TPTP Problem File: SLH0076^1.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 : Universal_Hash_Families/0033_Preliminary_Results/prob_00067_002413__18448904_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1575 ( 389 unt; 296 typ; 0 def)
% Number of atoms : 3976 ( 942 equ; 0 cnn)
% Maximal formula atoms : 36 ( 3 avg)
% Number of connectives : 12668 ( 223 ~; 22 |; 131 &;10195 @)
% ( 0 <=>;2097 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 7 avg)
% Number of types : 38 ( 37 usr)
% Number of type conns : 1064 (1064 >; 0 *; 0 +; 0 <<)
% Number of symbols : 262 ( 259 usr; 27 con; 0-5 aty)
% Number of variables : 3800 ( 170 ^;3600 !; 30 ?;3800 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:44:18.580
%------------------------------------------------------------------------------
% Could-be-implicit typings (37)
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Set__Oset_Itf__b_J_J,type,
sigma_measure_set_b: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Set__Oset_Itf__a_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__b_J_J_J,type,
set_set_set_b: $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_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Nat__Onat_J,type,
sigma_measure_nat: $tType ).
thf(ty_n_t__Extended____Nonnegative____Real__Oennreal,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__c_Mt__Nat__Onat_J_J,type,
set_c_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mtf__c_J_J,type,
set_nat_c: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_Mt__Nat__Onat_J_J,type,
set_o_nat: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_Itf__c_J,type,
sigma_measure_c: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_Itf__b_J,type,
sigma_measure_b: $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__b_J_J,type,
set_set_b: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_I_Eo_J,type,
sigma_measure_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__c_Mtf__c_J_J,type,
set_c_c: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__c_Mtf__a_J_J,type,
set_c_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__b_Mtf__b_J_J,type,
set_b_b: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__b_Mtf__a_J_J,type,
set_b_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
set_a_b: $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_I_062_Itf__c_M_Eo_J_J,type,
set_c_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_Mtf__c_J_J,type,
set_o_c: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
set_o_o: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_Itf__c_J,type,
set_c: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__c,type,
c: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (259)
thf(sy_c_Complete__Measure_Ocomplete__measure_001t__Set__Oset_Itf__a_J,type,
comple6693822263253554161_set_a: sigma_measure_set_a > $o ).
thf(sy_c_Complete__Measure_Ocomplete__measure_001t__Set__Oset_Itf__b_J,type,
comple6693822267556782962_set_b: sigma_measure_set_b > $o ).
thf(sy_c_Complete__Measure_Ocomplete__measure_001tf__a,type,
comple8155536527497655953sure_a: sigma_measure_a > $o ).
thf(sy_c_Complete__Measure_Ocomplete__measure_001tf__b,type,
comple8155536527497655954sure_b: sigma_measure_b > $o ).
thf(sy_c_Complete__Measure_Ocompletion_001t__Set__Oset_Itf__a_J,type,
comple8942076146008132200_set_a: sigma_measure_set_a > sigma_measure_set_a ).
thf(sy_c_Complete__Measure_Ocompletion_001t__Set__Oset_Itf__b_J,type,
comple8942076150311361001_set_b: sigma_measure_set_b > sigma_measure_set_b ).
thf(sy_c_Complete__Measure_Ocompletion_001tf__a,type,
comple3428971583294703880tion_a: sigma_measure_a > sigma_measure_a ).
thf(sy_c_Complete__Measure_Ocompletion_001tf__b,type,
comple3428971583294703881tion_b: sigma_measure_b > sigma_measure_b ).
thf(sy_c_Complete__Measure_Omain__part_001tf__a,type,
complete_main_part_a: sigma_measure_a > set_a > set_a ).
thf(sy_c_Complete__Measure_Omain__part_001tf__b,type,
complete_main_part_b: sigma_measure_b > set_b > set_b ).
thf(sy_c_Complete__Measure_Onull__part_001t__Set__Oset_Itf__a_J,type,
comple7912714278253110634_set_a: sigma_measure_set_a > set_set_a > set_set_a ).
thf(sy_c_Complete__Measure_Onull__part_001t__Set__Oset_Itf__b_J,type,
comple7912714282556339435_set_b: sigma_measure_set_b > set_set_b > set_set_b ).
thf(sy_c_Complete__Measure_Onull__part_001tf__a,type,
complete_null_part_a: sigma_measure_a > set_a > set_a ).
thf(sy_c_Complete__Measure_Onull__part_001tf__b,type,
complete_null_part_b: sigma_measure_b > set_b > set_b ).
thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity_001t__Extended____Nonnegative____Real__Oennreal,type,
extend2057119558705770725nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Extended__Nonnegative__Real_Oenn2real,type,
extend1669699412028896998n2real: extend8495563244428889912nnreal > real ).
thf(sy_c_Extended__Nonnegative__Real_Oennreal,type,
extend7643940197134561352nnreal: real > extend8495563244428889912nnreal ).
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_Ocomp_001tf__a_001tf__a_001tf__b,type,
comp_a_a_b: ( a > a ) > ( b > a ) > b > a ).
thf(sy_c_Fun_Ocomp_001tf__a_001tf__b_001tf__a,type,
comp_a_b_a: ( a > b ) > ( a > a ) > a > b ).
thf(sy_c_Fun_Ocomp_001tf__a_001tf__b_001tf__b,type,
comp_a_b_b: ( a > b ) > ( b > a ) > b > b ).
thf(sy_c_Fun_Ocomp_001tf__b_001tf__a_001tf__a,type,
comp_b_a_a: ( b > a ) > ( a > b ) > a > a ).
thf(sy_c_Fun_Ocomp_001tf__b_001tf__b_001tf__a,type,
comp_b_b_a: ( b > b ) > ( a > b ) > a > b ).
thf(sy_c_Giry__Monad_Osubprob__space_001tf__a,type,
giry_subprob_space_a: sigma_measure_a > $o ).
thf(sy_c_Giry__Monad_Osubprob__space_001tf__b,type,
giry_subprob_space_b: sigma_measure_b > $o ).
thf(sy_c_Giry__Monad_Osubprob__space__axioms_001tf__a,type,
giry_s1767857069175831631ioms_a: sigma_measure_a > $o ).
thf(sy_c_Giry__Monad_Osubprob__space__axioms_001tf__b,type,
giry_s1767857069175831632ioms_b: sigma_measure_b > $o ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
abs_abs_real: real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nonnegative____Real__Oennreal,type,
minus_8429688780609304081nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
minus_minus_set_a_a: set_a_a > set_a_a > set_a_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
minus_minus_set_a_b: set_a_b > set_a_b > set_a_b ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_Eo_J,type,
minus_minus_set_o: set_o > set_o > set_o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
minus_5736297505244876581_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
minus_5807331545291222566_set_b: set_set_b > set_set_b > set_set_b ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__b_J,type,
minus_minus_set_b: set_b > set_b > set_b ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__c_J,type,
minus_minus_set_c: set_c > set_c > set_c ).
thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nonnegative____Real__Oennreal,type,
one_on2969667320475766781nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nonnegative____Real__Oennreal,type,
plus_p1859984266308609217nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Nonnegative____Real__Oennreal,type,
times_1893300245718287421nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
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__Extended____Nonnegative____Real__Oennreal,type,
zero_z7100319975126383169nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__event_001tf__a,type,
indepe3567167809233210430vent_a: sigma_measure_a > set_a > set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__events_001tf__a_001_Eo,type,
indepe3695496658712714478ts_a_o: sigma_measure_a > ( $o > set_a ) > set_o > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__set_001tf__a,type,
indepe2041756565122539606_set_a: sigma_measure_a > set_set_a > set_set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__set_001tf__b,type,
indepe2041756565122539607_set_b: sigma_measure_b > set_set_b > set_set_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001_062_Itf__a_Mtf__a_J,type,
indepe1219004911012930679_a_a_a: sigma_measure_a > ( ( a > a ) > set_set_a ) > set_a_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001_062_Itf__a_Mtf__b_J,type,
indepe1219004915316159480_a_a_b: sigma_measure_a > ( ( a > b ) > set_set_a ) > set_a_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001_Eo,type,
indepe7780107833195774214ts_a_o: sigma_measure_a > ( $o > set_set_a ) > set_o > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001t__Nat__Onat,type,
indepe6267730027088848354_a_nat: sigma_measure_a > ( nat > set_set_a ) > set_nat > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001t__Set__Oset_Itf__a_J,type,
indepe4967106450811773644_set_a: sigma_measure_a > ( set_a > set_set_a ) > set_set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001t__Set__Oset_Itf__b_J,type,
indepe4967106455115002445_set_b: sigma_measure_a > ( set_b > set_set_a ) > set_set_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001tf__a,type,
indepe8927441866673418604ts_a_a: sigma_measure_a > ( a > set_set_a ) > set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001tf__c,type,
indepe8927441866673418606ts_a_c: sigma_measure_a > ( c > set_set_a ) > set_c > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001_062_Itf__a_Mtf__a_J,type,
indepe7072862680909553272_b_a_a: sigma_measure_b > ( ( a > a ) > set_set_b ) > set_a_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001_062_Itf__a_Mtf__b_J,type,
indepe7072862685212782073_b_a_b: sigma_measure_b > ( ( a > b ) > set_set_b ) > set_a_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001_Eo,type,
indepe4880885433731379909ts_b_o: sigma_measure_b > ( $o > set_set_b ) > set_o > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001t__Nat__Onat,type,
indepe7503174356045242851_b_nat: sigma_measure_b > ( nat > set_set_b ) > set_nat > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001t__Set__Oset_Itf__a_J,type,
indepe6311571487621261579_set_a: sigma_measure_b > ( set_a > set_set_b ) > set_set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001t__Set__Oset_Itf__b_J,type,
indepe6311571491924490380_set_b: sigma_measure_b > ( set_b > set_set_b ) > set_set_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001tf__a,type,
indepe6139986284700742571ts_b_a: sigma_measure_b > ( a > set_set_b ) > set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001tf__c,type,
indepe6139986284700742573ts_b_c: sigma_measure_b > ( c > set_set_b ) > set_c > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001tf__a_001tf__a,type,
indepe2440653194691626188ar_a_a: sigma_measure_a > sigma_measure_a > ( a > a ) > sigma_measure_a > ( a > a ) > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001tf__a_001tf__b,type,
indepe2440653194691626189ar_a_b: sigma_measure_a > sigma_measure_b > ( a > b ) > sigma_measure_b > ( a > b ) > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001tf__b_001tf__a,type,
indepe8876569649573725963ar_b_a: sigma_measure_b > sigma_measure_a > ( b > a ) > sigma_measure_a > ( b > a ) > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001tf__b_001tf__b,type,
indepe8876569649573725964ar_b_b: sigma_measure_b > sigma_measure_b > ( b > b ) > sigma_measure_b > ( b > b ) > $o ).
thf(sy_c_Independent__Family_Oprob__space_Otail__events_001tf__a_001t__Nat__Onat,type,
indepe7538416700049374166_a_nat: sigma_measure_a > ( nat > set_set_a ) > set_set_a ).
thf(sy_c_Independent__Family_Oprob__space_Otail__events_001tf__b_001t__Nat__Onat,type,
indepe8773861029005768663_b_nat: sigma_measure_b > ( nat > set_set_b ) > set_set_b ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Int__Oint,type,
inf_inf_int: int > int > int ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
inf_inf_set_a_a: set_a_a > set_a_a > set_a_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
inf_inf_set_a_b: set_a_b > set_a_b > set_a_b ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_Eo_J,type,
inf_inf_set_o: set_o > set_o > set_o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
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_It__Set__Oset_Itf__b_J_J,type,
inf_inf_set_set_b: set_set_b > set_set_b > set_set_b ).
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_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__b_J,type,
inf_inf_set_b: set_b > set_b > set_b ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__c_J,type,
inf_inf_set_c: set_c > set_c > set_c ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
sup_sup_int: int > int > int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
sup_sup_set_a_a: set_a_a > set_a_a > set_a_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
sup_sup_set_a_b: set_a_b > set_a_b > set_a_b ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_Eo_J,type,
sup_sup_set_o: set_o > set_o > set_o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
sup_sup_set_set_b: set_set_b > set_set_b > set_set_b ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__b_J,type,
sup_sup_set_b: set_b > set_b > set_b ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__c_J,type,
sup_sup_set_c: set_c > set_c > set_c ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Sigma____Algebra__Omeasure_Itf__a_J,type,
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thf(sy_c_Measure__Space_Odistr_001_Eo_001tf__a,type,
measure_distr_o_a: sigma_measure_o > sigma_measure_a > ( $o > a ) > sigma_measure_a ).
thf(sy_c_Measure__Space_Odistr_001_Eo_001tf__b,type,
measure_distr_o_b: sigma_measure_o > sigma_measure_b > ( $o > b ) > sigma_measure_b ).
thf(sy_c_Measure__Space_Odistr_001t__Nat__Onat_001tf__a,type,
measure_distr_nat_a: sigma_measure_nat > sigma_measure_a > ( nat > a ) > sigma_measure_a ).
thf(sy_c_Measure__Space_Odistr_001t__Nat__Onat_001tf__b,type,
measure_distr_nat_b: sigma_measure_nat > sigma_measure_b > ( nat > b ) > sigma_measure_b ).
thf(sy_c_Measure__Space_Odistr_001t__Set__Oset_Itf__a_J_001tf__a,type,
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thf(sy_c_Measure__Space_Odistr_001t__Set__Oset_Itf__a_J_001tf__b,type,
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thf(sy_c_Measure__Space_Odistr_001tf__a_001tf__a,type,
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thf(sy_c_Measure__Space_Odistr_001tf__a_001tf__b,type,
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thf(sy_c_Measure__Space_Odistr_001tf__b_001tf__a,type,
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thf(sy_c_Measure__Space_Odistr_001tf__b_001tf__b,type,
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thf(sy_c_Measure__Space_Odistr_001tf__c_001tf__a,type,
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thf(sy_c_Measure__Space_Odistr_001tf__c_001tf__b,type,
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thf(sy_c_Measure__Space_Ofinite__measure_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Measure__Space_Ofinite__measure_001t__Set__Oset_Itf__b_J,type,
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thf(sy_c_Measure__Space_Ofinite__measure_001tf__a,type,
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thf(sy_c_Measure__Space_Ofinite__measure_001tf__b,type,
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thf(sy_c_Measure__Space_Ofinite__measure__axioms_001tf__a,type,
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thf(sy_c_Measure__Space_Ofinite__measure__axioms_001tf__b,type,
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thf(sy_c_Measure__Space_Ofmeasurable_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Measure__Space_Ofmeasurable_001t__Set__Oset_Itf__b_J,type,
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thf(sy_c_Measure__Space_Ofmeasurable_001tf__a,type,
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thf(sy_c_Measure__Space_Ofmeasurable_001tf__b,type,
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thf(sy_c_Measure__Space_Oincreasing_001t__Set__Oset_Itf__a_J_001t__Int__Oint,type,
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thf(sy_c_Measure__Space_Oincreasing_001t__Set__Oset_Itf__a_J_001t__Nat__Onat,type,
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thf(sy_c_Measure__Space_Oincreasing_001tf__a_001t__Int__Oint,type,
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thf(sy_c_Measure__Space_Oincreasing_001tf__a_001t__Nat__Onat,type,
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thf(sy_c_Measure__Space_Oincreasing_001tf__a_001t__Real__Oreal,type,
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thf(sy_c_Measure__Space_Oincreasing_001tf__a_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Measure__Space_Onull__sets_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Measure__Space_Onull__sets_001tf__a,type,
measure_null_sets_a: sigma_measure_a > set_set_a ).
thf(sy_c_Measure__Space_Onull__sets_001tf__b,type,
measure_null_sets_b: sigma_measure_b > set_set_b ).
thf(sy_c_Measure__Space_Osigma__finite__measure_001tf__a,type,
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thf(sy_c_Measure__Space_Osigma__finite__measure_001tf__b,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
bot_bot_set_a_a: set_a_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
bot_bot_set_a_b: set_a_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
bot_bot_set_o: set_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
bot_bot_set_set_b: set_set_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__b_J,type,
bot_bot_set_b: set_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__c_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
ord_less_eq_set_a_a: set_a_a > set_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
ord_less_eq_set_a_b: set_a_b > set_a_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__b_Mtf__a_J_J,type,
ord_less_eq_set_b_a: set_b_a > set_b_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__b_Mtf__b_J_J,type,
ord_less_eq_set_b_b: set_b_b > set_b_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__b_J_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__b_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__c_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Probability__Measure_Oprob__space_001tf__a,type,
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thf(sy_c_Probability__Measure_Oprob__space_001tf__b,type,
probab7247484486040049090pace_b: sigma_measure_b > $o ).
thf(sy_c_Probability__Measure_Oprob__space__axioms_001tf__a,type,
probab8302655048591552734ioms_a: sigma_measure_a > $o ).
thf(sy_c_Probability__Measure_Oprob__space__axioms_001tf__b,type,
probab8302655048591552735ioms_b: sigma_measure_b > $o ).
thf(sy_c_Product__Type_Obool_Ocase__bool_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Product__Type_Obool_Ocase__bool_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Set_OCollect_001_062_Itf__a_Mtf__a_J,type,
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thf(sy_c_Set_OCollect_001_062_Itf__a_Mtf__b_J,type,
collect_a_b: ( ( a > b ) > $o ) > set_a_b ).
thf(sy_c_Set_OCollect_001_Eo,type,
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thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__b_J,type,
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thf(sy_c_Set_OCollect_001tf__a,type,
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thf(sy_c_Set_OCollect_001tf__c,type,
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thf(sy_c_Sigma__Algebra_Oemeasure_001tf__a,type,
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thf(sy_c_Sigma__Algebra_Omeasurable_001_Eo_001_Eo,type,
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thf(sy_c_Sigma__Algebra_Omeasurable_001_Eo_001t__Nat__Onat,type,
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thf(sy_c_Sigma__Algebra_Omeasurable_001t__Nat__Onat_001_Eo,type,
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thf(sy_c_Sigma__Algebra_Omeasurable_001tf__a_001tf__b,type,
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thf(sy_c_Sigma__Algebra_Omeasurable_001tf__b_001tf__a,type,
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thf(sy_c_Sigma__Algebra_Omeasurable_001tf__b_001tf__b,type,
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thf(sy_c_Sigma__Algebra_Omeasurable_001tf__c_001_Eo,type,
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thf(sy_c_Sigma__Algebra_Omeasurable_001tf__c_001t__Nat__Onat,type,
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thf(sy_c_Sigma__Algebra_Omeasurable_001tf__c_001tf__c,type,
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thf(sy_c_Sigma__Algebra_Omeasure_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Sigma__Algebra_Omeasure_001t__Set__Oset_Itf__b_J,type,
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thf(sy_c_Sigma__Algebra_Omeasure_001tf__a,type,
sigma_measure_a2: sigma_measure_a > set_a > real ).
thf(sy_c_Sigma__Algebra_Omeasure_001tf__b,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_Osets_001t__Set__Oset_Itf__b_J,type,
sigma_sets_set_b: sigma_measure_set_b > set_set_set_b ).
thf(sy_c_Sigma__Algebra_Osets_001tf__a,type,
sigma_sets_a: sigma_measure_a > set_set_a ).
thf(sy_c_Sigma__Algebra_Osets_001tf__b,type,
sigma_sets_b: sigma_measure_b > set_set_b ).
thf(sy_c_Sigma__Algebra_Osigma__algebra_001tf__a,type,
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thf(sy_c_Sigma__Algebra_Ospace_001_Eo,type,
sigma_space_o: sigma_measure_o > set_o ).
thf(sy_c_Sigma__Algebra_Ospace_001t__Nat__Onat,type,
sigma_space_nat: sigma_measure_nat > set_nat ).
thf(sy_c_Sigma__Algebra_Ospace_001t__Set__Oset_Itf__a_J,type,
sigma_space_set_a: sigma_measure_set_a > set_set_a ).
thf(sy_c_Sigma__Algebra_Ospace_001t__Set__Oset_Itf__b_J,type,
sigma_space_set_b: sigma_measure_set_b > set_set_b ).
thf(sy_c_Sigma__Algebra_Ospace_001tf__a,type,
sigma_space_a: sigma_measure_a > set_a ).
thf(sy_c_Sigma__Algebra_Ospace_001tf__b,type,
sigma_space_b: sigma_measure_b > set_b ).
thf(sy_c_Sigma__Algebra_Ospace_001tf__c,type,
sigma_space_c: sigma_measure_c > set_c ).
thf(sy_c_member_001_062_I_Eo_M_Eo_J,type,
member_o_o: ( $o > $o ) > set_o_o > $o ).
thf(sy_c_member_001_062_I_Eo_Mt__Nat__Onat_J,type,
member_o_nat: ( $o > nat ) > set_o_nat > $o ).
thf(sy_c_member_001_062_I_Eo_Mtf__c_J,type,
member_o_c: ( $o > c ) > set_o_c > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_Eo_J,type,
member_nat_o: ( nat > $o ) > set_nat_o > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__c_J,type,
member_nat_c: ( nat > c ) > set_nat_c > $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_001_062_Itf__a_Mtf__b_J,type,
member_a_b: ( a > b ) > set_a_b > $o ).
thf(sy_c_member_001_062_Itf__b_Mtf__a_J,type,
member_b_a: ( b > a ) > set_b_a > $o ).
thf(sy_c_member_001_062_Itf__b_Mtf__b_J,type,
member_b_b: ( b > b ) > set_b_b > $o ).
thf(sy_c_member_001_062_Itf__c_M_Eo_J,type,
member_c_o: ( c > $o ) > set_c_o > $o ).
thf(sy_c_member_001_062_Itf__c_Mt__Nat__Onat_J,type,
member_c_nat: ( c > nat ) > set_c_nat > $o ).
thf(sy_c_member_001_062_Itf__c_Mtf__a_J,type,
member_c_a: ( c > a ) > set_c_a > $o ).
thf(sy_c_member_001_062_Itf__c_Mtf__c_J,type,
member_c_c: ( c > c ) > set_c_c > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $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_It__Set__Oset_Itf__b_J_J,type,
member_set_set_b: set_set_b > set_set_set_b > $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_001t__Set__Oset_Itf__b_J,type,
member_set_b: set_b > set_set_b > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__c,type,
member_c: c > set_c > $o ).
thf(sy_v_A,type,
a2: c > set_set_b ).
thf(sy_v_F,type,
f: c > set_set_a ).
thf(sy_v_I,type,
i: set_c ).
thf(sy_v_M,type,
m: sigma_measure_a ).
thf(sy_v_N,type,
n: sigma_measure_b ).
thf(sy_v_f,type,
f2: a > b ).
% Relevant facts (1275)
thf(fact_0_prob__space__axioms,axiom,
probab7247484486040049089pace_a @ m ).
% prob_space_axioms
thf(fact_1_assms_I1_J,axiom,
member_a_b @ f2 @ ( sigma_measurable_a_b @ m @ n ) ).
% assms(1)
thf(fact_2_prob__space__distr,axiom,
! [F: a > b,M: sigma_measure_b] :
( ( member_a_b @ F @ ( sigma_measurable_a_b @ m @ M ) )
=> ( probab7247484486040049090pace_b @ ( measure_distr_a_b @ m @ M @ F ) ) ) ).
% prob_space_distr
thf(fact_3_prob__space__distr,axiom,
! [F: a > a,M: sigma_measure_a] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ m @ M ) )
=> ( probab7247484486040049089pace_a @ ( measure_distr_a_a @ m @ M @ F ) ) ) ).
% prob_space_distr
thf(fact_4_subprob__space__axioms,axiom,
giry_subprob_space_a @ m ).
% subprob_space_axioms
thf(fact_5_indep__sets__cong,axiom,
! [I: set_a_a,J: set_a_a,F2: ( a > a ) > set_set_a,G: ( a > a ) > set_set_a] :
( ( I = J )
=> ( ! [I2: a > a] :
( ( member_a_a @ I2 @ I )
=> ( ( F2 @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe1219004911012930679_a_a_a @ m @ F2 @ I )
= ( indepe1219004911012930679_a_a_a @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_6_indep__sets__cong,axiom,
! [I: set_a_b,J: set_a_b,F2: ( a > b ) > set_set_a,G: ( a > b ) > set_set_a] :
( ( I = J )
=> ( ! [I2: a > b] :
( ( member_a_b @ I2 @ I )
=> ( ( F2 @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe1219004915316159480_a_a_b @ m @ F2 @ I )
= ( indepe1219004915316159480_a_a_b @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_7_indep__sets__cong,axiom,
! [I: set_set_a,J: set_set_a,F2: set_a > set_set_a,G: set_a > set_set_a] :
( ( I = J )
=> ( ! [I2: set_a] :
( ( member_set_a @ I2 @ I )
=> ( ( F2 @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe4967106450811773644_set_a @ m @ F2 @ I )
= ( indepe4967106450811773644_set_a @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_8_indep__sets__cong,axiom,
! [I: set_c,J: set_c,F2: c > set_set_a,G: c > set_set_a] :
( ( I = J )
=> ( ! [I2: c] :
( ( member_c @ I2 @ I )
=> ( ( F2 @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe8927441866673418606ts_a_c @ m @ F2 @ I )
= ( indepe8927441866673418606ts_a_c @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_9_indep__sets__cong,axiom,
! [I: set_nat,J: set_nat,F2: nat > set_set_a,G: nat > set_set_a] :
( ( I = J )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ( F2 @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe6267730027088848354_a_nat @ m @ F2 @ I )
= ( indepe6267730027088848354_a_nat @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_10_indep__sets__cong,axiom,
! [I: set_o,J: set_o,F2: $o > set_set_a,G: $o > set_set_a] :
( ( I = J )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I )
=> ( ( F2 @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe7780107833195774214ts_a_o @ m @ F2 @ I )
= ( indepe7780107833195774214ts_a_o @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_11_finite__measure__axioms,axiom,
measur930452917991658466sure_a @ m ).
% finite_measure_axioms
thf(fact_12_sigma__finite__measure__axioms,axiom,
measur4308613598931908895sure_a @ m ).
% sigma_finite_measure_axioms
thf(fact_13_indep__F,axiom,
indepe8927441866673418606ts_a_c @ m @ f @ i ).
% indep_F
thf(fact_14_finite__measure__distr,axiom,
! [F: a > b,M: sigma_measure_b] :
( ( member_a_b @ F @ ( sigma_measurable_a_b @ m @ M ) )
=> ( measur930452917991658467sure_b @ ( measure_distr_a_b @ m @ M @ F ) ) ) ).
% finite_measure_distr
thf(fact_15_finite__measure__distr,axiom,
! [F: a > a,M: sigma_measure_a] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ m @ M ) )
=> ( measur930452917991658466sure_a @ ( measure_distr_a_a @ m @ M @ F ) ) ) ).
% finite_measure_distr
thf(fact_16_prob__space_Oprob__space__distr,axiom,
! [M2: sigma_measure_b,F: b > b,M: sigma_measure_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M2 @ M ) )
=> ( probab7247484486040049090pace_b @ ( measure_distr_b_b @ M2 @ M @ F ) ) ) ) ).
% prob_space.prob_space_distr
thf(fact_17_prob__space_Oprob__space__distr,axiom,
! [M2: sigma_measure_b,F: b > a,M: sigma_measure_a] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ M ) )
=> ( probab7247484486040049089pace_a @ ( measure_distr_b_a @ M2 @ M @ F ) ) ) ) ).
% prob_space.prob_space_distr
thf(fact_18_prob__space_Oprob__space__distr,axiom,
! [M2: sigma_measure_a,F: a > b,M: sigma_measure_b] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ M ) )
=> ( probab7247484486040049090pace_b @ ( measure_distr_a_b @ M2 @ M @ F ) ) ) ) ).
% prob_space.prob_space_distr
thf(fact_19_prob__space_Oprob__space__distr,axiom,
! [M2: sigma_measure_a,F: a > a,M: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ M ) )
=> ( probab7247484486040049089pace_a @ ( measure_distr_a_a @ M2 @ M @ F ) ) ) ) ).
% prob_space.prob_space_distr
thf(fact_20_prob__space__distrD,axiom,
! [F: b > b,M2: sigma_measure_b,N: sigma_measure_b] :
( ( member_b_b @ F @ ( sigma_measurable_b_b @ M2 @ N ) )
=> ( ( probab7247484486040049090pace_b @ ( measure_distr_b_b @ M2 @ N @ F ) )
=> ( probab7247484486040049090pace_b @ M2 ) ) ) ).
% prob_space_distrD
thf(fact_21_prob__space__distrD,axiom,
! [F: a > b,M2: sigma_measure_a,N: sigma_measure_b] :
( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ N ) )
=> ( ( probab7247484486040049090pace_b @ ( measure_distr_a_b @ M2 @ N @ F ) )
=> ( probab7247484486040049089pace_a @ M2 ) ) ) ).
% prob_space_distrD
thf(fact_22_prob__space__distrD,axiom,
! [F: b > a,M2: sigma_measure_b,N: sigma_measure_a] :
( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ N ) )
=> ( ( probab7247484486040049089pace_a @ ( measure_distr_b_a @ M2 @ N @ F ) )
=> ( probab7247484486040049090pace_b @ M2 ) ) ) ).
% prob_space_distrD
thf(fact_23_prob__space__distrD,axiom,
! [F: a > a,M2: sigma_measure_a,N: sigma_measure_a] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N ) )
=> ( ( probab7247484486040049089pace_a @ ( measure_distr_a_a @ M2 @ N @ F ) )
=> ( probab7247484486040049089pace_a @ M2 ) ) ) ).
% prob_space_distrD
thf(fact_24_indep__var__rv1,axiom,
! [S: sigma_measure_b,X: a > b,T: sigma_measure_b,Y: a > b] :
( ( indepe2440653194691626189ar_a_b @ m @ S @ X @ T @ Y )
=> ( member_a_b @ X @ ( sigma_measurable_a_b @ m @ S ) ) ) ).
% indep_var_rv1
thf(fact_25_indep__var__rv1,axiom,
! [S: sigma_measure_a,X: a > a,T: sigma_measure_a,Y: a > a] :
( ( indepe2440653194691626188ar_a_a @ m @ S @ X @ T @ Y )
=> ( member_a_a @ X @ ( sigma_measurable_a_a @ m @ S ) ) ) ).
% indep_var_rv1
thf(fact_26_indep__var__rv2,axiom,
! [S: sigma_measure_b,X: a > b,T: sigma_measure_b,Y: a > b] :
( ( indepe2440653194691626189ar_a_b @ m @ S @ X @ T @ Y )
=> ( member_a_b @ Y @ ( sigma_measurable_a_b @ m @ T ) ) ) ).
% indep_var_rv2
thf(fact_27_indep__var__rv2,axiom,
! [S: sigma_measure_a,X: a > a,T: sigma_measure_a,Y: a > a] :
( ( indepe2440653194691626188ar_a_a @ m @ S @ X @ T @ Y )
=> ( member_a_a @ Y @ ( sigma_measurable_a_a @ m @ T ) ) ) ).
% indep_var_rv2
thf(fact_28_prob__space__imp__sigma__finite,axiom,
! [M2: sigma_measure_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( measur4308613598931908896sure_b @ M2 ) ) ).
% prob_space_imp_sigma_finite
thf(fact_29_prob__space__imp__sigma__finite,axiom,
! [M2: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( measur4308613598931908895sure_a @ M2 ) ) ).
% prob_space_imp_sigma_finite
thf(fact_30_prob__space_Ofinite__measure,axiom,
! [M2: sigma_measure_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( measur930452917991658467sure_b @ M2 ) ) ).
% prob_space.finite_measure
thf(fact_31_prob__space_Ofinite__measure,axiom,
! [M2: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( measur930452917991658466sure_a @ M2 ) ) ).
% prob_space.finite_measure
thf(fact_32_measurable__distr__eq2,axiom,
! [Mg: sigma_measure_a,Mg2: sigma_measure_a,Ng: sigma_measure_b,G2: a > b] :
( ( sigma_measurable_a_b @ Mg @ ( measure_distr_a_b @ Mg2 @ Ng @ G2 ) )
= ( sigma_measurable_a_b @ Mg @ Ng ) ) ).
% measurable_distr_eq2
thf(fact_33_measurable__distr__eq2,axiom,
! [Mg: sigma_measure_a,Mg2: sigma_measure_a,Ng: sigma_measure_a,G2: a > a] :
( ( sigma_measurable_a_a @ Mg @ ( measure_distr_a_a @ Mg2 @ Ng @ G2 ) )
= ( sigma_measurable_a_a @ Mg @ Ng ) ) ).
% measurable_distr_eq2
thf(fact_34_measurable__distr__eq1,axiom,
! [Mf: sigma_measure_a,Nf: sigma_measure_a,F: a > a,Mf2: sigma_measure_b] :
( ( sigma_measurable_a_b @ ( measure_distr_a_a @ Mf @ Nf @ F ) @ Mf2 )
= ( sigma_measurable_a_b @ Nf @ Mf2 ) ) ).
% measurable_distr_eq1
thf(fact_35_measurable__distr__eq1,axiom,
! [Mf: sigma_measure_a,Nf: sigma_measure_a,F: a > a,Mf2: sigma_measure_a] :
( ( sigma_measurable_a_a @ ( measure_distr_a_a @ Mf @ Nf @ F ) @ Mf2 )
= ( sigma_measurable_a_a @ Nf @ Mf2 ) ) ).
% measurable_distr_eq1
thf(fact_36_prob__space_Oindep__var__rv1,axiom,
! [M2: sigma_measure_a,S: sigma_measure_b,X: a > b,T: sigma_measure_b,Y: a > b] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe2440653194691626189ar_a_b @ M2 @ S @ X @ T @ Y )
=> ( member_a_b @ X @ ( sigma_measurable_a_b @ M2 @ S ) ) ) ) ).
% prob_space.indep_var_rv1
thf(fact_37_prob__space_Oindep__var__rv1,axiom,
! [M2: sigma_measure_a,S: sigma_measure_a,X: a > a,T: sigma_measure_a,Y: a > a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe2440653194691626188ar_a_a @ M2 @ S @ X @ T @ Y )
=> ( member_a_a @ X @ ( sigma_measurable_a_a @ M2 @ S ) ) ) ) ).
% prob_space.indep_var_rv1
thf(fact_38_prob__space_Oindep__var__rv2,axiom,
! [M2: sigma_measure_a,S: sigma_measure_b,X: a > b,T: sigma_measure_b,Y: a > b] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe2440653194691626189ar_a_b @ M2 @ S @ X @ T @ Y )
=> ( member_a_b @ Y @ ( sigma_measurable_a_b @ M2 @ T ) ) ) ) ).
% prob_space.indep_var_rv2
thf(fact_39_prob__space_Oindep__var__rv2,axiom,
! [M2: sigma_measure_a,S: sigma_measure_a,X: a > a,T: sigma_measure_a,Y: a > a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe2440653194691626188ar_a_a @ M2 @ S @ X @ T @ Y )
=> ( member_a_a @ Y @ ( sigma_measurable_a_a @ M2 @ T ) ) ) ) ).
% prob_space.indep_var_rv2
thf(fact_40_finite__measure_Ofinite__measure__distr,axiom,
! [M2: sigma_measure_a,F: a > a,M: sigma_measure_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ M ) )
=> ( measur930452917991658466sure_a @ ( measure_distr_a_a @ M2 @ M @ F ) ) ) ) ).
% finite_measure.finite_measure_distr
thf(fact_41_finite__measure_Ofinite__measure__distr,axiom,
! [M2: sigma_measure_a,F: a > b,M: sigma_measure_b] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ M ) )
=> ( measur930452917991658467sure_b @ ( measure_distr_a_b @ M2 @ M @ F ) ) ) ) ).
% finite_measure.finite_measure_distr
thf(fact_42_finite__measure_Ofinite__measure__distr,axiom,
! [M2: sigma_measure_b,F: b > a,M: sigma_measure_a] :
( ( measur930452917991658467sure_b @ M2 )
=> ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ M ) )
=> ( measur930452917991658466sure_a @ ( measure_distr_b_a @ M2 @ M @ F ) ) ) ) ).
% finite_measure.finite_measure_distr
thf(fact_43_finite__measure_Ofinite__measure__distr,axiom,
! [M2: sigma_measure_b,F: b > b,M: sigma_measure_b] :
( ( measur930452917991658467sure_b @ M2 )
=> ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M2 @ M ) )
=> ( measur930452917991658467sure_b @ ( measure_distr_b_b @ M2 @ M @ F ) ) ) ) ).
% finite_measure.finite_measure_distr
thf(fact_44_sigma__finite__measure__distr,axiom,
! [M2: sigma_measure_a,N: sigma_measure_b,F: a > b] :
( ( measur4308613598931908896sure_b @ ( measure_distr_a_b @ M2 @ N @ F ) )
=> ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ N ) )
=> ( measur4308613598931908895sure_a @ M2 ) ) ) ).
% sigma_finite_measure_distr
thf(fact_45_sigma__finite__measure__distr,axiom,
! [M2: sigma_measure_a,N: sigma_measure_a,F: a > a] :
( ( measur4308613598931908895sure_a @ ( measure_distr_a_a @ M2 @ N @ F ) )
=> ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N ) )
=> ( measur4308613598931908895sure_a @ M2 ) ) ) ).
% sigma_finite_measure_distr
thf(fact_46_prob__space__completion,axiom,
probab7247484486040049089pace_a @ ( comple3428971583294703880tion_a @ m ) ).
% prob_space_completion
thf(fact_47_indep__var__compose,axiom,
! [M1: sigma_measure_a,X1: a > a,M22: sigma_measure_a,X2: a > a,Y1: a > b,N1: sigma_measure_b,Y2: a > b,N2: sigma_measure_b] :
( ( indepe2440653194691626188ar_a_a @ m @ M1 @ X1 @ M22 @ X2 )
=> ( ( member_a_b @ Y1 @ ( sigma_measurable_a_b @ M1 @ N1 ) )
=> ( ( member_a_b @ Y2 @ ( sigma_measurable_a_b @ M22 @ N2 ) )
=> ( indepe2440653194691626189ar_a_b @ m @ N1 @ ( comp_a_b_a @ Y1 @ X1 ) @ N2 @ ( comp_a_b_a @ Y2 @ X2 ) ) ) ) ) ).
% indep_var_compose
thf(fact_48_indep__var__compose,axiom,
! [M1: sigma_measure_a,X1: a > a,M22: sigma_measure_a,X2: a > a,Y1: a > a,N1: sigma_measure_a,Y2: a > a,N2: sigma_measure_a] :
( ( indepe2440653194691626188ar_a_a @ m @ M1 @ X1 @ M22 @ X2 )
=> ( ( member_a_a @ Y1 @ ( sigma_measurable_a_a @ M1 @ N1 ) )
=> ( ( member_a_a @ Y2 @ ( sigma_measurable_a_a @ M22 @ N2 ) )
=> ( indepe2440653194691626188ar_a_a @ m @ N1 @ ( comp_a_a_a @ Y1 @ X1 ) @ N2 @ ( comp_a_a_a @ Y2 @ X2 ) ) ) ) ) ).
% indep_var_compose
thf(fact_49_indep__sets__mono__sets,axiom,
! [F2: ( a > a ) > set_set_a,I: set_a_a,G: ( a > a ) > set_set_a] :
( ( indepe1219004911012930679_a_a_a @ m @ F2 @ I )
=> ( ! [I2: a > a] :
( ( member_a_a @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe1219004911012930679_a_a_a @ m @ G @ I ) ) ) ).
% indep_sets_mono_sets
thf(fact_50_indep__sets__mono__sets,axiom,
! [F2: c > set_set_a,I: set_c,G: c > set_set_a] :
( ( indepe8927441866673418606ts_a_c @ m @ F2 @ I )
=> ( ! [I2: c] :
( ( member_c @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe8927441866673418606ts_a_c @ m @ G @ I ) ) ) ).
% indep_sets_mono_sets
thf(fact_51_indep__sets__mono__sets,axiom,
! [F2: nat > set_set_a,I: set_nat,G: nat > set_set_a] :
( ( indepe6267730027088848354_a_nat @ m @ F2 @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe6267730027088848354_a_nat @ m @ G @ I ) ) ) ).
% indep_sets_mono_sets
thf(fact_52_indep__sets__mono__sets,axiom,
! [F2: $o > set_set_a,I: set_o,G: $o > set_set_a] :
( ( indepe7780107833195774214ts_a_o @ m @ F2 @ I )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe7780107833195774214ts_a_o @ m @ G @ I ) ) ) ).
% indep_sets_mono_sets
thf(fact_53_indep__sets__mono__sets,axiom,
! [F2: set_a > set_set_a,I: set_set_a,G: set_a > set_set_a] :
( ( indepe4967106450811773644_set_a @ m @ F2 @ I )
=> ( ! [I2: set_a] :
( ( member_set_a @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe4967106450811773644_set_a @ m @ G @ I ) ) ) ).
% indep_sets_mono_sets
thf(fact_54_indep__sets__mono__sets,axiom,
! [F2: ( a > b ) > set_set_a,I: set_a_b,G: ( a > b ) > set_set_a] :
( ( indepe1219004915316159480_a_a_b @ m @ F2 @ I )
=> ( ! [I2: a > b] :
( ( member_a_b @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe1219004915316159480_a_a_b @ m @ G @ I ) ) ) ).
% indep_sets_mono_sets
thf(fact_55_indep__sets__mono__index,axiom,
! [J: set_set_b,I: set_set_b,F2: set_b > set_set_a] :
( ( ord_le3795704787696855135_set_b @ J @ I )
=> ( ( indepe4967106455115002445_set_b @ m @ F2 @ I )
=> ( indepe4967106455115002445_set_b @ m @ F2 @ J ) ) ) ).
% indep_sets_mono_index
thf(fact_56_indep__sets__mono__index,axiom,
! [J: set_a,I: set_a,F2: a > set_set_a] :
( ( ord_less_eq_set_a @ J @ I )
=> ( ( indepe8927441866673418604ts_a_a @ m @ F2 @ I )
=> ( indepe8927441866673418604ts_a_a @ m @ F2 @ J ) ) ) ).
% indep_sets_mono_index
thf(fact_57_indep__sets__mono__index,axiom,
! [J: set_c,I: set_c,F2: c > set_set_a] :
( ( ord_less_eq_set_c @ J @ I )
=> ( ( indepe8927441866673418606ts_a_c @ m @ F2 @ I )
=> ( indepe8927441866673418606ts_a_c @ m @ F2 @ J ) ) ) ).
% indep_sets_mono_index
thf(fact_58_indep__sets__mono__index,axiom,
! [J: set_nat,I: set_nat,F2: nat > set_set_a] :
( ( ord_less_eq_set_nat @ J @ I )
=> ( ( indepe6267730027088848354_a_nat @ m @ F2 @ I )
=> ( indepe6267730027088848354_a_nat @ m @ F2 @ J ) ) ) ).
% indep_sets_mono_index
thf(fact_59_indep__sets__mono__index,axiom,
! [J: set_o,I: set_o,F2: $o > set_set_a] :
( ( ord_less_eq_set_o @ J @ I )
=> ( ( indepe7780107833195774214ts_a_o @ m @ F2 @ I )
=> ( indepe7780107833195774214ts_a_o @ m @ F2 @ J ) ) ) ).
% indep_sets_mono_index
thf(fact_60_indep__sets__mono__index,axiom,
! [J: set_set_a,I: set_set_a,F2: set_a > set_set_a] :
( ( ord_le3724670747650509150_set_a @ J @ I )
=> ( ( indepe4967106450811773644_set_a @ m @ F2 @ I )
=> ( indepe4967106450811773644_set_a @ m @ F2 @ J ) ) ) ).
% indep_sets_mono_index
thf(fact_61_indep__sets__mono__index,axiom,
! [J: set_a_b,I: set_a_b,F2: ( a > b ) > set_set_a] :
( ( ord_less_eq_set_a_b @ J @ I )
=> ( ( indepe1219004915316159480_a_a_b @ m @ F2 @ I )
=> ( indepe1219004915316159480_a_a_b @ m @ F2 @ J ) ) ) ).
% indep_sets_mono_index
thf(fact_62_subprob__space_Oaxioms_I1_J,axiom,
! [M2: sigma_measure_a] :
( ( giry_subprob_space_a @ M2 )
=> ( measur930452917991658466sure_a @ M2 ) ) ).
% subprob_space.axioms(1)
thf(fact_63_subprob__space_Oaxioms_I1_J,axiom,
! [M2: sigma_measure_b] :
( ( giry_subprob_space_b @ M2 )
=> ( measur930452917991658467sure_b @ M2 ) ) ).
% subprob_space.axioms(1)
thf(fact_64_indep__sets__mono,axiom,
! [F2: ( a > a ) > set_set_a,I: set_a_a,J: set_a_a,G: ( a > a ) > set_set_a] :
( ( indepe1219004911012930679_a_a_a @ m @ F2 @ I )
=> ( ( ord_less_eq_set_a_a @ J @ I )
=> ( ! [I2: a > a] :
( ( member_a_a @ I2 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe1219004911012930679_a_a_a @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_65_indep__sets__mono,axiom,
! [F2: set_b > set_set_a,I: set_set_b,J: set_set_b,G: set_b > set_set_a] :
( ( indepe4967106455115002445_set_b @ m @ F2 @ I )
=> ( ( ord_le3795704787696855135_set_b @ J @ I )
=> ( ! [I2: set_b] :
( ( member_set_b @ I2 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe4967106455115002445_set_b @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_66_indep__sets__mono,axiom,
! [F2: a > set_set_a,I: set_a,J: set_a,G: a > set_set_a] :
( ( indepe8927441866673418604ts_a_a @ m @ F2 @ I )
=> ( ( ord_less_eq_set_a @ J @ I )
=> ( ! [I2: a] :
( ( member_a @ I2 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe8927441866673418604ts_a_a @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_67_indep__sets__mono,axiom,
! [F2: c > set_set_a,I: set_c,J: set_c,G: c > set_set_a] :
( ( indepe8927441866673418606ts_a_c @ m @ F2 @ I )
=> ( ( ord_less_eq_set_c @ J @ I )
=> ( ! [I2: c] :
( ( member_c @ I2 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe8927441866673418606ts_a_c @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_68_indep__sets__mono,axiom,
! [F2: nat > set_set_a,I: set_nat,J: set_nat,G: nat > set_set_a] :
( ( indepe6267730027088848354_a_nat @ m @ F2 @ I )
=> ( ( ord_less_eq_set_nat @ J @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe6267730027088848354_a_nat @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_69_indep__sets__mono,axiom,
! [F2: $o > set_set_a,I: set_o,J: set_o,G: $o > set_set_a] :
( ( indepe7780107833195774214ts_a_o @ m @ F2 @ I )
=> ( ( ord_less_eq_set_o @ J @ I )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe7780107833195774214ts_a_o @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_70_indep__sets__mono,axiom,
! [F2: set_a > set_set_a,I: set_set_a,J: set_set_a,G: set_a > set_set_a] :
( ( indepe4967106450811773644_set_a @ m @ F2 @ I )
=> ( ( ord_le3724670747650509150_set_a @ J @ I )
=> ( ! [I2: set_a] :
( ( member_set_a @ I2 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe4967106450811773644_set_a @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_71_indep__sets__mono,axiom,
! [F2: ( a > b ) > set_set_a,I: set_a_b,J: set_a_b,G: ( a > b ) > set_set_a] :
( ( indepe1219004915316159480_a_a_b @ m @ F2 @ I )
=> ( ( ord_less_eq_set_a_b @ J @ I )
=> ( ! [I2: a > b] :
( ( member_a_b @ I2 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe1219004915316159480_a_a_b @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_72_prob__space_Oindep__sets__mono,axiom,
! [M2: sigma_measure_b,F2: c > set_set_b,I: set_c,J: set_c,G: c > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe6139986284700742573ts_b_c @ M2 @ F2 @ I )
=> ( ( ord_less_eq_set_c @ J @ I )
=> ( ! [I2: c] :
( ( member_c @ I2 @ J )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe6139986284700742573ts_b_c @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_73_prob__space_Oindep__sets__mono,axiom,
! [M2: sigma_measure_b,F2: $o > set_set_b,I: set_o,J: set_o,G: $o > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe4880885433731379909ts_b_o @ M2 @ F2 @ I )
=> ( ( ord_less_eq_set_o @ J @ I )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ J )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe4880885433731379909ts_b_o @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_74_prob__space_Oindep__sets__mono,axiom,
! [M2: sigma_measure_b,F2: nat > set_set_b,I: set_nat,J: set_nat,G: nat > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe7503174356045242851_b_nat @ M2 @ F2 @ I )
=> ( ( ord_less_eq_set_nat @ J @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ J )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe7503174356045242851_b_nat @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_75_prob__space_Oindep__sets__mono,axiom,
! [M2: sigma_measure_b,F2: a > set_set_b,I: set_a,J: set_a,G: a > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe6139986284700742571ts_b_a @ M2 @ F2 @ I )
=> ( ( ord_less_eq_set_a @ J @ I )
=> ( ! [I2: a] :
( ( member_a @ I2 @ J )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe6139986284700742571ts_b_a @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_76_prob__space_Oindep__sets__mono,axiom,
! [M2: sigma_measure_a,F2: a > set_set_a,I: set_a,J: set_a,G: a > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe8927441866673418604ts_a_a @ M2 @ F2 @ I )
=> ( ( ord_less_eq_set_a @ J @ I )
=> ( ! [I2: a] :
( ( member_a @ I2 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe8927441866673418604ts_a_a @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_77_prob__space_Oindep__sets__mono,axiom,
! [M2: sigma_measure_a,F2: c > set_set_a,I: set_c,J: set_c,G: c > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe8927441866673418606ts_a_c @ M2 @ F2 @ I )
=> ( ( ord_less_eq_set_c @ J @ I )
=> ( ! [I2: c] :
( ( member_c @ I2 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe8927441866673418606ts_a_c @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_78_prob__space_Oindep__sets__mono,axiom,
! [M2: sigma_measure_a,F2: nat > set_set_a,I: set_nat,J: set_nat,G: nat > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe6267730027088848354_a_nat @ M2 @ F2 @ I )
=> ( ( ord_less_eq_set_nat @ J @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe6267730027088848354_a_nat @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_79_prob__space_Oindep__sets__mono,axiom,
! [M2: sigma_measure_a,F2: $o > set_set_a,I: set_o,J: set_o,G: $o > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe7780107833195774214ts_a_o @ M2 @ F2 @ I )
=> ( ( ord_less_eq_set_o @ J @ I )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe7780107833195774214ts_a_o @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_80_prob__space_Oindep__sets__mono,axiom,
! [M2: sigma_measure_b,F2: set_b > set_set_b,I: set_set_b,J: set_set_b,G: set_b > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe6311571491924490380_set_b @ M2 @ F2 @ I )
=> ( ( ord_le3795704787696855135_set_b @ J @ I )
=> ( ! [I2: set_b] :
( ( member_set_b @ I2 @ J )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe6311571491924490380_set_b @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_81_prob__space_Oindep__sets__mono,axiom,
! [M2: sigma_measure_b,F2: set_a > set_set_b,I: set_set_a,J: set_set_a,G: set_a > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe6311571487621261579_set_a @ M2 @ F2 @ I )
=> ( ( ord_le3724670747650509150_set_a @ J @ I )
=> ( ! [I2: set_a] :
( ( member_set_a @ I2 @ J )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe6311571487621261579_set_a @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_82_prob__space_Oprob__space__completion,axiom,
! [M2: sigma_measure_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( probab7247484486040049090pace_b @ ( comple3428971583294703881tion_b @ M2 ) ) ) ).
% prob_space.prob_space_completion
thf(fact_83_prob__space_Oprob__space__completion,axiom,
! [M2: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( probab7247484486040049089pace_a @ ( comple3428971583294703880tion_a @ M2 ) ) ) ).
% prob_space.prob_space_completion
thf(fact_84_distr__distr,axiom,
! [G2: b > b,N: sigma_measure_b,L: sigma_measure_b,F: a > b,M2: sigma_measure_a] :
( ( member_b_b @ G2 @ ( sigma_measurable_b_b @ N @ L ) )
=> ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ N ) )
=> ( ( measure_distr_b_b @ ( measure_distr_a_b @ M2 @ N @ F ) @ L @ G2 )
= ( measure_distr_a_b @ M2 @ L @ ( comp_b_b_a @ G2 @ F ) ) ) ) ) ).
% distr_distr
thf(fact_85_distr__distr,axiom,
! [G2: b > a,N: sigma_measure_b,L: sigma_measure_a,F: a > b,M2: sigma_measure_a] :
( ( member_b_a @ G2 @ ( sigma_measurable_b_a @ N @ L ) )
=> ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ N ) )
=> ( ( measure_distr_b_a @ ( measure_distr_a_b @ M2 @ N @ F ) @ L @ G2 )
= ( measure_distr_a_a @ M2 @ L @ ( comp_b_a_a @ G2 @ F ) ) ) ) ) ).
% distr_distr
thf(fact_86_distr__distr,axiom,
! [G2: a > b,N: sigma_measure_a,L: sigma_measure_b,F: a > a,M2: sigma_measure_a] :
( ( member_a_b @ G2 @ ( sigma_measurable_a_b @ N @ L ) )
=> ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N ) )
=> ( ( measure_distr_a_b @ ( measure_distr_a_a @ M2 @ N @ F ) @ L @ G2 )
= ( measure_distr_a_b @ M2 @ L @ ( comp_a_b_a @ G2 @ F ) ) ) ) ) ).
% distr_distr
thf(fact_87_distr__distr,axiom,
! [G2: a > a,N: sigma_measure_a,L: sigma_measure_a,F: a > a,M2: sigma_measure_a] :
( ( member_a_a @ G2 @ ( sigma_measurable_a_a @ N @ L ) )
=> ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N ) )
=> ( ( measure_distr_a_a @ ( measure_distr_a_a @ M2 @ N @ F ) @ L @ G2 )
= ( measure_distr_a_a @ M2 @ L @ ( comp_a_a_a @ G2 @ F ) ) ) ) ) ).
% distr_distr
thf(fact_88_prob__space_Oindep__sets__mono__index,axiom,
! [M2: sigma_measure_b,J: set_set_b,I: set_set_b,F2: set_b > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( ord_le3795704787696855135_set_b @ J @ I )
=> ( ( indepe6311571491924490380_set_b @ M2 @ F2 @ I )
=> ( indepe6311571491924490380_set_b @ M2 @ F2 @ J ) ) ) ) ).
% prob_space.indep_sets_mono_index
thf(fact_89_prob__space_Oindep__sets__mono__index,axiom,
! [M2: sigma_measure_b,J: set_set_a,I: set_set_a,F2: set_a > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( ord_le3724670747650509150_set_a @ J @ I )
=> ( ( indepe6311571487621261579_set_a @ M2 @ F2 @ I )
=> ( indepe6311571487621261579_set_a @ M2 @ F2 @ J ) ) ) ) ).
% prob_space.indep_sets_mono_index
thf(fact_90_prob__space_Oindep__sets__mono__index,axiom,
! [M2: sigma_measure_b,J: set_a,I: set_a,F2: a > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( ord_less_eq_set_a @ J @ I )
=> ( ( indepe6139986284700742571ts_b_a @ M2 @ F2 @ I )
=> ( indepe6139986284700742571ts_b_a @ M2 @ F2 @ J ) ) ) ) ).
% prob_space.indep_sets_mono_index
thf(fact_91_prob__space_Oindep__sets__mono__index,axiom,
! [M2: sigma_measure_a,J: set_set_b,I: set_set_b,F2: set_b > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( ord_le3795704787696855135_set_b @ J @ I )
=> ( ( indepe4967106455115002445_set_b @ M2 @ F2 @ I )
=> ( indepe4967106455115002445_set_b @ M2 @ F2 @ J ) ) ) ) ).
% prob_space.indep_sets_mono_index
thf(fact_92_prob__space_Oindep__sets__mono__index,axiom,
! [M2: sigma_measure_a,J: set_a,I: set_a,F2: a > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( ord_less_eq_set_a @ J @ I )
=> ( ( indepe8927441866673418604ts_a_a @ M2 @ F2 @ I )
=> ( indepe8927441866673418604ts_a_a @ M2 @ F2 @ J ) ) ) ) ).
% prob_space.indep_sets_mono_index
thf(fact_93_prob__space_Oindep__sets__mono__index,axiom,
! [M2: sigma_measure_a,J: set_c,I: set_c,F2: c > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( ord_less_eq_set_c @ J @ I )
=> ( ( indepe8927441866673418606ts_a_c @ M2 @ F2 @ I )
=> ( indepe8927441866673418606ts_a_c @ M2 @ F2 @ J ) ) ) ) ).
% prob_space.indep_sets_mono_index
thf(fact_94_prob__space_Oindep__sets__mono__index,axiom,
! [M2: sigma_measure_a,J: set_nat,I: set_nat,F2: nat > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( ord_less_eq_set_nat @ J @ I )
=> ( ( indepe6267730027088848354_a_nat @ M2 @ F2 @ I )
=> ( indepe6267730027088848354_a_nat @ M2 @ F2 @ J ) ) ) ) ).
% prob_space.indep_sets_mono_index
thf(fact_95_prob__space_Oindep__sets__mono__index,axiom,
! [M2: sigma_measure_a,J: set_o,I: set_o,F2: $o > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( ord_less_eq_set_o @ J @ I )
=> ( ( indepe7780107833195774214ts_a_o @ M2 @ F2 @ I )
=> ( indepe7780107833195774214ts_a_o @ M2 @ F2 @ J ) ) ) ) ).
% prob_space.indep_sets_mono_index
thf(fact_96_prob__space_Oindep__sets__mono__index,axiom,
! [M2: sigma_measure_a,J: set_set_a,I: set_set_a,F2: set_a > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( ord_le3724670747650509150_set_a @ J @ I )
=> ( ( indepe4967106450811773644_set_a @ M2 @ F2 @ I )
=> ( indepe4967106450811773644_set_a @ M2 @ F2 @ J ) ) ) ) ).
% prob_space.indep_sets_mono_index
thf(fact_97_prob__space_Oindep__sets__mono__index,axiom,
! [M2: sigma_measure_a,J: set_a_b,I: set_a_b,F2: ( a > b ) > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( ord_less_eq_set_a_b @ J @ I )
=> ( ( indepe1219004915316159480_a_a_b @ M2 @ F2 @ I )
=> ( indepe1219004915316159480_a_a_b @ M2 @ F2 @ J ) ) ) ) ).
% prob_space.indep_sets_mono_index
thf(fact_98_prob__space_Oindep__sets__mono__sets,axiom,
! [M2: sigma_measure_b,F2: c > set_set_b,I: set_c,G: c > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe6139986284700742573ts_b_c @ M2 @ F2 @ I )
=> ( ! [I2: c] :
( ( member_c @ I2 @ I )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe6139986284700742573ts_b_c @ M2 @ G @ I ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_99_prob__space_Oindep__sets__mono__sets,axiom,
! [M2: sigma_measure_b,F2: $o > set_set_b,I: set_o,G: $o > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe4880885433731379909ts_b_o @ M2 @ F2 @ I )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe4880885433731379909ts_b_o @ M2 @ G @ I ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_100_prob__space_Oindep__sets__mono__sets,axiom,
! [M2: sigma_measure_b,F2: nat > set_set_b,I: set_nat,G: nat > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe7503174356045242851_b_nat @ M2 @ F2 @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe7503174356045242851_b_nat @ M2 @ G @ I ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_101_prob__space_Oindep__sets__mono__sets,axiom,
! [M2: sigma_measure_a,F2: c > set_set_a,I: set_c,G: c > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe8927441866673418606ts_a_c @ M2 @ F2 @ I )
=> ( ! [I2: c] :
( ( member_c @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe8927441866673418606ts_a_c @ M2 @ G @ I ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_102_prob__space_Oindep__sets__mono__sets,axiom,
! [M2: sigma_measure_a,F2: nat > set_set_a,I: set_nat,G: nat > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe6267730027088848354_a_nat @ M2 @ F2 @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe6267730027088848354_a_nat @ M2 @ G @ I ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_103_prob__space_Oindep__sets__mono__sets,axiom,
! [M2: sigma_measure_a,F2: $o > set_set_a,I: set_o,G: $o > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe7780107833195774214ts_a_o @ M2 @ F2 @ I )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe7780107833195774214ts_a_o @ M2 @ G @ I ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_104_prob__space_Oindep__sets__mono__sets,axiom,
! [M2: sigma_measure_b,F2: set_a > set_set_b,I: set_set_a,G: set_a > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe6311571487621261579_set_a @ M2 @ F2 @ I )
=> ( ! [I2: set_a] :
( ( member_set_a @ I2 @ I )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe6311571487621261579_set_a @ M2 @ G @ I ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_105_prob__space_Oindep__sets__mono__sets,axiom,
! [M2: sigma_measure_a,F2: set_a > set_set_a,I: set_set_a,G: set_a > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe4967106450811773644_set_a @ M2 @ F2 @ I )
=> ( ! [I2: set_a] :
( ( member_set_a @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe4967106450811773644_set_a @ M2 @ G @ I ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_106_prob__space_Oindep__sets__mono__sets,axiom,
! [M2: sigma_measure_b,F2: ( a > b ) > set_set_b,I: set_a_b,G: ( a > b ) > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe7072862685212782073_b_a_b @ M2 @ F2 @ I )
=> ( ! [I2: a > b] :
( ( member_a_b @ I2 @ I )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe7072862685212782073_b_a_b @ M2 @ G @ I ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_107_prob__space_Oindep__sets__mono__sets,axiom,
! [M2: sigma_measure_b,F2: ( a > a ) > set_set_b,I: set_a_a,G: ( a > a ) > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe7072862680909553272_b_a_a @ M2 @ F2 @ I )
=> ( ! [I2: a > a] :
( ( member_a_a @ I2 @ I )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
=> ( indepe7072862680909553272_b_a_a @ M2 @ G @ I ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_108_prob__space_Oindep__var__compose,axiom,
! [M2: sigma_measure_b,M1: sigma_measure_a,X1: b > a,M22: sigma_measure_a,X2: b > a,Y1: a > b,N1: sigma_measure_b,Y2: a > b,N2: sigma_measure_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe8876569649573725963ar_b_a @ M2 @ M1 @ X1 @ M22 @ X2 )
=> ( ( member_a_b @ Y1 @ ( sigma_measurable_a_b @ M1 @ N1 ) )
=> ( ( member_a_b @ Y2 @ ( sigma_measurable_a_b @ M22 @ N2 ) )
=> ( indepe8876569649573725964ar_b_b @ M2 @ N1 @ ( comp_a_b_b @ Y1 @ X1 ) @ N2 @ ( comp_a_b_b @ Y2 @ X2 ) ) ) ) ) ) ).
% prob_space.indep_var_compose
thf(fact_109_prob__space_Oindep__var__compose,axiom,
! [M2: sigma_measure_b,M1: sigma_measure_a,X1: b > a,M22: sigma_measure_a,X2: b > a,Y1: a > a,N1: sigma_measure_a,Y2: a > a,N2: sigma_measure_a] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe8876569649573725963ar_b_a @ M2 @ M1 @ X1 @ M22 @ X2 )
=> ( ( member_a_a @ Y1 @ ( sigma_measurable_a_a @ M1 @ N1 ) )
=> ( ( member_a_a @ Y2 @ ( sigma_measurable_a_a @ M22 @ N2 ) )
=> ( indepe8876569649573725963ar_b_a @ M2 @ N1 @ ( comp_a_a_b @ Y1 @ X1 ) @ N2 @ ( comp_a_a_b @ Y2 @ X2 ) ) ) ) ) ) ).
% prob_space.indep_var_compose
thf(fact_110_prob__space_Oindep__var__compose,axiom,
! [M2: sigma_measure_a,M1: sigma_measure_a,X1: a > a,M22: sigma_measure_a,X2: a > a,Y1: a > b,N1: sigma_measure_b,Y2: a > b,N2: sigma_measure_b] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe2440653194691626188ar_a_a @ M2 @ M1 @ X1 @ M22 @ X2 )
=> ( ( member_a_b @ Y1 @ ( sigma_measurable_a_b @ M1 @ N1 ) )
=> ( ( member_a_b @ Y2 @ ( sigma_measurable_a_b @ M22 @ N2 ) )
=> ( indepe2440653194691626189ar_a_b @ M2 @ N1 @ ( comp_a_b_a @ Y1 @ X1 ) @ N2 @ ( comp_a_b_a @ Y2 @ X2 ) ) ) ) ) ) ).
% prob_space.indep_var_compose
thf(fact_111_prob__space_Oindep__var__compose,axiom,
! [M2: sigma_measure_a,M1: sigma_measure_a,X1: a > a,M22: sigma_measure_a,X2: a > a,Y1: a > a,N1: sigma_measure_a,Y2: a > a,N2: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe2440653194691626188ar_a_a @ M2 @ M1 @ X1 @ M22 @ X2 )
=> ( ( member_a_a @ Y1 @ ( sigma_measurable_a_a @ M1 @ N1 ) )
=> ( ( member_a_a @ Y2 @ ( sigma_measurable_a_a @ M22 @ N2 ) )
=> ( indepe2440653194691626188ar_a_a @ M2 @ N1 @ ( comp_a_a_a @ Y1 @ X1 ) @ N2 @ ( comp_a_a_a @ Y2 @ X2 ) ) ) ) ) ) ).
% prob_space.indep_var_compose
thf(fact_112_prob__space_Oindep__sets_Ocong,axiom,
indepe8927441866673418606ts_a_c = indepe8927441866673418606ts_a_c ).
% prob_space.indep_sets.cong
thf(fact_113_prob__space_Oindep__sets_Ocong,axiom,
indepe6267730027088848354_a_nat = indepe6267730027088848354_a_nat ).
% prob_space.indep_sets.cong
thf(fact_114_prob__space_Oindep__sets_Ocong,axiom,
indepe7780107833195774214ts_a_o = indepe7780107833195774214ts_a_o ).
% prob_space.indep_sets.cong
thf(fact_115_prob__space_Oindep__sets_Ocong,axiom,
indepe4967106450811773644_set_a = indepe4967106450811773644_set_a ).
% prob_space.indep_sets.cong
thf(fact_116_prob__space_Oindep__sets_Ocong,axiom,
indepe1219004915316159480_a_a_b = indepe1219004915316159480_a_a_b ).
% prob_space.indep_sets.cong
thf(fact_117_finite__measure_Oaxioms_I1_J,axiom,
! [M2: sigma_measure_b] :
( ( measur930452917991658467sure_b @ M2 )
=> ( measur4308613598931908896sure_b @ M2 ) ) ).
% finite_measure.axioms(1)
thf(fact_118_finite__measure_Oaxioms_I1_J,axiom,
! [M2: sigma_measure_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( measur4308613598931908895sure_a @ M2 ) ) ).
% finite_measure.axioms(1)
thf(fact_119_prob__space_Oindep__sets__cong,axiom,
! [M2: sigma_measure_b,I: set_c,J: set_c,F2: c > set_set_b,G: c > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( I = J )
=> ( ! [I2: c] :
( ( member_c @ I2 @ I )
=> ( ( F2 @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe6139986284700742573ts_b_c @ M2 @ F2 @ I )
= ( indepe6139986284700742573ts_b_c @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_cong
thf(fact_120_prob__space_Oindep__sets__cong,axiom,
! [M2: sigma_measure_b,I: set_o,J: set_o,F2: $o > set_set_b,G: $o > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( I = J )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I )
=> ( ( F2 @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe4880885433731379909ts_b_o @ M2 @ F2 @ I )
= ( indepe4880885433731379909ts_b_o @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_cong
thf(fact_121_prob__space_Oindep__sets__cong,axiom,
! [M2: sigma_measure_b,I: set_nat,J: set_nat,F2: nat > set_set_b,G: nat > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( I = J )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ( F2 @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe7503174356045242851_b_nat @ M2 @ F2 @ I )
= ( indepe7503174356045242851_b_nat @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_cong
thf(fact_122_prob__space_Oindep__sets__cong,axiom,
! [M2: sigma_measure_a,I: set_c,J: set_c,F2: c > set_set_a,G: c > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( I = J )
=> ( ! [I2: c] :
( ( member_c @ I2 @ I )
=> ( ( F2 @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe8927441866673418606ts_a_c @ M2 @ F2 @ I )
= ( indepe8927441866673418606ts_a_c @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_cong
thf(fact_123_prob__space_Oindep__sets__cong,axiom,
! [M2: sigma_measure_a,I: set_nat,J: set_nat,F2: nat > set_set_a,G: nat > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( I = J )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ( F2 @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe6267730027088848354_a_nat @ M2 @ F2 @ I )
= ( indepe6267730027088848354_a_nat @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_cong
thf(fact_124_prob__space_Oindep__sets__cong,axiom,
! [M2: sigma_measure_a,I: set_o,J: set_o,F2: $o > set_set_a,G: $o > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( I = J )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I )
=> ( ( F2 @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe7780107833195774214ts_a_o @ M2 @ F2 @ I )
= ( indepe7780107833195774214ts_a_o @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_cong
thf(fact_125_prob__space_Oindep__sets__cong,axiom,
! [M2: sigma_measure_b,I: set_set_a,J: set_set_a,F2: set_a > set_set_b,G: set_a > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( I = J )
=> ( ! [I2: set_a] :
( ( member_set_a @ I2 @ I )
=> ( ( F2 @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe6311571487621261579_set_a @ M2 @ F2 @ I )
= ( indepe6311571487621261579_set_a @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_cong
thf(fact_126_prob__space_Oindep__sets__cong,axiom,
! [M2: sigma_measure_a,I: set_set_a,J: set_set_a,F2: set_a > set_set_a,G: set_a > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( I = J )
=> ( ! [I2: set_a] :
( ( member_set_a @ I2 @ I )
=> ( ( F2 @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe4967106450811773644_set_a @ M2 @ F2 @ I )
= ( indepe4967106450811773644_set_a @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_cong
thf(fact_127_prob__space_Oindep__sets__cong,axiom,
! [M2: sigma_measure_b,I: set_a_b,J: set_a_b,F2: ( a > b ) > set_set_b,G: ( a > b ) > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( I = J )
=> ( ! [I2: a > b] :
( ( member_a_b @ I2 @ I )
=> ( ( F2 @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe7072862685212782073_b_a_b @ M2 @ F2 @ I )
= ( indepe7072862685212782073_b_a_b @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_cong
thf(fact_128_prob__space_Oindep__sets__cong,axiom,
! [M2: sigma_measure_b,I: set_a_a,J: set_a_a,F2: ( a > a ) > set_set_b,G: ( a > a ) > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( I = J )
=> ( ! [I2: a > a] :
( ( member_a_a @ I2 @ I )
=> ( ( F2 @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe7072862680909553272_b_a_a @ M2 @ F2 @ I )
= ( indepe7072862680909553272_b_a_a @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_cong
thf(fact_129_prob__space__imp__subprob__space,axiom,
! [M2: sigma_measure_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( giry_subprob_space_b @ M2 ) ) ).
% prob_space_imp_subprob_space
thf(fact_130_prob__space__imp__subprob__space,axiom,
! [M2: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( giry_subprob_space_a @ M2 ) ) ).
% prob_space_imp_subprob_space
thf(fact_131_subprob__space__imp__sigma__finite,axiom,
! [M2: sigma_measure_a] :
( ( giry_subprob_space_a @ M2 )
=> ( measur4308613598931908895sure_a @ M2 ) ) ).
% subprob_space_imp_sigma_finite
thf(fact_132_distr__completion,axiom,
! [X: a > b,M2: sigma_measure_a,N: sigma_measure_b] :
( ( member_a_b @ X @ ( sigma_measurable_a_b @ M2 @ N ) )
=> ( ( measure_distr_a_b @ ( comple3428971583294703880tion_a @ M2 ) @ N @ X )
= ( measure_distr_a_b @ M2 @ N @ X ) ) ) ).
% distr_completion
thf(fact_133_distr__completion,axiom,
! [X: a > a,M2: sigma_measure_a,N: sigma_measure_a] :
( ( member_a_a @ X @ ( sigma_measurable_a_a @ M2 @ N ) )
=> ( ( measure_distr_a_a @ ( comple3428971583294703880tion_a @ M2 ) @ N @ X )
= ( measure_distr_a_a @ M2 @ N @ X ) ) ) ).
% distr_completion
thf(fact_134_mem__Collect__eq,axiom,
! [A: a > b,P: ( a > b ) > $o] :
( ( member_a_b @ A @ ( collect_a_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_135_mem__Collect__eq,axiom,
! [A: c,P: c > $o] :
( ( member_c @ A @ ( collect_c @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_136_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_137_mem__Collect__eq,axiom,
! [A: a > a,P: ( a > a ) > $o] :
( ( member_a_a @ A @ ( collect_a_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_138_mem__Collect__eq,axiom,
! [A: $o,P: $o > $o] :
( ( member_o @ A @ ( collect_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_139_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_140_Collect__mem__eq,axiom,
! [A2: set_a_b] :
( ( collect_a_b
@ ^ [X3: a > b] : ( member_a_b @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_141_Collect__mem__eq,axiom,
! [A2: set_c] :
( ( collect_c
@ ^ [X3: c] : ( member_c @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_142_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_143_Collect__mem__eq,axiom,
! [A2: set_a_a] :
( ( collect_a_a
@ ^ [X3: a > a] : ( member_a_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_144_Collect__mem__eq,axiom,
! [A2: set_o] :
( ( collect_o
@ ^ [X3: $o] : ( member_o @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_145_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_146_subset__antisym,axiom,
! [A2: set_set_b,B: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ B )
=> ( ( ord_le3795704787696855135_set_b @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_147_subset__antisym,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_148_subset__antisym,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_149_subsetI,axiom,
! [A2: set_a_b,B: set_a_b] :
( ! [X4: a > b] :
( ( member_a_b @ X4 @ A2 )
=> ( member_a_b @ X4 @ B ) )
=> ( ord_less_eq_set_a_b @ A2 @ B ) ) ).
% subsetI
thf(fact_150_subsetI,axiom,
! [A2: set_c,B: set_c] :
( ! [X4: c] :
( ( member_c @ X4 @ A2 )
=> ( member_c @ X4 @ B ) )
=> ( ord_less_eq_set_c @ A2 @ B ) ) ).
% subsetI
thf(fact_151_subsetI,axiom,
! [A2: set_a_a,B: set_a_a] :
( ! [X4: a > a] :
( ( member_a_a @ X4 @ A2 )
=> ( member_a_a @ X4 @ B ) )
=> ( ord_less_eq_set_a_a @ A2 @ B ) ) ).
% subsetI
thf(fact_152_subsetI,axiom,
! [A2: set_o,B: set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( member_o @ X4 @ B ) )
=> ( ord_less_eq_set_o @ A2 @ B ) ) ).
% subsetI
thf(fact_153_subsetI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat @ X4 @ B ) )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_154_subsetI,axiom,
! [A2: set_set_b,B: set_set_b] :
( ! [X4: set_b] :
( ( member_set_b @ X4 @ A2 )
=> ( member_set_b @ X4 @ B ) )
=> ( ord_le3795704787696855135_set_b @ A2 @ B ) ) ).
% subsetI
thf(fact_155_subsetI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ( member_set_a @ X4 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_156_subsetI,axiom,
! [A2: set_a,B: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( member_a @ X4 @ B ) )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_157_dual__order_Orefl,axiom,
! [A: set_set_b] : ( ord_le3795704787696855135_set_b @ A @ A ) ).
% dual_order.refl
thf(fact_158_dual__order_Orefl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_159_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_160_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_161_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_162_order__refl,axiom,
! [X5: set_set_b] : ( ord_le3795704787696855135_set_b @ X5 @ X5 ) ).
% order_refl
thf(fact_163_order__refl,axiom,
! [X5: set_set_a] : ( ord_le3724670747650509150_set_a @ X5 @ X5 ) ).
% order_refl
thf(fact_164_order__refl,axiom,
! [X5: set_a] : ( ord_less_eq_set_a @ X5 @ X5 ) ).
% order_refl
thf(fact_165_order__refl,axiom,
! [X5: nat] : ( ord_less_eq_nat @ X5 @ X5 ) ).
% order_refl
thf(fact_166_order__refl,axiom,
! [X5: int] : ( ord_less_eq_int @ X5 @ X5 ) ).
% order_refl
thf(fact_167_increasing__def,axiom,
( measur1776380161843274167a_real
= ( ^ [M3: set_set_a,Mu: set_a > real] :
! [X3: set_a] :
( ( member_set_a @ X3 @ M3 )
=> ! [Y3: set_a] :
( ( member_set_a @ Y3 @ M3 )
=> ( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( Mu @ X3 ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_168_increasing__def,axiom,
( measur8151441426001876059_a_nat
= ( ^ [M3: set_set_a,Mu: set_a > nat] :
! [X3: set_a] :
( ( member_set_a @ X3 @ M3 )
=> ! [Y3: set_a] :
( ( member_set_a @ Y3 @ M3 )
=> ( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( Mu @ X3 ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_169_increasing__def,axiom,
( measur8148950955492825783_a_int
= ( ^ [M3: set_set_a,Mu: set_a > int] :
! [X3: set_a] :
( ( member_set_a @ X3 @ M3 )
=> ! [Y3: set_a] :
( ( member_set_a @ Y3 @ M3 )
=> ( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( Mu @ X3 ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_170_increasing__def,axiom,
( measur6841170515247421756_b_nat
= ( ^ [M3: set_set_set_b,Mu: set_set_b > nat] :
! [X3: set_set_b] :
( ( member_set_set_b @ X3 @ M3 )
=> ! [Y3: set_set_b] :
( ( member_set_set_b @ Y3 @ M3 )
=> ( ( ord_le3795704787696855135_set_b @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( Mu @ X3 ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_171_increasing__def,axiom,
( measur6838680044738371480_b_int
= ( ^ [M3: set_set_set_b,Mu: set_set_b > int] :
! [X3: set_set_b] :
( ( member_set_set_b @ X3 @ M3 )
=> ! [Y3: set_set_b] :
( ( member_set_set_b @ Y3 @ M3 )
=> ( ( ord_le3795704787696855135_set_b @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( Mu @ X3 ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_172_increasing__def,axiom,
( measur1244951900059291067_a_nat
= ( ^ [M3: set_set_set_a,Mu: set_set_a > nat] :
! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ M3 )
=> ! [Y3: set_set_a] :
( ( member_set_set_a @ Y3 @ M3 )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( Mu @ X3 ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_173_increasing__def,axiom,
( measur1242461429550240791_a_int
= ( ^ [M3: set_set_set_a,Mu: set_set_a > int] :
! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ M3 )
=> ! [Y3: set_set_a] :
( ( member_set_set_a @ Y3 @ M3 )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( Mu @ X3 ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_174_increasing__def,axiom,
( measur7842569353079325843_set_a
= ( ^ [M3: set_set_a,Mu: set_a > set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ M3 )
=> ! [Y3: set_a] :
( ( member_set_a @ Y3 @ M3 )
=> ( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( Mu @ X3 ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_175_increasing__def,axiom,
( measur6668990528421247474_set_a
= ( ^ [M3: set_set_set_b,Mu: set_set_b > set_a] :
! [X3: set_set_b] :
( ( member_set_set_b @ X3 @ M3 )
=> ! [Y3: set_set_b] :
( ( member_set_set_b @ Y3 @ M3 )
=> ( ( ord_le3795704787696855135_set_b @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( Mu @ X3 ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_176_increasing__def,axiom,
( measur5181028491126448947_set_a
= ( ^ [M3: set_set_set_a,Mu: set_set_a > set_a] :
! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ M3 )
=> ! [Y3: set_set_a] :
( ( member_set_set_a @ Y3 @ M3 )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( Mu @ X3 ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_177_increasingD,axiom,
! [M2: set_set_a,F: set_a > real,X5: set_a,Y4: set_a] :
( ( measur1776380161843274167a_real @ M2 @ F )
=> ( ( ord_less_eq_set_a @ X5 @ Y4 )
=> ( ( member_set_a @ X5 @ M2 )
=> ( ( member_set_a @ Y4 @ M2 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% increasingD
thf(fact_178_increasingD,axiom,
! [M2: set_set_a,F: set_a > nat,X5: set_a,Y4: set_a] :
( ( measur8151441426001876059_a_nat @ M2 @ F )
=> ( ( ord_less_eq_set_a @ X5 @ Y4 )
=> ( ( member_set_a @ X5 @ M2 )
=> ( ( member_set_a @ Y4 @ M2 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% increasingD
thf(fact_179_increasingD,axiom,
! [M2: set_set_a,F: set_a > int,X5: set_a,Y4: set_a] :
( ( measur8148950955492825783_a_int @ M2 @ F )
=> ( ( ord_less_eq_set_a @ X5 @ Y4 )
=> ( ( member_set_a @ X5 @ M2 )
=> ( ( member_set_a @ Y4 @ M2 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% increasingD
thf(fact_180_increasingD,axiom,
! [M2: set_set_set_b,F: set_set_b > nat,X5: set_set_b,Y4: set_set_b] :
( ( measur6841170515247421756_b_nat @ M2 @ F )
=> ( ( ord_le3795704787696855135_set_b @ X5 @ Y4 )
=> ( ( member_set_set_b @ X5 @ M2 )
=> ( ( member_set_set_b @ Y4 @ M2 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% increasingD
thf(fact_181_increasingD,axiom,
! [M2: set_set_set_b,F: set_set_b > int,X5: set_set_b,Y4: set_set_b] :
( ( measur6838680044738371480_b_int @ M2 @ F )
=> ( ( ord_le3795704787696855135_set_b @ X5 @ Y4 )
=> ( ( member_set_set_b @ X5 @ M2 )
=> ( ( member_set_set_b @ Y4 @ M2 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% increasingD
thf(fact_182_increasingD,axiom,
! [M2: set_set_set_a,F: set_set_a > nat,X5: set_set_a,Y4: set_set_a] :
( ( measur1244951900059291067_a_nat @ M2 @ F )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ Y4 )
=> ( ( member_set_set_a @ X5 @ M2 )
=> ( ( member_set_set_a @ Y4 @ M2 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% increasingD
thf(fact_183_increasingD,axiom,
! [M2: set_set_set_a,F: set_set_a > int,X5: set_set_a,Y4: set_set_a] :
( ( measur1242461429550240791_a_int @ M2 @ F )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ Y4 )
=> ( ( member_set_set_a @ X5 @ M2 )
=> ( ( member_set_set_a @ Y4 @ M2 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% increasingD
thf(fact_184_increasingD,axiom,
! [M2: set_set_a,F: set_a > set_a,X5: set_a,Y4: set_a] :
( ( measur7842569353079325843_set_a @ M2 @ F )
=> ( ( ord_less_eq_set_a @ X5 @ Y4 )
=> ( ( member_set_a @ X5 @ M2 )
=> ( ( member_set_a @ Y4 @ M2 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% increasingD
thf(fact_185_increasingD,axiom,
! [M2: set_set_set_b,F: set_set_b > set_a,X5: set_set_b,Y4: set_set_b] :
( ( measur6668990528421247474_set_a @ M2 @ F )
=> ( ( ord_le3795704787696855135_set_b @ X5 @ Y4 )
=> ( ( member_set_set_b @ X5 @ M2 )
=> ( ( member_set_set_b @ Y4 @ M2 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% increasingD
thf(fact_186_increasingD,axiom,
! [M2: set_set_set_a,F: set_set_a > set_a,X5: set_set_a,Y4: set_set_a] :
( ( measur5181028491126448947_set_a @ M2 @ F )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ Y4 )
=> ( ( member_set_set_a @ X5 @ M2 )
=> ( ( member_set_set_a @ Y4 @ M2 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y4 ) ) ) ) ) ) ).
% increasingD
thf(fact_187_measurable__completion,axiom,
! [F: a > b,M2: sigma_measure_a,N: sigma_measure_b] :
( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ N ) )
=> ( member_a_b @ F @ ( sigma_measurable_a_b @ ( comple3428971583294703880tion_a @ M2 ) @ N ) ) ) ).
% measurable_completion
thf(fact_188_measurable__completion,axiom,
! [F: a > a,M2: sigma_measure_a,N: sigma_measure_a] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N ) )
=> ( member_a_a @ F @ ( sigma_measurable_a_a @ ( comple3428971583294703880tion_a @ M2 ) @ N ) ) ) ).
% measurable_completion
thf(fact_189_measurable__comp,axiom,
! [F: a > b,M2: sigma_measure_a,N: sigma_measure_b,G2: b > b,L: sigma_measure_b] :
( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ N ) )
=> ( ( member_b_b @ G2 @ ( sigma_measurable_b_b @ N @ L ) )
=> ( member_a_b @ ( comp_b_b_a @ G2 @ F ) @ ( sigma_measurable_a_b @ M2 @ L ) ) ) ) ).
% measurable_comp
thf(fact_190_measurable__comp,axiom,
! [F: a > b,M2: sigma_measure_a,N: sigma_measure_b,G2: b > a,L: sigma_measure_a] :
( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ N ) )
=> ( ( member_b_a @ G2 @ ( sigma_measurable_b_a @ N @ L ) )
=> ( member_a_a @ ( comp_b_a_a @ G2 @ F ) @ ( sigma_measurable_a_a @ M2 @ L ) ) ) ) ).
% measurable_comp
thf(fact_191_measurable__comp,axiom,
! [F: a > a,M2: sigma_measure_a,N: sigma_measure_a,G2: a > b,L: sigma_measure_b] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N ) )
=> ( ( member_a_b @ G2 @ ( sigma_measurable_a_b @ N @ L ) )
=> ( member_a_b @ ( comp_a_b_a @ G2 @ F ) @ ( sigma_measurable_a_b @ M2 @ L ) ) ) ) ).
% measurable_comp
thf(fact_192_measurable__comp,axiom,
! [F: a > a,M2: sigma_measure_a,N: sigma_measure_a,G2: a > a,L: sigma_measure_a] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N ) )
=> ( ( member_a_a @ G2 @ ( sigma_measurable_a_a @ N @ L ) )
=> ( member_a_a @ ( comp_a_a_a @ G2 @ F ) @ ( sigma_measurable_a_a @ M2 @ L ) ) ) ) ).
% measurable_comp
thf(fact_193_sets__A,axiom,
! [I3: c] :
( ( member_c @ I3 @ i )
=> ( ord_le3795704787696855135_set_b @ ( a2 @ I3 ) @ ( sigma_sets_b @ n ) ) ) ).
% sets_A
thf(fact_194_sets__completionI__sets,axiom,
! [A2: set_b,M2: sigma_measure_b] :
( ( member_set_b @ A2 @ ( sigma_sets_b @ M2 ) )
=> ( member_set_b @ A2 @ ( sigma_sets_b @ ( comple3428971583294703881tion_b @ M2 ) ) ) ) ).
% sets_completionI_sets
thf(fact_195_sets__completionI__sets,axiom,
! [A2: set_a,M2: sigma_measure_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ M2 ) )
=> ( member_set_a @ A2 @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) ) ) ).
% sets_completionI_sets
thf(fact_196_sets__distr,axiom,
! [M2: sigma_measure_a,N: sigma_measure_b,F: a > b] :
( ( sigma_sets_b @ ( measure_distr_a_b @ M2 @ N @ F ) )
= ( sigma_sets_b @ N ) ) ).
% sets_distr
thf(fact_197_sets__distr,axiom,
! [M2: sigma_measure_a,N: sigma_measure_a,F: a > a] :
( ( sigma_sets_a @ ( measure_distr_a_a @ M2 @ N @ F ) )
= ( sigma_sets_a @ N ) ) ).
% sets_distr
thf(fact_198_sets__eq__iff__bounded,axiom,
! [A2: sigma_measure_b,B: sigma_measure_b,C: sigma_measure_b] :
( ( ord_le254669799889008988sure_b @ A2 @ B )
=> ( ( ord_le254669799889008988sure_b @ B @ C )
=> ( ( ( sigma_sets_b @ A2 )
= ( sigma_sets_b @ C ) )
=> ( ( sigma_sets_b @ B )
= ( sigma_sets_b @ A2 ) ) ) ) ) ).
% sets_eq_iff_bounded
thf(fact_199_sets__eq__iff__bounded,axiom,
! [A2: sigma_measure_a,B: sigma_measure_a,C: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A2 @ B )
=> ( ( ord_le254669795585780187sure_a @ B @ C )
=> ( ( ( sigma_sets_a @ A2 )
= ( sigma_sets_a @ C ) )
=> ( ( sigma_sets_a @ B )
= ( sigma_sets_a @ A2 ) ) ) ) ) ).
% sets_eq_iff_bounded
thf(fact_200_measurable__cong__sets,axiom,
! [M2: sigma_measure_b,M: sigma_measure_b,N: sigma_measure_b,N3: sigma_measure_b] :
( ( ( sigma_sets_b @ M2 )
= ( sigma_sets_b @ M ) )
=> ( ( ( sigma_sets_b @ N )
= ( sigma_sets_b @ N3 ) )
=> ( ( sigma_measurable_b_b @ M2 @ N )
= ( sigma_measurable_b_b @ M @ N3 ) ) ) ) ).
% measurable_cong_sets
thf(fact_201_measurable__cong__sets,axiom,
! [M2: sigma_measure_b,M: sigma_measure_b,N: sigma_measure_a,N3: sigma_measure_a] :
( ( ( sigma_sets_b @ M2 )
= ( sigma_sets_b @ M ) )
=> ( ( ( sigma_sets_a @ N )
= ( sigma_sets_a @ N3 ) )
=> ( ( sigma_measurable_b_a @ M2 @ N )
= ( sigma_measurable_b_a @ M @ N3 ) ) ) ) ).
% measurable_cong_sets
thf(fact_202_measurable__cong__sets,axiom,
! [M2: sigma_measure_a,M: sigma_measure_a,N: sigma_measure_b,N3: sigma_measure_b] :
( ( ( sigma_sets_a @ M2 )
= ( sigma_sets_a @ M ) )
=> ( ( ( sigma_sets_b @ N )
= ( sigma_sets_b @ N3 ) )
=> ( ( sigma_measurable_a_b @ M2 @ N )
= ( sigma_measurable_a_b @ M @ N3 ) ) ) ) ).
% measurable_cong_sets
thf(fact_203_measurable__cong__sets,axiom,
! [M2: sigma_measure_a,M: sigma_measure_a,N: sigma_measure_a,N3: sigma_measure_a] :
( ( ( sigma_sets_a @ M2 )
= ( sigma_sets_a @ M ) )
=> ( ( ( sigma_sets_a @ N )
= ( sigma_sets_a @ N3 ) )
=> ( ( sigma_measurable_a_a @ M2 @ N )
= ( sigma_measurable_a_a @ M @ N3 ) ) ) ) ).
% measurable_cong_sets
thf(fact_204_nle__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_205_nle__le,axiom,
! [A: int,B2: int] :
( ( ~ ( ord_less_eq_int @ A @ B2 ) )
= ( ( ord_less_eq_int @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_206_le__cases3,axiom,
! [X5: nat,Y4: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X5 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X5 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y4 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ X5 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X5 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_207_le__cases3,axiom,
! [X5: int,Y4: int,Z: int] :
( ( ( ord_less_eq_int @ X5 @ Y4 )
=> ~ ( ord_less_eq_int @ Y4 @ Z ) )
=> ( ( ( ord_less_eq_int @ Y4 @ X5 )
=> ~ ( ord_less_eq_int @ X5 @ Z ) )
=> ( ( ( ord_less_eq_int @ X5 @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y4 ) )
=> ( ( ( ord_less_eq_int @ Z @ Y4 )
=> ~ ( ord_less_eq_int @ Y4 @ X5 ) )
=> ( ( ( ord_less_eq_int @ Y4 @ Z )
=> ~ ( ord_less_eq_int @ Z @ X5 ) )
=> ~ ( ( ord_less_eq_int @ Z @ X5 )
=> ~ ( ord_less_eq_int @ X5 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_208_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_b,Z2: set_set_b] : ( Y5 = Z2 ) )
= ( ^ [X3: set_set_b,Y3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X3 @ Y3 )
& ( ord_le3795704787696855135_set_b @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_209_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_a,Z2: set_set_a] : ( Y5 = Z2 ) )
= ( ^ [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
& ( ord_le3724670747650509150_set_a @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_210_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
& ( ord_less_eq_set_a @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_211_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_212_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
& ( ord_less_eq_int @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_213_ord__eq__le__trans,axiom,
! [A: set_set_b,B2: set_set_b,C2: set_set_b] :
( ( A = B2 )
=> ( ( ord_le3795704787696855135_set_b @ B2 @ C2 )
=> ( ord_le3795704787696855135_set_b @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_214_ord__eq__le__trans,axiom,
! [A: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( A = B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C2 )
=> ( ord_le3724670747650509150_set_a @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_215_ord__eq__le__trans,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( A = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_216_ord__eq__le__trans,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( A = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_217_ord__eq__le__trans,axiom,
! [A: int,B2: int,C2: int] :
( ( A = B2 )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_218_ord__le__eq__trans,axiom,
! [A: set_set_b,B2: set_set_b,C2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le3795704787696855135_set_b @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_219_ord__le__eq__trans,axiom,
! [A: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le3724670747650509150_set_a @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_220_ord__le__eq__trans,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_221_ord__le__eq__trans,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_222_ord__le__eq__trans,axiom,
! [A: int,B2: int,C2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_223_order__antisym,axiom,
! [X5: set_set_b,Y4: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X5 @ Y4 )
=> ( ( ord_le3795704787696855135_set_b @ Y4 @ X5 )
=> ( X5 = Y4 ) ) ) ).
% order_antisym
thf(fact_224_order__antisym,axiom,
! [X5: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ Y4 )
=> ( ( ord_le3724670747650509150_set_a @ Y4 @ X5 )
=> ( X5 = Y4 ) ) ) ).
% order_antisym
thf(fact_225_order__antisym,axiom,
! [X5: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y4 )
=> ( ( ord_less_eq_set_a @ Y4 @ X5 )
=> ( X5 = Y4 ) ) ) ).
% order_antisym
thf(fact_226_order__antisym,axiom,
! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ X5 )
=> ( X5 = Y4 ) ) ) ).
% order_antisym
thf(fact_227_order__antisym,axiom,
! [X5: int,Y4: int] :
( ( ord_less_eq_int @ X5 @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ X5 )
=> ( X5 = Y4 ) ) ) ).
% order_antisym
thf(fact_228_order_Otrans,axiom,
! [A: set_set_b,B2: set_set_b,C2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ B2 )
=> ( ( ord_le3795704787696855135_set_b @ B2 @ C2 )
=> ( ord_le3795704787696855135_set_b @ A @ C2 ) ) ) ).
% order.trans
thf(fact_229_order_Otrans,axiom,
! [A: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C2 )
=> ( ord_le3724670747650509150_set_a @ A @ C2 ) ) ) ).
% order.trans
thf(fact_230_order_Otrans,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% order.trans
thf(fact_231_order_Otrans,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_232_order_Otrans,axiom,
! [A: int,B2: int,C2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% order.trans
thf(fact_233_order__trans,axiom,
! [X5: set_set_b,Y4: set_set_b,Z: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X5 @ Y4 )
=> ( ( ord_le3795704787696855135_set_b @ Y4 @ Z )
=> ( ord_le3795704787696855135_set_b @ X5 @ Z ) ) ) ).
% order_trans
thf(fact_234_order__trans,axiom,
! [X5: set_set_a,Y4: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ Y4 )
=> ( ( ord_le3724670747650509150_set_a @ Y4 @ Z )
=> ( ord_le3724670747650509150_set_a @ X5 @ Z ) ) ) ).
% order_trans
thf(fact_235_order__trans,axiom,
! [X5: set_a,Y4: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y4 )
=> ( ( ord_less_eq_set_a @ Y4 @ Z )
=> ( ord_less_eq_set_a @ X5 @ Z ) ) ) ).
% order_trans
thf(fact_236_order__trans,axiom,
! [X5: nat,Y4: nat,Z: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z )
=> ( ord_less_eq_nat @ X5 @ Z ) ) ) ).
% order_trans
thf(fact_237_order__trans,axiom,
! [X5: int,Y4: int,Z: int] :
( ( ord_less_eq_int @ X5 @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ Z )
=> ( ord_less_eq_int @ X5 @ Z ) ) ) ).
% order_trans
thf(fact_238_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_239_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B2: int] :
( ! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: int,B3: int] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_240_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_set_b,Z2: set_set_b] : ( Y5 = Z2 ) )
= ( ^ [A4: set_set_b,B4: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B4 @ A4 )
& ( ord_le3795704787696855135_set_b @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_241_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_set_a,Z2: set_set_a] : ( Y5 = Z2 ) )
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A4 )
& ( ord_le3724670747650509150_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_242_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_243_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_244_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_245_dual__order_Oantisym,axiom,
! [B2: set_set_b,A: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B2 @ A )
=> ( ( ord_le3795704787696855135_set_b @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_246_dual__order_Oantisym,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_247_dual__order_Oantisym,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_248_dual__order_Oantisym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_249_dual__order_Oantisym,axiom,
! [B2: int,A: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( ord_less_eq_int @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_250_dual__order_Otrans,axiom,
! [B2: set_set_b,A: set_set_b,C2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B2 @ A )
=> ( ( ord_le3795704787696855135_set_b @ C2 @ B2 )
=> ( ord_le3795704787696855135_set_b @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_251_dual__order_Otrans,axiom,
! [B2: set_set_a,A: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( ( ord_le3724670747650509150_set_a @ C2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_252_dual__order_Otrans,axiom,
! [B2: set_a,A: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B2 )
=> ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_253_dual__order_Otrans,axiom,
! [B2: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_254_dual__order_Otrans,axiom,
! [B2: int,A: int,C2: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( ord_less_eq_int @ C2 @ B2 )
=> ( ord_less_eq_int @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_255_antisym,axiom,
! [A: set_set_b,B2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ B2 )
=> ( ( ord_le3795704787696855135_set_b @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_256_antisym,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_257_antisym,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_258_antisym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_259_antisym,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_260_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_b,Z2: set_set_b] : ( Y5 = Z2 ) )
= ( ^ [A4: set_set_b,B4: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A4 @ B4 )
& ( ord_le3795704787696855135_set_b @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_261_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_a,Z2: set_set_a] : ( Y5 = Z2 ) )
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_262_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_263_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_264_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_265_order__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_266_order__subst1,axiom,
! [A: nat,F: int > nat,B2: int,C2: int] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X4: int,Y6: int] :
( ( ord_less_eq_int @ X4 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_267_order__subst1,axiom,
! [A: int,F: nat > int,B2: nat,C2: nat] :
( ( ord_less_eq_int @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_268_order__subst1,axiom,
! [A: int,F: int > int,B2: int,C2: int] :
( ( ord_less_eq_int @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X4: int,Y6: int] :
( ( ord_less_eq_int @ X4 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_269_order__subst1,axiom,
! [A: set_a,F: nat > set_a,B2: nat,C2: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_270_order__subst1,axiom,
! [A: set_a,F: int > set_a,B2: int,C2: int] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X4: int,Y6: int] :
( ( ord_less_eq_int @ X4 @ Y6 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_271_order__subst1,axiom,
! [A: nat,F: set_a > nat,B2: set_a,C2: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X4: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_272_order__subst1,axiom,
! [A: int,F: set_a > int,B2: set_a,C2: set_a] :
( ( ord_less_eq_int @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X4: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_273_order__subst1,axiom,
! [A: set_set_b,F: nat > set_set_b,B2: nat,C2: nat] :
( ( ord_le3795704787696855135_set_b @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ord_le3795704787696855135_set_b @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_le3795704787696855135_set_b @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_274_order__subst1,axiom,
! [A: set_set_b,F: int > set_set_b,B2: int,C2: int] :
( ( ord_le3795704787696855135_set_b @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X4: int,Y6: int] :
( ( ord_less_eq_int @ X4 @ Y6 )
=> ( ord_le3795704787696855135_set_b @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_le3795704787696855135_set_b @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_275_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_276_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C2 )
=> ( ! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_277_order__subst2,axiom,
! [A: int,B2: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: int,Y6: int] :
( ( ord_less_eq_int @ X4 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_278_order__subst2,axiom,
! [A: int,B2: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C2 )
=> ( ! [X4: int,Y6: int] :
( ( ord_less_eq_int @ X4 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_279_order__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_280_order__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > int,C2: int] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C2 )
=> ( ! [X4: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_281_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C2 )
=> ( ! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_282_order__subst2,axiom,
! [A: int,B2: int,F: int > set_a,C2: set_a] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C2 )
=> ( ! [X4: int,Y6: int] :
( ( ord_less_eq_int @ X4 @ Y6 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_283_order__subst2,axiom,
! [A: set_set_b,B2: set_set_b,F: set_set_b > nat,C2: nat] :
( ( ord_le3795704787696855135_set_b @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: set_set_b,Y6: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X4 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_284_order__subst2,axiom,
! [A: set_set_b,B2: set_set_b,F: set_set_b > int,C2: int] :
( ( ord_le3795704787696855135_set_b @ A @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C2 )
=> ( ! [X4: set_set_b,Y6: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X4 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_285_order__eq__refl,axiom,
! [X5: set_set_b,Y4: set_set_b] :
( ( X5 = Y4 )
=> ( ord_le3795704787696855135_set_b @ X5 @ Y4 ) ) ).
% order_eq_refl
thf(fact_286_order__eq__refl,axiom,
! [X5: set_set_a,Y4: set_set_a] :
( ( X5 = Y4 )
=> ( ord_le3724670747650509150_set_a @ X5 @ Y4 ) ) ).
% order_eq_refl
thf(fact_287_order__eq__refl,axiom,
! [X5: set_a,Y4: set_a] :
( ( X5 = Y4 )
=> ( ord_less_eq_set_a @ X5 @ Y4 ) ) ).
% order_eq_refl
thf(fact_288_order__eq__refl,axiom,
! [X5: nat,Y4: nat] :
( ( X5 = Y4 )
=> ( ord_less_eq_nat @ X5 @ Y4 ) ) ).
% order_eq_refl
thf(fact_289_order__eq__refl,axiom,
! [X5: int,Y4: int] :
( ( X5 = Y4 )
=> ( ord_less_eq_int @ X5 @ Y4 ) ) ).
% order_eq_refl
thf(fact_290_linorder__linear,axiom,
! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
| ( ord_less_eq_nat @ Y4 @ X5 ) ) ).
% linorder_linear
thf(fact_291_linorder__linear,axiom,
! [X5: int,Y4: int] :
( ( ord_less_eq_int @ X5 @ Y4 )
| ( ord_less_eq_int @ Y4 @ X5 ) ) ).
% linorder_linear
thf(fact_292_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C2: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_293_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B2: nat,C2: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_294_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B2: int,C2: int] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X4: int,Y6: int] :
( ( ord_less_eq_int @ X4 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_295_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B2: int,C2: int] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X4: int,Y6: int] :
( ( ord_less_eq_int @ X4 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_296_ord__eq__le__subst,axiom,
! [A: nat,F: set_a > nat,B2: set_a,C2: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X4: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_297_ord__eq__le__subst,axiom,
! [A: int,F: set_a > int,B2: set_a,C2: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X4: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_298_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B2: nat,C2: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_299_ord__eq__le__subst,axiom,
! [A: set_a,F: int > set_a,B2: int,C2: int] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X4: int,Y6: int] :
( ( ord_less_eq_int @ X4 @ Y6 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_300_ord__eq__le__subst,axiom,
! [A: nat,F: set_set_b > nat,B2: set_set_b,C2: set_set_b] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le3795704787696855135_set_b @ B2 @ C2 )
=> ( ! [X4: set_set_b,Y6: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X4 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_301_ord__eq__le__subst,axiom,
! [A: int,F: set_set_b > int,B2: set_set_b,C2: set_set_b] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le3795704787696855135_set_b @ B2 @ C2 )
=> ( ! [X4: set_set_b,Y6: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X4 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_302_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_303_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_304_ord__le__eq__subst,axiom,
! [A: int,B2: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: int,Y6: int] :
( ( ord_less_eq_int @ X4 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_305_ord__le__eq__subst,axiom,
! [A: int,B2: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: int,Y6: int] :
( ( ord_less_eq_int @ X4 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_306_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_307_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > int,C2: int] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_308_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_309_ord__le__eq__subst,axiom,
! [A: int,B2: int,F: int > set_a,C2: set_a] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: int,Y6: int] :
( ( ord_less_eq_int @ X4 @ Y6 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_310_ord__le__eq__subst,axiom,
! [A: set_set_b,B2: set_set_b,F: set_set_b > nat,C2: nat] :
( ( ord_le3795704787696855135_set_b @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: set_set_b,Y6: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X4 @ Y6 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_311_ord__le__eq__subst,axiom,
! [A: set_set_b,B2: set_set_b,F: set_set_b > int,C2: int] :
( ( ord_le3795704787696855135_set_b @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: set_set_b,Y6: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X4 @ Y6 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_312_linorder__le__cases,axiom,
! [X5: nat,Y4: nat] :
( ~ ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X5 ) ) ).
% linorder_le_cases
thf(fact_313_linorder__le__cases,axiom,
! [X5: int,Y4: int] :
( ~ ( ord_less_eq_int @ X5 @ Y4 )
=> ( ord_less_eq_int @ Y4 @ X5 ) ) ).
% linorder_le_cases
thf(fact_314_order__antisym__conv,axiom,
! [Y4: set_set_b,X5: set_set_b] :
( ( ord_le3795704787696855135_set_b @ Y4 @ X5 )
=> ( ( ord_le3795704787696855135_set_b @ X5 @ Y4 )
= ( X5 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_315_order__antisym__conv,axiom,
! [Y4: set_set_a,X5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y4 @ X5 )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ Y4 )
= ( X5 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_316_order__antisym__conv,axiom,
! [Y4: set_a,X5: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X5 )
=> ( ( ord_less_eq_set_a @ X5 @ Y4 )
= ( X5 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_317_order__antisym__conv,axiom,
! [Y4: nat,X5: nat] :
( ( ord_less_eq_nat @ Y4 @ X5 )
=> ( ( ord_less_eq_nat @ X5 @ Y4 )
= ( X5 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_318_order__antisym__conv,axiom,
! [Y4: int,X5: int] :
( ( ord_less_eq_int @ Y4 @ X5 )
=> ( ( ord_less_eq_int @ X5 @ Y4 )
= ( X5 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_319_in__mono,axiom,
! [A2: set_a_b,B: set_a_b,X5: a > b] :
( ( ord_less_eq_set_a_b @ A2 @ B )
=> ( ( member_a_b @ X5 @ A2 )
=> ( member_a_b @ X5 @ B ) ) ) ).
% in_mono
thf(fact_320_in__mono,axiom,
! [A2: set_c,B: set_c,X5: c] :
( ( ord_less_eq_set_c @ A2 @ B )
=> ( ( member_c @ X5 @ A2 )
=> ( member_c @ X5 @ B ) ) ) ).
% in_mono
thf(fact_321_in__mono,axiom,
! [A2: set_a_a,B: set_a_a,X5: a > a] :
( ( ord_less_eq_set_a_a @ A2 @ B )
=> ( ( member_a_a @ X5 @ A2 )
=> ( member_a_a @ X5 @ B ) ) ) ).
% in_mono
thf(fact_322_in__mono,axiom,
! [A2: set_o,B: set_o,X5: $o] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ( member_o @ X5 @ A2 )
=> ( member_o @ X5 @ B ) ) ) ).
% in_mono
thf(fact_323_in__mono,axiom,
! [A2: set_nat,B: set_nat,X5: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ X5 @ A2 )
=> ( member_nat @ X5 @ B ) ) ) ).
% in_mono
thf(fact_324_in__mono,axiom,
! [A2: set_set_b,B: set_set_b,X5: set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ B )
=> ( ( member_set_b @ X5 @ A2 )
=> ( member_set_b @ X5 @ B ) ) ) ).
% in_mono
thf(fact_325_in__mono,axiom,
! [A2: set_set_a,B: set_set_a,X5: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_a @ X5 @ A2 )
=> ( member_set_a @ X5 @ B ) ) ) ).
% in_mono
thf(fact_326_in__mono,axiom,
! [A2: set_a,B: set_a,X5: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ X5 @ A2 )
=> ( member_a @ X5 @ B ) ) ) ).
% in_mono
thf(fact_327_subsetD,axiom,
! [A2: set_a_b,B: set_a_b,C2: a > b] :
( ( ord_less_eq_set_a_b @ A2 @ B )
=> ( ( member_a_b @ C2 @ A2 )
=> ( member_a_b @ C2 @ B ) ) ) ).
% subsetD
thf(fact_328_subsetD,axiom,
! [A2: set_c,B: set_c,C2: c] :
( ( ord_less_eq_set_c @ A2 @ B )
=> ( ( member_c @ C2 @ A2 )
=> ( member_c @ C2 @ B ) ) ) ).
% subsetD
thf(fact_329_subsetD,axiom,
! [A2: set_a_a,B: set_a_a,C2: a > a] :
( ( ord_less_eq_set_a_a @ A2 @ B )
=> ( ( member_a_a @ C2 @ A2 )
=> ( member_a_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_330_subsetD,axiom,
! [A2: set_o,B: set_o,C2: $o] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ( member_o @ C2 @ A2 )
=> ( member_o @ C2 @ B ) ) ) ).
% subsetD
thf(fact_331_subsetD,axiom,
! [A2: set_nat,B: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_332_subsetD,axiom,
! [A2: set_set_b,B: set_set_b,C2: set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ B )
=> ( ( member_set_b @ C2 @ A2 )
=> ( member_set_b @ C2 @ B ) ) ) ).
% subsetD
thf(fact_333_subsetD,axiom,
! [A2: set_set_a,B: set_set_a,C2: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_a @ C2 @ A2 )
=> ( member_set_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_334_subsetD,axiom,
! [A2: set_a,B: set_a,C2: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_335_equalityE,axiom,
! [A2: set_set_b,B: set_set_b] :
( ( A2 = B )
=> ~ ( ( ord_le3795704787696855135_set_b @ A2 @ B )
=> ~ ( ord_le3795704787696855135_set_b @ B @ A2 ) ) ) ).
% equalityE
thf(fact_336_equalityE,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( A2 = B )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ~ ( ord_le3724670747650509150_set_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_337_equalityE,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B )
=> ~ ( ord_less_eq_set_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_338_subset__eq,axiom,
( ord_less_eq_set_a_b
= ( ^ [A5: set_a_b,B5: set_a_b] :
! [X3: a > b] :
( ( member_a_b @ X3 @ A5 )
=> ( member_a_b @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_339_subset__eq,axiom,
( ord_less_eq_set_c
= ( ^ [A5: set_c,B5: set_c] :
! [X3: c] :
( ( member_c @ X3 @ A5 )
=> ( member_c @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_340_subset__eq,axiom,
( ord_less_eq_set_a_a
= ( ^ [A5: set_a_a,B5: set_a_a] :
! [X3: a > a] :
( ( member_a_a @ X3 @ A5 )
=> ( member_a_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_341_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A5: set_o,B5: set_o] :
! [X3: $o] :
( ( member_o @ X3 @ A5 )
=> ( member_o @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_342_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( member_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_343_subset__eq,axiom,
( ord_le3795704787696855135_set_b
= ( ^ [A5: set_set_b,B5: set_set_b] :
! [X3: set_b] :
( ( member_set_b @ X3 @ A5 )
=> ( member_set_b @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_344_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A5 )
=> ( member_set_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_345_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( member_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_346_equalityD1,axiom,
! [A2: set_set_b,B: set_set_b] :
( ( A2 = B )
=> ( ord_le3795704787696855135_set_b @ A2 @ B ) ) ).
% equalityD1
thf(fact_347_equalityD1,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( A2 = B )
=> ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_348_equalityD1,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_349_equalityD2,axiom,
! [A2: set_set_b,B: set_set_b] :
( ( A2 = B )
=> ( ord_le3795704787696855135_set_b @ B @ A2 ) ) ).
% equalityD2
thf(fact_350_equalityD2,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( A2 = B )
=> ( ord_le3724670747650509150_set_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_351_equalityD2,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_352_subset__iff,axiom,
( ord_less_eq_set_a_b
= ( ^ [A5: set_a_b,B5: set_a_b] :
! [T2: a > b] :
( ( member_a_b @ T2 @ A5 )
=> ( member_a_b @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_353_subset__iff,axiom,
( ord_less_eq_set_c
= ( ^ [A5: set_c,B5: set_c] :
! [T2: c] :
( ( member_c @ T2 @ A5 )
=> ( member_c @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_354_subset__iff,axiom,
( ord_less_eq_set_a_a
= ( ^ [A5: set_a_a,B5: set_a_a] :
! [T2: a > a] :
( ( member_a_a @ T2 @ A5 )
=> ( member_a_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_355_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A5: set_o,B5: set_o] :
! [T2: $o] :
( ( member_o @ T2 @ A5 )
=> ( member_o @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_356_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A5 )
=> ( member_nat @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_357_subset__iff,axiom,
( ord_le3795704787696855135_set_b
= ( ^ [A5: set_set_b,B5: set_set_b] :
! [T2: set_b] :
( ( member_set_b @ T2 @ A5 )
=> ( member_set_b @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_358_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
! [T2: set_a] :
( ( member_set_a @ T2 @ A5 )
=> ( member_set_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_359_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A5 )
=> ( member_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_360_subset__refl,axiom,
! [A2: set_set_b] : ( ord_le3795704787696855135_set_b @ A2 @ A2 ) ).
% subset_refl
thf(fact_361_subset__refl,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_362_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_363_Collect__mono,axiom,
! [P: set_b > $o,Q: set_b > $o] :
( ! [X4: set_b] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le3795704787696855135_set_b @ ( collect_set_b @ P ) @ ( collect_set_b @ Q ) ) ) ).
% Collect_mono
thf(fact_364_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X4: set_a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_365_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_366_subset__trans,axiom,
! [A2: set_set_b,B: set_set_b,C: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ B )
=> ( ( ord_le3795704787696855135_set_b @ B @ C )
=> ( ord_le3795704787696855135_set_b @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_367_subset__trans,axiom,
! [A2: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_368_subset__trans,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_369_set__eq__subset,axiom,
( ( ^ [Y5: set_set_b,Z2: set_set_b] : ( Y5 = Z2 ) )
= ( ^ [A5: set_set_b,B5: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A5 @ B5 )
& ( ord_le3795704787696855135_set_b @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_370_set__eq__subset,axiom,
( ( ^ [Y5: set_set_a,Z2: set_set_a] : ( Y5 = Z2 ) )
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_371_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_372_Collect__mono__iff,axiom,
! [P: set_b > $o,Q: set_b > $o] :
( ( ord_le3795704787696855135_set_b @ ( collect_set_b @ P ) @ ( collect_set_b @ Q ) )
= ( ! [X3: set_b] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_373_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X3: set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_374_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_375_prob__space_Otail__events__sets,axiom,
! [M2: sigma_measure_b,A2: nat > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ! [I2: nat] : ( ord_le3795704787696855135_set_b @ ( A2 @ I2 ) @ ( sigma_sets_b @ M2 ) )
=> ( ord_le3795704787696855135_set_b @ ( indepe8773861029005768663_b_nat @ M2 @ A2 ) @ ( sigma_sets_b @ M2 ) ) ) ) ).
% prob_space.tail_events_sets
thf(fact_376_prob__space_Otail__events__sets,axiom,
! [M2: sigma_measure_a,A2: nat > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ! [I2: nat] : ( ord_le3724670747650509150_set_a @ ( A2 @ I2 ) @ ( sigma_sets_a @ M2 ) )
=> ( ord_le3724670747650509150_set_a @ ( indepe7538416700049374166_a_nat @ M2 @ A2 ) @ ( sigma_sets_a @ M2 ) ) ) ) ).
% prob_space.tail_events_sets
thf(fact_377_subprob__space_Ointro,axiom,
! [M2: sigma_measure_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ( giry_s1767857069175831631ioms_a @ M2 )
=> ( giry_subprob_space_a @ M2 ) ) ) ).
% subprob_space.intro
thf(fact_378_subprob__space_Ointro,axiom,
! [M2: sigma_measure_b] :
( ( measur930452917991658467sure_b @ M2 )
=> ( ( giry_s1767857069175831632ioms_b @ M2 )
=> ( giry_subprob_space_b @ M2 ) ) ) ).
% subprob_space.intro
thf(fact_379_subprob__space__def,axiom,
( giry_subprob_space_a
= ( ^ [M3: sigma_measure_a] :
( ( measur930452917991658466sure_a @ M3 )
& ( giry_s1767857069175831631ioms_a @ M3 ) ) ) ) ).
% subprob_space_def
thf(fact_380_subprob__space__def,axiom,
( giry_subprob_space_b
= ( ^ [M3: sigma_measure_b] :
( ( measur930452917991658467sure_b @ M3 )
& ( giry_s1767857069175831632ioms_b @ M3 ) ) ) ) ).
% subprob_space_def
thf(fact_381_finite__measure_Ointro,axiom,
! [M2: sigma_measure_b] :
( ( measur4308613598931908896sure_b @ M2 )
=> ( ( measur2595372213310369024ioms_b @ M2 )
=> ( measur930452917991658467sure_b @ M2 ) ) ) ).
% finite_measure.intro
thf(fact_382_finite__measure_Ointro,axiom,
! [M2: sigma_measure_a] :
( ( measur4308613598931908895sure_a @ M2 )
=> ( ( measur2595372213310369023ioms_a @ M2 )
=> ( measur930452917991658466sure_a @ M2 ) ) ) ).
% finite_measure.intro
thf(fact_383_finite__measure__def,axiom,
( measur930452917991658467sure_b
= ( ^ [M3: sigma_measure_b] :
( ( measur4308613598931908896sure_b @ M3 )
& ( measur2595372213310369024ioms_b @ M3 ) ) ) ) ).
% finite_measure_def
thf(fact_384_finite__measure__def,axiom,
( measur930452917991658466sure_a
= ( ^ [M3: sigma_measure_a] :
( ( measur4308613598931908895sure_a @ M3 )
& ( measur2595372213310369023ioms_a @ M3 ) ) ) ) ).
% finite_measure_def
thf(fact_385_prob__space__def,axiom,
( probab7247484486040049090pace_b
= ( ^ [M3: sigma_measure_b] :
( ( measur930452917991658467sure_b @ M3 )
& ( probab8302655048591552735ioms_b @ M3 ) ) ) ) ).
% prob_space_def
thf(fact_386_prob__space__def,axiom,
( probab7247484486040049089pace_a
= ( ^ [M3: sigma_measure_a] :
( ( measur930452917991658466sure_a @ M3 )
& ( probab8302655048591552734ioms_a @ M3 ) ) ) ) ).
% prob_space_def
thf(fact_387_prob__space_Ointro,axiom,
! [M2: sigma_measure_b] :
( ( measur930452917991658467sure_b @ M2 )
=> ( ( probab8302655048591552735ioms_b @ M2 )
=> ( probab7247484486040049090pace_b @ M2 ) ) ) ).
% prob_space.intro
thf(fact_388_prob__space_Ointro,axiom,
! [M2: sigma_measure_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ( probab8302655048591552734ioms_a @ M2 )
=> ( probab7247484486040049089pace_a @ M2 ) ) ) ).
% prob_space.intro
thf(fact_389_prob__space_Oindep__setD__ev2,axiom,
! [M2: sigma_measure_b,A2: set_set_b,B: set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe2041756565122539607_set_b @ M2 @ A2 @ B )
=> ( ord_le3795704787696855135_set_b @ B @ ( sigma_sets_b @ M2 ) ) ) ) ).
% prob_space.indep_setD_ev2
thf(fact_390_prob__space_Oindep__setD__ev2,axiom,
! [M2: sigma_measure_a,A2: set_set_a,B: set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe2041756565122539606_set_a @ M2 @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ B @ ( sigma_sets_a @ M2 ) ) ) ) ).
% prob_space.indep_setD_ev2
thf(fact_391_prob__space_Oindep__setD__ev1,axiom,
! [M2: sigma_measure_b,A2: set_set_b,B: set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe2041756565122539607_set_b @ M2 @ A2 @ B )
=> ( ord_le3795704787696855135_set_b @ A2 @ ( sigma_sets_b @ M2 ) ) ) ) ).
% prob_space.indep_setD_ev1
thf(fact_392_prob__space_Oindep__setD__ev1,axiom,
! [M2: sigma_measure_a,A2: set_set_a,B: set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe2041756565122539606_set_a @ M2 @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( sigma_sets_a @ M2 ) ) ) ) ).
% prob_space.indep_setD_ev1
thf(fact_393_main__part__sets,axiom,
! [S: set_b,M2: sigma_measure_b] :
( ( member_set_b @ S @ ( sigma_sets_b @ ( comple3428971583294703881tion_b @ M2 ) ) )
=> ( member_set_b @ ( complete_main_part_b @ M2 @ S ) @ ( sigma_sets_b @ M2 ) ) ) ).
% main_part_sets
thf(fact_394_main__part__sets,axiom,
! [S: set_a,M2: sigma_measure_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) )
=> ( member_set_a @ ( complete_main_part_a @ M2 @ S ) @ ( sigma_sets_a @ M2 ) ) ) ).
% main_part_sets
thf(fact_395_completion_Omeasurable__completion2,axiom,
! [F: a > b,M2: sigma_measure_a,N: sigma_measure_b] :
( ( member_a_b @ F @ ( sigma_measurable_a_b @ ( comple3428971583294703880tion_a @ M2 ) @ N ) )
=> ( ( ord_le3795704787696855135_set_b @ ( measure_null_sets_b @ N ) @ ( measure_null_sets_b @ ( measure_distr_a_b @ ( comple3428971583294703880tion_a @ M2 ) @ N @ F ) ) )
=> ( member_a_b @ F @ ( sigma_measurable_a_b @ ( comple3428971583294703880tion_a @ M2 ) @ ( comple3428971583294703881tion_b @ N ) ) ) ) ) ).
% completion.measurable_completion2
thf(fact_396_completion_Omeasurable__completion2,axiom,
! [F: a > a,M2: sigma_measure_a,N: sigma_measure_a] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ ( comple3428971583294703880tion_a @ M2 ) @ N ) )
=> ( ( ord_le3724670747650509150_set_a @ ( measure_null_sets_a @ N ) @ ( measure_null_sets_a @ ( measure_distr_a_a @ ( comple3428971583294703880tion_a @ M2 ) @ N @ F ) ) )
=> ( member_a_a @ F @ ( sigma_measurable_a_a @ ( comple3428971583294703880tion_a @ M2 ) @ ( comple3428971583294703880tion_a @ N ) ) ) ) ) ).
% completion.measurable_completion2
thf(fact_397_indep__setD__ev1,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( indepe2041756565122539606_set_a @ m @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( sigma_sets_a @ m ) ) ) ).
% indep_setD_ev1
thf(fact_398_indep__setD__ev2,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( indepe2041756565122539606_set_a @ m @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ B @ ( sigma_sets_a @ m ) ) ) ).
% indep_setD_ev2
thf(fact_399_tail__events__sets,axiom,
! [A2: nat > set_set_a] :
( ! [I2: nat] : ( ord_le3724670747650509150_set_a @ ( A2 @ I2 ) @ ( sigma_sets_a @ m ) )
=> ( ord_le3724670747650509150_set_a @ ( indepe7538416700049374166_a_nat @ m @ A2 ) @ ( sigma_sets_a @ m ) ) ) ).
% tail_events_sets
thf(fact_400_main__part,axiom,
! [S: set_b,M2: sigma_measure_b] :
( ( member_set_b @ S @ ( sigma_sets_b @ M2 ) )
=> ( ( complete_main_part_b @ M2 @ S )
= S ) ) ).
% main_part
thf(fact_401_main__part,axiom,
! [S: set_a,M2: sigma_measure_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ M2 ) )
=> ( ( complete_main_part_a @ M2 @ S )
= S ) ) ).
% main_part
thf(fact_402_prob__space_Oindep__set_Ocong,axiom,
indepe2041756565122539606_set_a = indepe2041756565122539606_set_a ).
% prob_space.indep_set.cong
thf(fact_403_prob__space_Otail__events_Ocong,axiom,
indepe7538416700049374166_a_nat = indepe7538416700049374166_a_nat ).
% prob_space.tail_events.cong
thf(fact_404_null__setsD2,axiom,
! [A2: set_b,M2: sigma_measure_b] :
( ( member_set_b @ A2 @ ( measure_null_sets_b @ M2 ) )
=> ( member_set_b @ A2 @ ( sigma_sets_b @ M2 ) ) ) ).
% null_setsD2
thf(fact_405_null__setsD2,axiom,
! [A2: set_a,M2: sigma_measure_a] :
( ( member_set_a @ A2 @ ( measure_null_sets_a @ M2 ) )
=> ( member_set_a @ A2 @ ( sigma_sets_a @ M2 ) ) ) ).
% null_setsD2
thf(fact_406_null__sets__completionI,axiom,
! [N: set_a,M2: sigma_measure_a] :
( ( member_set_a @ N @ ( measure_null_sets_a @ M2 ) )
=> ( member_set_a @ N @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) ) ) ).
% null_sets_completionI
thf(fact_407_null__sets__subset,axiom,
! [B: set_b,M2: sigma_measure_b,A2: set_b] :
( ( member_set_b @ B @ ( measure_null_sets_b @ M2 ) )
=> ( ( member_set_b @ A2 @ ( sigma_sets_b @ M2 ) )
=> ( ( ord_less_eq_set_b @ A2 @ B )
=> ( member_set_b @ A2 @ ( measure_null_sets_b @ M2 ) ) ) ) ) ).
% null_sets_subset
thf(fact_408_null__sets__subset,axiom,
! [B: set_set_b,M2: sigma_measure_set_b,A2: set_set_b] :
( ( member_set_set_b @ B @ ( measur1516554132335629586_set_b @ M2 ) )
=> ( ( member_set_set_b @ A2 @ ( sigma_sets_set_b @ M2 ) )
=> ( ( ord_le3795704787696855135_set_b @ A2 @ B )
=> ( member_set_set_b @ A2 @ ( measur1516554132335629586_set_b @ M2 ) ) ) ) ) ).
% null_sets_subset
thf(fact_409_null__sets__subset,axiom,
! [B: set_set_a,M2: sigma_measure_set_a,A2: set_set_a] :
( ( member_set_set_a @ B @ ( measur1516554128032400785_set_a @ M2 ) )
=> ( ( member_set_set_a @ A2 @ ( sigma_sets_set_a @ M2 ) )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( member_set_set_a @ A2 @ ( measur1516554128032400785_set_a @ M2 ) ) ) ) ) ).
% null_sets_subset
thf(fact_410_null__sets__subset,axiom,
! [B: set_a,M2: sigma_measure_a,A2: set_a] :
( ( member_set_a @ B @ ( measure_null_sets_a @ M2 ) )
=> ( ( member_set_a @ A2 @ ( sigma_sets_a @ M2 ) )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( member_set_a @ A2 @ ( measure_null_sets_a @ M2 ) ) ) ) ) ).
% null_sets_subset
thf(fact_411_null__sets__completion__subset,axiom,
! [B: set_set_b,A2: set_set_b,M2: sigma_measure_set_b] :
( ( ord_le3795704787696855135_set_b @ B @ A2 )
=> ( ( member_set_set_b @ A2 @ ( measur1516554132335629586_set_b @ ( comple8942076150311361001_set_b @ M2 ) ) )
=> ( member_set_set_b @ B @ ( measur1516554132335629586_set_b @ ( comple8942076150311361001_set_b @ M2 ) ) ) ) ) ).
% null_sets_completion_subset
thf(fact_412_null__sets__completion__subset,axiom,
! [B: set_set_a,A2: set_set_a,M2: sigma_measure_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( ( member_set_set_a @ A2 @ ( measur1516554128032400785_set_a @ ( comple8942076146008132200_set_a @ M2 ) ) )
=> ( member_set_set_a @ B @ ( measur1516554128032400785_set_a @ ( comple8942076146008132200_set_a @ M2 ) ) ) ) ) ).
% null_sets_completion_subset
thf(fact_413_null__sets__completion__subset,axiom,
! [B: set_a,A2: set_a,M2: sigma_measure_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( member_set_a @ A2 @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) )
=> ( member_set_a @ B @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) ) ) ) ).
% null_sets_completion_subset
thf(fact_414_null__sets__completion__iff2,axiom,
! [A2: set_set_b,M2: sigma_measure_set_b] :
( ( member_set_set_b @ A2 @ ( measur1516554132335629586_set_b @ ( comple8942076150311361001_set_b @ M2 ) ) )
= ( ? [X3: set_set_b] :
( ( member_set_set_b @ X3 @ ( measur1516554132335629586_set_b @ M2 ) )
& ( ord_le3795704787696855135_set_b @ A2 @ X3 ) ) ) ) ).
% null_sets_completion_iff2
thf(fact_415_null__sets__completion__iff2,axiom,
! [A2: set_set_a,M2: sigma_measure_set_a] :
( ( member_set_set_a @ A2 @ ( measur1516554128032400785_set_a @ ( comple8942076146008132200_set_a @ M2 ) ) )
= ( ? [X3: set_set_a] :
( ( member_set_set_a @ X3 @ ( measur1516554128032400785_set_a @ M2 ) )
& ( ord_le3724670747650509150_set_a @ A2 @ X3 ) ) ) ) ).
% null_sets_completion_iff2
thf(fact_416_null__sets__completion__iff2,axiom,
! [A2: set_a,M2: sigma_measure_a] :
( ( member_set_a @ A2 @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) )
= ( ? [X3: set_a] :
( ( member_set_a @ X3 @ ( measure_null_sets_a @ M2 ) )
& ( ord_less_eq_set_a @ A2 @ X3 ) ) ) ) ).
% null_sets_completion_iff2
thf(fact_417_completion_Ocomplete2,axiom,
! [A2: set_set_b,B: set_set_b,M2: sigma_measure_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ B )
=> ( ( member_set_set_b @ B @ ( measur1516554132335629586_set_b @ ( comple8942076150311361001_set_b @ M2 ) ) )
=> ( member_set_set_b @ A2 @ ( measur1516554132335629586_set_b @ ( comple8942076150311361001_set_b @ M2 ) ) ) ) ) ).
% completion.complete2
thf(fact_418_completion_Ocomplete2,axiom,
! [A2: set_set_a,B: set_set_a,M2: sigma_measure_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_set_a @ B @ ( measur1516554128032400785_set_a @ ( comple8942076146008132200_set_a @ M2 ) ) )
=> ( member_set_set_a @ A2 @ ( measur1516554128032400785_set_a @ ( comple8942076146008132200_set_a @ M2 ) ) ) ) ) ).
% completion.complete2
thf(fact_419_completion_Ocomplete2,axiom,
! [A2: set_a,B: set_a,M2: sigma_measure_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_set_a @ B @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) )
=> ( member_set_a @ A2 @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) ) ) ) ).
% completion.complete2
thf(fact_420_null__sets__completion__iff,axiom,
! [N: set_b,M2: sigma_measure_b] :
( ( member_set_b @ N @ ( sigma_sets_b @ M2 ) )
=> ( ( member_set_b @ N @ ( measure_null_sets_b @ ( comple3428971583294703881tion_b @ M2 ) ) )
= ( member_set_b @ N @ ( measure_null_sets_b @ M2 ) ) ) ) ).
% null_sets_completion_iff
thf(fact_421_null__sets__completion__iff,axiom,
! [N: set_a,M2: sigma_measure_a] :
( ( member_set_a @ N @ ( sigma_sets_a @ M2 ) )
=> ( ( member_set_a @ N @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) )
= ( member_set_a @ N @ ( measure_null_sets_a @ M2 ) ) ) ) ).
% null_sets_completion_iff
thf(fact_422_sets__completionI__sub,axiom,
! [N3: set_b,M2: sigma_measure_b,N: set_b] :
( ( member_set_b @ N3 @ ( measure_null_sets_b @ M2 ) )
=> ( ( ord_less_eq_set_b @ N @ N3 )
=> ( member_set_b @ N @ ( sigma_sets_b @ ( comple3428971583294703881tion_b @ M2 ) ) ) ) ) ).
% sets_completionI_sub
thf(fact_423_sets__completionI__sub,axiom,
! [N3: set_set_b,M2: sigma_measure_set_b,N: set_set_b] :
( ( member_set_set_b @ N3 @ ( measur1516554132335629586_set_b @ M2 ) )
=> ( ( ord_le3795704787696855135_set_b @ N @ N3 )
=> ( member_set_set_b @ N @ ( sigma_sets_set_b @ ( comple8942076150311361001_set_b @ M2 ) ) ) ) ) ).
% sets_completionI_sub
thf(fact_424_sets__completionI__sub,axiom,
! [N3: set_set_a,M2: sigma_measure_set_a,N: set_set_a] :
( ( member_set_set_a @ N3 @ ( measur1516554128032400785_set_a @ M2 ) )
=> ( ( ord_le3724670747650509150_set_a @ N @ N3 )
=> ( member_set_set_a @ N @ ( sigma_sets_set_a @ ( comple8942076146008132200_set_a @ M2 ) ) ) ) ) ).
% sets_completionI_sub
thf(fact_425_sets__completionI__sub,axiom,
! [N3: set_a,M2: sigma_measure_a,N: set_a] :
( ( member_set_a @ N3 @ ( measure_null_sets_a @ M2 ) )
=> ( ( ord_less_eq_set_a @ N @ N3 )
=> ( member_set_a @ N @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) ) ) ) ).
% sets_completionI_sub
thf(fact_426_completion_Ocomplete,axiom,
! [B: set_b,A2: set_b,M2: sigma_measure_b] :
( ( ord_less_eq_set_b @ B @ A2 )
=> ( ( member_set_b @ A2 @ ( measure_null_sets_b @ ( comple3428971583294703881tion_b @ M2 ) ) )
=> ( member_set_b @ B @ ( sigma_sets_b @ ( comple3428971583294703881tion_b @ M2 ) ) ) ) ) ).
% completion.complete
thf(fact_427_completion_Ocomplete,axiom,
! [B: set_set_b,A2: set_set_b,M2: sigma_measure_set_b] :
( ( ord_le3795704787696855135_set_b @ B @ A2 )
=> ( ( member_set_set_b @ A2 @ ( measur1516554132335629586_set_b @ ( comple8942076150311361001_set_b @ M2 ) ) )
=> ( member_set_set_b @ B @ ( sigma_sets_set_b @ ( comple8942076150311361001_set_b @ M2 ) ) ) ) ) ).
% completion.complete
thf(fact_428_completion_Ocomplete,axiom,
! [B: set_set_a,A2: set_set_a,M2: sigma_measure_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( ( member_set_set_a @ A2 @ ( measur1516554128032400785_set_a @ ( comple8942076146008132200_set_a @ M2 ) ) )
=> ( member_set_set_a @ B @ ( sigma_sets_set_a @ ( comple8942076146008132200_set_a @ M2 ) ) ) ) ) ).
% completion.complete
thf(fact_429_completion_Ocomplete,axiom,
! [B: set_a,A2: set_a,M2: sigma_measure_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( member_set_a @ A2 @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) )
=> ( member_set_a @ B @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) ) ) ) ).
% completion.complete
thf(fact_430_completion_Ocompletion__distr__eq,axiom,
! [X: a > b,M2: sigma_measure_a,N: sigma_measure_b] :
( ( member_a_b @ X @ ( sigma_measurable_a_b @ ( comple3428971583294703880tion_a @ M2 ) @ N ) )
=> ( ( ( measure_null_sets_b @ ( measure_distr_a_b @ ( comple3428971583294703880tion_a @ M2 ) @ N @ X ) )
= ( measure_null_sets_b @ N ) )
=> ( ( comple3428971583294703881tion_b @ ( measure_distr_a_b @ ( comple3428971583294703880tion_a @ M2 ) @ N @ X ) )
= ( measure_distr_a_b @ ( comple3428971583294703880tion_a @ M2 ) @ ( comple3428971583294703881tion_b @ N ) @ X ) ) ) ) ).
% completion.completion_distr_eq
thf(fact_431_completion_Ocompletion__distr__eq,axiom,
! [X: a > a,M2: sigma_measure_a,N: sigma_measure_a] :
( ( member_a_a @ X @ ( sigma_measurable_a_a @ ( comple3428971583294703880tion_a @ M2 ) @ N ) )
=> ( ( ( measure_null_sets_a @ ( measure_distr_a_a @ ( comple3428971583294703880tion_a @ M2 ) @ N @ X ) )
= ( measure_null_sets_a @ N ) )
=> ( ( comple3428971583294703880tion_a @ ( measure_distr_a_a @ ( comple3428971583294703880tion_a @ M2 ) @ N @ X ) )
= ( measure_distr_a_a @ ( comple3428971583294703880tion_a @ M2 ) @ ( comple3428971583294703880tion_a @ N ) @ X ) ) ) ) ).
% completion.completion_distr_eq
thf(fact_432_prob__space_Oaxioms_I2_J,axiom,
! [M2: sigma_measure_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( probab8302655048591552735ioms_b @ M2 ) ) ).
% prob_space.axioms(2)
thf(fact_433_prob__space_Oaxioms_I2_J,axiom,
! [M2: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( probab8302655048591552734ioms_a @ M2 ) ) ).
% prob_space.axioms(2)
thf(fact_434_finite__measure_Oaxioms_I2_J,axiom,
! [M2: sigma_measure_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( measur2595372213310369023ioms_a @ M2 ) ) ).
% finite_measure.axioms(2)
thf(fact_435_finite__measure_Oaxioms_I2_J,axiom,
! [M2: sigma_measure_b] :
( ( measur930452917991658467sure_b @ M2 )
=> ( measur2595372213310369024ioms_b @ M2 ) ) ).
% finite_measure.axioms(2)
thf(fact_436_subprob__space_Oaxioms_I2_J,axiom,
! [M2: sigma_measure_a] :
( ( giry_subprob_space_a @ M2 )
=> ( giry_s1767857069175831631ioms_a @ M2 ) ) ).
% subprob_space.axioms(2)
thf(fact_437_complete__measure_Omeasurable__completion2,axiom,
! [M2: sigma_measure_a,F: a > b,N: sigma_measure_b] :
( ( comple8155536527497655953sure_a @ M2 )
=> ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ N ) )
=> ( ( ord_le3795704787696855135_set_b @ ( measure_null_sets_b @ N ) @ ( measure_null_sets_b @ ( measure_distr_a_b @ M2 @ N @ F ) ) )
=> ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ ( comple3428971583294703881tion_b @ N ) ) ) ) ) ) ).
% complete_measure.measurable_completion2
thf(fact_438_complete__measure_Omeasurable__completion2,axiom,
! [M2: sigma_measure_a,F: a > a,N: sigma_measure_a] :
( ( comple8155536527497655953sure_a @ M2 )
=> ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N ) )
=> ( ( ord_le3724670747650509150_set_a @ ( measure_null_sets_a @ N ) @ ( measure_null_sets_a @ ( measure_distr_a_a @ M2 @ N @ F ) ) )
=> ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ ( comple3428971583294703880tion_a @ N ) ) ) ) ) ) ).
% complete_measure.measurable_completion2
thf(fact_439_null__part,axiom,
! [S: set_b,M2: sigma_measure_b] :
( ( member_set_b @ S @ ( sigma_sets_b @ ( comple3428971583294703881tion_b @ M2 ) ) )
=> ? [N4: set_b] :
( ( member_set_b @ N4 @ ( measure_null_sets_b @ M2 ) )
& ( ord_less_eq_set_b @ ( complete_null_part_b @ M2 @ S ) @ N4 ) ) ) ).
% null_part
thf(fact_440_null__part,axiom,
! [S: set_set_b,M2: sigma_measure_set_b] :
( ( member_set_set_b @ S @ ( sigma_sets_set_b @ ( comple8942076150311361001_set_b @ M2 ) ) )
=> ? [N4: set_set_b] :
( ( member_set_set_b @ N4 @ ( measur1516554132335629586_set_b @ M2 ) )
& ( ord_le3795704787696855135_set_b @ ( comple7912714282556339435_set_b @ M2 @ S ) @ N4 ) ) ) ).
% null_part
thf(fact_441_null__part,axiom,
! [S: set_set_a,M2: sigma_measure_set_a] :
( ( member_set_set_a @ S @ ( sigma_sets_set_a @ ( comple8942076146008132200_set_a @ M2 ) ) )
=> ? [N4: set_set_a] :
( ( member_set_set_a @ N4 @ ( measur1516554128032400785_set_a @ M2 ) )
& ( ord_le3724670747650509150_set_a @ ( comple7912714278253110634_set_a @ M2 @ S ) @ N4 ) ) ) ).
% null_part
thf(fact_442_null__part,axiom,
! [S: set_a,M2: sigma_measure_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) )
=> ? [N4: set_a] :
( ( member_set_a @ N4 @ ( measure_null_sets_a @ M2 ) )
& ( ord_less_eq_set_a @ ( complete_null_part_a @ M2 @ S ) @ N4 ) ) ) ).
% null_part
thf(fact_443_complete__measure_Ocompletion__distr__eq,axiom,
! [M2: sigma_measure_a,X: a > b,N: sigma_measure_b] :
( ( comple8155536527497655953sure_a @ M2 )
=> ( ( member_a_b @ X @ ( sigma_measurable_a_b @ M2 @ N ) )
=> ( ( ( measure_null_sets_b @ ( measure_distr_a_b @ M2 @ N @ X ) )
= ( measure_null_sets_b @ N ) )
=> ( ( comple3428971583294703881tion_b @ ( measure_distr_a_b @ M2 @ N @ X ) )
= ( measure_distr_a_b @ M2 @ ( comple3428971583294703881tion_b @ N ) @ X ) ) ) ) ) ).
% complete_measure.completion_distr_eq
thf(fact_444_complete__measure_Ocompletion__distr__eq,axiom,
! [M2: sigma_measure_a,X: a > a,N: sigma_measure_a] :
( ( comple8155536527497655953sure_a @ M2 )
=> ( ( member_a_a @ X @ ( sigma_measurable_a_a @ M2 @ N ) )
=> ( ( ( measure_null_sets_a @ ( measure_distr_a_a @ M2 @ N @ X ) )
= ( measure_null_sets_a @ N ) )
=> ( ( comple3428971583294703880tion_a @ ( measure_distr_a_a @ M2 @ N @ X ) )
= ( measure_distr_a_a @ M2 @ ( comple3428971583294703880tion_a @ N ) @ X ) ) ) ) ) ).
% complete_measure.completion_distr_eq
thf(fact_445_null__part__null__sets,axiom,
! [S: set_b,M2: sigma_measure_b] :
( ( member_set_b @ S @ ( sigma_sets_b @ ( comple3428971583294703881tion_b @ M2 ) ) )
=> ( member_set_b @ ( complete_null_part_b @ M2 @ S ) @ ( measure_null_sets_b @ ( comple3428971583294703881tion_b @ M2 ) ) ) ) ).
% null_part_null_sets
thf(fact_446_null__part__null__sets,axiom,
! [S: set_a,M2: sigma_measure_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) )
=> ( member_set_a @ ( complete_null_part_a @ M2 @ S ) @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) ) ) ).
% null_part_null_sets
thf(fact_447_measure__increasing,axiom,
measur1776380161843274167a_real @ ( sigma_sets_a @ m ) @ ( sigma_measure_a2 @ m ) ).
% measure_increasing
thf(fact_448_finite__measure__mono,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ A2 ) @ ( sigma_measure_a2 @ m @ B ) ) ) ) ).
% finite_measure_mono
thf(fact_449_complete__measure_Ointro,axiom,
! [M2: sigma_measure_b] :
( ! [A6: set_b,B6: set_b] :
( ( ord_less_eq_set_b @ B6 @ A6 )
=> ( ( member_set_b @ A6 @ ( measure_null_sets_b @ M2 ) )
=> ( member_set_b @ B6 @ ( sigma_sets_b @ M2 ) ) ) )
=> ( comple8155536527497655954sure_b @ M2 ) ) ).
% complete_measure.intro
thf(fact_450_complete__measure_Ointro,axiom,
! [M2: sigma_measure_set_b] :
( ! [A6: set_set_b,B6: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B6 @ A6 )
=> ( ( member_set_set_b @ A6 @ ( measur1516554132335629586_set_b @ M2 ) )
=> ( member_set_set_b @ B6 @ ( sigma_sets_set_b @ M2 ) ) ) )
=> ( comple6693822267556782962_set_b @ M2 ) ) ).
% complete_measure.intro
thf(fact_451_complete__measure_Ointro,axiom,
! [M2: sigma_measure_set_a] :
( ! [A6: set_set_a,B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B6 @ A6 )
=> ( ( member_set_set_a @ A6 @ ( measur1516554128032400785_set_a @ M2 ) )
=> ( member_set_set_a @ B6 @ ( sigma_sets_set_a @ M2 ) ) ) )
=> ( comple6693822263253554161_set_a @ M2 ) ) ).
% complete_measure.intro
thf(fact_452_complete__measure_Ointro,axiom,
! [M2: sigma_measure_a] :
( ! [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ A6 )
=> ( ( member_set_a @ A6 @ ( measure_null_sets_a @ M2 ) )
=> ( member_set_a @ B6 @ ( sigma_sets_a @ M2 ) ) ) )
=> ( comple8155536527497655953sure_a @ M2 ) ) ).
% complete_measure.intro
thf(fact_453_complete__measure_Ocomplete,axiom,
! [M2: sigma_measure_b,B: set_b,A2: set_b] :
( ( comple8155536527497655954sure_b @ M2 )
=> ( ( ord_less_eq_set_b @ B @ A2 )
=> ( ( member_set_b @ A2 @ ( measure_null_sets_b @ M2 ) )
=> ( member_set_b @ B @ ( sigma_sets_b @ M2 ) ) ) ) ) ).
% complete_measure.complete
thf(fact_454_complete__measure_Ocomplete,axiom,
! [M2: sigma_measure_set_b,B: set_set_b,A2: set_set_b] :
( ( comple6693822267556782962_set_b @ M2 )
=> ( ( ord_le3795704787696855135_set_b @ B @ A2 )
=> ( ( member_set_set_b @ A2 @ ( measur1516554132335629586_set_b @ M2 ) )
=> ( member_set_set_b @ B @ ( sigma_sets_set_b @ M2 ) ) ) ) ) ).
% complete_measure.complete
thf(fact_455_complete__measure_Ocomplete,axiom,
! [M2: sigma_measure_set_a,B: set_set_a,A2: set_set_a] :
( ( comple6693822263253554161_set_a @ M2 )
=> ( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( ( member_set_set_a @ A2 @ ( measur1516554128032400785_set_a @ M2 ) )
=> ( member_set_set_a @ B @ ( sigma_sets_set_a @ M2 ) ) ) ) ) ).
% complete_measure.complete
thf(fact_456_complete__measure_Ocomplete,axiom,
! [M2: sigma_measure_a,B: set_a,A2: set_a] :
( ( comple8155536527497655953sure_a @ M2 )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( member_set_a @ A2 @ ( measure_null_sets_a @ M2 ) )
=> ( member_set_a @ B @ ( sigma_sets_a @ M2 ) ) ) ) ) ).
% complete_measure.complete
thf(fact_457_complete__measure__def,axiom,
( comple8155536527497655954sure_b
= ( ^ [M3: sigma_measure_b] :
! [A5: set_b,B5: set_b] :
( ( ord_less_eq_set_b @ B5 @ A5 )
=> ( ( member_set_b @ A5 @ ( measure_null_sets_b @ M3 ) )
=> ( member_set_b @ B5 @ ( sigma_sets_b @ M3 ) ) ) ) ) ) ).
% complete_measure_def
thf(fact_458_complete__measure__def,axiom,
( comple6693822267556782962_set_b
= ( ^ [M3: sigma_measure_set_b] :
! [A5: set_set_b,B5: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B5 @ A5 )
=> ( ( member_set_set_b @ A5 @ ( measur1516554132335629586_set_b @ M3 ) )
=> ( member_set_set_b @ B5 @ ( sigma_sets_set_b @ M3 ) ) ) ) ) ) ).
% complete_measure_def
thf(fact_459_complete__measure__def,axiom,
( comple6693822263253554161_set_a
= ( ^ [M3: sigma_measure_set_a] :
! [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ A5 )
=> ( ( member_set_set_a @ A5 @ ( measur1516554128032400785_set_a @ M3 ) )
=> ( member_set_set_a @ B5 @ ( sigma_sets_set_a @ M3 ) ) ) ) ) ) ).
% complete_measure_def
thf(fact_460_complete__measure__def,axiom,
( comple8155536527497655953sure_a
= ( ^ [M3: sigma_measure_a] :
! [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A5 )
=> ( ( member_set_a @ A5 @ ( measure_null_sets_a @ M3 ) )
=> ( member_set_a @ B5 @ ( sigma_sets_a @ M3 ) ) ) ) ) ) ).
% complete_measure_def
thf(fact_461_null__part__sets_I1_J,axiom,
! [S: set_b,M2: sigma_measure_b] :
( ( member_set_b @ S @ ( sigma_sets_b @ M2 ) )
=> ( member_set_b @ ( complete_null_part_b @ M2 @ S ) @ ( sigma_sets_b @ M2 ) ) ) ).
% null_part_sets(1)
thf(fact_462_null__part__sets_I1_J,axiom,
! [S: set_a,M2: sigma_measure_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ M2 ) )
=> ( member_set_a @ ( complete_null_part_a @ M2 @ S ) @ ( sigma_sets_a @ M2 ) ) ) ).
% null_part_sets(1)
thf(fact_463_measure__completion,axiom,
! [S: set_b,M2: sigma_measure_b] :
( ( member_set_b @ S @ ( sigma_sets_b @ M2 ) )
=> ( ( sigma_measure_b2 @ ( comple3428971583294703881tion_b @ M2 ) @ S )
= ( sigma_measure_b2 @ M2 @ S ) ) ) ).
% measure_completion
thf(fact_464_measure__completion,axiom,
! [S: set_a,M2: sigma_measure_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ M2 ) )
=> ( ( sigma_measure_a2 @ ( comple3428971583294703880tion_a @ M2 ) @ S )
= ( sigma_measure_a2 @ M2 @ S ) ) ) ).
% measure_completion
thf(fact_465_completion_Ocomplete__measure__axioms,axiom,
! [M2: sigma_measure_a] : ( comple8155536527497655953sure_a @ ( comple3428971583294703880tion_a @ M2 ) ) ).
% completion.complete_measure_axioms
thf(fact_466_finite__measure_Omeasure__increasing,axiom,
! [M2: sigma_measure_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( measur1776380161843274167a_real @ ( sigma_sets_a @ M2 ) @ ( sigma_measure_a2 @ M2 ) ) ) ).
% finite_measure.measure_increasing
thf(fact_467_finite__measure_Omeasure__increasing,axiom,
! [M2: sigma_measure_b] :
( ( measur930452917991658467sure_b @ M2 )
=> ( measur7372598777031404856b_real @ ( sigma_sets_b @ M2 ) @ ( sigma_measure_b2 @ M2 ) ) ) ).
% finite_measure.measure_increasing
thf(fact_468_complete__measure_Ocomplete2,axiom,
! [M2: sigma_measure_set_b,A2: set_set_b,B: set_set_b] :
( ( comple6693822267556782962_set_b @ M2 )
=> ( ( ord_le3795704787696855135_set_b @ A2 @ B )
=> ( ( member_set_set_b @ B @ ( measur1516554132335629586_set_b @ M2 ) )
=> ( member_set_set_b @ A2 @ ( measur1516554132335629586_set_b @ M2 ) ) ) ) ) ).
% complete_measure.complete2
thf(fact_469_complete__measure_Ocomplete2,axiom,
! [M2: sigma_measure_set_a,A2: set_set_a,B: set_set_a] :
( ( comple6693822263253554161_set_a @ M2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_set_a @ B @ ( measur1516554128032400785_set_a @ M2 ) )
=> ( member_set_set_a @ A2 @ ( measur1516554128032400785_set_a @ M2 ) ) ) ) ) ).
% complete_measure.complete2
thf(fact_470_complete__measure_Ocomplete2,axiom,
! [M2: sigma_measure_a,A2: set_a,B: set_a] :
( ( comple8155536527497655953sure_a @ M2 )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_set_a @ B @ ( measure_null_sets_a @ M2 ) )
=> ( member_set_a @ A2 @ ( measure_null_sets_a @ M2 ) ) ) ) ) ).
% complete_measure.complete2
thf(fact_471_finite__measure_Ofinite__measure__mono,axiom,
! [M2: sigma_measure_set_b,A2: set_set_b,B: set_set_b] :
( ( measur2212693997687831747_set_b @ M2 )
=> ( ( ord_le3795704787696855135_set_b @ A2 @ B )
=> ( ( member_set_set_b @ B @ ( sigma_sets_set_b @ M2 ) )
=> ( ord_less_eq_real @ ( sigma_measure_set_b2 @ M2 @ A2 ) @ ( sigma_measure_set_b2 @ M2 @ B ) ) ) ) ) ).
% finite_measure.finite_measure_mono
thf(fact_472_finite__measure_Ofinite__measure__mono,axiom,
! [M2: sigma_measure_set_a,A2: set_set_a,B: set_set_a] :
( ( measur2212693993384602946_set_a @ M2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_set_a @ B @ ( sigma_sets_set_a @ M2 ) )
=> ( ord_less_eq_real @ ( sigma_measure_set_a2 @ M2 @ A2 ) @ ( sigma_measure_set_a2 @ M2 @ B ) ) ) ) ) ).
% finite_measure.finite_measure_mono
thf(fact_473_finite__measure_Ofinite__measure__mono,axiom,
! [M2: sigma_measure_a,A2: set_a,B: set_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M2 ) )
=> ( ord_less_eq_real @ ( sigma_measure_a2 @ M2 @ A2 ) @ ( sigma_measure_a2 @ M2 @ B ) ) ) ) ) ).
% finite_measure.finite_measure_mono
thf(fact_474_finite__measure_Ofinite__measure__mono,axiom,
! [M2: sigma_measure_b,A2: set_b,B: set_b] :
( ( measur930452917991658467sure_b @ M2 )
=> ( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( member_set_b @ B @ ( sigma_sets_b @ M2 ) )
=> ( ord_less_eq_real @ ( sigma_measure_b2 @ M2 @ A2 ) @ ( sigma_measure_b2 @ M2 @ B ) ) ) ) ) ).
% finite_measure.finite_measure_mono
thf(fact_475_measure__ge__1__iff,axiom,
! [A2: set_a] :
( ( ord_less_eq_real @ one_one_real @ ( sigma_measure_a2 @ m @ A2 ) )
= ( ( sigma_measure_a2 @ m @ A2 )
= one_one_real ) ) ).
% measure_ge_1_iff
thf(fact_476_bounded__measure,axiom,
! [A2: set_a] : ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ A2 ) @ ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) ) ) ).
% bounded_measure
thf(fact_477_fmeasurable__eq__sets,axiom,
( ( measur3645360004775918570able_a @ m )
= ( sigma_sets_a @ m ) ) ).
% fmeasurable_eq_sets
thf(fact_478_main__part__null__part__Un,axiom,
! [S: set_b,M2: sigma_measure_b] :
( ( member_set_b @ S @ ( sigma_sets_b @ ( comple3428971583294703881tion_b @ M2 ) ) )
=> ( ( sup_sup_set_b @ ( complete_main_part_b @ M2 @ S ) @ ( complete_null_part_b @ M2 @ S ) )
= S ) ) ).
% main_part_null_part_Un
thf(fact_479_main__part__null__part__Un,axiom,
! [S: set_a,M2: sigma_measure_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) )
=> ( ( sup_sup_set_a @ ( complete_main_part_a @ M2 @ S ) @ ( complete_null_part_a @ M2 @ S ) )
= S ) ) ).
% main_part_null_part_Un
thf(fact_480_subprob__measure__le__1,axiom,
! [X: set_a] : ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ X ) @ one_one_real ) ).
% subprob_measure_le_1
thf(fact_481_prob__le__1,axiom,
! [A2: set_a] : ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ A2 ) @ one_one_real ) ).
% prob_le_1
thf(fact_482_indep__setI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ! [A3: set_a,B3: set_a] :
( ( member_set_a @ A3 @ A2 )
=> ( ( member_set_a @ B3 @ B )
=> ( ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ A3 @ B3 ) )
= ( times_times_real @ ( sigma_measure_a2 @ m @ A3 ) @ ( sigma_measure_a2 @ m @ B3 ) ) ) ) )
=> ( indepe2041756565122539606_set_a @ m @ A2 @ B ) ) ) ) ).
% indep_setI
thf(fact_483_indep__sets2__eq,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( indepe2041756565122539606_set_a @ m @ A2 @ B )
= ( ( ord_le3724670747650509150_set_a @ A2 @ ( sigma_sets_a @ m ) )
& ( ord_le3724670747650509150_set_a @ B @ ( sigma_sets_a @ m ) )
& ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ! [Y3: set_a] :
( ( member_set_a @ Y3 @ B )
=> ( ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ X3 @ Y3 ) )
= ( times_times_real @ ( sigma_measure_a2 @ m @ X3 ) @ ( sigma_measure_a2 @ m @ Y3 ) ) ) ) ) ) ) ).
% indep_sets2_eq
thf(fact_484_sets__completionI,axiom,
! [A2: set_b,S: set_b,N: set_b,N3: set_b,M2: sigma_measure_b] :
( ( A2
= ( sup_sup_set_b @ S @ N ) )
=> ( ( ord_less_eq_set_b @ N @ N3 )
=> ( ( member_set_b @ N3 @ ( measure_null_sets_b @ M2 ) )
=> ( ( member_set_b @ S @ ( sigma_sets_b @ M2 ) )
=> ( member_set_b @ A2 @ ( sigma_sets_b @ ( comple3428971583294703881tion_b @ M2 ) ) ) ) ) ) ) ).
% sets_completionI
thf(fact_485_sets__completionI,axiom,
! [A2: set_set_b,S: set_set_b,N: set_set_b,N3: set_set_b,M2: sigma_measure_set_b] :
( ( A2
= ( sup_sup_set_set_b @ S @ N ) )
=> ( ( ord_le3795704787696855135_set_b @ N @ N3 )
=> ( ( member_set_set_b @ N3 @ ( measur1516554132335629586_set_b @ M2 ) )
=> ( ( member_set_set_b @ S @ ( sigma_sets_set_b @ M2 ) )
=> ( member_set_set_b @ A2 @ ( sigma_sets_set_b @ ( comple8942076150311361001_set_b @ M2 ) ) ) ) ) ) ) ).
% sets_completionI
thf(fact_486_sets__completionI,axiom,
! [A2: set_set_a,S: set_set_a,N: set_set_a,N3: set_set_a,M2: sigma_measure_set_a] :
( ( A2
= ( sup_sup_set_set_a @ S @ N ) )
=> ( ( ord_le3724670747650509150_set_a @ N @ N3 )
=> ( ( member_set_set_a @ N3 @ ( measur1516554128032400785_set_a @ M2 ) )
=> ( ( member_set_set_a @ S @ ( sigma_sets_set_a @ M2 ) )
=> ( member_set_set_a @ A2 @ ( sigma_sets_set_a @ ( comple8942076146008132200_set_a @ M2 ) ) ) ) ) ) ) ).
% sets_completionI
thf(fact_487_sets__completionI,axiom,
! [A2: set_a,S: set_a,N: set_a,N3: set_a,M2: sigma_measure_a] :
( ( A2
= ( sup_sup_set_a @ S @ N ) )
=> ( ( ord_less_eq_set_a @ N @ N3 )
=> ( ( member_set_a @ N3 @ ( measure_null_sets_a @ M2 ) )
=> ( ( member_set_a @ S @ ( sigma_sets_a @ M2 ) )
=> ( member_set_a @ A2 @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) ) ) ) ) ) ).
% sets_completionI
thf(fact_488_IntI,axiom,
! [C2: a > b,A2: set_a_b,B: set_a_b] :
( ( member_a_b @ C2 @ A2 )
=> ( ( member_a_b @ C2 @ B )
=> ( member_a_b @ C2 @ ( inf_inf_set_a_b @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_489_IntI,axiom,
! [C2: c,A2: set_c,B: set_c] :
( ( member_c @ C2 @ A2 )
=> ( ( member_c @ C2 @ B )
=> ( member_c @ C2 @ ( inf_inf_set_c @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_490_IntI,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ A2 )
=> ( ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_491_IntI,axiom,
! [C2: a > a,A2: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ A2 )
=> ( ( member_a_a @ C2 @ B )
=> ( member_a_a @ C2 @ ( inf_inf_set_a_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_492_IntI,axiom,
! [C2: $o,A2: set_o,B: set_o] :
( ( member_o @ C2 @ A2 )
=> ( ( member_o @ C2 @ B )
=> ( member_o @ C2 @ ( inf_inf_set_o @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_493_IntI,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A2 )
=> ( ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_494_IntI,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ A2 )
=> ( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_495_Int__iff,axiom,
! [C2: a > b,A2: set_a_b,B: set_a_b] :
( ( member_a_b @ C2 @ ( inf_inf_set_a_b @ A2 @ B ) )
= ( ( member_a_b @ C2 @ A2 )
& ( member_a_b @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_496_Int__iff,axiom,
! [C2: c,A2: set_c,B: set_c] :
( ( member_c @ C2 @ ( inf_inf_set_c @ A2 @ B ) )
= ( ( member_c @ C2 @ A2 )
& ( member_c @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_497_Int__iff,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) )
= ( ( member_set_a @ C2 @ A2 )
& ( member_set_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_498_Int__iff,axiom,
! [C2: a > a,A2: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( inf_inf_set_a_a @ A2 @ B ) )
= ( ( member_a_a @ C2 @ A2 )
& ( member_a_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_499_Int__iff,axiom,
! [C2: $o,A2: set_o,B: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A2 @ B ) )
= ( ( member_o @ C2 @ A2 )
& ( member_o @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_500_Int__iff,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) )
= ( ( member_nat @ C2 @ A2 )
& ( member_nat @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_501_Int__iff,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( member_a @ C2 @ A2 )
& ( member_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_502_prob__space,axiom,
( ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) )
= one_one_real ) ).
% prob_space
thf(fact_503_UnCI,axiom,
! [C2: a > b,B: set_a_b,A2: set_a_b] :
( ( ~ ( member_a_b @ C2 @ B )
=> ( member_a_b @ C2 @ A2 ) )
=> ( member_a_b @ C2 @ ( sup_sup_set_a_b @ A2 @ B ) ) ) ).
% UnCI
thf(fact_504_UnCI,axiom,
! [C2: c,B: set_c,A2: set_c] :
( ( ~ ( member_c @ C2 @ B )
=> ( member_c @ C2 @ A2 ) )
=> ( member_c @ C2 @ ( sup_sup_set_c @ A2 @ B ) ) ) ).
% UnCI
thf(fact_505_UnCI,axiom,
! [C2: set_a,B: set_set_a,A2: set_set_a] :
( ( ~ ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ A2 ) )
=> ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B ) ) ) ).
% UnCI
thf(fact_506_UnCI,axiom,
! [C2: a > a,B: set_a_a,A2: set_a_a] :
( ( ~ ( member_a_a @ C2 @ B )
=> ( member_a_a @ C2 @ A2 ) )
=> ( member_a_a @ C2 @ ( sup_sup_set_a_a @ A2 @ B ) ) ) ).
% UnCI
thf(fact_507_UnCI,axiom,
! [C2: $o,B: set_o,A2: set_o] :
( ( ~ ( member_o @ C2 @ B )
=> ( member_o @ C2 @ A2 ) )
=> ( member_o @ C2 @ ( sup_sup_set_o @ A2 @ B ) ) ) ).
% UnCI
thf(fact_508_UnCI,axiom,
! [C2: nat,B: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ A2 ) )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_509_UnCI,axiom,
! [C2: a,B: set_a,A2: set_a] :
( ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ A2 ) )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnCI
thf(fact_510_Un__iff,axiom,
! [C2: a > b,A2: set_a_b,B: set_a_b] :
( ( member_a_b @ C2 @ ( sup_sup_set_a_b @ A2 @ B ) )
= ( ( member_a_b @ C2 @ A2 )
| ( member_a_b @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_511_Un__iff,axiom,
! [C2: c,A2: set_c,B: set_c] :
( ( member_c @ C2 @ ( sup_sup_set_c @ A2 @ B ) )
= ( ( member_c @ C2 @ A2 )
| ( member_c @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_512_Un__iff,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B ) )
= ( ( member_set_a @ C2 @ A2 )
| ( member_set_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_513_Un__iff,axiom,
! [C2: a > a,A2: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( sup_sup_set_a_a @ A2 @ B ) )
= ( ( member_a_a @ C2 @ A2 )
| ( member_a_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_514_Un__iff,axiom,
! [C2: $o,A2: set_o,B: set_o] :
( ( member_o @ C2 @ ( sup_sup_set_o @ A2 @ B ) )
= ( ( member_o @ C2 @ A2 )
| ( member_o @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_515_Un__iff,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) )
= ( ( member_nat @ C2 @ A2 )
| ( member_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_516_Un__iff,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) )
= ( ( member_a @ C2 @ A2 )
| ( member_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_517_measure__space__inter,axiom,
! [S2: set_a,T3: set_a] :
( ( member_set_a @ S2 @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ T3 @ ( sigma_sets_a @ m ) )
=> ( ( ( sigma_measure_a2 @ m @ T3 )
= ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) ) )
=> ( ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ S2 @ T3 ) )
= ( sigma_measure_a2 @ m @ S2 ) ) ) ) ) ).
% measure_space_inter
thf(fact_518_indep__setD,axiom,
! [A2: set_set_a,B: set_set_a,A: set_a,B2: set_a] :
( ( indepe2041756565122539606_set_a @ m @ A2 @ B )
=> ( ( member_set_a @ A @ A2 )
=> ( ( member_set_a @ B2 @ B )
=> ( ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ A @ B2 ) )
= ( times_times_real @ ( sigma_measure_a2 @ m @ A ) @ ( sigma_measure_a2 @ m @ B2 ) ) ) ) ) ) ).
% indep_setD
thf(fact_519_Int__subset__iff,axiom,
! [C: set_set_b,A2: set_set_b,B: set_set_b] :
( ( ord_le3795704787696855135_set_b @ C @ ( inf_inf_set_set_b @ A2 @ B ) )
= ( ( ord_le3795704787696855135_set_b @ C @ A2 )
& ( ord_le3795704787696855135_set_b @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_520_Int__subset__iff,axiom,
! [C: set_set_a,A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B ) )
= ( ( ord_le3724670747650509150_set_a @ C @ A2 )
& ( ord_le3724670747650509150_set_a @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_521_Int__subset__iff,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( ord_less_eq_set_a @ C @ A2 )
& ( ord_less_eq_set_a @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_522_sets_OInt,axiom,
! [A: set_b,M2: sigma_measure_b,B2: set_b] :
( ( member_set_b @ A @ ( sigma_sets_b @ M2 ) )
=> ( ( member_set_b @ B2 @ ( sigma_sets_b @ M2 ) )
=> ( member_set_b @ ( inf_inf_set_b @ A @ B2 ) @ ( sigma_sets_b @ M2 ) ) ) ) ).
% sets.Int
thf(fact_523_sets_OInt,axiom,
! [A: set_a,M2: sigma_measure_a,B2: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ M2 ) )
=> ( ( member_set_a @ B2 @ ( sigma_sets_a @ M2 ) )
=> ( member_set_a @ ( inf_inf_set_a @ A @ B2 ) @ ( sigma_sets_a @ M2 ) ) ) ) ).
% sets.Int
thf(fact_524_Un__subset__iff,axiom,
! [A2: set_set_b,B: set_set_b,C: set_set_b] :
( ( ord_le3795704787696855135_set_b @ ( sup_sup_set_set_b @ A2 @ B ) @ C )
= ( ( ord_le3795704787696855135_set_b @ A2 @ C )
& ( ord_le3795704787696855135_set_b @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_525_Un__subset__iff,axiom,
! [A2: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B ) @ C )
= ( ( ord_le3724670747650509150_set_a @ A2 @ C )
& ( ord_le3724670747650509150_set_a @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_526_Un__subset__iff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C )
= ( ( ord_less_eq_set_a @ A2 @ C )
& ( ord_less_eq_set_a @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_527_sets_Otop,axiom,
! [M2: sigma_measure_b] : ( member_set_b @ ( sigma_space_b @ M2 ) @ ( sigma_sets_b @ M2 ) ) ).
% sets.top
thf(fact_528_sets_Otop,axiom,
! [M2: sigma_measure_a] : ( member_set_a @ ( sigma_space_a @ M2 ) @ ( sigma_sets_a @ M2 ) ) ).
% sets.top
thf(fact_529_sets_OUn,axiom,
! [A: set_b,M2: sigma_measure_b,B2: set_b] :
( ( member_set_b @ A @ ( sigma_sets_b @ M2 ) )
=> ( ( member_set_b @ B2 @ ( sigma_sets_b @ M2 ) )
=> ( member_set_b @ ( sup_sup_set_b @ A @ B2 ) @ ( sigma_sets_b @ M2 ) ) ) ) ).
% sets.Un
thf(fact_530_sets_OUn,axiom,
! [A: set_a,M2: sigma_measure_a,B2: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ M2 ) )
=> ( ( member_set_a @ B2 @ ( sigma_sets_a @ M2 ) )
=> ( member_set_a @ ( sup_sup_set_a @ A @ B2 ) @ ( sigma_sets_a @ M2 ) ) ) ) ).
% sets.Un
thf(fact_531_Int__Un__eq_I4_J,axiom,
! [T: set_a,S: set_a] :
( ( sup_sup_set_a @ T @ ( inf_inf_set_a @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_532_Int__Un__eq_I3_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_533_Int__Un__eq_I2_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_534_Int__Un__eq_I1_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_535_Un__Int__eq_I4_J,axiom,
! [T: set_a,S: set_a] :
( ( inf_inf_set_a @ T @ ( sup_sup_set_a @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_536_Un__Int__eq_I3_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_537_Un__Int__eq_I2_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_538_Un__Int__eq_I1_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_539_space__completion,axiom,
! [M2: sigma_measure_a] :
( ( sigma_space_a @ ( comple3428971583294703880tion_a @ M2 ) )
= ( sigma_space_a @ M2 ) ) ).
% space_completion
thf(fact_540_fmeasurable_OInt,axiom,
! [A: set_a,M2: sigma_measure_a,B2: set_a] :
( ( member_set_a @ A @ ( measur3645360004775918570able_a @ M2 ) )
=> ( ( member_set_a @ B2 @ ( measur3645360004775918570able_a @ M2 ) )
=> ( member_set_a @ ( inf_inf_set_a @ A @ B2 ) @ ( measur3645360004775918570able_a @ M2 ) ) ) ) ).
% fmeasurable.Int
thf(fact_541_null__sets_OInt,axiom,
! [A: set_a,M2: sigma_measure_a,B2: set_a] :
( ( member_set_a @ A @ ( measure_null_sets_a @ M2 ) )
=> ( ( member_set_a @ B2 @ ( measure_null_sets_a @ M2 ) )
=> ( member_set_a @ ( inf_inf_set_a @ A @ B2 ) @ ( measure_null_sets_a @ M2 ) ) ) ) ).
% null_sets.Int
thf(fact_542_fmeasurable_OUn,axiom,
! [A: set_a,M2: sigma_measure_a,B2: set_a] :
( ( member_set_a @ A @ ( measur3645360004775918570able_a @ M2 ) )
=> ( ( member_set_a @ B2 @ ( measur3645360004775918570able_a @ M2 ) )
=> ( member_set_a @ ( sup_sup_set_a @ A @ B2 ) @ ( measur3645360004775918570able_a @ M2 ) ) ) ) ).
% fmeasurable.Un
thf(fact_543_null__sets_OUn,axiom,
! [A: set_a,M2: sigma_measure_a,B2: set_a] :
( ( member_set_a @ A @ ( measure_null_sets_a @ M2 ) )
=> ( ( member_set_a @ B2 @ ( measure_null_sets_a @ M2 ) )
=> ( member_set_a @ ( sup_sup_set_a @ A @ B2 ) @ ( measure_null_sets_a @ M2 ) ) ) ) ).
% null_sets.Un
thf(fact_544_space__distr,axiom,
! [M2: sigma_measure_a,N: sigma_measure_b,F: a > b] :
( ( sigma_space_b @ ( measure_distr_a_b @ M2 @ N @ F ) )
= ( sigma_space_b @ N ) ) ).
% space_distr
thf(fact_545_space__distr,axiom,
! [M2: sigma_measure_a,N: sigma_measure_a,F: a > a] :
( ( sigma_space_a @ ( measure_distr_a_a @ M2 @ N @ F ) )
= ( sigma_space_a @ N ) ) ).
% space_distr
thf(fact_546_sets_OInt__space__eq2,axiom,
! [X5: set_b,M2: sigma_measure_b] :
( ( member_set_b @ X5 @ ( sigma_sets_b @ M2 ) )
=> ( ( inf_inf_set_b @ X5 @ ( sigma_space_b @ M2 ) )
= X5 ) ) ).
% sets.Int_space_eq2
thf(fact_547_sets_OInt__space__eq2,axiom,
! [X5: set_a,M2: sigma_measure_a] :
( ( member_set_a @ X5 @ ( sigma_sets_a @ M2 ) )
=> ( ( inf_inf_set_a @ X5 @ ( sigma_space_a @ M2 ) )
= X5 ) ) ).
% sets.Int_space_eq2
thf(fact_548_sets_OInt__space__eq1,axiom,
! [X5: set_b,M2: sigma_measure_b] :
( ( member_set_b @ X5 @ ( sigma_sets_b @ M2 ) )
=> ( ( inf_inf_set_b @ ( sigma_space_b @ M2 ) @ X5 )
= X5 ) ) ).
% sets.Int_space_eq1
thf(fact_549_sets_OInt__space__eq1,axiom,
! [X5: set_a,M2: sigma_measure_a] :
( ( member_set_a @ X5 @ ( sigma_sets_a @ M2 ) )
=> ( ( inf_inf_set_a @ ( sigma_space_a @ M2 ) @ X5 )
= X5 ) ) ).
% sets.Int_space_eq1
thf(fact_550_fmeasurable_OInt__space__eq1,axiom,
! [X5: set_a,M2: sigma_measure_a] :
( ( member_set_a @ X5 @ ( measur3645360004775918570able_a @ M2 ) )
=> ( ( inf_inf_set_a @ ( sigma_space_a @ M2 ) @ X5 )
= X5 ) ) ).
% fmeasurable.Int_space_eq1
thf(fact_551_fmeasurable_OInt__space__eq2,axiom,
! [X5: set_a,M2: sigma_measure_a] :
( ( member_set_a @ X5 @ ( measur3645360004775918570able_a @ M2 ) )
=> ( ( inf_inf_set_a @ X5 @ ( sigma_space_a @ M2 ) )
= X5 ) ) ).
% fmeasurable.Int_space_eq2
thf(fact_552_null__sets_OInt__space__eq2,axiom,
! [X5: set_a,M2: sigma_measure_a] :
( ( member_set_a @ X5 @ ( measure_null_sets_a @ M2 ) )
=> ( ( inf_inf_set_a @ X5 @ ( sigma_space_a @ M2 ) )
= X5 ) ) ).
% null_sets.Int_space_eq2
thf(fact_553_null__sets_OInt__space__eq1,axiom,
! [X5: set_a,M2: sigma_measure_a] :
( ( member_set_a @ X5 @ ( measure_null_sets_a @ M2 ) )
=> ( ( inf_inf_set_a @ ( sigma_space_a @ M2 ) @ X5 )
= X5 ) ) ).
% null_sets.Int_space_eq1
thf(fact_554_fmeasurable_Osets__into__space,axiom,
! [X5: set_set_b,M2: sigma_measure_set_b] :
( ( member_set_set_b @ X5 @ ( measur7460903249514972363_set_b @ M2 ) )
=> ( ord_le3795704787696855135_set_b @ X5 @ ( sigma_space_set_b @ M2 ) ) ) ).
% fmeasurable.sets_into_space
thf(fact_555_fmeasurable_Osets__into__space,axiom,
! [X5: set_set_a,M2: sigma_measure_set_a] :
( ( member_set_set_a @ X5 @ ( measur7460903245211743562_set_a @ M2 ) )
=> ( ord_le3724670747650509150_set_a @ X5 @ ( sigma_space_set_a @ M2 ) ) ) ).
% fmeasurable.sets_into_space
thf(fact_556_fmeasurable_Osets__into__space,axiom,
! [X5: set_a,M2: sigma_measure_a] :
( ( member_set_a @ X5 @ ( measur3645360004775918570able_a @ M2 ) )
=> ( ord_less_eq_set_a @ X5 @ ( sigma_space_a @ M2 ) ) ) ).
% fmeasurable.sets_into_space
thf(fact_557_fmeasurable__Int__fmeasurable,axiom,
! [S: set_b,M2: sigma_measure_b,T: set_b] :
( ( member_set_b @ S @ ( measur3645360004775918571able_b @ M2 ) )
=> ( ( member_set_b @ T @ ( sigma_sets_b @ M2 ) )
=> ( member_set_b @ ( inf_inf_set_b @ S @ T ) @ ( measur3645360004775918571able_b @ M2 ) ) ) ) ).
% fmeasurable_Int_fmeasurable
thf(fact_558_fmeasurable__Int__fmeasurable,axiom,
! [S: set_a,M2: sigma_measure_a,T: set_a] :
( ( member_set_a @ S @ ( measur3645360004775918570able_a @ M2 ) )
=> ( ( member_set_a @ T @ ( sigma_sets_a @ M2 ) )
=> ( member_set_a @ ( inf_inf_set_a @ S @ T ) @ ( measur3645360004775918570able_a @ M2 ) ) ) ) ).
% fmeasurable_Int_fmeasurable
thf(fact_559_Un__Int__assoc__eq,axiom,
! [A2: set_set_b,B: set_set_b,C: set_set_b] :
( ( ( sup_sup_set_set_b @ ( inf_inf_set_set_b @ A2 @ B ) @ C )
= ( inf_inf_set_set_b @ A2 @ ( sup_sup_set_set_b @ B @ C ) ) )
= ( ord_le3795704787696855135_set_b @ C @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_560_Un__Int__assoc__eq,axiom,
! [A2: set_set_a,B: set_set_a,C: set_set_a] :
( ( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_set_a @ A2 @ ( sup_sup_set_set_a @ B @ C ) ) )
= ( ord_le3724670747650509150_set_a @ C @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_561_Un__Int__assoc__eq,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) ) )
= ( ord_less_eq_set_a @ C @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_562_UnE,axiom,
! [C2: a > b,A2: set_a_b,B: set_a_b] :
( ( member_a_b @ C2 @ ( sup_sup_set_a_b @ A2 @ B ) )
=> ( ~ ( member_a_b @ C2 @ A2 )
=> ( member_a_b @ C2 @ B ) ) ) ).
% UnE
thf(fact_563_UnE,axiom,
! [C2: c,A2: set_c,B: set_c] :
( ( member_c @ C2 @ ( sup_sup_set_c @ A2 @ B ) )
=> ( ~ ( member_c @ C2 @ A2 )
=> ( member_c @ C2 @ B ) ) ) ).
% UnE
thf(fact_564_UnE,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B ) )
=> ( ~ ( member_set_a @ C2 @ A2 )
=> ( member_set_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_565_UnE,axiom,
! [C2: a > a,A2: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( sup_sup_set_a_a @ A2 @ B ) )
=> ( ~ ( member_a_a @ C2 @ A2 )
=> ( member_a_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_566_UnE,axiom,
! [C2: $o,A2: set_o,B: set_o] :
( ( member_o @ C2 @ ( sup_sup_set_o @ A2 @ B ) )
=> ( ~ ( member_o @ C2 @ A2 )
=> ( member_o @ C2 @ B ) ) ) ).
% UnE
thf(fact_567_UnE,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) )
=> ( ~ ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_568_UnE,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) )
=> ( ~ ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_569_IntE,axiom,
! [C2: a > b,A2: set_a_b,B: set_a_b] :
( ( member_a_b @ C2 @ ( inf_inf_set_a_b @ A2 @ B ) )
=> ~ ( ( member_a_b @ C2 @ A2 )
=> ~ ( member_a_b @ C2 @ B ) ) ) ).
% IntE
thf(fact_570_IntE,axiom,
! [C2: c,A2: set_c,B: set_c] :
( ( member_c @ C2 @ ( inf_inf_set_c @ A2 @ B ) )
=> ~ ( ( member_c @ C2 @ A2 )
=> ~ ( member_c @ C2 @ B ) ) ) ).
% IntE
thf(fact_571_IntE,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ~ ( ( member_set_a @ C2 @ A2 )
=> ~ ( member_set_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_572_IntE,axiom,
! [C2: a > a,A2: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( inf_inf_set_a_a @ A2 @ B ) )
=> ~ ( ( member_a_a @ C2 @ A2 )
=> ~ ( member_a_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_573_IntE,axiom,
! [C2: $o,A2: set_o,B: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A2 @ B ) )
=> ~ ( ( member_o @ C2 @ A2 )
=> ~ ( member_o @ C2 @ B ) ) ) ).
% IntE
thf(fact_574_IntE,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) )
=> ~ ( ( member_nat @ C2 @ A2 )
=> ~ ( member_nat @ C2 @ B ) ) ) ).
% IntE
thf(fact_575_IntE,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
=> ~ ( ( member_a @ C2 @ A2 )
=> ~ ( member_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_576_UnI1,axiom,
! [C2: a > b,A2: set_a_b,B: set_a_b] :
( ( member_a_b @ C2 @ A2 )
=> ( member_a_b @ C2 @ ( sup_sup_set_a_b @ A2 @ B ) ) ) ).
% UnI1
thf(fact_577_UnI1,axiom,
! [C2: c,A2: set_c,B: set_c] :
( ( member_c @ C2 @ A2 )
=> ( member_c @ C2 @ ( sup_sup_set_c @ A2 @ B ) ) ) ).
% UnI1
thf(fact_578_UnI1,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ A2 )
=> ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B ) ) ) ).
% UnI1
thf(fact_579_UnI1,axiom,
! [C2: a > a,A2: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ A2 )
=> ( member_a_a @ C2 @ ( sup_sup_set_a_a @ A2 @ B ) ) ) ).
% UnI1
thf(fact_580_UnI1,axiom,
! [C2: $o,A2: set_o,B: set_o] :
( ( member_o @ C2 @ A2 )
=> ( member_o @ C2 @ ( sup_sup_set_o @ A2 @ B ) ) ) ).
% UnI1
thf(fact_581_UnI1,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_582_UnI1,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnI1
thf(fact_583_UnI2,axiom,
! [C2: a > b,B: set_a_b,A2: set_a_b] :
( ( member_a_b @ C2 @ B )
=> ( member_a_b @ C2 @ ( sup_sup_set_a_b @ A2 @ B ) ) ) ).
% UnI2
thf(fact_584_UnI2,axiom,
! [C2: c,B: set_c,A2: set_c] :
( ( member_c @ C2 @ B )
=> ( member_c @ C2 @ ( sup_sup_set_c @ A2 @ B ) ) ) ).
% UnI2
thf(fact_585_UnI2,axiom,
! [C2: set_a,B: set_set_a,A2: set_set_a] :
( ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B ) ) ) ).
% UnI2
thf(fact_586_UnI2,axiom,
! [C2: a > a,B: set_a_a,A2: set_a_a] :
( ( member_a_a @ C2 @ B )
=> ( member_a_a @ C2 @ ( sup_sup_set_a_a @ A2 @ B ) ) ) ).
% UnI2
thf(fact_587_UnI2,axiom,
! [C2: $o,B: set_o,A2: set_o] :
( ( member_o @ C2 @ B )
=> ( member_o @ C2 @ ( sup_sup_set_o @ A2 @ B ) ) ) ).
% UnI2
thf(fact_588_UnI2,axiom,
! [C2: nat,B: set_nat,A2: set_nat] :
( ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_589_UnI2,axiom,
! [C2: a,B: set_a,A2: set_a] :
( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnI2
thf(fact_590_IntD1,axiom,
! [C2: a > b,A2: set_a_b,B: set_a_b] :
( ( member_a_b @ C2 @ ( inf_inf_set_a_b @ A2 @ B ) )
=> ( member_a_b @ C2 @ A2 ) ) ).
% IntD1
thf(fact_591_IntD1,axiom,
! [C2: c,A2: set_c,B: set_c] :
( ( member_c @ C2 @ ( inf_inf_set_c @ A2 @ B ) )
=> ( member_c @ C2 @ A2 ) ) ).
% IntD1
thf(fact_592_IntD1,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ( member_set_a @ C2 @ A2 ) ) ).
% IntD1
thf(fact_593_IntD1,axiom,
! [C2: a > a,A2: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( inf_inf_set_a_a @ A2 @ B ) )
=> ( member_a_a @ C2 @ A2 ) ) ).
% IntD1
thf(fact_594_IntD1,axiom,
! [C2: $o,A2: set_o,B: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A2 @ B ) )
=> ( member_o @ C2 @ A2 ) ) ).
% IntD1
thf(fact_595_IntD1,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) )
=> ( member_nat @ C2 @ A2 ) ) ).
% IntD1
thf(fact_596_IntD1,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C2 @ A2 ) ) ).
% IntD1
thf(fact_597_IntD2,axiom,
! [C2: a > b,A2: set_a_b,B: set_a_b] :
( ( member_a_b @ C2 @ ( inf_inf_set_a_b @ A2 @ B ) )
=> ( member_a_b @ C2 @ B ) ) ).
% IntD2
thf(fact_598_IntD2,axiom,
! [C2: c,A2: set_c,B: set_c] :
( ( member_c @ C2 @ ( inf_inf_set_c @ A2 @ B ) )
=> ( member_c @ C2 @ B ) ) ).
% IntD2
thf(fact_599_IntD2,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ( member_set_a @ C2 @ B ) ) ).
% IntD2
thf(fact_600_IntD2,axiom,
! [C2: a > a,A2: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( inf_inf_set_a_a @ A2 @ B ) )
=> ( member_a_a @ C2 @ B ) ) ).
% IntD2
thf(fact_601_IntD2,axiom,
! [C2: $o,A2: set_o,B: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A2 @ B ) )
=> ( member_o @ C2 @ B ) ) ).
% IntD2
thf(fact_602_IntD2,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) )
=> ( member_nat @ C2 @ B ) ) ).
% IntD2
thf(fact_603_IntD2,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C2 @ B ) ) ).
% IntD2
thf(fact_604_bex__Un,axiom,
! [A2: set_a,B: set_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A2 @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ X3 ) )
| ? [X3: a] :
( ( member_a @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_605_ball__Un,axiom,
! [A2: set_a,B: set_a,P: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A2 @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( P @ X3 ) )
& ! [X3: a] :
( ( member_a @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_606_Un__assoc,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Un_assoc
thf(fact_607_Int__assoc,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) ) ) ).
% Int_assoc
thf(fact_608_Un__absorb,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_609_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_610_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A5: set_a,B5: set_a] : ( sup_sup_set_a @ B5 @ A5 ) ) ) ).
% Un_commute
thf(fact_611_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A5: set_a,B5: set_a] : ( inf_inf_set_a @ B5 @ A5 ) ) ) ).
% Int_commute
thf(fact_612_Un__Int__crazy,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ B @ C ) ) @ ( inf_inf_set_a @ C @ A2 ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ B @ C ) ) @ ( sup_sup_set_a @ C @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_613_Int__Un__distrib,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ A2 @ C ) ) ) ).
% Int_Un_distrib
thf(fact_614_Un__Int__distrib,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ A2 @ C ) ) ) ).
% Un_Int_distrib
thf(fact_615_Un__left__absorb,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_616_Int__Un__distrib2,axiom,
! [B: set_a,C: set_a,A2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B @ C ) @ A2 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B @ A2 ) @ ( inf_inf_set_a @ C @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_617_Int__left__absorb,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ).
% Int_left_absorb
thf(fact_618_Un__Int__distrib2,axiom,
! [B: set_a,C: set_a,A2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B @ C ) @ A2 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B @ A2 ) @ ( sup_sup_set_a @ C @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_619_Un__left__commute,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) )
= ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A2 @ C ) ) ) ).
% Un_left_commute
thf(fact_620_Int__left__commute,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A2 @ C ) ) ) ).
% Int_left_commute
thf(fact_621_measurable__Un__null__set,axiom,
! [B: set_b,M2: sigma_measure_b,A2: set_b] :
( ( member_set_b @ B @ ( measure_null_sets_b @ M2 ) )
=> ( ( ( member_set_b @ ( sup_sup_set_b @ A2 @ B ) @ ( measur3645360004775918571able_b @ M2 ) )
& ( member_set_b @ A2 @ ( sigma_sets_b @ M2 ) ) )
= ( member_set_b @ A2 @ ( measur3645360004775918571able_b @ M2 ) ) ) ) ).
% measurable_Un_null_set
thf(fact_622_measurable__Un__null__set,axiom,
! [B: set_a,M2: sigma_measure_a,A2: set_a] :
( ( member_set_a @ B @ ( measure_null_sets_a @ M2 ) )
=> ( ( ( member_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( measur3645360004775918570able_a @ M2 ) )
& ( member_set_a @ A2 @ ( sigma_sets_a @ M2 ) ) )
= ( member_set_a @ A2 @ ( measur3645360004775918570able_a @ M2 ) ) ) ) ).
% measurable_Un_null_set
thf(fact_623_prob__space_Oprob__space,axiom,
! [M2: sigma_measure_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( sigma_measure_b2 @ M2 @ ( sigma_space_b @ M2 ) )
= one_one_real ) ) ).
% prob_space.prob_space
thf(fact_624_prob__space_Oprob__space,axiom,
! [M2: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( sigma_measure_a2 @ M2 @ ( sigma_space_a @ M2 ) )
= one_one_real ) ) ).
% prob_space.prob_space
thf(fact_625_fmeasurableD,axiom,
! [A2: set_b,M2: sigma_measure_b] :
( ( member_set_b @ A2 @ ( measur3645360004775918571able_b @ M2 ) )
=> ( member_set_b @ A2 @ ( sigma_sets_b @ M2 ) ) ) ).
% fmeasurableD
thf(fact_626_fmeasurableD,axiom,
! [A2: set_a,M2: sigma_measure_a] :
( ( member_set_a @ A2 @ ( measur3645360004775918570able_a @ M2 ) )
=> ( member_set_a @ A2 @ ( sigma_sets_a @ M2 ) ) ) ).
% fmeasurableD
thf(fact_627_subset__Un__eq,axiom,
( ord_le3795704787696855135_set_b
= ( ^ [A5: set_set_b,B5: set_set_b] :
( ( sup_sup_set_set_b @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_628_subset__Un__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( sup_sup_set_set_a @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_629_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( sup_sup_set_a @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_630_subset__UnE,axiom,
! [C: set_set_b,A2: set_set_b,B: set_set_b] :
( ( ord_le3795704787696855135_set_b @ C @ ( sup_sup_set_set_b @ A2 @ B ) )
=> ~ ! [A7: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A7 @ A2 )
=> ! [B7: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B7 @ B )
=> ( C
!= ( sup_sup_set_set_b @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_631_subset__UnE,axiom,
! [C: set_set_a,A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B ) )
=> ~ ! [A7: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A7 @ A2 )
=> ! [B7: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B7 @ B )
=> ( C
!= ( sup_sup_set_set_a @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_632_subset__UnE,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B ) )
=> ~ ! [A7: set_a] :
( ( ord_less_eq_set_a @ A7 @ A2 )
=> ! [B7: set_a] :
( ( ord_less_eq_set_a @ B7 @ B )
=> ( C
!= ( sup_sup_set_a @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_633_Un__absorb2,axiom,
! [B: set_set_b,A2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B @ A2 )
=> ( ( sup_sup_set_set_b @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_634_Un__absorb2,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( ( sup_sup_set_set_a @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_635_Un__absorb2,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_636_Un__absorb1,axiom,
! [A2: set_set_b,B: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ B )
=> ( ( sup_sup_set_set_b @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_637_Un__absorb1,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( sup_sup_set_set_a @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_638_Un__absorb1,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( sup_sup_set_a @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_639_Un__upper2,axiom,
! [B: set_set_b,A2: set_set_b] : ( ord_le3795704787696855135_set_b @ B @ ( sup_sup_set_set_b @ A2 @ B ) ) ).
% Un_upper2
thf(fact_640_Un__upper2,axiom,
! [B: set_set_a,A2: set_set_a] : ( ord_le3724670747650509150_set_a @ B @ ( sup_sup_set_set_a @ A2 @ B ) ) ).
% Un_upper2
thf(fact_641_Un__upper2,axiom,
! [B: set_a,A2: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_upper2
thf(fact_642_Un__upper1,axiom,
! [A2: set_set_b,B: set_set_b] : ( ord_le3795704787696855135_set_b @ A2 @ ( sup_sup_set_set_b @ A2 @ B ) ) ).
% Un_upper1
thf(fact_643_Un__upper1,axiom,
! [A2: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ ( sup_sup_set_set_a @ A2 @ B ) ) ).
% Un_upper1
thf(fact_644_Un__upper1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_upper1
thf(fact_645_Un__least,axiom,
! [A2: set_set_b,C: set_set_b,B: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ C )
=> ( ( ord_le3795704787696855135_set_b @ B @ C )
=> ( ord_le3795704787696855135_set_b @ ( sup_sup_set_set_b @ A2 @ B ) @ C ) ) ) ).
% Un_least
thf(fact_646_Un__least,axiom,
! [A2: set_set_a,C: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B ) @ C ) ) ) ).
% Un_least
thf(fact_647_Un__least,axiom,
! [A2: set_a,C: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C ) ) ) ).
% Un_least
thf(fact_648_Un__mono,axiom,
! [A2: set_set_b,C: set_set_b,B: set_set_b,D: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ C )
=> ( ( ord_le3795704787696855135_set_b @ B @ D )
=> ( ord_le3795704787696855135_set_b @ ( sup_sup_set_set_b @ A2 @ B ) @ ( sup_sup_set_set_b @ C @ D ) ) ) ) ).
% Un_mono
thf(fact_649_Un__mono,axiom,
! [A2: set_set_a,C: set_set_a,B: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ( ord_le3724670747650509150_set_a @ B @ D )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B ) @ ( sup_sup_set_set_a @ C @ D ) ) ) ) ).
% Un_mono
thf(fact_650_Un__mono,axiom,
! [A2: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ C @ D ) ) ) ) ).
% Un_mono
thf(fact_651_sets__eq__imp__space__eq,axiom,
! [M2: sigma_measure_b,M: sigma_measure_b] :
( ( ( sigma_sets_b @ M2 )
= ( sigma_sets_b @ M ) )
=> ( ( sigma_space_b @ M2 )
= ( sigma_space_b @ M ) ) ) ).
% sets_eq_imp_space_eq
thf(fact_652_sets__eq__imp__space__eq,axiom,
! [M2: sigma_measure_a,M: sigma_measure_a] :
( ( ( sigma_sets_a @ M2 )
= ( sigma_sets_a @ M ) )
=> ( ( sigma_space_a @ M2 )
= ( sigma_space_a @ M ) ) ) ).
% sets_eq_imp_space_eq
thf(fact_653_fmeasurableI__null__sets,axiom,
! [A2: set_a,M2: sigma_measure_a] :
( ( member_set_a @ A2 @ ( measure_null_sets_a @ M2 ) )
=> ( member_set_a @ A2 @ ( measur3645360004775918570able_a @ M2 ) ) ) ).
% fmeasurableI_null_sets
thf(fact_654_Int__Collect__mono,axiom,
! [A2: set_a_b,B: set_a_b,P: ( a > b ) > $o,Q: ( a > b ) > $o] :
( ( ord_less_eq_set_a_b @ A2 @ B )
=> ( ! [X4: a > b] :
( ( member_a_b @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_a_b @ ( inf_inf_set_a_b @ A2 @ ( collect_a_b @ P ) ) @ ( inf_inf_set_a_b @ B @ ( collect_a_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_655_Int__Collect__mono,axiom,
! [A2: set_c,B: set_c,P: c > $o,Q: c > $o] :
( ( ord_less_eq_set_c @ A2 @ B )
=> ( ! [X4: c] :
( ( member_c @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_c @ ( inf_inf_set_c @ A2 @ ( collect_c @ P ) ) @ ( inf_inf_set_c @ B @ ( collect_c @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_656_Int__Collect__mono,axiom,
! [A2: set_a_a,B: set_a_a,P: ( a > a ) > $o,Q: ( a > a ) > $o] :
( ( ord_less_eq_set_a_a @ A2 @ B )
=> ( ! [X4: a > a] :
( ( member_a_a @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_a_a @ ( inf_inf_set_a_a @ A2 @ ( collect_a_a @ P ) ) @ ( inf_inf_set_a_a @ B @ ( collect_a_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_657_Int__Collect__mono,axiom,
! [A2: set_o,B: set_o,P: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A2 @ ( collect_o @ P ) ) @ ( inf_inf_set_o @ B @ ( collect_o @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_658_Int__Collect__mono,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_659_Int__Collect__mono,axiom,
! [A2: set_set_b,B: set_set_b,P: set_b > $o,Q: set_b > $o] :
( ( ord_le3795704787696855135_set_b @ A2 @ B )
=> ( ! [X4: set_b] :
( ( member_set_b @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A2 @ ( collect_set_b @ P ) ) @ ( inf_inf_set_set_b @ B @ ( collect_set_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_660_Int__Collect__mono,axiom,
! [A2: set_set_a,B: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_661_Int__Collect__mono,axiom,
! [A2: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_662_Int__greatest,axiom,
! [C: set_set_b,A2: set_set_b,B: set_set_b] :
( ( ord_le3795704787696855135_set_b @ C @ A2 )
=> ( ( ord_le3795704787696855135_set_b @ C @ B )
=> ( ord_le3795704787696855135_set_b @ C @ ( inf_inf_set_set_b @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_663_Int__greatest,axiom,
! [C: set_set_a,A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ C @ B )
=> ( ord_le3724670747650509150_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_664_Int__greatest,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_665_Int__absorb2,axiom,
! [A2: set_set_b,B: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ B )
=> ( ( inf_inf_set_set_b @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_666_Int__absorb2,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( inf_inf_set_set_a @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_667_Int__absorb2,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( inf_inf_set_a @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_668_Int__absorb1,axiom,
! [B: set_set_b,A2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B @ A2 )
=> ( ( inf_inf_set_set_b @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_669_Int__absorb1,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_670_Int__absorb1,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_671_Int__lower2,axiom,
! [A2: set_set_b,B: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_672_Int__lower2,axiom,
! [A2: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_673_Int__lower2,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_674_Int__lower1,axiom,
! [A2: set_set_b,B: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_675_Int__lower1,axiom,
! [A2: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_676_Int__lower1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_677_Int__mono,axiom,
! [A2: set_set_b,C: set_set_b,B: set_set_b,D: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ C )
=> ( ( ord_le3795704787696855135_set_b @ B @ D )
=> ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A2 @ B ) @ ( inf_inf_set_set_b @ C @ D ) ) ) ) ).
% Int_mono
thf(fact_678_Int__mono,axiom,
! [A2: set_set_a,C: set_set_a,B: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ( ord_le3724670747650509150_set_a @ B @ D )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ ( inf_inf_set_set_a @ C @ D ) ) ) ) ).
% Int_mono
thf(fact_679_Int__mono,axiom,
! [A2: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% Int_mono
thf(fact_680_measurable__cong__simp,axiom,
! [M2: sigma_measure_a,N: sigma_measure_a,M: sigma_measure_b,N3: sigma_measure_b,F: a > b,G2: a > b] :
( ( M2 = N )
=> ( ( M = N3 )
=> ( ! [W: a] :
( ( member_a @ W @ ( sigma_space_a @ M2 ) )
=> ( ( F @ W )
= ( G2 @ W ) ) )
=> ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ M ) )
= ( member_a_b @ G2 @ ( sigma_measurable_a_b @ N @ N3 ) ) ) ) ) ) ).
% measurable_cong_simp
thf(fact_681_measurable__cong__simp,axiom,
! [M2: sigma_measure_a,N: sigma_measure_a,M: sigma_measure_a,N3: sigma_measure_a,F: a > a,G2: a > a] :
( ( M2 = N )
=> ( ( M = N3 )
=> ( ! [W: a] :
( ( member_a @ W @ ( sigma_space_a @ M2 ) )
=> ( ( F @ W )
= ( G2 @ W ) ) )
=> ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ M ) )
= ( member_a_a @ G2 @ ( sigma_measurable_a_a @ N @ N3 ) ) ) ) ) ) ).
% measurable_cong_simp
thf(fact_682_measurable__space,axiom,
! [F: c > c,M2: sigma_measure_c,A2: sigma_measure_c,X5: c] :
( ( member_c_c @ F @ ( sigma_measurable_c_c @ M2 @ A2 ) )
=> ( ( member_c @ X5 @ ( sigma_space_c @ M2 ) )
=> ( member_c @ ( F @ X5 ) @ ( sigma_space_c @ A2 ) ) ) ) ).
% measurable_space
thf(fact_683_measurable__space,axiom,
! [F: c > $o,M2: sigma_measure_c,A2: sigma_measure_o,X5: c] :
( ( member_c_o @ F @ ( sigma_measurable_c_o @ M2 @ A2 ) )
=> ( ( member_c @ X5 @ ( sigma_space_c @ M2 ) )
=> ( member_o @ ( F @ X5 ) @ ( sigma_space_o @ A2 ) ) ) ) ).
% measurable_space
thf(fact_684_measurable__space,axiom,
! [F: c > nat,M2: sigma_measure_c,A2: sigma_measure_nat,X5: c] :
( ( member_c_nat @ F @ ( sigma_2544038740538346112_c_nat @ M2 @ A2 ) )
=> ( ( member_c @ X5 @ ( sigma_space_c @ M2 ) )
=> ( member_nat @ ( F @ X5 ) @ ( sigma_space_nat @ A2 ) ) ) ) ).
% measurable_space
thf(fact_685_measurable__space,axiom,
! [F: $o > c,M2: sigma_measure_o,A2: sigma_measure_c,X5: $o] :
( ( member_o_c @ F @ ( sigma_measurable_o_c @ M2 @ A2 ) )
=> ( ( member_o @ X5 @ ( sigma_space_o @ M2 ) )
=> ( member_c @ ( F @ X5 ) @ ( sigma_space_c @ A2 ) ) ) ) ).
% measurable_space
thf(fact_686_measurable__space,axiom,
! [F: $o > $o,M2: sigma_measure_o,A2: sigma_measure_o,X5: $o] :
( ( member_o_o @ F @ ( sigma_measurable_o_o @ M2 @ A2 ) )
=> ( ( member_o @ X5 @ ( sigma_space_o @ M2 ) )
=> ( member_o @ ( F @ X5 ) @ ( sigma_space_o @ A2 ) ) ) ) ).
% measurable_space
thf(fact_687_measurable__space,axiom,
! [F: $o > nat,M2: sigma_measure_o,A2: sigma_measure_nat,X5: $o] :
( ( member_o_nat @ F @ ( sigma_1999164137574644376_o_nat @ M2 @ A2 ) )
=> ( ( member_o @ X5 @ ( sigma_space_o @ M2 ) )
=> ( member_nat @ ( F @ X5 ) @ ( sigma_space_nat @ A2 ) ) ) ) ).
% measurable_space
thf(fact_688_measurable__space,axiom,
! [F: nat > c,M2: sigma_measure_nat,A2: sigma_measure_c,X5: nat] :
( ( member_nat_c @ F @ ( sigma_4105081583803843550_nat_c @ M2 @ A2 ) )
=> ( ( member_nat @ X5 @ ( sigma_space_nat @ M2 ) )
=> ( member_c @ ( F @ X5 ) @ ( sigma_space_c @ A2 ) ) ) ) ).
% measurable_space
thf(fact_689_measurable__space,axiom,
! [F: nat > $o,M2: sigma_measure_nat,A2: sigma_measure_o,X5: nat] :
( ( member_nat_o @ F @ ( sigma_5101835498682829686_nat_o @ M2 @ A2 ) )
=> ( ( member_nat @ X5 @ ( sigma_space_nat @ M2 ) )
=> ( member_o @ ( F @ X5 ) @ ( sigma_space_o @ A2 ) ) ) ) ).
% measurable_space
thf(fact_690_measurable__space,axiom,
! [F: nat > nat,M2: sigma_measure_nat,A2: sigma_measure_nat,X5: nat] :
( ( member_nat_nat @ F @ ( sigma_4350458207664084850at_nat @ M2 @ A2 ) )
=> ( ( member_nat @ X5 @ ( sigma_space_nat @ M2 ) )
=> ( member_nat @ ( F @ X5 ) @ ( sigma_space_nat @ A2 ) ) ) ) ).
% measurable_space
thf(fact_691_measurable__space,axiom,
! [F: c > a,M2: sigma_measure_c,A2: sigma_measure_a,X5: c] :
( ( member_c_a @ F @ ( sigma_measurable_c_a @ M2 @ A2 ) )
=> ( ( member_c @ X5 @ ( sigma_space_c @ M2 ) )
=> ( member_a @ ( F @ X5 ) @ ( sigma_space_a @ A2 ) ) ) ) ).
% measurable_space
thf(fact_692_measurable__cong,axiom,
! [M2: sigma_measure_a,F: a > b,G2: a > b,M: sigma_measure_b] :
( ! [W: a] :
( ( member_a @ W @ ( sigma_space_a @ M2 ) )
=> ( ( F @ W )
= ( G2 @ W ) ) )
=> ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ M ) )
= ( member_a_b @ G2 @ ( sigma_measurable_a_b @ M2 @ M ) ) ) ) ).
% measurable_cong
thf(fact_693_measurable__cong,axiom,
! [M2: sigma_measure_a,F: a > a,G2: a > a,M: sigma_measure_a] :
( ! [W: a] :
( ( member_a @ W @ ( sigma_space_a @ M2 ) )
=> ( ( F @ W )
= ( G2 @ W ) ) )
=> ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ M ) )
= ( member_a_a @ G2 @ ( sigma_measurable_a_a @ M2 @ M ) ) ) ) ).
% measurable_cong
thf(fact_694_finite__measure_Omeasure__space__inter,axiom,
! [M2: sigma_measure_a,S2: set_a,T3: set_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ( member_set_a @ S2 @ ( sigma_sets_a @ M2 ) )
=> ( ( member_set_a @ T3 @ ( sigma_sets_a @ M2 ) )
=> ( ( ( sigma_measure_a2 @ M2 @ T3 )
= ( sigma_measure_a2 @ M2 @ ( sigma_space_a @ M2 ) ) )
=> ( ( sigma_measure_a2 @ M2 @ ( inf_inf_set_a @ S2 @ T3 ) )
= ( sigma_measure_a2 @ M2 @ S2 ) ) ) ) ) ) ).
% finite_measure.measure_space_inter
thf(fact_695_finite__measure_Omeasure__space__inter,axiom,
! [M2: sigma_measure_b,S2: set_b,T3: set_b] :
( ( measur930452917991658467sure_b @ M2 )
=> ( ( member_set_b @ S2 @ ( sigma_sets_b @ M2 ) )
=> ( ( member_set_b @ T3 @ ( sigma_sets_b @ M2 ) )
=> ( ( ( sigma_measure_b2 @ M2 @ T3 )
= ( sigma_measure_b2 @ M2 @ ( sigma_space_b @ M2 ) ) )
=> ( ( sigma_measure_b2 @ M2 @ ( inf_inf_set_b @ S2 @ T3 ) )
= ( sigma_measure_b2 @ M2 @ S2 ) ) ) ) ) ) ).
% finite_measure.measure_space_inter
thf(fact_696_prob__space_Oindep__setD,axiom,
! [M2: sigma_measure_b,A2: set_set_b,B: set_set_b,A: set_b,B2: set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe2041756565122539607_set_b @ M2 @ A2 @ B )
=> ( ( member_set_b @ A @ A2 )
=> ( ( member_set_b @ B2 @ B )
=> ( ( sigma_measure_b2 @ M2 @ ( inf_inf_set_b @ A @ B2 ) )
= ( times_times_real @ ( sigma_measure_b2 @ M2 @ A ) @ ( sigma_measure_b2 @ M2 @ B2 ) ) ) ) ) ) ) ).
% prob_space.indep_setD
thf(fact_697_prob__space_Oindep__setD,axiom,
! [M2: sigma_measure_a,A2: set_set_a,B: set_set_a,A: set_a,B2: set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe2041756565122539606_set_a @ M2 @ A2 @ B )
=> ( ( member_set_a @ A @ A2 )
=> ( ( member_set_a @ B2 @ B )
=> ( ( sigma_measure_a2 @ M2 @ ( inf_inf_set_a @ A @ B2 ) )
= ( times_times_real @ ( sigma_measure_a2 @ M2 @ A ) @ ( sigma_measure_a2 @ M2 @ B2 ) ) ) ) ) ) ) ).
% prob_space.indep_setD
thf(fact_698_fmeasurableI2,axiom,
! [A2: set_b,M2: sigma_measure_b,B: set_b] :
( ( member_set_b @ A2 @ ( measur3645360004775918571able_b @ M2 ) )
=> ( ( ord_less_eq_set_b @ B @ A2 )
=> ( ( member_set_b @ B @ ( sigma_sets_b @ M2 ) )
=> ( member_set_b @ B @ ( measur3645360004775918571able_b @ M2 ) ) ) ) ) ).
% fmeasurableI2
thf(fact_699_fmeasurableI2,axiom,
! [A2: set_set_b,M2: sigma_measure_set_b,B: set_set_b] :
( ( member_set_set_b @ A2 @ ( measur7460903249514972363_set_b @ M2 ) )
=> ( ( ord_le3795704787696855135_set_b @ B @ A2 )
=> ( ( member_set_set_b @ B @ ( sigma_sets_set_b @ M2 ) )
=> ( member_set_set_b @ B @ ( measur7460903249514972363_set_b @ M2 ) ) ) ) ) ).
% fmeasurableI2
thf(fact_700_fmeasurableI2,axiom,
! [A2: set_set_a,M2: sigma_measure_set_a,B: set_set_a] :
( ( member_set_set_a @ A2 @ ( measur7460903245211743562_set_a @ M2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( ( member_set_set_a @ B @ ( sigma_sets_set_a @ M2 ) )
=> ( member_set_set_a @ B @ ( measur7460903245211743562_set_a @ M2 ) ) ) ) ) ).
% fmeasurableI2
thf(fact_701_fmeasurableI2,axiom,
! [A2: set_a,M2: sigma_measure_a,B: set_a] :
( ( member_set_a @ A2 @ ( measur3645360004775918570able_a @ M2 ) )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M2 ) )
=> ( member_set_a @ B @ ( measur3645360004775918570able_a @ M2 ) ) ) ) ) ).
% fmeasurableI2
thf(fact_702_sets_Osets__into__space,axiom,
! [X5: set_b,M2: sigma_measure_b] :
( ( member_set_b @ X5 @ ( sigma_sets_b @ M2 ) )
=> ( ord_less_eq_set_b @ X5 @ ( sigma_space_b @ M2 ) ) ) ).
% sets.sets_into_space
thf(fact_703_sets_Osets__into__space,axiom,
! [X5: set_set_b,M2: sigma_measure_set_b] :
( ( member_set_set_b @ X5 @ ( sigma_sets_set_b @ M2 ) )
=> ( ord_le3795704787696855135_set_b @ X5 @ ( sigma_space_set_b @ M2 ) ) ) ).
% sets.sets_into_space
thf(fact_704_sets_Osets__into__space,axiom,
! [X5: set_set_a,M2: sigma_measure_set_a] :
( ( member_set_set_a @ X5 @ ( sigma_sets_set_a @ M2 ) )
=> ( ord_le3724670747650509150_set_a @ X5 @ ( sigma_space_set_a @ M2 ) ) ) ).
% sets.sets_into_space
thf(fact_705_sets_Osets__into__space,axiom,
! [X5: set_a,M2: sigma_measure_a] :
( ( member_set_a @ X5 @ ( sigma_sets_a @ M2 ) )
=> ( ord_less_eq_set_a @ X5 @ ( sigma_space_a @ M2 ) ) ) ).
% sets.sets_into_space
thf(fact_706_finite__measure_Ofmeasurable__eq__sets,axiom,
! [M2: sigma_measure_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ( measur3645360004775918570able_a @ M2 )
= ( sigma_sets_a @ M2 ) ) ) ).
% finite_measure.fmeasurable_eq_sets
thf(fact_707_finite__measure_Ofmeasurable__eq__sets,axiom,
! [M2: sigma_measure_b] :
( ( measur930452917991658467sure_b @ M2 )
=> ( ( measur3645360004775918571able_b @ M2 )
= ( sigma_sets_b @ M2 ) ) ) ).
% finite_measure.fmeasurable_eq_sets
thf(fact_708_null__sets_Osets__into__space,axiom,
! [X5: set_set_b,M2: sigma_measure_set_b] :
( ( member_set_set_b @ X5 @ ( measur1516554132335629586_set_b @ M2 ) )
=> ( ord_le3795704787696855135_set_b @ X5 @ ( sigma_space_set_b @ M2 ) ) ) ).
% null_sets.sets_into_space
thf(fact_709_null__sets_Osets__into__space,axiom,
! [X5: set_set_a,M2: sigma_measure_set_a] :
( ( member_set_set_a @ X5 @ ( measur1516554128032400785_set_a @ M2 ) )
=> ( ord_le3724670747650509150_set_a @ X5 @ ( sigma_space_set_a @ M2 ) ) ) ).
% null_sets.sets_into_space
thf(fact_710_null__sets_Osets__into__space,axiom,
! [X5: set_a,M2: sigma_measure_a] :
( ( member_set_a @ X5 @ ( measure_null_sets_a @ M2 ) )
=> ( ord_less_eq_set_a @ X5 @ ( sigma_space_a @ M2 ) ) ) ).
% null_sets.sets_into_space
thf(fact_711_distr__cong,axiom,
! [M2: sigma_measure_c,K: sigma_measure_c,N: sigma_measure_b,L: sigma_measure_b,F: c > b,G2: c > b] :
( ( M2 = K )
=> ( ( ( sigma_sets_b @ N )
= ( sigma_sets_b @ L ) )
=> ( ! [X4: c] :
( ( member_c @ X4 @ ( sigma_space_c @ M2 ) )
=> ( ( F @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( measure_distr_c_b @ M2 @ N @ F )
= ( measure_distr_c_b @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_712_distr__cong,axiom,
! [M2: sigma_measure_o,K: sigma_measure_o,N: sigma_measure_b,L: sigma_measure_b,F: $o > b,G2: $o > b] :
( ( M2 = K )
=> ( ( ( sigma_sets_b @ N )
= ( sigma_sets_b @ L ) )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ ( sigma_space_o @ M2 ) )
=> ( ( F @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( measure_distr_o_b @ M2 @ N @ F )
= ( measure_distr_o_b @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_713_distr__cong,axiom,
! [M2: sigma_measure_nat,K: sigma_measure_nat,N: sigma_measure_b,L: sigma_measure_b,F: nat > b,G2: nat > b] :
( ( M2 = K )
=> ( ( ( sigma_sets_b @ N )
= ( sigma_sets_b @ L ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( sigma_space_nat @ M2 ) )
=> ( ( F @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( measure_distr_nat_b @ M2 @ N @ F )
= ( measure_distr_nat_b @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_714_distr__cong,axiom,
! [M2: sigma_measure_c,K: sigma_measure_c,N: sigma_measure_a,L: sigma_measure_a,F: c > a,G2: c > a] :
( ( M2 = K )
=> ( ( ( sigma_sets_a @ N )
= ( sigma_sets_a @ L ) )
=> ( ! [X4: c] :
( ( member_c @ X4 @ ( sigma_space_c @ M2 ) )
=> ( ( F @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( measure_distr_c_a @ M2 @ N @ F )
= ( measure_distr_c_a @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_715_distr__cong,axiom,
! [M2: sigma_measure_o,K: sigma_measure_o,N: sigma_measure_a,L: sigma_measure_a,F: $o > a,G2: $o > a] :
( ( M2 = K )
=> ( ( ( sigma_sets_a @ N )
= ( sigma_sets_a @ L ) )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ ( sigma_space_o @ M2 ) )
=> ( ( F @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( measure_distr_o_a @ M2 @ N @ F )
= ( measure_distr_o_a @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_716_distr__cong,axiom,
! [M2: sigma_measure_nat,K: sigma_measure_nat,N: sigma_measure_a,L: sigma_measure_a,F: nat > a,G2: nat > a] :
( ( M2 = K )
=> ( ( ( sigma_sets_a @ N )
= ( sigma_sets_a @ L ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( sigma_space_nat @ M2 ) )
=> ( ( F @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( measure_distr_nat_a @ M2 @ N @ F )
= ( measure_distr_nat_a @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_717_distr__cong,axiom,
! [M2: sigma_measure_a,K: sigma_measure_a,N: sigma_measure_b,L: sigma_measure_b,F: a > b,G2: a > b] :
( ( M2 = K )
=> ( ( ( sigma_sets_b @ N )
= ( sigma_sets_b @ L ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( sigma_space_a @ M2 ) )
=> ( ( F @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( measure_distr_a_b @ M2 @ N @ F )
= ( measure_distr_a_b @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_718_distr__cong,axiom,
! [M2: sigma_measure_a,K: sigma_measure_a,N: sigma_measure_a,L: sigma_measure_a,F: a > a,G2: a > a] :
( ( M2 = K )
=> ( ( ( sigma_sets_a @ N )
= ( sigma_sets_a @ L ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( sigma_space_a @ M2 ) )
=> ( ( F @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( measure_distr_a_a @ M2 @ N @ F )
= ( measure_distr_a_a @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_719_distr__cong,axiom,
! [M2: sigma_measure_set_a,K: sigma_measure_set_a,N: sigma_measure_b,L: sigma_measure_b,F: set_a > b,G2: set_a > b] :
( ( M2 = K )
=> ( ( ( sigma_sets_b @ N )
= ( sigma_sets_b @ L ) )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ ( sigma_space_set_a @ M2 ) )
=> ( ( F @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( measur7064479691503150873et_a_b @ M2 @ N @ F )
= ( measur7064479691503150873et_a_b @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_720_distr__cong,axiom,
! [M2: sigma_measure_set_a,K: sigma_measure_set_a,N: sigma_measure_a,L: sigma_measure_a,F: set_a > a,G2: set_a > a] :
( ( M2 = K )
=> ( ( ( sigma_sets_a @ N )
= ( sigma_sets_a @ L ) )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ ( sigma_space_set_a @ M2 ) )
=> ( ( F @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( measur7064479691503150872et_a_a @ M2 @ N @ F )
= ( measur7064479691503150872et_a_a @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_721_null__set__Int1,axiom,
! [B: set_b,M2: sigma_measure_b,A2: set_b] :
( ( member_set_b @ B @ ( measure_null_sets_b @ M2 ) )
=> ( ( member_set_b @ A2 @ ( sigma_sets_b @ M2 ) )
=> ( member_set_b @ ( inf_inf_set_b @ A2 @ B ) @ ( measure_null_sets_b @ M2 ) ) ) ) ).
% null_set_Int1
thf(fact_722_null__set__Int1,axiom,
! [B: set_a,M2: sigma_measure_a,A2: set_a] :
( ( member_set_a @ B @ ( measure_null_sets_a @ M2 ) )
=> ( ( member_set_a @ A2 @ ( sigma_sets_a @ M2 ) )
=> ( member_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( measure_null_sets_a @ M2 ) ) ) ) ).
% null_set_Int1
thf(fact_723_null__set__Int2,axiom,
! [B: set_b,M2: sigma_measure_b,A2: set_b] :
( ( member_set_b @ B @ ( measure_null_sets_b @ M2 ) )
=> ( ( member_set_b @ A2 @ ( sigma_sets_b @ M2 ) )
=> ( member_set_b @ ( inf_inf_set_b @ B @ A2 ) @ ( measure_null_sets_b @ M2 ) ) ) ) ).
% null_set_Int2
thf(fact_724_null__set__Int2,axiom,
! [B: set_a,M2: sigma_measure_a,A2: set_a] :
( ( member_set_a @ B @ ( measure_null_sets_a @ M2 ) )
=> ( ( member_set_a @ A2 @ ( sigma_sets_a @ M2 ) )
=> ( member_set_a @ ( inf_inf_set_a @ B @ A2 ) @ ( measure_null_sets_a @ M2 ) ) ) ) ).
% null_set_Int2
thf(fact_725_le__measureD1,axiom,
! [A2: sigma_measure_set_b,B: sigma_measure_set_b] :
( ( ord_le5713619651007674940_set_b @ A2 @ B )
=> ( ord_le3795704787696855135_set_b @ ( sigma_space_set_b @ A2 ) @ ( sigma_space_set_b @ B ) ) ) ).
% le_measureD1
thf(fact_726_le__measureD1,axiom,
! [A2: sigma_measure_set_a,B: sigma_measure_set_a] :
( ( ord_le5642585610961328955_set_a @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ ( sigma_space_set_a @ A2 ) @ ( sigma_space_set_a @ B ) ) ) ).
% le_measureD1
thf(fact_727_le__measureD1,axiom,
! [A2: sigma_measure_a,B: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A2 @ B )
=> ( ord_less_eq_set_a @ ( sigma_space_a @ A2 ) @ ( sigma_space_a @ B ) ) ) ).
% le_measureD1
thf(fact_728_prob__space_Oindep__sets2__eq,axiom,
! [M2: sigma_measure_b,A2: set_set_b,B: set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe2041756565122539607_set_b @ M2 @ A2 @ B )
= ( ( ord_le3795704787696855135_set_b @ A2 @ ( sigma_sets_b @ M2 ) )
& ( ord_le3795704787696855135_set_b @ B @ ( sigma_sets_b @ M2 ) )
& ! [X3: set_b] :
( ( member_set_b @ X3 @ A2 )
=> ! [Y3: set_b] :
( ( member_set_b @ Y3 @ B )
=> ( ( sigma_measure_b2 @ M2 @ ( inf_inf_set_b @ X3 @ Y3 ) )
= ( times_times_real @ ( sigma_measure_b2 @ M2 @ X3 ) @ ( sigma_measure_b2 @ M2 @ Y3 ) ) ) ) ) ) ) ) ).
% prob_space.indep_sets2_eq
thf(fact_729_prob__space_Oindep__sets2__eq,axiom,
! [M2: sigma_measure_a,A2: set_set_a,B: set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe2041756565122539606_set_a @ M2 @ A2 @ B )
= ( ( ord_le3724670747650509150_set_a @ A2 @ ( sigma_sets_a @ M2 ) )
& ( ord_le3724670747650509150_set_a @ B @ ( sigma_sets_a @ M2 ) )
& ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ! [Y3: set_a] :
( ( member_set_a @ Y3 @ B )
=> ( ( sigma_measure_a2 @ M2 @ ( inf_inf_set_a @ X3 @ Y3 ) )
= ( times_times_real @ ( sigma_measure_a2 @ M2 @ X3 ) @ ( sigma_measure_a2 @ M2 @ Y3 ) ) ) ) ) ) ) ) ).
% prob_space.indep_sets2_eq
thf(fact_730_prob__space_Oindep__setI,axiom,
! [M2: sigma_measure_b,A2: set_set_b,B: set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( ord_le3795704787696855135_set_b @ A2 @ ( sigma_sets_b @ M2 ) )
=> ( ( ord_le3795704787696855135_set_b @ B @ ( sigma_sets_b @ M2 ) )
=> ( ! [A3: set_b,B3: set_b] :
( ( member_set_b @ A3 @ A2 )
=> ( ( member_set_b @ B3 @ B )
=> ( ( sigma_measure_b2 @ M2 @ ( inf_inf_set_b @ A3 @ B3 ) )
= ( times_times_real @ ( sigma_measure_b2 @ M2 @ A3 ) @ ( sigma_measure_b2 @ M2 @ B3 ) ) ) ) )
=> ( indepe2041756565122539607_set_b @ M2 @ A2 @ B ) ) ) ) ) ).
% prob_space.indep_setI
thf(fact_731_prob__space_Oindep__setI,axiom,
! [M2: sigma_measure_a,A2: set_set_a,B: set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ ( sigma_sets_a @ M2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ ( sigma_sets_a @ M2 ) )
=> ( ! [A3: set_a,B3: set_a] :
( ( member_set_a @ A3 @ A2 )
=> ( ( member_set_a @ B3 @ B )
=> ( ( sigma_measure_a2 @ M2 @ ( inf_inf_set_a @ A3 @ B3 ) )
= ( times_times_real @ ( sigma_measure_a2 @ M2 @ A3 ) @ ( sigma_measure_a2 @ M2 @ B3 ) ) ) ) )
=> ( indepe2041756565122539606_set_a @ M2 @ A2 @ B ) ) ) ) ) ).
% prob_space.indep_setI
thf(fact_732_completion_Ocomplete__sets__sandwich__fmeasurable,axiom,
! [A2: set_set_b,M2: sigma_measure_set_b,C: set_set_b,B: set_set_b] :
( ( member_set_set_b @ A2 @ ( measur7460903249514972363_set_b @ ( comple8942076150311361001_set_b @ M2 ) ) )
=> ( ( member_set_set_b @ C @ ( measur7460903249514972363_set_b @ ( comple8942076150311361001_set_b @ M2 ) ) )
=> ( ( ord_le3795704787696855135_set_b @ A2 @ B )
=> ( ( ord_le3795704787696855135_set_b @ B @ C )
=> ( ( ( sigma_measure_set_b2 @ ( comple8942076150311361001_set_b @ M2 ) @ A2 )
= ( sigma_measure_set_b2 @ ( comple8942076150311361001_set_b @ M2 ) @ C ) )
=> ( member_set_set_b @ B @ ( measur7460903249514972363_set_b @ ( comple8942076150311361001_set_b @ M2 ) ) ) ) ) ) ) ) ).
% completion.complete_sets_sandwich_fmeasurable
thf(fact_733_completion_Ocomplete__sets__sandwich__fmeasurable,axiom,
! [A2: set_set_a,M2: sigma_measure_set_a,C: set_set_a,B: set_set_a] :
( ( member_set_set_a @ A2 @ ( measur7460903245211743562_set_a @ ( comple8942076146008132200_set_a @ M2 ) ) )
=> ( ( member_set_set_a @ C @ ( measur7460903245211743562_set_a @ ( comple8942076146008132200_set_a @ M2 ) ) )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ( ( sigma_measure_set_a2 @ ( comple8942076146008132200_set_a @ M2 ) @ A2 )
= ( sigma_measure_set_a2 @ ( comple8942076146008132200_set_a @ M2 ) @ C ) )
=> ( member_set_set_a @ B @ ( measur7460903245211743562_set_a @ ( comple8942076146008132200_set_a @ M2 ) ) ) ) ) ) ) ) ).
% completion.complete_sets_sandwich_fmeasurable
thf(fact_734_completion_Ocomplete__sets__sandwich__fmeasurable,axiom,
! [A2: set_a,M2: sigma_measure_a,C: set_a,B: set_a] :
( ( member_set_a @ A2 @ ( measur3645360004775918570able_a @ ( comple3428971583294703880tion_a @ M2 ) ) )
=> ( ( member_set_a @ C @ ( measur3645360004775918570able_a @ ( comple3428971583294703880tion_a @ M2 ) ) )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ( ( sigma_measure_a2 @ ( comple3428971583294703880tion_a @ M2 ) @ A2 )
= ( sigma_measure_a2 @ ( comple3428971583294703880tion_a @ M2 ) @ C ) )
=> ( member_set_a @ B @ ( measur3645360004775918570able_a @ ( comple3428971583294703880tion_a @ M2 ) ) ) ) ) ) ) ) ).
% completion.complete_sets_sandwich_fmeasurable
thf(fact_735_sets__le__imp__space__le,axiom,
! [A2: sigma_measure_set_b,B: sigma_measure_set_b] :
( ( ord_le3201067847557142847_set_b @ ( sigma_sets_set_b @ A2 ) @ ( sigma_sets_set_b @ B ) )
=> ( ord_le3795704787696855135_set_b @ ( sigma_space_set_b @ A2 ) @ ( sigma_space_set_b @ B ) ) ) ).
% sets_le_imp_space_le
thf(fact_736_sets__le__imp__space__le,axiom,
! [A2: sigma_measure_set_a,B: sigma_measure_set_a] :
( ( ord_le5722252365846178494_set_a @ ( sigma_sets_set_a @ A2 ) @ ( sigma_sets_set_a @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( sigma_space_set_a @ A2 ) @ ( sigma_space_set_a @ B ) ) ) ).
% sets_le_imp_space_le
thf(fact_737_sets__le__imp__space__le,axiom,
! [A2: sigma_measure_b,B: sigma_measure_b] :
( ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ A2 ) @ ( sigma_sets_b @ B ) )
=> ( ord_less_eq_set_b @ ( sigma_space_b @ A2 ) @ ( sigma_space_b @ B ) ) ) ).
% sets_le_imp_space_le
thf(fact_738_sets__le__imp__space__le,axiom,
! [A2: sigma_measure_a,B: sigma_measure_a] :
( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ A2 ) @ ( sigma_sets_a @ B ) )
=> ( ord_less_eq_set_a @ ( sigma_space_a @ A2 ) @ ( sigma_space_a @ B ) ) ) ).
% sets_le_imp_space_le
thf(fact_739_measurable__mono,axiom,
! [N3: sigma_measure_b,N: sigma_measure_b,M2: sigma_measure_b,M: sigma_measure_b] :
( ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ N3 ) @ ( sigma_sets_b @ N ) )
=> ( ( ( sigma_space_b @ N )
= ( sigma_space_b @ N3 ) )
=> ( ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ M2 ) @ ( sigma_sets_b @ M ) )
=> ( ( ( sigma_space_b @ M2 )
= ( sigma_space_b @ M ) )
=> ( ord_less_eq_set_b_b @ ( sigma_measurable_b_b @ M2 @ N ) @ ( sigma_measurable_b_b @ M @ N3 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_740_measurable__mono,axiom,
! [N3: sigma_measure_b,N: sigma_measure_b,M2: sigma_measure_a,M: sigma_measure_a] :
( ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ N3 ) @ ( sigma_sets_b @ N ) )
=> ( ( ( sigma_space_b @ N )
= ( sigma_space_b @ N3 ) )
=> ( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ M2 ) @ ( sigma_sets_a @ M ) )
=> ( ( ( sigma_space_a @ M2 )
= ( sigma_space_a @ M ) )
=> ( ord_less_eq_set_a_b @ ( sigma_measurable_a_b @ M2 @ N ) @ ( sigma_measurable_a_b @ M @ N3 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_741_measurable__mono,axiom,
! [N3: sigma_measure_a,N: sigma_measure_a,M2: sigma_measure_b,M: sigma_measure_b] :
( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ N3 ) @ ( sigma_sets_a @ N ) )
=> ( ( ( sigma_space_a @ N )
= ( sigma_space_a @ N3 ) )
=> ( ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ M2 ) @ ( sigma_sets_b @ M ) )
=> ( ( ( sigma_space_b @ M2 )
= ( sigma_space_b @ M ) )
=> ( ord_less_eq_set_b_a @ ( sigma_measurable_b_a @ M2 @ N ) @ ( sigma_measurable_b_a @ M @ N3 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_742_measurable__mono,axiom,
! [N3: sigma_measure_a,N: sigma_measure_a,M2: sigma_measure_a,M: sigma_measure_a] :
( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ N3 ) @ ( sigma_sets_a @ N ) )
=> ( ( ( sigma_space_a @ N )
= ( sigma_space_a @ N3 ) )
=> ( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ M2 ) @ ( sigma_sets_a @ M ) )
=> ( ( ( sigma_space_a @ M2 )
= ( sigma_space_a @ M ) )
=> ( ord_less_eq_set_a_a @ ( sigma_measurable_a_a @ M2 @ N ) @ ( sigma_measurable_a_a @ M @ N3 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_743_measure__Un__null__set,axiom,
! [A2: set_b,M2: sigma_measure_b,B: set_b] :
( ( member_set_b @ A2 @ ( sigma_sets_b @ M2 ) )
=> ( ( member_set_b @ B @ ( measure_null_sets_b @ M2 ) )
=> ( ( sigma_measure_b2 @ M2 @ ( sup_sup_set_b @ A2 @ B ) )
= ( sigma_measure_b2 @ M2 @ A2 ) ) ) ) ).
% measure_Un_null_set
thf(fact_744_measure__Un__null__set,axiom,
! [A2: set_a,M2: sigma_measure_a,B: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ M2 ) )
=> ( ( member_set_a @ B @ ( measure_null_sets_a @ M2 ) )
=> ( ( sigma_measure_a2 @ M2 @ ( sup_sup_set_a @ A2 @ B ) )
= ( sigma_measure_a2 @ M2 @ A2 ) ) ) ) ).
% measure_Un_null_set
thf(fact_745_complete__measure_Ocomplete__sets__sandwich__fmeasurable,axiom,
! [M2: sigma_measure_set_b,A2: set_set_b,C: set_set_b,B: set_set_b] :
( ( comple6693822267556782962_set_b @ M2 )
=> ( ( member_set_set_b @ A2 @ ( measur7460903249514972363_set_b @ M2 ) )
=> ( ( member_set_set_b @ C @ ( measur7460903249514972363_set_b @ M2 ) )
=> ( ( ord_le3795704787696855135_set_b @ A2 @ B )
=> ( ( ord_le3795704787696855135_set_b @ B @ C )
=> ( ( ( sigma_measure_set_b2 @ M2 @ A2 )
= ( sigma_measure_set_b2 @ M2 @ C ) )
=> ( member_set_set_b @ B @ ( measur7460903249514972363_set_b @ M2 ) ) ) ) ) ) ) ) ).
% complete_measure.complete_sets_sandwich_fmeasurable
thf(fact_746_complete__measure_Ocomplete__sets__sandwich__fmeasurable,axiom,
! [M2: sigma_measure_set_a,A2: set_set_a,C: set_set_a,B: set_set_a] :
( ( comple6693822263253554161_set_a @ M2 )
=> ( ( member_set_set_a @ A2 @ ( measur7460903245211743562_set_a @ M2 ) )
=> ( ( member_set_set_a @ C @ ( measur7460903245211743562_set_a @ M2 ) )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ( ( sigma_measure_set_a2 @ M2 @ A2 )
= ( sigma_measure_set_a2 @ M2 @ C ) )
=> ( member_set_set_a @ B @ ( measur7460903245211743562_set_a @ M2 ) ) ) ) ) ) ) ) ).
% complete_measure.complete_sets_sandwich_fmeasurable
thf(fact_747_complete__measure_Ocomplete__sets__sandwich__fmeasurable,axiom,
! [M2: sigma_measure_a,A2: set_a,C: set_a,B: set_a] :
( ( comple8155536527497655953sure_a @ M2 )
=> ( ( member_set_a @ A2 @ ( measur3645360004775918570able_a @ M2 ) )
=> ( ( member_set_a @ C @ ( measur3645360004775918570able_a @ M2 ) )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ( ( sigma_measure_a2 @ M2 @ A2 )
= ( sigma_measure_a2 @ M2 @ C ) )
=> ( member_set_a @ B @ ( measur3645360004775918570able_a @ M2 ) ) ) ) ) ) ) ) ).
% complete_measure.complete_sets_sandwich_fmeasurable
thf(fact_748_finite__measure_Obounded__measure,axiom,
! [M2: sigma_measure_a,A2: set_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ord_less_eq_real @ ( sigma_measure_a2 @ M2 @ A2 ) @ ( sigma_measure_a2 @ M2 @ ( sigma_space_a @ M2 ) ) ) ) ).
% finite_measure.bounded_measure
thf(fact_749_finite__measure_Obounded__measure,axiom,
! [M2: sigma_measure_b,A2: set_b] :
( ( measur930452917991658467sure_b @ M2 )
=> ( ord_less_eq_real @ ( sigma_measure_b2 @ M2 @ A2 ) @ ( sigma_measure_b2 @ M2 @ ( sigma_space_b @ M2 ) ) ) ) ).
% finite_measure.bounded_measure
thf(fact_750_le__measureD2,axiom,
! [A2: sigma_measure_b,B: sigma_measure_b] :
( ( ord_le254669799889008988sure_b @ A2 @ B )
=> ( ( ( sigma_space_b @ A2 )
= ( sigma_space_b @ B ) )
=> ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ A2 ) @ ( sigma_sets_b @ B ) ) ) ) ).
% le_measureD2
thf(fact_751_le__measureD2,axiom,
! [A2: sigma_measure_a,B: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A2 @ B )
=> ( ( ( sigma_space_a @ A2 )
= ( sigma_space_a @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ A2 ) @ ( sigma_sets_a @ B ) ) ) ) ).
% le_measureD2
thf(fact_752_prob__space_Oprob__le__1,axiom,
! [M2: sigma_measure_b,A2: set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ord_less_eq_real @ ( sigma_measure_b2 @ M2 @ A2 ) @ one_one_real ) ) ).
% prob_space.prob_le_1
thf(fact_753_prob__space_Oprob__le__1,axiom,
! [M2: sigma_measure_a,A2: set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ord_less_eq_real @ ( sigma_measure_a2 @ M2 @ A2 ) @ one_one_real ) ) ).
% prob_space.prob_le_1
thf(fact_754_prob__space_Omeasure__ge__1__iff,axiom,
! [M2: sigma_measure_b,A2: set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( ord_less_eq_real @ one_one_real @ ( sigma_measure_b2 @ M2 @ A2 ) )
= ( ( sigma_measure_b2 @ M2 @ A2 )
= one_one_real ) ) ) ).
% prob_space.measure_ge_1_iff
thf(fact_755_prob__space_Omeasure__ge__1__iff,axiom,
! [M2: sigma_measure_a,A2: set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( ord_less_eq_real @ one_one_real @ ( sigma_measure_a2 @ M2 @ A2 ) )
= ( ( sigma_measure_a2 @ M2 @ A2 )
= one_one_real ) ) ) ).
% prob_space.measure_ge_1_iff
thf(fact_756_subprob__space_Osubprob__measure__le__1,axiom,
! [M2: sigma_measure_a,X: set_a] :
( ( giry_subprob_space_a @ M2 )
=> ( ord_less_eq_real @ ( sigma_measure_a2 @ M2 @ X ) @ one_one_real ) ) ).
% subprob_space.subprob_measure_le_1
thf(fact_757_measure__mono__fmeasurable,axiom,
! [A2: set_b,B: set_b,M2: sigma_measure_b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( member_set_b @ A2 @ ( sigma_sets_b @ M2 ) )
=> ( ( member_set_b @ B @ ( measur3645360004775918571able_b @ M2 ) )
=> ( ord_less_eq_real @ ( sigma_measure_b2 @ M2 @ A2 ) @ ( sigma_measure_b2 @ M2 @ B ) ) ) ) ) ).
% measure_mono_fmeasurable
thf(fact_758_measure__mono__fmeasurable,axiom,
! [A2: set_set_b,B: set_set_b,M2: sigma_measure_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ B )
=> ( ( member_set_set_b @ A2 @ ( sigma_sets_set_b @ M2 ) )
=> ( ( member_set_set_b @ B @ ( measur7460903249514972363_set_b @ M2 ) )
=> ( ord_less_eq_real @ ( sigma_measure_set_b2 @ M2 @ A2 ) @ ( sigma_measure_set_b2 @ M2 @ B ) ) ) ) ) ).
% measure_mono_fmeasurable
thf(fact_759_measure__mono__fmeasurable,axiom,
! [A2: set_set_a,B: set_set_a,M2: sigma_measure_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_set_a @ A2 @ ( sigma_sets_set_a @ M2 ) )
=> ( ( member_set_set_a @ B @ ( measur7460903245211743562_set_a @ M2 ) )
=> ( ord_less_eq_real @ ( sigma_measure_set_a2 @ M2 @ A2 ) @ ( sigma_measure_set_a2 @ M2 @ B ) ) ) ) ) ).
% measure_mono_fmeasurable
thf(fact_760_measure__mono__fmeasurable,axiom,
! [A2: set_a,B: set_a,M2: sigma_measure_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_set_a @ A2 @ ( sigma_sets_a @ M2 ) )
=> ( ( member_set_a @ B @ ( measur3645360004775918570able_a @ M2 ) )
=> ( ord_less_eq_real @ ( sigma_measure_a2 @ M2 @ A2 ) @ ( sigma_measure_a2 @ M2 @ B ) ) ) ) ) ).
% measure_mono_fmeasurable
thf(fact_761_sets__completionE,axiom,
! [A2: set_b,M2: sigma_measure_b] :
( ( member_set_b @ A2 @ ( sigma_sets_b @ ( comple3428971583294703881tion_b @ M2 ) ) )
=> ~ ! [S3: set_b,N4: set_b] :
( ( A2
= ( sup_sup_set_b @ S3 @ N4 ) )
=> ! [N5: set_b] :
( ( ord_less_eq_set_b @ N4 @ N5 )
=> ( ( member_set_b @ N5 @ ( measure_null_sets_b @ M2 ) )
=> ~ ( member_set_b @ S3 @ ( sigma_sets_b @ M2 ) ) ) ) ) ) ).
% sets_completionE
thf(fact_762_sets__completionE,axiom,
! [A2: set_set_b,M2: sigma_measure_set_b] :
( ( member_set_set_b @ A2 @ ( sigma_sets_set_b @ ( comple8942076150311361001_set_b @ M2 ) ) )
=> ~ ! [S3: set_set_b,N4: set_set_b] :
( ( A2
= ( sup_sup_set_set_b @ S3 @ N4 ) )
=> ! [N5: set_set_b] :
( ( ord_le3795704787696855135_set_b @ N4 @ N5 )
=> ( ( member_set_set_b @ N5 @ ( measur1516554132335629586_set_b @ M2 ) )
=> ~ ( member_set_set_b @ S3 @ ( sigma_sets_set_b @ M2 ) ) ) ) ) ) ).
% sets_completionE
thf(fact_763_sets__completionE,axiom,
! [A2: set_set_a,M2: sigma_measure_set_a] :
( ( member_set_set_a @ A2 @ ( sigma_sets_set_a @ ( comple8942076146008132200_set_a @ M2 ) ) )
=> ~ ! [S3: set_set_a,N4: set_set_a] :
( ( A2
= ( sup_sup_set_set_a @ S3 @ N4 ) )
=> ! [N5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ N4 @ N5 )
=> ( ( member_set_set_a @ N5 @ ( measur1516554128032400785_set_a @ M2 ) )
=> ~ ( member_set_set_a @ S3 @ ( sigma_sets_set_a @ M2 ) ) ) ) ) ) ).
% sets_completionE
thf(fact_764_sets__completionE,axiom,
! [A2: set_a,M2: sigma_measure_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) )
=> ~ ! [S3: set_a,N4: set_a] :
( ( A2
= ( sup_sup_set_a @ S3 @ N4 ) )
=> ! [N5: set_a] :
( ( ord_less_eq_set_a @ N4 @ N5 )
=> ( ( member_set_a @ N5 @ ( measure_null_sets_a @ M2 ) )
=> ~ ( member_set_a @ S3 @ ( sigma_sets_a @ M2 ) ) ) ) ) ) ).
% sets_completionE
thf(fact_765_subprob__space__distr,axiom,
! [F: a > b,M: sigma_measure_b] :
( ( member_a_b @ F @ ( sigma_measurable_a_b @ m @ M ) )
=> ( ( ( sigma_space_b @ M )
!= bot_bot_set_b )
=> ( giry_subprob_space_b @ ( measure_distr_a_b @ m @ M @ F ) ) ) ) ).
% subprob_space_distr
thf(fact_766_subprob__space__distr,axiom,
! [F: a > a,M: sigma_measure_a] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ m @ M ) )
=> ( ( ( sigma_space_a @ M )
!= bot_bot_set_a )
=> ( giry_subprob_space_a @ ( measure_distr_a_a @ m @ M @ F ) ) ) ) ).
% subprob_space_distr
thf(fact_767_le__sup__iff,axiom,
! [X5: set_set_b,Y4: set_set_b,Z: set_set_b] :
( ( ord_le3795704787696855135_set_b @ ( sup_sup_set_set_b @ X5 @ Y4 ) @ Z )
= ( ( ord_le3795704787696855135_set_b @ X5 @ Z )
& ( ord_le3795704787696855135_set_b @ Y4 @ Z ) ) ) ).
% le_sup_iff
thf(fact_768_le__sup__iff,axiom,
! [X5: set_set_a,Y4: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ X5 @ Y4 ) @ Z )
= ( ( ord_le3724670747650509150_set_a @ X5 @ Z )
& ( ord_le3724670747650509150_set_a @ Y4 @ Z ) ) ) ).
% le_sup_iff
thf(fact_769_le__sup__iff,axiom,
! [X5: set_a,Y4: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X5 @ Y4 ) @ Z )
= ( ( ord_less_eq_set_a @ X5 @ Z )
& ( ord_less_eq_set_a @ Y4 @ Z ) ) ) ).
% le_sup_iff
thf(fact_770_le__sup__iff,axiom,
! [X5: nat,Y4: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X5 @ Y4 ) @ Z )
= ( ( ord_less_eq_nat @ X5 @ Z )
& ( ord_less_eq_nat @ Y4 @ Z ) ) ) ).
% le_sup_iff
thf(fact_771_le__sup__iff,axiom,
! [X5: int,Y4: int,Z: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ X5 @ Y4 ) @ Z )
= ( ( ord_less_eq_int @ X5 @ Z )
& ( ord_less_eq_int @ Y4 @ Z ) ) ) ).
% le_sup_iff
thf(fact_772_sup_Obounded__iff,axiom,
! [B2: set_set_b,C2: set_set_b,A: set_set_b] :
( ( ord_le3795704787696855135_set_b @ ( sup_sup_set_set_b @ B2 @ C2 ) @ A )
= ( ( ord_le3795704787696855135_set_b @ B2 @ A )
& ( ord_le3795704787696855135_set_b @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_773_sup_Obounded__iff,axiom,
! [B2: set_set_a,C2: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B2 @ C2 ) @ A )
= ( ( ord_le3724670747650509150_set_a @ B2 @ A )
& ( ord_le3724670747650509150_set_a @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_774_sup_Obounded__iff,axiom,
! [B2: set_a,C2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C2 ) @ A )
= ( ( ord_less_eq_set_a @ B2 @ A )
& ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_775_sup_Obounded__iff,axiom,
! [B2: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_776_sup_Obounded__iff,axiom,
! [B2: int,C2: int,A: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C2 ) @ A )
= ( ( ord_less_eq_int @ B2 @ A )
& ( ord_less_eq_int @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_777_le__inf__iff,axiom,
! [X5: set_set_b,Y4: set_set_b,Z: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X5 @ ( inf_inf_set_set_b @ Y4 @ Z ) )
= ( ( ord_le3795704787696855135_set_b @ X5 @ Y4 )
& ( ord_le3795704787696855135_set_b @ X5 @ Z ) ) ) ).
% le_inf_iff
thf(fact_778_le__inf__iff,axiom,
! [X5: set_set_a,Y4: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ ( inf_inf_set_set_a @ Y4 @ Z ) )
= ( ( ord_le3724670747650509150_set_a @ X5 @ Y4 )
& ( ord_le3724670747650509150_set_a @ X5 @ Z ) ) ) ).
% le_inf_iff
thf(fact_779_le__inf__iff,axiom,
! [X5: set_a,Y4: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X5 @ ( inf_inf_set_a @ Y4 @ Z ) )
= ( ( ord_less_eq_set_a @ X5 @ Y4 )
& ( ord_less_eq_set_a @ X5 @ Z ) ) ) ).
% le_inf_iff
thf(fact_780_le__inf__iff,axiom,
! [X5: nat,Y4: nat,Z: nat] :
( ( ord_less_eq_nat @ X5 @ ( inf_inf_nat @ Y4 @ Z ) )
= ( ( ord_less_eq_nat @ X5 @ Y4 )
& ( ord_less_eq_nat @ X5 @ Z ) ) ) ).
% le_inf_iff
thf(fact_781_le__inf__iff,axiom,
! [X5: int,Y4: int,Z: int] :
( ( ord_less_eq_int @ X5 @ ( inf_inf_int @ Y4 @ Z ) )
= ( ( ord_less_eq_int @ X5 @ Y4 )
& ( ord_less_eq_int @ X5 @ Z ) ) ) ).
% le_inf_iff
thf(fact_782_inf_Obounded__iff,axiom,
! [A: set_set_b,B2: set_set_b,C2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ ( inf_inf_set_set_b @ B2 @ C2 ) )
= ( ( ord_le3795704787696855135_set_b @ A @ B2 )
& ( ord_le3795704787696855135_set_b @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_783_inf_Obounded__iff,axiom,
! [A: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( inf_inf_set_set_a @ B2 @ C2 ) )
= ( ( ord_le3724670747650509150_set_a @ A @ B2 )
& ( ord_le3724670747650509150_set_a @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_784_inf_Obounded__iff,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B2 @ C2 ) )
= ( ( ord_less_eq_set_a @ A @ B2 )
& ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_785_inf_Obounded__iff,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B2 @ C2 ) )
= ( ( ord_less_eq_nat @ A @ B2 )
& ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_786_inf_Obounded__iff,axiom,
! [A: int,B2: int,C2: int] :
( ( ord_less_eq_int @ A @ ( inf_inf_int @ B2 @ C2 ) )
= ( ( ord_less_eq_int @ A @ B2 )
& ( ord_less_eq_int @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_787_measure__eq__compl,axiom,
! [S2: set_a,T3: set_a] :
( ( member_set_a @ S2 @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ T3 @ ( sigma_sets_a @ m ) )
=> ( ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ ( sigma_space_a @ m ) @ S2 ) )
= ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ ( sigma_space_a @ m ) @ T3 ) ) )
=> ( ( sigma_measure_a2 @ m @ S2 )
= ( sigma_measure_a2 @ m @ T3 ) ) ) ) ) ).
% measure_eq_compl
thf(fact_788_finite__measure__subadditive,axiom,
! [A2: set_a,B: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ ( sup_sup_set_a @ A2 @ B ) ) @ ( plus_plus_real @ ( sigma_measure_a2 @ m @ A2 ) @ ( sigma_measure_a2 @ m @ B ) ) ) ) ) ).
% finite_measure_subadditive
thf(fact_789_measure__zero__union,axiom,
! [S2: set_a,T3: set_a] :
( ( member_set_a @ S2 @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ T3 @ ( sigma_sets_a @ m ) )
=> ( ( ( sigma_measure_a2 @ m @ T3 )
= zero_zero_real )
=> ( ( sigma_measure_a2 @ m @ ( sup_sup_set_a @ S2 @ T3 ) )
= ( sigma_measure_a2 @ m @ S2 ) ) ) ) ) ).
% measure_zero_union
thf(fact_790_subprob__not__empty,axiom,
( ( sigma_space_a @ m )
!= bot_bot_set_a ) ).
% subprob_not_empty
thf(fact_791_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_792_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_793_all__not__in__conv,axiom,
! [A2: set_a_b] :
( ( ! [X3: a > b] :
~ ( member_a_b @ X3 @ A2 ) )
= ( A2 = bot_bot_set_a_b ) ) ).
% all_not_in_conv
thf(fact_794_all__not__in__conv,axiom,
! [A2: set_c] :
( ( ! [X3: c] :
~ ( member_c @ X3 @ A2 ) )
= ( A2 = bot_bot_set_c ) ) ).
% all_not_in_conv
thf(fact_795_all__not__in__conv,axiom,
! [A2: set_set_a] :
( ( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_796_all__not__in__conv,axiom,
! [A2: set_a_a] :
( ( ! [X3: a > a] :
~ ( member_a_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_a_a ) ) ).
% all_not_in_conv
thf(fact_797_all__not__in__conv,axiom,
! [A2: set_o] :
( ( ! [X3: $o] :
~ ( member_o @ X3 @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_798_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_799_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_800_empty__iff,axiom,
! [C2: a > b] :
~ ( member_a_b @ C2 @ bot_bot_set_a_b ) ).
% empty_iff
thf(fact_801_empty__iff,axiom,
! [C2: c] :
~ ( member_c @ C2 @ bot_bot_set_c ) ).
% empty_iff
thf(fact_802_empty__iff,axiom,
! [C2: set_a] :
~ ( member_set_a @ C2 @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_803_empty__iff,axiom,
! [C2: a > a] :
~ ( member_a_a @ C2 @ bot_bot_set_a_a ) ).
% empty_iff
thf(fact_804_empty__iff,axiom,
! [C2: $o] :
~ ( member_o @ C2 @ bot_bot_set_o ) ).
% empty_iff
thf(fact_805_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_806_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_807_Diff__idemp,axiom,
! [A2: set_a,B: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_808_Diff__iff,axiom,
! [C2: a > b,A2: set_a_b,B: set_a_b] :
( ( member_a_b @ C2 @ ( minus_minus_set_a_b @ A2 @ B ) )
= ( ( member_a_b @ C2 @ A2 )
& ~ ( member_a_b @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_809_Diff__iff,axiom,
! [C2: c,A2: set_c,B: set_c] :
( ( member_c @ C2 @ ( minus_minus_set_c @ A2 @ B ) )
= ( ( member_c @ C2 @ A2 )
& ~ ( member_c @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_810_Diff__iff,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
= ( ( member_set_a @ C2 @ A2 )
& ~ ( member_set_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_811_Diff__iff,axiom,
! [C2: a > a,A2: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( minus_minus_set_a_a @ A2 @ B ) )
= ( ( member_a_a @ C2 @ A2 )
& ~ ( member_a_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_812_Diff__iff,axiom,
! [C2: $o,A2: set_o,B: set_o] :
( ( member_o @ C2 @ ( minus_minus_set_o @ A2 @ B ) )
= ( ( member_o @ C2 @ A2 )
& ~ ( member_o @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_813_Diff__iff,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
= ( ( member_nat @ C2 @ A2 )
& ~ ( member_nat @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_814_Diff__iff,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
= ( ( member_a @ C2 @ A2 )
& ~ ( member_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_815_DiffI,axiom,
! [C2: a > b,A2: set_a_b,B: set_a_b] :
( ( member_a_b @ C2 @ A2 )
=> ( ~ ( member_a_b @ C2 @ B )
=> ( member_a_b @ C2 @ ( minus_minus_set_a_b @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_816_DiffI,axiom,
! [C2: c,A2: set_c,B: set_c] :
( ( member_c @ C2 @ A2 )
=> ( ~ ( member_c @ C2 @ B )
=> ( member_c @ C2 @ ( minus_minus_set_c @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_817_DiffI,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ A2 )
=> ( ~ ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_818_DiffI,axiom,
! [C2: a > a,A2: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ A2 )
=> ( ~ ( member_a_a @ C2 @ B )
=> ( member_a_a @ C2 @ ( minus_minus_set_a_a @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_819_DiffI,axiom,
! [C2: $o,A2: set_o,B: set_o] :
( ( member_o @ C2 @ A2 )
=> ( ~ ( member_o @ C2 @ B )
=> ( member_o @ C2 @ ( minus_minus_set_o @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_820_DiffI,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A2 )
=> ( ~ ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_821_DiffI,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ A2 )
=> ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_822_measure__exclude,axiom,
! [A2: set_a,B: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ( ( sigma_measure_a2 @ m @ A2 )
= ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) ) )
=> ( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
=> ( ( sigma_measure_a2 @ m @ B )
= zero_zero_real ) ) ) ) ) ).
% measure_exclude
thf(fact_823_finite__measure__Union,axiom,
! [A2: set_a,B: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
=> ( ( sigma_measure_a2 @ m @ ( sup_sup_set_a @ A2 @ B ) )
= ( plus_plus_real @ ( sigma_measure_a2 @ m @ A2 ) @ ( sigma_measure_a2 @ m @ B ) ) ) ) ) ) ).
% finite_measure_Union
thf(fact_824_finite__measure__Union_H,axiom,
! [A2: set_a,B: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ( sigma_measure_a2 @ m @ ( sup_sup_set_a @ A2 @ B ) )
= ( plus_plus_real @ ( sigma_measure_a2 @ m @ A2 ) @ ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ B @ A2 ) ) ) ) ) ) ).
% finite_measure_Union'
thf(fact_825_subset__empty,axiom,
! [A2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ bot_bot_set_set_b )
= ( A2 = bot_bot_set_set_b ) ) ).
% subset_empty
thf(fact_826_subset__empty,axiom,
! [A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a )
= ( A2 = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_827_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_828_empty__subsetI,axiom,
! [A2: set_set_b] : ( ord_le3795704787696855135_set_b @ bot_bot_set_set_b @ A2 ) ).
% empty_subsetI
thf(fact_829_empty__subsetI,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A2 ) ).
% empty_subsetI
thf(fact_830_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_831_sets_Oempty__sets,axiom,
! [M2: sigma_measure_b] : ( member_set_b @ bot_bot_set_b @ ( sigma_sets_b @ M2 ) ) ).
% sets.empty_sets
thf(fact_832_sets_Oempty__sets,axiom,
! [M2: sigma_measure_a] : ( member_set_a @ bot_bot_set_a @ ( sigma_sets_a @ M2 ) ) ).
% sets.empty_sets
thf(fact_833_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_834_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_835_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_836_Un__empty,axiom,
! [A2: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_837_sets_ODiff,axiom,
! [A: set_b,M2: sigma_measure_b,B2: set_b] :
( ( member_set_b @ A @ ( sigma_sets_b @ M2 ) )
=> ( ( member_set_b @ B2 @ ( sigma_sets_b @ M2 ) )
=> ( member_set_b @ ( minus_minus_set_b @ A @ B2 ) @ ( sigma_sets_b @ M2 ) ) ) ) ).
% sets.Diff
thf(fact_838_sets_ODiff,axiom,
! [A: set_a,M2: sigma_measure_a,B2: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ M2 ) )
=> ( ( member_set_a @ B2 @ ( sigma_sets_a @ M2 ) )
=> ( member_set_a @ ( minus_minus_set_a @ A @ B2 ) @ ( sigma_sets_a @ M2 ) ) ) ) ).
% sets.Diff
thf(fact_839_fmeasurable_Oempty__sets,axiom,
! [M2: sigma_measure_a] : ( member_set_a @ bot_bot_set_a @ ( measur3645360004775918570able_a @ M2 ) ) ).
% fmeasurable.empty_sets
thf(fact_840_Un__Diff__cancel2,axiom,
! [B: set_a,A2: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B @ A2 ) @ A2 )
= ( sup_sup_set_a @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_841_Un__Diff__cancel,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B @ A2 ) )
= ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_842_null__sets_Oempty__sets,axiom,
! [M2: sigma_measure_a] : ( member_set_a @ bot_bot_set_a @ ( measure_null_sets_a @ M2 ) ) ).
% null_sets.empty_sets
thf(fact_843_fmeasurable_ODiff,axiom,
! [A: set_a,M2: sigma_measure_a,B2: set_a] :
( ( member_set_a @ A @ ( measur3645360004775918570able_a @ M2 ) )
=> ( ( member_set_a @ B2 @ ( measur3645360004775918570able_a @ M2 ) )
=> ( member_set_a @ ( minus_minus_set_a @ A @ B2 ) @ ( measur3645360004775918570able_a @ M2 ) ) ) ) ).
% fmeasurable.Diff
thf(fact_844_null__sets_ODiff,axiom,
! [A: set_a,M2: sigma_measure_a,B2: set_a] :
( ( member_set_a @ A @ ( measure_null_sets_a @ M2 ) )
=> ( ( member_set_a @ B2 @ ( measure_null_sets_a @ M2 ) )
=> ( member_set_a @ ( minus_minus_set_a @ A @ B2 ) @ ( measure_null_sets_a @ M2 ) ) ) ) ).
% null_sets.Diff
thf(fact_845_Diff__eq__empty__iff,axiom,
! [A2: set_set_b,B: set_set_b] :
( ( ( minus_5807331545291222566_set_b @ A2 @ B )
= bot_bot_set_set_b )
= ( ord_le3795704787696855135_set_b @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_846_Diff__eq__empty__iff,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ( minus_5736297505244876581_set_a @ A2 @ B )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_847_Diff__eq__empty__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_848_measure__empty,axiom,
! [M2: sigma_measure_a] :
( ( sigma_measure_a2 @ M2 @ bot_bot_set_a )
= zero_zero_real ) ).
% measure_empty
thf(fact_849_Diff__disjoint,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B @ A2 ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_850_sets_Ocompl__sets,axiom,
! [A: set_b,M2: sigma_measure_b] :
( ( member_set_b @ A @ ( sigma_sets_b @ M2 ) )
=> ( member_set_b @ ( minus_minus_set_b @ ( sigma_space_b @ M2 ) @ A ) @ ( sigma_sets_b @ M2 ) ) ) ).
% sets.compl_sets
thf(fact_851_sets_Ocompl__sets,axiom,
! [A: set_a,M2: sigma_measure_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ M2 ) )
=> ( member_set_a @ ( minus_minus_set_a @ ( sigma_space_a @ M2 ) @ A ) @ ( sigma_sets_a @ M2 ) ) ) ).
% sets.compl_sets
thf(fact_852_main__part__null__part__Int,axiom,
! [S: set_b,M2: sigma_measure_b] :
( ( member_set_b @ S @ ( sigma_sets_b @ ( comple3428971583294703881tion_b @ M2 ) ) )
=> ( ( inf_inf_set_b @ ( complete_main_part_b @ M2 @ S ) @ ( complete_null_part_b @ M2 @ S ) )
= bot_bot_set_b ) ) ).
% main_part_null_part_Int
thf(fact_853_main__part__null__part__Int,axiom,
! [S: set_a,M2: sigma_measure_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) )
=> ( ( inf_inf_set_a @ ( complete_main_part_a @ M2 @ S ) @ ( complete_null_part_a @ M2 @ S ) )
= bot_bot_set_a ) ) ).
% main_part_null_part_Int
thf(fact_854_Diff__triv,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A2 @ B )
= A2 ) ) ).
% Diff_triv
thf(fact_855_Int__Diff__disjoint,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( minus_minus_set_a @ A2 @ B ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_856_ex__in__conv,axiom,
! [A2: set_a_b] :
( ( ? [X3: a > b] : ( member_a_b @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a_b ) ) ).
% ex_in_conv
thf(fact_857_ex__in__conv,axiom,
! [A2: set_c] :
( ( ? [X3: c] : ( member_c @ X3 @ A2 ) )
= ( A2 != bot_bot_set_c ) ) ).
% ex_in_conv
thf(fact_858_ex__in__conv,axiom,
! [A2: set_set_a] :
( ( ? [X3: set_a] : ( member_set_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_859_ex__in__conv,axiom,
! [A2: set_a_a] :
( ( ? [X3: a > a] : ( member_a_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a_a ) ) ).
% ex_in_conv
thf(fact_860_ex__in__conv,axiom,
! [A2: set_o] :
( ( ? [X3: $o] : ( member_o @ X3 @ A2 ) )
= ( A2 != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_861_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_862_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_863_equals0I,axiom,
! [A2: set_a_b] :
( ! [Y6: a > b] :
~ ( member_a_b @ Y6 @ A2 )
=> ( A2 = bot_bot_set_a_b ) ) ).
% equals0I
thf(fact_864_equals0I,axiom,
! [A2: set_c] :
( ! [Y6: c] :
~ ( member_c @ Y6 @ A2 )
=> ( A2 = bot_bot_set_c ) ) ).
% equals0I
thf(fact_865_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y6: set_a] :
~ ( member_set_a @ Y6 @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_866_equals0I,axiom,
! [A2: set_a_a] :
( ! [Y6: a > a] :
~ ( member_a_a @ Y6 @ A2 )
=> ( A2 = bot_bot_set_a_a ) ) ).
% equals0I
thf(fact_867_equals0I,axiom,
! [A2: set_o] :
( ! [Y6: $o] :
~ ( member_o @ Y6 @ A2 )
=> ( A2 = bot_bot_set_o ) ) ).
% equals0I
thf(fact_868_equals0I,axiom,
! [A2: set_nat] :
( ! [Y6: nat] :
~ ( member_nat @ Y6 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_869_equals0I,axiom,
! [A2: set_a] :
( ! [Y6: a] :
~ ( member_a @ Y6 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_870_equals0D,axiom,
! [A2: set_a_b,A: a > b] :
( ( A2 = bot_bot_set_a_b )
=> ~ ( member_a_b @ A @ A2 ) ) ).
% equals0D
thf(fact_871_equals0D,axiom,
! [A2: set_c,A: c] :
( ( A2 = bot_bot_set_c )
=> ~ ( member_c @ A @ A2 ) ) ).
% equals0D
thf(fact_872_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_873_equals0D,axiom,
! [A2: set_a_a,A: a > a] :
( ( A2 = bot_bot_set_a_a )
=> ~ ( member_a_a @ A @ A2 ) ) ).
% equals0D
thf(fact_874_equals0D,axiom,
! [A2: set_o,A: $o] :
( ( A2 = bot_bot_set_o )
=> ~ ( member_o @ A @ A2 ) ) ).
% equals0D
thf(fact_875_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_876_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_877_emptyE,axiom,
! [A: a > b] :
~ ( member_a_b @ A @ bot_bot_set_a_b ) ).
% emptyE
thf(fact_878_emptyE,axiom,
! [A: c] :
~ ( member_c @ A @ bot_bot_set_c ) ).
% emptyE
thf(fact_879_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_880_emptyE,axiom,
! [A: a > a] :
~ ( member_a_a @ A @ bot_bot_set_a_a ) ).
% emptyE
thf(fact_881_emptyE,axiom,
! [A: $o] :
~ ( member_o @ A @ bot_bot_set_o ) ).
% emptyE
thf(fact_882_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_883_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_884_DiffD2,axiom,
! [C2: a > b,A2: set_a_b,B: set_a_b] :
( ( member_a_b @ C2 @ ( minus_minus_set_a_b @ A2 @ B ) )
=> ~ ( member_a_b @ C2 @ B ) ) ).
% DiffD2
thf(fact_885_DiffD2,axiom,
! [C2: c,A2: set_c,B: set_c] :
( ( member_c @ C2 @ ( minus_minus_set_c @ A2 @ B ) )
=> ~ ( member_c @ C2 @ B ) ) ).
% DiffD2
thf(fact_886_DiffD2,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
=> ~ ( member_set_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_887_DiffD2,axiom,
! [C2: a > a,A2: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( minus_minus_set_a_a @ A2 @ B ) )
=> ~ ( member_a_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_888_DiffD2,axiom,
! [C2: $o,A2: set_o,B: set_o] :
( ( member_o @ C2 @ ( minus_minus_set_o @ A2 @ B ) )
=> ~ ( member_o @ C2 @ B ) ) ).
% DiffD2
thf(fact_889_DiffD2,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ~ ( member_nat @ C2 @ B ) ) ).
% DiffD2
thf(fact_890_DiffD2,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
=> ~ ( member_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_891_DiffD1,axiom,
! [C2: a > b,A2: set_a_b,B: set_a_b] :
( ( member_a_b @ C2 @ ( minus_minus_set_a_b @ A2 @ B ) )
=> ( member_a_b @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_892_DiffD1,axiom,
! [C2: c,A2: set_c,B: set_c] :
( ( member_c @ C2 @ ( minus_minus_set_c @ A2 @ B ) )
=> ( member_c @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_893_DiffD1,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
=> ( member_set_a @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_894_DiffD1,axiom,
! [C2: a > a,A2: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( minus_minus_set_a_a @ A2 @ B ) )
=> ( member_a_a @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_895_DiffD1,axiom,
! [C2: $o,A2: set_o,B: set_o] :
( ( member_o @ C2 @ ( minus_minus_set_o @ A2 @ B ) )
=> ( member_o @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_896_DiffD1,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ( member_nat @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_897_DiffD1,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
=> ( member_a @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_898_DiffE,axiom,
! [C2: a > b,A2: set_a_b,B: set_a_b] :
( ( member_a_b @ C2 @ ( minus_minus_set_a_b @ A2 @ B ) )
=> ~ ( ( member_a_b @ C2 @ A2 )
=> ( member_a_b @ C2 @ B ) ) ) ).
% DiffE
thf(fact_899_DiffE,axiom,
! [C2: c,A2: set_c,B: set_c] :
( ( member_c @ C2 @ ( minus_minus_set_c @ A2 @ B ) )
=> ~ ( ( member_c @ C2 @ A2 )
=> ( member_c @ C2 @ B ) ) ) ).
% DiffE
thf(fact_900_DiffE,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
=> ~ ( ( member_set_a @ C2 @ A2 )
=> ( member_set_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_901_DiffE,axiom,
! [C2: a > a,A2: set_a_a,B: set_a_a] :
( ( member_a_a @ C2 @ ( minus_minus_set_a_a @ A2 @ B ) )
=> ~ ( ( member_a_a @ C2 @ A2 )
=> ( member_a_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_902_DiffE,axiom,
! [C2: $o,A2: set_o,B: set_o] :
( ( member_o @ C2 @ ( minus_minus_set_o @ A2 @ B ) )
=> ~ ( ( member_o @ C2 @ A2 )
=> ( member_o @ C2 @ B ) ) ) ).
% DiffE
thf(fact_903_DiffE,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ~ ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B ) ) ) ).
% DiffE
thf(fact_904_DiffE,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
=> ~ ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_905_diff__left__imp__eq,axiom,
! [A: real,B2: real,C2: real] :
( ( ( minus_minus_real @ A @ B2 )
= ( minus_minus_real @ A @ C2 ) )
=> ( B2 = C2 ) ) ).
% diff_left_imp_eq
thf(fact_906_diff__left__imp__eq,axiom,
! [A: int,B2: int,C2: int] :
( ( ( minus_minus_int @ A @ B2 )
= ( minus_minus_int @ A @ C2 ) )
=> ( B2 = C2 ) ) ).
% diff_left_imp_eq
thf(fact_907_bot_Oextremum,axiom,
! [A: set_set_b] : ( ord_le3795704787696855135_set_b @ bot_bot_set_set_b @ A ) ).
% bot.extremum
thf(fact_908_bot_Oextremum,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% bot.extremum
thf(fact_909_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_910_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_911_bot_Oextremum__unique,axiom,
! [A: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ bot_bot_set_set_b )
= ( A = bot_bot_set_set_b ) ) ).
% bot.extremum_unique
thf(fact_912_bot_Oextremum__unique,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% bot.extremum_unique
thf(fact_913_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_914_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_915_bot_Oextremum__uniqueI,axiom,
! [A: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ bot_bot_set_set_b )
=> ( A = bot_bot_set_set_b ) ) ).
% bot.extremum_uniqueI
thf(fact_916_bot_Oextremum__uniqueI,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
=> ( A = bot_bot_set_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_917_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_918_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_919_sets__sup,axiom,
! [A2: sigma_measure_b,M2: sigma_measure_b,B: sigma_measure_b] :
( ( ( sigma_sets_b @ A2 )
= ( sigma_sets_b @ M2 ) )
=> ( ( ( sigma_sets_b @ B )
= ( sigma_sets_b @ M2 ) )
=> ( ( sigma_sets_b @ ( sup_su27664956689621032sure_b @ A2 @ B ) )
= ( sigma_sets_b @ M2 ) ) ) ) ).
% sets_sup
thf(fact_920_sets__sup,axiom,
! [A2: sigma_measure_a,M2: sigma_measure_a,B: sigma_measure_a] :
( ( ( sigma_sets_a @ A2 )
= ( sigma_sets_a @ M2 ) )
=> ( ( ( sigma_sets_a @ B )
= ( sigma_sets_a @ M2 ) )
=> ( ( sigma_sets_a @ ( sup_su27664952386392231sure_a @ A2 @ B ) )
= ( sigma_sets_a @ M2 ) ) ) ) ).
% sets_sup
thf(fact_921_Diff__mono,axiom,
! [A2: set_set_b,C: set_set_b,D: set_set_b,B: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ C )
=> ( ( ord_le3795704787696855135_set_b @ D @ B )
=> ( ord_le3795704787696855135_set_b @ ( minus_5807331545291222566_set_b @ A2 @ B ) @ ( minus_5807331545291222566_set_b @ C @ D ) ) ) ) ).
% Diff_mono
thf(fact_922_Diff__mono,axiom,
! [A2: set_set_a,C: set_set_a,D: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ( ord_le3724670747650509150_set_a @ D @ B )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) @ ( minus_5736297505244876581_set_a @ C @ D ) ) ) ) ).
% Diff_mono
thf(fact_923_Diff__mono,axiom,
! [A2: set_a,C: set_a,D: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ D @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( minus_minus_set_a @ C @ D ) ) ) ) ).
% Diff_mono
thf(fact_924_Diff__subset,axiom,
! [A2: set_set_b,B: set_set_b] : ( ord_le3795704787696855135_set_b @ ( minus_5807331545291222566_set_b @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_925_Diff__subset,axiom,
! [A2: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_926_Diff__subset,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_927_double__diff,axiom,
! [A2: set_set_b,B: set_set_b,C: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ B )
=> ( ( ord_le3795704787696855135_set_b @ B @ C )
=> ( ( minus_5807331545291222566_set_b @ B @ ( minus_5807331545291222566_set_b @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_928_double__diff,axiom,
! [A2: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ( minus_5736297505244876581_set_a @ B @ ( minus_5736297505244876581_set_a @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_929_double__diff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_930_Diff__Int__distrib2,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A2 @ B ) @ C )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ ( inf_inf_set_a @ B @ C ) ) ) ).
% Diff_Int_distrib2
thf(fact_931_Diff__Int__distrib,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( inf_inf_set_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C @ A2 ) @ ( inf_inf_set_a @ C @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_932_Diff__Diff__Int,axiom,
! [A2: set_a,B: set_a] :
( ( minus_minus_set_a @ A2 @ ( minus_minus_set_a @ A2 @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ).
% Diff_Diff_Int
thf(fact_933_Diff__Int2,axiom,
! [A2: set_a,C: set_a,B: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ ( inf_inf_set_a @ B @ C ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ B ) ) ).
% Diff_Int2
thf(fact_934_Int__Diff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C )
= ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B @ C ) ) ) ).
% Int_Diff
thf(fact_935_Un__Diff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A2 @ C ) @ ( minus_minus_set_a @ B @ C ) ) ) ).
% Un_Diff
thf(fact_936_disjoint__iff__not__equal,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ! [Y3: a] :
( ( member_a @ Y3 @ B )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_937_Int__empty__right,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_938_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_939_disjoint__iff,axiom,
! [A2: set_a_b,B: set_a_b] :
( ( ( inf_inf_set_a_b @ A2 @ B )
= bot_bot_set_a_b )
= ( ! [X3: a > b] :
( ( member_a_b @ X3 @ A2 )
=> ~ ( member_a_b @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_940_disjoint__iff,axiom,
! [A2: set_c,B: set_c] :
( ( ( inf_inf_set_c @ A2 @ B )
= bot_bot_set_c )
= ( ! [X3: c] :
( ( member_c @ X3 @ A2 )
=> ~ ( member_c @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_941_disjoint__iff,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ~ ( member_set_a @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_942_disjoint__iff,axiom,
! [A2: set_a_a,B: set_a_a] :
( ( ( inf_inf_set_a_a @ A2 @ B )
= bot_bot_set_a_a )
= ( ! [X3: a > a] :
( ( member_a_a @ X3 @ A2 )
=> ~ ( member_a_a @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_943_disjoint__iff,axiom,
! [A2: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A2 @ B )
= bot_bot_set_o )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ~ ( member_o @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_944_disjoint__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ~ ( member_nat @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_945_disjoint__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_a @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_946_Int__emptyI,axiom,
! [A2: set_a_b,B: set_a_b] :
( ! [X4: a > b] :
( ( member_a_b @ X4 @ A2 )
=> ~ ( member_a_b @ X4 @ B ) )
=> ( ( inf_inf_set_a_b @ A2 @ B )
= bot_bot_set_a_b ) ) ).
% Int_emptyI
thf(fact_947_Int__emptyI,axiom,
! [A2: set_c,B: set_c] :
( ! [X4: c] :
( ( member_c @ X4 @ A2 )
=> ~ ( member_c @ X4 @ B ) )
=> ( ( inf_inf_set_c @ A2 @ B )
= bot_bot_set_c ) ) ).
% Int_emptyI
thf(fact_948_Int__emptyI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ~ ( member_set_a @ X4 @ B ) )
=> ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_949_Int__emptyI,axiom,
! [A2: set_a_a,B: set_a_a] :
( ! [X4: a > a] :
( ( member_a_a @ X4 @ A2 )
=> ~ ( member_a_a @ X4 @ B ) )
=> ( ( inf_inf_set_a_a @ A2 @ B )
= bot_bot_set_a_a ) ) ).
% Int_emptyI
thf(fact_950_Int__emptyI,axiom,
! [A2: set_o,B: set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ~ ( member_o @ X4 @ B ) )
=> ( ( inf_inf_set_o @ A2 @ B )
= bot_bot_set_o ) ) ).
% Int_emptyI
thf(fact_951_Int__emptyI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ~ ( member_nat @ X4 @ B ) )
=> ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_952_Int__emptyI,axiom,
! [A2: set_a,B: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ~ ( member_a @ X4 @ B ) )
=> ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_953_Un__empty__right,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Un_empty_right
thf(fact_954_Un__empty__left,axiom,
! [B: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_955_measure__Un2,axiom,
! [A2: set_a,M2: sigma_measure_a,B: set_a] :
( ( member_set_a @ A2 @ ( measur3645360004775918570able_a @ M2 ) )
=> ( ( member_set_a @ B @ ( measur3645360004775918570able_a @ M2 ) )
=> ( ( sigma_measure_a2 @ M2 @ ( sup_sup_set_a @ A2 @ B ) )
= ( plus_plus_real @ ( sigma_measure_a2 @ M2 @ A2 ) @ ( sigma_measure_a2 @ M2 @ ( minus_minus_set_a @ B @ A2 ) ) ) ) ) ) ).
% measure_Un2
thf(fact_956_finite__measure_Ofinite__measure__Union_H,axiom,
! [M2: sigma_measure_a,A2: set_a,B: set_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ( member_set_a @ A2 @ ( sigma_sets_a @ M2 ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M2 ) )
=> ( ( sigma_measure_a2 @ M2 @ ( sup_sup_set_a @ A2 @ B ) )
= ( plus_plus_real @ ( sigma_measure_a2 @ M2 @ A2 ) @ ( sigma_measure_a2 @ M2 @ ( minus_minus_set_a @ B @ A2 ) ) ) ) ) ) ) ).
% finite_measure.finite_measure_Union'
thf(fact_957_finite__measure_Ofinite__measure__Union_H,axiom,
! [M2: sigma_measure_b,A2: set_b,B: set_b] :
( ( measur930452917991658467sure_b @ M2 )
=> ( ( member_set_b @ A2 @ ( sigma_sets_b @ M2 ) )
=> ( ( member_set_b @ B @ ( sigma_sets_b @ M2 ) )
=> ( ( sigma_measure_b2 @ M2 @ ( sup_sup_set_b @ A2 @ B ) )
= ( plus_plus_real @ ( sigma_measure_b2 @ M2 @ A2 ) @ ( sigma_measure_b2 @ M2 @ ( minus_minus_set_b @ B @ A2 ) ) ) ) ) ) ) ).
% finite_measure.finite_measure_Union'
thf(fact_958_Diff__subset__conv,axiom,
! [A2: set_set_b,B: set_set_b,C: set_set_b] :
( ( ord_le3795704787696855135_set_b @ ( minus_5807331545291222566_set_b @ A2 @ B ) @ C )
= ( ord_le3795704787696855135_set_b @ A2 @ ( sup_sup_set_set_b @ B @ C ) ) ) ).
% Diff_subset_conv
thf(fact_959_Diff__subset__conv,axiom,
! [A2: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) @ C )
= ( ord_le3724670747650509150_set_a @ A2 @ ( sup_sup_set_set_a @ B @ C ) ) ) ).
% Diff_subset_conv
thf(fact_960_Diff__subset__conv,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ C )
= ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Diff_subset_conv
thf(fact_961_Diff__partition,axiom,
! [A2: set_set_b,B: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ B )
=> ( ( sup_sup_set_set_b @ A2 @ ( minus_5807331545291222566_set_b @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_962_Diff__partition,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( sup_sup_set_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_963_Diff__partition,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_964_Un__Diff__Int,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( inf_inf_set_a @ A2 @ B ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_965_Int__Diff__Un,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( minus_minus_set_a @ A2 @ B ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_966_Diff__Int,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ A2 @ ( inf_inf_set_a @ B @ C ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( minus_minus_set_a @ A2 @ C ) ) ) ).
% Diff_Int
thf(fact_967_Diff__Un,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) )
= ( inf_inf_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( minus_minus_set_a @ A2 @ C ) ) ) ).
% Diff_Un
thf(fact_968_fmeasurable__Diff__D,axiom,
! [T: set_set_b,S: set_set_b,M2: sigma_measure_set_b] :
( ( member_set_set_b @ ( minus_5807331545291222566_set_b @ T @ S ) @ ( measur7460903249514972363_set_b @ M2 ) )
=> ( ( member_set_set_b @ S @ ( measur7460903249514972363_set_b @ M2 ) )
=> ( ( ord_le3795704787696855135_set_b @ S @ T )
=> ( member_set_set_b @ T @ ( measur7460903249514972363_set_b @ M2 ) ) ) ) ) ).
% fmeasurable_Diff_D
thf(fact_969_fmeasurable__Diff__D,axiom,
! [T: set_set_a,S: set_set_a,M2: sigma_measure_set_a] :
( ( member_set_set_a @ ( minus_5736297505244876581_set_a @ T @ S ) @ ( measur7460903245211743562_set_a @ M2 ) )
=> ( ( member_set_set_a @ S @ ( measur7460903245211743562_set_a @ M2 ) )
=> ( ( ord_le3724670747650509150_set_a @ S @ T )
=> ( member_set_set_a @ T @ ( measur7460903245211743562_set_a @ M2 ) ) ) ) ) ).
% fmeasurable_Diff_D
thf(fact_970_fmeasurable__Diff__D,axiom,
! [T: set_a,S: set_a,M2: sigma_measure_a] :
( ( member_set_a @ ( minus_minus_set_a @ T @ S ) @ ( measur3645360004775918570able_a @ M2 ) )
=> ( ( member_set_a @ S @ ( measur3645360004775918570able_a @ M2 ) )
=> ( ( ord_less_eq_set_a @ S @ T )
=> ( member_set_a @ T @ ( measur3645360004775918570able_a @ M2 ) ) ) ) ) ).
% fmeasurable_Diff_D
thf(fact_971_fmeasurable__Diff,axiom,
! [A2: set_b,M2: sigma_measure_b,B: set_b] :
( ( member_set_b @ A2 @ ( measur3645360004775918571able_b @ M2 ) )
=> ( ( member_set_b @ B @ ( sigma_sets_b @ M2 ) )
=> ( member_set_b @ ( minus_minus_set_b @ A2 @ B ) @ ( measur3645360004775918571able_b @ M2 ) ) ) ) ).
% fmeasurable_Diff
thf(fact_972_fmeasurable__Diff,axiom,
! [A2: set_a,M2: sigma_measure_a,B: set_a] :
( ( member_set_a @ A2 @ ( measur3645360004775918570able_a @ M2 ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M2 ) )
=> ( member_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( measur3645360004775918570able_a @ M2 ) ) ) ) ).
% fmeasurable_Diff
thf(fact_973_null__set__Diff,axiom,
! [B: set_b,M2: sigma_measure_b,A2: set_b] :
( ( member_set_b @ B @ ( measure_null_sets_b @ M2 ) )
=> ( ( member_set_b @ A2 @ ( sigma_sets_b @ M2 ) )
=> ( member_set_b @ ( minus_minus_set_b @ B @ A2 ) @ ( measure_null_sets_b @ M2 ) ) ) ) ).
% null_set_Diff
thf(fact_974_null__set__Diff,axiom,
! [B: set_a,M2: sigma_measure_a,A2: set_a] :
( ( member_set_a @ B @ ( measure_null_sets_a @ M2 ) )
=> ( ( member_set_a @ A2 @ ( sigma_sets_a @ M2 ) )
=> ( member_set_a @ ( minus_minus_set_a @ B @ A2 ) @ ( measure_null_sets_a @ M2 ) ) ) ) ).
% null_set_Diff
thf(fact_975_measurable__empty__iff,axiom,
! [N: sigma_measure_b,F: a > b,M2: sigma_measure_a] :
( ( ( sigma_space_b @ N )
= bot_bot_set_b )
=> ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ N ) )
= ( ( sigma_space_a @ M2 )
= bot_bot_set_a ) ) ) ).
% measurable_empty_iff
thf(fact_976_measurable__empty__iff,axiom,
! [N: sigma_measure_a,F: a > a,M2: sigma_measure_a] :
( ( ( sigma_space_a @ N )
= bot_bot_set_a )
=> ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N ) )
= ( ( sigma_space_a @ M2 )
= bot_bot_set_a ) ) ) ).
% measurable_empty_iff
thf(fact_977_measure__le__0__iff,axiom,
! [M2: sigma_measure_a,X: set_a] :
( ( ord_less_eq_real @ ( sigma_measure_a2 @ M2 @ X ) @ zero_zero_real )
= ( ( sigma_measure_a2 @ M2 @ X )
= zero_zero_real ) ) ).
% measure_le_0_iff
thf(fact_978_measure__nonneg,axiom,
! [M2: sigma_measure_a,A2: set_a] : ( ord_less_eq_real @ zero_zero_real @ ( sigma_measure_a2 @ M2 @ A2 ) ) ).
% measure_nonneg
thf(fact_979_measure__notin__sets,axiom,
! [A2: set_b,M2: sigma_measure_b] :
( ~ ( member_set_b @ A2 @ ( sigma_sets_b @ M2 ) )
=> ( ( sigma_measure_b2 @ M2 @ A2 )
= zero_zero_real ) ) ).
% measure_notin_sets
thf(fact_980_measure__notin__sets,axiom,
! [A2: set_a,M2: sigma_measure_a] :
( ~ ( member_set_a @ A2 @ ( sigma_sets_a @ M2 ) )
=> ( ( sigma_measure_a2 @ M2 @ A2 )
= zero_zero_real ) ) ).
% measure_notin_sets
thf(fact_981_prob__space_Onot__empty,axiom,
! [M2: sigma_measure_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( sigma_space_b @ M2 )
!= bot_bot_set_b ) ) ).
% prob_space.not_empty
thf(fact_982_prob__space_Onot__empty,axiom,
! [M2: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( sigma_space_a @ M2 )
!= bot_bot_set_a ) ) ).
% prob_space.not_empty
thf(fact_983_subprob__space_Osubprob__not__empty,axiom,
! [M2: sigma_measure_a] :
( ( giry_subprob_space_a @ M2 )
=> ( ( sigma_space_a @ M2 )
!= bot_bot_set_a ) ) ).
% subprob_space.subprob_not_empty
thf(fact_984_finite__measure_Ofinite__measure__Union,axiom,
! [M2: sigma_measure_a,A2: set_a,B: set_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ( member_set_a @ A2 @ ( sigma_sets_a @ M2 ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M2 ) )
=> ( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
=> ( ( sigma_measure_a2 @ M2 @ ( sup_sup_set_a @ A2 @ B ) )
= ( plus_plus_real @ ( sigma_measure_a2 @ M2 @ A2 ) @ ( sigma_measure_a2 @ M2 @ B ) ) ) ) ) ) ) ).
% finite_measure.finite_measure_Union
thf(fact_985_finite__measure_Ofinite__measure__Union,axiom,
! [M2: sigma_measure_b,A2: set_b,B: set_b] :
( ( measur930452917991658467sure_b @ M2 )
=> ( ( member_set_b @ A2 @ ( sigma_sets_b @ M2 ) )
=> ( ( member_set_b @ B @ ( sigma_sets_b @ M2 ) )
=> ( ( ( inf_inf_set_b @ A2 @ B )
= bot_bot_set_b )
=> ( ( sigma_measure_b2 @ M2 @ ( sup_sup_set_b @ A2 @ B ) )
= ( plus_plus_real @ ( sigma_measure_b2 @ M2 @ A2 ) @ ( sigma_measure_b2 @ M2 @ B ) ) ) ) ) ) ) ).
% finite_measure.finite_measure_Union
thf(fact_986_finite__measure_Omeasure__exclude,axiom,
! [M2: sigma_measure_a,A2: set_a,B: set_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ( member_set_a @ A2 @ ( sigma_sets_a @ M2 ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M2 ) )
=> ( ( ( sigma_measure_a2 @ M2 @ A2 )
= ( sigma_measure_a2 @ M2 @ ( sigma_space_a @ M2 ) ) )
=> ( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
=> ( ( sigma_measure_a2 @ M2 @ B )
= zero_zero_real ) ) ) ) ) ) ).
% finite_measure.measure_exclude
thf(fact_987_finite__measure_Omeasure__exclude,axiom,
! [M2: sigma_measure_b,A2: set_b,B: set_b] :
( ( measur930452917991658467sure_b @ M2 )
=> ( ( member_set_b @ A2 @ ( sigma_sets_b @ M2 ) )
=> ( ( member_set_b @ B @ ( sigma_sets_b @ M2 ) )
=> ( ( ( sigma_measure_b2 @ M2 @ A2 )
= ( sigma_measure_b2 @ M2 @ ( sigma_space_b @ M2 ) ) )
=> ( ( ( inf_inf_set_b @ A2 @ B )
= bot_bot_set_b )
=> ( ( sigma_measure_b2 @ M2 @ B )
= zero_zero_real ) ) ) ) ) ) ).
% finite_measure.measure_exclude
thf(fact_988_measure__Diff__null__set,axiom,
! [A2: set_b,M2: sigma_measure_b,B: set_b] :
( ( member_set_b @ A2 @ ( sigma_sets_b @ M2 ) )
=> ( ( member_set_b @ B @ ( measure_null_sets_b @ M2 ) )
=> ( ( sigma_measure_b2 @ M2 @ ( minus_minus_set_b @ A2 @ B ) )
= ( sigma_measure_b2 @ M2 @ A2 ) ) ) ) ).
% measure_Diff_null_set
thf(fact_989_measure__Diff__null__set,axiom,
! [A2: set_a,M2: sigma_measure_a,B: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ M2 ) )
=> ( ( member_set_a @ B @ ( measure_null_sets_a @ M2 ) )
=> ( ( sigma_measure_a2 @ M2 @ ( minus_minus_set_a @ A2 @ B ) )
= ( sigma_measure_a2 @ M2 @ A2 ) ) ) ) ).
% measure_Diff_null_set
thf(fact_990_measurable__Diff__null__set,axiom,
! [B: set_b,M2: sigma_measure_b,A2: set_b] :
( ( member_set_b @ B @ ( measure_null_sets_b @ M2 ) )
=> ( ( ( member_set_b @ ( minus_minus_set_b @ A2 @ B ) @ ( measur3645360004775918571able_b @ M2 ) )
& ( member_set_b @ A2 @ ( sigma_sets_b @ M2 ) ) )
= ( member_set_b @ A2 @ ( measur3645360004775918571able_b @ M2 ) ) ) ) ).
% measurable_Diff_null_set
thf(fact_991_measurable__Diff__null__set,axiom,
! [B: set_a,M2: sigma_measure_a,A2: set_a] :
( ( member_set_a @ B @ ( measure_null_sets_a @ M2 ) )
=> ( ( ( member_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( measur3645360004775918570able_a @ M2 ) )
& ( member_set_a @ A2 @ ( sigma_sets_a @ M2 ) ) )
= ( member_set_a @ A2 @ ( measur3645360004775918570able_a @ M2 ) ) ) ) ).
% measurable_Diff_null_set
thf(fact_992_measure__Un__le,axiom,
! [A2: set_b,M2: sigma_measure_b,B: set_b] :
( ( member_set_b @ A2 @ ( sigma_sets_b @ M2 ) )
=> ( ( member_set_b @ B @ ( sigma_sets_b @ M2 ) )
=> ( ord_less_eq_real @ ( sigma_measure_b2 @ M2 @ ( sup_sup_set_b @ A2 @ B ) ) @ ( plus_plus_real @ ( sigma_measure_b2 @ M2 @ A2 ) @ ( sigma_measure_b2 @ M2 @ B ) ) ) ) ) ).
% measure_Un_le
thf(fact_993_measure__Un__le,axiom,
! [A2: set_a,M2: sigma_measure_a,B: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ M2 ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M2 ) )
=> ( ord_less_eq_real @ ( sigma_measure_a2 @ M2 @ ( sup_sup_set_a @ A2 @ B ) ) @ ( plus_plus_real @ ( sigma_measure_a2 @ M2 @ A2 ) @ ( sigma_measure_a2 @ M2 @ B ) ) ) ) ) ).
% measure_Un_le
thf(fact_994_finite__measure_Omeasure__eq__compl,axiom,
! [M2: sigma_measure_a,S2: set_a,T3: set_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ( member_set_a @ S2 @ ( sigma_sets_a @ M2 ) )
=> ( ( member_set_a @ T3 @ ( sigma_sets_a @ M2 ) )
=> ( ( ( sigma_measure_a2 @ M2 @ ( minus_minus_set_a @ ( sigma_space_a @ M2 ) @ S2 ) )
= ( sigma_measure_a2 @ M2 @ ( minus_minus_set_a @ ( sigma_space_a @ M2 ) @ T3 ) ) )
=> ( ( sigma_measure_a2 @ M2 @ S2 )
= ( sigma_measure_a2 @ M2 @ T3 ) ) ) ) ) ) ).
% finite_measure.measure_eq_compl
thf(fact_995_finite__measure_Omeasure__eq__compl,axiom,
! [M2: sigma_measure_b,S2: set_b,T3: set_b] :
( ( measur930452917991658467sure_b @ M2 )
=> ( ( member_set_b @ S2 @ ( sigma_sets_b @ M2 ) )
=> ( ( member_set_b @ T3 @ ( sigma_sets_b @ M2 ) )
=> ( ( ( sigma_measure_b2 @ M2 @ ( minus_minus_set_b @ ( sigma_space_b @ M2 ) @ S2 ) )
= ( sigma_measure_b2 @ M2 @ ( minus_minus_set_b @ ( sigma_space_b @ M2 ) @ T3 ) ) )
=> ( ( sigma_measure_b2 @ M2 @ S2 )
= ( sigma_measure_b2 @ M2 @ T3 ) ) ) ) ) ) ).
% finite_measure.measure_eq_compl
thf(fact_996_finite__measure_Omeasure__zero__union,axiom,
! [M2: sigma_measure_a,S2: set_a,T3: set_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ( member_set_a @ S2 @ ( sigma_sets_a @ M2 ) )
=> ( ( member_set_a @ T3 @ ( sigma_sets_a @ M2 ) )
=> ( ( ( sigma_measure_a2 @ M2 @ T3 )
= zero_zero_real )
=> ( ( sigma_measure_a2 @ M2 @ ( sup_sup_set_a @ S2 @ T3 ) )
= ( sigma_measure_a2 @ M2 @ S2 ) ) ) ) ) ) ).
% finite_measure.measure_zero_union
thf(fact_997_finite__measure_Omeasure__zero__union,axiom,
! [M2: sigma_measure_b,S2: set_b,T3: set_b] :
( ( measur930452917991658467sure_b @ M2 )
=> ( ( member_set_b @ S2 @ ( sigma_sets_b @ M2 ) )
=> ( ( member_set_b @ T3 @ ( sigma_sets_b @ M2 ) )
=> ( ( ( sigma_measure_b2 @ M2 @ T3 )
= zero_zero_real )
=> ( ( sigma_measure_b2 @ M2 @ ( sup_sup_set_b @ S2 @ T3 ) )
= ( sigma_measure_b2 @ M2 @ S2 ) ) ) ) ) ) ).
% finite_measure.measure_zero_union
thf(fact_998_subprob__space_Osubprob__space__distr,axiom,
! [M2: sigma_measure_a,F: a > b,M: sigma_measure_b] :
( ( giry_subprob_space_a @ M2 )
=> ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ M ) )
=> ( ( ( sigma_space_b @ M )
!= bot_bot_set_b )
=> ( giry_subprob_space_b @ ( measure_distr_a_b @ M2 @ M @ F ) ) ) ) ) ).
% subprob_space.subprob_space_distr
thf(fact_999_subprob__space_Osubprob__space__distr,axiom,
! [M2: sigma_measure_a,F: a > a,M: sigma_measure_a] :
( ( giry_subprob_space_a @ M2 )
=> ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ M ) )
=> ( ( ( sigma_space_a @ M )
!= bot_bot_set_a )
=> ( giry_subprob_space_a @ ( measure_distr_a_a @ M2 @ M @ F ) ) ) ) ) ).
% subprob_space.subprob_space_distr
thf(fact_1000_finite__measure_Ofinite__measure__subadditive,axiom,
! [M2: sigma_measure_a,A2: set_a,B: set_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ( member_set_a @ A2 @ ( sigma_sets_a @ M2 ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M2 ) )
=> ( ord_less_eq_real @ ( sigma_measure_a2 @ M2 @ ( sup_sup_set_a @ A2 @ B ) ) @ ( plus_plus_real @ ( sigma_measure_a2 @ M2 @ A2 ) @ ( sigma_measure_a2 @ M2 @ B ) ) ) ) ) ) ).
% finite_measure.finite_measure_subadditive
thf(fact_1001_finite__measure_Ofinite__measure__subadditive,axiom,
! [M2: sigma_measure_b,A2: set_b,B: set_b] :
( ( measur930452917991658467sure_b @ M2 )
=> ( ( member_set_b @ A2 @ ( sigma_sets_b @ M2 ) )
=> ( ( member_set_b @ B @ ( sigma_sets_b @ M2 ) )
=> ( ord_less_eq_real @ ( sigma_measure_b2 @ M2 @ ( sup_sup_set_b @ A2 @ B ) ) @ ( plus_plus_real @ ( sigma_measure_b2 @ M2 @ A2 ) @ ( sigma_measure_b2 @ M2 @ B ) ) ) ) ) ) ).
% finite_measure.finite_measure_subadditive
thf(fact_1002_inf__sup__ord_I2_J,axiom,
! [X5: set_set_b,Y4: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ X5 @ Y4 ) @ Y4 ) ).
% inf_sup_ord(2)
thf(fact_1003_inf__sup__ord_I2_J,axiom,
! [X5: set_set_a,Y4: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X5 @ Y4 ) @ Y4 ) ).
% inf_sup_ord(2)
thf(fact_1004_inf__sup__ord_I2_J,axiom,
! [X5: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X5 @ Y4 ) @ Y4 ) ).
% inf_sup_ord(2)
thf(fact_1005_inf__sup__ord_I2_J,axiom,
! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X5 @ Y4 ) @ Y4 ) ).
% inf_sup_ord(2)
thf(fact_1006_inf__sup__ord_I2_J,axiom,
! [X5: int,Y4: int] : ( ord_less_eq_int @ ( inf_inf_int @ X5 @ Y4 ) @ Y4 ) ).
% inf_sup_ord(2)
thf(fact_1007_inf__sup__ord_I1_J,axiom,
! [X5: set_set_b,Y4: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ X5 @ Y4 ) @ X5 ) ).
% inf_sup_ord(1)
thf(fact_1008_inf__sup__ord_I1_J,axiom,
! [X5: set_set_a,Y4: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X5 @ Y4 ) @ X5 ) ).
% inf_sup_ord(1)
thf(fact_1009_inf__sup__ord_I1_J,axiom,
! [X5: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X5 @ Y4 ) @ X5 ) ).
% inf_sup_ord(1)
thf(fact_1010_inf__sup__ord_I1_J,axiom,
! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X5 @ Y4 ) @ X5 ) ).
% inf_sup_ord(1)
thf(fact_1011_inf__sup__ord_I1_J,axiom,
! [X5: int,Y4: int] : ( ord_less_eq_int @ ( inf_inf_int @ X5 @ Y4 ) @ X5 ) ).
% inf_sup_ord(1)
thf(fact_1012_inf__le1,axiom,
! [X5: set_set_b,Y4: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ X5 @ Y4 ) @ X5 ) ).
% inf_le1
thf(fact_1013_inf__le1,axiom,
! [X5: set_set_a,Y4: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X5 @ Y4 ) @ X5 ) ).
% inf_le1
thf(fact_1014_inf__le1,axiom,
! [X5: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X5 @ Y4 ) @ X5 ) ).
% inf_le1
thf(fact_1015_inf__le1,axiom,
! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X5 @ Y4 ) @ X5 ) ).
% inf_le1
thf(fact_1016_inf__le1,axiom,
! [X5: int,Y4: int] : ( ord_less_eq_int @ ( inf_inf_int @ X5 @ Y4 ) @ X5 ) ).
% inf_le1
thf(fact_1017_inf__le2,axiom,
! [X5: set_set_b,Y4: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ X5 @ Y4 ) @ Y4 ) ).
% inf_le2
thf(fact_1018_inf__le2,axiom,
! [X5: set_set_a,Y4: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X5 @ Y4 ) @ Y4 ) ).
% inf_le2
thf(fact_1019_inf__le2,axiom,
! [X5: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X5 @ Y4 ) @ Y4 ) ).
% inf_le2
thf(fact_1020_inf__le2,axiom,
! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X5 @ Y4 ) @ Y4 ) ).
% inf_le2
thf(fact_1021_inf__le2,axiom,
! [X5: int,Y4: int] : ( ord_less_eq_int @ ( inf_inf_int @ X5 @ Y4 ) @ Y4 ) ).
% inf_le2
thf(fact_1022_le__infE,axiom,
! [X5: set_set_b,A: set_set_b,B2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X5 @ ( inf_inf_set_set_b @ A @ B2 ) )
=> ~ ( ( ord_le3795704787696855135_set_b @ X5 @ A )
=> ~ ( ord_le3795704787696855135_set_b @ X5 @ B2 ) ) ) ).
% le_infE
thf(fact_1023_le__infE,axiom,
! [X5: set_set_a,A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ ( inf_inf_set_set_a @ A @ B2 ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ X5 @ A )
=> ~ ( ord_le3724670747650509150_set_a @ X5 @ B2 ) ) ) ).
% le_infE
thf(fact_1024_le__infE,axiom,
! [X5: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X5 @ ( inf_inf_set_a @ A @ B2 ) )
=> ~ ( ( ord_less_eq_set_a @ X5 @ A )
=> ~ ( ord_less_eq_set_a @ X5 @ B2 ) ) ) ).
% le_infE
thf(fact_1025_le__infE,axiom,
! [X5: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ X5 @ ( inf_inf_nat @ A @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X5 @ A )
=> ~ ( ord_less_eq_nat @ X5 @ B2 ) ) ) ).
% le_infE
thf(fact_1026_le__infE,axiom,
! [X5: int,A: int,B2: int] :
( ( ord_less_eq_int @ X5 @ ( inf_inf_int @ A @ B2 ) )
=> ~ ( ( ord_less_eq_int @ X5 @ A )
=> ~ ( ord_less_eq_int @ X5 @ B2 ) ) ) ).
% le_infE
thf(fact_1027_le__infI,axiom,
! [X5: set_set_b,A: set_set_b,B2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X5 @ A )
=> ( ( ord_le3795704787696855135_set_b @ X5 @ B2 )
=> ( ord_le3795704787696855135_set_b @ X5 @ ( inf_inf_set_set_b @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_1028_le__infI,axiom,
! [X5: set_set_a,A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ A )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ B2 )
=> ( ord_le3724670747650509150_set_a @ X5 @ ( inf_inf_set_set_a @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_1029_le__infI,axiom,
! [X5: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X5 @ A )
=> ( ( ord_less_eq_set_a @ X5 @ B2 )
=> ( ord_less_eq_set_a @ X5 @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_1030_le__infI,axiom,
! [X5: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ X5 @ A )
=> ( ( ord_less_eq_nat @ X5 @ B2 )
=> ( ord_less_eq_nat @ X5 @ ( inf_inf_nat @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_1031_le__infI,axiom,
! [X5: int,A: int,B2: int] :
( ( ord_less_eq_int @ X5 @ A )
=> ( ( ord_less_eq_int @ X5 @ B2 )
=> ( ord_less_eq_int @ X5 @ ( inf_inf_int @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_1032_inf__mono,axiom,
! [A: set_set_b,C2: set_set_b,B2: set_set_b,D2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ C2 )
=> ( ( ord_le3795704787696855135_set_b @ B2 @ D2 )
=> ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A @ B2 ) @ ( inf_inf_set_set_b @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1033_inf__mono,axiom,
! [A: set_set_a,C2: set_set_a,B2: set_set_a,D2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ ( inf_inf_set_set_a @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1034_inf__mono,axiom,
! [A: set_a,C2: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1035_inf__mono,axiom,
! [A: nat,C2: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ ( inf_inf_nat @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1036_inf__mono,axiom,
! [A: int,C2: int,B2: int,D2: int] :
( ( ord_less_eq_int @ A @ C2 )
=> ( ( ord_less_eq_int @ B2 @ D2 )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B2 ) @ ( inf_inf_int @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1037_le__infI1,axiom,
! [A: set_set_b,X5: set_set_b,B2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ X5 )
=> ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A @ B2 ) @ X5 ) ) ).
% le_infI1
thf(fact_1038_le__infI1,axiom,
! [A: set_set_a,X5: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ X5 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ X5 ) ) ).
% le_infI1
thf(fact_1039_le__infI1,axiom,
! [A: set_a,X5: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ X5 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ X5 ) ) ).
% le_infI1
thf(fact_1040_le__infI1,axiom,
! [A: nat,X5: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ X5 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ X5 ) ) ).
% le_infI1
thf(fact_1041_le__infI1,axiom,
! [A: int,X5: int,B2: int] :
( ( ord_less_eq_int @ A @ X5 )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B2 ) @ X5 ) ) ).
% le_infI1
thf(fact_1042_le__infI2,axiom,
! [B2: set_set_b,X5: set_set_b,A: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B2 @ X5 )
=> ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A @ B2 ) @ X5 ) ) ).
% le_infI2
thf(fact_1043_le__infI2,axiom,
! [B2: set_set_a,X5: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ X5 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ X5 ) ) ).
% le_infI2
thf(fact_1044_le__infI2,axiom,
! [B2: set_a,X5: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ X5 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ X5 ) ) ).
% le_infI2
thf(fact_1045_le__infI2,axiom,
! [B2: nat,X5: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ X5 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ X5 ) ) ).
% le_infI2
thf(fact_1046_le__infI2,axiom,
! [B2: int,X5: int,A: int] :
( ( ord_less_eq_int @ B2 @ X5 )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B2 ) @ X5 ) ) ).
% le_infI2
thf(fact_1047_inf_OorderE,axiom,
! [A: set_set_b,B2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ B2 )
=> ( A
= ( inf_inf_set_set_b @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_1048_inf_OorderE,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( A
= ( inf_inf_set_set_a @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_1049_inf_OorderE,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( A
= ( inf_inf_set_a @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_1050_inf_OorderE,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( A
= ( inf_inf_nat @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_1051_inf_OorderE,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( A
= ( inf_inf_int @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_1052_inf_OorderI,axiom,
! [A: set_set_b,B2: set_set_b] :
( ( A
= ( inf_inf_set_set_b @ A @ B2 ) )
=> ( ord_le3795704787696855135_set_b @ A @ B2 ) ) ).
% inf.orderI
thf(fact_1053_inf_OorderI,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( A
= ( inf_inf_set_set_a @ A @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A @ B2 ) ) ).
% inf.orderI
thf(fact_1054_inf_OorderI,axiom,
! [A: set_a,B2: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B2 ) )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% inf.orderI
thf(fact_1055_inf_OorderI,axiom,
! [A: nat,B2: nat] :
( ( A
= ( inf_inf_nat @ A @ B2 ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% inf.orderI
thf(fact_1056_inf_OorderI,axiom,
! [A: int,B2: int] :
( ( A
= ( inf_inf_int @ A @ B2 ) )
=> ( ord_less_eq_int @ A @ B2 ) ) ).
% inf.orderI
thf(fact_1057_inf__unique,axiom,
! [F: set_set_b > set_set_b > set_set_b,X5: set_set_b,Y4: set_set_b] :
( ! [X4: set_set_b,Y6: set_set_b] : ( ord_le3795704787696855135_set_b @ ( F @ X4 @ Y6 ) @ X4 )
=> ( ! [X4: set_set_b,Y6: set_set_b] : ( ord_le3795704787696855135_set_b @ ( F @ X4 @ Y6 ) @ Y6 )
=> ( ! [X4: set_set_b,Y6: set_set_b,Z3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X4 @ Y6 )
=> ( ( ord_le3795704787696855135_set_b @ X4 @ Z3 )
=> ( ord_le3795704787696855135_set_b @ X4 @ ( F @ Y6 @ Z3 ) ) ) )
=> ( ( inf_inf_set_set_b @ X5 @ Y4 )
= ( F @ X5 @ Y4 ) ) ) ) ) ).
% inf_unique
thf(fact_1058_inf__unique,axiom,
! [F: set_set_a > set_set_a > set_set_a,X5: set_set_a,Y4: set_set_a] :
( ! [X4: set_set_a,Y6: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F @ X4 @ Y6 ) @ X4 )
=> ( ! [X4: set_set_a,Y6: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F @ X4 @ Y6 ) @ Y6 )
=> ( ! [X4: set_set_a,Y6: set_set_a,Z3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y6 )
=> ( ( ord_le3724670747650509150_set_a @ X4 @ Z3 )
=> ( ord_le3724670747650509150_set_a @ X4 @ ( F @ Y6 @ Z3 ) ) ) )
=> ( ( inf_inf_set_set_a @ X5 @ Y4 )
= ( F @ X5 @ Y4 ) ) ) ) ) ).
% inf_unique
thf(fact_1059_inf__unique,axiom,
! [F: set_a > set_a > set_a,X5: set_a,Y4: set_a] :
( ! [X4: set_a,Y6: set_a] : ( ord_less_eq_set_a @ ( F @ X4 @ Y6 ) @ X4 )
=> ( ! [X4: set_a,Y6: set_a] : ( ord_less_eq_set_a @ ( F @ X4 @ Y6 ) @ Y6 )
=> ( ! [X4: set_a,Y6: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y6 )
=> ( ( ord_less_eq_set_a @ X4 @ Z3 )
=> ( ord_less_eq_set_a @ X4 @ ( F @ Y6 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X5 @ Y4 )
= ( F @ X5 @ Y4 ) ) ) ) ) ).
% inf_unique
thf(fact_1060_inf__unique,axiom,
! [F: nat > nat > nat,X5: nat,Y4: nat] :
( ! [X4: nat,Y6: nat] : ( ord_less_eq_nat @ ( F @ X4 @ Y6 ) @ X4 )
=> ( ! [X4: nat,Y6: nat] : ( ord_less_eq_nat @ ( F @ X4 @ Y6 ) @ Y6 )
=> ( ! [X4: nat,Y6: nat,Z3: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
=> ( ( ord_less_eq_nat @ X4 @ Z3 )
=> ( ord_less_eq_nat @ X4 @ ( F @ Y6 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X5 @ Y4 )
= ( F @ X5 @ Y4 ) ) ) ) ) ).
% inf_unique
thf(fact_1061_inf__unique,axiom,
! [F: int > int > int,X5: int,Y4: int] :
( ! [X4: int,Y6: int] : ( ord_less_eq_int @ ( F @ X4 @ Y6 ) @ X4 )
=> ( ! [X4: int,Y6: int] : ( ord_less_eq_int @ ( F @ X4 @ Y6 ) @ Y6 )
=> ( ! [X4: int,Y6: int,Z3: int] :
( ( ord_less_eq_int @ X4 @ Y6 )
=> ( ( ord_less_eq_int @ X4 @ Z3 )
=> ( ord_less_eq_int @ X4 @ ( F @ Y6 @ Z3 ) ) ) )
=> ( ( inf_inf_int @ X5 @ Y4 )
= ( F @ X5 @ Y4 ) ) ) ) ) ).
% inf_unique
thf(fact_1062_le__iff__inf,axiom,
( ord_le3795704787696855135_set_b
= ( ^ [X3: set_set_b,Y3: set_set_b] :
( ( inf_inf_set_set_b @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_1063_le__iff__inf,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X3: set_set_a,Y3: set_set_a] :
( ( inf_inf_set_set_a @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_1064_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_1065_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y3: nat] :
( ( inf_inf_nat @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_1066_le__iff__inf,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Y3: int] :
( ( inf_inf_int @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_1067_inf_Oabsorb1,axiom,
! [A: set_set_b,B2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ B2 )
=> ( ( inf_inf_set_set_b @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_1068_inf_Oabsorb1,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( inf_inf_set_set_a @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_1069_inf_Oabsorb1,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( inf_inf_set_a @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_1070_inf_Oabsorb1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( inf_inf_nat @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_1071_inf_Oabsorb1,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( inf_inf_int @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_1072_inf_Oabsorb2,axiom,
! [B2: set_set_b,A: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B2 @ A )
=> ( ( inf_inf_set_set_b @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_1073_inf_Oabsorb2,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( ( inf_inf_set_set_a @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_1074_inf_Oabsorb2,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( inf_inf_set_a @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_1075_inf_Oabsorb2,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( inf_inf_nat @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_1076_inf_Oabsorb2,axiom,
! [B2: int,A: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( inf_inf_int @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_1077_inf__absorb1,axiom,
! [X5: set_set_b,Y4: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X5 @ Y4 )
=> ( ( inf_inf_set_set_b @ X5 @ Y4 )
= X5 ) ) ).
% inf_absorb1
thf(fact_1078_inf__absorb1,axiom,
! [X5: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ Y4 )
=> ( ( inf_inf_set_set_a @ X5 @ Y4 )
= X5 ) ) ).
% inf_absorb1
thf(fact_1079_inf__absorb1,axiom,
! [X5: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y4 )
=> ( ( inf_inf_set_a @ X5 @ Y4 )
= X5 ) ) ).
% inf_absorb1
thf(fact_1080_inf__absorb1,axiom,
! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ( inf_inf_nat @ X5 @ Y4 )
= X5 ) ) ).
% inf_absorb1
thf(fact_1081_inf__absorb1,axiom,
! [X5: int,Y4: int] :
( ( ord_less_eq_int @ X5 @ Y4 )
=> ( ( inf_inf_int @ X5 @ Y4 )
= X5 ) ) ).
% inf_absorb1
thf(fact_1082_inf__absorb2,axiom,
! [Y4: set_set_b,X5: set_set_b] :
( ( ord_le3795704787696855135_set_b @ Y4 @ X5 )
=> ( ( inf_inf_set_set_b @ X5 @ Y4 )
= Y4 ) ) ).
% inf_absorb2
thf(fact_1083_inf__absorb2,axiom,
! [Y4: set_set_a,X5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y4 @ X5 )
=> ( ( inf_inf_set_set_a @ X5 @ Y4 )
= Y4 ) ) ).
% inf_absorb2
thf(fact_1084_inf__absorb2,axiom,
! [Y4: set_a,X5: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X5 )
=> ( ( inf_inf_set_a @ X5 @ Y4 )
= Y4 ) ) ).
% inf_absorb2
thf(fact_1085_inf__absorb2,axiom,
! [Y4: nat,X5: nat] :
( ( ord_less_eq_nat @ Y4 @ X5 )
=> ( ( inf_inf_nat @ X5 @ Y4 )
= Y4 ) ) ).
% inf_absorb2
thf(fact_1086_inf__absorb2,axiom,
! [Y4: int,X5: int] :
( ( ord_less_eq_int @ Y4 @ X5 )
=> ( ( inf_inf_int @ X5 @ Y4 )
= Y4 ) ) ).
% inf_absorb2
thf(fact_1087_inf_OboundedE,axiom,
! [A: set_set_b,B2: set_set_b,C2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ ( inf_inf_set_set_b @ B2 @ C2 ) )
=> ~ ( ( ord_le3795704787696855135_set_b @ A @ B2 )
=> ~ ( ord_le3795704787696855135_set_b @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_1088_inf_OboundedE,axiom,
! [A: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( inf_inf_set_set_a @ B2 @ C2 ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ~ ( ord_le3724670747650509150_set_a @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_1089_inf_OboundedE,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B2 )
=> ~ ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_1090_inf_OboundedE,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A @ B2 )
=> ~ ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_1091_inf_OboundedE,axiom,
! [A: int,B2: int,C2: int] :
( ( ord_less_eq_int @ A @ ( inf_inf_int @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_int @ A @ B2 )
=> ~ ( ord_less_eq_int @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_1092_inf_OboundedI,axiom,
! [A: set_set_b,B2: set_set_b,C2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ B2 )
=> ( ( ord_le3795704787696855135_set_b @ A @ C2 )
=> ( ord_le3795704787696855135_set_b @ A @ ( inf_inf_set_set_b @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_1093_inf_OboundedI,axiom,
! [A: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A @ C2 )
=> ( ord_le3724670747650509150_set_a @ A @ ( inf_inf_set_set_a @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_1094_inf_OboundedI,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ A @ C2 )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_1095_inf_OboundedI,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_1096_inf_OboundedI,axiom,
! [A: int,B2: int,C2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ A @ C2 )
=> ( ord_less_eq_int @ A @ ( inf_inf_int @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_1097_inf__greatest,axiom,
! [X5: set_set_b,Y4: set_set_b,Z: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X5 @ Y4 )
=> ( ( ord_le3795704787696855135_set_b @ X5 @ Z )
=> ( ord_le3795704787696855135_set_b @ X5 @ ( inf_inf_set_set_b @ Y4 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_1098_inf__greatest,axiom,
! [X5: set_set_a,Y4: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ Y4 )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ Z )
=> ( ord_le3724670747650509150_set_a @ X5 @ ( inf_inf_set_set_a @ Y4 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_1099_inf__greatest,axiom,
! [X5: set_a,Y4: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y4 )
=> ( ( ord_less_eq_set_a @ X5 @ Z )
=> ( ord_less_eq_set_a @ X5 @ ( inf_inf_set_a @ Y4 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_1100_inf__greatest,axiom,
! [X5: nat,Y4: nat,Z: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ( ord_less_eq_nat @ X5 @ Z )
=> ( ord_less_eq_nat @ X5 @ ( inf_inf_nat @ Y4 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_1101_inf__greatest,axiom,
! [X5: int,Y4: int,Z: int] :
( ( ord_less_eq_int @ X5 @ Y4 )
=> ( ( ord_less_eq_int @ X5 @ Z )
=> ( ord_less_eq_int @ X5 @ ( inf_inf_int @ Y4 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_1102_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( A4
= ( inf_inf_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_1103_inf_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B4: int] :
( A4
= ( inf_inf_int @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_1104_finite__measure__compl,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ m ) )
=> ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ ( sigma_space_a @ m ) @ S ) )
= ( minus_minus_real @ ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) ) @ ( sigma_measure_a2 @ m @ S ) ) ) ) ).
% finite_measure_compl
thf(fact_1105_finite__measure__Diff_H,axiom,
! [A2: set_a,B: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_minus_real @ ( sigma_measure_a2 @ m @ A2 ) @ ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ) ) ).
% finite_measure_Diff'
thf(fact_1106_finite__measure__Diff,axiom,
! [A2: set_a,B: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_minus_real @ ( sigma_measure_a2 @ m @ A2 ) @ ( sigma_measure_a2 @ m @ B ) ) ) ) ) ) ).
% finite_measure_Diff
thf(fact_1107_prob__compl,axiom,
! [A2: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ ( sigma_space_a @ m ) @ A2 ) )
= ( minus_minus_real @ one_one_real @ ( sigma_measure_a2 @ m @ A2 ) ) ) ) ).
% prob_compl
thf(fact_1108_segment__bound__lemma,axiom,
! [B: real,X5: real,Y4: real,U: real] :
( ( ord_less_eq_real @ B @ X5 )
=> ( ( ord_less_eq_real @ B @ Y4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ U @ one_one_real )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ one_one_real @ U ) @ X5 ) @ ( times_times_real @ U @ Y4 ) ) ) ) ) ) ) ).
% segment_bound_lemma
thf(fact_1109_sum__le__prod1,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ B2 @ one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ A @ B2 ) ) ) ) ) ).
% sum_le_prod1
thf(fact_1110_sigma__algebra__tail__events,axiom,
! [A2: nat > set_set_a] :
( ! [I2: nat] : ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ m ) @ ( A2 @ I2 ) )
=> ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ m ) @ ( indepe7538416700049374166_a_nat @ m @ A2 ) ) ) ).
% sigma_algebra_tail_events
thf(fact_1111_kolmogorov__0__1__law,axiom,
! [A2: nat > set_set_a,X: set_a] :
( ! [I2: nat] : ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ m ) @ ( A2 @ I2 ) )
=> ( ( indepe6267730027088848354_a_nat @ m @ A2 @ top_top_set_nat )
=> ( ( member_set_a @ X @ ( indepe7538416700049374166_a_nat @ m @ A2 ) )
=> ( ( ( sigma_measure_a2 @ m @ X )
= zero_zero_real )
| ( ( sigma_measure_a2 @ m @ X )
= one_one_real ) ) ) ) ) ).
% kolmogorov_0_1_law
thf(fact_1112_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X4: nat > real] :
( ( P @ X4 )
=> ( P @ ( F @ X4 ) ) )
=> ( ! [X4: nat > real] :
( ( P @ X4 )
=> ! [I2: nat] :
( ( Q @ I2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X4 @ I2 ) )
& ( ord_less_eq_real @ ( X4 @ I2 ) @ one_one_real ) ) ) )
=> ? [L2: ( nat > real ) > nat > nat] :
( ! [X6: nat > real,I4: nat] : ( ord_less_eq_nat @ ( L2 @ X6 @ I4 ) @ one_one_nat )
& ! [X6: nat > real,I4: nat] :
( ( ( P @ X6 )
& ( Q @ I4 )
& ( ( X6 @ I4 )
= zero_zero_real ) )
=> ( ( L2 @ X6 @ I4 )
= zero_zero_nat ) )
& ! [X6: nat > real,I4: nat] :
( ( ( P @ X6 )
& ( Q @ I4 )
& ( ( X6 @ I4 )
= one_one_real ) )
=> ( ( L2 @ X6 @ I4 )
= one_one_nat ) )
& ! [X6: nat > real,I4: nat] :
( ( ( P @ X6 )
& ( Q @ I4 )
& ( ( L2 @ X6 @ I4 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X6 @ I4 ) @ ( F @ X6 @ I4 ) ) )
& ! [X6: nat > real,I4: nat] :
( ( ( P @ X6 )
& ( Q @ I4 )
& ( ( L2 @ X6 @ I4 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X6 @ I4 ) @ ( X6 @ I4 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1113_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1114_le0,axiom,
! [N6: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N6 ) ).
% le0
thf(fact_1115_diff__is__0__eq_H,axiom,
! [M4: nat,N6: nat] :
( ( ord_less_eq_nat @ M4 @ N6 )
=> ( ( minus_minus_nat @ M4 @ N6 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1116_diff__is__0__eq,axiom,
! [M4: nat,N6: nat] :
( ( ( minus_minus_nat @ M4 @ N6 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M4 @ N6 ) ) ).
% diff_is_0_eq
thf(fact_1117_nat__add__left__cancel__le,axiom,
! [K2: nat,M4: nat,N6: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M4 ) @ ( plus_plus_nat @ K2 @ N6 ) )
= ( ord_less_eq_nat @ M4 @ N6 ) ) ).
% nat_add_left_cancel_le
thf(fact_1118_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J2 @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J2 ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1119_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K2 ) @ I3 )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I3 ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1120_Nat_Odiff__diff__right,axiom,
! [K2: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( minus_minus_nat @ I3 @ ( minus_minus_nat @ J2 @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K2 ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_1121_diff__diff__cancel,axiom,
! [I3: nat,N6: nat] :
( ( ord_less_eq_nat @ I3 @ N6 )
=> ( ( minus_minus_nat @ N6 @ ( minus_minus_nat @ N6 @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_1122_indep__set__def,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( indepe2041756565122539606_set_a @ m @ A2 @ B )
= ( indepe7780107833195774214ts_a_o @ m @ ( produc6113963288868236716_set_a @ A2 @ B ) @ top_top_set_o ) ) ).
% indep_set_def
thf(fact_1123_eq__diff__iff,axiom,
! [K2: nat,M4: nat,N6: nat] :
( ( ord_less_eq_nat @ K2 @ M4 )
=> ( ( ord_less_eq_nat @ K2 @ N6 )
=> ( ( ( minus_minus_nat @ M4 @ K2 )
= ( minus_minus_nat @ N6 @ K2 ) )
= ( M4 = N6 ) ) ) ) ).
% eq_diff_iff
thf(fact_1124_le__diff__iff,axiom,
! [K2: nat,M4: nat,N6: nat] :
( ( ord_less_eq_nat @ K2 @ M4 )
=> ( ( ord_less_eq_nat @ K2 @ N6 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ K2 ) @ ( minus_minus_nat @ N6 @ K2 ) )
= ( ord_less_eq_nat @ M4 @ N6 ) ) ) ) ).
% le_diff_iff
thf(fact_1125_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M4: nat,N6: nat] :
( ( ord_less_eq_nat @ K2 @ M4 )
=> ( ( ord_less_eq_nat @ K2 @ N6 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M4 @ K2 ) @ ( minus_minus_nat @ N6 @ K2 ) )
= ( minus_minus_nat @ M4 @ N6 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1126_diff__le__mono,axiom,
! [M4: nat,N6: nat,L3: nat] :
( ( ord_less_eq_nat @ M4 @ N6 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ L3 ) @ ( minus_minus_nat @ N6 @ L3 ) ) ) ).
% diff_le_mono
thf(fact_1127_diff__le__self,axiom,
! [M4: nat,N6: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ N6 ) @ M4 ) ).
% diff_le_self
thf(fact_1128_le__diff__conv,axiom,
! [J2: nat,K2: nat,I3: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K2 ) @ I3 )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I3 @ K2 ) ) ) ).
% le_diff_conv
thf(fact_1129_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1130_diff__le__mono2,axiom,
! [M4: nat,N6: nat,L3: nat] :
( ( ord_less_eq_nat @ M4 @ N6 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L3 @ N6 ) @ ( minus_minus_nat @ L3 @ M4 ) ) ) ).
% diff_le_mono2
thf(fact_1131_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J2 @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1132_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J2 ) @ K2 )
= ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J2 @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1133_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I3 ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K2 ) @ I3 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1134_Nat_Ole__imp__diff__is__add,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I3 )
= K2 )
= ( J2
= ( plus_plus_nat @ K2 @ I3 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1135_le__cube,axiom,
! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ) ).
% le_cube
thf(fact_1136_le__square,axiom,
! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ).
% le_square
thf(fact_1137_mult__le__mono,axiom,
! [I3: nat,J2: nat,K2: nat,L3: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ K2 @ L3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K2 ) @ ( times_times_nat @ J2 @ L3 ) ) ) ) ).
% mult_le_mono
thf(fact_1138_mult__le__mono1,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K2 ) @ ( times_times_nat @ J2 @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_1139_mult__le__mono2,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I3 ) @ ( times_times_nat @ K2 @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_1140_le__refl,axiom,
! [N6: nat] : ( ord_less_eq_nat @ N6 @ N6 ) ).
% le_refl
thf(fact_1141_le__trans,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K2 )
=> ( ord_less_eq_nat @ I3 @ K2 ) ) ) ).
% le_trans
thf(fact_1142_eq__imp__le,axiom,
! [M4: nat,N6: nat] :
( ( M4 = N6 )
=> ( ord_less_eq_nat @ M4 @ N6 ) ) ).
% eq_imp_le
thf(fact_1143_le__antisym,axiom,
! [M4: nat,N6: nat] :
( ( ord_less_eq_nat @ M4 @ N6 )
=> ( ( ord_less_eq_nat @ N6 @ M4 )
=> ( M4 = N6 ) ) ) ).
% le_antisym
thf(fact_1144_nat__le__linear,axiom,
! [M4: nat,N6: nat] :
( ( ord_less_eq_nat @ M4 @ N6 )
| ( ord_less_eq_nat @ N6 @ M4 ) ) ).
% nat_le_linear
thf(fact_1145_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B2: nat] :
( ( P @ K2 )
=> ( ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ B2 ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y7: nat] :
( ( P @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1146_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N7: nat] :
? [K3: nat] :
( N7
= ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1147_trans__le__add2,axiom,
! [I3: nat,J2: nat,M4: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ M4 @ J2 ) ) ) ).
% trans_le_add2
thf(fact_1148_trans__le__add1,axiom,
! [I3: nat,J2: nat,M4: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ J2 @ M4 ) ) ) ).
% trans_le_add1
thf(fact_1149_add__le__mono1,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J2 @ K2 ) ) ) ).
% add_le_mono1
thf(fact_1150_add__le__mono,axiom,
! [I3: nat,J2: nat,K2: nat,L3: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ K2 @ L3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J2 @ L3 ) ) ) ) ).
% add_le_mono
thf(fact_1151_le__Suc__ex,axiom,
! [K2: nat,L3: nat] :
( ( ord_less_eq_nat @ K2 @ L3 )
=> ? [N8: nat] :
( L3
= ( plus_plus_nat @ K2 @ N8 ) ) ) ).
% le_Suc_ex
thf(fact_1152_add__leD2,axiom,
! [M4: nat,K2: nat,N6: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K2 ) @ N6 )
=> ( ord_less_eq_nat @ K2 @ N6 ) ) ).
% add_leD2
thf(fact_1153_add__leD1,axiom,
! [M4: nat,K2: nat,N6: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K2 ) @ N6 )
=> ( ord_less_eq_nat @ M4 @ N6 ) ) ).
% add_leD1
thf(fact_1154_le__add2,axiom,
! [N6: nat,M4: nat] : ( ord_less_eq_nat @ N6 @ ( plus_plus_nat @ M4 @ N6 ) ) ).
% le_add2
thf(fact_1155_le__add1,axiom,
! [N6: nat,M4: nat] : ( ord_less_eq_nat @ N6 @ ( plus_plus_nat @ N6 @ M4 ) ) ).
% le_add1
thf(fact_1156_add__leE,axiom,
! [M4: nat,K2: nat,N6: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K2 ) @ N6 )
=> ~ ( ( ord_less_eq_nat @ M4 @ N6 )
=> ~ ( ord_less_eq_nat @ K2 @ N6 ) ) ) ).
% add_leE
thf(fact_1157_le__0__eq,axiom,
! [N6: nat] :
( ( ord_less_eq_nat @ N6 @ zero_zero_nat )
= ( N6 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1158_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1159_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1160_less__eq__nat_Osimps_I1_J,axiom,
! [N6: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N6 ) ).
% less_eq_nat.simps(1)
thf(fact_1161_nat__eq__add__iff1,axiom,
! [J2: nat,I3: nat,U: nat,M4: nat,N6: nat] :
( ( ord_less_eq_nat @ J2 @ I3 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M4 )
= ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N6 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J2 ) @ U ) @ M4 )
= N6 ) ) ) ).
% nat_eq_add_iff1
thf(fact_1162_nat__eq__add__iff2,axiom,
! [I3: nat,J2: nat,U: nat,M4: nat,N6: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M4 )
= ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N6 ) )
= ( M4
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I3 ) @ U ) @ N6 ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1163_nat__le__add__iff1,axiom,
! [J2: nat,I3: nat,U: nat,M4: nat,N6: nat] :
( ( ord_less_eq_nat @ J2 @ I3 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N6 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J2 ) @ U ) @ M4 ) @ N6 ) ) ) ).
% nat_le_add_iff1
thf(fact_1164_nat__diff__add__eq2,axiom,
! [I3: nat,J2: nat,U: nat,M4: nat,N6: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N6 ) )
= ( minus_minus_nat @ M4 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I3 ) @ U ) @ N6 ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1165_nat__diff__add__eq1,axiom,
! [J2: nat,I3: nat,U: nat,M4: nat,N6: nat] :
( ( ord_less_eq_nat @ J2 @ I3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N6 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J2 ) @ U ) @ M4 ) @ N6 ) ) ) ).
% nat_diff_add_eq1
thf(fact_1166_nat__le__add__iff2,axiom,
! [I3: nat,J2: nat,U: nat,M4: nat,N6: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N6 ) )
= ( ord_less_eq_nat @ M4 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I3 ) @ U ) @ N6 ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1167_indep__event__def,axiom,
! [A2: set_a,B: set_a] :
( ( indepe3567167809233210430vent_a @ m @ A2 @ B )
= ( indepe3695496658712714478ts_a_o @ m @ ( produc2496386666562076748_set_a @ A2 @ B ) @ top_top_set_o ) ) ).
% indep_event_def
thf(fact_1168_Bolzano,axiom,
! [A: real,B2: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ! [A3: real,B3: real,C3: real] :
( ( P @ A3 @ B3 )
=> ( ( P @ B3 @ C3 )
=> ( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C3 )
=> ( P @ A3 @ C3 ) ) ) ) )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B2 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [A3: real,B3: real] :
( ( ( ord_less_eq_real @ A3 @ X4 )
& ( ord_less_eq_real @ X4 @ B3 )
& ( ord_less_real @ ( minus_minus_real @ B3 @ A3 ) @ D3 ) )
=> ( P @ A3 @ B3 ) ) ) ) )
=> ( P @ A @ B2 ) ) ) ) ).
% Bolzano
thf(fact_1169_mult__le__cancel2,axiom,
! [M4: nat,K2: nat,N6: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M4 @ K2 ) @ ( times_times_nat @ N6 @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M4 @ N6 ) ) ) ).
% mult_le_cancel2
thf(fact_1170_nat__mult__le__cancel__disj,axiom,
! [K2: nat,M4: nat,N6: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M4 ) @ ( times_times_nat @ K2 @ N6 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M4 @ N6 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1171_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N7: nat] :
( ( ord_less_eq_nat @ M5 @ N7 )
& ( M5 != N7 ) ) ) ) ).
% nat_less_le
thf(fact_1172_less__imp__le__nat,axiom,
! [M4: nat,N6: nat] :
( ( ord_less_nat @ M4 @ N6 )
=> ( ord_less_eq_nat @ M4 @ N6 ) ) ).
% less_imp_le_nat
thf(fact_1173_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N7: nat] :
( ( ord_less_nat @ M5 @ N7 )
| ( M5 = N7 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1174_less__or__eq__imp__le,axiom,
! [M4: nat,N6: nat] :
( ( ( ord_less_nat @ M4 @ N6 )
| ( M4 = N6 ) )
=> ( ord_less_eq_nat @ M4 @ N6 ) ) ).
% less_or_eq_imp_le
thf(fact_1175_le__neq__implies__less,axiom,
! [M4: nat,N6: nat] :
( ( ord_less_eq_nat @ M4 @ N6 )
=> ( ( M4 != N6 )
=> ( ord_less_nat @ M4 @ N6 ) ) ) ).
% le_neq_implies_less
thf(fact_1176_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J2: nat] :
( ! [I2: nat,J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1177_ex__least__nat__le,axiom,
! [P: nat > $o,N6: nat] :
( ( P @ N6 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N6 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K4 )
=> ~ ( P @ I4 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1178_mono__nat__linear__lb,axiom,
! [F: nat > nat,M4: nat,K2: nat] :
( ! [M6: nat,N8: nat] :
( ( ord_less_nat @ M6 @ N8 )
=> ( ord_less_nat @ ( F @ M6 ) @ ( F @ N8 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M4 ) @ K2 ) @ ( F @ ( plus_plus_nat @ M4 @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1179_less__diff__iff,axiom,
! [K2: nat,M4: nat,N6: nat] :
( ( ord_less_eq_nat @ K2 @ M4 )
=> ( ( ord_less_eq_nat @ K2 @ N6 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M4 @ K2 ) @ ( minus_minus_nat @ N6 @ K2 ) )
= ( ord_less_nat @ M4 @ N6 ) ) ) ) ).
% less_diff_iff
thf(fact_1180_diff__less__mono,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B2 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1181_nat__mult__le__cancel1,axiom,
! [K2: nat,M4: nat,N6: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M4 ) @ ( times_times_nat @ K2 @ N6 ) )
= ( ord_less_eq_nat @ M4 @ N6 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1182_less__diff__conv2,axiom,
! [K2: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K2 ) @ I3 )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I3 @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_1183_kuhn__lemma,axiom,
! [P2: nat,N6: nat,Label: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ P2 )
=> ( ! [X4: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N6 )
=> ( ord_less_eq_nat @ ( X4 @ I4 ) @ P2 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N6 )
=> ( ( ( Label @ X4 @ I2 )
= zero_zero_nat )
| ( ( Label @ X4 @ I2 )
= one_one_nat ) ) ) )
=> ( ! [X4: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N6 )
=> ( ord_less_eq_nat @ ( X4 @ I4 ) @ P2 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N6 )
=> ( ( ( X4 @ I2 )
= zero_zero_nat )
=> ( ( Label @ X4 @ I2 )
= zero_zero_nat ) ) ) )
=> ( ! [X4: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N6 )
=> ( ord_less_eq_nat @ ( X4 @ I4 ) @ P2 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N6 )
=> ( ( ( X4 @ I2 )
= P2 )
=> ( ( Label @ X4 @ I2 )
= one_one_nat ) ) ) )
=> ~ ! [Q2: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N6 )
=> ( ord_less_nat @ ( Q2 @ I4 ) @ P2 ) )
=> ~ ! [I4: nat] :
( ( ord_less_nat @ I4 @ N6 )
=> ? [R: nat > nat] :
( ! [J4: nat] :
( ( ord_less_nat @ J4 @ N6 )
=> ( ( ord_less_eq_nat @ ( Q2 @ J4 ) @ ( R @ J4 ) )
& ( ord_less_eq_nat @ ( R @ J4 ) @ ( plus_plus_nat @ ( Q2 @ J4 ) @ one_one_nat ) ) ) )
& ? [S4: nat > nat] :
( ! [J4: nat] :
( ( ord_less_nat @ J4 @ N6 )
=> ( ( ord_less_eq_nat @ ( Q2 @ J4 ) @ ( S4 @ J4 ) )
& ( ord_less_eq_nat @ ( S4 @ J4 ) @ ( plus_plus_nat @ ( Q2 @ J4 ) @ one_one_nat ) ) ) )
& ( ( Label @ R @ I4 )
!= ( Label @ S4 @ I4 ) ) ) ) ) ) ) ) ) ) ).
% kuhn_lemma
thf(fact_1184_nat__less__add__iff1,axiom,
! [J2: nat,I3: nat,U: nat,M4: nat,N6: nat] :
( ( ord_less_eq_nat @ J2 @ I3 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N6 ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J2 ) @ U ) @ M4 ) @ N6 ) ) ) ).
% nat_less_add_iff1
thf(fact_1185_nat__less__add__iff2,axiom,
! [I3: nat,J2: nat,U: nat,M4: nat,N6: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N6 ) )
= ( ord_less_nat @ M4 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I3 ) @ U ) @ N6 ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1186_seq__mono__lemma,axiom,
! [M4: nat,D2: nat > real,E: nat > real] :
( ! [N8: nat] :
( ( ord_less_eq_nat @ M4 @ N8 )
=> ( ord_less_real @ ( D2 @ N8 ) @ ( E @ N8 ) ) )
=> ( ! [N8: nat] :
( ( ord_less_eq_nat @ M4 @ N8 )
=> ( ord_less_eq_real @ ( E @ N8 ) @ ( E @ M4 ) ) )
=> ! [N9: nat] :
( ( ord_less_eq_nat @ M4 @ N9 )
=> ( ord_less_real @ ( D2 @ N9 ) @ ( E @ M4 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_1187_lemma__interval,axiom,
! [A: real,X5: real,B2: real] :
( ( ord_less_real @ A @ X5 )
=> ( ( ord_less_real @ X5 @ B2 )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [Y7: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X5 @ Y7 ) ) @ D4 )
=> ( ( ord_less_eq_real @ A @ Y7 )
& ( ord_less_eq_real @ Y7 @ B2 ) ) ) ) ) ) ).
% lemma_interval
thf(fact_1188_sin__bound__lemma,axiom,
! [X5: real,Y4: real,U: real,V: real] :
( ( X5 = Y4 )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X5 @ U ) @ Y4 ) ) @ V ) ) ) ).
% sin_bound_lemma
thf(fact_1189_nat0__intermed__int__val,axiom,
! [N6: nat,F: nat > int,K2: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N6 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N6 ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N6 )
& ( ( F @ I2 )
= K2 ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1190_emeasure__space__1,axiom,
( ( sigma_emeasure_a @ m @ ( sigma_space_a @ m ) )
= one_on2969667320475766781nnreal ) ).
% emeasure_space_1
thf(fact_1191_finite__emeasure__space,axiom,
( ( sigma_emeasure_a @ m @ ( sigma_space_a @ m ) )
!= top_to1496364449551166952nnreal ) ).
% finite_emeasure_space
thf(fact_1192_subprob__emeasure__le__1,axiom,
! [X: set_a] : ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ m @ X ) @ one_on2969667320475766781nnreal ) ).
% subprob_emeasure_le_1
thf(fact_1193_emeasure__le__1,axiom,
! [S: set_a] : ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ m @ S ) @ one_on2969667320475766781nnreal ) ).
% emeasure_le_1
thf(fact_1194_emeasure__ge__1__iff,axiom,
! [A2: set_a] :
( ( ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ ( sigma_emeasure_a @ m @ A2 ) )
= ( ( sigma_emeasure_a @ m @ A2 )
= one_on2969667320475766781nnreal ) ) ).
% emeasure_ge_1_iff
thf(fact_1195_emeasure__space__le__1,axiom,
ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ m @ ( sigma_space_a @ m ) ) @ one_on2969667320475766781nnreal ).
% emeasure_space_le_1
thf(fact_1196_zle__add1__eq__le,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ).
% zle_add1_eq_le
thf(fact_1197_emeasure__subprob__space__less__top,axiom,
! [A2: set_a] :
( ( sigma_emeasure_a @ m @ A2 )
!= top_to1496364449551166952nnreal ) ).
% emeasure_subprob_space_less_top
thf(fact_1198_zle__diff1__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W2 @ Z ) ) ).
% zle_diff1_eq
thf(fact_1199_int__ge__induct,axiom,
! [K2: int,I3: int,P: int > $o] :
( ( ord_less_eq_int @ K2 @ I3 )
=> ( ( P @ K2 )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K2 @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_ge_induct
thf(fact_1200_int__induct,axiom,
! [P: int > $o,K2: int,I3: int] :
( ( P @ K2 )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K2 @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K2 )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_induct
thf(fact_1201_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1202_int__le__induct,axiom,
! [I3: int,K2: int,P: int > $o] :
( ( ord_less_eq_int @ I3 @ K2 )
=> ( ( P @ K2 )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K2 )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_le_induct
thf(fact_1203_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_1204_ennreal__approx__unit,axiom,
! [Z: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ! [A3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ A3 )
=> ( ( ord_le7381754540660121996nnreal @ A3 @ one_on2969667320475766781nnreal )
=> ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ A3 @ Z ) @ Y4 ) ) )
=> ( ord_le3935885782089961368nnreal @ Z @ Y4 ) ) ).
% ennreal_approx_unit
thf(fact_1205_add1__zle__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z )
= ( ord_less_int @ W2 @ Z ) ) ).
% add1_zle_eq
thf(fact_1206_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_1207_zless__imp__add1__zle,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ W2 @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_1208_emeasure__real,axiom,
! [A2: set_a] :
? [R: real] :
( ( ord_less_eq_real @ zero_zero_real @ R )
& ( ( sigma_emeasure_a @ m @ A2 )
= ( extend7643940197134561352nnreal @ R ) ) ) ).
% emeasure_real
thf(fact_1209_emeasure__eq__measure,axiom,
! [A2: set_a] :
( ( sigma_emeasure_a @ m @ A2 )
= ( extend7643940197134561352nnreal @ ( sigma_measure_a2 @ m @ A2 ) ) ) ).
% emeasure_eq_measure
thf(fact_1210_add__diff__eq__iff__ennreal,axiom,
! [X5: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ( plus_p1859984266308609217nnreal @ X5 @ ( minus_8429688780609304081nnreal @ Y4 @ X5 ) )
= Y4 )
= ( ord_le3935885782089961368nnreal @ X5 @ Y4 ) ) ).
% add_diff_eq_iff_ennreal
thf(fact_1211_ennreal__inj,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ( ( extend7643940197134561352nnreal @ A )
= ( extend7643940197134561352nnreal @ B2 ) )
= ( A = B2 ) ) ) ) ).
% ennreal_inj
thf(fact_1212_ennreal__eq__zero__iff,axiom,
! [X5: real] :
( ( ord_less_eq_real @ zero_zero_real @ X5 )
=> ( ( ( extend7643940197134561352nnreal @ X5 )
= zero_z7100319975126383169nnreal )
= ( X5 = zero_zero_real ) ) ) ).
% ennreal_eq_zero_iff
thf(fact_1213_ennreal__le__iff,axiom,
! [Y4: real,X5: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ( ord_le3935885782089961368nnreal @ ( extend7643940197134561352nnreal @ X5 ) @ ( extend7643940197134561352nnreal @ Y4 ) )
= ( ord_less_eq_real @ X5 @ Y4 ) ) ) ).
% ennreal_le_iff
thf(fact_1214_ennreal__plus,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ( extend7643940197134561352nnreal @ ( plus_plus_real @ A @ B2 ) )
= ( plus_p1859984266308609217nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B2 ) ) ) ) ) ).
% ennreal_plus
thf(fact_1215_ennreal__ge__1,axiom,
! [X5: real] :
( ( ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ ( extend7643940197134561352nnreal @ X5 ) )
= ( ord_less_eq_real @ one_one_real @ X5 ) ) ).
% ennreal_ge_1
thf(fact_1216_ennreal__le__1,axiom,
! [X5: real] :
( ( ord_le3935885782089961368nnreal @ ( extend7643940197134561352nnreal @ X5 ) @ one_on2969667320475766781nnreal )
= ( ord_less_eq_real @ X5 @ one_one_real ) ) ).
% ennreal_le_1
thf(fact_1217_add__diff__inverse__ennreal,axiom,
! [X5: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X5 @ Y4 )
=> ( ( plus_p1859984266308609217nnreal @ X5 @ ( minus_8429688780609304081nnreal @ Y4 @ X5 ) )
= Y4 ) ) ).
% add_diff_inverse_ennreal
thf(fact_1218_diff__add__cancel__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B2 )
=> ( ( plus_p1859984266308609217nnreal @ ( minus_8429688780609304081nnreal @ B2 @ A ) @ A )
= B2 ) ) ).
% diff_add_cancel_ennreal
thf(fact_1219_diff__add__assoc2__ennreal,axiom,
! [B2: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B2 @ A )
=> ( ( plus_p1859984266308609217nnreal @ ( minus_8429688780609304081nnreal @ A @ B2 ) @ C2 )
= ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ A @ C2 ) @ B2 ) ) ) ).
% diff_add_assoc2_ennreal
thf(fact_1220_ennreal__diff__add__assoc,axiom,
! [A: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B2 )
=> ( ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ C2 @ B2 ) @ A )
= ( plus_p1859984266308609217nnreal @ C2 @ ( minus_8429688780609304081nnreal @ B2 @ A ) ) ) ) ).
% ennreal_diff_add_assoc
thf(fact_1221_ennreal__ineq__diff__add,axiom,
! [B2: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B2 @ A )
=> ( A
= ( plus_p1859984266308609217nnreal @ B2 @ ( minus_8429688780609304081nnreal @ A @ B2 ) ) ) ) ).
% ennreal_ineq_diff_add
thf(fact_1222_diff__add__self__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ( ord_le3935885782089961368nnreal @ A @ B2 )
=> ( ( plus_p1859984266308609217nnreal @ ( minus_8429688780609304081nnreal @ B2 @ A ) @ A )
= B2 ) )
& ( ~ ( ord_le3935885782089961368nnreal @ A @ B2 )
=> ( ( plus_p1859984266308609217nnreal @ ( minus_8429688780609304081nnreal @ B2 @ A ) @ A )
= A ) ) ) ).
% diff_add_self_ennreal
thf(fact_1223_add__diff__self__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ( ord_le3935885782089961368nnreal @ A @ B2 )
=> ( ( plus_p1859984266308609217nnreal @ A @ ( minus_8429688780609304081nnreal @ B2 @ A ) )
= B2 ) )
& ( ~ ( ord_le3935885782089961368nnreal @ A @ B2 )
=> ( ( plus_p1859984266308609217nnreal @ A @ ( minus_8429688780609304081nnreal @ B2 @ A ) )
= A ) ) ) ).
% add_diff_self_ennreal
thf(fact_1224_ennreal__minus__le__iff,axiom,
! [A: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B2 ) @ C2 )
= ( ( ord_le3935885782089961368nnreal @ A @ ( plus_p1859984266308609217nnreal @ B2 @ C2 ) )
& ( ( ( A = top_to1496364449551166952nnreal )
& ( B2 = top_to1496364449551166952nnreal ) )
=> ( C2 = top_to1496364449551166952nnreal ) ) ) ) ).
% ennreal_minus_le_iff
thf(fact_1225_ennreal__le__minus__iff,axiom,
! [A: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ ( minus_8429688780609304081nnreal @ B2 @ C2 ) )
= ( ( ord_le3935885782089961368nnreal @ ( plus_p1859984266308609217nnreal @ A @ C2 ) @ B2 )
| ( ( A = zero_z7100319975126383169nnreal )
& ( ord_le3935885782089961368nnreal @ B2 @ C2 ) ) ) ) ).
% ennreal_le_minus_iff
thf(fact_1226_diff__diff__ennreal_H_H,axiom,
! [Z: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal,X5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Z @ Y4 )
=> ( ( ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ Y4 @ Z ) @ X5 )
=> ( ( minus_8429688780609304081nnreal @ X5 @ ( minus_8429688780609304081nnreal @ Y4 @ Z ) )
= ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ X5 @ Z ) @ Y4 ) ) )
& ( ~ ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ Y4 @ Z ) @ X5 )
=> ( ( minus_8429688780609304081nnreal @ X5 @ ( minus_8429688780609304081nnreal @ Y4 @ Z ) )
= zero_z7100319975126383169nnreal ) ) ) ) ).
% diff_diff_ennreal''
thf(fact_1227_add__diff__le__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ A @ B2 ) @ C2 ) @ ( plus_p1859984266308609217nnreal @ A @ ( minus_8429688780609304081nnreal @ B2 @ C2 ) ) ) ).
% add_diff_le_ennreal
thf(fact_1228_add__diff__eq__ennreal,axiom,
! [Z: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal,X5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Z @ Y4 )
=> ( ( plus_p1859984266308609217nnreal @ X5 @ ( minus_8429688780609304081nnreal @ Y4 @ Z ) )
= ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ X5 @ Y4 ) @ Z ) ) ) ).
% add_diff_eq_ennreal
thf(fact_1229_diff__diff__ennreal_H,axiom,
! [Z: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal,X5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Z @ Y4 )
=> ( ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ Y4 @ Z ) @ X5 )
=> ( ( minus_8429688780609304081nnreal @ X5 @ ( minus_8429688780609304081nnreal @ Y4 @ Z ) )
= ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ X5 @ Z ) @ Y4 ) ) ) ) ).
% diff_diff_ennreal'
thf(fact_1230_ennreal__neg,axiom,
! [X5: real] :
( ( ord_less_eq_real @ X5 @ zero_zero_real )
=> ( ( extend7643940197134561352nnreal @ X5 )
= zero_z7100319975126383169nnreal ) ) ).
% ennreal_neg
thf(fact_1231_ennreal__eq__0__iff,axiom,
! [X5: real] :
( ( ( extend7643940197134561352nnreal @ X5 )
= zero_z7100319975126383169nnreal )
= ( ord_less_eq_real @ X5 @ zero_zero_real ) ) ).
% ennreal_eq_0_iff
thf(fact_1232_ennreal__le__iff2,axiom,
! [X5: real,Y4: real] :
( ( ord_le3935885782089961368nnreal @ ( extend7643940197134561352nnreal @ X5 ) @ ( extend7643940197134561352nnreal @ Y4 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
& ( ord_less_eq_real @ X5 @ Y4 ) )
| ( ( ord_less_eq_real @ X5 @ zero_zero_real )
& ( ord_less_eq_real @ Y4 @ zero_zero_real ) ) ) ) ).
% ennreal_le_iff2
thf(fact_1233_le__ennreal__iff,axiom,
! [R2: real,X5: extend8495563244428889912nnreal] :
( ( ord_less_eq_real @ zero_zero_real @ R2 )
=> ( ( ord_le3935885782089961368nnreal @ X5 @ ( extend7643940197134561352nnreal @ R2 ) )
= ( ? [Q3: real] :
( ( ord_less_eq_real @ zero_zero_real @ Q3 )
& ( X5
= ( extend7643940197134561352nnreal @ Q3 ) )
& ( ord_less_eq_real @ Q3 @ R2 ) ) ) ) ) ).
% le_ennreal_iff
thf(fact_1234_mult__right__ennreal__cancel,axiom,
! [A: extend8495563244428889912nnreal,C2: real,B2: extend8495563244428889912nnreal] :
( ( ( times_1893300245718287421nnreal @ A @ ( extend7643940197134561352nnreal @ C2 ) )
= ( times_1893300245718287421nnreal @ B2 @ ( extend7643940197134561352nnreal @ C2 ) ) )
= ( ( A = B2 )
| ( ord_less_eq_real @ C2 @ zero_zero_real ) ) ) ).
% mult_right_ennreal_cancel
thf(fact_1235_ennreal__cases,axiom,
! [X5: extend8495563244428889912nnreal] :
( ! [R: real] :
( ( ord_less_eq_real @ zero_zero_real @ R )
=> ( X5
!= ( extend7643940197134561352nnreal @ R ) ) )
=> ( X5 = top_to1496364449551166952nnreal ) ) ).
% ennreal_cases
thf(fact_1236_ennreal2__cases,axiom,
! [X5: extend8495563244428889912nnreal,Xa: extend8495563244428889912nnreal] :
( ! [R: real] :
( ( ord_less_eq_real @ zero_zero_real @ R )
=> ( ( X5
= ( extend7643940197134561352nnreal @ R ) )
=> ! [Ra: real] :
( ( ord_less_eq_real @ zero_zero_real @ Ra )
=> ( Xa
!= ( extend7643940197134561352nnreal @ Ra ) ) ) ) )
=> ( ! [R: real] :
( ( ord_less_eq_real @ zero_zero_real @ R )
=> ( ( X5
= ( extend7643940197134561352nnreal @ R ) )
=> ( Xa != top_to1496364449551166952nnreal ) ) )
=> ( ( ( X5 = top_to1496364449551166952nnreal )
=> ! [R: real] :
( ( ord_less_eq_real @ zero_zero_real @ R )
=> ( Xa
!= ( extend7643940197134561352nnreal @ R ) ) ) )
=> ~ ( ( X5 = top_to1496364449551166952nnreal )
=> ( Xa != top_to1496364449551166952nnreal ) ) ) ) ) ).
% ennreal2_cases
thf(fact_1237_ennreal3__cases,axiom,
! [X5: extend8495563244428889912nnreal,Xa: extend8495563244428889912nnreal,Xaa: extend8495563244428889912nnreal] :
( ! [R: real] :
( ( ord_less_eq_real @ zero_zero_real @ R )
=> ( ( X5
= ( extend7643940197134561352nnreal @ R ) )
=> ! [Ra: real] :
( ( ord_less_eq_real @ zero_zero_real @ Ra )
=> ( ( Xa
= ( extend7643940197134561352nnreal @ Ra ) )
=> ! [Raa: real] :
( ( ord_less_eq_real @ zero_zero_real @ Raa )
=> ( Xaa
!= ( extend7643940197134561352nnreal @ Raa ) ) ) ) ) ) )
=> ( ! [R: real] :
( ( ord_less_eq_real @ zero_zero_real @ R )
=> ( ( X5
= ( extend7643940197134561352nnreal @ R ) )
=> ! [Ra: real] :
( ( ord_less_eq_real @ zero_zero_real @ Ra )
=> ( ( Xa
= ( extend7643940197134561352nnreal @ Ra ) )
=> ( Xaa != top_to1496364449551166952nnreal ) ) ) ) )
=> ( ! [R: real] :
( ( ord_less_eq_real @ zero_zero_real @ R )
=> ( ( X5
= ( extend7643940197134561352nnreal @ R ) )
=> ( ( Xa = top_to1496364449551166952nnreal )
=> ! [Ra: real] :
( ( ord_less_eq_real @ zero_zero_real @ Ra )
=> ( Xaa
!= ( extend7643940197134561352nnreal @ Ra ) ) ) ) ) )
=> ( ! [R: real] :
( ( ord_less_eq_real @ zero_zero_real @ R )
=> ( ( X5
= ( extend7643940197134561352nnreal @ R ) )
=> ( ( Xa = top_to1496364449551166952nnreal )
=> ( Xaa != top_to1496364449551166952nnreal ) ) ) )
=> ( ( ( X5 = top_to1496364449551166952nnreal )
=> ! [R: real] :
( ( ord_less_eq_real @ zero_zero_real @ R )
=> ( ( Xa
= ( extend7643940197134561352nnreal @ R ) )
=> ! [Ra: real] :
( ( ord_less_eq_real @ zero_zero_real @ Ra )
=> ( Xaa
!= ( extend7643940197134561352nnreal @ Ra ) ) ) ) ) )
=> ( ( ( X5 = top_to1496364449551166952nnreal )
=> ! [R: real] :
( ( ord_less_eq_real @ zero_zero_real @ R )
=> ( ( Xa
= ( extend7643940197134561352nnreal @ R ) )
=> ( Xaa != top_to1496364449551166952nnreal ) ) ) )
=> ( ( ( X5 = top_to1496364449551166952nnreal )
=> ( ( Xa = top_to1496364449551166952nnreal )
=> ! [R: real] :
( ( ord_less_eq_real @ zero_zero_real @ R )
=> ( Xaa
!= ( extend7643940197134561352nnreal @ R ) ) ) ) )
=> ~ ( ( X5 = top_to1496364449551166952nnreal )
=> ( ( Xa = top_to1496364449551166952nnreal )
=> ( Xaa != top_to1496364449551166952nnreal ) ) ) ) ) ) ) ) ) ) ).
% ennreal3_cases
thf(fact_1238_ennreal__minus__cancel,axiom,
! [C2: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( C2 != top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ A @ C2 )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ C2 )
=> ( ( ( minus_8429688780609304081nnreal @ C2 @ A )
= ( minus_8429688780609304081nnreal @ C2 @ B2 ) )
=> ( A = B2 ) ) ) ) ) ).
% ennreal_minus_cancel
thf(fact_1239_ennreal__minus__cancel__iff,axiom,
! [A: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A @ B2 )
= ( minus_8429688780609304081nnreal @ A @ C2 ) )
= ( ( B2 = C2 )
| ( ( ord_le3935885782089961368nnreal @ A @ B2 )
& ( ord_le3935885782089961368nnreal @ A @ C2 ) )
| ( A = top_to1496364449551166952nnreal ) ) ) ).
% ennreal_minus_cancel_iff
thf(fact_1240_neq__top__trans,axiom,
! [Y4: extend8495563244428889912nnreal,X5: extend8495563244428889912nnreal] :
( ( Y4 != top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ X5 @ Y4 )
=> ( X5 != top_to1496364449551166952nnreal ) ) ) ).
% neq_top_trans
thf(fact_1241_ennreal__leI,axiom,
! [X5: real,Y4: real] :
( ( ord_less_eq_real @ X5 @ Y4 )
=> ( ord_le3935885782089961368nnreal @ ( extend7643940197134561352nnreal @ X5 ) @ ( extend7643940197134561352nnreal @ Y4 ) ) ) ).
% ennreal_leI
thf(fact_1242_ennreal__diff__le__mono__left,axiom,
! [A: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B2 )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ C2 ) @ B2 ) ) ).
% ennreal_diff_le_mono_left
thf(fact_1243_diff__le__self__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B2 ) @ A ) ).
% diff_le_self_ennreal
thf(fact_1244_ennreal__mono__minus,axiom,
! [C2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ C2 @ B2 )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B2 ) @ ( minus_8429688780609304081nnreal @ A @ C2 ) ) ) ).
% ennreal_mono_minus
thf(fact_1245_ennreal__minus__mono,axiom,
! [A: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal,D2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ C2 )
=> ( ( ord_le3935885782089961368nnreal @ D2 @ B2 )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B2 ) @ ( minus_8429688780609304081nnreal @ C2 @ D2 ) ) ) ) ).
% ennreal_minus_mono
thf(fact_1246_ennreal__minus__eq__0,axiom,
! [A: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A @ B2 )
= zero_z7100319975126383169nnreal )
=> ( ord_le3935885782089961368nnreal @ A @ B2 ) ) ).
% ennreal_minus_eq_0
thf(fact_1247_ennreal__mult__le__mult__iff,axiom,
! [C2: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( C2 != zero_z7100319975126383169nnreal )
=> ( ( C2 != top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ C2 @ A ) @ ( times_1893300245718287421nnreal @ C2 @ B2 ) )
= ( ord_le3935885782089961368nnreal @ A @ B2 ) ) ) ) ).
% ennreal_mult_le_mult_iff
thf(fact_1248_less__top__ennreal,axiom,
! [X5: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X5 @ top_to1496364449551166952nnreal )
= ( ? [R3: real] :
( ( ord_less_eq_real @ zero_zero_real @ R3 )
& ( X5
= ( extend7643940197134561352nnreal @ R3 ) ) ) ) ) ).
% less_top_ennreal
thf(fact_1249_ennreal__le__epsilon,axiom,
! [Y4: extend8495563244428889912nnreal,X5: extend8495563244428889912nnreal] :
( ! [E2: real] :
( ( ord_le7381754540660121996nnreal @ Y4 @ top_to1496364449551166952nnreal )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( ord_le3935885782089961368nnreal @ X5 @ ( plus_p1859984266308609217nnreal @ Y4 @ ( extend7643940197134561352nnreal @ E2 ) ) ) ) )
=> ( ord_le3935885782089961368nnreal @ X5 @ Y4 ) ) ).
% ennreal_le_epsilon
thf(fact_1250_ennreal__mono__minus__cancel,axiom,
! [A: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B2 ) @ ( minus_8429688780609304081nnreal @ A @ C2 ) )
=> ( ( ord_le7381754540660121996nnreal @ A @ top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ A )
=> ( ( ord_le3935885782089961368nnreal @ C2 @ A )
=> ( ord_le3935885782089961368nnreal @ C2 @ B2 ) ) ) ) ) ).
% ennreal_mono_minus_cancel
thf(fact_1251_minus__less__iff__ennreal,axiom,
! [B2: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B2 @ top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ B2 @ A )
=> ( ( ord_le7381754540660121996nnreal @ ( minus_8429688780609304081nnreal @ A @ B2 ) @ C2 )
= ( ord_le7381754540660121996nnreal @ A @ ( plus_p1859984266308609217nnreal @ C2 @ B2 ) ) ) ) ) ).
% minus_less_iff_ennreal
thf(fact_1252_diff__eq__0__iff__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A @ B2 )
= zero_z7100319975126383169nnreal )
= ( ( ord_le7381754540660121996nnreal @ A @ top_to1496364449551166952nnreal )
& ( ord_le3935885782089961368nnreal @ A @ B2 ) ) ) ).
% diff_eq_0_iff_ennreal
thf(fact_1253_diff__eq__0__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ A @ B2 )
=> ( ( minus_8429688780609304081nnreal @ A @ B2 )
= zero_z7100319975126383169nnreal ) ) ) ).
% diff_eq_0_ennreal
thf(fact_1254_ennreal__less__iff,axiom,
! [R2: real,Q4: real] :
( ( ord_less_eq_real @ zero_zero_real @ R2 )
=> ( ( ord_le7381754540660121996nnreal @ ( extend7643940197134561352nnreal @ R2 ) @ ( extend7643940197134561352nnreal @ Q4 ) )
= ( ord_less_real @ R2 @ Q4 ) ) ) ).
% ennreal_less_iff
thf(fact_1255_ennreal__mult,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ( extend7643940197134561352nnreal @ ( times_times_real @ A @ B2 ) )
= ( times_1893300245718287421nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B2 ) ) ) ) ) ).
% ennreal_mult
thf(fact_1256_ennreal__mult_H,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( extend7643940197134561352nnreal @ ( times_times_real @ A @ B2 ) )
= ( times_1893300245718287421nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B2 ) ) ) ) ).
% ennreal_mult'
thf(fact_1257_ennreal__mult_H_H,axiom,
! [B2: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ( extend7643940197134561352nnreal @ ( times_times_real @ A @ B2 ) )
= ( times_1893300245718287421nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B2 ) ) ) ) ).
% ennreal_mult''
thf(fact_1258_ennreal__plus__if,axiom,
! [A: real,B2: real] :
( ( plus_p1859984266308609217nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B2 ) )
= ( extend7643940197134561352nnreal @ ( if_real @ ( ord_less_eq_real @ zero_zero_real @ A ) @ ( if_real @ ( ord_less_eq_real @ zero_zero_real @ B2 ) @ ( plus_plus_real @ A @ B2 ) @ A ) @ B2 ) ) ) ).
% ennreal_plus_if
thf(fact_1259_ennreal__minus__if,axiom,
! [A: real,B2: real] :
( ( minus_8429688780609304081nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B2 ) )
= ( extend7643940197134561352nnreal @ ( if_real @ ( ord_less_eq_real @ zero_zero_real @ B2 ) @ ( if_real @ ( ord_less_eq_real @ B2 @ A ) @ ( minus_minus_real @ A @ B2 ) @ zero_zero_real ) @ A ) ) ) ).
% ennreal_minus_if
thf(fact_1260_ennreal__minus,axiom,
! [Q4: real,R2: real] :
( ( ord_less_eq_real @ zero_zero_real @ Q4 )
=> ( ( minus_8429688780609304081nnreal @ ( extend7643940197134561352nnreal @ R2 ) @ ( extend7643940197134561352nnreal @ Q4 ) )
= ( extend7643940197134561352nnreal @ ( minus_minus_real @ R2 @ Q4 ) ) ) ) ).
% ennreal_minus
thf(fact_1261_approx__PInf__emeasure__with__finite,axiom,
! [W3: set_a,C: real] :
( ( member_set_a @ W3 @ ( sigma_sets_a @ m ) )
=> ( ( ( sigma_emeasure_a @ m @ W3 )
= extend2057119558705770725nnreal )
=> ~ ! [Z4: set_a] :
( ( member_set_a @ Z4 @ ( sigma_sets_a @ m ) )
=> ( ( ord_less_eq_set_a @ Z4 @ W3 )
=> ( ( ord_le7381754540660121996nnreal @ ( sigma_emeasure_a @ m @ Z4 ) @ extend2057119558705770725nnreal )
=> ~ ( ord_le7381754540660121996nnreal @ ( extend7643940197134561352nnreal @ C ) @ ( sigma_emeasure_a @ m @ Z4 ) ) ) ) ) ) ) ).
% approx_PInf_emeasure_with_finite
thf(fact_1262_decr__mult__lemma,axiom,
! [D2: int,P: int > $o,K2: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( minus_minus_int @ X4 @ D2 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ! [X6: int] :
( ( P @ X6 )
=> ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K2 @ D2 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1263_ennreal__add__left__cancel__le,axiom,
! [A: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ ( plus_p1859984266308609217nnreal @ A @ B2 ) @ ( plus_p1859984266308609217nnreal @ A @ C2 ) )
= ( ( A = extend2057119558705770725nnreal )
| ( ord_le3935885782089961368nnreal @ B2 @ C2 ) ) ) ).
% ennreal_add_left_cancel_le
thf(fact_1264_diff__diff__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B2 )
=> ( ( B2 != extend2057119558705770725nnreal )
=> ( ( minus_8429688780609304081nnreal @ B2 @ ( minus_8429688780609304081nnreal @ B2 @ A ) )
= A ) ) ) ).
% diff_diff_ennreal
thf(fact_1265_imp__le__cong,axiom,
! [X5: int,X7: int,P: $o,P3: $o] :
( ( X5 = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P3 ) ) ) ) ).
% imp_le_cong
thf(fact_1266_conj__le__cong,axiom,
! [X5: int,X7: int,P: $o,P3: $o] :
( ( X5 = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P3 ) ) ) ) ).
% conj_le_cong
thf(fact_1267_incr__mult__lemma,axiom,
! [D2: int,P: int > $o,K2: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( plus_plus_int @ X4 @ D2 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ! [X6: int] :
( ( P @ X6 )
=> ( P @ ( plus_plus_int @ X6 @ ( times_times_int @ K2 @ D2 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1268_enn2real__le__iff,axiom,
! [X5: extend8495563244428889912nnreal,C2: real] :
( ( ord_le7381754540660121996nnreal @ X5 @ top_to1496364449551166952nnreal )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ ( extend1669699412028896998n2real @ X5 ) @ C2 )
= ( ord_le3935885782089961368nnreal @ X5 @ ( extend7643940197134561352nnreal @ C2 ) ) ) ) ) ).
% enn2real_le_iff
thf(fact_1269_enn2real__ennreal,axiom,
! [R2: real] :
( ( ord_less_eq_real @ zero_zero_real @ R2 )
=> ( ( extend1669699412028896998n2real @ ( extend7643940197134561352nnreal @ R2 ) )
= R2 ) ) ).
% enn2real_ennreal
thf(fact_1270_enn2real__nonneg,axiom,
! [X5: extend8495563244428889912nnreal] : ( ord_less_eq_real @ zero_zero_real @ ( extend1669699412028896998n2real @ X5 ) ) ).
% enn2real_nonneg
thf(fact_1271_enn2real__leI,axiom,
! [B: real,X5: extend8495563244428889912nnreal] :
( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_le3935885782089961368nnreal @ X5 @ ( extend7643940197134561352nnreal @ B ) )
=> ( ord_less_eq_real @ ( extend1669699412028896998n2real @ X5 ) @ B ) ) ) ).
% enn2real_leI
thf(fact_1272_enn2real__le,axiom,
! [E: extend8495563244428889912nnreal,R2: real] :
( ( ord_less_eq_real @ ( extend1669699412028896998n2real @ E ) @ R2 )
=> ( ( E != top_to1496364449551166952nnreal )
=> ( ord_le3935885782089961368nnreal @ E @ ( extend7643940197134561352nnreal @ R2 ) ) ) ) ).
% enn2real_le
thf(fact_1273_enn2real__mono,axiom,
! [A: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B2 )
=> ( ( ord_le7381754540660121996nnreal @ B2 @ top_to1496364449551166952nnreal )
=> ( ord_less_eq_real @ ( extend1669699412028896998n2real @ A ) @ ( extend1669699412028896998n2real @ B2 ) ) ) ) ).
% enn2real_mono
thf(fact_1274_enn2real__plus,axiom,
! [A: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ top_to1496364449551166952nnreal )
=> ( ( ord_le7381754540660121996nnreal @ B2 @ top_to1496364449551166952nnreal )
=> ( ( extend1669699412028896998n2real @ ( plus_p1859984266308609217nnreal @ A @ B2 ) )
= ( plus_plus_real @ ( extend1669699412028896998n2real @ A ) @ ( extend1669699412028896998n2real @ B2 ) ) ) ) ) ).
% enn2real_plus
% Helper facts (3)
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X5: real,Y4: real] :
( ( if_real @ $false @ X5 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X5: real,Y4: real] :
( ( if_real @ $true @ X5 @ Y4 )
= X5 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
probab7247484486040049090pace_b @ ( measure_distr_a_b @ m @ n @ f2 ) ).
%------------------------------------------------------------------------------