TPTP Problem File: SLH0929^1.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 : Quasi_Borel_Spaces/0002_Measure_QuasiBorel_Adjunction/prob_00082_002980__15144152_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1639 ( 377 unt; 357 typ; 0 def)
% Number of atoms : 3870 ( 924 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 12570 ( 387 ~; 37 |; 232 &;9819 @)
% ( 0 <=>;2095 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 7 avg)
% Number of types : 52 ( 51 usr)
% Number of type conns : 1449 (1449 >; 0 *; 0 +; 0 <<)
% Number of symbols : 309 ( 306 usr; 19 con; 0-5 aty)
% Number of variables : 3619 ( 212 ^;3280 !; 127 ?;3619 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:08:28.555
%------------------------------------------------------------------------------
% Could-be-implicit typings (51)
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% Explicit typings (306)
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qbs_morphism_a_a_a2: quasi_borel_a > quasi_borel_a_a > set_a_a_a ).
thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__a_001t__Real__Oreal,type,
qbs_morphism_a_real: quasi_borel_a > quasi_borel_real > set_a_real ).
thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__a_001t__Set__Oset_Itf__a_J,type,
qbs_morphism_a_set_a: quasi_borel_a > quasi_borel_set_a > set_a_set_a ).
thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__a_001tf__a,type,
qbs_morphism_a_a: quasi_borel_a > quasi_borel_a > set_a_a ).
thf(sy_c_QuasiBorel_Oqbs__space_001_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J,type,
qbs_space_a_a_a: quasi_borel_a_a_a2 > set_a_a_a2 ).
thf(sy_c_QuasiBorel_Oqbs__space_001_062_It__Real__Oreal_Mtf__a_J,type,
qbs_space_real_a: quasi_borel_real_a > set_real_a ).
thf(sy_c_QuasiBorel_Oqbs__space_001_062_Itf__a_M_062_It__Real__Oreal_Mtf__a_J_J,type,
qbs_space_a_real_a: quasi_borel_a_real_a > set_a_real_a ).
thf(sy_c_QuasiBorel_Oqbs__space_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J,type,
qbs_space_a_a_a2: quasi_borel_a_a_a > set_a_a_a ).
thf(sy_c_QuasiBorel_Oqbs__space_001_062_Itf__a_Mt__Real__Oreal_J,type,
qbs_space_a_real: quasi_borel_a_real > set_a_real ).
thf(sy_c_QuasiBorel_Oqbs__space_001_062_Itf__a_Mt__Set__Oset_Itf__a_J_J,type,
qbs_space_a_set_a: quasi_borel_a_set_a > set_a_set_a ).
thf(sy_c_QuasiBorel_Oqbs__space_001_062_Itf__a_Mtf__a_J,type,
qbs_space_a_a: quasi_borel_a_a > set_a_a ).
thf(sy_c_QuasiBorel_Oqbs__space_001t__Real__Oreal,type,
qbs_space_real: quasi_borel_real > set_real ).
thf(sy_c_QuasiBorel_Oqbs__space_001t__Set__Oset_Itf__a_J,type,
qbs_space_set_a: quasi_borel_set_a > set_set_a ).
thf(sy_c_QuasiBorel_Oqbs__space_001tf__a,type,
qbs_space_a: quasi_borel_a > set_a ).
thf(sy_c_QuasiBorel_Osub__qbs_001tf__a,type,
sub_qbs_a: quasi_borel_a > set_a > quasi_borel_a ).
thf(sy_c_Set_OCollect_001_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J,type,
collect_a_a_a: ( ( ( a > a ) > a ) > $o ) > set_a_a_a2 ).
thf(sy_c_Set_OCollect_001_062_It__Real__Oreal_Mtf__a_J,type,
collect_real_a: ( ( real > a ) > $o ) > set_real_a ).
thf(sy_c_Set_OCollect_001_062_Itf__a_M_062_It__Real__Oreal_Mtf__a_J_J,type,
collect_a_real_a: ( ( a > real > a ) > $o ) > set_a_real_a ).
thf(sy_c_Set_OCollect_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J,type,
collect_a_a_a2: ( ( a > a > a ) > $o ) > set_a_a_a ).
thf(sy_c_Set_OCollect_001_062_Itf__a_Mt__Real__Oreal_J,type,
collect_a_real: ( ( a > real ) > $o ) > set_a_real ).
thf(sy_c_Set_OCollect_001_062_Itf__a_Mt__Set__Oset_Itf__a_J_J,type,
collect_a_set_a: ( ( a > set_a ) > $o ) > set_a_set_a ).
thf(sy_c_Set_OCollect_001_062_Itf__a_Mtf__a_J,type,
collect_a_a: ( ( a > a ) > $o ) > set_a_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OPow_001_062_It__Real__Oreal_Mtf__a_J,type,
pow_real_a: set_real_a > set_set_real_a ).
thf(sy_c_Set_OPow_001_062_Itf__a_Mtf__a_J,type,
pow_a_a: set_a_a > set_set_a_a ).
thf(sy_c_Set_OPow_001t__Extended____Nonnegative____Real__Oennreal,type,
pow_Ex5372160365422184283nnreal: set_Ex3793607809372303086nnreal > set_se4580700918925141924nnreal ).
thf(sy_c_Set_OPow_001t__Set__Oset_Itf__a_J,type,
pow_set_a: set_set_a > set_set_set_a ).
thf(sy_c_Set_OPow_001tf__a,type,
pow_a: set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001_062_It__Real__Oreal_Mtf__a_J,type,
insert_real_a: ( real > a ) > set_real_a > set_real_a ).
thf(sy_c_Set_Oinsert_001_062_Itf__a_Mtf__a_J,type,
insert_a_a: ( a > a ) > set_a_a > set_a_a ).
thf(sy_c_Set_Oinsert_001t__Extended____Nonnegative____Real__Oennreal,type,
insert7407984058720857448nnreal: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__singleton_001_062_It__Real__Oreal_Mtf__a_J,type,
is_singleton_real_a: set_real_a > $o ).
thf(sy_c_Set_Ois__singleton_001_062_Itf__a_Mtf__a_J,type,
is_singleton_a_a: set_a_a > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_Itf__a_J,type,
is_singleton_set_a: set_set_a > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Sigma__Algebra_ODynkin_001_062_It__Real__Oreal_Mtf__a_J,type,
sigma_Dynkin_real_a: set_real_a > set_set_real_a > set_set_real_a ).
thf(sy_c_Sigma__Algebra_ODynkin_001_062_Itf__a_Mtf__a_J,type,
sigma_Dynkin_a_a: set_a_a > set_set_a_a > set_set_a_a ).
thf(sy_c_Sigma__Algebra_ODynkin_001t__Set__Oset_Itf__a_J,type,
sigma_Dynkin_set_a: set_set_a > set_set_set_a > set_set_set_a ).
thf(sy_c_Sigma__Algebra_ODynkin_001tf__a,type,
sigma_Dynkin_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Sigma__Algebra_ODynkin__system_001_062_It__Real__Oreal_Mtf__a_J,type,
sigma_4207074038752281043real_a: set_real_a > set_set_real_a > $o ).
thf(sy_c_Sigma__Algebra_ODynkin__system_001_062_Itf__a_Mtf__a_J,type,
sigma_4503493975268142235em_a_a: set_a_a > set_set_a_a > $o ).
thf(sy_c_Sigma__Algebra_ODynkin__system_001t__Set__Oset_Itf__a_J,type,
sigma_6194451600871760808_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_Sigma__Algebra_ODynkin__system_001tf__a,type,
sigma_2757993544071651912stem_a: set_a > set_set_a > $o ).
thf(sy_c_Sigma__Algebra_ODynkin__system__axioms_001tf__a,type,
sigma_1042032682244386661ioms_a: set_a > set_set_a > $o ).
thf(sy_c_Sigma__Algebra_OInt__stable_001_062_It__Real__Oreal_Mtf__a_J,type,
sigma_5335421319604048045real_a: set_set_real_a > $o ).
thf(sy_c_Sigma__Algebra_OInt__stable_001_062_Itf__a_Mtf__a_J,type,
sigma_Int_stable_a_a: set_set_a_a > $o ).
thf(sy_c_Sigma__Algebra_OInt__stable_001t__Set__Oset_Itf__a_J,type,
sigma_5249310804214965890_set_a: set_set_set_a > $o ).
thf(sy_c_Sigma__Algebra_OInt__stable_001tf__a,type,
sigma_Int_stable_a: set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Oalgebra_001_062_It__Real__Oreal_Mtf__a_J,type,
sigma_algebra_real_a: set_real_a > set_set_real_a > $o ).
thf(sy_c_Sigma__Algebra_Oalgebra_001_062_Itf__a_Mtf__a_J,type,
sigma_algebra_a_a: set_a_a > set_set_a_a > $o ).
thf(sy_c_Sigma__Algebra_Oalgebra_001t__Set__Oset_Itf__a_J,type,
sigma_algebra_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Oalgebra_001tf__a,type,
sigma_algebra_a: set_a > set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Oalgebra__axioms_001tf__a,type,
sigma_1540858801969833537ioms_a: set_a > set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Obinary_001t__Set__Oset_Itf__a_J,type,
sigma_binary_set_a: set_a > set_a > nat > set_a ).
thf(sy_c_Sigma__Algebra_Oclosed__cdi_001tf__a,type,
sigma_closed_cdi_a: set_a > set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Ocountably__additive_001tf__a,type,
sigma_498350552340604931tive_a: set_set_a > ( set_a > extend8495563244428889912nnreal ) > $o ).
thf(sy_c_Sigma__Algebra_Oemeasure_001_062_It__Real__Oreal_Mtf__a_J,type,
sigma_6502373073922819808real_a: sigma_measure_real_a > set_real_a > extend8495563244428889912nnreal ).
thf(sy_c_Sigma__Algebra_Oemeasure_001_062_Itf__a_Mtf__a_J,type,
sigma_emeasure_a_a: sigma_measure_a_a > set_a_a > extend8495563244428889912nnreal ).
thf(sy_c_Sigma__Algebra_Oemeasure_001t__Set__Oset_Itf__a_J,type,
sigma_emeasure_set_a: sigma_measure_set_a > set_set_a > extend8495563244428889912nnreal ).
thf(sy_c_Sigma__Algebra_Oemeasure_001tf__a,type,
sigma_emeasure_a: sigma_measure_a > set_a > extend8495563244428889912nnreal ).
thf(sy_c_Sigma__Algebra_Omeasurable_001_062_It__Real__Oreal_Mtf__a_J_001tf__a,type,
sigma_5527925685445946117al_a_a: sigma_measure_real_a > sigma_measure_a > set_real_a_a2 ).
thf(sy_c_Sigma__Algebra_Omeasurable_001_062_Itf__a_Mtf__a_J_001t__Set__Oset_Itf__a_J,type,
sigma_8078287805088307175_set_a: sigma_measure_a_a > sigma_measure_set_a > set_a_a_set_a ).
thf(sy_c_Sigma__Algebra_Omeasurable_001_062_Itf__a_Mtf__a_J_001tf__a,type,
sigma_3107241577691976327_a_a_a: sigma_measure_a_a > sigma_measure_a > set_a_a_a2 ).
thf(sy_c_Sigma__Algebra_Omeasurable_001t__Real__Oreal_001t__Real__Oreal,type,
sigma_5267869275261027754l_real: sigma_measure_real > sigma_measure_real > set_real_real ).
thf(sy_c_Sigma__Algebra_Omeasurable_001t__Real__Oreal_001tf__a,type,
sigma_523072396149930112real_a: sigma_measure_real > sigma_measure_a > set_real_a ).
thf(sy_c_Sigma__Algebra_Omeasurable_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
sigma_5212894042034225104_set_a: sigma_measure_set_a > sigma_measure_set_a > set_set_a_set_a ).
thf(sy_c_Sigma__Algebra_Omeasurable_001t__Set__Oset_Itf__a_J_001tf__a,type,
sigma_3901645225212141168et_a_a: sigma_measure_set_a > sigma_measure_a > set_set_a_a2 ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__a_001_062_It__Real__Oreal_Mtf__a_J,type,
sigma_5590391210564117339real_a: sigma_measure_a > sigma_measure_real_a > set_a_real_a ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__a_001_062_Itf__a_Mtf__a_J,type,
sigma_3971135313551251731_a_a_a: sigma_measure_a > sigma_measure_a_a > set_a_a_a ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__a_001t__Real__Oreal,type,
sigma_9116425665531756122a_real: sigma_measure_a > sigma_measure_real > set_a_real ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__a_001t__Set__Oset_Itf__a_J,type,
sigma_3685133166752798000_set_a: sigma_measure_a > sigma_measure_set_a > set_a_set_a ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__a_001tf__a,type,
sigma_measurable_a_a: sigma_measure_a > sigma_measure_a > set_a_a ).
thf(sy_c_Sigma__Algebra_Omeasure__of_001tf__a,type,
sigma_measure_of_a: set_a > set_set_a > ( set_a > extend8495563244428889912nnreal ) > sigma_measure_a ).
thf(sy_c_Sigma__Algebra_Omeasure__space_001tf__a,type,
sigma_3179946494550678598pace_a: set_a > set_set_a > ( set_a > extend8495563244428889912nnreal ) > $o ).
thf(sy_c_Sigma__Algebra_Opositive_001tf__a,type,
sigma_positive_a: set_set_a > ( set_a > extend8495563244428889912nnreal ) > $o ).
thf(sy_c_Sigma__Algebra_Oring__of__sets_001tf__a,type,
sigma_ring_of_sets_a: set_a > set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Oring__of__sets__axioms_001tf__a,type,
sigma_7143023581266409363ioms_a: set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Osemiring__of__sets_001tf__a,type,
sigma_8461971822185508616sets_a: set_a > set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Osets_001_062_It__Real__Oreal_Mtf__a_J,type,
sigma_sets_real_a: sigma_measure_real_a > set_set_real_a ).
thf(sy_c_Sigma__Algebra_Osets_001_062_Itf__a_Mtf__a_J,type,
sigma_sets_a_a: sigma_measure_a_a > set_set_a_a ).
thf(sy_c_Sigma__Algebra_Osets_001t__Real__Oreal,type,
sigma_sets_real: sigma_measure_real > set_set_real ).
thf(sy_c_Sigma__Algebra_Osets_001t__Set__Oset_Itf__a_J,type,
sigma_sets_set_a: sigma_measure_set_a > set_set_set_a ).
thf(sy_c_Sigma__Algebra_Osets_001tf__a,type,
sigma_sets_a: sigma_measure_a > set_set_a ).
thf(sy_c_Sigma__Algebra_Osigma__algebra_001_062_It__Real__Oreal_Mtf__a_J,type,
sigma_6829682388519410934real_a: set_real_a > set_set_real_a > $o ).
thf(sy_c_Sigma__Algebra_Osigma__algebra_001_062_Itf__a_Mtf__a_J,type,
sigma_17020130125533880ra_a_a: set_a_a > set_set_a_a > $o ).
thf(sy_c_Sigma__Algebra_Osigma__algebra_001t__Set__Oset_Itf__a_J,type,
sigma_6575008089947538635_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Osigma__algebra_001tf__a,type,
sigma_4968961713055010667ebra_a: set_a > set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Osigma__algebra__axioms_001tf__a,type,
sigma_1832572188971243144ioms_a: set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Osigma__sets_001_062_It__Real__Oreal_Mtf__a_J,type,
sigma_136333207653772897real_a: set_real_a > set_set_real_a > set_set_real_a ).
thf(sy_c_Sigma__Algebra_Osigma__sets_001_062_Itf__a_Mtf__a_J,type,
sigma_sigma_sets_a_a: set_a_a > set_set_a_a > set_set_a_a ).
thf(sy_c_Sigma__Algebra_Osigma__sets_001t__Set__Oset_Itf__a_J,type,
sigma_2987359967864564790_set_a: set_set_a > set_set_set_a > set_set_set_a ).
thf(sy_c_Sigma__Algebra_Osigma__sets_001tf__a,type,
sigma_sigma_sets_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Sigma__Algebra_Osmallest__ccdi__sets_001tf__a,type,
sigma_5648178489087971417sets_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Sigma__Algebra_Ospace_001_062_It__Real__Oreal_Mtf__a_J,type,
sigma_space_real_a: sigma_measure_real_a > set_real_a ).
thf(sy_c_Sigma__Algebra_Ospace_001_062_Itf__a_Mtf__a_J,type,
sigma_space_a_a: sigma_measure_a_a > set_a_a ).
thf(sy_c_Sigma__Algebra_Ospace_001t__Real__Oreal,type,
sigma_space_real: sigma_measure_real > set_real ).
thf(sy_c_Sigma__Algebra_Ospace_001t__Set__Oset_Itf__a_J,type,
sigma_space_set_a: sigma_measure_set_a > set_set_a ).
thf(sy_c_Sigma__Algebra_Ospace_001tf__a,type,
sigma_space_a: sigma_measure_a > set_a ).
thf(sy_c_Sigma__Algebra_Osubset__class_001tf__a,type,
sigma_subset_class_a: set_a > set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Ovimage__algebra_001t__Real__Oreal_001tf__a,type,
sigma_7241109303225351740real_a: set_real > ( real > a ) > sigma_measure_a > sigma_measure_real ).
thf(sy_c_Sigma__Algebra_Ovimage__algebra_001tf__a_001tf__a,type,
sigma_2018398322970380052ra_a_a: set_a > ( a > a ) > sigma_measure_a > sigma_measure_a ).
thf(sy_c_member_001_062_I_062_It__Real__Oreal_Mtf__a_J_Mt__Real__Oreal_J,type,
member_real_a_real: ( ( real > a ) > real ) > set_real_a_real2 > $o ).
thf(sy_c_member_001_062_I_062_It__Real__Oreal_Mtf__a_J_Mtf__a_J,type,
member_real_a_a: ( ( real > a ) > a ) > set_real_a_a2 > $o ).
thf(sy_c_member_001_062_I_062_Itf__a_Mtf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
member_a_a_set_a: ( ( a > a ) > set_a ) > set_a_a_set_a > $o ).
thf(sy_c_member_001_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J,type,
member_a_a_a: ( ( a > a ) > a ) > set_a_a_a2 > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_M_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
member_real_a_a_a: ( real > ( a > a ) > a ) > set_real_a_a_a2 > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_M_062_It__Real__Oreal_Mtf__a_J_J,type,
member_real_real_a: ( real > real > a ) > set_real_real_a > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_M_062_Itf__a_M_062_It__Real__Oreal_Mtf__a_J_J_J,type,
member_real_a_real_a: ( real > a > real > a ) > set_real_a_real_a > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J,type,
member_real_a_a_a2: ( real > a > a > a ) > set_real_a_a_a > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_M_062_Itf__a_Mt__Real__Oreal_J_J,type,
member_real_a_real2: ( real > a > real ) > set_real_a_real > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_M_062_Itf__a_Mt__Set__Oset_Itf__a_J_J_J,type,
member_real_a_set_a: ( real > a > set_a ) > set_real_a_set_a > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_M_062_Itf__a_Mtf__a_J_J,type,
member_real_a_a2: ( real > a > a ) > set_real_a_a > $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_Itf__a_J_J,type,
member_real_set_a: ( real > set_a ) > set_real_set_a > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_Mtf__a_J,type,
member_real_a: ( real > a ) > set_real_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_Itf__a_J_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
member4180043592386426928nnreal: ( set_a > extend8495563244428889912nnreal ) > set_se9209621484078883815nnreal > $o ).
thf(sy_c_member_001_062_It__Set__Oset_Itf__a_J_Mt__Real__Oreal_J,type,
member_set_a_real: ( set_a > real ) > set_set_a_real > $o ).
thf(sy_c_member_001_062_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
member_set_a_set_a: ( set_a > set_a ) > set_set_a_set_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_Itf__a_J_Mtf__a_J,type,
member_set_a_a: ( set_a > a ) > set_set_a_a2 > $o ).
thf(sy_c_member_001_062_Itf__a_M_062_It__Real__Oreal_Mtf__a_J_J,type,
member_a_real_a: ( a > real > a ) > set_a_real_a > $o ).
thf(sy_c_member_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J,type,
member_a_a_a2: ( a > a > a ) > set_a_a_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Real__Oreal_J,type,
member_a_real: ( a > real ) > set_a_real > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Set__Oset_Itf__a_J_J,type,
member_a_set_a: ( a > set_a ) > set_a_set_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001t__Extended____Nonnegative____Real__Oennreal,type,
member7908768830364227535nnreal: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal > $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_Mtf__a_J_J,type,
member_set_real_a: set_real_a > set_set_real_a > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
member_set_a_a2: set_a_a > set_set_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_X,type,
x: quasi_borel_a ).
% Relevant facts (1276)
thf(fact_0_max__qbs__space,axiom,
! [X: set_real_a] :
( ( qbs_space_real_a @ ( max_qu1938530464164800784real_a @ X ) )
= X ) ).
% max_qbs_space
thf(fact_1_max__qbs__space,axiom,
! [X: set_set_a] :
( ( qbs_space_set_a @ ( max_qu9034415943339951333_set_a @ X ) )
= X ) ).
% max_qbs_space
thf(fact_2_max__qbs__space,axiom,
! [X: set_a_a] :
( ( qbs_space_a_a @ ( max_quasi_borel_a_a @ X ) )
= X ) ).
% max_qbs_space
thf(fact_3_max__qbs__space,axiom,
! [X: set_real] :
( ( qbs_space_real @ ( max_quasi_borel_real @ X ) )
= X ) ).
% max_qbs_space
thf(fact_4_max__qbs__space,axiom,
! [X: set_a] :
( ( qbs_space_a @ ( max_quasi_borel_a @ X ) )
= X ) ).
% max_qbs_space
thf(fact_5_sigma__algebra_Oaxioms_I2_J,axiom,
! [Omega: set_a,M: set_set_a] :
( ( sigma_4968961713055010667ebra_a @ Omega @ M )
=> ( sigma_1832572188971243144ioms_a @ M ) ) ).
% sigma_algebra.axioms(2)
thf(fact_6_qbs__morphismE_I2_J,axiom,
! [F: a > a,X: quasi_borel_a,Y: quasi_borel_a,X2: a] :
( ( member_a_a @ F @ ( qbs_morphism_a_a @ X @ Y ) )
=> ( ( member_a @ X2 @ ( qbs_space_a @ X ) )
=> ( member_a @ ( F @ X2 ) @ ( qbs_space_a @ Y ) ) ) ) ).
% qbs_morphismE(2)
thf(fact_7_qbs__morphismE_I2_J,axiom,
! [F: real > a,X: quasi_borel_real,Y: quasi_borel_a,X2: real] :
( ( member_real_a @ F @ ( qbs_morphism_real_a @ X @ Y ) )
=> ( ( member_real @ X2 @ ( qbs_space_real @ X ) )
=> ( member_a @ ( F @ X2 ) @ ( qbs_space_a @ Y ) ) ) ) ).
% qbs_morphismE(2)
thf(fact_8_qbs__morphismE_I2_J,axiom,
! [F: set_a > a,X: quasi_borel_set_a,Y: quasi_borel_a,X2: set_a] :
( ( member_set_a_a @ F @ ( qbs_morphism_set_a_a @ X @ Y ) )
=> ( ( member_set_a @ X2 @ ( qbs_space_set_a @ X ) )
=> ( member_a @ ( F @ X2 ) @ ( qbs_space_a @ Y ) ) ) ) ).
% qbs_morphismE(2)
thf(fact_9_qbs__morphismE_I2_J,axiom,
! [F: a > set_a,X: quasi_borel_a,Y: quasi_borel_set_a,X2: a] :
( ( member_a_set_a @ F @ ( qbs_morphism_a_set_a @ X @ Y ) )
=> ( ( member_a @ X2 @ ( qbs_space_a @ X ) )
=> ( member_set_a @ ( F @ X2 ) @ ( qbs_space_set_a @ Y ) ) ) ) ).
% qbs_morphismE(2)
thf(fact_10_qbs__morphismE_I2_J,axiom,
! [F: set_a > set_a,X: quasi_borel_set_a,Y: quasi_borel_set_a,X2: set_a] :
( ( member_set_a_set_a @ F @ ( qbs_mo7890142103032497016_set_a @ X @ Y ) )
=> ( ( member_set_a @ X2 @ ( qbs_space_set_a @ X ) )
=> ( member_set_a @ ( F @ X2 ) @ ( qbs_space_set_a @ Y ) ) ) ) ).
% qbs_morphismE(2)
thf(fact_11_qbs__morphismE_I2_J,axiom,
! [F: ( a > a ) > a,X: quasi_borel_a_a,Y: quasi_borel_a,X2: a > a] :
( ( member_a_a_a @ F @ ( qbs_morphism_a_a_a @ X @ Y ) )
=> ( ( member_a_a @ X2 @ ( qbs_space_a_a @ X ) )
=> ( member_a @ ( F @ X2 ) @ ( qbs_space_a @ Y ) ) ) ) ).
% qbs_morphismE(2)
thf(fact_12_qbs__morphismE_I2_J,axiom,
! [F: ( real > a ) > a,X: quasi_borel_real_a,Y: quasi_borel_a,X2: real > a] :
( ( member_real_a_a @ F @ ( qbs_mo2483107194261503661al_a_a @ X @ Y ) )
=> ( ( member_real_a @ X2 @ ( qbs_space_real_a @ X ) )
=> ( member_a @ ( F @ X2 ) @ ( qbs_space_a @ Y ) ) ) ) ).
% qbs_morphismE(2)
thf(fact_13_qbs__morphismE_I2_J,axiom,
! [F: a > a > a,X: quasi_borel_a,Y: quasi_borel_a_a,X2: a] :
( ( member_a_a_a2 @ F @ ( qbs_morphism_a_a_a2 @ X @ Y ) )
=> ( ( member_a @ X2 @ ( qbs_space_a @ X ) )
=> ( member_a_a @ ( F @ X2 ) @ ( qbs_space_a_a @ Y ) ) ) ) ).
% qbs_morphismE(2)
thf(fact_14_qbs__morphismE_I2_J,axiom,
! [F: a > real > a,X: quasi_borel_a,Y: quasi_borel_real_a,X2: a] :
( ( member_a_real_a @ F @ ( qbs_mo2545572719379674883real_a @ X @ Y ) )
=> ( ( member_a @ X2 @ ( qbs_space_a @ X ) )
=> ( member_real_a @ ( F @ X2 ) @ ( qbs_space_real_a @ Y ) ) ) ) ).
% qbs_morphismE(2)
thf(fact_15_qbs__morphismE_I2_J,axiom,
! [F: ( a > a ) > set_a,X: quasi_borel_a_a,Y: quasi_borel_set_a,X2: a > a] :
( ( member_a_a_set_a @ F @ ( qbs_mo5558031909227686463_set_a @ X @ Y ) )
=> ( ( member_a_a @ X2 @ ( qbs_space_a_a @ X ) )
=> ( member_set_a @ ( F @ X2 ) @ ( qbs_space_set_a @ Y ) ) ) ) ).
% qbs_morphismE(2)
thf(fact_16_qbs__morphism__cong,axiom,
! [X: quasi_borel_a_a,F: ( a > a ) > a,G: ( a > a ) > a,Y: quasi_borel_a] :
( ! [X3: a > a] :
( ( member_a_a @ X3 @ ( qbs_space_a_a @ X ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( member_a_a_a @ F @ ( qbs_morphism_a_a_a @ X @ Y ) )
=> ( member_a_a_a @ G @ ( qbs_morphism_a_a_a @ X @ Y ) ) ) ) ).
% qbs_morphism_cong
thf(fact_17_qbs__morphism__cong,axiom,
! [X: quasi_borel_a,F: a > real,G: a > real,Y: quasi_borel_real] :
( ! [X3: a] :
( ( member_a @ X3 @ ( qbs_space_a @ X ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( member_a_real @ F @ ( qbs_morphism_a_real @ X @ Y ) )
=> ( member_a_real @ G @ ( qbs_morphism_a_real @ X @ Y ) ) ) ) ).
% qbs_morphism_cong
thf(fact_18_qbs__morphism__cong,axiom,
! [X: quasi_borel_a,F: a > set_a,G: a > set_a,Y: quasi_borel_set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( qbs_space_a @ X ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( member_a_set_a @ F @ ( qbs_morphism_a_set_a @ X @ Y ) )
=> ( member_a_set_a @ G @ ( qbs_morphism_a_set_a @ X @ Y ) ) ) ) ).
% qbs_morphism_cong
thf(fact_19_qbs__morphism__cong,axiom,
! [X: quasi_borel_a,F: a > real > a,G: a > real > a,Y: quasi_borel_real_a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( qbs_space_a @ X ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( member_a_real_a @ F @ ( qbs_mo2545572719379674883real_a @ X @ Y ) )
=> ( member_a_real_a @ G @ ( qbs_mo2545572719379674883real_a @ X @ Y ) ) ) ) ).
% qbs_morphism_cong
thf(fact_20_qbs__morphism__cong,axiom,
! [X: quasi_borel_a,F: a > a > a,G: a > a > a,Y: quasi_borel_a_a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( qbs_space_a @ X ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( member_a_a_a2 @ F @ ( qbs_morphism_a_a_a2 @ X @ Y ) )
=> ( member_a_a_a2 @ G @ ( qbs_morphism_a_a_a2 @ X @ Y ) ) ) ) ).
% qbs_morphism_cong
thf(fact_21_qbs__morphism__cong,axiom,
! [X: quasi_borel_real,F: real > a,G: real > a,Y: quasi_borel_a] :
( ! [X3: real] :
( ( member_real @ X3 @ ( qbs_space_real @ X ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( member_real_a @ F @ ( qbs_morphism_real_a @ X @ Y ) )
=> ( member_real_a @ G @ ( qbs_morphism_real_a @ X @ Y ) ) ) ) ).
% qbs_morphism_cong
thf(fact_22_qbs__morphism__cong,axiom,
! [X: quasi_borel_a,F: a > a,G: a > a,Y: quasi_borel_a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( qbs_space_a @ X ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( member_a_a @ F @ ( qbs_morphism_a_a @ X @ Y ) )
=> ( member_a_a @ G @ ( qbs_morphism_a_a @ X @ Y ) ) ) ) ).
% qbs_morphism_cong
thf(fact_23_qbs__space__R,axiom,
! [X: sigma_measure_real_a] :
( ( qbs_space_real_a @ ( measur4579233246441984221real_a @ X ) )
= ( sigma_space_real_a @ X ) ) ).
% qbs_space_R
thf(fact_24_qbs__space__R,axiom,
! [X: sigma_measure_set_a] :
( ( qbs_space_set_a @ ( measur1138716014562805426_set_a @ X ) )
= ( sigma_space_set_a @ X ) ) ).
% qbs_space_R
thf(fact_25_qbs__space__R,axiom,
! [X: sigma_measure_a_a] :
( ( qbs_space_a_a @ ( measur4192826365014959889bs_a_a @ X ) )
= ( sigma_space_a_a @ X ) ) ).
% qbs_space_R
thf(fact_26_qbs__space__R,axiom,
! [X: sigma_measure_real] :
( ( qbs_space_real @ ( measur6875533127466166616s_real @ X ) )
= ( sigma_space_real @ X ) ) ).
% qbs_space_R
thf(fact_27_qbs__space__R,axiom,
! [X: sigma_measure_a] :
( ( qbs_space_a @ ( measur6507891955840068946_qbs_a @ X ) )
= ( sigma_space_a @ X ) ) ).
% qbs_space_R
thf(fact_28_sigma__algebra__imp__Dynkin__system,axiom,
! [Omega: set_a,M: set_set_a] :
( ( sigma_4968961713055010667ebra_a @ Omega @ M )
=> ( sigma_2757993544071651912stem_a @ Omega @ M ) ) ).
% sigma_algebra_imp_Dynkin_system
thf(fact_29_sigma__algebra_Oaxioms_I1_J,axiom,
! [Omega: set_a,M: set_set_a] :
( ( sigma_4968961713055010667ebra_a @ Omega @ M )
=> ( sigma_algebra_a @ Omega @ M ) ) ).
% sigma_algebra.axioms(1)
thf(fact_30_sigma__algebra_Ocountably__additive__eq,axiom,
! [Omega: set_a,M: set_set_a,Mu: set_a > extend8495563244428889912nnreal,Mu2: set_a > extend8495563244428889912nnreal] :
( ( sigma_4968961713055010667ebra_a @ Omega @ M )
=> ( ! [A: set_a] :
( ( member_set_a @ A @ M )
=> ( ( Mu @ A )
= ( Mu2 @ A ) ) )
=> ( ( sigma_498350552340604931tive_a @ M @ Mu )
= ( sigma_498350552340604931tive_a @ M @ Mu2 ) ) ) ) ).
% sigma_algebra.countably_additive_eq
thf(fact_31_qbs__Mx__to__X_I2_J,axiom,
! [Alpha: real > ( a > a ) > a,X: quasi_borel_a_a_a2,R: real] :
( ( member_real_a_a_a @ Alpha @ ( qbs_Mx_a_a_a @ X ) )
=> ( member_a_a_a @ ( Alpha @ R ) @ ( qbs_space_a_a_a @ X ) ) ) ).
% qbs_Mx_to_X(2)
thf(fact_32_qbs__Mx__to__X_I2_J,axiom,
! [Alpha: real > a > real,X: quasi_borel_a_real,R: real] :
( ( member_real_a_real2 @ Alpha @ ( qbs_Mx_a_real @ X ) )
=> ( member_a_real @ ( Alpha @ R ) @ ( qbs_space_a_real @ X ) ) ) ).
% qbs_Mx_to_X(2)
thf(fact_33_qbs__Mx__to__X_I2_J,axiom,
! [Alpha: real > a > set_a,X: quasi_borel_a_set_a,R: real] :
( ( member_real_a_set_a @ Alpha @ ( qbs_Mx_a_set_a @ X ) )
=> ( member_a_set_a @ ( Alpha @ R ) @ ( qbs_space_a_set_a @ X ) ) ) ).
% qbs_Mx_to_X(2)
thf(fact_34_qbs__Mx__to__X_I2_J,axiom,
! [Alpha: real > a > real > a,X: quasi_borel_a_real_a,R: real] :
( ( member_real_a_real_a @ Alpha @ ( qbs_Mx_a_real_a @ X ) )
=> ( member_a_real_a @ ( Alpha @ R ) @ ( qbs_space_a_real_a @ X ) ) ) ).
% qbs_Mx_to_X(2)
thf(fact_35_qbs__Mx__to__X_I2_J,axiom,
! [Alpha: real > a > a > a,X: quasi_borel_a_a_a,R: real] :
( ( member_real_a_a_a2 @ Alpha @ ( qbs_Mx_a_a_a2 @ X ) )
=> ( member_a_a_a2 @ ( Alpha @ R ) @ ( qbs_space_a_a_a2 @ X ) ) ) ).
% qbs_Mx_to_X(2)
thf(fact_36_qbs__Mx__to__X_I2_J,axiom,
! [Alpha: real > real,X: quasi_borel_real,R: real] :
( ( member_real_real @ Alpha @ ( qbs_Mx_real @ X ) )
=> ( member_real @ ( Alpha @ R ) @ ( qbs_space_real @ X ) ) ) ).
% qbs_Mx_to_X(2)
thf(fact_37_qbs__Mx__to__X_I2_J,axiom,
! [Alpha: real > a > a,X: quasi_borel_a_a,R: real] :
( ( member_real_a_a2 @ Alpha @ ( qbs_Mx_a_a @ X ) )
=> ( member_a_a @ ( Alpha @ R ) @ ( qbs_space_a_a @ X ) ) ) ).
% qbs_Mx_to_X(2)
thf(fact_38_qbs__Mx__to__X_I2_J,axiom,
! [Alpha: real > set_a,X: quasi_borel_set_a,R: real] :
( ( member_real_set_a @ Alpha @ ( qbs_Mx_set_a @ X ) )
=> ( member_set_a @ ( Alpha @ R ) @ ( qbs_space_set_a @ X ) ) ) ).
% qbs_Mx_to_X(2)
thf(fact_39_qbs__Mx__to__X_I2_J,axiom,
! [Alpha: real > real > a,X: quasi_borel_real_a,R: real] :
( ( member_real_real_a @ Alpha @ ( qbs_Mx_real_a @ X ) )
=> ( member_real_a @ ( Alpha @ R ) @ ( qbs_space_real_a @ X ) ) ) ).
% qbs_Mx_to_X(2)
thf(fact_40_qbs__Mx__to__X_I2_J,axiom,
! [Alpha: real > a,X: quasi_borel_a,R: real] :
( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X ) )
=> ( member_a @ ( Alpha @ R ) @ ( qbs_space_a @ X ) ) ) ).
% qbs_Mx_to_X(2)
thf(fact_41_qbs__space__eq__Mx,axiom,
! [X: quasi_borel_real_a,Y: quasi_borel_real_a] :
( ( ( qbs_Mx_real_a @ X )
= ( qbs_Mx_real_a @ Y ) )
=> ( ( qbs_space_real_a @ X )
= ( qbs_space_real_a @ Y ) ) ) ).
% qbs_space_eq_Mx
thf(fact_42_qbs__space__eq__Mx,axiom,
! [X: quasi_borel_set_a,Y: quasi_borel_set_a] :
( ( ( qbs_Mx_set_a @ X )
= ( qbs_Mx_set_a @ Y ) )
=> ( ( qbs_space_set_a @ X )
= ( qbs_space_set_a @ Y ) ) ) ).
% qbs_space_eq_Mx
thf(fact_43_qbs__space__eq__Mx,axiom,
! [X: quasi_borel_a_a,Y: quasi_borel_a_a] :
( ( ( qbs_Mx_a_a @ X )
= ( qbs_Mx_a_a @ Y ) )
=> ( ( qbs_space_a_a @ X )
= ( qbs_space_a_a @ Y ) ) ) ).
% qbs_space_eq_Mx
thf(fact_44_qbs__space__eq__Mx,axiom,
! [X: quasi_borel_real,Y: quasi_borel_real] :
( ( ( qbs_Mx_real @ X )
= ( qbs_Mx_real @ Y ) )
=> ( ( qbs_space_real @ X )
= ( qbs_space_real @ Y ) ) ) ).
% qbs_space_eq_Mx
thf(fact_45_qbs__space__eq__Mx,axiom,
! [X: quasi_borel_a,Y: quasi_borel_a] :
( ( ( qbs_Mx_a @ X )
= ( qbs_Mx_a @ Y ) )
=> ( ( qbs_space_a @ X )
= ( qbs_space_a @ Y ) ) ) ).
% qbs_space_eq_Mx
thf(fact_46_sigma__algebra__Pow,axiom,
! [Sp: set_real_a] : ( sigma_6829682388519410934real_a @ Sp @ ( pow_real_a @ Sp ) ) ).
% sigma_algebra_Pow
thf(fact_47_sigma__algebra__Pow,axiom,
! [Sp: set_set_a] : ( sigma_6575008089947538635_set_a @ Sp @ ( pow_set_a @ Sp ) ) ).
% sigma_algebra_Pow
thf(fact_48_sigma__algebra__Pow,axiom,
! [Sp: set_a_a] : ( sigma_17020130125533880ra_a_a @ Sp @ ( pow_a_a @ Sp ) ) ).
% sigma_algebra_Pow
thf(fact_49_sigma__algebra__Pow,axiom,
! [Sp: set_a] : ( sigma_4968961713055010667ebra_a @ Sp @ ( pow_a @ Sp ) ) ).
% sigma_algebra_Pow
thf(fact_50_sigma__algebra_Osigma__sets__eq,axiom,
! [Omega: set_a,M: set_set_a] :
( ( sigma_4968961713055010667ebra_a @ Omega @ M )
=> ( ( sigma_sigma_sets_a @ Omega @ M )
= M ) ) ).
% sigma_algebra.sigma_sets_eq
thf(fact_51_algebra_Otop,axiom,
! [Omega: set_a,M: set_set_a] :
( ( sigma_algebra_a @ Omega @ M )
=> ( member_set_a @ Omega @ M ) ) ).
% algebra.top
thf(fact_52_sigma__sets_OBasic,axiom,
! [A2: set_a,A3: set_set_a,Sp: set_a] :
( ( member_set_a @ A2 @ A3 )
=> ( member_set_a @ A2 @ ( sigma_sigma_sets_a @ Sp @ A3 ) ) ) ).
% sigma_sets.Basic
thf(fact_53_Dynkin__system_Ospace,axiom,
! [Omega: set_a,M: set_set_a] :
( ( sigma_2757993544071651912stem_a @ Omega @ M )
=> ( member_set_a @ Omega @ M ) ) ).
% Dynkin_system.space
thf(fact_54_qbs__eqI,axiom,
! [X: quasi_borel_real_a,Y: quasi_borel_real_a] :
( ( ( qbs_Mx_real_a @ X )
= ( qbs_Mx_real_a @ Y ) )
=> ( X = Y ) ) ).
% qbs_eqI
thf(fact_55_qbs__eqI,axiom,
! [X: quasi_borel_set_a,Y: quasi_borel_set_a] :
( ( ( qbs_Mx_set_a @ X )
= ( qbs_Mx_set_a @ Y ) )
=> ( X = Y ) ) ).
% qbs_eqI
thf(fact_56_qbs__eqI,axiom,
! [X: quasi_borel_a_a,Y: quasi_borel_a_a] :
( ( ( qbs_Mx_a_a @ X )
= ( qbs_Mx_a_a @ Y ) )
=> ( X = Y ) ) ).
% qbs_eqI
thf(fact_57_qbs__eqI,axiom,
! [X: quasi_borel_real,Y: quasi_borel_real] :
( ( ( qbs_Mx_real @ X )
= ( qbs_Mx_real @ Y ) )
=> ( X = Y ) ) ).
% qbs_eqI
thf(fact_58_qbs__eqI,axiom,
! [X: quasi_borel_a,Y: quasi_borel_a] :
( ( ( qbs_Mx_a @ X )
= ( qbs_Mx_a @ Y ) )
=> ( X = Y ) ) ).
% qbs_eqI
thf(fact_59_algebra__Pow,axiom,
! [Sp: set_real_a] : ( sigma_algebra_real_a @ Sp @ ( pow_real_a @ Sp ) ) ).
% algebra_Pow
thf(fact_60_algebra__Pow,axiom,
! [Sp: set_set_a] : ( sigma_algebra_set_a @ Sp @ ( pow_set_a @ Sp ) ) ).
% algebra_Pow
thf(fact_61_algebra__Pow,axiom,
! [Sp: set_a_a] : ( sigma_algebra_a_a @ Sp @ ( pow_a_a @ Sp ) ) ).
% algebra_Pow
thf(fact_62_algebra__Pow,axiom,
! [Sp: set_a] : ( sigma_algebra_a @ Sp @ ( pow_a @ Sp ) ) ).
% algebra_Pow
thf(fact_63_sigma__sets__eqI,axiom,
! [A3: set_set_a,M: set_a,B: set_set_a] :
( ! [A: set_a] :
( ( member_set_a @ A @ A3 )
=> ( member_set_a @ A @ ( sigma_sigma_sets_a @ M @ B ) ) )
=> ( ! [B2: set_a] :
( ( member_set_a @ B2 @ B )
=> ( member_set_a @ B2 @ ( sigma_sigma_sets_a @ M @ A3 ) ) )
=> ( ( sigma_sigma_sets_a @ M @ A3 )
= ( sigma_sigma_sets_a @ M @ B ) ) ) ) ).
% sigma_sets_eqI
thf(fact_64_sigma__sets__top,axiom,
! [Sp: set_a,A3: set_set_a] : ( member_set_a @ Sp @ ( sigma_sigma_sets_a @ Sp @ A3 ) ) ).
% sigma_sets_top
thf(fact_65_Dynkin__system__trivial,axiom,
! [A3: set_real_a] : ( sigma_4207074038752281043real_a @ A3 @ ( pow_real_a @ A3 ) ) ).
% Dynkin_system_trivial
thf(fact_66_Dynkin__system__trivial,axiom,
! [A3: set_set_a] : ( sigma_6194451600871760808_set_a @ A3 @ ( pow_set_a @ A3 ) ) ).
% Dynkin_system_trivial
thf(fact_67_Dynkin__system__trivial,axiom,
! [A3: set_a_a] : ( sigma_4503493975268142235em_a_a @ A3 @ ( pow_a_a @ A3 ) ) ).
% Dynkin_system_trivial
thf(fact_68_Dynkin__system__trivial,axiom,
! [A3: set_a] : ( sigma_2757993544071651912stem_a @ A3 @ ( pow_a @ A3 ) ) ).
% Dynkin_system_trivial
thf(fact_69_sigma__algebra__def,axiom,
( sigma_4968961713055010667ebra_a
= ( ^ [Omega2: set_a,M2: set_set_a] :
( ( sigma_algebra_a @ Omega2 @ M2 )
& ( sigma_1832572188971243144ioms_a @ M2 ) ) ) ) ).
% sigma_algebra_def
thf(fact_70_sigma__algebra_Ointro,axiom,
! [Omega: set_a,M: set_set_a] :
( ( sigma_algebra_a @ Omega @ M )
=> ( ( sigma_1832572188971243144ioms_a @ M )
=> ( sigma_4968961713055010667ebra_a @ Omega @ M ) ) ) ).
% sigma_algebra.intro
thf(fact_71_sigma__algebra__sigma__sets,axiom,
! [A2: set_set_real_a,Omega: set_real_a] :
( ( ord_le6426559262036278761real_a @ A2 @ ( pow_real_a @ Omega ) )
=> ( sigma_6829682388519410934real_a @ Omega @ ( sigma_136333207653772897real_a @ Omega @ A2 ) ) ) ).
% sigma_algebra_sigma_sets
thf(fact_72_sigma__algebra__sigma__sets,axiom,
! [A2: set_set_set_a,Omega: set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ ( pow_set_a @ Omega ) )
=> ( sigma_6575008089947538635_set_a @ Omega @ ( sigma_2987359967864564790_set_a @ Omega @ A2 ) ) ) ).
% sigma_algebra_sigma_sets
thf(fact_73_sigma__algebra__sigma__sets,axiom,
! [A2: set_set_a_a,Omega: set_a_a] :
( ( ord_le1342339564197316145et_a_a @ A2 @ ( pow_a_a @ Omega ) )
=> ( sigma_17020130125533880ra_a_a @ Omega @ ( sigma_sigma_sets_a_a @ Omega @ A2 ) ) ) ).
% sigma_algebra_sigma_sets
thf(fact_74_sigma__algebra__sigma__sets,axiom,
! [A2: set_set_a,Omega: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( pow_a @ Omega ) )
=> ( sigma_4968961713055010667ebra_a @ Omega @ ( sigma_sigma_sets_a @ Omega @ A2 ) ) ) ).
% sigma_algebra_sigma_sets
thf(fact_75_Pow__top,axiom,
! [A3: set_real_a] : ( member_set_real_a @ A3 @ ( pow_real_a @ A3 ) ) ).
% Pow_top
thf(fact_76_Pow__top,axiom,
! [A3: set_set_a] : ( member_set_set_a @ A3 @ ( pow_set_a @ A3 ) ) ).
% Pow_top
thf(fact_77_Pow__top,axiom,
! [A3: set_a_a] : ( member_set_a_a2 @ A3 @ ( pow_a_a @ A3 ) ) ).
% Pow_top
thf(fact_78_Pow__top,axiom,
! [A3: set_a] : ( member_set_a @ A3 @ ( pow_a @ A3 ) ) ).
% Pow_top
thf(fact_79_Dynkin__system_Oaxioms_I2_J,axiom,
! [Omega: set_a,M: set_set_a] :
( ( sigma_2757993544071651912stem_a @ Omega @ M )
=> ( sigma_1042032682244386661ioms_a @ Omega @ M ) ) ).
% Dynkin_system.axioms(2)
thf(fact_80_Dynkin__system_Osigma__algebra__eq__Int__stable,axiom,
! [Omega: set_a,M: set_set_a] :
( ( sigma_2757993544071651912stem_a @ Omega @ M )
=> ( ( sigma_4968961713055010667ebra_a @ Omega @ M )
= ( sigma_Int_stable_a @ M ) ) ) ).
% Dynkin_system.sigma_algebra_eq_Int_stable
thf(fact_81_algebra_Oaxioms_I2_J,axiom,
! [Omega: set_a,M: set_set_a] :
( ( sigma_algebra_a @ Omega @ M )
=> ( sigma_1540858801969833537ioms_a @ Omega @ M ) ) ).
% algebra.axioms(2)
thf(fact_82_binary__in__sigma__sets,axiom,
! [A2: set_a,Sp: set_a,A3: set_set_a,B3: set_a,I: nat] :
( ( member_set_a @ A2 @ ( sigma_sigma_sets_a @ Sp @ A3 ) )
=> ( ( member_set_a @ B3 @ ( sigma_sigma_sets_a @ Sp @ A3 ) )
=> ( member_set_a @ ( sigma_binary_set_a @ A2 @ B3 @ I ) @ ( sigma_sigma_sets_a @ Sp @ A3 ) ) ) ) ).
% binary_in_sigma_sets
thf(fact_83_generating__Mx__Mx,axiom,
! [X: quasi_borel_real_a] :
( ( generating_Mx_real_a @ ( qbs_space_real_a @ X ) @ ( qbs_Mx_real_a @ X ) )
= ( qbs_Mx_real_a @ X ) ) ).
% generating_Mx_Mx
thf(fact_84_generating__Mx__Mx,axiom,
! [X: quasi_borel_set_a] :
( ( generating_Mx_set_a @ ( qbs_space_set_a @ X ) @ ( qbs_Mx_set_a @ X ) )
= ( qbs_Mx_set_a @ X ) ) ).
% generating_Mx_Mx
thf(fact_85_generating__Mx__Mx,axiom,
! [X: quasi_borel_a_a] :
( ( generating_Mx_a_a @ ( qbs_space_a_a @ X ) @ ( qbs_Mx_a_a @ X ) )
= ( qbs_Mx_a_a @ X ) ) ).
% generating_Mx_Mx
thf(fact_86_generating__Mx__Mx,axiom,
! [X: quasi_borel_real] :
( ( generating_Mx_real @ ( qbs_space_real @ X ) @ ( qbs_Mx_real @ X ) )
= ( qbs_Mx_real @ X ) ) ).
% generating_Mx_Mx
thf(fact_87_generating__Mx__Mx,axiom,
! [X: quasi_borel_a] :
( ( generating_Mx_a @ ( qbs_space_a @ X ) @ ( qbs_Mx_a @ X ) )
= ( qbs_Mx_a @ X ) ) ).
% generating_Mx_Mx
thf(fact_88_Dynkin__system_ODynkin__idem,axiom,
! [Omega: set_a,M: set_set_a] :
( ( sigma_2757993544071651912stem_a @ Omega @ M )
=> ( ( sigma_Dynkin_a @ Omega @ M )
= M ) ) ).
% Dynkin_system.Dynkin_idem
thf(fact_89_algebra_Ois__sigma__algebra,axiom,
! [Omega: set_a,M: set_set_a] :
( ( sigma_algebra_a @ Omega @ M )
=> ( ( finite_finite_set_a @ M )
=> ( sigma_4968961713055010667ebra_a @ Omega @ M ) ) ) ).
% algebra.is_sigma_algebra
thf(fact_90_sigma__algebra_Osigma__sets__subset,axiom,
! [Omega: set_a,M: set_set_a,A2: set_set_a] :
( ( sigma_4968961713055010667ebra_a @ Omega @ M )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ M )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sigma_sets_a @ Omega @ A2 ) @ M ) ) ) ).
% sigma_algebra.sigma_sets_subset
thf(fact_91_qbs__morphismE_I3_J,axiom,
! [F: real > a,X: quasi_borel_real,Y: quasi_borel_a,Alpha: real > real] :
( ( member_real_a @ F @ ( qbs_morphism_real_a @ X @ Y ) )
=> ( ( member_real_real @ Alpha @ ( qbs_Mx_real @ X ) )
=> ( member_real_a @ ( comp_real_a_real @ F @ Alpha ) @ ( qbs_Mx_a @ Y ) ) ) ) ).
% qbs_morphismE(3)
thf(fact_92_qbs__morphismE_I3_J,axiom,
! [F: a > a,X: quasi_borel_a,Y: quasi_borel_a,Alpha: real > a] :
( ( member_a_a @ F @ ( qbs_morphism_a_a @ X @ Y ) )
=> ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X ) )
=> ( member_real_a @ ( comp_a_a_real @ F @ Alpha ) @ ( qbs_Mx_a @ Y ) ) ) ) ).
% qbs_morphismE(3)
thf(fact_93_qbs__morphismE_I3_J,axiom,
! [F: real > real,X: quasi_borel_real,Y: quasi_borel_real,Alpha: real > real] :
( ( member_real_real @ F @ ( qbs_mo5229770564518008146l_real @ X @ Y ) )
=> ( ( member_real_real @ Alpha @ ( qbs_Mx_real @ X ) )
=> ( member_real_real @ ( comp_real_real_real @ F @ Alpha ) @ ( qbs_Mx_real @ Y ) ) ) ) ).
% qbs_morphismE(3)
thf(fact_94_qbs__morphismE_I3_J,axiom,
! [F: a > real,X: quasi_borel_a,Y: quasi_borel_real,Alpha: real > a] :
( ( member_a_real @ F @ ( qbs_morphism_a_real @ X @ Y ) )
=> ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X ) )
=> ( member_real_real @ ( comp_a_real_real @ F @ Alpha ) @ ( qbs_Mx_real @ Y ) ) ) ) ).
% qbs_morphismE(3)
thf(fact_95_qbs__morphismE_I3_J,axiom,
! [F: set_a > a,X: quasi_borel_set_a,Y: quasi_borel_a,Alpha: real > set_a] :
( ( member_set_a_a @ F @ ( qbs_morphism_set_a_a @ X @ Y ) )
=> ( ( member_real_set_a @ Alpha @ ( qbs_Mx_set_a @ X ) )
=> ( member_real_a @ ( comp_set_a_a_real @ F @ Alpha ) @ ( qbs_Mx_a @ Y ) ) ) ) ).
% qbs_morphismE(3)
thf(fact_96_qbs__morphismE_I3_J,axiom,
! [F: set_a > real,X: quasi_borel_set_a,Y: quasi_borel_real,Alpha: real > set_a] :
( ( member_set_a_real @ F @ ( qbs_mo4864135009344410130a_real @ X @ Y ) )
=> ( ( member_real_set_a @ Alpha @ ( qbs_Mx_set_a @ X ) )
=> ( member_real_real @ ( comp_set_a_real_real @ F @ Alpha ) @ ( qbs_Mx_real @ Y ) ) ) ) ).
% qbs_morphismE(3)
thf(fact_97_qbs__morphismE_I3_J,axiom,
! [F: real > set_a,X: quasi_borel_real,Y: quasi_borel_set_a,Alpha: real > real] :
( ( member_real_set_a @ F @ ( qbs_mo8580062899995455544_set_a @ X @ Y ) )
=> ( ( member_real_real @ Alpha @ ( qbs_Mx_real @ X ) )
=> ( member_real_set_a @ ( comp_real_set_a_real @ F @ Alpha ) @ ( qbs_Mx_set_a @ Y ) ) ) ) ).
% qbs_morphismE(3)
thf(fact_98_qbs__morphismE_I3_J,axiom,
! [F: a > set_a,X: quasi_borel_a,Y: quasi_borel_set_a,Alpha: real > a] :
( ( member_a_set_a @ F @ ( qbs_morphism_a_set_a @ X @ Y ) )
=> ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X ) )
=> ( member_real_set_a @ ( comp_a_set_a_real @ F @ Alpha ) @ ( qbs_Mx_set_a @ Y ) ) ) ) ).
% qbs_morphismE(3)
thf(fact_99_qbs__morphismE_I3_J,axiom,
! [F: ( real > a ) > a,X: quasi_borel_real_a,Y: quasi_borel_a,Alpha: real > real > a] :
( ( member_real_a_a @ F @ ( qbs_mo2483107194261503661al_a_a @ X @ Y ) )
=> ( ( member_real_real_a @ Alpha @ ( qbs_Mx_real_a @ X ) )
=> ( member_real_a @ ( comp_real_a_a_real @ F @ Alpha ) @ ( qbs_Mx_a @ Y ) ) ) ) ).
% qbs_morphismE(3)
thf(fact_100_qbs__morphismE_I3_J,axiom,
! [F: ( real > a ) > real,X: quasi_borel_real_a,Y: quasi_borel_real,Alpha: real > real > a] :
( ( member_real_a_real @ F @ ( qbs_mo719586419469067453a_real @ X @ Y ) )
=> ( ( member_real_real_a @ Alpha @ ( qbs_Mx_real_a @ X ) )
=> ( member_real_real @ ( comp_r2324749603045630912l_real @ F @ Alpha ) @ ( qbs_Mx_real @ Y ) ) ) ) ).
% qbs_morphismE(3)
thf(fact_101_qbs__morphismI,axiom,
! [X: quasi_borel_real,F: real > a,Y: quasi_borel_a] :
( ! [Alpha2: real > real] :
( ( member_real_real @ Alpha2 @ ( qbs_Mx_real @ X ) )
=> ( member_real_a @ ( comp_real_a_real @ F @ Alpha2 ) @ ( qbs_Mx_a @ Y ) ) )
=> ( member_real_a @ F @ ( qbs_morphism_real_a @ X @ Y ) ) ) ).
% qbs_morphismI
thf(fact_102_qbs__morphismI,axiom,
! [X: quasi_borel_a,F: a > a,Y: quasi_borel_a] :
( ! [Alpha2: real > a] :
( ( member_real_a @ Alpha2 @ ( qbs_Mx_a @ X ) )
=> ( member_real_a @ ( comp_a_a_real @ F @ Alpha2 ) @ ( qbs_Mx_a @ Y ) ) )
=> ( member_a_a @ F @ ( qbs_morphism_a_a @ X @ Y ) ) ) ).
% qbs_morphismI
thf(fact_103_qbs__morphismI,axiom,
! [X: quasi_borel_real,F: real > real,Y: quasi_borel_real] :
( ! [Alpha2: real > real] :
( ( member_real_real @ Alpha2 @ ( qbs_Mx_real @ X ) )
=> ( member_real_real @ ( comp_real_real_real @ F @ Alpha2 ) @ ( qbs_Mx_real @ Y ) ) )
=> ( member_real_real @ F @ ( qbs_mo5229770564518008146l_real @ X @ Y ) ) ) ).
% qbs_morphismI
thf(fact_104_qbs__morphismI,axiom,
! [X: quasi_borel_a,F: a > real,Y: quasi_borel_real] :
( ! [Alpha2: real > a] :
( ( member_real_a @ Alpha2 @ ( qbs_Mx_a @ X ) )
=> ( member_real_real @ ( comp_a_real_real @ F @ Alpha2 ) @ ( qbs_Mx_real @ Y ) ) )
=> ( member_a_real @ F @ ( qbs_morphism_a_real @ X @ Y ) ) ) ).
% qbs_morphismI
thf(fact_105_qbs__morphismI,axiom,
! [X: quasi_borel_set_a,F: set_a > a,Y: quasi_borel_a] :
( ! [Alpha2: real > set_a] :
( ( member_real_set_a @ Alpha2 @ ( qbs_Mx_set_a @ X ) )
=> ( member_real_a @ ( comp_set_a_a_real @ F @ Alpha2 ) @ ( qbs_Mx_a @ Y ) ) )
=> ( member_set_a_a @ F @ ( qbs_morphism_set_a_a @ X @ Y ) ) ) ).
% qbs_morphismI
thf(fact_106_qbs__morphismI,axiom,
! [X: quasi_borel_set_a,F: set_a > real,Y: quasi_borel_real] :
( ! [Alpha2: real > set_a] :
( ( member_real_set_a @ Alpha2 @ ( qbs_Mx_set_a @ X ) )
=> ( member_real_real @ ( comp_set_a_real_real @ F @ Alpha2 ) @ ( qbs_Mx_real @ Y ) ) )
=> ( member_set_a_real @ F @ ( qbs_mo4864135009344410130a_real @ X @ Y ) ) ) ).
% qbs_morphismI
thf(fact_107_qbs__morphismI,axiom,
! [X: quasi_borel_real,F: real > set_a,Y: quasi_borel_set_a] :
( ! [Alpha2: real > real] :
( ( member_real_real @ Alpha2 @ ( qbs_Mx_real @ X ) )
=> ( member_real_set_a @ ( comp_real_set_a_real @ F @ Alpha2 ) @ ( qbs_Mx_set_a @ Y ) ) )
=> ( member_real_set_a @ F @ ( qbs_mo8580062899995455544_set_a @ X @ Y ) ) ) ).
% qbs_morphismI
thf(fact_108_qbs__morphismI,axiom,
! [X: quasi_borel_a,F: a > set_a,Y: quasi_borel_set_a] :
( ! [Alpha2: real > a] :
( ( member_real_a @ Alpha2 @ ( qbs_Mx_a @ X ) )
=> ( member_real_set_a @ ( comp_a_set_a_real @ F @ Alpha2 ) @ ( qbs_Mx_set_a @ Y ) ) )
=> ( member_a_set_a @ F @ ( qbs_morphism_a_set_a @ X @ Y ) ) ) ).
% qbs_morphismI
thf(fact_109_qbs__morphismI,axiom,
! [X: quasi_borel_real_a,F: ( real > a ) > a,Y: quasi_borel_a] :
( ! [Alpha2: real > real > a] :
( ( member_real_real_a @ Alpha2 @ ( qbs_Mx_real_a @ X ) )
=> ( member_real_a @ ( comp_real_a_a_real @ F @ Alpha2 ) @ ( qbs_Mx_a @ Y ) ) )
=> ( member_real_a_a @ F @ ( qbs_mo2483107194261503661al_a_a @ X @ Y ) ) ) ).
% qbs_morphismI
thf(fact_110_qbs__morphismI,axiom,
! [X: quasi_borel_real_a,F: ( real > a ) > real,Y: quasi_borel_real] :
( ! [Alpha2: real > real > a] :
( ( member_real_real_a @ Alpha2 @ ( qbs_Mx_real_a @ X ) )
=> ( member_real_real @ ( comp_r2324749603045630912l_real @ F @ Alpha2 ) @ ( qbs_Mx_real @ Y ) ) )
=> ( member_real_a_real @ F @ ( qbs_mo719586419469067453a_real @ X @ Y ) ) ) ).
% qbs_morphismI
thf(fact_111_subset__antisym,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_112_subset__antisym,axiom,
! [A3: set_real_a,B: set_real_a] :
( ( ord_le5743406823621094409real_a @ A3 @ B )
=> ( ( ord_le5743406823621094409real_a @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_113_subset__antisym,axiom,
! [A3: set_a_a,B: set_a_a] :
( ( ord_less_eq_set_a_a @ A3 @ B )
=> ( ( ord_less_eq_set_a_a @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_114_subset__antisym,axiom,
! [A3: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ( ord_less_eq_set_a @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_115_subsetI,axiom,
! [A3: set_a_a_a2,B: set_a_a_a2] :
( ! [X3: ( a > a ) > a] :
( ( member_a_a_a @ X3 @ A3 )
=> ( member_a_a_a @ X3 @ B ) )
=> ( ord_le7181591058469194768_a_a_a @ A3 @ B ) ) ).
% subsetI
thf(fact_116_subsetI,axiom,
! [A3: set_a_real,B: set_a_real] :
( ! [X3: a > real] :
( ( member_a_real @ X3 @ A3 )
=> ( member_a_real @ X3 @ B ) )
=> ( ord_le3334967407727675675a_real @ A3 @ B ) ) ).
% subsetI
thf(fact_117_subsetI,axiom,
! [A3: set_a_set_a,B: set_a_set_a] :
( ! [X3: a > set_a] :
( ( member_a_set_a @ X3 @ A3 )
=> ( member_a_set_a @ X3 @ B ) )
=> ( ord_le8982423316524089179_set_a @ A3 @ B ) ) ).
% subsetI
thf(fact_118_subsetI,axiom,
! [A3: set_a_real_a,B: set_a_real_a] :
( ! [X3: a > real > a] :
( ( member_a_real_a @ X3 @ A3 )
=> ( member_a_real_a @ X3 @ B ) )
=> ( ord_le792252046110693510real_a @ A3 @ B ) ) ).
% subsetI
thf(fact_119_subsetI,axiom,
! [A3: set_a_a_a,B: set_a_a_a] :
( ! [X3: a > a > a] :
( ( member_a_a_a2 @ X3 @ A3 )
=> ( member_a_a_a2 @ X3 @ B ) )
=> ( ord_le695056150018308692_a_a_a @ A3 @ B ) ) ).
% subsetI
thf(fact_120_subsetI,axiom,
! [A3: set_a_a,B: set_a_a] :
( ! [X3: a > a] :
( ( member_a_a @ X3 @ A3 )
=> ( member_a_a @ X3 @ B ) )
=> ( ord_less_eq_set_a_a @ A3 @ B ) ) ).
% subsetI
thf(fact_121_subsetI,axiom,
! [A3: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_a @ X3 @ B ) )
=> ( ord_less_eq_set_a @ A3 @ B ) ) ).
% subsetI
thf(fact_122_subsetI,axiom,
! [A3: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
=> ( member_set_a @ X3 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A3 @ B ) ) ).
% subsetI
thf(fact_123_subsetI,axiom,
! [A3: set_real_a,B: set_real_a] :
( ! [X3: real > a] :
( ( member_real_a @ X3 @ A3 )
=> ( member_real_a @ X3 @ B ) )
=> ( ord_le5743406823621094409real_a @ A3 @ B ) ) ).
% subsetI
thf(fact_124_Dynkin__Basic,axiom,
! [A3: set_a,M: set_set_a,Omega: set_a] :
( ( member_set_a @ A3 @ M )
=> ( member_set_a @ A3 @ ( sigma_Dynkin_a @ Omega @ M ) ) ) ).
% Dynkin_Basic
thf(fact_125_Pow__iff,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( member_set_set_a @ A3 @ ( pow_set_a @ B ) )
= ( ord_le3724670747650509150_set_a @ A3 @ B ) ) ).
% Pow_iff
thf(fact_126_Pow__iff,axiom,
! [A3: set_real_a,B: set_real_a] :
( ( member_set_real_a @ A3 @ ( pow_real_a @ B ) )
= ( ord_le5743406823621094409real_a @ A3 @ B ) ) ).
% Pow_iff
thf(fact_127_Pow__iff,axiom,
! [A3: set_a_a,B: set_a_a] :
( ( member_set_a_a2 @ A3 @ ( pow_a_a @ B ) )
= ( ord_less_eq_set_a_a @ A3 @ B ) ) ).
% Pow_iff
thf(fact_128_Pow__iff,axiom,
! [A3: set_a,B: set_a] :
( ( member_set_a @ A3 @ ( pow_a @ B ) )
= ( ord_less_eq_set_a @ A3 @ B ) ) ).
% Pow_iff
thf(fact_129_PowI,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( member_set_set_a @ A3 @ ( pow_set_a @ B ) ) ) ).
% PowI
thf(fact_130_PowI,axiom,
! [A3: set_real_a,B: set_real_a] :
( ( ord_le5743406823621094409real_a @ A3 @ B )
=> ( member_set_real_a @ A3 @ ( pow_real_a @ B ) ) ) ).
% PowI
thf(fact_131_PowI,axiom,
! [A3: set_a_a,B: set_a_a] :
( ( ord_less_eq_set_a_a @ A3 @ B )
=> ( member_set_a_a2 @ A3 @ ( pow_a_a @ B ) ) ) ).
% PowI
thf(fact_132_PowI,axiom,
! [A3: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( member_set_a @ A3 @ ( pow_a @ B ) ) ) ).
% PowI
thf(fact_133_generating__Mx_OBasic,axiom,
! [Alpha: real > a,Mx: set_real_a,X: set_a] :
( ( member_real_a @ Alpha @ Mx )
=> ( member_real_a @ Alpha @ ( generating_Mx_a @ X @ Mx ) ) ) ).
% generating_Mx.Basic
thf(fact_134_Pow__mono,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ord_le5722252365846178494_set_a @ ( pow_set_a @ A3 ) @ ( pow_set_a @ B ) ) ) ).
% Pow_mono
thf(fact_135_Pow__mono,axiom,
! [A3: set_real_a,B: set_real_a] :
( ( ord_le5743406823621094409real_a @ A3 @ B )
=> ( ord_le6426559262036278761real_a @ ( pow_real_a @ A3 ) @ ( pow_real_a @ B ) ) ) ).
% Pow_mono
thf(fact_136_Pow__mono,axiom,
! [A3: set_a_a,B: set_a_a] :
( ( ord_less_eq_set_a_a @ A3 @ B )
=> ( ord_le1342339564197316145et_a_a @ ( pow_a_a @ A3 ) @ ( pow_a_a @ B ) ) ) ).
% Pow_mono
thf(fact_137_Pow__mono,axiom,
! [A3: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ord_le3724670747650509150_set_a @ ( pow_a @ A3 ) @ ( pow_a @ B ) ) ) ).
% Pow_mono
thf(fact_138_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X4: set_a] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_139_Collect__mono__iff,axiom,
! [P: ( real > a ) > $o,Q: ( real > a ) > $o] :
( ( ord_le5743406823621094409real_a @ ( collect_real_a @ P ) @ ( collect_real_a @ Q ) )
= ( ! [X4: real > a] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_140_Collect__mono__iff,axiom,
! [P: ( a > a ) > $o,Q: ( a > a ) > $o] :
( ( ord_less_eq_set_a_a @ ( collect_a_a @ P ) @ ( collect_a_a @ Q ) )
= ( ! [X4: a > a] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_141_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_142_set__eq__subset,axiom,
( ( ^ [Y2: set_set_a,Z: set_set_a] : ( Y2 = Z ) )
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_143_set__eq__subset,axiom,
( ( ^ [Y2: set_real_a,Z: set_real_a] : ( Y2 = Z ) )
= ( ^ [A4: set_real_a,B4: set_real_a] :
( ( ord_le5743406823621094409real_a @ A4 @ B4 )
& ( ord_le5743406823621094409real_a @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_144_set__eq__subset,axiom,
( ( ^ [Y2: set_a_a,Z: set_a_a] : ( Y2 = Z ) )
= ( ^ [A4: set_a_a,B4: set_a_a] :
( ( ord_less_eq_set_a_a @ A4 @ B4 )
& ( ord_less_eq_set_a_a @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_145_set__eq__subset,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_146_subset__trans,axiom,
! [A3: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ord_le3724670747650509150_set_a @ A3 @ C ) ) ) ).
% subset_trans
thf(fact_147_subset__trans,axiom,
! [A3: set_real_a,B: set_real_a,C: set_real_a] :
( ( ord_le5743406823621094409real_a @ A3 @ B )
=> ( ( ord_le5743406823621094409real_a @ B @ C )
=> ( ord_le5743406823621094409real_a @ A3 @ C ) ) ) ).
% subset_trans
thf(fact_148_subset__trans,axiom,
! [A3: set_a_a,B: set_a_a,C: set_a_a] :
( ( ord_less_eq_set_a_a @ A3 @ B )
=> ( ( ord_less_eq_set_a_a @ B @ C )
=> ( ord_less_eq_set_a_a @ A3 @ C ) ) ) ).
% subset_trans
thf(fact_149_subset__trans,axiom,
! [A3: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A3 @ C ) ) ) ).
% subset_trans
thf(fact_150_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X3: set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_151_Collect__mono,axiom,
! [P: ( real > a ) > $o,Q: ( real > a ) > $o] :
( ! [X3: real > a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le5743406823621094409real_a @ ( collect_real_a @ P ) @ ( collect_real_a @ Q ) ) ) ).
% Collect_mono
thf(fact_152_Collect__mono,axiom,
! [P: ( a > a ) > $o,Q: ( a > a ) > $o] :
( ! [X3: a > a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a_a @ ( collect_a_a @ P ) @ ( collect_a_a @ Q ) ) ) ).
% Collect_mono
thf(fact_153_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_154_mem__Collect__eq,axiom,
! [A2: ( a > a ) > a,P: ( ( a > a ) > a ) > $o] :
( ( member_a_a_a @ A2 @ ( collect_a_a_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_155_mem__Collect__eq,axiom,
! [A2: a > real,P: ( a > real ) > $o] :
( ( member_a_real @ A2 @ ( collect_a_real @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_156_mem__Collect__eq,axiom,
! [A2: a > set_a,P: ( a > set_a ) > $o] :
( ( member_a_set_a @ A2 @ ( collect_a_set_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_157_mem__Collect__eq,axiom,
! [A2: a > real > a,P: ( a > real > a ) > $o] :
( ( member_a_real_a @ A2 @ ( collect_a_real_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_158_mem__Collect__eq,axiom,
! [A2: a > a > a,P: ( a > a > a ) > $o] :
( ( member_a_a_a2 @ A2 @ ( collect_a_a_a2 @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_159_mem__Collect__eq,axiom,
! [A2: a > a,P: ( a > a ) > $o] :
( ( member_a_a @ A2 @ ( collect_a_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_160_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_161_mem__Collect__eq,axiom,
! [A2: set_a,P: set_a > $o] :
( ( member_set_a @ A2 @ ( collect_set_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_162_mem__Collect__eq,axiom,
! [A2: real > a,P: ( real > a ) > $o] :
( ( member_real_a @ A2 @ ( collect_real_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_163_Collect__mem__eq,axiom,
! [A3: set_a_a_a2] :
( ( collect_a_a_a
@ ^ [X4: ( a > a ) > a] : ( member_a_a_a @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_164_Collect__mem__eq,axiom,
! [A3: set_a_real] :
( ( collect_a_real
@ ^ [X4: a > real] : ( member_a_real @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_165_Collect__mem__eq,axiom,
! [A3: set_a_set_a] :
( ( collect_a_set_a
@ ^ [X4: a > set_a] : ( member_a_set_a @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_166_Collect__mem__eq,axiom,
! [A3: set_a_real_a] :
( ( collect_a_real_a
@ ^ [X4: a > real > a] : ( member_a_real_a @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_167_Collect__mem__eq,axiom,
! [A3: set_a_a_a] :
( ( collect_a_a_a2
@ ^ [X4: a > a > a] : ( member_a_a_a2 @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_168_Collect__mem__eq,axiom,
! [A3: set_a_a] :
( ( collect_a_a
@ ^ [X4: a > a] : ( member_a_a @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_169_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_170_Collect__mem__eq,axiom,
! [A3: set_set_a] :
( ( collect_set_a
@ ^ [X4: set_a] : ( member_set_a @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_171_Collect__mem__eq,axiom,
! [A3: set_real_a] :
( ( collect_real_a
@ ^ [X4: real > a] : ( member_real_a @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_172_Collect__cong,axiom,
! [P: ( real > a ) > $o,Q: ( real > a ) > $o] :
( ! [X3: real > a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_real_a @ P )
= ( collect_real_a @ Q ) ) ) ).
% Collect_cong
thf(fact_173_Collect__cong,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X3: set_a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_set_a @ P )
= ( collect_set_a @ Q ) ) ) ).
% Collect_cong
thf(fact_174_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_175_Collect__cong,axiom,
! [P: ( a > a ) > $o,Q: ( a > a ) > $o] :
( ! [X3: a > a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_a_a @ P )
= ( collect_a_a @ Q ) ) ) ).
% Collect_cong
thf(fact_176_subset__refl,axiom,
! [A3: set_set_a] : ( ord_le3724670747650509150_set_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_177_subset__refl,axiom,
! [A3: set_real_a] : ( ord_le5743406823621094409real_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_178_subset__refl,axiom,
! [A3: set_a_a] : ( ord_less_eq_set_a_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_179_subset__refl,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_180_subset__iff,axiom,
( ord_le7181591058469194768_a_a_a
= ( ^ [A4: set_a_a_a2,B4: set_a_a_a2] :
! [T: ( a > a ) > a] :
( ( member_a_a_a @ T @ A4 )
=> ( member_a_a_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_181_subset__iff,axiom,
( ord_le3334967407727675675a_real
= ( ^ [A4: set_a_real,B4: set_a_real] :
! [T: a > real] :
( ( member_a_real @ T @ A4 )
=> ( member_a_real @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_182_subset__iff,axiom,
( ord_le8982423316524089179_set_a
= ( ^ [A4: set_a_set_a,B4: set_a_set_a] :
! [T: a > set_a] :
( ( member_a_set_a @ T @ A4 )
=> ( member_a_set_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_183_subset__iff,axiom,
( ord_le792252046110693510real_a
= ( ^ [A4: set_a_real_a,B4: set_a_real_a] :
! [T: a > real > a] :
( ( member_a_real_a @ T @ A4 )
=> ( member_a_real_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_184_subset__iff,axiom,
( ord_le695056150018308692_a_a_a
= ( ^ [A4: set_a_a_a,B4: set_a_a_a] :
! [T: a > a > a] :
( ( member_a_a_a2 @ T @ A4 )
=> ( member_a_a_a2 @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_185_subset__iff,axiom,
( ord_less_eq_set_a_a
= ( ^ [A4: set_a_a,B4: set_a_a] :
! [T: a > a] :
( ( member_a_a @ T @ A4 )
=> ( member_a_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_186_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [T: a] :
( ( member_a @ T @ A4 )
=> ( member_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_187_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A4 )
=> ( member_set_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_188_subset__iff,axiom,
( ord_le5743406823621094409real_a
= ( ^ [A4: set_real_a,B4: set_real_a] :
! [T: real > a] :
( ( member_real_a @ T @ A4 )
=> ( member_real_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_189_equalityD2,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( A3 = B )
=> ( ord_le3724670747650509150_set_a @ B @ A3 ) ) ).
% equalityD2
thf(fact_190_equalityD2,axiom,
! [A3: set_real_a,B: set_real_a] :
( ( A3 = B )
=> ( ord_le5743406823621094409real_a @ B @ A3 ) ) ).
% equalityD2
thf(fact_191_equalityD2,axiom,
! [A3: set_a_a,B: set_a_a] :
( ( A3 = B )
=> ( ord_less_eq_set_a_a @ B @ A3 ) ) ).
% equalityD2
thf(fact_192_equalityD2,axiom,
! [A3: set_a,B: set_a] :
( ( A3 = B )
=> ( ord_less_eq_set_a @ B @ A3 ) ) ).
% equalityD2
thf(fact_193_equalityD1,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( A3 = B )
=> ( ord_le3724670747650509150_set_a @ A3 @ B ) ) ).
% equalityD1
thf(fact_194_equalityD1,axiom,
! [A3: set_real_a,B: set_real_a] :
( ( A3 = B )
=> ( ord_le5743406823621094409real_a @ A3 @ B ) ) ).
% equalityD1
thf(fact_195_equalityD1,axiom,
! [A3: set_a_a,B: set_a_a] :
( ( A3 = B )
=> ( ord_less_eq_set_a_a @ A3 @ B ) ) ).
% equalityD1
thf(fact_196_equalityD1,axiom,
! [A3: set_a,B: set_a] :
( ( A3 = B )
=> ( ord_less_eq_set_a @ A3 @ B ) ) ).
% equalityD1
thf(fact_197_subset__eq,axiom,
( ord_le7181591058469194768_a_a_a
= ( ^ [A4: set_a_a_a2,B4: set_a_a_a2] :
! [X4: ( a > a ) > a] :
( ( member_a_a_a @ X4 @ A4 )
=> ( member_a_a_a @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_198_subset__eq,axiom,
( ord_le3334967407727675675a_real
= ( ^ [A4: set_a_real,B4: set_a_real] :
! [X4: a > real] :
( ( member_a_real @ X4 @ A4 )
=> ( member_a_real @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_199_subset__eq,axiom,
( ord_le8982423316524089179_set_a
= ( ^ [A4: set_a_set_a,B4: set_a_set_a] :
! [X4: a > set_a] :
( ( member_a_set_a @ X4 @ A4 )
=> ( member_a_set_a @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_200_subset__eq,axiom,
( ord_le792252046110693510real_a
= ( ^ [A4: set_a_real_a,B4: set_a_real_a] :
! [X4: a > real > a] :
( ( member_a_real_a @ X4 @ A4 )
=> ( member_a_real_a @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_201_subset__eq,axiom,
( ord_le695056150018308692_a_a_a
= ( ^ [A4: set_a_a_a,B4: set_a_a_a] :
! [X4: a > a > a] :
( ( member_a_a_a2 @ X4 @ A4 )
=> ( member_a_a_a2 @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_202_subset__eq,axiom,
( ord_less_eq_set_a_a
= ( ^ [A4: set_a_a,B4: set_a_a] :
! [X4: a > a] :
( ( member_a_a @ X4 @ A4 )
=> ( member_a_a @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_203_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [X4: a] :
( ( member_a @ X4 @ A4 )
=> ( member_a @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_204_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
! [X4: set_a] :
( ( member_set_a @ X4 @ A4 )
=> ( member_set_a @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_205_subset__eq,axiom,
( ord_le5743406823621094409real_a
= ( ^ [A4: set_real_a,B4: set_real_a] :
! [X4: real > a] :
( ( member_real_a @ X4 @ A4 )
=> ( member_real_a @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_206_equalityE,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( A3 = B )
=> ~ ( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ~ ( ord_le3724670747650509150_set_a @ B @ A3 ) ) ) ).
% equalityE
thf(fact_207_equalityE,axiom,
! [A3: set_real_a,B: set_real_a] :
( ( A3 = B )
=> ~ ( ( ord_le5743406823621094409real_a @ A3 @ B )
=> ~ ( ord_le5743406823621094409real_a @ B @ A3 ) ) ) ).
% equalityE
thf(fact_208_equalityE,axiom,
! [A3: set_a_a,B: set_a_a] :
( ( A3 = B )
=> ~ ( ( ord_less_eq_set_a_a @ A3 @ B )
=> ~ ( ord_less_eq_set_a_a @ B @ A3 ) ) ) ).
% equalityE
thf(fact_209_equalityE,axiom,
! [A3: set_a,B: set_a] :
( ( A3 = B )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B )
=> ~ ( ord_less_eq_set_a @ B @ A3 ) ) ) ).
% equalityE
thf(fact_210_subsetD,axiom,
! [A3: set_a_a_a2,B: set_a_a_a2,C2: ( a > a ) > a] :
( ( ord_le7181591058469194768_a_a_a @ A3 @ B )
=> ( ( member_a_a_a @ C2 @ A3 )
=> ( member_a_a_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_211_subsetD,axiom,
! [A3: set_a_real,B: set_a_real,C2: a > real] :
( ( ord_le3334967407727675675a_real @ A3 @ B )
=> ( ( member_a_real @ C2 @ A3 )
=> ( member_a_real @ C2 @ B ) ) ) ).
% subsetD
thf(fact_212_subsetD,axiom,
! [A3: set_a_set_a,B: set_a_set_a,C2: a > set_a] :
( ( ord_le8982423316524089179_set_a @ A3 @ B )
=> ( ( member_a_set_a @ C2 @ A3 )
=> ( member_a_set_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_213_subsetD,axiom,
! [A3: set_a_real_a,B: set_a_real_a,C2: a > real > a] :
( ( ord_le792252046110693510real_a @ A3 @ B )
=> ( ( member_a_real_a @ C2 @ A3 )
=> ( member_a_real_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_214_subsetD,axiom,
! [A3: set_a_a_a,B: set_a_a_a,C2: a > a > a] :
( ( ord_le695056150018308692_a_a_a @ A3 @ B )
=> ( ( member_a_a_a2 @ C2 @ A3 )
=> ( member_a_a_a2 @ C2 @ B ) ) ) ).
% subsetD
thf(fact_215_subsetD,axiom,
! [A3: set_a_a,B: set_a_a,C2: a > a] :
( ( ord_less_eq_set_a_a @ A3 @ B )
=> ( ( member_a_a @ C2 @ A3 )
=> ( member_a_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_216_subsetD,axiom,
! [A3: set_a,B: set_a,C2: a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ( member_a @ C2 @ A3 )
=> ( member_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_217_subsetD,axiom,
! [A3: set_set_a,B: set_set_a,C2: set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ( member_set_a @ C2 @ A3 )
=> ( member_set_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_218_subsetD,axiom,
! [A3: set_real_a,B: set_real_a,C2: real > a] :
( ( ord_le5743406823621094409real_a @ A3 @ B )
=> ( ( member_real_a @ C2 @ A3 )
=> ( member_real_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_219_in__mono,axiom,
! [A3: set_a_a_a2,B: set_a_a_a2,X2: ( a > a ) > a] :
( ( ord_le7181591058469194768_a_a_a @ A3 @ B )
=> ( ( member_a_a_a @ X2 @ A3 )
=> ( member_a_a_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_220_in__mono,axiom,
! [A3: set_a_real,B: set_a_real,X2: a > real] :
( ( ord_le3334967407727675675a_real @ A3 @ B )
=> ( ( member_a_real @ X2 @ A3 )
=> ( member_a_real @ X2 @ B ) ) ) ).
% in_mono
thf(fact_221_in__mono,axiom,
! [A3: set_a_set_a,B: set_a_set_a,X2: a > set_a] :
( ( ord_le8982423316524089179_set_a @ A3 @ B )
=> ( ( member_a_set_a @ X2 @ A3 )
=> ( member_a_set_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_222_in__mono,axiom,
! [A3: set_a_real_a,B: set_a_real_a,X2: a > real > a] :
( ( ord_le792252046110693510real_a @ A3 @ B )
=> ( ( member_a_real_a @ X2 @ A3 )
=> ( member_a_real_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_223_in__mono,axiom,
! [A3: set_a_a_a,B: set_a_a_a,X2: a > a > a] :
( ( ord_le695056150018308692_a_a_a @ A3 @ B )
=> ( ( member_a_a_a2 @ X2 @ A3 )
=> ( member_a_a_a2 @ X2 @ B ) ) ) ).
% in_mono
thf(fact_224_in__mono,axiom,
! [A3: set_a_a,B: set_a_a,X2: a > a] :
( ( ord_less_eq_set_a_a @ A3 @ B )
=> ( ( member_a_a @ X2 @ A3 )
=> ( member_a_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_225_in__mono,axiom,
! [A3: set_a,B: set_a,X2: a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ( member_a @ X2 @ A3 )
=> ( member_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_226_in__mono,axiom,
! [A3: set_set_a,B: set_set_a,X2: set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ( member_set_a @ X2 @ A3 )
=> ( member_set_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_227_in__mono,axiom,
! [A3: set_real_a,B: set_real_a,X2: real > a] :
( ( ord_le5743406823621094409real_a @ A3 @ B )
=> ( ( member_real_a @ X2 @ A3 )
=> ( member_real_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_228_PowD,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( member_set_set_a @ A3 @ ( pow_set_a @ B ) )
=> ( ord_le3724670747650509150_set_a @ A3 @ B ) ) ).
% PowD
thf(fact_229_PowD,axiom,
! [A3: set_real_a,B: set_real_a] :
( ( member_set_real_a @ A3 @ ( pow_real_a @ B ) )
=> ( ord_le5743406823621094409real_a @ A3 @ B ) ) ).
% PowD
thf(fact_230_PowD,axiom,
! [A3: set_a_a,B: set_a_a] :
( ( member_set_a_a2 @ A3 @ ( pow_a_a @ B ) )
=> ( ord_less_eq_set_a_a @ A3 @ B ) ) ).
% PowD
thf(fact_231_PowD,axiom,
! [A3: set_a,B: set_a] :
( ( member_set_a @ A3 @ ( pow_a @ B ) )
=> ( ord_less_eq_set_a @ A3 @ B ) ) ).
% PowD
thf(fact_232_algebra__axioms__def,axiom,
sigma_1540858801969833537ioms_a = member_set_a ).
% algebra_axioms_def
thf(fact_233_algebra__axioms_Ointro,axiom,
! [Omega: set_a,M: set_set_a] :
( ( member_set_a @ Omega @ M )
=> ( sigma_1540858801969833537ioms_a @ Omega @ M ) ) ).
% algebra_axioms.intro
thf(fact_234_sigma__eq__Dynkin,axiom,
! [M: set_set_real_a,Omega: set_real_a] :
( ( ord_le6426559262036278761real_a @ M @ ( pow_real_a @ Omega ) )
=> ( ( sigma_5335421319604048045real_a @ M )
=> ( ( sigma_136333207653772897real_a @ Omega @ M )
= ( sigma_Dynkin_real_a @ Omega @ M ) ) ) ) ).
% sigma_eq_Dynkin
thf(fact_235_sigma__eq__Dynkin,axiom,
! [M: set_set_set_a,Omega: set_set_a] :
( ( ord_le5722252365846178494_set_a @ M @ ( pow_set_a @ Omega ) )
=> ( ( sigma_5249310804214965890_set_a @ M )
=> ( ( sigma_2987359967864564790_set_a @ Omega @ M )
= ( sigma_Dynkin_set_a @ Omega @ M ) ) ) ) ).
% sigma_eq_Dynkin
thf(fact_236_sigma__eq__Dynkin,axiom,
! [M: set_set_a_a,Omega: set_a_a] :
( ( ord_le1342339564197316145et_a_a @ M @ ( pow_a_a @ Omega ) )
=> ( ( sigma_Int_stable_a_a @ M )
=> ( ( sigma_sigma_sets_a_a @ Omega @ M )
= ( sigma_Dynkin_a_a @ Omega @ M ) ) ) ) ).
% sigma_eq_Dynkin
thf(fact_237_sigma__eq__Dynkin,axiom,
! [M: set_set_a,Omega: set_a] :
( ( ord_le3724670747650509150_set_a @ M @ ( pow_a @ Omega ) )
=> ( ( sigma_Int_stable_a @ M )
=> ( ( sigma_sigma_sets_a @ Omega @ M )
= ( sigma_Dynkin_a @ Omega @ M ) ) ) ) ).
% sigma_eq_Dynkin
thf(fact_238_Dynkin__system_ODynkin__subset,axiom,
! [Omega: set_a,M: set_set_a,N: set_set_a] :
( ( sigma_2757993544071651912stem_a @ Omega @ M )
=> ( ( ord_le3724670747650509150_set_a @ N @ M )
=> ( ord_le3724670747650509150_set_a @ ( sigma_Dynkin_a @ Omega @ N ) @ M ) ) ) ).
% Dynkin_system.Dynkin_subset
thf(fact_239_sigma__sets__into__sp,axiom,
! [A3: set_set_set_a,Sp: set_set_a,X2: set_set_a] :
( ( ord_le5722252365846178494_set_a @ A3 @ ( pow_set_a @ Sp ) )
=> ( ( member_set_set_a @ X2 @ ( sigma_2987359967864564790_set_a @ Sp @ A3 ) )
=> ( ord_le3724670747650509150_set_a @ X2 @ Sp ) ) ) ).
% sigma_sets_into_sp
thf(fact_240_sigma__sets__into__sp,axiom,
! [A3: set_set_real_a,Sp: set_real_a,X2: set_real_a] :
( ( ord_le6426559262036278761real_a @ A3 @ ( pow_real_a @ Sp ) )
=> ( ( member_set_real_a @ X2 @ ( sigma_136333207653772897real_a @ Sp @ A3 ) )
=> ( ord_le5743406823621094409real_a @ X2 @ Sp ) ) ) ).
% sigma_sets_into_sp
thf(fact_241_sigma__sets__into__sp,axiom,
! [A3: set_set_a_a,Sp: set_a_a,X2: set_a_a] :
( ( ord_le1342339564197316145et_a_a @ A3 @ ( pow_a_a @ Sp ) )
=> ( ( member_set_a_a2 @ X2 @ ( sigma_sigma_sets_a_a @ Sp @ A3 ) )
=> ( ord_less_eq_set_a_a @ X2 @ Sp ) ) ) ).
% sigma_sets_into_sp
thf(fact_242_sigma__sets__into__sp,axiom,
! [A3: set_set_a,Sp: set_a,X2: set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ ( pow_a @ Sp ) )
=> ( ( member_set_a @ X2 @ ( sigma_sigma_sets_a @ Sp @ A3 ) )
=> ( ord_less_eq_set_a @ X2 @ Sp ) ) ) ).
% sigma_sets_into_sp
thf(fact_243_sigma__sets__superset__generator,axiom,
! [A3: set_set_a,X: set_a] : ( ord_le3724670747650509150_set_a @ A3 @ ( sigma_sigma_sets_a @ X @ A3 ) ) ).
% sigma_sets_superset_generator
thf(fact_244_sigma__sets__subseteq,axiom,
! [A3: set_set_a,B: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sigma_sets_a @ X @ A3 ) @ ( sigma_sigma_sets_a @ X @ B ) ) ) ).
% sigma_sets_subseteq
thf(fact_245_sigma__sets__mono,axiom,
! [A3: set_set_a,X: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ ( sigma_sigma_sets_a @ X @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sigma_sets_a @ X @ A3 ) @ ( sigma_sigma_sets_a @ X @ B ) ) ) ).
% sigma_sets_mono
thf(fact_246_Dynkin__system_ODynkin__lemma,axiom,
! [Omega: set_a,M: set_set_a,E: set_set_a] :
( ( sigma_2757993544071651912stem_a @ Omega @ M )
=> ( ( sigma_Int_stable_a @ E )
=> ( ( ord_le3724670747650509150_set_a @ E @ M )
=> ( ( ord_le3724670747650509150_set_a @ M @ ( sigma_sigma_sets_a @ Omega @ E ) )
=> ( ( sigma_sigma_sets_a @ Omega @ E )
= M ) ) ) ) ) ).
% Dynkin_system.Dynkin_lemma
thf(fact_247_qbs__morphism__comp,axiom,
! [F: real > real,X: quasi_borel_real,Y: quasi_borel_real,G: real > a,Z2: quasi_borel_a] :
( ( member_real_real @ F @ ( qbs_mo5229770564518008146l_real @ X @ Y ) )
=> ( ( member_real_a @ G @ ( qbs_morphism_real_a @ Y @ Z2 ) )
=> ( member_real_a @ ( comp_real_a_real @ G @ F ) @ ( qbs_morphism_real_a @ X @ Z2 ) ) ) ) ).
% qbs_morphism_comp
thf(fact_248_qbs__morphism__comp,axiom,
! [F: a > real,X: quasi_borel_a,Y: quasi_borel_real,G: real > a,Z2: quasi_borel_a] :
( ( member_a_real @ F @ ( qbs_morphism_a_real @ X @ Y ) )
=> ( ( member_real_a @ G @ ( qbs_morphism_real_a @ Y @ Z2 ) )
=> ( member_a_a @ ( comp_real_a_a @ G @ F ) @ ( qbs_morphism_a_a @ X @ Z2 ) ) ) ) ).
% qbs_morphism_comp
thf(fact_249_qbs__morphism__comp,axiom,
! [F: real > a,X: quasi_borel_real,Y: quasi_borel_a,G: a > a,Z2: quasi_borel_a] :
( ( member_real_a @ F @ ( qbs_morphism_real_a @ X @ Y ) )
=> ( ( member_a_a @ G @ ( qbs_morphism_a_a @ Y @ Z2 ) )
=> ( member_real_a @ ( comp_a_a_real @ G @ F ) @ ( qbs_morphism_real_a @ X @ Z2 ) ) ) ) ).
% qbs_morphism_comp
thf(fact_250_qbs__morphism__comp,axiom,
! [F: a > a,X: quasi_borel_a,Y: quasi_borel_a,G: a > a,Z2: quasi_borel_a] :
( ( member_a_a @ F @ ( qbs_morphism_a_a @ X @ Y ) )
=> ( ( member_a_a @ G @ ( qbs_morphism_a_a @ Y @ Z2 ) )
=> ( member_a_a @ ( comp_a_a_a @ G @ F ) @ ( qbs_morphism_a_a @ X @ Z2 ) ) ) ) ).
% qbs_morphism_comp
thf(fact_251_qbs__morphism__comp,axiom,
! [F: real > a,X: quasi_borel_real,Y: quasi_borel_a,G: a > real,Z2: quasi_borel_real] :
( ( member_real_a @ F @ ( qbs_morphism_real_a @ X @ Y ) )
=> ( ( member_a_real @ G @ ( qbs_morphism_a_real @ Y @ Z2 ) )
=> ( member_real_real @ ( comp_a_real_real @ G @ F ) @ ( qbs_mo5229770564518008146l_real @ X @ Z2 ) ) ) ) ).
% qbs_morphism_comp
thf(fact_252_qbs__morphism__comp,axiom,
! [F: a > a,X: quasi_borel_a,Y: quasi_borel_a,G: a > real,Z2: quasi_borel_real] :
( ( member_a_a @ F @ ( qbs_morphism_a_a @ X @ Y ) )
=> ( ( member_a_real @ G @ ( qbs_morphism_a_real @ Y @ Z2 ) )
=> ( member_a_real @ ( comp_a_real_a @ G @ F ) @ ( qbs_morphism_a_real @ X @ Z2 ) ) ) ) ).
% qbs_morphism_comp
thf(fact_253_qbs__morphism__comp,axiom,
! [F: a > real,X: quasi_borel_a,Y: quasi_borel_real,G: real > real,Z2: quasi_borel_real] :
( ( member_a_real @ F @ ( qbs_morphism_a_real @ X @ Y ) )
=> ( ( member_real_real @ G @ ( qbs_mo5229770564518008146l_real @ Y @ Z2 ) )
=> ( member_a_real @ ( comp_real_real_a @ G @ F ) @ ( qbs_morphism_a_real @ X @ Z2 ) ) ) ) ).
% qbs_morphism_comp
thf(fact_254_qbs__morphism__comp,axiom,
! [F: real > a,X: quasi_borel_real,Y: quasi_borel_a,G: a > set_a,Z2: quasi_borel_set_a] :
( ( member_real_a @ F @ ( qbs_morphism_real_a @ X @ Y ) )
=> ( ( member_a_set_a @ G @ ( qbs_morphism_a_set_a @ Y @ Z2 ) )
=> ( member_real_set_a @ ( comp_a_set_a_real @ G @ F ) @ ( qbs_mo8580062899995455544_set_a @ X @ Z2 ) ) ) ) ).
% qbs_morphism_comp
thf(fact_255_qbs__morphism__comp,axiom,
! [F: a > a,X: quasi_borel_a,Y: quasi_borel_a,G: a > set_a,Z2: quasi_borel_set_a] :
( ( member_a_a @ F @ ( qbs_morphism_a_a @ X @ Y ) )
=> ( ( member_a_set_a @ G @ ( qbs_morphism_a_set_a @ Y @ Z2 ) )
=> ( member_a_set_a @ ( comp_a_set_a_a @ G @ F ) @ ( qbs_morphism_a_set_a @ X @ Z2 ) ) ) ) ).
% qbs_morphism_comp
thf(fact_256_qbs__morphism__comp,axiom,
! [F: a > real,X: quasi_borel_a,Y: quasi_borel_real,G: real > set_a,Z2: quasi_borel_set_a] :
( ( member_a_real @ F @ ( qbs_morphism_a_real @ X @ Y ) )
=> ( ( member_real_set_a @ G @ ( qbs_mo8580062899995455544_set_a @ Y @ Z2 ) )
=> ( member_a_set_a @ ( comp_real_set_a_a @ G @ F ) @ ( qbs_morphism_a_set_a @ X @ Z2 ) ) ) ) ).
% qbs_morphism_comp
thf(fact_257_Dynkin__system__Dynkin,axiom,
! [M: set_set_real_a,Omega: set_real_a] :
( ( ord_le6426559262036278761real_a @ M @ ( pow_real_a @ Omega ) )
=> ( sigma_4207074038752281043real_a @ Omega @ ( sigma_Dynkin_real_a @ Omega @ M ) ) ) ).
% Dynkin_system_Dynkin
thf(fact_258_Dynkin__system__Dynkin,axiom,
! [M: set_set_set_a,Omega: set_set_a] :
( ( ord_le5722252365846178494_set_a @ M @ ( pow_set_a @ Omega ) )
=> ( sigma_6194451600871760808_set_a @ Omega @ ( sigma_Dynkin_set_a @ Omega @ M ) ) ) ).
% Dynkin_system_Dynkin
thf(fact_259_Dynkin__system__Dynkin,axiom,
! [M: set_set_a_a,Omega: set_a_a] :
( ( ord_le1342339564197316145et_a_a @ M @ ( pow_a_a @ Omega ) )
=> ( sigma_4503493975268142235em_a_a @ Omega @ ( sigma_Dynkin_a_a @ Omega @ M ) ) ) ).
% Dynkin_system_Dynkin
thf(fact_260_Dynkin__system__Dynkin,axiom,
! [M: set_set_a,Omega: set_a] :
( ( ord_le3724670747650509150_set_a @ M @ ( pow_a @ Omega ) )
=> ( sigma_2757993544071651912stem_a @ Omega @ ( sigma_Dynkin_a @ Omega @ M ) ) ) ).
% Dynkin_system_Dynkin
thf(fact_261_algebra_OInt__stable,axiom,
! [Omega: set_a,M: set_set_a] :
( ( sigma_algebra_a @ Omega @ M )
=> ( sigma_Int_stable_a @ M ) ) ).
% algebra.Int_stable
thf(fact_262_sigma__sets__sigma__sets__eq,axiom,
! [M: set_set_real_a,S: set_real_a] :
( ( ord_le6426559262036278761real_a @ M @ ( pow_real_a @ S ) )
=> ( ( sigma_136333207653772897real_a @ S @ ( sigma_136333207653772897real_a @ S @ M ) )
= ( sigma_136333207653772897real_a @ S @ M ) ) ) ).
% sigma_sets_sigma_sets_eq
thf(fact_263_sigma__sets__sigma__sets__eq,axiom,
! [M: set_set_set_a,S: set_set_a] :
( ( ord_le5722252365846178494_set_a @ M @ ( pow_set_a @ S ) )
=> ( ( sigma_2987359967864564790_set_a @ S @ ( sigma_2987359967864564790_set_a @ S @ M ) )
= ( sigma_2987359967864564790_set_a @ S @ M ) ) ) ).
% sigma_sets_sigma_sets_eq
thf(fact_264_sigma__sets__sigma__sets__eq,axiom,
! [M: set_set_a_a,S: set_a_a] :
( ( ord_le1342339564197316145et_a_a @ M @ ( pow_a_a @ S ) )
=> ( ( sigma_sigma_sets_a_a @ S @ ( sigma_sigma_sets_a_a @ S @ M ) )
= ( sigma_sigma_sets_a_a @ S @ M ) ) ) ).
% sigma_sets_sigma_sets_eq
thf(fact_265_sigma__sets__sigma__sets__eq,axiom,
! [M: set_set_a,S: set_a] :
( ( ord_le3724670747650509150_set_a @ M @ ( pow_a @ S ) )
=> ( ( sigma_sigma_sets_a @ S @ ( sigma_sigma_sets_a @ S @ M ) )
= ( sigma_sigma_sets_a @ S @ M ) ) ) ).
% sigma_sets_sigma_sets_eq
thf(fact_266_sigma__sets__mono_H_H,axiom,
! [A3: set_real_a,C: set_real_a,D: set_set_real_a,B: set_set_real_a] :
( ( member_set_real_a @ A3 @ ( sigma_136333207653772897real_a @ C @ D ) )
=> ( ( ord_le6426559262036278761real_a @ B @ D )
=> ( ( ord_le6426559262036278761real_a @ D @ ( pow_real_a @ C ) )
=> ( ord_le6426559262036278761real_a @ ( sigma_136333207653772897real_a @ A3 @ B ) @ ( sigma_136333207653772897real_a @ C @ D ) ) ) ) ) ).
% sigma_sets_mono''
thf(fact_267_sigma__sets__mono_H_H,axiom,
! [A3: set_set_a,C: set_set_a,D: set_set_set_a,B: set_set_set_a] :
( ( member_set_set_a @ A3 @ ( sigma_2987359967864564790_set_a @ C @ D ) )
=> ( ( ord_le5722252365846178494_set_a @ B @ D )
=> ( ( ord_le5722252365846178494_set_a @ D @ ( pow_set_a @ C ) )
=> ( ord_le5722252365846178494_set_a @ ( sigma_2987359967864564790_set_a @ A3 @ B ) @ ( sigma_2987359967864564790_set_a @ C @ D ) ) ) ) ) ).
% sigma_sets_mono''
thf(fact_268_sigma__sets__mono_H_H,axiom,
! [A3: set_a_a,C: set_a_a,D: set_set_a_a,B: set_set_a_a] :
( ( member_set_a_a2 @ A3 @ ( sigma_sigma_sets_a_a @ C @ D ) )
=> ( ( ord_le1342339564197316145et_a_a @ B @ D )
=> ( ( ord_le1342339564197316145et_a_a @ D @ ( pow_a_a @ C ) )
=> ( ord_le1342339564197316145et_a_a @ ( sigma_sigma_sets_a_a @ A3 @ B ) @ ( sigma_sigma_sets_a_a @ C @ D ) ) ) ) ) ).
% sigma_sets_mono''
thf(fact_269_sigma__sets__mono_H_H,axiom,
! [A3: set_a,C: set_a,D: set_set_a,B: set_set_a] :
( ( member_set_a @ A3 @ ( sigma_sigma_sets_a @ C @ D ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ D )
=> ( ( ord_le3724670747650509150_set_a @ D @ ( pow_a @ C ) )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sigma_sets_a @ A3 @ B ) @ ( sigma_sigma_sets_a @ C @ D ) ) ) ) ) ).
% sigma_sets_mono''
thf(fact_270_finite__Pow__iff,axiom,
! [A3: set_set_a] :
( ( finite7209287970140883943_set_a @ ( pow_set_a @ A3 ) )
= ( finite_finite_set_a @ A3 ) ) ).
% finite_Pow_iff
thf(fact_271_finite__Pow__iff,axiom,
! [A3: set_real_a] :
( ( finite1252724379825953938real_a @ ( pow_real_a @ A3 ) )
= ( finite_finite_real_a @ A3 ) ) ).
% finite_Pow_iff
thf(fact_272_finite__Pow__iff,axiom,
! [A3: set_a_a] :
( ( finite4645002659469139250et_a_a @ ( pow_a_a @ A3 ) )
= ( finite_finite_a_a @ A3 ) ) ).
% finite_Pow_iff
thf(fact_273_finite__Pow__iff,axiom,
! [A3: set_Ex3793607809372303086nnreal] :
( ( finite3583719589609615493nnreal @ ( pow_Ex5372160365422184283nnreal @ A3 ) )
= ( finite3782138982310603983nnreal @ A3 ) ) ).
% finite_Pow_iff
thf(fact_274_finite__Pow__iff,axiom,
! [A3: set_a] :
( ( finite_finite_set_a @ ( pow_a @ A3 ) )
= ( finite_finite_a @ A3 ) ) ).
% finite_Pow_iff
thf(fact_275_sigma__algebra_Osigma__sets__subset_H,axiom,
! [Omega: set_a,M: set_set_a,A2: set_set_a,Omega3: set_a] :
( ( sigma_4968961713055010667ebra_a @ Omega @ M )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ M )
=> ( ( member_set_a @ Omega3 @ M )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sigma_sets_a @ Omega3 @ A2 ) @ M ) ) ) ) ).
% sigma_algebra.sigma_sets_subset'
thf(fact_276_comp__apply,axiom,
( comp_a_a_real
= ( ^ [F2: a > a,G2: real > a,X4: real] : ( F2 @ ( G2 @ X4 ) ) ) ) ).
% comp_apply
thf(fact_277_comp__apply,axiom,
( comp_real_a_real
= ( ^ [F2: real > a,G2: real > real,X4: real] : ( F2 @ ( G2 @ X4 ) ) ) ) ).
% comp_apply
thf(fact_278_comp__apply,axiom,
( comp_a_a_a
= ( ^ [F2: a > a,G2: a > a,X4: a] : ( F2 @ ( G2 @ X4 ) ) ) ) ).
% comp_apply
thf(fact_279_comp__apply,axiom,
( comp_real_a_a
= ( ^ [F2: real > a,G2: a > real,X4: a] : ( F2 @ ( G2 @ X4 ) ) ) ) ).
% comp_apply
thf(fact_280_dual__order_Orefl,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_281_dual__order_Orefl,axiom,
! [A2: set_real_a] : ( ord_le5743406823621094409real_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_282_dual__order_Orefl,axiom,
! [A2: set_a_a] : ( ord_less_eq_set_a_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_283_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_284_dual__order_Orefl,axiom,
! [A2: set_a > extend8495563244428889912nnreal] : ( ord_le6700572704167691815nnreal @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_285_dual__order_Orefl,axiom,
! [A2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_286_order__refl,axiom,
! [X2: set_set_a] : ( ord_le3724670747650509150_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_287_order__refl,axiom,
! [X2: set_real_a] : ( ord_le5743406823621094409real_a @ X2 @ X2 ) ).
% order_refl
thf(fact_288_order__refl,axiom,
! [X2: set_a_a] : ( ord_less_eq_set_a_a @ X2 @ X2 ) ).
% order_refl
thf(fact_289_order__refl,axiom,
! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_290_order__refl,axiom,
! [X2: set_a > extend8495563244428889912nnreal] : ( ord_le6700572704167691815nnreal @ X2 @ X2 ) ).
% order_refl
thf(fact_291_order__refl,axiom,
! [X2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ X2 @ X2 ) ).
% order_refl
thf(fact_292_rev__finite__subset,axiom,
! [B: set_Ex3793607809372303086nnreal,A3: set_Ex3793607809372303086nnreal] :
( ( finite3782138982310603983nnreal @ B )
=> ( ( ord_le6787938422905777998nnreal @ A3 @ B )
=> ( finite3782138982310603983nnreal @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_293_rev__finite__subset,axiom,
! [B: set_set_a,A3: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( finite_finite_set_a @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_294_rev__finite__subset,axiom,
! [B: set_real_a,A3: set_real_a] :
( ( finite_finite_real_a @ B )
=> ( ( ord_le5743406823621094409real_a @ A3 @ B )
=> ( finite_finite_real_a @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_295_rev__finite__subset,axiom,
! [B: set_a_a,A3: set_a_a] :
( ( finite_finite_a_a @ B )
=> ( ( ord_less_eq_set_a_a @ A3 @ B )
=> ( finite_finite_a_a @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_296_rev__finite__subset,axiom,
! [B: set_a,A3: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A3 @ B )
=> ( finite_finite_a @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_297_infinite__super,axiom,
! [S: set_Ex3793607809372303086nnreal,T2: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ S @ T2 )
=> ( ~ ( finite3782138982310603983nnreal @ S )
=> ~ ( finite3782138982310603983nnreal @ T2 ) ) ) ).
% infinite_super
thf(fact_298_infinite__super,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ S @ T2 )
=> ( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ T2 ) ) ) ).
% infinite_super
thf(fact_299_infinite__super,axiom,
! [S: set_real_a,T2: set_real_a] :
( ( ord_le5743406823621094409real_a @ S @ T2 )
=> ( ~ ( finite_finite_real_a @ S )
=> ~ ( finite_finite_real_a @ T2 ) ) ) ).
% infinite_super
thf(fact_300_infinite__super,axiom,
! [S: set_a_a,T2: set_a_a] :
( ( ord_less_eq_set_a_a @ S @ T2 )
=> ( ~ ( finite_finite_a_a @ S )
=> ~ ( finite_finite_a_a @ T2 ) ) ) ).
% infinite_super
thf(fact_301_infinite__super,axiom,
! [S: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_302_finite__subset,axiom,
! [A3: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ A3 @ B )
=> ( ( finite3782138982310603983nnreal @ B )
=> ( finite3782138982310603983nnreal @ A3 ) ) ) ).
% finite_subset
thf(fact_303_finite__subset,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ( finite_finite_set_a @ B )
=> ( finite_finite_set_a @ A3 ) ) ) ).
% finite_subset
thf(fact_304_finite__subset,axiom,
! [A3: set_real_a,B: set_real_a] :
( ( ord_le5743406823621094409real_a @ A3 @ B )
=> ( ( finite_finite_real_a @ B )
=> ( finite_finite_real_a @ A3 ) ) ) ).
% finite_subset
thf(fact_305_finite__subset,axiom,
! [A3: set_a_a,B: set_a_a] :
( ( ord_less_eq_set_a_a @ A3 @ B )
=> ( ( finite_finite_a_a @ B )
=> ( finite_finite_a_a @ A3 ) ) ) ).
% finite_subset
thf(fact_306_finite__subset,axiom,
! [A3: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A3 ) ) ) ).
% finite_subset
thf(fact_307_finite__indexed__bound,axiom,
! [A3: set_a_a_a2,P: ( ( a > a ) > a ) > extend8495563244428889912nnreal > $o] :
( ( finite_finite_a_a_a @ A3 )
=> ( ! [X3: ( a > a ) > a] :
( ( member_a_a_a @ X3 @ A3 )
=> ? [X_1: extend8495563244428889912nnreal] : ( P @ X3 @ X_1 ) )
=> ? [M3: extend8495563244428889912nnreal] :
! [X5: ( a > a ) > a] :
( ( member_a_a_a @ X5 @ A3 )
=> ? [K: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ K @ M3 )
& ( P @ X5 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_308_finite__indexed__bound,axiom,
! [A3: set_a_real,P: ( a > real ) > extend8495563244428889912nnreal > $o] :
( ( finite_finite_a_real @ A3 )
=> ( ! [X3: a > real] :
( ( member_a_real @ X3 @ A3 )
=> ? [X_1: extend8495563244428889912nnreal] : ( P @ X3 @ X_1 ) )
=> ? [M3: extend8495563244428889912nnreal] :
! [X5: a > real] :
( ( member_a_real @ X5 @ A3 )
=> ? [K: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ K @ M3 )
& ( P @ X5 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_309_finite__indexed__bound,axiom,
! [A3: set_a_set_a,P: ( a > set_a ) > extend8495563244428889912nnreal > $o] :
( ( finite2844999134675159900_set_a @ A3 )
=> ( ! [X3: a > set_a] :
( ( member_a_set_a @ X3 @ A3 )
=> ? [X_1: extend8495563244428889912nnreal] : ( P @ X3 @ X_1 ) )
=> ? [M3: extend8495563244428889912nnreal] :
! [X5: a > set_a] :
( ( member_a_set_a @ X5 @ A3 )
=> ? [K: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ K @ M3 )
& ( P @ X5 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_310_finite__indexed__bound,axiom,
! [A3: set_a_real_a,P: ( a > real > a ) > extend8495563244428889912nnreal > $o] :
( ( finite5056320738175349255real_a @ A3 )
=> ( ! [X3: a > real > a] :
( ( member_a_real_a @ X3 @ A3 )
=> ? [X_1: extend8495563244428889912nnreal] : ( P @ X3 @ X_1 ) )
=> ? [M3: extend8495563244428889912nnreal] :
! [X5: a > real > a] :
( ( member_a_real_a @ X5 @ A3 )
=> ? [K: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ K @ M3 )
& ( P @ X5 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_311_finite__indexed__bound,axiom,
! [A3: set_a_a_a,P: ( a > a > a ) > extend8495563244428889912nnreal > $o] :
( ( finite_finite_a_a_a2 @ A3 )
=> ( ! [X3: a > a > a] :
( ( member_a_a_a2 @ X3 @ A3 )
=> ? [X_1: extend8495563244428889912nnreal] : ( P @ X3 @ X_1 ) )
=> ? [M3: extend8495563244428889912nnreal] :
! [X5: a > a > a] :
( ( member_a_a_a2 @ X5 @ A3 )
=> ? [K: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ K @ M3 )
& ( P @ X5 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_312_finite__indexed__bound,axiom,
! [A3: set_Ex3793607809372303086nnreal,P: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o] :
( ( finite3782138982310603983nnreal @ A3 )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A3 )
=> ? [X_1: extend8495563244428889912nnreal] : ( P @ X3 @ X_1 ) )
=> ? [M3: extend8495563244428889912nnreal] :
! [X5: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X5 @ A3 )
=> ? [K: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ K @ M3 )
& ( P @ X5 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_313_finite__indexed__bound,axiom,
! [A3: set_a_a,P: ( a > a ) > extend8495563244428889912nnreal > $o] :
( ( finite_finite_a_a @ A3 )
=> ( ! [X3: a > a] :
( ( member_a_a @ X3 @ A3 )
=> ? [X_1: extend8495563244428889912nnreal] : ( P @ X3 @ X_1 ) )
=> ? [M3: extend8495563244428889912nnreal] :
! [X5: a > a] :
( ( member_a_a @ X5 @ A3 )
=> ? [K: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ K @ M3 )
& ( P @ X5 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_314_finite__indexed__bound,axiom,
! [A3: set_a,P: a > extend8495563244428889912nnreal > $o] :
( ( finite_finite_a @ A3 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ? [X_1: extend8495563244428889912nnreal] : ( P @ X3 @ X_1 ) )
=> ? [M3: extend8495563244428889912nnreal] :
! [X5: a] :
( ( member_a @ X5 @ A3 )
=> ? [K: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ K @ M3 )
& ( P @ X5 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_315_finite__indexed__bound,axiom,
! [A3: set_set_a,P: set_a > extend8495563244428889912nnreal > $o] :
( ( finite_finite_set_a @ A3 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
=> ? [X_1: extend8495563244428889912nnreal] : ( P @ X3 @ X_1 ) )
=> ? [M3: extend8495563244428889912nnreal] :
! [X5: set_a] :
( ( member_set_a @ X5 @ A3 )
=> ? [K: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ K @ M3 )
& ( P @ X5 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_316_finite__indexed__bound,axiom,
! [A3: set_real_a,P: ( real > a ) > extend8495563244428889912nnreal > $o] :
( ( finite_finite_real_a @ A3 )
=> ( ! [X3: real > a] :
( ( member_real_a @ X3 @ A3 )
=> ? [X_1: extend8495563244428889912nnreal] : ( P @ X3 @ X_1 ) )
=> ? [M3: extend8495563244428889912nnreal] :
! [X5: real > a] :
( ( member_real_a @ X5 @ A3 )
=> ? [K: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ K @ M3 )
& ( P @ X5 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_317_finite__has__minimal2,axiom,
! [A3: set_a_real,A2: a > real] :
( ( finite_finite_a_real @ A3 )
=> ( ( member_a_real @ A2 @ A3 )
=> ? [X3: a > real] :
( ( member_a_real @ X3 @ A3 )
& ( ord_less_eq_a_real @ X3 @ A2 )
& ! [Xa: a > real] :
( ( member_a_real @ Xa @ A3 )
=> ( ( ord_less_eq_a_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_318_finite__has__minimal2,axiom,
! [A3: set_a_set_a,A2: a > set_a] :
( ( finite2844999134675159900_set_a @ A3 )
=> ( ( member_a_set_a @ A2 @ A3 )
=> ? [X3: a > set_a] :
( ( member_a_set_a @ X3 @ A3 )
& ( ord_less_eq_a_set_a @ X3 @ A2 )
& ! [Xa: a > set_a] :
( ( member_a_set_a @ Xa @ A3 )
=> ( ( ord_less_eq_a_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_319_finite__has__minimal2,axiom,
! [A3: set_set_set_a,A2: set_set_a] :
( ( finite7209287970140883943_set_a @ A3 )
=> ( ( member_set_set_a @ A2 @ A3 )
=> ? [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A3 )
& ( ord_le3724670747650509150_set_a @ X3 @ A2 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A3 )
=> ( ( ord_le3724670747650509150_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_320_finite__has__minimal2,axiom,
! [A3: set_set_real_a,A2: set_real_a] :
( ( finite1252724379825953938real_a @ A3 )
=> ( ( member_set_real_a @ A2 @ A3 )
=> ? [X3: set_real_a] :
( ( member_set_real_a @ X3 @ A3 )
& ( ord_le5743406823621094409real_a @ X3 @ A2 )
& ! [Xa: set_real_a] :
( ( member_set_real_a @ Xa @ A3 )
=> ( ( ord_le5743406823621094409real_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_321_finite__has__minimal2,axiom,
! [A3: set_set_a_a,A2: set_a_a] :
( ( finite4645002659469139250et_a_a @ A3 )
=> ( ( member_set_a_a2 @ A2 @ A3 )
=> ? [X3: set_a_a] :
( ( member_set_a_a2 @ X3 @ A3 )
& ( ord_less_eq_set_a_a @ X3 @ A2 )
& ! [Xa: set_a_a] :
( ( member_set_a_a2 @ Xa @ A3 )
=> ( ( ord_less_eq_set_a_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_322_finite__has__minimal2,axiom,
! [A3: set_se9209621484078883815nnreal,A2: set_a > extend8495563244428889912nnreal] :
( ( finite7792283316523934512nnreal @ A3 )
=> ( ( member4180043592386426928nnreal @ A2 @ A3 )
=> ? [X3: set_a > extend8495563244428889912nnreal] :
( ( member4180043592386426928nnreal @ X3 @ A3 )
& ( ord_le6700572704167691815nnreal @ X3 @ A2 )
& ! [Xa: set_a > extend8495563244428889912nnreal] :
( ( member4180043592386426928nnreal @ Xa @ A3 )
=> ( ( ord_le6700572704167691815nnreal @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_323_finite__has__minimal2,axiom,
! [A3: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( member_set_a @ A2 @ A3 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
& ( ord_less_eq_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A3 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_324_finite__has__minimal2,axiom,
! [A3: set_Ex3793607809372303086nnreal,A2: extend8495563244428889912nnreal] :
( ( finite3782138982310603983nnreal @ A3 )
=> ( ( member7908768830364227535nnreal @ A2 @ A3 )
=> ? [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A3 )
& ( ord_le3935885782089961368nnreal @ X3 @ A2 )
& ! [Xa: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ Xa @ A3 )
=> ( ( ord_le3935885782089961368nnreal @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_325_less__eq__quasi__borel_Ointros_I2_J,axiom,
! [X: quasi_borel_real_a,Y: quasi_borel_real_a] :
( ( ( qbs_space_real_a @ X )
= ( qbs_space_real_a @ Y ) )
=> ( ( ord_le2748211696382713492real_a @ ( qbs_Mx_real_a @ Y ) @ ( qbs_Mx_real_a @ X ) )
=> ( ord_le7221076938822796111real_a @ X @ Y ) ) ) ).
% less_eq_quasi_borel.intros(2)
thf(fact_326_less__eq__quasi__borel_Ointros_I2_J,axiom,
! [X: quasi_borel_set_a,Y: quasi_borel_set_a] :
( ( ( qbs_space_set_a @ X )
= ( qbs_space_set_a @ Y ) )
=> ( ( ord_le3830582842290364777_set_a @ ( qbs_Mx_set_a @ Y ) @ ( qbs_Mx_set_a @ X ) )
=> ( ord_le6167762037667146916_set_a @ X @ Y ) ) ) ).
% less_eq_quasi_borel.intros(2)
thf(fact_327_less__eq__quasi__borel_Ointros_I2_J,axiom,
! [X: quasi_borel_a_a,Y: quasi_borel_a_a] :
( ( ( qbs_space_a_a @ X )
= ( qbs_space_a_a @ Y ) )
=> ( ( ord_le7862321826480370886al_a_a @ ( qbs_Mx_a_a @ Y ) @ ( qbs_Mx_a_a @ X ) )
=> ( ord_le4071610493548526901el_a_a @ X @ Y ) ) ) ).
% less_eq_quasi_borel.intros(2)
thf(fact_328_less__eq__quasi__borel_Ointros_I2_J,axiom,
! [X: quasi_borel_real,Y: quasi_borel_real] :
( ( ( qbs_space_real @ X )
= ( qbs_space_real @ Y ) )
=> ( ( ord_le4198349162570665613l_real @ ( qbs_Mx_real @ Y ) @ ( qbs_Mx_real @ X ) )
=> ( ord_le1181865976659726716l_real @ X @ Y ) ) ) ).
% less_eq_quasi_borel.intros(2)
thf(fact_329_less__eq__quasi__borel_Ointros_I2_J,axiom,
! [X: quasi_borel_a,Y: quasi_borel_a] :
( ( ( qbs_space_a @ X )
= ( qbs_space_a @ Y ) )
=> ( ( ord_le5743406823621094409real_a @ ( qbs_Mx_a @ Y ) @ ( qbs_Mx_a @ X ) )
=> ( ord_le1843388692487544644orel_a @ X @ Y ) ) ) ).
% less_eq_quasi_borel.intros(2)
thf(fact_330_le__measureD1,axiom,
! [A3: sigma_measure_real,B: sigma_measure_real] :
( ( ord_le487379304121309861e_real @ A3 @ B )
=> ( ord_less_eq_set_real @ ( sigma_space_real @ A3 ) @ ( sigma_space_real @ B ) ) ) ).
% le_measureD1
thf(fact_331_le__measureD1,axiom,
! [A3: sigma_measure_set_a,B: sigma_measure_set_a] :
( ( ord_le5642585610961328955_set_a @ A3 @ B )
=> ( ord_le3724670747650509150_set_a @ ( sigma_space_set_a @ A3 ) @ ( sigma_space_set_a @ B ) ) ) ).
% le_measureD1
thf(fact_332_le__measureD1,axiom,
! [A3: sigma_measure_real_a,B: sigma_measure_real_a] :
( ( ord_le2499155701527069670real_a @ A3 @ B )
=> ( ord_le5743406823621094409real_a @ ( sigma_space_real_a @ A3 ) @ ( sigma_space_real_a @ B ) ) ) ).
% le_measureD1
thf(fact_333_le__measureD1,axiom,
! [A3: sigma_measure_a_a,B: sigma_measure_a_a] :
( ( ord_le296570614984210654re_a_a @ A3 @ B )
=> ( ord_less_eq_set_a_a @ ( sigma_space_a_a @ A3 ) @ ( sigma_space_a_a @ B ) ) ) ).
% le_measureD1
thf(fact_334_le__measureD1,axiom,
! [A3: sigma_measure_a,B: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A3 @ B )
=> ( ord_less_eq_set_a @ ( sigma_space_a @ A3 ) @ ( sigma_space_a @ B ) ) ) ).
% le_measureD1
thf(fact_335_nle__le,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ~ ( ord_le3935885782089961368nnreal @ A2 @ B3 ) )
= ( ( ord_le3935885782089961368nnreal @ B3 @ A2 )
& ( B3 != A2 ) ) ) ).
% nle_le
thf(fact_336_le__cases3,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] :
( ( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
=> ~ ( ord_le3935885782089961368nnreal @ Y3 @ Z3 ) )
=> ( ( ( ord_le3935885782089961368nnreal @ Y3 @ X2 )
=> ~ ( ord_le3935885782089961368nnreal @ X2 @ Z3 ) )
=> ( ( ( ord_le3935885782089961368nnreal @ X2 @ Z3 )
=> ~ ( ord_le3935885782089961368nnreal @ Z3 @ Y3 ) )
=> ( ( ( ord_le3935885782089961368nnreal @ Z3 @ Y3 )
=> ~ ( ord_le3935885782089961368nnreal @ Y3 @ X2 ) )
=> ( ( ( ord_le3935885782089961368nnreal @ Y3 @ Z3 )
=> ~ ( ord_le3935885782089961368nnreal @ Z3 @ X2 ) )
=> ~ ( ( ord_le3935885782089961368nnreal @ Z3 @ X2 )
=> ~ ( ord_le3935885782089961368nnreal @ X2 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_337_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_set_a,Z: set_set_a] : ( Y2 = Z ) )
= ( ^ [X4: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y4 )
& ( ord_le3724670747650509150_set_a @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_338_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_real_a,Z: set_real_a] : ( Y2 = Z ) )
= ( ^ [X4: set_real_a,Y4: set_real_a] :
( ( ord_le5743406823621094409real_a @ X4 @ Y4 )
& ( ord_le5743406823621094409real_a @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_339_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a_a,Z: set_a_a] : ( Y2 = Z ) )
= ( ^ [X4: set_a_a,Y4: set_a_a] :
( ( ord_less_eq_set_a_a @ X4 @ Y4 )
& ( ord_less_eq_set_a_a @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_340_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_341_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a > extend8495563244428889912nnreal,Z: set_a > extend8495563244428889912nnreal] : ( Y2 = Z ) )
= ( ^ [X4: set_a > extend8495563244428889912nnreal,Y4: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ X4 @ Y4 )
& ( ord_le6700572704167691815nnreal @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_342_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] : ( Y2 = Z ) )
= ( ^ [X4: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X4 @ Y4 )
& ( ord_le3935885782089961368nnreal @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_343_ord__eq__le__trans,axiom,
! [A2: set_set_a,B3: set_set_a,C2: set_set_a] :
( ( A2 = B3 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ C2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_344_ord__eq__le__trans,axiom,
! [A2: set_real_a,B3: set_real_a,C2: set_real_a] :
( ( A2 = B3 )
=> ( ( ord_le5743406823621094409real_a @ B3 @ C2 )
=> ( ord_le5743406823621094409real_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_345_ord__eq__le__trans,axiom,
! [A2: set_a_a,B3: set_a_a,C2: set_a_a] :
( ( A2 = B3 )
=> ( ( ord_less_eq_set_a_a @ B3 @ C2 )
=> ( ord_less_eq_set_a_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_346_ord__eq__le__trans,axiom,
! [A2: set_a,B3: set_a,C2: set_a] :
( ( A2 = B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_347_ord__eq__le__trans,axiom,
! [A2: set_a > extend8495563244428889912nnreal,B3: set_a > extend8495563244428889912nnreal,C2: set_a > extend8495563244428889912nnreal] :
( ( A2 = B3 )
=> ( ( ord_le6700572704167691815nnreal @ B3 @ C2 )
=> ( ord_le6700572704167691815nnreal @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_348_ord__eq__le__trans,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( A2 = B3 )
=> ( ( ord_le3935885782089961368nnreal @ B3 @ C2 )
=> ( ord_le3935885782089961368nnreal @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_349_ord__le__eq__trans,axiom,
! [A2: set_set_a,B3: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_350_ord__le__eq__trans,axiom,
! [A2: set_real_a,B3: set_real_a,C2: set_real_a] :
( ( ord_le5743406823621094409real_a @ A2 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_le5743406823621094409real_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_351_ord__le__eq__trans,axiom,
! [A2: set_a_a,B3: set_a_a,C2: set_a_a] :
( ( ord_less_eq_set_a_a @ A2 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq_set_a_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_352_ord__le__eq__trans,axiom,
! [A2: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_353_ord__le__eq__trans,axiom,
! [A2: set_a > extend8495563244428889912nnreal,B3: set_a > extend8495563244428889912nnreal,C2: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ A2 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_le6700572704167691815nnreal @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_354_ord__le__eq__trans,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_le3935885782089961368nnreal @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_355_order__antisym,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
=> ( ( ord_le3724670747650509150_set_a @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_356_order__antisym,axiom,
! [X2: set_real_a,Y3: set_real_a] :
( ( ord_le5743406823621094409real_a @ X2 @ Y3 )
=> ( ( ord_le5743406823621094409real_a @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_357_order__antisym,axiom,
! [X2: set_a_a,Y3: set_a_a] :
( ( ord_less_eq_set_a_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a_a @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_358_order__antisym,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_359_order__antisym,axiom,
! [X2: set_a > extend8495563244428889912nnreal,Y3: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ X2 @ Y3 )
=> ( ( ord_le6700572704167691815nnreal @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_360_order__antisym,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
=> ( ( ord_le3935885782089961368nnreal @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_361_order_Otrans,axiom,
! [A2: set_set_a,B3: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ C2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_362_order_Otrans,axiom,
! [A2: set_real_a,B3: set_real_a,C2: set_real_a] :
( ( ord_le5743406823621094409real_a @ A2 @ B3 )
=> ( ( ord_le5743406823621094409real_a @ B3 @ C2 )
=> ( ord_le5743406823621094409real_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_363_order_Otrans,axiom,
! [A2: set_a_a,B3: set_a_a,C2: set_a_a] :
( ( ord_less_eq_set_a_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a_a @ B3 @ C2 )
=> ( ord_less_eq_set_a_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_364_order_Otrans,axiom,
! [A2: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_365_order_Otrans,axiom,
! [A2: set_a > extend8495563244428889912nnreal,B3: set_a > extend8495563244428889912nnreal,C2: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ A2 @ B3 )
=> ( ( ord_le6700572704167691815nnreal @ B3 @ C2 )
=> ( ord_le6700572704167691815nnreal @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_366_order_Otrans,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B3 )
=> ( ( ord_le3935885782089961368nnreal @ B3 @ C2 )
=> ( ord_le3935885782089961368nnreal @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_367_order__trans,axiom,
! [X2: set_set_a,Y3: set_set_a,Z3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
=> ( ( ord_le3724670747650509150_set_a @ Y3 @ Z3 )
=> ( ord_le3724670747650509150_set_a @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_368_order__trans,axiom,
! [X2: set_real_a,Y3: set_real_a,Z3: set_real_a] :
( ( ord_le5743406823621094409real_a @ X2 @ Y3 )
=> ( ( ord_le5743406823621094409real_a @ Y3 @ Z3 )
=> ( ord_le5743406823621094409real_a @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_369_order__trans,axiom,
! [X2: set_a_a,Y3: set_a_a,Z3: set_a_a] :
( ( ord_less_eq_set_a_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a_a @ Y3 @ Z3 )
=> ( ord_less_eq_set_a_a @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_370_order__trans,axiom,
! [X2: set_a,Y3: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ Z3 )
=> ( ord_less_eq_set_a @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_371_order__trans,axiom,
! [X2: set_a > extend8495563244428889912nnreal,Y3: set_a > extend8495563244428889912nnreal,Z3: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ X2 @ Y3 )
=> ( ( ord_le6700572704167691815nnreal @ Y3 @ Z3 )
=> ( ord_le6700572704167691815nnreal @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_372_order__trans,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
=> ( ( ord_le3935885782089961368nnreal @ Y3 @ Z3 )
=> ( ord_le3935885782089961368nnreal @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_373_linorder__wlog,axiom,
! [P: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o,A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ! [A: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B2 )
=> ( P @ A @ B2 ) )
=> ( ! [A: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( P @ B2 @ A )
=> ( P @ A @ B2 ) )
=> ( P @ A2 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_374_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_set_a,Z: set_set_a] : ( Y2 = Z ) )
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ A5 )
& ( ord_le3724670747650509150_set_a @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_375_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_real_a,Z: set_real_a] : ( Y2 = Z ) )
= ( ^ [A5: set_real_a,B5: set_real_a] :
( ( ord_le5743406823621094409real_a @ B5 @ A5 )
& ( ord_le5743406823621094409real_a @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_376_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_a_a,Z: set_a_a] : ( Y2 = Z ) )
= ( ^ [A5: set_a_a,B5: set_a_a] :
( ( ord_less_eq_set_a_a @ B5 @ A5 )
& ( ord_less_eq_set_a_a @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_377_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A5 )
& ( ord_less_eq_set_a @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_378_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_a > extend8495563244428889912nnreal,Z: set_a > extend8495563244428889912nnreal] : ( Y2 = Z ) )
= ( ^ [A5: set_a > extend8495563244428889912nnreal,B5: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ B5 @ A5 )
& ( ord_le6700572704167691815nnreal @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_379_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] : ( Y2 = Z ) )
= ( ^ [A5: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B5 @ A5 )
& ( ord_le3935885782089961368nnreal @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_380_dual__order_Oantisym,axiom,
! [B3: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_381_dual__order_Oantisym,axiom,
! [B3: set_real_a,A2: set_real_a] :
( ( ord_le5743406823621094409real_a @ B3 @ A2 )
=> ( ( ord_le5743406823621094409real_a @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_382_dual__order_Oantisym,axiom,
! [B3: set_a_a,A2: set_a_a] :
( ( ord_less_eq_set_a_a @ B3 @ A2 )
=> ( ( ord_less_eq_set_a_a @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_383_dual__order_Oantisym,axiom,
! [B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_384_dual__order_Oantisym,axiom,
! [B3: set_a > extend8495563244428889912nnreal,A2: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ B3 @ A2 )
=> ( ( ord_le6700572704167691815nnreal @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_385_dual__order_Oantisym,axiom,
! [B3: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B3 @ A2 )
=> ( ( ord_le3935885782089961368nnreal @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_386_dual__order_Otrans,axiom,
! [B3: set_set_a,A2: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ C2 @ B3 )
=> ( ord_le3724670747650509150_set_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_387_dual__order_Otrans,axiom,
! [B3: set_real_a,A2: set_real_a,C2: set_real_a] :
( ( ord_le5743406823621094409real_a @ B3 @ A2 )
=> ( ( ord_le5743406823621094409real_a @ C2 @ B3 )
=> ( ord_le5743406823621094409real_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_388_dual__order_Otrans,axiom,
! [B3: set_a_a,A2: set_a_a,C2: set_a_a] :
( ( ord_less_eq_set_a_a @ B3 @ A2 )
=> ( ( ord_less_eq_set_a_a @ C2 @ B3 )
=> ( ord_less_eq_set_a_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_389_dual__order_Otrans,axiom,
! [B3: set_a,A2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ B3 )
=> ( ord_less_eq_set_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_390_dual__order_Otrans,axiom,
! [B3: set_a > extend8495563244428889912nnreal,A2: set_a > extend8495563244428889912nnreal,C2: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ B3 @ A2 )
=> ( ( ord_le6700572704167691815nnreal @ C2 @ B3 )
=> ( ord_le6700572704167691815nnreal @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_391_dual__order_Otrans,axiom,
! [B3: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B3 @ A2 )
=> ( ( ord_le3935885782089961368nnreal @ C2 @ B3 )
=> ( ord_le3935885782089961368nnreal @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_392_antisym,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_393_antisym,axiom,
! [A2: set_real_a,B3: set_real_a] :
( ( ord_le5743406823621094409real_a @ A2 @ B3 )
=> ( ( ord_le5743406823621094409real_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_394_antisym,axiom,
! [A2: set_a_a,B3: set_a_a] :
( ( ord_less_eq_set_a_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_395_antisym,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_396_antisym,axiom,
! [A2: set_a > extend8495563244428889912nnreal,B3: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ A2 @ B3 )
=> ( ( ord_le6700572704167691815nnreal @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_397_antisym,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B3 )
=> ( ( ord_le3935885782089961368nnreal @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_398_le__funD,axiom,
! [F: set_a > extend8495563244428889912nnreal,G: set_a > extend8495563244428889912nnreal,X2: set_a] :
( ( ord_le6700572704167691815nnreal @ F @ G )
=> ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( G @ X2 ) ) ) ).
% le_funD
thf(fact_399_le__funE,axiom,
! [F: set_a > extend8495563244428889912nnreal,G: set_a > extend8495563244428889912nnreal,X2: set_a] :
( ( ord_le6700572704167691815nnreal @ F @ G )
=> ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( G @ X2 ) ) ) ).
% le_funE
thf(fact_400_le__funI,axiom,
! [F: set_a > extend8495563244428889912nnreal,G: set_a > extend8495563244428889912nnreal] :
( ! [X3: set_a] : ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_le6700572704167691815nnreal @ F @ G ) ) ).
% le_funI
thf(fact_401_le__fun__def,axiom,
( ord_le6700572704167691815nnreal
= ( ^ [F2: set_a > extend8495563244428889912nnreal,G2: set_a > extend8495563244428889912nnreal] :
! [X4: set_a] : ( ord_le3935885782089961368nnreal @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ).
% le_fun_def
thf(fact_402_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a_a,Z: set_a_a] : ( Y2 = Z ) )
= ( ^ [A5: set_a_a,B5: set_a_a] :
( ( ord_less_eq_set_a_a @ A5 @ B5 )
& ( ord_less_eq_set_a_a @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_403_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_404_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a > extend8495563244428889912nnreal,Z: set_a > extend8495563244428889912nnreal] : ( Y2 = Z ) )
= ( ^ [A5: set_a > extend8495563244428889912nnreal,B5: set_a > extend8495563244428889912nnreal] :
( ( ord_le6700572704167691815nnreal @ A5 @ B5 )
& ( ord_le6700572704167691815nnreal @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_405_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] : ( Y2 = Z ) )
= ( ^ [A5: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A5 @ B5 )
& ( ord_le3935885782089961368nnreal @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_406_order__subst1,axiom,
! [A2: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ ( F @ B3 ) )
=> ( ( ord_le3935885782089961368nnreal @ B3 @ C2 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3935885782089961368nnreal @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_407_order__subst2,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B3 )
=> ( ( ord_le3935885782089961368nnreal @ ( F @ B3 ) @ C2 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_408_order__eq__refl,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( X2 = Y3 )
=> ( ord_le3935885782089961368nnreal @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_409_linorder__linear,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
| ( ord_le3935885782089961368nnreal @ Y3 @ X2 ) ) ).
% linorder_linear
thf(fact_410_ord__eq__le__subst,axiom,
! [A2: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_le3935885782089961368nnreal @ B3 @ C2 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3935885782089961368nnreal @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_411_ord__le__eq__subst,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_412_linorder__le__cases,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ~ ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ Y3 @ X2 ) ) ).
% linorder_le_cases
thf(fact_413_order__antisym__conv,axiom,
! [Y3: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Y3 @ X2 )
=> ( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_414_finite__has__maximal2,axiom,
! [A3: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( member_set_a @ A2 @ A3 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
& ( ord_less_eq_set_a @ A2 @ X3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A3 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_415_finite__has__maximal2,axiom,
! [A3: set_Ex3793607809372303086nnreal,A2: extend8495563244428889912nnreal] :
( ( finite3782138982310603983nnreal @ A3 )
=> ( ( member7908768830364227535nnreal @ A2 @ A3 )
=> ? [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A3 )
& ( ord_le3935885782089961368nnreal @ A2 @ X3 )
& ! [Xa: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ Xa @ A3 )
=> ( ( ord_le3935885782089961368nnreal @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_416_increasingD,axiom,
! [M: set_set_a,F: set_a > extend8495563244428889912nnreal,X2: set_a,Y3: set_a] :
( ( measur5393715408109795267nnreal @ M @ F )
=> ( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( member_set_a @ X2 @ M )
=> ( ( member_set_a @ Y3 @ M )
=> ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_417_algebra_Osigma__property__disjoint__lemma,axiom,
! [Omega: set_a,M: set_set_a,C: set_set_a] :
( ( sigma_algebra_a @ Omega @ M )
=> ( ( ord_le3724670747650509150_set_a @ M @ C )
=> ( ( sigma_closed_cdi_a @ Omega @ C )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sigma_sets_a @ Omega @ M ) @ C ) ) ) ) ).
% algebra.sigma_property_disjoint_lemma
thf(fact_418_Dynkin__system_Ointro,axiom,
! [Omega: set_a,M: set_set_a] :
( ( sigma_subset_class_a @ Omega @ M )
=> ( ( sigma_1042032682244386661ioms_a @ Omega @ M )
=> ( sigma_2757993544071651912stem_a @ Omega @ M ) ) ) ).
% Dynkin_system.intro
thf(fact_419_Dynkin__system__def,axiom,
( sigma_2757993544071651912stem_a
= ( ^ [Omega2: set_a,M2: set_set_a] :
( ( sigma_subset_class_a @ Omega2 @ M2 )
& ( sigma_1042032682244386661ioms_a @ Omega2 @ M2 ) ) ) ) ).
% Dynkin_system_def
thf(fact_420_measure__space__eq,axiom,
! [A3: set_set_a,Omega: set_a,Mu2: set_a > extend8495563244428889912nnreal,Mu: set_a > extend8495563244428889912nnreal] :
( ( ord_le3724670747650509150_set_a @ A3 @ ( pow_a @ Omega ) )
=> ( ! [A: set_a] :
( ( member_set_a @ A @ ( sigma_sigma_sets_a @ Omega @ A3 ) )
=> ( ( Mu2 @ A )
= ( Mu @ A ) ) )
=> ( ( sigma_3179946494550678598pace_a @ Omega @ ( sigma_sigma_sets_a @ Omega @ A3 ) @ Mu2 )
= ( sigma_3179946494550678598pace_a @ Omega @ ( sigma_sigma_sets_a @ Omega @ A3 ) @ Mu ) ) ) ) ).
% measure_space_eq
thf(fact_421_GreatestI2__order,axiom,
! [P: extend8495563244428889912nnreal > $o,X2: extend8495563244428889912nnreal,Q: extend8495563244428889912nnreal > $o] :
( ( P @ X2 )
=> ( ! [Y5: extend8495563244428889912nnreal] :
( ( P @ Y5 )
=> ( ord_le3935885782089961368nnreal @ Y5 @ X2 ) )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( P @ X3 )
=> ( ! [Y6: extend8495563244428889912nnreal] :
( ( P @ Y6 )
=> ( ord_le3935885782089961368nnreal @ Y6 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_7545170809120406815nnreal @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_422_subset__class_Osets__into__space,axiom,
! [Omega: set_a,M: set_set_a,X2: set_a] :
( ( sigma_subset_class_a @ Omega @ M )
=> ( ( member_set_a @ X2 @ M )
=> ( ord_less_eq_set_a @ X2 @ Omega ) ) ) ).
% subset_class.sets_into_space
thf(fact_423_Dynkin__system_Oaxioms_I1_J,axiom,
! [Omega: set_a,M: set_set_a] :
( ( sigma_2757993544071651912stem_a @ Omega @ M )
=> ( sigma_subset_class_a @ Omega @ M ) ) ).
% Dynkin_system.axioms(1)
thf(fact_424_Greatest__equality,axiom,
! [P: extend8495563244428889912nnreal > $o,X2: extend8495563244428889912nnreal] :
( ( P @ X2 )
=> ( ! [Y5: extend8495563244428889912nnreal] :
( ( P @ Y5 )
=> ( ord_le3935885782089961368nnreal @ Y5 @ X2 ) )
=> ( ( order_7545170809120406815nnreal @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_425_measure__down,axiom,
! [Omega: set_a,N: set_set_a,Mu2: set_a > extend8495563244428889912nnreal,M: set_set_a] :
( ( sigma_3179946494550678598pace_a @ Omega @ N @ Mu2 )
=> ( ( sigma_4968961713055010667ebra_a @ Omega @ M )
=> ( ( ord_le3724670747650509150_set_a @ M @ N )
=> ( sigma_3179946494550678598pace_a @ Omega @ M @ Mu2 ) ) ) ) ).
% measure_down
thf(fact_426_le__quasi__borel__iff,axiom,
( ord_le1843388692487544644orel_a
= ( ^ [X6: quasi_borel_a,Y7: quasi_borel_a] :
( ( ( ( qbs_space_a @ X6 )
= ( qbs_space_a @ Y7 ) )
=> ( ord_le5743406823621094409real_a @ ( qbs_Mx_a @ Y7 ) @ ( qbs_Mx_a @ X6 ) ) )
& ( ( ( qbs_space_a @ X6 )
!= ( qbs_space_a @ Y7 ) )
=> ( ord_less_set_a @ ( qbs_space_a @ X6 ) @ ( qbs_space_a @ Y7 ) ) ) ) ) ) ).
% le_quasi_borel_iff
thf(fact_427_less__eq__quasi__borel_Osimps,axiom,
( ord_le1843388692487544644orel_a
= ( ^ [A1: quasi_borel_a,A22: quasi_borel_a] :
( ? [X6: quasi_borel_a,Y7: quasi_borel_a] :
( ( A1 = X6 )
& ( A22 = Y7 )
& ( ord_less_set_a @ ( qbs_space_a @ X6 ) @ ( qbs_space_a @ Y7 ) ) )
| ? [X6: quasi_borel_a,Y7: quasi_borel_a] :
( ( A1 = X6 )
& ( A22 = Y7 )
& ( ( qbs_space_a @ X6 )
= ( qbs_space_a @ Y7 ) )
& ( ord_le5743406823621094409real_a @ ( qbs_Mx_a @ Y7 ) @ ( qbs_Mx_a @ X6 ) ) ) ) ) ) ).
% less_eq_quasi_borel.simps
thf(fact_428_less__eq__quasi__borel_Ocases,axiom,
! [A12: quasi_borel_a,A23: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ A12 @ A23 )
=> ( ~ ( ord_less_set_a @ ( qbs_space_a @ A12 ) @ ( qbs_space_a @ A23 ) )
=> ~ ( ( ( qbs_space_a @ A12 )
= ( qbs_space_a @ A23 ) )
=> ~ ( ord_le5743406823621094409real_a @ ( qbs_Mx_a @ A23 ) @ ( qbs_Mx_a @ A12 ) ) ) ) ) ).
% less_eq_quasi_borel.cases
thf(fact_429_sets_Osigma__property__disjoint__lemma,axiom,
! [M: sigma_measure_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ M ) @ C )
=> ( ( sigma_closed_cdi_a @ ( sigma_space_a @ M ) @ C )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sigma_sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) @ C ) ) ) ).
% sets.sigma_property_disjoint_lemma
thf(fact_430_r__preserves__morphisms,axiom,
! [X: sigma_measure_a,Y: sigma_measure_a] : ( ord_less_eq_set_a_a @ ( sigma_measurable_a_a @ X @ Y ) @ ( qbs_morphism_a_a @ ( measur6507891955840068946_qbs_a @ X ) @ ( measur6507891955840068946_qbs_a @ Y ) ) ) ).
% r_preserves_morphisms
thf(fact_431_le__measureD2,axiom,
! [A3: sigma_measure_a,B: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A3 @ B )
=> ( ( ( sigma_space_a @ A3 )
= ( sigma_space_a @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ A3 ) @ ( sigma_sets_a @ B ) ) ) ) ).
% le_measureD2
thf(fact_432_sets_Otop,axiom,
! [M: sigma_measure_a] : ( member_set_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) ).
% sets.top
thf(fact_433_psubsetD,axiom,
! [A3: set_a_a,B: set_a_a,C2: a > a] :
( ( ord_less_set_a_a @ A3 @ B )
=> ( ( member_a_a @ C2 @ A3 )
=> ( member_a_a @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_434_psubsetD,axiom,
! [A3: set_a,B: set_a,C2: a] :
( ( ord_less_set_a @ A3 @ B )
=> ( ( member_a @ C2 @ A3 )
=> ( member_a @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_435_psubsetD,axiom,
! [A3: set_set_a,B: set_set_a,C2: set_a] :
( ( ord_less_set_set_a @ A3 @ B )
=> ( ( member_set_a @ C2 @ A3 )
=> ( member_set_a @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_436_psubsetD,axiom,
! [A3: set_real_a,B: set_real_a,C2: real > a] :
( ( ord_less_set_real_a @ A3 @ B )
=> ( ( member_real_a @ C2 @ A3 )
=> ( member_real_a @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_437_smallest__ccdi__sets_OBasic,axiom,
! [A2: set_a,M: set_set_a,Omega: set_a] :
( ( member_set_a @ A2 @ M )
=> ( member_set_a @ A2 @ ( sigma_5648178489087971417sets_a @ Omega @ M ) ) ) ).
% smallest_ccdi_sets.Basic
thf(fact_438_order__less__imp__not__less,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y3 )
=> ~ ( ord_le7381754540660121996nnreal @ Y3 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_439_order__less__imp__not__eq2,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_440_order__less__imp__not__eq,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_441_linorder__less__linear,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y3 )
| ( X2 = Y3 )
| ( ord_le7381754540660121996nnreal @ Y3 @ X2 ) ) ).
% linorder_less_linear
thf(fact_442_order__less__imp__triv,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal,P: $o] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y3 )
=> ( ( ord_le7381754540660121996nnreal @ Y3 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_443_order__less__not__sym,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y3 )
=> ~ ( ord_le7381754540660121996nnreal @ Y3 @ X2 ) ) ).
% order_less_not_sym
thf(fact_444_order__less__subst2,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ B3 )
=> ( ( ord_le7381754540660121996nnreal @ ( F @ B3 ) @ C2 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y5 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_445_order__less__subst1,axiom,
! [A2: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ ( F @ B3 ) )
=> ( ( ord_le7381754540660121996nnreal @ B3 @ C2 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y5 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le7381754540660121996nnreal @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_446_order__less__irrefl,axiom,
! [X2: extend8495563244428889912nnreal] :
~ ( ord_le7381754540660121996nnreal @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_447_ord__less__eq__subst,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y5 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_448_ord__eq__less__subst,axiom,
! [A2: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_le7381754540660121996nnreal @ B3 @ C2 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y5 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le7381754540660121996nnreal @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_449_order__less__trans,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y3 )
=> ( ( ord_le7381754540660121996nnreal @ Y3 @ Z3 )
=> ( ord_le7381754540660121996nnreal @ X2 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_450_order__less__asym_H,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ B3 )
=> ~ ( ord_le7381754540660121996nnreal @ B3 @ A2 ) ) ).
% order_less_asym'
thf(fact_451_linorder__neq__iff,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( X2 != Y3 )
= ( ( ord_le7381754540660121996nnreal @ X2 @ Y3 )
| ( ord_le7381754540660121996nnreal @ Y3 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_452_order__less__asym,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y3 )
=> ~ ( ord_le7381754540660121996nnreal @ Y3 @ X2 ) ) ).
% order_less_asym
thf(fact_453_linorder__neqE,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( X2 != Y3 )
=> ( ~ ( ord_le7381754540660121996nnreal @ X2 @ Y3 )
=> ( ord_le7381754540660121996nnreal @ Y3 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_454_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B3 @ A2 )
=> ( A2 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_455_order_Ostrict__implies__not__eq,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ B3 )
=> ( A2 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_456_dual__order_Ostrict__trans,axiom,
! [B3: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B3 @ A2 )
=> ( ( ord_le7381754540660121996nnreal @ C2 @ B3 )
=> ( ord_le7381754540660121996nnreal @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_457_not__less__iff__gr__or__eq,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ~ ( ord_le7381754540660121996nnreal @ X2 @ Y3 ) )
= ( ( ord_le7381754540660121996nnreal @ Y3 @ X2 )
| ( X2 = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_458_order_Ostrict__trans,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ B3 )
=> ( ( ord_le7381754540660121996nnreal @ B3 @ C2 )
=> ( ord_le7381754540660121996nnreal @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_459_linorder__less__wlog,axiom,
! [P: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o,A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ! [A: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ B2 )
=> ( P @ A @ B2 ) )
=> ( ! [A: extend8495563244428889912nnreal] : ( P @ A @ A )
=> ( ! [A: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( P @ B2 @ A )
=> ( P @ A @ B2 ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_460_dual__order_Oirrefl,axiom,
! [A2: extend8495563244428889912nnreal] :
~ ( ord_le7381754540660121996nnreal @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_461_dual__order_Oasym,axiom,
! [B3: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B3 @ A2 )
=> ~ ( ord_le7381754540660121996nnreal @ A2 @ B3 ) ) ).
% dual_order.asym
thf(fact_462_linorder__cases,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ~ ( ord_le7381754540660121996nnreal @ X2 @ Y3 )
=> ( ( X2 != Y3 )
=> ( ord_le7381754540660121996nnreal @ Y3 @ X2 ) ) ) ).
% linorder_cases
thf(fact_463_antisym__conv3,axiom,
! [Y3: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ~ ( ord_le7381754540660121996nnreal @ Y3 @ X2 )
=> ( ( ~ ( ord_le7381754540660121996nnreal @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_464_ord__less__eq__trans,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_le7381754540660121996nnreal @ A2 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_465_ord__eq__less__trans,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( A2 = B3 )
=> ( ( ord_le7381754540660121996nnreal @ B3 @ C2 )
=> ( ord_le7381754540660121996nnreal @ A2 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_466_order_Oasym,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ B3 )
=> ~ ( ord_le7381754540660121996nnreal @ B3 @ A2 ) ) ).
% order.asym
thf(fact_467_less__imp__neq,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% less_imp_neq
thf(fact_468_dense,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y3 )
=> ? [Z4: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Z4 )
& ( ord_le7381754540660121996nnreal @ Z4 @ Y3 ) ) ) ).
% dense
thf(fact_469_sets_Osmallest__closed__cdi1,axiom,
! [M: sigma_measure_a] : ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ M ) @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) ) ).
% sets.smallest_closed_cdi1
thf(fact_470_leD,axiom,
! [Y3: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Y3 @ X2 )
=> ~ ( ord_le7381754540660121996nnreal @ X2 @ Y3 ) ) ).
% leD
thf(fact_471_leI,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ~ ( ord_le7381754540660121996nnreal @ X2 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ Y3 @ X2 ) ) ).
% leI
thf(fact_472_nless__le,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ~ ( ord_le7381754540660121996nnreal @ A2 @ B3 ) )
= ( ~ ( ord_le3935885782089961368nnreal @ A2 @ B3 )
| ( A2 = B3 ) ) ) ).
% nless_le
thf(fact_473_antisym__conv1,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ~ ( ord_le7381754540660121996nnreal @ X2 @ Y3 )
=> ( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_474_antisym__conv2,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
=> ( ( ~ ( ord_le7381754540660121996nnreal @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_475_dense__ge,axiom,
! [Z3: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ! [X3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z3 @ X3 )
=> ( ord_le3935885782089961368nnreal @ Y3 @ X3 ) )
=> ( ord_le3935885782089961368nnreal @ Y3 @ Z3 ) ) ).
% dense_ge
thf(fact_476_dense__le,axiom,
! [Y3: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] :
( ! [X3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ X3 @ Z3 ) )
=> ( ord_le3935885782089961368nnreal @ Y3 @ Z3 ) ) ).
% dense_le
thf(fact_477_less__le__not__le,axiom,
( ord_le7381754540660121996nnreal
= ( ^ [X4: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X4 @ Y4 )
& ~ ( ord_le3935885782089961368nnreal @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_478_not__le__imp__less,axiom,
! [Y3: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ~ ( ord_le3935885782089961368nnreal @ Y3 @ X2 )
=> ( ord_le7381754540660121996nnreal @ X2 @ Y3 ) ) ).
% not_le_imp_less
thf(fact_479_order_Oorder__iff__strict,axiom,
( ord_le3935885782089961368nnreal
= ( ^ [A5: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_480_order_Ostrict__iff__order,axiom,
( ord_le7381754540660121996nnreal
= ( ^ [A5: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_481_order_Ostrict__trans1,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B3 )
=> ( ( ord_le7381754540660121996nnreal @ B3 @ C2 )
=> ( ord_le7381754540660121996nnreal @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_482_order_Ostrict__trans2,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ B3 )
=> ( ( ord_le3935885782089961368nnreal @ B3 @ C2 )
=> ( ord_le7381754540660121996nnreal @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_483_order_Ostrict__iff__not,axiom,
( ord_le7381754540660121996nnreal
= ( ^ [A5: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A5 @ B5 )
& ~ ( ord_le3935885782089961368nnreal @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_484_dense__ge__bounded,axiom,
! [Z3: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z3 @ X2 )
=> ( ! [W: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z3 @ W )
=> ( ( ord_le7381754540660121996nnreal @ W @ X2 )
=> ( ord_le3935885782089961368nnreal @ Y3 @ W ) ) )
=> ( ord_le3935885782089961368nnreal @ Y3 @ Z3 ) ) ) ).
% dense_ge_bounded
thf(fact_485_dense__le__bounded,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y3 )
=> ( ! [W: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ W )
=> ( ( ord_le7381754540660121996nnreal @ W @ Y3 )
=> ( ord_le3935885782089961368nnreal @ W @ Z3 ) ) )
=> ( ord_le3935885782089961368nnreal @ Y3 @ Z3 ) ) ) ).
% dense_le_bounded
thf(fact_486_dual__order_Oorder__iff__strict,axiom,
( ord_le3935885782089961368nnreal
= ( ^ [B5: extend8495563244428889912nnreal,A5: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_487_dual__order_Ostrict__iff__order,axiom,
( ord_le7381754540660121996nnreal
= ( ^ [B5: extend8495563244428889912nnreal,A5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_488_dual__order_Ostrict__trans1,axiom,
! [B3: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B3 @ A2 )
=> ( ( ord_le7381754540660121996nnreal @ C2 @ B3 )
=> ( ord_le7381754540660121996nnreal @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_489_dual__order_Ostrict__trans2,axiom,
! [B3: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B3 @ A2 )
=> ( ( ord_le3935885782089961368nnreal @ C2 @ B3 )
=> ( ord_le7381754540660121996nnreal @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_490_dual__order_Ostrict__iff__not,axiom,
( ord_le7381754540660121996nnreal
= ( ^ [B5: extend8495563244428889912nnreal,A5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B5 @ A5 )
& ~ ( ord_le3935885782089961368nnreal @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_491_order_Ostrict__implies__order,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ B3 )
=> ( ord_le3935885782089961368nnreal @ A2 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_492_dual__order_Ostrict__implies__order,axiom,
! [B3: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B3 @ A2 )
=> ( ord_le3935885782089961368nnreal @ B3 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_493_order__le__less,axiom,
( ord_le3935885782089961368nnreal
= ( ^ [X4: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_494_order__less__le,axiom,
( ord_le7381754540660121996nnreal
= ( ^ [X4: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_495_linorder__not__le,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ~ ( ord_le3935885782089961368nnreal @ X2 @ Y3 ) )
= ( ord_le7381754540660121996nnreal @ Y3 @ X2 ) ) ).
% linorder_not_le
thf(fact_496_linorder__not__less,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ~ ( ord_le7381754540660121996nnreal @ X2 @ Y3 ) )
= ( ord_le3935885782089961368nnreal @ Y3 @ X2 ) ) ).
% linorder_not_less
thf(fact_497_order__less__imp__le,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_498_order__le__neq__trans,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_le7381754540660121996nnreal @ A2 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_499_order__neq__le__trans,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( A2 != B3 )
=> ( ( ord_le3935885782089961368nnreal @ A2 @ B3 )
=> ( ord_le7381754540660121996nnreal @ A2 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_500_order__le__less__trans,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
=> ( ( ord_le7381754540660121996nnreal @ Y3 @ Z3 )
=> ( ord_le7381754540660121996nnreal @ X2 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_501_order__less__le__trans,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y3 )
=> ( ( ord_le3935885782089961368nnreal @ Y3 @ Z3 )
=> ( ord_le7381754540660121996nnreal @ X2 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_502_order__le__less__subst1,axiom,
! [A2: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ ( F @ B3 ) )
=> ( ( ord_le7381754540660121996nnreal @ B3 @ C2 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y5 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le7381754540660121996nnreal @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_503_order__le__less__subst2,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B3 )
=> ( ( ord_le7381754540660121996nnreal @ ( F @ B3 ) @ C2 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_504_order__less__le__subst1,axiom,
! [A2: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ ( F @ B3 ) )
=> ( ( ord_le3935885782089961368nnreal @ B3 @ C2 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le7381754540660121996nnreal @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_505_order__less__le__subst2,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ B3 )
=> ( ( ord_le3935885782089961368nnreal @ ( F @ B3 ) @ C2 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Y5 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_506_linorder__le__less__linear,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
| ( ord_le7381754540660121996nnreal @ Y3 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_507_order__le__imp__less__or__eq,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
=> ( ( ord_le7381754540660121996nnreal @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_508_less__eq__measure_Ointros_I2_J,axiom,
! [M: sigma_measure_a,N: sigma_measure_a] :
( ( ( sigma_space_a @ M )
= ( sigma_space_a @ N ) )
=> ( ( ord_less_set_set_a @ ( sigma_sets_a @ M ) @ ( sigma_sets_a @ N ) )
=> ( ord_le254669795585780187sure_a @ M @ N ) ) ) ).
% less_eq_measure.intros(2)
thf(fact_509_measurable__cong,axiom,
! [M: sigma_measure_real,F: real > a,G: real > a,M4: sigma_measure_a] :
( ! [W: real] :
( ( member_real @ W @ ( sigma_space_real @ M ) )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ M4 ) )
= ( member_real_a @ G @ ( sigma_523072396149930112real_a @ M @ M4 ) ) ) ) ).
% measurable_cong
thf(fact_510_measurable__cong,axiom,
! [M: sigma_measure_a,F: a > a,G: a > a,M4: sigma_measure_a] :
( ! [W: a] :
( ( member_a @ W @ ( sigma_space_a @ M ) )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ M4 ) )
= ( member_a_a @ G @ ( sigma_measurable_a_a @ M @ M4 ) ) ) ) ).
% measurable_cong
thf(fact_511_measurable__space,axiom,
! [F: real > a,M: sigma_measure_real,A3: sigma_measure_a,X2: real] :
( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ A3 ) )
=> ( ( member_real @ X2 @ ( sigma_space_real @ M ) )
=> ( member_a @ ( F @ X2 ) @ ( sigma_space_a @ A3 ) ) ) ) ).
% measurable_space
thf(fact_512_measurable__space,axiom,
! [F: a > a,M: sigma_measure_a,A3: sigma_measure_a,X2: a] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ A3 ) )
=> ( ( member_a @ X2 @ ( sigma_space_a @ M ) )
=> ( member_a @ ( F @ X2 ) @ ( sigma_space_a @ A3 ) ) ) ) ).
% measurable_space
thf(fact_513_measurable__space,axiom,
! [F: set_a > a,M: sigma_measure_set_a,A3: sigma_measure_a,X2: set_a] :
( ( member_set_a_a @ F @ ( sigma_3901645225212141168et_a_a @ M @ A3 ) )
=> ( ( member_set_a @ X2 @ ( sigma_space_set_a @ M ) )
=> ( member_a @ ( F @ X2 ) @ ( sigma_space_a @ A3 ) ) ) ) ).
% measurable_space
thf(fact_514_measurable__space,axiom,
! [F: a > set_a,M: sigma_measure_a,A3: sigma_measure_set_a,X2: a] :
( ( member_a_set_a @ F @ ( sigma_3685133166752798000_set_a @ M @ A3 ) )
=> ( ( member_a @ X2 @ ( sigma_space_a @ M ) )
=> ( member_set_a @ ( F @ X2 ) @ ( sigma_space_set_a @ A3 ) ) ) ) ).
% measurable_space
thf(fact_515_measurable__space,axiom,
! [F: set_a > set_a,M: sigma_measure_set_a,A3: sigma_measure_set_a,X2: set_a] :
( ( member_set_a_set_a @ F @ ( sigma_5212894042034225104_set_a @ M @ A3 ) )
=> ( ( member_set_a @ X2 @ ( sigma_space_set_a @ M ) )
=> ( member_set_a @ ( F @ X2 ) @ ( sigma_space_set_a @ A3 ) ) ) ) ).
% measurable_space
thf(fact_516_measurable__space,axiom,
! [F: ( a > a ) > a,M: sigma_measure_a_a,A3: sigma_measure_a,X2: a > a] :
( ( member_a_a_a @ F @ ( sigma_3107241577691976327_a_a_a @ M @ A3 ) )
=> ( ( member_a_a @ X2 @ ( sigma_space_a_a @ M ) )
=> ( member_a @ ( F @ X2 ) @ ( sigma_space_a @ A3 ) ) ) ) ).
% measurable_space
thf(fact_517_measurable__space,axiom,
! [F: ( real > a ) > a,M: sigma_measure_real_a,A3: sigma_measure_a,X2: real > a] :
( ( member_real_a_a @ F @ ( sigma_5527925685445946117al_a_a @ M @ A3 ) )
=> ( ( member_real_a @ X2 @ ( sigma_space_real_a @ M ) )
=> ( member_a @ ( F @ X2 ) @ ( sigma_space_a @ A3 ) ) ) ) ).
% measurable_space
thf(fact_518_measurable__space,axiom,
! [F: a > a > a,M: sigma_measure_a,A3: sigma_measure_a_a,X2: a] :
( ( member_a_a_a2 @ F @ ( sigma_3971135313551251731_a_a_a @ M @ A3 ) )
=> ( ( member_a @ X2 @ ( sigma_space_a @ M ) )
=> ( member_a_a @ ( F @ X2 ) @ ( sigma_space_a_a @ A3 ) ) ) ) ).
% measurable_space
thf(fact_519_measurable__space,axiom,
! [F: a > real > a,M: sigma_measure_a,A3: sigma_measure_real_a,X2: a] :
( ( member_a_real_a @ F @ ( sigma_5590391210564117339real_a @ M @ A3 ) )
=> ( ( member_a @ X2 @ ( sigma_space_a @ M ) )
=> ( member_real_a @ ( F @ X2 ) @ ( sigma_space_real_a @ A3 ) ) ) ) ).
% measurable_space
thf(fact_520_measurable__space,axiom,
! [F: ( a > a ) > set_a,M: sigma_measure_a_a,A3: sigma_measure_set_a,X2: a > a] :
( ( member_a_a_set_a @ F @ ( sigma_8078287805088307175_set_a @ M @ A3 ) )
=> ( ( member_a_a @ X2 @ ( sigma_space_a_a @ M ) )
=> ( member_set_a @ ( F @ X2 ) @ ( sigma_space_set_a @ A3 ) ) ) ) ).
% measurable_space
thf(fact_521_measurable__cong__simp,axiom,
! [M: sigma_measure_real,N: sigma_measure_real,M4: sigma_measure_a,N2: sigma_measure_a,F: real > a,G: real > a] :
( ( M = N )
=> ( ( M4 = N2 )
=> ( ! [W: real] :
( ( member_real @ W @ ( sigma_space_real @ M ) )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ M4 ) )
= ( member_real_a @ G @ ( sigma_523072396149930112real_a @ N @ N2 ) ) ) ) ) ) ).
% measurable_cong_simp
thf(fact_522_measurable__cong__simp,axiom,
! [M: sigma_measure_a,N: sigma_measure_a,M4: sigma_measure_a,N2: sigma_measure_a,F: a > a,G: a > a] :
( ( M = N )
=> ( ( M4 = N2 )
=> ( ! [W: a] :
( ( member_a @ W @ ( sigma_space_a @ M ) )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ M4 ) )
= ( member_a_a @ G @ ( sigma_measurable_a_a @ N @ N2 ) ) ) ) ) ) ).
% measurable_cong_simp
thf(fact_523_measurable__comp,axiom,
! [F: a > real,M: sigma_measure_a,N: sigma_measure_real,G: real > a,L: sigma_measure_a] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ N ) )
=> ( ( member_real_a @ G @ ( sigma_523072396149930112real_a @ N @ L ) )
=> ( member_a_a @ ( comp_real_a_a @ G @ F ) @ ( sigma_measurable_a_a @ M @ L ) ) ) ) ).
% measurable_comp
thf(fact_524_measurable__comp,axiom,
! [F: real > real,M: sigma_measure_real,N: sigma_measure_real,G: real > a,L: sigma_measure_a] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) )
=> ( ( member_real_a @ G @ ( sigma_523072396149930112real_a @ N @ L ) )
=> ( member_real_a @ ( comp_real_a_real @ G @ F ) @ ( sigma_523072396149930112real_a @ M @ L ) ) ) ) ).
% measurable_comp
thf(fact_525_measurable__comp,axiom,
! [F: a > a,M: sigma_measure_a,N: sigma_measure_a,G: a > a,L: sigma_measure_a] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ N ) )
=> ( ( member_a_a @ G @ ( sigma_measurable_a_a @ N @ L ) )
=> ( member_a_a @ ( comp_a_a_a @ G @ F ) @ ( sigma_measurable_a_a @ M @ L ) ) ) ) ).
% measurable_comp
thf(fact_526_measurable__comp,axiom,
! [F: real > a,M: sigma_measure_real,N: sigma_measure_a,G: a > a,L: sigma_measure_a] :
( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ N ) )
=> ( ( member_a_a @ G @ ( sigma_measurable_a_a @ N @ L ) )
=> ( member_real_a @ ( comp_a_a_real @ G @ F ) @ ( sigma_523072396149930112real_a @ M @ L ) ) ) ) ).
% measurable_comp
thf(fact_527_sets__eq__imp__space__eq,axiom,
! [M: sigma_measure_a,M4: sigma_measure_a] :
( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ M4 ) )
=> ( ( sigma_space_a @ M )
= ( sigma_space_a @ M4 ) ) ) ).
% sets_eq_imp_space_eq
thf(fact_528_sets_Osmallest__closed__cdi2,axiom,
! [M: sigma_measure_a] : ( sigma_closed_cdi_a @ ( sigma_space_a @ M ) @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) ) ).
% sets.smallest_closed_cdi2
thf(fact_529_sets_Ocountably__additive__eq,axiom,
! [M: sigma_measure_a,Mu: set_a > extend8495563244428889912nnreal,Mu2: set_a > extend8495563244428889912nnreal] :
( ! [A: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ( Mu @ A )
= ( Mu2 @ A ) ) )
=> ( ( sigma_498350552340604931tive_a @ ( sigma_sets_a @ M ) @ Mu )
= ( sigma_498350552340604931tive_a @ ( sigma_sets_a @ M ) @ Mu2 ) ) ) ).
% sets.countably_additive_eq
thf(fact_530_measurable__mono,axiom,
! [N2: sigma_measure_a,N: sigma_measure_a,M: sigma_measure_a,M4: sigma_measure_a] :
( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ N2 ) @ ( sigma_sets_a @ N ) )
=> ( ( ( sigma_space_a @ N )
= ( sigma_space_a @ N2 ) )
=> ( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ M ) @ ( sigma_sets_a @ M4 ) )
=> ( ( ( sigma_space_a @ M )
= ( sigma_space_a @ M4 ) )
=> ( ord_less_eq_set_a_a @ ( sigma_measurable_a_a @ M @ N ) @ ( sigma_measurable_a_a @ M4 @ N2 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_531_sets_Osmallest__ccdi__sets,axiom,
! [M: sigma_measure_a] : ( ord_le3724670747650509150_set_a @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) @ ( pow_a @ ( sigma_space_a @ M ) ) ) ).
% sets.smallest_ccdi_sets
thf(fact_532_sets_Osets__into__space,axiom,
! [X2: set_a,M: sigma_measure_a] :
( ( member_set_a @ X2 @ ( sigma_sets_a @ M ) )
=> ( ord_less_eq_set_a @ X2 @ ( sigma_space_a @ M ) ) ) ).
% sets.sets_into_space
thf(fact_533_sets_Osigma__sets__eq,axiom,
! [M: sigma_measure_a] :
( ( sigma_sigma_sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) )
= ( sigma_sets_a @ M ) ) ).
% sets.sigma_sets_eq
thf(fact_534_sets_Osigma__sets__subset_H,axiom,
! [A2: set_set_a,M: sigma_measure_a,Omega3: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ Omega3 @ ( sigma_sets_a @ M ) )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sigma_sets_a @ Omega3 @ A2 ) @ ( sigma_sets_a @ M ) ) ) ) ).
% sets.sigma_sets_subset'
thf(fact_535_sets_Osigma__algebra__axioms,axiom,
! [M: sigma_measure_a] : ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) ).
% sets.sigma_algebra_axioms
thf(fact_536_sets_Oalgebra__axioms,axiom,
! [M: sigma_measure_a] : ( sigma_algebra_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) ).
% sets.algebra_axioms
thf(fact_537_less__eq__measure_Ointros_I1_J,axiom,
! [M: sigma_measure_a,N: sigma_measure_a] :
( ( ord_less_set_a @ ( sigma_space_a @ M ) @ ( sigma_space_a @ N ) )
=> ( ord_le254669795585780187sure_a @ M @ N ) ) ).
% less_eq_measure.intros(1)
thf(fact_538_sets_Osubset__class__axioms,axiom,
! [M: sigma_measure_a] : ( sigma_subset_class_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) ).
% sets.subset_class_axioms
thf(fact_539_less__eq__quasi__borel_Ointros_I1_J,axiom,
! [X: quasi_borel_a,Y: quasi_borel_a] :
( ( ord_less_set_a @ ( qbs_space_a @ X ) @ ( qbs_space_a @ Y ) )
=> ( ord_le1843388692487544644orel_a @ X @ Y ) ) ).
% less_eq_quasi_borel.intros(1)
thf(fact_540_sets__le__imp__space__le,axiom,
! [A3: sigma_measure_a,B: sigma_measure_a] :
( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ A3 ) @ ( sigma_sets_a @ B ) )
=> ( ord_less_eq_set_a @ ( sigma_space_a @ A3 ) @ ( sigma_space_a @ B ) ) ) ).
% sets_le_imp_space_le
thf(fact_541_sets_Ospace__closed,axiom,
! [M: sigma_measure_a] : ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ M ) @ ( pow_a @ ( sigma_space_a @ M ) ) ) ).
% sets.space_closed
thf(fact_542_sigma__sets__le__sets__iff,axiom,
! [X2: sigma_measure_a,A6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sigma_sigma_sets_a @ ( sigma_space_a @ X2 ) @ A6 ) @ ( sigma_sets_a @ X2 ) )
= ( ord_le3724670747650509150_set_a @ A6 @ ( sigma_sets_a @ X2 ) ) ) ).
% sigma_sets_le_sets_iff
thf(fact_543_sets_Osigma__sets__subset,axiom,
! [A2: set_set_a,M: sigma_measure_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( sigma_sets_a @ M ) )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sigma_sets_a @ ( sigma_space_a @ M ) @ A2 ) @ ( sigma_sets_a @ M ) ) ) ).
% sets.sigma_sets_subset
thf(fact_544_sets_Ois__sigma__algebra,axiom,
! [M: sigma_measure_a] :
( ( finite_finite_set_a @ ( sigma_sets_a @ M ) )
=> ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) ) ).
% sets.is_sigma_algebra
thf(fact_545_complete__interval,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,P: extend8495563244428889912nnreal > $o] :
( ( ord_le7381754540660121996nnreal @ A2 @ B3 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B3 )
=> ? [C3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ C3 )
& ( ord_le3935885782089961368nnreal @ C3 @ B3 )
& ! [X5: extend8495563244428889912nnreal] :
( ( ( ord_le3935885782089961368nnreal @ A2 @ X5 )
& ( ord_le7381754540660121996nnreal @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D2: extend8495563244428889912nnreal] :
( ! [X3: extend8495563244428889912nnreal] :
( ( ( ord_le3935885782089961368nnreal @ A2 @ X3 )
& ( ord_le7381754540660121996nnreal @ X3 @ D2 ) )
=> ( P @ X3 ) )
=> ( ord_le3935885782089961368nnreal @ D2 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_546_verit__comp__simplify1_I3_J,axiom,
! [B6: extend8495563244428889912nnreal,A7: extend8495563244428889912nnreal] :
( ( ~ ( ord_le3935885782089961368nnreal @ B6 @ A7 ) )
= ( ord_le7381754540660121996nnreal @ A7 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_547_pinf_I6_J,axiom,
! [T3: extend8495563244428889912nnreal] :
? [Z4: extend8495563244428889912nnreal] :
! [X5: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z4 @ X5 )
=> ~ ( ord_le3935885782089961368nnreal @ X5 @ T3 ) ) ).
% pinf(6)
thf(fact_548_pinf_I8_J,axiom,
! [T3: extend8495563244428889912nnreal] :
? [Z4: extend8495563244428889912nnreal] :
! [X5: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z4 @ X5 )
=> ( ord_le3935885782089961368nnreal @ T3 @ X5 ) ) ).
% pinf(8)
thf(fact_549_minf_I6_J,axiom,
! [T3: extend8495563244428889912nnreal] :
? [Z4: extend8495563244428889912nnreal] :
! [X5: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X5 @ Z4 )
=> ( ord_le3935885782089961368nnreal @ X5 @ T3 ) ) ).
% minf(6)
thf(fact_550_minf_I8_J,axiom,
! [T3: extend8495563244428889912nnreal] :
? [Z4: extend8495563244428889912nnreal] :
! [X5: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X5 @ Z4 )
=> ~ ( ord_le3935885782089961368nnreal @ T3 @ X5 ) ) ).
% minf(8)
thf(fact_551_space__sup__measure_H,axiom,
! [B: sigma_measure_a,A3: sigma_measure_a] :
( ( ( sigma_sets_a @ B )
= ( sigma_sets_a @ A3 ) )
=> ( ( sigma_space_a @ ( measur3004909623614618064sure_a @ A3 @ B ) )
= ( sigma_space_a @ A3 ) ) ) ).
% space_sup_measure'
thf(fact_552_le__measure__iff,axiom,
( ord_le254669795585780187sure_a
= ( ^ [M2: sigma_measure_a,N3: sigma_measure_a] :
( ( ( ( sigma_space_a @ M2 )
= ( sigma_space_a @ N3 ) )
=> ( ( ( ( sigma_sets_a @ M2 )
= ( sigma_sets_a @ N3 ) )
=> ( ord_le6700572704167691815nnreal @ ( sigma_emeasure_a @ M2 ) @ ( sigma_emeasure_a @ N3 ) ) )
& ( ( ( sigma_sets_a @ M2 )
!= ( sigma_sets_a @ N3 ) )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ M2 ) @ ( sigma_sets_a @ N3 ) ) ) ) )
& ( ( ( sigma_space_a @ M2 )
!= ( sigma_space_a @ N3 ) )
=> ( ord_less_eq_set_a @ ( sigma_space_a @ M2 ) @ ( sigma_space_a @ N3 ) ) ) ) ) ) ).
% le_measure_iff
thf(fact_553_le__emeasure__sup__measure_H1,axiom,
! [B: sigma_measure_a,A3: sigma_measure_a,X: set_a] :
( ( ( sigma_sets_a @ B )
= ( sigma_sets_a @ A3 ) )
=> ( ( member_set_a @ X @ ( sigma_sets_a @ A3 ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ A3 @ X ) @ ( sigma_emeasure_a @ ( measur3004909623614618064sure_a @ A3 @ B ) @ X ) ) ) ) ).
% le_emeasure_sup_measure'1
thf(fact_554_le__emeasure__sup__measure_H2,axiom,
! [B: sigma_measure_a,A3: sigma_measure_a,X: set_a] :
( ( ( sigma_sets_a @ B )
= ( sigma_sets_a @ A3 ) )
=> ( ( member_set_a @ X @ ( sigma_sets_a @ A3 ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ B @ X ) @ ( sigma_emeasure_a @ ( measur3004909623614618064sure_a @ A3 @ B ) @ X ) ) ) ) ).
% le_emeasure_sup_measure'2
thf(fact_555_measure__eqI,axiom,
! [M: sigma_measure_a,N: sigma_measure_a] :
( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ N ) )
=> ( ! [A8: set_a] :
( ( member_set_a @ A8 @ ( sigma_sets_a @ M ) )
=> ( ( sigma_emeasure_a @ M @ A8 )
= ( sigma_emeasure_a @ N @ A8 ) ) )
=> ( M = N ) ) ) ).
% measure_eqI
thf(fact_556_emeasure__space,axiom,
! [M: sigma_measure_a,A3: set_a] : ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ M @ A3 ) @ ( sigma_emeasure_a @ M @ ( sigma_space_a @ M ) ) ) ).
% emeasure_space
thf(fact_557_emeasure__sup__measure_H__le2,axiom,
! [B: sigma_measure_a,C: sigma_measure_a,A3: sigma_measure_a,X: set_a] :
( ( ( sigma_sets_a @ B )
= ( sigma_sets_a @ C ) )
=> ( ( ( sigma_sets_a @ A3 )
= ( sigma_sets_a @ C ) )
=> ( ( member_set_a @ X @ ( sigma_sets_a @ C ) )
=> ( ! [Y8: set_a] :
( ( ord_less_eq_set_a @ Y8 @ X )
=> ( ( member_set_a @ Y8 @ ( sigma_sets_a @ A3 ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ A3 @ Y8 ) @ ( sigma_emeasure_a @ C @ Y8 ) ) ) )
=> ( ! [Y8: set_a] :
( ( ord_less_eq_set_a @ Y8 @ X )
=> ( ( member_set_a @ Y8 @ ( sigma_sets_a @ A3 ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ B @ Y8 ) @ ( sigma_emeasure_a @ C @ Y8 ) ) ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ ( measur3004909623614618064sure_a @ A3 @ B ) @ X ) @ ( sigma_emeasure_a @ C @ X ) ) ) ) ) ) ) ).
% emeasure_sup_measure'_le2
thf(fact_558_emeasure__mono,axiom,
! [A2: set_a,B3: set_a,M: sigma_measure_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_set_a @ B3 @ ( sigma_sets_a @ M ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ M @ A2 ) @ ( sigma_emeasure_a @ M @ B3 ) ) ) ) ).
% emeasure_mono
thf(fact_559_emeasure__countably__additive,axiom,
! [M: sigma_measure_a] : ( sigma_498350552340604931tive_a @ ( sigma_sets_a @ M ) @ ( sigma_emeasure_a @ M ) ) ).
% emeasure_countably_additive
thf(fact_560_verit__la__disequality,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( A2 = B3 )
| ~ ( ord_le3935885782089961368nnreal @ A2 @ B3 )
| ~ ( ord_le3935885782089961368nnreal @ B3 @ A2 ) ) ).
% verit_la_disequality
thf(fact_561_verit__comp__simplify1_I2_J,axiom,
! [A2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_562_ex__gt__or__lt,axiom,
! [A2: extend8495563244428889912nnreal] :
? [B2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ B2 )
| ( ord_le7381754540660121996nnreal @ B2 @ A2 ) ) ).
% ex_gt_or_lt
thf(fact_563_verit__comp__simplify1_I1_J,axiom,
! [A2: extend8495563244428889912nnreal] :
~ ( ord_le7381754540660121996nnreal @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_564_pinf_I1_J,axiom,
! [P: extend8495563244428889912nnreal > $o,P2: extend8495563244428889912nnreal > $o,Q: extend8495563244428889912nnreal > $o,Q2: extend8495563244428889912nnreal > $o] :
( ? [Z5: extend8495563244428889912nnreal] :
! [X3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z5: extend8495563244428889912nnreal] :
! [X3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: extend8495563244428889912nnreal] :
! [X5: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z4 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P2 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_565_pinf_I2_J,axiom,
! [P: extend8495563244428889912nnreal > $o,P2: extend8495563244428889912nnreal > $o,Q: extend8495563244428889912nnreal > $o,Q2: extend8495563244428889912nnreal > $o] :
( ? [Z5: extend8495563244428889912nnreal] :
! [X3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z5: extend8495563244428889912nnreal] :
! [X3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: extend8495563244428889912nnreal] :
! [X5: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z4 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P2 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_566_pinf_I3_J,axiom,
! [T3: extend8495563244428889912nnreal] :
? [Z4: extend8495563244428889912nnreal] :
! [X5: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z4 @ X5 )
=> ( X5 != T3 ) ) ).
% pinf(3)
thf(fact_567_pinf_I4_J,axiom,
! [T3: extend8495563244428889912nnreal] :
? [Z4: extend8495563244428889912nnreal] :
! [X5: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z4 @ X5 )
=> ( X5 != T3 ) ) ).
% pinf(4)
thf(fact_568_pinf_I5_J,axiom,
! [T3: extend8495563244428889912nnreal] :
? [Z4: extend8495563244428889912nnreal] :
! [X5: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z4 @ X5 )
=> ~ ( ord_le7381754540660121996nnreal @ X5 @ T3 ) ) ).
% pinf(5)
thf(fact_569_pinf_I7_J,axiom,
! [T3: extend8495563244428889912nnreal] :
? [Z4: extend8495563244428889912nnreal] :
! [X5: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ Z4 @ X5 )
=> ( ord_le7381754540660121996nnreal @ T3 @ X5 ) ) ).
% pinf(7)
thf(fact_570_minf_I1_J,axiom,
! [P: extend8495563244428889912nnreal > $o,P2: extend8495563244428889912nnreal > $o,Q: extend8495563244428889912nnreal > $o,Q2: extend8495563244428889912nnreal > $o] :
( ? [Z5: extend8495563244428889912nnreal] :
! [X3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z5: extend8495563244428889912nnreal] :
! [X3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: extend8495563244428889912nnreal] :
! [X5: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X5 @ Z4 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P2 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_571_minf_I2_J,axiom,
! [P: extend8495563244428889912nnreal > $o,P2: extend8495563244428889912nnreal > $o,Q: extend8495563244428889912nnreal > $o,Q2: extend8495563244428889912nnreal > $o] :
( ? [Z5: extend8495563244428889912nnreal] :
! [X3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z5: extend8495563244428889912nnreal] :
! [X3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: extend8495563244428889912nnreal] :
! [X5: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X5 @ Z4 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P2 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_572_minf_I3_J,axiom,
! [T3: extend8495563244428889912nnreal] :
? [Z4: extend8495563244428889912nnreal] :
! [X5: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X5 @ Z4 )
=> ( X5 != T3 ) ) ).
% minf(3)
thf(fact_573_minf_I4_J,axiom,
! [T3: extend8495563244428889912nnreal] :
? [Z4: extend8495563244428889912nnreal] :
! [X5: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X5 @ Z4 )
=> ( X5 != T3 ) ) ).
% minf(4)
thf(fact_574_minf_I5_J,axiom,
! [T3: extend8495563244428889912nnreal] :
? [Z4: extend8495563244428889912nnreal] :
! [X5: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X5 @ Z4 )
=> ( ord_le7381754540660121996nnreal @ X5 @ T3 ) ) ).
% minf(5)
thf(fact_575_minf_I7_J,axiom,
! [T3: extend8495563244428889912nnreal] :
? [Z4: extend8495563244428889912nnreal] :
! [X5: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X5 @ Z4 )
=> ~ ( ord_le7381754540660121996nnreal @ T3 @ X5 ) ) ).
% minf(7)
thf(fact_576_measure__space,axiom,
! [M: sigma_measure_a] : ( sigma_3179946494550678598pace_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) @ ( sigma_emeasure_a @ M ) ) ).
% measure_space
thf(fact_577_less__eq__measure_Ointros_I3_J,axiom,
! [M: sigma_measure_a,N: sigma_measure_a] :
( ( ( sigma_space_a @ M )
= ( sigma_space_a @ N ) )
=> ( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ N ) )
=> ( ( ord_le6700572704167691815nnreal @ ( sigma_emeasure_a @ M ) @ ( sigma_emeasure_a @ N ) )
=> ( ord_le254669795585780187sure_a @ M @ N ) ) ) ) ).
% less_eq_measure.intros(3)
thf(fact_578_less__eq__measure_Osimps,axiom,
( ord_le254669795585780187sure_a
= ( ^ [A1: sigma_measure_a,A22: sigma_measure_a] :
( ? [M2: sigma_measure_a,N3: sigma_measure_a] :
( ( A1 = M2 )
& ( A22 = N3 )
& ( ord_less_set_a @ ( sigma_space_a @ M2 ) @ ( sigma_space_a @ N3 ) ) )
| ? [M2: sigma_measure_a,N3: sigma_measure_a] :
( ( A1 = M2 )
& ( A22 = N3 )
& ( ( sigma_space_a @ M2 )
= ( sigma_space_a @ N3 ) )
& ( ord_less_set_set_a @ ( sigma_sets_a @ M2 ) @ ( sigma_sets_a @ N3 ) ) )
| ? [M2: sigma_measure_a,N3: sigma_measure_a] :
( ( A1 = M2 )
& ( A22 = N3 )
& ( ( sigma_space_a @ M2 )
= ( sigma_space_a @ N3 ) )
& ( ( sigma_sets_a @ M2 )
= ( sigma_sets_a @ N3 ) )
& ( ord_le6700572704167691815nnreal @ ( sigma_emeasure_a @ M2 ) @ ( sigma_emeasure_a @ N3 ) ) ) ) ) ) ).
% less_eq_measure.simps
thf(fact_579_less__eq__measure_Ocases,axiom,
! [A12: sigma_measure_a,A23: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A12 @ A23 )
=> ( ~ ( ord_less_set_a @ ( sigma_space_a @ A12 ) @ ( sigma_space_a @ A23 ) )
=> ( ( ( ( sigma_space_a @ A12 )
= ( sigma_space_a @ A23 ) )
=> ~ ( ord_less_set_set_a @ ( sigma_sets_a @ A12 ) @ ( sigma_sets_a @ A23 ) ) )
=> ~ ( ( ( sigma_space_a @ A12 )
= ( sigma_space_a @ A23 ) )
=> ( ( ( sigma_sets_a @ A12 )
= ( sigma_sets_a @ A23 ) )
=> ~ ( ord_le6700572704167691815nnreal @ ( sigma_emeasure_a @ A12 ) @ ( sigma_emeasure_a @ A23 ) ) ) ) ) ) ) ).
% less_eq_measure.cases
thf(fact_580_sets__image__in__sets,axiom,
! [N: sigma_measure_real,X: set_real,F: real > a,M: sigma_measure_a] :
( ( ( sigma_space_real @ N )
= X )
=> ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ N @ M ) )
=> ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ ( sigma_7241109303225351740real_a @ X @ F @ M ) ) @ ( sigma_sets_real @ N ) ) ) ) ).
% sets_image_in_sets
thf(fact_581_sets__image__in__sets,axiom,
! [N: sigma_measure_a,X: set_a,F: a > a,M: sigma_measure_a] :
( ( ( sigma_space_a @ N )
= X )
=> ( ( member_a_a @ F @ ( sigma_measurable_a_a @ N @ M ) )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ ( sigma_2018398322970380052ra_a_a @ X @ F @ M ) ) @ ( sigma_sets_a @ N ) ) ) ) ).
% sets_image_in_sets
thf(fact_582_sets_Ocaratheodory_H,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_positive_a @ ( sigma_sets_a @ M ) @ F )
=> ( ( sigma_498350552340604931tive_a @ ( sigma_sets_a @ M ) @ F )
=> ? [Mu3: set_a > extend8495563244428889912nnreal] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ ( sigma_sets_a @ M ) )
=> ( ( Mu3 @ X5 )
= ( F @ X5 ) ) )
& ( sigma_3179946494550678598pace_a @ ( sigma_space_a @ M ) @ ( sigma_sigma_sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) @ Mu3 ) ) ) ) ).
% sets.caratheodory'
thf(fact_583_sets_Ocaratheodory,axiom,
! [M: sigma_measure_a,Mu2: set_a > extend8495563244428889912nnreal] :
( ( sigma_positive_a @ ( sigma_sets_a @ M ) @ Mu2 )
=> ( ( sigma_498350552340604931tive_a @ ( sigma_sets_a @ M ) @ Mu2 )
=> ? [Mu4: set_a > extend8495563244428889912nnreal] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ ( sigma_sets_a @ M ) )
=> ( ( Mu4 @ X5 )
= ( Mu2 @ X5 ) ) )
& ( sigma_3179946494550678598pace_a @ ( sigma_space_a @ M ) @ ( sigma_sigma_sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) @ Mu4 ) ) ) ) ).
% sets.caratheodory
thf(fact_584_le__sup__lexord,axiom,
! [K2: extend8495563244428889912nnreal > extend8495563244428889912nnreal,A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,Ca: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal,S2: extend8495563244428889912nnreal] :
( ( ( ord_le7381754540660121996nnreal @ ( K2 @ A3 ) @ ( K2 @ B ) )
=> ( ord_le3935885782089961368nnreal @ Ca @ B ) )
=> ( ( ( ord_le7381754540660121996nnreal @ ( K2 @ B ) @ ( K2 @ A3 ) )
=> ( ord_le3935885782089961368nnreal @ Ca @ A3 ) )
=> ( ( ( ( K2 @ A3 )
= ( K2 @ B ) )
=> ( ord_le3935885782089961368nnreal @ Ca @ C2 ) )
=> ( ( ~ ( ord_le3935885782089961368nnreal @ ( K2 @ B ) @ ( K2 @ A3 ) )
=> ( ~ ( ord_le3935885782089961368nnreal @ ( K2 @ A3 ) @ ( K2 @ B ) )
=> ( ord_le3935885782089961368nnreal @ Ca @ S2 ) ) )
=> ( ord_le3935885782089961368nnreal @ Ca @ ( measur8354350339198833422nnreal @ A3 @ B @ K2 @ S2 @ C2 ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_585_fmeasurable_Osmallest__ccdi__sets,axiom,
! [M: sigma_measure_a] : ( ord_le3724670747650509150_set_a @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( measur3645360004775918570able_a @ M ) ) @ ( pow_a @ ( sigma_space_a @ M ) ) ) ).
% fmeasurable.smallest_ccdi_sets
thf(fact_586_sets_Ospace__measure__of__eq,axiom,
! [M: sigma_measure_a,Mu2: set_a > extend8495563244428889912nnreal] :
( ( sigma_space_a @ ( sigma_measure_of_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) @ Mu2 ) )
= ( sigma_space_a @ M ) ) ).
% sets.space_measure_of_eq
thf(fact_587_sets_Osets__measure__of__eq,axiom,
! [M: sigma_measure_a,Mu2: set_a > extend8495563244428889912nnreal] :
( ( sigma_sets_a @ ( sigma_measure_of_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) @ Mu2 ) )
= ( sigma_sets_a @ M ) ) ).
% sets.sets_measure_of_eq
thf(fact_588_in__measure__of,axiom,
! [M: set_set_a,Omega: set_a,A3: set_a,Mu2: set_a > extend8495563244428889912nnreal] :
( ( ord_le3724670747650509150_set_a @ M @ ( pow_a @ Omega ) )
=> ( ( member_set_a @ A3 @ M )
=> ( member_set_a @ A3 @ ( sigma_sets_a @ ( sigma_measure_of_a @ Omega @ M @ Mu2 ) ) ) ) ) ).
% in_measure_of
thf(fact_589_space__measure__of,axiom,
! [A3: set_set_a,Omega: set_a,Mu2: set_a > extend8495563244428889912nnreal] :
( ( ord_le3724670747650509150_set_a @ A3 @ ( pow_a @ Omega ) )
=> ( ( sigma_space_a @ ( sigma_measure_of_a @ Omega @ A3 @ Mu2 ) )
= Omega ) ) ).
% space_measure_of
thf(fact_590_fmeasurable_Opositive__cong__eq,axiom,
! [M: sigma_measure_a,Mu: set_a > extend8495563244428889912nnreal,Mu2: set_a > extend8495563244428889912nnreal] :
( ! [A: set_a] :
( ( member_set_a @ A @ ( measur3645360004775918570able_a @ M ) )
=> ( ( Mu @ A )
= ( Mu2 @ A ) ) )
=> ( ( sigma_positive_a @ ( measur3645360004775918570able_a @ M ) @ Mu )
= ( sigma_positive_a @ ( measur3645360004775918570able_a @ M ) @ Mu2 ) ) ) ).
% fmeasurable.positive_cong_eq
thf(fact_591_space__in__measure__of,axiom,
! [Omega: set_a,M: set_set_a,Mu2: set_a > extend8495563244428889912nnreal] : ( member_set_a @ Omega @ ( sigma_sets_a @ ( sigma_measure_of_a @ Omega @ M @ Mu2 ) ) ) ).
% space_in_measure_of
thf(fact_592_sets_Opositive__cong__eq,axiom,
! [M: sigma_measure_a,Mu: set_a > extend8495563244428889912nnreal,Mu2: set_a > extend8495563244428889912nnreal] :
( ! [A: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ( Mu @ A )
= ( Mu2 @ A ) ) )
=> ( ( sigma_positive_a @ ( sigma_sets_a @ M ) @ Mu )
= ( sigma_positive_a @ ( sigma_sets_a @ M ) @ Mu2 ) ) ) ).
% sets.positive_cong_eq
thf(fact_593_space__measure__of__conv,axiom,
! [Omega: set_a,A3: set_set_a,Mu2: set_a > extend8495563244428889912nnreal] :
( ( sigma_space_a @ ( sigma_measure_of_a @ Omega @ A3 @ Mu2 ) )
= Omega ) ).
% space_measure_of_conv
thf(fact_594_fmeasurableD,axiom,
! [A3: set_a,M: sigma_measure_a] :
( ( member_set_a @ A3 @ ( measur3645360004775918570able_a @ M ) )
=> ( member_set_a @ A3 @ ( sigma_sets_a @ M ) ) ) ).
% fmeasurableD
thf(fact_595_emeasure__measure__of__sigma,axiom,
! [Omega: set_a,M: set_set_a,Mu2: set_a > extend8495563244428889912nnreal,A3: set_a] :
( ( sigma_4968961713055010667ebra_a @ Omega @ M )
=> ( ( sigma_positive_a @ M @ Mu2 )
=> ( ( sigma_498350552340604931tive_a @ M @ Mu2 )
=> ( ( member_set_a @ A3 @ M )
=> ( ( sigma_emeasure_a @ ( sigma_measure_of_a @ Omega @ M @ Mu2 ) @ A3 )
= ( Mu2 @ A3 ) ) ) ) ) ) ).
% emeasure_measure_of_sigma
thf(fact_596_sigma__algebra_Osets__measure__of__eq,axiom,
! [Omega: set_a,M: set_set_a,Mu2: set_a > extend8495563244428889912nnreal] :
( ( sigma_4968961713055010667ebra_a @ Omega @ M )
=> ( ( sigma_sets_a @ ( sigma_measure_of_a @ Omega @ M @ Mu2 ) )
= M ) ) ).
% sigma_algebra.sets_measure_of_eq
thf(fact_597_sigma__algebra_Ospace__measure__of__eq,axiom,
! [Omega: set_a,M: set_set_a,Mu2: set_a > extend8495563244428889912nnreal] :
( ( sigma_4968961713055010667ebra_a @ Omega @ M )
=> ( ( sigma_space_a @ ( sigma_measure_of_a @ Omega @ M @ Mu2 ) )
= Omega ) ) ).
% sigma_algebra.space_measure_of_eq
thf(fact_598_fmeasurable_Ocaratheodory,axiom,
! [M: sigma_measure_a,Mu2: set_a > extend8495563244428889912nnreal] :
( ( sigma_positive_a @ ( measur3645360004775918570able_a @ M ) @ Mu2 )
=> ( ( sigma_498350552340604931tive_a @ ( measur3645360004775918570able_a @ M ) @ Mu2 )
=> ? [Mu4: set_a > extend8495563244428889912nnreal] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ ( measur3645360004775918570able_a @ M ) )
=> ( ( Mu4 @ X5 )
= ( Mu2 @ X5 ) ) )
& ( sigma_3179946494550678598pace_a @ ( sigma_space_a @ M ) @ ( sigma_sigma_sets_a @ ( sigma_space_a @ M ) @ ( measur3645360004775918570able_a @ M ) ) @ Mu4 ) ) ) ) ).
% fmeasurable.caratheodory
thf(fact_599_fmeasurable_Ocaratheodory_H,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_positive_a @ ( measur3645360004775918570able_a @ M ) @ F )
=> ( ( sigma_498350552340604931tive_a @ ( measur3645360004775918570able_a @ M ) @ F )
=> ? [Mu3: set_a > extend8495563244428889912nnreal] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ ( measur3645360004775918570able_a @ M ) )
=> ( ( Mu3 @ X5 )
= ( F @ X5 ) ) )
& ( sigma_3179946494550678598pace_a @ ( sigma_space_a @ M ) @ ( sigma_sigma_sets_a @ ( sigma_space_a @ M ) @ ( measur3645360004775918570able_a @ M ) ) @ Mu3 ) ) ) ) ).
% fmeasurable.caratheodory'
thf(fact_600_fmeasurableI2,axiom,
! [A3: set_a,M: sigma_measure_a,B: set_a] :
( ( member_set_a @ A3 @ ( measur3645360004775918570able_a @ M ) )
=> ( ( ord_less_eq_set_a @ B @ A3 )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ B @ ( measur3645360004775918570able_a @ M ) ) ) ) ) ).
% fmeasurableI2
thf(fact_601_fmeasurable_Osets__into__space,axiom,
! [X2: set_a,M: sigma_measure_a] :
( ( member_set_a @ X2 @ ( measur3645360004775918570able_a @ M ) )
=> ( ord_less_eq_set_a @ X2 @ ( sigma_space_a @ M ) ) ) ).
% fmeasurable.sets_into_space
thf(fact_602_emeasure__measure__of,axiom,
! [M: sigma_measure_a,Omega: set_a,A3: set_set_a,Mu2: set_a > extend8495563244428889912nnreal,X: set_a] :
( ( M
= ( sigma_measure_of_a @ Omega @ A3 @ Mu2 ) )
=> ( ( ord_le3724670747650509150_set_a @ A3 @ ( pow_a @ Omega ) )
=> ( ( sigma_positive_a @ ( sigma_sets_a @ M ) @ Mu2 )
=> ( ( sigma_498350552340604931tive_a @ ( sigma_sets_a @ M ) @ Mu2 )
=> ( ( member_set_a @ X @ ( sigma_sets_a @ M ) )
=> ( ( sigma_emeasure_a @ M @ X )
= ( Mu2 @ X ) ) ) ) ) ) ) ).
% emeasure_measure_of
thf(fact_603_fmeasurable_Osubset__class__axioms,axiom,
! [M: sigma_measure_a] : ( sigma_subset_class_a @ ( sigma_space_a @ M ) @ ( measur3645360004775918570able_a @ M ) ) ).
% fmeasurable.subset_class_axioms
thf(fact_604_measure__of__of__measure,axiom,
! [M: sigma_measure_a] :
( ( sigma_measure_of_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) @ ( sigma_emeasure_a @ M ) )
= M ) ).
% measure_of_of_measure
thf(fact_605_measure__of__eq,axiom,
! [A3: set_set_a,Omega: set_a,Mu2: set_a > extend8495563244428889912nnreal,Mu: set_a > extend8495563244428889912nnreal] :
( ( ord_le3724670747650509150_set_a @ A3 @ ( pow_a @ Omega ) )
=> ( ! [A: set_a] :
( ( member_set_a @ A @ ( sigma_sigma_sets_a @ Omega @ A3 ) )
=> ( ( Mu2 @ A )
= ( Mu @ A ) ) )
=> ( ( sigma_measure_of_a @ Omega @ A3 @ Mu2 )
= ( sigma_measure_of_a @ Omega @ A3 @ Mu ) ) ) ) ).
% measure_of_eq
thf(fact_606_fmeasurable_Ospace__closed,axiom,
! [M: sigma_measure_a] : ( ord_le3724670747650509150_set_a @ ( measur3645360004775918570able_a @ M ) @ ( pow_a @ ( sigma_space_a @ M ) ) ) ).
% fmeasurable.space_closed
thf(fact_607_measure__space__def,axiom,
( sigma_3179946494550678598pace_a
= ( ^ [Omega2: set_a,A4: set_set_a,Mu5: set_a > extend8495563244428889912nnreal] :
( ( sigma_4968961713055010667ebra_a @ Omega2 @ A4 )
& ( sigma_positive_a @ A4 @ Mu5 )
& ( sigma_498350552340604931tive_a @ A4 @ Mu5 ) ) ) ) ).
% measure_space_def
thf(fact_608_fmeasurable_Osmallest__closed__cdi1,axiom,
! [M: sigma_measure_a] : ( ord_le3724670747650509150_set_a @ ( measur3645360004775918570able_a @ M ) @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( measur3645360004775918570able_a @ M ) ) ) ).
% fmeasurable.smallest_closed_cdi1
thf(fact_609_fmeasurable_Osmallest__closed__cdi2,axiom,
! [M: sigma_measure_a] : ( sigma_closed_cdi_a @ ( sigma_space_a @ M ) @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( measur3645360004775918570able_a @ M ) ) ) ).
% fmeasurable.smallest_closed_cdi2
thf(fact_610_semiring__of__sets_Ocaratheodory,axiom,
! [Omega: set_a,M: set_set_a,Mu2: set_a > extend8495563244428889912nnreal] :
( ( sigma_8461971822185508616sets_a @ Omega @ M )
=> ( ( sigma_positive_a @ M @ Mu2 )
=> ( ( sigma_498350552340604931tive_a @ M @ Mu2 )
=> ? [Mu4: set_a > extend8495563244428889912nnreal] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ M )
=> ( ( Mu4 @ X5 )
= ( Mu2 @ X5 ) ) )
& ( sigma_3179946494550678598pace_a @ Omega @ ( sigma_sigma_sets_a @ Omega @ M ) @ Mu4 ) ) ) ) ) ).
% semiring_of_sets.caratheodory
thf(fact_611_null__sets_Ocaratheodory,axiom,
! [M: sigma_measure_a,Mu2: set_a > extend8495563244428889912nnreal] :
( ( sigma_positive_a @ ( measure_null_sets_a @ M ) @ Mu2 )
=> ( ( sigma_498350552340604931tive_a @ ( measure_null_sets_a @ M ) @ Mu2 )
=> ? [Mu4: set_a > extend8495563244428889912nnreal] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ ( measure_null_sets_a @ M ) )
=> ( ( Mu4 @ X5 )
= ( Mu2 @ X5 ) ) )
& ( sigma_3179946494550678598pace_a @ ( sigma_space_a @ M ) @ ( sigma_sigma_sets_a @ ( sigma_space_a @ M ) @ ( measure_null_sets_a @ M ) ) @ Mu4 ) ) ) ) ).
% null_sets.caratheodory
thf(fact_612_null__sets_Ocaratheodory_H,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_positive_a @ ( measure_null_sets_a @ M ) @ F )
=> ( ( sigma_498350552340604931tive_a @ ( measure_null_sets_a @ M ) @ F )
=> ? [Mu3: set_a > extend8495563244428889912nnreal] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ ( measure_null_sets_a @ M ) )
=> ( ( Mu3 @ X5 )
= ( F @ X5 ) ) )
& ( sigma_3179946494550678598pace_a @ ( sigma_space_a @ M ) @ ( sigma_sigma_sets_a @ ( sigma_space_a @ M ) @ ( measure_null_sets_a @ M ) ) @ Mu3 ) ) ) ) ).
% null_sets.caratheodory'
thf(fact_613_fmeasurable_Ocountably__additiveI__finite,axiom,
! [M: sigma_measure_a,Mu2: set_a > extend8495563244428889912nnreal] :
( ( finite_finite_a @ ( sigma_space_a @ M ) )
=> ( ( sigma_positive_a @ ( measur3645360004775918570able_a @ M ) @ Mu2 )
=> ( ( measur2545585918067192918nnreal @ ( measur3645360004775918570able_a @ M ) @ Mu2 )
=> ( sigma_498350552340604931tive_a @ ( measur3645360004775918570able_a @ M ) @ Mu2 ) ) ) ) ).
% fmeasurable.countably_additiveI_finite
thf(fact_614_sets_Ocountably__additiveI__finite,axiom,
! [M: sigma_measure_a,Mu2: set_a > extend8495563244428889912nnreal] :
( ( finite_finite_a @ ( sigma_space_a @ M ) )
=> ( ( sigma_positive_a @ ( sigma_sets_a @ M ) @ Mu2 )
=> ( ( measur2545585918067192918nnreal @ ( sigma_sets_a @ M ) @ Mu2 )
=> ( sigma_498350552340604931tive_a @ ( sigma_sets_a @ M ) @ Mu2 ) ) ) ) ).
% sets.countably_additiveI_finite
thf(fact_615_ring__of__sets_Ocaratheodory_H,axiom,
! [Omega: set_a,M: set_set_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_ring_of_sets_a @ Omega @ M )
=> ( ( sigma_positive_a @ M @ F )
=> ( ( sigma_498350552340604931tive_a @ M @ F )
=> ? [Mu3: set_a > extend8495563244428889912nnreal] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ M )
=> ( ( Mu3 @ X5 )
= ( F @ X5 ) ) )
& ( sigma_3179946494550678598pace_a @ Omega @ ( sigma_sigma_sets_a @ Omega @ M ) @ Mu3 ) ) ) ) ) ).
% ring_of_sets.caratheodory'
thf(fact_616_null__sets_Oring__of__sets__axioms,axiom,
! [M: sigma_measure_a] : ( sigma_ring_of_sets_a @ ( sigma_space_a @ M ) @ ( measure_null_sets_a @ M ) ) ).
% null_sets.ring_of_sets_axioms
thf(fact_617_null__sets_Osemiring__of__sets__axioms,axiom,
! [M: sigma_measure_a] : ( sigma_8461971822185508616sets_a @ ( sigma_space_a @ M ) @ ( measure_null_sets_a @ M ) ) ).
% null_sets.semiring_of_sets_axioms
thf(fact_618_null__sets_Ocountably__additive__additive,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_positive_a @ ( measure_null_sets_a @ M ) @ F )
=> ( ( sigma_498350552340604931tive_a @ ( measure_null_sets_a @ M ) @ F )
=> ( measur2545585918067192918nnreal @ ( measure_null_sets_a @ M ) @ F ) ) ) ).
% null_sets.countably_additive_additive
thf(fact_619_null__setsD2,axiom,
! [A3: set_a,M: sigma_measure_a] :
( ( member_set_a @ A3 @ ( measure_null_sets_a @ M ) )
=> ( member_set_a @ A3 @ ( sigma_sets_a @ M ) ) ) ).
% null_setsD2
thf(fact_620_ring__of__sets_Ocountably__additive__additive,axiom,
! [Omega: set_a,M: set_set_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_ring_of_sets_a @ Omega @ M )
=> ( ( sigma_positive_a @ M @ F )
=> ( ( sigma_498350552340604931tive_a @ M @ F )
=> ( measur2545585918067192918nnreal @ M @ F ) ) ) ) ).
% ring_of_sets.countably_additive_additive
thf(fact_621_fmeasurableI__null__sets,axiom,
! [A3: set_a,M: sigma_measure_a] :
( ( member_set_a @ A3 @ ( measure_null_sets_a @ M ) )
=> ( member_set_a @ A3 @ ( measur3645360004775918570able_a @ M ) ) ) ).
% fmeasurableI_null_sets
thf(fact_622_null__sets_Opositive__cong__eq,axiom,
! [M: sigma_measure_a,Mu: set_a > extend8495563244428889912nnreal,Mu2: set_a > extend8495563244428889912nnreal] :
( ! [A: set_a] :
( ( member_set_a @ A @ ( measure_null_sets_a @ M ) )
=> ( ( Mu @ A )
= ( Mu2 @ A ) ) )
=> ( ( sigma_positive_a @ ( measure_null_sets_a @ M ) @ Mu )
= ( sigma_positive_a @ ( measure_null_sets_a @ M ) @ Mu2 ) ) ) ).
% null_sets.positive_cong_eq
thf(fact_623_ring__of__sets_Opositive__cong__eq,axiom,
! [Omega: set_a,M: set_set_a,Mu: set_a > extend8495563244428889912nnreal,Mu2: set_a > extend8495563244428889912nnreal] :
( ( sigma_ring_of_sets_a @ Omega @ M )
=> ( ! [A: set_a] :
( ( member_set_a @ A @ M )
=> ( ( Mu @ A )
= ( Mu2 @ A ) ) )
=> ( ( sigma_positive_a @ M @ Mu )
= ( sigma_positive_a @ M @ Mu2 ) ) ) ) ).
% ring_of_sets.positive_cong_eq
thf(fact_624_algebra_Oaxioms_I1_J,axiom,
! [Omega: set_a,M: set_set_a] :
( ( sigma_algebra_a @ Omega @ M )
=> ( sigma_ring_of_sets_a @ Omega @ M ) ) ).
% algebra.axioms(1)
thf(fact_625_ring__of__sets_Ocountably__additiveI__finite,axiom,
! [Omega: set_a,M: set_set_a,Mu2: set_a > extend8495563244428889912nnreal] :
( ( sigma_ring_of_sets_a @ Omega @ M )
=> ( ( finite_finite_a @ Omega )
=> ( ( sigma_positive_a @ M @ Mu2 )
=> ( ( measur2545585918067192918nnreal @ M @ Mu2 )
=> ( sigma_498350552340604931tive_a @ M @ Mu2 ) ) ) ) ) ).
% ring_of_sets.countably_additiveI_finite
thf(fact_626_null__sets_Osets__into__space,axiom,
! [X2: set_a,M: sigma_measure_a] :
( ( member_set_a @ X2 @ ( measure_null_sets_a @ M ) )
=> ( ord_less_eq_set_a @ X2 @ ( sigma_space_a @ M ) ) ) ).
% null_sets.sets_into_space
thf(fact_627_null__sets_Ocountably__additiveI__finite,axiom,
! [M: sigma_measure_a,Mu2: set_a > extend8495563244428889912nnreal] :
( ( finite_finite_a @ ( sigma_space_a @ M ) )
=> ( ( sigma_positive_a @ ( measure_null_sets_a @ M ) @ Mu2 )
=> ( ( measur2545585918067192918nnreal @ ( measure_null_sets_a @ M ) @ Mu2 )
=> ( sigma_498350552340604931tive_a @ ( measure_null_sets_a @ M ) @ Mu2 ) ) ) ) ).
% null_sets.countably_additiveI_finite
thf(fact_628_sets_Oring__of__sets__axioms,axiom,
! [M: sigma_measure_a] : ( sigma_ring_of_sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) ).
% sets.ring_of_sets_axioms
thf(fact_629_fmeasurable_Oring__of__sets__axioms,axiom,
! [M: sigma_measure_a] : ( sigma_ring_of_sets_a @ ( sigma_space_a @ M ) @ ( measur3645360004775918570able_a @ M ) ) ).
% fmeasurable.ring_of_sets_axioms
thf(fact_630_null__sets_Osubset__class__axioms,axiom,
! [M: sigma_measure_a] : ( sigma_subset_class_a @ ( sigma_space_a @ M ) @ ( measure_null_sets_a @ M ) ) ).
% null_sets.subset_class_axioms
thf(fact_631_sets_Osemiring__of__sets__axioms,axiom,
! [M: sigma_measure_a] : ( sigma_8461971822185508616sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) ).
% sets.semiring_of_sets_axioms
thf(fact_632_fmeasurable_Osemiring__of__sets__axioms,axiom,
! [M: sigma_measure_a] : ( sigma_8461971822185508616sets_a @ ( sigma_space_a @ M ) @ ( measur3645360004775918570able_a @ M ) ) ).
% fmeasurable.semiring_of_sets_axioms
thf(fact_633_null__sets_Ospace__closed,axiom,
! [M: sigma_measure_a] : ( ord_le3724670747650509150_set_a @ ( measure_null_sets_a @ M ) @ ( pow_a @ ( sigma_space_a @ M ) ) ) ).
% null_sets.space_closed
thf(fact_634_algebra__def,axiom,
( sigma_algebra_a
= ( ^ [Omega2: set_a,M2: set_set_a] :
( ( sigma_ring_of_sets_a @ Omega2 @ M2 )
& ( sigma_1540858801969833537ioms_a @ Omega2 @ M2 ) ) ) ) ).
% algebra_def
thf(fact_635_algebra_Ointro,axiom,
! [Omega: set_a,M: set_set_a] :
( ( sigma_ring_of_sets_a @ Omega @ M )
=> ( ( sigma_1540858801969833537ioms_a @ Omega @ M )
=> ( sigma_algebra_a @ Omega @ M ) ) ) ).
% algebra.intro
thf(fact_636_null__sets_Osmallest__closed__cdi1,axiom,
! [M: sigma_measure_a] : ( ord_le3724670747650509150_set_a @ ( measure_null_sets_a @ M ) @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( measure_null_sets_a @ M ) ) ) ).
% null_sets.smallest_closed_cdi1
thf(fact_637_sets_Ocountably__additive__additive,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_positive_a @ ( sigma_sets_a @ M ) @ F )
=> ( ( sigma_498350552340604931tive_a @ ( sigma_sets_a @ M ) @ F )
=> ( measur2545585918067192918nnreal @ ( sigma_sets_a @ M ) @ F ) ) ) ).
% sets.countably_additive_additive
thf(fact_638_fmeasurable_Ocountably__additive__additive,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_positive_a @ ( measur3645360004775918570able_a @ M ) @ F )
=> ( ( sigma_498350552340604931tive_a @ ( measur3645360004775918570able_a @ M ) @ F )
=> ( measur2545585918067192918nnreal @ ( measur3645360004775918570able_a @ M ) @ F ) ) ) ).
% fmeasurable.countably_additive_additive
thf(fact_639_null__sets_Osmallest__closed__cdi2,axiom,
! [M: sigma_measure_a] : ( sigma_closed_cdi_a @ ( sigma_space_a @ M ) @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( measure_null_sets_a @ M ) ) ) ).
% null_sets.smallest_closed_cdi2
thf(fact_640_null__sets_Osmallest__ccdi__sets,axiom,
! [M: sigma_measure_a] : ( ord_le3724670747650509150_set_a @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( measure_null_sets_a @ M ) ) @ ( pow_a @ ( sigma_space_a @ M ) ) ) ).
% null_sets.smallest_ccdi_sets
thf(fact_641_null__sets__subset,axiom,
! [B: set_a,M: sigma_measure_a,A3: set_a] :
( ( member_set_a @ B @ ( measure_null_sets_a @ M ) )
=> ( ( member_set_a @ A3 @ ( sigma_sets_a @ M ) )
=> ( ( ord_less_eq_set_a @ A3 @ B )
=> ( member_set_a @ A3 @ ( measure_null_sets_a @ M ) ) ) ) ) ).
% null_sets_subset
thf(fact_642_emeasure__measure__of__conv,axiom,
! [Omega: set_a,A3: set_set_a,Mu2: set_a > extend8495563244428889912nnreal] :
( ( sigma_emeasure_a @ ( sigma_measure_of_a @ Omega @ A3 @ Mu2 ) )
= ( ^ [B4: set_a] :
( if_Ext9135588136721118450nnreal
@ ( ( member_set_a @ B4 @ ( sigma_sigma_sets_a @ Omega @ A3 ) )
& ( sigma_3179946494550678598pace_a @ Omega @ ( sigma_sigma_sets_a @ Omega @ A3 ) @ Mu2 ) )
@ ( Mu2 @ B4 )
@ zero_z7100319975126383169nnreal ) ) ) ).
% emeasure_measure_of_conv
thf(fact_643_null__setsI,axiom,
! [M: sigma_measure_a,A3: set_a] :
( ( ( sigma_emeasure_a @ M @ A3 )
= zero_z7100319975126383169nnreal )
=> ( ( member_set_a @ A3 @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ A3 @ ( measure_null_sets_a @ M ) ) ) ) ).
% null_setsI
thf(fact_644_emeasure__neq__0__sets,axiom,
! [M: sigma_measure_a,A3: set_a] :
( ( ( sigma_emeasure_a @ M @ A3 )
!= zero_z7100319975126383169nnreal )
=> ( member_set_a @ A3 @ ( sigma_sets_a @ M ) ) ) ).
% emeasure_neq_0_sets
thf(fact_645_emeasure__notin__sets,axiom,
! [A3: set_a,M: sigma_measure_a] :
( ~ ( member_set_a @ A3 @ ( sigma_sets_a @ M ) )
=> ( ( sigma_emeasure_a @ M @ A3 )
= zero_z7100319975126383169nnreal ) ) ).
% emeasure_notin_sets
thf(fact_646_null__setsD1,axiom,
! [A3: set_a,M: sigma_measure_a] :
( ( member_set_a @ A3 @ ( measure_null_sets_a @ M ) )
=> ( ( sigma_emeasure_a @ M @ A3 )
= zero_z7100319975126383169nnreal ) ) ).
% null_setsD1
thf(fact_647_emeasure__eq__0,axiom,
! [N: set_a,M: sigma_measure_a,K3: set_a] :
( ( member_set_a @ N @ ( sigma_sets_a @ M ) )
=> ( ( ( sigma_emeasure_a @ M @ N )
= zero_z7100319975126383169nnreal )
=> ( ( ord_less_eq_set_a @ K3 @ N )
=> ( ( sigma_emeasure_a @ M @ K3 )
= zero_z7100319975126383169nnreal ) ) ) ) ).
% emeasure_eq_0
thf(fact_648_not__gr__zero,axiom,
! [N4: extend8495563244428889912nnreal] :
( ( ~ ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ N4 ) )
= ( N4 = zero_z7100319975126383169nnreal ) ) ).
% not_gr_zero
thf(fact_649_le__zero__eq,axiom,
! [N4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ N4 @ zero_z7100319975126383169nnreal )
= ( N4 = zero_z7100319975126383169nnreal ) ) ).
% le_zero_eq
thf(fact_650_null__part__sets_I2_J,axiom,
! [S: set_a,M: sigma_measure_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ M ) )
=> ( ( sigma_emeasure_a @ M @ ( complete_null_part_a @ M @ S ) )
= zero_z7100319975126383169nnreal ) ) ).
% null_part_sets(2)
thf(fact_651_zero__less__iff__neq__zero,axiom,
! [N4: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ N4 )
= ( N4 != zero_z7100319975126383169nnreal ) ) ).
% zero_less_iff_neq_zero
thf(fact_652_null__part__sets_I1_J,axiom,
! [S: set_a,M: sigma_measure_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( complete_null_part_a @ M @ S ) @ ( sigma_sets_a @ M ) ) ) ).
% null_part_sets(1)
thf(fact_653_zero__le,axiom,
! [X2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ X2 ) ).
% zero_le
thf(fact_654_gr__zeroI,axiom,
! [N4: extend8495563244428889912nnreal] :
( ( N4 != zero_z7100319975126383169nnreal )
=> ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ N4 ) ) ).
% gr_zeroI
thf(fact_655_not__less__zero,axiom,
! [N4: extend8495563244428889912nnreal] :
~ ( ord_le7381754540660121996nnreal @ N4 @ zero_z7100319975126383169nnreal ) ).
% not_less_zero
thf(fact_656_gr__implies__not__zero,axiom,
! [M5: extend8495563244428889912nnreal,N4: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ M5 @ N4 )
=> ( N4 != zero_z7100319975126383169nnreal ) ) ).
% gr_implies_not_zero
thf(fact_657_less__numeral__extra_I3_J,axiom,
~ ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ zero_z7100319975126383169nnreal ) ).
% less_numeral_extra(3)
thf(fact_658_le__numeral__extra_I3_J,axiom,
ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ zero_z7100319975126383169nnreal ).
% le_numeral_extra(3)
thf(fact_659_outer__measure__of__attain,axiom,
! [A3: set_a,M: sigma_measure_a] :
( ( ord_less_eq_set_a @ A3 @ ( sigma_space_a @ M ) )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ ( sigma_sets_a @ M ) )
& ( ord_less_eq_set_a @ A3 @ X3 )
& ( ( comple3326625653960060650e_of_a @ M @ A3 )
= ( sigma_emeasure_a @ M @ X3 ) ) ) ) ).
% outer_measure_of_attain
thf(fact_660_null__sets_Oincreasing__outer__measure,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] : ( measur5393715408109795267nnreal @ ( pow_a @ ( sigma_space_a @ M ) ) @ ( outer_measure_a @ ( measure_null_sets_a @ M ) @ F ) ) ).
% null_sets.increasing_outer_measure
thf(fact_661_fmeasurable_Oincreasing__outer__measure,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] : ( measur5393715408109795267nnreal @ ( pow_a @ ( sigma_space_a @ M ) ) @ ( outer_measure_a @ ( measur3645360004775918570able_a @ M ) @ F ) ) ).
% fmeasurable.increasing_outer_measure
thf(fact_662_outer__measure__of__eq,axiom,
! [A3: set_a,M: sigma_measure_a] :
( ( member_set_a @ A3 @ ( sigma_sets_a @ M ) )
=> ( ( comple3326625653960060650e_of_a @ M @ A3 )
= ( sigma_emeasure_a @ M @ A3 ) ) ) ).
% outer_measure_of_eq
thf(fact_663_sets_Oouter__measure__agrees,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal,S2: set_a] :
( ( sigma_positive_a @ ( sigma_sets_a @ M ) @ F )
=> ( ( sigma_498350552340604931tive_a @ ( sigma_sets_a @ M ) @ F )
=> ( ( member_set_a @ S2 @ ( sigma_sets_a @ M ) )
=> ( ( outer_measure_a @ ( sigma_sets_a @ M ) @ F @ S2 )
= ( F @ S2 ) ) ) ) ) ).
% sets.outer_measure_agrees
thf(fact_664_fmeasurable_Oouter__measure__agrees,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal,S2: set_a] :
( ( sigma_positive_a @ ( measur3645360004775918570able_a @ M ) @ F )
=> ( ( sigma_498350552340604931tive_a @ ( measur3645360004775918570able_a @ M ) @ F )
=> ( ( member_set_a @ S2 @ ( measur3645360004775918570able_a @ M ) )
=> ( ( outer_measure_a @ ( measur3645360004775918570able_a @ M ) @ F @ S2 )
= ( F @ S2 ) ) ) ) ) ).
% fmeasurable.outer_measure_agrees
thf(fact_665_null__sets_Oouter__measure__agrees,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal,S2: set_a] :
( ( sigma_positive_a @ ( measure_null_sets_a @ M ) @ F )
=> ( ( sigma_498350552340604931tive_a @ ( measure_null_sets_a @ M ) @ F )
=> ( ( member_set_a @ S2 @ ( measure_null_sets_a @ M ) )
=> ( ( outer_measure_a @ ( measure_null_sets_a @ M ) @ F @ S2 )
= ( F @ S2 ) ) ) ) ) ).
% null_sets.outer_measure_agrees
thf(fact_666_ring__of__sets_Oouter__measure__agrees,axiom,
! [Omega: set_a,M: set_set_a,F: set_a > extend8495563244428889912nnreal,S2: set_a] :
( ( sigma_ring_of_sets_a @ Omega @ M )
=> ( ( sigma_positive_a @ M @ F )
=> ( ( sigma_498350552340604931tive_a @ M @ F )
=> ( ( member_set_a @ S2 @ M )
=> ( ( outer_measure_a @ M @ F @ S2 )
= ( F @ S2 ) ) ) ) ) ) ).
% ring_of_sets.outer_measure_agrees
thf(fact_667_sets_Opositive__outer__measure,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_positive_a @ ( sigma_sets_a @ M ) @ F )
=> ( sigma_positive_a @ ( pow_a @ ( sigma_space_a @ M ) ) @ ( outer_measure_a @ ( sigma_sets_a @ M ) @ F ) ) ) ).
% sets.positive_outer_measure
thf(fact_668_fmeasurable_Opositive__outer__measure,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_positive_a @ ( measur3645360004775918570able_a @ M ) @ F )
=> ( sigma_positive_a @ ( pow_a @ ( sigma_space_a @ M ) ) @ ( outer_measure_a @ ( measur3645360004775918570able_a @ M ) @ F ) ) ) ).
% fmeasurable.positive_outer_measure
thf(fact_669_null__sets_Opositive__outer__measure,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_positive_a @ ( measure_null_sets_a @ M ) @ F )
=> ( sigma_positive_a @ ( pow_a @ ( sigma_space_a @ M ) ) @ ( outer_measure_a @ ( measure_null_sets_a @ M ) @ F ) ) ) ).
% null_sets.positive_outer_measure
thf(fact_670_sets_Oincreasing__outer__measure,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] : ( measur5393715408109795267nnreal @ ( pow_a @ ( sigma_space_a @ M ) ) @ ( outer_measure_a @ ( sigma_sets_a @ M ) @ F ) ) ).
% sets.increasing_outer_measure
thf(fact_671_null__sets_Oouter__measure__space__outer__measure,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_positive_a @ ( measure_null_sets_a @ M ) @ F )
=> ( ( measur5393715408109795267nnreal @ ( measure_null_sets_a @ M ) @ F )
=> ( outer_7793981174483491pace_a @ ( pow_a @ ( sigma_space_a @ M ) ) @ ( outer_measure_a @ ( measure_null_sets_a @ M ) @ F ) ) ) ) ).
% null_sets.outer_measure_space_outer_measure
thf(fact_672_fmeasurable_Oouter__measure__space__outer__measure,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_positive_a @ ( measur3645360004775918570able_a @ M ) @ F )
=> ( ( measur5393715408109795267nnreal @ ( measur3645360004775918570able_a @ M ) @ F )
=> ( outer_7793981174483491pace_a @ ( pow_a @ ( sigma_space_a @ M ) ) @ ( outer_measure_a @ ( measur3645360004775918570able_a @ M ) @ F ) ) ) ) ).
% fmeasurable.outer_measure_space_outer_measure
thf(fact_673_sets_Oouter__measure__space__outer__measure,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_positive_a @ ( sigma_sets_a @ M ) @ F )
=> ( ( measur5393715408109795267nnreal @ ( sigma_sets_a @ M ) @ F )
=> ( outer_7793981174483491pace_a @ ( pow_a @ ( sigma_space_a @ M ) ) @ ( outer_measure_a @ ( sigma_sets_a @ M ) @ F ) ) ) ) ).
% sets.outer_measure_space_outer_measure
thf(fact_674_null__sets_Ocountably__subadditive__outer__measure,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_positive_a @ ( measure_null_sets_a @ M ) @ F )
=> ( ( measur5393715408109795267nnreal @ ( measure_null_sets_a @ M ) @ F )
=> ( measur1003548731350908915nnreal @ ( pow_a @ ( sigma_space_a @ M ) ) @ ( outer_measure_a @ ( measure_null_sets_a @ M ) @ F ) ) ) ) ).
% null_sets.countably_subadditive_outer_measure
thf(fact_675_fmeasurable_Ocountably__subadditive__outer__measure,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_positive_a @ ( measur3645360004775918570able_a @ M ) @ F )
=> ( ( measur5393715408109795267nnreal @ ( measur3645360004775918570able_a @ M ) @ F )
=> ( measur1003548731350908915nnreal @ ( pow_a @ ( sigma_space_a @ M ) ) @ ( outer_measure_a @ ( measur3645360004775918570able_a @ M ) @ F ) ) ) ) ).
% fmeasurable.countably_subadditive_outer_measure
thf(fact_676_sets_Ocountably__subadditive__outer__measure,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_positive_a @ ( sigma_sets_a @ M ) @ F )
=> ( ( measur5393715408109795267nnreal @ ( sigma_sets_a @ M ) @ F )
=> ( measur1003548731350908915nnreal @ ( pow_a @ ( sigma_space_a @ M ) ) @ ( outer_measure_a @ ( sigma_sets_a @ M ) @ F ) ) ) ) ).
% sets.countably_subadditive_outer_measure
thf(fact_677_null__sets_Oalgebra__subset__lambda__system,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_positive_a @ ( measure_null_sets_a @ M ) @ F )
=> ( ( measur5393715408109795267nnreal @ ( measure_null_sets_a @ M ) @ F )
=> ( ( measur2545585918067192918nnreal @ ( measure_null_sets_a @ M ) @ F )
=> ( ord_le3724670747650509150_set_a @ ( measure_null_sets_a @ M ) @ ( lambda_system_a @ ( sigma_space_a @ M ) @ ( pow_a @ ( sigma_space_a @ M ) ) @ ( outer_measure_a @ ( measure_null_sets_a @ M ) @ F ) ) ) ) ) ) ).
% null_sets.algebra_subset_lambda_system
thf(fact_678_fmeasurable_Oalgebra__subset__lambda__system,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_positive_a @ ( measur3645360004775918570able_a @ M ) @ F )
=> ( ( measur5393715408109795267nnreal @ ( measur3645360004775918570able_a @ M ) @ F )
=> ( ( measur2545585918067192918nnreal @ ( measur3645360004775918570able_a @ M ) @ F )
=> ( ord_le3724670747650509150_set_a @ ( measur3645360004775918570able_a @ M ) @ ( lambda_system_a @ ( sigma_space_a @ M ) @ ( pow_a @ ( sigma_space_a @ M ) ) @ ( outer_measure_a @ ( measur3645360004775918570able_a @ M ) @ F ) ) ) ) ) ) ).
% fmeasurable.algebra_subset_lambda_system
thf(fact_679_sets_Oalgebra__subset__lambda__system,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_positive_a @ ( sigma_sets_a @ M ) @ F )
=> ( ( measur5393715408109795267nnreal @ ( sigma_sets_a @ M ) @ F )
=> ( ( measur2545585918067192918nnreal @ ( sigma_sets_a @ M ) @ F )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ M ) @ ( lambda_system_a @ ( sigma_space_a @ M ) @ ( pow_a @ ( sigma_space_a @ M ) ) @ ( outer_measure_a @ ( sigma_sets_a @ M ) @ F ) ) ) ) ) ) ).
% sets.algebra_subset_lambda_system
thf(fact_680_lambda__system__sets,axiom,
! [X2: set_a,Omega: set_a,M: set_set_a,F: set_a > extend8495563244428889912nnreal] :
( ( member_set_a @ X2 @ ( lambda_system_a @ Omega @ M @ F ) )
=> ( member_set_a @ X2 @ M ) ) ).
% lambda_system_sets
thf(fact_681_algebra_Olambda__system__algebra,axiom,
! [Omega: set_a,M: set_set_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_algebra_a @ Omega @ M )
=> ( ( sigma_positive_a @ M @ F )
=> ( sigma_algebra_a @ Omega @ ( lambda_system_a @ Omega @ M @ F ) ) ) ) ).
% algebra.lambda_system_algebra
thf(fact_682_algebra_Olambda__system__additive,axiom,
! [Omega: set_a,M: set_set_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_algebra_a @ Omega @ M )
=> ( measur2545585918067192918nnreal @ ( lambda_system_a @ Omega @ M @ F ) @ F ) ) ).
% algebra.lambda_system_additive
thf(fact_683_sets_Olambda__system__additive,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] : ( measur2545585918067192918nnreal @ ( lambda_system_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) @ F ) @ F ) ).
% sets.lambda_system_additive
thf(fact_684_sigma__algebra_Ocaratheodory__lemma,axiom,
! [Omega: set_a,M: set_set_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_4968961713055010667ebra_a @ Omega @ M )
=> ( ( outer_7793981174483491pace_a @ M @ F )
=> ( sigma_3179946494550678598pace_a @ Omega @ ( lambda_system_a @ Omega @ M @ F ) @ F ) ) ) ).
% sigma_algebra.caratheodory_lemma
thf(fact_685_sets_Olambda__system__algebra,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_positive_a @ ( sigma_sets_a @ M ) @ F )
=> ( sigma_algebra_a @ ( sigma_space_a @ M ) @ ( lambda_system_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) @ F ) ) ) ).
% sets.lambda_system_algebra
thf(fact_686_sets_Ocaratheodory__lemma,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] :
( ( outer_7793981174483491pace_a @ ( sigma_sets_a @ M ) @ F )
=> ( sigma_3179946494550678598pace_a @ ( sigma_space_a @ M ) @ ( lambda_system_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) @ F ) @ F ) ) ).
% sets.caratheodory_lemma
thf(fact_687_sets_Olambda__system__empty,axiom,
! [M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_positive_a @ ( sigma_sets_a @ M ) @ F )
=> ( member_set_a @ bot_bot_set_a @ ( lambda_system_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) @ F ) ) ) ).
% sets.lambda_system_empty
thf(fact_688_null__part,axiom,
! [S: set_a,M: sigma_measure_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ? [N5: set_a] :
( ( member_set_a @ N5 @ ( measure_null_sets_a @ M ) )
& ( ord_less_eq_set_a @ ( complete_null_part_a @ M @ S ) @ N5 ) ) ) ).
% null_part
thf(fact_689_complete__measure__def,axiom,
( comple8155536527497655953sure_a
= ( ^ [M2: sigma_measure_a] :
! [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
=> ( ( member_set_a @ A4 @ ( measure_null_sets_a @ M2 ) )
=> ( member_set_a @ B4 @ ( sigma_sets_a @ M2 ) ) ) ) ) ) ).
% complete_measure_def
thf(fact_690_complete__measure_Ocomplete,axiom,
! [M: sigma_measure_a,B: set_a,A3: set_a] :
( ( comple8155536527497655953sure_a @ M )
=> ( ( ord_less_eq_set_a @ B @ A3 )
=> ( ( member_set_a @ A3 @ ( measure_null_sets_a @ M ) )
=> ( member_set_a @ B @ ( sigma_sets_a @ M ) ) ) ) ) ).
% complete_measure.complete
thf(fact_691_complete__measure_Ointro,axiom,
! [M: sigma_measure_a] :
( ! [A8: set_a,B7: set_a] :
( ( ord_less_eq_set_a @ B7 @ A8 )
=> ( ( member_set_a @ A8 @ ( measure_null_sets_a @ M ) )
=> ( member_set_a @ B7 @ ( sigma_sets_a @ M ) ) ) )
=> ( comple8155536527497655953sure_a @ M ) ) ).
% complete_measure.intro
thf(fact_692_empty__iff,axiom,
! [C2: a > a] :
~ ( member_a_a @ C2 @ bot_bot_set_a_a ) ).
% empty_iff
thf(fact_693_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_694_empty__iff,axiom,
! [C2: set_a] :
~ ( member_set_a @ C2 @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_695_empty__iff,axiom,
! [C2: real > a] :
~ ( member_real_a @ C2 @ bot_bot_set_real_a ) ).
% empty_iff
thf(fact_696_all__not__in__conv,axiom,
! [A3: set_a_a] :
( ( ! [X4: a > a] :
~ ( member_a_a @ X4 @ A3 ) )
= ( A3 = bot_bot_set_a_a ) ) ).
% all_not_in_conv
thf(fact_697_all__not__in__conv,axiom,
! [A3: set_a] :
( ( ! [X4: a] :
~ ( member_a @ X4 @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_698_all__not__in__conv,axiom,
! [A3: set_set_a] :
( ( ! [X4: set_a] :
~ ( member_set_a @ X4 @ A3 ) )
= ( A3 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_699_all__not__in__conv,axiom,
! [A3: set_real_a] :
( ( ! [X4: real > a] :
~ ( member_real_a @ X4 @ A3 ) )
= ( A3 = bot_bot_set_real_a ) ) ).
% all_not_in_conv
thf(fact_700_sets_Oempty__sets,axiom,
! [M: sigma_measure_a] : ( member_set_a @ bot_bot_set_a @ ( sigma_sets_a @ M ) ) ).
% sets.empty_sets
thf(fact_701_sets__completionI__sets,axiom,
! [A3: set_a,M: sigma_measure_a] :
( ( member_set_a @ A3 @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ A3 @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) ) ) ).
% sets_completionI_sets
thf(fact_702_space__completion,axiom,
! [M: sigma_measure_a] :
( ( sigma_space_a @ ( comple3428971583294703880tion_a @ M ) )
= ( sigma_space_a @ M ) ) ).
% space_completion
thf(fact_703_fmeasurable_Oempty__sets,axiom,
! [M: sigma_measure_a] : ( member_set_a @ bot_bot_set_a @ ( measur3645360004775918570able_a @ M ) ) ).
% fmeasurable.empty_sets
thf(fact_704_null__sets_Oempty__sets,axiom,
! [M: sigma_measure_a] : ( member_set_a @ bot_bot_set_a @ ( measure_null_sets_a @ M ) ) ).
% null_sets.empty_sets
thf(fact_705_qbs__empty__equiv,axiom,
! [X: quasi_borel_a] :
( ( ( qbs_space_a @ X )
= bot_bot_set_a )
= ( ( qbs_Mx_a @ X )
= bot_bot_set_real_a ) ) ).
% qbs_empty_equiv
thf(fact_706_measurable__completion,axiom,
! [F: a > a,M: sigma_measure_a,N: sigma_measure_a] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ N ) )
=> ( member_a_a @ F @ ( sigma_measurable_a_a @ ( comple3428971583294703880tion_a @ M ) @ N ) ) ) ).
% measurable_completion
thf(fact_707_measurable__completion,axiom,
! [F: real > a,M: sigma_measure_real,N: sigma_measure_a] :
( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ N ) )
=> ( member_real_a @ F @ ( sigma_523072396149930112real_a @ ( comple3506806835435775778n_real @ M ) @ N ) ) ) ).
% measurable_completion
thf(fact_708_bot_Oextremum,axiom,
! [A2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ bot_bo841427958541957580nnreal @ A2 ) ).
% bot.extremum
thf(fact_709_bot_Oextremum__unique,axiom,
! [A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ bot_bo841427958541957580nnreal )
= ( A2 = bot_bo841427958541957580nnreal ) ) ).
% bot.extremum_unique
thf(fact_710_bot_Oextremum__uniqueI,axiom,
! [A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ bot_bo841427958541957580nnreal )
=> ( A2 = bot_bo841427958541957580nnreal ) ) ).
% bot.extremum_uniqueI
thf(fact_711_bot_Onot__eq__extremum,axiom,
! [A2: extend8495563244428889912nnreal] :
( ( A2 != bot_bo841427958541957580nnreal )
= ( ord_le7381754540660121996nnreal @ bot_bo841427958541957580nnreal @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_712_bot_Oextremum__strict,axiom,
! [A2: extend8495563244428889912nnreal] :
~ ( ord_le7381754540660121996nnreal @ A2 @ bot_bo841427958541957580nnreal ) ).
% bot.extremum_strict
thf(fact_713_emptyE,axiom,
! [A2: a > a] :
~ ( member_a_a @ A2 @ bot_bot_set_a_a ) ).
% emptyE
thf(fact_714_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_715_emptyE,axiom,
! [A2: set_a] :
~ ( member_set_a @ A2 @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_716_emptyE,axiom,
! [A2: real > a] :
~ ( member_real_a @ A2 @ bot_bot_set_real_a ) ).
% emptyE
thf(fact_717_equals0D,axiom,
! [A3: set_a_a,A2: a > a] :
( ( A3 = bot_bot_set_a_a )
=> ~ ( member_a_a @ A2 @ A3 ) ) ).
% equals0D
thf(fact_718_equals0D,axiom,
! [A3: set_a,A2: a] :
( ( A3 = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A3 ) ) ).
% equals0D
thf(fact_719_equals0D,axiom,
! [A3: set_set_a,A2: set_a] :
( ( A3 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A2 @ A3 ) ) ).
% equals0D
thf(fact_720_equals0D,axiom,
! [A3: set_real_a,A2: real > a] :
( ( A3 = bot_bot_set_real_a )
=> ~ ( member_real_a @ A2 @ A3 ) ) ).
% equals0D
thf(fact_721_equals0I,axiom,
! [A3: set_a_a] :
( ! [Y5: a > a] :
~ ( member_a_a @ Y5 @ A3 )
=> ( A3 = bot_bot_set_a_a ) ) ).
% equals0I
thf(fact_722_equals0I,axiom,
! [A3: set_a] :
( ! [Y5: a] :
~ ( member_a @ Y5 @ A3 )
=> ( A3 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_723_equals0I,axiom,
! [A3: set_set_a] :
( ! [Y5: set_a] :
~ ( member_set_a @ Y5 @ A3 )
=> ( A3 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_724_equals0I,axiom,
! [A3: set_real_a] :
( ! [Y5: real > a] :
~ ( member_real_a @ Y5 @ A3 )
=> ( A3 = bot_bot_set_real_a ) ) ).
% equals0I
thf(fact_725_ex__in__conv,axiom,
! [A3: set_a_a] :
( ( ? [X4: a > a] : ( member_a_a @ X4 @ A3 ) )
= ( A3 != bot_bot_set_a_a ) ) ).
% ex_in_conv
thf(fact_726_ex__in__conv,axiom,
! [A3: set_a] :
( ( ? [X4: a] : ( member_a @ X4 @ A3 ) )
= ( A3 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_727_ex__in__conv,axiom,
! [A3: set_set_a] :
( ( ? [X4: set_a] : ( member_set_a @ X4 @ A3 ) )
= ( A3 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_728_ex__in__conv,axiom,
! [A3: set_real_a] :
( ( ? [X4: real > a] : ( member_real_a @ X4 @ A3 ) )
= ( A3 != bot_bot_set_real_a ) ) ).
% ex_in_conv
thf(fact_729_Pow__bottom,axiom,
! [B: set_a] : ( member_set_a @ bot_bot_set_a @ ( pow_a @ B ) ) ).
% Pow_bottom
thf(fact_730_sigma__sets_OEmpty,axiom,
! [Sp: set_a,A3: set_set_a] : ( member_set_a @ bot_bot_set_a @ ( sigma_sigma_sets_a @ Sp @ A3 ) ) ).
% sigma_sets.Empty
thf(fact_731_semiring__of__sets_Oempty__sets,axiom,
! [Omega: set_a,M: set_set_a] :
( ( sigma_8461971822185508616sets_a @ Omega @ M )
=> ( member_set_a @ bot_bot_set_a @ M ) ) ).
% semiring_of_sets.empty_sets
thf(fact_732_Dynkin__system_Oempty,axiom,
! [Omega: set_a,M: set_set_a] :
( ( sigma_2757993544071651912stem_a @ Omega @ M )
=> ( member_set_a @ bot_bot_set_a @ M ) ) ).
% Dynkin_system.empty
thf(fact_733_empty__in__Fpow,axiom,
! [A3: set_a] : ( member_set_a @ bot_bot_set_a @ ( finite_Fpow_a @ A3 ) ) ).
% empty_in_Fpow
thf(fact_734_null__sets__completion__subset,axiom,
! [B: set_a,A3: set_a,M: sigma_measure_a] :
( ( ord_less_eq_set_a @ B @ A3 )
=> ( ( member_set_a @ A3 @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ( member_set_a @ B @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M ) ) ) ) ) ).
% null_sets_completion_subset
thf(fact_735_null__sets__completion__iff2,axiom,
! [A3: set_a,M: sigma_measure_a] :
( ( member_set_a @ A3 @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
= ( ? [X4: set_a] :
( ( member_set_a @ X4 @ ( measure_null_sets_a @ M ) )
& ( ord_less_eq_set_a @ A3 @ X4 ) ) ) ) ).
% null_sets_completion_iff2
thf(fact_736_completion_Ocomplete2,axiom,
! [A3: set_a,B: set_a,M: sigma_measure_a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ( member_set_a @ B @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ( member_set_a @ A3 @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M ) ) ) ) ) ).
% completion.complete2
thf(fact_737_null__sets__completion__iff,axiom,
! [N: set_a,M: sigma_measure_a] :
( ( member_set_a @ N @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ N @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
= ( member_set_a @ N @ ( measure_null_sets_a @ M ) ) ) ) ).
% null_sets_completion_iff
thf(fact_738_finite__has__minimal,axiom,
! [A3: set_Ex3793607809372303086nnreal] :
( ( finite3782138982310603983nnreal @ A3 )
=> ( ( A3 != bot_bo4854962954004695426nnreal )
=> ? [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A3 )
& ! [Xa: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ Xa @ A3 )
=> ( ( ord_le3935885782089961368nnreal @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_739_finite__has__maximal,axiom,
! [A3: set_Ex3793607809372303086nnreal] :
( ( finite3782138982310603983nnreal @ A3 )
=> ( ( A3 != bot_bo4854962954004695426nnreal )
=> ? [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A3 )
& ! [Xa: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ Xa @ A3 )
=> ( ( ord_le3935885782089961368nnreal @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_740_measurable__empty__iff,axiom,
! [N: sigma_measure_a,F: real > a,M: sigma_measure_real] :
( ( ( sigma_space_a @ N )
= bot_bot_set_a )
=> ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ N ) )
= ( ( sigma_space_real @ M )
= bot_bot_set_real ) ) ) ).
% measurable_empty_iff
thf(fact_741_measurable__empty__iff,axiom,
! [N: sigma_measure_a,F: a > a,M: sigma_measure_a] :
( ( ( sigma_space_a @ N )
= bot_bot_set_a )
=> ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ N ) )
= ( ( sigma_space_a @ M )
= bot_bot_set_a ) ) ) ).
% measurable_empty_iff
thf(fact_742_complete__measure_Ocomplete2,axiom,
! [M: sigma_measure_a,A3: set_a,B: set_a] :
( ( comple8155536527497655953sure_a @ M )
=> ( ( ord_less_eq_set_a @ A3 @ B )
=> ( ( member_set_a @ B @ ( measure_null_sets_a @ M ) )
=> ( member_set_a @ A3 @ ( measure_null_sets_a @ M ) ) ) ) ) ).
% complete_measure.complete2
thf(fact_743_Complete__Measure_Ocompletion__upper,axiom,
! [A3: set_a,M: sigma_measure_a] :
( ( member_set_a @ A3 @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ ( sigma_sets_a @ M ) )
& ( ord_less_eq_set_a @ A3 @ X3 )
& ( ( sigma_emeasure_a @ ( comple3428971583294703880tion_a @ M ) @ A3 )
= ( sigma_emeasure_a @ M @ X3 ) ) ) ) ).
% Complete_Measure.completion_upper
thf(fact_744_completion_Ocomplete,axiom,
! [B: set_a,A3: set_a,M: sigma_measure_a] :
( ( ord_less_eq_set_a @ B @ A3 )
=> ( ( member_set_a @ A3 @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ( member_set_a @ B @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) ) ) ) ).
% completion.complete
thf(fact_745_sets__completionI__sub,axiom,
! [N2: set_a,M: sigma_measure_a,N: set_a] :
( ( member_set_a @ N2 @ ( measure_null_sets_a @ M ) )
=> ( ( ord_less_eq_set_a @ N @ N2 )
=> ( member_set_a @ N @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) ) ) ) ).
% sets_completionI_sub
thf(fact_746_null__part__null__sets,axiom,
! [S: set_a,M: sigma_measure_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ( member_set_a @ ( complete_null_part_a @ M @ S ) @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M ) ) ) ) ).
% null_part_null_sets
thf(fact_747_outer__measure__empty,axiom,
! [M: set_set_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_positive_a @ M @ F )
=> ( ( member_set_a @ bot_bot_set_a @ M )
=> ( ( outer_measure_a @ M @ F @ bot_bot_set_a )
= zero_z7100319975126383169nnreal ) ) ) ).
% outer_measure_empty
thf(fact_748_algebra_Olambda__system__empty,axiom,
! [Omega: set_a,M: set_set_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_algebra_a @ Omega @ M )
=> ( ( sigma_positive_a @ M @ F )
=> ( member_set_a @ bot_bot_set_a @ ( lambda_system_a @ Omega @ M @ F ) ) ) ) ).
% algebra.lambda_system_empty
thf(fact_749_ex__min__if__finite,axiom,
! [S: set_Ex3793607809372303086nnreal] :
( ( finite3782138982310603983nnreal @ S )
=> ( ( S != bot_bo4854962954004695426nnreal )
=> ? [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ S )
& ~ ? [Xa: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ Xa @ S )
& ( ord_le7381754540660121996nnreal @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_750_infinite__growing,axiom,
! [X: set_Ex3793607809372303086nnreal] :
( ( X != bot_bo4854962954004695426nnreal )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ X )
=> ? [Xa: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ Xa @ X )
& ( ord_le7381754540660121996nnreal @ X3 @ Xa ) ) )
=> ~ ( finite3782138982310603983nnreal @ X ) ) ) ).
% infinite_growing
thf(fact_751_eqb__space,axiom,
( ( qbs_space_a @ empty_quasi_borel_a )
= bot_bot_set_a ) ).
% eqb_space
thf(fact_752_finite__transitivity__chain,axiom,
! [A3: set_a_a,R2: ( a > a ) > ( a > a ) > $o] :
( ( finite_finite_a_a @ A3 )
=> ( ! [X3: a > a] :
~ ( R2 @ X3 @ X3 )
=> ( ! [X3: a > a,Y5: a > a,Z4: a > a] :
( ( R2 @ X3 @ Y5 )
=> ( ( R2 @ Y5 @ Z4 )
=> ( R2 @ X3 @ Z4 ) ) )
=> ( ! [X3: a > a] :
( ( member_a_a @ X3 @ A3 )
=> ? [Y6: a > a] :
( ( member_a_a @ Y6 @ A3 )
& ( R2 @ X3 @ Y6 ) ) )
=> ( A3 = bot_bot_set_a_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_753_finite__transitivity__chain,axiom,
! [A3: set_a,R2: a > a > $o] :
( ( finite_finite_a @ A3 )
=> ( ! [X3: a] :
~ ( R2 @ X3 @ X3 )
=> ( ! [X3: a,Y5: a,Z4: a] :
( ( R2 @ X3 @ Y5 )
=> ( ( R2 @ Y5 @ Z4 )
=> ( R2 @ X3 @ Z4 ) ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ? [Y6: a] :
( ( member_a @ Y6 @ A3 )
& ( R2 @ X3 @ Y6 ) ) )
=> ( A3 = bot_bot_set_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_754_finite__transitivity__chain,axiom,
! [A3: set_set_a,R2: set_a > set_a > $o] :
( ( finite_finite_set_a @ A3 )
=> ( ! [X3: set_a] :
~ ( R2 @ X3 @ X3 )
=> ( ! [X3: set_a,Y5: set_a,Z4: set_a] :
( ( R2 @ X3 @ Y5 )
=> ( ( R2 @ Y5 @ Z4 )
=> ( R2 @ X3 @ Z4 ) ) )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
=> ? [Y6: set_a] :
( ( member_set_a @ Y6 @ A3 )
& ( R2 @ X3 @ Y6 ) ) )
=> ( A3 = bot_bot_set_set_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_755_finite__transitivity__chain,axiom,
! [A3: set_real_a,R2: ( real > a ) > ( real > a ) > $o] :
( ( finite_finite_real_a @ A3 )
=> ( ! [X3: real > a] :
~ ( R2 @ X3 @ X3 )
=> ( ! [X3: real > a,Y5: real > a,Z4: real > a] :
( ( R2 @ X3 @ Y5 )
=> ( ( R2 @ Y5 @ Z4 )
=> ( R2 @ X3 @ Z4 ) ) )
=> ( ! [X3: real > a] :
( ( member_real_a @ X3 @ A3 )
=> ? [Y6: real > a] :
( ( member_real_a @ Y6 @ A3 )
& ( R2 @ X3 @ Y6 ) ) )
=> ( A3 = bot_bot_set_real_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_756_space__bot,axiom,
( ( sigma_space_a @ bot_bo2108912051383640591sure_a )
= bot_bot_set_a ) ).
% space_bot
thf(fact_757_eqb__Mx,axiom,
( ( qbs_Mx_a @ empty_quasi_borel_a )
= bot_bot_set_real_a ) ).
% eqb_Mx
thf(fact_758_space__empty__eq__bot,axiom,
! [A2: sigma_measure_a] :
( ( ( sigma_space_a @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bo2108912051383640591sure_a ) ) ).
% space_empty_eq_bot
thf(fact_759_empty__quasi__borel__iff,axiom,
! [X: quasi_borel_a] :
( ( ( qbs_space_a @ X )
= bot_bot_set_a )
= ( X = empty_quasi_borel_a ) ) ).
% empty_quasi_borel_iff
thf(fact_760_arg__min__least,axiom,
! [S: set_a_a,Y3: a > a,F: ( a > a ) > extend8495563244428889912nnreal] :
( ( finite_finite_a_a @ S )
=> ( ( S != bot_bot_set_a_a )
=> ( ( member_a_a @ Y3 @ S )
=> ( ord_le3935885782089961368nnreal @ ( F @ ( lattic2420503671435095992nnreal @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% arg_min_least
thf(fact_761_arg__min__least,axiom,
! [S: set_a,Y3: a,F: a > extend8495563244428889912nnreal] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ( ( member_a @ Y3 @ S )
=> ( ord_le3935885782089961368nnreal @ ( F @ ( lattic4812999201339876317nnreal @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% arg_min_least
thf(fact_762_arg__min__least,axiom,
! [S: set_set_a,Y3: set_a,F: set_a > extend8495563244428889912nnreal] :
( ( finite_finite_set_a @ S )
=> ( ( S != bot_bot_set_set_a )
=> ( ( member_set_a @ Y3 @ S )
=> ( ord_le3935885782089961368nnreal @ ( F @ ( lattic1895148585496753469nnreal @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% arg_min_least
thf(fact_763_arg__min__least,axiom,
! [S: set_real_a,Y3: real > a,F: ( real > a ) > extend8495563244428889912nnreal] :
( ( finite_finite_real_a @ S )
=> ( ( S != bot_bot_set_real_a )
=> ( ( member_real_a @ Y3 @ S )
=> ( ord_le3935885782089961368nnreal @ ( F @ ( lattic6219155568527054696nnreal @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% arg_min_least
thf(fact_764_emeasure__completion,axiom,
! [S: set_a,M: sigma_measure_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ( ( sigma_emeasure_a @ ( comple3428971583294703880tion_a @ M ) @ S )
= ( sigma_emeasure_a @ M @ ( complete_main_part_a @ M @ S ) ) ) ) ).
% emeasure_completion
thf(fact_765_insertCI,axiom,
! [A2: a > a,B: set_a_a,B3: a > a] :
( ( ~ ( member_a_a @ A2 @ B )
=> ( A2 = B3 ) )
=> ( member_a_a @ A2 @ ( insert_a_a @ B3 @ B ) ) ) ).
% insertCI
thf(fact_766_insertCI,axiom,
! [A2: a,B: set_a,B3: a] :
( ( ~ ( member_a @ A2 @ B )
=> ( A2 = B3 ) )
=> ( member_a @ A2 @ ( insert_a @ B3 @ B ) ) ) ).
% insertCI
thf(fact_767_insertCI,axiom,
! [A2: set_a,B: set_set_a,B3: set_a] :
( ( ~ ( member_set_a @ A2 @ B )
=> ( A2 = B3 ) )
=> ( member_set_a @ A2 @ ( insert_set_a @ B3 @ B ) ) ) ).
% insertCI
thf(fact_768_insertCI,axiom,
! [A2: real > a,B: set_real_a,B3: real > a] :
( ( ~ ( member_real_a @ A2 @ B )
=> ( A2 = B3 ) )
=> ( member_real_a @ A2 @ ( insert_real_a @ B3 @ B ) ) ) ).
% insertCI
thf(fact_769_insert__iff,axiom,
! [A2: a > a,B3: a > a,A3: set_a_a] :
( ( member_a_a @ A2 @ ( insert_a_a @ B3 @ A3 ) )
= ( ( A2 = B3 )
| ( member_a_a @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_770_insert__iff,axiom,
! [A2: a,B3: a,A3: set_a] :
( ( member_a @ A2 @ ( insert_a @ B3 @ A3 ) )
= ( ( A2 = B3 )
| ( member_a @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_771_insert__iff,axiom,
! [A2: set_a,B3: set_a,A3: set_set_a] :
( ( member_set_a @ A2 @ ( insert_set_a @ B3 @ A3 ) )
= ( ( A2 = B3 )
| ( member_set_a @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_772_insert__iff,axiom,
! [A2: real > a,B3: real > a,A3: set_real_a] :
( ( member_real_a @ A2 @ ( insert_real_a @ B3 @ A3 ) )
= ( ( A2 = B3 )
| ( member_real_a @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_773_singletonI,axiom,
! [A2: a > a] : ( member_a_a @ A2 @ ( insert_a_a @ A2 @ bot_bot_set_a_a ) ) ).
% singletonI
thf(fact_774_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_775_singletonI,axiom,
! [A2: set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_776_singletonI,axiom,
! [A2: real > a] : ( member_real_a @ A2 @ ( insert_real_a @ A2 @ bot_bot_set_real_a ) ) ).
% singletonI
thf(fact_777_insert__subset,axiom,
! [X2: a > a,A3: set_a_a,B: set_a_a] :
( ( ord_less_eq_set_a_a @ ( insert_a_a @ X2 @ A3 ) @ B )
= ( ( member_a_a @ X2 @ B )
& ( ord_less_eq_set_a_a @ A3 @ B ) ) ) ).
% insert_subset
thf(fact_778_insert__subset,axiom,
! [X2: a,A3: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X2 @ A3 ) @ B )
= ( ( member_a @ X2 @ B )
& ( ord_less_eq_set_a @ A3 @ B ) ) ) ).
% insert_subset
thf(fact_779_insert__subset,axiom,
! [X2: set_a,A3: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X2 @ A3 ) @ B )
= ( ( member_set_a @ X2 @ B )
& ( ord_le3724670747650509150_set_a @ A3 @ B ) ) ) ).
% insert_subset
thf(fact_780_insert__subset,axiom,
! [X2: real > a,A3: set_real_a,B: set_real_a] :
( ( ord_le5743406823621094409real_a @ ( insert_real_a @ X2 @ A3 ) @ B )
= ( ( member_real_a @ X2 @ B )
& ( ord_le5743406823621094409real_a @ A3 @ B ) ) ) ).
% insert_subset
thf(fact_781_main__part,axiom,
! [S: set_a,M: sigma_measure_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ M ) )
=> ( ( complete_main_part_a @ M @ S )
= S ) ) ).
% main_part
thf(fact_782_main__part__sets,axiom,
! [S: set_a,M: sigma_measure_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ( member_set_a @ ( complete_main_part_a @ M @ S ) @ ( sigma_sets_a @ M ) ) ) ).
% main_part_sets
thf(fact_783_singletonD,axiom,
! [B3: a > a,A2: a > a] :
( ( member_a_a @ B3 @ ( insert_a_a @ A2 @ bot_bot_set_a_a ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_784_singletonD,axiom,
! [B3: a,A2: a] :
( ( member_a @ B3 @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_785_singletonD,axiom,
! [B3: set_a,A2: set_a] :
( ( member_set_a @ B3 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_786_singletonD,axiom,
! [B3: real > a,A2: real > a] :
( ( member_real_a @ B3 @ ( insert_real_a @ A2 @ bot_bot_set_real_a ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_787_singleton__iff,axiom,
! [B3: a > a,A2: a > a] :
( ( member_a_a @ B3 @ ( insert_a_a @ A2 @ bot_bot_set_a_a ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_788_singleton__iff,axiom,
! [B3: a,A2: a] :
( ( member_a @ B3 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_789_singleton__iff,axiom,
! [B3: set_a,A2: set_a] :
( ( member_set_a @ B3 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_790_singleton__iff,axiom,
! [B3: real > a,A2: real > a] :
( ( member_real_a @ B3 @ ( insert_real_a @ A2 @ bot_bot_set_real_a ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_791_insertE,axiom,
! [A2: a > a,B3: a > a,A3: set_a_a] :
( ( member_a_a @ A2 @ ( insert_a_a @ B3 @ A3 ) )
=> ( ( A2 != B3 )
=> ( member_a_a @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_792_insertE,axiom,
! [A2: a,B3: a,A3: set_a] :
( ( member_a @ A2 @ ( insert_a @ B3 @ A3 ) )
=> ( ( A2 != B3 )
=> ( member_a @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_793_insertE,axiom,
! [A2: set_a,B3: set_a,A3: set_set_a] :
( ( member_set_a @ A2 @ ( insert_set_a @ B3 @ A3 ) )
=> ( ( A2 != B3 )
=> ( member_set_a @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_794_insertE,axiom,
! [A2: real > a,B3: real > a,A3: set_real_a] :
( ( member_real_a @ A2 @ ( insert_real_a @ B3 @ A3 ) )
=> ( ( A2 != B3 )
=> ( member_real_a @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_795_insertI1,axiom,
! [A2: a > a,B: set_a_a] : ( member_a_a @ A2 @ ( insert_a_a @ A2 @ B ) ) ).
% insertI1
thf(fact_796_insertI1,axiom,
! [A2: a,B: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B ) ) ).
% insertI1
thf(fact_797_insertI1,axiom,
! [A2: set_a,B: set_set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ B ) ) ).
% insertI1
thf(fact_798_insertI1,axiom,
! [A2: real > a,B: set_real_a] : ( member_real_a @ A2 @ ( insert_real_a @ A2 @ B ) ) ).
% insertI1
thf(fact_799_insertI2,axiom,
! [A2: a > a,B: set_a_a,B3: a > a] :
( ( member_a_a @ A2 @ B )
=> ( member_a_a @ A2 @ ( insert_a_a @ B3 @ B ) ) ) ).
% insertI2
thf(fact_800_insertI2,axiom,
! [A2: a,B: set_a,B3: a] :
( ( member_a @ A2 @ B )
=> ( member_a @ A2 @ ( insert_a @ B3 @ B ) ) ) ).
% insertI2
thf(fact_801_insertI2,axiom,
! [A2: set_a,B: set_set_a,B3: set_a] :
( ( member_set_a @ A2 @ B )
=> ( member_set_a @ A2 @ ( insert_set_a @ B3 @ B ) ) ) ).
% insertI2
thf(fact_802_insertI2,axiom,
! [A2: real > a,B: set_real_a,B3: real > a] :
( ( member_real_a @ A2 @ B )
=> ( member_real_a @ A2 @ ( insert_real_a @ B3 @ B ) ) ) ).
% insertI2
thf(fact_803_Set_Oset__insert,axiom,
! [X2: a > a,A3: set_a_a] :
( ( member_a_a @ X2 @ A3 )
=> ~ ! [B7: set_a_a] :
( ( A3
= ( insert_a_a @ X2 @ B7 ) )
=> ( member_a_a @ X2 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_804_Set_Oset__insert,axiom,
! [X2: a,A3: set_a] :
( ( member_a @ X2 @ A3 )
=> ~ ! [B7: set_a] :
( ( A3
= ( insert_a @ X2 @ B7 ) )
=> ( member_a @ X2 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_805_Set_Oset__insert,axiom,
! [X2: set_a,A3: set_set_a] :
( ( member_set_a @ X2 @ A3 )
=> ~ ! [B7: set_set_a] :
( ( A3
= ( insert_set_a @ X2 @ B7 ) )
=> ( member_set_a @ X2 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_806_Set_Oset__insert,axiom,
! [X2: real > a,A3: set_real_a] :
( ( member_real_a @ X2 @ A3 )
=> ~ ! [B7: set_real_a] :
( ( A3
= ( insert_real_a @ X2 @ B7 ) )
=> ( member_real_a @ X2 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_807_insert__ident,axiom,
! [X2: a > a,A3: set_a_a,B: set_a_a] :
( ~ ( member_a_a @ X2 @ A3 )
=> ( ~ ( member_a_a @ X2 @ B )
=> ( ( ( insert_a_a @ X2 @ A3 )
= ( insert_a_a @ X2 @ B ) )
= ( A3 = B ) ) ) ) ).
% insert_ident
thf(fact_808_insert__ident,axiom,
! [X2: a,A3: set_a,B: set_a] :
( ~ ( member_a @ X2 @ A3 )
=> ( ~ ( member_a @ X2 @ B )
=> ( ( ( insert_a @ X2 @ A3 )
= ( insert_a @ X2 @ B ) )
= ( A3 = B ) ) ) ) ).
% insert_ident
thf(fact_809_insert__ident,axiom,
! [X2: set_a,A3: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X2 @ A3 )
=> ( ~ ( member_set_a @ X2 @ B )
=> ( ( ( insert_set_a @ X2 @ A3 )
= ( insert_set_a @ X2 @ B ) )
= ( A3 = B ) ) ) ) ).
% insert_ident
thf(fact_810_insert__ident,axiom,
! [X2: real > a,A3: set_real_a,B: set_real_a] :
( ~ ( member_real_a @ X2 @ A3 )
=> ( ~ ( member_real_a @ X2 @ B )
=> ( ( ( insert_real_a @ X2 @ A3 )
= ( insert_real_a @ X2 @ B ) )
= ( A3 = B ) ) ) ) ).
% insert_ident
thf(fact_811_insert__absorb,axiom,
! [A2: a > a,A3: set_a_a] :
( ( member_a_a @ A2 @ A3 )
=> ( ( insert_a_a @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_812_insert__absorb,axiom,
! [A2: a,A3: set_a] :
( ( member_a @ A2 @ A3 )
=> ( ( insert_a @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_813_insert__absorb,axiom,
! [A2: set_a,A3: set_set_a] :
( ( member_set_a @ A2 @ A3 )
=> ( ( insert_set_a @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_814_insert__absorb,axiom,
! [A2: real > a,A3: set_real_a] :
( ( member_real_a @ A2 @ A3 )
=> ( ( insert_real_a @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_815_insert__eq__iff,axiom,
! [A2: a > a,A3: set_a_a,B3: a > a,B: set_a_a] :
( ~ ( member_a_a @ A2 @ A3 )
=> ( ~ ( member_a_a @ B3 @ B )
=> ( ( ( insert_a_a @ A2 @ A3 )
= ( insert_a_a @ B3 @ B ) )
= ( ( ( A2 = B3 )
=> ( A3 = B ) )
& ( ( A2 != B3 )
=> ? [C4: set_a_a] :
( ( A3
= ( insert_a_a @ B3 @ C4 ) )
& ~ ( member_a_a @ B3 @ C4 )
& ( B
= ( insert_a_a @ A2 @ C4 ) )
& ~ ( member_a_a @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_816_insert__eq__iff,axiom,
! [A2: a,A3: set_a,B3: a,B: set_a] :
( ~ ( member_a @ A2 @ A3 )
=> ( ~ ( member_a @ B3 @ B )
=> ( ( ( insert_a @ A2 @ A3 )
= ( insert_a @ B3 @ B ) )
= ( ( ( A2 = B3 )
=> ( A3 = B ) )
& ( ( A2 != B3 )
=> ? [C4: set_a] :
( ( A3
= ( insert_a @ B3 @ C4 ) )
& ~ ( member_a @ B3 @ C4 )
& ( B
= ( insert_a @ A2 @ C4 ) )
& ~ ( member_a @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_817_insert__eq__iff,axiom,
! [A2: set_a,A3: set_set_a,B3: set_a,B: set_set_a] :
( ~ ( member_set_a @ A2 @ A3 )
=> ( ~ ( member_set_a @ B3 @ B )
=> ( ( ( insert_set_a @ A2 @ A3 )
= ( insert_set_a @ B3 @ B ) )
= ( ( ( A2 = B3 )
=> ( A3 = B ) )
& ( ( A2 != B3 )
=> ? [C4: set_set_a] :
( ( A3
= ( insert_set_a @ B3 @ C4 ) )
& ~ ( member_set_a @ B3 @ C4 )
& ( B
= ( insert_set_a @ A2 @ C4 ) )
& ~ ( member_set_a @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_818_insert__eq__iff,axiom,
! [A2: real > a,A3: set_real_a,B3: real > a,B: set_real_a] :
( ~ ( member_real_a @ A2 @ A3 )
=> ( ~ ( member_real_a @ B3 @ B )
=> ( ( ( insert_real_a @ A2 @ A3 )
= ( insert_real_a @ B3 @ B ) )
= ( ( ( A2 = B3 )
=> ( A3 = B ) )
& ( ( A2 != B3 )
=> ? [C4: set_real_a] :
( ( A3
= ( insert_real_a @ B3 @ C4 ) )
& ~ ( member_real_a @ B3 @ C4 )
& ( B
= ( insert_real_a @ A2 @ C4 ) )
& ~ ( member_real_a @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_819_mk__disjoint__insert,axiom,
! [A2: a > a,A3: set_a_a] :
( ( member_a_a @ A2 @ A3 )
=> ? [B7: set_a_a] :
( ( A3
= ( insert_a_a @ A2 @ B7 ) )
& ~ ( member_a_a @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_820_mk__disjoint__insert,axiom,
! [A2: a,A3: set_a] :
( ( member_a @ A2 @ A3 )
=> ? [B7: set_a] :
( ( A3
= ( insert_a @ A2 @ B7 ) )
& ~ ( member_a @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_821_mk__disjoint__insert,axiom,
! [A2: set_a,A3: set_set_a] :
( ( member_set_a @ A2 @ A3 )
=> ? [B7: set_set_a] :
( ( A3
= ( insert_set_a @ A2 @ B7 ) )
& ~ ( member_set_a @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_822_mk__disjoint__insert,axiom,
! [A2: real > a,A3: set_real_a] :
( ( member_real_a @ A2 @ A3 )
=> ? [B7: set_real_a] :
( ( A3
= ( insert_real_a @ A2 @ B7 ) )
& ~ ( member_real_a @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_823_subset__insert,axiom,
! [X2: a > a,A3: set_a_a,B: set_a_a] :
( ~ ( member_a_a @ X2 @ A3 )
=> ( ( ord_less_eq_set_a_a @ A3 @ ( insert_a_a @ X2 @ B ) )
= ( ord_less_eq_set_a_a @ A3 @ B ) ) ) ).
% subset_insert
thf(fact_824_subset__insert,axiom,
! [X2: a,A3: set_a,B: set_a] :
( ~ ( member_a @ X2 @ A3 )
=> ( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X2 @ B ) )
= ( ord_less_eq_set_a @ A3 @ B ) ) ) ).
% subset_insert
thf(fact_825_subset__insert,axiom,
! [X2: set_a,A3: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X2 @ A3 )
=> ( ( ord_le3724670747650509150_set_a @ A3 @ ( insert_set_a @ X2 @ B ) )
= ( ord_le3724670747650509150_set_a @ A3 @ B ) ) ) ).
% subset_insert
thf(fact_826_subset__insert,axiom,
! [X2: real > a,A3: set_real_a,B: set_real_a] :
( ~ ( member_real_a @ X2 @ A3 )
=> ( ( ord_le5743406823621094409real_a @ A3 @ ( insert_real_a @ X2 @ B ) )
= ( ord_le5743406823621094409real_a @ A3 @ B ) ) ) ).
% subset_insert
thf(fact_827_infinite__finite__induct,axiom,
! [P: set_a_a > $o,A3: set_a_a] :
( ! [A8: set_a_a] :
( ~ ( finite_finite_a_a @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_a_a )
=> ( ! [X3: a > a,F3: set_a_a] :
( ( finite_finite_a_a @ F3 )
=> ( ~ ( member_a_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a_a @ X3 @ F3 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_828_infinite__finite__induct,axiom,
! [P: set_a > $o,A3: set_a] :
( ! [A8: set_a] :
( ~ ( finite_finite_a @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_829_infinite__finite__induct,axiom,
! [P: set_set_a > $o,A3: set_set_a] :
( ! [A8: set_set_a] :
( ~ ( finite_finite_set_a @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X3: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X3 @ F3 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_830_infinite__finite__induct,axiom,
! [P: set_real_a > $o,A3: set_real_a] :
( ! [A8: set_real_a] :
( ~ ( finite_finite_real_a @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_real_a )
=> ( ! [X3: real > a,F3: set_real_a] :
( ( finite_finite_real_a @ F3 )
=> ( ~ ( member_real_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real_a @ X3 @ F3 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_831_finite__ne__induct,axiom,
! [F4: set_a_a,P: set_a_a > $o] :
( ( finite_finite_a_a @ F4 )
=> ( ( F4 != bot_bot_set_a_a )
=> ( ! [X3: a > a] : ( P @ ( insert_a_a @ X3 @ bot_bot_set_a_a ) )
=> ( ! [X3: a > a,F3: set_a_a] :
( ( finite_finite_a_a @ F3 )
=> ( ( F3 != bot_bot_set_a_a )
=> ( ~ ( member_a_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a_a @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_832_finite__ne__induct,axiom,
! [F4: set_a,P: set_a > $o] :
( ( finite_finite_a @ F4 )
=> ( ( F4 != bot_bot_set_a )
=> ( ! [X3: a] : ( P @ ( insert_a @ X3 @ bot_bot_set_a ) )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_833_finite__ne__induct,axiom,
! [F4: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F4 )
=> ( ( F4 != bot_bot_set_set_a )
=> ( ! [X3: set_a] : ( P @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
=> ( ! [X3: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( F3 != bot_bot_set_set_a )
=> ( ~ ( member_set_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_834_finite__ne__induct,axiom,
! [F4: set_real_a,P: set_real_a > $o] :
( ( finite_finite_real_a @ F4 )
=> ( ( F4 != bot_bot_set_real_a )
=> ( ! [X3: real > a] : ( P @ ( insert_real_a @ X3 @ bot_bot_set_real_a ) )
=> ( ! [X3: real > a,F3: set_real_a] :
( ( finite_finite_real_a @ F3 )
=> ( ( F3 != bot_bot_set_real_a )
=> ( ~ ( member_real_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real_a @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_835_finite__induct,axiom,
! [F4: set_a_a,P: set_a_a > $o] :
( ( finite_finite_a_a @ F4 )
=> ( ( P @ bot_bot_set_a_a )
=> ( ! [X3: a > a,F3: set_a_a] :
( ( finite_finite_a_a @ F3 )
=> ( ~ ( member_a_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a_a @ X3 @ F3 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_836_finite__induct,axiom,
! [F4: set_a,P: set_a > $o] :
( ( finite_finite_a @ F4 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_837_finite__induct,axiom,
! [F4: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F4 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X3: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X3 @ F3 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_838_finite__induct,axiom,
! [F4: set_real_a,P: set_real_a > $o] :
( ( finite_finite_real_a @ F4 )
=> ( ( P @ bot_bot_set_real_a )
=> ( ! [X3: real > a,F3: set_real_a] :
( ( finite_finite_real_a @ F3 )
=> ( ~ ( member_real_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real_a @ X3 @ F3 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_839_sets_Oinsert__in__sets,axiom,
! [X2: a,M: sigma_measure_a,A3: set_a] :
( ( member_set_a @ ( insert_a @ X2 @ bot_bot_set_a ) @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ A3 @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( insert_a @ X2 @ A3 ) @ ( sigma_sets_a @ M ) ) ) ) ).
% sets.insert_in_sets
thf(fact_840_fmeasurable_Oinsert__in__sets,axiom,
! [X2: a,M: sigma_measure_a,A3: set_a] :
( ( member_set_a @ ( insert_a @ X2 @ bot_bot_set_a ) @ ( measur3645360004775918570able_a @ M ) )
=> ( ( member_set_a @ A3 @ ( measur3645360004775918570able_a @ M ) )
=> ( member_set_a @ ( insert_a @ X2 @ A3 ) @ ( measur3645360004775918570able_a @ M ) ) ) ) ).
% fmeasurable.insert_in_sets
thf(fact_841_null__sets_Oinsert__in__sets,axiom,
! [X2: a,M: sigma_measure_a,A3: set_a] :
( ( member_set_a @ ( insert_a @ X2 @ bot_bot_set_a ) @ ( measure_null_sets_a @ M ) )
=> ( ( member_set_a @ A3 @ ( measure_null_sets_a @ M ) )
=> ( member_set_a @ ( insert_a @ X2 @ A3 ) @ ( measure_null_sets_a @ M ) ) ) ) ).
% null_sets.insert_in_sets
thf(fact_842_ring__of__sets_Oinsert__in__sets,axiom,
! [Omega: set_a,M: set_set_a,X2: a,A3: set_a] :
( ( sigma_ring_of_sets_a @ Omega @ M )
=> ( ( member_set_a @ ( insert_a @ X2 @ bot_bot_set_a ) @ M )
=> ( ( member_set_a @ A3 @ M )
=> ( member_set_a @ ( insert_a @ X2 @ A3 ) @ M ) ) ) ) ).
% ring_of_sets.insert_in_sets
thf(fact_843_finite__ranking__induct,axiom,
! [S: set_a_a,P: set_a_a > $o,F: ( a > a ) > extend8495563244428889912nnreal] :
( ( finite_finite_a_a @ S )
=> ( ( P @ bot_bot_set_a_a )
=> ( ! [X3: a > a,S3: set_a_a] :
( ( finite_finite_a_a @ S3 )
=> ( ! [Y6: a > a] :
( ( member_a_a @ Y6 @ S3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ Y6 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_a_a @ X3 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_844_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F: a > extend8495563244428889912nnreal] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,S3: set_a] :
( ( finite_finite_a @ S3 )
=> ( ! [Y6: a] :
( ( member_a @ Y6 @ S3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ Y6 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_a @ X3 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_845_finite__ranking__induct,axiom,
! [S: set_set_a,P: set_set_a > $o,F: set_a > extend8495563244428889912nnreal] :
( ( finite_finite_set_a @ S )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X3: set_a,S3: set_set_a] :
( ( finite_finite_set_a @ S3 )
=> ( ! [Y6: set_a] :
( ( member_set_a @ Y6 @ S3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ Y6 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_set_a @ X3 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_846_finite__ranking__induct,axiom,
! [S: set_real_a,P: set_real_a > $o,F: ( real > a ) > extend8495563244428889912nnreal] :
( ( finite_finite_real_a @ S )
=> ( ( P @ bot_bot_set_real_a )
=> ( ! [X3: real > a,S3: set_real_a] :
( ( finite_finite_real_a @ S3 )
=> ( ! [Y6: real > a] :
( ( member_real_a @ Y6 @ S3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ Y6 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_real_a @ X3 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_847_finite__linorder__min__induct,axiom,
! [A3: set_Ex3793607809372303086nnreal,P: set_Ex3793607809372303086nnreal > $o] :
( ( finite3782138982310603983nnreal @ A3 )
=> ( ( P @ bot_bo4854962954004695426nnreal )
=> ( ! [B2: extend8495563244428889912nnreal,A8: set_Ex3793607809372303086nnreal] :
( ( finite3782138982310603983nnreal @ A8 )
=> ( ! [X5: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X5 @ A8 )
=> ( ord_le7381754540660121996nnreal @ B2 @ X5 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert7407984058720857448nnreal @ B2 @ A8 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_848_finite__linorder__max__induct,axiom,
! [A3: set_Ex3793607809372303086nnreal,P: set_Ex3793607809372303086nnreal > $o] :
( ( finite3782138982310603983nnreal @ A3 )
=> ( ( P @ bot_bo4854962954004695426nnreal )
=> ( ! [B2: extend8495563244428889912nnreal,A8: set_Ex3793607809372303086nnreal] :
( ( finite3782138982310603983nnreal @ A8 )
=> ( ! [X5: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X5 @ A8 )
=> ( ord_le7381754540660121996nnreal @ X5 @ B2 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert7407984058720857448nnreal @ B2 @ A8 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_849_finite__subset__induct,axiom,
! [F4: set_a_a,A3: set_a_a,P: set_a_a > $o] :
( ( finite_finite_a_a @ F4 )
=> ( ( ord_less_eq_set_a_a @ F4 @ A3 )
=> ( ( P @ bot_bot_set_a_a )
=> ( ! [A: a > a,F3: set_a_a] :
( ( finite_finite_a_a @ F3 )
=> ( ( member_a_a @ A @ A3 )
=> ( ~ ( member_a_a @ A @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a_a @ A @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_850_finite__subset__induct,axiom,
! [F4: set_a,A3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F4 )
=> ( ( ord_less_eq_set_a @ F4 @ A3 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A @ A3 )
=> ( ~ ( member_a @ A @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_851_finite__subset__induct,axiom,
! [F4: set_set_a,A3: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F4 )
=> ( ( ord_le3724670747650509150_set_a @ F4 @ A3 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A @ A3 )
=> ( ~ ( member_set_a @ A @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ A @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_852_finite__subset__induct,axiom,
! [F4: set_real_a,A3: set_real_a,P: set_real_a > $o] :
( ( finite_finite_real_a @ F4 )
=> ( ( ord_le5743406823621094409real_a @ F4 @ A3 )
=> ( ( P @ bot_bot_set_real_a )
=> ( ! [A: real > a,F3: set_real_a] :
( ( finite_finite_real_a @ F3 )
=> ( ( member_real_a @ A @ A3 )
=> ( ~ ( member_real_a @ A @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real_a @ A @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_853_finite__subset__induct_H,axiom,
! [F4: set_a_a,A3: set_a_a,P: set_a_a > $o] :
( ( finite_finite_a_a @ F4 )
=> ( ( ord_less_eq_set_a_a @ F4 @ A3 )
=> ( ( P @ bot_bot_set_a_a )
=> ( ! [A: a > a,F3: set_a_a] :
( ( finite_finite_a_a @ F3 )
=> ( ( member_a_a @ A @ A3 )
=> ( ( ord_less_eq_set_a_a @ F3 @ A3 )
=> ( ~ ( member_a_a @ A @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a_a @ A @ F3 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_854_finite__subset__induct_H,axiom,
! [F4: set_a,A3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F4 )
=> ( ( ord_less_eq_set_a @ F4 @ A3 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A @ A3 )
=> ( ( ord_less_eq_set_a @ F3 @ A3 )
=> ( ~ ( member_a @ A @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A @ F3 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_855_finite__subset__induct_H,axiom,
! [F4: set_set_a,A3: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F4 )
=> ( ( ord_le3724670747650509150_set_a @ F4 @ A3 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A @ A3 )
=> ( ( ord_le3724670747650509150_set_a @ F3 @ A3 )
=> ( ~ ( member_set_a @ A @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ A @ F3 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_856_finite__subset__induct_H,axiom,
! [F4: set_real_a,A3: set_real_a,P: set_real_a > $o] :
( ( finite_finite_real_a @ F4 )
=> ( ( ord_le5743406823621094409real_a @ F4 @ A3 )
=> ( ( P @ bot_bot_set_real_a )
=> ( ! [A: real > a,F3: set_real_a] :
( ( finite_finite_real_a @ F3 )
=> ( ( member_real_a @ A @ A3 )
=> ( ( ord_le5743406823621094409real_a @ F3 @ A3 )
=> ( ~ ( member_real_a @ A @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real_a @ A @ F3 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_857_sigma__algebra__trivial,axiom,
! [Omega: set_a] : ( sigma_4968961713055010667ebra_a @ Omega @ ( insert_set_a @ bot_bot_set_a @ ( insert_set_a @ Omega @ bot_bot_set_set_a ) ) ) ).
% sigma_algebra_trivial
thf(fact_858_emeasure__single__in__space,axiom,
! [M: sigma_measure_a_a,X2: a > a] :
( ( ( sigma_emeasure_a_a @ M @ ( insert_a_a @ X2 @ bot_bot_set_a_a ) )
!= zero_z7100319975126383169nnreal )
=> ( member_a_a @ X2 @ ( sigma_space_a_a @ M ) ) ) ).
% emeasure_single_in_space
thf(fact_859_emeasure__single__in__space,axiom,
! [M: sigma_measure_set_a,X2: set_a] :
( ( ( sigma_emeasure_set_a @ M @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
!= zero_z7100319975126383169nnreal )
=> ( member_set_a @ X2 @ ( sigma_space_set_a @ M ) ) ) ).
% emeasure_single_in_space
thf(fact_860_emeasure__single__in__space,axiom,
! [M: sigma_measure_real_a,X2: real > a] :
( ( ( sigma_6502373073922819808real_a @ M @ ( insert_real_a @ X2 @ bot_bot_set_real_a ) )
!= zero_z7100319975126383169nnreal )
=> ( member_real_a @ X2 @ ( sigma_space_real_a @ M ) ) ) ).
% emeasure_single_in_space
thf(fact_861_emeasure__single__in__space,axiom,
! [M: sigma_measure_a,X2: a] :
( ( ( sigma_emeasure_a @ M @ ( insert_a @ X2 @ bot_bot_set_a ) )
!= zero_z7100319975126383169nnreal )
=> ( member_a @ X2 @ ( sigma_space_a @ M ) ) ) ).
% emeasure_single_in_space
thf(fact_862_space__empty__iff,axiom,
! [N: sigma_measure_a] :
( ( ( sigma_space_a @ N )
= bot_bot_set_a )
= ( ( sigma_sets_a @ N )
= ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) ) ) ).
% space_empty_iff
thf(fact_863_measure__eqI__finite,axiom,
! [M: sigma_measure_a_a,A3: set_a_a,N: sigma_measure_a_a] :
( ( ( sigma_sets_a_a @ M )
= ( pow_a_a @ A3 ) )
=> ( ( ( sigma_sets_a_a @ N )
= ( pow_a_a @ A3 ) )
=> ( ( finite_finite_a_a @ A3 )
=> ( ! [A: a > a] :
( ( member_a_a @ A @ A3 )
=> ( ( sigma_emeasure_a_a @ M @ ( insert_a_a @ A @ bot_bot_set_a_a ) )
= ( sigma_emeasure_a_a @ N @ ( insert_a_a @ A @ bot_bot_set_a_a ) ) ) )
=> ( M = N ) ) ) ) ) ).
% measure_eqI_finite
thf(fact_864_measure__eqI__finite,axiom,
! [M: sigma_measure_a,A3: set_a,N: sigma_measure_a] :
( ( ( sigma_sets_a @ M )
= ( pow_a @ A3 ) )
=> ( ( ( sigma_sets_a @ N )
= ( pow_a @ A3 ) )
=> ( ( finite_finite_a @ A3 )
=> ( ! [A: a] :
( ( member_a @ A @ A3 )
=> ( ( sigma_emeasure_a @ M @ ( insert_a @ A @ bot_bot_set_a ) )
= ( sigma_emeasure_a @ N @ ( insert_a @ A @ bot_bot_set_a ) ) ) )
=> ( M = N ) ) ) ) ) ).
% measure_eqI_finite
thf(fact_865_measure__eqI__finite,axiom,
! [M: sigma_measure_set_a,A3: set_set_a,N: sigma_measure_set_a] :
( ( ( sigma_sets_set_a @ M )
= ( pow_set_a @ A3 ) )
=> ( ( ( sigma_sets_set_a @ N )
= ( pow_set_a @ A3 ) )
=> ( ( finite_finite_set_a @ A3 )
=> ( ! [A: set_a] :
( ( member_set_a @ A @ A3 )
=> ( ( sigma_emeasure_set_a @ M @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( sigma_emeasure_set_a @ N @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) )
=> ( M = N ) ) ) ) ) ).
% measure_eqI_finite
thf(fact_866_measure__eqI__finite,axiom,
! [M: sigma_measure_real_a,A3: set_real_a,N: sigma_measure_real_a] :
( ( ( sigma_sets_real_a @ M )
= ( pow_real_a @ A3 ) )
=> ( ( ( sigma_sets_real_a @ N )
= ( pow_real_a @ A3 ) )
=> ( ( finite_finite_real_a @ A3 )
=> ( ! [A: real > a] :
( ( member_real_a @ A @ A3 )
=> ( ( sigma_6502373073922819808real_a @ M @ ( insert_real_a @ A @ bot_bot_set_real_a ) )
= ( sigma_6502373073922819808real_a @ N @ ( insert_real_a @ A @ bot_bot_set_real_a ) ) ) )
=> ( M = N ) ) ) ) ) ).
% measure_eqI_finite
thf(fact_867_main__part__null__part__Int,axiom,
! [S: set_a,M: sigma_measure_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ( ( inf_inf_set_a @ ( complete_main_part_a @ M @ S ) @ ( complete_null_part_a @ M @ S ) )
= bot_bot_set_a ) ) ).
% main_part_null_part_Int
thf(fact_868_insert__subsetI,axiom,
! [X2: a > a,A3: set_a_a,X: set_a_a] :
( ( member_a_a @ X2 @ A3 )
=> ( ( ord_less_eq_set_a_a @ X @ A3 )
=> ( ord_less_eq_set_a_a @ ( insert_a_a @ X2 @ X ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_869_insert__subsetI,axiom,
! [X2: a,A3: set_a,X: set_a] :
( ( member_a @ X2 @ A3 )
=> ( ( ord_less_eq_set_a @ X @ A3 )
=> ( ord_less_eq_set_a @ ( insert_a @ X2 @ X ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_870_insert__subsetI,axiom,
! [X2: set_a,A3: set_set_a,X: set_set_a] :
( ( member_set_a @ X2 @ A3 )
=> ( ( ord_le3724670747650509150_set_a @ X @ A3 )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X2 @ X ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_871_insert__subsetI,axiom,
! [X2: real > a,A3: set_real_a,X: set_real_a] :
( ( member_real_a @ X2 @ A3 )
=> ( ( ord_le5743406823621094409real_a @ X @ A3 )
=> ( ord_le5743406823621094409real_a @ ( insert_real_a @ X2 @ X ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_872_subset__emptyI,axiom,
! [A3: set_a_a] :
( ! [X3: a > a] :
~ ( member_a_a @ X3 @ A3 )
=> ( ord_less_eq_set_a_a @ A3 @ bot_bot_set_a_a ) ) ).
% subset_emptyI
thf(fact_873_subset__emptyI,axiom,
! [A3: set_a] :
( ! [X3: a] :
~ ( member_a @ X3 @ A3 )
=> ( ord_less_eq_set_a @ A3 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_874_subset__emptyI,axiom,
! [A3: set_set_a] :
( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A3 )
=> ( ord_le3724670747650509150_set_a @ A3 @ bot_bot_set_set_a ) ) ).
% subset_emptyI
thf(fact_875_subset__emptyI,axiom,
! [A3: set_real_a] :
( ! [X3: real > a] :
~ ( member_real_a @ X3 @ A3 )
=> ( ord_le5743406823621094409real_a @ A3 @ bot_bot_set_real_a ) ) ).
% subset_emptyI
thf(fact_876_Int__iff,axiom,
! [C2: a > a,A3: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( inf_inf_set_a_a @ A3 @ B ) )
= ( ( member_a_a @ C2 @ A3 )
& ( member_a_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_877_Int__iff,axiom,
! [C2: a,A3: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A3 @ B ) )
= ( ( member_a @ C2 @ A3 )
& ( member_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_878_Int__iff,axiom,
! [C2: set_a,A3: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A3 @ B ) )
= ( ( member_set_a @ C2 @ A3 )
& ( member_set_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_879_Int__iff,axiom,
! [C2: real > a,A3: set_real_a,B: set_real_a] :
( ( member_real_a @ C2 @ ( inf_inf_set_real_a @ A3 @ B ) )
= ( ( member_real_a @ C2 @ A3 )
& ( member_real_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_880_IntI,axiom,
! [C2: a > a,A3: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ A3 )
=> ( ( member_a_a @ C2 @ B )
=> ( member_a_a @ C2 @ ( inf_inf_set_a_a @ A3 @ B ) ) ) ) ).
% IntI
thf(fact_881_IntI,axiom,
! [C2: a,A3: set_a,B: set_a] :
( ( member_a @ C2 @ A3 )
=> ( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( inf_inf_set_a @ A3 @ B ) ) ) ) ).
% IntI
thf(fact_882_IntI,axiom,
! [C2: set_a,A3: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ A3 )
=> ( ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A3 @ B ) ) ) ) ).
% IntI
thf(fact_883_IntI,axiom,
! [C2: real > a,A3: set_real_a,B: set_real_a] :
( ( member_real_a @ C2 @ A3 )
=> ( ( member_real_a @ C2 @ B )
=> ( member_real_a @ C2 @ ( inf_inf_set_real_a @ A3 @ B ) ) ) ) ).
% IntI
thf(fact_884_Int__insert__left__if0,axiom,
! [A2: a > a,C: set_a_a,B: set_a_a] :
( ~ ( member_a_a @ A2 @ C )
=> ( ( inf_inf_set_a_a @ ( insert_a_a @ A2 @ B ) @ C )
= ( inf_inf_set_a_a @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_885_Int__insert__left__if0,axiom,
! [A2: a,C: set_a,B: set_a] :
( ~ ( member_a @ A2 @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_886_Int__insert__left__if0,axiom,
! [A2: set_a,C: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A2 @ C )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_set_a @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_887_Int__insert__left__if0,axiom,
! [A2: real > a,C: set_real_a,B: set_real_a] :
( ~ ( member_real_a @ A2 @ C )
=> ( ( inf_inf_set_real_a @ ( insert_real_a @ A2 @ B ) @ C )
= ( inf_inf_set_real_a @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_888_Int__insert__left__if1,axiom,
! [A2: a > a,C: set_a_a,B: set_a_a] :
( ( member_a_a @ A2 @ C )
=> ( ( inf_inf_set_a_a @ ( insert_a_a @ A2 @ B ) @ C )
= ( insert_a_a @ A2 @ ( inf_inf_set_a_a @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_889_Int__insert__left__if1,axiom,
! [A2: a,C: set_a,B: set_a] :
( ( member_a @ A2 @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_890_Int__insert__left__if1,axiom,
! [A2: set_a,C: set_set_a,B: set_set_a] :
( ( member_set_a @ A2 @ C )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_891_Int__insert__left__if1,axiom,
! [A2: real > a,C: set_real_a,B: set_real_a] :
( ( member_real_a @ A2 @ C )
=> ( ( inf_inf_set_real_a @ ( insert_real_a @ A2 @ B ) @ C )
= ( insert_real_a @ A2 @ ( inf_inf_set_real_a @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_892_Int__insert__right__if0,axiom,
! [A2: a > a,A3: set_a_a,B: set_a_a] :
( ~ ( member_a_a @ A2 @ A3 )
=> ( ( inf_inf_set_a_a @ A3 @ ( insert_a_a @ A2 @ B ) )
= ( inf_inf_set_a_a @ A3 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_893_Int__insert__right__if0,axiom,
! [A2: a,A3: set_a,B: set_a] :
( ~ ( member_a @ A2 @ A3 )
=> ( ( inf_inf_set_a @ A3 @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A3 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_894_Int__insert__right__if0,axiom,
! [A2: set_a,A3: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A2 @ A3 )
=> ( ( inf_inf_set_set_a @ A3 @ ( insert_set_a @ A2 @ B ) )
= ( inf_inf_set_set_a @ A3 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_895_Int__insert__right__if0,axiom,
! [A2: real > a,A3: set_real_a,B: set_real_a] :
( ~ ( member_real_a @ A2 @ A3 )
=> ( ( inf_inf_set_real_a @ A3 @ ( insert_real_a @ A2 @ B ) )
= ( inf_inf_set_real_a @ A3 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_896_Int__insert__right__if1,axiom,
! [A2: a > a,A3: set_a_a,B: set_a_a] :
( ( member_a_a @ A2 @ A3 )
=> ( ( inf_inf_set_a_a @ A3 @ ( insert_a_a @ A2 @ B ) )
= ( insert_a_a @ A2 @ ( inf_inf_set_a_a @ A3 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_897_Int__insert__right__if1,axiom,
! [A2: a,A3: set_a,B: set_a] :
( ( member_a @ A2 @ A3 )
=> ( ( inf_inf_set_a @ A3 @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A3 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_898_Int__insert__right__if1,axiom,
! [A2: set_a,A3: set_set_a,B: set_set_a] :
( ( member_set_a @ A2 @ A3 )
=> ( ( inf_inf_set_set_a @ A3 @ ( insert_set_a @ A2 @ B ) )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ A3 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_899_Int__insert__right__if1,axiom,
! [A2: real > a,A3: set_real_a,B: set_real_a] :
( ( member_real_a @ A2 @ A3 )
=> ( ( inf_inf_set_real_a @ A3 @ ( insert_real_a @ A2 @ B ) )
= ( insert_real_a @ A2 @ ( inf_inf_set_real_a @ A3 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_900_sets_OInt,axiom,
! [A2: set_a,M: sigma_measure_a,B3: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ B3 @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ ( sigma_sets_a @ M ) ) ) ) ).
% sets.Int
thf(fact_901_fmeasurable_OInt,axiom,
! [A2: set_a,M: sigma_measure_a,B3: set_a] :
( ( member_set_a @ A2 @ ( measur3645360004775918570able_a @ M ) )
=> ( ( member_set_a @ B3 @ ( measur3645360004775918570able_a @ M ) )
=> ( member_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ ( measur3645360004775918570able_a @ M ) ) ) ) ).
% fmeasurable.Int
thf(fact_902_null__sets_OInt,axiom,
! [A2: set_a,M: sigma_measure_a,B3: set_a] :
( ( member_set_a @ A2 @ ( measure_null_sets_a @ M ) )
=> ( ( member_set_a @ B3 @ ( measure_null_sets_a @ M ) )
=> ( member_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ ( measure_null_sets_a @ M ) ) ) ) ).
% null_sets.Int
thf(fact_903_insert__disjoint_I1_J,axiom,
! [A2: a > a,A3: set_a_a,B: set_a_a] :
( ( ( inf_inf_set_a_a @ ( insert_a_a @ A2 @ A3 ) @ B )
= bot_bot_set_a_a )
= ( ~ ( member_a_a @ A2 @ B )
& ( ( inf_inf_set_a_a @ A3 @ B )
= bot_bot_set_a_a ) ) ) ).
% insert_disjoint(1)
thf(fact_904_insert__disjoint_I1_J,axiom,
! [A2: a,A3: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A2 @ A3 ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ A3 @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_905_insert__disjoint_I1_J,axiom,
! [A2: set_a,A3: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ A3 ) @ B )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A2 @ B )
& ( ( inf_inf_set_set_a @ A3 @ B )
= bot_bot_set_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_906_insert__disjoint_I1_J,axiom,
! [A2: real > a,A3: set_real_a,B: set_real_a] :
( ( ( inf_inf_set_real_a @ ( insert_real_a @ A2 @ A3 ) @ B )
= bot_bot_set_real_a )
= ( ~ ( member_real_a @ A2 @ B )
& ( ( inf_inf_set_real_a @ A3 @ B )
= bot_bot_set_real_a ) ) ) ).
% insert_disjoint(1)
thf(fact_907_insert__disjoint_I2_J,axiom,
! [A2: a > a,A3: set_a_a,B: set_a_a] :
( ( bot_bot_set_a_a
= ( inf_inf_set_a_a @ ( insert_a_a @ A2 @ A3 ) @ B ) )
= ( ~ ( member_a_a @ A2 @ B )
& ( bot_bot_set_a_a
= ( inf_inf_set_a_a @ A3 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_908_insert__disjoint_I2_J,axiom,
! [A2: a,A3: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A2 @ A3 ) @ B ) )
= ( ~ ( member_a @ A2 @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A3 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_909_insert__disjoint_I2_J,axiom,
! [A2: set_a,A3: set_set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ A3 ) @ B ) )
= ( ~ ( member_set_a @ A2 @ B )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A3 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_910_insert__disjoint_I2_J,axiom,
! [A2: real > a,A3: set_real_a,B: set_real_a] :
( ( bot_bot_set_real_a
= ( inf_inf_set_real_a @ ( insert_real_a @ A2 @ A3 ) @ B ) )
= ( ~ ( member_real_a @ A2 @ B )
& ( bot_bot_set_real_a
= ( inf_inf_set_real_a @ A3 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_911_disjoint__insert_I1_J,axiom,
! [B: set_a_a,A2: a > a,A3: set_a_a] :
( ( ( inf_inf_set_a_a @ B @ ( insert_a_a @ A2 @ A3 ) )
= bot_bot_set_a_a )
= ( ~ ( member_a_a @ A2 @ B )
& ( ( inf_inf_set_a_a @ B @ A3 )
= bot_bot_set_a_a ) ) ) ).
% disjoint_insert(1)
thf(fact_912_disjoint__insert_I1_J,axiom,
! [B: set_a,A2: a,A3: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A2 @ A3 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ B @ A3 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_913_disjoint__insert_I1_J,axiom,
! [B: set_set_a,A2: set_a,A3: set_set_a] :
( ( ( inf_inf_set_set_a @ B @ ( insert_set_a @ A2 @ A3 ) )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A2 @ B )
& ( ( inf_inf_set_set_a @ B @ A3 )
= bot_bot_set_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_914_disjoint__insert_I1_J,axiom,
! [B: set_real_a,A2: real > a,A3: set_real_a] :
( ( ( inf_inf_set_real_a @ B @ ( insert_real_a @ A2 @ A3 ) )
= bot_bot_set_real_a )
= ( ~ ( member_real_a @ A2 @ B )
& ( ( inf_inf_set_real_a @ B @ A3 )
= bot_bot_set_real_a ) ) ) ).
% disjoint_insert(1)
thf(fact_915_disjoint__insert_I2_J,axiom,
! [A3: set_a_a,B3: a > a,B: set_a_a] :
( ( bot_bot_set_a_a
= ( inf_inf_set_a_a @ A3 @ ( insert_a_a @ B3 @ B ) ) )
= ( ~ ( member_a_a @ B3 @ A3 )
& ( bot_bot_set_a_a
= ( inf_inf_set_a_a @ A3 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_916_disjoint__insert_I2_J,axiom,
! [A3: set_a,B3: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A3 @ ( insert_a @ B3 @ B ) ) )
= ( ~ ( member_a @ B3 @ A3 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A3 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_917_disjoint__insert_I2_J,axiom,
! [A3: set_set_a,B3: set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A3 @ ( insert_set_a @ B3 @ B ) ) )
= ( ~ ( member_set_a @ B3 @ A3 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A3 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_918_disjoint__insert_I2_J,axiom,
! [A3: set_real_a,B3: real > a,B: set_real_a] :
( ( bot_bot_set_real_a
= ( inf_inf_set_real_a @ A3 @ ( insert_real_a @ B3 @ B ) ) )
= ( ~ ( member_real_a @ B3 @ A3 )
& ( bot_bot_set_real_a
= ( inf_inf_set_real_a @ A3 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_919_sets_OInt__space__eq2,axiom,
! [X2: set_a,M: sigma_measure_a] :
( ( member_set_a @ X2 @ ( sigma_sets_a @ M ) )
=> ( ( inf_inf_set_a @ X2 @ ( sigma_space_a @ M ) )
= X2 ) ) ).
% sets.Int_space_eq2
thf(fact_920_sets_OInt__space__eq1,axiom,
! [X2: set_a,M: sigma_measure_a] :
( ( member_set_a @ X2 @ ( sigma_sets_a @ M ) )
=> ( ( inf_inf_set_a @ ( sigma_space_a @ M ) @ X2 )
= X2 ) ) ).
% sets.Int_space_eq1
thf(fact_921_fmeasurable_OInt__space__eq1,axiom,
! [X2: set_a,M: sigma_measure_a] :
( ( member_set_a @ X2 @ ( measur3645360004775918570able_a @ M ) )
=> ( ( inf_inf_set_a @ ( sigma_space_a @ M ) @ X2 )
= X2 ) ) ).
% fmeasurable.Int_space_eq1
thf(fact_922_fmeasurable_OInt__space__eq2,axiom,
! [X2: set_a,M: sigma_measure_a] :
( ( member_set_a @ X2 @ ( measur3645360004775918570able_a @ M ) )
=> ( ( inf_inf_set_a @ X2 @ ( sigma_space_a @ M ) )
= X2 ) ) ).
% fmeasurable.Int_space_eq2
thf(fact_923_null__sets_OInt__space__eq1,axiom,
! [X2: set_a,M: sigma_measure_a] :
( ( member_set_a @ X2 @ ( measure_null_sets_a @ M ) )
=> ( ( inf_inf_set_a @ ( sigma_space_a @ M ) @ X2 )
= X2 ) ) ).
% null_sets.Int_space_eq1
thf(fact_924_null__sets_OInt__space__eq2,axiom,
! [X2: set_a,M: sigma_measure_a] :
( ( member_set_a @ X2 @ ( measure_null_sets_a @ M ) )
=> ( ( inf_inf_set_a @ X2 @ ( sigma_space_a @ M ) )
= X2 ) ) ).
% null_sets.Int_space_eq2
thf(fact_925_Int__emptyI,axiom,
! [A3: set_a_a,B: set_a_a] :
( ! [X3: a > a] :
( ( member_a_a @ X3 @ A3 )
=> ~ ( member_a_a @ X3 @ B ) )
=> ( ( inf_inf_set_a_a @ A3 @ B )
= bot_bot_set_a_a ) ) ).
% Int_emptyI
thf(fact_926_Int__emptyI,axiom,
! [A3: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ~ ( member_a @ X3 @ B ) )
=> ( ( inf_inf_set_a @ A3 @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_927_Int__emptyI,axiom,
! [A3: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
=> ~ ( member_set_a @ X3 @ B ) )
=> ( ( inf_inf_set_set_a @ A3 @ B )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_928_Int__emptyI,axiom,
! [A3: set_real_a,B: set_real_a] :
( ! [X3: real > a] :
( ( member_real_a @ X3 @ A3 )
=> ~ ( member_real_a @ X3 @ B ) )
=> ( ( inf_inf_set_real_a @ A3 @ B )
= bot_bot_set_real_a ) ) ).
% Int_emptyI
thf(fact_929_disjoint__iff,axiom,
! [A3: set_a_a,B: set_a_a] :
( ( ( inf_inf_set_a_a @ A3 @ B )
= bot_bot_set_a_a )
= ( ! [X4: a > a] :
( ( member_a_a @ X4 @ A3 )
=> ~ ( member_a_a @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_930_disjoint__iff,axiom,
! [A3: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A3 @ B )
= bot_bot_set_a )
= ( ! [X4: a] :
( ( member_a @ X4 @ A3 )
=> ~ ( member_a @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_931_disjoint__iff,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A3 @ B )
= bot_bot_set_set_a )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A3 )
=> ~ ( member_set_a @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_932_disjoint__iff,axiom,
! [A3: set_real_a,B: set_real_a] :
( ( ( inf_inf_set_real_a @ A3 @ B )
= bot_bot_set_real_a )
= ( ! [X4: real > a] :
( ( member_real_a @ X4 @ A3 )
=> ~ ( member_real_a @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_933_Int__insert__left,axiom,
! [A2: a > a,C: set_a_a,B: set_a_a] :
( ( ( member_a_a @ A2 @ C )
=> ( ( inf_inf_set_a_a @ ( insert_a_a @ A2 @ B ) @ C )
= ( insert_a_a @ A2 @ ( inf_inf_set_a_a @ B @ C ) ) ) )
& ( ~ ( member_a_a @ A2 @ C )
=> ( ( inf_inf_set_a_a @ ( insert_a_a @ A2 @ B ) @ C )
= ( inf_inf_set_a_a @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_934_Int__insert__left,axiom,
! [A2: a,C: set_a,B: set_a] :
( ( ( member_a @ A2 @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C ) ) ) )
& ( ~ ( member_a @ A2 @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_935_Int__insert__left,axiom,
! [A2: set_a,C: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A2 @ C )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C ) ) ) )
& ( ~ ( member_set_a @ A2 @ C )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_set_a @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_936_Int__insert__left,axiom,
! [A2: real > a,C: set_real_a,B: set_real_a] :
( ( ( member_real_a @ A2 @ C )
=> ( ( inf_inf_set_real_a @ ( insert_real_a @ A2 @ B ) @ C )
= ( insert_real_a @ A2 @ ( inf_inf_set_real_a @ B @ C ) ) ) )
& ( ~ ( member_real_a @ A2 @ C )
=> ( ( inf_inf_set_real_a @ ( insert_real_a @ A2 @ B ) @ C )
= ( inf_inf_set_real_a @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_937_Int__insert__right,axiom,
! [A2: a > a,A3: set_a_a,B: set_a_a] :
( ( ( member_a_a @ A2 @ A3 )
=> ( ( inf_inf_set_a_a @ A3 @ ( insert_a_a @ A2 @ B ) )
= ( insert_a_a @ A2 @ ( inf_inf_set_a_a @ A3 @ B ) ) ) )
& ( ~ ( member_a_a @ A2 @ A3 )
=> ( ( inf_inf_set_a_a @ A3 @ ( insert_a_a @ A2 @ B ) )
= ( inf_inf_set_a_a @ A3 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_938_Int__insert__right,axiom,
! [A2: a,A3: set_a,B: set_a] :
( ( ( member_a @ A2 @ A3 )
=> ( ( inf_inf_set_a @ A3 @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A3 @ B ) ) ) )
& ( ~ ( member_a @ A2 @ A3 )
=> ( ( inf_inf_set_a @ A3 @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A3 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_939_Int__insert__right,axiom,
! [A2: set_a,A3: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A2 @ A3 )
=> ( ( inf_inf_set_set_a @ A3 @ ( insert_set_a @ A2 @ B ) )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ A3 @ B ) ) ) )
& ( ~ ( member_set_a @ A2 @ A3 )
=> ( ( inf_inf_set_set_a @ A3 @ ( insert_set_a @ A2 @ B ) )
= ( inf_inf_set_set_a @ A3 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_940_Int__insert__right,axiom,
! [A2: real > a,A3: set_real_a,B: set_real_a] :
( ( ( member_real_a @ A2 @ A3 )
=> ( ( inf_inf_set_real_a @ A3 @ ( insert_real_a @ A2 @ B ) )
= ( insert_real_a @ A2 @ ( inf_inf_set_real_a @ A3 @ B ) ) ) )
& ( ~ ( member_real_a @ A2 @ A3 )
=> ( ( inf_inf_set_real_a @ A3 @ ( insert_real_a @ A2 @ B ) )
= ( inf_inf_set_real_a @ A3 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_941_Int__Collect__mono,axiom,
! [A3: set_a_a,B: set_a_a,P: ( a > a ) > $o,Q: ( a > a ) > $o] :
( ( ord_less_eq_set_a_a @ A3 @ B )
=> ( ! [X3: a > a] :
( ( member_a_a @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_a_a @ ( inf_inf_set_a_a @ A3 @ ( collect_a_a @ P ) ) @ ( inf_inf_set_a_a @ B @ ( collect_a_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_942_Int__Collect__mono,axiom,
! [A3: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_943_Int__Collect__mono,axiom,
! [A3: set_set_a,B: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A3 @ B )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A3 @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_944_Int__Collect__mono,axiom,
! [A3: set_real_a,B: set_real_a,P: ( real > a ) > $o,Q: ( real > a ) > $o] :
( ( ord_le5743406823621094409real_a @ A3 @ B )
=> ( ! [X3: real > a] :
( ( member_real_a @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le5743406823621094409real_a @ ( inf_inf_set_real_a @ A3 @ ( collect_real_a @ P ) ) @ ( inf_inf_set_real_a @ B @ ( collect_real_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_945_IntD2,axiom,
! [C2: a > a,A3: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( inf_inf_set_a_a @ A3 @ B ) )
=> ( member_a_a @ C2 @ B ) ) ).
% IntD2
thf(fact_946_IntD2,axiom,
! [C2: a,A3: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A3 @ B ) )
=> ( member_a @ C2 @ B ) ) ).
% IntD2
thf(fact_947_IntD2,axiom,
! [C2: set_a,A3: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A3 @ B ) )
=> ( member_set_a @ C2 @ B ) ) ).
% IntD2
thf(fact_948_IntD2,axiom,
! [C2: real > a,A3: set_real_a,B: set_real_a] :
( ( member_real_a @ C2 @ ( inf_inf_set_real_a @ A3 @ B ) )
=> ( member_real_a @ C2 @ B ) ) ).
% IntD2
thf(fact_949_IntD1,axiom,
! [C2: a > a,A3: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( inf_inf_set_a_a @ A3 @ B ) )
=> ( member_a_a @ C2 @ A3 ) ) ).
% IntD1
thf(fact_950_IntD1,axiom,
! [C2: a,A3: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A3 @ B ) )
=> ( member_a @ C2 @ A3 ) ) ).
% IntD1
thf(fact_951_IntD1,axiom,
! [C2: set_a,A3: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A3 @ B ) )
=> ( member_set_a @ C2 @ A3 ) ) ).
% IntD1
thf(fact_952_IntD1,axiom,
! [C2: real > a,A3: set_real_a,B: set_real_a] :
( ( member_real_a @ C2 @ ( inf_inf_set_real_a @ A3 @ B ) )
=> ( member_real_a @ C2 @ A3 ) ) ).
% IntD1
thf(fact_953_IntE,axiom,
! [C2: a > a,A3: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( inf_inf_set_a_a @ A3 @ B ) )
=> ~ ( ( member_a_a @ C2 @ A3 )
=> ~ ( member_a_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_954_IntE,axiom,
! [C2: a,A3: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A3 @ B ) )
=> ~ ( ( member_a @ C2 @ A3 )
=> ~ ( member_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_955_IntE,axiom,
! [C2: set_a,A3: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A3 @ B ) )
=> ~ ( ( member_set_a @ C2 @ A3 )
=> ~ ( member_set_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_956_IntE,axiom,
! [C2: real > a,A3: set_real_a,B: set_real_a] :
( ( member_real_a @ C2 @ ( inf_inf_set_real_a @ A3 @ B ) )
=> ~ ( ( member_real_a @ C2 @ A3 )
=> ~ ( member_real_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_957_Int__stableD,axiom,
! [M: set_set_a,A2: set_a,B3: set_a] :
( ( sigma_Int_stable_a @ M )
=> ( ( member_set_a @ A2 @ M )
=> ( ( member_set_a @ B3 @ M )
=> ( member_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ M ) ) ) ) ).
% Int_stableD
thf(fact_958_Int__stableI,axiom,
! [A3: set_set_a] :
( ! [A: set_a,B2: set_a] :
( ( member_set_a @ A @ A3 )
=> ( ( member_set_a @ B2 @ A3 )
=> ( member_set_a @ ( inf_inf_set_a @ A @ B2 ) @ A3 ) ) )
=> ( sigma_Int_stable_a @ A3 ) ) ).
% Int_stableI
thf(fact_959_Int__stable__def,axiom,
( sigma_Int_stable_a
= ( ^ [M2: set_set_a] :
! [X4: set_a] :
( ( member_set_a @ X4 @ M2 )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ M2 )
=> ( member_set_a @ ( inf_inf_set_a @ X4 @ Y4 ) @ M2 ) ) ) ) ) ).
% Int_stable_def
thf(fact_960_semiring__of__sets_OInt,axiom,
! [Omega: set_a,M: set_set_a,A2: set_a,B3: set_a] :
( ( sigma_8461971822185508616sets_a @ Omega @ M )
=> ( ( member_set_a @ A2 @ M )
=> ( ( member_set_a @ B3 @ M )
=> ( member_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ M ) ) ) ) ).
% semiring_of_sets.Int
thf(fact_961_semiring__of__sets_OInt__space__eq1,axiom,
! [Omega: set_a,M: set_set_a,X2: set_a] :
( ( sigma_8461971822185508616sets_a @ Omega @ M )
=> ( ( member_set_a @ X2 @ M )
=> ( ( inf_inf_set_a @ Omega @ X2 )
= X2 ) ) ) ).
% semiring_of_sets.Int_space_eq1
thf(fact_962_semiring__of__sets_OInt__space__eq2,axiom,
! [Omega: set_a,M: set_set_a,X2: set_a] :
( ( sigma_8461971822185508616sets_a @ Omega @ M )
=> ( ( member_set_a @ X2 @ M )
=> ( ( inf_inf_set_a @ X2 @ Omega )
= X2 ) ) ) ).
% semiring_of_sets.Int_space_eq2
thf(fact_963_fmeasurable__Int__fmeasurable,axiom,
! [S: set_a,M: sigma_measure_a,T2: set_a] :
( ( member_set_a @ S @ ( measur3645360004775918570able_a @ M ) )
=> ( ( member_set_a @ T2 @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( inf_inf_set_a @ S @ T2 ) @ ( measur3645360004775918570able_a @ M ) ) ) ) ).
% fmeasurable_Int_fmeasurable
thf(fact_964_null__set__Int1,axiom,
! [B: set_a,M: sigma_measure_a,A3: set_a] :
( ( member_set_a @ B @ ( measure_null_sets_a @ M ) )
=> ( ( member_set_a @ A3 @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( inf_inf_set_a @ A3 @ B ) @ ( measure_null_sets_a @ M ) ) ) ) ).
% null_set_Int1
thf(fact_965_null__set__Int2,axiom,
! [B: set_a,M: sigma_measure_a,A3: set_a] :
( ( member_set_a @ B @ ( measure_null_sets_a @ M ) )
=> ( ( member_set_a @ A3 @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( inf_inf_set_a @ B @ A3 ) @ ( measure_null_sets_a @ M ) ) ) ) ).
% null_set_Int2
thf(fact_966_algebra_Osmallest__ccdi__sets__Int,axiom,
! [Omega: set_a,M: set_set_a,B3: set_a,A2: set_a] :
( ( sigma_algebra_a @ Omega @ M )
=> ( ( member_set_a @ B3 @ ( sigma_5648178489087971417sets_a @ Omega @ M ) )
=> ( ( member_set_a @ A2 @ ( sigma_5648178489087971417sets_a @ Omega @ M ) )
=> ( member_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ ( sigma_5648178489087971417sets_a @ Omega @ M ) ) ) ) ) ).
% algebra.smallest_ccdi_sets_Int
thf(fact_967_algebra_Osmallest__ccdi__sets__Int1,axiom,
! [Omega: set_a,M: set_set_a,A2: set_a,B3: set_a] :
( ( sigma_algebra_a @ Omega @ M )
=> ( ( member_set_a @ A2 @ M )
=> ( ( member_set_a @ B3 @ ( sigma_5648178489087971417sets_a @ Omega @ M ) )
=> ( member_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ ( sigma_5648178489087971417sets_a @ Omega @ M ) ) ) ) ) ).
% algebra.smallest_ccdi_sets_Int1
thf(fact_968_algebra_Olambda__system__Int,axiom,
! [Omega: set_a,M: set_set_a,X2: set_a,F: set_a > extend8495563244428889912nnreal,Y3: set_a] :
( ( sigma_algebra_a @ Omega @ M )
=> ( ( member_set_a @ X2 @ ( lambda_system_a @ Omega @ M @ F ) )
=> ( ( member_set_a @ Y3 @ ( lambda_system_a @ Omega @ M @ F ) )
=> ( member_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ ( lambda_system_a @ Omega @ M @ F ) ) ) ) ) ).
% algebra.lambda_system_Int
thf(fact_969_is__singletonI_H,axiom,
! [A3: set_a_a] :
( ( A3 != bot_bot_set_a_a )
=> ( ! [X3: a > a,Y5: a > a] :
( ( member_a_a @ X3 @ A3 )
=> ( ( member_a_a @ Y5 @ A3 )
=> ( X3 = Y5 ) ) )
=> ( is_singleton_a_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_970_is__singletonI_H,axiom,
! [A3: set_a] :
( ( A3 != bot_bot_set_a )
=> ( ! [X3: a,Y5: a] :
( ( member_a @ X3 @ A3 )
=> ( ( member_a @ Y5 @ A3 )
=> ( X3 = Y5 ) ) )
=> ( is_singleton_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_971_is__singletonI_H,axiom,
! [A3: set_set_a] :
( ( A3 != bot_bot_set_set_a )
=> ( ! [X3: set_a,Y5: set_a] :
( ( member_set_a @ X3 @ A3 )
=> ( ( member_set_a @ Y5 @ A3 )
=> ( X3 = Y5 ) ) )
=> ( is_singleton_set_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_972_is__singletonI_H,axiom,
! [A3: set_real_a] :
( ( A3 != bot_bot_set_real_a )
=> ( ! [X3: real > a,Y5: real > a] :
( ( member_real_a @ X3 @ A3 )
=> ( ( member_real_a @ Y5 @ A3 )
=> ( X3 = Y5 ) ) )
=> ( is_singleton_real_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_973_sets_Osmallest__ccdi__sets__Int1,axiom,
! [A2: set_a,M: sigma_measure_a,B3: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ B3 @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) )
=> ( member_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) ) ) ) ).
% sets.smallest_ccdi_sets_Int1
thf(fact_974_sets_Osmallest__ccdi__sets__Int,axiom,
! [B3: set_a,M: sigma_measure_a,A2: set_a] :
( ( member_set_a @ B3 @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) )
=> ( ( member_set_a @ A2 @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) )
=> ( member_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) ) ) ) ).
% sets.smallest_ccdi_sets_Int
thf(fact_975_sets_Olambda__system__Int,axiom,
! [X2: set_a,M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal,Y3: set_a] :
( ( member_set_a @ X2 @ ( lambda_system_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) @ F ) )
=> ( ( member_set_a @ Y3 @ ( lambda_system_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) @ F ) )
=> ( member_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ ( lambda_system_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) @ F ) ) ) ) ).
% sets.lambda_system_Int
thf(fact_976_inf_Obounded__iff,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ ( inf_in7439215052339218890nnreal @ B3 @ C2 ) )
= ( ( ord_le3935885782089961368nnreal @ A2 @ B3 )
& ( ord_le3935885782089961368nnreal @ A2 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_977_le__inf__iff,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ ( inf_in7439215052339218890nnreal @ Y3 @ Z3 ) )
= ( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
& ( ord_le3935885782089961368nnreal @ X2 @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_978_inf__qbs__space,axiom,
! [X: quasi_borel_a,X7: quasi_borel_a] :
( ( qbs_space_a @ ( inf_quasi_borel_a @ X @ X7 ) )
= ( inf_inf_set_a @ ( qbs_space_a @ X ) @ ( qbs_space_a @ X7 ) ) ) ).
% inf_qbs_space
thf(fact_979_inf__qbs__Mx,axiom,
! [X: quasi_borel_a,X7: quasi_borel_a] :
( ( qbs_Mx_a @ ( inf_quasi_borel_a @ X @ X7 ) )
= ( inf_inf_set_real_a @ ( qbs_Mx_a @ X ) @ ( qbs_Mx_a @ X7 ) ) ) ).
% inf_qbs_Mx
thf(fact_980_inf_OcoboundedI2,axiom,
! [B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B3 @ C2 )
=> ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ A2 @ B3 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_981_inf_OcoboundedI1,axiom,
! [A2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ C2 )
=> ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ A2 @ B3 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_982_inf_Oabsorb__iff2,axiom,
( ord_le3935885782089961368nnreal
= ( ^ [B5: extend8495563244428889912nnreal,A5: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_983_inf_Oabsorb__iff1,axiom,
( ord_le3935885782089961368nnreal
= ( ^ [A5: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_984_inf_Ocobounded2,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ A2 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_985_inf_Ocobounded1,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ A2 @ B3 ) @ A2 ) ).
% inf.cobounded1
thf(fact_986_inf_Oorder__iff,axiom,
( ord_le3935885782089961368nnreal
= ( ^ [A5: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
( A5
= ( inf_in7439215052339218890nnreal @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_987_inf__greatest,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
=> ( ( ord_le3935885782089961368nnreal @ X2 @ Z3 )
=> ( ord_le3935885782089961368nnreal @ X2 @ ( inf_in7439215052339218890nnreal @ Y3 @ Z3 ) ) ) ) ).
% inf_greatest
thf(fact_988_inf_OboundedI,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B3 )
=> ( ( ord_le3935885782089961368nnreal @ A2 @ C2 )
=> ( ord_le3935885782089961368nnreal @ A2 @ ( inf_in7439215052339218890nnreal @ B3 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_989_inf_OboundedE,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ ( inf_in7439215052339218890nnreal @ B3 @ C2 ) )
=> ~ ( ( ord_le3935885782089961368nnreal @ A2 @ B3 )
=> ~ ( ord_le3935885782089961368nnreal @ A2 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_990_inf__absorb2,axiom,
! [Y3: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Y3 @ X2 )
=> ( ( inf_in7439215052339218890nnreal @ X2 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_991_inf__absorb1,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
=> ( ( inf_in7439215052339218890nnreal @ X2 @ Y3 )
= X2 ) ) ).
% inf_absorb1
thf(fact_992_inf_Oabsorb2,axiom,
! [B3: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B3 @ A2 )
=> ( ( inf_in7439215052339218890nnreal @ A2 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_993_inf_Oabsorb1,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B3 )
=> ( ( inf_in7439215052339218890nnreal @ A2 @ B3 )
= A2 ) ) ).
% inf.absorb1
thf(fact_994_le__iff__inf,axiom,
( ord_le3935885782089961368nnreal
= ( ^ [X4: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_995_inf__unique,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( F @ X3 @ Y5 ) @ X3 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( F @ X3 @ Y5 ) @ Y5 )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal,Z4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
=> ( ( ord_le3935885782089961368nnreal @ X3 @ Z4 )
=> ( ord_le3935885782089961368nnreal @ X3 @ ( F @ Y5 @ Z4 ) ) ) )
=> ( ( inf_in7439215052339218890nnreal @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_996_inf_OorderI,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( A2
= ( inf_in7439215052339218890nnreal @ A2 @ B3 ) )
=> ( ord_le3935885782089961368nnreal @ A2 @ B3 ) ) ).
% inf.orderI
thf(fact_997_inf_OorderE,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B3 )
=> ( A2
= ( inf_in7439215052339218890nnreal @ A2 @ B3 ) ) ) ).
% inf.orderE
thf(fact_998_le__infI2,axiom,
! [B3: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B3 @ X2 )
=> ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ A2 @ B3 ) @ X2 ) ) ).
% le_infI2
thf(fact_999_le__infI1,axiom,
! [A2: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ X2 )
=> ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ A2 @ B3 ) @ X2 ) ) ).
% le_infI1
thf(fact_1000_inf__mono,axiom,
! [A2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,D3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ C2 )
=> ( ( ord_le3935885782089961368nnreal @ B3 @ D3 )
=> ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ A2 @ B3 ) @ ( inf_in7439215052339218890nnreal @ C2 @ D3 ) ) ) ) ).
% inf_mono
thf(fact_1001_le__infI,axiom,
! [X2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ A2 )
=> ( ( ord_le3935885782089961368nnreal @ X2 @ B3 )
=> ( ord_le3935885782089961368nnreal @ X2 @ ( inf_in7439215052339218890nnreal @ A2 @ B3 ) ) ) ) ).
% le_infI
thf(fact_1002_le__infE,axiom,
! [X2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ ( inf_in7439215052339218890nnreal @ A2 @ B3 ) )
=> ~ ( ( ord_le3935885782089961368nnreal @ X2 @ A2 )
=> ~ ( ord_le3935885782089961368nnreal @ X2 @ B3 ) ) ) ).
% le_infE
thf(fact_1003_inf__le2,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ X2 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_1004_inf__le1,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ X2 @ Y3 ) @ X2 ) ).
% inf_le1
thf(fact_1005_inf__sup__ord_I1_J,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ X2 @ Y3 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_1006_inf__sup__ord_I2_J,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ X2 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_1007_less__infI1,axiom,
! [A2: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ X2 )
=> ( ord_le7381754540660121996nnreal @ ( inf_in7439215052339218890nnreal @ A2 @ B3 ) @ X2 ) ) ).
% less_infI1
thf(fact_1008_less__infI2,axiom,
! [B3: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B3 @ X2 )
=> ( ord_le7381754540660121996nnreal @ ( inf_in7439215052339218890nnreal @ A2 @ B3 ) @ X2 ) ) ).
% less_infI2
thf(fact_1009_inf_Oabsorb3,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ B3 )
=> ( ( inf_in7439215052339218890nnreal @ A2 @ B3 )
= A2 ) ) ).
% inf.absorb3
thf(fact_1010_inf_Oabsorb4,axiom,
! [B3: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B3 @ A2 )
=> ( ( inf_in7439215052339218890nnreal @ A2 @ B3 )
= B3 ) ) ).
% inf.absorb4
thf(fact_1011_inf_Ostrict__boundedE,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ ( inf_in7439215052339218890nnreal @ B3 @ C2 ) )
=> ~ ( ( ord_le7381754540660121996nnreal @ A2 @ B3 )
=> ~ ( ord_le7381754540660121996nnreal @ A2 @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_1012_inf_Ostrict__order__iff,axiom,
( ord_le7381754540660121996nnreal
= ( ^ [A5: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
( ( A5
= ( inf_in7439215052339218890nnreal @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_1013_inf_Ostrict__coboundedI1,axiom,
! [A2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ C2 )
=> ( ord_le7381754540660121996nnreal @ ( inf_in7439215052339218890nnreal @ A2 @ B3 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_1014_inf_Ostrict__coboundedI2,axiom,
! [B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B3 @ C2 )
=> ( ord_le7381754540660121996nnreal @ ( inf_in7439215052339218890nnreal @ A2 @ B3 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_1015_measurable__envelope__def,axiom,
( comple1317578422686860819lope_a
= ( ^ [M2: sigma_measure_a,A4: set_a,E2: set_a] :
( ( ord_less_eq_set_a @ A4 @ E2 )
& ( member_set_a @ E2 @ ( sigma_sets_a @ M2 ) )
& ! [X4: set_a] :
( ( member_set_a @ X4 @ ( sigma_sets_a @ M2 ) )
=> ( ( sigma_emeasure_a @ M2 @ ( inf_inf_set_a @ X4 @ E2 ) )
= ( comple3326625653960060650e_of_a @ M2 @ ( inf_inf_set_a @ X4 @ A4 ) ) ) ) ) ) ) ).
% measurable_envelope_def
thf(fact_1016_sub__qbs__space,axiom,
! [X: quasi_borel_a,U: set_a] :
( ( qbs_space_a @ ( sub_qbs_a @ X @ U ) )
= ( inf_inf_set_a @ ( qbs_space_a @ X ) @ U ) ) ).
% sub_qbs_space
thf(fact_1017_measurable__envelopeD_I3_J,axiom,
! [M: sigma_measure_a,A3: set_a,E: set_a,F4: set_a] :
( ( comple1317578422686860819lope_a @ M @ A3 @ E )
=> ( ( member_set_a @ F4 @ ( sigma_sets_a @ M ) )
=> ( ( sigma_emeasure_a @ M @ ( inf_inf_set_a @ F4 @ E ) )
= ( comple3326625653960060650e_of_a @ M @ ( inf_inf_set_a @ F4 @ A3 ) ) ) ) ) ).
% measurable_envelopeD(3)
thf(fact_1018_measurable__envelopeD_I2_J,axiom,
! [M: sigma_measure_a,A3: set_a,E: set_a] :
( ( comple1317578422686860819lope_a @ M @ A3 @ E )
=> ( member_set_a @ E @ ( sigma_sets_a @ M ) ) ) ).
% measurable_envelopeD(2)
thf(fact_1019_measurable__envelopeD_I4_J,axiom,
! [M: sigma_measure_a,A3: set_a,E: set_a] :
( ( comple1317578422686860819lope_a @ M @ A3 @ E )
=> ( ord_less_eq_set_a @ A3 @ ( sigma_space_a @ M ) ) ) ).
% measurable_envelopeD(4)
thf(fact_1020_unsigned__Hahn__decomposition,axiom,
! [N: sigma_measure_a,M: sigma_measure_a,A3: set_a] :
( ( ( sigma_sets_a @ N )
= ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ A3 @ ( sigma_sets_a @ M ) )
=> ( ( ( sigma_emeasure_a @ M @ A3 )
!= top_to1496364449551166952nnreal )
=> ( ( ( sigma_emeasure_a @ N @ A3 )
!= top_to1496364449551166952nnreal )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ ( sigma_sets_a @ M ) )
& ( ord_less_eq_set_a @ X3 @ A3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( sigma_sets_a @ M ) )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ N @ Xa ) @ ( sigma_emeasure_a @ M @ Xa ) ) ) )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( sigma_sets_a @ M ) )
=> ( ( ord_less_eq_set_a @ Xa @ A3 )
=> ( ( ( inf_inf_set_a @ Xa @ X3 )
= bot_bot_set_a )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ M @ Xa ) @ ( sigma_emeasure_a @ N @ Xa ) ) ) ) ) ) ) ) ) ) ).
% unsigned_Hahn_decomposition
thf(fact_1021_main__part__null__part__Un,axiom,
! [S: set_a,M: sigma_measure_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ( ( sup_sup_set_a @ ( complete_main_part_a @ M @ S ) @ ( complete_null_part_a @ M @ S ) )
= S ) ) ).
% main_part_null_part_Un
thf(fact_1022_UnCI,axiom,
! [C2: a > a,B: set_a_a,A3: set_a_a] :
( ( ~ ( member_a_a @ C2 @ B )
=> ( member_a_a @ C2 @ A3 ) )
=> ( member_a_a @ C2 @ ( sup_sup_set_a_a @ A3 @ B ) ) ) ).
% UnCI
thf(fact_1023_UnCI,axiom,
! [C2: a,B: set_a,A3: set_a] :
( ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ A3 ) )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B ) ) ) ).
% UnCI
thf(fact_1024_UnCI,axiom,
! [C2: set_a,B: set_set_a,A3: set_set_a] :
( ( ~ ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ A3 ) )
=> ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A3 @ B ) ) ) ).
% UnCI
thf(fact_1025_UnCI,axiom,
! [C2: real > a,B: set_real_a,A3: set_real_a] :
( ( ~ ( member_real_a @ C2 @ B )
=> ( member_real_a @ C2 @ A3 ) )
=> ( member_real_a @ C2 @ ( sup_sup_set_real_a @ A3 @ B ) ) ) ).
% UnCI
thf(fact_1026_Un__iff,axiom,
! [C2: a > a,A3: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( sup_sup_set_a_a @ A3 @ B ) )
= ( ( member_a_a @ C2 @ A3 )
| ( member_a_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_1027_Un__iff,axiom,
! [C2: a,A3: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B ) )
= ( ( member_a @ C2 @ A3 )
| ( member_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_1028_Un__iff,axiom,
! [C2: set_a,A3: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A3 @ B ) )
= ( ( member_set_a @ C2 @ A3 )
| ( member_set_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_1029_Un__iff,axiom,
! [C2: real > a,A3: set_real_a,B: set_real_a] :
( ( member_real_a @ C2 @ ( sup_sup_set_real_a @ A3 @ B ) )
= ( ( member_real_a @ C2 @ A3 )
| ( member_real_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_1030_sup_Obounded__iff,axiom,
! [B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ ( sup_su6922871097908479076nnreal @ B3 @ C2 ) @ A2 )
= ( ( ord_le3935885782089961368nnreal @ B3 @ A2 )
& ( ord_le3935885782089961368nnreal @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_1031_le__sup__iff,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ ( sup_su6922871097908479076nnreal @ X2 @ Y3 ) @ Z3 )
= ( ( ord_le3935885782089961368nnreal @ X2 @ Z3 )
& ( ord_le3935885782089961368nnreal @ Y3 @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_1032_sets_OUn,axiom,
! [A2: set_a,M: sigma_measure_a,B3: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ B3 @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( sup_sup_set_a @ A2 @ B3 ) @ ( sigma_sets_a @ M ) ) ) ) ).
% sets.Un
thf(fact_1033_fmeasurable_OUn,axiom,
! [A2: set_a,M: sigma_measure_a,B3: set_a] :
( ( member_set_a @ A2 @ ( measur3645360004775918570able_a @ M ) )
=> ( ( member_set_a @ B3 @ ( measur3645360004775918570able_a @ M ) )
=> ( member_set_a @ ( sup_sup_set_a @ A2 @ B3 ) @ ( measur3645360004775918570able_a @ M ) ) ) ) ).
% fmeasurable.Un
thf(fact_1034_null__sets_OUn,axiom,
! [A2: set_a,M: sigma_measure_a,B3: set_a] :
( ( member_set_a @ A2 @ ( measure_null_sets_a @ M ) )
=> ( ( member_set_a @ B3 @ ( measure_null_sets_a @ M ) )
=> ( member_set_a @ ( sup_sup_set_a @ A2 @ B3 ) @ ( measure_null_sets_a @ M ) ) ) ) ).
% null_sets.Un
thf(fact_1035_distrib__sup__le,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( sup_su6922871097908479076nnreal @ X2 @ ( inf_in7439215052339218890nnreal @ Y3 @ Z3 ) ) @ ( inf_in7439215052339218890nnreal @ ( sup_su6922871097908479076nnreal @ X2 @ Y3 ) @ ( sup_su6922871097908479076nnreal @ X2 @ Z3 ) ) ) ).
% distrib_sup_le
thf(fact_1036_distrib__inf__le,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( sup_su6922871097908479076nnreal @ ( inf_in7439215052339218890nnreal @ X2 @ Y3 ) @ ( inf_in7439215052339218890nnreal @ X2 @ Z3 ) ) @ ( inf_in7439215052339218890nnreal @ X2 @ ( sup_su6922871097908479076nnreal @ Y3 @ Z3 ) ) ) ).
% distrib_inf_le
thf(fact_1037_ring__of__sets_OUn,axiom,
! [Omega: set_a,M: set_set_a,A2: set_a,B3: set_a] :
( ( sigma_ring_of_sets_a @ Omega @ M )
=> ( ( member_set_a @ A2 @ M )
=> ( ( member_set_a @ B3 @ M )
=> ( member_set_a @ ( sup_sup_set_a @ A2 @ B3 ) @ M ) ) ) ) ).
% ring_of_sets.Un
thf(fact_1038_sigma__sets__Un,axiom,
! [A2: set_a,Sp: set_a,A3: set_set_a,B3: set_a] :
( ( member_set_a @ A2 @ ( sigma_sigma_sets_a @ Sp @ A3 ) )
=> ( ( member_set_a @ B3 @ ( sigma_sigma_sets_a @ Sp @ A3 ) )
=> ( member_set_a @ ( sup_sup_set_a @ A2 @ B3 ) @ ( sigma_sigma_sets_a @ Sp @ A3 ) ) ) ) ).
% sigma_sets_Un
thf(fact_1039_top__greatest,axiom,
! [A2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ A2 @ top_to1496364449551166952nnreal ) ).
% top_greatest
thf(fact_1040_top_Oextremum__unique,axiom,
! [A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ top_to1496364449551166952nnreal @ A2 )
= ( A2 = top_to1496364449551166952nnreal ) ) ).
% top.extremum_unique
thf(fact_1041_top_Oextremum__uniqueI,axiom,
! [A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ top_to1496364449551166952nnreal @ A2 )
=> ( A2 = top_to1496364449551166952nnreal ) ) ).
% top.extremum_uniqueI
thf(fact_1042_top_Onot__eq__extremum,axiom,
! [A2: extend8495563244428889912nnreal] :
( ( A2 != top_to1496364449551166952nnreal )
= ( ord_le7381754540660121996nnreal @ A2 @ top_to1496364449551166952nnreal ) ) ).
% top.not_eq_extremum
thf(fact_1043_top_Oextremum__strict,axiom,
! [A2: extend8495563244428889912nnreal] :
~ ( ord_le7381754540660121996nnreal @ top_to1496364449551166952nnreal @ A2 ) ).
% top.extremum_strict
thf(fact_1044_UnE,axiom,
! [C2: a > a,A3: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( sup_sup_set_a_a @ A3 @ B ) )
=> ( ~ ( member_a_a @ C2 @ A3 )
=> ( member_a_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_1045_UnE,axiom,
! [C2: a,A3: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B ) )
=> ( ~ ( member_a @ C2 @ A3 )
=> ( member_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_1046_UnE,axiom,
! [C2: set_a,A3: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A3 @ B ) )
=> ( ~ ( member_set_a @ C2 @ A3 )
=> ( member_set_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_1047_UnE,axiom,
! [C2: real > a,A3: set_real_a,B: set_real_a] :
( ( member_real_a @ C2 @ ( sup_sup_set_real_a @ A3 @ B ) )
=> ( ~ ( member_real_a @ C2 @ A3 )
=> ( member_real_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_1048_UnI1,axiom,
! [C2: a > a,A3: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ A3 )
=> ( member_a_a @ C2 @ ( sup_sup_set_a_a @ A3 @ B ) ) ) ).
% UnI1
thf(fact_1049_UnI1,axiom,
! [C2: a,A3: set_a,B: set_a] :
( ( member_a @ C2 @ A3 )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B ) ) ) ).
% UnI1
thf(fact_1050_UnI1,axiom,
! [C2: set_a,A3: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ A3 )
=> ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A3 @ B ) ) ) ).
% UnI1
thf(fact_1051_UnI1,axiom,
! [C2: real > a,A3: set_real_a,B: set_real_a] :
( ( member_real_a @ C2 @ A3 )
=> ( member_real_a @ C2 @ ( sup_sup_set_real_a @ A3 @ B ) ) ) ).
% UnI1
thf(fact_1052_UnI2,axiom,
! [C2: a > a,B: set_a_a,A3: set_a_a] :
( ( member_a_a @ C2 @ B )
=> ( member_a_a @ C2 @ ( sup_sup_set_a_a @ A3 @ B ) ) ) ).
% UnI2
thf(fact_1053_UnI2,axiom,
! [C2: a,B: set_a,A3: set_a] :
( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B ) ) ) ).
% UnI2
thf(fact_1054_UnI2,axiom,
! [C2: set_a,B: set_set_a,A3: set_set_a] :
( ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A3 @ B ) ) ) ).
% UnI2
thf(fact_1055_UnI2,axiom,
! [C2: real > a,B: set_real_a,A3: set_real_a] :
( ( member_real_a @ C2 @ B )
=> ( member_real_a @ C2 @ ( sup_sup_set_real_a @ A3 @ B ) ) ) ).
% UnI2
thf(fact_1056_ring__of__sets__axioms__def,axiom,
( sigma_7143023581266409363ioms_a
= ( ^ [M2: set_set_a] :
! [A5: set_a,B5: set_a] :
( ( member_set_a @ A5 @ M2 )
=> ( ( member_set_a @ B5 @ M2 )
=> ( member_set_a @ ( sup_sup_set_a @ A5 @ B5 ) @ M2 ) ) ) ) ) ).
% ring_of_sets_axioms_def
thf(fact_1057_ring__of__sets__axioms_Ointro,axiom,
! [M: set_set_a] :
( ! [A: set_a,B2: set_a] :
( ( member_set_a @ A @ M )
=> ( ( member_set_a @ B2 @ M )
=> ( member_set_a @ ( sup_sup_set_a @ A @ B2 ) @ M ) ) )
=> ( sigma_7143023581266409363ioms_a @ M ) ) ).
% ring_of_sets_axioms.intro
thf(fact_1058_sup_Ostrict__coboundedI2,axiom,
! [C2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ C2 @ B3 )
=> ( ord_le7381754540660121996nnreal @ C2 @ ( sup_su6922871097908479076nnreal @ A2 @ B3 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_1059_sup_Ostrict__coboundedI1,axiom,
! [C2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ C2 @ A2 )
=> ( ord_le7381754540660121996nnreal @ C2 @ ( sup_su6922871097908479076nnreal @ A2 @ B3 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_1060_sup_Ostrict__order__iff,axiom,
( ord_le7381754540660121996nnreal
= ( ^ [B5: extend8495563244428889912nnreal,A5: extend8495563244428889912nnreal] :
( ( A5
= ( sup_su6922871097908479076nnreal @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1061_sup_Ostrict__boundedE,axiom,
! [B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ ( sup_su6922871097908479076nnreal @ B3 @ C2 ) @ A2 )
=> ~ ( ( ord_le7381754540660121996nnreal @ B3 @ A2 )
=> ~ ( ord_le7381754540660121996nnreal @ C2 @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_1062_sup_Oabsorb4,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ B3 )
=> ( ( sup_su6922871097908479076nnreal @ A2 @ B3 )
= B3 ) ) ).
% sup.absorb4
thf(fact_1063_sup_Oabsorb3,axiom,
! [B3: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B3 @ A2 )
=> ( ( sup_su6922871097908479076nnreal @ A2 @ B3 )
= A2 ) ) ).
% sup.absorb3
thf(fact_1064_less__supI2,axiom,
! [X2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ B3 )
=> ( ord_le7381754540660121996nnreal @ X2 @ ( sup_su6922871097908479076nnreal @ A2 @ B3 ) ) ) ).
% less_supI2
thf(fact_1065_less__supI1,axiom,
! [X2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ A2 )
=> ( ord_le7381754540660121996nnreal @ X2 @ ( sup_su6922871097908479076nnreal @ A2 @ B3 ) ) ) ).
% less_supI1
thf(fact_1066_inf__sup__ord_I4_J,axiom,
! [Y3: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ Y3 @ ( sup_su6922871097908479076nnreal @ X2 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_1067_inf__sup__ord_I3_J,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ X2 @ ( sup_su6922871097908479076nnreal @ X2 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_1068_le__supE,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ ( sup_su6922871097908479076nnreal @ A2 @ B3 ) @ X2 )
=> ~ ( ( ord_le3935885782089961368nnreal @ A2 @ X2 )
=> ~ ( ord_le3935885782089961368nnreal @ B3 @ X2 ) ) ) ).
% le_supE
thf(fact_1069_le__supI,axiom,
! [A2: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ X2 )
=> ( ( ord_le3935885782089961368nnreal @ B3 @ X2 )
=> ( ord_le3935885782089961368nnreal @ ( sup_su6922871097908479076nnreal @ A2 @ B3 ) @ X2 ) ) ) ).
% le_supI
thf(fact_1070_sup__ge1,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ X2 @ ( sup_su6922871097908479076nnreal @ X2 @ Y3 ) ) ).
% sup_ge1
thf(fact_1071_sup__ge2,axiom,
! [Y3: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ Y3 @ ( sup_su6922871097908479076nnreal @ X2 @ Y3 ) ) ).
% sup_ge2
thf(fact_1072_le__supI1,axiom,
! [X2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ A2 )
=> ( ord_le3935885782089961368nnreal @ X2 @ ( sup_su6922871097908479076nnreal @ A2 @ B3 ) ) ) ).
% le_supI1
thf(fact_1073_le__supI2,axiom,
! [X2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ B3 )
=> ( ord_le3935885782089961368nnreal @ X2 @ ( sup_su6922871097908479076nnreal @ A2 @ B3 ) ) ) ).
% le_supI2
thf(fact_1074_sup_Omono,axiom,
! [C2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal,D3: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ C2 @ A2 )
=> ( ( ord_le3935885782089961368nnreal @ D3 @ B3 )
=> ( ord_le3935885782089961368nnreal @ ( sup_su6922871097908479076nnreal @ C2 @ D3 ) @ ( sup_su6922871097908479076nnreal @ A2 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_1075_sup__mono,axiom,
! [A2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,D3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ C2 )
=> ( ( ord_le3935885782089961368nnreal @ B3 @ D3 )
=> ( ord_le3935885782089961368nnreal @ ( sup_su6922871097908479076nnreal @ A2 @ B3 ) @ ( sup_su6922871097908479076nnreal @ C2 @ D3 ) ) ) ) ).
% sup_mono
thf(fact_1076_sup__least,axiom,
! [Y3: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Y3 @ X2 )
=> ( ( ord_le3935885782089961368nnreal @ Z3 @ X2 )
=> ( ord_le3935885782089961368nnreal @ ( sup_su6922871097908479076nnreal @ Y3 @ Z3 ) @ X2 ) ) ) ).
% sup_least
thf(fact_1077_le__iff__sup,axiom,
( ord_le3935885782089961368nnreal
= ( ^ [X4: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( sup_su6922871097908479076nnreal @ X4 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_1078_sup_OorderE,axiom,
! [B3: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B3 @ A2 )
=> ( A2
= ( sup_su6922871097908479076nnreal @ A2 @ B3 ) ) ) ).
% sup.orderE
thf(fact_1079_sup_OorderI,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( A2
= ( sup_su6922871097908479076nnreal @ A2 @ B3 ) )
=> ( ord_le3935885782089961368nnreal @ B3 @ A2 ) ) ).
% sup.orderI
thf(fact_1080_sup__unique,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ X3 @ ( F @ X3 @ Y5 ) )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ Y5 @ ( F @ X3 @ Y5 ) )
=> ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal,Z4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Y5 @ X3 )
=> ( ( ord_le3935885782089961368nnreal @ Z4 @ X3 )
=> ( ord_le3935885782089961368nnreal @ ( F @ Y5 @ Z4 ) @ X3 ) ) )
=> ( ( sup_su6922871097908479076nnreal @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_1081_sup_Oabsorb1,axiom,
! [B3: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B3 @ A2 )
=> ( ( sup_su6922871097908479076nnreal @ A2 @ B3 )
= A2 ) ) ).
% sup.absorb1
thf(fact_1082_sup_Oabsorb2,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B3 )
=> ( ( sup_su6922871097908479076nnreal @ A2 @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_1083_sup__absorb1,axiom,
! [Y3: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Y3 @ X2 )
=> ( ( sup_su6922871097908479076nnreal @ X2 @ Y3 )
= X2 ) ) ).
% sup_absorb1
thf(fact_1084_sup__absorb2,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
=> ( ( sup_su6922871097908479076nnreal @ X2 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_1085_sup_OboundedE,axiom,
! [B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ ( sup_su6922871097908479076nnreal @ B3 @ C2 ) @ A2 )
=> ~ ( ( ord_le3935885782089961368nnreal @ B3 @ A2 )
=> ~ ( ord_le3935885782089961368nnreal @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_1086_sup_OboundedI,axiom,
! [B3: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B3 @ A2 )
=> ( ( ord_le3935885782089961368nnreal @ C2 @ A2 )
=> ( ord_le3935885782089961368nnreal @ ( sup_su6922871097908479076nnreal @ B3 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_1087_sup_Oorder__iff,axiom,
( ord_le3935885782089961368nnreal
= ( ^ [B5: extend8495563244428889912nnreal,A5: extend8495563244428889912nnreal] :
( A5
= ( sup_su6922871097908479076nnreal @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_1088_sup_Ocobounded1,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ A2 @ ( sup_su6922871097908479076nnreal @ A2 @ B3 ) ) ).
% sup.cobounded1
thf(fact_1089_sup_Ocobounded2,axiom,
! [B3: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ B3 @ ( sup_su6922871097908479076nnreal @ A2 @ B3 ) ) ).
% sup.cobounded2
thf(fact_1090_sup_Oabsorb__iff1,axiom,
( ord_le3935885782089961368nnreal
= ( ^ [B5: extend8495563244428889912nnreal,A5: extend8495563244428889912nnreal] :
( ( sup_su6922871097908479076nnreal @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_1091_sup_Oabsorb__iff2,axiom,
( ord_le3935885782089961368nnreal
= ( ^ [A5: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
( ( sup_su6922871097908479076nnreal @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_1092_sup_OcoboundedI1,axiom,
! [C2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ C2 @ A2 )
=> ( ord_le3935885782089961368nnreal @ C2 @ ( sup_su6922871097908479076nnreal @ A2 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_1093_sup_OcoboundedI2,axiom,
! [C2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ C2 @ B3 )
=> ( ord_le3935885782089961368nnreal @ C2 @ ( sup_su6922871097908479076nnreal @ A2 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_1094_fmeasurableD2,axiom,
! [A3: set_a,M: sigma_measure_a] :
( ( member_set_a @ A3 @ ( measur3645360004775918570able_a @ M ) )
=> ( ( sigma_emeasure_a @ M @ A3 )
!= top_to1496364449551166952nnreal ) ) ).
% fmeasurableD2
thf(fact_1095_algebra_Olambda__system__Un,axiom,
! [Omega: set_a,M: set_set_a,X2: set_a,F: set_a > extend8495563244428889912nnreal,Y3: set_a] :
( ( sigma_algebra_a @ Omega @ M )
=> ( ( member_set_a @ X2 @ ( lambda_system_a @ Omega @ M @ F ) )
=> ( ( member_set_a @ Y3 @ ( lambda_system_a @ Omega @ M @ F ) )
=> ( member_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ ( lambda_system_a @ Omega @ M @ F ) ) ) ) ) ).
% algebra.lambda_system_Un
thf(fact_1096_emeasure__Un__null__set,axiom,
! [A3: set_a,M: sigma_measure_a,B: set_a] :
( ( member_set_a @ A3 @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ B @ ( measure_null_sets_a @ M ) )
=> ( ( sigma_emeasure_a @ M @ ( sup_sup_set_a @ A3 @ B ) )
= ( sigma_emeasure_a @ M @ A3 ) ) ) ) ).
% emeasure_Un_null_set
thf(fact_1097_measurable__Un__null__set,axiom,
! [B: set_a,M: sigma_measure_a,A3: set_a] :
( ( member_set_a @ B @ ( measure_null_sets_a @ M ) )
=> ( ( ( member_set_a @ ( sup_sup_set_a @ A3 @ B ) @ ( measur3645360004775918570able_a @ M ) )
& ( member_set_a @ A3 @ ( sigma_sets_a @ M ) ) )
= ( member_set_a @ A3 @ ( measur3645360004775918570able_a @ M ) ) ) ) ).
% measurable_Un_null_set
thf(fact_1098_sets_Olambda__system__Un,axiom,
! [X2: set_a,M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal,Y3: set_a] :
( ( member_set_a @ X2 @ ( lambda_system_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) @ F ) )
=> ( ( member_set_a @ Y3 @ ( lambda_system_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) @ F ) )
=> ( member_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ ( lambda_system_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) @ F ) ) ) ) ).
% sets.lambda_system_Un
thf(fact_1099_closed__cdi__Un,axiom,
! [Omega: set_a,M: set_set_a,A3: set_a,B: set_a] :
( ( sigma_closed_cdi_a @ Omega @ M )
=> ( ( member_set_a @ bot_bot_set_a @ M )
=> ( ( member_set_a @ A3 @ M )
=> ( ( member_set_a @ B @ M )
=> ( ( ( inf_inf_set_a @ A3 @ B )
= bot_bot_set_a )
=> ( member_set_a @ ( sup_sup_set_a @ A3 @ B ) @ M ) ) ) ) ) ) ).
% closed_cdi_Un
thf(fact_1100_sets__completionI,axiom,
! [A3: set_a,S: set_a,N: set_a,N2: set_a,M: sigma_measure_a] :
( ( A3
= ( sup_sup_set_a @ S @ N ) )
=> ( ( ord_less_eq_set_a @ N @ N2 )
=> ( ( member_set_a @ N2 @ ( measure_null_sets_a @ M ) )
=> ( ( member_set_a @ S @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ A3 @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) ) ) ) ) ) ).
% sets_completionI
thf(fact_1101_sets__completionE,axiom,
! [A3: set_a,M: sigma_measure_a] :
( ( member_set_a @ A3 @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ~ ! [S3: set_a,N5: set_a] :
( ( A3
= ( sup_sup_set_a @ S3 @ N5 ) )
=> ! [N6: set_a] :
( ( ord_less_eq_set_a @ N5 @ N6 )
=> ( ( member_set_a @ N6 @ ( measure_null_sets_a @ M ) )
=> ~ ( member_set_a @ S3 @ ( sigma_sets_a @ M ) ) ) ) ) ) ).
% sets_completionE
thf(fact_1102_algebra_Osmallest__ccdi__sets__Un,axiom,
! [Omega: set_a,M: set_set_a,A3: set_a,B: set_a] :
( ( sigma_algebra_a @ Omega @ M )
=> ( ( member_set_a @ A3 @ ( sigma_5648178489087971417sets_a @ Omega @ M ) )
=> ( ( member_set_a @ B @ ( sigma_5648178489087971417sets_a @ Omega @ M ) )
=> ( ( ( inf_inf_set_a @ A3 @ B )
= bot_bot_set_a )
=> ( member_set_a @ ( sup_sup_set_a @ A3 @ B ) @ ( sigma_5648178489087971417sets_a @ Omega @ M ) ) ) ) ) ) ).
% algebra.smallest_ccdi_sets_Un
thf(fact_1103_emeasure__completion__Un,axiom,
! [S: set_a,M: sigma_measure_a,T2: set_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ( ( member_set_a @ T2 @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ( ( sigma_emeasure_a @ ( comple3428971583294703880tion_a @ M ) @ ( sup_sup_set_a @ S @ T2 ) )
= ( sigma_emeasure_a @ M @ ( sup_sup_set_a @ ( complete_main_part_a @ M @ S ) @ ( complete_main_part_a @ M @ T2 ) ) ) ) ) ) ).
% emeasure_completion_Un
thf(fact_1104_sets_Osmallest__ccdi__sets__Un,axiom,
! [A3: set_a,M: sigma_measure_a,B: set_a] :
( ( member_set_a @ A3 @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) )
=> ( ( member_set_a @ B @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) )
=> ( ( ( inf_inf_set_a @ A3 @ B )
= bot_bot_set_a )
=> ( member_set_a @ ( sup_sup_set_a @ A3 @ B ) @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) ) ) ) ) ).
% sets.smallest_ccdi_sets_Un
thf(fact_1105_finite__measure__axioms_Ointro,axiom,
! [M: sigma_measure_a] :
( ( ( sigma_emeasure_a @ M @ ( sigma_space_a @ M ) )
!= top_to1496364449551166952nnreal )
=> ( measur2595372213310369023ioms_a @ M ) ) ).
% finite_measure_axioms.intro
thf(fact_1106_finite__measure__axioms__def,axiom,
( measur2595372213310369023ioms_a
= ( ^ [M2: sigma_measure_a] :
( ( sigma_emeasure_a @ M2 @ ( sigma_space_a @ M2 ) )
!= top_to1496364449551166952nnreal ) ) ) ).
% finite_measure_axioms_def
thf(fact_1107_neq__top__trans,axiom,
! [Y3: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ( Y3 != top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
=> ( X2 != top_to1496364449551166952nnreal ) ) ) ).
% neq_top_trans
thf(fact_1108_ex__new__if__finite,axiom,
! [A3: set_a_a] :
( ~ ( finite_finite_a_a @ top_top_set_a_a )
=> ( ( finite_finite_a_a @ A3 )
=> ? [A: a > a] :
~ ( member_a_a @ A @ A3 ) ) ) ).
% ex_new_if_finite
thf(fact_1109_ex__new__if__finite,axiom,
! [A3: set_a] :
( ~ ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_a @ A3 )
=> ? [A: a] :
~ ( member_a @ A @ A3 ) ) ) ).
% ex_new_if_finite
thf(fact_1110_ex__new__if__finite,axiom,
! [A3: set_set_a] :
( ~ ( finite_finite_set_a @ top_top_set_set_a )
=> ( ( finite_finite_set_a @ A3 )
=> ? [A: set_a] :
~ ( member_set_a @ A @ A3 ) ) ) ).
% ex_new_if_finite
thf(fact_1111_ex__new__if__finite,axiom,
! [A3: set_real_a] :
( ~ ( finite_finite_real_a @ top_top_set_real_a )
=> ( ( finite_finite_real_a @ A3 )
=> ? [A: real > a] :
~ ( member_real_a @ A @ A3 ) ) ) ).
% ex_new_if_finite
thf(fact_1112_sup__quasi__borel__def,axiom,
( sup_su6298519176299948920orel_a
= ( ^ [X6: quasi_borel_a,Y7: quasi_borel_a] :
( if_quasi_borel_a
@ ( ( qbs_space_a @ X6 )
= ( qbs_space_a @ Y7 ) )
@ ( inf_quasi_borel_a @ X6 @ Y7 )
@ ( if_quasi_borel_a @ ( ord_less_set_a @ ( qbs_space_a @ X6 ) @ ( qbs_space_a @ Y7 ) ) @ Y7 @ ( if_quasi_borel_a @ ( ord_less_set_a @ ( qbs_space_a @ Y7 ) @ ( qbs_space_a @ X6 ) ) @ X6 @ ( max_quasi_borel_a @ ( sup_sup_set_a @ ( qbs_space_a @ X6 ) @ ( qbs_space_a @ Y7 ) ) ) ) ) ) ) ) ).
% sup_quasi_borel_def
thf(fact_1113_ennreal__zero__less__top,axiom,
ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ top_to1496364449551166952nnreal ).
% ennreal_zero_less_top
thf(fact_1114_UNIV__I,axiom,
! [X2: a > a] : ( member_a_a @ X2 @ top_top_set_a_a ) ).
% UNIV_I
thf(fact_1115_UNIV__I,axiom,
! [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_I
thf(fact_1116_UNIV__I,axiom,
! [X2: set_a] : ( member_set_a @ X2 @ top_top_set_set_a ) ).
% UNIV_I
thf(fact_1117_UNIV__I,axiom,
! [X2: real > a] : ( member_real_a @ X2 @ top_top_set_real_a ) ).
% UNIV_I
thf(fact_1118_UNIV__eq__I,axiom,
! [A3: set_a_a] :
( ! [X3: a > a] : ( member_a_a @ X3 @ A3 )
=> ( top_top_set_a_a = A3 ) ) ).
% UNIV_eq_I
thf(fact_1119_UNIV__eq__I,axiom,
! [A3: set_a] :
( ! [X3: a] : ( member_a @ X3 @ A3 )
=> ( top_top_set_a = A3 ) ) ).
% UNIV_eq_I
thf(fact_1120_UNIV__eq__I,axiom,
! [A3: set_set_a] :
( ! [X3: set_a] : ( member_set_a @ X3 @ A3 )
=> ( top_top_set_set_a = A3 ) ) ).
% UNIV_eq_I
thf(fact_1121_UNIV__eq__I,axiom,
! [A3: set_real_a] :
( ! [X3: real > a] : ( member_real_a @ X3 @ A3 )
=> ( top_top_set_real_a = A3 ) ) ).
% UNIV_eq_I
thf(fact_1122_UNIV__witness,axiom,
? [X3: a > a] : ( member_a_a @ X3 @ top_top_set_a_a ) ).
% UNIV_witness
thf(fact_1123_UNIV__witness,axiom,
? [X3: a] : ( member_a @ X3 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_1124_UNIV__witness,axiom,
? [X3: set_a] : ( member_set_a @ X3 @ top_top_set_set_a ) ).
% UNIV_witness
thf(fact_1125_UNIV__witness,axiom,
? [X3: real > a] : ( member_real_a @ X3 @ top_top_set_real_a ) ).
% UNIV_witness
thf(fact_1126_measurable__envelope__eq2,axiom,
! [A3: set_a,E: set_a,M: sigma_measure_a] :
( ( ord_less_eq_set_a @ A3 @ E )
=> ( ( member_set_a @ E @ ( sigma_sets_a @ M ) )
=> ( ( ord_le7381754540660121996nnreal @ ( sigma_emeasure_a @ M @ E ) @ extend2057119558705770725nnreal )
=> ( ( comple1317578422686860819lope_a @ M @ A3 @ E )
= ( ( sigma_emeasure_a @ M @ E )
= ( comple3326625653960060650e_of_a @ M @ A3 ) ) ) ) ) ) ).
% measurable_envelope_eq2
thf(fact_1127_fmeasurableI,axiom,
! [A3: set_a,M: sigma_measure_a] :
( ( member_set_a @ A3 @ ( sigma_sets_a @ M ) )
=> ( ( ord_le7381754540660121996nnreal @ ( sigma_emeasure_a @ M @ A3 ) @ extend2057119558705770725nnreal )
=> ( member_set_a @ A3 @ ( measur3645360004775918570able_a @ M ) ) ) ) ).
% fmeasurableI
thf(fact_1128_completion_Ocomplete__sets__sandwich,axiom,
! [A3: set_a,M: sigma_measure_a,C: set_a,B: set_a] :
( ( member_set_a @ A3 @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ( ( member_set_a @ C @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ( ( ord_less_eq_set_a @ A3 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ( ( sigma_emeasure_a @ ( comple3428971583294703880tion_a @ M ) @ A3 )
= ( sigma_emeasure_a @ ( comple3428971583294703880tion_a @ M ) @ C ) )
=> ( ( ord_le7381754540660121996nnreal @ ( sigma_emeasure_a @ ( comple3428971583294703880tion_a @ M ) @ A3 ) @ extend2057119558705770725nnreal )
=> ( member_set_a @ B @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) ) ) ) ) ) ) ) ).
% completion.complete_sets_sandwich
thf(fact_1129_complete__measure_Ocomplete__sets__sandwich,axiom,
! [M: sigma_measure_a,A3: set_a,C: set_a,B: set_a] :
( ( comple8155536527497655953sure_a @ M )
=> ( ( member_set_a @ A3 @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ C @ ( sigma_sets_a @ M ) )
=> ( ( ord_less_eq_set_a @ A3 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ( ( sigma_emeasure_a @ M @ A3 )
= ( sigma_emeasure_a @ M @ C ) )
=> ( ( ord_le7381754540660121996nnreal @ ( sigma_emeasure_a @ M @ A3 ) @ extend2057119558705770725nnreal )
=> ( member_set_a @ B @ ( sigma_sets_a @ M ) ) ) ) ) ) ) ) ) ).
% complete_measure.complete_sets_sandwich
thf(fact_1130_locally__determined__measure__axioms__def,axiom,
( comple7282333422040910444ioms_a
= ( ^ [M2: sigma_measure_a] :
! [A4: set_a] :
( ( ord_less_eq_set_a @ A4 @ ( sigma_space_a @ M2 ) )
=> ( ! [B4: set_a] :
( ( member_set_a @ B4 @ ( sigma_sets_a @ M2 ) )
=> ( ( ord_le7381754540660121996nnreal @ ( sigma_emeasure_a @ M2 @ B4 ) @ extend2057119558705770725nnreal )
=> ( member_set_a @ ( inf_inf_set_a @ A4 @ B4 ) @ ( sigma_sets_a @ M2 ) ) ) )
=> ( member_set_a @ A4 @ ( sigma_sets_a @ M2 ) ) ) ) ) ) ).
% locally_determined_measure_axioms_def
thf(fact_1131_locally__determined__measure__axioms_Ointro,axiom,
! [M: sigma_measure_a] :
( ! [A8: set_a] :
( ( ord_less_eq_set_a @ A8 @ ( sigma_space_a @ M ) )
=> ( ! [B8: set_a] :
( ( member_set_a @ B8 @ ( sigma_sets_a @ M ) )
=> ( ( ord_le7381754540660121996nnreal @ ( sigma_emeasure_a @ M @ B8 ) @ extend2057119558705770725nnreal )
=> ( member_set_a @ ( inf_inf_set_a @ A8 @ B8 ) @ ( sigma_sets_a @ M ) ) ) )
=> ( member_set_a @ A8 @ ( sigma_sets_a @ M ) ) ) )
=> ( comple7282333422040910444ioms_a @ M ) ) ).
% locally_determined_measure_axioms.intro
thf(fact_1132_locally__determined__measure_Olocally__determined,axiom,
! [M: sigma_measure_a,A3: set_a] :
( ( comple7257097362500340559sure_a @ M )
=> ( ( ord_less_eq_set_a @ A3 @ ( sigma_space_a @ M ) )
=> ( ! [B7: set_a] :
( ( member_set_a @ B7 @ ( sigma_sets_a @ M ) )
=> ( ( ord_le7381754540660121996nnreal @ ( sigma_emeasure_a @ M @ B7 ) @ extend2057119558705770725nnreal )
=> ( member_set_a @ ( inf_inf_set_a @ A3 @ B7 ) @ ( sigma_sets_a @ M ) ) ) )
=> ( member_set_a @ A3 @ ( sigma_sets_a @ M ) ) ) ) ) ).
% locally_determined_measure.locally_determined
thf(fact_1133_semifinite__measure_Ointro,axiom,
! [M: sigma_measure_a] :
( ! [A8: set_a] :
( ( member_set_a @ A8 @ ( sigma_sets_a @ M ) )
=> ( ( ( sigma_emeasure_a @ M @ A8 )
= extend2057119558705770725nnreal )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ ( sigma_sets_a @ M ) )
& ( ord_less_eq_set_a @ X5 @ A8 )
& ( ord_le7381754540660121996nnreal @ ( sigma_emeasure_a @ M @ X5 ) @ extend2057119558705770725nnreal ) ) ) )
=> ( comple8962655729304856821sure_a @ M ) ) ).
% semifinite_measure.intro
thf(fact_1134_semifinite__measure__def,axiom,
( comple8962655729304856821sure_a
= ( ^ [M2: sigma_measure_a] :
! [A4: set_a] :
( ( member_set_a @ A4 @ ( sigma_sets_a @ M2 ) )
=> ( ( ( sigma_emeasure_a @ M2 @ A4 )
= extend2057119558705770725nnreal )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ ( sigma_sets_a @ M2 ) )
& ( ord_less_eq_set_a @ X4 @ A4 )
& ( ord_le7381754540660121996nnreal @ ( sigma_emeasure_a @ M2 @ X4 ) @ extend2057119558705770725nnreal ) ) ) ) ) ) ).
% semifinite_measure_def
thf(fact_1135_semifinite__measure_Osemifinite,axiom,
! [M: sigma_measure_a,A3: set_a] :
( ( comple8962655729304856821sure_a @ M )
=> ( ( member_set_a @ A3 @ ( sigma_sets_a @ M ) )
=> ( ( ( sigma_emeasure_a @ M @ A3 )
= extend2057119558705770725nnreal )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ ( sigma_sets_a @ M ) )
& ( ord_less_eq_set_a @ X3 @ A3 )
& ( ord_le7381754540660121996nnreal @ ( sigma_emeasure_a @ M @ X3 ) @ extend2057119558705770725nnreal ) ) ) ) ) ).
% semifinite_measure.semifinite
thf(fact_1136_cld__measure_Onotin__sets__outer__measure__of__cover,axiom,
! [M: sigma_measure_a,E: set_a] :
( ( comple4590207612277428097sure_a @ M )
=> ( ( ord_less_eq_set_a @ E @ ( sigma_space_a @ M ) )
=> ( ~ ( member_set_a @ E @ ( sigma_sets_a @ M ) )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ ( sigma_sets_a @ M ) )
& ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( sigma_emeasure_a @ M @ X3 ) )
& ( ord_le7381754540660121996nnreal @ ( sigma_emeasure_a @ M @ X3 ) @ extend2057119558705770725nnreal )
& ( ( comple3326625653960060650e_of_a @ M @ ( inf_inf_set_a @ X3 @ E ) )
= ( sigma_emeasure_a @ M @ X3 ) )
& ( ( comple3326625653960060650e_of_a @ M @ ( minus_minus_set_a @ X3 @ E ) )
= ( sigma_emeasure_a @ M @ X3 ) ) ) ) ) ) ).
% cld_measure.notin_sets_outer_measure_of_cover
thf(fact_1137_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila6505086053094918154nnreal @ inf_in7439215052339218890nnreal @ top_to1496364449551166952nnreal @ ord_le3935885782089961368nnreal @ ord_le7381754540660121996nnreal ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_1138_DiffI,axiom,
! [C2: a > a,A3: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ A3 )
=> ( ~ ( member_a_a @ C2 @ B )
=> ( member_a_a @ C2 @ ( minus_minus_set_a_a @ A3 @ B ) ) ) ) ).
% DiffI
thf(fact_1139_DiffI,axiom,
! [C2: a,A3: set_a,B: set_a] :
( ( member_a @ C2 @ A3 )
=> ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( minus_minus_set_a @ A3 @ B ) ) ) ) ).
% DiffI
thf(fact_1140_DiffI,axiom,
! [C2: set_a,A3: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ A3 )
=> ( ~ ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A3 @ B ) ) ) ) ).
% DiffI
thf(fact_1141_DiffI,axiom,
! [C2: real > a,A3: set_real_a,B: set_real_a] :
( ( member_real_a @ C2 @ A3 )
=> ( ~ ( member_real_a @ C2 @ B )
=> ( member_real_a @ C2 @ ( minus_6532636778494125008real_a @ A3 @ B ) ) ) ) ).
% DiffI
thf(fact_1142_Diff__iff,axiom,
! [C2: a > a,A3: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( minus_minus_set_a_a @ A3 @ B ) )
= ( ( member_a_a @ C2 @ A3 )
& ~ ( member_a_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_1143_Diff__iff,axiom,
! [C2: a,A3: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A3 @ B ) )
= ( ( member_a @ C2 @ A3 )
& ~ ( member_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_1144_Diff__iff,axiom,
! [C2: set_a,A3: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A3 @ B ) )
= ( ( member_set_a @ C2 @ A3 )
& ~ ( member_set_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_1145_Diff__iff,axiom,
! [C2: real > a,A3: set_real_a,B: set_real_a] :
( ( member_real_a @ C2 @ ( minus_6532636778494125008real_a @ A3 @ B ) )
= ( ( member_real_a @ C2 @ A3 )
& ~ ( member_real_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_1146_Diff__insert0,axiom,
! [X2: a > a,A3: set_a_a,B: set_a_a] :
( ~ ( member_a_a @ X2 @ A3 )
=> ( ( minus_minus_set_a_a @ A3 @ ( insert_a_a @ X2 @ B ) )
= ( minus_minus_set_a_a @ A3 @ B ) ) ) ).
% Diff_insert0
thf(fact_1147_Diff__insert0,axiom,
! [X2: a,A3: set_a,B: set_a] :
( ~ ( member_a @ X2 @ A3 )
=> ( ( minus_minus_set_a @ A3 @ ( insert_a @ X2 @ B ) )
= ( minus_minus_set_a @ A3 @ B ) ) ) ).
% Diff_insert0
thf(fact_1148_Diff__insert0,axiom,
! [X2: set_a,A3: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X2 @ A3 )
=> ( ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X2 @ B ) )
= ( minus_5736297505244876581_set_a @ A3 @ B ) ) ) ).
% Diff_insert0
thf(fact_1149_Diff__insert0,axiom,
! [X2: real > a,A3: set_real_a,B: set_real_a] :
( ~ ( member_real_a @ X2 @ A3 )
=> ( ( minus_6532636778494125008real_a @ A3 @ ( insert_real_a @ X2 @ B ) )
= ( minus_6532636778494125008real_a @ A3 @ B ) ) ) ).
% Diff_insert0
thf(fact_1150_insert__Diff1,axiom,
! [X2: a > a,B: set_a_a,A3: set_a_a] :
( ( member_a_a @ X2 @ B )
=> ( ( minus_minus_set_a_a @ ( insert_a_a @ X2 @ A3 ) @ B )
= ( minus_minus_set_a_a @ A3 @ B ) ) ) ).
% insert_Diff1
thf(fact_1151_insert__Diff1,axiom,
! [X2: a,B: set_a,A3: set_a] :
( ( member_a @ X2 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A3 ) @ B )
= ( minus_minus_set_a @ A3 @ B ) ) ) ).
% insert_Diff1
thf(fact_1152_insert__Diff1,axiom,
! [X2: set_a,B: set_set_a,A3: set_set_a] :
( ( member_set_a @ X2 @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X2 @ A3 ) @ B )
= ( minus_5736297505244876581_set_a @ A3 @ B ) ) ) ).
% insert_Diff1
thf(fact_1153_insert__Diff1,axiom,
! [X2: real > a,B: set_real_a,A3: set_real_a] :
( ( member_real_a @ X2 @ B )
=> ( ( minus_6532636778494125008real_a @ ( insert_real_a @ X2 @ A3 ) @ B )
= ( minus_6532636778494125008real_a @ A3 @ B ) ) ) ).
% insert_Diff1
thf(fact_1154_sets_ODiff,axiom,
! [A2: set_a,M: sigma_measure_a,B3: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ B3 @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ ( sigma_sets_a @ M ) ) ) ) ).
% sets.Diff
thf(fact_1155_fmeasurable_ODiff,axiom,
! [A2: set_a,M: sigma_measure_a,B3: set_a] :
( ( member_set_a @ A2 @ ( measur3645360004775918570able_a @ M ) )
=> ( ( member_set_a @ B3 @ ( measur3645360004775918570able_a @ M ) )
=> ( member_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ ( measur3645360004775918570able_a @ M ) ) ) ) ).
% fmeasurable.Diff
thf(fact_1156_null__sets_ODiff,axiom,
! [A2: set_a,M: sigma_measure_a,B3: set_a] :
( ( member_set_a @ A2 @ ( measure_null_sets_a @ M ) )
=> ( ( member_set_a @ B3 @ ( measure_null_sets_a @ M ) )
=> ( member_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ ( measure_null_sets_a @ M ) ) ) ) ).
% null_sets.Diff
thf(fact_1157_sets_Ocompl__sets,axiom,
! [A2: set_a,M: sigma_measure_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( minus_minus_set_a @ ( sigma_space_a @ M ) @ A2 ) @ ( sigma_sets_a @ M ) ) ) ).
% sets.compl_sets
thf(fact_1158_Diff__insert__absorb,axiom,
! [X2: a > a,A3: set_a_a] :
( ~ ( member_a_a @ X2 @ A3 )
=> ( ( minus_minus_set_a_a @ ( insert_a_a @ X2 @ A3 ) @ ( insert_a_a @ X2 @ bot_bot_set_a_a ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_1159_Diff__insert__absorb,axiom,
! [X2: a,A3: set_a] :
( ~ ( member_a @ X2 @ A3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A3 ) @ ( insert_a @ X2 @ bot_bot_set_a ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_1160_Diff__insert__absorb,axiom,
! [X2: set_a,A3: set_set_a] :
( ~ ( member_set_a @ X2 @ A3 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X2 @ A3 ) @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_1161_Diff__insert__absorb,axiom,
! [X2: real > a,A3: set_real_a] :
( ~ ( member_real_a @ X2 @ A3 )
=> ( ( minus_6532636778494125008real_a @ ( insert_real_a @ X2 @ A3 ) @ ( insert_real_a @ X2 @ bot_bot_set_real_a ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_1162_insert__Diff,axiom,
! [A2: a > a,A3: set_a_a] :
( ( member_a_a @ A2 @ A3 )
=> ( ( insert_a_a @ A2 @ ( minus_minus_set_a_a @ A3 @ ( insert_a_a @ A2 @ bot_bot_set_a_a ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_1163_insert__Diff,axiom,
! [A2: a,A3: set_a] :
( ( member_a @ A2 @ A3 )
=> ( ( insert_a @ A2 @ ( minus_minus_set_a @ A3 @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_1164_insert__Diff,axiom,
! [A2: set_a,A3: set_set_a] :
( ( member_set_a @ A2 @ A3 )
=> ( ( insert_set_a @ A2 @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_1165_insert__Diff,axiom,
! [A2: real > a,A3: set_real_a] :
( ( member_real_a @ A2 @ A3 )
=> ( ( insert_real_a @ A2 @ ( minus_6532636778494125008real_a @ A3 @ ( insert_real_a @ A2 @ bot_bot_set_real_a ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_1166_subset__Diff__insert,axiom,
! [A3: set_a_a,B: set_a_a,X2: a > a,C: set_a_a] :
( ( ord_less_eq_set_a_a @ A3 @ ( minus_minus_set_a_a @ B @ ( insert_a_a @ X2 @ C ) ) )
= ( ( ord_less_eq_set_a_a @ A3 @ ( minus_minus_set_a_a @ B @ C ) )
& ~ ( member_a_a @ X2 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1167_subset__Diff__insert,axiom,
! [A3: set_a,B: set_a,X2: a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( minus_minus_set_a @ B @ ( insert_a @ X2 @ C ) ) )
= ( ( ord_less_eq_set_a @ A3 @ ( minus_minus_set_a @ B @ C ) )
& ~ ( member_a @ X2 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1168_subset__Diff__insert,axiom,
! [A3: set_set_a,B: set_set_a,X2: set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ ( minus_5736297505244876581_set_a @ B @ ( insert_set_a @ X2 @ C ) ) )
= ( ( ord_le3724670747650509150_set_a @ A3 @ ( minus_5736297505244876581_set_a @ B @ C ) )
& ~ ( member_set_a @ X2 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1169_subset__Diff__insert,axiom,
! [A3: set_real_a,B: set_real_a,X2: real > a,C: set_real_a] :
( ( ord_le5743406823621094409real_a @ A3 @ ( minus_6532636778494125008real_a @ B @ ( insert_real_a @ X2 @ C ) ) )
= ( ( ord_le5743406823621094409real_a @ A3 @ ( minus_6532636778494125008real_a @ B @ C ) )
& ~ ( member_real_a @ X2 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1170_fmeasurable__Diff__D,axiom,
! [T2: set_a,S: set_a,M: sigma_measure_a] :
( ( member_set_a @ ( minus_minus_set_a @ T2 @ S ) @ ( measur3645360004775918570able_a @ M ) )
=> ( ( member_set_a @ S @ ( measur3645360004775918570able_a @ M ) )
=> ( ( ord_less_eq_set_a @ S @ T2 )
=> ( member_set_a @ T2 @ ( measur3645360004775918570able_a @ M ) ) ) ) ) ).
% fmeasurable_Diff_D
thf(fact_1171_fmeasurable__Diff,axiom,
! [A3: set_a,M: sigma_measure_a,B: set_a] :
( ( member_set_a @ A3 @ ( measur3645360004775918570able_a @ M ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( minus_minus_set_a @ A3 @ B ) @ ( measur3645360004775918570able_a @ M ) ) ) ) ).
% fmeasurable_Diff
thf(fact_1172_null__set__Diff,axiom,
! [B: set_a,M: sigma_measure_a,A3: set_a] :
( ( member_set_a @ B @ ( measure_null_sets_a @ M ) )
=> ( ( member_set_a @ A3 @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( minus_minus_set_a @ B @ A3 ) @ ( measure_null_sets_a @ M ) ) ) ) ).
% null_set_Diff
thf(fact_1173_Dynkin__system_Odiff,axiom,
! [Omega: set_a,M: set_set_a,D: set_a,E: set_a] :
( ( sigma_2757993544071651912stem_a @ Omega @ M )
=> ( ( member_set_a @ D @ M )
=> ( ( member_set_a @ E @ M )
=> ( ( ord_less_eq_set_a @ D @ E )
=> ( member_set_a @ ( minus_minus_set_a @ E @ D ) @ M ) ) ) ) ) ).
% Dynkin_system.diff
thf(fact_1174_insert__Diff__if,axiom,
! [X2: a > a,B: set_a_a,A3: set_a_a] :
( ( ( member_a_a @ X2 @ B )
=> ( ( minus_minus_set_a_a @ ( insert_a_a @ X2 @ A3 ) @ B )
= ( minus_minus_set_a_a @ A3 @ B ) ) )
& ( ~ ( member_a_a @ X2 @ B )
=> ( ( minus_minus_set_a_a @ ( insert_a_a @ X2 @ A3 ) @ B )
= ( insert_a_a @ X2 @ ( minus_minus_set_a_a @ A3 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1175_insert__Diff__if,axiom,
! [X2: a,B: set_a,A3: set_a] :
( ( ( member_a @ X2 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A3 ) @ B )
= ( minus_minus_set_a @ A3 @ B ) ) )
& ( ~ ( member_a @ X2 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A3 ) @ B )
= ( insert_a @ X2 @ ( minus_minus_set_a @ A3 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1176_insert__Diff__if,axiom,
! [X2: set_a,B: set_set_a,A3: set_set_a] :
( ( ( member_set_a @ X2 @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X2 @ A3 ) @ B )
= ( minus_5736297505244876581_set_a @ A3 @ B ) ) )
& ( ~ ( member_set_a @ X2 @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X2 @ A3 ) @ B )
= ( insert_set_a @ X2 @ ( minus_5736297505244876581_set_a @ A3 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1177_insert__Diff__if,axiom,
! [X2: real > a,B: set_real_a,A3: set_real_a] :
( ( ( member_real_a @ X2 @ B )
=> ( ( minus_6532636778494125008real_a @ ( insert_real_a @ X2 @ A3 ) @ B )
= ( minus_6532636778494125008real_a @ A3 @ B ) ) )
& ( ~ ( member_real_a @ X2 @ B )
=> ( ( minus_6532636778494125008real_a @ ( insert_real_a @ X2 @ A3 ) @ B )
= ( insert_real_a @ X2 @ ( minus_6532636778494125008real_a @ A3 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1178_closed__cdi__Compl,axiom,
! [Omega: set_a,M: set_set_a,S2: set_a] :
( ( sigma_closed_cdi_a @ Omega @ M )
=> ( ( member_set_a @ S2 @ M )
=> ( member_set_a @ ( minus_minus_set_a @ Omega @ S2 ) @ M ) ) ) ).
% closed_cdi_Compl
thf(fact_1179_ring__of__sets_ODiff,axiom,
! [Omega: set_a,M: set_set_a,A2: set_a,B3: set_a] :
( ( sigma_ring_of_sets_a @ Omega @ M )
=> ( ( member_set_a @ A2 @ M )
=> ( ( member_set_a @ B3 @ M )
=> ( member_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ M ) ) ) ) ).
% ring_of_sets.Diff
thf(fact_1180_smallest__ccdi__sets_OCompl,axiom,
! [A2: set_a,Omega: set_a,M: set_set_a] :
( ( member_set_a @ A2 @ ( sigma_5648178489087971417sets_a @ Omega @ M ) )
=> ( member_set_a @ ( minus_minus_set_a @ Omega @ A2 ) @ ( sigma_5648178489087971417sets_a @ Omega @ M ) ) ) ).
% smallest_ccdi_sets.Compl
thf(fact_1181_algebra_Ocompl__sets,axiom,
! [Omega: set_a,M: set_set_a,A2: set_a] :
( ( sigma_algebra_a @ Omega @ M )
=> ( ( member_set_a @ A2 @ M )
=> ( member_set_a @ ( minus_minus_set_a @ Omega @ A2 ) @ M ) ) ) ).
% algebra.compl_sets
thf(fact_1182_DiffE,axiom,
! [C2: a > a,A3: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( minus_minus_set_a_a @ A3 @ B ) )
=> ~ ( ( member_a_a @ C2 @ A3 )
=> ( member_a_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_1183_DiffE,axiom,
! [C2: a,A3: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A3 @ B ) )
=> ~ ( ( member_a @ C2 @ A3 )
=> ( member_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_1184_DiffE,axiom,
! [C2: set_a,A3: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A3 @ B ) )
=> ~ ( ( member_set_a @ C2 @ A3 )
=> ( member_set_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_1185_DiffE,axiom,
! [C2: real > a,A3: set_real_a,B: set_real_a] :
( ( member_real_a @ C2 @ ( minus_6532636778494125008real_a @ A3 @ B ) )
=> ~ ( ( member_real_a @ C2 @ A3 )
=> ( member_real_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_1186_DiffD1,axiom,
! [C2: a > a,A3: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( minus_minus_set_a_a @ A3 @ B ) )
=> ( member_a_a @ C2 @ A3 ) ) ).
% DiffD1
thf(fact_1187_DiffD1,axiom,
! [C2: a,A3: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A3 @ B ) )
=> ( member_a @ C2 @ A3 ) ) ).
% DiffD1
thf(fact_1188_DiffD1,axiom,
! [C2: set_a,A3: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A3 @ B ) )
=> ( member_set_a @ C2 @ A3 ) ) ).
% DiffD1
thf(fact_1189_DiffD1,axiom,
! [C2: real > a,A3: set_real_a,B: set_real_a] :
( ( member_real_a @ C2 @ ( minus_6532636778494125008real_a @ A3 @ B ) )
=> ( member_real_a @ C2 @ A3 ) ) ).
% DiffD1
thf(fact_1190_DiffD2,axiom,
! [C2: a > a,A3: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( minus_minus_set_a_a @ A3 @ B ) )
=> ~ ( member_a_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_1191_DiffD2,axiom,
! [C2: a,A3: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A3 @ B ) )
=> ~ ( member_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_1192_DiffD2,axiom,
! [C2: set_a,A3: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A3 @ B ) )
=> ~ ( member_set_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_1193_DiffD2,axiom,
! [C2: real > a,A3: set_real_a,B: set_real_a] :
( ( member_real_a @ C2 @ ( minus_6532636778494125008real_a @ A3 @ B ) )
=> ~ ( member_real_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_1194_sigma__sets_OCompl,axiom,
! [A2: set_a,Sp: set_a,A3: set_set_a] :
( ( member_set_a @ A2 @ ( sigma_sigma_sets_a @ Sp @ A3 ) )
=> ( member_set_a @ ( minus_minus_set_a @ Sp @ A2 ) @ ( sigma_sigma_sets_a @ Sp @ A3 ) ) ) ).
% sigma_sets.Compl
thf(fact_1195_Dynkin__system_Ocompl,axiom,
! [Omega: set_a,M: set_set_a,A3: set_a] :
( ( sigma_2757993544071651912stem_a @ Omega @ M )
=> ( ( member_set_a @ A3 @ M )
=> ( member_set_a @ ( minus_minus_set_a @ Omega @ A3 ) @ M ) ) ) ).
% Dynkin_system.compl
thf(fact_1196_algebra_Olambda__system__Compl,axiom,
! [Omega: set_a,M: set_set_a,X2: set_a,F: set_a > extend8495563244428889912nnreal] :
( ( sigma_algebra_a @ Omega @ M )
=> ( ( member_set_a @ X2 @ ( lambda_system_a @ Omega @ M @ F ) )
=> ( member_set_a @ ( minus_minus_set_a @ Omega @ X2 ) @ ( lambda_system_a @ Omega @ M @ F ) ) ) ) ).
% algebra.lambda_system_Compl
thf(fact_1197_psubset__imp__ex__mem,axiom,
! [A3: set_a_a,B: set_a_a] :
( ( ord_less_set_a_a @ A3 @ B )
=> ? [B2: a > a] : ( member_a_a @ B2 @ ( minus_minus_set_a_a @ B @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1198_psubset__imp__ex__mem,axiom,
! [A3: set_a,B: set_a] :
( ( ord_less_set_a @ A3 @ B )
=> ? [B2: a] : ( member_a @ B2 @ ( minus_minus_set_a @ B @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1199_psubset__imp__ex__mem,axiom,
! [A3: set_set_a,B: set_set_a] :
( ( ord_less_set_set_a @ A3 @ B )
=> ? [B2: set_a] : ( member_set_a @ B2 @ ( minus_5736297505244876581_set_a @ B @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1200_psubset__imp__ex__mem,axiom,
! [A3: set_real_a,B: set_real_a] :
( ( ord_less_set_real_a @ A3 @ B )
=> ? [B2: real > a] : ( member_real_a @ B2 @ ( minus_6532636778494125008real_a @ B @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1201_subset__insert__iff,axiom,
! [A3: set_a_a,X2: a > a,B: set_a_a] :
( ( ord_less_eq_set_a_a @ A3 @ ( insert_a_a @ X2 @ B ) )
= ( ( ( member_a_a @ X2 @ A3 )
=> ( ord_less_eq_set_a_a @ ( minus_minus_set_a_a @ A3 @ ( insert_a_a @ X2 @ bot_bot_set_a_a ) ) @ B ) )
& ( ~ ( member_a_a @ X2 @ A3 )
=> ( ord_less_eq_set_a_a @ A3 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1202_subset__insert__iff,axiom,
! [A3: set_a,X2: a,B: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X2 @ B ) )
= ( ( ( member_a @ X2 @ A3 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X2 @ A3 )
=> ( ord_less_eq_set_a @ A3 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1203_subset__insert__iff,axiom,
! [A3: set_set_a,X2: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ ( insert_set_a @ X2 @ B ) )
= ( ( ( member_set_a @ X2 @ A3 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) @ B ) )
& ( ~ ( member_set_a @ X2 @ A3 )
=> ( ord_le3724670747650509150_set_a @ A3 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1204_subset__insert__iff,axiom,
! [A3: set_real_a,X2: real > a,B: set_real_a] :
( ( ord_le5743406823621094409real_a @ A3 @ ( insert_real_a @ X2 @ B ) )
= ( ( ( member_real_a @ X2 @ A3 )
=> ( ord_le5743406823621094409real_a @ ( minus_6532636778494125008real_a @ A3 @ ( insert_real_a @ X2 @ bot_bot_set_real_a ) ) @ B ) )
& ( ~ ( member_real_a @ X2 @ A3 )
=> ( ord_le5743406823621094409real_a @ A3 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1205_finite__empty__induct,axiom,
! [A3: set_a_a,P: set_a_a > $o] :
( ( finite_finite_a_a @ A3 )
=> ( ( P @ A3 )
=> ( ! [A: a > a,A8: set_a_a] :
( ( finite_finite_a_a @ A8 )
=> ( ( member_a_a @ A @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_minus_set_a_a @ A8 @ ( insert_a_a @ A @ bot_bot_set_a_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1206_finite__empty__induct,axiom,
! [A3: set_a,P: set_a > $o] :
( ( finite_finite_a @ A3 )
=> ( ( P @ A3 )
=> ( ! [A: a,A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( member_a @ A @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1207_finite__empty__induct,axiom,
! [A3: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ A3 )
=> ( ( P @ A3 )
=> ( ! [A: set_a,A8: set_set_a] :
( ( finite_finite_set_a @ A8 )
=> ( ( member_set_a @ A @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_5736297505244876581_set_a @ A8 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1208_finite__empty__induct,axiom,
! [A3: set_real_a,P: set_real_a > $o] :
( ( finite_finite_real_a @ A3 )
=> ( ( P @ A3 )
=> ( ! [A: real > a,A8: set_real_a] :
( ( finite_finite_real_a @ A8 )
=> ( ( member_real_a @ A @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_6532636778494125008real_a @ A8 @ ( insert_real_a @ A @ bot_bot_set_real_a ) ) ) ) ) )
=> ( P @ bot_bot_set_real_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1209_emeasure__Diff__null__set,axiom,
! [B: set_a,M: sigma_measure_a,A3: set_a] :
( ( member_set_a @ B @ ( measure_null_sets_a @ M ) )
=> ( ( member_set_a @ A3 @ ( sigma_sets_a @ M ) )
=> ( ( sigma_emeasure_a @ M @ ( minus_minus_set_a @ A3 @ B ) )
= ( sigma_emeasure_a @ M @ A3 ) ) ) ) ).
% emeasure_Diff_null_set
thf(fact_1210_measurable__Diff__null__set,axiom,
! [B: set_a,M: sigma_measure_a,A3: set_a] :
( ( member_set_a @ B @ ( measure_null_sets_a @ M ) )
=> ( ( ( member_set_a @ ( minus_minus_set_a @ A3 @ B ) @ ( measur3645360004775918570able_a @ M ) )
& ( member_set_a @ A3 @ ( sigma_sets_a @ M ) ) )
= ( member_set_a @ A3 @ ( measur3645360004775918570able_a @ M ) ) ) ) ).
% measurable_Diff_null_set
thf(fact_1211_sets_Olambda__system__Compl,axiom,
! [X2: set_a,M: sigma_measure_a,F: set_a > extend8495563244428889912nnreal] :
( ( member_set_a @ X2 @ ( lambda_system_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) @ F ) )
=> ( member_set_a @ ( minus_minus_set_a @ ( sigma_space_a @ M ) @ X2 ) @ ( lambda_system_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) @ F ) ) ) ).
% sets.lambda_system_Compl
thf(fact_1212_remove__induct,axiom,
! [P: set_a_a > $o,B: set_a_a] :
( ( P @ bot_bot_set_a_a )
=> ( ( ~ ( finite_finite_a_a @ B )
=> ( P @ B ) )
=> ( ! [A8: set_a_a] :
( ( finite_finite_a_a @ A8 )
=> ( ( A8 != bot_bot_set_a_a )
=> ( ( ord_less_eq_set_a_a @ A8 @ B )
=> ( ! [X5: a > a] :
( ( member_a_a @ X5 @ A8 )
=> ( P @ ( minus_minus_set_a_a @ A8 @ ( insert_a_a @ X5 @ bot_bot_set_a_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_1213_remove__induct,axiom,
! [P: set_a > $o,B: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B )
=> ( P @ B ) )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A8 @ B )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_1214_remove__induct,axiom,
! [P: set_set_a > $o,B: set_set_a] :
( ( P @ bot_bot_set_set_a )
=> ( ( ~ ( finite_finite_set_a @ B )
=> ( P @ B ) )
=> ( ! [A8: set_set_a] :
( ( finite_finite_set_a @ A8 )
=> ( ( A8 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A8 @ B )
=> ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A8 )
=> ( P @ ( minus_5736297505244876581_set_a @ A8 @ ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_1215_remove__induct,axiom,
! [P: set_real_a > $o,B: set_real_a] :
( ( P @ bot_bot_set_real_a )
=> ( ( ~ ( finite_finite_real_a @ B )
=> ( P @ B ) )
=> ( ! [A8: set_real_a] :
( ( finite_finite_real_a @ A8 )
=> ( ( A8 != bot_bot_set_real_a )
=> ( ( ord_le5743406823621094409real_a @ A8 @ B )
=> ( ! [X5: real > a] :
( ( member_real_a @ X5 @ A8 )
=> ( P @ ( minus_6532636778494125008real_a @ A8 @ ( insert_real_a @ X5 @ bot_bot_set_real_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_1216_finite__remove__induct,axiom,
! [B: set_a_a,P: set_a_a > $o] :
( ( finite_finite_a_a @ B )
=> ( ( P @ bot_bot_set_a_a )
=> ( ! [A8: set_a_a] :
( ( finite_finite_a_a @ A8 )
=> ( ( A8 != bot_bot_set_a_a )
=> ( ( ord_less_eq_set_a_a @ A8 @ B )
=> ( ! [X5: a > a] :
( ( member_a_a @ X5 @ A8 )
=> ( P @ ( minus_minus_set_a_a @ A8 @ ( insert_a_a @ X5 @ bot_bot_set_a_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1217_finite__remove__induct,axiom,
! [B: set_a,P: set_a > $o] :
( ( finite_finite_a @ B )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A8: set_a] :
( ( finite_finite_a @ A8 )
=> ( ( A8 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A8 @ B )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A8 )
=> ( P @ ( minus_minus_set_a @ A8 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1218_finite__remove__induct,axiom,
! [B: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ B )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A8: set_set_a] :
( ( finite_finite_set_a @ A8 )
=> ( ( A8 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A8 @ B )
=> ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A8 )
=> ( P @ ( minus_5736297505244876581_set_a @ A8 @ ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1219_finite__remove__induct,axiom,
! [B: set_real_a,P: set_real_a > $o] :
( ( finite_finite_real_a @ B )
=> ( ( P @ bot_bot_set_real_a )
=> ( ! [A8: set_real_a] :
( ( finite_finite_real_a @ A8 )
=> ( ( A8 != bot_bot_set_real_a )
=> ( ( ord_le5743406823621094409real_a @ A8 @ B )
=> ( ! [X5: real > a] :
( ( member_real_a @ X5 @ A8 )
=> ( P @ ( minus_6532636778494125008real_a @ A8 @ ( insert_real_a @ X5 @ bot_bot_set_real_a ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1220_psubset__insert__iff,axiom,
! [A3: set_a_a,X2: a > a,B: set_a_a] :
( ( ord_less_set_a_a @ A3 @ ( insert_a_a @ X2 @ B ) )
= ( ( ( member_a_a @ X2 @ B )
=> ( ord_less_set_a_a @ A3 @ B ) )
& ( ~ ( member_a_a @ X2 @ B )
=> ( ( ( member_a_a @ X2 @ A3 )
=> ( ord_less_set_a_a @ ( minus_minus_set_a_a @ A3 @ ( insert_a_a @ X2 @ bot_bot_set_a_a ) ) @ B ) )
& ( ~ ( member_a_a @ X2 @ A3 )
=> ( ord_less_eq_set_a_a @ A3 @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1221_psubset__insert__iff,axiom,
! [A3: set_a,X2: a,B: set_a] :
( ( ord_less_set_a @ A3 @ ( insert_a @ X2 @ B ) )
= ( ( ( member_a @ X2 @ B )
=> ( ord_less_set_a @ A3 @ B ) )
& ( ~ ( member_a @ X2 @ B )
=> ( ( ( member_a @ X2 @ A3 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X2 @ A3 )
=> ( ord_less_eq_set_a @ A3 @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1222_psubset__insert__iff,axiom,
! [A3: set_set_a,X2: set_a,B: set_set_a] :
( ( ord_less_set_set_a @ A3 @ ( insert_set_a @ X2 @ B ) )
= ( ( ( member_set_a @ X2 @ B )
=> ( ord_less_set_set_a @ A3 @ B ) )
& ( ~ ( member_set_a @ X2 @ B )
=> ( ( ( member_set_a @ X2 @ A3 )
=> ( ord_less_set_set_a @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) @ B ) )
& ( ~ ( member_set_a @ X2 @ A3 )
=> ( ord_le3724670747650509150_set_a @ A3 @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1223_psubset__insert__iff,axiom,
! [A3: set_real_a,X2: real > a,B: set_real_a] :
( ( ord_less_set_real_a @ A3 @ ( insert_real_a @ X2 @ B ) )
= ( ( ( member_real_a @ X2 @ B )
=> ( ord_less_set_real_a @ A3 @ B ) )
& ( ~ ( member_real_a @ X2 @ B )
=> ( ( ( member_real_a @ X2 @ A3 )
=> ( ord_less_set_real_a @ ( minus_6532636778494125008real_a @ A3 @ ( insert_real_a @ X2 @ bot_bot_set_real_a ) ) @ B ) )
& ( ~ ( member_real_a @ X2 @ A3 )
=> ( ord_le5743406823621094409real_a @ A3 @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1224_algebra__iff__Int,axiom,
( sigma_algebra_a
= ( ^ [Omega2: set_a,M2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ M2 @ ( pow_a @ Omega2 ) )
& ( member_set_a @ bot_bot_set_a @ M2 )
& ! [X4: set_a] :
( ( member_set_a @ X4 @ M2 )
=> ( member_set_a @ ( minus_minus_set_a @ Omega2 @ X4 ) @ M2 ) )
& ! [X4: set_a] :
( ( member_set_a @ X4 @ M2 )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ M2 )
=> ( member_set_a @ ( inf_inf_set_a @ X4 @ Y4 ) @ M2 ) ) ) ) ) ) ).
% algebra_iff_Int
thf(fact_1225_ring__of__setsI,axiom,
! [M: set_set_a,Omega: set_a] :
( ( ord_le3724670747650509150_set_a @ M @ ( pow_a @ Omega ) )
=> ( ( member_set_a @ bot_bot_set_a @ M )
=> ( ! [A: set_a,B2: set_a] :
( ( member_set_a @ A @ M )
=> ( ( member_set_a @ B2 @ M )
=> ( member_set_a @ ( sup_sup_set_a @ A @ B2 ) @ M ) ) )
=> ( ! [A: set_a,B2: set_a] :
( ( member_set_a @ A @ M )
=> ( ( member_set_a @ B2 @ M )
=> ( member_set_a @ ( minus_minus_set_a @ A @ B2 ) @ M ) ) )
=> ( sigma_ring_of_sets_a @ Omega @ M ) ) ) ) ) ).
% ring_of_setsI
thf(fact_1226_ring__of__sets__iff,axiom,
( sigma_ring_of_sets_a
= ( ^ [Omega2: set_a,M2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ M2 @ ( pow_a @ Omega2 ) )
& ( member_set_a @ bot_bot_set_a @ M2 )
& ! [X4: set_a] :
( ( member_set_a @ X4 @ M2 )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ M2 )
=> ( member_set_a @ ( sup_sup_set_a @ X4 @ Y4 ) @ M2 ) ) )
& ! [X4: set_a] :
( ( member_set_a @ X4 @ M2 )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ M2 )
=> ( member_set_a @ ( minus_minus_set_a @ X4 @ Y4 ) @ M2 ) ) ) ) ) ) ).
% ring_of_sets_iff
thf(fact_1227_algebra__iff__Un,axiom,
( sigma_algebra_a
= ( ^ [Omega2: set_a,M2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ M2 @ ( pow_a @ Omega2 ) )
& ( member_set_a @ bot_bot_set_a @ M2 )
& ! [X4: set_a] :
( ( member_set_a @ X4 @ M2 )
=> ( member_set_a @ ( minus_minus_set_a @ Omega2 @ X4 ) @ M2 ) )
& ! [X4: set_a] :
( ( member_set_a @ X4 @ M2 )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ M2 )
=> ( member_set_a @ ( sup_sup_set_a @ X4 @ Y4 ) @ M2 ) ) ) ) ) ) ).
% algebra_iff_Un
thf(fact_1228_sigma__algebra__single__set,axiom,
! [X: set_a,S: set_a] :
( ( ord_less_eq_set_a @ X @ S )
=> ( sigma_4968961713055010667ebra_a @ S @ ( insert_set_a @ bot_bot_set_a @ ( insert_set_a @ X @ ( insert_set_a @ ( minus_minus_set_a @ S @ X ) @ ( insert_set_a @ S @ bot_bot_set_set_a ) ) ) ) ) ) ).
% sigma_algebra_single_set
thf(fact_1229_algebra__single__set,axiom,
! [X: set_a,S: set_a] :
( ( ord_less_eq_set_a @ X @ S )
=> ( sigma_algebra_a @ S @ ( insert_set_a @ bot_bot_set_a @ ( insert_set_a @ X @ ( insert_set_a @ ( minus_minus_set_a @ S @ X ) @ ( insert_set_a @ S @ bot_bot_set_set_a ) ) ) ) ) ) ).
% algebra_single_set
thf(fact_1230_measurable__envelope__eq1,axiom,
! [A3: set_a,E: set_a,M: sigma_measure_a] :
( ( ord_less_eq_set_a @ A3 @ E )
=> ( ( member_set_a @ E @ ( sigma_sets_a @ M ) )
=> ( ( comple1317578422686860819lope_a @ M @ A3 @ E )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ ( sigma_sets_a @ M ) )
=> ( ( ord_less_eq_set_a @ X4 @ ( minus_minus_set_a @ E @ A3 ) )
=> ( ( sigma_emeasure_a @ M @ X4 )
= zero_z7100319975126383169nnreal ) ) ) ) ) ) ) ).
% measurable_envelope_eq1
thf(fact_1231_measurable__envelopeD1,axiom,
! [M: sigma_measure_a,A3: set_a,E: set_a,F4: set_a] :
( ( comple1317578422686860819lope_a @ M @ A3 @ E )
=> ( ( member_set_a @ F4 @ ( sigma_sets_a @ M ) )
=> ( ( ord_less_eq_set_a @ F4 @ ( minus_minus_set_a @ E @ A3 ) )
=> ( ( sigma_emeasure_a @ M @ F4 )
= zero_z7100319975126383169nnreal ) ) ) ) ).
% measurable_envelopeD1
thf(fact_1232_Lebesgue__Measure_Ocompletion__upper,axiom,
! [A3: set_a,M: sigma_measure_a] :
( ( member_set_a @ A3 @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ~ ! [A9: set_a] :
( ( ord_less_eq_set_a @ A3 @ A9 )
=> ( ( member_set_a @ A9 @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ ( minus_minus_set_a @ A9 @ A3 ) @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ( ( sigma_emeasure_a @ ( comple3428971583294703880tion_a @ M ) @ A3 )
!= ( sigma_emeasure_a @ M @ A9 ) ) ) ) ) ) ).
% Lebesgue_Measure.completion_upper
thf(fact_1233_ennreal__minus__cancel__iff,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A2 @ B3 )
= ( minus_8429688780609304081nnreal @ A2 @ C2 ) )
= ( ( B3 = C2 )
| ( ( ord_le3935885782089961368nnreal @ A2 @ B3 )
& ( ord_le3935885782089961368nnreal @ A2 @ C2 ) )
| ( A2 = top_to1496364449551166952nnreal ) ) ) ).
% ennreal_minus_cancel_iff
thf(fact_1234_ennreal__minus__cancel,axiom,
! [C2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( C2 != top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ A2 @ C2 )
=> ( ( ord_le3935885782089961368nnreal @ B3 @ C2 )
=> ( ( ( minus_8429688780609304081nnreal @ C2 @ A2 )
= ( minus_8429688780609304081nnreal @ C2 @ B3 ) )
=> ( A2 = B3 ) ) ) ) ) ).
% ennreal_minus_cancel
thf(fact_1235_diff__less__top__ennreal,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ ( minus_8429688780609304081nnreal @ A2 @ B3 ) @ top_to1496364449551166952nnreal )
= ( ord_le7381754540660121996nnreal @ A2 @ top_to1496364449551166952nnreal ) ) ).
% diff_less_top_ennreal
thf(fact_1236_diff__diff__ennreal,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B3 )
=> ( ( B3 != extend2057119558705770725nnreal )
=> ( ( minus_8429688780609304081nnreal @ B3 @ ( minus_8429688780609304081nnreal @ B3 @ A2 ) )
= A2 ) ) ) ).
% diff_diff_ennreal
thf(fact_1237_ennreal__minus__mono,axiom,
! [A2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal,D3: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ C2 )
=> ( ( ord_le3935885782089961368nnreal @ D3 @ B3 )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A2 @ B3 ) @ ( minus_8429688780609304081nnreal @ C2 @ D3 ) ) ) ) ).
% ennreal_minus_mono
thf(fact_1238_ennreal__mono__minus,axiom,
! [C2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ C2 @ B3 )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A2 @ B3 ) @ ( minus_8429688780609304081nnreal @ A2 @ C2 ) ) ) ).
% ennreal_mono_minus
thf(fact_1239_diff__le__self__ennreal,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A2 @ B3 ) @ A2 ) ).
% diff_le_self_ennreal
thf(fact_1240_ennreal__diff__le__mono__left,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B3 )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A2 @ C2 ) @ B3 ) ) ).
% ennreal_diff_le_mono_left
thf(fact_1241_ennreal__minus__eq__0,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A2 @ B3 )
= zero_z7100319975126383169nnreal )
=> ( ord_le3935885782089961368nnreal @ A2 @ B3 ) ) ).
% ennreal_minus_eq_0
thf(fact_1242_diff__gr0__ennreal,axiom,
! [B3: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B3 @ A2 )
=> ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( minus_8429688780609304081nnreal @ A2 @ B3 ) ) ) ).
% diff_gr0_ennreal
thf(fact_1243_diff__gt__0__iff__gt__ennreal,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( minus_8429688780609304081nnreal @ A2 @ B3 ) )
= ( ( ( A2 = top_to1496364449551166952nnreal )
& ( B3 = top_to1496364449551166952nnreal ) )
| ( ord_le7381754540660121996nnreal @ B3 @ A2 ) ) ) ).
% diff_gt_0_iff_gt_ennreal
thf(fact_1244_ennreal__minus__pos__iff,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ( ord_le7381754540660121996nnreal @ A2 @ top_to1496364449551166952nnreal )
| ( ord_le7381754540660121996nnreal @ B3 @ top_to1496364449551166952nnreal ) )
=> ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( minus_8429688780609304081nnreal @ A2 @ B3 ) )
=> ( ord_le7381754540660121996nnreal @ B3 @ A2 ) ) ) ).
% ennreal_minus_pos_iff
thf(fact_1245_ennreal__between,axiom,
! [E3: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ E3 )
=> ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ X2 )
=> ( ( ord_le7381754540660121996nnreal @ X2 @ top_to1496364449551166952nnreal )
=> ( ord_le7381754540660121996nnreal @ ( minus_8429688780609304081nnreal @ X2 @ E3 ) @ X2 ) ) ) ) ).
% ennreal_between
thf(fact_1246_ennreal__mono__minus__cancel,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A2 @ B3 ) @ ( minus_8429688780609304081nnreal @ A2 @ C2 ) )
=> ( ( ord_le7381754540660121996nnreal @ A2 @ top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ B3 @ A2 )
=> ( ( ord_le3935885782089961368nnreal @ C2 @ A2 )
=> ( ord_le3935885782089961368nnreal @ C2 @ B3 ) ) ) ) ) ).
% ennreal_mono_minus_cancel
thf(fact_1247_diff__eq__0__ennreal,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ A2 @ B3 )
=> ( ( minus_8429688780609304081nnreal @ A2 @ B3 )
= zero_z7100319975126383169nnreal ) ) ) ).
% diff_eq_0_ennreal
thf(fact_1248_diff__eq__0__iff__ennreal,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A2 @ B3 )
= zero_z7100319975126383169nnreal )
= ( ( ord_le7381754540660121996nnreal @ A2 @ top_to1496364449551166952nnreal )
& ( ord_le3935885782089961368nnreal @ A2 @ B3 ) ) ) ).
% diff_eq_0_iff_ennreal
thf(fact_1249_emeasure__Diff,axiom,
! [M: sigma_measure_a,B: set_a,A3: set_a] :
( ( ( sigma_emeasure_a @ M @ B )
!= extend2057119558705770725nnreal )
=> ( ( member_set_a @ A3 @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M ) )
=> ( ( ord_less_eq_set_a @ B @ A3 )
=> ( ( sigma_emeasure_a @ M @ ( minus_minus_set_a @ A3 @ B ) )
= ( minus_8429688780609304081nnreal @ ( sigma_emeasure_a @ M @ A3 ) @ ( sigma_emeasure_a @ M @ B ) ) ) ) ) ) ) ).
% emeasure_Diff
thf(fact_1250_emeasure__compl,axiom,
! [S2: set_a,M: sigma_measure_a] :
( ( member_set_a @ S2 @ ( sigma_sets_a @ M ) )
=> ( ( ( sigma_emeasure_a @ M @ S2 )
!= extend2057119558705770725nnreal )
=> ( ( sigma_emeasure_a @ M @ ( minus_minus_set_a @ ( sigma_space_a @ M ) @ S2 ) )
= ( minus_8429688780609304081nnreal @ ( sigma_emeasure_a @ M @ ( sigma_space_a @ M ) ) @ ( sigma_emeasure_a @ M @ S2 ) ) ) ) ) ).
% emeasure_compl
thf(fact_1251_sets_Olambda__system__strong__additive,axiom,
! [Z3: set_a,M: sigma_measure_a,X2: set_a,Y3: set_a,F: set_a > extend8495563244428889912nnreal] :
( ( member_set_a @ Z3 @ ( sigma_sets_a @ M ) )
=> ( ( ( inf_inf_set_a @ X2 @ Y3 )
= bot_bot_set_a )
=> ( ( member_set_a @ X2 @ ( lambda_system_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) @ F ) )
=> ( ( member_set_a @ Y3 @ ( lambda_system_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) @ F ) )
=> ( ( F @ ( inf_inf_set_a @ Z3 @ ( sup_sup_set_a @ X2 @ Y3 ) ) )
= ( plus_p1859984266308609217nnreal @ ( F @ ( inf_inf_set_a @ Z3 @ X2 ) ) @ ( F @ ( inf_inf_set_a @ Z3 @ Y3 ) ) ) ) ) ) ) ) ).
% sets.lambda_system_strong_additive
thf(fact_1252_ennreal__add__less__top,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ ( plus_p1859984266308609217nnreal @ A2 @ B3 ) @ top_to1496364449551166952nnreal )
= ( ( ord_le7381754540660121996nnreal @ A2 @ top_to1496364449551166952nnreal )
& ( ord_le7381754540660121996nnreal @ B3 @ top_to1496364449551166952nnreal ) ) ) ).
% ennreal_add_less_top
thf(fact_1253_add__diff__eq__iff__ennreal,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ( plus_p1859984266308609217nnreal @ X2 @ ( minus_8429688780609304081nnreal @ Y3 @ X2 ) )
= Y3 )
= ( ord_le3935885782089961368nnreal @ X2 @ Y3 ) ) ).
% add_diff_eq_iff_ennreal
thf(fact_1254_ennreal__minus__le__iff,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A2 @ B3 ) @ C2 )
= ( ( ord_le3935885782089961368nnreal @ A2 @ ( plus_p1859984266308609217nnreal @ B3 @ C2 ) )
& ( ( ( A2 = top_to1496364449551166952nnreal )
& ( B3 = top_to1496364449551166952nnreal ) )
=> ( C2 = top_to1496364449551166952nnreal ) ) ) ) ).
% ennreal_minus_le_iff
thf(fact_1255_less__diff__eq__ennreal,axiom,
! [B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ( ord_le7381754540660121996nnreal @ B3 @ top_to1496364449551166952nnreal )
| ( ord_le7381754540660121996nnreal @ C2 @ top_to1496364449551166952nnreal ) )
=> ( ( ord_le7381754540660121996nnreal @ A2 @ ( minus_8429688780609304081nnreal @ B3 @ C2 ) )
= ( ord_le7381754540660121996nnreal @ ( plus_p1859984266308609217nnreal @ A2 @ C2 ) @ B3 ) ) ) ).
% less_diff_eq_ennreal
thf(fact_1256_diff__diff__ennreal_H_H,axiom,
! [Z3: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Z3 @ Y3 )
=> ( ( ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ Y3 @ Z3 ) @ X2 )
=> ( ( minus_8429688780609304081nnreal @ X2 @ ( minus_8429688780609304081nnreal @ Y3 @ Z3 ) )
= ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ X2 @ Z3 ) @ Y3 ) ) )
& ( ~ ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ Y3 @ Z3 ) @ X2 )
=> ( ( minus_8429688780609304081nnreal @ X2 @ ( minus_8429688780609304081nnreal @ Y3 @ Z3 ) )
= zero_z7100319975126383169nnreal ) ) ) ) ).
% diff_diff_ennreal''
thf(fact_1257_ennreal__le__minus__iff,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ ( minus_8429688780609304081nnreal @ B3 @ C2 ) )
= ( ( ord_le3935885782089961368nnreal @ ( plus_p1859984266308609217nnreal @ A2 @ C2 ) @ B3 )
| ( ( A2 = zero_z7100319975126383169nnreal )
& ( ord_le3935885782089961368nnreal @ B3 @ C2 ) ) ) ) ).
% ennreal_le_minus_iff
thf(fact_1258_diff__diff__ennreal_H,axiom,
! [Z3: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Z3 @ Y3 )
=> ( ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ Y3 @ Z3 ) @ X2 )
=> ( ( minus_8429688780609304081nnreal @ X2 @ ( minus_8429688780609304081nnreal @ Y3 @ Z3 ) )
= ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ X2 @ Z3 ) @ Y3 ) ) ) ) ).
% diff_diff_ennreal'
thf(fact_1259_add__diff__eq__ennreal,axiom,
! [Z3: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Z3 @ Y3 )
=> ( ( plus_p1859984266308609217nnreal @ X2 @ ( minus_8429688780609304081nnreal @ Y3 @ Z3 ) )
= ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ X2 @ Y3 ) @ Z3 ) ) ) ).
% add_diff_eq_ennreal
thf(fact_1260_add__diff__le__ennreal,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ A2 @ B3 ) @ C2 ) @ ( plus_p1859984266308609217nnreal @ A2 @ ( minus_8429688780609304081nnreal @ B3 @ C2 ) ) ) ).
% add_diff_le_ennreal
thf(fact_1261_add__diff__self__ennreal,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ( ord_le3935885782089961368nnreal @ A2 @ B3 )
=> ( ( plus_p1859984266308609217nnreal @ A2 @ ( minus_8429688780609304081nnreal @ B3 @ A2 ) )
= B3 ) )
& ( ~ ( ord_le3935885782089961368nnreal @ A2 @ B3 )
=> ( ( plus_p1859984266308609217nnreal @ A2 @ ( minus_8429688780609304081nnreal @ B3 @ A2 ) )
= A2 ) ) ) ).
% add_diff_self_ennreal
thf(fact_1262_diff__add__self__ennreal,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ( ord_le3935885782089961368nnreal @ A2 @ B3 )
=> ( ( plus_p1859984266308609217nnreal @ ( minus_8429688780609304081nnreal @ B3 @ A2 ) @ A2 )
= B3 ) )
& ( ~ ( ord_le3935885782089961368nnreal @ A2 @ B3 )
=> ( ( plus_p1859984266308609217nnreal @ ( minus_8429688780609304081nnreal @ B3 @ A2 ) @ A2 )
= A2 ) ) ) ).
% diff_add_self_ennreal
thf(fact_1263_ennreal__ineq__diff__add,axiom,
! [B3: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B3 @ A2 )
=> ( A2
= ( plus_p1859984266308609217nnreal @ B3 @ ( minus_8429688780609304081nnreal @ A2 @ B3 ) ) ) ) ).
% ennreal_ineq_diff_add
thf(fact_1264_ennreal__diff__add__assoc,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B3 )
=> ( ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ C2 @ B3 ) @ A2 )
= ( plus_p1859984266308609217nnreal @ C2 @ ( minus_8429688780609304081nnreal @ B3 @ A2 ) ) ) ) ).
% ennreal_diff_add_assoc
thf(fact_1265_diff__add__assoc2__ennreal,axiom,
! [B3: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B3 @ A2 )
=> ( ( plus_p1859984266308609217nnreal @ ( minus_8429688780609304081nnreal @ A2 @ B3 ) @ C2 )
= ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ A2 @ C2 ) @ B3 ) ) ) ).
% diff_add_assoc2_ennreal
thf(fact_1266_diff__add__cancel__ennreal,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ B3 )
=> ( ( plus_p1859984266308609217nnreal @ ( minus_8429688780609304081nnreal @ B3 @ A2 ) @ A2 )
= B3 ) ) ).
% diff_add_cancel_ennreal
thf(fact_1267_add__diff__inverse__ennreal,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
=> ( ( plus_p1859984266308609217nnreal @ X2 @ ( minus_8429688780609304081nnreal @ Y3 @ X2 ) )
= Y3 ) ) ).
% add_diff_inverse_ennreal
thf(fact_1268_add__pos__nonneg,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ A2 )
=> ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ B3 )
=> ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( plus_p1859984266308609217nnreal @ A2 @ B3 ) ) ) ) ).
% add_pos_nonneg
thf(fact_1269_add__nonpos__neg,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A2 @ zero_z7100319975126383169nnreal )
=> ( ( ord_le7381754540660121996nnreal @ B3 @ zero_z7100319975126383169nnreal )
=> ( ord_le7381754540660121996nnreal @ ( plus_p1859984266308609217nnreal @ A2 @ B3 ) @ zero_z7100319975126383169nnreal ) ) ) ).
% add_nonpos_neg
thf(fact_1270_add__nonneg__pos,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ A2 )
=> ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ B3 )
=> ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( plus_p1859984266308609217nnreal @ A2 @ B3 ) ) ) ) ).
% add_nonneg_pos
thf(fact_1271_add__neg__nonpos,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ zero_z7100319975126383169nnreal )
=> ( ( ord_le3935885782089961368nnreal @ B3 @ zero_z7100319975126383169nnreal )
=> ( ord_le7381754540660121996nnreal @ ( plus_p1859984266308609217nnreal @ A2 @ B3 ) @ zero_z7100319975126383169nnreal ) ) ) ).
% add_neg_nonpos
thf(fact_1272_top__add,axiom,
! [X2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ X2 )
=> ( ( plus_p1859984266308609217nnreal @ top_to1496364449551166952nnreal @ X2 )
= top_to1496364449551166952nnreal ) ) ).
% top_add
thf(fact_1273_add__top,axiom,
! [X2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ X2 )
=> ( ( plus_p1859984266308609217nnreal @ X2 @ top_to1496364449551166952nnreal )
= top_to1496364449551166952nnreal ) ) ).
% add_top
thf(fact_1274_ennreal__add__left__cancel__less,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ ( plus_p1859984266308609217nnreal @ A2 @ B3 ) @ ( plus_p1859984266308609217nnreal @ A2 @ C2 ) )
= ( ( A2 != extend2057119558705770725nnreal )
& ( ord_le7381754540660121996nnreal @ B3 @ C2 ) ) ) ).
% ennreal_add_left_cancel_less
thf(fact_1275_ennreal__add__left__cancel__le,axiom,
! [A2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ ( plus_p1859984266308609217nnreal @ A2 @ B3 ) @ ( plus_p1859984266308609217nnreal @ A2 @ C2 ) )
= ( ( A2 = extend2057119558705770725nnreal )
| ( ord_le3935885782089961368nnreal @ B3 @ C2 ) ) ) ).
% ennreal_add_left_cancel_le
% Helper facts (5)
thf(help_If_2_1_If_001t__QuasiBorel__Oquasi____borel_Itf__a_J_T,axiom,
! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( if_quasi_borel_a @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__QuasiBorel__Oquasi____borel_Itf__a_J_T,axiom,
! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( if_quasi_borel_a @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_3_1_If_001t__Extended____Nonnegative____Real__Oennreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Extended____Nonnegative____Real__Oennreal_T,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( if_Ext9135588136721118450nnreal @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Extended____Nonnegative____Real__Oennreal_T,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( if_Ext9135588136721118450nnreal @ $true @ X2 @ Y3 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
sigma_4968961713055010667ebra_a @ ( qbs_space_a @ x ) @ ( measur1355555235234291375a_Mx_a @ x ) ).
%------------------------------------------------------------------------------