TPTP Problem File: SLH0653^1.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_00033_001125__18388602_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1558 ( 530 unt; 277 typ; 0 def)
% Number of atoms : 3711 (1135 equ; 0 cnn)
% Maximal formula atoms : 26 ( 2 avg)
% Number of connectives : 10985 ( 199 ~; 37 |; 115 &;8861 @)
% ( 0 <=>;1773 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 7 avg)
% Number of types : 40 ( 39 usr)
% Number of type conns : 864 ( 864 >; 0 *; 0 +; 0 <<)
% Number of symbols : 241 ( 238 usr; 33 con; 0-5 aty)
% Number of variables : 3474 ( 201 ^;3247 !; 26 ?;3474 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:43:24.816
%------------------------------------------------------------------------------
% Could-be-implicit typings (39)
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,
sigma_measure_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
set_real_real: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__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__Real__Oreal_Mt__Nat__Onat_J_J,type,
set_real_nat: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
sigma_measure_real: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Nat__Onat_J,type,
sigma_measure_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__c_Mt__Real__Oreal_J_J,type,
set_c_real: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mtf__c_J_J,type,
set_real_c: $tType ).
thf(ty_n_t__Extended____Nonnegative____Real__Oennreal,type,
extend8495563244428889912nnreal: $tType ).
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__Real__Oreal_M_Eo_J_J,type,
set_real_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_Mt__Real__Oreal_J_J,type,
set_o_real: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_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__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_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__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 (238)
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_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__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_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__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__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__Real__Oreal_J,type,
minus_minus_set_real: set_real > set_real > set_real ).
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__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__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
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__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__Real__Oreal,type,
indepe2510005150887255998a_real: sigma_measure_a > ( real > set_set_a ) > set_real > $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_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__Real__Oreal,type,
indepe8106223766075386687b_real: sigma_measure_b > ( real > set_set_b ) > set_real > $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_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_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__Extended____Nonnegative____Real__Oennreal,type,
inf_in7439215052339218890nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
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__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__Real__Oreal_J,type,
inf_inf_set_real: set_real > set_real > set_real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
inf_inf_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_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__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
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__Real__Oreal_001tf__a,type,
measure_distr_real_a: sigma_measure_real > sigma_measure_a > ( real > a ) > sigma_measure_a ).
thf(sy_c_Measure__Space_Odistr_001t__Real__Oreal_001tf__b,type,
measure_distr_real_b: sigma_measure_real > sigma_measure_b > ( real > b ) > sigma_measure_b ).
thf(sy_c_Measure__Space_Odistr_001tf__a_001tf__a,type,
measure_distr_a_a: sigma_measure_a > sigma_measure_a > ( a > a ) > sigma_measure_a ).
thf(sy_c_Measure__Space_Odistr_001tf__a_001tf__b,type,
measure_distr_a_b: sigma_measure_a > sigma_measure_b > ( a > b ) > sigma_measure_b ).
thf(sy_c_Measure__Space_Odistr_001tf__b_001tf__b,type,
measure_distr_b_b: sigma_measure_b > sigma_measure_b > ( b > b ) > sigma_measure_b ).
thf(sy_c_Measure__Space_Odistr_001tf__c_001tf__a,type,
measure_distr_c_a: sigma_measure_c > sigma_measure_a > ( c > a ) > sigma_measure_a ).
thf(sy_c_Measure__Space_Odistr_001tf__c_001tf__b,type,
measure_distr_c_b: sigma_measure_c > sigma_measure_b > ( c > b ) > sigma_measure_b ).
thf(sy_c_Measure__Space_Ofinite__measure_001t__Set__Oset_Itf__a_J,type,
measur2212693993384602946_set_a: sigma_measure_set_a > $o ).
thf(sy_c_Measure__Space_Ofinite__measure_001t__Set__Oset_Itf__b_J,type,
measur2212693997687831747_set_b: sigma_measure_set_b > $o ).
thf(sy_c_Measure__Space_Ofinite__measure_001tf__a,type,
measur930452917991658466sure_a: sigma_measure_a > $o ).
thf(sy_c_Measure__Space_Ofinite__measure_001tf__b,type,
measur930452917991658467sure_b: sigma_measure_b > $o ).
thf(sy_c_Measure__Space_Ofinite__measure__axioms_001tf__a,type,
measur2595372213310369023ioms_a: sigma_measure_a > $o ).
thf(sy_c_Measure__Space_Ofmeasurable_001t__Set__Oset_Itf__a_J,type,
measur7460903245211743562_set_a: sigma_measure_set_a > set_set_set_a ).
thf(sy_c_Measure__Space_Ofmeasurable_001t__Set__Oset_Itf__b_J,type,
measur7460903249514972363_set_b: sigma_measure_set_b > set_set_set_b ).
thf(sy_c_Measure__Space_Ofmeasurable_001tf__a,type,
measur3645360004775918570able_a: sigma_measure_a > set_set_a ).
thf(sy_c_Measure__Space_Ofmeasurable_001tf__b,type,
measur3645360004775918571able_b: sigma_measure_b > set_set_b ).
thf(sy_c_Measure__Space_Oincreasing_001t__Set__Oset_Itf__a_J_001t__Int__Oint,type,
measur1242461429550240791_a_int: set_set_set_a > ( set_set_a > int ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001t__Set__Oset_Itf__a_J_001t__Nat__Onat,type,
measur1244951900059291067_a_nat: set_set_set_a > ( set_set_a > nat ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
measur5181028491126448947_set_a: set_set_set_a > ( set_set_a > set_a ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001t__Set__Oset_Itf__b_J_001t__Int__Oint,type,
measur6838680044738371480_b_int: set_set_set_b > ( set_set_b > int ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001t__Set__Oset_Itf__b_J_001t__Nat__Onat,type,
measur6841170515247421756_b_nat: set_set_set_b > ( set_set_b > nat ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__a_J,type,
measur6668990528421247474_set_a: set_set_set_b > ( set_set_b > set_a ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001tf__a_001t__Int__Oint,type,
measur8148950955492825783_a_int: set_set_a > ( set_a > int ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001tf__a_001t__Nat__Onat,type,
measur8151441426001876059_a_nat: set_set_a > ( set_a > nat ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001tf__a_001t__Real__Oreal,type,
measur1776380161843274167a_real: set_set_a > ( set_a > real ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001tf__a_001t__Set__Oset_Itf__a_J,type,
measur7842569353079325843_set_a: set_set_a > ( set_a > set_a ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001tf__b_001t__Real__Oreal,type,
measur7372598777031404856b_real: set_set_b > ( set_b > real ) > $o ).
thf(sy_c_Measure__Space_Onull__sets_001t__Set__Oset_Itf__a_J,type,
measur1516554128032400785_set_a: sigma_measure_set_a > set_set_set_a ).
thf(sy_c_Measure__Space_Onull__sets_001t__Set__Oset_Itf__b_J,type,
measur1516554132335629586_set_b: sigma_measure_set_b > set_set_set_b ).
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,
measur4308613598931908895sure_a: sigma_measure_a > $o ).
thf(sy_c_Measure__Space_Osigma__finite__measure_001tf__b,type,
measur4308613598931908896sure_b: sigma_measure_b > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
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__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
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,
bot_bot_set_c: set_c ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Sigma____Algebra__Omeasure_Itf__a_J,type,
bot_bo2108912051383640591sure_a: sigma_measure_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nonnegative____Real__Oennreal,type,
ord_le7381754540660121996nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ).
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,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nonnegative____Real__Oennreal,type,
ord_le3935885782089961368nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__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__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
ord_le5722252365846178494_set_a: set_set_set_a > set_set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__b_J_J_J,type,
ord_le3201067847557142847_set_b: set_set_set_b > set_set_set_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
ord_le3795704787696855135_set_b: set_set_b > set_set_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__b_J,type,
ord_less_eq_set_b: set_b > set_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__c_J,type,
ord_less_eq_set_c: set_c > set_c > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Sigma____Algebra__Omeasure_It__Set__Oset_Itf__a_J_J,type,
ord_le5642585610961328955_set_a: sigma_measure_set_a > sigma_measure_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Sigma____Algebra__Omeasure_It__Set__Oset_Itf__b_J_J,type,
ord_le5713619651007674940_set_b: sigma_measure_set_b > sigma_measure_set_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Sigma____Algebra__Omeasure_Itf__a_J,type,
ord_le254669795585780187sure_a: sigma_measure_a > sigma_measure_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Sigma____Algebra__Omeasure_Itf__b_J,type,
ord_le254669799889008988sure_b: sigma_measure_b > sigma_measure_b > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Extended____Nonnegative____Real__Oennreal,type,
top_to1496364449551166952nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
top_top_set_a_b: set_a_b ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
top_top_set_o: set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
top_top_set_real: set_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
top_top_set_set_a: set_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
top_top_set_set_b: set_set_b ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__c_J,type,
top_top_set_c: set_c ).
thf(sy_c_Probability__Measure_Oprob__space_001tf__a,type,
probab7247484486040049089pace_a: sigma_measure_a > $o ).
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_Product__Type_Obool_Ocase__bool_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
produc6113963288868236716_set_a: set_set_a > set_set_a > $o > set_set_a ).
thf(sy_c_Product__Type_Obool_Ocase__bool_001t__Set__Oset_Itf__a_J,type,
produc2496386666562076748_set_a: set_a > set_a > $o > set_a ).
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,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__b_J,type,
collect_set_b: ( set_b > $o ) > set_set_b ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OCollect_001tf__c,type,
collect_c: ( c > $o ) > set_c ).
thf(sy_c_Sigma__Algebra_Oemeasure_001tf__a,type,
sigma_emeasure_a: sigma_measure_a > set_a > extend8495563244428889912nnreal ).
thf(sy_c_Sigma__Algebra_Omeasurable_001_Eo_001t__Real__Oreal,type,
sigma_2430008634441611636o_real: sigma_measure_o > sigma_measure_real > set_o_real ).
thf(sy_c_Sigma__Algebra_Omeasurable_001_Eo_001tf__c,type,
sigma_measurable_o_c: sigma_measure_o > sigma_measure_c > set_o_c ).
thf(sy_c_Sigma__Algebra_Omeasurable_001t__Real__Oreal_001_Eo,type,
sigma_3939073009482781210real_o: sigma_measure_real > sigma_measure_o > set_real_o ).
thf(sy_c_Sigma__Algebra_Omeasurable_001t__Real__Oreal_001t__Nat__Onat,type,
sigma_6315060578831106510al_nat: sigma_measure_real > sigma_measure_nat > set_real_nat ).
thf(sy_c_Sigma__Algebra_Omeasurable_001t__Real__Oreal_001t__Real__Oreal,type,
sigma_5267869275261027754l_real: sigma_measure_real > sigma_measure_real > set_real_real ).
thf(sy_c_Sigma__Algebra_Omeasurable_001t__Real__Oreal_001tf__c,type,
sigma_523072396149930114real_c: sigma_measure_real > sigma_measure_c > set_real_c ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__a_001tf__a,type,
sigma_measurable_a_a: sigma_measure_a > sigma_measure_a > set_a_a ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__a_001tf__b,type,
sigma_measurable_a_b: sigma_measure_a > sigma_measure_b > set_a_b ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__b_001tf__a,type,
sigma_measurable_b_a: sigma_measure_b > sigma_measure_a > set_b_a ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__b_001tf__b,type,
sigma_measurable_b_b: sigma_measure_b > sigma_measure_b > set_b_b ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__c_001_Eo,type,
sigma_measurable_c_o: sigma_measure_c > sigma_measure_o > set_c_o ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__c_001t__Nat__Onat,type,
sigma_2544038740538346112_c_nat: sigma_measure_c > sigma_measure_nat > set_c_nat ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__c_001t__Real__Oreal,type,
sigma_1862118822198465884c_real: sigma_measure_c > sigma_measure_real > set_c_real ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__c_001tf__c,type,
sigma_measurable_c_c: sigma_measure_c > sigma_measure_c > set_c_c ).
thf(sy_c_Sigma__Algebra_Omeasure_001t__Set__Oset_Itf__a_J,type,
sigma_measure_set_a2: sigma_measure_set_a > set_set_a > real ).
thf(sy_c_Sigma__Algebra_Omeasure_001t__Set__Oset_Itf__b_J,type,
sigma_measure_set_b2: sigma_measure_set_b > set_set_b > real ).
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,
sigma_measure_b2: sigma_measure_b > set_b > real ).
thf(sy_c_Sigma__Algebra_Osets_001t__Set__Oset_Itf__a_J,type,
sigma_sets_set_a: sigma_measure_set_a > set_set_set_a ).
thf(sy_c_Sigma__Algebra_Osets_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,
sigma_4968961713055010667ebra_a: set_a > set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Osigma__algebra_001tf__b,type,
sigma_4968961713055010668ebra_b: set_b > set_set_b > $o ).
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__Real__Oreal,type,
sigma_space_real: sigma_measure_real > set_real ).
thf(sy_c_Sigma__Algebra_Ospace_001t__Set__Oset_Itf__a_J,type,
sigma_space_set_a: sigma_measure_set_a > set_set_a ).
thf(sy_c_Sigma__Algebra_Ospace_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_Mt__Real__Oreal_J,type,
member_o_real: ( $o > real ) > set_o_real > $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__Real__Oreal_M_Eo_J,type,
member_real_o: ( real > $o ) > set_real_o > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
member_real_nat: ( real > nat ) > set_real_nat > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
member_real_real: ( real > real ) > set_real_real > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_Mtf__c_J,type,
member_real_c: ( real > c ) > set_real_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__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_Mt__Real__Oreal_J,type,
member_c_real: ( c > real ) > set_c_real > $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__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_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 (1277)
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_indep__F,axiom,
indepe8927441866673418606ts_a_c @ m @ f @ i ).
% indep_F
thf(fact_3_sets__A,axiom,
! [I: c] :
( ( member_c @ I @ i )
=> ( ord_le3795704787696855135_set_b @ ( a2 @ I ) @ ( sigma_sets_b @ n ) ) ) ).
% sets_A
thf(fact_4_subprob__space__axioms,axiom,
giry_subprob_space_a @ m ).
% subprob_space_axioms
thf(fact_5_indep__sets__cong,axiom,
! [I2: set_a_b,J: set_a_b,F: ( a > b ) > set_set_a,G: ( a > b ) > set_set_a] :
( ( I2 = J )
=> ( ! [I3: a > b] :
( ( member_a_b @ I3 @ I2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( indepe1219004915316159480_a_a_b @ m @ F @ I2 )
= ( indepe1219004915316159480_a_a_b @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_6_indep__sets__cong,axiom,
! [I2: set_set_a,J: set_set_a,F: set_a > set_set_a,G: set_a > set_set_a] :
( ( I2 = J )
=> ( ! [I3: set_a] :
( ( member_set_a @ I3 @ I2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( indepe4967106450811773644_set_a @ m @ F @ I2 )
= ( indepe4967106450811773644_set_a @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_7_indep__sets__cong,axiom,
! [I2: set_real,J: set_real,F: real > set_set_a,G: real > set_set_a] :
( ( I2 = J )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( indepe2510005150887255998a_real @ m @ F @ I2 )
= ( indepe2510005150887255998a_real @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_8_indep__sets__cong,axiom,
! [I2: set_c,J: set_c,F: c > set_set_a,G: c > set_set_a] :
( ( I2 = J )
=> ( ! [I3: c] :
( ( member_c @ I3 @ I2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( indepe8927441866673418606ts_a_c @ m @ F @ I2 )
= ( indepe8927441866673418606ts_a_c @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_9_indep__sets__cong,axiom,
! [I2: set_nat,J: set_nat,F: nat > set_set_a,G: nat > set_set_a] :
( ( I2 = J )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( indepe6267730027088848354_a_nat @ m @ F @ I2 )
= ( indepe6267730027088848354_a_nat @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_10_indep__sets__cong,axiom,
! [I2: set_o,J: set_o,F: $o > set_set_a,G: $o > set_set_a] :
( ( I2 = J )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( indepe7780107833195774214ts_a_o @ m @ F @ I2 )
= ( indepe7780107833195774214ts_a_o @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_11_prob__space_Oindep__sets_Ocong,axiom,
indepe2510005150887255998a_real = indepe2510005150887255998a_real ).
% prob_space.indep_sets.cong
thf(fact_12_prob__space_Oindep__sets_Ocong,axiom,
indepe4967106450811773644_set_a = indepe4967106450811773644_set_a ).
% prob_space.indep_sets.cong
thf(fact_13_prob__space_Oindep__sets_Ocong,axiom,
indepe1219004915316159480_a_a_b = indepe1219004915316159480_a_a_b ).
% prob_space.indep_sets.cong
thf(fact_14_prob__space_Oindep__sets_Ocong,axiom,
indepe6139986284700742573ts_b_c = indepe6139986284700742573ts_b_c ).
% prob_space.indep_sets.cong
thf(fact_15_prob__space_Oindep__sets_Ocong,axiom,
indepe8927441866673418606ts_a_c = indepe8927441866673418606ts_a_c ).
% prob_space.indep_sets.cong
thf(fact_16_prob__space_Oindep__sets_Ocong,axiom,
indepe6267730027088848354_a_nat = indepe6267730027088848354_a_nat ).
% prob_space.indep_sets.cong
thf(fact_17_prob__space_Oindep__sets_Ocong,axiom,
indepe7780107833195774214ts_a_o = indepe7780107833195774214ts_a_o ).
% prob_space.indep_sets.cong
thf(fact_18_finite__measure__axioms,axiom,
measur930452917991658466sure_a @ m ).
% finite_measure_axioms
thf(fact_19_sigma__finite__measure__axioms,axiom,
measur4308613598931908895sure_a @ m ).
% sigma_finite_measure_axioms
thf(fact_20_prob__space__distr,axiom,
! [F2: a > b,M: sigma_measure_b] :
( ( member_a_b @ F2 @ ( sigma_measurable_a_b @ m @ M ) )
=> ( probab7247484486040049090pace_b @ ( measure_distr_a_b @ m @ M @ F2 ) ) ) ).
% prob_space_distr
thf(fact_21_prob__space__distr,axiom,
! [F2: a > a,M: sigma_measure_a] :
( ( member_a_a @ F2 @ ( sigma_measurable_a_a @ m @ M ) )
=> ( probab7247484486040049089pace_a @ ( measure_distr_a_a @ m @ M @ F2 ) ) ) ).
% prob_space_distr
thf(fact_22_finite__measure__distr,axiom,
! [F2: a > b,M: sigma_measure_b] :
( ( member_a_b @ F2 @ ( sigma_measurable_a_b @ m @ M ) )
=> ( measur930452917991658467sure_b @ ( measure_distr_a_b @ m @ M @ F2 ) ) ) ).
% finite_measure_distr
thf(fact_23_finite__measure__distr,axiom,
! [F2: a > a,M: sigma_measure_a] :
( ( member_a_a @ F2 @ ( sigma_measurable_a_a @ m @ M ) )
=> ( measur930452917991658466sure_a @ ( measure_distr_a_a @ m @ M @ F2 ) ) ) ).
% finite_measure_distr
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__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_26_indep__sets__mono,axiom,
! [F: set_b > set_set_a,I2: set_set_b,J: set_set_b,G: set_b > set_set_a] :
( ( indepe4967106455115002445_set_b @ m @ F @ I2 )
=> ( ( ord_le3795704787696855135_set_b @ J @ I2 )
=> ( ! [I3: set_b] :
( ( member_set_b @ I3 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe4967106455115002445_set_b @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_27_indep__sets__mono,axiom,
! [F: a > set_set_a,I2: set_a,J: set_a,G: a > set_set_a] :
( ( indepe8927441866673418604ts_a_a @ m @ F @ I2 )
=> ( ( ord_less_eq_set_a @ J @ I2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe8927441866673418604ts_a_a @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_28_indep__sets__mono,axiom,
! [F: c > set_set_a,I2: set_c,J: set_c,G: c > set_set_a] :
( ( indepe8927441866673418606ts_a_c @ m @ F @ I2 )
=> ( ( ord_less_eq_set_c @ J @ I2 )
=> ( ! [I3: c] :
( ( member_c @ I3 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe8927441866673418606ts_a_c @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_29_indep__sets__mono,axiom,
! [F: nat > set_set_a,I2: set_nat,J: set_nat,G: nat > set_set_a] :
( ( indepe6267730027088848354_a_nat @ m @ F @ I2 )
=> ( ( ord_less_eq_set_nat @ J @ I2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe6267730027088848354_a_nat @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_30_indep__sets__mono,axiom,
! [F: $o > set_set_a,I2: set_o,J: set_o,G: $o > set_set_a] :
( ( indepe7780107833195774214ts_a_o @ m @ F @ I2 )
=> ( ( ord_less_eq_set_o @ J @ I2 )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe7780107833195774214ts_a_o @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_31_indep__sets__mono,axiom,
! [F: real > set_set_a,I2: set_real,J: set_real,G: real > set_set_a] :
( ( indepe2510005150887255998a_real @ m @ F @ I2 )
=> ( ( ord_less_eq_set_real @ J @ I2 )
=> ( ! [I3: real] :
( ( member_real @ I3 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe2510005150887255998a_real @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_32_indep__sets__mono,axiom,
! [F: set_a > set_set_a,I2: set_set_a,J: set_set_a,G: set_a > set_set_a] :
( ( indepe4967106450811773644_set_a @ m @ F @ I2 )
=> ( ( ord_le3724670747650509150_set_a @ J @ I2 )
=> ( ! [I3: set_a] :
( ( member_set_a @ I3 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe4967106450811773644_set_a @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_33_indep__sets__mono,axiom,
! [F: ( a > b ) > set_set_a,I2: set_a_b,J: set_a_b,G: ( a > b ) > set_set_a] :
( ( indepe1219004915316159480_a_a_b @ m @ F @ I2 )
=> ( ( ord_less_eq_set_a_b @ J @ I2 )
=> ( ! [I3: a > b] :
( ( member_a_b @ I3 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe1219004915316159480_a_a_b @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_34_indep__sets__mono__index,axiom,
! [J: set_set_b,I2: set_set_b,F: set_b > set_set_a] :
( ( ord_le3795704787696855135_set_b @ J @ I2 )
=> ( ( indepe4967106455115002445_set_b @ m @ F @ I2 )
=> ( indepe4967106455115002445_set_b @ m @ F @ J ) ) ) ).
% indep_sets_mono_index
thf(fact_35_indep__sets__mono__index,axiom,
! [J: set_a,I2: set_a,F: a > set_set_a] :
( ( ord_less_eq_set_a @ J @ I2 )
=> ( ( indepe8927441866673418604ts_a_a @ m @ F @ I2 )
=> ( indepe8927441866673418604ts_a_a @ m @ F @ J ) ) ) ).
% indep_sets_mono_index
thf(fact_36_indep__sets__mono__index,axiom,
! [J: set_c,I2: set_c,F: c > set_set_a] :
( ( ord_less_eq_set_c @ J @ I2 )
=> ( ( indepe8927441866673418606ts_a_c @ m @ F @ I2 )
=> ( indepe8927441866673418606ts_a_c @ m @ F @ J ) ) ) ).
% indep_sets_mono_index
thf(fact_37_indep__sets__mono__index,axiom,
! [J: set_nat,I2: set_nat,F: nat > set_set_a] :
( ( ord_less_eq_set_nat @ J @ I2 )
=> ( ( indepe6267730027088848354_a_nat @ m @ F @ I2 )
=> ( indepe6267730027088848354_a_nat @ m @ F @ J ) ) ) ).
% indep_sets_mono_index
thf(fact_38_indep__sets__mono__index,axiom,
! [J: set_o,I2: set_o,F: $o > set_set_a] :
( ( ord_less_eq_set_o @ J @ I2 )
=> ( ( indepe7780107833195774214ts_a_o @ m @ F @ I2 )
=> ( indepe7780107833195774214ts_a_o @ m @ F @ J ) ) ) ).
% indep_sets_mono_index
thf(fact_39_indep__sets__mono__index,axiom,
! [J: set_real,I2: set_real,F: real > set_set_a] :
( ( ord_less_eq_set_real @ J @ I2 )
=> ( ( indepe2510005150887255998a_real @ m @ F @ I2 )
=> ( indepe2510005150887255998a_real @ m @ F @ J ) ) ) ).
% indep_sets_mono_index
thf(fact_40_indep__sets__mono__index,axiom,
! [J: set_set_a,I2: set_set_a,F: set_a > set_set_a] :
( ( ord_le3724670747650509150_set_a @ J @ I2 )
=> ( ( indepe4967106450811773644_set_a @ m @ F @ I2 )
=> ( indepe4967106450811773644_set_a @ m @ F @ J ) ) ) ).
% indep_sets_mono_index
thf(fact_41_indep__sets__mono__index,axiom,
! [J: set_a_b,I2: set_a_b,F: ( a > b ) > set_set_a] :
( ( ord_less_eq_set_a_b @ J @ I2 )
=> ( ( indepe1219004915316159480_a_a_b @ m @ F @ I2 )
=> ( indepe1219004915316159480_a_a_b @ m @ F @ J ) ) ) ).
% indep_sets_mono_index
thf(fact_42_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_43_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_44_prob__space_Oindep__sets__mono,axiom,
! [M2: sigma_measure_b,F: real > set_set_b,I2: set_real,J: set_real,G: real > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe8106223766075386687b_real @ M2 @ F @ I2 )
=> ( ( ord_less_eq_set_real @ J @ I2 )
=> ( ! [I3: real] :
( ( member_real @ I3 @ J )
=> ( ord_le3795704787696855135_set_b @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe8106223766075386687b_real @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_45_prob__space_Oindep__sets__mono,axiom,
! [M2: sigma_measure_b,F: $o > set_set_b,I2: set_o,J: set_o,G: $o > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe4880885433731379909ts_b_o @ M2 @ F @ I2 )
=> ( ( ord_less_eq_set_o @ J @ I2 )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ J )
=> ( ord_le3795704787696855135_set_b @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe4880885433731379909ts_b_o @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_46_prob__space_Oindep__sets__mono,axiom,
! [M2: sigma_measure_b,F: nat > set_set_b,I2: set_nat,J: set_nat,G: nat > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe7503174356045242851_b_nat @ M2 @ F @ I2 )
=> ( ( ord_less_eq_set_nat @ J @ I2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ J )
=> ( ord_le3795704787696855135_set_b @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe7503174356045242851_b_nat @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_47_prob__space_Oindep__sets__mono,axiom,
! [M2: sigma_measure_b,F: a > set_set_b,I2: set_a,J: set_a,G: a > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe6139986284700742571ts_b_a @ M2 @ F @ I2 )
=> ( ( ord_less_eq_set_a @ J @ I2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ J )
=> ( ord_le3795704787696855135_set_b @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe6139986284700742571ts_b_a @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_48_prob__space_Oindep__sets__mono,axiom,
! [M2: sigma_measure_a,F: a > set_set_a,I2: set_a,J: set_a,G: a > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe8927441866673418604ts_a_a @ M2 @ F @ I2 )
=> ( ( ord_less_eq_set_a @ J @ I2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe8927441866673418604ts_a_a @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_49_prob__space_Oindep__sets__mono,axiom,
! [M2: sigma_measure_b,F: c > set_set_b,I2: set_c,J: set_c,G: c > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe6139986284700742573ts_b_c @ M2 @ F @ I2 )
=> ( ( ord_less_eq_set_c @ J @ I2 )
=> ( ! [I3: c] :
( ( member_c @ I3 @ J )
=> ( ord_le3795704787696855135_set_b @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe6139986284700742573ts_b_c @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_50_prob__space_Oindep__sets__mono,axiom,
! [M2: sigma_measure_a,F: c > set_set_a,I2: set_c,J: set_c,G: c > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe8927441866673418606ts_a_c @ M2 @ F @ I2 )
=> ( ( ord_less_eq_set_c @ J @ I2 )
=> ( ! [I3: c] :
( ( member_c @ I3 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe8927441866673418606ts_a_c @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_51_prob__space_Oindep__sets__mono,axiom,
! [M2: sigma_measure_a,F: nat > set_set_a,I2: set_nat,J: set_nat,G: nat > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe6267730027088848354_a_nat @ M2 @ F @ I2 )
=> ( ( ord_less_eq_set_nat @ J @ I2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe6267730027088848354_a_nat @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_52_prob__space_Oindep__sets__mono,axiom,
! [M2: sigma_measure_a,F: $o > set_set_a,I2: set_o,J: set_o,G: $o > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe7780107833195774214ts_a_o @ M2 @ F @ I2 )
=> ( ( ord_less_eq_set_o @ J @ I2 )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe7780107833195774214ts_a_o @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_53_prob__space_Oindep__sets__mono,axiom,
! [M2: sigma_measure_a,F: real > set_set_a,I2: set_real,J: set_real,G: real > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe2510005150887255998a_real @ M2 @ F @ I2 )
=> ( ( ord_less_eq_set_real @ J @ I2 )
=> ( ! [I3: real] :
( ( member_real @ I3 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe2510005150887255998a_real @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_54_prob__space_Oindep__sets__mono__sets,axiom,
! [M2: sigma_measure_b,F: real > set_set_b,I2: set_real,G: real > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe8106223766075386687b_real @ M2 @ F @ I2 )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I2 )
=> ( ord_le3795704787696855135_set_b @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe8106223766075386687b_real @ M2 @ G @ I2 ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_55_prob__space_Oindep__sets__mono__sets,axiom,
! [M2: sigma_measure_b,F: $o > set_set_b,I2: set_o,G: $o > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe4880885433731379909ts_b_o @ M2 @ F @ I2 )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ord_le3795704787696855135_set_b @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe4880885433731379909ts_b_o @ M2 @ G @ I2 ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_56_prob__space_Oindep__sets__mono__sets,axiom,
! [M2: sigma_measure_b,F: nat > set_set_b,I2: set_nat,G: nat > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe7503174356045242851_b_nat @ M2 @ F @ I2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ord_le3795704787696855135_set_b @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe7503174356045242851_b_nat @ M2 @ G @ I2 ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_57_prob__space_Oindep__sets__mono__sets,axiom,
! [M2: sigma_measure_b,F: c > set_set_b,I2: set_c,G: c > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe6139986284700742573ts_b_c @ M2 @ F @ I2 )
=> ( ! [I3: c] :
( ( member_c @ I3 @ I2 )
=> ( ord_le3795704787696855135_set_b @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe6139986284700742573ts_b_c @ M2 @ G @ I2 ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_58_prob__space_Oindep__sets__mono__sets,axiom,
! [M2: sigma_measure_a,F: c > set_set_a,I2: set_c,G: c > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe8927441866673418606ts_a_c @ M2 @ F @ I2 )
=> ( ! [I3: c] :
( ( member_c @ I3 @ I2 )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe8927441866673418606ts_a_c @ M2 @ G @ I2 ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_59_prob__space_Oindep__sets__mono__sets,axiom,
! [M2: sigma_measure_a,F: nat > set_set_a,I2: set_nat,G: nat > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe6267730027088848354_a_nat @ M2 @ F @ I2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe6267730027088848354_a_nat @ M2 @ G @ I2 ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_60_prob__space_Oindep__sets__mono__sets,axiom,
! [M2: sigma_measure_a,F: $o > set_set_a,I2: set_o,G: $o > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe7780107833195774214ts_a_o @ M2 @ F @ I2 )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe7780107833195774214ts_a_o @ M2 @ G @ I2 ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_61_prob__space_Oindep__sets__mono__sets,axiom,
! [M2: sigma_measure_a,F: real > set_set_a,I2: set_real,G: real > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe2510005150887255998a_real @ M2 @ F @ I2 )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I2 )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe2510005150887255998a_real @ M2 @ G @ I2 ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_62_prob__space_Oindep__sets__mono__sets,axiom,
! [M2: sigma_measure_b,F: set_a > set_set_b,I2: set_set_a,G: set_a > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe6311571487621261579_set_a @ M2 @ F @ I2 )
=> ( ! [I3: set_a] :
( ( member_set_a @ I3 @ I2 )
=> ( ord_le3795704787696855135_set_b @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe6311571487621261579_set_a @ M2 @ G @ I2 ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_63_prob__space_Oindep__sets__mono__sets,axiom,
! [M2: sigma_measure_a,F: set_a > set_set_a,I2: set_set_a,G: set_a > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe4967106450811773644_set_a @ M2 @ F @ I2 )
=> ( ! [I3: set_a] :
( ( member_set_a @ I3 @ I2 )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe4967106450811773644_set_a @ M2 @ G @ I2 ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_64_prob__space_Oindep__sets__mono__index,axiom,
! [M2: sigma_measure_a,J: set_set_b,I2: set_set_b,F: set_b > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( ord_le3795704787696855135_set_b @ J @ I2 )
=> ( ( indepe4967106455115002445_set_b @ M2 @ F @ I2 )
=> ( indepe4967106455115002445_set_b @ M2 @ F @ J ) ) ) ) ).
% prob_space.indep_sets_mono_index
thf(fact_65_prob__space_Oindep__sets__mono__index,axiom,
! [M2: sigma_measure_a,J: set_a,I2: set_a,F: a > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( ord_less_eq_set_a @ J @ I2 )
=> ( ( indepe8927441866673418604ts_a_a @ M2 @ F @ I2 )
=> ( indepe8927441866673418604ts_a_a @ M2 @ F @ J ) ) ) ) ).
% prob_space.indep_sets_mono_index
thf(fact_66_prob__space_Oindep__sets__mono__index,axiom,
! [M2: sigma_measure_b,J: set_c,I2: set_c,F: c > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( ord_less_eq_set_c @ J @ I2 )
=> ( ( indepe6139986284700742573ts_b_c @ M2 @ F @ I2 )
=> ( indepe6139986284700742573ts_b_c @ M2 @ F @ J ) ) ) ) ).
% prob_space.indep_sets_mono_index
thf(fact_67_prob__space_Oindep__sets__mono__index,axiom,
! [M2: sigma_measure_a,J: set_c,I2: set_c,F: c > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( ord_less_eq_set_c @ J @ I2 )
=> ( ( indepe8927441866673418606ts_a_c @ M2 @ F @ I2 )
=> ( indepe8927441866673418606ts_a_c @ M2 @ F @ J ) ) ) ) ).
% prob_space.indep_sets_mono_index
thf(fact_68_prob__space_Oindep__sets__mono__index,axiom,
! [M2: sigma_measure_a,J: set_nat,I2: set_nat,F: nat > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( ord_less_eq_set_nat @ J @ I2 )
=> ( ( indepe6267730027088848354_a_nat @ M2 @ F @ I2 )
=> ( indepe6267730027088848354_a_nat @ M2 @ F @ J ) ) ) ) ).
% prob_space.indep_sets_mono_index
thf(fact_69_prob__space_Oindep__sets__mono__index,axiom,
! [M2: sigma_measure_a,J: set_o,I2: set_o,F: $o > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( ord_less_eq_set_o @ J @ I2 )
=> ( ( indepe7780107833195774214ts_a_o @ M2 @ F @ I2 )
=> ( indepe7780107833195774214ts_a_o @ M2 @ F @ J ) ) ) ) ).
% prob_space.indep_sets_mono_index
thf(fact_70_prob__space_Oindep__sets__mono__index,axiom,
! [M2: sigma_measure_a,J: set_real,I2: set_real,F: real > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( ord_less_eq_set_real @ J @ I2 )
=> ( ( indepe2510005150887255998a_real @ M2 @ F @ I2 )
=> ( indepe2510005150887255998a_real @ M2 @ F @ J ) ) ) ) ).
% prob_space.indep_sets_mono_index
thf(fact_71_prob__space_Oindep__sets__mono__index,axiom,
! [M2: sigma_measure_a,J: set_set_a,I2: set_set_a,F: set_a > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( ord_le3724670747650509150_set_a @ J @ I2 )
=> ( ( indepe4967106450811773644_set_a @ M2 @ F @ I2 )
=> ( indepe4967106450811773644_set_a @ M2 @ F @ J ) ) ) ) ).
% prob_space.indep_sets_mono_index
thf(fact_72_prob__space_Oindep__sets__mono__index,axiom,
! [M2: sigma_measure_a,J: set_a_b,I2: set_a_b,F: ( a > b ) > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( ord_less_eq_set_a_b @ J @ I2 )
=> ( ( indepe1219004915316159480_a_a_b @ M2 @ F @ I2 )
=> ( indepe1219004915316159480_a_a_b @ M2 @ F @ J ) ) ) ) ).
% prob_space.indep_sets_mono_index
thf(fact_73_prob__space_Oindep__sets__cong,axiom,
! [M2: sigma_measure_b,I2: set_c,J: set_c,F: c > set_set_b,G: c > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( I2 = J )
=> ( ! [I3: c] :
( ( member_c @ I3 @ I2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( indepe6139986284700742573ts_b_c @ M2 @ F @ I2 )
= ( indepe6139986284700742573ts_b_c @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_cong
thf(fact_74_prob__space_Oindep__sets__cong,axiom,
! [M2: sigma_measure_a,I2: set_c,J: set_c,F: c > set_set_a,G: c > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( I2 = J )
=> ( ! [I3: c] :
( ( member_c @ I3 @ I2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( indepe8927441866673418606ts_a_c @ M2 @ F @ I2 )
= ( indepe8927441866673418606ts_a_c @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_cong
thf(fact_75_prob__space_Oindep__sets__cong,axiom,
! [M2: sigma_measure_a,I2: set_nat,J: set_nat,F: nat > set_set_a,G: nat > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( I2 = J )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( indepe6267730027088848354_a_nat @ M2 @ F @ I2 )
= ( indepe6267730027088848354_a_nat @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_cong
thf(fact_76_prob__space_Oindep__sets__cong,axiom,
! [M2: sigma_measure_a,I2: set_o,J: set_o,F: $o > set_set_a,G: $o > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( I2 = J )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( indepe7780107833195774214ts_a_o @ M2 @ F @ I2 )
= ( indepe7780107833195774214ts_a_o @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_cong
thf(fact_77_prob__space_Oindep__sets__cong,axiom,
! [M2: sigma_measure_a,I2: set_real,J: set_real,F: real > set_set_a,G: real > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( I2 = J )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( indepe2510005150887255998a_real @ M2 @ F @ I2 )
= ( indepe2510005150887255998a_real @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_cong
thf(fact_78_prob__space_Oindep__sets__cong,axiom,
! [M2: sigma_measure_a,I2: set_set_a,J: set_set_a,F: set_a > set_set_a,G: set_a > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( I2 = J )
=> ( ! [I3: set_a] :
( ( member_set_a @ I3 @ I2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( indepe4967106450811773644_set_a @ M2 @ F @ I2 )
= ( indepe4967106450811773644_set_a @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_cong
thf(fact_79_prob__space_Oindep__sets__cong,axiom,
! [M2: sigma_measure_a,I2: set_a_b,J: set_a_b,F: ( a > b ) > set_set_a,G: ( a > b ) > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( I2 = J )
=> ( ! [I3: a > b] :
( ( member_a_b @ I3 @ I2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( indepe1219004915316159480_a_a_b @ M2 @ F @ I2 )
= ( indepe1219004915316159480_a_a_b @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_cong
thf(fact_80_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_81_sets__distr,axiom,
! [M2: sigma_measure_a,N: sigma_measure_b,F2: a > b] :
( ( sigma_sets_b @ ( measure_distr_a_b @ M2 @ N @ F2 ) )
= ( sigma_sets_b @ N ) ) ).
% sets_distr
thf(fact_82_finite__measure_Ofinite__measure__distr,axiom,
! [M2: sigma_measure_a,F2: a > b,M: sigma_measure_b] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ( member_a_b @ F2 @ ( sigma_measurable_a_b @ M2 @ M ) )
=> ( measur930452917991658467sure_b @ ( measure_distr_a_b @ M2 @ M @ F2 ) ) ) ) ).
% finite_measure.finite_measure_distr
thf(fact_83_finite__measure_Ofinite__measure__distr,axiom,
! [M2: sigma_measure_a,F2: a > a,M: sigma_measure_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ( member_a_a @ F2 @ ( sigma_measurable_a_a @ M2 @ M ) )
=> ( measur930452917991658466sure_a @ ( measure_distr_a_a @ M2 @ M @ F2 ) ) ) ) ).
% finite_measure.finite_measure_distr
thf(fact_84_sigma__finite__measure__distr,axiom,
! [M2: sigma_measure_a,N: sigma_measure_b,F2: a > b] :
( ( measur4308613598931908896sure_b @ ( measure_distr_a_b @ M2 @ N @ F2 ) )
=> ( ( member_a_b @ F2 @ ( sigma_measurable_a_b @ M2 @ N ) )
=> ( measur4308613598931908895sure_a @ M2 ) ) ) ).
% sigma_finite_measure_distr
thf(fact_85_sigma__finite__measure__distr,axiom,
! [M2: sigma_measure_a,N: sigma_measure_a,F2: a > a] :
( ( measur4308613598931908895sure_a @ ( measure_distr_a_a @ M2 @ N @ F2 ) )
=> ( ( member_a_a @ F2 @ ( sigma_measurable_a_a @ M2 @ N ) )
=> ( measur4308613598931908895sure_a @ M2 ) ) ) ).
% sigma_finite_measure_distr
thf(fact_86_prob__space_Oprob__space__distr,axiom,
! [M2: sigma_measure_a,F2: a > b,M: sigma_measure_b] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( member_a_b @ F2 @ ( sigma_measurable_a_b @ M2 @ M ) )
=> ( probab7247484486040049090pace_b @ ( measure_distr_a_b @ M2 @ M @ F2 ) ) ) ) ).
% prob_space.prob_space_distr
thf(fact_87_prob__space_Oprob__space__distr,axiom,
! [M2: sigma_measure_a,F2: a > a,M: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( member_a_a @ F2 @ ( sigma_measurable_a_a @ M2 @ M ) )
=> ( probab7247484486040049089pace_a @ ( measure_distr_a_a @ M2 @ M @ F2 ) ) ) ) ).
% prob_space.prob_space_distr
thf(fact_88_prob__space__distrD,axiom,
! [F2: a > b,M2: sigma_measure_a,N: sigma_measure_b] :
( ( member_a_b @ F2 @ ( sigma_measurable_a_b @ M2 @ N ) )
=> ( ( probab7247484486040049090pace_b @ ( measure_distr_a_b @ M2 @ N @ F2 ) )
=> ( probab7247484486040049089pace_a @ M2 ) ) ) ).
% prob_space_distrD
thf(fact_89_prob__space__distrD,axiom,
! [F2: a > a,M2: sigma_measure_a,N: sigma_measure_a] :
( ( member_a_a @ F2 @ ( sigma_measurable_a_a @ M2 @ N ) )
=> ( ( probab7247484486040049089pace_a @ ( measure_distr_a_a @ M2 @ N @ F2 ) )
=> ( probab7247484486040049089pace_a @ M2 ) ) ) ).
% prob_space_distrD
thf(fact_90_prob__space__completion,axiom,
probab7247484486040049089pace_a @ ( comple3428971583294703880tion_a @ m ) ).
% prob_space_completion
thf(fact_91_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_92_indep__sets__mono__sets,axiom,
! [F: c > set_set_a,I2: set_c,G: c > set_set_a] :
( ( indepe8927441866673418606ts_a_c @ m @ F @ I2 )
=> ( ! [I3: c] :
( ( member_c @ I3 @ I2 )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe8927441866673418606ts_a_c @ m @ G @ I2 ) ) ) ).
% indep_sets_mono_sets
thf(fact_93_indep__sets__mono__sets,axiom,
! [F: nat > set_set_a,I2: set_nat,G: nat > set_set_a] :
( ( indepe6267730027088848354_a_nat @ m @ F @ I2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe6267730027088848354_a_nat @ m @ G @ I2 ) ) ) ).
% indep_sets_mono_sets
thf(fact_94_indep__sets__mono__sets,axiom,
! [F: $o > set_set_a,I2: set_o,G: $o > set_set_a] :
( ( indepe7780107833195774214ts_a_o @ m @ F @ I2 )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe7780107833195774214ts_a_o @ m @ G @ I2 ) ) ) ).
% indep_sets_mono_sets
thf(fact_95_indep__sets__mono__sets,axiom,
! [F: real > set_set_a,I2: set_real,G: real > set_set_a] :
( ( indepe2510005150887255998a_real @ m @ F @ I2 )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I2 )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe2510005150887255998a_real @ m @ G @ I2 ) ) ) ).
% indep_sets_mono_sets
thf(fact_96_indep__sets__mono__sets,axiom,
! [F: set_a > set_set_a,I2: set_set_a,G: set_a > set_set_a] :
( ( indepe4967106450811773644_set_a @ m @ F @ I2 )
=> ( ! [I3: set_a] :
( ( member_set_a @ I3 @ I2 )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe4967106450811773644_set_a @ m @ G @ I2 ) ) ) ).
% indep_sets_mono_sets
thf(fact_97_indep__sets__mono__sets,axiom,
! [F: ( a > b ) > set_set_a,I2: set_a_b,G: ( a > b ) > set_set_a] :
( ( indepe1219004915316159480_a_a_b @ m @ F @ I2 )
=> ( ! [I3: a > b] :
( ( member_a_b @ I3 @ I2 )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe1219004915316159480_a_a_b @ m @ G @ I2 ) ) ) ).
% indep_sets_mono_sets
thf(fact_98_subsetI,axiom,
! [A: set_a_b,B: set_a_b] :
( ! [X3: a > b] :
( ( member_a_b @ X3 @ A )
=> ( member_a_b @ X3 @ B ) )
=> ( ord_less_eq_set_a_b @ A @ B ) ) ).
% subsetI
thf(fact_99_subsetI,axiom,
! [A: set_c,B: set_c] :
( ! [X3: c] :
( ( member_c @ X3 @ A )
=> ( member_c @ X3 @ B ) )
=> ( ord_less_eq_set_c @ A @ B ) ) ).
% subsetI
thf(fact_100_subsetI,axiom,
! [A: set_real,B: set_real] :
( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( member_real @ X3 @ B ) )
=> ( ord_less_eq_set_real @ A @ B ) ) ).
% subsetI
thf(fact_101_subsetI,axiom,
! [A: set_o,B: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_o @ X3 @ B ) )
=> ( ord_less_eq_set_o @ A @ B ) ) ).
% subsetI
thf(fact_102_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_103_subsetI,axiom,
! [A: set_set_b,B: set_set_b] :
( ! [X3: set_b] :
( ( member_set_b @ X3 @ A )
=> ( member_set_b @ X3 @ B ) )
=> ( ord_le3795704787696855135_set_b @ A @ B ) ) ).
% subsetI
thf(fact_104_subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( member_set_a @ X3 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% subsetI
thf(fact_105_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_106_subset__antisym,axiom,
! [A: set_set_b,B: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( ( ord_le3795704787696855135_set_b @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_107_subset__antisym,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_108_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_109_tail__events__sets,axiom,
! [A: nat > set_set_a] :
( ! [I3: nat] : ( ord_le3724670747650509150_set_a @ ( A @ I3 ) @ ( sigma_sets_a @ m ) )
=> ( ord_le3724670747650509150_set_a @ ( indepe7538416700049374166_a_nat @ m @ A ) @ ( sigma_sets_a @ m ) ) ) ).
% tail_events_sets
thf(fact_110_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_111_distr__distr,axiom,
! [G2: b > b,N: sigma_measure_b,L: sigma_measure_b,F2: a > b,M2: sigma_measure_a] :
( ( member_b_b @ G2 @ ( sigma_measurable_b_b @ N @ L ) )
=> ( ( member_a_b @ F2 @ ( sigma_measurable_a_b @ M2 @ N ) )
=> ( ( measure_distr_b_b @ ( measure_distr_a_b @ M2 @ N @ F2 ) @ L @ G2 )
= ( measure_distr_a_b @ M2 @ L @ ( comp_b_b_a @ G2 @ F2 ) ) ) ) ) ).
% distr_distr
thf(fact_112_distr__distr,axiom,
! [G2: a > b,N: sigma_measure_a,L: sigma_measure_b,F2: a > a,M2: sigma_measure_a] :
( ( member_a_b @ G2 @ ( sigma_measurable_a_b @ N @ L ) )
=> ( ( member_a_a @ F2 @ ( sigma_measurable_a_a @ M2 @ N ) )
=> ( ( measure_distr_a_b @ ( measure_distr_a_a @ M2 @ N @ F2 ) @ L @ G2 )
= ( measure_distr_a_b @ M2 @ L @ ( comp_a_b_a @ G2 @ F2 ) ) ) ) ) ).
% distr_distr
thf(fact_113_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_114_Collect__mono__iff,axiom,
! [P: set_b > $o,Q: set_b > $o] :
( ( ord_le3795704787696855135_set_b @ ( collect_set_b @ P ) @ ( collect_set_b @ Q ) )
= ( ! [X4: set_b] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_115_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X4: set_a] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_116_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_117_set__eq__subset,axiom,
( ( ^ [Y3: set_set_b,Z: set_set_b] : ( Y3 = Z ) )
= ( ^ [A2: set_set_b,B2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
& ( ord_le3795704787696855135_set_b @ B2 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_118_set__eq__subset,axiom,
( ( ^ [Y3: set_set_a,Z: set_set_a] : ( Y3 = Z ) )
= ( ^ [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
& ( ord_le3724670747650509150_set_a @ B2 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_119_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
& ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_120_mem__Collect__eq,axiom,
! [A3: a > b,P: ( a > b ) > $o] :
( ( member_a_b @ A3 @ ( collect_a_b @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_121_mem__Collect__eq,axiom,
! [A3: c,P: c > $o] :
( ( member_c @ A3 @ ( collect_c @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_122_mem__Collect__eq,axiom,
! [A3: set_a,P: set_a > $o] :
( ( member_set_a @ A3 @ ( collect_set_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_123_mem__Collect__eq,axiom,
! [A3: real,P: real > $o] :
( ( member_real @ A3 @ ( collect_real @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_124_mem__Collect__eq,axiom,
! [A3: $o,P: $o > $o] :
( ( member_o @ A3 @ ( collect_o @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_125_mem__Collect__eq,axiom,
! [A3: nat,P: nat > $o] :
( ( member_nat @ A3 @ ( collect_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_126_Collect__mem__eq,axiom,
! [A: set_a_b] :
( ( collect_a_b
@ ^ [X4: a > b] : ( member_a_b @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_127_Collect__mem__eq,axiom,
! [A: set_c] :
( ( collect_c
@ ^ [X4: c] : ( member_c @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_128_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X4: set_a] : ( member_set_a @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_129_Collect__mem__eq,axiom,
! [A: set_real] :
( ( collect_real
@ ^ [X4: real] : ( member_real @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_130_Collect__mem__eq,axiom,
! [A: set_o] :
( ( collect_o
@ ^ [X4: $o] : ( member_o @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_131_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_132_subset__trans,axiom,
! [A: set_set_b,B: set_set_b,C: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( ( ord_le3795704787696855135_set_b @ B @ C )
=> ( ord_le3795704787696855135_set_b @ A @ C ) ) ) ).
% subset_trans
thf(fact_133_subset__trans,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% subset_trans
thf(fact_134_subset__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% subset_trans
thf(fact_135_Collect__mono,axiom,
! [P: set_b > $o,Q: set_b > $o] :
( ! [X3: set_b] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le3795704787696855135_set_b @ ( collect_set_b @ P ) @ ( collect_set_b @ Q ) ) ) ).
% Collect_mono
thf(fact_136_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X3: set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_137_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_138_subset__refl,axiom,
! [A: set_set_b] : ( ord_le3795704787696855135_set_b @ A @ A ) ).
% subset_refl
thf(fact_139_subset__refl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% subset_refl
thf(fact_140_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_141_subset__iff,axiom,
( ord_less_eq_set_a_b
= ( ^ [A2: set_a_b,B2: set_a_b] :
! [T2: a > b] :
( ( member_a_b @ T2 @ A2 )
=> ( member_a_b @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_142_subset__iff,axiom,
( ord_less_eq_set_c
= ( ^ [A2: set_c,B2: set_c] :
! [T2: c] :
( ( member_c @ T2 @ A2 )
=> ( member_c @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_143_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A2: set_real,B2: set_real] :
! [T2: real] :
( ( member_real @ T2 @ A2 )
=> ( member_real @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_144_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A2: set_o,B2: set_o] :
! [T2: $o] :
( ( member_o @ T2 @ A2 )
=> ( member_o @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_145_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A2 )
=> ( member_nat @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_146_subset__iff,axiom,
( ord_le3795704787696855135_set_b
= ( ^ [A2: set_set_b,B2: set_set_b] :
! [T2: set_b] :
( ( member_set_b @ T2 @ A2 )
=> ( member_set_b @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_147_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A2: set_set_a,B2: set_set_a] :
! [T2: set_a] :
( ( member_set_a @ T2 @ A2 )
=> ( member_set_a @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_148_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B2: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A2 )
=> ( member_a @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_149_equalityD2,axiom,
! [A: set_set_b,B: set_set_b] :
( ( A = B )
=> ( ord_le3795704787696855135_set_b @ B @ A ) ) ).
% equalityD2
thf(fact_150_equalityD2,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le3724670747650509150_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_151_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_152_equalityD1,axiom,
! [A: set_set_b,B: set_set_b] :
( ( A = B )
=> ( ord_le3795704787696855135_set_b @ A @ B ) ) ).
% equalityD1
thf(fact_153_equalityD1,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_154_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_155_subset__eq,axiom,
( ord_less_eq_set_a_b
= ( ^ [A2: set_a_b,B2: set_a_b] :
! [X4: a > b] :
( ( member_a_b @ X4 @ A2 )
=> ( member_a_b @ X4 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_156_subset__eq,axiom,
( ord_less_eq_set_c
= ( ^ [A2: set_c,B2: set_c] :
! [X4: c] :
( ( member_c @ X4 @ A2 )
=> ( member_c @ X4 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_157_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A2: set_real,B2: set_real] :
! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( member_real @ X4 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_158_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A2: set_o,B2: set_o] :
! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( member_o @ X4 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_159_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat @ X4 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_160_subset__eq,axiom,
( ord_le3795704787696855135_set_b
= ( ^ [A2: set_set_b,B2: set_set_b] :
! [X4: set_b] :
( ( member_set_b @ X4 @ A2 )
=> ( member_set_b @ X4 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_161_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A2: set_set_a,B2: set_set_a] :
! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ( member_set_a @ X4 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_162_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B2: set_a] :
! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( member_a @ X4 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_163_equalityE,axiom,
! [A: set_set_b,B: set_set_b] :
( ( A = B )
=> ~ ( ( ord_le3795704787696855135_set_b @ A @ B )
=> ~ ( ord_le3795704787696855135_set_b @ B @ A ) ) ) ).
% equalityE
thf(fact_164_equalityE,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ~ ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ~ ( ord_le3724670747650509150_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_165_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_166_subsetD,axiom,
! [A: set_a_b,B: set_a_b,C2: a > b] :
( ( ord_less_eq_set_a_b @ A @ B )
=> ( ( member_a_b @ C2 @ A )
=> ( member_a_b @ C2 @ B ) ) ) ).
% subsetD
thf(fact_167_subsetD,axiom,
! [A: set_c,B: set_c,C2: c] :
( ( ord_less_eq_set_c @ A @ B )
=> ( ( member_c @ C2 @ A )
=> ( member_c @ C2 @ B ) ) ) ).
% subsetD
thf(fact_168_subsetD,axiom,
! [A: set_real,B: set_real,C2: real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( member_real @ C2 @ A )
=> ( member_real @ C2 @ B ) ) ) ).
% subsetD
thf(fact_169_subsetD,axiom,
! [A: set_o,B: set_o,C2: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ C2 @ A )
=> ( member_o @ C2 @ B ) ) ) ).
% subsetD
thf(fact_170_subsetD,axiom,
! [A: set_nat,B: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_171_subsetD,axiom,
! [A: set_set_b,B: set_set_b,C2: set_b] :
( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( ( member_set_b @ C2 @ A )
=> ( member_set_b @ C2 @ B ) ) ) ).
% subsetD
thf(fact_172_subsetD,axiom,
! [A: set_set_a,B: set_set_a,C2: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ C2 @ A )
=> ( member_set_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_173_subsetD,axiom,
! [A: set_a,B: set_a,C2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_174_in__mono,axiom,
! [A: set_a_b,B: set_a_b,X5: a > b] :
( ( ord_less_eq_set_a_b @ A @ B )
=> ( ( member_a_b @ X5 @ A )
=> ( member_a_b @ X5 @ B ) ) ) ).
% in_mono
thf(fact_175_in__mono,axiom,
! [A: set_c,B: set_c,X5: c] :
( ( ord_less_eq_set_c @ A @ B )
=> ( ( member_c @ X5 @ A )
=> ( member_c @ X5 @ B ) ) ) ).
% in_mono
thf(fact_176_in__mono,axiom,
! [A: set_real,B: set_real,X5: real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( member_real @ X5 @ A )
=> ( member_real @ X5 @ B ) ) ) ).
% in_mono
thf(fact_177_in__mono,axiom,
! [A: set_o,B: set_o,X5: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ X5 @ A )
=> ( member_o @ X5 @ B ) ) ) ).
% in_mono
thf(fact_178_in__mono,axiom,
! [A: set_nat,B: set_nat,X5: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X5 @ A )
=> ( member_nat @ X5 @ B ) ) ) ).
% in_mono
thf(fact_179_in__mono,axiom,
! [A: set_set_b,B: set_set_b,X5: set_b] :
( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( ( member_set_b @ X5 @ A )
=> ( member_set_b @ X5 @ B ) ) ) ).
% in_mono
thf(fact_180_in__mono,axiom,
! [A: set_set_a,B: set_set_a,X5: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ X5 @ A )
=> ( member_set_a @ X5 @ B ) ) ) ).
% in_mono
thf(fact_181_in__mono,axiom,
! [A: set_a,B: set_a,X5: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X5 @ A )
=> ( member_a @ X5 @ B ) ) ) ).
% in_mono
thf(fact_182_prob__space__imp__sigma__finite,axiom,
! [M2: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( measur4308613598931908895sure_a @ M2 ) ) ).
% prob_space_imp_sigma_finite
thf(fact_183_prob__space_Ofinite__measure,axiom,
! [M2: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( measur930452917991658466sure_a @ M2 ) ) ).
% prob_space.finite_measure
thf(fact_184_finite__measure_Oaxioms_I1_J,axiom,
! [M2: sigma_measure_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( measur4308613598931908895sure_a @ M2 ) ) ).
% finite_measure.axioms(1)
thf(fact_185_indep__setD__ev1,axiom,
! [A: set_set_a,B: set_set_a] :
( ( indepe2041756565122539606_set_a @ m @ A @ B )
=> ( ord_le3724670747650509150_set_a @ A @ ( sigma_sets_a @ m ) ) ) ).
% indep_setD_ev1
thf(fact_186_indep__setD__ev2,axiom,
! [A: set_set_a,B: set_set_a] :
( ( indepe2041756565122539606_set_a @ m @ A @ B )
=> ( ord_le3724670747650509150_set_a @ B @ ( sigma_sets_a @ m ) ) ) ).
% indep_setD_ev2
thf(fact_187_sets__completionI__sets,axiom,
! [A: set_b,M2: sigma_measure_b] :
( ( member_set_b @ A @ ( sigma_sets_b @ M2 ) )
=> ( member_set_b @ A @ ( sigma_sets_b @ ( comple3428971583294703881tion_b @ M2 ) ) ) ) ).
% sets_completionI_sets
thf(fact_188_sets__completionI__sets,axiom,
! [A: set_a,M2: sigma_measure_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ M2 ) )
=> ( member_set_a @ A @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) ) ) ).
% sets_completionI_sets
thf(fact_189_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_190_dual__order_Orefl,axiom,
! [A3: set_set_b] : ( ord_le3795704787696855135_set_b @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_191_dual__order_Orefl,axiom,
! [A3: set_set_a] : ( ord_le3724670747650509150_set_a @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_192_dual__order_Orefl,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_193_dual__order_Orefl,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_194_dual__order_Orefl,axiom,
! [A3: int] : ( ord_less_eq_int @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_195_order__refl,axiom,
! [X5: set_set_b] : ( ord_le3795704787696855135_set_b @ X5 @ X5 ) ).
% order_refl
thf(fact_196_order__refl,axiom,
! [X5: set_set_a] : ( ord_le3724670747650509150_set_a @ X5 @ X5 ) ).
% order_refl
thf(fact_197_order__refl,axiom,
! [X5: set_a] : ( ord_less_eq_set_a @ X5 @ X5 ) ).
% order_refl
thf(fact_198_order__refl,axiom,
! [X5: nat] : ( ord_less_eq_nat @ X5 @ X5 ) ).
% order_refl
thf(fact_199_order__refl,axiom,
! [X5: int] : ( ord_less_eq_int @ X5 @ X5 ) ).
% order_refl
thf(fact_200_increasing__def,axiom,
( measur1776380161843274167a_real
= ( ^ [M3: set_set_a,Mu: set_a > real] :
! [X4: set_a] :
( ( member_set_a @ X4 @ M3 )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ M3 )
=> ( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( Mu @ X4 ) @ ( Mu @ Y4 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_201_increasing__def,axiom,
( measur8151441426001876059_a_nat
= ( ^ [M3: set_set_a,Mu: set_a > nat] :
! [X4: set_a] :
( ( member_set_a @ X4 @ M3 )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ M3 )
=> ( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( Mu @ X4 ) @ ( Mu @ Y4 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_202_increasing__def,axiom,
( measur8148950955492825783_a_int
= ( ^ [M3: set_set_a,Mu: set_a > int] :
! [X4: set_a] :
( ( member_set_a @ X4 @ M3 )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ M3 )
=> ( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( Mu @ X4 ) @ ( Mu @ Y4 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_203_increasing__def,axiom,
( measur6841170515247421756_b_nat
= ( ^ [M3: set_set_set_b,Mu: set_set_b > nat] :
! [X4: set_set_b] :
( ( member_set_set_b @ X4 @ M3 )
=> ! [Y4: set_set_b] :
( ( member_set_set_b @ Y4 @ M3 )
=> ( ( ord_le3795704787696855135_set_b @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( Mu @ X4 ) @ ( Mu @ Y4 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_204_increasing__def,axiom,
( measur6838680044738371480_b_int
= ( ^ [M3: set_set_set_b,Mu: set_set_b > int] :
! [X4: set_set_b] :
( ( member_set_set_b @ X4 @ M3 )
=> ! [Y4: set_set_b] :
( ( member_set_set_b @ Y4 @ M3 )
=> ( ( ord_le3795704787696855135_set_b @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( Mu @ X4 ) @ ( Mu @ Y4 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_205_increasing__def,axiom,
( measur1244951900059291067_a_nat
= ( ^ [M3: set_set_set_a,Mu: set_set_a > nat] :
! [X4: set_set_a] :
( ( member_set_set_a @ X4 @ M3 )
=> ! [Y4: set_set_a] :
( ( member_set_set_a @ Y4 @ M3 )
=> ( ( ord_le3724670747650509150_set_a @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( Mu @ X4 ) @ ( Mu @ Y4 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_206_increasing__def,axiom,
( measur1242461429550240791_a_int
= ( ^ [M3: set_set_set_a,Mu: set_set_a > int] :
! [X4: set_set_a] :
( ( member_set_set_a @ X4 @ M3 )
=> ! [Y4: set_set_a] :
( ( member_set_set_a @ Y4 @ M3 )
=> ( ( ord_le3724670747650509150_set_a @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( Mu @ X4 ) @ ( Mu @ Y4 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_207_increasing__def,axiom,
( measur7842569353079325843_set_a
= ( ^ [M3: set_set_a,Mu: set_a > set_a] :
! [X4: set_a] :
( ( member_set_a @ X4 @ M3 )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ M3 )
=> ( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ord_less_eq_set_a @ ( Mu @ X4 ) @ ( Mu @ Y4 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_208_increasing__def,axiom,
( measur6668990528421247474_set_a
= ( ^ [M3: set_set_set_b,Mu: set_set_b > set_a] :
! [X4: set_set_b] :
( ( member_set_set_b @ X4 @ M3 )
=> ! [Y4: set_set_b] :
( ( member_set_set_b @ Y4 @ M3 )
=> ( ( ord_le3795704787696855135_set_b @ X4 @ Y4 )
=> ( ord_less_eq_set_a @ ( Mu @ X4 ) @ ( Mu @ Y4 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_209_increasing__def,axiom,
( measur5181028491126448947_set_a
= ( ^ [M3: set_set_set_a,Mu: set_set_a > set_a] :
! [X4: set_set_a] :
( ( member_set_set_a @ X4 @ M3 )
=> ! [Y4: set_set_a] :
( ( member_set_set_a @ Y4 @ M3 )
=> ( ( ord_le3724670747650509150_set_a @ X4 @ Y4 )
=> ( ord_less_eq_set_a @ ( Mu @ X4 ) @ ( Mu @ Y4 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_210_increasingD,axiom,
! [M2: set_set_a,F2: set_a > real,X5: set_a,Y5: set_a] :
( ( measur1776380161843274167a_real @ M2 @ F2 )
=> ( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ( member_set_a @ X5 @ M2 )
=> ( ( member_set_a @ Y5 @ M2 )
=> ( ord_less_eq_real @ ( F2 @ X5 ) @ ( F2 @ Y5 ) ) ) ) ) ) ).
% increasingD
thf(fact_211_increasingD,axiom,
! [M2: set_set_a,F2: set_a > nat,X5: set_a,Y5: set_a] :
( ( measur8151441426001876059_a_nat @ M2 @ F2 )
=> ( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ( member_set_a @ X5 @ M2 )
=> ( ( member_set_a @ Y5 @ M2 )
=> ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y5 ) ) ) ) ) ) ).
% increasingD
thf(fact_212_increasingD,axiom,
! [M2: set_set_a,F2: set_a > int,X5: set_a,Y5: set_a] :
( ( measur8148950955492825783_a_int @ M2 @ F2 )
=> ( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ( member_set_a @ X5 @ M2 )
=> ( ( member_set_a @ Y5 @ M2 )
=> ( ord_less_eq_int @ ( F2 @ X5 ) @ ( F2 @ Y5 ) ) ) ) ) ) ).
% increasingD
thf(fact_213_increasingD,axiom,
! [M2: set_set_set_b,F2: set_set_b > nat,X5: set_set_b,Y5: set_set_b] :
( ( measur6841170515247421756_b_nat @ M2 @ F2 )
=> ( ( ord_le3795704787696855135_set_b @ X5 @ Y5 )
=> ( ( member_set_set_b @ X5 @ M2 )
=> ( ( member_set_set_b @ Y5 @ M2 )
=> ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y5 ) ) ) ) ) ) ).
% increasingD
thf(fact_214_increasingD,axiom,
! [M2: set_set_set_b,F2: set_set_b > int,X5: set_set_b,Y5: set_set_b] :
( ( measur6838680044738371480_b_int @ M2 @ F2 )
=> ( ( ord_le3795704787696855135_set_b @ X5 @ Y5 )
=> ( ( member_set_set_b @ X5 @ M2 )
=> ( ( member_set_set_b @ Y5 @ M2 )
=> ( ord_less_eq_int @ ( F2 @ X5 ) @ ( F2 @ Y5 ) ) ) ) ) ) ).
% increasingD
thf(fact_215_increasingD,axiom,
! [M2: set_set_set_a,F2: set_set_a > nat,X5: set_set_a,Y5: set_set_a] :
( ( measur1244951900059291067_a_nat @ M2 @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ Y5 )
=> ( ( member_set_set_a @ X5 @ M2 )
=> ( ( member_set_set_a @ Y5 @ M2 )
=> ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y5 ) ) ) ) ) ) ).
% increasingD
thf(fact_216_increasingD,axiom,
! [M2: set_set_set_a,F2: set_set_a > int,X5: set_set_a,Y5: set_set_a] :
( ( measur1242461429550240791_a_int @ M2 @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ Y5 )
=> ( ( member_set_set_a @ X5 @ M2 )
=> ( ( member_set_set_a @ Y5 @ M2 )
=> ( ord_less_eq_int @ ( F2 @ X5 ) @ ( F2 @ Y5 ) ) ) ) ) ) ).
% increasingD
thf(fact_217_increasingD,axiom,
! [M2: set_set_a,F2: set_a > set_a,X5: set_a,Y5: set_a] :
( ( measur7842569353079325843_set_a @ M2 @ F2 )
=> ( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ( member_set_a @ X5 @ M2 )
=> ( ( member_set_a @ Y5 @ M2 )
=> ( ord_less_eq_set_a @ ( F2 @ X5 ) @ ( F2 @ Y5 ) ) ) ) ) ) ).
% increasingD
thf(fact_218_increasingD,axiom,
! [M2: set_set_set_b,F2: set_set_b > set_a,X5: set_set_b,Y5: set_set_b] :
( ( measur6668990528421247474_set_a @ M2 @ F2 )
=> ( ( ord_le3795704787696855135_set_b @ X5 @ Y5 )
=> ( ( member_set_set_b @ X5 @ M2 )
=> ( ( member_set_set_b @ Y5 @ M2 )
=> ( ord_less_eq_set_a @ ( F2 @ X5 ) @ ( F2 @ Y5 ) ) ) ) ) ) ).
% increasingD
thf(fact_219_increasingD,axiom,
! [M2: set_set_set_a,F2: set_set_a > set_a,X5: set_set_a,Y5: set_set_a] :
( ( measur5181028491126448947_set_a @ M2 @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ Y5 )
=> ( ( member_set_set_a @ X5 @ M2 )
=> ( ( member_set_set_a @ Y5 @ M2 )
=> ( ord_less_eq_set_a @ ( F2 @ X5 ) @ ( F2 @ Y5 ) ) ) ) ) ) ).
% increasingD
thf(fact_220_subprob__space_Oaxioms_I1_J,axiom,
! [M2: sigma_measure_a] :
( ( giry_subprob_space_a @ M2 )
=> ( measur930452917991658466sure_a @ M2 ) ) ).
% subprob_space.axioms(1)
thf(fact_221_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_222_prob__space_Oindep__set_Ocong,axiom,
indepe2041756565122539606_set_a = indepe2041756565122539606_set_a ).
% prob_space.indep_set.cong
thf(fact_223_prob__space_Otail__events_Ocong,axiom,
indepe7538416700049374166_a_nat = indepe7538416700049374166_a_nat ).
% prob_space.tail_events.cong
thf(fact_224_sets__eq__iff__bounded,axiom,
! [A: sigma_measure_b,B: sigma_measure_b,C: sigma_measure_b] :
( ( ord_le254669799889008988sure_b @ A @ B )
=> ( ( ord_le254669799889008988sure_b @ B @ C )
=> ( ( ( sigma_sets_b @ A )
= ( sigma_sets_b @ C ) )
=> ( ( sigma_sets_b @ B )
= ( sigma_sets_b @ A ) ) ) ) ) ).
% sets_eq_iff_bounded
thf(fact_225_sets__eq__iff__bounded,axiom,
! [A: sigma_measure_a,B: sigma_measure_a,C: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A @ B )
=> ( ( ord_le254669795585780187sure_a @ B @ C )
=> ( ( ( sigma_sets_a @ A )
= ( sigma_sets_a @ C ) )
=> ( ( sigma_sets_a @ B )
= ( sigma_sets_a @ A ) ) ) ) ) ).
% sets_eq_iff_bounded
thf(fact_226_nle__le,axiom,
! [A3: nat,B3: nat] :
( ( ~ ( ord_less_eq_nat @ A3 @ B3 ) )
= ( ( ord_less_eq_nat @ B3 @ A3 )
& ( B3 != A3 ) ) ) ).
% nle_le
thf(fact_227_nle__le,axiom,
! [A3: int,B3: int] :
( ( ~ ( ord_less_eq_int @ A3 @ B3 ) )
= ( ( ord_less_eq_int @ B3 @ A3 )
& ( B3 != A3 ) ) ) ).
% nle_le
thf(fact_228_le__cases3,axiom,
! [X5: nat,Y5: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X5 @ Y5 )
=> ~ ( ord_less_eq_nat @ Y5 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y5 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X5 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y5 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y5 )
=> ~ ( ord_less_eq_nat @ Y5 @ X5 ) )
=> ( ( ( ord_less_eq_nat @ Y5 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X5 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ Y5 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_229_le__cases3,axiom,
! [X5: int,Y5: int,Z2: int] :
( ( ( ord_less_eq_int @ X5 @ Y5 )
=> ~ ( ord_less_eq_int @ Y5 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y5 @ X5 )
=> ~ ( ord_less_eq_int @ X5 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X5 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y5 ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y5 )
=> ~ ( ord_less_eq_int @ Y5 @ X5 ) )
=> ( ( ( ord_less_eq_int @ Y5 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X5 ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X5 )
=> ~ ( ord_less_eq_int @ X5 @ Y5 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_230_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_set_b,Z: set_set_b] : ( Y3 = Z ) )
= ( ^ [X4: set_set_b,Y4: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X4 @ Y4 )
& ( ord_le3795704787696855135_set_b @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_231_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_set_a,Z: set_set_a] : ( Y3 = Z ) )
= ( ^ [X4: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y4 )
& ( ord_le3724670747650509150_set_a @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_232_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_233_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_234_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_235_ord__eq__le__trans,axiom,
! [A3: set_set_b,B3: set_set_b,C2: set_set_b] :
( ( A3 = B3 )
=> ( ( ord_le3795704787696855135_set_b @ B3 @ C2 )
=> ( ord_le3795704787696855135_set_b @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_236_ord__eq__le__trans,axiom,
! [A3: set_set_a,B3: set_set_a,C2: set_set_a] :
( ( A3 = B3 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ C2 )
=> ( ord_le3724670747650509150_set_a @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_237_ord__eq__le__trans,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( A3 = B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_238_ord__eq__le__trans,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( A3 = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_239_ord__eq__le__trans,axiom,
! [A3: int,B3: int,C2: int] :
( ( A3 = B3 )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ord_less_eq_int @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_240_ord__le__eq__trans,axiom,
! [A3: set_set_b,B3: set_set_b,C2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_le3795704787696855135_set_b @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_241_ord__le__eq__trans,axiom,
! [A3: set_set_a,B3: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_le3724670747650509150_set_a @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_242_ord__le__eq__trans,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_243_ord__le__eq__trans,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_244_ord__le__eq__trans,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq_int @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_245_order__antisym,axiom,
! [X5: set_set_b,Y5: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X5 @ Y5 )
=> ( ( ord_le3795704787696855135_set_b @ Y5 @ X5 )
=> ( X5 = Y5 ) ) ) ).
% order_antisym
thf(fact_246_order__antisym,axiom,
! [X5: set_set_a,Y5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ Y5 )
=> ( ( ord_le3724670747650509150_set_a @ Y5 @ X5 )
=> ( X5 = Y5 ) ) ) ).
% order_antisym
thf(fact_247_order__antisym,axiom,
! [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ( ord_less_eq_set_a @ Y5 @ X5 )
=> ( X5 = Y5 ) ) ) ).
% order_antisym
thf(fact_248_order__antisym,axiom,
! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ( ord_less_eq_nat @ Y5 @ X5 )
=> ( X5 = Y5 ) ) ) ).
% order_antisym
thf(fact_249_order__antisym,axiom,
! [X5: int,Y5: int] :
( ( ord_less_eq_int @ X5 @ Y5 )
=> ( ( ord_less_eq_int @ Y5 @ X5 )
=> ( X5 = Y5 ) ) ) ).
% order_antisym
thf(fact_250_order_Otrans,axiom,
! [A3: set_set_b,B3: set_set_b,C2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A3 @ B3 )
=> ( ( ord_le3795704787696855135_set_b @ B3 @ C2 )
=> ( ord_le3795704787696855135_set_b @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_251_order_Otrans,axiom,
! [A3: set_set_a,B3: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ C2 )
=> ( ord_le3724670747650509150_set_a @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_252_order_Otrans,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_253_order_Otrans,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_254_order_Otrans,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ord_less_eq_int @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_255_order__trans,axiom,
! [X5: set_set_b,Y5: set_set_b,Z2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X5 @ Y5 )
=> ( ( ord_le3795704787696855135_set_b @ Y5 @ Z2 )
=> ( ord_le3795704787696855135_set_b @ X5 @ Z2 ) ) ) ).
% order_trans
thf(fact_256_order__trans,axiom,
! [X5: set_set_a,Y5: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ Y5 )
=> ( ( ord_le3724670747650509150_set_a @ Y5 @ Z2 )
=> ( ord_le3724670747650509150_set_a @ X5 @ Z2 ) ) ) ).
% order_trans
thf(fact_257_order__trans,axiom,
! [X5: set_a,Y5: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ( ord_less_eq_set_a @ Y5 @ Z2 )
=> ( ord_less_eq_set_a @ X5 @ Z2 ) ) ) ).
% order_trans
thf(fact_258_order__trans,axiom,
! [X5: nat,Y5: nat,Z2: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ( ord_less_eq_nat @ Y5 @ Z2 )
=> ( ord_less_eq_nat @ X5 @ Z2 ) ) ) ).
% order_trans
thf(fact_259_order__trans,axiom,
! [X5: int,Y5: int,Z2: int] :
( ( ord_less_eq_int @ X5 @ Y5 )
=> ( ( ord_less_eq_int @ Y5 @ Z2 )
=> ( ord_less_eq_int @ X5 @ Z2 ) ) ) ).
% order_trans
thf(fact_260_linorder__wlog,axiom,
! [P: nat > nat > $o,A3: nat,B3: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A3 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_261_linorder__wlog,axiom,
! [P: int > int > $o,A3: int,B3: int] :
( ! [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A3 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_262_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_set_b,Z: set_set_b] : ( Y3 = Z ) )
= ( ^ [A5: set_set_b,B5: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B5 @ A5 )
& ( ord_le3795704787696855135_set_b @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_263_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_set_a,Z: set_set_a] : ( Y3 = Z ) )
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ A5 )
& ( ord_le3724670747650509150_set_a @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_264_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A5 )
& ( ord_less_eq_set_a @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_265_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_266_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ B5 @ A5 )
& ( ord_less_eq_int @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_267_dual__order_Oantisym,axiom,
! [B3: set_set_b,A3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B3 @ A3 )
=> ( ( ord_le3795704787696855135_set_b @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_268_dual__order_Oantisym,axiom,
! [B3: set_set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A3 )
=> ( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_269_dual__order_Oantisym,axiom,
! [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_270_dual__order_Oantisym,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_271_dual__order_Oantisym,axiom,
! [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
=> ( ( ord_less_eq_int @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_272_dual__order_Otrans,axiom,
! [B3: set_set_b,A3: set_set_b,C2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B3 @ A3 )
=> ( ( ord_le3795704787696855135_set_b @ C2 @ B3 )
=> ( ord_le3795704787696855135_set_b @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_273_dual__order_Otrans,axiom,
! [B3: set_set_a,A3: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A3 )
=> ( ( ord_le3724670747650509150_set_a @ C2 @ B3 )
=> ( ord_le3724670747650509150_set_a @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_274_dual__order_Otrans,axiom,
! [B3: set_a,A3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( ( ord_less_eq_set_a @ C2 @ B3 )
=> ( ord_less_eq_set_a @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_275_dual__order_Otrans,axiom,
! [B3: nat,A3: nat,C2: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ C2 @ B3 )
=> ( ord_less_eq_nat @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_276_dual__order_Otrans,axiom,
! [B3: int,A3: int,C2: int] :
( ( ord_less_eq_int @ B3 @ A3 )
=> ( ( ord_less_eq_int @ C2 @ B3 )
=> ( ord_less_eq_int @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_277_antisym,axiom,
! [A3: set_set_b,B3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A3 @ B3 )
=> ( ( ord_le3795704787696855135_set_b @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_278_antisym,axiom,
! [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_279_antisym,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_280_antisym,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_281_antisym,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_282_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_set_b,Z: set_set_b] : ( Y3 = Z ) )
= ( ^ [A5: set_set_b,B5: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A5 @ B5 )
& ( ord_le3795704787696855135_set_b @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_283_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_set_a,Z: set_set_a] : ( Y3 = Z ) )
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_284_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_285_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_286_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
& ( ord_less_eq_int @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_287_order__subst1,axiom,
! [A3: nat,F2: nat > nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ ( F2 @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_288_order__subst1,axiom,
! [A3: nat,F2: int > nat,B3: int,C2: int] :
( ( ord_less_eq_nat @ A3 @ ( F2 @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X3: int,Y6: int] :
( ( ord_less_eq_int @ X3 @ Y6 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_289_order__subst1,axiom,
! [A3: int,F2: nat > int,B3: nat,C2: nat] :
( ( ord_less_eq_int @ A3 @ ( F2 @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_290_order__subst1,axiom,
! [A3: int,F2: int > int,B3: int,C2: int] :
( ( ord_less_eq_int @ A3 @ ( F2 @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X3: int,Y6: int] :
( ( ord_less_eq_int @ X3 @ Y6 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_291_order__subst1,axiom,
! [A3: set_a,F2: nat > set_a,B3: nat,C2: nat] :
( ( ord_less_eq_set_a @ A3 @ ( F2 @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
=> ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_set_a @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_292_order__subst1,axiom,
! [A3: set_a,F2: int > set_a,B3: int,C2: int] :
( ( ord_less_eq_set_a @ A3 @ ( F2 @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X3: int,Y6: int] :
( ( ord_less_eq_int @ X3 @ Y6 )
=> ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_set_a @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_293_order__subst1,axiom,
! [A3: nat,F2: set_a > nat,B3: set_a,C2: set_a] :
( ( ord_less_eq_nat @ A3 @ ( F2 @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ! [X3: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y6 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_294_order__subst1,axiom,
! [A3: int,F2: set_a > int,B3: set_a,C2: set_a] :
( ( ord_less_eq_int @ A3 @ ( F2 @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ! [X3: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y6 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_295_order__subst1,axiom,
! [A3: set_set_b,F2: nat > set_set_b,B3: nat,C2: nat] :
( ( ord_le3795704787696855135_set_b @ A3 @ ( F2 @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
=> ( ord_le3795704787696855135_set_b @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_le3795704787696855135_set_b @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_296_order__subst1,axiom,
! [A3: set_set_b,F2: int > set_set_b,B3: int,C2: int] :
( ( ord_le3795704787696855135_set_b @ A3 @ ( F2 @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X3: int,Y6: int] :
( ( ord_less_eq_int @ X3 @ Y6 )
=> ( ord_le3795704787696855135_set_b @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_le3795704787696855135_set_b @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_297_order__subst2,axiom,
! [A3: nat,B3: nat,F2: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F2 @ B3 ) @ C2 )
=> ( ! [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_298_order__subst2,axiom,
! [A3: nat,B3: nat,F2: nat > int,C2: int] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_int @ ( F2 @ B3 ) @ C2 )
=> ( ! [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_299_order__subst2,axiom,
! [A3: int,B3: int,F2: int > nat,C2: nat] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F2 @ B3 ) @ C2 )
=> ( ! [X3: int,Y6: int] :
( ( ord_less_eq_int @ X3 @ Y6 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_300_order__subst2,axiom,
! [A3: int,B3: int,F2: int > int,C2: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ ( F2 @ B3 ) @ C2 )
=> ( ! [X3: int,Y6: int] :
( ( ord_less_eq_int @ X3 @ Y6 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_301_order__subst2,axiom,
! [A3: set_a,B3: set_a,F2: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F2 @ B3 ) @ C2 )
=> ( ! [X3: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y6 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_302_order__subst2,axiom,
! [A3: set_a,B3: set_a,F2: set_a > int,C2: int] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_int @ ( F2 @ B3 ) @ C2 )
=> ( ! [X3: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y6 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_303_order__subst2,axiom,
! [A3: nat,B3: nat,F2: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ ( F2 @ B3 ) @ C2 )
=> ( ! [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
=> ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_set_a @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_304_order__subst2,axiom,
! [A3: int,B3: int,F2: int > set_a,C2: set_a] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ ( F2 @ B3 ) @ C2 )
=> ( ! [X3: int,Y6: int] :
( ( ord_less_eq_int @ X3 @ Y6 )
=> ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_set_a @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_305_order__subst2,axiom,
! [A3: set_set_b,B3: set_set_b,F2: set_set_b > nat,C2: nat] :
( ( ord_le3795704787696855135_set_b @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F2 @ B3 ) @ C2 )
=> ( ! [X3: set_set_b,Y6: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X3 @ Y6 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_306_order__subst2,axiom,
! [A3: set_set_b,B3: set_set_b,F2: set_set_b > int,C2: int] :
( ( ord_le3795704787696855135_set_b @ A3 @ B3 )
=> ( ( ord_less_eq_int @ ( F2 @ B3 ) @ C2 )
=> ( ! [X3: set_set_b,Y6: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X3 @ Y6 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_307_order__eq__refl,axiom,
! [X5: set_set_b,Y5: set_set_b] :
( ( X5 = Y5 )
=> ( ord_le3795704787696855135_set_b @ X5 @ Y5 ) ) ).
% order_eq_refl
thf(fact_308_order__eq__refl,axiom,
! [X5: set_set_a,Y5: set_set_a] :
( ( X5 = Y5 )
=> ( ord_le3724670747650509150_set_a @ X5 @ Y5 ) ) ).
% order_eq_refl
thf(fact_309_order__eq__refl,axiom,
! [X5: set_a,Y5: set_a] :
( ( X5 = Y5 )
=> ( ord_less_eq_set_a @ X5 @ Y5 ) ) ).
% order_eq_refl
thf(fact_310_order__eq__refl,axiom,
! [X5: nat,Y5: nat] :
( ( X5 = Y5 )
=> ( ord_less_eq_nat @ X5 @ Y5 ) ) ).
% order_eq_refl
thf(fact_311_order__eq__refl,axiom,
! [X5: int,Y5: int] :
( ( X5 = Y5 )
=> ( ord_less_eq_int @ X5 @ Y5 ) ) ).
% order_eq_refl
thf(fact_312_linorder__linear,axiom,
! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
| ( ord_less_eq_nat @ Y5 @ X5 ) ) ).
% linorder_linear
thf(fact_313_linorder__linear,axiom,
! [X5: int,Y5: int] :
( ( ord_less_eq_int @ X5 @ Y5 )
| ( ord_less_eq_int @ Y5 @ X5 ) ) ).
% linorder_linear
thf(fact_314_ord__eq__le__subst,axiom,
! [A3: nat,F2: nat > nat,B3: nat,C2: nat] :
( ( A3
= ( F2 @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_315_ord__eq__le__subst,axiom,
! [A3: int,F2: nat > int,B3: nat,C2: nat] :
( ( A3
= ( F2 @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_316_ord__eq__le__subst,axiom,
! [A3: nat,F2: int > nat,B3: int,C2: int] :
( ( A3
= ( F2 @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X3: int,Y6: int] :
( ( ord_less_eq_int @ X3 @ Y6 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_317_ord__eq__le__subst,axiom,
! [A3: int,F2: int > int,B3: int,C2: int] :
( ( A3
= ( F2 @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X3: int,Y6: int] :
( ( ord_less_eq_int @ X3 @ Y6 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_318_ord__eq__le__subst,axiom,
! [A3: nat,F2: set_a > nat,B3: set_a,C2: set_a] :
( ( A3
= ( F2 @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ! [X3: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y6 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_319_ord__eq__le__subst,axiom,
! [A3: int,F2: set_a > int,B3: set_a,C2: set_a] :
( ( A3
= ( F2 @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ! [X3: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y6 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_320_ord__eq__le__subst,axiom,
! [A3: set_a,F2: nat > set_a,B3: nat,C2: nat] :
( ( A3
= ( F2 @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
=> ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_set_a @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_321_ord__eq__le__subst,axiom,
! [A3: set_a,F2: int > set_a,B3: int,C2: int] :
( ( A3
= ( F2 @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X3: int,Y6: int] :
( ( ord_less_eq_int @ X3 @ Y6 )
=> ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_set_a @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_322_ord__eq__le__subst,axiom,
! [A3: nat,F2: set_set_b > nat,B3: set_set_b,C2: set_set_b] :
( ( A3
= ( F2 @ B3 ) )
=> ( ( ord_le3795704787696855135_set_b @ B3 @ C2 )
=> ( ! [X3: set_set_b,Y6: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X3 @ Y6 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_323_ord__eq__le__subst,axiom,
! [A3: int,F2: set_set_b > int,B3: set_set_b,C2: set_set_b] :
( ( A3
= ( F2 @ B3 ) )
=> ( ( ord_le3795704787696855135_set_b @ B3 @ C2 )
=> ( ! [X3: set_set_b,Y6: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X3 @ Y6 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_324_ord__le__eq__subst,axiom,
! [A3: nat,B3: nat,F2: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F2 @ B3 )
= C2 )
=> ( ! [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_325_ord__le__eq__subst,axiom,
! [A3: nat,B3: nat,F2: nat > int,C2: int] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F2 @ B3 )
= C2 )
=> ( ! [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_326_ord__le__eq__subst,axiom,
! [A3: int,B3: int,F2: int > nat,C2: nat] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ( F2 @ B3 )
= C2 )
=> ( ! [X3: int,Y6: int] :
( ( ord_less_eq_int @ X3 @ Y6 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_327_ord__le__eq__subst,axiom,
! [A3: int,B3: int,F2: int > int,C2: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ( F2 @ B3 )
= C2 )
=> ( ! [X3: int,Y6: int] :
( ( ord_less_eq_int @ X3 @ Y6 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_328_ord__le__eq__subst,axiom,
! [A3: set_a,B3: set_a,F2: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ( F2 @ B3 )
= C2 )
=> ( ! [X3: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y6 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_329_ord__le__eq__subst,axiom,
! [A3: set_a,B3: set_a,F2: set_a > int,C2: int] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ( F2 @ B3 )
= C2 )
=> ( ! [X3: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y6 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_330_ord__le__eq__subst,axiom,
! [A3: nat,B3: nat,F2: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F2 @ B3 )
= C2 )
=> ( ! [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
=> ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_set_a @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_331_ord__le__eq__subst,axiom,
! [A3: int,B3: int,F2: int > set_a,C2: set_a] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ( F2 @ B3 )
= C2 )
=> ( ! [X3: int,Y6: int] :
( ( ord_less_eq_int @ X3 @ Y6 )
=> ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_set_a @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_332_ord__le__eq__subst,axiom,
! [A3: set_set_b,B3: set_set_b,F2: set_set_b > nat,C2: nat] :
( ( ord_le3795704787696855135_set_b @ A3 @ B3 )
=> ( ( ( F2 @ B3 )
= C2 )
=> ( ! [X3: set_set_b,Y6: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X3 @ Y6 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_333_ord__le__eq__subst,axiom,
! [A3: set_set_b,B3: set_set_b,F2: set_set_b > int,C2: int] :
( ( ord_le3795704787696855135_set_b @ A3 @ B3 )
=> ( ( ( F2 @ B3 )
= C2 )
=> ( ! [X3: set_set_b,Y6: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X3 @ Y6 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y6 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_334_linorder__le__cases,axiom,
! [X5: nat,Y5: nat] :
( ~ ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X5 ) ) ).
% linorder_le_cases
thf(fact_335_linorder__le__cases,axiom,
! [X5: int,Y5: int] :
( ~ ( ord_less_eq_int @ X5 @ Y5 )
=> ( ord_less_eq_int @ Y5 @ X5 ) ) ).
% linorder_le_cases
thf(fact_336_order__antisym__conv,axiom,
! [Y5: set_set_b,X5: set_set_b] :
( ( ord_le3795704787696855135_set_b @ Y5 @ X5 )
=> ( ( ord_le3795704787696855135_set_b @ X5 @ Y5 )
= ( X5 = Y5 ) ) ) ).
% order_antisym_conv
thf(fact_337_order__antisym__conv,axiom,
! [Y5: set_set_a,X5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y5 @ X5 )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ Y5 )
= ( X5 = Y5 ) ) ) ).
% order_antisym_conv
thf(fact_338_order__antisym__conv,axiom,
! [Y5: set_a,X5: set_a] :
( ( ord_less_eq_set_a @ Y5 @ X5 )
=> ( ( ord_less_eq_set_a @ X5 @ Y5 )
= ( X5 = Y5 ) ) ) ).
% order_antisym_conv
thf(fact_339_order__antisym__conv,axiom,
! [Y5: nat,X5: nat] :
( ( ord_less_eq_nat @ Y5 @ X5 )
=> ( ( ord_less_eq_nat @ X5 @ Y5 )
= ( X5 = Y5 ) ) ) ).
% order_antisym_conv
thf(fact_340_order__antisym__conv,axiom,
! [Y5: int,X5: int] :
( ( ord_less_eq_int @ Y5 @ X5 )
=> ( ( ord_less_eq_int @ X5 @ Y5 )
= ( X5 = Y5 ) ) ) ).
% order_antisym_conv
thf(fact_341_prob__space_Oindep__setD__ev1,axiom,
! [M2: sigma_measure_b,A: set_set_b,B: set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe2041756565122539607_set_b @ M2 @ A @ B )
=> ( ord_le3795704787696855135_set_b @ A @ ( sigma_sets_b @ M2 ) ) ) ) ).
% prob_space.indep_setD_ev1
thf(fact_342_prob__space_Oindep__setD__ev1,axiom,
! [M2: sigma_measure_a,A: set_set_a,B: set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe2041756565122539606_set_a @ M2 @ A @ B )
=> ( ord_le3724670747650509150_set_a @ A @ ( sigma_sets_a @ M2 ) ) ) ) ).
% prob_space.indep_setD_ev1
thf(fact_343_prob__space_Oindep__setD__ev2,axiom,
! [M2: sigma_measure_b,A: set_set_b,B: set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe2041756565122539607_set_b @ M2 @ A @ B )
=> ( ord_le3795704787696855135_set_b @ B @ ( sigma_sets_b @ M2 ) ) ) ) ).
% prob_space.indep_setD_ev2
thf(fact_344_prob__space_Oindep__setD__ev2,axiom,
! [M2: sigma_measure_a,A: set_set_a,B: set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe2041756565122539606_set_a @ M2 @ A @ B )
=> ( ord_le3724670747650509150_set_a @ B @ ( sigma_sets_a @ M2 ) ) ) ) ).
% prob_space.indep_setD_ev2
thf(fact_345_prob__space_Otail__events__sets,axiom,
! [M2: sigma_measure_b,A: nat > set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ! [I3: nat] : ( ord_le3795704787696855135_set_b @ ( A @ I3 ) @ ( sigma_sets_b @ M2 ) )
=> ( ord_le3795704787696855135_set_b @ ( indepe8773861029005768663_b_nat @ M2 @ A ) @ ( sigma_sets_b @ M2 ) ) ) ) ).
% prob_space.tail_events_sets
thf(fact_346_prob__space_Otail__events__sets,axiom,
! [M2: sigma_measure_a,A: nat > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ! [I3: nat] : ( ord_le3724670747650509150_set_a @ ( A @ I3 ) @ ( sigma_sets_a @ M2 ) )
=> ( ord_le3724670747650509150_set_a @ ( indepe7538416700049374166_a_nat @ M2 @ A ) @ ( sigma_sets_a @ M2 ) ) ) ) ).
% prob_space.tail_events_sets
thf(fact_347_measurable__completion,axiom,
! [F2: a > b,M2: sigma_measure_a,N: sigma_measure_b] :
( ( member_a_b @ F2 @ ( sigma_measurable_a_b @ M2 @ N ) )
=> ( member_a_b @ F2 @ ( sigma_measurable_a_b @ ( comple3428971583294703880tion_a @ M2 ) @ N ) ) ) ).
% measurable_completion
thf(fact_348_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_349_finite__measure__mono,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ A ) @ ( sigma_measure_a2 @ m @ B ) ) ) ) ).
% finite_measure_mono
thf(fact_350_measure__increasing,axiom,
measur1776380161843274167a_real @ ( sigma_sets_a @ m ) @ ( sigma_measure_a2 @ m ) ).
% measure_increasing
thf(fact_351_measurable__comp,axiom,
! [F2: a > a,M2: sigma_measure_a,N: sigma_measure_a,G2: a > b,L: sigma_measure_b] :
( ( member_a_a @ F2 @ ( sigma_measurable_a_a @ M2 @ N ) )
=> ( ( member_a_b @ G2 @ ( sigma_measurable_a_b @ N @ L ) )
=> ( member_a_b @ ( comp_a_b_a @ G2 @ F2 ) @ ( sigma_measurable_a_b @ M2 @ L ) ) ) ) ).
% measurable_comp
thf(fact_352_measurable__comp,axiom,
! [F2: a > b,M2: sigma_measure_a,N: sigma_measure_b,G2: b > b,L: sigma_measure_b] :
( ( member_a_b @ F2 @ ( sigma_measurable_a_b @ M2 @ N ) )
=> ( ( member_b_b @ G2 @ ( sigma_measurable_b_b @ N @ L ) )
=> ( member_a_b @ ( comp_b_b_a @ G2 @ F2 ) @ ( sigma_measurable_a_b @ M2 @ L ) ) ) ) ).
% measurable_comp
thf(fact_353_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_354_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_355_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_356_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_357_fmeasurable__eq__sets,axiom,
( ( measur3645360004775918570able_a @ m )
= ( sigma_sets_a @ m ) ) ).
% fmeasurable_eq_sets
thf(fact_358_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_359_subprob__space__def,axiom,
( giry_subprob_space_a
= ( ^ [M3: sigma_measure_a] :
( ( measur930452917991658466sure_a @ M3 )
& ( giry_s1767857069175831631ioms_a @ M3 ) ) ) ) ).
% subprob_space_def
thf(fact_360_finite__measure_Ointro,axiom,
! [M2: sigma_measure_a] :
( ( measur4308613598931908895sure_a @ M2 )
=> ( ( measur2595372213310369023ioms_a @ M2 )
=> ( measur930452917991658466sure_a @ M2 ) ) ) ).
% finite_measure.intro
thf(fact_361_finite__measure__def,axiom,
( measur930452917991658466sure_a
= ( ^ [M3: sigma_measure_a] :
( ( measur4308613598931908895sure_a @ M3 )
& ( measur2595372213310369023ioms_a @ M3 ) ) ) ) ).
% finite_measure_def
thf(fact_362_prob__space_Ointro,axiom,
! [M2: sigma_measure_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ( probab8302655048591552734ioms_a @ M2 )
=> ( probab7247484486040049089pace_a @ M2 ) ) ) ).
% prob_space.intro
thf(fact_363_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_364_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_365_measure__mono__fmeasurable,axiom,
! [A: set_b,B: set_b,M2: sigma_measure_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( member_set_b @ A @ ( sigma_sets_b @ M2 ) )
=> ( ( member_set_b @ B @ ( measur3645360004775918571able_b @ M2 ) )
=> ( ord_less_eq_real @ ( sigma_measure_b2 @ M2 @ A ) @ ( sigma_measure_b2 @ M2 @ B ) ) ) ) ) ).
% measure_mono_fmeasurable
thf(fact_366_measure__mono__fmeasurable,axiom,
! [A: set_set_b,B: set_set_b,M2: sigma_measure_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( ( member_set_set_b @ A @ ( sigma_sets_set_b @ M2 ) )
=> ( ( member_set_set_b @ B @ ( measur7460903249514972363_set_b @ M2 ) )
=> ( ord_less_eq_real @ ( sigma_measure_set_b2 @ M2 @ A ) @ ( sigma_measure_set_b2 @ M2 @ B ) ) ) ) ) ).
% measure_mono_fmeasurable
thf(fact_367_measure__mono__fmeasurable,axiom,
! [A: set_set_a,B: set_set_a,M2: sigma_measure_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_set_a @ A @ ( sigma_sets_set_a @ M2 ) )
=> ( ( member_set_set_a @ B @ ( measur7460903245211743562_set_a @ M2 ) )
=> ( ord_less_eq_real @ ( sigma_measure_set_a2 @ M2 @ A ) @ ( sigma_measure_set_a2 @ M2 @ B ) ) ) ) ) ).
% measure_mono_fmeasurable
thf(fact_368_measure__mono__fmeasurable,axiom,
! [A: set_a,B: set_a,M2: sigma_measure_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M2 ) )
=> ( ( member_set_a @ B @ ( measur3645360004775918570able_a @ M2 ) )
=> ( ord_less_eq_real @ ( sigma_measure_a2 @ M2 @ A ) @ ( sigma_measure_a2 @ M2 @ B ) ) ) ) ) ).
% measure_mono_fmeasurable
thf(fact_369_completion_Ocomplete__sets__sandwich__fmeasurable,axiom,
! [A: set_set_b,M2: sigma_measure_set_b,C: set_set_b,B: set_set_b] :
( ( member_set_set_b @ A @ ( measur7460903249514972363_set_b @ ( comple8942076150311361001_set_b @ M2 ) ) )
=> ( ( member_set_set_b @ C @ ( measur7460903249514972363_set_b @ ( comple8942076150311361001_set_b @ M2 ) ) )
=> ( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( ( ord_le3795704787696855135_set_b @ B @ C )
=> ( ( ( sigma_measure_set_b2 @ ( comple8942076150311361001_set_b @ M2 ) @ A )
= ( 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_370_completion_Ocomplete__sets__sandwich__fmeasurable,axiom,
! [A: set_set_a,M2: sigma_measure_set_a,C: set_set_a,B: set_set_a] :
( ( member_set_set_a @ A @ ( measur7460903245211743562_set_a @ ( comple8942076146008132200_set_a @ M2 ) ) )
=> ( ( member_set_set_a @ C @ ( measur7460903245211743562_set_a @ ( comple8942076146008132200_set_a @ M2 ) ) )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ( ( sigma_measure_set_a2 @ ( comple8942076146008132200_set_a @ M2 ) @ A )
= ( 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_371_completion_Ocomplete__sets__sandwich__fmeasurable,axiom,
! [A: set_a,M2: sigma_measure_a,C: set_a,B: set_a] :
( ( member_set_a @ A @ ( measur3645360004775918570able_a @ ( comple3428971583294703880tion_a @ M2 ) ) )
=> ( ( member_set_a @ C @ ( measur3645360004775918570able_a @ ( comple3428971583294703880tion_a @ M2 ) ) )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ( ( sigma_measure_a2 @ ( comple3428971583294703880tion_a @ M2 ) @ A )
= ( sigma_measure_a2 @ ( comple3428971583294703880tion_a @ M2 ) @ C ) )
=> ( member_set_a @ B @ ( measur3645360004775918570able_a @ ( comple3428971583294703880tion_a @ M2 ) ) ) ) ) ) ) ) ).
% completion.complete_sets_sandwich_fmeasurable
thf(fact_372_fmeasurableD,axiom,
! [A: set_b,M2: sigma_measure_b] :
( ( member_set_b @ A @ ( measur3645360004775918571able_b @ M2 ) )
=> ( member_set_b @ A @ ( sigma_sets_b @ M2 ) ) ) ).
% fmeasurableD
thf(fact_373_fmeasurableD,axiom,
! [A: set_a,M2: sigma_measure_a] :
( ( member_set_a @ A @ ( measur3645360004775918570able_a @ M2 ) )
=> ( member_set_a @ A @ ( sigma_sets_a @ M2 ) ) ) ).
% fmeasurableD
thf(fact_374_fmeasurableI2,axiom,
! [A: set_b,M2: sigma_measure_b,B: set_b] :
( ( member_set_b @ A @ ( measur3645360004775918571able_b @ M2 ) )
=> ( ( ord_less_eq_set_b @ B @ A )
=> ( ( member_set_b @ B @ ( sigma_sets_b @ M2 ) )
=> ( member_set_b @ B @ ( measur3645360004775918571able_b @ M2 ) ) ) ) ) ).
% fmeasurableI2
thf(fact_375_fmeasurableI2,axiom,
! [A: set_set_b,M2: sigma_measure_set_b,B: set_set_b] :
( ( member_set_set_b @ A @ ( measur7460903249514972363_set_b @ M2 ) )
=> ( ( ord_le3795704787696855135_set_b @ B @ A )
=> ( ( member_set_set_b @ B @ ( sigma_sets_set_b @ M2 ) )
=> ( member_set_set_b @ B @ ( measur7460903249514972363_set_b @ M2 ) ) ) ) ) ).
% fmeasurableI2
thf(fact_376_fmeasurableI2,axiom,
! [A: set_set_a,M2: sigma_measure_set_a,B: set_set_a] :
( ( member_set_set_a @ A @ ( measur7460903245211743562_set_a @ M2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( member_set_set_a @ B @ ( sigma_sets_set_a @ M2 ) )
=> ( member_set_set_a @ B @ ( measur7460903245211743562_set_a @ M2 ) ) ) ) ) ).
% fmeasurableI2
thf(fact_377_fmeasurableI2,axiom,
! [A: set_a,M2: sigma_measure_a,B: set_a] :
( ( member_set_a @ A @ ( measur3645360004775918570able_a @ M2 ) )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M2 ) )
=> ( member_set_a @ B @ ( measur3645360004775918570able_a @ M2 ) ) ) ) ) ).
% fmeasurableI2
thf(fact_378_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_379_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_380_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_381_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_382_prob__space_Oaxioms_I2_J,axiom,
! [M2: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( probab8302655048591552734ioms_a @ M2 ) ) ).
% prob_space.axioms(2)
thf(fact_383_finite__measure_Ofinite__measure__mono,axiom,
! [M2: sigma_measure_b,A: set_b,B: set_b] :
( ( measur930452917991658467sure_b @ M2 )
=> ( ( ord_less_eq_set_b @ A @ B )
=> ( ( member_set_b @ B @ ( sigma_sets_b @ M2 ) )
=> ( ord_less_eq_real @ ( sigma_measure_b2 @ M2 @ A ) @ ( sigma_measure_b2 @ M2 @ B ) ) ) ) ) ).
% finite_measure.finite_measure_mono
thf(fact_384_finite__measure_Ofinite__measure__mono,axiom,
! [M2: sigma_measure_set_b,A: set_set_b,B: set_set_b] :
( ( measur2212693997687831747_set_b @ M2 )
=> ( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( ( member_set_set_b @ B @ ( sigma_sets_set_b @ M2 ) )
=> ( ord_less_eq_real @ ( sigma_measure_set_b2 @ M2 @ A ) @ ( sigma_measure_set_b2 @ M2 @ B ) ) ) ) ) ).
% finite_measure.finite_measure_mono
thf(fact_385_finite__measure_Ofinite__measure__mono,axiom,
! [M2: sigma_measure_set_a,A: set_set_a,B: set_set_a] :
( ( measur2212693993384602946_set_a @ M2 )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_set_a @ B @ ( sigma_sets_set_a @ M2 ) )
=> ( ord_less_eq_real @ ( sigma_measure_set_a2 @ M2 @ A ) @ ( sigma_measure_set_a2 @ M2 @ B ) ) ) ) ) ).
% finite_measure.finite_measure_mono
thf(fact_386_finite__measure_Ofinite__measure__mono,axiom,
! [M2: sigma_measure_a,A: set_a,B: set_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M2 ) )
=> ( ord_less_eq_real @ ( sigma_measure_a2 @ M2 @ A ) @ ( sigma_measure_a2 @ M2 @ B ) ) ) ) ) ).
% finite_measure.finite_measure_mono
thf(fact_387_finite__measure_Oaxioms_I2_J,axiom,
! [M2: sigma_measure_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( measur2595372213310369023ioms_a @ M2 ) ) ).
% finite_measure.axioms(2)
thf(fact_388_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_389_prob__space__def,axiom,
( probab7247484486040049089pace_a
= ( ^ [M3: sigma_measure_a] :
( ( measur930452917991658466sure_a @ M3 )
& ( probab8302655048591552734ioms_a @ M3 ) ) ) ) ).
% prob_space_def
thf(fact_390_measure__ge__1__iff,axiom,
! [A: set_a] :
( ( ord_less_eq_real @ one_one_real @ ( sigma_measure_a2 @ m @ A ) )
= ( ( sigma_measure_a2 @ m @ A )
= one_one_real ) ) ).
% measure_ge_1_iff
thf(fact_391_bounded__measure,axiom,
! [A: set_a] : ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ A ) @ ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) ) ) ).
% bounded_measure
thf(fact_392_prob__le__1,axiom,
! [A: set_a] : ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ A ) @ one_one_real ) ).
% prob_le_1
thf(fact_393_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_394_indep__setI,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( sigma_sets_a @ m ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ! [A4: set_a,B4: set_a] :
( ( member_set_a @ A4 @ A )
=> ( ( member_set_a @ B4 @ B )
=> ( ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ A4 @ B4 ) )
= ( times_times_real @ ( sigma_measure_a2 @ m @ A4 ) @ ( sigma_measure_a2 @ m @ B4 ) ) ) ) )
=> ( indepe2041756565122539606_set_a @ m @ A @ B ) ) ) ) ).
% indep_setI
thf(fact_395_indep__sets2__eq,axiom,
! [A: set_set_a,B: set_set_a] :
( ( indepe2041756565122539606_set_a @ m @ A @ B )
= ( ( ord_le3724670747650509150_set_a @ A @ ( sigma_sets_a @ m ) )
& ( ord_le3724670747650509150_set_a @ B @ ( sigma_sets_a @ m ) )
& ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ B )
=> ( ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ X4 @ Y4 ) )
= ( times_times_real @ ( sigma_measure_a2 @ m @ X4 ) @ ( sigma_measure_a2 @ m @ Y4 ) ) ) ) ) ) ) ).
% indep_sets2_eq
thf(fact_396_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_397_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_398_complete__measure_Ocomplete__sets__sandwich__fmeasurable,axiom,
! [M2: sigma_measure_set_b,A: set_set_b,C: set_set_b,B: set_set_b] :
( ( comple6693822267556782962_set_b @ M2 )
=> ( ( member_set_set_b @ A @ ( measur7460903249514972363_set_b @ M2 ) )
=> ( ( member_set_set_b @ C @ ( measur7460903249514972363_set_b @ M2 ) )
=> ( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( ( ord_le3795704787696855135_set_b @ B @ C )
=> ( ( ( sigma_measure_set_b2 @ M2 @ A )
= ( sigma_measure_set_b2 @ M2 @ C ) )
=> ( member_set_set_b @ B @ ( measur7460903249514972363_set_b @ M2 ) ) ) ) ) ) ) ) ).
% complete_measure.complete_sets_sandwich_fmeasurable
thf(fact_399_complete__measure_Ocomplete__sets__sandwich__fmeasurable,axiom,
! [M2: sigma_measure_set_a,A: set_set_a,C: set_set_a,B: set_set_a] :
( ( comple6693822263253554161_set_a @ M2 )
=> ( ( member_set_set_a @ A @ ( measur7460903245211743562_set_a @ M2 ) )
=> ( ( member_set_set_a @ C @ ( measur7460903245211743562_set_a @ M2 ) )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ( ( sigma_measure_set_a2 @ M2 @ A )
= ( sigma_measure_set_a2 @ M2 @ C ) )
=> ( member_set_set_a @ B @ ( measur7460903245211743562_set_a @ M2 ) ) ) ) ) ) ) ) ).
% complete_measure.complete_sets_sandwich_fmeasurable
thf(fact_400_complete__measure_Ocomplete__sets__sandwich__fmeasurable,axiom,
! [M2: sigma_measure_a,A: set_a,C: set_a,B: set_a] :
( ( comple8155536527497655953sure_a @ M2 )
=> ( ( member_set_a @ A @ ( measur3645360004775918570able_a @ M2 ) )
=> ( ( member_set_a @ C @ ( measur3645360004775918570able_a @ M2 ) )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ( ( sigma_measure_a2 @ M2 @ A )
= ( sigma_measure_a2 @ M2 @ C ) )
=> ( member_set_a @ B @ ( measur3645360004775918570able_a @ M2 ) ) ) ) ) ) ) ) ).
% complete_measure.complete_sets_sandwich_fmeasurable
thf(fact_401_completion_Omeasurable__completion2,axiom,
! [F2: a > b,M2: sigma_measure_a,N: sigma_measure_b] :
( ( member_a_b @ F2 @ ( 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 @ F2 ) ) )
=> ( member_a_b @ F2 @ ( sigma_measurable_a_b @ ( comple3428971583294703880tion_a @ M2 ) @ ( comple3428971583294703881tion_b @ N ) ) ) ) ) ).
% completion.measurable_completion2
thf(fact_402_completion_Omeasurable__completion2,axiom,
! [F2: a > a,M2: sigma_measure_a,N: sigma_measure_a] :
( ( member_a_a @ F2 @ ( 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 @ F2 ) ) )
=> ( member_a_a @ F2 @ ( sigma_measurable_a_a @ ( comple3428971583294703880tion_a @ M2 ) @ ( comple3428971583294703880tion_a @ N ) ) ) ) ) ).
% completion.measurable_completion2
thf(fact_403_Int__iff,axiom,
! [C2: a > b,A: set_a_b,B: set_a_b] :
( ( member_a_b @ C2 @ ( inf_inf_set_a_b @ A @ B ) )
= ( ( member_a_b @ C2 @ A )
& ( member_a_b @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_404_Int__iff,axiom,
! [C2: c,A: set_c,B: set_c] :
( ( member_c @ C2 @ ( inf_inf_set_c @ A @ B ) )
= ( ( member_c @ C2 @ A )
& ( member_c @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_405_Int__iff,axiom,
! [C2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A @ B ) )
= ( ( member_set_a @ C2 @ A )
& ( member_set_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_406_Int__iff,axiom,
! [C2: real,A: set_real,B: set_real] :
( ( member_real @ C2 @ ( inf_inf_set_real @ A @ B ) )
= ( ( member_real @ C2 @ A )
& ( member_real @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_407_Int__iff,axiom,
! [C2: $o,A: set_o,B: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A @ B ) )
= ( ( member_o @ C2 @ A )
& ( member_o @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_408_Int__iff,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
= ( ( member_nat @ C2 @ A )
& ( member_nat @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_409_Int__iff,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
= ( ( member_a @ C2 @ A )
& ( member_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_410_IntI,axiom,
! [C2: a > b,A: set_a_b,B: set_a_b] :
( ( member_a_b @ C2 @ A )
=> ( ( member_a_b @ C2 @ B )
=> ( member_a_b @ C2 @ ( inf_inf_set_a_b @ A @ B ) ) ) ) ).
% IntI
thf(fact_411_IntI,axiom,
! [C2: c,A: set_c,B: set_c] :
( ( member_c @ C2 @ A )
=> ( ( member_c @ C2 @ B )
=> ( member_c @ C2 @ ( inf_inf_set_c @ A @ B ) ) ) ) ).
% IntI
thf(fact_412_IntI,axiom,
! [C2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ A )
=> ( ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_413_IntI,axiom,
! [C2: real,A: set_real,B: set_real] :
( ( member_real @ C2 @ A )
=> ( ( member_real @ C2 @ B )
=> ( member_real @ C2 @ ( inf_inf_set_real @ A @ B ) ) ) ) ).
% IntI
thf(fact_414_IntI,axiom,
! [C2: $o,A: set_o,B: set_o] :
( ( member_o @ C2 @ A )
=> ( ( member_o @ C2 @ B )
=> ( member_o @ C2 @ ( inf_inf_set_o @ A @ B ) ) ) ) ).
% IntI
thf(fact_415_IntI,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A )
=> ( ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_416_IntI,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ A )
=> ( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_417_prob__space,axiom,
( ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) )
= one_one_real ) ).
% prob_space
thf(fact_418_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_419_indep__setD,axiom,
! [A: set_set_a,B: set_set_a,A3: set_a,B3: set_a] :
( ( indepe2041756565122539606_set_a @ m @ A @ B )
=> ( ( member_set_a @ A3 @ A )
=> ( ( 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 ) ) ) ) ) ) ).
% indep_setD
thf(fact_420_Int__subset__iff,axiom,
! [C: set_set_b,A: set_set_b,B: set_set_b] :
( ( ord_le3795704787696855135_set_b @ C @ ( inf_inf_set_set_b @ A @ B ) )
= ( ( ord_le3795704787696855135_set_b @ C @ A )
& ( ord_le3795704787696855135_set_b @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_421_Int__subset__iff,axiom,
! [C: set_set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
= ( ( ord_le3724670747650509150_set_a @ C @ A )
& ( ord_le3724670747650509150_set_a @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_422_Int__subset__iff,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A @ B ) )
= ( ( ord_less_eq_set_a @ C @ A )
& ( ord_less_eq_set_a @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_423_sets_OInt,axiom,
! [A3: set_b,M2: sigma_measure_b,B3: set_b] :
( ( member_set_b @ A3 @ ( sigma_sets_b @ M2 ) )
=> ( ( member_set_b @ B3 @ ( sigma_sets_b @ M2 ) )
=> ( member_set_b @ ( inf_inf_set_b @ A3 @ B3 ) @ ( sigma_sets_b @ M2 ) ) ) ) ).
% sets.Int
thf(fact_424_sets_OInt,axiom,
! [A3: set_a,M2: sigma_measure_a,B3: set_a] :
( ( member_set_a @ A3 @ ( sigma_sets_a @ M2 ) )
=> ( ( member_set_a @ B3 @ ( sigma_sets_a @ M2 ) )
=> ( member_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ ( sigma_sets_a @ M2 ) ) ) ) ).
% sets.Int
thf(fact_425_sets_Otop,axiom,
! [M2: sigma_measure_b] : ( member_set_b @ ( sigma_space_b @ M2 ) @ ( sigma_sets_b @ M2 ) ) ).
% sets.top
thf(fact_426_sets_Otop,axiom,
! [M2: sigma_measure_a] : ( member_set_a @ ( sigma_space_a @ M2 ) @ ( sigma_sets_a @ M2 ) ) ).
% sets.top
thf(fact_427_space__completion,axiom,
! [M2: sigma_measure_a] :
( ( sigma_space_a @ ( comple3428971583294703880tion_a @ M2 ) )
= ( sigma_space_a @ M2 ) ) ).
% space_completion
thf(fact_428_fmeasurable_OInt,axiom,
! [A3: set_a,M2: sigma_measure_a,B3: set_a] :
( ( member_set_a @ A3 @ ( measur3645360004775918570able_a @ M2 ) )
=> ( ( member_set_a @ B3 @ ( measur3645360004775918570able_a @ M2 ) )
=> ( member_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ ( measur3645360004775918570able_a @ M2 ) ) ) ) ).
% fmeasurable.Int
thf(fact_429_null__sets_OInt,axiom,
! [A3: set_a,M2: sigma_measure_a,B3: set_a] :
( ( member_set_a @ A3 @ ( measure_null_sets_a @ M2 ) )
=> ( ( member_set_a @ B3 @ ( measure_null_sets_a @ M2 ) )
=> ( member_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ ( measure_null_sets_a @ M2 ) ) ) ) ).
% null_sets.Int
thf(fact_430_space__distr,axiom,
! [M2: sigma_measure_a,N: sigma_measure_b,F2: a > b] :
( ( sigma_space_b @ ( measure_distr_a_b @ M2 @ N @ F2 ) )
= ( sigma_space_b @ N ) ) ).
% space_distr
thf(fact_431_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_432_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_433_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_434_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_435_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_436_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_437_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_438_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_439_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_440_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_441_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_442_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_443_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_444_complete__measure_Ocomplete2,axiom,
! [M2: sigma_measure_set_b,A: set_set_b,B: set_set_b] :
( ( comple6693822267556782962_set_b @ M2 )
=> ( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( ( member_set_set_b @ B @ ( measur1516554132335629586_set_b @ M2 ) )
=> ( member_set_set_b @ A @ ( measur1516554132335629586_set_b @ M2 ) ) ) ) ) ).
% complete_measure.complete2
thf(fact_445_complete__measure_Ocomplete2,axiom,
! [M2: sigma_measure_set_a,A: set_set_a,B: set_set_a] :
( ( comple6693822263253554161_set_a @ M2 )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_set_a @ B @ ( measur1516554128032400785_set_a @ M2 ) )
=> ( member_set_set_a @ A @ ( measur1516554128032400785_set_a @ M2 ) ) ) ) ) ).
% complete_measure.complete2
thf(fact_446_complete__measure_Ocomplete2,axiom,
! [M2: sigma_measure_a,A: set_a,B: set_a] :
( ( comple8155536527497655953sure_a @ M2 )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_set_a @ B @ ( measure_null_sets_a @ M2 ) )
=> ( member_set_a @ A @ ( measure_null_sets_a @ M2 ) ) ) ) ) ).
% complete_measure.complete2
thf(fact_447_Int__left__commute,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A @ C ) ) ) ).
% Int_left_commute
thf(fact_448_Int__left__absorb,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% Int_left_absorb
thf(fact_449_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A2: set_a,B2: set_a] : ( inf_inf_set_a @ B2 @ A2 ) ) ) ).
% Int_commute
thf(fact_450_Int__absorb,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_451_Int__assoc,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ C )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ).
% Int_assoc
thf(fact_452_IntD2,axiom,
! [C2: a > b,A: set_a_b,B: set_a_b] :
( ( member_a_b @ C2 @ ( inf_inf_set_a_b @ A @ B ) )
=> ( member_a_b @ C2 @ B ) ) ).
% IntD2
thf(fact_453_IntD2,axiom,
! [C2: c,A: set_c,B: set_c] :
( ( member_c @ C2 @ ( inf_inf_set_c @ A @ B ) )
=> ( member_c @ C2 @ B ) ) ).
% IntD2
thf(fact_454_IntD2,axiom,
! [C2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A @ B ) )
=> ( member_set_a @ C2 @ B ) ) ).
% IntD2
thf(fact_455_IntD2,axiom,
! [C2: real,A: set_real,B: set_real] :
( ( member_real @ C2 @ ( inf_inf_set_real @ A @ B ) )
=> ( member_real @ C2 @ B ) ) ).
% IntD2
thf(fact_456_IntD2,axiom,
! [C2: $o,A: set_o,B: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A @ B ) )
=> ( member_o @ C2 @ B ) ) ).
% IntD2
thf(fact_457_IntD2,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C2 @ B ) ) ).
% IntD2
thf(fact_458_IntD2,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C2 @ B ) ) ).
% IntD2
thf(fact_459_IntD1,axiom,
! [C2: a > b,A: set_a_b,B: set_a_b] :
( ( member_a_b @ C2 @ ( inf_inf_set_a_b @ A @ B ) )
=> ( member_a_b @ C2 @ A ) ) ).
% IntD1
thf(fact_460_IntD1,axiom,
! [C2: c,A: set_c,B: set_c] :
( ( member_c @ C2 @ ( inf_inf_set_c @ A @ B ) )
=> ( member_c @ C2 @ A ) ) ).
% IntD1
thf(fact_461_IntD1,axiom,
! [C2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A @ B ) )
=> ( member_set_a @ C2 @ A ) ) ).
% IntD1
thf(fact_462_IntD1,axiom,
! [C2: real,A: set_real,B: set_real] :
( ( member_real @ C2 @ ( inf_inf_set_real @ A @ B ) )
=> ( member_real @ C2 @ A ) ) ).
% IntD1
thf(fact_463_IntD1,axiom,
! [C2: $o,A: set_o,B: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A @ B ) )
=> ( member_o @ C2 @ A ) ) ).
% IntD1
thf(fact_464_IntD1,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C2 @ A ) ) ).
% IntD1
thf(fact_465_IntD1,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C2 @ A ) ) ).
% IntD1
thf(fact_466_IntE,axiom,
! [C2: a > b,A: set_a_b,B: set_a_b] :
( ( member_a_b @ C2 @ ( inf_inf_set_a_b @ A @ B ) )
=> ~ ( ( member_a_b @ C2 @ A )
=> ~ ( member_a_b @ C2 @ B ) ) ) ).
% IntE
thf(fact_467_IntE,axiom,
! [C2: c,A: set_c,B: set_c] :
( ( member_c @ C2 @ ( inf_inf_set_c @ A @ B ) )
=> ~ ( ( member_c @ C2 @ A )
=> ~ ( member_c @ C2 @ B ) ) ) ).
% IntE
thf(fact_468_IntE,axiom,
! [C2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A @ B ) )
=> ~ ( ( member_set_a @ C2 @ A )
=> ~ ( member_set_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_469_IntE,axiom,
! [C2: real,A: set_real,B: set_real] :
( ( member_real @ C2 @ ( inf_inf_set_real @ A @ B ) )
=> ~ ( ( member_real @ C2 @ A )
=> ~ ( member_real @ C2 @ B ) ) ) ).
% IntE
thf(fact_470_IntE,axiom,
! [C2: $o,A: set_o,B: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A @ B ) )
=> ~ ( ( member_o @ C2 @ A )
=> ~ ( member_o @ C2 @ B ) ) ) ).
% IntE
thf(fact_471_IntE,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C2 @ A )
=> ~ ( member_nat @ C2 @ B ) ) ) ).
% IntE
thf(fact_472_IntE,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( member_a @ C2 @ A )
=> ~ ( member_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_473_null__set__Int1,axiom,
! [B: set_b,M2: sigma_measure_b,A: set_b] :
( ( member_set_b @ B @ ( measure_null_sets_b @ M2 ) )
=> ( ( member_set_b @ A @ ( sigma_sets_b @ M2 ) )
=> ( member_set_b @ ( inf_inf_set_b @ A @ B ) @ ( measure_null_sets_b @ M2 ) ) ) ) ).
% null_set_Int1
thf(fact_474_null__set__Int1,axiom,
! [B: set_a,M2: sigma_measure_a,A: set_a] :
( ( member_set_a @ B @ ( measure_null_sets_a @ M2 ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M2 ) )
=> ( member_set_a @ ( inf_inf_set_a @ A @ B ) @ ( measure_null_sets_a @ M2 ) ) ) ) ).
% null_set_Int1
thf(fact_475_null__set__Int2,axiom,
! [B: set_b,M2: sigma_measure_b,A: set_b] :
( ( member_set_b @ B @ ( measure_null_sets_b @ M2 ) )
=> ( ( member_set_b @ A @ ( sigma_sets_b @ M2 ) )
=> ( member_set_b @ ( inf_inf_set_b @ B @ A ) @ ( measure_null_sets_b @ M2 ) ) ) ) ).
% null_set_Int2
thf(fact_476_null__set__Int2,axiom,
! [B: set_a,M2: sigma_measure_a,A: set_a] :
( ( member_set_a @ B @ ( measure_null_sets_a @ M2 ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M2 ) )
=> ( member_set_a @ ( inf_inf_set_a @ B @ A ) @ ( measure_null_sets_a @ M2 ) ) ) ) ).
% null_set_Int2
thf(fact_477_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_478_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_479_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_480_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_481_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_482_complete__measure_Ocomplete,axiom,
! [M2: sigma_measure_b,B: set_b,A: set_b] :
( ( comple8155536527497655954sure_b @ M2 )
=> ( ( ord_less_eq_set_b @ B @ A )
=> ( ( member_set_b @ A @ ( measure_null_sets_b @ M2 ) )
=> ( member_set_b @ B @ ( sigma_sets_b @ M2 ) ) ) ) ) ).
% complete_measure.complete
thf(fact_483_complete__measure_Ocomplete,axiom,
! [M2: sigma_measure_set_b,B: set_set_b,A: set_set_b] :
( ( comple6693822267556782962_set_b @ M2 )
=> ( ( ord_le3795704787696855135_set_b @ B @ A )
=> ( ( member_set_set_b @ A @ ( measur1516554132335629586_set_b @ M2 ) )
=> ( member_set_set_b @ B @ ( sigma_sets_set_b @ M2 ) ) ) ) ) ).
% complete_measure.complete
thf(fact_484_complete__measure_Ocomplete,axiom,
! [M2: sigma_measure_set_a,B: set_set_a,A: set_set_a] :
( ( comple6693822263253554161_set_a @ M2 )
=> ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( member_set_set_a @ A @ ( measur1516554128032400785_set_a @ M2 ) )
=> ( member_set_set_a @ B @ ( sigma_sets_set_a @ M2 ) ) ) ) ) ).
% complete_measure.complete
thf(fact_485_complete__measure_Ocomplete,axiom,
! [M2: sigma_measure_a,B: set_a,A: set_a] :
( ( comple8155536527497655953sure_a @ M2 )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( ( member_set_a @ A @ ( measure_null_sets_a @ M2 ) )
=> ( member_set_a @ B @ ( sigma_sets_a @ M2 ) ) ) ) ) ).
% complete_measure.complete
thf(fact_486_complete__measure__def,axiom,
( comple8155536527497655954sure_b
= ( ^ [M3: sigma_measure_b] :
! [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( ( member_set_b @ A2 @ ( measure_null_sets_b @ M3 ) )
=> ( member_set_b @ B2 @ ( sigma_sets_b @ M3 ) ) ) ) ) ) ).
% complete_measure_def
thf(fact_487_complete__measure__def,axiom,
( comple6693822267556782962_set_b
= ( ^ [M3: sigma_measure_set_b] :
! [A2: set_set_b,B2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B2 @ A2 )
=> ( ( member_set_set_b @ A2 @ ( measur1516554132335629586_set_b @ M3 ) )
=> ( member_set_set_b @ B2 @ ( sigma_sets_set_b @ M3 ) ) ) ) ) ) ).
% complete_measure_def
thf(fact_488_complete__measure__def,axiom,
( comple6693822263253554161_set_a
= ( ^ [M3: sigma_measure_set_a] :
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( member_set_set_a @ A2 @ ( measur1516554128032400785_set_a @ M3 ) )
=> ( member_set_set_a @ B2 @ ( sigma_sets_set_a @ M3 ) ) ) ) ) ) ).
% complete_measure_def
thf(fact_489_complete__measure__def,axiom,
( comple8155536527497655953sure_a
= ( ^ [M3: sigma_measure_a] :
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( member_set_a @ A2 @ ( measure_null_sets_a @ M3 ) )
=> ( member_set_a @ B2 @ ( sigma_sets_a @ M3 ) ) ) ) ) ) ).
% complete_measure_def
thf(fact_490_null__setsD2,axiom,
! [A: set_b,M2: sigma_measure_b] :
( ( member_set_b @ A @ ( measure_null_sets_b @ M2 ) )
=> ( member_set_b @ A @ ( sigma_sets_b @ M2 ) ) ) ).
% null_setsD2
thf(fact_491_null__setsD2,axiom,
! [A: set_a,M2: sigma_measure_a] :
( ( member_set_a @ A @ ( measure_null_sets_a @ M2 ) )
=> ( member_set_a @ A @ ( sigma_sets_a @ M2 ) ) ) ).
% null_setsD2
thf(fact_492_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_493_fmeasurableI__null__sets,axiom,
! [A: set_a,M2: sigma_measure_a] :
( ( member_set_a @ A @ ( measure_null_sets_a @ M2 ) )
=> ( member_set_a @ A @ ( measur3645360004775918570able_a @ M2 ) ) ) ).
% fmeasurableI_null_sets
thf(fact_494_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_495_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_496_Int__mono,axiom,
! [A: set_set_b,C: set_set_b,B: set_set_b,D: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ C )
=> ( ( ord_le3795704787696855135_set_b @ B @ D )
=> ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A @ B ) @ ( inf_inf_set_set_b @ C @ D ) ) ) ) ).
% Int_mono
thf(fact_497_Int__mono,axiom,
! [A: set_set_a,C: set_set_a,B: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C )
=> ( ( ord_le3724670747650509150_set_a @ B @ D )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ ( inf_inf_set_set_a @ C @ D ) ) ) ) ).
% Int_mono
thf(fact_498_Int__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% Int_mono
thf(fact_499_Int__lower1,axiom,
! [A: set_set_b,B: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_500_Int__lower1,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_501_Int__lower1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_502_Int__lower2,axiom,
! [A: set_set_b,B: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_503_Int__lower2,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_504_Int__lower2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_505_Int__absorb1,axiom,
! [B: set_set_b,A: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B @ A )
=> ( ( inf_inf_set_set_b @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_506_Int__absorb1,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( inf_inf_set_set_a @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_507_Int__absorb1,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_508_Int__absorb2,axiom,
! [A: set_set_b,B: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( ( inf_inf_set_set_b @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_509_Int__absorb2,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( inf_inf_set_set_a @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_510_Int__absorb2,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_511_Int__greatest,axiom,
! [C: set_set_b,A: set_set_b,B: set_set_b] :
( ( ord_le3795704787696855135_set_b @ C @ A )
=> ( ( ord_le3795704787696855135_set_b @ C @ B )
=> ( ord_le3795704787696855135_set_b @ C @ ( inf_inf_set_set_b @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_512_Int__greatest,axiom,
! [C: set_set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ A )
=> ( ( ord_le3724670747650509150_set_a @ C @ B )
=> ( ord_le3724670747650509150_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_513_Int__greatest,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_514_Int__Collect__mono,axiom,
! [A: set_a_b,B: set_a_b,P: ( a > b ) > $o,Q: ( a > b ) > $o] :
( ( ord_less_eq_set_a_b @ A @ B )
=> ( ! [X3: a > b] :
( ( member_a_b @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_a_b @ ( inf_inf_set_a_b @ A @ ( collect_a_b @ P ) ) @ ( inf_inf_set_a_b @ B @ ( collect_a_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_515_Int__Collect__mono,axiom,
! [A: set_c,B: set_c,P: c > $o,Q: c > $o] :
( ( ord_less_eq_set_c @ A @ B )
=> ( ! [X3: c] :
( ( member_c @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_c @ ( inf_inf_set_c @ A @ ( collect_c @ P ) ) @ ( inf_inf_set_c @ B @ ( collect_c @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_516_Int__Collect__mono,axiom,
! [A: set_real,B: set_real,P: real > $o,Q: real > $o] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_real @ ( inf_inf_set_real @ A @ ( collect_real @ P ) ) @ ( inf_inf_set_real @ B @ ( collect_real @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_517_Int__Collect__mono,axiom,
! [A: set_o,B: set_o,P: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A @ ( collect_o @ P ) ) @ ( inf_inf_set_o @ B @ ( collect_o @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_518_Int__Collect__mono,axiom,
! [A: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_519_Int__Collect__mono,axiom,
! [A: set_set_b,B: set_set_b,P: set_b > $o,Q: set_b > $o] :
( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( ! [X3: set_b] :
( ( member_set_b @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A @ ( collect_set_b @ P ) ) @ ( inf_inf_set_set_b @ B @ ( collect_set_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_520_Int__Collect__mono,axiom,
! [A: set_set_a,B: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_521_Int__Collect__mono,axiom,
! [A: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_522_measurable__cong,axiom,
! [M2: sigma_measure_a,F2: a > b,G2: a > b,M: sigma_measure_b] :
( ! [W: a] :
( ( member_a @ W @ ( sigma_space_a @ M2 ) )
=> ( ( F2 @ W )
= ( G2 @ W ) ) )
=> ( ( member_a_b @ F2 @ ( sigma_measurable_a_b @ M2 @ M ) )
= ( member_a_b @ G2 @ ( sigma_measurable_a_b @ M2 @ M ) ) ) ) ).
% measurable_cong
thf(fact_523_measurable__space,axiom,
! [F2: c > c,M2: sigma_measure_c,A: sigma_measure_c,X5: c] :
( ( member_c_c @ F2 @ ( sigma_measurable_c_c @ M2 @ A ) )
=> ( ( member_c @ X5 @ ( sigma_space_c @ M2 ) )
=> ( member_c @ ( F2 @ X5 ) @ ( sigma_space_c @ A ) ) ) ) ).
% measurable_space
thf(fact_524_measurable__space,axiom,
! [F2: c > real,M2: sigma_measure_c,A: sigma_measure_real,X5: c] :
( ( member_c_real @ F2 @ ( sigma_1862118822198465884c_real @ M2 @ A ) )
=> ( ( member_c @ X5 @ ( sigma_space_c @ M2 ) )
=> ( member_real @ ( F2 @ X5 ) @ ( sigma_space_real @ A ) ) ) ) ).
% measurable_space
thf(fact_525_measurable__space,axiom,
! [F2: c > $o,M2: sigma_measure_c,A: sigma_measure_o,X5: c] :
( ( member_c_o @ F2 @ ( sigma_measurable_c_o @ M2 @ A ) )
=> ( ( member_c @ X5 @ ( sigma_space_c @ M2 ) )
=> ( member_o @ ( F2 @ X5 ) @ ( sigma_space_o @ A ) ) ) ) ).
% measurable_space
thf(fact_526_measurable__space,axiom,
! [F2: c > nat,M2: sigma_measure_c,A: sigma_measure_nat,X5: c] :
( ( member_c_nat @ F2 @ ( sigma_2544038740538346112_c_nat @ M2 @ A ) )
=> ( ( member_c @ X5 @ ( sigma_space_c @ M2 ) )
=> ( member_nat @ ( F2 @ X5 ) @ ( sigma_space_nat @ A ) ) ) ) ).
% measurable_space
thf(fact_527_measurable__space,axiom,
! [F2: real > c,M2: sigma_measure_real,A: sigma_measure_c,X5: real] :
( ( member_real_c @ F2 @ ( sigma_523072396149930114real_c @ M2 @ A ) )
=> ( ( member_real @ X5 @ ( sigma_space_real @ M2 ) )
=> ( member_c @ ( F2 @ X5 ) @ ( sigma_space_c @ A ) ) ) ) ).
% measurable_space
thf(fact_528_measurable__space,axiom,
! [F2: real > real,M2: sigma_measure_real,A: sigma_measure_real,X5: real] :
( ( member_real_real @ F2 @ ( sigma_5267869275261027754l_real @ M2 @ A ) )
=> ( ( member_real @ X5 @ ( sigma_space_real @ M2 ) )
=> ( member_real @ ( F2 @ X5 ) @ ( sigma_space_real @ A ) ) ) ) ).
% measurable_space
thf(fact_529_measurable__space,axiom,
! [F2: real > $o,M2: sigma_measure_real,A: sigma_measure_o,X5: real] :
( ( member_real_o @ F2 @ ( sigma_3939073009482781210real_o @ M2 @ A ) )
=> ( ( member_real @ X5 @ ( sigma_space_real @ M2 ) )
=> ( member_o @ ( F2 @ X5 ) @ ( sigma_space_o @ A ) ) ) ) ).
% measurable_space
thf(fact_530_measurable__space,axiom,
! [F2: real > nat,M2: sigma_measure_real,A: sigma_measure_nat,X5: real] :
( ( member_real_nat @ F2 @ ( sigma_6315060578831106510al_nat @ M2 @ A ) )
=> ( ( member_real @ X5 @ ( sigma_space_real @ M2 ) )
=> ( member_nat @ ( F2 @ X5 ) @ ( sigma_space_nat @ A ) ) ) ) ).
% measurable_space
thf(fact_531_measurable__space,axiom,
! [F2: $o > c,M2: sigma_measure_o,A: sigma_measure_c,X5: $o] :
( ( member_o_c @ F2 @ ( sigma_measurable_o_c @ M2 @ A ) )
=> ( ( member_o @ X5 @ ( sigma_space_o @ M2 ) )
=> ( member_c @ ( F2 @ X5 ) @ ( sigma_space_c @ A ) ) ) ) ).
% measurable_space
thf(fact_532_measurable__space,axiom,
! [F2: $o > real,M2: sigma_measure_o,A: sigma_measure_real,X5: $o] :
( ( member_o_real @ F2 @ ( sigma_2430008634441611636o_real @ M2 @ A ) )
=> ( ( member_o @ X5 @ ( sigma_space_o @ M2 ) )
=> ( member_real @ ( F2 @ X5 ) @ ( sigma_space_real @ A ) ) ) ) ).
% measurable_space
thf(fact_533_measurable__cong__simp,axiom,
! [M2: sigma_measure_a,N: sigma_measure_a,M: sigma_measure_b,N3: sigma_measure_b,F2: a > b,G2: a > b] :
( ( M2 = N )
=> ( ( M = N3 )
=> ( ! [W: a] :
( ( member_a @ W @ ( sigma_space_a @ M2 ) )
=> ( ( F2 @ W )
= ( G2 @ W ) ) )
=> ( ( member_a_b @ F2 @ ( sigma_measurable_a_b @ M2 @ M ) )
= ( member_a_b @ G2 @ ( sigma_measurable_a_b @ N @ N3 ) ) ) ) ) ) ).
% measurable_cong_simp
thf(fact_534_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_535_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_536_completion_Ocomplete__measure__axioms,axiom,
! [M2: sigma_measure_a] : ( comple8155536527497655953sure_a @ ( comple3428971583294703880tion_a @ M2 ) ) ).
% completion.complete_measure_axioms
thf(fact_537_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_538_prob__space_Oindep__setD,axiom,
! [M2: sigma_measure_a,A: set_set_a,B: set_set_a,A3: set_a,B3: set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe2041756565122539606_set_a @ M2 @ A @ B )
=> ( ( member_set_a @ A3 @ A )
=> ( ( 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 ) ) ) ) ) ) ) ).
% prob_space.indep_setD
thf(fact_539_null__sets__subset,axiom,
! [B: set_b,M2: sigma_measure_b,A: set_b] :
( ( member_set_b @ B @ ( measure_null_sets_b @ M2 ) )
=> ( ( member_set_b @ A @ ( sigma_sets_b @ M2 ) )
=> ( ( ord_less_eq_set_b @ A @ B )
=> ( member_set_b @ A @ ( measure_null_sets_b @ M2 ) ) ) ) ) ).
% null_sets_subset
thf(fact_540_null__sets__subset,axiom,
! [B: set_set_b,M2: sigma_measure_set_b,A: set_set_b] :
( ( member_set_set_b @ B @ ( measur1516554132335629586_set_b @ M2 ) )
=> ( ( member_set_set_b @ A @ ( sigma_sets_set_b @ M2 ) )
=> ( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( member_set_set_b @ A @ ( measur1516554132335629586_set_b @ M2 ) ) ) ) ) ).
% null_sets_subset
thf(fact_541_null__sets__subset,axiom,
! [B: set_set_a,M2: sigma_measure_set_a,A: set_set_a] :
( ( member_set_set_a @ B @ ( measur1516554128032400785_set_a @ M2 ) )
=> ( ( member_set_set_a @ A @ ( sigma_sets_set_a @ M2 ) )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( member_set_set_a @ A @ ( measur1516554128032400785_set_a @ M2 ) ) ) ) ) ).
% null_sets_subset
thf(fact_542_null__sets__subset,axiom,
! [B: set_a,M2: sigma_measure_a,A: set_a] :
( ( member_set_a @ B @ ( measure_null_sets_a @ M2 ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M2 ) )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( member_set_a @ A @ ( measure_null_sets_a @ M2 ) ) ) ) ) ).
% null_sets_subset
thf(fact_543_null__sets__completion__subset,axiom,
! [B: set_set_b,A: set_set_b,M2: sigma_measure_set_b] :
( ( ord_le3795704787696855135_set_b @ B @ A )
=> ( ( member_set_set_b @ A @ ( 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_544_null__sets__completion__subset,axiom,
! [B: set_set_a,A: set_set_a,M2: sigma_measure_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( member_set_set_a @ A @ ( 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_545_null__sets__completion__subset,axiom,
! [B: set_a,A: set_a,M2: sigma_measure_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( member_set_a @ A @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) )
=> ( member_set_a @ B @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) ) ) ) ).
% null_sets_completion_subset
thf(fact_546_null__sets__completion__iff2,axiom,
! [A: set_set_b,M2: sigma_measure_set_b] :
( ( member_set_set_b @ A @ ( measur1516554132335629586_set_b @ ( comple8942076150311361001_set_b @ M2 ) ) )
= ( ? [X4: set_set_b] :
( ( member_set_set_b @ X4 @ ( measur1516554132335629586_set_b @ M2 ) )
& ( ord_le3795704787696855135_set_b @ A @ X4 ) ) ) ) ).
% null_sets_completion_iff2
thf(fact_547_null__sets__completion__iff2,axiom,
! [A: set_set_a,M2: sigma_measure_set_a] :
( ( member_set_set_a @ A @ ( measur1516554128032400785_set_a @ ( comple8942076146008132200_set_a @ M2 ) ) )
= ( ? [X4: set_set_a] :
( ( member_set_set_a @ X4 @ ( measur1516554128032400785_set_a @ M2 ) )
& ( ord_le3724670747650509150_set_a @ A @ X4 ) ) ) ) ).
% null_sets_completion_iff2
thf(fact_548_null__sets__completion__iff2,axiom,
! [A: set_a,M2: sigma_measure_a] :
( ( member_set_a @ A @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) )
= ( ? [X4: set_a] :
( ( member_set_a @ X4 @ ( measure_null_sets_a @ M2 ) )
& ( ord_less_eq_set_a @ A @ X4 ) ) ) ) ).
% null_sets_completion_iff2
thf(fact_549_completion_Ocomplete2,axiom,
! [A: set_set_b,B: set_set_b,M2: sigma_measure_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( ( member_set_set_b @ B @ ( measur1516554132335629586_set_b @ ( comple8942076150311361001_set_b @ M2 ) ) )
=> ( member_set_set_b @ A @ ( measur1516554132335629586_set_b @ ( comple8942076150311361001_set_b @ M2 ) ) ) ) ) ).
% completion.complete2
thf(fact_550_completion_Ocomplete2,axiom,
! [A: set_set_a,B: set_set_a,M2: sigma_measure_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_set_a @ B @ ( measur1516554128032400785_set_a @ ( comple8942076146008132200_set_a @ M2 ) ) )
=> ( member_set_set_a @ A @ ( measur1516554128032400785_set_a @ ( comple8942076146008132200_set_a @ M2 ) ) ) ) ) ).
% completion.complete2
thf(fact_551_completion_Ocomplete2,axiom,
! [A: set_a,B: set_a,M2: sigma_measure_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_set_a @ B @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) )
=> ( member_set_a @ A @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) ) ) ) ).
% completion.complete2
thf(fact_552_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_553_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_554_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_555_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_556_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_557_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_558_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_559_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_560_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_561_distr__cong,axiom,
! [M2: sigma_measure_c,K: sigma_measure_c,N: sigma_measure_b,L: sigma_measure_b,F2: c > b,G2: c > b] :
( ( M2 = K )
=> ( ( ( sigma_sets_b @ N )
= ( sigma_sets_b @ L ) )
=> ( ! [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ M2 ) )
=> ( ( F2 @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( measure_distr_c_b @ M2 @ N @ F2 )
= ( measure_distr_c_b @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_562_distr__cong,axiom,
! [M2: sigma_measure_real,K: sigma_measure_real,N: sigma_measure_b,L: sigma_measure_b,F2: real > b,G2: real > b] :
( ( M2 = K )
=> ( ( ( sigma_sets_b @ N )
= ( sigma_sets_b @ L ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( sigma_space_real @ M2 ) )
=> ( ( F2 @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( measure_distr_real_b @ M2 @ N @ F2 )
= ( measure_distr_real_b @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_563_distr__cong,axiom,
! [M2: sigma_measure_o,K: sigma_measure_o,N: sigma_measure_b,L: sigma_measure_b,F2: $o > b,G2: $o > b] :
( ( M2 = K )
=> ( ( ( sigma_sets_b @ N )
= ( sigma_sets_b @ L ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ M2 ) )
=> ( ( F2 @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( measure_distr_o_b @ M2 @ N @ F2 )
= ( measure_distr_o_b @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_564_distr__cong,axiom,
! [M2: sigma_measure_nat,K: sigma_measure_nat,N: sigma_measure_b,L: sigma_measure_b,F2: nat > b,G2: nat > b] :
( ( M2 = K )
=> ( ( ( sigma_sets_b @ N )
= ( sigma_sets_b @ L ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( sigma_space_nat @ M2 ) )
=> ( ( F2 @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( measure_distr_nat_b @ M2 @ N @ F2 )
= ( measure_distr_nat_b @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_565_distr__cong,axiom,
! [M2: sigma_measure_c,K: sigma_measure_c,N: sigma_measure_a,L: sigma_measure_a,F2: c > a,G2: c > a] :
( ( M2 = K )
=> ( ( ( sigma_sets_a @ N )
= ( sigma_sets_a @ L ) )
=> ( ! [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ M2 ) )
=> ( ( F2 @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( measure_distr_c_a @ M2 @ N @ F2 )
= ( measure_distr_c_a @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_566_distr__cong,axiom,
! [M2: sigma_measure_real,K: sigma_measure_real,N: sigma_measure_a,L: sigma_measure_a,F2: real > a,G2: real > a] :
( ( M2 = K )
=> ( ( ( sigma_sets_a @ N )
= ( sigma_sets_a @ L ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( sigma_space_real @ M2 ) )
=> ( ( F2 @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( measure_distr_real_a @ M2 @ N @ F2 )
= ( measure_distr_real_a @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_567_distr__cong,axiom,
! [M2: sigma_measure_o,K: sigma_measure_o,N: sigma_measure_a,L: sigma_measure_a,F2: $o > a,G2: $o > a] :
( ( M2 = K )
=> ( ( ( sigma_sets_a @ N )
= ( sigma_sets_a @ L ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ M2 ) )
=> ( ( F2 @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( measure_distr_o_a @ M2 @ N @ F2 )
= ( measure_distr_o_a @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_568_distr__cong,axiom,
! [M2: sigma_measure_nat,K: sigma_measure_nat,N: sigma_measure_a,L: sigma_measure_a,F2: nat > a,G2: nat > a] :
( ( M2 = K )
=> ( ( ( sigma_sets_a @ N )
= ( sigma_sets_a @ L ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( sigma_space_nat @ M2 ) )
=> ( ( F2 @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( measure_distr_nat_a @ M2 @ N @ F2 )
= ( measure_distr_nat_a @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_569_distr__cong,axiom,
! [M2: sigma_measure_a,K: sigma_measure_a,N: sigma_measure_a,L: sigma_measure_a,F2: a > a,G2: a > a] :
( ( M2 = K )
=> ( ( ( sigma_sets_a @ N )
= ( sigma_sets_a @ L ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
=> ( ( F2 @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( measure_distr_a_a @ M2 @ N @ F2 )
= ( measure_distr_a_a @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_570_distr__cong,axiom,
! [M2: sigma_measure_a,K: sigma_measure_a,N: sigma_measure_b,L: sigma_measure_b,F2: a > b,G2: a > b] :
( ( M2 = K )
=> ( ( ( sigma_sets_b @ N )
= ( sigma_sets_b @ L ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
=> ( ( F2 @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( measure_distr_a_b @ M2 @ N @ F2 )
= ( measure_distr_a_b @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_571_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_572_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_573_complete__measure_Omeasurable__completion2,axiom,
! [M2: sigma_measure_a,F2: a > b,N: sigma_measure_b] :
( ( comple8155536527497655953sure_a @ M2 )
=> ( ( member_a_b @ F2 @ ( 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 @ F2 ) ) )
=> ( member_a_b @ F2 @ ( sigma_measurable_a_b @ M2 @ ( comple3428971583294703881tion_b @ N ) ) ) ) ) ) ).
% complete_measure.measurable_completion2
thf(fact_574_le__measureD1,axiom,
! [A: sigma_measure_set_b,B: sigma_measure_set_b] :
( ( ord_le5713619651007674940_set_b @ A @ B )
=> ( ord_le3795704787696855135_set_b @ ( sigma_space_set_b @ A ) @ ( sigma_space_set_b @ B ) ) ) ).
% le_measureD1
thf(fact_575_le__measureD1,axiom,
! [A: sigma_measure_set_a,B: sigma_measure_set_a] :
( ( ord_le5642585610961328955_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ ( sigma_space_set_a @ A ) @ ( sigma_space_set_a @ B ) ) ) ).
% le_measureD1
thf(fact_576_le__measureD1,axiom,
! [A: sigma_measure_a,B: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A @ B )
=> ( ord_less_eq_set_a @ ( sigma_space_a @ A ) @ ( sigma_space_a @ B ) ) ) ).
% le_measureD1
thf(fact_577_prob__space_Oindep__setI,axiom,
! [M2: sigma_measure_b,A: set_set_b,B: set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( ord_le3795704787696855135_set_b @ A @ ( sigma_sets_b @ M2 ) )
=> ( ( ord_le3795704787696855135_set_b @ B @ ( sigma_sets_b @ M2 ) )
=> ( ! [A4: set_b,B4: set_b] :
( ( member_set_b @ A4 @ A )
=> ( ( member_set_b @ B4 @ B )
=> ( ( sigma_measure_b2 @ M2 @ ( inf_inf_set_b @ A4 @ B4 ) )
= ( times_times_real @ ( sigma_measure_b2 @ M2 @ A4 ) @ ( sigma_measure_b2 @ M2 @ B4 ) ) ) ) )
=> ( indepe2041756565122539607_set_b @ M2 @ A @ B ) ) ) ) ) ).
% prob_space.indep_setI
thf(fact_578_prob__space_Oindep__setI,axiom,
! [M2: sigma_measure_a,A: set_set_a,B: set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( ord_le3724670747650509150_set_a @ A @ ( sigma_sets_a @ M2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ ( sigma_sets_a @ M2 ) )
=> ( ! [A4: set_a,B4: set_a] :
( ( member_set_a @ A4 @ A )
=> ( ( member_set_a @ B4 @ B )
=> ( ( sigma_measure_a2 @ M2 @ ( inf_inf_set_a @ A4 @ B4 ) )
= ( times_times_real @ ( sigma_measure_a2 @ M2 @ A4 ) @ ( sigma_measure_a2 @ M2 @ B4 ) ) ) ) )
=> ( indepe2041756565122539606_set_a @ M2 @ A @ B ) ) ) ) ) ).
% prob_space.indep_setI
thf(fact_579_prob__space_Oindep__sets2__eq,axiom,
! [M2: sigma_measure_b,A: set_set_b,B: set_set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe2041756565122539607_set_b @ M2 @ A @ B )
= ( ( ord_le3795704787696855135_set_b @ A @ ( sigma_sets_b @ M2 ) )
& ( ord_le3795704787696855135_set_b @ B @ ( sigma_sets_b @ M2 ) )
& ! [X4: set_b] :
( ( member_set_b @ X4 @ A )
=> ! [Y4: set_b] :
( ( member_set_b @ Y4 @ B )
=> ( ( sigma_measure_b2 @ M2 @ ( inf_inf_set_b @ X4 @ Y4 ) )
= ( times_times_real @ ( sigma_measure_b2 @ M2 @ X4 ) @ ( sigma_measure_b2 @ M2 @ Y4 ) ) ) ) ) ) ) ) ).
% prob_space.indep_sets2_eq
thf(fact_580_prob__space_Oindep__sets2__eq,axiom,
! [M2: sigma_measure_a,A: set_set_a,B: set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe2041756565122539606_set_a @ M2 @ A @ B )
= ( ( ord_le3724670747650509150_set_a @ A @ ( sigma_sets_a @ M2 ) )
& ( ord_le3724670747650509150_set_a @ B @ ( sigma_sets_a @ M2 ) )
& ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ B )
=> ( ( sigma_measure_a2 @ M2 @ ( inf_inf_set_a @ X4 @ Y4 ) )
= ( times_times_real @ ( sigma_measure_a2 @ M2 @ X4 ) @ ( sigma_measure_a2 @ M2 @ Y4 ) ) ) ) ) ) ) ) ).
% prob_space.indep_sets2_eq
thf(fact_581_completion_Ocomplete,axiom,
! [B: set_b,A: set_b,M2: sigma_measure_b] :
( ( ord_less_eq_set_b @ B @ A )
=> ( ( member_set_b @ A @ ( measure_null_sets_b @ ( comple3428971583294703881tion_b @ M2 ) ) )
=> ( member_set_b @ B @ ( sigma_sets_b @ ( comple3428971583294703881tion_b @ M2 ) ) ) ) ) ).
% completion.complete
thf(fact_582_completion_Ocomplete,axiom,
! [B: set_set_b,A: set_set_b,M2: sigma_measure_set_b] :
( ( ord_le3795704787696855135_set_b @ B @ A )
=> ( ( member_set_set_b @ A @ ( measur1516554132335629586_set_b @ ( comple8942076150311361001_set_b @ M2 ) ) )
=> ( member_set_set_b @ B @ ( sigma_sets_set_b @ ( comple8942076150311361001_set_b @ M2 ) ) ) ) ) ).
% completion.complete
thf(fact_583_completion_Ocomplete,axiom,
! [B: set_set_a,A: set_set_a,M2: sigma_measure_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( member_set_set_a @ A @ ( measur1516554128032400785_set_a @ ( comple8942076146008132200_set_a @ M2 ) ) )
=> ( member_set_set_a @ B @ ( sigma_sets_set_a @ ( comple8942076146008132200_set_a @ M2 ) ) ) ) ) ).
% completion.complete
thf(fact_584_completion_Ocomplete,axiom,
! [B: set_a,A: set_a,M2: sigma_measure_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( member_set_a @ A @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) )
=> ( member_set_a @ B @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) ) ) ) ).
% completion.complete
thf(fact_585_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_586_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_587_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_588_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_589_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_590_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_591_sets__le__imp__space__le,axiom,
! [A: sigma_measure_set_b,B: sigma_measure_set_b] :
( ( ord_le3201067847557142847_set_b @ ( sigma_sets_set_b @ A ) @ ( sigma_sets_set_b @ B ) )
=> ( ord_le3795704787696855135_set_b @ ( sigma_space_set_b @ A ) @ ( sigma_space_set_b @ B ) ) ) ).
% sets_le_imp_space_le
thf(fact_592_sets__le__imp__space__le,axiom,
! [A: sigma_measure_set_a,B: sigma_measure_set_a] :
( ( ord_le5722252365846178494_set_a @ ( sigma_sets_set_a @ A ) @ ( sigma_sets_set_a @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( sigma_space_set_a @ A ) @ ( sigma_space_set_a @ B ) ) ) ).
% sets_le_imp_space_le
thf(fact_593_sets__le__imp__space__le,axiom,
! [A: sigma_measure_b,B: sigma_measure_b] :
( ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ A ) @ ( sigma_sets_b @ B ) )
=> ( ord_less_eq_set_b @ ( sigma_space_b @ A ) @ ( sigma_space_b @ B ) ) ) ).
% sets_le_imp_space_le
thf(fact_594_sets__le__imp__space__le,axiom,
! [A: sigma_measure_a,B: sigma_measure_a] :
( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ A ) @ ( sigma_sets_a @ B ) )
=> ( ord_less_eq_set_a @ ( sigma_space_a @ A ) @ ( sigma_space_a @ B ) ) ) ).
% sets_le_imp_space_le
thf(fact_595_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_596_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_597_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_598_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_599_finite__measure_Obounded__measure,axiom,
! [M2: sigma_measure_a,A: set_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ord_less_eq_real @ ( sigma_measure_a2 @ M2 @ A ) @ ( sigma_measure_a2 @ M2 @ ( sigma_space_a @ M2 ) ) ) ) ).
% finite_measure.bounded_measure
thf(fact_600_le__measureD2,axiom,
! [A: sigma_measure_b,B: sigma_measure_b] :
( ( ord_le254669799889008988sure_b @ A @ B )
=> ( ( ( sigma_space_b @ A )
= ( sigma_space_b @ B ) )
=> ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ A ) @ ( sigma_sets_b @ B ) ) ) ) ).
% le_measureD2
thf(fact_601_le__measureD2,axiom,
! [A: sigma_measure_a,B: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A @ B )
=> ( ( ( sigma_space_a @ A )
= ( sigma_space_a @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ A ) @ ( sigma_sets_a @ B ) ) ) ) ).
% le_measureD2
thf(fact_602_prob__space_Oprob__le__1,axiom,
! [M2: sigma_measure_a,A: set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ord_less_eq_real @ ( sigma_measure_a2 @ M2 @ A ) @ one_one_real ) ) ).
% prob_space.prob_le_1
thf(fact_603_prob__space_Omeasure__ge__1__iff,axiom,
! [M2: sigma_measure_a,A: set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( ord_less_eq_real @ one_one_real @ ( sigma_measure_a2 @ M2 @ A ) )
= ( ( sigma_measure_a2 @ M2 @ A )
= one_one_real ) ) ) ).
% prob_space.measure_ge_1_iff
thf(fact_604_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_605_subprob__space__distr,axiom,
! [F2: a > a,M: sigma_measure_a] :
( ( member_a_a @ F2 @ ( sigma_measurable_a_a @ m @ M ) )
=> ( ( ( sigma_space_a @ M )
!= bot_bot_set_a )
=> ( giry_subprob_space_a @ ( measure_distr_a_a @ m @ M @ F2 ) ) ) ) ).
% subprob_space_distr
thf(fact_606_subprob__space__distr,axiom,
! [F2: a > b,M: sigma_measure_b] :
( ( member_a_b @ F2 @ ( sigma_measurable_a_b @ m @ M ) )
=> ( ( ( sigma_space_b @ M )
!= bot_bot_set_b )
=> ( giry_subprob_space_b @ ( measure_distr_a_b @ m @ M @ F2 ) ) ) ) ).
% subprob_space_distr
thf(fact_607_le__inf__iff,axiom,
! [X5: set_set_b,Y5: set_set_b,Z2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X5 @ ( inf_inf_set_set_b @ Y5 @ Z2 ) )
= ( ( ord_le3795704787696855135_set_b @ X5 @ Y5 )
& ( ord_le3795704787696855135_set_b @ X5 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_608_le__inf__iff,axiom,
! [X5: set_set_a,Y5: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ ( inf_inf_set_set_a @ Y5 @ Z2 ) )
= ( ( ord_le3724670747650509150_set_a @ X5 @ Y5 )
& ( ord_le3724670747650509150_set_a @ X5 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_609_le__inf__iff,axiom,
! [X5: set_a,Y5: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X5 @ ( inf_inf_set_a @ Y5 @ Z2 ) )
= ( ( ord_less_eq_set_a @ X5 @ Y5 )
& ( ord_less_eq_set_a @ X5 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_610_le__inf__iff,axiom,
! [X5: nat,Y5: nat,Z2: nat] :
( ( ord_less_eq_nat @ X5 @ ( inf_inf_nat @ Y5 @ Z2 ) )
= ( ( ord_less_eq_nat @ X5 @ Y5 )
& ( ord_less_eq_nat @ X5 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_611_le__inf__iff,axiom,
! [X5: int,Y5: int,Z2: int] :
( ( ord_less_eq_int @ X5 @ ( inf_inf_int @ Y5 @ Z2 ) )
= ( ( ord_less_eq_int @ X5 @ Y5 )
& ( ord_less_eq_int @ X5 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_612_inf_Obounded__iff,axiom,
! [A3: set_set_b,B3: set_set_b,C2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A3 @ ( inf_inf_set_set_b @ B3 @ C2 ) )
= ( ( ord_le3795704787696855135_set_b @ A3 @ B3 )
& ( ord_le3795704787696855135_set_b @ A3 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_613_inf_Obounded__iff,axiom,
! [A3: set_set_a,B3: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ ( inf_inf_set_set_a @ B3 @ C2 ) )
= ( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
& ( ord_le3724670747650509150_set_a @ A3 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_614_inf_Obounded__iff,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( inf_inf_set_a @ B3 @ C2 ) )
= ( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_615_inf_Obounded__iff,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ ( inf_inf_nat @ B3 @ C2 ) )
= ( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_616_inf_Obounded__iff,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_eq_int @ A3 @ ( inf_inf_int @ B3 @ C2 ) )
= ( ( ord_less_eq_int @ A3 @ B3 )
& ( ord_less_eq_int @ A3 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_617_mult_Oright__neutral,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ one_one_real )
= A3 ) ).
% mult.right_neutral
thf(fact_618_mult_Oright__neutral,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% mult.right_neutral
thf(fact_619_mult_Oright__neutral,axiom,
! [A3: int] :
( ( times_times_int @ A3 @ one_one_int )
= A3 ) ).
% mult.right_neutral
thf(fact_620_mult_Oright__neutral,axiom,
! [A3: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ A3 @ one_on2969667320475766781nnreal )
= A3 ) ).
% mult.right_neutral
thf(fact_621_vector__space__over__itself_Oscale__one,axiom,
! [X5: real] :
( ( times_times_real @ one_one_real @ X5 )
= X5 ) ).
% vector_space_over_itself.scale_one
thf(fact_622_mult__1,axiom,
! [A3: real] :
( ( times_times_real @ one_one_real @ A3 )
= A3 ) ).
% mult_1
thf(fact_623_mult__1,axiom,
! [A3: nat] :
( ( times_times_nat @ one_one_nat @ A3 )
= A3 ) ).
% mult_1
thf(fact_624_mult__1,axiom,
! [A3: int] :
( ( times_times_int @ one_one_int @ A3 )
= A3 ) ).
% mult_1
thf(fact_625_mult__1,axiom,
! [A3: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ one_on2969667320475766781nnreal @ A3 )
= A3 ) ).
% mult_1
thf(fact_626_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_627_sigma__algebra__tail__events,axiom,
! [A: nat > set_set_a] :
( ! [I3: nat] : ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ m ) @ ( A @ I3 ) )
=> ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ m ) @ ( indepe7538416700049374166_a_nat @ m @ A ) ) ) ).
% sigma_algebra_tail_events
thf(fact_628_subprob__not__empty,axiom,
( ( sigma_space_a @ m )
!= bot_bot_set_a ) ).
% subprob_not_empty
thf(fact_629_empty__iff,axiom,
! [C2: a > b] :
~ ( member_a_b @ C2 @ bot_bot_set_a_b ) ).
% empty_iff
thf(fact_630_empty__iff,axiom,
! [C2: c] :
~ ( member_c @ C2 @ bot_bot_set_c ) ).
% empty_iff
thf(fact_631_empty__iff,axiom,
! [C2: set_a] :
~ ( member_set_a @ C2 @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_632_empty__iff,axiom,
! [C2: real] :
~ ( member_real @ C2 @ bot_bot_set_real ) ).
% empty_iff
thf(fact_633_empty__iff,axiom,
! [C2: $o] :
~ ( member_o @ C2 @ bot_bot_set_o ) ).
% empty_iff
thf(fact_634_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_635_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_636_all__not__in__conv,axiom,
! [A: set_a_b] :
( ( ! [X4: a > b] :
~ ( member_a_b @ X4 @ A ) )
= ( A = bot_bot_set_a_b ) ) ).
% all_not_in_conv
thf(fact_637_all__not__in__conv,axiom,
! [A: set_c] :
( ( ! [X4: c] :
~ ( member_c @ X4 @ A ) )
= ( A = bot_bot_set_c ) ) ).
% all_not_in_conv
thf(fact_638_all__not__in__conv,axiom,
! [A: set_set_a] :
( ( ! [X4: set_a] :
~ ( member_set_a @ X4 @ A ) )
= ( A = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_639_all__not__in__conv,axiom,
! [A: set_real] :
( ( ! [X4: real] :
~ ( member_real @ X4 @ A ) )
= ( A = bot_bot_set_real ) ) ).
% all_not_in_conv
thf(fact_640_all__not__in__conv,axiom,
! [A: set_o] :
( ( ! [X4: $o] :
~ ( member_o @ X4 @ A ) )
= ( A = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_641_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X4: nat] :
~ ( member_nat @ X4 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_642_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X4: a] :
~ ( member_a @ X4 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_643_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X4: a] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_644_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X4: a] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_645_inf__right__idem,axiom,
! [X5: set_a,Y5: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X5 @ Y5 ) @ Y5 )
= ( inf_inf_set_a @ X5 @ Y5 ) ) ).
% inf_right_idem
thf(fact_646_inf_Oright__idem,axiom,
! [A3: set_a,B3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ B3 )
= ( inf_inf_set_a @ A3 @ B3 ) ) ).
% inf.right_idem
thf(fact_647_inf__left__idem,axiom,
! [X5: set_a,Y5: set_a] :
( ( inf_inf_set_a @ X5 @ ( inf_inf_set_a @ X5 @ Y5 ) )
= ( inf_inf_set_a @ X5 @ Y5 ) ) ).
% inf_left_idem
thf(fact_648_inf_Oleft__idem,axiom,
! [A3: set_a,B3: set_a] :
( ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ A3 @ B3 ) )
= ( inf_inf_set_a @ A3 @ B3 ) ) ).
% inf.left_idem
thf(fact_649_inf__idem,axiom,
! [X5: set_a] :
( ( inf_inf_set_a @ X5 @ X5 )
= X5 ) ).
% inf_idem
thf(fact_650_inf_Oidem,axiom,
! [A3: set_a] :
( ( inf_inf_set_a @ A3 @ A3 )
= A3 ) ).
% inf.idem
thf(fact_651_DiffI,axiom,
! [C2: a > b,A: set_a_b,B: set_a_b] :
( ( member_a_b @ C2 @ A )
=> ( ~ ( member_a_b @ C2 @ B )
=> ( member_a_b @ C2 @ ( minus_minus_set_a_b @ A @ B ) ) ) ) ).
% DiffI
thf(fact_652_DiffI,axiom,
! [C2: c,A: set_c,B: set_c] :
( ( member_c @ C2 @ A )
=> ( ~ ( member_c @ C2 @ B )
=> ( member_c @ C2 @ ( minus_minus_set_c @ A @ B ) ) ) ) ).
% DiffI
thf(fact_653_DiffI,axiom,
! [C2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ A )
=> ( ~ ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_654_DiffI,axiom,
! [C2: real,A: set_real,B: set_real] :
( ( member_real @ C2 @ A )
=> ( ~ ( member_real @ C2 @ B )
=> ( member_real @ C2 @ ( minus_minus_set_real @ A @ B ) ) ) ) ).
% DiffI
thf(fact_655_DiffI,axiom,
! [C2: $o,A: set_o,B: set_o] :
( ( member_o @ C2 @ A )
=> ( ~ ( member_o @ C2 @ B )
=> ( member_o @ C2 @ ( minus_minus_set_o @ A @ B ) ) ) ) ).
% DiffI
thf(fact_656_DiffI,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A )
=> ( ~ ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_657_DiffI,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ A )
=> ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_658_Diff__iff,axiom,
! [C2: a > b,A: set_a_b,B: set_a_b] :
( ( member_a_b @ C2 @ ( minus_minus_set_a_b @ A @ B ) )
= ( ( member_a_b @ C2 @ A )
& ~ ( member_a_b @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_659_Diff__iff,axiom,
! [C2: c,A: set_c,B: set_c] :
( ( member_c @ C2 @ ( minus_minus_set_c @ A @ B ) )
= ( ( member_c @ C2 @ A )
& ~ ( member_c @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_660_Diff__iff,axiom,
! [C2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A @ B ) )
= ( ( member_set_a @ C2 @ A )
& ~ ( member_set_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_661_Diff__iff,axiom,
! [C2: real,A: set_real,B: set_real] :
( ( member_real @ C2 @ ( minus_minus_set_real @ A @ B ) )
= ( ( member_real @ C2 @ A )
& ~ ( member_real @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_662_Diff__iff,axiom,
! [C2: $o,A: set_o,B: set_o] :
( ( member_o @ C2 @ ( minus_minus_set_o @ A @ B ) )
= ( ( member_o @ C2 @ A )
& ~ ( member_o @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_663_Diff__iff,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) )
= ( ( member_nat @ C2 @ A )
& ~ ( member_nat @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_664_Diff__iff,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
= ( ( member_a @ C2 @ A )
& ~ ( member_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_665_Diff__idemp,axiom,
! [A: set_a,B: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ).
% Diff_idemp
thf(fact_666_empty__subsetI,axiom,
! [A: set_set_b] : ( ord_le3795704787696855135_set_b @ bot_bot_set_set_b @ A ) ).
% empty_subsetI
thf(fact_667_empty__subsetI,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% empty_subsetI
thf(fact_668_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_669_subset__empty,axiom,
! [A: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ bot_bot_set_set_b )
= ( A = bot_bot_set_set_b ) ) ).
% subset_empty
thf(fact_670_subset__empty,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_671_subset__empty,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_672_inf__bot__right,axiom,
! [X5: set_a] :
( ( inf_inf_set_a @ X5 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_673_inf__bot__left,axiom,
! [X5: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X5 )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_674_sets_Oempty__sets,axiom,
! [M2: sigma_measure_b] : ( member_set_b @ bot_bot_set_b @ ( sigma_sets_b @ M2 ) ) ).
% sets.empty_sets
thf(fact_675_sets_Oempty__sets,axiom,
! [M2: sigma_measure_a] : ( member_set_a @ bot_bot_set_a @ ( sigma_sets_a @ M2 ) ) ).
% sets.empty_sets
thf(fact_676_Diff__empty,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ bot_bot_set_a )
= A ) ).
% Diff_empty
thf(fact_677_empty__Diff,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_678_Diff__cancel,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ A )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_679_sets_ODiff,axiom,
! [A3: set_b,M2: sigma_measure_b,B3: set_b] :
( ( member_set_b @ A3 @ ( sigma_sets_b @ M2 ) )
=> ( ( member_set_b @ B3 @ ( sigma_sets_b @ M2 ) )
=> ( member_set_b @ ( minus_minus_set_b @ A3 @ B3 ) @ ( sigma_sets_b @ M2 ) ) ) ) ).
% sets.Diff
thf(fact_680_sets_ODiff,axiom,
! [A3: set_a,M2: sigma_measure_a,B3: set_a] :
( ( member_set_a @ A3 @ ( sigma_sets_a @ M2 ) )
=> ( ( member_set_a @ B3 @ ( sigma_sets_a @ M2 ) )
=> ( member_set_a @ ( minus_minus_set_a @ A3 @ B3 ) @ ( sigma_sets_a @ M2 ) ) ) ) ).
% sets.Diff
thf(fact_681_fmeasurable_Oempty__sets,axiom,
! [M2: sigma_measure_a] : ( member_set_a @ bot_bot_set_a @ ( measur3645360004775918570able_a @ M2 ) ) ).
% fmeasurable.empty_sets
thf(fact_682_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_683_fmeasurable_ODiff,axiom,
! [A3: set_a,M2: sigma_measure_a,B3: set_a] :
( ( member_set_a @ A3 @ ( measur3645360004775918570able_a @ M2 ) )
=> ( ( member_set_a @ B3 @ ( measur3645360004775918570able_a @ M2 ) )
=> ( member_set_a @ ( minus_minus_set_a @ A3 @ B3 ) @ ( measur3645360004775918570able_a @ M2 ) ) ) ) ).
% fmeasurable.Diff
thf(fact_684_null__sets_ODiff,axiom,
! [A3: set_a,M2: sigma_measure_a,B3: set_a] :
( ( member_set_a @ A3 @ ( measure_null_sets_a @ M2 ) )
=> ( ( member_set_a @ B3 @ ( measure_null_sets_a @ M2 ) )
=> ( member_set_a @ ( minus_minus_set_a @ A3 @ B3 ) @ ( measure_null_sets_a @ M2 ) ) ) ) ).
% null_sets.Diff
thf(fact_685_Diff__eq__empty__iff,axiom,
! [A: set_set_b,B: set_set_b] :
( ( ( minus_5807331545291222566_set_b @ A @ B )
= bot_bot_set_set_b )
= ( ord_le3795704787696855135_set_b @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_686_Diff__eq__empty__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( minus_5736297505244876581_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_687_Diff__eq__empty__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_688_Diff__disjoint,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_689_sets_Ocompl__sets,axiom,
! [A3: set_b,M2: sigma_measure_b] :
( ( member_set_b @ A3 @ ( sigma_sets_b @ M2 ) )
=> ( member_set_b @ ( minus_minus_set_b @ ( sigma_space_b @ M2 ) @ A3 ) @ ( sigma_sets_b @ M2 ) ) ) ).
% sets.compl_sets
thf(fact_690_sets_Ocompl__sets,axiom,
! [A3: set_a,M2: sigma_measure_a] :
( ( member_set_a @ A3 @ ( sigma_sets_a @ M2 ) )
=> ( member_set_a @ ( minus_minus_set_a @ ( sigma_space_a @ M2 ) @ A3 ) @ ( sigma_sets_a @ M2 ) ) ) ).
% sets.compl_sets
thf(fact_691_Int__Diff__disjoint,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ B ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_692_Diff__triv,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_693_vector__space__over__itself_Oscale__left__diff__distrib,axiom,
! [A3: real,B3: real,X5: real] :
( ( times_times_real @ ( minus_minus_real @ A3 @ B3 ) @ X5 )
= ( minus_minus_real @ ( times_times_real @ A3 @ X5 ) @ ( times_times_real @ B3 @ X5 ) ) ) ).
% vector_space_over_itself.scale_left_diff_distrib
thf(fact_694_vector__space__over__itself_Oscale__right__diff__distrib,axiom,
! [A3: real,X5: real,Y5: real] :
( ( times_times_real @ A3 @ ( minus_minus_real @ X5 @ Y5 ) )
= ( minus_minus_real @ ( times_times_real @ A3 @ X5 ) @ ( times_times_real @ A3 @ Y5 ) ) ) ).
% vector_space_over_itself.scale_right_diff_distrib
thf(fact_695_diff__left__imp__eq,axiom,
! [A3: real,B3: real,C2: real] :
( ( ( minus_minus_real @ A3 @ B3 )
= ( minus_minus_real @ A3 @ C2 ) )
=> ( B3 = C2 ) ) ).
% diff_left_imp_eq
thf(fact_696_diff__left__imp__eq,axiom,
! [A3: int,B3: int,C2: int] :
( ( ( minus_minus_int @ A3 @ B3 )
= ( minus_minus_int @ A3 @ C2 ) )
=> ( B3 = C2 ) ) ).
% diff_left_imp_eq
thf(fact_697_DiffE,axiom,
! [C2: a > b,A: set_a_b,B: set_a_b] :
( ( member_a_b @ C2 @ ( minus_minus_set_a_b @ A @ B ) )
=> ~ ( ( member_a_b @ C2 @ A )
=> ( member_a_b @ C2 @ B ) ) ) ).
% DiffE
thf(fact_698_DiffE,axiom,
! [C2: c,A: set_c,B: set_c] :
( ( member_c @ C2 @ ( minus_minus_set_c @ A @ B ) )
=> ~ ( ( member_c @ C2 @ A )
=> ( member_c @ C2 @ B ) ) ) ).
% DiffE
thf(fact_699_DiffE,axiom,
! [C2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A @ B ) )
=> ~ ( ( member_set_a @ C2 @ A )
=> ( member_set_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_700_DiffE,axiom,
! [C2: real,A: set_real,B: set_real] :
( ( member_real @ C2 @ ( minus_minus_set_real @ A @ B ) )
=> ~ ( ( member_real @ C2 @ A )
=> ( member_real @ C2 @ B ) ) ) ).
% DiffE
thf(fact_701_DiffE,axiom,
! [C2: $o,A: set_o,B: set_o] :
( ( member_o @ C2 @ ( minus_minus_set_o @ A @ B ) )
=> ~ ( ( member_o @ C2 @ A )
=> ( member_o @ C2 @ B ) ) ) ).
% DiffE
thf(fact_702_DiffE,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% DiffE
thf(fact_703_DiffE,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_704_DiffD1,axiom,
! [C2: a > b,A: set_a_b,B: set_a_b] :
( ( member_a_b @ C2 @ ( minus_minus_set_a_b @ A @ B ) )
=> ( member_a_b @ C2 @ A ) ) ).
% DiffD1
thf(fact_705_DiffD1,axiom,
! [C2: c,A: set_c,B: set_c] :
( ( member_c @ C2 @ ( minus_minus_set_c @ A @ B ) )
=> ( member_c @ C2 @ A ) ) ).
% DiffD1
thf(fact_706_DiffD1,axiom,
! [C2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A @ B ) )
=> ( member_set_a @ C2 @ A ) ) ).
% DiffD1
thf(fact_707_DiffD1,axiom,
! [C2: real,A: set_real,B: set_real] :
( ( member_real @ C2 @ ( minus_minus_set_real @ A @ B ) )
=> ( member_real @ C2 @ A ) ) ).
% DiffD1
thf(fact_708_DiffD1,axiom,
! [C2: $o,A: set_o,B: set_o] :
( ( member_o @ C2 @ ( minus_minus_set_o @ A @ B ) )
=> ( member_o @ C2 @ A ) ) ).
% DiffD1
thf(fact_709_DiffD1,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) )
=> ( member_nat @ C2 @ A ) ) ).
% DiffD1
thf(fact_710_DiffD1,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
=> ( member_a @ C2 @ A ) ) ).
% DiffD1
thf(fact_711_DiffD2,axiom,
! [C2: a > b,A: set_a_b,B: set_a_b] :
( ( member_a_b @ C2 @ ( minus_minus_set_a_b @ A @ B ) )
=> ~ ( member_a_b @ C2 @ B ) ) ).
% DiffD2
thf(fact_712_DiffD2,axiom,
! [C2: c,A: set_c,B: set_c] :
( ( member_c @ C2 @ ( minus_minus_set_c @ A @ B ) )
=> ~ ( member_c @ C2 @ B ) ) ).
% DiffD2
thf(fact_713_DiffD2,axiom,
! [C2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A @ B ) )
=> ~ ( member_set_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_714_DiffD2,axiom,
! [C2: real,A: set_real,B: set_real] :
( ( member_real @ C2 @ ( minus_minus_set_real @ A @ B ) )
=> ~ ( member_real @ C2 @ B ) ) ).
% DiffD2
thf(fact_715_DiffD2,axiom,
! [C2: $o,A: set_o,B: set_o] :
( ( member_o @ C2 @ ( minus_minus_set_o @ A @ B ) )
=> ~ ( member_o @ C2 @ B ) ) ).
% DiffD2
thf(fact_716_DiffD2,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( member_nat @ C2 @ B ) ) ).
% DiffD2
thf(fact_717_DiffD2,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( member_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_718_emptyE,axiom,
! [A3: a > b] :
~ ( member_a_b @ A3 @ bot_bot_set_a_b ) ).
% emptyE
thf(fact_719_emptyE,axiom,
! [A3: c] :
~ ( member_c @ A3 @ bot_bot_set_c ) ).
% emptyE
thf(fact_720_emptyE,axiom,
! [A3: set_a] :
~ ( member_set_a @ A3 @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_721_emptyE,axiom,
! [A3: real] :
~ ( member_real @ A3 @ bot_bot_set_real ) ).
% emptyE
thf(fact_722_emptyE,axiom,
! [A3: $o] :
~ ( member_o @ A3 @ bot_bot_set_o ) ).
% emptyE
thf(fact_723_emptyE,axiom,
! [A3: nat] :
~ ( member_nat @ A3 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_724_emptyE,axiom,
! [A3: a] :
~ ( member_a @ A3 @ bot_bot_set_a ) ).
% emptyE
thf(fact_725_equals0D,axiom,
! [A: set_a_b,A3: a > b] :
( ( A = bot_bot_set_a_b )
=> ~ ( member_a_b @ A3 @ A ) ) ).
% equals0D
thf(fact_726_equals0D,axiom,
! [A: set_c,A3: c] :
( ( A = bot_bot_set_c )
=> ~ ( member_c @ A3 @ A ) ) ).
% equals0D
thf(fact_727_equals0D,axiom,
! [A: set_set_a,A3: set_a] :
( ( A = bot_bot_set_set_a )
=> ~ ( member_set_a @ A3 @ A ) ) ).
% equals0D
thf(fact_728_equals0D,axiom,
! [A: set_real,A3: real] :
( ( A = bot_bot_set_real )
=> ~ ( member_real @ A3 @ A ) ) ).
% equals0D
thf(fact_729_equals0D,axiom,
! [A: set_o,A3: $o] :
( ( A = bot_bot_set_o )
=> ~ ( member_o @ A3 @ A ) ) ).
% equals0D
thf(fact_730_equals0D,axiom,
! [A: set_nat,A3: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A3 @ A ) ) ).
% equals0D
thf(fact_731_equals0D,axiom,
! [A: set_a,A3: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A3 @ A ) ) ).
% equals0D
thf(fact_732_equals0I,axiom,
! [A: set_a_b] :
( ! [Y6: a > b] :
~ ( member_a_b @ Y6 @ A )
=> ( A = bot_bot_set_a_b ) ) ).
% equals0I
thf(fact_733_equals0I,axiom,
! [A: set_c] :
( ! [Y6: c] :
~ ( member_c @ Y6 @ A )
=> ( A = bot_bot_set_c ) ) ).
% equals0I
thf(fact_734_equals0I,axiom,
! [A: set_set_a] :
( ! [Y6: set_a] :
~ ( member_set_a @ Y6 @ A )
=> ( A = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_735_equals0I,axiom,
! [A: set_real] :
( ! [Y6: real] :
~ ( member_real @ Y6 @ A )
=> ( A = bot_bot_set_real ) ) ).
% equals0I
thf(fact_736_equals0I,axiom,
! [A: set_o] :
( ! [Y6: $o] :
~ ( member_o @ Y6 @ A )
=> ( A = bot_bot_set_o ) ) ).
% equals0I
thf(fact_737_equals0I,axiom,
! [A: set_nat] :
( ! [Y6: nat] :
~ ( member_nat @ Y6 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_738_equals0I,axiom,
! [A: set_a] :
( ! [Y6: a] :
~ ( member_a @ Y6 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_739_ex__in__conv,axiom,
! [A: set_a_b] :
( ( ? [X4: a > b] : ( member_a_b @ X4 @ A ) )
= ( A != bot_bot_set_a_b ) ) ).
% ex_in_conv
thf(fact_740_ex__in__conv,axiom,
! [A: set_c] :
( ( ? [X4: c] : ( member_c @ X4 @ A ) )
= ( A != bot_bot_set_c ) ) ).
% ex_in_conv
thf(fact_741_ex__in__conv,axiom,
! [A: set_set_a] :
( ( ? [X4: set_a] : ( member_set_a @ X4 @ A ) )
= ( A != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_742_ex__in__conv,axiom,
! [A: set_real] :
( ( ? [X4: real] : ( member_real @ X4 @ A ) )
= ( A != bot_bot_set_real ) ) ).
% ex_in_conv
thf(fact_743_ex__in__conv,axiom,
! [A: set_o] :
( ( ? [X4: $o] : ( member_o @ X4 @ A ) )
= ( A != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_744_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X4: nat] : ( member_nat @ X4 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_745_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X4: a] : ( member_a @ X4 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_746_diff__mono,axiom,
! [A3: real,B3: real,D2: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ D2 @ C2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ A3 @ C2 ) @ ( minus_minus_real @ B3 @ D2 ) ) ) ) ).
% diff_mono
thf(fact_747_diff__mono,axiom,
! [A3: int,B3: int,D2: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ D2 @ C2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A3 @ C2 ) @ ( minus_minus_int @ B3 @ D2 ) ) ) ) ).
% diff_mono
thf(fact_748_diff__left__mono,axiom,
! [B3: real,A3: real,C2: real] :
( ( ord_less_eq_real @ B3 @ A3 )
=> ( ord_less_eq_real @ ( minus_minus_real @ C2 @ A3 ) @ ( minus_minus_real @ C2 @ B3 ) ) ) ).
% diff_left_mono
thf(fact_749_diff__left__mono,axiom,
! [B3: int,A3: int,C2: int] :
( ( ord_less_eq_int @ B3 @ A3 )
=> ( ord_less_eq_int @ ( minus_minus_int @ C2 @ A3 ) @ ( minus_minus_int @ C2 @ B3 ) ) ) ).
% diff_left_mono
thf(fact_750_diff__right__mono,axiom,
! [A3: real,B3: real,C2: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ord_less_eq_real @ ( minus_minus_real @ A3 @ C2 ) @ ( minus_minus_real @ B3 @ C2 ) ) ) ).
% diff_right_mono
thf(fact_751_diff__right__mono,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A3 @ C2 ) @ ( minus_minus_int @ B3 @ C2 ) ) ) ).
% diff_right_mono
thf(fact_752_diff__eq__diff__less__eq,axiom,
! [A3: real,B3: real,C2: real,D2: real] :
( ( ( minus_minus_real @ A3 @ B3 )
= ( minus_minus_real @ C2 @ D2 ) )
=> ( ( ord_less_eq_real @ A3 @ B3 )
= ( ord_less_eq_real @ C2 @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_753_diff__eq__diff__less__eq,axiom,
! [A3: int,B3: int,C2: int,D2: int] :
( ( ( minus_minus_int @ A3 @ B3 )
= ( minus_minus_int @ C2 @ D2 ) )
=> ( ( ord_less_eq_int @ A3 @ B3 )
= ( ord_less_eq_int @ C2 @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_754_bot_Oextremum__uniqueI,axiom,
! [A3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A3 @ bot_bot_set_set_b )
=> ( A3 = bot_bot_set_set_b ) ) ).
% bot.extremum_uniqueI
thf(fact_755_bot_Oextremum__uniqueI,axiom,
! [A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ bot_bot_set_set_a )
=> ( A3 = bot_bot_set_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_756_bot_Oextremum__uniqueI,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
=> ( A3 = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_757_bot_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ bot_bot_nat )
=> ( A3 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_758_bot_Oextremum__unique,axiom,
! [A3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A3 @ bot_bot_set_set_b )
= ( A3 = bot_bot_set_set_b ) ) ).
% bot.extremum_unique
thf(fact_759_bot_Oextremum__unique,axiom,
! [A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ bot_bot_set_set_a )
= ( A3 = bot_bot_set_set_a ) ) ).
% bot.extremum_unique
thf(fact_760_bot_Oextremum__unique,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
= ( A3 = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_761_bot_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ bot_bot_nat )
= ( A3 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_762_bot_Oextremum,axiom,
! [A3: set_set_b] : ( ord_le3795704787696855135_set_b @ bot_bot_set_set_b @ A3 ) ).
% bot.extremum
thf(fact_763_bot_Oextremum,axiom,
! [A3: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A3 ) ).
% bot.extremum
thf(fact_764_bot_Oextremum,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% bot.extremum
thf(fact_765_bot_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A3 ) ).
% bot.extremum
thf(fact_766_double__diff,axiom,
! [A: set_set_b,B: set_set_b,C: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( ( ord_le3795704787696855135_set_b @ B @ C )
=> ( ( minus_5807331545291222566_set_b @ B @ ( minus_5807331545291222566_set_b @ C @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_767_double__diff,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ( minus_5736297505244876581_set_a @ B @ ( minus_5736297505244876581_set_a @ C @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_768_double__diff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_769_Diff__subset,axiom,
! [A: set_set_b,B: set_set_b] : ( ord_le3795704787696855135_set_b @ ( minus_5807331545291222566_set_b @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_770_Diff__subset,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_771_Diff__subset,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_772_Diff__mono,axiom,
! [A: set_set_b,C: set_set_b,D: set_set_b,B: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ C )
=> ( ( ord_le3795704787696855135_set_b @ D @ B )
=> ( ord_le3795704787696855135_set_b @ ( minus_5807331545291222566_set_b @ A @ B ) @ ( minus_5807331545291222566_set_b @ C @ D ) ) ) ) ).
% Diff_mono
thf(fact_773_Diff__mono,axiom,
! [A: set_set_a,C: set_set_a,D: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C )
=> ( ( ord_le3724670747650509150_set_a @ D @ B )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) @ ( minus_5736297505244876581_set_a @ C @ D ) ) ) ) ).
% Diff_mono
thf(fact_774_Diff__mono,axiom,
! [A: set_a,C: set_a,D: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ D @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ C @ D ) ) ) ) ).
% Diff_mono
thf(fact_775_Int__Diff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A @ B ) @ C )
= ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B @ C ) ) ) ).
% Int_Diff
thf(fact_776_Diff__Int2,axiom,
! [A: set_a,C: set_a,B: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C ) @ ( inf_inf_set_a @ B @ C ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C ) @ B ) ) ).
% Diff_Int2
thf(fact_777_Diff__Diff__Int,axiom,
! [A: set_a,B: set_a] :
( ( minus_minus_set_a @ A @ ( minus_minus_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_778_Diff__Int__distrib,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ C @ ( minus_minus_set_a @ A @ B ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C @ A ) @ ( inf_inf_set_a @ C @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_779_Diff__Int__distrib2,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A @ B ) @ C )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C ) @ ( inf_inf_set_a @ B @ C ) ) ) ).
% Diff_Int_distrib2
thf(fact_780_Int__emptyI,axiom,
! [A: set_a_b,B: set_a_b] :
( ! [X3: a > b] :
( ( member_a_b @ X3 @ A )
=> ~ ( member_a_b @ X3 @ B ) )
=> ( ( inf_inf_set_a_b @ A @ B )
= bot_bot_set_a_b ) ) ).
% Int_emptyI
thf(fact_781_Int__emptyI,axiom,
! [A: set_c,B: set_c] :
( ! [X3: c] :
( ( member_c @ X3 @ A )
=> ~ ( member_c @ X3 @ B ) )
=> ( ( inf_inf_set_c @ A @ B )
= bot_bot_set_c ) ) ).
% Int_emptyI
thf(fact_782_Int__emptyI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ~ ( member_set_a @ X3 @ B ) )
=> ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_783_Int__emptyI,axiom,
! [A: set_real,B: set_real] :
( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ~ ( member_real @ X3 @ B ) )
=> ( ( inf_inf_set_real @ A @ B )
= bot_bot_set_real ) ) ).
% Int_emptyI
thf(fact_784_Int__emptyI,axiom,
! [A: set_o,B: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ~ ( member_o @ X3 @ B ) )
=> ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o ) ) ).
% Int_emptyI
thf(fact_785_Int__emptyI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ~ ( member_nat @ X3 @ B ) )
=> ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_786_Int__emptyI,axiom,
! [A: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ~ ( member_a @ X3 @ B ) )
=> ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_787_disjoint__iff,axiom,
! [A: set_a_b,B: set_a_b] :
( ( ( inf_inf_set_a_b @ A @ B )
= bot_bot_set_a_b )
= ( ! [X4: a > b] :
( ( member_a_b @ X4 @ A )
=> ~ ( member_a_b @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_788_disjoint__iff,axiom,
! [A: set_c,B: set_c] :
( ( ( inf_inf_set_c @ A @ B )
= bot_bot_set_c )
= ( ! [X4: c] :
( ( member_c @ X4 @ A )
=> ~ ( member_c @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_789_disjoint__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ~ ( member_set_a @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_790_disjoint__iff,axiom,
! [A: set_real,B: set_real] :
( ( ( inf_inf_set_real @ A @ B )
= bot_bot_set_real )
= ( ! [X4: real] :
( ( member_real @ X4 @ A )
=> ~ ( member_real @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_791_disjoint__iff,axiom,
! [A: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o )
= ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ~ ( member_o @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_792_disjoint__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ~ ( member_nat @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_793_disjoint__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ~ ( member_a @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_794_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_795_Int__empty__right,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_796_disjoint__iff__not__equal,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ! [Y4: a] :
( ( member_a @ Y4 @ B )
=> ( X4 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_797_sets_Osigma__algebra__axioms,axiom,
! [M2: sigma_measure_b] : ( sigma_4968961713055010668ebra_b @ ( sigma_space_b @ M2 ) @ ( sigma_sets_b @ M2 ) ) ).
% sets.sigma_algebra_axioms
thf(fact_798_sets_Osigma__algebra__axioms,axiom,
! [M2: sigma_measure_a] : ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ M2 ) @ ( sigma_sets_a @ M2 ) ) ).
% sets.sigma_algebra_axioms
thf(fact_799_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_800_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_801_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_802_fmeasurable__Diff,axiom,
! [A: set_b,M2: sigma_measure_b,B: set_b] :
( ( member_set_b @ A @ ( measur3645360004775918571able_b @ M2 ) )
=> ( ( member_set_b @ B @ ( sigma_sets_b @ M2 ) )
=> ( member_set_b @ ( minus_minus_set_b @ A @ B ) @ ( measur3645360004775918571able_b @ M2 ) ) ) ) ).
% fmeasurable_Diff
thf(fact_803_fmeasurable__Diff,axiom,
! [A: set_a,M2: sigma_measure_a,B: set_a] :
( ( member_set_a @ A @ ( measur3645360004775918570able_a @ M2 ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M2 ) )
=> ( member_set_a @ ( minus_minus_set_a @ A @ B ) @ ( measur3645360004775918570able_a @ M2 ) ) ) ) ).
% fmeasurable_Diff
thf(fact_804_null__set__Diff,axiom,
! [B: set_b,M2: sigma_measure_b,A: set_b] :
( ( member_set_b @ B @ ( measure_null_sets_b @ M2 ) )
=> ( ( member_set_b @ A @ ( sigma_sets_b @ M2 ) )
=> ( member_set_b @ ( minus_minus_set_b @ B @ A ) @ ( measure_null_sets_b @ M2 ) ) ) ) ).
% null_set_Diff
thf(fact_805_null__set__Diff,axiom,
! [B: set_a,M2: sigma_measure_a,A: set_a] :
( ( member_set_a @ B @ ( measure_null_sets_a @ M2 ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M2 ) )
=> ( member_set_a @ ( minus_minus_set_a @ B @ A ) @ ( measure_null_sets_a @ M2 ) ) ) ) ).
% null_set_Diff
thf(fact_806_measurable__empty__iff,axiom,
! [N: sigma_measure_a,F2: a > a,M2: sigma_measure_a] :
( ( ( sigma_space_a @ N )
= bot_bot_set_a )
=> ( ( member_a_a @ F2 @ ( sigma_measurable_a_a @ M2 @ N ) )
= ( ( sigma_space_a @ M2 )
= bot_bot_set_a ) ) ) ).
% measurable_empty_iff
thf(fact_807_measurable__empty__iff,axiom,
! [N: sigma_measure_b,F2: a > b,M2: sigma_measure_a] :
( ( ( sigma_space_b @ N )
= bot_bot_set_b )
=> ( ( member_a_b @ F2 @ ( sigma_measurable_a_b @ M2 @ N ) )
= ( ( sigma_space_a @ M2 )
= bot_bot_set_a ) ) ) ).
% measurable_empty_iff
thf(fact_808_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_809_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_810_vector__space__over__itself_Oscale__left__commute,axiom,
! [A3: real,B3: real,X5: real] :
( ( times_times_real @ A3 @ ( times_times_real @ B3 @ X5 ) )
= ( times_times_real @ B3 @ ( times_times_real @ A3 @ X5 ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_811_vector__space__over__itself_Oscale__scale,axiom,
! [A3: real,B3: real,X5: real] :
( ( times_times_real @ A3 @ ( times_times_real @ B3 @ X5 ) )
= ( times_times_real @ ( times_times_real @ A3 @ B3 ) @ X5 ) ) ).
% vector_space_over_itself.scale_scale
thf(fact_812_mult_Oleft__commute,axiom,
! [B3: real,A3: real,C2: real] :
( ( times_times_real @ B3 @ ( times_times_real @ A3 @ C2 ) )
= ( times_times_real @ A3 @ ( times_times_real @ B3 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_813_mult_Oleft__commute,axiom,
! [B3: nat,A3: nat,C2: nat] :
( ( times_times_nat @ B3 @ ( times_times_nat @ A3 @ C2 ) )
= ( times_times_nat @ A3 @ ( times_times_nat @ B3 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_814_mult_Oleft__commute,axiom,
! [B3: int,A3: int,C2: int] :
( ( times_times_int @ B3 @ ( times_times_int @ A3 @ C2 ) )
= ( times_times_int @ A3 @ ( times_times_int @ B3 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_815_mult_Oleft__commute,axiom,
! [B3: extend8495563244428889912nnreal,A3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ B3 @ ( times_1893300245718287421nnreal @ A3 @ C2 ) )
= ( times_1893300245718287421nnreal @ A3 @ ( times_1893300245718287421nnreal @ B3 @ C2 ) ) ) ).
% mult.left_commute
thf(fact_816_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A5: real,B5: real] : ( times_times_real @ B5 @ A5 ) ) ) ).
% mult.commute
thf(fact_817_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A5: nat,B5: nat] : ( times_times_nat @ B5 @ A5 ) ) ) ).
% mult.commute
thf(fact_818_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A5: int,B5: int] : ( times_times_int @ B5 @ A5 ) ) ) ).
% mult.commute
thf(fact_819_mult_Ocommute,axiom,
( times_1893300245718287421nnreal
= ( ^ [A5: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] : ( times_1893300245718287421nnreal @ B5 @ A5 ) ) ) ).
% mult.commute
thf(fact_820_mult_Oassoc,axiom,
! [A3: real,B3: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A3 @ B3 ) @ C2 )
= ( times_times_real @ A3 @ ( times_times_real @ B3 @ C2 ) ) ) ).
% mult.assoc
thf(fact_821_mult_Oassoc,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B3 ) @ C2 )
= ( times_times_nat @ A3 @ ( times_times_nat @ B3 @ C2 ) ) ) ).
% mult.assoc
thf(fact_822_mult_Oassoc,axiom,
! [A3: int,B3: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A3 @ B3 ) @ C2 )
= ( times_times_int @ A3 @ ( times_times_int @ B3 @ C2 ) ) ) ).
% mult.assoc
thf(fact_823_mult_Oassoc,axiom,
! [A3: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ ( times_1893300245718287421nnreal @ A3 @ B3 ) @ C2 )
= ( times_1893300245718287421nnreal @ A3 @ ( times_1893300245718287421nnreal @ B3 @ C2 ) ) ) ).
% mult.assoc
thf(fact_824_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: real,B3: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A3 @ B3 ) @ C2 )
= ( times_times_real @ A3 @ ( times_times_real @ B3 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_825_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B3 ) @ C2 )
= ( times_times_nat @ A3 @ ( times_times_nat @ B3 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_826_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: int,B3: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A3 @ B3 ) @ C2 )
= ( times_times_int @ A3 @ ( times_times_int @ B3 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_827_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ ( times_1893300245718287421nnreal @ A3 @ B3 ) @ C2 )
= ( times_1893300245718287421nnreal @ A3 @ ( times_1893300245718287421nnreal @ B3 @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_828_one__reorient,axiom,
! [X5: real] :
( ( one_one_real = X5 )
= ( X5 = one_one_real ) ) ).
% one_reorient
thf(fact_829_one__reorient,axiom,
! [X5: nat] :
( ( one_one_nat = X5 )
= ( X5 = one_one_nat ) ) ).
% one_reorient
thf(fact_830_one__reorient,axiom,
! [X5: int] :
( ( one_one_int = X5 )
= ( X5 = one_one_int ) ) ).
% one_reorient
thf(fact_831_one__reorient,axiom,
! [X5: extend8495563244428889912nnreal] :
( ( one_on2969667320475766781nnreal = X5 )
= ( X5 = one_on2969667320475766781nnreal ) ) ).
% one_reorient
thf(fact_832_measure__Diff__null__set,axiom,
! [A: set_b,M2: sigma_measure_b,B: set_b] :
( ( member_set_b @ A @ ( sigma_sets_b @ M2 ) )
=> ( ( member_set_b @ B @ ( measure_null_sets_b @ M2 ) )
=> ( ( sigma_measure_b2 @ M2 @ ( minus_minus_set_b @ A @ B ) )
= ( sigma_measure_b2 @ M2 @ A ) ) ) ) ).
% measure_Diff_null_set
thf(fact_833_measure__Diff__null__set,axiom,
! [A: set_a,M2: sigma_measure_a,B: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ M2 ) )
=> ( ( member_set_a @ B @ ( measure_null_sets_a @ M2 ) )
=> ( ( sigma_measure_a2 @ M2 @ ( minus_minus_set_a @ A @ B ) )
= ( sigma_measure_a2 @ M2 @ A ) ) ) ) ).
% measure_Diff_null_set
thf(fact_834_inf__left__commute,axiom,
! [X5: set_a,Y5: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X5 @ ( inf_inf_set_a @ Y5 @ Z2 ) )
= ( inf_inf_set_a @ Y5 @ ( inf_inf_set_a @ X5 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_835_inf_Oleft__commute,axiom,
! [B3: set_a,A3: set_a,C2: set_a] :
( ( inf_inf_set_a @ B3 @ ( inf_inf_set_a @ A3 @ C2 ) )
= ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ B3 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_836_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X4: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X4 ) ) ) ).
% inf_commute
thf(fact_837_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A5: set_a,B5: set_a] : ( inf_inf_set_a @ B5 @ A5 ) ) ) ).
% inf.commute
thf(fact_838_inf__assoc,axiom,
! [X5: set_a,Y5: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X5 @ Y5 ) @ Z2 )
= ( inf_inf_set_a @ X5 @ ( inf_inf_set_a @ Y5 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_839_inf_Oassoc,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ C2 )
= ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ B3 @ C2 ) ) ) ).
% inf.assoc
thf(fact_840_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X4: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X4 ) ) ) ).
% inf_sup_aci(1)
thf(fact_841_inf__sup__aci_I2_J,axiom,
! [X5: set_a,Y5: set_a,Z2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X5 @ Y5 ) @ Z2 )
= ( inf_inf_set_a @ X5 @ ( inf_inf_set_a @ Y5 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_842_inf__sup__aci_I3_J,axiom,
! [X5: set_a,Y5: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X5 @ ( inf_inf_set_a @ Y5 @ Z2 ) )
= ( inf_inf_set_a @ Y5 @ ( inf_inf_set_a @ X5 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_843_inf__sup__aci_I4_J,axiom,
! [X5: set_a,Y5: set_a] :
( ( inf_inf_set_a @ X5 @ ( inf_inf_set_a @ X5 @ Y5 ) )
= ( inf_inf_set_a @ X5 @ Y5 ) ) ).
% inf_sup_aci(4)
thf(fact_844_measurable__Diff__null__set,axiom,
! [B: set_b,M2: sigma_measure_b,A: set_b] :
( ( member_set_b @ B @ ( measure_null_sets_b @ M2 ) )
=> ( ( ( member_set_b @ ( minus_minus_set_b @ A @ B ) @ ( measur3645360004775918571able_b @ M2 ) )
& ( member_set_b @ A @ ( sigma_sets_b @ M2 ) ) )
= ( member_set_b @ A @ ( measur3645360004775918571able_b @ M2 ) ) ) ) ).
% measurable_Diff_null_set
thf(fact_845_measurable__Diff__null__set,axiom,
! [B: set_a,M2: sigma_measure_a,A: set_a] :
( ( member_set_a @ B @ ( measure_null_sets_a @ M2 ) )
=> ( ( ( member_set_a @ ( minus_minus_set_a @ A @ B ) @ ( measur3645360004775918570able_a @ M2 ) )
& ( member_set_a @ A @ ( sigma_sets_a @ M2 ) ) )
= ( member_set_a @ A @ ( measur3645360004775918570able_a @ M2 ) ) ) ) ).
% measurable_Diff_null_set
thf(fact_846_prob__space_Osigma__algebra__tail__events,axiom,
! [M2: sigma_measure_a,A: nat > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ! [I3: nat] : ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ M2 ) @ ( A @ I3 ) )
=> ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ M2 ) @ ( indepe7538416700049374166_a_nat @ M2 @ A ) ) ) ) ).
% prob_space.sigma_algebra_tail_events
thf(fact_847_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_848_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_849_subprob__space_Osubprob__space__distr,axiom,
! [M2: sigma_measure_a,F2: a > a,M: sigma_measure_a] :
( ( giry_subprob_space_a @ M2 )
=> ( ( member_a_a @ F2 @ ( sigma_measurable_a_a @ M2 @ M ) )
=> ( ( ( sigma_space_a @ M )
!= bot_bot_set_a )
=> ( giry_subprob_space_a @ ( measure_distr_a_a @ M2 @ M @ F2 ) ) ) ) ) ).
% subprob_space.subprob_space_distr
thf(fact_850_subprob__space_Osubprob__space__distr,axiom,
! [M2: sigma_measure_a,F2: a > b,M: sigma_measure_b] :
( ( giry_subprob_space_a @ M2 )
=> ( ( member_a_b @ F2 @ ( sigma_measurable_a_b @ M2 @ M ) )
=> ( ( ( sigma_space_b @ M )
!= bot_bot_set_b )
=> ( giry_subprob_space_b @ ( measure_distr_a_b @ M2 @ M @ F2 ) ) ) ) ) ).
% subprob_space.subprob_space_distr
thf(fact_851_mult_Ocomm__neutral,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ one_one_real )
= A3 ) ).
% mult.comm_neutral
thf(fact_852_mult_Ocomm__neutral,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% mult.comm_neutral
thf(fact_853_mult_Ocomm__neutral,axiom,
! [A3: int] :
( ( times_times_int @ A3 @ one_one_int )
= A3 ) ).
% mult.comm_neutral
thf(fact_854_mult_Ocomm__neutral,axiom,
! [A3: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ A3 @ one_on2969667320475766781nnreal )
= A3 ) ).
% mult.comm_neutral
thf(fact_855_comm__monoid__mult__class_Omult__1,axiom,
! [A3: real] :
( ( times_times_real @ one_one_real @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_856_comm__monoid__mult__class_Omult__1,axiom,
! [A3: nat] :
( ( times_times_nat @ one_one_nat @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_857_comm__monoid__mult__class_Omult__1,axiom,
! [A3: int] :
( ( times_times_int @ one_one_int @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_858_comm__monoid__mult__class_Omult__1,axiom,
! [A3: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ one_on2969667320475766781nnreal @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_859_inf__sup__ord_I2_J,axiom,
! [X5: set_set_b,Y5: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ X5 @ Y5 ) @ Y5 ) ).
% inf_sup_ord(2)
thf(fact_860_inf__sup__ord_I2_J,axiom,
! [X5: set_set_a,Y5: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X5 @ Y5 ) @ Y5 ) ).
% inf_sup_ord(2)
thf(fact_861_inf__sup__ord_I2_J,axiom,
! [X5: set_a,Y5: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X5 @ Y5 ) @ Y5 ) ).
% inf_sup_ord(2)
thf(fact_862_inf__sup__ord_I2_J,axiom,
! [X5: nat,Y5: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X5 @ Y5 ) @ Y5 ) ).
% inf_sup_ord(2)
thf(fact_863_inf__sup__ord_I2_J,axiom,
! [X5: int,Y5: int] : ( ord_less_eq_int @ ( inf_inf_int @ X5 @ Y5 ) @ Y5 ) ).
% inf_sup_ord(2)
thf(fact_864_inf__sup__ord_I1_J,axiom,
! [X5: set_set_b,Y5: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ X5 @ Y5 ) @ X5 ) ).
% inf_sup_ord(1)
thf(fact_865_inf__sup__ord_I1_J,axiom,
! [X5: set_set_a,Y5: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X5 @ Y5 ) @ X5 ) ).
% inf_sup_ord(1)
thf(fact_866_inf__sup__ord_I1_J,axiom,
! [X5: set_a,Y5: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X5 @ Y5 ) @ X5 ) ).
% inf_sup_ord(1)
thf(fact_867_inf__sup__ord_I1_J,axiom,
! [X5: nat,Y5: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X5 @ Y5 ) @ X5 ) ).
% inf_sup_ord(1)
thf(fact_868_inf__sup__ord_I1_J,axiom,
! [X5: int,Y5: int] : ( ord_less_eq_int @ ( inf_inf_int @ X5 @ Y5 ) @ X5 ) ).
% inf_sup_ord(1)
thf(fact_869_inf__le1,axiom,
! [X5: set_set_b,Y5: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ X5 @ Y5 ) @ X5 ) ).
% inf_le1
thf(fact_870_inf__le1,axiom,
! [X5: set_set_a,Y5: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X5 @ Y5 ) @ X5 ) ).
% inf_le1
thf(fact_871_inf__le1,axiom,
! [X5: set_a,Y5: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X5 @ Y5 ) @ X5 ) ).
% inf_le1
thf(fact_872_inf__le1,axiom,
! [X5: nat,Y5: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X5 @ Y5 ) @ X5 ) ).
% inf_le1
thf(fact_873_inf__le1,axiom,
! [X5: int,Y5: int] : ( ord_less_eq_int @ ( inf_inf_int @ X5 @ Y5 ) @ X5 ) ).
% inf_le1
thf(fact_874_inf__le2,axiom,
! [X5: set_set_b,Y5: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ X5 @ Y5 ) @ Y5 ) ).
% inf_le2
thf(fact_875_inf__le2,axiom,
! [X5: set_set_a,Y5: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X5 @ Y5 ) @ Y5 ) ).
% inf_le2
thf(fact_876_inf__le2,axiom,
! [X5: set_a,Y5: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X5 @ Y5 ) @ Y5 ) ).
% inf_le2
thf(fact_877_inf__le2,axiom,
! [X5: nat,Y5: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X5 @ Y5 ) @ Y5 ) ).
% inf_le2
thf(fact_878_inf__le2,axiom,
! [X5: int,Y5: int] : ( ord_less_eq_int @ ( inf_inf_int @ X5 @ Y5 ) @ Y5 ) ).
% inf_le2
thf(fact_879_le__infE,axiom,
! [X5: set_set_b,A3: set_set_b,B3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X5 @ ( inf_inf_set_set_b @ A3 @ B3 ) )
=> ~ ( ( ord_le3795704787696855135_set_b @ X5 @ A3 )
=> ~ ( ord_le3795704787696855135_set_b @ X5 @ B3 ) ) ) ).
% le_infE
thf(fact_880_le__infE,axiom,
! [X5: set_set_a,A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ ( inf_inf_set_set_a @ A3 @ B3 ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ X5 @ A3 )
=> ~ ( ord_le3724670747650509150_set_a @ X5 @ B3 ) ) ) ).
% le_infE
thf(fact_881_le__infE,axiom,
! [X5: set_a,A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ X5 @ ( inf_inf_set_a @ A3 @ B3 ) )
=> ~ ( ( ord_less_eq_set_a @ X5 @ A3 )
=> ~ ( ord_less_eq_set_a @ X5 @ B3 ) ) ) ).
% le_infE
thf(fact_882_le__infE,axiom,
! [X5: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ X5 @ ( inf_inf_nat @ A3 @ B3 ) )
=> ~ ( ( ord_less_eq_nat @ X5 @ A3 )
=> ~ ( ord_less_eq_nat @ X5 @ B3 ) ) ) ).
% le_infE
thf(fact_883_le__infE,axiom,
! [X5: int,A3: int,B3: int] :
( ( ord_less_eq_int @ X5 @ ( inf_inf_int @ A3 @ B3 ) )
=> ~ ( ( ord_less_eq_int @ X5 @ A3 )
=> ~ ( ord_less_eq_int @ X5 @ B3 ) ) ) ).
% le_infE
thf(fact_884_le__infI,axiom,
! [X5: set_set_b,A3: set_set_b,B3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X5 @ A3 )
=> ( ( ord_le3795704787696855135_set_b @ X5 @ B3 )
=> ( ord_le3795704787696855135_set_b @ X5 @ ( inf_inf_set_set_b @ A3 @ B3 ) ) ) ) ).
% le_infI
thf(fact_885_le__infI,axiom,
! [X5: set_set_a,A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ A3 )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ B3 )
=> ( ord_le3724670747650509150_set_a @ X5 @ ( inf_inf_set_set_a @ A3 @ B3 ) ) ) ) ).
% le_infI
thf(fact_886_le__infI,axiom,
! [X5: set_a,A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ X5 @ A3 )
=> ( ( ord_less_eq_set_a @ X5 @ B3 )
=> ( ord_less_eq_set_a @ X5 @ ( inf_inf_set_a @ A3 @ B3 ) ) ) ) ).
% le_infI
thf(fact_887_le__infI,axiom,
! [X5: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ X5 @ A3 )
=> ( ( ord_less_eq_nat @ X5 @ B3 )
=> ( ord_less_eq_nat @ X5 @ ( inf_inf_nat @ A3 @ B3 ) ) ) ) ).
% le_infI
thf(fact_888_le__infI,axiom,
! [X5: int,A3: int,B3: int] :
( ( ord_less_eq_int @ X5 @ A3 )
=> ( ( ord_less_eq_int @ X5 @ B3 )
=> ( ord_less_eq_int @ X5 @ ( inf_inf_int @ A3 @ B3 ) ) ) ) ).
% le_infI
thf(fact_889_inf__mono,axiom,
! [A3: set_set_b,C2: set_set_b,B3: set_set_b,D2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A3 @ C2 )
=> ( ( ord_le3795704787696855135_set_b @ B3 @ D2 )
=> ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A3 @ B3 ) @ ( inf_inf_set_set_b @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_890_inf__mono,axiom,
! [A3: set_set_a,C2: set_set_a,B3: set_set_a,D2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A3 @ B3 ) @ ( inf_inf_set_set_a @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_891_inf__mono,axiom,
! [A3: set_a,C2: set_a,B3: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ( ord_less_eq_set_a @ B3 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_892_inf__mono,axiom,
! [A3: nat,C2: nat,B3: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ C2 )
=> ( ( ord_less_eq_nat @ B3 @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B3 ) @ ( inf_inf_nat @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_893_inf__mono,axiom,
! [A3: int,C2: int,B3: int,D2: int] :
( ( ord_less_eq_int @ A3 @ C2 )
=> ( ( ord_less_eq_int @ B3 @ D2 )
=> ( ord_less_eq_int @ ( inf_inf_int @ A3 @ B3 ) @ ( inf_inf_int @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_894_le__infI1,axiom,
! [A3: set_set_b,X5: set_set_b,B3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A3 @ X5 )
=> ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A3 @ B3 ) @ X5 ) ) ).
% le_infI1
thf(fact_895_le__infI1,axiom,
! [A3: set_set_a,X5: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ X5 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A3 @ B3 ) @ X5 ) ) ).
% le_infI1
thf(fact_896_le__infI1,axiom,
! [A3: set_a,X5: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ X5 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ X5 ) ) ).
% le_infI1
thf(fact_897_le__infI1,axiom,
! [A3: nat,X5: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ X5 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B3 ) @ X5 ) ) ).
% le_infI1
thf(fact_898_le__infI1,axiom,
! [A3: int,X5: int,B3: int] :
( ( ord_less_eq_int @ A3 @ X5 )
=> ( ord_less_eq_int @ ( inf_inf_int @ A3 @ B3 ) @ X5 ) ) ).
% le_infI1
thf(fact_899_le__infI2,axiom,
! [B3: set_set_b,X5: set_set_b,A3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B3 @ X5 )
=> ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A3 @ B3 ) @ X5 ) ) ).
% le_infI2
thf(fact_900_le__infI2,axiom,
! [B3: set_set_a,X5: set_set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ X5 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A3 @ B3 ) @ X5 ) ) ).
% le_infI2
thf(fact_901_le__infI2,axiom,
! [B3: set_a,X5: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ X5 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ X5 ) ) ).
% le_infI2
thf(fact_902_le__infI2,axiom,
! [B3: nat,X5: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ X5 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B3 ) @ X5 ) ) ).
% le_infI2
thf(fact_903_le__infI2,axiom,
! [B3: int,X5: int,A3: int] :
( ( ord_less_eq_int @ B3 @ X5 )
=> ( ord_less_eq_int @ ( inf_inf_int @ A3 @ B3 ) @ X5 ) ) ).
% le_infI2
thf(fact_904_inf_OorderE,axiom,
! [A3: set_set_b,B3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A3 @ B3 )
=> ( A3
= ( inf_inf_set_set_b @ A3 @ B3 ) ) ) ).
% inf.orderE
thf(fact_905_inf_OorderE,axiom,
! [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
=> ( A3
= ( inf_inf_set_set_a @ A3 @ B3 ) ) ) ).
% inf.orderE
thf(fact_906_inf_OorderE,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( A3
= ( inf_inf_set_a @ A3 @ B3 ) ) ) ).
% inf.orderE
thf(fact_907_inf_OorderE,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( A3
= ( inf_inf_nat @ A3 @ B3 ) ) ) ).
% inf.orderE
thf(fact_908_inf_OorderE,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( A3
= ( inf_inf_int @ A3 @ B3 ) ) ) ).
% inf.orderE
thf(fact_909_inf_OorderI,axiom,
! [A3: set_set_b,B3: set_set_b] :
( ( A3
= ( inf_inf_set_set_b @ A3 @ B3 ) )
=> ( ord_le3795704787696855135_set_b @ A3 @ B3 ) ) ).
% inf.orderI
thf(fact_910_inf_OorderI,axiom,
! [A3: set_set_a,B3: set_set_a] :
( ( A3
= ( inf_inf_set_set_a @ A3 @ B3 ) )
=> ( ord_le3724670747650509150_set_a @ A3 @ B3 ) ) ).
% inf.orderI
thf(fact_911_inf_OorderI,axiom,
! [A3: set_a,B3: set_a] :
( ( A3
= ( inf_inf_set_a @ A3 @ B3 ) )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% inf.orderI
thf(fact_912_inf_OorderI,axiom,
! [A3: nat,B3: nat] :
( ( A3
= ( inf_inf_nat @ A3 @ B3 ) )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% inf.orderI
thf(fact_913_inf_OorderI,axiom,
! [A3: int,B3: int] :
( ( A3
= ( inf_inf_int @ A3 @ B3 ) )
=> ( ord_less_eq_int @ A3 @ B3 ) ) ).
% inf.orderI
thf(fact_914_inf__unique,axiom,
! [F2: set_set_b > set_set_b > set_set_b,X5: set_set_b,Y5: set_set_b] :
( ! [X3: set_set_b,Y6: set_set_b] : ( ord_le3795704787696855135_set_b @ ( F2 @ X3 @ Y6 ) @ X3 )
=> ( ! [X3: set_set_b,Y6: set_set_b] : ( ord_le3795704787696855135_set_b @ ( F2 @ X3 @ Y6 ) @ Y6 )
=> ( ! [X3: set_set_b,Y6: set_set_b,Z3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X3 @ Y6 )
=> ( ( ord_le3795704787696855135_set_b @ X3 @ Z3 )
=> ( ord_le3795704787696855135_set_b @ X3 @ ( F2 @ Y6 @ Z3 ) ) ) )
=> ( ( inf_inf_set_set_b @ X5 @ Y5 )
= ( F2 @ X5 @ Y5 ) ) ) ) ) ).
% inf_unique
thf(fact_915_inf__unique,axiom,
! [F2: set_set_a > set_set_a > set_set_a,X5: set_set_a,Y5: set_set_a] :
( ! [X3: set_set_a,Y6: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F2 @ X3 @ Y6 ) @ X3 )
=> ( ! [X3: set_set_a,Y6: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F2 @ X3 @ Y6 ) @ Y6 )
=> ( ! [X3: set_set_a,Y6: set_set_a,Z3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y6 )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Z3 )
=> ( ord_le3724670747650509150_set_a @ X3 @ ( F2 @ Y6 @ Z3 ) ) ) )
=> ( ( inf_inf_set_set_a @ X5 @ Y5 )
= ( F2 @ X5 @ Y5 ) ) ) ) ) ).
% inf_unique
thf(fact_916_inf__unique,axiom,
! [F2: set_a > set_a > set_a,X5: set_a,Y5: set_a] :
( ! [X3: set_a,Y6: set_a] : ( ord_less_eq_set_a @ ( F2 @ X3 @ Y6 ) @ X3 )
=> ( ! [X3: set_a,Y6: set_a] : ( ord_less_eq_set_a @ ( F2 @ X3 @ Y6 ) @ Y6 )
=> ( ! [X3: set_a,Y6: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y6 )
=> ( ( ord_less_eq_set_a @ X3 @ Z3 )
=> ( ord_less_eq_set_a @ X3 @ ( F2 @ Y6 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X5 @ Y5 )
= ( F2 @ X5 @ Y5 ) ) ) ) ) ).
% inf_unique
thf(fact_917_inf__unique,axiom,
! [F2: nat > nat > nat,X5: nat,Y5: nat] :
( ! [X3: nat,Y6: nat] : ( ord_less_eq_nat @ ( F2 @ X3 @ Y6 ) @ X3 )
=> ( ! [X3: nat,Y6: nat] : ( ord_less_eq_nat @ ( F2 @ X3 @ Y6 ) @ Y6 )
=> ( ! [X3: nat,Y6: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
=> ( ( ord_less_eq_nat @ X3 @ Z3 )
=> ( ord_less_eq_nat @ X3 @ ( F2 @ Y6 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X5 @ Y5 )
= ( F2 @ X5 @ Y5 ) ) ) ) ) ).
% inf_unique
thf(fact_918_inf__unique,axiom,
! [F2: int > int > int,X5: int,Y5: int] :
( ! [X3: int,Y6: int] : ( ord_less_eq_int @ ( F2 @ X3 @ Y6 ) @ X3 )
=> ( ! [X3: int,Y6: int] : ( ord_less_eq_int @ ( F2 @ X3 @ Y6 ) @ Y6 )
=> ( ! [X3: int,Y6: int,Z3: int] :
( ( ord_less_eq_int @ X3 @ Y6 )
=> ( ( ord_less_eq_int @ X3 @ Z3 )
=> ( ord_less_eq_int @ X3 @ ( F2 @ Y6 @ Z3 ) ) ) )
=> ( ( inf_inf_int @ X5 @ Y5 )
= ( F2 @ X5 @ Y5 ) ) ) ) ) ).
% inf_unique
thf(fact_919_le__iff__inf,axiom,
( ord_le3795704787696855135_set_b
= ( ^ [X4: set_set_b,Y4: set_set_b] :
( ( inf_inf_set_set_b @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_920_le__iff__inf,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X4: set_set_a,Y4: set_set_a] :
( ( inf_inf_set_set_a @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_921_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X4: set_a,Y4: set_a] :
( ( inf_inf_set_a @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_922_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( inf_inf_nat @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_923_le__iff__inf,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y4: int] :
( ( inf_inf_int @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_924_inf_Oabsorb1,axiom,
! [A3: set_set_b,B3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A3 @ B3 )
=> ( ( inf_inf_set_set_b @ A3 @ B3 )
= A3 ) ) ).
% inf.absorb1
thf(fact_925_inf_Oabsorb1,axiom,
! [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
=> ( ( inf_inf_set_set_a @ A3 @ B3 )
= A3 ) ) ).
% inf.absorb1
thf(fact_926_inf_Oabsorb1,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( inf_inf_set_a @ A3 @ B3 )
= A3 ) ) ).
% inf.absorb1
thf(fact_927_inf_Oabsorb1,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( inf_inf_nat @ A3 @ B3 )
= A3 ) ) ).
% inf.absorb1
thf(fact_928_inf_Oabsorb1,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( inf_inf_int @ A3 @ B3 )
= A3 ) ) ).
% inf.absorb1
thf(fact_929_inf_Oabsorb2,axiom,
! [B3: set_set_b,A3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B3 @ A3 )
=> ( ( inf_inf_set_set_b @ A3 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_930_inf_Oabsorb2,axiom,
! [B3: set_set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A3 )
=> ( ( inf_inf_set_set_a @ A3 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_931_inf_Oabsorb2,axiom,
! [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( ( inf_inf_set_a @ A3 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_932_inf_Oabsorb2,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( inf_inf_nat @ A3 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_933_inf_Oabsorb2,axiom,
! [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
=> ( ( inf_inf_int @ A3 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_934_inf__absorb1,axiom,
! [X5: set_set_b,Y5: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X5 @ Y5 )
=> ( ( inf_inf_set_set_b @ X5 @ Y5 )
= X5 ) ) ).
% inf_absorb1
thf(fact_935_inf__absorb1,axiom,
! [X5: set_set_a,Y5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ Y5 )
=> ( ( inf_inf_set_set_a @ X5 @ Y5 )
= X5 ) ) ).
% inf_absorb1
thf(fact_936_inf__absorb1,axiom,
! [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ( inf_inf_set_a @ X5 @ Y5 )
= X5 ) ) ).
% inf_absorb1
thf(fact_937_inf__absorb1,axiom,
! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ( inf_inf_nat @ X5 @ Y5 )
= X5 ) ) ).
% inf_absorb1
thf(fact_938_inf__absorb1,axiom,
! [X5: int,Y5: int] :
( ( ord_less_eq_int @ X5 @ Y5 )
=> ( ( inf_inf_int @ X5 @ Y5 )
= X5 ) ) ).
% inf_absorb1
thf(fact_939_inf__absorb2,axiom,
! [Y5: set_set_b,X5: set_set_b] :
( ( ord_le3795704787696855135_set_b @ Y5 @ X5 )
=> ( ( inf_inf_set_set_b @ X5 @ Y5 )
= Y5 ) ) ).
% inf_absorb2
thf(fact_940_inf__absorb2,axiom,
! [Y5: set_set_a,X5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y5 @ X5 )
=> ( ( inf_inf_set_set_a @ X5 @ Y5 )
= Y5 ) ) ).
% inf_absorb2
thf(fact_941_inf__absorb2,axiom,
! [Y5: set_a,X5: set_a] :
( ( ord_less_eq_set_a @ Y5 @ X5 )
=> ( ( inf_inf_set_a @ X5 @ Y5 )
= Y5 ) ) ).
% inf_absorb2
thf(fact_942_inf__absorb2,axiom,
! [Y5: nat,X5: nat] :
( ( ord_less_eq_nat @ Y5 @ X5 )
=> ( ( inf_inf_nat @ X5 @ Y5 )
= Y5 ) ) ).
% inf_absorb2
thf(fact_943_inf__absorb2,axiom,
! [Y5: int,X5: int] :
( ( ord_less_eq_int @ Y5 @ X5 )
=> ( ( inf_inf_int @ X5 @ Y5 )
= Y5 ) ) ).
% inf_absorb2
thf(fact_944_inf_OboundedE,axiom,
! [A3: set_set_b,B3: set_set_b,C2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A3 @ ( inf_inf_set_set_b @ B3 @ C2 ) )
=> ~ ( ( ord_le3795704787696855135_set_b @ A3 @ B3 )
=> ~ ( ord_le3795704787696855135_set_b @ A3 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_945_inf_OboundedE,axiom,
! [A3: set_set_a,B3: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ ( inf_inf_set_set_a @ B3 @ C2 ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
=> ~ ( ord_le3724670747650509150_set_a @ A3 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_946_inf_OboundedE,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( inf_inf_set_a @ B3 @ C2 ) )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B3 )
=> ~ ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_947_inf_OboundedE,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ ( inf_inf_nat @ B3 @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A3 @ B3 )
=> ~ ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_948_inf_OboundedE,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_eq_int @ A3 @ ( inf_inf_int @ B3 @ C2 ) )
=> ~ ( ( ord_less_eq_int @ A3 @ B3 )
=> ~ ( ord_less_eq_int @ A3 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_949_inf_OboundedI,axiom,
! [A3: set_set_b,B3: set_set_b,C2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A3 @ B3 )
=> ( ( ord_le3795704787696855135_set_b @ A3 @ C2 )
=> ( ord_le3795704787696855135_set_b @ A3 @ ( inf_inf_set_set_b @ B3 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_950_inf_OboundedI,axiom,
! [A3: set_set_a,B3: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ A3 @ C2 )
=> ( ord_le3724670747650509150_set_a @ A3 @ ( inf_inf_set_set_a @ B3 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_951_inf_OboundedI,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ord_less_eq_set_a @ A3 @ ( inf_inf_set_a @ B3 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_952_inf_OboundedI,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ A3 @ C2 )
=> ( ord_less_eq_nat @ A3 @ ( inf_inf_nat @ B3 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_953_inf_OboundedI,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ A3 @ C2 )
=> ( ord_less_eq_int @ A3 @ ( inf_inf_int @ B3 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_954_inf__greatest,axiom,
! [X5: set_set_b,Y5: set_set_b,Z2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X5 @ Y5 )
=> ( ( ord_le3795704787696855135_set_b @ X5 @ Z2 )
=> ( ord_le3795704787696855135_set_b @ X5 @ ( inf_inf_set_set_b @ Y5 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_955_inf__greatest,axiom,
! [X5: set_set_a,Y5: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ Y5 )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ Z2 )
=> ( ord_le3724670747650509150_set_a @ X5 @ ( inf_inf_set_set_a @ Y5 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_956_inf__greatest,axiom,
! [X5: set_a,Y5: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ( ord_less_eq_set_a @ X5 @ Z2 )
=> ( ord_less_eq_set_a @ X5 @ ( inf_inf_set_a @ Y5 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_957_inf__greatest,axiom,
! [X5: nat,Y5: nat,Z2: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ( ord_less_eq_nat @ X5 @ Z2 )
=> ( ord_less_eq_nat @ X5 @ ( inf_inf_nat @ Y5 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_958_inf__greatest,axiom,
! [X5: int,Y5: int,Z2: int] :
( ( ord_less_eq_int @ X5 @ Y5 )
=> ( ( ord_less_eq_int @ X5 @ Z2 )
=> ( ord_less_eq_int @ X5 @ ( inf_inf_int @ Y5 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_959_inf_Oorder__iff,axiom,
( ord_le3795704787696855135_set_b
= ( ^ [A5: set_set_b,B5: set_set_b] :
( A5
= ( inf_inf_set_set_b @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_960_inf_Oorder__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
( A5
= ( inf_inf_set_set_a @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_961_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( A5
= ( inf_inf_set_a @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_962_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( A5
= ( inf_inf_nat @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_963_inf_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B5: int] :
( A5
= ( inf_inf_int @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_964_inf_Ocobounded1,axiom,
! [A3: set_set_b,B3: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A3 @ B3 ) @ A3 ) ).
% inf.cobounded1
thf(fact_965_inf_Ocobounded1,axiom,
! [A3: set_set_a,B3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A3 @ B3 ) @ A3 ) ).
% inf.cobounded1
thf(fact_966_inf_Ocobounded1,axiom,
! [A3: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ A3 ) ).
% inf.cobounded1
thf(fact_967_inf_Ocobounded1,axiom,
! [A3: nat,B3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B3 ) @ A3 ) ).
% inf.cobounded1
thf(fact_968_inf_Ocobounded1,axiom,
! [A3: int,B3: int] : ( ord_less_eq_int @ ( inf_inf_int @ A3 @ B3 ) @ A3 ) ).
% inf.cobounded1
thf(fact_969_inf_Ocobounded2,axiom,
! [A3: set_set_b,B3: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A3 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_970_inf_Ocobounded2,axiom,
! [A3: set_set_a,B3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A3 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_971_inf_Ocobounded2,axiom,
! [A3: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_972_inf_Ocobounded2,axiom,
! [A3: nat,B3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_973_inf_Ocobounded2,axiom,
! [A3: int,B3: int] : ( ord_less_eq_int @ ( inf_inf_int @ A3 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_974_inf_Oabsorb__iff1,axiom,
( ord_le3795704787696855135_set_b
= ( ^ [A5: set_set_b,B5: set_set_b] :
( ( inf_inf_set_set_b @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_975_inf_Oabsorb__iff1,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( inf_inf_set_set_a @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_976_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( inf_inf_set_a @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_977_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( inf_inf_nat @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_978_inf_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B5: int] :
( ( inf_inf_int @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_979_inf_Oabsorb__iff2,axiom,
( ord_le3795704787696855135_set_b
= ( ^ [B5: set_set_b,A5: set_set_b] :
( ( inf_inf_set_set_b @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_980_inf_Oabsorb__iff2,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B5: set_set_a,A5: set_set_a] :
( ( inf_inf_set_set_a @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_981_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B5: set_a,A5: set_a] :
( ( inf_inf_set_a @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_982_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( inf_inf_nat @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_983_inf_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [B5: int,A5: int] :
( ( inf_inf_int @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_984_inf_OcoboundedI1,axiom,
! [A3: set_set_b,C2: set_set_b,B3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A3 @ C2 )
=> ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A3 @ B3 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_985_inf_OcoboundedI1,axiom,
! [A3: set_set_a,C2: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ C2 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A3 @ B3 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_986_inf_OcoboundedI1,axiom,
! [A3: set_a,C2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_987_inf_OcoboundedI1,axiom,
! [A3: nat,C2: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B3 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_988_inf_OcoboundedI1,axiom,
! [A3: int,C2: int,B3: int] :
( ( ord_less_eq_int @ A3 @ C2 )
=> ( ord_less_eq_int @ ( inf_inf_int @ A3 @ B3 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_989_inf_OcoboundedI2,axiom,
! [B3: set_set_b,C2: set_set_b,A3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B3 @ C2 )
=> ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A3 @ B3 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_990_inf_OcoboundedI2,axiom,
! [B3: set_set_a,C2: set_set_a,A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ C2 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A3 @ B3 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_991_inf_OcoboundedI2,axiom,
! [B3: set_a,C2: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_992_inf_OcoboundedI2,axiom,
! [B3: nat,C2: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B3 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_993_inf_OcoboundedI2,axiom,
! [B3: int,C2: int,A3: int] :
( ( ord_less_eq_int @ B3 @ C2 )
=> ( ord_less_eq_int @ ( inf_inf_int @ A3 @ B3 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_994_measure__exclude,axiom,
! [A: set_a,B: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ( ( sigma_measure_a2 @ m @ A )
= ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) ) )
=> ( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
=> ( ( sigma_measure_a2 @ m @ B )
= zero_zero_real ) ) ) ) ) ).
% measure_exclude
thf(fact_995_prob__compl,axiom,
! [A: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ m ) )
=> ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ ( sigma_space_a @ m ) @ A ) )
= ( minus_minus_real @ one_one_real @ ( sigma_measure_a2 @ m @ A ) ) ) ) ).
% prob_compl
thf(fact_996_finite__measure__Diff,axiom,
! [A: set_a,B: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ A @ B ) )
= ( minus_minus_real @ ( sigma_measure_a2 @ m @ A ) @ ( sigma_measure_a2 @ m @ B ) ) ) ) ) ) ).
% finite_measure_Diff
thf(fact_997_finite__measure__Diff_H,axiom,
! [A: set_a,B: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ A @ B ) )
= ( minus_minus_real @ ( sigma_measure_a2 @ m @ A ) @ ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ A @ B ) ) ) ) ) ) ).
% finite_measure_Diff'
thf(fact_998_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_999_boolean__algebra_Oconj__zero__right,axiom,
! [X5: set_a] :
( ( inf_inf_set_a @ X5 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_1000_boolean__algebra_Oconj__zero__left,axiom,
! [X5: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X5 )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_1001_le__zero__eq,axiom,
! [N4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ N4 @ zero_z7100319975126383169nnreal )
= ( N4 = zero_z7100319975126383169nnreal ) ) ).
% le_zero_eq
thf(fact_1002_le__zero__eq,axiom,
! [N4: nat] :
( ( ord_less_eq_nat @ N4 @ zero_zero_nat )
= ( N4 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1003_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A3: real,X5: real] :
( ( ( times_times_real @ A3 @ X5 )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( X5 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_1004_vector__space__over__itself_Oscale__zero__left,axiom,
! [X5: real] :
( ( times_times_real @ zero_zero_real @ X5 )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_1005_vector__space__over__itself_Oscale__zero__right,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_1006_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A3: real,X5: real,Y5: real] :
( ( ( times_times_real @ A3 @ X5 )
= ( times_times_real @ A3 @ Y5 ) )
= ( ( X5 = Y5 )
| ( A3 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_1007_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A3: real,X5: real,B3: real] :
( ( ( times_times_real @ A3 @ X5 )
= ( times_times_real @ B3 @ X5 ) )
= ( ( A3 = B3 )
| ( X5 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_1008_space__bot,axiom,
( ( sigma_space_a @ bot_bo2108912051383640591sure_a )
= bot_bot_set_a ) ).
% space_bot
thf(fact_1009_diff__ge__0__iff__ge,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A3 @ B3 ) )
= ( ord_less_eq_real @ B3 @ A3 ) ) ).
% diff_ge_0_iff_ge
thf(fact_1010_diff__ge__0__iff__ge,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A3 @ B3 ) )
= ( ord_less_eq_int @ B3 @ A3 ) ) ).
% diff_ge_0_iff_ge
thf(fact_1011_measure__empty,axiom,
! [M2: sigma_measure_a] :
( ( sigma_measure_a2 @ M2 @ bot_bot_set_a )
= zero_zero_real ) ).
% measure_empty
thf(fact_1012_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_1013_zero__le,axiom,
! [X5: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ X5 ) ).
% zero_le
thf(fact_1014_zero__le,axiom,
! [X5: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X5 ) ).
% zero_le
thf(fact_1015_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X5: real,A3: real,B3: real] :
( ( X5 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ X5 )
= ( times_times_real @ B3 @ X5 ) )
=> ( A3 = B3 ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_1016_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A3: real,X5: real,Y5: real] :
( ( A3 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ X5 )
= ( times_times_real @ A3 @ Y5 ) )
=> ( X5 = Y5 ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_1017_space__empty__eq__bot,axiom,
! [A3: sigma_measure_a] :
( ( ( sigma_space_a @ A3 )
= bot_bot_set_a )
= ( A3 = bot_bo2108912051383640591sure_a ) ) ).
% space_empty_eq_bot
thf(fact_1018_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B5: real] : ( ord_less_eq_real @ ( minus_minus_real @ A5 @ B5 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_1019_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B5: int] : ( ord_less_eq_int @ ( minus_minus_int @ A5 @ B5 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_1020_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_1021_measure__nonneg,axiom,
! [M2: sigma_measure_a,A: set_a] : ( ord_less_eq_real @ zero_zero_real @ ( sigma_measure_a2 @ M2 @ A ) ) ).
% measure_nonneg
thf(fact_1022_measure__notin__sets,axiom,
! [A: set_b,M2: sigma_measure_b] :
( ~ ( member_set_b @ A @ ( sigma_sets_b @ M2 ) )
=> ( ( sigma_measure_b2 @ M2 @ A )
= zero_zero_real ) ) ).
% measure_notin_sets
thf(fact_1023_measure__notin__sets,axiom,
! [A: set_a,M2: sigma_measure_a] :
( ~ ( member_set_a @ A @ ( sigma_sets_a @ M2 ) )
=> ( ( sigma_measure_a2 @ M2 @ A )
= zero_zero_real ) ) ).
% measure_notin_sets
thf(fact_1024_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_a,K2: set_a,A3: set_a,B3: set_a] :
( ( A
= ( inf_inf_set_a @ K2 @ A3 ) )
=> ( ( inf_inf_set_a @ A @ B3 )
= ( inf_inf_set_a @ K2 @ ( inf_inf_set_a @ A3 @ B3 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_1025_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_a,K2: set_a,B3: set_a,A3: set_a] :
( ( B
= ( inf_inf_set_a @ K2 @ B3 ) )
=> ( ( inf_inf_set_a @ A3 @ B )
= ( inf_inf_set_a @ K2 @ ( inf_inf_set_a @ A3 @ B3 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_1026_measure__diff__le__measure__setdiff,axiom,
! [S: set_a,M2: sigma_measure_a,T: set_a] :
( ( member_set_a @ S @ ( measur3645360004775918570able_a @ M2 ) )
=> ( ( member_set_a @ T @ ( measur3645360004775918570able_a @ M2 ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ ( sigma_measure_a2 @ M2 @ S ) @ ( sigma_measure_a2 @ M2 @ T ) ) @ ( sigma_measure_a2 @ M2 @ ( minus_minus_set_a @ S @ T ) ) ) ) ) ).
% measure_diff_le_measure_setdiff
thf(fact_1027_measurable__measure__Diff,axiom,
! [A: set_b,M2: sigma_measure_b,B: set_b] :
( ( member_set_b @ A @ ( measur3645360004775918571able_b @ M2 ) )
=> ( ( member_set_b @ B @ ( sigma_sets_b @ M2 ) )
=> ( ( ord_less_eq_set_b @ B @ A )
=> ( ( sigma_measure_b2 @ M2 @ ( minus_minus_set_b @ A @ B ) )
= ( minus_minus_real @ ( sigma_measure_b2 @ M2 @ A ) @ ( sigma_measure_b2 @ M2 @ B ) ) ) ) ) ) ).
% measurable_measure_Diff
thf(fact_1028_measurable__measure__Diff,axiom,
! [A: set_set_b,M2: sigma_measure_set_b,B: set_set_b] :
( ( member_set_set_b @ A @ ( measur7460903249514972363_set_b @ M2 ) )
=> ( ( member_set_set_b @ B @ ( sigma_sets_set_b @ M2 ) )
=> ( ( ord_le3795704787696855135_set_b @ B @ A )
=> ( ( sigma_measure_set_b2 @ M2 @ ( minus_5807331545291222566_set_b @ A @ B ) )
= ( minus_minus_real @ ( sigma_measure_set_b2 @ M2 @ A ) @ ( sigma_measure_set_b2 @ M2 @ B ) ) ) ) ) ) ).
% measurable_measure_Diff
thf(fact_1029_measurable__measure__Diff,axiom,
! [A: set_set_a,M2: sigma_measure_set_a,B: set_set_a] :
( ( member_set_set_a @ A @ ( measur7460903245211743562_set_a @ M2 ) )
=> ( ( member_set_set_a @ B @ ( sigma_sets_set_a @ M2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( sigma_measure_set_a2 @ M2 @ ( minus_5736297505244876581_set_a @ A @ B ) )
= ( minus_minus_real @ ( sigma_measure_set_a2 @ M2 @ A ) @ ( sigma_measure_set_a2 @ M2 @ B ) ) ) ) ) ) ).
% measurable_measure_Diff
thf(fact_1030_measurable__measure__Diff,axiom,
! [A: set_a,M2: sigma_measure_a,B: set_a] :
( ( member_set_a @ A @ ( measur3645360004775918570able_a @ M2 ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M2 ) )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( ( sigma_measure_a2 @ M2 @ ( minus_minus_set_a @ A @ B ) )
= ( minus_minus_real @ ( sigma_measure_a2 @ M2 @ A ) @ ( sigma_measure_a2 @ M2 @ B ) ) ) ) ) ) ).
% measurable_measure_Diff
thf(fact_1031_finite__measure_Ofinite__measure__Diff,axiom,
! [M2: sigma_measure_b,A: set_b,B: set_b] :
( ( measur930452917991658467sure_b @ M2 )
=> ( ( member_set_b @ A @ ( sigma_sets_b @ M2 ) )
=> ( ( member_set_b @ B @ ( sigma_sets_b @ M2 ) )
=> ( ( ord_less_eq_set_b @ B @ A )
=> ( ( sigma_measure_b2 @ M2 @ ( minus_minus_set_b @ A @ B ) )
= ( minus_minus_real @ ( sigma_measure_b2 @ M2 @ A ) @ ( sigma_measure_b2 @ M2 @ B ) ) ) ) ) ) ) ).
% finite_measure.finite_measure_Diff
thf(fact_1032_finite__measure_Ofinite__measure__Diff,axiom,
! [M2: sigma_measure_set_b,A: set_set_b,B: set_set_b] :
( ( measur2212693997687831747_set_b @ M2 )
=> ( ( member_set_set_b @ A @ ( sigma_sets_set_b @ M2 ) )
=> ( ( member_set_set_b @ B @ ( sigma_sets_set_b @ M2 ) )
=> ( ( ord_le3795704787696855135_set_b @ B @ A )
=> ( ( sigma_measure_set_b2 @ M2 @ ( minus_5807331545291222566_set_b @ A @ B ) )
= ( minus_minus_real @ ( sigma_measure_set_b2 @ M2 @ A ) @ ( sigma_measure_set_b2 @ M2 @ B ) ) ) ) ) ) ) ).
% finite_measure.finite_measure_Diff
thf(fact_1033_finite__measure_Ofinite__measure__Diff,axiom,
! [M2: sigma_measure_set_a,A: set_set_a,B: set_set_a] :
( ( measur2212693993384602946_set_a @ M2 )
=> ( ( member_set_set_a @ A @ ( sigma_sets_set_a @ M2 ) )
=> ( ( member_set_set_a @ B @ ( sigma_sets_set_a @ M2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( sigma_measure_set_a2 @ M2 @ ( minus_5736297505244876581_set_a @ A @ B ) )
= ( minus_minus_real @ ( sigma_measure_set_a2 @ M2 @ A ) @ ( sigma_measure_set_a2 @ M2 @ B ) ) ) ) ) ) ) ).
% finite_measure.finite_measure_Diff
thf(fact_1034_finite__measure_Ofinite__measure__Diff,axiom,
! [M2: sigma_measure_a,A: set_a,B: set_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M2 ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M2 ) )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( ( sigma_measure_a2 @ M2 @ ( minus_minus_set_a @ A @ B ) )
= ( minus_minus_real @ ( sigma_measure_a2 @ M2 @ A ) @ ( sigma_measure_a2 @ M2 @ B ) ) ) ) ) ) ) ).
% finite_measure.finite_measure_Diff
thf(fact_1035_finite__measure_Ofinite__measure__Diff_H,axiom,
! [M2: sigma_measure_b,A: set_b,B: set_b] :
( ( measur930452917991658467sure_b @ M2 )
=> ( ( member_set_b @ A @ ( sigma_sets_b @ M2 ) )
=> ( ( member_set_b @ B @ ( sigma_sets_b @ M2 ) )
=> ( ( sigma_measure_b2 @ M2 @ ( minus_minus_set_b @ A @ B ) )
= ( minus_minus_real @ ( sigma_measure_b2 @ M2 @ A ) @ ( sigma_measure_b2 @ M2 @ ( inf_inf_set_b @ A @ B ) ) ) ) ) ) ) ).
% finite_measure.finite_measure_Diff'
thf(fact_1036_finite__measure_Ofinite__measure__Diff_H,axiom,
! [M2: sigma_measure_a,A: set_a,B: set_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M2 ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M2 ) )
=> ( ( sigma_measure_a2 @ M2 @ ( minus_minus_set_a @ A @ B ) )
= ( minus_minus_real @ ( sigma_measure_a2 @ M2 @ A ) @ ( sigma_measure_a2 @ M2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ) ) ) ).
% finite_measure.finite_measure_Diff'
thf(fact_1037_finite__measure_Ofinite__measure__compl,axiom,
! [M2: sigma_measure_b,S: set_b] :
( ( measur930452917991658467sure_b @ M2 )
=> ( ( member_set_b @ S @ ( sigma_sets_b @ M2 ) )
=> ( ( sigma_measure_b2 @ M2 @ ( minus_minus_set_b @ ( sigma_space_b @ M2 ) @ S ) )
= ( minus_minus_real @ ( sigma_measure_b2 @ M2 @ ( sigma_space_b @ M2 ) ) @ ( sigma_measure_b2 @ M2 @ S ) ) ) ) ) ).
% finite_measure.finite_measure_compl
thf(fact_1038_finite__measure_Ofinite__measure__compl,axiom,
! [M2: sigma_measure_a,S: set_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ( member_set_a @ S @ ( sigma_sets_a @ M2 ) )
=> ( ( sigma_measure_a2 @ M2 @ ( minus_minus_set_a @ ( sigma_space_a @ M2 ) @ S ) )
= ( minus_minus_real @ ( sigma_measure_a2 @ M2 @ ( sigma_space_a @ M2 ) ) @ ( sigma_measure_a2 @ M2 @ S ) ) ) ) ) ).
% finite_measure.finite_measure_compl
thf(fact_1039_prob__space_Oprob__compl,axiom,
! [M2: sigma_measure_b,A: set_b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( member_set_b @ A @ ( sigma_sets_b @ M2 ) )
=> ( ( sigma_measure_b2 @ M2 @ ( minus_minus_set_b @ ( sigma_space_b @ M2 ) @ A ) )
= ( minus_minus_real @ one_one_real @ ( sigma_measure_b2 @ M2 @ A ) ) ) ) ) ).
% prob_space.prob_compl
thf(fact_1040_prob__space_Oprob__compl,axiom,
! [M2: sigma_measure_a,A: set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M2 ) )
=> ( ( sigma_measure_a2 @ M2 @ ( minus_minus_set_a @ ( sigma_space_a @ M2 ) @ A ) )
= ( minus_minus_real @ one_one_real @ ( sigma_measure_a2 @ M2 @ A ) ) ) ) ) ).
% prob_space.prob_compl
thf(fact_1041_finite__measure_Omeasure__exclude,axiom,
! [M2: sigma_measure_b,A: set_b,B: set_b] :
( ( measur930452917991658467sure_b @ M2 )
=> ( ( member_set_b @ A @ ( sigma_sets_b @ M2 ) )
=> ( ( member_set_b @ B @ ( sigma_sets_b @ M2 ) )
=> ( ( ( sigma_measure_b2 @ M2 @ A )
= ( sigma_measure_b2 @ M2 @ ( sigma_space_b @ M2 ) ) )
=> ( ( ( inf_inf_set_b @ A @ B )
= bot_bot_set_b )
=> ( ( sigma_measure_b2 @ M2 @ B )
= zero_zero_real ) ) ) ) ) ) ).
% finite_measure.measure_exclude
thf(fact_1042_finite__measure_Omeasure__exclude,axiom,
! [M2: sigma_measure_a,A: set_a,B: set_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M2 ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M2 ) )
=> ( ( ( sigma_measure_a2 @ M2 @ A )
= ( sigma_measure_a2 @ M2 @ ( sigma_space_a @ M2 ) ) )
=> ( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
=> ( ( sigma_measure_a2 @ M2 @ B )
= zero_zero_real ) ) ) ) ) ) ).
% finite_measure.measure_exclude
thf(fact_1043_diff__shunt__var,axiom,
! [X5: set_set_b,Y5: set_set_b] :
( ( ( minus_5807331545291222566_set_b @ X5 @ Y5 )
= bot_bot_set_set_b )
= ( ord_le3795704787696855135_set_b @ X5 @ Y5 ) ) ).
% diff_shunt_var
thf(fact_1044_diff__shunt__var,axiom,
! [X5: set_set_a,Y5: set_set_a] :
( ( ( minus_5736297505244876581_set_a @ X5 @ Y5 )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ X5 @ Y5 ) ) ).
% diff_shunt_var
thf(fact_1045_diff__shunt__var,axiom,
! [X5: set_a,Y5: set_a] :
( ( ( minus_minus_set_a @ X5 @ Y5 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X5 @ Y5 ) ) ).
% diff_shunt_var
thf(fact_1046_kolmogorov__0__1__law,axiom,
! [A: nat > set_set_a,X: set_a] :
( ! [I3: nat] : ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ m ) @ ( A @ I3 ) )
=> ( ( indepe6267730027088848354_a_nat @ m @ A @ top_top_set_nat )
=> ( ( member_set_a @ X @ ( indepe7538416700049374166_a_nat @ m @ A ) )
=> ( ( ( sigma_measure_a2 @ m @ X )
= zero_zero_real )
| ( ( sigma_measure_a2 @ m @ X )
= one_one_real ) ) ) ) ) ).
% kolmogorov_0_1_law
thf(fact_1047_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_1048_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_1049_mult__cancel__left1,axiom,
! [C2: real,B3: real] :
( ( C2
= ( times_times_real @ C2 @ B3 ) )
= ( ( C2 = zero_zero_real )
| ( B3 = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_1050_mult__cancel__left1,axiom,
! [C2: int,B3: int] :
( ( C2
= ( times_times_int @ C2 @ B3 ) )
= ( ( C2 = zero_zero_int )
| ( B3 = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_1051_mult__cancel__left2,axiom,
! [C2: real,A3: real] :
( ( ( times_times_real @ C2 @ A3 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A3 = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_1052_mult__cancel__left2,axiom,
! [C2: int,A3: int] :
( ( ( times_times_int @ C2 @ A3 )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A3 = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_1053_mult__cancel__right1,axiom,
! [C2: real,B3: real] :
( ( C2
= ( times_times_real @ B3 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( B3 = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_1054_mult__cancel__right1,axiom,
! [C2: int,B3: int] :
( ( C2
= ( times_times_int @ B3 @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( B3 = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_1055_mult__cancel__right2,axiom,
! [A3: real,C2: real] :
( ( ( times_times_real @ A3 @ C2 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A3 = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_1056_mult__cancel__right2,axiom,
! [A3: int,C2: int] :
( ( ( times_times_int @ A3 @ C2 )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A3 = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_1057_UNIV__I,axiom,
! [X5: a > b] : ( member_a_b @ X5 @ top_top_set_a_b ) ).
% UNIV_I
thf(fact_1058_UNIV__I,axiom,
! [X5: c] : ( member_c @ X5 @ top_top_set_c ) ).
% UNIV_I
thf(fact_1059_UNIV__I,axiom,
! [X5: set_a] : ( member_set_a @ X5 @ top_top_set_set_a ) ).
% UNIV_I
thf(fact_1060_UNIV__I,axiom,
! [X5: real] : ( member_real @ X5 @ top_top_set_real ) ).
% UNIV_I
thf(fact_1061_UNIV__I,axiom,
! [X5: nat] : ( member_nat @ X5 @ top_top_set_nat ) ).
% UNIV_I
thf(fact_1062_UNIV__I,axiom,
! [X5: $o] : ( member_o @ X5 @ top_top_set_o ) ).
% UNIV_I
thf(fact_1063_mult__zero__left,axiom,
! [A3: real] :
( ( times_times_real @ zero_zero_real @ A3 )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_1064_mult__zero__left,axiom,
! [A3: nat] :
( ( times_times_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1065_mult__zero__left,axiom,
! [A3: int] :
( ( times_times_int @ zero_zero_int @ A3 )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_1066_mult__zero__left,axiom,
! [A3: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ zero_z7100319975126383169nnreal @ A3 )
= zero_z7100319975126383169nnreal ) ).
% mult_zero_left
thf(fact_1067_mult__zero__right,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_1068_mult__zero__right,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1069_mult__zero__right,axiom,
! [A3: int] :
( ( times_times_int @ A3 @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_1070_mult__zero__right,axiom,
! [A3: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ A3 @ zero_z7100319975126383169nnreal )
= zero_z7100319975126383169nnreal ) ).
% mult_zero_right
thf(fact_1071_mult__eq__0__iff,axiom,
! [A3: real,B3: real] :
( ( ( times_times_real @ A3 @ B3 )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( B3 = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_1072_mult__eq__0__iff,axiom,
! [A3: nat,B3: nat] :
( ( ( times_times_nat @ A3 @ B3 )
= zero_zero_nat )
= ( ( A3 = zero_zero_nat )
| ( B3 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_1073_mult__eq__0__iff,axiom,
! [A3: int,B3: int] :
( ( ( times_times_int @ A3 @ B3 )
= zero_zero_int )
= ( ( A3 = zero_zero_int )
| ( B3 = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_1074_mult__eq__0__iff,axiom,
! [A3: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ( times_1893300245718287421nnreal @ A3 @ B3 )
= zero_z7100319975126383169nnreal )
= ( ( A3 = zero_z7100319975126383169nnreal )
| ( B3 = zero_z7100319975126383169nnreal ) ) ) ).
% mult_eq_0_iff
thf(fact_1075_mult__cancel__left,axiom,
! [C2: real,A3: real,B3: real] :
( ( ( times_times_real @ C2 @ A3 )
= ( times_times_real @ C2 @ B3 ) )
= ( ( C2 = zero_zero_real )
| ( A3 = B3 ) ) ) ).
% mult_cancel_left
thf(fact_1076_mult__cancel__left,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( ( times_times_nat @ C2 @ A3 )
= ( times_times_nat @ C2 @ B3 ) )
= ( ( C2 = zero_zero_nat )
| ( A3 = B3 ) ) ) ).
% mult_cancel_left
thf(fact_1077_mult__cancel__left,axiom,
! [C2: int,A3: int,B3: int] :
( ( ( times_times_int @ C2 @ A3 )
= ( times_times_int @ C2 @ B3 ) )
= ( ( C2 = zero_zero_int )
| ( A3 = B3 ) ) ) ).
% mult_cancel_left
thf(fact_1078_mult__cancel__right,axiom,
! [A3: real,C2: real,B3: real] :
( ( ( times_times_real @ A3 @ C2 )
= ( times_times_real @ B3 @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A3 = B3 ) ) ) ).
% mult_cancel_right
thf(fact_1079_mult__cancel__right,axiom,
! [A3: nat,C2: nat,B3: nat] :
( ( ( times_times_nat @ A3 @ C2 )
= ( times_times_nat @ B3 @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A3 = B3 ) ) ) ).
% mult_cancel_right
thf(fact_1080_mult__cancel__right,axiom,
! [A3: int,C2: int,B3: int] :
( ( ( times_times_int @ A3 @ C2 )
= ( times_times_int @ B3 @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( A3 = B3 ) ) ) ).
% mult_cancel_right
thf(fact_1081_inf__top_Oright__neutral,axiom,
! [A3: set_a] :
( ( inf_inf_set_a @ A3 @ top_top_set_a )
= A3 ) ).
% inf_top.right_neutral
thf(fact_1082_inf__top_Oright__neutral,axiom,
! [A3: set_nat] :
( ( inf_inf_set_nat @ A3 @ top_top_set_nat )
= A3 ) ).
% inf_top.right_neutral
thf(fact_1083_inf__top_Oright__neutral,axiom,
! [A3: set_o] :
( ( inf_inf_set_o @ A3 @ top_top_set_o )
= A3 ) ).
% inf_top.right_neutral
thf(fact_1084_inf__top_Oright__neutral,axiom,
! [A3: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ A3 @ top_to1496364449551166952nnreal )
= A3 ) ).
% inf_top.right_neutral
thf(fact_1085_inf__top_Oneutr__eq__iff,axiom,
! [A3: set_a,B3: set_a] :
( ( top_top_set_a
= ( inf_inf_set_a @ A3 @ B3 ) )
= ( ( A3 = top_top_set_a )
& ( B3 = top_top_set_a ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1086_inf__top_Oneutr__eq__iff,axiom,
! [A3: set_nat,B3: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ A3 @ B3 ) )
= ( ( A3 = top_top_set_nat )
& ( B3 = top_top_set_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1087_inf__top_Oneutr__eq__iff,axiom,
! [A3: set_o,B3: set_o] :
( ( top_top_set_o
= ( inf_inf_set_o @ A3 @ B3 ) )
= ( ( A3 = top_top_set_o )
& ( B3 = top_top_set_o ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1088_inf__top_Oneutr__eq__iff,axiom,
! [A3: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( top_to1496364449551166952nnreal
= ( inf_in7439215052339218890nnreal @ A3 @ B3 ) )
= ( ( A3 = top_to1496364449551166952nnreal )
& ( B3 = top_to1496364449551166952nnreal ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1089_inf__top_Oleft__neutral,axiom,
! [A3: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ A3 )
= A3 ) ).
% inf_top.left_neutral
thf(fact_1090_inf__top_Oleft__neutral,axiom,
! [A3: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ A3 )
= A3 ) ).
% inf_top.left_neutral
thf(fact_1091_inf__top_Oleft__neutral,axiom,
! [A3: set_o] :
( ( inf_inf_set_o @ top_top_set_o @ A3 )
= A3 ) ).
% inf_top.left_neutral
thf(fact_1092_inf__top_Oleft__neutral,axiom,
! [A3: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ top_to1496364449551166952nnreal @ A3 )
= A3 ) ).
% inf_top.left_neutral
thf(fact_1093_inf__top_Oeq__neutr__iff,axiom,
! [A3: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A3 @ B3 )
= top_top_set_a )
= ( ( A3 = top_top_set_a )
& ( B3 = top_top_set_a ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1094_inf__top_Oeq__neutr__iff,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ( inf_inf_set_nat @ A3 @ B3 )
= top_top_set_nat )
= ( ( A3 = top_top_set_nat )
& ( B3 = top_top_set_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1095_inf__top_Oeq__neutr__iff,axiom,
! [A3: set_o,B3: set_o] :
( ( ( inf_inf_set_o @ A3 @ B3 )
= top_top_set_o )
= ( ( A3 = top_top_set_o )
& ( B3 = top_top_set_o ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1096_inf__top_Oeq__neutr__iff,axiom,
! [A3: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ( inf_in7439215052339218890nnreal @ A3 @ B3 )
= top_to1496364449551166952nnreal )
= ( ( A3 = top_to1496364449551166952nnreal )
& ( B3 = top_to1496364449551166952nnreal ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1097_top__eq__inf__iff,axiom,
! [X5: set_a,Y5: set_a] :
( ( top_top_set_a
= ( inf_inf_set_a @ X5 @ Y5 ) )
= ( ( X5 = top_top_set_a )
& ( Y5 = top_top_set_a ) ) ) ).
% top_eq_inf_iff
thf(fact_1098_top__eq__inf__iff,axiom,
! [X5: set_nat,Y5: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ X5 @ Y5 ) )
= ( ( X5 = top_top_set_nat )
& ( Y5 = top_top_set_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_1099_top__eq__inf__iff,axiom,
! [X5: set_o,Y5: set_o] :
( ( top_top_set_o
= ( inf_inf_set_o @ X5 @ Y5 ) )
= ( ( X5 = top_top_set_o )
& ( Y5 = top_top_set_o ) ) ) ).
% top_eq_inf_iff
thf(fact_1100_top__eq__inf__iff,axiom,
! [X5: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ( top_to1496364449551166952nnreal
= ( inf_in7439215052339218890nnreal @ X5 @ Y5 ) )
= ( ( X5 = top_to1496364449551166952nnreal )
& ( Y5 = top_to1496364449551166952nnreal ) ) ) ).
% top_eq_inf_iff
thf(fact_1101_inf__eq__top__iff,axiom,
! [X5: set_a,Y5: set_a] :
( ( ( inf_inf_set_a @ X5 @ Y5 )
= top_top_set_a )
= ( ( X5 = top_top_set_a )
& ( Y5 = top_top_set_a ) ) ) ).
% inf_eq_top_iff
thf(fact_1102_inf__eq__top__iff,axiom,
! [X5: set_nat,Y5: set_nat] :
( ( ( inf_inf_set_nat @ X5 @ Y5 )
= top_top_set_nat )
= ( ( X5 = top_top_set_nat )
& ( Y5 = top_top_set_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_1103_inf__eq__top__iff,axiom,
! [X5: set_o,Y5: set_o] :
( ( ( inf_inf_set_o @ X5 @ Y5 )
= top_top_set_o )
= ( ( X5 = top_top_set_o )
& ( Y5 = top_top_set_o ) ) ) ).
% inf_eq_top_iff
thf(fact_1104_inf__eq__top__iff,axiom,
! [X5: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ( ( inf_in7439215052339218890nnreal @ X5 @ Y5 )
= top_to1496364449551166952nnreal )
= ( ( X5 = top_to1496364449551166952nnreal )
& ( Y5 = top_to1496364449551166952nnreal ) ) ) ).
% inf_eq_top_iff
thf(fact_1105_inf__top__right,axiom,
! [X5: set_a] :
( ( inf_inf_set_a @ X5 @ top_top_set_a )
= X5 ) ).
% inf_top_right
thf(fact_1106_inf__top__right,axiom,
! [X5: set_nat] :
( ( inf_inf_set_nat @ X5 @ top_top_set_nat )
= X5 ) ).
% inf_top_right
thf(fact_1107_inf__top__right,axiom,
! [X5: set_o] :
( ( inf_inf_set_o @ X5 @ top_top_set_o )
= X5 ) ).
% inf_top_right
thf(fact_1108_inf__top__right,axiom,
! [X5: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ X5 @ top_to1496364449551166952nnreal )
= X5 ) ).
% inf_top_right
thf(fact_1109_inf__top__left,axiom,
! [X5: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ X5 )
= X5 ) ).
% inf_top_left
thf(fact_1110_inf__top__left,axiom,
! [X5: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ X5 )
= X5 ) ).
% inf_top_left
thf(fact_1111_inf__top__left,axiom,
! [X5: set_o] :
( ( inf_inf_set_o @ top_top_set_o @ X5 )
= X5 ) ).
% inf_top_left
thf(fact_1112_inf__top__left,axiom,
! [X5: extend8495563244428889912nnreal] :
( ( inf_in7439215052339218890nnreal @ top_to1496364449551166952nnreal @ X5 )
= X5 ) ).
% inf_top_left
thf(fact_1113_Int__UNIV,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= top_top_set_a )
= ( ( A = top_top_set_a )
& ( B = top_top_set_a ) ) ) ).
% Int_UNIV
thf(fact_1114_Int__UNIV,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= top_top_set_nat )
= ( ( A = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_1115_Int__UNIV,axiom,
! [A: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A @ B )
= top_top_set_o )
= ( ( A = top_top_set_o )
& ( B = top_top_set_o ) ) ) ).
% Int_UNIV
thf(fact_1116_Diff__UNIV,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Diff_UNIV
thf(fact_1117_Diff__UNIV,axiom,
! [A: set_o] :
( ( minus_minus_set_o @ A @ top_top_set_o )
= bot_bot_set_o ) ).
% Diff_UNIV
thf(fact_1118_Diff__UNIV,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ top_top_set_a )
= bot_bot_set_a ) ).
% Diff_UNIV
thf(fact_1119_top__greatest,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ top_top_set_nat ) ).
% top_greatest
thf(fact_1120_top__greatest,axiom,
! [A3: set_o] : ( ord_less_eq_set_o @ A3 @ top_top_set_o ) ).
% top_greatest
thf(fact_1121_top__greatest,axiom,
! [A3: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ A3 @ top_to1496364449551166952nnreal ) ).
% top_greatest
thf(fact_1122_top__greatest,axiom,
! [A3: set_set_b] : ( ord_le3795704787696855135_set_b @ A3 @ top_top_set_set_b ) ).
% top_greatest
thf(fact_1123_top__greatest,axiom,
! [A3: set_set_a] : ( ord_le3724670747650509150_set_a @ A3 @ top_top_set_set_a ) ).
% top_greatest
thf(fact_1124_top__greatest,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ top_top_set_a ) ).
% top_greatest
thf(fact_1125_top_Oextremum__unique,axiom,
! [A3: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A3 )
= ( A3 = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_1126_top_Oextremum__unique,axiom,
! [A3: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A3 )
= ( A3 = top_top_set_o ) ) ).
% top.extremum_unique
thf(fact_1127_top_Oextremum__unique,axiom,
! [A3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ top_to1496364449551166952nnreal @ A3 )
= ( A3 = top_to1496364449551166952nnreal ) ) ).
% top.extremum_unique
thf(fact_1128_top_Oextremum__unique,axiom,
! [A3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ top_top_set_set_b @ A3 )
= ( A3 = top_top_set_set_b ) ) ).
% top.extremum_unique
thf(fact_1129_top_Oextremum__unique,axiom,
! [A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ top_top_set_set_a @ A3 )
= ( A3 = top_top_set_set_a ) ) ).
% top.extremum_unique
thf(fact_1130_top_Oextremum__unique,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A3 )
= ( A3 = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_1131_top_Oextremum__uniqueI,axiom,
! [A3: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A3 )
=> ( A3 = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_1132_top_Oextremum__uniqueI,axiom,
! [A3: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A3 )
=> ( A3 = top_top_set_o ) ) ).
% top.extremum_uniqueI
thf(fact_1133_top_Oextremum__uniqueI,axiom,
! [A3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ top_to1496364449551166952nnreal @ A3 )
=> ( A3 = top_to1496364449551166952nnreal ) ) ).
% top.extremum_uniqueI
thf(fact_1134_top_Oextremum__uniqueI,axiom,
! [A3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ top_top_set_set_b @ A3 )
=> ( A3 = top_top_set_set_b ) ) ).
% top.extremum_uniqueI
thf(fact_1135_top_Oextremum__uniqueI,axiom,
! [A3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ top_top_set_set_a @ A3 )
=> ( A3 = top_top_set_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_1136_top_Oextremum__uniqueI,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A3 )
=> ( A3 = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_1137_UNIV__eq__I,axiom,
! [A: set_a_b] :
( ! [X3: a > b] : ( member_a_b @ X3 @ A )
=> ( top_top_set_a_b = A ) ) ).
% UNIV_eq_I
thf(fact_1138_UNIV__eq__I,axiom,
! [A: set_c] :
( ! [X3: c] : ( member_c @ X3 @ A )
=> ( top_top_set_c = A ) ) ).
% UNIV_eq_I
thf(fact_1139_UNIV__eq__I,axiom,
! [A: set_set_a] :
( ! [X3: set_a] : ( member_set_a @ X3 @ A )
=> ( top_top_set_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_1140_UNIV__eq__I,axiom,
! [A: set_real] :
( ! [X3: real] : ( member_real @ X3 @ A )
=> ( top_top_set_real = A ) ) ).
% UNIV_eq_I
thf(fact_1141_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X3: nat] : ( member_nat @ X3 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_1142_UNIV__eq__I,axiom,
! [A: set_o] :
( ! [X3: $o] : ( member_o @ X3 @ A )
=> ( top_top_set_o = A ) ) ).
% UNIV_eq_I
thf(fact_1143_UNIV__witness,axiom,
? [X3: a > b] : ( member_a_b @ X3 @ top_top_set_a_b ) ).
% UNIV_witness
thf(fact_1144_UNIV__witness,axiom,
? [X3: c] : ( member_c @ X3 @ top_top_set_c ) ).
% UNIV_witness
thf(fact_1145_UNIV__witness,axiom,
? [X3: set_a] : ( member_set_a @ X3 @ top_top_set_set_a ) ).
% UNIV_witness
thf(fact_1146_UNIV__witness,axiom,
? [X3: real] : ( member_real @ X3 @ top_top_set_real ) ).
% UNIV_witness
thf(fact_1147_UNIV__witness,axiom,
? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_1148_UNIV__witness,axiom,
? [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% UNIV_witness
thf(fact_1149_boolean__algebra_Oconj__one__right,axiom,
! [X5: set_a] :
( ( inf_inf_set_a @ X5 @ top_top_set_a )
= X5 ) ).
% boolean_algebra.conj_one_right
thf(fact_1150_boolean__algebra_Oconj__one__right,axiom,
! [X5: set_nat] :
( ( inf_inf_set_nat @ X5 @ top_top_set_nat )
= X5 ) ).
% boolean_algebra.conj_one_right
thf(fact_1151_boolean__algebra_Oconj__one__right,axiom,
! [X5: set_o] :
( ( inf_inf_set_o @ X5 @ top_top_set_o )
= X5 ) ).
% boolean_algebra.conj_one_right
thf(fact_1152_subset__UNIV,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_1153_subset__UNIV,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ A @ top_top_set_o ) ).
% subset_UNIV
thf(fact_1154_subset__UNIV,axiom,
! [A: set_set_b] : ( ord_le3795704787696855135_set_b @ A @ top_top_set_set_b ) ).
% subset_UNIV
thf(fact_1155_subset__UNIV,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ top_top_set_set_a ) ).
% subset_UNIV
thf(fact_1156_subset__UNIV,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% subset_UNIV
thf(fact_1157_empty__not__UNIV,axiom,
bot_bot_set_a != top_top_set_a ).
% empty_not_UNIV
thf(fact_1158_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_1159_empty__not__UNIV,axiom,
bot_bot_set_o != top_top_set_o ).
% empty_not_UNIV
thf(fact_1160_Int__UNIV__left,axiom,
! [B: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ B )
= B ) ).
% Int_UNIV_left
thf(fact_1161_Int__UNIV__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ B )
= B ) ).
% Int_UNIV_left
thf(fact_1162_Int__UNIV__left,axiom,
! [B: set_o] :
( ( inf_inf_set_o @ top_top_set_o @ B )
= B ) ).
% Int_UNIV_left
thf(fact_1163_Int__UNIV__right,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ top_top_set_a )
= A ) ).
% Int_UNIV_right
thf(fact_1164_Int__UNIV__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ top_top_set_nat )
= A ) ).
% Int_UNIV_right
thf(fact_1165_Int__UNIV__right,axiom,
! [A: set_o] :
( ( inf_inf_set_o @ A @ top_top_set_o )
= A ) ).
% Int_UNIV_right
thf(fact_1166_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_1167_le__numeral__extra_I3_J,axiom,
ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ zero_z7100319975126383169nnreal ).
% le_numeral_extra(3)
thf(fact_1168_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1169_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_1170_mult__right__cancel,axiom,
! [C2: real,A3: real,B3: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ C2 )
= ( times_times_real @ B3 @ C2 ) )
= ( A3 = B3 ) ) ) ).
% mult_right_cancel
thf(fact_1171_mult__right__cancel,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A3 @ C2 )
= ( times_times_nat @ B3 @ C2 ) )
= ( A3 = B3 ) ) ) ).
% mult_right_cancel
thf(fact_1172_mult__right__cancel,axiom,
! [C2: int,A3: int,B3: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ A3 @ C2 )
= ( times_times_int @ B3 @ C2 ) )
= ( A3 = B3 ) ) ) ).
% mult_right_cancel
thf(fact_1173_mult__left__cancel,axiom,
! [C2: real,A3: real,B3: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ C2 @ A3 )
= ( times_times_real @ C2 @ B3 ) )
= ( A3 = B3 ) ) ) ).
% mult_left_cancel
thf(fact_1174_mult__left__cancel,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A3 )
= ( times_times_nat @ C2 @ B3 ) )
= ( A3 = B3 ) ) ) ).
% mult_left_cancel
thf(fact_1175_mult__left__cancel,axiom,
! [C2: int,A3: int,B3: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ C2 @ A3 )
= ( times_times_int @ C2 @ B3 ) )
= ( A3 = B3 ) ) ) ).
% mult_left_cancel
thf(fact_1176_no__zero__divisors,axiom,
! [A3: real,B3: real] :
( ( A3 != zero_zero_real )
=> ( ( B3 != zero_zero_real )
=> ( ( times_times_real @ A3 @ B3 )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_1177_no__zero__divisors,axiom,
! [A3: nat,B3: nat] :
( ( A3 != zero_zero_nat )
=> ( ( B3 != zero_zero_nat )
=> ( ( times_times_nat @ A3 @ B3 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_1178_no__zero__divisors,axiom,
! [A3: int,B3: int] :
( ( A3 != zero_zero_int )
=> ( ( B3 != zero_zero_int )
=> ( ( times_times_int @ A3 @ B3 )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_1179_no__zero__divisors,axiom,
! [A3: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( A3 != zero_z7100319975126383169nnreal )
=> ( ( B3 != zero_z7100319975126383169nnreal )
=> ( ( times_1893300245718287421nnreal @ A3 @ B3 )
!= zero_z7100319975126383169nnreal ) ) ) ).
% no_zero_divisors
thf(fact_1180_divisors__zero,axiom,
! [A3: real,B3: real] :
( ( ( times_times_real @ A3 @ B3 )
= zero_zero_real )
=> ( ( A3 = zero_zero_real )
| ( B3 = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_1181_divisors__zero,axiom,
! [A3: nat,B3: nat] :
( ( ( times_times_nat @ A3 @ B3 )
= zero_zero_nat )
=> ( ( A3 = zero_zero_nat )
| ( B3 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_1182_divisors__zero,axiom,
! [A3: int,B3: int] :
( ( ( times_times_int @ A3 @ B3 )
= zero_zero_int )
=> ( ( A3 = zero_zero_int )
| ( B3 = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_1183_divisors__zero,axiom,
! [A3: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ( times_1893300245718287421nnreal @ A3 @ B3 )
= zero_z7100319975126383169nnreal )
=> ( ( A3 = zero_z7100319975126383169nnreal )
| ( B3 = zero_z7100319975126383169nnreal ) ) ) ).
% divisors_zero
thf(fact_1184_mult__not__zero,axiom,
! [A3: real,B3: real] :
( ( ( times_times_real @ A3 @ B3 )
!= zero_zero_real )
=> ( ( A3 != zero_zero_real )
& ( B3 != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_1185_mult__not__zero,axiom,
! [A3: nat,B3: nat] :
( ( ( times_times_nat @ A3 @ B3 )
!= zero_zero_nat )
=> ( ( A3 != zero_zero_nat )
& ( B3 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_1186_mult__not__zero,axiom,
! [A3: int,B3: int] :
( ( ( times_times_int @ A3 @ B3 )
!= zero_zero_int )
=> ( ( A3 != zero_zero_int )
& ( B3 != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_1187_mult__not__zero,axiom,
! [A3: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ( times_1893300245718287421nnreal @ A3 @ B3 )
!= zero_z7100319975126383169nnreal )
=> ( ( A3 != zero_z7100319975126383169nnreal )
& ( B3 != zero_z7100319975126383169nnreal ) ) ) ).
% mult_not_zero
thf(fact_1188_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_1189_indep__set__def,axiom,
! [A: set_set_a,B: set_set_a] :
( ( indepe2041756565122539606_set_a @ m @ A @ B )
= ( indepe7780107833195774214ts_a_o @ m @ ( produc6113963288868236716_set_a @ A @ B ) @ top_top_set_o ) ) ).
% indep_set_def
thf(fact_1190_finite__measure__Union,axiom,
! [A: set_a,B: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
=> ( ( sigma_measure_a2 @ m @ ( sup_sup_set_a @ A @ B ) )
= ( plus_plus_real @ ( sigma_measure_a2 @ m @ A ) @ ( sigma_measure_a2 @ m @ B ) ) ) ) ) ) ).
% finite_measure_Union
thf(fact_1191_finite__measure__Union_H,axiom,
! [A: set_a,B: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ( sigma_measure_a2 @ m @ ( sup_sup_set_a @ A @ B ) )
= ( plus_plus_real @ ( sigma_measure_a2 @ m @ A ) @ ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ B @ A ) ) ) ) ) ) ).
% finite_measure_Union'
thf(fact_1192_finite__measure__subadditive,axiom,
! [A: set_a,B: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ ( sup_sup_set_a @ A @ B ) ) @ ( plus_plus_real @ ( sigma_measure_a2 @ m @ A ) @ ( sigma_measure_a2 @ m @ B ) ) ) ) ) ).
% finite_measure_subadditive
thf(fact_1193_segment__bound__lemma,axiom,
! [B: real,X5: real,Y5: real,U: real] :
( ( ord_less_eq_real @ B @ X5 )
=> ( ( ord_less_eq_real @ B @ Y5 )
=> ( ( 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 @ Y5 ) ) ) ) ) ) ) ).
% segment_bound_lemma
thf(fact_1194_sum__le__prod1,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ( ord_less_eq_real @ B3 @ one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A3 @ B3 ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ A3 @ B3 ) ) ) ) ) ).
% sum_le_prod1
thf(fact_1195_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F2: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X3: nat > real] :
( ( P @ X3 )
=> ( P @ ( F2 @ X3 ) ) )
=> ( ! [X3: nat > real] :
( ( P @ X3 )
=> ! [I3: nat] :
( ( Q @ I3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I3 ) )
& ( ord_less_eq_real @ ( X3 @ I3 ) @ 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 ) @ ( F2 @ X6 @ I4 ) ) )
& ! [X6: nat > real,I4: nat] :
( ( ( P @ X6 )
& ( Q @ I4 )
& ( ( L2 @ X6 @ I4 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F2 @ X6 @ I4 ) @ ( X6 @ I4 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1196_indep__event__def,axiom,
! [A: set_a,B: set_a] :
( ( indepe3567167809233210430vent_a @ m @ A @ B )
= ( indepe3695496658712714478ts_a_o @ m @ ( produc2496386666562076748_set_a @ A @ B ) @ top_top_set_o ) ) ).
% indep_event_def
thf(fact_1197_complete__real,axiom,
! [S: set_real] :
( ? [X6: real] : ( member_real @ X6 @ S )
=> ( ? [Z4: real] :
! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ? [Y6: real] :
( ! [X6: real] :
( ( member_real @ X6 @ S )
=> ( ord_less_eq_real @ X6 @ Y6 ) )
& ! [Z4: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ( ord_less_eq_real @ Y6 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_1198_nat__mult__eq__1__iff,axiom,
! [M4: nat,N4: nat] :
( ( ( times_times_nat @ M4 @ N4 )
= one_one_nat )
= ( ( M4 = one_one_nat )
& ( N4 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1199_mult__is__0,axiom,
! [M4: nat,N4: nat] :
( ( ( times_times_nat @ M4 @ N4 )
= zero_zero_nat )
= ( ( M4 = zero_zero_nat )
| ( N4 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1200_mult__0__right,axiom,
! [M4: nat] :
( ( times_times_nat @ M4 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1201_mult__cancel1,axiom,
! [K2: nat,M4: nat,N4: nat] :
( ( ( times_times_nat @ K2 @ M4 )
= ( times_times_nat @ K2 @ N4 ) )
= ( ( M4 = N4 )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1202_mult__cancel2,axiom,
! [M4: nat,K2: nat,N4: nat] :
( ( ( times_times_nat @ M4 @ K2 )
= ( times_times_nat @ N4 @ K2 ) )
= ( ( M4 = N4 )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1203_nat__1__eq__mult__iff,axiom,
! [M4: nat,N4: nat] :
( ( one_one_nat
= ( times_times_nat @ M4 @ N4 ) )
= ( ( M4 = one_one_nat )
& ( N4 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1204_diff__mult__distrib2,axiom,
! [K2: nat,M4: nat,N4: nat] :
( ( times_times_nat @ K2 @ ( minus_minus_nat @ M4 @ N4 ) )
= ( minus_minus_nat @ ( times_times_nat @ K2 @ M4 ) @ ( times_times_nat @ K2 @ N4 ) ) ) ).
% diff_mult_distrib2
thf(fact_1205_diff__mult__distrib,axiom,
! [M4: nat,N4: nat,K2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M4 @ N4 ) @ K2 )
= ( minus_minus_nat @ ( times_times_nat @ M4 @ K2 ) @ ( times_times_nat @ N4 @ K2 ) ) ) ).
% diff_mult_distrib
thf(fact_1206_add__mult__distrib,axiom,
! [M4: nat,N4: nat,K2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M4 @ N4 ) @ K2 )
= ( plus_plus_nat @ ( times_times_nat @ M4 @ K2 ) @ ( times_times_nat @ N4 @ K2 ) ) ) ).
% add_mult_distrib
thf(fact_1207_add__mult__distrib2,axiom,
! [K2: nat,M4: nat,N4: nat] :
( ( times_times_nat @ K2 @ ( plus_plus_nat @ M4 @ N4 ) )
= ( plus_plus_nat @ ( times_times_nat @ K2 @ M4 ) @ ( times_times_nat @ K2 @ N4 ) ) ) ).
% add_mult_distrib2
thf(fact_1208_mult__0,axiom,
! [N4: nat] :
( ( times_times_nat @ zero_zero_nat @ N4 )
= zero_zero_nat ) ).
% mult_0
thf(fact_1209_le__cube,axiom,
! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ) ).
% le_cube
thf(fact_1210_le__square,axiom,
! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ).
% le_square
thf(fact_1211_mult__le__mono,axiom,
! [I: nat,J2: nat,K2: nat,L3: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ K2 @ L3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J2 @ L3 ) ) ) ) ).
% mult_le_mono
thf(fact_1212_mult__le__mono1,axiom,
! [I: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J2 @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_1213_mult__le__mono2,axiom,
! [I: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_1214_nat__mult__1,axiom,
! [N4: nat] :
( ( times_times_nat @ one_one_nat @ N4 )
= N4 ) ).
% nat_mult_1
thf(fact_1215_nat__mult__1__right,axiom,
! [N4: nat] :
( ( times_times_nat @ N4 @ one_one_nat )
= N4 ) ).
% nat_mult_1_right
thf(fact_1216_mult__eq__self__implies__10,axiom,
! [M4: nat,N4: nat] :
( ( M4
= ( times_times_nat @ M4 @ N4 ) )
=> ( ( N4 = one_one_nat )
| ( M4 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1217_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M5: nat,N5: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N5 @ ( times_times_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N5 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1218_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_1219_not__real__square__gt__zero,axiom,
! [X5: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X5 @ X5 ) ) )
= ( X5 = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1220_nat__eq__add__iff1,axiom,
! [J2: nat,I: nat,U: nat,M4: nat,N4: nat] :
( ( ord_less_eq_nat @ J2 @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M4 )
= ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N4 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J2 ) @ U ) @ M4 )
= N4 ) ) ) ).
% nat_eq_add_iff1
thf(fact_1221_less__one,axiom,
! [N4: nat] :
( ( ord_less_nat @ N4 @ one_one_nat )
= ( N4 = zero_zero_nat ) ) ).
% less_one
thf(fact_1222_nat__mult__less__cancel__disj,axiom,
! [K2: nat,M4: nat,N4: nat] :
( ( ord_less_nat @ ( times_times_nat @ K2 @ M4 ) @ ( times_times_nat @ K2 @ N4 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M4 @ N4 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1223_nat__0__less__mult__iff,axiom,
! [M4: nat,N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M4 @ N4 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M4 )
& ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1224_mult__less__cancel2,axiom,
! [M4: nat,K2: nat,N4: nat] :
( ( ord_less_nat @ ( times_times_nat @ M4 @ K2 ) @ ( times_times_nat @ N4 @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M4 @ N4 ) ) ) ).
% mult_less_cancel2
thf(fact_1225_nat__mult__le__cancel__disj,axiom,
! [K2: nat,M4: nat,N4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M4 ) @ ( times_times_nat @ K2 @ N4 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M4 @ N4 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1226_mult__le__cancel2,axiom,
! [M4: nat,K2: nat,N4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M4 @ K2 ) @ ( times_times_nat @ N4 @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M4 @ N4 ) ) ) ).
% mult_le_cancel2
thf(fact_1227_nat__mult__less__cancel1,axiom,
! [K2: nat,M4: nat,N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_nat @ ( times_times_nat @ K2 @ M4 ) @ ( times_times_nat @ K2 @ N4 ) )
= ( ord_less_nat @ M4 @ N4 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1228_nat__mult__eq__cancel1,axiom,
! [K2: nat,M4: nat,N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ( times_times_nat @ K2 @ M4 )
= ( times_times_nat @ K2 @ N4 ) )
= ( M4 = N4 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1229_nat__mult__le__cancel1,axiom,
! [K2: nat,M4: nat,N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M4 ) @ ( times_times_nat @ K2 @ N4 ) )
= ( ord_less_eq_nat @ M4 @ N4 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1230_mult__less__mono2,axiom,
! [I: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_1231_mult__less__mono1,axiom,
! [I: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J2 @ K2 ) ) ) ) ).
% mult_less_mono1
thf(fact_1232_nat__less__add__iff2,axiom,
! [I: nat,J2: nat,U: nat,M4: nat,N4: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N4 ) )
= ( ord_less_nat @ M4 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I ) @ U ) @ N4 ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1233_nat__less__add__iff1,axiom,
! [J2: nat,I: nat,U: nat,M4: nat,N4: nat] :
( ( ord_less_eq_nat @ J2 @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N4 ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J2 ) @ U ) @ M4 ) @ N4 ) ) ) ).
% nat_less_add_iff1
thf(fact_1234_kuhn__lemma,axiom,
! [P2: nat,N4: nat,Label: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ P2 )
=> ( ! [X3: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N4 )
=> ( ord_less_eq_nat @ ( X3 @ I4 ) @ P2 ) )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ N4 )
=> ( ( ( Label @ X3 @ I3 )
= zero_zero_nat )
| ( ( Label @ X3 @ I3 )
= one_one_nat ) ) ) )
=> ( ! [X3: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N4 )
=> ( ord_less_eq_nat @ ( X3 @ I4 ) @ P2 ) )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ N4 )
=> ( ( ( X3 @ I3 )
= zero_zero_nat )
=> ( ( Label @ X3 @ I3 )
= zero_zero_nat ) ) ) )
=> ( ! [X3: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N4 )
=> ( ord_less_eq_nat @ ( X3 @ I4 ) @ P2 ) )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ N4 )
=> ( ( ( X3 @ I3 )
= P2 )
=> ( ( Label @ X3 @ I3 )
= one_one_nat ) ) ) )
=> ~ ! [Q2: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N4 )
=> ( ord_less_nat @ ( Q2 @ I4 ) @ P2 ) )
=> ~ ! [I4: nat] :
( ( ord_less_nat @ I4 @ N4 )
=> ? [R: nat > nat] :
( ! [J3: nat] :
( ( ord_less_nat @ J3 @ N4 )
=> ( ( ord_less_eq_nat @ ( Q2 @ J3 ) @ ( R @ J3 ) )
& ( ord_less_eq_nat @ ( R @ J3 ) @ ( plus_plus_nat @ ( Q2 @ J3 ) @ one_one_nat ) ) ) )
& ? [S3: nat > nat] :
( ! [J3: nat] :
( ( ord_less_nat @ J3 @ N4 )
=> ( ( ord_less_eq_nat @ ( Q2 @ J3 ) @ ( S3 @ J3 ) )
& ( ord_less_eq_nat @ ( S3 @ J3 ) @ ( plus_plus_nat @ ( Q2 @ J3 ) @ one_one_nat ) ) ) )
& ( ( Label @ R @ I4 )
!= ( Label @ S3 @ I4 ) ) ) ) ) ) ) ) ) ) ).
% kuhn_lemma
thf(fact_1235_nat__mult__eq__cancel__disj,axiom,
! [K2: nat,M4: nat,N4: nat] :
( ( ( times_times_nat @ K2 @ M4 )
= ( times_times_nat @ K2 @ N4 ) )
= ( ( K2 = zero_zero_nat )
| ( M4 = N4 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1236_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J2: nat,K2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ K2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J2 ) @ U ) @ K2 ) ) ).
% left_add_mult_distrib
thf(fact_1237_nat__diff__add__eq2,axiom,
! [I: nat,J2: nat,U: nat,M4: nat,N4: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N4 ) )
= ( minus_minus_nat @ M4 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I ) @ U ) @ N4 ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1238_nat__diff__add__eq1,axiom,
! [J2: nat,I: nat,U: nat,M4: nat,N4: nat] :
( ( ord_less_eq_nat @ J2 @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N4 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J2 ) @ U ) @ M4 ) @ N4 ) ) ) ).
% nat_diff_add_eq1
thf(fact_1239_nat__le__add__iff2,axiom,
! [I: nat,J2: nat,U: nat,M4: nat,N4: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N4 ) )
= ( ord_less_eq_nat @ M4 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I ) @ U ) @ N4 ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1240_nat__le__add__iff1,axiom,
! [J2: nat,I: nat,U: nat,M4: nat,N4: nat] :
( ( ord_less_eq_nat @ J2 @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N4 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J2 ) @ U ) @ M4 ) @ N4 ) ) ) ).
% nat_le_add_iff1
thf(fact_1241_nat__eq__add__iff2,axiom,
! [I: nat,J2: nat,U: nat,M4: nat,N4: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M4 )
= ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N4 ) )
= ( M4
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I ) @ U ) @ N4 ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1242_square__bound__lemma,axiom,
! [X5: real] : ( ord_less_real @ X5 @ ( times_times_real @ ( plus_plus_real @ one_one_real @ X5 ) @ ( plus_plus_real @ one_one_real @ X5 ) ) ) ).
% square_bound_lemma
thf(fact_1243_Bolzano,axiom,
! [A3: real,B3: real,P: real > real > $o] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ! [A4: real,B4: real,C3: real] :
( ( P @ A4 @ B4 )
=> ( ( P @ B4 @ C3 )
=> ( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_eq_real @ B4 @ C3 )
=> ( P @ A4 @ C3 ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A3 @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B3 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [A4: real,B4: real] :
( ( ( ord_less_eq_real @ A4 @ X3 )
& ( ord_less_eq_real @ X3 @ B4 )
& ( ord_less_real @ ( minus_minus_real @ B4 @ A4 ) @ D3 ) )
=> ( P @ A4 @ B4 ) ) ) ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% Bolzano
thf(fact_1244_seq__mono__lemma,axiom,
! [M4: nat,D2: nat > real,E: nat > real] :
( ! [N6: nat] :
( ( ord_less_eq_nat @ M4 @ N6 )
=> ( ord_less_real @ ( D2 @ N6 ) @ ( E @ N6 ) ) )
=> ( ! [N6: nat] :
( ( ord_less_eq_nat @ M4 @ N6 )
=> ( ord_less_eq_real @ ( E @ N6 ) @ ( E @ M4 ) ) )
=> ! [N7: nat] :
( ( ord_less_eq_nat @ M4 @ N7 )
=> ( ord_less_real @ ( D2 @ N7 ) @ ( E @ M4 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_1245_square__continuous,axiom,
! [E: real,X5: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [Y7: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ Y7 @ X5 ) ) @ D4 )
=> ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( times_times_real @ Y7 @ Y7 ) @ ( times_times_real @ X5 @ X5 ) ) ) @ E ) ) ) ) ).
% square_continuous
thf(fact_1246_lemma__interval,axiom,
! [A3: real,X5: real,B3: real] :
( ( ord_less_real @ A3 @ X5 )
=> ( ( ord_less_real @ X5 @ B3 )
=> ? [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 @ A3 @ Y7 )
& ( ord_less_eq_real @ Y7 @ B3 ) ) ) ) ) ) ).
% lemma_interval
thf(fact_1247_sin__bound__lemma,axiom,
! [X5: real,Y5: real,U: real,V: real] :
( ( X5 = Y5 )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X5 @ U ) @ Y5 ) ) @ V ) ) ) ).
% sin_bound_lemma
thf(fact_1248_nat0__intermed__int__val,axiom,
! [N4: nat,F2: nat > int,K2: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N4 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F2 @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F2 @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F2 @ zero_zero_nat ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F2 @ N4 ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N4 )
& ( ( F2 @ I3 )
= K2 ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1249_emeasure__space__1,axiom,
( ( sigma_emeasure_a @ m @ ( sigma_space_a @ m ) )
= one_on2969667320475766781nnreal ) ).
% emeasure_space_1
thf(fact_1250_finite__emeasure__space,axiom,
( ( sigma_emeasure_a @ m @ ( sigma_space_a @ m ) )
!= top_to1496364449551166952nnreal ) ).
% finite_emeasure_space
thf(fact_1251_subprob__emeasure__le__1,axiom,
! [X: set_a] : ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ m @ X ) @ one_on2969667320475766781nnreal ) ).
% subprob_emeasure_le_1
thf(fact_1252_emeasure__le__1,axiom,
! [S: set_a] : ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ m @ S ) @ one_on2969667320475766781nnreal ) ).
% emeasure_le_1
thf(fact_1253_emeasure__ge__1__iff,axiom,
! [A: set_a] :
( ( ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ ( sigma_emeasure_a @ m @ A ) )
= ( ( sigma_emeasure_a @ m @ A )
= one_on2969667320475766781nnreal ) ) ).
% emeasure_ge_1_iff
thf(fact_1254_emeasure__space__le__1,axiom,
ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ m @ ( sigma_space_a @ m ) ) @ one_on2969667320475766781nnreal ).
% emeasure_space_le_1
thf(fact_1255_zle__add1__eq__le,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_1256_zabs__less__one__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
= ( Z2 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1257_emeasure__subprob__space__less__top,axiom,
! [A: set_a] :
( ( sigma_emeasure_a @ m @ A )
!= top_to1496364449551166952nnreal ) ).
% emeasure_subprob_space_less_top
thf(fact_1258_zle__diff1__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_1259_abs__zmult__eq__1,axiom,
! [M4: int,N4: int] :
( ( ( abs_abs_int @ ( times_times_int @ M4 @ N4 ) )
= one_one_int )
=> ( ( abs_abs_int @ M4 )
= one_one_int ) ) ).
% abs_zmult_eq_1
thf(fact_1260_ennreal__approx__unit,axiom,
! [Z2: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
( ! [A4: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ A4 )
=> ( ( ord_le7381754540660121996nnreal @ A4 @ one_on2969667320475766781nnreal )
=> ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ A4 @ Z2 ) @ Y5 ) ) )
=> ( ord_le3935885782089961368nnreal @ Z2 @ Y5 ) ) ).
% ennreal_approx_unit
thf(fact_1261_pos__zmult__eq__1__iff,axiom,
! [M4: int,N4: int] :
( ( ord_less_int @ zero_zero_int @ M4 )
=> ( ( ( times_times_int @ M4 @ N4 )
= one_one_int )
= ( ( M4 = one_one_int )
& ( N4 = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1262_ennreal__one__less__top,axiom,
ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ top_to1496364449551166952nnreal ).
% ennreal_one_less_top
thf(fact_1263_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1264_int__less__induct,axiom,
! [I: int,K2: int,P: int > $o] :
( ( ord_less_int @ I @ K2 )
=> ( ( P @ ( minus_minus_int @ K2 @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ I3 @ K2 )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1265_ennreal__zero__less__one,axiom,
ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ one_on2969667320475766781nnreal ).
% ennreal_zero_less_one
thf(fact_1266_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_1267_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1268_add1__zle__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% add1_zle_eq
thf(fact_1269_zless__imp__add1__zle,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ Z2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_1270_int__gr__induct,axiom,
! [K2: int,I: int,P: int > $o] :
( ( ord_less_int @ K2 @ I )
=> ( ( P @ ( plus_plus_int @ K2 @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K2 @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1271_zless__add1__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W2 @ Z2 )
| ( W2 = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_1272_int__le__induct,axiom,
! [I: int,K2: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K2 )
=> ( ( P @ K2 )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K2 )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1273_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1274_int__ge__induct,axiom,
! [K2: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K2 @ I )
=> ( ( P @ K2 )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K2 @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1275_int__induct,axiom,
! [P: int > $o,K2: int,I: int] :
( ( P @ K2 )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K2 @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K2 )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1276_ennreal__top__neq__one,axiom,
top_to1496364449551166952nnreal != one_on2969667320475766781nnreal ).
% ennreal_top_neq_one
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X5: nat,Y5: nat] :
( ( if_nat @ $false @ X5 @ Y5 )
= Y5 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X5: nat,Y5: nat] :
( ( if_nat @ $true @ X5 @ Y5 )
= X5 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
indepe6139986284700742573ts_b_c @ ( measure_distr_a_b @ m @ n @ f2 ) @ a2 @ i ).
%------------------------------------------------------------------------------