TPTP Problem File: SLH0057^1.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 : Youngs_Inequality/0000_Youngs/prob_00117_004741__12915202_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1629 ( 442 unt; 356 typ; 0 def)
% Number of atoms : 4388 ( 863 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 13077 ( 168 ~; 35 |; 308 &;10683 @)
% ( 0 <=>;1883 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 42 ( 41 usr)
% Number of type conns : 2362 (2362 >; 0 *; 0 +; 0 <<)
% Number of symbols : 318 ( 315 usr; 34 con; 0-4 aty)
% Number of variables : 4297 ( 593 ^;3597 !; 107 ?;4297 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:26:46.874
%------------------------------------------------------------------------------
% Could-be-implicit typings (41)
thf(ty_n_t__Set__Oset_I_062_I_062_It__Real__Oreal_Mt__Real__Oreal_J_Mt__Real__Oreal_J_J,type,
set_real_real_real: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Extended____Nonnegative____Real__Oennreal_J,type,
sigma_7234349610311085201nnreal: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_J,type,
set_se4580700918925141924nnreal: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
sigma_4258434043392614480l_real: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J_J,type,
set_set_real_real: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__Real__Oreal_J_Mt__Real__Oreal_J_J,type,
set_set_real_real2: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Set__Oset_It__Real__Oreal_J_J_J,type,
set_real_set_real: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Extended____Real__Oereal_J_J,type,
set_re5986766091859030457_ereal: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_M_062_It__Real__Oreal_Mt__Real__Oreal_J_J_J,type,
set_o_real_real: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Real__Oreal_Mt__Real__Oreal_J_M_Eo_J_J,type,
set_real_real_o: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Set__Oset_It__Real__Oreal_J_J,type,
sigma_3733394171116455995t_real: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J_J,type,
set_set_set_real: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Extended____Real__Oereal_J,type,
sigma_7227684458468523851_ereal: $tType ).
thf(ty_n_t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
set_Ex3793607809372303086nnreal: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Complex__Ocomplex_J_J,type,
set_real_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Extended____Real__Oereal_J_J,type,
set_se6634062954251873166_ereal: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__Real__Oreal_J_M_Eo_J_J,type,
set_set_real_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_Mt__Set__Oset_It__Real__Oreal_J_J_J,type,
set_o_set_real: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
set_real_real: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Complex__Ocomplex_J,type,
sigma_3077487657436305159omplex: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
set_set_complex: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
set_real_nat: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
sigma_measure_real: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
set_set_real: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Nat__Onat_J,type,
sigma_measure_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Extended____Nonnegative____Real__Oennreal,type,
extend8495563244428889912nnreal: $tType ).
thf(ty_n_t__Set__Oset_It__Extended____Real__Oereal_J,type,
set_Extended_ereal: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_M_Eo_J_J,type,
set_real_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_Mt__Real__Oreal_J_J,type,
set_o_real: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_I_Eo_J,type,
sigma_measure_o: $tType ).
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
set_o_o: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $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__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (315)
thf(sy_c_Bochner__Integration_Ointegrable_001t__Real__Oreal_001t__Real__Oreal,type,
bochne3340023020068487468l_real: sigma_measure_real > ( real > real ) > $o ).
thf(sy_c_Borel__Space_Ois__borel_001t__Real__Oreal_001t__Real__Oreal,type,
borel_236569967776329622l_real: ( real > real ) > sigma_measure_real > $o ).
thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Complex__Ocomplex,type,
borel_1392132677378845456omplex: sigma_3077487657436305159omplex ).
thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Extended____Real__Oereal,type,
borel_2631802743099733228_ereal: sigma_7227684458468523851_ereal ).
thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Nat__Onat,type,
borel_8449730974584783410el_nat: sigma_measure_nat ).
thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Real__Oreal,type,
borel_5078946678739801102l_real: sigma_measure_real ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Extended____Nonnegative____Real__Oennreal,type,
comple7330758040695736817nnreal: set_Ex3793607809372303086nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
comple8933463103962640202l_real: set_real_real > real > real ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Extended____Nonnegative____Real__Oennreal,type,
comple6814414086264997003nnreal: set_Ex3793607809372303086nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Extended____Real__Oereal,type,
comple8415311339701865915_ereal: set_Extended_ereal > extended_ereal ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
comple1385675409528146559p_real: set_real > real ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Complex__Ocomplex_J,type,
comple8424636186594484919omplex: set_set_complex > set_complex ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
comple4226387801268262977nnreal: set_se4580700918925141924nnreal > set_Ex3793607809372303086nnreal ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
comple4319282863272126363_ereal: set_se6634062954251873166_ereal > set_Extended_ereal ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Real__Oreal_J,type,
comple3096694443085538997t_real: set_set_real > set_real ).
thf(sy_c_Complete__Measure_Ocompletion_001t__Complex__Ocomplex,type,
comple7748144648682430500omplex: sigma_3077487657436305159omplex > sigma_3077487657436305159omplex ).
thf(sy_c_Complete__Measure_Ocompletion_001t__Real__Oreal,type,
comple3506806835435775778n_real: sigma_measure_real > sigma_measure_real ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Real__Oreal,type,
condit7084745239686109512e_real: set_real > $o ).
thf(sy_c_Disjoint__Sets_Odisjoint__family__on_001t__Nat__Onat_001t__Real__Oreal,type,
disjoi2035185749148668758t_real: ( nat > set_real ) > set_nat > $o ).
thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity_001t__Extended____Nonnegative____Real__Oennreal,type,
extend2057119558705770725nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Fun_Oid_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
id_real_real: ( real > real ) > real > real ).
thf(sy_c_Fun_Oid_001_Eo,type,
id_o: $o > $o ).
thf(sy_c_Fun_Oid_001t__Complex__Ocomplex,type,
id_complex: complex > complex ).
thf(sy_c_Fun_Oid_001t__Extended____Real__Oereal,type,
id_Extended_ereal: extended_ereal > extended_ereal ).
thf(sy_c_Fun_Oid_001t__Nat__Onat,type,
id_nat: nat > nat ).
thf(sy_c_Fun_Oid_001t__Real__Oreal,type,
id_real: real > real ).
thf(sy_c_Fun_Oid_001t__Set__Oset_It__Real__Oreal_J,type,
id_set_real: set_real > set_real ).
thf(sy_c_Fun_Omonotone__on_001_062_It__Real__Oreal_Mt__Real__Oreal_J_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
monoto4255458463005331543l_real: set_real_real > ( ( real > real ) > ( real > real ) > $o ) > ( ( real > real ) > ( real > real ) > $o ) > ( ( real > real ) > real > real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001_Eo_001_Eo,type,
monotone_on_o_o: set_o > ( $o > $o > $o ) > ( $o > $o > $o ) > ( $o > $o ) > $o ).
thf(sy_c_Fun_Omonotone__on_001_Eo_001t__Extended____Real__Oereal,type,
monoto3224395847736110621_ereal: set_o > ( $o > $o > $o ) > ( extended_ereal > extended_ereal > $o ) > ( $o > extended_ereal ) > $o ).
thf(sy_c_Fun_Omonotone__on_001_Eo_001t__Nat__Onat,type,
monotone_on_o_nat: set_o > ( $o > $o > $o ) > ( nat > nat > $o ) > ( $o > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001_Eo_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
monoto2235544329619742815nnreal: set_o > ( $o > $o > $o ) > ( set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal > $o ) > ( $o > set_Ex3793607809372303086nnreal ) > $o ).
thf(sy_c_Fun_Omonotone__on_001_Eo_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
monoto1696596325804605437_ereal: set_o > ( $o > $o > $o ) > ( set_Extended_ereal > set_Extended_ereal > $o ) > ( $o > set_Extended_ereal ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Complex__Ocomplex_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
monoto35072715882444302l_real: set_complex > ( complex > complex > $o ) > ( ( real > real ) > ( real > real ) > $o ) > ( complex > real > real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Complex__Ocomplex_001t__Nat__Onat,type,
monoto2406513391651152359ex_nat: set_complex > ( complex > complex > $o ) > ( nat > nat > $o ) > ( complex > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Complex__Ocomplex_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
monoto3400374153095604485nnreal: set_complex > ( complex > complex > $o ) > ( set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal > $o ) > ( complex > set_Ex3793607809372303086nnreal ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Extended____Nonnegative____Real__Oennreal_001t__Nat__Onat,type,
monoto7065424282079713393al_nat: set_Ex3793607809372303086nnreal > ( extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ) > ( nat > nat > $o ) > ( extend8495563244428889912nnreal > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Extended____Nonnegative____Real__Oennreal_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
monoto5979632021179393935nnreal: set_Ex3793607809372303086nnreal > ( extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ) > ( set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal > $o ) > ( extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Extended____Real__Oereal_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
monoto4113897589229538718l_real: set_Extended_ereal > ( extended_ereal > extended_ereal > $o ) > ( ( real > real ) > ( real > real ) > $o ) > ( extended_ereal > real > real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Extended____Real__Oereal_001_Eo,type,
monoto5670193684399400497real_o: set_Extended_ereal > ( extended_ereal > extended_ereal > $o ) > ( $o > $o > $o ) > ( extended_ereal > $o ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
monoto2923698811514177639_ereal: set_Extended_ereal > ( extended_ereal > extended_ereal > $o ) > ( extended_ereal > extended_ereal > $o ) > ( extended_ereal > extended_ereal ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
monoto2580034644210098551al_nat: set_Extended_ereal > ( extended_ereal > extended_ereal > $o ) > ( nat > nat > $o ) > ( extended_ereal > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Extended____Real__Oereal_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
monoto6742655004523227285nnreal: set_Extended_ereal > ( extended_ereal > extended_ereal > $o ) > ( set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal > $o ) > ( extended_ereal > set_Ex3793607809372303086nnreal ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Extended____Real__Oereal_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
monoto1076656197419758151_ereal: set_Extended_ereal > ( extended_ereal > extended_ereal > $o ) > ( set_Extended_ereal > set_Extended_ereal > $o ) > ( extended_ereal > set_Extended_ereal ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Extended____Real__Oereal_001t__Set__Oset_It__Real__Oreal_J,type,
monoto8952765597322587401t_real: set_Extended_ereal > ( extended_ereal > extended_ereal > $o ) > ( set_real > set_real > $o ) > ( extended_ereal > set_real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
monoto2824216093323351088l_real: set_nat > ( nat > nat > $o ) > ( ( real > real ) > ( real > real ) > $o ) > ( nat > real > real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Extended____Nonnegative____Real__Oennreal,type,
monoto2291723841412853873nnreal: set_nat > ( nat > nat > $o ) > ( extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ) > ( nat > extend8495563244428889912nnreal ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Extended____Real__Oereal,type,
monoto8452838292781035605_ereal: set_nat > ( nat > nat > $o ) > ( extended_ereal > extended_ereal > $o ) > ( nat > extended_ereal ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Nat__Onat,type,
monotone_on_nat_nat: set_nat > ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
monoto4660286046138248231nnreal: set_nat > ( nat > nat > $o ) > ( set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal > $o ) > ( nat > set_Ex3793607809372303086nnreal ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
monoto6788471982328799797_ereal: set_nat > ( nat > nat > $o ) > ( set_Extended_ereal > set_Extended_ereal > $o ) > ( nat > set_Extended_ereal ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Set__Oset_It__Real__Oreal_J,type,
monoto7274299666542614427t_real: set_nat > ( nat > nat > $o ) > ( set_real > set_real > $o ) > ( nat > set_real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Real__Oreal_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
monoto8965231823629880588l_real: set_real > ( real > real > $o ) > ( ( real > real ) > ( real > real ) > $o ) > ( real > real > real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Real__Oreal_001t__Nat__Onat,type,
monotone_on_real_nat: set_real > ( real > real > $o ) > ( nat > nat > $o ) > ( real > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Real__Oreal_001t__Real__Oreal,type,
monoto4017252874604999745l_real: set_real > ( real > real > $o ) > ( real > real > $o ) > ( real > real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Real__Oreal_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
monoto2626391617355967235nnreal: set_real > ( real > real > $o ) > ( set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal > $o ) > ( real > set_Ex3793607809372303086nnreal ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Real__Oreal_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
monoto1556369633277000793_ereal: set_real > ( real > real > $o ) > ( set_Extended_ereal > set_Extended_ereal > $o ) > ( real > set_Extended_ereal ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
monoto6196239568472585294l_real: set_se4580700918925141924nnreal > ( set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal > $o ) > ( ( real > real ) > ( real > real ) > $o ) > ( set_Ex3793607809372303086nnreal > real > real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_001_Eo,type,
monoto3509817954306571137real_o: set_se4580700918925141924nnreal > ( set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal > $o ) > ( $o > $o > $o ) > ( set_Ex3793607809372303086nnreal > $o ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_001t__Extended____Real__Oereal,type,
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topolo6736312753545056449_ereal: ( nat > set_Extended_ereal ) > $o ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Real__Oreal_J,type,
topolo2489691266198938127t_real: ( nat > set_real ) > $o ).
thf(sy_c_member_001_062_I_062_It__Real__Oreal_Mt__Real__Oreal_J_M_Eo_J,type,
member_real_real_o: ( ( real > real ) > $o ) > set_real_real_o > $o ).
thf(sy_c_member_001_062_I_062_It__Real__Oreal_Mt__Real__Oreal_J_Mt__Real__Oreal_J,type,
member5749659578190367193l_real: ( ( real > real ) > real ) > set_real_real_real > $o ).
thf(sy_c_member_001_062_I_Eo_M_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
member_o_real_real: ( $o > real > real ) > set_o_real_real > $o ).
thf(sy_c_member_001_062_I_Eo_M_Eo_J,type,
member_o_o: ( $o > $o ) > set_o_o > $o ).
thf(sy_c_member_001_062_I_Eo_Mt__Real__Oreal_J,type,
member_o_real: ( $o > real ) > set_o_real > $o ).
thf(sy_c_member_001_062_I_Eo_Mt__Set__Oset_It__Real__Oreal_J_J,type,
member_o_set_real: ( $o > set_real ) > set_o_set_real > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_M_Eo_J,type,
member_real_o: ( real > $o ) > set_real_o > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_Mt__Complex__Ocomplex_J,type,
member_real_complex: ( real > complex ) > set_real_complex > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_Mt__Extended____Real__Oereal_J,type,
member7593515600531736322_ereal: ( real > extended_ereal ) > set_re5986766091859030457_ereal > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
member_real_nat: ( real > nat ) > set_real_nat > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
member_real_real: ( real > real ) > set_real_real > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_Mt__Set__Oset_It__Real__Oreal_J_J,type,
member_real_set_real: ( real > set_real ) > set_real_set_real > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Real__Oreal_J_M_Eo_J,type,
member_set_real_o: ( set_real > $o ) > set_set_real_o > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Real__Oreal_J_Mt__Real__Oreal_J,type,
member_set_real_real: ( set_real > real ) > set_set_real_real2 > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__Extended____Nonnegative____Real__Oennreal,type,
member7908768830364227535nnreal: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal > $o ).
thf(sy_c_member_001t__Extended____Real__Oereal,type,
member2350847679896131959_ereal: extended_ereal > set_Extended_ereal > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
member_set_real_real2: set_real_real > set_set_real_real > $o ).
thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__Complex__Ocomplex_J,type,
member_set_complex: set_complex > set_set_complex > $o ).
thf(sy_c_member_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
member603777416030116741nnreal: set_Ex3793607809372303086nnreal > set_se4580700918925141924nnreal > $o ).
thf(sy_c_member_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
member5519481007471526743_ereal: set_Extended_ereal > set_se6634062954251873166_ereal > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Real__Oreal_J,type,
member_set_real: set_real > set_set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
member_set_set_real: set_set_real > set_set_set_real > $o ).
thf(sy_v_a,type,
a: real ).
thf(sy_v_b,type,
b: real ).
thf(sy_v_f,type,
f: real > real ).
thf(sy_v_g____,type,
g: nat > real > real ).
% Relevant facts (1269)
thf(fact_0_fborel,axiom,
member_real_real @ f @ ( sigma_5267869275261027754l_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ ( set_or1222579329274155063t_real @ a @ b ) ) @ borel_5078946678739801102l_real ) ).
% fborel
thf(fact_1_measurable__lborel1,axiom,
! [M: sigma_measure_real] :
( ( sigma_5267869275261027754l_real @ M @ lebesgue_lborel_real )
= ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ).
% measurable_lborel1
thf(fact_2_measurable__lborel2,axiom,
! [M: sigma_measure_real] :
( ( sigma_5267869275261027754l_real @ lebesgue_lborel_real @ M )
= ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ M ) ) ).
% measurable_lborel2
thf(fact_3_measurable__lebesgue__cong,axiom,
! [S: set_real,F: real > real,G: real > real,M: sigma_measure_real] :
( ! [X: real] :
( ( member_real @ X @ S )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ S ) @ M ) )
= ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ S ) @ M ) ) ) ) ).
% measurable_lebesgue_cong
thf(fact_4_mon,axiom,
monoto4017252874604999745l_real @ ( set_or1222579329274155063t_real @ a @ b ) @ ord_less_eq_real @ ord_less_eq_real @ f ).
% mon
thf(fact_5_measurable__restrict__space1,axiom,
! [F: real > real,M: sigma_measure_real,N: sigma_measure_real,Omega: set_real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) )
=> ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ ( sigma_5414646170262037096e_real @ M @ Omega ) @ N ) ) ) ).
% measurable_restrict_space1
thf(fact_6_measurable__completion,axiom,
! [F: real > real,M: sigma_measure_real,N: sigma_measure_real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) )
=> ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ ( comple3506806835435775778n_real @ M ) @ N ) ) ) ).
% measurable_completion
thf(fact_7_simple,axiom,
! [I: nat] : ( nonneg485563716852976898l_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ ( set_or1222579329274155063t_real @ a @ b ) ) @ ( g @ I ) ) ).
% simple
thf(fact_8_gfb,axiom,
! [X2: real,I: nat] :
( ( member_real @ X2 @ ( set_or1222579329274155063t_real @ a @ b ) )
=> ( ord_less_eq_real @ ( g @ I @ X2 ) @ ( f @ b ) ) ) ).
% gfb
thf(fact_9__092_060open_062space_Alborel_A_061_Aspace_Alebesgue_092_060close_062,axiom,
( ( sigma_space_real @ lebesgue_lborel_real )
= ( sigma_space_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) ) ) ).
% \<open>space lborel = space lebesgue\<close>
thf(fact_10__092_060open_062f_A_096_A_123a_O_Ob_125_A_092_060subseteq_062_A_123f_Aa_O_Of_Ab_125_092_060close_062,axiom,
ord_less_eq_set_real @ ( image_real_real @ f @ ( set_or1222579329274155063t_real @ a @ b ) ) @ ( set_or1222579329274155063t_real @ ( f @ a ) @ ( f @ b ) ) ).
% \<open>f ` {a..b} \<subseteq> {f a..f b}\<close>
thf(fact_11_is__borel__def,axiom,
( borel_236569967776329622l_real
= ( ^ [F2: real > real,M2: sigma_measure_real] : ( member_real_real @ F2 @ ( sigma_5267869275261027754l_real @ M2 @ borel_5078946678739801102l_real ) ) ) ) ).
% is_borel_def
thf(fact_12_id__borel__measurable__lebesgue__on,axiom,
! [S: set_real] : ( member_real_real @ id_real @ ( sigma_5267869275261027754l_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ S ) @ borel_5078946678739801102l_real ) ) ).
% id_borel_measurable_lebesgue_on
thf(fact_13_g__le__f,axiom,
! [I: nat,X2: real] : ( ord_less_eq_real @ ( g @ I @ X2 ) @ ( f @ X2 ) ) ).
% g_le_f
thf(fact_14_nonneg,axiom,
! [I2: nat,X3: real] : ( ord_less_eq_real @ zero_zero_real @ ( g @ I2 @ X3 ) ) ).
% nonneg
thf(fact_15_space__completion,axiom,
! [M: sigma_measure_real] :
( ( sigma_space_real @ ( comple3506806835435775778n_real @ M ) )
= ( sigma_space_real @ M ) ) ).
% space_completion
thf(fact_16_space__lborel,axiom,
( ( sigma_space_real @ lebesgue_lborel_real )
= ( sigma_space_real @ borel_5078946678739801102l_real ) ) ).
% space_lborel
thf(fact_17_space__lebesgue__on,axiom,
! [S: set_real] :
( ( sigma_space_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ S ) )
= S ) ).
% space_lebesgue_on
thf(fact_18_id__borel__measurable__lebesgue,axiom,
member_real_real @ id_real @ ( sigma_5267869275261027754l_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ borel_5078946678739801102l_real ) ).
% id_borel_measurable_lebesgue
thf(fact_19_simple__function__completion,axiom,
! [M: sigma_measure_real,F: real > real] :
( ( nonneg485563716852976898l_real @ M @ F )
=> ( nonneg485563716852976898l_real @ ( comple3506806835435775778n_real @ M ) @ F ) ) ).
% simple_function_completion
thf(fact_20_measurable__ident,axiom,
! [M: sigma_measure_real] : ( member_real_real @ id_real @ ( sigma_5267869275261027754l_real @ M @ M ) ) ).
% measurable_ident
thf(fact_21_measurable__cong__simp,axiom,
! [M: sigma_measure_real,N: sigma_measure_real,M3: sigma_measure_real,N2: sigma_measure_real,F: real > real,G: real > real] :
( ( M = N )
=> ( ( M3 = N2 )
=> ( ! [W: real] :
( ( member_real @ W @ ( sigma_space_real @ M ) )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ M3 ) )
= ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ N @ N2 ) ) ) ) ) ) ).
% measurable_cong_simp
thf(fact_22_measurable__space,axiom,
! [F: $o > $o,M: sigma_measure_o,A: sigma_measure_o,X2: $o] :
( ( member_o_o @ F @ ( sigma_measurable_o_o @ M @ A ) )
=> ( ( member_o @ X2 @ ( sigma_space_o @ M ) )
=> ( member_o @ ( F @ X2 ) @ ( sigma_space_o @ A ) ) ) ) ).
% measurable_space
thf(fact_23_measurable__space,axiom,
! [F: $o > real,M: sigma_measure_o,A: sigma_measure_real,X2: $o] :
( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ A ) )
=> ( ( member_o @ X2 @ ( sigma_space_o @ M ) )
=> ( member_real @ ( F @ X2 ) @ ( sigma_space_real @ A ) ) ) ) ).
% measurable_space
thf(fact_24_measurable__space,axiom,
! [F: real > $o,M: sigma_measure_real,A: sigma_measure_o,X2: real] :
( ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ M @ A ) )
=> ( ( member_real @ X2 @ ( sigma_space_real @ M ) )
=> ( member_o @ ( F @ X2 ) @ ( sigma_space_o @ A ) ) ) ) ).
% measurable_space
thf(fact_25_measurable__space,axiom,
! [F: real > real,M: sigma_measure_real,A: sigma_measure_real,X2: real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ A ) )
=> ( ( member_real @ X2 @ ( sigma_space_real @ M ) )
=> ( member_real @ ( F @ X2 ) @ ( sigma_space_real @ A ) ) ) ) ).
% measurable_space
thf(fact_26_measurable__space,axiom,
! [F: $o > set_real,M: sigma_measure_o,A: sigma_3733394171116455995t_real,X2: $o] :
( ( member_o_set_real @ F @ ( sigma_4088809687498539434t_real @ M @ A ) )
=> ( ( member_o @ X2 @ ( sigma_space_o @ M ) )
=> ( member_set_real @ ( F @ X2 ) @ ( sigma_space_set_real @ A ) ) ) ) ).
% measurable_space
thf(fact_27_measurable__space,axiom,
! [F: set_real > $o,M: sigma_3733394171116455995t_real,A: sigma_measure_o,X2: set_real] :
( ( member_set_real_o @ F @ ( sigma_6120279872303721572real_o @ M @ A ) )
=> ( ( member_set_real @ X2 @ ( sigma_space_set_real @ M ) )
=> ( member_o @ ( F @ X2 ) @ ( sigma_space_o @ A ) ) ) ) ).
% measurable_space
thf(fact_28_measurable__space,axiom,
! [F: set_real > real,M: sigma_3733394171116455995t_real,A: sigma_measure_real,X2: set_real] :
( ( member_set_real_real @ F @ ( sigma_397049400287467232l_real @ M @ A ) )
=> ( ( member_set_real @ X2 @ ( sigma_space_set_real @ M ) )
=> ( member_real @ ( F @ X2 ) @ ( sigma_space_real @ A ) ) ) ) ).
% measurable_space
thf(fact_29_measurable__space,axiom,
! [F: real > set_real,M: sigma_measure_real,A: sigma_3733394171116455995t_real,X2: real] :
( ( member_real_set_real @ F @ ( sigma_6606012509476713952t_real @ M @ A ) )
=> ( ( member_real @ X2 @ ( sigma_space_real @ M ) )
=> ( member_set_real @ ( F @ X2 ) @ ( sigma_space_set_real @ A ) ) ) ) ).
% measurable_space
thf(fact_30_measurable__space,axiom,
! [F: ( real > real ) > $o,M: sigma_4258434043392614480l_real,A: sigma_measure_o,X2: real > real] :
( ( member_real_real_o @ F @ ( sigma_3869891368983680143real_o @ M @ A ) )
=> ( ( member_real_real @ X2 @ ( sigma_3619470280215722479l_real @ M ) )
=> ( member_o @ ( F @ X2 ) @ ( sigma_space_o @ A ) ) ) ) ).
% measurable_space
thf(fact_31_measurable__space,axiom,
! [F: $o > real > real,M: sigma_measure_o,A: sigma_4258434043392614480l_real,X2: $o] :
( ( member_o_real_real @ F @ ( sigma_1778443616946741311l_real @ M @ A ) )
=> ( ( member_o @ X2 @ ( sigma_space_o @ M ) )
=> ( member_real_real @ ( F @ X2 ) @ ( sigma_3619470280215722479l_real @ A ) ) ) ) ).
% measurable_space
thf(fact_32_measurable__cong,axiom,
! [M: sigma_measure_real,F: real > real,G: real > real,M3: sigma_measure_real] :
( ! [W: real] :
( ( member_real @ W @ ( sigma_space_real @ M ) )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ M3 ) )
= ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ M3 ) ) ) ) ).
% measurable_cong
thf(fact_33_borel__measurable__mono__on__fnc,axiom,
! [A: set_real,F: real > real] :
( ( monoto4017252874604999745l_real @ A @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ ( sigma_5414646170262037096e_real @ borel_5078946678739801102l_real @ A ) @ borel_5078946678739801102l_real ) ) ) ).
% borel_measurable_mono_on_fnc
thf(fact_34_measurable__restrict__mono,axiom,
! [F: real > real,M: sigma_measure_real,A: set_real,N: sigma_measure_real,B: set_real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ ( sigma_5414646170262037096e_real @ M @ A ) @ N ) )
=> ( ( ord_less_eq_set_real @ B @ A )
=> ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ ( sigma_5414646170262037096e_real @ M @ B ) @ N ) ) ) ) ).
% measurable_restrict_mono
thf(fact_35_atLeastatMost__subset__iff,axiom,
! [A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,C: set_Ex3793607809372303086nnreal,D: set_Ex3793607809372303086nnreal] :
( ( ord_le3366939622266546180nnreal @ ( set_or4467169092781218297nnreal @ A2 @ B2 ) @ ( set_or4467169092781218297nnreal @ C @ D ) )
= ( ~ ( ord_le6787938422905777998nnreal @ A2 @ B2 )
| ( ( ord_le6787938422905777998nnreal @ C @ A2 )
& ( ord_le6787938422905777998nnreal @ B2 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_36_atLeastatMost__subset__iff,axiom,
! [A2: set_Extended_ereal,B2: set_Extended_ereal,C: set_Extended_ereal,D: set_Extended_ereal] :
( ( ord_le5287700718633833262_ereal @ ( set_or814338393153804259_ereal @ A2 @ B2 ) @ ( set_or814338393153804259_ereal @ C @ D ) )
= ( ~ ( ord_le1644982726543182158_ereal @ A2 @ B2 )
| ( ( ord_le1644982726543182158_ereal @ C @ A2 )
& ( ord_le1644982726543182158_ereal @ B2 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_37_atLeastatMost__subset__iff,axiom,
! [A2: set_real,B2: set_real,C: set_real,D: set_real] :
( ( ord_le3558479182127378552t_real @ ( set_or7743017856606604397t_real @ A2 @ B2 ) @ ( set_or7743017856606604397t_real @ C @ D ) )
= ( ~ ( ord_less_eq_set_real @ A2 @ B2 )
| ( ( ord_less_eq_set_real @ C @ A2 )
& ( ord_less_eq_set_real @ B2 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_38_atLeastatMost__subset__iff,axiom,
! [A2: real > real,B2: real > real,C: real > real,D: real > real] :
( ( ord_le4198349162570665613l_real @ ( set_or6435548056685896962l_real @ A2 @ B2 ) @ ( set_or6435548056685896962l_real @ C @ D ) )
= ( ~ ( ord_le6948328307412524503l_real @ A2 @ B2 )
| ( ( ord_le6948328307412524503l_real @ C @ A2 )
& ( ord_le6948328307412524503l_real @ B2 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_39_atLeastatMost__subset__iff,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D: extend8495563244428889912nnreal] :
( ( ord_le6787938422905777998nnreal @ ( set_or6180253510304638531nnreal @ A2 @ B2 ) @ ( set_or6180253510304638531nnreal @ C @ D ) )
= ( ~ ( ord_le3935885782089961368nnreal @ A2 @ B2 )
| ( ( ord_le3935885782089961368nnreal @ C @ A2 )
& ( ord_le3935885782089961368nnreal @ B2 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_40_atLeastatMost__subset__iff,axiom,
! [A2: extended_ereal,B2: extended_ereal,C: extended_ereal,D: extended_ereal] :
( ( ord_le1644982726543182158_ereal @ ( set_or2336185686312672771_ereal @ A2 @ B2 ) @ ( set_or2336185686312672771_ereal @ C @ D ) )
= ( ~ ( ord_le1083603963089353582_ereal @ A2 @ B2 )
| ( ( ord_le1083603963089353582_ereal @ C @ A2 )
& ( ord_le1083603963089353582_ereal @ B2 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_41_atLeastatMost__subset__iff,axiom,
! [A2: real,B2: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ~ ( ord_less_eq_real @ A2 @ B2 )
| ( ( ord_less_eq_real @ C @ A2 )
& ( ord_less_eq_real @ B2 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_42_atLeastatMost__subset__iff,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( ( ord_less_eq_nat @ C @ A2 )
& ( ord_less_eq_nat @ B2 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_43_atLeastAtMost__iff,axiom,
! [I: $o,L: $o,U: $o] :
( ( member_o @ I @ ( set_or8904488021354931149Most_o @ L @ U ) )
= ( ( ord_less_eq_o @ L @ I )
& ( ord_less_eq_o @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_44_atLeastAtMost__iff,axiom,
! [I: set_Ex3793607809372303086nnreal,L: set_Ex3793607809372303086nnreal,U: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ I @ ( set_or4467169092781218297nnreal @ L @ U ) )
= ( ( ord_le6787938422905777998nnreal @ L @ I )
& ( ord_le6787938422905777998nnreal @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_45_atLeastAtMost__iff,axiom,
! [I: set_Extended_ereal,L: set_Extended_ereal,U: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ I @ ( set_or814338393153804259_ereal @ L @ U ) )
= ( ( ord_le1644982726543182158_ereal @ L @ I )
& ( ord_le1644982726543182158_ereal @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_46_atLeastAtMost__iff,axiom,
! [I: set_real,L: set_real,U: set_real] :
( ( member_set_real @ I @ ( set_or7743017856606604397t_real @ L @ U ) )
= ( ( ord_less_eq_set_real @ L @ I )
& ( ord_less_eq_set_real @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_47_atLeastAtMost__iff,axiom,
! [I: real > real,L: real > real,U: real > real] :
( ( member_real_real @ I @ ( set_or6435548056685896962l_real @ L @ U ) )
= ( ( ord_le6948328307412524503l_real @ L @ I )
& ( ord_le6948328307412524503l_real @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_48_atLeastAtMost__iff,axiom,
! [I: real,L: real,U: real] :
( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U ) )
= ( ( ord_less_eq_real @ L @ I )
& ( ord_less_eq_real @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_49_atLeastAtMost__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_eq_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_50_Icc__eq__Icc,axiom,
! [L: set_Ex3793607809372303086nnreal,H: set_Ex3793607809372303086nnreal,L2: set_Ex3793607809372303086nnreal,H2: set_Ex3793607809372303086nnreal] :
( ( ( set_or4467169092781218297nnreal @ L @ H )
= ( set_or4467169092781218297nnreal @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_le6787938422905777998nnreal @ L @ H )
& ~ ( ord_le6787938422905777998nnreal @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_51_Icc__eq__Icc,axiom,
! [L: set_Extended_ereal,H: set_Extended_ereal,L2: set_Extended_ereal,H2: set_Extended_ereal] :
( ( ( set_or814338393153804259_ereal @ L @ H )
= ( set_or814338393153804259_ereal @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_le1644982726543182158_ereal @ L @ H )
& ~ ( ord_le1644982726543182158_ereal @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_52_Icc__eq__Icc,axiom,
! [L: set_real,H: set_real,L2: set_real,H2: set_real] :
( ( ( set_or7743017856606604397t_real @ L @ H )
= ( set_or7743017856606604397t_real @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_set_real @ L @ H )
& ~ ( ord_less_eq_set_real @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_53_Icc__eq__Icc,axiom,
! [L: real > real,H: real > real,L2: real > real,H2: real > real] :
( ( ( set_or6435548056685896962l_real @ L @ H )
= ( set_or6435548056685896962l_real @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_le6948328307412524503l_real @ L @ H )
& ~ ( ord_le6948328307412524503l_real @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_54_Icc__eq__Icc,axiom,
! [L: real,H: real,L2: real,H2: real] :
( ( ( set_or1222579329274155063t_real @ L @ H )
= ( set_or1222579329274155063t_real @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_real @ L @ H )
& ~ ( ord_less_eq_real @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_55_Icc__eq__Icc,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or1269000886237332187st_nat @ L @ H )
= ( set_or1269000886237332187st_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_nat @ L @ H )
& ~ ( ord_less_eq_nat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_56__C0_C,axiom,
! [X2: real] : ( ord_less_eq_real @ zero_zero_real @ ( f @ X2 ) ) ).
% "0"
thf(fact_57_borel__measurable__simple__function,axiom,
! [M: sigma_measure_real,F: real > real] :
( ( nonneg485563716852976898l_real @ M @ F )
=> ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ).
% borel_measurable_simple_function
thf(fact_58_ord_Omono__on__subset,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > extended_ereal,B: set_nat] :
( ( monoto8452838292781035605_ereal @ A @ Less_eq @ ord_le1083603963089353582_ereal @ F )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( monoto8452838292781035605_ereal @ B @ Less_eq @ ord_le1083603963089353582_ereal @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_59_ord_Omono__on__subset,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > extend8495563244428889912nnreal,B: set_nat] :
( ( monoto2291723841412853873nnreal @ A @ Less_eq @ ord_le3935885782089961368nnreal @ F )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( monoto2291723841412853873nnreal @ B @ Less_eq @ ord_le3935885782089961368nnreal @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_60_ord_Omono__on__subset,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > real,B: set_real] :
( ( monoto4017252874604999745l_real @ A @ Less_eq @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_set_real @ B @ A )
=> ( monoto4017252874604999745l_real @ B @ Less_eq @ ord_less_eq_real @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_61_ord_Omono__on__subset,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > nat,B: set_nat] :
( ( monotone_on_nat_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( monotone_on_nat_nat @ B @ Less_eq @ ord_less_eq_nat @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_62_ord_Omono__on__subset,axiom,
! [A: set_Ex3793607809372303086nnreal,Less_eq: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o,F: extend8495563244428889912nnreal > nat,B: set_Ex3793607809372303086nnreal] :
( ( monoto7065424282079713393al_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( ord_le6787938422905777998nnreal @ B @ A )
=> ( monoto7065424282079713393al_nat @ B @ Less_eq @ ord_less_eq_nat @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_63_ord_Omono__on__subset,axiom,
! [A: set_Extended_ereal,Less_eq: extended_ereal > extended_ereal > $o,F: extended_ereal > nat,B: set_Extended_ereal] :
( ( monoto2580034644210098551al_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( ord_le1644982726543182158_ereal @ B @ A )
=> ( monoto2580034644210098551al_nat @ B @ Less_eq @ ord_less_eq_nat @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_64_ord_Omono__on__subset,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > nat,B: set_real] :
( ( monotone_on_real_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_real @ B @ A )
=> ( monotone_on_real_nat @ B @ Less_eq @ ord_less_eq_nat @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_65_ord_Omono__on__subset,axiom,
! [A: set_Ex3793607809372303086nnreal,Less_eq: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o,F: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
( ( monoto5979632021179393935nnreal @ A @ Less_eq @ ord_le6787938422905777998nnreal @ F )
=> ( ( ord_le6787938422905777998nnreal @ B @ A )
=> ( monoto5979632021179393935nnreal @ B @ Less_eq @ ord_le6787938422905777998nnreal @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_66_ord_Omono__on__subset,axiom,
! [A: set_Extended_ereal,Less_eq: extended_ereal > extended_ereal > $o,F: extended_ereal > set_Ex3793607809372303086nnreal,B: set_Extended_ereal] :
( ( monoto6742655004523227285nnreal @ A @ Less_eq @ ord_le6787938422905777998nnreal @ F )
=> ( ( ord_le1644982726543182158_ereal @ B @ A )
=> ( monoto6742655004523227285nnreal @ B @ Less_eq @ ord_le6787938422905777998nnreal @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_67_ord_Omono__on__subset,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > set_Ex3793607809372303086nnreal,B: set_real] :
( ( monoto2626391617355967235nnreal @ A @ Less_eq @ ord_le6787938422905777998nnreal @ F )
=> ( ( ord_less_eq_set_real @ B @ A )
=> ( monoto2626391617355967235nnreal @ B @ Less_eq @ ord_le6787938422905777998nnreal @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_68_mono__on__subset,axiom,
! [A: set_real,F: real > real,B: set_real] :
( ( monoto4017252874604999745l_real @ A @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_set_real @ B @ A )
=> ( monoto4017252874604999745l_real @ B @ ord_less_eq_real @ ord_less_eq_real @ F ) ) ) ).
% mono_on_subset
thf(fact_69_mono__on__subset,axiom,
! [A: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > nat,B: set_Ex3793607809372303086nnreal] :
( ( monoto7065424282079713393al_nat @ A @ ord_le3935885782089961368nnreal @ ord_less_eq_nat @ F )
=> ( ( ord_le6787938422905777998nnreal @ B @ A )
=> ( monoto7065424282079713393al_nat @ B @ ord_le3935885782089961368nnreal @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_70_mono__on__subset,axiom,
! [A: set_Extended_ereal,F: extended_ereal > nat,B: set_Extended_ereal] :
( ( monoto2580034644210098551al_nat @ A @ ord_le1083603963089353582_ereal @ ord_less_eq_nat @ F )
=> ( ( ord_le1644982726543182158_ereal @ B @ A )
=> ( monoto2580034644210098551al_nat @ B @ ord_le1083603963089353582_ereal @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_71_mono__on__subset,axiom,
! [A: set_real,F: real > nat,B: set_real] :
( ( monotone_on_real_nat @ A @ ord_less_eq_real @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_real @ B @ A )
=> ( monotone_on_real_nat @ B @ ord_less_eq_real @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_72_mono__on__subset,axiom,
! [A: set_nat,F: nat > extended_ereal,B: set_nat] :
( ( monoto8452838292781035605_ereal @ A @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ F )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( monoto8452838292781035605_ereal @ B @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ F ) ) ) ).
% mono_on_subset
thf(fact_73_mono__on__subset,axiom,
! [A: set_nat,F: nat > extend8495563244428889912nnreal,B: set_nat] :
( ( monoto2291723841412853873nnreal @ A @ ord_less_eq_nat @ ord_le3935885782089961368nnreal @ F )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( monoto2291723841412853873nnreal @ B @ ord_less_eq_nat @ ord_le3935885782089961368nnreal @ F ) ) ) ).
% mono_on_subset
thf(fact_74_mono__on__subset,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( monotone_on_nat_nat @ B @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_75_mono__on__subset,axiom,
! [A: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
( ( monoto5979632021179393935nnreal @ A @ ord_le3935885782089961368nnreal @ ord_le6787938422905777998nnreal @ F )
=> ( ( ord_le6787938422905777998nnreal @ B @ A )
=> ( monoto5979632021179393935nnreal @ B @ ord_le3935885782089961368nnreal @ ord_le6787938422905777998nnreal @ F ) ) ) ).
% mono_on_subset
thf(fact_76_mono__on__subset,axiom,
! [A: set_Extended_ereal,F: extended_ereal > set_Ex3793607809372303086nnreal,B: set_Extended_ereal] :
( ( monoto6742655004523227285nnreal @ A @ ord_le1083603963089353582_ereal @ ord_le6787938422905777998nnreal @ F )
=> ( ( ord_le1644982726543182158_ereal @ B @ A )
=> ( monoto6742655004523227285nnreal @ B @ ord_le1083603963089353582_ereal @ ord_le6787938422905777998nnreal @ F ) ) ) ).
% mono_on_subset
thf(fact_77_mono__on__subset,axiom,
! [A: set_real,F: real > set_Ex3793607809372303086nnreal,B: set_real] :
( ( monoto2626391617355967235nnreal @ A @ ord_less_eq_real @ ord_le6787938422905777998nnreal @ F )
=> ( ( ord_less_eq_set_real @ B @ A )
=> ( monoto2626391617355967235nnreal @ B @ ord_less_eq_real @ ord_le6787938422905777998nnreal @ F ) ) ) ).
% mono_on_subset
thf(fact_78_g__le,axiom,
! [I: nat,J: nat,X2: real] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_real @ ( g @ I @ X2 ) @ ( g @ J @ X2 ) ) ) ).
% g_le
thf(fact_79_subsetI,axiom,
! [A: set_real_real,B: set_real_real] :
( ! [X: real > real] :
( ( member_real_real @ X @ A )
=> ( member_real_real @ X @ B ) )
=> ( ord_le4198349162570665613l_real @ A @ B ) ) ).
% subsetI
thf(fact_80_subsetI,axiom,
! [A: set_o,B: set_o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member_o @ X @ B ) )
=> ( ord_less_eq_set_o @ A @ B ) ) ).
% subsetI
thf(fact_81_subsetI,axiom,
! [A: set_set_real,B: set_set_real] :
( ! [X: set_real] :
( ( member_set_real @ X @ A )
=> ( member_set_real @ X @ B ) )
=> ( ord_le3558479182127378552t_real @ A @ B ) ) ).
% subsetI
thf(fact_82_subsetI,axiom,
! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ A )
=> ( member7908768830364227535nnreal @ X @ B ) )
=> ( ord_le6787938422905777998nnreal @ A @ B ) ) ).
% subsetI
thf(fact_83_subsetI,axiom,
! [A: set_Extended_ereal,B: set_Extended_ereal] :
( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A )
=> ( member2350847679896131959_ereal @ X @ B ) )
=> ( ord_le1644982726543182158_ereal @ A @ B ) ) ).
% subsetI
thf(fact_84_subsetI,axiom,
! [A: set_real,B: set_real] :
( ! [X: real] :
( ( member_real @ X @ A )
=> ( member_real @ X @ B ) )
=> ( ord_less_eq_set_real @ A @ B ) ) ).
% subsetI
thf(fact_85_subset__antisym,axiom,
! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ A @ B )
=> ( ( ord_le6787938422905777998nnreal @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_86_subset__antisym,axiom,
! [A: set_Extended_ereal,B: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A @ B )
=> ( ( ord_le1644982726543182158_ereal @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_87_subset__antisym,axiom,
! [A: set_real,B: set_real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( ord_less_eq_set_real @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_88_image__eqI,axiom,
! [B2: extended_ereal,F: nat > extended_ereal,X2: nat,A: set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member2350847679896131959_ereal @ B2 @ ( image_4309273772856505399_ereal @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_89_image__eqI,axiom,
! [B2: complex,F: nat > complex,X2: nat,A: set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_complex @ B2 @ ( image_nat_complex @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_90_image__eqI,axiom,
! [B2: real,F: nat > real,X2: nat,A: set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_real @ B2 @ ( image_nat_real @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_91_image__eqI,axiom,
! [B2: real,F: real > real,X2: real,A: set_real] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_real @ X2 @ A )
=> ( member_real @ B2 @ ( image_real_real @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_92_image__eqI,axiom,
! [B2: $o,F: real > $o,X2: real,A: set_real] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_real @ X2 @ A )
=> ( member_o @ B2 @ ( image_real_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_93_image__eqI,axiom,
! [B2: real,F: $o > real,X2: $o,A: set_o] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A )
=> ( member_real @ B2 @ ( image_o_real @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_94_image__eqI,axiom,
! [B2: $o,F: $o > $o,X2: $o,A: set_o] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A )
=> ( member_o @ B2 @ ( image_o_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_95_image__eqI,axiom,
! [B2: set_real,F: nat > set_real,X2: nat,A: set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_set_real @ B2 @ ( image_nat_set_real @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_96_image__eqI,axiom,
! [B2: set_real,F: real > set_real,X2: real,A: set_real] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_real @ X2 @ A )
=> ( member_set_real @ B2 @ ( image_real_set_real @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_97_image__eqI,axiom,
! [B2: set_real,F: $o > set_real,X2: $o,A: set_o] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A )
=> ( member_set_real @ B2 @ ( image_o_set_real @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_98_g,axiom,
monoto2824216093323351088l_real @ top_top_set_nat @ ord_less_eq_nat @ ord_le6948328307412524503l_real @ g ).
% g
thf(fact_99_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M: nat] :
( ( P @ X2 )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_100_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B2: extended_ereal,F: nat > extended_ereal] :
( ( member_nat @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member2350847679896131959_ereal @ B2 @ ( image_4309273772856505399_ereal @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_101_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B2: complex,F: nat > complex] :
( ( member_nat @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_complex @ B2 @ ( image_nat_complex @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_102_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B2: real,F: nat > real] :
( ( member_nat @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_real @ B2 @ ( image_nat_real @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_103_rev__image__eqI,axiom,
! [X2: real,A: set_real,B2: real,F: real > real] :
( ( member_real @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_real @ B2 @ ( image_real_real @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_104_rev__image__eqI,axiom,
! [X2: real,A: set_real,B2: $o,F: real > $o] :
( ( member_real @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_o @ B2 @ ( image_real_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_105_rev__image__eqI,axiom,
! [X2: $o,A: set_o,B2: real,F: $o > real] :
( ( member_o @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_real @ B2 @ ( image_o_real @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_106_rev__image__eqI,axiom,
! [X2: $o,A: set_o,B2: $o,F: $o > $o] :
( ( member_o @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_o @ B2 @ ( image_o_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_107_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B2: set_real,F: nat > set_real] :
( ( member_nat @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_set_real @ B2 @ ( image_nat_set_real @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_108_rev__image__eqI,axiom,
! [X2: real,A: set_real,B2: set_real,F: real > set_real] :
( ( member_real @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_set_real @ B2 @ ( image_real_set_real @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_109_rev__image__eqI,axiom,
! [X2: $o,A: set_o,B2: set_real,F: $o > set_real] :
( ( member_o @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_set_real @ B2 @ ( image_o_set_real @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_110_ball__imageD,axiom,
! [F: nat > extended_ereal,A: set_nat,P: extended_ereal > $o] :
( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ ( image_4309273772856505399_ereal @ F @ A ) )
=> ( P @ X ) )
=> ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( P @ ( F @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_111_ball__imageD,axiom,
! [F: nat > complex,A: set_nat,P: complex > $o] :
( ! [X: complex] :
( ( member_complex @ X @ ( image_nat_complex @ F @ A ) )
=> ( P @ X ) )
=> ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( P @ ( F @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_112_ball__imageD,axiom,
! [F: nat > real,A: set_nat,P: real > $o] :
( ! [X: real] :
( ( member_real @ X @ ( image_nat_real @ F @ A ) )
=> ( P @ X ) )
=> ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( P @ ( F @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_113_ball__imageD,axiom,
! [F: nat > set_real,A: set_nat,P: set_real > $o] :
( ! [X: set_real] :
( ( member_set_real @ X @ ( image_nat_set_real @ F @ A ) )
=> ( P @ X ) )
=> ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( P @ ( F @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_114_ball__imageD,axiom,
! [F: nat > real > real,A: set_nat,P: ( real > real ) > $o] :
( ! [X: real > real] :
( ( member_real_real @ X @ ( image_nat_real_real @ F @ A ) )
=> ( P @ X ) )
=> ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( P @ ( F @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_115_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > extended_ereal,G: nat > extended_ereal] :
( ( M = N )
=> ( ! [X: nat] :
( ( member_nat @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_4309273772856505399_ereal @ F @ M )
= ( image_4309273772856505399_ereal @ G @ N ) ) ) ) ).
% image_cong
thf(fact_116_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > complex,G: nat > complex] :
( ( M = N )
=> ( ! [X: nat] :
( ( member_nat @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_complex @ F @ M )
= ( image_nat_complex @ G @ N ) ) ) ) ).
% image_cong
thf(fact_117_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > real,G: nat > real] :
( ( M = N )
=> ( ! [X: nat] :
( ( member_nat @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_real @ F @ M )
= ( image_nat_real @ G @ N ) ) ) ) ).
% image_cong
thf(fact_118_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > set_real,G: nat > set_real] :
( ( M = N )
=> ( ! [X: nat] :
( ( member_nat @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_set_real @ F @ M )
= ( image_nat_set_real @ G @ N ) ) ) ) ).
% image_cong
thf(fact_119_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > real > real,G: nat > real > real] :
( ( M = N )
=> ( ! [X: nat] :
( ( member_nat @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_real_real @ F @ M )
= ( image_nat_real_real @ G @ N ) ) ) ) ).
% image_cong
thf(fact_120_bex__imageD,axiom,
! [F: nat > extended_ereal,A: set_nat,P: extended_ereal > $o] :
( ? [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ ( image_4309273772856505399_ereal @ F @ A ) )
& ( P @ X3 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_121_bex__imageD,axiom,
! [F: nat > complex,A: set_nat,P: complex > $o] :
( ? [X3: complex] :
( ( member_complex @ X3 @ ( image_nat_complex @ F @ A ) )
& ( P @ X3 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_122_bex__imageD,axiom,
! [F: nat > real,A: set_nat,P: real > $o] :
( ? [X3: real] :
( ( member_real @ X3 @ ( image_nat_real @ F @ A ) )
& ( P @ X3 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_123_bex__imageD,axiom,
! [F: nat > set_real,A: set_nat,P: set_real > $o] :
( ? [X3: set_real] :
( ( member_set_real @ X3 @ ( image_nat_set_real @ F @ A ) )
& ( P @ X3 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_124_bex__imageD,axiom,
! [F: nat > real > real,A: set_nat,P: ( real > real ) > $o] :
( ? [X3: real > real] :
( ( member_real_real @ X3 @ ( image_nat_real_real @ F @ A ) )
& ( P @ X3 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_125_mem__Collect__eq,axiom,
! [A2: real > real,P: ( real > real ) > $o] :
( ( member_real_real @ A2 @ ( collect_real_real @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_126_mem__Collect__eq,axiom,
! [A2: real,P: real > $o] :
( ( member_real @ A2 @ ( collect_real @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_127_mem__Collect__eq,axiom,
! [A2: $o,P: $o > $o] :
( ( member_o @ A2 @ ( collect_o @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_128_mem__Collect__eq,axiom,
! [A2: set_real,P: set_real > $o] :
( ( member_set_real @ A2 @ ( collect_set_real @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_129_Collect__mem__eq,axiom,
! [A: set_real_real] :
( ( collect_real_real
@ ^ [X4: real > real] : ( member_real_real @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_130_Collect__mem__eq,axiom,
! [A: set_real] :
( ( collect_real
@ ^ [X4: real] : ( member_real @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_131_Collect__mem__eq,axiom,
! [A: set_o] :
( ( collect_o
@ ^ [X4: $o] : ( member_o @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_132_Collect__mem__eq,axiom,
! [A: set_set_real] :
( ( collect_set_real
@ ^ [X4: set_real] : ( member_set_real @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_133_image__iff,axiom,
! [Z: extended_ereal,F: nat > extended_ereal,A: set_nat] :
( ( member2350847679896131959_ereal @ Z @ ( image_4309273772856505399_ereal @ F @ A ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( Z
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_134_image__iff,axiom,
! [Z: complex,F: nat > complex,A: set_nat] :
( ( member_complex @ Z @ ( image_nat_complex @ F @ A ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( Z
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_135_image__iff,axiom,
! [Z: real > real,F: nat > real > real,A: set_nat] :
( ( member_real_real @ Z @ ( image_nat_real_real @ F @ A ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( Z
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_136_image__iff,axiom,
! [Z: real,F: nat > real,A: set_nat] :
( ( member_real @ Z @ ( image_nat_real @ F @ A ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( Z
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_137_image__iff,axiom,
! [Z: set_real,F: nat > set_real,A: set_nat] :
( ( member_set_real @ Z @ ( image_nat_set_real @ F @ A ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( Z
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_138_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > extended_ereal] :
( ( member_nat @ X2 @ A )
=> ( member2350847679896131959_ereal @ ( F @ X2 ) @ ( image_4309273772856505399_ereal @ F @ A ) ) ) ).
% imageI
thf(fact_139_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > complex] :
( ( member_nat @ X2 @ A )
=> ( member_complex @ ( F @ X2 ) @ ( image_nat_complex @ F @ A ) ) ) ).
% imageI
thf(fact_140_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > real] :
( ( member_nat @ X2 @ A )
=> ( member_real @ ( F @ X2 ) @ ( image_nat_real @ F @ A ) ) ) ).
% imageI
thf(fact_141_imageI,axiom,
! [X2: real,A: set_real,F: real > real] :
( ( member_real @ X2 @ A )
=> ( member_real @ ( F @ X2 ) @ ( image_real_real @ F @ A ) ) ) ).
% imageI
thf(fact_142_imageI,axiom,
! [X2: real,A: set_real,F: real > $o] :
( ( member_real @ X2 @ A )
=> ( member_o @ ( F @ X2 ) @ ( image_real_o @ F @ A ) ) ) ).
% imageI
thf(fact_143_imageI,axiom,
! [X2: $o,A: set_o,F: $o > real] :
( ( member_o @ X2 @ A )
=> ( member_real @ ( F @ X2 ) @ ( image_o_real @ F @ A ) ) ) ).
% imageI
thf(fact_144_imageI,axiom,
! [X2: $o,A: set_o,F: $o > $o] :
( ( member_o @ X2 @ A )
=> ( member_o @ ( F @ X2 ) @ ( image_o_o @ F @ A ) ) ) ).
% imageI
thf(fact_145_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > set_real] :
( ( member_nat @ X2 @ A )
=> ( member_set_real @ ( F @ X2 ) @ ( image_nat_set_real @ F @ A ) ) ) ).
% imageI
thf(fact_146_imageI,axiom,
! [X2: real,A: set_real,F: real > set_real] :
( ( member_real @ X2 @ A )
=> ( member_set_real @ ( F @ X2 ) @ ( image_real_set_real @ F @ A ) ) ) ).
% imageI
thf(fact_147_imageI,axiom,
! [X2: $o,A: set_o,F: $o > set_real] :
( ( member_o @ X2 @ A )
=> ( member_set_real @ ( F @ X2 ) @ ( image_o_set_real @ F @ A ) ) ) ).
% imageI
thf(fact_148_Collect__mono__iff,axiom,
! [P: extend8495563244428889912nnreal > $o,Q: extend8495563244428889912nnreal > $o] :
( ( ord_le6787938422905777998nnreal @ ( collec6648975593938027277nnreal @ P ) @ ( collec6648975593938027277nnreal @ Q ) )
= ( ! [X4: extend8495563244428889912nnreal] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_149_Collect__mono__iff,axiom,
! [P: extended_ereal > $o,Q: extended_ereal > $o] :
( ( ord_le1644982726543182158_ereal @ ( collec5835592288176408249_ereal @ P ) @ ( collec5835592288176408249_ereal @ Q ) )
= ( ! [X4: extended_ereal] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_150_Collect__mono__iff,axiom,
! [P: real > $o,Q: real > $o] :
( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
= ( ! [X4: real] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_151_set__eq__subset,axiom,
( ( ^ [Y: set_Ex3793607809372303086nnreal,Z2: set_Ex3793607809372303086nnreal] : ( Y = Z2 ) )
= ( ^ [A3: set_Ex3793607809372303086nnreal,B3: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ A3 @ B3 )
& ( ord_le6787938422905777998nnreal @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_152_set__eq__subset,axiom,
( ( ^ [Y: set_Extended_ereal,Z2: set_Extended_ereal] : ( Y = Z2 ) )
= ( ^ [A3: set_Extended_ereal,B3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A3 @ B3 )
& ( ord_le1644982726543182158_ereal @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_153_set__eq__subset,axiom,
( ( ^ [Y: set_real,Z2: set_real] : ( Y = Z2 ) )
= ( ^ [A3: set_real,B3: set_real] :
( ( ord_less_eq_set_real @ A3 @ B3 )
& ( ord_less_eq_set_real @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_154_subset__trans,axiom,
! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal,C2: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ A @ B )
=> ( ( ord_le6787938422905777998nnreal @ B @ C2 )
=> ( ord_le6787938422905777998nnreal @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_155_subset__trans,axiom,
! [A: set_Extended_ereal,B: set_Extended_ereal,C2: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A @ B )
=> ( ( ord_le1644982726543182158_ereal @ B @ C2 )
=> ( ord_le1644982726543182158_ereal @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_156_subset__trans,axiom,
! [A: set_real,B: set_real,C2: set_real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( ord_less_eq_set_real @ B @ C2 )
=> ( ord_less_eq_set_real @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_157_Collect__mono,axiom,
! [P: extend8495563244428889912nnreal > $o,Q: extend8495563244428889912nnreal > $o] :
( ! [X: extend8495563244428889912nnreal] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le6787938422905777998nnreal @ ( collec6648975593938027277nnreal @ P ) @ ( collec6648975593938027277nnreal @ Q ) ) ) ).
% Collect_mono
thf(fact_158_Collect__mono,axiom,
! [P: extended_ereal > $o,Q: extended_ereal > $o] :
( ! [X: extended_ereal] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le1644982726543182158_ereal @ ( collec5835592288176408249_ereal @ P ) @ ( collec5835592288176408249_ereal @ Q ) ) ) ).
% Collect_mono
thf(fact_159_Collect__mono,axiom,
! [P: real > $o,Q: real > $o] :
( ! [X: real] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% Collect_mono
thf(fact_160_subset__refl,axiom,
! [A: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ A @ A ) ).
% subset_refl
thf(fact_161_subset__refl,axiom,
! [A: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ A @ A ) ).
% subset_refl
thf(fact_162_subset__refl,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ A @ A ) ).
% subset_refl
thf(fact_163_subset__iff,axiom,
( ord_le4198349162570665613l_real
= ( ^ [A3: set_real_real,B3: set_real_real] :
! [T: real > real] :
( ( member_real_real @ T @ A3 )
=> ( member_real_real @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_164_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
! [T: $o] :
( ( member_o @ T @ A3 )
=> ( member_o @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_165_subset__iff,axiom,
( ord_le3558479182127378552t_real
= ( ^ [A3: set_set_real,B3: set_set_real] :
! [T: set_real] :
( ( member_set_real @ T @ A3 )
=> ( member_set_real @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_166_subset__iff,axiom,
( ord_le6787938422905777998nnreal
= ( ^ [A3: set_Ex3793607809372303086nnreal,B3: set_Ex3793607809372303086nnreal] :
! [T: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ T @ A3 )
=> ( member7908768830364227535nnreal @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_167_subset__iff,axiom,
( ord_le1644982726543182158_ereal
= ( ^ [A3: set_Extended_ereal,B3: set_Extended_ereal] :
! [T: extended_ereal] :
( ( member2350847679896131959_ereal @ T @ A3 )
=> ( member2350847679896131959_ereal @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_168_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B3: set_real] :
! [T: real] :
( ( member_real @ T @ A3 )
=> ( member_real @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_169_equalityD2,axiom,
! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
( ( A = B )
=> ( ord_le6787938422905777998nnreal @ B @ A ) ) ).
% equalityD2
thf(fact_170_equalityD2,axiom,
! [A: set_Extended_ereal,B: set_Extended_ereal] :
( ( A = B )
=> ( ord_le1644982726543182158_ereal @ B @ A ) ) ).
% equalityD2
thf(fact_171_equalityD2,axiom,
! [A: set_real,B: set_real] :
( ( A = B )
=> ( ord_less_eq_set_real @ B @ A ) ) ).
% equalityD2
thf(fact_172_equalityD1,axiom,
! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
( ( A = B )
=> ( ord_le6787938422905777998nnreal @ A @ B ) ) ).
% equalityD1
thf(fact_173_equalityD1,axiom,
! [A: set_Extended_ereal,B: set_Extended_ereal] :
( ( A = B )
=> ( ord_le1644982726543182158_ereal @ A @ B ) ) ).
% equalityD1
thf(fact_174_equalityD1,axiom,
! [A: set_real,B: set_real] :
( ( A = B )
=> ( ord_less_eq_set_real @ A @ B ) ) ).
% equalityD1
thf(fact_175_subset__eq,axiom,
( ord_le4198349162570665613l_real
= ( ^ [A3: set_real_real,B3: set_real_real] :
! [X4: real > real] :
( ( member_real_real @ X4 @ A3 )
=> ( member_real_real @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_176_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
! [X4: $o] :
( ( member_o @ X4 @ A3 )
=> ( member_o @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_177_subset__eq,axiom,
( ord_le3558479182127378552t_real
= ( ^ [A3: set_set_real,B3: set_set_real] :
! [X4: set_real] :
( ( member_set_real @ X4 @ A3 )
=> ( member_set_real @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_178_subset__eq,axiom,
( ord_le6787938422905777998nnreal
= ( ^ [A3: set_Ex3793607809372303086nnreal,B3: set_Ex3793607809372303086nnreal] :
! [X4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X4 @ A3 )
=> ( member7908768830364227535nnreal @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_179_subset__eq,axiom,
( ord_le1644982726543182158_ereal
= ( ^ [A3: set_Extended_ereal,B3: set_Extended_ereal] :
! [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ A3 )
=> ( member2350847679896131959_ereal @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_180_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B3: set_real] :
! [X4: real] :
( ( member_real @ X4 @ A3 )
=> ( member_real @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_181_equalityE,axiom,
! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
( ( A = B )
=> ~ ( ( ord_le6787938422905777998nnreal @ A @ B )
=> ~ ( ord_le6787938422905777998nnreal @ B @ A ) ) ) ).
% equalityE
thf(fact_182_equalityE,axiom,
! [A: set_Extended_ereal,B: set_Extended_ereal] :
( ( A = B )
=> ~ ( ( ord_le1644982726543182158_ereal @ A @ B )
=> ~ ( ord_le1644982726543182158_ereal @ B @ A ) ) ) ).
% equalityE
thf(fact_183_equalityE,axiom,
! [A: set_real,B: set_real] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_real @ A @ B )
=> ~ ( ord_less_eq_set_real @ B @ A ) ) ) ).
% equalityE
thf(fact_184_subsetD,axiom,
! [A: set_real_real,B: set_real_real,C: real > real] :
( ( ord_le4198349162570665613l_real @ A @ B )
=> ( ( member_real_real @ C @ A )
=> ( member_real_real @ C @ B ) ) ) ).
% subsetD
thf(fact_185_subsetD,axiom,
! [A: set_o,B: set_o,C: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ C @ A )
=> ( member_o @ C @ B ) ) ) ).
% subsetD
thf(fact_186_subsetD,axiom,
! [A: set_set_real,B: set_set_real,C: set_real] :
( ( ord_le3558479182127378552t_real @ A @ B )
=> ( ( member_set_real @ C @ A )
=> ( member_set_real @ C @ B ) ) ) ).
% subsetD
thf(fact_187_subsetD,axiom,
! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le6787938422905777998nnreal @ A @ B )
=> ( ( member7908768830364227535nnreal @ C @ A )
=> ( member7908768830364227535nnreal @ C @ B ) ) ) ).
% subsetD
thf(fact_188_subsetD,axiom,
! [A: set_Extended_ereal,B: set_Extended_ereal,C: extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A @ B )
=> ( ( member2350847679896131959_ereal @ C @ A )
=> ( member2350847679896131959_ereal @ C @ B ) ) ) ).
% subsetD
thf(fact_189_subsetD,axiom,
! [A: set_real,B: set_real,C: real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( member_real @ C @ A )
=> ( member_real @ C @ B ) ) ) ).
% subsetD
thf(fact_190_in__mono,axiom,
! [A: set_real_real,B: set_real_real,X2: real > real] :
( ( ord_le4198349162570665613l_real @ A @ B )
=> ( ( member_real_real @ X2 @ A )
=> ( member_real_real @ X2 @ B ) ) ) ).
% in_mono
thf(fact_191_in__mono,axiom,
! [A: set_o,B: set_o,X2: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ X2 @ A )
=> ( member_o @ X2 @ B ) ) ) ).
% in_mono
thf(fact_192_in__mono,axiom,
! [A: set_set_real,B: set_set_real,X2: set_real] :
( ( ord_le3558479182127378552t_real @ A @ B )
=> ( ( member_set_real @ X2 @ A )
=> ( member_set_real @ X2 @ B ) ) ) ).
% in_mono
thf(fact_193_in__mono,axiom,
! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal,X2: extend8495563244428889912nnreal] :
( ( ord_le6787938422905777998nnreal @ A @ B )
=> ( ( member7908768830364227535nnreal @ X2 @ A )
=> ( member7908768830364227535nnreal @ X2 @ B ) ) ) ).
% in_mono
thf(fact_194_in__mono,axiom,
! [A: set_Extended_ereal,B: set_Extended_ereal,X2: extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A @ B )
=> ( ( member2350847679896131959_ereal @ X2 @ A )
=> ( member2350847679896131959_ereal @ X2 @ B ) ) ) ).
% in_mono
thf(fact_195_in__mono,axiom,
! [A: set_real,B: set_real,X2: real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( member_real @ X2 @ A )
=> ( member_real @ X2 @ B ) ) ) ).
% in_mono
thf(fact_196_monotone__on__def,axiom,
( monoto4017252874604999745l_real
= ( ^ [A3: set_real,Orda: real > real > $o,Ordb: real > real > $o,F2: real > real] :
! [X4: real] :
( ( member_real @ X4 @ A3 )
=> ! [Y2: real] :
( ( member_real @ Y2 @ A3 )
=> ( ( Orda @ X4 @ Y2 )
=> ( Ordb @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) ) ) ) ) ) ).
% monotone_on_def
thf(fact_197_monotone__on__def,axiom,
( monoto2824216093323351088l_real
= ( ^ [A3: set_nat,Orda: nat > nat > $o,Ordb: ( real > real ) > ( real > real ) > $o,F2: nat > real > real] :
! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ A3 )
=> ( ( Orda @ X4 @ Y2 )
=> ( Ordb @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) ) ) ) ) ) ).
% monotone_on_def
thf(fact_198_monotone__on__def,axiom,
( monoto8452838292781035605_ereal
= ( ^ [A3: set_nat,Orda: nat > nat > $o,Ordb: extended_ereal > extended_ereal > $o,F2: nat > extended_ereal] :
! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ A3 )
=> ( ( Orda @ X4 @ Y2 )
=> ( Ordb @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) ) ) ) ) ) ).
% monotone_on_def
thf(fact_199_monotone__on__def,axiom,
( monoto2291723841412853873nnreal
= ( ^ [A3: set_nat,Orda: nat > nat > $o,Ordb: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o,F2: nat > extend8495563244428889912nnreal] :
! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ A3 )
=> ( ( Orda @ X4 @ Y2 )
=> ( Ordb @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) ) ) ) ) ) ).
% monotone_on_def
thf(fact_200_monotone__on__def,axiom,
( monotone_on_nat_nat
= ( ^ [A3: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F2: nat > nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ A3 )
=> ( ( Orda @ X4 @ Y2 )
=> ( Ordb @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) ) ) ) ) ) ).
% monotone_on_def
thf(fact_201_monotone__onI,axiom,
! [A: set_real,Orda2: real > real > $o,Ordb2: real > real > $o,F: real > real] :
( ! [X: real,Y3: real] :
( ( member_real @ X @ A )
=> ( ( member_real @ Y3 @ A )
=> ( ( Orda2 @ X @ Y3 )
=> ( Ordb2 @ ( F @ X ) @ ( F @ Y3 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A @ Orda2 @ Ordb2 @ F ) ) ).
% monotone_onI
thf(fact_202_monotone__onI,axiom,
! [A: set_nat,Orda2: nat > nat > $o,Ordb2: ( real > real ) > ( real > real ) > $o,F: nat > real > real] :
( ! [X: nat,Y3: nat] :
( ( member_nat @ X @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( Orda2 @ X @ Y3 )
=> ( Ordb2 @ ( F @ X ) @ ( F @ Y3 ) ) ) ) )
=> ( monoto2824216093323351088l_real @ A @ Orda2 @ Ordb2 @ F ) ) ).
% monotone_onI
thf(fact_203_monotone__onI,axiom,
! [A: set_nat,Orda2: nat > nat > $o,Ordb2: extended_ereal > extended_ereal > $o,F: nat > extended_ereal] :
( ! [X: nat,Y3: nat] :
( ( member_nat @ X @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( Orda2 @ X @ Y3 )
=> ( Ordb2 @ ( F @ X ) @ ( F @ Y3 ) ) ) ) )
=> ( monoto8452838292781035605_ereal @ A @ Orda2 @ Ordb2 @ F ) ) ).
% monotone_onI
thf(fact_204_monotone__onI,axiom,
! [A: set_nat,Orda2: nat > nat > $o,Ordb2: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o,F: nat > extend8495563244428889912nnreal] :
( ! [X: nat,Y3: nat] :
( ( member_nat @ X @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( Orda2 @ X @ Y3 )
=> ( Ordb2 @ ( F @ X ) @ ( F @ Y3 ) ) ) ) )
=> ( monoto2291723841412853873nnreal @ A @ Orda2 @ Ordb2 @ F ) ) ).
% monotone_onI
thf(fact_205_monotone__onI,axiom,
! [A: set_nat,Orda2: nat > nat > $o,Ordb2: nat > nat > $o,F: nat > nat] :
( ! [X: nat,Y3: nat] :
( ( member_nat @ X @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( Orda2 @ X @ Y3 )
=> ( Ordb2 @ ( F @ X ) @ ( F @ Y3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A @ Orda2 @ Ordb2 @ F ) ) ).
% monotone_onI
thf(fact_206_monotone__onD,axiom,
! [A: set_real,Orda2: real > real > $o,Ordb2: real > real > $o,F: real > real,X2: real,Y4: real] :
( ( monoto4017252874604999745l_real @ A @ Orda2 @ Ordb2 @ F )
=> ( ( member_real @ X2 @ A )
=> ( ( member_real @ Y4 @ A )
=> ( ( Orda2 @ X2 @ Y4 )
=> ( Ordb2 @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% monotone_onD
thf(fact_207_monotone__onD,axiom,
! [A: set_nat,Orda2: nat > nat > $o,Ordb2: ( real > real ) > ( real > real ) > $o,F: nat > real > real,X2: nat,Y4: nat] :
( ( monoto2824216093323351088l_real @ A @ Orda2 @ Ordb2 @ F )
=> ( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y4 @ A )
=> ( ( Orda2 @ X2 @ Y4 )
=> ( Ordb2 @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% monotone_onD
thf(fact_208_monotone__onD,axiom,
! [A: set_nat,Orda2: nat > nat > $o,Ordb2: extended_ereal > extended_ereal > $o,F: nat > extended_ereal,X2: nat,Y4: nat] :
( ( monoto8452838292781035605_ereal @ A @ Orda2 @ Ordb2 @ F )
=> ( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y4 @ A )
=> ( ( Orda2 @ X2 @ Y4 )
=> ( Ordb2 @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% monotone_onD
thf(fact_209_monotone__onD,axiom,
! [A: set_nat,Orda2: nat > nat > $o,Ordb2: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o,F: nat > extend8495563244428889912nnreal,X2: nat,Y4: nat] :
( ( monoto2291723841412853873nnreal @ A @ Orda2 @ Ordb2 @ F )
=> ( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y4 @ A )
=> ( ( Orda2 @ X2 @ Y4 )
=> ( Ordb2 @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% monotone_onD
thf(fact_210_monotone__onD,axiom,
! [A: set_nat,Orda2: nat > nat > $o,Ordb2: nat > nat > $o,F: nat > nat,X2: nat,Y4: nat] :
( ( monotone_on_nat_nat @ A @ Orda2 @ Ordb2 @ F )
=> ( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y4 @ A )
=> ( ( Orda2 @ X2 @ Y4 )
=> ( Ordb2 @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% monotone_onD
thf(fact_211_subset__image__iff,axiom,
! [B: set_complex,F: nat > complex,A: set_nat] :
( ( ord_le211207098394363844omplex @ B @ ( image_nat_complex @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_complex @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_212_subset__image__iff,axiom,
! [B: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ B @ ( image_8394674774369097847nnreal @ F @ A ) )
= ( ? [AA: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ AA @ A )
& ( B
= ( image_8394674774369097847nnreal @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_213_subset__image__iff,axiom,
! [B: set_Ex3793607809372303086nnreal,F: extended_ereal > extend8495563244428889912nnreal,A: set_Extended_ereal] :
( ( ord_le6787938422905777998nnreal @ B @ ( image_8614087454967683265nnreal @ F @ A ) )
= ( ? [AA: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ AA @ A )
& ( B
= ( image_8614087454967683265nnreal @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_214_subset__image__iff,axiom,
! [B: set_Ex3793607809372303086nnreal,F: real > extend8495563244428889912nnreal,A: set_real] :
( ( ord_le6787938422905777998nnreal @ B @ ( image_7616191137145695467nnreal @ F @ A ) )
= ( ? [AA: set_real] :
( ( ord_less_eq_set_real @ AA @ A )
& ( B
= ( image_7616191137145695467nnreal @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_215_subset__image__iff,axiom,
! [B: set_Extended_ereal,F: nat > extended_ereal,A: set_nat] :
( ( ord_le1644982726543182158_ereal @ B @ ( image_4309273772856505399_ereal @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_4309273772856505399_ereal @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_216_subset__image__iff,axiom,
! [B: set_Extended_ereal,F: extend8495563244428889912nnreal > extended_ereal,A: set_Ex3793607809372303086nnreal] :
( ( ord_le1644982726543182158_ereal @ B @ ( image_6393943237584228047_ereal @ F @ A ) )
= ( ? [AA: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ AA @ A )
& ( B
= ( image_6393943237584228047_ereal @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_217_subset__image__iff,axiom,
! [B: set_Extended_ereal,F: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B @ ( image_6042159593519690757_ereal @ F @ A ) )
= ( ? [AA: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ AA @ A )
& ( B
= ( image_6042159593519690757_ereal @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_218_subset__image__iff,axiom,
! [B: set_Extended_ereal,F: real > extended_ereal,A: set_real] :
( ( ord_le1644982726543182158_ereal @ B @ ( image_7147107595568778587_ereal @ F @ A ) )
= ( ? [AA: set_real] :
( ( ord_less_eq_set_real @ AA @ A )
& ( B
= ( image_7147107595568778587_ereal @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_219_subset__image__iff,axiom,
! [B: set_real,F: nat > real,A: set_nat] :
( ( ord_less_eq_set_real @ B @ ( image_nat_real @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_real @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_220_subset__image__iff,axiom,
! [B: set_real,F: extend8495563244428889912nnreal > real,A: set_Ex3793607809372303086nnreal] :
( ( ord_less_eq_set_real @ B @ ( image_5648444867695151211l_real @ F @ A ) )
= ( ? [AA: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ AA @ A )
& ( B
= ( image_5648444867695151211l_real @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_221_image__subset__iff,axiom,
! [F: nat > complex,A: set_nat,B: set_complex] :
( ( ord_le211207098394363844omplex @ ( image_nat_complex @ F @ A ) @ B )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_complex @ ( F @ X4 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_222_image__subset__iff,axiom,
! [F: nat > real > real,A: set_nat,B: set_real_real] :
( ( ord_le4198349162570665613l_real @ ( image_nat_real_real @ F @ A ) @ B )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_real_real @ ( F @ X4 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_223_image__subset__iff,axiom,
! [F: nat > set_real,A: set_nat,B: set_set_real] :
( ( ord_le3558479182127378552t_real @ ( image_nat_set_real @ F @ A ) @ B )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_set_real @ ( F @ X4 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_224_image__subset__iff,axiom,
! [F: nat > extended_ereal,A: set_nat,B: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F @ A ) @ B )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member2350847679896131959_ereal @ ( F @ X4 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_225_image__subset__iff,axiom,
! [F: nat > real,A: set_nat,B: set_real] :
( ( ord_less_eq_set_real @ ( image_nat_real @ F @ A ) @ B )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_real @ ( F @ X4 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_226_subset__imageE,axiom,
! [B: set_complex,F: nat > complex,A: set_nat] :
( ( ord_le211207098394363844omplex @ B @ ( image_nat_complex @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_nat_complex @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_227_subset__imageE,axiom,
! [B: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ B @ ( image_8394674774369097847nnreal @ F @ A ) )
=> ~ ! [C3: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ C3 @ A )
=> ( B
!= ( image_8394674774369097847nnreal @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_228_subset__imageE,axiom,
! [B: set_Ex3793607809372303086nnreal,F: extended_ereal > extend8495563244428889912nnreal,A: set_Extended_ereal] :
( ( ord_le6787938422905777998nnreal @ B @ ( image_8614087454967683265nnreal @ F @ A ) )
=> ~ ! [C3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ C3 @ A )
=> ( B
!= ( image_8614087454967683265nnreal @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_229_subset__imageE,axiom,
! [B: set_Ex3793607809372303086nnreal,F: real > extend8495563244428889912nnreal,A: set_real] :
( ( ord_le6787938422905777998nnreal @ B @ ( image_7616191137145695467nnreal @ F @ A ) )
=> ~ ! [C3: set_real] :
( ( ord_less_eq_set_real @ C3 @ A )
=> ( B
!= ( image_7616191137145695467nnreal @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_230_subset__imageE,axiom,
! [B: set_Extended_ereal,F: nat > extended_ereal,A: set_nat] :
( ( ord_le1644982726543182158_ereal @ B @ ( image_4309273772856505399_ereal @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_4309273772856505399_ereal @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_231_subset__imageE,axiom,
! [B: set_Extended_ereal,F: extend8495563244428889912nnreal > extended_ereal,A: set_Ex3793607809372303086nnreal] :
( ( ord_le1644982726543182158_ereal @ B @ ( image_6393943237584228047_ereal @ F @ A ) )
=> ~ ! [C3: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ C3 @ A )
=> ( B
!= ( image_6393943237584228047_ereal @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_232_subset__imageE,axiom,
! [B: set_Extended_ereal,F: extended_ereal > extended_ereal,A: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B @ ( image_6042159593519690757_ereal @ F @ A ) )
=> ~ ! [C3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ C3 @ A )
=> ( B
!= ( image_6042159593519690757_ereal @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_233_subset__imageE,axiom,
! [B: set_Extended_ereal,F: real > extended_ereal,A: set_real] :
( ( ord_le1644982726543182158_ereal @ B @ ( image_7147107595568778587_ereal @ F @ A ) )
=> ~ ! [C3: set_real] :
( ( ord_less_eq_set_real @ C3 @ A )
=> ( B
!= ( image_7147107595568778587_ereal @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_234_subset__imageE,axiom,
! [B: set_real,F: nat > real,A: set_nat] :
( ( ord_less_eq_set_real @ B @ ( image_nat_real @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_nat_real @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_235_subset__imageE,axiom,
! [B: set_real,F: extend8495563244428889912nnreal > real,A: set_Ex3793607809372303086nnreal] :
( ( ord_less_eq_set_real @ B @ ( image_5648444867695151211l_real @ F @ A ) )
=> ~ ! [C3: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ C3 @ A )
=> ( B
!= ( image_5648444867695151211l_real @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_236_image__subsetI,axiom,
! [A: set_nat,F: nat > complex,B: set_complex] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_complex @ ( F @ X ) @ B ) )
=> ( ord_le211207098394363844omplex @ ( image_nat_complex @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_237_image__subsetI,axiom,
! [A: set_real,F: real > $o,B: set_o] :
( ! [X: real] :
( ( member_real @ X @ A )
=> ( member_o @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_real_o @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_238_image__subsetI,axiom,
! [A: set_o,F: $o > $o,B: set_o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member_o @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_239_image__subsetI,axiom,
! [A: set_real,F: real > extend8495563244428889912nnreal,B: set_Ex3793607809372303086nnreal] :
( ! [X: real] :
( ( member_real @ X @ A )
=> ( member7908768830364227535nnreal @ ( F @ X ) @ B ) )
=> ( ord_le6787938422905777998nnreal @ ( image_7616191137145695467nnreal @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_240_image__subsetI,axiom,
! [A: set_o,F: $o > extend8495563244428889912nnreal,B: set_Ex3793607809372303086nnreal] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member7908768830364227535nnreal @ ( F @ X ) @ B ) )
=> ( ord_le6787938422905777998nnreal @ ( image_3342735880743421067nnreal @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_241_image__subsetI,axiom,
! [A: set_nat,F: nat > extended_ereal,B: set_Extended_ereal] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member2350847679896131959_ereal @ ( F @ X ) @ B ) )
=> ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_242_image__subsetI,axiom,
! [A: set_real,F: real > extended_ereal,B: set_Extended_ereal] :
( ! [X: real] :
( ( member_real @ X @ A )
=> ( member2350847679896131959_ereal @ ( F @ X ) @ B ) )
=> ( ord_le1644982726543182158_ereal @ ( image_7147107595568778587_ereal @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_243_image__subsetI,axiom,
! [A: set_o,F: $o > extended_ereal,B: set_Extended_ereal] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member2350847679896131959_ereal @ ( F @ X ) @ B ) )
=> ( ord_le1644982726543182158_ereal @ ( image_7729549296133164475_ereal @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_244_image__subsetI,axiom,
! [A: set_nat,F: nat > real,B: set_real] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_real @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_real @ ( image_nat_real @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_245_image__subsetI,axiom,
! [A: set_real,F: real > real,B: set_real] :
( ! [X: real] :
( ( member_real @ X @ A )
=> ( member_real @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_real @ ( image_real_real @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_246_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > complex] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le211207098394363844omplex @ ( image_nat_complex @ F @ A ) @ ( image_nat_complex @ F @ B ) ) ) ).
% image_mono
thf(fact_247_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > extended_ereal] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F @ A ) @ ( image_4309273772856505399_ereal @ F @ B ) ) ) ).
% image_mono
thf(fact_248_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > real] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_real @ ( image_nat_real @ F @ A ) @ ( image_nat_real @ F @ B ) ) ) ).
% image_mono
thf(fact_249_image__mono,axiom,
! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( ord_le6787938422905777998nnreal @ A @ B )
=> ( ord_le6787938422905777998nnreal @ ( image_8394674774369097847nnreal @ F @ A ) @ ( image_8394674774369097847nnreal @ F @ B ) ) ) ).
% image_mono
thf(fact_250_image__mono,axiom,
! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > extended_ereal] :
( ( ord_le6787938422905777998nnreal @ A @ B )
=> ( ord_le1644982726543182158_ereal @ ( image_6393943237584228047_ereal @ F @ A ) @ ( image_6393943237584228047_ereal @ F @ B ) ) ) ).
% image_mono
thf(fact_251_image__mono,axiom,
! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > real] :
( ( ord_le6787938422905777998nnreal @ A @ B )
=> ( ord_less_eq_set_real @ ( image_5648444867695151211l_real @ F @ A ) @ ( image_5648444867695151211l_real @ F @ B ) ) ) ).
% image_mono
thf(fact_252_image__mono,axiom,
! [A: set_Extended_ereal,B: set_Extended_ereal,F: extended_ereal > extend8495563244428889912nnreal] :
( ( ord_le1644982726543182158_ereal @ A @ B )
=> ( ord_le6787938422905777998nnreal @ ( image_8614087454967683265nnreal @ F @ A ) @ ( image_8614087454967683265nnreal @ F @ B ) ) ) ).
% image_mono
thf(fact_253_image__mono,axiom,
! [A: set_Extended_ereal,B: set_Extended_ereal,F: extended_ereal > extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A @ B )
=> ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F @ A ) @ ( image_6042159593519690757_ereal @ F @ B ) ) ) ).
% image_mono
thf(fact_254_image__mono,axiom,
! [A: set_Extended_ereal,B: set_Extended_ereal,F: extended_ereal > real] :
( ( ord_le1644982726543182158_ereal @ A @ B )
=> ( ord_less_eq_set_real @ ( image_2321174223038010293l_real @ F @ A ) @ ( image_2321174223038010293l_real @ F @ B ) ) ) ).
% image_mono
thf(fact_255_image__mono,axiom,
! [A: set_real,B: set_real,F: real > extend8495563244428889912nnreal] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ord_le6787938422905777998nnreal @ ( image_7616191137145695467nnreal @ F @ A ) @ ( image_7616191137145695467nnreal @ F @ B ) ) ) ).
% image_mono
thf(fact_256_mono__onI,axiom,
! [A: set_real,F: real > real] :
( ! [R: real,S2: real] :
( ( member_real @ R @ A )
=> ( ( member_real @ S2 @ A )
=> ( ( ord_less_eq_real @ R @ S2 )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S2 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A @ ord_less_eq_real @ ord_less_eq_real @ F ) ) ).
% mono_onI
thf(fact_257_mono__onI,axiom,
! [A: set_real,F: real > nat] :
( ! [R: real,S2: real] :
( ( member_real @ R @ A )
=> ( ( member_real @ S2 @ A )
=> ( ( ord_less_eq_real @ R @ S2 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_real_nat @ A @ ord_less_eq_real @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_258_mono__onI,axiom,
! [A: set_o,F: $o > nat] :
( ! [R: $o,S2: $o] :
( ( member_o @ R @ A )
=> ( ( member_o @ S2 @ A )
=> ( ( ord_less_eq_o @ R @ S2 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_o_nat @ A @ ord_less_eq_o @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_259_mono__onI,axiom,
! [A: set_nat,F: nat > extended_ereal] :
( ! [R: nat,S2: nat] :
( ( member_nat @ R @ A )
=> ( ( member_nat @ S2 @ A )
=> ( ( ord_less_eq_nat @ R @ S2 )
=> ( ord_le1083603963089353582_ereal @ ( F @ R ) @ ( F @ S2 ) ) ) ) )
=> ( monoto8452838292781035605_ereal @ A @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ F ) ) ).
% mono_onI
thf(fact_260_mono__onI,axiom,
! [A: set_nat,F: nat > extend8495563244428889912nnreal] :
( ! [R: nat,S2: nat] :
( ( member_nat @ R @ A )
=> ( ( member_nat @ S2 @ A )
=> ( ( ord_less_eq_nat @ R @ S2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ R ) @ ( F @ S2 ) ) ) ) )
=> ( monoto2291723841412853873nnreal @ A @ ord_less_eq_nat @ ord_le3935885782089961368nnreal @ F ) ) ).
% mono_onI
thf(fact_261_mono__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [R: nat,S2: nat] :
( ( member_nat @ R @ A )
=> ( ( member_nat @ S2 @ A )
=> ( ( ord_less_eq_nat @ R @ S2 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_262_mono__onI,axiom,
! [A: set_real,F: real > set_Ex3793607809372303086nnreal] :
( ! [R: real,S2: real] :
( ( member_real @ R @ A )
=> ( ( member_real @ S2 @ A )
=> ( ( ord_less_eq_real @ R @ S2 )
=> ( ord_le6787938422905777998nnreal @ ( F @ R ) @ ( F @ S2 ) ) ) ) )
=> ( monoto2626391617355967235nnreal @ A @ ord_less_eq_real @ ord_le6787938422905777998nnreal @ F ) ) ).
% mono_onI
thf(fact_263_mono__onI,axiom,
! [A: set_o,F: $o > set_Ex3793607809372303086nnreal] :
( ! [R: $o,S2: $o] :
( ( member_o @ R @ A )
=> ( ( member_o @ S2 @ A )
=> ( ( ord_less_eq_o @ R @ S2 )
=> ( ord_le6787938422905777998nnreal @ ( F @ R ) @ ( F @ S2 ) ) ) ) )
=> ( monoto2235544329619742815nnreal @ A @ ord_less_eq_o @ ord_le6787938422905777998nnreal @ F ) ) ).
% mono_onI
thf(fact_264_mono__onI,axiom,
! [A: set_real,F: real > set_Extended_ereal] :
( ! [R: real,S2: real] :
( ( member_real @ R @ A )
=> ( ( member_real @ S2 @ A )
=> ( ( ord_less_eq_real @ R @ S2 )
=> ( ord_le1644982726543182158_ereal @ ( F @ R ) @ ( F @ S2 ) ) ) ) )
=> ( monoto1556369633277000793_ereal @ A @ ord_less_eq_real @ ord_le1644982726543182158_ereal @ F ) ) ).
% mono_onI
thf(fact_265_mono__onI,axiom,
! [A: set_o,F: $o > set_Extended_ereal] :
( ! [R: $o,S2: $o] :
( ( member_o @ R @ A )
=> ( ( member_o @ S2 @ A )
=> ( ( ord_less_eq_o @ R @ S2 )
=> ( ord_le1644982726543182158_ereal @ ( F @ R ) @ ( F @ S2 ) ) ) ) )
=> ( monoto1696596325804605437_ereal @ A @ ord_less_eq_o @ ord_le1644982726543182158_ereal @ F ) ) ).
% mono_onI
thf(fact_266_mono__onD,axiom,
! [A: set_real,F: real > real,R2: real,S3: real] :
( ( monoto4017252874604999745l_real @ A @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( ord_less_eq_real @ R2 @ S3 )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_267_mono__onD,axiom,
! [A: set_real,F: real > nat,R2: real,S3: real] :
( ( monotone_on_real_nat @ A @ ord_less_eq_real @ ord_less_eq_nat @ F )
=> ( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( ord_less_eq_real @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_268_mono__onD,axiom,
! [A: set_o,F: $o > nat,R2: $o,S3: $o] :
( ( monotone_on_o_nat @ A @ ord_less_eq_o @ ord_less_eq_nat @ F )
=> ( ( member_o @ R2 @ A )
=> ( ( member_o @ S3 @ A )
=> ( ( ord_less_eq_o @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_269_mono__onD,axiom,
! [A: set_nat,F: nat > extended_ereal,R2: nat,S3: nat] :
( ( monoto8452838292781035605_ereal @ A @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ F )
=> ( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S3 @ A )
=> ( ( ord_less_eq_nat @ R2 @ S3 )
=> ( ord_le1083603963089353582_ereal @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_270_mono__onD,axiom,
! [A: set_nat,F: nat > extend8495563244428889912nnreal,R2: nat,S3: nat] :
( ( monoto2291723841412853873nnreal @ A @ ord_less_eq_nat @ ord_le3935885782089961368nnreal @ F )
=> ( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S3 @ A )
=> ( ( ord_less_eq_nat @ R2 @ S3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_271_mono__onD,axiom,
! [A: set_nat,F: nat > nat,R2: nat,S3: nat] :
( ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S3 @ A )
=> ( ( ord_less_eq_nat @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_272_mono__onD,axiom,
! [A: set_real,F: real > set_Ex3793607809372303086nnreal,R2: real,S3: real] :
( ( monoto2626391617355967235nnreal @ A @ ord_less_eq_real @ ord_le6787938422905777998nnreal @ F )
=> ( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( ord_less_eq_real @ R2 @ S3 )
=> ( ord_le6787938422905777998nnreal @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_273_mono__onD,axiom,
! [A: set_o,F: $o > set_Ex3793607809372303086nnreal,R2: $o,S3: $o] :
( ( monoto2235544329619742815nnreal @ A @ ord_less_eq_o @ ord_le6787938422905777998nnreal @ F )
=> ( ( member_o @ R2 @ A )
=> ( ( member_o @ S3 @ A )
=> ( ( ord_less_eq_o @ R2 @ S3 )
=> ( ord_le6787938422905777998nnreal @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_274_mono__onD,axiom,
! [A: set_real,F: real > set_Extended_ereal,R2: real,S3: real] :
( ( monoto1556369633277000793_ereal @ A @ ord_less_eq_real @ ord_le1644982726543182158_ereal @ F )
=> ( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( ord_less_eq_real @ R2 @ S3 )
=> ( ord_le1644982726543182158_ereal @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_275_mono__onD,axiom,
! [A: set_o,F: $o > set_Extended_ereal,R2: $o,S3: $o] :
( ( monoto1696596325804605437_ereal @ A @ ord_less_eq_o @ ord_le1644982726543182158_ereal @ F )
=> ( ( member_o @ R2 @ A )
=> ( ( member_o @ S3 @ A )
=> ( ( ord_less_eq_o @ R2 @ S3 )
=> ( ord_le1644982726543182158_ereal @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_276_ord_Omono__on__def,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > real] :
( ( monoto4017252874604999745l_real @ A @ Less_eq @ ord_less_eq_real @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A )
& ( member_real @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_277_ord_Omono__on__def,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > extended_ereal] :
( ( monoto8452838292781035605_ereal @ A @ Less_eq @ ord_le1083603963089353582_ereal @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A )
& ( member_nat @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_le1083603963089353582_ereal @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_278_ord_Omono__on__def,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > extend8495563244428889912nnreal] :
( ( monoto2291723841412853873nnreal @ A @ Less_eq @ ord_le3935885782089961368nnreal @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A )
& ( member_nat @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_279_ord_Omono__on__def,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > nat] :
( ( monotone_on_real_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A )
& ( member_real @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_280_ord_Omono__on__def,axiom,
! [A: set_o,Less_eq: $o > $o > $o,F: $o > nat] :
( ( monotone_on_o_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: $o,S4: $o] :
( ( ( member_o @ R3 @ A )
& ( member_o @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_281_ord_Omono__on__def,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ( monotone_on_nat_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A )
& ( member_nat @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_282_ord_Omono__on__def,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > set_Ex3793607809372303086nnreal] :
( ( monoto2626391617355967235nnreal @ A @ Less_eq @ ord_le6787938422905777998nnreal @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A )
& ( member_real @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_le6787938422905777998nnreal @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_283_ord_Omono__on__def,axiom,
! [A: set_o,Less_eq: $o > $o > $o,F: $o > set_Ex3793607809372303086nnreal] :
( ( monoto2235544329619742815nnreal @ A @ Less_eq @ ord_le6787938422905777998nnreal @ F )
= ( ! [R3: $o,S4: $o] :
( ( ( member_o @ R3 @ A )
& ( member_o @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_le6787938422905777998nnreal @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_284_ord_Omono__on__def,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > set_Extended_ereal] :
( ( monoto1556369633277000793_ereal @ A @ Less_eq @ ord_le1644982726543182158_ereal @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A )
& ( member_real @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_le1644982726543182158_ereal @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_285_ord_Omono__on__def,axiom,
! [A: set_o,Less_eq: $o > $o > $o,F: $o > set_Extended_ereal] :
( ( monoto1696596325804605437_ereal @ A @ Less_eq @ ord_le1644982726543182158_ereal @ F )
= ( ! [R3: $o,S4: $o] :
( ( ( member_o @ R3 @ A )
& ( member_o @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_le1644982726543182158_ereal @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_286_ord_Omono__onI,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > real] :
( ! [R: real,S2: real] :
( ( member_real @ R @ A )
=> ( ( member_real @ S2 @ A )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S2 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A @ Less_eq @ ord_less_eq_real @ F ) ) ).
% ord.mono_onI
thf(fact_287_ord_Omono__onI,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > extended_ereal] :
( ! [R: nat,S2: nat] :
( ( member_nat @ R @ A )
=> ( ( member_nat @ S2 @ A )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_le1083603963089353582_ereal @ ( F @ R ) @ ( F @ S2 ) ) ) ) )
=> ( monoto8452838292781035605_ereal @ A @ Less_eq @ ord_le1083603963089353582_ereal @ F ) ) ).
% ord.mono_onI
thf(fact_288_ord_Omono__onI,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > extend8495563244428889912nnreal] :
( ! [R: nat,S2: nat] :
( ( member_nat @ R @ A )
=> ( ( member_nat @ S2 @ A )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ R ) @ ( F @ S2 ) ) ) ) )
=> ( monoto2291723841412853873nnreal @ A @ Less_eq @ ord_le3935885782089961368nnreal @ F ) ) ).
% ord.mono_onI
thf(fact_289_ord_Omono__onI,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > nat] :
( ! [R: real,S2: real] :
( ( member_real @ R @ A )
=> ( ( member_real @ S2 @ A )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_real_nat @ A @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_290_ord_Omono__onI,axiom,
! [A: set_o,Less_eq: $o > $o > $o,F: $o > nat] :
( ! [R: $o,S2: $o] :
( ( member_o @ R @ A )
=> ( ( member_o @ S2 @ A )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_o_nat @ A @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_291_ord_Omono__onI,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ! [R: nat,S2: nat] :
( ( member_nat @ R @ A )
=> ( ( member_nat @ S2 @ A )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) )
=> ( monotone_on_nat_nat @ A @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_292_ord_Omono__onI,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > set_Ex3793607809372303086nnreal] :
( ! [R: real,S2: real] :
( ( member_real @ R @ A )
=> ( ( member_real @ S2 @ A )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_le6787938422905777998nnreal @ ( F @ R ) @ ( F @ S2 ) ) ) ) )
=> ( monoto2626391617355967235nnreal @ A @ Less_eq @ ord_le6787938422905777998nnreal @ F ) ) ).
% ord.mono_onI
thf(fact_293_ord_Omono__onI,axiom,
! [A: set_o,Less_eq: $o > $o > $o,F: $o > set_Ex3793607809372303086nnreal] :
( ! [R: $o,S2: $o] :
( ( member_o @ R @ A )
=> ( ( member_o @ S2 @ A )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_le6787938422905777998nnreal @ ( F @ R ) @ ( F @ S2 ) ) ) ) )
=> ( monoto2235544329619742815nnreal @ A @ Less_eq @ ord_le6787938422905777998nnreal @ F ) ) ).
% ord.mono_onI
thf(fact_294_ord_Omono__onI,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > set_Extended_ereal] :
( ! [R: real,S2: real] :
( ( member_real @ R @ A )
=> ( ( member_real @ S2 @ A )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_le1644982726543182158_ereal @ ( F @ R ) @ ( F @ S2 ) ) ) ) )
=> ( monoto1556369633277000793_ereal @ A @ Less_eq @ ord_le1644982726543182158_ereal @ F ) ) ).
% ord.mono_onI
thf(fact_295_ord_Omono__onI,axiom,
! [A: set_o,Less_eq: $o > $o > $o,F: $o > set_Extended_ereal] :
( ! [R: $o,S2: $o] :
( ( member_o @ R @ A )
=> ( ( member_o @ S2 @ A )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_le1644982726543182158_ereal @ ( F @ R ) @ ( F @ S2 ) ) ) ) )
=> ( monoto1696596325804605437_ereal @ A @ Less_eq @ ord_le1644982726543182158_ereal @ F ) ) ).
% ord.mono_onI
thf(fact_296_ord_Omono__onD,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > real,R2: real,S3: real] :
( ( monoto4017252874604999745l_real @ A @ Less_eq @ ord_less_eq_real @ F )
=> ( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_297_ord_Omono__onD,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > extended_ereal,R2: nat,S3: nat] :
( ( monoto8452838292781035605_ereal @ A @ Less_eq @ ord_le1083603963089353582_ereal @ F )
=> ( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_le1083603963089353582_ereal @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_298_ord_Omono__onD,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > extend8495563244428889912nnreal,R2: nat,S3: nat] :
( ( monoto2291723841412853873nnreal @ A @ Less_eq @ ord_le3935885782089961368nnreal @ F )
=> ( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_299_ord_Omono__onD,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > nat,R2: real,S3: real] :
( ( monotone_on_real_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_300_ord_Omono__onD,axiom,
! [A: set_o,Less_eq: $o > $o > $o,F: $o > nat,R2: $o,S3: $o] :
( ( monotone_on_o_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_o @ R2 @ A )
=> ( ( member_o @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_301_ord_Omono__onD,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > nat,R2: nat,S3: nat] :
( ( monotone_on_nat_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_302_ord_Omono__onD,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > set_Ex3793607809372303086nnreal,R2: real,S3: real] :
( ( monoto2626391617355967235nnreal @ A @ Less_eq @ ord_le6787938422905777998nnreal @ F )
=> ( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_le6787938422905777998nnreal @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_303_ord_Omono__onD,axiom,
! [A: set_o,Less_eq: $o > $o > $o,F: $o > set_Ex3793607809372303086nnreal,R2: $o,S3: $o] :
( ( monoto2235544329619742815nnreal @ A @ Less_eq @ ord_le6787938422905777998nnreal @ F )
=> ( ( member_o @ R2 @ A )
=> ( ( member_o @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_le6787938422905777998nnreal @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_304_ord_Omono__onD,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > set_Extended_ereal,R2: real,S3: real] :
( ( monoto1556369633277000793_ereal @ A @ Less_eq @ ord_le1644982726543182158_ereal @ F )
=> ( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_le1644982726543182158_ereal @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_305_ord_Omono__onD,axiom,
! [A: set_o,Less_eq: $o > $o > $o,F: $o > set_Extended_ereal,R2: $o,S3: $o] :
( ( monoto1696596325804605437_ereal @ A @ Less_eq @ ord_le1644982726543182158_ereal @ F )
=> ( ( member_o @ R2 @ A )
=> ( ( member_o @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_le1644982726543182158_ereal @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_306_monotone__on__subset,axiom,
! [A: set_nat,Orda2: nat > nat > $o,Ordb2: ( real > real ) > ( real > real ) > $o,F: nat > real > real,B: set_nat] :
( ( monoto2824216093323351088l_real @ A @ Orda2 @ Ordb2 @ F )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( monoto2824216093323351088l_real @ B @ Orda2 @ Ordb2 @ F ) ) ) ).
% monotone_on_subset
thf(fact_307_monotone__on__subset,axiom,
! [A: set_nat,Orda2: nat > nat > $o,Ordb2: extended_ereal > extended_ereal > $o,F: nat > extended_ereal,B: set_nat] :
( ( monoto8452838292781035605_ereal @ A @ Orda2 @ Ordb2 @ F )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( monoto8452838292781035605_ereal @ B @ Orda2 @ Ordb2 @ F ) ) ) ).
% monotone_on_subset
thf(fact_308_monotone__on__subset,axiom,
! [A: set_nat,Orda2: nat > nat > $o,Ordb2: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o,F: nat > extend8495563244428889912nnreal,B: set_nat] :
( ( monoto2291723841412853873nnreal @ A @ Orda2 @ Ordb2 @ F )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( monoto2291723841412853873nnreal @ B @ Orda2 @ Ordb2 @ F ) ) ) ).
% monotone_on_subset
thf(fact_309_monotone__on__subset,axiom,
! [A: set_nat,Orda2: nat > nat > $o,Ordb2: nat > nat > $o,F: nat > nat,B: set_nat] :
( ( monotone_on_nat_nat @ A @ Orda2 @ Ordb2 @ F )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( monotone_on_nat_nat @ B @ Orda2 @ Ordb2 @ F ) ) ) ).
% monotone_on_subset
thf(fact_310_monotone__on__subset,axiom,
! [A: set_real,Orda2: real > real > $o,Ordb2: real > real > $o,F: real > real,B: set_real] :
( ( monoto4017252874604999745l_real @ A @ Orda2 @ Ordb2 @ F )
=> ( ( ord_less_eq_set_real @ B @ A )
=> ( monoto4017252874604999745l_real @ B @ Orda2 @ Ordb2 @ F ) ) ) ).
% monotone_on_subset
thf(fact_311_simple__function__cong,axiom,
! [M: sigma_measure_real,F: real > real,G: real > real] :
( ! [T2: real] :
( ( member_real @ T2 @ ( sigma_space_real @ M ) )
=> ( ( F @ T2 )
= ( G @ T2 ) ) )
=> ( ( nonneg485563716852976898l_real @ M @ F )
= ( nonneg485563716852976898l_real @ M @ G ) ) ) ).
% simple_function_cong
thf(fact_312_le__zero__eq,axiom,
! [N3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ N3 @ zero_z7100319975126383169nnreal )
= ( N3 = zero_z7100319975126383169nnreal ) ) ).
% le_zero_eq
thf(fact_313_le__zero__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_314__092_060open_062sets_Aborel_A_092_060subseteq_062_Asets_Alebesgue_092_060close_062,axiom,
ord_le3558479182127378552t_real @ ( sigma_sets_real @ borel_5078946678739801102l_real ) @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) ) ).
% \<open>sets borel \<subseteq> sets lebesgue\<close>
thf(fact_315_dual__order_Orefl,axiom,
! [A2: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_316_dual__order_Orefl,axiom,
! [A2: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_317_dual__order_Orefl,axiom,
! [A2: set_real] : ( ord_less_eq_set_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_318_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_319_dual__order_Orefl,axiom,
! [A2: real > real] : ( ord_le6948328307412524503l_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_320_order__refl,axiom,
! [X2: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ X2 @ X2 ) ).
% order_refl
thf(fact_321_order__refl,axiom,
! [X2: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ X2 @ X2 ) ).
% order_refl
thf(fact_322_order__refl,axiom,
! [X2: set_real] : ( ord_less_eq_set_real @ X2 @ X2 ) ).
% order_refl
thf(fact_323_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_324_order__refl,axiom,
! [X2: real > real] : ( ord_le6948328307412524503l_real @ X2 @ X2 ) ).
% order_refl
thf(fact_325_le__measureD1,axiom,
! [A: sigma_7234349610311085201nnreal,B: sigma_7234349610311085201nnreal] :
( ( ord_le1854472233513649201nnreal @ A @ B )
=> ( ord_le6787938422905777998nnreal @ ( sigma_3147302497200244656nnreal @ A ) @ ( sigma_3147302497200244656nnreal @ B ) ) ) ).
% le_measureD1
thf(fact_326_le__measureD1,axiom,
! [A: sigma_7227684458468523851_ereal,B: sigma_7227684458468523851_ereal] :
( ( ord_le998643035147222955_ereal @ A @ B )
=> ( ord_le1644982726543182158_ereal @ ( sigma_3843066954113031510_ereal @ A ) @ ( sigma_3843066954113031510_ereal @ B ) ) ) ).
% le_measureD1
thf(fact_327_le__measureD1,axiom,
! [A: sigma_measure_real,B: sigma_measure_real] :
( ( ord_le487379304121309861e_real @ A @ B )
=> ( ord_less_eq_set_real @ ( sigma_space_real @ A ) @ ( sigma_space_real @ B ) ) ) ).
% le_measureD1
thf(fact_328_all__subset__image,axiom,
! [F: nat > complex,A: set_nat,P: set_complex > $o] :
( ( ! [B3: set_complex] :
( ( ord_le211207098394363844omplex @ B3 @ ( image_nat_complex @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A )
=> ( P @ ( image_nat_complex @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_329_all__subset__image,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal,P: set_Ex3793607809372303086nnreal > $o] :
( ( ! [B3: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ B3 @ ( image_8394674774369097847nnreal @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ B3 @ A )
=> ( P @ ( image_8394674774369097847nnreal @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_330_all__subset__image,axiom,
! [F: extended_ereal > extend8495563244428889912nnreal,A: set_Extended_ereal,P: set_Ex3793607809372303086nnreal > $o] :
( ( ! [B3: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ B3 @ ( image_8614087454967683265nnreal @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B3 @ A )
=> ( P @ ( image_8614087454967683265nnreal @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_331_all__subset__image,axiom,
! [F: real > extend8495563244428889912nnreal,A: set_real,P: set_Ex3793607809372303086nnreal > $o] :
( ( ! [B3: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ B3 @ ( image_7616191137145695467nnreal @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_real] :
( ( ord_less_eq_set_real @ B3 @ A )
=> ( P @ ( image_7616191137145695467nnreal @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_332_all__subset__image,axiom,
! [F: nat > extended_ereal,A: set_nat,P: set_Extended_ereal > $o] :
( ( ! [B3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B3 @ ( image_4309273772856505399_ereal @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A )
=> ( P @ ( image_4309273772856505399_ereal @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_333_all__subset__image,axiom,
! [F: extend8495563244428889912nnreal > extended_ereal,A: set_Ex3793607809372303086nnreal,P: set_Extended_ereal > $o] :
( ( ! [B3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B3 @ ( image_6393943237584228047_ereal @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ B3 @ A )
=> ( P @ ( image_6393943237584228047_ereal @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_334_all__subset__image,axiom,
! [F: extended_ereal > extended_ereal,A: set_Extended_ereal,P: set_Extended_ereal > $o] :
( ( ! [B3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B3 @ ( image_6042159593519690757_ereal @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B3 @ A )
=> ( P @ ( image_6042159593519690757_ereal @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_335_all__subset__image,axiom,
! [F: real > extended_ereal,A: set_real,P: set_Extended_ereal > $o] :
( ( ! [B3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B3 @ ( image_7147107595568778587_ereal @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_real] :
( ( ord_less_eq_set_real @ B3 @ A )
=> ( P @ ( image_7147107595568778587_ereal @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_336_all__subset__image,axiom,
! [F: nat > real,A: set_nat,P: set_real > $o] :
( ( ! [B3: set_real] :
( ( ord_less_eq_set_real @ B3 @ ( image_nat_real @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A )
=> ( P @ ( image_nat_real @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_337_all__subset__image,axiom,
! [F: extend8495563244428889912nnreal > real,A: set_Ex3793607809372303086nnreal,P: set_real > $o] :
( ( ! [B3: set_real] :
( ( ord_less_eq_set_real @ B3 @ ( image_5648444867695151211l_real @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ B3 @ A )
=> ( P @ ( image_5648444867695151211l_real @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_338_zero__le,axiom,
! [X2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ X2 ) ).
% zero_le
thf(fact_339_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_340_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_341_le__numeral__extra_I3_J,axiom,
ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ zero_z7100319975126383169nnreal ).
% le_numeral_extra(3)
thf(fact_342_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_343_UNIV__I,axiom,
! [X2: real > real] : ( member_real_real @ X2 @ top_to2071711978144146653l_real ) ).
% UNIV_I
thf(fact_344_UNIV__I,axiom,
! [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).
% UNIV_I
thf(fact_345_UNIV__I,axiom,
! [X2: set_real] : ( member_set_real @ X2 @ top_top_set_set_real ) ).
% UNIV_I
thf(fact_346_UNIV__I,axiom,
! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_I
thf(fact_347_UNIV__I,axiom,
! [X2: real] : ( member_real @ X2 @ top_top_set_real ) ).
% UNIV_I
thf(fact_348_UNIV__I,axiom,
! [X2: complex] : ( member_complex @ X2 @ top_top_set_complex ) ).
% UNIV_I
thf(fact_349_UNIV__I,axiom,
! [X2: extended_ereal] : ( member2350847679896131959_ereal @ X2 @ top_to5683747375963461374_ereal ) ).
% UNIV_I
thf(fact_350_sets_Otop,axiom,
! [M: sigma_measure_real] : ( member_set_real @ ( sigma_space_real @ M ) @ ( sigma_sets_real @ M ) ) ).
% sets.top
thf(fact_351_sets__completionI__sets,axiom,
! [A: set_real,M: sigma_measure_real] :
( ( member_set_real @ A @ ( sigma_sets_real @ M ) )
=> ( member_set_real @ A @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ M ) ) ) ) ).
% sets_completionI_sets
thf(fact_352_space__borel,axiom,
( ( sigma_space_nat @ borel_8449730974584783410el_nat )
= top_top_set_nat ) ).
% space_borel
thf(fact_353_space__borel,axiom,
( ( sigma_space_complex @ borel_1392132677378845456omplex )
= top_top_set_complex ) ).
% space_borel
thf(fact_354_space__borel,axiom,
( ( sigma_3843066954113031510_ereal @ borel_2631802743099733228_ereal )
= top_to5683747375963461374_ereal ) ).
% space_borel
thf(fact_355_space__borel,axiom,
( ( sigma_space_real @ borel_5078946678739801102l_real )
= top_top_set_real ) ).
% space_borel
thf(fact_356_sets__lborel,axiom,
( ( sigma_sets_real @ lebesgue_lborel_real )
= ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ).
% sets_lborel
thf(fact_357_sets__restrict__UNIV,axiom,
! [M: sigma_measure_nat] :
( ( sigma_sets_nat @ ( sigma_744083341818469772ce_nat @ M @ top_top_set_nat ) )
= ( sigma_sets_nat @ M ) ) ).
% sets_restrict_UNIV
thf(fact_358_sets__restrict__UNIV,axiom,
! [M: sigma_3077487657436305159omplex] :
( ( sigma_sets_complex @ ( sigma_216592511309337194omplex @ M @ top_top_set_complex ) )
= ( sigma_sets_complex @ M ) ) ).
% sets_restrict_UNIV
thf(fact_359_sets__restrict__UNIV,axiom,
! [M: sigma_7227684458468523851_ereal] :
( ( sigma_6858886609720962221_ereal @ ( sigma_2834138478934969106_ereal @ M @ top_to5683747375963461374_ereal ) )
= ( sigma_6858886609720962221_ereal @ M ) ) ).
% sets_restrict_UNIV
thf(fact_360_sets__restrict__UNIV,axiom,
! [M: sigma_measure_real] :
( ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ M @ top_top_set_real ) )
= ( sigma_sets_real @ M ) ) ).
% sets_restrict_UNIV
thf(fact_361_space__restrict__space2,axiom,
! [Omega: set_real,M: sigma_measure_real] :
( ( member_set_real @ Omega @ ( sigma_sets_real @ M ) )
=> ( ( sigma_space_real @ ( sigma_5414646170262037096e_real @ M @ Omega ) )
= Omega ) ) ).
% space_restrict_space2
thf(fact_362_sets__lebesgue__on__refl,axiom,
! [S: set_real] : ( member_set_real @ S @ ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ S ) ) ) ).
% sets_lebesgue_on_refl
thf(fact_363_sets__eq__iff__bounded,axiom,
! [A: sigma_measure_real,B: sigma_measure_real,C2: sigma_measure_real] :
( ( ord_le487379304121309861e_real @ A @ B )
=> ( ( ord_le487379304121309861e_real @ B @ C2 )
=> ( ( ( sigma_sets_real @ A )
= ( sigma_sets_real @ C2 ) )
=> ( ( sigma_sets_real @ B )
= ( sigma_sets_real @ A ) ) ) ) ) ).
% sets_eq_iff_bounded
thf(fact_364_UNIV__witness,axiom,
? [X: real > real] : ( member_real_real @ X @ top_to2071711978144146653l_real ) ).
% UNIV_witness
thf(fact_365_UNIV__witness,axiom,
? [X: $o] : ( member_o @ X @ top_top_set_o ) ).
% UNIV_witness
thf(fact_366_UNIV__witness,axiom,
? [X: set_real] : ( member_set_real @ X @ top_top_set_set_real ) ).
% UNIV_witness
thf(fact_367_UNIV__witness,axiom,
? [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_368_UNIV__witness,axiom,
? [X: real] : ( member_real @ X @ top_top_set_real ) ).
% UNIV_witness
thf(fact_369_UNIV__witness,axiom,
? [X: complex] : ( member_complex @ X @ top_top_set_complex ) ).
% UNIV_witness
thf(fact_370_UNIV__witness,axiom,
? [X: extended_ereal] : ( member2350847679896131959_ereal @ X @ top_to5683747375963461374_ereal ) ).
% UNIV_witness
thf(fact_371_UNIV__eq__I,axiom,
! [A: set_real_real] :
( ! [X: real > real] : ( member_real_real @ X @ A )
=> ( top_to2071711978144146653l_real = A ) ) ).
% UNIV_eq_I
thf(fact_372_UNIV__eq__I,axiom,
! [A: set_o] :
( ! [X: $o] : ( member_o @ X @ A )
=> ( top_top_set_o = A ) ) ).
% UNIV_eq_I
thf(fact_373_UNIV__eq__I,axiom,
! [A: set_set_real] :
( ! [X: set_real] : ( member_set_real @ X @ A )
=> ( top_top_set_set_real = A ) ) ).
% UNIV_eq_I
thf(fact_374_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X: nat] : ( member_nat @ X @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_375_UNIV__eq__I,axiom,
! [A: set_real] :
( ! [X: real] : ( member_real @ X @ A )
=> ( top_top_set_real = A ) ) ).
% UNIV_eq_I
thf(fact_376_UNIV__eq__I,axiom,
! [A: set_complex] :
( ! [X: complex] : ( member_complex @ X @ A )
=> ( top_top_set_complex = A ) ) ).
% UNIV_eq_I
thf(fact_377_UNIV__eq__I,axiom,
! [A: set_Extended_ereal] :
( ! [X: extended_ereal] : ( member2350847679896131959_ereal @ X @ A )
=> ( top_to5683747375963461374_ereal = A ) ) ).
% UNIV_eq_I
thf(fact_378_top__greatest,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% top_greatest
thf(fact_379_top__greatest,axiom,
! [A2: set_complex] : ( ord_le211207098394363844omplex @ A2 @ top_top_set_complex ) ).
% top_greatest
thf(fact_380_top__greatest,axiom,
! [A2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ A2 @ top_to1496364449551166952nnreal ) ).
% top_greatest
thf(fact_381_top__greatest,axiom,
! [A2: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ A2 @ top_to7994903218803871134nnreal ) ).
% top_greatest
thf(fact_382_top__greatest,axiom,
! [A2: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ A2 @ top_to5683747375963461374_ereal ) ).
% top_greatest
thf(fact_383_top__greatest,axiom,
! [A2: set_real] : ( ord_less_eq_set_real @ A2 @ top_top_set_real ) ).
% top_greatest
thf(fact_384_top_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
= ( A2 = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_385_top_Oextremum__unique,axiom,
! [A2: set_complex] :
( ( ord_le211207098394363844omplex @ top_top_set_complex @ A2 )
= ( A2 = top_top_set_complex ) ) ).
% top.extremum_unique
thf(fact_386_top_Oextremum__unique,axiom,
! [A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ top_to1496364449551166952nnreal @ A2 )
= ( A2 = top_to1496364449551166952nnreal ) ) ).
% top.extremum_unique
thf(fact_387_top_Oextremum__unique,axiom,
! [A2: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ top_to7994903218803871134nnreal @ A2 )
= ( A2 = top_to7994903218803871134nnreal ) ) ).
% top.extremum_unique
thf(fact_388_top_Oextremum__unique,axiom,
! [A2: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ top_to5683747375963461374_ereal @ A2 )
= ( A2 = top_to5683747375963461374_ereal ) ) ).
% top.extremum_unique
thf(fact_389_top_Oextremum__unique,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ top_top_set_real @ A2 )
= ( A2 = top_top_set_real ) ) ).
% top.extremum_unique
thf(fact_390_top_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
=> ( A2 = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_391_top_Oextremum__uniqueI,axiom,
! [A2: set_complex] :
( ( ord_le211207098394363844omplex @ top_top_set_complex @ A2 )
=> ( A2 = top_top_set_complex ) ) ).
% top.extremum_uniqueI
thf(fact_392_top_Oextremum__uniqueI,axiom,
! [A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ top_to1496364449551166952nnreal @ A2 )
=> ( A2 = top_to1496364449551166952nnreal ) ) ).
% top.extremum_uniqueI
thf(fact_393_top_Oextremum__uniqueI,axiom,
! [A2: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ top_to7994903218803871134nnreal @ A2 )
=> ( A2 = top_to7994903218803871134nnreal ) ) ).
% top.extremum_uniqueI
thf(fact_394_top_Oextremum__uniqueI,axiom,
! [A2: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ top_to5683747375963461374_ereal @ A2 )
=> ( A2 = top_to5683747375963461374_ereal ) ) ).
% top.extremum_uniqueI
thf(fact_395_top_Oextremum__uniqueI,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ top_top_set_real @ A2 )
=> ( A2 = top_top_set_real ) ) ).
% top.extremum_uniqueI
thf(fact_396_space__in__borel,axiom,
member_set_nat @ top_top_set_nat @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ).
% space_in_borel
thf(fact_397_space__in__borel,axiom,
member_set_complex @ top_top_set_complex @ ( sigma_sets_complex @ borel_1392132677378845456omplex ) ).
% space_in_borel
thf(fact_398_space__in__borel,axiom,
member5519481007471526743_ereal @ top_to5683747375963461374_ereal @ ( sigma_6858886609720962221_ereal @ borel_2631802743099733228_ereal ) ).
% space_in_borel
thf(fact_399_space__in__borel,axiom,
member_set_real @ top_top_set_real @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ).
% space_in_borel
thf(fact_400_le__measureD2,axiom,
! [A: sigma_measure_real,B: sigma_measure_real] :
( ( ord_le487379304121309861e_real @ A @ B )
=> ( ( ( sigma_space_real @ A )
= ( sigma_space_real @ B ) )
=> ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ A ) @ ( sigma_sets_real @ B ) ) ) ) ).
% le_measureD2
thf(fact_401_mono__restrict__space,axiom,
! [M: sigma_measure_real,N: sigma_measure_real,X5: set_real] :
( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ M ) @ ( sigma_sets_real @ N ) )
=> ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ M @ X5 ) ) @ ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ N @ X5 ) ) ) ) ).
% mono_restrict_space
thf(fact_402_measurable__cong__sets,axiom,
! [M: sigma_measure_real,M3: sigma_measure_real,N: sigma_measure_real,N2: sigma_measure_real] :
( ( ( sigma_sets_real @ M )
= ( sigma_sets_real @ M3 ) )
=> ( ( ( sigma_sets_real @ N )
= ( sigma_sets_real @ N2 ) )
=> ( ( sigma_5267869275261027754l_real @ M @ N )
= ( sigma_5267869275261027754l_real @ M3 @ N2 ) ) ) ) ).
% measurable_cong_sets
thf(fact_403_sets__eq__imp__space__eq,axiom,
! [M: sigma_measure_real,M3: sigma_measure_real] :
( ( ( sigma_sets_real @ M )
= ( sigma_sets_real @ M3 ) )
=> ( ( sigma_space_real @ M )
= ( sigma_space_real @ M3 ) ) ) ).
% sets_eq_imp_space_eq
thf(fact_404_surjD,axiom,
! [F: nat > nat,Y4: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X: nat] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_405_surjD,axiom,
! [F: nat > real,Y4: real] :
( ( ( image_nat_real @ F @ top_top_set_nat )
= top_top_set_real )
=> ? [X: nat] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_406_surjD,axiom,
! [F: nat > complex,Y4: complex] :
( ( ( image_nat_complex @ F @ top_top_set_nat )
= top_top_set_complex )
=> ? [X: nat] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_407_surjD,axiom,
! [F: nat > extended_ereal,Y4: extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F @ top_top_set_nat )
= top_to5683747375963461374_ereal )
=> ? [X: nat] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_408_surjD,axiom,
! [F: real > nat,Y4: nat] :
( ( ( image_real_nat @ F @ top_top_set_real )
= top_top_set_nat )
=> ? [X: real] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_409_surjD,axiom,
! [F: real > real,Y4: real] :
( ( ( image_real_real @ F @ top_top_set_real )
= top_top_set_real )
=> ? [X: real] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_410_surjD,axiom,
! [F: real > complex,Y4: complex] :
( ( ( image_real_complex @ F @ top_top_set_real )
= top_top_set_complex )
=> ? [X: real] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_411_surjD,axiom,
! [F: real > extended_ereal,Y4: extended_ereal] :
( ( ( image_7147107595568778587_ereal @ F @ top_top_set_real )
= top_to5683747375963461374_ereal )
=> ? [X: real] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_412_surjD,axiom,
! [F: complex > nat,Y4: nat] :
( ( ( image_complex_nat @ F @ top_top_set_complex )
= top_top_set_nat )
=> ? [X: complex] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_413_surjD,axiom,
! [F: complex > real,Y4: real] :
( ( ( image_complex_real @ F @ top_top_set_complex )
= top_top_set_real )
=> ? [X: complex] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_414_surjE,axiom,
! [F: nat > nat,Y4: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X: nat] :
( Y4
!= ( F @ X ) ) ) ).
% surjE
thf(fact_415_surjE,axiom,
! [F: nat > real,Y4: real] :
( ( ( image_nat_real @ F @ top_top_set_nat )
= top_top_set_real )
=> ~ ! [X: nat] :
( Y4
!= ( F @ X ) ) ) ).
% surjE
thf(fact_416_surjE,axiom,
! [F: nat > complex,Y4: complex] :
( ( ( image_nat_complex @ F @ top_top_set_nat )
= top_top_set_complex )
=> ~ ! [X: nat] :
( Y4
!= ( F @ X ) ) ) ).
% surjE
thf(fact_417_surjE,axiom,
! [F: nat > extended_ereal,Y4: extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F @ top_top_set_nat )
= top_to5683747375963461374_ereal )
=> ~ ! [X: nat] :
( Y4
!= ( F @ X ) ) ) ).
% surjE
thf(fact_418_surjE,axiom,
! [F: real > nat,Y4: nat] :
( ( ( image_real_nat @ F @ top_top_set_real )
= top_top_set_nat )
=> ~ ! [X: real] :
( Y4
!= ( F @ X ) ) ) ).
% surjE
thf(fact_419_surjE,axiom,
! [F: real > real,Y4: real] :
( ( ( image_real_real @ F @ top_top_set_real )
= top_top_set_real )
=> ~ ! [X: real] :
( Y4
!= ( F @ X ) ) ) ).
% surjE
thf(fact_420_surjE,axiom,
! [F: real > complex,Y4: complex] :
( ( ( image_real_complex @ F @ top_top_set_real )
= top_top_set_complex )
=> ~ ! [X: real] :
( Y4
!= ( F @ X ) ) ) ).
% surjE
thf(fact_421_surjE,axiom,
! [F: real > extended_ereal,Y4: extended_ereal] :
( ( ( image_7147107595568778587_ereal @ F @ top_top_set_real )
= top_to5683747375963461374_ereal )
=> ~ ! [X: real] :
( Y4
!= ( F @ X ) ) ) ).
% surjE
thf(fact_422_surjE,axiom,
! [F: complex > nat,Y4: nat] :
( ( ( image_complex_nat @ F @ top_top_set_complex )
= top_top_set_nat )
=> ~ ! [X: complex] :
( Y4
!= ( F @ X ) ) ) ).
% surjE
thf(fact_423_surjE,axiom,
! [F: complex > real,Y4: real] :
( ( ( image_complex_real @ F @ top_top_set_complex )
= top_top_set_real )
=> ~ ! [X: complex] :
( Y4
!= ( F @ X ) ) ) ).
% surjE
thf(fact_424_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X: nat] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_425_surjI,axiom,
! [G: nat > real,F: real > nat] :
( ! [X: real] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_nat_real @ G @ top_top_set_nat )
= top_top_set_real ) ) ).
% surjI
thf(fact_426_surjI,axiom,
! [G: nat > complex,F: complex > nat] :
( ! [X: complex] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_nat_complex @ G @ top_top_set_nat )
= top_top_set_complex ) ) ).
% surjI
thf(fact_427_surjI,axiom,
! [G: nat > extended_ereal,F: extended_ereal > nat] :
( ! [X: extended_ereal] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_4309273772856505399_ereal @ G @ top_top_set_nat )
= top_to5683747375963461374_ereal ) ) ).
% surjI
thf(fact_428_surjI,axiom,
! [G: real > nat,F: nat > real] :
( ! [X: nat] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_real_nat @ G @ top_top_set_real )
= top_top_set_nat ) ) ).
% surjI
thf(fact_429_surjI,axiom,
! [G: real > real,F: real > real] :
( ! [X: real] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_real_real @ G @ top_top_set_real )
= top_top_set_real ) ) ).
% surjI
thf(fact_430_surjI,axiom,
! [G: real > complex,F: complex > real] :
( ! [X: complex] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_real_complex @ G @ top_top_set_real )
= top_top_set_complex ) ) ).
% surjI
thf(fact_431_surjI,axiom,
! [G: real > extended_ereal,F: extended_ereal > real] :
( ! [X: extended_ereal] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_7147107595568778587_ereal @ G @ top_top_set_real )
= top_to5683747375963461374_ereal ) ) ).
% surjI
thf(fact_432_surjI,axiom,
! [G: complex > nat,F: nat > complex] :
( ! [X: nat] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_complex_nat @ G @ top_top_set_complex )
= top_top_set_nat ) ) ).
% surjI
thf(fact_433_surjI,axiom,
! [G: complex > real,F: real > complex] :
( ! [X: real] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_complex_real @ G @ top_top_set_complex )
= top_top_set_real ) ) ).
% surjI
thf(fact_434_rangeI,axiom,
! [F: nat > extended_ereal,X2: nat] : ( member2350847679896131959_ereal @ ( F @ X2 ) @ ( image_4309273772856505399_ereal @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_435_rangeI,axiom,
! [F: nat > complex,X2: nat] : ( member_complex @ ( F @ X2 ) @ ( image_nat_complex @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_436_rangeI,axiom,
! [F: nat > real,X2: nat] : ( member_real @ ( F @ X2 ) @ ( image_nat_real @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_437_rangeI,axiom,
! [F: nat > $o,X2: nat] : ( member_o @ ( F @ X2 ) @ ( image_nat_o @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_438_rangeI,axiom,
! [F: real > real,X2: real] : ( member_real @ ( F @ X2 ) @ ( image_real_real @ F @ top_top_set_real ) ) ).
% rangeI
thf(fact_439_rangeI,axiom,
! [F: real > $o,X2: real] : ( member_o @ ( F @ X2 ) @ ( image_real_o @ F @ top_top_set_real ) ) ).
% rangeI
thf(fact_440_rangeI,axiom,
! [F: complex > real,X2: complex] : ( member_real @ ( F @ X2 ) @ ( image_complex_real @ F @ top_top_set_complex ) ) ).
% rangeI
thf(fact_441_rangeI,axiom,
! [F: complex > $o,X2: complex] : ( member_o @ ( F @ X2 ) @ ( image_complex_o @ F @ top_top_set_complex ) ) ).
% rangeI
thf(fact_442_rangeI,axiom,
! [F: extended_ereal > real,X2: extended_ereal] : ( member_real @ ( F @ X2 ) @ ( image_2321174223038010293l_real @ F @ top_to5683747375963461374_ereal ) ) ).
% rangeI
thf(fact_443_rangeI,axiom,
! [F: extended_ereal > $o,X2: extended_ereal] : ( member_o @ ( F @ X2 ) @ ( image_951975095941678543real_o @ F @ top_to5683747375963461374_ereal ) ) ).
% rangeI
thf(fact_444_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y2: nat] :
? [X4: nat] :
( Y2
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_445_surj__def,axiom,
! [F: nat > real] :
( ( ( image_nat_real @ F @ top_top_set_nat )
= top_top_set_real )
= ( ! [Y2: real] :
? [X4: nat] :
( Y2
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_446_surj__def,axiom,
! [F: nat > complex] :
( ( ( image_nat_complex @ F @ top_top_set_nat )
= top_top_set_complex )
= ( ! [Y2: complex] :
? [X4: nat] :
( Y2
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_447_surj__def,axiom,
! [F: nat > extended_ereal] :
( ( ( image_4309273772856505399_ereal @ F @ top_top_set_nat )
= top_to5683747375963461374_ereal )
= ( ! [Y2: extended_ereal] :
? [X4: nat] :
( Y2
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_448_surj__def,axiom,
! [F: real > nat] :
( ( ( image_real_nat @ F @ top_top_set_real )
= top_top_set_nat )
= ( ! [Y2: nat] :
? [X4: real] :
( Y2
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_449_surj__def,axiom,
! [F: real > real] :
( ( ( image_real_real @ F @ top_top_set_real )
= top_top_set_real )
= ( ! [Y2: real] :
? [X4: real] :
( Y2
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_450_surj__def,axiom,
! [F: real > complex] :
( ( ( image_real_complex @ F @ top_top_set_real )
= top_top_set_complex )
= ( ! [Y2: complex] :
? [X4: real] :
( Y2
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_451_surj__def,axiom,
! [F: real > extended_ereal] :
( ( ( image_7147107595568778587_ereal @ F @ top_top_set_real )
= top_to5683747375963461374_ereal )
= ( ! [Y2: extended_ereal] :
? [X4: real] :
( Y2
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_452_surj__def,axiom,
! [F: complex > nat] :
( ( ( image_complex_nat @ F @ top_top_set_complex )
= top_top_set_nat )
= ( ! [Y2: nat] :
? [X4: complex] :
( Y2
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_453_surj__def,axiom,
! [F: complex > real] :
( ( ( image_complex_real @ F @ top_top_set_complex )
= top_top_set_real )
= ( ! [Y2: real] :
? [X4: complex] :
( Y2
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_454_range__eqI,axiom,
! [B2: extended_ereal,F: nat > extended_ereal,X2: nat] :
( ( B2
= ( F @ X2 ) )
=> ( member2350847679896131959_ereal @ B2 @ ( image_4309273772856505399_ereal @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_455_range__eqI,axiom,
! [B2: complex,F: nat > complex,X2: nat] :
( ( B2
= ( F @ X2 ) )
=> ( member_complex @ B2 @ ( image_nat_complex @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_456_range__eqI,axiom,
! [B2: real,F: nat > real,X2: nat] :
( ( B2
= ( F @ X2 ) )
=> ( member_real @ B2 @ ( image_nat_real @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_457_range__eqI,axiom,
! [B2: $o,F: nat > $o,X2: nat] :
( ( B2
= ( F @ X2 ) )
=> ( member_o @ B2 @ ( image_nat_o @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_458_range__eqI,axiom,
! [B2: real,F: real > real,X2: real] :
( ( B2
= ( F @ X2 ) )
=> ( member_real @ B2 @ ( image_real_real @ F @ top_top_set_real ) ) ) ).
% range_eqI
thf(fact_459_range__eqI,axiom,
! [B2: $o,F: real > $o,X2: real] :
( ( B2
= ( F @ X2 ) )
=> ( member_o @ B2 @ ( image_real_o @ F @ top_top_set_real ) ) ) ).
% range_eqI
thf(fact_460_range__eqI,axiom,
! [B2: real,F: complex > real,X2: complex] :
( ( B2
= ( F @ X2 ) )
=> ( member_real @ B2 @ ( image_complex_real @ F @ top_top_set_complex ) ) ) ).
% range_eqI
thf(fact_461_range__eqI,axiom,
! [B2: $o,F: complex > $o,X2: complex] :
( ( B2
= ( F @ X2 ) )
=> ( member_o @ B2 @ ( image_complex_o @ F @ top_top_set_complex ) ) ) ).
% range_eqI
thf(fact_462_range__eqI,axiom,
! [B2: real,F: extended_ereal > real,X2: extended_ereal] :
( ( B2
= ( F @ X2 ) )
=> ( member_real @ B2 @ ( image_2321174223038010293l_real @ F @ top_to5683747375963461374_ereal ) ) ) ).
% range_eqI
thf(fact_463_range__eqI,axiom,
! [B2: $o,F: extended_ereal > $o,X2: extended_ereal] :
( ( B2
= ( F @ X2 ) )
=> ( member_o @ B2 @ ( image_951975095941678543real_o @ F @ top_to5683747375963461374_ereal ) ) ) ).
% range_eqI
thf(fact_464_subset__UNIV,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_465_subset__UNIV,axiom,
! [A: set_complex] : ( ord_le211207098394363844omplex @ A @ top_top_set_complex ) ).
% subset_UNIV
thf(fact_466_subset__UNIV,axiom,
! [A: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ A @ top_to7994903218803871134nnreal ) ).
% subset_UNIV
thf(fact_467_subset__UNIV,axiom,
! [A: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ A @ top_to5683747375963461374_ereal ) ).
% subset_UNIV
thf(fact_468_subset__UNIV,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ A @ top_top_set_real ) ).
% subset_UNIV
thf(fact_469_sets__restrict__space__cong,axiom,
! [M: sigma_measure_real,N: sigma_measure_real,Omega: set_real] :
( ( ( sigma_sets_real @ M )
= ( sigma_sets_real @ N ) )
=> ( ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ M @ Omega ) )
= ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ N @ Omega ) ) ) ) ).
% sets_restrict_space_cong
thf(fact_470_restrict__space__sets__cong,axiom,
! [A: set_real,B: set_real,M: sigma_measure_real,N: sigma_measure_real] :
( ( A = B )
=> ( ( ( sigma_sets_real @ M )
= ( sigma_sets_real @ N ) )
=> ( ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ M @ A ) )
= ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ N @ B ) ) ) ) ) ).
% restrict_space_sets_cong
thf(fact_471_not__UNIV__eq__Icc,axiom,
! [L2: real,H2: real] :
( top_top_set_real
!= ( set_or1222579329274155063t_real @ L2 @ H2 ) ) ).
% not_UNIV_eq_Icc
thf(fact_472_not__UNIV__eq__Icc,axiom,
! [L2: nat,H2: nat] :
( top_top_set_nat
!= ( set_or1269000886237332187st_nat @ L2 @ H2 ) ) ).
% not_UNIV_eq_Icc
thf(fact_473_monotoneD,axiom,
! [Orda2: nat > nat > $o,Ordb2: ( real > real ) > ( real > real ) > $o,F: nat > real > real,X2: nat,Y4: nat] :
( ( monoto2824216093323351088l_real @ top_top_set_nat @ Orda2 @ Ordb2 @ F )
=> ( ( Orda2 @ X2 @ Y4 )
=> ( Ordb2 @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% monotoneD
thf(fact_474_monotoneD,axiom,
! [Orda2: nat > nat > $o,Ordb2: extended_ereal > extended_ereal > $o,F: nat > extended_ereal,X2: nat,Y4: nat] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ Orda2 @ Ordb2 @ F )
=> ( ( Orda2 @ X2 @ Y4 )
=> ( Ordb2 @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% monotoneD
thf(fact_475_monotoneD,axiom,
! [Orda2: nat > nat > $o,Ordb2: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o,F: nat > extend8495563244428889912nnreal,X2: nat,Y4: nat] :
( ( monoto2291723841412853873nnreal @ top_top_set_nat @ Orda2 @ Ordb2 @ F )
=> ( ( Orda2 @ X2 @ Y4 )
=> ( Ordb2 @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% monotoneD
thf(fact_476_monotoneD,axiom,
! [Orda2: nat > nat > $o,Ordb2: nat > nat > $o,F: nat > nat,X2: nat,Y4: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ Orda2 @ Ordb2 @ F )
=> ( ( Orda2 @ X2 @ Y4 )
=> ( Ordb2 @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% monotoneD
thf(fact_477_monotoneD,axiom,
! [Orda2: real > real > $o,Ordb2: real > real > $o,F: real > real,X2: real,Y4: real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ Orda2 @ Ordb2 @ F )
=> ( ( Orda2 @ X2 @ Y4 )
=> ( Ordb2 @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% monotoneD
thf(fact_478_monotoneI,axiom,
! [Orda2: nat > nat > $o,Ordb2: ( real > real ) > ( real > real ) > $o,F: nat > real > real] :
( ! [X: nat,Y3: nat] :
( ( Orda2 @ X @ Y3 )
=> ( Ordb2 @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( monoto2824216093323351088l_real @ top_top_set_nat @ Orda2 @ Ordb2 @ F ) ) ).
% monotoneI
thf(fact_479_monotoneI,axiom,
! [Orda2: nat > nat > $o,Ordb2: extended_ereal > extended_ereal > $o,F: nat > extended_ereal] :
( ! [X: nat,Y3: nat] :
( ( Orda2 @ X @ Y3 )
=> ( Ordb2 @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( monoto8452838292781035605_ereal @ top_top_set_nat @ Orda2 @ Ordb2 @ F ) ) ).
% monotoneI
thf(fact_480_monotoneI,axiom,
! [Orda2: nat > nat > $o,Ordb2: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o,F: nat > extend8495563244428889912nnreal] :
( ! [X: nat,Y3: nat] :
( ( Orda2 @ X @ Y3 )
=> ( Ordb2 @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( monoto2291723841412853873nnreal @ top_top_set_nat @ Orda2 @ Ordb2 @ F ) ) ).
% monotoneI
thf(fact_481_monotoneI,axiom,
! [Orda2: nat > nat > $o,Ordb2: nat > nat > $o,F: nat > nat] :
( ! [X: nat,Y3: nat] :
( ( Orda2 @ X @ Y3 )
=> ( Ordb2 @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ Orda2 @ Ordb2 @ F ) ) ).
% monotoneI
thf(fact_482_monotoneI,axiom,
! [Orda2: real > real > $o,Ordb2: real > real > $o,F: real > real] :
( ! [X: real,Y3: real] :
( ( Orda2 @ X @ Y3 )
=> ( Ordb2 @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( monoto4017252874604999745l_real @ top_top_set_real @ Orda2 @ Ordb2 @ F ) ) ).
% monotoneI
thf(fact_483_measurable__mono,axiom,
! [N2: sigma_measure_real,N: sigma_measure_real,M: sigma_measure_real,M3: sigma_measure_real] :
( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ N2 ) @ ( sigma_sets_real @ N ) )
=> ( ( ( sigma_space_real @ N )
= ( sigma_space_real @ N2 ) )
=> ( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ M ) @ ( sigma_sets_real @ M3 ) )
=> ( ( ( sigma_space_real @ M )
= ( sigma_space_real @ M3 ) )
=> ( ord_le4198349162570665613l_real @ ( sigma_5267869275261027754l_real @ M @ N ) @ ( sigma_5267869275261027754l_real @ M3 @ N2 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_484_sets_Osets__into__space,axiom,
! [X2: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ X2 @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ord_le6787938422905777998nnreal @ X2 @ ( sigma_3147302497200244656nnreal @ M ) ) ) ).
% sets.sets_into_space
thf(fact_485_sets_Osets__into__space,axiom,
! [X2: set_Extended_ereal,M: sigma_7227684458468523851_ereal] :
( ( member5519481007471526743_ereal @ X2 @ ( sigma_6858886609720962221_ereal @ M ) )
=> ( ord_le1644982726543182158_ereal @ X2 @ ( sigma_3843066954113031510_ereal @ M ) ) ) ).
% sets.sets_into_space
thf(fact_486_sets_Osets__into__space,axiom,
! [X2: set_real,M: sigma_measure_real] :
( ( member_set_real @ X2 @ ( sigma_sets_real @ M ) )
=> ( ord_less_eq_set_real @ X2 @ ( sigma_space_real @ M ) ) ) ).
% sets.sets_into_space
thf(fact_487_sets__le__imp__space__le,axiom,
! [A: sigma_7234349610311085201nnreal,B: sigma_7234349610311085201nnreal] :
( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ A ) @ ( sigma_5465916536984168985nnreal @ B ) )
=> ( ord_le6787938422905777998nnreal @ ( sigma_3147302497200244656nnreal @ A ) @ ( sigma_3147302497200244656nnreal @ B ) ) ) ).
% sets_le_imp_space_le
thf(fact_488_sets__le__imp__space__le,axiom,
! [A: sigma_7227684458468523851_ereal,B: sigma_7227684458468523851_ereal] :
( ( ord_le5287700718633833262_ereal @ ( sigma_6858886609720962221_ereal @ A ) @ ( sigma_6858886609720962221_ereal @ B ) )
=> ( ord_le1644982726543182158_ereal @ ( sigma_3843066954113031510_ereal @ A ) @ ( sigma_3843066954113031510_ereal @ B ) ) ) ).
% sets_le_imp_space_le
thf(fact_489_sets__le__imp__space__le,axiom,
! [A: sigma_measure_real,B: sigma_measure_real] :
( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ A ) @ ( sigma_sets_real @ B ) )
=> ( ord_less_eq_set_real @ ( sigma_space_real @ A ) @ ( sigma_space_real @ B ) ) ) ).
% sets_le_imp_space_le
thf(fact_490_lborelD,axiom,
! [A: set_real] :
( ( member_set_real @ A @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
=> ( member_set_real @ A @ ( sigma_sets_real @ lebesgue_lborel_real ) ) ) ).
% lborelD
thf(fact_491_atLeastAtMost__borel,axiom,
! [A2: nat,B2: nat] : ( member_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ).
% atLeastAtMost_borel
thf(fact_492_atLeastAtMost__borel,axiom,
! [A2: real,B2: real] : ( member_set_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ).
% atLeastAtMost_borel
thf(fact_493_eucl__ivals_I5_J,axiom,
! [A2: real,B2: real] : ( member_set_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ).
% eucl_ivals(5)
thf(fact_494_sets__restrict__space__subset,axiom,
! [S: set_real,M: sigma_measure_real] :
( ( member_set_real @ S @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ M ) ) )
=> ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ M ) @ S ) ) @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ M ) ) ) ) ).
% sets_restrict_space_subset
thf(fact_495_range__subsetD,axiom,
! [F: nat > complex,B: set_complex,I: nat] :
( ( ord_le211207098394363844omplex @ ( image_nat_complex @ F @ top_top_set_nat ) @ B )
=> ( member_complex @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_496_range__subsetD,axiom,
! [F: nat > $o,B: set_o,I: nat] :
( ( ord_less_eq_set_o @ ( image_nat_o @ F @ top_top_set_nat ) @ B )
=> ( member_o @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_497_range__subsetD,axiom,
! [F: real > $o,B: set_o,I: real] :
( ( ord_less_eq_set_o @ ( image_real_o @ F @ top_top_set_real ) @ B )
=> ( member_o @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_498_range__subsetD,axiom,
! [F: complex > $o,B: set_o,I: complex] :
( ( ord_less_eq_set_o @ ( image_complex_o @ F @ top_top_set_complex ) @ B )
=> ( member_o @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_499_range__subsetD,axiom,
! [F: extended_ereal > $o,B: set_o,I: extended_ereal] :
( ( ord_less_eq_set_o @ ( image_951975095941678543real_o @ F @ top_to5683747375963461374_ereal ) @ B )
=> ( member_o @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_500_range__subsetD,axiom,
! [F: nat > extend8495563244428889912nnreal,B: set_Ex3793607809372303086nnreal,I: nat] :
( ( ord_le6787938422905777998nnreal @ ( image_8459861568512453903nnreal @ F @ top_top_set_nat ) @ B )
=> ( member7908768830364227535nnreal @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_501_range__subsetD,axiom,
! [F: real > extend8495563244428889912nnreal,B: set_Ex3793607809372303086nnreal,I: real] :
( ( ord_le6787938422905777998nnreal @ ( image_7616191137145695467nnreal @ F @ top_top_set_real ) @ B )
=> ( member7908768830364227535nnreal @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_502_range__subsetD,axiom,
! [F: complex > extend8495563244428889912nnreal,B: set_Ex3793607809372303086nnreal,I: complex] :
( ( ord_le6787938422905777998nnreal @ ( image_4927658817219388909nnreal @ F @ top_top_set_complex ) @ B )
=> ( member7908768830364227535nnreal @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_503_range__subsetD,axiom,
! [F: extended_ereal > extend8495563244428889912nnreal,B: set_Ex3793607809372303086nnreal,I: extended_ereal] :
( ( ord_le6787938422905777998nnreal @ ( image_8614087454967683265nnreal @ F @ top_to5683747375963461374_ereal ) @ B )
=> ( member7908768830364227535nnreal @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_504_range__subsetD,axiom,
! [F: nat > extended_ereal,B: set_Extended_ereal,I: nat] :
( ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F @ top_top_set_nat ) @ B )
=> ( member2350847679896131959_ereal @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_505_simple__function__cong__algebra,axiom,
! [N: sigma_measure_real,M: sigma_measure_real,F: real > real] :
( ( ( sigma_sets_real @ N )
= ( sigma_sets_real @ M ) )
=> ( ( ( sigma_space_real @ N )
= ( sigma_space_real @ M ) )
=> ( ( nonneg485563716852976898l_real @ M @ F )
= ( nonneg485563716852976898l_real @ N @ F ) ) ) ) ).
% simple_function_cong_algebra
thf(fact_506_simple__function__subalgebra,axiom,
! [N: sigma_measure_real,F: real > real,M: sigma_measure_real] :
( ( nonneg485563716852976898l_real @ N @ F )
=> ( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ N ) @ ( sigma_sets_real @ M ) )
=> ( ( ( sigma_space_real @ N )
= ( sigma_space_real @ M ) )
=> ( nonneg485563716852976898l_real @ M @ F ) ) ) ) ).
% simple_function_subalgebra
thf(fact_507_not__UNIV__le__Icc,axiom,
! [L: real,H: real] :
~ ( ord_less_eq_set_real @ top_top_set_real @ ( set_or1222579329274155063t_real @ L @ H ) ) ).
% not_UNIV_le_Icc
thf(fact_508_not__UNIV__le__Icc,axiom,
! [L: nat,H: nat] :
~ ( ord_less_eq_set_nat @ top_top_set_nat @ ( set_or1269000886237332187st_nat @ L @ H ) ) ).
% not_UNIV_le_Icc
thf(fact_509_monoD,axiom,
! [F: real > real,X2: real,Y4: real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% monoD
thf(fact_510_monoD,axiom,
! [F: real > nat,X2: real,Y4: real] :
( ( monotone_on_real_nat @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% monoD
thf(fact_511_monoD,axiom,
! [F: complex > nat,X2: complex,Y4: complex] :
( ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_complex @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% monoD
thf(fact_512_monoD,axiom,
! [F: extended_ereal > nat,X2: extended_ereal,Y4: extended_ereal] :
( ( monoto2580034644210098551al_nat @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_less_eq_nat @ F )
=> ( ( ord_le1083603963089353582_ereal @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% monoD
thf(fact_513_monoD,axiom,
! [F: nat > extended_ereal,X2: nat,Y4: nat] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ F )
=> ( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% monoD
thf(fact_514_monoD,axiom,
! [F: nat > extend8495563244428889912nnreal,X2: nat,Y4: nat] :
( ( monoto2291723841412853873nnreal @ top_top_set_nat @ ord_less_eq_nat @ ord_le3935885782089961368nnreal @ F )
=> ( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% monoD
thf(fact_515_monoD,axiom,
! [F: nat > nat,X2: nat,Y4: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% monoD
thf(fact_516_monoD,axiom,
! [F: real > set_Ex3793607809372303086nnreal,X2: real,Y4: real] :
( ( monoto2626391617355967235nnreal @ top_top_set_real @ ord_less_eq_real @ ord_le6787938422905777998nnreal @ F )
=> ( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_le6787938422905777998nnreal @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% monoD
thf(fact_517_monoD,axiom,
! [F: complex > set_Ex3793607809372303086nnreal,X2: complex,Y4: complex] :
( ( monoto3400374153095604485nnreal @ top_top_set_complex @ ord_less_eq_complex @ ord_le6787938422905777998nnreal @ F )
=> ( ( ord_less_eq_complex @ X2 @ Y4 )
=> ( ord_le6787938422905777998nnreal @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% monoD
thf(fact_518_monoD,axiom,
! [F: extended_ereal > set_Ex3793607809372303086nnreal,X2: extended_ereal,Y4: extended_ereal] :
( ( monoto6742655004523227285nnreal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_le6787938422905777998nnreal @ F )
=> ( ( ord_le1083603963089353582_ereal @ X2 @ Y4 )
=> ( ord_le6787938422905777998nnreal @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% monoD
thf(fact_519_monoE,axiom,
! [F: real > real,X2: real,Y4: real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% monoE
thf(fact_520_monoE,axiom,
! [F: real > nat,X2: real,Y4: real] :
( ( monotone_on_real_nat @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% monoE
thf(fact_521_monoE,axiom,
! [F: complex > nat,X2: complex,Y4: complex] :
( ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_complex @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% monoE
thf(fact_522_monoE,axiom,
! [F: extended_ereal > nat,X2: extended_ereal,Y4: extended_ereal] :
( ( monoto2580034644210098551al_nat @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_less_eq_nat @ F )
=> ( ( ord_le1083603963089353582_ereal @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% monoE
thf(fact_523_monoE,axiom,
! [F: nat > extended_ereal,X2: nat,Y4: nat] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ F )
=> ( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% monoE
thf(fact_524_monoE,axiom,
! [F: nat > extend8495563244428889912nnreal,X2: nat,Y4: nat] :
( ( monoto2291723841412853873nnreal @ top_top_set_nat @ ord_less_eq_nat @ ord_le3935885782089961368nnreal @ F )
=> ( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% monoE
thf(fact_525_monoE,axiom,
! [F: nat > nat,X2: nat,Y4: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% monoE
thf(fact_526_monoE,axiom,
! [F: real > set_Ex3793607809372303086nnreal,X2: real,Y4: real] :
( ( monoto2626391617355967235nnreal @ top_top_set_real @ ord_less_eq_real @ ord_le6787938422905777998nnreal @ F )
=> ( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_le6787938422905777998nnreal @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% monoE
thf(fact_527_monoE,axiom,
! [F: complex > set_Ex3793607809372303086nnreal,X2: complex,Y4: complex] :
( ( monoto3400374153095604485nnreal @ top_top_set_complex @ ord_less_eq_complex @ ord_le6787938422905777998nnreal @ F )
=> ( ( ord_less_eq_complex @ X2 @ Y4 )
=> ( ord_le6787938422905777998nnreal @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% monoE
thf(fact_528_monoE,axiom,
! [F: extended_ereal > set_Ex3793607809372303086nnreal,X2: extended_ereal,Y4: extended_ereal] :
( ( monoto6742655004523227285nnreal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_le6787938422905777998nnreal @ F )
=> ( ( ord_le1083603963089353582_ereal @ X2 @ Y4 )
=> ( ord_le6787938422905777998nnreal @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% monoE
thf(fact_529_monoI,axiom,
! [F: real > real] :
( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_real @ F ) ) ).
% monoI
thf(fact_530_monoI,axiom,
! [F: real > nat] :
( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( monotone_on_real_nat @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_531_monoI,axiom,
! [F: complex > nat] :
( ! [X: complex,Y3: complex] :
( ( ord_less_eq_complex @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_532_monoI,axiom,
! [F: extended_ereal > nat] :
( ! [X: extended_ereal,Y3: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( monoto2580034644210098551al_nat @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_533_monoI,axiom,
! [F: nat > extended_ereal] :
( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le1083603963089353582_ereal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ F ) ) ).
% monoI
thf(fact_534_monoI,axiom,
! [F: nat > extend8495563244428889912nnreal] :
( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( monoto2291723841412853873nnreal @ top_top_set_nat @ ord_less_eq_nat @ ord_le3935885782089961368nnreal @ F ) ) ).
% monoI
thf(fact_535_monoI,axiom,
! [F: nat > nat] :
( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_536_monoI,axiom,
! [F: real > set_Ex3793607809372303086nnreal] :
( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_le6787938422905777998nnreal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( monoto2626391617355967235nnreal @ top_top_set_real @ ord_less_eq_real @ ord_le6787938422905777998nnreal @ F ) ) ).
% monoI
thf(fact_537_monoI,axiom,
! [F: complex > set_Ex3793607809372303086nnreal] :
( ! [X: complex,Y3: complex] :
( ( ord_less_eq_complex @ X @ Y3 )
=> ( ord_le6787938422905777998nnreal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( monoto3400374153095604485nnreal @ top_top_set_complex @ ord_less_eq_complex @ ord_le6787938422905777998nnreal @ F ) ) ).
% monoI
thf(fact_538_monoI,axiom,
! [F: extended_ereal > set_Ex3793607809372303086nnreal] :
( ! [X: extended_ereal,Y3: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ Y3 )
=> ( ord_le6787938422905777998nnreal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( monoto6742655004523227285nnreal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_le6787938422905777998nnreal @ F ) ) ).
% monoI
thf(fact_539_mono__imp__mono__on,axiom,
! [F: real > real,A: set_real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( monoto4017252874604999745l_real @ A @ ord_less_eq_real @ ord_less_eq_real @ F ) ) ).
% mono_imp_mono_on
thf(fact_540_mono__imp__mono__on,axiom,
! [F: real > nat,A: set_real] :
( ( monotone_on_real_nat @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_nat @ F )
=> ( monotone_on_real_nat @ A @ ord_less_eq_real @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_541_mono__imp__mono__on,axiom,
! [F: complex > nat,A: set_complex] :
( ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_nat @ F )
=> ( monoto2406513391651152359ex_nat @ A @ ord_less_eq_complex @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_542_mono__imp__mono__on,axiom,
! [F: extended_ereal > nat,A: set_Extended_ereal] :
( ( monoto2580034644210098551al_nat @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_less_eq_nat @ F )
=> ( monoto2580034644210098551al_nat @ A @ ord_le1083603963089353582_ereal @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_543_mono__imp__mono__on,axiom,
! [F: nat > extended_ereal,A: set_nat] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ F )
=> ( monoto8452838292781035605_ereal @ A @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ F ) ) ).
% mono_imp_mono_on
thf(fact_544_mono__imp__mono__on,axiom,
! [F: nat > extend8495563244428889912nnreal,A: set_nat] :
( ( monoto2291723841412853873nnreal @ top_top_set_nat @ ord_less_eq_nat @ ord_le3935885782089961368nnreal @ F )
=> ( monoto2291723841412853873nnreal @ A @ ord_less_eq_nat @ ord_le3935885782089961368nnreal @ F ) ) ).
% mono_imp_mono_on
thf(fact_545_mono__imp__mono__on,axiom,
! [F: nat > nat,A: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_546_mono__imp__mono__on,axiom,
! [F: real > set_Ex3793607809372303086nnreal,A: set_real] :
( ( monoto2626391617355967235nnreal @ top_top_set_real @ ord_less_eq_real @ ord_le6787938422905777998nnreal @ F )
=> ( monoto2626391617355967235nnreal @ A @ ord_less_eq_real @ ord_le6787938422905777998nnreal @ F ) ) ).
% mono_imp_mono_on
thf(fact_547_mono__imp__mono__on,axiom,
! [F: complex > set_Ex3793607809372303086nnreal,A: set_complex] :
( ( monoto3400374153095604485nnreal @ top_top_set_complex @ ord_less_eq_complex @ ord_le6787938422905777998nnreal @ F )
=> ( monoto3400374153095604485nnreal @ A @ ord_less_eq_complex @ ord_le6787938422905777998nnreal @ F ) ) ).
% mono_imp_mono_on
thf(fact_548_mono__imp__mono__on,axiom,
! [F: extended_ereal > set_Ex3793607809372303086nnreal,A: set_Extended_ereal] :
( ( monoto6742655004523227285nnreal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_le6787938422905777998nnreal @ F )
=> ( monoto6742655004523227285nnreal @ A @ ord_le1083603963089353582_ereal @ ord_le6787938422905777998nnreal @ F ) ) ).
% mono_imp_mono_on
thf(fact_549_surj__id,axiom,
( ( image_nat_nat @ id_nat @ top_top_set_nat )
= top_top_set_nat ) ).
% surj_id
thf(fact_550_surj__id,axiom,
( ( image_real_real @ id_real @ top_top_set_real )
= top_top_set_real ) ).
% surj_id
thf(fact_551_surj__id,axiom,
( ( image_1468599708987790691omplex @ id_complex @ top_top_set_complex )
= top_top_set_complex ) ).
% surj_id
thf(fact_552_surj__id,axiom,
( ( image_6042159593519690757_ereal @ id_Extended_ereal @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal ) ).
% surj_id
thf(fact_553_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_554_le__cases3,axiom,
! [X2: nat,Y4: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y4 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_555_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_Ex3793607809372303086nnreal,Z2: set_Ex3793607809372303086nnreal] : ( Y = Z2 ) )
= ( ^ [X4: set_Ex3793607809372303086nnreal,Y2: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ X4 @ Y2 )
& ( ord_le6787938422905777998nnreal @ Y2 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_556_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_Extended_ereal,Z2: set_Extended_ereal] : ( Y = Z2 ) )
= ( ^ [X4: set_Extended_ereal,Y2: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ X4 @ Y2 )
& ( ord_le1644982726543182158_ereal @ Y2 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_557_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_real,Z2: set_real] : ( Y = Z2 ) )
= ( ^ [X4: set_real,Y2: set_real] :
( ( ord_less_eq_set_real @ X4 @ Y2 )
& ( ord_less_eq_set_real @ Y2 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_558_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
= ( ^ [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_559_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: real > real,Z2: real > real] : ( Y = Z2 ) )
= ( ^ [X4: real > real,Y2: real > real] :
( ( ord_le6948328307412524503l_real @ X4 @ Y2 )
& ( ord_le6948328307412524503l_real @ Y2 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_560_ord__eq__le__trans,axiom,
! [A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,C: set_Ex3793607809372303086nnreal] :
( ( A2 = B2 )
=> ( ( ord_le6787938422905777998nnreal @ B2 @ C )
=> ( ord_le6787938422905777998nnreal @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_561_ord__eq__le__trans,axiom,
! [A2: set_Extended_ereal,B2: set_Extended_ereal,C: set_Extended_ereal] :
( ( A2 = B2 )
=> ( ( ord_le1644982726543182158_ereal @ B2 @ C )
=> ( ord_le1644982726543182158_ereal @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_562_ord__eq__le__trans,axiom,
! [A2: set_real,B2: set_real,C: set_real] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_real @ B2 @ C )
=> ( ord_less_eq_set_real @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_563_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_564_ord__eq__le__trans,axiom,
! [A2: real > real,B2: real > real,C: real > real] :
( ( A2 = B2 )
=> ( ( ord_le6948328307412524503l_real @ B2 @ C )
=> ( ord_le6948328307412524503l_real @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_565_ord__le__eq__trans,axiom,
! [A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,C: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le6787938422905777998nnreal @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_566_ord__le__eq__trans,axiom,
! [A2: set_Extended_ereal,B2: set_Extended_ereal,C: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le1644982726543182158_ereal @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_567_ord__le__eq__trans,axiom,
! [A2: set_real,B2: set_real,C: set_real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_real @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_568_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_569_ord__le__eq__trans,axiom,
! [A2: real > real,B2: real > real,C: real > real] :
( ( ord_le6948328307412524503l_real @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le6948328307412524503l_real @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_570_order__antisym,axiom,
! [X2: set_Ex3793607809372303086nnreal,Y4: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ X2 @ Y4 )
=> ( ( ord_le6787938422905777998nnreal @ Y4 @ X2 )
=> ( X2 = Y4 ) ) ) ).
% order_antisym
thf(fact_571_order__antisym,axiom,
! [X2: set_Extended_ereal,Y4: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ X2 @ Y4 )
=> ( ( ord_le1644982726543182158_ereal @ Y4 @ X2 )
=> ( X2 = Y4 ) ) ) ).
% order_antisym
thf(fact_572_order__antisym,axiom,
! [X2: set_real,Y4: set_real] :
( ( ord_less_eq_set_real @ X2 @ Y4 )
=> ( ( ord_less_eq_set_real @ Y4 @ X2 )
=> ( X2 = Y4 ) ) ) ).
% order_antisym
thf(fact_573_order__antisym,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ X2 )
=> ( X2 = Y4 ) ) ) ).
% order_antisym
thf(fact_574_order__antisym,axiom,
! [X2: real > real,Y4: real > real] :
( ( ord_le6948328307412524503l_real @ X2 @ Y4 )
=> ( ( ord_le6948328307412524503l_real @ Y4 @ X2 )
=> ( X2 = Y4 ) ) ) ).
% order_antisym
thf(fact_575_order_Otrans,axiom,
! [A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,C: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ A2 @ B2 )
=> ( ( ord_le6787938422905777998nnreal @ B2 @ C )
=> ( ord_le6787938422905777998nnreal @ A2 @ C ) ) ) ).
% order.trans
thf(fact_576_order_Otrans,axiom,
! [A2: set_Extended_ereal,B2: set_Extended_ereal,C: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A2 @ B2 )
=> ( ( ord_le1644982726543182158_ereal @ B2 @ C )
=> ( ord_le1644982726543182158_ereal @ A2 @ C ) ) ) ).
% order.trans
thf(fact_577_order_Otrans,axiom,
! [A2: set_real,B2: set_real,C: set_real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( ord_less_eq_set_real @ B2 @ C )
=> ( ord_less_eq_set_real @ A2 @ C ) ) ) ).
% order.trans
thf(fact_578_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_579_order_Otrans,axiom,
! [A2: real > real,B2: real > real,C: real > real] :
( ( ord_le6948328307412524503l_real @ A2 @ B2 )
=> ( ( ord_le6948328307412524503l_real @ B2 @ C )
=> ( ord_le6948328307412524503l_real @ A2 @ C ) ) ) ).
% order.trans
thf(fact_580_order__trans,axiom,
! [X2: set_Ex3793607809372303086nnreal,Y4: set_Ex3793607809372303086nnreal,Z: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ X2 @ Y4 )
=> ( ( ord_le6787938422905777998nnreal @ Y4 @ Z )
=> ( ord_le6787938422905777998nnreal @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_581_order__trans,axiom,
! [X2: set_Extended_ereal,Y4: set_Extended_ereal,Z: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ X2 @ Y4 )
=> ( ( ord_le1644982726543182158_ereal @ Y4 @ Z )
=> ( ord_le1644982726543182158_ereal @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_582_order__trans,axiom,
! [X2: set_real,Y4: set_real,Z: set_real] :
( ( ord_less_eq_set_real @ X2 @ Y4 )
=> ( ( ord_less_eq_set_real @ Y4 @ Z )
=> ( ord_less_eq_set_real @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_583_order__trans,axiom,
! [X2: nat,Y4: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z )
=> ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_584_order__trans,axiom,
! [X2: real > real,Y4: real > real,Z: real > real] :
( ( ord_le6948328307412524503l_real @ X2 @ Y4 )
=> ( ( ord_le6948328307412524503l_real @ Y4 @ Z )
=> ( ord_le6948328307412524503l_real @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_585_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_586_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_Ex3793607809372303086nnreal,Z2: set_Ex3793607809372303086nnreal] : ( Y = Z2 ) )
= ( ^ [A5: set_Ex3793607809372303086nnreal,B5: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ B5 @ A5 )
& ( ord_le6787938422905777998nnreal @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_587_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_Extended_ereal,Z2: set_Extended_ereal] : ( Y = Z2 ) )
= ( ^ [A5: set_Extended_ereal,B5: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B5 @ A5 )
& ( ord_le1644982726543182158_ereal @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_588_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_real,Z2: set_real] : ( Y = Z2 ) )
= ( ^ [A5: set_real,B5: set_real] :
( ( ord_less_eq_set_real @ B5 @ A5 )
& ( ord_less_eq_set_real @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_589_dual__order_Oeq__iff,axiom,
( ( ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_590_dual__order_Oeq__iff,axiom,
( ( ^ [Y: real > real,Z2: real > real] : ( Y = Z2 ) )
= ( ^ [A5: real > real,B5: real > real] :
( ( ord_le6948328307412524503l_real @ B5 @ A5 )
& ( ord_le6948328307412524503l_real @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_591_dual__order_Oantisym,axiom,
! [B2: set_Ex3793607809372303086nnreal,A2: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ B2 @ A2 )
=> ( ( ord_le6787938422905777998nnreal @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_592_dual__order_Oantisym,axiom,
! [B2: set_Extended_ereal,A2: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B2 @ A2 )
=> ( ( ord_le1644982726543182158_ereal @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_593_dual__order_Oantisym,axiom,
! [B2: set_real,A2: set_real] :
( ( ord_less_eq_set_real @ B2 @ A2 )
=> ( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_594_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_595_dual__order_Oantisym,axiom,
! [B2: real > real,A2: real > real] :
( ( ord_le6948328307412524503l_real @ B2 @ A2 )
=> ( ( ord_le6948328307412524503l_real @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_596_dual__order_Otrans,axiom,
! [B2: set_Ex3793607809372303086nnreal,A2: set_Ex3793607809372303086nnreal,C: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ B2 @ A2 )
=> ( ( ord_le6787938422905777998nnreal @ C @ B2 )
=> ( ord_le6787938422905777998nnreal @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_597_dual__order_Otrans,axiom,
! [B2: set_Extended_ereal,A2: set_Extended_ereal,C: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B2 @ A2 )
=> ( ( ord_le1644982726543182158_ereal @ C @ B2 )
=> ( ord_le1644982726543182158_ereal @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_598_dual__order_Otrans,axiom,
! [B2: set_real,A2: set_real,C: set_real] :
( ( ord_less_eq_set_real @ B2 @ A2 )
=> ( ( ord_less_eq_set_real @ C @ B2 )
=> ( ord_less_eq_set_real @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_599_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_600_dual__order_Otrans,axiom,
! [B2: real > real,A2: real > real,C: real > real] :
( ( ord_le6948328307412524503l_real @ B2 @ A2 )
=> ( ( ord_le6948328307412524503l_real @ C @ B2 )
=> ( ord_le6948328307412524503l_real @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_601_antisym,axiom,
! [A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ A2 @ B2 )
=> ( ( ord_le6787938422905777998nnreal @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_602_antisym,axiom,
! [A2: set_Extended_ereal,B2: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A2 @ B2 )
=> ( ( ord_le1644982726543182158_ereal @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_603_antisym,axiom,
! [A2: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( ord_less_eq_set_real @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_604_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_605_antisym,axiom,
! [A2: real > real,B2: real > real] :
( ( ord_le6948328307412524503l_real @ A2 @ B2 )
=> ( ( ord_le6948328307412524503l_real @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_606_le__funD,axiom,
! [F: real > real,G: real > real,X2: real] :
( ( ord_le6948328307412524503l_real @ F @ G )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( G @ X2 ) ) ) ).
% le_funD
thf(fact_607_le__funE,axiom,
! [F: real > real,G: real > real,X2: real] :
( ( ord_le6948328307412524503l_real @ F @ G )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( G @ X2 ) ) ) ).
% le_funE
thf(fact_608_le__funI,axiom,
! [F: real > real,G: real > real] :
( ! [X: real] : ( ord_less_eq_real @ ( F @ X ) @ ( G @ X ) )
=> ( ord_le6948328307412524503l_real @ F @ G ) ) ).
% le_funI
thf(fact_609_le__fun__def,axiom,
( ord_le6948328307412524503l_real
= ( ^ [F2: real > real,G2: real > real] :
! [X4: real] : ( ord_less_eq_real @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ).
% le_fun_def
thf(fact_610_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_Ex3793607809372303086nnreal,Z2: set_Ex3793607809372303086nnreal] : ( Y = Z2 ) )
= ( ^ [A5: set_Ex3793607809372303086nnreal,B5: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ A5 @ B5 )
& ( ord_le6787938422905777998nnreal @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_611_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_Extended_ereal,Z2: set_Extended_ereal] : ( Y = Z2 ) )
= ( ^ [A5: set_Extended_ereal,B5: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A5 @ B5 )
& ( ord_le1644982726543182158_ereal @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_612_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_real,Z2: set_real] : ( Y = Z2 ) )
= ( ^ [A5: set_real,B5: set_real] :
( ( ord_less_eq_set_real @ A5 @ B5 )
& ( ord_less_eq_set_real @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_613_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_614_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: real > real,Z2: real > real] : ( Y = Z2 ) )
= ( ^ [A5: real > real,B5: real > real] :
( ( ord_le6948328307412524503l_real @ A5 @ B5 )
& ( ord_le6948328307412524503l_real @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_615_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_616_order__subst1,axiom,
! [A2: set_Ex3793607809372303086nnreal,F: nat > set_Ex3793607809372303086nnreal,B2: nat,C: nat] :
( ( ord_le6787938422905777998nnreal @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le6787938422905777998nnreal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le6787938422905777998nnreal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_617_order__subst1,axiom,
! [A2: set_Extended_ereal,F: nat > set_Extended_ereal,B2: nat,C: nat] :
( ( ord_le1644982726543182158_ereal @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le1644982726543182158_ereal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le1644982726543182158_ereal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_618_order__subst1,axiom,
! [A2: set_real,F: nat > set_real,B2: nat,C: nat] :
( ( ord_less_eq_set_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_619_order__subst1,axiom,
! [A2: nat,F: set_Ex3793607809372303086nnreal > nat,B2: set_Ex3793607809372303086nnreal,C: set_Ex3793607809372303086nnreal] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le6787938422905777998nnreal @ B2 @ C )
=> ( ! [X: set_Ex3793607809372303086nnreal,Y3: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_620_order__subst1,axiom,
! [A2: nat,F: set_Extended_ereal > nat,B2: set_Extended_ereal,C: set_Extended_ereal] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le1644982726543182158_ereal @ B2 @ C )
=> ( ! [X: set_Extended_ereal,Y3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_621_order__subst1,axiom,
! [A2: nat,F: set_real > nat,B2: set_real,C: set_real] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_real @ B2 @ C )
=> ( ! [X: set_real,Y3: set_real] :
( ( ord_less_eq_set_real @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_622_order__subst1,axiom,
! [A2: set_Ex3793607809372303086nnreal,F: set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,C: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ A2 @ ( F @ B2 ) )
=> ( ( ord_le6787938422905777998nnreal @ B2 @ C )
=> ( ! [X: set_Ex3793607809372303086nnreal,Y3: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ X @ Y3 )
=> ( ord_le6787938422905777998nnreal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le6787938422905777998nnreal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_623_order__subst1,axiom,
! [A2: set_Ex3793607809372303086nnreal,F: set_Extended_ereal > set_Ex3793607809372303086nnreal,B2: set_Extended_ereal,C: set_Extended_ereal] :
( ( ord_le6787938422905777998nnreal @ A2 @ ( F @ B2 ) )
=> ( ( ord_le1644982726543182158_ereal @ B2 @ C )
=> ( ! [X: set_Extended_ereal,Y3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ X @ Y3 )
=> ( ord_le6787938422905777998nnreal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le6787938422905777998nnreal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_624_order__subst1,axiom,
! [A2: set_Ex3793607809372303086nnreal,F: set_real > set_Ex3793607809372303086nnreal,B2: set_real,C: set_real] :
( ( ord_le6787938422905777998nnreal @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_real @ B2 @ C )
=> ( ! [X: set_real,Y3: set_real] :
( ( ord_less_eq_set_real @ X @ Y3 )
=> ( ord_le6787938422905777998nnreal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le6787938422905777998nnreal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_625_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_626_order__subst2,axiom,
! [A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,F: set_Ex3793607809372303086nnreal > nat,C: nat] :
( ( ord_le6787938422905777998nnreal @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X: set_Ex3793607809372303086nnreal,Y3: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_627_order__subst2,axiom,
! [A2: set_Extended_ereal,B2: set_Extended_ereal,F: set_Extended_ereal > nat,C: nat] :
( ( ord_le1644982726543182158_ereal @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X: set_Extended_ereal,Y3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_628_order__subst2,axiom,
! [A2: set_real,B2: set_real,F: set_real > nat,C: nat] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X: set_real,Y3: set_real] :
( ( ord_less_eq_set_real @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_629_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_Ex3793607809372303086nnreal,C: set_Ex3793607809372303086nnreal] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_le6787938422905777998nnreal @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le6787938422905777998nnreal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le6787938422905777998nnreal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_630_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_Extended_ereal,C: set_Extended_ereal] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_le1644982726543182158_ereal @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le1644982726543182158_ereal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le1644982726543182158_ereal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_631_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_real,C: set_real] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_real @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_632_order__subst2,axiom,
! [A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,F: set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal,C: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ A2 @ B2 )
=> ( ( ord_le6787938422905777998nnreal @ ( F @ B2 ) @ C )
=> ( ! [X: set_Ex3793607809372303086nnreal,Y3: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ X @ Y3 )
=> ( ord_le6787938422905777998nnreal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le6787938422905777998nnreal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_633_order__subst2,axiom,
! [A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,F: set_Ex3793607809372303086nnreal > set_Extended_ereal,C: set_Extended_ereal] :
( ( ord_le6787938422905777998nnreal @ A2 @ B2 )
=> ( ( ord_le1644982726543182158_ereal @ ( F @ B2 ) @ C )
=> ( ! [X: set_Ex3793607809372303086nnreal,Y3: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ X @ Y3 )
=> ( ord_le1644982726543182158_ereal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le1644982726543182158_ereal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_634_order__subst2,axiom,
! [A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,F: set_Ex3793607809372303086nnreal > set_real,C: set_real] :
( ( ord_le6787938422905777998nnreal @ A2 @ B2 )
=> ( ( ord_less_eq_set_real @ ( F @ B2 ) @ C )
=> ( ! [X: set_Ex3793607809372303086nnreal,Y3: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ X @ Y3 )
=> ( ord_less_eq_set_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_635_order__eq__refl,axiom,
! [X2: set_Ex3793607809372303086nnreal,Y4: set_Ex3793607809372303086nnreal] :
( ( X2 = Y4 )
=> ( ord_le6787938422905777998nnreal @ X2 @ Y4 ) ) ).
% order_eq_refl
thf(fact_636_order__eq__refl,axiom,
! [X2: set_Extended_ereal,Y4: set_Extended_ereal] :
( ( X2 = Y4 )
=> ( ord_le1644982726543182158_ereal @ X2 @ Y4 ) ) ).
% order_eq_refl
thf(fact_637_order__eq__refl,axiom,
! [X2: set_real,Y4: set_real] :
( ( X2 = Y4 )
=> ( ord_less_eq_set_real @ X2 @ Y4 ) ) ).
% order_eq_refl
thf(fact_638_order__eq__refl,axiom,
! [X2: nat,Y4: nat] :
( ( X2 = Y4 )
=> ( ord_less_eq_nat @ X2 @ Y4 ) ) ).
% order_eq_refl
thf(fact_639_order__eq__refl,axiom,
! [X2: real > real,Y4: real > real] :
( ( X2 = Y4 )
=> ( ord_le6948328307412524503l_real @ X2 @ Y4 ) ) ).
% order_eq_refl
thf(fact_640_linorder__linear,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
| ( ord_less_eq_nat @ Y4 @ X2 ) ) ).
% linorder_linear
thf(fact_641_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_642_ord__eq__le__subst,axiom,
! [A2: nat,F: set_Ex3793607809372303086nnreal > nat,B2: set_Ex3793607809372303086nnreal,C: set_Ex3793607809372303086nnreal] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le6787938422905777998nnreal @ B2 @ C )
=> ( ! [X: set_Ex3793607809372303086nnreal,Y3: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_643_ord__eq__le__subst,axiom,
! [A2: nat,F: set_Extended_ereal > nat,B2: set_Extended_ereal,C: set_Extended_ereal] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le1644982726543182158_ereal @ B2 @ C )
=> ( ! [X: set_Extended_ereal,Y3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_644_ord__eq__le__subst,axiom,
! [A2: nat,F: set_real > nat,B2: set_real,C: set_real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_real @ B2 @ C )
=> ( ! [X: set_real,Y3: set_real] :
( ( ord_less_eq_set_real @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_645_ord__eq__le__subst,axiom,
! [A2: set_Ex3793607809372303086nnreal,F: nat > set_Ex3793607809372303086nnreal,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le6787938422905777998nnreal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le6787938422905777998nnreal @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_646_ord__eq__le__subst,axiom,
! [A2: set_Extended_ereal,F: nat > set_Extended_ereal,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le1644982726543182158_ereal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le1644982726543182158_ereal @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_647_ord__eq__le__subst,axiom,
! [A2: set_real,F: nat > set_real,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_648_ord__eq__le__subst,axiom,
! [A2: set_Ex3793607809372303086nnreal,F: set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,C: set_Ex3793607809372303086nnreal] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le6787938422905777998nnreal @ B2 @ C )
=> ( ! [X: set_Ex3793607809372303086nnreal,Y3: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ X @ Y3 )
=> ( ord_le6787938422905777998nnreal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le6787938422905777998nnreal @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_649_ord__eq__le__subst,axiom,
! [A2: set_Extended_ereal,F: set_Ex3793607809372303086nnreal > set_Extended_ereal,B2: set_Ex3793607809372303086nnreal,C: set_Ex3793607809372303086nnreal] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le6787938422905777998nnreal @ B2 @ C )
=> ( ! [X: set_Ex3793607809372303086nnreal,Y3: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ X @ Y3 )
=> ( ord_le1644982726543182158_ereal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le1644982726543182158_ereal @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_650_ord__eq__le__subst,axiom,
! [A2: set_real,F: set_Ex3793607809372303086nnreal > set_real,B2: set_Ex3793607809372303086nnreal,C: set_Ex3793607809372303086nnreal] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le6787938422905777998nnreal @ B2 @ C )
=> ( ! [X: set_Ex3793607809372303086nnreal,Y3: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ X @ Y3 )
=> ( ord_less_eq_set_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_651_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_652_ord__le__eq__subst,axiom,
! [A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,F: set_Ex3793607809372303086nnreal > nat,C: nat] :
( ( ord_le6787938422905777998nnreal @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_Ex3793607809372303086nnreal,Y3: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_653_ord__le__eq__subst,axiom,
! [A2: set_Extended_ereal,B2: set_Extended_ereal,F: set_Extended_ereal > nat,C: nat] :
( ( ord_le1644982726543182158_ereal @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_Extended_ereal,Y3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_654_ord__le__eq__subst,axiom,
! [A2: set_real,B2: set_real,F: set_real > nat,C: nat] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_real,Y3: set_real] :
( ( ord_less_eq_set_real @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_655_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_Ex3793607809372303086nnreal,C: set_Ex3793607809372303086nnreal] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le6787938422905777998nnreal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le6787938422905777998nnreal @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_656_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_Extended_ereal,C: set_Extended_ereal] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le1644982726543182158_ereal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le1644982726543182158_ereal @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_657_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_real,C: set_real] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_658_ord__le__eq__subst,axiom,
! [A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,F: set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal,C: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_Ex3793607809372303086nnreal,Y3: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ X @ Y3 )
=> ( ord_le6787938422905777998nnreal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le6787938422905777998nnreal @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_659_ord__le__eq__subst,axiom,
! [A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,F: set_Ex3793607809372303086nnreal > set_Extended_ereal,C: set_Extended_ereal] :
( ( ord_le6787938422905777998nnreal @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_Ex3793607809372303086nnreal,Y3: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ X @ Y3 )
=> ( ord_le1644982726543182158_ereal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le1644982726543182158_ereal @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_660_ord__le__eq__subst,axiom,
! [A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,F: set_Ex3793607809372303086nnreal > set_real,C: set_real] :
( ( ord_le6787938422905777998nnreal @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_Ex3793607809372303086nnreal,Y3: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ X @ Y3 )
=> ( ord_less_eq_set_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_661_linorder__le__cases,axiom,
! [X2: nat,Y4: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) ) ).
% linorder_le_cases
thf(fact_662_order__antisym__conv,axiom,
! [Y4: set_Ex3793607809372303086nnreal,X2: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ Y4 @ X2 )
=> ( ( ord_le6787938422905777998nnreal @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_663_order__antisym__conv,axiom,
! [Y4: set_Extended_ereal,X2: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ Y4 @ X2 )
=> ( ( ord_le1644982726543182158_ereal @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_664_order__antisym__conv,axiom,
! [Y4: set_real,X2: set_real] :
( ( ord_less_eq_set_real @ Y4 @ X2 )
=> ( ( ord_less_eq_set_real @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_665_order__antisym__conv,axiom,
! [Y4: nat,X2: nat] :
( ( ord_less_eq_nat @ Y4 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_666_order__antisym__conv,axiom,
! [Y4: real > real,X2: real > real] :
( ( ord_le6948328307412524503l_real @ Y4 @ X2 )
=> ( ( ord_le6948328307412524503l_real @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_667_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_668_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_669_zero__reorient,axiom,
! [X2: extend8495563244428889912nnreal] :
( ( zero_z7100319975126383169nnreal = X2 )
= ( X2 = zero_z7100319975126383169nnreal ) ) ).
% zero_reorient
thf(fact_670_borel__measurable__subalgebra,axiom,
! [N: sigma_measure_real,M: sigma_measure_real,F: real > real] :
( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ N ) @ ( sigma_sets_real @ M ) )
=> ( ( ( sigma_space_real @ N )
= ( sigma_space_real @ M ) )
=> ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ N @ borel_5078946678739801102l_real ) )
=> ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ) ).
% borel_measurable_subalgebra
thf(fact_671_Inf_OINF__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > extended_ereal,D2: nat > extended_ereal,Inf: set_Extended_ereal > extended_ereal] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( Inf @ ( image_4309273772856505399_ereal @ C2 @ A ) )
= ( Inf @ ( image_4309273772856505399_ereal @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_672_Inf_OINF__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > complex,D2: nat > complex,Inf: set_complex > complex] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( Inf @ ( image_nat_complex @ C2 @ A ) )
= ( Inf @ ( image_nat_complex @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_673_Inf_OINF__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > real,D2: nat > real,Inf: set_real > real] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( Inf @ ( image_nat_real @ C2 @ A ) )
= ( Inf @ ( image_nat_real @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_674_Inf_OINF__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > set_real,D2: nat > set_real,Inf: set_set_real > set_real] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( Inf @ ( image_nat_set_real @ C2 @ A ) )
= ( Inf @ ( image_nat_set_real @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_675_Inf_OINF__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > real > real,D2: nat > real > real,Inf: set_real_real > real > real] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( Inf @ ( image_nat_real_real @ C2 @ A ) )
= ( Inf @ ( image_nat_real_real @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_676_Sup_OSUP__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > extended_ereal,D2: nat > extended_ereal,Sup: set_Extended_ereal > extended_ereal] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( Sup @ ( image_4309273772856505399_ereal @ C2 @ A ) )
= ( Sup @ ( image_4309273772856505399_ereal @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_677_Sup_OSUP__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > complex,D2: nat > complex,Sup: set_complex > complex] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( Sup @ ( image_nat_complex @ C2 @ A ) )
= ( Sup @ ( image_nat_complex @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_678_Sup_OSUP__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > real,D2: nat > real,Sup: set_real > real] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( Sup @ ( image_nat_real @ C2 @ A ) )
= ( Sup @ ( image_nat_real @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_679_Sup_OSUP__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > set_real,D2: nat > set_real,Sup: set_set_real > set_real] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( Sup @ ( image_nat_set_real @ C2 @ A ) )
= ( Sup @ ( image_nat_set_real @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_680_Sup_OSUP__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > real > real,D2: nat > real > real,Sup: set_real_real > real > real] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( Sup @ ( image_nat_real_real @ C2 @ A ) )
= ( Sup @ ( image_nat_real_real @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_681_lebesgue__on__UNIV__eq,axiom,
( ( sigma_216592511309337194omplex @ ( comple7748144648682430500omplex @ lebesg5555883192292225345omplex ) @ top_top_set_complex )
= ( comple7748144648682430500omplex @ lebesg5555883192292225345omplex ) ) ).
% lebesgue_on_UNIV_eq
thf(fact_682_lebesgue__on__UNIV__eq,axiom,
( ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ top_top_set_real )
= ( comple3506806835435775778n_real @ lebesgue_lborel_real ) ) ).
% lebesgue_on_UNIV_eq
thf(fact_683_incseq__def,axiom,
! [X5: nat > extended_ereal] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ X5 )
= ( ! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_le1083603963089353582_ereal @ ( X5 @ M5 ) @ ( X5 @ N4 ) ) ) ) ) ).
% incseq_def
thf(fact_684_incseq__def,axiom,
! [X5: nat > extend8495563244428889912nnreal] :
( ( monoto2291723841412853873nnreal @ top_top_set_nat @ ord_less_eq_nat @ ord_le3935885782089961368nnreal @ X5 )
= ( ! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_le3935885782089961368nnreal @ ( X5 @ M5 ) @ ( X5 @ N4 ) ) ) ) ) ).
% incseq_def
thf(fact_685_incseq__def,axiom,
! [X5: nat > set_Ex3793607809372303086nnreal] :
( ( monoto4660286046138248231nnreal @ top_top_set_nat @ ord_less_eq_nat @ ord_le6787938422905777998nnreal @ X5 )
= ( ! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_le6787938422905777998nnreal @ ( X5 @ M5 ) @ ( X5 @ N4 ) ) ) ) ) ).
% incseq_def
thf(fact_686_incseq__def,axiom,
! [X5: nat > set_Extended_ereal] :
( ( monoto6788471982328799797_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1644982726543182158_ereal @ X5 )
= ( ! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_le1644982726543182158_ereal @ ( X5 @ M5 ) @ ( X5 @ N4 ) ) ) ) ) ).
% incseq_def
thf(fact_687_incseq__def,axiom,
! [X5: nat > set_real] :
( ( monoto7274299666542614427t_real @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_real @ X5 )
= ( ! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_less_eq_set_real @ ( X5 @ M5 ) @ ( X5 @ N4 ) ) ) ) ) ).
% incseq_def
thf(fact_688_incseq__def,axiom,
! [X5: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ X5 )
= ( ! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_less_eq_nat @ ( X5 @ M5 ) @ ( X5 @ N4 ) ) ) ) ) ).
% incseq_def
thf(fact_689_incseq__def,axiom,
! [X5: nat > real > real] :
( ( monoto2824216093323351088l_real @ top_top_set_nat @ ord_less_eq_nat @ ord_le6948328307412524503l_real @ X5 )
= ( ! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_le6948328307412524503l_real @ ( X5 @ M5 ) @ ( X5 @ N4 ) ) ) ) ) ).
% incseq_def
thf(fact_690_incseqD,axiom,
! [F: nat > extended_ereal,I: nat,J: nat] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ F )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_le1083603963089353582_ereal @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% incseqD
thf(fact_691_incseqD,axiom,
! [F: nat > extend8495563244428889912nnreal,I: nat,J: nat] :
( ( monoto2291723841412853873nnreal @ top_top_set_nat @ ord_less_eq_nat @ ord_le3935885782089961368nnreal @ F )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_le3935885782089961368nnreal @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% incseqD
thf(fact_692_incseqD,axiom,
! [F: nat > set_Ex3793607809372303086nnreal,I: nat,J: nat] :
( ( monoto4660286046138248231nnreal @ top_top_set_nat @ ord_less_eq_nat @ ord_le6787938422905777998nnreal @ F )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_le6787938422905777998nnreal @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% incseqD
thf(fact_693_incseqD,axiom,
! [F: nat > set_Extended_ereal,I: nat,J: nat] :
( ( monoto6788471982328799797_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1644982726543182158_ereal @ F )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_le1644982726543182158_ereal @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% incseqD
thf(fact_694_incseqD,axiom,
! [F: nat > set_real,I: nat,J: nat] :
( ( monoto7274299666542614427t_real @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_real @ F )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_set_real @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% incseqD
thf(fact_695_incseqD,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% incseqD
thf(fact_696_incseqD,axiom,
! [F: nat > real > real,I: nat,J: nat] :
( ( monoto2824216093323351088l_real @ top_top_set_nat @ ord_less_eq_nat @ ord_le6948328307412524503l_real @ F )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_le6948328307412524503l_real @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% incseqD
thf(fact_697_le0,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% le0
thf(fact_698_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_699_iso__tuple__UNIV__I,axiom,
! [X2: real > real] : ( member_real_real @ X2 @ top_to2071711978144146653l_real ) ).
% iso_tuple_UNIV_I
thf(fact_700_iso__tuple__UNIV__I,axiom,
! [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).
% iso_tuple_UNIV_I
thf(fact_701_iso__tuple__UNIV__I,axiom,
! [X2: set_real] : ( member_set_real @ X2 @ top_top_set_set_real ) ).
% iso_tuple_UNIV_I
thf(fact_702_iso__tuple__UNIV__I,axiom,
! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_703_iso__tuple__UNIV__I,axiom,
! [X2: real] : ( member_real @ X2 @ top_top_set_real ) ).
% iso_tuple_UNIV_I
thf(fact_704_iso__tuple__UNIV__I,axiom,
! [X2: complex] : ( member_complex @ X2 @ top_top_set_complex ) ).
% iso_tuple_UNIV_I
thf(fact_705_iso__tuple__UNIV__I,axiom,
! [X2: extended_ereal] : ( member2350847679896131959_ereal @ X2 @ top_to5683747375963461374_ereal ) ).
% iso_tuple_UNIV_I
thf(fact_706_increasingD,axiom,
! [M: set_se4580700918925141924nnreal,F: set_Ex3793607809372303086nnreal > nat,X2: set_Ex3793607809372303086nnreal,Y4: set_Ex3793607809372303086nnreal] :
( ( measur2312693293899776957al_nat @ M @ F )
=> ( ( ord_le6787938422905777998nnreal @ X2 @ Y4 )
=> ( ( member603777416030116741nnreal @ X2 @ M )
=> ( ( member603777416030116741nnreal @ Y4 @ M )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% increasingD
thf(fact_707_increasingD,axiom,
! [M: set_se6634062954251873166_ereal,F: set_Extended_ereal > nat,X2: set_Extended_ereal,Y4: set_Extended_ereal] :
( ( measur1341731035474441003al_nat @ M @ F )
=> ( ( ord_le1644982726543182158_ereal @ X2 @ Y4 )
=> ( ( member5519481007471526743_ereal @ X2 @ M )
=> ( ( member5519481007471526743_ereal @ Y4 @ M )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% increasingD
thf(fact_708_increasingD,axiom,
! [M: set_set_real,F: set_real > nat,X2: set_real,Y4: set_real] :
( ( measur1249124728302858545al_nat @ M @ F )
=> ( ( ord_less_eq_set_real @ X2 @ Y4 )
=> ( ( member_set_real @ X2 @ M )
=> ( ( member_set_real @ Y4 @ M )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% increasingD
thf(fact_709_increasingD,axiom,
! [M: set_se4580700918925141924nnreal,F: set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal,X2: set_Ex3793607809372303086nnreal,Y4: set_Ex3793607809372303086nnreal] :
( ( measur3398826073623134939nnreal @ M @ F )
=> ( ( ord_le6787938422905777998nnreal @ X2 @ Y4 )
=> ( ( member603777416030116741nnreal @ X2 @ M )
=> ( ( member603777416030116741nnreal @ Y4 @ M )
=> ( ord_le6787938422905777998nnreal @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% increasingD
thf(fact_710_increasingD,axiom,
! [M: set_se4580700918925141924nnreal,F: set_Ex3793607809372303086nnreal > set_Extended_ereal,X2: set_Ex3793607809372303086nnreal,Y4: set_Ex3793607809372303086nnreal] :
( ( measur8371784536020471425_ereal @ M @ F )
=> ( ( ord_le6787938422905777998nnreal @ X2 @ Y4 )
=> ( ( member603777416030116741nnreal @ X2 @ M )
=> ( ( member603777416030116741nnreal @ Y4 @ M )
=> ( ord_le1644982726543182158_ereal @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% increasingD
thf(fact_711_increasingD,axiom,
! [M: set_se4580700918925141924nnreal,F: set_Ex3793607809372303086nnreal > set_real,X2: set_Ex3793607809372303086nnreal,Y4: set_Ex3793607809372303086nnreal] :
( ( measur335931744288126031t_real @ M @ F )
=> ( ( ord_le6787938422905777998nnreal @ X2 @ Y4 )
=> ( ( member603777416030116741nnreal @ X2 @ M )
=> ( ( member603777416030116741nnreal @ Y4 @ M )
=> ( ord_less_eq_set_real @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% increasingD
thf(fact_712_increasingD,axiom,
! [M: set_se6634062954251873166_ereal,F: set_Extended_ereal > set_Ex3793607809372303086nnreal,X2: set_Extended_ereal,Y4: set_Extended_ereal] :
( ( measur9031206749974399561nnreal @ M @ F )
=> ( ( ord_le1644982726543182158_ereal @ X2 @ Y4 )
=> ( ( member5519481007471526743_ereal @ X2 @ M )
=> ( ( member5519481007471526743_ereal @ Y4 @ M )
=> ( ord_le6787938422905777998nnreal @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% increasingD
thf(fact_713_increasingD,axiom,
! [M: set_se6634062954251873166_ereal,F: set_Extended_ereal > set_Extended_ereal,X2: set_Extended_ereal,Y4: set_Extended_ereal] :
( ( measur4166458732864689043_ereal @ M @ F )
=> ( ( ord_le1644982726543182158_ereal @ X2 @ Y4 )
=> ( ( member5519481007471526743_ereal @ X2 @ M )
=> ( ( member5519481007471526743_ereal @ Y4 @ M )
=> ( ord_le1644982726543182158_ereal @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% increasingD
thf(fact_714_increasingD,axiom,
! [M: set_se6634062954251873166_ereal,F: set_Extended_ereal > set_real,X2: set_Extended_ereal,Y4: set_Extended_ereal] :
( ( measur2974261746804184253t_real @ M @ F )
=> ( ( ord_le1644982726543182158_ereal @ X2 @ Y4 )
=> ( ( member5519481007471526743_ereal @ X2 @ M )
=> ( ( member5519481007471526743_ereal @ Y4 @ M )
=> ( ord_less_eq_set_real @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% increasingD
thf(fact_715_increasingD,axiom,
! [M: set_set_real,F: set_real > set_Ex3793607809372303086nnreal,X2: set_real,Y4: set_real] :
( ( measur6783772439937024079nnreal @ M @ F )
=> ( ( ord_less_eq_set_real @ X2 @ Y4 )
=> ( ( member_set_real @ X2 @ M )
=> ( ( member_set_real @ Y4 @ M )
=> ( ord_le6787938422905777998nnreal @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% increasingD
thf(fact_716_increasing__def,axiom,
( measur2312693293899776957al_nat
= ( ^ [M2: set_se4580700918925141924nnreal,Mu: set_Ex3793607809372303086nnreal > nat] :
! [X4: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X4 @ M2 )
=> ! [Y2: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ Y2 @ M2 )
=> ( ( ord_le6787938422905777998nnreal @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( Mu @ X4 ) @ ( Mu @ Y2 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_717_increasing__def,axiom,
( measur1341731035474441003al_nat
= ( ^ [M2: set_se6634062954251873166_ereal,Mu: set_Extended_ereal > nat] :
! [X4: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ X4 @ M2 )
=> ! [Y2: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ Y2 @ M2 )
=> ( ( ord_le1644982726543182158_ereal @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( Mu @ X4 ) @ ( Mu @ Y2 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_718_increasing__def,axiom,
( measur1249124728302858545al_nat
= ( ^ [M2: set_set_real,Mu: set_real > nat] :
! [X4: set_real] :
( ( member_set_real @ X4 @ M2 )
=> ! [Y2: set_real] :
( ( member_set_real @ Y2 @ M2 )
=> ( ( ord_less_eq_set_real @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( Mu @ X4 ) @ ( Mu @ Y2 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_719_increasing__def,axiom,
( measur3398826073623134939nnreal
= ( ^ [M2: set_se4580700918925141924nnreal,Mu: set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal] :
! [X4: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X4 @ M2 )
=> ! [Y2: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ Y2 @ M2 )
=> ( ( ord_le6787938422905777998nnreal @ X4 @ Y2 )
=> ( ord_le6787938422905777998nnreal @ ( Mu @ X4 ) @ ( Mu @ Y2 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_720_increasing__def,axiom,
( measur8371784536020471425_ereal
= ( ^ [M2: set_se4580700918925141924nnreal,Mu: set_Ex3793607809372303086nnreal > set_Extended_ereal] :
! [X4: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X4 @ M2 )
=> ! [Y2: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ Y2 @ M2 )
=> ( ( ord_le6787938422905777998nnreal @ X4 @ Y2 )
=> ( ord_le1644982726543182158_ereal @ ( Mu @ X4 ) @ ( Mu @ Y2 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_721_increasing__def,axiom,
( measur335931744288126031t_real
= ( ^ [M2: set_se4580700918925141924nnreal,Mu: set_Ex3793607809372303086nnreal > set_real] :
! [X4: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X4 @ M2 )
=> ! [Y2: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ Y2 @ M2 )
=> ( ( ord_le6787938422905777998nnreal @ X4 @ Y2 )
=> ( ord_less_eq_set_real @ ( Mu @ X4 ) @ ( Mu @ Y2 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_722_increasing__def,axiom,
( measur9031206749974399561nnreal
= ( ^ [M2: set_se6634062954251873166_ereal,Mu: set_Extended_ereal > set_Ex3793607809372303086nnreal] :
! [X4: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ X4 @ M2 )
=> ! [Y2: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ Y2 @ M2 )
=> ( ( ord_le1644982726543182158_ereal @ X4 @ Y2 )
=> ( ord_le6787938422905777998nnreal @ ( Mu @ X4 ) @ ( Mu @ Y2 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_723_increasing__def,axiom,
( measur4166458732864689043_ereal
= ( ^ [M2: set_se6634062954251873166_ereal,Mu: set_Extended_ereal > set_Extended_ereal] :
! [X4: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ X4 @ M2 )
=> ! [Y2: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ Y2 @ M2 )
=> ( ( ord_le1644982726543182158_ereal @ X4 @ Y2 )
=> ( ord_le1644982726543182158_ereal @ ( Mu @ X4 ) @ ( Mu @ Y2 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_724_increasing__def,axiom,
( measur2974261746804184253t_real
= ( ^ [M2: set_se6634062954251873166_ereal,Mu: set_Extended_ereal > set_real] :
! [X4: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ X4 @ M2 )
=> ! [Y2: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ Y2 @ M2 )
=> ( ( ord_le1644982726543182158_ereal @ X4 @ Y2 )
=> ( ord_less_eq_set_real @ ( Mu @ X4 ) @ ( Mu @ Y2 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_725_increasing__def,axiom,
( measur6783772439937024079nnreal
= ( ^ [M2: set_set_real,Mu: set_real > set_Ex3793607809372303086nnreal] :
! [X4: set_real] :
( ( member_set_real @ X4 @ M2 )
=> ! [Y2: set_real] :
( ( member_set_real @ Y2 @ M2 )
=> ( ( ord_less_eq_set_real @ X4 @ Y2 )
=> ( ord_le6787938422905777998nnreal @ ( Mu @ X4 ) @ ( Mu @ Y2 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_726_sets__range,axiom,
! [A: nat > set_real,I3: set_nat,M: sigma_measure_real,I: nat] :
( ( ord_le3558479182127378552t_real @ ( image_nat_set_real @ A @ I3 ) @ ( sigma_sets_real @ M ) )
=> ( ( member_nat @ I @ I3 )
=> ( member_set_real @ ( A @ I ) @ ( sigma_sets_real @ M ) ) ) ) ).
% sets_range
thf(fact_727_sets__range,axiom,
! [A: ( real > real ) > set_real,I3: set_real_real,M: sigma_measure_real,I: real > real] :
( ( ord_le3558479182127378552t_real @ ( image_6663718904102175840t_real @ A @ I3 ) @ ( sigma_sets_real @ M ) )
=> ( ( member_real_real @ I @ I3 )
=> ( member_set_real @ ( A @ I ) @ ( sigma_sets_real @ M ) ) ) ) ).
% sets_range
thf(fact_728_sets__range,axiom,
! [A: real > set_real,I3: set_real,M: sigma_measure_real,I: real] :
( ( ord_le3558479182127378552t_real @ ( image_real_set_real @ A @ I3 ) @ ( sigma_sets_real @ M ) )
=> ( ( member_real @ I @ I3 )
=> ( member_set_real @ ( A @ I ) @ ( sigma_sets_real @ M ) ) ) ) ).
% sets_range
thf(fact_729_sets__range,axiom,
! [A: $o > set_real,I3: set_o,M: sigma_measure_real,I: $o] :
( ( ord_le3558479182127378552t_real @ ( image_o_set_real @ A @ I3 ) @ ( sigma_sets_real @ M ) )
=> ( ( member_o @ I @ I3 )
=> ( member_set_real @ ( A @ I ) @ ( sigma_sets_real @ M ) ) ) ) ).
% sets_range
thf(fact_730_sets__range,axiom,
! [A: set_real > set_real,I3: set_set_real,M: sigma_measure_real,I: set_real] :
( ( ord_le3558479182127378552t_real @ ( image_2436557299294012491t_real @ A @ I3 ) @ ( sigma_sets_real @ M ) )
=> ( ( member_set_real @ I @ I3 )
=> ( member_set_real @ ( A @ I ) @ ( sigma_sets_real @ M ) ) ) ) ).
% sets_range
thf(fact_731_DEADID_Oin__rel,axiom,
( ( ^ [Y: real > real,Z2: real > real] : ( Y = Z2 ) )
= ( ^ [A5: real > real,B5: real > real] :
? [Z3: real > real] :
( ( member_real_real @ Z3 @ top_to2071711978144146653l_real )
& ( ( id_real_real @ Z3 )
= A5 )
& ( ( id_real_real @ Z3 )
= B5 ) ) ) ) ).
% DEADID.in_rel
thf(fact_732_DEADID_Oin__rel,axiom,
( ( ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
= ( ^ [A5: $o,B5: $o] :
? [Z3: $o] :
( ( member_o @ Z3 @ top_top_set_o )
& ( ( id_o @ Z3 )
= A5 )
& ( ( id_o @ Z3 )
= B5 ) ) ) ) ).
% DEADID.in_rel
thf(fact_733_DEADID_Oin__rel,axiom,
( ( ^ [Y: set_real,Z2: set_real] : ( Y = Z2 ) )
= ( ^ [A5: set_real,B5: set_real] :
? [Z3: set_real] :
( ( member_set_real @ Z3 @ top_top_set_set_real )
& ( ( id_set_real @ Z3 )
= A5 )
& ( ( id_set_real @ Z3 )
= B5 ) ) ) ) ).
% DEADID.in_rel
thf(fact_734_DEADID_Oin__rel,axiom,
( ( ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
= ( ^ [A5: nat,B5: nat] :
? [Z3: nat] :
( ( member_nat @ Z3 @ top_top_set_nat )
& ( ( id_nat @ Z3 )
= A5 )
& ( ( id_nat @ Z3 )
= B5 ) ) ) ) ).
% DEADID.in_rel
thf(fact_735_DEADID_Oin__rel,axiom,
( ( ^ [Y: real,Z2: real] : ( Y = Z2 ) )
= ( ^ [A5: real,B5: real] :
? [Z3: real] :
( ( member_real @ Z3 @ top_top_set_real )
& ( ( id_real @ Z3 )
= A5 )
& ( ( id_real @ Z3 )
= B5 ) ) ) ) ).
% DEADID.in_rel
thf(fact_736_DEADID_Oin__rel,axiom,
( ( ^ [Y: complex,Z2: complex] : ( Y = Z2 ) )
= ( ^ [A5: complex,B5: complex] :
? [Z3: complex] :
( ( member_complex @ Z3 @ top_top_set_complex )
& ( ( id_complex @ Z3 )
= A5 )
& ( ( id_complex @ Z3 )
= B5 ) ) ) ) ).
% DEADID.in_rel
thf(fact_737_DEADID_Oin__rel,axiom,
( ( ^ [Y: extended_ereal,Z2: extended_ereal] : ( Y = Z2 ) )
= ( ^ [A5: extended_ereal,B5: extended_ereal] :
? [Z3: extended_ereal] :
( ( member2350847679896131959_ereal @ Z3 @ top_to5683747375963461374_ereal )
& ( ( id_Extended_ereal @ Z3 )
= A5 )
& ( ( id_Extended_ereal @ Z3 )
= B5 ) ) ) ) ).
% DEADID.in_rel
thf(fact_738_space__sup__measure_H,axiom,
! [B: sigma_measure_real,A: sigma_measure_real] :
( ( ( sigma_sets_real @ B )
= ( sigma_sets_real @ A ) )
=> ( ( sigma_space_real @ ( measur2147279183506585690e_real @ A @ B ) )
= ( sigma_space_real @ A ) ) ) ).
% space_sup_measure'
thf(fact_739_sets__sup__measure_H,axiom,
! [B: sigma_measure_real,A: sigma_measure_real] :
( ( ( sigma_sets_real @ B )
= ( sigma_sets_real @ A ) )
=> ( ( sigma_sets_real @ ( measur2147279183506585690e_real @ A @ B ) )
= ( sigma_sets_real @ A ) ) ) ).
% sets_sup_measure'
thf(fact_740_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_741_top__set__def,axiom,
( top_top_set_real
= ( collect_real @ top_top_real_o ) ) ).
% top_set_def
thf(fact_742_top__set__def,axiom,
( top_top_set_complex
= ( collect_complex @ top_top_complex_o ) ) ).
% top_set_def
thf(fact_743_top__set__def,axiom,
( top_to5683747375963461374_ereal
= ( collec5835592288176408249_ereal @ top_to6999531812125281119real_o ) ) ).
% top_set_def
thf(fact_744_sets__Ball,axiom,
! [I3: set_real_real,A: ( real > real ) > set_real,M: ( real > real ) > sigma_measure_real,I: real > real] :
( ! [X: real > real] :
( ( member_real_real @ X @ I3 )
=> ( member_set_real @ ( A @ X ) @ ( sigma_sets_real @ ( M @ X ) ) ) )
=> ( ( member_real_real @ I @ I3 )
=> ( member_set_real @ ( A @ I ) @ ( sigma_sets_real @ ( M @ I ) ) ) ) ) ).
% sets_Ball
thf(fact_745_sets__Ball,axiom,
! [I3: set_real,A: real > set_real,M: real > sigma_measure_real,I: real] :
( ! [X: real] :
( ( member_real @ X @ I3 )
=> ( member_set_real @ ( A @ X ) @ ( sigma_sets_real @ ( M @ X ) ) ) )
=> ( ( member_real @ I @ I3 )
=> ( member_set_real @ ( A @ I ) @ ( sigma_sets_real @ ( M @ I ) ) ) ) ) ).
% sets_Ball
thf(fact_746_sets__Ball,axiom,
! [I3: set_o,A: $o > set_real,M: $o > sigma_measure_real,I: $o] :
( ! [X: $o] :
( ( member_o @ X @ I3 )
=> ( member_set_real @ ( A @ X ) @ ( sigma_sets_real @ ( M @ X ) ) ) )
=> ( ( member_o @ I @ I3 )
=> ( member_set_real @ ( A @ I ) @ ( sigma_sets_real @ ( M @ I ) ) ) ) ) ).
% sets_Ball
thf(fact_747_sets__Ball,axiom,
! [I3: set_set_real,A: set_real > set_real,M: set_real > sigma_measure_real,I: set_real] :
( ! [X: set_real] :
( ( member_set_real @ X @ I3 )
=> ( member_set_real @ ( A @ X ) @ ( sigma_sets_real @ ( M @ X ) ) ) )
=> ( ( member_set_real @ I @ I3 )
=> ( member_set_real @ ( A @ I ) @ ( sigma_sets_real @ ( M @ I ) ) ) ) ) ).
% sets_Ball
thf(fact_748_le__refl,axiom,
! [N3: nat] : ( ord_less_eq_nat @ N3 @ N3 ) ).
% le_refl
thf(fact_749_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_750_eq__imp__le,axiom,
! [M6: nat,N3: nat] :
( ( M6 = N3 )
=> ( ord_less_eq_nat @ M6 @ N3 ) ) ).
% eq_imp_le
thf(fact_751_le__antisym,axiom,
! [M6: nat,N3: nat] :
( ( ord_less_eq_nat @ M6 @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ M6 )
=> ( M6 = N3 ) ) ) ).
% le_antisym
thf(fact_752_nat__le__linear,axiom,
! [M6: nat,N3: nat] :
( ( ord_less_eq_nat @ M6 @ N3 )
| ( ord_less_eq_nat @ N3 @ M6 ) ) ).
% nat_le_linear
thf(fact_753_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B2 ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_754_less__eq__nat_Osimps_I1_J,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% less_eq_nat.simps(1)
thf(fact_755_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_756_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_757_le__0__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_758_borel__measurable__mono,axiom,
! [F: real > real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) ) ).
% borel_measurable_mono
thf(fact_759_ex__nat__less,axiom,
! [N3: nat,P: nat > $o] :
( ( ? [M5: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
& ( P @ M5 ) ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) )
& ( P @ X4 ) ) ) ) ).
% ex_nat_less
thf(fact_760_all__nat__less,axiom,
! [N3: nat,P: nat > $o] :
( ( ! [M5: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( P @ M5 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) )
=> ( P @ X4 ) ) ) ) ).
% all_nat_less
thf(fact_761_incseq__imp__monoseq,axiom,
! [X5: nat > extended_ereal] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ X5 )
=> ( topolo608505905947791073_ereal @ X5 ) ) ).
% incseq_imp_monoseq
thf(fact_762_incseq__imp__monoseq,axiom,
! [X5: nat > extend8495563244428889912nnreal] :
( ( monoto2291723841412853873nnreal @ top_top_set_nat @ ord_less_eq_nat @ ord_le3935885782089961368nnreal @ X5 )
=> ( topolo2569500529754793189nnreal @ X5 ) ) ).
% incseq_imp_monoseq
thf(fact_763_incseq__imp__monoseq,axiom,
! [X5: nat > set_Ex3793607809372303086nnreal] :
( ( monoto4660286046138248231nnreal @ top_top_set_nat @ ord_less_eq_nat @ ord_le6787938422905777998nnreal @ X5 )
=> ( topolo1981301258947743899nnreal @ X5 ) ) ).
% incseq_imp_monoseq
thf(fact_764_incseq__imp__monoseq,axiom,
! [X5: nat > set_Extended_ereal] :
( ( monoto6788471982328799797_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1644982726543182158_ereal @ X5 )
=> ( topolo6736312753545056449_ereal @ X5 ) ) ).
% incseq_imp_monoseq
thf(fact_765_incseq__imp__monoseq,axiom,
! [X5: nat > set_real] :
( ( monoto7274299666542614427t_real @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_real @ X5 )
=> ( topolo2489691266198938127t_real @ X5 ) ) ).
% incseq_imp_monoseq
thf(fact_766_incseq__imp__monoseq,axiom,
! [X5: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ X5 )
=> ( topolo4902158794631467389eq_nat @ X5 ) ) ).
% incseq_imp_monoseq
thf(fact_767_incseq__imp__monoseq,axiom,
! [X5: nat > real > real] :
( ( monoto2824216093323351088l_real @ top_top_set_nat @ ord_less_eq_nat @ ord_le6948328307412524503l_real @ X5 )
=> ( topolo6106433626167785124l_real @ X5 ) ) ).
% incseq_imp_monoseq
thf(fact_768_continuous__imp__measurable__on__sets__lebesgue,axiom,
! [S: set_real,F: real > real] :
( ( topolo5044208981011980120l_real @ S @ F )
=> ( ( member_set_real @ S @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) ) )
=> ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ S ) @ borel_5078946678739801102l_real ) ) ) ) ).
% continuous_imp_measurable_on_sets_lebesgue
thf(fact_769_sets__uniform__count__measure__eq__UNIV_I2_J,axiom,
( top_top_set_set_nat
= ( sigma_sets_nat @ ( nonneg7031465154080143958re_nat @ top_top_set_nat ) ) ) ).
% sets_uniform_count_measure_eq_UNIV(2)
thf(fact_770_sets__uniform__count__measure__eq__UNIV_I2_J,axiom,
( top_top_set_set_real
= ( sigma_sets_real @ ( nonneg387815094551837234e_real @ top_top_set_real ) ) ) ).
% sets_uniform_count_measure_eq_UNIV(2)
thf(fact_771_sets__uniform__count__measure__eq__UNIV_I2_J,axiom,
( top_to4650676778325599690omplex
= ( sigma_sets_complex @ ( nonneg5466461028601436980omplex @ top_top_set_complex ) ) ) ).
% sets_uniform_count_measure_eq_UNIV(2)
thf(fact_772_sets__uniform__count__measure__eq__UNIV_I2_J,axiom,
( top_to4757929550322229470_ereal
= ( sigma_6858886609720962221_ereal @ ( nonneg528171625637462984_ereal @ top_to5683747375963461374_ereal ) ) ) ).
% sets_uniform_count_measure_eq_UNIV(2)
thf(fact_773_sets__uniform__count__measure__eq__UNIV_I1_J,axiom,
( ( sigma_sets_nat @ ( nonneg7031465154080143958re_nat @ top_top_set_nat ) )
= top_top_set_set_nat ) ).
% sets_uniform_count_measure_eq_UNIV(1)
thf(fact_774_sets__uniform__count__measure__eq__UNIV_I1_J,axiom,
( ( sigma_sets_real @ ( nonneg387815094551837234e_real @ top_top_set_real ) )
= top_top_set_set_real ) ).
% sets_uniform_count_measure_eq_UNIV(1)
thf(fact_775_sets__uniform__count__measure__eq__UNIV_I1_J,axiom,
( ( sigma_sets_complex @ ( nonneg5466461028601436980omplex @ top_top_set_complex ) )
= top_to4650676778325599690omplex ) ).
% sets_uniform_count_measure_eq_UNIV(1)
thf(fact_776_sets__uniform__count__measure__eq__UNIV_I1_J,axiom,
( ( sigma_6858886609720962221_ereal @ ( nonneg528171625637462984_ereal @ top_to5683747375963461374_ereal ) )
= top_to4757929550322229470_ereal ) ).
% sets_uniform_count_measure_eq_UNIV(1)
thf(fact_777__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062g_O_A_092_060lbrakk_062incseq_Ag_059_A_092_060And_062i_O_Asimple__function_A_Ilebesgue__on_A_123a_O_Ob_125_J_A_Ig_Ai_J_059_A_092_060forall_062x_O_Abdd__above_A_Irange_A_I_092_060lambda_062i_O_Ag_Ai_Ax_J_J_059_A_092_060forall_062i_Ax_O_A0_A_092_060le_062_Ag_Ai_Ax_059_Af_A_061_ASup_A_Irange_Ag_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [G3: nat > real > real] :
( ( monoto2824216093323351088l_real @ top_top_set_nat @ ord_less_eq_nat @ ord_le6948328307412524503l_real @ G3 )
=> ( ! [I2: nat] : ( nonneg485563716852976898l_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ ( set_or1222579329274155063t_real @ a @ b ) ) @ ( G3 @ I2 ) )
=> ( ! [X3: real] :
( condit7084745239686109512e_real
@ ( image_nat_real
@ ^ [I4: nat] : ( G3 @ I4 @ X3 )
@ top_top_set_nat ) )
=> ( ! [I2: nat,X3: real] : ( ord_less_eq_real @ zero_zero_real @ ( G3 @ I2 @ X3 ) )
=> ( f
!= ( comple8933463103962640202l_real @ ( image_nat_real_real @ G3 @ top_top_set_nat ) ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>g. \<lbrakk>incseq g; \<And>i. simple_function (lebesgue_on {a..b}) (g i); \<forall>x. bdd_above (range (\<lambda>i. g i x)); \<forall>i x. 0 \<le> g i x; f = Sup (range g)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_778_integrable__mono__on__nonneg,axiom,
! [A2: real,B2: real,F: real > real] :
( ( monoto4017252874604999745l_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) )
=> ( bochne3340023020068487468l_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ ( set_or1222579329274155063t_real @ A2 @ B2 ) ) @ F ) ) ) ).
% integrable_mono_on_nonneg
thf(fact_779_bdd,axiom,
! [X3: real] :
( condit7084745239686109512e_real
@ ( image_nat_real
@ ^ [I4: nat] : ( g @ I4 @ X3 )
@ top_top_set_nat ) ) ).
% bdd
thf(fact_780_fsup,axiom,
( f
= ( comple8933463103962640202l_real @ ( image_nat_real_real @ g @ top_top_set_nat ) ) ) ).
% fsup
thf(fact_781_simple__function__const,axiom,
! [M: sigma_measure_real,C: real] :
( nonneg485563716852976898l_real @ M
@ ^ [X4: real] : C ) ).
% simple_function_const
thf(fact_782_SUP__identity__eq,axiom,
! [A: set_real] :
( ( comple1385675409528146559p_real
@ ( image_real_real
@ ^ [X4: real] : X4
@ A ) )
= ( comple1385675409528146559p_real @ A ) ) ).
% SUP_identity_eq
thf(fact_783_SUP__identity__eq,axiom,
! [A: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X4: $o] : X4
@ A ) )
= ( complete_Sup_Sup_o @ A ) ) ).
% SUP_identity_eq
thf(fact_784_SUP__identity__eq,axiom,
! [A: set_set_real] :
( ( comple3096694443085538997t_real
@ ( image_2436557299294012491t_real
@ ^ [X4: set_real] : X4
@ A ) )
= ( comple3096694443085538997t_real @ A ) ) ).
% SUP_identity_eq
thf(fact_785_SUP__identity__eq,axiom,
! [A: set_nat] :
( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X4: nat] : X4
@ A ) )
= ( complete_Sup_Sup_nat @ A ) ) ).
% SUP_identity_eq
thf(fact_786_SUP__identity__eq,axiom,
! [A: set_real_real] :
( ( comple8933463103962640202l_real
@ ( image_745864523092522741l_real
@ ^ [X4: real > real] : X4
@ A ) )
= ( comple8933463103962640202l_real @ A ) ) ).
% SUP_identity_eq
thf(fact_787_Sup__apply,axiom,
( comple8933463103962640202l_real
= ( ^ [A3: set_real_real,X4: real] :
( comple1385675409528146559p_real
@ ( image_real_real_real
@ ^ [F2: real > real] : ( F2 @ X4 )
@ A3 ) ) ) ) ).
% Sup_apply
thf(fact_788_Sup__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Sup_UNIV
thf(fact_789_Sup__UNIV,axiom,
( ( comple8424636186594484919omplex @ top_to4650676778325599690omplex )
= top_top_set_complex ) ).
% Sup_UNIV
thf(fact_790_Sup__UNIV,axiom,
( ( comple6814414086264997003nnreal @ top_to7994903218803871134nnreal )
= top_to1496364449551166952nnreal ) ).
% Sup_UNIV
thf(fact_791_Sup__UNIV,axiom,
( ( comple4319282863272126363_ereal @ top_to4757929550322229470_ereal )
= top_to5683747375963461374_ereal ) ).
% Sup_UNIV
thf(fact_792_Sup__UNIV,axiom,
( ( comple8415311339701865915_ereal @ top_to5683747375963461374_ereal )
= top_to6662034908053899550_ereal ) ).
% Sup_UNIV
thf(fact_793_Sup__UNIV,axiom,
( ( complete_Sup_Sup_o @ top_top_set_o )
= top_top_o ) ).
% Sup_UNIV
thf(fact_794_Sup__UNIV,axiom,
( ( comple3096694443085538997t_real @ top_top_set_set_real )
= top_top_set_real ) ).
% Sup_UNIV
thf(fact_795_Sup__atLeastAtMost,axiom,
! [X2: set_Ex3793607809372303086nnreal,Y4: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ X2 @ Y4 )
=> ( ( comple4226387801268262977nnreal @ ( set_or4467169092781218297nnreal @ X2 @ Y4 ) )
= Y4 ) ) ).
% Sup_atLeastAtMost
thf(fact_796_Sup__atLeastAtMost,axiom,
! [X2: set_Extended_ereal,Y4: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ X2 @ Y4 )
=> ( ( comple4319282863272126363_ereal @ ( set_or814338393153804259_ereal @ X2 @ Y4 ) )
= Y4 ) ) ).
% Sup_atLeastAtMost
thf(fact_797_Sup__atLeastAtMost,axiom,
! [X2: $o,Y4: $o] :
( ( ord_less_eq_o @ X2 @ Y4 )
=> ( ( complete_Sup_Sup_o @ ( set_or8904488021354931149Most_o @ X2 @ Y4 ) )
= Y4 ) ) ).
% Sup_atLeastAtMost
thf(fact_798_Sup__atLeastAtMost,axiom,
! [X2: set_real,Y4: set_real] :
( ( ord_less_eq_set_real @ X2 @ Y4 )
=> ( ( comple3096694443085538997t_real @ ( set_or7743017856606604397t_real @ X2 @ Y4 ) )
= Y4 ) ) ).
% Sup_atLeastAtMost
thf(fact_799_SUP__id__eq,axiom,
! [A: set_real] :
( ( comple1385675409528146559p_real @ ( image_real_real @ id_real @ A ) )
= ( comple1385675409528146559p_real @ A ) ) ).
% SUP_id_eq
thf(fact_800_SUP__id__eq,axiom,
! [A: set_o] :
( ( complete_Sup_Sup_o @ ( image_o_o @ id_o @ A ) )
= ( complete_Sup_Sup_o @ A ) ) ).
% SUP_id_eq
thf(fact_801_SUP__id__eq,axiom,
! [A: set_set_real] :
( ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ id_set_real @ A ) )
= ( comple3096694443085538997t_real @ A ) ) ).
% SUP_id_eq
thf(fact_802_SUP__id__eq,axiom,
! [A: set_nat] :
( ( complete_Sup_Sup_nat @ ( image_nat_nat @ id_nat @ A ) )
= ( complete_Sup_Sup_nat @ A ) ) ).
% SUP_id_eq
thf(fact_803_SUP__id__eq,axiom,
! [A: set_real_real] :
( ( comple8933463103962640202l_real @ ( image_745864523092522741l_real @ id_real_real @ A ) )
= ( comple8933463103962640202l_real @ A ) ) ).
% SUP_id_eq
thf(fact_804_decseq__const,axiom,
! [K: extended_ereal] :
( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: extended_ereal,Y2: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y2 @ X4 )
@ ^ [X4: nat] : K ) ).
% decseq_const
thf(fact_805_decseq__const,axiom,
! [K: extend8495563244428889912nnreal] :
( monoto2291723841412853873nnreal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ Y2 @ X4 )
@ ^ [X4: nat] : K ) ).
% decseq_const
thf(fact_806_decseq__const,axiom,
! [K: set_Ex3793607809372303086nnreal] :
( monoto4660286046138248231nnreal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: set_Ex3793607809372303086nnreal,Y2: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ Y2 @ X4 )
@ ^ [X4: nat] : K ) ).
% decseq_const
thf(fact_807_decseq__const,axiom,
! [K: set_Extended_ereal] :
( monoto6788471982328799797_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: set_Extended_ereal,Y2: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ Y2 @ X4 )
@ ^ [X4: nat] : K ) ).
% decseq_const
thf(fact_808_decseq__const,axiom,
! [K: set_real] :
( monoto7274299666542614427t_real @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: set_real,Y2: set_real] : ( ord_less_eq_set_real @ Y2 @ X4 )
@ ^ [X4: nat] : K ) ).
% decseq_const
thf(fact_809_decseq__const,axiom,
! [K: nat] :
( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X4 )
@ ^ [X4: nat] : K ) ).
% decseq_const
thf(fact_810_decseq__const,axiom,
! [K: real > real] :
( monoto2824216093323351088l_real @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: real > real,Y2: real > real] : ( ord_le6948328307412524503l_real @ Y2 @ X4 )
@ ^ [X4: nat] : K ) ).
% decseq_const
thf(fact_811_incseq__const,axiom,
! [K: extended_ereal] :
( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal
@ ^ [X4: nat] : K ) ).
% incseq_const
thf(fact_812_incseq__const,axiom,
! [K: extend8495563244428889912nnreal] :
( monoto2291723841412853873nnreal @ top_top_set_nat @ ord_less_eq_nat @ ord_le3935885782089961368nnreal
@ ^ [X4: nat] : K ) ).
% incseq_const
thf(fact_813_incseq__const,axiom,
! [K: set_Ex3793607809372303086nnreal] :
( monoto4660286046138248231nnreal @ top_top_set_nat @ ord_less_eq_nat @ ord_le6787938422905777998nnreal
@ ^ [X4: nat] : K ) ).
% incseq_const
thf(fact_814_incseq__const,axiom,
! [K: set_Extended_ereal] :
( monoto6788471982328799797_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1644982726543182158_ereal
@ ^ [X4: nat] : K ) ).
% incseq_const
thf(fact_815_incseq__const,axiom,
! [K: set_real] :
( monoto7274299666542614427t_real @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_real
@ ^ [X4: nat] : K ) ).
% incseq_const
thf(fact_816_incseq__const,axiom,
! [K: nat] :
( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat
@ ^ [X4: nat] : K ) ).
% incseq_const
thf(fact_817_incseq__const,axiom,
! [K: real > real] :
( monoto2824216093323351088l_real @ top_top_set_nat @ ord_less_eq_nat @ ord_le6948328307412524503l_real
@ ^ [X4: nat] : K ) ).
% incseq_const
thf(fact_818_SUP__apply,axiom,
! [F: nat > real > real,A: set_nat,X2: real] :
( ( comple8933463103962640202l_real @ ( image_nat_real_real @ F @ A ) @ X2 )
= ( comple1385675409528146559p_real
@ ( image_nat_real
@ ^ [Y2: nat] : ( F @ Y2 @ X2 )
@ A ) ) ) ).
% SUP_apply
thf(fact_819_Lebesgue__Measure_Ointegrable__const__ivl,axiom,
! [A2: real,B2: real,C: real] :
( bochne3340023020068487468l_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ ( set_or1222579329274155063t_real @ A2 @ B2 ) )
@ ^ [X4: real] : C ) ).
% Lebesgue_Measure.integrable_const_ivl
thf(fact_820_SUP__eqI,axiom,
! [A: set_nat,F: nat > extended_ereal,X2: extended_ereal] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A )
=> ( ord_le1083603963089353582_ereal @ ( F @ I5 ) @ X2 ) )
=> ( ! [Y3: extended_ereal] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A )
=> ( ord_le1083603963089353582_ereal @ ( F @ I2 ) @ Y3 ) )
=> ( ord_le1083603963089353582_ereal @ X2 @ Y3 ) )
=> ( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F @ A ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_821_SUP__eqI,axiom,
! [A: set_real,F: real > $o,X2: $o] :
( ! [I5: real] :
( ( member_real @ I5 @ A )
=> ( ord_less_eq_o @ ( F @ I5 ) @ X2 ) )
=> ( ! [Y3: $o] :
( ! [I2: real] :
( ( member_real @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ Y3 ) )
=> ( ord_less_eq_o @ X2 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_real_o @ F @ A ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_822_SUP__eqI,axiom,
! [A: set_o,F: $o > $o,X2: $o] :
( ! [I5: $o] :
( ( member_o @ I5 @ A )
=> ( ord_less_eq_o @ ( F @ I5 ) @ X2 ) )
=> ( ! [Y3: $o] :
( ! [I2: $o] :
( ( member_o @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ Y3 ) )
=> ( ord_less_eq_o @ X2 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_823_SUP__eqI,axiom,
! [A: set_real,F: real > set_Ex3793607809372303086nnreal,X2: set_Ex3793607809372303086nnreal] :
( ! [I5: real] :
( ( member_real @ I5 @ A )
=> ( ord_le6787938422905777998nnreal @ ( F @ I5 ) @ X2 ) )
=> ( ! [Y3: set_Ex3793607809372303086nnreal] :
( ! [I2: real] :
( ( member_real @ I2 @ A )
=> ( ord_le6787938422905777998nnreal @ ( F @ I2 ) @ Y3 ) )
=> ( ord_le6787938422905777998nnreal @ X2 @ Y3 ) )
=> ( ( comple4226387801268262977nnreal @ ( image_6044265115298037921nnreal @ F @ A ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_824_SUP__eqI,axiom,
! [A: set_o,F: $o > set_Ex3793607809372303086nnreal,X2: set_Ex3793607809372303086nnreal] :
( ! [I5: $o] :
( ( member_o @ I5 @ A )
=> ( ord_le6787938422905777998nnreal @ ( F @ I5 ) @ X2 ) )
=> ( ! [Y3: set_Ex3793607809372303086nnreal] :
( ! [I2: $o] :
( ( member_o @ I2 @ A )
=> ( ord_le6787938422905777998nnreal @ ( F @ I2 ) @ Y3 ) )
=> ( ord_le6787938422905777998nnreal @ X2 @ Y3 ) )
=> ( ( comple4226387801268262977nnreal @ ( image_1679811975146592321nnreal @ F @ A ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_825_SUP__eqI,axiom,
! [A: set_real,F: real > set_Extended_ereal,X2: set_Extended_ereal] :
( ! [I5: real] :
( ( member_real @ I5 @ A )
=> ( ord_le1644982726543182158_ereal @ ( F @ I5 ) @ X2 ) )
=> ( ! [Y3: set_Extended_ereal] :
( ! [I2: real] :
( ( member_real @ I2 @ A )
=> ( ord_le1644982726543182158_ereal @ ( F @ I2 ) @ Y3 ) )
=> ( ord_le1644982726543182158_ereal @ X2 @ Y3 ) )
=> ( ( comple4319282863272126363_ereal @ ( image_5115244857779447611_ereal @ F @ A ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_826_SUP__eqI,axiom,
! [A: set_o,F: $o > set_Extended_ereal,X2: set_Extended_ereal] :
( ! [I5: $o] :
( ( member_o @ I5 @ A )
=> ( ord_le1644982726543182158_ereal @ ( F @ I5 ) @ X2 ) )
=> ( ! [Y3: set_Extended_ereal] :
( ! [I2: $o] :
( ( member_o @ I2 @ A )
=> ( ord_le1644982726543182158_ereal @ ( F @ I2 ) @ Y3 ) )
=> ( ord_le1644982726543182158_ereal @ X2 @ Y3 ) )
=> ( ( comple4319282863272126363_ereal @ ( image_6375117163256653723_ereal @ F @ A ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_827_SUP__eqI,axiom,
! [A: set_set_real,F: set_real > $o,X2: $o] :
( ! [I5: set_real] :
( ( member_set_real @ I5 @ A )
=> ( ord_less_eq_o @ ( F @ I5 ) @ X2 ) )
=> ( ! [Y3: $o] :
( ! [I2: set_real] :
( ( member_set_real @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ Y3 ) )
=> ( ord_less_eq_o @ X2 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_real_o @ F @ A ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_828_SUP__eqI,axiom,
! [A: set_nat,F: nat > set_real,X2: set_real] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A )
=> ( ord_less_eq_set_real @ ( F @ I5 ) @ X2 ) )
=> ( ! [Y3: set_real] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A )
=> ( ord_less_eq_set_real @ ( F @ I2 ) @ Y3 ) )
=> ( ord_less_eq_set_real @ X2 @ Y3 ) )
=> ( ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_829_SUP__eqI,axiom,
! [A: set_real,F: real > set_real,X2: set_real] :
( ! [I5: real] :
( ( member_real @ I5 @ A )
=> ( ord_less_eq_set_real @ ( F @ I5 ) @ X2 ) )
=> ( ! [Y3: set_real] :
( ! [I2: real] :
( ( member_real @ I2 @ A )
=> ( ord_less_eq_set_real @ ( F @ I2 ) @ Y3 ) )
=> ( ord_less_eq_set_real @ X2 @ Y3 ) )
=> ( ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_830_SUP__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > extended_ereal,G: nat > extended_ereal] :
( ! [N5: nat] :
( ( member_nat @ N5 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ( ord_le1083603963089353582_ereal @ ( F @ N5 ) @ ( G @ X3 ) ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_831_SUP__mono,axiom,
! [A: set_real_real,B: set_nat,F: ( real > real ) > extended_ereal,G: nat > extended_ereal] :
( ! [N5: real > real] :
( ( member_real_real @ N5 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ( ord_le1083603963089353582_ereal @ ( F @ N5 ) @ ( G @ X3 ) ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_2479213647061651408_ereal @ F @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_832_SUP__mono,axiom,
! [A: set_real,B: set_nat,F: real > extended_ereal,G: nat > extended_ereal] :
( ! [N5: real] :
( ( member_real @ N5 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ( ord_le1083603963089353582_ereal @ ( F @ N5 ) @ ( G @ X3 ) ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_7147107595568778587_ereal @ F @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_833_SUP__mono,axiom,
! [A: set_o,B: set_nat,F: $o > extended_ereal,G: nat > extended_ereal] :
( ! [N5: $o] :
( ( member_o @ N5 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ( ord_le1083603963089353582_ereal @ ( F @ N5 ) @ ( G @ X3 ) ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_7729549296133164475_ereal @ F @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_834_SUP__mono,axiom,
! [A: set_set_real,B: set_nat,F: set_real > extended_ereal,G: nat > extended_ereal] :
( ! [N5: set_real] :
( ( member_set_real @ N5 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ( ord_le1083603963089353582_ereal @ ( F @ N5 ) @ ( G @ X3 ) ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_1826965545724592037_ereal @ F @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_835_SUP__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_real,G: nat > set_real] :
( ! [N5: nat] :
( ( member_nat @ N5 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ( ord_less_eq_set_real @ ( F @ N5 ) @ ( G @ X3 ) ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A ) ) @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_836_SUP__mono,axiom,
! [A: set_real_real,B: set_nat,F: ( real > real ) > set_real,G: nat > set_real] :
( ! [N5: real > real] :
( ( member_real_real @ N5 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ( ord_less_eq_set_real @ ( F @ N5 ) @ ( G @ X3 ) ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_6663718904102175840t_real @ F @ A ) ) @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_837_SUP__mono,axiom,
! [A: set_real,B: set_nat,F: real > set_real,G: nat > set_real] :
( ! [N5: real] :
( ( member_real @ N5 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ( ord_less_eq_set_real @ ( F @ N5 ) @ ( G @ X3 ) ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A ) ) @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_838_SUP__mono,axiom,
! [A: set_o,B: set_nat,F: $o > set_real,G: nat > set_real] :
( ! [N5: $o] :
( ( member_o @ N5 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ( ord_less_eq_set_real @ ( F @ N5 ) @ ( G @ X3 ) ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ A ) ) @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_839_SUP__mono,axiom,
! [A: set_set_real,B: set_nat,F: set_real > set_real,G: nat > set_real] :
( ! [N5: set_real] :
( ( member_set_real @ N5 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ( ord_less_eq_set_real @ ( F @ N5 ) @ ( G @ X3 ) ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ F @ A ) ) @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_840_SUP__least,axiom,
! [A: set_nat,F: nat > extended_ereal,U: extended_ereal] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A )
=> ( ord_le1083603963089353582_ereal @ ( F @ I5 ) @ U ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_841_SUP__least,axiom,
! [A: set_real,F: real > $o,U: $o] :
( ! [I5: real] :
( ( member_real @ I5 @ A )
=> ( ord_less_eq_o @ ( F @ I5 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_real_o @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_842_SUP__least,axiom,
! [A: set_o,F: $o > $o,U: $o] :
( ! [I5: $o] :
( ( member_o @ I5 @ A )
=> ( ord_less_eq_o @ ( F @ I5 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_843_SUP__least,axiom,
! [A: set_real,F: real > set_Ex3793607809372303086nnreal,U: set_Ex3793607809372303086nnreal] :
( ! [I5: real] :
( ( member_real @ I5 @ A )
=> ( ord_le6787938422905777998nnreal @ ( F @ I5 ) @ U ) )
=> ( ord_le6787938422905777998nnreal @ ( comple4226387801268262977nnreal @ ( image_6044265115298037921nnreal @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_844_SUP__least,axiom,
! [A: set_o,F: $o > set_Ex3793607809372303086nnreal,U: set_Ex3793607809372303086nnreal] :
( ! [I5: $o] :
( ( member_o @ I5 @ A )
=> ( ord_le6787938422905777998nnreal @ ( F @ I5 ) @ U ) )
=> ( ord_le6787938422905777998nnreal @ ( comple4226387801268262977nnreal @ ( image_1679811975146592321nnreal @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_845_SUP__least,axiom,
! [A: set_real,F: real > set_Extended_ereal,U: set_Extended_ereal] :
( ! [I5: real] :
( ( member_real @ I5 @ A )
=> ( ord_le1644982726543182158_ereal @ ( F @ I5 ) @ U ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ ( image_5115244857779447611_ereal @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_846_SUP__least,axiom,
! [A: set_o,F: $o > set_Extended_ereal,U: set_Extended_ereal] :
( ! [I5: $o] :
( ( member_o @ I5 @ A )
=> ( ord_le1644982726543182158_ereal @ ( F @ I5 ) @ U ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ ( image_6375117163256653723_ereal @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_847_SUP__least,axiom,
! [A: set_set_real,F: set_real > $o,U: $o] :
( ! [I5: set_real] :
( ( member_set_real @ I5 @ A )
=> ( ord_less_eq_o @ ( F @ I5 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_set_real_o @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_848_SUP__least,axiom,
! [A: set_nat,F: nat > set_real,U: set_real] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A )
=> ( ord_less_eq_set_real @ ( F @ I5 ) @ U ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_849_SUP__least,axiom,
! [A: set_real,F: real > set_real,U: set_real] :
( ! [I5: real] :
( ( member_real @ I5 @ A )
=> ( ord_less_eq_set_real @ ( F @ I5 ) @ U ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_850_SUP__mono_H,axiom,
! [F: nat > extended_ereal,G: nat > extended_ereal,A: set_nat] :
( ! [X: nat] : ( ord_le1083603963089353582_ereal @ ( F @ X ) @ ( G @ X ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ A ) ) ) ) ).
% SUP_mono'
thf(fact_851_SUP__mono_H,axiom,
! [F: nat > set_real,G: nat > set_real,A: set_nat] :
( ! [X: nat] : ( ord_less_eq_set_real @ ( F @ X ) @ ( G @ X ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A ) ) @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ G @ A ) ) ) ) ).
% SUP_mono'
thf(fact_852_SUP__upper,axiom,
! [I: nat,A: set_nat,F: nat > extended_ereal] :
( ( member_nat @ I @ A )
=> ( ord_le1083603963089353582_ereal @ ( F @ I ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_853_SUP__upper,axiom,
! [I: real,A: set_real,F: real > $o] :
( ( member_real @ I @ A )
=> ( ord_less_eq_o @ ( F @ I ) @ ( complete_Sup_Sup_o @ ( image_real_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_854_SUP__upper,axiom,
! [I: $o,A: set_o,F: $o > $o] :
( ( member_o @ I @ A )
=> ( ord_less_eq_o @ ( F @ I ) @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_855_SUP__upper,axiom,
! [I: real,A: set_real,F: real > set_Ex3793607809372303086nnreal] :
( ( member_real @ I @ A )
=> ( ord_le6787938422905777998nnreal @ ( F @ I ) @ ( comple4226387801268262977nnreal @ ( image_6044265115298037921nnreal @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_856_SUP__upper,axiom,
! [I: $o,A: set_o,F: $o > set_Ex3793607809372303086nnreal] :
( ( member_o @ I @ A )
=> ( ord_le6787938422905777998nnreal @ ( F @ I ) @ ( comple4226387801268262977nnreal @ ( image_1679811975146592321nnreal @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_857_SUP__upper,axiom,
! [I: real,A: set_real,F: real > set_Extended_ereal] :
( ( member_real @ I @ A )
=> ( ord_le1644982726543182158_ereal @ ( F @ I ) @ ( comple4319282863272126363_ereal @ ( image_5115244857779447611_ereal @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_858_SUP__upper,axiom,
! [I: $o,A: set_o,F: $o > set_Extended_ereal] :
( ( member_o @ I @ A )
=> ( ord_le1644982726543182158_ereal @ ( F @ I ) @ ( comple4319282863272126363_ereal @ ( image_6375117163256653723_ereal @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_859_SUP__upper,axiom,
! [I: set_real,A: set_set_real,F: set_real > $o] :
( ( member_set_real @ I @ A )
=> ( ord_less_eq_o @ ( F @ I ) @ ( complete_Sup_Sup_o @ ( image_set_real_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_860_SUP__upper,axiom,
! [I: nat,A: set_nat,F: nat > set_real] :
( ( member_nat @ I @ A )
=> ( ord_less_eq_set_real @ ( F @ I ) @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_861_SUP__upper,axiom,
! [I: real,A: set_real,F: real > set_real] :
( ( member_real @ I @ A )
=> ( ord_less_eq_set_real @ ( F @ I ) @ ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_862_SUP__le__iff,axiom,
! [F: nat > extended_ereal,A: set_nat,U: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F @ A ) ) @ U )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_863_SUP__le__iff,axiom,
! [F: nat > set_real,A: set_nat,U: set_real] :
( ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A ) ) @ U )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_less_eq_set_real @ ( F @ X4 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_864_SUP__upper2,axiom,
! [I: nat,A: set_nat,U: extended_ereal,F: nat > extended_ereal] :
( ( member_nat @ I @ A )
=> ( ( ord_le1083603963089353582_ereal @ U @ ( F @ I ) )
=> ( ord_le1083603963089353582_ereal @ U @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_865_SUP__upper2,axiom,
! [I: real,A: set_real,U: $o,F: real > $o] :
( ( member_real @ I @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ I ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_real_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_866_SUP__upper2,axiom,
! [I: $o,A: set_o,U: $o,F: $o > $o] :
( ( member_o @ I @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ I ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_867_SUP__upper2,axiom,
! [I: real,A: set_real,U: set_Ex3793607809372303086nnreal,F: real > set_Ex3793607809372303086nnreal] :
( ( member_real @ I @ A )
=> ( ( ord_le6787938422905777998nnreal @ U @ ( F @ I ) )
=> ( ord_le6787938422905777998nnreal @ U @ ( comple4226387801268262977nnreal @ ( image_6044265115298037921nnreal @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_868_SUP__upper2,axiom,
! [I: $o,A: set_o,U: set_Ex3793607809372303086nnreal,F: $o > set_Ex3793607809372303086nnreal] :
( ( member_o @ I @ A )
=> ( ( ord_le6787938422905777998nnreal @ U @ ( F @ I ) )
=> ( ord_le6787938422905777998nnreal @ U @ ( comple4226387801268262977nnreal @ ( image_1679811975146592321nnreal @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_869_SUP__upper2,axiom,
! [I: real,A: set_real,U: set_Extended_ereal,F: real > set_Extended_ereal] :
( ( member_real @ I @ A )
=> ( ( ord_le1644982726543182158_ereal @ U @ ( F @ I ) )
=> ( ord_le1644982726543182158_ereal @ U @ ( comple4319282863272126363_ereal @ ( image_5115244857779447611_ereal @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_870_SUP__upper2,axiom,
! [I: $o,A: set_o,U: set_Extended_ereal,F: $o > set_Extended_ereal] :
( ( member_o @ I @ A )
=> ( ( ord_le1644982726543182158_ereal @ U @ ( F @ I ) )
=> ( ord_le1644982726543182158_ereal @ U @ ( comple4319282863272126363_ereal @ ( image_6375117163256653723_ereal @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_871_SUP__upper2,axiom,
! [I: set_real,A: set_set_real,U: $o,F: set_real > $o] :
( ( member_set_real @ I @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ I ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_set_real_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_872_SUP__upper2,axiom,
! [I: nat,A: set_nat,U: set_real,F: nat > set_real] :
( ( member_nat @ I @ A )
=> ( ( ord_less_eq_set_real @ U @ ( F @ I ) )
=> ( ord_less_eq_set_real @ U @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_873_SUP__upper2,axiom,
! [I: real,A: set_real,U: set_real,F: real > set_real] :
( ( member_real @ I @ A )
=> ( ( ord_less_eq_set_real @ U @ ( F @ I ) )
=> ( ord_less_eq_set_real @ U @ ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_874_SUP__commute,axiom,
! [F: nat > nat > extended_ereal,B: set_nat,A: set_nat] :
( ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I4: nat] : ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ ( F @ I4 ) @ B ) )
@ A ) )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [J2: nat] :
( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [I4: nat] : ( F @ I4 @ J2 )
@ A ) )
@ B ) ) ) ).
% SUP_commute
thf(fact_875_SUP__commute,axiom,
! [F: nat > nat > set_real,B: set_nat,A: set_nat] :
( ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [I4: nat] : ( comple3096694443085538997t_real @ ( image_nat_set_real @ ( F @ I4 ) @ B ) )
@ A ) )
= ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [J2: nat] :
( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [I4: nat] : ( F @ I4 @ J2 )
@ A ) )
@ B ) ) ) ).
% SUP_commute
thf(fact_876_Collect__subset,axiom,
! [A: set_real_real,P: ( real > real ) > $o] :
( ord_le4198349162570665613l_real
@ ( collect_real_real
@ ^ [X4: real > real] :
( ( member_real_real @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_877_Collect__subset,axiom,
! [A: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_878_Collect__subset,axiom,
! [A: set_set_real,P: set_real > $o] :
( ord_le3558479182127378552t_real
@ ( collect_set_real
@ ^ [X4: set_real] :
( ( member_set_real @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_879_Collect__subset,axiom,
! [A: set_Ex3793607809372303086nnreal,P: extend8495563244428889912nnreal > $o] :
( ord_le6787938422905777998nnreal
@ ( collec6648975593938027277nnreal
@ ^ [X4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_880_Collect__subset,axiom,
! [A: set_Extended_ereal,P: extended_ereal > $o] :
( ord_le1644982726543182158_ereal
@ ( collec5835592288176408249_ereal
@ ^ [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_881_Collect__subset,axiom,
! [A: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_882_less__eq__set__def,axiom,
( ord_le4198349162570665613l_real
= ( ^ [A3: set_real_real,B3: set_real_real] :
( ord_le5273791883478943800real_o
@ ^ [X4: real > real] : ( member_real_real @ X4 @ A3 )
@ ^ [X4: real > real] : ( member_real_real @ X4 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_883_less__eq__set__def,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
( ord_less_eq_o_o
@ ^ [X4: $o] : ( member_o @ X4 @ A3 )
@ ^ [X4: $o] : ( member_o @ X4 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_884_less__eq__set__def,axiom,
( ord_le3558479182127378552t_real
= ( ^ [A3: set_set_real,B3: set_set_real] :
( ord_le2392157289819280397real_o
@ ^ [X4: set_real] : ( member_set_real @ X4 @ A3 )
@ ^ [X4: set_real] : ( member_set_real @ X4 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_885_less__eq__set__def,axiom,
( ord_le6787938422905777998nnreal
= ( ^ [A3: set_Ex3793607809372303086nnreal,B3: set_Ex3793607809372303086nnreal] :
( ord_le7025323315894483639real_o
@ ^ [X4: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X4 @ A3 )
@ ^ [X4: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X4 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_886_less__eq__set__def,axiom,
( ord_le1644982726543182158_ereal
= ( ^ [A3: set_Extended_ereal,B3: set_Extended_ereal] :
( ord_le6694447793465728271real_o
@ ^ [X4: extended_ereal] : ( member2350847679896131959_ereal @ X4 @ A3 )
@ ^ [X4: extended_ereal] : ( member2350847679896131959_ereal @ X4 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_887_less__eq__set__def,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B3: set_real] :
( ord_less_eq_real_o
@ ^ [X4: real] : ( member_real @ X4 @ A3 )
@ ^ [X4: real] : ( member_real @ X4 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_888_SUP__UNION,axiom,
! [F: nat > extended_ereal,G: nat > set_nat,A: set_nat] :
( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A ) ) ) )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [Y2: nat] : ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F @ ( G @ Y2 ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_889_SUP__UNION,axiom,
! [F: real > extended_ereal,G: nat > set_real,A: set_nat] :
( ( comple8415311339701865915_ereal @ ( image_7147107595568778587_ereal @ F @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ G @ A ) ) ) )
= ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [Y2: nat] : ( comple8415311339701865915_ereal @ ( image_7147107595568778587_ereal @ F @ ( G @ Y2 ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_890_SUP__UNION,axiom,
! [F: real > $o,G: nat > set_real,A: set_nat] :
( ( complete_Sup_Sup_o @ ( image_real_o @ F @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ G @ A ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y2: nat] : ( complete_Sup_Sup_o @ ( image_real_o @ F @ ( G @ Y2 ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_891_SUP__UNION,axiom,
! [F: nat > set_real,G: nat > set_nat,A: set_nat] :
( ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A ) ) ) )
= ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [Y2: nat] : ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ ( G @ Y2 ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_892_SUP__UNION,axiom,
! [F: real > set_real,G: nat > set_real,A: set_nat] :
( ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ G @ A ) ) ) )
= ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [Y2: nat] : ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ ( G @ Y2 ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_893_Sup__fun__def,axiom,
( comple8933463103962640202l_real
= ( ^ [A3: set_real_real,X4: real] :
( comple1385675409528146559p_real
@ ( image_real_real_real
@ ^ [F2: real > real] : ( F2 @ X4 )
@ A3 ) ) ) ) ).
% Sup_fun_def
thf(fact_894_conj__subset__def,axiom,
! [A: set_Ex3793607809372303086nnreal,P: extend8495563244428889912nnreal > $o,Q: extend8495563244428889912nnreal > $o] :
( ( ord_le6787938422905777998nnreal @ A
@ ( collec6648975593938027277nnreal
@ ^ [X4: extend8495563244428889912nnreal] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) )
= ( ( ord_le6787938422905777998nnreal @ A @ ( collec6648975593938027277nnreal @ P ) )
& ( ord_le6787938422905777998nnreal @ A @ ( collec6648975593938027277nnreal @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_895_conj__subset__def,axiom,
! [A: set_Extended_ereal,P: extended_ereal > $o,Q: extended_ereal > $o] :
( ( ord_le1644982726543182158_ereal @ A
@ ( collec5835592288176408249_ereal
@ ^ [X4: extended_ereal] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) )
= ( ( ord_le1644982726543182158_ereal @ A @ ( collec5835592288176408249_ereal @ P ) )
& ( ord_le1644982726543182158_ereal @ A @ ( collec5835592288176408249_ereal @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_896_conj__subset__def,axiom,
! [A: set_real,P: real > $o,Q: real > $o] :
( ( ord_less_eq_set_real @ A
@ ( collect_real
@ ^ [X4: real] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) )
= ( ( ord_less_eq_set_real @ A @ ( collect_real @ P ) )
& ( ord_less_eq_set_real @ A @ ( collect_real @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_897_continuous__on__compose2,axiom,
! [T3: set_Ex3793607809372303086nnreal,G: extend8495563244428889912nnreal > real,S3: set_real,F: real > extend8495563244428889912nnreal] :
( ( topolo6030940669772499300l_real @ T3 @ G )
=> ( ( topolo7998686939223043556nnreal @ S3 @ F )
=> ( ( ord_le6787938422905777998nnreal @ ( image_7616191137145695467nnreal @ F @ S3 ) @ T3 )
=> ( topolo5044208981011980120l_real @ S3
@ ^ [X4: real] : ( G @ ( F @ X4 ) ) ) ) ) ) ).
% continuous_on_compose2
thf(fact_898_continuous__on__compose2,axiom,
! [T3: set_Extended_ereal,G: extended_ereal > real,S3: set_real,F: real > extended_ereal] :
( ( topolo1945291691832713340l_real @ T3 @ G )
=> ( ( topolo6771225064363481634_ereal @ S3 @ F )
=> ( ( ord_le1644982726543182158_ereal @ ( image_7147107595568778587_ereal @ F @ S3 ) @ T3 )
=> ( topolo5044208981011980120l_real @ S3
@ ^ [X4: real] : ( G @ ( F @ X4 ) ) ) ) ) ) ).
% continuous_on_compose2
thf(fact_899_continuous__on__compose2,axiom,
! [T3: set_Ex3793607809372303086nnreal,G: extend8495563244428889912nnreal > extended_ereal,S3: set_Extended_ereal,F: extended_ereal > extend8495563244428889912nnreal] :
( ( topolo5759031612838008598_ereal @ T3 @ G )
=> ( ( topolo7979175830221463816nnreal @ S3 @ F )
=> ( ( ord_le6787938422905777998nnreal @ ( image_8614087454967683265nnreal @ F @ S3 ) @ T3 )
=> ( topolo6777079828818185726_ereal @ S3
@ ^ [X4: extended_ereal] : ( G @ ( F @ X4 ) ) ) ) ) ) ).
% continuous_on_compose2
thf(fact_900_continuous__on__compose2,axiom,
! [T3: set_real,G: real > extended_ereal,S3: set_Extended_ereal,F: extended_ereal > real] :
( ( topolo6771225064363481634_ereal @ T3 @ G )
=> ( ( topolo1945291691832713340l_real @ S3 @ F )
=> ( ( ord_less_eq_set_real @ ( image_2321174223038010293l_real @ F @ S3 ) @ T3 )
=> ( topolo6777079828818185726_ereal @ S3
@ ^ [X4: extended_ereal] : ( G @ ( F @ X4 ) ) ) ) ) ) ).
% continuous_on_compose2
thf(fact_901_continuous__on__compose2,axiom,
! [T3: set_real,G: real > real,S3: set_nat,F: nat > real] :
( ( topolo5044208981011980120l_real @ T3 @ G )
=> ( ( topolo6943266826644216316t_real @ S3 @ F )
=> ( ( ord_less_eq_set_real @ ( image_nat_real @ F @ S3 ) @ T3 )
=> ( topolo6943266826644216316t_real @ S3
@ ^ [X4: nat] : ( G @ ( F @ X4 ) ) ) ) ) ) ).
% continuous_on_compose2
thf(fact_902_continuous__on__compose2,axiom,
! [T3: set_real,G: real > real,S3: set_real,F: real > real] :
( ( topolo5044208981011980120l_real @ T3 @ G )
=> ( ( topolo5044208981011980120l_real @ S3 @ F )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ F @ S3 ) @ T3 )
=> ( topolo5044208981011980120l_real @ S3
@ ^ [X4: real] : ( G @ ( F @ X4 ) ) ) ) ) ) ).
% continuous_on_compose2
thf(fact_903_continuous__on__compose2,axiom,
! [T3: set_Extended_ereal,G: extended_ereal > extended_ereal,S3: set_nat,F: nat > extended_ereal] :
( ( topolo6777079828818185726_ereal @ T3 @ G )
=> ( ( topolo7559974011338912894_ereal @ S3 @ F )
=> ( ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F @ S3 ) @ T3 )
=> ( topolo7559974011338912894_ereal @ S3
@ ^ [X4: nat] : ( G @ ( F @ X4 ) ) ) ) ) ) ).
% continuous_on_compose2
thf(fact_904_continuous__on__compose2,axiom,
! [T3: set_Extended_ereal,G: extended_ereal > extended_ereal,S3: set_Extended_ereal,F: extended_ereal > extended_ereal] :
( ( topolo6777079828818185726_ereal @ T3 @ G )
=> ( ( topolo6777079828818185726_ereal @ S3 @ F )
=> ( ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F @ S3 ) @ T3 )
=> ( topolo6777079828818185726_ereal @ S3
@ ^ [X4: extended_ereal] : ( G @ ( F @ X4 ) ) ) ) ) ) ).
% continuous_on_compose2
thf(fact_905_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X4: nat] : $true ) ) ).
% UNIV_def
thf(fact_906_UNIV__def,axiom,
( top_top_set_real
= ( collect_real
@ ^ [X4: real] : $true ) ) ).
% UNIV_def
thf(fact_907_UNIV__def,axiom,
( top_top_set_complex
= ( collect_complex
@ ^ [X4: complex] : $true ) ) ).
% UNIV_def
thf(fact_908_UNIV__def,axiom,
( top_to5683747375963461374_ereal
= ( collec5835592288176408249_ereal
@ ^ [X4: extended_ereal] : $true ) ) ).
% UNIV_def
thf(fact_909_continuous__on__const,axiom,
! [S3: set_real,C: real] :
( topolo5044208981011980120l_real @ S3
@ ^ [X4: real] : C ) ).
% continuous_on_const
thf(fact_910_continuous__on__const,axiom,
! [S3: set_Extended_ereal,C: extended_ereal] :
( topolo6777079828818185726_ereal @ S3
@ ^ [X4: extended_ereal] : C ) ).
% continuous_on_const
thf(fact_911_continuous__on__cong,axiom,
! [S3: set_real,T3: set_real,F: real > real,G: real > real] :
( ( S3 = T3 )
=> ( ! [X: real] :
( ( member_real @ X @ T3 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( topolo5044208981011980120l_real @ S3 @ F )
= ( topolo5044208981011980120l_real @ T3 @ G ) ) ) ) ).
% continuous_on_cong
thf(fact_912_continuous__on__cong,axiom,
! [S3: set_Extended_ereal,T3: set_Extended_ereal,F: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal] :
( ( S3 = T3 )
=> ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ T3 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( topolo6777079828818185726_ereal @ S3 @ F )
= ( topolo6777079828818185726_ereal @ T3 @ G ) ) ) ) ).
% continuous_on_cong
thf(fact_913_continuous__on__id,axiom,
! [S3: set_real] :
( topolo5044208981011980120l_real @ S3
@ ^ [X4: real] : X4 ) ).
% continuous_on_id
thf(fact_914_continuous__on__id,axiom,
! [S3: set_Extended_ereal] :
( topolo6777079828818185726_ereal @ S3
@ ^ [X4: extended_ereal] : X4 ) ).
% continuous_on_id
thf(fact_915_Compr__image__eq,axiom,
! [F: nat > extended_ereal,A: set_nat,P: extended_ereal > $o] :
( ( collec5835592288176408249_ereal
@ ^ [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ ( image_4309273772856505399_ereal @ F @ A ) )
& ( P @ X4 ) ) )
= ( image_4309273772856505399_ereal @ F
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_916_Compr__image__eq,axiom,
! [F: nat > complex,A: set_nat,P: complex > $o] :
( ( collect_complex
@ ^ [X4: complex] :
( ( member_complex @ X4 @ ( image_nat_complex @ F @ A ) )
& ( P @ X4 ) ) )
= ( image_nat_complex @ F
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_917_Compr__image__eq,axiom,
! [F: nat > real,A: set_nat,P: real > $o] :
( ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( image_nat_real @ F @ A ) )
& ( P @ X4 ) ) )
= ( image_nat_real @ F
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_918_Compr__image__eq,axiom,
! [F: real > real,A: set_real,P: real > $o] :
( ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( image_real_real @ F @ A ) )
& ( P @ X4 ) ) )
= ( image_real_real @ F
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_919_Compr__image__eq,axiom,
! [F: $o > real,A: set_o,P: real > $o] :
( ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( image_o_real @ F @ A ) )
& ( P @ X4 ) ) )
= ( image_o_real @ F
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_920_Compr__image__eq,axiom,
! [F: real > $o,A: set_real,P: $o > $o] :
( ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( image_real_o @ F @ A ) )
& ( P @ X4 ) ) )
= ( image_real_o @ F
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_921_Compr__image__eq,axiom,
! [F: $o > $o,A: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( image_o_o @ F @ A ) )
& ( P @ X4 ) ) )
= ( image_o_o @ F
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_922_Compr__image__eq,axiom,
! [F: set_real > real,A: set_set_real,P: real > $o] :
( ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( image_set_real_real @ F @ A ) )
& ( P @ X4 ) ) )
= ( image_set_real_real @ F
@ ( collect_set_real
@ ^ [X4: set_real] :
( ( member_set_real @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_923_Compr__image__eq,axiom,
! [F: set_real > $o,A: set_set_real,P: $o > $o] :
( ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( image_set_real_o @ F @ A ) )
& ( P @ X4 ) ) )
= ( image_set_real_o @ F
@ ( collect_set_real
@ ^ [X4: set_real] :
( ( member_set_real @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_924_Compr__image__eq,axiom,
! [F: nat > set_real,A: set_nat,P: set_real > $o] :
( ( collect_set_real
@ ^ [X4: set_real] :
( ( member_set_real @ X4 @ ( image_nat_set_real @ F @ A ) )
& ( P @ X4 ) ) )
= ( image_nat_set_real @ F
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_925_image__image,axiom,
! [F: extended_ereal > extended_ereal,G: nat > extended_ereal,A: set_nat] :
( ( image_6042159593519690757_ereal @ F @ ( image_4309273772856505399_ereal @ G @ A ) )
= ( image_4309273772856505399_ereal
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ A ) ) ).
% image_image
thf(fact_926_image__image,axiom,
! [F: extended_ereal > complex,G: nat > extended_ereal,A: set_nat] :
( ( image_2124648478513430583omplex @ F @ ( image_4309273772856505399_ereal @ G @ A ) )
= ( image_nat_complex
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ A ) ) ).
% image_image
thf(fact_927_image__image,axiom,
! [F: extended_ereal > real,G: nat > extended_ereal,A: set_nat] :
( ( image_2321174223038010293l_real @ F @ ( image_4309273772856505399_ereal @ G @ A ) )
= ( image_nat_real
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ A ) ) ).
% image_image
thf(fact_928_image__image,axiom,
! [F: complex > extended_ereal,G: nat > complex,A: set_nat] :
( ( image_6025055836837866457_ereal @ F @ ( image_nat_complex @ G @ A ) )
= ( image_4309273772856505399_ereal
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ A ) ) ).
% image_image
thf(fact_929_image__image,axiom,
! [F: complex > complex,G: nat > complex,A: set_nat] :
( ( image_1468599708987790691omplex @ F @ ( image_nat_complex @ G @ A ) )
= ( image_nat_complex
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ A ) ) ).
% image_image
thf(fact_930_image__image,axiom,
! [F: complex > real,G: nat > complex,A: set_nat] :
( ( image_complex_real @ F @ ( image_nat_complex @ G @ A ) )
= ( image_nat_real
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ A ) ) ).
% image_image
thf(fact_931_image__image,axiom,
! [F: real > extended_ereal,G: nat > real,A: set_nat] :
( ( image_7147107595568778587_ereal @ F @ ( image_nat_real @ G @ A ) )
= ( image_4309273772856505399_ereal
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ A ) ) ).
% image_image
thf(fact_932_image__image,axiom,
! [F: real > complex,G: nat > real,A: set_nat] :
( ( image_real_complex @ F @ ( image_nat_real @ G @ A ) )
= ( image_nat_complex
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ A ) ) ).
% image_image
thf(fact_933_image__image,axiom,
! [F: real > real,G: nat > real,A: set_nat] :
( ( image_real_real @ F @ ( image_nat_real @ G @ A ) )
= ( image_nat_real
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ A ) ) ).
% image_image
thf(fact_934_image__image,axiom,
! [F: nat > extended_ereal,G: nat > nat,A: set_nat] :
( ( image_4309273772856505399_ereal @ F @ ( image_nat_nat @ G @ A ) )
= ( image_4309273772856505399_ereal
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ A ) ) ).
% image_image
thf(fact_935_imageE,axiom,
! [B2: extended_ereal,F: nat > extended_ereal,A: set_nat] :
( ( member2350847679896131959_ereal @ B2 @ ( image_4309273772856505399_ereal @ F @ A ) )
=> ~ ! [X: nat] :
( ( B2
= ( F @ X ) )
=> ~ ( member_nat @ X @ A ) ) ) ).
% imageE
thf(fact_936_imageE,axiom,
! [B2: complex,F: nat > complex,A: set_nat] :
( ( member_complex @ B2 @ ( image_nat_complex @ F @ A ) )
=> ~ ! [X: nat] :
( ( B2
= ( F @ X ) )
=> ~ ( member_nat @ X @ A ) ) ) ).
% imageE
thf(fact_937_imageE,axiom,
! [B2: real,F: nat > real,A: set_nat] :
( ( member_real @ B2 @ ( image_nat_real @ F @ A ) )
=> ~ ! [X: nat] :
( ( B2
= ( F @ X ) )
=> ~ ( member_nat @ X @ A ) ) ) ).
% imageE
thf(fact_938_imageE,axiom,
! [B2: real,F: real > real,A: set_real] :
( ( member_real @ B2 @ ( image_real_real @ F @ A ) )
=> ~ ! [X: real] :
( ( B2
= ( F @ X ) )
=> ~ ( member_real @ X @ A ) ) ) ).
% imageE
thf(fact_939_imageE,axiom,
! [B2: real,F: $o > real,A: set_o] :
( ( member_real @ B2 @ ( image_o_real @ F @ A ) )
=> ~ ! [X: $o] :
( ( B2
= ( F @ X ) )
=> ~ ( member_o @ X @ A ) ) ) ).
% imageE
thf(fact_940_imageE,axiom,
! [B2: $o,F: real > $o,A: set_real] :
( ( member_o @ B2 @ ( image_real_o @ F @ A ) )
=> ~ ! [X: real] :
( ( B2
= ( F @ X ) )
=> ~ ( member_real @ X @ A ) ) ) ).
% imageE
thf(fact_941_imageE,axiom,
! [B2: $o,F: $o > $o,A: set_o] :
( ( member_o @ B2 @ ( image_o_o @ F @ A ) )
=> ~ ! [X: $o] :
( ( B2
= ( F @ X ) )
=> ~ ( member_o @ X @ A ) ) ) ).
% imageE
thf(fact_942_imageE,axiom,
! [B2: real,F: set_real > real,A: set_set_real] :
( ( member_real @ B2 @ ( image_set_real_real @ F @ A ) )
=> ~ ! [X: set_real] :
( ( B2
= ( F @ X ) )
=> ~ ( member_set_real @ X @ A ) ) ) ).
% imageE
thf(fact_943_imageE,axiom,
! [B2: $o,F: set_real > $o,A: set_set_real] :
( ( member_o @ B2 @ ( image_set_real_o @ F @ A ) )
=> ~ ! [X: set_real] :
( ( B2
= ( F @ X ) )
=> ~ ( member_set_real @ X @ A ) ) ) ).
% imageE
thf(fact_944_imageE,axiom,
! [B2: set_real,F: nat > set_real,A: set_nat] :
( ( member_set_real @ B2 @ ( image_nat_set_real @ F @ A ) )
=> ~ ! [X: nat] :
( ( B2
= ( F @ X ) )
=> ~ ( member_nat @ X @ A ) ) ) ).
% imageE
thf(fact_945_measurable__compose__rev,axiom,
! [F: real > real,L3: sigma_measure_real,N: sigma_measure_real,G: real > real,M: sigma_measure_real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ L3 @ N ) )
=> ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ L3 ) )
=> ( member_real_real
@ ^ [X4: real] : ( F @ ( G @ X4 ) )
@ ( sigma_5267869275261027754l_real @ M @ N ) ) ) ) ).
% measurable_compose_rev
thf(fact_946_measurable__compose,axiom,
! [F: real > real,M: sigma_measure_real,N: sigma_measure_real,G: real > real,L3: sigma_measure_real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) )
=> ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ N @ L3 ) )
=> ( member_real_real
@ ^ [X4: real] : ( G @ ( F @ X4 ) )
@ ( sigma_5267869275261027754l_real @ M @ L3 ) ) ) ) ).
% measurable_compose
thf(fact_947_measurable__id,axiom,
! [M: sigma_measure_real] :
( member_real_real
@ ^ [X4: real] : X4
@ ( sigma_5267869275261027754l_real @ M @ M ) ) ).
% measurable_id
thf(fact_948_simple__function__compose1,axiom,
! [M: sigma_measure_real,F: real > real,G: real > real] :
( ( nonneg485563716852976898l_real @ M @ F )
=> ( nonneg485563716852976898l_real @ M
@ ^ [X4: real] : ( G @ ( F @ X4 ) ) ) ) ).
% simple_function_compose1
thf(fact_949_simple__function__compose2,axiom,
! [M: sigma_measure_real,F: real > real,G: real > real,H: real > real > real] :
( ( nonneg485563716852976898l_real @ M @ F )
=> ( ( nonneg485563716852976898l_real @ M @ G )
=> ( nonneg485563716852976898l_real @ M
@ ^ [X4: real] : ( H @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ) ).
% simple_function_compose2
thf(fact_950_SUP__subset__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > extended_ereal,G: nat > extended_ereal] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_le1083603963089353582_ereal @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le1083603963089353582_ereal @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F @ A ) ) @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_951_SUP__subset__mono,axiom,
! [A: set_o,B: set_o,F: $o > $o,G: $o > $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_o_o @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_952_SUP__subset__mono,axiom,
! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > $o,G: extend8495563244428889912nnreal > $o] :
( ( ord_le6787938422905777998nnreal @ A @ B )
=> ( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ A )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_3162942742313426073real_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_3162942742313426073real_o @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_953_SUP__subset__mono,axiom,
! [A: set_Extended_ereal,B: set_Extended_ereal,F: extended_ereal > $o,G: extended_ereal > $o] :
( ( ord_le1644982726543182158_ereal @ A @ B )
=> ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_951975095941678543real_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_951975095941678543real_o @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_954_SUP__subset__mono,axiom,
! [A: set_real,B: set_real,F: real > $o,G: real > $o] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ! [X: real] :
( ( member_real @ X @ A )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_real_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_real_o @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_955_SUP__subset__mono,axiom,
! [A: set_o,B: set_o,F: $o > set_Ex3793607809372303086nnreal,G: $o > set_Ex3793607809372303086nnreal] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_le6787938422905777998nnreal @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le6787938422905777998nnreal @ ( comple4226387801268262977nnreal @ ( image_1679811975146592321nnreal @ F @ A ) ) @ ( comple4226387801268262977nnreal @ ( image_1679811975146592321nnreal @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_956_SUP__subset__mono,axiom,
! [A: set_o,B: set_o,F: $o > set_Extended_ereal,G: $o > set_Extended_ereal] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_le1644982726543182158_ereal @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ ( image_6375117163256653723_ereal @ F @ A ) ) @ ( comple4319282863272126363_ereal @ ( image_6375117163256653723_ereal @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_957_SUP__subset__mono,axiom,
! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal,G: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ A @ B )
=> ( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ A )
=> ( ord_le6787938422905777998nnreal @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le6787938422905777998nnreal @ ( comple4226387801268262977nnreal @ ( image_205196257943321645nnreal @ F @ A ) ) @ ( comple4226387801268262977nnreal @ ( image_205196257943321645nnreal @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_958_SUP__subset__mono,axiom,
! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > set_Extended_ereal,G: extend8495563244428889912nnreal > set_Extended_ereal] :
( ( ord_le6787938422905777998nnreal @ A @ B )
=> ( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ A )
=> ( ord_le1644982726543182158_ereal @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ ( image_5929344197358196911_ereal @ F @ A ) ) @ ( comple4319282863272126363_ereal @ ( image_5929344197358196911_ereal @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_959_SUP__subset__mono,axiom,
! [A: set_Extended_ereal,B: set_Extended_ereal,F: extended_ereal > set_Ex3793607809372303086nnreal,G: extended_ereal > set_Ex3793607809372303086nnreal] :
( ( ord_le1644982726543182158_ereal @ A @ B )
=> ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A )
=> ( ord_le6787938422905777998nnreal @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le6787938422905777998nnreal @ ( comple4226387801268262977nnreal @ ( image_6588766411312125047nnreal @ F @ A ) ) @ ( comple4226387801268262977nnreal @ ( image_6588766411312125047nnreal @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_960_borel__measurable__continuous__on,axiom,
! [F: nat > real,G: real > nat,M: sigma_measure_real] :
( ( topolo6943266826644216316t_real @ top_top_set_nat @ F )
=> ( ( member_real_nat @ G @ ( sigma_6315060578831106510al_nat @ M @ borel_8449730974584783410el_nat ) )
=> ( member_real_real
@ ^ [X4: real] : ( F @ ( G @ X4 ) )
@ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).
% borel_measurable_continuous_on
thf(fact_961_borel__measurable__continuous__on,axiom,
! [F: complex > real,G: real > complex,M: sigma_measure_real] :
( ( topolo8674095878704923098x_real @ top_top_set_complex @ F )
=> ( ( member_real_complex @ G @ ( sigma_9111916201866572460omplex @ M @ borel_1392132677378845456omplex ) )
=> ( member_real_real
@ ^ [X4: real] : ( F @ ( G @ X4 ) )
@ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).
% borel_measurable_continuous_on
thf(fact_962_borel__measurable__continuous__on,axiom,
! [F: extended_ereal > real,G: real > extended_ereal,M: sigma_measure_real] :
( ( topolo1945291691832713340l_real @ top_to5683747375963461374_ereal @ F )
=> ( ( member7593515600531736322_ereal @ G @ ( sigma_3635921994777574992_ereal @ M @ borel_2631802743099733228_ereal ) )
=> ( member_real_real
@ ^ [X4: real] : ( F @ ( G @ X4 ) )
@ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).
% borel_measurable_continuous_on
thf(fact_963_borel__measurable__continuous__on,axiom,
! [F: real > real,G: real > real,M: sigma_measure_real] :
( ( topolo5044208981011980120l_real @ top_top_set_real @ F )
=> ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
=> ( member_real_real
@ ^ [X4: real] : ( F @ ( G @ X4 ) )
@ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).
% borel_measurable_continuous_on
thf(fact_964_rangeE,axiom,
! [B2: extended_ereal,F: nat > extended_ereal] :
( ( member2350847679896131959_ereal @ B2 @ ( image_4309273772856505399_ereal @ F @ top_top_set_nat ) )
=> ~ ! [X: nat] :
( B2
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_965_rangeE,axiom,
! [B2: complex,F: nat > complex] :
( ( member_complex @ B2 @ ( image_nat_complex @ F @ top_top_set_nat ) )
=> ~ ! [X: nat] :
( B2
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_966_rangeE,axiom,
! [B2: real,F: nat > real] :
( ( member_real @ B2 @ ( image_nat_real @ F @ top_top_set_nat ) )
=> ~ ! [X: nat] :
( B2
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_967_rangeE,axiom,
! [B2: $o,F: nat > $o] :
( ( member_o @ B2 @ ( image_nat_o @ F @ top_top_set_nat ) )
=> ~ ! [X: nat] :
( B2
= ( ~ ( F @ X ) ) ) ) ).
% rangeE
thf(fact_968_rangeE,axiom,
! [B2: real,F: real > real] :
( ( member_real @ B2 @ ( image_real_real @ F @ top_top_set_real ) )
=> ~ ! [X: real] :
( B2
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_969_rangeE,axiom,
! [B2: $o,F: real > $o] :
( ( member_o @ B2 @ ( image_real_o @ F @ top_top_set_real ) )
=> ~ ! [X: real] :
( B2
= ( ~ ( F @ X ) ) ) ) ).
% rangeE
thf(fact_970_rangeE,axiom,
! [B2: real,F: complex > real] :
( ( member_real @ B2 @ ( image_complex_real @ F @ top_top_set_complex ) )
=> ~ ! [X: complex] :
( B2
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_971_rangeE,axiom,
! [B2: $o,F: complex > $o] :
( ( member_o @ B2 @ ( image_complex_o @ F @ top_top_set_complex ) )
=> ~ ! [X: complex] :
( B2
= ( ~ ( F @ X ) ) ) ) ).
% rangeE
thf(fact_972_rangeE,axiom,
! [B2: real,F: extended_ereal > real] :
( ( member_real @ B2 @ ( image_2321174223038010293l_real @ F @ top_to5683747375963461374_ereal ) )
=> ~ ! [X: extended_ereal] :
( B2
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_973_rangeE,axiom,
! [B2: $o,F: extended_ereal > $o] :
( ( member_o @ B2 @ ( image_951975095941678543real_o @ F @ top_to5683747375963461374_ereal ) )
=> ~ ! [X: extended_ereal] :
( B2
= ( ~ ( F @ X ) ) ) ) ).
% rangeE
thf(fact_974_range__composition,axiom,
! [F: extended_ereal > extended_ereal,G: nat > extended_ereal] :
( ( image_4309273772856505399_ereal
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ top_top_set_nat )
= ( image_6042159593519690757_ereal @ F @ ( image_4309273772856505399_ereal @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_975_range__composition,axiom,
! [F: complex > extended_ereal,G: nat > complex] :
( ( image_4309273772856505399_ereal
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ top_top_set_nat )
= ( image_6025055836837866457_ereal @ F @ ( image_nat_complex @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_976_range__composition,axiom,
! [F: real > extended_ereal,G: nat > real] :
( ( image_4309273772856505399_ereal
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ top_top_set_nat )
= ( image_7147107595568778587_ereal @ F @ ( image_nat_real @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_977_range__composition,axiom,
! [F: nat > extended_ereal,G: nat > nat] :
( ( image_4309273772856505399_ereal
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ top_top_set_nat )
= ( image_4309273772856505399_ereal @ F @ ( image_nat_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_978_range__composition,axiom,
! [F: extended_ereal > complex,G: nat > extended_ereal] :
( ( image_nat_complex
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ top_top_set_nat )
= ( image_2124648478513430583omplex @ F @ ( image_4309273772856505399_ereal @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_979_range__composition,axiom,
! [F: complex > complex,G: nat > complex] :
( ( image_nat_complex
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ top_top_set_nat )
= ( image_1468599708987790691omplex @ F @ ( image_nat_complex @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_980_range__composition,axiom,
! [F: real > complex,G: nat > real] :
( ( image_nat_complex
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ top_top_set_nat )
= ( image_real_complex @ F @ ( image_nat_real @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_981_range__composition,axiom,
! [F: nat > complex,G: nat > nat] :
( ( image_nat_complex
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ top_top_set_nat )
= ( image_nat_complex @ F @ ( image_nat_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_982_range__composition,axiom,
! [F: extended_ereal > real,G: nat > extended_ereal] :
( ( image_nat_real
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ top_top_set_nat )
= ( image_2321174223038010293l_real @ F @ ( image_4309273772856505399_ereal @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_983_range__composition,axiom,
! [F: complex > real,G: nat > complex] :
( ( image_nat_real
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ top_top_set_nat )
= ( image_complex_real @ F @ ( image_nat_complex @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_984_measurable__ident__sets,axiom,
! [M: sigma_measure_real,M3: sigma_measure_real] :
( ( ( sigma_sets_real @ M )
= ( sigma_sets_real @ M3 ) )
=> ( member_real_real
@ ^ [X4: real] : X4
@ ( sigma_5267869275261027754l_real @ M @ M3 ) ) ) ).
% measurable_ident_sets
thf(fact_985_sets_Osets__Collect__const,axiom,
! [M: sigma_4258434043392614480l_real,P: $o] :
( member_set_real_real2
@ ( collect_real_real
@ ^ [X4: real > real] :
( ( member_real_real @ X4 @ ( sigma_3619470280215722479l_real @ M ) )
& P ) )
@ ( sigma_sets_real_real @ M ) ) ).
% sets.sets_Collect_const
thf(fact_986_sets_Osets__Collect__const,axiom,
! [M: sigma_measure_o,P: $o] :
( member_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( sigma_space_o @ M ) )
& P ) )
@ ( sigma_sets_o @ M ) ) ).
% sets.sets_Collect_const
thf(fact_987_sets_Osets__Collect__const,axiom,
! [M: sigma_3733394171116455995t_real,P: $o] :
( member_set_set_real
@ ( collect_set_real
@ ^ [X4: set_real] :
( ( member_set_real @ X4 @ ( sigma_space_set_real @ M ) )
& P ) )
@ ( sigma_sets_set_real @ M ) ) ).
% sets.sets_Collect_const
thf(fact_988_sets_Osets__Collect__const,axiom,
! [M: sigma_measure_real,P: $o] :
( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ M ) )
& P ) )
@ ( sigma_sets_real @ M ) ) ).
% sets.sets_Collect_const
thf(fact_989_sets_Osets__Collect__disj,axiom,
! [M: sigma_4258434043392614480l_real,P: ( real > real ) > $o,Q: ( real > real ) > $o] :
( ( member_set_real_real2
@ ( collect_real_real
@ ^ [X4: real > real] :
( ( member_real_real @ X4 @ ( sigma_3619470280215722479l_real @ M ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_real_real @ M ) )
=> ( ( member_set_real_real2
@ ( collect_real_real
@ ^ [X4: real > real] :
( ( member_real_real @ X4 @ ( sigma_3619470280215722479l_real @ M ) )
& ( Q @ X4 ) ) )
@ ( sigma_sets_real_real @ M ) )
=> ( member_set_real_real2
@ ( collect_real_real
@ ^ [X4: real > real] :
( ( member_real_real @ X4 @ ( sigma_3619470280215722479l_real @ M ) )
& ( ( Q @ X4 )
| ( P @ X4 ) ) ) )
@ ( sigma_sets_real_real @ M ) ) ) ) ).
% sets.sets_Collect_disj
thf(fact_990_sets_Osets__Collect__disj,axiom,
! [M: sigma_measure_o,P: $o > $o,Q: $o > $o] :
( ( member_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( sigma_space_o @ M ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_o @ M ) )
=> ( ( member_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( sigma_space_o @ M ) )
& ( Q @ X4 ) ) )
@ ( sigma_sets_o @ M ) )
=> ( member_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( sigma_space_o @ M ) )
& ( ( Q @ X4 )
| ( P @ X4 ) ) ) )
@ ( sigma_sets_o @ M ) ) ) ) ).
% sets.sets_Collect_disj
thf(fact_991_sets_Osets__Collect__disj,axiom,
! [M: sigma_3733394171116455995t_real,P: set_real > $o,Q: set_real > $o] :
( ( member_set_set_real
@ ( collect_set_real
@ ^ [X4: set_real] :
( ( member_set_real @ X4 @ ( sigma_space_set_real @ M ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_set_real @ M ) )
=> ( ( member_set_set_real
@ ( collect_set_real
@ ^ [X4: set_real] :
( ( member_set_real @ X4 @ ( sigma_space_set_real @ M ) )
& ( Q @ X4 ) ) )
@ ( sigma_sets_set_real @ M ) )
=> ( member_set_set_real
@ ( collect_set_real
@ ^ [X4: set_real] :
( ( member_set_real @ X4 @ ( sigma_space_set_real @ M ) )
& ( ( Q @ X4 )
| ( P @ X4 ) ) ) )
@ ( sigma_sets_set_real @ M ) ) ) ) ).
% sets.sets_Collect_disj
thf(fact_992_sets_Osets__Collect__disj,axiom,
! [M: sigma_measure_real,P: real > $o,Q: real > $o] :
( ( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ M ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_real @ M ) )
=> ( ( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ M ) )
& ( Q @ X4 ) ) )
@ ( sigma_sets_real @ M ) )
=> ( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ M ) )
& ( ( Q @ X4 )
| ( P @ X4 ) ) ) )
@ ( sigma_sets_real @ M ) ) ) ) ).
% sets.sets_Collect_disj
thf(fact_993_sets_Osets__Collect__conj,axiom,
! [M: sigma_4258434043392614480l_real,P: ( real > real ) > $o,Q: ( real > real ) > $o] :
( ( member_set_real_real2
@ ( collect_real_real
@ ^ [X4: real > real] :
( ( member_real_real @ X4 @ ( sigma_3619470280215722479l_real @ M ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_real_real @ M ) )
=> ( ( member_set_real_real2
@ ( collect_real_real
@ ^ [X4: real > real] :
( ( member_real_real @ X4 @ ( sigma_3619470280215722479l_real @ M ) )
& ( Q @ X4 ) ) )
@ ( sigma_sets_real_real @ M ) )
=> ( member_set_real_real2
@ ( collect_real_real
@ ^ [X4: real > real] :
( ( member_real_real @ X4 @ ( sigma_3619470280215722479l_real @ M ) )
& ( Q @ X4 )
& ( P @ X4 ) ) )
@ ( sigma_sets_real_real @ M ) ) ) ) ).
% sets.sets_Collect_conj
thf(fact_994_sets_Osets__Collect__conj,axiom,
! [M: sigma_measure_o,P: $o > $o,Q: $o > $o] :
( ( member_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( sigma_space_o @ M ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_o @ M ) )
=> ( ( member_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( sigma_space_o @ M ) )
& ( Q @ X4 ) ) )
@ ( sigma_sets_o @ M ) )
=> ( member_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( sigma_space_o @ M ) )
& ( Q @ X4 )
& ( P @ X4 ) ) )
@ ( sigma_sets_o @ M ) ) ) ) ).
% sets.sets_Collect_conj
thf(fact_995_sets_Osets__Collect__conj,axiom,
! [M: sigma_3733394171116455995t_real,P: set_real > $o,Q: set_real > $o] :
( ( member_set_set_real
@ ( collect_set_real
@ ^ [X4: set_real] :
( ( member_set_real @ X4 @ ( sigma_space_set_real @ M ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_set_real @ M ) )
=> ( ( member_set_set_real
@ ( collect_set_real
@ ^ [X4: set_real] :
( ( member_set_real @ X4 @ ( sigma_space_set_real @ M ) )
& ( Q @ X4 ) ) )
@ ( sigma_sets_set_real @ M ) )
=> ( member_set_set_real
@ ( collect_set_real
@ ^ [X4: set_real] :
( ( member_set_real @ X4 @ ( sigma_space_set_real @ M ) )
& ( Q @ X4 )
& ( P @ X4 ) ) )
@ ( sigma_sets_set_real @ M ) ) ) ) ).
% sets.sets_Collect_conj
thf(fact_996_sets_Osets__Collect__conj,axiom,
! [M: sigma_measure_real,P: real > $o,Q: real > $o] :
( ( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ M ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_real @ M ) )
=> ( ( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ M ) )
& ( Q @ X4 ) ) )
@ ( sigma_sets_real @ M ) )
=> ( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ M ) )
& ( Q @ X4 )
& ( P @ X4 ) ) )
@ ( sigma_sets_real @ M ) ) ) ) ).
% sets.sets_Collect_conj
thf(fact_997_sets_Osets__Collect__neg,axiom,
! [M: sigma_4258434043392614480l_real,P: ( real > real ) > $o] :
( ( member_set_real_real2
@ ( collect_real_real
@ ^ [X4: real > real] :
( ( member_real_real @ X4 @ ( sigma_3619470280215722479l_real @ M ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_real_real @ M ) )
=> ( member_set_real_real2
@ ( collect_real_real
@ ^ [X4: real > real] :
( ( member_real_real @ X4 @ ( sigma_3619470280215722479l_real @ M ) )
& ~ ( P @ X4 ) ) )
@ ( sigma_sets_real_real @ M ) ) ) ).
% sets.sets_Collect_neg
thf(fact_998_sets_Osets__Collect__neg,axiom,
! [M: sigma_measure_o,P: $o > $o] :
( ( member_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( sigma_space_o @ M ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_o @ M ) )
=> ( member_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( sigma_space_o @ M ) )
& ~ ( P @ X4 ) ) )
@ ( sigma_sets_o @ M ) ) ) ).
% sets.sets_Collect_neg
thf(fact_999_sets_Osets__Collect__neg,axiom,
! [M: sigma_3733394171116455995t_real,P: set_real > $o] :
( ( member_set_set_real
@ ( collect_set_real
@ ^ [X4: set_real] :
( ( member_set_real @ X4 @ ( sigma_space_set_real @ M ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_set_real @ M ) )
=> ( member_set_set_real
@ ( collect_set_real
@ ^ [X4: set_real] :
( ( member_set_real @ X4 @ ( sigma_space_set_real @ M ) )
& ~ ( P @ X4 ) ) )
@ ( sigma_sets_set_real @ M ) ) ) ).
% sets.sets_Collect_neg
thf(fact_1000_sets_Osets__Collect__neg,axiom,
! [M: sigma_measure_real,P: real > $o] :
( ( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ M ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_real @ M ) )
=> ( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ M ) )
& ~ ( P @ X4 ) ) )
@ ( sigma_sets_real @ M ) ) ) ).
% sets.sets_Collect_neg
thf(fact_1001_sets_Osets__Collect__imp,axiom,
! [M: sigma_4258434043392614480l_real,P: ( real > real ) > $o,Q: ( real > real ) > $o] :
( ( member_set_real_real2
@ ( collect_real_real
@ ^ [X4: real > real] :
( ( member_real_real @ X4 @ ( sigma_3619470280215722479l_real @ M ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_real_real @ M ) )
=> ( ( member_set_real_real2
@ ( collect_real_real
@ ^ [X4: real > real] :
( ( member_real_real @ X4 @ ( sigma_3619470280215722479l_real @ M ) )
& ( Q @ X4 ) ) )
@ ( sigma_sets_real_real @ M ) )
=> ( member_set_real_real2
@ ( collect_real_real
@ ^ [X4: real > real] :
( ( member_real_real @ X4 @ ( sigma_3619470280215722479l_real @ M ) )
& ( ( Q @ X4 )
=> ( P @ X4 ) ) ) )
@ ( sigma_sets_real_real @ M ) ) ) ) ).
% sets.sets_Collect_imp
thf(fact_1002_sets_Osets__Collect__imp,axiom,
! [M: sigma_measure_o,P: $o > $o,Q: $o > $o] :
( ( member_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( sigma_space_o @ M ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_o @ M ) )
=> ( ( member_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( sigma_space_o @ M ) )
& ( Q @ X4 ) ) )
@ ( sigma_sets_o @ M ) )
=> ( member_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( sigma_space_o @ M ) )
& ( ( Q @ X4 )
=> ( P @ X4 ) ) ) )
@ ( sigma_sets_o @ M ) ) ) ) ).
% sets.sets_Collect_imp
thf(fact_1003_sets_Osets__Collect__imp,axiom,
! [M: sigma_3733394171116455995t_real,P: set_real > $o,Q: set_real > $o] :
( ( member_set_set_real
@ ( collect_set_real
@ ^ [X4: set_real] :
( ( member_set_real @ X4 @ ( sigma_space_set_real @ M ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_set_real @ M ) )
=> ( ( member_set_set_real
@ ( collect_set_real
@ ^ [X4: set_real] :
( ( member_set_real @ X4 @ ( sigma_space_set_real @ M ) )
& ( Q @ X4 ) ) )
@ ( sigma_sets_set_real @ M ) )
=> ( member_set_set_real
@ ( collect_set_real
@ ^ [X4: set_real] :
( ( member_set_real @ X4 @ ( sigma_space_set_real @ M ) )
& ( ( Q @ X4 )
=> ( P @ X4 ) ) ) )
@ ( sigma_sets_set_real @ M ) ) ) ) ).
% sets.sets_Collect_imp
thf(fact_1004_sets_Osets__Collect__imp,axiom,
! [M: sigma_measure_real,P: real > $o,Q: real > $o] :
( ( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ M ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_real @ M ) )
=> ( ( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ M ) )
& ( Q @ X4 ) ) )
@ ( sigma_sets_real @ M ) )
=> ( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ M ) )
& ( ( Q @ X4 )
=> ( P @ X4 ) ) ) )
@ ( sigma_sets_real @ M ) ) ) ) ).
% sets.sets_Collect_imp
thf(fact_1005_sets_Osets__Collect_I5_J,axiom,
! [M: sigma_4258434043392614480l_real,Pb: $o] :
( member_set_real_real2
@ ( collect_real_real
@ ^ [X4: real > real] :
( ( member_real_real @ X4 @ ( sigma_3619470280215722479l_real @ M ) )
& Pb ) )
@ ( sigma_sets_real_real @ M ) ) ).
% sets.sets_Collect(5)
thf(fact_1006_sets_Osets__Collect_I5_J,axiom,
! [M: sigma_measure_o,Pb: $o] :
( member_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( sigma_space_o @ M ) )
& Pb ) )
@ ( sigma_sets_o @ M ) ) ).
% sets.sets_Collect(5)
thf(fact_1007_sets_Osets__Collect_I5_J,axiom,
! [M: sigma_3733394171116455995t_real,Pb: $o] :
( member_set_set_real
@ ( collect_set_real
@ ^ [X4: set_real] :
( ( member_set_real @ X4 @ ( sigma_space_set_real @ M ) )
& Pb ) )
@ ( sigma_sets_set_real @ M ) ) ).
% sets.sets_Collect(5)
thf(fact_1008_sets_Osets__Collect_I5_J,axiom,
! [M: sigma_measure_real,Pb: $o] :
( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ M ) )
& Pb ) )
@ ( sigma_sets_real @ M ) ) ).
% sets.sets_Collect(5)
thf(fact_1009_borel__measurable__const,axiom,
! [C: real,M: sigma_measure_real] :
( member_real_real
@ ^ [X4: real] : C
@ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ).
% borel_measurable_const
thf(fact_1010_measurable__const,axiom,
! [C: real,M3: sigma_measure_real,M: sigma_measure_real] :
( ( member_real @ C @ ( sigma_space_real @ M3 ) )
=> ( member_real_real
@ ^ [X4: real] : C
@ ( sigma_5267869275261027754l_real @ M @ M3 ) ) ) ).
% measurable_const
thf(fact_1011_simple__function__comp,axiom,
! [T4: real > real,M: sigma_measure_real,M3: sigma_measure_real,F: real > real] :
( ( member_real_real @ T4 @ ( sigma_5267869275261027754l_real @ M @ M3 ) )
=> ( ( nonneg485563716852976898l_real @ M3 @ F )
=> ( nonneg485563716852976898l_real @ M
@ ^ [X4: real] : ( F @ ( T4 @ X4 ) ) ) ) ) ).
% simple_function_comp
thf(fact_1012_continuous__on__subset,axiom,
! [S3: set_real,F: real > real,T3: set_real] :
( ( topolo5044208981011980120l_real @ S3 @ F )
=> ( ( ord_less_eq_set_real @ T3 @ S3 )
=> ( topolo5044208981011980120l_real @ T3 @ F ) ) ) ).
% continuous_on_subset
thf(fact_1013_continuous__on__subset,axiom,
! [S3: set_Extended_ereal,F: extended_ereal > extended_ereal,T3: set_Extended_ereal] :
( ( topolo6777079828818185726_ereal @ S3 @ F )
=> ( ( ord_le1644982726543182158_ereal @ T3 @ S3 )
=> ( topolo6777079828818185726_ereal @ T3 @ F ) ) ) ).
% continuous_on_subset
thf(fact_1014_Sup__eqI,axiom,
! [A: set_se4580700918925141924nnreal,X2: set_Ex3793607809372303086nnreal] :
( ! [Y3: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ Y3 @ A )
=> ( ord_le6787938422905777998nnreal @ Y3 @ X2 ) )
=> ( ! [Y3: set_Ex3793607809372303086nnreal] :
( ! [Z4: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ Z4 @ A )
=> ( ord_le6787938422905777998nnreal @ Z4 @ Y3 ) )
=> ( ord_le6787938422905777998nnreal @ X2 @ Y3 ) )
=> ( ( comple4226387801268262977nnreal @ A )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_1015_Sup__eqI,axiom,
! [A: set_se6634062954251873166_ereal,X2: set_Extended_ereal] :
( ! [Y3: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ Y3 @ A )
=> ( ord_le1644982726543182158_ereal @ Y3 @ X2 ) )
=> ( ! [Y3: set_Extended_ereal] :
( ! [Z4: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ Z4 @ A )
=> ( ord_le1644982726543182158_ereal @ Z4 @ Y3 ) )
=> ( ord_le1644982726543182158_ereal @ X2 @ Y3 ) )
=> ( ( comple4319282863272126363_ereal @ A )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_1016_Sup__eqI,axiom,
! [A: set_o,X2: $o] :
( ! [Y3: $o] :
( ( member_o @ Y3 @ A )
=> ( ord_less_eq_o @ Y3 @ X2 ) )
=> ( ! [Y3: $o] :
( ! [Z4: $o] :
( ( member_o @ Z4 @ A )
=> ( ord_less_eq_o @ Z4 @ Y3 ) )
=> ( ord_less_eq_o @ X2 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ A )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_1017_Sup__eqI,axiom,
! [A: set_set_real,X2: set_real] :
( ! [Y3: set_real] :
( ( member_set_real @ Y3 @ A )
=> ( ord_less_eq_set_real @ Y3 @ X2 ) )
=> ( ! [Y3: set_real] :
( ! [Z4: set_real] :
( ( member_set_real @ Z4 @ A )
=> ( ord_less_eq_set_real @ Z4 @ Y3 ) )
=> ( ord_less_eq_set_real @ X2 @ Y3 ) )
=> ( ( comple3096694443085538997t_real @ A )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_1018_Sup__mono,axiom,
! [A: set_se4580700918925141924nnreal,B: set_se4580700918925141924nnreal] :
( ! [A4: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ A4 @ A )
=> ? [X3: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X3 @ B )
& ( ord_le6787938422905777998nnreal @ A4 @ X3 ) ) )
=> ( ord_le6787938422905777998nnreal @ ( comple4226387801268262977nnreal @ A ) @ ( comple4226387801268262977nnreal @ B ) ) ) ).
% Sup_mono
thf(fact_1019_Sup__mono,axiom,
! [A: set_se6634062954251873166_ereal,B: set_se6634062954251873166_ereal] :
( ! [A4: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ A4 @ A )
=> ? [X3: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ X3 @ B )
& ( ord_le1644982726543182158_ereal @ A4 @ X3 ) ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ A ) @ ( comple4319282863272126363_ereal @ B ) ) ) ).
% Sup_mono
thf(fact_1020_Sup__mono,axiom,
! [A: set_o,B: set_o] :
( ! [A4: $o] :
( ( member_o @ A4 @ A )
=> ? [X3: $o] :
( ( member_o @ X3 @ B )
& ( ord_less_eq_o @ A4 @ X3 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_mono
thf(fact_1021_Sup__mono,axiom,
! [A: set_set_real,B: set_set_real] :
( ! [A4: set_real] :
( ( member_set_real @ A4 @ A )
=> ? [X3: set_real] :
( ( member_set_real @ X3 @ B )
& ( ord_less_eq_set_real @ A4 @ X3 ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ A ) @ ( comple3096694443085538997t_real @ B ) ) ) ).
% Sup_mono
thf(fact_1022_Sup__least,axiom,
! [A: set_se4580700918925141924nnreal,Z: set_Ex3793607809372303086nnreal] :
( ! [X: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X @ A )
=> ( ord_le6787938422905777998nnreal @ X @ Z ) )
=> ( ord_le6787938422905777998nnreal @ ( comple4226387801268262977nnreal @ A ) @ Z ) ) ).
% Sup_least
thf(fact_1023_Sup__least,axiom,
! [A: set_se6634062954251873166_ereal,Z: set_Extended_ereal] :
( ! [X: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ X @ A )
=> ( ord_le1644982726543182158_ereal @ X @ Z ) )
=> ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ A ) @ Z ) ) ).
% Sup_least
thf(fact_1024_Sup__least,axiom,
! [A: set_o,Z: $o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_o @ X @ Z ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ Z ) ) ).
% Sup_least
thf(fact_1025_Sup__least,axiom,
! [A: set_set_real,Z: set_real] :
( ! [X: set_real] :
( ( member_set_real @ X @ A )
=> ( ord_less_eq_set_real @ X @ Z ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ A ) @ Z ) ) ).
% Sup_least
thf(fact_1026_Sup__upper,axiom,
! [X2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ X2 @ A )
=> ( ord_le6787938422905777998nnreal @ X2 @ ( comple4226387801268262977nnreal @ A ) ) ) ).
% Sup_upper
thf(fact_1027_Sup__upper,axiom,
! [X2: set_Extended_ereal,A: set_se6634062954251873166_ereal] :
( ( member5519481007471526743_ereal @ X2 @ A )
=> ( ord_le1644982726543182158_ereal @ X2 @ ( comple4319282863272126363_ereal @ A ) ) ) ).
% Sup_upper
thf(fact_1028_Sup__upper,axiom,
! [X2: $o,A: set_o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_o @ X2 @ ( complete_Sup_Sup_o @ A ) ) ) ).
% Sup_upper
thf(fact_1029_Sup__upper,axiom,
! [X2: set_real,A: set_set_real] :
( ( member_set_real @ X2 @ A )
=> ( ord_less_eq_set_real @ X2 @ ( comple3096694443085538997t_real @ A ) ) ) ).
% Sup_upper
thf(fact_1030_Sup__le__iff,axiom,
! [A: set_se4580700918925141924nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ ( comple4226387801268262977nnreal @ A ) @ B2 )
= ( ! [X4: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X4 @ A )
=> ( ord_le6787938422905777998nnreal @ X4 @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_1031_Sup__le__iff,axiom,
! [A: set_se6634062954251873166_ereal,B2: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ A ) @ B2 )
= ( ! [X4: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ X4 @ A )
=> ( ord_le1644982726543182158_ereal @ X4 @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_1032_Sup__le__iff,axiom,
! [A: set_o,B2: $o] :
( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ B2 )
= ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_o @ X4 @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_1033_Sup__le__iff,axiom,
! [A: set_set_real,B2: set_real] :
( ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ A ) @ B2 )
= ( ! [X4: set_real] :
( ( member_set_real @ X4 @ A )
=> ( ord_less_eq_set_real @ X4 @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_1034_Sup__upper2,axiom,
! [U: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,V: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ U @ A )
=> ( ( ord_le6787938422905777998nnreal @ V @ U )
=> ( ord_le6787938422905777998nnreal @ V @ ( comple4226387801268262977nnreal @ A ) ) ) ) ).
% Sup_upper2
thf(fact_1035_Sup__upper2,axiom,
! [U: set_Extended_ereal,A: set_se6634062954251873166_ereal,V: set_Extended_ereal] :
( ( member5519481007471526743_ereal @ U @ A )
=> ( ( ord_le1644982726543182158_ereal @ V @ U )
=> ( ord_le1644982726543182158_ereal @ V @ ( comple4319282863272126363_ereal @ A ) ) ) ) ).
% Sup_upper2
thf(fact_1036_Sup__upper2,axiom,
! [U: $o,A: set_o,V: $o] :
( ( member_o @ U @ A )
=> ( ( ord_less_eq_o @ V @ U )
=> ( ord_less_eq_o @ V @ ( complete_Sup_Sup_o @ A ) ) ) ) ).
% Sup_upper2
thf(fact_1037_Sup__upper2,axiom,
! [U: set_real,A: set_set_real,V: set_real] :
( ( member_set_real @ U @ A )
=> ( ( ord_less_eq_set_real @ V @ U )
=> ( ord_less_eq_set_real @ V @ ( comple3096694443085538997t_real @ A ) ) ) ) ).
% Sup_upper2
thf(fact_1038_SUP__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > extended_ereal,D2: nat > extended_ereal] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ C2 @ A ) )
= ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_1039_SUP__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > real,D2: nat > real] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( comple1385675409528146559p_real @ ( image_nat_real @ C2 @ A ) )
= ( comple1385675409528146559p_real @ ( image_nat_real @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_1040_SUP__cong,axiom,
! [A: set_real,B: set_real,C2: real > real,D2: real > real] :
( ( A = B )
=> ( ! [X: real] :
( ( member_real @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( comple1385675409528146559p_real @ ( image_real_real @ C2 @ A ) )
= ( comple1385675409528146559p_real @ ( image_real_real @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_1041_SUP__cong,axiom,
! [A: set_o,B: set_o,C2: $o > real,D2: $o > real] :
( ( A = B )
=> ( ! [X: $o] :
( ( member_o @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( comple1385675409528146559p_real @ ( image_o_real @ C2 @ A ) )
= ( comple1385675409528146559p_real @ ( image_o_real @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_1042_SUP__cong,axiom,
! [A: set_real,B: set_real,C2: real > $o,D2: real > $o] :
( ( A = B )
=> ( ! [X: real] :
( ( member_real @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_real_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_real_o @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_1043_SUP__cong,axiom,
! [A: set_o,B: set_o,C2: $o > $o,D2: $o > $o] :
( ( A = B )
=> ( ! [X: $o] :
( ( member_o @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_1044_SUP__cong,axiom,
! [A: set_real,B: set_real,C2: real > nat,D2: real > nat] :
( ( A = B )
=> ( ! [X: real] :
( ( member_real @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_real_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_real_nat @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_1045_SUP__cong,axiom,
! [A: set_o,B: set_o,C2: $o > nat,D2: $o > nat] :
( ( A = B )
=> ( ! [X: $o] :
( ( member_o @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_o_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_o_nat @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_1046_SUP__cong,axiom,
! [A: set_set_real,B: set_set_real,C2: set_real > real,D2: set_real > real] :
( ( A = B )
=> ( ! [X: set_real] :
( ( member_set_real @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( comple1385675409528146559p_real @ ( image_set_real_real @ C2 @ A ) )
= ( comple1385675409528146559p_real @ ( image_set_real_real @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_1047_SUP__cong,axiom,
! [A: set_set_real,B: set_set_real,C2: set_real > $o,D2: set_real > $o] :
( ( A = B )
=> ( ! [X: set_real] :
( ( member_set_real @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_real_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_set_real_o @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_1048_mono__SUP,axiom,
! [F: extended_ereal > extended_ereal,A: nat > extended_ereal,I3: set_nat] :
( ( monoto2923698811514177639_ereal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_le1083603963089353582_ereal @ F )
=> ( ord_le1083603963089353582_ereal
@ ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X4: nat] : ( F @ ( A @ X4 ) )
@ I3 ) )
@ ( F @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ A @ I3 ) ) ) ) ) ).
% mono_SUP
thf(fact_1049_mono__SUP,axiom,
! [F: $o > extended_ereal,A: nat > $o,I3: set_nat] :
( ( monoto3224395847736110621_ereal @ top_top_set_o @ ord_less_eq_o @ ord_le1083603963089353582_ereal @ F )
=> ( ord_le1083603963089353582_ereal
@ ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X4: nat] : ( F @ ( A @ X4 ) )
@ I3 ) )
@ ( F @ ( complete_Sup_Sup_o @ ( image_nat_o @ A @ I3 ) ) ) ) ) ).
% mono_SUP
thf(fact_1050_mono__SUP,axiom,
! [F: extended_ereal > $o,A: nat > extended_ereal,I3: set_nat] :
( ( monoto5670193684399400497real_o @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_less_eq_o @ F )
=> ( ord_less_eq_o
@ ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [X4: nat] : ( F @ ( A @ X4 ) )
@ I3 ) )
@ ( F @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ A @ I3 ) ) ) ) ) ).
% mono_SUP
thf(fact_1051_mono__SUP,axiom,
! [F: extended_ereal > set_Ex3793607809372303086nnreal,A: nat > extended_ereal,I3: set_nat] :
( ( monoto6742655004523227285nnreal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_le6787938422905777998nnreal @ F )
=> ( ord_le6787938422905777998nnreal
@ ( comple4226387801268262977nnreal
@ ( image_3394822847079329989nnreal
@ ^ [X4: nat] : ( F @ ( A @ X4 ) )
@ I3 ) )
@ ( F @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ A @ I3 ) ) ) ) ) ).
% mono_SUP
thf(fact_1052_mono__SUP,axiom,
! [F: extended_ereal > set_Extended_ereal,A: nat > extended_ereal,I3: set_nat] :
( ( monoto1076656197419758151_ereal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_le1644982726543182158_ereal @ F )
=> ( ord_le1644982726543182158_ereal
@ ( comple4319282863272126363_ereal
@ ( image_305533323056406039_ereal
@ ^ [X4: nat] : ( F @ ( A @ X4 ) )
@ I3 ) )
@ ( F @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ A @ I3 ) ) ) ) ) ).
% mono_SUP
thf(fact_1053_mono__SUP,axiom,
! [F: set_Ex3793607809372303086nnreal > extended_ereal,A: nat > set_Ex3793607809372303086nnreal,I3: set_nat] :
( ( monoto2256078941756194231_ereal @ top_to3356475028079756884nnreal @ ord_le6787938422905777998nnreal @ ord_le1083603963089353582_ereal @ F )
=> ( ord_le1083603963089353582_ereal
@ ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X4: nat] : ( F @ ( A @ X4 ) )
@ I3 ) )
@ ( F @ ( comple4226387801268262977nnreal @ ( image_3394822847079329989nnreal @ A @ I3 ) ) ) ) ) ).
% mono_SUP
thf(fact_1054_mono__SUP,axiom,
! [F: set_Extended_ereal > extended_ereal,A: nat > set_Extended_ereal,I3: set_nat] :
( ( monoto5942119339478196871_ereal @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal @ ord_le1083603963089353582_ereal @ F )
=> ( ord_le1083603963089353582_ereal
@ ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X4: nat] : ( F @ ( A @ X4 ) )
@ I3 ) )
@ ( F @ ( comple4319282863272126363_ereal @ ( image_305533323056406039_ereal @ A @ I3 ) ) ) ) ) ).
% mono_SUP
thf(fact_1055_mono__SUP,axiom,
! [F: set_real > extended_ereal,A: nat > set_real,I3: set_nat] :
( ( monoto7491462358076921027_ereal @ top_top_set_set_real @ ord_less_eq_set_real @ ord_le1083603963089353582_ereal @ F )
=> ( ord_le1083603963089353582_ereal
@ ( comple8415311339701865915_ereal
@ ( image_4309273772856505399_ereal
@ ^ [X4: nat] : ( F @ ( A @ X4 ) )
@ I3 ) )
@ ( F @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ A @ I3 ) ) ) ) ) ).
% mono_SUP
thf(fact_1056_mono__SUP,axiom,
! [F: set_real > $o,A: nat > set_real,I3: set_nat] :
( ( monoto8032444043680750221real_o @ top_top_set_set_real @ ord_less_eq_set_real @ ord_less_eq_o @ F )
=> ( ord_less_eq_o
@ ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [X4: nat] : ( F @ ( A @ X4 ) )
@ I3 ) )
@ ( F @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ A @ I3 ) ) ) ) ) ).
% mono_SUP
thf(fact_1057_mono__SUP,axiom,
! [F: extended_ereal > set_real,A: nat > extended_ereal,I3: set_nat] :
( ( monoto8952765597322587401t_real @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_less_eq_set_real @ F )
=> ( ord_less_eq_set_real
@ ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [X4: nat] : ( F @ ( A @ X4 ) )
@ I3 ) )
@ ( F @ ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ A @ I3 ) ) ) ) ) ).
% mono_SUP
thf(fact_1058_mono__Sup,axiom,
! [F: extended_ereal > $o,A: set_Extended_ereal] :
( ( monoto5670193684399400497real_o @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_less_eq_o @ F )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_951975095941678543real_o @ F @ A ) ) @ ( F @ ( comple8415311339701865915_ereal @ A ) ) ) ) ).
% mono_Sup
thf(fact_1059_mono__Sup,axiom,
! [F: $o > $o,A: set_o] :
( ( monotone_on_o_o @ top_top_set_o @ ord_less_eq_o @ ord_less_eq_o @ F )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) @ ( F @ ( complete_Sup_Sup_o @ A ) ) ) ) ).
% mono_Sup
thf(fact_1060_mono__Sup,axiom,
! [F: extended_ereal > set_Ex3793607809372303086nnreal,A: set_Extended_ereal] :
( ( monoto6742655004523227285nnreal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_le6787938422905777998nnreal @ F )
=> ( ord_le6787938422905777998nnreal @ ( comple4226387801268262977nnreal @ ( image_6588766411312125047nnreal @ F @ A ) ) @ ( F @ ( comple8415311339701865915_ereal @ A ) ) ) ) ).
% mono_Sup
thf(fact_1061_mono__Sup,axiom,
! [F: extended_ereal > set_Extended_ereal,A: set_Extended_ereal] :
( ( monoto1076656197419758151_ereal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_le1644982726543182158_ereal @ F )
=> ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ ( image_5562094264469218789_ereal @ F @ A ) ) @ ( F @ ( comple8415311339701865915_ereal @ A ) ) ) ) ).
% mono_Sup
thf(fact_1062_mono__Sup,axiom,
! [F: $o > set_Ex3793607809372303086nnreal,A: set_o] :
( ( monoto2235544329619742815nnreal @ top_top_set_o @ ord_less_eq_o @ ord_le6787938422905777998nnreal @ F )
=> ( ord_le6787938422905777998nnreal @ ( comple4226387801268262977nnreal @ ( image_1679811975146592321nnreal @ F @ A ) ) @ ( F @ ( complete_Sup_Sup_o @ A ) ) ) ) ).
% mono_Sup
thf(fact_1063_mono__Sup,axiom,
! [F: $o > set_Extended_ereal,A: set_o] :
( ( monoto1696596325804605437_ereal @ top_top_set_o @ ord_less_eq_o @ ord_le1644982726543182158_ereal @ F )
=> ( ord_le1644982726543182158_ereal @ ( comple4319282863272126363_ereal @ ( image_6375117163256653723_ereal @ F @ A ) ) @ ( F @ ( complete_Sup_Sup_o @ A ) ) ) ) ).
% mono_Sup
thf(fact_1064_mono__Sup,axiom,
! [F: set_Ex3793607809372303086nnreal > $o,A: set_se4580700918925141924nnreal] :
( ( monoto3509817954306571137real_o @ top_to3356475028079756884nnreal @ ord_le6787938422905777998nnreal @ ord_less_eq_o @ F )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_2954085599833420643real_o @ F @ A ) ) @ ( F @ ( comple4226387801268262977nnreal @ A ) ) ) ) ).
% mono_Sup
thf(fact_1065_mono__Sup,axiom,
! [F: set_Extended_ereal > $o,A: set_se6634062954251873166_ereal] :
( ( monoto6491474119614906449real_o @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal @ ord_less_eq_o @ F )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_1946622920212178927real_o @ F @ A ) ) @ ( F @ ( comple4319282863272126363_ereal @ A ) ) ) ) ).
% mono_Sup
thf(fact_1066_mono__Sup,axiom,
! [F: set_real > $o,A: set_set_real] :
( ( monoto8032444043680750221real_o @ top_top_set_set_real @ ord_less_eq_set_real @ ord_less_eq_o @ F )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_set_real_o @ F @ A ) ) @ ( F @ ( comple3096694443085538997t_real @ A ) ) ) ) ).
% mono_Sup
thf(fact_1067_mono__Sup,axiom,
! [F: extended_ereal > set_real,A: set_Extended_ereal] :
( ( monoto8952765597322587401t_real @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_less_eq_set_real @ F )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_3288268784970258411t_real @ F @ A ) ) @ ( F @ ( comple8415311339701865915_ereal @ A ) ) ) ) ).
% mono_Sup
thf(fact_1068_continuous__on__id_H,axiom,
! [S3: set_real] : ( topolo5044208981011980120l_real @ S3 @ id_real ) ).
% continuous_on_id'
thf(fact_1069_continuous__on__id_H,axiom,
! [S3: set_Extended_ereal] : ( topolo6777079828818185726_ereal @ S3 @ id_Extended_ereal ) ).
% continuous_on_id'
thf(fact_1070_antimonoI,axiom,
! [F: real > real] :
( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X ) ) )
=> ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_eq_real
@ ^ [X4: real,Y2: real] : ( ord_less_eq_real @ Y2 @ X4 )
@ F ) ) ).
% antimonoI
thf(fact_1071_antimonoI,axiom,
! [F: real > nat] :
( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X ) ) )
=> ( monotone_on_real_nat @ top_top_set_real @ ord_less_eq_real
@ ^ [X4: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X4 )
@ F ) ) ).
% antimonoI
thf(fact_1072_antimonoI,axiom,
! [F: complex > nat] :
( ! [X: complex,Y3: complex] :
( ( ord_less_eq_complex @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X ) ) )
=> ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_eq_complex
@ ^ [X4: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X4 )
@ F ) ) ).
% antimonoI
thf(fact_1073_antimonoI,axiom,
! [F: extended_ereal > nat] :
( ! [X: extended_ereal,Y3: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X ) ) )
=> ( monoto2580034644210098551al_nat @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal
@ ^ [X4: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X4 )
@ F ) ) ).
% antimonoI
thf(fact_1074_antimonoI,axiom,
! [F: nat > extended_ereal] :
( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le1083603963089353582_ereal @ ( F @ Y3 ) @ ( F @ X ) ) )
=> ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: extended_ereal,Y2: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y2 @ X4 )
@ F ) ) ).
% antimonoI
thf(fact_1075_antimonoI,axiom,
! [F: nat > extend8495563244428889912nnreal] :
( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ Y3 ) @ ( F @ X ) ) )
=> ( monoto2291723841412853873nnreal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ Y2 @ X4 )
@ F ) ) ).
% antimonoI
thf(fact_1076_antimonoI,axiom,
! [F: nat > nat] :
( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X4 )
@ F ) ) ).
% antimonoI
thf(fact_1077_antimonoI,axiom,
! [F: real > set_Ex3793607809372303086nnreal] :
( ! [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
=> ( ord_le6787938422905777998nnreal @ ( F @ Y3 ) @ ( F @ X ) ) )
=> ( monoto2626391617355967235nnreal @ top_top_set_real @ ord_less_eq_real
@ ^ [X4: set_Ex3793607809372303086nnreal,Y2: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ Y2 @ X4 )
@ F ) ) ).
% antimonoI
thf(fact_1078_antimonoI,axiom,
! [F: complex > set_Ex3793607809372303086nnreal] :
( ! [X: complex,Y3: complex] :
( ( ord_less_eq_complex @ X @ Y3 )
=> ( ord_le6787938422905777998nnreal @ ( F @ Y3 ) @ ( F @ X ) ) )
=> ( monoto3400374153095604485nnreal @ top_top_set_complex @ ord_less_eq_complex
@ ^ [X4: set_Ex3793607809372303086nnreal,Y2: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ Y2 @ X4 )
@ F ) ) ).
% antimonoI
thf(fact_1079_antimonoI,axiom,
! [F: extended_ereal > set_Ex3793607809372303086nnreal] :
( ! [X: extended_ereal,Y3: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ Y3 )
=> ( ord_le6787938422905777998nnreal @ ( F @ Y3 ) @ ( F @ X ) ) )
=> ( monoto6742655004523227285nnreal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal
@ ^ [X4: set_Ex3793607809372303086nnreal,Y2: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ Y2 @ X4 )
@ F ) ) ).
% antimonoI
thf(fact_1080_antimonoE,axiom,
! [F: real > real,X2: real,Y4: real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_eq_real
@ ^ [X4: real,Y2: real] : ( ord_less_eq_real @ Y2 @ X4 )
@ F )
=> ( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X2 ) ) ) ) ).
% antimonoE
thf(fact_1081_antimonoE,axiom,
! [F: real > nat,X2: real,Y4: real] :
( ( monotone_on_real_nat @ top_top_set_real @ ord_less_eq_real
@ ^ [X4: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X4 )
@ F )
=> ( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X2 ) ) ) ) ).
% antimonoE
thf(fact_1082_antimonoE,axiom,
! [F: complex > nat,X2: complex,Y4: complex] :
( ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_eq_complex
@ ^ [X4: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X4 )
@ F )
=> ( ( ord_less_eq_complex @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X2 ) ) ) ) ).
% antimonoE
thf(fact_1083_antimonoE,axiom,
! [F: extended_ereal > nat,X2: extended_ereal,Y4: extended_ereal] :
( ( monoto2580034644210098551al_nat @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal
@ ^ [X4: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X4 )
@ F )
=> ( ( ord_le1083603963089353582_ereal @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X2 ) ) ) ) ).
% antimonoE
thf(fact_1084_antimonoE,axiom,
! [F: nat > extended_ereal,X2: nat,Y4: nat] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: extended_ereal,Y2: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y2 @ X4 )
@ F )
=> ( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_le1083603963089353582_ereal @ ( F @ Y4 ) @ ( F @ X2 ) ) ) ) ).
% antimonoE
thf(fact_1085_antimonoE,axiom,
! [F: nat > extend8495563244428889912nnreal,X2: nat,Y4: nat] :
( ( monoto2291723841412853873nnreal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ Y2 @ X4 )
@ F )
=> ( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_le3935885782089961368nnreal @ ( F @ Y4 ) @ ( F @ X2 ) ) ) ) ).
% antimonoE
thf(fact_1086_antimonoE,axiom,
! [F: nat > nat,X2: nat,Y4: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X4 )
@ F )
=> ( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X2 ) ) ) ) ).
% antimonoE
thf(fact_1087_antimonoE,axiom,
! [F: real > set_Ex3793607809372303086nnreal,X2: real,Y4: real] :
( ( monoto2626391617355967235nnreal @ top_top_set_real @ ord_less_eq_real
@ ^ [X4: set_Ex3793607809372303086nnreal,Y2: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ Y2 @ X4 )
@ F )
=> ( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_le6787938422905777998nnreal @ ( F @ Y4 ) @ ( F @ X2 ) ) ) ) ).
% antimonoE
thf(fact_1088_antimonoE,axiom,
! [F: complex > set_Ex3793607809372303086nnreal,X2: complex,Y4: complex] :
( ( monoto3400374153095604485nnreal @ top_top_set_complex @ ord_less_eq_complex
@ ^ [X4: set_Ex3793607809372303086nnreal,Y2: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ Y2 @ X4 )
@ F )
=> ( ( ord_less_eq_complex @ X2 @ Y4 )
=> ( ord_le6787938422905777998nnreal @ ( F @ Y4 ) @ ( F @ X2 ) ) ) ) ).
% antimonoE
thf(fact_1089_antimonoE,axiom,
! [F: extended_ereal > set_Ex3793607809372303086nnreal,X2: extended_ereal,Y4: extended_ereal] :
( ( monoto6742655004523227285nnreal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal
@ ^ [X4: set_Ex3793607809372303086nnreal,Y2: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ Y2 @ X4 )
@ F )
=> ( ( ord_le1083603963089353582_ereal @ X2 @ Y4 )
=> ( ord_le6787938422905777998nnreal @ ( F @ Y4 ) @ ( F @ X2 ) ) ) ) ).
% antimonoE
thf(fact_1090_antimonoD,axiom,
! [F: real > real,X2: real,Y4: real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_eq_real
@ ^ [X4: real,Y2: real] : ( ord_less_eq_real @ Y2 @ X4 )
@ F )
=> ( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X2 ) ) ) ) ).
% antimonoD
thf(fact_1091_antimonoD,axiom,
! [F: real > nat,X2: real,Y4: real] :
( ( monotone_on_real_nat @ top_top_set_real @ ord_less_eq_real
@ ^ [X4: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X4 )
@ F )
=> ( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X2 ) ) ) ) ).
% antimonoD
thf(fact_1092_antimonoD,axiom,
! [F: complex > nat,X2: complex,Y4: complex] :
( ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_eq_complex
@ ^ [X4: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X4 )
@ F )
=> ( ( ord_less_eq_complex @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X2 ) ) ) ) ).
% antimonoD
thf(fact_1093_antimonoD,axiom,
! [F: extended_ereal > nat,X2: extended_ereal,Y4: extended_ereal] :
( ( monoto2580034644210098551al_nat @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal
@ ^ [X4: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X4 )
@ F )
=> ( ( ord_le1083603963089353582_ereal @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X2 ) ) ) ) ).
% antimonoD
thf(fact_1094_antimonoD,axiom,
! [F: nat > extended_ereal,X2: nat,Y4: nat] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: extended_ereal,Y2: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y2 @ X4 )
@ F )
=> ( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_le1083603963089353582_ereal @ ( F @ Y4 ) @ ( F @ X2 ) ) ) ) ).
% antimonoD
thf(fact_1095_antimonoD,axiom,
! [F: nat > extend8495563244428889912nnreal,X2: nat,Y4: nat] :
( ( monoto2291723841412853873nnreal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ Y2 @ X4 )
@ F )
=> ( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_le3935885782089961368nnreal @ ( F @ Y4 ) @ ( F @ X2 ) ) ) ) ).
% antimonoD
thf(fact_1096_antimonoD,axiom,
! [F: nat > nat,X2: nat,Y4: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X4 )
@ F )
=> ( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X2 ) ) ) ) ).
% antimonoD
thf(fact_1097_antimonoD,axiom,
! [F: real > set_Ex3793607809372303086nnreal,X2: real,Y4: real] :
( ( monoto2626391617355967235nnreal @ top_top_set_real @ ord_less_eq_real
@ ^ [X4: set_Ex3793607809372303086nnreal,Y2: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ Y2 @ X4 )
@ F )
=> ( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ord_le6787938422905777998nnreal @ ( F @ Y4 ) @ ( F @ X2 ) ) ) ) ).
% antimonoD
thf(fact_1098_antimonoD,axiom,
! [F: complex > set_Ex3793607809372303086nnreal,X2: complex,Y4: complex] :
( ( monoto3400374153095604485nnreal @ top_top_set_complex @ ord_less_eq_complex
@ ^ [X4: set_Ex3793607809372303086nnreal,Y2: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ Y2 @ X4 )
@ F )
=> ( ( ord_less_eq_complex @ X2 @ Y4 )
=> ( ord_le6787938422905777998nnreal @ ( F @ Y4 ) @ ( F @ X2 ) ) ) ) ).
% antimonoD
thf(fact_1099_antimonoD,axiom,
! [F: extended_ereal > set_Ex3793607809372303086nnreal,X2: extended_ereal,Y4: extended_ereal] :
( ( monoto6742655004523227285nnreal @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal
@ ^ [X4: set_Ex3793607809372303086nnreal,Y2: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ Y2 @ X4 )
@ F )
=> ( ( ord_le1083603963089353582_ereal @ X2 @ Y4 )
=> ( ord_le6787938422905777998nnreal @ ( F @ Y4 ) @ ( F @ X2 ) ) ) ) ).
% antimonoD
thf(fact_1100_monoseq__iff,axiom,
( topolo608505905947791073_ereal
= ( ^ [X6: nat > extended_ereal] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ X6 )
| ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: extended_ereal,Y2: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y2 @ X4 )
@ X6 ) ) ) ) ).
% monoseq_iff
thf(fact_1101_monoseq__iff,axiom,
( topolo2569500529754793189nnreal
= ( ^ [X6: nat > extend8495563244428889912nnreal] :
( ( monoto2291723841412853873nnreal @ top_top_set_nat @ ord_less_eq_nat @ ord_le3935885782089961368nnreal @ X6 )
| ( monoto2291723841412853873nnreal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ Y2 @ X4 )
@ X6 ) ) ) ) ).
% monoseq_iff
thf(fact_1102_monoseq__iff,axiom,
( topolo1981301258947743899nnreal
= ( ^ [X6: nat > set_Ex3793607809372303086nnreal] :
( ( monoto4660286046138248231nnreal @ top_top_set_nat @ ord_less_eq_nat @ ord_le6787938422905777998nnreal @ X6 )
| ( monoto4660286046138248231nnreal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: set_Ex3793607809372303086nnreal,Y2: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ Y2 @ X4 )
@ X6 ) ) ) ) ).
% monoseq_iff
thf(fact_1103_monoseq__iff,axiom,
( topolo6736312753545056449_ereal
= ( ^ [X6: nat > set_Extended_ereal] :
( ( monoto6788471982328799797_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1644982726543182158_ereal @ X6 )
| ( monoto6788471982328799797_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: set_Extended_ereal,Y2: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ Y2 @ X4 )
@ X6 ) ) ) ) ).
% monoseq_iff
thf(fact_1104_monoseq__iff,axiom,
( topolo2489691266198938127t_real
= ( ^ [X6: nat > set_real] :
( ( monoto7274299666542614427t_real @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_real @ X6 )
| ( monoto7274299666542614427t_real @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: set_real,Y2: set_real] : ( ord_less_eq_set_real @ Y2 @ X4 )
@ X6 ) ) ) ) ).
% monoseq_iff
thf(fact_1105_monoseq__iff,axiom,
( topolo4902158794631467389eq_nat
= ( ^ [X6: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ X6 )
| ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X4 )
@ X6 ) ) ) ) ).
% monoseq_iff
thf(fact_1106_monoseq__iff,axiom,
( topolo6106433626167785124l_real
= ( ^ [X6: nat > real > real] :
( ( monoto2824216093323351088l_real @ top_top_set_nat @ ord_less_eq_nat @ ord_le6948328307412524503l_real @ X6 )
| ( monoto2824216093323351088l_real @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: real > real,Y2: real > real] : ( ord_le6948328307412524503l_real @ Y2 @ X4 )
@ X6 ) ) ) ) ).
% monoseq_iff
thf(fact_1107_decseq__imp__monoseq,axiom,
! [X5: nat > extended_ereal] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: extended_ereal,Y2: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y2 @ X4 )
@ X5 )
=> ( topolo608505905947791073_ereal @ X5 ) ) ).
% decseq_imp_monoseq
thf(fact_1108_decseq__imp__monoseq,axiom,
! [X5: nat > extend8495563244428889912nnreal] :
( ( monoto2291723841412853873nnreal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ Y2 @ X4 )
@ X5 )
=> ( topolo2569500529754793189nnreal @ X5 ) ) ).
% decseq_imp_monoseq
thf(fact_1109_decseq__imp__monoseq,axiom,
! [X5: nat > set_Ex3793607809372303086nnreal] :
( ( monoto4660286046138248231nnreal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: set_Ex3793607809372303086nnreal,Y2: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ Y2 @ X4 )
@ X5 )
=> ( topolo1981301258947743899nnreal @ X5 ) ) ).
% decseq_imp_monoseq
thf(fact_1110_decseq__imp__monoseq,axiom,
! [X5: nat > set_Extended_ereal] :
( ( monoto6788471982328799797_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: set_Extended_ereal,Y2: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ Y2 @ X4 )
@ X5 )
=> ( topolo6736312753545056449_ereal @ X5 ) ) ).
% decseq_imp_monoseq
thf(fact_1111_decseq__imp__monoseq,axiom,
! [X5: nat > set_real] :
( ( monoto7274299666542614427t_real @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: set_real,Y2: set_real] : ( ord_less_eq_set_real @ Y2 @ X4 )
@ X5 )
=> ( topolo2489691266198938127t_real @ X5 ) ) ).
% decseq_imp_monoseq
thf(fact_1112_decseq__imp__monoseq,axiom,
! [X5: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X4 )
@ X5 )
=> ( topolo4902158794631467389eq_nat @ X5 ) ) ).
% decseq_imp_monoseq
thf(fact_1113_decseq__imp__monoseq,axiom,
! [X5: nat > real > real] :
( ( monoto2824216093323351088l_real @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: real > real,Y2: real > real] : ( ord_le6948328307412524503l_real @ Y2 @ X4 )
@ X5 )
=> ( topolo6106433626167785124l_real @ X5 ) ) ).
% decseq_imp_monoseq
thf(fact_1114_measurable__If,axiom,
! [F: real > real,M: sigma_measure_real,M3: sigma_measure_real,G: real > real,P: real > $o] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ M3 ) )
=> ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ M3 ) )
=> ( ( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ M ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_real @ M ) )
=> ( member_real_real
@ ^ [X4: real] : ( if_real @ ( P @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
@ ( sigma_5267869275261027754l_real @ M @ M3 ) ) ) ) ) ).
% measurable_If
thf(fact_1115_measurable__sets__Collect,axiom,
! [F: $o > $o,M: sigma_measure_o,N: sigma_measure_o,P: $o > $o] :
( ( member_o_o @ F @ ( sigma_measurable_o_o @ M @ N ) )
=> ( ( member_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( sigma_space_o @ N ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_o @ N ) )
=> ( member_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( sigma_space_o @ M ) )
& ( P @ ( F @ X4 ) ) ) )
@ ( sigma_sets_o @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_1116_measurable__sets__Collect,axiom,
! [F: real > $o,M: sigma_measure_real,N: sigma_measure_o,P: $o > $o] :
( ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ M @ N ) )
=> ( ( member_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( sigma_space_o @ N ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_o @ N ) )
=> ( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ M ) )
& ( P @ ( F @ X4 ) ) ) )
@ ( sigma_sets_real @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_1117_measurable__sets__Collect,axiom,
! [F: $o > real,M: sigma_measure_o,N: sigma_measure_real,P: real > $o] :
( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ N ) )
=> ( ( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ N ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_real @ N ) )
=> ( member_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( sigma_space_o @ M ) )
& ( P @ ( F @ X4 ) ) ) )
@ ( sigma_sets_o @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_1118_measurable__sets__Collect,axiom,
! [F: real > real,M: sigma_measure_real,N: sigma_measure_real,P: real > $o] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) )
=> ( ( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ N ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_real @ N ) )
=> ( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ M ) )
& ( P @ ( F @ X4 ) ) ) )
@ ( sigma_sets_real @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_1119_measurable__sets__Collect,axiom,
! [F: set_real > $o,M: sigma_3733394171116455995t_real,N: sigma_measure_o,P: $o > $o] :
( ( member_set_real_o @ F @ ( sigma_6120279872303721572real_o @ M @ N ) )
=> ( ( member_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( sigma_space_o @ N ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_o @ N ) )
=> ( member_set_set_real
@ ( collect_set_real
@ ^ [X4: set_real] :
( ( member_set_real @ X4 @ ( sigma_space_set_real @ M ) )
& ( P @ ( F @ X4 ) ) ) )
@ ( sigma_sets_set_real @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_1120_measurable__sets__Collect,axiom,
! [F: $o > set_real,M: sigma_measure_o,N: sigma_3733394171116455995t_real,P: set_real > $o] :
( ( member_o_set_real @ F @ ( sigma_4088809687498539434t_real @ M @ N ) )
=> ( ( member_set_set_real
@ ( collect_set_real
@ ^ [X4: set_real] :
( ( member_set_real @ X4 @ ( sigma_space_set_real @ N ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_set_real @ N ) )
=> ( member_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( sigma_space_o @ M ) )
& ( P @ ( F @ X4 ) ) ) )
@ ( sigma_sets_o @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_1121_measurable__sets__Collect,axiom,
! [F: real > set_real,M: sigma_measure_real,N: sigma_3733394171116455995t_real,P: set_real > $o] :
( ( member_real_set_real @ F @ ( sigma_6606012509476713952t_real @ M @ N ) )
=> ( ( member_set_set_real
@ ( collect_set_real
@ ^ [X4: set_real] :
( ( member_set_real @ X4 @ ( sigma_space_set_real @ N ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_set_real @ N ) )
=> ( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ M ) )
& ( P @ ( F @ X4 ) ) ) )
@ ( sigma_sets_real @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_1122_measurable__sets__Collect,axiom,
! [F: set_real > real,M: sigma_3733394171116455995t_real,N: sigma_measure_real,P: real > $o] :
( ( member_set_real_real @ F @ ( sigma_397049400287467232l_real @ M @ N ) )
=> ( ( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ N ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_real @ N ) )
=> ( member_set_set_real
@ ( collect_set_real
@ ^ [X4: set_real] :
( ( member_set_real @ X4 @ ( sigma_space_set_real @ M ) )
& ( P @ ( F @ X4 ) ) ) )
@ ( sigma_sets_set_real @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_1123_measurable__sets__Collect,axiom,
! [F: $o > real > real,M: sigma_measure_o,N: sigma_4258434043392614480l_real,P: ( real > real ) > $o] :
( ( member_o_real_real @ F @ ( sigma_1778443616946741311l_real @ M @ N ) )
=> ( ( member_set_real_real2
@ ( collect_real_real
@ ^ [X4: real > real] :
( ( member_real_real @ X4 @ ( sigma_3619470280215722479l_real @ N ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_real_real @ N ) )
=> ( member_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( sigma_space_o @ M ) )
& ( P @ ( F @ X4 ) ) ) )
@ ( sigma_sets_o @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_1124_measurable__sets__Collect,axiom,
! [F: ( real > real ) > $o,M: sigma_4258434043392614480l_real,N: sigma_measure_o,P: $o > $o] :
( ( member_real_real_o @ F @ ( sigma_3869891368983680143real_o @ M @ N ) )
=> ( ( member_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( sigma_space_o @ N ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_o @ N ) )
=> ( member_set_real_real2
@ ( collect_real_real
@ ^ [X4: real > real] :
( ( member_real_real @ X4 @ ( sigma_3619470280215722479l_real @ M ) )
& ( P @ ( F @ X4 ) ) ) )
@ ( sigma_sets_real_real @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_1125_sets__Collect__restrict__space__iff,axiom,
! [S: set_real_real,M: sigma_4258434043392614480l_real,P: ( real > real ) > $o] :
( ( member_set_real_real2 @ S @ ( sigma_sets_real_real @ M ) )
=> ( ( member_set_real_real2
@ ( collect_real_real
@ ^ [X4: real > real] :
( ( member_real_real @ X4 @ ( sigma_3619470280215722479l_real @ ( sigma_1174395894077247795l_real @ M @ S ) ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_real_real @ ( sigma_1174395894077247795l_real @ M @ S ) ) )
= ( member_set_real_real2
@ ( collect_real_real
@ ^ [X4: real > real] :
( ( member_real_real @ X4 @ ( sigma_3619470280215722479l_real @ M ) )
& ( member_real_real @ X4 @ S )
& ( P @ X4 ) ) )
@ ( sigma_sets_real_real @ M ) ) ) ) ).
% sets_Collect_restrict_space_iff
thf(fact_1126_sets__Collect__restrict__space__iff,axiom,
! [S: set_o,M: sigma_measure_o,P: $o > $o] :
( ( member_set_o @ S @ ( sigma_sets_o @ M ) )
=> ( ( member_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( sigma_space_o @ ( sigma_8520893325391096540pace_o @ M @ S ) ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_o @ ( sigma_8520893325391096540pace_o @ M @ S ) ) )
= ( member_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( sigma_space_o @ M ) )
& ( member_o @ X4 @ S )
& ( P @ X4 ) ) )
@ ( sigma_sets_o @ M ) ) ) ) ).
% sets_Collect_restrict_space_iff
thf(fact_1127_sets__Collect__restrict__space__iff,axiom,
! [S: set_set_real,M: sigma_3733394171116455995t_real,P: set_real > $o] :
( ( member_set_set_real @ S @ ( sigma_sets_set_real @ M ) )
=> ( ( member_set_set_real
@ ( collect_set_real
@ ^ [X4: set_real] :
( ( member_set_real @ X4 @ ( sigma_space_set_real @ ( sigma_8303079596608756382t_real @ M @ S ) ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_set_real @ ( sigma_8303079596608756382t_real @ M @ S ) ) )
= ( member_set_set_real
@ ( collect_set_real
@ ^ [X4: set_real] :
( ( member_set_real @ X4 @ ( sigma_space_set_real @ M ) )
& ( member_set_real @ X4 @ S )
& ( P @ X4 ) ) )
@ ( sigma_sets_set_real @ M ) ) ) ) ).
% sets_Collect_restrict_space_iff
thf(fact_1128_sets__Collect__restrict__space__iff,axiom,
! [S: set_real,M: sigma_measure_real,P: real > $o] :
( ( member_set_real @ S @ ( sigma_sets_real @ M ) )
=> ( ( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ ( sigma_5414646170262037096e_real @ M @ S ) ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ M @ S ) ) )
= ( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ M ) )
& ( member_real @ X4 @ S )
& ( P @ X4 ) ) )
@ ( sigma_sets_real @ M ) ) ) ) ).
% sets_Collect_restrict_space_iff
thf(fact_1129_simple__function__If,axiom,
! [M: sigma_measure_real,F: real > real,G: real > real,P: real > $o] :
( ( nonneg485563716852976898l_real @ M @ F )
=> ( ( nonneg485563716852976898l_real @ M @ G )
=> ( ( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ M ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_real @ M ) )
=> ( nonneg485563716852976898l_real @ M
@ ^ [X4: real] : ( if_real @ ( P @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% simple_function_If
thf(fact_1130_Lebesgue__Measure_Ointegrable__restrict__UNIV,axiom,
! [S: set_real,F: real > real] :
( ( member_set_real @ S @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) ) )
=> ( ( bochne3340023020068487468l_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real )
@ ^ [X4: real] : ( if_real @ ( member_real @ X4 @ S ) @ ( F @ X4 ) @ zero_zero_real ) )
= ( bochne3340023020068487468l_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ S ) @ F ) ) ) ).
% Lebesgue_Measure.integrable_restrict_UNIV
thf(fact_1131_lborelD__Collect,axiom,
! [P: real > $o] :
( ( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ borel_5078946678739801102l_real ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
=> ( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ lebesgue_lborel_real ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_real @ lebesgue_lborel_real ) ) ) ).
% lborelD_Collect
thf(fact_1132_borel__measurable__eq,axiom,
! [F: ( real > real ) > real,M: sigma_4258434043392614480l_real,G: ( real > real ) > real] :
( ( member5749659578190367193l_real @ F @ ( sigma_314226851082786037l_real @ M @ borel_5078946678739801102l_real ) )
=> ( ( member5749659578190367193l_real @ G @ ( sigma_314226851082786037l_real @ M @ borel_5078946678739801102l_real ) )
=> ( member_set_real_real2
@ ( collect_real_real
@ ^ [W2: real > real] :
( ( member_real_real @ W2 @ ( sigma_3619470280215722479l_real @ M ) )
& ( ( F @ W2 )
= ( G @ W2 ) ) ) )
@ ( sigma_sets_real_real @ M ) ) ) ) ).
% borel_measurable_eq
thf(fact_1133_borel__measurable__eq,axiom,
! [F: $o > real,M: sigma_measure_o,G: $o > real] :
( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ borel_5078946678739801102l_real ) )
=> ( ( member_o_real @ G @ ( sigma_2430008634441611636o_real @ M @ borel_5078946678739801102l_real ) )
=> ( member_set_o
@ ( collect_o
@ ^ [W2: $o] :
( ( member_o @ W2 @ ( sigma_space_o @ M ) )
& ( ( F @ W2 )
= ( G @ W2 ) ) ) )
@ ( sigma_sets_o @ M ) ) ) ) ).
% borel_measurable_eq
thf(fact_1134_borel__measurable__eq,axiom,
! [F: set_real > real,M: sigma_3733394171116455995t_real,G: set_real > real] :
( ( member_set_real_real @ F @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) )
=> ( ( member_set_real_real @ G @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) )
=> ( member_set_set_real
@ ( collect_set_real
@ ^ [W2: set_real] :
( ( member_set_real @ W2 @ ( sigma_space_set_real @ M ) )
& ( ( F @ W2 )
= ( G @ W2 ) ) ) )
@ ( sigma_sets_set_real @ M ) ) ) ) ).
% borel_measurable_eq
thf(fact_1135_borel__measurable__eq,axiom,
! [F: real > real,M: sigma_measure_real,G: real > real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
=> ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
=> ( member_set_real
@ ( collect_real
@ ^ [W2: real] :
( ( member_real @ W2 @ ( sigma_space_real @ M ) )
& ( ( F @ W2 )
= ( G @ W2 ) ) ) )
@ ( sigma_sets_real @ M ) ) ) ) ).
% borel_measurable_eq
thf(fact_1136_borel__measurable__neq,axiom,
! [F: ( real > real ) > real,M: sigma_4258434043392614480l_real,G: ( real > real ) > real] :
( ( member5749659578190367193l_real @ F @ ( sigma_314226851082786037l_real @ M @ borel_5078946678739801102l_real ) )
=> ( ( member5749659578190367193l_real @ G @ ( sigma_314226851082786037l_real @ M @ borel_5078946678739801102l_real ) )
=> ( member_set_real_real2
@ ( collect_real_real
@ ^ [W2: real > real] :
( ( member_real_real @ W2 @ ( sigma_3619470280215722479l_real @ M ) )
& ( ( F @ W2 )
!= ( G @ W2 ) ) ) )
@ ( sigma_sets_real_real @ M ) ) ) ) ).
% borel_measurable_neq
thf(fact_1137_borel__measurable__neq,axiom,
! [F: $o > real,M: sigma_measure_o,G: $o > real] :
( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ borel_5078946678739801102l_real ) )
=> ( ( member_o_real @ G @ ( sigma_2430008634441611636o_real @ M @ borel_5078946678739801102l_real ) )
=> ( member_set_o
@ ( collect_o
@ ^ [W2: $o] :
( ( member_o @ W2 @ ( sigma_space_o @ M ) )
& ( ( F @ W2 )
!= ( G @ W2 ) ) ) )
@ ( sigma_sets_o @ M ) ) ) ) ).
% borel_measurable_neq
thf(fact_1138_borel__measurable__neq,axiom,
! [F: set_real > real,M: sigma_3733394171116455995t_real,G: set_real > real] :
( ( member_set_real_real @ F @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) )
=> ( ( member_set_real_real @ G @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) )
=> ( member_set_set_real
@ ( collect_set_real
@ ^ [W2: set_real] :
( ( member_set_real @ W2 @ ( sigma_space_set_real @ M ) )
& ( ( F @ W2 )
!= ( G @ W2 ) ) ) )
@ ( sigma_sets_set_real @ M ) ) ) ) ).
% borel_measurable_neq
thf(fact_1139_borel__measurable__neq,axiom,
! [F: real > real,M: sigma_measure_real,G: real > real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
=> ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
=> ( member_set_real
@ ( collect_real
@ ^ [W2: real] :
( ( member_real @ W2 @ ( sigma_space_real @ M ) )
& ( ( F @ W2 )
!= ( G @ W2 ) ) ) )
@ ( sigma_sets_real @ M ) ) ) ) ).
% borel_measurable_neq
thf(fact_1140_measurable__equality__set,axiom,
! [F: ( real > real ) > real,M: sigma_4258434043392614480l_real,G: ( real > real ) > real] :
( ( member5749659578190367193l_real @ F @ ( sigma_314226851082786037l_real @ M @ borel_5078946678739801102l_real ) )
=> ( ( member5749659578190367193l_real @ G @ ( sigma_314226851082786037l_real @ M @ borel_5078946678739801102l_real ) )
=> ( member_set_real_real2
@ ( collect_real_real
@ ^ [X4: real > real] :
( ( member_real_real @ X4 @ ( sigma_3619470280215722479l_real @ M ) )
& ( ( F @ X4 )
= ( G @ X4 ) ) ) )
@ ( sigma_sets_real_real @ M ) ) ) ) ).
% measurable_equality_set
thf(fact_1141_measurable__equality__set,axiom,
! [F: $o > real,M: sigma_measure_o,G: $o > real] :
( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ borel_5078946678739801102l_real ) )
=> ( ( member_o_real @ G @ ( sigma_2430008634441611636o_real @ M @ borel_5078946678739801102l_real ) )
=> ( member_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( sigma_space_o @ M ) )
& ( ( F @ X4 )
= ( G @ X4 ) ) ) )
@ ( sigma_sets_o @ M ) ) ) ) ).
% measurable_equality_set
thf(fact_1142_measurable__equality__set,axiom,
! [F: set_real > real,M: sigma_3733394171116455995t_real,G: set_real > real] :
( ( member_set_real_real @ F @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) )
=> ( ( member_set_real_real @ G @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) )
=> ( member_set_set_real
@ ( collect_set_real
@ ^ [X4: set_real] :
( ( member_set_real @ X4 @ ( sigma_space_set_real @ M ) )
& ( ( F @ X4 )
= ( G @ X4 ) ) ) )
@ ( sigma_sets_set_real @ M ) ) ) ) ).
% measurable_equality_set
thf(fact_1143_measurable__equality__set,axiom,
! [F: real > real,M: sigma_measure_real,G: real > real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
=> ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
=> ( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ M ) )
& ( ( F @ X4 )
= ( G @ X4 ) ) ) )
@ ( sigma_sets_real @ M ) ) ) ) ).
% measurable_equality_set
thf(fact_1144_decseqD,axiom,
! [F: nat > extended_ereal,I: nat,J: nat] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: extended_ereal,Y2: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y2 @ X4 )
@ F )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_le1083603963089353582_ereal @ ( F @ J ) @ ( F @ I ) ) ) ) ).
% decseqD
thf(fact_1145_decseqD,axiom,
! [F: nat > extend8495563244428889912nnreal,I: nat,J: nat] :
( ( monoto2291723841412853873nnreal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ Y2 @ X4 )
@ F )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_le3935885782089961368nnreal @ ( F @ J ) @ ( F @ I ) ) ) ) ).
% decseqD
thf(fact_1146_decseqD,axiom,
! [F: nat > set_Ex3793607809372303086nnreal,I: nat,J: nat] :
( ( monoto4660286046138248231nnreal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: set_Ex3793607809372303086nnreal,Y2: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ Y2 @ X4 )
@ F )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_le6787938422905777998nnreal @ ( F @ J ) @ ( F @ I ) ) ) ) ).
% decseqD
thf(fact_1147_decseqD,axiom,
! [F: nat > set_Extended_ereal,I: nat,J: nat] :
( ( monoto6788471982328799797_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: set_Extended_ereal,Y2: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ Y2 @ X4 )
@ F )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_le1644982726543182158_ereal @ ( F @ J ) @ ( F @ I ) ) ) ) ).
% decseqD
thf(fact_1148_decseqD,axiom,
! [F: nat > set_real,I: nat,J: nat] :
( ( monoto7274299666542614427t_real @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: set_real,Y2: set_real] : ( ord_less_eq_set_real @ Y2 @ X4 )
@ F )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_set_real @ ( F @ J ) @ ( F @ I ) ) ) ) ).
% decseqD
thf(fact_1149_decseqD,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X4 )
@ F )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ J ) @ ( F @ I ) ) ) ) ).
% decseqD
thf(fact_1150_decseqD,axiom,
! [F: nat > real > real,I: nat,J: nat] :
( ( monoto2824216093323351088l_real @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: real > real,Y2: real > real] : ( ord_le6948328307412524503l_real @ Y2 @ X4 )
@ F )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_le6948328307412524503l_real @ ( F @ J ) @ ( F @ I ) ) ) ) ).
% decseqD
thf(fact_1151_decseq__def,axiom,
! [X5: nat > extended_ereal] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: extended_ereal,Y2: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y2 @ X4 )
@ X5 )
= ( ! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_le1083603963089353582_ereal @ ( X5 @ N4 ) @ ( X5 @ M5 ) ) ) ) ) ).
% decseq_def
thf(fact_1152_decseq__def,axiom,
! [X5: nat > extend8495563244428889912nnreal] :
( ( monoto2291723841412853873nnreal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ Y2 @ X4 )
@ X5 )
= ( ! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_le3935885782089961368nnreal @ ( X5 @ N4 ) @ ( X5 @ M5 ) ) ) ) ) ).
% decseq_def
thf(fact_1153_decseq__def,axiom,
! [X5: nat > set_Ex3793607809372303086nnreal] :
( ( monoto4660286046138248231nnreal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: set_Ex3793607809372303086nnreal,Y2: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ Y2 @ X4 )
@ X5 )
= ( ! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_le6787938422905777998nnreal @ ( X5 @ N4 ) @ ( X5 @ M5 ) ) ) ) ) ).
% decseq_def
thf(fact_1154_decseq__def,axiom,
! [X5: nat > set_Extended_ereal] :
( ( monoto6788471982328799797_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: set_Extended_ereal,Y2: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ Y2 @ X4 )
@ X5 )
= ( ! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_le1644982726543182158_ereal @ ( X5 @ N4 ) @ ( X5 @ M5 ) ) ) ) ) ).
% decseq_def
thf(fact_1155_decseq__def,axiom,
! [X5: nat > set_real] :
( ( monoto7274299666542614427t_real @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: set_real,Y2: set_real] : ( ord_less_eq_set_real @ Y2 @ X4 )
@ X5 )
= ( ! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_less_eq_set_real @ ( X5 @ N4 ) @ ( X5 @ M5 ) ) ) ) ) ).
% decseq_def
thf(fact_1156_decseq__def,axiom,
! [X5: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X4 )
@ X5 )
= ( ! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_less_eq_nat @ ( X5 @ N4 ) @ ( X5 @ M5 ) ) ) ) ) ).
% decseq_def
thf(fact_1157_decseq__def,axiom,
! [X5: nat > real > real] :
( ( monoto2824216093323351088l_real @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: real > real,Y2: real > real] : ( ord_le6948328307412524503l_real @ Y2 @ X4 )
@ X5 )
= ( ! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_le6948328307412524503l_real @ ( X5 @ N4 ) @ ( X5 @ M5 ) ) ) ) ) ).
% decseq_def
thf(fact_1158_measurable__If__restrict__space__iff,axiom,
! [M: sigma_measure_real,P: real > $o,F: real > real,G: real > real,N: sigma_measure_real] :
( ( member_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ ( sigma_space_real @ M ) )
& ( P @ X4 ) ) )
@ ( sigma_sets_real @ M ) )
=> ( ( member_real_real
@ ^ [X4: real] : ( if_real @ ( P @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
@ ( sigma_5267869275261027754l_real @ M @ N ) )
= ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ ( sigma_5414646170262037096e_real @ M @ ( collect_real @ P ) ) @ N ) )
& ( member_real_real @ G
@ ( sigma_5267869275261027754l_real
@ ( sigma_5414646170262037096e_real @ M
@ ( collect_real
@ ^ [X4: real] :
~ ( P @ X4 ) ) )
@ N ) ) ) ) ) ).
% measurable_If_restrict_space_iff
thf(fact_1159_mono__compose,axiom,
! [Q: real > real > real,F: real > real] :
( ( monoto8965231823629880588l_real @ top_top_set_real @ ord_less_eq_real @ ord_le6948328307412524503l_real @ Q )
=> ( monoto8965231823629880588l_real @ top_top_set_real @ ord_less_eq_real @ ord_le6948328307412524503l_real
@ ^ [I4: real,X4: real] : ( Q @ I4 @ ( F @ X4 ) ) ) ) ).
% mono_compose
thf(fact_1160_mono__compose,axiom,
! [Q: complex > real > real,F: real > real] :
( ( monoto35072715882444302l_real @ top_top_set_complex @ ord_less_eq_complex @ ord_le6948328307412524503l_real @ Q )
=> ( monoto35072715882444302l_real @ top_top_set_complex @ ord_less_eq_complex @ ord_le6948328307412524503l_real
@ ^ [I4: complex,X4: real] : ( Q @ I4 @ ( F @ X4 ) ) ) ) ).
% mono_compose
thf(fact_1161_mono__compose,axiom,
! [Q: extended_ereal > real > real,F: real > real] :
( ( monoto4113897589229538718l_real @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_le6948328307412524503l_real @ Q )
=> ( monoto4113897589229538718l_real @ top_to5683747375963461374_ereal @ ord_le1083603963089353582_ereal @ ord_le6948328307412524503l_real
@ ^ [I4: extended_ereal,X4: real] : ( Q @ I4 @ ( F @ X4 ) ) ) ) ).
% mono_compose
thf(fact_1162_mono__compose,axiom,
! [Q: set_Ex3793607809372303086nnreal > real > real,F: real > real] :
( ( monoto6196239568472585294l_real @ top_to3356475028079756884nnreal @ ord_le6787938422905777998nnreal @ ord_le6948328307412524503l_real @ Q )
=> ( monoto6196239568472585294l_real @ top_to3356475028079756884nnreal @ ord_le6787938422905777998nnreal @ ord_le6948328307412524503l_real
@ ^ [I4: set_Ex3793607809372303086nnreal,X4: real] : ( Q @ I4 @ ( F @ X4 ) ) ) ) ).
% mono_compose
thf(fact_1163_mono__compose,axiom,
! [Q: set_Extended_ereal > real > real,F: real > real] :
( ( monoto4211386846447127422l_real @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal @ ord_le6948328307412524503l_real @ Q )
=> ( monoto4211386846447127422l_real @ top_to4757929550322229470_ereal @ ord_le1644982726543182158_ereal @ ord_le6948328307412524503l_real
@ ^ [I4: set_Extended_ereal,X4: real] : ( Q @ I4 @ ( F @ X4 ) ) ) ) ).
% mono_compose
thf(fact_1164_mono__compose,axiom,
! [Q: set_real > real > real,F: real > real] :
( ( monoto7370776611129179714l_real @ top_top_set_set_real @ ord_less_eq_set_real @ ord_le6948328307412524503l_real @ Q )
=> ( monoto7370776611129179714l_real @ top_top_set_set_real @ ord_less_eq_set_real @ ord_le6948328307412524503l_real
@ ^ [I4: set_real,X4: real] : ( Q @ I4 @ ( F @ X4 ) ) ) ) ).
% mono_compose
thf(fact_1165_mono__compose,axiom,
! [Q: nat > real > real,F: real > real] :
( ( monoto2824216093323351088l_real @ top_top_set_nat @ ord_less_eq_nat @ ord_le6948328307412524503l_real @ Q )
=> ( monoto2824216093323351088l_real @ top_top_set_nat @ ord_less_eq_nat @ ord_le6948328307412524503l_real
@ ^ [I4: nat,X4: real] : ( Q @ I4 @ ( F @ X4 ) ) ) ) ).
% mono_compose
thf(fact_1166_mono__compose,axiom,
! [Q: ( real > real ) > real > real,F: real > real] :
( ( monoto4255458463005331543l_real @ top_to2071711978144146653l_real @ ord_le6948328307412524503l_real @ ord_le6948328307412524503l_real @ Q )
=> ( monoto4255458463005331543l_real @ top_to2071711978144146653l_real @ ord_le6948328307412524503l_real @ ord_le6948328307412524503l_real
@ ^ [I4: real > real,X4: real] : ( Q @ I4 @ ( F @ X4 ) ) ) ) ).
% mono_compose
thf(fact_1167_simple__function__lebesgue__if,axiom,
! [F: real > real,S: set_real] :
( ( nonneg485563716852976898l_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ F )
=> ( ( member_set_real @ S @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) ) )
=> ( nonneg485563716852976898l_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real )
@ ^ [X4: real] : ( if_real @ ( member_real @ X4 @ S ) @ ( F @ X4 ) @ zero_zero_real ) ) ) ) ).
% simple_function_lebesgue_if
thf(fact_1168_IVT_H,axiom,
! [F: extended_ereal > extended_ereal,A2: extended_ereal,Y4: extended_ereal,B2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( F @ A2 ) @ Y4 )
=> ( ( ord_le1083603963089353582_ereal @ Y4 @ ( F @ B2 ) )
=> ( ( ord_le1083603963089353582_ereal @ A2 @ B2 )
=> ( ( topolo6777079828818185726_ereal @ ( set_or2336185686312672771_ereal @ A2 @ B2 ) @ F )
=> ? [X: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A2 @ X )
& ( ord_le1083603963089353582_ereal @ X @ B2 )
& ( ( F @ X )
= Y4 ) ) ) ) ) ) ).
% IVT'
thf(fact_1169_IVT_H,axiom,
! [F: real > nat,A2: real,Y4: nat,B2: real] :
( ( ord_less_eq_nat @ ( F @ A2 ) @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( topolo2287203362918339196al_nat @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ F )
=> ? [X: real] :
( ( ord_less_eq_real @ A2 @ X )
& ( ord_less_eq_real @ X @ B2 )
& ( ( F @ X )
= Y4 ) ) ) ) ) ) ).
% IVT'
thf(fact_1170_IVT_H,axiom,
! [F: real > real,A2: real,Y4: real,B2: real] :
( ( ord_less_eq_real @ ( F @ A2 ) @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ F )
=> ? [X: real] :
( ( ord_less_eq_real @ A2 @ X )
& ( ord_less_eq_real @ X @ B2 )
& ( ( F @ X )
= Y4 ) ) ) ) ) ) ).
% IVT'
thf(fact_1171_IVT2_H,axiom,
! [F: extended_ereal > extended_ereal,B2: extended_ereal,Y4: extended_ereal,A2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( F @ B2 ) @ Y4 )
=> ( ( ord_le1083603963089353582_ereal @ Y4 @ ( F @ A2 ) )
=> ( ( ord_le1083603963089353582_ereal @ A2 @ B2 )
=> ( ( topolo6777079828818185726_ereal @ ( set_or2336185686312672771_ereal @ A2 @ B2 ) @ F )
=> ? [X: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A2 @ X )
& ( ord_le1083603963089353582_ereal @ X @ B2 )
& ( ( F @ X )
= Y4 ) ) ) ) ) ) ).
% IVT2'
thf(fact_1172_IVT2_H,axiom,
! [F: real > nat,B2: real,Y4: nat,A2: real] :
( ( ord_less_eq_nat @ ( F @ B2 ) @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ ( F @ A2 ) )
=> ( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( topolo2287203362918339196al_nat @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ F )
=> ? [X: real] :
( ( ord_less_eq_real @ A2 @ X )
& ( ord_less_eq_real @ X @ B2 )
& ( ( F @ X )
= Y4 ) ) ) ) ) ) ).
% IVT2'
thf(fact_1173_IVT2_H,axiom,
! [F: real > real,B2: real,Y4: real,A2: real] :
( ( ord_less_eq_real @ ( F @ B2 ) @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ ( F @ A2 ) )
=> ( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ F )
=> ? [X: real] :
( ( ord_less_eq_real @ A2 @ X )
& ( ord_less_eq_real @ X @ B2 )
& ( ( F @ X )
= Y4 ) ) ) ) ) ) ).
% IVT2'
thf(fact_1174_SUP__eq,axiom,
! [A: set_o,B: set_o,F: $o > set_real,G: $o > set_real] :
( ! [I5: $o] :
( ( member_o @ I5 @ A )
=> ? [X3: $o] :
( ( member_o @ X3 @ B )
& ( ord_less_eq_set_real @ ( F @ I5 ) @ ( G @ X3 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B )
=> ? [X3: $o] :
( ( member_o @ X3 @ A )
& ( ord_less_eq_set_real @ ( G @ J3 ) @ ( F @ X3 ) ) ) )
=> ( ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ A ) )
= ( comple3096694443085538997t_real @ ( image_o_set_real @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_1175_SUP__eq,axiom,
! [A: set_o,B: set_set_real,F: $o > set_real,G: set_real > set_real] :
( ! [I5: $o] :
( ( member_o @ I5 @ A )
=> ? [X3: set_real] :
( ( member_set_real @ X3 @ B )
& ( ord_less_eq_set_real @ ( F @ I5 ) @ ( G @ X3 ) ) ) )
=> ( ! [J3: set_real] :
( ( member_set_real @ J3 @ B )
=> ? [X3: $o] :
( ( member_o @ X3 @ A )
& ( ord_less_eq_set_real @ ( G @ J3 ) @ ( F @ X3 ) ) ) )
=> ( ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ A ) )
= ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_1176_SUP__eq,axiom,
! [A: set_set_real,B: set_nat,F: set_real > set_real,G: nat > set_real] :
( ! [I5: set_real] :
( ( member_set_real @ I5 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ( ord_less_eq_set_real @ ( F @ I5 ) @ ( G @ X3 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B )
=> ? [X3: set_real] :
( ( member_set_real @ X3 @ A )
& ( ord_less_eq_set_real @ ( G @ J3 ) @ ( F @ X3 ) ) ) )
=> ( ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ F @ A ) )
= ( comple3096694443085538997t_real @ ( image_nat_set_real @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_1177_SUP__eq,axiom,
! [A: set_set_real,B: set_real_real,F: set_real > set_real,G: ( real > real ) > set_real] :
( ! [I5: set_real] :
( ( member_set_real @ I5 @ A )
=> ? [X3: real > real] :
( ( member_real_real @ X3 @ B )
& ( ord_less_eq_set_real @ ( F @ I5 ) @ ( G @ X3 ) ) ) )
=> ( ! [J3: real > real] :
( ( member_real_real @ J3 @ B )
=> ? [X3: set_real] :
( ( member_set_real @ X3 @ A )
& ( ord_less_eq_set_real @ ( G @ J3 ) @ ( F @ X3 ) ) ) )
=> ( ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ F @ A ) )
= ( comple3096694443085538997t_real @ ( image_6663718904102175840t_real @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_1178_SUP__eq,axiom,
! [A: set_set_real,B: set_real,F: set_real > set_real,G: real > set_real] :
( ! [I5: set_real] :
( ( member_set_real @ I5 @ A )
=> ? [X3: real] :
( ( member_real @ X3 @ B )
& ( ord_less_eq_set_real @ ( F @ I5 ) @ ( G @ X3 ) ) ) )
=> ( ! [J3: real] :
( ( member_real @ J3 @ B )
=> ? [X3: set_real] :
( ( member_set_real @ X3 @ A )
& ( ord_less_eq_set_real @ ( G @ J3 ) @ ( F @ X3 ) ) ) )
=> ( ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ F @ A ) )
= ( comple3096694443085538997t_real @ ( image_real_set_real @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_1179_SUP__eq,axiom,
! [A: set_set_real,B: set_o,F: set_real > set_real,G: $o > set_real] :
( ! [I5: set_real] :
( ( member_set_real @ I5 @ A )
=> ? [X3: $o] :
( ( member_o @ X3 @ B )
& ( ord_less_eq_set_real @ ( F @ I5 ) @ ( G @ X3 ) ) ) )
=> ( ! [J3: $o] :
( ( member_o @ J3 @ B )
=> ? [X3: set_real] :
( ( member_set_real @ X3 @ A )
& ( ord_less_eq_set_real @ ( G @ J3 ) @ ( F @ X3 ) ) ) )
=> ( ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ F @ A ) )
= ( comple3096694443085538997t_real @ ( image_o_set_real @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_1180_SUP__eq,axiom,
! [A: set_set_real,B: set_set_real,F: set_real > set_real,G: set_real > set_real] :
( ! [I5: set_real] :
( ( member_set_real @ I5 @ A )
=> ? [X3: set_real] :
( ( member_set_real @ X3 @ B )
& ( ord_less_eq_set_real @ ( F @ I5 ) @ ( G @ X3 ) ) ) )
=> ( ! [J3: set_real] :
( ( member_set_real @ J3 @ B )
=> ? [X3: set_real] :
( ( member_set_real @ X3 @ A )
& ( ord_less_eq_set_real @ ( G @ J3 ) @ ( F @ X3 ) ) ) )
=> ( ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ F @ A ) )
= ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_1181_integrable__mono__on,axiom,
! [A2: real,B2: real,F: real > real] :
( ( monoto4017252874604999745l_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( bochne3340023020068487468l_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ ( set_or1222579329274155063t_real @ A2 @ B2 ) ) @ F ) ) ).
% integrable_mono_on
thf(fact_1182_real__non__denum,axiom,
~ ? [F3: nat > real] :
( ( image_nat_real @ F3 @ top_top_set_nat )
= top_top_set_real ) ).
% real_non_denum
thf(fact_1183_continuous__image__closed__interval,axiom,
! [A2: real,B2: real,F: real > real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ F )
=> ? [C4: real,D3: real] :
( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A2 @ B2 ) )
= ( set_or1222579329274155063t_real @ C4 @ D3 ) )
& ( ord_less_eq_real @ C4 @ D3 ) ) ) ) ).
% continuous_image_closed_interval
thf(fact_1184_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_1185_complex__non__denum,axiom,
~ ? [F3: nat > complex] :
( ( image_nat_complex @ F3 @ top_top_set_nat )
= top_top_set_complex ) ).
% complex_non_denum
thf(fact_1186_real__eq__0__iff__le__ge__0,axiom,
! [X2: real] :
( ( X2 = zero_zero_real )
= ( ( ord_less_eq_real @ zero_zero_real @ X2 )
& ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ X2 ) ) ) ) ).
% real_eq_0_iff_le_ge_0
thf(fact_1187_diff__0__eq__0,axiom,
! [N3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N3 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1188_diff__self__eq__0,axiom,
! [M6: nat] :
( ( minus_minus_nat @ M6 @ M6 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1189_diff__diff__cancel,axiom,
! [I: nat,N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( minus_minus_nat @ N3 @ ( minus_minus_nat @ N3 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1190_diff__is__0__eq,axiom,
! [M6: nat,N3: nat] :
( ( ( minus_minus_nat @ M6 @ N3 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M6 @ N3 ) ) ).
% diff_is_0_eq
thf(fact_1191_diff__is__0__eq_H,axiom,
! [M6: nat,N3: nat] :
( ( ord_less_eq_nat @ M6 @ N3 )
=> ( ( minus_minus_nat @ M6 @ N3 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1192_ereal__decseq__uminus,axiom,
! [F: nat > extended_ereal] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: extended_ereal,Y2: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y2 @ X4 )
@ ^ [X4: nat] : ( uminus27091377158695749_ereal @ ( F @ X4 ) ) )
= ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ F ) ) ).
% ereal_decseq_uminus
thf(fact_1193_ereal__incseq__uminus,axiom,
! [F: nat > extended_ereal] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal
@ ^ [X4: nat] : ( uminus27091377158695749_ereal @ ( F @ X4 ) ) )
= ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: extended_ereal,Y2: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y2 @ X4 )
@ F ) ) ).
% ereal_incseq_uminus
thf(fact_1194_eq__diff__iff,axiom,
! [K: nat,M6: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M6 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( ( minus_minus_nat @ M6 @ K )
= ( minus_minus_nat @ N3 @ K ) )
= ( M6 = N3 ) ) ) ) ).
% eq_diff_iff
thf(fact_1195_le__diff__iff,axiom,
! [K: nat,M6: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M6 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M6 @ K ) @ ( minus_minus_nat @ N3 @ K ) )
= ( ord_less_eq_nat @ M6 @ N3 ) ) ) ) ).
% le_diff_iff
thf(fact_1196_Nat_Odiff__diff__eq,axiom,
! [K: nat,M6: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M6 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M6 @ K ) @ ( minus_minus_nat @ N3 @ K ) )
= ( minus_minus_nat @ M6 @ N3 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1197_diff__le__mono,axiom,
! [M6: nat,N3: nat,L: nat] :
( ( ord_less_eq_nat @ M6 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M6 @ L ) @ ( minus_minus_nat @ N3 @ L ) ) ) ).
% diff_le_mono
thf(fact_1198_diff__le__self,axiom,
! [M6: nat,N3: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M6 @ N3 ) @ M6 ) ).
% diff_le_self
thf(fact_1199_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1200_diff__le__mono2,axiom,
! [M6: nat,N3: nat,L: nat] :
( ( ord_less_eq_nat @ M6 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N3 ) @ ( minus_minus_nat @ L @ M6 ) ) ) ).
% diff_le_mono2
thf(fact_1201_diffs0__imp__equal,axiom,
! [M6: nat,N3: nat] :
( ( ( minus_minus_nat @ M6 @ N3 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N3 @ M6 )
= zero_zero_nat )
=> ( M6 = N3 ) ) ) ).
% diffs0_imp_equal
thf(fact_1202_minus__nat_Odiff__0,axiom,
! [M6: nat] :
( ( minus_minus_nat @ M6 @ zero_zero_nat )
= M6 ) ).
% minus_nat.diff_0
thf(fact_1203_ennreal__diff__self,axiom,
! [A2: extend8495563244428889912nnreal] :
( ( A2 != top_to1496364449551166952nnreal )
=> ( ( minus_8429688780609304081nnreal @ A2 @ A2 )
= zero_z7100319975126383169nnreal ) ) ).
% ennreal_diff_self
thf(fact_1204_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1205_ereal__minus__minus__image,axiom,
! [S: set_Extended_ereal] :
( ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S ) )
= S ) ).
% ereal_minus_minus_image
thf(fact_1206_ereal__range__uminus,axiom,
( ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ top_to5683747375963461374_ereal )
= top_to5683747375963461374_ereal ) ).
% ereal_range_uminus
thf(fact_1207_ennreal__minus__eq__top,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A2 @ B2 )
= top_to1496364449551166952nnreal )
= ( A2 = top_to1496364449551166952nnreal ) ) ).
% ennreal_minus_eq_top
thf(fact_1208_ennreal__top__minus,axiom,
! [X2: extend8495563244428889912nnreal] :
( ( minus_8429688780609304081nnreal @ top_to1496364449551166952nnreal @ X2 )
= top_to1496364449551166952nnreal ) ).
% ennreal_top_minus
thf(fact_1209_ereal__complete__uminus__eq,axiom,
! [S: set_Extended_ereal,X2: extended_ereal] :
( ( ! [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S ) )
=> ( ord_le1083603963089353582_ereal @ X4 @ X2 ) )
& ! [Z3: extended_ereal] :
( ! [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S ) )
=> ( ord_le1083603963089353582_ereal @ X4 @ Z3 ) )
=> ( ord_le1083603963089353582_ereal @ X2 @ Z3 ) ) )
= ( ! [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ S )
=> ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ X2 ) @ X4 ) )
& ! [Z3: extended_ereal] :
( ! [X4: extended_ereal] :
( ( member2350847679896131959_ereal @ X4 @ S )
=> ( ord_le1083603963089353582_ereal @ Z3 @ X4 ) )
=> ( ord_le1083603963089353582_ereal @ Z3 @ ( uminus27091377158695749_ereal @ X2 ) ) ) ) ) ).
% ereal_complete_uminus_eq
thf(fact_1210_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1211_continuous__uminus__ereal,axiom,
! [A: set_Extended_ereal] : ( topolo6777079828818185726_ereal @ A @ uminus27091377158695749_ereal ) ).
% continuous_uminus_ereal
thf(fact_1212_ereal__uminus__complement,axiom,
! [S: set_Extended_ereal] :
( ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ ( uminus5895154729394068773_ereal @ S ) )
= ( uminus5895154729394068773_ereal @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S ) ) ) ).
% ereal_uminus_complement
thf(fact_1213_ereal__image__uminus__shift,axiom,
! [X5: set_Extended_ereal,Y6: set_Extended_ereal] :
( ( ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ X5 )
= Y6 )
= ( X5
= ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ Y6 ) ) ) ).
% ereal_image_uminus_shift
thf(fact_1214_Sup__countable__SUP,axiom,
! [A: set_Extended_ereal] :
( ( A != bot_bo8367695208629047834_ereal )
=> ? [F4: nat > extended_ereal] :
( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ F4 )
& ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F4 @ top_top_set_nat ) @ A )
& ( ( comple8415311339701865915_ereal @ A )
= ( comple8415311339701865915_ereal @ ( image_4309273772856505399_ereal @ F4 @ top_top_set_nat ) ) ) ) ) ).
% Sup_countable_SUP
thf(fact_1215_ennreal__zero__neq__top,axiom,
zero_z7100319975126383169nnreal != top_to1496364449551166952nnreal ).
% ennreal_zero_neq_top
thf(fact_1216_minus__top__ennreal,axiom,
! [X2: extend8495563244428889912nnreal] :
( ( ( X2 = top_to1496364449551166952nnreal )
=> ( ( minus_8429688780609304081nnreal @ X2 @ top_to1496364449551166952nnreal )
= top_to1496364449551166952nnreal ) )
& ( ( X2 != top_to1496364449551166952nnreal )
=> ( ( minus_8429688780609304081nnreal @ X2 @ top_to1496364449551166952nnreal )
= zero_z7100319975126383169nnreal ) ) ) ).
% minus_top_ennreal
thf(fact_1217_neq__top__trans,axiom,
! [Y4: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ( Y4 != top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ X2 @ Y4 )
=> ( X2 != top_to1496364449551166952nnreal ) ) ) ).
% neq_top_trans
thf(fact_1218_ennreal__minus__cancel__iff,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A2 @ B2 )
= ( minus_8429688780609304081nnreal @ A2 @ C ) )
= ( ( B2 = C )
| ( ( ord_le3935885782089961368nnreal @ A2 @ B2 )
& ( ord_le3935885782089961368nnreal @ A2 @ C ) )
| ( A2 = top_to1496364449551166952nnreal ) ) ) ).
% ennreal_minus_cancel_iff
thf(fact_1219_ennreal__minus__cancel,axiom,
! [C: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( C != top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ A2 @ C )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ C )
=> ( ( ( minus_8429688780609304081nnreal @ C @ A2 )
= ( minus_8429688780609304081nnreal @ C @ B2 ) )
=> ( A2 = B2 ) ) ) ) ) ).
% ennreal_minus_cancel
thf(fact_1220_infinity__ennreal__def,axiom,
extend2057119558705770725nnreal = top_to1496364449551166952nnreal ).
% infinity_ennreal_def
thf(fact_1221_borel__set__induct,axiom,
! [A: set_real,P: set_real > $o] :
( ( member_set_real @ A @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
=> ( ( P @ bot_bot_set_real )
=> ( ! [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( P @ ( set_or1222579329274155063t_real @ A4 @ B4 ) ) )
=> ( ! [A6: set_real] :
( ( member_set_real @ A6 @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
=> ( ( P @ A6 )
=> ( P @ ( uminus612125837232591019t_real @ A6 ) ) ) )
=> ( ! [F4: nat > set_real] :
( ( disjoi2035185749148668758t_real @ F4 @ top_top_set_nat )
=> ( ! [I2: nat] : ( member_set_real @ ( F4 @ I2 ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
=> ( ! [I2: nat] : ( P @ ( F4 @ I2 ) )
=> ( P @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ F4 @ top_top_set_nat ) ) ) ) ) )
=> ( P @ A ) ) ) ) ) ) ).
% borel_set_induct
thf(fact_1222_ennreal__Sup__countable__SUP,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( A != bot_bo4854962954004695426nnreal )
=> ? [F4: nat > extend8495563244428889912nnreal] :
( ( monoto2291723841412853873nnreal @ top_top_set_nat @ ord_less_eq_nat @ ord_le3935885782089961368nnreal @ F4 )
& ( ord_le6787938422905777998nnreal @ ( image_8459861568512453903nnreal @ F4 @ top_top_set_nat ) @ A )
& ( ( comple6814414086264997003nnreal @ A )
= ( comple6814414086264997003nnreal @ ( image_8459861568512453903nnreal @ F4 @ top_top_set_nat ) ) ) ) ) ).
% ennreal_Sup_countable_SUP
thf(fact_1223_negligible__atLeastAtMostI,axiom,
! [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( hensto2219138694998437081e_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) ) ) ).
% negligible_atLeastAtMostI
thf(fact_1224_interval__bounds__real_I1_J,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( comple1385675409528146559p_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) )
= B2 ) ) ).
% interval_bounds_real(1)
thf(fact_1225_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1226_ennreal__zero__less__top,axiom,
ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ top_to1496364449551166952nnreal ).
% ennreal_zero_less_top
thf(fact_1227_diff__less__top__ennreal,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ ( minus_8429688780609304081nnreal @ A2 @ B2 ) @ top_to1496364449551166952nnreal )
= ( ord_le7381754540660121996nnreal @ A2 @ top_to1496364449551166952nnreal ) ) ).
% diff_less_top_ennreal
thf(fact_1228_ennreal__between,axiom,
! [E: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ E )
=> ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ X2 )
=> ( ( ord_le7381754540660121996nnreal @ X2 @ top_to1496364449551166952nnreal )
=> ( ord_le7381754540660121996nnreal @ ( minus_8429688780609304081nnreal @ X2 @ E ) @ X2 ) ) ) ) ).
% ennreal_between
thf(fact_1229_ennreal__minus__pos__iff,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ( ord_le7381754540660121996nnreal @ A2 @ top_to1496364449551166952nnreal )
| ( ord_le7381754540660121996nnreal @ B2 @ top_to1496364449551166952nnreal ) )
=> ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( minus_8429688780609304081nnreal @ A2 @ B2 ) )
=> ( ord_le7381754540660121996nnreal @ B2 @ A2 ) ) ) ).
% ennreal_minus_pos_iff
thf(fact_1230_diff__gt__0__iff__gt__ennreal,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( minus_8429688780609304081nnreal @ A2 @ B2 ) )
= ( ( ( A2 = top_to1496364449551166952nnreal )
& ( B2 = top_to1496364449551166952nnreal ) )
| ( ord_le7381754540660121996nnreal @ B2 @ A2 ) ) ) ).
% diff_gt_0_iff_gt_ennreal
thf(fact_1231_ennreal__mono__minus__cancel,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A2 @ B2 ) @ ( minus_8429688780609304081nnreal @ A2 @ C ) )
=> ( ( ord_le7381754540660121996nnreal @ A2 @ top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ A2 )
=> ( ( ord_le3935885782089961368nnreal @ C @ A2 )
=> ( ord_le3935885782089961368nnreal @ C @ B2 ) ) ) ) ) ).
% ennreal_mono_minus_cancel
thf(fact_1232_diff__eq__0__iff__ennreal,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A2 @ B2 )
= zero_z7100319975126383169nnreal )
= ( ( ord_le7381754540660121996nnreal @ A2 @ top_to1496364449551166952nnreal )
& ( ord_le3935885782089961368nnreal @ A2 @ B2 ) ) ) ).
% diff_eq_0_iff_ennreal
thf(fact_1233_diff__eq__0__ennreal,axiom,
! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ A2 @ B2 )
=> ( ( minus_8429688780609304081nnreal @ A2 @ B2 )
= zero_z7100319975126383169nnreal ) ) ) ).
% diff_eq_0_ennreal
thf(fact_1234_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1235_neq0__conv,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% neq0_conv
thf(fact_1236_less__nat__zero__code,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1237_zero__less__diff,axiom,
! [N3: nat,M6: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N3 @ M6 ) )
= ( ord_less_nat @ M6 @ N3 ) ) ).
% zero_less_diff
thf(fact_1238_less__imp__diff__less,axiom,
! [J: nat,K: nat,N3: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N3 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1239_diff__less__mono2,axiom,
! [M6: nat,N3: nat,L: nat] :
( ( ord_less_nat @ M6 @ N3 )
=> ( ( ord_less_nat @ M6 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N3 ) @ ( minus_minus_nat @ L @ M6 ) ) ) ) ).
% diff_less_mono2
thf(fact_1240_strict__mono__imp__increasing,axiom,
! [F: nat > nat,N3: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ord_less_eq_nat @ N3 @ ( F @ N3 ) ) ) ).
% strict_mono_imp_increasing
thf(fact_1241_diff__less,axiom,
! [N3: nat,M6: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ M6 )
=> ( ord_less_nat @ ( minus_minus_nat @ M6 @ N3 ) @ M6 ) ) ) ).
% diff_less
thf(fact_1242_less__diff__iff,axiom,
! [K: nat,M6: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M6 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M6 @ K ) @ ( minus_minus_nat @ N3 @ K ) )
= ( ord_less_nat @ M6 @ N3 ) ) ) ) ).
% less_diff_iff
thf(fact_1243_diff__less__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1244_ex__least__nat__le,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ N3 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N3 )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ~ ( P @ I2 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1245_infinite__descent0,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
=> ( ~ ( P @ N5 )
=> ? [M7: nat] :
( ( ord_less_nat @ M7 @ N5 )
& ~ ( P @ M7 ) ) ) )
=> ( P @ N3 ) ) ) ).
% infinite_descent0
thf(fact_1246_gr__implies__not0,axiom,
! [M6: nat,N3: nat] :
( ( ord_less_nat @ M6 @ N3 )
=> ( N3 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1247_less__zeroE,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1248_not__less0,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% not_less0
thf(fact_1249_not__gr0,axiom,
! [N3: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
= ( N3 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1250_gr0I,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% gr0I
thf(fact_1251_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1252_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I5: nat,J3: nat] :
( ( ord_less_nat @ I5 @ J3 )
=> ( ord_less_nat @ ( F @ I5 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1253_le__neq__implies__less,axiom,
! [M6: nat,N3: nat] :
( ( ord_less_eq_nat @ M6 @ N3 )
=> ( ( M6 != N3 )
=> ( ord_less_nat @ M6 @ N3 ) ) ) ).
% le_neq_implies_less
thf(fact_1254_less__or__eq__imp__le,axiom,
! [M6: nat,N3: nat] :
( ( ( ord_less_nat @ M6 @ N3 )
| ( M6 = N3 ) )
=> ( ord_less_eq_nat @ M6 @ N3 ) ) ).
% less_or_eq_imp_le
thf(fact_1255_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
| ( M5 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1256_less__imp__le__nat,axiom,
! [M6: nat,N3: nat] :
( ( ord_less_nat @ M6 @ N3 )
=> ( ord_less_eq_nat @ M6 @ N3 ) ) ).
% less_imp_le_nat
thf(fact_1257_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
& ( M5 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_1258_Bolzano,axiom,
! [A2: real,B2: real,P: real > real > $o] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ! [A4: real,B4: real,C4: real] :
( ( P @ A4 @ B4 )
=> ( ( P @ B4 @ C4 )
=> ( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_eq_real @ B4 @ C4 )
=> ( P @ A4 @ C4 ) ) ) ) )
=> ( ! [X: real] :
( ( ord_less_eq_real @ A2 @ X )
=> ( ( ord_less_eq_real @ X @ B2 )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [A4: real,B4: real] :
( ( ( ord_less_eq_real @ A4 @ X )
& ( ord_less_eq_real @ X @ B4 )
& ( ord_less_real @ ( minus_minus_real @ B4 @ A4 ) @ D4 ) )
=> ( P @ A4 @ B4 ) ) ) ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% Bolzano
thf(fact_1259_linorder__neqE__nat,axiom,
! [X2: nat,Y4: nat] :
( ( X2 != Y4 )
=> ( ~ ( ord_less_nat @ X2 @ Y4 )
=> ( ord_less_nat @ Y4 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_1260_infinite__descent,axiom,
! [P: nat > $o,N3: nat] :
( ! [N5: nat] :
( ~ ( P @ N5 )
=> ? [M7: nat] :
( ( ord_less_nat @ M7 @ N5 )
& ~ ( P @ M7 ) ) )
=> ( P @ N3 ) ) ).
% infinite_descent
thf(fact_1261_nat__less__induct,axiom,
! [P: nat > $o,N3: nat] :
( ! [N5: nat] :
( ! [M7: nat] :
( ( ord_less_nat @ M7 @ N5 )
=> ( P @ M7 ) )
=> ( P @ N5 ) )
=> ( P @ N3 ) ) ).
% nat_less_induct
thf(fact_1262_less__irrefl__nat,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ N3 ) ).
% less_irrefl_nat
thf(fact_1263_less__not__refl3,axiom,
! [S3: nat,T3: nat] :
( ( ord_less_nat @ S3 @ T3 )
=> ( S3 != T3 ) ) ).
% less_not_refl3
thf(fact_1264_less__not__refl2,axiom,
! [N3: nat,M6: nat] :
( ( ord_less_nat @ N3 @ M6 )
=> ( M6 != N3 ) ) ).
% less_not_refl2
thf(fact_1265_less__not__refl,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ N3 ) ).
% less_not_refl
thf(fact_1266_nat__neq__iff,axiom,
! [M6: nat,N3: nat] :
( ( M6 != N3 )
= ( ( ord_less_nat @ M6 @ N3 )
| ( ord_less_nat @ N3 @ M6 ) ) ) ).
% nat_neq_iff
thf(fact_1267_seq__mono__lemma,axiom,
! [M6: nat,D: nat > real,E: nat > real] :
( ! [N5: nat] :
( ( ord_less_eq_nat @ M6 @ N5 )
=> ( ord_less_real @ ( D @ N5 ) @ ( E @ N5 ) ) )
=> ( ! [N5: nat] :
( ( ord_less_eq_nat @ M6 @ N5 )
=> ( ord_less_eq_real @ ( E @ N5 ) @ ( E @ M6 ) ) )
=> ! [N6: nat] :
( ( ord_less_eq_nat @ M6 @ N6 )
=> ( ord_less_real @ ( D @ N6 ) @ ( E @ M6 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_1268_ennreal__Inf__countable__INF,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( A != bot_bo4854962954004695426nnreal )
=> ? [F4: nat > extend8495563244428889912nnreal] :
( ( monoto2291723841412853873nnreal @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X4: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ Y2 @ X4 )
@ F4 )
& ( ord_le6787938422905777998nnreal @ ( image_8459861568512453903nnreal @ F4 @ top_top_set_nat ) @ A )
& ( ( comple7330758040695736817nnreal @ A )
= ( comple7330758040695736817nnreal @ ( image_8459861568512453903nnreal @ F4 @ top_top_set_nat ) ) ) ) ) ).
% ennreal_Inf_countable_INF
% Helper facts (3)
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y4: real] :
( ( if_real @ $false @ X2 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y4: real] :
( ( if_real @ $true @ X2 @ Y4 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member_real_real @ f @ ( sigma_5267869275261027754l_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ ( set_or1222579329274155063t_real @ a @ b ) ) @ borel_5078946678739801102l_real ) ).
%------------------------------------------------------------------------------