_a,F2: a > extended_ereal,U2: extended_ereal] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I3 ) @ U2 ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ F2 @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_728_SUP__mono_H,axiom,
! [F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ! [X3: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( F2 @ X3 ) @ ( G @ X3 ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ G @ A ) ) ) ) ).
% SUP_mono'
thf(fact_729_SUP__mono_H,axiom,
! [F2: nat > extended_ereal,G: nat > extended_ereal,A: set_nat] :
( ! [X3: nat] : ( ord_le1083603963089353582_ereal @ ( F2 @ X3 ) @ ( G @ X3 ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ A ) ) ) ) ).
% SUP_mono'
thf(fact_730_SUP__upper,axiom,
! [I: a,A: set_a,F2: a > set_Extended_ereal] :
( ( member_a @ I @ A )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ I ) @ ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ F2 @ A ) ) ) ) ).
% SUP_upper
thf(fact_731_SUP__upper,axiom,
! [I: extended_ereal,A: set_Extended_ereal,F2: extended_ereal > extended_ereal] :
( ( member2350847679896131959_ereal @ I @ A )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I ) @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) ) ) ).
% SUP_upper
thf(fact_732_SUP__upper,axiom,
! [I: nat,A: set_nat,F2: nat > extended_ereal] :
( ( member_nat @ I @ A )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) ) ) ).
% SUP_upper
thf(fact_733_SUP__upper,axiom,
! [I: a,A: set_a,F2: a > extended_ereal] :
( ( member_a @ I @ A )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I ) @ ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ F2 @ A ) ) ) ) ).
% SUP_upper
thf(fact_734_SUP__le__iff,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,U2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) @ U2 )
= ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ X ) @ U2 ) ) ) ) ).
% SUP_le_iff
thf(fact_735_SUP__le__iff,axiom,
! [F2: nat > extended_ereal,A: set_nat,U2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) @ U2 )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ X ) @ U2 ) ) ) ) ).
% SUP_le_iff
thf(fact_736_SUP__upper2,axiom,
! [I: a,A: set_a,U2: set_Extended_ereal,F2: a > set_Extended_ereal] :
( ( member_a @ I @ A )
=> ( ( ord_le1644982726543182158_ereal @ U2 @ ( F2 @ I ) )
=> ( ord_le1644982726543182158_ereal @ U2 @ ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ F2 @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_737_SUP__upper2,axiom,
! [I: extended_ereal,A: set_Extended_ereal,U2: extended_ereal,F2: extended_ereal > extended_ereal] :
( ( member2350847679896131959_ereal @ I @ A )
=> ( ( ord_le1083603963089353582_ereal @ U2 @ ( F2 @ I ) )
=> ( ord_le1083603963089353582_ereal @ U2 @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_738_SUP__upper2,axiom,
! [I: nat,A: set_nat,U2: extended_ereal,F2: nat > extended_ereal] :
( ( member_nat @ I @ A )
=> ( ( ord_le1083603963089353582_ereal @ U2 @ ( F2 @ I ) )
=> ( ord_le1083603963089353582_ereal @ U2 @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_739_SUP__upper2,axiom,
! [I: a,A: set_a,U2: extended_ereal,F2: a > extended_ereal] :
( ( member_a @ I @ A )
=> ( ( ord_le1083603963089353582_ereal @ U2 @ ( F2 @ I ) )
=> ( ord_le1083603963089353582_ereal @ U2 @ ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ F2 @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_740_complete__lattice__class_OSUP__sup__distrib,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,G: extended_ereal > extended_ereal] :
( ( sup_su7653423775389492130_ereal @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ G @ A ) ) )
= ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [A6: extended_ereal] : ( sup_su7653423775389492130_ereal @ ( F2 @ A6 ) @ ( G @ A6 ) )
@ A ) ) ) ).
% complete_lattice_class.SUP_sup_distrib
thf(fact_741_complete__lattice__class_OSUP__sup__distrib,axiom,
! [F2: nat > extended_ereal,A: set_nat,G: nat > extended_ereal] :
( ( sup_su7653423775389492130_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ A ) ) )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [A6: nat] : ( sup_su7653423775389492130_ereal @ ( F2 @ A6 ) @ ( G @ A6 ) )
@ A ) ) ) ).
% complete_lattice_class.SUP_sup_distrib
thf(fact_742_SUP__absorb,axiom,
! [K: extended_ereal,I5: set_Extended_ereal,A: extended_ereal > extended_ereal] :
( ( member2350847679896131959_ereal @ K @ I5 )
=> ( ( sup_su7653423775389492130_ereal @ ( A @ K ) @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ A @ I5 ) ) )
= ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ A @ I5 ) ) ) ) ).
% SUP_absorb
thf(fact_743_SUP__absorb,axiom,
! [K: nat,I5: set_nat,A: nat > extended_ereal] :
( ( member_nat @ K @ I5 )
=> ( ( sup_su7653423775389492130_ereal @ ( A @ K ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ A @ I5 ) ) )
= ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ A @ I5 ) ) ) ) ).
% SUP_absorb
thf(fact_744_SUP__absorb,axiom,
! [K: a,I5: set_a,A: a > extended_ereal] :
( ( member_a @ K @ I5 )
=> ( ( sup_su7653423775389492130_ereal @ ( A @ K ) @ ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ A @ I5 ) ) )
= ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ A @ I5 ) ) ) ) ).
% SUP_absorb
thf(fact_745_inj__on__UNION__chain,axiom,
! [I5: set_a,A: a > set_Extended_ereal,F2: extended_ereal > extended_ereal] :
( ! [I3: a,J2: a] :
( ( member_a @ I3 @ I5 )
=> ( ( member_a @ J2 @ I5 )
=> ( ( ord_le1644982726543182158_ereal @ ( A @ I3 ) @ ( A @ J2 ) )
| ( ord_le1644982726543182158_ereal @ ( A @ J2 ) @ ( A @ I3 ) ) ) ) )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I5 )
=> ( inj_on7162434037990268785_ereal @ F2 @ ( A @ I3 ) ) )
=> ( inj_on7162434037990268785_ereal @ F2 @ ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ A @ I5 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_746_SUP__subset__mono,axiom,
! [A: set_a,B5: set_a,F2: a > set_Extended_ereal,G: a > set_Extended_ereal] :
( ( ord_less_eq_set_a @ A @ B5 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ F2 @ A ) ) @ ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ G @ B5 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_747_SUP__subset__mono,axiom,
! [A: set_Extended_ereal,B5: set_Extended_ereal,F2: extended_ereal > set_Extended_ereal,G: extended_ereal > set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A @ B5 )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ F2 @ A ) ) @ ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ G @ B5 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_748_SUP__subset__mono,axiom,
! [A: set_nat,B5: set_nat,F2: nat > extended_ereal,G: nat > extended_ereal] :
( ( ord_less_eq_set_nat @ A @ B5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ B5 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_749_SUP__subset__mono,axiom,
! [A: set_a,B5: set_a,F2: a > extended_ereal,G: a > extended_ereal] :
( ( ord_less_eq_set_a @ A @ B5 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ F2 @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ G @ B5 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_750_SUP__subset__mono,axiom,
! [A: set_Extended_ereal,B5: set_Extended_ereal,F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A @ B5 )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ G @ B5 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_751_SUP__union,axiom,
! [M3: extended_ereal > extended_ereal,A: set_Extended_ereal,B5: set_Extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ M3 @ ( sup_su2680283192902082946_ereal @ A @ B5 ) ) )
= ( sup_su7653423775389492130_ereal @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ M3 @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ M3 @ B5 ) ) ) ) ).
% SUP_union
thf(fact_752_SUP__union,axiom,
! [M3: nat > extended_ereal,A: set_nat,B5: set_nat] :
( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ M3 @ ( sup_sup_set_nat @ A @ B5 ) ) )
= ( sup_su7653423775389492130_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ M3 @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ M3 @ B5 ) ) ) ) ).
% SUP_union
thf(fact_753_inj__on__image,axiom,
! [F2: nat > extended_ereal,A: set_set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ ( comple7399068483239264473et_nat @ A ) )
=> ( inj_on1463964778812548213_ereal @ ( image_4309273772856505399_ereal @ F2 ) @ A ) ) ).
% inj_on_image
thf(fact_754_inj__on__image,axiom,
! [F2: extended_ereal > extended_ereal,A: set_se6634062954251873166_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ ( comple4319282863272126363_ereal @ A ) )
=> ( inj_on5406440306785145713_ereal @ ( image_6042159593519690757_ereal @ F2 ) @ A ) ) ).
% inj_on_image
thf(fact_755_snd__image__Sigma,axiom,
! [A: set_a,B5: a > set_b] :
( ( image_2802296252294471260_a_b_b @ product_snd_a_b @ ( product_Sigma_a_b @ A @ B5 ) )
= ( comple2307003614231284044_set_b @ ( image_a_set_b @ B5 @ A ) ) ) ).
% snd_image_Sigma
thf(fact_756_SUP__pair,axiom,
! [F2: extended_ereal > extended_ereal > extended_ereal,B5: set_Extended_ereal,A: set_Extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ ( F2 @ I2 ) @ B5 ) )
@ A ) )
= ( comple8415311339701865915_ereal
@ ( image_8011834658935918572_ereal
@ ^ [P3: produc5501587555545223847_ereal] : ( F2 @ ( produc8000661846298591107_ereal @ P3 ) @ ( produc7555089578395445445_ereal @ P3 ) )
@ ( produc8095709571603465288_ereal @ A
@ ^ [Uu: extended_ereal] : B5 ) ) ) ) ).
% SUP_pair
thf(fact_757_SUP__pair,axiom,
! [F2: extended_ereal > nat > extended_ereal,B5: set_nat,A: set_Extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ ( F2 @ I2 ) @ B5 ) )
@ A ) )
= ( comple8415311339701865915_ereal
@ ( image_3315421938599550714_ereal
@ ^ [P3: produc5547956294148499661al_nat] : ( F2 @ ( produc3755854452330217819al_nat @ P3 ) @ ( produc5370844913463222425al_nat @ P3 ) )
@ ( produc4220900302008768982al_nat @ A
@ ^ [Uu: extended_ereal] : B5 ) ) ) ) ).
% SUP_pair
thf(fact_758_SUP__pair,axiom,
! [F2: nat > extended_ereal > extended_ereal,B5: set_Extended_ereal,A: set_nat] :
( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ ( F2 @ I2 ) @ B5 ) )
@ A ) )
= ( comple8415311339701865915_ereal
@ ( image_4404873737357082836_ereal
@ ^ [P3: produc7896515489273245043_ereal] : ( F2 @ ( produc405286064046379065_ereal @ P3 ) @ ( produc2020276525179383671_ereal @ P3 ) )
@ ( produc870331913724930228_ereal @ A
@ ^ [Uu: nat] : B5 ) ) ) ) ).
% SUP_pair
thf(fact_759_SUP__pair,axiom,
! [F2: nat > nat > extended_ereal,B5: set_nat,A: set_nat] :
( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ ( F2 @ I2 ) @ B5 ) )
@ A ) )
= ( comple8415311339701865915_ereal
@ ( image_3495945180199258130_ereal
@ ^ [P3: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ P3 ) @ ( product_snd_nat_nat @ P3 ) )
@ ( produc457027306803732586at_nat @ A
@ ^ [Uu: nat] : B5 ) ) ) ) ).
% SUP_pair
thf(fact_760_SUP__pair,axiom,
! [F2: a > b > extended_ereal,B5: set_b,A: set_a] :
( ( comple8415311339701865915_ereal
@ ( image_8405481351990995413_ereal
@ ^ [I2: a] : ( comple8415311339701865915_ereal @ ( image_5319725110001000852_ereal @ ( F2 @ I2 ) @ B5 ) )
@ A ) )
= ( comple8415311339701865915_ereal
@ ( image_5361069211733868459_ereal
@ ^ [P3: product_prod_a_b] : ( F2 @ ( product_fst_a_b @ P3 ) @ ( product_snd_a_b @ P3 ) )
@ ( product_Sigma_a_b @ A
@ ^ [Uu: a] : B5 ) ) ) ) ).
% SUP_pair
thf(fact_761_SUP__combine,axiom,
! [F2: extended_ereal > extended_ereal > set_Extended_ereal] :
( ! [A2: extended_ereal,B3: extended_ereal,C4: extended_ereal,D3: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A2 @ B3 )
=> ( ( ord_le1083603963089353582_ereal @ C4 @ D3 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ A2 @ C4 ) @ ( F2 @ B3 @ D3 ) ) ) )
=> ( ( comple4319282863272126363_ereal
@ ( image_5562094264469218789_ereal
@ ^ [I2: extended_ereal] : ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ ( F2 @ I2 ) @ top_to5683747375963461374_ereal ) )
@ top_to5683747375963461374_ereal ) )
= ( comple4319282863272126363_ereal
@ ( image_5562094264469218789_ereal
@ ^ [I2: extended_ereal] : ( F2 @ I2 @ I2 )
@ top_to5683747375963461374_ereal ) ) ) ) ).
% SUP_combine
thf(fact_762_SUP__combine,axiom,
! [F2: nat > nat > set_Extended_ereal] :
( ! [A2: nat,B3: nat,C4: nat,D3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ C4 @ D3 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ A2 @ C4 ) @ ( F2 @ B3 @ D3 ) ) ) )
=> ( ( comple4319282863272126363_ereal
@ ( image_305533323056406039_ereal
@ ^ [I2: nat] : ( comple4319282863272126363_ereal @ ( image_305533323056406039_ereal @ ( F2 @ I2 ) @ top_top_set_nat ) )
@ top_top_set_nat ) )
= ( comple4319282863272126363_ereal
@ ( image_305533323056406039_ereal
@ ^ [I2: nat] : ( F2 @ I2 @ I2 )
@ top_top_set_nat ) ) ) ) ).
% SUP_combine
thf(fact_763_SUP__combine,axiom,
! [F2: set_Extended_ereal > set_Extended_ereal > set_Extended_ereal] :
( ! [A2: set_Extended_ereal,B3: set_Extended_ereal,C4: set_Extended_ereal,D3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A2 @ B3 )
=> ( ( ord_le1644982726543182158_ereal @ C4 @ D3 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ A2 @ C4 ) @ ( F2 @ B3 @ D3 ) ) ) )
=> ( ( comple4319282863272126363_ereal
@ ( image_6293272304431515653_ereal
@ ^ [I2: set_Extended_ereal] : ( comple4319282863272126363_ereal @ ( image_6293272304431515653_ereal @ ( F2 @ I2 ) @ top_to4757929550322229470_ereal ) )
@ top_to4757929550322229470_ereal ) )
= ( comple4319282863272126363_ereal
@ ( image_6293272304431515653_ereal
@ ^ [I2: set_Extended_ereal] : ( F2 @ I2 @ I2 )
@ top_to4757929550322229470_ereal ) ) ) ) ).
% SUP_combine
thf(fact_764_SUP__combine,axiom,
! [F2: extended_ereal > extended_ereal > extended_ereal] :
( ! [A2: extended_ereal,B3: extended_ereal,C4: extended_ereal,D3: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A2 @ B3 )
=> ( ( ord_le1083603963089353582_ereal @ C4 @ D3 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ A2 @ C4 ) @ ( F2 @ B3 @ D3 ) ) ) )
=> ( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ ( F2 @ I2 ) @ top_to5683747375963461374_ereal ) )
@ top_to5683747375963461374_ereal ) )
= ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( F2 @ I2 @ I2 )
@ top_to5683747375963461374_ereal ) ) ) ) ).
% SUP_combine
thf(fact_765_SUP__combine,axiom,
! [F2: nat > nat > extended_ereal] :
( ! [A2: nat,B3: nat,C4: nat,D3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ C4 @ D3 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ A2 @ C4 ) @ ( F2 @ B3 @ D3 ) ) ) )
=> ( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ ( F2 @ I2 ) @ top_top_set_nat ) )
@ top_top_set_nat ) )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( F2 @ I2 @ I2 )
@ top_top_set_nat ) ) ) ) ).
% SUP_combine
thf(fact_766_SUP__combine,axiom,
! [F2: set_Extended_ereal > set_Extended_ereal > extended_ereal] :
( ! [A2: set_Extended_ereal,B3: set_Extended_ereal,C4: set_Extended_ereal,D3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A2 @ B3 )
=> ( ( ord_le1644982726543182158_ereal @ C4 @ D3 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ A2 @ C4 ) @ ( F2 @ B3 @ D3 ) ) ) )
=> ( ( comple8415311339701865915_ereal
@ ( image_1204185369672881701_ereal
@ ^ [I2: set_Extended_ereal] : ( comple8415311339701865915_ereal @ ( image_1204185369672881701_ereal @ ( F2 @ I2 ) @ top_to4757929550322229470_ereal ) )
@ top_to4757929550322229470_ereal ) )
= ( comple8415311339701865915_ereal
@ ( image_1204185369672881701_ereal
@ ^ [I2: set_Extended_ereal] : ( F2 @ I2 @ I2 )
@ top_to4757929550322229470_ereal ) ) ) ) ).
% SUP_combine
thf(fact_767_type__definition_OAbs__image,axiom,
! [Rep2: extended_ereal > extended_ereal,Abs2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( type_d4948314182666096300_ereal @ Rep2 @ Abs2 @ A )
=> ( ( image_6042159593519690757_ereal @ Abs2 @ A )
= top_to5683747375963461374_ereal ) ) ).
% type_definition.Abs_image
thf(fact_768_type__definition_OAbs__image,axiom,
! [Rep2: extended_ereal > nat,Abs2: nat > extended_ereal,A: set_nat] :
( ( type_d5406521743509674098al_nat @ Rep2 @ Abs2 @ A )
=> ( ( image_4309273772856505399_ereal @ Abs2 @ A )
= top_to5683747375963461374_ereal ) ) ).
% type_definition.Abs_image
thf(fact_769_type__definition_ORep__range,axiom,
! [Rep2: extended_ereal > extended_ereal,Abs2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( type_d4948314182666096300_ereal @ Rep2 @ Abs2 @ A )
=> ( ( image_6042159593519690757_ereal @ Rep2 @ top_to5683747375963461374_ereal )
= A ) ) ).
% type_definition.Rep_range
thf(fact_770_type__definition_ORep__range,axiom,
! [Rep2: nat > extended_ereal,Abs2: extended_ereal > nat,A: set_Extended_ereal] :
( ( type_d2055953355225835344_ereal @ Rep2 @ Abs2 @ A )
=> ( ( image_4309273772856505399_ereal @ Rep2 @ top_top_set_nat )
= A ) ) ).
% type_definition.Rep_range
thf(fact_771_Sup__set__def,axiom,
( comple2307003609928055243_set_a
= ( ^ [A3: set_set_a] :
( collect_a
@ ^ [X: a] : ( complete_Sup_Sup_o @ ( image_set_a_o @ ( member_a @ X ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_772_SUP__Sup__eq2,axiom,
! [S4: set_se8915165130606451687list_o] :
( ( comple5276478977970185563st_o_o
@ ( image_6785746108194547372st_o_o
@ ^ [I2: set_Pr7497825696620840711list_o,X: option_list_o,Y: option_list_o] : ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X @ Y ) @ I2 )
@ S4 ) )
= ( ^ [X: option_list_o,Y: option_list_o] : ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X @ Y ) @ ( comple1150810575505393780list_o @ S4 ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_773_SUP__Sup__eq,axiom,
! [S4: set_set_a] :
( ( complete_Sup_Sup_a_o
@ ( image_set_a_a_o
@ ^ [I2: set_a,X: a] : ( member_a @ X @ I2 )
@ S4 ) )
= ( ^ [X: a] : ( member_a @ X @ ( comple2307003609928055243_set_a @ S4 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_774_Sup__SUP__eq,axiom,
( complete_Sup_Sup_a_o
= ( ^ [S5: set_a_o,X: a] : ( member_a @ X @ ( comple2307003609928055243_set_a @ ( image_a_o_set_a @ collect_a @ S5 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_775_Sup__SUP__eq2,axiom,
( comple5276478977970185563st_o_o
= ( ^ [S5: set_op7420493560124892494st_o_o,X: option_list_o,Y: option_list_o] : ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X @ Y ) @ ( comple1150810575505393780list_o @ ( image_7941296275600097962list_o @ collec8383569672281745810list_o @ ( image_132443102827437503st_o_o @ produc2885628514414960748st_o_o @ S5 ) ) ) ) ) ) ).
% Sup_SUP_eq2
thf(fact_776_SUP__INF,axiom,
! [P2: extended_ereal > extended_ereal > extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [Y: extended_ereal] :
( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : ( P2 @ X @ Y )
@ top_to5683747375963461374_ereal ) )
@ top_to5683747375963461374_ereal ) )
= ( comple3556804143462414037_ereal
@ ( image_3247130582775387874_ereal
@ ^ [X: extended_ereal > extended_ereal] :
( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [Y: extended_ereal] : ( P2 @ ( X @ Y ) @ Y )
@ top_to5683747375963461374_ereal ) )
@ top_to908700840774984395_ereal ) ) ) ).
% SUP_INF
thf(fact_777_SUP__INF,axiom,
! [P2: extended_ereal > nat > extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [Y: nat] :
( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : ( P2 @ X @ Y )
@ top_to5683747375963461374_ereal ) )
@ top_top_set_nat ) )
= ( comple3556804143462414037_ereal
@ ( image_8489574568685215582_ereal
@ ^ [X: nat > extended_ereal] :
( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [Y: nat] : ( P2 @ ( X @ Y ) @ Y )
@ top_top_set_nat ) )
@ top_to1136031370110204389_ereal ) ) ) ).
% SUP_INF
thf(fact_778_SUP__INF,axiom,
! [P2: nat > extended_ereal > extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [Y: extended_ereal] :
( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X: nat] : ( P2 @ X @ Y )
@ top_top_set_nat ) )
@ top_to5683747375963461374_ereal ) )
= ( comple3556804143462414037_ereal
@ ( image_7400122769927683460_ereal
@ ^ [X: extended_ereal > nat] :
( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [Y: extended_ereal] : ( P2 @ ( X @ Y ) @ Y )
@ top_to5683747375963461374_ereal ) )
@ top_to2398772004365740095al_nat ) ) ) ).
% SUP_INF
thf(fact_779_SUP__INF,axiom,
! [P2: nat > nat > extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [Y: nat] :
( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X: nat] : ( P2 @ X @ Y )
@ top_top_set_nat ) )
@ top_top_set_nat ) )
= ( comple3556804143462414037_ereal
@ ( image_2873892944974324744_ereal
@ ^ [X: nat > nat] :
( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [Y: nat] : ( P2 @ ( X @ Y ) @ Y )
@ top_top_set_nat ) )
@ top_top_set_nat_nat ) ) ) ).
% SUP_INF
thf(fact_780_INF__SUP,axiom,
! [P2: extended_ereal > extended_ereal > extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [Y: extended_ereal] :
( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : ( P2 @ X @ Y )
@ top_to5683747375963461374_ereal ) )
@ top_to5683747375963461374_ereal ) )
= ( comple8415311339701865915_ereal
@ ( image_3247130582775387874_ereal
@ ^ [F: extended_ereal > extended_ereal] :
( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : ( P2 @ ( F @ X ) @ X )
@ top_to5683747375963461374_ereal ) )
@ top_to908700840774984395_ereal ) ) ) ).
% INF_SUP
thf(fact_781_INF__SUP,axiom,
! [P2: extended_ereal > nat > extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [Y: nat] :
( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : ( P2 @ X @ Y )
@ top_to5683747375963461374_ereal ) )
@ top_top_set_nat ) )
= ( comple8415311339701865915_ereal
@ ( image_8489574568685215582_ereal
@ ^ [F: nat > extended_ereal] :
( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X: nat] : ( P2 @ ( F @ X ) @ X )
@ top_top_set_nat ) )
@ top_to1136031370110204389_ereal ) ) ) ).
% INF_SUP
thf(fact_782_INF__SUP,axiom,
! [P2: nat > extended_ereal > extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [Y: extended_ereal] :
( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X: nat] : ( P2 @ X @ Y )
@ top_top_set_nat ) )
@ top_to5683747375963461374_ereal ) )
= ( comple8415311339701865915_ereal
@ ( image_7400122769927683460_ereal
@ ^ [F: extended_ereal > nat] :
( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : ( P2 @ ( F @ X ) @ X )
@ top_to5683747375963461374_ereal ) )
@ top_to2398772004365740095al_nat ) ) ) ).
% INF_SUP
thf(fact_783_INF__SUP,axiom,
! [P2: nat > nat > extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [Y: nat] :
( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X: nat] : ( P2 @ X @ Y )
@ top_top_set_nat ) )
@ top_top_set_nat ) )
= ( comple8415311339701865915_ereal
@ ( image_2873892944974324744_ereal
@ ^ [F: nat > nat] :
( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X: nat] : ( P2 @ ( F @ X ) @ X )
@ top_top_set_nat ) )
@ top_top_set_nat_nat ) ) ) ).
% INF_SUP
thf(fact_784_INF__pair,axiom,
! [F2: extended_ereal > extended_ereal > extended_ereal,B5: set_Extended_ereal,A: set_Extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ ( F2 @ I2 ) @ B5 ) )
@ A ) )
= ( comple3556804143462414037_ereal
@ ( image_8011834658935918572_ereal
@ ^ [P3: produc5501587555545223847_ereal] : ( F2 @ ( produc8000661846298591107_ereal @ P3 ) @ ( produc7555089578395445445_ereal @ P3 ) )
@ ( produc8095709571603465288_ereal @ A
@ ^ [Uu: extended_ereal] : B5 ) ) ) ) ).
% INF_pair
thf(fact_785_INF__pair,axiom,
! [F2: extended_ereal > nat > extended_ereal,B5: set_nat,A: set_Extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ ( F2 @ I2 ) @ B5 ) )
@ A ) )
= ( comple3556804143462414037_ereal
@ ( image_3315421938599550714_ereal
@ ^ [P3: produc5547956294148499661al_nat] : ( F2 @ ( produc3755854452330217819al_nat @ P3 ) @ ( produc5370844913463222425al_nat @ P3 ) )
@ ( produc4220900302008768982al_nat @ A
@ ^ [Uu: extended_ereal] : B5 ) ) ) ) ).
% INF_pair
thf(fact_786_INF__pair,axiom,
! [F2: nat > extended_ereal > extended_ereal,B5: set_Extended_ereal,A: set_nat] :
( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ ( F2 @ I2 ) @ B5 ) )
@ A ) )
= ( comple3556804143462414037_ereal
@ ( image_4404873737357082836_ereal
@ ^ [P3: produc7896515489273245043_ereal] : ( F2 @ ( produc405286064046379065_ereal @ P3 ) @ ( produc2020276525179383671_ereal @ P3 ) )
@ ( produc870331913724930228_ereal @ A
@ ^ [Uu: nat] : B5 ) ) ) ) ).
% INF_pair
thf(fact_787_INF__pair,axiom,
! [F2: nat > nat > extended_ereal,B5: set_nat,A: set_nat] :
( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ ( F2 @ I2 ) @ B5 ) )
@ A ) )
= ( comple3556804143462414037_ereal
@ ( image_3495945180199258130_ereal
@ ^ [P3: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ P3 ) @ ( product_snd_nat_nat @ P3 ) )
@ ( produc457027306803732586at_nat @ A
@ ^ [Uu: nat] : B5 ) ) ) ) ).
% INF_pair
thf(fact_788_INF__pair,axiom,
! [F2: a > b > extended_ereal,B5: set_b,A: set_a] :
( ( comple3556804143462414037_ereal
@ ( image_8405481351990995413_ereal
@ ^ [I2: a] : ( comple3556804143462414037_ereal @ ( image_5319725110001000852_ereal @ ( F2 @ I2 ) @ B5 ) )
@ A ) )
= ( comple3556804143462414037_ereal
@ ( image_5361069211733868459_ereal
@ ^ [P3: product_prod_a_b] : ( F2 @ ( product_fst_a_b @ P3 ) @ ( product_snd_a_b @ P3 ) )
@ ( product_Sigma_a_b @ A
@ ^ [Uu: a] : B5 ) ) ) ) ).
% INF_pair
thf(fact_789_INT__I,axiom,
! [A: set_a,B: a,B5: a > set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ B @ ( B5 @ X3 ) ) )
=> ( member_a @ B @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ B5 @ A ) ) ) ) ).
% INT_I
thf(fact_790_INF__identity__eq,axiom,
! [A: set_Extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : X
@ A ) )
= ( comple3556804143462414037_ereal @ A ) ) ).
% INF_identity_eq
thf(fact_791_INF__id__eq,axiom,
! [A: set_Extended_ereal] :
( ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ id_Extended_ereal @ A ) )
= ( comple3556804143462414037_ereal @ A ) ) ).
% INF_id_eq
thf(fact_792_INF__top__conv_I2_J,axiom,
! [B5: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( top_to6662034908053899550_ereal
= ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ B5 @ A ) ) )
= ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A )
=> ( ( B5 @ X )
= top_to6662034908053899550_ereal ) ) ) ) ).
% INF_top_conv(2)
thf(fact_793_INF__top__conv_I2_J,axiom,
! [B5: nat > extended_ereal,A: set_nat] :
( ( top_to6662034908053899550_ereal
= ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ B5 @ A ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( B5 @ X )
= top_to6662034908053899550_ereal ) ) ) ) ).
% INF_top_conv(2)
thf(fact_794_INF__top__conv_I1_J,axiom,
! [B5: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ B5 @ A ) )
= top_to6662034908053899550_ereal )
= ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A )
=> ( ( B5 @ X )
= top_to6662034908053899550_ereal ) ) ) ) ).
% INF_top_conv(1)
thf(fact_795_INF__top__conv_I1_J,axiom,
! [B5: nat > extended_ereal,A: set_nat] :
( ( ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ B5 @ A ) )
= top_to6662034908053899550_ereal )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( B5 @ X )
= top_to6662034908053899550_ereal ) ) ) ) ).
% INF_top_conv(1)
thf(fact_796_INF__top,axiom,
! [A: set_Extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : top_to6662034908053899550_ereal
@ A ) )
= top_to6662034908053899550_ereal ) ).
% INF_top
thf(fact_797_INF__top,axiom,
! [A: set_nat] :
( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X: nat] : top_to6662034908053899550_ereal
@ A ) )
= top_to6662034908053899550_ereal ) ).
% INF_top
thf(fact_798_INT__D,axiom,
! [B: a,B5: a > set_a,A: set_a,A4: a] :
( ( member_a @ B @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ B5 @ A ) ) )
=> ( ( member_a @ A4 @ A )
=> ( member_a @ B @ ( B5 @ A4 ) ) ) ) ).
% INT_D
thf(fact_799_INT__E,axiom,
! [B: a,B5: a > set_a,A: set_a,A4: a] :
( ( member_a @ B @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ B5 @ A ) ) )
=> ( ~ ( member_a @ B @ ( B5 @ A4 ) )
=> ~ ( member_a @ A4 @ A ) ) ) ).
% INT_E
thf(fact_800_Inf__set__def,axiom,
( comple6135023378680113637_set_a
= ( ^ [A3: set_set_a] :
( collect_a
@ ^ [X: a] : ( complete_Inf_Inf_o @ ( image_set_a_o @ ( member_a @ X ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_801_INF__Int__eq2,axiom,
! [S4: set_se8915165130606451687list_o] :
( ( comple1974978180206336373st_o_o
@ ( image_6785746108194547372st_o_o
@ ^ [I2: set_Pr7497825696620840711list_o,X: option_list_o,Y: option_list_o] : ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X @ Y ) @ I2 )
@ S4 ) )
= ( ^ [X: option_list_o,Y: option_list_o] : ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X @ Y ) @ ( comple3647107874392957070list_o @ S4 ) ) ) ) ).
% INF_Int_eq2
thf(fact_802_INF__Int__eq,axiom,
! [S4: set_set_a] :
( ( complete_Inf_Inf_a_o
@ ( image_set_a_a_o
@ ^ [I2: set_a,X: a] : ( member_a @ X @ I2 )
@ S4 ) )
= ( ^ [X: a] : ( member_a @ X @ ( comple6135023378680113637_set_a @ S4 ) ) ) ) ).
% INF_Int_eq
thf(fact_803_Inf__INT__eq,axiom,
( complete_Inf_Inf_a_o
= ( ^ [S5: set_a_o,X: a] : ( member_a @ X @ ( comple6135023378680113637_set_a @ ( image_a_o_set_a @ collect_a @ S5 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_804_INF__cong,axiom,
! [A: set_Extended_ereal,B5: set_Extended_ereal,C2: extended_ereal > extended_ereal,D: extended_ereal > extended_ereal] :
( ( A = B5 )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ B5 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ C2 @ A ) )
= ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ D @ B5 ) ) ) ) ) ).
% INF_cong
thf(fact_805_INF__cong,axiom,
! [A: set_nat,B5: set_nat,C2: nat > extended_ereal,D: nat > extended_ereal] :
( ( A = B5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B5 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ C2 @ A ) )
= ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ D @ B5 ) ) ) ) ) ).
% INF_cong
thf(fact_806_INF__cong,axiom,
! [A: set_a,B5: set_a,C2: a > extended_ereal,D: a > extended_ereal] :
( ( A = B5 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B5 )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple3556804143462414037_ereal @ ( image_8405481351990995413_ereal @ C2 @ A ) )
= ( comple3556804143462414037_ereal @ ( image_8405481351990995413_ereal @ D @ B5 ) ) ) ) ) ).
% INF_cong
thf(fact_807_INF__commute,axiom,
! [F2: extended_ereal > extended_ereal > extended_ereal,B5: set_Extended_ereal,A: set_Extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ ( F2 @ I2 ) @ B5 ) )
@ A ) )
= ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [J: extended_ereal] :
( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( F2 @ I2 @ J )
@ A ) )
@ B5 ) ) ) ).
% INF_commute
thf(fact_808_INF__commute,axiom,
! [F2: extended_ereal > nat > extended_ereal,B5: set_nat,A: set_Extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ ( F2 @ I2 ) @ B5 ) )
@ A ) )
= ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [J: nat] :
( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( F2 @ I2 @ J )
@ A ) )
@ B5 ) ) ) ).
% INF_commute
thf(fact_809_INF__commute,axiom,
! [F2: nat > extended_ereal > extended_ereal,B5: set_Extended_ereal,A: set_nat] :
( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ ( F2 @ I2 ) @ B5 ) )
@ A ) )
= ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [J: extended_ereal] :
( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( F2 @ I2 @ J )
@ A ) )
@ B5 ) ) ) ).
% INF_commute
thf(fact_810_INF__commute,axiom,
! [F2: nat > nat > extended_ereal,B5: set_nat,A: set_nat] :
( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ ( F2 @ I2 ) @ B5 ) )
@ A ) )
= ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [J: nat] :
( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( F2 @ I2 @ J )
@ A ) )
@ B5 ) ) ) ).
% INF_commute
thf(fact_811_INF__eq,axiom,
! [A: set_a,B5: set_a,G: a > set_Extended_ereal,F2: a > set_Extended_ereal] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X4: a] :
( ( member_a @ X4 @ B5 )
& ( ord_le1644982726543182158_ereal @ ( G @ X4 ) @ ( F2 @ I3 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B5 )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ord_le1644982726543182158_ereal @ ( F2 @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( comple4418415374894819509_ereal @ ( image_886028290468338613_ereal @ F2 @ A ) )
= ( comple4418415374894819509_ereal @ ( image_886028290468338613_ereal @ G @ B5 ) ) ) ) ) ).
% INF_eq
thf(fact_812_INF__eq,axiom,
! [A: set_Extended_ereal,B5: set_Extended_ereal,G: extended_ereal > extended_ereal,F2: extended_ereal > extended_ereal] :
( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ A )
=> ? [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ B5 )
& ( ord_le1083603963089353582_ereal @ ( G @ X4 ) @ ( F2 @ I3 ) ) ) )
=> ( ! [J2: extended_ereal] :
( ( member2350847679896131959_ereal @ J2 @ B5 )
=> ? [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ A )
& ( ord_le1083603963089353582_ereal @ ( F2 @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) )
= ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ G @ B5 ) ) ) ) ) ).
% INF_eq
thf(fact_813_INF__eq,axiom,
! [A: set_Extended_ereal,B5: set_nat,G: nat > extended_ereal,F2: extended_ereal > extended_ereal] :
( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B5 )
& ( ord_le1083603963089353582_ereal @ ( G @ X4 ) @ ( F2 @ I3 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B5 )
=> ? [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ A )
& ( ord_le1083603963089353582_ereal @ ( F2 @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) )
= ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ G @ B5 ) ) ) ) ) ).
% INF_eq
thf(fact_814_INF__eq,axiom,
! [A: set_nat,B5: set_Extended_ereal,G: extended_ereal > extended_ereal,F2: nat > extended_ereal] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ B5 )
& ( ord_le1083603963089353582_ereal @ ( G @ X4 ) @ ( F2 @ I3 ) ) ) )
=> ( ! [J2: extended_ereal] :
( ( member2350847679896131959_ereal @ J2 @ B5 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_le1083603963089353582_ereal @ ( F2 @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) )
= ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ G @ B5 ) ) ) ) ) ).
% INF_eq
thf(fact_815_INF__eq,axiom,
! [A: set_nat,B5: set_nat,G: nat > extended_ereal,F2: nat > extended_ereal] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B5 )
& ( ord_le1083603963089353582_ereal @ ( G @ X4 ) @ ( F2 @ I3 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B5 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_le1083603963089353582_ereal @ ( F2 @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) )
= ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ G @ B5 ) ) ) ) ) ).
% INF_eq
thf(fact_816_INF__eq,axiom,
! [A: set_Extended_ereal,B5: set_a,G: a > extended_ereal,F2: extended_ereal > extended_ereal] :
( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ A )
=> ? [X4: a] :
( ( member_a @ X4 @ B5 )
& ( ord_le1083603963089353582_ereal @ ( G @ X4 ) @ ( F2 @ I3 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B5 )
=> ? [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ A )
& ( ord_le1083603963089353582_ereal @ ( F2 @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) )
= ( comple3556804143462414037_ereal @ ( image_8405481351990995413_ereal @ G @ B5 ) ) ) ) ) ).
% INF_eq
thf(fact_817_INF__eq,axiom,
! [A: set_nat,B5: set_a,G: a > extended_ereal,F2: nat > extended_ereal] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X4: a] :
( ( member_a @ X4 @ B5 )
& ( ord_le1083603963089353582_ereal @ ( G @ X4 ) @ ( F2 @ I3 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B5 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_le1083603963089353582_ereal @ ( F2 @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) )
= ( comple3556804143462414037_ereal @ ( image_8405481351990995413_ereal @ G @ B5 ) ) ) ) ) ).
% INF_eq
thf(fact_818_INF__eq,axiom,
! [A: set_a,B5: set_Extended_ereal,G: extended_ereal > extended_ereal,F2: a > extended_ereal] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ B5 )
& ( ord_le1083603963089353582_ereal @ ( G @ X4 ) @ ( F2 @ I3 ) ) ) )
=> ( ! [J2: extended_ereal] :
( ( member2350847679896131959_ereal @ J2 @ B5 )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ord_le1083603963089353582_ereal @ ( F2 @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( comple3556804143462414037_ereal @ ( image_8405481351990995413_ereal @ F2 @ A ) )
= ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ G @ B5 ) ) ) ) ) ).
% INF_eq
thf(fact_819_INF__eq,axiom,
! [A: set_a,B5: set_nat,G: nat > extended_ereal,F2: a > extended_ereal] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B5 )
& ( ord_le1083603963089353582_ereal @ ( G @ X4 ) @ ( F2 @ I3 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B5 )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ord_le1083603963089353582_ereal @ ( F2 @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( comple3556804143462414037_ereal @ ( image_8405481351990995413_ereal @ F2 @ A ) )
= ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ G @ B5 ) ) ) ) ) ).
% INF_eq
thf(fact_820_INF__eq,axiom,
! [A: set_a,B5: set_a,G: a > extended_ereal,F2: a > extended_ereal] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X4: a] :
( ( member_a @ X4 @ B5 )
& ( ord_le1083603963089353582_ereal @ ( G @ X4 ) @ ( F2 @ I3 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B5 )
=> ? [X4: a] :
( ( member_a @ X4 @ A )
& ( ord_le1083603963089353582_ereal @ ( F2 @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( comple3556804143462414037_ereal @ ( image_8405481351990995413_ereal @ F2 @ A ) )
= ( comple3556804143462414037_ereal @ ( image_8405481351990995413_ereal @ G @ B5 ) ) ) ) ) ).
% INF_eq
thf(fact_821_Inf__sup__eq__top__iff,axiom,
! [B5: set_se6634062954251873166_ereal,A4: set_Extended_ereal] :
( ( ( sup_su2680283192902082946_ereal @ ( comple4418415374894819509_ereal @ B5 ) @ A4 )
= top_to5683747375963461374_ereal )
= ( ! [X: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ X @ B5 )
=> ( ( sup_su2680283192902082946_ereal @ X @ A4 )
= top_to5683747375963461374_ereal ) ) ) ) ).
% Inf_sup_eq_top_iff
thf(fact_822_Inf__sup__eq__top__iff,axiom,
! [B5: set_set_nat,A4: set_nat] :
( ( ( sup_sup_set_nat @ ( comple7806235888213564991et_nat @ B5 ) @ A4 )
= top_top_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ B5 )
=> ( ( sup_sup_set_nat @ X @ A4 )
= top_top_set_nat ) ) ) ) ).
% Inf_sup_eq_top_iff
thf(fact_823_Inf__sup__eq__top__iff,axiom,
! [B5: set_Extended_ereal,A4: extended_ereal] :
( ( ( sup_su7653423775389492130_ereal @ ( comple3556804143462414037_ereal @ B5 ) @ A4 )
= top_to6662034908053899550_ereal )
= ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ B5 )
=> ( ( sup_su7653423775389492130_ereal @ X @ A4 )
= top_to6662034908053899550_ereal ) ) ) ) ).
% Inf_sup_eq_top_iff
thf(fact_824_sup__Inf,axiom,
! [A4: extended_ereal,B5: set_Extended_ereal] :
( ( sup_su7653423775389492130_ereal @ A4 @ ( comple3556804143462414037_ereal @ B5 ) )
= ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ ( sup_su7653423775389492130_ereal @ A4 ) @ B5 ) ) ) ).
% sup_Inf
thf(fact_825_INF__image,axiom,
! [G: extended_ereal > extended_ereal,F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ G @ ( image_6042159593519690757_ereal @ F2 @ A ) ) )
= ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ ( comp_E9177254828515427499_ereal @ G @ F2 ) @ A ) ) ) ).
% INF_image
thf(fact_826_INF__image,axiom,
! [G: extended_ereal > extended_ereal,F2: nat > extended_ereal,A: set_nat] :
( ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ G @ ( image_4309273772856505399_ereal @ F2 @ A ) ) )
= ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ ( comp_E3726099860353345075al_nat @ G @ F2 ) @ A ) ) ) ).
% INF_image
thf(fact_827_INF__image,axiom,
! [G: nat > extended_ereal,F2: extended_ereal > nat,A: set_Extended_ereal] :
( ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ G @ ( image_7659842161140344153al_nat @ F2 @ A ) ) )
= ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ ( comp_n261702227720650419_ereal @ G @ F2 ) @ A ) ) ) ).
% INF_image
thf(fact_828_INF__image,axiom,
! [G: nat > extended_ereal,F2: nat > nat,A: set_nat] :
( ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ G @ ( image_nat_nat @ F2 @ A ) ) )
= ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ ( comp_n13370146242399787al_nat @ G @ F2 ) @ A ) ) ) ).
% INF_image
thf(fact_829_INF__greatest,axiom,
! [A: set_a,U2: set_Extended_ereal,F2: a > set_Extended_ereal] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_le1644982726543182158_ereal @ U2 @ ( F2 @ I3 ) ) )
=> ( ord_le1644982726543182158_ereal @ U2 @ ( comple4418415374894819509_ereal @ ( image_886028290468338613_ereal @ F2 @ A ) ) ) ) ).
% INF_greatest
thf(fact_830_INF__greatest,axiom,
! [A: set_Extended_ereal,U2: extended_ereal,F2: extended_ereal > extended_ereal] :
( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ A )
=> ( ord_le1083603963089353582_ereal @ U2 @ ( F2 @ I3 ) ) )
=> ( ord_le1083603963089353582_ereal @ U2 @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) ) ) ).
% INF_greatest
thf(fact_831_INF__greatest,axiom,
! [A: set_nat,U2: extended_ereal,F2: nat > extended_ereal] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_le1083603963089353582_ereal @ U2 @ ( F2 @ I3 ) ) )
=> ( ord_le1083603963089353582_ereal @ U2 @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) ) ) ).
% INF_greatest
thf(fact_832_INF__greatest,axiom,
! [A: set_a,U2: extended_ereal,F2: a > extended_ereal] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_le1083603963089353582_ereal @ U2 @ ( F2 @ I3 ) ) )
=> ( ord_le1083603963089353582_ereal @ U2 @ ( comple3556804143462414037_ereal @ ( image_8405481351990995413_ereal @ F2 @ A ) ) ) ) ).
% INF_greatest
thf(fact_833_le__INF__iff,axiom,
! [U2: extended_ereal,F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( ord_le1083603963089353582_ereal @ U2 @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) )
= ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A )
=> ( ord_le1083603963089353582_ereal @ U2 @ ( F2 @ X ) ) ) ) ) ).
% le_INF_iff
thf(fact_834_le__INF__iff,axiom,
! [U2: extended_ereal,F2: nat > extended_ereal,A: set_nat] :
( ( ord_le1083603963089353582_ereal @ U2 @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_le1083603963089353582_ereal @ U2 @ ( F2 @ X ) ) ) ) ) ).
% le_INF_iff
thf(fact_835_INF__lower2,axiom,
! [I: a,A: set_a,F2: a > set_Extended_ereal,U2: set_Extended_ereal] :
( ( member_a @ I @ A )
=> ( ( ord_le1644982726543182158_ereal @ ( F2 @ I ) @ U2 )
=> ( ord_le1644982726543182158_ereal @ ( comple4418415374894819509_ereal @ ( image_886028290468338613_ereal @ F2 @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_836_INF__lower2,axiom,
! [I: extended_ereal,A: set_Extended_ereal,F2: extended_ereal > extended_ereal,U2: extended_ereal] :
( ( member2350847679896131959_ereal @ I @ A )
=> ( ( ord_le1083603963089353582_ereal @ ( F2 @ I ) @ U2 )
=> ( ord_le1083603963089353582_ereal @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_837_INF__lower2,axiom,
! [I: nat,A: set_nat,F2: nat > extended_ereal,U2: extended_ereal] :
( ( member_nat @ I @ A )
=> ( ( ord_le1083603963089353582_ereal @ ( F2 @ I ) @ U2 )
=> ( ord_le1083603963089353582_ereal @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_838_INF__lower2,axiom,
! [I: a,A: set_a,F2: a > extended_ereal,U2: extended_ereal] :
( ( member_a @ I @ A )
=> ( ( ord_le1083603963089353582_ereal @ ( F2 @ I ) @ U2 )
=> ( ord_le1083603963089353582_ereal @ ( comple3556804143462414037_ereal @ ( image_8405481351990995413_ereal @ F2 @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_839_INF__mono_H,axiom,
! [F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ! [X3: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( F2 @ X3 ) @ ( G @ X3 ) )
=> ( ord_le1083603963089353582_ereal @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ G @ A ) ) ) ) ).
% INF_mono'
thf(fact_840_INF__mono_H,axiom,
! [F2: nat > extended_ereal,G: nat > extended_ereal,A: set_nat] :
( ! [X3: nat] : ( ord_le1083603963089353582_ereal @ ( F2 @ X3 ) @ ( G @ X3 ) )
=> ( ord_le1083603963089353582_ereal @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ G @ A ) ) ) ) ).
% INF_mono'
thf(fact_841_INF__lower,axiom,
! [I: a,A: set_a,F2: a > set_Extended_ereal] :
( ( member_a @ I @ A )
=> ( ord_le1644982726543182158_ereal @ ( comple4418415374894819509_ereal @ ( image_886028290468338613_ereal @ F2 @ A ) ) @ ( F2 @ I ) ) ) ).
% INF_lower
thf(fact_842_INF__lower,axiom,
! [I: extended_ereal,A: set_Extended_ereal,F2: extended_ereal > extended_ereal] :
( ( member2350847679896131959_ereal @ I @ A )
=> ( ord_le1083603963089353582_ereal @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) @ ( F2 @ I ) ) ) ).
% INF_lower
thf(fact_843_INF__lower,axiom,
! [I: nat,A: set_nat,F2: nat > extended_ereal] :
( ( member_nat @ I @ A )
=> ( ord_le1083603963089353582_ereal @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) @ ( F2 @ I ) ) ) ).
% INF_lower
thf(fact_844_INF__lower,axiom,
! [I: a,A: set_a,F2: a > extended_ereal] :
( ( member_a @ I @ A )
=> ( ord_le1083603963089353582_ereal @ ( comple3556804143462414037_ereal @ ( image_8405481351990995413_ereal @ F2 @ A ) ) @ ( F2 @ I ) ) ) ).
% INF_lower
thf(fact_845_INF__mono,axiom,
! [B5: set_Extended_ereal,A: set_Extended_ereal,F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal] :
( ! [M5: extended_ereal] :
( ( member2350847679896131959_ereal @ M5 @ B5 )
=> ? [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ A )
& ( ord_le1083603963089353582_ereal @ ( F2 @ X4 ) @ ( G @ M5 ) ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ G @ B5 ) ) ) ) ).
% INF_mono
thf(fact_846_INF__mono,axiom,
! [B5: set_nat,A: set_Extended_ereal,F2: extended_ereal > extended_ereal,G: nat > extended_ereal] :
( ! [M5: nat] :
( ( member_nat @ M5 @ B5 )
=> ? [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ A )
& ( ord_le1083603963089353582_ereal @ ( F2 @ X4 ) @ ( G @ M5 ) ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ G @ B5 ) ) ) ) ).
% INF_mono
thf(fact_847_INF__mono,axiom,
! [B5: set_Extended_ereal,A: set_nat,F2: nat > extended_ereal,G: extended_ereal > extended_ereal] :
( ! [M5: extended_ereal] :
( ( member2350847679896131959_ereal @ M5 @ B5 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_le1083603963089353582_ereal @ ( F2 @ X4 ) @ ( G @ M5 ) ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ G @ B5 ) ) ) ) ).
% INF_mono
thf(fact_848_INF__mono,axiom,
! [B5: set_nat,A: set_nat,F2: nat > extended_ereal,G: nat > extended_ereal] :
( ! [M5: nat] :
( ( member_nat @ M5 @ B5 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_le1083603963089353582_ereal @ ( F2 @ X4 ) @ ( G @ M5 ) ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ G @ B5 ) ) ) ) ).
% INF_mono
thf(fact_849_INF__mono,axiom,
! [B5: set_a,A: set_Extended_ereal,F2: extended_ereal > extended_ereal,G: a > extended_ereal] :
( ! [M5: a] :
( ( member_a @ M5 @ B5 )
=> ? [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ A )
& ( ord_le1083603963089353582_ereal @ ( F2 @ X4 ) @ ( G @ M5 ) ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) @ ( comple3556804143462414037_ereal @ ( image_8405481351990995413_ereal @ G @ B5 ) ) ) ) ).
% INF_mono
thf(fact_850_INF__mono,axiom,
! [B5: set_a,A: set_nat,F2: nat > extended_ereal,G: a > extended_ereal] :
( ! [M5: a] :
( ( member_a @ M5 @ B5 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_le1083603963089353582_ereal @ ( F2 @ X4 ) @ ( G @ M5 ) ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) @ ( comple3556804143462414037_ereal @ ( image_8405481351990995413_ereal @ G @ B5 ) ) ) ) ).
% INF_mono
thf(fact_851_INF__eqI,axiom,
! [A: set_a,X2: set_Extended_ereal,F2: a > set_Extended_ereal] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_le1644982726543182158_ereal @ X2 @ ( F2 @ I3 ) ) )
=> ( ! [Y3: set_Extended_ereal] :
( ! [I4: a] :
( ( member_a @ I4 @ A )
=> ( ord_le1644982726543182158_ereal @ Y3 @ ( F2 @ I4 ) ) )
=> ( ord_le1644982726543182158_ereal @ Y3 @ X2 ) )
=> ( ( comple4418415374894819509_ereal @ ( image_886028290468338613_ereal @ F2 @ A ) )
= X2 ) ) ) ).
% INF_eqI
thf(fact_852_INF__eqI,axiom,
! [A: set_Extended_ereal,X2: extended_ereal,F2: extended_ereal > extended_ereal] :
( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ A )
=> ( ord_le1083603963089353582_ereal @ X2 @ ( F2 @ I3 ) ) )
=> ( ! [Y3: extended_ereal] :
( ! [I4: extended_ereal] :
( ( member2350847679896131959_ereal @ I4 @ A )
=> ( ord_le1083603963089353582_ereal @ Y3 @ ( F2 @ I4 ) ) )
=> ( ord_le1083603963089353582_ereal @ Y3 @ X2 ) )
=> ( ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) )
= X2 ) ) ) ).
% INF_eqI
thf(fact_853_INF__eqI,axiom,
! [A: set_nat,X2: extended_ereal,F2: nat > extended_ereal] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_le1083603963089353582_ereal @ X2 @ ( F2 @ I3 ) ) )
=> ( ! [Y3: extended_ereal] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ( ord_le1083603963089353582_ereal @ Y3 @ ( F2 @ I4 ) ) )
=> ( ord_le1083603963089353582_ereal @ Y3 @ X2 ) )
=> ( ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) )
= X2 ) ) ) ).
% INF_eqI
thf(fact_854_INF__eqI,axiom,
! [A: set_a,X2: extended_ereal,F2: a > extended_ereal] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_le1083603963089353582_ereal @ X2 @ ( F2 @ I3 ) ) )
=> ( ! [Y3: extended_ereal] :
( ! [I4: a] :
( ( member_a @ I4 @ A )
=> ( ord_le1083603963089353582_ereal @ Y3 @ ( F2 @ I4 ) ) )
=> ( ord_le1083603963089353582_ereal @ Y3 @ X2 ) )
=> ( ( comple3556804143462414037_ereal @ ( image_8405481351990995413_ereal @ F2 @ A ) )
= X2 ) ) ) ).
% INF_eqI
thf(fact_855_INF__sup,axiom,
! [F2: extended_ereal > extended_ereal,B5: set_Extended_ereal,A4: extended_ereal] :
( ( sup_su7653423775389492130_ereal @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ B5 ) ) @ A4 )
= ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [B4: extended_ereal] : ( sup_su7653423775389492130_ereal @ ( F2 @ B4 ) @ A4 )
@ B5 ) ) ) ).
% INF_sup
thf(fact_856_INF__sup,axiom,
! [F2: nat > extended_ereal,B5: set_nat,A4: extended_ereal] :
( ( sup_su7653423775389492130_ereal @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ B5 ) ) @ A4 )
= ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [B4: nat] : ( sup_su7653423775389492130_ereal @ ( F2 @ B4 ) @ A4 )
@ B5 ) ) ) ).
% INF_sup
thf(fact_857_Inf__sup,axiom,
! [B5: set_Extended_ereal,A4: extended_ereal] :
( ( sup_su7653423775389492130_ereal @ ( comple3556804143462414037_ereal @ B5 ) @ A4 )
= ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [B4: extended_ereal] : ( sup_su7653423775389492130_ereal @ B4 @ A4 )
@ B5 ) ) ) ).
% Inf_sup
thf(fact_858_sup__INF,axiom,
! [A4: extended_ereal,F2: extended_ereal > extended_ereal,B5: set_Extended_ereal] :
( ( sup_su7653423775389492130_ereal @ A4 @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ B5 ) ) )
= ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [B4: extended_ereal] : ( sup_su7653423775389492130_ereal @ A4 @ ( F2 @ B4 ) )
@ B5 ) ) ) ).
% sup_INF
thf(fact_859_sup__INF,axiom,
! [A4: extended_ereal,F2: nat > extended_ereal,B5: set_nat] :
( ( sup_su7653423775389492130_ereal @ A4 @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ B5 ) ) )
= ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [B4: nat] : ( sup_su7653423775389492130_ereal @ A4 @ ( F2 @ B4 ) )
@ B5 ) ) ) ).
% sup_INF
thf(fact_860_INF__sup__distrib2,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,G: extended_ereal > extended_ereal,B5: set_Extended_ereal] :
( ( sup_su7653423775389492130_ereal @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ G @ B5 ) ) )
= ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [A6: extended_ereal] :
( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [B4: extended_ereal] : ( sup_su7653423775389492130_ereal @ ( F2 @ A6 ) @ ( G @ B4 ) )
@ B5 ) )
@ A ) ) ) ).
% INF_sup_distrib2
thf(fact_861_INF__sup__distrib2,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,G: nat > extended_ereal,B5: set_nat] :
( ( sup_su7653423775389492130_ereal @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ G @ B5 ) ) )
= ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [A6: extended_ereal] :
( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [B4: nat] : ( sup_su7653423775389492130_ereal @ ( F2 @ A6 ) @ ( G @ B4 ) )
@ B5 ) )
@ A ) ) ) ).
% INF_sup_distrib2
thf(fact_862_INF__sup__distrib2,axiom,
! [F2: nat > extended_ereal,A: set_nat,G: extended_ereal > extended_ereal,B5: set_Extended_ereal] :
( ( sup_su7653423775389492130_ereal @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ G @ B5 ) ) )
= ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [A6: nat] :
( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [B4: extended_ereal] : ( sup_su7653423775389492130_ereal @ ( F2 @ A6 ) @ ( G @ B4 ) )
@ B5 ) )
@ A ) ) ) ).
% INF_sup_distrib2
thf(fact_863_INF__sup__distrib2,axiom,
! [F2: nat > extended_ereal,A: set_nat,G: nat > extended_ereal,B5: set_nat] :
( ( sup_su7653423775389492130_ereal @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ G @ B5 ) ) )
= ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [A6: nat] :
( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [B4: nat] : ( sup_su7653423775389492130_ereal @ ( F2 @ A6 ) @ ( G @ B4 ) )
@ B5 ) )
@ A ) ) ) ).
% INF_sup_distrib2
thf(fact_864_INT__anti__mono,axiom,
! [A: set_a,B5: set_a,F2: a > set_Extended_ereal,G: a > set_Extended_ereal] :
( ( ord_less_eq_set_a @ A @ B5 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4418415374894819509_ereal @ ( image_886028290468338613_ereal @ F2 @ B5 ) ) @ ( comple4418415374894819509_ereal @ ( image_886028290468338613_ereal @ G @ A ) ) ) ) ) ).
% INT_anti_mono
thf(fact_865_INT__anti__mono,axiom,
! [A: set_Extended_ereal,B5: set_Extended_ereal,F2: extended_ereal > set_Extended_ereal,G: extended_ereal > set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A @ B5 )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4418415374894819509_ereal @ ( image_5562094264469218789_ereal @ F2 @ B5 ) ) @ ( comple4418415374894819509_ereal @ ( image_5562094264469218789_ereal @ G @ A ) ) ) ) ) ).
% INT_anti_mono
thf(fact_866_INT__greatest,axiom,
! [A: set_a,C2: set_Extended_ereal,B5: a > set_Extended_ereal] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ord_le1644982726543182158_ereal @ C2 @ ( B5 @ X3 ) ) )
=> ( ord_le1644982726543182158_ereal @ C2 @ ( comple4418415374894819509_ereal @ ( image_886028290468338613_ereal @ B5 @ A ) ) ) ) ).
% INT_greatest
thf(fact_867_INT__lower,axiom,
! [A4: a,A: set_a,B5: a > set_Extended_ereal] :
( ( member_a @ A4 @ A )
=> ( ord_le1644982726543182158_ereal @ ( comple4418415374894819509_ereal @ ( image_886028290468338613_ereal @ B5 @ A ) ) @ ( B5 @ A4 ) ) ) ).
% INT_lower
thf(fact_868_uminus__Inf,axiom,
! [A: set_se6634062954251873166_ereal] :
( ( uminus5895154729394068773_ereal @ ( comple4418415374894819509_ereal @ A ) )
= ( comple4319282863272126363_ereal @ ( image_6293272304431515653_ereal @ uminus5895154729394068773_ereal @ A ) ) ) ).
% uminus_Inf
thf(fact_869_uminus__Sup,axiom,
! [A: set_se6634062954251873166_ereal] :
( ( uminus5895154729394068773_ereal @ ( comple4319282863272126363_ereal @ A ) )
= ( comple4418415374894819509_ereal @ ( image_6293272304431515653_ereal @ uminus5895154729394068773_ereal @ A ) ) ) ).
% uminus_Sup
thf(fact_870_Inf__INT__eq2,axiom,
( comple1974978180206336373st_o_o
= ( ^ [S5: set_op7420493560124892494st_o_o,X: option_list_o,Y: option_list_o] : ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X @ Y ) @ ( comple3647107874392957070list_o @ ( image_7941296275600097962list_o @ collec8383569672281745810list_o @ ( image_132443102827437503st_o_o @ produc2885628514414960748st_o_o @ S5 ) ) ) ) ) ) ).
% Inf_INT_eq2
thf(fact_871_INF__superset__mono,axiom,
! [B5: set_a,A: set_a,F2: a > set_Extended_ereal,G: a > set_Extended_ereal] :
( ( ord_less_eq_set_a @ B5 @ A )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B5 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4418415374894819509_ereal @ ( image_886028290468338613_ereal @ F2 @ A ) ) @ ( comple4418415374894819509_ereal @ ( image_886028290468338613_ereal @ G @ B5 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_872_INF__superset__mono,axiom,
! [B5: set_Extended_ereal,A: set_Extended_ereal,F2: extended_ereal > set_Extended_ereal,G: extended_ereal > set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B5 @ A )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ B5 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4418415374894819509_ereal @ ( image_5562094264469218789_ereal @ F2 @ A ) ) @ ( comple4418415374894819509_ereal @ ( image_5562094264469218789_ereal @ G @ B5 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_873_INF__superset__mono,axiom,
! [B5: set_nat,A: set_nat,F2: nat > extended_ereal,G: nat > extended_ereal] :
( ( ord_less_eq_set_nat @ B5 @ A )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B5 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ G @ B5 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_874_INF__superset__mono,axiom,
! [B5: set_a,A: set_a,F2: a > extended_ereal,G: a > extended_ereal] :
( ( ord_less_eq_set_a @ B5 @ A )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B5 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple3556804143462414037_ereal @ ( image_8405481351990995413_ereal @ F2 @ A ) ) @ ( comple3556804143462414037_ereal @ ( image_8405481351990995413_ereal @ G @ B5 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_875_INF__superset__mono,axiom,
! [B5: set_Extended_ereal,A: set_Extended_ereal,F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B5 @ A )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ B5 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ G @ B5 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_876_image__INT,axiom,
! [F2: nat > extended_ereal,C2: set_nat,A: set_a,B5: a > set_nat,J3: a] :
( ( inj_on6191532827271902155_ereal @ F2 @ C2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( B5 @ X3 ) @ C2 ) )
=> ( ( member_a @ J3 @ A )
=> ( ( image_4309273772856505399_ereal @ F2 @ ( comple7806235888213564991et_nat @ ( image_a_set_nat @ B5 @ A ) ) )
= ( comple4418415374894819509_ereal
@ ( image_886028290468338613_ereal
@ ^ [X: a] : ( image_4309273772856505399_ereal @ F2 @ ( B5 @ X ) )
@ A ) ) ) ) ) ) ).
% image_INT
thf(fact_877_image__INT,axiom,
! [F2: extended_ereal > extended_ereal,C2: set_Extended_ereal,A: set_a,B5: a > set_Extended_ereal,J3: a] :
( ( inj_on7162434037990268785_ereal @ F2 @ C2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ord_le1644982726543182158_ereal @ ( B5 @ X3 ) @ C2 ) )
=> ( ( member_a @ J3 @ A )
=> ( ( image_6042159593519690757_ereal @ F2 @ ( comple4418415374894819509_ereal @ ( image_886028290468338613_ereal @ B5 @ A ) ) )
= ( comple4418415374894819509_ereal
@ ( image_886028290468338613_ereal
@ ^ [X: a] : ( image_6042159593519690757_ereal @ F2 @ ( B5 @ X ) )
@ A ) ) ) ) ) ) ).
% image_INT
thf(fact_878_INF__Sigma,axiom,
! [F2: extended_ereal > extended_ereal > extended_ereal,B5: extended_ereal > set_Extended_ereal,A: set_Extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ ( F2 @ I2 ) @ ( B5 @ I2 ) ) )
@ A ) )
= ( comple3556804143462414037_ereal
@ ( image_8011834658935918572_ereal
@ ^ [P3: produc5501587555545223847_ereal] : ( F2 @ ( produc8000661846298591107_ereal @ P3 ) @ ( produc7555089578395445445_ereal @ P3 ) )
@ ( produc8095709571603465288_ereal @ A @ B5 ) ) ) ) ).
% INF_Sigma
thf(fact_879_INF__Sigma,axiom,
! [F2: extended_ereal > nat > extended_ereal,B5: extended_ereal > set_nat,A: set_Extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ ( F2 @ I2 ) @ ( B5 @ I2 ) ) )
@ A ) )
= ( comple3556804143462414037_ereal
@ ( image_3315421938599550714_ereal
@ ^ [P3: produc5547956294148499661al_nat] : ( F2 @ ( produc3755854452330217819al_nat @ P3 ) @ ( produc5370844913463222425al_nat @ P3 ) )
@ ( produc4220900302008768982al_nat @ A @ B5 ) ) ) ) ).
% INF_Sigma
thf(fact_880_INF__Sigma,axiom,
! [F2: nat > extended_ereal > extended_ereal,B5: nat > set_Extended_ereal,A: set_nat] :
( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ ( F2 @ I2 ) @ ( B5 @ I2 ) ) )
@ A ) )
= ( comple3556804143462414037_ereal
@ ( image_4404873737357082836_ereal
@ ^ [P3: produc7896515489273245043_ereal] : ( F2 @ ( produc405286064046379065_ereal @ P3 ) @ ( produc2020276525179383671_ereal @ P3 ) )
@ ( produc870331913724930228_ereal @ A @ B5 ) ) ) ) ).
% INF_Sigma
thf(fact_881_INF__Sigma,axiom,
! [F2: nat > nat > extended_ereal,B5: nat > set_nat,A: set_nat] :
( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ ( F2 @ I2 ) @ ( B5 @ I2 ) ) )
@ A ) )
= ( comple3556804143462414037_ereal
@ ( image_3495945180199258130_ereal
@ ^ [P3: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ P3 ) @ ( product_snd_nat_nat @ P3 ) )
@ ( produc457027306803732586at_nat @ A @ B5 ) ) ) ) ).
% INF_Sigma
thf(fact_882_INF__Sigma,axiom,
! [F2: a > b > extended_ereal,B5: a > set_b,A: set_a] :
( ( comple3556804143462414037_ereal
@ ( image_8405481351990995413_ereal
@ ^ [I2: a] : ( comple3556804143462414037_ereal @ ( image_5319725110001000852_ereal @ ( F2 @ I2 ) @ ( B5 @ I2 ) ) )
@ A ) )
= ( comple3556804143462414037_ereal
@ ( image_5361069211733868459_ereal
@ ^ [P3: product_prod_a_b] : ( F2 @ ( product_fst_a_b @ P3 ) @ ( product_snd_a_b @ P3 ) )
@ ( product_Sigma_a_b @ A @ B5 ) ) ) ) ).
% INF_Sigma
thf(fact_883_ccINF__top,axiom,
! [A: set_Extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : top_to6662034908053899550_ereal
@ A ) )
= top_to6662034908053899550_ereal ) ).
% ccINF_top
thf(fact_884_ccINF__top,axiom,
! [A: set_nat] :
( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X: nat] : top_to6662034908053899550_ereal
@ A ) )
= top_to6662034908053899550_ereal ) ).
% ccINF_top
thf(fact_885_snd__image__times,axiom,
! [A: set_a,B5: set_b] :
( ( ( A = bot_bot_set_a )
=> ( ( image_2802296252294471260_a_b_b @ product_snd_a_b
@ ( product_Sigma_a_b @ A
@ ^ [Uu: a] : B5 ) )
= bot_bot_set_b ) )
& ( ( A != bot_bot_set_a )
=> ( ( image_2802296252294471260_a_b_b @ product_snd_a_b
@ ( product_Sigma_a_b @ A
@ ^ [Uu: a] : B5 ) )
= B5 ) ) ) ).
% snd_image_times
thf(fact_886_snd__image__times,axiom,
! [A: set_Extended_ereal,B5: set_Extended_ereal] :
( ( ( A = bot_bo8367695208629047834_ereal )
=> ( ( image_8011834658935918572_ereal @ produc7555089578395445445_ereal
@ ( produc8095709571603465288_ereal @ A
@ ^ [Uu: extended_ereal] : B5 ) )
= bot_bo8367695208629047834_ereal ) )
& ( ( A != bot_bo8367695208629047834_ereal )
=> ( ( image_8011834658935918572_ereal @ produc7555089578395445445_ereal
@ ( produc8095709571603465288_ereal @ A
@ ^ [Uu: extended_ereal] : B5 ) )
= B5 ) ) ) ).
% snd_image_times
thf(fact_887_fst__image__times,axiom,
! [B5: set_b,A: set_a] :
( ( ( B5 = bot_bot_set_b )
=> ( ( image_2802296252294471259_a_b_a @ product_fst_a_b
@ ( product_Sigma_a_b @ A
@ ^ [Uu: a] : B5 ) )
= bot_bot_set_a ) )
& ( ( B5 != bot_bot_set_b )
=> ( ( image_2802296252294471259_a_b_a @ product_fst_a_b
@ ( product_Sigma_a_b @ A
@ ^ [Uu: a] : B5 ) )
= A ) ) ) ).
% fst_image_times
thf(fact_888_fst__image__times,axiom,
! [B5: set_Extended_ereal,A: set_Extended_ereal] :
( ( ( B5 = bot_bo8367695208629047834_ereal )
=> ( ( image_8011834658935918572_ereal @ produc8000661846298591107_ereal
@ ( produc8095709571603465288_ereal @ A
@ ^ [Uu: extended_ereal] : B5 ) )
= bot_bo8367695208629047834_ereal ) )
& ( ( B5 != bot_bo8367695208629047834_ereal )
=> ( ( image_8011834658935918572_ereal @ produc8000661846298591107_ereal
@ ( produc8095709571603465288_ereal @ A
@ ^ [Uu: extended_ereal] : B5 ) )
= A ) ) ) ).
% fst_image_times
thf(fact_889_image__is__empty,axiom,
! [F2: nat > extended_ereal,A: set_nat] :
( ( ( image_4309273772856505399_ereal @ F2 @ A )
= bot_bo8367695208629047834_ereal )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_890_image__is__empty,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ A )
= bot_bo8367695208629047834_ereal )
= ( A = bot_bo8367695208629047834_ereal ) ) ).
% image_is_empty
thf(fact_891_empty__is__image,axiom,
! [F2: nat > extended_ereal,A: set_nat] :
( ( bot_bo8367695208629047834_ereal
= ( image_4309273772856505399_ereal @ F2 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_892_empty__is__image,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( bot_bo8367695208629047834_ereal
= ( image_6042159593519690757_ereal @ F2 @ A ) )
= ( A = bot_bo8367695208629047834_ereal ) ) ).
% empty_is_image
thf(fact_893_image__empty,axiom,
! [F2: nat > extended_ereal] :
( ( image_4309273772856505399_ereal @ F2 @ bot_bot_set_nat )
= bot_bo8367695208629047834_ereal ) ).
% image_empty
thf(fact_894_image__empty,axiom,
! [F2: extended_ereal > extended_ereal] :
( ( image_6042159593519690757_ereal @ F2 @ bot_bo8367695208629047834_ereal )
= bot_bo8367695208629047834_ereal ) ).
% image_empty
thf(fact_895_sup__bot_Oright__neutral,axiom,
! [A4: set_Extended_ereal] :
( ( sup_su2680283192902082946_ereal @ A4 @ bot_bo8367695208629047834_ereal )
= A4 ) ).
% sup_bot.right_neutral
thf(fact_896_sup__bot_Oneutr__eq__iff,axiom,
! [A4: set_Extended_ereal,B: set_Extended_ereal] :
( ( bot_bo8367695208629047834_ereal
= ( sup_su2680283192902082946_ereal @ A4 @ B ) )
= ( ( A4 = bot_bo8367695208629047834_ereal )
& ( B = bot_bo8367695208629047834_ereal ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_897_sup__bot_Oleft__neutral,axiom,
! [A4: set_Extended_ereal] :
( ( sup_su2680283192902082946_ereal @ bot_bo8367695208629047834_ereal @ A4 )
= A4 ) ).
% sup_bot.left_neutral
thf(fact_898_sup__bot_Oeq__neutr__iff,axiom,
! [A4: set_Extended_ereal,B: set_Extended_ereal] :
( ( ( sup_su2680283192902082946_ereal @ A4 @ B )
= bot_bo8367695208629047834_ereal )
= ( ( A4 = bot_bo8367695208629047834_ereal )
& ( B = bot_bo8367695208629047834_ereal ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_899_sup__eq__bot__iff,axiom,
! [X2: set_Extended_ereal,Y2: set_Extended_ereal] :
( ( ( sup_su2680283192902082946_ereal @ X2 @ Y2 )
= bot_bo8367695208629047834_ereal )
= ( ( X2 = bot_bo8367695208629047834_ereal )
& ( Y2 = bot_bo8367695208629047834_ereal ) ) ) ).
% sup_eq_bot_iff
thf(fact_900_bot__eq__sup__iff,axiom,
! [X2: set_Extended_ereal,Y2: set_Extended_ereal] :
( ( bot_bo8367695208629047834_ereal
= ( sup_su2680283192902082946_ereal @ X2 @ Y2 ) )
= ( ( X2 = bot_bo8367695208629047834_ereal )
& ( Y2 = bot_bo8367695208629047834_ereal ) ) ) ).
% bot_eq_sup_iff
thf(fact_901_sup__bot__right,axiom,
! [X2: set_Extended_ereal] :
( ( sup_su2680283192902082946_ereal @ X2 @ bot_bo8367695208629047834_ereal )
= X2 ) ).
% sup_bot_right
thf(fact_902_sup__bot__left,axiom,
! [X2: set_Extended_ereal] :
( ( sup_su2680283192902082946_ereal @ bot_bo8367695208629047834_ereal @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_903_inj__on__empty,axiom,
! [F2: extended_ereal > extended_ereal] : ( inj_on7162434037990268785_ereal @ F2 @ bot_bo8367695208629047834_ereal ) ).
% inj_on_empty
thf(fact_904_Un__empty,axiom,
! [A: set_Extended_ereal,B5: set_Extended_ereal] :
( ( ( sup_su2680283192902082946_ereal @ A @ B5 )
= bot_bo8367695208629047834_ereal )
= ( ( A = bot_bo8367695208629047834_ereal )
& ( B5 = bot_bo8367695208629047834_ereal ) ) ) ).
% Un_empty
thf(fact_905_Times__empty,axiom,
! [A: set_Extended_ereal,B5: set_Extended_ereal] :
( ( ( produc8095709571603465288_ereal @ A
@ ^ [Uu: extended_ereal] : B5 )
= bot_bo4002835157671732723_ereal )
= ( ( A = bot_bo8367695208629047834_ereal )
| ( B5 = bot_bo8367695208629047834_ereal ) ) ) ).
% Times_empty
thf(fact_906_ccSUP__bot,axiom,
! [A: set_Extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : bot_bo2710585358178759738_ereal
@ A ) )
= bot_bo2710585358178759738_ereal ) ).
% ccSUP_bot
thf(fact_907_ccSUP__bot,axiom,
! [A: set_nat] :
( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X: nat] : bot_bo2710585358178759738_ereal
@ A ) )
= bot_bo2710585358178759738_ereal ) ).
% ccSUP_bot
thf(fact_908_SUP__bot__conv_I2_J,axiom,
! [B5: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( bot_bo2710585358178759738_ereal
= ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ B5 @ A ) ) )
= ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A )
=> ( ( B5 @ X )
= bot_bo2710585358178759738_ereal ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_909_SUP__bot__conv_I2_J,axiom,
! [B5: nat > extended_ereal,A: set_nat] :
( ( bot_bo2710585358178759738_ereal
= ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ B5 @ A ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( B5 @ X )
= bot_bo2710585358178759738_ereal ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_910_SUP__bot__conv_I1_J,axiom,
! [B5: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ B5 @ A ) )
= bot_bo2710585358178759738_ereal )
= ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A )
=> ( ( B5 @ X )
= bot_bo2710585358178759738_ereal ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_911_SUP__bot__conv_I1_J,axiom,
! [B5: nat > extended_ereal,A: set_nat] :
( ( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ B5 @ A ) )
= bot_bo2710585358178759738_ereal )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( B5 @ X )
= bot_bo2710585358178759738_ereal ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_912_SUP__bot,axiom,
! [A: set_Extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : bot_bo2710585358178759738_ereal
@ A ) )
= bot_bo2710585358178759738_ereal ) ).
% SUP_bot
thf(fact_913_SUP__bot,axiom,
! [A: set_nat] :
( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X: nat] : bot_bo2710585358178759738_ereal
@ A ) )
= bot_bo2710585358178759738_ereal ) ).
% SUP_bot
thf(fact_914_ccSUP__const,axiom,
! [A: set_nat,F2: extended_ereal] :
( ( A != bot_bot_set_nat )
=> ( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : F2
@ A ) )
= F2 ) ) ).
% ccSUP_const
thf(fact_915_ccSUP__const,axiom,
! [A: set_Extended_ereal,F2: extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : F2
@ A ) )
= F2 ) ) ).
% ccSUP_const
thf(fact_916_SUP__const,axiom,
! [A: set_nat,F2: extended_ereal] :
( ( A != bot_bot_set_nat )
=> ( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : F2
@ A ) )
= F2 ) ) ).
% SUP_const
thf(fact_917_SUP__const,axiom,
! [A: set_Extended_ereal,F2: extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : F2
@ A ) )
= F2 ) ) ).
% SUP_const
thf(fact_918_ccINF__const,axiom,
! [A: set_nat,F2: extended_ereal] :
( ( A != bot_bot_set_nat )
=> ( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : F2
@ A ) )
= F2 ) ) ).
% ccINF_const
thf(fact_919_ccINF__const,axiom,
! [A: set_Extended_ereal,F2: extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : F2
@ A ) )
= F2 ) ) ).
% ccINF_const
thf(fact_920_INF__const,axiom,
! [A: set_nat,F2: extended_ereal] :
( ( A != bot_bot_set_nat )
=> ( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : F2
@ A ) )
= F2 ) ) ).
% INF_const
thf(fact_921_INF__const,axiom,
! [A: set_Extended_ereal,F2: extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : F2
@ A ) )
= F2 ) ) ).
% INF_const
thf(fact_922_UN__constant,axiom,
! [A: set_Extended_ereal,C: set_Extended_ereal] :
( ( ( A = bot_bo8367695208629047834_ereal )
=> ( ( comple4319282863272126363_ereal
@ ( image_5562094264469218789_ereal
@ ^ [Y: extended_ereal] : C
@ A ) )
= bot_bo8367695208629047834_ereal ) )
& ( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple4319282863272126363_ereal
@ ( image_5562094264469218789_ereal
@ ^ [Y: extended_ereal] : C
@ A ) )
= C ) ) ) ).
% UN_constant
thf(fact_923_ccSUP__empty,axiom,
! [F2: extended_ereal > set_Extended_ereal] :
( ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ F2 @ bot_bo8367695208629047834_ereal ) )
= bot_bo8367695208629047834_ereal ) ).
% ccSUP_empty
thf(fact_924_ccSUP__empty,axiom,
! [F2: nat > extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ bot_bot_set_nat ) )
= bot_bo2710585358178759738_ereal ) ).
% ccSUP_empty
thf(fact_925_ccSUP__empty,axiom,
! [F2: extended_ereal > extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ bot_bo8367695208629047834_ereal ) )
= bot_bo2710585358178759738_ereal ) ).
% ccSUP_empty
thf(fact_926_ccINF__empty,axiom,
! [F2: extended_ereal > set_Extended_ereal] :
( ( comple4418415374894819509_ereal @ ( image_5562094264469218789_ereal @ F2 @ bot_bo8367695208629047834_ereal ) )
= top_to5683747375963461374_ereal ) ).
% ccINF_empty
thf(fact_927_ccINF__empty,axiom,
! [F2: extended_ereal > set_nat] :
( ( comple7806235888213564991et_nat @ ( image_3090908713637162255et_nat @ F2 @ bot_bo8367695208629047834_ereal ) )
= top_top_set_nat ) ).
% ccINF_empty
thf(fact_928_ccINF__empty,axiom,
! [F2: nat > extended_ereal] :
( ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ bot_bot_set_nat ) )
= top_to6662034908053899550_ereal ) ).
% ccINF_empty
thf(fact_929_ccINF__empty,axiom,
! [F2: extended_ereal > extended_ereal] :
( ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ bot_bo8367695208629047834_ereal ) )
= top_to6662034908053899550_ereal ) ).
% ccINF_empty
thf(fact_930_INT__constant,axiom,
! [A: set_Extended_ereal,C: set_Extended_ereal] :
( ( ( A = bot_bo8367695208629047834_ereal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5562094264469218789_ereal
@ ^ [Y: extended_ereal] : C
@ A ) )
= top_to5683747375963461374_ereal ) )
& ( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5562094264469218789_ereal
@ ^ [Y: extended_ereal] : C
@ A ) )
= C ) ) ) ).
% INT_constant
thf(fact_931_INT__constant,axiom,
! [A: set_Extended_ereal,C: set_nat] :
( ( ( A = bot_bo8367695208629047834_ereal )
=> ( ( comple7806235888213564991et_nat
@ ( image_3090908713637162255et_nat
@ ^ [Y: extended_ereal] : C
@ A ) )
= top_top_set_nat ) )
& ( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple7806235888213564991et_nat
@ ( image_3090908713637162255et_nat
@ ^ [Y: extended_ereal] : C
@ A ) )
= C ) ) ) ).
% INT_constant
thf(fact_932_UN__simps_I2_J,axiom,
! [C2: set_Extended_ereal,A: extended_ereal > set_Extended_ereal,B5: set_Extended_ereal] :
( ( ( C2 = bot_bo8367695208629047834_ereal )
=> ( ( comple4319282863272126363_ereal
@ ( image_5562094264469218789_ereal
@ ^ [X: extended_ereal] : ( sup_su2680283192902082946_ereal @ ( A @ X ) @ B5 )
@ C2 ) )
= bot_bo8367695208629047834_ereal ) )
& ( ( C2 != bot_bo8367695208629047834_ereal )
=> ( ( comple4319282863272126363_ereal
@ ( image_5562094264469218789_ereal
@ ^ [X: extended_ereal] : ( sup_su2680283192902082946_ereal @ ( A @ X ) @ B5 )
@ C2 ) )
= ( sup_su2680283192902082946_ereal @ ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ A @ C2 ) ) @ B5 ) ) ) ) ).
% UN_simps(2)
thf(fact_933_UN__simps_I3_J,axiom,
! [C2: set_Extended_ereal,A: set_Extended_ereal,B5: extended_ereal > set_Extended_ereal] :
( ( ( C2 = bot_bo8367695208629047834_ereal )
=> ( ( comple4319282863272126363_ereal
@ ( image_5562094264469218789_ereal
@ ^ [X: extended_ereal] : ( sup_su2680283192902082946_ereal @ A @ ( B5 @ X ) )
@ C2 ) )
= bot_bo8367695208629047834_ereal ) )
& ( ( C2 != bot_bo8367695208629047834_ereal )
=> ( ( comple4319282863272126363_ereal
@ ( image_5562094264469218789_ereal
@ ^ [X: extended_ereal] : ( sup_su2680283192902082946_ereal @ A @ ( B5 @ X ) )
@ C2 ) )
= ( sup_su2680283192902082946_ereal @ A @ ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ B5 @ C2 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_934_SUP__countable__SUP,axiom,
! [A: set_nat,G: nat > extended_ereal] :
( ( A != bot_bot_set_nat )
=> ? [F4: nat > extended_ereal] :
( ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F4 @ top_top_set_nat ) @ ( image_4309273772856505399_ereal @ G @ A ) )
& ( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ A ) )
= ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F4 @ top_top_set_nat ) ) ) ) ) ).
% SUP_countable_SUP
thf(fact_935_SUP__countable__SUP,axiom,
! [A: set_Extended_ereal,G: extended_ereal > extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ? [F4: nat > extended_ereal] :
( ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F4 @ top_top_set_nat ) @ ( image_6042159593519690757_ereal @ G @ A ) )
& ( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ G @ A ) )
= ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F4 @ top_top_set_nat ) ) ) ) ) ).
% SUP_countable_SUP
thf(fact_936_SUP__empty,axiom,
! [F2: extended_ereal > set_Extended_ereal] :
( ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ F2 @ bot_bo8367695208629047834_ereal ) )
= bot_bo8367695208629047834_ereal ) ).
% SUP_empty
thf(fact_937_SUP__empty,axiom,
! [F2: nat > extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ bot_bot_set_nat ) )
= bot_bo2710585358178759738_ereal ) ).
% SUP_empty
thf(fact_938_SUP__empty,axiom,
! [F2: extended_ereal > extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ bot_bo8367695208629047834_ereal ) )
= bot_bo2710585358178759738_ereal ) ).
% SUP_empty
thf(fact_939_SUP__constant,axiom,
! [A: set_Extended_ereal,C: set_Extended_ereal] :
( ( ( A = bot_bo8367695208629047834_ereal )
=> ( ( comple4319282863272126363_ereal
@ ( image_5562094264469218789_ereal
@ ^ [Y: extended_ereal] : C
@ A ) )
= bot_bo8367695208629047834_ereal ) )
& ( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple4319282863272126363_ereal
@ ( image_5562094264469218789_ereal
@ ^ [Y: extended_ereal] : C
@ A ) )
= C ) ) ) ).
% SUP_constant
thf(fact_940_SUP__constant,axiom,
! [A: set_nat,C: extended_ereal] :
( ( ( A = bot_bot_set_nat )
=> ( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [Y: nat] : C
@ A ) )
= bot_bo2710585358178759738_ereal ) )
& ( ( A != bot_bot_set_nat )
=> ( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [Y: nat] : C
@ A ) )
= C ) ) ) ).
% SUP_constant
thf(fact_941_SUP__constant,axiom,
! [A: set_Extended_ereal,C: extended_ereal] :
( ( ( A = bot_bo8367695208629047834_ereal )
=> ( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [Y: extended_ereal] : C
@ A ) )
= bot_bo2710585358178759738_ereal ) )
& ( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [Y: extended_ereal] : C
@ A ) )
= C ) ) ) ).
% SUP_constant
thf(fact_942_inj__on__Inter,axiom,
! [S4: set_se6634062954251873166_ereal,F2: extended_ereal > extended_ereal] :
( ( S4 != bot_bo7400643019497942010_ereal )
=> ( ! [A9: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ A9 @ S4 )
=> ( inj_on7162434037990268785_ereal @ F2 @ A9 ) )
=> ( inj_on7162434037990268785_ereal @ F2 @ ( comple4418415374894819509_ereal @ S4 ) ) ) ) ).
% inj_on_Inter
thf(fact_943_Un__empty__right,axiom,
! [A: set_Extended_ereal] :
( ( sup_su2680283192902082946_ereal @ A @ bot_bo8367695208629047834_ereal )
= A ) ).
% Un_empty_right
thf(fact_944_Un__empty__left,axiom,
! [B5: set_Extended_ereal] :
( ( sup_su2680283192902082946_ereal @ bot_bo8367695208629047834_ereal @ B5 )
= B5 ) ).
% Un_empty_left
thf(fact_945_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_Extended_ereal] :
( ( sup_su2680283192902082946_ereal @ X2 @ bot_bo8367695208629047834_ereal )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_946_times__eq__iff,axiom,
! [A: set_Extended_ereal,B5: set_Extended_ereal,C2: set_Extended_ereal,D: set_Extended_ereal] :
( ( ( produc8095709571603465288_ereal @ A
@ ^ [Uu: extended_ereal] : B5 )
= ( produc8095709571603465288_ereal @ C2
@ ^ [Uu: extended_ereal] : D ) )
= ( ( ( A = C2 )
& ( B5 = D ) )
| ( ( ( A = bot_bo8367695208629047834_ereal )
| ( B5 = bot_bo8367695208629047834_ereal ) )
& ( ( C2 = bot_bo8367695208629047834_ereal )
| ( D = bot_bo8367695208629047834_ereal ) ) ) ) ) ).
% times_eq_iff
thf(fact_947_subset__emptyI,axiom,
! [A: set_a] :
( ! [X3: a] :
~ ( member_a @ X3 @ A )
=> ( ord_less_eq_set_a @ A @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_948_subset__emptyI,axiom,
! [A: set_Extended_ereal] :
( ! [X3: extended_ereal] :
~ ( member2350847679896131959_ereal @ X3 @ A )
=> ( ord_le1644982726543182158_ereal @ A @ bot_bo8367695208629047834_ereal ) ) ).
% subset_emptyI
thf(fact_949_SUP__eq__const,axiom,
! [I5: set_nat,F2: nat > extended_ereal,X2: extended_ereal] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( F2 @ I3 )
= X2 ) )
=> ( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ I5 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_950_SUP__eq__const,axiom,
! [I5: set_a,F2: a > extended_ereal,X2: extended_ereal] :
( ( I5 != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I5 )
=> ( ( F2 @ I3 )
= X2 ) )
=> ( ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ F2 @ I5 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_951_SUP__eq__const,axiom,
! [I5: set_Extended_ereal,F2: extended_ereal > extended_ereal,X2: extended_ereal] :
( ( I5 != bot_bo8367695208629047834_ereal )
=> ( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ I5 )
=> ( ( F2 @ I3 )
= X2 ) )
=> ( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ I5 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_952_INF__eq__const,axiom,
! [I5: set_nat,F2: nat > extended_ereal,X2: extended_ereal] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( F2 @ I3 )
= X2 ) )
=> ( ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ I5 ) )
= X2 ) ) ) ).
% INF_eq_const
thf(fact_953_INF__eq__const,axiom,
! [I5: set_a,F2: a > extended_ereal,X2: extended_ereal] :
( ( I5 != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I5 )
=> ( ( F2 @ I3 )
= X2 ) )
=> ( ( comple3556804143462414037_ereal @ ( image_8405481351990995413_ereal @ F2 @ I5 ) )
= X2 ) ) ) ).
% INF_eq_const
thf(fact_954_INF__eq__const,axiom,
! [I5: set_Extended_ereal,F2: extended_ereal > extended_ereal,X2: extended_ereal] :
( ( I5 != bot_bo8367695208629047834_ereal )
=> ( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ I5 )
=> ( ( F2 @ I3 )
= X2 ) )
=> ( ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ I5 ) )
= X2 ) ) ) ).
% INF_eq_const
thf(fact_955_INT__empty,axiom,
! [B5: extended_ereal > set_Extended_ereal] :
( ( comple4418415374894819509_ereal @ ( image_5562094264469218789_ereal @ B5 @ bot_bo8367695208629047834_ereal ) )
= top_to5683747375963461374_ereal ) ).
% INT_empty
thf(fact_956_INT__empty,axiom,
! [B5: extended_ereal > set_nat] :
( ( comple7806235888213564991et_nat @ ( image_3090908713637162255et_nat @ B5 @ bot_bo8367695208629047834_ereal ) )
= top_top_set_nat ) ).
% INT_empty
thf(fact_957_inj__on__INTER,axiom,
! [I5: set_a,F2: extended_ereal > extended_ereal,A: a > set_Extended_ereal] :
( ( I5 != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I5 )
=> ( inj_on7162434037990268785_ereal @ F2 @ ( A @ I3 ) ) )
=> ( inj_on7162434037990268785_ereal @ F2 @ ( comple4418415374894819509_ereal @ ( image_886028290468338613_ereal @ A @ I5 ) ) ) ) ) ).
% inj_on_INTER
thf(fact_958_inj__on__INTER,axiom,
! [I5: set_Extended_ereal,F2: extended_ereal > extended_ereal,A: extended_ereal > set_Extended_ereal] :
( ( I5 != bot_bo8367695208629047834_ereal )
=> ( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ I5 )
=> ( inj_on7162434037990268785_ereal @ F2 @ ( A @ I3 ) ) )
=> ( inj_on7162434037990268785_ereal @ F2 @ ( comple4418415374894819509_ereal @ ( image_5562094264469218789_ereal @ A @ I5 ) ) ) ) ) ).
% inj_on_INTER
thf(fact_959_UN__empty,axiom,
! [B5: extended_ereal > set_Extended_ereal] :
( ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ B5 @ bot_bo8367695208629047834_ereal ) )
= bot_bo8367695208629047834_ereal ) ).
% UN_empty
thf(fact_960_SUP__eq__iff,axiom,
! [I5: set_a,C: set_Extended_ereal,F2: a > set_Extended_ereal] :
( ( I5 != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I5 )
=> ( ord_le1644982726543182158_ereal @ C @ ( F2 @ I3 ) ) )
=> ( ( ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ F2 @ I5 ) )
= C )
= ( ! [X: a] :
( ( member_a @ X @ I5 )
=> ( ( F2 @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_961_SUP__eq__iff,axiom,
! [I5: set_Extended_ereal,C: set_Extended_ereal,F2: extended_ereal > set_Extended_ereal] :
( ( I5 != bot_bo8367695208629047834_ereal )
=> ( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ I5 )
=> ( ord_le1644982726543182158_ereal @ C @ ( F2 @ I3 ) ) )
=> ( ( ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ F2 @ I5 ) )
= C )
= ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ I5 )
=> ( ( F2 @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_962_SUP__eq__iff,axiom,
! [I5: set_nat,C: extended_ereal,F2: nat > extended_ereal] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_le1083603963089353582_ereal @ C @ ( F2 @ I3 ) ) )
=> ( ( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ I5 ) )
= C )
= ( ! [X: nat] :
( ( member_nat @ X @ I5 )
=> ( ( F2 @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_963_SUP__eq__iff,axiom,
! [I5: set_a,C: extended_ereal,F2: a > extended_ereal] :
( ( I5 != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I5 )
=> ( ord_le1083603963089353582_ereal @ C @ ( F2 @ I3 ) ) )
=> ( ( ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ F2 @ I5 ) )
= C )
= ( ! [X: a] :
( ( member_a @ X @ I5 )
=> ( ( F2 @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_964_SUP__eq__iff,axiom,
! [I5: set_Extended_ereal,C: extended_ereal,F2: extended_ereal > extended_ereal] :
( ( I5 != bot_bo8367695208629047834_ereal )
=> ( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ I5 )
=> ( ord_le1083603963089353582_ereal @ C @ ( F2 @ I3 ) ) )
=> ( ( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ I5 ) )
= C )
= ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ I5 )
=> ( ( F2 @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_965_INF__eq__iff,axiom,
! [I5: set_a,F2: a > set_Extended_ereal,C: set_Extended_ereal] :
( ( I5 != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I5 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ I3 ) @ C ) )
=> ( ( ( comple4418415374894819509_ereal @ ( image_886028290468338613_ereal @ F2 @ I5 ) )
= C )
= ( ! [X: a] :
( ( member_a @ X @ I5 )
=> ( ( F2 @ X )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_966_INF__eq__iff,axiom,
! [I5: set_Extended_ereal,F2: extended_ereal > set_Extended_ereal,C: set_Extended_ereal] :
( ( I5 != bot_bo8367695208629047834_ereal )
=> ( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ I5 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ I3 ) @ C ) )
=> ( ( ( comple4418415374894819509_ereal @ ( image_5562094264469218789_ereal @ F2 @ I5 ) )
= C )
= ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ I5 )
=> ( ( F2 @ X )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_967_INF__eq__iff,axiom,
! [I5: set_nat,F2: nat > extended_ereal,C: extended_ereal] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I3 ) @ C ) )
=> ( ( ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ I5 ) )
= C )
= ( ! [X: nat] :
( ( member_nat @ X @ I5 )
=> ( ( F2 @ X )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_968_INF__eq__iff,axiom,
! [I5: set_a,F2: a > extended_ereal,C: extended_ereal] :
( ( I5 != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I5 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I3 ) @ C ) )
=> ( ( ( comple3556804143462414037_ereal @ ( image_8405481351990995413_ereal @ F2 @ I5 ) )
= C )
= ( ! [X: a] :
( ( member_a @ X @ I5 )
=> ( ( F2 @ X )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_969_INF__eq__iff,axiom,
! [I5: set_Extended_ereal,F2: extended_ereal > extended_ereal,C: extended_ereal] :
( ( I5 != bot_bo8367695208629047834_ereal )
=> ( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ I5 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I3 ) @ C ) )
=> ( ( ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ I5 ) )
= C )
= ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ I5 )
=> ( ( F2 @ X )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_970_inj__on__iff__surj,axiom,
! [A: set_Extended_ereal,A8: set_nat] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( ? [F: extended_ereal > nat] :
( ( inj_on318729178700965101al_nat @ F @ A )
& ( ord_less_eq_set_nat @ ( image_7659842161140344153al_nat @ F @ A ) @ A8 ) ) )
= ( ? [G2: nat > extended_ereal] :
( ( image_4309273772856505399_ereal @ G2 @ A8 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_971_inj__on__iff__surj,axiom,
! [A: set_nat,A8: set_Extended_ereal] :
( ( A != bot_bot_set_nat )
=> ( ( ? [F: nat > extended_ereal] :
( ( inj_on6191532827271902155_ereal @ F @ A )
& ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F @ A ) @ A8 ) ) )
= ( ? [G2: extended_ereal > nat] :
( ( image_7659842161140344153al_nat @ G2 @ A8 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_972_inj__on__iff__surj,axiom,
! [A: set_Extended_ereal,A8: set_Extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( ? [F: extended_ereal > extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F @ A )
& ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F @ A ) @ A8 ) ) )
= ( ? [G2: extended_ereal > extended_ereal] :
( ( image_6042159593519690757_ereal @ G2 @ A8 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_973_INF__empty,axiom,
! [F2: extended_ereal > set_Extended_ereal] :
( ( comple4418415374894819509_ereal @ ( image_5562094264469218789_ereal @ F2 @ bot_bo8367695208629047834_ereal ) )
= top_to5683747375963461374_ereal ) ).
% INF_empty
thf(fact_974_INF__empty,axiom,
! [F2: extended_ereal > set_nat] :
( ( comple7806235888213564991et_nat @ ( image_3090908713637162255et_nat @ F2 @ bot_bo8367695208629047834_ereal ) )
= top_top_set_nat ) ).
% INF_empty
thf(fact_975_INF__empty,axiom,
! [F2: nat > extended_ereal] :
( ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ bot_bot_set_nat ) )
= top_to6662034908053899550_ereal ) ).
% INF_empty
thf(fact_976_INF__empty,axiom,
! [F2: extended_ereal > extended_ereal] :
( ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ bot_bo8367695208629047834_ereal ) )
= top_to6662034908053899550_ereal ) ).
% INF_empty
thf(fact_977_INF__constant,axiom,
! [A: set_Extended_ereal,C: set_Extended_ereal] :
( ( ( A = bot_bo8367695208629047834_ereal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5562094264469218789_ereal
@ ^ [Y: extended_ereal] : C
@ A ) )
= top_to5683747375963461374_ereal ) )
& ( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5562094264469218789_ereal
@ ^ [Y: extended_ereal] : C
@ A ) )
= C ) ) ) ).
% INF_constant
thf(fact_978_INF__constant,axiom,
! [A: set_Extended_ereal,C: set_nat] :
( ( ( A = bot_bo8367695208629047834_ereal )
=> ( ( comple7806235888213564991et_nat
@ ( image_3090908713637162255et_nat
@ ^ [Y: extended_ereal] : C
@ A ) )
= top_top_set_nat ) )
& ( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple7806235888213564991et_nat
@ ( image_3090908713637162255et_nat
@ ^ [Y: extended_ereal] : C
@ A ) )
= C ) ) ) ).
% INF_constant
thf(fact_979_INF__constant,axiom,
! [A: set_nat,C: extended_ereal] :
( ( ( A = bot_bot_set_nat )
=> ( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [Y: nat] : C
@ A ) )
= top_to6662034908053899550_ereal ) )
& ( ( A != bot_bot_set_nat )
=> ( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [Y: nat] : C
@ A ) )
= C ) ) ) ).
% INF_constant
thf(fact_980_INF__constant,axiom,
! [A: set_Extended_ereal,C: extended_ereal] :
( ( ( A = bot_bo8367695208629047834_ereal )
=> ( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [Y: extended_ereal] : C
@ A ) )
= top_to6662034908053899550_ereal ) )
& ( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [Y: extended_ereal] : C
@ A ) )
= C ) ) ) ).
% INF_constant
thf(fact_981_times__subset__iff,axiom,
! [A: set_Extended_ereal,C2: set_Extended_ereal,B5: set_Extended_ereal,D: set_Extended_ereal] :
( ( ord_le8239133294219471655_ereal
@ ( produc8095709571603465288_ereal @ A
@ ^ [Uu: extended_ereal] : C2 )
@ ( produc8095709571603465288_ereal @ B5
@ ^ [Uu: extended_ereal] : D ) )
= ( ( A = bot_bo8367695208629047834_ereal )
| ( C2 = bot_bo8367695208629047834_ereal )
| ( ( ord_le1644982726543182158_ereal @ A @ B5 )
& ( ord_le1644982726543182158_ereal @ C2 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_982_fst__image__Sigma,axiom,
! [A: set_a,B5: a > set_b] :
( ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( product_Sigma_a_b @ A @ B5 ) )
= ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A )
& ( ( B5 @ X )
!= bot_bot_set_b ) ) ) ) ).
% fst_image_Sigma
thf(fact_983_fst__image__Sigma,axiom,
! [A: set_a,B5: a > set_Extended_ereal] :
( ( image_625819584623090380real_a @ produc3019788807843334739_ereal @ ( produc7264369102385879704_ereal @ A @ B5 ) )
= ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A )
& ( ( B5 @ X )
!= bot_bo8367695208629047834_ereal ) ) ) ) ).
% fst_image_Sigma
thf(fact_984_INF__le__SUP,axiom,
! [A: set_Extended_ereal,F2: extended_ereal > set_Extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ord_le1644982726543182158_ereal @ ( comple4418415374894819509_ereal @ ( image_5562094264469218789_ereal @ F2 @ A ) ) @ ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ F2 @ A ) ) ) ) ).
% INF_le_SUP
thf(fact_985_INF__le__SUP,axiom,
! [A: set_nat,F2: nat > extended_ereal] :
( ( A != bot_bot_set_nat )
=> ( ord_le1083603963089353582_ereal @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) ) ) ).
% INF_le_SUP
thf(fact_986_INF__le__SUP,axiom,
! [A: set_Extended_ereal,F2: extended_ereal > extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ord_le1083603963089353582_ereal @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) ) ) ).
% INF_le_SUP
thf(fact_987_cINF__const,axiom,
! [A: set_nat,C: extended_ereal] :
( ( A != bot_bot_set_nat )
=> ( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X: nat] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_988_cINF__const,axiom,
! [A: set_Extended_ereal,C: extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : C
@ A ) )
= C ) ) ).
% cINF_const
thf(fact_989_cSUP__const,axiom,
! [A: set_nat,C: extended_ereal] :
( ( A != bot_bot_set_nat )
=> ( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X: nat] : C
@ A ) )
= C ) ) ).
% cSUP_const
thf(fact_990_cSUP__const,axiom,
! [A: set_Extended_ereal,C: extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : C
@ A ) )
= C ) ) ).
% cSUP_const
thf(fact_991_cINF__greatest,axiom,
! [A: set_a,M: nat,F2: a > nat] :
( ( A != bot_bot_set_a )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ord_less_eq_nat @ M @ ( F2 @ X3 ) ) )
=> ( ord_less_eq_nat @ M @ ( complete_Inf_Inf_nat @ ( image_a_nat @ F2 @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_992_cINF__greatest,axiom,
! [A: set_Extended_ereal,M: nat,F2: extended_ereal > nat] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
=> ( ord_less_eq_nat @ M @ ( F2 @ X3 ) ) )
=> ( ord_less_eq_nat @ M @ ( complete_Inf_Inf_nat @ ( image_7659842161140344153al_nat @ F2 @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_993_cINF__greatest,axiom,
! [A: set_a,M: set_Extended_ereal,F2: a > set_Extended_ereal] :
( ( A != bot_bot_set_a )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ord_le1644982726543182158_ereal @ M @ ( F2 @ X3 ) ) )
=> ( ord_le1644982726543182158_ereal @ M @ ( comple4418415374894819509_ereal @ ( image_886028290468338613_ereal @ F2 @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_994_cINF__greatest,axiom,
! [A: set_Extended_ereal,M: set_Extended_ereal,F2: extended_ereal > set_Extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
=> ( ord_le1644982726543182158_ereal @ M @ ( F2 @ X3 ) ) )
=> ( ord_le1644982726543182158_ereal @ M @ ( comple4418415374894819509_ereal @ ( image_5562094264469218789_ereal @ F2 @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_995_cINF__greatest,axiom,
! [A: set_nat,M: extended_ereal,F2: nat > extended_ereal] :
( ( A != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_le1083603963089353582_ereal @ M @ ( F2 @ X3 ) ) )
=> ( ord_le1083603963089353582_ereal @ M @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_996_cINF__greatest,axiom,
! [A: set_a,M: extended_ereal,F2: a > extended_ereal] :
( ( A != bot_bot_set_a )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ord_le1083603963089353582_ereal @ M @ ( F2 @ X3 ) ) )
=> ( ord_le1083603963089353582_ereal @ M @ ( comple3556804143462414037_ereal @ ( image_8405481351990995413_ereal @ F2 @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_997_cINF__greatest,axiom,
! [A: set_Extended_ereal,M: extended_ereal,F2: extended_ereal > extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
=> ( ord_le1083603963089353582_ereal @ M @ ( F2 @ X3 ) ) )
=> ( ord_le1083603963089353582_ereal @ M @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_998_ereal__SUP__uminus__eq,axiom,
! [F2: extended_ereal > extended_ereal,S4: set_Extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : ( uminus27091377158695749_ereal @ ( F2 @ X ) )
@ S4 ) )
= ( uminus27091377158695749_ereal @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ S4 ) ) ) ) ).
% ereal_SUP_uminus_eq
thf(fact_999_ereal__SUP__uminus__eq,axiom,
! [F2: nat > extended_ereal,S4: set_nat] :
( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X: nat] : ( uminus27091377158695749_ereal @ ( F2 @ X ) )
@ S4 ) )
= ( uminus27091377158695749_ereal @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ S4 ) ) ) ) ).
% ereal_SUP_uminus_eq
thf(fact_1000_ereal__INF__uminus__eq,axiom,
! [F2: extended_ereal > extended_ereal,S4: set_Extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : ( uminus27091377158695749_ereal @ ( F2 @ X ) )
@ S4 ) )
= ( uminus27091377158695749_ereal @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ S4 ) ) ) ) ).
% ereal_INF_uminus_eq
thf(fact_1001_ereal__INF__uminus__eq,axiom,
! [F2: nat > extended_ereal,S4: set_nat] :
( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X: nat] : ( uminus27091377158695749_ereal @ ( F2 @ X ) )
@ S4 ) )
= ( uminus27091377158695749_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ S4 ) ) ) ) ).
% ereal_INF_uminus_eq
thf(fact_1002_ereal__SUP__uminus,axiom,
! [F2: extended_ereal > extended_ereal,R2: set_Extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( uminus27091377158695749_ereal @ ( F2 @ I2 ) )
@ R2 ) )
= ( uminus27091377158695749_ereal @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ R2 ) ) ) ) ).
% ereal_SUP_uminus
thf(fact_1003_ereal__SUP__uminus,axiom,
! [F2: nat > extended_ereal,R2: set_nat] :
( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( uminus27091377158695749_ereal @ ( F2 @ I2 ) )
@ R2 ) )
= ( uminus27091377158695749_ereal @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ R2 ) ) ) ) ).
% ereal_SUP_uminus
thf(fact_1004_ereal__Inf__uminus__image__eq,axiom,
! [S4: set_Extended_ereal] :
( ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S4 ) )
= ( uminus27091377158695749_ereal @ ( comple8415311339701865915_ereal @ S4 ) ) ) ).
% ereal_Inf_uminus_image_eq
thf(fact_1005_ereal__Sup__uminus__image__eq,axiom,
! [S4: set_Extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S4 ) )
= ( uminus27091377158695749_ereal @ ( comple3556804143462414037_ereal @ S4 ) ) ) ).
% ereal_Sup_uminus_image_eq
thf(fact_1006_bot__empty__eq2,axiom,
( bot_bo2150971494257102938st_o_o
= ( ^ [X: option_list_o,Y: option_list_o] : ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X @ Y ) @ bot_bo6949037146550090099list_o ) ) ) ).
% bot_empty_eq2
thf(fact_1007_cSUP__least,axiom,
! [A: set_a,F2: a > nat,M3: nat] :
( ( A != bot_bot_set_a )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ M3 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F2 @ A ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_1008_cSUP__least,axiom,
! [A: set_Extended_ereal,F2: extended_ereal > nat,M3: nat] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ M3 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_7659842161140344153al_nat @ F2 @ A ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_1009_cSUP__least,axiom,
! [A: set_a,F2: a > set_Extended_ereal,M3: set_Extended_ereal] :
( ( A != bot_bot_set_a )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ X3 ) @ M3 ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ F2 @ A ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_1010_cSUP__least,axiom,
! [A: set_Extended_ereal,F2: extended_ereal > set_Extended_ereal,M3: set_Extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ X3 ) @ M3 ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ F2 @ A ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_1011_cSUP__least,axiom,
! [A: set_nat,F2: nat > extended_ereal,M3: extended_ereal] :
( ( A != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ X3 ) @ M3 ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_1012_cSUP__least,axiom,
! [A: set_a,F2: a > extended_ereal,M3: extended_ereal] :
( ( A != bot_bot_set_a )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ X3 ) @ M3 ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ F2 @ A ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_1013_cSUP__least,axiom,
! [A: set_Extended_ereal,F2: extended_ereal > extended_ereal,M3: extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ X3 ) @ M3 ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_1014_INT__simps_I4_J,axiom,
! [C2: set_Extended_ereal,A: set_Extended_ereal,B5: extended_ereal > set_Extended_ereal] :
( ( ( C2 = bot_bo8367695208629047834_ereal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5562094264469218789_ereal
@ ^ [X: extended_ereal] : ( minus_1264018925008434325_ereal @ A @ ( B5 @ X ) )
@ C2 ) )
= top_to5683747375963461374_ereal ) )
& ( ( C2 != bot_bo8367695208629047834_ereal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5562094264469218789_ereal
@ ^ [X: extended_ereal] : ( minus_1264018925008434325_ereal @ A @ ( B5 @ X ) )
@ C2 ) )
= ( minus_1264018925008434325_ereal @ A @ ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ B5 @ C2 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_1015_INT__simps_I4_J,axiom,
! [C2: set_Extended_ereal,A: set_nat,B5: extended_ereal > set_nat] :
( ( ( C2 = bot_bo8367695208629047834_ereal )
=> ( ( comple7806235888213564991et_nat
@ ( image_3090908713637162255et_nat
@ ^ [X: extended_ereal] : ( minus_minus_set_nat @ A @ ( B5 @ X ) )
@ C2 ) )
= top_top_set_nat ) )
& ( ( C2 != bot_bo8367695208629047834_ereal )
=> ( ( comple7806235888213564991et_nat
@ ( image_3090908713637162255et_nat
@ ^ [X: extended_ereal] : ( minus_minus_set_nat @ A @ ( B5 @ X ) )
@ C2 ) )
= ( minus_minus_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_3090908713637162255et_nat @ B5 @ C2 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_1016_Sigma__def,axiom,
( produc7548132086704134856list_o
= ( ^ [A3: set_option_list_o,B7: option_list_o > set_option_list_o] :
( comple1150810575505393780list_o
@ ( image_3844918200759902248list_o
@ ^ [X: option_list_o] :
( comple1150810575505393780list_o
@ ( image_3844918200759902248list_o
@ ^ [Y: option_list_o] : ( insert7989621605229100023list_o @ ( produc5745850523778858007list_o @ X @ Y ) @ bot_bo6949037146550090099list_o )
@ ( B7 @ X ) ) )
@ A3 ) ) ) ) ).
% Sigma_def
thf(fact_1017_ereal__inj__on__uminus,axiom,
! [A: set_Extended_ereal] : ( inj_on7162434037990268785_ereal @ uminus27091377158695749_ereal @ A ) ).
% ereal_inj_on_uminus
thf(fact_1018_insert__image,axiom,
! [X2: extended_ereal,A: set_Extended_ereal,F2: extended_ereal > extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A )
=> ( ( insert8967887681552722334_ereal @ ( F2 @ X2 ) @ ( image_6042159593519690757_ereal @ F2 @ A ) )
= ( image_6042159593519690757_ereal @ F2 @ A ) ) ) ).
% insert_image
thf(fact_1019_insert__image,axiom,
! [X2: nat,A: set_nat,F2: nat > extended_ereal] :
( ( member_nat @ X2 @ A )
=> ( ( insert8967887681552722334_ereal @ ( F2 @ X2 ) @ ( image_4309273772856505399_ereal @ F2 @ A ) )
= ( image_4309273772856505399_ereal @ F2 @ A ) ) ) ).
% insert_image
thf(fact_1020_image__insert,axiom,
! [F2: extended_ereal > extended_ereal,A4: extended_ereal,B5: set_Extended_ereal] :
( ( image_6042159593519690757_ereal @ F2 @ ( insert8967887681552722334_ereal @ A4 @ B5 ) )
= ( insert8967887681552722334_ereal @ ( F2 @ A4 ) @ ( image_6042159593519690757_ereal @ F2 @ B5 ) ) ) ).
% image_insert
thf(fact_1021_image__insert,axiom,
! [F2: nat > extended_ereal,A4: nat,B5: set_nat] :
( ( image_4309273772856505399_ereal @ F2 @ ( insert_nat @ A4 @ B5 ) )
= ( insert8967887681552722334_ereal @ ( F2 @ A4 ) @ ( image_4309273772856505399_ereal @ F2 @ B5 ) ) ) ).
% image_insert
thf(fact_1022_ereal__minus__minus__image,axiom,
! [S4: set_Extended_ereal] :
( ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S4 ) )
= S4 ) ).
% ereal_minus_minus_image
thf(fact_1023_ereal__range__uminus,axiom,
( ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal ) ).
% ereal_range_uminus
thf(fact_1024_Sup__insert,axiom,
! [A4: extended_ereal,A: set_Extended_ereal] :
( ( comple8415311339701865915_ereal @ ( insert8967887681552722334_ereal @ A4 @ A ) )
= ( sup_su7653423775389492130_ereal @ A4 @ ( comple8415311339701865915_ereal @ A ) ) ) ).
% Sup_insert
thf(fact_1025_Compl__Diff__eq,axiom,
! [A: set_Extended_ereal,B5: set_Extended_ereal] :
( ( uminus5895154729394068773_ereal @ ( minus_1264018925008434325_ereal @ A @ B5 ) )
= ( sup_su2680283192902082946_ereal @ ( uminus5895154729394068773_ereal @ A ) @ B5 ) ) ).
% Compl_Diff_eq
thf(fact_1026_range__constant,axiom,
! [X2: extended_ereal] :
( ( image_6042159593519690757_ereal
@ ^ [Uu: extended_ereal] : X2
@ top_to5683747375963461374_ereal )
= ( insert8967887681552722334_ereal @ X2 @ bot_bo8367695208629047834_ereal ) ) ).
% range_constant
thf(fact_1027_range__constant,axiom,
! [X2: extended_ereal] :
( ( image_4309273772856505399_ereal
@ ^ [Uu: nat] : X2
@ top_top_set_nat )
= ( insert8967887681552722334_ereal @ X2 @ bot_bo8367695208629047834_ereal ) ) ).
% range_constant
thf(fact_1028_UN__simps_I1_J,axiom,
! [C2: set_Extended_ereal,A4: extended_ereal,B5: extended_ereal > set_Extended_ereal] :
( ( ( C2 = bot_bo8367695208629047834_ereal )
=> ( ( comple4319282863272126363_ereal
@ ( image_5562094264469218789_ereal
@ ^ [X: extended_ereal] : ( insert8967887681552722334_ereal @ A4 @ ( B5 @ X ) )
@ C2 ) )
= bot_bo8367695208629047834_ereal ) )
& ( ( C2 != bot_bo8367695208629047834_ereal )
=> ( ( comple4319282863272126363_ereal
@ ( image_5562094264469218789_ereal
@ ^ [X: extended_ereal] : ( insert8967887681552722334_ereal @ A4 @ ( B5 @ X ) )
@ C2 ) )
= ( insert8967887681552722334_ereal @ A4 @ ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ B5 @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_1029_UN__singleton,axiom,
! [A: set_Extended_ereal] :
( ( comple4319282863272126363_ereal
@ ( image_5562094264469218789_ereal
@ ^ [X: extended_ereal] : ( insert8967887681552722334_ereal @ X @ bot_bo8367695208629047834_ereal )
@ A ) )
= A ) ).
% UN_singleton
thf(fact_1030_inj__on__insert,axiom,
! [F2: nat > extended_ereal,A4: nat,A: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ ( insert_nat @ A4 @ A ) )
= ( ( inj_on6191532827271902155_ereal @ F2 @ A )
& ~ ( member2350847679896131959_ereal @ ( F2 @ A4 ) @ ( image_4309273772856505399_ereal @ F2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_1031_inj__on__insert,axiom,
! [F2: extended_ereal > extended_ereal,A4: extended_ereal,A: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ ( insert8967887681552722334_ereal @ A4 @ A ) )
= ( ( inj_on7162434037990268785_ereal @ F2 @ A )
& ~ ( member2350847679896131959_ereal @ ( F2 @ A4 ) @ ( image_6042159593519690757_ereal @ F2 @ ( minus_1264018925008434325_ereal @ A @ ( insert8967887681552722334_ereal @ A4 @ bot_bo8367695208629047834_ereal ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_1032_inj__on__insert,axiom,
! [F2: extended_ereal > a,A4: extended_ereal,A: set_Extended_ereal] :
( ( inj_on8242634198667403041real_a @ F2 @ ( insert8967887681552722334_ereal @ A4 @ A ) )
= ( ( inj_on8242634198667403041real_a @ F2 @ A )
& ~ ( member_a @ ( F2 @ A4 ) @ ( image_3724615099042636213real_a @ F2 @ ( minus_1264018925008434325_ereal @ A @ ( insert8967887681552722334_ereal @ A4 @ bot_bo8367695208629047834_ereal ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_1033_INT__simps_I3_J,axiom,
! [C2: set_Extended_ereal,A: extended_ereal > set_Extended_ereal,B5: set_Extended_ereal] :
( ( ( C2 = bot_bo8367695208629047834_ereal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5562094264469218789_ereal
@ ^ [X: extended_ereal] : ( minus_1264018925008434325_ereal @ ( A @ X ) @ B5 )
@ C2 ) )
= top_to5683747375963461374_ereal ) )
& ( ( C2 != bot_bo8367695208629047834_ereal )
=> ( ( comple4418415374894819509_ereal
@ ( image_5562094264469218789_ereal
@ ^ [X: extended_ereal] : ( minus_1264018925008434325_ereal @ ( A @ X ) @ B5 )
@ C2 ) )
= ( minus_1264018925008434325_ereal @ ( comple4418415374894819509_ereal @ ( image_5562094264469218789_ereal @ A @ C2 ) ) @ B5 ) ) ) ) ).
% INT_simps(3)
thf(fact_1034_INT__simps_I3_J,axiom,
! [C2: set_Extended_ereal,A: extended_ereal > set_nat,B5: set_nat] :
( ( ( C2 = bot_bo8367695208629047834_ereal )
=> ( ( comple7806235888213564991et_nat
@ ( image_3090908713637162255et_nat
@ ^ [X: extended_ereal] : ( minus_minus_set_nat @ ( A @ X ) @ B5 )
@ C2 ) )
= top_top_set_nat ) )
& ( ( C2 != bot_bo8367695208629047834_ereal )
=> ( ( comple7806235888213564991et_nat
@ ( image_3090908713637162255et_nat
@ ^ [X: extended_ereal] : ( minus_minus_set_nat @ ( A @ X ) @ B5 )
@ C2 ) )
= ( minus_minus_set_nat @ ( comple7806235888213564991et_nat @ ( image_3090908713637162255et_nat @ A @ C2 ) ) @ B5 ) ) ) ) ).
% INT_simps(3)
thf(fact_1035_insert__Times__insert,axiom,
! [A4: option_list_o,A: set_option_list_o,B: option_list_o,B5: set_option_list_o] :
( ( produc7548132086704134856list_o @ ( insert_option_list_o @ A4 @ A )
@ ^ [Uu: option_list_o] : ( insert_option_list_o @ B @ B5 ) )
= ( insert7989621605229100023list_o @ ( produc5745850523778858007list_o @ A4 @ B )
@ ( sup_su2413444806983345371list_o
@ ( produc7548132086704134856list_o @ A
@ ^ [Uu: option_list_o] : ( insert_option_list_o @ B @ B5 ) )
@ ( produc7548132086704134856list_o @ ( insert_option_list_o @ A4 @ A )
@ ^ [Uu: option_list_o] : B5 ) ) ) ) ).
% insert_Times_insert
thf(fact_1036_in__image__insert__iff,axiom,
! [B5: set_set_a,X2: a,A: set_a] :
( ! [C3: set_a] :
( ( member_set_a @ C3 @ B5 )
=> ~ ( member_a @ X2 @ C3 ) )
=> ( ( member_set_a @ A @ ( image_set_a_set_a @ ( insert_a @ X2 ) @ B5 ) )
= ( ( member_a @ X2 @ A )
& ( member_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B5 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1037_in__image__insert__iff,axiom,
! [B5: set_se6634062954251873166_ereal,X2: extended_ereal,A: set_Extended_ereal] :
( ! [C3: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ C3 @ B5 )
=> ~ ( member2350847679896131959_ereal @ X2 @ C3 ) )
=> ( ( member5519481007471526743_ereal @ A @ ( image_6293272304431515653_ereal @ ( insert8967887681552722334_ereal @ X2 ) @ B5 ) )
= ( ( member2350847679896131959_ereal @ X2 @ A )
& ( member5519481007471526743_ereal @ ( minus_1264018925008434325_ereal @ A @ ( insert8967887681552722334_ereal @ X2 @ bot_bo8367695208629047834_ereal ) ) @ B5 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1038_insert__subsetI,axiom,
! [X2: a,A: set_a,X5: set_a] :
( ( member_a @ X2 @ A )
=> ( ( ord_less_eq_set_a @ X5 @ A )
=> ( ord_less_eq_set_a @ ( insert_a @ X2 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1039_insert__subsetI,axiom,
! [X2: extended_ereal,A: set_Extended_ereal,X5: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A )
=> ( ( ord_le1644982726543182158_ereal @ X5 @ A )
=> ( ord_le1644982726543182158_ereal @ ( insert8967887681552722334_ereal @ X2 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1040_ereal__complete__uminus__eq,axiom,
! [S4: set_Extended_ereal,X2: extended_ereal] :
( ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S4 ) )
=> ( ord_le1083603963089353582_ereal @ X @ X2 ) )
& ! [Z4: extended_ereal] :
( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S4 ) )
=> ( ord_le1083603963089353582_ereal @ X @ Z4 ) )
=> ( ord_le1083603963089353582_ereal @ X2 @ Z4 ) ) )
= ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ S4 )
=> ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ X2 ) @ X ) )
& ! [Z4: extended_ereal] :
( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ S4 )
=> ( ord_le1083603963089353582_ereal @ Z4 @ X ) )
=> ( ord_le1083603963089353582_ereal @ Z4 @ ( uminus27091377158695749_ereal @ X2 ) ) ) ) ) ).
% ereal_complete_uminus_eq
thf(fact_1041_ereal__image__uminus__shift,axiom,
! [X5: set_Extended_ereal,Y5: set_Extended_ereal] :
( ( ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ X5 )
= Y5 )
= ( X5
= ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ Y5 ) ) ) ).
% ereal_image_uminus_shift
thf(fact_1042_ereal__uminus__complement,axiom,
! [S4: set_Extended_ereal] :
( ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ ( uminus5895154729394068773_ereal @ S4 ) )
= ( uminus5895154729394068773_ereal @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S4 ) ) ) ).
% ereal_uminus_complement
thf(fact_1043_image__diff__subset,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,B5: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ ( minus_1264018925008434325_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) @ ( image_6042159593519690757_ereal @ F2 @ B5 ) ) @ ( image_6042159593519690757_ereal @ F2 @ ( minus_1264018925008434325_ereal @ A @ B5 ) ) ) ).
% image_diff_subset
thf(fact_1044_image__diff__subset,axiom,
! [F2: nat > extended_ereal,A: set_nat,B5: set_nat] : ( ord_le1644982726543182158_ereal @ ( minus_1264018925008434325_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) @ ( image_4309273772856505399_ereal @ F2 @ B5 ) ) @ ( image_4309273772856505399_ereal @ F2 @ ( minus_minus_set_nat @ A @ B5 ) ) ) ).
% image_diff_subset
thf(fact_1045_inj__img__insertE,axiom,
! [F2: nat > extended_ereal,A: set_nat,X2: extended_ereal,B5: set_Extended_ereal] :
( ( inj_on6191532827271902155_ereal @ F2 @ A )
=> ( ~ ( member2350847679896131959_ereal @ X2 @ B5 )
=> ( ( ( insert8967887681552722334_ereal @ X2 @ B5 )
= ( image_4309273772856505399_ereal @ F2 @ A ) )
=> ~ ! [X6: nat,A7: set_nat] :
( ~ ( member_nat @ X6 @ A7 )
=> ( ( A
= ( insert_nat @ X6 @ A7 ) )
=> ( ( X2
= ( F2 @ X6 ) )
=> ( B5
!= ( image_4309273772856505399_ereal @ F2 @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1046_inj__img__insertE,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,X2: extended_ereal,B5: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A )
=> ( ~ ( member2350847679896131959_ereal @ X2 @ B5 )
=> ( ( ( insert8967887681552722334_ereal @ X2 @ B5 )
= ( image_6042159593519690757_ereal @ F2 @ A ) )
=> ~ ! [X6: extended_ereal,A7: set_Extended_ereal] :
( ~ ( member2350847679896131959_ereal @ X6 @ A7 )
=> ( ( A
= ( insert8967887681552722334_ereal @ X6 @ A7 ) )
=> ( ( X2
= ( F2 @ X6 ) )
=> ( B5
!= ( image_6042159593519690757_ereal @ F2 @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1047_inj__img__insertE,axiom,
! [F2: a > a,A: set_a,X2: a,B5: set_a] :
( ( inj_on_a_a @ F2 @ A )
=> ( ~ ( member_a @ X2 @ B5 )
=> ( ( ( insert_a @ X2 @ B5 )
= ( image_a_a @ F2 @ A ) )
=> ~ ! [X6: a,A7: set_a] :
( ~ ( member_a @ X6 @ A7 )
=> ( ( A
= ( insert_a @ X6 @ A7 ) )
=> ( ( X2
= ( F2 @ X6 ) )
=> ( B5
!= ( image_a_a @ F2 @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1048_singleton__Un__iff,axiom,
! [X2: extended_ereal,A: set_Extended_ereal,B5: set_Extended_ereal] :
( ( ( insert8967887681552722334_ereal @ X2 @ bot_bo8367695208629047834_ereal )
= ( sup_su2680283192902082946_ereal @ A @ B5 ) )
= ( ( ( A = bot_bo8367695208629047834_ereal )
& ( B5
= ( insert8967887681552722334_ereal @ X2 @ bot_bo8367695208629047834_ereal ) ) )
| ( ( A
= ( insert8967887681552722334_ereal @ X2 @ bot_bo8367695208629047834_ereal ) )
& ( B5 = bot_bo8367695208629047834_ereal ) )
| ( ( A
= ( insert8967887681552722334_ereal @ X2 @ bot_bo8367695208629047834_ereal ) )
& ( B5
= ( insert8967887681552722334_ereal @ X2 @ bot_bo8367695208629047834_ereal ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1049_Un__singleton__iff,axiom,
! [A: set_Extended_ereal,B5: set_Extended_ereal,X2: extended_ereal] :
( ( ( sup_su2680283192902082946_ereal @ A @ B5 )
= ( insert8967887681552722334_ereal @ X2 @ bot_bo8367695208629047834_ereal ) )
= ( ( ( A = bot_bo8367695208629047834_ereal )
& ( B5
= ( insert8967887681552722334_ereal @ X2 @ bot_bo8367695208629047834_ereal ) ) )
| ( ( A
= ( insert8967887681552722334_ereal @ X2 @ bot_bo8367695208629047834_ereal ) )
& ( B5 = bot_bo8367695208629047834_ereal ) )
| ( ( A
= ( insert8967887681552722334_ereal @ X2 @ bot_bo8367695208629047834_ereal ) )
& ( B5
= ( insert8967887681552722334_ereal @ X2 @ bot_bo8367695208629047834_ereal ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1050_insert__is__Un,axiom,
( insert8967887681552722334_ereal
= ( ^ [A6: extended_ereal] : ( sup_su2680283192902082946_ereal @ ( insert8967887681552722334_ereal @ A6 @ bot_bo8367695208629047834_ereal ) ) ) ) ).
% insert_is_Un
thf(fact_1051_Diff__partition,axiom,
! [A: set_Extended_ereal,B5: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A @ B5 )
=> ( ( sup_su2680283192902082946_ereal @ A @ ( minus_1264018925008434325_ereal @ B5 @ A ) )
= B5 ) ) ).
% Diff_partition
thf(fact_1052_Diff__subset__conv,axiom,
! [A: set_Extended_ereal,B5: set_Extended_ereal,C2: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ ( minus_1264018925008434325_ereal @ A @ B5 ) @ C2 )
= ( ord_le1644982726543182158_ereal @ A @ ( sup_su2680283192902082946_ereal @ B5 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1053_inj__on__diff,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,B5: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A )
=> ( inj_on7162434037990268785_ereal @ F2 @ ( minus_1264018925008434325_ereal @ A @ B5 ) ) ) ).
% inj_on_diff
thf(fact_1054_image__constant__conv,axiom,
! [A: set_nat,C: extended_ereal] :
( ( ( A = bot_bot_set_nat )
=> ( ( image_4309273772856505399_ereal
@ ^ [X: nat] : C
@ A )
= bot_bo8367695208629047834_ereal ) )
& ( ( A != bot_bot_set_nat )
=> ( ( image_4309273772856505399_ereal
@ ^ [X: nat] : C
@ A )
= ( insert8967887681552722334_ereal @ C @ bot_bo8367695208629047834_ereal ) ) ) ) ).
% image_constant_conv
thf(fact_1055_image__constant__conv,axiom,
! [A: set_Extended_ereal,C: extended_ereal] :
( ( ( A = bot_bo8367695208629047834_ereal )
=> ( ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : C
@ A )
= bot_bo8367695208629047834_ereal ) )
& ( ( A != bot_bo8367695208629047834_ereal )
=> ( ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : C
@ A )
= ( insert8967887681552722334_ereal @ C @ bot_bo8367695208629047834_ereal ) ) ) ) ).
% image_constant_conv
thf(fact_1056_image__constant,axiom,
! [X2: extended_ereal,A: set_Extended_ereal,C: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A )
=> ( ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : C
@ A )
= ( insert8967887681552722334_ereal @ C @ bot_bo8367695208629047834_ereal ) ) ) ).
% image_constant
thf(fact_1057_image__constant,axiom,
! [X2: nat,A: set_nat,C: extended_ereal] :
( ( member_nat @ X2 @ A )
=> ( ( image_4309273772856505399_ereal
@ ^ [X: nat] : C
@ A )
= ( insert8967887681552722334_ereal @ C @ bot_bo8367695208629047834_ereal ) ) ) ).
% image_constant
thf(fact_1058_image__constant,axiom,
! [X2: a,A: set_a,C: extended_ereal] :
( ( member_a @ X2 @ A )
=> ( ( image_8405481351990995413_ereal
@ ^ [X: a] : C
@ A )
= ( insert8967887681552722334_ereal @ C @ bot_bo8367695208629047834_ereal ) ) ) ).
% image_constant
thf(fact_1059_inj__singleton,axiom,
! [A: set_Extended_ereal] :
( inj_on3063056893380862289_ereal
@ ^ [X: extended_ereal] : ( insert8967887681552722334_ereal @ X @ bot_bo8367695208629047834_ereal )
@ A ) ).
% inj_singleton
thf(fact_1060_range__eq__singletonD,axiom,
! [F2: extended_ereal > extended_ereal,A4: extended_ereal,X2: extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= ( insert8967887681552722334_ereal @ A4 @ bot_bo8367695208629047834_ereal ) )
=> ( ( F2 @ X2 )
= A4 ) ) ).
% range_eq_singletonD
thf(fact_1061_range__eq__singletonD,axiom,
! [F2: nat > extended_ereal,A4: extended_ereal,X2: nat] :
( ( ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat )
= ( insert8967887681552722334_ereal @ A4 @ bot_bo8367695208629047834_ereal ) )
=> ( ( F2 @ X2 )
= A4 ) ) ).
% range_eq_singletonD
thf(fact_1062_image__set__diff,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,B5: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ( image_6042159593519690757_ereal @ F2 @ ( minus_1264018925008434325_ereal @ A @ B5 ) )
= ( minus_1264018925008434325_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) @ ( image_6042159593519690757_ereal @ F2 @ B5 ) ) ) ) ).
% image_set_diff
thf(fact_1063_image__set__diff,axiom,
! [F2: nat > extended_ereal,A: set_nat,B5: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ top_top_set_nat )
=> ( ( image_4309273772856505399_ereal @ F2 @ ( minus_minus_set_nat @ A @ B5 ) )
= ( minus_1264018925008434325_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) @ ( image_4309273772856505399_ereal @ F2 @ B5 ) ) ) ) ).
% image_set_diff
thf(fact_1064_inj__on__image__set__diff,axiom,
! [F2: nat > extended_ereal,C2: set_nat,A: set_nat,B5: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ C2 )
=> ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B5 ) @ C2 )
=> ( ( ord_less_eq_set_nat @ B5 @ C2 )
=> ( ( image_4309273772856505399_ereal @ F2 @ ( minus_minus_set_nat @ A @ B5 ) )
= ( minus_1264018925008434325_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) @ ( image_4309273772856505399_ereal @ F2 @ B5 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1065_inj__on__image__set__diff,axiom,
! [F2: extended_ereal > extended_ereal,C2: set_Extended_ereal,A: set_Extended_ereal,B5: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ C2 )
=> ( ( ord_le1644982726543182158_ereal @ ( minus_1264018925008434325_ereal @ A @ B5 ) @ C2 )
=> ( ( ord_le1644982726543182158_ereal @ B5 @ C2 )
=> ( ( image_6042159593519690757_ereal @ F2 @ ( minus_1264018925008434325_ereal @ A @ B5 ) )
= ( minus_1264018925008434325_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) @ ( image_6042159593519690757_ereal @ F2 @ B5 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1066_SUP__insert,axiom,
! [F2: extended_ereal > extended_ereal,A4: extended_ereal,A: set_Extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ ( insert8967887681552722334_ereal @ A4 @ A ) ) )
= ( sup_su7653423775389492130_ereal @ ( F2 @ A4 ) @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) ) ) ).
% SUP_insert
thf(fact_1067_SUP__insert,axiom,
! [F2: nat > extended_ereal,A4: nat,A: set_nat] :
( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ ( insert_nat @ A4 @ A ) ) )
= ( sup_su7653423775389492130_ereal @ ( F2 @ A4 ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) ) ) ).
% SUP_insert
thf(fact_1068_UN__extend__simps_I1_J,axiom,
! [C2: set_Extended_ereal,A4: extended_ereal,B5: extended_ereal > set_Extended_ereal] :
( ( ( C2 = bot_bo8367695208629047834_ereal )
=> ( ( insert8967887681552722334_ereal @ A4 @ ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ B5 @ C2 ) ) )
= ( insert8967887681552722334_ereal @ A4 @ bot_bo8367695208629047834_ereal ) ) )
& ( ( C2 != bot_bo8367695208629047834_ereal )
=> ( ( insert8967887681552722334_ereal @ A4 @ ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ B5 @ C2 ) ) )
= ( comple4319282863272126363_ereal
@ ( image_5562094264469218789_ereal
@ ^ [X: extended_ereal] : ( insert8967887681552722334_ereal @ A4 @ ( B5 @ X ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_1069_UNION__singleton__eq__range,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( comple4319282863272126363_ereal
@ ( image_5562094264469218789_ereal
@ ^ [X: extended_ereal] : ( insert8967887681552722334_ereal @ ( F2 @ X ) @ bot_bo8367695208629047834_ereal )
@ A ) )
= ( image_6042159593519690757_ereal @ F2 @ A ) ) ).
% UNION_singleton_eq_range
thf(fact_1070_UNION__singleton__eq__range,axiom,
! [F2: nat > extended_ereal,A: set_nat] :
( ( comple4319282863272126363_ereal
@ ( image_305533323056406039_ereal
@ ^ [X: nat] : ( insert8967887681552722334_ereal @ ( F2 @ X ) @ bot_bo8367695208629047834_ereal )
@ A ) )
= ( image_4309273772856505399_ereal @ F2 @ A ) ) ).
% UNION_singleton_eq_range
thf(fact_1071_INT__extend__simps_I3_J,axiom,
! [C2: set_Extended_ereal,A: extended_ereal > set_Extended_ereal,B5: set_Extended_ereal] :
( ( ( C2 = bot_bo8367695208629047834_ereal )
=> ( ( minus_1264018925008434325_ereal @ ( comple4418415374894819509_ereal @ ( image_5562094264469218789_ereal @ A @ C2 ) ) @ B5 )
= ( minus_1264018925008434325_ereal @ top_to5683747375963461374_ereal @ B5 ) ) )
& ( ( C2 != bot_bo8367695208629047834_ereal )
=> ( ( minus_1264018925008434325_ereal @ ( comple4418415374894819509_ereal @ ( image_5562094264469218789_ereal @ A @ C2 ) ) @ B5 )
= ( comple4418415374894819509_ereal
@ ( image_5562094264469218789_ereal
@ ^ [X: extended_ereal] : ( minus_1264018925008434325_ereal @ ( A @ X ) @ B5 )
@ C2 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_1072_INT__extend__simps_I3_J,axiom,
! [C2: set_Extended_ereal,A: extended_ereal > set_nat,B5: set_nat] :
( ( ( C2 = bot_bo8367695208629047834_ereal )
=> ( ( minus_minus_set_nat @ ( comple7806235888213564991et_nat @ ( image_3090908713637162255et_nat @ A @ C2 ) ) @ B5 )
= ( minus_minus_set_nat @ top_top_set_nat @ B5 ) ) )
& ( ( C2 != bot_bo8367695208629047834_ereal )
=> ( ( minus_minus_set_nat @ ( comple7806235888213564991et_nat @ ( image_3090908713637162255et_nat @ A @ C2 ) ) @ B5 )
= ( comple7806235888213564991et_nat
@ ( image_3090908713637162255et_nat
@ ^ [X: extended_ereal] : ( minus_minus_set_nat @ ( A @ X ) @ B5 )
@ C2 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_1073_If__the__inv__into__in__Func,axiom,
! [G: nat > extended_ereal,C2: set_nat,B5: set_nat,X2: nat] :
( ( inj_on6191532827271902155_ereal @ G @ C2 )
=> ( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ B5 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
=> ( member8206335426146891064al_nat
@ ^ [I2: extended_ereal] : ( if_nat @ ( member2350847679896131959_ereal @ I2 @ ( image_4309273772856505399_ereal @ G @ C2 ) ) @ ( the_in5959796611709155849_ereal @ C2 @ G @ I2 ) @ X2 )
@ ( bNF_We489741039315895306al_nat @ top_to5683747375963461374_ereal @ ( sup_sup_set_nat @ B5 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_1074_If__the__inv__into__in__Func,axiom,
! [G: extended_ereal > a,C2: set_Extended_ereal,B5: set_Extended_ereal,X2: extended_ereal] :
( ( inj_on8242634198667403041real_a @ G @ C2 )
=> ( ( ord_le1644982726543182158_ereal @ C2 @ ( sup_su2680283192902082946_ereal @ B5 @ ( insert8967887681552722334_ereal @ X2 @ bot_bo8367695208629047834_ereal ) ) )
=> ( member5529087853344971084_ereal
@ ^ [I2: a] : ( if_Extended_ereal @ ( member_a @ I2 @ ( image_3724615099042636213real_a @ G @ C2 ) ) @ ( the_in377810665034427491real_a @ C2 @ G @ I2 ) @ X2 )
@ ( bNF_We381379198075229540_ereal @ top_top_set_a @ ( sup_su2680283192902082946_ereal @ B5 @ ( insert8967887681552722334_ereal @ X2 @ bot_bo8367695208629047834_ereal ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_1075_If__the__inv__into__in__Func,axiom,
! [G: extended_ereal > extended_ereal,C2: set_Extended_ereal,B5: set_Extended_ereal,X2: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ G @ C2 )
=> ( ( ord_le1644982726543182158_ereal @ C2 @ ( sup_su2680283192902082946_ereal @ B5 @ ( insert8967887681552722334_ereal @ X2 @ bot_bo8367695208629047834_ereal ) ) )
=> ( member9177544111580882812_ereal
@ ^ [I2: extended_ereal] : ( if_Extended_ereal @ ( member2350847679896131959_ereal @ I2 @ ( image_6042159593519690757_ereal @ G @ C2 ) ) @ ( the_in1141389326992810419_ereal @ C2 @ G @ I2 ) @ X2 )
@ ( bNF_We7694149557721242900_ereal @ top_to5683747375963461374_ereal @ ( sup_su2680283192902082946_ereal @ B5 @ ( insert8967887681552722334_ereal @ X2 @ bot_bo8367695208629047834_ereal ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_1076_If__the__inv__into__in__Func,axiom,
! [G: extended_ereal > nat,C2: set_Extended_ereal,B5: set_Extended_ereal,X2: extended_ereal] :
( ( inj_on318729178700965101al_nat @ G @ C2 )
=> ( ( ord_le1644982726543182158_ereal @ C2 @ ( sup_su2680283192902082946_ereal @ B5 @ ( insert8967887681552722334_ereal @ X2 @ bot_bo8367695208629047834_ereal ) ) )
=> ( member1331522584416860638_ereal
@ ^ [I2: nat] : ( if_Extended_ereal @ ( member_nat @ I2 @ ( image_7659842161140344153al_nat @ G @ C2 ) ) @ ( the_in86992963138218795al_nat @ C2 @ G @ I2 ) @ X2 )
@ ( bNF_We6362544687886832360_ereal @ top_top_set_nat @ ( sup_su2680283192902082946_ereal @ B5 @ ( insert8967887681552722334_ereal @ X2 @ bot_bo8367695208629047834_ereal ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_1077_Func__map__surj,axiom,
! [F1: extended_ereal > extended_ereal,A1: set_Extended_ereal,B1: set_Extended_ereal,F22: nat > extended_ereal,B22: set_nat,A22: set_Extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F1 @ A1 )
= B1 )
=> ( ( inj_on6191532827271902155_ereal @ F22 @ B22 )
=> ( ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F22 @ B22 ) @ A22 )
=> ( ( ( B22 = bot_bot_set_nat )
=> ( A22 = bot_bo8367695208629047834_ereal ) )
=> ( ( bNF_We6362544687886832360_ereal @ B22 @ B1 )
= ( image_5998279428054481609_ereal @ ( bNF_We5573648512502812116_ereal @ B22 @ F1 @ F22 ) @ ( bNF_We7694149557721242900_ereal @ A22 @ A1 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_1078_Func__map__surj,axiom,
! [F1: nat > extended_ereal,A1: set_nat,B1: set_Extended_ereal,F22: nat > extended_ereal,B22: set_nat,A22: set_Extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F1 @ A1 )
= B1 )
=> ( ( inj_on6191532827271902155_ereal @ F22 @ B22 )
=> ( ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F22 @ B22 ) @ A22 )
=> ( ( ( B22 = bot_bot_set_nat )
=> ( A22 = bot_bo8367695208629047834_ereal ) )
=> ( ( bNF_We6362544687886832360_ereal @ B22 @ B1 )
= ( image_6775788289030506091_ereal @ ( bNF_We7365622343019011018_ereal @ B22 @ F1 @ F22 ) @ ( bNF_We489741039315895306al_nat @ A22 @ A1 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_1079_Func__map__surj,axiom,
! [F1: extended_ereal > extended_ereal,A1: set_Extended_ereal,B1: set_Extended_ereal,F22: extended_ereal > extended_ereal,B22: set_Extended_ereal,A22: set_Extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F1 @ A1 )
= B1 )
=> ( ( inj_on7162434037990268785_ereal @ F22 @ B22 )
=> ( ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F22 @ B22 ) @ A22 )
=> ( ( ( B22 = bot_bo8367695208629047834_ereal )
=> ( A22 = bot_bo8367695208629047834_ereal ) )
=> ( ( bNF_We7694149557721242900_ereal @ B22 @ B1 )
= ( image_8079139767002015697_ereal @ ( bNF_We4175038070499737256_ereal @ B22 @ F1 @ F22 ) @ ( bNF_We7694149557721242900_ereal @ A22 @ A1 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_1080_Func__map__surj,axiom,
! [F1: nat > extended_ereal,A1: set_nat,B1: set_Extended_ereal,F22: extended_ereal > extended_ereal,B22: set_Extended_ereal,A22: set_Extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F1 @ A1 )
= B1 )
=> ( ( inj_on7162434037990268785_ereal @ F22 @ B22 )
=> ( ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F22 @ B22 ) @ A22 )
=> ( ( ( B22 = bot_bo8367695208629047834_ereal )
=> ( A22 = bot_bo8367695208629047834_ereal ) )
=> ( ( bNF_We7694149557721242900_ereal @ B22 @ B1 )
= ( image_7820330447883983791_ereal @ ( bNF_We3429895033966940150_ereal @ B22 @ F1 @ F22 ) @ ( bNF_We489741039315895306al_nat @ A22 @ A1 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_1081_Func__map,axiom,
! [G: extended_ereal > extended_ereal,A22: set_Extended_ereal,A1: set_Extended_ereal,F1: extended_ereal > extended_ereal,B1: set_Extended_ereal,F22: extended_ereal > extended_ereal,B22: set_Extended_ereal] :
( ( member9177544111580882812_ereal @ G @ ( bNF_We7694149557721242900_ereal @ A22 @ A1 ) )
=> ( ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F1 @ A1 ) @ B1 )
=> ( ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F22 @ B22 ) @ A22 )
=> ( member9177544111580882812_ereal @ ( bNF_We4175038070499737256_ereal @ B22 @ F1 @ F22 @ G ) @ ( bNF_We7694149557721242900_ereal @ B22 @ B1 ) ) ) ) ) ).
% Func_map
thf(fact_1082_Func__map,axiom,
! [G: extended_ereal > extended_ereal,A22: set_Extended_ereal,A1: set_Extended_ereal,F1: extended_ereal > extended_ereal,B1: set_Extended_ereal,F22: nat > extended_ereal,B22: set_nat] :
( ( member9177544111580882812_ereal @ G @ ( bNF_We7694149557721242900_ereal @ A22 @ A1 ) )
=> ( ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F1 @ A1 ) @ B1 )
=> ( ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F22 @ B22 ) @ A22 )
=> ( member1331522584416860638_ereal @ ( bNF_We5573648512502812116_ereal @ B22 @ F1 @ F22 @ G ) @ ( bNF_We6362544687886832360_ereal @ B22 @ B1 ) ) ) ) ) ).
% Func_map
thf(fact_1083_Func__map,axiom,
! [G: extended_ereal > nat,A22: set_Extended_ereal,A1: set_nat,F1: nat > extended_ereal,B1: set_Extended_ereal,F22: extended_ereal > extended_ereal,B22: set_Extended_ereal] :
( ( member8206335426146891064al_nat @ G @ ( bNF_We489741039315895306al_nat @ A22 @ A1 ) )
=> ( ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F1 @ A1 ) @ B1 )
=> ( ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F22 @ B22 ) @ A22 )
=> ( member9177544111580882812_ereal @ ( bNF_We3429895033966940150_ereal @ B22 @ F1 @ F22 @ G ) @ ( bNF_We7694149557721242900_ereal @ B22 @ B1 ) ) ) ) ) ).
% Func_map
thf(fact_1084_Func__map,axiom,
! [G: extended_ereal > nat,A22: set_Extended_ereal,A1: set_nat,F1: nat > extended_ereal,B1: set_Extended_ereal,F22: nat > extended_ereal,B22: set_nat] :
( ( member8206335426146891064al_nat @ G @ ( bNF_We489741039315895306al_nat @ A22 @ A1 ) )
=> ( ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F1 @ A1 ) @ B1 )
=> ( ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F22 @ B22 ) @ A22 )
=> ( member1331522584416860638_ereal @ ( bNF_We7365622343019011018_ereal @ B22 @ F1 @ F22 @ G ) @ ( bNF_We6362544687886832360_ereal @ B22 @ B1 ) ) ) ) ) ).
% Func_map
thf(fact_1085_Id__on__def,axiom,
( id_on_option_list_o
= ( ^ [A3: set_option_list_o] :
( comple1150810575505393780list_o
@ ( image_3844918200759902248list_o
@ ^ [X: option_list_o] : ( insert7989621605229100023list_o @ ( produc5745850523778858007list_o @ X @ X ) @ bot_bo6949037146550090099list_o )
@ A3 ) ) ) ) ).
% Id_on_def
thf(fact_1086_UNION__fun__upd,axiom,
! [A: a > set_Extended_ereal,I: a,B5: set_Extended_ereal,J4: set_a] :
( ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ ( fun_up1988461528155868205_ereal @ A @ I @ B5 ) @ J4 ) )
= ( sup_su2680283192902082946_ereal @ ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ A @ ( minus_minus_set_a @ J4 @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) @ ( if_set2528325808569986420_ereal @ ( member_a @ I @ J4 ) @ B5 @ bot_bo8367695208629047834_ereal ) ) ) ).
% UNION_fun_upd
thf(fact_1087_UNION__fun__upd,axiom,
! [A: extended_ereal > set_Extended_ereal,I: extended_ereal,B5: set_Extended_ereal,J4: set_Extended_ereal] :
( ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ ( fun_up8329262840412750429_ereal @ A @ I @ B5 ) @ J4 ) )
= ( sup_su2680283192902082946_ereal @ ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ A @ ( minus_1264018925008434325_ereal @ J4 @ ( insert8967887681552722334_ereal @ I @ bot_bo8367695208629047834_ereal ) ) ) ) @ ( if_set2528325808569986420_ereal @ ( member2350847679896131959_ereal @ I @ J4 ) @ B5 @ bot_bo8367695208629047834_ereal ) ) ) ).
% UNION_fun_upd
thf(fact_1088_Id__onI,axiom,
! [A4: a,A: set_a] :
( ( member_a @ A4 @ A )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ A4 ) @ ( id_on_a @ A ) ) ) ).
% Id_onI
thf(fact_1089_Id__onI,axiom,
! [A4: option_list_o,A: set_option_list_o] :
( ( member_option_list_o @ A4 @ A )
=> ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ A4 @ A4 ) @ ( id_on_option_list_o @ A ) ) ) ).
% Id_onI
thf(fact_1090_inj__on__fun__updI,axiom,
! [F2: nat > extended_ereal,A: set_nat,Y2: extended_ereal,X2: nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ A )
=> ( ~ ( member2350847679896131959_ereal @ Y2 @ ( image_4309273772856505399_ereal @ F2 @ A ) )
=> ( inj_on6191532827271902155_ereal @ ( fun_up483666046869009855_ereal @ F2 @ X2 @ Y2 ) @ A ) ) ) ).
% inj_on_fun_updI
thf(fact_1091_inj__on__fun__updI,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal,Y2: extended_ereal,X2: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A )
=> ( ~ ( member2350847679896131959_ereal @ Y2 @ ( image_6042159593519690757_ereal @ F2 @ A ) )
=> ( inj_on7162434037990268785_ereal @ ( fun_up8905698756362530941_ereal @ F2 @ X2 @ Y2 ) @ A ) ) ) ).
% inj_on_fun_updI
thf(fact_1092_Id__on__iff,axiom,
! [X2: a,Y2: a,A: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ ( id_on_a @ A ) )
= ( ( X2 = Y2 )
& ( member_a @ X2 @ A ) ) ) ).
% Id_on_iff
thf(fact_1093_Id__on__iff,axiom,
! [X2: option_list_o,Y2: option_list_o,A: set_option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X2 @ Y2 ) @ ( id_on_option_list_o @ A ) )
= ( ( X2 = Y2 )
& ( member_option_list_o @ X2 @ A ) ) ) ).
% Id_on_iff
thf(fact_1094_Id__on__eqI,axiom,
! [A4: a,B: a,A: set_a] :
( ( A4 = B )
=> ( ( member_a @ A4 @ A )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ B ) @ ( id_on_a @ A ) ) ) ) ).
% Id_on_eqI
thf(fact_1095_Id__on__eqI,axiom,
! [A4: option_list_o,B: option_list_o,A: set_option_list_o] :
( ( A4 = B )
=> ( ( member_option_list_o @ A4 @ A )
=> ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ A4 @ B ) @ ( id_on_option_list_o @ A ) ) ) ) ).
% Id_on_eqI
thf(fact_1096_Id__onE,axiom,
! [C: product_prod_a_a,A: set_a] :
( ( member1426531477525435216od_a_a @ C @ ( id_on_a @ A ) )
=> ~ ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( C
!= ( product_Pair_a_a @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_1097_Id__onE,axiom,
! [C: produc4882884732533091879list_o,A: set_option_list_o] :
( ( member1589324699396745552list_o @ C @ ( id_on_option_list_o @ A ) )
=> ~ ! [X3: option_list_o] :
( ( member_option_list_o @ X3 @ A )
=> ( C
!= ( produc5745850523778858007list_o @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_1098_fun__upd__image,axiom,
! [X2: nat,A: set_nat,F2: nat > extended_ereal,Y2: extended_ereal] :
( ( ( member_nat @ X2 @ A )
=> ( ( image_4309273772856505399_ereal @ ( fun_up483666046869009855_ereal @ F2 @ X2 @ Y2 ) @ A )
= ( insert8967887681552722334_ereal @ Y2 @ ( image_4309273772856505399_ereal @ F2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) )
& ( ~ ( member_nat @ X2 @ A )
=> ( ( image_4309273772856505399_ereal @ ( fun_up483666046869009855_ereal @ F2 @ X2 @ Y2 ) @ A )
= ( image_4309273772856505399_ereal @ F2 @ A ) ) ) ) ).
% fun_upd_image
thf(fact_1099_fun__upd__image,axiom,
! [X2: extended_ereal,A: set_Extended_ereal,F2: extended_ereal > extended_ereal,Y2: extended_ereal] :
( ( ( member2350847679896131959_ereal @ X2 @ A )
=> ( ( image_6042159593519690757_ereal @ ( fun_up8905698756362530941_ereal @ F2 @ X2 @ Y2 ) @ A )
= ( insert8967887681552722334_ereal @ Y2 @ ( image_6042159593519690757_ereal @ F2 @ ( minus_1264018925008434325_ereal @ A @ ( insert8967887681552722334_ereal @ X2 @ bot_bo8367695208629047834_ereal ) ) ) ) ) )
& ( ~ ( member2350847679896131959_ereal @ X2 @ A )
=> ( ( image_6042159593519690757_ereal @ ( fun_up8905698756362530941_ereal @ F2 @ X2 @ Y2 ) @ A )
= ( image_6042159593519690757_ereal @ F2 @ A ) ) ) ) ).
% fun_upd_image
thf(fact_1100_Field__insert,axiom,
! [A4: option_list_o,B: option_list_o,R: set_Pr7497825696620840711list_o] :
( ( field_option_list_o @ ( insert7989621605229100023list_o @ ( produc5745850523778858007list_o @ A4 @ B ) @ R ) )
= ( sup_su6641994874487927192list_o @ ( insert_option_list_o @ A4 @ ( insert_option_list_o @ B @ bot_bo3275880064859030064list_o ) ) @ ( field_option_list_o @ R ) ) ) ).
% Field_insert
thf(fact_1101_Field__insert,axiom,
! [A4: extended_ereal,B: extended_ereal,R: set_Pr2129990008675586951_ereal] :
( ( field_Extended_ereal @ ( insert3238486352267927671_ereal @ ( produc7614594614994623895_ereal @ A4 @ B ) @ R ) )
= ( sup_su2680283192902082946_ereal @ ( insert8967887681552722334_ereal @ A4 @ ( insert8967887681552722334_ereal @ B @ bot_bo8367695208629047834_ereal ) ) @ ( field_Extended_ereal @ R ) ) ) ).
% Field_insert
thf(fact_1102_SUP__ereal__minus__right,axiom,
! [I5: set_nat,C: extended_ereal,F2: nat > extended_ereal] :
( ( I5 != bot_bot_set_nat )
=> ( ( C
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( minus_2816186181549245109_ereal @ C @ ( F2 @ I2 ) )
@ I5 ) )
= ( minus_2816186181549245109_ereal @ C @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ I5 ) ) ) ) ) ) ).
% SUP_ereal_minus_right
thf(fact_1103_SUP__ereal__minus__right,axiom,
! [I5: set_Extended_ereal,C: extended_ereal,F2: extended_ereal > extended_ereal] :
( ( I5 != bot_bo8367695208629047834_ereal )
=> ( ( C
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( minus_2816186181549245109_ereal @ C @ ( F2 @ I2 ) )
@ I5 ) )
= ( minus_2816186181549245109_ereal @ C @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ I5 ) ) ) ) ) ) ).
% SUP_ereal_minus_right
thf(fact_1104_FieldI2,axiom,
! [I: a,J3: a,R2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J3 ) @ R2 )
=> ( member_a @ J3 @ ( field_a @ R2 ) ) ) ).
% FieldI2
thf(fact_1105_FieldI2,axiom,
! [I: option_list_o,J3: option_list_o,R2: set_Pr7497825696620840711list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ I @ J3 ) @ R2 )
=> ( member_option_list_o @ J3 @ ( field_option_list_o @ R2 ) ) ) ).
% FieldI2
thf(fact_1106_FieldI1,axiom,
! [I: a,J3: a,R2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J3 ) @ R2 )
=> ( member_a @ I @ ( field_a @ R2 ) ) ) ).
% FieldI1
thf(fact_1107_FieldI1,axiom,
! [I: option_list_o,J3: option_list_o,R2: set_Pr7497825696620840711list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ I @ J3 ) @ R2 )
=> ( member_option_list_o @ I @ ( field_option_list_o @ R2 ) ) ) ).
% FieldI1
thf(fact_1108_SUP__ereal__minus__left,axiom,
! [I5: set_nat,C: extended_ereal,F2: nat > extended_ereal] :
( ( I5 != bot_bot_set_nat )
=> ( ( C != extend1530274965995635425_ereal )
=> ( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( minus_2816186181549245109_ereal @ ( F2 @ I2 ) @ C )
@ I5 ) )
= ( minus_2816186181549245109_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ I5 ) ) @ C ) ) ) ) ).
% SUP_ereal_minus_left
thf(fact_1109_SUP__ereal__minus__left,axiom,
! [I5: set_Extended_ereal,C: extended_ereal,F2: extended_ereal > extended_ereal] :
( ( I5 != bot_bo8367695208629047834_ereal )
=> ( ( C != extend1530274965995635425_ereal )
=> ( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( minus_2816186181549245109_ereal @ ( F2 @ I2 ) @ C )
@ I5 ) )
= ( minus_2816186181549245109_ereal @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ I5 ) ) @ C ) ) ) ) ).
% SUP_ereal_minus_left
thf(fact_1110_cofinal__def,axiom,
( bNF_Ca7786835802997198645list_o
= ( ^ [A3: set_option_list_o,R3: set_Pr7497825696620840711list_o] :
! [X: option_list_o] :
( ( member_option_list_o @ X @ ( field_option_list_o @ R3 ) )
=> ? [Y: option_list_o] :
( ( member_option_list_o @ Y @ A3 )
& ( X != Y )
& ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X @ Y ) @ R3 ) ) ) ) ) ).
% cofinal_def
thf(fact_1111_ran__map__upd__Some,axiom,
! [M: a > option_list_o,X2: a,Y2: list_o,Z2: list_o] :
( ( ( M @ X2 )
= ( some_list_o @ Y2 ) )
=> ( ( inj_on374126998980950615list_o @ M @ ( dom_a_list_o @ M ) )
=> ( ~ ( member_list_o @ Z2 @ ( ran_a_list_o @ M ) )
=> ( ( ran_a_list_o @ ( fun_up6258215873833770339list_o @ M @ X2 @ ( some_list_o @ Z2 ) ) )
= ( sup_sup_set_list_o @ ( minus_8912710245716896613list_o @ ( ran_a_list_o @ M ) @ ( insert_list_o @ Y2 @ bot_bot_set_list_o ) ) @ ( insert_list_o @ Z2 @ bot_bot_set_list_o ) ) ) ) ) ) ).
% ran_map_upd_Some
thf(fact_1112_image__map__upd,axiom,
! [X2: a,A: set_a,M: a > option_list_o,Y2: list_o] :
( ~ ( member_a @ X2 @ A )
=> ( ( image_5155782636146240747list_o @ ( fun_up6258215873833770339list_o @ M @ X2 @ ( some_list_o @ Y2 ) ) @ A )
= ( image_5155782636146240747list_o @ M @ A ) ) ) ).
% image_map_upd
thf(fact_1113_dom__const,axiom,
! [F2: a > list_o] :
( ( dom_a_list_o
@ ^ [X: a] : ( some_list_o @ ( F2 @ X ) ) )
= top_top_set_a ) ).
% dom_const
thf(fact_1114_dom__const,axiom,
! [F2: extended_ereal > list_o] :
( ( dom_Ex1305233509568788642list_o
@ ^ [X: extended_ereal] : ( some_list_o @ ( F2 @ X ) ) )
= top_to5683747375963461374_ereal ) ).
% dom_const
thf(fact_1115_dom__const,axiom,
! [F2: nat > list_o] :
( ( dom_nat_list_o
@ ^ [X: nat] : ( some_list_o @ ( F2 @ X ) ) )
= top_top_set_nat ) ).
% dom_const
thf(fact_1116_map__add__upd__left,axiom,
! [M: a,E22: a > option_list_o,E1: a > option_list_o,U1: list_o] :
( ~ ( member_a @ M @ ( dom_a_list_o @ E22 ) )
=> ( ( map_add_a_list_o @ ( fun_up6258215873833770339list_o @ E1 @ M @ ( some_list_o @ U1 ) ) @ E22 )
= ( fun_up6258215873833770339list_o @ ( map_add_a_list_o @ E1 @ E22 ) @ M @ ( some_list_o @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_1117_insert__dom,axiom,
! [F2: a > option_list_o,X2: a,Y2: list_o] :
( ( ( F2 @ X2 )
= ( some_list_o @ Y2 ) )
=> ( ( insert_a @ X2 @ ( dom_a_list_o @ F2 ) )
= ( dom_a_list_o @ F2 ) ) ) ).
% insert_dom
thf(fact_1118_domI,axiom,
! [M: a > option_list_o,A4: a,B: list_o] :
( ( ( M @ A4 )
= ( some_list_o @ B ) )
=> ( member_a @ A4 @ ( dom_a_list_o @ M ) ) ) ).
% domI
thf(fact_1119_domD,axiom,
! [A4: a,M: a > option_list_o] :
( ( member_a @ A4 @ ( dom_a_list_o @ M ) )
=> ? [B3: list_o] :
( ( M @ A4 )
= ( some_list_o @ B3 ) ) ) ).
% domD
thf(fact_1120_in__graphI,axiom,
! [M: option_list_o > option_option_list_o,K: option_list_o,V: option_list_o] :
( ( ( M @ K )
= ( some_option_list_o @ V ) )
=> ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ K @ V ) @ ( graph_1138927715465675390list_o @ M ) ) ) ).
% in_graphI
thf(fact_1121_in__graphD,axiom,
! [K: option_list_o,V: option_list_o,M: option_list_o > option_option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ K @ V ) @ ( graph_1138927715465675390list_o @ M ) )
=> ( ( M @ K )
= ( some_option_list_o @ V ) ) ) ).
% in_graphD
thf(fact_1122_inj__Some,axiom,
! [A: set_list_o] : ( inj_on2239224958054982839list_o @ some_list_o @ A ) ).
% inj_Some
thf(fact_1123_restrict__upd__same,axiom,
! [M: extended_ereal > option_list_o,X2: extended_ereal,Y2: list_o] :
( ( restri4853605820953162368list_o @ ( fun_up3375645149235876755list_o @ M @ X2 @ ( some_list_o @ Y2 ) ) @ ( uminus5895154729394068773_ereal @ ( insert8967887681552722334_ereal @ X2 @ bot_bo8367695208629047834_ereal ) ) )
= ( restri4853605820953162368list_o @ M @ ( uminus5895154729394068773_ereal @ ( insert8967887681552722334_ereal @ X2 @ bot_bo8367695208629047834_ereal ) ) ) ) ).
% restrict_upd_same
thf(fact_1124_graph__restrictD_I1_J,axiom,
! [K: option_list_o,V: option_list_o,M: option_list_o > option_option_list_o,A: set_option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ K @ V ) @ ( graph_1138927715465675390list_o @ ( restri411196559532073264list_o @ M @ A ) ) )
=> ( member_option_list_o @ K @ A ) ) ).
% graph_restrictD(1)
thf(fact_1125_graph__restrictD_I2_J,axiom,
! [K: option_list_o,V: option_list_o,M: option_list_o > option_option_list_o,A: set_option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ K @ V ) @ ( graph_1138927715465675390list_o @ ( restri411196559532073264list_o @ M @ A ) ) )
=> ( ( M @ K )
= ( some_option_list_o @ V ) ) ) ).
% graph_restrictD(2)
thf(fact_1126_graph__map__upd,axiom,
! [M: option_list_o > option_option_list_o,K: option_list_o,V: option_list_o] :
( ( graph_1138927715465675390list_o @ ( fun_up4915027116007172163list_o @ M @ K @ ( some_option_list_o @ V ) ) )
= ( insert7989621605229100023list_o @ ( produc5745850523778858007list_o @ K @ V ) @ ( graph_1138927715465675390list_o @ ( fun_up4915027116007172163list_o @ M @ K @ none_option_list_o ) ) ) ) ).
% graph_map_upd
thf(fact_1127_INF__ereal__minus__right,axiom,
! [I5: set_nat,C: extended_ereal,F2: nat > extended_ereal] :
( ( I5 != bot_bot_set_nat )
=> ( ( ( abs_ab7465543570706387889_ereal @ C )
!= extend1530274965995635425_ereal )
=> ( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( minus_2816186181549245109_ereal @ C @ ( F2 @ I2 ) )
@ I5 ) )
= ( minus_2816186181549245109_ereal @ C @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ I5 ) ) ) ) ) ) ).
% INF_ereal_minus_right
thf(fact_1128_INF__ereal__minus__right,axiom,
! [I5: set_Extended_ereal,C: extended_ereal,F2: extended_ereal > extended_ereal] :
( ( I5 != bot_bo8367695208629047834_ereal )
=> ( ( ( abs_ab7465543570706387889_ereal @ C )
!= extend1530274965995635425_ereal )
=> ( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( minus_2816186181549245109_ereal @ C @ ( F2 @ I2 ) )
@ I5 ) )
= ( minus_2816186181549245109_ereal @ C @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ I5 ) ) ) ) ) ) ).
% INF_ereal_minus_right
thf(fact_1129_restrict__out,axiom,
! [X2: a,A: set_a,M: a > option_list_o] :
( ~ ( member_a @ X2 @ A )
=> ( ( restri1321886973334633680list_o @ M @ A @ X2 )
= none_list_o ) ) ).
% restrict_out
thf(fact_1130_dom__eq__empty__conv,axiom,
! [F2: a > option_list_o] :
( ( ( dom_a_list_o @ F2 )
= bot_bot_set_a )
= ( F2
= ( ^ [X: a] : none_list_o ) ) ) ).
% dom_eq_empty_conv
thf(fact_1131_dom__eq__empty__conv,axiom,
! [F2: extended_ereal > option_list_o] :
( ( ( dom_Ex1305233509568788642list_o @ F2 )
= bot_bo8367695208629047834_ereal )
= ( F2
= ( ^ [X: extended_ereal] : none_list_o ) ) ) ).
% dom_eq_empty_conv
thf(fact_1132_restrict__map__to__empty,axiom,
! [M: extended_ereal > option_list_o] :
( ( restri4853605820953162368list_o @ M @ bot_bo8367695208629047834_ereal )
= ( ^ [X: extended_ereal] : none_list_o ) ) ).
% restrict_map_to_empty
thf(fact_1133_fun__upd__None__if__notin__dom,axiom,
! [K: a,M: a > option_list_o] :
( ~ ( member_a @ K @ ( dom_a_list_o @ M ) )
=> ( ( fun_up6258215873833770339list_o @ M @ K @ none_list_o )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_1134_dom__empty,axiom,
( ( dom_a_list_o
@ ^ [X: a] : none_list_o )
= bot_bot_set_a ) ).
% dom_empty
thf(fact_1135_dom__empty,axiom,
( ( dom_Ex1305233509568788642list_o
@ ^ [X: extended_ereal] : none_list_o )
= bot_bo8367695208629047834_ereal ) ).
% dom_empty
thf(fact_1136_dom__fun__upd,axiom,
! [Y2: option_list_o,F2: a > option_list_o,X2: a] :
( ( ( Y2 = none_list_o )
=> ( ( dom_a_list_o @ ( fun_up6258215873833770339list_o @ F2 @ X2 @ Y2 ) )
= ( minus_minus_set_a @ ( dom_a_list_o @ F2 ) @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) )
& ( ( Y2 != none_list_o )
=> ( ( dom_a_list_o @ ( fun_up6258215873833770339list_o @ F2 @ X2 @ Y2 ) )
= ( insert_a @ X2 @ ( dom_a_list_o @ F2 ) ) ) ) ) ).
% dom_fun_upd
thf(fact_1137_dom__fun__upd,axiom,
! [Y2: option_list_o,F2: extended_ereal > option_list_o,X2: extended_ereal] :
( ( ( Y2 = none_list_o )
=> ( ( dom_Ex1305233509568788642list_o @ ( fun_up3375645149235876755list_o @ F2 @ X2 @ Y2 ) )
= ( minus_1264018925008434325_ereal @ ( dom_Ex1305233509568788642list_o @ F2 ) @ ( insert8967887681552722334_ereal @ X2 @ bot_bo8367695208629047834_ereal ) ) ) )
& ( ( Y2 != none_list_o )
=> ( ( dom_Ex1305233509568788642list_o @ ( fun_up3375645149235876755list_o @ F2 @ X2 @ Y2 ) )
= ( insert8967887681552722334_ereal @ X2 @ ( dom_Ex1305233509568788642list_o @ F2 ) ) ) ) ) ).
% dom_fun_upd
thf(fact_1138_fun__upd__None__restrict,axiom,
! [X2: a,D: set_a,M: a > option_list_o] :
( ( ( member_a @ X2 @ D )
=> ( ( fun_up6258215873833770339list_o @ ( restri1321886973334633680list_o @ M @ D ) @ X2 @ none_list_o )
= ( restri1321886973334633680list_o @ M @ ( minus_minus_set_a @ D @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) )
& ( ~ ( member_a @ X2 @ D )
=> ( ( fun_up6258215873833770339list_o @ ( restri1321886973334633680list_o @ M @ D ) @ X2 @ none_list_o )
= ( restri1321886973334633680list_o @ M @ D ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_1139_fun__upd__None__restrict,axiom,
! [X2: extended_ereal,D: set_Extended_ereal,M: extended_ereal > option_list_o] :
( ( ( member2350847679896131959_ereal @ X2 @ D )
=> ( ( fun_up3375645149235876755list_o @ ( restri4853605820953162368list_o @ M @ D ) @ X2 @ none_list_o )
= ( restri4853605820953162368list_o @ M @ ( minus_1264018925008434325_ereal @ D @ ( insert8967887681552722334_ereal @ X2 @ bot_bo8367695208629047834_ereal ) ) ) ) )
& ( ~ ( member2350847679896131959_ereal @ X2 @ D )
=> ( ( fun_up3375645149235876755list_o @ ( restri4853605820953162368list_o @ M @ D ) @ X2 @ none_list_o )
= ( restri4853605820953162368list_o @ M @ D ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_1140_None__notin__image__Some,axiom,
! [A: set_list_o] :
~ ( member_option_list_o @ none_list_o @ ( image_3961372166222858059list_o @ some_list_o @ A ) ) ).
% None_notin_image_Some
thf(fact_1141_restrict__map__def,axiom,
( restri1321886973334633680list_o
= ( ^ [M4: a > option_list_o,A3: set_a,X: a] : ( if_option_list_o @ ( member_a @ X @ A3 ) @ ( M4 @ X ) @ none_list_o ) ) ) ).
% restrict_map_def
thf(fact_1142_encoding__triv,axiom,
( prefix7485107378405021920ding_a
@ ^ [X: a] : none_list_o ) ).
% encoding_triv
thf(fact_1143_encoding__triv,axiom,
( prefix7485107378405021921ding_b
@ ^ [X: b] : none_list_o ) ).
% encoding_triv
thf(fact_1144_opt__prefix_Osimps_I2_J,axiom,
! [Uv: option_list_o] :
~ ( prefix8824957607401505554efix_o @ none_list_o @ Uv ) ).
% opt_prefix.simps(2)
thf(fact_1145_opt__prefix_Osimps_I3_J,axiom,
! [Uu2: option_list_o] :
~ ( prefix8824957607401505554efix_o @ Uu2 @ none_list_o ) ).
% opt_prefix.simps(3)
thf(fact_1146_opt__append_Osimps_I2_J,axiom,
! [Uv: option_list_o] :
( ( prefix5314359684614007693append @ none_list_o @ Uv )
= none_list_o ) ).
% opt_append.simps(2)
thf(fact_1147_opt__append_Osimps_I3_J,axiom,
! [Uu2: option_list_o] :
( ( prefix5314359684614007693append @ Uu2 @ none_list_o )
= none_list_o ) ).
% opt_append.simps(3)
thf(fact_1148_domIff,axiom,
! [A4: a,M: a > option_list_o] :
( ( member_a @ A4 @ ( dom_a_list_o @ M ) )
= ( ( M @ A4 )
!= none_list_o ) ) ).
% domIff
thf(fact_1149_dom__def,axiom,
( dom_a_list_o
= ( ^ [M4: a > option_list_o] :
( collect_a
@ ^ [A6: a] :
( ( M4 @ A6 )
!= none_list_o ) ) ) ) ).
% dom_def
thf(fact_1150_dom__minus,axiom,
! [F2: a > option_list_o,X2: a,A: set_a] :
( ( ( F2 @ X2 )
= none_list_o )
=> ( ( minus_minus_set_a @ ( dom_a_list_o @ F2 ) @ ( insert_a @ X2 @ A ) )
= ( minus_minus_set_a @ ( dom_a_list_o @ F2 ) @ A ) ) ) ).
% dom_minus
thf(fact_1151_notin__range__Some,axiom,
! [X2: option_list_o] :
( ( ~ ( member_option_list_o @ X2 @ ( image_3961372166222858059list_o @ some_list_o @ top_top_set_list_o ) ) )
= ( X2 = none_list_o ) ) ).
% notin_range_Some
thf(fact_1152_notin__range__Some,axiom,
! [X2: option2379537754664488340_ereal] :
( ( ~ ( member4803645955555566013_ereal @ X2 @ ( image_7938633928507707211_ereal @ some_Extended_ereal @ top_to5683747375963461374_ereal ) ) )
= ( X2 = none_Extended_ereal ) ) ).
% notin_range_Some
thf(fact_1153_notin__range__Some,axiom,
! [X2: option_nat] :
( ( ~ ( member_option_nat @ X2 @ ( image_nat_option_nat @ some_nat @ top_top_set_nat ) ) )
= ( X2 = none_nat ) ) ).
% notin_range_Some
thf(fact_1154_UNIV__option__conv,axiom,
( top_to633166595683317524list_o
= ( insert_option_list_o @ none_list_o @ ( image_3961372166222858059list_o @ some_list_o @ top_top_set_list_o ) ) ) ).
% UNIV_option_conv
thf(fact_1155_UNIV__option__conv,axiom,
( top_to8408571170889687620_ereal
= ( insert294909853350783844_ereal @ none_Extended_ereal @ ( image_7938633928507707211_ereal @ some_Extended_ereal @ top_to5683747375963461374_ereal ) ) ) ).
% UNIV_option_conv
thf(fact_1156_UNIV__option__conv,axiom,
( top_to8920198386146353926on_nat
= ( insert_option_nat @ none_nat @ ( image_nat_option_nat @ some_nat @ top_top_set_nat ) ) ) ).
% UNIV_option_conv
thf(fact_1157_graph__fun__upd__None,axiom,
! [M: a > option_b,K: a] :
( ( graph_a_b @ ( fun_upd_a_option_b @ M @ K @ none_b ) )
= ( collec3336397801687681299od_a_b
@ ^ [E2: product_prod_a_b] :
( ( member1426531481828664017od_a_b @ E2 @ ( graph_a_b @ M ) )
& ( ( product_fst_a_b @ E2 )
!= K ) ) ) ) ).
% graph_fun_upd_None
thf(fact_1158_restrict__complement__singleton__eq,axiom,
! [F2: extended_ereal > option_list_o,X2: extended_ereal] :
( ( restri4853605820953162368list_o @ F2 @ ( uminus5895154729394068773_ereal @ ( insert8967887681552722334_ereal @ X2 @ bot_bo8367695208629047834_ereal ) ) )
= ( fun_up3375645149235876755list_o @ F2 @ X2 @ none_list_o ) ) ).
% restrict_complement_singleton_eq
thf(fact_1159_dom__eq__singleton__conv,axiom,
! [F2: a > option_list_o,X2: a] :
( ( ( dom_a_list_o @ F2 )
= ( insert_a @ X2 @ bot_bot_set_a ) )
= ( ? [V2: list_o] :
( F2
= ( fun_up6258215873833770339list_o
@ ^ [X: a] : none_list_o
@ X2
@ ( some_list_o @ V2 ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_1160_dom__eq__singleton__conv,axiom,
! [F2: extended_ereal > option_list_o,X2: extended_ereal] :
( ( ( dom_Ex1305233509568788642list_o @ F2 )
= ( insert8967887681552722334_ereal @ X2 @ bot_bo8367695208629047834_ereal ) )
= ( ? [V2: list_o] :
( F2
= ( fun_up3375645149235876755list_o
@ ^ [X: extended_ereal] : none_list_o
@ X2
@ ( some_list_o @ V2 ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_1161_ereal__SUP__not__infty,axiom,
! [A: set_nat,L: extended_ereal,U2: extended_ereal,F2: nat > extended_ereal] :
( ( A != bot_bot_set_nat )
=> ( ( L
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( U2 != extend1530274965995635425_ereal )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( ord_le1083603963089353582_ereal @ L @ ( F2 @ X3 ) )
& ( ord_le1083603963089353582_ereal @ ( F2 @ X3 ) @ U2 ) ) )
=> ( ( abs_ab7465543570706387889_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) )
!= extend1530274965995635425_ereal ) ) ) ) ) ).
% ereal_SUP_not_infty
thf(fact_1162_ereal__SUP__not__infty,axiom,
! [A: set_Extended_ereal,L: extended_ereal,U2: extended_ereal,F2: extended_ereal > extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( L
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( U2 != extend1530274965995635425_ereal )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
=> ( ( ord_le1083603963089353582_ereal @ L @ ( F2 @ X3 ) )
& ( ord_le1083603963089353582_ereal @ ( F2 @ X3 ) @ U2 ) ) )
=> ( ( abs_ab7465543570706387889_ereal @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) )
!= extend1530274965995635425_ereal ) ) ) ) ) ).
% ereal_SUP_not_infty
thf(fact_1163_ereal__INF__not__infty,axiom,
! [A: set_nat,L: extended_ereal,U2: extended_ereal,F2: nat > extended_ereal] :
( ( A != bot_bot_set_nat )
=> ( ( L
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( U2 != extend1530274965995635425_ereal )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( ord_le1083603963089353582_ereal @ L @ ( F2 @ X3 ) )
& ( ord_le1083603963089353582_ereal @ ( F2 @ X3 ) @ U2 ) ) )
=> ( ( abs_ab7465543570706387889_ereal @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) ) )
!= extend1530274965995635425_ereal ) ) ) ) ) ).
% ereal_INF_not_infty
thf(fact_1164_ereal__INF__not__infty,axiom,
! [A: set_Extended_ereal,L: extended_ereal,U2: extended_ereal,F2: extended_ereal > extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ( ( L
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( U2 != extend1530274965995635425_ereal )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
=> ( ( ord_le1083603963089353582_ereal @ L @ ( F2 @ X3 ) )
& ( ord_le1083603963089353582_ereal @ ( F2 @ X3 ) @ U2 ) ) )
=> ( ( abs_ab7465543570706387889_ereal @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) ) )
!= extend1530274965995635425_ereal ) ) ) ) ) ).
% ereal_INF_not_infty
thf(fact_1165_Some__image__these__eq,axiom,
! [A: set_option_list_o] :
( ( image_3961372166222858059list_o @ some_list_o @ ( these_list_o @ A ) )
= ( collec4355076819549272527list_o
@ ^ [X: option_list_o] :
( ( member_option_list_o @ X @ A )
& ( X != none_list_o ) ) ) ) ).
% Some_image_these_eq
thf(fact_1166_these__image__Some__eq,axiom,
! [A: set_list_o] :
( ( these_list_o @ ( image_3961372166222858059list_o @ some_list_o @ A ) )
= A ) ).
% these_image_Some_eq
thf(fact_1167_bit__count_Ocases,axiom,
! [X2: option_list_o] :
( ( X2 != none_list_o )
=> ~ ! [X3: list_o] :
( X2
!= ( some_list_o @ X3 ) ) ) ).
% bit_count.cases
thf(fact_1168_opt__prefix_Ocases,axiom,
! [X2: produc4882884732533091879list_o] :
( ! [X3: list_o,Y3: list_o] :
( X2
!= ( produc5745850523778858007list_o @ ( some_list_o @ X3 ) @ ( some_list_o @ Y3 ) ) )
=> ( ! [Uv2: option_list_o] :
( X2
!= ( produc5745850523778858007list_o @ none_list_o @ Uv2 ) )
=> ~ ! [Uu3: option_list_o] :
( X2
!= ( produc5745850523778858007list_o @ Uu3 @ none_list_o ) ) ) ) ).
% opt_prefix.cases
thf(fact_1169_opt__append_Ocases,axiom,
! [X2: produc4882884732533091879list_o] :
( ! [X3: list_o,Y3: list_o] :
( X2
!= ( produc5745850523778858007list_o @ ( some_list_o @ X3 ) @ ( some_list_o @ Y3 ) ) )
=> ( ! [Uv2: option_list_o] :
( X2
!= ( produc5745850523778858007list_o @ none_list_o @ Uv2 ) )
=> ~ ! [Uu3: option_list_o] :
( X2
!= ( produc5745850523778858007list_o @ Uu3 @ none_list_o ) ) ) ) ).
% opt_append.cases
thf(fact_1170_Option_Othese__def,axiom,
( these_list_o
= ( ^ [A3: set_option_list_o] :
( image_6070160259998645695list_o @ the_list_o
@ ( collec4355076819549272527list_o
@ ^ [X: option_list_o] :
( ( member_option_list_o @ X @ A3 )
& ( X != none_list_o ) ) ) ) ) ) ).
% Option.these_def
thf(fact_1171_SUP__ereal__le__addI,axiom,
! [F2: extended_ereal > extended_ereal,Y2: extended_ereal,Z2: extended_ereal] :
( ! [I3: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ ( F2 @ I3 ) @ Y2 ) @ Z2 )
=> ( ( Y2
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal ) ) @ Y2 ) @ Z2 ) ) ) ).
% SUP_ereal_le_addI
thf(fact_1172_SUP__ereal__le__addI,axiom,
! [F2: nat > extended_ereal,Y2: extended_ereal,Z2: extended_ereal] :
( ! [I3: nat] : ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ ( F2 @ I3 ) @ Y2 ) @ Z2 )
=> ( ( Y2
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat ) ) @ Y2 ) @ Z2 ) ) ) ).
% SUP_ereal_le_addI
thf(fact_1173_SUP__ereal__add__left,axiom,
! [I5: set_nat,C: extended_ereal,F2: nat > extended_ereal] :
( ( I5 != bot_bot_set_nat )
=> ( ( C
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( plus_p7876563987511257093_ereal @ ( F2 @ I2 ) @ C )
@ I5 ) )
= ( plus_p7876563987511257093_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ I5 ) ) @ C ) ) ) ) ).
% SUP_ereal_add_left
thf(fact_1174_SUP__ereal__add__left,axiom,
! [I5: set_Extended_ereal,C: extended_ereal,F2: extended_ereal > extended_ereal] :
( ( I5 != bot_bo8367695208629047834_ereal )
=> ( ( C
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( plus_p7876563987511257093_ereal @ ( F2 @ I2 ) @ C )
@ I5 ) )
= ( plus_p7876563987511257093_ereal @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ I5 ) ) @ C ) ) ) ) ).
% SUP_ereal_add_left
thf(fact_1175_SUP__ereal__add__right,axiom,
! [I5: set_nat,C: extended_ereal,F2: nat > extended_ereal] :
( ( I5 != bot_bot_set_nat )
=> ( ( C
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( plus_p7876563987511257093_ereal @ C @ ( F2 @ I2 ) )
@ I5 ) )
= ( plus_p7876563987511257093_ereal @ C @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ I5 ) ) ) ) ) ) ).
% SUP_ereal_add_right
thf(fact_1176_SUP__ereal__add__right,axiom,
! [I5: set_Extended_ereal,C: extended_ereal,F2: extended_ereal > extended_ereal] :
( ( I5 != bot_bo8367695208629047834_ereal )
=> ( ( C
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( plus_p7876563987511257093_ereal @ C @ ( F2 @ I2 ) )
@ I5 ) )
= ( plus_p7876563987511257093_ereal @ C @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ I5 ) ) ) ) ) ) ).
% SUP_ereal_add_right
thf(fact_1177_INF__ereal__add__left,axiom,
! [I5: set_nat,C: extended_ereal,F2: nat > extended_ereal] :
( ( I5 != bot_bot_set_nat )
=> ( ( C
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ I5 )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F2 @ X3 ) ) )
=> ( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( plus_p7876563987511257093_ereal @ ( F2 @ I2 ) @ C )
@ I5 ) )
= ( plus_p7876563987511257093_ereal @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ I5 ) ) @ C ) ) ) ) ) ).
% INF_ereal_add_left
thf(fact_1178_INF__ereal__add__left,axiom,
! [I5: set_a,C: extended_ereal,F2: a > extended_ereal] :
( ( I5 != bot_bot_set_a )
=> ( ( C
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ I5 )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F2 @ X3 ) ) )
=> ( ( comple3556804143462414037_ereal
@ ( image_8405481351990995413_ereal
@ ^ [I2: a] : ( plus_p7876563987511257093_ereal @ ( F2 @ I2 ) @ C )
@ I5 ) )
= ( plus_p7876563987511257093_ereal @ ( comple3556804143462414037_ereal @ ( image_8405481351990995413_ereal @ F2 @ I5 ) ) @ C ) ) ) ) ) ).
% INF_ereal_add_left
thf(fact_1179_INF__ereal__add__left,axiom,
! [I5: set_Extended_ereal,C: extended_ereal,F2: extended_ereal > extended_ereal] :
( ( I5 != bot_bo8367695208629047834_ereal )
=> ( ( C
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ I5 )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F2 @ X3 ) ) )
=> ( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( plus_p7876563987511257093_ereal @ ( F2 @ I2 ) @ C )
@ I5 ) )
= ( plus_p7876563987511257093_ereal @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ I5 ) ) @ C ) ) ) ) ) ).
% INF_ereal_add_left
thf(fact_1180_INF__ereal__add__right,axiom,
! [I5: set_nat,C: extended_ereal,F2: nat > extended_ereal] :
( ( I5 != bot_bot_set_nat )
=> ( ( C
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ I5 )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F2 @ X3 ) ) )
=> ( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( plus_p7876563987511257093_ereal @ C @ ( F2 @ I2 ) )
@ I5 ) )
= ( plus_p7876563987511257093_ereal @ C @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ I5 ) ) ) ) ) ) ) ).
% INF_ereal_add_right
thf(fact_1181_INF__ereal__add__right,axiom,
! [I5: set_a,C: extended_ereal,F2: a > extended_ereal] :
( ( I5 != bot_bot_set_a )
=> ( ( C
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ I5 )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F2 @ X3 ) ) )
=> ( ( comple3556804143462414037_ereal
@ ( image_8405481351990995413_ereal
@ ^ [I2: a] : ( plus_p7876563987511257093_ereal @ C @ ( F2 @ I2 ) )
@ I5 ) )
= ( plus_p7876563987511257093_ereal @ C @ ( comple3556804143462414037_ereal @ ( image_8405481351990995413_ereal @ F2 @ I5 ) ) ) ) ) ) ) ).
% INF_ereal_add_right
thf(fact_1182_INF__ereal__add__right,axiom,
! [I5: set_Extended_ereal,C: extended_ereal,F2: extended_ereal > extended_ereal] :
( ( I5 != bot_bo8367695208629047834_ereal )
=> ( ( C
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ I5 )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F2 @ X3 ) ) )
=> ( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( plus_p7876563987511257093_ereal @ C @ ( F2 @ I2 ) )
@ I5 ) )
= ( plus_p7876563987511257093_ereal @ C @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ I5 ) ) ) ) ) ) ) ).
% INF_ereal_add_right
thf(fact_1183_image__add__0,axiom,
! [S4: set_Extended_ereal] :
( ( image_6042159593519690757_ereal @ ( plus_p7876563987511257093_ereal @ zero_z2744965634713055877_ereal ) @ S4 )
= S4 ) ).
% image_add_0
thf(fact_1184_SUP__ereal__eq__0__iff__nonneg,axiom,
! [A: set_nat,F2: nat > extended_ereal] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F2 @ X3 ) ) )
=> ( ( A != bot_bot_set_nat )
=> ( ( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A ) )
= zero_z2744965634713055877_ereal )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( F2 @ X )
= zero_z2744965634713055877_ereal ) ) ) ) ) ) ).
% SUP_ereal_eq_0_iff_nonneg
thf(fact_1185_SUP__ereal__eq__0__iff__nonneg,axiom,
! [A: set_Extended_ereal,F2: extended_ereal > extended_ereal] :
( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F2 @ X3 ) ) )
=> ( ( A != bot_bo8367695208629047834_ereal )
=> ( ( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A ) )
= zero_z2744965634713055877_ereal )
= ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A )
=> ( ( F2 @ X )
= zero_z2744965634713055877_ereal ) ) ) ) ) ) ).
% SUP_ereal_eq_0_iff_nonneg
thf(fact_1186_SUP__ereal__add__directed,axiom,
! [I5: set_Extended_ereal,F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal] :
( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ I5 )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F2 @ I3 ) ) )
=> ( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ I5 )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( G @ I3 ) ) )
=> ( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ I5 )
=> ! [J2: extended_ereal] :
( ( member2350847679896131959_ereal @ J2 @ I5 )
=> ? [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ I5 )
& ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ ( F2 @ I3 ) @ ( G @ J2 ) ) @ ( plus_p7876563987511257093_ereal @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ) )
=> ( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( plus_p7876563987511257093_ereal @ ( F2 @ I2 ) @ ( G @ I2 ) )
@ I5 ) )
= ( plus_p7876563987511257093_ereal @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ I5 ) ) @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ G @ I5 ) ) ) ) ) ) ) ).
% SUP_ereal_add_directed
thf(fact_1187_SUP__ereal__add__directed,axiom,
! [I5: set_nat,F2: nat > extended_ereal,G: nat > extended_ereal] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F2 @ I3 ) ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( G @ I3 ) ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ! [J2: nat] :
( ( member_nat @ J2 @ I5 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ I5 )
& ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ ( F2 @ I3 ) @ ( G @ J2 ) ) @ ( plus_p7876563987511257093_ereal @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ) )
=> ( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( plus_p7876563987511257093_ereal @ ( F2 @ I2 ) @ ( G @ I2 ) )
@ I5 ) )
= ( plus_p7876563987511257093_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ I5 ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ I5 ) ) ) ) ) ) ) ).
% SUP_ereal_add_directed
thf(fact_1188_SUP__ereal__add__directed,axiom,
! [I5: set_a,F2: a > extended_ereal,G: a > extended_ereal] :
( ! [I3: a] :
( ( member_a @ I3 @ I5 )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F2 @ I3 ) ) )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I5 )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( G @ I3 ) ) )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I5 )
=> ! [J2: a] :
( ( member_a @ J2 @ I5 )
=> ? [X4: a] :
( ( member_a @ X4 @ I5 )
& ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ ( F2 @ I3 ) @ ( G @ J2 ) ) @ ( plus_p7876563987511257093_ereal @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ) )
=> ( ( comple8415311339701865915_ereal
@ ( image_8405481351990995413_ereal
@ ^ [I2: a] : ( plus_p7876563987511257093_ereal @ ( F2 @ I2 ) @ ( G @ I2 ) )
@ I5 ) )
= ( plus_p7876563987511257093_ereal @ ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ F2 @ I5 ) ) @ ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ G @ I5 ) ) ) ) ) ) ) ).
% SUP_ereal_add_directed
thf(fact_1189_INF__ereal__add__directed,axiom,
! [I5: set_Extended_ereal,F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal] :
( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ I5 )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F2 @ I3 ) ) )
=> ( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ I5 )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( G @ I3 ) ) )
=> ( ! [I3: extended_ereal] :
( ( member2350847679896131959_ereal @ I3 @ I5 )
=> ! [J2: extended_ereal] :
( ( member2350847679896131959_ereal @ J2 @ I5 )
=> ? [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ I5 )
& ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ ( F2 @ X4 ) @ ( G @ X4 ) ) @ ( plus_p7876563987511257093_ereal @ ( F2 @ I3 ) @ ( G @ J2 ) ) ) ) ) )
=> ( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( plus_p7876563987511257093_ereal @ ( F2 @ I2 ) @ ( G @ I2 ) )
@ I5 ) )
= ( plus_p7876563987511257093_ereal @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ F2 @ I5 ) ) @ ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ G @ I5 ) ) ) ) ) ) ) ).
% INF_ereal_add_directed
thf(fact_1190_INF__ereal__add__directed,axiom,
! [I5: set_nat,F2: nat > extended_ereal,G: nat > extended_ereal] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F2 @ I3 ) ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( G @ I3 ) ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ! [J2: nat] :
( ( member_nat @ J2 @ I5 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ I5 )
& ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ ( F2 @ X4 ) @ ( G @ X4 ) ) @ ( plus_p7876563987511257093_ereal @ ( F2 @ I3 ) @ ( G @ J2 ) ) ) ) ) )
=> ( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( plus_p7876563987511257093_ereal @ ( F2 @ I2 ) @ ( G @ I2 ) )
@ I5 ) )
= ( plus_p7876563987511257093_ereal @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ I5 ) ) @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ G @ I5 ) ) ) ) ) ) ) ).
% INF_ereal_add_directed
thf(fact_1191_INF__ereal__add__directed,axiom,
! [I5: set_a,F2: a > extended_ereal,G: a > extended_ereal] :
( ! [I3: a] :
( ( member_a @ I3 @ I5 )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F2 @ I3 ) ) )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I5 )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( G @ I3 ) ) )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I5 )
=> ! [J2: a] :
( ( member_a @ J2 @ I5 )
=> ? [X4: a] :
( ( member_a @ X4 @ I5 )
& ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ ( F2 @ X4 ) @ ( G @ X4 ) ) @ ( plus_p7876563987511257093_ereal @ ( F2 @ I3 ) @ ( G @ J2 ) ) ) ) ) )
=> ( ( comple3556804143462414037_ereal
@ ( image_8405481351990995413_ereal
@ ^ [I2: a] : ( plus_p7876563987511257093_ereal @ ( F2 @ I2 ) @ ( G @ I2 ) )
@ I5 ) )
= ( plus_p7876563987511257093_ereal @ ( comple3556804143462414037_ereal @ ( image_8405481351990995413_ereal @ F2 @ I5 ) ) @ ( comple3556804143462414037_ereal @ ( image_8405481351990995413_ereal @ G @ I5 ) ) ) ) ) ) ) ).
% INF_ereal_add_directed
thf(fact_1192_SUP__ereal__add,axiom,
! [F2: nat > extended_ereal,G: nat > extended_ereal] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ F2 )
=> ( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ G )
=> ( ! [I3: nat] :
( ( F2 @ I3 )
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ! [I3: nat] :
( ( G @ I3 )
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( plus_p7876563987511257093_ereal @ ( F2 @ I2 ) @ ( G @ I2 ) )
@ top_top_set_nat ) )
= ( plus_p7876563987511257093_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ top_top_set_nat ) ) ) ) ) ) ) ) ).
% SUP_ereal_add
thf(fact_1193_ereal__inj__affinity,axiom,
! [M: extended_ereal,T2: extended_ereal,A: set_Extended_ereal] :
( ( ( abs_ab7465543570706387889_ereal @ M )
!= extend1530274965995635425_ereal )
=> ( ( M != zero_z2744965634713055877_ereal )
=> ( ( ( abs_ab7465543570706387889_ereal @ T2 )
!= extend1530274965995635425_ereal )
=> ( inj_on7162434037990268785_ereal
@ ^ [X: extended_ereal] : ( plus_p7876563987511257093_ereal @ ( times_7703590493115627913_ereal @ M @ X ) @ T2 )
@ A ) ) ) ) ).
% ereal_inj_affinity
thf(fact_1194_mono__add,axiom,
! [A4: extended_ereal] : ( monoto2923698811514177639_ereal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ A4 ) ) ).
% mono_add
thf(fact_1195_mono__add,axiom,
! [A4: nat] : ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ ( plus_plus_nat @ A4 ) ) ).
% mono_add
thf(fact_1196_antimonoI,axiom,
! [F2: extended_ereal > extended_ereal] :
( ! [X3: extended_ereal,Y3: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X3 @ Y3 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ Y3 ) @ ( F2 @ X3 ) ) )
=> ( monoto2923698811514177639_ereal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal
@ ^ [X: extended_ereal,Y: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y @ X )
@ F2 ) ) ).
% antimonoI
thf(fact_1197_antimonoI,axiom,
! [F2: extended_ereal > nat] :
( ! [X3: extended_ereal,Y3: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ Y3 ) @ ( F2 @ X3 ) ) )
=> ( monoto2580034644210098551al_nat @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal
@ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
@ F2 ) ) ).
% antimonoI
thf(fact_1198_antimonoI,axiom,
! [F2: extended_ereal > set_Extended_ereal] :
( ! [X3: extended_ereal,Y3: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X3 @ Y3 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ Y3 ) @ ( F2 @ X3 ) ) )
=> ( monoto1076656197419758151_ereal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal
@ ^ [X: set_Extended_ereal,Y: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ Y @ X )
@ F2 ) ) ).
% antimonoI
thf(fact_1199_antimonoI,axiom,
! [F2: nat > extended_ereal] :
( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ Y3 ) @ ( F2 @ X3 ) ) )
=> ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X: extended_ereal,Y: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y @ X )
@ F2 ) ) ).
% antimonoI
thf(fact_1200_antimonoI,axiom,
! [F2: nat > nat] :
( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ Y3 ) @ ( F2 @ X3 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
@ F2 ) ) ).
% antimonoI
thf(fact_1201_antimonoI,axiom,
! [F2: nat > set_Extended_ereal] :
( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ Y3 ) @ ( F2 @ X3 ) ) )
=> ( monoto6788471982328799797_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X: set_Extended_ereal,Y: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ Y @ X )
@ F2 ) ) ).
% antimonoI
thf(fact_1202_antimonoI,axiom,
! [F2: set_Extended_ereal > extended_ereal] :
( ! [X3: set_Extended_ereal,Y3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ X3 @ Y3 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ Y3 ) @ ( F2 @ X3 ) ) )
=> ( monoto5942119339478196871_ereal @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal
@ ^ [X: extended_ereal,Y: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y @ X )
@ F2 ) ) ).
% antimonoI
thf(fact_1203_antimonoI,axiom,
! [F2: set_Extended_ereal > nat] :
( ! [X3: set_Extended_ereal,Y3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ Y3 ) @ ( F2 @ X3 ) ) )
=> ( monoto375287072342888279al_nat @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal
@ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
@ F2 ) ) ).
% antimonoI
thf(fact_1204_antimonoI,axiom,
! [F2: set_Extended_ereal > set_Extended_ereal] :
( ! [X3: set_Extended_ereal,Y3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ X3 @ Y3 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ Y3 ) @ ( F2 @ X3 ) ) )
=> ( monoto2181219000246062183_ereal @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal
@ ^ [X: set_Extended_ereal,Y: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ Y @ X )
@ F2 ) ) ).
% antimonoI
thf(fact_1205_antimonoE,axiom,
! [F2: extended_ereal > extended_ereal,X2: extended_ereal,Y2: extended_ereal] :
( ( monoto2923698811514177639_ereal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal
@ ^ [X: extended_ereal,Y: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y @ X )
@ F2 )
=> ( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ Y2 ) @ ( F2 @ X2 ) ) ) ) ).
% antimonoE
thf(fact_1206_antimonoE,axiom,
! [F2: extended_ereal > nat,X2: extended_ereal,Y2: extended_ereal] :
( ( monoto2580034644210098551al_nat @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal
@ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
@ F2 )
=> ( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ Y2 ) @ ( F2 @ X2 ) ) ) ) ).
% antimonoE
thf(fact_1207_antimonoE,axiom,
! [F2: extended_ereal > set_Extended_ereal,X2: extended_ereal,Y2: extended_ereal] :
( ( monoto1076656197419758151_ereal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal
@ ^ [X: set_Extended_ereal,Y: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ Y @ X )
@ F2 )
=> ( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ Y2 ) @ ( F2 @ X2 ) ) ) ) ).
% antimonoE
thf(fact_1208_antimonoE,axiom,
! [F2: nat > extended_ereal,X2: nat,Y2: nat] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X: extended_ereal,Y: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y @ X )
@ F2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ Y2 ) @ ( F2 @ X2 ) ) ) ) ).
% antimonoE
thf(fact_1209_antimonoE,axiom,
! [F2: nat > nat,X2: nat,Y2: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
@ F2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ Y2 ) @ ( F2 @ X2 ) ) ) ) ).
% antimonoE
thf(fact_1210_antimonoE,axiom,
! [F2: nat > set_Extended_ereal,X2: nat,Y2: nat] :
( ( monoto6788471982328799797_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X: set_Extended_ereal,Y: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ Y @ X )
@ F2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ Y2 ) @ ( F2 @ X2 ) ) ) ) ).
% antimonoE
thf(fact_1211_antimonoE,axiom,
! [F2: set_Extended_ereal > extended_ereal,X2: set_Extended_ereal,Y2: set_Extended_ereal] :
( ( monoto5942119339478196871_ereal @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal
@ ^ [X: extended_ereal,Y: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y @ X )
@ F2 )
=> ( ( ord_le1644982726543182158_ereal @ X2 @ Y2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ Y2 ) @ ( F2 @ X2 ) ) ) ) ).
% antimonoE
thf(fact_1212_antimonoE,axiom,
! [F2: set_Extended_ereal > nat,X2: set_Extended_ereal,Y2: set_Extended_ereal] :
( ( monoto375287072342888279al_nat @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal
@ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
@ F2 )
=> ( ( ord_le1644982726543182158_ereal @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ Y2 ) @ ( F2 @ X2 ) ) ) ) ).
% antimonoE
thf(fact_1213_antimonoE,axiom,
! [F2: set_Extended_ereal > set_Extended_ereal,X2: set_Extended_ereal,Y2: set_Extended_ereal] :
( ( monoto2181219000246062183_ereal @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal
@ ^ [X: set_Extended_ereal,Y: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ Y @ X )
@ F2 )
=> ( ( ord_le1644982726543182158_ereal @ X2 @ Y2 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ Y2 ) @ ( F2 @ X2 ) ) ) ) ).
% antimonoE
thf(fact_1214_antimonoD,axiom,
! [F2: extended_ereal > extended_ereal,X2: extended_ereal,Y2: extended_ereal] :
( ( monoto2923698811514177639_ereal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal
@ ^ [X: extended_ereal,Y: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y @ X )
@ F2 )
=> ( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ Y2 ) @ ( F2 @ X2 ) ) ) ) ).
% antimonoD
thf(fact_1215_antimonoD,axiom,
! [F2: extended_ereal > nat,X2: extended_ereal,Y2: extended_ereal] :
( ( monoto2580034644210098551al_nat @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal
@ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
@ F2 )
=> ( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ Y2 ) @ ( F2 @ X2 ) ) ) ) ).
% antimonoD
thf(fact_1216_antimonoD,axiom,
! [F2: extended_ereal > set_Extended_ereal,X2: extended_ereal,Y2: extended_ereal] :
( ( monoto1076656197419758151_ereal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal
@ ^ [X: set_Extended_ereal,Y: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ Y @ X )
@ F2 )
=> ( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ Y2 ) @ ( F2 @ X2 ) ) ) ) ).
% antimonoD
thf(fact_1217_antimonoD,axiom,
! [F2: nat > extended_ereal,X2: nat,Y2: nat] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X: extended_ereal,Y: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y @ X )
@ F2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ Y2 ) @ ( F2 @ X2 ) ) ) ) ).
% antimonoD
thf(fact_1218_antimonoD,axiom,
! [F2: nat > nat,X2: nat,Y2: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
@ F2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ Y2 ) @ ( F2 @ X2 ) ) ) ) ).
% antimonoD
thf(fact_1219_antimonoD,axiom,
! [F2: nat > set_Extended_ereal,X2: nat,Y2: nat] :
( ( monoto6788471982328799797_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X: set_Extended_ereal,Y: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ Y @ X )
@ F2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ Y2 ) @ ( F2 @ X2 ) ) ) ) ).
% antimonoD
thf(fact_1220_antimonoD,axiom,
! [F2: set_Extended_ereal > extended_ereal,X2: set_Extended_ereal,Y2: set_Extended_ereal] :
( ( monoto5942119339478196871_ereal @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal
@ ^ [X: extended_ereal,Y: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y @ X )
@ F2 )
=> ( ( ord_le1644982726543182158_ereal @ X2 @ Y2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ Y2 ) @ ( F2 @ X2 ) ) ) ) ).
% antimonoD
thf(fact_1221_antimonoD,axiom,
! [F2: set_Extended_ereal > nat,X2: set_Extended_ereal,Y2: set_Extended_ereal] :
( ( monoto375287072342888279al_nat @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal
@ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
@ F2 )
=> ( ( ord_le1644982726543182158_ereal @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ Y2 ) @ ( F2 @ X2 ) ) ) ) ).
% antimonoD
thf(fact_1222_antimonoD,axiom,
! [F2: set_Extended_ereal > set_Extended_ereal,X2: set_Extended_ereal,Y2: set_Extended_ereal] :
( ( monoto2181219000246062183_ereal @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal
@ ^ [X: set_Extended_ereal,Y: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ Y @ X )
@ F2 )
=> ( ( ord_le1644982726543182158_ereal @ X2 @ Y2 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ Y2 ) @ ( F2 @ X2 ) ) ) ) ).
% antimonoD
thf(fact_1223_mono__imp__mono__on,axiom,
! [F2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( monoto2923698811514177639_ereal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_le1083603963089353582_ereal @ F2 )
=> ( monoto2923698811514177639_ereal @ A @ ord_le1083603963089353582_ereal @ ord_le1083603963089353582_ereal @ F2 ) ) ).
% mono_imp_mono_on
thf(fact_1224_mono__imp__mono__on,axiom,
! [F2: extended_ereal > nat,A: set_Extended_ereal] :
( ( monoto2580034644210098551al_nat @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_less_eq_nat @ F2 )
=> ( monoto2580034644210098551al_nat @ A @ ord_le1083603963089353582_ereal @ ord_less_eq_nat @ F2 ) ) ).
% mono_imp_mono_on
thf(fact_1225_mono__imp__mono__on,axiom,
! [F2: extended_ereal > set_Extended_ereal,A: set_Extended_ereal] :
( ( monoto1076656197419758151_ereal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_le1644982726543182158_ereal @ F2 )
=> ( monoto1076656197419758151_ereal @ A @ ord_le1083603963089353582_ereal @ ord_le1644982726543182158_ereal @ F2 ) ) ).
% mono_imp_mono_on
thf(fact_1226_mono__imp__mono__on,axiom,
! [F2: nat > extended_ereal,A: set_nat] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ F2 )
=> ( monoto8452838292781035605_ereal @ A @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ F2 ) ) ).
% mono_imp_mono_on
thf(fact_1227_mono__imp__mono__on,axiom,
! [F2: nat > nat,A: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F2 )
=> ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F2 ) ) ).
% mono_imp_mono_on
thf(fact_1228_mono__imp__mono__on,axiom,
! [F2: nat > set_Extended_ereal,A: set_nat] :
( ( monoto6788471982328799797_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1644982726543182158_ereal @ F2 )
=> ( monoto6788471982328799797_ereal @ A @ ord_less_eq_nat @ ord_le1644982726543182158_ereal @ F2 ) ) ).
% mono_imp_mono_on
thf(fact_1229_mono__imp__mono__on,axiom,
! [F2: set_Extended_ereal > extended_ereal,A: set_se6634062954251873166_ereal] :
( ( monoto5942119339478196871_ereal @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal @ ord_le1083603963089353582_ereal @ F2 )
=> ( monoto5942119339478196871_ereal @ A @ ord_le1644982726543182158_ereal @ ord_le1083603963089353582_ereal @ F2 ) ) ).
% mono_imp_mono_on
thf(fact_1230_mono__imp__mono__on,axiom,
! [F2: set_Extended_ereal > nat,A: set_se6634062954251873166_ereal] :
( ( monoto375287072342888279al_nat @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal @ ord_less_eq_nat @ F2 )
=> ( monoto375287072342888279al_nat @ A @ ord_le1644982726543182158_ereal @ ord_less_eq_nat @ F2 ) ) ).
% mono_imp_mono_on
thf(fact_1231_mono__imp__mono__on,axiom,
! [F2: set_Extended_ereal > set_Extended_ereal,A: set_se6634062954251873166_ereal] :
( ( monoto2181219000246062183_ereal @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal @ ord_le1644982726543182158_ereal @ F2 )
=> ( monoto2181219000246062183_ereal @ A @ ord_le1644982726543182158_ereal @ ord_le1644982726543182158_ereal @ F2 ) ) ).
% mono_imp_mono_on
thf(fact_1232_monoI,axiom,
! [F2: extended_ereal > extended_ereal] :
( ! [X3: extended_ereal,Y3: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X3 @ Y3 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( monoto2923698811514177639_ereal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_le1083603963089353582_ereal @ F2 ) ) ).
% monoI
thf(fact_1233_monoI,axiom,
! [F2: extended_ereal > nat] :
( ! [X3: extended_ereal,Y3: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( monoto2580034644210098551al_nat @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_less_eq_nat @ F2 ) ) ).
% monoI
thf(fact_1234_monoI,axiom,
! [F2: extended_ereal > set_Extended_ereal] :
( ! [X3: extended_ereal,Y3: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X3 @ Y3 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( monoto1076656197419758151_ereal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_le1644982726543182158_ereal @ F2 ) ) ).
% monoI
thf(fact_1235_monoI,axiom,
! [F2: nat > extended_ereal] :
( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ F2 ) ) ).
% monoI
thf(fact_1236_monoI,axiom,
! [F2: nat > nat] :
( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F2 ) ) ).
% monoI
thf(fact_1237_monoI,axiom,
! [F2: nat > set_Extended_ereal] :
( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( monoto6788471982328799797_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1644982726543182158_ereal @ F2 ) ) ).
% monoI
thf(fact_1238_monoI,axiom,
! [F2: set_Extended_ereal > extended_ereal] :
( ! [X3: set_Extended_ereal,Y3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ X3 @ Y3 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( monoto5942119339478196871_ereal @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal @ ord_le1083603963089353582_ereal @ F2 ) ) ).
% monoI
thf(fact_1239_monoI,axiom,
! [F2: set_Extended_ereal > nat] :
( ! [X3: set_Extended_ereal,Y3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( monoto375287072342888279al_nat @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal @ ord_less_eq_nat @ F2 ) ) ).
% monoI
thf(fact_1240_monoI,axiom,
! [F2: set_Extended_ereal > set_Extended_ereal] :
( ! [X3: set_Extended_ereal,Y3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ X3 @ Y3 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( monoto2181219000246062183_ereal @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal @ ord_le1644982726543182158_ereal @ F2 ) ) ).
% monoI
thf(fact_1241_monoE,axiom,
! [F2: extended_ereal > extended_ereal,X2: extended_ereal,Y2: extended_ereal] :
( ( monoto2923698811514177639_ereal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_le1083603963089353582_ereal @ F2 )
=> ( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) ) ) ).
% monoE
thf(fact_1242_monoE,axiom,
! [F2: extended_ereal > nat,X2: extended_ereal,Y2: extended_ereal] :
( ( monoto2580034644210098551al_nat @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_less_eq_nat @ F2 )
=> ( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) ) ) ).
% monoE
thf(fact_1243_monoE,axiom,
! [F2: extended_ereal > set_Extended_ereal,X2: extended_ereal,Y2: extended_ereal] :
( ( monoto1076656197419758151_ereal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_le1644982726543182158_ereal @ F2 )
=> ( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) ) ) ).
% monoE
thf(fact_1244_monoE,axiom,
! [F2: nat > extended_ereal,X2: nat,Y2: nat] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ F2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) ) ) ).
% monoE
thf(fact_1245_monoE,axiom,
! [F2: nat > nat,X2: nat,Y2: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) ) ) ).
% monoE
thf(fact_1246_monoE,axiom,
! [F2: nat > set_Extended_ereal,X2: nat,Y2: nat] :
( ( monoto6788471982328799797_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1644982726543182158_ereal @ F2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) ) ) ).
% monoE
thf(fact_1247_monoE,axiom,
! [F2: set_Extended_ereal > extended_ereal,X2: set_Extended_ereal,Y2: set_Extended_ereal] :
( ( monoto5942119339478196871_ereal @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal @ ord_le1083603963089353582_ereal @ F2 )
=> ( ( ord_le1644982726543182158_ereal @ X2 @ Y2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) ) ) ).
% monoE
thf(fact_1248_monoE,axiom,
! [F2: set_Extended_ereal > nat,X2: set_Extended_ereal,Y2: set_Extended_ereal] :
( ( monoto375287072342888279al_nat @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal @ ord_less_eq_nat @ F2 )
=> ( ( ord_le1644982726543182158_ereal @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) ) ) ).
% monoE
thf(fact_1249_monoE,axiom,
! [F2: set_Extended_ereal > set_Extended_ereal,X2: set_Extended_ereal,Y2: set_Extended_ereal] :
( ( monoto2181219000246062183_ereal @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal @ ord_le1644982726543182158_ereal @ F2 )
=> ( ( ord_le1644982726543182158_ereal @ X2 @ Y2 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) ) ) ).
% monoE
thf(fact_1250_monoD,axiom,
! [F2: extended_ereal > extended_ereal,X2: extended_ereal,Y2: extended_ereal] :
( ( monoto2923698811514177639_ereal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_le1083603963089353582_ereal @ F2 )
=> ( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) ) ) ).
% monoD
thf(fact_1251_monoD,axiom,
! [F2: extended_ereal > nat,X2: extended_ereal,Y2: extended_ereal] :
( ( monoto2580034644210098551al_nat @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_less_eq_nat @ F2 )
=> ( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) ) ) ).
% monoD
thf(fact_1252_monoD,axiom,
! [F2: extended_ereal > set_Extended_ereal,X2: extended_ereal,Y2: extended_ereal] :
( ( monoto1076656197419758151_ereal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_le1644982726543182158_ereal @ F2 )
=> ( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) ) ) ).
% monoD
thf(fact_1253_monoD,axiom,
! [F2: nat > extended_ereal,X2: nat,Y2: nat] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ F2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) ) ) ).
% monoD
thf(fact_1254_monoD,axiom,
! [F2: nat > nat,X2: nat,Y2: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) ) ) ).
% monoD
thf(fact_1255_monoD,axiom,
! [F2: nat > set_Extended_ereal,X2: nat,Y2: nat] :
( ( monoto6788471982328799797_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1644982726543182158_ereal @ F2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) ) ) ).
% monoD
thf(fact_1256_monoD,axiom,
! [F2: set_Extended_ereal > extended_ereal,X2: set_Extended_ereal,Y2: set_Extended_ereal] :
( ( monoto5942119339478196871_ereal @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal @ ord_le1083603963089353582_ereal @ F2 )
=> ( ( ord_le1644982726543182158_ereal @ X2 @ Y2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) ) ) ).
% monoD
thf(fact_1257_monoD,axiom,
! [F2: set_Extended_ereal > nat,X2: set_Extended_ereal,Y2: set_Extended_ereal] :
( ( monoto375287072342888279al_nat @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal @ ord_less_eq_nat @ F2 )
=> ( ( ord_le1644982726543182158_ereal @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) ) ) ).
% monoD
thf(fact_1258_monoD,axiom,
! [F2: set_Extended_ereal > set_Extended_ereal,X2: set_Extended_ereal,Y2: set_Extended_ereal] :
( ( monoto2181219000246062183_ereal @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal @ ord_le1644982726543182158_ereal @ F2 )
=> ( ( ord_le1644982726543182158_ereal @ X2 @ Y2 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) ) ) ).
% monoD
thf(fact_1259_monotoneI,axiom,
! [Orda: nat > nat > $o,Ordb: extended_ereal > extended_ereal > $o,F2: nat > extended_ereal] :
( ! [X3: nat,Y3: nat] :
( ( Orda @ X3 @ Y3 )
=> ( Ordb @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( monoto8452838292781035605_ereal @ top_top_set_nat @ Orda @ Ordb @ F2 ) ) ).
% monotoneI
thf(fact_1260_monotoneD,axiom,
! [Orda: nat > nat > $o,Ordb: extended_ereal > extended_ereal > $o,F2: nat > extended_ereal,X2: nat,Y2: nat] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ Orda @ Ordb @ F2 )
=> ( ( Orda @ X2 @ Y2 )
=> ( Ordb @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) ) ) ).
% monotoneD
thf(fact_1261_mono__sup,axiom,
! [F2: nat > extended_ereal,A: nat,B5: nat] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ F2 )
=> ( ord_le1083603963089353582_ereal @ ( sup_su7653423775389492130_ereal @ ( F2 @ A ) @ ( F2 @ B5 ) ) @ ( F2 @ ( sup_sup_nat @ A @ B5 ) ) ) ) ).
% mono_sup
thf(fact_1262_mono__sup,axiom,
! [F2: nat > nat,A: nat,B5: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ ( F2 @ A ) @ ( F2 @ B5 ) ) @ ( F2 @ ( sup_sup_nat @ A @ B5 ) ) ) ) ).
% mono_sup
thf(fact_1263_mono__sup,axiom,
! [F2: nat > set_Extended_ereal,A: nat,B5: nat] :
( ( monoto6788471982328799797_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1644982726543182158_ereal @ F2 )
=> ( ord_le1644982726543182158_ereal @ ( sup_su2680283192902082946_ereal @ ( F2 @ A ) @ ( F2 @ B5 ) ) @ ( F2 @ ( sup_sup_nat @ A @ B5 ) ) ) ) ).
% mono_sup
thf(fact_1264_mono__sup,axiom,
! [F2: set_Extended_ereal > extended_ereal,A: set_Extended_ereal,B5: set_Extended_ereal] :
( ( monoto5942119339478196871_ereal @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal @ ord_le1083603963089353582_ereal @ F2 )
=> ( ord_le1083603963089353582_ereal @ ( sup_su7653423775389492130_ereal @ ( F2 @ A ) @ ( F2 @ B5 ) ) @ ( F2 @ ( sup_su2680283192902082946_ereal @ A @ B5 ) ) ) ) ).
% mono_sup
thf(fact_1265_mono__sup,axiom,
! [F2: set_Extended_ereal > nat,A: set_Extended_ereal,B5: set_Extended_ereal] :
( ( monoto375287072342888279al_nat @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal @ ord_less_eq_nat @ F2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ ( F2 @ A ) @ ( F2 @ B5 ) ) @ ( F2 @ ( sup_su2680283192902082946_ereal @ A @ B5 ) ) ) ) ).
% mono_sup
thf(fact_1266_mono__sup,axiom,
! [F2: set_Extended_ereal > set_Extended_ereal,A: set_Extended_ereal,B5: set_Extended_ereal] :
( ( monoto2181219000246062183_ereal @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal @ ord_le1644982726543182158_ereal @ F2 )
=> ( ord_le1644982726543182158_ereal @ ( sup_su2680283192902082946_ereal @ ( F2 @ A ) @ ( F2 @ B5 ) ) @ ( F2 @ ( sup_su2680283192902082946_ereal @ A @ B5 ) ) ) ) ).
% mono_sup
thf(fact_1267_Sup__countable__SUP,axiom,
! [A: set_Extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ? [F4: nat > extended_ereal] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ F4 )
& ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F4 @ top_top_set_nat ) @ A )
& ( ( comple8415311339701865915_ereal @ A )
= ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F4 @ top_top_set_nat ) ) ) ) ) ).
% Sup_countable_SUP
thf(fact_1268_Inf__countable__INF,axiom,
! [A: set_Extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ? [F4: nat > extended_ereal] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X: extended_ereal,Y: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y @ X )
@ F4 )
& ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F4 @ top_top_set_nat ) @ A )
& ( ( comple3556804143462414037_ereal @ A )
= ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F4 @ top_top_set_nat ) ) ) ) ) ).
% Inf_countable_INF
thf(fact_1269_SUP__ereal__add__pos,axiom,
! [F2: nat > extended_ereal,G: nat > extended_ereal] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ F2 )
=> ( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ G )
=> ( ! [I3: nat] : ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F2 @ I3 ) )
=> ( ! [I3: nat] : ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( G @ I3 ) )
=> ( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( plus_p7876563987511257093_ereal @ ( F2 @ I2 ) @ ( G @ I2 ) )
@ top_top_set_nat ) )
= ( plus_p7876563987511257093_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ top_top_set_nat ) ) ) ) ) ) ) ) ).
% SUP_ereal_add_pos
thf(fact_1270_INF__ereal__add,axiom,
! [F2: nat > extended_ereal,G: nat > extended_ereal] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X: extended_ereal,Y: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y @ X )
@ F2 )
=> ( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X: extended_ereal,Y: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y @ X )
@ G )
=> ( ! [I3: nat] :
( ( F2 @ I3 )
!= extend1530274965995635425_ereal )
=> ( ! [I3: nat] :
( ( G @ I3 )
!= extend1530274965995635425_ereal )
=> ( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( plus_p7876563987511257093_ereal @ ( F2 @ I2 ) @ ( G @ I2 ) )
@ top_top_set_nat ) )
= ( plus_p7876563987511257093_ereal @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat ) ) @ ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ G @ top_top_set_nat ) ) ) ) ) ) ) ) ).
% INF_ereal_add
thf(fact_1271_suminf__SUP__eq,axiom,
! [F2: nat > nat > extended_ereal] :
( ! [I3: nat] :
( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal
@ ^ [N4: nat] : ( F2 @ N4 @ I3 ) )
=> ( ! [N3: nat,I3: nat] : ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F2 @ N3 @ I3 ) )
=> ( ( suminf4411151127299490740_ereal
@ ^ [I2: nat] :
( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [N4: nat] : ( F2 @ N4 @ I2 )
@ top_top_set_nat ) ) )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [N4: nat] : ( suminf4411151127299490740_ereal @ ( F2 @ N4 ) )
@ top_top_set_nat ) ) ) ) ) ).
% suminf_SUP_eq
thf(fact_1272_ereal__open__affinity,axiom,
! [S4: set_Extended_ereal,M: extended_ereal,T2: extended_ereal] :
( ( topolo9005793602937862549_ereal @ S4 )
=> ( ( ( abs_ab7465543570706387889_ereal @ M )
!= extend1530274965995635425_ereal )
=> ( ( M != zero_z2744965634713055877_ereal )
=> ( ( ( abs_ab7465543570706387889_ereal @ T2 )
!= extend1530274965995635425_ereal )
=> ( topolo9005793602937862549_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : ( plus_p7876563987511257093_ereal @ ( times_7703590493115627913_ereal @ M @ X ) @ T2 )
@ S4 ) ) ) ) ) ) ).
% ereal_open_affinity
thf(fact_1273_open__uminus__iff,axiom,
! [S4: set_Extended_ereal] :
( ( topolo9005793602937862549_ereal @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S4 ) )
= ( topolo9005793602937862549_ereal @ S4 ) ) ).
% open_uminus_iff
thf(fact_1274_ereal__open__uminus,axiom,
! [S4: set_Extended_ereal] :
( ( topolo9005793602937862549_ereal @ S4 )
=> ( topolo9005793602937862549_ereal @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S4 ) ) ) ).
% ereal_open_uminus
thf(fact_1275_ereal__open__affinity__pos,axiom,
! [S4: set_Extended_ereal,M: extended_ereal,T2: extended_ereal] :
( ( topolo9005793602937862549_ereal @ S4 )
=> ( ( M != extend1530274965995635425_ereal )
=> ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ M )
=> ( ( ( abs_ab7465543570706387889_ereal @ T2 )
!= extend1530274965995635425_ereal )
=> ( topolo9005793602937862549_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : ( plus_p7876563987511257093_ereal @ ( times_7703590493115627913_ereal @ M @ X ) @ T2 )
@ S4 ) ) ) ) ) ) ).
% ereal_open_affinity_pos
% Helper facts (11)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X2: a,Y2: a] :
( ( if_a @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X2: a,Y2: a] :
( ( if_a @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_2_1_If_001t__Extended____Real__Oereal_T,axiom,
! [X2: extended_ereal,Y2: extended_ereal] :
( ( if_Extended_ereal @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Extended____Real__Oereal_T,axiom,
! [X2: extended_ereal,Y2: extended_ereal] :
( ( if_Extended_ereal @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_2_1_If_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_T,axiom,
! [X2: option_list_o,Y2: option_list_o] :
( ( if_option_list_o @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_T,axiom,
! [X2: option_list_o,Y2: option_list_o] :
( ( if_option_list_o @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__Extended____Real__Oereal_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Extended____Real__Oereal_J_T,axiom,
! [X2: set_Extended_ereal,Y2: set_Extended_ereal] :
( ( if_set2528325808569986420_ereal @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Extended____Real__Oereal_J_T,axiom,
! [X2: set_Extended_ereal,Y2: set_Extended_ereal] :
( ( if_set2528325808569986420_ereal @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
x = y ).
%------------------------------------------------------------------------------