TPTP Problem File: SLH0605^1.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/0000_StandardBorel/prob_00175_005745__15064616_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1557 ( 511 unt; 278 typ; 0 def)
% Number of atoms : 3590 (1404 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 11428 ( 282 ~; 14 |; 175 &;9401 @)
% ( 0 <=>;1556 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 7 avg)
% Number of types : 33 ( 32 usr)
% Number of type conns : 2006 (2006 >; 0 *; 0 +; 0 <<)
% Number of symbols : 247 ( 246 usr; 19 con; 0-4 aty)
% Number of variables : 3753 ( 123 ^;3544 !; 86 ?;3753 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:01:19.793
%------------------------------------------------------------------------------
% Could-be-implicit typings (32)
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_M_062_It__Real__Oreal_Mt__Real__Oreal_J_J_J,type,
set_real_real_real: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Real__Oreal_Mt__Real__Oreal_J_Mt__Real__Oreal_J_J,type,
set_real_real_real2: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
sigma_4258434043392614480l_real: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J_J,type,
set_set_real_real: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_M_062_Itf__a_Mt__Real__Oreal_J_J_J,type,
set_real_a_real: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_M_062_It__Real__Oreal_Mtf__a_J_J_J,type,
set_real_real_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_Mt__Real__Oreal_J_Mt__Real__Oreal_J_J,type,
set_a_real_real: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Real__Oreal_Mtf__a_J_Mt__Real__Oreal_J_J,type,
set_real_a_real2: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Real__Oreal_Mt__Real__Oreal_J_Mtf__a_J_J,type,
set_real_real_a2: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_J,type,
set_set_a_set_a: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_I_062_Itf__a_Mt__Real__Oreal_J_J,type,
sigma_measure_a_real: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_I_062_It__Real__Oreal_Mtf__a_J_J,type,
sigma_measure_real_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_Itf__a_Mt__Real__Oreal_J_J_J,type,
set_set_a_real: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J_J,type,
set_set_real_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_Itf__a_J_Mt__Real__Oreal_J_J,type,
set_set_a_real2: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Set__Oset_Itf__a_J_J_J,type,
set_real_set_a: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Set__Oset_Itf__a_J_J,type,
sigma_measure_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
set_real_real: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Set__Oset_Itf__a_J_J_J,type,
set_a_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_Itf__a_J_Mtf__a_J_J,type,
set_set_a_a: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
sigma_measure_real: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
set_set_real: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Real__Oreal_J_J,type,
set_a_real: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J,type,
set_real_a: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_Itf__a_J,type,
sigma_measure_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
set_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (246)
thf(sy_c_Borel__Space_Ois__borel_001t__Real__Oreal_001t__Real__Oreal,type,
borel_236569967776329622l_real: ( real > real ) > sigma_measure_real > $o ).
thf(sy_c_Borel__Space_Ois__borel_001tf__a_001t__Real__Oreal,type,
borel_4993665998515044718a_real: ( a > real ) > sigma_measure_a > $o ).
thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Real__Oreal,type,
borel_5078946678739801102l_real: sigma_measure_real ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001_062_It__Real__Oreal_Mt__Real__Oreal_J_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
comp_r2210161833511426559l_real: ( real > real > real ) > ( ( real > real ) > real ) > ( real > real ) > real > real ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001_062_It__Real__Oreal_Mtf__a_J_001_062_It__Real__Oreal_Mtf__a_J,type,
comp_r4421311556806241359real_a: ( real > real > a ) > ( ( real > a ) > real ) > ( real > a ) > real > a ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001_062_Itf__a_Mt__Real__Oreal_J_001_062_Itf__a_Mt__Real__Oreal_J,type,
comp_r2811049936330031311a_real: ( real > a > real ) > ( ( a > real ) > real ) > ( a > real ) > a > real ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal,type,
comp_real_real_real: ( real > real ) > ( real > real ) > real > real ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001tf__a,type,
comp_real_real_a: ( real > real ) > ( a > real ) > a > real ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
comp_r8810651240769227663_set_a: ( real > set_a ) > ( set_a > real ) > set_a > set_a ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001tf__a_001t__Real__Oreal,type,
comp_real_a_real: ( real > a ) > ( real > real ) > real > a ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001tf__a_001tf__a,type,
comp_real_a_a: ( real > a ) > ( a > real ) > a > a ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_It__Real__Oreal_J,type,
comp_s5292247282120234379t_real: ( set_real > set_real ) > ( set_real > set_real ) > set_real > set_real ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_Itf__a_J,type,
comp_s762043476418857333_set_a: ( set_real > set_real ) > ( set_a > set_real ) > set_a > set_real ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Real__Oreal_J,type,
comp_s6687884485137378875t_real: ( set_real > set_a ) > ( set_real > set_real ) > set_real > set_a ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
comp_s535997479508436165_set_a: ( set_real > set_a ) > ( set_a > set_real ) > set_a > set_a ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_It__Real__Oreal_J,type,
comp_s6092362911880814593t_real: ( set_a > set_real ) > ( set_real > set_a ) > set_real > set_real ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_Itf__a_J,type,
comp_s5765971254739342719_set_a: ( set_a > set_real ) > ( set_a > set_a ) > set_a > set_real ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Real__Oreal_J,type,
comp_s2468440226603088453t_real: ( set_a > set_a ) > ( set_real > set_a ) > set_real > set_a ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
comp_s9136116056826389051_set_a: ( set_a > set_a ) > ( set_a > set_a ) > set_a > set_a ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Real__Oreal_001t__Real__Oreal,type,
comp_a_real_real: ( a > real ) > ( real > a ) > real > real ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Real__Oreal_001t__Set__Oset_Itf__a_J,type,
comp_a_real_set_a: ( a > real ) > ( set_a > a ) > set_a > real ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Real__Oreal_001tf__a,type,
comp_a_real_a: ( a > real ) > ( a > a ) > a > real ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Set__Oset_Itf__a_J_001t__Real__Oreal,type,
comp_a_set_a_real: ( a > set_a ) > ( real > a ) > real > set_a ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Set__Oset_Itf__a_J_001tf__a,type,
comp_a_set_a_a: ( a > set_a ) > ( a > a ) > a > set_a ).
thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001t__Real__Oreal,type,
comp_a_a_real: ( a > a ) > ( real > a ) > real > a ).
thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001t__Set__Oset_Itf__a_J,type,
comp_a_a_set_a: ( a > a ) > ( set_a > a ) > set_a > a ).
thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001tf__a,type,
comp_a_a_a: ( a > a ) > ( a > a ) > a > a ).
thf(sy_c_Fun_Oid_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
id_real_real: ( real > real ) > real > real ).
thf(sy_c_Fun_Oid_001t__Real__Oreal,type,
id_real: real > real ).
thf(sy_c_Fun_Oid_001t__Set__Oset_It__Real__Oreal_J,type,
id_set_real: set_real > set_real ).
thf(sy_c_Fun_Oid_001t__Set__Oset_Itf__a_J,type,
id_set_a: set_a > set_a ).
thf(sy_c_Fun_Oid_001tf__a,type,
id_a: a > a ).
thf(sy_c_Fun_Oinj__on_001_062_It__Real__Oreal_Mt__Real__Oreal_J_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
inj_on838481142967259233l_real: ( ( real > real ) > real > real ) > set_real_real > $o ).
thf(sy_c_Fun_Oinj__on_001_062_It__Real__Oreal_Mt__Real__Oreal_J_001t__Real__Oreal,type,
inj_on1257811008913012118l_real: ( ( real > real ) > real ) > set_real_real > $o ).
thf(sy_c_Fun_Oinj__on_001_062_It__Real__Oreal_Mt__Real__Oreal_J_001tf__a,type,
inj_on_real_real_a: ( ( real > real ) > a ) > set_real_real > $o ).
thf(sy_c_Fun_Oinj__on_001_062_It__Real__Oreal_Mtf__a_J_001t__Real__Oreal,type,
inj_on_real_a_real: ( ( real > a ) > real ) > set_real_a > $o ).
thf(sy_c_Fun_Oinj__on_001_062_It__Real__Oreal_Mtf__a_J_001tf__a,type,
inj_on_real_a_a: ( ( real > a ) > a ) > set_real_a > $o ).
thf(sy_c_Fun_Oinj__on_001_062_Itf__a_Mt__Real__Oreal_J_001_062_Itf__a_Mtf__a_J,type,
inj_on_a_real_a_a: ( ( a > real ) > a > a ) > set_a_real > $o ).
thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
inj_on_real_real: ( real > real ) > set_real > $o ).
thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Set__Oset_Itf__a_J,type,
inj_on_real_set_a: ( real > set_a ) > set_real > $o ).
thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001tf__a,type,
inj_on_real_a: ( real > a ) > set_real > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_Itf__a_J_001t__Real__Oreal,type,
inj_on_set_a_real: ( set_a > real ) > set_set_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
inj_on_set_a_set_a: ( set_a > set_a ) > set_set_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_Itf__a_J_001tf__a,type,
inj_on_set_a_a: ( set_a > a ) > set_set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001t__Real__Oreal,type,
inj_on_a_real: ( a > real ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001t__Set__Oset_Itf__a_J,type,
inj_on_a_set_a: ( a > set_a ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
inj_on_a_a: ( a > a ) > set_a > $o ).
thf(sy_c_Fun_Othe__inv__into_001_062_It__Real__Oreal_Mt__Real__Oreal_J_001tf__a,type,
the_in6219261096523627986real_a: set_real_real > ( ( real > real ) > a ) > a > real > real ).
thf(sy_c_Fun_Othe__inv__into_001_062_It__Real__Oreal_Mtf__a_J_001tf__a,type,
the_in3864030383226952808al_a_a: set_real_a > ( ( real > a ) > a ) > a > real > a ).
thf(sy_c_Fun_Othe__inv__into_001t__Real__Oreal_001t__Real__Oreal,type,
the_in5290026491893676941l_real: set_real > ( real > real ) > real > real ).
thf(sy_c_Fun_Othe__inv__into_001t__Real__Oreal_001t__Set__Oset_Itf__a_J,type,
the_in3800189856886063421_set_a: set_real > ( real > set_a ) > set_a > real ).
thf(sy_c_Fun_Othe__inv__into_001t__Real__Oreal_001tf__a,type,
the_inv_into_real_a: set_real > ( real > a ) > a > real ).
thf(sy_c_Fun_Othe__inv__into_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
the_in3497175299774311219_set_a: set_set_a > ( set_a > set_a ) > set_a > set_a ).
thf(sy_c_Fun_Othe__inv__into_001t__Set__Oset_Itf__a_J_001tf__a,type,
the_inv_into_set_a_a: set_set_a > ( set_a > a ) > a > set_a ).
thf(sy_c_Fun_Othe__inv__into_001tf__a_001t__Real__Oreal,type,
the_inv_into_a_real: set_a > ( a > real ) > real > a ).
thf(sy_c_Fun_Othe__inv__into_001tf__a_001t__Set__Oset_Itf__a_J,type,
the_inv_into_a_set_a: set_a > ( a > set_a ) > set_a > a ).
thf(sy_c_Fun_Othe__inv__into_001tf__a_001tf__a,type,
the_inv_into_a_a: set_a > ( a > a ) > a > a ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
inf_in5166753670444608447l_real: set_real_real > set_real_real > set_real_real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J,type,
inf_inf_set_real_a: set_real_a > set_real_a > set_real_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_Itf__a_Mt__Real__Oreal_J_J,type,
inf_inf_set_a_real: set_a_real > set_a_real > set_a_real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Real__Oreal_J,type,
inf_inf_set_real: set_real > set_real > set_real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
inf_inf_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Measure__Space_Oincreasing_001t__Real__Oreal_001t__Set__Oset_It__Real__Oreal_J,type,
measur7695573945809229763t_real: set_set_real > ( set_real > set_real ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001t__Real__Oreal_001t__Set__Oset_Itf__a_J,type,
measur6047784561759668541_set_a: set_set_real > ( set_real > set_a ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001tf__a_001t__Set__Oset_It__Real__Oreal_J,type,
measur2503387328813080045t_real: set_set_a > ( set_a > set_real ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001tf__a_001t__Set__Oset_Itf__a_J,type,
measur7842569353079325843_set_a: set_set_a > ( set_a > set_a ) > $o ).
thf(sy_c_Measure__Space_Osup__measure_H_001t__Real__Oreal,type,
measur2147279183506585690e_real: sigma_measure_real > sigma_measure_real > sigma_measure_real ).
thf(sy_c_Measure__Space_Osup__measure_H_001tf__a,type,
measur3004909623614618064sure_a: sigma_measure_a > sigma_measure_a > sigma_measure_a ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Real__Oreal_Mt__Real__Oreal_J_M_Eo_J,type,
bot_bot_real_real_o: ( real > real ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Real__Oreal_Mtf__a_J_M_Eo_J,type,
bot_bot_real_a_o: ( real > a ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_Itf__a_Mt__Real__Oreal_J_M_Eo_J,type,
bot_bot_a_real_o: ( a > real ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Real__Oreal_M_Eo_J,type,
bot_bot_real_o: real > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
bot_bot_set_a_o: set_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
bot_bo6767488733719836353l_real: set_real_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J,type,
bot_bot_set_real_a: set_real_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_Itf__a_Mt__Real__Oreal_J_J,type,
bot_bot_set_a_real: set_a_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
bot_bot_set_set_real: set_set_real ).
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thf(sy_c_Sigma__Algebra_Osets_001t__Real__Oreal,type,
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thf(sy_c_Sigma__Algebra_Osets_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Sigma__Algebra_Ospace_001t__Set__Oset_Itf__a_J,type,
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sigma_space_a: sigma_measure_a > set_a ).
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standa4474357433005041006a_real: sigma_measure_a_real > $o ).
thf(sy_c_StandardBorel_Ostandard__borel_001t__Real__Oreal,type,
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thf(sy_c_StandardBorel_Ostandard__borel_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_StandardBorel_Ostandard__borel_001tf__a,type,
standard_borel_a: sigma_measure_a > $o ).
thf(sy_c_StandardBorel_Ostandard__borel_Of_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
standard_f_real_real: sigma_4258434043392614480l_real > ( real > real ) > real ).
thf(sy_c_StandardBorel_Ostandard__borel_Of_001_062_It__Real__Oreal_Mtf__a_J,type,
standard_f_real_a: sigma_measure_real_a > ( real > a ) > real ).
thf(sy_c_StandardBorel_Ostandard__borel_Of_001_062_Itf__a_Mt__Real__Oreal_J,type,
standard_f_a_real: sigma_measure_a_real > ( a > real ) > real ).
thf(sy_c_StandardBorel_Ostandard__borel_Of_001t__Real__Oreal,type,
standard_f_real: sigma_measure_real > real > real ).
thf(sy_c_StandardBorel_Ostandard__borel_Of_001t__Set__Oset_Itf__a_J,type,
standard_f_set_a: sigma_measure_set_a > set_a > real ).
thf(sy_c_StandardBorel_Ostandard__borel_Of_001tf__a,type,
standard_f_a: sigma_measure_a > a > real ).
thf(sy_c_StandardBorel_Ostandard__borel_Og_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
standard_g_real_real: sigma_4258434043392614480l_real > real > real > real ).
thf(sy_c_StandardBorel_Ostandard__borel_Og_001_062_It__Real__Oreal_Mtf__a_J,type,
standard_g_real_a: sigma_measure_real_a > real > real > a ).
thf(sy_c_StandardBorel_Ostandard__borel_Og_001_062_Itf__a_Mt__Real__Oreal_J,type,
standard_g_a_real: sigma_measure_a_real > real > a > real ).
thf(sy_c_StandardBorel_Ostandard__borel_Og_001t__Real__Oreal,type,
standard_g_real: sigma_measure_real > real > real ).
thf(sy_c_StandardBorel_Ostandard__borel_Og_001t__Set__Oset_Itf__a_J,type,
standard_g_set_a: sigma_measure_set_a > real > set_a ).
thf(sy_c_StandardBorel_Ostandard__borel_Og_001tf__a,type,
standard_g_a: sigma_measure_a > real > a ).
thf(sy_c_StandardBorel_Ostandard__borel__space__UNIV_001t__Real__Oreal,type,
standa1306199911732814765V_real: sigma_measure_real > $o ).
thf(sy_c_StandardBorel_Ostandard__borel__space__UNIV_001tf__a,type,
standa5776109378963170237UNIV_a: sigma_measure_a > $o ).
thf(sy_c_StandardBorel_Ostandard__borel__space__UNIV__axioms_001t__Real__Oreal,type,
standa1498722272452280784s_real: sigma_measure_real > $o ).
thf(sy_c_StandardBorel_Ostandard__borel__space__UNIV__axioms_001tf__a,type,
standa2153564630574221018ioms_a: sigma_measure_a > $o ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Real__Oreal,type,
topolo4860482606490270245n_real: set_real > $o ).
thf(sy_c_member_001_062_I_062_It__Real__Oreal_Mt__Real__Oreal_J_Mt__Real__Oreal_J,type,
member5749659578190367193l_real: ( ( real > real ) > real ) > set_real_real_real2 > $o ).
thf(sy_c_member_001_062_I_062_It__Real__Oreal_Mt__Real__Oreal_J_Mtf__a_J,type,
member_real_real_a: ( ( real > real ) > a ) > set_real_real_a2 > $o ).
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_Itf__a_Mt__Real__Oreal_J_Mt__Real__Oreal_J,type,
member_a_real_real: ( ( a > real ) > real ) > set_a_real_real > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_M_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
member8878224140454985689l_real: ( real > real > real ) > set_real_real_real > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_M_062_It__Real__Oreal_Mtf__a_J_J,type,
member_real_real_a2: ( real > real > a ) > set_real_real_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_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__Real__Oreal_J,type,
member_set_a_real: ( set_a > real ) > set_set_a_real2 > $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_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__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
member_set_real_real: set_real_real > set_set_real_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_Mt__Real__Oreal_J_J,type,
member_set_a_real2: set_a_real > set_set_a_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Real__Oreal_J,type,
member_set_real: set_real > set_set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_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_M,type,
m: sigma_measure_a ).
thf(sy_v_U,type,
u: set_a ).
% Relevant facts (1278)
thf(fact_0__092_060open_062U_A_092_060subseteq_062_Aspace_AM_092_060close_062,axiom,
ord_less_eq_set_a @ u @ ( sigma_space_a @ m ) ).
% \<open>U \<subseteq> space M\<close>
thf(fact_1_standard__borel__axioms,axiom,
standard_borel_a @ m ).
% standard_borel_axioms
thf(fact_2_assms_I2_J,axiom,
u != bot_bot_set_a ).
% assms(2)
thf(fact_3_assms_I1_J,axiom,
member_set_a @ u @ ( sigma_sets_a @ m ) ).
% assms(1)
thf(fact_4_standard__borel_Of_Ocong,axiom,
standard_f_a = standard_f_a ).
% standard_borel.f.cong
thf(fact_5_standard__borel_Of_Ocong,axiom,
standard_f_real = standard_f_real ).
% standard_borel.f.cong
thf(fact_6_standard__borel_Og_Ocong,axiom,
standard_g_a = standard_g_a ).
% standard_borel.g.cong
thf(fact_7_standard__borel_Og_Ocong,axiom,
standard_g_real = standard_g_real ).
% standard_borel.g.cong
thf(fact_8_gf__comp__id_I2_J,axiom,
! [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
=> ( ( standard_g_a @ m @ ( standard_f_a @ m @ X ) )
= X ) ) ).
% gf_comp_id(2)
thf(fact_9_gf__comp__id_I1_J,axiom,
! [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
=> ( ( comp_real_a_a @ ( standard_g_a @ m ) @ ( standard_f_a @ m ) @ X )
= X ) ) ).
% gf_comp_id(1)
thf(fact_10_image__vimage__subset,axiom,
! [F: a > a,A: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F @ ( vimage_a_a @ F @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_11_image__vimage__subset,axiom,
! [F: a > real,A: set_real] : ( ord_less_eq_set_real @ ( image_a_real @ F @ ( vimage_a_real @ F @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_12_image__vimage__subset,axiom,
! [F: real > real,A: set_real] : ( ord_less_eq_set_real @ ( image_real_real @ F @ ( vimage_real_real @ F @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_13_image__vimage__subset,axiom,
! [F: real > a,A: set_a] : ( ord_less_eq_set_a @ ( image_real_a @ F @ ( vimage_real_a @ F @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_14_image__subset__iff__subset__vimage,axiom,
! [F: real > real,A: set_real,B: set_real] :
( ( ord_less_eq_set_real @ ( image_real_real @ F @ A ) @ B )
= ( ord_less_eq_set_real @ A @ ( vimage_real_real @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_15_image__subset__iff__subset__vimage,axiom,
! [F: a > real,A: set_a,B: set_real] :
( ( ord_less_eq_set_real @ ( image_a_real @ F @ A ) @ B )
= ( ord_less_eq_set_a @ A @ ( vimage_a_real @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_16_image__subset__iff__subset__vimage,axiom,
! [F: real > a,A: set_real,B: set_a] :
( ( ord_less_eq_set_a @ ( image_real_a @ F @ A ) @ B )
= ( ord_less_eq_set_real @ A @ ( vimage_real_a @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_17_image__subset__iff__subset__vimage,axiom,
! [F: a > a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B )
= ( ord_less_eq_set_a @ A @ ( vimage_a_a @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_18_vimageI,axiom,
! [F: real > a,A2: real,B2: a,B: set_a] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_a @ B2 @ B )
=> ( member_real @ A2 @ ( vimage_real_a @ F @ B ) ) ) ) ).
% vimageI
thf(fact_19_vimageI,axiom,
! [F: real > real,A2: real,B2: real,B: set_real] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_real @ B2 @ B )
=> ( member_real @ A2 @ ( vimage_real_real @ F @ B ) ) ) ) ).
% vimageI
thf(fact_20_vimageI,axiom,
! [F: a > real,A2: a,B2: real,B: set_real] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_real @ B2 @ B )
=> ( member_a @ A2 @ ( vimage_a_real @ F @ B ) ) ) ) ).
% vimageI
thf(fact_21_vimageI,axiom,
! [F: a > a,A2: a,B2: a,B: set_a] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_a @ B2 @ B )
=> ( member_a @ A2 @ ( vimage_a_a @ F @ B ) ) ) ) ).
% vimageI
thf(fact_22_vimageI,axiom,
! [F: a > set_a,A2: a,B2: set_a,B: set_set_a] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_set_a @ B2 @ B )
=> ( member_a @ A2 @ ( vimage_a_set_a @ F @ B ) ) ) ) ).
% vimageI
thf(fact_23_vimageI,axiom,
! [F: set_a > a,A2: set_a,B2: a,B: set_a] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_a @ B2 @ B )
=> ( member_set_a @ A2 @ ( vimage_set_a_a @ F @ B ) ) ) ) ).
% vimageI
thf(fact_24_vimageI,axiom,
! [F: set_a > set_a,A2: set_a,B2: set_a,B: set_set_a] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_set_a @ B2 @ B )
=> ( member_set_a @ A2 @ ( vimage_set_a_set_a @ F @ B ) ) ) ) ).
% vimageI
thf(fact_25_vimageI,axiom,
! [F: a > real > real,A2: a,B2: real > real,B: set_real_real] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_real_real @ B2 @ B )
=> ( member_a @ A2 @ ( vimage_a_real_real @ F @ B ) ) ) ) ).
% vimageI
thf(fact_26_vimageI,axiom,
! [F: a > real > a,A2: a,B2: real > a,B: set_real_a] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_real_a @ B2 @ B )
=> ( member_a @ A2 @ ( vimage_a_real_a2 @ F @ B ) ) ) ) ).
% vimageI
thf(fact_27_vimageI,axiom,
! [F: a > a > real,A2: a,B2: a > real,B: set_a_real] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_a_real @ B2 @ B )
=> ( member_a @ A2 @ ( vimage_a_a_real @ F @ B ) ) ) ) ).
% vimageI
thf(fact_28_vimage__eq,axiom,
! [A2: real,F: real > a,B: set_a] :
( ( member_real @ A2 @ ( vimage_real_a @ F @ B ) )
= ( member_a @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_29_vimage__eq,axiom,
! [A2: real,F: real > real,B: set_real] :
( ( member_real @ A2 @ ( vimage_real_real @ F @ B ) )
= ( member_real @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_30_vimage__eq,axiom,
! [A2: a,F: a > real,B: set_real] :
( ( member_a @ A2 @ ( vimage_a_real @ F @ B ) )
= ( member_real @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_31_vimage__eq,axiom,
! [A2: a,F: a > a,B: set_a] :
( ( member_a @ A2 @ ( vimage_a_a @ F @ B ) )
= ( member_a @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_32_vimage__eq,axiom,
! [A2: set_a,F: set_a > a,B: set_a] :
( ( member_set_a @ A2 @ ( vimage_set_a_a @ F @ B ) )
= ( member_a @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_33_vimage__eq,axiom,
! [A2: a,F: a > set_a,B: set_set_a] :
( ( member_a @ A2 @ ( vimage_a_set_a @ F @ B ) )
= ( member_set_a @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_34_vimage__eq,axiom,
! [A2: set_a,F: set_a > set_a,B: set_set_a] :
( ( member_set_a @ A2 @ ( vimage_set_a_set_a @ F @ B ) )
= ( member_set_a @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_35_vimage__eq,axiom,
! [A2: real > real,F: ( real > real ) > a,B: set_a] :
( ( member_real_real @ A2 @ ( vimage_real_real_a @ F @ B ) )
= ( member_a @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_36_vimage__eq,axiom,
! [A2: real > a,F: ( real > a ) > a,B: set_a] :
( ( member_real_a @ A2 @ ( vimage_real_a_a @ F @ B ) )
= ( member_a @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_37_vimage__eq,axiom,
! [A2: a > real,F: ( a > real ) > a,B: set_a] :
( ( member_a_real @ A2 @ ( vimage_a_real_a @ F @ B ) )
= ( member_a @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_38_subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( member_set_a @ X2 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% subsetI
thf(fact_39_subsetI,axiom,
! [A: set_real_real,B: set_real_real] :
( ! [X2: real > real] :
( ( member_real_real @ X2 @ A )
=> ( member_real_real @ X2 @ B ) )
=> ( ord_le4198349162570665613l_real @ A @ B ) ) ).
% subsetI
thf(fact_40_subsetI,axiom,
! [A: set_real_a,B: set_real_a] :
( ! [X2: real > a] :
( ( member_real_a @ X2 @ A )
=> ( member_real_a @ X2 @ B ) )
=> ( ord_le5743406823621094409real_a @ A @ B ) ) ).
% subsetI
thf(fact_41_subsetI,axiom,
! [A: set_a_real,B: set_a_real] :
( ! [X2: a > real] :
( ( member_a_real @ X2 @ A )
=> ( member_a_real @ X2 @ B ) )
=> ( ord_le3334967407727675675a_real @ A @ B ) ) ).
% subsetI
thf(fact_42_subsetI,axiom,
! [A: set_real,B: set_real] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( member_real @ X2 @ B ) )
=> ( ord_less_eq_set_real @ A @ B ) ) ).
% subsetI
thf(fact_43_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_44_subset__antisym,axiom,
! [A: set_real,B: set_real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( ord_less_eq_set_real @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_45_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_46_image__eqI,axiom,
! [B2: real,F: real > real,X: real,A: set_real] :
( ( B2
= ( F @ X ) )
=> ( ( member_real @ X @ A )
=> ( member_real @ B2 @ ( image_real_real @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_47_image__eqI,axiom,
! [B2: a,F: real > a,X: real,A: set_real] :
( ( B2
= ( F @ X ) )
=> ( ( member_real @ X @ A )
=> ( member_a @ B2 @ ( image_real_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_48_image__eqI,axiom,
! [B2: real,F: a > real,X: a,A: set_a] :
( ( B2
= ( F @ X ) )
=> ( ( member_a @ X @ A )
=> ( member_real @ B2 @ ( image_a_real @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_49_image__eqI,axiom,
! [B2: a,F: a > a,X: a,A: set_a] :
( ( B2
= ( F @ X ) )
=> ( ( member_a @ X @ A )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_50_image__eqI,axiom,
! [B2: a,F: set_a > a,X: set_a,A: set_set_a] :
( ( B2
= ( F @ X ) )
=> ( ( member_set_a @ X @ A )
=> ( member_a @ B2 @ ( image_set_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_51_image__eqI,axiom,
! [B2: set_a,F: a > set_a,X: a,A: set_a] :
( ( B2
= ( F @ X ) )
=> ( ( member_a @ X @ A )
=> ( member_set_a @ B2 @ ( image_a_set_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_52_image__eqI,axiom,
! [B2: set_a,F: set_a > set_a,X: set_a,A: set_set_a] :
( ( B2
= ( F @ X ) )
=> ( ( member_set_a @ X @ A )
=> ( member_set_a @ B2 @ ( image_set_a_set_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_53_image__eqI,axiom,
! [B2: a,F: ( real > real ) > a,X: real > real,A: set_real_real] :
( ( B2
= ( F @ X ) )
=> ( ( member_real_real @ X @ A )
=> ( member_a @ B2 @ ( image_real_real_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_54_image__eqI,axiom,
! [B2: a,F: ( real > a ) > a,X: real > a,A: set_real_a] :
( ( B2
= ( F @ X ) )
=> ( ( member_real_a @ X @ A )
=> ( member_a @ B2 @ ( image_real_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_55_image__eqI,axiom,
! [B2: a,F: ( a > real ) > a,X: a > real,A: set_a_real] :
( ( B2
= ( F @ X ) )
=> ( ( member_a_real @ X @ A )
=> ( member_a @ B2 @ ( image_a_real_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_56_order__refl,axiom,
! [X: set_real] : ( ord_less_eq_set_real @ X @ X ) ).
% order_refl
thf(fact_57_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_58_dual__order_Orefl,axiom,
! [A2: set_real] : ( ord_less_eq_set_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_59_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_60_vimage__mono,axiom,
! [A: set_real,B: set_real,F: real > real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ord_less_eq_set_real @ ( vimage_real_real @ F @ A ) @ ( vimage_real_real @ F @ B ) ) ) ).
% vimage_mono
thf(fact_61_vimage__mono,axiom,
! [A: set_real,B: set_real,F: a > real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ord_less_eq_set_a @ ( vimage_a_real @ F @ A ) @ ( vimage_a_real @ F @ B ) ) ) ).
% vimage_mono
thf(fact_62_vimage__mono,axiom,
! [A: set_a,B: set_a,F: real > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_real @ ( vimage_real_a @ F @ A ) @ ( vimage_real_a @ F @ B ) ) ) ).
% vimage_mono
thf(fact_63_vimage__mono,axiom,
! [A: set_a,B: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( vimage_a_a @ F @ A ) @ ( vimage_a_a @ F @ B ) ) ) ).
% vimage_mono
thf(fact_64_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_65_empty__iff,axiom,
! [C: real > real] :
~ ( member_real_real @ C @ bot_bo6767488733719836353l_real ) ).
% empty_iff
thf(fact_66_empty__iff,axiom,
! [C: real > a] :
~ ( member_real_a @ C @ bot_bot_set_real_a ) ).
% empty_iff
thf(fact_67_empty__iff,axiom,
! [C: a > real] :
~ ( member_a_real @ C @ bot_bot_set_a_real ) ).
% empty_iff
thf(fact_68_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_69_empty__iff,axiom,
! [C: real] :
~ ( member_real @ C @ bot_bot_set_real ) ).
% empty_iff
thf(fact_70_all__not__in__conv,axiom,
! [A: set_set_a] :
( ( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A ) )
= ( A = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_71_all__not__in__conv,axiom,
! [A: set_real_real] :
( ( ! [X3: real > real] :
~ ( member_real_real @ X3 @ A ) )
= ( A = bot_bo6767488733719836353l_real ) ) ).
% all_not_in_conv
thf(fact_72_all__not__in__conv,axiom,
! [A: set_real_a] :
( ( ! [X3: real > a] :
~ ( member_real_a @ X3 @ A ) )
= ( A = bot_bot_set_real_a ) ) ).
% all_not_in_conv
thf(fact_73_all__not__in__conv,axiom,
! [A: set_a_real] :
( ( ! [X3: a > real] :
~ ( member_a_real @ X3 @ A ) )
= ( A = bot_bot_set_a_real ) ) ).
% all_not_in_conv
thf(fact_74_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_75_all__not__in__conv,axiom,
! [A: set_real] :
( ( ! [X3: real] :
~ ( member_real @ X3 @ A ) )
= ( A = bot_bot_set_real ) ) ).
% all_not_in_conv
thf(fact_76_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_77_Collect__empty__eq,axiom,
! [P: real > $o] :
( ( ( collect_real @ P )
= bot_bot_set_real )
= ( ! [X3: real] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_78_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_79_empty__Collect__eq,axiom,
! [P: real > $o] :
( ( bot_bot_set_real
= ( collect_real @ P ) )
= ( ! [X3: real] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_80_standard__borel__sets,axiom,
! [Y: sigma_measure_a] :
( ( ( sigma_sets_a @ m )
= ( sigma_sets_a @ Y ) )
=> ( standard_borel_a @ Y ) ) ).
% standard_borel_sets
thf(fact_81_image__is__empty,axiom,
! [F: a > a,A: set_a] :
( ( ( image_a_a @ F @ A )
= bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_82_image__is__empty,axiom,
! [F: real > a,A: set_real] :
( ( ( image_real_a @ F @ A )
= bot_bot_set_a )
= ( A = bot_bot_set_real ) ) ).
% image_is_empty
thf(fact_83_image__is__empty,axiom,
! [F: a > real,A: set_a] :
( ( ( image_a_real @ F @ A )
= bot_bot_set_real )
= ( A = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_84_image__is__empty,axiom,
! [F: real > real,A: set_real] :
( ( ( image_real_real @ F @ A )
= bot_bot_set_real )
= ( A = bot_bot_set_real ) ) ).
% image_is_empty
thf(fact_85_empty__is__image,axiom,
! [F: a > a,A: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F @ A ) )
= ( A = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_86_empty__is__image,axiom,
! [F: real > a,A: set_real] :
( ( bot_bot_set_a
= ( image_real_a @ F @ A ) )
= ( A = bot_bot_set_real ) ) ).
% empty_is_image
thf(fact_87_empty__is__image,axiom,
! [F: a > real,A: set_a] :
( ( bot_bot_set_real
= ( image_a_real @ F @ A ) )
= ( A = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_88_empty__is__image,axiom,
! [F: real > real,A: set_real] :
( ( bot_bot_set_real
= ( image_real_real @ F @ A ) )
= ( A = bot_bot_set_real ) ) ).
% empty_is_image
thf(fact_89_image__empty,axiom,
! [F: a > a] :
( ( image_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_90_image__empty,axiom,
! [F: a > real] :
( ( image_a_real @ F @ bot_bot_set_a )
= bot_bot_set_real ) ).
% image_empty
thf(fact_91_image__empty,axiom,
! [F: real > a] :
( ( image_real_a @ F @ bot_bot_set_real )
= bot_bot_set_a ) ).
% image_empty
thf(fact_92_image__empty,axiom,
! [F: real > real] :
( ( image_real_real @ F @ bot_bot_set_real )
= bot_bot_set_real ) ).
% image_empty
thf(fact_93_empty__subsetI,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).
% empty_subsetI
thf(fact_94_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_95_subset__empty,axiom,
! [A: set_real] :
( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
= ( A = bot_bot_set_real ) ) ).
% subset_empty
thf(fact_96_subset__empty,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_97_vimage__empty,axiom,
! [F: a > a] :
( ( vimage_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% vimage_empty
thf(fact_98_vimage__empty,axiom,
! [F: real > a] :
( ( vimage_real_a @ F @ bot_bot_set_a )
= bot_bot_set_real ) ).
% vimage_empty
thf(fact_99_vimage__empty,axiom,
! [F: a > real] :
( ( vimage_a_real @ F @ bot_bot_set_real )
= bot_bot_set_a ) ).
% vimage_empty
thf(fact_100_vimage__empty,axiom,
! [F: real > real] :
( ( vimage_real_real @ F @ bot_bot_set_real )
= bot_bot_set_real ) ).
% vimage_empty
thf(fact_101_standard__borel_Ostandard__borel__sets,axiom,
! [M: sigma_measure_a,Y: sigma_measure_a] :
( ( standard_borel_a @ M )
=> ( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ Y ) )
=> ( standard_borel_a @ Y ) ) ) ).
% standard_borel.standard_borel_sets
thf(fact_102_standard__borel_Ostandard__borel__sets,axiom,
! [M: sigma_measure_real,Y: sigma_measure_real] :
( ( standard_borel_real @ M )
=> ( ( ( sigma_sets_real @ M )
= ( sigma_sets_real @ Y ) )
=> ( standard_borel_real @ Y ) ) ) ).
% standard_borel.standard_borel_sets
thf(fact_103_emptyE,axiom,
! [A2: set_a] :
~ ( member_set_a @ A2 @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_104_emptyE,axiom,
! [A2: real > real] :
~ ( member_real_real @ A2 @ bot_bo6767488733719836353l_real ) ).
% emptyE
thf(fact_105_emptyE,axiom,
! [A2: real > a] :
~ ( member_real_a @ A2 @ bot_bot_set_real_a ) ).
% emptyE
thf(fact_106_emptyE,axiom,
! [A2: a > real] :
~ ( member_a_real @ A2 @ bot_bot_set_a_real ) ).
% emptyE
thf(fact_107_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_108_emptyE,axiom,
! [A2: real] :
~ ( member_real @ A2 @ bot_bot_set_real ) ).
% emptyE
thf(fact_109_equals0D,axiom,
! [A: set_set_a,A2: set_a] :
( ( A = bot_bot_set_set_a )
=> ~ ( member_set_a @ A2 @ A ) ) ).
% equals0D
thf(fact_110_equals0D,axiom,
! [A: set_real_real,A2: real > real] :
( ( A = bot_bo6767488733719836353l_real )
=> ~ ( member_real_real @ A2 @ A ) ) ).
% equals0D
thf(fact_111_equals0D,axiom,
! [A: set_real_a,A2: real > a] :
( ( A = bot_bot_set_real_a )
=> ~ ( member_real_a @ A2 @ A ) ) ).
% equals0D
thf(fact_112_equals0D,axiom,
! [A: set_a_real,A2: a > real] :
( ( A = bot_bot_set_a_real )
=> ~ ( member_a_real @ A2 @ A ) ) ).
% equals0D
thf(fact_113_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_114_equals0D,axiom,
! [A: set_real,A2: real] :
( ( A = bot_bot_set_real )
=> ~ ( member_real @ A2 @ A ) ) ).
% equals0D
thf(fact_115_equals0I,axiom,
! [A: set_set_a] :
( ! [Y2: set_a] :
~ ( member_set_a @ Y2 @ A )
=> ( A = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_116_equals0I,axiom,
! [A: set_real_real] :
( ! [Y2: real > real] :
~ ( member_real_real @ Y2 @ A )
=> ( A = bot_bo6767488733719836353l_real ) ) ).
% equals0I
thf(fact_117_equals0I,axiom,
! [A: set_real_a] :
( ! [Y2: real > a] :
~ ( member_real_a @ Y2 @ A )
=> ( A = bot_bot_set_real_a ) ) ).
% equals0I
thf(fact_118_equals0I,axiom,
! [A: set_a_real] :
( ! [Y2: a > real] :
~ ( member_a_real @ Y2 @ A )
=> ( A = bot_bot_set_a_real ) ) ).
% equals0I
thf(fact_119_equals0I,axiom,
! [A: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_120_equals0I,axiom,
! [A: set_real] :
( ! [Y2: real] :
~ ( member_real @ Y2 @ A )
=> ( A = bot_bot_set_real ) ) ).
% equals0I
thf(fact_121_ex__in__conv,axiom,
! [A: set_set_a] :
( ( ? [X3: set_a] : ( member_set_a @ X3 @ A ) )
= ( A != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_122_ex__in__conv,axiom,
! [A: set_real_real] :
( ( ? [X3: real > real] : ( member_real_real @ X3 @ A ) )
= ( A != bot_bo6767488733719836353l_real ) ) ).
% ex_in_conv
thf(fact_123_ex__in__conv,axiom,
! [A: set_real_a] :
( ( ? [X3: real > a] : ( member_real_a @ X3 @ A ) )
= ( A != bot_bot_set_real_a ) ) ).
% ex_in_conv
thf(fact_124_ex__in__conv,axiom,
! [A: set_a_real] :
( ( ? [X3: a > real] : ( member_a_real @ X3 @ A ) )
= ( A != bot_bot_set_a_real ) ) ).
% ex_in_conv
thf(fact_125_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_126_ex__in__conv,axiom,
! [A: set_real] :
( ( ? [X3: real] : ( member_real @ X3 @ A ) )
= ( A != bot_bot_set_real ) ) ).
% ex_in_conv
thf(fact_127_bot_Oextremum__uniqueI,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
=> ( A2 = bot_bot_set_real ) ) ).
% bot.extremum_uniqueI
thf(fact_128_bot_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
=> ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_129_bot_Oextremum__unique,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
= ( A2 = bot_bot_set_real ) ) ).
% bot.extremum_unique
thf(fact_130_bot_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_131_bot_Oextremum,axiom,
! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).
% bot.extremum
thf(fact_132_bot_Oextremum,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% bot.extremum
thf(fact_133_standard__borel_Ogf__comp__id_I1_J,axiom,
! [M: sigma_measure_set_a,X: set_a] :
( ( standard_borel_set_a @ M )
=> ( ( member_set_a @ X @ ( sigma_space_set_a @ M ) )
=> ( ( comp_r8810651240769227663_set_a @ ( standard_g_set_a @ M ) @ ( standard_f_set_a @ M ) @ X )
= X ) ) ) ).
% standard_borel.gf_comp_id(1)
thf(fact_134_standard__borel_Ogf__comp__id_I1_J,axiom,
! [M: sigma_4258434043392614480l_real,X: real > real] :
( ( standa6209843396170762084l_real @ M )
=> ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
=> ( ( comp_r2210161833511426559l_real @ ( standard_g_real_real @ M ) @ ( standard_f_real_real @ M ) @ X )
= X ) ) ) ).
% standard_borel.gf_comp_id(1)
thf(fact_135_standard__borel_Ogf__comp__id_I1_J,axiom,
! [M: sigma_measure_real_a,X: real > a] :
( ( standa4991623935272378204real_a @ M )
=> ( ( member_real_a @ X @ ( sigma_space_real_a @ M ) )
=> ( ( comp_r4421311556806241359real_a @ ( standard_g_real_a @ M ) @ ( standard_f_real_a @ M ) @ X )
= X ) ) ) ).
% standard_borel.gf_comp_id(1)
thf(fact_136_standard__borel_Ogf__comp__id_I1_J,axiom,
! [M: sigma_measure_a_real,X: a > real] :
( ( standa4474357433005041006a_real @ M )
=> ( ( member_a_real @ X @ ( sigma_space_a_real @ M ) )
=> ( ( comp_r2811049936330031311a_real @ ( standard_g_a_real @ M ) @ ( standard_f_a_real @ M ) @ X )
= X ) ) ) ).
% standard_borel.gf_comp_id(1)
thf(fact_137_standard__borel_Ogf__comp__id_I1_J,axiom,
! [M: sigma_measure_a,X: a] :
( ( standard_borel_a @ M )
=> ( ( member_a @ X @ ( sigma_space_a @ M ) )
=> ( ( comp_real_a_a @ ( standard_g_a @ M ) @ ( standard_f_a @ M ) @ X )
= X ) ) ) ).
% standard_borel.gf_comp_id(1)
thf(fact_138_standard__borel_Ogf__comp__id_I1_J,axiom,
! [M: sigma_measure_real,X: real] :
( ( standard_borel_real @ M )
=> ( ( member_real @ X @ ( sigma_space_real @ M ) )
=> ( ( comp_real_real_real @ ( standard_g_real @ M ) @ ( standard_f_real @ M ) @ X )
= X ) ) ) ).
% standard_borel.gf_comp_id(1)
thf(fact_139_standard__borel_Ogf__comp__id_I2_J,axiom,
! [M: sigma_measure_set_a,X: set_a] :
( ( standard_borel_set_a @ M )
=> ( ( member_set_a @ X @ ( sigma_space_set_a @ M ) )
=> ( ( standard_g_set_a @ M @ ( standard_f_set_a @ M @ X ) )
= X ) ) ) ).
% standard_borel.gf_comp_id(2)
thf(fact_140_standard__borel_Ogf__comp__id_I2_J,axiom,
! [M: sigma_4258434043392614480l_real,X: real > real] :
( ( standa6209843396170762084l_real @ M )
=> ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
=> ( ( standard_g_real_real @ M @ ( standard_f_real_real @ M @ X ) )
= X ) ) ) ).
% standard_borel.gf_comp_id(2)
thf(fact_141_standard__borel_Ogf__comp__id_I2_J,axiom,
! [M: sigma_measure_real_a,X: real > a] :
( ( standa4991623935272378204real_a @ M )
=> ( ( member_real_a @ X @ ( sigma_space_real_a @ M ) )
=> ( ( standard_g_real_a @ M @ ( standard_f_real_a @ M @ X ) )
= X ) ) ) ).
% standard_borel.gf_comp_id(2)
thf(fact_142_standard__borel_Ogf__comp__id_I2_J,axiom,
! [M: sigma_measure_a_real,X: a > real] :
( ( standa4474357433005041006a_real @ M )
=> ( ( member_a_real @ X @ ( sigma_space_a_real @ M ) )
=> ( ( standard_g_a_real @ M @ ( standard_f_a_real @ M @ X ) )
= X ) ) ) ).
% standard_borel.gf_comp_id(2)
thf(fact_143_standard__borel_Ogf__comp__id_I2_J,axiom,
! [M: sigma_measure_a,X: a] :
( ( standard_borel_a @ M )
=> ( ( member_a @ X @ ( sigma_space_a @ M ) )
=> ( ( standard_g_a @ M @ ( standard_f_a @ M @ X ) )
= X ) ) ) ).
% standard_borel.gf_comp_id(2)
thf(fact_144_standard__borel_Ogf__comp__id_I2_J,axiom,
! [M: sigma_measure_real,X: real] :
( ( standard_borel_real @ M )
=> ( ( member_real @ X @ ( sigma_space_real @ M ) )
=> ( ( standard_g_real @ M @ ( standard_f_real @ M @ X ) )
= X ) ) ) ).
% standard_borel.gf_comp_id(2)
thf(fact_145_order__antisym__conv,axiom,
! [Y3: set_real,X: set_real] :
( ( ord_less_eq_set_real @ Y3 @ X )
=> ( ( ord_less_eq_set_real @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_146_order__antisym__conv,axiom,
! [Y3: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X )
=> ( ( ord_less_eq_set_a @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_147_ord__le__eq__subst,axiom,
! [A2: set_real,B2: set_real,F: set_real > set_real,C: set_real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_real,Y2: set_real] :
( ( ord_less_eq_set_real @ X2 @ Y2 )
=> ( ord_less_eq_set_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_148_ord__le__eq__subst,axiom,
! [A2: set_real,B2: set_real,F: set_real > set_a,C: set_a] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_real,Y2: set_real] :
( ( ord_less_eq_set_real @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_149_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_real,C: set_real] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_150_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_151_ord__eq__le__subst,axiom,
! [A2: set_real,F: set_real > set_real,B2: set_real,C: set_real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_real @ B2 @ C )
=> ( ! [X2: set_real,Y2: set_real] :
( ( ord_less_eq_set_real @ X2 @ Y2 )
=> ( ord_less_eq_set_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_152_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_real > set_a,B2: set_real,C: set_real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_real @ B2 @ C )
=> ( ! [X2: set_real,Y2: set_real] :
( ( ord_less_eq_set_real @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_153_ord__eq__le__subst,axiom,
! [A2: set_real,F: set_a > set_real,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_154_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_155_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_156_mem__Collect__eq,axiom,
! [A2: real > real,P: ( real > real ) > $o] :
( ( member_real_real @ A2 @ ( collect_real_real @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_157_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_158_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_159_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_160_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_161_Collect__mem__eq,axiom,
! [A: set_real_real] :
( ( collect_real_real
@ ^ [X3: real > real] : ( member_real_real @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_162_Collect__mem__eq,axiom,
! [A: set_real_a] :
( ( collect_real_a
@ ^ [X3: real > a] : ( member_real_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_163_Collect__mem__eq,axiom,
! [A: set_a_real] :
( ( collect_a_real
@ ^ [X3: a > real] : ( member_a_real @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_164_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_165_order__eq__refl,axiom,
! [X: set_real,Y3: set_real] :
( ( X = Y3 )
=> ( ord_less_eq_set_real @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_166_order__eq__refl,axiom,
! [X: set_a,Y3: set_a] :
( ( X = Y3 )
=> ( ord_less_eq_set_a @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_167_order__subst2,axiom,
! [A2: set_real,B2: set_real,F: set_real > set_real,C: set_real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( ord_less_eq_set_real @ ( F @ B2 ) @ C )
=> ( ! [X2: set_real,Y2: set_real] :
( ( ord_less_eq_set_real @ X2 @ Y2 )
=> ( ord_less_eq_set_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_168_order__subst2,axiom,
! [A2: set_real,B2: set_real,F: set_real > set_a,C: set_a] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_real,Y2: set_real] :
( ( ord_less_eq_set_real @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_169_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_real,C: set_real] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_real @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_170_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_171_order__subst1,axiom,
! [A2: set_real,F: set_real > set_real,B2: set_real,C: set_real] :
( ( ord_less_eq_set_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_real @ B2 @ C )
=> ( ! [X2: set_real,Y2: set_real] :
( ( ord_less_eq_set_real @ X2 @ Y2 )
=> ( ord_less_eq_set_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_172_order__subst1,axiom,
! [A2: set_real,F: set_a > set_real,B2: set_a,C: set_a] :
( ( ord_less_eq_set_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_173_order__subst1,axiom,
! [A2: set_a,F: set_real > set_a,B2: set_real,C: set_real] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_real @ B2 @ C )
=> ( ! [X2: set_real,Y2: set_real] :
( ( ord_less_eq_set_real @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_174_order__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_175_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_real,Z: set_real] : ( Y4 = Z ) )
= ( ^ [A3: set_real,B3: set_real] :
( ( ord_less_eq_set_real @ A3 @ B3 )
& ( ord_less_eq_set_real @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_176_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_177_antisym,axiom,
! [A2: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( ord_less_eq_set_real @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_178_antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_179_dual__order_Otrans,axiom,
! [B2: set_real,A2: set_real,C: set_real] :
( ( ord_less_eq_set_real @ B2 @ A2 )
=> ( ( ord_less_eq_set_real @ C @ B2 )
=> ( ord_less_eq_set_real @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_180_dual__order_Otrans,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_181_dual__order_Oantisym,axiom,
! [B2: set_real,A2: set_real] :
( ( ord_less_eq_set_real @ B2 @ A2 )
=> ( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_182_dual__order_Oantisym,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_183_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_real,Z: set_real] : ( Y4 = Z ) )
= ( ^ [A3: set_real,B3: set_real] :
( ( ord_less_eq_set_real @ B3 @ A3 )
& ( ord_less_eq_set_real @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_184_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_185_order__trans,axiom,
! [X: set_real,Y3: set_real,Z2: set_real] :
( ( ord_less_eq_set_real @ X @ Y3 )
=> ( ( ord_less_eq_set_real @ Y3 @ Z2 )
=> ( ord_less_eq_set_real @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_186_order__trans,axiom,
! [X: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ Z2 )
=> ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_187_order_Otrans,axiom,
! [A2: set_real,B2: set_real,C: set_real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( ord_less_eq_set_real @ B2 @ C )
=> ( ord_less_eq_set_real @ A2 @ C ) ) ) ).
% order.trans
thf(fact_188_order_Otrans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_189_order__antisym,axiom,
! [X: set_real,Y3: set_real] :
( ( ord_less_eq_set_real @ X @ Y3 )
=> ( ( ord_less_eq_set_real @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_190_order__antisym,axiom,
! [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_191_ord__le__eq__trans,axiom,
! [A2: set_real,B2: set_real,C: set_real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_real @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_192_ord__le__eq__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_193_ord__eq__le__trans,axiom,
! [A2: set_real,B2: set_real,C: set_real] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_real @ B2 @ C )
=> ( ord_less_eq_set_real @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_194_ord__eq__le__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_195_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_real,Z: set_real] : ( Y4 = Z ) )
= ( ^ [X3: set_real,Y5: set_real] :
( ( ord_less_eq_set_real @ X3 @ Y5 )
& ( ord_less_eq_set_real @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_196_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
& ( ord_less_eq_set_a @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_197_rev__image__eqI,axiom,
! [X: real,A: set_real,B2: real,F: real > real] :
( ( member_real @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_real @ B2 @ ( image_real_real @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_198_rev__image__eqI,axiom,
! [X: real,A: set_real,B2: a,F: real > a] :
( ( member_real @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_a @ B2 @ ( image_real_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_199_rev__image__eqI,axiom,
! [X: a,A: set_a,B2: real,F: a > real] :
( ( member_a @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_real @ B2 @ ( image_a_real @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_200_rev__image__eqI,axiom,
! [X: a,A: set_a,B2: a,F: a > a] :
( ( member_a @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_201_rev__image__eqI,axiom,
! [X: set_a,A: set_set_a,B2: a,F: set_a > a] :
( ( member_set_a @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_a @ B2 @ ( image_set_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_202_rev__image__eqI,axiom,
! [X: a,A: set_a,B2: set_a,F: a > set_a] :
( ( member_a @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_set_a @ B2 @ ( image_a_set_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_203_rev__image__eqI,axiom,
! [X: set_a,A: set_set_a,B2: set_a,F: set_a > set_a] :
( ( member_set_a @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_set_a @ B2 @ ( image_set_a_set_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_204_rev__image__eqI,axiom,
! [X: real > real,A: set_real_real,B2: a,F: ( real > real ) > a] :
( ( member_real_real @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_a @ B2 @ ( image_real_real_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_205_rev__image__eqI,axiom,
! [X: real > a,A: set_real_a,B2: a,F: ( real > a ) > a] :
( ( member_real_a @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_a @ B2 @ ( image_real_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_206_rev__image__eqI,axiom,
! [X: a > real,A: set_a_real,B2: a,F: ( a > real ) > a] :
( ( member_a_real @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_a @ B2 @ ( image_a_real_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_207_ball__imageD,axiom,
! [F: a > real,A: set_a,P: real > $o] :
( ! [X2: real] :
( ( member_real @ X2 @ ( image_a_real @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_208_ball__imageD,axiom,
! [F: real > real,A: set_real,P: real > $o] :
( ! [X2: real] :
( ( member_real @ X2 @ ( image_real_real @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X4: real] :
( ( member_real @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_209_ball__imageD,axiom,
! [F: real > a,A: set_real,P: a > $o] :
( ! [X2: a] :
( ( member_a @ X2 @ ( image_real_a @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X4: real] :
( ( member_real @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_210_ball__imageD,axiom,
! [F: a > a,A: set_a,P: a > $o] :
( ! [X2: a] :
( ( member_a @ X2 @ ( image_a_a @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_211_image__cong,axiom,
! [M: set_real,N: set_real,F: real > real,G: real > real] :
( ( M = N )
=> ( ! [X2: real] :
( ( member_real @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_real_real @ F @ M )
= ( image_real_real @ G @ N ) ) ) ) ).
% image_cong
thf(fact_212_image__cong,axiom,
! [M: set_real,N: set_real,F: real > a,G: real > a] :
( ( M = N )
=> ( ! [X2: real] :
( ( member_real @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_real_a @ F @ M )
= ( image_real_a @ G @ N ) ) ) ) ).
% image_cong
thf(fact_213_image__cong,axiom,
! [M: set_a,N: set_a,F: a > real,G: a > real] :
( ( M = N )
=> ( ! [X2: a] :
( ( member_a @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_a_real @ F @ M )
= ( image_a_real @ G @ N ) ) ) ) ).
% image_cong
thf(fact_214_image__cong,axiom,
! [M: set_a,N: set_a,F: a > a,G: a > a] :
( ( M = N )
=> ( ! [X2: a] :
( ( member_a @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_a_a @ F @ M )
= ( image_a_a @ G @ N ) ) ) ) ).
% image_cong
thf(fact_215_bex__imageD,axiom,
! [F: a > real,A: set_a,P: real > $o] :
( ? [X4: real] :
( ( member_real @ X4 @ ( image_a_real @ F @ A ) )
& ( P @ X4 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_216_bex__imageD,axiom,
! [F: real > real,A: set_real,P: real > $o] :
( ? [X4: real] :
( ( member_real @ X4 @ ( image_real_real @ F @ A ) )
& ( P @ X4 ) )
=> ? [X2: real] :
( ( member_real @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_217_bex__imageD,axiom,
! [F: real > a,A: set_real,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( image_real_a @ F @ A ) )
& ( P @ X4 ) )
=> ? [X2: real] :
( ( member_real @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_218_bex__imageD,axiom,
! [F: a > a,A: set_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( image_a_a @ F @ A ) )
& ( P @ X4 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_219_image__iff,axiom,
! [Z2: real,F: a > real,A: set_a] :
( ( member_real @ Z2 @ ( image_a_real @ F @ A ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_220_image__iff,axiom,
! [Z2: real,F: real > real,A: set_real] :
( ( member_real @ Z2 @ ( image_real_real @ F @ A ) )
= ( ? [X3: real] :
( ( member_real @ X3 @ A )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_221_image__iff,axiom,
! [Z2: a,F: real > a,A: set_real] :
( ( member_a @ Z2 @ ( image_real_a @ F @ A ) )
= ( ? [X3: real] :
( ( member_real @ X3 @ A )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_222_image__iff,axiom,
! [Z2: a,F: a > a,A: set_a] :
( ( member_a @ Z2 @ ( image_a_a @ F @ A ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_223_imageI,axiom,
! [X: real,A: set_real,F: real > real] :
( ( member_real @ X @ A )
=> ( member_real @ ( F @ X ) @ ( image_real_real @ F @ A ) ) ) ).
% imageI
thf(fact_224_imageI,axiom,
! [X: real,A: set_real,F: real > a] :
( ( member_real @ X @ A )
=> ( member_a @ ( F @ X ) @ ( image_real_a @ F @ A ) ) ) ).
% imageI
thf(fact_225_imageI,axiom,
! [X: a,A: set_a,F: a > real] :
( ( member_a @ X @ A )
=> ( member_real @ ( F @ X ) @ ( image_a_real @ F @ A ) ) ) ).
% imageI
thf(fact_226_imageI,axiom,
! [X: a,A: set_a,F: a > a] :
( ( member_a @ X @ A )
=> ( member_a @ ( F @ X ) @ ( image_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_227_imageI,axiom,
! [X: set_a,A: set_set_a,F: set_a > a] :
( ( member_set_a @ X @ A )
=> ( member_a @ ( F @ X ) @ ( image_set_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_228_imageI,axiom,
! [X: a,A: set_a,F: a > set_a] :
( ( member_a @ X @ A )
=> ( member_set_a @ ( F @ X ) @ ( image_a_set_a @ F @ A ) ) ) ).
% imageI
thf(fact_229_imageI,axiom,
! [X: set_a,A: set_set_a,F: set_a > set_a] :
( ( member_set_a @ X @ A )
=> ( member_set_a @ ( F @ X ) @ ( image_set_a_set_a @ F @ A ) ) ) ).
% imageI
thf(fact_230_imageI,axiom,
! [X: real > real,A: set_real_real,F: ( real > real ) > a] :
( ( member_real_real @ X @ A )
=> ( member_a @ ( F @ X ) @ ( image_real_real_a @ F @ A ) ) ) ).
% imageI
thf(fact_231_imageI,axiom,
! [X: real > a,A: set_real_a,F: ( real > a ) > a] :
( ( member_real_a @ X @ A )
=> ( member_a @ ( F @ X ) @ ( image_real_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_232_imageI,axiom,
! [X: a > real,A: set_a_real,F: ( a > real ) > a] :
( ( member_a_real @ X @ A )
=> ( member_a @ ( F @ X ) @ ( image_a_real_a @ F @ A ) ) ) ).
% imageI
thf(fact_233_Collect__mono__iff,axiom,
! [P: real > $o,Q: real > $o] :
( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
= ( ! [X3: real] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_234_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_235_set__eq__subset,axiom,
( ( ^ [Y4: set_real,Z: set_real] : ( Y4 = Z ) )
= ( ^ [A4: set_real,B4: set_real] :
( ( ord_less_eq_set_real @ A4 @ B4 )
& ( ord_less_eq_set_real @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_236_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = 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_237_subset__trans,axiom,
! [A: set_real,B: set_real,C2: set_real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( ord_less_eq_set_real @ B @ C2 )
=> ( ord_less_eq_set_real @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_238_subset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_239_Collect__mono,axiom,
! [P: real > $o,Q: real > $o] :
( ! [X2: real] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% Collect_mono
thf(fact_240_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_241_subset__refl,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ A @ A ) ).
% subset_refl
thf(fact_242_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_243_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_244_subset__iff,axiom,
( ord_le4198349162570665613l_real
= ( ^ [A4: set_real_real,B4: set_real_real] :
! [T: real > real] :
( ( member_real_real @ T @ A4 )
=> ( member_real_real @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_245_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_246_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_247_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A4: set_real,B4: set_real] :
! [T: real] :
( ( member_real @ T @ A4 )
=> ( member_real @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_248_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_249_equalityD2,axiom,
! [A: set_real,B: set_real] :
( ( A = B )
=> ( ord_less_eq_set_real @ B @ A ) ) ).
% equalityD2
thf(fact_250_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_251_equalityD1,axiom,
! [A: set_real,B: set_real] :
( ( A = B )
=> ( ord_less_eq_set_real @ A @ B ) ) ).
% equalityD1
thf(fact_252_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_253_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A4 )
=> ( member_set_a @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_254_subset__eq,axiom,
( ord_le4198349162570665613l_real
= ( ^ [A4: set_real_real,B4: set_real_real] :
! [X3: real > real] :
( ( member_real_real @ X3 @ A4 )
=> ( member_real_real @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_255_subset__eq,axiom,
( ord_le5743406823621094409real_a
= ( ^ [A4: set_real_a,B4: set_real_a] :
! [X3: real > a] :
( ( member_real_a @ X3 @ A4 )
=> ( member_real_a @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_256_subset__eq,axiom,
( ord_le3334967407727675675a_real
= ( ^ [A4: set_a_real,B4: set_a_real] :
! [X3: a > real] :
( ( member_a_real @ X3 @ A4 )
=> ( member_a_real @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_257_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A4: set_real,B4: set_real] :
! [X3: real] :
( ( member_real @ X3 @ A4 )
=> ( member_real @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_258_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( member_a @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_259_equalityE,axiom,
! [A: set_real,B: set_real] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_real @ A @ B )
=> ~ ( ord_less_eq_set_real @ B @ A ) ) ) ).
% equalityE
thf(fact_260_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_261_subsetD,axiom,
! [A: set_set_a,B: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% subsetD
thf(fact_262_subsetD,axiom,
! [A: set_real_real,B: set_real_real,C: real > real] :
( ( ord_le4198349162570665613l_real @ A @ B )
=> ( ( member_real_real @ C @ A )
=> ( member_real_real @ C @ B ) ) ) ).
% subsetD
thf(fact_263_subsetD,axiom,
! [A: set_real_a,B: set_real_a,C: real > a] :
( ( ord_le5743406823621094409real_a @ A @ B )
=> ( ( member_real_a @ C @ A )
=> ( member_real_a @ C @ B ) ) ) ).
% subsetD
thf(fact_264_subsetD,axiom,
! [A: set_a_real,B: set_a_real,C: a > real] :
( ( ord_le3334967407727675675a_real @ A @ B )
=> ( ( member_a_real @ C @ A )
=> ( member_a_real @ C @ B ) ) ) ).
% subsetD
thf(fact_265_subsetD,axiom,
! [A: set_real,B: set_real,C: real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( member_real @ C @ A )
=> ( member_real @ C @ B ) ) ) ).
% subsetD
thf(fact_266_subsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_267_in__mono,axiom,
! [A: set_set_a,B: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ X @ A )
=> ( member_set_a @ X @ B ) ) ) ).
% in_mono
thf(fact_268_in__mono,axiom,
! [A: set_real_real,B: set_real_real,X: real > real] :
( ( ord_le4198349162570665613l_real @ A @ B )
=> ( ( member_real_real @ X @ A )
=> ( member_real_real @ X @ B ) ) ) ).
% in_mono
thf(fact_269_in__mono,axiom,
! [A: set_real_a,B: set_real_a,X: real > a] :
( ( ord_le5743406823621094409real_a @ A @ B )
=> ( ( member_real_a @ X @ A )
=> ( member_real_a @ X @ B ) ) ) ).
% in_mono
thf(fact_270_in__mono,axiom,
! [A: set_a_real,B: set_a_real,X: a > real] :
( ( ord_le3334967407727675675a_real @ A @ B )
=> ( ( member_a_real @ X @ A )
=> ( member_a_real @ X @ B ) ) ) ).
% in_mono
thf(fact_271_in__mono,axiom,
! [A: set_real,B: set_real,X: real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( member_real @ X @ A )
=> ( member_real @ X @ B ) ) ) ).
% in_mono
thf(fact_272_in__mono,axiom,
! [A: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X @ A )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_273_vimage__Collect,axiom,
! [P: a > $o,F: real > a,Q: real > $o] :
( ! [X2: real] :
( ( P @ ( F @ X2 ) )
= ( Q @ X2 ) )
=> ( ( vimage_real_a @ F @ ( collect_a @ P ) )
= ( collect_real @ Q ) ) ) ).
% vimage_Collect
thf(fact_274_vimage__Collect,axiom,
! [P: real > $o,F: real > real,Q: real > $o] :
( ! [X2: real] :
( ( P @ ( F @ X2 ) )
= ( Q @ X2 ) )
=> ( ( vimage_real_real @ F @ ( collect_real @ P ) )
= ( collect_real @ Q ) ) ) ).
% vimage_Collect
thf(fact_275_vimage__Collect,axiom,
! [P: real > $o,F: a > real,Q: a > $o] :
( ! [X2: a] :
( ( P @ ( F @ X2 ) )
= ( Q @ X2 ) )
=> ( ( vimage_a_real @ F @ ( collect_real @ P ) )
= ( collect_a @ Q ) ) ) ).
% vimage_Collect
thf(fact_276_vimage__Collect,axiom,
! [P: a > $o,F: a > a,Q: a > $o] :
( ! [X2: a] :
( ( P @ ( F @ X2 ) )
= ( Q @ X2 ) )
=> ( ( vimage_a_a @ F @ ( collect_a @ P ) )
= ( collect_a @ Q ) ) ) ).
% vimage_Collect
thf(fact_277_vimageI2,axiom,
! [F: real > a,A2: real,A: set_a] :
( ( member_a @ ( F @ A2 ) @ A )
=> ( member_real @ A2 @ ( vimage_real_a @ F @ A ) ) ) ).
% vimageI2
thf(fact_278_vimageI2,axiom,
! [F: real > real,A2: real,A: set_real] :
( ( member_real @ ( F @ A2 ) @ A )
=> ( member_real @ A2 @ ( vimage_real_real @ F @ A ) ) ) ).
% vimageI2
thf(fact_279_vimageI2,axiom,
! [F: a > real,A2: a,A: set_real] :
( ( member_real @ ( F @ A2 ) @ A )
=> ( member_a @ A2 @ ( vimage_a_real @ F @ A ) ) ) ).
% vimageI2
thf(fact_280_vimageI2,axiom,
! [F: a > a,A2: a,A: set_a] :
( ( member_a @ ( F @ A2 ) @ A )
=> ( member_a @ A2 @ ( vimage_a_a @ F @ A ) ) ) ).
% vimageI2
thf(fact_281_vimageI2,axiom,
! [F: a > set_a,A2: a,A: set_set_a] :
( ( member_set_a @ ( F @ A2 ) @ A )
=> ( member_a @ A2 @ ( vimage_a_set_a @ F @ A ) ) ) ).
% vimageI2
thf(fact_282_vimageI2,axiom,
! [F: set_a > a,A2: set_a,A: set_a] :
( ( member_a @ ( F @ A2 ) @ A )
=> ( member_set_a @ A2 @ ( vimage_set_a_a @ F @ A ) ) ) ).
% vimageI2
thf(fact_283_vimageI2,axiom,
! [F: set_a > set_a,A2: set_a,A: set_set_a] :
( ( member_set_a @ ( F @ A2 ) @ A )
=> ( member_set_a @ A2 @ ( vimage_set_a_set_a @ F @ A ) ) ) ).
% vimageI2
thf(fact_284_vimageI2,axiom,
! [F: a > real > real,A2: a,A: set_real_real] :
( ( member_real_real @ ( F @ A2 ) @ A )
=> ( member_a @ A2 @ ( vimage_a_real_real @ F @ A ) ) ) ).
% vimageI2
thf(fact_285_vimageI2,axiom,
! [F: a > real > a,A2: a,A: set_real_a] :
( ( member_real_a @ ( F @ A2 ) @ A )
=> ( member_a @ A2 @ ( vimage_a_real_a2 @ F @ A ) ) ) ).
% vimageI2
thf(fact_286_vimageI2,axiom,
! [F: a > a > real,A2: a,A: set_a_real] :
( ( member_a_real @ ( F @ A2 ) @ A )
=> ( member_a @ A2 @ ( vimage_a_a_real @ F @ A ) ) ) ).
% vimageI2
thf(fact_287_vimageE,axiom,
! [A2: real,F: real > a,B: set_a] :
( ( member_real @ A2 @ ( vimage_real_a @ F @ B ) )
=> ( member_a @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_288_vimageE,axiom,
! [A2: real,F: real > real,B: set_real] :
( ( member_real @ A2 @ ( vimage_real_real @ F @ B ) )
=> ( member_real @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_289_vimageE,axiom,
! [A2: a,F: a > real,B: set_real] :
( ( member_a @ A2 @ ( vimage_a_real @ F @ B ) )
=> ( member_real @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_290_vimageE,axiom,
! [A2: a,F: a > a,B: set_a] :
( ( member_a @ A2 @ ( vimage_a_a @ F @ B ) )
=> ( member_a @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_291_vimageE,axiom,
! [A2: set_a,F: set_a > a,B: set_a] :
( ( member_set_a @ A2 @ ( vimage_set_a_a @ F @ B ) )
=> ( member_a @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_292_vimageE,axiom,
! [A2: a,F: a > set_a,B: set_set_a] :
( ( member_a @ A2 @ ( vimage_a_set_a @ F @ B ) )
=> ( member_set_a @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_293_vimageE,axiom,
! [A2: set_a,F: set_a > set_a,B: set_set_a] :
( ( member_set_a @ A2 @ ( vimage_set_a_set_a @ F @ B ) )
=> ( member_set_a @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_294_vimageE,axiom,
! [A2: real > real,F: ( real > real ) > a,B: set_a] :
( ( member_real_real @ A2 @ ( vimage_real_real_a @ F @ B ) )
=> ( member_a @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_295_vimageE,axiom,
! [A2: real > a,F: ( real > a ) > a,B: set_a] :
( ( member_real_a @ A2 @ ( vimage_real_a_a @ F @ B ) )
=> ( member_a @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_296_vimageE,axiom,
! [A2: a > real,F: ( a > real ) > a,B: set_a] :
( ( member_a_real @ A2 @ ( vimage_a_real_a @ F @ B ) )
=> ( member_a @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_297_vimageD,axiom,
! [A2: real,F: real > a,A: set_a] :
( ( member_real @ A2 @ ( vimage_real_a @ F @ A ) )
=> ( member_a @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_298_vimageD,axiom,
! [A2: real,F: real > real,A: set_real] :
( ( member_real @ A2 @ ( vimage_real_real @ F @ A ) )
=> ( member_real @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_299_vimageD,axiom,
! [A2: a,F: a > real,A: set_real] :
( ( member_a @ A2 @ ( vimage_a_real @ F @ A ) )
=> ( member_real @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_300_vimageD,axiom,
! [A2: a,F: a > a,A: set_a] :
( ( member_a @ A2 @ ( vimage_a_a @ F @ A ) )
=> ( member_a @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_301_vimageD,axiom,
! [A2: set_a,F: set_a > a,A: set_a] :
( ( member_set_a @ A2 @ ( vimage_set_a_a @ F @ A ) )
=> ( member_a @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_302_vimageD,axiom,
! [A2: a,F: a > set_a,A: set_set_a] :
( ( member_a @ A2 @ ( vimage_a_set_a @ F @ A ) )
=> ( member_set_a @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_303_vimageD,axiom,
! [A2: set_a,F: set_a > set_a,A: set_set_a] :
( ( member_set_a @ A2 @ ( vimage_set_a_set_a @ F @ A ) )
=> ( member_set_a @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_304_vimageD,axiom,
! [A2: real > real,F: ( real > real ) > a,A: set_a] :
( ( member_real_real @ A2 @ ( vimage_real_real_a @ F @ A ) )
=> ( member_a @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_305_vimageD,axiom,
! [A2: real > a,F: ( real > a ) > a,A: set_a] :
( ( member_real_a @ A2 @ ( vimage_real_a_a @ F @ A ) )
=> ( member_a @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_306_vimageD,axiom,
! [A2: a > real,F: ( a > real ) > a,A: set_a] :
( ( member_a_real @ A2 @ ( vimage_a_real_a @ F @ A ) )
=> ( member_a @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_307_subset__image__iff,axiom,
! [B: set_real,F: real > real,A: set_real] :
( ( ord_less_eq_set_real @ B @ ( image_real_real @ F @ A ) )
= ( ? [AA: set_real] :
( ( ord_less_eq_set_real @ AA @ A )
& ( B
= ( image_real_real @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_308_subset__image__iff,axiom,
! [B: set_real,F: a > real,A: set_a] :
( ( ord_less_eq_set_real @ B @ ( image_a_real @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_real @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_309_subset__image__iff,axiom,
! [B: set_a,F: real > a,A: set_real] :
( ( ord_less_eq_set_a @ B @ ( image_real_a @ F @ A ) )
= ( ? [AA: set_real] :
( ( ord_less_eq_set_real @ AA @ A )
& ( B
= ( image_real_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_310_subset__image__iff,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_311_image__subset__iff,axiom,
! [F: a > real,A: set_a,B: set_real] :
( ( ord_less_eq_set_real @ ( image_a_real @ F @ A ) @ B )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_real @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_312_image__subset__iff,axiom,
! [F: real > real,A: set_real,B: set_real] :
( ( ord_less_eq_set_real @ ( image_real_real @ F @ A ) @ B )
= ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( member_real @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_313_image__subset__iff,axiom,
! [F: real > a,A: set_real,B: set_a] :
( ( ord_less_eq_set_a @ ( image_real_a @ F @ A ) @ B )
= ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_314_image__subset__iff,axiom,
! [F: a > a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_315_subset__imageE,axiom,
! [B: set_real,F: real > real,A: set_real] :
( ( ord_less_eq_set_real @ B @ ( image_real_real @ F @ A ) )
=> ~ ! [C3: set_real] :
( ( ord_less_eq_set_real @ C3 @ A )
=> ( B
!= ( image_real_real @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_316_subset__imageE,axiom,
! [B: set_real,F: a > real,A: set_a] :
( ( ord_less_eq_set_real @ B @ ( image_a_real @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B
!= ( image_a_real @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_317_subset__imageE,axiom,
! [B: set_a,F: real > a,A: set_real] :
( ( ord_less_eq_set_a @ B @ ( image_real_a @ F @ A ) )
=> ~ ! [C3: set_real] :
( ( ord_less_eq_set_real @ C3 @ A )
=> ( B
!= ( image_real_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_318_subset__imageE,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B
!= ( image_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_319_image__subsetI,axiom,
! [A: set_real,F: real > real,B: set_real] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( member_real @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_real @ ( image_real_real @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_320_image__subsetI,axiom,
! [A: set_a,F: a > real,B: set_real] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_real @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_real @ ( image_a_real @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_321_image__subsetI,axiom,
! [A: set_real,F: real > a,B: set_a] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_real_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_322_image__subsetI,axiom,
! [A: set_a,F: a > a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_323_image__subsetI,axiom,
! [A: set_a,F: a > set_a,B: set_set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_set_a @ ( F @ X2 ) @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_324_image__subsetI,axiom,
! [A: set_set_a,F: set_a > real,B: set_real] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( member_real @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_real @ ( image_set_a_real @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_325_image__subsetI,axiom,
! [A: set_set_a,F: set_a > a,B: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_326_image__subsetI,axiom,
! [A: set_set_a,F: set_a > set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( member_set_a @ ( F @ X2 ) @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_327_image__subsetI,axiom,
! [A: set_a,F: a > real > real,B: set_real_real] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_real_real @ ( F @ X2 ) @ B ) )
=> ( ord_le4198349162570665613l_real @ ( image_a_real_real @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_328_image__subsetI,axiom,
! [A: set_a,F: a > real > a,B: set_real_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_real_a @ ( F @ X2 ) @ B ) )
=> ( ord_le5743406823621094409real_a @ ( image_a_real_a2 @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_329_image__mono,axiom,
! [A: set_real,B: set_real,F: real > real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ord_less_eq_set_real @ ( image_real_real @ F @ A ) @ ( image_real_real @ F @ B ) ) ) ).
% image_mono
thf(fact_330_image__mono,axiom,
! [A: set_real,B: set_real,F: real > a] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ord_less_eq_set_a @ ( image_real_a @ F @ A ) @ ( image_real_a @ F @ B ) ) ) ).
% image_mono
thf(fact_331_image__mono,axiom,
! [A: set_a,B: set_a,F: a > real] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_real @ ( image_a_real @ F @ A ) @ ( image_a_real @ F @ B ) ) ) ).
% image_mono
thf(fact_332_image__mono,axiom,
! [A: set_a,B: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_333_subset__vimage__iff,axiom,
! [A: set_real,F: real > a,B: set_a] :
( ( ord_less_eq_set_real @ A @ ( vimage_real_a @ F @ B ) )
= ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_334_subset__vimage__iff,axiom,
! [A: set_real,F: real > real,B: set_real] :
( ( ord_less_eq_set_real @ A @ ( vimage_real_real @ F @ B ) )
= ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( member_real @ ( F @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_335_subset__vimage__iff,axiom,
! [A: set_a,F: a > real,B: set_real] :
( ( ord_less_eq_set_a @ A @ ( vimage_a_real @ F @ B ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_real @ ( F @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_336_subset__vimage__iff,axiom,
! [A: set_a,F: a > a,B: set_a] :
( ( ord_less_eq_set_a @ A @ ( vimage_a_a @ F @ B ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_337_subset__vimage__iff,axiom,
! [A: set_real,F: real > set_a,B: set_set_a] :
( ( ord_less_eq_set_real @ A @ ( vimage_real_set_a @ F @ B ) )
= ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( member_set_a @ ( F @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_338_subset__vimage__iff,axiom,
! [A: set_a,F: a > set_a,B: set_set_a] :
( ( ord_less_eq_set_a @ A @ ( vimage_a_set_a @ F @ B ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_set_a @ ( F @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_339_subset__vimage__iff,axiom,
! [A: set_real,F: real > real > real,B: set_real_real] :
( ( ord_less_eq_set_real @ A @ ( vimage6512082192006080096l_real @ F @ B ) )
= ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( member_real_real @ ( F @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_340_subset__vimage__iff,axiom,
! [A: set_real,F: real > real > a,B: set_real_a] :
( ( ord_less_eq_set_real @ A @ ( vimage_real_real_a2 @ F @ B ) )
= ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( member_real_a @ ( F @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_341_subset__vimage__iff,axiom,
! [A: set_real,F: real > a > real,B: set_a_real] :
( ( ord_less_eq_set_real @ A @ ( vimage_real_a_real @ F @ B ) )
= ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( member_a_real @ ( F @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_342_subset__vimage__iff,axiom,
! [A: set_a,F: a > real > real,B: set_real_real] :
( ( ord_less_eq_set_a @ A @ ( vimage_a_real_real @ F @ B ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_real_real @ ( F @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_343_singleton__sets,axiom,
! [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
=> ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ ( sigma_sets_a @ m ) ) ) ).
% singleton_sets
thf(fact_344_sets_Otop,axiom,
! [M: sigma_measure_a] : ( member_set_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) ).
% sets.top
thf(fact_345_sets_Otop,axiom,
! [M: sigma_measure_real] : ( member_set_real @ ( sigma_space_real @ M ) @ ( sigma_sets_real @ M ) ) ).
% sets.top
thf(fact_346_sets_Oempty__sets,axiom,
! [M: sigma_measure_a] : ( member_set_a @ bot_bot_set_a @ ( sigma_sets_a @ M ) ) ).
% sets.empty_sets
thf(fact_347_sets_Oempty__sets,axiom,
! [M: sigma_measure_real] : ( member_set_real @ bot_bot_set_real @ ( sigma_sets_real @ M ) ) ).
% sets.empty_sets
thf(fact_348_f__inj,axiom,
inj_on_a_real @ ( standard_f_a @ m ) @ ( sigma_space_a @ m ) ).
% f_inj
thf(fact_349_sets__le__imp__space__le,axiom,
! [A: sigma_measure_real,B: sigma_measure_real] :
( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ A ) @ ( sigma_sets_real @ B ) )
=> ( ord_less_eq_set_real @ ( sigma_space_real @ A ) @ ( sigma_space_real @ B ) ) ) ).
% sets_le_imp_space_le
thf(fact_350_sets__le__imp__space__le,axiom,
! [A: sigma_measure_a,B: sigma_measure_a] :
( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ A ) @ ( sigma_sets_a @ B ) )
=> ( ord_less_eq_set_a @ ( sigma_space_a @ A ) @ ( sigma_space_a @ B ) ) ) ).
% sets_le_imp_space_le
thf(fact_351_sets_Osets__into__space,axiom,
! [X: set_real,M: sigma_measure_real] :
( ( member_set_real @ X @ ( sigma_sets_real @ M ) )
=> ( ord_less_eq_set_real @ X @ ( sigma_space_real @ M ) ) ) ).
% sets.sets_into_space
thf(fact_352_sets_Osets__into__space,axiom,
! [X: set_a,M: sigma_measure_a] :
( ( member_set_a @ X @ ( sigma_sets_a @ M ) )
=> ( ord_less_eq_set_a @ X @ ( sigma_space_a @ M ) ) ) ).
% sets.sets_into_space
thf(fact_353_comp__apply,axiom,
( comp_real_a_a
= ( ^ [F2: real > a,G2: a > real,X3: a] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% comp_apply
thf(fact_354_comp__apply,axiom,
( comp_real_real_real
= ( ^ [F2: real > real,G2: real > real,X3: real] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% comp_apply
thf(fact_355_set_Ocomp,axiom,
! [F: real > a,G: a > real] :
( ( comp_s6092362911880814593t_real @ ( vimage_real_a @ F ) @ ( vimage_a_real @ G ) )
= ( vimage_real_real @ ( comp_a_real_real @ G @ F ) ) ) ).
% set.comp
thf(fact_356_set_Ocomp,axiom,
! [F: real > a,G: a > a] :
( ( comp_s5765971254739342719_set_a @ ( vimage_real_a @ F ) @ ( vimage_a_a @ G ) )
= ( vimage_real_a @ ( comp_a_a_real @ G @ F ) ) ) ).
% set.comp
thf(fact_357_set_Ocomp,axiom,
! [F: real > real,G: real > a] :
( ( comp_s762043476418857333_set_a @ ( vimage_real_real @ F ) @ ( vimage_real_a @ G ) )
= ( vimage_real_a @ ( comp_real_a_real @ G @ F ) ) ) ).
% set.comp
thf(fact_358_set_Ocomp,axiom,
! [F: real > real,G: real > real] :
( ( comp_s5292247282120234379t_real @ ( vimage_real_real @ F ) @ ( vimage_real_real @ G ) )
= ( vimage_real_real @ ( comp_real_real_real @ G @ F ) ) ) ).
% set.comp
thf(fact_359_set_Ocomp,axiom,
! [F: a > real,G: real > a] :
( ( comp_s535997479508436165_set_a @ ( vimage_a_real @ F ) @ ( vimage_real_a @ G ) )
= ( vimage_a_a @ ( comp_real_a_a @ G @ F ) ) ) ).
% set.comp
thf(fact_360_set_Ocomp,axiom,
! [F: a > real,G: real > real] :
( ( comp_s6687884485137378875t_real @ ( vimage_a_real @ F ) @ ( vimage_real_real @ G ) )
= ( vimage_a_real @ ( comp_real_real_a @ G @ F ) ) ) ).
% set.comp
thf(fact_361_set_Ocomp,axiom,
! [F: a > a,G: a > real] :
( ( comp_s2468440226603088453t_real @ ( vimage_a_a @ F ) @ ( vimage_a_real @ G ) )
= ( vimage_a_real @ ( comp_a_real_a @ G @ F ) ) ) ).
% set.comp
thf(fact_362_set_Ocomp,axiom,
! [F: a > a,G: a > a] :
( ( comp_s9136116056826389051_set_a @ ( vimage_a_a @ F ) @ ( vimage_a_a @ G ) )
= ( vimage_a_a @ ( comp_a_a_a @ G @ F ) ) ) ).
% set.comp
thf(fact_363_vimage__comp,axiom,
! [F: real > a,G: a > real,X: set_real] :
( ( vimage_real_a @ F @ ( vimage_a_real @ G @ X ) )
= ( vimage_real_real @ ( comp_a_real_real @ G @ F ) @ X ) ) ).
% vimage_comp
thf(fact_364_vimage__comp,axiom,
! [F: real > a,G: a > a,X: set_a] :
( ( vimage_real_a @ F @ ( vimage_a_a @ G @ X ) )
= ( vimage_real_a @ ( comp_a_a_real @ G @ F ) @ X ) ) ).
% vimage_comp
thf(fact_365_vimage__comp,axiom,
! [F: real > real,G: real > a,X: set_a] :
( ( vimage_real_real @ F @ ( vimage_real_a @ G @ X ) )
= ( vimage_real_a @ ( comp_real_a_real @ G @ F ) @ X ) ) ).
% vimage_comp
thf(fact_366_vimage__comp,axiom,
! [F: real > real,G: real > real,X: set_real] :
( ( vimage_real_real @ F @ ( vimage_real_real @ G @ X ) )
= ( vimage_real_real @ ( comp_real_real_real @ G @ F ) @ X ) ) ).
% vimage_comp
thf(fact_367_vimage__comp,axiom,
! [F: a > real,G: real > a,X: set_a] :
( ( vimage_a_real @ F @ ( vimage_real_a @ G @ X ) )
= ( vimage_a_a @ ( comp_real_a_a @ G @ F ) @ X ) ) ).
% vimage_comp
thf(fact_368_vimage__comp,axiom,
! [F: a > real,G: real > real,X: set_real] :
( ( vimage_a_real @ F @ ( vimage_real_real @ G @ X ) )
= ( vimage_a_real @ ( comp_real_real_a @ G @ F ) @ X ) ) ).
% vimage_comp
thf(fact_369_vimage__comp,axiom,
! [F: a > a,G: a > real,X: set_real] :
( ( vimage_a_a @ F @ ( vimage_a_real @ G @ X ) )
= ( vimage_a_real @ ( comp_a_real_a @ G @ F ) @ X ) ) ).
% vimage_comp
thf(fact_370_vimage__comp,axiom,
! [F: a > a,G: a > a,X: set_a] :
( ( vimage_a_a @ F @ ( vimage_a_a @ G @ X ) )
= ( vimage_a_a @ ( comp_a_a_a @ G @ F ) @ X ) ) ).
% vimage_comp
thf(fact_371_set_Ocompositionality,axiom,
! [F: real > a,G: a > real,Set: set_real] :
( ( vimage_real_a @ F @ ( vimage_a_real @ G @ Set ) )
= ( vimage_real_real @ ( comp_a_real_real @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_372_set_Ocompositionality,axiom,
! [F: real > a,G: a > a,Set: set_a] :
( ( vimage_real_a @ F @ ( vimage_a_a @ G @ Set ) )
= ( vimage_real_a @ ( comp_a_a_real @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_373_set_Ocompositionality,axiom,
! [F: real > real,G: real > a,Set: set_a] :
( ( vimage_real_real @ F @ ( vimage_real_a @ G @ Set ) )
= ( vimage_real_a @ ( comp_real_a_real @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_374_set_Ocompositionality,axiom,
! [F: real > real,G: real > real,Set: set_real] :
( ( vimage_real_real @ F @ ( vimage_real_real @ G @ Set ) )
= ( vimage_real_real @ ( comp_real_real_real @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_375_set_Ocompositionality,axiom,
! [F: a > real,G: real > a,Set: set_a] :
( ( vimage_a_real @ F @ ( vimage_real_a @ G @ Set ) )
= ( vimage_a_a @ ( comp_real_a_a @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_376_set_Ocompositionality,axiom,
! [F: a > real,G: real > real,Set: set_real] :
( ( vimage_a_real @ F @ ( vimage_real_real @ G @ Set ) )
= ( vimage_a_real @ ( comp_real_real_a @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_377_set_Ocompositionality,axiom,
! [F: a > a,G: a > real,Set: set_real] :
( ( vimage_a_a @ F @ ( vimage_a_real @ G @ Set ) )
= ( vimage_a_real @ ( comp_a_real_a @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_378_set_Ocompositionality,axiom,
! [F: a > a,G: a > a,Set: set_a] :
( ( vimage_a_a @ F @ ( vimage_a_a @ G @ Set ) )
= ( vimage_a_a @ ( comp_a_a_a @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_379_hgu1,axiom,
member_set_real @ ( vimage_real_a @ ( standard_g_a @ m ) @ u ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ).
% hgu1
thf(fact_380_le__measureD2,axiom,
! [A: sigma_measure_a,B: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A @ B )
=> ( ( ( sigma_space_a @ A )
= ( sigma_space_a @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ A ) @ ( sigma_sets_a @ B ) ) ) ) ).
% le_measureD2
thf(fact_381_le__measureD2,axiom,
! [A: sigma_measure_real,B: sigma_measure_real] :
( ( ord_le487379304121309861e_real @ A @ B )
=> ( ( ( sigma_space_real @ A )
= ( sigma_space_real @ B ) )
=> ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ A ) @ ( sigma_sets_real @ B ) ) ) ) ).
% le_measureD2
thf(fact_382_insertCI,axiom,
! [A2: real,B: set_real,B2: real] :
( ( ~ ( member_real @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_real @ A2 @ ( insert_real @ B2 @ B ) ) ) ).
% insertCI
thf(fact_383_insertCI,axiom,
! [A2: set_a,B: set_set_a,B2: set_a] :
( ( ~ ( member_set_a @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_set_a @ A2 @ ( insert_set_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_384_insertCI,axiom,
! [A2: real > real,B: set_real_real,B2: real > real] :
( ( ~ ( member_real_real @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_real_real @ A2 @ ( insert_real_real @ B2 @ B ) ) ) ).
% insertCI
thf(fact_385_insertCI,axiom,
! [A2: real > a,B: set_real_a,B2: real > a] :
( ( ~ ( member_real_a @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_real_a @ A2 @ ( insert_real_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_386_insertCI,axiom,
! [A2: a > real,B: set_a_real,B2: a > real] :
( ( ~ ( member_a_real @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_a_real @ A2 @ ( insert_a_real @ B2 @ B ) ) ) ).
% insertCI
thf(fact_387_insertCI,axiom,
! [A2: a,B: set_a,B2: a] :
( ( ~ ( member_a @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_388_insert__iff,axiom,
! [A2: real,B2: real,A: set_real] :
( ( member_real @ A2 @ ( insert_real @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_real @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_389_insert__iff,axiom,
! [A2: set_a,B2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ ( insert_set_a @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_set_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_390_insert__iff,axiom,
! [A2: real > real,B2: real > real,A: set_real_real] :
( ( member_real_real @ A2 @ ( insert_real_real @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_real_real @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_391_insert__iff,axiom,
! [A2: real > a,B2: real > a,A: set_real_a] :
( ( member_real_a @ A2 @ ( insert_real_a @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_real_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_392_insert__iff,axiom,
! [A2: a > real,B2: a > real,A: set_a_real] :
( ( member_a_real @ A2 @ ( insert_a_real @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_a_real @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_393_insert__iff,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_394_insert__absorb2,axiom,
! [X: a,A: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A ) )
= ( insert_a @ X @ A ) ) ).
% insert_absorb2
thf(fact_395_insert__absorb2,axiom,
! [X: real,A: set_real] :
( ( insert_real @ X @ ( insert_real @ X @ A ) )
= ( insert_real @ X @ A ) ) ).
% insert_absorb2
thf(fact_396_real_Ogf__comp__id_I2_J,axiom,
! [X: real] :
( ( member_real @ X @ ( sigma_space_real @ borel_5078946678739801102l_real ) )
=> ( ( standard_g_real @ borel_5078946678739801102l_real @ ( standard_f_real @ borel_5078946678739801102l_real @ X ) )
= X ) ) ).
% real.gf_comp_id(2)
thf(fact_397_real_Ogf__comp__id_I1_J,axiom,
! [X: real] :
( ( member_real @ X @ ( sigma_space_real @ borel_5078946678739801102l_real ) )
=> ( ( comp_real_real_real @ ( standard_g_real @ borel_5078946678739801102l_real ) @ ( standard_f_real @ borel_5078946678739801102l_real ) @ X )
= X ) ) ).
% real.gf_comp_id(1)
thf(fact_398_real_Ogf__comp__id_H_I2_J,axiom,
! [X: real] :
( ( standard_g_real @ borel_5078946678739801102l_real @ ( standard_f_real @ borel_5078946678739801102l_real @ X ) )
= X ) ).
% real.gf_comp_id'(2)
thf(fact_399_image__insert,axiom,
! [F: a > a,A2: a,B: set_a] :
( ( image_a_a @ F @ ( insert_a @ A2 @ B ) )
= ( insert_a @ ( F @ A2 ) @ ( image_a_a @ F @ B ) ) ) ).
% image_insert
thf(fact_400_image__insert,axiom,
! [F: a > real,A2: a,B: set_a] :
( ( image_a_real @ F @ ( insert_a @ A2 @ B ) )
= ( insert_real @ ( F @ A2 ) @ ( image_a_real @ F @ B ) ) ) ).
% image_insert
thf(fact_401_image__insert,axiom,
! [F: real > a,A2: real,B: set_real] :
( ( image_real_a @ F @ ( insert_real @ A2 @ B ) )
= ( insert_a @ ( F @ A2 ) @ ( image_real_a @ F @ B ) ) ) ).
% image_insert
thf(fact_402_image__insert,axiom,
! [F: real > real,A2: real,B: set_real] :
( ( image_real_real @ F @ ( insert_real @ A2 @ B ) )
= ( insert_real @ ( F @ A2 ) @ ( image_real_real @ F @ B ) ) ) ).
% image_insert
thf(fact_403_insert__image,axiom,
! [X: real,A: set_real,F: real > a] :
( ( member_real @ X @ A )
=> ( ( insert_a @ ( F @ X ) @ ( image_real_a @ F @ A ) )
= ( image_real_a @ F @ A ) ) ) ).
% insert_image
thf(fact_404_insert__image,axiom,
! [X: real,A: set_real,F: real > real] :
( ( member_real @ X @ A )
=> ( ( insert_real @ ( F @ X ) @ ( image_real_real @ F @ A ) )
= ( image_real_real @ F @ A ) ) ) ).
% insert_image
thf(fact_405_insert__image,axiom,
! [X: a,A: set_a,F: a > a] :
( ( member_a @ X @ A )
=> ( ( insert_a @ ( F @ X ) @ ( image_a_a @ F @ A ) )
= ( image_a_a @ F @ A ) ) ) ).
% insert_image
thf(fact_406_insert__image,axiom,
! [X: a,A: set_a,F: a > real] :
( ( member_a @ X @ A )
=> ( ( insert_real @ ( F @ X ) @ ( image_a_real @ F @ A ) )
= ( image_a_real @ F @ A ) ) ) ).
% insert_image
thf(fact_407_insert__image,axiom,
! [X: set_a,A: set_set_a,F: set_a > a] :
( ( member_set_a @ X @ A )
=> ( ( insert_a @ ( F @ X ) @ ( image_set_a_a @ F @ A ) )
= ( image_set_a_a @ F @ A ) ) ) ).
% insert_image
thf(fact_408_insert__image,axiom,
! [X: set_a,A: set_set_a,F: set_a > real] :
( ( member_set_a @ X @ A )
=> ( ( insert_real @ ( F @ X ) @ ( image_set_a_real @ F @ A ) )
= ( image_set_a_real @ F @ A ) ) ) ).
% insert_image
thf(fact_409_insert__image,axiom,
! [X: real > real,A: set_real_real,F: ( real > real ) > a] :
( ( member_real_real @ X @ A )
=> ( ( insert_a @ ( F @ X ) @ ( image_real_real_a @ F @ A ) )
= ( image_real_real_a @ F @ A ) ) ) ).
% insert_image
thf(fact_410_insert__image,axiom,
! [X: real > real,A: set_real_real,F: ( real > real ) > real] :
( ( member_real_real @ X @ A )
=> ( ( insert_real @ ( F @ X ) @ ( image_real_real_real @ F @ A ) )
= ( image_real_real_real @ F @ A ) ) ) ).
% insert_image
thf(fact_411_insert__image,axiom,
! [X: real > a,A: set_real_a,F: ( real > a ) > a] :
( ( member_real_a @ X @ A )
=> ( ( insert_a @ ( F @ X ) @ ( image_real_a_a @ F @ A ) )
= ( image_real_a_a @ F @ A ) ) ) ).
% insert_image
thf(fact_412_insert__image,axiom,
! [X: real > a,A: set_real_a,F: ( real > a ) > real] :
( ( member_real_a @ X @ A )
=> ( ( insert_real @ ( F @ X ) @ ( image_real_a_real @ F @ A ) )
= ( image_real_a_real @ F @ A ) ) ) ).
% insert_image
thf(fact_413_singletonI,axiom,
! [A2: set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_414_singletonI,axiom,
! [A2: real > real] : ( member_real_real @ A2 @ ( insert_real_real @ A2 @ bot_bo6767488733719836353l_real ) ) ).
% singletonI
thf(fact_415_singletonI,axiom,
! [A2: real > a] : ( member_real_a @ A2 @ ( insert_real_a @ A2 @ bot_bot_set_real_a ) ) ).
% singletonI
thf(fact_416_singletonI,axiom,
! [A2: a > real] : ( member_a_real @ A2 @ ( insert_a_real @ A2 @ bot_bot_set_a_real ) ) ).
% singletonI
thf(fact_417_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_418_singletonI,axiom,
! [A2: real] : ( member_real @ A2 @ ( insert_real @ A2 @ bot_bot_set_real ) ) ).
% singletonI
thf(fact_419_insert__subset,axiom,
! [X: set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A ) @ B )
= ( ( member_set_a @ X @ B )
& ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_420_insert__subset,axiom,
! [X: real > real,A: set_real_real,B: set_real_real] :
( ( ord_le4198349162570665613l_real @ ( insert_real_real @ X @ A ) @ B )
= ( ( member_real_real @ X @ B )
& ( ord_le4198349162570665613l_real @ A @ B ) ) ) ).
% insert_subset
thf(fact_421_insert__subset,axiom,
! [X: real > a,A: set_real_a,B: set_real_a] :
( ( ord_le5743406823621094409real_a @ ( insert_real_a @ X @ A ) @ B )
= ( ( member_real_a @ X @ B )
& ( ord_le5743406823621094409real_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_422_insert__subset,axiom,
! [X: a > real,A: set_a_real,B: set_a_real] :
( ( ord_le3334967407727675675a_real @ ( insert_a_real @ X @ A ) @ B )
= ( ( member_a_real @ X @ B )
& ( ord_le3334967407727675675a_real @ A @ B ) ) ) ).
% insert_subset
thf(fact_423_insert__subset,axiom,
! [X: real,A: set_real,B: set_real] :
( ( ord_less_eq_set_real @ ( insert_real @ X @ A ) @ B )
= ( ( member_real @ X @ B )
& ( ord_less_eq_set_real @ A @ B ) ) ) ).
% insert_subset
thf(fact_424_insert__subset,axiom,
! [X: a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A ) @ B )
= ( ( member_a @ X @ B )
& ( ord_less_eq_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_425_space__bot,axiom,
( ( sigma_space_a @ bot_bo2108912051383640591sure_a )
= bot_bot_set_a ) ).
% space_bot
thf(fact_426_space__bot,axiom,
( ( sigma_space_real @ bot_bo5982154664989874033e_real )
= bot_bot_set_real ) ).
% space_bot
thf(fact_427_inj__on__empty,axiom,
! [F: a > real] : ( inj_on_a_real @ F @ bot_bot_set_a ) ).
% inj_on_empty
thf(fact_428_inj__on__empty,axiom,
! [F: real > real] : ( inj_on_real_real @ F @ bot_bot_set_real ) ).
% inj_on_empty
thf(fact_429_singleton__insert__inj__eq_H,axiom,
! [A2: real,A: set_real,B2: real] :
( ( ( insert_real @ A2 @ A )
= ( insert_real @ B2 @ bot_bot_set_real ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_real @ A @ ( insert_real @ B2 @ bot_bot_set_real ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_430_singleton__insert__inj__eq_H,axiom,
! [A2: a,A: set_a,B2: a] :
( ( ( insert_a @ A2 @ A )
= ( insert_a @ B2 @ bot_bot_set_a ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_431_singleton__insert__inj__eq,axiom,
! [B2: real,A2: real,A: set_real] :
( ( ( insert_real @ B2 @ bot_bot_set_real )
= ( insert_real @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_real @ A @ ( insert_real @ B2 @ bot_bot_set_real ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_432_singleton__insert__inj__eq,axiom,
! [B2: a,A2: a,A: set_a] :
( ( ( insert_a @ B2 @ bot_bot_set_a )
= ( insert_a @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_433_real_Ostandard__borel__sets,axiom,
! [Y: sigma_measure_real] :
( ( ( sigma_sets_real @ borel_5078946678739801102l_real )
= ( sigma_sets_real @ Y ) )
=> ( standard_borel_real @ Y ) ) ).
% real.standard_borel_sets
thf(fact_434_real_Ostandard__borel__axioms,axiom,
standard_borel_real @ borel_5078946678739801102l_real ).
% real.standard_borel_axioms
thf(fact_435_insertE,axiom,
! [A2: real,B2: real,A: set_real] :
( ( member_real @ A2 @ ( insert_real @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_real @ A2 @ A ) ) ) ).
% insertE
thf(fact_436_insertE,axiom,
! [A2: set_a,B2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ ( insert_set_a @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_set_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_437_insertE,axiom,
! [A2: real > real,B2: real > real,A: set_real_real] :
( ( member_real_real @ A2 @ ( insert_real_real @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_real_real @ A2 @ A ) ) ) ).
% insertE
thf(fact_438_insertE,axiom,
! [A2: real > a,B2: real > a,A: set_real_a] :
( ( member_real_a @ A2 @ ( insert_real_a @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_real_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_439_insertE,axiom,
! [A2: a > real,B2: a > real,A: set_a_real] :
( ( member_a_real @ A2 @ ( insert_a_real @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_a_real @ A2 @ A ) ) ) ).
% insertE
thf(fact_440_insertE,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_441_insertI1,axiom,
! [A2: real,B: set_real] : ( member_real @ A2 @ ( insert_real @ A2 @ B ) ) ).
% insertI1
thf(fact_442_insertI1,axiom,
! [A2: set_a,B: set_set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ B ) ) ).
% insertI1
thf(fact_443_insertI1,axiom,
! [A2: real > real,B: set_real_real] : ( member_real_real @ A2 @ ( insert_real_real @ A2 @ B ) ) ).
% insertI1
thf(fact_444_insertI1,axiom,
! [A2: real > a,B: set_real_a] : ( member_real_a @ A2 @ ( insert_real_a @ A2 @ B ) ) ).
% insertI1
thf(fact_445_insertI1,axiom,
! [A2: a > real,B: set_a_real] : ( member_a_real @ A2 @ ( insert_a_real @ A2 @ B ) ) ).
% insertI1
thf(fact_446_insertI1,axiom,
! [A2: a,B: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B ) ) ).
% insertI1
thf(fact_447_insertI2,axiom,
! [A2: real,B: set_real,B2: real] :
( ( member_real @ A2 @ B )
=> ( member_real @ A2 @ ( insert_real @ B2 @ B ) ) ) ).
% insertI2
thf(fact_448_insertI2,axiom,
! [A2: set_a,B: set_set_a,B2: set_a] :
( ( member_set_a @ A2 @ B )
=> ( member_set_a @ A2 @ ( insert_set_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_449_insertI2,axiom,
! [A2: real > real,B: set_real_real,B2: real > real] :
( ( member_real_real @ A2 @ B )
=> ( member_real_real @ A2 @ ( insert_real_real @ B2 @ B ) ) ) ).
% insertI2
thf(fact_450_insertI2,axiom,
! [A2: real > a,B: set_real_a,B2: real > a] :
( ( member_real_a @ A2 @ B )
=> ( member_real_a @ A2 @ ( insert_real_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_451_insertI2,axiom,
! [A2: a > real,B: set_a_real,B2: a > real] :
( ( member_a_real @ A2 @ B )
=> ( member_a_real @ A2 @ ( insert_a_real @ B2 @ B ) ) ) ).
% insertI2
thf(fact_452_insertI2,axiom,
! [A2: a,B: set_a,B2: a] :
( ( member_a @ A2 @ B )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_453_Set_Oset__insert,axiom,
! [X: real,A: set_real] :
( ( member_real @ X @ A )
=> ~ ! [B5: set_real] :
( ( A
= ( insert_real @ X @ B5 ) )
=> ( member_real @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_454_Set_Oset__insert,axiom,
! [X: set_a,A: set_set_a] :
( ( member_set_a @ X @ A )
=> ~ ! [B5: set_set_a] :
( ( A
= ( insert_set_a @ X @ B5 ) )
=> ( member_set_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_455_Set_Oset__insert,axiom,
! [X: real > real,A: set_real_real] :
( ( member_real_real @ X @ A )
=> ~ ! [B5: set_real_real] :
( ( A
= ( insert_real_real @ X @ B5 ) )
=> ( member_real_real @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_456_Set_Oset__insert,axiom,
! [X: real > a,A: set_real_a] :
( ( member_real_a @ X @ A )
=> ~ ! [B5: set_real_a] :
( ( A
= ( insert_real_a @ X @ B5 ) )
=> ( member_real_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_457_Set_Oset__insert,axiom,
! [X: a > real,A: set_a_real] :
( ( member_a_real @ X @ A )
=> ~ ! [B5: set_a_real] :
( ( A
= ( insert_a_real @ X @ B5 ) )
=> ( member_a_real @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_458_Set_Oset__insert,axiom,
! [X: a,A: set_a] :
( ( member_a @ X @ A )
=> ~ ! [B5: set_a] :
( ( A
= ( insert_a @ X @ B5 ) )
=> ( member_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_459_insert__ident,axiom,
! [X: real,A: set_real,B: set_real] :
( ~ ( member_real @ X @ A )
=> ( ~ ( member_real @ X @ B )
=> ( ( ( insert_real @ X @ A )
= ( insert_real @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_460_insert__ident,axiom,
! [X: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X @ A )
=> ( ~ ( member_set_a @ X @ B )
=> ( ( ( insert_set_a @ X @ A )
= ( insert_set_a @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_461_insert__ident,axiom,
! [X: real > real,A: set_real_real,B: set_real_real] :
( ~ ( member_real_real @ X @ A )
=> ( ~ ( member_real_real @ X @ B )
=> ( ( ( insert_real_real @ X @ A )
= ( insert_real_real @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_462_insert__ident,axiom,
! [X: real > a,A: set_real_a,B: set_real_a] :
( ~ ( member_real_a @ X @ A )
=> ( ~ ( member_real_a @ X @ B )
=> ( ( ( insert_real_a @ X @ A )
= ( insert_real_a @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_463_insert__ident,axiom,
! [X: a > real,A: set_a_real,B: set_a_real] :
( ~ ( member_a_real @ X @ A )
=> ( ~ ( member_a_real @ X @ B )
=> ( ( ( insert_a_real @ X @ A )
= ( insert_a_real @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_464_insert__ident,axiom,
! [X: a,A: set_a,B: set_a] :
( ~ ( member_a @ X @ A )
=> ( ~ ( member_a @ X @ B )
=> ( ( ( insert_a @ X @ A )
= ( insert_a @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_465_insert__absorb,axiom,
! [A2: real,A: set_real] :
( ( member_real @ A2 @ A )
=> ( ( insert_real @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_466_insert__absorb,axiom,
! [A2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ( ( insert_set_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_467_insert__absorb,axiom,
! [A2: real > real,A: set_real_real] :
( ( member_real_real @ A2 @ A )
=> ( ( insert_real_real @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_468_insert__absorb,axiom,
! [A2: real > a,A: set_real_a] :
( ( member_real_a @ A2 @ A )
=> ( ( insert_real_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_469_insert__absorb,axiom,
! [A2: a > real,A: set_a_real] :
( ( member_a_real @ A2 @ A )
=> ( ( insert_a_real @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_470_insert__absorb,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_471_insert__eq__iff,axiom,
! [A2: real,A: set_real,B2: real,B: set_real] :
( ~ ( member_real @ A2 @ A )
=> ( ~ ( member_real @ B2 @ B )
=> ( ( ( insert_real @ A2 @ A )
= ( insert_real @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C4: set_real] :
( ( A
= ( insert_real @ B2 @ C4 ) )
& ~ ( member_real @ B2 @ C4 )
& ( B
= ( insert_real @ A2 @ C4 ) )
& ~ ( member_real @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_472_insert__eq__iff,axiom,
! [A2: set_a,A: set_set_a,B2: set_a,B: set_set_a] :
( ~ ( member_set_a @ A2 @ A )
=> ( ~ ( member_set_a @ B2 @ B )
=> ( ( ( insert_set_a @ A2 @ A )
= ( insert_set_a @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C4: set_set_a] :
( ( A
= ( insert_set_a @ B2 @ C4 ) )
& ~ ( member_set_a @ B2 @ C4 )
& ( B
= ( insert_set_a @ A2 @ C4 ) )
& ~ ( member_set_a @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_473_insert__eq__iff,axiom,
! [A2: real > real,A: set_real_real,B2: real > real,B: set_real_real] :
( ~ ( member_real_real @ A2 @ A )
=> ( ~ ( member_real_real @ B2 @ B )
=> ( ( ( insert_real_real @ A2 @ A )
= ( insert_real_real @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C4: set_real_real] :
( ( A
= ( insert_real_real @ B2 @ C4 ) )
& ~ ( member_real_real @ B2 @ C4 )
& ( B
= ( insert_real_real @ A2 @ C4 ) )
& ~ ( member_real_real @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_474_insert__eq__iff,axiom,
! [A2: real > a,A: set_real_a,B2: real > a,B: set_real_a] :
( ~ ( member_real_a @ A2 @ A )
=> ( ~ ( member_real_a @ B2 @ B )
=> ( ( ( insert_real_a @ A2 @ A )
= ( insert_real_a @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C4: set_real_a] :
( ( A
= ( insert_real_a @ B2 @ C4 ) )
& ~ ( member_real_a @ B2 @ C4 )
& ( B
= ( insert_real_a @ A2 @ C4 ) )
& ~ ( member_real_a @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_475_insert__eq__iff,axiom,
! [A2: a > real,A: set_a_real,B2: a > real,B: set_a_real] :
( ~ ( member_a_real @ A2 @ A )
=> ( ~ ( member_a_real @ B2 @ B )
=> ( ( ( insert_a_real @ A2 @ A )
= ( insert_a_real @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C4: set_a_real] :
( ( A
= ( insert_a_real @ B2 @ C4 ) )
& ~ ( member_a_real @ B2 @ C4 )
& ( B
= ( insert_a_real @ A2 @ C4 ) )
& ~ ( member_a_real @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_476_insert__eq__iff,axiom,
! [A2: a,A: set_a,B2: a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ~ ( member_a @ B2 @ B )
=> ( ( ( insert_a @ A2 @ A )
= ( insert_a @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C4: set_a] :
( ( A
= ( insert_a @ B2 @ C4 ) )
& ~ ( member_a @ B2 @ C4 )
& ( B
= ( insert_a @ A2 @ C4 ) )
& ~ ( member_a @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_477_insert__commute,axiom,
! [X: a,Y3: a,A: set_a] :
( ( insert_a @ X @ ( insert_a @ Y3 @ A ) )
= ( insert_a @ Y3 @ ( insert_a @ X @ A ) ) ) ).
% insert_commute
thf(fact_478_insert__commute,axiom,
! [X: real,Y3: real,A: set_real] :
( ( insert_real @ X @ ( insert_real @ Y3 @ A ) )
= ( insert_real @ Y3 @ ( insert_real @ X @ A ) ) ) ).
% insert_commute
thf(fact_479_inj__img__insertE,axiom,
! [F: real > real,A: set_real,X: real,B: set_real] :
( ( inj_on_real_real @ F @ A )
=> ( ~ ( member_real @ X @ B )
=> ( ( ( insert_real @ X @ B )
= ( image_real_real @ F @ A ) )
=> ~ ! [X5: real,A5: set_real] :
( ~ ( member_real @ X5 @ A5 )
=> ( ( A
= ( insert_real @ X5 @ A5 ) )
=> ( ( X
= ( F @ X5 ) )
=> ( B
!= ( image_real_real @ F @ A5 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_480_inj__img__insertE,axiom,
! [F: a > real,A: set_a,X: real,B: set_real] :
( ( inj_on_a_real @ F @ A )
=> ( ~ ( member_real @ X @ B )
=> ( ( ( insert_real @ X @ B )
= ( image_a_real @ F @ A ) )
=> ~ ! [X5: a,A5: set_a] :
( ~ ( member_a @ X5 @ A5 )
=> ( ( A
= ( insert_a @ X5 @ A5 ) )
=> ( ( X
= ( F @ X5 ) )
=> ( B
!= ( image_a_real @ F @ A5 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_481_inj__img__insertE,axiom,
! [F: real > a,A: set_real,X: a,B: set_a] :
( ( inj_on_real_a @ F @ A )
=> ( ~ ( member_a @ X @ B )
=> ( ( ( insert_a @ X @ B )
= ( image_real_a @ F @ A ) )
=> ~ ! [X5: real,A5: set_real] :
( ~ ( member_real @ X5 @ A5 )
=> ( ( A
= ( insert_real @ X5 @ A5 ) )
=> ( ( X
= ( F @ X5 ) )
=> ( B
!= ( image_real_a @ F @ A5 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_482_inj__img__insertE,axiom,
! [F: a > a,A: set_a,X: a,B: set_a] :
( ( inj_on_a_a @ F @ A )
=> ( ~ ( member_a @ X @ B )
=> ( ( ( insert_a @ X @ B )
= ( image_a_a @ F @ A ) )
=> ~ ! [X5: a,A5: set_a] :
( ~ ( member_a @ X5 @ A5 )
=> ( ( A
= ( insert_a @ X5 @ A5 ) )
=> ( ( X
= ( F @ X5 ) )
=> ( B
!= ( image_a_a @ F @ A5 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_483_inj__img__insertE,axiom,
! [F: set_a > real,A: set_set_a,X: real,B: set_real] :
( ( inj_on_set_a_real @ F @ A )
=> ( ~ ( member_real @ X @ B )
=> ( ( ( insert_real @ X @ B )
= ( image_set_a_real @ F @ A ) )
=> ~ ! [X5: set_a,A5: set_set_a] :
( ~ ( member_set_a @ X5 @ A5 )
=> ( ( A
= ( insert_set_a @ X5 @ A5 ) )
=> ( ( X
= ( F @ X5 ) )
=> ( B
!= ( image_set_a_real @ F @ A5 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_484_inj__img__insertE,axiom,
! [F: real > set_a,A: set_real,X: set_a,B: set_set_a] :
( ( inj_on_real_set_a @ F @ A )
=> ( ~ ( member_set_a @ X @ B )
=> ( ( ( insert_set_a @ X @ B )
= ( image_real_set_a @ F @ A ) )
=> ~ ! [X5: real,A5: set_real] :
( ~ ( member_real @ X5 @ A5 )
=> ( ( A
= ( insert_real @ X5 @ A5 ) )
=> ( ( X
= ( F @ X5 ) )
=> ( B
!= ( image_real_set_a @ F @ A5 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_485_inj__img__insertE,axiom,
! [F: a > set_a,A: set_a,X: set_a,B: set_set_a] :
( ( inj_on_a_set_a @ F @ A )
=> ( ~ ( member_set_a @ X @ B )
=> ( ( ( insert_set_a @ X @ B )
= ( image_a_set_a @ F @ A ) )
=> ~ ! [X5: a,A5: set_a] :
( ~ ( member_a @ X5 @ A5 )
=> ( ( A
= ( insert_a @ X5 @ A5 ) )
=> ( ( X
= ( F @ X5 ) )
=> ( B
!= ( image_a_set_a @ F @ A5 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_486_inj__img__insertE,axiom,
! [F: set_a > a,A: set_set_a,X: a,B: set_a] :
( ( inj_on_set_a_a @ F @ A )
=> ( ~ ( member_a @ X @ B )
=> ( ( ( insert_a @ X @ B )
= ( image_set_a_a @ F @ A ) )
=> ~ ! [X5: set_a,A5: set_set_a] :
( ~ ( member_set_a @ X5 @ A5 )
=> ( ( A
= ( insert_set_a @ X5 @ A5 ) )
=> ( ( X
= ( F @ X5 ) )
=> ( B
!= ( image_set_a_a @ F @ A5 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_487_inj__img__insertE,axiom,
! [F: ( real > real ) > real,A: set_real_real,X: real,B: set_real] :
( ( inj_on1257811008913012118l_real @ F @ A )
=> ( ~ ( member_real @ X @ B )
=> ( ( ( insert_real @ X @ B )
= ( image_real_real_real @ F @ A ) )
=> ~ ! [X5: real > real,A5: set_real_real] :
( ~ ( member_real_real @ X5 @ A5 )
=> ( ( A
= ( insert_real_real @ X5 @ A5 ) )
=> ( ( X
= ( F @ X5 ) )
=> ( B
!= ( image_real_real_real @ F @ A5 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_488_inj__img__insertE,axiom,
! [F: ( real > a ) > real,A: set_real_a,X: real,B: set_real] :
( ( inj_on_real_a_real @ F @ A )
=> ( ~ ( member_real @ X @ B )
=> ( ( ( insert_real @ X @ B )
= ( image_real_a_real @ F @ A ) )
=> ~ ! [X5: real > a,A5: set_real_a] :
( ~ ( member_real_a @ X5 @ A5 )
=> ( ( A
= ( insert_real_a @ X5 @ A5 ) )
=> ( ( X
= ( F @ X5 ) )
=> ( B
!= ( image_real_a_real @ F @ A5 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_489_mk__disjoint__insert,axiom,
! [A2: real,A: set_real] :
( ( member_real @ A2 @ A )
=> ? [B5: set_real] :
( ( A
= ( insert_real @ A2 @ B5 ) )
& ~ ( member_real @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_490_mk__disjoint__insert,axiom,
! [A2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ? [B5: set_set_a] :
( ( A
= ( insert_set_a @ A2 @ B5 ) )
& ~ ( member_set_a @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_491_mk__disjoint__insert,axiom,
! [A2: real > real,A: set_real_real] :
( ( member_real_real @ A2 @ A )
=> ? [B5: set_real_real] :
( ( A
= ( insert_real_real @ A2 @ B5 ) )
& ~ ( member_real_real @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_492_mk__disjoint__insert,axiom,
! [A2: real > a,A: set_real_a] :
( ( member_real_a @ A2 @ A )
=> ? [B5: set_real_a] :
( ( A
= ( insert_real_a @ A2 @ B5 ) )
& ~ ( member_real_a @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_493_mk__disjoint__insert,axiom,
! [A2: a > real,A: set_a_real] :
( ( member_a_real @ A2 @ A )
=> ? [B5: set_a_real] :
( ( A
= ( insert_a_real @ A2 @ B5 ) )
& ~ ( member_a_real @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_494_mk__disjoint__insert,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ? [B5: set_a] :
( ( A
= ( insert_a @ A2 @ B5 ) )
& ~ ( member_a @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_495_inj__on__inverseI,axiom,
! [A: set_real,G: real > real,F: real > real] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( G @ ( F @ X2 ) )
= X2 ) )
=> ( inj_on_real_real @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_496_inj__on__inverseI,axiom,
! [A: set_a,G: real > a,F: a > real] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( G @ ( F @ X2 ) )
= X2 ) )
=> ( inj_on_a_real @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_497_inj__on__contraD,axiom,
! [F: real > real,A: set_real,X: real,Y3: real] :
( ( inj_on_real_real @ F @ A )
=> ( ( X != Y3 )
=> ( ( member_real @ X @ A )
=> ( ( member_real @ Y3 @ A )
=> ( ( F @ X )
!= ( F @ Y3 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_498_inj__on__contraD,axiom,
! [F: a > real,A: set_a,X: a,Y3: a] :
( ( inj_on_a_real @ F @ A )
=> ( ( X != Y3 )
=> ( ( member_a @ X @ A )
=> ( ( member_a @ Y3 @ A )
=> ( ( F @ X )
!= ( F @ Y3 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_499_inj__on__eq__iff,axiom,
! [F: real > real,A: set_real,X: real,Y3: real] :
( ( inj_on_real_real @ F @ A )
=> ( ( member_real @ X @ A )
=> ( ( member_real @ Y3 @ A )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
= ( X = Y3 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_500_inj__on__eq__iff,axiom,
! [F: a > real,A: set_a,X: a,Y3: a] :
( ( inj_on_a_real @ F @ A )
=> ( ( member_a @ X @ A )
=> ( ( member_a @ Y3 @ A )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
= ( X = Y3 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_501_inj__on__cong,axiom,
! [A: set_real,F: real > real,G: real > real] :
( ! [A6: real] :
( ( member_real @ A6 @ A )
=> ( ( F @ A6 )
= ( G @ A6 ) ) )
=> ( ( inj_on_real_real @ F @ A )
= ( inj_on_real_real @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_502_inj__on__cong,axiom,
! [A: set_a,F: a > real,G: a > real] :
( ! [A6: a] :
( ( member_a @ A6 @ A )
=> ( ( F @ A6 )
= ( G @ A6 ) ) )
=> ( ( inj_on_a_real @ F @ A )
= ( inj_on_a_real @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_503_inj__on__def,axiom,
( inj_on_a_real
= ( ^ [F2: a > real,A4: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ! [Y5: a] :
( ( member_a @ Y5 @ A4 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y5 ) )
=> ( X3 = Y5 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_504_inj__on__def,axiom,
( inj_on_real_real
= ( ^ [F2: real > real,A4: set_real] :
! [X3: real] :
( ( member_real @ X3 @ A4 )
=> ! [Y5: real] :
( ( member_real @ Y5 @ A4 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y5 ) )
=> ( X3 = Y5 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_505_inj__onI,axiom,
! [A: set_real,F: real > real] :
( ! [X2: real,Y2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_real @ Y2 @ A )
=> ( ( ( F @ X2 )
= ( F @ Y2 ) )
=> ( X2 = Y2 ) ) ) )
=> ( inj_on_real_real @ F @ A ) ) ).
% inj_onI
thf(fact_506_inj__onI,axiom,
! [A: set_a,F: a > real] :
( ! [X2: a,Y2: a] :
( ( member_a @ X2 @ A )
=> ( ( member_a @ Y2 @ A )
=> ( ( ( F @ X2 )
= ( F @ Y2 ) )
=> ( X2 = Y2 ) ) ) )
=> ( inj_on_a_real @ F @ A ) ) ).
% inj_onI
thf(fact_507_inj__onD,axiom,
! [F: real > real,A: set_real,X: real,Y3: real] :
( ( inj_on_real_real @ F @ A )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( ( member_real @ X @ A )
=> ( ( member_real @ Y3 @ A )
=> ( X = Y3 ) ) ) ) ) ).
% inj_onD
thf(fact_508_inj__onD,axiom,
! [F: a > real,A: set_a,X: a,Y3: a] :
( ( inj_on_a_real @ F @ A )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( ( member_a @ X @ A )
=> ( ( member_a @ Y3 @ A )
=> ( X = Y3 ) ) ) ) ) ).
% inj_onD
thf(fact_509_inj__on__image__iff,axiom,
! [A: set_a,G: a > real,F: a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ! [Xa: a] :
( ( member_a @ Xa @ A )
=> ( ( ( G @ ( F @ X2 ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X2 )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on_a_a @ F @ A )
=> ( ( inj_on_a_real @ G @ ( image_a_a @ F @ A ) )
= ( inj_on_a_real @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_510_inj__on__image__iff,axiom,
! [A: set_real,G: real > real,F: real > real] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ! [Xa: real] :
( ( member_real @ Xa @ A )
=> ( ( ( G @ ( F @ X2 ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X2 )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on_real_real @ F @ A )
=> ( ( inj_on_real_real @ G @ ( image_real_real @ F @ A ) )
= ( inj_on_real_real @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_511_inj__on__subset,axiom,
! [F: real > real,A: set_real,B: set_real] :
( ( inj_on_real_real @ F @ A )
=> ( ( ord_less_eq_set_real @ B @ A )
=> ( inj_on_real_real @ F @ B ) ) ) ).
% inj_on_subset
thf(fact_512_inj__on__subset,axiom,
! [F: a > real,A: set_a,B: set_a] :
( ( inj_on_a_real @ F @ A )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( inj_on_a_real @ F @ B ) ) ) ).
% inj_on_subset
thf(fact_513_subset__inj__on,axiom,
! [F: real > real,B: set_real,A: set_real] :
( ( inj_on_real_real @ F @ B )
=> ( ( ord_less_eq_set_real @ A @ B )
=> ( inj_on_real_real @ F @ A ) ) ) ).
% subset_inj_on
thf(fact_514_subset__inj__on,axiom,
! [F: a > real,B: set_a,A: set_a] :
( ( inj_on_a_real @ F @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( inj_on_a_real @ F @ A ) ) ) ).
% subset_inj_on
thf(fact_515_inj__on__imageI2,axiom,
! [F3: real > a,F: a > real,A: set_a] :
( ( inj_on_a_a @ ( comp_real_a_a @ F3 @ F ) @ A )
=> ( inj_on_a_real @ F @ A ) ) ).
% inj_on_imageI2
thf(fact_516_inj__on__imageI2,axiom,
! [F3: real > real,F: a > real,A: set_a] :
( ( inj_on_a_real @ ( comp_real_real_a @ F3 @ F ) @ A )
=> ( inj_on_a_real @ F @ A ) ) ).
% inj_on_imageI2
thf(fact_517_inj__on__imageI2,axiom,
! [F3: real > real,F: real > real,A: set_real] :
( ( inj_on_real_real @ ( comp_real_real_real @ F3 @ F ) @ A )
=> ( inj_on_real_real @ F @ A ) ) ).
% inj_on_imageI2
thf(fact_518_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_519_bot__set__def,axiom,
( bot_bot_set_real
= ( collect_real @ bot_bot_real_o ) ) ).
% bot_set_def
thf(fact_520_singleton__inject,axiom,
! [A2: a,B2: a] :
( ( ( insert_a @ A2 @ bot_bot_set_a )
= ( insert_a @ B2 @ bot_bot_set_a ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_521_singleton__inject,axiom,
! [A2: real,B2: real] :
( ( ( insert_real @ A2 @ bot_bot_set_real )
= ( insert_real @ B2 @ bot_bot_set_real ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_522_insert__not__empty,axiom,
! [A2: a,A: set_a] :
( ( insert_a @ A2 @ A )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_523_insert__not__empty,axiom,
! [A2: real,A: set_real] :
( ( insert_real @ A2 @ A )
!= bot_bot_set_real ) ).
% insert_not_empty
thf(fact_524_doubleton__eq__iff,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ( insert_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_525_doubleton__eq__iff,axiom,
! [A2: real,B2: real,C: real,D: real] :
( ( ( insert_real @ A2 @ ( insert_real @ B2 @ bot_bot_set_real ) )
= ( insert_real @ C @ ( insert_real @ D @ bot_bot_set_real ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_526_singleton__iff,axiom,
! [B2: set_a,A2: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_527_singleton__iff,axiom,
! [B2: real > real,A2: real > real] :
( ( member_real_real @ B2 @ ( insert_real_real @ A2 @ bot_bo6767488733719836353l_real ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_528_singleton__iff,axiom,
! [B2: real > a,A2: real > a] :
( ( member_real_a @ B2 @ ( insert_real_a @ A2 @ bot_bot_set_real_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_529_singleton__iff,axiom,
! [B2: a > real,A2: a > real] :
( ( member_a_real @ B2 @ ( insert_a_real @ A2 @ bot_bot_set_a_real ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_530_singleton__iff,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_531_singleton__iff,axiom,
! [B2: real,A2: real] :
( ( member_real @ B2 @ ( insert_real @ A2 @ bot_bot_set_real ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_532_singletonD,axiom,
! [B2: set_a,A2: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_533_singletonD,axiom,
! [B2: real > real,A2: real > real] :
( ( member_real_real @ B2 @ ( insert_real_real @ A2 @ bot_bo6767488733719836353l_real ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_534_singletonD,axiom,
! [B2: real > a,A2: real > a] :
( ( member_real_a @ B2 @ ( insert_real_a @ A2 @ bot_bot_set_real_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_535_singletonD,axiom,
! [B2: a > real,A2: a > real] :
( ( member_a_real @ B2 @ ( insert_a_real @ A2 @ bot_bot_set_a_real ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_536_singletonD,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_537_singletonD,axiom,
! [B2: real,A2: real] :
( ( member_real @ B2 @ ( insert_real @ A2 @ bot_bot_set_real ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_538_Set_Oinsert__mono,axiom,
! [C2: set_real,D2: set_real,A2: real] :
( ( ord_less_eq_set_real @ C2 @ D2 )
=> ( ord_less_eq_set_real @ ( insert_real @ A2 @ C2 ) @ ( insert_real @ A2 @ D2 ) ) ) ).
% Set.insert_mono
thf(fact_539_Set_Oinsert__mono,axiom,
! [C2: set_a,D2: set_a,A2: a] :
( ( ord_less_eq_set_a @ C2 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A2 @ C2 ) @ ( insert_a @ A2 @ D2 ) ) ) ).
% Set.insert_mono
thf(fact_540_subset__insert,axiom,
! [X: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X @ B ) )
= ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_541_subset__insert,axiom,
! [X: real > real,A: set_real_real,B: set_real_real] :
( ~ ( member_real_real @ X @ A )
=> ( ( ord_le4198349162570665613l_real @ A @ ( insert_real_real @ X @ B ) )
= ( ord_le4198349162570665613l_real @ A @ B ) ) ) ).
% subset_insert
thf(fact_542_subset__insert,axiom,
! [X: real > a,A: set_real_a,B: set_real_a] :
( ~ ( member_real_a @ X @ A )
=> ( ( ord_le5743406823621094409real_a @ A @ ( insert_real_a @ X @ B ) )
= ( ord_le5743406823621094409real_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_543_subset__insert,axiom,
! [X: a > real,A: set_a_real,B: set_a_real] :
( ~ ( member_a_real @ X @ A )
=> ( ( ord_le3334967407727675675a_real @ A @ ( insert_a_real @ X @ B ) )
= ( ord_le3334967407727675675a_real @ A @ B ) ) ) ).
% subset_insert
thf(fact_544_subset__insert,axiom,
! [X: real,A: set_real,B: set_real] :
( ~ ( member_real @ X @ A )
=> ( ( ord_less_eq_set_real @ A @ ( insert_real @ X @ B ) )
= ( ord_less_eq_set_real @ A @ B ) ) ) ).
% subset_insert
thf(fact_545_subset__insert,axiom,
! [X: a,A: set_a,B: set_a] :
( ~ ( member_a @ X @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_546_subset__insertI,axiom,
! [B: set_real,A2: real] : ( ord_less_eq_set_real @ B @ ( insert_real @ A2 @ B ) ) ).
% subset_insertI
thf(fact_547_subset__insertI,axiom,
! [B: set_a,A2: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A2 @ B ) ) ).
% subset_insertI
thf(fact_548_subset__insertI2,axiom,
! [A: set_real,B: set_real,B2: real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ord_less_eq_set_real @ A @ ( insert_real @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_549_subset__insertI2,axiom,
! [A: set_a,B: set_a,B2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_550_inj__on__image__eq__iff,axiom,
! [F: real > a,C2: set_real,A: set_real,B: set_real] :
( ( inj_on_real_a @ F @ C2 )
=> ( ( ord_less_eq_set_real @ A @ C2 )
=> ( ( ord_less_eq_set_real @ B @ C2 )
=> ( ( ( image_real_a @ F @ A )
= ( image_real_a @ F @ B ) )
= ( A = B ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_551_inj__on__image__eq__iff,axiom,
! [F: real > real,C2: set_real,A: set_real,B: set_real] :
( ( inj_on_real_real @ F @ C2 )
=> ( ( ord_less_eq_set_real @ A @ C2 )
=> ( ( ord_less_eq_set_real @ B @ C2 )
=> ( ( ( image_real_real @ F @ A )
= ( image_real_real @ F @ B ) )
= ( A = B ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_552_inj__on__image__eq__iff,axiom,
! [F: a > a,C2: set_a,A: set_a,B: set_a] :
( ( inj_on_a_a @ F @ C2 )
=> ( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ( ( image_a_a @ F @ A )
= ( image_a_a @ F @ B ) )
= ( A = B ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_553_inj__on__image__eq__iff,axiom,
! [F: a > real,C2: set_a,A: set_a,B: set_a] :
( ( inj_on_a_real @ F @ C2 )
=> ( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ( ( image_a_real @ F @ A )
= ( image_a_real @ F @ B ) )
= ( A = B ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_554_inj__on__image__mem__iff,axiom,
! [F: real > real,B: set_real,A2: real,A: set_real] :
( ( inj_on_real_real @ F @ B )
=> ( ( member_real @ A2 @ B )
=> ( ( ord_less_eq_set_real @ A @ B )
=> ( ( member_real @ ( F @ A2 ) @ ( image_real_real @ F @ A ) )
= ( member_real @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_555_inj__on__image__mem__iff,axiom,
! [F: real > a,B: set_real,A2: real,A: set_real] :
( ( inj_on_real_a @ F @ B )
=> ( ( member_real @ A2 @ B )
=> ( ( ord_less_eq_set_real @ A @ B )
=> ( ( member_a @ ( F @ A2 ) @ ( image_real_a @ F @ A ) )
= ( member_real @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_556_inj__on__image__mem__iff,axiom,
! [F: a > real,B: set_a,A2: a,A: set_a] :
( ( inj_on_a_real @ F @ B )
=> ( ( member_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_real @ ( F @ A2 ) @ ( image_a_real @ F @ A ) )
= ( member_a @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_557_inj__on__image__mem__iff,axiom,
! [F: a > a,B: set_a,A2: a,A: set_a] :
( ( inj_on_a_a @ F @ B )
=> ( ( member_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ ( F @ A2 ) @ ( image_a_a @ F @ A ) )
= ( member_a @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_558_inj__on__image__mem__iff,axiom,
! [F: set_a > a,B: set_set_a,A2: set_a,A: set_set_a] :
( ( inj_on_set_a_a @ F @ B )
=> ( ( member_set_a @ A2 @ B )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_a @ ( F @ A2 ) @ ( image_set_a_a @ F @ A ) )
= ( member_set_a @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_559_inj__on__image__mem__iff,axiom,
! [F: real > set_a,B: set_real,A2: real,A: set_real] :
( ( inj_on_real_set_a @ F @ B )
=> ( ( member_real @ A2 @ B )
=> ( ( ord_less_eq_set_real @ A @ B )
=> ( ( member_set_a @ ( F @ A2 ) @ ( image_real_set_a @ F @ A ) )
= ( member_real @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_560_inj__on__image__mem__iff,axiom,
! [F: a > set_a,B: set_a,A2: a,A: set_a] :
( ( inj_on_a_set_a @ F @ B )
=> ( ( member_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_set_a @ ( F @ A2 ) @ ( image_a_set_a @ F @ A ) )
= ( member_a @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_561_inj__on__image__mem__iff,axiom,
! [F: set_a > set_a,B: set_set_a,A2: set_a,A: set_set_a] :
( ( inj_on_set_a_set_a @ F @ B )
=> ( ( member_set_a @ A2 @ B )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ ( F @ A2 ) @ ( image_set_a_set_a @ F @ A ) )
= ( member_set_a @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_562_inj__on__image__mem__iff,axiom,
! [F: ( real > real ) > a,B: set_real_real,A2: real > real,A: set_real_real] :
( ( inj_on_real_real_a @ F @ B )
=> ( ( member_real_real @ A2 @ B )
=> ( ( ord_le4198349162570665613l_real @ A @ B )
=> ( ( member_a @ ( F @ A2 ) @ ( image_real_real_a @ F @ A ) )
= ( member_real_real @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_563_inj__on__image__mem__iff,axiom,
! [F: ( real > a ) > a,B: set_real_a,A2: real > a,A: set_real_a] :
( ( inj_on_real_a_a @ F @ B )
=> ( ( member_real_a @ A2 @ B )
=> ( ( ord_le5743406823621094409real_a @ A @ B )
=> ( ( member_a @ ( F @ A2 ) @ ( image_real_a_a @ F @ A ) )
= ( member_real_a @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_564_comp__inj__on,axiom,
! [F: a > a,A: set_a,G: a > real] :
( ( inj_on_a_a @ F @ A )
=> ( ( inj_on_a_real @ G @ ( image_a_a @ F @ A ) )
=> ( inj_on_a_real @ ( comp_a_real_a @ G @ F ) @ A ) ) ) ).
% comp_inj_on
thf(fact_565_comp__inj__on,axiom,
! [F: real > a,A: set_real,G: a > real] :
( ( inj_on_real_a @ F @ A )
=> ( ( inj_on_a_real @ G @ ( image_real_a @ F @ A ) )
=> ( inj_on_real_real @ ( comp_a_real_real @ G @ F ) @ A ) ) ) ).
% comp_inj_on
thf(fact_566_comp__inj__on,axiom,
! [F: a > real,A: set_a,G: real > a] :
( ( inj_on_a_real @ F @ A )
=> ( ( inj_on_real_a @ G @ ( image_a_real @ F @ A ) )
=> ( inj_on_a_a @ ( comp_real_a_a @ G @ F ) @ A ) ) ) ).
% comp_inj_on
thf(fact_567_comp__inj__on,axiom,
! [F: a > real,A: set_a,G: real > real] :
( ( inj_on_a_real @ F @ A )
=> ( ( inj_on_real_real @ G @ ( image_a_real @ F @ A ) )
=> ( inj_on_a_real @ ( comp_real_real_a @ G @ F ) @ A ) ) ) ).
% comp_inj_on
thf(fact_568_comp__inj__on,axiom,
! [F: real > real,A: set_real,G: real > real] :
( ( inj_on_real_real @ F @ A )
=> ( ( inj_on_real_real @ G @ ( image_real_real @ F @ A ) )
=> ( inj_on_real_real @ ( comp_real_real_real @ G @ F ) @ A ) ) ) ).
% comp_inj_on
thf(fact_569_inj__on__imageI,axiom,
! [G: real > a,F: a > real,A: set_a] :
( ( inj_on_a_a @ ( comp_real_a_a @ G @ F ) @ A )
=> ( inj_on_real_a @ G @ ( image_a_real @ F @ A ) ) ) ).
% inj_on_imageI
thf(fact_570_inj__on__imageI,axiom,
! [G: a > real,F: a > a,A: set_a] :
( ( inj_on_a_real @ ( comp_a_real_a @ G @ F ) @ A )
=> ( inj_on_a_real @ G @ ( image_a_a @ F @ A ) ) ) ).
% inj_on_imageI
thf(fact_571_inj__on__imageI,axiom,
! [G: real > real,F: a > real,A: set_a] :
( ( inj_on_a_real @ ( comp_real_real_a @ G @ F ) @ A )
=> ( inj_on_real_real @ G @ ( image_a_real @ F @ A ) ) ) ).
% inj_on_imageI
thf(fact_572_inj__on__imageI,axiom,
! [G: a > real,F: real > a,A: set_real] :
( ( inj_on_real_real @ ( comp_a_real_real @ G @ F ) @ A )
=> ( inj_on_a_real @ G @ ( image_real_a @ F @ A ) ) ) ).
% inj_on_imageI
thf(fact_573_inj__on__imageI,axiom,
! [G: real > real,F: real > real,A: set_real] :
( ( inj_on_real_real @ ( comp_real_real_real @ G @ F ) @ A )
=> ( inj_on_real_real @ G @ ( image_real_real @ F @ A ) ) ) ).
% inj_on_imageI
thf(fact_574_comp__inj__on__iff,axiom,
! [F: a > a,A: set_a,F3: a > real] :
( ( inj_on_a_a @ F @ A )
=> ( ( inj_on_a_real @ F3 @ ( image_a_a @ F @ A ) )
= ( inj_on_a_real @ ( comp_a_real_a @ F3 @ F ) @ A ) ) ) ).
% comp_inj_on_iff
thf(fact_575_comp__inj__on__iff,axiom,
! [F: real > a,A: set_real,F3: a > real] :
( ( inj_on_real_a @ F @ A )
=> ( ( inj_on_a_real @ F3 @ ( image_real_a @ F @ A ) )
= ( inj_on_real_real @ ( comp_a_real_real @ F3 @ F ) @ A ) ) ) ).
% comp_inj_on_iff
thf(fact_576_comp__inj__on__iff,axiom,
! [F: a > real,A: set_a,F3: real > a] :
( ( inj_on_a_real @ F @ A )
=> ( ( inj_on_real_a @ F3 @ ( image_a_real @ F @ A ) )
= ( inj_on_a_a @ ( comp_real_a_a @ F3 @ F ) @ A ) ) ) ).
% comp_inj_on_iff
thf(fact_577_comp__inj__on__iff,axiom,
! [F: a > real,A: set_a,F3: real > real] :
( ( inj_on_a_real @ F @ A )
=> ( ( inj_on_real_real @ F3 @ ( image_a_real @ F @ A ) )
= ( inj_on_a_real @ ( comp_real_real_a @ F3 @ F ) @ A ) ) ) ).
% comp_inj_on_iff
thf(fact_578_comp__inj__on__iff,axiom,
! [F: real > real,A: set_real,F3: real > real] :
( ( inj_on_real_real @ F @ A )
=> ( ( inj_on_real_real @ F3 @ ( image_real_real @ F @ A ) )
= ( inj_on_real_real @ ( comp_real_real_real @ F3 @ F ) @ A ) ) ) ).
% comp_inj_on_iff
thf(fact_579_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_580_space__empty__eq__bot,axiom,
! [A2: sigma_measure_real] :
( ( ( sigma_space_real @ A2 )
= bot_bot_set_real )
= ( A2 = bot_bo5982154664989874033e_real ) ) ).
% space_empty_eq_bot
thf(fact_581_sets_Oinsert__in__sets,axiom,
! [X: a,M: sigma_measure_a,A: set_a] :
( ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( insert_a @ X @ A ) @ ( sigma_sets_a @ M ) ) ) ) ).
% sets.insert_in_sets
thf(fact_582_sets_Oinsert__in__sets,axiom,
! [X: real,M: sigma_measure_real,A: set_real] :
( ( member_set_real @ ( insert_real @ X @ bot_bot_set_real ) @ ( sigma_sets_real @ M ) )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ M ) )
=> ( member_set_real @ ( insert_real @ X @ A ) @ ( sigma_sets_real @ M ) ) ) ) ).
% sets.insert_in_sets
thf(fact_583_subset__singleton__iff,axiom,
! [X6: set_real,A2: real] :
( ( ord_less_eq_set_real @ X6 @ ( insert_real @ A2 @ bot_bot_set_real ) )
= ( ( X6 = bot_bot_set_real )
| ( X6
= ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ).
% subset_singleton_iff
thf(fact_584_subset__singleton__iff,axiom,
! [X6: set_a,A2: a] :
( ( ord_less_eq_set_a @ X6 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( ( X6 = bot_bot_set_a )
| ( X6
= ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_585_subset__singletonD,axiom,
! [A: set_real,X: real] :
( ( ord_less_eq_set_real @ A @ ( insert_real @ X @ bot_bot_set_real ) )
=> ( ( A = bot_bot_set_real )
| ( A
= ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).
% subset_singletonD
thf(fact_586_subset__singletonD,axiom,
! [A: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A = bot_bot_set_a )
| ( A
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_587_vimage__singleton__eq,axiom,
! [A2: real,F: real > a,B2: a] :
( ( member_real @ A2 @ ( vimage_real_a @ F @ ( insert_a @ B2 @ bot_bot_set_a ) ) )
= ( ( F @ A2 )
= B2 ) ) ).
% vimage_singleton_eq
thf(fact_588_vimage__singleton__eq,axiom,
! [A2: a,F: a > a,B2: a] :
( ( member_a @ A2 @ ( vimage_a_a @ F @ ( insert_a @ B2 @ bot_bot_set_a ) ) )
= ( ( F @ A2 )
= B2 ) ) ).
% vimage_singleton_eq
thf(fact_589_vimage__singleton__eq,axiom,
! [A2: real,F: real > real,B2: real] :
( ( member_real @ A2 @ ( vimage_real_real @ F @ ( insert_real @ B2 @ bot_bot_set_real ) ) )
= ( ( F @ A2 )
= B2 ) ) ).
% vimage_singleton_eq
thf(fact_590_vimage__singleton__eq,axiom,
! [A2: a,F: a > real,B2: real] :
( ( member_a @ A2 @ ( vimage_a_real @ F @ ( insert_real @ B2 @ bot_bot_set_real ) ) )
= ( ( F @ A2 )
= B2 ) ) ).
% vimage_singleton_eq
thf(fact_591_vimage__singleton__eq,axiom,
! [A2: set_a,F: set_a > a,B2: a] :
( ( member_set_a @ A2 @ ( vimage_set_a_a @ F @ ( insert_a @ B2 @ bot_bot_set_a ) ) )
= ( ( F @ A2 )
= B2 ) ) ).
% vimage_singleton_eq
thf(fact_592_vimage__singleton__eq,axiom,
! [A2: set_a,F: set_a > real,B2: real] :
( ( member_set_a @ A2 @ ( vimage_set_a_real @ F @ ( insert_real @ B2 @ bot_bot_set_real ) ) )
= ( ( F @ A2 )
= B2 ) ) ).
% vimage_singleton_eq
thf(fact_593_vimage__singleton__eq,axiom,
! [A2: real > real,F: ( real > real ) > a,B2: a] :
( ( member_real_real @ A2 @ ( vimage_real_real_a @ F @ ( insert_a @ B2 @ bot_bot_set_a ) ) )
= ( ( F @ A2 )
= B2 ) ) ).
% vimage_singleton_eq
thf(fact_594_vimage__singleton__eq,axiom,
! [A2: real > a,F: ( real > a ) > a,B2: a] :
( ( member_real_a @ A2 @ ( vimage_real_a_a @ F @ ( insert_a @ B2 @ bot_bot_set_a ) ) )
= ( ( F @ A2 )
= B2 ) ) ).
% vimage_singleton_eq
thf(fact_595_vimage__singleton__eq,axiom,
! [A2: a > real,F: ( a > real ) > a,B2: a] :
( ( member_a_real @ A2 @ ( vimage_a_real_a @ F @ ( insert_a @ B2 @ bot_bot_set_a ) ) )
= ( ( F @ A2 )
= B2 ) ) ).
% vimage_singleton_eq
thf(fact_596_vimage__singleton__eq,axiom,
! [A2: real > real,F: ( real > real ) > real,B2: real] :
( ( member_real_real @ A2 @ ( vimage1639110810821759712l_real @ F @ ( insert_real @ B2 @ bot_bot_set_real ) ) )
= ( ( F @ A2 )
= B2 ) ) ).
% vimage_singleton_eq
thf(fact_597_sets__eq__iff__bounded,axiom,
! [A: sigma_measure_a,B: sigma_measure_a,C2: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A @ B )
=> ( ( ord_le254669795585780187sure_a @ B @ C2 )
=> ( ( ( sigma_sets_a @ A )
= ( sigma_sets_a @ C2 ) )
=> ( ( sigma_sets_a @ B )
= ( sigma_sets_a @ A ) ) ) ) ) ).
% sets_eq_iff_bounded
thf(fact_598_sets__eq__iff__bounded,axiom,
! [A: sigma_measure_real,B: sigma_measure_real,C2: sigma_measure_real] :
( ( ord_le487379304121309861e_real @ A @ B )
=> ( ( ord_le487379304121309861e_real @ B @ C2 )
=> ( ( ( sigma_sets_real @ A )
= ( sigma_sets_real @ C2 ) )
=> ( ( sigma_sets_real @ B )
= ( sigma_sets_real @ A ) ) ) ) ) ).
% sets_eq_iff_bounded
thf(fact_599_comp__eq__dest__lhs,axiom,
! [A2: real > a,B2: a > real,C: a > a,V: a] :
( ( ( comp_real_a_a @ A2 @ B2 )
= C )
=> ( ( A2 @ ( B2 @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_600_comp__eq__dest__lhs,axiom,
! [A2: real > real,B2: real > real,C: real > real,V: real] :
( ( ( comp_real_real_real @ A2 @ B2 )
= C )
=> ( ( A2 @ ( B2 @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_601_comp__eq__elim,axiom,
! [A2: real > a,B2: a > real,C: real > a,D: a > real] :
( ( ( comp_real_a_a @ A2 @ B2 )
= ( comp_real_a_a @ C @ D ) )
=> ! [V2: a] :
( ( A2 @ ( B2 @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_602_comp__eq__elim,axiom,
! [A2: real > real,B2: real > real,C: real > real,D: real > real] :
( ( ( comp_real_real_real @ A2 @ B2 )
= ( comp_real_real_real @ C @ D ) )
=> ! [V2: real] :
( ( A2 @ ( B2 @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_603_comp__eq__dest,axiom,
! [A2: real > a,B2: a > real,C: real > a,D: a > real,V: a] :
( ( ( comp_real_a_a @ A2 @ B2 )
= ( comp_real_a_a @ C @ D ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_604_comp__eq__dest,axiom,
! [A2: real > real,B2: real > real,C: real > real,D: real > real,V: real] :
( ( ( comp_real_real_real @ A2 @ B2 )
= ( comp_real_real_real @ C @ D ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_605_comp__assoc,axiom,
! [F: real > a,G: a > real,H: a > a] :
( ( comp_a_a_a @ ( comp_real_a_a @ F @ G ) @ H )
= ( comp_real_a_a @ F @ ( comp_a_real_a @ G @ H ) ) ) ).
% comp_assoc
thf(fact_606_comp__assoc,axiom,
! [F: a > a,G: real > a,H: a > real] :
( ( comp_real_a_a @ ( comp_a_a_real @ F @ G ) @ H )
= ( comp_a_a_a @ F @ ( comp_real_a_a @ G @ H ) ) ) ).
% comp_assoc
thf(fact_607_comp__assoc,axiom,
! [F: real > a,G: real > real,H: a > real] :
( ( comp_real_a_a @ ( comp_real_a_real @ F @ G ) @ H )
= ( comp_real_a_a @ F @ ( comp_real_real_a @ G @ H ) ) ) ).
% comp_assoc
thf(fact_608_comp__assoc,axiom,
! [F: real > real,G: real > real,H: real > real] :
( ( comp_real_real_real @ ( comp_real_real_real @ F @ G ) @ H )
= ( comp_real_real_real @ F @ ( comp_real_real_real @ G @ H ) ) ) ).
% comp_assoc
thf(fact_609_comp__def,axiom,
( comp_real_a_a
= ( ^ [F2: real > a,G2: a > real,X3: a] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% comp_def
thf(fact_610_comp__def,axiom,
( comp_real_real_real
= ( ^ [F2: real > real,G2: real > real,X3: real] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% comp_def
thf(fact_611_standard__borel_Of__inj,axiom,
! [M: sigma_measure_a] :
( ( standard_borel_a @ M )
=> ( inj_on_a_real @ ( standard_f_a @ M ) @ ( sigma_space_a @ M ) ) ) ).
% standard_borel.f_inj
thf(fact_612_standard__borel_Of__inj,axiom,
! [M: sigma_measure_real] :
( ( standard_borel_real @ M )
=> ( inj_on_real_real @ ( standard_f_real @ M ) @ ( sigma_space_real @ M ) ) ) ).
% standard_borel.f_inj
thf(fact_613_standard__borel_Osingleton__sets,axiom,
! [M: sigma_measure_set_a,X: set_a] :
( ( standard_borel_set_a @ M )
=> ( ( member_set_a @ X @ ( sigma_space_set_a @ M ) )
=> ( member_set_set_a @ ( insert_set_a @ X @ bot_bot_set_set_a ) @ ( sigma_sets_set_a @ M ) ) ) ) ).
% standard_borel.singleton_sets
thf(fact_614_standard__borel_Osingleton__sets,axiom,
! [M: sigma_4258434043392614480l_real,X: real > real] :
( ( standa6209843396170762084l_real @ M )
=> ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
=> ( member_set_real_real @ ( insert_real_real @ X @ bot_bo6767488733719836353l_real ) @ ( sigma_sets_real_real @ M ) ) ) ) ).
% standard_borel.singleton_sets
thf(fact_615_standard__borel_Osingleton__sets,axiom,
! [M: sigma_measure_real_a,X: real > a] :
( ( standa4991623935272378204real_a @ M )
=> ( ( member_real_a @ X @ ( sigma_space_real_a @ M ) )
=> ( member_set_real_a @ ( insert_real_a @ X @ bot_bot_set_real_a ) @ ( sigma_sets_real_a @ M ) ) ) ) ).
% standard_borel.singleton_sets
thf(fact_616_standard__borel_Osingleton__sets,axiom,
! [M: sigma_measure_a_real,X: a > real] :
( ( standa4474357433005041006a_real @ M )
=> ( ( member_a_real @ X @ ( sigma_space_a_real @ M ) )
=> ( member_set_a_real2 @ ( insert_a_real @ X @ bot_bot_set_a_real ) @ ( sigma_sets_a_real @ M ) ) ) ) ).
% standard_borel.singleton_sets
thf(fact_617_standard__borel_Osingleton__sets,axiom,
! [M: sigma_measure_a,X: a] :
( ( standard_borel_a @ M )
=> ( ( member_a @ X @ ( sigma_space_a @ M ) )
=> ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ ( sigma_sets_a @ M ) ) ) ) ).
% standard_borel.singleton_sets
thf(fact_618_standard__borel_Osingleton__sets,axiom,
! [M: sigma_measure_real,X: real] :
( ( standard_borel_real @ M )
=> ( ( member_real @ X @ ( sigma_space_real @ M ) )
=> ( member_set_real @ ( insert_real @ X @ bot_bot_set_real ) @ ( sigma_sets_real @ M ) ) ) ) ).
% standard_borel.singleton_sets
thf(fact_619_le__measureD1,axiom,
! [A: sigma_measure_real,B: sigma_measure_real] :
( ( ord_le487379304121309861e_real @ A @ B )
=> ( ord_less_eq_set_real @ ( sigma_space_real @ A ) @ ( sigma_space_real @ B ) ) ) ).
% le_measureD1
thf(fact_620_le__measureD1,axiom,
! [A: sigma_measure_a,B: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A @ B )
=> ( ord_less_eq_set_a @ ( sigma_space_a @ A ) @ ( sigma_space_a @ B ) ) ) ).
% le_measureD1
thf(fact_621_image__comp,axiom,
! [F: a > real,G: real > a,R: set_real] :
( ( image_a_real @ F @ ( image_real_a @ G @ R ) )
= ( image_real_real @ ( comp_a_real_real @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_622_image__comp,axiom,
! [F: a > real,G: a > a,R: set_a] :
( ( image_a_real @ F @ ( image_a_a @ G @ R ) )
= ( image_a_real @ ( comp_a_real_a @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_623_image__comp,axiom,
! [F: real > real,G: a > real,R: set_a] :
( ( image_real_real @ F @ ( image_a_real @ G @ R ) )
= ( image_a_real @ ( comp_real_real_a @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_624_image__comp,axiom,
! [F: real > real,G: real > real,R: set_real] :
( ( image_real_real @ F @ ( image_real_real @ G @ R ) )
= ( image_real_real @ ( comp_real_real_real @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_625_image__comp,axiom,
! [F: real > a,G: a > real,R: set_a] :
( ( image_real_a @ F @ ( image_a_real @ G @ R ) )
= ( image_a_a @ ( comp_real_a_a @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_626_image__comp,axiom,
! [F: real > a,G: real > real,R: set_real] :
( ( image_real_a @ F @ ( image_real_real @ G @ R ) )
= ( image_real_a @ ( comp_real_a_real @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_627_image__comp,axiom,
! [F: a > a,G: real > a,R: set_real] :
( ( image_a_a @ F @ ( image_real_a @ G @ R ) )
= ( image_real_a @ ( comp_a_a_real @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_628_image__comp,axiom,
! [F: a > a,G: a > a,R: set_a] :
( ( image_a_a @ F @ ( image_a_a @ G @ R ) )
= ( image_a_a @ ( comp_a_a_a @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_629_image__eq__imp__comp,axiom,
! [F: a > real,A: set_a,G: a > real,B: set_a,H: real > real] :
( ( ( image_a_real @ F @ A )
= ( image_a_real @ G @ B ) )
=> ( ( image_a_real @ ( comp_real_real_a @ H @ F ) @ A )
= ( image_a_real @ ( comp_real_real_a @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_630_image__eq__imp__comp,axiom,
! [F: a > real,A: set_a,G: a > real,B: set_a,H: real > a] :
( ( ( image_a_real @ F @ A )
= ( image_a_real @ G @ B ) )
=> ( ( image_a_a @ ( comp_real_a_a @ H @ F ) @ A )
= ( image_a_a @ ( comp_real_a_a @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_631_image__eq__imp__comp,axiom,
! [F: a > real,A: set_a,G: real > real,B: set_real,H: real > real] :
( ( ( image_a_real @ F @ A )
= ( image_real_real @ G @ B ) )
=> ( ( image_a_real @ ( comp_real_real_a @ H @ F ) @ A )
= ( image_real_real @ ( comp_real_real_real @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_632_image__eq__imp__comp,axiom,
! [F: a > real,A: set_a,G: real > real,B: set_real,H: real > a] :
( ( ( image_a_real @ F @ A )
= ( image_real_real @ G @ B ) )
=> ( ( image_a_a @ ( comp_real_a_a @ H @ F ) @ A )
= ( image_real_a @ ( comp_real_a_real @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_633_image__eq__imp__comp,axiom,
! [F: real > real,A: set_real,G: a > real,B: set_a,H: real > real] :
( ( ( image_real_real @ F @ A )
= ( image_a_real @ G @ B ) )
=> ( ( image_real_real @ ( comp_real_real_real @ H @ F ) @ A )
= ( image_a_real @ ( comp_real_real_a @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_634_image__eq__imp__comp,axiom,
! [F: real > real,A: set_real,G: a > real,B: set_a,H: real > a] :
( ( ( image_real_real @ F @ A )
= ( image_a_real @ G @ B ) )
=> ( ( image_real_a @ ( comp_real_a_real @ H @ F ) @ A )
= ( image_a_a @ ( comp_real_a_a @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_635_image__eq__imp__comp,axiom,
! [F: real > real,A: set_real,G: real > real,B: set_real,H: real > real] :
( ( ( image_real_real @ F @ A )
= ( image_real_real @ G @ B ) )
=> ( ( image_real_real @ ( comp_real_real_real @ H @ F ) @ A )
= ( image_real_real @ ( comp_real_real_real @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_636_image__eq__imp__comp,axiom,
! [F: real > real,A: set_real,G: real > real,B: set_real,H: real > a] :
( ( ( image_real_real @ F @ A )
= ( image_real_real @ G @ B ) )
=> ( ( image_real_a @ ( comp_real_a_real @ H @ F ) @ A )
= ( image_real_a @ ( comp_real_a_real @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_637_image__eq__imp__comp,axiom,
! [F: real > a,A: set_real,G: real > a,B: set_real,H: a > real] :
( ( ( image_real_a @ F @ A )
= ( image_real_a @ G @ B ) )
=> ( ( image_real_real @ ( comp_a_real_real @ H @ F ) @ A )
= ( image_real_real @ ( comp_a_real_real @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_638_image__eq__imp__comp,axiom,
! [F: real > a,A: set_real,G: real > a,B: set_real,H: a > a] :
( ( ( image_real_a @ F @ A )
= ( image_real_a @ G @ B ) )
=> ( ( image_real_a @ ( comp_a_a_real @ H @ F ) @ A )
= ( image_real_a @ ( comp_a_a_real @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_639_sets__eq__imp__space__eq,axiom,
! [M: sigma_measure_a,M2: sigma_measure_a] :
( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ M2 ) )
=> ( ( sigma_space_a @ M )
= ( sigma_space_a @ M2 ) ) ) ).
% sets_eq_imp_space_eq
thf(fact_640_sets__eq__imp__space__eq,axiom,
! [M: sigma_measure_real,M2: sigma_measure_real] :
( ( ( sigma_sets_real @ M )
= ( sigma_sets_real @ M2 ) )
=> ( ( sigma_space_real @ M )
= ( sigma_space_real @ M2 ) ) ) ).
% sets_eq_imp_space_eq
thf(fact_641_inj__on__iff__surj,axiom,
! [A: set_a,A7: set_real] :
( ( A != bot_bot_set_a )
=> ( ( ? [F2: a > real] :
( ( inj_on_a_real @ F2 @ A )
& ( ord_less_eq_set_real @ ( image_a_real @ F2 @ A ) @ A7 ) ) )
= ( ? [G2: real > a] :
( ( image_real_a @ G2 @ A7 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_642_inj__on__iff__surj,axiom,
! [A: set_real,A7: set_real] :
( ( A != bot_bot_set_real )
=> ( ( ? [F2: real > real] :
( ( inj_on_real_real @ F2 @ A )
& ( ord_less_eq_set_real @ ( image_real_real @ F2 @ A ) @ A7 ) ) )
= ( ? [G2: real > real] :
( ( image_real_real @ G2 @ A7 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_643_inj__on__iff__surj,axiom,
! [A: set_a,A7: set_a] :
( ( A != bot_bot_set_a )
=> ( ( ? [F2: a > a] :
( ( inj_on_a_a @ F2 @ A )
& ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A ) @ A7 ) ) )
= ( ? [G2: a > a] :
( ( image_a_a @ G2 @ A7 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_644_inj__on__iff__surj,axiom,
! [A: set_real,A7: set_a] :
( ( A != bot_bot_set_real )
=> ( ( ? [F2: real > a] :
( ( inj_on_real_a @ F2 @ A )
& ( ord_less_eq_set_a @ ( image_real_a @ F2 @ A ) @ A7 ) ) )
= ( ? [G2: a > real] :
( ( image_a_real @ G2 @ A7 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_645_f__meas,axiom,
member_a_real @ ( standard_f_a @ m ) @ ( sigma_9116425665531756122a_real @ m @ borel_5078946678739801102l_real ) ).
% f_meas
thf(fact_646_g__meas,axiom,
member_real_a @ ( standard_g_a @ m ) @ ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ m ) ).
% g_meas
thf(fact_647_borel__singleton,axiom,
! [A: set_real,X: real] :
( ( member_set_real @ A @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
=> ( member_set_real @ ( insert_real @ X @ A ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ).
% borel_singleton
thf(fact_648_subset__image__inj,axiom,
! [S: set_real,F: real > real,T2: set_real] :
( ( ord_less_eq_set_real @ S @ ( image_real_real @ F @ T2 ) )
= ( ? [U: set_real] :
( ( ord_less_eq_set_real @ U @ T2 )
& ( inj_on_real_real @ F @ U )
& ( S
= ( image_real_real @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_649_subset__image__inj,axiom,
! [S: set_real,F: a > real,T2: set_a] :
( ( ord_less_eq_set_real @ S @ ( image_a_real @ F @ T2 ) )
= ( ? [U: set_a] :
( ( ord_less_eq_set_a @ U @ T2 )
& ( inj_on_a_real @ F @ U )
& ( S
= ( image_a_real @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_650_subset__image__inj,axiom,
! [S: set_a,F: real > a,T2: set_real] :
( ( ord_less_eq_set_a @ S @ ( image_real_a @ F @ T2 ) )
= ( ? [U: set_real] :
( ( ord_less_eq_set_real @ U @ T2 )
& ( inj_on_real_a @ F @ U )
& ( S
= ( image_real_a @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_651_subset__image__inj,axiom,
! [S: set_a,F: a > a,T2: set_a] :
( ( ord_less_eq_set_a @ S @ ( image_a_a @ F @ T2 ) )
= ( ? [U: set_a] :
( ( ord_less_eq_set_a @ U @ T2 )
& ( inj_on_a_a @ F @ U )
& ( S
= ( image_a_a @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_652_ex__subset__image__inj,axiom,
! [F: real > real,S: set_real,P: set_real > $o] :
( ( ? [T3: set_real] :
( ( ord_less_eq_set_real @ T3 @ ( image_real_real @ F @ S ) )
& ( P @ T3 ) ) )
= ( ? [T3: set_real] :
( ( ord_less_eq_set_real @ T3 @ S )
& ( inj_on_real_real @ F @ T3 )
& ( P @ ( image_real_real @ F @ T3 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_653_ex__subset__image__inj,axiom,
! [F: a > real,S: set_a,P: set_real > $o] :
( ( ? [T3: set_real] :
( ( ord_less_eq_set_real @ T3 @ ( image_a_real @ F @ S ) )
& ( P @ T3 ) ) )
= ( ? [T3: set_a] :
( ( ord_less_eq_set_a @ T3 @ S )
& ( inj_on_a_real @ F @ T3 )
& ( P @ ( image_a_real @ F @ T3 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_654_ex__subset__image__inj,axiom,
! [F: real > a,S: set_real,P: set_a > $o] :
( ( ? [T3: set_a] :
( ( ord_less_eq_set_a @ T3 @ ( image_real_a @ F @ S ) )
& ( P @ T3 ) ) )
= ( ? [T3: set_real] :
( ( ord_less_eq_set_real @ T3 @ S )
& ( inj_on_real_a @ F @ T3 )
& ( P @ ( image_real_a @ F @ T3 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_655_ex__subset__image__inj,axiom,
! [F: a > a,S: set_a,P: set_a > $o] :
( ( ? [T3: set_a] :
( ( ord_less_eq_set_a @ T3 @ ( image_a_a @ F @ S ) )
& ( P @ T3 ) ) )
= ( ? [T3: set_a] :
( ( ord_less_eq_set_a @ T3 @ S )
& ( inj_on_a_a @ F @ T3 )
& ( P @ ( image_a_a @ F @ T3 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_656_all__subset__image__inj,axiom,
! [F: real > real,S: set_real,P: set_real > $o] :
( ( ! [T3: set_real] :
( ( ord_less_eq_set_real @ T3 @ ( image_real_real @ F @ S ) )
=> ( P @ T3 ) ) )
= ( ! [T3: set_real] :
( ( ( ord_less_eq_set_real @ T3 @ S )
& ( inj_on_real_real @ F @ T3 ) )
=> ( P @ ( image_real_real @ F @ T3 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_657_all__subset__image__inj,axiom,
! [F: a > real,S: set_a,P: set_real > $o] :
( ( ! [T3: set_real] :
( ( ord_less_eq_set_real @ T3 @ ( image_a_real @ F @ S ) )
=> ( P @ T3 ) ) )
= ( ! [T3: set_a] :
( ( ( ord_less_eq_set_a @ T3 @ S )
& ( inj_on_a_real @ F @ T3 ) )
=> ( P @ ( image_a_real @ F @ T3 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_658_all__subset__image__inj,axiom,
! [F: real > a,S: set_real,P: set_a > $o] :
( ( ! [T3: set_a] :
( ( ord_less_eq_set_a @ T3 @ ( image_real_a @ F @ S ) )
=> ( P @ T3 ) ) )
= ( ! [T3: set_real] :
( ( ( ord_less_eq_set_real @ T3 @ S )
& ( inj_on_real_a @ F @ T3 ) )
=> ( P @ ( image_real_a @ F @ T3 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_659_all__subset__image__inj,axiom,
! [F: a > a,S: set_a,P: set_a > $o] :
( ( ! [T3: set_a] :
( ( ord_less_eq_set_a @ T3 @ ( image_a_a @ F @ S ) )
=> ( P @ T3 ) ) )
= ( ! [T3: set_a] :
( ( ( ord_less_eq_set_a @ T3 @ S )
& ( inj_on_a_a @ F @ T3 ) )
=> ( P @ ( image_a_a @ F @ T3 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_660_inj__on__image__subset__iff,axiom,
! [F: real > real,C2: set_real,A: set_real,B: set_real] :
( ( inj_on_real_real @ F @ C2 )
=> ( ( ord_less_eq_set_real @ A @ C2 )
=> ( ( ord_less_eq_set_real @ B @ C2 )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ F @ A ) @ ( image_real_real @ F @ B ) )
= ( ord_less_eq_set_real @ A @ B ) ) ) ) ) ).
% inj_on_image_subset_iff
thf(fact_661_inj__on__image__subset__iff,axiom,
! [F: real > a,C2: set_real,A: set_real,B: set_real] :
( ( inj_on_real_a @ F @ C2 )
=> ( ( ord_less_eq_set_real @ A @ C2 )
=> ( ( ord_less_eq_set_real @ B @ C2 )
=> ( ( ord_less_eq_set_a @ ( image_real_a @ F @ A ) @ ( image_real_a @ F @ B ) )
= ( ord_less_eq_set_real @ A @ B ) ) ) ) ) ).
% inj_on_image_subset_iff
thf(fact_662_inj__on__image__subset__iff,axiom,
! [F: a > real,C2: set_a,A: set_a,B: set_a] :
( ( inj_on_a_real @ F @ C2 )
=> ( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ( ord_less_eq_set_real @ ( image_a_real @ F @ A ) @ ( image_a_real @ F @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ) ) ).
% inj_on_image_subset_iff
thf(fact_663_inj__on__image__subset__iff,axiom,
! [F: a > a,C2: set_a,A: set_a,B: set_a] :
( ( inj_on_a_a @ F @ C2 )
=> ( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ) ) ).
% inj_on_image_subset_iff
thf(fact_664_ball__insert,axiom,
! [A2: a,B: set_a,P: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( insert_a @ A2 @ B ) )
=> ( P @ X3 ) ) )
= ( ( P @ A2 )
& ! [X3: a] :
( ( member_a @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_insert
thf(fact_665_ball__insert,axiom,
! [A2: real,B: set_real,P: real > $o] :
( ( ! [X3: real] :
( ( member_real @ X3 @ ( insert_real @ A2 @ B ) )
=> ( P @ X3 ) ) )
= ( ( P @ A2 )
& ! [X3: real] :
( ( member_real @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_insert
thf(fact_666_exist__fg,axiom,
? [X2: a > real] :
( ( member_a_real @ X2 @ ( sigma_9116425665531756122a_real @ m @ borel_5078946678739801102l_real ) )
& ? [Xa: real > a] :
( ( member_real_a @ Xa @ ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ m ) )
& ! [Xb: a] :
( ( member_a @ Xb @ ( sigma_space_a @ m ) )
=> ( ( comp_real_a_a @ Xa @ X2 @ Xb )
= Xb ) ) ) ) ).
% exist_fg
thf(fact_667_sets__bot,axiom,
( ( sigma_sets_a @ bot_bo2108912051383640591sure_a )
= ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) ) ).
% sets_bot
thf(fact_668_sets__bot,axiom,
( ( sigma_sets_real @ bot_bo5982154664989874033e_real )
= ( insert_set_real @ bot_bot_set_real @ bot_bot_set_set_real ) ) ).
% sets_bot
thf(fact_669_real_Of__inj,axiom,
inj_on_real_real @ ( standard_f_real @ borel_5078946678739801102l_real ) @ ( sigma_space_real @ borel_5078946678739801102l_real ) ).
% real.f_inj
thf(fact_670_measurable__cong__sets,axiom,
! [M: sigma_measure_a,M2: sigma_measure_a,N: sigma_measure_a,N2: sigma_measure_a] :
( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ M2 ) )
=> ( ( ( sigma_sets_a @ N )
= ( sigma_sets_a @ N2 ) )
=> ( ( sigma_measurable_a_a @ M @ N )
= ( sigma_measurable_a_a @ M2 @ N2 ) ) ) ) ).
% measurable_cong_sets
thf(fact_671_measurable__cong__sets,axiom,
! [M: sigma_measure_a,M2: sigma_measure_a,N: sigma_measure_real,N2: sigma_measure_real] :
( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ M2 ) )
=> ( ( ( sigma_sets_real @ N )
= ( sigma_sets_real @ N2 ) )
=> ( ( sigma_9116425665531756122a_real @ M @ N )
= ( sigma_9116425665531756122a_real @ M2 @ N2 ) ) ) ) ).
% measurable_cong_sets
thf(fact_672_measurable__cong__sets,axiom,
! [M: sigma_measure_real,M2: sigma_measure_real,N: sigma_measure_a,N2: sigma_measure_a] :
( ( ( sigma_sets_real @ M )
= ( sigma_sets_real @ M2 ) )
=> ( ( ( sigma_sets_a @ N )
= ( sigma_sets_a @ N2 ) )
=> ( ( sigma_523072396149930112real_a @ M @ N )
= ( sigma_523072396149930112real_a @ M2 @ N2 ) ) ) ) ).
% measurable_cong_sets
thf(fact_673_measurable__cong__sets,axiom,
! [M: sigma_measure_real,M2: sigma_measure_real,N: sigma_measure_real,N2: sigma_measure_real] :
( ( ( sigma_sets_real @ M )
= ( sigma_sets_real @ M2 ) )
=> ( ( ( sigma_sets_real @ N )
= ( sigma_sets_real @ N2 ) )
=> ( ( sigma_5267869275261027754l_real @ M @ N )
= ( sigma_5267869275261027754l_real @ M2 @ N2 ) ) ) ) ).
% measurable_cong_sets
thf(fact_674_measurable__cong,axiom,
! [M: sigma_measure_a,F: a > real,G: a > real,M2: sigma_measure_real] :
( ! [W: a] :
( ( member_a @ W @ ( sigma_space_a @ M ) )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ M2 ) )
= ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ M @ M2 ) ) ) ) ).
% measurable_cong
thf(fact_675_measurable__cong,axiom,
! [M: sigma_measure_real,F: real > a,G: real > a,M2: sigma_measure_a] :
( ! [W: real] :
( ( member_real @ W @ ( sigma_space_real @ M ) )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ M2 ) )
= ( member_real_a @ G @ ( sigma_523072396149930112real_a @ M @ M2 ) ) ) ) ).
% measurable_cong
thf(fact_676_measurable__cong,axiom,
! [M: sigma_measure_real,F: real > real,G: real > real,M2: sigma_measure_real] :
( ! [W: real] :
( ( member_real @ W @ ( sigma_space_real @ M ) )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ M2 ) )
= ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ M2 ) ) ) ) ).
% measurable_cong
thf(fact_677_measurable__space,axiom,
! [F: a > a,M: sigma_measure_a,A: sigma_measure_a,X: a] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ A ) )
=> ( ( member_a @ X @ ( sigma_space_a @ M ) )
=> ( member_a @ ( F @ X ) @ ( sigma_space_a @ A ) ) ) ) ).
% measurable_space
thf(fact_678_measurable__space,axiom,
! [F: a > real,M: sigma_measure_a,A: sigma_measure_real,X: a] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ A ) )
=> ( ( member_a @ X @ ( sigma_space_a @ M ) )
=> ( member_real @ ( F @ X ) @ ( sigma_space_real @ A ) ) ) ) ).
% measurable_space
thf(fact_679_measurable__space,axiom,
! [F: real > a,M: sigma_measure_real,A: sigma_measure_a,X: real] :
( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ A ) )
=> ( ( member_real @ X @ ( sigma_space_real @ M ) )
=> ( member_a @ ( F @ X ) @ ( sigma_space_a @ A ) ) ) ) ).
% measurable_space
thf(fact_680_measurable__space,axiom,
! [F: real > real,M: sigma_measure_real,A: sigma_measure_real,X: real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ A ) )
=> ( ( member_real @ X @ ( sigma_space_real @ M ) )
=> ( member_real @ ( F @ X ) @ ( sigma_space_real @ A ) ) ) ) ).
% measurable_space
thf(fact_681_measurable__space,axiom,
! [F: set_a > a,M: sigma_measure_set_a,A: sigma_measure_a,X: set_a] :
( ( member_set_a_a @ F @ ( sigma_3901645225212141168et_a_a @ M @ A ) )
=> ( ( member_set_a @ X @ ( sigma_space_set_a @ M ) )
=> ( member_a @ ( F @ X ) @ ( sigma_space_a @ A ) ) ) ) ).
% measurable_space
thf(fact_682_measurable__space,axiom,
! [F: set_a > real,M: sigma_measure_set_a,A: sigma_measure_real,X: set_a] :
( ( member_set_a_real @ F @ ( sigma_567508090560183226a_real @ M @ A ) )
=> ( ( member_set_a @ X @ ( sigma_space_set_a @ M ) )
=> ( member_real @ ( F @ X ) @ ( sigma_space_real @ A ) ) ) ) ).
% measurable_space
thf(fact_683_measurable__space,axiom,
! [F: a > set_a,M: sigma_measure_a,A: sigma_measure_set_a,X: a] :
( ( member_a_set_a @ F @ ( sigma_3685133166752798000_set_a @ M @ A ) )
=> ( ( member_a @ X @ ( sigma_space_a @ M ) )
=> ( member_set_a @ ( F @ X ) @ ( sigma_space_set_a @ A ) ) ) ) ).
% measurable_space
thf(fact_684_measurable__space,axiom,
! [F: real > set_a,M: sigma_measure_real,A: sigma_measure_set_a,X: real] :
( ( member_real_set_a @ F @ ( sigma_4283435981211228640_set_a @ M @ A ) )
=> ( ( member_real @ X @ ( sigma_space_real @ M ) )
=> ( member_set_a @ ( F @ X ) @ ( sigma_space_set_a @ A ) ) ) ) ).
% measurable_space
thf(fact_685_measurable__space,axiom,
! [F: set_a > set_a,M: sigma_measure_set_a,A: sigma_measure_set_a,X: set_a] :
( ( member_set_a_set_a @ F @ ( sigma_5212894042034225104_set_a @ M @ A ) )
=> ( ( member_set_a @ X @ ( sigma_space_set_a @ M ) )
=> ( member_set_a @ ( F @ X ) @ ( sigma_space_set_a @ A ) ) ) ) ).
% measurable_space
thf(fact_686_measurable__space,axiom,
! [F: ( real > real ) > a,M: sigma_4258434043392614480l_real,A: sigma_measure_a,X: real > real] :
( ( member_real_real_a @ F @ ( sigma_2844068865785385077real_a @ M @ A ) )
=> ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
=> ( member_a @ ( F @ X ) @ ( sigma_space_a @ A ) ) ) ) ).
% measurable_space
thf(fact_687_measurable__cong__simp,axiom,
! [M: sigma_measure_a,N: sigma_measure_a,M2: sigma_measure_real,N2: sigma_measure_real,F: a > real,G: a > real] :
( ( M = N )
=> ( ( M2 = N2 )
=> ( ! [W: a] :
( ( member_a @ W @ ( sigma_space_a @ M ) )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ M2 ) )
= ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ N @ N2 ) ) ) ) ) ) ).
% measurable_cong_simp
thf(fact_688_measurable__cong__simp,axiom,
! [M: sigma_measure_real,N: sigma_measure_real,M2: sigma_measure_a,N2: sigma_measure_a,F: real > a,G: real > a] :
( ( M = N )
=> ( ( M2 = N2 )
=> ( ! [W: real] :
( ( member_real @ W @ ( sigma_space_real @ M ) )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ M2 ) )
= ( member_real_a @ G @ ( sigma_523072396149930112real_a @ N @ N2 ) ) ) ) ) ) ).
% measurable_cong_simp
thf(fact_689_measurable__cong__simp,axiom,
! [M: sigma_measure_real,N: sigma_measure_real,M2: sigma_measure_real,N2: sigma_measure_real,F: real > real,G: real > real] :
( ( M = N )
=> ( ( M2 = N2 )
=> ( ! [W: real] :
( ( member_real @ W @ ( sigma_space_real @ M ) )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ M2 ) )
= ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ N @ N2 ) ) ) ) ) ) ).
% measurable_cong_simp
thf(fact_690_measurable__comp,axiom,
! [F: a > a,M: sigma_measure_a,N: sigma_measure_a,G: a > real,L: sigma_measure_real] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ N ) )
=> ( ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ N @ L ) )
=> ( member_a_real @ ( comp_a_real_a @ G @ F ) @ ( sigma_9116425665531756122a_real @ M @ L ) ) ) ) ).
% measurable_comp
thf(fact_691_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_692_measurable__comp,axiom,
! [F: a > real,M: sigma_measure_a,N: sigma_measure_real,G: real > real,L: sigma_measure_real] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ N ) )
=> ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ N @ L ) )
=> ( member_a_real @ ( comp_real_real_a @ G @ F ) @ ( sigma_9116425665531756122a_real @ M @ L ) ) ) ) ).
% measurable_comp
thf(fact_693_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_694_measurable__comp,axiom,
! [F: real > a,M: sigma_measure_real,N: sigma_measure_a,G: a > real,L: sigma_measure_real] :
( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ N ) )
=> ( ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ N @ L ) )
=> ( member_real_real @ ( comp_a_real_real @ G @ F ) @ ( sigma_5267869275261027754l_real @ M @ L ) ) ) ) ).
% measurable_comp
thf(fact_695_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_696_measurable__comp,axiom,
! [F: real > real,M: sigma_measure_real,N: sigma_measure_real,G: real > real,L: sigma_measure_real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) )
=> ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ N @ L ) )
=> ( member_real_real @ ( comp_real_real_real @ G @ F ) @ ( sigma_5267869275261027754l_real @ M @ L ) ) ) ) ).
% measurable_comp
thf(fact_697_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_698_measurable__empty__iff,axiom,
! [N: sigma_measure_real,F: a > real,M: sigma_measure_a] :
( ( ( sigma_space_real @ N )
= bot_bot_set_real )
=> ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ N ) )
= ( ( sigma_space_a @ M )
= bot_bot_set_a ) ) ) ).
% measurable_empty_iff
thf(fact_699_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_700_measurable__empty__iff,axiom,
! [N: sigma_measure_real,F: real > real,M: sigma_measure_real] :
( ( ( sigma_space_real @ N )
= bot_bot_set_real )
=> ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) )
= ( ( sigma_space_real @ M )
= bot_bot_set_real ) ) ) ).
% measurable_empty_iff
thf(fact_701_measurable__sets__borel,axiom,
! [F: real > a,M: sigma_measure_a,A: set_a] :
( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ M ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( member_set_real @ ( vimage_real_a @ F @ A ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ) ).
% measurable_sets_borel
thf(fact_702_measurable__sets__borel,axiom,
! [F: real > real,M: sigma_measure_real,A: set_real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ M ) )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ M ) )
=> ( member_set_real @ ( vimage_real_real @ F @ A ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ) ).
% measurable_sets_borel
thf(fact_703_sets__eq__bot,axiom,
! [M: sigma_measure_a] :
( ( ( sigma_sets_a @ M )
= ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) )
= ( M = bot_bo2108912051383640591sure_a ) ) ).
% sets_eq_bot
thf(fact_704_sets__eq__bot,axiom,
! [M: sigma_measure_real] :
( ( ( sigma_sets_real @ M )
= ( insert_set_real @ bot_bot_set_real @ bot_bot_set_set_real ) )
= ( M = bot_bo5982154664989874033e_real ) ) ).
% sets_eq_bot
thf(fact_705_sets__eq__bot2,axiom,
! [M: sigma_measure_a] :
( ( ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a )
= ( sigma_sets_a @ M ) )
= ( M = bot_bo2108912051383640591sure_a ) ) ).
% sets_eq_bot2
thf(fact_706_sets__eq__bot2,axiom,
! [M: sigma_measure_real] :
( ( ( insert_set_real @ bot_bot_set_real @ bot_bot_set_set_real )
= ( sigma_sets_real @ M ) )
= ( M = bot_bo5982154664989874033e_real ) ) ).
% sets_eq_bot2
thf(fact_707_standard__borel_Ointro,axiom,
! [M: sigma_measure_a] :
( ? [X4: a > real] :
( ( member_a_real @ X4 @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
& ? [Xa2: real > a] :
( ( member_real_a @ Xa2 @ ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ M ) )
& ! [Xb2: a] :
( ( member_a @ Xb2 @ ( sigma_space_a @ M ) )
=> ( ( comp_real_a_a @ Xa2 @ X4 @ Xb2 )
= Xb2 ) ) ) )
=> ( standard_borel_a @ M ) ) ).
% standard_borel.intro
thf(fact_708_standard__borel_Ointro,axiom,
! [M: sigma_measure_real] :
( ? [X4: real > real] :
( ( member_real_real @ X4 @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
& ? [Xa2: real > real] :
( ( member_real_real @ Xa2 @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ M ) )
& ! [Xb2: real] :
( ( member_real @ Xb2 @ ( sigma_space_real @ M ) )
=> ( ( comp_real_real_real @ Xa2 @ X4 @ Xb2 )
= Xb2 ) ) ) )
=> ( standard_borel_real @ M ) ) ).
% standard_borel.intro
thf(fact_709_standard__borel_Oexist__fg,axiom,
! [M: sigma_measure_a] :
( ( standard_borel_a @ M )
=> ? [X2: a > real] :
( ( member_a_real @ X2 @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
& ? [Xa: real > a] :
( ( member_real_a @ Xa @ ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ M ) )
& ! [Xb: a] :
( ( member_a @ Xb @ ( sigma_space_a @ M ) )
=> ( ( comp_real_a_a @ Xa @ X2 @ Xb )
= Xb ) ) ) ) ) ).
% standard_borel.exist_fg
thf(fact_710_standard__borel_Oexist__fg,axiom,
! [M: sigma_measure_real] :
( ( standard_borel_real @ M )
=> ? [X2: real > real] :
( ( member_real_real @ X2 @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
& ? [Xa: real > real] :
( ( member_real_real @ Xa @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ M ) )
& ! [Xb: real] :
( ( member_real @ Xb @ ( sigma_space_real @ M ) )
=> ( ( comp_real_real_real @ Xa @ X2 @ Xb )
= Xb ) ) ) ) ) ).
% standard_borel.exist_fg
thf(fact_711_standard__borelI,axiom,
! [F: set_a > real,Y: sigma_measure_set_a,G: real > set_a] :
( ( member_set_a_real @ F @ ( sigma_567508090560183226a_real @ Y @ borel_5078946678739801102l_real ) )
=> ( ( member_real_set_a @ G @ ( sigma_4283435981211228640_set_a @ borel_5078946678739801102l_real @ Y ) )
=> ( ! [Y2: set_a] :
( ( member_set_a @ Y2 @ ( sigma_space_set_a @ Y ) )
=> ( ( comp_r8810651240769227663_set_a @ G @ F @ Y2 )
= Y2 ) )
=> ( standard_borel_set_a @ Y ) ) ) ) ).
% standard_borelI
thf(fact_712_standard__borelI,axiom,
! [F: ( real > real ) > real,Y: sigma_4258434043392614480l_real,G: real > real > real] :
( ( member5749659578190367193l_real @ F @ ( sigma_314226851082786037l_real @ Y @ borel_5078946678739801102l_real ) )
=> ( ( member8878224140454985689l_real @ G @ ( sigma_5187198232267106421l_real @ borel_5078946678739801102l_real @ Y ) )
=> ( ! [Y2: real > real] :
( ( member_real_real @ Y2 @ ( sigma_3619470280215722479l_real @ Y ) )
=> ( ( comp_r2210161833511426559l_real @ G @ F @ Y2 )
= Y2 ) )
=> ( standa6209843396170762084l_real @ Y ) ) ) ) ).
% standard_borelI
thf(fact_713_standard__borelI,axiom,
! [F: ( real > a ) > real,Y: sigma_measure_real_a,G: real > real > a] :
( ( member_real_a_real @ F @ ( sigma_7374358782470848101a_real @ Y @ borel_5078946678739801102l_real ) )
=> ( ( member_real_real_a2 @ G @ ( sigma_5735160441797593099real_a @ borel_5078946678739801102l_real @ Y ) )
=> ( ! [Y2: real > a] :
( ( member_real_a @ Y2 @ ( sigma_space_real_a @ Y ) )
=> ( ( comp_r4421311556806241359real_a @ G @ F @ Y2 )
= Y2 ) )
=> ( standa4991623935272378204real_a @ Y ) ) ) ) ).
% standard_borelI
thf(fact_714_standard__borelI,axiom,
! [F: ( a > real ) > real,Y: sigma_measure_a_real,G: real > a > real] :
( ( member_a_real_real @ F @ ( sigma_1638934570974270327l_real @ Y @ borel_5078946678739801102l_real ) )
=> ( ( member_real_a_real2 @ G @ ( sigma_5217893939530255901a_real @ borel_5078946678739801102l_real @ Y ) )
=> ( ! [Y2: a > real] :
( ( member_a_real @ Y2 @ ( sigma_space_a_real @ Y ) )
=> ( ( comp_r2811049936330031311a_real @ G @ F @ Y2 )
= Y2 ) )
=> ( standa4474357433005041006a_real @ Y ) ) ) ) ).
% standard_borelI
thf(fact_715_standard__borelI,axiom,
! [F: a > real,Y: sigma_measure_a,G: real > a] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ Y @ borel_5078946678739801102l_real ) )
=> ( ( member_real_a @ G @ ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ Y ) )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ ( sigma_space_a @ Y ) )
=> ( ( comp_real_a_a @ G @ F @ Y2 )
= Y2 ) )
=> ( standard_borel_a @ Y ) ) ) ) ).
% standard_borelI
thf(fact_716_standard__borelI,axiom,
! [F: real > real,Y: sigma_measure_real,G: real > real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ Y @ borel_5078946678739801102l_real ) )
=> ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ Y ) )
=> ( ! [Y2: real] :
( ( member_real @ Y2 @ ( sigma_space_real @ Y ) )
=> ( ( comp_real_real_real @ G @ F @ Y2 )
= Y2 ) )
=> ( standard_borel_real @ Y ) ) ) ) ).
% standard_borelI
thf(fact_717_standard__borel__def,axiom,
( standard_borel_a
= ( ^ [M3: sigma_measure_a] :
? [X3: a > real] :
( ( member_a_real @ X3 @ ( sigma_9116425665531756122a_real @ M3 @ borel_5078946678739801102l_real ) )
& ? [Y5: real > a] :
( ( member_real_a @ Y5 @ ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ M3 ) )
& ! [Z3: a] :
( ( member_a @ Z3 @ ( sigma_space_a @ M3 ) )
=> ( ( comp_real_a_a @ Y5 @ X3 @ Z3 )
= Z3 ) ) ) ) ) ) ).
% standard_borel_def
thf(fact_718_standard__borel__def,axiom,
( standard_borel_real
= ( ^ [M3: sigma_measure_real] :
? [X3: real > real] :
( ( member_real_real @ X3 @ ( sigma_5267869275261027754l_real @ M3 @ borel_5078946678739801102l_real ) )
& ? [Y5: real > real] :
( ( member_real_real @ Y5 @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ M3 ) )
& ! [Z3: real] :
( ( member_real @ Z3 @ ( sigma_space_real @ M3 ) )
=> ( ( comp_real_real_real @ Y5 @ X3 @ Z3 )
= Z3 ) ) ) ) ) ) ).
% standard_borel_def
thf(fact_719_real_Osingleton__sets,axiom,
! [X: real] :
( ( member_real @ X @ ( sigma_space_real @ borel_5078946678739801102l_real ) )
=> ( member_set_real @ ( insert_real @ X @ bot_bot_set_real ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ).
% real.singleton_sets
thf(fact_720_borel__measurable__subalgebra,axiom,
! [N: sigma_measure_a,M: sigma_measure_a,F: a > real] :
( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ N ) @ ( sigma_sets_a @ M ) )
=> ( ( ( sigma_space_a @ N )
= ( sigma_space_a @ M ) )
=> ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ N @ borel_5078946678739801102l_real ) )
=> ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) ) ) ) ).
% borel_measurable_subalgebra
thf(fact_721_borel__measurable__subalgebra,axiom,
! [N: sigma_measure_real,M: sigma_measure_real,F: real > real] :
( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ N ) @ ( sigma_sets_real @ M ) )
=> ( ( ( sigma_space_real @ N )
= ( sigma_space_real @ M ) )
=> ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ N @ borel_5078946678739801102l_real ) )
=> ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ) ).
% borel_measurable_subalgebra
thf(fact_722_measurable__mono,axiom,
! [N2: sigma_measure_a,N: sigma_measure_a,M: sigma_measure_a,M2: 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 @ M2 ) )
=> ( ( ( sigma_space_a @ M )
= ( sigma_space_a @ M2 ) )
=> ( ord_less_eq_set_a_a @ ( sigma_measurable_a_a @ M @ N ) @ ( sigma_measurable_a_a @ M2 @ N2 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_723_measurable__mono,axiom,
! [N2: sigma_measure_real,N: sigma_measure_real,M: sigma_measure_a,M2: sigma_measure_a] :
( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ N2 ) @ ( sigma_sets_real @ N ) )
=> ( ( ( sigma_space_real @ N )
= ( sigma_space_real @ N2 ) )
=> ( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ M ) @ ( sigma_sets_a @ M2 ) )
=> ( ( ( sigma_space_a @ M )
= ( sigma_space_a @ M2 ) )
=> ( ord_le3334967407727675675a_real @ ( sigma_9116425665531756122a_real @ M @ N ) @ ( sigma_9116425665531756122a_real @ M2 @ N2 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_724_measurable__mono,axiom,
! [N2: sigma_measure_a,N: sigma_measure_a,M: sigma_measure_real,M2: sigma_measure_real] :
( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ N2 ) @ ( sigma_sets_a @ N ) )
=> ( ( ( sigma_space_a @ N )
= ( sigma_space_a @ N2 ) )
=> ( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ M ) @ ( sigma_sets_real @ M2 ) )
=> ( ( ( sigma_space_real @ M )
= ( sigma_space_real @ M2 ) )
=> ( ord_le5743406823621094409real_a @ ( sigma_523072396149930112real_a @ M @ N ) @ ( sigma_523072396149930112real_a @ M2 @ N2 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_725_measurable__mono,axiom,
! [N2: sigma_measure_real,N: sigma_measure_real,M: sigma_measure_real,M2: sigma_measure_real] :
( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ N2 ) @ ( sigma_sets_real @ N ) )
=> ( ( ( sigma_space_real @ N )
= ( sigma_space_real @ N2 ) )
=> ( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ M ) @ ( sigma_sets_real @ M2 ) )
=> ( ( ( sigma_space_real @ M )
= ( sigma_space_real @ M2 ) )
=> ( ord_le4198349162570665613l_real @ ( sigma_5267869275261027754l_real @ M @ N ) @ ( sigma_5267869275261027754l_real @ M2 @ N2 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_726_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_727_space__empty__iff,axiom,
! [N: sigma_measure_real] :
( ( ( sigma_space_real @ N )
= bot_bot_set_real )
= ( ( sigma_sets_real @ N )
= ( insert_set_real @ bot_bot_set_real @ bot_bot_set_set_real ) ) ) ).
% space_empty_iff
thf(fact_728_standard__borel_Og__meas,axiom,
! [M: sigma_measure_a] :
( ( standard_borel_a @ M )
=> ( member_real_a @ ( standard_g_a @ M ) @ ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ M ) ) ) ).
% standard_borel.g_meas
thf(fact_729_standard__borel_Og__meas,axiom,
! [M: sigma_measure_real] :
( ( standard_borel_real @ M )
=> ( member_real_real @ ( standard_g_real @ M ) @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ M ) ) ) ).
% standard_borel.g_meas
thf(fact_730_standard__borel_Of__meas,axiom,
! [M: sigma_measure_a] :
( ( standard_borel_a @ M )
=> ( member_a_real @ ( standard_f_a @ M ) @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) ) ).
% standard_borel.f_meas
thf(fact_731_standard__borel_Of__meas,axiom,
! [M: sigma_measure_real] :
( ( standard_borel_real @ M )
=> ( member_real_real @ ( standard_f_real @ M ) @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ).
% standard_borel.f_meas
thf(fact_732_is__borel__def,axiom,
( borel_4993665998515044718a_real
= ( ^ [F2: a > real,M3: sigma_measure_a] : ( member_a_real @ F2 @ ( sigma_9116425665531756122a_real @ M3 @ borel_5078946678739801102l_real ) ) ) ) ).
% is_borel_def
thf(fact_733_is__borel__def,axiom,
( borel_236569967776329622l_real
= ( ^ [F2: real > real,M3: sigma_measure_real] : ( member_real_real @ F2 @ ( sigma_5267869275261027754l_real @ M3 @ borel_5078946678739801102l_real ) ) ) ) ).
% is_borel_def
thf(fact_734_increasing__def,axiom,
( measur7695573945809229763t_real
= ( ^ [M3: set_set_real,Mu: set_real > set_real] :
! [X3: set_real] :
( ( member_set_real @ X3 @ M3 )
=> ! [Y5: set_real] :
( ( member_set_real @ Y5 @ M3 )
=> ( ( ord_less_eq_set_real @ X3 @ Y5 )
=> ( ord_less_eq_set_real @ ( Mu @ X3 ) @ ( Mu @ Y5 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_735_increasing__def,axiom,
( measur6047784561759668541_set_a
= ( ^ [M3: set_set_real,Mu: set_real > set_a] :
! [X3: set_real] :
( ( member_set_real @ X3 @ M3 )
=> ! [Y5: set_real] :
( ( member_set_real @ Y5 @ M3 )
=> ( ( ord_less_eq_set_real @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( Mu @ X3 ) @ ( Mu @ Y5 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_736_increasing__def,axiom,
( measur2503387328813080045t_real
= ( ^ [M3: set_set_a,Mu: set_a > set_real] :
! [X3: set_a] :
( ( member_set_a @ X3 @ M3 )
=> ! [Y5: set_a] :
( ( member_set_a @ Y5 @ M3 )
=> ( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_set_real @ ( Mu @ X3 ) @ ( Mu @ Y5 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_737_increasing__def,axiom,
( measur7842569353079325843_set_a
= ( ^ [M3: set_set_a,Mu: set_a > set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ M3 )
=> ! [Y5: set_a] :
( ( member_set_a @ Y5 @ M3 )
=> ( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( Mu @ X3 ) @ ( Mu @ Y5 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_738_increasingD,axiom,
! [M: set_set_real,F: set_real > set_real,X: set_real,Y3: set_real] :
( ( measur7695573945809229763t_real @ M @ F )
=> ( ( ord_less_eq_set_real @ X @ Y3 )
=> ( ( member_set_real @ X @ M )
=> ( ( member_set_real @ Y3 @ M )
=> ( ord_less_eq_set_real @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_739_increasingD,axiom,
! [M: set_set_real,F: set_real > set_a,X: set_real,Y3: set_real] :
( ( measur6047784561759668541_set_a @ M @ F )
=> ( ( ord_less_eq_set_real @ X @ Y3 )
=> ( ( member_set_real @ X @ M )
=> ( ( member_set_real @ Y3 @ M )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_740_increasingD,axiom,
! [M: set_set_a,F: set_a > set_real,X: set_a,Y3: set_a] :
( ( measur2503387328813080045t_real @ M @ F )
=> ( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ( member_set_a @ X @ M )
=> ( ( member_set_a @ Y3 @ M )
=> ( ord_less_eq_set_real @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_741_increasingD,axiom,
! [M: set_set_a,F: set_a > set_a,X: set_a,Y3: set_a] :
( ( measur7842569353079325843_set_a @ M @ F )
=> ( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ( member_set_a @ X @ M )
=> ( ( member_set_a @ Y3 @ M )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_742_bot__empty__eq,axiom,
( bot_bot_set_a_o
= ( ^ [X3: set_a] : ( member_set_a @ X3 @ bot_bot_set_set_a ) ) ) ).
% bot_empty_eq
thf(fact_743_bot__empty__eq,axiom,
( bot_bot_real_real_o
= ( ^ [X3: real > real] : ( member_real_real @ X3 @ bot_bo6767488733719836353l_real ) ) ) ).
% bot_empty_eq
thf(fact_744_bot__empty__eq,axiom,
( bot_bot_real_a_o
= ( ^ [X3: real > a] : ( member_real_a @ X3 @ bot_bot_set_real_a ) ) ) ).
% bot_empty_eq
thf(fact_745_bot__empty__eq,axiom,
( bot_bot_a_real_o
= ( ^ [X3: a > real] : ( member_a_real @ X3 @ bot_bot_set_a_real ) ) ) ).
% bot_empty_eq
thf(fact_746_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_747_bot__empty__eq,axiom,
( bot_bot_real_o
= ( ^ [X3: real] : ( member_real @ X3 @ bot_bot_set_real ) ) ) ).
% bot_empty_eq
thf(fact_748_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_749_Collect__empty__eq__bot,axiom,
! [P: real > $o] :
( ( ( collect_real @ P )
= bot_bot_set_real )
= ( P = bot_bot_real_o ) ) ).
% Collect_empty_eq_bot
thf(fact_750_is__singletonI,axiom,
! [X: a] : ( is_singleton_a @ ( insert_a @ X @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_751_is__singletonI,axiom,
! [X: real] : ( is_singleton_real @ ( insert_real @ X @ bot_bot_set_real ) ) ).
% is_singletonI
thf(fact_752_borel__measurable__vimage,axiom,
! [F: a > real,M: sigma_measure_a,X: real] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_real @ F @ ( insert_real @ X @ bot_bot_set_real ) ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) ) ).
% borel_measurable_vimage
thf(fact_753_borel__measurable__vimage,axiom,
! [F: real > real,M: sigma_measure_real,X: real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_real @ F @ ( insert_real @ X @ bot_bot_set_real ) ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) ) ).
% borel_measurable_vimage
thf(fact_754_sets__range,axiom,
! [A: set_a > set_a,I: set_set_a,M: sigma_measure_a,I2: set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ A @ I ) @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ I2 @ I )
=> ( member_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ M ) ) ) ) ).
% sets_range
thf(fact_755_sets__range,axiom,
! [A: ( real > real ) > set_a,I: set_real_real,M: sigma_measure_a,I2: real > real] :
( ( ord_le3724670747650509150_set_a @ ( image_6696214103625886560_set_a @ A @ I ) @ ( sigma_sets_a @ M ) )
=> ( ( member_real_real @ I2 @ I )
=> ( member_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ M ) ) ) ) ).
% sets_range
thf(fact_756_sets__range,axiom,
! [A: ( real > a ) > set_a,I: set_real_a,M: sigma_measure_a,I2: real > a] :
( ( ord_le3724670747650509150_set_a @ ( image_real_a_set_a @ A @ I ) @ ( sigma_sets_a @ M ) )
=> ( ( member_real_a @ I2 @ I )
=> ( member_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ M ) ) ) ) ).
% sets_range
thf(fact_757_sets__range,axiom,
! [A: ( a > real ) > set_a,I: set_a_real,M: sigma_measure_a,I2: a > real] :
( ( ord_le3724670747650509150_set_a @ ( image_a_real_set_a @ A @ I ) @ ( sigma_sets_a @ M ) )
=> ( ( member_a_real @ I2 @ I )
=> ( member_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ M ) ) ) ) ).
% sets_range
thf(fact_758_sets__range,axiom,
! [A: a > set_a,I: set_a,M: sigma_measure_a,I2: a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ A @ I ) @ ( sigma_sets_a @ M ) )
=> ( ( member_a @ I2 @ I )
=> ( member_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ M ) ) ) ) ).
% sets_range
thf(fact_759_sets__range,axiom,
! [A: set_a > set_real,I: set_set_a,M: sigma_measure_real,I2: set_a] :
( ( ord_le3558479182127378552t_real @ ( image_set_a_set_real @ A @ I ) @ ( sigma_sets_real @ M ) )
=> ( ( member_set_a @ I2 @ I )
=> ( member_set_real @ ( A @ I2 ) @ ( sigma_sets_real @ M ) ) ) ) ).
% sets_range
thf(fact_760_sets__range,axiom,
! [A: ( real > real ) > set_real,I: set_real_real,M: sigma_measure_real,I2: real > real] :
( ( ord_le3558479182127378552t_real @ ( image_6663718904102175840t_real @ A @ I ) @ ( sigma_sets_real @ M ) )
=> ( ( member_real_real @ I2 @ I )
=> ( member_set_real @ ( A @ I2 ) @ ( sigma_sets_real @ M ) ) ) ) ).
% sets_range
thf(fact_761_sets__range,axiom,
! [A: ( real > a ) > set_real,I: set_real_a,M: sigma_measure_real,I2: real > a] :
( ( ord_le3558479182127378552t_real @ ( image_3398683075489632294t_real @ A @ I ) @ ( sigma_sets_real @ M ) )
=> ( ( member_real_a @ I2 @ I )
=> ( member_set_real @ ( A @ I2 ) @ ( sigma_sets_real @ M ) ) ) ) ).
% sets_range
thf(fact_762_sets__range,axiom,
! [A: ( a > real ) > set_real,I: set_a_real,M: sigma_measure_real,I2: a > real] :
( ( ord_le3558479182127378552t_real @ ( image_6506138200666676536t_real @ A @ I ) @ ( sigma_sets_real @ M ) )
=> ( ( member_a_real @ I2 @ I )
=> ( member_set_real @ ( A @ I2 ) @ ( sigma_sets_real @ M ) ) ) ) ).
% sets_range
thf(fact_763_sets__range,axiom,
! [A: a > set_real,I: set_a,M: sigma_measure_real,I2: a] :
( ( ord_le3558479182127378552t_real @ ( image_a_set_real @ A @ I ) @ ( sigma_sets_real @ M ) )
=> ( ( member_a @ I2 @ I )
=> ( member_set_real @ ( A @ I2 ) @ ( sigma_sets_real @ M ) ) ) ) ).
% sets_range
thf(fact_764_IntI,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A )
=> ( ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_765_IntI,axiom,
! [C: real > real,A: set_real_real,B: set_real_real] :
( ( member_real_real @ C @ A )
=> ( ( member_real_real @ C @ B )
=> ( member_real_real @ C @ ( inf_in5166753670444608447l_real @ A @ B ) ) ) ) ).
% IntI
thf(fact_766_IntI,axiom,
! [C: real > a,A: set_real_a,B: set_real_a] :
( ( member_real_a @ C @ A )
=> ( ( member_real_a @ C @ B )
=> ( member_real_a @ C @ ( inf_inf_set_real_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_767_IntI,axiom,
! [C: a > real,A: set_a_real,B: set_a_real] :
( ( member_a_real @ C @ A )
=> ( ( member_a_real @ C @ B )
=> ( member_a_real @ C @ ( inf_inf_set_a_real @ A @ B ) ) ) ) ).
% IntI
thf(fact_768_IntI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ( member_a @ C @ B )
=> ( member_a @ C @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_769_Int__iff,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
= ( ( member_set_a @ C @ A )
& ( member_set_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_770_Int__iff,axiom,
! [C: real > real,A: set_real_real,B: set_real_real] :
( ( member_real_real @ C @ ( inf_in5166753670444608447l_real @ A @ B ) )
= ( ( member_real_real @ C @ A )
& ( member_real_real @ C @ B ) ) ) ).
% Int_iff
thf(fact_771_Int__iff,axiom,
! [C: real > a,A: set_real_a,B: set_real_a] :
( ( member_real_a @ C @ ( inf_inf_set_real_a @ A @ B ) )
= ( ( member_real_a @ C @ A )
& ( member_real_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_772_Int__iff,axiom,
! [C: a > real,A: set_a_real,B: set_a_real] :
( ( member_a_real @ C @ ( inf_inf_set_a_real @ A @ B ) )
= ( ( member_a_real @ C @ A )
& ( member_a_real @ C @ B ) ) ) ).
% Int_iff
thf(fact_773_Int__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ( member_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_774_Int__subset__iff,axiom,
! [C2: set_real,A: set_real,B: set_real] :
( ( ord_less_eq_set_real @ C2 @ ( inf_inf_set_real @ A @ B ) )
= ( ( ord_less_eq_set_real @ C2 @ A )
& ( ord_less_eq_set_real @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_775_Int__subset__iff,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
= ( ( ord_less_eq_set_a @ C2 @ A )
& ( ord_less_eq_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_776_Int__insert__left__if0,axiom,
! [A2: real,C2: set_real,B: set_real] :
( ~ ( member_real @ A2 @ C2 )
=> ( ( inf_inf_set_real @ ( insert_real @ A2 @ B ) @ C2 )
= ( inf_inf_set_real @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_777_Int__insert__left__if0,axiom,
! [A2: set_a,C2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A2 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_778_Int__insert__left__if0,axiom,
! [A2: real > real,C2: set_real_real,B: set_real_real] :
( ~ ( member_real_real @ A2 @ C2 )
=> ( ( inf_in5166753670444608447l_real @ ( insert_real_real @ A2 @ B ) @ C2 )
= ( inf_in5166753670444608447l_real @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_779_Int__insert__left__if0,axiom,
! [A2: real > a,C2: set_real_a,B: set_real_a] :
( ~ ( member_real_a @ A2 @ C2 )
=> ( ( inf_inf_set_real_a @ ( insert_real_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_real_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_780_Int__insert__left__if0,axiom,
! [A2: a > real,C2: set_a_real,B: set_a_real] :
( ~ ( member_a_real @ A2 @ C2 )
=> ( ( inf_inf_set_a_real @ ( insert_a_real @ A2 @ B ) @ C2 )
= ( inf_inf_set_a_real @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_781_Int__insert__left__if0,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_782_Int__insert__left__if1,axiom,
! [A2: real,C2: set_real,B: set_real] :
( ( member_real @ A2 @ C2 )
=> ( ( inf_inf_set_real @ ( insert_real @ A2 @ B ) @ C2 )
= ( insert_real @ A2 @ ( inf_inf_set_real @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_783_Int__insert__left__if1,axiom,
! [A2: set_a,C2: set_set_a,B: set_set_a] :
( ( member_set_a @ A2 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C2 )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_784_Int__insert__left__if1,axiom,
! [A2: real > real,C2: set_real_real,B: set_real_real] :
( ( member_real_real @ A2 @ C2 )
=> ( ( inf_in5166753670444608447l_real @ ( insert_real_real @ A2 @ B ) @ C2 )
= ( insert_real_real @ A2 @ ( inf_in5166753670444608447l_real @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_785_Int__insert__left__if1,axiom,
! [A2: real > a,C2: set_real_a,B: set_real_a] :
( ( member_real_a @ A2 @ C2 )
=> ( ( inf_inf_set_real_a @ ( insert_real_a @ A2 @ B ) @ C2 )
= ( insert_real_a @ A2 @ ( inf_inf_set_real_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_786_Int__insert__left__if1,axiom,
! [A2: a > real,C2: set_a_real,B: set_a_real] :
( ( member_a_real @ A2 @ C2 )
=> ( ( inf_inf_set_a_real @ ( insert_a_real @ A2 @ B ) @ C2 )
= ( insert_a_real @ A2 @ ( inf_inf_set_a_real @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_787_Int__insert__left__if1,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_788_insert__inter__insert,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_789_insert__inter__insert,axiom,
! [A2: real,A: set_real,B: set_real] :
( ( inf_inf_set_real @ ( insert_real @ A2 @ A ) @ ( insert_real @ A2 @ B ) )
= ( insert_real @ A2 @ ( inf_inf_set_real @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_790_Int__insert__right__if0,axiom,
! [A2: real,A: set_real,B: set_real] :
( ~ ( member_real @ A2 @ A )
=> ( ( inf_inf_set_real @ A @ ( insert_real @ A2 @ B ) )
= ( inf_inf_set_real @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_791_Int__insert__right__if0,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( inf_inf_set_set_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_792_Int__insert__right__if0,axiom,
! [A2: real > real,A: set_real_real,B: set_real_real] :
( ~ ( member_real_real @ A2 @ A )
=> ( ( inf_in5166753670444608447l_real @ A @ ( insert_real_real @ A2 @ B ) )
= ( inf_in5166753670444608447l_real @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_793_Int__insert__right__if0,axiom,
! [A2: real > a,A: set_real_a,B: set_real_a] :
( ~ ( member_real_a @ A2 @ A )
=> ( ( inf_inf_set_real_a @ A @ ( insert_real_a @ A2 @ B ) )
= ( inf_inf_set_real_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_794_Int__insert__right__if0,axiom,
! [A2: a > real,A: set_a_real,B: set_a_real] :
( ~ ( member_a_real @ A2 @ A )
=> ( ( inf_inf_set_a_real @ A @ ( insert_a_real @ A2 @ B ) )
= ( inf_inf_set_a_real @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_795_Int__insert__right__if0,axiom,
! [A2: a,A: set_a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_796_Int__insert__right__if1,axiom,
! [A2: real,A: set_real,B: set_real] :
( ( member_real @ A2 @ A )
=> ( ( inf_inf_set_real @ A @ ( insert_real @ A2 @ B ) )
= ( insert_real @ A2 @ ( inf_inf_set_real @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_797_Int__insert__right__if1,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_798_Int__insert__right__if1,axiom,
! [A2: real > real,A: set_real_real,B: set_real_real] :
( ( member_real_real @ A2 @ A )
=> ( ( inf_in5166753670444608447l_real @ A @ ( insert_real_real @ A2 @ B ) )
= ( insert_real_real @ A2 @ ( inf_in5166753670444608447l_real @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_799_Int__insert__right__if1,axiom,
! [A2: real > a,A: set_real_a,B: set_real_a] :
( ( member_real_a @ A2 @ A )
=> ( ( inf_inf_set_real_a @ A @ ( insert_real_a @ A2 @ B ) )
= ( insert_real_a @ A2 @ ( inf_inf_set_real_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_800_Int__insert__right__if1,axiom,
! [A2: a > real,A: set_a_real,B: set_a_real] :
( ( member_a_real @ A2 @ A )
=> ( ( inf_inf_set_a_real @ A @ ( insert_a_real @ A2 @ B ) )
= ( insert_a_real @ A2 @ ( inf_inf_set_a_real @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_801_Int__insert__right__if1,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_802_sets_OInt,axiom,
! [A2: set_a,M: sigma_measure_a,B2: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ B2 @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( sigma_sets_a @ M ) ) ) ) ).
% sets.Int
thf(fact_803_sets_OInt,axiom,
! [A2: set_real,M: sigma_measure_real,B2: set_real] :
( ( member_set_real @ A2 @ ( sigma_sets_real @ M ) )
=> ( ( member_set_real @ B2 @ ( sigma_sets_real @ M ) )
=> ( member_set_real @ ( inf_inf_set_real @ A2 @ B2 ) @ ( sigma_sets_real @ M ) ) ) ) ).
% sets.Int
thf(fact_804_vimage__Int,axiom,
! [F: real > a,A: set_a,B: set_a] :
( ( vimage_real_a @ F @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_real @ ( vimage_real_a @ F @ A ) @ ( vimage_real_a @ F @ B ) ) ) ).
% vimage_Int
thf(fact_805_vimage__Int,axiom,
! [F: real > real,A: set_real,B: set_real] :
( ( vimage_real_real @ F @ ( inf_inf_set_real @ A @ B ) )
= ( inf_inf_set_real @ ( vimage_real_real @ F @ A ) @ ( vimage_real_real @ F @ B ) ) ) ).
% vimage_Int
thf(fact_806_vimage__Int,axiom,
! [F: a > real,A: set_real,B: set_real] :
( ( vimage_a_real @ F @ ( inf_inf_set_real @ A @ B ) )
= ( inf_inf_set_a @ ( vimage_a_real @ F @ A ) @ ( vimage_a_real @ F @ B ) ) ) ).
% vimage_Int
thf(fact_807_vimage__Int,axiom,
! [F: a > a,A: set_a,B: set_a] :
( ( vimage_a_a @ F @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_a @ ( vimage_a_a @ F @ A ) @ ( vimage_a_a @ F @ B ) ) ) ).
% vimage_Int
thf(fact_808_insert__disjoint_I1_J,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ A ) @ B )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A2 @ B )
& ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_809_insert__disjoint_I1_J,axiom,
! [A2: real > real,A: set_real_real,B: set_real_real] :
( ( ( inf_in5166753670444608447l_real @ ( insert_real_real @ A2 @ A ) @ B )
= bot_bo6767488733719836353l_real )
= ( ~ ( member_real_real @ A2 @ B )
& ( ( inf_in5166753670444608447l_real @ A @ B )
= bot_bo6767488733719836353l_real ) ) ) ).
% insert_disjoint(1)
thf(fact_810_insert__disjoint_I1_J,axiom,
! [A2: real > a,A: set_real_a,B: set_real_a] :
( ( ( inf_inf_set_real_a @ ( insert_real_a @ A2 @ A ) @ B )
= bot_bot_set_real_a )
= ( ~ ( member_real_a @ A2 @ B )
& ( ( inf_inf_set_real_a @ A @ B )
= bot_bot_set_real_a ) ) ) ).
% insert_disjoint(1)
thf(fact_811_insert__disjoint_I1_J,axiom,
! [A2: a > real,A: set_a_real,B: set_a_real] :
( ( ( inf_inf_set_a_real @ ( insert_a_real @ A2 @ A ) @ B )
= bot_bot_set_a_real )
= ( ~ ( member_a_real @ A2 @ B )
& ( ( inf_inf_set_a_real @ A @ B )
= bot_bot_set_a_real ) ) ) ).
% insert_disjoint(1)
thf(fact_812_insert__disjoint_I1_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_813_insert__disjoint_I1_J,axiom,
! [A2: real,A: set_real,B: set_real] :
( ( ( inf_inf_set_real @ ( insert_real @ A2 @ A ) @ B )
= bot_bot_set_real )
= ( ~ ( member_real @ A2 @ B )
& ( ( inf_inf_set_real @ A @ B )
= bot_bot_set_real ) ) ) ).
% insert_disjoint(1)
thf(fact_814_insert__disjoint_I2_J,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ A ) @ B ) )
= ( ~ ( member_set_a @ A2 @ B )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_815_insert__disjoint_I2_J,axiom,
! [A2: real > real,A: set_real_real,B: set_real_real] :
( ( bot_bo6767488733719836353l_real
= ( inf_in5166753670444608447l_real @ ( insert_real_real @ A2 @ A ) @ B ) )
= ( ~ ( member_real_real @ A2 @ B )
& ( bot_bo6767488733719836353l_real
= ( inf_in5166753670444608447l_real @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_816_insert__disjoint_I2_J,axiom,
! [A2: real > a,A: set_real_a,B: set_real_a] :
( ( bot_bot_set_real_a
= ( inf_inf_set_real_a @ ( insert_real_a @ A2 @ A ) @ B ) )
= ( ~ ( member_real_a @ A2 @ B )
& ( bot_bot_set_real_a
= ( inf_inf_set_real_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_817_insert__disjoint_I2_J,axiom,
! [A2: a > real,A: set_a_real,B: set_a_real] :
( ( bot_bot_set_a_real
= ( inf_inf_set_a_real @ ( insert_a_real @ A2 @ A ) @ B ) )
= ( ~ ( member_a_real @ A2 @ B )
& ( bot_bot_set_a_real
= ( inf_inf_set_a_real @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_818_insert__disjoint_I2_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B ) )
= ( ~ ( member_a @ A2 @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_819_insert__disjoint_I2_J,axiom,
! [A2: real,A: set_real,B: set_real] :
( ( bot_bot_set_real
= ( inf_inf_set_real @ ( insert_real @ A2 @ A ) @ B ) )
= ( ~ ( member_real @ A2 @ B )
& ( bot_bot_set_real
= ( inf_inf_set_real @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_820_disjoint__insert_I1_J,axiom,
! [B: set_set_a,A2: set_a,A: set_set_a] :
( ( ( inf_inf_set_set_a @ B @ ( insert_set_a @ A2 @ A ) )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A2 @ B )
& ( ( inf_inf_set_set_a @ B @ A )
= bot_bot_set_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_821_disjoint__insert_I1_J,axiom,
! [B: set_real_real,A2: real > real,A: set_real_real] :
( ( ( inf_in5166753670444608447l_real @ B @ ( insert_real_real @ A2 @ A ) )
= bot_bo6767488733719836353l_real )
= ( ~ ( member_real_real @ A2 @ B )
& ( ( inf_in5166753670444608447l_real @ B @ A )
= bot_bo6767488733719836353l_real ) ) ) ).
% disjoint_insert(1)
thf(fact_822_disjoint__insert_I1_J,axiom,
! [B: set_real_a,A2: real > a,A: set_real_a] :
( ( ( inf_inf_set_real_a @ B @ ( insert_real_a @ A2 @ A ) )
= bot_bot_set_real_a )
= ( ~ ( member_real_a @ A2 @ B )
& ( ( inf_inf_set_real_a @ B @ A )
= bot_bot_set_real_a ) ) ) ).
% disjoint_insert(1)
thf(fact_823_disjoint__insert_I1_J,axiom,
! [B: set_a_real,A2: a > real,A: set_a_real] :
( ( ( inf_inf_set_a_real @ B @ ( insert_a_real @ A2 @ A ) )
= bot_bot_set_a_real )
= ( ~ ( member_a_real @ A2 @ B )
& ( ( inf_inf_set_a_real @ B @ A )
= bot_bot_set_a_real ) ) ) ).
% disjoint_insert(1)
thf(fact_824_disjoint__insert_I1_J,axiom,
! [B: set_a,A2: a,A: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A2 @ A ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ B @ A )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_825_disjoint__insert_I1_J,axiom,
! [B: set_real,A2: real,A: set_real] :
( ( ( inf_inf_set_real @ B @ ( insert_real @ A2 @ A ) )
= bot_bot_set_real )
= ( ~ ( member_real @ A2 @ B )
& ( ( inf_inf_set_real @ B @ A )
= bot_bot_set_real ) ) ) ).
% disjoint_insert(1)
thf(fact_826_disjoint__insert_I2_J,axiom,
! [A: set_set_a,B2: set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A @ ( insert_set_a @ B2 @ B ) ) )
= ( ~ ( member_set_a @ B2 @ A )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_827_disjoint__insert_I2_J,axiom,
! [A: set_real_real,B2: real > real,B: set_real_real] :
( ( bot_bo6767488733719836353l_real
= ( inf_in5166753670444608447l_real @ A @ ( insert_real_real @ B2 @ B ) ) )
= ( ~ ( member_real_real @ B2 @ A )
& ( bot_bo6767488733719836353l_real
= ( inf_in5166753670444608447l_real @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_828_disjoint__insert_I2_J,axiom,
! [A: set_real_a,B2: real > a,B: set_real_a] :
( ( bot_bot_set_real_a
= ( inf_inf_set_real_a @ A @ ( insert_real_a @ B2 @ B ) ) )
= ( ~ ( member_real_a @ B2 @ A )
& ( bot_bot_set_real_a
= ( inf_inf_set_real_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_829_disjoint__insert_I2_J,axiom,
! [A: set_a_real,B2: a > real,B: set_a_real] :
( ( bot_bot_set_a_real
= ( inf_inf_set_a_real @ A @ ( insert_a_real @ B2 @ B ) ) )
= ( ~ ( member_a_real @ B2 @ A )
& ( bot_bot_set_a_real
= ( inf_inf_set_a_real @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_830_disjoint__insert_I2_J,axiom,
! [A: set_a,B2: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A @ ( insert_a @ B2 @ B ) ) )
= ( ~ ( member_a @ B2 @ A )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_831_disjoint__insert_I2_J,axiom,
! [A: set_real,B2: real,B: set_real] :
( ( bot_bot_set_real
= ( inf_inf_set_real @ A @ ( insert_real @ B2 @ B ) ) )
= ( ~ ( member_real @ B2 @ A )
& ( bot_bot_set_real
= ( inf_inf_set_real @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_832_sets_OInt__space__eq2,axiom,
! [X: set_a,M: sigma_measure_a] :
( ( member_set_a @ X @ ( sigma_sets_a @ M ) )
=> ( ( inf_inf_set_a @ X @ ( sigma_space_a @ M ) )
= X ) ) ).
% sets.Int_space_eq2
thf(fact_833_sets_OInt__space__eq2,axiom,
! [X: set_real,M: sigma_measure_real] :
( ( member_set_real @ X @ ( sigma_sets_real @ M ) )
=> ( ( inf_inf_set_real @ X @ ( sigma_space_real @ M ) )
= X ) ) ).
% sets.Int_space_eq2
thf(fact_834_sets_OInt__space__eq1,axiom,
! [X: set_a,M: sigma_measure_a] :
( ( member_set_a @ X @ ( sigma_sets_a @ M ) )
=> ( ( inf_inf_set_a @ ( sigma_space_a @ M ) @ X )
= X ) ) ).
% sets.Int_space_eq1
thf(fact_835_sets_OInt__space__eq1,axiom,
! [X: set_real,M: sigma_measure_real] :
( ( member_set_real @ X @ ( sigma_sets_real @ M ) )
=> ( ( inf_inf_set_real @ ( sigma_space_real @ M ) @ X )
= X ) ) ).
% sets.Int_space_eq1
thf(fact_836_IntE,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ~ ( ( member_set_a @ C @ A )
=> ~ ( member_set_a @ C @ B ) ) ) ).
% IntE
thf(fact_837_IntE,axiom,
! [C: real > real,A: set_real_real,B: set_real_real] :
( ( member_real_real @ C @ ( inf_in5166753670444608447l_real @ A @ B ) )
=> ~ ( ( member_real_real @ C @ A )
=> ~ ( member_real_real @ C @ B ) ) ) ).
% IntE
thf(fact_838_IntE,axiom,
! [C: real > a,A: set_real_a,B: set_real_a] :
( ( member_real_a @ C @ ( inf_inf_set_real_a @ A @ B ) )
=> ~ ( ( member_real_a @ C @ A )
=> ~ ( member_real_a @ C @ B ) ) ) ).
% IntE
thf(fact_839_IntE,axiom,
! [C: a > real,A: set_a_real,B: set_a_real] :
( ( member_a_real @ C @ ( inf_inf_set_a_real @ A @ B ) )
=> ~ ( ( member_a_real @ C @ A )
=> ~ ( member_a_real @ C @ B ) ) ) ).
% IntE
thf(fact_840_IntE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ~ ( member_a @ C @ B ) ) ) ).
% IntE
thf(fact_841_IntD1,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ( member_set_a @ C @ A ) ) ).
% IntD1
thf(fact_842_IntD1,axiom,
! [C: real > real,A: set_real_real,B: set_real_real] :
( ( member_real_real @ C @ ( inf_in5166753670444608447l_real @ A @ B ) )
=> ( member_real_real @ C @ A ) ) ).
% IntD1
thf(fact_843_IntD1,axiom,
! [C: real > a,A: set_real_a,B: set_real_a] :
( ( member_real_a @ C @ ( inf_inf_set_real_a @ A @ B ) )
=> ( member_real_a @ C @ A ) ) ).
% IntD1
thf(fact_844_IntD1,axiom,
! [C: a > real,A: set_a_real,B: set_a_real] :
( ( member_a_real @ C @ ( inf_inf_set_a_real @ A @ B ) )
=> ( member_a_real @ C @ A ) ) ).
% IntD1
thf(fact_845_IntD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% IntD1
thf(fact_846_IntD2,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ( member_set_a @ C @ B ) ) ).
% IntD2
thf(fact_847_IntD2,axiom,
! [C: real > real,A: set_real_real,B: set_real_real] :
( ( member_real_real @ C @ ( inf_in5166753670444608447l_real @ A @ B ) )
=> ( member_real_real @ C @ B ) ) ).
% IntD2
thf(fact_848_IntD2,axiom,
! [C: real > a,A: set_real_a,B: set_real_a] :
( ( member_real_a @ C @ ( inf_inf_set_real_a @ A @ B ) )
=> ( member_real_a @ C @ B ) ) ).
% IntD2
thf(fact_849_IntD2,axiom,
! [C: a > real,A: set_a_real,B: set_a_real] :
( ( member_a_real @ C @ ( inf_inf_set_a_real @ A @ B ) )
=> ( member_a_real @ C @ B ) ) ).
% IntD2
thf(fact_850_IntD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ B ) ) ).
% IntD2
thf(fact_851_disjoint__iff__not__equal,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ! [Y5: a] :
( ( member_a @ Y5 @ B )
=> ( X3 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_852_disjoint__iff__not__equal,axiom,
! [A: set_real,B: set_real] :
( ( ( inf_inf_set_real @ A @ B )
= bot_bot_set_real )
= ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ! [Y5: real] :
( ( member_real @ Y5 @ B )
=> ( X3 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_853_Int__empty__right,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_854_Int__empty__right,axiom,
! [A: set_real] :
( ( inf_inf_set_real @ A @ bot_bot_set_real )
= bot_bot_set_real ) ).
% Int_empty_right
thf(fact_855_Int__empty__left,axiom,
! [B: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_856_Int__empty__left,axiom,
! [B: set_real] :
( ( inf_inf_set_real @ bot_bot_set_real @ B )
= bot_bot_set_real ) ).
% Int_empty_left
thf(fact_857_disjoint__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ~ ( member_set_a @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_858_disjoint__iff,axiom,
! [A: set_real_real,B: set_real_real] :
( ( ( inf_in5166753670444608447l_real @ A @ B )
= bot_bo6767488733719836353l_real )
= ( ! [X3: real > real] :
( ( member_real_real @ X3 @ A )
=> ~ ( member_real_real @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_859_disjoint__iff,axiom,
! [A: set_real_a,B: set_real_a] :
( ( ( inf_inf_set_real_a @ A @ B )
= bot_bot_set_real_a )
= ( ! [X3: real > a] :
( ( member_real_a @ X3 @ A )
=> ~ ( member_real_a @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_860_disjoint__iff,axiom,
! [A: set_a_real,B: set_a_real] :
( ( ( inf_inf_set_a_real @ A @ B )
= bot_bot_set_a_real )
= ( ! [X3: a > real] :
( ( member_a_real @ X3 @ A )
=> ~ ( member_a_real @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_861_disjoint__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ~ ( member_a @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_862_disjoint__iff,axiom,
! [A: set_real,B: set_real] :
( ( ( inf_inf_set_real @ A @ B )
= bot_bot_set_real )
= ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ~ ( member_real @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_863_Int__emptyI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ~ ( member_set_a @ X2 @ B ) )
=> ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_864_Int__emptyI,axiom,
! [A: set_real_real,B: set_real_real] :
( ! [X2: real > real] :
( ( member_real_real @ X2 @ A )
=> ~ ( member_real_real @ X2 @ B ) )
=> ( ( inf_in5166753670444608447l_real @ A @ B )
= bot_bo6767488733719836353l_real ) ) ).
% Int_emptyI
thf(fact_865_Int__emptyI,axiom,
! [A: set_real_a,B: set_real_a] :
( ! [X2: real > a] :
( ( member_real_a @ X2 @ A )
=> ~ ( member_real_a @ X2 @ B ) )
=> ( ( inf_inf_set_real_a @ A @ B )
= bot_bot_set_real_a ) ) ).
% Int_emptyI
thf(fact_866_Int__emptyI,axiom,
! [A: set_a_real,B: set_a_real] :
( ! [X2: a > real] :
( ( member_a_real @ X2 @ A )
=> ~ ( member_a_real @ X2 @ B ) )
=> ( ( inf_inf_set_a_real @ A @ B )
= bot_bot_set_a_real ) ) ).
% Int_emptyI
thf(fact_867_Int__emptyI,axiom,
! [A: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ~ ( member_a @ X2 @ B ) )
=> ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_868_Int__emptyI,axiom,
! [A: set_real,B: set_real] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ~ ( member_real @ X2 @ B ) )
=> ( ( inf_inf_set_real @ A @ B )
= bot_bot_set_real ) ) ).
% Int_emptyI
thf(fact_869_Int__mono,axiom,
! [A: set_real,C2: set_real,B: set_real,D2: set_real] :
( ( ord_less_eq_set_real @ A @ C2 )
=> ( ( ord_less_eq_set_real @ B @ D2 )
=> ( ord_less_eq_set_real @ ( inf_inf_set_real @ A @ B ) @ ( inf_inf_set_real @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_870_Int__mono,axiom,
! [A: set_a,C2: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_871_Int__lower1,axiom,
! [A: set_real,B: set_real] : ( ord_less_eq_set_real @ ( inf_inf_set_real @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_872_Int__lower1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_873_Int__lower2,axiom,
! [A: set_real,B: set_real] : ( ord_less_eq_set_real @ ( inf_inf_set_real @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_874_Int__lower2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_875_Int__absorb1,axiom,
! [B: set_real,A: set_real] :
( ( ord_less_eq_set_real @ B @ A )
=> ( ( inf_inf_set_real @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_876_Int__absorb1,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_877_Int__absorb2,axiom,
! [A: set_real,B: set_real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( inf_inf_set_real @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_878_Int__absorb2,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_879_Int__greatest,axiom,
! [C2: set_real,A: set_real,B: set_real] :
( ( ord_less_eq_set_real @ C2 @ A )
=> ( ( ord_less_eq_set_real @ C2 @ B )
=> ( ord_less_eq_set_real @ C2 @ ( inf_inf_set_real @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_880_Int__greatest,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_881_Int__Collect__mono,axiom,
! [A: set_set_a,B: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_882_Int__Collect__mono,axiom,
! [A: set_real_real,B: set_real_real,P: ( real > real ) > $o,Q: ( real > real ) > $o] :
( ( ord_le4198349162570665613l_real @ A @ B )
=> ( ! [X2: real > real] :
( ( member_real_real @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le4198349162570665613l_real @ ( inf_in5166753670444608447l_real @ A @ ( collect_real_real @ P ) ) @ ( inf_in5166753670444608447l_real @ B @ ( collect_real_real @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_883_Int__Collect__mono,axiom,
! [A: set_real_a,B: set_real_a,P: ( real > a ) > $o,Q: ( real > a ) > $o] :
( ( ord_le5743406823621094409real_a @ A @ B )
=> ( ! [X2: real > a] :
( ( member_real_a @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le5743406823621094409real_a @ ( inf_inf_set_real_a @ A @ ( collect_real_a @ P ) ) @ ( inf_inf_set_real_a @ B @ ( collect_real_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_884_Int__Collect__mono,axiom,
! [A: set_a_real,B: set_a_real,P: ( a > real ) > $o,Q: ( a > real ) > $o] :
( ( ord_le3334967407727675675a_real @ A @ B )
=> ( ! [X2: a > real] :
( ( member_a_real @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le3334967407727675675a_real @ ( inf_inf_set_a_real @ A @ ( collect_a_real @ P ) ) @ ( inf_inf_set_a_real @ B @ ( collect_a_real @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_885_Int__Collect__mono,axiom,
! [A: set_real,B: set_real,P: real > $o,Q: real > $o] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_real @ ( inf_inf_set_real @ A @ ( collect_real @ P ) ) @ ( inf_inf_set_real @ B @ ( collect_real @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_886_Int__Collect__mono,axiom,
! [A: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_887_Int__insert__left,axiom,
! [A2: real,C2: set_real,B: set_real] :
( ( ( member_real @ A2 @ C2 )
=> ( ( inf_inf_set_real @ ( insert_real @ A2 @ B ) @ C2 )
= ( insert_real @ A2 @ ( inf_inf_set_real @ B @ C2 ) ) ) )
& ( ~ ( member_real @ A2 @ C2 )
=> ( ( inf_inf_set_real @ ( insert_real @ A2 @ B ) @ C2 )
= ( inf_inf_set_real @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_888_Int__insert__left,axiom,
! [A2: set_a,C2: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A2 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C2 )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_set_a @ A2 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_889_Int__insert__left,axiom,
! [A2: real > real,C2: set_real_real,B: set_real_real] :
( ( ( member_real_real @ A2 @ C2 )
=> ( ( inf_in5166753670444608447l_real @ ( insert_real_real @ A2 @ B ) @ C2 )
= ( insert_real_real @ A2 @ ( inf_in5166753670444608447l_real @ B @ C2 ) ) ) )
& ( ~ ( member_real_real @ A2 @ C2 )
=> ( ( inf_in5166753670444608447l_real @ ( insert_real_real @ A2 @ B ) @ C2 )
= ( inf_in5166753670444608447l_real @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_890_Int__insert__left,axiom,
! [A2: real > a,C2: set_real_a,B: set_real_a] :
( ( ( member_real_a @ A2 @ C2 )
=> ( ( inf_inf_set_real_a @ ( insert_real_a @ A2 @ B ) @ C2 )
= ( insert_real_a @ A2 @ ( inf_inf_set_real_a @ B @ C2 ) ) ) )
& ( ~ ( member_real_a @ A2 @ C2 )
=> ( ( inf_inf_set_real_a @ ( insert_real_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_real_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_891_Int__insert__left,axiom,
! [A2: a > real,C2: set_a_real,B: set_a_real] :
( ( ( member_a_real @ A2 @ C2 )
=> ( ( inf_inf_set_a_real @ ( insert_a_real @ A2 @ B ) @ C2 )
= ( insert_a_real @ A2 @ ( inf_inf_set_a_real @ B @ C2 ) ) ) )
& ( ~ ( member_a_real @ A2 @ C2 )
=> ( ( inf_inf_set_a_real @ ( insert_a_real @ A2 @ B ) @ C2 )
= ( inf_inf_set_a_real @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_892_Int__insert__left,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_893_Int__insert__right,axiom,
! [A2: real,A: set_real,B: set_real] :
( ( ( member_real @ A2 @ A )
=> ( ( inf_inf_set_real @ A @ ( insert_real @ A2 @ B ) )
= ( insert_real @ A2 @ ( inf_inf_set_real @ A @ B ) ) ) )
& ( ~ ( member_real @ A2 @ A )
=> ( ( inf_inf_set_real @ A @ ( insert_real @ A2 @ B ) )
= ( inf_inf_set_real @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_894_Int__insert__right,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ A @ B ) ) ) )
& ( ~ ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_895_Int__insert__right,axiom,
! [A2: real > real,A: set_real_real,B: set_real_real] :
( ( ( member_real_real @ A2 @ A )
=> ( ( inf_in5166753670444608447l_real @ A @ ( insert_real_real @ A2 @ B ) )
= ( insert_real_real @ A2 @ ( inf_in5166753670444608447l_real @ A @ B ) ) ) )
& ( ~ ( member_real_real @ A2 @ A )
=> ( ( inf_in5166753670444608447l_real @ A @ ( insert_real_real @ A2 @ B ) )
= ( inf_in5166753670444608447l_real @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_896_Int__insert__right,axiom,
! [A2: real > a,A: set_real_a,B: set_real_a] :
( ( ( member_real_a @ A2 @ A )
=> ( ( inf_inf_set_real_a @ A @ ( insert_real_a @ A2 @ B ) )
= ( insert_real_a @ A2 @ ( inf_inf_set_real_a @ A @ B ) ) ) )
& ( ~ ( member_real_a @ A2 @ A )
=> ( ( inf_inf_set_real_a @ A @ ( insert_real_a @ A2 @ B ) )
= ( inf_inf_set_real_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_897_Int__insert__right,axiom,
! [A2: a > real,A: set_a_real,B: set_a_real] :
( ( ( member_a_real @ A2 @ A )
=> ( ( inf_inf_set_a_real @ A @ ( insert_a_real @ A2 @ B ) )
= ( insert_a_real @ A2 @ ( inf_inf_set_a_real @ A @ B ) ) ) )
& ( ~ ( member_a_real @ A2 @ A )
=> ( ( inf_inf_set_a_real @ A @ ( insert_a_real @ A2 @ B ) )
= ( inf_inf_set_a_real @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_898_Int__insert__right,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) )
& ( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_899_inj__on__Int,axiom,
! [F: a > real,A: set_a,B: set_a] :
( ( ( inj_on_a_real @ F @ A )
| ( inj_on_a_real @ F @ B ) )
=> ( inj_on_a_real @ F @ ( inf_inf_set_a @ A @ B ) ) ) ).
% inj_on_Int
thf(fact_900_inj__on__Int,axiom,
! [F: real > real,A: set_real,B: set_real] :
( ( ( inj_on_real_real @ F @ A )
| ( inj_on_real_real @ F @ B ) )
=> ( inj_on_real_real @ F @ ( inf_inf_set_real @ A @ B ) ) ) ).
% inj_on_Int
thf(fact_901_vimage__inter__cong,axiom,
! [S: set_real,F: real > a,G: real > a,Y3: set_a] :
( ! [W: real] :
( ( member_real @ W @ S )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( inf_inf_set_real @ ( vimage_real_a @ F @ Y3 ) @ S )
= ( inf_inf_set_real @ ( vimage_real_a @ G @ Y3 ) @ S ) ) ) ).
% vimage_inter_cong
thf(fact_902_vimage__inter__cong,axiom,
! [S: set_real,F: real > real,G: real > real,Y3: set_real] :
( ! [W: real] :
( ( member_real @ W @ S )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( inf_inf_set_real @ ( vimage_real_real @ F @ Y3 ) @ S )
= ( inf_inf_set_real @ ( vimage_real_real @ G @ Y3 ) @ S ) ) ) ).
% vimage_inter_cong
thf(fact_903_vimage__inter__cong,axiom,
! [S: set_a,F: a > real,G: a > real,Y3: set_real] :
( ! [W: a] :
( ( member_a @ W @ S )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( inf_inf_set_a @ ( vimage_a_real @ F @ Y3 ) @ S )
= ( inf_inf_set_a @ ( vimage_a_real @ G @ Y3 ) @ S ) ) ) ).
% vimage_inter_cong
thf(fact_904_vimage__inter__cong,axiom,
! [S: set_a,F: a > a,G: a > a,Y3: set_a] :
( ! [W: a] :
( ( member_a @ W @ S )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( inf_inf_set_a @ ( vimage_a_a @ F @ Y3 ) @ S )
= ( inf_inf_set_a @ ( vimage_a_a @ G @ Y3 ) @ S ) ) ) ).
% vimage_inter_cong
thf(fact_905_image__Int__subset,axiom,
! [F: a > real,A: set_a,B: set_a] : ( ord_less_eq_set_real @ ( image_a_real @ F @ ( inf_inf_set_a @ A @ B ) ) @ ( inf_inf_set_real @ ( image_a_real @ F @ A ) @ ( image_a_real @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_906_image__Int__subset,axiom,
! [F: real > real,A: set_real,B: set_real] : ( ord_less_eq_set_real @ ( image_real_real @ F @ ( inf_inf_set_real @ A @ B ) ) @ ( inf_inf_set_real @ ( image_real_real @ F @ A ) @ ( image_real_real @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_907_image__Int__subset,axiom,
! [F: real > a,A: set_real,B: set_real] : ( ord_less_eq_set_a @ ( image_real_a @ F @ ( inf_inf_set_real @ A @ B ) ) @ ( inf_inf_set_a @ ( image_real_a @ F @ A ) @ ( image_real_a @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_908_image__Int__subset,axiom,
! [F: a > a,A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F @ ( inf_inf_set_a @ A @ B ) ) @ ( inf_inf_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_909_is__singletonI_H,axiom,
! [A: set_set_a] :
( ( A != bot_bot_set_set_a )
=> ( ! [X2: set_a,Y2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ( member_set_a @ Y2 @ A )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_set_a @ A ) ) ) ).
% is_singletonI'
thf(fact_910_is__singletonI_H,axiom,
! [A: set_real_real] :
( ( A != bot_bo6767488733719836353l_real )
=> ( ! [X2: real > real,Y2: real > real] :
( ( member_real_real @ X2 @ A )
=> ( ( member_real_real @ Y2 @ A )
=> ( X2 = Y2 ) ) )
=> ( is_sin8515336128161245059l_real @ A ) ) ) ).
% is_singletonI'
thf(fact_911_is__singletonI_H,axiom,
! [A: set_real_a] :
( ( A != bot_bot_set_real_a )
=> ( ! [X2: real > a,Y2: real > a] :
( ( member_real_a @ X2 @ A )
=> ( ( member_real_a @ Y2 @ A )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_real_a @ A ) ) ) ).
% is_singletonI'
thf(fact_912_is__singletonI_H,axiom,
! [A: set_a_real] :
( ( A != bot_bot_set_a_real )
=> ( ! [X2: a > real,Y2: a > real] :
( ( member_a_real @ X2 @ A )
=> ( ( member_a_real @ Y2 @ A )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_a_real @ A ) ) ) ).
% is_singletonI'
thf(fact_913_is__singletonI_H,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
=> ( ! [X2: a,Y2: a] :
( ( member_a @ X2 @ A )
=> ( ( member_a @ Y2 @ A )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_a @ A ) ) ) ).
% is_singletonI'
thf(fact_914_is__singletonI_H,axiom,
! [A: set_real] :
( ( A != bot_bot_set_real )
=> ( ! [X2: real,Y2: real] :
( ( member_real @ X2 @ A )
=> ( ( member_real @ Y2 @ A )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_real @ A ) ) ) ).
% is_singletonI'
thf(fact_915_real_Of__meas,axiom,
member_real_real @ ( standard_f_real @ borel_5078946678739801102l_real ) @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ).
% real.f_meas
thf(fact_916_real_Og__meas,axiom,
member_real_real @ ( standard_g_real @ borel_5078946678739801102l_real ) @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ).
% real.g_meas
thf(fact_917_inj__on__image__Int,axiom,
! [F: real > a,C2: set_real,A: set_real,B: set_real] :
( ( inj_on_real_a @ F @ C2 )
=> ( ( ord_less_eq_set_real @ A @ C2 )
=> ( ( ord_less_eq_set_real @ B @ C2 )
=> ( ( image_real_a @ F @ ( inf_inf_set_real @ A @ B ) )
= ( inf_inf_set_a @ ( image_real_a @ F @ A ) @ ( image_real_a @ F @ B ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_918_inj__on__image__Int,axiom,
! [F: real > real,C2: set_real,A: set_real,B: set_real] :
( ( inj_on_real_real @ F @ C2 )
=> ( ( ord_less_eq_set_real @ A @ C2 )
=> ( ( ord_less_eq_set_real @ B @ C2 )
=> ( ( image_real_real @ F @ ( inf_inf_set_real @ A @ B ) )
= ( inf_inf_set_real @ ( image_real_real @ F @ A ) @ ( image_real_real @ F @ B ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_919_inj__on__image__Int,axiom,
! [F: a > a,C2: set_a,A: set_a,B: set_a] :
( ( inj_on_a_a @ F @ C2 )
=> ( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ( image_a_a @ F @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_920_inj__on__image__Int,axiom,
! [F: a > real,C2: set_a,A: set_a,B: set_a] :
( ( inj_on_a_real @ F @ C2 )
=> ( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ( image_a_real @ F @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_real @ ( image_a_real @ F @ A ) @ ( image_a_real @ F @ B ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_921_sets__Ball,axiom,
! [I: set_set_a,A: set_a > set_a,M: set_a > sigma_measure_a,I2: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ I )
=> ( member_set_a @ ( A @ X2 ) @ ( sigma_sets_a @ ( M @ X2 ) ) ) )
=> ( ( member_set_a @ I2 @ I )
=> ( member_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_922_sets__Ball,axiom,
! [I: set_real_real,A: ( real > real ) > set_a,M: ( real > real ) > sigma_measure_a,I2: real > real] :
( ! [X2: real > real] :
( ( member_real_real @ X2 @ I )
=> ( member_set_a @ ( A @ X2 ) @ ( sigma_sets_a @ ( M @ X2 ) ) ) )
=> ( ( member_real_real @ I2 @ I )
=> ( member_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_923_sets__Ball,axiom,
! [I: set_real_a,A: ( real > a ) > set_a,M: ( real > a ) > sigma_measure_a,I2: real > a] :
( ! [X2: real > a] :
( ( member_real_a @ X2 @ I )
=> ( member_set_a @ ( A @ X2 ) @ ( sigma_sets_a @ ( M @ X2 ) ) ) )
=> ( ( member_real_a @ I2 @ I )
=> ( member_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_924_sets__Ball,axiom,
! [I: set_a_real,A: ( a > real ) > set_a,M: ( a > real ) > sigma_measure_a,I2: a > real] :
( ! [X2: a > real] :
( ( member_a_real @ X2 @ I )
=> ( member_set_a @ ( A @ X2 ) @ ( sigma_sets_a @ ( M @ X2 ) ) ) )
=> ( ( member_a_real @ I2 @ I )
=> ( member_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_925_sets__Ball,axiom,
! [I: set_a,A: a > set_a,M: a > sigma_measure_a,I2: a] :
( ! [X2: a] :
( ( member_a @ X2 @ I )
=> ( member_set_a @ ( A @ X2 ) @ ( sigma_sets_a @ ( M @ X2 ) ) ) )
=> ( ( member_a @ I2 @ I )
=> ( member_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_926_sets__Ball,axiom,
! [I: set_set_a,A: set_a > set_real,M: set_a > sigma_measure_real,I2: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ I )
=> ( member_set_real @ ( A @ X2 ) @ ( sigma_sets_real @ ( M @ X2 ) ) ) )
=> ( ( member_set_a @ I2 @ I )
=> ( member_set_real @ ( A @ I2 ) @ ( sigma_sets_real @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_927_sets__Ball,axiom,
! [I: set_real_real,A: ( real > real ) > set_real,M: ( real > real ) > sigma_measure_real,I2: real > real] :
( ! [X2: real > real] :
( ( member_real_real @ X2 @ I )
=> ( member_set_real @ ( A @ X2 ) @ ( sigma_sets_real @ ( M @ X2 ) ) ) )
=> ( ( member_real_real @ I2 @ I )
=> ( member_set_real @ ( A @ I2 ) @ ( sigma_sets_real @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_928_sets__Ball,axiom,
! [I: set_real_a,A: ( real > a ) > set_real,M: ( real > a ) > sigma_measure_real,I2: real > a] :
( ! [X2: real > a] :
( ( member_real_a @ X2 @ I )
=> ( member_set_real @ ( A @ X2 ) @ ( sigma_sets_real @ ( M @ X2 ) ) ) )
=> ( ( member_real_a @ I2 @ I )
=> ( member_set_real @ ( A @ I2 ) @ ( sigma_sets_real @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_929_sets__Ball,axiom,
! [I: set_a_real,A: ( a > real ) > set_real,M: ( a > real ) > sigma_measure_real,I2: a > real] :
( ! [X2: a > real] :
( ( member_a_real @ X2 @ I )
=> ( member_set_real @ ( A @ X2 ) @ ( sigma_sets_real @ ( M @ X2 ) ) ) )
=> ( ( member_a_real @ I2 @ I )
=> ( member_set_real @ ( A @ I2 ) @ ( sigma_sets_real @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_930_sets__Ball,axiom,
! [I: set_a,A: a > set_real,M: a > sigma_measure_real,I2: a] :
( ! [X2: a] :
( ( member_a @ X2 @ I )
=> ( member_set_real @ ( A @ X2 ) @ ( sigma_sets_real @ ( M @ X2 ) ) ) )
=> ( ( member_a @ I2 @ I )
=> ( member_set_real @ ( A @ I2 ) @ ( sigma_sets_real @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_931_real_Oexist__fg,axiom,
? [X2: real > real] :
( ( member_real_real @ X2 @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
& ? [Xa: real > real] :
( ( member_real_real @ Xa @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
& ! [Xb: real] :
( ( member_real @ Xb @ ( sigma_space_real @ borel_5078946678739801102l_real ) )
=> ( ( comp_real_real_real @ Xa @ X2 @ Xb )
= Xb ) ) ) ) ).
% real.exist_fg
thf(fact_932_measurable__sets,axiom,
! [F: a > a,M: sigma_measure_a,A: sigma_measure_a,S: set_a] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ A ) )
=> ( ( member_set_a @ S @ ( sigma_sets_a @ A ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_a @ F @ S ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) ) ) ).
% measurable_sets
thf(fact_933_measurable__sets,axiom,
! [F: a > real,M: sigma_measure_a,A: sigma_measure_real,S: set_real] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ A ) )
=> ( ( member_set_real @ S @ ( sigma_sets_real @ A ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_real @ F @ S ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) ) ) ).
% measurable_sets
thf(fact_934_measurable__sets,axiom,
! [F: real > a,M: sigma_measure_real,A: sigma_measure_a,S: set_a] :
( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ A ) )
=> ( ( member_set_a @ S @ ( sigma_sets_a @ A ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_a @ F @ S ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) ) ) ).
% measurable_sets
thf(fact_935_measurable__sets,axiom,
! [F: real > real,M: sigma_measure_real,A: sigma_measure_real,S: set_real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ A ) )
=> ( ( member_set_real @ S @ ( sigma_sets_real @ A ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_real @ F @ S ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) ) ) ).
% measurable_sets
thf(fact_936_measurableI,axiom,
! [M: sigma_measure_a,F: a > a,N: sigma_measure_a] :
( ! [X2: a] :
( ( member_a @ X2 @ ( sigma_space_a @ M ) )
=> ( member_a @ ( F @ X2 ) @ ( sigma_space_a @ N ) ) )
=> ( ! [A8: set_a] :
( ( member_set_a @ A8 @ ( sigma_sets_a @ N ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_a @ F @ A8 ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) )
=> ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ N ) ) ) ) ).
% measurableI
thf(fact_937_measurableI,axiom,
! [M: sigma_measure_a,F: a > real,N: sigma_measure_real] :
( ! [X2: a] :
( ( member_a @ X2 @ ( sigma_space_a @ M ) )
=> ( member_real @ ( F @ X2 ) @ ( sigma_space_real @ N ) ) )
=> ( ! [A8: set_real] :
( ( member_set_real @ A8 @ ( sigma_sets_real @ N ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_real @ F @ A8 ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) )
=> ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ N ) ) ) ) ).
% measurableI
thf(fact_938_measurableI,axiom,
! [M: sigma_measure_real,F: real > a,N: sigma_measure_a] :
( ! [X2: real] :
( ( member_real @ X2 @ ( sigma_space_real @ M ) )
=> ( member_a @ ( F @ X2 ) @ ( sigma_space_a @ N ) ) )
=> ( ! [A8: set_a] :
( ( member_set_a @ A8 @ ( sigma_sets_a @ N ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_a @ F @ A8 ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) )
=> ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ N ) ) ) ) ).
% measurableI
thf(fact_939_measurableI,axiom,
! [M: sigma_measure_real,F: real > real,N: sigma_measure_real] :
( ! [X2: real] :
( ( member_real @ X2 @ ( sigma_space_real @ M ) )
=> ( member_real @ ( F @ X2 ) @ ( sigma_space_real @ N ) ) )
=> ( ! [A8: set_real] :
( ( member_set_real @ A8 @ ( sigma_sets_real @ N ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_real @ F @ A8 ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) )
=> ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) ) ) ) ).
% measurableI
thf(fact_940_measurableI,axiom,
! [M: sigma_measure_set_a,F: set_a > a,N: sigma_measure_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( sigma_space_set_a @ M ) )
=> ( member_a @ ( F @ X2 ) @ ( sigma_space_a @ N ) ) )
=> ( ! [A8: set_a] :
( ( member_set_a @ A8 @ ( sigma_sets_a @ N ) )
=> ( member_set_set_a @ ( inf_inf_set_set_a @ ( vimage_set_a_a @ F @ A8 ) @ ( sigma_space_set_a @ M ) ) @ ( sigma_sets_set_a @ M ) ) )
=> ( member_set_a_a @ F @ ( sigma_3901645225212141168et_a_a @ M @ N ) ) ) ) ).
% measurableI
thf(fact_941_measurableI,axiom,
! [M: sigma_measure_set_a,F: set_a > real,N: sigma_measure_real] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( sigma_space_set_a @ M ) )
=> ( member_real @ ( F @ X2 ) @ ( sigma_space_real @ N ) ) )
=> ( ! [A8: set_real] :
( ( member_set_real @ A8 @ ( sigma_sets_real @ N ) )
=> ( member_set_set_a @ ( inf_inf_set_set_a @ ( vimage_set_a_real @ F @ A8 ) @ ( sigma_space_set_a @ M ) ) @ ( sigma_sets_set_a @ M ) ) )
=> ( member_set_a_real @ F @ ( sigma_567508090560183226a_real @ M @ N ) ) ) ) ).
% measurableI
thf(fact_942_measurableI,axiom,
! [M: sigma_measure_a,F: a > set_a,N: sigma_measure_set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ ( sigma_space_a @ M ) )
=> ( member_set_a @ ( F @ X2 ) @ ( sigma_space_set_a @ N ) ) )
=> ( ! [A8: set_set_a] :
( ( member_set_set_a @ A8 @ ( sigma_sets_set_a @ N ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_set_a @ F @ A8 ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) )
=> ( member_a_set_a @ F @ ( sigma_3685133166752798000_set_a @ M @ N ) ) ) ) ).
% measurableI
thf(fact_943_measurableI,axiom,
! [M: sigma_measure_real,F: real > set_a,N: sigma_measure_set_a] :
( ! [X2: real] :
( ( member_real @ X2 @ ( sigma_space_real @ M ) )
=> ( member_set_a @ ( F @ X2 ) @ ( sigma_space_set_a @ N ) ) )
=> ( ! [A8: set_set_a] :
( ( member_set_set_a @ A8 @ ( sigma_sets_set_a @ N ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_set_a @ F @ A8 ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) )
=> ( member_real_set_a @ F @ ( sigma_4283435981211228640_set_a @ M @ N ) ) ) ) ).
% measurableI
thf(fact_944_measurableI,axiom,
! [M: sigma_measure_set_a,F: set_a > set_a,N: sigma_measure_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( sigma_space_set_a @ M ) )
=> ( member_set_a @ ( F @ X2 ) @ ( sigma_space_set_a @ N ) ) )
=> ( ! [A8: set_set_a] :
( ( member_set_set_a @ A8 @ ( sigma_sets_set_a @ N ) )
=> ( member_set_set_a @ ( inf_inf_set_set_a @ ( vimage_set_a_set_a @ F @ A8 ) @ ( sigma_space_set_a @ M ) ) @ ( sigma_sets_set_a @ M ) ) )
=> ( member_set_a_set_a @ F @ ( sigma_5212894042034225104_set_a @ M @ N ) ) ) ) ).
% measurableI
thf(fact_945_measurableI,axiom,
! [M: sigma_4258434043392614480l_real,F: ( real > real ) > a,N: sigma_measure_a] :
( ! [X2: real > real] :
( ( member_real_real @ X2 @ ( sigma_3619470280215722479l_real @ M ) )
=> ( member_a @ ( F @ X2 ) @ ( sigma_space_a @ N ) ) )
=> ( ! [A8: set_a] :
( ( member_set_a @ A8 @ ( sigma_sets_a @ N ) )
=> ( member_set_real_real @ ( inf_in5166753670444608447l_real @ ( vimage_real_real_a @ F @ A8 ) @ ( sigma_3619470280215722479l_real @ M ) ) @ ( sigma_sets_real_real @ M ) ) )
=> ( member_real_real_a @ F @ ( sigma_2844068865785385077real_a @ M @ N ) ) ) ) ).
% measurableI
thf(fact_946_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A4: set_a] :
? [X3: a] :
( A4
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_947_is__singleton__def,axiom,
( is_singleton_real
= ( ^ [A4: set_real] :
? [X3: real] :
( A4
= ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ).
% is_singleton_def
thf(fact_948_is__singletonE,axiom,
! [A: set_a] :
( ( is_singleton_a @ A )
=> ~ ! [X2: a] :
( A
!= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_949_is__singletonE,axiom,
! [A: set_real] :
( ( is_singleton_real @ A )
=> ~ ! [X2: real] :
( A
!= ( insert_real @ X2 @ bot_bot_set_real ) ) ) ).
% is_singletonE
thf(fact_950_in__borel__measurable__borel,axiom,
! [F: a > real,M: sigma_measure_a] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
= ( ! [X3: set_real] :
( ( member_set_real @ X3 @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_real @ F @ X3 ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) ) ) ) ).
% in_borel_measurable_borel
thf(fact_951_in__borel__measurable__borel,axiom,
! [F: real > real,M: sigma_measure_real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
= ( ! [X3: set_real] :
( ( member_set_real @ X3 @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_real @ F @ X3 ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) ) ) ) ).
% in_borel_measurable_borel
thf(fact_952_inf__bot__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_953_inf__bot__left,axiom,
! [X: set_real] :
( ( inf_inf_set_real @ bot_bot_set_real @ X )
= bot_bot_set_real ) ).
% inf_bot_left
thf(fact_954_inf__bot__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_955_inf__bot__right,axiom,
! [X: set_real] :
( ( inf_inf_set_real @ X @ bot_bot_set_real )
= bot_bot_set_real ) ).
% inf_bot_right
thf(fact_956_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_957_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_real] :
( ( inf_inf_set_real @ bot_bot_set_real @ X )
= bot_bot_set_real ) ).
% boolean_algebra.conj_zero_left
thf(fact_958_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_959_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_real] :
( ( inf_inf_set_real @ X @ bot_bot_set_real )
= bot_bot_set_real ) ).
% boolean_algebra.conj_zero_right
thf(fact_960_inf_Obounded__iff,axiom,
! [A2: set_real,B2: set_real,C: set_real] :
( ( ord_less_eq_set_real @ A2 @ ( inf_inf_set_real @ B2 @ C ) )
= ( ( ord_less_eq_set_real @ A2 @ B2 )
& ( ord_less_eq_set_real @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_961_inf_Obounded__iff,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
= ( ( ord_less_eq_set_a @ A2 @ B2 )
& ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_962_le__inf__iff,axiom,
! [X: set_real,Y3: set_real,Z2: set_real] :
( ( ord_less_eq_set_real @ X @ ( inf_inf_set_real @ Y3 @ Z2 ) )
= ( ( ord_less_eq_set_real @ X @ Y3 )
& ( ord_less_eq_set_real @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_963_le__inf__iff,axiom,
! [X: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y3 @ Z2 ) )
= ( ( ord_less_eq_set_a @ X @ Y3 )
& ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_964_inf__sup__ord_I2_J,axiom,
! [X: set_real,Y3: set_real] : ( ord_less_eq_set_real @ ( inf_inf_set_real @ X @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_965_inf__sup__ord_I2_J,axiom,
! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_966_inf__sup__ord_I1_J,axiom,
! [X: set_real,Y3: set_real] : ( ord_less_eq_set_real @ ( inf_inf_set_real @ X @ Y3 ) @ X ) ).
% inf_sup_ord(1)
thf(fact_967_inf__sup__ord_I1_J,axiom,
! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y3 ) @ X ) ).
% inf_sup_ord(1)
thf(fact_968_inf__le1,axiom,
! [X: set_real,Y3: set_real] : ( ord_less_eq_set_real @ ( inf_inf_set_real @ X @ Y3 ) @ X ) ).
% inf_le1
thf(fact_969_inf__le1,axiom,
! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y3 ) @ X ) ).
% inf_le1
thf(fact_970_inf__le2,axiom,
! [X: set_real,Y3: set_real] : ( ord_less_eq_set_real @ ( inf_inf_set_real @ X @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_971_inf__le2,axiom,
! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_972_le__infE,axiom,
! [X: set_real,A2: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ X @ ( inf_inf_set_real @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_real @ X @ A2 )
=> ~ ( ord_less_eq_set_real @ X @ B2 ) ) ) ).
% le_infE
thf(fact_973_le__infE,axiom,
! [X: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A2 )
=> ~ ( ord_less_eq_set_a @ X @ B2 ) ) ) ).
% le_infE
thf(fact_974_le__infI,axiom,
! [X: set_real,A2: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ X @ A2 )
=> ( ( ord_less_eq_set_real @ X @ B2 )
=> ( ord_less_eq_set_real @ X @ ( inf_inf_set_real @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_975_le__infI,axiom,
! [X: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ X @ B2 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_976_inf__mono,axiom,
! [A2: set_real,C: set_real,B2: set_real,D: set_real] :
( ( ord_less_eq_set_real @ A2 @ C )
=> ( ( ord_less_eq_set_real @ B2 @ D )
=> ( ord_less_eq_set_real @ ( inf_inf_set_real @ A2 @ B2 ) @ ( inf_inf_set_real @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_977_inf__mono,axiom,
! [A2: set_a,C: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_978_le__infI1,axiom,
! [A2: set_real,X: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ A2 @ X )
=> ( ord_less_eq_set_real @ ( inf_inf_set_real @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_979_le__infI1,axiom,
! [A2: set_a,X: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_980_le__infI2,axiom,
! [B2: set_real,X: set_real,A2: set_real] :
( ( ord_less_eq_set_real @ B2 @ X )
=> ( ord_less_eq_set_real @ ( inf_inf_set_real @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_981_le__infI2,axiom,
! [B2: set_a,X: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_982_inf_OorderE,axiom,
! [A2: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_real @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_983_inf_OorderE,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_984_inf_OorderI,axiom,
! [A2: set_real,B2: set_real] :
( ( A2
= ( inf_inf_set_real @ A2 @ B2 ) )
=> ( ord_less_eq_set_real @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_985_inf_OorderI,axiom,
! [A2: set_a,B2: set_a] :
( ( A2
= ( inf_inf_set_a @ A2 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_986_inf__unique,axiom,
! [F: set_real > set_real > set_real,X: set_real,Y3: set_real] :
( ! [X2: set_real,Y2: set_real] : ( ord_less_eq_set_real @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: set_real,Y2: set_real] : ( ord_less_eq_set_real @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: set_real,Y2: set_real,Z4: set_real] :
( ( ord_less_eq_set_real @ X2 @ Y2 )
=> ( ( ord_less_eq_set_real @ X2 @ Z4 )
=> ( ord_less_eq_set_real @ X2 @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( inf_inf_set_real @ X @ Y3 )
= ( F @ X @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_987_inf__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y3: set_a] :
( ! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: set_a,Y2: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ( ord_less_eq_set_a @ X2 @ Z4 )
=> ( ord_less_eq_set_a @ X2 @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y3 )
= ( F @ X @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_988_le__iff__inf,axiom,
( ord_less_eq_set_real
= ( ^ [X3: set_real,Y5: set_real] :
( ( inf_inf_set_real @ X3 @ Y5 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_989_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y5: set_a] :
( ( inf_inf_set_a @ X3 @ Y5 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_990_inf_Oabsorb1,axiom,
! [A2: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( inf_inf_set_real @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_991_inf_Oabsorb1,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_992_inf_Oabsorb2,axiom,
! [B2: set_real,A2: set_real] :
( ( ord_less_eq_set_real @ B2 @ A2 )
=> ( ( inf_inf_set_real @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_993_inf_Oabsorb2,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_994_inf__absorb1,axiom,
! [X: set_real,Y3: set_real] :
( ( ord_less_eq_set_real @ X @ Y3 )
=> ( ( inf_inf_set_real @ X @ Y3 )
= X ) ) ).
% inf_absorb1
thf(fact_995_inf__absorb1,axiom,
! [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ( inf_inf_set_a @ X @ Y3 )
= X ) ) ).
% inf_absorb1
thf(fact_996_inf__absorb2,axiom,
! [Y3: set_real,X: set_real] :
( ( ord_less_eq_set_real @ Y3 @ X )
=> ( ( inf_inf_set_real @ X @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_997_inf__absorb2,axiom,
! [Y3: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X )
=> ( ( inf_inf_set_a @ X @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_998_inf_OboundedE,axiom,
! [A2: set_real,B2: set_real,C: set_real] :
( ( ord_less_eq_set_real @ A2 @ ( inf_inf_set_real @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_real @ A2 @ B2 )
=> ~ ( ord_less_eq_set_real @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_999_inf_OboundedE,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_1000_inf_OboundedI,axiom,
! [A2: set_real,B2: set_real,C: set_real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( ord_less_eq_set_real @ A2 @ C )
=> ( ord_less_eq_set_real @ A2 @ ( inf_inf_set_real @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1001_inf_OboundedI,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ C )
=> ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1002_inf__greatest,axiom,
! [X: set_real,Y3: set_real,Z2: set_real] :
( ( ord_less_eq_set_real @ X @ Y3 )
=> ( ( ord_less_eq_set_real @ X @ Z2 )
=> ( ord_less_eq_set_real @ X @ ( inf_inf_set_real @ Y3 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_1003_inf__greatest,axiom,
! [X: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ( ord_less_eq_set_a @ X @ Z2 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y3 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_1004_inf_Oorder__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B3: set_real] :
( A3
= ( inf_inf_set_real @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_1005_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( A3
= ( inf_inf_set_a @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_1006_inf_Ocobounded1,axiom,
! [A2: set_real,B2: set_real] : ( ord_less_eq_set_real @ ( inf_inf_set_real @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_1007_inf_Ocobounded1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_1008_inf_Ocobounded2,axiom,
! [A2: set_real,B2: set_real] : ( ord_less_eq_set_real @ ( inf_inf_set_real @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_1009_inf_Ocobounded2,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_1010_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B3: set_real] :
( ( inf_inf_set_real @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_1011_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( inf_inf_set_a @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_1012_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_real
= ( ^ [B3: set_real,A3: set_real] :
( ( inf_inf_set_real @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_1013_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( inf_inf_set_a @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_1014_inf_OcoboundedI1,axiom,
! [A2: set_real,C: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ A2 @ C )
=> ( ord_less_eq_set_real @ ( inf_inf_set_real @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1015_inf_OcoboundedI1,axiom,
! [A2: set_a,C: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1016_inf_OcoboundedI2,axiom,
! [B2: set_real,C: set_real,A2: set_real] :
( ( ord_less_eq_set_real @ B2 @ C )
=> ( ord_less_eq_set_real @ ( inf_inf_set_real @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1017_inf_OcoboundedI2,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1018_Sup_OSUP__image,axiom,
! [Sup: set_real > real,G: a > real,F: real > a,A: set_real] :
( ( Sup @ ( image_a_real @ G @ ( image_real_a @ F @ A ) ) )
= ( Sup @ ( image_real_real @ ( comp_a_real_real @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_1019_Sup_OSUP__image,axiom,
! [Sup: set_real > real,G: a > real,F: a > a,A: set_a] :
( ( Sup @ ( image_a_real @ G @ ( image_a_a @ F @ A ) ) )
= ( Sup @ ( image_a_real @ ( comp_a_real_a @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_1020_Sup_OSUP__image,axiom,
! [Sup: set_real > real,G: real > real,F: a > real,A: set_a] :
( ( Sup @ ( image_real_real @ G @ ( image_a_real @ F @ A ) ) )
= ( Sup @ ( image_a_real @ ( comp_real_real_a @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_1021_Sup_OSUP__image,axiom,
! [Sup: set_real > real,G: real > real,F: real > real,A: set_real] :
( ( Sup @ ( image_real_real @ G @ ( image_real_real @ F @ A ) ) )
= ( Sup @ ( image_real_real @ ( comp_real_real_real @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_1022_Sup_OSUP__image,axiom,
! [Sup: set_a > a,G: real > a,F: a > real,A: set_a] :
( ( Sup @ ( image_real_a @ G @ ( image_a_real @ F @ A ) ) )
= ( Sup @ ( image_a_a @ ( comp_real_a_a @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_1023_Sup_OSUP__image,axiom,
! [Sup: set_a > a,G: real > a,F: real > real,A: set_real] :
( ( Sup @ ( image_real_a @ G @ ( image_real_real @ F @ A ) ) )
= ( Sup @ ( image_real_a @ ( comp_real_a_real @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_1024_Sup_OSUP__image,axiom,
! [Sup: set_a > a,G: a > a,F: real > a,A: set_real] :
( ( Sup @ ( image_a_a @ G @ ( image_real_a @ F @ A ) ) )
= ( Sup @ ( image_real_a @ ( comp_a_a_real @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_1025_Sup_OSUP__image,axiom,
! [Sup: set_a > a,G: a > a,F: a > a,A: set_a] :
( ( Sup @ ( image_a_a @ G @ ( image_a_a @ F @ A ) ) )
= ( Sup @ ( image_a_a @ ( comp_a_a_a @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_1026_Inf_OINF__image,axiom,
! [Inf: set_real > real,G: a > real,F: real > a,A: set_real] :
( ( Inf @ ( image_a_real @ G @ ( image_real_a @ F @ A ) ) )
= ( Inf @ ( image_real_real @ ( comp_a_real_real @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_1027_Inf_OINF__image,axiom,
! [Inf: set_real > real,G: a > real,F: a > a,A: set_a] :
( ( Inf @ ( image_a_real @ G @ ( image_a_a @ F @ A ) ) )
= ( Inf @ ( image_a_real @ ( comp_a_real_a @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_1028_Inf_OINF__image,axiom,
! [Inf: set_real > real,G: real > real,F: a > real,A: set_a] :
( ( Inf @ ( image_real_real @ G @ ( image_a_real @ F @ A ) ) )
= ( Inf @ ( image_a_real @ ( comp_real_real_a @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_1029_Inf_OINF__image,axiom,
! [Inf: set_real > real,G: real > real,F: real > real,A: set_real] :
( ( Inf @ ( image_real_real @ G @ ( image_real_real @ F @ A ) ) )
= ( Inf @ ( image_real_real @ ( comp_real_real_real @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_1030_Inf_OINF__image,axiom,
! [Inf: set_a > a,G: real > a,F: a > real,A: set_a] :
( ( Inf @ ( image_real_a @ G @ ( image_a_real @ F @ A ) ) )
= ( Inf @ ( image_a_a @ ( comp_real_a_a @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_1031_Inf_OINF__image,axiom,
! [Inf: set_a > a,G: real > a,F: real > real,A: set_real] :
( ( Inf @ ( image_real_a @ G @ ( image_real_real @ F @ A ) ) )
= ( Inf @ ( image_real_a @ ( comp_real_a_real @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_1032_Inf_OINF__image,axiom,
! [Inf: set_a > a,G: a > a,F: real > a,A: set_real] :
( ( Inf @ ( image_a_a @ G @ ( image_real_a @ F @ A ) ) )
= ( Inf @ ( image_real_a @ ( comp_a_a_real @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_1033_Inf_OINF__image,axiom,
! [Inf: set_a > a,G: a > a,F: a > a,A: set_a] :
( ( Inf @ ( image_a_a @ G @ ( image_a_a @ F @ A ) ) )
= ( Inf @ ( image_a_a @ ( comp_a_a_a @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_1034_insert__subsetI,axiom,
! [X: set_a,A: set_set_a,X6: set_set_a] :
( ( member_set_a @ X @ A )
=> ( ( ord_le3724670747650509150_set_a @ X6 @ A )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1035_insert__subsetI,axiom,
! [X: real > real,A: set_real_real,X6: set_real_real] :
( ( member_real_real @ X @ A )
=> ( ( ord_le4198349162570665613l_real @ X6 @ A )
=> ( ord_le4198349162570665613l_real @ ( insert_real_real @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1036_insert__subsetI,axiom,
! [X: real > a,A: set_real_a,X6: set_real_a] :
( ( member_real_a @ X @ A )
=> ( ( ord_le5743406823621094409real_a @ X6 @ A )
=> ( ord_le5743406823621094409real_a @ ( insert_real_a @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1037_insert__subsetI,axiom,
! [X: a > real,A: set_a_real,X6: set_a_real] :
( ( member_a_real @ X @ A )
=> ( ( ord_le3334967407727675675a_real @ X6 @ A )
=> ( ord_le3334967407727675675a_real @ ( insert_a_real @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1038_insert__subsetI,axiom,
! [X: real,A: set_real,X6: set_real] :
( ( member_real @ X @ A )
=> ( ( ord_less_eq_set_real @ X6 @ A )
=> ( ord_less_eq_set_real @ ( insert_real @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1039_insert__subsetI,axiom,
! [X: a,A: set_a,X6: set_a] :
( ( member_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X6 @ A )
=> ( ord_less_eq_set_a @ ( insert_a @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1040_the__inv__into__vimage,axiom,
! [F: real > real,X6: set_real,A: set_real] :
( ( inj_on_real_real @ F @ X6 )
=> ( ( ord_less_eq_set_real @ A @ X6 )
=> ( ( inf_inf_set_real @ ( vimage_real_real @ ( the_in5290026491893676941l_real @ X6 @ F ) @ A ) @ ( image_real_real @ F @ X6 ) )
= ( image_real_real @ F @ A ) ) ) ) ).
% the_inv_into_vimage
thf(fact_1041_the__inv__into__vimage,axiom,
! [F: real > a,X6: set_real,A: set_real] :
( ( inj_on_real_a @ F @ X6 )
=> ( ( ord_less_eq_set_real @ A @ X6 )
=> ( ( inf_inf_set_a @ ( vimage_a_real @ ( the_inv_into_real_a @ X6 @ F ) @ A ) @ ( image_real_a @ F @ X6 ) )
= ( image_real_a @ F @ A ) ) ) ) ).
% the_inv_into_vimage
thf(fact_1042_the__inv__into__vimage,axiom,
! [F: a > real,X6: set_a,A: set_a] :
( ( inj_on_a_real @ F @ X6 )
=> ( ( ord_less_eq_set_a @ A @ X6 )
=> ( ( inf_inf_set_real @ ( vimage_real_a @ ( the_inv_into_a_real @ X6 @ F ) @ A ) @ ( image_a_real @ F @ X6 ) )
= ( image_a_real @ F @ A ) ) ) ) ).
% the_inv_into_vimage
thf(fact_1043_the__inv__into__vimage,axiom,
! [F: a > a,X6: set_a,A: set_a] :
( ( inj_on_a_a @ F @ X6 )
=> ( ( ord_less_eq_set_a @ A @ X6 )
=> ( ( inf_inf_set_a @ ( vimage_a_a @ ( the_inv_into_a_a @ X6 @ F ) @ A ) @ ( image_a_a @ F @ X6 ) )
= ( image_a_a @ F @ A ) ) ) ) ).
% the_inv_into_vimage
thf(fact_1044_subset__emptyI,axiom,
! [A: set_set_a] :
( ! [X2: set_a] :
~ ( member_set_a @ X2 @ A )
=> ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a ) ) ).
% subset_emptyI
thf(fact_1045_subset__emptyI,axiom,
! [A: set_real_real] :
( ! [X2: real > real] :
~ ( member_real_real @ X2 @ A )
=> ( ord_le4198349162570665613l_real @ A @ bot_bo6767488733719836353l_real ) ) ).
% subset_emptyI
thf(fact_1046_subset__emptyI,axiom,
! [A: set_real_a] :
( ! [X2: real > a] :
~ ( member_real_a @ X2 @ A )
=> ( ord_le5743406823621094409real_a @ A @ bot_bot_set_real_a ) ) ).
% subset_emptyI
thf(fact_1047_subset__emptyI,axiom,
! [A: set_a_real] :
( ! [X2: a > real] :
~ ( member_a_real @ X2 @ A )
=> ( ord_le3334967407727675675a_real @ A @ bot_bot_set_a_real ) ) ).
% subset_emptyI
thf(fact_1048_subset__emptyI,axiom,
! [A: set_real] :
( ! [X2: real] :
~ ( member_real @ X2 @ A )
=> ( ord_less_eq_set_real @ A @ bot_bot_set_real ) ) ).
% subset_emptyI
thf(fact_1049_subset__emptyI,axiom,
! [A: set_a] :
( ! [X2: a] :
~ ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ A @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_1050_all__subset__image,axiom,
! [F: real > real,A: set_real,P: set_real > $o] :
( ( ! [B4: set_real] :
( ( ord_less_eq_set_real @ B4 @ ( image_real_real @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_real] :
( ( ord_less_eq_set_real @ B4 @ A )
=> ( P @ ( image_real_real @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1051_all__subset__image,axiom,
! [F: a > real,A: set_a,P: set_real > $o] :
( ( ! [B4: set_real] :
( ( ord_less_eq_set_real @ B4 @ ( image_a_real @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( P @ ( image_a_real @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1052_all__subset__image,axiom,
! [F: real > a,A: set_real,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_real_a @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_real] :
( ( ord_less_eq_set_real @ B4 @ A )
=> ( P @ ( image_real_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1053_all__subset__image,axiom,
! [F: a > a,A: set_a,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( P @ ( image_a_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1054_real_Ogf__comp__id_H_I1_J,axiom,
( ( comp_real_real_real @ ( standard_g_real @ borel_5078946678739801102l_real ) @ ( standard_f_real @ borel_5078946678739801102l_real ) )
= id_real ) ).
% real.gf_comp_id'(1)
thf(fact_1055_id__apply,axiom,
( id_real
= ( ^ [X3: real] : X3 ) ) ).
% id_apply
thf(fact_1056_image__id,axiom,
( ( image_real_real @ id_real )
= id_set_real ) ).
% image_id
thf(fact_1057_image__id,axiom,
( ( image_a_a @ id_a )
= id_set_a ) ).
% image_id
thf(fact_1058_id__comp,axiom,
! [G: real > real] :
( ( comp_real_real_real @ id_real @ G )
= G ) ).
% id_comp
thf(fact_1059_comp__id,axiom,
! [F: real > real] :
( ( comp_real_real_real @ F @ id_real )
= F ) ).
% comp_id
thf(fact_1060_vimage__id,axiom,
( ( vimage_real_real @ id_real )
= id_set_real ) ).
% vimage_id
thf(fact_1061_vimage__id,axiom,
( ( vimage_a_a @ id_a )
= id_set_a ) ).
% vimage_id
thf(fact_1062_the__inv__into__onto,axiom,
! [F: real > a,A: set_real] :
( ( inj_on_real_a @ F @ A )
=> ( ( image_a_real @ ( the_inv_into_real_a @ A @ F ) @ ( image_real_a @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_1063_the__inv__into__onto,axiom,
! [F: a > a,A: set_a] :
( ( inj_on_a_a @ F @ A )
=> ( ( image_a_a @ ( the_inv_into_a_a @ A @ F ) @ ( image_a_a @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_1064_the__inv__into__onto,axiom,
! [F: a > real,A: set_a] :
( ( inj_on_a_real @ F @ A )
=> ( ( image_real_a @ ( the_inv_into_a_real @ A @ F ) @ ( image_a_real @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_1065_the__inv__into__onto,axiom,
! [F: real > real,A: set_real] :
( ( inj_on_real_real @ F @ A )
=> ( ( image_real_real @ ( the_in5290026491893676941l_real @ A @ F ) @ ( image_real_real @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_1066_id__def,axiom,
( id_real
= ( ^ [X3: real] : X3 ) ) ).
% id_def
thf(fact_1067_eq__id__iff,axiom,
! [F: real > real] :
( ( ! [X3: real] :
( ( F @ X3 )
= X3 ) )
= ( F = id_real ) ) ).
% eq_id_iff
thf(fact_1068_Inf_OINF__id__eq,axiom,
! [Inf: set_real > real,A: set_real] :
( ( Inf @ ( image_real_real @ id_real @ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_id_eq
thf(fact_1069_Inf_OINF__id__eq,axiom,
! [Inf: set_a > a,A: set_a] :
( ( Inf @ ( image_a_a @ id_a @ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_id_eq
thf(fact_1070_Sup_OSUP__id__eq,axiom,
! [Sup: set_real > real,A: set_real] :
( ( Sup @ ( image_real_real @ id_real @ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_id_eq
thf(fact_1071_Sup_OSUP__id__eq,axiom,
! [Sup: set_a > a,A: set_a] :
( ( Sup @ ( image_a_a @ id_a @ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_id_eq
thf(fact_1072_measurable__ident,axiom,
! [M: sigma_measure_real] : ( member_real_real @ id_real @ ( sigma_5267869275261027754l_real @ M @ M ) ) ).
% measurable_ident
thf(fact_1073_comp__eq__id__dest,axiom,
! [A2: real > a,B2: a > real,C: a > a,V: a] :
( ( ( comp_real_a_a @ A2 @ B2 )
= ( comp_a_a_a @ id_a @ C ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_1074_comp__eq__id__dest,axiom,
! [A2: real > real,B2: real > real,C: real > real,V: real] :
( ( ( comp_real_real_real @ A2 @ B2 )
= ( comp_real_real_real @ id_real @ C ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_1075_inj__on__id,axiom,
! [A: set_real] : ( inj_on_real_real @ id_real @ A ) ).
% inj_on_id
thf(fact_1076_the__inv__into__f__eq,axiom,
! [F: real > real,A: set_real,X: real,Y3: real] :
( ( inj_on_real_real @ F @ A )
=> ( ( ( F @ X )
= Y3 )
=> ( ( member_real @ X @ A )
=> ( ( the_in5290026491893676941l_real @ A @ F @ Y3 )
= X ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_1077_the__inv__into__f__eq,axiom,
! [F: a > real,A: set_a,X: a,Y3: real] :
( ( inj_on_a_real @ F @ A )
=> ( ( ( F @ X )
= Y3 )
=> ( ( member_a @ X @ A )
=> ( ( the_inv_into_a_real @ A @ F @ Y3 )
= X ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_1078_the__inv__into__f__f,axiom,
! [F: real > real,A: set_real,X: real] :
( ( inj_on_real_real @ F @ A )
=> ( ( member_real @ X @ A )
=> ( ( the_in5290026491893676941l_real @ A @ F @ ( F @ X ) )
= X ) ) ) ).
% the_inv_into_f_f
thf(fact_1079_the__inv__into__f__f,axiom,
! [F: a > real,A: set_a,X: a] :
( ( inj_on_a_real @ F @ A )
=> ( ( member_a @ X @ A )
=> ( ( the_inv_into_a_real @ A @ F @ ( F @ X ) )
= X ) ) ) ).
% the_inv_into_f_f
thf(fact_1080_f__the__inv__into__f,axiom,
! [F: a > real,A: set_a,Y3: real] :
( ( inj_on_a_real @ F @ A )
=> ( ( member_real @ Y3 @ ( image_a_real @ F @ A ) )
=> ( ( F @ ( the_inv_into_a_real @ A @ F @ Y3 ) )
= Y3 ) ) ) ).
% f_the_inv_into_f
thf(fact_1081_f__the__inv__into__f,axiom,
! [F: real > real,A: set_real,Y3: real] :
( ( inj_on_real_real @ F @ A )
=> ( ( member_real @ Y3 @ ( image_real_real @ F @ A ) )
=> ( ( F @ ( the_in5290026491893676941l_real @ A @ F @ Y3 ) )
= Y3 ) ) ) ).
% f_the_inv_into_f
thf(fact_1082_f__the__inv__into__f,axiom,
! [F: real > a,A: set_real,Y3: a] :
( ( inj_on_real_a @ F @ A )
=> ( ( member_a @ Y3 @ ( image_real_a @ F @ A ) )
=> ( ( F @ ( the_inv_into_real_a @ A @ F @ Y3 ) )
= Y3 ) ) ) ).
% f_the_inv_into_f
thf(fact_1083_f__the__inv__into__f,axiom,
! [F: a > a,A: set_a,Y3: a] :
( ( inj_on_a_a @ F @ A )
=> ( ( member_a @ Y3 @ ( image_a_a @ F @ A ) )
=> ( ( F @ ( the_inv_into_a_a @ A @ F @ Y3 ) )
= Y3 ) ) ) ).
% f_the_inv_into_f
thf(fact_1084_inj__on__the__inv__into,axiom,
! [F: a > a,A: set_a] :
( ( inj_on_a_a @ F @ A )
=> ( inj_on_a_a @ ( the_inv_into_a_a @ A @ F ) @ ( image_a_a @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_1085_inj__on__the__inv__into,axiom,
! [F: real > a,A: set_real] :
( ( inj_on_real_a @ F @ A )
=> ( inj_on_a_real @ ( the_inv_into_real_a @ A @ F ) @ ( image_real_a @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_1086_inj__on__the__inv__into,axiom,
! [F: a > real,A: set_a] :
( ( inj_on_a_real @ F @ A )
=> ( inj_on_real_a @ ( the_inv_into_a_real @ A @ F ) @ ( image_a_real @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_1087_inj__on__the__inv__into,axiom,
! [F: real > real,A: set_real] :
( ( inj_on_real_real @ F @ A )
=> ( inj_on_real_real @ ( the_in5290026491893676941l_real @ A @ F ) @ ( image_real_real @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_1088_Inf_OINF__cong,axiom,
! [A: set_real,B: set_real,C2: real > real,D2: real > real,Inf: set_real > real] :
( ( A = B )
=> ( ! [X2: real] :
( ( member_real @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Inf @ ( image_real_real @ C2 @ A ) )
= ( Inf @ ( image_real_real @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_1089_Inf_OINF__cong,axiom,
! [A: set_real,B: set_real,C2: real > a,D2: real > a,Inf: set_a > a] :
( ( A = B )
=> ( ! [X2: real] :
( ( member_real @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Inf @ ( image_real_a @ C2 @ A ) )
= ( Inf @ ( image_real_a @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_1090_Inf_OINF__cong,axiom,
! [A: set_a,B: set_a,C2: a > real,D2: a > real,Inf: set_real > real] :
( ( A = B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Inf @ ( image_a_real @ C2 @ A ) )
= ( Inf @ ( image_a_real @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_1091_Inf_OINF__cong,axiom,
! [A: set_a,B: set_a,C2: a > a,D2: a > a,Inf: set_a > a] :
( ( A = B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Inf @ ( image_a_a @ C2 @ A ) )
= ( Inf @ ( image_a_a @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_1092_Sup_OSUP__cong,axiom,
! [A: set_real,B: set_real,C2: real > real,D2: real > real,Sup: set_real > real] :
( ( A = B )
=> ( ! [X2: real] :
( ( member_real @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Sup @ ( image_real_real @ C2 @ A ) )
= ( Sup @ ( image_real_real @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_1093_Sup_OSUP__cong,axiom,
! [A: set_real,B: set_real,C2: real > a,D2: real > a,Sup: set_a > a] :
( ( A = B )
=> ( ! [X2: real] :
( ( member_real @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Sup @ ( image_real_a @ C2 @ A ) )
= ( Sup @ ( image_real_a @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_1094_Sup_OSUP__cong,axiom,
! [A: set_a,B: set_a,C2: a > real,D2: a > real,Sup: set_real > real] :
( ( A = B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Sup @ ( image_a_real @ C2 @ A ) )
= ( Sup @ ( image_a_real @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_1095_Sup_OSUP__cong,axiom,
! [A: set_a,B: set_a,C2: a > a,D2: a > a,Sup: set_a > a] :
( ( A = B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Sup @ ( image_a_a @ C2 @ A ) )
= ( Sup @ ( image_a_a @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_1096_the__inv__into__into,axiom,
! [F: real > real,A: set_real,X: real,B: set_real] :
( ( inj_on_real_real @ F @ A )
=> ( ( member_real @ X @ ( image_real_real @ F @ A ) )
=> ( ( ord_less_eq_set_real @ A @ B )
=> ( member_real @ ( the_in5290026491893676941l_real @ A @ F @ X ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_1097_the__inv__into__into,axiom,
! [F: real > a,A: set_real,X: a,B: set_real] :
( ( inj_on_real_a @ F @ A )
=> ( ( member_a @ X @ ( image_real_a @ F @ A ) )
=> ( ( ord_less_eq_set_real @ A @ B )
=> ( member_real @ ( the_inv_into_real_a @ A @ F @ X ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_1098_the__inv__into__into,axiom,
! [F: a > real,A: set_a,X: real,B: set_a] :
( ( inj_on_a_real @ F @ A )
=> ( ( member_real @ X @ ( image_a_real @ F @ A ) )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( member_a @ ( the_inv_into_a_real @ A @ F @ X ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_1099_the__inv__into__into,axiom,
! [F: a > a,A: set_a,X: a,B: set_a] :
( ( inj_on_a_a @ F @ A )
=> ( ( member_a @ X @ ( image_a_a @ F @ A ) )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( member_a @ ( the_inv_into_a_a @ A @ F @ X ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_1100_the__inv__into__into,axiom,
! [F: set_a > a,A: set_set_a,X: a,B: set_set_a] :
( ( inj_on_set_a_a @ F @ A )
=> ( ( member_a @ X @ ( image_set_a_a @ F @ A ) )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( member_set_a @ ( the_inv_into_set_a_a @ A @ F @ X ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_1101_the__inv__into__into,axiom,
! [F: real > set_a,A: set_real,X: set_a,B: set_real] :
( ( inj_on_real_set_a @ F @ A )
=> ( ( member_set_a @ X @ ( image_real_set_a @ F @ A ) )
=> ( ( ord_less_eq_set_real @ A @ B )
=> ( member_real @ ( the_in3800189856886063421_set_a @ A @ F @ X ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_1102_the__inv__into__into,axiom,
! [F: a > set_a,A: set_a,X: set_a,B: set_a] :
( ( inj_on_a_set_a @ F @ A )
=> ( ( member_set_a @ X @ ( image_a_set_a @ F @ A ) )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( member_a @ ( the_inv_into_a_set_a @ A @ F @ X ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_1103_the__inv__into__into,axiom,
! [F: set_a > set_a,A: set_set_a,X: set_a,B: set_set_a] :
( ( inj_on_set_a_set_a @ F @ A )
=> ( ( member_set_a @ X @ ( image_set_a_set_a @ F @ A ) )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( member_set_a @ ( the_in3497175299774311219_set_a @ A @ F @ X ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_1104_the__inv__into__into,axiom,
! [F: ( real > real ) > a,A: set_real_real,X: a,B: set_real_real] :
( ( inj_on_real_real_a @ F @ A )
=> ( ( member_a @ X @ ( image_real_real_a @ F @ A ) )
=> ( ( ord_le4198349162570665613l_real @ A @ B )
=> ( member_real_real @ ( the_in6219261096523627986real_a @ A @ F @ X ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_1105_the__inv__into__into,axiom,
! [F: ( real > a ) > a,A: set_real_a,X: a,B: set_real_a] :
( ( inj_on_real_a_a @ F @ A )
=> ( ( member_a @ X @ ( image_real_a_a @ F @ A ) )
=> ( ( ord_le5743406823621094409real_a @ A @ B )
=> ( member_real_a @ ( the_in3864030383226952808al_a_a @ A @ F @ X ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_1106_the__inv__into__comp,axiom,
! [F: a > real,G: real > a,A: set_real,X: real] :
( ( inj_on_a_real @ F @ ( image_real_a @ G @ A ) )
=> ( ( inj_on_real_a @ G @ A )
=> ( ( member_real @ X @ ( image_a_real @ F @ ( image_real_a @ G @ A ) ) )
=> ( ( the_in5290026491893676941l_real @ A @ ( comp_a_real_real @ F @ G ) @ X )
= ( comp_a_real_real @ ( the_inv_into_real_a @ A @ G ) @ ( the_inv_into_a_real @ ( image_real_a @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_1107_the__inv__into__comp,axiom,
! [F: a > real,G: a > a,A: set_a,X: real] :
( ( inj_on_a_real @ F @ ( image_a_a @ G @ A ) )
=> ( ( inj_on_a_a @ G @ A )
=> ( ( member_real @ X @ ( image_a_real @ F @ ( image_a_a @ G @ A ) ) )
=> ( ( the_inv_into_a_real @ A @ ( comp_a_real_a @ F @ G ) @ X )
= ( comp_a_a_real @ ( the_inv_into_a_a @ A @ G ) @ ( the_inv_into_a_real @ ( image_a_a @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_1108_the__inv__into__comp,axiom,
! [F: real > real,G: a > real,A: set_a,X: real] :
( ( inj_on_real_real @ F @ ( image_a_real @ G @ A ) )
=> ( ( inj_on_a_real @ G @ A )
=> ( ( member_real @ X @ ( image_real_real @ F @ ( image_a_real @ G @ A ) ) )
=> ( ( the_inv_into_a_real @ A @ ( comp_real_real_a @ F @ G ) @ X )
= ( comp_real_a_real @ ( the_inv_into_a_real @ A @ G ) @ ( the_in5290026491893676941l_real @ ( image_a_real @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_1109_the__inv__into__comp,axiom,
! [F: real > real,G: real > real,A: set_real,X: real] :
( ( inj_on_real_real @ F @ ( image_real_real @ G @ A ) )
=> ( ( inj_on_real_real @ G @ A )
=> ( ( member_real @ X @ ( image_real_real @ F @ ( image_real_real @ G @ A ) ) )
=> ( ( the_in5290026491893676941l_real @ A @ ( comp_real_real_real @ F @ G ) @ X )
= ( comp_real_real_real @ ( the_in5290026491893676941l_real @ A @ G ) @ ( the_in5290026491893676941l_real @ ( image_real_real @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_1110_the__inv__into__comp,axiom,
! [F: a > a,G: real > a,A: set_real,X: a] :
( ( inj_on_a_a @ F @ ( image_real_a @ G @ A ) )
=> ( ( inj_on_real_a @ G @ A )
=> ( ( member_a @ X @ ( image_a_a @ F @ ( image_real_a @ G @ A ) ) )
=> ( ( the_inv_into_real_a @ A @ ( comp_a_a_real @ F @ G ) @ X )
= ( comp_a_real_a @ ( the_inv_into_real_a @ A @ G ) @ ( the_inv_into_a_a @ ( image_real_a @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_1111_the__inv__into__comp,axiom,
! [F: a > a,G: a > a,A: set_a,X: a] :
( ( inj_on_a_a @ F @ ( image_a_a @ G @ A ) )
=> ( ( inj_on_a_a @ G @ A )
=> ( ( member_a @ X @ ( image_a_a @ F @ ( image_a_a @ G @ A ) ) )
=> ( ( the_inv_into_a_a @ A @ ( comp_a_a_a @ F @ G ) @ X )
= ( comp_a_a_a @ ( the_inv_into_a_a @ A @ G ) @ ( the_inv_into_a_a @ ( image_a_a @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_1112_the__inv__into__comp,axiom,
! [F: real > a,G: a > real,A: set_a,X: a] :
( ( inj_on_real_a @ F @ ( image_a_real @ G @ A ) )
=> ( ( inj_on_a_real @ G @ A )
=> ( ( member_a @ X @ ( image_real_a @ F @ ( image_a_real @ G @ A ) ) )
=> ( ( the_inv_into_a_a @ A @ ( comp_real_a_a @ F @ G ) @ X )
= ( comp_real_a_a @ ( the_inv_into_a_real @ A @ G ) @ ( the_inv_into_real_a @ ( image_a_real @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_1113_the__inv__into__comp,axiom,
! [F: real > a,G: real > real,A: set_real,X: a] :
( ( inj_on_real_a @ F @ ( image_real_real @ G @ A ) )
=> ( ( inj_on_real_real @ G @ A )
=> ( ( member_a @ X @ ( image_real_a @ F @ ( image_real_real @ G @ A ) ) )
=> ( ( the_inv_into_real_a @ A @ ( comp_real_a_real @ F @ G ) @ X )
= ( comp_real_real_a @ ( the_in5290026491893676941l_real @ A @ G ) @ ( the_inv_into_real_a @ ( image_real_real @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_1114_the__inv__into__comp,axiom,
! [F: a > set_a,G: real > a,A: set_real,X: set_a] :
( ( inj_on_a_set_a @ F @ ( image_real_a @ G @ A ) )
=> ( ( inj_on_real_a @ G @ A )
=> ( ( member_set_a @ X @ ( image_a_set_a @ F @ ( image_real_a @ G @ A ) ) )
=> ( ( the_in3800189856886063421_set_a @ A @ ( comp_a_set_a_real @ F @ G ) @ X )
= ( comp_a_real_set_a @ ( the_inv_into_real_a @ A @ G ) @ ( the_inv_into_a_set_a @ ( image_real_a @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_1115_the__inv__into__comp,axiom,
! [F: a > set_a,G: a > a,A: set_a,X: set_a] :
( ( inj_on_a_set_a @ F @ ( image_a_a @ G @ A ) )
=> ( ( inj_on_a_a @ G @ A )
=> ( ( member_set_a @ X @ ( image_a_set_a @ F @ ( image_a_a @ G @ A ) ) )
=> ( ( the_inv_into_a_set_a @ A @ ( comp_a_set_a_a @ F @ G ) @ X )
= ( comp_a_a_set_a @ ( the_inv_into_a_a @ A @ G ) @ ( the_inv_into_a_set_a @ ( image_a_a @ G @ A ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_1116_fun_Omap__id,axiom,
! [T4: real > real] :
( ( comp_real_real_real @ id_real @ T4 )
= T4 ) ).
% fun.map_id
thf(fact_1117_standard__borel__space__UNIV_Ogf__comp__id_H_I1_J,axiom,
! [M: sigma_measure_a] :
( ( standa5776109378963170237UNIV_a @ M )
=> ( ( comp_real_a_a @ ( standard_g_a @ M ) @ ( standard_f_a @ M ) )
= id_a ) ) ).
% standard_borel_space_UNIV.gf_comp_id'(1)
thf(fact_1118_standard__borel__space__UNIV_Ogf__comp__id_H_I1_J,axiom,
! [M: sigma_measure_real] :
( ( standa1306199911732814765V_real @ M )
=> ( ( comp_real_real_real @ ( standard_g_real @ M ) @ ( standard_f_real @ M ) )
= id_real ) ) ).
% standard_borel_space_UNIV.gf_comp_id'(1)
thf(fact_1119_fun_Omap__id0,axiom,
( ( comp_real_real_real @ id_real )
= id_real_real ) ).
% fun.map_id0
thf(fact_1120_left__right__inverse__eq,axiom,
! [F: a > real,G: real > a,H: a > real] :
( ( ( comp_a_real_real @ F @ G )
= id_real )
=> ( ( ( comp_real_a_a @ G @ H )
= id_a )
=> ( F = H ) ) ) ).
% left_right_inverse_eq
thf(fact_1121_left__right__inverse__eq,axiom,
! [F: real > a,G: a > real,H: real > a] :
( ( ( comp_real_a_a @ F @ G )
= id_a )
=> ( ( ( comp_a_real_real @ G @ H )
= id_real )
=> ( F = H ) ) ) ).
% left_right_inverse_eq
thf(fact_1122_left__right__inverse__eq,axiom,
! [F: real > real,G: real > real,H: real > real] :
( ( ( comp_real_real_real @ F @ G )
= id_real )
=> ( ( ( comp_real_real_real @ G @ H )
= id_real )
=> ( F = H ) ) ) ).
% left_right_inverse_eq
thf(fact_1123_isomorphism__expand,axiom,
! [F: a > real,G: real > a] :
( ( ( ( comp_a_real_real @ F @ G )
= id_real )
& ( ( comp_real_a_a @ G @ F )
= id_a ) )
= ( ! [X3: real] :
( ( F @ ( G @ X3 ) )
= X3 )
& ! [X3: a] :
( ( G @ ( F @ X3 ) )
= X3 ) ) ) ).
% isomorphism_expand
thf(fact_1124_isomorphism__expand,axiom,
! [F: real > a,G: a > real] :
( ( ( ( comp_real_a_a @ F @ G )
= id_a )
& ( ( comp_a_real_real @ G @ F )
= id_real ) )
= ( ! [X3: a] :
( ( F @ ( G @ X3 ) )
= X3 )
& ! [X3: real] :
( ( G @ ( F @ X3 ) )
= X3 ) ) ) ).
% isomorphism_expand
thf(fact_1125_isomorphism__expand,axiom,
! [F: real > real,G: real > real] :
( ( ( ( comp_real_real_real @ F @ G )
= id_real )
& ( ( comp_real_real_real @ G @ F )
= id_real ) )
= ( ! [X3: real] :
( ( F @ ( G @ X3 ) )
= X3 )
& ! [X3: real] :
( ( G @ ( F @ X3 ) )
= X3 ) ) ) ).
% isomorphism_expand
thf(fact_1126_pointfree__idE,axiom,
! [F: real > a,G: a > real,X: a] :
( ( ( comp_real_a_a @ F @ G )
= id_a )
=> ( ( F @ ( G @ X ) )
= X ) ) ).
% pointfree_idE
thf(fact_1127_pointfree__idE,axiom,
! [F: real > real,G: real > real,X: real] :
( ( ( comp_real_real_real @ F @ G )
= id_real )
=> ( ( F @ ( G @ X ) )
= X ) ) ).
% pointfree_idE
thf(fact_1128_real_Ostandard__borel__space__UNIV__axioms,axiom,
standa1306199911732814765V_real @ borel_5078946678739801102l_real ).
% real.standard_borel_space_UNIV_axioms
thf(fact_1129_standard__borel__space__UNIV_Oaxioms_I1_J,axiom,
! [M: sigma_measure_a] :
( ( standa5776109378963170237UNIV_a @ M )
=> ( standard_borel_a @ M ) ) ).
% standard_borel_space_UNIV.axioms(1)
thf(fact_1130_standard__borel__space__UNIV_Oaxioms_I1_J,axiom,
! [M: sigma_measure_real] :
( ( standa1306199911732814765V_real @ M )
=> ( standard_borel_real @ M ) ) ).
% standard_borel_space_UNIV.axioms(1)
thf(fact_1131_fun_Omap__comp,axiom,
! [G: a > a,F: real > a,V: a > real] :
( ( comp_a_a_a @ G @ ( comp_real_a_a @ F @ V ) )
= ( comp_real_a_a @ ( comp_a_a_real @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_1132_fun_Omap__comp,axiom,
! [G: real > a,F: a > real,V: a > a] :
( ( comp_real_a_a @ G @ ( comp_a_real_a @ F @ V ) )
= ( comp_a_a_a @ ( comp_real_a_a @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_1133_fun_Omap__comp,axiom,
! [G: real > a,F: real > real,V: a > real] :
( ( comp_real_a_a @ G @ ( comp_real_real_a @ F @ V ) )
= ( comp_real_a_a @ ( comp_real_a_real @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_1134_fun_Omap__comp,axiom,
! [G: real > real,F: real > real,V: real > real] :
( ( comp_real_real_real @ G @ ( comp_real_real_real @ F @ V ) )
= ( comp_real_real_real @ ( comp_real_real_real @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_1135_rewriteL__comp__comp,axiom,
! [F: a > a,G: real > a,L2: real > a,H: a > real] :
( ( ( comp_a_a_real @ F @ G )
= L2 )
=> ( ( comp_a_a_a @ F @ ( comp_real_a_a @ G @ H ) )
= ( comp_real_a_a @ L2 @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_1136_rewriteL__comp__comp,axiom,
! [F: real > a,G: real > real,L2: real > a,H: a > real] :
( ( ( comp_real_a_real @ F @ G )
= L2 )
=> ( ( comp_real_a_a @ F @ ( comp_real_real_a @ G @ H ) )
= ( comp_real_a_a @ L2 @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_1137_rewriteL__comp__comp,axiom,
! [F: real > a,G: a > real,L2: a > a,H: a > a] :
( ( ( comp_real_a_a @ F @ G )
= L2 )
=> ( ( comp_real_a_a @ F @ ( comp_a_real_a @ G @ H ) )
= ( comp_a_a_a @ L2 @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_1138_rewriteL__comp__comp,axiom,
! [F: real > real,G: real > real,L2: real > real,H: real > real] :
( ( ( comp_real_real_real @ F @ G )
= L2 )
=> ( ( comp_real_real_real @ F @ ( comp_real_real_real @ G @ H ) )
= ( comp_real_real_real @ L2 @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_1139_rewriteR__comp__comp,axiom,
! [G: a > real,H: a > a,R: a > real,F: real > a] :
( ( ( comp_a_real_a @ G @ H )
= R )
=> ( ( comp_a_a_a @ ( comp_real_a_a @ F @ G ) @ H )
= ( comp_real_a_a @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_1140_rewriteR__comp__comp,axiom,
! [G: real > real,H: a > real,R: a > real,F: real > a] :
( ( ( comp_real_real_a @ G @ H )
= R )
=> ( ( comp_real_a_a @ ( comp_real_a_real @ F @ G ) @ H )
= ( comp_real_a_a @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_1141_rewriteR__comp__comp,axiom,
! [G: real > a,H: a > real,R: a > a,F: a > a] :
( ( ( comp_real_a_a @ G @ H )
= R )
=> ( ( comp_real_a_a @ ( comp_a_a_real @ F @ G ) @ H )
= ( comp_a_a_a @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_1142_rewriteR__comp__comp,axiom,
! [G: real > real,H: real > real,R: real > real,F: real > real] :
( ( ( comp_real_real_real @ G @ H )
= R )
=> ( ( comp_real_real_real @ ( comp_real_real_real @ F @ G ) @ H )
= ( comp_real_real_real @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_1143_rewriteL__comp__comp2,axiom,
! [F: a > a,G: real > a,L1: real > a,L22: real > real,H: a > real,R: a > real] :
( ( ( comp_a_a_real @ F @ G )
= ( comp_real_a_real @ L1 @ L22 ) )
=> ( ( ( comp_real_real_a @ L22 @ H )
= R )
=> ( ( comp_a_a_a @ F @ ( comp_real_a_a @ G @ H ) )
= ( comp_real_a_a @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_1144_rewriteL__comp__comp2,axiom,
! [F: real > a,G: real > real,L1: a > a,L22: real > a,H: a > real,R: a > a] :
( ( ( comp_real_a_real @ F @ G )
= ( comp_a_a_real @ L1 @ L22 ) )
=> ( ( ( comp_real_a_a @ L22 @ H )
= R )
=> ( ( comp_real_a_a @ F @ ( comp_real_real_a @ G @ H ) )
= ( comp_a_a_a @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_1145_rewriteL__comp__comp2,axiom,
! [F: a > real,G: real > a,L1: real > real,L22: real > real,H: a > real,R: a > real] :
( ( ( comp_a_real_real @ F @ G )
= ( comp_real_real_real @ L1 @ L22 ) )
=> ( ( ( comp_real_real_a @ L22 @ H )
= R )
=> ( ( comp_a_real_a @ F @ ( comp_real_a_a @ G @ H ) )
= ( comp_real_real_a @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_1146_rewriteL__comp__comp2,axiom,
! [F: real > a,G: a > real,L1: real > a,L22: a > real,H: a > a,R: a > real] :
( ( ( comp_real_a_a @ F @ G )
= ( comp_real_a_a @ L1 @ L22 ) )
=> ( ( ( comp_a_real_a @ L22 @ H )
= R )
=> ( ( comp_real_a_a @ F @ ( comp_a_real_a @ G @ H ) )
= ( comp_real_a_a @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_1147_rewriteL__comp__comp2,axiom,
! [F: real > real,G: real > real,L1: a > real,L22: real > a,H: a > real,R: a > a] :
( ( ( comp_real_real_real @ F @ G )
= ( comp_a_real_real @ L1 @ L22 ) )
=> ( ( ( comp_real_a_a @ L22 @ H )
= R )
=> ( ( comp_real_real_a @ F @ ( comp_real_real_a @ G @ H ) )
= ( comp_a_real_a @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_1148_rewriteL__comp__comp2,axiom,
! [F: real > real,G: real > real,L1: real > real,L22: real > real,H: real > real,R: real > real] :
( ( ( comp_real_real_real @ F @ G )
= ( comp_real_real_real @ L1 @ L22 ) )
=> ( ( ( comp_real_real_real @ L22 @ H )
= R )
=> ( ( comp_real_real_real @ F @ ( comp_real_real_real @ G @ H ) )
= ( comp_real_real_real @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_1149_rewriteR__comp__comp2,axiom,
! [G: a > real,H: a > a,R1: real > real,R2: a > real,F: real > a,L2: real > a] :
( ( ( comp_a_real_a @ G @ H )
= ( comp_real_real_a @ R1 @ R2 ) )
=> ( ( ( comp_real_a_real @ F @ R1 )
= L2 )
=> ( ( comp_a_a_a @ ( comp_real_a_a @ F @ G ) @ H )
= ( comp_real_a_a @ L2 @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_1150_rewriteR__comp__comp2,axiom,
! [G: real > real,H: a > real,R1: a > real,R2: a > a,F: real > a,L2: a > a] :
( ( ( comp_real_real_a @ G @ H )
= ( comp_a_real_a @ R1 @ R2 ) )
=> ( ( ( comp_real_a_a @ F @ R1 )
= L2 )
=> ( ( comp_real_a_a @ ( comp_real_a_real @ F @ G ) @ H )
= ( comp_a_a_a @ L2 @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_1151_rewriteR__comp__comp2,axiom,
! [G: a > real,H: real > a,R1: real > real,R2: real > real,F: real > a,L2: real > a] :
( ( ( comp_a_real_real @ G @ H )
= ( comp_real_real_real @ R1 @ R2 ) )
=> ( ( ( comp_real_a_real @ F @ R1 )
= L2 )
=> ( ( comp_a_a_real @ ( comp_real_a_a @ F @ G ) @ H )
= ( comp_real_a_real @ L2 @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_1152_rewriteR__comp__comp2,axiom,
! [G: real > a,H: a > real,R1: real > a,R2: a > real,F: a > a,L2: real > a] :
( ( ( comp_real_a_a @ G @ H )
= ( comp_real_a_a @ R1 @ R2 ) )
=> ( ( ( comp_a_a_real @ F @ R1 )
= L2 )
=> ( ( comp_real_a_a @ ( comp_a_a_real @ F @ G ) @ H )
= ( comp_real_a_a @ L2 @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_1153_rewriteR__comp__comp2,axiom,
! [G: real > real,H: real > real,R1: a > real,R2: real > a,F: real > a,L2: a > a] :
( ( ( comp_real_real_real @ G @ H )
= ( comp_a_real_real @ R1 @ R2 ) )
=> ( ( ( comp_real_a_a @ F @ R1 )
= L2 )
=> ( ( comp_real_a_real @ ( comp_real_a_real @ F @ G ) @ H )
= ( comp_a_a_real @ L2 @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_1154_rewriteR__comp__comp2,axiom,
! [G: real > real,H: real > real,R1: real > real,R2: real > real,F: real > real,L2: real > real] :
( ( ( comp_real_real_real @ G @ H )
= ( comp_real_real_real @ R1 @ R2 ) )
=> ( ( ( comp_real_real_real @ F @ R1 )
= L2 )
=> ( ( comp_real_real_real @ ( comp_real_real_real @ F @ G ) @ H )
= ( comp_real_real_real @ L2 @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_1155_standard__borel__space__UNIV_Ogf__comp__id_H_I2_J,axiom,
! [M: sigma_measure_a,X: a] :
( ( standa5776109378963170237UNIV_a @ M )
=> ( ( standard_g_a @ M @ ( standard_f_a @ M @ X ) )
= X ) ) ).
% standard_borel_space_UNIV.gf_comp_id'(2)
thf(fact_1156_standard__borel__space__UNIV_Ogf__comp__id_H_I2_J,axiom,
! [M: sigma_measure_real,X: real] :
( ( standa1306199911732814765V_real @ M )
=> ( ( standard_g_real @ M @ ( standard_f_real @ M @ X ) )
= X ) ) ).
% standard_borel_space_UNIV.gf_comp_id'(2)
thf(fact_1157_standard__borel__space__UNIVI,axiom,
! [F: a > real,Y: sigma_measure_a,G: real > a] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ Y @ borel_5078946678739801102l_real ) )
=> ( ( member_real_a @ G @ ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ Y ) )
=> ( ( ( comp_real_a_a @ G @ F )
= id_a )
=> ( ( ( sigma_space_a @ Y )
= top_top_set_a )
=> ( standa5776109378963170237UNIV_a @ Y ) ) ) ) ) ).
% standard_borel_space_UNIVI
thf(fact_1158_standard__borel__space__UNIVI,axiom,
! [F: real > real,Y: sigma_measure_real,G: real > real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ Y @ borel_5078946678739801102l_real ) )
=> ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ Y ) )
=> ( ( ( comp_real_real_real @ G @ F )
= id_real )
=> ( ( ( sigma_space_real @ Y )
= top_top_set_real )
=> ( standa1306199911732814765V_real @ Y ) ) ) ) ) ).
% standard_borel_space_UNIVI
thf(fact_1159_standard__borel__space__UNIV_Ointro,axiom,
! [M: sigma_measure_a] :
( ( standard_borel_a @ M )
=> ( ( standa2153564630574221018ioms_a @ M )
=> ( standa5776109378963170237UNIV_a @ M ) ) ) ).
% standard_borel_space_UNIV.intro
thf(fact_1160_standard__borel__space__UNIV_Ointro,axiom,
! [M: sigma_measure_real] :
( ( standard_borel_real @ M )
=> ( ( standa1498722272452280784s_real @ M )
=> ( standa1306199911732814765V_real @ M ) ) ) ).
% standard_borel_space_UNIV.intro
thf(fact_1161_standard__borel__space__UNIV__def,axiom,
( standa5776109378963170237UNIV_a
= ( ^ [M3: sigma_measure_a] :
( ( standard_borel_a @ M3 )
& ( standa2153564630574221018ioms_a @ M3 ) ) ) ) ).
% standard_borel_space_UNIV_def
thf(fact_1162_standard__borel__space__UNIV__def,axiom,
( standa1306199911732814765V_real
= ( ^ [M3: sigma_measure_real] :
( ( standard_borel_real @ M3 )
& ( standa1498722272452280784s_real @ M3 ) ) ) ) ).
% standard_borel_space_UNIV_def
thf(fact_1163_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A4: set_a] :
( A4
= ( insert_a @ ( the_elem_a @ A4 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1164_is__singleton__the__elem,axiom,
( is_singleton_real
= ( ^ [A4: set_real] :
( A4
= ( insert_real @ ( the_elem_real @ A4 ) @ bot_bot_set_real ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1165_space__sup__measure_H,axiom,
! [B: sigma_measure_a,A: sigma_measure_a] :
( ( ( sigma_sets_a @ B )
= ( sigma_sets_a @ A ) )
=> ( ( sigma_space_a @ ( measur3004909623614618064sure_a @ A @ B ) )
= ( sigma_space_a @ A ) ) ) ).
% space_sup_measure'
thf(fact_1166_space__sup__measure_H,axiom,
! [B: sigma_measure_real,A: sigma_measure_real] :
( ( ( sigma_sets_real @ B )
= ( sigma_sets_real @ A ) )
=> ( ( sigma_space_real @ ( measur2147279183506585690e_real @ A @ B ) )
= ( sigma_space_real @ A ) ) ) ).
% space_sup_measure'
thf(fact_1167_borel__measurableI,axiom,
! [F: a > real,M: sigma_measure_a] :
( ! [S2: set_real] :
( ( topolo4860482606490270245n_real @ S2 )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_real @ F @ S2 ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) )
=> ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) ) ).
% borel_measurableI
thf(fact_1168_borel__measurableI,axiom,
! [F: real > real,M: sigma_measure_real] :
( ! [S2: set_real] :
( ( topolo4860482606490270245n_real @ S2 )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_real @ F @ S2 ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) )
=> ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ).
% borel_measurableI
thf(fact_1169_UNIV__I,axiom,
! [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ).
% UNIV_I
thf(fact_1170_UNIV__I,axiom,
! [X: real > real] : ( member_real_real @ X @ top_to2071711978144146653l_real ) ).
% UNIV_I
thf(fact_1171_UNIV__I,axiom,
! [X: real > a] : ( member_real_a @ X @ top_top_set_real_a ) ).
% UNIV_I
thf(fact_1172_UNIV__I,axiom,
! [X: a > real] : ( member_a_real @ X @ top_top_set_a_real ) ).
% UNIV_I
thf(fact_1173_UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_I
thf(fact_1174_UNIV__I,axiom,
! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% UNIV_I
thf(fact_1175_inf__top__left,axiom,
! [X: set_real] :
( ( inf_inf_set_real @ top_top_set_real @ X )
= X ) ).
% inf_top_left
thf(fact_1176_inf__top__right,axiom,
! [X: set_real] :
( ( inf_inf_set_real @ X @ top_top_set_real )
= X ) ).
% inf_top_right
thf(fact_1177_inf__eq__top__iff,axiom,
! [X: set_real,Y3: set_real] :
( ( ( inf_inf_set_real @ X @ Y3 )
= top_top_set_real )
= ( ( X = top_top_set_real )
& ( Y3 = top_top_set_real ) ) ) ).
% inf_eq_top_iff
thf(fact_1178_top__eq__inf__iff,axiom,
! [X: set_real,Y3: set_real] :
( ( top_top_set_real
= ( inf_inf_set_real @ X @ Y3 ) )
= ( ( X = top_top_set_real )
& ( Y3 = top_top_set_real ) ) ) ).
% top_eq_inf_iff
thf(fact_1179_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_real,B2: set_real] :
( ( ( inf_inf_set_real @ A2 @ B2 )
= top_top_set_real )
= ( ( A2 = top_top_set_real )
& ( B2 = top_top_set_real ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1180_inf__top_Oleft__neutral,axiom,
! [A2: set_real] :
( ( inf_inf_set_real @ top_top_set_real @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_1181_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_real,B2: set_real] :
( ( top_top_set_real
= ( inf_inf_set_real @ A2 @ B2 ) )
= ( ( A2 = top_top_set_real )
& ( B2 = top_top_set_real ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1182_inf__top_Oright__neutral,axiom,
! [A2: set_real] :
( ( inf_inf_set_real @ A2 @ top_top_set_real )
= A2 ) ).
% inf_top.right_neutral
thf(fact_1183_Int__UNIV,axiom,
! [A: set_real,B: set_real] :
( ( ( inf_inf_set_real @ A @ B )
= top_top_set_real )
= ( ( A = top_top_set_real )
& ( B = top_top_set_real ) ) ) ).
% Int_UNIV
thf(fact_1184_vimage__UNIV,axiom,
! [F: a > a] :
( ( vimage_a_a @ F @ top_top_set_a )
= top_top_set_a ) ).
% vimage_UNIV
thf(fact_1185_vimage__UNIV,axiom,
! [F: real > a] :
( ( vimage_real_a @ F @ top_top_set_a )
= top_top_set_real ) ).
% vimage_UNIV
thf(fact_1186_vimage__UNIV,axiom,
! [F: a > real] :
( ( vimage_a_real @ F @ top_top_set_real )
= top_top_set_a ) ).
% vimage_UNIV
thf(fact_1187_vimage__UNIV,axiom,
! [F: real > real] :
( ( vimage_real_real @ F @ top_top_set_real )
= top_top_set_real ) ).
% vimage_UNIV
thf(fact_1188_sets__sup__measure_H,axiom,
! [B: sigma_measure_a,A: sigma_measure_a] :
( ( ( sigma_sets_a @ B )
= ( sigma_sets_a @ A ) )
=> ( ( sigma_sets_a @ ( measur3004909623614618064sure_a @ A @ B ) )
= ( sigma_sets_a @ A ) ) ) ).
% sets_sup_measure'
thf(fact_1189_sets__sup__measure_H,axiom,
! [B: sigma_measure_real,A: sigma_measure_real] :
( ( ( sigma_sets_real @ B )
= ( sigma_sets_real @ A ) )
=> ( ( sigma_sets_real @ ( measur2147279183506585690e_real @ A @ B ) )
= ( sigma_sets_real @ A ) ) ) ).
% sets_sup_measure'
thf(fact_1190_space__borel,axiom,
( ( sigma_space_real @ borel_5078946678739801102l_real )
= top_top_set_real ) ).
% space_borel
thf(fact_1191_the__elem__eq,axiom,
! [X: a] :
( ( the_elem_a @ ( insert_a @ X @ bot_bot_set_a ) )
= X ) ).
% the_elem_eq
thf(fact_1192_the__elem__eq,axiom,
! [X: real] :
( ( the_elem_real @ ( insert_real @ X @ bot_bot_set_real ) )
= X ) ).
% the_elem_eq
thf(fact_1193_image__vimage__eq,axiom,
! [F: a > real,A: set_real] :
( ( image_a_real @ F @ ( vimage_a_real @ F @ A ) )
= ( inf_inf_set_real @ A @ ( image_a_real @ F @ top_top_set_a ) ) ) ).
% image_vimage_eq
thf(fact_1194_image__vimage__eq,axiom,
! [F: a > a,A: set_a] :
( ( image_a_a @ F @ ( vimage_a_a @ F @ A ) )
= ( inf_inf_set_a @ A @ ( image_a_a @ F @ top_top_set_a ) ) ) ).
% image_vimage_eq
thf(fact_1195_image__vimage__eq,axiom,
! [F: real > a,A: set_a] :
( ( image_real_a @ F @ ( vimage_real_a @ F @ A ) )
= ( inf_inf_set_a @ A @ ( image_real_a @ F @ top_top_set_real ) ) ) ).
% image_vimage_eq
thf(fact_1196_image__vimage__eq,axiom,
! [F: real > real,A: set_real] :
( ( image_real_real @ F @ ( vimage_real_real @ F @ A ) )
= ( inf_inf_set_real @ A @ ( image_real_real @ F @ top_top_set_real ) ) ) ).
% image_vimage_eq
thf(fact_1197_fun_Oinj__map,axiom,
! [F: real > a] :
( ( inj_on_real_a @ F @ top_top_set_real )
=> ( inj_on_a_real_a_a @ ( comp_real_a_a @ F ) @ top_top_set_a_real ) ) ).
% fun.inj_map
thf(fact_1198_fun_Oinj__map,axiom,
! [F: real > real] :
( ( inj_on_real_real @ F @ top_top_set_real )
=> ( inj_on838481142967259233l_real @ ( comp_real_real_real @ F ) @ top_to2071711978144146653l_real ) ) ).
% fun.inj_map
thf(fact_1199_top_Oextremum__uniqueI,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ top_top_set_real @ A2 )
=> ( A2 = top_top_set_real ) ) ).
% top.extremum_uniqueI
thf(fact_1200_top_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
=> ( A2 = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_1201_top_Oextremum__unique,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ top_top_set_real @ A2 )
= ( A2 = top_top_set_real ) ) ).
% top.extremum_unique
thf(fact_1202_top_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
= ( A2 = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_1203_top__greatest,axiom,
! [A2: set_real] : ( ord_less_eq_set_real @ A2 @ top_top_set_real ) ).
% top_greatest
thf(fact_1204_top__greatest,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ top_top_set_a ) ).
% top_greatest
thf(fact_1205_standard__borel__space__UNIV__axioms_Ointro,axiom,
! [M: sigma_measure_a] :
( ( ( sigma_space_a @ M )
= top_top_set_a )
=> ( standa2153564630574221018ioms_a @ M ) ) ).
% standard_borel_space_UNIV_axioms.intro
thf(fact_1206_standard__borel__space__UNIV__axioms_Ointro,axiom,
! [M: sigma_measure_real] :
( ( ( sigma_space_real @ M )
= top_top_set_real )
=> ( standa1498722272452280784s_real @ M ) ) ).
% standard_borel_space_UNIV_axioms.intro
thf(fact_1207_standard__borel__space__UNIV__axioms__def,axiom,
( standa2153564630574221018ioms_a
= ( ^ [M3: sigma_measure_a] :
( ( sigma_space_a @ M3 )
= top_top_set_a ) ) ) ).
% standard_borel_space_UNIV_axioms_def
thf(fact_1208_standard__borel__space__UNIV__axioms__def,axiom,
( standa1498722272452280784s_real
= ( ^ [M3: sigma_measure_real] :
( ( sigma_space_real @ M3 )
= top_top_set_real ) ) ) ).
% standard_borel_space_UNIV_axioms_def
thf(fact_1209_UNIV__eq__I,axiom,
! [A: set_set_a] :
( ! [X2: set_a] : ( member_set_a @ X2 @ A )
=> ( top_top_set_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_1210_UNIV__eq__I,axiom,
! [A: set_real_real] :
( ! [X2: real > real] : ( member_real_real @ X2 @ A )
=> ( top_to2071711978144146653l_real = A ) ) ).
% UNIV_eq_I
thf(fact_1211_UNIV__eq__I,axiom,
! [A: set_real_a] :
( ! [X2: real > a] : ( member_real_a @ X2 @ A )
=> ( top_top_set_real_a = A ) ) ).
% UNIV_eq_I
thf(fact_1212_UNIV__eq__I,axiom,
! [A: set_a_real] :
( ! [X2: a > real] : ( member_a_real @ X2 @ A )
=> ( top_top_set_a_real = A ) ) ).
% UNIV_eq_I
thf(fact_1213_UNIV__eq__I,axiom,
! [A: set_a] :
( ! [X2: a] : ( member_a @ X2 @ A )
=> ( top_top_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_1214_UNIV__eq__I,axiom,
! [A: set_real] :
( ! [X2: real] : ( member_real @ X2 @ A )
=> ( top_top_set_real = A ) ) ).
% UNIV_eq_I
thf(fact_1215_UNIV__witness,axiom,
? [X2: set_a] : ( member_set_a @ X2 @ top_top_set_set_a ) ).
% UNIV_witness
thf(fact_1216_UNIV__witness,axiom,
? [X2: real > real] : ( member_real_real @ X2 @ top_to2071711978144146653l_real ) ).
% UNIV_witness
thf(fact_1217_UNIV__witness,axiom,
? [X2: real > a] : ( member_real_a @ X2 @ top_top_set_real_a ) ).
% UNIV_witness
thf(fact_1218_UNIV__witness,axiom,
? [X2: a > real] : ( member_a_real @ X2 @ top_top_set_a_real ) ).
% UNIV_witness
thf(fact_1219_UNIV__witness,axiom,
? [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_1220_UNIV__witness,axiom,
? [X2: real] : ( member_real @ X2 @ top_top_set_real ) ).
% UNIV_witness
thf(fact_1221_boolean__algebra_Oconj__one__right,axiom,
! [X: set_real] :
( ( inf_inf_set_real @ X @ top_top_set_real )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_1222_surjD,axiom,
! [F: a > a,Y3: a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ? [X2: a] :
( Y3
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_1223_surjD,axiom,
! [F: a > real,Y3: real] :
( ( ( image_a_real @ F @ top_top_set_a )
= top_top_set_real )
=> ? [X2: a] :
( Y3
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_1224_surjD,axiom,
! [F: real > a,Y3: a] :
( ( ( image_real_a @ F @ top_top_set_real )
= top_top_set_a )
=> ? [X2: real] :
( Y3
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_1225_surjD,axiom,
! [F: real > real,Y3: real] :
( ( ( image_real_real @ F @ top_top_set_real )
= top_top_set_real )
=> ? [X2: real] :
( Y3
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_1226_surjE,axiom,
! [F: a > a,Y3: a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ~ ! [X2: a] :
( Y3
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_1227_surjE,axiom,
! [F: a > real,Y3: real] :
( ( ( image_a_real @ F @ top_top_set_a )
= top_top_set_real )
=> ~ ! [X2: a] :
( Y3
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_1228_surjE,axiom,
! [F: real > a,Y3: a] :
( ( ( image_real_a @ F @ top_top_set_real )
= top_top_set_a )
=> ~ ! [X2: real] :
( Y3
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_1229_surjE,axiom,
! [F: real > real,Y3: real] :
( ( ( image_real_real @ F @ top_top_set_real )
= top_top_set_real )
=> ~ ! [X2: real] :
( Y3
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_1230_surjI,axiom,
! [G: a > a,F: a > a] :
( ! [X2: a] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_a_a @ G @ top_top_set_a )
= top_top_set_a ) ) ).
% surjI
thf(fact_1231_surjI,axiom,
! [G: a > real,F: real > a] :
( ! [X2: real] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_a_real @ G @ top_top_set_a )
= top_top_set_real ) ) ).
% surjI
thf(fact_1232_surjI,axiom,
! [G: real > a,F: a > real] :
( ! [X2: a] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_real_a @ G @ top_top_set_real )
= top_top_set_a ) ) ).
% surjI
thf(fact_1233_surjI,axiom,
! [G: real > real,F: real > real] :
( ! [X2: real] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_real_real @ G @ top_top_set_real )
= top_top_set_real ) ) ).
% surjI
thf(fact_1234_surj__def,axiom,
! [F: a > a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
= ( ! [Y5: a] :
? [X3: a] :
( Y5
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_1235_surj__def,axiom,
! [F: a > real] :
( ( ( image_a_real @ F @ top_top_set_a )
= top_top_set_real )
= ( ! [Y5: real] :
? [X3: a] :
( Y5
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_1236_surj__def,axiom,
! [F: real > a] :
( ( ( image_real_a @ F @ top_top_set_real )
= top_top_set_a )
= ( ! [Y5: a] :
? [X3: real] :
( Y5
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_1237_surj__def,axiom,
! [F: real > real] :
( ( ( image_real_real @ F @ top_top_set_real )
= top_top_set_real )
= ( ! [Y5: real] :
? [X3: real] :
( Y5
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_1238_rangeI,axiom,
! [F: a > real,X: a] : ( member_real @ ( F @ X ) @ ( image_a_real @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_1239_rangeI,axiom,
! [F: a > a,X: a] : ( member_a @ ( F @ X ) @ ( image_a_a @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_1240_rangeI,axiom,
! [F: real > real,X: real] : ( member_real @ ( F @ X ) @ ( image_real_real @ F @ top_top_set_real ) ) ).
% rangeI
thf(fact_1241_rangeI,axiom,
! [F: real > set_a,X: real] : ( member_set_a @ ( F @ X ) @ ( image_real_set_a @ F @ top_top_set_real ) ) ).
% rangeI
thf(fact_1242_rangeI,axiom,
! [F: real > real > real,X: real] : ( member_real_real @ ( F @ X ) @ ( image_real_real_real2 @ F @ top_top_set_real ) ) ).
% rangeI
thf(fact_1243_rangeI,axiom,
! [F: real > real > a,X: real] : ( member_real_a @ ( F @ X ) @ ( image_real_real_a2 @ F @ top_top_set_real ) ) ).
% rangeI
thf(fact_1244_rangeI,axiom,
! [F: real > a > real,X: real] : ( member_a_real @ ( F @ X ) @ ( image_real_a_real2 @ F @ top_top_set_real ) ) ).
% rangeI
thf(fact_1245_rangeI,axiom,
! [F: real > a,X: real] : ( member_a @ ( F @ X ) @ ( image_real_a @ F @ top_top_set_real ) ) ).
% rangeI
thf(fact_1246_range__eqI,axiom,
! [B2: real,F: a > real,X: a] :
( ( B2
= ( F @ X ) )
=> ( member_real @ B2 @ ( image_a_real @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_1247_range__eqI,axiom,
! [B2: a,F: a > a,X: a] :
( ( B2
= ( F @ X ) )
=> ( member_a @ B2 @ ( image_a_a @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_1248_range__eqI,axiom,
! [B2: real,F: real > real,X: real] :
( ( B2
= ( F @ X ) )
=> ( member_real @ B2 @ ( image_real_real @ F @ top_top_set_real ) ) ) ).
% range_eqI
thf(fact_1249_range__eqI,axiom,
! [B2: set_a,F: real > set_a,X: real] :
( ( B2
= ( F @ X ) )
=> ( member_set_a @ B2 @ ( image_real_set_a @ F @ top_top_set_real ) ) ) ).
% range_eqI
thf(fact_1250_range__eqI,axiom,
! [B2: real > real,F: real > real > real,X: real] :
( ( B2
= ( F @ X ) )
=> ( member_real_real @ B2 @ ( image_real_real_real2 @ F @ top_top_set_real ) ) ) ).
% range_eqI
thf(fact_1251_range__eqI,axiom,
! [B2: real > a,F: real > real > a,X: real] :
( ( B2
= ( F @ X ) )
=> ( member_real_a @ B2 @ ( image_real_real_a2 @ F @ top_top_set_real ) ) ) ).
% range_eqI
thf(fact_1252_range__eqI,axiom,
! [B2: a > real,F: real > a > real,X: real] :
( ( B2
= ( F @ X ) )
=> ( member_a_real @ B2 @ ( image_real_a_real2 @ F @ top_top_set_real ) ) ) ).
% range_eqI
thf(fact_1253_range__eqI,axiom,
! [B2: a,F: real > a,X: real] :
( ( B2
= ( F @ X ) )
=> ( member_a @ B2 @ ( image_real_a @ F @ top_top_set_real ) ) ) ).
% range_eqI
thf(fact_1254_empty__not__UNIV,axiom,
bot_bot_set_a != top_top_set_a ).
% empty_not_UNIV
thf(fact_1255_empty__not__UNIV,axiom,
bot_bot_set_real != top_top_set_real ).
% empty_not_UNIV
thf(fact_1256_subset__UNIV,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ A @ top_top_set_real ) ).
% subset_UNIV
thf(fact_1257_subset__UNIV,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% subset_UNIV
thf(fact_1258_insert__UNIV,axiom,
! [X: a] :
( ( insert_a @ X @ top_top_set_a )
= top_top_set_a ) ).
% insert_UNIV
thf(fact_1259_insert__UNIV,axiom,
! [X: real] :
( ( insert_real @ X @ top_top_set_real )
= top_top_set_real ) ).
% insert_UNIV
thf(fact_1260_Int__UNIV__left,axiom,
! [B: set_real] :
( ( inf_inf_set_real @ top_top_set_real @ B )
= B ) ).
% Int_UNIV_left
thf(fact_1261_Int__UNIV__right,axiom,
! [A: set_real] :
( ( inf_inf_set_real @ A @ top_top_set_real )
= A ) ).
% Int_UNIV_right
thf(fact_1262_inj__def,axiom,
! [F: a > real] :
( ( inj_on_a_real @ F @ top_top_set_a )
= ( ! [X3: a,Y5: a] :
( ( ( F @ X3 )
= ( F @ Y5 ) )
=> ( X3 = Y5 ) ) ) ) ).
% inj_def
thf(fact_1263_inj__def,axiom,
! [F: real > real] :
( ( inj_on_real_real @ F @ top_top_set_real )
= ( ! [X3: real,Y5: real] :
( ( ( F @ X3 )
= ( F @ Y5 ) )
=> ( X3 = Y5 ) ) ) ) ).
% inj_def
thf(fact_1264_inj__eq,axiom,
! [F: a > real,X: a,Y3: a] :
( ( inj_on_a_real @ F @ top_top_set_a )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
= ( X = Y3 ) ) ) ).
% inj_eq
thf(fact_1265_inj__eq,axiom,
! [F: real > real,X: real,Y3: real] :
( ( inj_on_real_real @ F @ top_top_set_real )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
= ( X = Y3 ) ) ) ).
% inj_eq
thf(fact_1266_injI,axiom,
! [F: a > real] :
( ! [X2: a,Y2: a] :
( ( ( F @ X2 )
= ( F @ Y2 ) )
=> ( X2 = Y2 ) )
=> ( inj_on_a_real @ F @ top_top_set_a ) ) ).
% injI
thf(fact_1267_injI,axiom,
! [F: real > real] :
( ! [X2: real,Y2: real] :
( ( ( F @ X2 )
= ( F @ Y2 ) )
=> ( X2 = Y2 ) )
=> ( inj_on_real_real @ F @ top_top_set_real ) ) ).
% injI
thf(fact_1268_injD,axiom,
! [F: a > real,X: a,Y3: a] :
( ( inj_on_a_real @ F @ top_top_set_a )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( X = Y3 ) ) ) ).
% injD
thf(fact_1269_injD,axiom,
! [F: real > real,X: real,Y3: real] :
( ( inj_on_real_real @ F @ top_top_set_real )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( X = Y3 ) ) ) ).
% injD
thf(fact_1270_standard__borel__space__UNIV_Og__surj_H,axiom,
! [M: sigma_measure_a] :
( ( standa5776109378963170237UNIV_a @ M )
=> ( ( image_real_a @ ( standard_g_a @ M ) @ top_top_set_real )
= top_top_set_a ) ) ).
% standard_borel_space_UNIV.g_surj'
thf(fact_1271_standard__borel__space__UNIV_Og__surj_H,axiom,
! [M: sigma_measure_real] :
( ( standa1306199911732814765V_real @ M )
=> ( ( image_real_real @ ( standard_g_real @ M ) @ top_top_set_real )
= top_top_set_real ) ) ).
% standard_borel_space_UNIV.g_surj'
thf(fact_1272_borel__open,axiom,
! [A: set_real] :
( ( topolo4860482606490270245n_real @ A )
=> ( member_set_real @ A @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ).
% borel_open
thf(fact_1273_range__subsetD,axiom,
! [F: real > a,B: set_a,I2: real] :
( ( ord_less_eq_set_a @ ( image_real_a @ F @ top_top_set_real ) @ B )
=> ( member_a @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_1274_real_Ospace__UNIV,axiom,
( ( sigma_space_real @ borel_5078946678739801102l_real )
= top_top_set_real ) ).
% real.space_UNIV
thf(fact_1275_real_Og__surj_H,axiom,
( ( image_real_real @ ( standard_g_real @ borel_5078946678739801102l_real ) @ top_top_set_real )
= top_top_set_real ) ).
% real.g_surj'
thf(fact_1276_real_Of__inj_H,axiom,
inj_on_real_real @ ( standard_f_real @ borel_5078946678739801102l_real ) @ top_top_set_real ).
% real.f_inj'
thf(fact_1277_range__mult,axiom,
! [A2: real] :
( ( ( A2 = zero_zero_real )
=> ( ( image_real_real @ ( times_times_real @ A2 ) @ top_top_set_real )
= ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
& ( ( A2 != zero_zero_real )
=> ( ( image_real_real @ ( times_times_real @ A2 ) @ top_top_set_real )
= top_top_set_real ) ) ) ).
% range_mult
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_set_real @ ( image_a_real @ ( standard_f_a @ m ) @ u ) @ ( vimage_real_a @ ( standard_g_a @ m ) @ u ) ).
%------------------------------------------------------------------------------