TPTP Problem File: SLH0507^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Prefix_Free_Code_Combinators/0000_Prefix_Free_Code_Combinators/prob_00152_005480__11797746_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1947 ( 459 unt; 663 typ; 0 def)
% Number of atoms : 3711 (1566 equ; 0 cnn)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 14157 ( 111 ~; 6 |; 171 &;12188 @)
% ( 0 <=>;1681 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 66 ( 65 usr)
% Number of type conns : 4424 (4424 >; 0 *; 0 +; 0 <<)
% Number of symbols : 601 ( 598 usr; 35 con; 0-5 aty)
% Number of variables : 4661 ( 636 ^;3916 !; 109 ?;4661 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:57:18.509
%------------------------------------------------------------------------------
% Could-be-implicit typings (65)
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__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_se7855581050983116737at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
set_Pr400265656397884439et_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Rat__Orat_Mt__Extended____Real__Oereal_J_J,type,
set_Pr856337538289292321_ereal: $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__Rat__Orat_J_J,type,
set_Pr3294083815370073227al_rat: $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,
set_Pr594667477129265975real_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Rat__Orat_Mt__Product____Type__Oprod_Itf__a_Mtf__b_J_J_J,type,
set_ra7925906563089340391od_a_b: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__a_Mtf__b_J_J_J,type,
set_na3619395881858068463od_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Rat__Orat_Mt__Rat__Orat_J_J,type,
set_Pr8928021450653196913at_rat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Rat__Orat_Mt__Nat__Onat_J_J,type,
set_Pr6084635751276098665at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
set_Pr4105333604307423337at_rat: $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__Rat__Orat_Mt__Extended____Real__Oereal_J,type,
produc6179351041441756523_ereal: $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__Rat__Orat_J,type,
produc2039727735755102933al_rat: $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_I_062_It__Rat__Orat_Mt__Extended____Real__Oereal_J_J,type,
set_ra8820423881963243661_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__Rat__Orat_J_J,type,
set_Ex2034798122189248759al_rat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Extended____Real__Oereal_Mt__Nat__Onat_J_J,type,
set_Ex8414784459666926319al_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
set_Pr4934435412358123699_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_J,type,
set_Pr4193341848836149977_nat_a: $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__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $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__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_It__Rat__Orat_Mt__Nat__Onat_J,type,
product_prod_rat_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Rat__Orat_J,type,
product_prod_nat_rat: $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_I_062_It__Rat__Orat_Mt__Rat__Orat_J_J,type,
set_rat_rat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Rat__Orat_Mt__Nat__Onat_J_J,type,
set_rat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
set_nat_rat: $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__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
product_prod_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
product_prod_nat_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
set_set_rat: $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__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_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_I_062_Itf__a_M_Eo_J_J,type,
set_a_o: $tType ).
thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
set_rat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Extended____Real__Oereal,type,
extended_ereal: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Rat__Orat,type,
rat: $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 (598)
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__Nat__Onat_001t__Nat__Onat,type,
bNF_Gr_nat_nat: set_nat > ( nat > nat ) > set_Pr1261947904930325089at_nat ).
thf(sy_c_BNF__Def_OGr_001t__Nat__Onat_001t__Rat__Orat,type,
bNF_Gr_nat_rat: set_nat > ( nat > rat ) > set_Pr4105333604307423337at_rat ).
thf(sy_c_BNF__Def_OGr_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
bNF_Gr_nat_set_nat: set_nat > ( nat > set_nat ) > set_Pr400265656397884439et_nat ).
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__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
bNF_Gr3847987472475283150at_nat: set_nat_nat > ( ( nat > nat ) > nat > nat ) > ( nat > nat ) > ( nat > nat ) > $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_001_062_It__Rat__Orat_Mt__Product____Type__Oprod_Itf__a_Mtf__b_J_J_001_062_It__Rat__Orat_Mtf__a_J,type,
bNF_Gr7582414742221077988_rat_a: set_ra7925906563089340391od_a_b > ( ( rat > product_prod_a_b ) > rat > a ) > ( rat > product_prod_a_b ) > ( rat > a ) > $o ).
thf(sy_c_BNF__Def_OGrp_001_062_It__Rat__Orat_Mt__Product____Type__Oprod_Itf__a_Mtf__b_J_J_001_062_It__Rat__Orat_Mtf__b_J,type,
bNF_Gr7582414746524306789_rat_b: set_ra7925906563089340391od_a_b > ( ( rat > product_prod_a_b ) > rat > b ) > ( rat > product_prod_a_b ) > ( rat > 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__Extended____Real__Oereal_001t__Rat__Orat,type,
bNF_Gr3211092294157423064al_rat: set_Extended_ereal > ( extended_ereal > rat ) > extended_ereal > rat > $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__Nat__Onat_001t__Rat__Orat,type,
bNF_Grp_nat_rat: set_nat > ( nat > rat ) > nat > rat > $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_001t__Rat__Orat_001t__Extended____Real__Oereal,type,
bNF_Gr8001861554983367334_ereal: set_rat > ( rat > extended_ereal ) > rat > extended_ereal > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Rat__Orat_001t__Nat__Onat,type,
bNF_Grp_rat_nat: set_rat > ( rat > nat ) > rat > nat > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Rat__Orat_001t__Rat__Orat,type,
bNF_Grp_rat_rat: set_rat > ( rat > rat ) > rat > rat > $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_Oconvol_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_co805650143699787099at_nat: ( nat > nat ) > ( nat > nat ) > nat > product_prod_nat_nat ).
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_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bNF_re3262823321055862553at_nat: ( ( nat > nat ) > ( nat > nat ) > $o ) > ( ( ( nat > nat ) > nat > nat ) > ( ( nat > nat ) > nat > nat ) > $o ) > ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
bNF_re239970166668089693at_nat: ( ( nat > nat ) > ( nat > nat ) > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( ( nat > nat ) > nat > nat ) > ( ( nat > nat ) > nat > nat ) > $o ).
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_001t__Nat__Onat,type,
bNF_re1087668796686660353al_nat: ( extended_ereal > extended_ereal > $o ) > ( extended_ereal > nat > $o ) > ( extended_ereal > extended_ereal ) > ( extended_ereal > nat ) > $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_001t__Nat__Onat_001t__Extended____Real__Oereal,type,
bNF_re6960472445257597407_ereal: ( extended_ereal > extended_ereal > $o ) > ( nat > extended_ereal > $o ) > ( extended_ereal > nat ) > ( extended_ereal > extended_ereal ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_re4327621139402860031at_nat: ( extended_ereal > extended_ereal > $o ) > ( nat > nat > $o ) > ( extended_ereal > nat ) > ( extended_ereal > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal_001t__Nat__Onat_001tf__a,type,
bNF_re5750240445021093519_nat_a: ( extended_ereal > extended_ereal > $o ) > ( nat > a > $o ) > ( extended_ereal > nat ) > ( 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_001t__Nat__Onat,type,
bNF_re1718308943842807089_a_nat: ( extended_ereal > extended_ereal > $o ) > ( a > nat > $o ) > ( extended_ereal > a ) > ( extended_ereal > nat ) > $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 ).
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the_in5057678521256355851et_nat: set_nat > ( nat > set_nat ) > set_nat > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001tf__a,type,
the_inv_into_nat_a: set_nat > ( nat > a ) > a > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Rat__Orat_001t__Nat__Onat,type,
the_inv_into_rat_nat: set_rat > ( rat > nat ) > nat > rat ).
thf(sy_c_Fun_Othe__inv__into_001t__Rat__Orat_001tf__a,type,
the_inv_into_rat_a: set_rat > ( rat > a ) > a > rat ).
thf(sy_c_Fun_Othe__inv__into_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
the_in1492043616600986635at_nat: set_set_nat > ( set_nat > nat ) > nat > set_nat ).
thf(sy_c_Fun_Othe__inv__into_001tf__a_001t__Nat__Onat,type,
the_inv_into_a_nat: set_a > ( a > nat ) > nat > a ).
thf(sy_c_Fun_Othe__inv__into_001tf__a_001tf__a,type,
the_inv_into_a_a: set_a > ( a > a ) > a > a ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Extended____Real__Oereal,type,
abs_ab7465543570706387889_ereal: extended_ereal > extended_ereal ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Real__Oereal,type,
plus_p7876563987511257093_ereal: extended_ereal > extended_ereal > extended_ereal ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Real__Oereal,type,
times_7703590493115627913_ereal: extended_ereal > extended_ereal > extended_ereal ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Extended____Real__Oereal,type,
uminus27091377158695749_ereal: extended_ereal > extended_ereal ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Rat__Orat,type,
uminus_uminus_rat: rat > rat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
uminus5895154729394068773_ereal: set_Extended_ereal > set_Extended_ereal ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Product____Type__Oprod_It__Extended____Real__Oereal_Mt__Extended____Real__Oereal_J_J,type,
uminus6085167613793993086_ereal: set_Pr2129990008675586951_ereal > set_Pr2129990008675586951_ereal ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Product____Type__Oprod_It__Extended____Real__Oereal_Mt__Nat__Onat_J_J,type,
uminus6262216577377593932al_nat: set_Pr450698115992974979al_nat > set_Pr450698115992974979al_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Product____Type__Oprod_It__Extended____Real__Oereal_Mt__Rat__Orat_J_J,type,
uminus9105602276754692180al_rat: set_Pr3294083815370073227al_rat > set_Pr3294083815370073227al_rat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Extended____Real__Oereal_J_J,type,
uminus4999475943122058226_ereal: set_Pr8411329518592215081_ereal > set_Pr8411329518592215081_ereal ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Product____Type__Oprod_It__Rat__Orat_Mt__Extended____Real__Oereal_J_J,type,
uminus6667855999673911274_ereal: set_Pr856337538289292321_ereal > set_Pr856337538289292321_ereal ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Rat__Orat_J,type,
uminus2201863774496077783et_rat: set_rat > set_rat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
uminus613421341184616069et_nat: set_set_nat > set_set_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Real__Oereal,type,
zero_z2744965634713055877_ereal: extended_ereal ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Extended____Real__Oereal,type,
groups5544561043438645954_ereal: ( nat > extended_ereal ) > set_nat > extended_ereal ).
thf(sy_c_Hilbert__Choice_Obijection_001t__Extended____Real__Oereal,type,
hilber6088754731438466237_ereal: ( extended_ereal > extended_ereal ) > $o ).
thf(sy_c_Hilbert__Choice_Obijection_001t__Nat__Onat,type,
hilber5277034221543178913on_nat: ( nat > nat ) > $o ).
thf(sy_c_Hilbert__Choice_Obijection_001t__Rat__Orat,type,
hilber4641904161456683177on_rat: ( rat > rat ) > $o ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
hilber7422611030134141132_ereal: set_Extended_ereal > ( extended_ereal > extended_ereal ) > extended_ereal > extended_ereal ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
hilber949482391279124050al_nat: set_Extended_ereal > ( extended_ereal > nat ) > nat > extended_ereal ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Extended____Real__Oereal_001t__Rat__Orat,type,
hilber314352331192628314al_rat: set_Extended_ereal > ( extended_ereal > rat ) > rat > extended_ereal ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Extended____Real__Oereal_001tf__a,type,
hilber3276319488169350396real_a: set_Extended_ereal > ( extended_ereal > a ) > a > extended_ereal ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Nat__Onat_001t__Extended____Real__Oereal,type,
hilber6822286039850061104_ereal: set_nat > ( nat > extended_ereal ) > extended_ereal > nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Nat__Onat_001t__Nat__Onat,type,
hilber3633877196798814958at_nat: set_nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Nat__Onat_001t__Rat__Orat,type,
hilber2998747136712319222at_rat: set_nat > ( nat > rat ) > rat > nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
hilber7337601069305372324et_nat: set_nat > ( nat > set_nat ) > set_nat > nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Nat__Onat_001tf__a,type,
hilber2795491120104822624_nat_a: set_nat > ( nat > a ) > a > nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Rat__Orat_001t__Extended____Real__Oereal,type,
hilber5105121592018572584_ereal: set_rat > ( rat > extended_ereal ) > extended_ereal > rat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Rat__Orat_001t__Nat__Onat,type,
hilber3317322552863949046at_nat: set_rat > ( rat > nat ) > nat > rat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Rat__Orat_001t__Rat__Orat,type,
hilber2682192492777453310at_rat: set_rat > ( rat > rat ) > rat > rat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Rat__Orat_001tf__a,type,
hilber2046056624969986904_rat_a: set_rat > ( rat > a ) > a > rat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
hilber3771966164650003108at_nat: set_set_nat > ( set_nat > nat ) > nat > set_nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001tf__a_001t__Nat__Onat,type,
hilber7986931655781312002_a_nat: set_a > ( a > nat ) > nat > a ).
thf(sy_c_Hilbert__Choice_Oinv__into_001tf__a_001tf__a,type,
hilbert_inv_into_a_a: set_a > ( a > a ) > a > a ).
thf(sy_c_If_001t__Extended____Real__Oereal,type,
if_Extended_ereal: $o > extended_ereal > extended_ereal > extended_ereal ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_Lifting_OQuotient_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
quotie5409145007053189782_ereal: ( extended_ereal > extended_ereal > $o ) > ( extended_ereal > extended_ereal ) > ( extended_ereal > extended_ereal ) > ( extended_ereal > extended_ereal > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
quotie1598486591011391240al_nat: ( extended_ereal > extended_ereal > $o ) > ( extended_ereal > nat ) > ( nat > extended_ereal ) > ( extended_ereal > nat > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__Extended____Real__Oereal_001t__Rat__Orat,type,
quotie963356530924895504al_rat: ( extended_ereal > extended_ereal > $o ) > ( extended_ereal > rat ) > ( rat > extended_ereal ) > ( extended_ereal > rat > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__Nat__Onat_001t__Extended____Real__Oereal,type,
quotie7471290239582328294_ereal: ( nat > nat > $o ) > ( nat > extended_ereal ) > ( extended_ereal > nat ) > ( nat > extended_ereal > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__Nat__Onat_001t__Nat__Onat,type,
quotient_nat_nat: ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__Nat__Onat_001t__Rat__Orat,type,
quotient_nat_rat: ( nat > nat > $o ) > ( nat > rat ) > ( rat > nat ) > ( nat > rat > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__Rat__Orat_001t__Extended____Real__Oereal,type,
quotie5754125791750839774_ereal: ( rat > rat > $o ) > ( rat > extended_ereal ) > ( extended_ereal > rat ) > ( rat > extended_ereal > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__Rat__Orat_001t__Nat__Onat,type,
quotient_rat_nat: ( rat > rat > $o ) > ( rat > nat ) > ( nat > rat ) > ( rat > nat > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__Rat__Orat_001t__Rat__Orat,type,
quotient_rat_rat: ( rat > rat > $o ) > ( rat > rat ) > ( rat > rat ) > ( rat > rat > $o ) > $o ).
thf(sy_c_Map_Odom_001tf__a_001t__List__Olist_I_Eo_J,type,
dom_a_list_o: ( a > option_list_o ) > set_a ).
thf(sy_c_Map_Odom_001tf__b_001t__List__Olist_I_Eo_J,type,
dom_b_list_o: ( b > option_list_o ) > set_b ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
bot_bo8367695208629047834_ereal: set_Extended_ereal ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Real__Oereal,type,
ord_le1188267648640031866_ereal: extended_ereal > extended_ereal > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Extended____Real__Oereal_M_062_It__Extended____Real__Oereal_M_Eo_J_J,type,
ord_le6654028770825229838real_o: ( extended_ereal > extended_ereal > $o ) > ( extended_ereal > extended_ereal > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Extended____Real__Oereal_M_062_It__Nat__Onat_M_Eo_J_J,type,
ord_le3354276052480486518_nat_o: ( extended_ereal > nat > $o ) > ( extended_ereal > nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Extended____Real__Oereal_M_062_It__Rat__Orat_M_Eo_J_J,type,
ord_le2604841557345650798_rat_o: ( extended_ereal > rat > $o ) > ( extended_ereal > rat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Extended____Real__Oereal_M_Eo_J_J,type,
ord_le3392000996217782014real_o: ( nat > extended_ereal > $o ) > ( nat > extended_ereal > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
ord_le2646555220125990790_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Rat__Orat_M_Eo_J_J,type,
ord_le1897120724991155070_rat_o: ( nat > rat > $o ) > ( nat > rat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Rat__Orat_M_062_It__Extended____Real__Oereal_M_Eo_J_J,type,
ord_le9156709395967171846real_o: ( rat > extended_ereal > $o ) > ( rat > extended_ereal > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Rat__Orat_M_062_It__Nat__Onat_M_Eo_J_J,type,
ord_le5467402850006352766_nat_o: ( rat > nat > $o ) > ( rat > nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Rat__Orat_M_062_It__Rat__Orat_M_Eo_J_J,type,
ord_le4717968354871517046_rat_o: ( rat > rat > $o ) > ( rat > rat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Real__Oereal,type,
ord_le1083603963089353582_ereal: extended_ereal > extended_ereal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
ord_less_eq_rat: rat > rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
ord_le1644982726543182158_ereal: set_Extended_ereal > set_Extended_ereal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Extended____Real__Oereal_Mt__Extended____Real__Oereal_J_J,type,
ord_le8239133294219471655_ereal: set_Pr2129990008675586951_ereal > set_Pr2129990008675586951_ereal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Extended____Real__Oereal_Mt__Nat__Onat_J_J,type,
ord_le5182862553726584547al_nat: set_Pr450698115992974979al_nat > set_Pr450698115992974979al_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Extended____Real__Oereal_Mtf__a_J_J,type,
ord_le7790192137282702039real_a: set_Pr594667477129265975real_a > set_Pr594667477129265975real_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Extended____Real__Oereal_J_J,type,
ord_le3920121919471048841_ereal: set_Pr8411329518592215081_ereal > set_Pr8411329518592215081_ereal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
ord_le5989899228261996553at_rat: set_Pr4105333604307423337at_rat > set_Pr4105333604307423337at_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le3845944159117341623et_nat: set_Pr400265656397884439et_nat > set_Pr400265656397884439et_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Extended____Real__Oereal_J_J,type,
ord_le8132973195874727159_ereal: set_Pr937448535721291095_ereal > set_Pr937448535721291095_ereal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J,type,
ord_le817736998455962536od_a_b: set_Product_prod_a_b > set_Product_prod_a_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
ord_less_eq_set_rat: set_rat > set_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Extended____Real__Oereal_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
ord_le513522071413781156et_rat: set_set_rat > set_set_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le9131159989063066194et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__b_J,type,
ord_less_eq_set_b: set_b > set_b > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Extended____Real__Oereal,type,
top_to6662034908053899550_ereal: extended_ereal ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Extended____Real__Oereal_Mt__Extended____Real__Oereal_J_J,type,
top_to908700840774984395_ereal: set_Ex2354994561656779803_ereal ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Extended____Real__Oereal_Mt__Nat__Onat_J_J,type,
top_to2398772004365740095al_nat: set_Ex8414784459666926319al_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Extended____Real__Oereal_Mt__Rat__Orat_J_J,type,
top_to5242157703742838343al_rat: set_Ex2034798122189248759al_rat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Extended____Real__Oereal_J_J,type,
top_to1136031370110204389_ereal: set_na7152043825411390613_ereal ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_top_set_nat_nat: set_nat_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
top_top_set_nat_rat: set_nat_rat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Rat__Orat_Mt__Extended____Real__Oereal_J_J,type,
top_to2804411426662057437_ereal: set_ra8820423881963243661_ereal ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Rat__Orat_Mt__Nat__Onat_J_J,type,
top_top_set_rat_nat: set_rat_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Rat__Orat_Mt__Rat__Orat_J_J,type,
top_top_set_rat_rat: set_rat_rat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
top_to5683747375963461374_ereal: set_Extended_ereal ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
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% Relevant facts (1276)
thf(fact_0_assms_I1_J,axiom,
prefix7485107378405021920ding_a @ e1 ).
% assms(1)
thf(fact_1_b,axiom,
( ( product_fst_a_b @ x )
= ( product_fst_a_b @ y ) ) ).
% b
thf(fact_2_a,axiom,
prefix454693708527911765comp_o @ ( prefix6990428588352057089od_a_b @ e1 @ e2 @ x ) @ ( prefix6990428588352057089od_a_b @ e1 @ e2 @ y ) ).
% a
thf(fact_3_d,axiom,
prefix454693708527911765comp_o @ ( e1 @ ( product_fst_a_b @ x ) ) @ ( e1 @ ( product_fst_a_b @ y ) ) ).
% d
thf(fact_4_opt__comp__sym,axiom,
( prefix454693708527911765comp_o
= ( ^ [X: option_list_o,Y: option_list_o] : ( prefix454693708527911765comp_o @ Y @ X ) ) ) ).
% opt_comp_sym
thf(fact_5_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_6_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_7_exE__realizer_H,axiom,
! [P2: b > a > $o,P: product_prod_a_b] :
( ( P2 @ ( product_snd_a_b @ P ) @ ( product_fst_a_b @ P ) )
=> ~ ! [X2: a,Y2: b] :
~ ( P2 @ Y2 @ X2 ) ) ).
% exE_realizer'
thf(fact_8_prod__eq__iff,axiom,
( ( ^ [Y3: product_prod_a_b,Z: product_prod_a_b] : ( Y3 = 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_9_encode__dependent__prod__def,axiom,
( prefix6990428588352057089od_a_b
= ( ^ [E: a > option_list_o,F: a > b > option_list_o,X: product_prod_a_b] : ( prefix5314359684614007693append @ ( E @ ( product_fst_a_b @ X ) ) @ ( F @ ( product_fst_a_b @ X ) @ ( product_snd_a_b @ X ) ) ) ) ) ).
% encode_dependent_prod_def
thf(fact_10_is__encodingD,axiom,
! [E2: a > option_list_o,X3: a,Y4: a] :
( ( prefix7485107378405021920ding_a @ E2 )
=> ( ( prefix454693708527911765comp_o @ ( E2 @ X3 ) @ ( E2 @ Y4 ) )
=> ( X3 = Y4 ) ) ) ).
% is_encodingD
thf(fact_11_is__encodingD,axiom,
! [E2: b > option_list_o,X3: b,Y4: b] :
( ( prefix7485107378405021921ding_b @ E2 )
=> ( ( prefix454693708527911765comp_o @ ( E2 @ X3 ) @ ( E2 @ Y4 ) )
=> ( X3 = Y4 ) ) ) ).
% is_encodingD
thf(fact_12_is__encodingI__2,axiom,
! [E2: a > option_list_o] :
( ! [X2: a,Y2: a] :
( ( prefix454693708527911765comp_o @ ( E2 @ X2 ) @ ( E2 @ Y2 ) )
=> ( X2 = Y2 ) )
=> ( prefix7485107378405021920ding_a @ E2 ) ) ).
% is_encodingI_2
thf(fact_13_is__encodingI__2,axiom,
! [E2: b > option_list_o] :
( ! [X2: b,Y2: b] :
( ( prefix454693708527911765comp_o @ ( E2 @ X2 ) @ ( E2 @ Y2 ) )
=> ( X2 = Y2 ) )
=> ( prefix7485107378405021921ding_b @ E2 ) ) ).
% is_encodingI_2
thf(fact_14_opt__comp__append,axiom,
! [X3: option_list_o,Y4: option_list_o,Z2: option_list_o] :
( ( prefix454693708527911765comp_o @ ( prefix5314359684614007693append @ X3 @ Y4 ) @ Z2 )
=> ( prefix454693708527911765comp_o @ X3 @ Z2 ) ) ).
% opt_comp_append
thf(fact_15_opt__comp__append__2,axiom,
! [X3: option_list_o,Y4: option_list_o,Z2: option_list_o] :
( ( prefix454693708527911765comp_o @ X3 @ ( prefix5314359684614007693append @ Y4 @ Z2 ) )
=> ( prefix454693708527911765comp_o @ X3 @ Y4 ) ) ).
% opt_comp_append_2
thf(fact_16_opt__comp__append__3,axiom,
! [X3: option_list_o,Y4: option_list_o,Z2: option_list_o] :
( ( prefix454693708527911765comp_o @ ( prefix5314359684614007693append @ X3 @ Y4 ) @ ( prefix5314359684614007693append @ X3 @ Z2 ) )
=> ( prefix454693708527911765comp_o @ Y4 @ Z2 ) ) ).
% opt_comp_append_3
thf(fact_17_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_18_assms_I2_J,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( dom_a_list_o @ e1 ) )
=> ( prefix7485107378405021921ding_b @ ( e2 @ X3 ) ) ) ).
% assms(2)
thf(fact_19_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_20_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_21_snd__swap,axiom,
! [X3: product_prod_a_b] :
( ( product_snd_b_a @ ( product_swap_a_b @ X3 ) )
= ( product_fst_a_b @ X3 ) ) ).
% snd_swap
thf(fact_22_snd__swap,axiom,
! [X3: product_prod_b_a] :
( ( product_snd_a_b @ ( product_swap_b_a @ X3 ) )
= ( product_fst_b_a @ X3 ) ) ).
% snd_swap
thf(fact_23_fst__swap,axiom,
! [X3: product_prod_b_a] :
( ( product_fst_a_b @ ( product_swap_b_a @ X3 ) )
= ( product_snd_b_a @ X3 ) ) ).
% fst_swap
thf(fact_24_fst__swap,axiom,
! [X3: product_prod_a_b] :
( ( product_fst_b_a @ ( product_swap_a_b @ X3 ) )
= ( product_snd_a_b @ X3 ) ) ).
% fst_swap
thf(fact_25_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_26_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_27_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_28_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_29_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_30_exI__realizer,axiom,
! [P2: b > a > $o,Y4: b,X3: a] :
( ( P2 @ Y4 @ X3 )
=> ( P2 @ ( product_snd_a_b @ ( product_Pair_a_b @ X3 @ Y4 ) ) @ ( product_fst_a_b @ ( product_Pair_a_b @ X3 @ Y4 ) ) ) ) ).
% exI_realizer
thf(fact_31_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_32_snd__apsnd,axiom,
! [F2: b > b,X3: product_prod_a_b] :
( ( product_snd_a_b @ ( product_apsnd_b_b_a @ F2 @ X3 ) )
= ( F2 @ ( product_snd_a_b @ X3 ) ) ) ).
% snd_apsnd
thf(fact_33_fst__apsnd,axiom,
! [F2: b > b,X3: product_prod_a_b] :
( ( product_fst_a_b @ ( product_apsnd_b_b_a @ F2 @ X3 ) )
= ( product_fst_a_b @ X3 ) ) ).
% fst_apsnd
thf(fact_34_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_35_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_36_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_37_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_38_fst__eqD,axiom,
! [X3: a,Y4: b,A: a] :
( ( ( product_fst_a_b @ ( product_Pair_a_b @ X3 @ Y4 ) )
= A )
=> ( X3 = A ) ) ).
% fst_eqD
thf(fact_39_fst__conv,axiom,
! [X1: a,X22: b] :
( ( product_fst_a_b @ ( product_Pair_a_b @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_40_snd__eqD,axiom,
! [X3: a,Y4: b,A: b] :
( ( ( product_snd_a_b @ ( product_Pair_a_b @ X3 @ Y4 ) )
= A )
=> ( Y4 = A ) ) ).
% snd_eqD
thf(fact_41_snd__conv,axiom,
! [X1: a,X22: b] :
( ( product_snd_a_b @ ( product_Pair_a_b @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_42_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_43_Product__Type_OCollect__case__prodD,axiom,
! [X3: product_prod_a_b,A2: a > b > $o] :
( ( member1426531481828664017od_a_b @ X3 @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ A2 ) ) )
=> ( A2 @ ( product_fst_a_b @ X3 ) @ ( product_snd_a_b @ X3 ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_44_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P2: a > b > $o,X3: a,Y4: b,A: product_prod_a_b] :
( ( P2 @ X3 @ Y4 )
=> ( ( A
= ( product_Pair_a_b @ X3 @ Y4 ) )
=> ( P2 @ ( product_fst_a_b @ A ) @ ( product_snd_a_b @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_45_sndI,axiom,
! [X3: product_prod_a_b,Y4: a,Z2: b] :
( ( X3
= ( product_Pair_a_b @ Y4 @ Z2 ) )
=> ( ( product_snd_a_b @ X3 )
= Z2 ) ) ).
% sndI
thf(fact_46_eq__snd__iff,axiom,
! [B: b,P: product_prod_a_b] :
( ( B
= ( product_snd_a_b @ P ) )
= ( ? [A3: a] :
( P
= ( product_Pair_a_b @ A3 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_47_fstI,axiom,
! [X3: product_prod_a_b,Y4: a,Z2: b] :
( ( X3
= ( product_Pair_a_b @ Y4 @ Z2 ) )
=> ( ( product_fst_a_b @ X3 )
= Y4 ) ) ).
% fstI
thf(fact_48_eq__fst__iff,axiom,
! [A: a,P: product_prod_a_b] :
( ( A
= ( product_fst_a_b @ P ) )
= ( ? [B2: b] :
( P
= ( product_Pair_a_b @ A @ B2 ) ) ) ) ).
% eq_fst_iff
thf(fact_49_fst__apfst,axiom,
! [F2: a > a,X3: product_prod_a_b] :
( ( product_fst_a_b @ ( product_apfst_a_a_b @ F2 @ X3 ) )
= ( F2 @ ( product_fst_a_b @ X3 ) ) ) ).
% fst_apfst
thf(fact_50_snd__apfst,axiom,
! [F2: a > a,X3: product_prod_a_b] :
( ( product_snd_a_b @ ( product_apfst_a_a_b @ F2 @ X3 ) )
= ( product_snd_a_b @ X3 ) ) ).
% snd_apfst
thf(fact_51_inj__onD,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,X3: extended_ereal,Y4: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A2 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y4 ) )
=> ( ( member2350847679896131959_ereal @ X3 @ A2 )
=> ( ( member2350847679896131959_ereal @ Y4 @ A2 )
=> ( X3 = Y4 ) ) ) ) ) ).
% inj_onD
thf(fact_52_inj__onD,axiom,
! [F2: nat > nat,A2: set_nat,X3: nat,Y4: nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y4 ) )
=> ( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ Y4 @ A2 )
=> ( X3 = Y4 ) ) ) ) ) ).
% inj_onD
thf(fact_53_inj__onI,axiom,
! [A2: set_Extended_ereal,F2: extended_ereal > extended_ereal] :
( ! [X2: extended_ereal,Y2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A2 )
=> ( ( member2350847679896131959_ereal @ Y2 @ A2 )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y2 ) )
=> ( X2 = Y2 ) ) ) )
=> ( inj_on7162434037990268785_ereal @ F2 @ A2 ) ) ).
% inj_onI
thf(fact_54_inj__onI,axiom,
! [A2: set_nat,F2: nat > nat] :
( ! [X2: nat,Y2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y2 @ A2 )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y2 ) )
=> ( X2 = Y2 ) ) ) )
=> ( inj_on_nat_nat @ F2 @ A2 ) ) ).
% inj_onI
thf(fact_55_inj__on__def,axiom,
( inj_on7162434037990268785_ereal
= ( ^ [F: extended_ereal > extended_ereal,A4: set_Extended_ereal] :
! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A4 )
=> ! [Y: extended_ereal] :
( ( member2350847679896131959_ereal @ Y @ A4 )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ) ) ) ).
% inj_on_def
thf(fact_56_inj__on__def,axiom,
( inj_on_nat_nat
= ( ^ [F: nat > nat,A4: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A4 )
=> ! [Y: nat] :
( ( member_nat @ Y @ A4 )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ) ) ) ).
% inj_on_def
thf(fact_57_inj__on__cong,axiom,
! [A2: set_Extended_ereal,F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal] :
( ! [A5: extended_ereal] :
( ( member2350847679896131959_ereal @ A5 @ A2 )
=> ( ( F2 @ A5 )
= ( G @ A5 ) ) )
=> ( ( inj_on7162434037990268785_ereal @ F2 @ A2 )
= ( inj_on7162434037990268785_ereal @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_58_inj__on__cong,axiom,
! [A2: set_nat,F2: nat > nat,G: nat > nat] :
( ! [A5: nat] :
( ( member_nat @ A5 @ A2 )
=> ( ( F2 @ A5 )
= ( G @ A5 ) ) )
=> ( ( inj_on_nat_nat @ F2 @ A2 )
= ( inj_on_nat_nat @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_59_inj__on__eq__iff,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,X3: extended_ereal,Y4: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A2 )
=> ( ( member2350847679896131959_ereal @ X3 @ A2 )
=> ( ( member2350847679896131959_ereal @ Y4 @ A2 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y4 ) )
= ( X3 = Y4 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_60_inj__on__eq__iff,axiom,
! [F2: nat > nat,A2: set_nat,X3: nat,Y4: nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ Y4 @ A2 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y4 ) )
= ( X3 = Y4 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_61_inj__on__contraD,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,X3: extended_ereal,Y4: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A2 )
=> ( ( X3 != Y4 )
=> ( ( member2350847679896131959_ereal @ X3 @ A2 )
=> ( ( member2350847679896131959_ereal @ Y4 @ A2 )
=> ( ( F2 @ X3 )
!= ( F2 @ Y4 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_62_inj__on__contraD,axiom,
! [F2: nat > nat,A2: set_nat,X3: nat,Y4: nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( X3 != Y4 )
=> ( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ Y4 @ A2 )
=> ( ( F2 @ X3 )
!= ( F2 @ Y4 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_63_inj__on__inverseI,axiom,
! [A2: set_Extended_ereal,G: extended_ereal > extended_ereal,F2: extended_ereal > extended_ereal] :
( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A2 )
=> ( ( G @ ( F2 @ X2 ) )
= X2 ) )
=> ( inj_on7162434037990268785_ereal @ F2 @ A2 ) ) ).
% inj_on_inverseI
thf(fact_64_inj__on__inverseI,axiom,
! [A2: set_nat,G: nat > nat,F2: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( G @ ( F2 @ X2 ) )
= X2 ) )
=> ( inj_on_nat_nat @ F2 @ A2 ) ) ).
% inj_on_inverseI
thf(fact_65_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_66_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_67_comp__apply,axiom,
( comp_nat_nat_nat
= ( ^ [F: nat > nat,G2: nat > nat,X: nat] : ( F @ ( G2 @ X ) ) ) ) ).
% comp_apply
thf(fact_68_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_69_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_70_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_71_image__eq__imp__comp,axiom,
! [F2: extended_ereal > nat,A2: set_Extended_ereal,G: nat > nat,B3: set_nat,H: nat > extended_ereal] :
( ( ( image_7659842161140344153al_nat @ F2 @ A2 )
= ( image_nat_nat @ G @ B3 ) )
=> ( ( image_6042159593519690757_ereal @ ( comp_n261702227720650419_ereal @ H @ F2 ) @ A2 )
= ( image_4309273772856505399_ereal @ ( comp_n13370146242399787al_nat @ H @ G ) @ B3 ) ) ) ).
% image_eq_imp_comp
thf(fact_72_image__eq__imp__comp,axiom,
! [F2: extended_ereal > rat,A2: set_Extended_ereal,G: nat > rat,B3: set_nat,H: rat > extended_ereal] :
( ( ( image_7024712101053848417al_rat @ F2 @ A2 )
= ( image_nat_rat @ G @ B3 ) )
=> ( ( image_6042159593519690757_ereal @ ( comp_r2952691998189091003_ereal @ H @ F2 ) @ A2 )
= ( image_4309273772856505399_ereal @ ( comp_r4319880827473671715al_nat @ H @ G ) @ B3 ) ) ) ).
% image_eq_imp_comp
thf(fact_73_image__eq__imp__comp,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,G: extended_ereal > extended_ereal,B3: set_Extended_ereal,H: extended_ereal > extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ A2 )
= ( image_6042159593519690757_ereal @ G @ B3 ) )
=> ( ( image_6042159593519690757_ereal @ ( comp_E9177254828515427499_ereal @ H @ F2 ) @ A2 )
= ( image_6042159593519690757_ereal @ ( comp_E9177254828515427499_ereal @ H @ G ) @ B3 ) ) ) ).
% image_eq_imp_comp
thf(fact_74_image__eq__imp__comp,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,G: nat > extended_ereal,B3: set_nat,H: extended_ereal > nat] :
( ( ( image_6042159593519690757_ereal @ F2 @ A2 )
= ( image_4309273772856505399_ereal @ G @ B3 ) )
=> ( ( image_7659842161140344153al_nat @ ( comp_E375531472069506321_ereal @ H @ F2 ) @ A2 )
= ( image_nat_nat @ ( comp_E7502005551946643277at_nat @ H @ G ) @ B3 ) ) ) ).
% image_eq_imp_comp
thf(fact_75_image__eq__imp__comp,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,G: nat > extended_ereal,B3: set_nat,H: extended_ereal > rat] :
( ( ( image_6042159593519690757_ereal @ F2 @ A2 )
= ( image_4309273772856505399_ereal @ G @ B3 ) )
=> ( ( image_7024712101053848417al_rat @ ( comp_E7881739061092793609_ereal @ H @ F2 ) @ A2 )
= ( image_nat_rat @ ( comp_E7185450908011777365at_nat @ H @ G ) @ B3 ) ) ) ).
% image_eq_imp_comp
thf(fact_76_image__eq__imp__comp,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,G: nat > extended_ereal,B3: set_nat,H: extended_ereal > extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ A2 )
= ( image_4309273772856505399_ereal @ G @ B3 ) )
=> ( ( image_6042159593519690757_ereal @ ( comp_E9177254828515427499_ereal @ H @ F2 ) @ A2 )
= ( image_4309273772856505399_ereal @ ( comp_E3726099860353345075al_nat @ H @ G ) @ B3 ) ) ) ).
% image_eq_imp_comp
thf(fact_77_image__eq__imp__comp,axiom,
! [F2: nat > extended_ereal,A2: set_nat,G: extended_ereal > extended_ereal,B3: set_Extended_ereal,H: extended_ereal > extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F2 @ A2 )
= ( image_6042159593519690757_ereal @ G @ B3 ) )
=> ( ( image_4309273772856505399_ereal @ ( comp_E3726099860353345075al_nat @ H @ F2 ) @ A2 )
= ( image_6042159593519690757_ereal @ ( comp_E9177254828515427499_ereal @ H @ G ) @ B3 ) ) ) ).
% image_eq_imp_comp
thf(fact_78_image__eq__imp__comp,axiom,
! [F2: nat > extended_ereal,A2: set_nat,G: extended_ereal > extended_ereal,B3: set_Extended_ereal,H: extended_ereal > nat] :
( ( ( image_4309273772856505399_ereal @ F2 @ A2 )
= ( image_6042159593519690757_ereal @ G @ B3 ) )
=> ( ( image_nat_nat @ ( comp_E7502005551946643277at_nat @ H @ F2 ) @ A2 )
= ( image_7659842161140344153al_nat @ ( comp_E375531472069506321_ereal @ H @ G ) @ B3 ) ) ) ).
% image_eq_imp_comp
thf(fact_79_image__eq__imp__comp,axiom,
! [F2: nat > extended_ereal,A2: set_nat,G: extended_ereal > extended_ereal,B3: set_Extended_ereal,H: extended_ereal > rat] :
( ( ( image_4309273772856505399_ereal @ F2 @ A2 )
= ( image_6042159593519690757_ereal @ G @ B3 ) )
=> ( ( image_nat_rat @ ( comp_E7185450908011777365at_nat @ H @ F2 ) @ A2 )
= ( image_7024712101053848417al_rat @ ( comp_E7881739061092793609_ereal @ H @ G ) @ B3 ) ) ) ).
% image_eq_imp_comp
thf(fact_80_image__eq__imp__comp,axiom,
! [F2: nat > extended_ereal,A2: set_nat,G: nat > extended_ereal,B3: set_nat,H: extended_ereal > extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F2 @ A2 )
= ( image_4309273772856505399_ereal @ G @ B3 ) )
=> ( ( image_4309273772856505399_ereal @ ( comp_E3726099860353345075al_nat @ H @ F2 ) @ A2 )
= ( image_4309273772856505399_ereal @ ( comp_E3726099860353345075al_nat @ H @ G ) @ B3 ) ) ) ).
% image_eq_imp_comp
thf(fact_81_comp__eq__dest__lhs,axiom,
! [A: nat > nat,B: nat > nat,C: nat > nat,V: nat] :
( ( ( comp_nat_nat_nat @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_82_comp__eq__elim,axiom,
! [A: nat > nat,B: nat > nat,C: nat > nat,D: nat > nat] :
( ( ( comp_nat_nat_nat @ A @ B )
= ( comp_nat_nat_nat @ C @ D ) )
=> ! [V2: nat] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_83_comp__eq__dest,axiom,
! [A: nat > nat,B: nat > nat,C: nat > nat,D: nat > nat,V: nat] :
( ( ( comp_nat_nat_nat @ A @ B )
= ( comp_nat_nat_nat @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_84_image__comp,axiom,
! [F2: extended_ereal > nat,G: nat > extended_ereal,R: set_nat] :
( ( image_7659842161140344153al_nat @ F2 @ ( image_4309273772856505399_ereal @ G @ R ) )
= ( image_nat_nat @ ( comp_E7502005551946643277at_nat @ F2 @ G ) @ R ) ) ).
% image_comp
thf(fact_85_image__comp,axiom,
! [F2: extended_ereal > rat,G: nat > extended_ereal,R: set_nat] :
( ( image_7024712101053848417al_rat @ F2 @ ( image_4309273772856505399_ereal @ G @ R ) )
= ( image_nat_rat @ ( comp_E7185450908011777365at_nat @ F2 @ G ) @ R ) ) ).
% image_comp
thf(fact_86_image__comp,axiom,
! [F2: rat > extended_ereal,G: nat > rat,R: set_nat] :
( ( image_2592109325025016879_ereal @ F2 @ ( image_nat_rat @ G @ R ) )
= ( image_4309273772856505399_ereal @ ( comp_r4319880827473671715al_nat @ F2 @ G ) @ R ) ) ).
% image_comp
thf(fact_87_image__comp,axiom,
! [F2: rat > nat,G: nat > rat,R: set_nat] :
( ( image_rat_nat @ F2 @ ( image_nat_rat @ G @ R ) )
= ( image_nat_nat @ ( comp_rat_nat_nat @ F2 @ G ) @ R ) ) ).
% image_comp
thf(fact_88_image__comp,axiom,
! [F2: rat > rat,G: nat > rat,R: set_nat] :
( ( image_rat_rat @ F2 @ ( image_nat_rat @ G @ R ) )
= ( image_nat_rat @ ( comp_rat_rat_nat @ F2 @ G ) @ R ) ) ).
% image_comp
thf(fact_89_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_90_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_91_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_92_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_93_image__comp,axiom,
! [F2: nat > nat,G: nat > nat,R: set_nat] :
( ( image_nat_nat @ F2 @ ( image_nat_nat @ G @ R ) )
= ( image_nat_nat @ ( comp_nat_nat_nat @ F2 @ G ) @ R ) ) ).
% image_comp
thf(fact_94_comp__assoc,axiom,
! [F2: nat > nat,G: nat > nat,H: nat > nat] :
( ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ F2 @ G ) @ H )
= ( comp_nat_nat_nat @ F2 @ ( comp_nat_nat_nat @ G @ H ) ) ) ).
% comp_assoc
thf(fact_95_comp__def,axiom,
( comp_nat_nat_nat
= ( ^ [F: nat > nat,G2: nat > nat,X: nat] : ( F @ ( G2 @ X ) ) ) ) ).
% comp_def
thf(fact_96_comp__inj__on,axiom,
! [F2: nat > extended_ereal,A2: set_nat,G: extended_ereal > nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ A2 )
=> ( ( inj_on318729178700965101al_nat @ G @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
=> ( inj_on_nat_nat @ ( comp_E7502005551946643277at_nat @ G @ F2 ) @ A2 ) ) ) ).
% comp_inj_on
thf(fact_97_comp__inj__on,axiom,
! [F2: nat > set_nat,A2: set_nat,G: set_nat > nat] :
( ( inj_on_nat_set_nat @ F2 @ A2 )
=> ( ( inj_on_set_nat_nat @ G @ ( image_nat_set_nat @ F2 @ A2 ) )
=> ( inj_on_nat_nat @ ( comp_set_nat_nat_nat @ G @ F2 ) @ A2 ) ) ) ).
% comp_inj_on
thf(fact_98_comp__inj__on,axiom,
! [F2: nat > rat,A2: set_nat,G: rat > nat] :
( ( inj_on_nat_rat @ F2 @ A2 )
=> ( ( inj_on_rat_nat @ G @ ( image_nat_rat @ F2 @ A2 ) )
=> ( inj_on_nat_nat @ ( comp_rat_nat_nat @ G @ F2 ) @ A2 ) ) ) ).
% comp_inj_on
thf(fact_99_comp__inj__on,axiom,
! [F2: nat > extended_ereal,A2: set_nat,G: extended_ereal > extended_ereal] :
( ( inj_on6191532827271902155_ereal @ F2 @ A2 )
=> ( ( inj_on7162434037990268785_ereal @ G @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
=> ( inj_on6191532827271902155_ereal @ ( comp_E3726099860353345075al_nat @ G @ F2 ) @ A2 ) ) ) ).
% comp_inj_on
thf(fact_100_comp__inj__on,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,G: extended_ereal > extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A2 )
=> ( ( inj_on7162434037990268785_ereal @ G @ ( image_6042159593519690757_ereal @ F2 @ A2 ) )
=> ( inj_on7162434037990268785_ereal @ ( comp_E9177254828515427499_ereal @ G @ F2 ) @ A2 ) ) ) ).
% comp_inj_on
thf(fact_101_comp__inj__on,axiom,
! [F2: nat > nat,A2: set_nat,G: nat > nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( inj_on_nat_nat @ G @ ( image_nat_nat @ F2 @ A2 ) )
=> ( inj_on_nat_nat @ ( comp_nat_nat_nat @ G @ F2 ) @ A2 ) ) ) ).
% comp_inj_on
thf(fact_102_inj__on__imageI,axiom,
! [G: extended_ereal > extended_ereal,F2: nat > extended_ereal,A2: set_nat] :
( ( inj_on6191532827271902155_ereal @ ( comp_E3726099860353345075al_nat @ G @ F2 ) @ A2 )
=> ( inj_on7162434037990268785_ereal @ G @ ( image_4309273772856505399_ereal @ F2 @ A2 ) ) ) ).
% inj_on_imageI
thf(fact_103_inj__on__imageI,axiom,
! [G: extended_ereal > extended_ereal,F2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ ( comp_E9177254828515427499_ereal @ G @ F2 ) @ A2 )
=> ( inj_on7162434037990268785_ereal @ G @ ( image_6042159593519690757_ereal @ F2 @ A2 ) ) ) ).
% inj_on_imageI
thf(fact_104_inj__on__imageI,axiom,
! [G: extended_ereal > nat,F2: nat > extended_ereal,A2: set_nat] :
( ( inj_on_nat_nat @ ( comp_E7502005551946643277at_nat @ G @ F2 ) @ A2 )
=> ( inj_on318729178700965101al_nat @ G @ ( image_4309273772856505399_ereal @ F2 @ A2 ) ) ) ).
% inj_on_imageI
thf(fact_105_inj__on__imageI,axiom,
! [G: set_nat > nat,F2: nat > set_nat,A2: set_nat] :
( ( inj_on_nat_nat @ ( comp_set_nat_nat_nat @ G @ F2 ) @ A2 )
=> ( inj_on_set_nat_nat @ G @ ( image_nat_set_nat @ F2 @ A2 ) ) ) ).
% inj_on_imageI
thf(fact_106_inj__on__imageI,axiom,
! [G: rat > nat,F2: nat > rat,A2: set_nat] :
( ( inj_on_nat_nat @ ( comp_rat_nat_nat @ G @ F2 ) @ A2 )
=> ( inj_on_rat_nat @ G @ ( image_nat_rat @ F2 @ A2 ) ) ) ).
% inj_on_imageI
thf(fact_107_inj__on__imageI,axiom,
! [G: nat > nat,F2: nat > nat,A2: set_nat] :
( ( inj_on_nat_nat @ ( comp_nat_nat_nat @ G @ F2 ) @ A2 )
=> ( inj_on_nat_nat @ G @ ( image_nat_nat @ F2 @ A2 ) ) ) ).
% inj_on_imageI
thf(fact_108_comp__inj__on__iff,axiom,
! [F2: nat > extended_ereal,A2: set_nat,F3: extended_ereal > nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ A2 )
=> ( ( inj_on318729178700965101al_nat @ F3 @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
= ( inj_on_nat_nat @ ( comp_E7502005551946643277at_nat @ F3 @ F2 ) @ A2 ) ) ) ).
% comp_inj_on_iff
thf(fact_109_comp__inj__on__iff,axiom,
! [F2: nat > set_nat,A2: set_nat,F3: set_nat > nat] :
( ( inj_on_nat_set_nat @ F2 @ A2 )
=> ( ( inj_on_set_nat_nat @ F3 @ ( image_nat_set_nat @ F2 @ A2 ) )
= ( inj_on_nat_nat @ ( comp_set_nat_nat_nat @ F3 @ F2 ) @ A2 ) ) ) ).
% comp_inj_on_iff
thf(fact_110_comp__inj__on__iff,axiom,
! [F2: nat > rat,A2: set_nat,F3: rat > nat] :
( ( inj_on_nat_rat @ F2 @ A2 )
=> ( ( inj_on_rat_nat @ F3 @ ( image_nat_rat @ F2 @ A2 ) )
= ( inj_on_nat_nat @ ( comp_rat_nat_nat @ F3 @ F2 ) @ A2 ) ) ) ).
% comp_inj_on_iff
thf(fact_111_comp__inj__on__iff,axiom,
! [F2: nat > extended_ereal,A2: set_nat,F3: extended_ereal > extended_ereal] :
( ( inj_on6191532827271902155_ereal @ F2 @ A2 )
=> ( ( inj_on7162434037990268785_ereal @ F3 @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
= ( inj_on6191532827271902155_ereal @ ( comp_E3726099860353345075al_nat @ F3 @ F2 ) @ A2 ) ) ) ).
% comp_inj_on_iff
thf(fact_112_comp__inj__on__iff,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,F3: extended_ereal > extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A2 )
=> ( ( inj_on7162434037990268785_ereal @ F3 @ ( image_6042159593519690757_ereal @ F2 @ A2 ) )
= ( inj_on7162434037990268785_ereal @ ( comp_E9177254828515427499_ereal @ F3 @ F2 ) @ A2 ) ) ) ).
% comp_inj_on_iff
thf(fact_113_comp__inj__on__iff,axiom,
! [F2: nat > nat,A2: set_nat,F3: nat > nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( inj_on_nat_nat @ F3 @ ( image_nat_nat @ F2 @ A2 ) )
= ( inj_on_nat_nat @ ( comp_nat_nat_nat @ F3 @ F2 ) @ A2 ) ) ) ).
% comp_inj_on_iff
thf(fact_114_comp__apply__eq,axiom,
! [F2: nat > nat,G: nat > nat,X3: nat,H: nat > nat,K: nat > nat] :
( ( ( F2 @ ( G @ X3 ) )
= ( H @ ( K @ X3 ) ) )
=> ( ( comp_nat_nat_nat @ F2 @ G @ X3 )
= ( comp_nat_nat_nat @ H @ K @ X3 ) ) ) ).
% comp_apply_eq
thf(fact_115_comp__cong,axiom,
! [F2: nat > nat,G: nat > nat,X3: nat,F3: nat > nat,G3: nat > nat,X4: nat] :
( ( ( F2 @ ( G @ X3 ) )
= ( F3 @ ( G3 @ X4 ) ) )
=> ( ( comp_nat_nat_nat @ F2 @ G @ X3 )
= ( comp_nat_nat_nat @ F3 @ G3 @ X4 ) ) ) ).
% comp_cong
thf(fact_116_inj__on__imageI2,axiom,
! [F3: extended_ereal > extended_ereal,F2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ ( comp_E9177254828515427499_ereal @ F3 @ F2 ) @ A2 )
=> ( inj_on7162434037990268785_ereal @ F2 @ A2 ) ) ).
% inj_on_imageI2
thf(fact_117_inj__on__imageI2,axiom,
! [F3: nat > nat,F2: nat > nat,A2: set_nat] :
( ( inj_on_nat_nat @ ( comp_nat_nat_nat @ F3 @ F2 ) @ A2 )
=> ( inj_on_nat_nat @ F2 @ A2 ) ) ).
% inj_on_imageI2
thf(fact_118_inj__on__image__iff,axiom,
! [A2: set_Extended_ereal,G: extended_ereal > extended_ereal,F2: extended_ereal > extended_ereal] :
( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A2 )
=> ! [Xa: extended_ereal] :
( ( member2350847679896131959_ereal @ Xa @ A2 )
=> ( ( ( G @ ( F2 @ X2 ) )
= ( G @ ( F2 @ Xa ) ) )
= ( ( G @ X2 )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on7162434037990268785_ereal @ F2 @ A2 )
=> ( ( inj_on7162434037990268785_ereal @ G @ ( image_6042159593519690757_ereal @ F2 @ A2 ) )
= ( inj_on7162434037990268785_ereal @ G @ A2 ) ) ) ) ).
% inj_on_image_iff
thf(fact_119_inj__on__image__iff,axiom,
! [A2: set_nat,G: nat > nat,F2: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ( G @ ( F2 @ X2 ) )
= ( G @ ( F2 @ Xa ) ) )
= ( ( G @ X2 )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( inj_on_nat_nat @ G @ ( image_nat_nat @ F2 @ A2 ) )
= ( inj_on_nat_nat @ G @ A2 ) ) ) ) ).
% inj_on_image_iff
thf(fact_120_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_121_image__eqI,axiom,
! [B: extended_ereal,F2: extended_ereal > extended_ereal,X3: extended_ereal,A2: set_Extended_ereal] :
( ( B
= ( F2 @ X3 ) )
=> ( ( member2350847679896131959_ereal @ X3 @ A2 )
=> ( member2350847679896131959_ereal @ B @ ( image_6042159593519690757_ereal @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_122_image__eqI,axiom,
! [B: a,F2: a > a,X3: a,A2: set_a] :
( ( B
= ( F2 @ X3 ) )
=> ( ( member_a @ X3 @ A2 )
=> ( member_a @ B @ ( image_a_a @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_123_image__eqI,axiom,
! [B: nat,F2: a > nat,X3: a,A2: set_a] :
( ( B
= ( F2 @ X3 ) )
=> ( ( member_a @ X3 @ A2 )
=> ( member_nat @ B @ ( image_a_nat @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_124_image__eqI,axiom,
! [B: extended_ereal,F2: nat > extended_ereal,X3: nat,A2: set_nat] :
( ( B
= ( F2 @ X3 ) )
=> ( ( member_nat @ X3 @ A2 )
=> ( member2350847679896131959_ereal @ B @ ( image_4309273772856505399_ereal @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_125_image__eqI,axiom,
! [B: set_nat,F2: nat > set_nat,X3: nat,A2: set_nat] :
( ( B
= ( F2 @ X3 ) )
=> ( ( member_nat @ X3 @ A2 )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_126_image__eqI,axiom,
! [B: rat,F2: nat > rat,X3: nat,A2: set_nat] :
( ( B
= ( F2 @ X3 ) )
=> ( ( member_nat @ X3 @ A2 )
=> ( member_rat @ B @ ( image_nat_rat @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_127_image__eqI,axiom,
! [B: a,F2: nat > a,X3: nat,A2: set_nat] :
( ( B
= ( F2 @ X3 ) )
=> ( ( member_nat @ X3 @ A2 )
=> ( member_a @ B @ ( image_nat_a @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_128_image__eqI,axiom,
! [B: nat,F2: nat > nat,X3: nat,A2: set_nat] :
( ( B
= ( F2 @ X3 ) )
=> ( ( member_nat @ X3 @ A2 )
=> ( member_nat @ B @ ( image_nat_nat @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_129_Inf_OINF__image,axiom,
! [Inf: set_nat > nat,G: extended_ereal > nat,F2: nat > extended_ereal,A2: set_nat] :
( ( Inf @ ( image_7659842161140344153al_nat @ G @ ( image_4309273772856505399_ereal @ F2 @ A2 ) ) )
= ( Inf @ ( image_nat_nat @ ( comp_E7502005551946643277at_nat @ G @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_130_Inf_OINF__image,axiom,
! [Inf: set_rat > rat,G: extended_ereal > rat,F2: nat > extended_ereal,A2: set_nat] :
( ( Inf @ ( image_7024712101053848417al_rat @ G @ ( image_4309273772856505399_ereal @ F2 @ A2 ) ) )
= ( Inf @ ( image_nat_rat @ ( comp_E7185450908011777365at_nat @ G @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_131_Inf_OINF__image,axiom,
! [Inf: set_Extended_ereal > extended_ereal,G: rat > extended_ereal,F2: nat > rat,A2: set_nat] :
( ( Inf @ ( image_2592109325025016879_ereal @ G @ ( image_nat_rat @ F2 @ A2 ) ) )
= ( Inf @ ( image_4309273772856505399_ereal @ ( comp_r4319880827473671715al_nat @ G @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_132_Inf_OINF__image,axiom,
! [Inf: set_nat > nat,G: rat > nat,F2: nat > rat,A2: set_nat] :
( ( Inf @ ( image_rat_nat @ G @ ( image_nat_rat @ F2 @ A2 ) ) )
= ( Inf @ ( image_nat_nat @ ( comp_rat_nat_nat @ G @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_133_Inf_OINF__image,axiom,
! [Inf: set_rat > rat,G: rat > rat,F2: nat > rat,A2: set_nat] :
( ( Inf @ ( image_rat_rat @ G @ ( image_nat_rat @ F2 @ A2 ) ) )
= ( Inf @ ( image_nat_rat @ ( comp_rat_rat_nat @ G @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_134_Inf_OINF__image,axiom,
! [Inf: set_Extended_ereal > extended_ereal,G: extended_ereal > extended_ereal,F2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( Inf @ ( image_6042159593519690757_ereal @ G @ ( image_6042159593519690757_ereal @ F2 @ A2 ) ) )
= ( Inf @ ( image_6042159593519690757_ereal @ ( comp_E9177254828515427499_ereal @ G @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_135_Inf_OINF__image,axiom,
! [Inf: set_Extended_ereal > extended_ereal,G: extended_ereal > extended_ereal,F2: nat > extended_ereal,A2: set_nat] :
( ( Inf @ ( image_6042159593519690757_ereal @ G @ ( image_4309273772856505399_ereal @ F2 @ A2 ) ) )
= ( Inf @ ( image_4309273772856505399_ereal @ ( comp_E3726099860353345075al_nat @ G @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_136_Inf_OINF__image,axiom,
! [Inf: set_Extended_ereal > extended_ereal,G: nat > extended_ereal,F2: extended_ereal > nat,A2: set_Extended_ereal] :
( ( Inf @ ( image_4309273772856505399_ereal @ G @ ( image_7659842161140344153al_nat @ F2 @ A2 ) ) )
= ( Inf @ ( image_6042159593519690757_ereal @ ( comp_n261702227720650419_ereal @ G @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_137_Inf_OINF__image,axiom,
! [Inf: set_Extended_ereal > extended_ereal,G: nat > extended_ereal,F2: nat > nat,A2: set_nat] :
( ( Inf @ ( image_4309273772856505399_ereal @ G @ ( image_nat_nat @ F2 @ A2 ) ) )
= ( Inf @ ( image_4309273772856505399_ereal @ ( comp_n13370146242399787al_nat @ G @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_138_Inf_OINF__image,axiom,
! [Inf: set_nat > nat,G: nat > nat,F2: nat > nat,A2: set_nat] :
( ( Inf @ ( image_nat_nat @ G @ ( image_nat_nat @ F2 @ A2 ) ) )
= ( Inf @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_139_Sup_OSUP__image,axiom,
! [Sup: set_nat > nat,G: extended_ereal > nat,F2: nat > extended_ereal,A2: set_nat] :
( ( Sup @ ( image_7659842161140344153al_nat @ G @ ( image_4309273772856505399_ereal @ F2 @ A2 ) ) )
= ( Sup @ ( image_nat_nat @ ( comp_E7502005551946643277at_nat @ G @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_140_Sup_OSUP__image,axiom,
! [Sup: set_rat > rat,G: extended_ereal > rat,F2: nat > extended_ereal,A2: set_nat] :
( ( Sup @ ( image_7024712101053848417al_rat @ G @ ( image_4309273772856505399_ereal @ F2 @ A2 ) ) )
= ( Sup @ ( image_nat_rat @ ( comp_E7185450908011777365at_nat @ G @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_141_Sup_OSUP__image,axiom,
! [Sup: set_Extended_ereal > extended_ereal,G: rat > extended_ereal,F2: nat > rat,A2: set_nat] :
( ( Sup @ ( image_2592109325025016879_ereal @ G @ ( image_nat_rat @ F2 @ A2 ) ) )
= ( Sup @ ( image_4309273772856505399_ereal @ ( comp_r4319880827473671715al_nat @ G @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_142_Sup_OSUP__image,axiom,
! [Sup: set_nat > nat,G: rat > nat,F2: nat > rat,A2: set_nat] :
( ( Sup @ ( image_rat_nat @ G @ ( image_nat_rat @ F2 @ A2 ) ) )
= ( Sup @ ( image_nat_nat @ ( comp_rat_nat_nat @ G @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_143_Sup_OSUP__image,axiom,
! [Sup: set_rat > rat,G: rat > rat,F2: nat > rat,A2: set_nat] :
( ( Sup @ ( image_rat_rat @ G @ ( image_nat_rat @ F2 @ A2 ) ) )
= ( Sup @ ( image_nat_rat @ ( comp_rat_rat_nat @ G @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_144_Sup_OSUP__image,axiom,
! [Sup: set_Extended_ereal > extended_ereal,G: extended_ereal > extended_ereal,F2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( Sup @ ( image_6042159593519690757_ereal @ G @ ( image_6042159593519690757_ereal @ F2 @ A2 ) ) )
= ( Sup @ ( image_6042159593519690757_ereal @ ( comp_E9177254828515427499_ereal @ G @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_145_Sup_OSUP__image,axiom,
! [Sup: set_Extended_ereal > extended_ereal,G: extended_ereal > extended_ereal,F2: nat > extended_ereal,A2: set_nat] :
( ( Sup @ ( image_6042159593519690757_ereal @ G @ ( image_4309273772856505399_ereal @ F2 @ A2 ) ) )
= ( Sup @ ( image_4309273772856505399_ereal @ ( comp_E3726099860353345075al_nat @ G @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_146_Sup_OSUP__image,axiom,
! [Sup: set_Extended_ereal > extended_ereal,G: nat > extended_ereal,F2: extended_ereal > nat,A2: set_Extended_ereal] :
( ( Sup @ ( image_4309273772856505399_ereal @ G @ ( image_7659842161140344153al_nat @ F2 @ A2 ) ) )
= ( Sup @ ( image_6042159593519690757_ereal @ ( comp_n261702227720650419_ereal @ G @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_147_Sup_OSUP__image,axiom,
! [Sup: set_Extended_ereal > extended_ereal,G: nat > extended_ereal,F2: nat > nat,A2: set_nat] :
( ( Sup @ ( image_4309273772856505399_ereal @ G @ ( image_nat_nat @ F2 @ A2 ) ) )
= ( Sup @ ( image_4309273772856505399_ereal @ ( comp_n13370146242399787al_nat @ G @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_148_Sup_OSUP__image,axiom,
! [Sup: set_nat > nat,G: nat > nat,F2: nat > nat,A2: set_nat] :
( ( Sup @ ( image_nat_nat @ G @ ( image_nat_nat @ F2 @ A2 ) ) )
= ( Sup @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_149_Collect__case__prod__Grp__eqD,axiom,
! [Z2: product_prod_a_b,A2: set_a,F2: a > b] :
( ( member1426531481828664017od_a_b @ Z2 @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ ( bNF_Grp_a_b @ A2 @ F2 ) ) ) )
=> ( ( comp_a9170378079104387268od_a_b @ F2 @ product_fst_a_b @ Z2 )
= ( product_snd_a_b @ Z2 ) ) ) ).
% Collect_case_prod_Grp_eqD
thf(fact_150_Collect__split__mono__strong,axiom,
! [X5: set_a,A2: 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 @ A2 ) )
=> ( ( Y5
= ( image_2802296252294471260_a_b_b @ product_snd_a_b @ A2 ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ X5 )
=> ! [Xa: b] :
( ( member_b @ Xa @ Y5 )
=> ( ( P2 @ X2 @ Xa )
=> ( Q2 @ X2 @ Xa ) ) ) )
=> ( ( ord_le817736998455962536od_a_b @ A2 @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ P2 ) ) )
=> ( ord_le817736998455962536od_a_b @ A2 @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ Q2 ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_151_the__inv__into__comp,axiom,
! [F2: nat > extended_ereal,G: nat > nat,A2: set_nat,X3: extended_ereal] :
( ( inj_on6191532827271902155_ereal @ F2 @ ( image_nat_nat @ G @ A2 ) )
=> ( ( inj_on_nat_nat @ G @ A2 )
=> ( ( member2350847679896131959_ereal @ X3 @ ( image_4309273772856505399_ereal @ F2 @ ( image_nat_nat @ G @ A2 ) ) )
=> ( ( the_in5959796611709155849_ereal @ A2 @ ( comp_n13370146242399787al_nat @ F2 @ G ) @ X3 )
= ( comp_n5886173794813336841_ereal @ ( the_inv_into_nat_nat @ A2 @ G ) @ ( the_in5959796611709155849_ereal @ ( image_nat_nat @ G @ A2 ) @ F2 ) @ X3 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_152_the__inv__into__comp,axiom,
! [F2: nat > rat,G: nat > nat,A2: set_nat,X3: rat] :
( ( inj_on_nat_rat @ F2 @ ( image_nat_nat @ G @ A2 ) )
=> ( ( inj_on_nat_nat @ G @ A2 )
=> ( ( member_rat @ X3 @ ( image_nat_rat @ F2 @ ( image_nat_nat @ G @ A2 ) ) )
=> ( ( the_inv_into_nat_rat @ A2 @ ( comp_nat_rat_nat @ F2 @ G ) @ X3 )
= ( comp_nat_nat_rat @ ( the_inv_into_nat_nat @ A2 @ G ) @ ( the_inv_into_nat_rat @ ( image_nat_nat @ G @ A2 ) @ F2 ) @ X3 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_153_the__inv__into__comp,axiom,
! [F2: extended_ereal > extended_ereal,G: nat > extended_ereal,A2: set_nat,X3: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ ( image_4309273772856505399_ereal @ G @ A2 ) )
=> ( ( inj_on6191532827271902155_ereal @ G @ A2 )
=> ( ( member2350847679896131959_ereal @ X3 @ ( image_6042159593519690757_ereal @ F2 @ ( image_4309273772856505399_ereal @ G @ A2 ) ) )
=> ( ( the_in5959796611709155849_ereal @ A2 @ ( comp_E3726099860353345075al_nat @ F2 @ G ) @ X3 )
= ( comp_E375531472069506321_ereal @ ( the_in5959796611709155849_ereal @ A2 @ G ) @ ( the_in1141389326992810419_ereal @ ( image_4309273772856505399_ereal @ G @ A2 ) @ F2 ) @ X3 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_154_the__inv__into__comp,axiom,
! [F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal,A2: set_Extended_ereal,X3: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ ( image_6042159593519690757_ereal @ G @ A2 ) )
=> ( ( inj_on7162434037990268785_ereal @ G @ A2 )
=> ( ( member2350847679896131959_ereal @ X3 @ ( image_6042159593519690757_ereal @ F2 @ ( image_6042159593519690757_ereal @ G @ A2 ) ) )
=> ( ( the_in1141389326992810419_ereal @ A2 @ ( comp_E9177254828515427499_ereal @ F2 @ G ) @ X3 )
= ( comp_E9177254828515427499_ereal @ ( the_in1141389326992810419_ereal @ A2 @ G ) @ ( the_in1141389326992810419_ereal @ ( image_6042159593519690757_ereal @ G @ A2 ) @ F2 ) @ X3 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_155_the__inv__into__comp,axiom,
! [F2: extended_ereal > a,G: nat > extended_ereal,A2: set_nat,X3: a] :
( ( inj_on8242634198667403041real_a @ F2 @ ( image_4309273772856505399_ereal @ G @ A2 ) )
=> ( ( inj_on6191532827271902155_ereal @ G @ A2 )
=> ( ( member_a @ X3 @ ( image_3724615099042636213real_a @ F2 @ ( image_4309273772856505399_ereal @ G @ A2 ) ) )
=> ( ( the_inv_into_nat_a @ A2 @ ( comp_E5637448798707004259_a_nat @ F2 @ G ) @ X3 )
= ( comp_E446008263030514881_nat_a @ ( the_in5959796611709155849_ereal @ A2 @ G ) @ ( the_in377810665034427491real_a @ ( image_4309273772856505399_ereal @ G @ A2 ) @ F2 ) @ X3 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_156_the__inv__into__comp,axiom,
! [F2: rat > a,G: nat > rat,A2: set_nat,X3: a] :
( ( inj_on_rat_a @ F2 @ ( image_nat_rat @ G @ A2 ) )
=> ( ( inj_on_nat_rat @ G @ A2 )
=> ( ( member_a @ X3 @ ( image_rat_a @ F2 @ ( image_nat_rat @ G @ A2 ) ) )
=> ( ( the_inv_into_nat_a @ A2 @ ( comp_rat_a_nat @ F2 @ G ) @ X3 )
= ( comp_rat_nat_a @ ( the_inv_into_nat_rat @ A2 @ G ) @ ( the_inv_into_rat_a @ ( image_nat_rat @ G @ A2 ) @ F2 ) @ X3 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_157_the__inv__into__comp,axiom,
! [F2: extended_ereal > a,G: extended_ereal > extended_ereal,A2: set_Extended_ereal,X3: a] :
( ( inj_on8242634198667403041real_a @ F2 @ ( image_6042159593519690757_ereal @ G @ A2 ) )
=> ( ( inj_on7162434037990268785_ereal @ G @ A2 )
=> ( ( member_a @ X3 @ ( image_3724615099042636213real_a @ F2 @ ( image_6042159593519690757_ereal @ G @ A2 ) ) )
=> ( ( the_in377810665034427491real_a @ A2 @ ( comp_E6551704282591734651_ereal @ F2 @ G ) @ X3 )
= ( comp_E1870838029643375451real_a @ ( the_in1141389326992810419_ereal @ A2 @ G ) @ ( the_in377810665034427491real_a @ ( image_6042159593519690757_ereal @ G @ A2 ) @ F2 ) @ X3 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_158_the__inv__into__comp,axiom,
! [F2: nat > a,G: nat > nat,A2: set_nat,X3: a] :
( ( inj_on_nat_a @ F2 @ ( image_nat_nat @ G @ A2 ) )
=> ( ( inj_on_nat_nat @ G @ A2 )
=> ( ( member_a @ X3 @ ( image_nat_a @ F2 @ ( image_nat_nat @ G @ A2 ) ) )
=> ( ( the_inv_into_nat_a @ A2 @ ( comp_nat_a_nat @ F2 @ G ) @ X3 )
= ( comp_nat_nat_a @ ( the_inv_into_nat_nat @ A2 @ G ) @ ( the_inv_into_nat_a @ ( image_nat_nat @ G @ A2 ) @ F2 ) @ X3 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_159_the__inv__into__comp,axiom,
! [F2: extended_ereal > nat,G: nat > extended_ereal,A2: set_nat,X3: nat] :
( ( inj_on318729178700965101al_nat @ F2 @ ( image_4309273772856505399_ereal @ G @ A2 ) )
=> ( ( inj_on6191532827271902155_ereal @ G @ A2 )
=> ( ( member_nat @ X3 @ ( image_7659842161140344153al_nat @ F2 @ ( image_4309273772856505399_ereal @ G @ A2 ) ) )
=> ( ( the_inv_into_nat_nat @ A2 @ ( comp_E7502005551946643277at_nat @ F2 @ G ) @ X3 )
= ( comp_E7502005551946643277at_nat @ ( the_in5959796611709155849_ereal @ A2 @ G ) @ ( the_in86992963138218795al_nat @ ( image_4309273772856505399_ereal @ G @ A2 ) @ F2 ) @ X3 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_160_the__inv__into__comp,axiom,
! [F2: rat > nat,G: nat > rat,A2: set_nat,X3: nat] :
( ( inj_on_rat_nat @ F2 @ ( image_nat_rat @ G @ A2 ) )
=> ( ( inj_on_nat_rat @ G @ A2 )
=> ( ( member_nat @ X3 @ ( image_rat_nat @ F2 @ ( image_nat_rat @ G @ A2 ) ) )
=> ( ( the_inv_into_nat_nat @ A2 @ ( comp_rat_nat_nat @ F2 @ G ) @ X3 )
= ( comp_rat_nat_nat @ ( the_inv_into_nat_rat @ A2 @ G ) @ ( the_inv_into_rat_nat @ ( image_nat_rat @ G @ A2 ) @ F2 ) @ X3 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_161_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_162_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_163_the__inv__into__onto,axiom,
! [F2: nat > extended_ereal,A2: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ A2 )
=> ( ( image_7659842161140344153al_nat @ ( the_in5959796611709155849_ereal @ A2 @ F2 ) @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
= A2 ) ) ).
% the_inv_into_onto
thf(fact_164_the__inv__into__onto,axiom,
! [F2: nat > set_nat,A2: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ A2 )
=> ( ( image_set_nat_nat @ ( the_in5057678521256355851et_nat @ A2 @ F2 ) @ ( image_nat_set_nat @ F2 @ A2 ) )
= A2 ) ) ).
% the_inv_into_onto
thf(fact_165_the__inv__into__onto,axiom,
! [F2: nat > rat,A2: set_nat] :
( ( inj_on_nat_rat @ F2 @ A2 )
=> ( ( image_rat_nat @ ( the_inv_into_nat_rat @ A2 @ F2 ) @ ( image_nat_rat @ F2 @ A2 ) )
= A2 ) ) ).
% the_inv_into_onto
thf(fact_166_the__inv__into__onto,axiom,
! [F2: extended_ereal > nat,A2: set_Extended_ereal] :
( ( inj_on318729178700965101al_nat @ F2 @ A2 )
=> ( ( image_4309273772856505399_ereal @ ( the_in86992963138218795al_nat @ A2 @ F2 ) @ ( image_7659842161140344153al_nat @ F2 @ A2 ) )
= A2 ) ) ).
% the_inv_into_onto
thf(fact_167_the__inv__into__onto,axiom,
! [F2: set_nat > nat,A2: set_set_nat] :
( ( inj_on_set_nat_nat @ F2 @ A2 )
=> ( ( image_nat_set_nat @ ( the_in1492043616600986635at_nat @ A2 @ F2 ) @ ( image_set_nat_nat @ F2 @ A2 ) )
= A2 ) ) ).
% the_inv_into_onto
thf(fact_168_the__inv__into__onto,axiom,
! [F2: rat > nat,A2: set_rat] :
( ( inj_on_rat_nat @ F2 @ A2 )
=> ( ( image_nat_rat @ ( the_inv_into_rat_nat @ A2 @ F2 ) @ ( image_rat_nat @ F2 @ A2 ) )
= A2 ) ) ).
% the_inv_into_onto
thf(fact_169_the__inv__into__onto,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A2 )
=> ( ( image_6042159593519690757_ereal @ ( the_in1141389326992810419_ereal @ A2 @ F2 ) @ ( image_6042159593519690757_ereal @ F2 @ A2 ) )
= A2 ) ) ).
% the_inv_into_onto
thf(fact_170_the__inv__into__onto,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( image_nat_nat @ ( the_inv_into_nat_nat @ A2 @ F2 ) @ ( image_nat_nat @ F2 @ A2 ) )
= A2 ) ) ).
% the_inv_into_onto
thf(fact_171_fst__map__prod,axiom,
! [F2: a > a,G: b > b,X3: product_prod_a_b] :
( ( product_fst_a_b @ ( produc1231242867592151606_a_b_b @ F2 @ G @ X3 ) )
= ( F2 @ ( product_fst_a_b @ X3 ) ) ) ).
% fst_map_prod
thf(fact_172_snd__map__prod,axiom,
! [F2: a > a,G: b > b,X3: product_prod_a_b] :
( ( product_snd_a_b @ ( produc1231242867592151606_a_b_b @ F2 @ G @ X3 ) )
= ( G @ ( product_snd_a_b @ X3 ) ) ) ).
% snd_map_prod
thf(fact_173_range__subsetD,axiom,
! [F2: extended_ereal > a,B3: set_a,I: extended_ereal] :
( ( ord_less_eq_set_a @ ( image_3724615099042636213real_a @ F2 @ top_to5683747375963461374_ereal ) @ B3 )
=> ( member_a @ ( F2 @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_174_range__subsetD,axiom,
! [F2: extended_ereal > nat,B3: set_nat,I: extended_ereal] :
( ( ord_less_eq_set_nat @ ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal ) @ B3 )
=> ( member_nat @ ( F2 @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_175_range__subsetD,axiom,
! [F2: nat > rat,B3: set_rat,I: nat] :
( ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ top_top_set_nat ) @ B3 )
=> ( member_rat @ ( F2 @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_176_range__subsetD,axiom,
! [F2: nat > a,B3: set_a,I: nat] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F2 @ top_top_set_nat ) @ B3 )
=> ( member_a @ ( F2 @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_177_range__subsetD,axiom,
! [F2: nat > nat,B3: set_nat,I: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ top_top_set_nat ) @ B3 )
=> ( member_nat @ ( F2 @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_178_range__subsetD,axiom,
! [F2: rat > a,B3: set_a,I: rat] :
( ( ord_less_eq_set_a @ ( image_rat_a @ F2 @ top_top_set_rat ) @ B3 )
=> ( member_a @ ( F2 @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_179_range__subsetD,axiom,
! [F2: rat > nat,B3: set_nat,I: rat] :
( ( ord_less_eq_set_nat @ ( image_rat_nat @ F2 @ top_top_set_rat ) @ B3 )
=> ( member_nat @ ( F2 @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_180_range__subsetD,axiom,
! [F2: extended_ereal > extended_ereal,B3: set_Extended_ereal,I: extended_ereal] :
( ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal ) @ B3 )
=> ( member2350847679896131959_ereal @ ( F2 @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_181_range__subsetD,axiom,
! [F2: nat > extended_ereal,B3: set_Extended_ereal,I: nat] :
( ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat ) @ B3 )
=> ( member2350847679896131959_ereal @ ( F2 @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_182_range__subsetD,axiom,
! [F2: rat > extended_ereal,B3: set_Extended_ereal,I: rat] :
( ( ord_le1644982726543182158_ereal @ ( image_2592109325025016879_ereal @ F2 @ top_top_set_rat ) @ B3 )
=> ( member2350847679896131959_ereal @ ( F2 @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_183_subset__image__iff,axiom,
! [B3: set_set_nat,F2: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ ( image_nat_set_nat @ F2 @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B3
= ( image_nat_set_nat @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_184_subset__image__iff,axiom,
! [B3: set_nat,F2: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F2 @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B3
= ( image_nat_nat @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_185_subset__image__iff,axiom,
! [B3: set_rat,F2: nat > rat,A2: set_nat] :
( ( ord_less_eq_set_rat @ B3 @ ( image_nat_rat @ F2 @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B3
= ( image_nat_rat @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_186_subset__image__iff,axiom,
! [B3: set_Extended_ereal,F2: nat > extended_ereal,A2: set_nat] :
( ( ord_le1644982726543182158_ereal @ B3 @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B3
= ( image_4309273772856505399_ereal @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_187_subset__image__iff,axiom,
! [B3: set_Extended_ereal,F2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B3 @ ( image_6042159593519690757_ereal @ F2 @ A2 ) )
= ( ? [AA: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ AA @ A2 )
& ( B3
= ( image_6042159593519690757_ereal @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_188_image__subset__iff,axiom,
! [F2: nat > set_nat,A2: set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ A2 ) @ B3 )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_set_nat @ ( F2 @ X ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_189_image__subset__iff,axiom,
! [F2: nat > rat,A2: set_nat,B3: set_rat] :
( ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A2 ) @ B3 )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_rat @ ( F2 @ X ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_190_image__subset__iff,axiom,
! [F2: nat > nat,A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ B3 )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F2 @ X ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_191_image__subset__iff,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,B3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F2 @ A2 ) @ B3 )
= ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A2 )
=> ( member2350847679896131959_ereal @ ( F2 @ X ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_192_image__subset__iff,axiom,
! [F2: nat > extended_ereal,A2: set_nat,B3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F2 @ A2 ) @ B3 )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member2350847679896131959_ereal @ ( F2 @ X ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_193_subset__imageE,axiom,
! [B3: set_set_nat,F2: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ ( image_nat_set_nat @ F2 @ A2 ) )
=> ~ ! [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( B3
!= ( image_nat_set_nat @ F2 @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_194_subset__imageE,axiom,
! [B3: set_nat,F2: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F2 @ A2 ) )
=> ~ ! [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( B3
!= ( image_nat_nat @ F2 @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_195_subset__imageE,axiom,
! [B3: set_rat,F2: nat > rat,A2: set_nat] :
( ( ord_less_eq_set_rat @ B3 @ ( image_nat_rat @ F2 @ A2 ) )
=> ~ ! [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( B3
!= ( image_nat_rat @ F2 @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_196_subset__imageE,axiom,
! [B3: set_Extended_ereal,F2: nat > extended_ereal,A2: set_nat] :
( ( ord_le1644982726543182158_ereal @ B3 @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
=> ~ ! [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( B3
!= ( image_4309273772856505399_ereal @ F2 @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_197_subset__imageE,axiom,
! [B3: set_Extended_ereal,F2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B3 @ ( image_6042159593519690757_ereal @ F2 @ A2 ) )
=> ~ ! [C2: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ C2 @ A2 )
=> ( B3
!= ( image_6042159593519690757_ereal @ F2 @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_198_image__subsetI,axiom,
! [A2: set_a,F2: a > a,B3: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ ( F2 @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_199_image__subsetI,axiom,
! [A2: set_a,F2: a > nat,B3: set_nat] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_nat @ ( F2 @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F2 @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_200_image__subsetI,axiom,
! [A2: set_nat,F2: nat > set_nat,B3: set_set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_set_nat @ ( F2 @ X2 ) @ B3 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_201_image__subsetI,axiom,
! [A2: set_nat,F2: nat > rat,B3: set_rat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_rat @ ( F2 @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_202_image__subsetI,axiom,
! [A2: set_nat,F2: nat > a,B3: set_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_a @ ( F2 @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F2 @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_203_image__subsetI,axiom,
! [A2: set_nat,F2: nat > nat,B3: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F2 @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_204_image__subsetI,axiom,
! [A2: set_Extended_ereal,F2: extended_ereal > extended_ereal,B3: set_Extended_ereal] :
( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A2 )
=> ( member2350847679896131959_ereal @ ( F2 @ X2 ) @ B3 ) )
=> ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F2 @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_205_image__subsetI,axiom,
! [A2: set_a,F2: a > extended_ereal,B3: set_Extended_ereal] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member2350847679896131959_ereal @ ( F2 @ X2 ) @ B3 ) )
=> ( ord_le1644982726543182158_ereal @ ( image_8405481351990995413_ereal @ F2 @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_206_image__subsetI,axiom,
! [A2: set_nat,F2: nat > extended_ereal,B3: set_Extended_ereal] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member2350847679896131959_ereal @ ( F2 @ X2 ) @ B3 ) )
=> ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F2 @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_207_image__mono,axiom,
! [A2: set_nat,B3: set_nat,F2: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ A2 ) @ ( image_nat_set_nat @ F2 @ B3 ) ) ) ).
% image_mono
thf(fact_208_image__mono,axiom,
! [A2: set_nat,B3: set_nat,F2: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ ( image_nat_nat @ F2 @ B3 ) ) ) ).
% image_mono
thf(fact_209_image__mono,axiom,
! [A2: set_nat,B3: set_nat,F2: nat > rat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A2 ) @ ( image_nat_rat @ F2 @ B3 ) ) ) ).
% image_mono
thf(fact_210_image__mono,axiom,
! [A2: set_nat,B3: set_nat,F2: nat > extended_ereal] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F2 @ A2 ) @ ( image_4309273772856505399_ereal @ F2 @ B3 ) ) ) ).
% image_mono
thf(fact_211_image__mono,axiom,
! [A2: set_Extended_ereal,B3: set_Extended_ereal,F2: extended_ereal > extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A2 @ B3 )
=> ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F2 @ A2 ) @ ( image_6042159593519690757_ereal @ F2 @ B3 ) ) ) ).
% image_mono
thf(fact_212_range__eqI,axiom,
! [B: extended_ereal,F2: extended_ereal > extended_ereal,X3: extended_ereal] :
( ( B
= ( F2 @ X3 ) )
=> ( member2350847679896131959_ereal @ B @ ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal ) ) ) ).
% range_eqI
thf(fact_213_range__eqI,axiom,
! [B: a,F2: extended_ereal > a,X3: extended_ereal] :
( ( B
= ( F2 @ X3 ) )
=> ( member_a @ B @ ( image_3724615099042636213real_a @ F2 @ top_to5683747375963461374_ereal ) ) ) ).
% range_eqI
thf(fact_214_range__eqI,axiom,
! [B: nat,F2: extended_ereal > nat,X3: extended_ereal] :
( ( B
= ( F2 @ X3 ) )
=> ( member_nat @ B @ ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal ) ) ) ).
% range_eqI
thf(fact_215_range__eqI,axiom,
! [B: extended_ereal,F2: nat > extended_ereal,X3: nat] :
( ( B
= ( F2 @ X3 ) )
=> ( member2350847679896131959_ereal @ B @ ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_216_range__eqI,axiom,
! [B: set_nat,F2: nat > set_nat,X3: nat] :
( ( B
= ( F2 @ X3 ) )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F2 @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_217_range__eqI,axiom,
! [B: rat,F2: nat > rat,X3: nat] :
( ( B
= ( F2 @ X3 ) )
=> ( member_rat @ B @ ( image_nat_rat @ F2 @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_218_range__eqI,axiom,
! [B: a,F2: nat > a,X3: nat] :
( ( B
= ( F2 @ X3 ) )
=> ( member_a @ B @ ( image_nat_a @ F2 @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_219_range__eqI,axiom,
! [B: nat,F2: nat > nat,X3: nat] :
( ( B
= ( F2 @ X3 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F2 @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_220_range__eqI,axiom,
! [B: a,F2: rat > a,X3: rat] :
( ( B
= ( F2 @ X3 ) )
=> ( member_a @ B @ ( image_rat_a @ F2 @ top_top_set_rat ) ) ) ).
% range_eqI
thf(fact_221_range__eqI,axiom,
! [B: nat,F2: rat > nat,X3: rat] :
( ( B
= ( F2 @ X3 ) )
=> ( member_nat @ B @ ( image_rat_nat @ F2 @ top_top_set_rat ) ) ) ).
% range_eqI
thf(fact_222_rangeI,axiom,
! [F2: extended_ereal > extended_ereal,X3: extended_ereal] : ( member2350847679896131959_ereal @ ( F2 @ X3 ) @ ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal ) ) ).
% rangeI
thf(fact_223_rangeI,axiom,
! [F2: extended_ereal > a,X3: extended_ereal] : ( member_a @ ( F2 @ X3 ) @ ( image_3724615099042636213real_a @ F2 @ top_to5683747375963461374_ereal ) ) ).
% rangeI
thf(fact_224_rangeI,axiom,
! [F2: extended_ereal > nat,X3: extended_ereal] : ( member_nat @ ( F2 @ X3 ) @ ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal ) ) ).
% rangeI
thf(fact_225_rangeI,axiom,
! [F2: nat > extended_ereal,X3: nat] : ( member2350847679896131959_ereal @ ( F2 @ X3 ) @ ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat ) ) ).
% rangeI
thf(fact_226_rangeI,axiom,
! [F2: nat > set_nat,X3: nat] : ( member_set_nat @ ( F2 @ X3 ) @ ( image_nat_set_nat @ F2 @ top_top_set_nat ) ) ).
% rangeI
thf(fact_227_rangeI,axiom,
! [F2: nat > rat,X3: nat] : ( member_rat @ ( F2 @ X3 ) @ ( image_nat_rat @ F2 @ top_top_set_nat ) ) ).
% rangeI
thf(fact_228_rangeI,axiom,
! [F2: nat > a,X3: nat] : ( member_a @ ( F2 @ X3 ) @ ( image_nat_a @ F2 @ top_top_set_nat ) ) ).
% rangeI
thf(fact_229_rangeI,axiom,
! [F2: nat > nat,X3: nat] : ( member_nat @ ( F2 @ X3 ) @ ( image_nat_nat @ F2 @ top_top_set_nat ) ) ).
% rangeI
thf(fact_230_rangeI,axiom,
! [F2: rat > a,X3: rat] : ( member_a @ ( F2 @ X3 ) @ ( image_rat_a @ F2 @ top_top_set_rat ) ) ).
% rangeI
thf(fact_231_rangeI,axiom,
! [F2: rat > nat,X3: rat] : ( member_nat @ ( F2 @ X3 ) @ ( image_rat_nat @ F2 @ top_top_set_rat ) ) ).
% rangeI
thf(fact_232_the__inv__f__f,axiom,
! [F2: extended_ereal > extended_ereal,X3: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ( the_in1141389326992810419_ereal @ top_to5683747375963461374_ereal @ F2 @ ( F2 @ X3 ) )
= X3 ) ) ).
% the_inv_f_f
thf(fact_233_the__inv__f__f,axiom,
! [F2: nat > nat,X3: nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( the_inv_into_nat_nat @ top_top_set_nat @ F2 @ ( F2 @ X3 ) )
= X3 ) ) ).
% the_inv_f_f
thf(fact_234_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_235_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_236_map__prod__surj,axiom,
! [F2: extended_ereal > extended_ereal,G: extended_ereal > rat] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ( ( ( image_7024712101053848417al_rat @ G @ top_to5683747375963461374_ereal )
= top_top_set_rat )
=> ( ( image_1515364090743046455al_rat @ ( produc7976008252921656496al_rat @ F2 @ G ) @ top_to3798671025730093271_ereal )
= top_to1516866950439144251al_rat ) ) ) ).
% map_prod_surj
thf(fact_237_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_238_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_239_map__prod__surj,axiom,
! [F2: extended_ereal > extended_ereal,G: nat > rat] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ( ( ( image_nat_rat @ G @ top_top_set_nat )
= top_top_set_rat )
=> ( ( image_8441009968994543785al_rat @ ( produc8043076864036019744at_rat @ F2 @ G ) @ top_to7896853287916821811al_nat )
= top_to1516866950439144251al_rat ) ) ) ).
% map_prod_surj
thf(fact_240_map__prod__surj,axiom,
! [F2: extended_ereal > extended_ereal,G: rat > extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ( ( ( image_2592109325025016879_ereal @ G @ top_top_set_rat )
= top_to5683747375963461374_ereal )
=> ( ( image_9136506195567255499_ereal @ ( produc3543405476892824958_ereal @ F2 @ G ) @ top_to1516866950439144251al_rat )
= top_to3798671025730093271_ereal ) ) ) ).
% map_prod_surj
thf(fact_241_map__prod__surj,axiom,
! [F2: extended_ereal > extended_ereal,G: rat > nat] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ( ( ( image_rat_nat @ G @ top_top_set_rat )
= top_top_set_nat )
=> ( ( image_8795816601219519657al_nat @ ( produc8361652280187649568at_nat @ F2 @ G ) @ top_to1516866950439144251al_rat )
= top_to7896853287916821811al_nat ) ) ) ).
% map_prod_surj
thf(fact_242_map__prod__surj,axiom,
! [F2: extended_ereal > extended_ereal,G: rat > rat] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ( ( ( image_rat_rat @ G @ top_top_set_rat )
= top_top_set_rat )
=> ( ( image_5287588042826122929al_rat @ ( produc7726522220101153832at_rat @ F2 @ G ) @ top_to1516866950439144251al_rat )
= top_to1516866950439144251al_rat ) ) ) ).
% map_prod_surj
thf(fact_243_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_244_inj__image__subset__iff,axiom,
! [F2: nat > set_nat,A2: set_nat,B3: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ top_top_set_nat )
=> ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ A2 ) @ ( image_nat_set_nat @ F2 @ B3 ) )
= ( ord_less_eq_set_nat @ A2 @ B3 ) ) ) ).
% inj_image_subset_iff
thf(fact_245_inj__image__subset__iff,axiom,
! [F2: nat > rat,A2: set_nat,B3: set_nat] :
( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
=> ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A2 ) @ ( image_nat_rat @ F2 @ B3 ) )
= ( ord_less_eq_set_nat @ A2 @ B3 ) ) ) ).
% inj_image_subset_iff
thf(fact_246_inj__image__subset__iff,axiom,
! [F2: nat > nat,A2: set_nat,B3: set_nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ ( image_nat_nat @ F2 @ B3 ) )
= ( ord_less_eq_set_nat @ A2 @ B3 ) ) ) ).
% inj_image_subset_iff
thf(fact_247_inj__image__subset__iff,axiom,
! [F2: nat > extended_ereal,A2: set_nat,B3: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ top_top_set_nat )
=> ( ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F2 @ A2 ) @ ( image_4309273772856505399_ereal @ F2 @ B3 ) )
= ( ord_less_eq_set_nat @ A2 @ B3 ) ) ) ).
% inj_image_subset_iff
thf(fact_248_inj__image__subset__iff,axiom,
! [F2: rat > extended_ereal,A2: set_rat,B3: set_rat] :
( ( inj_on4474368379440413635_ereal @ F2 @ top_top_set_rat )
=> ( ( ord_le1644982726543182158_ereal @ ( image_2592109325025016879_ereal @ F2 @ A2 ) @ ( image_2592109325025016879_ereal @ F2 @ B3 ) )
= ( ord_less_eq_set_rat @ A2 @ B3 ) ) ) ).
% inj_image_subset_iff
thf(fact_249_inj__image__subset__iff,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,B3: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F2 @ A2 ) @ ( image_6042159593519690757_ereal @ F2 @ B3 ) )
= ( ord_le1644982726543182158_ereal @ A2 @ B3 ) ) ) ).
% inj_image_subset_iff
thf(fact_250_map__prod_Ocomp,axiom,
! [F2: nat > nat,G: nat > nat,H: nat > nat,I: nat > nat] :
( ( comp_P2240441846945064862at_nat @ ( produc6977886695330630970at_nat @ F2 @ G ) @ ( produc6977886695330630970at_nat @ H @ I ) )
= ( produc6977886695330630970at_nat @ ( comp_nat_nat_nat @ F2 @ H ) @ ( comp_nat_nat_nat @ G @ I ) ) ) ).
% map_prod.comp
thf(fact_251_map__prod_Ocompositionality,axiom,
! [F2: nat > nat,G: nat > nat,H: nat > nat,I: nat > nat,Prod: product_prod_nat_nat] :
( ( produc6977886695330630970at_nat @ F2 @ G @ ( produc6977886695330630970at_nat @ H @ I @ Prod ) )
= ( produc6977886695330630970at_nat @ ( comp_nat_nat_nat @ F2 @ H ) @ ( comp_nat_nat_nat @ G @ I ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_252_map__prod__compose,axiom,
! [F1: nat > nat,F22: nat > nat,G1: nat > nat,G22: nat > nat] :
( ( produc6977886695330630970at_nat @ ( comp_nat_nat_nat @ F1 @ F22 ) @ ( comp_nat_nat_nat @ G1 @ G22 ) )
= ( comp_P2240441846945064862at_nat @ ( produc6977886695330630970at_nat @ F1 @ G1 ) @ ( produc6977886695330630970at_nat @ F22 @ G22 ) ) ) ).
% map_prod_compose
thf(fact_253_the__inv__into__into,axiom,
! [F2: nat > extended_ereal,A2: set_nat,X3: extended_ereal,B3: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ A2 )
=> ( ( member2350847679896131959_ereal @ X3 @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( member_nat @ ( the_in5959796611709155849_ereal @ A2 @ F2 @ X3 ) @ B3 ) ) ) ) ).
% the_inv_into_into
thf(fact_254_the__inv__into__into,axiom,
! [F2: nat > set_nat,A2: set_nat,X3: set_nat,B3: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ A2 )
=> ( ( member_set_nat @ X3 @ ( image_nat_set_nat @ F2 @ A2 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( member_nat @ ( the_in5057678521256355851et_nat @ A2 @ F2 @ X3 ) @ B3 ) ) ) ) ).
% the_inv_into_into
thf(fact_255_the__inv__into__into,axiom,
! [F2: nat > rat,A2: set_nat,X3: rat,B3: set_nat] :
( ( inj_on_nat_rat @ F2 @ A2 )
=> ( ( member_rat @ X3 @ ( image_nat_rat @ F2 @ A2 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( member_nat @ ( the_inv_into_nat_rat @ A2 @ F2 @ X3 ) @ B3 ) ) ) ) ).
% the_inv_into_into
thf(fact_256_the__inv__into__into,axiom,
! [F2: a > a,A2: set_a,X3: a,B3: set_a] :
( ( inj_on_a_a @ F2 @ A2 )
=> ( ( member_a @ X3 @ ( image_a_a @ F2 @ A2 ) )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( member_a @ ( the_inv_into_a_a @ A2 @ F2 @ X3 ) @ B3 ) ) ) ) ).
% the_inv_into_into
thf(fact_257_the__inv__into__into,axiom,
! [F2: nat > a,A2: set_nat,X3: a,B3: set_nat] :
( ( inj_on_nat_a @ F2 @ A2 )
=> ( ( member_a @ X3 @ ( image_nat_a @ F2 @ A2 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( member_nat @ ( the_inv_into_nat_a @ A2 @ F2 @ X3 ) @ B3 ) ) ) ) ).
% the_inv_into_into
thf(fact_258_the__inv__into__into,axiom,
! [F2: a > nat,A2: set_a,X3: nat,B3: set_a] :
( ( inj_on_a_nat @ F2 @ A2 )
=> ( ( member_nat @ X3 @ ( image_a_nat @ F2 @ A2 ) )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( member_a @ ( the_inv_into_a_nat @ A2 @ F2 @ X3 ) @ B3 ) ) ) ) ).
% the_inv_into_into
thf(fact_259_the__inv__into__into,axiom,
! [F2: nat > nat,A2: set_nat,X3: nat,B3: set_nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( member_nat @ X3 @ ( image_nat_nat @ F2 @ A2 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( member_nat @ ( the_inv_into_nat_nat @ A2 @ F2 @ X3 ) @ B3 ) ) ) ) ).
% the_inv_into_into
thf(fact_260_the__inv__into__into,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,X3: extended_ereal,B3: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A2 )
=> ( ( member2350847679896131959_ereal @ X3 @ ( image_6042159593519690757_ereal @ F2 @ A2 ) )
=> ( ( ord_le1644982726543182158_ereal @ A2 @ B3 )
=> ( member2350847679896131959_ereal @ ( the_in1141389326992810419_ereal @ A2 @ F2 @ X3 ) @ B3 ) ) ) ) ).
% the_inv_into_into
thf(fact_261_the__inv__into__into,axiom,
! [F2: extended_ereal > a,A2: set_Extended_ereal,X3: a,B3: set_Extended_ereal] :
( ( inj_on8242634198667403041real_a @ F2 @ A2 )
=> ( ( member_a @ X3 @ ( image_3724615099042636213real_a @ F2 @ A2 ) )
=> ( ( ord_le1644982726543182158_ereal @ A2 @ B3 )
=> ( member2350847679896131959_ereal @ ( the_in377810665034427491real_a @ A2 @ F2 @ X3 ) @ B3 ) ) ) ) ).
% the_inv_into_into
thf(fact_262_the__inv__into__into,axiom,
! [F2: extended_ereal > nat,A2: set_Extended_ereal,X3: nat,B3: set_Extended_ereal] :
( ( inj_on318729178700965101al_nat @ F2 @ A2 )
=> ( ( member_nat @ X3 @ ( image_7659842161140344153al_nat @ F2 @ A2 ) )
=> ( ( ord_le1644982726543182158_ereal @ A2 @ B3 )
=> ( member2350847679896131959_ereal @ ( the_in86992963138218795al_nat @ A2 @ F2 @ X3 ) @ B3 ) ) ) ) ).
% the_inv_into_into
thf(fact_263_surjD,axiom,
! [F2: extended_ereal > extended_ereal,Y4: extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ? [X2: extended_ereal] :
( Y4
= ( F2 @ X2 ) ) ) ).
% surjD
thf(fact_264_surjD,axiom,
! [F2: extended_ereal > nat,Y4: nat] :
( ( ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_nat )
=> ? [X2: extended_ereal] :
( Y4
= ( F2 @ X2 ) ) ) ).
% surjD
thf(fact_265_surjD,axiom,
! [F2: extended_ereal > rat,Y4: rat] :
( ( ( image_7024712101053848417al_rat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_rat )
=> ? [X2: extended_ereal] :
( Y4
= ( F2 @ X2 ) ) ) ).
% surjD
thf(fact_266_surjD,axiom,
! [F2: nat > set_nat,Y4: set_nat] :
( ( ( image_nat_set_nat @ F2 @ top_top_set_nat )
= top_top_set_set_nat )
=> ? [X2: nat] :
( Y4
= ( F2 @ X2 ) ) ) ).
% surjD
thf(fact_267_surjD,axiom,
! [F2: nat > extended_ereal,Y4: extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat )
= top_to5683747375963461374_ereal )
=> ? [X2: nat] :
( Y4
= ( F2 @ X2 ) ) ) ).
% surjD
thf(fact_268_surjD,axiom,
! [F2: nat > nat,Y4: nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ? [X2: nat] :
( Y4
= ( F2 @ X2 ) ) ) ).
% surjD
thf(fact_269_surjD,axiom,
! [F2: nat > rat,Y4: rat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
=> ? [X2: nat] :
( Y4
= ( F2 @ X2 ) ) ) ).
% surjD
thf(fact_270_surjD,axiom,
! [F2: rat > extended_ereal,Y4: extended_ereal] :
( ( ( image_2592109325025016879_ereal @ F2 @ top_top_set_rat )
= top_to5683747375963461374_ereal )
=> ? [X2: rat] :
( Y4
= ( F2 @ X2 ) ) ) ).
% surjD
thf(fact_271_surjD,axiom,
! [F2: rat > nat,Y4: nat] :
( ( ( image_rat_nat @ F2 @ top_top_set_rat )
= top_top_set_nat )
=> ? [X2: rat] :
( Y4
= ( F2 @ X2 ) ) ) ).
% surjD
thf(fact_272_surjD,axiom,
! [F2: rat > rat,Y4: rat] :
( ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat )
=> ? [X2: rat] :
( Y4
= ( F2 @ X2 ) ) ) ).
% surjD
thf(fact_273_surjE,axiom,
! [F2: extended_ereal > extended_ereal,Y4: extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ~ ! [X2: extended_ereal] :
( Y4
!= ( F2 @ X2 ) ) ) ).
% surjE
thf(fact_274_surjE,axiom,
! [F2: extended_ereal > nat,Y4: nat] :
( ( ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_nat )
=> ~ ! [X2: extended_ereal] :
( Y4
!= ( F2 @ X2 ) ) ) ).
% surjE
thf(fact_275_surjE,axiom,
! [F2: extended_ereal > rat,Y4: rat] :
( ( ( image_7024712101053848417al_rat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_rat )
=> ~ ! [X2: extended_ereal] :
( Y4
!= ( F2 @ X2 ) ) ) ).
% surjE
thf(fact_276_surjE,axiom,
! [F2: nat > set_nat,Y4: set_nat] :
( ( ( image_nat_set_nat @ F2 @ top_top_set_nat )
= top_top_set_set_nat )
=> ~ ! [X2: nat] :
( Y4
!= ( F2 @ X2 ) ) ) ).
% surjE
thf(fact_277_surjE,axiom,
! [F2: nat > extended_ereal,Y4: extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat )
= top_to5683747375963461374_ereal )
=> ~ ! [X2: nat] :
( Y4
!= ( F2 @ X2 ) ) ) ).
% surjE
thf(fact_278_surjE,axiom,
! [F2: nat > nat,Y4: nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X2: nat] :
( Y4
!= ( F2 @ X2 ) ) ) ).
% surjE
thf(fact_279_surjE,axiom,
! [F2: nat > rat,Y4: rat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
=> ~ ! [X2: nat] :
( Y4
!= ( F2 @ X2 ) ) ) ).
% surjE
thf(fact_280_surjE,axiom,
! [F2: rat > extended_ereal,Y4: extended_ereal] :
( ( ( image_2592109325025016879_ereal @ F2 @ top_top_set_rat )
= top_to5683747375963461374_ereal )
=> ~ ! [X2: rat] :
( Y4
!= ( F2 @ X2 ) ) ) ).
% surjE
thf(fact_281_surjE,axiom,
! [F2: rat > nat,Y4: nat] :
( ( ( image_rat_nat @ F2 @ top_top_set_rat )
= top_top_set_nat )
=> ~ ! [X2: rat] :
( Y4
!= ( F2 @ X2 ) ) ) ).
% surjE
thf(fact_282_surjE,axiom,
! [F2: rat > rat,Y4: rat] :
( ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat )
=> ~ ! [X2: rat] :
( Y4
!= ( F2 @ X2 ) ) ) ).
% surjE
thf(fact_283_surjI,axiom,
! [G: extended_ereal > extended_ereal,F2: extended_ereal > extended_ereal] :
( ! [X2: extended_ereal] :
( ( G @ ( F2 @ X2 ) )
= X2 )
=> ( ( image_6042159593519690757_ereal @ G @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal ) ) ).
% surjI
thf(fact_284_surjI,axiom,
! [G: extended_ereal > nat,F2: nat > extended_ereal] :
( ! [X2: nat] :
( ( G @ ( F2 @ X2 ) )
= X2 )
=> ( ( image_7659842161140344153al_nat @ G @ top_to5683747375963461374_ereal )
= top_top_set_nat ) ) ).
% surjI
thf(fact_285_surjI,axiom,
! [G: extended_ereal > rat,F2: rat > extended_ereal] :
( ! [X2: rat] :
( ( G @ ( F2 @ X2 ) )
= X2 )
=> ( ( image_7024712101053848417al_rat @ G @ top_to5683747375963461374_ereal )
= top_top_set_rat ) ) ).
% surjI
thf(fact_286_surjI,axiom,
! [G: nat > set_nat,F2: set_nat > nat] :
( ! [X2: set_nat] :
( ( G @ ( F2 @ X2 ) )
= X2 )
=> ( ( image_nat_set_nat @ G @ top_top_set_nat )
= top_top_set_set_nat ) ) ).
% surjI
thf(fact_287_surjI,axiom,
! [G: nat > extended_ereal,F2: extended_ereal > nat] :
( ! [X2: extended_ereal] :
( ( G @ ( F2 @ X2 ) )
= X2 )
=> ( ( image_4309273772856505399_ereal @ G @ top_top_set_nat )
= top_to5683747375963461374_ereal ) ) ).
% surjI
thf(fact_288_surjI,axiom,
! [G: nat > nat,F2: nat > nat] :
( ! [X2: nat] :
( ( G @ ( F2 @ X2 ) )
= X2 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_289_surjI,axiom,
! [G: nat > rat,F2: rat > nat] :
( ! [X2: rat] :
( ( G @ ( F2 @ X2 ) )
= X2 )
=> ( ( image_nat_rat @ G @ top_top_set_nat )
= top_top_set_rat ) ) ).
% surjI
thf(fact_290_surjI,axiom,
! [G: rat > extended_ereal,F2: extended_ereal > rat] :
( ! [X2: extended_ereal] :
( ( G @ ( F2 @ X2 ) )
= X2 )
=> ( ( image_2592109325025016879_ereal @ G @ top_top_set_rat )
= top_to5683747375963461374_ereal ) ) ).
% surjI
thf(fact_291_surjI,axiom,
! [G: rat > nat,F2: nat > rat] :
( ! [X2: nat] :
( ( G @ ( F2 @ X2 ) )
= X2 )
=> ( ( image_rat_nat @ G @ top_top_set_rat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_292_surjI,axiom,
! [G: rat > rat,F2: rat > rat] :
( ! [X2: rat] :
( ( G @ ( F2 @ X2 ) )
= X2 )
=> ( ( image_rat_rat @ G @ top_top_set_rat )
= top_top_set_rat ) ) ).
% surjI
thf(fact_293_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_294_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_295_surj__def,axiom,
! [F2: extended_ereal > rat] :
( ( ( image_7024712101053848417al_rat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_rat )
= ( ! [Y: rat] :
? [X: extended_ereal] :
( Y
= ( F2 @ X ) ) ) ) ).
% surj_def
thf(fact_296_surj__def,axiom,
! [F2: nat > set_nat] :
( ( ( image_nat_set_nat @ F2 @ top_top_set_nat )
= top_top_set_set_nat )
= ( ! [Y: set_nat] :
? [X: nat] :
( Y
= ( F2 @ X ) ) ) ) ).
% surj_def
thf(fact_297_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_298_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_299_surj__def,axiom,
! [F2: nat > rat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
= ( ! [Y: rat] :
? [X: nat] :
( Y
= ( F2 @ X ) ) ) ) ).
% surj_def
thf(fact_300_surj__def,axiom,
! [F2: rat > extended_ereal] :
( ( ( image_2592109325025016879_ereal @ F2 @ top_top_set_rat )
= top_to5683747375963461374_ereal )
= ( ! [Y: extended_ereal] :
? [X: rat] :
( Y
= ( F2 @ X ) ) ) ) ).
% surj_def
thf(fact_301_surj__def,axiom,
! [F2: rat > nat] :
( ( ( image_rat_nat @ F2 @ top_top_set_rat )
= top_top_set_nat )
= ( ! [Y: nat] :
? [X: rat] :
( Y
= ( F2 @ X ) ) ) ) ).
% surj_def
thf(fact_302_surj__def,axiom,
! [F2: rat > rat] :
( ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat )
= ( ! [Y: rat] :
? [X: rat] :
( Y
= ( F2 @ X ) ) ) ) ).
% surj_def
thf(fact_303_subset__inj__on,axiom,
! [F2: nat > nat,B3: set_nat,A2: set_nat] :
( ( inj_on_nat_nat @ F2 @ B3 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( inj_on_nat_nat @ F2 @ A2 ) ) ) ).
% subset_inj_on
thf(fact_304_subset__inj__on,axiom,
! [F2: extended_ereal > extended_ereal,B3: set_Extended_ereal,A2: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ B3 )
=> ( ( ord_le1644982726543182158_ereal @ A2 @ B3 )
=> ( inj_on7162434037990268785_ereal @ F2 @ A2 ) ) ) ).
% subset_inj_on
thf(fact_305_inj__on__subset,axiom,
! [F2: nat > nat,A2: set_nat,B3: set_nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( inj_on_nat_nat @ F2 @ B3 ) ) ) ).
% inj_on_subset
thf(fact_306_inj__on__subset,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,B3: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A2 )
=> ( ( ord_le1644982726543182158_ereal @ B3 @ A2 )
=> ( inj_on7162434037990268785_ereal @ F2 @ B3 ) ) ) ).
% inj_on_subset
thf(fact_307_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_308_inj__def,axiom,
! [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
= ( ! [X: nat,Y: nat] :
( ( ( F2 @ X )
= ( F2 @ Y ) )
=> ( X = Y ) ) ) ) ).
% inj_def
thf(fact_309_inj__eq,axiom,
! [F2: extended_ereal > extended_ereal,X3: extended_ereal,Y4: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% inj_eq
thf(fact_310_inj__eq,axiom,
! [F2: nat > nat,X3: nat,Y4: nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% inj_eq
thf(fact_311_injI,axiom,
! [F2: extended_ereal > extended_ereal] :
( ! [X2: extended_ereal,Y2: extended_ereal] :
( ( ( F2 @ X2 )
= ( F2 @ Y2 ) )
=> ( X2 = Y2 ) )
=> ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal ) ) ).
% injI
thf(fact_312_injI,axiom,
! [F2: nat > nat] :
( ! [X2: nat,Y2: nat] :
( ( ( F2 @ X2 )
= ( F2 @ Y2 ) )
=> ( X2 = Y2 ) )
=> ( inj_on_nat_nat @ F2 @ top_top_set_nat ) ) ).
% injI
thf(fact_313_injD,axiom,
! [F2: extended_ereal > extended_ereal,X3: extended_ereal,Y4: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y4 ) )
=> ( X3 = Y4 ) ) ) ).
% injD
thf(fact_314_injD,axiom,
! [F2: nat > nat,X3: nat,Y4: nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y4 ) )
=> ( X3 = Y4 ) ) ) ).
% injD
thf(fact_315_the__inv__into__f__eq,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,X3: extended_ereal,Y4: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A2 )
=> ( ( ( F2 @ X3 )
= Y4 )
=> ( ( member2350847679896131959_ereal @ X3 @ A2 )
=> ( ( the_in1141389326992810419_ereal @ A2 @ F2 @ Y4 )
= X3 ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_316_the__inv__into__f__eq,axiom,
! [F2: nat > nat,A2: set_nat,X3: nat,Y4: nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( ( F2 @ X3 )
= Y4 )
=> ( ( member_nat @ X3 @ A2 )
=> ( ( the_inv_into_nat_nat @ A2 @ F2 @ Y4 )
= X3 ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_317_the__inv__into__f__f,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,X3: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A2 )
=> ( ( member2350847679896131959_ereal @ X3 @ A2 )
=> ( ( the_in1141389326992810419_ereal @ A2 @ F2 @ ( F2 @ X3 ) )
= X3 ) ) ) ).
% the_inv_into_f_f
thf(fact_318_the__inv__into__f__f,axiom,
! [F2: nat > nat,A2: set_nat,X3: nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( member_nat @ X3 @ A2 )
=> ( ( the_inv_into_nat_nat @ A2 @ F2 @ ( F2 @ X3 ) )
= X3 ) ) ) ).
% the_inv_into_f_f
thf(fact_319_inj__on__image__mem__iff,axiom,
! [F2: a > a,B3: set_a,A: a,A2: set_a] :
( ( inj_on_a_a @ F2 @ B3 )
=> ( ( member_a @ A @ B3 )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_a @ ( F2 @ A ) @ ( image_a_a @ F2 @ A2 ) )
= ( member_a @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_320_inj__on__image__mem__iff,axiom,
! [F2: a > nat,B3: set_a,A: a,A2: set_a] :
( ( inj_on_a_nat @ F2 @ B3 )
=> ( ( member_a @ A @ B3 )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_nat @ ( F2 @ A ) @ ( image_a_nat @ F2 @ A2 ) )
= ( member_a @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_321_inj__on__image__mem__iff,axiom,
! [F2: nat > extended_ereal,B3: set_nat,A: nat,A2: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ B3 )
=> ( ( member_nat @ A @ B3 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( member2350847679896131959_ereal @ ( F2 @ A ) @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
= ( member_nat @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_322_inj__on__image__mem__iff,axiom,
! [F2: nat > set_nat,B3: set_nat,A: nat,A2: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ B3 )
=> ( ( member_nat @ A @ B3 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( member_set_nat @ ( F2 @ A ) @ ( image_nat_set_nat @ F2 @ A2 ) )
= ( member_nat @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_323_inj__on__image__mem__iff,axiom,
! [F2: nat > rat,B3: set_nat,A: nat,A2: set_nat] :
( ( inj_on_nat_rat @ F2 @ B3 )
=> ( ( member_nat @ A @ B3 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( member_rat @ ( F2 @ A ) @ ( image_nat_rat @ F2 @ A2 ) )
= ( member_nat @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_324_inj__on__image__mem__iff,axiom,
! [F2: nat > a,B3: set_nat,A: nat,A2: set_nat] :
( ( inj_on_nat_a @ F2 @ B3 )
=> ( ( member_nat @ A @ B3 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( member_a @ ( F2 @ A ) @ ( image_nat_a @ F2 @ A2 ) )
= ( member_nat @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_325_inj__on__image__mem__iff,axiom,
! [F2: nat > nat,B3: set_nat,A: nat,A2: set_nat] :
( ( inj_on_nat_nat @ F2 @ B3 )
=> ( ( member_nat @ A @ B3 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( member_nat @ ( F2 @ A ) @ ( image_nat_nat @ F2 @ A2 ) )
= ( member_nat @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_326_inj__on__image__mem__iff,axiom,
! [F2: extended_ereal > extended_ereal,B3: set_Extended_ereal,A: extended_ereal,A2: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ B3 )
=> ( ( member2350847679896131959_ereal @ A @ B3 )
=> ( ( ord_le1644982726543182158_ereal @ A2 @ B3 )
=> ( ( member2350847679896131959_ereal @ ( F2 @ A ) @ ( image_6042159593519690757_ereal @ F2 @ A2 ) )
= ( member2350847679896131959_ereal @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_327_inj__on__image__mem__iff,axiom,
! [F2: extended_ereal > a,B3: set_Extended_ereal,A: extended_ereal,A2: set_Extended_ereal] :
( ( inj_on8242634198667403041real_a @ F2 @ B3 )
=> ( ( member2350847679896131959_ereal @ A @ B3 )
=> ( ( ord_le1644982726543182158_ereal @ A2 @ B3 )
=> ( ( member_a @ ( F2 @ A ) @ ( image_3724615099042636213real_a @ F2 @ A2 ) )
= ( member2350847679896131959_ereal @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_328_inj__on__image__mem__iff,axiom,
! [F2: extended_ereal > nat,B3: set_Extended_ereal,A: extended_ereal,A2: set_Extended_ereal] :
( ( inj_on318729178700965101al_nat @ F2 @ B3 )
=> ( ( member2350847679896131959_ereal @ A @ B3 )
=> ( ( ord_le1644982726543182158_ereal @ A2 @ B3 )
=> ( ( member_nat @ ( F2 @ A ) @ ( image_7659842161140344153al_nat @ F2 @ A2 ) )
= ( member2350847679896131959_ereal @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_329_inj__on__image__eq__iff,axiom,
! [F2: nat > extended_ereal,C3: set_nat,A2: set_nat,B3: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ C3 )
=> ( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C3 )
=> ( ( ( image_4309273772856505399_ereal @ F2 @ A2 )
= ( image_4309273772856505399_ereal @ F2 @ B3 ) )
= ( A2 = B3 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_330_inj__on__image__eq__iff,axiom,
! [F2: nat > set_nat,C3: set_nat,A2: set_nat,B3: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ C3 )
=> ( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C3 )
=> ( ( ( image_nat_set_nat @ F2 @ A2 )
= ( image_nat_set_nat @ F2 @ B3 ) )
= ( A2 = B3 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_331_inj__on__image__eq__iff,axiom,
! [F2: nat > rat,C3: set_nat,A2: set_nat,B3: set_nat] :
( ( inj_on_nat_rat @ F2 @ C3 )
=> ( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C3 )
=> ( ( ( image_nat_rat @ F2 @ A2 )
= ( image_nat_rat @ F2 @ B3 ) )
= ( A2 = B3 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_332_inj__on__image__eq__iff,axiom,
! [F2: nat > nat,C3: set_nat,A2: set_nat,B3: set_nat] :
( ( inj_on_nat_nat @ F2 @ C3 )
=> ( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C3 )
=> ( ( ( image_nat_nat @ F2 @ A2 )
= ( image_nat_nat @ F2 @ B3 ) )
= ( A2 = B3 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_333_inj__on__image__eq__iff,axiom,
! [F2: extended_ereal > extended_ereal,C3: set_Extended_ereal,A2: set_Extended_ereal,B3: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ C3 )
=> ( ( ord_le1644982726543182158_ereal @ A2 @ C3 )
=> ( ( ord_le1644982726543182158_ereal @ B3 @ C3 )
=> ( ( ( image_6042159593519690757_ereal @ F2 @ A2 )
= ( image_6042159593519690757_ereal @ F2 @ B3 ) )
= ( A2 = B3 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_334_surj__fun__eq,axiom,
! [F2: nat > nat,X5: set_nat,G1: nat > nat,G22: nat > nat] :
( ( ( image_nat_nat @ F2 @ X5 )
= top_top_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X5 )
=> ( ( comp_nat_nat_nat @ G1 @ F2 @ X2 )
= ( comp_nat_nat_nat @ G22 @ F2 @ X2 ) ) )
=> ( G1 = G22 ) ) ) ).
% surj_fun_eq
thf(fact_335_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_336_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_337_comp__surj,axiom,
! [F2: extended_ereal > extended_ereal,G: extended_ereal > rat] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ( ( ( image_7024712101053848417al_rat @ G @ top_to5683747375963461374_ereal )
= top_top_set_rat )
=> ( ( image_7024712101053848417al_rat @ ( comp_E7881739061092793609_ereal @ G @ F2 ) @ top_to5683747375963461374_ereal )
= top_top_set_rat ) ) ) ).
% comp_surj
thf(fact_338_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_339_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_340_comp__surj,axiom,
! [F2: extended_ereal > nat,G: nat > rat] :
( ( ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_nat )
=> ( ( ( image_nat_rat @ G @ top_top_set_nat )
= top_top_set_rat )
=> ( ( image_7024712101053848417al_rat @ ( comp_n4169009346981848321_ereal @ G @ F2 ) @ top_to5683747375963461374_ereal )
= top_top_set_rat ) ) ) ).
% comp_surj
thf(fact_341_comp__surj,axiom,
! [F2: extended_ereal > rat,G: rat > extended_ereal] :
( ( ( image_7024712101053848417al_rat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_rat )
=> ( ( ( image_2592109325025016879_ereal @ G @ top_top_set_rat )
= top_to5683747375963461374_ereal )
=> ( ( image_6042159593519690757_ereal @ ( comp_r2952691998189091003_ereal @ G @ F2 ) @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal ) ) ) ).
% comp_surj
thf(fact_342_comp__surj,axiom,
! [F2: extended_ereal > rat,G: rat > nat] :
( ( ( image_7024712101053848417al_rat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_rat )
=> ( ( ( image_rat_nat @ G @ top_top_set_rat )
= top_top_set_nat )
=> ( ( image_7659842161140344153al_nat @ ( comp_r969312439189832961_ereal @ G @ F2 ) @ top_to5683747375963461374_ereal )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_343_comp__surj,axiom,
! [F2: extended_ereal > rat,G: rat > rat] :
( ( ( image_7024712101053848417al_rat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_rat )
=> ( ( ( image_rat_rat @ G @ top_top_set_rat )
= top_top_set_rat )
=> ( ( image_7024712101053848417al_rat @ ( comp_r8475520028213120249_ereal @ G @ F2 ) @ top_to5683747375963461374_ereal )
= top_top_set_rat ) ) ) ).
% comp_surj
thf(fact_344_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_345_inj__image__mem__iff,axiom,
! [F2: a > a,A: a,A2: set_a] :
( ( inj_on_a_a @ F2 @ top_top_set_a )
=> ( ( member_a @ ( F2 @ A ) @ ( image_a_a @ F2 @ A2 ) )
= ( member_a @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_346_inj__image__mem__iff,axiom,
! [F2: a > nat,A: a,A2: set_a] :
( ( inj_on_a_nat @ F2 @ top_top_set_a )
=> ( ( member_nat @ ( F2 @ A ) @ ( image_a_nat @ F2 @ A2 ) )
= ( member_a @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_347_inj__image__mem__iff,axiom,
! [F2: extended_ereal > extended_ereal,A: extended_ereal,A2: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ( member2350847679896131959_ereal @ ( F2 @ A ) @ ( image_6042159593519690757_ereal @ F2 @ A2 ) )
= ( member2350847679896131959_ereal @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_348_inj__image__mem__iff,axiom,
! [F2: extended_ereal > a,A: extended_ereal,A2: set_Extended_ereal] :
( ( inj_on8242634198667403041real_a @ F2 @ top_to5683747375963461374_ereal )
=> ( ( member_a @ ( F2 @ A ) @ ( image_3724615099042636213real_a @ F2 @ A2 ) )
= ( member2350847679896131959_ereal @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_349_inj__image__mem__iff,axiom,
! [F2: extended_ereal > nat,A: extended_ereal,A2: set_Extended_ereal] :
( ( inj_on318729178700965101al_nat @ F2 @ top_to5683747375963461374_ereal )
=> ( ( member_nat @ ( F2 @ A ) @ ( image_7659842161140344153al_nat @ F2 @ A2 ) )
= ( member2350847679896131959_ereal @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_350_inj__image__mem__iff,axiom,
! [F2: nat > extended_ereal,A: nat,A2: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ top_top_set_nat )
=> ( ( member2350847679896131959_ereal @ ( F2 @ A ) @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
= ( member_nat @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_351_inj__image__mem__iff,axiom,
! [F2: nat > rat,A: nat,A2: set_nat] :
( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
=> ( ( member_rat @ ( F2 @ A ) @ ( image_nat_rat @ F2 @ A2 ) )
= ( member_nat @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_352_inj__image__mem__iff,axiom,
! [F2: nat > a,A: nat,A2: set_nat] :
( ( inj_on_nat_a @ F2 @ top_top_set_nat )
=> ( ( member_a @ ( F2 @ A ) @ ( image_nat_a @ F2 @ A2 ) )
= ( member_nat @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_353_inj__image__mem__iff,axiom,
! [F2: nat > nat,A: nat,A2: set_nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( member_nat @ ( F2 @ A ) @ ( image_nat_nat @ F2 @ A2 ) )
= ( member_nat @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_354_inj__image__mem__iff,axiom,
! [F2: rat > a,A: rat,A2: set_rat] :
( ( inj_on_rat_a @ F2 @ top_top_set_rat )
=> ( ( member_a @ ( F2 @ A ) @ ( image_rat_a @ F2 @ A2 ) )
= ( member_rat @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_355_inj__image__eq__iff,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,B3: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ( ( image_6042159593519690757_ereal @ F2 @ A2 )
= ( image_6042159593519690757_ereal @ F2 @ B3 ) )
= ( A2 = B3 ) ) ) ).
% inj_image_eq_iff
thf(fact_356_inj__image__eq__iff,axiom,
! [F2: nat > extended_ereal,A2: set_nat,B3: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ top_top_set_nat )
=> ( ( ( image_4309273772856505399_ereal @ F2 @ A2 )
= ( image_4309273772856505399_ereal @ F2 @ B3 ) )
= ( A2 = B3 ) ) ) ).
% inj_image_eq_iff
thf(fact_357_inj__image__eq__iff,axiom,
! [F2: nat > set_nat,A2: set_nat,B3: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ top_top_set_nat )
=> ( ( ( image_nat_set_nat @ F2 @ A2 )
= ( image_nat_set_nat @ F2 @ B3 ) )
= ( A2 = B3 ) ) ) ).
% inj_image_eq_iff
thf(fact_358_inj__image__eq__iff,axiom,
! [F2: nat > rat,A2: set_nat,B3: set_nat] :
( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
=> ( ( ( image_nat_rat @ F2 @ A2 )
= ( image_nat_rat @ F2 @ B3 ) )
= ( A2 = B3 ) ) ) ).
% inj_image_eq_iff
thf(fact_359_inj__image__eq__iff,axiom,
! [F2: nat > nat,A2: set_nat,B3: set_nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F2 @ A2 )
= ( image_nat_nat @ F2 @ B3 ) )
= ( A2 = B3 ) ) ) ).
% inj_image_eq_iff
thf(fact_360_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_361_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_362_range__ex1__eq,axiom,
! [F2: extended_ereal > nat,B: nat] :
( ( inj_on318729178700965101al_nat @ F2 @ top_to5683747375963461374_ereal )
=> ( ( member_nat @ B @ ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal ) )
= ( ? [X: extended_ereal] :
( ( B
= ( F2 @ X ) )
& ! [Y: extended_ereal] :
( ( B
= ( F2 @ Y ) )
=> ( Y = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_363_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_364_range__ex1__eq,axiom,
! [F2: nat > set_nat,B: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ top_top_set_nat )
=> ( ( member_set_nat @ B @ ( image_nat_set_nat @ F2 @ top_top_set_nat ) )
= ( ? [X: nat] :
( ( B
= ( F2 @ X ) )
& ! [Y: nat] :
( ( B
= ( F2 @ Y ) )
=> ( Y = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_365_range__ex1__eq,axiom,
! [F2: nat > rat,B: rat] :
( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
=> ( ( member_rat @ B @ ( image_nat_rat @ F2 @ top_top_set_nat ) )
= ( ? [X: nat] :
( ( B
= ( F2 @ X ) )
& ! [Y: nat] :
( ( B
= ( F2 @ Y ) )
=> ( Y = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_366_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_367_range__ex1__eq,axiom,
! [F2: nat > nat,B: nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( member_nat @ B @ ( image_nat_nat @ F2 @ top_top_set_nat ) )
= ( ? [X: nat] :
( ( B
= ( F2 @ X ) )
& ! [Y: nat] :
( ( B
= ( F2 @ Y ) )
=> ( Y = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_368_range__ex1__eq,axiom,
! [F2: rat > a,B: a] :
( ( inj_on_rat_a @ F2 @ top_top_set_rat )
=> ( ( member_a @ B @ ( image_rat_a @ F2 @ top_top_set_rat ) )
= ( ? [X: rat] :
( ( B
= ( F2 @ X ) )
& ! [Y: rat] :
( ( B
= ( F2 @ Y ) )
=> ( Y = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_369_range__ex1__eq,axiom,
! [F2: rat > nat,B: nat] :
( ( inj_on_rat_nat @ F2 @ top_top_set_rat )
=> ( ( member_nat @ B @ ( image_rat_nat @ F2 @ top_top_set_rat ) )
= ( ? [X: rat] :
( ( B
= ( F2 @ X ) )
& ! [Y: rat] :
( ( B
= ( F2 @ Y ) )
=> ( Y = X ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_370_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_371_inj__compose,axiom,
! [F2: extended_ereal > nat,G: nat > extended_ereal] :
( ( inj_on318729178700965101al_nat @ F2 @ top_to5683747375963461374_ereal )
=> ( ( inj_on6191532827271902155_ereal @ G @ top_top_set_nat )
=> ( inj_on_nat_nat @ ( comp_E7502005551946643277at_nat @ F2 @ G ) @ top_top_set_nat ) ) ) ).
% inj_compose
thf(fact_372_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_373_inj__compose,axiom,
! [F2: extended_ereal > extended_ereal,G: rat > extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ( inj_on4474368379440413635_ereal @ G @ top_top_set_rat )
=> ( inj_on4474368379440413635_ereal @ ( comp_E3090969800266849339al_rat @ F2 @ G ) @ top_top_set_rat ) ) ) ).
% inj_compose
thf(fact_374_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_375_inj__compose,axiom,
! [F2: nat > nat,G: extended_ereal > nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( inj_on318729178700965101al_nat @ G @ top_to5683747375963461374_ereal )
=> ( inj_on318729178700965101al_nat @ ( comp_n5886173794813336841_ereal @ F2 @ G ) @ top_to5683747375963461374_ereal ) ) ) ).
% inj_compose
thf(fact_376_inj__compose,axiom,
! [F2: nat > nat,G: nat > nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( inj_on_nat_nat @ G @ top_top_set_nat )
=> ( inj_on_nat_nat @ ( comp_nat_nat_nat @ F2 @ G ) @ top_top_set_nat ) ) ) ).
% inj_compose
thf(fact_377_inj__compose,axiom,
! [F2: nat > nat,G: rat > nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( inj_on_rat_nat @ G @ top_top_set_rat )
=> ( inj_on_rat_nat @ ( comp_nat_nat_rat @ F2 @ G ) @ top_top_set_rat ) ) ) ).
% inj_compose
thf(fact_378_inj__compose,axiom,
! [F2: rat > extended_ereal,G: extended_ereal > rat] :
( ( inj_on4474368379440413635_ereal @ F2 @ top_top_set_rat )
=> ( ( inj_on8906971155469245173al_rat @ G @ top_to5683747375963461374_ereal )
=> ( inj_on7162434037990268785_ereal @ ( comp_r2952691998189091003_ereal @ F2 @ G ) @ top_to5683747375963461374_ereal ) ) ) ).
% inj_compose
thf(fact_379_inj__compose,axiom,
! [F2: rat > nat,G: nat > rat] :
( ( inj_on_rat_nat @ F2 @ top_top_set_rat )
=> ( ( inj_on_nat_rat @ G @ top_top_set_nat )
=> ( inj_on_nat_nat @ ( comp_rat_nat_nat @ F2 @ G ) @ top_top_set_nat ) ) ) ).
% inj_compose
thf(fact_380_Collect__case__prod__Grp__in,axiom,
! [Z2: product_prod_a_b,A2: set_a,F2: a > b] :
( ( member1426531481828664017od_a_b @ Z2 @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ ( bNF_Grp_a_b @ A2 @ F2 ) ) ) )
=> ( member_a @ ( product_fst_a_b @ Z2 ) @ A2 ) ) ).
% Collect_case_prod_Grp_in
thf(fact_381_f__the__inv__into__f,axiom,
! [F2: nat > extended_ereal,A2: set_nat,Y4: extended_ereal] :
( ( inj_on6191532827271902155_ereal @ F2 @ A2 )
=> ( ( member2350847679896131959_ereal @ Y4 @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
=> ( ( F2 @ ( the_in5959796611709155849_ereal @ A2 @ F2 @ Y4 ) )
= Y4 ) ) ) ).
% f_the_inv_into_f
thf(fact_382_f__the__inv__into__f,axiom,
! [F2: nat > set_nat,A2: set_nat,Y4: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ A2 )
=> ( ( member_set_nat @ Y4 @ ( image_nat_set_nat @ F2 @ A2 ) )
=> ( ( F2 @ ( the_in5057678521256355851et_nat @ A2 @ F2 @ Y4 ) )
= Y4 ) ) ) ).
% f_the_inv_into_f
thf(fact_383_f__the__inv__into__f,axiom,
! [F2: nat > rat,A2: set_nat,Y4: rat] :
( ( inj_on_nat_rat @ F2 @ A2 )
=> ( ( member_rat @ Y4 @ ( image_nat_rat @ F2 @ A2 ) )
=> ( ( F2 @ ( the_inv_into_nat_rat @ A2 @ F2 @ Y4 ) )
= Y4 ) ) ) ).
% f_the_inv_into_f
thf(fact_384_f__the__inv__into__f,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,Y4: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A2 )
=> ( ( member2350847679896131959_ereal @ Y4 @ ( image_6042159593519690757_ereal @ F2 @ A2 ) )
=> ( ( F2 @ ( the_in1141389326992810419_ereal @ A2 @ F2 @ Y4 ) )
= Y4 ) ) ) ).
% f_the_inv_into_f
thf(fact_385_f__the__inv__into__f,axiom,
! [F2: nat > nat,A2: set_nat,Y4: nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( member_nat @ Y4 @ ( image_nat_nat @ F2 @ A2 ) )
=> ( ( F2 @ ( the_inv_into_nat_nat @ A2 @ F2 @ Y4 ) )
= Y4 ) ) ) ).
% f_the_inv_into_f
thf(fact_386_inj__on__the__inv__into,axiom,
! [F2: nat > extended_ereal,A2: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ A2 )
=> ( inj_on318729178700965101al_nat @ ( the_in5959796611709155849_ereal @ A2 @ F2 ) @ ( image_4309273772856505399_ereal @ F2 @ A2 ) ) ) ).
% inj_on_the_inv_into
thf(fact_387_inj__on__the__inv__into,axiom,
! [F2: nat > set_nat,A2: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ A2 )
=> ( inj_on_set_nat_nat @ ( the_in5057678521256355851et_nat @ A2 @ F2 ) @ ( image_nat_set_nat @ F2 @ A2 ) ) ) ).
% inj_on_the_inv_into
thf(fact_388_inj__on__the__inv__into,axiom,
! [F2: nat > rat,A2: set_nat] :
( ( inj_on_nat_rat @ F2 @ A2 )
=> ( inj_on_rat_nat @ ( the_inv_into_nat_rat @ A2 @ F2 ) @ ( image_nat_rat @ F2 @ A2 ) ) ) ).
% inj_on_the_inv_into
thf(fact_389_inj__on__the__inv__into,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A2 )
=> ( inj_on7162434037990268785_ereal @ ( the_in1141389326992810419_ereal @ A2 @ F2 ) @ ( image_6042159593519690757_ereal @ F2 @ A2 ) ) ) ).
% inj_on_the_inv_into
thf(fact_390_inj__on__the__inv__into,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( inj_on_nat_nat @ ( the_inv_into_nat_nat @ A2 @ F2 ) @ ( image_nat_nat @ F2 @ A2 ) ) ) ).
% inj_on_the_inv_into
thf(fact_391_rev__image__eqI,axiom,
! [X3: extended_ereal,A2: set_Extended_ereal,B: extended_ereal,F2: extended_ereal > extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A2 )
=> ( ( B
= ( F2 @ X3 ) )
=> ( member2350847679896131959_ereal @ B @ ( image_6042159593519690757_ereal @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_392_rev__image__eqI,axiom,
! [X3: a,A2: set_a,B: a,F2: a > a] :
( ( member_a @ X3 @ A2 )
=> ( ( B
= ( F2 @ X3 ) )
=> ( member_a @ B @ ( image_a_a @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_393_rev__image__eqI,axiom,
! [X3: a,A2: set_a,B: nat,F2: a > nat] :
( ( member_a @ X3 @ A2 )
=> ( ( B
= ( F2 @ X3 ) )
=> ( member_nat @ B @ ( image_a_nat @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_394_rev__image__eqI,axiom,
! [X3: nat,A2: set_nat,B: extended_ereal,F2: nat > extended_ereal] :
( ( member_nat @ X3 @ A2 )
=> ( ( B
= ( F2 @ X3 ) )
=> ( member2350847679896131959_ereal @ B @ ( image_4309273772856505399_ereal @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_395_rev__image__eqI,axiom,
! [X3: nat,A2: set_nat,B: set_nat,F2: nat > set_nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( B
= ( F2 @ X3 ) )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_396_rev__image__eqI,axiom,
! [X3: nat,A2: set_nat,B: rat,F2: nat > rat] :
( ( member_nat @ X3 @ A2 )
=> ( ( B
= ( F2 @ X3 ) )
=> ( member_rat @ B @ ( image_nat_rat @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_397_rev__image__eqI,axiom,
! [X3: nat,A2: set_nat,B: a,F2: nat > a] :
( ( member_nat @ X3 @ A2 )
=> ( ( B
= ( F2 @ X3 ) )
=> ( member_a @ B @ ( image_nat_a @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_398_rev__image__eqI,axiom,
! [X3: nat,A2: set_nat,B: nat,F2: nat > nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( B
= ( F2 @ X3 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_399_ball__imageD,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,P2: extended_ereal > $o] :
( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ ( image_6042159593519690757_ereal @ F2 @ A2 ) )
=> ( P2 @ X2 ) )
=> ! [X6: extended_ereal] :
( ( member2350847679896131959_ereal @ X6 @ A2 )
=> ( P2 @ ( F2 @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_400_ball__imageD,axiom,
! [F2: nat > extended_ereal,A2: set_nat,P2: extended_ereal > $o] :
( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
=> ( P2 @ X2 ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A2 )
=> ( P2 @ ( F2 @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_401_ball__imageD,axiom,
! [F2: nat > set_nat,A2: set_nat,P2: set_nat > $o] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( image_nat_set_nat @ F2 @ A2 ) )
=> ( P2 @ X2 ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A2 )
=> ( P2 @ ( F2 @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_402_ball__imageD,axiom,
! [F2: nat > nat,A2: set_nat,P2: nat > $o] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F2 @ A2 ) )
=> ( P2 @ X2 ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A2 )
=> ( P2 @ ( F2 @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_403_ball__imageD,axiom,
! [F2: nat > rat,A2: set_nat,P2: rat > $o] :
( ! [X2: rat] :
( ( member_rat @ X2 @ ( image_nat_rat @ F2 @ A2 ) )
=> ( P2 @ X2 ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A2 )
=> ( P2 @ ( F2 @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_404_image__cong,axiom,
! [M: set_Extended_ereal,N: set_Extended_ereal,F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal] :
( ( M = N )
=> ( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ N )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_6042159593519690757_ereal @ F2 @ M )
= ( image_6042159593519690757_ereal @ G @ N ) ) ) ) ).
% image_cong
thf(fact_405_image__cong,axiom,
! [M: set_nat,N: set_nat,F2: nat > extended_ereal,G: nat > extended_ereal] :
( ( M = N )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_4309273772856505399_ereal @ F2 @ M )
= ( image_4309273772856505399_ereal @ G @ N ) ) ) ) ).
% image_cong
thf(fact_406_image__cong,axiom,
! [M: set_nat,N: set_nat,F2: nat > set_nat,G: nat > set_nat] :
( ( M = N )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_set_nat @ F2 @ M )
= ( image_nat_set_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_407_image__cong,axiom,
! [M: set_nat,N: set_nat,F2: nat > nat,G: nat > nat] :
( ( M = N )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_nat @ F2 @ M )
= ( image_nat_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_408_image__cong,axiom,
! [M: set_nat,N: set_nat,F2: nat > rat,G: nat > rat] :
( ( M = N )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_rat @ F2 @ M )
= ( image_nat_rat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_409_bex__imageD,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,P2: extended_ereal > $o] :
( ? [X6: extended_ereal] :
( ( member2350847679896131959_ereal @ X6 @ ( image_6042159593519690757_ereal @ F2 @ A2 ) )
& ( P2 @ X6 ) )
=> ? [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A2 )
& ( P2 @ ( F2 @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_410_bex__imageD,axiom,
! [F2: nat > extended_ereal,A2: set_nat,P2: extended_ereal > $o] :
( ? [X6: extended_ereal] :
( ( member2350847679896131959_ereal @ X6 @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
& ( P2 @ X6 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P2 @ ( F2 @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_411_bex__imageD,axiom,
! [F2: nat > set_nat,A2: set_nat,P2: set_nat > $o] :
( ? [X6: set_nat] :
( ( member_set_nat @ X6 @ ( image_nat_set_nat @ F2 @ A2 ) )
& ( P2 @ X6 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P2 @ ( F2 @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_412_bex__imageD,axiom,
! [F2: nat > nat,A2: set_nat,P2: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( image_nat_nat @ F2 @ A2 ) )
& ( P2 @ X6 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P2 @ ( F2 @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_413_bex__imageD,axiom,
! [F2: nat > rat,A2: set_nat,P2: rat > $o] :
( ? [X6: rat] :
( ( member_rat @ X6 @ ( image_nat_rat @ F2 @ A2 ) )
& ( P2 @ X6 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P2 @ ( F2 @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_414_image__iff,axiom,
! [Z2: extended_ereal,F2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ Z2 @ ( image_6042159593519690757_ereal @ F2 @ A2 ) )
= ( ? [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A2 )
& ( Z2
= ( F2 @ X ) ) ) ) ) ).
% image_iff
thf(fact_415_image__iff,axiom,
! [Z2: extended_ereal,F2: nat > extended_ereal,A2: set_nat] :
( ( member2350847679896131959_ereal @ Z2 @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( Z2
= ( F2 @ X ) ) ) ) ) ).
% image_iff
thf(fact_416_image__iff,axiom,
! [Z2: set_nat,F2: nat > set_nat,A2: set_nat] :
( ( member_set_nat @ Z2 @ ( image_nat_set_nat @ F2 @ A2 ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( Z2
= ( F2 @ X ) ) ) ) ) ).
% image_iff
thf(fact_417_image__iff,axiom,
! [Z2: rat,F2: nat > rat,A2: set_nat] :
( ( member_rat @ Z2 @ ( image_nat_rat @ F2 @ A2 ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( Z2
= ( F2 @ X ) ) ) ) ) ).
% image_iff
thf(fact_418_image__iff,axiom,
! [Z2: nat,F2: nat > nat,A2: set_nat] :
( ( member_nat @ Z2 @ ( image_nat_nat @ F2 @ A2 ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( Z2
= ( F2 @ X ) ) ) ) ) ).
% image_iff
thf(fact_419_imageI,axiom,
! [X3: extended_ereal,A2: set_Extended_ereal,F2: extended_ereal > extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A2 )
=> ( member2350847679896131959_ereal @ ( F2 @ X3 ) @ ( image_6042159593519690757_ereal @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_420_imageI,axiom,
! [X3: a,A2: set_a,F2: a > a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ ( F2 @ X3 ) @ ( image_a_a @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_421_imageI,axiom,
! [X3: a,A2: set_a,F2: a > nat] :
( ( member_a @ X3 @ A2 )
=> ( member_nat @ ( F2 @ X3 ) @ ( image_a_nat @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_422_imageI,axiom,
! [X3: nat,A2: set_nat,F2: nat > extended_ereal] :
( ( member_nat @ X3 @ A2 )
=> ( member2350847679896131959_ereal @ ( F2 @ X3 ) @ ( image_4309273772856505399_ereal @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_423_imageI,axiom,
! [X3: nat,A2: set_nat,F2: nat > set_nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_set_nat @ ( F2 @ X3 ) @ ( image_nat_set_nat @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_424_imageI,axiom,
! [X3: nat,A2: set_nat,F2: nat > rat] :
( ( member_nat @ X3 @ A2 )
=> ( member_rat @ ( F2 @ X3 ) @ ( image_nat_rat @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_425_imageI,axiom,
! [X3: nat,A2: set_nat,F2: nat > a] :
( ( member_nat @ X3 @ A2 )
=> ( member_a @ ( F2 @ X3 ) @ ( image_nat_a @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_426_imageI,axiom,
! [X3: nat,A2: set_nat,F2: nat > nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ ( F2 @ X3 ) @ ( image_nat_nat @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_427_Sup_OSUP__cong,axiom,
! [A2: set_Extended_ereal,B3: set_Extended_ereal,C3: extended_ereal > extended_ereal,D2: extended_ereal > extended_ereal,Sup: set_Extended_ereal > extended_ereal] :
( ( A2 = B3 )
=> ( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ B3 )
=> ( ( C3 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Sup @ ( image_6042159593519690757_ereal @ C3 @ A2 ) )
= ( Sup @ ( image_6042159593519690757_ereal @ D2 @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_428_Sup_OSUP__cong,axiom,
! [A2: set_nat,B3: set_nat,C3: nat > extended_ereal,D2: nat > extended_ereal,Sup: set_Extended_ereal > extended_ereal] :
( ( A2 = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C3 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Sup @ ( image_4309273772856505399_ereal @ C3 @ A2 ) )
= ( Sup @ ( image_4309273772856505399_ereal @ D2 @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_429_Sup_OSUP__cong,axiom,
! [A2: set_nat,B3: set_nat,C3: nat > set_nat,D2: nat > set_nat,Sup: set_set_nat > set_nat] :
( ( A2 = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C3 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Sup @ ( image_nat_set_nat @ C3 @ A2 ) )
= ( Sup @ ( image_nat_set_nat @ D2 @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_430_Sup_OSUP__cong,axiom,
! [A2: set_nat,B3: set_nat,C3: nat > nat,D2: nat > nat,Sup: set_nat > nat] :
( ( A2 = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C3 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C3 @ A2 ) )
= ( Sup @ ( image_nat_nat @ D2 @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_431_Sup_OSUP__cong,axiom,
! [A2: set_nat,B3: set_nat,C3: nat > rat,D2: nat > rat,Sup: set_rat > rat] :
( ( A2 = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C3 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Sup @ ( image_nat_rat @ C3 @ A2 ) )
= ( Sup @ ( image_nat_rat @ D2 @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_432_Inf_OINF__cong,axiom,
! [A2: set_Extended_ereal,B3: set_Extended_ereal,C3: extended_ereal > extended_ereal,D2: extended_ereal > extended_ereal,Inf: set_Extended_ereal > extended_ereal] :
( ( A2 = B3 )
=> ( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ B3 )
=> ( ( C3 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Inf @ ( image_6042159593519690757_ereal @ C3 @ A2 ) )
= ( Inf @ ( image_6042159593519690757_ereal @ D2 @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_433_Inf_OINF__cong,axiom,
! [A2: set_nat,B3: set_nat,C3: nat > extended_ereal,D2: nat > extended_ereal,Inf: set_Extended_ereal > extended_ereal] :
( ( A2 = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C3 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Inf @ ( image_4309273772856505399_ereal @ C3 @ A2 ) )
= ( Inf @ ( image_4309273772856505399_ereal @ D2 @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_434_Inf_OINF__cong,axiom,
! [A2: set_nat,B3: set_nat,C3: nat > set_nat,D2: nat > set_nat,Inf: set_set_nat > set_nat] :
( ( A2 = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C3 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Inf @ ( image_nat_set_nat @ C3 @ A2 ) )
= ( Inf @ ( image_nat_set_nat @ D2 @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_435_Inf_OINF__cong,axiom,
! [A2: set_nat,B3: set_nat,C3: nat > nat,D2: nat > nat,Inf: set_nat > nat] :
( ( A2 = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C3 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C3 @ A2 ) )
= ( Inf @ ( image_nat_nat @ D2 @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_436_Inf_OINF__cong,axiom,
! [A2: set_nat,B3: set_nat,C3: nat > rat,D2: nat > rat,Inf: set_rat > rat] :
( ( A2 = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C3 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Inf @ ( image_nat_rat @ C3 @ A2 ) )
= ( Inf @ ( image_nat_rat @ D2 @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_437_range__snd,axiom,
( ( image_2802296252294471260_a_b_b @ product_snd_a_b @ top_to8134405472303993176od_a_b )
= top_top_set_b ) ).
% range_snd
thf(fact_438_range__fst,axiom,
( ( image_2802296252294471259_a_b_a @ product_fst_a_b @ top_to8134405472303993176od_a_b )
= top_top_set_a ) ).
% range_fst
thf(fact_439_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_440_prod_Oinj__map,axiom,
! [F1: extended_ereal > extended_ereal,F22: nat > nat] :
( ( inj_on7162434037990268785_ereal @ F1 @ top_to5683747375963461374_ereal )
=> ( ( inj_on_nat_nat @ F22 @ top_top_set_nat )
=> ( inj_on3854950389080018957al_nat @ ( produc8678206924122515480at_nat @ F1 @ F22 ) @ top_to7896853287916821811al_nat ) ) ) ).
% prod.inj_map
thf(fact_441_prod_Oinj__map,axiom,
! [F1: nat > nat,F22: extended_ereal > extended_ereal] :
( ( inj_on_nat_nat @ F1 @ top_top_set_nat )
=> ( ( inj_on7162434037990268785_ereal @ F22 @ top_to5683747375963461374_ereal )
=> ( inj_on113668994391194201_ereal @ ( produc464594342652746008_ereal @ F1 @ F22 ) @ top_to6634112653661286105_ereal ) ) ) ).
% prod.inj_map
thf(fact_442_prod_Oinj__map,axiom,
! [F1: nat > nat,F22: nat > nat] :
( ( inj_on_nat_nat @ F1 @ top_top_set_nat )
=> ( ( inj_on_nat_nat @ F22 @ top_top_set_nat )
=> ( inj_on8969904277767023793at_nat @ ( produc6977886695330630970at_nat @ F1 @ F22 ) @ top_to4669805908274784177at_nat ) ) ) ).
% prod.inj_map
thf(fact_443_fun_Oinj__map,axiom,
! [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( inj_on2461717442902640625at_nat @ ( comp_nat_nat_nat @ F2 ) @ top_top_set_nat_nat ) ) ).
% fun.inj_map
thf(fact_444_all__subset__image__inj,axiom,
! [F2: nat > set_nat,S2: set_nat,P2: set_set_nat > $o] :
( ( ! [T3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ T3 @ ( image_nat_set_nat @ F2 @ S2 ) )
=> ( P2 @ T3 ) ) )
= ( ! [T3: set_nat] :
( ( ( ord_less_eq_set_nat @ T3 @ S2 )
& ( inj_on_nat_set_nat @ F2 @ T3 ) )
=> ( P2 @ ( image_nat_set_nat @ F2 @ T3 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_445_all__subset__image__inj,axiom,
! [F2: nat > rat,S2: set_nat,P2: set_rat > $o] :
( ( ! [T3: set_rat] :
( ( ord_less_eq_set_rat @ T3 @ ( image_nat_rat @ F2 @ S2 ) )
=> ( P2 @ T3 ) ) )
= ( ! [T3: set_nat] :
( ( ( ord_less_eq_set_nat @ T3 @ S2 )
& ( inj_on_nat_rat @ F2 @ T3 ) )
=> ( P2 @ ( image_nat_rat @ F2 @ T3 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_446_all__subset__image__inj,axiom,
! [F2: nat > nat,S2: set_nat,P2: set_nat > $o] :
( ( ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ ( image_nat_nat @ F2 @ S2 ) )
=> ( P2 @ T3 ) ) )
= ( ! [T3: set_nat] :
( ( ( ord_less_eq_set_nat @ T3 @ S2 )
& ( inj_on_nat_nat @ F2 @ T3 ) )
=> ( P2 @ ( image_nat_nat @ F2 @ T3 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_447_all__subset__image__inj,axiom,
! [F2: nat > extended_ereal,S2: set_nat,P2: set_Extended_ereal > $o] :
( ( ! [T3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ T3 @ ( image_4309273772856505399_ereal @ F2 @ S2 ) )
=> ( P2 @ T3 ) ) )
= ( ! [T3: set_nat] :
( ( ( ord_less_eq_set_nat @ T3 @ S2 )
& ( inj_on6191532827271902155_ereal @ F2 @ T3 ) )
=> ( P2 @ ( image_4309273772856505399_ereal @ F2 @ T3 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_448_all__subset__image__inj,axiom,
! [F2: extended_ereal > extended_ereal,S2: set_Extended_ereal,P2: set_Extended_ereal > $o] :
( ( ! [T3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ T3 @ ( image_6042159593519690757_ereal @ F2 @ S2 ) )
=> ( P2 @ T3 ) ) )
= ( ! [T3: set_Extended_ereal] :
( ( ( ord_le1644982726543182158_ereal @ T3 @ S2 )
& ( inj_on7162434037990268785_ereal @ F2 @ T3 ) )
=> ( P2 @ ( image_6042159593519690757_ereal @ F2 @ T3 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_449_ex__subset__image__inj,axiom,
! [F2: nat > set_nat,S2: set_nat,P2: set_set_nat > $o] :
( ( ? [T3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ T3 @ ( image_nat_set_nat @ F2 @ S2 ) )
& ( P2 @ T3 ) ) )
= ( ? [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S2 )
& ( inj_on_nat_set_nat @ F2 @ T3 )
& ( P2 @ ( image_nat_set_nat @ F2 @ T3 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_450_ex__subset__image__inj,axiom,
! [F2: nat > rat,S2: set_nat,P2: set_rat > $o] :
( ( ? [T3: set_rat] :
( ( ord_less_eq_set_rat @ T3 @ ( image_nat_rat @ F2 @ S2 ) )
& ( P2 @ T3 ) ) )
= ( ? [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S2 )
& ( inj_on_nat_rat @ F2 @ T3 )
& ( P2 @ ( image_nat_rat @ F2 @ T3 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_451_ex__subset__image__inj,axiom,
! [F2: nat > nat,S2: set_nat,P2: set_nat > $o] :
( ( ? [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ ( image_nat_nat @ F2 @ S2 ) )
& ( P2 @ T3 ) ) )
= ( ? [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S2 )
& ( inj_on_nat_nat @ F2 @ T3 )
& ( P2 @ ( image_nat_nat @ F2 @ T3 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_452_ex__subset__image__inj,axiom,
! [F2: nat > extended_ereal,S2: set_nat,P2: set_Extended_ereal > $o] :
( ( ? [T3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ T3 @ ( image_4309273772856505399_ereal @ F2 @ S2 ) )
& ( P2 @ T3 ) ) )
= ( ? [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S2 )
& ( inj_on6191532827271902155_ereal @ F2 @ T3 )
& ( P2 @ ( image_4309273772856505399_ereal @ F2 @ T3 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_453_ex__subset__image__inj,axiom,
! [F2: extended_ereal > extended_ereal,S2: set_Extended_ereal,P2: set_Extended_ereal > $o] :
( ( ? [T3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ T3 @ ( image_6042159593519690757_ereal @ F2 @ S2 ) )
& ( P2 @ T3 ) ) )
= ( ? [T3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ T3 @ S2 )
& ( inj_on7162434037990268785_ereal @ F2 @ T3 )
& ( P2 @ ( image_6042159593519690757_ereal @ F2 @ T3 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_454_subset__image__inj,axiom,
! [S2: set_set_nat,F2: nat > set_nat,T4: set_nat] :
( ( ord_le6893508408891458716et_nat @ S2 @ ( image_nat_set_nat @ F2 @ T4 ) )
= ( ? [U: set_nat] :
( ( ord_less_eq_set_nat @ U @ T4 )
& ( inj_on_nat_set_nat @ F2 @ U )
& ( S2
= ( image_nat_set_nat @ F2 @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_455_subset__image__inj,axiom,
! [S2: set_rat,F2: nat > rat,T4: set_nat] :
( ( ord_less_eq_set_rat @ S2 @ ( image_nat_rat @ F2 @ T4 ) )
= ( ? [U: set_nat] :
( ( ord_less_eq_set_nat @ U @ T4 )
& ( inj_on_nat_rat @ F2 @ U )
& ( S2
= ( image_nat_rat @ F2 @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_456_subset__image__inj,axiom,
! [S2: set_nat,F2: nat > nat,T4: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ ( image_nat_nat @ F2 @ T4 ) )
= ( ? [U: set_nat] :
( ( ord_less_eq_set_nat @ U @ T4 )
& ( inj_on_nat_nat @ F2 @ U )
& ( S2
= ( image_nat_nat @ F2 @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_457_subset__image__inj,axiom,
! [S2: set_Extended_ereal,F2: nat > extended_ereal,T4: set_nat] :
( ( ord_le1644982726543182158_ereal @ S2 @ ( image_4309273772856505399_ereal @ F2 @ T4 ) )
= ( ? [U: set_nat] :
( ( ord_less_eq_set_nat @ U @ T4 )
& ( inj_on6191532827271902155_ereal @ F2 @ U )
& ( S2
= ( image_4309273772856505399_ereal @ F2 @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_458_subset__image__inj,axiom,
! [S2: set_Extended_ereal,F2: extended_ereal > extended_ereal,T4: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ S2 @ ( image_6042159593519690757_ereal @ F2 @ T4 ) )
= ( ? [U: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ U @ T4 )
& ( inj_on7162434037990268785_ereal @ F2 @ U )
& ( S2
= ( image_6042159593519690757_ereal @ F2 @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_459_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_460_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_461_fun_Omap__ident__strong,axiom,
! [T2: extended_ereal > nat,F2: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( image_7659842161140344153al_nat @ T2 @ top_to5683747375963461374_ereal ) )
=> ( ( F2 @ Z3 )
= Z3 ) )
=> ( ( comp_n5886173794813336841_ereal @ F2 @ T2 )
= T2 ) ) ).
% fun.map_ident_strong
thf(fact_462_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_463_fun_Omap__ident__strong,axiom,
! [T2: nat > set_nat,F2: set_nat > set_nat] :
( ! [Z3: set_nat] :
( ( member_set_nat @ Z3 @ ( image_nat_set_nat @ T2 @ top_top_set_nat ) )
=> ( ( F2 @ Z3 )
= Z3 ) )
=> ( ( comp_s3433241188411525313at_nat @ F2 @ T2 )
= T2 ) ) ).
% fun.map_ident_strong
thf(fact_464_fun_Omap__ident__strong,axiom,
! [T2: nat > rat,F2: rat > rat] :
( ! [Z3: rat] :
( ( member_rat @ Z3 @ ( image_nat_rat @ T2 @ top_top_set_nat ) )
=> ( ( F2 @ Z3 )
= Z3 ) )
=> ( ( comp_rat_rat_nat @ F2 @ T2 )
= T2 ) ) ).
% fun.map_ident_strong
thf(fact_465_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_466_fun_Omap__ident__strong,axiom,
! [T2: nat > nat,F2: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( image_nat_nat @ T2 @ top_top_set_nat ) )
=> ( ( F2 @ Z3 )
= Z3 ) )
=> ( ( comp_nat_nat_nat @ F2 @ T2 )
= T2 ) ) ).
% fun.map_ident_strong
thf(fact_467_fun_Omap__ident__strong,axiom,
! [T2: rat > a,F2: a > a] :
( ! [Z3: a] :
( ( member_a @ Z3 @ ( image_rat_a @ T2 @ top_top_set_rat ) )
=> ( ( F2 @ Z3 )
= Z3 ) )
=> ( ( comp_a_a_rat @ F2 @ T2 )
= T2 ) ) ).
% fun.map_ident_strong
thf(fact_468_fun_Omap__ident__strong,axiom,
! [T2: rat > nat,F2: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( image_rat_nat @ T2 @ top_top_set_rat ) )
=> ( ( F2 @ Z3 )
= Z3 ) )
=> ( ( comp_nat_nat_rat @ F2 @ T2 )
= T2 ) ) ).
% fun.map_ident_strong
thf(fact_469_fun_Oinj__map__strong,axiom,
! [X3: nat > nat,Xa2: nat > nat,F2: nat > nat,Fa: nat > nat] :
( ! [Z3: nat,Za: nat] :
( ( member_nat @ Z3 @ ( image_nat_nat @ X3 @ top_top_set_nat ) )
=> ( ( member_nat @ Za @ ( image_nat_nat @ Xa2 @ top_top_set_nat ) )
=> ( ( ( F2 @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( comp_nat_nat_nat @ F2 @ X3 )
= ( comp_nat_nat_nat @ Fa @ Xa2 ) )
=> ( X3 = Xa2 ) ) ) ).
% fun.inj_map_strong
thf(fact_470_fun_Omap__comp,axiom,
! [G: nat > nat,F2: nat > nat,V: nat > nat] :
( ( comp_nat_nat_nat @ G @ ( comp_nat_nat_nat @ F2 @ V ) )
= ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ G @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_471_prod_Omap__comp,axiom,
! [G1: nat > nat,G22: nat > nat,F1: nat > nat,F22: nat > nat,V: product_prod_nat_nat] :
( ( produc6977886695330630970at_nat @ G1 @ G22 @ ( produc6977886695330630970at_nat @ F1 @ F22 @ V ) )
= ( produc6977886695330630970at_nat @ ( comp_nat_nat_nat @ G1 @ F1 ) @ ( comp_nat_nat_nat @ G22 @ F22 ) @ V ) ) ).
% prod.map_comp
thf(fact_472_fun_Oset__map,axiom,
! [F2: nat > nat,V: extended_ereal > nat] :
( ( image_7659842161140344153al_nat @ ( comp_n5886173794813336841_ereal @ F2 @ V ) @ top_to5683747375963461374_ereal )
= ( image_nat_nat @ F2 @ ( image_7659842161140344153al_nat @ V @ top_to5683747375963461374_ereal ) ) ) ).
% fun.set_map
thf(fact_473_fun_Oset__map,axiom,
! [F2: nat > rat,V: extended_ereal > nat] :
( ( image_7024712101053848417al_rat @ ( comp_n4169009346981848321_ereal @ F2 @ V ) @ top_to5683747375963461374_ereal )
= ( image_nat_rat @ F2 @ ( image_7659842161140344153al_nat @ V @ top_to5683747375963461374_ereal ) ) ) ).
% fun.set_map
thf(fact_474_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_475_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_476_fun_Oset__map,axiom,
! [F2: rat > extended_ereal,V: nat > rat] :
( ( image_4309273772856505399_ereal @ ( comp_r4319880827473671715al_nat @ F2 @ V ) @ top_top_set_nat )
= ( image_2592109325025016879_ereal @ F2 @ ( image_nat_rat @ V @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_477_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_478_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_479_fun_Oset__map,axiom,
! [F2: extended_ereal > nat,V: nat > extended_ereal] :
( ( image_nat_nat @ ( comp_E7502005551946643277at_nat @ F2 @ V ) @ top_top_set_nat )
= ( image_7659842161140344153al_nat @ F2 @ ( image_4309273772856505399_ereal @ V @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_480_fun_Oset__map,axiom,
! [F2: rat > nat,V: nat > rat] :
( ( image_nat_nat @ ( comp_rat_nat_nat @ F2 @ V ) @ top_top_set_nat )
= ( image_rat_nat @ F2 @ ( image_nat_rat @ V @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_481_fun_Oset__map,axiom,
! [F2: nat > nat,V: nat > nat] :
( ( image_nat_nat @ ( comp_nat_nat_nat @ F2 @ V ) @ top_top_set_nat )
= ( image_nat_nat @ F2 @ ( image_nat_nat @ V @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_482_fun_Omap__cong,axiom,
! [X3: nat > nat,Ya: nat > nat,F2: nat > nat,G: nat > nat] :
( ( X3 = Ya )
=> ( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( image_nat_nat @ Ya @ top_top_set_nat ) )
=> ( ( F2 @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( comp_nat_nat_nat @ F2 @ X3 )
= ( comp_nat_nat_nat @ G @ Ya ) ) ) ) ).
% fun.map_cong
thf(fact_483_fun_Omap__cong0,axiom,
! [X3: nat > nat,F2: nat > nat,G: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( image_nat_nat @ X3 @ top_top_set_nat ) )
=> ( ( F2 @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( comp_nat_nat_nat @ F2 @ X3 )
= ( comp_nat_nat_nat @ G @ X3 ) ) ) ).
% fun.map_cong0
thf(fact_484_all__subset__image,axiom,
! [F2: nat > set_nat,A2: set_nat,P2: set_set_nat > $o] :
( ( ! [B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ ( image_nat_set_nat @ F2 @ A2 ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( P2 @ ( image_nat_set_nat @ F2 @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_485_all__subset__image,axiom,
! [F2: nat > nat,A2: set_nat,P2: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F2 @ A2 ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( P2 @ ( image_nat_nat @ F2 @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_486_all__subset__image,axiom,
! [F2: nat > rat,A2: set_nat,P2: set_rat > $o] :
( ( ! [B4: set_rat] :
( ( ord_less_eq_set_rat @ B4 @ ( image_nat_rat @ F2 @ A2 ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( P2 @ ( image_nat_rat @ F2 @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_487_all__subset__image,axiom,
! [F2: nat > extended_ereal,A2: set_nat,P2: set_Extended_ereal > $o] :
( ( ! [B4: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B4 @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( P2 @ ( image_4309273772856505399_ereal @ F2 @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_488_all__subset__image,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,P2: set_Extended_ereal > $o] :
( ( ! [B4: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B4 @ ( image_6042159593519690757_ereal @ F2 @ A2 ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B4 @ A2 )
=> ( P2 @ ( image_6042159593519690757_ereal @ F2 @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_489_prod_Osize__gen__o__map,axiom,
! [F2: nat > nat,Fa: nat > nat,G: nat > nat,Ga: nat > nat] :
( ( comp_P1567445206330693457at_nat @ ( basic_876126793109182934at_nat @ F2 @ Fa ) @ ( produc6977886695330630970at_nat @ G @ Ga ) )
= ( basic_876126793109182934at_nat @ ( comp_nat_nat_nat @ F2 @ G ) @ ( comp_nat_nat_nat @ Fa @ Ga ) ) ) ).
% prod.size_gen_o_map
thf(fact_490_inj__image__Compl__subset,axiom,
! [F2: nat > set_nat,A2: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ top_top_set_nat )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ ( uminus5710092332889474511et_nat @ A2 ) ) @ ( uminus613421341184616069et_nat @ ( image_nat_set_nat @ F2 @ A2 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_491_inj__image__Compl__subset,axiom,
! [F2: nat > rat,A2: set_nat] :
( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
=> ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ ( uminus5710092332889474511et_nat @ A2 ) ) @ ( uminus2201863774496077783et_rat @ ( image_nat_rat @ F2 @ A2 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_492_inj__image__Compl__subset,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ ( uminus5710092332889474511et_nat @ A2 ) ) @ ( uminus5710092332889474511et_nat @ ( image_nat_nat @ F2 @ A2 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_493_inj__image__Compl__subset,axiom,
! [F2: nat > extended_ereal,A2: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ top_top_set_nat )
=> ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F2 @ ( uminus5710092332889474511et_nat @ A2 ) ) @ ( uminus5895154729394068773_ereal @ ( image_4309273772856505399_ereal @ F2 @ A2 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_494_inj__image__Compl__subset,axiom,
! [F2: rat > extended_ereal,A2: set_rat] :
( ( inj_on4474368379440413635_ereal @ F2 @ top_top_set_rat )
=> ( ord_le1644982726543182158_ereal @ ( image_2592109325025016879_ereal @ F2 @ ( uminus2201863774496077783et_rat @ A2 ) ) @ ( uminus5895154729394068773_ereal @ ( image_2592109325025016879_ereal @ F2 @ A2 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_495_inj__image__Compl__subset,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F2 @ ( uminus5895154729394068773_ereal @ A2 ) ) @ ( uminus5895154729394068773_ereal @ ( image_6042159593519690757_ereal @ F2 @ A2 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_496_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_497_the__inv__f__o__f__id,axiom,
! [F2: nat > nat,Z2: nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( comp_nat_nat_nat @ ( the_inv_into_nat_nat @ top_top_set_nat @ F2 ) @ F2 @ Z2 )
= ( id_nat @ Z2 ) ) ) ).
% the_inv_f_o_f_id
thf(fact_498_image__id,axiom,
( ( image_6042159593519690757_ereal @ id_Extended_ereal )
= id_set1423601066293951391_ereal ) ).
% image_id
thf(fact_499_image__id,axiom,
( ( image_nat_nat @ id_nat )
= id_set_nat ) ).
% image_id
thf(fact_500_comp__id,axiom,
! [F2: nat > nat] :
( ( comp_nat_nat_nat @ F2 @ id_nat )
= F2 ) ).
% comp_id
thf(fact_501_id__comp,axiom,
! [G: nat > nat] :
( ( comp_nat_nat_nat @ id_nat @ G )
= G ) ).
% id_comp
thf(fact_502_fun_Omap__id,axiom,
! [T2: nat > nat] :
( ( comp_nat_nat_nat @ id_nat @ T2 )
= T2 ) ).
% fun.map_id
thf(fact_503_surj__uminus,axiom,
( ( image_rat_rat @ uminus_uminus_rat @ top_top_set_rat )
= top_top_set_rat ) ).
% surj_uminus
thf(fact_504_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_505_fun_Omap__id0,axiom,
( ( comp_nat_nat_nat @ id_nat )
= id_nat_nat ) ).
% fun.map_id0
thf(fact_506_fun_Omap__transfer,axiom,
! [Rb: nat > nat > $o,Sd: nat > nat > $o] :
( bNF_re3262823321055862553at_nat @ ( bNF_re5653821019739307937at_nat @ Rb @ Sd )
@ ( bNF_re239970166668089693at_nat
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
@ Rb )
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
@ Sd ) )
@ comp_nat_nat_nat
@ comp_nat_nat_nat ) ).
% fun.map_transfer
thf(fact_507_Inf_OINF__id__eq,axiom,
! [Inf: set_Extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( Inf @ ( image_6042159593519690757_ereal @ id_Extended_ereal @ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_id_eq
thf(fact_508_Inf_OINF__id__eq,axiom,
! [Inf: set_nat > nat,A2: set_nat] :
( ( Inf @ ( image_nat_nat @ id_nat @ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_id_eq
thf(fact_509_Sup_OSUP__id__eq,axiom,
! [Sup: set_Extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( Sup @ ( image_6042159593519690757_ereal @ id_Extended_ereal @ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_id_eq
thf(fact_510_Sup_OSUP__id__eq,axiom,
! [Sup: set_nat > nat,A2: set_nat] :
( ( Sup @ ( image_nat_nat @ id_nat @ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_id_eq
thf(fact_511_comp__eq__id__dest,axiom,
! [A: nat > nat,B: nat > nat,C: nat > nat,V: nat] :
( ( ( comp_nat_nat_nat @ A @ B )
= ( comp_nat_nat_nat @ id_nat @ C ) )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_512_inj__on__id,axiom,
! [A2: set_Extended_ereal] : ( inj_on7162434037990268785_ereal @ id_Extended_ereal @ A2 ) ).
% inj_on_id
thf(fact_513_inj__on__id,axiom,
! [A2: set_nat] : ( inj_on_nat_nat @ id_nat @ A2 ) ).
% inj_on_id
thf(fact_514_fun_Orel__cong,axiom,
! [X3: extended_ereal > extended_ereal,Ya: extended_ereal > extended_ereal,Y4: extended_ereal > extended_ereal,Xa2: extended_ereal > extended_ereal,R2: extended_ereal > extended_ereal > $o,Ra: extended_ereal > extended_ereal > $o] :
( ( X3 = Ya )
=> ( ( Y4 = 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
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ R2
@ X3
@ Y4 )
= ( bNF_re3416630401399921757_ereal
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_515_fun_Orel__cong,axiom,
! [X3: extended_ereal > extended_ereal,Ya: extended_ereal > extended_ereal,Y4: extended_ereal > a,Xa2: extended_ereal > a,R2: extended_ereal > a > $o,Ra: extended_ereal > a > $o] :
( ( X3 = Ya )
=> ( ( Y4 = 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
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ R2
@ X3
@ Y4 )
= ( bNF_re5691446141301026317real_a
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_516_fun_Orel__cong,axiom,
! [X3: extended_ereal > extended_ereal,Ya: extended_ereal > extended_ereal,Y4: extended_ereal > nat,Xa2: extended_ereal > nat,R2: extended_ereal > nat > $o,Ra: extended_ereal > nat > $o] :
( ( X3 = Ya )
=> ( ( Y4 = Xa2 )
=> ( ! [Z3: extended_ereal,Yb: nat] :
( ( member2350847679896131959_ereal @ Z3 @ ( image_6042159593519690757_ereal @ Ya @ top_to5683747375963461374_ereal ) )
=> ( ( member_nat @ Yb @ ( image_7659842161140344153al_nat @ Xa2 @ top_to5683747375963461374_ereal ) )
=> ( ( R2 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( bNF_re1087668796686660353al_nat
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ R2
@ X3
@ Y4 )
= ( bNF_re1087668796686660353al_nat
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_517_fun_Orel__cong,axiom,
! [X3: extended_ereal > a,Ya: extended_ereal > a,Y4: extended_ereal > extended_ereal,Xa2: extended_ereal > extended_ereal,R2: a > extended_ereal > $o,Ra: a > extended_ereal > $o] :
( ( X3 = Ya )
=> ( ( Y4 = 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
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ R2
@ X3
@ Y4 )
= ( bNF_re1148940357394609709_ereal
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_518_fun_Orel__cong,axiom,
! [X3: extended_ereal > a,Ya: extended_ereal > a,Y4: extended_ereal > a,Xa2: extended_ereal > a,R2: a > a > $o,Ra: a > a > $o] :
( ( X3 = Ya )
=> ( ( Y4 = 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
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ R2
@ X3
@ Y4 )
= ( bNF_re4205385778126815197al_a_a
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_519_fun_Orel__cong,axiom,
! [X3: extended_ereal > a,Ya: extended_ereal > a,Y4: extended_ereal > nat,Xa2: extended_ereal > nat,R2: a > nat > $o,Ra: a > nat > $o] :
( ( X3 = Ya )
=> ( ( Y4 = Xa2 )
=> ( ! [Z3: a,Yb: nat] :
( ( member_a @ Z3 @ ( image_3724615099042636213real_a @ Ya @ top_to5683747375963461374_ereal ) )
=> ( ( member_nat @ Yb @ ( image_7659842161140344153al_nat @ Xa2 @ top_to5683747375963461374_ereal ) )
=> ( ( R2 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( bNF_re1718308943842807089_a_nat
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ R2
@ X3
@ Y4 )
= ( bNF_re1718308943842807089_a_nat
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_520_fun_Orel__cong,axiom,
! [X3: extended_ereal > nat,Ya: extended_ereal > nat,Y4: extended_ereal > extended_ereal,Xa2: extended_ereal > extended_ereal,R2: nat > extended_ereal > $o,Ra: nat > extended_ereal > $o] :
( ( X3 = Ya )
=> ( ( Y4 = Xa2 )
=> ( ! [Z3: nat,Yb: extended_ereal] :
( ( member_nat @ Z3 @ ( image_7659842161140344153al_nat @ Ya @ top_to5683747375963461374_ereal ) )
=> ( ( member2350847679896131959_ereal @ Yb @ ( image_6042159593519690757_ereal @ Xa2 @ top_to5683747375963461374_ereal ) )
=> ( ( R2 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( bNF_re6960472445257597407_ereal
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ R2
@ X3
@ Y4 )
= ( bNF_re6960472445257597407_ereal
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_521_fun_Orel__cong,axiom,
! [X3: extended_ereal > nat,Ya: extended_ereal > nat,Y4: extended_ereal > a,Xa2: extended_ereal > a,R2: nat > a > $o,Ra: nat > a > $o] :
( ( X3 = Ya )
=> ( ( Y4 = Xa2 )
=> ( ! [Z3: nat,Yb: a] :
( ( member_nat @ Z3 @ ( image_7659842161140344153al_nat @ Ya @ top_to5683747375963461374_ereal ) )
=> ( ( member_a @ Yb @ ( image_3724615099042636213real_a @ Xa2 @ top_to5683747375963461374_ereal ) )
=> ( ( R2 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( bNF_re5750240445021093519_nat_a
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ R2
@ X3
@ Y4 )
= ( bNF_re5750240445021093519_nat_a
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_522_fun_Orel__cong,axiom,
! [X3: extended_ereal > nat,Ya: extended_ereal > nat,Y4: extended_ereal > nat,Xa2: extended_ereal > nat,R2: nat > nat > $o,Ra: nat > nat > $o] :
( ( X3 = Ya )
=> ( ( Y4 = Xa2 )
=> ( ! [Z3: nat,Yb: nat] :
( ( member_nat @ Z3 @ ( image_7659842161140344153al_nat @ Ya @ top_to5683747375963461374_ereal ) )
=> ( ( member_nat @ Yb @ ( image_7659842161140344153al_nat @ Xa2 @ top_to5683747375963461374_ereal ) )
=> ( ( R2 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( bNF_re4327621139402860031at_nat
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ R2
@ X3
@ Y4 )
= ( bNF_re4327621139402860031at_nat
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_523_fun_Orel__cong,axiom,
! [X3: nat > extended_ereal,Ya: nat > extended_ereal,Y4: nat > extended_ereal,Xa2: nat > extended_ereal,R2: extended_ereal > extended_ereal > $o,Ra: extended_ereal > extended_ereal > $o] :
( ( X3 = Ya )
=> ( ( Y4 = 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
@ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
@ R2
@ X3
@ Y4 )
= ( bNF_re5337380594787866367_ereal
@ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_524_fun_Orel__mono__strong,axiom,
! [R2: extended_ereal > extended_ereal > $o,X3: extended_ereal > extended_ereal,Y4: extended_ereal > extended_ereal,Ra: extended_ereal > extended_ereal > $o] :
( ( bNF_re3416630401399921757_ereal
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ R2
@ X3
@ Y4 )
=> ( ! [Z3: extended_ereal,Yb: extended_ereal] :
( ( member2350847679896131959_ereal @ Z3 @ ( image_6042159593519690757_ereal @ X3 @ top_to5683747375963461374_ereal ) )
=> ( ( member2350847679896131959_ereal @ Yb @ ( image_6042159593519690757_ereal @ Y4 @ top_to5683747375963461374_ereal ) )
=> ( ( R2 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re3416630401399921757_ereal
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ Ra
@ X3
@ Y4 ) ) ) ).
% fun.rel_mono_strong
thf(fact_525_fun_Orel__mono__strong,axiom,
! [R2: extended_ereal > a > $o,X3: extended_ereal > extended_ereal,Y4: extended_ereal > a,Ra: extended_ereal > a > $o] :
( ( bNF_re5691446141301026317real_a
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ R2
@ X3
@ Y4 )
=> ( ! [Z3: extended_ereal,Yb: a] :
( ( member2350847679896131959_ereal @ Z3 @ ( image_6042159593519690757_ereal @ X3 @ top_to5683747375963461374_ereal ) )
=> ( ( member_a @ Yb @ ( image_3724615099042636213real_a @ Y4 @ top_to5683747375963461374_ereal ) )
=> ( ( R2 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re5691446141301026317real_a
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ Ra
@ X3
@ Y4 ) ) ) ).
% fun.rel_mono_strong
thf(fact_526_fun_Orel__mono__strong,axiom,
! [R2: extended_ereal > nat > $o,X3: extended_ereal > extended_ereal,Y4: extended_ereal > nat,Ra: extended_ereal > nat > $o] :
( ( bNF_re1087668796686660353al_nat
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ R2
@ X3
@ Y4 )
=> ( ! [Z3: extended_ereal,Yb: nat] :
( ( member2350847679896131959_ereal @ Z3 @ ( image_6042159593519690757_ereal @ X3 @ top_to5683747375963461374_ereal ) )
=> ( ( member_nat @ Yb @ ( image_7659842161140344153al_nat @ Y4 @ top_to5683747375963461374_ereal ) )
=> ( ( R2 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re1087668796686660353al_nat
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ Ra
@ X3
@ Y4 ) ) ) ).
% fun.rel_mono_strong
thf(fact_527_fun_Orel__mono__strong,axiom,
! [R2: a > extended_ereal > $o,X3: extended_ereal > a,Y4: extended_ereal > extended_ereal,Ra: a > extended_ereal > $o] :
( ( bNF_re1148940357394609709_ereal
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ R2
@ X3
@ Y4 )
=> ( ! [Z3: a,Yb: extended_ereal] :
( ( member_a @ Z3 @ ( image_3724615099042636213real_a @ X3 @ top_to5683747375963461374_ereal ) )
=> ( ( member2350847679896131959_ereal @ Yb @ ( image_6042159593519690757_ereal @ Y4 @ top_to5683747375963461374_ereal ) )
=> ( ( R2 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re1148940357394609709_ereal
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ Ra
@ X3
@ Y4 ) ) ) ).
% fun.rel_mono_strong
thf(fact_528_fun_Orel__mono__strong,axiom,
! [R2: a > a > $o,X3: extended_ereal > a,Y4: extended_ereal > a,Ra: a > a > $o] :
( ( bNF_re4205385778126815197al_a_a
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ R2
@ X3
@ Y4 )
=> ( ! [Z3: a,Yb: a] :
( ( member_a @ Z3 @ ( image_3724615099042636213real_a @ X3 @ top_to5683747375963461374_ereal ) )
=> ( ( member_a @ Yb @ ( image_3724615099042636213real_a @ Y4 @ top_to5683747375963461374_ereal ) )
=> ( ( R2 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re4205385778126815197al_a_a
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ Ra
@ X3
@ Y4 ) ) ) ).
% fun.rel_mono_strong
thf(fact_529_fun_Orel__mono__strong,axiom,
! [R2: a > nat > $o,X3: extended_ereal > a,Y4: extended_ereal > nat,Ra: a > nat > $o] :
( ( bNF_re1718308943842807089_a_nat
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ R2
@ X3
@ Y4 )
=> ( ! [Z3: a,Yb: nat] :
( ( member_a @ Z3 @ ( image_3724615099042636213real_a @ X3 @ top_to5683747375963461374_ereal ) )
=> ( ( member_nat @ Yb @ ( image_7659842161140344153al_nat @ Y4 @ top_to5683747375963461374_ereal ) )
=> ( ( R2 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re1718308943842807089_a_nat
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ Ra
@ X3
@ Y4 ) ) ) ).
% fun.rel_mono_strong
thf(fact_530_fun_Orel__mono__strong,axiom,
! [R2: nat > extended_ereal > $o,X3: extended_ereal > nat,Y4: extended_ereal > extended_ereal,Ra: nat > extended_ereal > $o] :
( ( bNF_re6960472445257597407_ereal
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ R2
@ X3
@ Y4 )
=> ( ! [Z3: nat,Yb: extended_ereal] :
( ( member_nat @ Z3 @ ( image_7659842161140344153al_nat @ X3 @ top_to5683747375963461374_ereal ) )
=> ( ( member2350847679896131959_ereal @ Yb @ ( image_6042159593519690757_ereal @ Y4 @ top_to5683747375963461374_ereal ) )
=> ( ( R2 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re6960472445257597407_ereal
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ Ra
@ X3
@ Y4 ) ) ) ).
% fun.rel_mono_strong
thf(fact_531_fun_Orel__mono__strong,axiom,
! [R2: nat > a > $o,X3: extended_ereal > nat,Y4: extended_ereal > a,Ra: nat > a > $o] :
( ( bNF_re5750240445021093519_nat_a
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ R2
@ X3
@ Y4 )
=> ( ! [Z3: nat,Yb: a] :
( ( member_nat @ Z3 @ ( image_7659842161140344153al_nat @ X3 @ top_to5683747375963461374_ereal ) )
=> ( ( member_a @ Yb @ ( image_3724615099042636213real_a @ Y4 @ top_to5683747375963461374_ereal ) )
=> ( ( R2 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re5750240445021093519_nat_a
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ Ra
@ X3
@ Y4 ) ) ) ).
% fun.rel_mono_strong
thf(fact_532_fun_Orel__mono__strong,axiom,
! [R2: nat > nat > $o,X3: extended_ereal > nat,Y4: extended_ereal > nat,Ra: nat > nat > $o] :
( ( bNF_re4327621139402860031at_nat
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ R2
@ X3
@ Y4 )
=> ( ! [Z3: nat,Yb: nat] :
( ( member_nat @ Z3 @ ( image_7659842161140344153al_nat @ X3 @ top_to5683747375963461374_ereal ) )
=> ( ( member_nat @ Yb @ ( image_7659842161140344153al_nat @ Y4 @ top_to5683747375963461374_ereal ) )
=> ( ( R2 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re4327621139402860031at_nat
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ Ra
@ X3
@ Y4 ) ) ) ).
% fun.rel_mono_strong
thf(fact_533_fun_Orel__mono__strong,axiom,
! [R2: extended_ereal > extended_ereal > $o,X3: nat > extended_ereal,Y4: nat > extended_ereal,Ra: extended_ereal > extended_ereal > $o] :
( ( bNF_re5337380594787866367_ereal
@ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
@ R2
@ X3
@ Y4 )
=> ( ! [Z3: extended_ereal,Yb: extended_ereal] :
( ( member2350847679896131959_ereal @ Z3 @ ( image_4309273772856505399_ereal @ X3 @ top_top_set_nat ) )
=> ( ( member2350847679896131959_ereal @ Yb @ ( image_4309273772856505399_ereal @ Y4 @ top_top_set_nat ) )
=> ( ( R2 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re5337380594787866367_ereal
@ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
@ Ra
@ X3
@ Y4 ) ) ) ).
% fun.rel_mono_strong
thf(fact_534_fun_Orel__refl__strong,axiom,
! [X3: extended_ereal > extended_ereal,Ra: extended_ereal > extended_ereal > $o] :
( ! [Z3: extended_ereal] :
( ( member2350847679896131959_ereal @ Z3 @ ( image_6042159593519690757_ereal @ X3 @ top_to5683747375963461374_ereal ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_re3416630401399921757_ereal
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ Ra
@ X3
@ X3 ) ) ).
% fun.rel_refl_strong
thf(fact_535_fun_Orel__refl__strong,axiom,
! [X3: extended_ereal > a,Ra: a > a > $o] :
( ! [Z3: a] :
( ( member_a @ Z3 @ ( image_3724615099042636213real_a @ X3 @ top_to5683747375963461374_ereal ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_re4205385778126815197al_a_a
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ Ra
@ X3
@ X3 ) ) ).
% fun.rel_refl_strong
thf(fact_536_fun_Orel__refl__strong,axiom,
! [X3: extended_ereal > nat,Ra: nat > nat > $o] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( image_7659842161140344153al_nat @ X3 @ top_to5683747375963461374_ereal ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_re4327621139402860031at_nat
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ Ra
@ X3
@ X3 ) ) ).
% fun.rel_refl_strong
thf(fact_537_fun_Orel__refl__strong,axiom,
! [X3: nat > extended_ereal,Ra: extended_ereal > extended_ereal > $o] :
( ! [Z3: extended_ereal] :
( ( member2350847679896131959_ereal @ Z3 @ ( image_4309273772856505399_ereal @ X3 @ top_top_set_nat ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_re5337380594787866367_ereal
@ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
@ Ra
@ X3
@ X3 ) ) ).
% fun.rel_refl_strong
thf(fact_538_fun_Orel__refl__strong,axiom,
! [X3: nat > set_nat,Ra: set_nat > set_nat > $o] :
( ! [Z3: set_nat] :
( ( member_set_nat @ Z3 @ ( image_nat_set_nat @ X3 @ top_top_set_nat ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_re4683002380766094093et_nat
@ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
@ Ra
@ X3
@ X3 ) ) ).
% fun.rel_refl_strong
thf(fact_539_fun_Orel__refl__strong,axiom,
! [X3: nat > rat,Ra: rat > rat > $o] :
( ! [Z3: rat] :
( ( member_rat @ Z3 @ ( image_nat_rat @ X3 @ top_top_set_nat ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_re4702136315717946289at_rat
@ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
@ Ra
@ X3
@ X3 ) ) ).
% fun.rel_refl_strong
thf(fact_540_fun_Orel__refl__strong,axiom,
! [X3: nat > a,Ra: a > a > $o] :
( ! [Z3: a] :
( ( member_a @ Z3 @ ( image_nat_a @ X3 @ top_top_set_nat ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_re4153754443986628735at_a_a
@ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
@ Ra
@ X3
@ X3 ) ) ).
% fun.rel_refl_strong
thf(fact_541_fun_Orel__refl__strong,axiom,
! [X3: nat > nat,Ra: nat > nat > $o] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( image_nat_nat @ X3 @ top_top_set_nat ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_re5653821019739307937at_nat
@ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
@ Ra
@ X3
@ X3 ) ) ).
% fun.rel_refl_strong
thf(fact_542_fun_Orel__refl__strong,axiom,
! [X3: rat > a,Ra: a > a > $o] :
( ! [Z3: a] :
( ( member_a @ Z3 @ ( image_rat_a @ X3 @ top_top_set_rat ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_re8507182716570760335at_a_a
@ ^ [Y3: rat,Z: rat] : ( Y3 = Z )
@ Ra
@ X3
@ X3 ) ) ).
% fun.rel_refl_strong
thf(fact_543_fun_Orel__refl__strong,axiom,
! [X3: rat > nat,Ra: nat > nat > $o] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( image_rat_nat @ X3 @ top_top_set_rat ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_re6557955094579809201at_nat
@ ^ [Y3: rat,Z: rat] : ( Y3 = Z )
@ Ra
@ X3
@ X3 ) ) ).
% fun.rel_refl_strong
thf(fact_544_surj__id,axiom,
( ( image_6042159593519690757_ereal @ id_Extended_ereal @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal ) ).
% surj_id
thf(fact_545_surj__id,axiom,
( ( image_nat_nat @ id_nat @ top_top_set_nat )
= top_top_set_nat ) ).
% surj_id
thf(fact_546_surj__id,axiom,
( ( image_rat_rat @ id_rat @ top_top_set_rat )
= top_top_set_rat ) ).
% surj_id
thf(fact_547_eq__alt,axiom,
( ( ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z ) )
= ( bNF_Gr4257726243395767246_ereal @ top_to5683747375963461374_ereal @ id_Extended_ereal ) ) ).
% eq_alt
thf(fact_548_eq__alt,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( bNF_Grp_nat_nat @ top_top_set_nat @ id_nat ) ) ).
% eq_alt
thf(fact_549_eq__alt,axiom,
( ( ^ [Y3: rat,Z: rat] : ( Y3 = Z ) )
= ( bNF_Grp_rat_rat @ top_top_set_rat @ id_rat ) ) ).
% eq_alt
thf(fact_550_Grp__UNIV__idI,axiom,
! [X3: extended_ereal,Y4: extended_ereal] :
( ( X3 = Y4 )
=> ( bNF_Gr4257726243395767246_ereal @ top_to5683747375963461374_ereal @ id_Extended_ereal @ X3 @ Y4 ) ) ).
% Grp_UNIV_idI
thf(fact_551_Grp__UNIV__idI,axiom,
! [X3: nat,Y4: nat] :
( ( X3 = Y4 )
=> ( bNF_Grp_nat_nat @ top_top_set_nat @ id_nat @ X3 @ Y4 ) ) ).
% Grp_UNIV_idI
thf(fact_552_Grp__UNIV__idI,axiom,
! [X3: rat,Y4: rat] :
( ( X3 = Y4 )
=> ( bNF_Grp_rat_rat @ top_top_set_rat @ id_rat @ X3 @ Y4 ) ) ).
% Grp_UNIV_idI
thf(fact_553_surj__Compl__image__subset,axiom,
! [F2: nat > set_nat,A2: set_nat] :
( ( ( image_nat_set_nat @ F2 @ top_top_set_nat )
= top_top_set_set_nat )
=> ( ord_le6893508408891458716et_nat @ ( uminus613421341184616069et_nat @ ( image_nat_set_nat @ F2 @ A2 ) ) @ ( image_nat_set_nat @ F2 @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_554_surj__Compl__image__subset,axiom,
! [F2: nat > nat,A2: 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 @ A2 ) ) @ ( image_nat_nat @ F2 @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_555_surj__Compl__image__subset,axiom,
! [F2: nat > rat,A2: set_nat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
=> ( ord_less_eq_set_rat @ ( uminus2201863774496077783et_rat @ ( image_nat_rat @ F2 @ A2 ) ) @ ( image_nat_rat @ F2 @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_556_surj__Compl__image__subset,axiom,
! [F2: rat > nat,A2: set_rat] :
( ( ( image_rat_nat @ F2 @ top_top_set_rat )
= top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_rat_nat @ F2 @ A2 ) ) @ ( image_rat_nat @ F2 @ ( uminus2201863774496077783et_rat @ A2 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_557_surj__Compl__image__subset,axiom,
! [F2: rat > rat,A2: set_rat] :
( ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat )
=> ( ord_less_eq_set_rat @ ( uminus2201863774496077783et_rat @ ( image_rat_rat @ F2 @ A2 ) ) @ ( image_rat_rat @ F2 @ ( uminus2201863774496077783et_rat @ A2 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_558_surj__Compl__image__subset,axiom,
! [F2: extended_ereal > nat,A2: 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 @ A2 ) ) @ ( image_7659842161140344153al_nat @ F2 @ ( uminus5895154729394068773_ereal @ A2 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_559_surj__Compl__image__subset,axiom,
! [F2: extended_ereal > rat,A2: set_Extended_ereal] :
( ( ( image_7024712101053848417al_rat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_rat )
=> ( ord_less_eq_set_rat @ ( uminus2201863774496077783et_rat @ ( image_7024712101053848417al_rat @ F2 @ A2 ) ) @ ( image_7024712101053848417al_rat @ F2 @ ( uminus5895154729394068773_ereal @ A2 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_560_surj__Compl__image__subset,axiom,
! [F2: nat > extended_ereal,A2: set_nat] :
( ( ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat )
= top_to5683747375963461374_ereal )
=> ( ord_le1644982726543182158_ereal @ ( uminus5895154729394068773_ereal @ ( image_4309273772856505399_ereal @ F2 @ A2 ) ) @ ( image_4309273772856505399_ereal @ F2 @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_561_surj__Compl__image__subset,axiom,
! [F2: rat > extended_ereal,A2: set_rat] :
( ( ( image_2592109325025016879_ereal @ F2 @ top_top_set_rat )
= top_to5683747375963461374_ereal )
=> ( ord_le1644982726543182158_ereal @ ( uminus5895154729394068773_ereal @ ( image_2592109325025016879_ereal @ F2 @ A2 ) ) @ ( image_2592109325025016879_ereal @ F2 @ ( uminus2201863774496077783et_rat @ A2 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_562_surj__Compl__image__subset,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ( ord_le1644982726543182158_ereal @ ( uminus5895154729394068773_ereal @ ( image_6042159593519690757_ereal @ F2 @ A2 ) ) @ ( image_6042159593519690757_ereal @ F2 @ ( uminus5895154729394068773_ereal @ A2 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_563_map__prod__o__convol,axiom,
! [H1: nat > nat,H2: nat > nat,F2: nat > nat,G: nat > nat] :
( ( comp_P7430769850142754163at_nat @ ( produc6977886695330630970at_nat @ H1 @ H2 ) @ ( bNF_co805650143699787099at_nat @ F2 @ G ) )
= ( bNF_co805650143699787099at_nat @ ( comp_nat_nat_nat @ H1 @ F2 ) @ ( comp_nat_nat_nat @ H2 @ G ) ) ) ).
% map_prod_o_convol
thf(fact_564_convol__o,axiom,
! [F2: nat > nat,G: nat > nat,H: nat > nat] :
( ( comp_n8574565218330151774at_nat @ ( bNF_co805650143699787099at_nat @ F2 @ G ) @ H )
= ( bNF_co805650143699787099at_nat @ ( comp_nat_nat_nat @ F2 @ H ) @ ( comp_nat_nat_nat @ G @ H ) ) ) ).
% convol_o
thf(fact_565_type__copy__map__cong0,axiom,
! [M: nat > nat,G: nat > nat,X3: nat,N: nat > nat,H: nat > nat,F2: nat > nat] :
( ( ( M @ ( G @ X3 ) )
= ( N @ ( H @ X3 ) ) )
=> ( ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ F2 @ M ) @ G @ X3 )
= ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ F2 @ N ) @ H @ X3 ) ) ) ).
% type_copy_map_cong0
thf(fact_566_rewriteL__comp__comp,axiom,
! [F2: nat > nat,G: nat > nat,L: nat > nat,H: nat > nat] :
( ( ( comp_nat_nat_nat @ F2 @ G )
= L )
=> ( ( comp_nat_nat_nat @ F2 @ ( comp_nat_nat_nat @ G @ H ) )
= ( comp_nat_nat_nat @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_567_rewriteR__comp__comp,axiom,
! [G: nat > nat,H: nat > nat,R: nat > nat,F2: nat > nat] :
( ( ( comp_nat_nat_nat @ G @ H )
= R )
=> ( ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ F2 @ G ) @ H )
= ( comp_nat_nat_nat @ F2 @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_568_rewriteL__comp__comp2,axiom,
! [F2: nat > nat,G: nat > nat,L1: nat > nat,L2: nat > nat,H: nat > nat,R: nat > nat] :
( ( ( comp_nat_nat_nat @ F2 @ G )
= ( comp_nat_nat_nat @ L1 @ L2 ) )
=> ( ( ( comp_nat_nat_nat @ L2 @ H )
= R )
=> ( ( comp_nat_nat_nat @ F2 @ ( comp_nat_nat_nat @ G @ H ) )
= ( comp_nat_nat_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_569_rewriteR__comp__comp2,axiom,
! [G: nat > nat,H: nat > nat,R1: nat > nat,R22: nat > nat,F2: nat > nat,L: nat > nat] :
( ( ( comp_nat_nat_nat @ G @ H )
= ( comp_nat_nat_nat @ R1 @ R22 ) )
=> ( ( ( comp_nat_nat_nat @ F2 @ R1 )
= L )
=> ( ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ F2 @ G ) @ H )
= ( comp_nat_nat_nat @ L @ R22 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_570_comp__transfer,axiom,
! [B3: nat > nat > $o,C3: nat > nat > $o,A2: nat > nat > $o] : ( bNF_re3262823321055862553at_nat @ ( bNF_re5653821019739307937at_nat @ B3 @ C3 ) @ ( bNF_re239970166668089693at_nat @ ( bNF_re5653821019739307937at_nat @ A2 @ B3 ) @ ( bNF_re5653821019739307937at_nat @ A2 @ C3 ) ) @ comp_nat_nat_nat @ comp_nat_nat_nat ) ).
% comp_transfer
thf(fact_571_pointfree__idE,axiom,
! [F2: nat > nat,G: nat > nat,X3: nat] :
( ( ( comp_nat_nat_nat @ F2 @ G )
= id_nat )
=> ( ( F2 @ ( G @ X3 ) )
= X3 ) ) ).
% pointfree_idE
thf(fact_572_o__rsp_I1_J,axiom,
! [R23: nat > nat > $o,R3: nat > nat > $o,R12: nat > nat > $o] : ( bNF_re3262823321055862553at_nat @ ( bNF_re5653821019739307937at_nat @ R23 @ R3 ) @ ( bNF_re239970166668089693at_nat @ ( bNF_re5653821019739307937at_nat @ R12 @ R23 ) @ ( bNF_re5653821019739307937at_nat @ R12 @ R3 ) ) @ comp_nat_nat_nat @ comp_nat_nat_nat ) ).
% o_rsp(1)
thf(fact_573_o__rsp_I2_J,axiom,
! [R12: nat > nat > $o] :
( bNF_re3262823321055862553at_nat
@ ^ [Y3: nat > nat,Z: nat > nat] : ( Y3 = Z )
@ ( bNF_re239970166668089693at_nat
@ ( bNF_re5653821019739307937at_nat @ R12
@ ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
@ ( bNF_re5653821019739307937at_nat @ R12
@ ^ [Y3: nat,Z: nat] : ( Y3 = Z ) ) )
@ comp_nat_nat_nat
@ comp_nat_nat_nat ) ).
% o_rsp(2)
thf(fact_574_isomorphism__expand,axiom,
! [F2: nat > nat,G: nat > nat] :
( ( ( ( comp_nat_nat_nat @ F2 @ G )
= id_nat )
& ( ( comp_nat_nat_nat @ G @ F2 )
= id_nat ) )
= ( ! [X: nat] :
( ( F2 @ ( G @ X ) )
= X )
& ! [X: nat] :
( ( G @ ( F2 @ X ) )
= X ) ) ) ).
% isomorphism_expand
thf(fact_575_left__right__inverse__eq,axiom,
! [F2: nat > nat,G: nat > nat,H: nat > nat] :
( ( ( comp_nat_nat_nat @ F2 @ G )
= id_nat )
=> ( ( ( comp_nat_nat_nat @ G @ H )
= id_nat )
=> ( F2 = H ) ) ) ).
% left_right_inverse_eq
thf(fact_576_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_577_inv__o__cancel,axiom,
! [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( comp_nat_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 ) @ F2 )
= id_nat ) ) ).
% inv_o_cancel
thf(fact_578_inv__into__f__f,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,X3: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A2 )
=> ( ( member2350847679896131959_ereal @ X3 @ A2 )
=> ( ( hilber7422611030134141132_ereal @ A2 @ F2 @ ( F2 @ X3 ) )
= X3 ) ) ) ).
% inv_into_f_f
thf(fact_579_inv__into__f__f,axiom,
! [F2: nat > nat,A2: set_nat,X3: nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( member_nat @ X3 @ A2 )
=> ( ( hilber3633877196798814958at_nat @ A2 @ F2 @ ( F2 @ X3 ) )
= X3 ) ) ) ).
% inv_into_f_f
thf(fact_580_inv__into__image__cancel,axiom,
! [F2: nat > extended_ereal,A2: set_nat,S2: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ A2 )
=> ( ( ord_less_eq_set_nat @ S2 @ A2 )
=> ( ( image_7659842161140344153al_nat @ ( hilber6822286039850061104_ereal @ A2 @ F2 ) @ ( image_4309273772856505399_ereal @ F2 @ S2 ) )
= S2 ) ) ) ).
% inv_into_image_cancel
thf(fact_581_inv__into__image__cancel,axiom,
! [F2: nat > set_nat,A2: set_nat,S2: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ A2 )
=> ( ( ord_less_eq_set_nat @ S2 @ A2 )
=> ( ( image_set_nat_nat @ ( hilber7337601069305372324et_nat @ A2 @ F2 ) @ ( image_nat_set_nat @ F2 @ S2 ) )
= S2 ) ) ) ).
% inv_into_image_cancel
thf(fact_582_inv__into__image__cancel,axiom,
! [F2: nat > rat,A2: set_nat,S2: set_nat] :
( ( inj_on_nat_rat @ F2 @ A2 )
=> ( ( ord_less_eq_set_nat @ S2 @ A2 )
=> ( ( image_rat_nat @ ( hilber2998747136712319222at_rat @ A2 @ F2 ) @ ( image_nat_rat @ F2 @ S2 ) )
= S2 ) ) ) ).
% inv_into_image_cancel
thf(fact_583_inv__into__image__cancel,axiom,
! [F2: set_nat > nat,A2: set_set_nat,S2: set_set_nat] :
( ( inj_on_set_nat_nat @ F2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ S2 @ A2 )
=> ( ( image_nat_set_nat @ ( hilber3771966164650003108at_nat @ A2 @ F2 ) @ ( image_set_nat_nat @ F2 @ S2 ) )
= S2 ) ) ) ).
% inv_into_image_cancel
thf(fact_584_inv__into__image__cancel,axiom,
! [F2: rat > nat,A2: set_rat,S2: set_rat] :
( ( inj_on_rat_nat @ F2 @ A2 )
=> ( ( ord_less_eq_set_rat @ S2 @ A2 )
=> ( ( image_nat_rat @ ( hilber3317322552863949046at_nat @ A2 @ F2 ) @ ( image_rat_nat @ F2 @ S2 ) )
= S2 ) ) ) ).
% inv_into_image_cancel
thf(fact_585_inv__into__image__cancel,axiom,
! [F2: nat > nat,A2: set_nat,S2: set_nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( ord_less_eq_set_nat @ S2 @ A2 )
=> ( ( image_nat_nat @ ( hilber3633877196798814958at_nat @ A2 @ F2 ) @ ( image_nat_nat @ F2 @ S2 ) )
= S2 ) ) ) ).
% inv_into_image_cancel
thf(fact_586_inv__into__image__cancel,axiom,
! [F2: extended_ereal > nat,A2: set_Extended_ereal,S2: set_Extended_ereal] :
( ( inj_on318729178700965101al_nat @ F2 @ A2 )
=> ( ( ord_le1644982726543182158_ereal @ S2 @ A2 )
=> ( ( image_4309273772856505399_ereal @ ( hilber949482391279124050al_nat @ A2 @ F2 ) @ ( image_7659842161140344153al_nat @ F2 @ S2 ) )
= S2 ) ) ) ).
% inv_into_image_cancel
thf(fact_587_inv__into__image__cancel,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,S2: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A2 )
=> ( ( ord_le1644982726543182158_ereal @ S2 @ A2 )
=> ( ( image_6042159593519690757_ereal @ ( hilber7422611030134141132_ereal @ A2 @ F2 ) @ ( image_6042159593519690757_ereal @ F2 @ S2 ) )
= S2 ) ) ) ).
% inv_into_image_cancel
thf(fact_588_o__inv__o__cancel,axiom,
! [F2: nat > nat,G: nat > nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ G @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 ) ) @ F2 )
= G ) ) ).
% o_inv_o_cancel
thf(fact_589_f__inv__into__f,axiom,
! [Y4: extended_ereal,F2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ Y4 @ ( image_6042159593519690757_ereal @ F2 @ A2 ) )
=> ( ( F2 @ ( hilber7422611030134141132_ereal @ A2 @ F2 @ Y4 ) )
= Y4 ) ) ).
% f_inv_into_f
thf(fact_590_f__inv__into__f,axiom,
! [Y4: extended_ereal,F2: nat > extended_ereal,A2: set_nat] :
( ( member2350847679896131959_ereal @ Y4 @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
=> ( ( F2 @ ( hilber6822286039850061104_ereal @ A2 @ F2 @ Y4 ) )
= Y4 ) ) ).
% f_inv_into_f
thf(fact_591_f__inv__into__f,axiom,
! [Y4: set_nat,F2: nat > set_nat,A2: set_nat] :
( ( member_set_nat @ Y4 @ ( image_nat_set_nat @ F2 @ A2 ) )
=> ( ( F2 @ ( hilber7337601069305372324et_nat @ A2 @ F2 @ Y4 ) )
= Y4 ) ) ).
% f_inv_into_f
thf(fact_592_f__inv__into__f,axiom,
! [Y4: rat,F2: nat > rat,A2: set_nat] :
( ( member_rat @ Y4 @ ( image_nat_rat @ F2 @ A2 ) )
=> ( ( F2 @ ( hilber2998747136712319222at_rat @ A2 @ F2 @ Y4 ) )
= Y4 ) ) ).
% f_inv_into_f
thf(fact_593_f__inv__into__f,axiom,
! [Y4: nat,F2: nat > nat,A2: set_nat] :
( ( member_nat @ Y4 @ ( image_nat_nat @ F2 @ A2 ) )
=> ( ( F2 @ ( hilber3633877196798814958at_nat @ A2 @ F2 @ Y4 ) )
= Y4 ) ) ).
% f_inv_into_f
thf(fact_594_inv__into__into,axiom,
! [X3: extended_ereal,F2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ ( image_6042159593519690757_ereal @ F2 @ A2 ) )
=> ( member2350847679896131959_ereal @ ( hilber7422611030134141132_ereal @ A2 @ F2 @ X3 ) @ A2 ) ) ).
% inv_into_into
thf(fact_595_inv__into__into,axiom,
! [X3: extended_ereal,F2: nat > extended_ereal,A2: set_nat] :
( ( member2350847679896131959_ereal @ X3 @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
=> ( member_nat @ ( hilber6822286039850061104_ereal @ A2 @ F2 @ X3 ) @ A2 ) ) ).
% inv_into_into
thf(fact_596_inv__into__into,axiom,
! [X3: set_nat,F2: nat > set_nat,A2: set_nat] :
( ( member_set_nat @ X3 @ ( image_nat_set_nat @ F2 @ A2 ) )
=> ( member_nat @ ( hilber7337601069305372324et_nat @ A2 @ F2 @ X3 ) @ A2 ) ) ).
% inv_into_into
thf(fact_597_inv__into__into,axiom,
! [X3: rat,F2: nat > rat,A2: set_nat] :
( ( member_rat @ X3 @ ( image_nat_rat @ F2 @ A2 ) )
=> ( member_nat @ ( hilber2998747136712319222at_rat @ A2 @ F2 @ X3 ) @ A2 ) ) ).
% inv_into_into
thf(fact_598_inv__into__into,axiom,
! [X3: a,F2: a > a,A2: set_a] :
( ( member_a @ X3 @ ( image_a_a @ F2 @ A2 ) )
=> ( member_a @ ( hilbert_inv_into_a_a @ A2 @ F2 @ X3 ) @ A2 ) ) ).
% inv_into_into
thf(fact_599_inv__into__into,axiom,
! [X3: a,F2: nat > a,A2: set_nat] :
( ( member_a @ X3 @ ( image_nat_a @ F2 @ A2 ) )
=> ( member_nat @ ( hilber2795491120104822624_nat_a @ A2 @ F2 @ X3 ) @ A2 ) ) ).
% inv_into_into
thf(fact_600_inv__into__into,axiom,
! [X3: nat,F2: a > nat,A2: set_a] :
( ( member_nat @ X3 @ ( image_a_nat @ F2 @ A2 ) )
=> ( member_a @ ( hilber7986931655781312002_a_nat @ A2 @ F2 @ X3 ) @ A2 ) ) ).
% inv_into_into
thf(fact_601_inv__into__into,axiom,
! [X3: nat,F2: nat > nat,A2: set_nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F2 @ A2 ) )
=> ( member_nat @ ( hilber3633877196798814958at_nat @ A2 @ F2 @ X3 ) @ A2 ) ) ).
% inv_into_into
thf(fact_602_inv__into__injective,axiom,
! [A2: set_Extended_ereal,F2: extended_ereal > extended_ereal,X3: extended_ereal,Y4: extended_ereal] :
( ( ( hilber7422611030134141132_ereal @ A2 @ F2 @ X3 )
= ( hilber7422611030134141132_ereal @ A2 @ F2 @ Y4 ) )
=> ( ( member2350847679896131959_ereal @ X3 @ ( image_6042159593519690757_ereal @ F2 @ A2 ) )
=> ( ( member2350847679896131959_ereal @ Y4 @ ( image_6042159593519690757_ereal @ F2 @ A2 ) )
=> ( X3 = Y4 ) ) ) ) ).
% inv_into_injective
thf(fact_603_inv__into__injective,axiom,
! [A2: set_nat,F2: nat > extended_ereal,X3: extended_ereal,Y4: extended_ereal] :
( ( ( hilber6822286039850061104_ereal @ A2 @ F2 @ X3 )
= ( hilber6822286039850061104_ereal @ A2 @ F2 @ Y4 ) )
=> ( ( member2350847679896131959_ereal @ X3 @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
=> ( ( member2350847679896131959_ereal @ Y4 @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
=> ( X3 = Y4 ) ) ) ) ).
% inv_into_injective
thf(fact_604_inv__into__injective,axiom,
! [A2: set_nat,F2: nat > set_nat,X3: set_nat,Y4: set_nat] :
( ( ( hilber7337601069305372324et_nat @ A2 @ F2 @ X3 )
= ( hilber7337601069305372324et_nat @ A2 @ F2 @ Y4 ) )
=> ( ( member_set_nat @ X3 @ ( image_nat_set_nat @ F2 @ A2 ) )
=> ( ( member_set_nat @ Y4 @ ( image_nat_set_nat @ F2 @ A2 ) )
=> ( X3 = Y4 ) ) ) ) ).
% inv_into_injective
thf(fact_605_inv__into__injective,axiom,
! [A2: set_nat,F2: nat > rat,X3: rat,Y4: rat] :
( ( ( hilber2998747136712319222at_rat @ A2 @ F2 @ X3 )
= ( hilber2998747136712319222at_rat @ A2 @ F2 @ Y4 ) )
=> ( ( member_rat @ X3 @ ( image_nat_rat @ F2 @ A2 ) )
=> ( ( member_rat @ Y4 @ ( image_nat_rat @ F2 @ A2 ) )
=> ( X3 = Y4 ) ) ) ) ).
% inv_into_injective
thf(fact_606_inv__into__injective,axiom,
! [A2: set_nat,F2: nat > nat,X3: nat,Y4: nat] :
( ( ( hilber3633877196798814958at_nat @ A2 @ F2 @ X3 )
= ( hilber3633877196798814958at_nat @ A2 @ F2 @ Y4 ) )
=> ( ( member_nat @ X3 @ ( image_nat_nat @ F2 @ A2 ) )
=> ( ( member_nat @ Y4 @ ( image_nat_nat @ F2 @ A2 ) )
=> ( X3 = Y4 ) ) ) ) ).
% inv_into_injective
thf(fact_607_inv__into__f__eq,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,X3: extended_ereal,Y4: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A2 )
=> ( ( member2350847679896131959_ereal @ X3 @ A2 )
=> ( ( ( F2 @ X3 )
= Y4 )
=> ( ( hilber7422611030134141132_ereal @ A2 @ F2 @ Y4 )
= X3 ) ) ) ) ).
% inv_into_f_eq
thf(fact_608_inv__into__f__eq,axiom,
! [F2: nat > nat,A2: set_nat,X3: nat,Y4: nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( member_nat @ X3 @ A2 )
=> ( ( ( F2 @ X3 )
= Y4 )
=> ( ( hilber3633877196798814958at_nat @ A2 @ F2 @ Y4 )
= X3 ) ) ) ) ).
% inv_into_f_eq
thf(fact_609_comp__fun__commute__on__def,axiom,
( finite6263006670422244734_a_nat
= ( ^ [S3: set_a,F: a > nat > nat] :
! [X: a,Y: a] :
( ( member_a @ X @ S3 )
=> ( ( member_a @ Y @ S3 )
=> ( ( comp_nat_nat_nat @ ( F @ Y ) @ ( F @ X ) )
= ( comp_nat_nat_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ) ).
% comp_fun_commute_on_def
thf(fact_610_comp__fun__commute__on__def,axiom,
( finite3582905537739598962at_nat
= ( ^ [S3: set_nat,F: nat > nat > nat] :
! [X: nat,Y: nat] :
( ( member_nat @ X @ S3 )
=> ( ( member_nat @ Y @ S3 )
=> ( ( comp_nat_nat_nat @ ( F @ Y ) @ ( F @ X ) )
= ( comp_nat_nat_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ) ).
% comp_fun_commute_on_def
thf(fact_611_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
! [S2: set_a,F2: a > nat > nat,X3: a,Y4: a] :
( ( finite6263006670422244734_a_nat @ S2 @ F2 )
=> ( ( member_a @ X3 @ S2 )
=> ( ( member_a @ Y4 @ S2 )
=> ( ( comp_nat_nat_nat @ ( F2 @ Y4 ) @ ( F2 @ X3 ) )
= ( comp_nat_nat_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) ) ) ) ) ).
% comp_fun_commute_on.comp_fun_commute_on
thf(fact_612_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
! [S2: set_nat,F2: nat > nat > nat,X3: nat,Y4: nat] :
( ( finite3582905537739598962at_nat @ S2 @ F2 )
=> ( ( member_nat @ X3 @ S2 )
=> ( ( member_nat @ Y4 @ S2 )
=> ( ( comp_nat_nat_nat @ ( F2 @ Y4 ) @ ( F2 @ X3 ) )
= ( comp_nat_nat_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) ) ) ) ) ).
% comp_fun_commute_on.comp_fun_commute_on
thf(fact_613_comp__fun__commute__on_Ocommute__left__comp,axiom,
! [S2: set_a,F2: a > nat > nat,X3: a,Y4: a,G: nat > nat] :
( ( finite6263006670422244734_a_nat @ S2 @ F2 )
=> ( ( member_a @ X3 @ S2 )
=> ( ( member_a @ Y4 @ S2 )
=> ( ( comp_nat_nat_nat @ ( F2 @ Y4 ) @ ( comp_nat_nat_nat @ ( F2 @ X3 ) @ G ) )
= ( comp_nat_nat_nat @ ( F2 @ X3 ) @ ( comp_nat_nat_nat @ ( F2 @ Y4 ) @ G ) ) ) ) ) ) ).
% comp_fun_commute_on.commute_left_comp
thf(fact_614_comp__fun__commute__on_Ocommute__left__comp,axiom,
! [S2: set_nat,F2: nat > nat > nat,X3: nat,Y4: nat,G: nat > nat] :
( ( finite3582905537739598962at_nat @ S2 @ F2 )
=> ( ( member_nat @ X3 @ S2 )
=> ( ( member_nat @ Y4 @ S2 )
=> ( ( comp_nat_nat_nat @ ( F2 @ Y4 ) @ ( comp_nat_nat_nat @ ( F2 @ X3 ) @ G ) )
= ( comp_nat_nat_nat @ ( F2 @ X3 ) @ ( comp_nat_nat_nat @ ( F2 @ Y4 ) @ G ) ) ) ) ) ) ).
% comp_fun_commute_on.commute_left_comp
thf(fact_615_comp__fun__commute__on_Ointro,axiom,
! [S2: set_a,F2: a > nat > nat] :
( ! [X2: a,Y2: a] :
( ( member_a @ X2 @ S2 )
=> ( ( member_a @ Y2 @ S2 )
=> ( ( comp_nat_nat_nat @ ( F2 @ Y2 ) @ ( F2 @ X2 ) )
= ( comp_nat_nat_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) ) ) )
=> ( finite6263006670422244734_a_nat @ S2 @ F2 ) ) ).
% comp_fun_commute_on.intro
thf(fact_616_comp__fun__commute__on_Ointro,axiom,
! [S2: set_nat,F2: nat > nat > nat] :
( ! [X2: nat,Y2: nat] :
( ( member_nat @ X2 @ S2 )
=> ( ( member_nat @ Y2 @ S2 )
=> ( ( comp_nat_nat_nat @ ( F2 @ Y2 ) @ ( F2 @ X2 ) )
= ( comp_nat_nat_nat @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) ) ) )
=> ( finite3582905537739598962at_nat @ S2 @ F2 ) ) ).
% comp_fun_commute_on.intro
thf(fact_617_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 )
=> ( ! [X2: extended_ereal] :
( ( G @ ( F2 @ X2 ) )
= X2 )
=> ( ( hilber7422611030134141132_ereal @ top_to5683747375963461374_ereal @ F2 )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_618_surj__imp__inv__eq,axiom,
! [F2: extended_ereal > nat,G: nat > extended_ereal] :
( ( ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_nat )
=> ( ! [X2: extended_ereal] :
( ( G @ ( F2 @ X2 ) )
= X2 )
=> ( ( hilber949482391279124050al_nat @ top_to5683747375963461374_ereal @ F2 )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_619_surj__imp__inv__eq,axiom,
! [F2: extended_ereal > rat,G: rat > extended_ereal] :
( ( ( image_7024712101053848417al_rat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_rat )
=> ( ! [X2: extended_ereal] :
( ( G @ ( F2 @ X2 ) )
= X2 )
=> ( ( hilber314352331192628314al_rat @ top_to5683747375963461374_ereal @ F2 )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_620_surj__imp__inv__eq,axiom,
! [F2: nat > set_nat,G: set_nat > nat] :
( ( ( image_nat_set_nat @ F2 @ top_top_set_nat )
= top_top_set_set_nat )
=> ( ! [X2: nat] :
( ( G @ ( F2 @ X2 ) )
= X2 )
=> ( ( hilber7337601069305372324et_nat @ top_top_set_nat @ F2 )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_621_surj__imp__inv__eq,axiom,
! [F2: nat > extended_ereal,G: extended_ereal > nat] :
( ( ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat )
= top_to5683747375963461374_ereal )
=> ( ! [X2: nat] :
( ( G @ ( F2 @ X2 ) )
= X2 )
=> ( ( hilber6822286039850061104_ereal @ top_top_set_nat @ F2 )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_622_surj__imp__inv__eq,axiom,
! [F2: nat > nat,G: nat > nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ! [X2: nat] :
( ( G @ ( F2 @ X2 ) )
= X2 )
=> ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_623_surj__imp__inv__eq,axiom,
! [F2: nat > rat,G: rat > nat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
=> ( ! [X2: nat] :
( ( G @ ( F2 @ X2 ) )
= X2 )
=> ( ( hilber2998747136712319222at_rat @ top_top_set_nat @ F2 )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_624_surj__imp__inv__eq,axiom,
! [F2: rat > extended_ereal,G: extended_ereal > rat] :
( ( ( image_2592109325025016879_ereal @ F2 @ top_top_set_rat )
= top_to5683747375963461374_ereal )
=> ( ! [X2: rat] :
( ( G @ ( F2 @ X2 ) )
= X2 )
=> ( ( hilber5105121592018572584_ereal @ top_top_set_rat @ F2 )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_625_surj__imp__inv__eq,axiom,
! [F2: rat > nat,G: nat > rat] :
( ( ( image_rat_nat @ F2 @ top_top_set_rat )
= top_top_set_nat )
=> ( ! [X2: rat] :
( ( G @ ( F2 @ X2 ) )
= X2 )
=> ( ( hilber3317322552863949046at_nat @ top_top_set_rat @ F2 )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_626_surj__imp__inv__eq,axiom,
! [F2: rat > rat,G: rat > rat] :
( ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat )
=> ( ! [X2: rat] :
( ( G @ ( F2 @ X2 ) )
= X2 )
=> ( ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_627_image__f__inv__f,axiom,
! [F2: extended_ereal > extended_ereal,A2: 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 ) @ A2 ) )
= A2 ) ) ).
% image_f_inv_f
thf(fact_628_image__f__inv__f,axiom,
! [F2: extended_ereal > nat,A2: 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 ) @ A2 ) )
= A2 ) ) ).
% image_f_inv_f
thf(fact_629_image__f__inv__f,axiom,
! [F2: extended_ereal > rat,A2: set_rat] :
( ( ( image_7024712101053848417al_rat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_rat )
=> ( ( image_7024712101053848417al_rat @ F2 @ ( image_2592109325025016879_ereal @ ( hilber314352331192628314al_rat @ top_to5683747375963461374_ereal @ F2 ) @ A2 ) )
= A2 ) ) ).
% image_f_inv_f
thf(fact_630_image__f__inv__f,axiom,
! [F2: nat > extended_ereal,A2: 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 ) @ A2 ) )
= A2 ) ) ).
% image_f_inv_f
thf(fact_631_image__f__inv__f,axiom,
! [F2: nat > nat,A2: 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 ) @ A2 ) )
= A2 ) ) ).
% image_f_inv_f
thf(fact_632_image__f__inv__f,axiom,
! [F2: nat > rat,A2: set_rat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
=> ( ( image_nat_rat @ F2 @ ( image_rat_nat @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F2 ) @ A2 ) )
= A2 ) ) ).
% image_f_inv_f
thf(fact_633_image__f__inv__f,axiom,
! [F2: rat > extended_ereal,A2: set_Extended_ereal] :
( ( ( image_2592109325025016879_ereal @ F2 @ top_top_set_rat )
= top_to5683747375963461374_ereal )
=> ( ( image_2592109325025016879_ereal @ F2 @ ( image_7024712101053848417al_rat @ ( hilber5105121592018572584_ereal @ top_top_set_rat @ F2 ) @ A2 ) )
= A2 ) ) ).
% image_f_inv_f
thf(fact_634_image__f__inv__f,axiom,
! [F2: rat > nat,A2: set_nat] :
( ( ( image_rat_nat @ F2 @ top_top_set_rat )
= top_top_set_nat )
=> ( ( image_rat_nat @ F2 @ ( image_nat_rat @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F2 ) @ A2 ) )
= A2 ) ) ).
% image_f_inv_f
thf(fact_635_image__f__inv__f,axiom,
! [F2: rat > rat,A2: set_rat] :
( ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat )
=> ( ( image_rat_rat @ F2 @ ( image_rat_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 ) @ A2 ) )
= A2 ) ) ).
% image_f_inv_f
thf(fact_636_image__f__inv__f,axiom,
! [F2: set_nat > nat,A2: set_nat] :
( ( ( image_set_nat_nat @ F2 @ top_top_set_set_nat )
= top_top_set_nat )
=> ( ( image_set_nat_nat @ F2 @ ( image_nat_set_nat @ ( hilber3771966164650003108at_nat @ top_top_set_set_nat @ F2 ) @ A2 ) )
= A2 ) ) ).
% image_f_inv_f
thf(fact_637_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_638_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_639_surj__iff__all,axiom,
! [F2: extended_ereal > rat] :
( ( ( image_7024712101053848417al_rat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_rat )
= ( ! [X: rat] :
( ( F2 @ ( hilber314352331192628314al_rat @ top_to5683747375963461374_ereal @ F2 @ X ) )
= X ) ) ) ).
% surj_iff_all
thf(fact_640_surj__iff__all,axiom,
! [F2: nat > set_nat] :
( ( ( image_nat_set_nat @ F2 @ top_top_set_nat )
= top_top_set_set_nat )
= ( ! [X: set_nat] :
( ( F2 @ ( hilber7337601069305372324et_nat @ top_top_set_nat @ F2 @ X ) )
= X ) ) ) ).
% surj_iff_all
thf(fact_641_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_642_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_643_surj__iff__all,axiom,
! [F2: nat > rat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
= ( ! [X: rat] :
( ( F2 @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F2 @ X ) )
= X ) ) ) ).
% surj_iff_all
thf(fact_644_surj__iff__all,axiom,
! [F2: rat > extended_ereal] :
( ( ( image_2592109325025016879_ereal @ F2 @ top_top_set_rat )
= top_to5683747375963461374_ereal )
= ( ! [X: extended_ereal] :
( ( F2 @ ( hilber5105121592018572584_ereal @ top_top_set_rat @ F2 @ X ) )
= X ) ) ) ).
% surj_iff_all
thf(fact_645_surj__iff__all,axiom,
! [F2: rat > nat] :
( ( ( image_rat_nat @ F2 @ top_top_set_rat )
= top_top_set_nat )
= ( ! [X: nat] :
( ( F2 @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F2 @ X ) )
= X ) ) ) ).
% surj_iff_all
thf(fact_646_surj__iff__all,axiom,
! [F2: rat > rat] :
( ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat )
= ( ! [X: rat] :
( ( F2 @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 @ X ) )
= X ) ) ) ).
% surj_iff_all
thf(fact_647_surj__f__inv__f,axiom,
! [F2: extended_ereal > extended_ereal,Y4: extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal )
=> ( ( F2 @ ( hilber7422611030134141132_ereal @ top_to5683747375963461374_ereal @ F2 @ Y4 ) )
= Y4 ) ) ).
% surj_f_inv_f
thf(fact_648_surj__f__inv__f,axiom,
! [F2: extended_ereal > nat,Y4: nat] :
( ( ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_nat )
=> ( ( F2 @ ( hilber949482391279124050al_nat @ top_to5683747375963461374_ereal @ F2 @ Y4 ) )
= Y4 ) ) ).
% surj_f_inv_f
thf(fact_649_surj__f__inv__f,axiom,
! [F2: extended_ereal > rat,Y4: rat] :
( ( ( image_7024712101053848417al_rat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_rat )
=> ( ( F2 @ ( hilber314352331192628314al_rat @ top_to5683747375963461374_ereal @ F2 @ Y4 ) )
= Y4 ) ) ).
% surj_f_inv_f
thf(fact_650_surj__f__inv__f,axiom,
! [F2: nat > set_nat,Y4: set_nat] :
( ( ( image_nat_set_nat @ F2 @ top_top_set_nat )
= top_top_set_set_nat )
=> ( ( F2 @ ( hilber7337601069305372324et_nat @ top_top_set_nat @ F2 @ Y4 ) )
= Y4 ) ) ).
% surj_f_inv_f
thf(fact_651_surj__f__inv__f,axiom,
! [F2: nat > extended_ereal,Y4: extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat )
= top_to5683747375963461374_ereal )
=> ( ( F2 @ ( hilber6822286039850061104_ereal @ top_top_set_nat @ F2 @ Y4 ) )
= Y4 ) ) ).
% surj_f_inv_f
thf(fact_652_surj__f__inv__f,axiom,
! [F2: nat > nat,Y4: nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( F2 @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 @ Y4 ) )
= Y4 ) ) ).
% surj_f_inv_f
thf(fact_653_surj__f__inv__f,axiom,
! [F2: nat > rat,Y4: rat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
=> ( ( F2 @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F2 @ Y4 ) )
= Y4 ) ) ).
% surj_f_inv_f
thf(fact_654_surj__f__inv__f,axiom,
! [F2: rat > extended_ereal,Y4: extended_ereal] :
( ( ( image_2592109325025016879_ereal @ F2 @ top_top_set_rat )
= top_to5683747375963461374_ereal )
=> ( ( F2 @ ( hilber5105121592018572584_ereal @ top_top_set_rat @ F2 @ Y4 ) )
= Y4 ) ) ).
% surj_f_inv_f
thf(fact_655_surj__f__inv__f,axiom,
! [F2: rat > nat,Y4: nat] :
( ( ( image_rat_nat @ F2 @ top_top_set_rat )
= top_top_set_nat )
=> ( ( F2 @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F2 @ Y4 ) )
= Y4 ) ) ).
% surj_f_inv_f
thf(fact_656_surj__f__inv__f,axiom,
! [F2: rat > rat,Y4: rat] :
( ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat )
=> ( ( F2 @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 @ Y4 ) )
= Y4 ) ) ).
% surj_f_inv_f
thf(fact_657_image__inv__into__cancel,axiom,
! [F2: extended_ereal > nat,A2: set_Extended_ereal,A6: set_nat,B5: set_nat] :
( ( ( image_7659842161140344153al_nat @ F2 @ A2 )
= A6 )
=> ( ( ord_less_eq_set_nat @ B5 @ A6 )
=> ( ( image_7659842161140344153al_nat @ F2 @ ( image_4309273772856505399_ereal @ ( hilber949482391279124050al_nat @ A2 @ F2 ) @ B5 ) )
= B5 ) ) ) ).
% image_inv_into_cancel
thf(fact_658_image__inv__into__cancel,axiom,
! [F2: set_nat > nat,A2: set_set_nat,A6: set_nat,B5: set_nat] :
( ( ( image_set_nat_nat @ F2 @ A2 )
= A6 )
=> ( ( ord_less_eq_set_nat @ B5 @ A6 )
=> ( ( image_set_nat_nat @ F2 @ ( image_nat_set_nat @ ( hilber3771966164650003108at_nat @ A2 @ F2 ) @ B5 ) )
= B5 ) ) ) ).
% image_inv_into_cancel
thf(fact_659_image__inv__into__cancel,axiom,
! [F2: rat > nat,A2: set_rat,A6: set_nat,B5: set_nat] :
( ( ( image_rat_nat @ F2 @ A2 )
= A6 )
=> ( ( ord_less_eq_set_nat @ B5 @ A6 )
=> ( ( image_rat_nat @ F2 @ ( image_nat_rat @ ( hilber3317322552863949046at_nat @ A2 @ F2 ) @ B5 ) )
= B5 ) ) ) ).
% image_inv_into_cancel
thf(fact_660_image__inv__into__cancel,axiom,
! [F2: nat > set_nat,A2: set_nat,A6: set_set_nat,B5: set_set_nat] :
( ( ( image_nat_set_nat @ F2 @ A2 )
= A6 )
=> ( ( ord_le6893508408891458716et_nat @ B5 @ A6 )
=> ( ( image_nat_set_nat @ F2 @ ( image_set_nat_nat @ ( hilber7337601069305372324et_nat @ A2 @ F2 ) @ B5 ) )
= B5 ) ) ) ).
% image_inv_into_cancel
thf(fact_661_image__inv__into__cancel,axiom,
! [F2: nat > nat,A2: set_nat,A6: set_nat,B5: set_nat] :
( ( ( image_nat_nat @ F2 @ A2 )
= A6 )
=> ( ( ord_less_eq_set_nat @ B5 @ A6 )
=> ( ( image_nat_nat @ F2 @ ( image_nat_nat @ ( hilber3633877196798814958at_nat @ A2 @ F2 ) @ B5 ) )
= B5 ) ) ) ).
% image_inv_into_cancel
thf(fact_662_image__inv__into__cancel,axiom,
! [F2: nat > rat,A2: set_nat,A6: set_rat,B5: set_rat] :
( ( ( image_nat_rat @ F2 @ A2 )
= A6 )
=> ( ( ord_less_eq_set_rat @ B5 @ A6 )
=> ( ( image_nat_rat @ F2 @ ( image_rat_nat @ ( hilber2998747136712319222at_rat @ A2 @ F2 ) @ B5 ) )
= B5 ) ) ) ).
% image_inv_into_cancel
thf(fact_663_image__inv__into__cancel,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,A6: set_Extended_ereal,B5: set_Extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ A2 )
= A6 )
=> ( ( ord_le1644982726543182158_ereal @ B5 @ A6 )
=> ( ( image_6042159593519690757_ereal @ F2 @ ( image_6042159593519690757_ereal @ ( hilber7422611030134141132_ereal @ A2 @ F2 ) @ B5 ) )
= B5 ) ) ) ).
% image_inv_into_cancel
thf(fact_664_image__inv__into__cancel,axiom,
! [F2: nat > extended_ereal,A2: set_nat,A6: set_Extended_ereal,B5: set_Extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F2 @ A2 )
= A6 )
=> ( ( ord_le1644982726543182158_ereal @ B5 @ A6 )
=> ( ( image_4309273772856505399_ereal @ F2 @ ( image_7659842161140344153al_nat @ ( hilber6822286039850061104_ereal @ A2 @ F2 ) @ B5 ) )
= B5 ) ) ) ).
% image_inv_into_cancel
thf(fact_665_inj__imp__inv__eq,axiom,
! [F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ! [X2: extended_ereal] :
( ( F2 @ ( G @ X2 ) )
= X2 )
=> ( ( hilber7422611030134141132_ereal @ top_to5683747375963461374_ereal @ F2 )
= G ) ) ) ).
% inj_imp_inv_eq
thf(fact_666_inj__imp__inv__eq,axiom,
! [F2: nat > nat,G: nat > nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ! [X2: nat] :
( ( F2 @ ( G @ X2 ) )
= X2 )
=> ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 )
= G ) ) ) ).
% inj_imp_inv_eq
thf(fact_667_inv__f__eq,axiom,
! [F2: extended_ereal > extended_ereal,X3: extended_ereal,Y4: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ( ( F2 @ X3 )
= Y4 )
=> ( ( hilber7422611030134141132_ereal @ top_to5683747375963461374_ereal @ F2 @ Y4 )
= X3 ) ) ) ).
% inv_f_eq
thf(fact_668_inv__f__eq,axiom,
! [F2: nat > nat,X3: nat,Y4: nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( ( F2 @ X3 )
= Y4 )
=> ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 @ Y4 )
= X3 ) ) ) ).
% inv_f_eq
thf(fact_669_inv__f__f,axiom,
! [F2: extended_ereal > extended_ereal,X3: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ( hilber7422611030134141132_ereal @ top_to5683747375963461374_ereal @ F2 @ ( F2 @ X3 ) )
= X3 ) ) ).
% inv_f_f
thf(fact_670_inv__f__f,axiom,
! [F2: nat > nat,X3: nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 @ ( F2 @ X3 ) )
= X3 ) ) ).
% inv_f_f
thf(fact_671_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
! [S2: set_a,F2: a > nat > nat,X3: a] :
( ( finite1847142123255318723_a_nat @ S2 @ F2 )
=> ( ( member_a @ X3 @ S2 )
=> ( ( comp_nat_nat_nat @ ( F2 @ X3 ) @ ( F2 @ X3 ) )
= ( F2 @ X3 ) ) ) ) ).
% comp_fun_idem_on.comp_fun_idem_on
thf(fact_672_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
! [S2: set_nat,F2: nat > nat > nat,X3: nat] :
( ( finite7982400111564556781at_nat @ S2 @ F2 )
=> ( ( member_nat @ X3 @ S2 )
=> ( ( comp_nat_nat_nat @ ( F2 @ X3 ) @ ( F2 @ X3 ) )
= ( F2 @ X3 ) ) ) ) ).
% comp_fun_idem_on.comp_fun_idem_on
thf(fact_673_inj__transfer,axiom,
! [F2: extended_ereal > extended_ereal,P2: extended_ereal > $o,X3: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ! [Y2: extended_ereal] :
( ( member2350847679896131959_ereal @ Y2 @ ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal ) )
=> ( P2 @ ( hilber7422611030134141132_ereal @ top_to5683747375963461374_ereal @ F2 @ Y2 ) ) )
=> ( P2 @ X3 ) ) ) ).
% inj_transfer
thf(fact_674_inj__transfer,axiom,
! [F2: extended_ereal > a,P2: extended_ereal > $o,X3: extended_ereal] :
( ( inj_on8242634198667403041real_a @ F2 @ top_to5683747375963461374_ereal )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ ( image_3724615099042636213real_a @ F2 @ top_to5683747375963461374_ereal ) )
=> ( P2 @ ( hilber3276319488169350396real_a @ top_to5683747375963461374_ereal @ F2 @ Y2 ) ) )
=> ( P2 @ X3 ) ) ) ).
% inj_transfer
thf(fact_675_inj__transfer,axiom,
! [F2: extended_ereal > nat,P2: extended_ereal > $o,X3: extended_ereal] :
( ( inj_on318729178700965101al_nat @ F2 @ top_to5683747375963461374_ereal )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal ) )
=> ( P2 @ ( hilber949482391279124050al_nat @ top_to5683747375963461374_ereal @ F2 @ Y2 ) ) )
=> ( P2 @ X3 ) ) ) ).
% inj_transfer
thf(fact_676_inj__transfer,axiom,
! [F2: nat > extended_ereal,P2: nat > $o,X3: nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ top_top_set_nat )
=> ( ! [Y2: extended_ereal] :
( ( member2350847679896131959_ereal @ Y2 @ ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat ) )
=> ( P2 @ ( hilber6822286039850061104_ereal @ top_top_set_nat @ F2 @ Y2 ) ) )
=> ( P2 @ X3 ) ) ) ).
% inj_transfer
thf(fact_677_inj__transfer,axiom,
! [F2: nat > set_nat,P2: nat > $o,X3: nat] :
( ( inj_on_nat_set_nat @ F2 @ top_top_set_nat )
=> ( ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ ( image_nat_set_nat @ F2 @ top_top_set_nat ) )
=> ( P2 @ ( hilber7337601069305372324et_nat @ top_top_set_nat @ F2 @ Y2 ) ) )
=> ( P2 @ X3 ) ) ) ).
% inj_transfer
thf(fact_678_inj__transfer,axiom,
! [F2: nat > rat,P2: nat > $o,X3: nat] :
( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
=> ( ! [Y2: rat] :
( ( member_rat @ Y2 @ ( image_nat_rat @ F2 @ top_top_set_nat ) )
=> ( P2 @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F2 @ Y2 ) ) )
=> ( P2 @ X3 ) ) ) ).
% inj_transfer
thf(fact_679_inj__transfer,axiom,
! [F2: nat > a,P2: nat > $o,X3: nat] :
( ( inj_on_nat_a @ F2 @ top_top_set_nat )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ ( image_nat_a @ F2 @ top_top_set_nat ) )
=> ( P2 @ ( hilber2795491120104822624_nat_a @ top_top_set_nat @ F2 @ Y2 ) ) )
=> ( P2 @ X3 ) ) ) ).
% inj_transfer
thf(fact_680_inj__transfer,axiom,
! [F2: nat > nat,P2: nat > $o,X3: nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ ( image_nat_nat @ F2 @ top_top_set_nat ) )
=> ( P2 @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 @ Y2 ) ) )
=> ( P2 @ X3 ) ) ) ).
% inj_transfer
thf(fact_681_inj__transfer,axiom,
! [F2: rat > a,P2: rat > $o,X3: rat] :
( ( inj_on_rat_a @ F2 @ top_top_set_rat )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ ( image_rat_a @ F2 @ top_top_set_rat ) )
=> ( P2 @ ( hilber2046056624969986904_rat_a @ top_top_set_rat @ F2 @ Y2 ) ) )
=> ( P2 @ X3 ) ) ) ).
% inj_transfer
thf(fact_682_inj__transfer,axiom,
! [F2: rat > nat,P2: rat > $o,X3: rat] :
( ( inj_on_rat_nat @ F2 @ top_top_set_rat )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ ( image_rat_nat @ F2 @ top_top_set_rat ) )
=> ( P2 @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F2 @ Y2 ) ) )
=> ( P2 @ X3 ) ) ) ).
% inj_transfer
thf(fact_683_image__inv__f__f,axiom,
! [F2: set_nat > nat,A2: set_set_nat] :
( ( inj_on_set_nat_nat @ F2 @ top_top_set_set_nat )
=> ( ( image_nat_set_nat @ ( hilber3771966164650003108at_nat @ top_top_set_set_nat @ F2 ) @ ( image_set_nat_nat @ F2 @ A2 ) )
= A2 ) ) ).
% image_inv_f_f
thf(fact_684_image__inv__f__f,axiom,
! [F2: extended_ereal > nat,A2: set_Extended_ereal] :
( ( inj_on318729178700965101al_nat @ F2 @ top_to5683747375963461374_ereal )
=> ( ( image_4309273772856505399_ereal @ ( hilber949482391279124050al_nat @ top_to5683747375963461374_ereal @ F2 ) @ ( image_7659842161140344153al_nat @ F2 @ A2 ) )
= A2 ) ) ).
% image_inv_f_f
thf(fact_685_image__inv__f__f,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal )
=> ( ( image_6042159593519690757_ereal @ ( hilber7422611030134141132_ereal @ top_to5683747375963461374_ereal @ F2 ) @ ( image_6042159593519690757_ereal @ F2 @ A2 ) )
= A2 ) ) ).
% image_inv_f_f
thf(fact_686_image__inv__f__f,axiom,
! [F2: nat > extended_ereal,A2: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ top_top_set_nat )
=> ( ( image_7659842161140344153al_nat @ ( hilber6822286039850061104_ereal @ top_top_set_nat @ F2 ) @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
= A2 ) ) ).
% image_inv_f_f
thf(fact_687_image__inv__f__f,axiom,
! [F2: nat > set_nat,A2: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ top_top_set_nat )
=> ( ( image_set_nat_nat @ ( hilber7337601069305372324et_nat @ top_top_set_nat @ F2 ) @ ( image_nat_set_nat @ F2 @ A2 ) )
= A2 ) ) ).
% image_inv_f_f
thf(fact_688_image__inv__f__f,axiom,
! [F2: nat > rat,A2: set_nat] :
( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
=> ( ( image_rat_nat @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F2 ) @ ( image_nat_rat @ F2 @ A2 ) )
= A2 ) ) ).
% image_inv_f_f
thf(fact_689_image__inv__f__f,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( image_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 ) @ ( image_nat_nat @ F2 @ A2 ) )
= A2 ) ) ).
% image_inv_f_f
thf(fact_690_image__inv__f__f,axiom,
! [F2: rat > nat,A2: set_rat] :
( ( inj_on_rat_nat @ F2 @ top_top_set_rat )
=> ( ( image_nat_rat @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F2 ) @ ( image_rat_nat @ F2 @ A2 ) )
= A2 ) ) ).
% image_inv_f_f
thf(fact_691_inj__imp__surj__inv,axiom,
! [F2: set_nat > nat] :
( ( inj_on_set_nat_nat @ F2 @ top_top_set_set_nat )
=> ( ( image_nat_set_nat @ ( hilber3771966164650003108at_nat @ top_top_set_set_nat @ F2 ) @ top_top_set_nat )
= top_top_set_set_nat ) ) ).
% inj_imp_surj_inv
thf(fact_692_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_693_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_694_inj__imp__surj__inv,axiom,
! [F2: extended_ereal > rat] :
( ( inj_on8906971155469245173al_rat @ F2 @ top_to5683747375963461374_ereal )
=> ( ( image_2592109325025016879_ereal @ ( hilber314352331192628314al_rat @ top_to5683747375963461374_ereal @ F2 ) @ top_top_set_rat )
= top_to5683747375963461374_ereal ) ) ).
% inj_imp_surj_inv
thf(fact_695_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_696_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_697_inj__imp__surj__inv,axiom,
! [F2: nat > rat] :
( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
=> ( ( image_rat_nat @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F2 ) @ top_top_set_rat )
= top_top_set_nat ) ) ).
% inj_imp_surj_inv
thf(fact_698_inj__imp__surj__inv,axiom,
! [F2: rat > extended_ereal] :
( ( inj_on4474368379440413635_ereal @ F2 @ top_top_set_rat )
=> ( ( image_7024712101053848417al_rat @ ( hilber5105121592018572584_ereal @ top_top_set_rat @ F2 ) @ top_to5683747375963461374_ereal )
= top_top_set_rat ) ) ).
% inj_imp_surj_inv
thf(fact_699_inj__imp__surj__inv,axiom,
! [F2: rat > nat] :
( ( inj_on_rat_nat @ F2 @ top_top_set_rat )
=> ( ( image_nat_rat @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F2 ) @ top_top_set_nat )
= top_top_set_rat ) ) ).
% inj_imp_surj_inv
thf(fact_700_inj__imp__surj__inv,axiom,
! [F2: rat > rat] :
( ( inj_on_rat_rat @ F2 @ top_top_set_rat )
=> ( ( image_rat_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 ) @ top_top_set_rat )
= top_top_set_rat ) ) ).
% inj_imp_surj_inv
thf(fact_701_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_702_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_703_surj__imp__inj__inv,axiom,
! [F2: extended_ereal > rat] :
( ( ( image_7024712101053848417al_rat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_rat )
=> ( inj_on4474368379440413635_ereal @ ( hilber314352331192628314al_rat @ top_to5683747375963461374_ereal @ F2 ) @ top_top_set_rat ) ) ).
% surj_imp_inj_inv
thf(fact_704_surj__imp__inj__inv,axiom,
! [F2: nat > set_nat] :
( ( ( image_nat_set_nat @ F2 @ top_top_set_nat )
= top_top_set_set_nat )
=> ( inj_on_set_nat_nat @ ( hilber7337601069305372324et_nat @ top_top_set_nat @ F2 ) @ top_top_set_set_nat ) ) ).
% surj_imp_inj_inv
thf(fact_705_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_706_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_707_surj__imp__inj__inv,axiom,
! [F2: nat > rat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
=> ( inj_on_rat_nat @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F2 ) @ top_top_set_rat ) ) ).
% surj_imp_inj_inv
thf(fact_708_surj__imp__inj__inv,axiom,
! [F2: rat > extended_ereal] :
( ( ( image_2592109325025016879_ereal @ F2 @ top_top_set_rat )
= top_to5683747375963461374_ereal )
=> ( inj_on8906971155469245173al_rat @ ( hilber5105121592018572584_ereal @ top_top_set_rat @ F2 ) @ top_to5683747375963461374_ereal ) ) ).
% surj_imp_inj_inv
thf(fact_709_surj__imp__inj__inv,axiom,
! [F2: rat > nat] :
( ( ( image_rat_nat @ F2 @ top_top_set_rat )
= top_top_set_nat )
=> ( inj_on_nat_rat @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F2 ) @ top_top_set_nat ) ) ).
% surj_imp_inj_inv
thf(fact_710_surj__imp__inj__inv,axiom,
! [F2: rat > rat] :
( ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat )
=> ( inj_on_rat_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 ) @ top_top_set_rat ) ) ).
% surj_imp_inj_inv
thf(fact_711_inj__on__inv__into,axiom,
! [B3: set_set_nat,F2: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ ( image_nat_set_nat @ F2 @ A2 ) )
=> ( inj_on_set_nat_nat @ ( hilber7337601069305372324et_nat @ A2 @ F2 ) @ B3 ) ) ).
% inj_on_inv_into
thf(fact_712_inj__on__inv__into,axiom,
! [B3: set_rat,F2: nat > rat,A2: set_nat] :
( ( ord_less_eq_set_rat @ B3 @ ( image_nat_rat @ F2 @ A2 ) )
=> ( inj_on_rat_nat @ ( hilber2998747136712319222at_rat @ A2 @ F2 ) @ B3 ) ) ).
% inj_on_inv_into
thf(fact_713_inj__on__inv__into,axiom,
! [B3: set_nat,F2: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F2 @ A2 ) )
=> ( inj_on_nat_nat @ ( hilber3633877196798814958at_nat @ A2 @ F2 ) @ B3 ) ) ).
% inj_on_inv_into
thf(fact_714_inj__on__inv__into,axiom,
! [B3: set_Extended_ereal,F2: nat > extended_ereal,A2: set_nat] :
( ( ord_le1644982726543182158_ereal @ B3 @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
=> ( inj_on318729178700965101al_nat @ ( hilber6822286039850061104_ereal @ A2 @ F2 ) @ B3 ) ) ).
% inj_on_inv_into
thf(fact_715_inj__on__inv__into,axiom,
! [B3: set_Extended_ereal,F2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B3 @ ( image_6042159593519690757_ereal @ F2 @ A2 ) )
=> ( inj_on7162434037990268785_ereal @ ( hilber7422611030134141132_ereal @ A2 @ F2 ) @ B3 ) ) ).
% inj_on_inv_into
thf(fact_716_inv__into__comp,axiom,
! [F2: nat > extended_ereal,G: nat > nat,A2: set_nat,X3: extended_ereal] :
( ( inj_on6191532827271902155_ereal @ F2 @ ( image_nat_nat @ G @ A2 ) )
=> ( ( inj_on_nat_nat @ G @ A2 )
=> ( ( member2350847679896131959_ereal @ X3 @ ( image_4309273772856505399_ereal @ F2 @ ( image_nat_nat @ G @ A2 ) ) )
=> ( ( hilber6822286039850061104_ereal @ A2 @ ( comp_n13370146242399787al_nat @ F2 @ G ) @ X3 )
= ( comp_n5886173794813336841_ereal @ ( hilber3633877196798814958at_nat @ A2 @ G ) @ ( hilber6822286039850061104_ereal @ ( image_nat_nat @ G @ A2 ) @ F2 ) @ X3 ) ) ) ) ) ).
% inv_into_comp
thf(fact_717_inv__into__comp,axiom,
! [F2: nat > rat,G: nat > nat,A2: set_nat,X3: rat] :
( ( inj_on_nat_rat @ F2 @ ( image_nat_nat @ G @ A2 ) )
=> ( ( inj_on_nat_nat @ G @ A2 )
=> ( ( member_rat @ X3 @ ( image_nat_rat @ F2 @ ( image_nat_nat @ G @ A2 ) ) )
=> ( ( hilber2998747136712319222at_rat @ A2 @ ( comp_nat_rat_nat @ F2 @ G ) @ X3 )
= ( comp_nat_nat_rat @ ( hilber3633877196798814958at_nat @ A2 @ G ) @ ( hilber2998747136712319222at_rat @ ( image_nat_nat @ G @ A2 ) @ F2 ) @ X3 ) ) ) ) ) ).
% inv_into_comp
thf(fact_718_inv__into__comp,axiom,
! [F2: extended_ereal > extended_ereal,G: nat > extended_ereal,A2: set_nat,X3: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ ( image_4309273772856505399_ereal @ G @ A2 ) )
=> ( ( inj_on6191532827271902155_ereal @ G @ A2 )
=> ( ( member2350847679896131959_ereal @ X3 @ ( image_6042159593519690757_ereal @ F2 @ ( image_4309273772856505399_ereal @ G @ A2 ) ) )
=> ( ( hilber6822286039850061104_ereal @ A2 @ ( comp_E3726099860353345075al_nat @ F2 @ G ) @ X3 )
= ( comp_E375531472069506321_ereal @ ( hilber6822286039850061104_ereal @ A2 @ G ) @ ( hilber7422611030134141132_ereal @ ( image_4309273772856505399_ereal @ G @ A2 ) @ F2 ) @ X3 ) ) ) ) ) ).
% inv_into_comp
thf(fact_719_inv__into__comp,axiom,
! [F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal,A2: set_Extended_ereal,X3: extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ ( image_6042159593519690757_ereal @ G @ A2 ) )
=> ( ( inj_on7162434037990268785_ereal @ G @ A2 )
=> ( ( member2350847679896131959_ereal @ X3 @ ( image_6042159593519690757_ereal @ F2 @ ( image_6042159593519690757_ereal @ G @ A2 ) ) )
=> ( ( hilber7422611030134141132_ereal @ A2 @ ( comp_E9177254828515427499_ereal @ F2 @ G ) @ X3 )
= ( comp_E9177254828515427499_ereal @ ( hilber7422611030134141132_ereal @ A2 @ G ) @ ( hilber7422611030134141132_ereal @ ( image_6042159593519690757_ereal @ G @ A2 ) @ F2 ) @ X3 ) ) ) ) ) ).
% inv_into_comp
thf(fact_720_inv__into__comp,axiom,
! [F2: extended_ereal > a,G: nat > extended_ereal,A2: set_nat,X3: a] :
( ( inj_on8242634198667403041real_a @ F2 @ ( image_4309273772856505399_ereal @ G @ A2 ) )
=> ( ( inj_on6191532827271902155_ereal @ G @ A2 )
=> ( ( member_a @ X3 @ ( image_3724615099042636213real_a @ F2 @ ( image_4309273772856505399_ereal @ G @ A2 ) ) )
=> ( ( hilber2795491120104822624_nat_a @ A2 @ ( comp_E5637448798707004259_a_nat @ F2 @ G ) @ X3 )
= ( comp_E446008263030514881_nat_a @ ( hilber6822286039850061104_ereal @ A2 @ G ) @ ( hilber3276319488169350396real_a @ ( image_4309273772856505399_ereal @ G @ A2 ) @ F2 ) @ X3 ) ) ) ) ) ).
% inv_into_comp
thf(fact_721_inv__into__comp,axiom,
! [F2: rat > a,G: nat > rat,A2: set_nat,X3: a] :
( ( inj_on_rat_a @ F2 @ ( image_nat_rat @ G @ A2 ) )
=> ( ( inj_on_nat_rat @ G @ A2 )
=> ( ( member_a @ X3 @ ( image_rat_a @ F2 @ ( image_nat_rat @ G @ A2 ) ) )
=> ( ( hilber2795491120104822624_nat_a @ A2 @ ( comp_rat_a_nat @ F2 @ G ) @ X3 )
= ( comp_rat_nat_a @ ( hilber2998747136712319222at_rat @ A2 @ G ) @ ( hilber2046056624969986904_rat_a @ ( image_nat_rat @ G @ A2 ) @ F2 ) @ X3 ) ) ) ) ) ).
% inv_into_comp
thf(fact_722_inv__into__comp,axiom,
! [F2: extended_ereal > a,G: extended_ereal > extended_ereal,A2: set_Extended_ereal,X3: a] :
( ( inj_on8242634198667403041real_a @ F2 @ ( image_6042159593519690757_ereal @ G @ A2 ) )
=> ( ( inj_on7162434037990268785_ereal @ G @ A2 )
=> ( ( member_a @ X3 @ ( image_3724615099042636213real_a @ F2 @ ( image_6042159593519690757_ereal @ G @ A2 ) ) )
=> ( ( hilber3276319488169350396real_a @ A2 @ ( comp_E6551704282591734651_ereal @ F2 @ G ) @ X3 )
= ( comp_E1870838029643375451real_a @ ( hilber7422611030134141132_ereal @ A2 @ G ) @ ( hilber3276319488169350396real_a @ ( image_6042159593519690757_ereal @ G @ A2 ) @ F2 ) @ X3 ) ) ) ) ) ).
% inv_into_comp
thf(fact_723_inv__into__comp,axiom,
! [F2: nat > a,G: nat > nat,A2: set_nat,X3: a] :
( ( inj_on_nat_a @ F2 @ ( image_nat_nat @ G @ A2 ) )
=> ( ( inj_on_nat_nat @ G @ A2 )
=> ( ( member_a @ X3 @ ( image_nat_a @ F2 @ ( image_nat_nat @ G @ A2 ) ) )
=> ( ( hilber2795491120104822624_nat_a @ A2 @ ( comp_nat_a_nat @ F2 @ G ) @ X3 )
= ( comp_nat_nat_a @ ( hilber3633877196798814958at_nat @ A2 @ G ) @ ( hilber2795491120104822624_nat_a @ ( image_nat_nat @ G @ A2 ) @ F2 ) @ X3 ) ) ) ) ) ).
% inv_into_comp
thf(fact_724_inv__into__comp,axiom,
! [F2: extended_ereal > nat,G: nat > extended_ereal,A2: set_nat,X3: nat] :
( ( inj_on318729178700965101al_nat @ F2 @ ( image_4309273772856505399_ereal @ G @ A2 ) )
=> ( ( inj_on6191532827271902155_ereal @ G @ A2 )
=> ( ( member_nat @ X3 @ ( image_7659842161140344153al_nat @ F2 @ ( image_4309273772856505399_ereal @ G @ A2 ) ) )
=> ( ( hilber3633877196798814958at_nat @ A2 @ ( comp_E7502005551946643277at_nat @ F2 @ G ) @ X3 )
= ( comp_E7502005551946643277at_nat @ ( hilber6822286039850061104_ereal @ A2 @ G ) @ ( hilber949482391279124050al_nat @ ( image_4309273772856505399_ereal @ G @ A2 ) @ F2 ) @ X3 ) ) ) ) ) ).
% inv_into_comp
thf(fact_725_inv__into__comp,axiom,
! [F2: rat > nat,G: nat > rat,A2: set_nat,X3: nat] :
( ( inj_on_rat_nat @ F2 @ ( image_nat_rat @ G @ A2 ) )
=> ( ( inj_on_nat_rat @ G @ A2 )
=> ( ( member_nat @ X3 @ ( image_rat_nat @ F2 @ ( image_nat_rat @ G @ A2 ) ) )
=> ( ( hilber3633877196798814958at_nat @ A2 @ ( comp_rat_nat_nat @ F2 @ G ) @ X3 )
= ( comp_rat_nat_nat @ ( hilber2998747136712319222at_rat @ A2 @ G ) @ ( hilber3317322552863949046at_nat @ ( image_nat_rat @ G @ A2 ) @ F2 ) @ X3 ) ) ) ) ) ).
% inv_into_comp
thf(fact_726_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_727_inv__unique__comp,axiom,
! [F2: nat > nat,G: nat > nat] :
( ( ( comp_nat_nat_nat @ F2 @ G )
= id_nat )
=> ( ( ( comp_nat_nat_nat @ G @ F2 )
= id_nat )
=> ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 )
= G ) ) ) ).
% inv_unique_comp
thf(fact_728_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_729_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_730_surj__iff,axiom,
! [F2: extended_ereal > rat] :
( ( ( image_7024712101053848417al_rat @ F2 @ top_to5683747375963461374_ereal )
= top_top_set_rat )
= ( ( comp_E6550320847925281629at_rat @ F2 @ ( hilber314352331192628314al_rat @ top_to5683747375963461374_ereal @ F2 ) )
= id_rat ) ) ).
% surj_iff
thf(fact_731_surj__iff,axiom,
! [F2: nat > set_nat] :
( ( ( image_nat_set_nat @ F2 @ top_top_set_nat )
= top_top_set_set_nat )
= ( ( comp_n1099043700422570177et_nat @ F2 @ ( hilber7337601069305372324et_nat @ top_top_set_nat @ F2 ) )
= id_set_nat ) ) ).
% surj_iff
thf(fact_732_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_733_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_734_surj__iff,axiom,
! [F2: nat > rat] :
( ( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat )
= ( ( comp_nat_rat_rat @ F2 @ ( hilber2998747136712319222at_rat @ top_top_set_nat @ F2 ) )
= id_rat ) ) ).
% surj_iff
thf(fact_735_surj__iff,axiom,
! [F2: rat > extended_ereal] :
( ( ( image_2592109325025016879_ereal @ F2 @ top_top_set_rat )
= top_to5683747375963461374_ereal )
= ( ( comp_r2952691998189091003_ereal @ F2 @ ( hilber5105121592018572584_ereal @ top_top_set_rat @ F2 ) )
= id_Extended_ereal ) ) ).
% surj_iff
thf(fact_736_surj__iff,axiom,
! [F2: rat > nat] :
( ( ( image_rat_nat @ F2 @ top_top_set_rat )
= top_top_set_nat )
= ( ( comp_rat_nat_nat @ F2 @ ( hilber3317322552863949046at_nat @ top_top_set_rat @ F2 ) )
= id_nat ) ) ).
% surj_iff
thf(fact_737_surj__iff,axiom,
! [F2: rat > rat] :
( ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat )
= ( ( comp_rat_rat_rat @ F2 @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 ) )
= id_rat ) ) ).
% surj_iff
thf(fact_738_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_739_inj__iff,axiom,
! [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
= ( ( comp_nat_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 ) @ F2 )
= id_nat ) ) ).
% inj_iff
thf(fact_740_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_741_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_742_Grp__UNIV__id,axiom,
! [F2: rat > rat] :
( ( F2 = id_rat )
=> ( ( relcompp_rat_rat_rat @ ( conversep_rat_rat @ ( bNF_Grp_rat_rat @ top_top_set_rat @ F2 ) ) @ ( bNF_Grp_rat_rat @ top_top_set_rat @ F2 ) )
= ( bNF_Grp_rat_rat @ top_top_set_rat @ F2 ) ) ) ).
% Grp_UNIV_id
thf(fact_743_bijection_Oinv__comp__right,axiom,
! [F2: extended_ereal > extended_ereal] :
( ( hilber6088754731438466237_ereal @ F2 )
=> ( ( comp_E9177254828515427499_ereal @ F2 @ ( hilber7422611030134141132_ereal @ top_to5683747375963461374_ereal @ F2 ) )
= id_Extended_ereal ) ) ).
% bijection.inv_comp_right
thf(fact_744_bijection_Oinv__comp__right,axiom,
! [F2: nat > nat] :
( ( hilber5277034221543178913on_nat @ F2 )
=> ( ( comp_nat_nat_nat @ F2 @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 ) )
= id_nat ) ) ).
% bijection.inv_comp_right
thf(fact_745_bijection_Oinv__comp__right,axiom,
! [F2: rat > rat] :
( ( hilber4641904161456683177on_rat @ F2 )
=> ( ( comp_rat_rat_rat @ F2 @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 ) )
= id_rat ) ) ).
% bijection.inv_comp_right
thf(fact_746_bijection_Oinv__comp__left,axiom,
! [F2: extended_ereal > extended_ereal] :
( ( hilber6088754731438466237_ereal @ F2 )
=> ( ( comp_E9177254828515427499_ereal @ ( hilber7422611030134141132_ereal @ top_to5683747375963461374_ereal @ F2 ) @ F2 )
= id_Extended_ereal ) ) ).
% bijection.inv_comp_left
thf(fact_747_bijection_Oinv__comp__left,axiom,
! [F2: nat > nat] :
( ( hilber5277034221543178913on_nat @ F2 )
=> ( ( comp_nat_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F2 ) @ F2 )
= id_nat ) ) ).
% bijection.inv_comp_left
thf(fact_748_bijection_Oinv__comp__left,axiom,
! [F2: rat > rat] :
( ( hilber4641904161456683177on_rat @ F2 )
=> ( ( comp_rat_rat_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 ) @ F2 )
= id_rat ) ) ).
% bijection.inv_comp_left
thf(fact_749_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_750_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_751_Quotient__alt__def5,axiom,
( quotie963356530924895504al_rat
= ( ^ [R4: extended_ereal > extended_ereal > $o,Abs: extended_ereal > rat,Rep: rat > extended_ereal,T3: extended_ereal > rat > $o] :
( ( ord_le2604841557345650798_rat_o @ T3 @ ( bNF_Gr3211092294157423064al_rat @ top_to5683747375963461374_ereal @ Abs ) )
& ( ord_le9156709395967171846real_o @ ( bNF_Gr8001861554983367334_ereal @ top_top_set_rat @ Rep ) @ ( conver283822580028012401al_rat @ T3 ) )
& ( R4
= ( relcom3127755349049835863_ereal @ T3 @ ( conver283822580028012401al_rat @ T3 ) ) ) ) ) ) ).
% Quotient_alt_def5
thf(fact_752_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_753_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_754_Quotient__alt__def5,axiom,
( quotient_nat_rat
= ( ^ [R4: nat > nat > $o,Abs: nat > rat,Rep: rat > nat,T3: nat > rat > $o] :
( ( ord_le1897120724991155070_rat_o @ T3 @ ( bNF_Grp_nat_rat @ top_top_set_nat @ Abs ) )
& ( ord_le5467402850006352766_nat_o @ ( bNF_Grp_rat_nat @ top_top_set_rat @ Rep ) @ ( conversep_nat_rat @ T3 ) )
& ( R4
= ( relcompp_nat_rat_nat @ T3 @ ( conversep_nat_rat @ T3 ) ) ) ) ) ) ).
% Quotient_alt_def5
thf(fact_755_Quotient__alt__def5,axiom,
( quotie5754125791750839774_ereal
= ( ^ [R4: rat > rat > $o,Abs: rat > extended_ereal,Rep: extended_ereal > rat,T3: rat > extended_ereal > $o] :
( ( ord_le9156709395967171846real_o @ T3 @ ( bNF_Gr8001861554983367334_ereal @ top_top_set_rat @ Abs ) )
& ( ord_le2604841557345650798_rat_o @ ( bNF_Gr3211092294157423064al_rat @ top_to5683747375963461374_ereal @ Rep ) @ ( conver5074591840853956671_ereal @ T3 ) )
& ( R4
= ( relcom7653232278902604637al_rat @ T3 @ ( conver5074591840853956671_ereal @ T3 ) ) ) ) ) ) ).
% Quotient_alt_def5
thf(fact_756_Quotient__alt__def5,axiom,
( quotient_rat_nat
= ( ^ [R4: rat > rat > $o,Abs: rat > nat,Rep: nat > rat,T3: rat > nat > $o] :
( ( ord_le5467402850006352766_nat_o @ T3 @ ( bNF_Grp_rat_nat @ top_top_set_rat @ Abs ) )
& ( ord_le1897120724991155070_rat_o @ ( bNF_Grp_nat_rat @ top_top_set_nat @ Rep ) @ ( conversep_rat_nat @ T3 ) )
& ( R4
= ( relcompp_rat_nat_rat @ T3 @ ( conversep_rat_nat @ T3 ) ) ) ) ) ) ).
% Quotient_alt_def5
thf(fact_757_Quotient__alt__def5,axiom,
( quotient_rat_rat
= ( ^ [R4: rat > rat > $o,Abs: rat > rat,Rep: rat > rat,T3: rat > rat > $o] :
( ( ord_le4717968354871517046_rat_o @ T3 @ ( bNF_Grp_rat_rat @ top_top_set_rat @ Abs ) )
& ( ord_le4717968354871517046_rat_o @ ( bNF_Grp_rat_rat @ top_top_set_rat @ Rep ) @ ( conversep_rat_rat @ T3 ) )
& ( R4
= ( relcompp_rat_rat_rat @ T3 @ ( conversep_rat_rat @ T3 ) ) ) ) ) ) ).
% Quotient_alt_def5
thf(fact_758_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_759_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_760_bijection_Oinj__inv,axiom,
! [F2: rat > rat] :
( ( hilber4641904161456683177on_rat @ F2 )
=> ( inj_on_rat_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 ) @ top_top_set_rat ) ) ).
% bijection.inj_inv
thf(fact_761_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_762_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_763_bijection_Osurj__inv,axiom,
! [F2: rat > rat] :
( ( hilber4641904161456683177on_rat @ F2 )
=> ( ( image_rat_rat @ ( hilber2682192492777453310at_rat @ top_top_set_rat @ F2 ) @ top_top_set_rat )
= top_top_set_rat ) ) ).
% bijection.surj_inv
thf(fact_764_type__copy__map__comp0,axiom,
! [Rep2: nat > extended_ereal,Abs2: extended_ereal > nat,M: nat > nat,M1: extended_ereal > nat,M2: nat > extended_ereal,F2: nat > nat,G: nat > nat] :
( ( type_d2055953355225835344_ereal @ Rep2 @ Abs2 @ top_to5683747375963461374_ereal )
=> ( ( M
= ( comp_E7502005551946643277at_nat @ M1 @ M2 ) )
=> ( ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ F2 @ M ) @ G )
= ( comp_nat_nat_nat @ ( comp_E7502005551946643277at_nat @ ( comp_n5886173794813336841_ereal @ F2 @ M1 ) @ Rep2 ) @ ( comp_nat_nat_nat @ ( comp_E7502005551946643277at_nat @ Abs2 @ M2 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_765_type__copy__map__comp0,axiom,
! [Rep2: nat > nat,Abs2: nat > nat,M: nat > nat,M1: nat > nat,M2: nat > nat,F2: nat > nat,G: nat > nat] :
( ( type_d6250493948777748686at_nat @ Rep2 @ Abs2 @ top_top_set_nat )
=> ( ( M
= ( comp_nat_nat_nat @ M1 @ M2 ) )
=> ( ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ F2 @ M ) @ G )
= ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ F2 @ M1 ) @ Rep2 ) @ ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ Abs2 @ M2 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_766_type__copy__map__comp0,axiom,
! [Rep2: nat > rat,Abs2: rat > nat,M: nat > nat,M1: rat > nat,M2: nat > rat,F2: nat > nat,G: nat > nat] :
( ( type_d5615363888691252950at_rat @ Rep2 @ Abs2 @ top_top_set_rat )
=> ( ( M
= ( comp_rat_nat_nat @ M1 @ M2 ) )
=> ( ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ F2 @ M ) @ G )
= ( comp_nat_nat_nat @ ( comp_rat_nat_nat @ ( comp_nat_nat_rat @ F2 @ M1 ) @ Rep2 ) @ ( comp_nat_nat_nat @ ( comp_rat_nat_nat @ Abs2 @ M2 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_767_type__copy__map__comp0__undo,axiom,
! [Rep2: nat > extended_ereal,Abs2: extended_ereal > nat,Rep3: nat > extended_ereal,Abs3: extended_ereal > nat,Rep4: nat > extended_ereal,Abs4: extended_ereal > nat,M: extended_ereal > extended_ereal,M1: extended_ereal > extended_ereal,M2: extended_ereal > extended_ereal] :
( ( type_d2055953355225835344_ereal @ Rep2 @ Abs2 @ top_to5683747375963461374_ereal )
=> ( ( type_d2055953355225835344_ereal @ Rep3 @ Abs3 @ top_to5683747375963461374_ereal )
=> ( ( type_d2055953355225835344_ereal @ Rep4 @ Abs4 @ top_to5683747375963461374_ereal )
=> ( ( ( comp_E7502005551946643277at_nat @ ( comp_E375531472069506321_ereal @ Abs3 @ M ) @ Rep4 )
= ( comp_nat_nat_nat @ ( comp_E7502005551946643277at_nat @ ( comp_E375531472069506321_ereal @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_E7502005551946643277at_nat @ ( comp_E375531472069506321_ereal @ Abs2 @ M2 ) @ Rep4 ) ) )
=> ( ( comp_E9177254828515427499_ereal @ M1 @ M2 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_768_type__copy__map__comp0__undo,axiom,
! [Rep2: nat > extended_ereal,Abs2: extended_ereal > nat,Rep3: nat > extended_ereal,Abs3: extended_ereal > nat,Rep4: nat > nat,Abs4: nat > nat,M: nat > extended_ereal,M1: extended_ereal > extended_ereal,M2: nat > extended_ereal] :
( ( type_d2055953355225835344_ereal @ Rep2 @ Abs2 @ top_to5683747375963461374_ereal )
=> ( ( type_d2055953355225835344_ereal @ Rep3 @ Abs3 @ top_to5683747375963461374_ereal )
=> ( ( type_d6250493948777748686at_nat @ Rep4 @ Abs4 @ top_top_set_nat )
=> ( ( ( comp_nat_nat_nat @ ( comp_E7502005551946643277at_nat @ Abs3 @ M ) @ Rep4 )
= ( comp_nat_nat_nat @ ( comp_E7502005551946643277at_nat @ ( comp_E375531472069506321_ereal @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_nat_nat_nat @ ( comp_E7502005551946643277at_nat @ Abs2 @ M2 ) @ Rep4 ) ) )
=> ( ( comp_E3726099860353345075al_nat @ M1 @ M2 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_769_type__copy__map__comp0__undo,axiom,
! [Rep2: nat > extended_ereal,Abs2: extended_ereal > nat,Rep3: nat > extended_ereal,Abs3: extended_ereal > nat,Rep4: nat > rat,Abs4: rat > nat,M: rat > extended_ereal,M1: extended_ereal > extended_ereal,M2: rat > extended_ereal] :
( ( type_d2055953355225835344_ereal @ Rep2 @ Abs2 @ top_to5683747375963461374_ereal )
=> ( ( type_d2055953355225835344_ereal @ Rep3 @ Abs3 @ top_to5683747375963461374_ereal )
=> ( ( type_d5615363888691252950at_rat @ Rep4 @ Abs4 @ top_top_set_rat )
=> ( ( ( comp_rat_nat_nat @ ( comp_E6866875491860147541at_rat @ Abs3 @ M ) @ Rep4 )
= ( comp_nat_nat_nat @ ( comp_E7502005551946643277at_nat @ ( comp_E375531472069506321_ereal @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_rat_nat_nat @ ( comp_E6866875491860147541at_rat @ Abs2 @ M2 ) @ Rep4 ) ) )
=> ( ( comp_E3090969800266849339al_rat @ M1 @ M2 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_770_type__copy__map__comp0__undo,axiom,
! [Rep2: nat > extended_ereal,Abs2: extended_ereal > nat,Rep3: nat > nat,Abs3: nat > nat,Rep4: nat > extended_ereal,Abs4: extended_ereal > nat,M: extended_ereal > nat,M1: extended_ereal > nat,M2: extended_ereal > extended_ereal] :
( ( type_d2055953355225835344_ereal @ Rep2 @ Abs2 @ top_to5683747375963461374_ereal )
=> ( ( type_d6250493948777748686at_nat @ Rep3 @ Abs3 @ top_top_set_nat )
=> ( ( type_d2055953355225835344_ereal @ Rep4 @ Abs4 @ top_to5683747375963461374_ereal )
=> ( ( ( comp_E7502005551946643277at_nat @ ( comp_n5886173794813336841_ereal @ Abs3 @ M ) @ Rep4 )
= ( comp_nat_nat_nat @ ( comp_E7502005551946643277at_nat @ ( comp_n5886173794813336841_ereal @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_E7502005551946643277at_nat @ ( comp_E375531472069506321_ereal @ Abs2 @ M2 ) @ Rep4 ) ) )
=> ( ( comp_E375531472069506321_ereal @ M1 @ M2 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_771_type__copy__map__comp0__undo,axiom,
! [Rep2: nat > extended_ereal,Abs2: extended_ereal > nat,Rep3: nat > nat,Abs3: nat > nat,Rep4: nat > nat,Abs4: nat > nat,M: nat > nat,M1: extended_ereal > nat,M2: nat > extended_ereal] :
( ( type_d2055953355225835344_ereal @ Rep2 @ Abs2 @ top_to5683747375963461374_ereal )
=> ( ( type_d6250493948777748686at_nat @ Rep3 @ Abs3 @ top_top_set_nat )
=> ( ( type_d6250493948777748686at_nat @ Rep4 @ Abs4 @ top_top_set_nat )
=> ( ( ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ Abs3 @ M ) @ Rep4 )
= ( comp_nat_nat_nat @ ( comp_E7502005551946643277at_nat @ ( comp_n5886173794813336841_ereal @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_nat_nat_nat @ ( comp_E7502005551946643277at_nat @ Abs2 @ M2 ) @ Rep4 ) ) )
=> ( ( comp_E7502005551946643277at_nat @ M1 @ M2 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_772_type__copy__map__comp0__undo,axiom,
! [Rep2: nat > extended_ereal,Abs2: extended_ereal > nat,Rep3: nat > nat,Abs3: nat > nat,Rep4: nat > rat,Abs4: rat > nat,M: rat > nat,M1: extended_ereal > nat,M2: rat > extended_ereal] :
( ( type_d2055953355225835344_ereal @ Rep2 @ Abs2 @ top_to5683747375963461374_ereal )
=> ( ( type_d6250493948777748686at_nat @ Rep3 @ Abs3 @ top_top_set_nat )
=> ( ( type_d5615363888691252950at_rat @ Rep4 @ Abs4 @ top_top_set_rat )
=> ( ( ( comp_rat_nat_nat @ ( comp_nat_nat_rat @ Abs3 @ M ) @ Rep4 )
= ( comp_nat_nat_nat @ ( comp_E7502005551946643277at_nat @ ( comp_n5886173794813336841_ereal @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_rat_nat_nat @ ( comp_E6866875491860147541at_rat @ Abs2 @ M2 ) @ Rep4 ) ) )
=> ( ( comp_E6866875491860147541at_rat @ M1 @ M2 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_773_type__copy__map__comp0__undo,axiom,
! [Rep2: nat > extended_ereal,Abs2: extended_ereal > nat,Rep3: nat > rat,Abs3: rat > nat,Rep4: nat > extended_ereal,Abs4: extended_ereal > nat,M: extended_ereal > rat,M1: extended_ereal > rat,M2: extended_ereal > extended_ereal] :
( ( type_d2055953355225835344_ereal @ Rep2 @ Abs2 @ top_to5683747375963461374_ereal )
=> ( ( type_d5615363888691252950at_rat @ Rep3 @ Abs3 @ top_top_set_rat )
=> ( ( type_d2055953355225835344_ereal @ Rep4 @ Abs4 @ top_to5683747375963461374_ereal )
=> ( ( ( comp_E7502005551946643277at_nat @ ( comp_r969312439189832961_ereal @ Abs3 @ M ) @ Rep4 )
= ( comp_nat_nat_nat @ ( comp_E7502005551946643277at_nat @ ( comp_r969312439189832961_ereal @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_E7502005551946643277at_nat @ ( comp_E375531472069506321_ereal @ Abs2 @ M2 ) @ Rep4 ) ) )
=> ( ( comp_E7881739061092793609_ereal @ M1 @ M2 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_774_type__copy__map__comp0__undo,axiom,
! [Rep2: nat > extended_ereal,Abs2: extended_ereal > nat,Rep3: nat > rat,Abs3: rat > nat,Rep4: nat > nat,Abs4: nat > nat,M: nat > rat,M1: extended_ereal > rat,M2: nat > extended_ereal] :
( ( type_d2055953355225835344_ereal @ Rep2 @ Abs2 @ top_to5683747375963461374_ereal )
=> ( ( type_d5615363888691252950at_rat @ Rep3 @ Abs3 @ top_top_set_rat )
=> ( ( type_d6250493948777748686at_nat @ Rep4 @ Abs4 @ top_top_set_nat )
=> ( ( ( comp_nat_nat_nat @ ( comp_rat_nat_nat @ Abs3 @ M ) @ Rep4 )
= ( comp_nat_nat_nat @ ( comp_E7502005551946643277at_nat @ ( comp_r969312439189832961_ereal @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_nat_nat_nat @ ( comp_E7502005551946643277at_nat @ Abs2 @ M2 ) @ Rep4 ) ) )
=> ( ( comp_E7185450908011777365at_nat @ M1 @ M2 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_775_type__copy__map__comp0__undo,axiom,
! [Rep2: nat > extended_ereal,Abs2: extended_ereal > nat,Rep3: nat > rat,Abs3: rat > nat,Rep4: nat > rat,Abs4: rat > nat,M: rat > rat,M1: extended_ereal > rat,M2: rat > extended_ereal] :
( ( type_d2055953355225835344_ereal @ Rep2 @ Abs2 @ top_to5683747375963461374_ereal )
=> ( ( type_d5615363888691252950at_rat @ Rep3 @ Abs3 @ top_top_set_rat )
=> ( ( type_d5615363888691252950at_rat @ Rep4 @ Abs4 @ top_top_set_rat )
=> ( ( ( comp_rat_nat_nat @ ( comp_rat_nat_rat @ Abs3 @ M ) @ Rep4 )
= ( comp_nat_nat_nat @ ( comp_E7502005551946643277at_nat @ ( comp_r969312439189832961_ereal @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_rat_nat_nat @ ( comp_E6866875491860147541at_rat @ Abs2 @ M2 ) @ Rep4 ) ) )
=> ( ( comp_E6550320847925281629at_rat @ M1 @ M2 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_776_type__copy__map__comp0__undo,axiom,
! [Rep2: nat > nat,Abs2: nat > nat,Rep3: nat > extended_ereal,Abs3: extended_ereal > nat,Rep4: nat > extended_ereal,Abs4: extended_ereal > nat,M: extended_ereal > extended_ereal,M1: nat > extended_ereal,M2: extended_ereal > nat] :
( ( type_d6250493948777748686at_nat @ Rep2 @ Abs2 @ top_top_set_nat )
=> ( ( type_d2055953355225835344_ereal @ Rep3 @ Abs3 @ top_to5683747375963461374_ereal )
=> ( ( type_d2055953355225835344_ereal @ Rep4 @ Abs4 @ top_to5683747375963461374_ereal )
=> ( ( ( comp_E7502005551946643277at_nat @ ( comp_E375531472069506321_ereal @ Abs3 @ M ) @ Rep4 )
= ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ ( comp_E7502005551946643277at_nat @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_E7502005551946643277at_nat @ ( comp_n5886173794813336841_ereal @ Abs2 @ M2 ) @ Rep4 ) ) )
=> ( ( comp_n261702227720650419_ereal @ M1 @ M2 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_777_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_778_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_779_bijection_Osurj,axiom,
! [F2: rat > rat] :
( ( hilber4641904161456683177on_rat @ F2 )
=> ( ( image_rat_rat @ F2 @ top_top_set_rat )
= top_top_set_rat ) ) ).
% bijection.surj
thf(fact_780_bijection_Oinj,axiom,
! [F2: extended_ereal > extended_ereal] :
( ( hilber6088754731438466237_ereal @ F2 )
=> ( inj_on7162434037990268785_ereal @ F2 @ top_to5683747375963461374_ereal ) ) ).
% bijection.inj
thf(fact_781_bijection_Oinj,axiom,
! [F2: nat > nat] :
( ( hilber5277034221543178913on_nat @ F2 )
=> ( inj_on_nat_nat @ F2 @ top_top_set_nat ) ) ).
% bijection.inj
thf(fact_782_bijection_Oinj,axiom,
! [F2: rat > rat] :
( ( hilber4641904161456683177on_rat @ F2 )
=> ( inj_on_rat_rat @ F2 @ top_top_set_rat ) ) ).
% bijection.inj
thf(fact_783_Quotient__compose,axiom,
! [R12: nat > nat > $o,Abs1: nat > nat,Rep1: nat > nat,T1: nat > nat > $o,R23: nat > nat > $o,Abs22: nat > nat,Rep22: nat > nat,T22: nat > nat > $o] :
( ( quotient_nat_nat @ R12 @ Abs1 @ Rep1 @ T1 )
=> ( ( quotient_nat_nat @ R23 @ Abs22 @ Rep22 @ T22 )
=> ( quotient_nat_nat @ ( relcompp_nat_nat_nat @ T1 @ ( relcompp_nat_nat_nat @ R23 @ ( conversep_nat_nat @ T1 ) ) ) @ ( comp_nat_nat_nat @ Abs22 @ Abs1 ) @ ( comp_nat_nat_nat @ Rep1 @ Rep22 ) @ ( relcompp_nat_nat_nat @ T1 @ T22 ) ) ) ) ).
% Quotient_compose
thf(fact_784_type__copy__set__map0,axiom,
! [Rep2: nat > extended_ereal,Abs2: extended_ereal > nat,S2: extended_ereal > set_Extended_ereal,M: nat > extended_ereal,F2: extended_ereal > extended_ereal,S4: nat > set_Extended_ereal,G: nat > nat] :
( ( type_d2055953355225835344_ereal @ Rep2 @ Abs2 @ top_to5683747375963461374_ereal )
=> ( ( ( comp_E8701735184323617811al_nat @ S2 @ M )
= ( comp_s6650310590247965235al_nat @ ( image_6042159593519690757_ereal @ F2 ) @ S4 ) )
=> ( ( comp_n2164160572221156363al_nat @ ( comp_E8701735184323617811al_nat @ S2 @ Rep2 ) @ ( comp_nat_nat_nat @ ( comp_E7502005551946643277at_nat @ Abs2 @ M ) @ G ) )
= ( comp_s6650310590247965235al_nat @ ( image_6042159593519690757_ereal @ F2 ) @ ( comp_n2164160572221156363al_nat @ S4 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_785_type__copy__set__map0,axiom,
! [Rep2: nat > extended_ereal,Abs2: extended_ereal > nat,S2: extended_ereal > set_Extended_ereal,M: nat > extended_ereal,F2: nat > extended_ereal,S4: nat > set_nat,G: nat > nat] :
( ( type_d2055953355225835344_ereal @ Rep2 @ Abs2 @ top_to5683747375963461374_ereal )
=> ( ( ( comp_E8701735184323617811al_nat @ S2 @ M )
= ( comp_s2020303692887056085al_nat @ ( image_4309273772856505399_ereal @ F2 ) @ S4 ) )
=> ( ( comp_n2164160572221156363al_nat @ ( comp_E8701735184323617811al_nat @ S2 @ Rep2 ) @ ( comp_nat_nat_nat @ ( comp_E7502005551946643277at_nat @ Abs2 @ M ) @ G ) )
= ( comp_s2020303692887056085al_nat @ ( image_4309273772856505399_ereal @ F2 ) @ ( comp_nat_set_nat_nat @ S4 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_786_type__copy__set__map0,axiom,
! [Rep2: nat > extended_ereal,Abs2: extended_ereal > nat,S2: extended_ereal > set_nat,M: nat > extended_ereal,F2: nat > nat,S4: nat > set_nat,G: nat > nat] :
( ( type_d2055953355225835344_ereal @ Rep2 @ Abs2 @ top_to5683747375963461374_ereal )
=> ( ( ( comp_E3950336156362365699at_nat @ S2 @ M )
= ( comp_s3433241188411525313at_nat @ ( image_nat_nat @ F2 ) @ S4 ) )
=> ( ( comp_nat_set_nat_nat @ ( comp_E3950336156362365699at_nat @ S2 @ Rep2 ) @ ( comp_nat_nat_nat @ ( comp_E7502005551946643277at_nat @ Abs2 @ M ) @ G ) )
= ( comp_s3433241188411525313at_nat @ ( image_nat_nat @ F2 ) @ ( comp_nat_set_nat_nat @ S4 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_787_type__copy__set__map0,axiom,
! [Rep2: nat > extended_ereal,Abs2: extended_ereal > nat,S2: extended_ereal > set_rat,M: nat > extended_ereal,F2: nat > rat,S4: nat > set_nat,G: nat > nat] :
( ( type_d2055953355225835344_ereal @ Rep2 @ Abs2 @ top_to5683747375963461374_ereal )
=> ( ( ( comp_E8773024002708139275at_nat @ S2 @ M )
= ( comp_s8255929034757298889at_nat @ ( image_nat_rat @ F2 ) @ S4 ) )
=> ( ( comp_nat_set_rat_nat @ ( comp_E8773024002708139275at_nat @ S2 @ Rep2 ) @ ( comp_nat_nat_nat @ ( comp_E7502005551946643277at_nat @ Abs2 @ M ) @ G ) )
= ( comp_s8255929034757298889at_nat @ ( image_nat_rat @ F2 ) @ ( comp_nat_set_nat_nat @ S4 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_788_type__copy__set__map0,axiom,
! [Rep2: nat > nat,Abs2: nat > nat,S2: nat > set_Extended_ereal,M: nat > nat,F2: extended_ereal > extended_ereal,S4: nat > set_Extended_ereal,G: nat > nat] :
( ( type_d6250493948777748686at_nat @ Rep2 @ Abs2 @ top_top_set_nat )
=> ( ( ( comp_n2164160572221156363al_nat @ S2 @ M )
= ( comp_s6650310590247965235al_nat @ ( image_6042159593519690757_ereal @ F2 ) @ S4 ) )
=> ( ( comp_n2164160572221156363al_nat @ ( comp_n2164160572221156363al_nat @ S2 @ Rep2 ) @ ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ Abs2 @ M ) @ G ) )
= ( comp_s6650310590247965235al_nat @ ( image_6042159593519690757_ereal @ F2 ) @ ( comp_n2164160572221156363al_nat @ S4 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_789_type__copy__set__map0,axiom,
! [Rep2: nat > nat,Abs2: nat > nat,S2: nat > set_Extended_ereal,M: nat > nat,F2: nat > extended_ereal,S4: nat > set_nat,G: nat > nat] :
( ( type_d6250493948777748686at_nat @ Rep2 @ Abs2 @ top_top_set_nat )
=> ( ( ( comp_n2164160572221156363al_nat @ S2 @ M )
= ( comp_s2020303692887056085al_nat @ ( image_4309273772856505399_ereal @ F2 ) @ S4 ) )
=> ( ( comp_n2164160572221156363al_nat @ ( comp_n2164160572221156363al_nat @ S2 @ Rep2 ) @ ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ Abs2 @ M ) @ G ) )
= ( comp_s2020303692887056085al_nat @ ( image_4309273772856505399_ereal @ F2 ) @ ( comp_nat_set_nat_nat @ S4 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_790_type__copy__set__map0,axiom,
! [Rep2: nat > nat,Abs2: nat > nat,S2: nat > set_nat,M: nat > nat,F2: nat > nat,S4: nat > set_nat,G: nat > nat] :
( ( type_d6250493948777748686at_nat @ Rep2 @ Abs2 @ top_top_set_nat )
=> ( ( ( comp_nat_set_nat_nat @ S2 @ M )
= ( comp_s3433241188411525313at_nat @ ( image_nat_nat @ F2 ) @ S4 ) )
=> ( ( comp_nat_set_nat_nat @ ( comp_nat_set_nat_nat @ S2 @ Rep2 ) @ ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ Abs2 @ M ) @ G ) )
= ( comp_s3433241188411525313at_nat @ ( image_nat_nat @ F2 ) @ ( comp_nat_set_nat_nat @ S4 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_791_type__copy__set__map0,axiom,
! [Rep2: nat > nat,Abs2: nat > nat,S2: nat > set_rat,M: nat > nat,F2: nat > rat,S4: nat > set_nat,G: nat > nat] :
( ( type_d6250493948777748686at_nat @ Rep2 @ Abs2 @ top_top_set_nat )
=> ( ( ( comp_nat_set_rat_nat @ S2 @ M )
= ( comp_s8255929034757298889at_nat @ ( image_nat_rat @ F2 ) @ S4 ) )
=> ( ( comp_nat_set_rat_nat @ ( comp_nat_set_rat_nat @ S2 @ Rep2 ) @ ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ Abs2 @ M ) @ G ) )
= ( comp_s8255929034757298889at_nat @ ( image_nat_rat @ F2 ) @ ( comp_nat_set_nat_nat @ S4 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_792_type__copy__set__map0,axiom,
! [Rep2: nat > rat,Abs2: rat > nat,S2: rat > set_Extended_ereal,M: nat > rat,F2: extended_ereal > extended_ereal,S4: nat > set_Extended_ereal,G: nat > nat] :
( ( type_d5615363888691252950at_rat @ Rep2 @ Abs2 @ top_top_set_rat )
=> ( ( ( comp_r2284700167900836867al_nat @ S2 @ M )
= ( comp_s6650310590247965235al_nat @ ( image_6042159593519690757_ereal @ F2 ) @ S4 ) )
=> ( ( comp_n2164160572221156363al_nat @ ( comp_r2284700167900836867al_nat @ S2 @ Rep2 ) @ ( comp_nat_nat_nat @ ( comp_rat_nat_nat @ Abs2 @ M ) @ G ) )
= ( comp_s6650310590247965235al_nat @ ( image_6042159593519690757_ereal @ F2 ) @ ( comp_n2164160572221156363al_nat @ S4 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_793_type__copy__set__map0,axiom,
! [Rep2: nat > rat,Abs2: rat > nat,S2: rat > set_Extended_ereal,M: nat > rat,F2: nat > extended_ereal,S4: nat > set_nat,G: nat > nat] :
( ( type_d5615363888691252950at_rat @ Rep2 @ Abs2 @ top_top_set_rat )
=> ( ( ( comp_r2284700167900836867al_nat @ S2 @ M )
= ( comp_s2020303692887056085al_nat @ ( image_4309273772856505399_ereal @ F2 ) @ S4 ) )
=> ( ( comp_n2164160572221156363al_nat @ ( comp_r2284700167900836867al_nat @ S2 @ Rep2 ) @ ( comp_nat_nat_nat @ ( comp_rat_nat_nat @ Abs2 @ M ) @ G ) )
= ( comp_s2020303692887056085al_nat @ ( image_4309273772856505399_ereal @ F2 ) @ ( comp_nat_set_nat_nat @ S4 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_794_type__copy__Rep__o__Abs,axiom,
! [Rep2: nat > nat,Abs2: nat > nat] :
( ( type_d6250493948777748686at_nat @ Rep2 @ Abs2 @ top_top_set_nat )
=> ( ( comp_nat_nat_nat @ Rep2 @ Abs2 )
= id_nat ) ) ).
% type_copy_Rep_o_Abs
thf(fact_795_type__copy__Abs__o__Rep,axiom,
! [Rep2: nat > nat,Abs2: nat > nat] :
( ( type_d6250493948777748686at_nat @ Rep2 @ Abs2 @ top_top_set_nat )
=> ( ( comp_nat_nat_nat @ Abs2 @ Rep2 )
= id_nat ) ) ).
% type_copy_Abs_o_Rep
thf(fact_796_type__copy__map__id0,axiom,
! [Rep2: nat > nat,Abs2: nat > nat,M: nat > nat] :
( ( type_d6250493948777748686at_nat @ Rep2 @ Abs2 @ top_top_set_nat )
=> ( ( M = id_nat )
=> ( ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ Abs2 @ M ) @ Rep2 )
= id_nat ) ) ) ).
% type_copy_map_id0
thf(fact_797_type__definition_OAbs__image,axiom,
! [Rep2: set_nat > nat,Abs2: nat > set_nat,A2: set_nat] :
( ( type_d202864840109488260at_nat @ Rep2 @ Abs2 @ A2 )
=> ( ( image_nat_set_nat @ Abs2 @ A2 )
= top_top_set_set_nat ) ) ).
% type_definition.Abs_image
thf(fact_798_type__definition_OAbs__image,axiom,
! [Rep2: extended_ereal > extended_ereal,Abs2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( type_d4948314182666096300_ereal @ Rep2 @ Abs2 @ A2 )
=> ( ( image_6042159593519690757_ereal @ Abs2 @ A2 )
= top_to5683747375963461374_ereal ) ) ).
% type_definition.Abs_image
thf(fact_799_type__definition_OAbs__image,axiom,
! [Rep2: extended_ereal > nat,Abs2: nat > extended_ereal,A2: set_nat] :
( ( type_d5406521743509674098al_nat @ Rep2 @ Abs2 @ A2 )
=> ( ( image_4309273772856505399_ereal @ Abs2 @ A2 )
= top_to5683747375963461374_ereal ) ) ).
% type_definition.Abs_image
thf(fact_800_type__definition_OAbs__image,axiom,
! [Rep2: nat > nat,Abs2: nat > nat,A2: set_nat] :
( ( type_d6250493948777748686at_nat @ Rep2 @ Abs2 @ A2 )
=> ( ( image_nat_nat @ Abs2 @ A2 )
= top_top_set_nat ) ) ).
% type_definition.Abs_image
thf(fact_801_type__definition_OAbs__image,axiom,
! [Rep2: rat > nat,Abs2: nat > rat,A2: set_nat] :
( ( type_d5933939304842882774at_nat @ Rep2 @ Abs2 @ A2 )
=> ( ( image_nat_rat @ Abs2 @ A2 )
= top_top_set_rat ) ) ).
% type_definition.Abs_image
thf(fact_802_type__definition_ORep__range,axiom,
! [Rep2: extended_ereal > extended_ereal,Abs2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( type_d4948314182666096300_ereal @ Rep2 @ Abs2 @ A2 )
=> ( ( image_6042159593519690757_ereal @ Rep2 @ top_to5683747375963461374_ereal )
= A2 ) ) ).
% type_definition.Rep_range
thf(fact_803_type__definition_ORep__range,axiom,
! [Rep2: nat > extended_ereal,Abs2: extended_ereal > nat,A2: set_Extended_ereal] :
( ( type_d2055953355225835344_ereal @ Rep2 @ Abs2 @ A2 )
=> ( ( image_4309273772856505399_ereal @ Rep2 @ top_top_set_nat )
= A2 ) ) ).
% type_definition.Rep_range
thf(fact_804_type__definition_ORep__range,axiom,
! [Rep2: nat > set_nat,Abs2: set_nat > nat,A2: set_set_nat] :
( ( type_d3768499744764857476et_nat @ Rep2 @ Abs2 @ A2 )
=> ( ( image_nat_set_nat @ Rep2 @ top_top_set_nat )
= A2 ) ) ).
% type_definition.Rep_range
thf(fact_805_type__definition_ORep__range,axiom,
! [Rep2: nat > nat,Abs2: nat > nat,A2: set_nat] :
( ( type_d6250493948777748686at_nat @ Rep2 @ Abs2 @ A2 )
=> ( ( image_nat_nat @ Rep2 @ top_top_set_nat )
= A2 ) ) ).
% type_definition.Rep_range
thf(fact_806_type__definition_ORep__range,axiom,
! [Rep2: nat > rat,Abs2: rat > nat,A2: set_rat] :
( ( type_d5615363888691252950at_rat @ Rep2 @ Abs2 @ A2 )
=> ( ( image_nat_rat @ Rep2 @ top_top_set_nat )
= A2 ) ) ).
% type_definition.Rep_range
thf(fact_807_fun_Oin__rel,axiom,
! [R2: a > b > $o,A: extended_ereal > a,B: extended_ereal > b] :
( ( bNF_re4205385778126815198al_a_b
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = Z )
@ R2
@ A
@ 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 )
= A )
& ( ( comp_P2119009801111474836_ereal @ product_snd_a_b @ Z4 )
= B ) ) ) ) ).
% fun.in_rel
thf(fact_808_fun_Oin__rel,axiom,
! [R2: a > b > $o,A: nat > a,B: nat > b] :
( ( bNF_re4153754443986628736at_a_b
@ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
@ R2
@ A
@ 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 )
= A )
& ( ( comp_P3598376862245727882_b_nat @ product_snd_a_b @ Z4 )
= B ) ) ) ) ).
% fun.in_rel
thf(fact_809_fun_Oin__rel,axiom,
! [R2: a > b > $o,A: rat > a,B: rat > b] :
( ( bNF_re8507182716570760336at_a_b
@ ^ [Y3: rat,Z: rat] : ( Y3 = Z )
@ R2
@ A
@ B )
= ( ? [Z4: rat > product_prod_a_b] :
( ( member1833076429153978416od_a_b @ Z4
@ ( collec3304177515406664562od_a_b
@ ^ [X: rat > product_prod_a_b] : ( ord_le817736998455962536od_a_b @ ( image_1225746006951428489od_a_b @ X @ top_top_set_rat ) @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R2 ) ) ) ) )
& ( ( comp_P1727802473202837649_a_rat @ product_fst_a_b @ Z4 )
= A )
& ( ( comp_P2963246802159232146_b_rat @ product_snd_a_b @ Z4 )
= B ) ) ) ) ).
% fun.in_rel
thf(fact_810_fun_Orel__compp__Grp,axiom,
! [R2: a > b > $o] :
( ( bNF_re4205385778126815198al_a_b
@ ^ [Y3: extended_ereal,Z: extended_ereal] : ( Y3 = 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_811_fun_Orel__compp__Grp,axiom,
! [R2: a > b > $o] :
( ( bNF_re4153754443986628736at_a_b
@ ^ [Y3: nat,Z: nat] : ( Y3 = 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_812_fun_Orel__compp__Grp,axiom,
! [R2: a > b > $o] :
( ( bNF_re8507182716570760336at_a_b
@ ^ [Y3: rat,Z: rat] : ( Y3 = Z )
@ R2 )
= ( relcom4384204519082135299_rat_b
@ ( conver3336998252722865931_rat_a
@ ( bNF_Gr7582414742221077988_rat_a
@ ( collec3304177515406664562od_a_b
@ ^ [X: rat > product_prod_a_b] : ( ord_le817736998455962536od_a_b @ ( image_1225746006951428489od_a_b @ X @ top_top_set_rat ) @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R2 ) ) ) )
@ ( comp_P1727802473202837649_a_rat @ product_fst_a_b ) ) )
@ ( bNF_Gr7582414746524306789_rat_b
@ ( collec3304177515406664562od_a_b
@ ^ [X: rat > product_prod_a_b] : ( ord_le817736998455962536od_a_b @ ( image_1225746006951428489od_a_b @ X @ top_top_set_rat ) @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R2 ) ) ) )
@ ( comp_P2963246802159232146_b_rat @ product_snd_a_b ) ) ) ) ).
% fun.rel_compp_Grp
thf(fact_813_image__ident,axiom,
! [Y5: set_Extended_ereal] :
( ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : X
@ Y5 )
= Y5 ) ).
% image_ident
thf(fact_814_image__ident,axiom,
! [Y5: set_nat] :
( ( image_nat_nat
@ ^ [X: nat] : X
@ Y5 )
= Y5 ) ).
% image_ident
thf(fact_815_prop__restrict,axiom,
! [X3: a,Z5: set_a,X5: set_a,P2: a > $o] :
( ( member_a @ X3 @ Z5 )
=> ( ( ord_less_eq_set_a @ Z5
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ X5 )
& ( P2 @ X ) ) ) )
=> ( P2 @ X3 ) ) ) ).
% prop_restrict
thf(fact_816_prop__restrict,axiom,
! [X3: nat,Z5: set_nat,X5: set_nat,P2: nat > $o] :
( ( member_nat @ X3 @ Z5 )
=> ( ( ord_less_eq_set_nat @ Z5
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X5 )
& ( P2 @ X ) ) ) )
=> ( P2 @ X3 ) ) ) ).
% prop_restrict
thf(fact_817_prop__restrict,axiom,
! [X3: extended_ereal,Z5: set_Extended_ereal,X5: set_Extended_ereal,P2: extended_ereal > $o] :
( ( member2350847679896131959_ereal @ X3 @ Z5 )
=> ( ( ord_le1644982726543182158_ereal @ Z5
@ ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ X5 )
& ( P2 @ X ) ) ) )
=> ( P2 @ X3 ) ) ) ).
% prop_restrict
thf(fact_818_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_819_Collect__restrict,axiom,
! [X5: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X5 )
& ( P2 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_820_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_821_subset__CollectI,axiom,
! [B3: set_a,A2: set_a,Q2: a > $o,P2: a > $o] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B3 )
=> ( ( Q2 @ X2 )
=> ( P2 @ X2 ) ) )
=> ( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ B3 )
& ( Q2 @ X ) ) )
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
& ( P2 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_822_subset__CollectI,axiom,
! [B3: set_nat,A2: set_nat,Q2: nat > $o,P2: nat > $o] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( Q2 @ X2 )
=> ( P2 @ X2 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ B3 )
& ( Q2 @ X ) ) )
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P2 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_823_subset__CollectI,axiom,
! [B3: set_Extended_ereal,A2: set_Extended_ereal,Q2: extended_ereal > $o,P2: extended_ereal > $o] :
( ( ord_le1644982726543182158_ereal @ B3 @ A2 )
=> ( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ B3 )
=> ( ( Q2 @ X2 )
=> ( P2 @ X2 ) ) )
=> ( ord_le1644982726543182158_ereal
@ ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ B3 )
& ( Q2 @ X ) ) )
@ ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A2 )
& ( P2 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_824_subset__Collect__iff,axiom,
! [B3: set_a,A2: set_a,P2: a > $o] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( ord_less_eq_set_a @ B3
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
& ( P2 @ X ) ) ) )
= ( ! [X: a] :
( ( member_a @ X @ B3 )
=> ( P2 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_825_subset__Collect__iff,axiom,
! [B3: set_nat,A2: set_nat,P2: nat > $o] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( ord_less_eq_set_nat @ B3
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P2 @ X ) ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ B3 )
=> ( P2 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_826_subset__Collect__iff,axiom,
! [B3: set_Extended_ereal,A2: set_Extended_ereal,P2: extended_ereal > $o] :
( ( ord_le1644982726543182158_ereal @ B3 @ A2 )
=> ( ( ord_le1644982726543182158_ereal @ B3
@ ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A2 )
& ( P2 @ X ) ) ) )
= ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ B3 )
=> ( P2 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_827_image__Collect__subsetI,axiom,
! [P2: nat > $o,F2: nat > set_nat,B3: set_set_nat] :
( ! [X2: nat] :
( ( P2 @ X2 )
=> ( member_set_nat @ ( F2 @ X2 ) @ B3 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ ( collect_nat @ P2 ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_828_image__Collect__subsetI,axiom,
! [P2: nat > $o,F2: nat > rat,B3: set_rat] :
( ! [X2: nat] :
( ( P2 @ X2 )
=> ( member_rat @ ( F2 @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ ( collect_nat @ P2 ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_829_image__Collect__subsetI,axiom,
! [P2: nat > $o,F2: nat > nat,B3: set_nat] :
( ! [X2: nat] :
( ( P2 @ X2 )
=> ( member_nat @ ( F2 @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ ( collect_nat @ P2 ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_830_image__Collect__subsetI,axiom,
! [P2: extended_ereal > $o,F2: extended_ereal > extended_ereal,B3: set_Extended_ereal] :
( ! [X2: extended_ereal] :
( ( P2 @ X2 )
=> ( member2350847679896131959_ereal @ ( F2 @ X2 ) @ B3 ) )
=> ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F2 @ ( collec5835592288176408249_ereal @ P2 ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_831_image__Collect__subsetI,axiom,
! [P2: nat > $o,F2: nat > extended_ereal,B3: set_Extended_ereal] :
( ! [X2: nat] :
( ( P2 @ X2 )
=> ( member2350847679896131959_ereal @ ( F2 @ X2 ) @ B3 ) )
=> ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F2 @ ( collect_nat @ P2 ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_832_fst__def,axiom,
( product_fst_a_b
= ( produc6028431345588019473_a_b_a
@ ^ [X12: a,X23: b] : X12 ) ) ).
% fst_def
thf(fact_833_snd__def,axiom,
( product_snd_a_b
= ( produc6028431345588019474_a_b_b
@ ^ [X12: a,X23: b] : X23 ) ) ).
% snd_def
thf(fact_834_Inf_OINF__identity__eq,axiom,
! [Inf: set_Extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( Inf
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : X
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_835_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat > nat,A2: set_nat] :
( ( Inf
@ ( image_nat_nat
@ ^ [X: nat] : X
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_836_Sup_OSUP__identity__eq,axiom,
! [Sup: set_Extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( Sup
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : X
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_837_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat > nat,A2: set_nat] :
( ( Sup
@ ( image_nat_nat
@ ^ [X: nat] : X
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_838_imageE,axiom,
! [B: extended_ereal,F2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ B @ ( image_6042159593519690757_ereal @ F2 @ A2 ) )
=> ~ ! [X2: extended_ereal] :
( ( B
= ( F2 @ X2 ) )
=> ~ ( member2350847679896131959_ereal @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_839_imageE,axiom,
! [B: extended_ereal,F2: nat > extended_ereal,A2: set_nat] :
( ( member2350847679896131959_ereal @ B @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
=> ~ ! [X2: nat] :
( ( B
= ( F2 @ X2 ) )
=> ~ ( member_nat @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_840_imageE,axiom,
! [B: set_nat,F2: nat > set_nat,A2: set_nat] :
( ( member_set_nat @ B @ ( image_nat_set_nat @ F2 @ A2 ) )
=> ~ ! [X2: nat] :
( ( B
= ( F2 @ X2 ) )
=> ~ ( member_nat @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_841_imageE,axiom,
! [B: rat,F2: nat > rat,A2: set_nat] :
( ( member_rat @ B @ ( image_nat_rat @ F2 @ A2 ) )
=> ~ ! [X2: nat] :
( ( B
= ( F2 @ X2 ) )
=> ~ ( member_nat @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_842_imageE,axiom,
! [B: a,F2: a > a,A2: set_a] :
( ( member_a @ B @ ( image_a_a @ F2 @ A2 ) )
=> ~ ! [X2: a] :
( ( B
= ( F2 @ X2 ) )
=> ~ ( member_a @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_843_imageE,axiom,
! [B: a,F2: nat > a,A2: set_nat] :
( ( member_a @ B @ ( image_nat_a @ F2 @ A2 ) )
=> ~ ! [X2: nat] :
( ( B
= ( F2 @ X2 ) )
=> ~ ( member_nat @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_844_imageE,axiom,
! [B: nat,F2: a > nat,A2: set_a] :
( ( member_nat @ B @ ( image_a_nat @ F2 @ A2 ) )
=> ~ ! [X2: a] :
( ( B
= ( F2 @ X2 ) )
=> ~ ( member_a @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_845_imageE,axiom,
! [B: nat,F2: nat > nat,A2: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ F2 @ A2 ) )
=> ~ ! [X2: nat] :
( ( B
= ( F2 @ X2 ) )
=> ~ ( member_nat @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_846_image__image,axiom,
! [F2: extended_ereal > nat,G: nat > extended_ereal,A2: set_nat] :
( ( image_7659842161140344153al_nat @ F2 @ ( image_4309273772856505399_ereal @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X: nat] : ( F2 @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_847_image__image,axiom,
! [F2: extended_ereal > rat,G: nat > extended_ereal,A2: set_nat] :
( ( image_7024712101053848417al_rat @ F2 @ ( image_4309273772856505399_ereal @ G @ A2 ) )
= ( image_nat_rat
@ ^ [X: nat] : ( F2 @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_848_image__image,axiom,
! [F2: rat > extended_ereal,G: nat > rat,A2: set_nat] :
( ( image_2592109325025016879_ereal @ F2 @ ( image_nat_rat @ G @ A2 ) )
= ( image_4309273772856505399_ereal
@ ^ [X: nat] : ( F2 @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_849_image__image,axiom,
! [F2: rat > nat,G: nat > rat,A2: set_nat] :
( ( image_rat_nat @ F2 @ ( image_nat_rat @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X: nat] : ( F2 @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_850_image__image,axiom,
! [F2: rat > rat,G: nat > rat,A2: set_nat] :
( ( image_rat_rat @ F2 @ ( image_nat_rat @ G @ A2 ) )
= ( image_nat_rat
@ ^ [X: nat] : ( F2 @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_851_image__image,axiom,
! [F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( image_6042159593519690757_ereal @ F2 @ ( image_6042159593519690757_ereal @ G @ A2 ) )
= ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : ( F2 @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_852_image__image,axiom,
! [F2: extended_ereal > extended_ereal,G: nat > extended_ereal,A2: set_nat] :
( ( image_6042159593519690757_ereal @ F2 @ ( image_4309273772856505399_ereal @ G @ A2 ) )
= ( image_4309273772856505399_ereal
@ ^ [X: nat] : ( F2 @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_853_image__image,axiom,
! [F2: nat > extended_ereal,G: extended_ereal > nat,A2: set_Extended_ereal] :
( ( image_4309273772856505399_ereal @ F2 @ ( image_7659842161140344153al_nat @ G @ A2 ) )
= ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : ( F2 @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_854_image__image,axiom,
! [F2: nat > extended_ereal,G: nat > nat,A2: set_nat] :
( ( image_4309273772856505399_ereal @ F2 @ ( image_nat_nat @ G @ A2 ) )
= ( image_4309273772856505399_ereal
@ ^ [X: nat] : ( F2 @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_855_image__image,axiom,
! [F2: nat > nat,G: nat > nat,A2: set_nat] :
( ( image_nat_nat @ F2 @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X: nat] : ( F2 @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_856_Compr__image__eq,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,P2: extended_ereal > $o] :
( ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ ( image_6042159593519690757_ereal @ F2 @ A2 ) )
& ( P2 @ X ) ) )
= ( image_6042159593519690757_ereal @ F2
@ ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A2 )
& ( P2 @ ( F2 @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_857_Compr__image__eq,axiom,
! [F2: nat > extended_ereal,A2: set_nat,P2: extended_ereal > $o] :
( ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
& ( P2 @ X ) ) )
= ( image_4309273772856505399_ereal @ F2
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P2 @ ( F2 @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_858_Compr__image__eq,axiom,
! [F2: nat > set_nat,A2: set_nat,P2: set_nat > $o] :
( ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ ( image_nat_set_nat @ F2 @ A2 ) )
& ( P2 @ X ) ) )
= ( image_nat_set_nat @ F2
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P2 @ ( F2 @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_859_Compr__image__eq,axiom,
! [F2: nat > rat,A2: set_nat,P2: rat > $o] :
( ( collect_rat
@ ^ [X: rat] :
( ( member_rat @ X @ ( image_nat_rat @ F2 @ A2 ) )
& ( P2 @ X ) ) )
= ( image_nat_rat @ F2
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P2 @ ( F2 @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_860_Compr__image__eq,axiom,
! [F2: a > a,A2: set_a,P2: a > $o] :
( ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( image_a_a @ F2 @ A2 ) )
& ( P2 @ X ) ) )
= ( image_a_a @ F2
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
& ( P2 @ ( F2 @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_861_Compr__image__eq,axiom,
! [F2: nat > a,A2: set_nat,P2: a > $o] :
( ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( image_nat_a @ F2 @ A2 ) )
& ( P2 @ X ) ) )
= ( image_nat_a @ F2
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P2 @ ( F2 @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_862_Compr__image__eq,axiom,
! [F2: a > nat,A2: set_a,P2: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_a_nat @ F2 @ A2 ) )
& ( P2 @ X ) ) )
= ( image_a_nat @ F2
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
& ( P2 @ ( F2 @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_863_Compr__image__eq,axiom,
! [F2: nat > nat,A2: set_nat,P2: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F2 @ A2 ) )
& ( P2 @ X ) ) )
= ( image_nat_nat @ F2
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P2 @ ( F2 @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_864_inj__on__id2,axiom,
! [A2: set_Extended_ereal] :
( inj_on7162434037990268785_ereal
@ ^ [X: extended_ereal] : X
@ A2 ) ).
% inj_on_id2
thf(fact_865_inj__on__id2,axiom,
! [A2: set_nat] :
( inj_on_nat_nat
@ ^ [X: nat] : X
@ A2 ) ).
% inj_on_id2
thf(fact_866_rangeE,axiom,
! [B: extended_ereal,F2: extended_ereal > extended_ereal] :
( ( member2350847679896131959_ereal @ B @ ( image_6042159593519690757_ereal @ F2 @ top_to5683747375963461374_ereal ) )
=> ~ ! [X2: extended_ereal] :
( B
!= ( F2 @ X2 ) ) ) ).
% rangeE
thf(fact_867_rangeE,axiom,
! [B: a,F2: extended_ereal > a] :
( ( member_a @ B @ ( image_3724615099042636213real_a @ F2 @ top_to5683747375963461374_ereal ) )
=> ~ ! [X2: extended_ereal] :
( B
!= ( F2 @ X2 ) ) ) ).
% rangeE
thf(fact_868_rangeE,axiom,
! [B: nat,F2: extended_ereal > nat] :
( ( member_nat @ B @ ( image_7659842161140344153al_nat @ F2 @ top_to5683747375963461374_ereal ) )
=> ~ ! [X2: extended_ereal] :
( B
!= ( F2 @ X2 ) ) ) ).
% rangeE
thf(fact_869_rangeE,axiom,
! [B: extended_ereal,F2: nat > extended_ereal] :
( ( member2350847679896131959_ereal @ B @ ( image_4309273772856505399_ereal @ F2 @ top_top_set_nat ) )
=> ~ ! [X2: nat] :
( B
!= ( F2 @ X2 ) ) ) ).
% rangeE
thf(fact_870_rangeE,axiom,
! [B: set_nat,F2: nat > set_nat] :
( ( member_set_nat @ B @ ( image_nat_set_nat @ F2 @ top_top_set_nat ) )
=> ~ ! [X2: nat] :
( B
!= ( F2 @ X2 ) ) ) ).
% rangeE
thf(fact_871_rangeE,axiom,
! [B: rat,F2: nat > rat] :
( ( member_rat @ B @ ( image_nat_rat @ F2 @ top_top_set_nat ) )
=> ~ ! [X2: nat] :
( B
!= ( F2 @ X2 ) ) ) ).
% rangeE
thf(fact_872_rangeE,axiom,
! [B: a,F2: nat > a] :
( ( member_a @ B @ ( image_nat_a @ F2 @ top_top_set_nat ) )
=> ~ ! [X2: nat] :
( B
!= ( F2 @ X2 ) ) ) ).
% rangeE
thf(fact_873_rangeE,axiom,
! [B: nat,F2: nat > nat] :
( ( member_nat @ B @ ( image_nat_nat @ F2 @ top_top_set_nat ) )
=> ~ ! [X2: nat] :
( B
!= ( F2 @ X2 ) ) ) ).
% rangeE
thf(fact_874_rangeE,axiom,
! [B: a,F2: rat > a] :
( ( member_a @ B @ ( image_rat_a @ F2 @ top_top_set_rat ) )
=> ~ ! [X2: rat] :
( B
!= ( F2 @ X2 ) ) ) ).
% rangeE
thf(fact_875_rangeE,axiom,
! [B: nat,F2: rat > nat] :
( ( member_nat @ B @ ( image_rat_nat @ F2 @ top_top_set_rat ) )
=> ~ ! [X2: rat] :
( B
!= ( F2 @ X2 ) ) ) ).
% rangeE
thf(fact_876_range__composition,axiom,
! [F2: nat > nat,G: extended_ereal > nat] :
( ( image_7659842161140344153al_nat
@ ^ [X: extended_ereal] : ( F2 @ ( G @ X ) )
@ top_to5683747375963461374_ereal )
= ( image_nat_nat @ F2 @ ( image_7659842161140344153al_nat @ G @ top_to5683747375963461374_ereal ) ) ) ).
% range_composition
thf(fact_877_range__composition,axiom,
! [F2: nat > rat,G: extended_ereal > nat] :
( ( image_7024712101053848417al_rat
@ ^ [X: extended_ereal] : ( F2 @ ( G @ X ) )
@ top_to5683747375963461374_ereal )
= ( image_nat_rat @ F2 @ ( image_7659842161140344153al_nat @ G @ top_to5683747375963461374_ereal ) ) ) ).
% range_composition
thf(fact_878_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_879_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_880_range__composition,axiom,
! [F2: rat > extended_ereal,G: nat > rat] :
( ( image_4309273772856505399_ereal
@ ^ [X: nat] : ( F2 @ ( G @ X ) )
@ top_top_set_nat )
= ( image_2592109325025016879_ereal @ F2 @ ( image_nat_rat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_881_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_882_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_883_range__composition,axiom,
! [F2: extended_ereal > nat,G: nat > extended_ereal] :
( ( image_nat_nat
@ ^ [X: nat] : ( F2 @ ( G @ X ) )
@ top_top_set_nat )
= ( image_7659842161140344153al_nat @ F2 @ ( image_4309273772856505399_ereal @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_884_range__composition,axiom,
! [F2: rat > nat,G: nat > rat] :
( ( image_nat_nat
@ ^ [X: nat] : ( F2 @ ( G @ X ) )
@ top_top_set_nat )
= ( image_rat_nat @ F2 @ ( image_nat_rat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_885_range__composition,axiom,
! [F2: nat > nat,G: nat > nat] :
( ( image_nat_nat
@ ^ [X: nat] : ( F2 @ ( G @ X ) )
@ top_top_set_nat )
= ( image_nat_nat @ F2 @ ( image_nat_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_886_fun_Omap__ident,axiom,
! [T2: nat > nat] :
( ( comp_nat_nat_nat
@ ^ [X: nat] : X
@ T2 )
= T2 ) ).
% fun.map_ident
thf(fact_887_predicate2__transferD,axiom,
! [R12: a > b > $o,R23: a > b > $o,P2: a > a > $o,Q2: b > b > $o,A: product_prod_a_b,A2: set_Product_prod_a_b,B: product_prod_a_b,B3: set_Product_prod_a_b] :
( ( bNF_re5830743871565202077_o_b_o @ R12
@ ( bNF_rel_fun_a_b_o_o @ R23
@ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
@ P2
@ Q2 )
=> ( ( member1426531481828664017od_a_b @ A @ A2 )
=> ( ( member1426531481828664017od_a_b @ B @ B3 )
=> ( ( ord_le817736998455962536od_a_b @ A2 @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R12 ) ) )
=> ( ( ord_le817736998455962536od_a_b @ B3 @ ( collec3336397801687681299od_a_b @ ( produc3537405659489547051_a_b_o @ R23 ) ) )
=> ( ( P2 @ ( product_fst_a_b @ A ) @ ( product_fst_a_b @ B ) )
= ( Q2 @ ( product_snd_a_b @ A ) @ ( product_snd_a_b @ B ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_888_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_889_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_890_If__the__inv__into__f__f,axiom,
! [I: extended_ereal,C3: set_Extended_ereal,G: extended_ereal > extended_ereal,X3: extended_ereal] :
( ( member2350847679896131959_ereal @ I @ C3 )
=> ( ( inj_on7162434037990268785_ereal @ G @ C3 )
=> ( ( comp_E9177254828515427499_ereal
@ ^ [I2: extended_ereal] : ( if_Extended_ereal @ ( member2350847679896131959_ereal @ I2 @ ( image_6042159593519690757_ereal @ G @ C3 ) ) @ ( the_in1141389326992810419_ereal @ C3 @ G @ I2 ) @ X3 )
@ G
@ I )
= ( id_Extended_ereal @ I ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_891_If__the__inv__into__f__f,axiom,
! [I: a,C3: set_a,G: a > a,X3: a] :
( ( member_a @ I @ C3 )
=> ( ( inj_on_a_a @ G @ C3 )
=> ( ( comp_a_a_a
@ ^ [I2: a] : ( if_a @ ( member_a @ I2 @ ( image_a_a @ G @ C3 ) ) @ ( the_inv_into_a_a @ C3 @ G @ I2 ) @ X3 )
@ G
@ I )
= ( id_a @ I ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_892_If__the__inv__into__f__f,axiom,
! [I: a,C3: set_a,G: a > nat,X3: a] :
( ( member_a @ I @ C3 )
=> ( ( inj_on_a_nat @ G @ C3 )
=> ( ( comp_nat_a_a
@ ^ [I2: nat] : ( if_a @ ( member_nat @ I2 @ ( image_a_nat @ G @ C3 ) ) @ ( the_inv_into_a_nat @ C3 @ G @ I2 ) @ X3 )
@ G
@ I )
= ( id_a @ I ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_893_If__the__inv__into__f__f,axiom,
! [I: nat,C3: set_nat,G: nat > extended_ereal,X3: nat] :
( ( member_nat @ I @ C3 )
=> ( ( inj_on6191532827271902155_ereal @ G @ C3 )
=> ( ( comp_E7502005551946643277at_nat
@ ^ [I2: extended_ereal] : ( if_nat @ ( member2350847679896131959_ereal @ I2 @ ( image_4309273772856505399_ereal @ G @ C3 ) ) @ ( the_in5959796611709155849_ereal @ C3 @ G @ I2 ) @ X3 )
@ G
@ I )
= ( id_nat @ I ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_894_If__the__inv__into__f__f,axiom,
! [I: nat,C3: set_nat,G: nat > set_nat,X3: nat] :
( ( member_nat @ I @ C3 )
=> ( ( inj_on_nat_set_nat @ G @ C3 )
=> ( ( comp_set_nat_nat_nat
@ ^ [I2: set_nat] : ( if_nat @ ( member_set_nat @ I2 @ ( image_nat_set_nat @ G @ C3 ) ) @ ( the_in5057678521256355851et_nat @ C3 @ G @ I2 ) @ X3 )
@ G
@ I )
= ( id_nat @ I ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_895_If__the__inv__into__f__f,axiom,
! [I: nat,C3: set_nat,G: nat > rat,X3: nat] :
( ( member_nat @ I @ C3 )
=> ( ( inj_on_nat_rat @ G @ C3 )
=> ( ( comp_rat_nat_nat
@ ^ [I2: rat] : ( if_nat @ ( member_rat @ I2 @ ( image_nat_rat @ G @ C3 ) ) @ ( the_inv_into_nat_rat @ C3 @ G @ I2 ) @ X3 )
@ G
@ I )
= ( id_nat @ I ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_896_If__the__inv__into__f__f,axiom,
! [I: nat,C3: set_nat,G: nat > a,X3: nat] :
( ( member_nat @ I @ C3 )
=> ( ( inj_on_nat_a @ G @ C3 )
=> ( ( comp_a_nat_nat
@ ^ [I2: a] : ( if_nat @ ( member_a @ I2 @ ( image_nat_a @ G @ C3 ) ) @ ( the_inv_into_nat_a @ C3 @ G @ I2 ) @ X3 )
@ G
@ I )
= ( id_nat @ I ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_897_If__the__inv__into__f__f,axiom,
! [I: nat,C3: set_nat,G: nat > nat,X3: nat] :
( ( member_nat @ I @ C3 )
=> ( ( inj_on_nat_nat @ G @ C3 )
=> ( ( comp_nat_nat_nat
@ ^ [I2: nat] : ( if_nat @ ( member_nat @ I2 @ ( image_nat_nat @ G @ C3 ) ) @ ( the_inv_into_nat_nat @ C3 @ G @ I2 ) @ X3 )
@ G
@ I )
= ( id_nat @ I ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_898_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_899_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_900_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_901_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_902_type__copy__vimage2p__Grp__Rep,axiom,
! [Rep2: nat > extended_ereal,Abs2: extended_ereal > nat,F2: nat > nat,P2: nat > $o,H: nat > extended_ereal] :
( ( type_d2055953355225835344_ereal @ Rep2 @ Abs2 @ top_to5683747375963461374_ereal )
=> ( ( bNF_vi3772184142234500727real_o @ F2 @ Rep2 @ ( bNF_Gr495653965960080046_ereal @ ( collect_nat @ P2 ) @ H ) )
= ( bNF_Grp_nat_nat
@ ( collect_nat
@ ^ [X: nat] : ( P2 @ ( F2 @ X ) ) )
@ ( comp_nat_nat_nat @ ( comp_E7502005551946643277at_nat @ Abs2 @ H ) @ F2 ) ) ) ) ).
% type_copy_vimage2p_Grp_Rep
thf(fact_903_type__copy__vimage2p__Grp__Rep,axiom,
! [Rep2: nat > nat,Abs2: nat > nat,F2: nat > nat,P2: nat > $o,H: nat > nat] :
( ( type_d6250493948777748686at_nat @ Rep2 @ Abs2 @ top_top_set_nat )
=> ( ( bNF_vi6667456707345531737_nat_o @ F2 @ Rep2 @ ( bNF_Grp_nat_nat @ ( collect_nat @ P2 ) @ H ) )
= ( bNF_Grp_nat_nat
@ ( collect_nat
@ ^ [X: nat] : ( P2 @ ( F2 @ X ) ) )
@ ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ Abs2 @ H ) @ F2 ) ) ) ) ).
% type_copy_vimage2p_Grp_Rep
thf(fact_904_type__copy__vimage2p__Grp__Rep,axiom,
! [Rep2: nat > rat,Abs2: rat > nat,F2: nat > nat,P2: nat > $o,H: nat > rat] :
( ( type_d5615363888691252950at_rat @ Rep2 @ Abs2 @ top_top_set_rat )
=> ( ( bNF_vi2930123780028963665_rat_o @ F2 @ Rep2 @ ( bNF_Grp_nat_rat @ ( collect_nat @ P2 ) @ H ) )
= ( bNF_Grp_nat_nat
@ ( collect_nat
@ ^ [X: nat] : ( P2 @ ( F2 @ X ) ) )
@ ( comp_nat_nat_nat @ ( comp_rat_nat_nat @ Abs2 @ H ) @ F2 ) ) ) ) ).
% type_copy_vimage2p_Grp_Rep
thf(fact_905_type__copy__vimage2p__Grp__Abs,axiom,
! [Rep2: nat > nat,Abs2: nat > nat,G: nat > nat,P2: nat > $o,H: nat > nat] :
( ( type_d6250493948777748686at_nat @ Rep2 @ Abs2 @ top_top_set_nat )
=> ( ( bNF_vi6667456707345531737_nat_o @ G @ Abs2 @ ( bNF_Grp_nat_nat @ ( collect_nat @ P2 ) @ H ) )
= ( bNF_Grp_nat_nat
@ ( collect_nat
@ ^ [X: nat] : ( P2 @ ( G @ X ) ) )
@ ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ Rep2 @ H ) @ G ) ) ) ) ).
% type_copy_vimage2p_Grp_Abs
thf(fact_906_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_907_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_908_fun_Orel__Grp,axiom,
! [A2: set_nat,F2: nat > nat] :
( ( bNF_re5653821019739307937at_nat
@ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
@ ( bNF_Grp_nat_nat @ A2 @ F2 ) )
= ( bNF_Gr3847987472475283150at_nat
@ ( collect_nat_nat
@ ^ [X: nat > nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ X @ top_top_set_nat ) @ A2 ) )
@ ( comp_nat_nat_nat @ F2 ) ) ) ).
% fun.rel_Grp
thf(fact_909_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_910_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_911_sorted__list__of__set_Oinj__on,axiom,
( inj_on_rat_rat
@ ^ [X: rat] : X
@ top_top_set_rat ) ).
% sorted_list_of_set.inj_on
thf(fact_912_inj__on__apsnd,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( inj_on7007621943451069233_ereal @ ( produc5246002025674687312_ereal @ F2 )
@ ( produc8095709571603465288_ereal @ top_to5683747375963461374_ereal
@ ^ [Uu: extended_ereal] : A2 ) )
= ( inj_on7162434037990268785_ereal @ F2 @ A2 ) ) ).
% inj_on_apsnd
thf(fact_913_inj__on__apsnd,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( inj_on3854950389080018957al_nat @ ( produc8025838815002378542_ereal @ F2 )
@ ( produc4220900302008768982al_nat @ top_to5683747375963461374_ereal
@ ^ [Uu: extended_ereal] : A2 ) )
= ( inj_on_nat_nat @ F2 @ A2 ) ) ).
% inj_on_apsnd
thf(fact_914_inj__on__apsnd,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( inj_on113668994391194201_ereal @ ( produc6756617771423349966al_nat @ F2 )
@ ( produc870331913724930228_ereal @ top_top_set_nat
@ ^ [Uu: nat] : A2 ) )
= ( inj_on7162434037990268785_ereal @ F2 @ A2 ) ) ).
% inj_on_apsnd
thf(fact_915_inj__on__apsnd,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( inj_on8969904277767023793at_nat @ ( produc3094765310956027504at_nat @ F2 )
@ ( produc457027306803732586at_nat @ top_top_set_nat
@ ^ [Uu: nat] : A2 ) )
= ( inj_on_nat_nat @ F2 @ A2 ) ) ).
% inj_on_apsnd
thf(fact_916_inj__on__apsnd,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( inj_on2578515524460076617_ereal @ ( produc6121487711336854230al_rat @ F2 )
@ ( produc8376539502748217516_ereal @ top_top_set_rat
@ ^ [Uu: rat] : A2 ) )
= ( inj_on7162434037990268785_ereal @ F2 @ A2 ) ) ).
% inj_on_apsnd
thf(fact_917_inj__on__apsnd,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( inj_on1295019391013151409at_nat @ ( produc2459635250869531768at_rat @ F2 )
@ ( produc140472662868866674at_nat @ top_top_set_rat
@ ^ [Uu: rat] : A2 ) )
= ( inj_on_nat_nat @ F2 @ A2 ) ) ).
% inj_on_apsnd
thf(fact_918_inj__on__apfst,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( inj_on7007621943451069233_ereal @ ( produc7707886444459085074_ereal @ F2 )
@ ( produc8095709571603465288_ereal @ A2
@ ^ [Uu: extended_ereal] : top_to5683747375963461374_ereal ) )
= ( inj_on7162434037990268785_ereal @ F2 @ A2 ) ) ).
% inj_on_apfst
thf(fact_919_inj__on__apfst,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( inj_on113668994391194201_ereal @ ( produc8255056200670124528_ereal @ F2 )
@ ( produc870331913724930228_ereal @ A2
@ ^ [Uu: nat] : top_to5683747375963461374_ereal ) )
= ( inj_on_nat_nat @ F2 @ A2 ) ) ).
% inj_on_apfst
thf(fact_920_inj__on__apfst,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( inj_on3854950389080018957al_nat @ ( produc6531249619332995980al_nat @ F2 )
@ ( produc4220900302008768982al_nat @ A2
@ ^ [Uu: extended_ereal] : top_top_set_nat ) )
= ( inj_on7162434037990268785_ereal @ F2 @ A2 ) ) ).
% inj_on_apfst
thf(fact_921_inj__on__apfst,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( inj_on8969904277767023793at_nat @ ( produc986720760941809198at_nat @ F2 )
@ ( produc457027306803732586at_nat @ A2
@ ^ [Uu: nat] : top_top_set_nat ) )
= ( inj_on_nat_nat @ F2 @ A2 ) ) ).
% inj_on_apfst
thf(fact_922_inj__on__apfst,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( inj_on6416671941372977181al_rat @ ( produc5896119559246500244al_rat @ F2 )
@ ( produc3585770241922273246al_rat @ A2
@ ^ [Uu: extended_ereal] : top_top_set_rat ) )
= ( inj_on7162434037990268785_ereal @ F2 @ A2 ) ) ).
% inj_on_apfst
thf(fact_923_inj__on__apfst,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( inj_on7952486960743965361at_rat @ ( produc351590700855313462at_rat @ F2 )
@ ( produc9045269283572012658at_rat @ A2
@ ^ [Uu: nat] : top_top_set_rat ) )
= ( inj_on_nat_nat @ F2 @ A2 ) ) ).
% inj_on_apfst
thf(fact_924_K__record__comp,axiom,
! [C: nat,F2: nat > nat] :
( ( comp_nat_nat_nat
@ ^ [X: nat] : C
@ F2 )
= ( ^ [X: nat] : C ) ) ).
% K_record_comp
thf(fact_925_Gr__incl,axiom,
! [A2: set_nat,F2: nat > set_nat,B3: set_set_nat] :
( ( ord_le3845944159117341623et_nat @ ( bNF_Gr_nat_set_nat @ A2 @ F2 )
@ ( produc8883945523214541856et_nat @ A2
@ ^ [Uu: nat] : B3 ) )
= ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ A2 ) @ B3 ) ) ).
% Gr_incl
thf(fact_926_Gr__incl,axiom,
! [A2: set_nat,F2: nat > nat,B3: set_nat] :
( ( ord_le3146513528884898305at_nat @ ( bNF_Gr_nat_nat @ A2 @ F2 )
@ ( produc457027306803732586at_nat @ A2
@ ^ [Uu: nat] : B3 ) )
= ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ B3 ) ) ).
% Gr_incl
thf(fact_927_Gr__incl,axiom,
! [A2: set_nat,F2: nat > rat,B3: set_rat] :
( ( ord_le5989899228261996553at_rat @ ( bNF_Gr_nat_rat @ A2 @ F2 )
@ ( produc9045269283572012658at_rat @ A2
@ ^ [Uu: nat] : B3 ) )
= ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A2 ) @ B3 ) ) ).
% Gr_incl
thf(fact_928_Gr__incl,axiom,
! [A2: set_Extended_ereal,F2: extended_ereal > extended_ereal,B3: set_Extended_ereal] :
( ( ord_le8239133294219471655_ereal @ ( bNF_Gr945911885070196386_ereal @ A2 @ F2 )
@ ( produc8095709571603465288_ereal @ A2
@ ^ [Uu: extended_ereal] : B3 ) )
= ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F2 @ A2 ) @ B3 ) ) ).
% Gr_incl
thf(fact_929_Gr__incl,axiom,
! [A2: set_nat,F2: nat > extended_ereal,B3: set_Extended_ereal] :
( ( ord_le3920121919471048841_ereal @ ( bNF_Gr8352086143671499994_ereal @ A2 @ F2 )
@ ( produc870331913724930228_ereal @ A2
@ ^ [Uu: nat] : B3 ) )
= ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F2 @ A2 ) @ B3 ) ) ).
% Gr_incl
thf(fact_930_SigmaI,axiom,
! [A: a,A2: set_a,B: a,B3: a > set_a] :
( ( member_a @ A @ A2 )
=> ( ( member_a @ B @ ( B3 @ A ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( product_Sigma_a_a @ A2 @ B3 ) ) ) ) ).
% SigmaI
thf(fact_931_SigmaI,axiom,
! [A: a,A2: set_a,B: nat,B3: a > set_nat] :
( ( member_a @ A @ A2 )
=> ( ( member_nat @ B @ ( B3 @ A ) )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ B ) @ ( product_Sigma_a_nat @ A2 @ B3 ) ) ) ) ).
% SigmaI
thf(fact_932_SigmaI,axiom,
! [A: nat,A2: set_nat,B: a,B3: nat > set_a] :
( ( member_nat @ A @ A2 )
=> ( ( member_a @ B @ ( B3 @ A ) )
=> ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A @ B ) @ ( product_Sigma_nat_a @ A2 @ B3 ) ) ) ) ).
% SigmaI
thf(fact_933_SigmaI,axiom,
! [A: nat,A2: set_nat,B: nat,B3: nat > set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat @ B @ ( B3 @ A ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A2 @ B3 ) ) ) ) ).
% SigmaI
thf(fact_934_mem__Sigma__iff,axiom,
! [A: a,B: a,A2: set_a,B3: a > set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( product_Sigma_a_a @ A2 @ B3 ) )
= ( ( member_a @ A @ A2 )
& ( member_a @ B @ ( B3 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_935_mem__Sigma__iff,axiom,
! [A: a,B: nat,A2: set_a,B3: a > set_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ B ) @ ( product_Sigma_a_nat @ A2 @ B3 ) )
= ( ( member_a @ A @ A2 )
& ( member_nat @ B @ ( B3 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_936_mem__Sigma__iff,axiom,
! [A: nat,B: a,A2: set_nat,B3: nat > set_a] :
( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A @ B ) @ ( product_Sigma_nat_a @ A2 @ B3 ) )
= ( ( member_nat @ A @ A2 )
& ( member_a @ B @ ( B3 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_937_mem__Sigma__iff,axiom,
! [A: nat,B: nat,A2: set_nat,B3: nat > set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A2 @ B3 ) )
= ( ( member_nat @ A @ A2 )
& ( member_nat @ B @ ( B3 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_938_Compl__Times__UNIV2,axiom,
! [A2: set_Extended_ereal] :
( ( uminus6085167613793993086_ereal
@ ( produc8095709571603465288_ereal @ A2
@ ^ [Uu: extended_ereal] : top_to5683747375963461374_ereal ) )
= ( produc8095709571603465288_ereal @ ( uminus5895154729394068773_ereal @ A2 )
@ ^ [Uu: extended_ereal] : top_to5683747375963461374_ereal ) ) ).
% Compl_Times_UNIV2
thf(fact_939_Compl__Times__UNIV2,axiom,
! [A2: set_Extended_ereal] :
( ( uminus6262216577377593932al_nat
@ ( produc4220900302008768982al_nat @ A2
@ ^ [Uu: extended_ereal] : top_top_set_nat ) )
= ( produc4220900302008768982al_nat @ ( uminus5895154729394068773_ereal @ A2 )
@ ^ [Uu: extended_ereal] : top_top_set_nat ) ) ).
% Compl_Times_UNIV2
thf(fact_940_Compl__Times__UNIV2,axiom,
! [A2: set_Extended_ereal] :
( ( uminus9105602276754692180al_rat
@ ( produc3585770241922273246al_rat @ A2
@ ^ [Uu: extended_ereal] : top_top_set_rat ) )
= ( produc3585770241922273246al_rat @ ( uminus5895154729394068773_ereal @ A2 )
@ ^ [Uu: extended_ereal] : top_top_set_rat ) ) ).
% Compl_Times_UNIV2
thf(fact_941_Compl__Times__UNIV1,axiom,
! [A2: set_Extended_ereal] :
( ( uminus6085167613793993086_ereal
@ ( produc8095709571603465288_ereal @ top_to5683747375963461374_ereal
@ ^ [Uu: extended_ereal] : A2 ) )
= ( produc8095709571603465288_ereal @ top_to5683747375963461374_ereal
@ ^ [Uu: extended_ereal] : ( uminus5895154729394068773_ereal @ A2 ) ) ) ).
% Compl_Times_UNIV1
thf(fact_942_Compl__Times__UNIV1,axiom,
! [A2: set_Extended_ereal] :
( ( uminus4999475943122058226_ereal
@ ( produc870331913724930228_ereal @ top_top_set_nat
@ ^ [Uu: nat] : A2 ) )
= ( produc870331913724930228_ereal @ top_top_set_nat
@ ^ [Uu: nat] : ( uminus5895154729394068773_ereal @ A2 ) ) ) ).
% Compl_Times_UNIV1
thf(fact_943_Compl__Times__UNIV1,axiom,
! [A2: set_Extended_ereal] :
( ( uminus6667855999673911274_ereal
@ ( produc8376539502748217516_ereal @ top_top_set_rat
@ ^ [Uu: rat] : A2 ) )
= ( produc8376539502748217516_ereal @ top_top_set_rat
@ ^ [Uu: rat] : ( uminus5895154729394068773_ereal @ A2 ) ) ) ).
% Compl_Times_UNIV1
thf(fact_944_UNIV__Times__UNIV,axiom,
( ( produc8095709571603465288_ereal @ top_to5683747375963461374_ereal
@ ^ [Uu: extended_ereal] : top_to5683747375963461374_ereal )
= top_to3798671025730093271_ereal ) ).
% UNIV_Times_UNIV
thf(fact_945_UNIV__Times__UNIV,axiom,
( ( produc4220900302008768982al_nat @ top_to5683747375963461374_ereal
@ ^ [Uu: extended_ereal] : top_top_set_nat )
= top_to7896853287916821811al_nat ) ).
% UNIV_Times_UNIV
thf(fact_946_UNIV__Times__UNIV,axiom,
( ( produc3585770241922273246al_rat @ top_to5683747375963461374_ereal
@ ^ [Uu: extended_ereal] : top_top_set_rat )
= top_to1516866950439144251al_rat ) ).
% UNIV_Times_UNIV
thf(fact_947_UNIV__Times__UNIV,axiom,
( ( produc870331913724930228_ereal @ top_top_set_nat
@ ^ [Uu: nat] : top_to5683747375963461374_ereal )
= top_to6634112653661286105_ereal ) ).
% UNIV_Times_UNIV
thf(fact_948_UNIV__Times__UNIV,axiom,
( ( produc457027306803732586at_nat @ top_top_set_nat
@ ^ [Uu: nat] : top_top_set_nat )
= top_to4669805908274784177at_nat ) ).
% UNIV_Times_UNIV
thf(fact_949_UNIV__Times__UNIV,axiom,
( ( produc9045269283572012658at_rat @ top_top_set_nat
@ ^ [Uu: nat] : top_top_set_rat )
= top_to7513191607651882425at_rat ) ).
% UNIV_Times_UNIV
thf(fact_950_UNIV__Times__UNIV,axiom,
( ( produc8376539502748217516_ereal @ top_top_set_rat
@ ^ [Uu: rat] : top_to5683747375963461374_ereal )
= top_to8302492710213139153_ereal ) ).
% UNIV_Times_UNIV
thf(fact_951_UNIV__Times__UNIV,axiom,
( ( produc140472662868866674at_nat @ top_top_set_rat
@ ^ [Uu: rat] : top_top_set_nat )
= top_to269121717765781945at_nat ) ).
% UNIV_Times_UNIV
thf(fact_952_UNIV__Times__UNIV,axiom,
( ( produc8728714639637146746at_rat @ top_top_set_rat
@ ^ [Uu: rat] : top_top_set_rat )
= top_to3112507417142880193at_rat ) ).
% UNIV_Times_UNIV
thf(fact_953_SigmaE,axiom,
! [C: product_prod_a_a,A2: set_a,B3: a > set_a] :
( ( member1426531477525435216od_a_a @ C @ ( product_Sigma_a_a @ A2 @ B3 ) )
=> ~ ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ ( B3 @ X2 ) )
=> ( C
!= ( product_Pair_a_a @ X2 @ Y2 ) ) ) ) ) ).
% SigmaE
thf(fact_954_SigmaE,axiom,
! [C: product_prod_a_nat,A2: set_a,B3: a > set_nat] :
( ( member5724188588386418708_a_nat @ C @ ( product_Sigma_a_nat @ A2 @ B3 ) )
=> ~ ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( B3 @ X2 ) )
=> ( C
!= ( product_Pair_a_nat @ X2 @ Y2 ) ) ) ) ) ).
% SigmaE
thf(fact_955_SigmaE,axiom,
! [C: product_prod_nat_a,A2: set_nat,B3: nat > set_a] :
( ( member8962352052110095674_nat_a @ C @ ( product_Sigma_nat_a @ A2 @ B3 ) )
=> ~ ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ ( B3 @ X2 ) )
=> ( C
!= ( product_Pair_nat_a @ X2 @ Y2 ) ) ) ) ) ).
% SigmaE
thf(fact_956_SigmaE,axiom,
! [C: product_prod_nat_nat,A2: set_nat,B3: nat > set_nat] :
( ( member8440522571783428010at_nat @ C @ ( produc457027306803732586at_nat @ A2 @ B3 ) )
=> ~ ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( B3 @ X2 ) )
=> ( C
!= ( product_Pair_nat_nat @ X2 @ Y2 ) ) ) ) ) ).
% SigmaE
thf(fact_957_SigmaE2,axiom,
! [A: a,B: a,A2: set_a,B3: a > set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( product_Sigma_a_a @ A2 @ B3 ) )
=> ~ ( ( member_a @ A @ A2 )
=> ~ ( member_a @ B @ ( B3 @ A ) ) ) ) ).
% SigmaE2
thf(fact_958_SigmaE2,axiom,
! [A: a,B: nat,A2: set_a,B3: a > set_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ B ) @ ( product_Sigma_a_nat @ A2 @ B3 ) )
=> ~ ( ( member_a @ A @ A2 )
=> ~ ( member_nat @ B @ ( B3 @ A ) ) ) ) ).
% SigmaE2
thf(fact_959_SigmaE2,axiom,
! [A: nat,B: a,A2: set_nat,B3: nat > set_a] :
( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A @ B ) @ ( product_Sigma_nat_a @ A2 @ B3 ) )
=> ~ ( ( member_nat @ A @ A2 )
=> ~ ( member_a @ B @ ( B3 @ A ) ) ) ) ).
% SigmaE2
thf(fact_960_SigmaE2,axiom,
! [A: nat,B: nat,A2: set_nat,B3: nat > set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A2 @ B3 ) )
=> ~ ( ( member_nat @ A @ A2 )
=> ~ ( member_nat @ B @ ( B3 @ A ) ) ) ) ).
% SigmaE2
thf(fact_961_Sigma__mono,axiom,
! [A2: set_a,C3: set_a,B3: a > set_Extended_ereal,D2: a > set_Extended_ereal] :
( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_le1644982726543182158_ereal @ ( B3 @ X2 ) @ ( D2 @ X2 ) ) )
=> ( ord_le8132973195874727159_ereal @ ( produc7264369102385879704_ereal @ A2 @ B3 ) @ ( produc7264369102385879704_ereal @ C3 @ D2 ) ) ) ) ).
% Sigma_mono
thf(fact_962_Sigma__mono,axiom,
! [A2: set_nat,C3: set_nat,B3: nat > set_Extended_ereal,D2: nat > set_Extended_ereal] :
( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_le1644982726543182158_ereal @ ( B3 @ X2 ) @ ( D2 @ X2 ) ) )
=> ( ord_le3920121919471048841_ereal @ ( produc870331913724930228_ereal @ A2 @ B3 ) @ ( produc870331913724930228_ereal @ C3 @ D2 ) ) ) ) ).
% Sigma_mono
thf(fact_963_Sigma__mono,axiom,
! [A2: set_Extended_ereal,C3: set_Extended_ereal,B3: extended_ereal > set_Extended_ereal,D2: extended_ereal > set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A2 @ C3 )
=> ( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A2 )
=> ( ord_le1644982726543182158_ereal @ ( B3 @ X2 ) @ ( D2 @ X2 ) ) )
=> ( ord_le8239133294219471655_ereal @ ( produc8095709571603465288_ereal @ A2 @ B3 ) @ ( produc8095709571603465288_ereal @ C3 @ D2 ) ) ) ) ).
% Sigma_mono
thf(fact_964_Times__subset__cancel2,axiom,
! [X3: a,C3: set_a,A2: set_Extended_ereal,B3: set_Extended_ereal] :
( ( member_a @ X3 @ C3 )
=> ( ( ord_le7790192137282702039real_a
@ ( produc2583502849437520504real_a @ A2
@ ^ [Uu: extended_ereal] : C3 )
@ ( produc2583502849437520504real_a @ B3
@ ^ [Uu: extended_ereal] : C3 ) )
= ( ord_le1644982726543182158_ereal @ A2 @ B3 ) ) ) ).
% Times_subset_cancel2
thf(fact_965_Times__subset__cancel2,axiom,
! [X3: nat,C3: set_nat,A2: set_Extended_ereal,B3: set_Extended_ereal] :
( ( member_nat @ X3 @ C3 )
=> ( ( ord_le5182862553726584547al_nat
@ ( produc4220900302008768982al_nat @ A2
@ ^ [Uu: extended_ereal] : C3 )
@ ( produc4220900302008768982al_nat @ B3
@ ^ [Uu: extended_ereal] : C3 ) )
= ( ord_le1644982726543182158_ereal @ A2 @ B3 ) ) ) ).
% Times_subset_cancel2
thf(fact_966_mem__Times__iff,axiom,
! [X3: product_prod_a_a,A2: set_a,B3: set_a] :
( ( member1426531477525435216od_a_a @ X3
@ ( product_Sigma_a_a @ A2
@ ^ [Uu: a] : B3 ) )
= ( ( member_a @ ( product_fst_a_a @ X3 ) @ A2 )
& ( member_a @ ( product_snd_a_a @ X3 ) @ B3 ) ) ) ).
% mem_Times_iff
thf(fact_967_mem__Times__iff,axiom,
! [X3: product_prod_a_nat,A2: set_a,B3: set_nat] :
( ( member5724188588386418708_a_nat @ X3
@ ( product_Sigma_a_nat @ A2
@ ^ [Uu: a] : B3 ) )
= ( ( member_a @ ( product_fst_a_nat @ X3 ) @ A2 )
& ( member_nat @ ( product_snd_a_nat @ X3 ) @ B3 ) ) ) ).
% mem_Times_iff
thf(fact_968_mem__Times__iff,axiom,
! [X3: product_prod_nat_a,A2: set_nat,B3: set_a] :
( ( member8962352052110095674_nat_a @ X3
@ ( product_Sigma_nat_a @ A2
@ ^ [Uu: nat] : B3 ) )
= ( ( member_nat @ ( product_fst_nat_a @ X3 ) @ A2 )
& ( member_a @ ( product_snd_nat_a @ X3 ) @ B3 ) ) ) ).
% mem_Times_iff
thf(fact_969_mem__Times__iff,axiom,
! [X3: product_prod_nat_nat,A2: set_nat,B3: set_nat] :
( ( member8440522571783428010at_nat @ X3
@ ( produc457027306803732586at_nat @ A2
@ ^ [Uu: nat] : B3 ) )
= ( ( member_nat @ ( product_fst_nat_nat @ X3 ) @ A2 )
& ( member_nat @ ( product_snd_nat_nat @ X3 ) @ B3 ) ) ) ).
% mem_Times_iff
thf(fact_970_mem__Times__iff,axiom,
! [X3: product_prod_a_b,A2: set_a,B3: set_b] :
( ( member1426531481828664017od_a_b @ X3
@ ( product_Sigma_a_b @ A2
@ ^ [Uu: a] : B3 ) )
= ( ( member_a @ ( product_fst_a_b @ X3 ) @ A2 )
& ( member_b @ ( product_snd_a_b @ X3 ) @ B3 ) ) ) ).
% mem_Times_iff
thf(fact_971_map__prod__surj__on,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,A6: set_Extended_ereal,G: extended_ereal > extended_ereal,B3: set_Extended_ereal,B5: set_Extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ A2 )
= A6 )
=> ( ( ( image_6042159593519690757_ereal @ G @ B3 )
= B5 )
=> ( ( image_959328165755419589_ereal @ ( produc7788783332699689718_ereal @ F2 @ G )
@ ( produc8095709571603465288_ereal @ A2
@ ^ [Uu: extended_ereal] : B3 ) )
= ( produc8095709571603465288_ereal @ A6
@ ^ [Uu: extended_ereal] : B5 ) ) ) ) ).
% map_prod_surj_on
thf(fact_972_map__prod__surj__on,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,A6: set_Extended_ereal,G: nat > extended_ereal,B3: set_nat,B5: set_Extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F2 @ A2 )
= A6 )
=> ( ( ( image_4309273772856505399_ereal @ G @ B3 )
= B5 )
=> ( ( image_6926512632986509267_ereal @ ( produc5260569924724313478_ereal @ F2 @ G )
@ ( produc4220900302008768982al_nat @ A2
@ ^ [Uu: extended_ereal] : B3 ) )
= ( produc8095709571603465288_ereal @ A6
@ ^ [Uu: extended_ereal] : B5 ) ) ) ) ).
% map_prod_surj_on
thf(fact_973_map__prod__surj__on,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,A6: set_Extended_ereal,G: nat > nat,B3: set_nat,B5: set_nat] :
( ( ( image_6042159593519690757_ereal @ F2 @ A2 )
= A6 )
=> ( ( ( image_nat_nat @ G @ B3 )
= B5 )
=> ( ( image_2725866490533164705al_nat @ ( produc8678206924122515480at_nat @ F2 @ G )
@ ( produc4220900302008768982al_nat @ A2
@ ^ [Uu: extended_ereal] : B3 ) )
= ( produc4220900302008768982al_nat @ A6
@ ^ [Uu: extended_ereal] : B5 ) ) ) ) ).
% map_prod_surj_on
thf(fact_974_map__prod__surj__on,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,A6: set_Extended_ereal,G: nat > rat,B3: set_nat,B5: set_rat] :
( ( ( image_6042159593519690757_ereal @ F2 @ A2 )
= A6 )
=> ( ( ( image_nat_rat @ G @ B3 )
= B5 )
=> ( ( image_8441009968994543785al_rat @ ( produc8043076864036019744at_rat @ F2 @ G )
@ ( produc4220900302008768982al_nat @ A2
@ ^ [Uu: extended_ereal] : B3 ) )
= ( produc3585770241922273246al_rat @ A6
@ ^ [Uu: extended_ereal] : B5 ) ) ) ) ).
% map_prod_surj_on
thf(fact_975_map__prod__surj__on,axiom,
! [F2: nat > extended_ereal,A2: set_nat,A6: set_Extended_ereal,G: extended_ereal > extended_ereal,B3: set_Extended_ereal,B5: set_Extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F2 @ A2 )
= A6 )
=> ( ( ( image_6042159593519690757_ereal @ G @ B3 )
= B5 )
=> ( ( image_3845515196057536685_ereal @ ( produc7290494158911329542_ereal @ F2 @ G )
@ ( produc870331913724930228_ereal @ A2
@ ^ [Uu: nat] : B3 ) )
= ( produc8095709571603465288_ereal @ A6
@ ^ [Uu: extended_ereal] : B5 ) ) ) ) ).
% map_prod_surj_on
thf(fact_976_map__prod__surj__on,axiom,
! [F2: nat > extended_ereal,A2: set_nat,A6: set_Extended_ereal,G: nat > extended_ereal,B3: set_nat,B5: set_Extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F2 @ A2 )
= A6 )
=> ( ( ( image_4309273772856505399_ereal @ G @ B3 )
= B5 )
=> ( ( image_7762317921424436459_ereal @ ( produc578423587001601910_ereal @ F2 @ G )
@ ( produc457027306803732586at_nat @ A2
@ ^ [Uu: nat] : B3 ) )
= ( produc8095709571603465288_ereal @ A6
@ ^ [Uu: extended_ereal] : B5 ) ) ) ) ).
% map_prod_surj_on
thf(fact_977_map__prod__surj__on,axiom,
! [F2: nat > extended_ereal,A2: set_nat,A6: set_Extended_ereal,G: nat > nat,B3: set_nat,B5: set_nat] :
( ( ( image_4309273772856505399_ereal @ F2 @ A2 )
= A6 )
=> ( ( ( image_nat_nat @ G @ B3 )
= B5 )
=> ( ( image_3171559211387932553al_nat @ ( produc4195411439541084712at_nat @ F2 @ G )
@ ( produc457027306803732586at_nat @ A2
@ ^ [Uu: nat] : B3 ) )
= ( produc4220900302008768982al_nat @ A6
@ ^ [Uu: extended_ereal] : B5 ) ) ) ) ).
% map_prod_surj_on
thf(fact_978_map__prod__surj__on,axiom,
! [F2: nat > extended_ereal,A2: set_nat,A6: set_Extended_ereal,G: nat > rat,B3: set_nat,B5: set_rat] :
( ( ( image_4309273772856505399_ereal @ F2 @ A2 )
= A6 )
=> ( ( ( image_nat_rat @ G @ B3 )
= B5 )
=> ( ( image_8886702689849311633al_rat @ ( produc3560281379454588976at_rat @ F2 @ G )
@ ( produc457027306803732586at_nat @ A2
@ ^ [Uu: nat] : B3 ) )
= ( produc3585770241922273246al_rat @ A6
@ ^ [Uu: extended_ereal] : B5 ) ) ) ) ).
% map_prod_surj_on
thf(fact_979_map__prod__surj__on,axiom,
! [F2: nat > nat,A2: set_nat,A6: set_nat,G: extended_ereal > extended_ereal,B3: set_Extended_ereal,B5: set_Extended_ereal] :
( ( ( image_nat_nat @ F2 @ A2 )
= A6 )
=> ( ( ( image_6042159593519690757_ereal @ G @ B3 )
= B5 )
=> ( ( image_8207957132699115757_ereal @ ( produc464594342652746008_ereal @ F2 @ G )
@ ( produc870331913724930228_ereal @ A2
@ ^ [Uu: nat] : B3 ) )
= ( produc870331913724930228_ereal @ A6
@ ^ [Uu: nat] : B5 ) ) ) ) ).
% map_prod_surj_on
thf(fact_980_map__prod__surj__on,axiom,
! [F2: nat > nat,A2: set_nat,A6: set_nat,G: nat > extended_ereal,B3: set_nat,B5: set_Extended_ereal] :
( ( ( image_nat_nat @ F2 @ A2 )
= A6 )
=> ( ( ( image_4309273772856505399_ereal @ G @ B3 )
= B5 )
=> ( ( image_5520118406512677935_ereal @ ( produc2579579682407778276_ereal @ F2 @ G )
@ ( produc457027306803732586at_nat @ A2
@ ^ [Uu: nat] : B3 ) )
= ( produc870331913724930228_ereal @ A6
@ ^ [Uu: nat] : B5 ) ) ) ) ).
% map_prod_surj_on
thf(fact_981_map__prod__inj__on,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,G: extended_ereal > extended_ereal,B3: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A2 )
=> ( ( inj_on7162434037990268785_ereal @ G @ B3 )
=> ( inj_on7007621943451069233_ereal @ ( produc7788783332699689718_ereal @ F2 @ G )
@ ( produc8095709571603465288_ereal @ A2
@ ^ [Uu: extended_ereal] : B3 ) ) ) ) ).
% map_prod_inj_on
thf(fact_982_map__prod__inj__on,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,G: nat > nat,B3: set_nat] :
( ( inj_on7162434037990268785_ereal @ F2 @ A2 )
=> ( ( inj_on_nat_nat @ G @ B3 )
=> ( inj_on3854950389080018957al_nat @ ( produc8678206924122515480at_nat @ F2 @ G )
@ ( produc4220900302008768982al_nat @ A2
@ ^ [Uu: extended_ereal] : B3 ) ) ) ) ).
% map_prod_inj_on
thf(fact_983_map__prod__inj__on,axiom,
! [F2: nat > nat,A2: set_nat,G: extended_ereal > extended_ereal,B3: set_Extended_ereal] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( inj_on7162434037990268785_ereal @ G @ B3 )
=> ( inj_on113668994391194201_ereal @ ( produc464594342652746008_ereal @ F2 @ G )
@ ( produc870331913724930228_ereal @ A2
@ ^ [Uu: nat] : B3 ) ) ) ) ).
% map_prod_inj_on
thf(fact_984_map__prod__inj__on,axiom,
! [F2: nat > nat,A2: set_nat,G: nat > nat,B3: set_nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( inj_on_nat_nat @ G @ B3 )
=> ( inj_on8969904277767023793at_nat @ ( produc6977886695330630970at_nat @ F2 @ G )
@ ( produc457027306803732586at_nat @ A2
@ ^ [Uu: nat] : B3 ) ) ) ) ).
% map_prod_inj_on
thf(fact_985_subset__fst__snd,axiom,
! [A2: set_Product_prod_a_b] :
( ord_le817736998455962536od_a_b @ A2
@ ( product_Sigma_a_b @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ A2 )
@ ^ [Uu: a] : ( image_2802296252294471260_a_b_b @ product_snd_a_b @ A2 ) ) ) ).
% subset_fst_snd
thf(fact_986_image__paired__Times,axiom,
! [F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal,A2: set_Extended_ereal,B3: set_Extended_ereal] :
( ( image_959328165755419589_ereal
@ ( produc1215357860076780539_ereal
@ ^ [X: extended_ereal,Y: extended_ereal] : ( produc7614594614994623895_ereal @ ( F2 @ X ) @ ( G @ Y ) ) )
@ ( produc8095709571603465288_ereal @ A2
@ ^ [Uu: extended_ereal] : B3 ) )
= ( produc8095709571603465288_ereal @ ( image_6042159593519690757_ereal @ F2 @ A2 )
@ ^ [Uu: extended_ereal] : ( image_6042159593519690757_ereal @ G @ B3 ) ) ) ).
% image_paired_Times
thf(fact_987_image__paired__Times,axiom,
! [F2: extended_ereal > extended_ereal,G: nat > extended_ereal,A2: set_Extended_ereal,B3: set_nat] :
( ( image_6926512632986509267_ereal
@ ( produc976730521393319219_ereal
@ ^ [X: extended_ereal,Y: nat] : ( produc7614594614994623895_ereal @ ( F2 @ X ) @ ( G @ Y ) ) )
@ ( produc4220900302008768982al_nat @ A2
@ ^ [Uu: extended_ereal] : B3 ) )
= ( produc8095709571603465288_ereal @ ( image_6042159593519690757_ereal @ F2 @ A2 )
@ ^ [Uu: extended_ereal] : ( image_4309273772856505399_ereal @ G @ B3 ) ) ) ).
% image_paired_Times
thf(fact_988_image__paired__Times,axiom,
! [F2: extended_ereal > extended_ereal,G: nat > nat,A2: set_Extended_ereal,B3: set_nat] :
( ( image_2725866490533164705al_nat
@ ( produc4493823603722215745al_nat
@ ^ [X: extended_ereal,Y: nat] : ( produc5853503232740825159al_nat @ ( F2 @ X ) @ ( G @ Y ) ) )
@ ( produc4220900302008768982al_nat @ A2
@ ^ [Uu: extended_ereal] : B3 ) )
= ( produc4220900302008768982al_nat @ ( image_6042159593519690757_ereal @ F2 @ A2 )
@ ^ [Uu: extended_ereal] : ( image_nat_nat @ G @ B3 ) ) ) ).
% image_paired_Times
thf(fact_989_image__paired__Times,axiom,
! [F2: extended_ereal > extended_ereal,G: nat > rat,A2: set_Extended_ereal,B3: set_nat] :
( ( image_8441009968994543785al_rat
@ ( produc985595045328819017al_rat
@ ^ [X: extended_ereal,Y: nat] : ( produc5218373172654329423al_rat @ ( F2 @ X ) @ ( G @ Y ) ) )
@ ( produc4220900302008768982al_nat @ A2
@ ^ [Uu: extended_ereal] : B3 ) )
= ( produc3585770241922273246al_rat @ ( image_6042159593519690757_ereal @ F2 @ A2 )
@ ^ [Uu: extended_ereal] : ( image_nat_rat @ G @ B3 ) ) ) ).
% image_paired_Times
thf(fact_990_image__paired__Times,axiom,
! [F2: nat > extended_ereal,G: extended_ereal > extended_ereal,A2: set_nat,B3: set_Extended_ereal] :
( ( image_3845515196057536685_ereal
@ ( produc7030432313249384917_ereal
@ ^ [X: nat,Y: extended_ereal] : ( produc7614594614994623895_ereal @ ( F2 @ X ) @ ( G @ Y ) ) )
@ ( produc870331913724930228_ereal @ A2
@ ^ [Uu: nat] : B3 ) )
= ( produc8095709571603465288_ereal @ ( image_4309273772856505399_ereal @ F2 @ A2 )
@ ^ [Uu: extended_ereal] : ( image_6042159593519690757_ereal @ G @ B3 ) ) ) ).
% image_paired_Times
thf(fact_991_image__paired__Times,axiom,
! [F2: nat > extended_ereal,G: nat > extended_ereal,A2: set_nat,B3: set_nat] :
( ( image_7762317921424436459_ereal
@ ( produc3135378909069315545_ereal
@ ^ [X: nat,Y: nat] : ( produc7614594614994623895_ereal @ ( F2 @ X ) @ ( G @ Y ) ) )
@ ( produc457027306803732586at_nat @ A2
@ ^ [Uu: nat] : B3 ) )
= ( produc8095709571603465288_ereal @ ( image_4309273772856505399_ereal @ F2 @ A2 )
@ ^ [Uu: extended_ereal] : ( image_4309273772856505399_ereal @ G @ B3 ) ) ) ).
% image_paired_Times
thf(fact_992_image__paired__Times,axiom,
! [F2: nat > extended_ereal,G: nat > nat,A2: set_nat,B3: set_nat] :
( ( image_3171559211387932553al_nat
@ ( produc7146148067317756443al_nat
@ ^ [X: nat,Y: nat] : ( produc5853503232740825159al_nat @ ( F2 @ X ) @ ( G @ Y ) ) )
@ ( produc457027306803732586at_nat @ A2
@ ^ [Uu: nat] : B3 ) )
= ( produc4220900302008768982al_nat @ ( image_4309273772856505399_ereal @ F2 @ A2 )
@ ^ [Uu: extended_ereal] : ( image_nat_nat @ G @ B3 ) ) ) ).
% image_paired_Times
thf(fact_993_image__paired__Times,axiom,
! [F2: nat > extended_ereal,G: nat > rat,A2: set_nat,B3: set_nat] :
( ( image_8886702689849311633al_rat
@ ( produc3637919508924359715al_rat
@ ^ [X: nat,Y: nat] : ( produc5218373172654329423al_rat @ ( F2 @ X ) @ ( G @ Y ) ) )
@ ( produc457027306803732586at_nat @ A2
@ ^ [Uu: nat] : B3 ) )
= ( produc3585770241922273246al_rat @ ( image_4309273772856505399_ereal @ F2 @ A2 )
@ ^ [Uu: extended_ereal] : ( image_nat_rat @ G @ B3 ) ) ) ).
% image_paired_Times
thf(fact_994_image__paired__Times,axiom,
! [F2: nat > nat,G: extended_ereal > extended_ereal,A2: set_nat,B3: set_Extended_ereal] :
( ( image_8207957132699115757_ereal
@ ( produc1296637566603872709_ereal
@ ^ [X: nat,Y: extended_ereal] : ( produc2502934844456986405_ereal @ ( F2 @ X ) @ ( G @ Y ) ) )
@ ( produc870331913724930228_ereal @ A2
@ ^ [Uu: nat] : B3 ) )
= ( produc870331913724930228_ereal @ ( image_nat_nat @ F2 @ A2 )
@ ^ [Uu: nat] : ( image_6042159593519690757_ereal @ G @ B3 ) ) ) ).
% image_paired_Times
thf(fact_995_image__paired__Times,axiom,
! [F2: nat > nat,G: nat > extended_ereal,A2: set_nat,B3: set_nat] :
( ( image_5520118406512677935_ereal
@ ( produc271335225587726017_ereal
@ ^ [X: nat,Y: nat] : ( produc2502934844456986405_ereal @ ( F2 @ X ) @ ( G @ Y ) ) )
@ ( produc457027306803732586at_nat @ A2
@ ^ [Uu: nat] : B3 ) )
= ( produc870331913724930228_ereal @ ( image_nat_nat @ F2 @ A2 )
@ ^ [Uu: nat] : ( image_4309273772856505399_ereal @ G @ B3 ) ) ) ).
% image_paired_Times
thf(fact_996_subset__snd__imageI,axiom,
! [A2: set_a,B3: set_b,S2: set_Product_prod_a_b,X3: a] :
( ( ord_le817736998455962536od_a_b
@ ( product_Sigma_a_b @ A2
@ ^ [Uu: a] : B3 )
@ S2 )
=> ( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_set_b @ B3 @ ( image_2802296252294471260_a_b_b @ product_snd_a_b @ S2 ) ) ) ) ).
% subset_snd_imageI
thf(fact_997_subset__snd__imageI,axiom,
! [A2: set_a,B3: set_Extended_ereal,S2: set_Pr937448535721291095_ereal,X3: a] :
( ( ord_le8132973195874727159_ereal
@ ( produc7264369102385879704_ereal @ A2
@ ^ [Uu: a] : B3 )
@ S2 )
=> ( ( member_a @ X3 @ A2 )
=> ( ord_le1644982726543182158_ereal @ B3 @ ( image_5059165124122689180_ereal @ produc2480566465808596629_ereal @ S2 ) ) ) ) ).
% subset_snd_imageI
thf(fact_998_subset__snd__imageI,axiom,
! [A2: set_nat,B3: set_Extended_ereal,S2: set_Pr8411329518592215081_ereal,X3: nat] :
( ( ord_le3920121919471048841_ereal
@ ( produc870331913724930228_ereal @ A2
@ ^ [Uu: nat] : B3 )
@ S2 )
=> ( ( member_nat @ X3 @ A2 )
=> ( ord_le1644982726543182158_ereal @ B3 @ ( image_4404873737357082836_ereal @ produc2020276525179383671_ereal @ S2 ) ) ) ) ).
% subset_snd_imageI
thf(fact_999_subset__fst__imageI,axiom,
! [A2: set_a,B3: set_b,S2: set_Product_prod_a_b,Y4: b] :
( ( ord_le817736998455962536od_a_b
@ ( product_Sigma_a_b @ A2
@ ^ [Uu: a] : B3 )
@ S2 )
=> ( ( member_b @ Y4 @ B3 )
=> ( ord_less_eq_set_a @ A2 @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ S2 ) ) ) ) ).
% subset_fst_imageI
thf(fact_1000_subset__fst__imageI,axiom,
! [A2: set_Extended_ereal,B3: set_a,S2: set_Pr594667477129265975real_a,Y4: a] :
( ( ord_le7790192137282702039real_a
@ ( produc2583502849437520504real_a @ A2
@ ^ [Uu: extended_ereal] : B3 )
@ S2 )
=> ( ( member_a @ Y4 @ B3 )
=> ( ord_le1644982726543182158_ereal @ A2 @ ( image_5786011877969985724_ereal @ produc7562294591749751347real_a @ S2 ) ) ) ) ).
% subset_fst_imageI
thf(fact_1001_subset__fst__imageI,axiom,
! [A2: set_Extended_ereal,B3: set_nat,S2: set_Pr450698115992974979al_nat,Y4: nat] :
( ( ord_le5182862553726584547al_nat
@ ( produc4220900302008768982al_nat @ A2
@ ^ [Uu: extended_ereal] : B3 )
@ S2 )
=> ( ( member_nat @ Y4 @ B3 )
=> ( ord_le1644982726543182158_ereal @ A2 @ ( image_3315421938599550714_ereal @ produc3755854452330217819al_nat @ S2 ) ) ) ) ).
% subset_fst_imageI
thf(fact_1002_Ex__inj__on__UNION__Sigma,axiom,
! [A2: nat > set_nat,I3: set_nat] :
? [F4: nat > product_prod_nat_nat] :
( ( inj_on5538052773655684606at_nat @ F4 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ I3 ) ) )
& ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ F4 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ I3 ) ) ) @ ( produc457027306803732586at_nat @ I3 @ A2 ) ) ) ).
% Ex_inj_on_UNION_Sigma
thf(fact_1003_ball__UN,axiom,
! [B3: nat > set_nat,A2: set_nat,P2: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) )
=> ( P2 @ X ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ! [Y: nat] :
( ( member_nat @ Y @ ( B3 @ X ) )
=> ( P2 @ Y ) ) ) ) ) ).
% ball_UN
thf(fact_1004_bex__UN,axiom,
! [B3: nat > set_nat,A2: set_nat,P2: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) )
& ( P2 @ X ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ? [Y: nat] :
( ( member_nat @ Y @ ( B3 @ X ) )
& ( P2 @ Y ) ) ) ) ) ).
% bex_UN
thf(fact_1005_UN__ball__bex__simps_I2_J,axiom,
! [B3: nat > set_nat,A2: set_nat,P2: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) )
=> ( P2 @ X ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ! [Y: nat] :
( ( member_nat @ Y @ ( B3 @ X ) )
=> ( P2 @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_1006_UN__ball__bex__simps_I4_J,axiom,
! [B3: nat > set_nat,A2: set_nat,P2: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) )
& ( P2 @ X ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ? [Y: nat] :
( ( member_nat @ Y @ ( B3 @ X ) )
& ( P2 @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_1007_SUP__identity__eq,axiom,
! [A2: set_nat] :
( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X: nat] : X
@ A2 ) )
= ( complete_Sup_Sup_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_1008_SUP__identity__eq,axiom,
! [A2: set_Extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : X
@ A2 ) )
= ( comple8415311339701865915_ereal @ A2 ) ) ).
% SUP_identity_eq
thf(fact_1009_SUP__identity__eq,axiom,
! [A2: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [X: set_nat] : X
@ A2 ) )
= ( comple7399068483239264473et_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_1010_UN__I,axiom,
! [A: a,A2: set_a,B: a,B3: a > set_a] :
( ( member_a @ A @ A2 )
=> ( ( member_a @ B @ ( B3 @ A ) )
=> ( member_a @ B @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B3 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_1011_UN__I,axiom,
! [A: nat,A2: set_nat,B: a,B3: nat > set_a] :
( ( member_nat @ A @ A2 )
=> ( ( member_a @ B @ ( B3 @ A ) )
=> ( member_a @ B @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B3 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_1012_UN__I,axiom,
! [A: a,A2: set_a,B: nat,B3: a > set_nat] :
( ( member_a @ A @ A2 )
=> ( ( member_nat @ B @ ( B3 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B3 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_1013_UN__I,axiom,
! [A: nat,A2: set_nat,B: nat,B3: nat > set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat @ B @ ( B3 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_1014_UN__iff,axiom,
! [B: nat,B3: nat > set_nat,A2: set_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( member_nat @ B @ ( B3 @ X ) ) ) ) ) ).
% UN_iff
thf(fact_1015_SUP__id__eq,axiom,
! [A2: set_nat] :
( ( complete_Sup_Sup_nat @ ( image_nat_nat @ id_nat @ A2 ) )
= ( complete_Sup_Sup_nat @ A2 ) ) ).
% SUP_id_eq
thf(fact_1016_SUP__id__eq,axiom,
! [A2: set_Extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ id_Extended_ereal @ A2 ) )
= ( comple8415311339701865915_ereal @ A2 ) ) ).
% SUP_id_eq
thf(fact_1017_SUP__id__eq,axiom,
! [A2: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ id_set_nat @ A2 ) )
= ( comple7399068483239264473et_nat @ A2 ) ) ).
% SUP_id_eq
thf(fact_1018_UN__extend__simps_I9_J,axiom,
! [C3: nat > set_nat,B3: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C3 @ ( B3 @ X ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_1019_UN__extend__simps_I8_J,axiom,
! [B3: nat > set_nat,A2: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [Y: set_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ Y ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_1020_UN__E,axiom,
! [B: a,B3: a > set_a,A2: set_a] :
( ( member_a @ B @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B3 @ A2 ) ) )
=> ~ ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ~ ( member_a @ B @ ( B3 @ X2 ) ) ) ) ).
% UN_E
thf(fact_1021_UN__E,axiom,
! [B: a,B3: nat > set_a,A2: set_nat] :
( ( member_a @ B @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B3 @ A2 ) ) )
=> ~ ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ( member_a @ B @ ( B3 @ X2 ) ) ) ) ).
% UN_E
thf(fact_1022_UN__E,axiom,
! [B: nat,B3: a > set_nat,A2: set_a] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B3 @ A2 ) ) )
=> ~ ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ~ ( member_nat @ B @ ( B3 @ X2 ) ) ) ) ).
% UN_E
thf(fact_1023_UN__E,axiom,
! [B: nat,B3: nat > set_nat,A2: set_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) )
=> ~ ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ( member_nat @ B @ ( B3 @ X2 ) ) ) ) ).
% UN_E
thf(fact_1024_SUP__UNION,axiom,
! [F2: extended_ereal > extended_ereal,G: extended_ereal > set_Extended_ereal,A2: set_Extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ G @ A2 ) ) ) )
= ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [Y: extended_ereal] : ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_1025_SUP__UNION,axiom,
! [F2: extended_ereal > extended_ereal,G: nat > set_Extended_ereal,A2: set_nat] :
( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ ( comple4319282863272126363_ereal @ ( image_305533323056406039_ereal @ G @ A2 ) ) ) )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [Y: nat] : ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_1026_SUP__UNION,axiom,
! [F2: nat > extended_ereal,G: extended_ereal > set_nat,A2: set_Extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ ( comple7399068483239264473et_nat @ ( image_3090908713637162255et_nat @ G @ A2 ) ) ) )
= ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [Y: extended_ereal] : ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_1027_SUP__UNION,axiom,
! [F2: nat > extended_ereal,G: nat > set_nat,A2: set_nat] :
( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A2 ) ) ) )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [Y: nat] : ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_1028_SUP__UNION,axiom,
! [F2: nat > set_nat,G: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_1029_UN__UN__flatten,axiom,
! [C3: nat > set_nat,B3: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C3 @ ( B3 @ Y ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_1030_UN__Times__distrib,axiom,
! [E3: nat > set_nat,F5: nat > set_nat,A2: set_nat,B3: set_nat] :
( ( comple5685304695842803022at_nat
@ ( image_5423882813909314213at_nat
@ ( produc8197505143624133779at_nat
@ ^ [A3: nat,B2: nat] :
( produc457027306803732586at_nat @ ( E3 @ A3 )
@ ^ [Uu: nat] : ( F5 @ B2 ) ) )
@ ( produc457027306803732586at_nat @ A2
@ ^ [Uu: nat] : B3 ) ) )
= ( produc457027306803732586at_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ E3 @ A2 ) )
@ ^ [Uu: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F5 @ B3 ) ) ) ) ).
% UN_Times_distrib
thf(fact_1031_SUP__cong,axiom,
! [A2: set_nat,B3: set_nat,C3: nat > nat,D2: nat > nat] :
( ( A2 = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C3 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C3 @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D2 @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_1032_SUP__cong,axiom,
! [A2: set_Extended_ereal,B3: set_Extended_ereal,C3: extended_ereal > extended_ereal,D2: extended_ereal > extended_ereal] :
( ( A2 = B3 )
=> ( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ B3 )
=> ( ( C3 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ C3 @ A2 ) )
= ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ D2 @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_1033_SUP__cong,axiom,
! [A2: set_a,B3: set_a,C3: a > extended_ereal,D2: a > extended_ereal] :
( ( A2 = B3 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B3 )
=> ( ( C3 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ C3 @ A2 ) )
= ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ D2 @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_1034_SUP__cong,axiom,
! [A2: set_nat,B3: set_nat,C3: nat > extended_ereal,D2: nat > extended_ereal] :
( ( A2 = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C3 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ C3 @ A2 ) )
= ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ D2 @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_1035_SUP__cong,axiom,
! [A2: set_a,B3: set_a,C3: a > set_nat,D2: a > set_nat] :
( ( A2 = B3 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B3 )
=> ( ( C3 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_a_set_nat @ C3 @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_a_set_nat @ D2 @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_1036_SUP__cong,axiom,
! [A2: set_nat,B3: set_nat,C3: nat > set_nat,D2: nat > set_nat] :
( ( A2 = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C3 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C3 @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ D2 @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_1037_SUP__commute,axiom,
! [F2: extended_ereal > extended_ereal > extended_ereal,B3: set_Extended_ereal,A2: set_Extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ ( F2 @ I2 ) @ B3 ) )
@ A2 ) )
= ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [J: extended_ereal] :
( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( F2 @ I2 @ J )
@ A2 ) )
@ B3 ) ) ) ).
% SUP_commute
thf(fact_1038_SUP__commute,axiom,
! [F2: extended_ereal > nat > extended_ereal,B3: set_nat,A2: set_Extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ ( F2 @ I2 ) @ B3 ) )
@ A2 ) )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [J: nat] :
( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( F2 @ I2 @ J )
@ A2 ) )
@ B3 ) ) ) ).
% SUP_commute
thf(fact_1039_SUP__commute,axiom,
! [F2: nat > extended_ereal > extended_ereal,B3: set_Extended_ereal,A2: set_nat] :
( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ ( F2 @ I2 ) @ B3 ) )
@ A2 ) )
= ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [J: extended_ereal] :
( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( F2 @ I2 @ J )
@ A2 ) )
@ B3 ) ) ) ).
% SUP_commute
thf(fact_1040_SUP__commute,axiom,
! [F2: nat > nat > extended_ereal,B3: set_nat,A2: set_nat] :
( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ ( F2 @ I2 ) @ B3 ) )
@ A2 ) )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [J: nat] :
( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( F2 @ I2 @ J )
@ A2 ) )
@ B3 ) ) ) ).
% SUP_commute
thf(fact_1041_SUP__commute,axiom,
! [F2: nat > nat > set_nat,B3: set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I2: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F2 @ I2 ) @ B3 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [J: nat] :
( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I2: nat] : ( F2 @ I2 @ J )
@ A2 ) )
@ B3 ) ) ) ).
% SUP_commute
thf(fact_1042_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_1043_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_1044_Union__natural,axiom,
! [F2: nat > set_nat] :
( ( comp_s5357857527062720591et_nat @ comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F2 ) ) )
= ( comp_s1376360633444582883et_nat @ ( image_nat_set_nat @ F2 ) @ comple7399068483239264473et_nat ) ) ).
% Union_natural
thf(fact_1045_Union__natural,axiom,
! [F2: nat > rat] :
( ( comp_s703482200030188019et_nat @ comple3890839924845867745et_rat @ ( image_4408659257933336347et_rat @ ( image_nat_rat @ F2 ) ) )
= ( comp_s6435139744279249717et_nat @ ( image_nat_rat @ F2 ) @ comple7399068483239264473et_nat ) ) ).
% Union_natural
thf(fact_1046_Union__natural,axiom,
! [F2: nat > nat] :
( ( comp_s174380336271864291et_nat @ comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F2 ) ) )
= ( comp_s6993074788030935341et_nat @ ( image_nat_nat @ F2 ) @ comple7399068483239264473et_nat ) ) ).
% Union_natural
thf(fact_1047_image__Union,axiom,
! [F2: extended_ereal > extended_ereal,S2: set_se6634062954251873166_ereal] :
( ( image_6042159593519690757_ereal @ F2 @ ( comple4319282863272126363_ereal @ S2 ) )
= ( comple4319282863272126363_ereal @ ( image_6293272304431515653_ereal @ ( image_6042159593519690757_ereal @ F2 ) @ S2 ) ) ) ).
% image_Union
thf(fact_1048_image__Union,axiom,
! [F2: nat > extended_ereal,S2: set_set_nat] :
( ( image_4309273772856505399_ereal @ F2 @ ( comple7399068483239264473et_nat @ S2 ) )
= ( comple4319282863272126363_ereal @ ( image_8825259783980156129_ereal @ ( image_4309273772856505399_ereal @ F2 ) @ S2 ) ) ) ).
% image_Union
thf(fact_1049_image__Union,axiom,
! [F2: nat > set_nat,S2: set_set_nat] :
( ( image_nat_set_nat @ F2 @ ( comple7399068483239264473et_nat @ S2 ) )
= ( comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F2 ) @ S2 ) ) ) ).
% image_Union
thf(fact_1050_image__Union,axiom,
! [F2: nat > rat,S2: set_set_nat] :
( ( image_nat_rat @ F2 @ ( comple7399068483239264473et_nat @ S2 ) )
= ( comple3890839924845867745et_rat @ ( image_4408659257933336347et_rat @ ( image_nat_rat @ F2 ) @ S2 ) ) ) ).
% image_Union
thf(fact_1051_image__Union,axiom,
! [F2: nat > nat,S2: set_set_nat] :
( ( image_nat_nat @ F2 @ ( comple7399068483239264473et_nat @ S2 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F2 ) @ S2 ) ) ) ).
% image_Union
thf(fact_1052_image__UN,axiom,
! [F2: nat > extended_ereal,B3: nat > set_nat,A2: set_nat] :
( ( image_4309273772856505399_ereal @ F2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) )
= ( comple4319282863272126363_ereal
@ ( image_305533323056406039_ereal
@ ^ [X: nat] : ( image_4309273772856505399_ereal @ F2 @ ( B3 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_1053_image__UN,axiom,
! [F2: nat > set_nat,B3: nat > set_nat,A2: set_nat] :
( ( image_nat_set_nat @ F2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) )
= ( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [X: nat] : ( image_nat_set_nat @ F2 @ ( B3 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_1054_image__UN,axiom,
! [F2: nat > rat,B3: nat > set_nat,A2: set_nat] :
( ( image_nat_rat @ F2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) )
= ( comple3890839924845867745et_rat
@ ( image_nat_set_rat
@ ^ [X: nat] : ( image_nat_rat @ F2 @ ( B3 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_1055_image__UN,axiom,
! [F2: nat > nat,B3: nat > set_nat,A2: set_nat] :
( ( image_nat_nat @ F2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( image_nat_nat @ F2 @ ( B3 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_1056_UN__extend__simps_I10_J,axiom,
! [B3: extended_ereal > set_nat,F2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( comple7399068483239264473et_nat
@ ( image_3090908713637162255et_nat
@ ^ [A3: extended_ereal] : ( B3 @ ( F2 @ A3 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_3090908713637162255et_nat @ B3 @ ( image_6042159593519690757_ereal @ F2 @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_1057_UN__extend__simps_I10_J,axiom,
! [B3: extended_ereal > set_nat,F2: nat > extended_ereal,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A3: nat] : ( B3 @ ( F2 @ A3 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_3090908713637162255et_nat @ B3 @ ( image_4309273772856505399_ereal @ F2 @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_1058_UN__extend__simps_I10_J,axiom,
! [B3: set_nat > set_nat,F2: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A3: nat] : ( B3 @ ( F2 @ A3 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B3 @ ( image_nat_set_nat @ F2 @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_1059_UN__extend__simps_I10_J,axiom,
! [B3: rat > set_nat,F2: nat > rat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A3: nat] : ( B3 @ ( F2 @ A3 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_rat_set_nat @ B3 @ ( image_nat_rat @ F2 @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_1060_UN__extend__simps_I10_J,axiom,
! [B3: nat > set_nat,F2: nat > nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A3: nat] : ( B3 @ ( F2 @ A3 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ ( image_nat_nat @ F2 @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_1061_UN__subset__iff,axiom,
! [A2: nat > set_nat,I3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ I3 ) ) @ B3 )
= ( ! [X: nat] :
( ( member_nat @ X @ I3 )
=> ( ord_less_eq_set_nat @ ( A2 @ X ) @ B3 ) ) ) ) ).
% UN_subset_iff
thf(fact_1062_UN__upper,axiom,
! [A: a,A2: set_a,B3: a > set_Extended_ereal] :
( ( member_a @ A @ A2 )
=> ( ord_le1644982726543182158_ereal @ ( B3 @ A ) @ ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ B3 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_1063_UN__upper,axiom,
! [A: nat,A2: set_nat,B3: nat > set_Extended_ereal] :
( ( member_nat @ A @ A2 )
=> ( ord_le1644982726543182158_ereal @ ( B3 @ A ) @ ( comple4319282863272126363_ereal @ ( image_305533323056406039_ereal @ B3 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_1064_UN__upper,axiom,
! [A: a,A2: set_a,B3: a > set_nat] :
( ( member_a @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( B3 @ A ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B3 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_1065_UN__upper,axiom,
! [A: nat,A2: set_nat,B3: nat > set_nat] :
( ( member_nat @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( B3 @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_1066_UN__least,axiom,
! [A2: set_a,B3: a > set_Extended_ereal,C3: set_Extended_ereal] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_le1644982726543182158_ereal @ ( B3 @ X2 ) @ C3 ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ B3 @ A2 ) ) @ C3 ) ) ).
% UN_least
thf(fact_1067_UN__least,axiom,
! [A2: set_nat,B3: nat > set_Extended_ereal,C3: set_Extended_ereal] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_le1644982726543182158_ereal @ ( B3 @ X2 ) @ C3 ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ ( image_305533323056406039_ereal @ B3 @ A2 ) ) @ C3 ) ) ).
% UN_least
thf(fact_1068_UN__least,axiom,
! [A2: set_a,B3: a > set_nat,C3: set_nat] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( B3 @ X2 ) @ C3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B3 @ A2 ) ) @ C3 ) ) ).
% UN_least
thf(fact_1069_UN__least,axiom,
! [A2: set_nat,B3: nat > set_nat,C3: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( B3 @ X2 ) @ C3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) @ C3 ) ) ).
% UN_least
thf(fact_1070_UN__mono,axiom,
! [A2: set_a,B3: set_a,F2: a > set_Extended_ereal,G: a > set_Extended_ereal] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ F2 @ A2 ) ) @ ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ G @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_1071_UN__mono,axiom,
! [A2: set_nat,B3: set_nat,F2: nat > set_Extended_ereal,G: nat > set_Extended_ereal] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ ( image_305533323056406039_ereal @ F2 @ A2 ) ) @ ( comple4319282863272126363_ereal @ ( image_305533323056406039_ereal @ G @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_1072_UN__mono,axiom,
! [A2: set_Extended_ereal,B3: set_Extended_ereal,F2: extended_ereal > set_Extended_ereal,G: extended_ereal > set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A2 @ B3 )
=> ( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A2 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ F2 @ A2 ) ) @ ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ G @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_1073_UN__mono,axiom,
! [A2: set_a,B3: set_a,F2: a > set_nat,G: a > set_nat] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F2 @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F2 @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ G @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_1074_UN__mono,axiom,
! [A2: set_nat,B3: set_nat,F2: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F2 @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_1075_UN__mono,axiom,
! [A2: set_Extended_ereal,B3: set_Extended_ereal,F2: extended_ereal > set_nat,G: extended_ereal > set_nat] :
( ( ord_le1644982726543182158_ereal @ A2 @ B3 )
=> ( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F2 @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_3090908713637162255et_nat @ F2 @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_3090908713637162255et_nat @ G @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_1076_SUP__eq,axiom,
! [A2: set_Extended_ereal,B3: set_Extended_ereal,F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal] :
( ! [I4: extended_ereal] :
( ( member2350847679896131959_ereal @ I4 @ A2 )
=> ? [X6: extended_ereal] :
( ( member2350847679896131959_ereal @ X6 @ B3 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: extended_ereal] :
( ( member2350847679896131959_ereal @ J2 @ B3 )
=> ? [X6: extended_ereal] :
( ( member2350847679896131959_ereal @ X6 @ A2 )
& ( ord_le1083603963089353582_ereal @ ( G @ J2 ) @ ( F2 @ X6 ) ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A2 ) )
= ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_1077_SUP__eq,axiom,
! [A2: set_Extended_ereal,B3: set_a,F2: extended_ereal > extended_ereal,G: a > extended_ereal] :
( ! [I4: extended_ereal] :
( ( member2350847679896131959_ereal @ I4 @ A2 )
=> ? [X6: a] :
( ( member_a @ X6 @ B3 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B3 )
=> ? [X6: extended_ereal] :
( ( member2350847679896131959_ereal @ X6 @ A2 )
& ( ord_le1083603963089353582_ereal @ ( G @ J2 ) @ ( F2 @ X6 ) ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A2 ) )
= ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_1078_SUP__eq,axiom,
! [A2: set_Extended_ereal,B3: set_nat,F2: extended_ereal > extended_ereal,G: nat > extended_ereal] :
( ! [I4: extended_ereal] :
( ( member2350847679896131959_ereal @ I4 @ A2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B3 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B3 )
=> ? [X6: extended_ereal] :
( ( member2350847679896131959_ereal @ X6 @ A2 )
& ( ord_le1083603963089353582_ereal @ ( G @ J2 ) @ ( F2 @ X6 ) ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A2 ) )
= ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_1079_SUP__eq,axiom,
! [A2: set_a,B3: set_Extended_ereal,F2: a > extended_ereal,G: extended_ereal > extended_ereal] :
( ! [I4: a] :
( ( member_a @ I4 @ A2 )
=> ? [X6: extended_ereal] :
( ( member2350847679896131959_ereal @ X6 @ B3 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: extended_ereal] :
( ( member2350847679896131959_ereal @ J2 @ B3 )
=> ? [X6: a] :
( ( member_a @ X6 @ A2 )
& ( ord_le1083603963089353582_ereal @ ( G @ J2 ) @ ( F2 @ X6 ) ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ F2 @ A2 ) )
= ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_1080_SUP__eq,axiom,
! [A2: set_a,B3: set_a,F2: a > extended_ereal,G: a > extended_ereal] :
( ! [I4: a] :
( ( member_a @ I4 @ A2 )
=> ? [X6: a] :
( ( member_a @ X6 @ B3 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B3 )
=> ? [X6: a] :
( ( member_a @ X6 @ A2 )
& ( ord_le1083603963089353582_ereal @ ( G @ J2 ) @ ( F2 @ X6 ) ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ F2 @ A2 ) )
= ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_1081_SUP__eq,axiom,
! [A2: set_a,B3: set_nat,F2: a > extended_ereal,G: nat > extended_ereal] :
( ! [I4: a] :
( ( member_a @ I4 @ A2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B3 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B3 )
=> ? [X6: a] :
( ( member_a @ X6 @ A2 )
& ( ord_le1083603963089353582_ereal @ ( G @ J2 ) @ ( F2 @ X6 ) ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ F2 @ A2 ) )
= ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_1082_SUP__eq,axiom,
! [A2: set_nat,B3: set_Extended_ereal,F2: nat > extended_ereal,G: extended_ereal > extended_ereal] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ? [X6: extended_ereal] :
( ( member2350847679896131959_ereal @ X6 @ B3 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: extended_ereal] :
( ( member2350847679896131959_ereal @ J2 @ B3 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A2 )
& ( ord_le1083603963089353582_ereal @ ( G @ J2 ) @ ( F2 @ X6 ) ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
= ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_1083_SUP__eq,axiom,
! [A2: set_nat,B3: set_a,F2: nat > extended_ereal,G: a > extended_ereal] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ? [X6: a] :
( ( member_a @ X6 @ B3 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B3 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A2 )
& ( ord_le1083603963089353582_ereal @ ( G @ J2 ) @ ( F2 @ X6 ) ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
= ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_1084_SUP__eq,axiom,
! [A2: set_nat,B3: set_nat,F2: nat > extended_ereal,G: nat > extended_ereal] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B3 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B3 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A2 )
& ( ord_le1083603963089353582_ereal @ ( G @ J2 ) @ ( F2 @ X6 ) ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
= ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_1085_SUP__eq,axiom,
! [A2: set_a,B3: set_a,F2: a > set_Extended_ereal,G: a > set_Extended_ereal] :
( ! [I4: a] :
( ( member_a @ I4 @ A2 )
=> ? [X6: a] :
( ( member_a @ X6 @ B3 )
& ( ord_le1644982726543182158_ereal @ ( F2 @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B3 )
=> ? [X6: a] :
( ( member_a @ X6 @ A2 )
& ( ord_le1644982726543182158_ereal @ ( G @ J2 ) @ ( F2 @ X6 ) ) ) )
=> ( ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ F2 @ A2 ) )
= ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_1086_SUP__image,axiom,
! [G: extended_ereal > nat,F2: nat > extended_ereal,A2: set_nat] :
( ( complete_Sup_Sup_nat @ ( image_7659842161140344153al_nat @ G @ ( image_4309273772856505399_ereal @ F2 @ A2 ) ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ ( comp_E7502005551946643277at_nat @ G @ F2 ) @ A2 ) ) ) ).
% SUP_image
thf(fact_1087_SUP__image,axiom,
! [G: rat > nat,F2: nat > rat,A2: set_nat] :
( ( complete_Sup_Sup_nat @ ( image_rat_nat @ G @ ( image_nat_rat @ F2 @ A2 ) ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ ( comp_rat_nat_nat @ G @ F2 ) @ A2 ) ) ) ).
% SUP_image
thf(fact_1088_SUP__image,axiom,
! [G: nat > nat,F2: nat > nat,A2: set_nat] :
( ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ ( image_nat_nat @ F2 @ A2 ) ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F2 ) @ A2 ) ) ) ).
% SUP_image
thf(fact_1089_SUP__image,axiom,
! [G: rat > extended_ereal,F2: nat > rat,A2: set_nat] :
( ( comple8415311339701865915_ereal @ ( image_2592109325025016879_ereal @ G @ ( image_nat_rat @ F2 @ A2 ) ) )
= ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ ( comp_r4319880827473671715al_nat @ G @ F2 ) @ A2 ) ) ) ).
% SUP_image
thf(fact_1090_SUP__image,axiom,
! [G: extended_ereal > extended_ereal,F2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ G @ ( image_6042159593519690757_ereal @ F2 @ A2 ) ) )
= ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ ( comp_E9177254828515427499_ereal @ G @ F2 ) @ A2 ) ) ) ).
% SUP_image
thf(fact_1091_SUP__image,axiom,
! [G: extended_ereal > extended_ereal,F2: nat > extended_ereal,A2: set_nat] :
( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ G @ ( image_4309273772856505399_ereal @ F2 @ A2 ) ) )
= ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ ( comp_E3726099860353345075al_nat @ G @ F2 ) @ A2 ) ) ) ).
% SUP_image
thf(fact_1092_SUP__image,axiom,
! [G: nat > extended_ereal,F2: extended_ereal > nat,A2: set_Extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ ( image_7659842161140344153al_nat @ F2 @ A2 ) ) )
= ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ ( comp_n261702227720650419_ereal @ G @ F2 ) @ A2 ) ) ) ).
% SUP_image
thf(fact_1093_SUP__image,axiom,
! [G: nat > extended_ereal,F2: nat > nat,A2: set_nat] :
( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ ( image_nat_nat @ F2 @ A2 ) ) )
= ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ ( comp_n13370146242399787al_nat @ G @ F2 ) @ A2 ) ) ) ).
% SUP_image
thf(fact_1094_SUP__image,axiom,
! [G: set_nat > nat,F2: nat > set_nat,A2: set_nat] :
( ( complete_Sup_Sup_nat @ ( image_set_nat_nat @ G @ ( image_nat_set_nat @ F2 @ A2 ) ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ ( comp_set_nat_nat_nat @ G @ F2 ) @ A2 ) ) ) ).
% SUP_image
thf(fact_1095_SUP__image,axiom,
! [G: set_nat > extended_ereal,F2: nat > set_nat,A2: set_nat] :
( ( comple8415311339701865915_ereal @ ( image_390863237321605889_ereal @ G @ ( image_nat_set_nat @ F2 @ A2 ) ) )
= ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ ( comp_s7405936075137034485al_nat @ G @ F2 ) @ A2 ) ) ) ).
% SUP_image
thf(fact_1096_inj__on__image,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_se6634062954251873166_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ ( comple4319282863272126363_ereal @ A2 ) )
=> ( inj_on5406440306785145713_ereal @ ( image_6042159593519690757_ereal @ F2 ) @ A2 ) ) ).
% inj_on_image
thf(fact_1097_inj__on__image,axiom,
! [F2: nat > extended_ereal,A2: set_set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ ( comple7399068483239264473et_nat @ A2 ) )
=> ( inj_on1463964778812548213_ereal @ ( image_4309273772856505399_ereal @ F2 ) @ A2 ) ) ).
% inj_on_image
thf(fact_1098_inj__on__image,axiom,
! [F2: nat > set_nat,A2: set_set_nat] :
( ( inj_on_nat_set_nat @ F2 @ ( comple7399068483239264473et_nat @ A2 ) )
=> ( inj_on2776966659131765557et_nat @ ( image_nat_set_nat @ F2 ) @ A2 ) ) ).
% inj_on_image
thf(fact_1099_inj__on__image,axiom,
! [F2: nat > rat,A2: set_set_nat] :
( ( inj_on_nat_rat @ F2 @ ( comple7399068483239264473et_nat @ A2 ) )
=> ( inj_on1096178645466186887et_rat @ ( image_nat_rat @ F2 ) @ A2 ) ) ).
% inj_on_image
thf(fact_1100_inj__on__image,axiom,
! [F2: nat > nat,A2: set_set_nat] :
( ( inj_on_nat_nat @ F2 @ ( comple7399068483239264473et_nat @ A2 ) )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F2 ) @ A2 ) ) ).
% inj_on_image
thf(fact_1101_SUP__upper2,axiom,
! [I: a,A2: set_a,U2: set_Extended_ereal,F2: a > set_Extended_ereal] :
( ( member_a @ I @ A2 )
=> ( ( ord_le1644982726543182158_ereal @ U2 @ ( F2 @ I ) )
=> ( ord_le1644982726543182158_ereal @ U2 @ ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ F2 @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_1102_SUP__upper2,axiom,
! [I: nat,A2: set_nat,U2: set_Extended_ereal,F2: nat > set_Extended_ereal] :
( ( member_nat @ I @ A2 )
=> ( ( ord_le1644982726543182158_ereal @ U2 @ ( F2 @ I ) )
=> ( ord_le1644982726543182158_ereal @ U2 @ ( comple4319282863272126363_ereal @ ( image_305533323056406039_ereal @ F2 @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_1103_SUP__upper2,axiom,
! [I: extended_ereal,A2: set_Extended_ereal,U2: extended_ereal,F2: extended_ereal > extended_ereal] :
( ( member2350847679896131959_ereal @ I @ A2 )
=> ( ( ord_le1083603963089353582_ereal @ U2 @ ( F2 @ I ) )
=> ( ord_le1083603963089353582_ereal @ U2 @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_1104_SUP__upper2,axiom,
! [I: a,A2: set_a,U2: extended_ereal,F2: a > extended_ereal] :
( ( member_a @ I @ A2 )
=> ( ( ord_le1083603963089353582_ereal @ U2 @ ( F2 @ I ) )
=> ( ord_le1083603963089353582_ereal @ U2 @ ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ F2 @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_1105_SUP__upper2,axiom,
! [I: nat,A2: set_nat,U2: extended_ereal,F2: nat > extended_ereal] :
( ( member_nat @ I @ A2 )
=> ( ( ord_le1083603963089353582_ereal @ U2 @ ( F2 @ I ) )
=> ( ord_le1083603963089353582_ereal @ U2 @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_1106_SUP__upper2,axiom,
! [I: a,A2: set_a,U2: set_nat,F2: a > set_nat] :
( ( member_a @ I @ A2 )
=> ( ( ord_less_eq_set_nat @ U2 @ ( F2 @ I ) )
=> ( ord_less_eq_set_nat @ U2 @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F2 @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_1107_SUP__upper2,axiom,
! [I: nat,A2: set_nat,U2: set_nat,F2: nat > set_nat] :
( ( member_nat @ I @ A2 )
=> ( ( ord_less_eq_set_nat @ U2 @ ( F2 @ I ) )
=> ( ord_less_eq_set_nat @ U2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_1108_SUP__le__iff,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,U2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A2 ) ) @ U2 )
= ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ X ) @ U2 ) ) ) ) ).
% SUP_le_iff
thf(fact_1109_SUP__le__iff,axiom,
! [F2: nat > extended_ereal,A2: set_nat,U2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A2 ) ) @ U2 )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ X ) @ U2 ) ) ) ) ).
% SUP_le_iff
thf(fact_1110_SUP__le__iff,axiom,
! [F2: nat > set_nat,A2: set_nat,U2: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A2 ) ) @ U2 )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F2 @ X ) @ U2 ) ) ) ) ).
% SUP_le_iff
thf(fact_1111_SUP__upper,axiom,
! [I: a,A2: set_a,F2: a > set_Extended_ereal] :
( ( member_a @ I @ A2 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ I ) @ ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ F2 @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1112_SUP__upper,axiom,
! [I: nat,A2: set_nat,F2: nat > set_Extended_ereal] :
( ( member_nat @ I @ A2 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ I ) @ ( comple4319282863272126363_ereal @ ( image_305533323056406039_ereal @ F2 @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1113_SUP__upper,axiom,
! [I: extended_ereal,A2: set_Extended_ereal,F2: extended_ereal > extended_ereal] :
( ( member2350847679896131959_ereal @ I @ A2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I ) @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1114_SUP__upper,axiom,
! [I: a,A2: set_a,F2: a > extended_ereal] :
( ( member_a @ I @ A2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I ) @ ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ F2 @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1115_SUP__upper,axiom,
! [I: nat,A2: set_nat,F2: nat > extended_ereal] :
( ( member_nat @ I @ A2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1116_SUP__upper,axiom,
! [I: a,A2: set_a,F2: a > set_nat] :
( ( member_a @ I @ A2 )
=> ( ord_less_eq_set_nat @ ( F2 @ I ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F2 @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1117_SUP__upper,axiom,
! [I: nat,A2: set_nat,F2: nat > set_nat] :
( ( member_nat @ I @ A2 )
=> ( ord_less_eq_set_nat @ ( F2 @ I ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1118_SUP__mono_H,axiom,
! [F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ! [X2: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( F2 @ X2 ) @ ( G @ X2 ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A2 ) ) @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ G @ A2 ) ) ) ) ).
% SUP_mono'
thf(fact_1119_SUP__mono_H,axiom,
! [F2: nat > extended_ereal,G: nat > extended_ereal,A2: set_nat] :
( ! [X2: nat] : ( ord_le1083603963089353582_ereal @ ( F2 @ X2 ) @ ( G @ X2 ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A2 ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ A2 ) ) ) ) ).
% SUP_mono'
thf(fact_1120_SUP__mono_H,axiom,
! [F2: nat > set_nat,G: nat > set_nat,A2: set_nat] :
( ! [X2: nat] : ( ord_less_eq_set_nat @ ( F2 @ X2 ) @ ( G @ X2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A2 ) ) ) ) ).
% SUP_mono'
thf(fact_1121_SUP__least,axiom,
! [A2: set_a,F2: a > set_Extended_ereal,U2: set_Extended_ereal] :
( ! [I4: a] :
( ( member_a @ I4 @ A2 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ I4 ) @ U2 ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ F2 @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1122_SUP__least,axiom,
! [A2: set_nat,F2: nat > set_Extended_ereal,U2: set_Extended_ereal] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ I4 ) @ U2 ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ ( image_305533323056406039_ereal @ F2 @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1123_SUP__least,axiom,
! [A2: set_Extended_ereal,F2: extended_ereal > extended_ereal,U2: extended_ereal] :
( ! [I4: extended_ereal] :
( ( member2350847679896131959_ereal @ I4 @ A2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I4 ) @ U2 ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1124_SUP__least,axiom,
! [A2: set_a,F2: a > extended_ereal,U2: extended_ereal] :
( ! [I4: a] :
( ( member_a @ I4 @ A2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I4 ) @ U2 ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ F2 @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1125_SUP__least,axiom,
! [A2: set_nat,F2: nat > extended_ereal,U2: extended_ereal] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I4 ) @ U2 ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1126_SUP__least,axiom,
! [A2: set_a,F2: a > set_nat,U2: set_nat] :
( ! [I4: a] :
( ( member_a @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F2 @ I4 ) @ U2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F2 @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1127_SUP__least,axiom,
! [A2: set_nat,F2: nat > set_nat,U2: set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F2 @ I4 ) @ U2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1128_SUP__mono,axiom,
! [A2: set_Extended_ereal,B3: set_Extended_ereal,F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal] :
( ! [N2: extended_ereal] :
( ( member2350847679896131959_ereal @ N2 @ A2 )
=> ? [X6: extended_ereal] :
( ( member2350847679896131959_ereal @ X6 @ B3 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ N2 ) @ ( G @ X6 ) ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A2 ) ) @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ G @ B3 ) ) ) ) ).
% SUP_mono
thf(fact_1129_SUP__mono,axiom,
! [A2: set_Extended_ereal,B3: set_nat,F2: extended_ereal > extended_ereal,G: nat > extended_ereal] :
( ! [N2: extended_ereal] :
( ( member2350847679896131959_ereal @ N2 @ A2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B3 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ N2 ) @ ( G @ X6 ) ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A2 ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ B3 ) ) ) ) ).
% SUP_mono
thf(fact_1130_SUP__mono,axiom,
! [A2: set_a,B3: set_Extended_ereal,F2: a > extended_ereal,G: extended_ereal > extended_ereal] :
( ! [N2: a] :
( ( member_a @ N2 @ A2 )
=> ? [X6: extended_ereal] :
( ( member2350847679896131959_ereal @ X6 @ B3 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ N2 ) @ ( G @ X6 ) ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ F2 @ A2 ) ) @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ G @ B3 ) ) ) ) ).
% SUP_mono
thf(fact_1131_SUP__mono,axiom,
! [A2: set_a,B3: set_nat,F2: a > extended_ereal,G: nat > extended_ereal] :
( ! [N2: a] :
( ( member_a @ N2 @ A2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B3 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ N2 ) @ ( G @ X6 ) ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ F2 @ A2 ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ B3 ) ) ) ) ).
% SUP_mono
thf(fact_1132_SUP__mono,axiom,
! [A2: set_nat,B3: set_Extended_ereal,F2: nat > extended_ereal,G: extended_ereal > extended_ereal] :
( ! [N2: nat] :
( ( member_nat @ N2 @ A2 )
=> ? [X6: extended_ereal] :
( ( member2350847679896131959_ereal @ X6 @ B3 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ N2 ) @ ( G @ X6 ) ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A2 ) ) @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ G @ B3 ) ) ) ) ).
% SUP_mono
thf(fact_1133_SUP__mono,axiom,
! [A2: set_nat,B3: set_nat,F2: nat > extended_ereal,G: nat > extended_ereal] :
( ! [N2: nat] :
( ( member_nat @ N2 @ A2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B3 )
& ( ord_le1083603963089353582_ereal @ ( F2 @ N2 ) @ ( G @ X6 ) ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A2 ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ B3 ) ) ) ) ).
% SUP_mono
thf(fact_1134_SUP__mono,axiom,
! [A2: set_a,B3: set_nat,F2: a > set_nat,G: nat > set_nat] :
( ! [N2: a] :
( ( member_a @ N2 @ A2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B3 )
& ( ord_less_eq_set_nat @ ( F2 @ N2 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F2 @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ).
% SUP_mono
thf(fact_1135_SUP__mono,axiom,
! [A2: set_nat,B3: set_nat,F2: nat > set_nat,G: nat > set_nat] :
( ! [N2: nat] :
( ( member_nat @ N2 @ A2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B3 )
& ( ord_less_eq_set_nat @ ( F2 @ N2 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ).
% SUP_mono
thf(fact_1136_SUP__eqI,axiom,
! [A2: set_a,F2: a > set_Extended_ereal,X3: set_Extended_ereal] :
( ! [I4: a] :
( ( member_a @ I4 @ A2 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ I4 ) @ X3 ) )
=> ( ! [Y2: set_Extended_ereal] :
( ! [I5: a] :
( ( member_a @ I5 @ A2 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ I5 ) @ Y2 ) )
=> ( ord_le1644982726543182158_ereal @ X3 @ Y2 ) )
=> ( ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ F2 @ A2 ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_1137_SUP__eqI,axiom,
! [A2: set_nat,F2: nat > set_Extended_ereal,X3: set_Extended_ereal] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ I4 ) @ X3 ) )
=> ( ! [Y2: set_Extended_ereal] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A2 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ I5 ) @ Y2 ) )
=> ( ord_le1644982726543182158_ereal @ X3 @ Y2 ) )
=> ( ( comple4319282863272126363_ereal @ ( image_305533323056406039_ereal @ F2 @ A2 ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_1138_SUP__eqI,axiom,
! [A2: set_Extended_ereal,F2: extended_ereal > extended_ereal,X3: extended_ereal] :
( ! [I4: extended_ereal] :
( ( member2350847679896131959_ereal @ I4 @ A2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I4 ) @ X3 ) )
=> ( ! [Y2: extended_ereal] :
( ! [I5: extended_ereal] :
( ( member2350847679896131959_ereal @ I5 @ A2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I5 ) @ Y2 ) )
=> ( ord_le1083603963089353582_ereal @ X3 @ Y2 ) )
=> ( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A2 ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_1139_SUP__eqI,axiom,
! [A2: set_a,F2: a > extended_ereal,X3: extended_ereal] :
( ! [I4: a] :
( ( member_a @ I4 @ A2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I4 ) @ X3 ) )
=> ( ! [Y2: extended_ereal] :
( ! [I5: a] :
( ( member_a @ I5 @ A2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I5 ) @ Y2 ) )
=> ( ord_le1083603963089353582_ereal @ X3 @ Y2 ) )
=> ( ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ F2 @ A2 ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_1140_SUP__eqI,axiom,
! [A2: set_nat,F2: nat > extended_ereal,X3: extended_ereal] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I4 ) @ X3 ) )
=> ( ! [Y2: extended_ereal] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ I5 ) @ Y2 ) )
=> ( ord_le1083603963089353582_ereal @ X3 @ Y2 ) )
=> ( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A2 ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_1141_SUP__eqI,axiom,
! [A2: set_a,F2: a > set_nat,X3: set_nat] :
( ! [I4: a] :
( ( member_a @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F2 @ I4 ) @ X3 ) )
=> ( ! [Y2: set_nat] :
( ! [I5: a] :
( ( member_a @ I5 @ A2 )
=> ( ord_less_eq_set_nat @ ( F2 @ I5 ) @ Y2 ) )
=> ( ord_less_eq_set_nat @ X3 @ Y2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F2 @ A2 ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_1142_SUP__eqI,axiom,
! [A2: set_nat,F2: nat > set_nat,X3: set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F2 @ I4 ) @ X3 ) )
=> ( ! [Y2: set_nat] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A2 )
=> ( ord_less_eq_set_nat @ ( F2 @ I5 ) @ Y2 ) )
=> ( ord_less_eq_set_nat @ X3 @ Y2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A2 ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_1143_inj__on__UNION__chain,axiom,
! [I3: set_a,A2: a > set_Extended_ereal,F2: extended_ereal > extended_ereal] :
( ! [I4: a,J2: a] :
( ( member_a @ I4 @ I3 )
=> ( ( member_a @ J2 @ I3 )
=> ( ( ord_le1644982726543182158_ereal @ ( A2 @ I4 ) @ ( A2 @ J2 ) )
| ( ord_le1644982726543182158_ereal @ ( A2 @ J2 ) @ ( A2 @ I4 ) ) ) ) )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( inj_on7162434037990268785_ereal @ F2 @ ( A2 @ I4 ) ) )
=> ( inj_on7162434037990268785_ereal @ F2 @ ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ A2 @ I3 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_1144_inj__on__UNION__chain,axiom,
! [I3: set_nat,A2: nat > set_Extended_ereal,F2: extended_ereal > extended_ereal] :
( ! [I4: nat,J2: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( member_nat @ J2 @ I3 )
=> ( ( ord_le1644982726543182158_ereal @ ( A2 @ I4 ) @ ( A2 @ J2 ) )
| ( ord_le1644982726543182158_ereal @ ( A2 @ J2 ) @ ( A2 @ I4 ) ) ) ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( inj_on7162434037990268785_ereal @ F2 @ ( A2 @ I4 ) ) )
=> ( inj_on7162434037990268785_ereal @ F2 @ ( comple4319282863272126363_ereal @ ( image_305533323056406039_ereal @ A2 @ I3 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_1145_inj__on__UNION__chain,axiom,
! [I3: set_a,A2: a > set_nat,F2: nat > nat] :
( ! [I4: a,J2: a] :
( ( member_a @ I4 @ I3 )
=> ( ( member_a @ J2 @ I3 )
=> ( ( ord_less_eq_set_nat @ ( A2 @ I4 ) @ ( A2 @ J2 ) )
| ( ord_less_eq_set_nat @ ( A2 @ J2 ) @ ( A2 @ I4 ) ) ) ) )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( inj_on_nat_nat @ F2 @ ( A2 @ I4 ) ) )
=> ( inj_on_nat_nat @ F2 @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ A2 @ I3 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_1146_inj__on__UNION__chain,axiom,
! [I3: set_nat,A2: nat > set_nat,F2: nat > nat] :
( ! [I4: nat,J2: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( member_nat @ J2 @ I3 )
=> ( ( ord_less_eq_set_nat @ ( A2 @ I4 ) @ ( A2 @ J2 ) )
| ( ord_less_eq_set_nat @ ( A2 @ J2 ) @ ( A2 @ I4 ) ) ) ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( inj_on_nat_nat @ F2 @ ( A2 @ I4 ) ) )
=> ( inj_on_nat_nat @ F2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ I3 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_1147_SUP__subset__mono,axiom,
! [A2: set_a,B3: set_a,F2: a > set_Extended_ereal,G: a > set_Extended_ereal] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ F2 @ A2 ) ) @ ( comple4319282863272126363_ereal @ ( image_886028290468338613_ereal @ G @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1148_SUP__subset__mono,axiom,
! [A2: set_nat,B3: set_nat,F2: nat > set_Extended_ereal,G: nat > set_Extended_ereal] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ ( image_305533323056406039_ereal @ F2 @ A2 ) ) @ ( comple4319282863272126363_ereal @ ( image_305533323056406039_ereal @ G @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1149_SUP__subset__mono,axiom,
! [A2: set_Extended_ereal,B3: set_Extended_ereal,F2: extended_ereal > set_Extended_ereal,G: extended_ereal > set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A2 @ B3 )
=> ( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A2 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ F2 @ A2 ) ) @ ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ G @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1150_SUP__subset__mono,axiom,
! [A2: set_a,B3: set_a,F2: a > extended_ereal,G: a > extended_ereal] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ F2 @ A2 ) ) @ ( comple8415311339701865915_ereal @ ( image_8405481351990995413_ereal @ G @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1151_SUP__subset__mono,axiom,
! [A2: set_nat,B3: set_nat,F2: nat > extended_ereal,G: nat > extended_ereal] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F2 @ A2 ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1152_SUP__subset__mono,axiom,
! [A2: set_Extended_ereal,B3: set_Extended_ereal,F2: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A2 @ B3 )
=> ( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A2 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ F2 @ A2 ) ) @ ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ G @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1153_SUP__subset__mono,axiom,
! [A2: set_a,B3: set_a,F2: a > set_nat,G: a > set_nat] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F2 @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F2 @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ G @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1154_SUP__subset__mono,axiom,
! [A2: set_nat,B3: set_nat,F2: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F2 @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1155_SUP__subset__mono,axiom,
! [A2: set_Extended_ereal,B3: set_Extended_ereal,F2: extended_ereal > set_nat,G: extended_ereal > set_nat] :
( ( ord_le1644982726543182158_ereal @ A2 @ B3 )
=> ( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F2 @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_3090908713637162255et_nat @ F2 @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_3090908713637162255et_nat @ G @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1156_snd__image__Sigma,axiom,
! [A2: set_a,B3: a > set_b] :
( ( image_2802296252294471260_a_b_b @ product_snd_a_b @ ( product_Sigma_a_b @ A2 @ B3 ) )
= ( comple2307003614231284044_set_b @ ( image_a_set_b @ B3 @ A2 ) ) ) ).
% snd_image_Sigma
thf(fact_1157_snd__image__Sigma,axiom,
! [A2: set_nat,B3: nat > set_nat] :
( ( image_2486076414777270412at_nat @ product_snd_nat_nat @ ( produc457027306803732586at_nat @ A2 @ B3 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) ) ).
% snd_image_Sigma
thf(fact_1158_inj__setminus,axiom,
! [A2: set_se6634062954251873166_ereal] : ( inj_on5406440306785145713_ereal @ uminus5895154729394068773_ereal @ A2 ) ).
% inj_setminus
thf(fact_1159_SUP__pair,axiom,
! [F2: extended_ereal > extended_ereal > extended_ereal,B3: set_Extended_ereal,A2: set_Extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ ( F2 @ I2 ) @ B3 ) )
@ A2 ) )
= ( comple8415311339701865915_ereal
@ ( image_8011834658935918572_ereal
@ ^ [P3: produc5501587555545223847_ereal] : ( F2 @ ( produc8000661846298591107_ereal @ P3 ) @ ( produc7555089578395445445_ereal @ P3 ) )
@ ( produc8095709571603465288_ereal @ A2
@ ^ [Uu: extended_ereal] : B3 ) ) ) ) ).
% SUP_pair
thf(fact_1160_SUP__pair,axiom,
! [F2: extended_ereal > nat > extended_ereal,B3: set_nat,A2: set_Extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [I2: extended_ereal] : ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ ( F2 @ I2 ) @ B3 ) )
@ A2 ) )
= ( comple8415311339701865915_ereal
@ ( image_3315421938599550714_ereal
@ ^ [P3: produc5547956294148499661al_nat] : ( F2 @ ( produc3755854452330217819al_nat @ P3 ) @ ( produc5370844913463222425al_nat @ P3 ) )
@ ( produc4220900302008768982al_nat @ A2
@ ^ [Uu: extended_ereal] : B3 ) ) ) ) ).
% SUP_pair
thf(fact_1161_SUP__pair,axiom,
! [F2: nat > extended_ereal > extended_ereal,B3: set_Extended_ereal,A2: set_nat] :
( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ ( F2 @ I2 ) @ B3 ) )
@ A2 ) )
= ( comple8415311339701865915_ereal
@ ( image_4404873737357082836_ereal
@ ^ [P3: produc7896515489273245043_ereal] : ( F2 @ ( produc405286064046379065_ereal @ P3 ) @ ( produc2020276525179383671_ereal @ P3 ) )
@ ( produc870331913724930228_ereal @ A2
@ ^ [Uu: nat] : B3 ) ) ) ) ).
% SUP_pair
thf(fact_1162_SUP__pair,axiom,
! [F2: nat > nat > extended_ereal,B3: set_nat,A2: set_nat] :
( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I2: nat] : ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ ( F2 @ I2 ) @ B3 ) )
@ A2 ) )
= ( comple8415311339701865915_ereal
@ ( image_3495945180199258130_ereal
@ ^ [P3: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ P3 ) @ ( product_snd_nat_nat @ P3 ) )
@ ( produc457027306803732586at_nat @ A2
@ ^ [Uu: nat] : B3 ) ) ) ) ).
% SUP_pair
thf(fact_1163_SUP__pair,axiom,
! [F2: a > b > extended_ereal,B3: set_b,A2: set_a] :
( ( comple8415311339701865915_ereal
@ ( image_8405481351990995413_ereal
@ ^ [I2: a] : ( comple8415311339701865915_ereal @ ( image_5319725110001000852_ereal @ ( F2 @ I2 ) @ B3 ) )
@ A2 ) )
= ( 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 @ A2
@ ^ [Uu: a] : B3 ) ) ) ) ).
% SUP_pair
thf(fact_1164_SUP__pair,axiom,
! [F2: nat > nat > set_nat,B3: set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I2: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F2 @ I2 ) @ B3 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_15824709712370754et_nat
@ ^ [P3: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ P3 ) @ ( product_snd_nat_nat @ P3 ) )
@ ( produc457027306803732586at_nat @ A2
@ ^ [Uu: nat] : B3 ) ) ) ) ).
% SUP_pair
thf(fact_1165_SUP__pair,axiom,
! [F2: a > b > set_nat,B3: set_b,A2: set_a] :
( ( comple7399068483239264473et_nat
@ ( image_a_set_nat
@ ^ [I2: a] : ( comple7399068483239264473et_nat @ ( image_b_set_nat @ ( F2 @ I2 ) @ B3 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_2257100470014249705et_nat
@ ^ [P3: product_prod_a_b] : ( F2 @ ( product_fst_a_b @ P3 ) @ ( product_snd_a_b @ P3 ) )
@ ( product_Sigma_a_b @ A2
@ ^ [Uu: a] : B3 ) ) ) ) ).
% SUP_pair
thf(fact_1166_SUP__combine,axiom,
! [F2: rat > rat > extended_ereal] :
( ! [A5: rat,B6: rat,C4: rat,D3: rat] :
( ( ord_less_eq_rat @ A5 @ B6 )
=> ( ( ord_less_eq_rat @ C4 @ D3 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ A5 @ C4 ) @ ( F2 @ B6 @ D3 ) ) ) )
=> ( ( comple8415311339701865915_ereal
@ ( image_2592109325025016879_ereal
@ ^ [I2: rat] : ( comple8415311339701865915_ereal @ ( image_2592109325025016879_ereal @ ( F2 @ I2 ) @ top_top_set_rat ) )
@ top_top_set_rat ) )
= ( comple8415311339701865915_ereal
@ ( image_2592109325025016879_ereal
@ ^ [I2: rat] : ( F2 @ I2 @ I2 )
@ top_top_set_rat ) ) ) ) ).
% SUP_combine
thf(fact_1167_SUP__combine,axiom,
! [F2: extended_ereal > extended_ereal > extended_ereal] :
( ! [A5: extended_ereal,B6: extended_ereal,C4: extended_ereal,D3: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A5 @ B6 )
=> ( ( ord_le1083603963089353582_ereal @ C4 @ D3 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ A5 @ C4 ) @ ( F2 @ B6 @ 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_1168_SUP__combine,axiom,
! [F2: nat > nat > extended_ereal] :
( ! [A5: nat,B6: nat,C4: nat,D3: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
=> ( ( ord_less_eq_nat @ C4 @ D3 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ A5 @ C4 ) @ ( F2 @ B6 @ 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_1169_SUP__combine,axiom,
! [F2: rat > rat > set_Extended_ereal] :
( ! [A5: rat,B6: rat,C4: rat,D3: rat] :
( ( ord_less_eq_rat @ A5 @ B6 )
=> ( ( ord_less_eq_rat @ C4 @ D3 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ A5 @ C4 ) @ ( F2 @ B6 @ D3 ) ) ) )
=> ( ( comple4319282863272126363_ereal
@ ( image_4612044004287677967_ereal
@ ^ [I2: rat] : ( comple4319282863272126363_ereal @ ( image_4612044004287677967_ereal @ ( F2 @ I2 ) @ top_top_set_rat ) )
@ top_top_set_rat ) )
= ( comple4319282863272126363_ereal
@ ( image_4612044004287677967_ereal
@ ^ [I2: rat] : ( F2 @ I2 @ I2 )
@ top_top_set_rat ) ) ) ) ).
% SUP_combine
thf(fact_1170_SUP__combine,axiom,
! [F2: extended_ereal > extended_ereal > set_Extended_ereal] :
( ! [A5: extended_ereal,B6: extended_ereal,C4: extended_ereal,D3: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A5 @ B6 )
=> ( ( ord_le1083603963089353582_ereal @ C4 @ D3 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ A5 @ C4 ) @ ( F2 @ B6 @ 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_1171_SUP__combine,axiom,
! [F2: nat > nat > set_Extended_ereal] :
( ! [A5: nat,B6: nat,C4: nat,D3: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
=> ( ( ord_less_eq_nat @ C4 @ D3 )
=> ( ord_le1644982726543182158_ereal @ ( F2 @ A5 @ C4 ) @ ( F2 @ B6 @ 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_1172_SUP__combine,axiom,
! [F2: set_Extended_ereal > set_Extended_ereal > extended_ereal] :
( ! [A5: set_Extended_ereal,B6: set_Extended_ereal,C4: set_Extended_ereal,D3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A5 @ B6 )
=> ( ( ord_le1644982726543182158_ereal @ C4 @ D3 )
=> ( ord_le1083603963089353582_ereal @ ( F2 @ A5 @ C4 ) @ ( F2 @ B6 @ 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_1173_SUP__combine,axiom,
! [F2: rat > rat > set_nat] :
( ! [A5: rat,B6: rat,C4: rat,D3: rat] :
( ( ord_less_eq_rat @ A5 @ B6 )
=> ( ( ord_less_eq_rat @ C4 @ D3 )
=> ( ord_less_eq_set_nat @ ( F2 @ A5 @ C4 ) @ ( F2 @ B6 @ D3 ) ) ) )
=> ( ( comple7399068483239264473et_nat
@ ( image_rat_set_nat
@ ^ [I2: rat] : ( comple7399068483239264473et_nat @ ( image_rat_set_nat @ ( F2 @ I2 ) @ top_top_set_rat ) )
@ top_top_set_rat ) )
= ( comple7399068483239264473et_nat
@ ( image_rat_set_nat
@ ^ [I2: rat] : ( F2 @ I2 @ I2 )
@ top_top_set_rat ) ) ) ) ).
% SUP_combine
thf(fact_1174_SUP__combine,axiom,
! [F2: extended_ereal > extended_ereal > set_nat] :
( ! [A5: extended_ereal,B6: extended_ereal,C4: extended_ereal,D3: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A5 @ B6 )
=> ( ( ord_le1083603963089353582_ereal @ C4 @ D3 )
=> ( ord_less_eq_set_nat @ ( F2 @ A5 @ C4 ) @ ( F2 @ B6 @ D3 ) ) ) )
=> ( ( comple7399068483239264473et_nat
@ ( image_3090908713637162255et_nat
@ ^ [I2: extended_ereal] : ( comple7399068483239264473et_nat @ ( image_3090908713637162255et_nat @ ( F2 @ I2 ) @ top_to5683747375963461374_ereal ) )
@ top_to5683747375963461374_ereal ) )
= ( comple7399068483239264473et_nat
@ ( image_3090908713637162255et_nat
@ ^ [I2: extended_ereal] : ( F2 @ I2 @ I2 )
@ top_to5683747375963461374_ereal ) ) ) ) ).
% SUP_combine
thf(fact_1175_SUP__combine,axiom,
! [F2: nat > nat > set_nat] :
( ! [A5: nat,B6: nat,C4: nat,D3: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
=> ( ( ord_less_eq_nat @ C4 @ D3 )
=> ( ord_less_eq_set_nat @ ( F2 @ A5 @ C4 ) @ ( F2 @ B6 @ D3 ) ) ) )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I2: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F2 @ I2 ) @ top_top_set_nat ) )
@ top_top_set_nat ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I2: nat] : ( F2 @ I2 @ I2 )
@ top_top_set_nat ) ) ) ) ).
% SUP_combine
thf(fact_1176_UN__constant__eq,axiom,
! [A: a,A2: set_a,F2: a > set_nat,C: set_nat] :
( ( member_a @ A @ A2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( F2 @ X2 )
= C ) )
=> ( ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F2 @ A2 ) )
= C ) ) ) ).
% UN_constant_eq
thf(fact_1177_UN__constant__eq,axiom,
! [A: nat,A2: set_nat,F2: nat > set_nat,C: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( F2 @ X2 )
= C ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A2 ) )
= C ) ) ) ).
% UN_constant_eq
thf(fact_1178_Sup__SUP__eq,axiom,
( complete_Sup_Sup_a_o
= ( ^ [S3: set_a_o,X: a] : ( member_a @ X @ ( comple2307003609928055243_set_a @ ( image_a_o_set_a @ collect_a @ S3 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1179_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S3: set_nat_o,X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S3 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1180_SUP__Sup__eq,axiom,
! [S2: set_set_a] :
( ( complete_Sup_Sup_a_o
@ ( image_set_a_a_o
@ ^ [I2: set_a,X: a] : ( member_a @ X @ I2 )
@ S2 ) )
= ( ^ [X: a] : ( member_a @ X @ ( comple2307003609928055243_set_a @ S2 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1181_SUP__Sup__eq,axiom,
! [S2: set_set_nat] :
( ( comple8317665133742190828_nat_o
@ ( image_set_nat_nat_o
@ ^ [I2: set_nat,X: nat] : ( member_nat @ X @ I2 )
@ S2 ) )
= ( ^ [X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ S2 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1182_Sup__set__def,axiom,
( comple2307003609928055243_set_a
= ( ^ [A4: set_set_a] :
( collect_a
@ ^ [X: a] : ( complete_Sup_Sup_o @ ( image_set_a_o @ ( member_a @ X ) @ A4 ) ) ) ) ) ).
% Sup_set_def
thf(fact_1183_Sup__set__def,axiom,
( comple7399068483239264473et_nat
= ( ^ [A4: set_set_nat] :
( collect_nat
@ ^ [X: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X ) @ A4 ) ) ) ) ) ).
% Sup_set_def
thf(fact_1184_SUP__UN__eq,axiom,
! [R: nat > set_nat,S2: set_nat] :
( ( comple8317665133742190828_nat_o
@ ( image_nat_nat_o
@ ^ [I2: nat,X: nat] : ( member_nat @ X @ ( R @ I2 ) )
@ S2 ) )
= ( ^ [X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ R @ S2 ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_1185_inj__on__image__Fpow,axiom,
! [F2: nat > extended_ereal,A2: set_nat] :
( ( inj_on6191532827271902155_ereal @ F2 @ A2 )
=> ( inj_on1463964778812548213_ereal @ ( image_4309273772856505399_ereal @ F2 ) @ ( finite_Fpow_nat @ A2 ) ) ) ).
% inj_on_image_Fpow
thf(fact_1186_inj__on__image__Fpow,axiom,
! [F2: nat > set_nat,A2: set_nat] :
( ( inj_on_nat_set_nat @ F2 @ A2 )
=> ( inj_on2776966659131765557et_nat @ ( image_nat_set_nat @ F2 ) @ ( finite_Fpow_nat @ A2 ) ) ) ).
% inj_on_image_Fpow
thf(fact_1187_inj__on__image__Fpow,axiom,
! [F2: nat > rat,A2: set_nat] :
( ( inj_on_nat_rat @ F2 @ A2 )
=> ( inj_on1096178645466186887et_rat @ ( image_nat_rat @ F2 ) @ ( finite_Fpow_nat @ A2 ) ) ) ).
% inj_on_image_Fpow
thf(fact_1188_inj__on__image__Fpow,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F2 @ A2 )
=> ( inj_on5406440306785145713_ereal @ ( image_6042159593519690757_ereal @ F2 ) @ ( finite2137394461708460340_ereal @ A2 ) ) ) ).
% inj_on_image_Fpow
thf(fact_1189_inj__on__image__Fpow,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F2 ) @ ( finite_Fpow_nat @ A2 ) ) ) ).
% inj_on_image_Fpow
thf(fact_1190_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_1191_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_1192_SUP__INF,axiom,
! [P2: extended_ereal > rat > extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_2592109325025016879_ereal
@ ^ [Y: rat] :
( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : ( P2 @ X @ Y )
@ top_to5683747375963461374_ereal ) )
@ top_top_set_rat ) )
= ( comple3556804143462414037_ereal
@ ( image_1957192302298880358_ereal
@ ^ [X: rat > extended_ereal] :
( comple8415311339701865915_ereal
@ ( image_2592109325025016879_ereal
@ ^ [Y: rat] : ( P2 @ ( X @ Y ) @ Y )
@ top_top_set_rat ) )
@ top_to2804411426662057437_ereal ) ) ) ).
% SUP_INF
thf(fact_1193_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_1194_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_1195_SUP__INF,axiom,
! [P2: nat > rat > extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_2592109325025016879_ereal
@ ^ [Y: rat] :
( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X: nat] : ( P2 @ X @ Y )
@ top_top_set_nat ) )
@ top_top_set_rat ) )
= ( comple3556804143462414037_ereal
@ ( image_7180403626205596672_ereal
@ ^ [X: rat > nat] :
( comple8415311339701865915_ereal
@ ( image_2592109325025016879_ereal
@ ^ [Y: rat] : ( P2 @ ( X @ Y ) @ Y )
@ top_top_set_rat ) )
@ top_top_set_rat_nat ) ) ) ).
% SUP_INF
thf(fact_1196_SUP__INF,axiom,
! [P2: rat > extended_ereal > extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [Y: extended_ereal] :
( comple3556804143462414037_ereal
@ ( image_2592109325025016879_ereal
@ ^ [X: rat] : ( P2 @ X @ Y )
@ top_top_set_rat ) )
@ top_to5683747375963461374_ereal ) )
= ( comple3556804143462414037_ereal
@ ( image_9068502826479536508_ereal
@ ^ [X: extended_ereal > rat] :
( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [Y: extended_ereal] : ( P2 @ ( X @ Y ) @ Y )
@ top_to5683747375963461374_ereal ) )
@ top_to5242157703742838343al_rat ) ) ) ).
% SUP_INF
thf(fact_1197_SUP__INF,axiom,
! [P2: rat > nat > extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [Y: nat] :
( comple3556804143462414037_ereal
@ ( image_2592109325025016879_ereal
@ ^ [X: rat] : ( P2 @ X @ Y )
@ top_top_set_rat ) )
@ top_top_set_nat ) )
= ( comple3556804143462414037_ereal
@ ( image_4542273001526177792_ereal
@ ^ [X: nat > rat] :
( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [Y: nat] : ( P2 @ ( X @ Y ) @ Y )
@ top_top_set_nat ) )
@ top_top_set_nat_rat ) ) ) ).
% SUP_INF
thf(fact_1198_SUP__INF,axiom,
! [P2: rat > rat > extended_ereal] :
( ( comple8415311339701865915_ereal
@ ( image_2592109325025016879_ereal
@ ^ [Y: rat] :
( comple3556804143462414037_ereal
@ ( image_2592109325025016879_ereal
@ ^ [X: rat] : ( P2 @ X @ Y )
@ top_top_set_rat ) )
@ top_top_set_rat ) )
= ( comple3556804143462414037_ereal
@ ( image_8848783682757449720_ereal
@ ^ [X: rat > rat] :
( comple8415311339701865915_ereal
@ ( image_2592109325025016879_ereal
@ ^ [Y: rat] : ( P2 @ ( X @ Y ) @ Y )
@ top_top_set_rat ) )
@ top_top_set_rat_rat ) ) ) ).
% SUP_INF
thf(fact_1199_SUP__INF,axiom,
! [P2: extended_ereal > extended_ereal > set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_3090908713637162255et_nat
@ ^ [Y: extended_ereal] :
( comple7806235888213564991et_nat
@ ( image_3090908713637162255et_nat
@ ^ [X: extended_ereal] : ( P2 @ X @ Y )
@ top_to5683747375963461374_ereal ) )
@ top_to5683747375963461374_ereal ) )
= ( comple7806235888213564991et_nat
@ ( image_3395553534200627570et_nat
@ ^ [X: extended_ereal > extended_ereal] :
( comple7399068483239264473et_nat
@ ( image_3090908713637162255et_nat
@ ^ [Y: extended_ereal] : ( P2 @ ( X @ Y ) @ Y )
@ top_to5683747375963461374_ereal ) )
@ top_to908700840774984395_ereal ) ) ) ).
% SUP_INF
thf(fact_1200_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_1201_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_1202_INF__SUP,axiom,
! [P2: extended_ereal > rat > extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_2592109325025016879_ereal
@ ^ [Y: rat] :
( comple8415311339701865915_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : ( P2 @ X @ Y )
@ top_to5683747375963461374_ereal ) )
@ top_top_set_rat ) )
= ( comple8415311339701865915_ereal
@ ( image_1957192302298880358_ereal
@ ^ [F: rat > extended_ereal] :
( comple3556804143462414037_ereal
@ ( image_2592109325025016879_ereal
@ ^ [X: rat] : ( P2 @ ( F @ X ) @ X )
@ top_top_set_rat ) )
@ top_to2804411426662057437_ereal ) ) ) ).
% INF_SUP
thf(fact_1203_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_1204_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_1205_INF__SUP,axiom,
! [P2: nat > rat > extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_2592109325025016879_ereal
@ ^ [Y: rat] :
( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X: nat] : ( P2 @ X @ Y )
@ top_top_set_nat ) )
@ top_top_set_rat ) )
= ( comple8415311339701865915_ereal
@ ( image_7180403626205596672_ereal
@ ^ [F: rat > nat] :
( comple3556804143462414037_ereal
@ ( image_2592109325025016879_ereal
@ ^ [X: rat] : ( P2 @ ( F @ X ) @ X )
@ top_top_set_rat ) )
@ top_top_set_rat_nat ) ) ) ).
% INF_SUP
thf(fact_1206_INF__SUP,axiom,
! [P2: rat > extended_ereal > extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [Y: extended_ereal] :
( comple8415311339701865915_ereal
@ ( image_2592109325025016879_ereal
@ ^ [X: rat] : ( P2 @ X @ Y )
@ top_top_set_rat ) )
@ top_to5683747375963461374_ereal ) )
= ( comple8415311339701865915_ereal
@ ( image_9068502826479536508_ereal
@ ^ [F: extended_ereal > rat] :
( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : ( P2 @ ( F @ X ) @ X )
@ top_to5683747375963461374_ereal ) )
@ top_to5242157703742838343al_rat ) ) ) ).
% INF_SUP
thf(fact_1207_INF__SUP,axiom,
! [P2: rat > nat > extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [Y: nat] :
( comple8415311339701865915_ereal
@ ( image_2592109325025016879_ereal
@ ^ [X: rat] : ( P2 @ X @ Y )
@ top_top_set_rat ) )
@ top_top_set_nat ) )
= ( comple8415311339701865915_ereal
@ ( image_4542273001526177792_ereal
@ ^ [F: nat > rat] :
( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X: nat] : ( P2 @ ( F @ X ) @ X )
@ top_top_set_nat ) )
@ top_top_set_nat_rat ) ) ) ).
% INF_SUP
thf(fact_1208_INF__SUP,axiom,
! [P2: rat > rat > extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_2592109325025016879_ereal
@ ^ [Y: rat] :
( comple8415311339701865915_ereal
@ ( image_2592109325025016879_ereal
@ ^ [X: rat] : ( P2 @ X @ Y )
@ top_top_set_rat ) )
@ top_top_set_rat ) )
= ( comple8415311339701865915_ereal
@ ( image_8848783682757449720_ereal
@ ^ [F: rat > rat] :
( comple3556804143462414037_ereal
@ ( image_2592109325025016879_ereal
@ ^ [X: rat] : ( P2 @ ( F @ X ) @ X )
@ top_top_set_rat ) )
@ top_top_set_rat_rat ) ) ) ).
% INF_SUP
thf(fact_1209_INF__SUP,axiom,
! [P2: extended_ereal > extended_ereal > set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_3090908713637162255et_nat
@ ^ [Y: extended_ereal] :
( comple7399068483239264473et_nat
@ ( image_3090908713637162255et_nat
@ ^ [X: extended_ereal] : ( P2 @ X @ Y )
@ top_to5683747375963461374_ereal ) )
@ top_to5683747375963461374_ereal ) )
= ( comple7399068483239264473et_nat
@ ( image_3395553534200627570et_nat
@ ^ [F: extended_ereal > extended_ereal] :
( comple7806235888213564991et_nat
@ ( image_3090908713637162255et_nat
@ ^ [X: extended_ereal] : ( P2 @ ( F @ X ) @ X )
@ top_to5683747375963461374_ereal ) )
@ top_to908700840774984395_ereal ) ) ) ).
% INF_SUP
thf(fact_1210_image__Fpow__mono,axiom,
! [F2: nat > set_nat,A2: set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F2 @ A2 ) @ B3 )
=> ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F2 ) @ ( finite_Fpow_nat @ A2 ) ) @ ( finite_Fpow_set_nat @ B3 ) ) ) ).
% image_Fpow_mono
thf(fact_1211_image__Fpow__mono,axiom,
! [F2: nat > nat,A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ B3 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F2 ) @ ( finite_Fpow_nat @ A2 ) ) @ ( finite_Fpow_nat @ B3 ) ) ) ).
% image_Fpow_mono
thf(fact_1212_image__Fpow__mono,axiom,
! [F2: nat > rat,A2: set_nat,B3: set_rat] :
( ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A2 ) @ B3 )
=> ( ord_le513522071413781156et_rat @ ( image_4408659257933336347et_rat @ ( image_nat_rat @ F2 ) @ ( finite_Fpow_nat @ A2 ) ) @ ( finite_Fpow_rat @ B3 ) ) ) ).
% image_Fpow_mono
thf(fact_1213_image__Fpow__mono,axiom,
! [F2: extended_ereal > extended_ereal,A2: set_Extended_ereal,B3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F2 @ A2 ) @ B3 )
=> ( ord_le5287700718633833262_ereal @ ( image_6293272304431515653_ereal @ ( image_6042159593519690757_ereal @ F2 ) @ ( finite2137394461708460340_ereal @ A2 ) ) @ ( finite2137394461708460340_ereal @ B3 ) ) ) ).
% image_Fpow_mono
thf(fact_1214_image__Fpow__mono,axiom,
! [F2: nat > extended_ereal,A2: set_nat,B3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F2 @ A2 ) @ B3 )
=> ( ord_le5287700718633833262_ereal @ ( image_8825259783980156129_ereal @ ( image_4309273772856505399_ereal @ F2 ) @ ( finite_Fpow_nat @ A2 ) ) @ ( finite2137394461708460340_ereal @ B3 ) ) ) ).
% image_Fpow_mono
thf(fact_1215_INT__I,axiom,
! [A2: set_a,B: a,B3: a > set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ B @ ( B3 @ X2 ) ) )
=> ( member_a @ B @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ B3 @ A2 ) ) ) ) ).
% INT_I
thf(fact_1216_INT__I,axiom,
! [A2: set_a,B: nat,B3: a > set_nat] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_nat @ B @ ( B3 @ X2 ) ) )
=> ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_a_set_nat @ B3 @ A2 ) ) ) ) ).
% INT_I
thf(fact_1217_INT__I,axiom,
! [A2: set_nat,B: a,B3: nat > set_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_a @ B @ ( B3 @ X2 ) ) )
=> ( member_a @ B @ ( comple6135023378680113637_set_a @ ( image_nat_set_a @ B3 @ A2 ) ) ) ) ).
% INT_I
thf(fact_1218_INT__I,axiom,
! [A2: set_nat,B: nat,B3: nat > set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ B @ ( B3 @ X2 ) ) )
=> ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) ) ) ).
% INT_I
thf(fact_1219_INT__iff,axiom,
! [B: nat,B3: nat > set_nat,A2: set_nat] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ B @ ( B3 @ X ) ) ) ) ) ).
% INT_iff
thf(fact_1220_INF__identity__eq,axiom,
! [A2: set_nat] :
( ( complete_Inf_Inf_nat
@ ( image_nat_nat
@ ^ [X: nat] : X
@ A2 ) )
= ( complete_Inf_Inf_nat @ A2 ) ) ).
% INF_identity_eq
thf(fact_1221_INF__identity__eq,axiom,
! [A2: set_Extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : X
@ A2 ) )
= ( comple3556804143462414037_ereal @ A2 ) ) ).
% INF_identity_eq
thf(fact_1222_INF__id__eq,axiom,
! [A2: set_nat] :
( ( complete_Inf_Inf_nat @ ( image_nat_nat @ id_nat @ A2 ) )
= ( complete_Inf_Inf_nat @ A2 ) ) ).
% INF_id_eq
thf(fact_1223_INF__id__eq,axiom,
! [A2: set_Extended_ereal] :
( ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ id_Extended_ereal @ A2 ) )
= ( comple3556804143462414037_ereal @ A2 ) ) ).
% INF_id_eq
thf(fact_1224_INF__top__conv_I2_J,axiom,
! [B3: nat > set_nat,A2: set_nat] :
( ( top_top_set_nat
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( B3 @ X )
= top_top_set_nat ) ) ) ) ).
% INF_top_conv(2)
thf(fact_1225_INF__top__conv_I2_J,axiom,
! [B3: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( top_to6662034908053899550_ereal
= ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ B3 @ A2 ) ) )
= ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A2 )
=> ( ( B3 @ X )
= top_to6662034908053899550_ereal ) ) ) ) ).
% INF_top_conv(2)
thf(fact_1226_INF__top__conv_I2_J,axiom,
! [B3: nat > extended_ereal,A2: set_nat] :
( ( top_to6662034908053899550_ereal
= ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ B3 @ A2 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( B3 @ X )
= top_to6662034908053899550_ereal ) ) ) ) ).
% INF_top_conv(2)
thf(fact_1227_INF__top__conv_I1_J,axiom,
! [B3: nat > set_nat,A2: set_nat] :
( ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A2 ) )
= top_top_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( B3 @ X )
= top_top_set_nat ) ) ) ) ).
% INF_top_conv(1)
thf(fact_1228_INF__top__conv_I1_J,axiom,
! [B3: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ B3 @ A2 ) )
= top_to6662034908053899550_ereal )
= ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A2 )
=> ( ( B3 @ X )
= top_to6662034908053899550_ereal ) ) ) ) ).
% INF_top_conv(1)
thf(fact_1229_INF__top__conv_I1_J,axiom,
! [B3: nat > extended_ereal,A2: set_nat] :
( ( ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ B3 @ A2 ) )
= top_to6662034908053899550_ereal )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( B3 @ X )
= top_to6662034908053899550_ereal ) ) ) ) ).
% INF_top_conv(1)
thf(fact_1230_INF__top,axiom,
! [A2: set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : top_top_set_nat
@ A2 ) )
= top_top_set_nat ) ).
% INF_top
thf(fact_1231_INF__top,axiom,
! [A2: set_Extended_ereal] :
( ( comple3556804143462414037_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : top_to6662034908053899550_ereal
@ A2 ) )
= top_to6662034908053899550_ereal ) ).
% INF_top
thf(fact_1232_INF__top,axiom,
! [A2: set_nat] :
( ( comple3556804143462414037_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X: nat] : top_to6662034908053899550_ereal
@ A2 ) )
= top_to6662034908053899550_ereal ) ).
% INF_top
thf(fact_1233_Compl__UN,axiom,
! [B3: nat > set_nat,A2: set_nat] :
( ( uminus5710092332889474511et_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) )
= ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( uminus5710092332889474511et_nat @ ( B3 @ X ) )
@ A2 ) ) ) ).
% Compl_UN
thf(fact_1234_Compl__INT,axiom,
! [B3: nat > set_nat,A2: set_nat] :
( ( uminus5710092332889474511et_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( uminus5710092332889474511et_nat @ ( B3 @ X ) )
@ A2 ) ) ) ).
% Compl_INT
thf(fact_1235_Inf__set__def,axiom,
( comple6135023378680113637_set_a
= ( ^ [A4: set_set_a] :
( collect_a
@ ^ [X: a] : ( complete_Inf_Inf_o @ ( image_set_a_o @ ( member_a @ X ) @ A4 ) ) ) ) ) ).
% Inf_set_def
thf(fact_1236_Inf__set__def,axiom,
( comple7806235888213564991et_nat
= ( ^ [A4: set_set_nat] :
( collect_nat
@ ^ [X: nat] : ( complete_Inf_Inf_o @ ( image_set_nat_o @ ( member_nat @ X ) @ A4 ) ) ) ) ) ).
% Inf_set_def
thf(fact_1237_INF__Int__eq,axiom,
! [S2: set_set_a] :
( ( complete_Inf_Inf_a_o
@ ( image_set_a_a_o
@ ^ [I2: set_a,X: a] : ( member_a @ X @ I2 )
@ S2 ) )
= ( ^ [X: a] : ( member_a @ X @ ( comple6135023378680113637_set_a @ S2 ) ) ) ) ).
% INF_Int_eq
thf(fact_1238_INF__Int__eq,axiom,
! [S2: set_set_nat] :
( ( comple6214475593288795910_nat_o
@ ( image_set_nat_nat_o
@ ^ [I2: set_nat,X: nat] : ( member_nat @ X @ I2 )
@ S2 ) )
= ( ^ [X: nat] : ( member_nat @ X @ ( comple7806235888213564991et_nat @ S2 ) ) ) ) ).
% INF_Int_eq
thf(fact_1239_INF__INT__eq,axiom,
! [R: nat > set_nat,S2: set_nat] :
( ( comple6214475593288795910_nat_o
@ ( image_nat_nat_o
@ ^ [I2: nat,X: nat] : ( member_nat @ X @ ( R @ I2 ) )
@ S2 ) )
= ( ^ [X: nat] : ( member_nat @ X @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ R @ S2 ) ) ) ) ) ).
% INF_INT_eq
thf(fact_1240_INF__cong,axiom,
! [A2: set_Extended_ereal,B3: set_Extended_ereal,C3: extended_ereal > extended_ereal,D2: extended_ereal > extended_ereal] :
( ( A2 = B3 )
=> ( ! [X2: extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ B3 )
=> ( ( C3 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ C3 @ A2 ) )
= ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ D2 @ B3 ) ) ) ) ) ).
% INF_cong
thf(fact_1241_INF__cong,axiom,
! [A2: set_a,B3: set_a,C3: a > extended_ereal,D2: a > extended_ereal] :
( ( A2 = B3 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B3 )
=> ( ( C3 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( comple3556804143462414037_ereal @ ( image_8405481351990995413_ereal @ C3 @ A2 ) )
= ( comple3556804143462414037_ereal @ ( image_8405481351990995413_ereal @ D2 @ B3 ) ) ) ) ) ).
% INF_cong
thf(fact_1242_INF__cong,axiom,
! [A2: set_nat,B3: set_nat,C3: nat > extended_ereal,D2: nat > extended_ereal] :
( ( A2 = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C3 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ C3 @ A2 ) )
= ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ D2 @ B3 ) ) ) ) ) ).
% INF_cong
thf(fact_1243_ereal__Sup__uminus__image__eq,axiom,
! [S2: set_Extended_ereal] :
( ( comple8415311339701865915_ereal @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S2 ) )
= ( uminus27091377158695749_ereal @ ( comple3556804143462414037_ereal @ S2 ) ) ) ).
% ereal_Sup_uminus_image_eq
thf(fact_1244_ereal__Inf__uminus__image__eq,axiom,
! [S2: set_Extended_ereal] :
( ( comple3556804143462414037_ereal @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S2 ) )
= ( uminus27091377158695749_ereal @ ( comple8415311339701865915_ereal @ S2 ) ) ) ).
% ereal_Inf_uminus_image_eq
thf(fact_1245_ereal__inj__on__uminus,axiom,
! [A2: set_Extended_ereal] : ( inj_on7162434037990268785_ereal @ uminus27091377158695749_ereal @ A2 ) ).
% ereal_inj_on_uminus
thf(fact_1246_ereal__range__uminus,axiom,
( ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal ) ).
% ereal_range_uminus
thf(fact_1247_ereal__minus__minus__image,axiom,
! [S2: set_Extended_ereal] :
( ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S2 ) )
= S2 ) ).
% ereal_minus_minus_image
thf(fact_1248_ereal__complete__uminus__eq,axiom,
! [S2: set_Extended_ereal,X3: extended_ereal] :
( ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S2 ) )
=> ( ord_le1083603963089353582_ereal @ X @ X3 ) )
& ! [Z4: extended_ereal] :
( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S2 ) )
=> ( ord_le1083603963089353582_ereal @ X @ Z4 ) )
=> ( ord_le1083603963089353582_ereal @ X3 @ Z4 ) ) )
= ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ S2 )
=> ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ X3 ) @ X ) )
& ! [Z4: extended_ereal] :
( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ S2 )
=> ( ord_le1083603963089353582_ereal @ Z4 @ X ) )
=> ( ord_le1083603963089353582_ereal @ Z4 @ ( uminus27091377158695749_ereal @ X3 ) ) ) ) ) ).
% ereal_complete_uminus_eq
thf(fact_1249_ereal__uminus__complement,axiom,
! [S2: set_Extended_ereal] :
( ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ ( uminus5895154729394068773_ereal @ S2 ) )
= ( uminus5895154729394068773_ereal @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S2 ) ) ) ).
% ereal_uminus_complement
thf(fact_1250_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_1251_Inf__countable__INF,axiom,
! [A2: set_Extended_ereal] :
( ( A2 != 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 ) @ A2 )
& ( ( comple3556804143462414037_ereal @ A2 )
= ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F4 @ top_top_set_nat ) ) ) ) ) ).
% Inf_countable_INF
thf(fact_1252_Sup__countable__SUP,axiom,
! [A2: set_Extended_ereal] :
( ( A2 != 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 ) @ A2 )
& ( ( comple8415311339701865915_ereal @ A2 )
= ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F4 @ top_top_set_nat ) ) ) ) ) ).
% Sup_countable_SUP
thf(fact_1253_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 )
=> ( ! [I4: nat] :
( ( F2 @ I4 )
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ! [I4: nat] :
( ( G @ I4 )
!= ( 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_1254_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 )
=> ( ! [I4: nat] :
( ( F2 @ I4 )
!= extend1530274965995635425_ereal )
=> ( ! [I4: nat] :
( ( G @ I4 )
!= 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_1255_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 )
=> ( ! [I4: nat] : ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F2 @ I4 ) )
=> ( ! [I4: nat] : ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( G @ I4 ) )
=> ( ( 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_1256_suminf__SUP__eq,axiom,
! [F2: nat > nat > extended_ereal] :
( ! [I4: nat] :
( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal
@ ^ [N3: nat] : ( F2 @ N3 @ I4 ) )
=> ( ! [N2: nat,I4: nat] : ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F2 @ N2 @ I4 ) )
=> ( ( suminf4411151127299490740_ereal
@ ^ [I2: nat] :
( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [N3: nat] : ( F2 @ N3 @ I2 )
@ top_top_set_nat ) ) )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [N3: nat] : ( suminf4411151127299490740_ereal @ ( F2 @ N3 ) )
@ top_top_set_nat ) ) ) ) ) ).
% suminf_SUP_eq
thf(fact_1257_ereal__inj__affinity,axiom,
! [M3: extended_ereal,T2: extended_ereal,A2: set_Extended_ereal] :
( ( ( abs_ab7465543570706387889_ereal @ M3 )
!= extend1530274965995635425_ereal )
=> ( ( M3 != zero_z2744965634713055877_ereal )
=> ( ( ( abs_ab7465543570706387889_ereal @ T2 )
!= extend1530274965995635425_ereal )
=> ( inj_on7162434037990268785_ereal
@ ^ [X: extended_ereal] : ( plus_p7876563987511257093_ereal @ ( times_7703590493115627913_ereal @ M3 @ X ) @ T2 )
@ A2 ) ) ) ) ).
% ereal_inj_affinity
thf(fact_1258_suminf__ereal__eq__SUP,axiom,
! [F2: nat > extended_ereal] :
( ! [I4: nat] : ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F2 @ I4 ) )
=> ( ( suminf4411151127299490740_ereal @ F2 )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [N3: nat] : ( groups5544561043438645954_ereal @ F2 @ ( set_ord_lessThan_nat @ N3 ) )
@ top_top_set_nat ) ) ) ) ).
% suminf_ereal_eq_SUP
thf(fact_1259_UN__lessThan__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_lessThan_UNIV
thf(fact_1260_sums__ereal__positive,axiom,
! [F2: nat > extended_ereal] :
( ! [I4: nat] : ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F2 @ I4 ) )
=> ( sums_Extended_ereal @ F2
@ ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [N3: nat] : ( groups5544561043438645954_ereal @ F2 @ ( set_ord_lessThan_nat @ N3 ) )
@ top_top_set_nat ) ) ) ) ).
% sums_ereal_positive
thf(fact_1261_ereal__open__affinity,axiom,
! [S2: set_Extended_ereal,M3: extended_ereal,T2: extended_ereal] :
( ( topolo9005793602937862549_ereal @ S2 )
=> ( ( ( abs_ab7465543570706387889_ereal @ M3 )
!= extend1530274965995635425_ereal )
=> ( ( M3 != zero_z2744965634713055877_ereal )
=> ( ( ( abs_ab7465543570706387889_ereal @ T2 )
!= extend1530274965995635425_ereal )
=> ( topolo9005793602937862549_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : ( plus_p7876563987511257093_ereal @ ( times_7703590493115627913_ereal @ M3 @ X ) @ T2 )
@ S2 ) ) ) ) ) ) ).
% ereal_open_affinity
thf(fact_1262_open__uminus__iff,axiom,
! [S2: set_Extended_ereal] :
( ( topolo9005793602937862549_ereal @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S2 ) )
= ( topolo9005793602937862549_ereal @ S2 ) ) ).
% open_uminus_iff
thf(fact_1263_ereal__open__uminus,axiom,
! [S2: set_Extended_ereal] :
( ( topolo9005793602937862549_ereal @ S2 )
=> ( topolo9005793602937862549_ereal @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S2 ) ) ) ).
% ereal_open_uminus
thf(fact_1264_ereal__open__affinity__pos,axiom,
! [S2: set_Extended_ereal,M3: extended_ereal,T2: extended_ereal] :
( ( topolo9005793602937862549_ereal @ S2 )
=> ( ( M3 != extend1530274965995635425_ereal )
=> ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ M3 )
=> ( ( ( abs_ab7465543570706387889_ereal @ T2 )
!= extend1530274965995635425_ereal )
=> ( topolo9005793602937862549_ereal
@ ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : ( plus_p7876563987511257093_ereal @ ( times_7703590493115627913_ereal @ M3 @ X ) @ T2 )
@ S2 ) ) ) ) ) ) ).
% ereal_open_affinity_pos
thf(fact_1265_nat__descend__induct,axiom,
! [N4: nat,P2: nat > $o,M3: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N4 @ K2 )
=> ( P2 @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N4 )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K2 @ I5 )
=> ( P2 @ I5 ) )
=> ( P2 @ K2 ) ) )
=> ( P2 @ M3 ) ) ) ).
% nat_descend_induct
thf(fact_1266_ereal__closed__uminus,axiom,
! [S2: set_Extended_ereal] :
( ( topolo4677514683155976768_ereal @ S2 )
=> ( topolo4677514683155976768_ereal @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S2 ) ) ) ).
% ereal_closed_uminus
thf(fact_1267_bit__count__append,axiom,
! [X3: option_list_o,Y4: option_list_o] :
( ( prefix3213528784805800034_count @ ( prefix5314359684614007693append @ X3 @ Y4 ) )
= ( plus_p7876563987511257093_ereal @ ( prefix3213528784805800034_count @ X3 ) @ ( prefix3213528784805800034_count @ Y4 ) ) ) ).
% bit_count_append
thf(fact_1268_zero__notin__Suc__image,axiom,
! [A2: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_1269_card_Ocomp__fun__commute__on,axiom,
( ( comp_nat_nat_nat @ suc @ suc )
= ( comp_nat_nat_nat @ suc @ suc ) ) ).
% card.comp_fun_commute_on
thf(fact_1270_UNIV__nat__eq,axiom,
( top_top_set_nat
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).
% UNIV_nat_eq
thf(fact_1271_lessThan__Suc__eq__insert__0,axiom,
! [N4: nat] :
( ( set_ord_lessThan_nat @ ( suc @ N4 ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N4 ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_1272_rat__denum,axiom,
? [F4: nat > rat] :
( ( image_nat_rat @ F4 @ top_top_set_nat )
= top_top_set_rat ) ).
% rat_denum
thf(fact_1273_surj__nat__to__rat__surj,axiom,
( ( image_nat_rat @ nat_to_rat_surj @ top_top_set_nat )
= top_top_set_rat ) ).
% surj_nat_to_rat_surj
thf(fact_1274_inj__Suc,axiom,
! [N: set_nat] : ( inj_on_nat_nat @ suc @ N ) ).
% inj_Suc
thf(fact_1275_inj__on__diff__nat,axiom,
! [N: set_nat,K: nat] :
( ! [N2: nat] :
( ( member_nat @ N2 @ N )
=> ( ord_less_eq_nat @ K @ N2 ) )
=> ( inj_on_nat_nat
@ ^ [N3: nat] : ( minus_minus_nat @ N3 @ K )
@ N ) ) ).
% inj_on_diff_nat
% Helper facts (7)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X3: a,Y4: a] :
( ( if_a @ $false @ X3 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X3: a,Y4: a] :
( ( if_a @ $true @ X3 @ Y4 )
= X3 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y4: nat] :
( ( if_nat @ $false @ X3 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y4: nat] :
( ( if_nat @ $true @ X3 @ Y4 )
= X3 ) ).
thf(help_If_3_1_If_001t__Extended____Real__Oereal_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Extended____Real__Oereal_T,axiom,
! [X3: extended_ereal,Y4: extended_ereal] :
( ( if_Extended_ereal @ $false @ X3 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Extended____Real__Oereal_T,axiom,
! [X3: extended_ereal,Y4: extended_ereal] :
( ( if_Extended_ereal @ $true @ X3 @ Y4 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
prefix454693708527911765comp_o @ ( e2 @ ( product_fst_a_b @ x ) @ ( product_snd_a_b @ x ) ) @ ( e2 @ ( product_fst_a_b @ x ) @ ( product_snd_a_b @ y ) ) ).
%------------------------------------------------------------------------------