TPTP Problem File: SLH0654^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% 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_00019_000531__18391206_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1891 ( 388 unt; 614 typ; 0 def)
% Number of atoms : 4323 ( 952 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 13694 ( 318 ~; 40 |; 325 &;10952 @)
% ( 0 <=>;2059 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg)
% Number of types : 111 ( 110 usr)
% Number of type conns : 2677 (2677 >; 0 *; 0 +; 0 <<)
% Number of symbols : 507 ( 504 usr; 74 con; 0-5 aty)
% Number of variables : 4060 ( 690 ^;3297 !; 73 ?;4060 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:43:29.530
%------------------------------------------------------------------------------
% Could-be-implicit typings (110)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
set_Pr4334478416066269672t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
set_Su7539578257924484756t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__String__Oliteral_Mt__Nat__Onat_J_J,type,
set_Pr2368984730104727433al_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__String__Oliteral_J_J,type,
set_Pr4449086587592242223iteral: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Product____Type__Ounit_J_J,type,
set_Pr6671815714118170108t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__String__Oliteral_Mt__Nat__Onat_J_J,type,
set_Su2755845247344502325al_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__String__Oliteral_J_J,type,
set_Su4835947104832017115iteral: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_I_Eo_Mt__Product____Type__Ounit_J_J,type,
set_Su4100838662412113488t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__String__Oliteral_M_Eo_J_J,type,
set_Pr5001933180752110795eral_o: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__String__Oliteral_J_J,type,
set_Pr1504714482567607875iteral: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Sum_sum_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__String__Oliteral_M_Eo_J_J,type,
set_Su2489366904224875295eral_o: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_I_Eo_Mt__String__Oliteral_J_J,type,
set_Su8215520242895148183iteral: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Product____Type__Ounit_J_J,type,
set_op3165557761946182707t_unit: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_I_062_Itf__a_Mt__Real__Oreal_J_J,type,
sigma_measure_a_real: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
set_Pr3149072824959771635_nat_o: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
set_Pr2101469702781467981_o_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
set_Ex3793607809372303086nnreal: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_Mt__Real__Oreal_J_Mtf__a_J_J,type,
set_a_real_a: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Product____Type__Ounit_J,type,
sigma_3917854644531351306t_unit: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_Mt__Real__Oreal_J_M_Eo_J_J,type,
set_a_real_o: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_M_Eo_J_J,type,
set_Sum_sum_nat_o: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_I_Eo_Mt__Nat__Onat_J_J,type,
set_Sum_sum_o_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Ounit_J_J,type,
set_set_Product_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__String__Oliteral_J_J,type,
set_option_literal: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__c_Mt__Product____Type__Ounit_J_J,type,
set_c_Product_unit: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Product____Type__Ounit_J_J,type,
set_a_Product_unit: $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_I_062_I_Eo_Mt__Product____Type__Ounit_J_J,type,
set_o_Product_unit: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_I_062_Itf__c_Mtf__d_J_J,type,
sigma_measure_c_d: $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_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
set_Product_prod_o_o: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_It__String__Oliteral_J,type,
sigma_4743690205272893521iteral: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_I_062_Itf__a_M_Eo_J_J,type,
sigma_measure_a_o: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__String__Oliteral_J_J,type,
set_set_literal: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mtf__a_J_J,type,
set_c_d_a: $tType ).
thf(ty_n_t__Filter__Ofilter_I_062_Itf__a_Mt__Real__Oreal_J_J,type,
filter_a_real: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
set_option_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__c_Mt__String__Oliteral_J_J,type,
set_c_literal: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__String__Oliteral_J_J,type,
set_a_literal: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_M_Eo_J_J,type,
set_c_d_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_M_Eo_J_Mtf__a_J_J,type,
set_a_o_a: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_I_Eo_M_Eo_J_J,type,
set_Sum_sum_o_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_Mt__String__Oliteral_J_J,type,
set_o_literal: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
sigma_measure_real: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_M_Eo_J_M_Eo_J_J,type,
set_a_o_o: $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__a_Mt__Real__Oreal_J_J,type,
set_a_real: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Extended____Nonnegative____Real__Oennreal,type,
extend8495563244428889912nnreal: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__c_Mt__Nat__Onat_J_J,type,
set_c_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__b_Mt__Nat__Onat_J_J,type,
set_b_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Nat__Onat_J_J,type,
set_a_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_It__Nat__Onat_Mtf__b_J_J,type,
set_nat_b: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
set_nat_a: $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__Filter__Ofilter_I_062_Itf__c_Mtf__d_J_J,type,
filter_c_d: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
set_Product_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_I_Eo_J_J,type,
set_option_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_Mt__Nat__Onat_J_J,type,
set_o_nat: $tType ).
thf(ty_n_t__Filter__Ofilter_I_062_Itf__a_M_Eo_J_J,type,
filter_a_o: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_Itf__d_J,type,
sigma_measure_d: $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__d_Mtf__d_J_J,type,
set_d_d: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__c_Mtf__d_J_J,type,
set_c_d: $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__Filter__Ofilter_It__Real__Oreal_J,type,
filter_real: $tType ).
thf(ty_n_t__Set__Oset_It__String__Oliteral_J,type,
set_literal: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__c_M_Eo_J_J,type,
set_c_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__b_M_Eo_J_J,type,
set_b_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
set_a_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_Mtf__b_J_J,type,
set_o_b: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_Mtf__a_J_J,type,
set_o_a: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
filter_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
set_o_o: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Filter__Ofilter_Itf__d_J,type,
filter_d: $tType ).
thf(ty_n_t__Filter__Ofilter_Itf__c_J,type,
filter_c: $tType ).
thf(ty_n_t__Filter__Ofilter_Itf__b_J,type,
filter_b: $tType ).
thf(ty_n_t__Filter__Ofilter_Itf__a_J,type,
filter_a: $tType ).
thf(ty_n_t__Product____Type__Ounit,type,
product_unit: $tType ).
thf(ty_n_t__Filter__Ofilter_I_Eo_J,type,
filter_o: $tType ).
thf(ty_n_t__Set__Oset_Itf__d_J,type,
set_d: $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__String__Oliteral,type,
literal: $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_tf__d,type,
d: $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 (504)
thf(sy_c_Bochner__Integration_Ointegrable_001_Eo_001t__Real__Oreal,type,
bochne661340805755426930o_real: sigma_measure_o > ( $o > real ) > $o ).
thf(sy_c_Bochner__Integration_Ointegrable_001t__Real__Oreal_001t__Real__Oreal,type,
bochne3340023020068487468l_real: sigma_measure_real > ( real > real ) > $o ).
thf(sy_c_Bochner__Integration_Ointegrable_001tf__a_001t__Real__Oreal,type,
bochne2139062162225249880a_real: sigma_measure_a > ( a > real ) > $o ).
thf(sy_c_Bochner__Integration_Olebesgue__integral_001tf__a_001t__Real__Oreal,type,
bochne378719280626478695a_real: sigma_measure_a > ( a > real ) > real ).
thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Real__Oreal,type,
borel_5078946678739801102l_real: sigma_measure_real ).
thf(sy_c_Complete__Measure_Ocompletion_001tf__a,type,
comple3428971583294703880tion_a: sigma_measure_a > sigma_measure_a ).
thf(sy_c_Complete__Measure_Ocompletion_001tf__c,type,
comple3428971583294703882tion_c: sigma_measure_c > sigma_measure_c ).
thf(sy_c_Definitions_Oprob__space_Ok__universal_001tf__a_001tf__b_001tf__c,type,
prob_k5927988752154431057_a_b_c: sigma_measure_a > nat > ( b > a > c ) > set_b > set_c > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__universal_001tf__a_001tf__b_001tf__d,type,
prob_k5927988752154431058_a_b_d: sigma_measure_a > nat > ( b > a > d ) > set_b > set_d > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001tf__b_001tf__c,type,
prob_k6574085460301583242_a_b_c: sigma_measure_a > nat > ( b > sigma_measure_c ) > ( b > a > c ) > set_b > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001tf__b_001tf__d,type,
prob_k6574085460301583243_a_b_d: sigma_measure_a > nat > ( b > sigma_measure_d ) > ( b > a > d ) > set_b > $o ).
thf(sy_c_Definitions_Oprob__space_Ouniform__on_001tf__a_001_Eo,type,
prob_uniform_on_a_o: sigma_measure_a > ( a > $o ) > set_o > $o ).
thf(sy_c_Definitions_Oprob__space_Ouniform__on_001tf__a_001t__Nat__Onat,type,
prob_u5224086380619908136_a_nat: sigma_measure_a > ( a > nat ) > set_nat > $o ).
thf(sy_c_Definitions_Oprob__space_Ouniform__on_001tf__a_001t__Product____Type__Ounit,type,
prob_u3984251234878355381t_unit: sigma_measure_a > ( a > product_unit ) > set_Product_unit > $o ).
thf(sy_c_Definitions_Oprob__space_Ouniform__on_001tf__a_001t__Real__Oreal,type,
prob_u8923871485046717188a_real: sigma_measure_a > ( a > real ) > set_real > $o ).
thf(sy_c_Definitions_Oprob__space_Ouniform__on_001tf__a_001t__String__Oliteral,type,
prob_u1119184700330386812iteral: sigma_measure_a > ( a > literal ) > set_literal > $o ).
thf(sy_c_Definitions_Oprob__space_Ouniform__on_001tf__a_001tf__a,type,
prob_uniform_on_a_a: sigma_measure_a > ( a > a ) > set_a > $o ).
thf(sy_c_Definitions_Oprob__space_Ouniform__on_001tf__a_001tf__c,type,
prob_uniform_on_a_c: sigma_measure_a > ( a > c ) > set_c > $o ).
thf(sy_c_Definitions_Oprob__space_Ouniform__on_001tf__a_001tf__d,type,
prob_uniform_on_a_d: sigma_measure_a > ( a > d ) > set_d > $o ).
thf(sy_c_Definitions_Oprob__space_Ouniform__on_001tf__c_001tf__d,type,
prob_uniform_on_c_d: sigma_measure_c > ( c > d ) > set_d > $o ).
thf(sy_c_Filter_Oeventually_001_062_Itf__a_M_Eo_J,type,
eventually_a_o: ( ( a > $o ) > $o ) > filter_a_o > $o ).
thf(sy_c_Filter_Oeventually_001_062_Itf__a_Mt__Real__Oreal_J,type,
eventually_a_real: ( ( a > real ) > $o ) > filter_a_real > $o ).
thf(sy_c_Filter_Oeventually_001_062_Itf__c_Mtf__d_J,type,
eventually_c_d: ( ( c > d ) > $o ) > filter_c_d > $o ).
thf(sy_c_Filter_Oeventually_001_Eo,type,
eventually_o: ( $o > $o ) > filter_o > $o ).
thf(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
eventually_nat: ( nat > $o ) > filter_nat > $o ).
thf(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
eventually_real: ( real > $o ) > filter_real > $o ).
thf(sy_c_Filter_Oeventually_001tf__a,type,
eventually_a: ( a > $o ) > filter_a > $o ).
thf(sy_c_Filter_Oeventually_001tf__b,type,
eventually_b: ( b > $o ) > filter_b > $o ).
thf(sy_c_Filter_Oeventually_001tf__c,type,
eventually_c: ( c > $o ) > filter_c > $o ).
thf(sy_c_Filter_Oeventually_001tf__d,type,
eventually_d: ( d > $o ) > filter_d > $o ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Ounit,type,
finite410649719033368117t_unit: set_Product_unit > nat ).
thf(sy_c_Finite__Set_Ocard_001t__String__Oliteral,type,
finite_card_literal: set_literal > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__b,type,
finite_card_b: set_b > nat ).
thf(sy_c_Finite__Set_Ofinite_001_062_Itf__a_M_Eo_J,type,
finite_finite_a_o: set_a_o > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_Itf__a_Mt__Real__Oreal_J,type,
finite_finite_a_real: set_a_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_Itf__c_Mtf__d_J,type,
finite_finite_c_d: set_c_d > $o ).
thf(sy_c_Finite__Set_Ofinite_001_Eo,type,
finite_finite_o: set_o > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Extended____Nonnegative____Real__Oennreal,type,
finite3782138982310603983nnreal: set_Ex3793607809372303086nnreal > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_I_Eo_J,type,
finite4093902646404507527tion_o: set_option_o > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_It__Nat__Onat_J,type,
finite5523153139673422903on_nat: set_option_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_It__Product____Type__Ounit_J,type,
finite1445617369574913404t_unit: set_op3165557761946182707t_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_It__String__Oliteral_J,type,
finite5071707688241699267iteral: set_option_literal > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
finite6120865539452801872od_o_o: set_Product_prod_o_o > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
finite1846473907987936430_o_nat: set_Pr2101469702781467981_o_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_I_Eo_Mt__Product____Type__Ounit_J,type,
finite5959031561554722757t_unit: set_Pr6671815714118170108t_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_I_Eo_Mt__String__Oliteral_J,type,
finite8502321148819841420iteral: set_Pr1504714482567607875iteral > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
finite5355008432043429460_nat_o: set_Pr3149072824959771635_nat_o > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite6177210948735845034at_nat: set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
finite5113082511001691337t_unit: set_Pr4334478416066269672t_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__String__Oliteral_J,type,
finite211349803975347344iteral: set_Pr4449086587592242223iteral > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__String__Oliteral_M_Eo_J,type,
finite6349991929781392404eral_o: set_Pr5001933180752110795eral_o > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__String__Oliteral_Mt__Nat__Onat_J,type,
finite4899330813501287658al_nat: set_Pr2368984730104727433al_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Ounit,type,
finite4290736615968046902t_unit: set_Product_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
finite_finite_real: set_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_I_Eo_J,type,
finite_finite_set_o: set_set_o > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Product____Type__Ounit_J,type,
finite1772178364199683094t_unit: set_set_Product_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
finite7209287970140883943_set_a: set_set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__String__Oliteral_J,type,
finite2869373537460367197iteral: set_set_literal > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__String__Oliteral,type,
finite5847741373460823677iteral: set_literal > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_I_Eo_M_Eo_J,type,
finite6699802884135759036um_o_o: set_Sum_sum_o_o > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_I_Eo_Mt__Nat__Onat_J,type,
finite5809725721784815170_o_nat: set_Sum_sum_o_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_I_Eo_Mt__Product____Type__Ounit_J,type,
finite2105581108028245809t_unit: set_Su4100838662412113488t_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_I_Eo_Mt__String__Oliteral_J,type,
finite5240078910061808376iteral: set_Su8215520242895148183iteral > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Nat__Onat_M_Eo_J,type,
finite94888208985532392_nat_o: set_Sum_sum_nat_o > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite6187706683773761046at_nat: set_Sum_sum_nat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
finite4327512606132785245t_unit: set_Su7539578257924484756t_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Nat__Onat_Mt__String__Oliteral_J,type,
finite7336130560110450212iteral: set_Su4835947104832017115iteral > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__String__Oliteral_M_Eo_J,type,
finite3087749691023359360eral_o: set_Su2489366904224875295eral_o > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__String__Oliteral_Mt__Nat__Onat_J,type,
finite2800739532781614718al_nat: set_Su2755845247344502325al_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__b,type,
finite_finite_b: set_b > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__d,type,
finite_finite_d: set_d > $o ).
thf(sy_c_Fun_Ocomp_001_Eo_001_Eo_001tf__a,type,
comp_o_o_a: ( $o > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Fun_Ocomp_001_Eo_001t__Real__Oreal_001tf__a,type,
comp_o_real_a: ( $o > real ) > ( a > $o ) > a > real ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001_Eo_001tf__a,type,
comp_real_o_a: ( real > $o ) > ( a > real ) > a > $o ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001tf__a,type,
comp_real_real_a: ( real > real ) > ( a > real ) > a > real ).
thf(sy_c_Fun_Ocomp_001tf__a_001_Eo_001tf__a,type,
comp_a_o_a: ( a > $o ) > ( a > a ) > a > $o ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Real__Oreal_001tf__a,type,
comp_a_real_a: ( a > real ) > ( a > a ) > a > real ).
thf(sy_c_Fun_Ocomp_001tf__c_001tf__d_001tf__a,type,
comp_c_d_a: ( c > d ) > ( a > c ) > a > d ).
thf(sy_c_Fun_Ocomp_001tf__c_001tf__d_001tf__c,type,
comp_c_d_c: ( c > d ) > ( c > c ) > c > d ).
thf(sy_c_Fun_Ocomp_001tf__d_001tf__d_001tf__c,type,
comp_d_d_c: ( d > d ) > ( c > d ) > c > d ).
thf(sy_c_Giry__Monad_Oreturn_001_062_Itf__a_M_Eo_J,type,
giry_return_a_o: sigma_measure_a_o > ( a > $o ) > sigma_measure_a_o ).
thf(sy_c_Giry__Monad_Oreturn_001_062_Itf__a_Mt__Real__Oreal_J,type,
giry_return_a_real: sigma_measure_a_real > ( a > real ) > sigma_measure_a_real ).
thf(sy_c_Giry__Monad_Oreturn_001_062_Itf__c_Mtf__d_J,type,
giry_return_c_d: sigma_measure_c_d > ( c > d ) > sigma_measure_c_d ).
thf(sy_c_Giry__Monad_Oreturn_001_Eo,type,
giry_return_o: sigma_measure_o > $o > sigma_measure_o ).
thf(sy_c_Giry__Monad_Oreturn_001t__Nat__Onat,type,
giry_return_nat: sigma_measure_nat > nat > sigma_measure_nat ).
thf(sy_c_Giry__Monad_Oreturn_001t__Real__Oreal,type,
giry_return_real: sigma_measure_real > real > sigma_measure_real ).
thf(sy_c_Giry__Monad_Oreturn_001tf__a,type,
giry_return_a: sigma_measure_a > a > sigma_measure_a ).
thf(sy_c_Giry__Monad_Oreturn_001tf__b,type,
giry_return_b: sigma_measure_b > b > sigma_measure_b ).
thf(sy_c_Giry__Monad_Oreturn_001tf__c,type,
giry_return_c: sigma_measure_c > c > sigma_measure_c ).
thf(sy_c_Giry__Monad_Oreturn_001tf__d,type,
giry_return_d: sigma_measure_d > d > sigma_measure_d ).
thf(sy_c_Giry__Monad_Osubprob__space_001_062_Itf__a_M_Eo_J,type,
giry_s2201070112685910219ce_a_o: sigma_measure_a_o > $o ).
thf(sy_c_Giry__Monad_Osubprob__space_001_062_Itf__a_Mt__Real__Oreal_J,type,
giry_s8245238245569539279a_real: sigma_measure_a_real > $o ).
thf(sy_c_Giry__Monad_Osubprob__space_001_062_Itf__c_Mtf__d_J,type,
giry_s3243706009590697010ce_c_d: sigma_measure_c_d > $o ).
thf(sy_c_Giry__Monad_Osubprob__space_001_Eo,type,
giry_subprob_space_o: sigma_measure_o > $o ).
thf(sy_c_Giry__Monad_Osubprob__space_001t__Nat__Onat,type,
giry_s459323515522551452ce_nat: sigma_measure_nat > $o ).
thf(sy_c_Giry__Monad_Osubprob__space_001t__Real__Oreal,type,
giry_s8208748868292234104e_real: sigma_measure_real > $o ).
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_001tf__c,type,
giry_subprob_space_c: sigma_measure_c > $o ).
thf(sy_c_Giry__Monad_Osubprob__space_001tf__d,type,
giry_subprob_space_d: sigma_measure_d > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nonnegative____Real__Oennreal,type,
minus_8429688780609304081nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__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_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nonnegative____Real__Oennreal,type,
one_on2969667320475766781nnreal: extend8495563244428889912nnreal ).
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_Otimes__class_Otimes_001t__Extended____Nonnegative____Real__Oennreal,type,
times_1893300245718287421nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).
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__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_Independent__Family_Oprob__space_Oindep__events_001_062_Itf__a_M_Eo_J_001t__Nat__Onat,type,
indepe7519481913767141285_o_nat: sigma_measure_a_o > ( nat > set_a_o ) > set_nat > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__events_001_062_Itf__a_Mt__Real__Oreal_J_001t__Nat__Onat,type,
indepe3916917635824975767al_nat: sigma_measure_a_real > ( nat > set_a_real ) > set_nat > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__events_001_062_Itf__c_Mtf__d_J_001t__Nat__Onat,type,
indepe4353200269427704460_d_nat: sigma_measure_c_d > ( nat > set_c_d ) > set_nat > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__events_001_Eo_001t__Nat__Onat,type,
indepe8054989831391459860_o_nat: sigma_measure_o > ( nat > set_o ) > set_nat > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__events_001t__Nat__Onat_001t__Nat__Onat,type,
indepe6019118560394178294at_nat: sigma_measure_nat > ( nat > set_nat ) > set_nat > $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__events_001tf__a_001t__Nat__Onat,type,
indepe1551197314001032186_a_nat: sigma_measure_a > ( nat > set_a ) > set_nat > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__events_001tf__a_001t__Set__Oset_Itf__a_J,type,
indepe451107616305015476_set_a: sigma_measure_a > ( set_a > set_a ) > set_set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__events_001tf__a_001tf__a,type,
indepe1948846426091821396ts_a_a: sigma_measure_a > ( a > set_a ) > set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__events_001tf__a_001tf__b,type,
indepe1948846426091821397ts_a_b: sigma_measure_a > ( b > set_a ) > set_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__events_001tf__b_001t__Nat__Onat,type,
indepe2786641642957426683_b_nat: sigma_measure_b > ( nat > set_b ) > set_nat > $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__sets_001tf__a_001_062_Itf__a_M_Eo_J,type,
indepe7801696130336798481_a_a_o: sigma_measure_a > ( ( a > $o ) > set_set_a ) > set_a_o > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001_062_Itf__a_Mt__Real__Oreal_J,type,
indepe4749599203615801097a_real: sigma_measure_a > ( ( a > real ) > set_set_a ) > set_a_real > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001_062_Itf__c_Mtf__d_J,type,
indepe4867465796832040824_a_c_d: sigma_measure_a > ( ( c > d ) > set_set_a ) > set_c_d > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001_Eo,type,
indepe7780107833195774214ts_a_o: sigma_measure_a > ( $o > set_set_a ) > set_o > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001t__Nat__Onat,type,
indepe6267730027088848354_a_nat: sigma_measure_a > ( nat > set_set_a ) > set_nat > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001t__Set__Oset_Itf__a_J,type,
indepe4967106450811773644_set_a: sigma_measure_a > ( set_a > set_set_a ) > set_set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_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__b,type,
indepe8927441866673418605ts_a_b: sigma_measure_a > ( b > set_set_a ) > set_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001tf__a_001_Eo,type,
indepe2772234535145391206ar_a_o: sigma_measure_a > sigma_measure_o > ( a > $o ) > sigma_measure_o > ( a > $o ) > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001tf__a_001t__Real__Oreal,type,
indepe8958435565499147358a_real: sigma_measure_a > sigma_measure_real > ( a > real ) > sigma_measure_real > ( a > real ) > $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__c,type,
indepe2440653194691626190ar_a_c: sigma_measure_a > sigma_measure_c > ( a > c ) > sigma_measure_c > ( a > c ) > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001tf__a_001tf__d,type,
indepe2440653194691626191ar_a_d: sigma_measure_a > sigma_measure_d > ( a > d ) > sigma_measure_d > ( a > d ) > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001tf__c_001tf__d,type,
indepe6089114067601049933ar_c_d: sigma_measure_c > sigma_measure_d > ( c > d ) > sigma_measure_d > ( c > d ) > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001_Eo_001_Eo,type,
indepe9162428965118168502_a_o_o: sigma_measure_a > ( $o > sigma_measure_o ) > ( $o > a > $o ) > set_o > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001_Eo_001t__Nat__Onat,type,
indepe3518881924824081714_o_nat: sigma_measure_a > ( $o > sigma_measure_nat ) > ( $o > a > nat ) > set_o > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001_Eo_001tf__a,type,
indepe3252683823613847068_a_o_a: sigma_measure_a > ( $o > sigma_measure_a ) > ( $o > a > a ) > set_o > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001_Eo_001tf__b,type,
indepe3252683823613847069_a_o_b: sigma_measure_a > ( $o > sigma_measure_b ) > ( $o > a > b ) > set_o > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001_Eo_001tf__c,type,
indepe3252683823613847070_a_o_c: sigma_measure_a > ( $o > sigma_measure_c ) > ( $o > a > c ) > set_o > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001_Eo_001tf__d,type,
indepe3252683823613847071_a_o_d: sigma_measure_a > ( $o > sigma_measure_d ) > ( $o > a > d ) > set_o > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001t__Nat__Onat_001_Eo,type,
indepe6621553285932267024_nat_o: sigma_measure_a > ( nat > sigma_measure_o ) > ( nat > a > $o ) > set_nat > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001t__Nat__Onat_001tf__a,type,
indepe3245197900929106294_nat_a: sigma_measure_a > ( nat > sigma_measure_a ) > ( nat > a > a ) > set_nat > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001t__Nat__Onat_001tf__c,type,
indepe3245197900929106296_nat_c: sigma_measure_a > ( nat > sigma_measure_c ) > ( nat > a > c ) > set_nat > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001t__Nat__Onat_001tf__d,type,
indepe3245197900929106297_nat_d: sigma_measure_a > ( nat > sigma_measure_d ) > ( nat > a > d ) > set_nat > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001tf__a_001_Eo,type,
indepe3332163980079594832_a_a_o: sigma_measure_a > ( a > sigma_measure_o ) > ( a > a > $o ) > set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001tf__a_001t__Nat__Onat,type,
indepe8436638436605595672_a_nat: sigma_measure_a > ( a > sigma_measure_nat ) > ( a > a > nat ) > set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001tf__a_001t__Real__Oreal,type,
indepe2669223931359383284a_real: sigma_measure_a > ( a > sigma_measure_real ) > ( a > a > real ) > set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001tf__a_001tf__a,type,
indepe1203440900223019190_a_a_a: sigma_measure_a > ( a > sigma_measure_a ) > ( a > a > a ) > set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001tf__a_001tf__b,type,
indepe1203440900223019191_a_a_b: sigma_measure_a > ( a > sigma_measure_b ) > ( a > a > b ) > set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001tf__a_001tf__c,type,
indepe1203440900223019192_a_a_c: sigma_measure_a > ( a > sigma_measure_c ) > ( a > a > c ) > set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001tf__a_001tf__d,type,
indepe1203440900223019193_a_a_d: sigma_measure_a > ( a > sigma_measure_d ) > ( a > a > d ) > set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001tf__b_001_Eo,type,
indepe432941580615200527_a_b_o: sigma_measure_a > ( b > sigma_measure_o ) > ( b > a > $o ) > set_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001tf__b_001t__Nat__Onat,type,
indepe448710728707214361_b_nat: sigma_measure_a > ( b > sigma_measure_nat ) > ( b > a > nat ) > set_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001tf__b_001t__Real__Oreal,type,
indepe8265442546547513973b_real: sigma_measure_a > ( b > sigma_measure_real ) > ( b > a > real ) > set_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001tf__b_001tf__a,type,
indepe7639357355105118965_a_b_a: sigma_measure_a > ( b > sigma_measure_a ) > ( b > a > a ) > set_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001tf__b_001tf__b,type,
indepe7639357355105118966_a_b_b: sigma_measure_a > ( b > sigma_measure_b ) > ( b > a > b ) > set_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001tf__b_001tf__c,type,
indepe7639357355105118967_a_b_c: sigma_measure_a > ( b > sigma_measure_c ) > ( b > a > c ) > set_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001tf__b_001tf__d,type,
indepe7639357355105118968_a_b_d: sigma_measure_a > ( b > sigma_measure_d ) > ( b > a > d ) > set_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_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Measure__Space_Oae__filter_001_062_Itf__a_M_Eo_J,type,
measur1950989848205109360er_a_o: sigma_measure_a_o > filter_a_o ).
thf(sy_c_Measure__Space_Oae__filter_001_062_Itf__a_Mt__Real__Oreal_J,type,
measur8946935934331175658a_real: sigma_measure_a_real > filter_a_real ).
thf(sy_c_Measure__Space_Oae__filter_001_062_Itf__c_Mtf__d_J,type,
measur253975988404078935er_c_d: sigma_measure_c_d > filter_c_d ).
thf(sy_c_Measure__Space_Oae__filter_001_Eo,type,
measure_ae_filter_o: sigma_measure_o > filter_o ).
thf(sy_c_Measure__Space_Oae__filter_001t__Nat__Onat,type,
measur6539087422748349889er_nat: sigma_measure_nat > filter_nat ).
thf(sy_c_Measure__Space_Oae__filter_001t__Real__Oreal,type,
measur1097577823623106589r_real: sigma_measure_real > filter_real ).
thf(sy_c_Measure__Space_Oae__filter_001tf__a,type,
measure_ae_filter_a: sigma_measure_a > filter_a ).
thf(sy_c_Measure__Space_Oae__filter_001tf__b,type,
measure_ae_filter_b: sigma_measure_b > filter_b ).
thf(sy_c_Measure__Space_Oae__filter_001tf__c,type,
measure_ae_filter_c: sigma_measure_c > filter_c ).
thf(sy_c_Measure__Space_Oae__filter_001tf__d,type,
measure_ae_filter_d: sigma_measure_d > filter_d ).
thf(sy_c_Measure__Space_Odistr_001_062_Itf__a_M_Eo_J_001tf__a,type,
measure_distr_a_o_a: sigma_measure_a_o > sigma_measure_a > ( ( a > $o ) > a ) > sigma_measure_a ).
thf(sy_c_Measure__Space_Odistr_001_062_Itf__a_Mt__Real__Oreal_J_001tf__a,type,
measur1246039135095686683real_a: sigma_measure_a_real > sigma_measure_a > ( ( a > real ) > a ) > sigma_measure_a ).
thf(sy_c_Measure__Space_Odistr_001_062_Itf__c_Mtf__d_J_001tf__a,type,
measure_distr_c_d_a: sigma_measure_c_d > sigma_measure_a > ( ( c > d ) > a ) > sigma_measure_a ).
thf(sy_c_Measure__Space_Odistr_001_Eo_001_Eo,type,
measure_distr_o_o: sigma_measure_o > sigma_measure_o > ( $o > $o ) > sigma_measure_o ).
thf(sy_c_Measure__Space_Odistr_001_Eo_001t__Nat__Onat,type,
measure_distr_o_nat: sigma_measure_o > sigma_measure_nat > ( $o > nat ) > sigma_measure_nat ).
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_001_Eo,type,
measure_distr_nat_o: sigma_measure_nat > sigma_measure_o > ( nat > $o ) > sigma_measure_o ).
thf(sy_c_Measure__Space_Odistr_001t__Nat__Onat_001t__Nat__Onat,type,
measur36828333294957594at_nat: sigma_measure_nat > sigma_measure_nat > ( nat > nat ) > sigma_measure_nat ).
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_001tf__a_001_062_Itf__a_M_Eo_J,type,
measure_distr_a_a_o: sigma_measure_a > sigma_measure_a_o > ( a > a > $o ) > sigma_measure_a_o ).
thf(sy_c_Measure__Space_Odistr_001tf__a_001_062_Itf__a_Mt__Real__Oreal_J,type,
measur6806378183764566549a_real: sigma_measure_a > sigma_measure_a_real > ( a > a > real ) > sigma_measure_a_real ).
thf(sy_c_Measure__Space_Odistr_001tf__a_001_062_Itf__c_Mtf__d_J,type,
measure_distr_a_c_d: sigma_measure_a > sigma_measure_c_d > ( a > c > d ) > sigma_measure_c_d ).
thf(sy_c_Measure__Space_Odistr_001tf__a_001_Eo,type,
measure_distr_a_o: sigma_measure_a > sigma_measure_o > ( a > $o ) > sigma_measure_o ).
thf(sy_c_Measure__Space_Odistr_001tf__a_001t__Nat__Onat,type,
measure_distr_a_nat: sigma_measure_a > sigma_measure_nat > ( a > nat ) > sigma_measure_nat ).
thf(sy_c_Measure__Space_Odistr_001tf__a_001t__Product____Type__Ounit,type,
measur2774250482188116359t_unit: sigma_measure_a > sigma_3917854644531351306t_unit > ( a > product_unit ) > sigma_3917854644531351306t_unit ).
thf(sy_c_Measure__Space_Odistr_001tf__a_001t__Real__Oreal,type,
measure_distr_a_real: sigma_measure_a > sigma_measure_real > ( a > real ) > sigma_measure_real ).
thf(sy_c_Measure__Space_Odistr_001tf__a_001t__String__Oliteral,type,
measur2089735993314667086iteral: sigma_measure_a > sigma_4743690205272893521iteral > ( a > literal ) > sigma_4743690205272893521iteral ).
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_001_Eo,type,
measure_distr_b_o: sigma_measure_b > sigma_measure_o > ( b > $o ) > sigma_measure_o ).
thf(sy_c_Measure__Space_Odistr_001tf__b_001t__Nat__Onat,type,
measure_distr_b_nat: sigma_measure_b > sigma_measure_nat > ( b > nat ) > sigma_measure_nat ).
thf(sy_c_Measure__Space_Odistr_001tf__b_001tf__a,type,
measure_distr_b_a: sigma_measure_b > sigma_measure_a > ( b > a ) > sigma_measure_a ).
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__d,type,
measure_distr_c_d: sigma_measure_c > sigma_measure_d > ( c > d ) > sigma_measure_d ).
thf(sy_c_Measure__Space_Ofinite__measure_001_Eo,type,
measur2447921437955784316sure_o: sigma_measure_o > $o ).
thf(sy_c_Measure__Space_Ofinite__measure_001t__Nat__Onat,type,
measur8338831127414845932re_nat: sigma_measure_nat > $o ).
thf(sy_c_Measure__Space_Ofinite__measure_001t__Real__Oreal,type,
measur3606880022600206024e_real: sigma_measure_real > $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__c,type,
measur930452917991658468sure_c: sigma_measure_c > $o ).
thf(sy_c_Measure__Space_Ofinite__measure_001tf__d,type,
measur930452917991658469sure_d: sigma_measure_d > $o ).
thf(sy_c_Measure__Space_Oincreasing_001t__Set__Oset_Itf__a_J_001t__Extended____Nonnegative____Real__Oennreal,type,
measur1771626496591458595nnreal: set_set_set_a > ( set_set_a > extend8495563244428889912nnreal ) > $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__Real__Oreal,type,
measur2331856671108623127a_real: set_set_set_a > ( set_set_a > real ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
measur2197171192767378579_set_a: set_set_set_a > ( set_set_a > set_set_a ) > $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_001tf__a_001t__Extended____Nonnegative____Real__Oennreal,type,
measur5393715408109795267nnreal: set_set_a > ( set_a > extend8495563244428889912nnreal ) > $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_It__Set__Oset_Itf__a_J_J,type,
measur8202069185322079731_set_a: set_set_a > ( set_a > set_set_a ) > $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_Osigma__finite__measure_001_Eo,type,
measur1827666076404920889sure_o: sigma_measure_o > $o ).
thf(sy_c_Measure__Space_Osigma__finite__measure_001t__Nat__Onat,type,
measur8258956421386577775re_nat: sigma_measure_nat > $o ).
thf(sy_c_Measure__Space_Osigma__finite__measure_001t__Real__Oreal,type,
measur487378040549452491e_real: sigma_measure_real > $o ).
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__c,type,
measur4308613598931908897sure_c: sigma_measure_c > $o ).
thf(sy_c_Measure__Space_Osigma__finite__measure_001tf__d,type,
measur4308613598931908898sure_d: sigma_measure_d > $o ).
thf(sy_c_Nonnegative__Lebesgue__Integration_Onn__integral_001tf__a,type,
nonneg2725512125972007571gral_a: sigma_measure_a > ( a > extend8495563244428889912nnreal ) > extend8495563244428889912nnreal ).
thf(sy_c_Nonnegative__Lebesgue__Integration_Ouniform__measure_001_Eo,type,
nonneg5638544851443855887sure_o: sigma_measure_o > set_o > sigma_measure_o ).
thf(sy_c_Nonnegative__Lebesgue__Integration_Ouniform__measure_001t__Nat__Onat,type,
nonneg5218579358776314137re_nat: sigma_measure_nat > set_nat > sigma_measure_nat ).
thf(sy_c_Nonnegative__Lebesgue__Integration_Ouniform__measure_001t__Product____Type__Ounit,type,
nonneg8606312109825204740t_unit: sigma_3917854644531351306t_unit > set_Product_unit > sigma_3917854644531351306t_unit ).
thf(sy_c_Nonnegative__Lebesgue__Integration_Ouniform__measure_001t__Real__Oreal,type,
nonneg3404582084018902901e_real: sigma_measure_real > set_real > sigma_measure_real ).
thf(sy_c_Nonnegative__Lebesgue__Integration_Ouniform__measure_001t__String__Oliteral,type,
nonneg4334240857813153355iteral: sigma_4743690205272893521iteral > set_literal > sigma_4743690205272893521iteral ).
thf(sy_c_Nonnegative__Lebesgue__Integration_Ouniform__measure_001tf__a,type,
nonneg6757527617543859701sure_a: sigma_measure_a > set_a > sigma_measure_a ).
thf(sy_c_Nonnegative__Lebesgue__Integration_Ouniform__measure_001tf__d,type,
nonneg6757527617543859704sure_d: sigma_measure_d > set_d > sigma_measure_d ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $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__Extended____Nonnegative____Real__Oennreal,type,
bot_bo841427958541957580nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_Itf__a_J,type,
bot_bot_filter_a: filter_a ).
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_M_Eo_J_J,type,
bot_bot_set_a_o: set_a_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_Itf__a_Mt__Real__Oreal_J_J,type,
bot_bot_set_a_real: set_a_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_Itf__c_Mtf__d_J_J,type,
bot_bot_set_c_d: set_c_d ).
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__Extended____Nonnegative____Real__Oennreal_J,type,
bot_bo4854962954004695426nnreal: set_Ex3793607809372303086nnreal ).
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__Product____Type__Ounit_J,type,
bot_bo3957492148770167129t_unit: set_Product_unit ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
bot_bo3380559777022489994_set_a: set_set_set_a ).
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__String__Oliteral_J,type,
bot_bot_set_literal: set_literal ).
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__Set__Oset_Itf__d_J,type,
bot_bot_set_d: set_d ).
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__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_001_062_I_062_Itf__a_M_Eo_J_M_Eo_J,type,
ord_less_eq_a_o_o: ( ( a > $o ) > $o ) > ( ( a > $o ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_Itf__a_Mt__Real__Oreal_J_M_Eo_J,type,
ord_less_eq_a_real_o: ( ( a > real ) > $o ) > ( ( a > real ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_Itf__c_Mtf__d_J_M_Eo_J,type,
ord_less_eq_c_d_o: ( ( c > d ) > $o ) > ( ( c > d ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_Eo_J,type,
ord_less_eq_o_o: ( $o > $o ) > ( $o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
ord_less_eq_set_a_o: ( set_a > $o ) > ( set_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_Mt__Real__Oreal_J,type,
ord_less_eq_a_real: ( a > real ) > ( a > real ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__b_M_Eo_J,type,
ord_less_eq_b_o: ( b > $o ) > ( b > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
ord_less_eq_o: $o > $o > $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__Filter__Ofilter_Itf__a_J,type,
ord_less_eq_filter_a: filter_a > filter_a > $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_M_Eo_J_J,type,
ord_less_eq_set_a_o2: set_a_o > set_a_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__a_Mt__Real__Oreal_J_J,type,
ord_le3334967407727675675a_real: set_a_real > set_a_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__c_Mtf__d_J_J,type,
ord_less_eq_set_c_d: set_c_d > set_c_d > $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__Product____Type__Ounit_J,type,
ord_le3507040750410214029t_unit: set_Product_unit > set_Product_unit > $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__String__Oliteral_J,type,
ord_le7307670543136651348iteral: set_literal > set_literal > $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__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_Itf__a_J,type,
ord_le254669795585780187sure_a: sigma_measure_a > sigma_measure_a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_Eo_M_Eo_J,type,
top_top_o_o: $o > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Product____Type__Ounit_M_Eo_J,type,
top_to2465898995584390880unit_o: product_unit > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__String__Oliteral_M_Eo_J,type,
top_top_literal_o: literal > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_Itf__a_M_Eo_J,type,
top_top_a_o: a > $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__Filter__Ofilter_Itf__a_J,type,
top_top_filter_a: filter_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
top_top_set_a_o: set_a_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__a_Mt__Real__Oreal_J_J,type,
top_top_set_a_real: set_a_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__c_Mtf__d_J_J,type,
top_top_set_c_d: set_c_d ).
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__Option__Ooption_I_Eo_J_J,type,
top_top_set_option_o: set_option_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
top_to8920198386146353926on_nat: set_option_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Product____Type__Ounit_J_J,type,
top_to2690860209552263555t_unit: set_op3165557761946182707t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__String__Oliteral_J_J,type,
top_to8248435444729185354iteral: set_option_literal ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
top_to7721136755696657239od_o_o: set_Product_prod_o_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
top_to7022684507342537725_o_nat: set_Pr2101469702781467981_o_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Product____Type__Ounit_J_J,type,
top_to1629657009876642636t_unit: set_Pr6671815714118170108t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__String__Oliteral_J_J,type,
top_to2491963433743229843iteral: set_Pr1504714482567607875iteral ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
top_to8070287629520841379_nat_o: set_Pr3149072824959771635_nat_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
top_to8544742955230171288t_unit: set_Pr4334478416066269672t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__String__Oliteral_J_J,type,
top_to6658620532179778271iteral: set_Pr4449086587592242223iteral ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__String__Oliteral_M_Eo_J_J,type,
top_to5989182131927732763eral_o: set_Pr5001933180752110795eral_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__String__Oliteral_Mt__Nat__Onat_J_J,type,
top_to4578518674692263481al_nat: set_Pr2368984730104727433al_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
top_to1996260823553986621t_unit: set_Product_unit ).
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_I_Eo_J_J,type,
top_top_set_set_o: set_set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
top_top_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Product____Type__Ounit_J_J,type,
top_to1767297665138865437t_unit: set_set_Product_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__String__Oliteral_J_J,type,
top_to5694933271948605156iteral: set_set_literal ).
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__String__Oliteral_J,type,
top_top_set_literal: set_literal ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_I_Eo_M_Eo_J_J,type,
top_to1686961084667892491um_o_o: set_Sum_sum_o_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_I_Eo_Mt__Nat__Onat_J_J,type,
top_to6072511757011528009_o_nat: set_Sum_sum_o_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_I_Eo_Mt__Product____Type__Ounit_J_J,type,
top_to126344037801868544t_unit: set_Su4100838662412113488t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_I_Eo_Mt__String__Oliteral_J_J,type,
top_to6634777175915823943iteral: set_Su8215520242895148183iteral ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_M_Eo_J_J,type,
top_to7120114879189831663_nat_o: set_Sum_sum_nat_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_to6661820994512907621at_nat: set_Sum_sum_nat_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
top_to5465250082899874788t_unit: set_Su7539578257924484756t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__String__Oliteral_J_J,type,
top_to148093990134820907iteral: set_Su4835947104832017115iteral ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__String__Oliteral_M_Eo_J_J,type,
top_to908623837245551055eral_o: set_Su2489366904224875295eral_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__String__Oliteral_Mt__Nat__Onat_J_J,type,
top_to7291364169502081925al_nat: set_Su2755845247344502325al_nat ).
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__b_J,type,
top_top_set_b: set_b ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__c_J,type,
top_top_set_c: set_c ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__d_J,type,
top_top_set_d: set_d ).
thf(sy_c_Probability__Measure_Ocond__prob_001tf__a,type,
probab5246806142355027601prob_a: sigma_measure_a > ( a > $o ) > ( a > $o ) > real ).
thf(sy_c_Probability__Measure_Odistributed_001_Eo_001_Eo,type,
probab6865799537298334222ed_o_o: sigma_measure_o > sigma_measure_o > ( $o > $o ) > ( $o > extend8495563244428889912nnreal ) > $o ).
thf(sy_c_Probability__Measure_Odistributed_001_Eo_001t__Real__Oreal,type,
probab5069348595295671478o_real: sigma_measure_o > sigma_measure_real > ( $o > real ) > ( real > extend8495563244428889912nnreal ) > $o ).
thf(sy_c_Probability__Measure_Odistributed_001t__Real__Oreal_001_Eo,type,
probab6578412970336841052real_o: sigma_measure_real > sigma_measure_o > ( real > $o ) > ( $o > extend8495563244428889912nnreal ) > $o ).
thf(sy_c_Probability__Measure_Odistributed_001t__Real__Oreal_001t__Real__Oreal,type,
probab1340766270110547944l_real: sigma_measure_real > sigma_measure_real > ( real > real ) > ( real > extend8495563244428889912nnreal ) > $o ).
thf(sy_c_Probability__Measure_Odistributed_001tf__a_001_062_Itf__a_M_Eo_J,type,
probab3024911352429335023_a_a_o: sigma_measure_a > sigma_measure_a_o > ( a > a > $o ) > ( ( a > $o ) > extend8495563244428889912nnreal ) > $o ).
thf(sy_c_Probability__Measure_Odistributed_001tf__a_001_062_Itf__a_Mt__Real__Oreal_J,type,
probab1295962615871907499a_real: sigma_measure_a > sigma_measure_a_real > ( a > a > real ) > ( ( a > real ) > extend8495563244428889912nnreal ) > $o ).
thf(sy_c_Probability__Measure_Odistributed_001tf__a_001_062_Itf__c_Mtf__d_J,type,
probab7909271913183257430_a_c_d: sigma_measure_a > sigma_measure_c_d > ( a > c > d ) > ( ( c > d ) > extend8495563244428889912nnreal ) > $o ).
thf(sy_c_Probability__Measure_Odistributed_001tf__a_001_Eo,type,
probab177417448539121064ed_a_o: sigma_measure_a > sigma_measure_o > ( a > $o ) > ( $o > extend8495563244428889912nnreal ) > $o ).
thf(sy_c_Probability__Measure_Odistributed_001tf__a_001t__Nat__Onat,type,
probab2712490043479616960_a_nat: sigma_measure_a > sigma_measure_nat > ( a > nat ) > ( nat > extend8495563244428889912nnreal ) > $o ).
thf(sy_c_Probability__Measure_Odistributed_001tf__a_001t__Real__Oreal,type,
probab9192628068011464860a_real: sigma_measure_a > sigma_measure_real > ( a > real ) > ( real > extend8495563244428889912nnreal ) > $o ).
thf(sy_c_Probability__Measure_Odistributed_001tf__a_001tf__a,type,
probab8876900357271971086ed_a_a: sigma_measure_a > sigma_measure_a > ( a > a ) > ( a > extend8495563244428889912nnreal ) > $o ).
thf(sy_c_Probability__Measure_Odistributed_001tf__a_001tf__b,type,
probab8876900357271971087ed_a_b: sigma_measure_a > sigma_measure_b > ( a > b ) > ( b > extend8495563244428889912nnreal ) > $o ).
thf(sy_c_Probability__Measure_Odistributed_001tf__c_001tf__d,type,
probab3301989193326619023ed_c_d: sigma_measure_c > sigma_measure_d > ( c > d ) > ( d > extend8495563244428889912nnreal ) > $o ).
thf(sy_c_Probability__Measure_Odistributed_001tf__d_001tf__d,type,
probab514533611353942990ed_d_d: sigma_measure_d > sigma_measure_d > ( d > d ) > ( d > extend8495563244428889912nnreal ) > $o ).
thf(sy_c_Probability__Measure_Oprob__space_001_062_Itf__a_M_Eo_J,type,
probab7249042050958952188ce_a_o: sigma_measure_a_o > $o ).
thf(sy_c_Probability__Measure_Oprob__space_001_062_Itf__a_Mt__Real__Oreal_J,type,
probab2024454272037758302a_real: sigma_measure_a_real > $o ).
thf(sy_c_Probability__Measure_Oprob__space_001_062_Itf__c_Mtf__d_J,type,
probab3693743499390067171ce_c_d: sigma_measure_c_d > $o ).
thf(sy_c_Probability__Measure_Oprob__space_001_Eo,type,
probab1190487603588612059pace_o: sigma_measure_o > $o ).
thf(sy_c_Probability__Measure_Oprob__space_001t__Nat__Onat,type,
probab2904919403188438605ce_nat: sigma_measure_nat > $o ).
thf(sy_c_Probability__Measure_Oprob__space_001t__Real__Oreal,type,
probab535871623910865577e_real: sigma_measure_real > $o ).
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_001tf__c,type,
probab7247484486040049091pace_c: sigma_measure_c > $o ).
thf(sy_c_Probability__Measure_Oprob__space_001tf__d,type,
probab7247484486040049092pace_d: sigma_measure_d > $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_Set_OCollect_001_062_Itf__a_M_Eo_J,type,
collect_a_o: ( ( a > $o ) > $o ) > set_a_o ).
thf(sy_c_Set_OCollect_001_062_Itf__a_Mt__Real__Oreal_J,type,
collect_a_real: ( ( a > real ) > $o ) > set_a_real ).
thf(sy_c_Set_OCollect_001_062_Itf__c_Mtf__d_J,type,
collect_c_d: ( ( c > d ) > $o ) > set_c_d ).
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__Product____Type__Ounit,type,
collect_Product_unit: ( product_unit > $o ) > set_Product_unit ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
collect_set_set_a: ( set_set_a > $o ) > set_set_set_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001t__String__Oliteral,type,
collect_literal: ( literal > $o ) > set_literal ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
thf(sy_c_Sigma__Algebra_ODynkin_001tf__a,type,
sigma_Dynkin_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Sigma__Algebra_Ocount__space_001_Eo,type,
sigma_count_space_o: set_o > sigma_measure_o ).
thf(sy_c_Sigma__Algebra_Ocount__space_001t__Nat__Onat,type,
sigma_7685844798829912695ce_nat: set_nat > sigma_measure_nat ).
thf(sy_c_Sigma__Algebra_Ocount__space_001t__Product____Type__Ounit,type,
sigma_2593334793047783974t_unit: set_Product_unit > sigma_3917854644531351306t_unit ).
thf(sy_c_Sigma__Algebra_Ocount__space_001t__Real__Oreal,type,
sigma_8508918144308765139e_real: set_real > sigma_measure_real ).
thf(sy_c_Sigma__Algebra_Ocount__space_001t__Set__Oset_Itf__a_J,type,
sigma_1106005778614564215_set_a: set_set_a > sigma_measure_set_a ).
thf(sy_c_Sigma__Algebra_Ocount__space_001t__String__Oliteral,type,
sigma_7122031993496068461iteral: set_literal > sigma_4743690205272893521iteral ).
thf(sy_c_Sigma__Algebra_Ocount__space_001tf__a,type,
sigma_count_space_a: set_a > sigma_measure_a ).
thf(sy_c_Sigma__Algebra_Ocount__space_001tf__c,type,
sigma_count_space_c: set_c > sigma_measure_c ).
thf(sy_c_Sigma__Algebra_Ocount__space_001tf__d,type,
sigma_count_space_d: set_d > sigma_measure_d ).
thf(sy_c_Sigma__Algebra_Oemeasure_001_062_Itf__a_M_Eo_J,type,
sigma_emeasure_a_o: sigma_measure_a_o > set_a_o > extend8495563244428889912nnreal ).
thf(sy_c_Sigma__Algebra_Oemeasure_001_062_Itf__a_Mt__Real__Oreal_J,type,
sigma_5985106571655482610a_real: sigma_measure_a_real > set_a_real > extend8495563244428889912nnreal ).
thf(sy_c_Sigma__Algebra_Oemeasure_001_062_Itf__c_Mtf__d_J,type,
sigma_emeasure_c_d: sigma_measure_c_d > set_c_d > extend8495563244428889912nnreal ).
thf(sy_c_Sigma__Algebra_Oemeasure_001_Eo,type,
sigma_emeasure_o: sigma_measure_o > set_o > extend8495563244428889912nnreal ).
thf(sy_c_Sigma__Algebra_Oemeasure_001t__Nat__Onat,type,
sigma_emeasure_nat: sigma_measure_nat > set_nat > extend8495563244428889912nnreal ).
thf(sy_c_Sigma__Algebra_Oemeasure_001t__Set__Oset_Itf__a_J,type,
sigma_emeasure_set_a: sigma_measure_set_a > set_set_a > extend8495563244428889912nnreal ).
thf(sy_c_Sigma__Algebra_Oemeasure_001tf__a,type,
sigma_emeasure_a: sigma_measure_a > set_a > extend8495563244428889912nnreal ).
thf(sy_c_Sigma__Algebra_Oemeasure_001tf__b,type,
sigma_emeasure_b: sigma_measure_b > set_b > extend8495563244428889912nnreal ).
thf(sy_c_Sigma__Algebra_Omeasurable_001_062_Itf__a_M_Eo_J_001_Eo,type,
sigma_1195952539894209287_a_o_o: sigma_measure_a_o > sigma_measure_o > set_a_o_o ).
thf(sy_c_Sigma__Algebra_Omeasurable_001_062_Itf__a_M_Eo_J_001tf__a,type,
sigma_754132208923876077_a_o_a: sigma_measure_a_o > sigma_measure_a > set_a_o_a ).
thf(sy_c_Sigma__Algebra_Omeasurable_001_062_Itf__a_Mt__Real__Oreal_J_001_Eo,type,
sigma_9085598459323199629real_o: sigma_measure_a_real > sigma_measure_o > set_a_real_o ).
thf(sy_c_Sigma__Algebra_Omeasurable_001_062_Itf__a_Mt__Real__Oreal_J_001tf__a,type,
sigma_8736157696482676083real_a: sigma_measure_a_real > sigma_measure_a > set_a_real_a ).
thf(sy_c_Sigma__Algebra_Omeasurable_001_062_Itf__c_Mtf__d_J_001_Eo,type,
sigma_1714064210060623456_c_d_o: sigma_measure_c_d > sigma_measure_o > set_c_d_o ).
thf(sy_c_Sigma__Algebra_Omeasurable_001_062_Itf__c_Mtf__d_J_001tf__a,type,
sigma_3169359769398501830_c_d_a: sigma_measure_c_d > sigma_measure_a > set_c_d_a ).
thf(sy_c_Sigma__Algebra_Omeasurable_001_Eo_001_Eo,type,
sigma_measurable_o_o: sigma_measure_o > sigma_measure_o > set_o_o ).
thf(sy_c_Sigma__Algebra_Omeasurable_001_Eo_001t__Nat__Onat,type,
sigma_1999164137574644376_o_nat: sigma_measure_o > sigma_measure_nat > set_o_nat ).
thf(sy_c_Sigma__Algebra_Omeasurable_001_Eo_001t__Product____Type__Ounit,type,
sigma_4658569928750460229t_unit: sigma_measure_o > sigma_3917854644531351306t_unit > set_o_Product_unit ).
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_001t__String__Oliteral,type,
sigma_4705034698148213516iteral: sigma_measure_o > sigma_4743690205272893521iteral > set_o_literal ).
thf(sy_c_Sigma__Algebra_Omeasurable_001_Eo_001tf__a,type,
sigma_measurable_o_a: sigma_measure_o > sigma_measure_a > set_o_a ).
thf(sy_c_Sigma__Algebra_Omeasurable_001_Eo_001tf__b,type,
sigma_measurable_o_b: sigma_measure_o > sigma_measure_b > set_o_b ).
thf(sy_c_Sigma__Algebra_Omeasurable_001t__Nat__Onat_001_Eo,type,
sigma_5101835498682829686_nat_o: sigma_measure_nat > sigma_measure_o > set_nat_o ).
thf(sy_c_Sigma__Algebra_Omeasurable_001t__Nat__Onat_001t__Nat__Onat,type,
sigma_4350458207664084850at_nat: sigma_measure_nat > sigma_measure_nat > set_nat_nat ).
thf(sy_c_Sigma__Algebra_Omeasurable_001t__Nat__Onat_001tf__a,type,
sigma_4105081583803843548_nat_a: sigma_measure_nat > sigma_measure_a > set_nat_a ).
thf(sy_c_Sigma__Algebra_Omeasurable_001t__Nat__Onat_001tf__b,type,
sigma_4105081583803843549_nat_b: sigma_measure_nat > sigma_measure_b > set_nat_b ).
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__Real__Oreal,type,
sigma_5267869275261027754l_real: sigma_measure_real > sigma_measure_real > set_real_real ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__a_001_Eo,type,
sigma_measurable_a_o: sigma_measure_a > sigma_measure_o > set_a_o ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__a_001t__Nat__Onat,type,
sigma_73150082625557118_a_nat: sigma_measure_a > sigma_measure_nat > set_a_nat ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__a_001t__Product____Type__Ounit,type,
sigma_1804883157691777247t_unit: sigma_measure_a > sigma_3917854644531351306t_unit > set_a_Product_unit ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__a_001t__Real__Oreal,type,
sigma_9116425665531756122a_real: sigma_measure_a > sigma_measure_real > set_a_real ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__a_001t__String__Oliteral,type,
sigma_6403365867683794342iteral: sigma_measure_a > sigma_4743690205272893521iteral > set_a_literal ).
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_001_Eo,type,
sigma_measurable_b_o: sigma_measure_b > sigma_measure_o > set_b_o ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__b_001t__Nat__Onat,type,
sigma_1308594411581951615_b_nat: sigma_measure_b > sigma_measure_nat > set_b_nat ).
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__Product____Type__Ounit,type,
sigma_483294392074178397t_unit: sigma_measure_c > sigma_3917854644531351306t_unit > set_c_Product_unit ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__c_001t__String__Oliteral,type,
sigma_633224849116739876iteral: sigma_measure_c > sigma_4743690205272893521iteral > set_c_literal ).
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_Omeasurable_001tf__c_001tf__d,type,
sigma_measurable_c_d: sigma_measure_c > sigma_measure_d > set_c_d ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__d_001tf__d,type,
sigma_measurable_d_d: sigma_measure_d > sigma_measure_d > set_d_d ).
thf(sy_c_Sigma__Algebra_Omeasure_001tf__a,type,
sigma_measure_a2: sigma_measure_a > set_a > real ).
thf(sy_c_Sigma__Algebra_Osets_001_Eo,type,
sigma_sets_o: sigma_measure_o > set_set_o ).
thf(sy_c_Sigma__Algebra_Osets_001t__Nat__Onat,type,
sigma_sets_nat: sigma_measure_nat > set_set_nat ).
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_Ospace_001_062_Itf__a_M_Eo_J,type,
sigma_space_a_o: sigma_measure_a_o > set_a_o ).
thf(sy_c_Sigma__Algebra_Ospace_001_062_Itf__a_Mt__Real__Oreal_J,type,
sigma_space_a_real: sigma_measure_a_real > set_a_real ).
thf(sy_c_Sigma__Algebra_Ospace_001_062_Itf__c_Mtf__d_J,type,
sigma_space_c_d: sigma_measure_c_d > set_c_d ).
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_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_Sigma__Algebra_Ospace_001tf__d,type,
sigma_space_d: sigma_measure_d > set_d ).
thf(sy_c_member_001_062_I_062_Itf__a_M_Eo_J_M_Eo_J,type,
member_a_o_o: ( ( a > $o ) > $o ) > set_a_o_o > $o ).
thf(sy_c_member_001_062_I_062_Itf__a_M_Eo_J_Mtf__a_J,type,
member_a_o_a: ( ( a > $o ) > a ) > set_a_o_a > $o ).
thf(sy_c_member_001_062_I_062_Itf__a_Mt__Real__Oreal_J_M_Eo_J,type,
member_a_real_o: ( ( a > real ) > $o ) > set_a_real_o > $o ).
thf(sy_c_member_001_062_I_062_Itf__a_Mt__Real__Oreal_J_Mtf__a_J,type,
member_a_real_a: ( ( a > real ) > a ) > set_a_real_a > $o ).
thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_M_Eo_J,type,
member_c_d_o: ( ( c > d ) > $o ) > set_c_d_o > $o ).
thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_Mtf__a_J,type,
member_c_d_a: ( ( c > d ) > a ) > set_c_d_a > $o ).
thf(sy_c_member_001_062_I_Eo_M_Eo_J,type,
member_o_o: ( $o > $o ) > set_o_o > $o ).
thf(sy_c_member_001_062_I_Eo_Mt__Nat__Onat_J,type,
member_o_nat: ( $o > nat ) > set_o_nat > $o ).
thf(sy_c_member_001_062_I_Eo_Mt__Product____Type__Ounit_J,type,
member2370919827131729009t_unit: ( $o > product_unit ) > set_o_Product_unit > $o ).
thf(sy_c_member_001_062_I_Eo_Mt__Real__Oreal_J,type,
member_o_real: ( $o > real ) > set_o_real > $o ).
thf(sy_c_member_001_062_I_Eo_Mt__String__Oliteral_J,type,
member_o_literal: ( $o > literal ) > set_o_literal > $o ).
thf(sy_c_member_001_062_I_Eo_Mtf__a_J,type,
member_o_a: ( $o > a ) > set_o_a > $o ).
thf(sy_c_member_001_062_I_Eo_Mtf__b_J,type,
member_o_b: ( $o > b ) > set_o_b > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_Eo_J,type,
member_nat_o: ( nat > $o ) > set_nat_o > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__a_J,type,
member_nat_a: ( nat > a ) > set_nat_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__b_J,type,
member_nat_b: ( nat > b ) > set_nat_b > $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__Real__Oreal_J,type,
member_real_real: ( real > real ) > set_real_real > $o ).
thf(sy_c_member_001_062_Itf__a_M_Eo_J,type,
member_a_o: ( a > $o ) > set_a_o > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Nat__Onat_J,type,
member_a_nat: ( a > nat ) > set_a_nat > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Product____Type__Ounit_J,type,
member3545396013249599883t_unit: ( a > product_unit ) > set_a_Product_unit > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Real__Oreal_J,type,
member_a_real: ( a > real ) > set_a_real > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__String__Oliteral_J,type,
member_a_literal: ( a > literal ) > set_a_literal > $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_M_Eo_J,type,
member_b_o: ( b > $o ) > set_b_o > $o ).
thf(sy_c_member_001_062_Itf__b_Mt__Nat__Onat_J,type,
member_b_nat: ( b > nat ) > set_b_nat > $o ).
thf(sy_c_member_001_062_Itf__b_Mtf__a_J,type,
member_b_a: ( b > a ) > set_b_a > $o ).
thf(sy_c_member_001_062_Itf__b_Mtf__b_J,type,
member_b_b: ( b > b ) > set_b_b > $o ).
thf(sy_c_member_001_062_Itf__c_M_Eo_J,type,
member_c_o: ( c > $o ) > set_c_o > $o ).
thf(sy_c_member_001_062_Itf__c_Mt__Nat__Onat_J,type,
member_c_nat: ( c > nat ) > set_c_nat > $o ).
thf(sy_c_member_001_062_Itf__c_Mt__Product____Type__Ounit_J,type,
member2223807247632001033t_unit: ( c > product_unit ) > set_c_Product_unit > $o ).
thf(sy_c_member_001_062_Itf__c_Mt__String__Oliteral_J,type,
member_c_literal: ( c > literal ) > set_c_literal > $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_062_Itf__c_Mtf__d_J,type,
member_c_d: ( c > d ) > set_c_d > $o ).
thf(sy_c_member_001_062_Itf__d_Mtf__d_J,type,
member_d_d: ( d > d ) > set_d_d > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Extended____Nonnegative____Real__Oennreal,type,
member7908768830364227535nnreal: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Ounit,type,
member_Product_unit: product_unit > set_Product_unit > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_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_001t__String__Oliteral,type,
member_literal: literal > set_literal > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_c_member_001tf__c,type,
member_c: c > set_c > $o ).
thf(sy_c_member_001tf__d,type,
member_d: d > set_d > $o ).
thf(sy_v_I,type,
i: set_b ).
thf(sy_v_M,type,
m: sigma_measure_a ).
thf(sy_v_M_H,type,
m2: b > sigma_measure_c ).
thf(sy_v_N,type,
n: b > sigma_measure_d ).
thf(sy_v_X,type,
x: b > a > c ).
thf(sy_v_Y,type,
y: b > c > d ).
thf(sy_v_k,type,
k: nat ).
% Relevant facts (1276)
thf(fact_0_prob__space__axioms,axiom,
probab7247484486040049089pace_a @ m ).
% prob_space_axioms
thf(fact_1_assms_I1_J,axiom,
prob_k6574085460301583242_a_b_c @ m @ k @ m2 @ x @ i ).
% assms(1)
thf(fact_2_assms_I2_J,axiom,
! [I: b] :
( ( member_b @ I @ i )
=> ( member_c_d @ ( y @ I ) @ ( sigma_measurable_c_d @ ( m2 @ I ) @ ( n @ I ) ) ) ) ).
% assms(2)
thf(fact_3_subprob__space__axioms,axiom,
giry_subprob_space_a @ m ).
% subprob_space_axioms
thf(fact_4_prob__space_Ok__wise__indep__vars_Ocong,axiom,
prob_k6574085460301583243_a_b_d = prob_k6574085460301583243_a_b_d ).
% prob_space.k_wise_indep_vars.cong
thf(fact_5_prob__space_Ok__wise__indep__vars_Ocong,axiom,
prob_k6574085460301583242_a_b_c = prob_k6574085460301583242_a_b_c ).
% prob_space.k_wise_indep_vars.cong
thf(fact_6_indep__sets__cong,axiom,
! [I2: set_c_d,J: set_c_d,F: ( c > d ) > set_set_a,G: ( c > d ) > set_set_a] :
( ( I2 = J )
=> ( ! [I3: c > d] :
( ( member_c_d @ I3 @ I2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( indepe4867465796832040824_a_c_d @ m @ F @ I2 )
= ( indepe4867465796832040824_a_c_d @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_7_indep__sets__cong,axiom,
! [I2: set_a_real,J: set_a_real,F: ( a > real ) > set_set_a,G: ( a > real ) > set_set_a] :
( ( I2 = J )
=> ( ! [I3: a > real] :
( ( member_a_real @ I3 @ I2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( indepe4749599203615801097a_real @ m @ F @ I2 )
= ( indepe4749599203615801097a_real @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_8_indep__sets__cong,axiom,
! [I2: set_a_o,J: set_a_o,F: ( a > $o ) > set_set_a,G: ( a > $o ) > set_set_a] :
( ( I2 = J )
=> ( ! [I3: a > $o] :
( ( member_a_o @ I3 @ I2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( indepe7801696130336798481_a_a_o @ m @ F @ I2 )
= ( indepe7801696130336798481_a_a_o @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_9_indep__sets__cong,axiom,
! [I2: set_b,J: set_b,F: b > set_set_a,G: b > set_set_a] :
( ( I2 = J )
=> ( ! [I3: b] :
( ( member_b @ I3 @ I2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( indepe8927441866673418605ts_a_b @ m @ F @ I2 )
= ( indepe8927441866673418605ts_a_b @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_10_indep__sets__cong,axiom,
! [I2: set_a,J: set_a,F: a > set_set_a,G: a > set_set_a] :
( ( I2 = J )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( indepe8927441866673418604ts_a_a @ m @ F @ I2 )
= ( indepe8927441866673418604ts_a_a @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_11_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_12_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_13_finite__measure__axioms,axiom,
measur930452917991658466sure_a @ m ).
% finite_measure_axioms
thf(fact_14_sigma__finite__measure__axioms,axiom,
measur4308613598931908895sure_a @ m ).
% sigma_finite_measure_axioms
thf(fact_15_local_Ointegrable__const,axiom,
! [A: real] :
( bochne2139062162225249880a_real @ m
@ ^ [X: a] : A ) ).
% local.integrable_const
thf(fact_16_k__universal__def,axiom,
! [K: nat,X2: b > a > d,D: set_b,R: set_d] :
( ( prob_k5927988752154431058_a_b_d @ m @ K @ X2 @ D @ R )
= ( ( prob_k6574085460301583243_a_b_d @ m @ K
@ ^ [Uu: b] : ( sigma_count_space_d @ top_top_set_d )
@ X2
@ D )
& ! [X: b] :
( ( member_b @ X @ D )
=> ( prob_uniform_on_a_d @ m @ ( X2 @ X ) @ R ) ) ) ) ).
% k_universal_def
thf(fact_17_k__universal__def,axiom,
! [K: nat,X2: b > a > c,D: set_b,R: set_c] :
( ( prob_k5927988752154431057_a_b_c @ m @ K @ X2 @ D @ R )
= ( ( prob_k6574085460301583242_a_b_c @ m @ K
@ ^ [Uu: b] : ( sigma_count_space_c @ top_top_set_c )
@ X2
@ D )
& ! [X: b] :
( ( member_b @ X @ D )
=> ( prob_uniform_on_a_c @ m @ ( X2 @ X ) @ R ) ) ) ) ).
% k_universal_def
thf(fact_18_indep__vars__compose2,axiom,
! [M: b > sigma_measure_c,X2: b > a > c,I2: set_b,Y: b > c > d,N: b > sigma_measure_d] :
( ( indepe7639357355105118967_a_b_c @ m @ M @ X2 @ I2 )
=> ( ! [I3: b] :
( ( member_b @ I3 @ I2 )
=> ( member_c_d @ ( Y @ I3 ) @ ( sigma_measurable_c_d @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe7639357355105118968_a_b_d @ m @ N
@ ^ [I4: b,X: a] : ( Y @ I4 @ ( X2 @ I4 @ X ) )
@ I2 ) ) ) ).
% indep_vars_compose2
thf(fact_19_indep__vars__compose2,axiom,
! [M: a > sigma_measure_c,X2: a > a > c,I2: set_a,Y: a > c > d,N: a > sigma_measure_d] :
( ( indepe1203440900223019192_a_a_c @ m @ M @ X2 @ I2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I2 )
=> ( member_c_d @ ( Y @ I3 ) @ ( sigma_measurable_c_d @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe1203440900223019193_a_a_d @ m @ N
@ ^ [I4: a,X: a] : ( Y @ I4 @ ( X2 @ I4 @ X ) )
@ I2 ) ) ) ).
% indep_vars_compose2
thf(fact_20_indep__vars__compose2,axiom,
! [M: nat > sigma_measure_c,X2: nat > a > c,I2: set_nat,Y: nat > c > d,N: nat > sigma_measure_d] :
( ( indepe3245197900929106296_nat_c @ m @ M @ X2 @ I2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( member_c_d @ ( Y @ I3 ) @ ( sigma_measurable_c_d @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe3245197900929106297_nat_d @ m @ N
@ ^ [I4: nat,X: a] : ( Y @ I4 @ ( X2 @ I4 @ X ) )
@ I2 ) ) ) ).
% indep_vars_compose2
thf(fact_21_indep__vars__compose2,axiom,
! [M: $o > sigma_measure_c,X2: $o > a > c,I2: set_o,Y: $o > c > d,N: $o > sigma_measure_d] :
( ( indepe3252683823613847070_a_o_c @ m @ M @ X2 @ I2 )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( member_c_d @ ( Y @ I3 ) @ ( sigma_measurable_c_d @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe3252683823613847071_a_o_d @ m @ N
@ ^ [I4: $o,X: a] : ( Y @ I4 @ ( X2 @ I4 @ X ) )
@ I2 ) ) ) ).
% indep_vars_compose2
thf(fact_22_indep__vars__compose2,axiom,
! [M: b > sigma_measure_a,X2: b > a > a,I2: set_b,Y: b > a > $o,N: b > sigma_measure_o] :
( ( indepe7639357355105118965_a_b_a @ m @ M @ X2 @ I2 )
=> ( ! [I3: b] :
( ( member_b @ I3 @ I2 )
=> ( member_a_o @ ( Y @ I3 ) @ ( sigma_measurable_a_o @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe432941580615200527_a_b_o @ m @ N
@ ^ [I4: b,X: a] : ( Y @ I4 @ ( X2 @ I4 @ X ) )
@ I2 ) ) ) ).
% indep_vars_compose2
thf(fact_23_indep__vars__compose2,axiom,
! [M: a > sigma_measure_a,X2: a > a > a,I2: set_a,Y: a > a > $o,N: a > sigma_measure_o] :
( ( indepe1203440900223019190_a_a_a @ m @ M @ X2 @ I2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I2 )
=> ( member_a_o @ ( Y @ I3 ) @ ( sigma_measurable_a_o @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe3332163980079594832_a_a_o @ m @ N
@ ^ [I4: a,X: a] : ( Y @ I4 @ ( X2 @ I4 @ X ) )
@ I2 ) ) ) ).
% indep_vars_compose2
thf(fact_24_indep__vars__compose2,axiom,
! [M: nat > sigma_measure_a,X2: nat > a > a,I2: set_nat,Y: nat > a > $o,N: nat > sigma_measure_o] :
( ( indepe3245197900929106294_nat_a @ m @ M @ X2 @ I2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( member_a_o @ ( Y @ I3 ) @ ( sigma_measurable_a_o @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe6621553285932267024_nat_o @ m @ N
@ ^ [I4: nat,X: a] : ( Y @ I4 @ ( X2 @ I4 @ X ) )
@ I2 ) ) ) ).
% indep_vars_compose2
thf(fact_25_indep__vars__compose2,axiom,
! [M: $o > sigma_measure_a,X2: $o > a > a,I2: set_o,Y: $o > a > $o,N: $o > sigma_measure_o] :
( ( indepe3252683823613847068_a_o_a @ m @ M @ X2 @ I2 )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( member_a_o @ ( Y @ I3 ) @ ( sigma_measurable_a_o @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe9162428965118168502_a_o_o @ m @ N
@ ^ [I4: $o,X: a] : ( Y @ I4 @ ( X2 @ I4 @ X ) )
@ I2 ) ) ) ).
% indep_vars_compose2
thf(fact_26_indep__vars__compose2,axiom,
! [M: b > sigma_measure_a,X2: b > a > a,I2: set_b,Y: b > a > real,N: b > sigma_measure_real] :
( ( indepe7639357355105118965_a_b_a @ m @ M @ X2 @ I2 )
=> ( ! [I3: b] :
( ( member_b @ I3 @ I2 )
=> ( member_a_real @ ( Y @ I3 ) @ ( sigma_9116425665531756122a_real @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe8265442546547513973b_real @ m @ N
@ ^ [I4: b,X: a] : ( Y @ I4 @ ( X2 @ I4 @ X ) )
@ I2 ) ) ) ).
% indep_vars_compose2
thf(fact_27_indep__vars__compose2,axiom,
! [M: a > sigma_measure_a,X2: a > a > a,I2: set_a,Y: a > a > real,N: a > sigma_measure_real] :
( ( indepe1203440900223019190_a_a_a @ m @ M @ X2 @ I2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I2 )
=> ( member_a_real @ ( Y @ I3 ) @ ( sigma_9116425665531756122a_real @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe2669223931359383284a_real @ m @ N
@ ^ [I4: a,X: a] : ( Y @ I4 @ ( X2 @ I4 @ X ) )
@ I2 ) ) ) ).
% indep_vars_compose2
thf(fact_28_AE__False,axiom,
~ ( eventually_a
@ ^ [X: a] : $false
@ ( measure_ae_filter_a @ m ) ) ).
% AE_False
thf(fact_29_AE__contr,axiom,
! [P: a > $o] :
( ( eventually_a @ P @ ( measure_ae_filter_a @ m ) )
=> ~ ( eventually_a
@ ^ [Omega: a] :
~ ( P @ Omega )
@ ( measure_ae_filter_a @ m ) ) ) ).
% AE_contr
thf(fact_30_AE__const,axiom,
! [P: $o] :
( ( eventually_a
@ ^ [X: a] : P
@ ( measure_ae_filter_a @ m ) )
= P ) ).
% AE_const
thf(fact_31_prob__space_Ok__universal__def,axiom,
! [M2: sigma_measure_a,K: nat,X2: b > a > d,D: set_b,R: set_d] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( prob_k5927988752154431058_a_b_d @ M2 @ K @ X2 @ D @ R )
= ( ( prob_k6574085460301583243_a_b_d @ M2 @ K
@ ^ [Uu: b] : ( sigma_count_space_d @ top_top_set_d )
@ X2
@ D )
& ! [X: b] :
( ( member_b @ X @ D )
=> ( prob_uniform_on_a_d @ M2 @ ( X2 @ X ) @ R ) ) ) ) ) ).
% prob_space.k_universal_def
thf(fact_32_prob__space_Ok__universal__def,axiom,
! [M2: sigma_measure_a,K: nat,X2: b > a > c,D: set_b,R: set_c] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( prob_k5927988752154431057_a_b_c @ M2 @ K @ X2 @ D @ R )
= ( ( prob_k6574085460301583242_a_b_c @ M2 @ K
@ ^ [Uu: b] : ( sigma_count_space_c @ top_top_set_c )
@ X2
@ D )
& ! [X: b] :
( ( member_b @ X @ D )
=> ( prob_uniform_on_a_c @ M2 @ ( X2 @ X ) @ R ) ) ) ) ) ).
% prob_space.k_universal_def
thf(fact_33_AE__conj__iff,axiom,
! [P: a > $o,Q: a > $o,M2: sigma_measure_a] :
( ( eventually_a
@ ^ [X: a] :
( ( P @ X )
& ( Q @ X ) )
@ ( measure_ae_filter_a @ M2 ) )
= ( ( eventually_a @ P @ ( measure_ae_filter_a @ M2 ) )
& ( eventually_a @ Q @ ( measure_ae_filter_a @ M2 ) ) ) ) ).
% AE_conj_iff
thf(fact_34_AE__eq__constD_I1_J,axiom,
! [Y2: a] :
( ( eventually_a
@ ^ [X: a] : ( X = Y2 )
@ ( measure_ae_filter_a @ m ) )
=> ( m
= ( giry_return_a @ m @ Y2 ) ) ) ).
% AE_eq_constD(1)
thf(fact_35_prob__space_Oindep__vars__compose2,axiom,
! [M2: sigma_measure_a,M: b > sigma_measure_c,X2: b > a > c,I2: set_b,Y: b > c > d,N: b > sigma_measure_d] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe7639357355105118967_a_b_c @ M2 @ M @ X2 @ I2 )
=> ( ! [I3: b] :
( ( member_b @ I3 @ I2 )
=> ( member_c_d @ ( Y @ I3 ) @ ( sigma_measurable_c_d @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe7639357355105118968_a_b_d @ M2 @ N
@ ^ [I4: b,X: a] : ( Y @ I4 @ ( X2 @ I4 @ X ) )
@ I2 ) ) ) ) ).
% prob_space.indep_vars_compose2
thf(fact_36_prob__space_Oindep__vars__compose2,axiom,
! [M2: sigma_measure_a,M: a > sigma_measure_c,X2: a > a > c,I2: set_a,Y: a > c > d,N: a > sigma_measure_d] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe1203440900223019192_a_a_c @ M2 @ M @ X2 @ I2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I2 )
=> ( member_c_d @ ( Y @ I3 ) @ ( sigma_measurable_c_d @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe1203440900223019193_a_a_d @ M2 @ N
@ ^ [I4: a,X: a] : ( Y @ I4 @ ( X2 @ I4 @ X ) )
@ I2 ) ) ) ) ).
% prob_space.indep_vars_compose2
thf(fact_37_prob__space_Oindep__vars__compose2,axiom,
! [M2: sigma_measure_a,M: nat > sigma_measure_c,X2: nat > a > c,I2: set_nat,Y: nat > c > d,N: nat > sigma_measure_d] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe3245197900929106296_nat_c @ M2 @ M @ X2 @ I2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( member_c_d @ ( Y @ I3 ) @ ( sigma_measurable_c_d @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe3245197900929106297_nat_d @ M2 @ N
@ ^ [I4: nat,X: a] : ( Y @ I4 @ ( X2 @ I4 @ X ) )
@ I2 ) ) ) ) ).
% prob_space.indep_vars_compose2
thf(fact_38_prob__space_Oindep__vars__compose2,axiom,
! [M2: sigma_measure_a,M: $o > sigma_measure_c,X2: $o > a > c,I2: set_o,Y: $o > c > d,N: $o > sigma_measure_d] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe3252683823613847070_a_o_c @ M2 @ M @ X2 @ I2 )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( member_c_d @ ( Y @ I3 ) @ ( sigma_measurable_c_d @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe3252683823613847071_a_o_d @ M2 @ N
@ ^ [I4: $o,X: a] : ( Y @ I4 @ ( X2 @ I4 @ X ) )
@ I2 ) ) ) ) ).
% prob_space.indep_vars_compose2
thf(fact_39_prob__space_Oindep__vars__compose2,axiom,
! [M2: sigma_measure_a,M: b > sigma_measure_a,X2: b > a > a,I2: set_b,Y: b > a > $o,N: b > sigma_measure_o] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe7639357355105118965_a_b_a @ M2 @ M @ X2 @ I2 )
=> ( ! [I3: b] :
( ( member_b @ I3 @ I2 )
=> ( member_a_o @ ( Y @ I3 ) @ ( sigma_measurable_a_o @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe432941580615200527_a_b_o @ M2 @ N
@ ^ [I4: b,X: a] : ( Y @ I4 @ ( X2 @ I4 @ X ) )
@ I2 ) ) ) ) ).
% prob_space.indep_vars_compose2
thf(fact_40_prob__space_Oindep__vars__compose2,axiom,
! [M2: sigma_measure_a,M: a > sigma_measure_a,X2: a > a > a,I2: set_a,Y: a > a > $o,N: a > sigma_measure_o] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe1203440900223019190_a_a_a @ M2 @ M @ X2 @ I2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I2 )
=> ( member_a_o @ ( Y @ I3 ) @ ( sigma_measurable_a_o @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe3332163980079594832_a_a_o @ M2 @ N
@ ^ [I4: a,X: a] : ( Y @ I4 @ ( X2 @ I4 @ X ) )
@ I2 ) ) ) ) ).
% prob_space.indep_vars_compose2
thf(fact_41_prob__space_Oindep__vars__compose2,axiom,
! [M2: sigma_measure_a,M: nat > sigma_measure_a,X2: nat > a > a,I2: set_nat,Y: nat > a > $o,N: nat > sigma_measure_o] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe3245197900929106294_nat_a @ M2 @ M @ X2 @ I2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( member_a_o @ ( Y @ I3 ) @ ( sigma_measurable_a_o @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe6621553285932267024_nat_o @ M2 @ N
@ ^ [I4: nat,X: a] : ( Y @ I4 @ ( X2 @ I4 @ X ) )
@ I2 ) ) ) ) ).
% prob_space.indep_vars_compose2
thf(fact_42_prob__space_Oindep__vars__compose2,axiom,
! [M2: sigma_measure_a,M: $o > sigma_measure_a,X2: $o > a > a,I2: set_o,Y: $o > a > $o,N: $o > sigma_measure_o] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe3252683823613847068_a_o_a @ M2 @ M @ X2 @ I2 )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( member_a_o @ ( Y @ I3 ) @ ( sigma_measurable_a_o @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe9162428965118168502_a_o_o @ M2 @ N
@ ^ [I4: $o,X: a] : ( Y @ I4 @ ( X2 @ I4 @ X ) )
@ I2 ) ) ) ) ).
% prob_space.indep_vars_compose2
thf(fact_43_prob__space_Oindep__vars__compose2,axiom,
! [M2: sigma_measure_a,M: b > sigma_measure_a,X2: b > a > a,I2: set_b,Y: b > a > real,N: b > sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe7639357355105118965_a_b_a @ M2 @ M @ X2 @ I2 )
=> ( ! [I3: b] :
( ( member_b @ I3 @ I2 )
=> ( member_a_real @ ( Y @ I3 ) @ ( sigma_9116425665531756122a_real @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe8265442546547513973b_real @ M2 @ N
@ ^ [I4: b,X: a] : ( Y @ I4 @ ( X2 @ I4 @ X ) )
@ I2 ) ) ) ) ).
% prob_space.indep_vars_compose2
thf(fact_44_prob__space_Oindep__vars__compose2,axiom,
! [M2: sigma_measure_a,M: a > sigma_measure_a,X2: a > a > a,I2: set_a,Y: a > a > real,N: a > sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe1203440900223019190_a_a_a @ M2 @ M @ X2 @ I2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I2 )
=> ( member_a_real @ ( Y @ I3 ) @ ( sigma_9116425665531756122a_real @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe2669223931359383284a_real @ M2 @ N
@ ^ [I4: a,X: a] : ( Y @ I4 @ ( X2 @ I4 @ X ) )
@ I2 ) ) ) ) ).
% prob_space.indep_vars_compose2
thf(fact_45_prob__space_OAE__False,axiom,
! [M2: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ~ ( eventually_a
@ ^ [X: a] : $false
@ ( measure_ae_filter_a @ M2 ) ) ) ).
% prob_space.AE_False
thf(fact_46_prob__space_OAE__const,axiom,
! [M2: sigma_measure_a,P: $o] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( eventually_a
@ ^ [X: a] : P
@ ( measure_ae_filter_a @ M2 ) )
= P ) ) ).
% prob_space.AE_const
thf(fact_47_prob__space_OAE__contr,axiom,
! [M2: sigma_measure_a,P: a > $o] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( eventually_a @ P @ ( measure_ae_filter_a @ M2 ) )
=> ~ ( eventually_a
@ ^ [Omega: a] :
~ ( P @ Omega )
@ ( measure_ae_filter_a @ M2 ) ) ) ) ).
% prob_space.AE_contr
thf(fact_48_AE__count__space,axiom,
! [P: a > $o,A2: set_a] :
( ( eventually_a @ P @ ( measure_ae_filter_a @ ( sigma_count_space_a @ A2 ) ) )
= ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( P @ X ) ) ) ) ).
% AE_count_space
thf(fact_49_AE__count__space,axiom,
! [P: $o > $o,A2: set_o] :
( ( eventually_o @ P @ ( measure_ae_filter_o @ ( sigma_count_space_o @ A2 ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( P @ X ) ) ) ) ).
% AE_count_space
thf(fact_50_pred__count__space__const1,axiom,
! [F2: a > nat,M2: sigma_measure_a,C: nat] :
( ( member_a_nat @ F2 @ ( sigma_73150082625557118_a_nat @ M2 @ ( sigma_7685844798829912695ce_nat @ top_top_set_nat ) ) )
=> ( member_a_o
@ ^ [X: a] :
( ( F2 @ X )
= C )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ).
% pred_count_space_const1
thf(fact_51_pred__count__space__const1,axiom,
! [F2: a > literal,M2: sigma_measure_a,C: literal] :
( ( member_a_literal @ F2 @ ( sigma_6403365867683794342iteral @ M2 @ ( sigma_7122031993496068461iteral @ top_top_set_literal ) ) )
=> ( member_a_o
@ ^ [X: a] :
( ( F2 @ X )
= C )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ).
% pred_count_space_const1
thf(fact_52_pred__count__space__const1,axiom,
! [F2: a > product_unit,M2: sigma_measure_a,C: product_unit] :
( ( member3545396013249599883t_unit @ F2 @ ( sigma_1804883157691777247t_unit @ M2 @ ( sigma_2593334793047783974t_unit @ top_to1996260823553986621t_unit ) ) )
=> ( member_a_o
@ ^ [X: a] :
( ( F2 @ X )
= C )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ).
% pred_count_space_const1
thf(fact_53_pred__count__space__const1,axiom,
! [F2: c > d,M2: sigma_measure_c,C: d] :
( ( member_c_d @ F2 @ ( sigma_measurable_c_d @ M2 @ ( sigma_count_space_d @ top_top_set_d ) ) )
=> ( member_c_o
@ ^ [X: c] :
( ( F2 @ X )
= C )
@ ( sigma_measurable_c_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ).
% pred_count_space_const1
thf(fact_54_pred__count__space__const1,axiom,
! [F2: a > real,M2: sigma_measure_a,C: real] :
( ( member_a_real @ F2 @ ( sigma_9116425665531756122a_real @ M2 @ ( sigma_8508918144308765139e_real @ top_top_set_real ) ) )
=> ( member_a_o
@ ^ [X: a] :
( ( F2 @ X )
= C )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ).
% pred_count_space_const1
thf(fact_55_pred__count__space__const1,axiom,
! [F2: a > $o,M2: sigma_measure_a,C: $o] :
( ( member_a_o @ F2 @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( member_a_o
@ ^ [X: a] :
( ( F2 @ X )
= C )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ).
% pred_count_space_const1
thf(fact_56_pred__count__space__const2,axiom,
! [F2: a > nat,M2: sigma_measure_a,C: nat] :
( ( member_a_nat @ F2 @ ( sigma_73150082625557118_a_nat @ M2 @ ( sigma_7685844798829912695ce_nat @ top_top_set_nat ) ) )
=> ( member_a_o
@ ^ [X: a] :
( C
= ( F2 @ X ) )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ).
% pred_count_space_const2
thf(fact_57_pred__count__space__const2,axiom,
! [F2: a > literal,M2: sigma_measure_a,C: literal] :
( ( member_a_literal @ F2 @ ( sigma_6403365867683794342iteral @ M2 @ ( sigma_7122031993496068461iteral @ top_top_set_literal ) ) )
=> ( member_a_o
@ ^ [X: a] :
( C
= ( F2 @ X ) )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ).
% pred_count_space_const2
thf(fact_58_pred__count__space__const2,axiom,
! [F2: a > product_unit,M2: sigma_measure_a,C: product_unit] :
( ( member3545396013249599883t_unit @ F2 @ ( sigma_1804883157691777247t_unit @ M2 @ ( sigma_2593334793047783974t_unit @ top_to1996260823553986621t_unit ) ) )
=> ( member_a_o
@ ^ [X: a] :
( C
= ( F2 @ X ) )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ).
% pred_count_space_const2
thf(fact_59_pred__count__space__const2,axiom,
! [F2: c > d,M2: sigma_measure_c,C: d] :
( ( member_c_d @ F2 @ ( sigma_measurable_c_d @ M2 @ ( sigma_count_space_d @ top_top_set_d ) ) )
=> ( member_c_o
@ ^ [X: c] :
( C
= ( F2 @ X ) )
@ ( sigma_measurable_c_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ).
% pred_count_space_const2
thf(fact_60_pred__count__space__const2,axiom,
! [F2: a > real,M2: sigma_measure_a,C: real] :
( ( member_a_real @ F2 @ ( sigma_9116425665531756122a_real @ M2 @ ( sigma_8508918144308765139e_real @ top_top_set_real ) ) )
=> ( member_a_o
@ ^ [X: a] :
( C
= ( F2 @ X ) )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ).
% pred_count_space_const2
thf(fact_61_pred__count__space__const2,axiom,
! [F2: a > $o,M2: sigma_measure_a,C: $o] :
( ( member_a_o @ F2 @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( member_a_o
@ ^ [X: a] :
( C
= ( F2 @ X ) )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ).
% pred_count_space_const2
thf(fact_62_measurable__compose__countable,axiom,
! [F2: nat > c > d,M2: sigma_measure_c,N: sigma_measure_d,G2: c > nat] :
( ! [I3: nat] : ( member_c_d @ ( F2 @ I3 ) @ ( sigma_measurable_c_d @ M2 @ N ) )
=> ( ( member_c_nat @ G2 @ ( sigma_2544038740538346112_c_nat @ M2 @ ( sigma_7685844798829912695ce_nat @ top_top_set_nat ) ) )
=> ( member_c_d
@ ^ [X: c] : ( F2 @ ( G2 @ X ) @ X )
@ ( sigma_measurable_c_d @ M2 @ N ) ) ) ) ).
% measurable_compose_countable
thf(fact_63_measurable__compose__countable,axiom,
! [F2: literal > c > d,M2: sigma_measure_c,N: sigma_measure_d,G2: c > literal] :
( ! [I3: literal] : ( member_c_d @ ( F2 @ I3 ) @ ( sigma_measurable_c_d @ M2 @ N ) )
=> ( ( member_c_literal @ G2 @ ( sigma_633224849116739876iteral @ M2 @ ( sigma_7122031993496068461iteral @ top_top_set_literal ) ) )
=> ( member_c_d
@ ^ [X: c] : ( F2 @ ( G2 @ X ) @ X )
@ ( sigma_measurable_c_d @ M2 @ N ) ) ) ) ).
% measurable_compose_countable
thf(fact_64_measurable__compose__countable,axiom,
! [F2: product_unit > c > d,M2: sigma_measure_c,N: sigma_measure_d,G2: c > product_unit] :
( ! [I3: product_unit] : ( member_c_d @ ( F2 @ I3 ) @ ( sigma_measurable_c_d @ M2 @ N ) )
=> ( ( member2223807247632001033t_unit @ G2 @ ( sigma_483294392074178397t_unit @ M2 @ ( sigma_2593334793047783974t_unit @ top_to1996260823553986621t_unit ) ) )
=> ( member_c_d
@ ^ [X: c] : ( F2 @ ( G2 @ X ) @ X )
@ ( sigma_measurable_c_d @ M2 @ N ) ) ) ) ).
% measurable_compose_countable
thf(fact_65_measurable__compose__countable,axiom,
! [F2: nat > a > $o,M2: sigma_measure_a,N: sigma_measure_o,G2: a > nat] :
( ! [I3: nat] : ( member_a_o @ ( F2 @ I3 ) @ ( sigma_measurable_a_o @ M2 @ N ) )
=> ( ( member_a_nat @ G2 @ ( sigma_73150082625557118_a_nat @ M2 @ ( sigma_7685844798829912695ce_nat @ top_top_set_nat ) ) )
=> ( member_a_o
@ ^ [X: a] : ( F2 @ ( G2 @ X ) @ X )
@ ( sigma_measurable_a_o @ M2 @ N ) ) ) ) ).
% measurable_compose_countable
thf(fact_66_measurable__compose__countable,axiom,
! [F2: literal > a > $o,M2: sigma_measure_a,N: sigma_measure_o,G2: a > literal] :
( ! [I3: literal] : ( member_a_o @ ( F2 @ I3 ) @ ( sigma_measurable_a_o @ M2 @ N ) )
=> ( ( member_a_literal @ G2 @ ( sigma_6403365867683794342iteral @ M2 @ ( sigma_7122031993496068461iteral @ top_top_set_literal ) ) )
=> ( member_a_o
@ ^ [X: a] : ( F2 @ ( G2 @ X ) @ X )
@ ( sigma_measurable_a_o @ M2 @ N ) ) ) ) ).
% measurable_compose_countable
thf(fact_67_measurable__compose__countable,axiom,
! [F2: product_unit > a > $o,M2: sigma_measure_a,N: sigma_measure_o,G2: a > product_unit] :
( ! [I3: product_unit] : ( member_a_o @ ( F2 @ I3 ) @ ( sigma_measurable_a_o @ M2 @ N ) )
=> ( ( member3545396013249599883t_unit @ G2 @ ( sigma_1804883157691777247t_unit @ M2 @ ( sigma_2593334793047783974t_unit @ top_to1996260823553986621t_unit ) ) )
=> ( member_a_o
@ ^ [X: a] : ( F2 @ ( G2 @ X ) @ X )
@ ( sigma_measurable_a_o @ M2 @ N ) ) ) ) ).
% measurable_compose_countable
thf(fact_68_measurable__compose__countable,axiom,
! [F2: nat > a > real,M2: sigma_measure_a,N: sigma_measure_real,G2: a > nat] :
( ! [I3: nat] : ( member_a_real @ ( F2 @ I3 ) @ ( sigma_9116425665531756122a_real @ M2 @ N ) )
=> ( ( member_a_nat @ G2 @ ( sigma_73150082625557118_a_nat @ M2 @ ( sigma_7685844798829912695ce_nat @ top_top_set_nat ) ) )
=> ( member_a_real
@ ^ [X: a] : ( F2 @ ( G2 @ X ) @ X )
@ ( sigma_9116425665531756122a_real @ M2 @ N ) ) ) ) ).
% measurable_compose_countable
thf(fact_69_measurable__compose__countable,axiom,
! [F2: literal > a > real,M2: sigma_measure_a,N: sigma_measure_real,G2: a > literal] :
( ! [I3: literal] : ( member_a_real @ ( F2 @ I3 ) @ ( sigma_9116425665531756122a_real @ M2 @ N ) )
=> ( ( member_a_literal @ G2 @ ( sigma_6403365867683794342iteral @ M2 @ ( sigma_7122031993496068461iteral @ top_top_set_literal ) ) )
=> ( member_a_real
@ ^ [X: a] : ( F2 @ ( G2 @ X ) @ X )
@ ( sigma_9116425665531756122a_real @ M2 @ N ) ) ) ) ).
% measurable_compose_countable
thf(fact_70_measurable__compose__countable,axiom,
! [F2: product_unit > a > real,M2: sigma_measure_a,N: sigma_measure_real,G2: a > product_unit] :
( ! [I3: product_unit] : ( member_a_real @ ( F2 @ I3 ) @ ( sigma_9116425665531756122a_real @ M2 @ N ) )
=> ( ( member3545396013249599883t_unit @ G2 @ ( sigma_1804883157691777247t_unit @ M2 @ ( sigma_2593334793047783974t_unit @ top_to1996260823553986621t_unit ) ) )
=> ( member_a_real
@ ^ [X: a] : ( F2 @ ( G2 @ X ) @ X )
@ ( sigma_9116425665531756122a_real @ M2 @ N ) ) ) ) ).
% measurable_compose_countable
thf(fact_71_measurable__compose__countable,axiom,
! [F2: $o > c > d,M2: sigma_measure_c,N: sigma_measure_d,G2: c > $o] :
( ! [I3: $o] : ( member_c_d @ ( F2 @ I3 ) @ ( sigma_measurable_c_d @ M2 @ N ) )
=> ( ( member_c_o @ G2 @ ( sigma_measurable_c_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( member_c_d
@ ^ [X: c] : ( F2 @ ( G2 @ X ) @ X )
@ ( sigma_measurable_c_d @ M2 @ N ) ) ) ) ).
% measurable_compose_countable
thf(fact_72_measurable__count__space__const,axiom,
! [C: d,M2: sigma_measure_c] :
( member_c_d
@ ^ [X: c] : C
@ ( sigma_measurable_c_d @ M2 @ ( sigma_count_space_d @ top_top_set_d ) ) ) ).
% measurable_count_space_const
thf(fact_73_measurable__count__space__const,axiom,
! [C: real,M2: sigma_measure_a] :
( member_a_real
@ ^ [X: a] : C
@ ( sigma_9116425665531756122a_real @ M2 @ ( sigma_8508918144308765139e_real @ top_top_set_real ) ) ) ).
% measurable_count_space_const
thf(fact_74_measurable__count__space__const,axiom,
! [C: $o,M2: sigma_measure_a] :
( member_a_o
@ ^ [X: a] : C
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% measurable_count_space_const
thf(fact_75_AE__eq__constD_I2_J,axiom,
! [Y2: a] :
( ( eventually_a
@ ^ [X: a] : ( X = Y2 )
@ ( measure_ae_filter_a @ m ) )
=> ( member_a @ Y2 @ ( sigma_space_a @ m ) ) ) ).
% AE_eq_constD(2)
thf(fact_76_space__count__space,axiom,
! [Omega2: set_a] :
( ( sigma_space_a @ ( sigma_count_space_a @ Omega2 ) )
= Omega2 ) ).
% space_count_space
thf(fact_77_space__count__space,axiom,
! [Omega2: set_o] :
( ( sigma_space_o @ ( sigma_count_space_o @ Omega2 ) )
= Omega2 ) ).
% space_count_space
thf(fact_78_AE__I2,axiom,
! [M2: sigma_measure_c_d,P: ( c > d ) > $o] :
( ! [X3: c > d] :
( ( member_c_d @ X3 @ ( sigma_space_c_d @ M2 ) )
=> ( P @ X3 ) )
=> ( eventually_c_d @ P @ ( measur253975988404078935er_c_d @ M2 ) ) ) ).
% AE_I2
thf(fact_79_AE__I2,axiom,
! [M2: sigma_measure_a_real,P: ( a > real ) > $o] :
( ! [X3: a > real] :
( ( member_a_real @ X3 @ ( sigma_space_a_real @ M2 ) )
=> ( P @ X3 ) )
=> ( eventually_a_real @ P @ ( measur8946935934331175658a_real @ M2 ) ) ) ).
% AE_I2
thf(fact_80_AE__I2,axiom,
! [M2: sigma_measure_a_o,P: ( a > $o ) > $o] :
( ! [X3: a > $o] :
( ( member_a_o @ X3 @ ( sigma_space_a_o @ M2 ) )
=> ( P @ X3 ) )
=> ( eventually_a_o @ P @ ( measur1950989848205109360er_a_o @ M2 ) ) ) ).
% AE_I2
thf(fact_81_AE__I2,axiom,
! [M2: sigma_measure_b,P: b > $o] :
( ! [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ M2 ) )
=> ( P @ X3 ) )
=> ( eventually_b @ P @ ( measure_ae_filter_b @ M2 ) ) ) ).
% AE_I2
thf(fact_82_AE__I2,axiom,
! [M2: sigma_measure_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( sigma_space_nat @ M2 ) )
=> ( P @ X3 ) )
=> ( eventually_nat @ P @ ( measur6539087422748349889er_nat @ M2 ) ) ) ).
% AE_I2
thf(fact_83_AE__I2,axiom,
! [M2: sigma_measure_o,P: $o > $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ M2 ) )
=> ( P @ X3 ) )
=> ( eventually_o @ P @ ( measure_ae_filter_o @ M2 ) ) ) ).
% AE_I2
thf(fact_84_AE__I2,axiom,
! [M2: sigma_measure_a,P: a > $o] :
( ! [X3: a] :
( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
=> ( P @ X3 ) )
=> ( eventually_a @ P @ ( measure_ae_filter_a @ M2 ) ) ) ).
% AE_I2
thf(fact_85_indep__sets__Dynkin,axiom,
! [F: $o > set_set_a,I2: set_o] :
( ( indepe7780107833195774214ts_a_o @ m @ F @ I2 )
=> ( indepe7780107833195774214ts_a_o @ m
@ ^ [I4: $o] : ( sigma_Dynkin_a @ ( sigma_space_a @ m ) @ ( F @ I4 ) )
@ I2 ) ) ).
% indep_sets_Dynkin
thf(fact_86_indep__sets__Dynkin,axiom,
! [F: nat > set_set_a,I2: set_nat] :
( ( indepe6267730027088848354_a_nat @ m @ F @ I2 )
=> ( indepe6267730027088848354_a_nat @ m
@ ^ [I4: nat] : ( sigma_Dynkin_a @ ( sigma_space_a @ m ) @ ( F @ I4 ) )
@ I2 ) ) ).
% indep_sets_Dynkin
thf(fact_87_indep__sets__Dynkin,axiom,
! [F: a > set_set_a,I2: set_a] :
( ( indepe8927441866673418604ts_a_a @ m @ F @ I2 )
=> ( indepe8927441866673418604ts_a_a @ m
@ ^ [I4: a] : ( sigma_Dynkin_a @ ( sigma_space_a @ m ) @ ( F @ I4 ) )
@ I2 ) ) ).
% indep_sets_Dynkin
thf(fact_88_indep__sets__Dynkin,axiom,
! [F: b > set_set_a,I2: set_b] :
( ( indepe8927441866673418605ts_a_b @ m @ F @ I2 )
=> ( indepe8927441866673418605ts_a_b @ m
@ ^ [I4: b] : ( sigma_Dynkin_a @ ( sigma_space_a @ m ) @ ( F @ I4 ) )
@ I2 ) ) ).
% indep_sets_Dynkin
thf(fact_89_indep__sets__Dynkin,axiom,
! [F: ( a > $o ) > set_set_a,I2: set_a_o] :
( ( indepe7801696130336798481_a_a_o @ m @ F @ I2 )
=> ( indepe7801696130336798481_a_a_o @ m
@ ^ [I4: a > $o] : ( sigma_Dynkin_a @ ( sigma_space_a @ m ) @ ( F @ I4 ) )
@ I2 ) ) ).
% indep_sets_Dynkin
thf(fact_90_indep__sets__Dynkin,axiom,
! [F: ( a > real ) > set_set_a,I2: set_a_real] :
( ( indepe4749599203615801097a_real @ m @ F @ I2 )
=> ( indepe4749599203615801097a_real @ m
@ ^ [I4: a > real] : ( sigma_Dynkin_a @ ( sigma_space_a @ m ) @ ( F @ I4 ) )
@ I2 ) ) ).
% indep_sets_Dynkin
thf(fact_91_indep__sets__Dynkin,axiom,
! [F: ( c > d ) > set_set_a,I2: set_c_d] :
( ( indepe4867465796832040824_a_c_d @ m @ F @ I2 )
=> ( indepe4867465796832040824_a_c_d @ m
@ ^ [I4: c > d] : ( sigma_Dynkin_a @ ( sigma_space_a @ m ) @ ( F @ I4 ) )
@ I2 ) ) ).
% indep_sets_Dynkin
thf(fact_92_pred__intros__logic_I9_J,axiom,
! [P2: ( c > d ) > a > $o,F2: a > c > d,M2: sigma_measure_a] :
( ( member_a_o
@ ^ [X: a] : ( P2 @ ( F2 @ X ) @ X )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( member_a_o
@ ^ [X: a] :
( member_c_d @ ( F2 @ X )
@ ( collect_c_d
@ ^ [Y3: c > d] : ( P2 @ Y3 @ X ) ) )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ).
% pred_intros_logic(9)
thf(fact_93_pred__intros__logic_I9_J,axiom,
! [P2: ( a > real ) > a > $o,F2: a > a > real,M2: sigma_measure_a] :
( ( member_a_o
@ ^ [X: a] : ( P2 @ ( F2 @ X ) @ X )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( member_a_o
@ ^ [X: a] :
( member_a_real @ ( F2 @ X )
@ ( collect_a_real
@ ^ [Y3: a > real] : ( P2 @ Y3 @ X ) ) )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ).
% pred_intros_logic(9)
thf(fact_94_pred__intros__logic_I9_J,axiom,
! [P2: ( a > $o ) > a > $o,F2: a > a > $o,M2: sigma_measure_a] :
( ( member_a_o
@ ^ [X: a] : ( P2 @ ( F2 @ X ) @ X )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( member_a_o
@ ^ [X: a] :
( member_a_o @ ( F2 @ X )
@ ( collect_a_o
@ ^ [Y3: a > $o] : ( P2 @ Y3 @ X ) ) )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ).
% pred_intros_logic(9)
thf(fact_95_pred__intros__logic_I9_J,axiom,
! [P2: b > a > $o,F2: a > b,M2: sigma_measure_a] :
( ( member_a_o
@ ^ [X: a] : ( P2 @ ( F2 @ X ) @ X )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( member_a_o
@ ^ [X: a] :
( member_b @ ( F2 @ X )
@ ( collect_b
@ ^ [Y3: b] : ( P2 @ Y3 @ X ) ) )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ).
% pred_intros_logic(9)
thf(fact_96_pred__intros__logic_I9_J,axiom,
! [P2: $o > a > $o,F2: a > $o,M2: sigma_measure_a] :
( ( member_a_o
@ ^ [X: a] : ( P2 @ ( F2 @ X ) @ X )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( member_a_o
@ ^ [X: a] :
( member_o @ ( F2 @ X )
@ ( collect_o
@ ^ [Y3: $o] : ( P2 @ Y3 @ X ) ) )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ).
% pred_intros_logic(9)
thf(fact_97_pred__intros__logic_I9_J,axiom,
! [P2: a > a > $o,F2: a > a,M2: sigma_measure_a] :
( ( member_a_o
@ ^ [X: a] : ( P2 @ ( F2 @ X ) @ X )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( member_a_o
@ ^ [X: a] :
( member_a @ ( F2 @ X )
@ ( collect_a
@ ^ [Y3: a] : ( P2 @ Y3 @ X ) ) )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ).
% pred_intros_logic(9)
thf(fact_98_pred__intros__logic_I9_J,axiom,
! [P2: nat > a > $o,F2: a > nat,M2: sigma_measure_a] :
( ( member_a_o
@ ^ [X: a] : ( P2 @ ( F2 @ X ) @ X )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( member_a_o
@ ^ [X: a] :
( member_nat @ ( F2 @ X )
@ ( collect_nat
@ ^ [Y3: nat] : ( P2 @ Y3 @ X ) ) )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ).
% pred_intros_logic(9)
thf(fact_99_pred__intros__logic_I6_J,axiom,
! [Q: a > $o,M2: sigma_measure_a,P: a > $o] :
( ( member_a_o @ Q @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( member_a_o @ P @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( member_a_o
@ ^ [X: a] :
( ( Q @ X )
= ( P @ X ) )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ) ).
% pred_intros_logic(6)
thf(fact_100_pred__intros__logic_I5_J,axiom,
! [Q: a > $o,M2: sigma_measure_a,P: a > $o] :
( ( member_a_o @ Q @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( member_a_o @ P @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( member_a_o
@ ^ [X: a] :
( ( Q @ X )
| ( P @ X ) )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ) ).
% pred_intros_logic(5)
thf(fact_101_pred__intros__logic_I4_J,axiom,
! [Q: a > $o,M2: sigma_measure_a,P: a > $o] :
( ( member_a_o @ Q @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( member_a_o @ P @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( member_a_o
@ ^ [X: a] :
( ( Q @ X )
=> ( P @ X ) )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ) ).
% pred_intros_logic(4)
thf(fact_102_pred__intros__logic_I3_J,axiom,
! [Q: a > $o,M2: sigma_measure_a,P: a > $o] :
( ( member_a_o @ Q @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( member_a_o @ P @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( member_a_o
@ ^ [X: a] :
( ( Q @ X )
& ( P @ X ) )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ) ).
% pred_intros_logic(3)
thf(fact_103_pred__intros__logic_I2_J,axiom,
! [P: a > $o,M2: sigma_measure_a] :
( ( member_a_o @ P @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( member_a_o
@ ^ [X: a] :
~ ( P @ X )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ).
% pred_intros_logic(2)
thf(fact_104_pred__intros__logic_I1_J,axiom,
! [M2: sigma_measure_c_d] :
( member_c_d_o
@ ^ [X: c > d] : ( member_c_d @ X @ ( sigma_space_c_d @ M2 ) )
@ ( sigma_1714064210060623456_c_d_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% pred_intros_logic(1)
thf(fact_105_pred__intros__logic_I1_J,axiom,
! [M2: sigma_measure_a_real] :
( member_a_real_o
@ ^ [X: a > real] : ( member_a_real @ X @ ( sigma_space_a_real @ M2 ) )
@ ( sigma_9085598459323199629real_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% pred_intros_logic(1)
thf(fact_106_pred__intros__logic_I1_J,axiom,
! [M2: sigma_measure_a_o] :
( member_a_o_o
@ ^ [X: a > $o] : ( member_a_o @ X @ ( sigma_space_a_o @ M2 ) )
@ ( sigma_1195952539894209287_a_o_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% pred_intros_logic(1)
thf(fact_107_pred__intros__logic_I1_J,axiom,
! [M2: sigma_measure_b] :
( member_b_o
@ ^ [X: b] : ( member_b @ X @ ( sigma_space_b @ M2 ) )
@ ( sigma_measurable_b_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% pred_intros_logic(1)
thf(fact_108_pred__intros__logic_I1_J,axiom,
! [M2: sigma_measure_nat] :
( member_nat_o
@ ^ [X: nat] : ( member_nat @ X @ ( sigma_space_nat @ M2 ) )
@ ( sigma_5101835498682829686_nat_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% pred_intros_logic(1)
thf(fact_109_pred__intros__logic_I1_J,axiom,
! [M2: sigma_measure_o] :
( member_o_o
@ ^ [X: $o] : ( member_o @ X @ ( sigma_space_o @ M2 ) )
@ ( sigma_measurable_o_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% pred_intros_logic(1)
thf(fact_110_pred__intros__logic_I1_J,axiom,
! [M2: sigma_measure_a] :
( member_a_o
@ ^ [X: a] : ( member_a @ X @ ( sigma_space_a @ M2 ) )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% pred_intros_logic(1)
thf(fact_111_pred__in__If,axiom,
! [P: $o,A2: ( c > d ) > set_c_d,M2: sigma_measure_c_d,B: ( c > d ) > set_c_d] :
( ( P
=> ( member_c_d_o
@ ^ [X: c > d] : ( member_c_d @ X @ ( A2 @ X ) )
@ ( sigma_1714064210060623456_c_d_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) )
=> ( ( ~ P
=> ( member_c_d_o
@ ^ [X: c > d] : ( member_c_d @ X @ ( B @ X ) )
@ ( sigma_1714064210060623456_c_d_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) )
=> ( member_c_d_o
@ ^ [X: c > d] :
( ( P
=> ( member_c_d @ X @ ( A2 @ X ) ) )
& ( ~ P
=> ( member_c_d @ X @ ( B @ X ) ) ) )
@ ( sigma_1714064210060623456_c_d_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ) ).
% pred_in_If
thf(fact_112_pred__in__If,axiom,
! [P: $o,A2: ( a > real ) > set_a_real,M2: sigma_measure_a_real,B: ( a > real ) > set_a_real] :
( ( P
=> ( member_a_real_o
@ ^ [X: a > real] : ( member_a_real @ X @ ( A2 @ X ) )
@ ( sigma_9085598459323199629real_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) )
=> ( ( ~ P
=> ( member_a_real_o
@ ^ [X: a > real] : ( member_a_real @ X @ ( B @ X ) )
@ ( sigma_9085598459323199629real_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) )
=> ( member_a_real_o
@ ^ [X: a > real] :
( ( P
=> ( member_a_real @ X @ ( A2 @ X ) ) )
& ( ~ P
=> ( member_a_real @ X @ ( B @ X ) ) ) )
@ ( sigma_9085598459323199629real_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ) ).
% pred_in_If
thf(fact_113_pred__in__If,axiom,
! [P: $o,A2: ( a > $o ) > set_a_o,M2: sigma_measure_a_o,B: ( a > $o ) > set_a_o] :
( ( P
=> ( member_a_o_o
@ ^ [X: a > $o] : ( member_a_o @ X @ ( A2 @ X ) )
@ ( sigma_1195952539894209287_a_o_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) )
=> ( ( ~ P
=> ( member_a_o_o
@ ^ [X: a > $o] : ( member_a_o @ X @ ( B @ X ) )
@ ( sigma_1195952539894209287_a_o_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) )
=> ( member_a_o_o
@ ^ [X: a > $o] :
( ( P
=> ( member_a_o @ X @ ( A2 @ X ) ) )
& ( ~ P
=> ( member_a_o @ X @ ( B @ X ) ) ) )
@ ( sigma_1195952539894209287_a_o_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ) ).
% pred_in_If
thf(fact_114_pred__in__If,axiom,
! [P: $o,A2: b > set_b,M2: sigma_measure_b,B: b > set_b] :
( ( P
=> ( member_b_o
@ ^ [X: b] : ( member_b @ X @ ( A2 @ X ) )
@ ( sigma_measurable_b_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) )
=> ( ( ~ P
=> ( member_b_o
@ ^ [X: b] : ( member_b @ X @ ( B @ X ) )
@ ( sigma_measurable_b_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) )
=> ( member_b_o
@ ^ [X: b] :
( ( P
=> ( member_b @ X @ ( A2 @ X ) ) )
& ( ~ P
=> ( member_b @ X @ ( B @ X ) ) ) )
@ ( sigma_measurable_b_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ) ).
% pred_in_If
thf(fact_115_pred__in__If,axiom,
! [P: $o,A2: nat > set_nat,M2: sigma_measure_nat,B: nat > set_nat] :
( ( P
=> ( member_nat_o
@ ^ [X: nat] : ( member_nat @ X @ ( A2 @ X ) )
@ ( sigma_5101835498682829686_nat_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) )
=> ( ( ~ P
=> ( member_nat_o
@ ^ [X: nat] : ( member_nat @ X @ ( B @ X ) )
@ ( sigma_5101835498682829686_nat_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) )
=> ( member_nat_o
@ ^ [X: nat] :
( ( P
=> ( member_nat @ X @ ( A2 @ X ) ) )
& ( ~ P
=> ( member_nat @ X @ ( B @ X ) ) ) )
@ ( sigma_5101835498682829686_nat_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ) ).
% pred_in_If
thf(fact_116_pred__in__If,axiom,
! [P: $o,A2: $o > set_o,M2: sigma_measure_o,B: $o > set_o] :
( ( P
=> ( member_o_o
@ ^ [X: $o] : ( member_o @ X @ ( A2 @ X ) )
@ ( sigma_measurable_o_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) )
=> ( ( ~ P
=> ( member_o_o
@ ^ [X: $o] : ( member_o @ X @ ( B @ X ) )
@ ( sigma_measurable_o_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) )
=> ( member_o_o
@ ^ [X: $o] :
( ( P
=> ( member_o @ X @ ( A2 @ X ) ) )
& ( ~ P
=> ( member_o @ X @ ( B @ X ) ) ) )
@ ( sigma_measurable_o_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ) ).
% pred_in_If
thf(fact_117_pred__in__If,axiom,
! [P: $o,A2: a > set_a,M2: sigma_measure_a,B: a > set_a] :
( ( P
=> ( member_a_o
@ ^ [X: a] : ( member_a @ X @ ( A2 @ X ) )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) )
=> ( ( ~ P
=> ( member_a_o
@ ^ [X: a] : ( member_a @ X @ ( B @ X ) )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) )
=> ( member_a_o
@ ^ [X: a] :
( ( P
=> ( member_a @ X @ ( A2 @ X ) ) )
& ( ~ P
=> ( member_a @ X @ ( B @ X ) ) ) )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ) ).
% pred_in_If
thf(fact_118_pred__intros__imp_H,axiom,
! [K2: $o,P: a > $o,M2: sigma_measure_a] :
( ( K2
=> ( member_a_o @ P @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) )
=> ( member_a_o
@ ^ [X: a] :
( K2
=> ( P @ X ) )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ).
% pred_intros_imp'
thf(fact_119_pred__intros__conj1_H,axiom,
! [K2: $o,P: a > $o,M2: sigma_measure_a] :
( ( K2
=> ( member_a_o @ P @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) )
=> ( member_a_o
@ ^ [X: a] :
( K2
& ( P @ X ) )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ).
% pred_intros_conj1'
thf(fact_120_pred__intros__conj2_H,axiom,
! [K2: $o,P: a > $o,M2: sigma_measure_a] :
( ( K2
=> ( member_a_o @ P @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) )
=> ( member_a_o
@ ^ [X: a] :
( ( P @ X )
& K2 )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ).
% pred_intros_conj2'
thf(fact_121_pred__intros__disj1_H,axiom,
! [K2: $o,P: a > $o,M2: sigma_measure_a] :
( ( ~ K2
=> ( member_a_o @ P @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) )
=> ( member_a_o
@ ^ [X: a] :
( K2
| ( P @ X ) )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ).
% pred_intros_disj1'
thf(fact_122_mem__Collect__eq,axiom,
! [A: c > d,P: ( c > d ) > $o] :
( ( member_c_d @ A @ ( collect_c_d @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_123_mem__Collect__eq,axiom,
! [A: a > real,P: ( a > real ) > $o] :
( ( member_a_real @ A @ ( collect_a_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_124_mem__Collect__eq,axiom,
! [A: a > $o,P: ( a > $o ) > $o] :
( ( member_a_o @ A @ ( collect_a_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_125_mem__Collect__eq,axiom,
! [A: b,P: b > $o] :
( ( member_b @ A @ ( collect_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_126_mem__Collect__eq,axiom,
! [A: $o,P: $o > $o] :
( ( member_o @ A @ ( collect_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_127_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_128_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_129_Collect__mem__eq,axiom,
! [A2: set_c_d] :
( ( collect_c_d
@ ^ [X: c > d] : ( member_c_d @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_130_Collect__mem__eq,axiom,
! [A2: set_a_real] :
( ( collect_a_real
@ ^ [X: a > real] : ( member_a_real @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_131_Collect__mem__eq,axiom,
! [A2: set_a_o] :
( ( collect_a_o
@ ^ [X: a > $o] : ( member_a_o @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_132_Collect__mem__eq,axiom,
! [A2: set_b] :
( ( collect_b
@ ^ [X: b] : ( member_b @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_133_Collect__mem__eq,axiom,
! [A2: set_o] :
( ( collect_o
@ ^ [X: $o] : ( member_o @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_134_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_135_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_136_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_137_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_138_pred__intros__disj2_H,axiom,
! [K2: $o,P: a > $o,M2: sigma_measure_a] :
( ( ~ K2
=> ( member_a_o @ P @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) )
=> ( member_a_o
@ ^ [X: a] :
( ( P @ X )
| K2 )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ).
% pred_intros_disj2'
thf(fact_139_measurable__if__split,axiom,
! [C: $o,F2: a > $o,M2: sigma_measure_a,G2: a > $o] :
( ( C
=> ( member_a_o @ F2 @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) )
=> ( ( ~ C
=> ( member_a_o @ G2 @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) )
=> ( ( C
=> ( member_a_o @ F2 @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) )
& ( ~ C
=> ( member_a_o @ G2 @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ) ) ) ).
% measurable_if_split
thf(fact_140_measurable__cong__simp,axiom,
! [M2: sigma_measure_c,N: sigma_measure_c,M: sigma_measure_d,N2: sigma_measure_d,F2: c > d,G2: c > d] :
( ( M2 = N )
=> ( ( M = N2 )
=> ( ! [W: c] :
( ( member_c @ W @ ( sigma_space_c @ M2 ) )
=> ( ( F2 @ W )
= ( G2 @ W ) ) )
=> ( ( member_c_d @ F2 @ ( sigma_measurable_c_d @ M2 @ M ) )
= ( member_c_d @ G2 @ ( sigma_measurable_c_d @ N @ N2 ) ) ) ) ) ) ).
% measurable_cong_simp
thf(fact_141_measurable__cong__simp,axiom,
! [M2: sigma_measure_a,N: sigma_measure_a,M: sigma_measure_o,N2: sigma_measure_o,F2: a > $o,G2: a > $o] :
( ( M2 = N )
=> ( ( M = N2 )
=> ( ! [W: a] :
( ( member_a @ W @ ( sigma_space_a @ M2 ) )
=> ( ( F2 @ W )
= ( G2 @ W ) ) )
=> ( ( member_a_o @ F2 @ ( sigma_measurable_a_o @ M2 @ M ) )
= ( member_a_o @ G2 @ ( sigma_measurable_a_o @ N @ N2 ) ) ) ) ) ) ).
% measurable_cong_simp
thf(fact_142_measurable__cong__simp,axiom,
! [M2: sigma_measure_a,N: sigma_measure_a,M: sigma_measure_real,N2: sigma_measure_real,F2: a > real,G2: a > real] :
( ( M2 = N )
=> ( ( M = N2 )
=> ( ! [W: a] :
( ( member_a @ W @ ( sigma_space_a @ M2 ) )
=> ( ( F2 @ W )
= ( G2 @ W ) ) )
=> ( ( member_a_real @ F2 @ ( sigma_9116425665531756122a_real @ M2 @ M ) )
= ( member_a_real @ G2 @ ( sigma_9116425665531756122a_real @ N @ N2 ) ) ) ) ) ) ).
% measurable_cong_simp
thf(fact_143_measurable__space,axiom,
! [F2: b > b,M2: sigma_measure_b,A2: sigma_measure_b,X4: b] :
( ( member_b_b @ F2 @ ( sigma_measurable_b_b @ M2 @ A2 ) )
=> ( ( member_b @ X4 @ ( sigma_space_b @ M2 ) )
=> ( member_b @ ( F2 @ X4 ) @ ( sigma_space_b @ A2 ) ) ) ) ).
% measurable_space
thf(fact_144_measurable__space,axiom,
! [F2: b > nat,M2: sigma_measure_b,A2: sigma_measure_nat,X4: b] :
( ( member_b_nat @ F2 @ ( sigma_1308594411581951615_b_nat @ M2 @ A2 ) )
=> ( ( member_b @ X4 @ ( sigma_space_b @ M2 ) )
=> ( member_nat @ ( F2 @ X4 ) @ ( sigma_space_nat @ A2 ) ) ) ) ).
% measurable_space
thf(fact_145_measurable__space,axiom,
! [F2: b > $o,M2: sigma_measure_b,A2: sigma_measure_o,X4: b] :
( ( member_b_o @ F2 @ ( sigma_measurable_b_o @ M2 @ A2 ) )
=> ( ( member_b @ X4 @ ( sigma_space_b @ M2 ) )
=> ( member_o @ ( F2 @ X4 ) @ ( sigma_space_o @ A2 ) ) ) ) ).
% measurable_space
thf(fact_146_measurable__space,axiom,
! [F2: nat > b,M2: sigma_measure_nat,A2: sigma_measure_b,X4: nat] :
( ( member_nat_b @ F2 @ ( sigma_4105081583803843549_nat_b @ M2 @ A2 ) )
=> ( ( member_nat @ X4 @ ( sigma_space_nat @ M2 ) )
=> ( member_b @ ( F2 @ X4 ) @ ( sigma_space_b @ A2 ) ) ) ) ).
% measurable_space
thf(fact_147_measurable__space,axiom,
! [F2: nat > nat,M2: sigma_measure_nat,A2: sigma_measure_nat,X4: nat] :
( ( member_nat_nat @ F2 @ ( sigma_4350458207664084850at_nat @ M2 @ A2 ) )
=> ( ( member_nat @ X4 @ ( sigma_space_nat @ M2 ) )
=> ( member_nat @ ( F2 @ X4 ) @ ( sigma_space_nat @ A2 ) ) ) ) ).
% measurable_space
thf(fact_148_measurable__space,axiom,
! [F2: nat > $o,M2: sigma_measure_nat,A2: sigma_measure_o,X4: nat] :
( ( member_nat_o @ F2 @ ( sigma_5101835498682829686_nat_o @ M2 @ A2 ) )
=> ( ( member_nat @ X4 @ ( sigma_space_nat @ M2 ) )
=> ( member_o @ ( F2 @ X4 ) @ ( sigma_space_o @ A2 ) ) ) ) ).
% measurable_space
thf(fact_149_measurable__space,axiom,
! [F2: $o > b,M2: sigma_measure_o,A2: sigma_measure_b,X4: $o] :
( ( member_o_b @ F2 @ ( sigma_measurable_o_b @ M2 @ A2 ) )
=> ( ( member_o @ X4 @ ( sigma_space_o @ M2 ) )
=> ( member_b @ ( F2 @ X4 ) @ ( sigma_space_b @ A2 ) ) ) ) ).
% measurable_space
thf(fact_150_measurable__space,axiom,
! [F2: $o > nat,M2: sigma_measure_o,A2: sigma_measure_nat,X4: $o] :
( ( member_o_nat @ F2 @ ( sigma_1999164137574644376_o_nat @ M2 @ A2 ) )
=> ( ( member_o @ X4 @ ( sigma_space_o @ M2 ) )
=> ( member_nat @ ( F2 @ X4 ) @ ( sigma_space_nat @ A2 ) ) ) ) ).
% measurable_space
thf(fact_151_measurable__space,axiom,
! [F2: $o > $o,M2: sigma_measure_o,A2: sigma_measure_o,X4: $o] :
( ( member_o_o @ F2 @ ( sigma_measurable_o_o @ M2 @ A2 ) )
=> ( ( member_o @ X4 @ ( sigma_space_o @ M2 ) )
=> ( member_o @ ( F2 @ X4 ) @ ( sigma_space_o @ A2 ) ) ) ) ).
% measurable_space
thf(fact_152_measurable__space,axiom,
! [F2: b > a,M2: sigma_measure_b,A2: sigma_measure_a,X4: b] :
( ( member_b_a @ F2 @ ( sigma_measurable_b_a @ M2 @ A2 ) )
=> ( ( member_b @ X4 @ ( sigma_space_b @ M2 ) )
=> ( member_a @ ( F2 @ X4 ) @ ( sigma_space_a @ A2 ) ) ) ) ).
% measurable_space
thf(fact_153_measurable__cong,axiom,
! [M2: sigma_measure_c,F2: c > d,G2: c > d,M: sigma_measure_d] :
( ! [W: c] :
( ( member_c @ W @ ( sigma_space_c @ M2 ) )
=> ( ( F2 @ W )
= ( G2 @ W ) ) )
=> ( ( member_c_d @ F2 @ ( sigma_measurable_c_d @ M2 @ M ) )
= ( member_c_d @ G2 @ ( sigma_measurable_c_d @ M2 @ M ) ) ) ) ).
% measurable_cong
thf(fact_154_measurable__cong,axiom,
! [M2: sigma_measure_a,F2: a > $o,G2: a > $o,M: sigma_measure_o] :
( ! [W: a] :
( ( member_a @ W @ ( sigma_space_a @ M2 ) )
=> ( ( F2 @ W )
= ( G2 @ W ) ) )
=> ( ( member_a_o @ F2 @ ( sigma_measurable_a_o @ M2 @ M ) )
= ( member_a_o @ G2 @ ( sigma_measurable_a_o @ M2 @ M ) ) ) ) ).
% measurable_cong
thf(fact_155_measurable__cong,axiom,
! [M2: sigma_measure_a,F2: a > real,G2: a > real,M: sigma_measure_real] :
( ! [W: a] :
( ( member_a @ W @ ( sigma_space_a @ M2 ) )
=> ( ( F2 @ W )
= ( G2 @ W ) ) )
=> ( ( member_a_real @ F2 @ ( sigma_9116425665531756122a_real @ M2 @ M ) )
= ( member_a_real @ G2 @ ( sigma_9116425665531756122a_real @ M2 @ M ) ) ) ) ).
% measurable_cong
thf(fact_156_measurable__const,axiom,
! [C: d,M: sigma_measure_d,M2: sigma_measure_c] :
( ( member_d @ C @ ( sigma_space_d @ M ) )
=> ( member_c_d
@ ^ [X: c] : C
@ ( sigma_measurable_c_d @ M2 @ M ) ) ) ).
% measurable_const
thf(fact_157_measurable__const,axiom,
! [C: $o,M: sigma_measure_o,M2: sigma_measure_a] :
( ( member_o @ C @ ( sigma_space_o @ M ) )
=> ( member_a_o
@ ^ [X: a] : C
@ ( sigma_measurable_a_o @ M2 @ M ) ) ) ).
% measurable_const
thf(fact_158_measurable__const,axiom,
! [C: real,M: sigma_measure_real,M2: sigma_measure_a] :
( ( member_real @ C @ ( sigma_space_real @ M ) )
=> ( member_a_real
@ ^ [X: a] : C
@ ( sigma_9116425665531756122a_real @ M2 @ M ) ) ) ).
% measurable_const
thf(fact_159_AE__Ball__mp,axiom,
! [M2: sigma_measure_a,P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
=> ( P @ X3 ) )
=> ( ( eventually_a
@ ^ [X: a] :
( ( P @ X )
=> ( Q @ X ) )
@ ( measure_ae_filter_a @ M2 ) )
=> ( eventually_a @ Q @ ( measure_ae_filter_a @ M2 ) ) ) ) ).
% AE_Ball_mp
thf(fact_160_AE__space,axiom,
! [M2: sigma_measure_c_d] :
( eventually_c_d
@ ^ [X: c > d] : ( member_c_d @ X @ ( sigma_space_c_d @ M2 ) )
@ ( measur253975988404078935er_c_d @ M2 ) ) ).
% AE_space
thf(fact_161_AE__space,axiom,
! [M2: sigma_measure_a_real] :
( eventually_a_real
@ ^ [X: a > real] : ( member_a_real @ X @ ( sigma_space_a_real @ M2 ) )
@ ( measur8946935934331175658a_real @ M2 ) ) ).
% AE_space
thf(fact_162_AE__space,axiom,
! [M2: sigma_measure_a_o] :
( eventually_a_o
@ ^ [X: a > $o] : ( member_a_o @ X @ ( sigma_space_a_o @ M2 ) )
@ ( measur1950989848205109360er_a_o @ M2 ) ) ).
% AE_space
thf(fact_163_AE__space,axiom,
! [M2: sigma_measure_b] :
( eventually_b
@ ^ [X: b] : ( member_b @ X @ ( sigma_space_b @ M2 ) )
@ ( measure_ae_filter_b @ M2 ) ) ).
% AE_space
thf(fact_164_AE__space,axiom,
! [M2: sigma_measure_nat] :
( eventually_nat
@ ^ [X: nat] : ( member_nat @ X @ ( sigma_space_nat @ M2 ) )
@ ( measur6539087422748349889er_nat @ M2 ) ) ).
% AE_space
thf(fact_165_AE__space,axiom,
! [M2: sigma_measure_o] :
( eventually_o
@ ^ [X: $o] : ( member_o @ X @ ( sigma_space_o @ M2 ) )
@ ( measure_ae_filter_o @ M2 ) ) ).
% AE_space
thf(fact_166_AE__space,axiom,
! [M2: sigma_measure_a] :
( eventually_a
@ ^ [X: a] : ( member_a @ X @ ( sigma_space_a @ M2 ) )
@ ( measure_ae_filter_a @ M2 ) ) ).
% AE_space
thf(fact_167_AE__cong,axiom,
! [M2: sigma_measure_c_d,P: ( c > d ) > $o,Q: ( c > d ) > $o] :
( ! [X3: c > d] :
( ( member_c_d @ X3 @ ( sigma_space_c_d @ M2 ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( eventually_c_d @ P @ ( measur253975988404078935er_c_d @ M2 ) )
= ( eventually_c_d @ Q @ ( measur253975988404078935er_c_d @ M2 ) ) ) ) ).
% AE_cong
thf(fact_168_AE__cong,axiom,
! [M2: sigma_measure_a_real,P: ( a > real ) > $o,Q: ( a > real ) > $o] :
( ! [X3: a > real] :
( ( member_a_real @ X3 @ ( sigma_space_a_real @ M2 ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( eventually_a_real @ P @ ( measur8946935934331175658a_real @ M2 ) )
= ( eventually_a_real @ Q @ ( measur8946935934331175658a_real @ M2 ) ) ) ) ).
% AE_cong
thf(fact_169_AE__cong,axiom,
! [M2: sigma_measure_a_o,P: ( a > $o ) > $o,Q: ( a > $o ) > $o] :
( ! [X3: a > $o] :
( ( member_a_o @ X3 @ ( sigma_space_a_o @ M2 ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( eventually_a_o @ P @ ( measur1950989848205109360er_a_o @ M2 ) )
= ( eventually_a_o @ Q @ ( measur1950989848205109360er_a_o @ M2 ) ) ) ) ).
% AE_cong
thf(fact_170_AE__cong,axiom,
! [M2: sigma_measure_b,P: b > $o,Q: b > $o] :
( ! [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ M2 ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( eventually_b @ P @ ( measure_ae_filter_b @ M2 ) )
= ( eventually_b @ Q @ ( measure_ae_filter_b @ M2 ) ) ) ) ).
% AE_cong
thf(fact_171_AE__cong,axiom,
! [M2: sigma_measure_nat,P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( sigma_space_nat @ M2 ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( eventually_nat @ P @ ( measur6539087422748349889er_nat @ M2 ) )
= ( eventually_nat @ Q @ ( measur6539087422748349889er_nat @ M2 ) ) ) ) ).
% AE_cong
thf(fact_172_AE__cong,axiom,
! [M2: sigma_measure_o,P: $o > $o,Q: $o > $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ M2 ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( eventually_o @ P @ ( measure_ae_filter_o @ M2 ) )
= ( eventually_o @ Q @ ( measure_ae_filter_o @ M2 ) ) ) ) ).
% AE_cong
thf(fact_173_AE__cong,axiom,
! [M2: sigma_measure_a,P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( eventually_a @ P @ ( measure_ae_filter_a @ M2 ) )
= ( eventually_a @ Q @ ( measure_ae_filter_a @ M2 ) ) ) ) ).
% AE_cong
thf(fact_174_prob__space_Oindep__sets_Ocong,axiom,
indepe7780107833195774214ts_a_o = indepe7780107833195774214ts_a_o ).
% prob_space.indep_sets.cong
thf(fact_175_prob__space_Oindep__sets_Ocong,axiom,
indepe6267730027088848354_a_nat = indepe6267730027088848354_a_nat ).
% prob_space.indep_sets.cong
thf(fact_176_prob__space_Oindep__sets_Ocong,axiom,
indepe8927441866673418604ts_a_a = indepe8927441866673418604ts_a_a ).
% prob_space.indep_sets.cong
thf(fact_177_prob__space_Oindep__sets_Ocong,axiom,
indepe8927441866673418605ts_a_b = indepe8927441866673418605ts_a_b ).
% prob_space.indep_sets.cong
thf(fact_178_prob__space_Oindep__sets_Ocong,axiom,
indepe7801696130336798481_a_a_o = indepe7801696130336798481_a_a_o ).
% prob_space.indep_sets.cong
thf(fact_179_prob__space_Oindep__sets_Ocong,axiom,
indepe4749599203615801097a_real = indepe4749599203615801097a_real ).
% prob_space.indep_sets.cong
thf(fact_180_prob__space_Oindep__sets_Ocong,axiom,
indepe4867465796832040824_a_c_d = indepe4867465796832040824_a_c_d ).
% prob_space.indep_sets.cong
thf(fact_181_pred__intros__logic_I7_J,axiom,
! [F2: a > c > d,M2: sigma_measure_a] :
( member_a_o
@ ^ [X: a] : ( member_c_d @ ( F2 @ X ) @ top_top_set_c_d )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% pred_intros_logic(7)
thf(fact_182_pred__intros__logic_I7_J,axiom,
! [F2: a > a > real,M2: sigma_measure_a] :
( member_a_o
@ ^ [X: a] : ( member_a_real @ ( F2 @ X ) @ top_top_set_a_real )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% pred_intros_logic(7)
thf(fact_183_pred__intros__logic_I7_J,axiom,
! [F2: a > a > $o,M2: sigma_measure_a] :
( member_a_o
@ ^ [X: a] : ( member_a_o @ ( F2 @ X ) @ top_top_set_a_o )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% pred_intros_logic(7)
thf(fact_184_pred__intros__logic_I7_J,axiom,
! [F2: a > b,M2: sigma_measure_a] :
( member_a_o
@ ^ [X: a] : ( member_b @ ( F2 @ X ) @ top_top_set_b )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% pred_intros_logic(7)
thf(fact_185_pred__intros__logic_I7_J,axiom,
! [F2: a > a,M2: sigma_measure_a] :
( member_a_o
@ ^ [X: a] : ( member_a @ ( F2 @ X ) @ top_top_set_a )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% pred_intros_logic(7)
thf(fact_186_pred__intros__logic_I7_J,axiom,
! [F2: a > nat,M2: sigma_measure_a] :
( member_a_o
@ ^ [X: a] : ( member_nat @ ( F2 @ X ) @ top_top_set_nat )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% pred_intros_logic(7)
thf(fact_187_pred__intros__logic_I7_J,axiom,
! [F2: a > $o,M2: sigma_measure_a] :
( member_a_o
@ ^ [X: a] : ( member_o @ ( F2 @ X ) @ top_top_set_o )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% pred_intros_logic(7)
thf(fact_188_pred__intros__logic_I7_J,axiom,
! [F2: a > literal,M2: sigma_measure_a] :
( member_a_o
@ ^ [X: a] : ( member_literal @ ( F2 @ X ) @ top_top_set_literal )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% pred_intros_logic(7)
thf(fact_189_pred__intros__logic_I7_J,axiom,
! [F2: a > product_unit,M2: sigma_measure_a] :
( member_a_o
@ ^ [X: a] : ( member_Product_unit @ ( F2 @ X ) @ top_to1996260823553986621t_unit )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% pred_intros_logic(7)
thf(fact_190_measurable__compose__rev,axiom,
! [F2: d > d,L: sigma_measure_d,N: sigma_measure_d,G2: c > d,M2: sigma_measure_c] :
( ( member_d_d @ F2 @ ( sigma_measurable_d_d @ L @ N ) )
=> ( ( member_c_d @ G2 @ ( sigma_measurable_c_d @ M2 @ L ) )
=> ( member_c_d
@ ^ [X: c] : ( F2 @ ( G2 @ X ) )
@ ( sigma_measurable_c_d @ M2 @ N ) ) ) ) ).
% measurable_compose_rev
thf(fact_191_measurable__compose__rev,axiom,
! [F2: $o > $o,L: sigma_measure_o,N: sigma_measure_o,G2: a > $o,M2: sigma_measure_a] :
( ( member_o_o @ F2 @ ( sigma_measurable_o_o @ L @ N ) )
=> ( ( member_a_o @ G2 @ ( sigma_measurable_a_o @ M2 @ L ) )
=> ( member_a_o
@ ^ [X: a] : ( F2 @ ( G2 @ X ) )
@ ( sigma_measurable_a_o @ M2 @ N ) ) ) ) ).
% measurable_compose_rev
thf(fact_192_measurable__compose__rev,axiom,
! [F2: $o > real,L: sigma_measure_o,N: sigma_measure_real,G2: a > $o,M2: sigma_measure_a] :
( ( member_o_real @ F2 @ ( sigma_2430008634441611636o_real @ L @ N ) )
=> ( ( member_a_o @ G2 @ ( sigma_measurable_a_o @ M2 @ L ) )
=> ( member_a_real
@ ^ [X: a] : ( F2 @ ( G2 @ X ) )
@ ( sigma_9116425665531756122a_real @ M2 @ N ) ) ) ) ).
% measurable_compose_rev
thf(fact_193_measurable__compose__rev,axiom,
! [F2: real > $o,L: sigma_measure_real,N: sigma_measure_o,G2: a > real,M2: sigma_measure_a] :
( ( member_real_o @ F2 @ ( sigma_3939073009482781210real_o @ L @ N ) )
=> ( ( member_a_real @ G2 @ ( sigma_9116425665531756122a_real @ M2 @ L ) )
=> ( member_a_o
@ ^ [X: a] : ( F2 @ ( G2 @ X ) )
@ ( sigma_measurable_a_o @ M2 @ N ) ) ) ) ).
% measurable_compose_rev
thf(fact_194_measurable__compose__rev,axiom,
! [F2: real > real,L: sigma_measure_real,N: sigma_measure_real,G2: a > real,M2: sigma_measure_a] :
( ( member_real_real @ F2 @ ( sigma_5267869275261027754l_real @ L @ N ) )
=> ( ( member_a_real @ G2 @ ( sigma_9116425665531756122a_real @ M2 @ L ) )
=> ( member_a_real
@ ^ [X: a] : ( F2 @ ( G2 @ X ) )
@ ( sigma_9116425665531756122a_real @ M2 @ N ) ) ) ) ).
% measurable_compose_rev
thf(fact_195_measurable__compose__rev,axiom,
! [F2: c > d,L: sigma_measure_c,N: sigma_measure_d,G2: c > c,M2: sigma_measure_c] :
( ( member_c_d @ F2 @ ( sigma_measurable_c_d @ L @ N ) )
=> ( ( member_c_c @ G2 @ ( sigma_measurable_c_c @ M2 @ L ) )
=> ( member_c_d
@ ^ [X: c] : ( F2 @ ( G2 @ X ) )
@ ( sigma_measurable_c_d @ M2 @ N ) ) ) ) ).
% measurable_compose_rev
thf(fact_196_measurable__compose__rev,axiom,
! [F2: a > $o,L: sigma_measure_a,N: sigma_measure_o,G2: a > a,M2: sigma_measure_a] :
( ( member_a_o @ F2 @ ( sigma_measurable_a_o @ L @ N ) )
=> ( ( member_a_a @ G2 @ ( sigma_measurable_a_a @ M2 @ L ) )
=> ( member_a_o
@ ^ [X: a] : ( F2 @ ( G2 @ X ) )
@ ( sigma_measurable_a_o @ M2 @ N ) ) ) ) ).
% measurable_compose_rev
thf(fact_197_measurable__compose__rev,axiom,
! [F2: a > real,L: sigma_measure_a,N: sigma_measure_real,G2: a > a,M2: sigma_measure_a] :
( ( member_a_real @ F2 @ ( sigma_9116425665531756122a_real @ L @ N ) )
=> ( ( member_a_a @ G2 @ ( sigma_measurable_a_a @ M2 @ L ) )
=> ( member_a_real
@ ^ [X: a] : ( F2 @ ( G2 @ X ) )
@ ( sigma_9116425665531756122a_real @ M2 @ N ) ) ) ) ).
% measurable_compose_rev
thf(fact_198_measurable__compose,axiom,
! [F2: c > c,M2: sigma_measure_c,N: sigma_measure_c,G2: c > d,L: sigma_measure_d] :
( ( member_c_c @ F2 @ ( sigma_measurable_c_c @ M2 @ N ) )
=> ( ( member_c_d @ G2 @ ( sigma_measurable_c_d @ N @ L ) )
=> ( member_c_d
@ ^ [X: c] : ( G2 @ ( F2 @ X ) )
@ ( sigma_measurable_c_d @ M2 @ L ) ) ) ) ).
% measurable_compose
thf(fact_199_measurable__compose,axiom,
! [F2: a > a,M2: sigma_measure_a,N: sigma_measure_a,G2: a > $o,L: sigma_measure_o] :
( ( member_a_a @ F2 @ ( sigma_measurable_a_a @ M2 @ N ) )
=> ( ( member_a_o @ G2 @ ( sigma_measurable_a_o @ N @ L ) )
=> ( member_a_o
@ ^ [X: a] : ( G2 @ ( F2 @ X ) )
@ ( sigma_measurable_a_o @ M2 @ L ) ) ) ) ).
% measurable_compose
thf(fact_200_measurable__compose,axiom,
! [F2: a > a,M2: sigma_measure_a,N: sigma_measure_a,G2: a > real,L: sigma_measure_real] :
( ( member_a_a @ F2 @ ( sigma_measurable_a_a @ M2 @ N ) )
=> ( ( member_a_real @ G2 @ ( sigma_9116425665531756122a_real @ N @ L ) )
=> ( member_a_real
@ ^ [X: a] : ( G2 @ ( F2 @ X ) )
@ ( sigma_9116425665531756122a_real @ M2 @ L ) ) ) ) ).
% measurable_compose
thf(fact_201_measurable__compose,axiom,
! [F2: c > d,M2: sigma_measure_c,N: sigma_measure_d,G2: d > d,L: sigma_measure_d] :
( ( member_c_d @ F2 @ ( sigma_measurable_c_d @ M2 @ N ) )
=> ( ( member_d_d @ G2 @ ( sigma_measurable_d_d @ N @ L ) )
=> ( member_c_d
@ ^ [X: c] : ( G2 @ ( F2 @ X ) )
@ ( sigma_measurable_c_d @ M2 @ L ) ) ) ) ).
% measurable_compose
thf(fact_202_measurable__compose,axiom,
! [F2: a > $o,M2: sigma_measure_a,N: sigma_measure_o,G2: $o > $o,L: sigma_measure_o] :
( ( member_a_o @ F2 @ ( sigma_measurable_a_o @ M2 @ N ) )
=> ( ( member_o_o @ G2 @ ( sigma_measurable_o_o @ N @ L ) )
=> ( member_a_o
@ ^ [X: a] : ( G2 @ ( F2 @ X ) )
@ ( sigma_measurable_a_o @ M2 @ L ) ) ) ) ).
% measurable_compose
thf(fact_203_measurable__compose,axiom,
! [F2: a > $o,M2: sigma_measure_a,N: sigma_measure_o,G2: $o > real,L: sigma_measure_real] :
( ( member_a_o @ F2 @ ( sigma_measurable_a_o @ M2 @ N ) )
=> ( ( member_o_real @ G2 @ ( sigma_2430008634441611636o_real @ N @ L ) )
=> ( member_a_real
@ ^ [X: a] : ( G2 @ ( F2 @ X ) )
@ ( sigma_9116425665531756122a_real @ M2 @ L ) ) ) ) ).
% measurable_compose
thf(fact_204_measurable__compose,axiom,
! [F2: a > real,M2: sigma_measure_a,N: sigma_measure_real,G2: real > $o,L: sigma_measure_o] :
( ( member_a_real @ F2 @ ( sigma_9116425665531756122a_real @ M2 @ N ) )
=> ( ( member_real_o @ G2 @ ( sigma_3939073009482781210real_o @ N @ L ) )
=> ( member_a_o
@ ^ [X: a] : ( G2 @ ( F2 @ X ) )
@ ( sigma_measurable_a_o @ M2 @ L ) ) ) ) ).
% measurable_compose
thf(fact_205_measurable__compose,axiom,
! [F2: a > real,M2: sigma_measure_a,N: sigma_measure_real,G2: real > real,L: sigma_measure_real] :
( ( member_a_real @ F2 @ ( sigma_9116425665531756122a_real @ M2 @ N ) )
=> ( ( member_real_real @ G2 @ ( sigma_5267869275261027754l_real @ N @ L ) )
=> ( member_a_real
@ ^ [X: a] : ( G2 @ ( F2 @ X ) )
@ ( sigma_9116425665531756122a_real @ M2 @ L ) ) ) ) ).
% measurable_compose
thf(fact_206_sigma__finite__measure__count__space,axiom,
! [A2: set_o] : ( measur1827666076404920889sure_o @ ( sigma_count_space_o @ A2 ) ) ).
% sigma_finite_measure_count_space
thf(fact_207_prob__space__imp__sigma__finite,axiom,
! [M2: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( measur4308613598931908895sure_a @ M2 ) ) ).
% prob_space_imp_sigma_finite
thf(fact_208_prob__space_Ofinite__measure,axiom,
! [M2: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( measur930452917991658466sure_a @ M2 ) ) ).
% prob_space.finite_measure
thf(fact_209_finite__measure_Oaxioms_I1_J,axiom,
! [M2: sigma_measure_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( measur4308613598931908895sure_a @ M2 ) ) ).
% finite_measure.axioms(1)
thf(fact_210_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_211_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_212_prob__space_Oindep__sets__cong,axiom,
! [M2: sigma_measure_a,I2: set_a,J: set_a,F: a > set_set_a,G: a > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( I2 = J )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( indepe8927441866673418604ts_a_a @ M2 @ F @ I2 )
= ( indepe8927441866673418604ts_a_a @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_cong
thf(fact_213_prob__space_Oindep__sets__cong,axiom,
! [M2: sigma_measure_a,I2: set_b,J: set_b,F: b > set_set_a,G: b > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( I2 = J )
=> ( ! [I3: b] :
( ( member_b @ I3 @ I2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( indepe8927441866673418605ts_a_b @ M2 @ F @ I2 )
= ( indepe8927441866673418605ts_a_b @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_cong
thf(fact_214_prob__space_Oindep__sets__cong,axiom,
! [M2: sigma_measure_a,I2: set_a_o,J: set_a_o,F: ( a > $o ) > set_set_a,G: ( a > $o ) > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( I2 = J )
=> ( ! [I3: a > $o] :
( ( member_a_o @ I3 @ I2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( indepe7801696130336798481_a_a_o @ M2 @ F @ I2 )
= ( indepe7801696130336798481_a_a_o @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_cong
thf(fact_215_prob__space_Oindep__sets__cong,axiom,
! [M2: sigma_measure_a,I2: set_a_real,J: set_a_real,F: ( a > real ) > set_set_a,G: ( a > real ) > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( I2 = J )
=> ( ! [I3: a > real] :
( ( member_a_real @ I3 @ I2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( indepe4749599203615801097a_real @ M2 @ F @ I2 )
= ( indepe4749599203615801097a_real @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_cong
thf(fact_216_prob__space_Oindep__sets__cong,axiom,
! [M2: sigma_measure_a,I2: set_c_d,J: set_c_d,F: ( c > d ) > set_set_a,G: ( c > d ) > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( I2 = J )
=> ( ! [I3: c > d] :
( ( member_c_d @ I3 @ I2 )
=> ( ( F @ I3 )
= ( G @ I3 ) ) )
=> ( ( indepe4867465796832040824_a_c_d @ M2 @ F @ I2 )
= ( indepe4867465796832040824_a_c_d @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_cong
thf(fact_217_AE__disjI2,axiom,
! [Q: a > $o,M2: sigma_measure_a,P: a > $o] :
( ( eventually_a @ Q @ ( measure_ae_filter_a @ M2 ) )
=> ( eventually_a
@ ^ [X: a] :
( ( P @ X )
| ( Q @ X ) )
@ ( measure_ae_filter_a @ M2 ) ) ) ).
% AE_disjI2
thf(fact_218_AE__disjI1,axiom,
! [P: a > $o,M2: sigma_measure_a,Q: a > $o] :
( ( eventually_a @ P @ ( measure_ae_filter_a @ M2 ) )
=> ( eventually_a
@ ^ [X: a] :
( ( P @ X )
| ( Q @ X ) )
@ ( measure_ae_filter_a @ M2 ) ) ) ).
% AE_disjI1
thf(fact_219_AE__conjI,axiom,
! [P: a > $o,M2: sigma_measure_a,Q: a > $o] :
( ( eventually_a @ P @ ( measure_ae_filter_a @ M2 ) )
=> ( ( eventually_a @ Q @ ( measure_ae_filter_a @ M2 ) )
=> ( eventually_a
@ ^ [X: a] :
( ( P @ X )
& ( Q @ X ) )
@ ( measure_ae_filter_a @ M2 ) ) ) ) ).
% AE_conjI
thf(fact_220_AE__impI,axiom,
! [P: $o,Q: a > $o,M2: sigma_measure_a] :
( ( P
=> ( eventually_a @ Q @ ( measure_ae_filter_a @ M2 ) ) )
=> ( eventually_a
@ ^ [X: a] :
( P
=> ( Q @ X ) )
@ ( measure_ae_filter_a @ M2 ) ) ) ).
% AE_impI
thf(fact_221_AE__iffI,axiom,
! [P: a > $o,M2: sigma_measure_a,Q: a > $o] :
( ( eventually_a @ P @ ( measure_ae_filter_a @ M2 ) )
=> ( ( eventually_a
@ ^ [X: a] :
( ( P @ X )
= ( Q @ X ) )
@ ( measure_ae_filter_a @ M2 ) )
=> ( eventually_a @ Q @ ( measure_ae_filter_a @ M2 ) ) ) ) ).
% AE_iffI
thf(fact_222_AE__mp,axiom,
! [P: a > $o,M2: sigma_measure_a,Q: a > $o] :
( ( eventually_a @ P @ ( measure_ae_filter_a @ M2 ) )
=> ( ( eventually_a
@ ^ [X: a] :
( ( P @ X )
=> ( Q @ X ) )
@ ( measure_ae_filter_a @ M2 ) )
=> ( eventually_a @ Q @ ( measure_ae_filter_a @ M2 ) ) ) ) ).
% AE_mp
thf(fact_223_measurable__count__space,axiom,
! [F2: c > d,A2: set_c] : ( member_c_d @ F2 @ ( sigma_measurable_c_d @ ( sigma_count_space_c @ A2 ) @ ( sigma_count_space_d @ top_top_set_d ) ) ) ).
% measurable_count_space
thf(fact_224_measurable__count__space,axiom,
! [F2: a > real,A2: set_a] : ( member_a_real @ F2 @ ( sigma_9116425665531756122a_real @ ( sigma_count_space_a @ A2 ) @ ( sigma_8508918144308765139e_real @ top_top_set_real ) ) ) ).
% measurable_count_space
thf(fact_225_measurable__count__space,axiom,
! [F2: a > $o,A2: set_a] : ( member_a_o @ F2 @ ( sigma_measurable_a_o @ ( sigma_count_space_a @ A2 ) @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% measurable_count_space
thf(fact_226_measurable__count__space,axiom,
! [F2: $o > nat,A2: set_o] : ( member_o_nat @ F2 @ ( sigma_1999164137574644376_o_nat @ ( sigma_count_space_o @ A2 ) @ ( sigma_7685844798829912695ce_nat @ top_top_set_nat ) ) ) ).
% measurable_count_space
thf(fact_227_measurable__count__space,axiom,
! [F2: $o > literal,A2: set_o] : ( member_o_literal @ F2 @ ( sigma_4705034698148213516iteral @ ( sigma_count_space_o @ A2 ) @ ( sigma_7122031993496068461iteral @ top_top_set_literal ) ) ) ).
% measurable_count_space
thf(fact_228_measurable__count__space,axiom,
! [F2: $o > product_unit,A2: set_o] : ( member2370919827131729009t_unit @ F2 @ ( sigma_4658569928750460229t_unit @ ( sigma_count_space_o @ A2 ) @ ( sigma_2593334793047783974t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% measurable_count_space
thf(fact_229_measurable__count__space,axiom,
! [F2: $o > $o,A2: set_o] : ( member_o_o @ F2 @ ( sigma_measurable_o_o @ ( sigma_count_space_o @ A2 ) @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% measurable_count_space
thf(fact_230_AE__return,axiom,
! [X4: c > d,M2: sigma_measure_c_d,P: ( c > d ) > $o] :
( ( member_c_d @ X4 @ ( sigma_space_c_d @ M2 ) )
=> ( ( member_c_d_o @ P @ ( sigma_1714064210060623456_c_d_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( eventually_c_d @ P @ ( measur253975988404078935er_c_d @ ( giry_return_c_d @ M2 @ X4 ) ) )
= ( P @ X4 ) ) ) ) ).
% AE_return
thf(fact_231_AE__return,axiom,
! [X4: a > real,M2: sigma_measure_a_real,P: ( a > real ) > $o] :
( ( member_a_real @ X4 @ ( sigma_space_a_real @ M2 ) )
=> ( ( member_a_real_o @ P @ ( sigma_9085598459323199629real_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( eventually_a_real @ P @ ( measur8946935934331175658a_real @ ( giry_return_a_real @ M2 @ X4 ) ) )
= ( P @ X4 ) ) ) ) ).
% AE_return
thf(fact_232_AE__return,axiom,
! [X4: a > $o,M2: sigma_measure_a_o,P: ( a > $o ) > $o] :
( ( member_a_o @ X4 @ ( sigma_space_a_o @ M2 ) )
=> ( ( member_a_o_o @ P @ ( sigma_1195952539894209287_a_o_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( eventually_a_o @ P @ ( measur1950989848205109360er_a_o @ ( giry_return_a_o @ M2 @ X4 ) ) )
= ( P @ X4 ) ) ) ) ).
% AE_return
thf(fact_233_AE__return,axiom,
! [X4: b,M2: sigma_measure_b,P: b > $o] :
( ( member_b @ X4 @ ( sigma_space_b @ M2 ) )
=> ( ( member_b_o @ P @ ( sigma_measurable_b_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( eventually_b @ P @ ( measure_ae_filter_b @ ( giry_return_b @ M2 @ X4 ) ) )
= ( P @ X4 ) ) ) ) ).
% AE_return
thf(fact_234_AE__return,axiom,
! [X4: nat,M2: sigma_measure_nat,P: nat > $o] :
( ( member_nat @ X4 @ ( sigma_space_nat @ M2 ) )
=> ( ( member_nat_o @ P @ ( sigma_5101835498682829686_nat_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( eventually_nat @ P @ ( measur6539087422748349889er_nat @ ( giry_return_nat @ M2 @ X4 ) ) )
= ( P @ X4 ) ) ) ) ).
% AE_return
thf(fact_235_AE__return,axiom,
! [X4: $o,M2: sigma_measure_o,P: $o > $o] :
( ( member_o @ X4 @ ( sigma_space_o @ M2 ) )
=> ( ( member_o_o @ P @ ( sigma_measurable_o_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( eventually_o @ P @ ( measure_ae_filter_o @ ( giry_return_o @ M2 @ X4 ) ) )
= ( P @ X4 ) ) ) ) ).
% AE_return
thf(fact_236_AE__return,axiom,
! [X4: a,M2: sigma_measure_a,P: a > $o] :
( ( member_a @ X4 @ ( sigma_space_a @ M2 ) )
=> ( ( member_a_o @ P @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( eventually_a @ P @ ( measure_ae_filter_a @ ( giry_return_a @ M2 @ X4 ) ) )
= ( P @ X4 ) ) ) ) ).
% AE_return
thf(fact_237_space__return,axiom,
! [M2: sigma_measure_a,X4: a] :
( ( sigma_space_a @ ( giry_return_a @ M2 @ X4 ) )
= ( sigma_space_a @ M2 ) ) ).
% space_return
thf(fact_238_measurable__return2,axiom,
! [L: sigma_measure_c,N: sigma_measure_d,X4: d] :
( ( sigma_measurable_c_d @ L @ ( giry_return_d @ N @ X4 ) )
= ( sigma_measurable_c_d @ L @ N ) ) ).
% measurable_return2
thf(fact_239_measurable__return2,axiom,
! [L: sigma_measure_a,N: sigma_measure_o,X4: $o] :
( ( sigma_measurable_a_o @ L @ ( giry_return_o @ N @ X4 ) )
= ( sigma_measurable_a_o @ L @ N ) ) ).
% measurable_return2
thf(fact_240_measurable__return2,axiom,
! [L: sigma_measure_a,N: sigma_measure_real,X4: real] :
( ( sigma_9116425665531756122a_real @ L @ ( giry_return_real @ N @ X4 ) )
= ( sigma_9116425665531756122a_real @ L @ N ) ) ).
% measurable_return2
thf(fact_241_measurable__return1,axiom,
! [N: sigma_measure_c,X4: c,L: sigma_measure_d] :
( ( sigma_measurable_c_d @ ( giry_return_c @ N @ X4 ) @ L )
= ( sigma_measurable_c_d @ N @ L ) ) ).
% measurable_return1
thf(fact_242_measurable__return1,axiom,
! [N: sigma_measure_a,X4: a,L: sigma_measure_o] :
( ( sigma_measurable_a_o @ ( giry_return_a @ N @ X4 ) @ L )
= ( sigma_measurable_a_o @ N @ L ) ) ).
% measurable_return1
thf(fact_243_measurable__return1,axiom,
! [N: sigma_measure_a,X4: a,L: sigma_measure_real] :
( ( sigma_9116425665531756122a_real @ ( giry_return_a @ N @ X4 ) @ L )
= ( sigma_9116425665531756122a_real @ N @ L ) ) ).
% measurable_return1
thf(fact_244_prob__space_OAE__eq__constD_I1_J,axiom,
! [M2: sigma_measure_a,Y2: a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( eventually_a
@ ^ [X: a] : ( X = Y2 )
@ ( measure_ae_filter_a @ M2 ) )
=> ( M2
= ( giry_return_a @ M2 @ Y2 ) ) ) ) ).
% prob_space.AE_eq_constD(1)
thf(fact_245_prob__space_OAE__eq__constD_I2_J,axiom,
! [M2: sigma_measure_c_d,Y2: c > d] :
( ( probab3693743499390067171ce_c_d @ M2 )
=> ( ( eventually_c_d
@ ^ [X: c > d] : ( X = Y2 )
@ ( measur253975988404078935er_c_d @ M2 ) )
=> ( member_c_d @ Y2 @ ( sigma_space_c_d @ M2 ) ) ) ) ).
% prob_space.AE_eq_constD(2)
thf(fact_246_prob__space_OAE__eq__constD_I2_J,axiom,
! [M2: sigma_measure_a_real,Y2: a > real] :
( ( probab2024454272037758302a_real @ M2 )
=> ( ( eventually_a_real
@ ^ [X: a > real] : ( X = Y2 )
@ ( measur8946935934331175658a_real @ M2 ) )
=> ( member_a_real @ Y2 @ ( sigma_space_a_real @ M2 ) ) ) ) ).
% prob_space.AE_eq_constD(2)
thf(fact_247_prob__space_OAE__eq__constD_I2_J,axiom,
! [M2: sigma_measure_a_o,Y2: a > $o] :
( ( probab7249042050958952188ce_a_o @ M2 )
=> ( ( eventually_a_o
@ ^ [X: a > $o] : ( X = Y2 )
@ ( measur1950989848205109360er_a_o @ M2 ) )
=> ( member_a_o @ Y2 @ ( sigma_space_a_o @ M2 ) ) ) ) ).
% prob_space.AE_eq_constD(2)
thf(fact_248_prob__space_OAE__eq__constD_I2_J,axiom,
! [M2: sigma_measure_b,Y2: b] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( eventually_b
@ ^ [X: b] : ( X = Y2 )
@ ( measure_ae_filter_b @ M2 ) )
=> ( member_b @ Y2 @ ( sigma_space_b @ M2 ) ) ) ) ).
% prob_space.AE_eq_constD(2)
thf(fact_249_prob__space_OAE__eq__constD_I2_J,axiom,
! [M2: sigma_measure_nat,Y2: nat] :
( ( probab2904919403188438605ce_nat @ M2 )
=> ( ( eventually_nat
@ ^ [X: nat] : ( X = Y2 )
@ ( measur6539087422748349889er_nat @ M2 ) )
=> ( member_nat @ Y2 @ ( sigma_space_nat @ M2 ) ) ) ) ).
% prob_space.AE_eq_constD(2)
thf(fact_250_prob__space_OAE__eq__constD_I2_J,axiom,
! [M2: sigma_measure_o,Y2: $o] :
( ( probab1190487603588612059pace_o @ M2 )
=> ( ( eventually_o
@ ^ [X: $o] : ( X = Y2 )
@ ( measure_ae_filter_o @ M2 ) )
=> ( member_o @ Y2 @ ( sigma_space_o @ M2 ) ) ) ) ).
% prob_space.AE_eq_constD(2)
thf(fact_251_prob__space_OAE__eq__constD_I2_J,axiom,
! [M2: sigma_measure_a,Y2: a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( eventually_a
@ ^ [X: a] : ( X = Y2 )
@ ( measure_ae_filter_a @ M2 ) )
=> ( member_a @ Y2 @ ( sigma_space_a @ M2 ) ) ) ) ).
% prob_space.AE_eq_constD(2)
thf(fact_252_subprob__space__return,axiom,
! [X4: c > d,M2: sigma_measure_c_d] :
( ( member_c_d @ X4 @ ( sigma_space_c_d @ M2 ) )
=> ( giry_s3243706009590697010ce_c_d @ ( giry_return_c_d @ M2 @ X4 ) ) ) ).
% subprob_space_return
thf(fact_253_subprob__space__return,axiom,
! [X4: a > real,M2: sigma_measure_a_real] :
( ( member_a_real @ X4 @ ( sigma_space_a_real @ M2 ) )
=> ( giry_s8245238245569539279a_real @ ( giry_return_a_real @ M2 @ X4 ) ) ) ).
% subprob_space_return
thf(fact_254_subprob__space__return,axiom,
! [X4: a > $o,M2: sigma_measure_a_o] :
( ( member_a_o @ X4 @ ( sigma_space_a_o @ M2 ) )
=> ( giry_s2201070112685910219ce_a_o @ ( giry_return_a_o @ M2 @ X4 ) ) ) ).
% subprob_space_return
thf(fact_255_subprob__space__return,axiom,
! [X4: b,M2: sigma_measure_b] :
( ( member_b @ X4 @ ( sigma_space_b @ M2 ) )
=> ( giry_subprob_space_b @ ( giry_return_b @ M2 @ X4 ) ) ) ).
% subprob_space_return
thf(fact_256_subprob__space__return,axiom,
! [X4: nat,M2: sigma_measure_nat] :
( ( member_nat @ X4 @ ( sigma_space_nat @ M2 ) )
=> ( giry_s459323515522551452ce_nat @ ( giry_return_nat @ M2 @ X4 ) ) ) ).
% subprob_space_return
thf(fact_257_subprob__space__return,axiom,
! [X4: $o,M2: sigma_measure_o] :
( ( member_o @ X4 @ ( sigma_space_o @ M2 ) )
=> ( giry_subprob_space_o @ ( giry_return_o @ M2 @ X4 ) ) ) ).
% subprob_space_return
thf(fact_258_subprob__space__return,axiom,
! [X4: a,M2: sigma_measure_a] :
( ( member_a @ X4 @ ( sigma_space_a @ M2 ) )
=> ( giry_subprob_space_a @ ( giry_return_a @ M2 @ X4 ) ) ) ).
% subprob_space_return
thf(fact_259_prob__space__return,axiom,
! [X4: c > d,M2: sigma_measure_c_d] :
( ( member_c_d @ X4 @ ( sigma_space_c_d @ M2 ) )
=> ( probab3693743499390067171ce_c_d @ ( giry_return_c_d @ M2 @ X4 ) ) ) ).
% prob_space_return
thf(fact_260_prob__space__return,axiom,
! [X4: a > real,M2: sigma_measure_a_real] :
( ( member_a_real @ X4 @ ( sigma_space_a_real @ M2 ) )
=> ( probab2024454272037758302a_real @ ( giry_return_a_real @ M2 @ X4 ) ) ) ).
% prob_space_return
thf(fact_261_prob__space__return,axiom,
! [X4: a > $o,M2: sigma_measure_a_o] :
( ( member_a_o @ X4 @ ( sigma_space_a_o @ M2 ) )
=> ( probab7249042050958952188ce_a_o @ ( giry_return_a_o @ M2 @ X4 ) ) ) ).
% prob_space_return
thf(fact_262_prob__space__return,axiom,
! [X4: b,M2: sigma_measure_b] :
( ( member_b @ X4 @ ( sigma_space_b @ M2 ) )
=> ( probab7247484486040049090pace_b @ ( giry_return_b @ M2 @ X4 ) ) ) ).
% prob_space_return
thf(fact_263_prob__space__return,axiom,
! [X4: nat,M2: sigma_measure_nat] :
( ( member_nat @ X4 @ ( sigma_space_nat @ M2 ) )
=> ( probab2904919403188438605ce_nat @ ( giry_return_nat @ M2 @ X4 ) ) ) ).
% prob_space_return
thf(fact_264_prob__space__return,axiom,
! [X4: $o,M2: sigma_measure_o] :
( ( member_o @ X4 @ ( sigma_space_o @ M2 ) )
=> ( probab1190487603588612059pace_o @ ( giry_return_o @ M2 @ X4 ) ) ) ).
% prob_space_return
thf(fact_265_prob__space__return,axiom,
! [X4: a,M2: sigma_measure_a] :
( ( member_a @ X4 @ ( sigma_space_a @ M2 ) )
=> ( probab7247484486040049089pace_a @ ( giry_return_a @ M2 @ X4 ) ) ) ).
% prob_space_return
thf(fact_266_prob__space__completion,axiom,
probab7247484486040049089pace_a @ ( comple3428971583294703880tion_a @ m ) ).
% prob_space_completion
thf(fact_267_indep__vars__compose,axiom,
! [M: b > sigma_measure_c,X2: b > a > c,I2: set_b,Y: b > c > d,N: b > sigma_measure_d] :
( ( indepe7639357355105118967_a_b_c @ m @ M @ X2 @ I2 )
=> ( ! [I3: b] :
( ( member_b @ I3 @ I2 )
=> ( member_c_d @ ( Y @ I3 ) @ ( sigma_measurable_c_d @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe7639357355105118968_a_b_d @ m @ N
@ ^ [I4: b] : ( comp_c_d_a @ ( Y @ I4 ) @ ( X2 @ I4 ) )
@ I2 ) ) ) ).
% indep_vars_compose
thf(fact_268_indep__vars__compose,axiom,
! [M: a > sigma_measure_c,X2: a > a > c,I2: set_a,Y: a > c > d,N: a > sigma_measure_d] :
( ( indepe1203440900223019192_a_a_c @ m @ M @ X2 @ I2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I2 )
=> ( member_c_d @ ( Y @ I3 ) @ ( sigma_measurable_c_d @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe1203440900223019193_a_a_d @ m @ N
@ ^ [I4: a] : ( comp_c_d_a @ ( Y @ I4 ) @ ( X2 @ I4 ) )
@ I2 ) ) ) ).
% indep_vars_compose
thf(fact_269_indep__vars__compose,axiom,
! [M: nat > sigma_measure_c,X2: nat > a > c,I2: set_nat,Y: nat > c > d,N: nat > sigma_measure_d] :
( ( indepe3245197900929106296_nat_c @ m @ M @ X2 @ I2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( member_c_d @ ( Y @ I3 ) @ ( sigma_measurable_c_d @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe3245197900929106297_nat_d @ m @ N
@ ^ [I4: nat] : ( comp_c_d_a @ ( Y @ I4 ) @ ( X2 @ I4 ) )
@ I2 ) ) ) ).
% indep_vars_compose
thf(fact_270_indep__vars__compose,axiom,
! [M: $o > sigma_measure_c,X2: $o > a > c,I2: set_o,Y: $o > c > d,N: $o > sigma_measure_d] :
( ( indepe3252683823613847070_a_o_c @ m @ M @ X2 @ I2 )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( member_c_d @ ( Y @ I3 ) @ ( sigma_measurable_c_d @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe3252683823613847071_a_o_d @ m @ N
@ ^ [I4: $o] : ( comp_c_d_a @ ( Y @ I4 ) @ ( X2 @ I4 ) )
@ I2 ) ) ) ).
% indep_vars_compose
thf(fact_271_indep__vars__compose,axiom,
! [M: b > sigma_measure_a,X2: b > a > a,I2: set_b,Y: b > a > $o,N: b > sigma_measure_o] :
( ( indepe7639357355105118965_a_b_a @ m @ M @ X2 @ I2 )
=> ( ! [I3: b] :
( ( member_b @ I3 @ I2 )
=> ( member_a_o @ ( Y @ I3 ) @ ( sigma_measurable_a_o @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe432941580615200527_a_b_o @ m @ N
@ ^ [I4: b] : ( comp_a_o_a @ ( Y @ I4 ) @ ( X2 @ I4 ) )
@ I2 ) ) ) ).
% indep_vars_compose
thf(fact_272_indep__vars__compose,axiom,
! [M: a > sigma_measure_a,X2: a > a > a,I2: set_a,Y: a > a > $o,N: a > sigma_measure_o] :
( ( indepe1203440900223019190_a_a_a @ m @ M @ X2 @ I2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I2 )
=> ( member_a_o @ ( Y @ I3 ) @ ( sigma_measurable_a_o @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe3332163980079594832_a_a_o @ m @ N
@ ^ [I4: a] : ( comp_a_o_a @ ( Y @ I4 ) @ ( X2 @ I4 ) )
@ I2 ) ) ) ).
% indep_vars_compose
thf(fact_273_indep__vars__compose,axiom,
! [M: nat > sigma_measure_a,X2: nat > a > a,I2: set_nat,Y: nat > a > $o,N: nat > sigma_measure_o] :
( ( indepe3245197900929106294_nat_a @ m @ M @ X2 @ I2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( member_a_o @ ( Y @ I3 ) @ ( sigma_measurable_a_o @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe6621553285932267024_nat_o @ m @ N
@ ^ [I4: nat] : ( comp_a_o_a @ ( Y @ I4 ) @ ( X2 @ I4 ) )
@ I2 ) ) ) ).
% indep_vars_compose
thf(fact_274_indep__vars__compose,axiom,
! [M: $o > sigma_measure_a,X2: $o > a > a,I2: set_o,Y: $o > a > $o,N: $o > sigma_measure_o] :
( ( indepe3252683823613847068_a_o_a @ m @ M @ X2 @ I2 )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( member_a_o @ ( Y @ I3 ) @ ( sigma_measurable_a_o @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe9162428965118168502_a_o_o @ m @ N
@ ^ [I4: $o] : ( comp_a_o_a @ ( Y @ I4 ) @ ( X2 @ I4 ) )
@ I2 ) ) ) ).
% indep_vars_compose
thf(fact_275_indep__vars__compose,axiom,
! [M: b > sigma_measure_a,X2: b > a > a,I2: set_b,Y: b > a > real,N: b > sigma_measure_real] :
( ( indepe7639357355105118965_a_b_a @ m @ M @ X2 @ I2 )
=> ( ! [I3: b] :
( ( member_b @ I3 @ I2 )
=> ( member_a_real @ ( Y @ I3 ) @ ( sigma_9116425665531756122a_real @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe8265442546547513973b_real @ m @ N
@ ^ [I4: b] : ( comp_a_real_a @ ( Y @ I4 ) @ ( X2 @ I4 ) )
@ I2 ) ) ) ).
% indep_vars_compose
thf(fact_276_indep__vars__compose,axiom,
! [M: a > sigma_measure_a,X2: a > a > a,I2: set_a,Y: a > a > real,N: a > sigma_measure_real] :
( ( indepe1203440900223019190_a_a_a @ m @ M @ X2 @ I2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I2 )
=> ( member_a_real @ ( Y @ I3 ) @ ( sigma_9116425665531756122a_real @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe2669223931359383284a_real @ m @ N
@ ^ [I4: a] : ( comp_a_real_a @ ( Y @ I4 ) @ ( X2 @ I4 ) )
@ I2 ) ) ) ).
% indep_vars_compose
thf(fact_277_subprob__not__empty,axiom,
( ( sigma_space_a @ m )
!= bot_bot_set_a ) ).
% subprob_not_empty
thf(fact_278_measurable__comp,axiom,
! [F2: c > c,M2: sigma_measure_c,N: sigma_measure_c,G2: c > d,L: sigma_measure_d] :
( ( member_c_c @ F2 @ ( sigma_measurable_c_c @ M2 @ N ) )
=> ( ( member_c_d @ G2 @ ( sigma_measurable_c_d @ N @ L ) )
=> ( member_c_d @ ( comp_c_d_c @ G2 @ F2 ) @ ( sigma_measurable_c_d @ M2 @ L ) ) ) ) ).
% measurable_comp
thf(fact_279_measurable__comp,axiom,
! [F2: a > a,M2: sigma_measure_a,N: sigma_measure_a,G2: a > $o,L: sigma_measure_o] :
( ( member_a_a @ F2 @ ( sigma_measurable_a_a @ M2 @ N ) )
=> ( ( member_a_o @ G2 @ ( sigma_measurable_a_o @ N @ L ) )
=> ( member_a_o @ ( comp_a_o_a @ G2 @ F2 ) @ ( sigma_measurable_a_o @ M2 @ L ) ) ) ) ).
% measurable_comp
thf(fact_280_measurable__comp,axiom,
! [F2: a > a,M2: sigma_measure_a,N: sigma_measure_a,G2: a > real,L: sigma_measure_real] :
( ( member_a_a @ F2 @ ( sigma_measurable_a_a @ M2 @ N ) )
=> ( ( member_a_real @ G2 @ ( sigma_9116425665531756122a_real @ N @ L ) )
=> ( member_a_real @ ( comp_a_real_a @ G2 @ F2 ) @ ( sigma_9116425665531756122a_real @ M2 @ L ) ) ) ) ).
% measurable_comp
thf(fact_281_measurable__comp,axiom,
! [F2: c > d,M2: sigma_measure_c,N: sigma_measure_d,G2: d > d,L: sigma_measure_d] :
( ( member_c_d @ F2 @ ( sigma_measurable_c_d @ M2 @ N ) )
=> ( ( member_d_d @ G2 @ ( sigma_measurable_d_d @ N @ L ) )
=> ( member_c_d @ ( comp_d_d_c @ G2 @ F2 ) @ ( sigma_measurable_c_d @ M2 @ L ) ) ) ) ).
% measurable_comp
thf(fact_282_measurable__comp,axiom,
! [F2: a > $o,M2: sigma_measure_a,N: sigma_measure_o,G2: $o > $o,L: sigma_measure_o] :
( ( member_a_o @ F2 @ ( sigma_measurable_a_o @ M2 @ N ) )
=> ( ( member_o_o @ G2 @ ( sigma_measurable_o_o @ N @ L ) )
=> ( member_a_o @ ( comp_o_o_a @ G2 @ F2 ) @ ( sigma_measurable_a_o @ M2 @ L ) ) ) ) ).
% measurable_comp
thf(fact_283_measurable__comp,axiom,
! [F2: a > $o,M2: sigma_measure_a,N: sigma_measure_o,G2: $o > real,L: sigma_measure_real] :
( ( member_a_o @ F2 @ ( sigma_measurable_a_o @ M2 @ N ) )
=> ( ( member_o_real @ G2 @ ( sigma_2430008634441611636o_real @ N @ L ) )
=> ( member_a_real @ ( comp_o_real_a @ G2 @ F2 ) @ ( sigma_9116425665531756122a_real @ M2 @ L ) ) ) ) ).
% measurable_comp
thf(fact_284_measurable__comp,axiom,
! [F2: a > real,M2: sigma_measure_a,N: sigma_measure_real,G2: real > $o,L: sigma_measure_o] :
( ( member_a_real @ F2 @ ( sigma_9116425665531756122a_real @ M2 @ N ) )
=> ( ( member_real_o @ G2 @ ( sigma_3939073009482781210real_o @ N @ L ) )
=> ( member_a_o @ ( comp_real_o_a @ G2 @ F2 ) @ ( sigma_measurable_a_o @ M2 @ L ) ) ) ) ).
% measurable_comp
thf(fact_285_measurable__comp,axiom,
! [F2: a > real,M2: sigma_measure_a,N: sigma_measure_real,G2: real > real,L: sigma_measure_real] :
( ( member_a_real @ F2 @ ( sigma_9116425665531756122a_real @ M2 @ N ) )
=> ( ( member_real_real @ G2 @ ( sigma_5267869275261027754l_real @ N @ L ) )
=> ( member_a_real @ ( comp_real_real_a @ G2 @ F2 ) @ ( sigma_9116425665531756122a_real @ M2 @ L ) ) ) ) ).
% measurable_comp
thf(fact_286_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_287_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_288_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_289_measurable__empty__iff,axiom,
! [N: sigma_measure_d,F2: c > d,M2: sigma_measure_c] :
( ( ( sigma_space_d @ N )
= bot_bot_set_d )
=> ( ( member_c_d @ F2 @ ( sigma_measurable_c_d @ M2 @ N ) )
= ( ( sigma_space_c @ M2 )
= bot_bot_set_c ) ) ) ).
% measurable_empty_iff
thf(fact_290_measurable__empty__iff,axiom,
! [N: sigma_measure_o,F2: a > $o,M2: sigma_measure_a] :
( ( ( sigma_space_o @ N )
= bot_bot_set_o )
=> ( ( member_a_o @ F2 @ ( sigma_measurable_a_o @ M2 @ N ) )
= ( ( sigma_space_a @ M2 )
= bot_bot_set_a ) ) ) ).
% measurable_empty_iff
thf(fact_291_measurable__empty__iff,axiom,
! [N: sigma_measure_real,F2: a > real,M2: sigma_measure_a] :
( ( ( sigma_space_real @ N )
= bot_bot_set_real )
=> ( ( member_a_real @ F2 @ ( sigma_9116425665531756122a_real @ M2 @ N ) )
= ( ( sigma_space_a @ M2 )
= bot_bot_set_a ) ) ) ).
% measurable_empty_iff
thf(fact_292_space__empty,axiom,
! [M2: sigma_measure_a] :
( ( ( sigma_space_a @ M2 )
= bot_bot_set_a )
=> ( M2
= ( sigma_count_space_a @ bot_bot_set_a ) ) ) ).
% space_empty
thf(fact_293_space__empty,axiom,
! [M2: sigma_measure_o] :
( ( ( sigma_space_o @ M2 )
= bot_bot_set_o )
=> ( M2
= ( sigma_count_space_o @ bot_bot_set_o ) ) ) ).
% space_empty
thf(fact_294_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_295_subprob__space__return__ne,axiom,
! [M2: sigma_measure_a,X4: a] :
( ( ( sigma_space_a @ M2 )
!= bot_bot_set_a )
=> ( giry_subprob_space_a @ ( giry_return_a @ M2 @ X4 ) ) ) ).
% subprob_space_return_ne
thf(fact_296_prob__space_Oindep__vars__compose,axiom,
! [M2: sigma_measure_a,M: b > sigma_measure_c,X2: b > a > c,I2: set_b,Y: b > c > d,N: b > sigma_measure_d] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe7639357355105118967_a_b_c @ M2 @ M @ X2 @ I2 )
=> ( ! [I3: b] :
( ( member_b @ I3 @ I2 )
=> ( member_c_d @ ( Y @ I3 ) @ ( sigma_measurable_c_d @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe7639357355105118968_a_b_d @ M2 @ N
@ ^ [I4: b] : ( comp_c_d_a @ ( Y @ I4 ) @ ( X2 @ I4 ) )
@ I2 ) ) ) ) ).
% prob_space.indep_vars_compose
thf(fact_297_prob__space_Oindep__vars__compose,axiom,
! [M2: sigma_measure_a,M: a > sigma_measure_c,X2: a > a > c,I2: set_a,Y: a > c > d,N: a > sigma_measure_d] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe1203440900223019192_a_a_c @ M2 @ M @ X2 @ I2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I2 )
=> ( member_c_d @ ( Y @ I3 ) @ ( sigma_measurable_c_d @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe1203440900223019193_a_a_d @ M2 @ N
@ ^ [I4: a] : ( comp_c_d_a @ ( Y @ I4 ) @ ( X2 @ I4 ) )
@ I2 ) ) ) ) ).
% prob_space.indep_vars_compose
thf(fact_298_prob__space_Oindep__vars__compose,axiom,
! [M2: sigma_measure_a,M: nat > sigma_measure_c,X2: nat > a > c,I2: set_nat,Y: nat > c > d,N: nat > sigma_measure_d] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe3245197900929106296_nat_c @ M2 @ M @ X2 @ I2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( member_c_d @ ( Y @ I3 ) @ ( sigma_measurable_c_d @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe3245197900929106297_nat_d @ M2 @ N
@ ^ [I4: nat] : ( comp_c_d_a @ ( Y @ I4 ) @ ( X2 @ I4 ) )
@ I2 ) ) ) ) ).
% prob_space.indep_vars_compose
thf(fact_299_prob__space_Oindep__vars__compose,axiom,
! [M2: sigma_measure_a,M: $o > sigma_measure_c,X2: $o > a > c,I2: set_o,Y: $o > c > d,N: $o > sigma_measure_d] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe3252683823613847070_a_o_c @ M2 @ M @ X2 @ I2 )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( member_c_d @ ( Y @ I3 ) @ ( sigma_measurable_c_d @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe3252683823613847071_a_o_d @ M2 @ N
@ ^ [I4: $o] : ( comp_c_d_a @ ( Y @ I4 ) @ ( X2 @ I4 ) )
@ I2 ) ) ) ) ).
% prob_space.indep_vars_compose
thf(fact_300_prob__space_Oindep__vars__compose,axiom,
! [M2: sigma_measure_a,M: b > sigma_measure_a,X2: b > a > a,I2: set_b,Y: b > a > $o,N: b > sigma_measure_o] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe7639357355105118965_a_b_a @ M2 @ M @ X2 @ I2 )
=> ( ! [I3: b] :
( ( member_b @ I3 @ I2 )
=> ( member_a_o @ ( Y @ I3 ) @ ( sigma_measurable_a_o @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe432941580615200527_a_b_o @ M2 @ N
@ ^ [I4: b] : ( comp_a_o_a @ ( Y @ I4 ) @ ( X2 @ I4 ) )
@ I2 ) ) ) ) ).
% prob_space.indep_vars_compose
thf(fact_301_prob__space_Oindep__vars__compose,axiom,
! [M2: sigma_measure_a,M: a > sigma_measure_a,X2: a > a > a,I2: set_a,Y: a > a > $o,N: a > sigma_measure_o] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe1203440900223019190_a_a_a @ M2 @ M @ X2 @ I2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I2 )
=> ( member_a_o @ ( Y @ I3 ) @ ( sigma_measurable_a_o @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe3332163980079594832_a_a_o @ M2 @ N
@ ^ [I4: a] : ( comp_a_o_a @ ( Y @ I4 ) @ ( X2 @ I4 ) )
@ I2 ) ) ) ) ).
% prob_space.indep_vars_compose
thf(fact_302_prob__space_Oindep__vars__compose,axiom,
! [M2: sigma_measure_a,M: nat > sigma_measure_a,X2: nat > a > a,I2: set_nat,Y: nat > a > $o,N: nat > sigma_measure_o] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe3245197900929106294_nat_a @ M2 @ M @ X2 @ I2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( member_a_o @ ( Y @ I3 ) @ ( sigma_measurable_a_o @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe6621553285932267024_nat_o @ M2 @ N
@ ^ [I4: nat] : ( comp_a_o_a @ ( Y @ I4 ) @ ( X2 @ I4 ) )
@ I2 ) ) ) ) ).
% prob_space.indep_vars_compose
thf(fact_303_prob__space_Oindep__vars__compose,axiom,
! [M2: sigma_measure_a,M: $o > sigma_measure_a,X2: $o > a > a,I2: set_o,Y: $o > a > $o,N: $o > sigma_measure_o] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe3252683823613847068_a_o_a @ M2 @ M @ X2 @ I2 )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( member_a_o @ ( Y @ I3 ) @ ( sigma_measurable_a_o @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe9162428965118168502_a_o_o @ M2 @ N
@ ^ [I4: $o] : ( comp_a_o_a @ ( Y @ I4 ) @ ( X2 @ I4 ) )
@ I2 ) ) ) ) ).
% prob_space.indep_vars_compose
thf(fact_304_prob__space_Oindep__vars__compose,axiom,
! [M2: sigma_measure_a,M: b > sigma_measure_a,X2: b > a > a,I2: set_b,Y: b > a > real,N: b > sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe7639357355105118965_a_b_a @ M2 @ M @ X2 @ I2 )
=> ( ! [I3: b] :
( ( member_b @ I3 @ I2 )
=> ( member_a_real @ ( Y @ I3 ) @ ( sigma_9116425665531756122a_real @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe8265442546547513973b_real @ M2 @ N
@ ^ [I4: b] : ( comp_a_real_a @ ( Y @ I4 ) @ ( X2 @ I4 ) )
@ I2 ) ) ) ) ).
% prob_space.indep_vars_compose
thf(fact_305_prob__space_Oindep__vars__compose,axiom,
! [M2: sigma_measure_a,M: a > sigma_measure_a,X2: a > a > a,I2: set_a,Y: a > a > real,N: a > sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe1203440900223019190_a_a_a @ M2 @ M @ X2 @ I2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I2 )
=> ( member_a_real @ ( Y @ I3 ) @ ( sigma_9116425665531756122a_real @ ( M @ I3 ) @ ( N @ I3 ) ) ) )
=> ( indepe2669223931359383284a_real @ M2 @ N
@ ^ [I4: a] : ( comp_a_real_a @ ( Y @ I4 ) @ ( X2 @ I4 ) )
@ I2 ) ) ) ) ).
% prob_space.indep_vars_compose
thf(fact_306_pred__intros__logic_I8_J,axiom,
! [F2: a > c > d,M2: sigma_measure_a] :
( member_a_o
@ ^ [X: a] : ( member_c_d @ ( F2 @ X ) @ bot_bot_set_c_d )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% pred_intros_logic(8)
thf(fact_307_pred__intros__logic_I8_J,axiom,
! [F2: a > a > real,M2: sigma_measure_a] :
( member_a_o
@ ^ [X: a] : ( member_a_real @ ( F2 @ X ) @ bot_bot_set_a_real )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% pred_intros_logic(8)
thf(fact_308_pred__intros__logic_I8_J,axiom,
! [F2: a > a > $o,M2: sigma_measure_a] :
( member_a_o
@ ^ [X: a] : ( member_a_o @ ( F2 @ X ) @ bot_bot_set_a_o )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% pred_intros_logic(8)
thf(fact_309_pred__intros__logic_I8_J,axiom,
! [F2: a > b,M2: sigma_measure_a] :
( member_a_o
@ ^ [X: a] : ( member_b @ ( F2 @ X ) @ bot_bot_set_b )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% pred_intros_logic(8)
thf(fact_310_pred__intros__logic_I8_J,axiom,
! [F2: a > nat,M2: sigma_measure_a] :
( member_a_o
@ ^ [X: a] : ( member_nat @ ( F2 @ X ) @ bot_bot_set_nat )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% pred_intros_logic(8)
thf(fact_311_pred__intros__logic_I8_J,axiom,
! [F2: a > $o,M2: sigma_measure_a] :
( member_a_o
@ ^ [X: a] : ( member_o @ ( F2 @ X ) @ bot_bot_set_o )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% pred_intros_logic(8)
thf(fact_312_pred__intros__logic_I8_J,axiom,
! [F2: a > a,M2: sigma_measure_a] :
( member_a_o
@ ^ [X: a] : ( member_a @ ( F2 @ X ) @ bot_bot_set_a )
@ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% pred_intros_logic(8)
thf(fact_313_prob__space_Oindep__sets__Dynkin,axiom,
! [M2: sigma_measure_a,F: $o > set_set_a,I2: set_o] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe7780107833195774214ts_a_o @ M2 @ F @ I2 )
=> ( indepe7780107833195774214ts_a_o @ M2
@ ^ [I4: $o] : ( sigma_Dynkin_a @ ( sigma_space_a @ M2 ) @ ( F @ I4 ) )
@ I2 ) ) ) ).
% prob_space.indep_sets_Dynkin
thf(fact_314_prob__space_Oindep__sets__Dynkin,axiom,
! [M2: sigma_measure_a,F: nat > set_set_a,I2: set_nat] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe6267730027088848354_a_nat @ M2 @ F @ I2 )
=> ( indepe6267730027088848354_a_nat @ M2
@ ^ [I4: nat] : ( sigma_Dynkin_a @ ( sigma_space_a @ M2 ) @ ( F @ I4 ) )
@ I2 ) ) ) ).
% prob_space.indep_sets_Dynkin
thf(fact_315_prob__space_Oindep__sets__Dynkin,axiom,
! [M2: sigma_measure_a,F: a > set_set_a,I2: set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe8927441866673418604ts_a_a @ M2 @ F @ I2 )
=> ( indepe8927441866673418604ts_a_a @ M2
@ ^ [I4: a] : ( sigma_Dynkin_a @ ( sigma_space_a @ M2 ) @ ( F @ I4 ) )
@ I2 ) ) ) ).
% prob_space.indep_sets_Dynkin
thf(fact_316_prob__space_Oindep__sets__Dynkin,axiom,
! [M2: sigma_measure_a,F: b > set_set_a,I2: set_b] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe8927441866673418605ts_a_b @ M2 @ F @ I2 )
=> ( indepe8927441866673418605ts_a_b @ M2
@ ^ [I4: b] : ( sigma_Dynkin_a @ ( sigma_space_a @ M2 ) @ ( F @ I4 ) )
@ I2 ) ) ) ).
% prob_space.indep_sets_Dynkin
thf(fact_317_prob__space_Oindep__sets__Dynkin,axiom,
! [M2: sigma_measure_a,F: ( a > $o ) > set_set_a,I2: set_a_o] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe7801696130336798481_a_a_o @ M2 @ F @ I2 )
=> ( indepe7801696130336798481_a_a_o @ M2
@ ^ [I4: a > $o] : ( sigma_Dynkin_a @ ( sigma_space_a @ M2 ) @ ( F @ I4 ) )
@ I2 ) ) ) ).
% prob_space.indep_sets_Dynkin
thf(fact_318_prob__space_Oindep__sets__Dynkin,axiom,
! [M2: sigma_measure_a,F: ( a > real ) > set_set_a,I2: set_a_real] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe4749599203615801097a_real @ M2 @ F @ I2 )
=> ( indepe4749599203615801097a_real @ M2
@ ^ [I4: a > real] : ( sigma_Dynkin_a @ ( sigma_space_a @ M2 ) @ ( F @ I4 ) )
@ I2 ) ) ) ).
% prob_space.indep_sets_Dynkin
thf(fact_319_prob__space_Oindep__sets__Dynkin,axiom,
! [M2: sigma_measure_a,F: ( c > d ) > set_set_a,I2: set_c_d] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe4867465796832040824_a_c_d @ M2 @ F @ I2 )
=> ( indepe4867465796832040824_a_c_d @ M2
@ ^ [I4: c > d] : ( sigma_Dynkin_a @ ( sigma_space_a @ M2 ) @ ( F @ I4 ) )
@ I2 ) ) ) ).
% prob_space.indep_sets_Dynkin
thf(fact_320_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_321_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_322_subprob__space_Oaxioms_I1_J,axiom,
! [M2: sigma_measure_a] :
( ( giry_subprob_space_a @ M2 )
=> ( measur930452917991658466sure_a @ M2 ) ) ).
% subprob_space.axioms(1)
thf(fact_323_sets__completion__AE,axiom,
! [P: a > $o,M2: sigma_measure_a] :
( ( eventually_a
@ ^ [X: a] :
~ ( P @ X )
@ ( measure_ae_filter_a @ M2 ) )
=> ( member_a_o @ P @ ( sigma_measurable_a_o @ ( comple3428971583294703880tion_a @ M2 ) @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ).
% sets_completion_AE
thf(fact_324_indep__var__compose,axiom,
! [M1: sigma_measure_c,X1: a > c,M22: sigma_measure_c,X22: a > c,Y1: c > d,N1: sigma_measure_d,Y22: c > d,N22: sigma_measure_d] :
( ( indepe2440653194691626190ar_a_c @ m @ M1 @ X1 @ M22 @ X22 )
=> ( ( member_c_d @ Y1 @ ( sigma_measurable_c_d @ M1 @ N1 ) )
=> ( ( member_c_d @ Y22 @ ( sigma_measurable_c_d @ M22 @ N22 ) )
=> ( indepe2440653194691626191ar_a_d @ m @ N1 @ ( comp_c_d_a @ Y1 @ X1 ) @ N22 @ ( comp_c_d_a @ Y22 @ X22 ) ) ) ) ) ).
% indep_var_compose
thf(fact_325_indep__var__compose,axiom,
! [M1: sigma_measure_a,X1: a > a,M22: sigma_measure_a,X22: a > a,Y1: a > $o,N1: sigma_measure_o,Y22: a > $o,N22: sigma_measure_o] :
( ( indepe2440653194691626188ar_a_a @ m @ M1 @ X1 @ M22 @ X22 )
=> ( ( member_a_o @ Y1 @ ( sigma_measurable_a_o @ M1 @ N1 ) )
=> ( ( member_a_o @ Y22 @ ( sigma_measurable_a_o @ M22 @ N22 ) )
=> ( indepe2772234535145391206ar_a_o @ m @ N1 @ ( comp_a_o_a @ Y1 @ X1 ) @ N22 @ ( comp_a_o_a @ Y22 @ X22 ) ) ) ) ) ).
% indep_var_compose
thf(fact_326_indep__var__compose,axiom,
! [M1: sigma_measure_a,X1: a > a,M22: sigma_measure_a,X22: a > a,Y1: a > real,N1: sigma_measure_real,Y22: a > real,N22: sigma_measure_real] :
( ( indepe2440653194691626188ar_a_a @ m @ M1 @ X1 @ M22 @ X22 )
=> ( ( member_a_real @ Y1 @ ( sigma_9116425665531756122a_real @ M1 @ N1 ) )
=> ( ( member_a_real @ Y22 @ ( sigma_9116425665531756122a_real @ M22 @ N22 ) )
=> ( indepe8958435565499147358a_real @ m @ N1 @ ( comp_a_real_a @ Y1 @ X1 ) @ N22 @ ( comp_a_real_a @ Y22 @ X22 ) ) ) ) ) ).
% indep_var_compose
thf(fact_327_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_a
@ ^ [S: a] : P )
= top_top_set_a ) )
& ( ~ P
=> ( ( collect_a
@ ^ [S: a] : P )
= bot_bot_set_a ) ) ) ).
% Collect_const
thf(fact_328_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_nat
@ ^ [S: nat] : P )
= top_top_set_nat ) )
& ( ~ P
=> ( ( collect_nat
@ ^ [S: nat] : P )
= bot_bot_set_nat ) ) ) ).
% Collect_const
thf(fact_329_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_o
@ ^ [S: $o] : P )
= top_top_set_o ) )
& ( ~ P
=> ( ( collect_o
@ ^ [S: $o] : P )
= bot_bot_set_o ) ) ) ).
% Collect_const
thf(fact_330_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_literal
@ ^ [S: literal] : P )
= top_top_set_literal ) )
& ( ~ P
=> ( ( collect_literal
@ ^ [S: literal] : P )
= bot_bot_set_literal ) ) ) ).
% Collect_const
thf(fact_331_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_Product_unit
@ ^ [S: product_unit] : P )
= top_to1996260823553986621t_unit ) )
& ( ~ P
=> ( ( collect_Product_unit
@ ^ [S: product_unit] : P )
= bot_bo3957492148770167129t_unit ) ) ) ).
% Collect_const
thf(fact_332_space__completion,axiom,
! [M2: sigma_measure_a] :
( ( sigma_space_a @ ( comple3428971583294703880tion_a @ M2 ) )
= ( sigma_space_a @ M2 ) ) ).
% space_completion
thf(fact_333_AE__completion,axiom,
! [P: a > $o,M2: sigma_measure_a] :
( ( eventually_a @ P @ ( measure_ae_filter_a @ M2 ) )
=> ( eventually_a @ P @ ( measure_ae_filter_a @ ( comple3428971583294703880tion_a @ M2 ) ) ) ) ).
% AE_completion
thf(fact_334_AE__completion__iff,axiom,
! [P: a > $o,M2: sigma_measure_a] :
( ( eventually_a @ P @ ( measure_ae_filter_a @ ( comple3428971583294703880tion_a @ M2 ) ) )
= ( eventually_a @ P @ ( measure_ae_filter_a @ M2 ) ) ) ).
% AE_completion_iff
thf(fact_335_distributed__unique,axiom,
! [S2: sigma_measure_a,X2: a > a,Px: a > extend8495563244428889912nnreal,Py: a > extend8495563244428889912nnreal] :
( ( probab8876900357271971086ed_a_a @ m @ S2 @ X2 @ Px )
=> ( ( probab8876900357271971086ed_a_a @ m @ S2 @ X2 @ Py )
=> ( eventually_a
@ ^ [X: a] :
( ( Px @ X )
= ( Py @ X ) )
@ ( measure_ae_filter_a @ S2 ) ) ) ) ).
% distributed_unique
thf(fact_336_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_337_subprob__space__distr,axiom,
! [F2: a > $o,M: sigma_measure_o] :
( ( member_a_o @ F2 @ ( sigma_measurable_a_o @ m @ M ) )
=> ( ( ( sigma_space_o @ M )
!= bot_bot_set_o )
=> ( giry_subprob_space_o @ ( measure_distr_a_o @ m @ M @ F2 ) ) ) ) ).
% subprob_space_distr
thf(fact_338_subprob__space__distr,axiom,
! [F2: a > real,M: sigma_measure_real] :
( ( member_a_real @ F2 @ ( sigma_9116425665531756122a_real @ m @ M ) )
=> ( ( ( sigma_space_real @ M )
!= bot_bot_set_real )
=> ( giry_s8208748868292234104e_real @ ( measure_distr_a_real @ m @ M @ F2 ) ) ) ) ).
% subprob_space_distr
thf(fact_339_indep__var__rv1,axiom,
! [S2: sigma_measure_o,X2: a > $o,T: sigma_measure_o,Y: a > $o] :
( ( indepe2772234535145391206ar_a_o @ m @ S2 @ X2 @ T @ Y )
=> ( member_a_o @ X2 @ ( sigma_measurable_a_o @ m @ S2 ) ) ) ).
% indep_var_rv1
thf(fact_340_indep__var__rv1,axiom,
! [S2: sigma_measure_real,X2: a > real,T: sigma_measure_real,Y: a > real] :
( ( indepe8958435565499147358a_real @ m @ S2 @ X2 @ T @ Y )
=> ( member_a_real @ X2 @ ( sigma_9116425665531756122a_real @ m @ S2 ) ) ) ).
% indep_var_rv1
thf(fact_341_indep__var__rv2,axiom,
! [S2: sigma_measure_o,X2: a > $o,T: sigma_measure_o,Y: a > $o] :
( ( indepe2772234535145391206ar_a_o @ m @ S2 @ X2 @ T @ Y )
=> ( member_a_o @ Y @ ( sigma_measurable_a_o @ m @ T ) ) ) ).
% indep_var_rv2
thf(fact_342_indep__var__rv2,axiom,
! [S2: sigma_measure_real,X2: a > real,T: sigma_measure_real,Y: a > real] :
( ( indepe8958435565499147358a_real @ m @ S2 @ X2 @ T @ Y )
=> ( member_a_real @ Y @ ( sigma_9116425665531756122a_real @ m @ T ) ) ) ).
% indep_var_rv2
thf(fact_343_iso__tuple__UNIV__I,axiom,
! [X4: c > d] : ( member_c_d @ X4 @ top_top_set_c_d ) ).
% iso_tuple_UNIV_I
thf(fact_344_iso__tuple__UNIV__I,axiom,
! [X4: a > real] : ( member_a_real @ X4 @ top_top_set_a_real ) ).
% iso_tuple_UNIV_I
thf(fact_345_iso__tuple__UNIV__I,axiom,
! [X4: a > $o] : ( member_a_o @ X4 @ top_top_set_a_o ) ).
% iso_tuple_UNIV_I
thf(fact_346_iso__tuple__UNIV__I,axiom,
! [X4: b] : ( member_b @ X4 @ top_top_set_b ) ).
% iso_tuple_UNIV_I
thf(fact_347_iso__tuple__UNIV__I,axiom,
! [X4: a] : ( member_a @ X4 @ top_top_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_348_iso__tuple__UNIV__I,axiom,
! [X4: nat] : ( member_nat @ X4 @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_349_iso__tuple__UNIV__I,axiom,
! [X4: $o] : ( member_o @ X4 @ top_top_set_o ) ).
% iso_tuple_UNIV_I
thf(fact_350_iso__tuple__UNIV__I,axiom,
! [X4: literal] : ( member_literal @ X4 @ top_top_set_literal ) ).
% iso_tuple_UNIV_I
thf(fact_351_iso__tuple__UNIV__I,axiom,
! [X4: product_unit] : ( member_Product_unit @ X4 @ top_to1996260823553986621t_unit ) ).
% iso_tuple_UNIV_I
thf(fact_352_UNIV__I,axiom,
! [X4: c > d] : ( member_c_d @ X4 @ top_top_set_c_d ) ).
% UNIV_I
thf(fact_353_UNIV__I,axiom,
! [X4: a > real] : ( member_a_real @ X4 @ top_top_set_a_real ) ).
% UNIV_I
thf(fact_354_UNIV__I,axiom,
! [X4: a > $o] : ( member_a_o @ X4 @ top_top_set_a_o ) ).
% UNIV_I
thf(fact_355_UNIV__I,axiom,
! [X4: b] : ( member_b @ X4 @ top_top_set_b ) ).
% UNIV_I
thf(fact_356_UNIV__I,axiom,
! [X4: a] : ( member_a @ X4 @ top_top_set_a ) ).
% UNIV_I
thf(fact_357_UNIV__I,axiom,
! [X4: nat] : ( member_nat @ X4 @ top_top_set_nat ) ).
% UNIV_I
thf(fact_358_UNIV__I,axiom,
! [X4: $o] : ( member_o @ X4 @ top_top_set_o ) ).
% UNIV_I
thf(fact_359_UNIV__I,axiom,
! [X4: literal] : ( member_literal @ X4 @ top_top_set_literal ) ).
% UNIV_I
thf(fact_360_UNIV__I,axiom,
! [X4: product_unit] : ( member_Product_unit @ X4 @ top_to1996260823553986621t_unit ) ).
% UNIV_I
thf(fact_361_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_362_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X: a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_363_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_364_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X: a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_365_all__not__in__conv,axiom,
! [A2: set_c_d] :
( ( ! [X: c > d] :
~ ( member_c_d @ X @ A2 ) )
= ( A2 = bot_bot_set_c_d ) ) ).
% all_not_in_conv
thf(fact_366_all__not__in__conv,axiom,
! [A2: set_a_real] :
( ( ! [X: a > real] :
~ ( member_a_real @ X @ A2 ) )
= ( A2 = bot_bot_set_a_real ) ) ).
% all_not_in_conv
thf(fact_367_all__not__in__conv,axiom,
! [A2: set_a_o] :
( ( ! [X: a > $o] :
~ ( member_a_o @ X @ A2 ) )
= ( A2 = bot_bot_set_a_o ) ) ).
% all_not_in_conv
thf(fact_368_all__not__in__conv,axiom,
! [A2: set_b] :
( ( ! [X: b] :
~ ( member_b @ X @ A2 ) )
= ( A2 = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_369_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X: nat] :
~ ( member_nat @ X @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_370_all__not__in__conv,axiom,
! [A2: set_o] :
( ( ! [X: $o] :
~ ( member_o @ X @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_371_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X: a] :
~ ( member_a @ X @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_372_empty__iff,axiom,
! [C: c > d] :
~ ( member_c_d @ C @ bot_bot_set_c_d ) ).
% empty_iff
thf(fact_373_empty__iff,axiom,
! [C: a > real] :
~ ( member_a_real @ C @ bot_bot_set_a_real ) ).
% empty_iff
thf(fact_374_empty__iff,axiom,
! [C: a > $o] :
~ ( member_a_o @ C @ bot_bot_set_a_o ) ).
% empty_iff
thf(fact_375_empty__iff,axiom,
! [C: b] :
~ ( member_b @ C @ bot_bot_set_b ) ).
% empty_iff
thf(fact_376_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_377_empty__iff,axiom,
! [C: $o] :
~ ( member_o @ C @ bot_bot_set_o ) ).
% empty_iff
thf(fact_378_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_379_prob__space__distr,axiom,
! [F2: a > $o,M: sigma_measure_o] :
( ( member_a_o @ F2 @ ( sigma_measurable_a_o @ m @ M ) )
=> ( probab1190487603588612059pace_o @ ( measure_distr_a_o @ m @ M @ F2 ) ) ) ).
% prob_space_distr
thf(fact_380_prob__space__distr,axiom,
! [F2: a > real,M: sigma_measure_real] :
( ( member_a_real @ F2 @ ( sigma_9116425665531756122a_real @ m @ M ) )
=> ( probab535871623910865577e_real @ ( measure_distr_a_real @ m @ M @ F2 ) ) ) ).
% prob_space_distr
thf(fact_381_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_382_finite__measure__distr,axiom,
! [F2: a > $o,M: sigma_measure_o] :
( ( member_a_o @ F2 @ ( sigma_measurable_a_o @ m @ M ) )
=> ( measur2447921437955784316sure_o @ ( measure_distr_a_o @ m @ M @ F2 ) ) ) ).
% finite_measure_distr
thf(fact_383_finite__measure__distr,axiom,
! [F2: a > real,M: sigma_measure_real] :
( ( member_a_real @ F2 @ ( sigma_9116425665531756122a_real @ m @ M ) )
=> ( measur3606880022600206024e_real @ ( measure_distr_a_real @ m @ M @ F2 ) ) ) ).
% finite_measure_distr
thf(fact_384_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_385_distr__const,axiom,
! [C: c > d,N: sigma_measure_c_d] :
( ( member_c_d @ C @ ( sigma_space_c_d @ N ) )
=> ( ( measure_distr_a_c_d @ m @ N
@ ^ [X: a] : C )
= ( giry_return_c_d @ N @ C ) ) ) ).
% distr_const
thf(fact_386_distr__const,axiom,
! [C: a > real,N: sigma_measure_a_real] :
( ( member_a_real @ C @ ( sigma_space_a_real @ N ) )
=> ( ( measur6806378183764566549a_real @ m @ N
@ ^ [X: a] : C )
= ( giry_return_a_real @ N @ C ) ) ) ).
% distr_const
thf(fact_387_distr__const,axiom,
! [C: a > $o,N: sigma_measure_a_o] :
( ( member_a_o @ C @ ( sigma_space_a_o @ N ) )
=> ( ( measure_distr_a_a_o @ m @ N
@ ^ [X: a] : C )
= ( giry_return_a_o @ N @ C ) ) ) ).
% distr_const
thf(fact_388_distr__const,axiom,
! [C: b,N: sigma_measure_b] :
( ( member_b @ C @ ( sigma_space_b @ N ) )
=> ( ( measure_distr_a_b @ m @ N
@ ^ [X: a] : C )
= ( giry_return_b @ N @ C ) ) ) ).
% distr_const
thf(fact_389_distr__const,axiom,
! [C: nat,N: sigma_measure_nat] :
( ( member_nat @ C @ ( sigma_space_nat @ N ) )
=> ( ( measure_distr_a_nat @ m @ N
@ ^ [X: a] : C )
= ( giry_return_nat @ N @ C ) ) ) ).
% distr_const
thf(fact_390_distr__const,axiom,
! [C: $o,N: sigma_measure_o] :
( ( member_o @ C @ ( sigma_space_o @ N ) )
=> ( ( measure_distr_a_o @ m @ N
@ ^ [X: a] : C )
= ( giry_return_o @ N @ C ) ) ) ).
% distr_const
thf(fact_391_distr__const,axiom,
! [C: a,N: sigma_measure_a] :
( ( member_a @ C @ ( sigma_space_a @ N ) )
=> ( ( measure_distr_a_a @ m @ N
@ ^ [X: a] : C )
= ( giry_return_a @ N @ C ) ) ) ).
% distr_const
thf(fact_392_distributed__measurable,axiom,
! [M2: sigma_measure_c,N: sigma_measure_d,X2: c > d,F2: d > extend8495563244428889912nnreal] :
( ( probab3301989193326619023ed_c_d @ M2 @ N @ X2 @ F2 )
=> ( member_c_d @ X2 @ ( sigma_measurable_c_d @ M2 @ N ) ) ) ).
% distributed_measurable
thf(fact_393_distributed__measurable,axiom,
! [M2: sigma_measure_a,N: sigma_measure_o,X2: a > $o,F2: $o > extend8495563244428889912nnreal] :
( ( probab177417448539121064ed_a_o @ M2 @ N @ X2 @ F2 )
=> ( member_a_o @ X2 @ ( sigma_measurable_a_o @ M2 @ N ) ) ) ).
% distributed_measurable
thf(fact_394_distributed__measurable,axiom,
! [M2: sigma_measure_a,N: sigma_measure_real,X2: a > real,F2: real > extend8495563244428889912nnreal] :
( ( probab9192628068011464860a_real @ M2 @ N @ X2 @ F2 )
=> ( member_a_real @ X2 @ ( sigma_9116425665531756122a_real @ M2 @ N ) ) ) ).
% distributed_measurable
thf(fact_395_distr__completion,axiom,
! [X2: c > d,M2: sigma_measure_c,N: sigma_measure_d] :
( ( member_c_d @ X2 @ ( sigma_measurable_c_d @ M2 @ N ) )
=> ( ( measure_distr_c_d @ ( comple3428971583294703882tion_c @ M2 ) @ N @ X2 )
= ( measure_distr_c_d @ M2 @ N @ X2 ) ) ) ).
% distr_completion
thf(fact_396_distr__completion,axiom,
! [X2: a > $o,M2: sigma_measure_a,N: sigma_measure_o] :
( ( member_a_o @ X2 @ ( sigma_measurable_a_o @ M2 @ N ) )
=> ( ( measure_distr_a_o @ ( comple3428971583294703880tion_a @ M2 ) @ N @ X2 )
= ( measure_distr_a_o @ M2 @ N @ X2 ) ) ) ).
% distr_completion
thf(fact_397_distr__completion,axiom,
! [X2: a > real,M2: sigma_measure_a,N: sigma_measure_real] :
( ( member_a_real @ X2 @ ( sigma_9116425665531756122a_real @ M2 @ N ) )
=> ( ( measure_distr_a_real @ ( comple3428971583294703880tion_a @ M2 ) @ N @ X2 )
= ( measure_distr_a_real @ M2 @ N @ X2 ) ) ) ).
% distr_completion
thf(fact_398_distributed__measurable_H,axiom,
! [M2: sigma_measure_d,N: sigma_measure_d,X2: d > d,F2: d > extend8495563244428889912nnreal,G2: c > d,L: sigma_measure_c] :
( ( probab514533611353942990ed_d_d @ M2 @ N @ X2 @ F2 )
=> ( ( member_c_d @ G2 @ ( sigma_measurable_c_d @ L @ M2 ) )
=> ( member_c_d
@ ^ [X: c] : ( X2 @ ( G2 @ X ) )
@ ( sigma_measurable_c_d @ L @ N ) ) ) ) ).
% distributed_measurable'
thf(fact_399_distributed__measurable_H,axiom,
! [M2: sigma_measure_o,N: sigma_measure_o,X2: $o > $o,F2: $o > extend8495563244428889912nnreal,G2: a > $o,L: sigma_measure_a] :
( ( probab6865799537298334222ed_o_o @ M2 @ N @ X2 @ F2 )
=> ( ( member_a_o @ G2 @ ( sigma_measurable_a_o @ L @ M2 ) )
=> ( member_a_o
@ ^ [X: a] : ( X2 @ ( G2 @ X ) )
@ ( sigma_measurable_a_o @ L @ N ) ) ) ) ).
% distributed_measurable'
thf(fact_400_distributed__measurable_H,axiom,
! [M2: sigma_measure_o,N: sigma_measure_real,X2: $o > real,F2: real > extend8495563244428889912nnreal,G2: a > $o,L: sigma_measure_a] :
( ( probab5069348595295671478o_real @ M2 @ N @ X2 @ F2 )
=> ( ( member_a_o @ G2 @ ( sigma_measurable_a_o @ L @ M2 ) )
=> ( member_a_real
@ ^ [X: a] : ( X2 @ ( G2 @ X ) )
@ ( sigma_9116425665531756122a_real @ L @ N ) ) ) ) ).
% distributed_measurable'
thf(fact_401_distributed__measurable_H,axiom,
! [M2: sigma_measure_real,N: sigma_measure_o,X2: real > $o,F2: $o > extend8495563244428889912nnreal,G2: a > real,L: sigma_measure_a] :
( ( probab6578412970336841052real_o @ M2 @ N @ X2 @ F2 )
=> ( ( member_a_real @ G2 @ ( sigma_9116425665531756122a_real @ L @ M2 ) )
=> ( member_a_o
@ ^ [X: a] : ( X2 @ ( G2 @ X ) )
@ ( sigma_measurable_a_o @ L @ N ) ) ) ) ).
% distributed_measurable'
thf(fact_402_distributed__measurable_H,axiom,
! [M2: sigma_measure_real,N: sigma_measure_real,X2: real > real,F2: real > extend8495563244428889912nnreal,G2: a > real,L: sigma_measure_a] :
( ( probab1340766270110547944l_real @ M2 @ N @ X2 @ F2 )
=> ( ( member_a_real @ G2 @ ( sigma_9116425665531756122a_real @ L @ M2 ) )
=> ( member_a_real
@ ^ [X: a] : ( X2 @ ( G2 @ X ) )
@ ( sigma_9116425665531756122a_real @ L @ N ) ) ) ) ).
% distributed_measurable'
thf(fact_403_prob__space__distrD,axiom,
! [F2: c > d,M2: sigma_measure_c,N: sigma_measure_d] :
( ( member_c_d @ F2 @ ( sigma_measurable_c_d @ M2 @ N ) )
=> ( ( probab7247484486040049092pace_d @ ( measure_distr_c_d @ M2 @ N @ F2 ) )
=> ( probab7247484486040049091pace_c @ M2 ) ) ) ).
% prob_space_distrD
thf(fact_404_prob__space__distrD,axiom,
! [F2: a > $o,M2: sigma_measure_a,N: sigma_measure_o] :
( ( member_a_o @ F2 @ ( sigma_measurable_a_o @ M2 @ N ) )
=> ( ( probab1190487603588612059pace_o @ ( measure_distr_a_o @ M2 @ N @ F2 ) )
=> ( probab7247484486040049089pace_a @ M2 ) ) ) ).
% prob_space_distrD
thf(fact_405_prob__space__distrD,axiom,
! [F2: a > real,M2: sigma_measure_a,N: sigma_measure_real] :
( ( member_a_real @ F2 @ ( sigma_9116425665531756122a_real @ M2 @ N ) )
=> ( ( probab535871623910865577e_real @ ( measure_distr_a_real @ M2 @ N @ F2 ) )
=> ( probab7247484486040049089pace_a @ M2 ) ) ) ).
% prob_space_distrD
thf(fact_406_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_407_prob__space_Oprob__space__distr,axiom,
! [M2: sigma_measure_c,F2: c > d,M: sigma_measure_d] :
( ( probab7247484486040049091pace_c @ M2 )
=> ( ( member_c_d @ F2 @ ( sigma_measurable_c_d @ M2 @ M ) )
=> ( probab7247484486040049092pace_d @ ( measure_distr_c_d @ M2 @ M @ F2 ) ) ) ) ).
% prob_space.prob_space_distr
thf(fact_408_prob__space_Oprob__space__distr,axiom,
! [M2: sigma_measure_a,F2: a > $o,M: sigma_measure_o] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( member_a_o @ F2 @ ( sigma_measurable_a_o @ M2 @ M ) )
=> ( probab1190487603588612059pace_o @ ( measure_distr_a_o @ M2 @ M @ F2 ) ) ) ) ).
% prob_space.prob_space_distr
thf(fact_409_prob__space_Oprob__space__distr,axiom,
! [M2: sigma_measure_a,F2: a > real,M: sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( member_a_real @ F2 @ ( sigma_9116425665531756122a_real @ M2 @ M ) )
=> ( probab535871623910865577e_real @ ( measure_distr_a_real @ M2 @ M @ F2 ) ) ) ) ).
% prob_space.prob_space_distr
thf(fact_410_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_411_finite__measure_Ofinite__measure__distr,axiom,
! [M2: sigma_measure_c,F2: c > d,M: sigma_measure_d] :
( ( measur930452917991658468sure_c @ M2 )
=> ( ( member_c_d @ F2 @ ( sigma_measurable_c_d @ M2 @ M ) )
=> ( measur930452917991658469sure_d @ ( measure_distr_c_d @ M2 @ M @ F2 ) ) ) ) ).
% finite_measure.finite_measure_distr
thf(fact_412_finite__measure_Ofinite__measure__distr,axiom,
! [M2: sigma_measure_a,F2: a > $o,M: sigma_measure_o] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ( member_a_o @ F2 @ ( sigma_measurable_a_o @ M2 @ M ) )
=> ( measur2447921437955784316sure_o @ ( measure_distr_a_o @ M2 @ M @ F2 ) ) ) ) ).
% finite_measure.finite_measure_distr
thf(fact_413_finite__measure_Ofinite__measure__distr,axiom,
! [M2: sigma_measure_a,F2: a > real,M: sigma_measure_real] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ( member_a_real @ F2 @ ( sigma_9116425665531756122a_real @ M2 @ M ) )
=> ( measur3606880022600206024e_real @ ( measure_distr_a_real @ M2 @ M @ F2 ) ) ) ) ).
% finite_measure.finite_measure_distr
thf(fact_414_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_415_distr__return,axiom,
! [F2: ( c > d ) > a,M2: sigma_measure_c_d,N: sigma_measure_a,X4: c > d] :
( ( member_c_d_a @ F2 @ ( sigma_3169359769398501830_c_d_a @ M2 @ N ) )
=> ( ( member_c_d @ X4 @ ( sigma_space_c_d @ M2 ) )
=> ( ( measure_distr_c_d_a @ ( giry_return_c_d @ M2 @ X4 ) @ N @ F2 )
= ( giry_return_a @ N @ ( F2 @ X4 ) ) ) ) ) ).
% distr_return
thf(fact_416_distr__return,axiom,
! [F2: ( a > real ) > a,M2: sigma_measure_a_real,N: sigma_measure_a,X4: a > real] :
( ( member_a_real_a @ F2 @ ( sigma_8736157696482676083real_a @ M2 @ N ) )
=> ( ( member_a_real @ X4 @ ( sigma_space_a_real @ M2 ) )
=> ( ( measur1246039135095686683real_a @ ( giry_return_a_real @ M2 @ X4 ) @ N @ F2 )
= ( giry_return_a @ N @ ( F2 @ X4 ) ) ) ) ) ).
% distr_return
thf(fact_417_distr__return,axiom,
! [F2: ( a > $o ) > a,M2: sigma_measure_a_o,N: sigma_measure_a,X4: a > $o] :
( ( member_a_o_a @ F2 @ ( sigma_754132208923876077_a_o_a @ M2 @ N ) )
=> ( ( member_a_o @ X4 @ ( sigma_space_a_o @ M2 ) )
=> ( ( measure_distr_a_o_a @ ( giry_return_a_o @ M2 @ X4 ) @ N @ F2 )
= ( giry_return_a @ N @ ( F2 @ X4 ) ) ) ) ) ).
% distr_return
thf(fact_418_distr__return,axiom,
! [F2: b > a,M2: sigma_measure_b,N: sigma_measure_a,X4: b] :
( ( member_b_a @ F2 @ ( sigma_measurable_b_a @ M2 @ N ) )
=> ( ( member_b @ X4 @ ( sigma_space_b @ M2 ) )
=> ( ( measure_distr_b_a @ ( giry_return_b @ M2 @ X4 ) @ N @ F2 )
= ( giry_return_a @ N @ ( F2 @ X4 ) ) ) ) ) ).
% distr_return
thf(fact_419_distr__return,axiom,
! [F2: nat > a,M2: sigma_measure_nat,N: sigma_measure_a,X4: nat] :
( ( member_nat_a @ F2 @ ( sigma_4105081583803843548_nat_a @ M2 @ N ) )
=> ( ( member_nat @ X4 @ ( sigma_space_nat @ M2 ) )
=> ( ( measure_distr_nat_a @ ( giry_return_nat @ M2 @ X4 ) @ N @ F2 )
= ( giry_return_a @ N @ ( F2 @ X4 ) ) ) ) ) ).
% distr_return
thf(fact_420_distr__return,axiom,
! [F2: $o > a,M2: sigma_measure_o,N: sigma_measure_a,X4: $o] :
( ( member_o_a @ F2 @ ( sigma_measurable_o_a @ M2 @ N ) )
=> ( ( member_o @ X4 @ ( sigma_space_o @ M2 ) )
=> ( ( measure_distr_o_a @ ( giry_return_o @ M2 @ X4 ) @ N @ F2 )
= ( giry_return_a @ N @ ( F2 @ X4 ) ) ) ) ) ).
% distr_return
thf(fact_421_distr__return,axiom,
! [F2: a > a,M2: sigma_measure_a,N: sigma_measure_a,X4: a] :
( ( member_a_a @ F2 @ ( sigma_measurable_a_a @ M2 @ N ) )
=> ( ( member_a @ X4 @ ( sigma_space_a @ M2 ) )
=> ( ( measure_distr_a_a @ ( giry_return_a @ M2 @ X4 ) @ N @ F2 )
= ( giry_return_a @ N @ ( F2 @ X4 ) ) ) ) ) ).
% distr_return
thf(fact_422_distr__return,axiom,
! [F2: c > d,M2: sigma_measure_c,N: sigma_measure_d,X4: c] :
( ( member_c_d @ F2 @ ( sigma_measurable_c_d @ M2 @ N ) )
=> ( ( member_c @ X4 @ ( sigma_space_c @ M2 ) )
=> ( ( measure_distr_c_d @ ( giry_return_c @ M2 @ X4 ) @ N @ F2 )
= ( giry_return_d @ N @ ( F2 @ X4 ) ) ) ) ) ).
% distr_return
thf(fact_423_distr__return,axiom,
! [F2: a > $o,M2: sigma_measure_a,N: sigma_measure_o,X4: a] :
( ( member_a_o @ F2 @ ( sigma_measurable_a_o @ M2 @ N ) )
=> ( ( member_a @ X4 @ ( sigma_space_a @ M2 ) )
=> ( ( measure_distr_a_o @ ( giry_return_a @ M2 @ X4 ) @ N @ F2 )
= ( giry_return_o @ N @ ( F2 @ X4 ) ) ) ) ) ).
% distr_return
thf(fact_424_distr__return,axiom,
! [F2: a > real,M2: sigma_measure_a,N: sigma_measure_real,X4: a] :
( ( member_a_real @ F2 @ ( sigma_9116425665531756122a_real @ M2 @ N ) )
=> ( ( member_a @ X4 @ ( sigma_space_a @ M2 ) )
=> ( ( measure_distr_a_real @ ( giry_return_a @ M2 @ X4 ) @ N @ F2 )
= ( giry_return_real @ N @ ( F2 @ X4 ) ) ) ) ) ).
% distr_return
thf(fact_425_AE__distrD,axiom,
! [F2: a > a,M2: sigma_measure_a,M: sigma_measure_a,P: a > $o] :
( ( member_a_a @ F2 @ ( sigma_measurable_a_a @ M2 @ M ) )
=> ( ( eventually_a @ P @ ( measure_ae_filter_a @ ( measure_distr_a_a @ M2 @ M @ F2 ) ) )
=> ( eventually_a
@ ^ [X: a] : ( P @ ( F2 @ X ) )
@ ( measure_ae_filter_a @ M2 ) ) ) ) ).
% AE_distrD
thf(fact_426_AE__distrD,axiom,
! [F2: c > d,M2: sigma_measure_c,M: sigma_measure_d,P: d > $o] :
( ( member_c_d @ F2 @ ( sigma_measurable_c_d @ M2 @ M ) )
=> ( ( eventually_d @ P @ ( measure_ae_filter_d @ ( measure_distr_c_d @ M2 @ M @ F2 ) ) )
=> ( eventually_c
@ ^ [X: c] : ( P @ ( F2 @ X ) )
@ ( measure_ae_filter_c @ M2 ) ) ) ) ).
% AE_distrD
thf(fact_427_AE__distrD,axiom,
! [F2: a > $o,M2: sigma_measure_a,M: sigma_measure_o,P: $o > $o] :
( ( member_a_o @ F2 @ ( sigma_measurable_a_o @ M2 @ M ) )
=> ( ( eventually_o @ P @ ( measure_ae_filter_o @ ( measure_distr_a_o @ M2 @ M @ F2 ) ) )
=> ( eventually_a
@ ^ [X: a] : ( P @ ( F2 @ X ) )
@ ( measure_ae_filter_a @ M2 ) ) ) ) ).
% AE_distrD
thf(fact_428_AE__distrD,axiom,
! [F2: a > real,M2: sigma_measure_a,M: sigma_measure_real,P: real > $o] :
( ( member_a_real @ F2 @ ( sigma_9116425665531756122a_real @ M2 @ M ) )
=> ( ( eventually_real @ P @ ( measur1097577823623106589r_real @ ( measure_distr_a_real @ M2 @ M @ F2 ) ) )
=> ( eventually_a
@ ^ [X: a] : ( P @ ( F2 @ X ) )
@ ( measure_ae_filter_a @ M2 ) ) ) ) ).
% AE_distrD
thf(fact_429_prob__space_Odistr__const,axiom,
! [M2: sigma_measure_a,C: c > d,N: sigma_measure_c_d] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( member_c_d @ C @ ( sigma_space_c_d @ N ) )
=> ( ( measure_distr_a_c_d @ M2 @ N
@ ^ [X: a] : C )
= ( giry_return_c_d @ N @ C ) ) ) ) ).
% prob_space.distr_const
thf(fact_430_prob__space_Odistr__const,axiom,
! [M2: sigma_measure_a,C: a > real,N: sigma_measure_a_real] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( member_a_real @ C @ ( sigma_space_a_real @ N ) )
=> ( ( measur6806378183764566549a_real @ M2 @ N
@ ^ [X: a] : C )
= ( giry_return_a_real @ N @ C ) ) ) ) ).
% prob_space.distr_const
thf(fact_431_prob__space_Odistr__const,axiom,
! [M2: sigma_measure_a,C: a > $o,N: sigma_measure_a_o] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( member_a_o @ C @ ( sigma_space_a_o @ N ) )
=> ( ( measure_distr_a_a_o @ M2 @ N
@ ^ [X: a] : C )
= ( giry_return_a_o @ N @ C ) ) ) ) ).
% prob_space.distr_const
thf(fact_432_prob__space_Odistr__const,axiom,
! [M2: sigma_measure_a,C: b,N: sigma_measure_b] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( member_b @ C @ ( sigma_space_b @ N ) )
=> ( ( measure_distr_a_b @ M2 @ N
@ ^ [X: a] : C )
= ( giry_return_b @ N @ C ) ) ) ) ).
% prob_space.distr_const
thf(fact_433_prob__space_Odistr__const,axiom,
! [M2: sigma_measure_a,C: nat,N: sigma_measure_nat] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( member_nat @ C @ ( sigma_space_nat @ N ) )
=> ( ( measure_distr_a_nat @ M2 @ N
@ ^ [X: a] : C )
= ( giry_return_nat @ N @ C ) ) ) ) ).
% prob_space.distr_const
thf(fact_434_prob__space_Odistr__const,axiom,
! [M2: sigma_measure_a,C: $o,N: sigma_measure_o] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( member_o @ C @ ( sigma_space_o @ N ) )
=> ( ( measure_distr_a_o @ M2 @ N
@ ^ [X: a] : C )
= ( giry_return_o @ N @ C ) ) ) ) ).
% prob_space.distr_const
thf(fact_435_prob__space_Odistr__const,axiom,
! [M2: sigma_measure_a,C: a,N: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( member_a @ C @ ( sigma_space_a @ N ) )
=> ( ( measure_distr_a_a @ M2 @ N
@ ^ [X: a] : C )
= ( giry_return_a @ N @ C ) ) ) ) ).
% prob_space.distr_const
thf(fact_436_UNIV__witness,axiom,
? [X3: c > d] : ( member_c_d @ X3 @ top_top_set_c_d ) ).
% UNIV_witness
thf(fact_437_UNIV__witness,axiom,
? [X3: a > real] : ( member_a_real @ X3 @ top_top_set_a_real ) ).
% UNIV_witness
thf(fact_438_UNIV__witness,axiom,
? [X3: a > $o] : ( member_a_o @ X3 @ top_top_set_a_o ) ).
% UNIV_witness
thf(fact_439_UNIV__witness,axiom,
? [X3: b] : ( member_b @ X3 @ top_top_set_b ) ).
% UNIV_witness
thf(fact_440_UNIV__witness,axiom,
? [X3: a] : ( member_a @ X3 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_441_UNIV__witness,axiom,
? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_442_UNIV__witness,axiom,
? [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% UNIV_witness
thf(fact_443_UNIV__witness,axiom,
? [X3: literal] : ( member_literal @ X3 @ top_top_set_literal ) ).
% UNIV_witness
thf(fact_444_UNIV__witness,axiom,
? [X3: product_unit] : ( member_Product_unit @ X3 @ top_to1996260823553986621t_unit ) ).
% UNIV_witness
thf(fact_445_UNIV__eq__I,axiom,
! [A2: set_c_d] :
( ! [X3: c > d] : ( member_c_d @ X3 @ A2 )
=> ( top_top_set_c_d = A2 ) ) ).
% UNIV_eq_I
thf(fact_446_UNIV__eq__I,axiom,
! [A2: set_a_real] :
( ! [X3: a > real] : ( member_a_real @ X3 @ A2 )
=> ( top_top_set_a_real = A2 ) ) ).
% UNIV_eq_I
thf(fact_447_UNIV__eq__I,axiom,
! [A2: set_a_o] :
( ! [X3: a > $o] : ( member_a_o @ X3 @ A2 )
=> ( top_top_set_a_o = A2 ) ) ).
% UNIV_eq_I
thf(fact_448_UNIV__eq__I,axiom,
! [A2: set_b] :
( ! [X3: b] : ( member_b @ X3 @ A2 )
=> ( top_top_set_b = A2 ) ) ).
% UNIV_eq_I
thf(fact_449_UNIV__eq__I,axiom,
! [A2: set_a] :
( ! [X3: a] : ( member_a @ X3 @ A2 )
=> ( top_top_set_a = A2 ) ) ).
% UNIV_eq_I
thf(fact_450_UNIV__eq__I,axiom,
! [A2: set_nat] :
( ! [X3: nat] : ( member_nat @ X3 @ A2 )
=> ( top_top_set_nat = A2 ) ) ).
% UNIV_eq_I
thf(fact_451_UNIV__eq__I,axiom,
! [A2: set_o] :
( ! [X3: $o] : ( member_o @ X3 @ A2 )
=> ( top_top_set_o = A2 ) ) ).
% UNIV_eq_I
thf(fact_452_UNIV__eq__I,axiom,
! [A2: set_literal] :
( ! [X3: literal] : ( member_literal @ X3 @ A2 )
=> ( top_top_set_literal = A2 ) ) ).
% UNIV_eq_I
thf(fact_453_UNIV__eq__I,axiom,
! [A2: set_Product_unit] :
( ! [X3: product_unit] : ( member_Product_unit @ X3 @ A2 )
=> ( top_to1996260823553986621t_unit = A2 ) ) ).
% UNIV_eq_I
thf(fact_454_ex__in__conv,axiom,
! [A2: set_c_d] :
( ( ? [X: c > d] : ( member_c_d @ X @ A2 ) )
= ( A2 != bot_bot_set_c_d ) ) ).
% ex_in_conv
thf(fact_455_ex__in__conv,axiom,
! [A2: set_a_real] :
( ( ? [X: a > real] : ( member_a_real @ X @ A2 ) )
= ( A2 != bot_bot_set_a_real ) ) ).
% ex_in_conv
thf(fact_456_ex__in__conv,axiom,
! [A2: set_a_o] :
( ( ? [X: a > $o] : ( member_a_o @ X @ A2 ) )
= ( A2 != bot_bot_set_a_o ) ) ).
% ex_in_conv
thf(fact_457_ex__in__conv,axiom,
! [A2: set_b] :
( ( ? [X: b] : ( member_b @ X @ A2 ) )
= ( A2 != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_458_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_459_ex__in__conv,axiom,
! [A2: set_o] :
( ( ? [X: $o] : ( member_o @ X @ A2 ) )
= ( A2 != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_460_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X: a] : ( member_a @ X @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_461_equals0I,axiom,
! [A2: set_c_d] :
( ! [Y4: c > d] :
~ ( member_c_d @ Y4 @ A2 )
=> ( A2 = bot_bot_set_c_d ) ) ).
% equals0I
thf(fact_462_equals0I,axiom,
! [A2: set_a_real] :
( ! [Y4: a > real] :
~ ( member_a_real @ Y4 @ A2 )
=> ( A2 = bot_bot_set_a_real ) ) ).
% equals0I
thf(fact_463_equals0I,axiom,
! [A2: set_a_o] :
( ! [Y4: a > $o] :
~ ( member_a_o @ Y4 @ A2 )
=> ( A2 = bot_bot_set_a_o ) ) ).
% equals0I
thf(fact_464_equals0I,axiom,
! [A2: set_b] :
( ! [Y4: b] :
~ ( member_b @ Y4 @ A2 )
=> ( A2 = bot_bot_set_b ) ) ).
% equals0I
thf(fact_465_equals0I,axiom,
! [A2: set_nat] :
( ! [Y4: nat] :
~ ( member_nat @ Y4 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_466_equals0I,axiom,
! [A2: set_o] :
( ! [Y4: $o] :
~ ( member_o @ Y4 @ A2 )
=> ( A2 = bot_bot_set_o ) ) ).
% equals0I
thf(fact_467_equals0I,axiom,
! [A2: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_468_equals0D,axiom,
! [A2: set_c_d,A: c > d] :
( ( A2 = bot_bot_set_c_d )
=> ~ ( member_c_d @ A @ A2 ) ) ).
% equals0D
thf(fact_469_equals0D,axiom,
! [A2: set_a_real,A: a > real] :
( ( A2 = bot_bot_set_a_real )
=> ~ ( member_a_real @ A @ A2 ) ) ).
% equals0D
thf(fact_470_equals0D,axiom,
! [A2: set_a_o,A: a > $o] :
( ( A2 = bot_bot_set_a_o )
=> ~ ( member_a_o @ A @ A2 ) ) ).
% equals0D
thf(fact_471_equals0D,axiom,
! [A2: set_b,A: b] :
( ( A2 = bot_bot_set_b )
=> ~ ( member_b @ A @ A2 ) ) ).
% equals0D
thf(fact_472_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_473_equals0D,axiom,
! [A2: set_o,A: $o] :
( ( A2 = bot_bot_set_o )
=> ~ ( member_o @ A @ A2 ) ) ).
% equals0D
thf(fact_474_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_475_emptyE,axiom,
! [A: c > d] :
~ ( member_c_d @ A @ bot_bot_set_c_d ) ).
% emptyE
thf(fact_476_emptyE,axiom,
! [A: a > real] :
~ ( member_a_real @ A @ bot_bot_set_a_real ) ).
% emptyE
thf(fact_477_emptyE,axiom,
! [A: a > $o] :
~ ( member_a_o @ A @ bot_bot_set_a_o ) ).
% emptyE
thf(fact_478_emptyE,axiom,
! [A: b] :
~ ( member_b @ A @ bot_bot_set_b ) ).
% emptyE
thf(fact_479_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_480_emptyE,axiom,
! [A: $o] :
~ ( member_o @ A @ bot_bot_set_o ) ).
% emptyE
thf(fact_481_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_482_prob__space_Oindep__var__rv2,axiom,
! [M2: sigma_measure_c,S2: sigma_measure_d,X2: c > d,T: sigma_measure_d,Y: c > d] :
( ( probab7247484486040049091pace_c @ M2 )
=> ( ( indepe6089114067601049933ar_c_d @ M2 @ S2 @ X2 @ T @ Y )
=> ( member_c_d @ Y @ ( sigma_measurable_c_d @ M2 @ T ) ) ) ) ).
% prob_space.indep_var_rv2
thf(fact_483_prob__space_Oindep__var__rv2,axiom,
! [M2: sigma_measure_a,S2: sigma_measure_o,X2: a > $o,T: sigma_measure_o,Y: a > $o] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe2772234535145391206ar_a_o @ M2 @ S2 @ X2 @ T @ Y )
=> ( member_a_o @ Y @ ( sigma_measurable_a_o @ M2 @ T ) ) ) ) ).
% prob_space.indep_var_rv2
thf(fact_484_prob__space_Oindep__var__rv2,axiom,
! [M2: sigma_measure_a,S2: sigma_measure_real,X2: a > real,T: sigma_measure_real,Y: a > real] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe8958435565499147358a_real @ M2 @ S2 @ X2 @ T @ Y )
=> ( member_a_real @ Y @ ( sigma_9116425665531756122a_real @ M2 @ T ) ) ) ) ).
% prob_space.indep_var_rv2
thf(fact_485_prob__space_Oindep__var__rv1,axiom,
! [M2: sigma_measure_c,S2: sigma_measure_d,X2: c > d,T: sigma_measure_d,Y: c > d] :
( ( probab7247484486040049091pace_c @ M2 )
=> ( ( indepe6089114067601049933ar_c_d @ M2 @ S2 @ X2 @ T @ Y )
=> ( member_c_d @ X2 @ ( sigma_measurable_c_d @ M2 @ S2 ) ) ) ) ).
% prob_space.indep_var_rv1
thf(fact_486_prob__space_Oindep__var__rv1,axiom,
! [M2: sigma_measure_a,S2: sigma_measure_o,X2: a > $o,T: sigma_measure_o,Y: a > $o] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe2772234535145391206ar_a_o @ M2 @ S2 @ X2 @ T @ Y )
=> ( member_a_o @ X2 @ ( sigma_measurable_a_o @ M2 @ S2 ) ) ) ) ).
% prob_space.indep_var_rv1
thf(fact_487_prob__space_Oindep__var__rv1,axiom,
! [M2: sigma_measure_a,S2: sigma_measure_real,X2: a > real,T: sigma_measure_real,Y: a > real] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe8958435565499147358a_real @ M2 @ S2 @ X2 @ T @ Y )
=> ( member_a_real @ X2 @ ( sigma_9116425665531756122a_real @ M2 @ S2 ) ) ) ) ).
% prob_space.indep_var_rv1
thf(fact_488_prob__space_Odistributed__unique,axiom,
! [M2: sigma_measure_a,S2: sigma_measure_a,X2: a > a,Px: a > extend8495563244428889912nnreal,Py: a > extend8495563244428889912nnreal] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( probab8876900357271971086ed_a_a @ M2 @ S2 @ X2 @ Px )
=> ( ( probab8876900357271971086ed_a_a @ M2 @ S2 @ X2 @ Py )
=> ( eventually_a
@ ^ [X: a] :
( ( Px @ X )
= ( Py @ X ) )
@ ( measure_ae_filter_a @ S2 ) ) ) ) ) ).
% prob_space.distributed_unique
thf(fact_489_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_490_subprob__space_Osubprob__space__distr,axiom,
! [M2: sigma_measure_c,F2: c > d,M: sigma_measure_d] :
( ( giry_subprob_space_c @ M2 )
=> ( ( member_c_d @ F2 @ ( sigma_measurable_c_d @ M2 @ M ) )
=> ( ( ( sigma_space_d @ M )
!= bot_bot_set_d )
=> ( giry_subprob_space_d @ ( measure_distr_c_d @ M2 @ M @ F2 ) ) ) ) ) ).
% subprob_space.subprob_space_distr
thf(fact_491_subprob__space_Osubprob__space__distr,axiom,
! [M2: sigma_measure_a,F2: a > $o,M: sigma_measure_o] :
( ( giry_subprob_space_a @ M2 )
=> ( ( member_a_o @ F2 @ ( sigma_measurable_a_o @ M2 @ M ) )
=> ( ( ( sigma_space_o @ M )
!= bot_bot_set_o )
=> ( giry_subprob_space_o @ ( measure_distr_a_o @ M2 @ M @ F2 ) ) ) ) ) ).
% subprob_space.subprob_space_distr
thf(fact_492_subprob__space_Osubprob__space__distr,axiom,
! [M2: sigma_measure_a,F2: a > real,M: sigma_measure_real] :
( ( giry_subprob_space_a @ M2 )
=> ( ( member_a_real @ F2 @ ( sigma_9116425665531756122a_real @ M2 @ M ) )
=> ( ( ( sigma_space_real @ M )
!= bot_bot_set_real )
=> ( giry_s8208748868292234104e_real @ ( measure_distr_a_real @ M2 @ M @ F2 ) ) ) ) ) ).
% subprob_space.subprob_space_distr
thf(fact_493_UNIV__def,axiom,
( top_top_set_a
= ( collect_a
@ ^ [X: a] : $true ) ) ).
% UNIV_def
thf(fact_494_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X: nat] : $true ) ) ).
% UNIV_def
thf(fact_495_UNIV__def,axiom,
( top_top_set_o
= ( collect_o
@ ^ [X: $o] : $true ) ) ).
% UNIV_def
thf(fact_496_UNIV__def,axiom,
( top_top_set_literal
= ( collect_literal
@ ^ [X: literal] : $true ) ) ).
% UNIV_def
thf(fact_497_UNIV__def,axiom,
( top_to1996260823553986621t_unit
= ( collect_Product_unit
@ ^ [X: product_unit] : $true ) ) ).
% UNIV_def
thf(fact_498_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X: nat] : $false ) ) ).
% empty_def
thf(fact_499_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X: a] : $false ) ) ).
% empty_def
thf(fact_500_prob__space_Oindep__var__compose,axiom,
! [M2: sigma_measure_a,M1: sigma_measure_c,X1: a > c,M22: sigma_measure_c,X22: a > c,Y1: c > d,N1: sigma_measure_d,Y22: c > d,N22: sigma_measure_d] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe2440653194691626190ar_a_c @ M2 @ M1 @ X1 @ M22 @ X22 )
=> ( ( member_c_d @ Y1 @ ( sigma_measurable_c_d @ M1 @ N1 ) )
=> ( ( member_c_d @ Y22 @ ( sigma_measurable_c_d @ M22 @ N22 ) )
=> ( indepe2440653194691626191ar_a_d @ M2 @ N1 @ ( comp_c_d_a @ Y1 @ X1 ) @ N22 @ ( comp_c_d_a @ Y22 @ X22 ) ) ) ) ) ) ).
% prob_space.indep_var_compose
thf(fact_501_prob__space_Oindep__var__compose,axiom,
! [M2: sigma_measure_a,M1: sigma_measure_a,X1: a > a,M22: sigma_measure_a,X22: a > a,Y1: a > $o,N1: sigma_measure_o,Y22: a > $o,N22: sigma_measure_o] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe2440653194691626188ar_a_a @ M2 @ M1 @ X1 @ M22 @ X22 )
=> ( ( member_a_o @ Y1 @ ( sigma_measurable_a_o @ M1 @ N1 ) )
=> ( ( member_a_o @ Y22 @ ( sigma_measurable_a_o @ M22 @ N22 ) )
=> ( indepe2772234535145391206ar_a_o @ M2 @ N1 @ ( comp_a_o_a @ Y1 @ X1 ) @ N22 @ ( comp_a_o_a @ Y22 @ X22 ) ) ) ) ) ) ).
% prob_space.indep_var_compose
thf(fact_502_prob__space_Oindep__var__compose,axiom,
! [M2: sigma_measure_a,M1: sigma_measure_a,X1: a > a,M22: sigma_measure_a,X22: a > a,Y1: a > real,N1: sigma_measure_real,Y22: a > real,N22: sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe2440653194691626188ar_a_a @ M2 @ M1 @ X1 @ M22 @ X22 )
=> ( ( member_a_real @ Y1 @ ( sigma_9116425665531756122a_real @ M1 @ N1 ) )
=> ( ( member_a_real @ Y22 @ ( sigma_9116425665531756122a_real @ M22 @ N22 ) )
=> ( indepe8958435565499147358a_real @ M2 @ N1 @ ( comp_a_real_a @ Y1 @ X1 ) @ N22 @ ( comp_a_real_a @ Y22 @ X22 ) ) ) ) ) ) ).
% prob_space.indep_var_compose
thf(fact_503_empty__not__UNIV,axiom,
bot_bot_set_a != top_top_set_a ).
% empty_not_UNIV
thf(fact_504_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_505_empty__not__UNIV,axiom,
bot_bot_set_o != top_top_set_o ).
% empty_not_UNIV
thf(fact_506_empty__not__UNIV,axiom,
bot_bot_set_literal != top_top_set_literal ).
% empty_not_UNIV
thf(fact_507_empty__not__UNIV,axiom,
bot_bo3957492148770167129t_unit != top_to1996260823553986621t_unit ).
% empty_not_UNIV
thf(fact_508_measurable__completion,axiom,
! [F2: c > d,M2: sigma_measure_c,N: sigma_measure_d] :
( ( member_c_d @ F2 @ ( sigma_measurable_c_d @ M2 @ N ) )
=> ( member_c_d @ F2 @ ( sigma_measurable_c_d @ ( comple3428971583294703882tion_c @ M2 ) @ N ) ) ) ).
% measurable_completion
thf(fact_509_measurable__completion,axiom,
! [F2: a > $o,M2: sigma_measure_a,N: sigma_measure_o] :
( ( member_a_o @ F2 @ ( sigma_measurable_a_o @ M2 @ N ) )
=> ( member_a_o @ F2 @ ( sigma_measurable_a_o @ ( comple3428971583294703880tion_a @ M2 ) @ N ) ) ) ).
% measurable_completion
thf(fact_510_measurable__completion,axiom,
! [F2: a > real,M2: sigma_measure_a,N: sigma_measure_real] :
( ( member_a_real @ F2 @ ( sigma_9116425665531756122a_real @ M2 @ N ) )
=> ( member_a_real @ F2 @ ( sigma_9116425665531756122a_real @ ( comple3428971583294703880tion_a @ M2 ) @ N ) ) ) ).
% measurable_completion
thf(fact_511_integrable__distr,axiom,
! [T: a > $o,M2: sigma_measure_a,M: sigma_measure_o,F2: $o > real] :
( ( member_a_o @ T @ ( sigma_measurable_a_o @ M2 @ M ) )
=> ( ( bochne661340805755426930o_real @ ( measure_distr_a_o @ M2 @ M @ T ) @ F2 )
=> ( bochne2139062162225249880a_real @ M2
@ ^ [X: a] : ( F2 @ ( T @ X ) ) ) ) ) ).
% integrable_distr
thf(fact_512_integrable__distr,axiom,
! [T: a > real,M2: sigma_measure_a,M: sigma_measure_real,F2: real > real] :
( ( member_a_real @ T @ ( sigma_9116425665531756122a_real @ M2 @ M ) )
=> ( ( bochne3340023020068487468l_real @ ( measure_distr_a_real @ M2 @ M @ T ) @ F2 )
=> ( bochne2139062162225249880a_real @ M2
@ ^ [X: a] : ( F2 @ ( T @ X ) ) ) ) ) ).
% integrable_distr
thf(fact_513_integrable__distr,axiom,
! [T: a > a,M2: sigma_measure_a,M: sigma_measure_a,F2: a > real] :
( ( member_a_a @ T @ ( sigma_measurable_a_a @ M2 @ M ) )
=> ( ( bochne2139062162225249880a_real @ ( measure_distr_a_a @ M2 @ M @ T ) @ F2 )
=> ( bochne2139062162225249880a_real @ M2
@ ^ [X: a] : ( F2 @ ( T @ X ) ) ) ) ) ).
% integrable_distr
thf(fact_514_sigma__finite__measure__distr,axiom,
! [M2: sigma_measure_c,N: sigma_measure_d,F2: c > d] :
( ( measur4308613598931908898sure_d @ ( measure_distr_c_d @ M2 @ N @ F2 ) )
=> ( ( member_c_d @ F2 @ ( sigma_measurable_c_d @ M2 @ N ) )
=> ( measur4308613598931908897sure_c @ M2 ) ) ) ).
% sigma_finite_measure_distr
thf(fact_515_sigma__finite__measure__distr,axiom,
! [M2: sigma_measure_a,N: sigma_measure_o,F2: a > $o] :
( ( measur1827666076404920889sure_o @ ( measure_distr_a_o @ M2 @ N @ F2 ) )
=> ( ( member_a_o @ F2 @ ( sigma_measurable_a_o @ M2 @ N ) )
=> ( measur4308613598931908895sure_a @ M2 ) ) ) ).
% sigma_finite_measure_distr
thf(fact_516_sigma__finite__measure__distr,axiom,
! [M2: sigma_measure_a,N: sigma_measure_real,F2: a > real] :
( ( measur487378040549452491e_real @ ( measure_distr_a_real @ M2 @ N @ F2 ) )
=> ( ( member_a_real @ F2 @ ( sigma_9116425665531756122a_real @ M2 @ N ) )
=> ( measur4308613598931908895sure_a @ M2 ) ) ) ).
% sigma_finite_measure_distr
thf(fact_517_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_518_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_519_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_520_indep__sets__mono__sets,axiom,
! [F: a > set_set_a,I2: set_a,G: a > set_set_a] :
( ( indepe8927441866673418604ts_a_a @ m @ F @ I2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I2 )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe8927441866673418604ts_a_a @ m @ G @ I2 ) ) ) ).
% indep_sets_mono_sets
thf(fact_521_indep__sets__mono__sets,axiom,
! [F: b > set_set_a,I2: set_b,G: b > set_set_a] :
( ( indepe8927441866673418605ts_a_b @ m @ F @ I2 )
=> ( ! [I3: b] :
( ( member_b @ I3 @ I2 )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe8927441866673418605ts_a_b @ m @ G @ I2 ) ) ) ).
% indep_sets_mono_sets
thf(fact_522_indep__sets__mono__sets,axiom,
! [F: ( a > $o ) > set_set_a,I2: set_a_o,G: ( a > $o ) > set_set_a] :
( ( indepe7801696130336798481_a_a_o @ m @ F @ I2 )
=> ( ! [I3: a > $o] :
( ( member_a_o @ I3 @ I2 )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe7801696130336798481_a_a_o @ m @ G @ I2 ) ) ) ).
% indep_sets_mono_sets
thf(fact_523_indep__sets__mono__sets,axiom,
! [F: ( a > real ) > set_set_a,I2: set_a_real,G: ( a > real ) > set_set_a] :
( ( indepe4749599203615801097a_real @ m @ F @ I2 )
=> ( ! [I3: a > real] :
( ( member_a_real @ I3 @ I2 )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe4749599203615801097a_real @ m @ G @ I2 ) ) ) ).
% indep_sets_mono_sets
thf(fact_524_indep__sets__mono__sets,axiom,
! [F: ( c > d ) > set_set_a,I2: set_c_d,G: ( c > d ) > set_set_a] :
( ( indepe4867465796832040824_a_c_d @ m @ F @ I2 )
=> ( ! [I3: c > d] :
( ( member_c_d @ I3 @ I2 )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe4867465796832040824_a_c_d @ m @ G @ I2 ) ) ) ).
% indep_sets_mono_sets
thf(fact_525_finite__measure_Ointegrable__const,axiom,
! [M2: sigma_measure_a,A: real] :
( ( measur930452917991658466sure_a @ M2 )
=> ( bochne2139062162225249880a_real @ M2
@ ^ [X: a] : A ) ) ).
% finite_measure.integrable_const
thf(fact_526_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_527_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_528_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_529_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_530_indep__sets__mono__index,axiom,
! [J: set_b,I2: set_b,F: b > set_set_a] :
( ( ord_less_eq_set_b @ J @ I2 )
=> ( ( indepe8927441866673418605ts_a_b @ m @ F @ I2 )
=> ( indepe8927441866673418605ts_a_b @ m @ F @ J ) ) ) ).
% indep_sets_mono_index
thf(fact_531_indep__sets__mono__index,axiom,
! [J: set_a_o,I2: set_a_o,F: ( a > $o ) > set_set_a] :
( ( ord_less_eq_set_a_o2 @ J @ I2 )
=> ( ( indepe7801696130336798481_a_a_o @ m @ F @ I2 )
=> ( indepe7801696130336798481_a_a_o @ m @ F @ J ) ) ) ).
% indep_sets_mono_index
thf(fact_532_indep__sets__mono__index,axiom,
! [J: set_a_real,I2: set_a_real,F: ( a > real ) > set_set_a] :
( ( ord_le3334967407727675675a_real @ J @ I2 )
=> ( ( indepe4749599203615801097a_real @ m @ F @ I2 )
=> ( indepe4749599203615801097a_real @ m @ F @ J ) ) ) ).
% indep_sets_mono_index
thf(fact_533_indep__sets__mono__index,axiom,
! [J: set_c_d,I2: set_c_d,F: ( c > d ) > set_set_a] :
( ( ord_less_eq_set_c_d @ J @ I2 )
=> ( ( indepe4867465796832040824_a_c_d @ m @ F @ I2 )
=> ( indepe4867465796832040824_a_c_d @ m @ F @ J ) ) ) ).
% indep_sets_mono_index
thf(fact_534_ae__filter__bot,axiom,
( ( measure_ae_filter_a @ m )
!= bot_bot_filter_a ) ).
% ae_filter_bot
thf(fact_535_order__refl,axiom,
! [X4: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ X4 @ X4 ) ).
% order_refl
thf(fact_536_order__refl,axiom,
! [X4: nat] : ( ord_less_eq_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_537_order__refl,axiom,
! [X4: set_set_a] : ( ord_le3724670747650509150_set_a @ X4 @ X4 ) ).
% order_refl
thf(fact_538_order__refl,axiom,
! [X4: real] : ( ord_less_eq_real @ X4 @ X4 ) ).
% order_refl
thf(fact_539_order__refl,axiom,
! [X4: set_a] : ( ord_less_eq_set_a @ X4 @ X4 ) ).
% order_refl
thf(fact_540_dual__order_Orefl,axiom,
! [A: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ A @ A ) ).
% dual_order.refl
thf(fact_541_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_542_dual__order_Orefl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_543_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_544_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_545_subset__antisym,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_546_subset__antisym,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_547_subsetI,axiom,
! [A2: set_c_d,B: set_c_d] :
( ! [X3: c > d] :
( ( member_c_d @ X3 @ A2 )
=> ( member_c_d @ X3 @ B ) )
=> ( ord_less_eq_set_c_d @ A2 @ B ) ) ).
% subsetI
thf(fact_548_subsetI,axiom,
! [A2: set_a_real,B: set_a_real] :
( ! [X3: a > real] :
( ( member_a_real @ X3 @ A2 )
=> ( member_a_real @ X3 @ B ) )
=> ( ord_le3334967407727675675a_real @ A2 @ B ) ) ).
% subsetI
thf(fact_549_subsetI,axiom,
! [A2: set_a_o,B: set_a_o] :
( ! [X3: a > $o] :
( ( member_a_o @ X3 @ A2 )
=> ( member_a_o @ X3 @ B ) )
=> ( ord_less_eq_set_a_o2 @ A2 @ B ) ) ).
% subsetI
thf(fact_550_subsetI,axiom,
! [A2: set_b,B: set_b] :
( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( member_b @ X3 @ B ) )
=> ( ord_less_eq_set_b @ A2 @ B ) ) ).
% subsetI
thf(fact_551_subsetI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B ) )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_552_subsetI,axiom,
! [A2: set_o,B: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( member_o @ X3 @ B ) )
=> ( ord_less_eq_set_o @ A2 @ B ) ) ).
% subsetI
thf(fact_553_subsetI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( member_set_a @ X3 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_554_subsetI,axiom,
! [A2: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B ) )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_555_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_556_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_557_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_558_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_559_indep__sets__mono,axiom,
! [F: b > set_set_a,I2: set_b,J: set_b,G: b > set_set_a] :
( ( indepe8927441866673418605ts_a_b @ m @ F @ I2 )
=> ( ( ord_less_eq_set_b @ J @ I2 )
=> ( ! [I3: b] :
( ( member_b @ I3 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe8927441866673418605ts_a_b @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_560_indep__sets__mono,axiom,
! [F: ( a > $o ) > set_set_a,I2: set_a_o,J: set_a_o,G: ( a > $o ) > set_set_a] :
( ( indepe7801696130336798481_a_a_o @ m @ F @ I2 )
=> ( ( ord_less_eq_set_a_o2 @ J @ I2 )
=> ( ! [I3: a > $o] :
( ( member_a_o @ I3 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe7801696130336798481_a_a_o @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_561_indep__sets__mono,axiom,
! [F: ( a > real ) > set_set_a,I2: set_a_real,J: set_a_real,G: ( a > real ) > set_set_a] :
( ( indepe4749599203615801097a_real @ m @ F @ I2 )
=> ( ( ord_le3334967407727675675a_real @ J @ I2 )
=> ( ! [I3: a > real] :
( ( member_a_real @ I3 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe4749599203615801097a_real @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_562_indep__sets__mono,axiom,
! [F: ( c > d ) > set_set_a,I2: set_c_d,J: set_c_d,G: ( c > d ) > set_set_a] :
( ( indepe4867465796832040824_a_c_d @ m @ F @ I2 )
=> ( ( ord_less_eq_set_c_d @ J @ I2 )
=> ( ! [I3: c > d] :
( ( member_c_d @ I3 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe4867465796832040824_a_c_d @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_563_subset__empty,axiom,
! [A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a )
= ( A2 = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_564_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_565_empty__subsetI,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A2 ) ).
% empty_subsetI
thf(fact_566_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_567_space__bot,axiom,
( ( sigma_space_a @ bot_bo2108912051383640591sure_a )
= bot_bot_set_a ) ).
% space_bot
thf(fact_568_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_569_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X: set_a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_570_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_571_set__eq__subset,axiom,
( ( ^ [Y5: set_set_a,Z: set_set_a] : ( Y5 = Z ) )
= ( ^ [A3: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B2 )
& ( ord_le3724670747650509150_set_a @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_572_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z: set_a] : ( Y5 = Z ) )
= ( ^ [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
& ( ord_less_eq_set_a @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_573_subset__trans,axiom,
! [A2: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_574_subset__trans,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_575_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_576_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_577_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_578_subset__refl,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_579_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_580_subset__iff,axiom,
( ord_less_eq_set_c_d
= ( ^ [A3: set_c_d,B2: set_c_d] :
! [T2: c > d] :
( ( member_c_d @ T2 @ A3 )
=> ( member_c_d @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_581_subset__iff,axiom,
( ord_le3334967407727675675a_real
= ( ^ [A3: set_a_real,B2: set_a_real] :
! [T2: a > real] :
( ( member_a_real @ T2 @ A3 )
=> ( member_a_real @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_582_subset__iff,axiom,
( ord_less_eq_set_a_o2
= ( ^ [A3: set_a_o,B2: set_a_o] :
! [T2: a > $o] :
( ( member_a_o @ T2 @ A3 )
=> ( member_a_o @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_583_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A3: set_b,B2: set_b] :
! [T2: b] :
( ( member_b @ T2 @ A3 )
=> ( member_b @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_584_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A3 )
=> ( member_nat @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_585_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B2: set_o] :
! [T2: $o] :
( ( member_o @ T2 @ A3 )
=> ( member_o @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_586_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B2: set_set_a] :
! [T2: set_a] :
( ( member_set_a @ T2 @ A3 )
=> ( member_set_a @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_587_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B2: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A3 )
=> ( member_a @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_588_equalityD2,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( A2 = B )
=> ( ord_le3724670747650509150_set_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_589_equalityD2,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_590_equalityD1,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( A2 = B )
=> ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_591_equalityD1,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_592_subset__eq,axiom,
( ord_less_eq_set_c_d
= ( ^ [A3: set_c_d,B2: set_c_d] :
! [X: c > d] :
( ( member_c_d @ X @ A3 )
=> ( member_c_d @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_593_subset__eq,axiom,
( ord_le3334967407727675675a_real
= ( ^ [A3: set_a_real,B2: set_a_real] :
! [X: a > real] :
( ( member_a_real @ X @ A3 )
=> ( member_a_real @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_594_subset__eq,axiom,
( ord_less_eq_set_a_o2
= ( ^ [A3: set_a_o,B2: set_a_o] :
! [X: a > $o] :
( ( member_a_o @ X @ A3 )
=> ( member_a_o @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_595_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A3: set_b,B2: set_b] :
! [X: b] :
( ( member_b @ X @ A3 )
=> ( member_b @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_596_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( member_nat @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_597_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B2: set_o] :
! [X: $o] :
( ( member_o @ X @ A3 )
=> ( member_o @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_598_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B2: set_set_a] :
! [X: set_a] :
( ( member_set_a @ X @ A3 )
=> ( member_set_a @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_599_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B2: set_a] :
! [X: a] :
( ( member_a @ X @ A3 )
=> ( member_a @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_600_equalityE,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( A2 = B )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ~ ( ord_le3724670747650509150_set_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_601_equalityE,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B )
=> ~ ( ord_less_eq_set_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_602_subsetD,axiom,
! [A2: set_c_d,B: set_c_d,C: c > d] :
( ( ord_less_eq_set_c_d @ A2 @ B )
=> ( ( member_c_d @ C @ A2 )
=> ( member_c_d @ C @ B ) ) ) ).
% subsetD
thf(fact_603_subsetD,axiom,
! [A2: set_a_real,B: set_a_real,C: a > real] :
( ( ord_le3334967407727675675a_real @ A2 @ B )
=> ( ( member_a_real @ C @ A2 )
=> ( member_a_real @ C @ B ) ) ) ).
% subsetD
thf(fact_604_subsetD,axiom,
! [A2: set_a_o,B: set_a_o,C: a > $o] :
( ( ord_less_eq_set_a_o2 @ A2 @ B )
=> ( ( member_a_o @ C @ A2 )
=> ( member_a_o @ C @ B ) ) ) ).
% subsetD
thf(fact_605_subsetD,axiom,
! [A2: set_b,B: set_b,C: b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( member_b @ C @ A2 )
=> ( member_b @ C @ B ) ) ) ).
% subsetD
thf(fact_606_subsetD,axiom,
! [A2: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_607_subsetD,axiom,
! [A2: set_o,B: set_o,C: $o] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ( member_o @ C @ A2 )
=> ( member_o @ C @ B ) ) ) ).
% subsetD
thf(fact_608_subsetD,axiom,
! [A2: set_set_a,B: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B ) ) ) ).
% subsetD
thf(fact_609_subsetD,axiom,
! [A2: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_610_in__mono,axiom,
! [A2: set_c_d,B: set_c_d,X4: c > d] :
( ( ord_less_eq_set_c_d @ A2 @ B )
=> ( ( member_c_d @ X4 @ A2 )
=> ( member_c_d @ X4 @ B ) ) ) ).
% in_mono
thf(fact_611_in__mono,axiom,
! [A2: set_a_real,B: set_a_real,X4: a > real] :
( ( ord_le3334967407727675675a_real @ A2 @ B )
=> ( ( member_a_real @ X4 @ A2 )
=> ( member_a_real @ X4 @ B ) ) ) ).
% in_mono
thf(fact_612_in__mono,axiom,
! [A2: set_a_o,B: set_a_o,X4: a > $o] :
( ( ord_less_eq_set_a_o2 @ A2 @ B )
=> ( ( member_a_o @ X4 @ A2 )
=> ( member_a_o @ X4 @ B ) ) ) ).
% in_mono
thf(fact_613_in__mono,axiom,
! [A2: set_b,B: set_b,X4: b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( member_b @ X4 @ A2 )
=> ( member_b @ X4 @ B ) ) ) ).
% in_mono
thf(fact_614_in__mono,axiom,
! [A2: set_nat,B: set_nat,X4: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ X4 @ A2 )
=> ( member_nat @ X4 @ B ) ) ) ).
% in_mono
thf(fact_615_in__mono,axiom,
! [A2: set_o,B: set_o,X4: $o] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ( member_o @ X4 @ A2 )
=> ( member_o @ X4 @ B ) ) ) ).
% in_mono
thf(fact_616_in__mono,axiom,
! [A2: set_set_a,B: set_set_a,X4: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_a @ X4 @ A2 )
=> ( member_set_a @ X4 @ B ) ) ) ).
% in_mono
thf(fact_617_in__mono,axiom,
! [A2: set_a,B: set_a,X4: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ X4 @ A2 )
=> ( member_a @ X4 @ B ) ) ) ).
% in_mono
thf(fact_618_nle__le,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ~ ( ord_le3935885782089961368nnreal @ A @ B3 ) )
= ( ( ord_le3935885782089961368nnreal @ B3 @ A )
& ( B3 != A ) ) ) ).
% nle_le
thf(fact_619_nle__le,axiom,
! [A: nat,B3: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B3 ) )
= ( ( ord_less_eq_nat @ B3 @ A )
& ( B3 != A ) ) ) ).
% nle_le
thf(fact_620_nle__le,axiom,
! [A: real,B3: real] :
( ( ~ ( ord_less_eq_real @ A @ B3 ) )
= ( ( ord_less_eq_real @ B3 @ A )
& ( B3 != A ) ) ) ).
% nle_le
thf(fact_621_le__cases3,axiom,
! [X4: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] :
( ( ( ord_le3935885782089961368nnreal @ X4 @ Y2 )
=> ~ ( ord_le3935885782089961368nnreal @ Y2 @ Z2 ) )
=> ( ( ( ord_le3935885782089961368nnreal @ Y2 @ X4 )
=> ~ ( ord_le3935885782089961368nnreal @ X4 @ Z2 ) )
=> ( ( ( ord_le3935885782089961368nnreal @ X4 @ Z2 )
=> ~ ( ord_le3935885782089961368nnreal @ Z2 @ Y2 ) )
=> ( ( ( ord_le3935885782089961368nnreal @ Z2 @ Y2 )
=> ~ ( ord_le3935885782089961368nnreal @ Y2 @ X4 ) )
=> ( ( ( ord_le3935885782089961368nnreal @ Y2 @ Z2 )
=> ~ ( ord_le3935885782089961368nnreal @ Z2 @ X4 ) )
=> ~ ( ( ord_le3935885782089961368nnreal @ Z2 @ X4 )
=> ~ ( ord_le3935885782089961368nnreal @ X4 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_622_le__cases3,axiom,
! [X4: nat,Y2: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X4 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X4 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X4 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X4 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_623_le__cases3,axiom,
! [X4: real,Y2: real,Z2: real] :
( ( ( ord_less_eq_real @ X4 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ X4 )
=> ~ ( ord_less_eq_real @ X4 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X4 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ X4 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X4 ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X4 )
=> ~ ( ord_less_eq_real @ X4 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_624_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] : ( Y5 = Z ) )
= ( ^ [X: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X @ Y3 )
& ( ord_le3935885782089961368nnreal @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_625_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_626_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_a,Z: set_set_a] : ( Y5 = Z ) )
= ( ^ [X: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y3 )
& ( ord_le3724670747650509150_set_a @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_627_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z: real] : ( Y5 = Z ) )
= ( ^ [X: real,Y3: real] :
( ( ord_less_eq_real @ X @ Y3 )
& ( ord_less_eq_real @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_628_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z: set_a] : ( Y5 = Z ) )
= ( ^ [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
& ( ord_less_eq_set_a @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_629_ord__eq__le__trans,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( A = B3 )
=> ( ( ord_le3935885782089961368nnreal @ B3 @ C )
=> ( ord_le3935885782089961368nnreal @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_630_ord__eq__le__trans,axiom,
! [A: nat,B3: nat,C: nat] :
( ( A = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_631_ord__eq__le__trans,axiom,
! [A: set_set_a,B3: set_set_a,C: set_set_a] :
( ( A = B3 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_632_ord__eq__le__trans,axiom,
! [A: real,B3: real,C: real] :
( ( A = B3 )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_633_ord__eq__le__trans,axiom,
! [A: set_a,B3: set_a,C: set_a] :
( ( A = B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_634_ord__le__eq__trans,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B3 )
=> ( ( B3 = C )
=> ( ord_le3935885782089961368nnreal @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_635_ord__le__eq__trans,axiom,
! [A: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_636_ord__le__eq__trans,axiom,
! [A: set_set_a,B3: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B3 )
=> ( ( B3 = C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_637_ord__le__eq__trans,axiom,
! [A: real,B3: real,C: real] :
( ( ord_less_eq_real @ A @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_638_ord__le__eq__trans,axiom,
! [A: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_639_order__antisym,axiom,
! [X4: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X4 @ Y2 )
=> ( ( ord_le3935885782089961368nnreal @ Y2 @ X4 )
=> ( X4 = Y2 ) ) ) ).
% order_antisym
thf(fact_640_order__antisym,axiom,
! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X4 )
=> ( X4 = Y2 ) ) ) ).
% order_antisym
thf(fact_641_order__antisym,axiom,
! [X4: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y2 )
=> ( ( ord_le3724670747650509150_set_a @ Y2 @ X4 )
=> ( X4 = Y2 ) ) ) ).
% order_antisym
thf(fact_642_order__antisym,axiom,
! [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ X4 )
=> ( X4 = Y2 ) ) ) ).
% order_antisym
thf(fact_643_order__antisym,axiom,
! [X4: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ X4 )
=> ( X4 = Y2 ) ) ) ).
% order_antisym
thf(fact_644_order_Otrans,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B3 )
=> ( ( ord_le3935885782089961368nnreal @ B3 @ C )
=> ( ord_le3935885782089961368nnreal @ A @ C ) ) ) ).
% order.trans
thf(fact_645_order_Otrans,axiom,
! [A: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_646_order_Otrans,axiom,
! [A: set_set_a,B3: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_647_order_Otrans,axiom,
! [A: real,B3: real,C: real] :
( ( ord_less_eq_real @ A @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_648_order_Otrans,axiom,
! [A: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_649_order__trans,axiom,
! [X4: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X4 @ Y2 )
=> ( ( ord_le3935885782089961368nnreal @ Y2 @ Z2 )
=> ( ord_le3935885782089961368nnreal @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_650_order__trans,axiom,
! [X4: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_eq_nat @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_651_order__trans,axiom,
! [X4: set_set_a,Y2: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y2 )
=> ( ( ord_le3724670747650509150_set_a @ Y2 @ Z2 )
=> ( ord_le3724670747650509150_set_a @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_652_order__trans,axiom,
! [X4: real,Y2: real,Z2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ( ord_less_eq_real @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_653_order__trans,axiom,
! [X4: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ Z2 )
=> ( ord_less_eq_set_a @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_654_linorder__wlog,axiom,
! [P: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o,A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ! [A4: extend8495563244428889912nnreal,B4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: extend8495563244428889912nnreal,B4: extend8495563244428889912nnreal] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B3 ) ) ) ).
% linorder_wlog
thf(fact_655_linorder__wlog,axiom,
! [P: nat > nat > $o,A: 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 @ A @ B3 ) ) ) ).
% linorder_wlog
thf(fact_656_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B3: real] :
( ! [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: real,B4: real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B3 ) ) ) ).
% linorder_wlog
thf(fact_657_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] : ( Y5 = Z ) )
= ( ^ [A5: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B5 @ A5 )
& ( ord_le3935885782089961368nnreal @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_658_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_659_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_set_a,Z: set_set_a] : ( Y5 = 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_660_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: real,Z: real] : ( Y5 = Z ) )
= ( ^ [A5: real,B5: real] :
( ( ord_less_eq_real @ B5 @ A5 )
& ( ord_less_eq_real @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_661_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z: set_a] : ( Y5 = 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_662_dual__order_Oantisym,axiom,
! [B3: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B3 @ A )
=> ( ( ord_le3935885782089961368nnreal @ A @ B3 )
=> ( A = B3 ) ) ) ).
% dual_order.antisym
thf(fact_663_dual__order_Oantisym,axiom,
! [B3: nat,A: nat] :
( ( ord_less_eq_nat @ B3 @ A )
=> ( ( ord_less_eq_nat @ A @ B3 )
=> ( A = B3 ) ) ) ).
% dual_order.antisym
thf(fact_664_dual__order_Oantisym,axiom,
! [B3: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ B3 )
=> ( A = B3 ) ) ) ).
% dual_order.antisym
thf(fact_665_dual__order_Oantisym,axiom,
! [B3: real,A: real] :
( ( ord_less_eq_real @ B3 @ A )
=> ( ( ord_less_eq_real @ A @ B3 )
=> ( A = B3 ) ) ) ).
% dual_order.antisym
thf(fact_666_dual__order_Oantisym,axiom,
! [B3: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B3 @ A )
=> ( ( ord_less_eq_set_a @ A @ B3 )
=> ( A = B3 ) ) ) ).
% dual_order.antisym
thf(fact_667_dual__order_Otrans,axiom,
! [B3: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B3 @ A )
=> ( ( ord_le3935885782089961368nnreal @ C @ B3 )
=> ( ord_le3935885782089961368nnreal @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_668_dual__order_Otrans,axiom,
! [B3: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_669_dual__order_Otrans,axiom,
! [B3: set_set_a,A: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A )
=> ( ( ord_le3724670747650509150_set_a @ C @ B3 )
=> ( ord_le3724670747650509150_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_670_dual__order_Otrans,axiom,
! [B3: real,A: real,C: real] :
( ( ord_less_eq_real @ B3 @ A )
=> ( ( ord_less_eq_real @ C @ B3 )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_671_dual__order_Otrans,axiom,
! [B3: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B3 @ A )
=> ( ( ord_less_eq_set_a @ C @ B3 )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_672_antisym,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B3 )
=> ( ( ord_le3935885782089961368nnreal @ B3 @ A )
=> ( A = B3 ) ) ) ).
% antisym
thf(fact_673_antisym,axiom,
! [A: nat,B3: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A )
=> ( A = B3 ) ) ) ).
% antisym
thf(fact_674_antisym,axiom,
! [A: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ A )
=> ( A = B3 ) ) ) ).
% antisym
thf(fact_675_antisym,axiom,
! [A: real,B3: real] :
( ( ord_less_eq_real @ A @ B3 )
=> ( ( ord_less_eq_real @ B3 @ A )
=> ( A = B3 ) ) ) ).
% antisym
thf(fact_676_antisym,axiom,
! [A: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A )
=> ( A = B3 ) ) ) ).
% antisym
thf(fact_677_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] : ( Y5 = Z ) )
= ( ^ [A5: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A5 @ B5 )
& ( ord_le3935885782089961368nnreal @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_678_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_679_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_a,Z: set_set_a] : ( Y5 = 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_680_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z: real] : ( Y5 = Z ) )
= ( ^ [A5: real,B5: real] :
( ( ord_less_eq_real @ A5 @ B5 )
& ( ord_less_eq_real @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_681_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z: set_a] : ( Y5 = 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_682_order__subst1,axiom,
! [A: extend8495563244428889912nnreal,F2: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ ( F2 @ B3 ) )
=> ( ( ord_le3935885782089961368nnreal @ B3 @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
=> ( ord_le3935885782089961368nnreal @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le3935885782089961368nnreal @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_683_order__subst1,axiom,
! [A: extend8495563244428889912nnreal,F2: nat > extend8495563244428889912nnreal,B3: nat,C: nat] :
( ( ord_le3935885782089961368nnreal @ A @ ( F2 @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le3935885782089961368nnreal @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le3935885782089961368nnreal @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_684_order__subst1,axiom,
! [A: extend8495563244428889912nnreal,F2: real > extend8495563244428889912nnreal,B3: real,C: real] :
( ( ord_le3935885782089961368nnreal @ A @ ( F2 @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_le3935885782089961368nnreal @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le3935885782089961368nnreal @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_685_order__subst1,axiom,
! [A: nat,F2: extend8495563244428889912nnreal > nat,B3: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_nat @ A @ ( F2 @ B3 ) )
=> ( ( ord_le3935885782089961368nnreal @ B3 @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_686_order__subst1,axiom,
! [A: nat,F2: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_687_order__subst1,axiom,
! [A: nat,F2: real > nat,B3: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F2 @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_688_order__subst1,axiom,
! [A: real,F2: extend8495563244428889912nnreal > real,B3: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_real @ A @ ( F2 @ B3 ) )
=> ( ( ord_le3935885782089961368nnreal @ B3 @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_689_order__subst1,axiom,
! [A: real,F2: nat > real,B3: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F2 @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_690_order__subst1,axiom,
! [A: real,F2: real > real,B3: real,C: real] :
( ( ord_less_eq_real @ A @ ( F2 @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_691_order__subst1,axiom,
! [A: extend8495563244428889912nnreal,F2: set_a > extend8495563244428889912nnreal,B3: set_a,C: set_a] :
( ( ord_le3935885782089961368nnreal @ A @ ( F2 @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
=> ( ord_le3935885782089961368nnreal @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le3935885782089961368nnreal @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_692_order__subst2,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,F2: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B3 )
=> ( ( ord_le3935885782089961368nnreal @ ( F2 @ B3 ) @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
=> ( ord_le3935885782089961368nnreal @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_693_order__subst2,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,F2: extend8495563244428889912nnreal > nat,C: nat] :
( ( ord_le3935885782089961368nnreal @ A @ B3 )
=> ( ( ord_less_eq_nat @ ( F2 @ B3 ) @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_694_order__subst2,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,F2: extend8495563244428889912nnreal > real,C: real] :
( ( ord_le3935885782089961368nnreal @ A @ B3 )
=> ( ( ord_less_eq_real @ ( F2 @ B3 ) @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_695_order__subst2,axiom,
! [A: nat,B3: nat,F2: nat > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_le3935885782089961368nnreal @ ( F2 @ B3 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le3935885782089961368nnreal @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_696_order__subst2,axiom,
! [A: nat,B3: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ ( F2 @ B3 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_697_order__subst2,axiom,
! [A: nat,B3: nat,F2: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_real @ ( F2 @ B3 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_698_order__subst2,axiom,
! [A: real,B3: real,F2: real > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_real @ A @ B3 )
=> ( ( ord_le3935885782089961368nnreal @ ( F2 @ B3 ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_le3935885782089961368nnreal @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_699_order__subst2,axiom,
! [A: real,B3: real,F2: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B3 )
=> ( ( ord_less_eq_nat @ ( F2 @ B3 ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_700_order__subst2,axiom,
! [A: real,B3: real,F2: real > real,C: real] :
( ( ord_less_eq_real @ A @ B3 )
=> ( ( ord_less_eq_real @ ( F2 @ B3 ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_701_order__subst2,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,F2: extend8495563244428889912nnreal > set_a,C: set_a] :
( ( ord_le3935885782089961368nnreal @ A @ B3 )
=> ( ( ord_less_eq_set_a @ ( F2 @ B3 ) @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_702_order__eq__refl,axiom,
! [X4: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( X4 = Y2 )
=> ( ord_le3935885782089961368nnreal @ X4 @ Y2 ) ) ).
% order_eq_refl
thf(fact_703_order__eq__refl,axiom,
! [X4: nat,Y2: nat] :
( ( X4 = Y2 )
=> ( ord_less_eq_nat @ X4 @ Y2 ) ) ).
% order_eq_refl
thf(fact_704_order__eq__refl,axiom,
! [X4: set_set_a,Y2: set_set_a] :
( ( X4 = Y2 )
=> ( ord_le3724670747650509150_set_a @ X4 @ Y2 ) ) ).
% order_eq_refl
thf(fact_705_order__eq__refl,axiom,
! [X4: real,Y2: real] :
( ( X4 = Y2 )
=> ( ord_less_eq_real @ X4 @ Y2 ) ) ).
% order_eq_refl
thf(fact_706_order__eq__refl,axiom,
! [X4: set_a,Y2: set_a] :
( ( X4 = Y2 )
=> ( ord_less_eq_set_a @ X4 @ Y2 ) ) ).
% order_eq_refl
thf(fact_707_linorder__linear,axiom,
! [X4: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X4 @ Y2 )
| ( ord_le3935885782089961368nnreal @ Y2 @ X4 ) ) ).
% linorder_linear
thf(fact_708_linorder__linear,axiom,
! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X4 ) ) ).
% linorder_linear
thf(fact_709_linorder__linear,axiom,
! [X4: real,Y2: real] :
( ( ord_less_eq_real @ X4 @ Y2 )
| ( ord_less_eq_real @ Y2 @ X4 ) ) ).
% linorder_linear
thf(fact_710_ord__eq__le__subst,axiom,
! [A: extend8495563244428889912nnreal,F2: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( A
= ( F2 @ B3 ) )
=> ( ( ord_le3935885782089961368nnreal @ B3 @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
=> ( ord_le3935885782089961368nnreal @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le3935885782089961368nnreal @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_711_ord__eq__le__subst,axiom,
! [A: nat,F2: extend8495563244428889912nnreal > nat,B3: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( A
= ( F2 @ B3 ) )
=> ( ( ord_le3935885782089961368nnreal @ B3 @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_712_ord__eq__le__subst,axiom,
! [A: real,F2: extend8495563244428889912nnreal > real,B3: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( A
= ( F2 @ B3 ) )
=> ( ( ord_le3935885782089961368nnreal @ B3 @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_713_ord__eq__le__subst,axiom,
! [A: extend8495563244428889912nnreal,F2: nat > extend8495563244428889912nnreal,B3: nat,C: nat] :
( ( A
= ( F2 @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le3935885782089961368nnreal @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le3935885782089961368nnreal @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_714_ord__eq__le__subst,axiom,
! [A: nat,F2: nat > nat,B3: nat,C: nat] :
( ( A
= ( F2 @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_715_ord__eq__le__subst,axiom,
! [A: real,F2: nat > real,B3: nat,C: nat] :
( ( A
= ( F2 @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_716_ord__eq__le__subst,axiom,
! [A: extend8495563244428889912nnreal,F2: real > extend8495563244428889912nnreal,B3: real,C: real] :
( ( A
= ( F2 @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_le3935885782089961368nnreal @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le3935885782089961368nnreal @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_717_ord__eq__le__subst,axiom,
! [A: nat,F2: real > nat,B3: real,C: real] :
( ( A
= ( F2 @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_718_ord__eq__le__subst,axiom,
! [A: real,F2: real > real,B3: real,C: real] :
( ( A
= ( F2 @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_719_ord__eq__le__subst,axiom,
! [A: set_a,F2: extend8495563244428889912nnreal > set_a,B3: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( A
= ( F2 @ B3 ) )
=> ( ( ord_le3935885782089961368nnreal @ B3 @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_720_ord__le__eq__subst,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,F2: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B3 )
=> ( ( ( F2 @ B3 )
= C )
=> ( ! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
=> ( ord_le3935885782089961368nnreal @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_721_ord__le__eq__subst,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,F2: extend8495563244428889912nnreal > nat,C: nat] :
( ( ord_le3935885782089961368nnreal @ A @ B3 )
=> ( ( ( F2 @ B3 )
= C )
=> ( ! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_722_ord__le__eq__subst,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,F2: extend8495563244428889912nnreal > real,C: real] :
( ( ord_le3935885782089961368nnreal @ A @ B3 )
=> ( ( ( F2 @ B3 )
= C )
=> ( ! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_723_ord__le__eq__subst,axiom,
! [A: nat,B3: nat,F2: nat > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ( F2 @ B3 )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le3935885782089961368nnreal @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_724_ord__le__eq__subst,axiom,
! [A: nat,B3: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ( F2 @ B3 )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_725_ord__le__eq__subst,axiom,
! [A: nat,B3: nat,F2: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ( F2 @ B3 )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_726_ord__le__eq__subst,axiom,
! [A: real,B3: real,F2: real > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_real @ A @ B3 )
=> ( ( ( F2 @ B3 )
= C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_le3935885782089961368nnreal @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_727_ord__le__eq__subst,axiom,
! [A: real,B3: real,F2: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B3 )
=> ( ( ( F2 @ B3 )
= C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_728_ord__le__eq__subst,axiom,
! [A: real,B3: real,F2: real > real,C: real] :
( ( ord_less_eq_real @ A @ B3 )
=> ( ( ( F2 @ B3 )
= C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_729_ord__le__eq__subst,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,F2: extend8495563244428889912nnreal > set_a,C: set_a] :
( ( ord_le3935885782089961368nnreal @ A @ B3 )
=> ( ( ( F2 @ B3 )
= C )
=> ( ! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
=> ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_730_linorder__le__cases,axiom,
! [X4: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ~ ( ord_le3935885782089961368nnreal @ X4 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ Y2 @ X4 ) ) ).
% linorder_le_cases
thf(fact_731_linorder__le__cases,axiom,
! [X4: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X4 ) ) ).
% linorder_le_cases
thf(fact_732_linorder__le__cases,axiom,
! [X4: real,Y2: real] :
( ~ ( ord_less_eq_real @ X4 @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X4 ) ) ).
% linorder_le_cases
thf(fact_733_order__antisym__conv,axiom,
! [Y2: extend8495563244428889912nnreal,X4: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Y2 @ X4 )
=> ( ( ord_le3935885782089961368nnreal @ X4 @ Y2 )
= ( X4 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_734_order__antisym__conv,axiom,
! [Y2: nat,X4: nat] :
( ( ord_less_eq_nat @ Y2 @ X4 )
=> ( ( ord_less_eq_nat @ X4 @ Y2 )
= ( X4 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_735_order__antisym__conv,axiom,
! [Y2: set_set_a,X4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y2 @ X4 )
=> ( ( ord_le3724670747650509150_set_a @ X4 @ Y2 )
= ( X4 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_736_order__antisym__conv,axiom,
! [Y2: real,X4: real] :
( ( ord_less_eq_real @ Y2 @ X4 )
=> ( ( ord_less_eq_real @ X4 @ Y2 )
= ( X4 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_737_order__antisym__conv,axiom,
! [Y2: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X4 )
=> ( ( ord_less_eq_set_a @ X4 @ Y2 )
= ( X4 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_738_top__greatest,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% top_greatest
thf(fact_739_top__greatest,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ A @ top_top_set_o ) ).
% top_greatest
thf(fact_740_top__greatest,axiom,
! [A: set_literal] : ( ord_le7307670543136651348iteral @ A @ top_top_set_literal ) ).
% top_greatest
thf(fact_741_top__greatest,axiom,
! [A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A @ top_to1996260823553986621t_unit ) ).
% top_greatest
thf(fact_742_top__greatest,axiom,
! [A: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ A @ top_to1496364449551166952nnreal ) ).
% top_greatest
thf(fact_743_top__greatest,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ top_top_set_set_a ) ).
% top_greatest
thf(fact_744_top__greatest,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% top_greatest
thf(fact_745_top_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
= ( A = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_746_top_Oextremum__unique,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A )
= ( A = top_top_set_o ) ) ).
% top.extremum_unique
thf(fact_747_top_Oextremum__unique,axiom,
! [A: set_literal] :
( ( ord_le7307670543136651348iteral @ top_top_set_literal @ A )
= ( A = top_top_set_literal ) ) ).
% top.extremum_unique
thf(fact_748_top_Oextremum__unique,axiom,
! [A: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A )
= ( A = top_to1996260823553986621t_unit ) ) ).
% top.extremum_unique
thf(fact_749_top_Oextremum__unique,axiom,
! [A: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ top_to1496364449551166952nnreal @ A )
= ( A = top_to1496364449551166952nnreal ) ) ).
% top.extremum_unique
thf(fact_750_top_Oextremum__unique,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ top_top_set_set_a @ A )
= ( A = top_top_set_set_a ) ) ).
% top.extremum_unique
thf(fact_751_top_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
= ( A = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_752_top_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
=> ( A = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_753_top_Oextremum__uniqueI,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A )
=> ( A = top_top_set_o ) ) ).
% top.extremum_uniqueI
thf(fact_754_top_Oextremum__uniqueI,axiom,
! [A: set_literal] :
( ( ord_le7307670543136651348iteral @ top_top_set_literal @ A )
=> ( A = top_top_set_literal ) ) ).
% top.extremum_uniqueI
thf(fact_755_top_Oextremum__uniqueI,axiom,
! [A: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A )
=> ( A = top_to1996260823553986621t_unit ) ) ).
% top.extremum_uniqueI
thf(fact_756_top_Oextremum__uniqueI,axiom,
! [A: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ top_to1496364449551166952nnreal @ A )
=> ( A = top_to1496364449551166952nnreal ) ) ).
% top.extremum_uniqueI
thf(fact_757_top_Oextremum__uniqueI,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ top_top_set_set_a @ A )
=> ( A = top_top_set_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_758_top_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
=> ( A = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_759_bot_Oextremum,axiom,
! [A: filter_a] : ( ord_less_eq_filter_a @ bot_bot_filter_a @ A ) ).
% bot.extremum
thf(fact_760_bot_Oextremum,axiom,
! [A: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ bot_bo841427958541957580nnreal @ A ) ).
% bot.extremum
thf(fact_761_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_762_bot_Oextremum,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% bot.extremum
thf(fact_763_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_764_bot_Oextremum__unique,axiom,
! [A: filter_a] :
( ( ord_less_eq_filter_a @ A @ bot_bot_filter_a )
= ( A = bot_bot_filter_a ) ) ).
% bot.extremum_unique
thf(fact_765_bot_Oextremum__unique,axiom,
! [A: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ bot_bo841427958541957580nnreal )
= ( A = bot_bo841427958541957580nnreal ) ) ).
% bot.extremum_unique
thf(fact_766_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_767_bot_Oextremum__unique,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% bot.extremum_unique
thf(fact_768_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_769_bot_Oextremum__uniqueI,axiom,
! [A: filter_a] :
( ( ord_less_eq_filter_a @ A @ bot_bot_filter_a )
=> ( A = bot_bot_filter_a ) ) ).
% bot.extremum_uniqueI
thf(fact_770_bot_Oextremum__uniqueI,axiom,
! [A: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ bot_bo841427958541957580nnreal )
=> ( A = bot_bo841427958541957580nnreal ) ) ).
% bot.extremum_uniqueI
thf(fact_771_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_772_bot_Oextremum__uniqueI,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
=> ( A = bot_bot_set_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_773_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_774_Collect__subset,axiom,
! [A2: set_c_d,P: ( c > d ) > $o] :
( ord_less_eq_set_c_d
@ ( collect_c_d
@ ^ [X: c > d] :
( ( member_c_d @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_775_Collect__subset,axiom,
! [A2: set_a_real,P: ( a > real ) > $o] :
( ord_le3334967407727675675a_real
@ ( collect_a_real
@ ^ [X: a > real] :
( ( member_a_real @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_776_Collect__subset,axiom,
! [A2: set_a_o,P: ( a > $o ) > $o] :
( ord_less_eq_set_a_o2
@ ( collect_a_o
@ ^ [X: a > $o] :
( ( member_a_o @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_777_Collect__subset,axiom,
! [A2: set_b,P: b > $o] :
( ord_less_eq_set_b
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_778_Collect__subset,axiom,
! [A2: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_779_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_780_Collect__subset,axiom,
! [A2: set_set_a,P: set_a > $o] :
( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ^ [X: set_a] :
( ( member_set_a @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_781_Collect__subset,axiom,
! [A2: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_782_top__set__def,axiom,
( top_top_set_a
= ( collect_a @ top_top_a_o ) ) ).
% top_set_def
thf(fact_783_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_784_top__set__def,axiom,
( top_top_set_o
= ( collect_o @ top_top_o_o ) ) ).
% top_set_def
thf(fact_785_top__set__def,axiom,
( top_top_set_literal
= ( collect_literal @ top_top_literal_o ) ) ).
% top_set_def
thf(fact_786_top__set__def,axiom,
( top_to1996260823553986621t_unit
= ( collect_Product_unit @ top_to2465898995584390880unit_o ) ) ).
% top_set_def
thf(fact_787_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_788_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_789_prob__space_Oindep__sets__mono,axiom,
! [M2: sigma_measure_a,F: set_a > set_set_a,I2: set_set_a,J: set_set_a,G: set_a > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe4967106450811773644_set_a @ M2 @ 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 @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_790_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_791_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_792_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_793_prob__space_Oindep__sets__mono,axiom,
! [M2: sigma_measure_a,F: b > set_set_a,I2: set_b,J: set_b,G: b > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe8927441866673418605ts_a_b @ M2 @ F @ I2 )
=> ( ( ord_less_eq_set_b @ J @ I2 )
=> ( ! [I3: b] :
( ( member_b @ I3 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe8927441866673418605ts_a_b @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_794_prob__space_Oindep__sets__mono,axiom,
! [M2: sigma_measure_a,F: ( a > $o ) > set_set_a,I2: set_a_o,J: set_a_o,G: ( a > $o ) > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe7801696130336798481_a_a_o @ M2 @ F @ I2 )
=> ( ( ord_less_eq_set_a_o2 @ J @ I2 )
=> ( ! [I3: a > $o] :
( ( member_a_o @ I3 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe7801696130336798481_a_a_o @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_795_prob__space_Oindep__sets__mono,axiom,
! [M2: sigma_measure_a,F: ( a > real ) > set_set_a,I2: set_a_real,J: set_a_real,G: ( a > real ) > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe4749599203615801097a_real @ M2 @ F @ I2 )
=> ( ( ord_le3334967407727675675a_real @ J @ I2 )
=> ( ! [I3: a > real] :
( ( member_a_real @ I3 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe4749599203615801097a_real @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_796_prob__space_Oindep__sets__mono,axiom,
! [M2: sigma_measure_a,F: ( c > d ) > set_set_a,I2: set_c_d,J: set_c_d,G: ( c > d ) > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe4867465796832040824_a_c_d @ M2 @ F @ I2 )
=> ( ( ord_less_eq_set_c_d @ J @ I2 )
=> ( ! [I3: c > d] :
( ( member_c_d @ I3 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe4867465796832040824_a_c_d @ M2 @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_797_subset__UNIV,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_798_subset__UNIV,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ A2 @ top_top_set_o ) ).
% subset_UNIV
thf(fact_799_subset__UNIV,axiom,
! [A2: set_literal] : ( ord_le7307670543136651348iteral @ A2 @ top_top_set_literal ) ).
% subset_UNIV
thf(fact_800_subset__UNIV,axiom,
! [A2: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A2 @ top_to1996260823553986621t_unit ) ).
% subset_UNIV
thf(fact_801_subset__UNIV,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ top_top_set_set_a ) ).
% subset_UNIV
thf(fact_802_subset__UNIV,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ top_top_set_a ) ).
% subset_UNIV
thf(fact_803_space__empty__eq__bot,axiom,
! [A: sigma_measure_a] :
( ( ( sigma_space_a @ A )
= bot_bot_set_a )
= ( A = bot_bo2108912051383640591sure_a ) ) ).
% space_empty_eq_bot
thf(fact_804_prob__space_Oae__filter__bot,axiom,
! [M2: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( measure_ae_filter_a @ M2 )
!= bot_bot_filter_a ) ) ).
% prob_space.ae_filter_bot
thf(fact_805_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_806_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_807_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_808_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_809_prob__space_Oindep__sets__mono__index,axiom,
! [M2: sigma_measure_a,J: set_b,I2: set_b,F: b > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( ord_less_eq_set_b @ J @ I2 )
=> ( ( indepe8927441866673418605ts_a_b @ M2 @ F @ I2 )
=> ( indepe8927441866673418605ts_a_b @ M2 @ F @ J ) ) ) ) ).
% prob_space.indep_sets_mono_index
thf(fact_810_prob__space_Oindep__sets__mono__index,axiom,
! [M2: sigma_measure_a,J: set_a_o,I2: set_a_o,F: ( a > $o ) > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( ord_less_eq_set_a_o2 @ J @ I2 )
=> ( ( indepe7801696130336798481_a_a_o @ M2 @ F @ I2 )
=> ( indepe7801696130336798481_a_a_o @ M2 @ F @ J ) ) ) ) ).
% prob_space.indep_sets_mono_index
thf(fact_811_prob__space_Oindep__sets__mono__index,axiom,
! [M2: sigma_measure_a,J: set_a_real,I2: set_a_real,F: ( a > real ) > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( ord_le3334967407727675675a_real @ J @ I2 )
=> ( ( indepe4749599203615801097a_real @ M2 @ F @ I2 )
=> ( indepe4749599203615801097a_real @ M2 @ F @ J ) ) ) ) ).
% prob_space.indep_sets_mono_index
thf(fact_812_prob__space_Oindep__sets__mono__index,axiom,
! [M2: sigma_measure_a,J: set_c_d,I2: set_c_d,F: ( c > d ) > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( ord_less_eq_set_c_d @ J @ I2 )
=> ( ( indepe4867465796832040824_a_c_d @ M2 @ F @ I2 )
=> ( indepe4867465796832040824_a_c_d @ M2 @ F @ J ) ) ) ) ).
% prob_space.indep_sets_mono_index
thf(fact_813_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_814_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_815_prob__space_Oindep__sets__mono__sets,axiom,
! [M2: sigma_measure_a,F: a > set_set_a,I2: set_a,G: a > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe8927441866673418604ts_a_a @ M2 @ F @ I2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I2 )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe8927441866673418604ts_a_a @ M2 @ G @ I2 ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_816_prob__space_Oindep__sets__mono__sets,axiom,
! [M2: sigma_measure_a,F: b > set_set_a,I2: set_b,G: b > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe8927441866673418605ts_a_b @ M2 @ F @ I2 )
=> ( ! [I3: b] :
( ( member_b @ I3 @ I2 )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe8927441866673418605ts_a_b @ M2 @ G @ I2 ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_817_prob__space_Oindep__sets__mono__sets,axiom,
! [M2: sigma_measure_a,F: ( a > $o ) > set_set_a,I2: set_a_o,G: ( a > $o ) > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe7801696130336798481_a_a_o @ M2 @ F @ I2 )
=> ( ! [I3: a > $o] :
( ( member_a_o @ I3 @ I2 )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe7801696130336798481_a_a_o @ M2 @ G @ I2 ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_818_prob__space_Oindep__sets__mono__sets,axiom,
! [M2: sigma_measure_a,F: ( a > real ) > set_set_a,I2: set_a_real,G: ( a > real ) > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe4749599203615801097a_real @ M2 @ F @ I2 )
=> ( ! [I3: a > real] :
( ( member_a_real @ I3 @ I2 )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe4749599203615801097a_real @ M2 @ G @ I2 ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_819_prob__space_Oindep__sets__mono__sets,axiom,
! [M2: sigma_measure_a,F: ( c > d ) > set_set_a,I2: set_c_d,G: ( c > d ) > set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe4867465796832040824_a_c_d @ M2 @ F @ I2 )
=> ( ! [I3: c > d] :
( ( member_c_d @ I3 @ I2 )
=> ( ord_le3724670747650509150_set_a @ ( G @ I3 ) @ ( F @ I3 ) ) )
=> ( indepe4867465796832040824_a_c_d @ M2 @ G @ I2 ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_820_measurable__bot,axiom,
! [M2: sigma_measure_a] : ( member_a_o @ bot_bot_a_o @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% measurable_bot
thf(fact_821_measurable__top,axiom,
! [M2: sigma_measure_a] : ( member_a_o @ top_top_a_o @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ).
% measurable_top
thf(fact_822_Bochner__Integration_Ointegrable__cong,axiom,
! [M2: sigma_measure_a,N: sigma_measure_a,F2: a > real,G2: a > real] :
( ( M2 = N )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( sigma_space_a @ N ) )
=> ( ( F2 @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( bochne2139062162225249880a_real @ M2 @ F2 )
= ( bochne2139062162225249880a_real @ N @ G2 ) ) ) ) ).
% Bochner_Integration.integrable_cong
thf(fact_823_indep__sets__finite__index__sets,axiom,
! [F: set_a > set_set_a,I2: set_set_a] :
( ( indepe4967106450811773644_set_a @ m @ F @ I2 )
= ( ! [J2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ J2 @ I2 )
=> ( ( J2 != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ J2 )
=> ( indepe4967106450811773644_set_a @ m @ F @ J2 ) ) ) ) ) ) ).
% indep_sets_finite_index_sets
thf(fact_824_indep__sets__finite__index__sets,axiom,
! [F: $o > set_set_a,I2: set_o] :
( ( indepe7780107833195774214ts_a_o @ m @ F @ I2 )
= ( ! [J2: set_o] :
( ( ord_less_eq_set_o @ J2 @ I2 )
=> ( ( J2 != bot_bot_set_o )
=> ( ( finite_finite_o @ J2 )
=> ( indepe7780107833195774214ts_a_o @ m @ F @ J2 ) ) ) ) ) ) ).
% indep_sets_finite_index_sets
thf(fact_825_indep__sets__finite__index__sets,axiom,
! [F: nat > set_set_a,I2: set_nat] :
( ( indepe6267730027088848354_a_nat @ m @ F @ I2 )
= ( ! [J2: set_nat] :
( ( ord_less_eq_set_nat @ J2 @ I2 )
=> ( ( J2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ J2 )
=> ( indepe6267730027088848354_a_nat @ m @ F @ J2 ) ) ) ) ) ) ).
% indep_sets_finite_index_sets
thf(fact_826_indep__sets__finite__index__sets,axiom,
! [F: a > set_set_a,I2: set_a] :
( ( indepe8927441866673418604ts_a_a @ m @ F @ I2 )
= ( ! [J2: set_a] :
( ( ord_less_eq_set_a @ J2 @ I2 )
=> ( ( J2 != bot_bot_set_a )
=> ( ( finite_finite_a @ J2 )
=> ( indepe8927441866673418604ts_a_a @ m @ F @ J2 ) ) ) ) ) ) ).
% indep_sets_finite_index_sets
thf(fact_827_indep__sets__finite__index__sets,axiom,
! [F: b > set_set_a,I2: set_b] :
( ( indepe8927441866673418605ts_a_b @ m @ F @ I2 )
= ( ! [J2: set_b] :
( ( ord_less_eq_set_b @ J2 @ I2 )
=> ( ( J2 != bot_bot_set_b )
=> ( ( finite_finite_b @ J2 )
=> ( indepe8927441866673418605ts_a_b @ m @ F @ J2 ) ) ) ) ) ) ).
% indep_sets_finite_index_sets
thf(fact_828_indep__sets__finite__index__sets,axiom,
! [F: ( a > $o ) > set_set_a,I2: set_a_o] :
( ( indepe7801696130336798481_a_a_o @ m @ F @ I2 )
= ( ! [J2: set_a_o] :
( ( ord_less_eq_set_a_o2 @ J2 @ I2 )
=> ( ( J2 != bot_bot_set_a_o )
=> ( ( finite_finite_a_o @ J2 )
=> ( indepe7801696130336798481_a_a_o @ m @ F @ J2 ) ) ) ) ) ) ).
% indep_sets_finite_index_sets
thf(fact_829_indep__sets__finite__index__sets,axiom,
! [F: ( a > real ) > set_set_a,I2: set_a_real] :
( ( indepe4749599203615801097a_real @ m @ F @ I2 )
= ( ! [J2: set_a_real] :
( ( ord_le3334967407727675675a_real @ J2 @ I2 )
=> ( ( J2 != bot_bot_set_a_real )
=> ( ( finite_finite_a_real @ J2 )
=> ( indepe4749599203615801097a_real @ m @ F @ J2 ) ) ) ) ) ) ).
% indep_sets_finite_index_sets
thf(fact_830_indep__sets__finite__index__sets,axiom,
! [F: ( c > d ) > set_set_a,I2: set_c_d] :
( ( indepe4867465796832040824_a_c_d @ m @ F @ I2 )
= ( ! [J2: set_c_d] :
( ( ord_less_eq_set_c_d @ J2 @ I2 )
=> ( ( J2 != bot_bot_set_c_d )
=> ( ( finite_finite_c_d @ J2 )
=> ( indepe4867465796832040824_a_c_d @ m @ F @ J2 ) ) ) ) ) ) ).
% indep_sets_finite_index_sets
thf(fact_831_eventually__const,axiom,
! [F: filter_a,P: $o] :
( ( F != bot_bot_filter_a )
=> ( ( eventually_a
@ ^ [X: a] : P
@ F )
= P ) ) ).
% eventually_const
thf(fact_832_eventually__top,axiom,
! [P: a > $o] :
( ( eventually_a @ P @ top_top_filter_a )
= ( ! [X5: a] : ( P @ X5 ) ) ) ).
% eventually_top
thf(fact_833_increasing__def,axiom,
( measur1771626496591458595nnreal
= ( ^ [M3: set_set_set_a,Mu: set_set_a > extend8495563244428889912nnreal] :
! [X: set_set_a] :
( ( member_set_set_a @ X @ M3 )
=> ! [Y3: set_set_a] :
( ( member_set_set_a @ Y3 @ M3 )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( Mu @ X ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_834_increasing__def,axiom,
( measur1244951900059291067_a_nat
= ( ^ [M3: set_set_set_a,Mu: set_set_a > nat] :
! [X: set_set_a] :
( ( member_set_set_a @ X @ M3 )
=> ! [Y3: set_set_a] :
( ( member_set_set_a @ Y3 @ M3 )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y3 )
=> ( ord_less_eq_nat @ ( Mu @ X ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_835_increasing__def,axiom,
( measur2197171192767378579_set_a
= ( ^ [M3: set_set_set_a,Mu: set_set_a > set_set_a] :
! [X: set_set_a] :
( ( member_set_set_a @ X @ M3 )
=> ! [Y3: set_set_a] :
( ( member_set_set_a @ Y3 @ M3 )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( Mu @ X ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_836_increasing__def,axiom,
( measur2331856671108623127a_real
= ( ^ [M3: set_set_set_a,Mu: set_set_a > real] :
! [X: set_set_a] :
( ( member_set_set_a @ X @ M3 )
=> ! [Y3: set_set_a] :
( ( member_set_set_a @ Y3 @ M3 )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y3 )
=> ( ord_less_eq_real @ ( Mu @ X ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_837_increasing__def,axiom,
( measur5181028491126448947_set_a
= ( ^ [M3: set_set_set_a,Mu: set_set_a > set_a] :
! [X: set_set_a] :
( ( member_set_set_a @ X @ M3 )
=> ! [Y3: set_set_a] :
( ( member_set_set_a @ Y3 @ M3 )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y3 )
=> ( ord_less_eq_set_a @ ( Mu @ X ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_838_increasing__def,axiom,
( measur5393715408109795267nnreal
= ( ^ [M3: set_set_a,Mu: set_a > extend8495563244428889912nnreal] :
! [X: set_a] :
( ( member_set_a @ X @ M3 )
=> ! [Y3: set_a] :
( ( member_set_a @ Y3 @ M3 )
=> ( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( Mu @ X ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_839_increasing__def,axiom,
( measur8151441426001876059_a_nat
= ( ^ [M3: set_set_a,Mu: set_a > nat] :
! [X: set_a] :
( ( member_set_a @ X @ M3 )
=> ! [Y3: set_a] :
( ( member_set_a @ Y3 @ M3 )
=> ( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ord_less_eq_nat @ ( Mu @ X ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_840_increasing__def,axiom,
( measur8202069185322079731_set_a
= ( ^ [M3: set_set_a,Mu: set_a > set_set_a] :
! [X: set_a] :
( ( member_set_a @ X @ M3 )
=> ! [Y3: set_a] :
( ( member_set_a @ Y3 @ M3 )
=> ( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( Mu @ X ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_841_increasing__def,axiom,
( measur1776380161843274167a_real
= ( ^ [M3: set_set_a,Mu: set_a > real] :
! [X: set_a] :
( ( member_set_a @ X @ M3 )
=> ! [Y3: set_a] :
( ( member_set_a @ Y3 @ M3 )
=> ( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ord_less_eq_real @ ( Mu @ X ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_842_increasing__def,axiom,
( measur7842569353079325843_set_a
= ( ^ [M3: set_set_a,Mu: set_a > set_a] :
! [X: set_a] :
( ( member_set_a @ X @ M3 )
=> ! [Y3: set_a] :
( ( member_set_a @ Y3 @ M3 )
=> ( ( ord_less_eq_set_a @ X @ Y3 )
=> ( ord_less_eq_set_a @ ( Mu @ X ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_843_increasingD,axiom,
! [M2: set_set_set_a,F2: set_set_a > extend8495563244428889912nnreal,X4: set_set_a,Y2: set_set_a] :
( ( measur1771626496591458595nnreal @ M2 @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ X4 @ Y2 )
=> ( ( member_set_set_a @ X4 @ M2 )
=> ( ( member_set_set_a @ Y2 @ M2 )
=> ( ord_le3935885782089961368nnreal @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_844_increasingD,axiom,
! [M2: set_set_set_a,F2: set_set_a > nat,X4: set_set_a,Y2: set_set_a] :
( ( measur1244951900059291067_a_nat @ M2 @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ X4 @ Y2 )
=> ( ( member_set_set_a @ X4 @ M2 )
=> ( ( member_set_set_a @ Y2 @ M2 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_845_increasingD,axiom,
! [M2: set_set_set_a,F2: set_set_a > set_set_a,X4: set_set_a,Y2: set_set_a] :
( ( measur2197171192767378579_set_a @ M2 @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ X4 @ Y2 )
=> ( ( member_set_set_a @ X4 @ M2 )
=> ( ( member_set_set_a @ Y2 @ M2 )
=> ( ord_le3724670747650509150_set_a @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_846_increasingD,axiom,
! [M2: set_set_set_a,F2: set_set_a > real,X4: set_set_a,Y2: set_set_a] :
( ( measur2331856671108623127a_real @ M2 @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ X4 @ Y2 )
=> ( ( member_set_set_a @ X4 @ M2 )
=> ( ( member_set_set_a @ Y2 @ M2 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_847_increasingD,axiom,
! [M2: set_set_set_a,F2: set_set_a > set_a,X4: set_set_a,Y2: set_set_a] :
( ( measur5181028491126448947_set_a @ M2 @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ X4 @ Y2 )
=> ( ( member_set_set_a @ X4 @ M2 )
=> ( ( member_set_set_a @ Y2 @ M2 )
=> ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_848_increasingD,axiom,
! [M2: set_set_a,F2: set_a > extend8495563244428889912nnreal,X4: set_a,Y2: set_a] :
( ( measur5393715408109795267nnreal @ M2 @ F2 )
=> ( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ( member_set_a @ X4 @ M2 )
=> ( ( member_set_a @ Y2 @ M2 )
=> ( ord_le3935885782089961368nnreal @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_849_increasingD,axiom,
! [M2: set_set_a,F2: set_a > nat,X4: set_a,Y2: set_a] :
( ( measur8151441426001876059_a_nat @ M2 @ F2 )
=> ( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ( member_set_a @ X4 @ M2 )
=> ( ( member_set_a @ Y2 @ M2 )
=> ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_850_increasingD,axiom,
! [M2: set_set_a,F2: set_a > set_set_a,X4: set_a,Y2: set_a] :
( ( measur8202069185322079731_set_a @ M2 @ F2 )
=> ( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ( member_set_a @ X4 @ M2 )
=> ( ( member_set_a @ Y2 @ M2 )
=> ( ord_le3724670747650509150_set_a @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_851_increasingD,axiom,
! [M2: set_set_a,F2: set_a > real,X4: set_a,Y2: set_a] :
( ( measur1776380161843274167a_real @ M2 @ F2 )
=> ( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ( member_set_a @ X4 @ M2 )
=> ( ( member_set_a @ Y2 @ M2 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_852_increasingD,axiom,
! [M2: set_set_a,F2: set_a > set_a,X4: set_a,Y2: set_a] :
( ( measur7842569353079325843_set_a @ M2 @ F2 )
=> ( ( ord_less_eq_set_a @ X4 @ Y2 )
=> ( ( member_set_a @ X4 @ M2 )
=> ( ( member_set_a @ Y2 @ M2 )
=> ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_853_uniform__on__def,axiom,
! [X2: a > nat,A2: set_nat] :
( ( prob_u5224086380619908136_a_nat @ m @ X2 @ A2 )
= ( ( ( measure_distr_a_nat @ m @ ( sigma_7685844798829912695ce_nat @ top_top_set_nat ) @ X2 )
= ( nonneg5218579358776314137re_nat @ ( sigma_7685844798829912695ce_nat @ top_top_set_nat ) @ A2 ) )
& ( A2 != bot_bot_set_nat )
& ( finite_finite_nat @ A2 )
& ( member_a_nat @ X2 @ ( sigma_73150082625557118_a_nat @ m @ ( sigma_7685844798829912695ce_nat @ top_top_set_nat ) ) ) ) ) ).
% uniform_on_def
thf(fact_854_uniform__on__def,axiom,
! [X2: a > literal,A2: set_literal] :
( ( prob_u1119184700330386812iteral @ m @ X2 @ A2 )
= ( ( ( measur2089735993314667086iteral @ m @ ( sigma_7122031993496068461iteral @ top_top_set_literal ) @ X2 )
= ( nonneg4334240857813153355iteral @ ( sigma_7122031993496068461iteral @ top_top_set_literal ) @ A2 ) )
& ( A2 != bot_bot_set_literal )
& ( finite5847741373460823677iteral @ A2 )
& ( member_a_literal @ X2 @ ( sigma_6403365867683794342iteral @ m @ ( sigma_7122031993496068461iteral @ top_top_set_literal ) ) ) ) ) ).
% uniform_on_def
thf(fact_855_uniform__on__def,axiom,
! [X2: a > product_unit,A2: set_Product_unit] :
( ( prob_u3984251234878355381t_unit @ m @ X2 @ A2 )
= ( ( ( measur2774250482188116359t_unit @ m @ ( sigma_2593334793047783974t_unit @ top_to1996260823553986621t_unit ) @ X2 )
= ( nonneg8606312109825204740t_unit @ ( sigma_2593334793047783974t_unit @ top_to1996260823553986621t_unit ) @ A2 ) )
& ( A2 != bot_bo3957492148770167129t_unit )
& ( finite4290736615968046902t_unit @ A2 )
& ( member3545396013249599883t_unit @ X2 @ ( sigma_1804883157691777247t_unit @ m @ ( sigma_2593334793047783974t_unit @ top_to1996260823553986621t_unit ) ) ) ) ) ).
% uniform_on_def
thf(fact_856_uniform__on__def,axiom,
! [X2: a > real,A2: set_real] :
( ( prob_u8923871485046717188a_real @ m @ X2 @ A2 )
= ( ( ( measure_distr_a_real @ m @ ( sigma_8508918144308765139e_real @ top_top_set_real ) @ X2 )
= ( nonneg3404582084018902901e_real @ ( sigma_8508918144308765139e_real @ top_top_set_real ) @ A2 ) )
& ( A2 != bot_bot_set_real )
& ( finite_finite_real @ A2 )
& ( member_a_real @ X2 @ ( sigma_9116425665531756122a_real @ m @ ( sigma_8508918144308765139e_real @ top_top_set_real ) ) ) ) ) ).
% uniform_on_def
thf(fact_857_uniform__on__def,axiom,
! [X2: a > $o,A2: set_o] :
( ( prob_uniform_on_a_o @ m @ X2 @ A2 )
= ( ( ( measure_distr_a_o @ m @ ( sigma_count_space_o @ top_top_set_o ) @ X2 )
= ( nonneg5638544851443855887sure_o @ ( sigma_count_space_o @ top_top_set_o ) @ A2 ) )
& ( A2 != bot_bot_set_o )
& ( finite_finite_o @ A2 )
& ( member_a_o @ X2 @ ( sigma_measurable_a_o @ m @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ) ).
% uniform_on_def
thf(fact_858_uniform__on__def,axiom,
! [X2: a > a,A2: set_a] :
( ( prob_uniform_on_a_a @ m @ X2 @ A2 )
= ( ( ( measure_distr_a_a @ m @ ( sigma_count_space_a @ top_top_set_a ) @ X2 )
= ( nonneg6757527617543859701sure_a @ ( sigma_count_space_a @ top_top_set_a ) @ A2 ) )
& ( A2 != bot_bot_set_a )
& ( finite_finite_a @ A2 )
& ( member_a_a @ X2 @ ( sigma_measurable_a_a @ m @ ( sigma_count_space_a @ top_top_set_a ) ) ) ) ) ).
% uniform_on_def
thf(fact_859_finite__emeasure__space,axiom,
( ( sigma_emeasure_a @ m @ ( sigma_space_a @ m ) )
!= top_to1496364449551166952nnreal ) ).
% finite_emeasure_space
thf(fact_860_emeasure__subprob__space__less__top,axiom,
! [A2: set_a] :
( ( sigma_emeasure_a @ m @ A2 )
!= top_to1496364449551166952nnreal ) ).
% emeasure_subprob_space_less_top
thf(fact_861_emeasure__space,axiom,
! [M2: sigma_measure_a,A2: set_a] : ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ M2 @ A2 ) @ ( sigma_emeasure_a @ M2 @ ( sigma_space_a @ M2 ) ) ) ).
% emeasure_space
thf(fact_862_filter__leD,axiom,
! [F: filter_a,F3: filter_a,P: a > $o] :
( ( ord_less_eq_filter_a @ F @ F3 )
=> ( ( eventually_a @ P @ F3 )
=> ( eventually_a @ P @ F ) ) ) ).
% filter_leD
thf(fact_863_filter__leI,axiom,
! [F3: filter_a,F: filter_a] :
( ! [P3: a > $o] :
( ( eventually_a @ P3 @ F3 )
=> ( eventually_a @ P3 @ F ) )
=> ( ord_less_eq_filter_a @ F @ F3 ) ) ).
% filter_leI
thf(fact_864_le__filter__def,axiom,
( ord_less_eq_filter_a
= ( ^ [F4: filter_a,F5: filter_a] :
! [P4: a > $o] :
( ( eventually_a @ P4 @ F5 )
=> ( eventually_a @ P4 @ F4 ) ) ) ) ).
% le_filter_def
thf(fact_865_finite__measure_Oemeasure__finite,axiom,
! [M2: sigma_measure_a,A2: set_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ( sigma_emeasure_a @ M2 @ A2 )
!= top_to1496364449551166952nnreal ) ) ).
% finite_measure.emeasure_finite
thf(fact_866_subprob__space_Oemeasure__subprob__space__less__top,axiom,
! [M2: sigma_measure_a,A2: set_a] :
( ( giry_subprob_space_a @ M2 )
=> ( ( sigma_emeasure_a @ M2 @ A2 )
!= top_to1496364449551166952nnreal ) ) ).
% subprob_space.emeasure_subprob_space_less_top
thf(fact_867_less__eq__set__def,axiom,
( ord_less_eq_set_c_d
= ( ^ [A3: set_c_d,B2: set_c_d] :
( ord_less_eq_c_d_o
@ ^ [X: c > d] : ( member_c_d @ X @ A3 )
@ ^ [X: c > d] : ( member_c_d @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_868_less__eq__set__def,axiom,
( ord_le3334967407727675675a_real
= ( ^ [A3: set_a_real,B2: set_a_real] :
( ord_less_eq_a_real_o
@ ^ [X: a > real] : ( member_a_real @ X @ A3 )
@ ^ [X: a > real] : ( member_a_real @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_869_less__eq__set__def,axiom,
( ord_less_eq_set_a_o2
= ( ^ [A3: set_a_o,B2: set_a_o] :
( ord_less_eq_a_o_o
@ ^ [X: a > $o] : ( member_a_o @ X @ A3 )
@ ^ [X: a > $o] : ( member_a_o @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_870_less__eq__set__def,axiom,
( ord_less_eq_set_b
= ( ^ [A3: set_b,B2: set_b] :
( ord_less_eq_b_o
@ ^ [X: b] : ( member_b @ X @ A3 )
@ ^ [X: b] : ( member_b @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_871_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A3 )
@ ^ [X: nat] : ( member_nat @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_872_less__eq__set__def,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B2: set_o] :
( ord_less_eq_o_o
@ ^ [X: $o] : ( member_o @ X @ A3 )
@ ^ [X: $o] : ( member_o @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_873_less__eq__set__def,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B2: set_set_a] :
( ord_less_eq_set_a_o
@ ^ [X: set_a] : ( member_set_a @ X @ A3 )
@ ^ [X: set_a] : ( member_set_a @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_874_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B2: set_a] :
( ord_less_eq_a_o
@ ^ [X: a] : ( member_a @ X @ A3 )
@ ^ [X: a] : ( member_a @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_875_le__measureD1,axiom,
! [A2: sigma_measure_set_a,B: sigma_measure_set_a] :
( ( ord_le5642585610961328955_set_a @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ ( sigma_space_set_a @ A2 ) @ ( sigma_space_set_a @ B ) ) ) ).
% le_measureD1
thf(fact_876_le__measureD1,axiom,
! [A2: sigma_measure_a,B: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A2 @ B )
=> ( ord_less_eq_set_a @ ( sigma_space_a @ A2 ) @ ( sigma_space_a @ B ) ) ) ).
% le_measureD1
thf(fact_877_finite__measure_Ofinite__emeasure__space,axiom,
! [M2: sigma_measure_a] :
( ( measur930452917991658466sure_a @ M2 )
=> ( ( sigma_emeasure_a @ M2 @ ( sigma_space_a @ M2 ) )
!= top_to1496364449551166952nnreal ) ) ).
% finite_measure.finite_emeasure_space
thf(fact_878_integrable__count__space,axiom,
! [X2: set_a,F2: a > real] :
( ( finite_finite_a @ X2 )
=> ( bochne2139062162225249880a_real @ ( sigma_count_space_a @ X2 ) @ F2 ) ) ).
% integrable_count_space
thf(fact_879_sigma__finite__measure__count__space__finite,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( measur8258956421386577775re_nat @ ( sigma_7685844798829912695ce_nat @ A2 ) ) ) ).
% sigma_finite_measure_count_space_finite
thf(fact_880_sigma__finite__measure__count__space__finite,axiom,
! [A2: set_o] :
( ( finite_finite_o @ A2 )
=> ( measur1827666076404920889sure_o @ ( sigma_count_space_o @ A2 ) ) ) ).
% sigma_finite_measure_count_space_finite
thf(fact_881_sigma__finite__measure__count__space__finite,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( measur4308613598931908895sure_a @ ( sigma_count_space_a @ A2 ) ) ) ).
% sigma_finite_measure_count_space_finite
thf(fact_882_finite__measure__count__space,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( measur8338831127414845932re_nat @ ( sigma_7685844798829912695ce_nat @ A2 ) ) ) ).
% finite_measure_count_space
thf(fact_883_finite__measure__count__space,axiom,
! [A2: set_o] :
( ( finite_finite_o @ A2 )
=> ( measur2447921437955784316sure_o @ ( sigma_count_space_o @ A2 ) ) ) ).
% finite_measure_count_space
thf(fact_884_finite__measure__count__space,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( measur930452917991658466sure_a @ ( sigma_count_space_a @ A2 ) ) ) ).
% finite_measure_count_space
thf(fact_885_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_886_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_887_always__eventually,axiom,
! [P: a > $o,F: filter_a] :
( ! [X_1: a] : ( P @ X_1 )
=> ( eventually_a @ P @ F ) ) ).
% always_eventually
thf(fact_888_not__eventuallyD,axiom,
! [P: a > $o,F: filter_a] :
( ~ ( eventually_a @ P @ F )
=> ? [X3: a] :
~ ( P @ X3 ) ) ).
% not_eventuallyD
thf(fact_889_eventually__mono,axiom,
! [P: a > $o,F: filter_a,Q: a > $o] :
( ( eventually_a @ P @ F )
=> ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( eventually_a @ Q @ F ) ) ) ).
% eventually_mono
thf(fact_890_filter__eq__iff,axiom,
( ( ^ [Y5: filter_a,Z: filter_a] : ( Y5 = Z ) )
= ( ^ [F4: filter_a,F5: filter_a] :
! [P4: a > $o] :
( ( eventually_a @ P4 @ F4 )
= ( eventually_a @ P4 @ F5 ) ) ) ) ).
% filter_eq_iff
thf(fact_891_eventuallyI,axiom,
! [P: a > $o,F: filter_a] :
( ! [X_1: a] : ( P @ X_1 )
=> ( eventually_a @ P @ F ) ) ).
% eventuallyI
thf(fact_892_eventually__happens_H,axiom,
! [F: filter_a,P: a > $o] :
( ( F != bot_bot_filter_a )
=> ( ( eventually_a @ P @ F )
=> ? [X_1: a] : ( P @ X_1 ) ) ) ).
% eventually_happens'
thf(fact_893_eventually__happens,axiom,
! [P: a > $o,Net: filter_a] :
( ( eventually_a @ P @ Net )
=> ( ( Net = bot_bot_filter_a )
| ? [X_1: a] : ( P @ X_1 ) ) ) ).
% eventually_happens
thf(fact_894_eventually__bot,axiom,
! [P: a > $o] : ( eventually_a @ P @ bot_bot_filter_a ) ).
% eventually_bot
thf(fact_895_prob__space_Ouniform__on__def,axiom,
! [M2: sigma_measure_c,X2: c > d,A2: set_d] :
( ( probab7247484486040049091pace_c @ M2 )
=> ( ( prob_uniform_on_c_d @ M2 @ X2 @ A2 )
= ( ( ( measure_distr_c_d @ M2 @ ( sigma_count_space_d @ top_top_set_d ) @ X2 )
= ( nonneg6757527617543859704sure_d @ ( sigma_count_space_d @ top_top_set_d ) @ A2 ) )
& ( A2 != bot_bot_set_d )
& ( finite_finite_d @ A2 )
& ( member_c_d @ X2 @ ( sigma_measurable_c_d @ M2 @ ( sigma_count_space_d @ top_top_set_d ) ) ) ) ) ) ).
% prob_space.uniform_on_def
thf(fact_896_prob__space_Ouniform__on__def,axiom,
! [M2: sigma_measure_a,X2: a > nat,A2: set_nat] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( prob_u5224086380619908136_a_nat @ M2 @ X2 @ A2 )
= ( ( ( measure_distr_a_nat @ M2 @ ( sigma_7685844798829912695ce_nat @ top_top_set_nat ) @ X2 )
= ( nonneg5218579358776314137re_nat @ ( sigma_7685844798829912695ce_nat @ top_top_set_nat ) @ A2 ) )
& ( A2 != bot_bot_set_nat )
& ( finite_finite_nat @ A2 )
& ( member_a_nat @ X2 @ ( sigma_73150082625557118_a_nat @ M2 @ ( sigma_7685844798829912695ce_nat @ top_top_set_nat ) ) ) ) ) ) ).
% prob_space.uniform_on_def
thf(fact_897_prob__space_Ouniform__on__def,axiom,
! [M2: sigma_measure_a,X2: a > literal,A2: set_literal] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( prob_u1119184700330386812iteral @ M2 @ X2 @ A2 )
= ( ( ( measur2089735993314667086iteral @ M2 @ ( sigma_7122031993496068461iteral @ top_top_set_literal ) @ X2 )
= ( nonneg4334240857813153355iteral @ ( sigma_7122031993496068461iteral @ top_top_set_literal ) @ A2 ) )
& ( A2 != bot_bot_set_literal )
& ( finite5847741373460823677iteral @ A2 )
& ( member_a_literal @ X2 @ ( sigma_6403365867683794342iteral @ M2 @ ( sigma_7122031993496068461iteral @ top_top_set_literal ) ) ) ) ) ) ).
% prob_space.uniform_on_def
thf(fact_898_prob__space_Ouniform__on__def,axiom,
! [M2: sigma_measure_a,X2: a > product_unit,A2: set_Product_unit] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( prob_u3984251234878355381t_unit @ M2 @ X2 @ A2 )
= ( ( ( measur2774250482188116359t_unit @ M2 @ ( sigma_2593334793047783974t_unit @ top_to1996260823553986621t_unit ) @ X2 )
= ( nonneg8606312109825204740t_unit @ ( sigma_2593334793047783974t_unit @ top_to1996260823553986621t_unit ) @ A2 ) )
& ( A2 != bot_bo3957492148770167129t_unit )
& ( finite4290736615968046902t_unit @ A2 )
& ( member3545396013249599883t_unit @ X2 @ ( sigma_1804883157691777247t_unit @ M2 @ ( sigma_2593334793047783974t_unit @ top_to1996260823553986621t_unit ) ) ) ) ) ) ).
% prob_space.uniform_on_def
thf(fact_899_prob__space_Ouniform__on__def,axiom,
! [M2: sigma_measure_a,X2: a > real,A2: set_real] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( prob_u8923871485046717188a_real @ M2 @ X2 @ A2 )
= ( ( ( measure_distr_a_real @ M2 @ ( sigma_8508918144308765139e_real @ top_top_set_real ) @ X2 )
= ( nonneg3404582084018902901e_real @ ( sigma_8508918144308765139e_real @ top_top_set_real ) @ A2 ) )
& ( A2 != bot_bot_set_real )
& ( finite_finite_real @ A2 )
& ( member_a_real @ X2 @ ( sigma_9116425665531756122a_real @ M2 @ ( sigma_8508918144308765139e_real @ top_top_set_real ) ) ) ) ) ) ).
% prob_space.uniform_on_def
thf(fact_900_prob__space_Ouniform__on__def,axiom,
! [M2: sigma_measure_a,X2: a > $o,A2: set_o] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( prob_uniform_on_a_o @ M2 @ X2 @ A2 )
= ( ( ( measure_distr_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) @ X2 )
= ( nonneg5638544851443855887sure_o @ ( sigma_count_space_o @ top_top_set_o ) @ A2 ) )
& ( A2 != bot_bot_set_o )
& ( finite_finite_o @ A2 )
& ( member_a_o @ X2 @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) ) ) ) ) ).
% prob_space.uniform_on_def
thf(fact_901_prob__space_Ouniform__on__def,axiom,
! [M2: sigma_measure_a,X2: a > a,A2: set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( prob_uniform_on_a_a @ M2 @ X2 @ A2 )
= ( ( ( measure_distr_a_a @ M2 @ ( sigma_count_space_a @ top_top_set_a ) @ X2 )
= ( nonneg6757527617543859701sure_a @ ( sigma_count_space_a @ top_top_set_a ) @ A2 ) )
& ( A2 != bot_bot_set_a )
& ( finite_finite_a @ A2 )
& ( member_a_a @ X2 @ ( sigma_measurable_a_a @ M2 @ ( sigma_count_space_a @ top_top_set_a ) ) ) ) ) ) ).
% prob_space.uniform_on_def
thf(fact_902_prob__space_Oindep__sets__finite__index__sets,axiom,
! [M2: sigma_measure_a,F: set_a > set_set_a,I2: set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe4967106450811773644_set_a @ M2 @ F @ I2 )
= ( ! [J2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ J2 @ I2 )
=> ( ( J2 != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ J2 )
=> ( indepe4967106450811773644_set_a @ M2 @ F @ J2 ) ) ) ) ) ) ) ).
% prob_space.indep_sets_finite_index_sets
thf(fact_903_prob__space_Oindep__sets__finite__index__sets,axiom,
! [M2: sigma_measure_a,F: $o > set_set_a,I2: set_o] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe7780107833195774214ts_a_o @ M2 @ F @ I2 )
= ( ! [J2: set_o] :
( ( ord_less_eq_set_o @ J2 @ I2 )
=> ( ( J2 != bot_bot_set_o )
=> ( ( finite_finite_o @ J2 )
=> ( indepe7780107833195774214ts_a_o @ M2 @ F @ J2 ) ) ) ) ) ) ) ).
% prob_space.indep_sets_finite_index_sets
thf(fact_904_prob__space_Oindep__sets__finite__index__sets,axiom,
! [M2: sigma_measure_a,F: nat > set_set_a,I2: set_nat] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe6267730027088848354_a_nat @ M2 @ F @ I2 )
= ( ! [J2: set_nat] :
( ( ord_less_eq_set_nat @ J2 @ I2 )
=> ( ( J2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ J2 )
=> ( indepe6267730027088848354_a_nat @ M2 @ F @ J2 ) ) ) ) ) ) ) ).
% prob_space.indep_sets_finite_index_sets
thf(fact_905_prob__space_Oindep__sets__finite__index__sets,axiom,
! [M2: sigma_measure_a,F: a > set_set_a,I2: set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe8927441866673418604ts_a_a @ M2 @ F @ I2 )
= ( ! [J2: set_a] :
( ( ord_less_eq_set_a @ J2 @ I2 )
=> ( ( J2 != bot_bot_set_a )
=> ( ( finite_finite_a @ J2 )
=> ( indepe8927441866673418604ts_a_a @ M2 @ F @ J2 ) ) ) ) ) ) ) ).
% prob_space.indep_sets_finite_index_sets
thf(fact_906_prob__space_Oindep__sets__finite__index__sets,axiom,
! [M2: sigma_measure_a,F: b > set_set_a,I2: set_b] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe8927441866673418605ts_a_b @ M2 @ F @ I2 )
= ( ! [J2: set_b] :
( ( ord_less_eq_set_b @ J2 @ I2 )
=> ( ( J2 != bot_bot_set_b )
=> ( ( finite_finite_b @ J2 )
=> ( indepe8927441866673418605ts_a_b @ M2 @ F @ J2 ) ) ) ) ) ) ) ).
% prob_space.indep_sets_finite_index_sets
thf(fact_907_prob__space_Oindep__sets__finite__index__sets,axiom,
! [M2: sigma_measure_a,F: ( a > $o ) > set_set_a,I2: set_a_o] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe7801696130336798481_a_a_o @ M2 @ F @ I2 )
= ( ! [J2: set_a_o] :
( ( ord_less_eq_set_a_o2 @ J2 @ I2 )
=> ( ( J2 != bot_bot_set_a_o )
=> ( ( finite_finite_a_o @ J2 )
=> ( indepe7801696130336798481_a_a_o @ M2 @ F @ J2 ) ) ) ) ) ) ) ).
% prob_space.indep_sets_finite_index_sets
thf(fact_908_prob__space_Oindep__sets__finite__index__sets,axiom,
! [M2: sigma_measure_a,F: ( a > real ) > set_set_a,I2: set_a_real] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe4749599203615801097a_real @ M2 @ F @ I2 )
= ( ! [J2: set_a_real] :
( ( ord_le3334967407727675675a_real @ J2 @ I2 )
=> ( ( J2 != bot_bot_set_a_real )
=> ( ( finite_finite_a_real @ J2 )
=> ( indepe4749599203615801097a_real @ M2 @ F @ J2 ) ) ) ) ) ) ) ).
% prob_space.indep_sets_finite_index_sets
thf(fact_909_prob__space_Oindep__sets__finite__index__sets,axiom,
! [M2: sigma_measure_a,F: ( c > d ) > set_set_a,I2: set_c_d] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe4867465796832040824_a_c_d @ M2 @ F @ I2 )
= ( ! [J2: set_c_d] :
( ( ord_less_eq_set_c_d @ J2 @ I2 )
=> ( ( J2 != bot_bot_set_c_d )
=> ( ( finite_finite_c_d @ J2 )
=> ( indepe4867465796832040824_a_c_d @ M2 @ F @ J2 ) ) ) ) ) ) ) ).
% prob_space.indep_sets_finite_index_sets
thf(fact_910_not__eventually__impI,axiom,
! [P: a > $o,F: filter_a,Q: a > $o] :
( ( eventually_a @ P @ F )
=> ( ~ ( eventually_a @ Q @ F )
=> ~ ( eventually_a
@ ^ [X: a] :
( ( P @ X )
=> ( Q @ X ) )
@ F ) ) ) ).
% not_eventually_impI
thf(fact_911_eventually__conj__iff,axiom,
! [P: a > $o,Q: a > $o,F: filter_a] :
( ( eventually_a
@ ^ [X: a] :
( ( P @ X )
& ( Q @ X ) )
@ F )
= ( ( eventually_a @ P @ F )
& ( eventually_a @ Q @ F ) ) ) ).
% eventually_conj_iff
thf(fact_912_eventually__rev__mp,axiom,
! [P: a > $o,F: filter_a,Q: a > $o] :
( ( eventually_a @ P @ F )
=> ( ( eventually_a
@ ^ [X: a] :
( ( P @ X )
=> ( Q @ X ) )
@ F )
=> ( eventually_a @ Q @ F ) ) ) ).
% eventually_rev_mp
thf(fact_913_eventually__subst,axiom,
! [P: a > $o,Q: a > $o,F: filter_a] :
( ( eventually_a
@ ^ [N3: a] :
( ( P @ N3 )
= ( Q @ N3 ) )
@ F )
=> ( ( eventually_a @ P @ F )
= ( eventually_a @ Q @ F ) ) ) ).
% eventually_subst
thf(fact_914_eventually__elim2,axiom,
! [P: a > $o,F: filter_a,Q: a > $o,R: a > $o] :
( ( eventually_a @ P @ F )
=> ( ( eventually_a @ Q @ F )
=> ( ! [I3: a] :
( ( P @ I3 )
=> ( ( Q @ I3 )
=> ( R @ I3 ) ) )
=> ( eventually_a @ R @ F ) ) ) ) ).
% eventually_elim2
thf(fact_915_eventually__conj,axiom,
! [P: a > $o,F: filter_a,Q: a > $o] :
( ( eventually_a @ P @ F )
=> ( ( eventually_a @ Q @ F )
=> ( eventually_a
@ ^ [X: a] :
( ( P @ X )
& ( Q @ X ) )
@ F ) ) ) ).
% eventually_conj
thf(fact_916_eventually__True,axiom,
! [F: filter_a] :
( eventually_a
@ ^ [X: a] : $true
@ F ) ).
% eventually_True
thf(fact_917_eventually__mp,axiom,
! [P: a > $o,Q: a > $o,F: filter_a] :
( ( eventually_a
@ ^ [X: a] :
( ( P @ X )
=> ( Q @ X ) )
@ F )
=> ( ( eventually_a @ P @ F )
=> ( eventually_a @ Q @ F ) ) ) ).
% eventually_mp
thf(fact_918_eventually__frequently__const__simps_I3_J,axiom,
! [P: a > $o,C2: $o,F: filter_a] :
( ( eventually_a
@ ^ [X: a] :
( ( P @ X )
| C2 )
@ F )
= ( ( eventually_a @ P @ F )
| C2 ) ) ).
% eventually_frequently_const_simps(3)
thf(fact_919_eventually__frequently__const__simps_I4_J,axiom,
! [C2: $o,P: a > $o,F: filter_a] :
( ( eventually_a
@ ^ [X: a] :
( C2
| ( P @ X ) )
@ F )
= ( C2
| ( eventually_a @ P @ F ) ) ) ).
% eventually_frequently_const_simps(4)
thf(fact_920_eventually__frequently__const__simps_I6_J,axiom,
! [C2: $o,P: a > $o,F: filter_a] :
( ( eventually_a
@ ^ [X: a] :
( C2
=> ( P @ X ) )
@ F )
= ( C2
=> ( eventually_a @ P @ F ) ) ) ).
% eventually_frequently_const_simps(6)
thf(fact_921_False__imp__not__eventually,axiom,
! [P: a > $o,Net: filter_a] :
( ! [X3: a] :
~ ( P @ X3 )
=> ( ( Net != bot_bot_filter_a )
=> ~ ( eventually_a @ P @ Net ) ) ) ).
% False_imp_not_eventually
thf(fact_922_eventually__const__iff,axiom,
! [P: $o,F: filter_a] :
( ( eventually_a
@ ^ [X: a] : P
@ F )
= ( P
| ( F = bot_bot_filter_a ) ) ) ).
% eventually_const_iff
thf(fact_923_trivial__limit__def,axiom,
! [F: filter_a] :
( ( F = bot_bot_filter_a )
= ( eventually_a
@ ^ [X: a] : $false
@ F ) ) ).
% trivial_limit_def
thf(fact_924_emeasure__Collect__eq__AE,axiom,
! [P: ( c > d ) > $o,Q: ( c > d ) > $o,M2: sigma_measure_c_d] :
( ( eventually_c_d
@ ^ [X: c > d] :
( ( P @ X )
= ( Q @ X ) )
@ ( measur253975988404078935er_c_d @ M2 ) )
=> ( ( member_c_d_o @ Q @ ( sigma_1714064210060623456_c_d_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( member_c_d_o @ P @ ( sigma_1714064210060623456_c_d_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( sigma_emeasure_c_d @ M2
@ ( collect_c_d
@ ^ [X: c > d] :
( ( member_c_d @ X @ ( sigma_space_c_d @ M2 ) )
& ( P @ X ) ) ) )
= ( sigma_emeasure_c_d @ M2
@ ( collect_c_d
@ ^ [X: c > d] :
( ( member_c_d @ X @ ( sigma_space_c_d @ M2 ) )
& ( Q @ X ) ) ) ) ) ) ) ) ).
% emeasure_Collect_eq_AE
thf(fact_925_emeasure__Collect__eq__AE,axiom,
! [P: ( a > real ) > $o,Q: ( a > real ) > $o,M2: sigma_measure_a_real] :
( ( eventually_a_real
@ ^ [X: a > real] :
( ( P @ X )
= ( Q @ X ) )
@ ( measur8946935934331175658a_real @ M2 ) )
=> ( ( member_a_real_o @ Q @ ( sigma_9085598459323199629real_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( member_a_real_o @ P @ ( sigma_9085598459323199629real_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( sigma_5985106571655482610a_real @ M2
@ ( collect_a_real
@ ^ [X: a > real] :
( ( member_a_real @ X @ ( sigma_space_a_real @ M2 ) )
& ( P @ X ) ) ) )
= ( sigma_5985106571655482610a_real @ M2
@ ( collect_a_real
@ ^ [X: a > real] :
( ( member_a_real @ X @ ( sigma_space_a_real @ M2 ) )
& ( Q @ X ) ) ) ) ) ) ) ) ).
% emeasure_Collect_eq_AE
thf(fact_926_emeasure__Collect__eq__AE,axiom,
! [P: ( a > $o ) > $o,Q: ( a > $o ) > $o,M2: sigma_measure_a_o] :
( ( eventually_a_o
@ ^ [X: a > $o] :
( ( P @ X )
= ( Q @ X ) )
@ ( measur1950989848205109360er_a_o @ M2 ) )
=> ( ( member_a_o_o @ Q @ ( sigma_1195952539894209287_a_o_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( member_a_o_o @ P @ ( sigma_1195952539894209287_a_o_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( sigma_emeasure_a_o @ M2
@ ( collect_a_o
@ ^ [X: a > $o] :
( ( member_a_o @ X @ ( sigma_space_a_o @ M2 ) )
& ( P @ X ) ) ) )
= ( sigma_emeasure_a_o @ M2
@ ( collect_a_o
@ ^ [X: a > $o] :
( ( member_a_o @ X @ ( sigma_space_a_o @ M2 ) )
& ( Q @ X ) ) ) ) ) ) ) ) ).
% emeasure_Collect_eq_AE
thf(fact_927_emeasure__Collect__eq__AE,axiom,
! [P: b > $o,Q: b > $o,M2: sigma_measure_b] :
( ( eventually_b
@ ^ [X: b] :
( ( P @ X )
= ( Q @ X ) )
@ ( measure_ae_filter_b @ M2 ) )
=> ( ( member_b_o @ Q @ ( sigma_measurable_b_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( member_b_o @ P @ ( sigma_measurable_b_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( sigma_emeasure_b @ M2
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ ( sigma_space_b @ M2 ) )
& ( P @ X ) ) ) )
= ( sigma_emeasure_b @ M2
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ ( sigma_space_b @ M2 ) )
& ( Q @ X ) ) ) ) ) ) ) ) ).
% emeasure_Collect_eq_AE
thf(fact_928_emeasure__Collect__eq__AE,axiom,
! [P: $o > $o,Q: $o > $o,M2: sigma_measure_o] :
( ( eventually_o
@ ^ [X: $o] :
( ( P @ X )
= ( Q @ X ) )
@ ( measure_ae_filter_o @ M2 ) )
=> ( ( member_o_o @ Q @ ( sigma_measurable_o_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( member_o_o @ P @ ( sigma_measurable_o_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( sigma_emeasure_o @ M2
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( sigma_space_o @ M2 ) )
& ( P @ X ) ) ) )
= ( sigma_emeasure_o @ M2
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( sigma_space_o @ M2 ) )
& ( Q @ X ) ) ) ) ) ) ) ) ).
% emeasure_Collect_eq_AE
thf(fact_929_emeasure__Collect__eq__AE,axiom,
! [P: nat > $o,Q: nat > $o,M2: sigma_measure_nat] :
( ( eventually_nat
@ ^ [X: nat] :
( ( P @ X )
= ( Q @ X ) )
@ ( measur6539087422748349889er_nat @ M2 ) )
=> ( ( member_nat_o @ Q @ ( sigma_5101835498682829686_nat_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( member_nat_o @ P @ ( sigma_5101835498682829686_nat_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( sigma_emeasure_nat @ M2
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( sigma_space_nat @ M2 ) )
& ( P @ X ) ) ) )
= ( sigma_emeasure_nat @ M2
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( sigma_space_nat @ M2 ) )
& ( Q @ X ) ) ) ) ) ) ) ) ).
% emeasure_Collect_eq_AE
thf(fact_930_emeasure__Collect__eq__AE,axiom,
! [P: a > $o,Q: a > $o,M2: sigma_measure_a] :
( ( eventually_a
@ ^ [X: a] :
( ( P @ X )
= ( Q @ X ) )
@ ( measure_ae_filter_a @ M2 ) )
=> ( ( member_a_o @ Q @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( member_a_o @ P @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( sigma_emeasure_a @ M2
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ M2 ) )
& ( P @ X ) ) ) )
= ( sigma_emeasure_a @ M2
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ M2 ) )
& ( Q @ X ) ) ) ) ) ) ) ) ).
% emeasure_Collect_eq_AE
thf(fact_931_emeasure__Collect__distr,axiom,
! [X2: b > b,M2: sigma_measure_b,N: sigma_measure_b,P: b > $o] :
( ( member_b_b @ X2 @ ( sigma_measurable_b_b @ M2 @ N ) )
=> ( ( member_b_o @ P @ ( sigma_measurable_b_o @ N @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( sigma_emeasure_b @ ( measure_distr_b_b @ M2 @ N @ X2 )
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ ( sigma_space_b @ N ) )
& ( P @ X ) ) ) )
= ( sigma_emeasure_b @ M2
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ ( sigma_space_b @ M2 ) )
& ( P @ ( X2 @ X ) ) ) ) ) ) ) ) ).
% emeasure_Collect_distr
thf(fact_932_emeasure__Collect__distr,axiom,
! [X2: $o > b,M2: sigma_measure_o,N: sigma_measure_b,P: b > $o] :
( ( member_o_b @ X2 @ ( sigma_measurable_o_b @ M2 @ N ) )
=> ( ( member_b_o @ P @ ( sigma_measurable_b_o @ N @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( sigma_emeasure_b @ ( measure_distr_o_b @ M2 @ N @ X2 )
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ ( sigma_space_b @ N ) )
& ( P @ X ) ) ) )
= ( sigma_emeasure_o @ M2
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( sigma_space_o @ M2 ) )
& ( P @ ( X2 @ X ) ) ) ) ) ) ) ) ).
% emeasure_Collect_distr
thf(fact_933_emeasure__Collect__distr,axiom,
! [X2: b > $o,M2: sigma_measure_b,N: sigma_measure_o,P: $o > $o] :
( ( member_b_o @ X2 @ ( sigma_measurable_b_o @ M2 @ N ) )
=> ( ( member_o_o @ P @ ( sigma_measurable_o_o @ N @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( sigma_emeasure_o @ ( measure_distr_b_o @ M2 @ N @ X2 )
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( sigma_space_o @ N ) )
& ( P @ X ) ) ) )
= ( sigma_emeasure_b @ M2
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ ( sigma_space_b @ M2 ) )
& ( P @ ( X2 @ X ) ) ) ) ) ) ) ) ).
% emeasure_Collect_distr
thf(fact_934_emeasure__Collect__distr,axiom,
! [X2: $o > $o,M2: sigma_measure_o,N: sigma_measure_o,P: $o > $o] :
( ( member_o_o @ X2 @ ( sigma_measurable_o_o @ M2 @ N ) )
=> ( ( member_o_o @ P @ ( sigma_measurable_o_o @ N @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( sigma_emeasure_o @ ( measure_distr_o_o @ M2 @ N @ X2 )
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( sigma_space_o @ N ) )
& ( P @ X ) ) ) )
= ( sigma_emeasure_o @ M2
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( sigma_space_o @ M2 ) )
& ( P @ ( X2 @ X ) ) ) ) ) ) ) ) ).
% emeasure_Collect_distr
thf(fact_935_emeasure__Collect__distr,axiom,
! [X2: nat > b,M2: sigma_measure_nat,N: sigma_measure_b,P: b > $o] :
( ( member_nat_b @ X2 @ ( sigma_4105081583803843549_nat_b @ M2 @ N ) )
=> ( ( member_b_o @ P @ ( sigma_measurable_b_o @ N @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( sigma_emeasure_b @ ( measure_distr_nat_b @ M2 @ N @ X2 )
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ ( sigma_space_b @ N ) )
& ( P @ X ) ) ) )
= ( sigma_emeasure_nat @ M2
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( sigma_space_nat @ M2 ) )
& ( P @ ( X2 @ X ) ) ) ) ) ) ) ) ).
% emeasure_Collect_distr
thf(fact_936_emeasure__Collect__distr,axiom,
! [X2: nat > $o,M2: sigma_measure_nat,N: sigma_measure_o,P: $o > $o] :
( ( member_nat_o @ X2 @ ( sigma_5101835498682829686_nat_o @ M2 @ N ) )
=> ( ( member_o_o @ P @ ( sigma_measurable_o_o @ N @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( sigma_emeasure_o @ ( measure_distr_nat_o @ M2 @ N @ X2 )
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( sigma_space_o @ N ) )
& ( P @ X ) ) ) )
= ( sigma_emeasure_nat @ M2
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( sigma_space_nat @ M2 ) )
& ( P @ ( X2 @ X ) ) ) ) ) ) ) ) ).
% emeasure_Collect_distr
thf(fact_937_emeasure__Collect__distr,axiom,
! [X2: b > nat,M2: sigma_measure_b,N: sigma_measure_nat,P: nat > $o] :
( ( member_b_nat @ X2 @ ( sigma_1308594411581951615_b_nat @ M2 @ N ) )
=> ( ( member_nat_o @ P @ ( sigma_5101835498682829686_nat_o @ N @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( sigma_emeasure_nat @ ( measure_distr_b_nat @ M2 @ N @ X2 )
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( sigma_space_nat @ N ) )
& ( P @ X ) ) ) )
= ( sigma_emeasure_b @ M2
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ ( sigma_space_b @ M2 ) )
& ( P @ ( X2 @ X ) ) ) ) ) ) ) ) ).
% emeasure_Collect_distr
thf(fact_938_emeasure__Collect__distr,axiom,
! [X2: $o > nat,M2: sigma_measure_o,N: sigma_measure_nat,P: nat > $o] :
( ( member_o_nat @ X2 @ ( sigma_1999164137574644376_o_nat @ M2 @ N ) )
=> ( ( member_nat_o @ P @ ( sigma_5101835498682829686_nat_o @ N @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( sigma_emeasure_nat @ ( measure_distr_o_nat @ M2 @ N @ X2 )
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( sigma_space_nat @ N ) )
& ( P @ X ) ) ) )
= ( sigma_emeasure_o @ M2
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( sigma_space_o @ M2 ) )
& ( P @ ( X2 @ X ) ) ) ) ) ) ) ) ).
% emeasure_Collect_distr
thf(fact_939_emeasure__Collect__distr,axiom,
! [X2: nat > nat,M2: sigma_measure_nat,N: sigma_measure_nat,P: nat > $o] :
( ( member_nat_nat @ X2 @ ( sigma_4350458207664084850at_nat @ M2 @ N ) )
=> ( ( member_nat_o @ P @ ( sigma_5101835498682829686_nat_o @ N @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( sigma_emeasure_nat @ ( measur36828333294957594at_nat @ M2 @ N @ X2 )
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( sigma_space_nat @ N ) )
& ( P @ X ) ) ) )
= ( sigma_emeasure_nat @ M2
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( sigma_space_nat @ M2 ) )
& ( P @ ( X2 @ X ) ) ) ) ) ) ) ) ).
% emeasure_Collect_distr
thf(fact_940_emeasure__Collect__distr,axiom,
! [X2: a > b,M2: sigma_measure_a,N: sigma_measure_b,P: b > $o] :
( ( member_a_b @ X2 @ ( sigma_measurable_a_b @ M2 @ N ) )
=> ( ( member_b_o @ P @ ( sigma_measurable_b_o @ N @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( sigma_emeasure_b @ ( measure_distr_a_b @ M2 @ N @ X2 )
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ ( sigma_space_b @ N ) )
& ( P @ X ) ) ) )
= ( sigma_emeasure_a @ M2
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ M2 ) )
& ( P @ ( X2 @ X ) ) ) ) ) ) ) ) ).
% emeasure_Collect_distr
thf(fact_941_indep__events__finite__index__events,axiom,
! [F: set_a > set_a,I2: set_set_a] :
( ( indepe451107616305015476_set_a @ m @ F @ I2 )
= ( ! [J2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ J2 @ I2 )
=> ( ( J2 != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ J2 )
=> ( indepe451107616305015476_set_a @ m @ F @ J2 ) ) ) ) ) ) ).
% indep_events_finite_index_events
thf(fact_942_indep__events__finite__index__events,axiom,
! [F: a > set_a,I2: set_a] :
( ( indepe1948846426091821396ts_a_a @ m @ F @ I2 )
= ( ! [J2: set_a] :
( ( ord_less_eq_set_a @ J2 @ I2 )
=> ( ( J2 != bot_bot_set_a )
=> ( ( finite_finite_a @ J2 )
=> ( indepe1948846426091821396ts_a_a @ m @ F @ J2 ) ) ) ) ) ) ).
% indep_events_finite_index_events
thf(fact_943_indep__events__finite__index__events,axiom,
! [F: nat > set_a,I2: set_nat] :
( ( indepe1551197314001032186_a_nat @ m @ F @ I2 )
= ( ! [J2: set_nat] :
( ( ord_less_eq_set_nat @ J2 @ I2 )
=> ( ( J2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ J2 )
=> ( indepe1551197314001032186_a_nat @ m @ F @ J2 ) ) ) ) ) ) ).
% indep_events_finite_index_events
thf(fact_944_borel__0__1__law__AE,axiom,
! [P: nat > a > $o] :
( ( indepe1551197314001032186_a_nat @ m
@ ^ [M4: nat] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ M4 @ X ) ) )
@ top_top_set_nat )
=> ( ( eventually_a
@ ^ [X: a] :
~ ( finite_finite_nat
@ ( collect_nat
@ ^ [M4: nat] : ( P @ M4 @ X ) ) )
@ ( measure_ae_filter_a @ m ) )
| ( eventually_a
@ ^ [X: a] :
( finite_finite_nat
@ ( collect_nat
@ ^ [M4: nat] : ( P @ M4 @ X ) ) )
@ ( measure_ae_filter_a @ m ) ) ) ) ).
% borel_0_1_law_AE
thf(fact_945_AE__iff__emeasure__eq__1,axiom,
! [P: a > $o] :
( ( member_a_o @ P @ ( sigma_measurable_a_o @ m @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( eventually_a @ P @ ( measure_ae_filter_a @ m ) )
= ( ( sigma_emeasure_a @ m
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ X ) ) ) )
= one_on2969667320475766781nnreal ) ) ) ).
% AE_iff_emeasure_eq_1
thf(fact_946_finite__Collect__subsets,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B2: set_nat] : ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_947_finite__Collect__subsets,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( finite7209287970140883943_set_a
@ ( collect_set_set_a
@ ^ [B2: set_set_a] : ( ord_le3724670747650509150_set_a @ B2 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_948_finite__Collect__subsets,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B2: set_a] : ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_949_finite__Collect__not,axiom,
! [P: a > $o] :
( ( finite_finite_a @ ( collect_a @ P ) )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
~ ( P @ X ) ) )
= ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Collect_not
thf(fact_950_finite__Collect__not,axiom,
! [P: nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
~ ( P @ X ) ) )
= ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Collect_not
thf(fact_951_finite__Collect__not,axiom,
! [P: $o > $o] :
( ( finite_finite_o @ ( collect_o @ P ) )
=> ( ( finite_finite_o
@ ( collect_o
@ ^ [X: $o] :
~ ( P @ X ) ) )
= ( finite_finite_o @ top_top_set_o ) ) ) ).
% finite_Collect_not
thf(fact_952_finite__Collect__not,axiom,
! [P: literal > $o] :
( ( finite5847741373460823677iteral @ ( collect_literal @ P ) )
=> ( ( finite5847741373460823677iteral
@ ( collect_literal
@ ^ [X: literal] :
~ ( P @ X ) ) )
= ( finite5847741373460823677iteral @ top_top_set_literal ) ) ) ).
% finite_Collect_not
thf(fact_953_finite__Collect__not,axiom,
! [P: product_unit > $o] :
( ( finite4290736615968046902t_unit @ ( collect_Product_unit @ P ) )
=> ( ( finite4290736615968046902t_unit
@ ( collect_Product_unit
@ ^ [X: product_unit] :
~ ( P @ X ) ) )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Collect_not
thf(fact_954_space__uniform__measure,axiom,
! [M2: sigma_measure_a,A2: set_a] :
( ( sigma_space_a @ ( nonneg6757527617543859701sure_a @ M2 @ A2 ) )
= ( sigma_space_a @ M2 ) ) ).
% space_uniform_measure
thf(fact_955_distributed__imp__emeasure__nonzero,axiom,
! [MX: sigma_measure_c_d,X2: a > c > d,Px: ( c > d ) > extend8495563244428889912nnreal] :
( ( probab7909271913183257430_a_c_d @ m @ MX @ X2 @ Px )
=> ( ( sigma_emeasure_c_d @ MX
@ ( collect_c_d
@ ^ [X: c > d] :
( ( member_c_d @ X @ ( sigma_space_c_d @ MX ) )
& ( ( Px @ X )
!= zero_z7100319975126383169nnreal ) ) ) )
!= zero_z7100319975126383169nnreal ) ) ).
% distributed_imp_emeasure_nonzero
thf(fact_956_distributed__imp__emeasure__nonzero,axiom,
! [MX: sigma_measure_a_real,X2: a > a > real,Px: ( a > real ) > extend8495563244428889912nnreal] :
( ( probab1295962615871907499a_real @ m @ MX @ X2 @ Px )
=> ( ( sigma_5985106571655482610a_real @ MX
@ ( collect_a_real
@ ^ [X: a > real] :
( ( member_a_real @ X @ ( sigma_space_a_real @ MX ) )
& ( ( Px @ X )
!= zero_z7100319975126383169nnreal ) ) ) )
!= zero_z7100319975126383169nnreal ) ) ).
% distributed_imp_emeasure_nonzero
thf(fact_957_distributed__imp__emeasure__nonzero,axiom,
! [MX: sigma_measure_a_o,X2: a > a > $o,Px: ( a > $o ) > extend8495563244428889912nnreal] :
( ( probab3024911352429335023_a_a_o @ m @ MX @ X2 @ Px )
=> ( ( sigma_emeasure_a_o @ MX
@ ( collect_a_o
@ ^ [X: a > $o] :
( ( member_a_o @ X @ ( sigma_space_a_o @ MX ) )
& ( ( Px @ X )
!= zero_z7100319975126383169nnreal ) ) ) )
!= zero_z7100319975126383169nnreal ) ) ).
% distributed_imp_emeasure_nonzero
thf(fact_958_distributed__imp__emeasure__nonzero,axiom,
! [MX: sigma_measure_b,X2: a > b,Px: b > extend8495563244428889912nnreal] :
( ( probab8876900357271971087ed_a_b @ m @ MX @ X2 @ Px )
=> ( ( sigma_emeasure_b @ MX
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ ( sigma_space_b @ MX ) )
& ( ( Px @ X )
!= zero_z7100319975126383169nnreal ) ) ) )
!= zero_z7100319975126383169nnreal ) ) ).
% distributed_imp_emeasure_nonzero
thf(fact_959_distributed__imp__emeasure__nonzero,axiom,
! [MX: sigma_measure_o,X2: a > $o,Px: $o > extend8495563244428889912nnreal] :
( ( probab177417448539121064ed_a_o @ m @ MX @ X2 @ Px )
=> ( ( sigma_emeasure_o @ MX
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( sigma_space_o @ MX ) )
& ( ( Px @ X )
!= zero_z7100319975126383169nnreal ) ) ) )
!= zero_z7100319975126383169nnreal ) ) ).
% distributed_imp_emeasure_nonzero
thf(fact_960_distributed__imp__emeasure__nonzero,axiom,
! [MX: sigma_measure_nat,X2: a > nat,Px: nat > extend8495563244428889912nnreal] :
( ( probab2712490043479616960_a_nat @ m @ MX @ X2 @ Px )
=> ( ( sigma_emeasure_nat @ MX
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( sigma_space_nat @ MX ) )
& ( ( Px @ X )
!= zero_z7100319975126383169nnreal ) ) ) )
!= zero_z7100319975126383169nnreal ) ) ).
% distributed_imp_emeasure_nonzero
thf(fact_961_distributed__imp__emeasure__nonzero,axiom,
! [MX: sigma_measure_a,X2: a > a,Px: a > extend8495563244428889912nnreal] :
( ( probab8876900357271971086ed_a_a @ m @ MX @ X2 @ Px )
=> ( ( sigma_emeasure_a @ MX
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ MX ) )
& ( ( Px @ X )
!= zero_z7100319975126383169nnreal ) ) ) )
!= zero_z7100319975126383169nnreal ) ) ).
% distributed_imp_emeasure_nonzero
thf(fact_962_emeasure__ge__1__iff,axiom,
! [A2: set_a] :
( ( ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ ( sigma_emeasure_a @ m @ A2 ) )
= ( ( sigma_emeasure_a @ m @ A2 )
= one_on2969667320475766781nnreal ) ) ).
% emeasure_ge_1_iff
thf(fact_963_emeasure__le__1,axiom,
! [S2: set_a] : ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ m @ S2 ) @ one_on2969667320475766781nnreal ) ).
% emeasure_le_1
thf(fact_964_subprob__emeasure__le__1,axiom,
! [X2: set_a] : ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ m @ X2 ) @ one_on2969667320475766781nnreal ) ).
% subprob_emeasure_le_1
thf(fact_965_emeasure__space__1,axiom,
( ( sigma_emeasure_a @ m @ ( sigma_space_a @ m ) )
= one_on2969667320475766781nnreal ) ).
% emeasure_space_1
thf(fact_966_emeasure__space__le__1,axiom,
ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ m @ ( sigma_space_a @ m ) ) @ one_on2969667320475766781nnreal ).
% emeasure_space_le_1
thf(fact_967_finite__Collect__conjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P ) )
| ( finite_finite_a @ ( collect_a @ Q ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_968_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_969_finite__Collect__disjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P ) )
& ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_970_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_971_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_972_finite__Plus__UNIV__iff,axiom,
( ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_973_finite__Plus__UNIV__iff,axiom,
( ( finite94888208985532392_nat_o @ top_to7120114879189831663_nat_o )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_o @ top_top_set_o ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_974_finite__Plus__UNIV__iff,axiom,
( ( finite7336130560110450212iteral @ top_to148093990134820907iteral )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite5847741373460823677iteral @ top_top_set_literal ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_975_finite__Plus__UNIV__iff,axiom,
( ( finite4327512606132785245t_unit @ top_to5465250082899874788t_unit )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_976_finite__Plus__UNIV__iff,axiom,
( ( finite5809725721784815170_o_nat @ top_to6072511757011528009_o_nat )
= ( ( finite_finite_o @ top_top_set_o )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_977_finite__Plus__UNIV__iff,axiom,
( ( finite6699802884135759036um_o_o @ top_to1686961084667892491um_o_o )
= ( ( finite_finite_o @ top_top_set_o )
& ( finite_finite_o @ top_top_set_o ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_978_finite__Plus__UNIV__iff,axiom,
( ( finite5240078910061808376iteral @ top_to6634777175915823943iteral )
= ( ( finite_finite_o @ top_top_set_o )
& ( finite5847741373460823677iteral @ top_top_set_literal ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_979_finite__Plus__UNIV__iff,axiom,
( ( finite2105581108028245809t_unit @ top_to126344037801868544t_unit )
= ( ( finite_finite_o @ top_top_set_o )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_980_finite__Plus__UNIV__iff,axiom,
( ( finite2800739532781614718al_nat @ top_to7291364169502081925al_nat )
= ( ( finite5847741373460823677iteral @ top_top_set_literal )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_981_finite__Plus__UNIV__iff,axiom,
( ( finite3087749691023359360eral_o @ top_to908623837245551055eral_o )
= ( ( finite5847741373460823677iteral @ top_top_set_literal )
& ( finite_finite_o @ top_top_set_o ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_982_integrable__zero,axiom,
! [M2: sigma_measure_a] :
( bochne2139062162225249880a_real @ M2
@ ^ [X: a] : zero_zero_real ) ).
% integrable_zero
thf(fact_983_emeasure__empty,axiom,
! [M2: sigma_measure_a] :
( ( sigma_emeasure_a @ M2 @ bot_bot_set_a )
= zero_z7100319975126383169nnreal ) ).
% emeasure_empty
thf(fact_984_emeasure__bot,axiom,
! [X2: set_a] :
( ( sigma_emeasure_a @ bot_bo2108912051383640591sure_a @ X2 )
= zero_z7100319975126383169nnreal ) ).
% emeasure_bot
thf(fact_985_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_986_prob__space_Oindep__events_Ocong,axiom,
indepe1551197314001032186_a_nat = indepe1551197314001032186_a_nat ).
% prob_space.indep_events.cong
thf(fact_987_prob__space_Oemeasure__ge__1__iff,axiom,
! [M2: sigma_measure_a,A2: set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ ( sigma_emeasure_a @ M2 @ A2 ) )
= ( ( sigma_emeasure_a @ M2 @ A2 )
= one_on2969667320475766781nnreal ) ) ) ).
% prob_space.emeasure_ge_1_iff
thf(fact_988_prob__space_Oemeasure__le__1,axiom,
! [M2: sigma_measure_a,S2: set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ M2 @ S2 ) @ one_on2969667320475766781nnreal ) ) ).
% prob_space.emeasure_le_1
thf(fact_989_prob__space_Omeasure__le__1,axiom,
! [M2: sigma_measure_a,X2: set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ M2 @ X2 ) @ one_on2969667320475766781nnreal ) ) ).
% prob_space.measure_le_1
thf(fact_990_subprob__space_Osubprob__emeasure__le__1,axiom,
! [M2: sigma_measure_a,X2: set_a] :
( ( giry_subprob_space_a @ M2 )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ M2 @ X2 ) @ one_on2969667320475766781nnreal ) ) ).
% subprob_space.subprob_emeasure_le_1
thf(fact_991_prob__spaceI,axiom,
! [M2: sigma_measure_a] :
( ( ( sigma_emeasure_a @ M2 @ ( sigma_space_a @ M2 ) )
= one_on2969667320475766781nnreal )
=> ( probab7247484486040049089pace_a @ M2 ) ) ).
% prob_spaceI
thf(fact_992_prob__space_Oemeasure__space__1,axiom,
! [M2: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( sigma_emeasure_a @ M2 @ ( sigma_space_a @ M2 ) )
= one_on2969667320475766781nnreal ) ) ).
% prob_space.emeasure_space_1
thf(fact_993_subprob__space_Oemeasure__space__le__1,axiom,
! [M2: sigma_measure_a] :
( ( giry_subprob_space_a @ M2 )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ M2 @ ( sigma_space_a @ M2 ) ) @ one_on2969667320475766781nnreal ) ) ).
% subprob_space.emeasure_space_le_1
thf(fact_994_emeasure__count__space__eq__0,axiom,
! [A2: set_o,X2: set_o] :
( ( ( sigma_emeasure_o @ ( sigma_count_space_o @ A2 ) @ X2 )
= zero_z7100319975126383169nnreal )
= ( ( ord_less_eq_set_o @ X2 @ A2 )
=> ( X2 = bot_bot_set_o ) ) ) ).
% emeasure_count_space_eq_0
thf(fact_995_emeasure__count__space__eq__0,axiom,
! [A2: set_set_a,X2: set_set_a] :
( ( ( sigma_emeasure_set_a @ ( sigma_1106005778614564215_set_a @ A2 ) @ X2 )
= zero_z7100319975126383169nnreal )
= ( ( ord_le3724670747650509150_set_a @ X2 @ A2 )
=> ( X2 = bot_bot_set_set_a ) ) ) ).
% emeasure_count_space_eq_0
thf(fact_996_emeasure__count__space__eq__0,axiom,
! [A2: set_a,X2: set_a] :
( ( ( sigma_emeasure_a @ ( sigma_count_space_a @ A2 ) @ X2 )
= zero_z7100319975126383169nnreal )
= ( ( ord_less_eq_set_a @ X2 @ A2 )
=> ( X2 = bot_bot_set_a ) ) ) ).
% emeasure_count_space_eq_0
thf(fact_997_subprob__spaceI,axiom,
! [M2: sigma_measure_a] :
( ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ M2 @ ( sigma_space_a @ M2 ) ) @ one_on2969667320475766781nnreal )
=> ( ( ( sigma_space_a @ M2 )
!= bot_bot_set_a )
=> ( giry_subprob_space_a @ M2 ) ) ) ).
% subprob_spaceI
thf(fact_998_not__finite__existsD,axiom,
! [P: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P ) )
=> ? [X_1: a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_999_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_1000_pigeonhole__infinite__rel,axiom,
! [A2: set_c_d,B: set_nat,R: ( c > d ) > nat > $o] :
( ~ ( finite_finite_c_d @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: c > d] :
( ( member_c_d @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_c_d
@ ( collect_c_d
@ ^ [A5: c > d] :
( ( member_c_d @ A5 @ A2 )
& ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1001_pigeonhole__infinite__rel,axiom,
! [A2: set_a_real,B: set_nat,R: ( a > real ) > nat > $o] :
( ~ ( finite_finite_a_real @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: a > real] :
( ( member_a_real @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_a_real
@ ( collect_a_real
@ ^ [A5: a > real] :
( ( member_a_real @ A5 @ A2 )
& ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1002_pigeonhole__infinite__rel,axiom,
! [A2: set_a_o,B: set_nat,R: ( a > $o ) > nat > $o] :
( ~ ( finite_finite_a_o @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: a > $o] :
( ( member_a_o @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_a_o
@ ( collect_a_o
@ ^ [A5: a > $o] :
( ( member_a_o @ A5 @ A2 )
& ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1003_pigeonhole__infinite__rel,axiom,
! [A2: set_b,B: set_nat,R: b > nat > $o] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A5: b] :
( ( member_b @ A5 @ A2 )
& ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1004_pigeonhole__infinite__rel,axiom,
! [A2: set_o,B: set_nat,R: $o > nat > $o] :
( ~ ( finite_finite_o @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_o
@ ( collect_o
@ ^ [A5: $o] :
( ( member_o @ A5 @ A2 )
& ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1005_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B: set_nat,R: a > nat > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A5: a] :
( ( member_a @ A5 @ A2 )
& ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1006_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A2 )
& ( R @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1007_emeasure__0__AE,axiom,
! [M2: sigma_measure_a,P: a > $o] :
( ( ( sigma_emeasure_a @ M2 @ ( sigma_space_a @ M2 ) )
= zero_z7100319975126383169nnreal )
=> ( eventually_a @ P @ ( measure_ae_filter_a @ M2 ) ) ) ).
% emeasure_0_AE
thf(fact_1008_emeasure__eq__0__AE,axiom,
! [P: ( c > d ) > $o,M2: sigma_measure_c_d] :
( ( eventually_c_d
@ ^ [X: c > d] :
~ ( P @ X )
@ ( measur253975988404078935er_c_d @ M2 ) )
=> ( ( sigma_emeasure_c_d @ M2
@ ( collect_c_d
@ ^ [X: c > d] :
( ( member_c_d @ X @ ( sigma_space_c_d @ M2 ) )
& ( P @ X ) ) ) )
= zero_z7100319975126383169nnreal ) ) ).
% emeasure_eq_0_AE
thf(fact_1009_emeasure__eq__0__AE,axiom,
! [P: ( a > real ) > $o,M2: sigma_measure_a_real] :
( ( eventually_a_real
@ ^ [X: a > real] :
~ ( P @ X )
@ ( measur8946935934331175658a_real @ M2 ) )
=> ( ( sigma_5985106571655482610a_real @ M2
@ ( collect_a_real
@ ^ [X: a > real] :
( ( member_a_real @ X @ ( sigma_space_a_real @ M2 ) )
& ( P @ X ) ) ) )
= zero_z7100319975126383169nnreal ) ) ).
% emeasure_eq_0_AE
thf(fact_1010_emeasure__eq__0__AE,axiom,
! [P: ( a > $o ) > $o,M2: sigma_measure_a_o] :
( ( eventually_a_o
@ ^ [X: a > $o] :
~ ( P @ X )
@ ( measur1950989848205109360er_a_o @ M2 ) )
=> ( ( sigma_emeasure_a_o @ M2
@ ( collect_a_o
@ ^ [X: a > $o] :
( ( member_a_o @ X @ ( sigma_space_a_o @ M2 ) )
& ( P @ X ) ) ) )
= zero_z7100319975126383169nnreal ) ) ).
% emeasure_eq_0_AE
thf(fact_1011_emeasure__eq__0__AE,axiom,
! [P: b > $o,M2: sigma_measure_b] :
( ( eventually_b
@ ^ [X: b] :
~ ( P @ X )
@ ( measure_ae_filter_b @ M2 ) )
=> ( ( sigma_emeasure_b @ M2
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ ( sigma_space_b @ M2 ) )
& ( P @ X ) ) ) )
= zero_z7100319975126383169nnreal ) ) ).
% emeasure_eq_0_AE
thf(fact_1012_emeasure__eq__0__AE,axiom,
! [P: $o > $o,M2: sigma_measure_o] :
( ( eventually_o
@ ^ [X: $o] :
~ ( P @ X )
@ ( measure_ae_filter_o @ M2 ) )
=> ( ( sigma_emeasure_o @ M2
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( sigma_space_o @ M2 ) )
& ( P @ X ) ) ) )
= zero_z7100319975126383169nnreal ) ) ).
% emeasure_eq_0_AE
thf(fact_1013_emeasure__eq__0__AE,axiom,
! [P: nat > $o,M2: sigma_measure_nat] :
( ( eventually_nat
@ ^ [X: nat] :
~ ( P @ X )
@ ( measur6539087422748349889er_nat @ M2 ) )
=> ( ( sigma_emeasure_nat @ M2
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( sigma_space_nat @ M2 ) )
& ( P @ X ) ) ) )
= zero_z7100319975126383169nnreal ) ) ).
% emeasure_eq_0_AE
thf(fact_1014_emeasure__eq__0__AE,axiom,
! [P: a > $o,M2: sigma_measure_a] :
( ( eventually_a
@ ^ [X: a] :
~ ( P @ X )
@ ( measure_ae_filter_a @ M2 ) )
=> ( ( sigma_emeasure_a @ M2
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ M2 ) )
& ( P @ X ) ) ) )
= zero_z7100319975126383169nnreal ) ) ).
% emeasure_eq_0_AE
thf(fact_1015_prob__space_Odistributed__imp__emeasure__nonzero,axiom,
! [M2: sigma_measure_a,MX: sigma_measure_c_d,X2: a > c > d,Px: ( c > d ) > extend8495563244428889912nnreal] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( probab7909271913183257430_a_c_d @ M2 @ MX @ X2 @ Px )
=> ( ( sigma_emeasure_c_d @ MX
@ ( collect_c_d
@ ^ [X: c > d] :
( ( member_c_d @ X @ ( sigma_space_c_d @ MX ) )
& ( ( Px @ X )
!= zero_z7100319975126383169nnreal ) ) ) )
!= zero_z7100319975126383169nnreal ) ) ) ).
% prob_space.distributed_imp_emeasure_nonzero
thf(fact_1016_prob__space_Odistributed__imp__emeasure__nonzero,axiom,
! [M2: sigma_measure_a,MX: sigma_measure_a_real,X2: a > a > real,Px: ( a > real ) > extend8495563244428889912nnreal] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( probab1295962615871907499a_real @ M2 @ MX @ X2 @ Px )
=> ( ( sigma_5985106571655482610a_real @ MX
@ ( collect_a_real
@ ^ [X: a > real] :
( ( member_a_real @ X @ ( sigma_space_a_real @ MX ) )
& ( ( Px @ X )
!= zero_z7100319975126383169nnreal ) ) ) )
!= zero_z7100319975126383169nnreal ) ) ) ).
% prob_space.distributed_imp_emeasure_nonzero
thf(fact_1017_prob__space_Odistributed__imp__emeasure__nonzero,axiom,
! [M2: sigma_measure_a,MX: sigma_measure_a_o,X2: a > a > $o,Px: ( a > $o ) > extend8495563244428889912nnreal] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( probab3024911352429335023_a_a_o @ M2 @ MX @ X2 @ Px )
=> ( ( sigma_emeasure_a_o @ MX
@ ( collect_a_o
@ ^ [X: a > $o] :
( ( member_a_o @ X @ ( sigma_space_a_o @ MX ) )
& ( ( Px @ X )
!= zero_z7100319975126383169nnreal ) ) ) )
!= zero_z7100319975126383169nnreal ) ) ) ).
% prob_space.distributed_imp_emeasure_nonzero
thf(fact_1018_prob__space_Odistributed__imp__emeasure__nonzero,axiom,
! [M2: sigma_measure_a,MX: sigma_measure_b,X2: a > b,Px: b > extend8495563244428889912nnreal] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( probab8876900357271971087ed_a_b @ M2 @ MX @ X2 @ Px )
=> ( ( sigma_emeasure_b @ MX
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ ( sigma_space_b @ MX ) )
& ( ( Px @ X )
!= zero_z7100319975126383169nnreal ) ) ) )
!= zero_z7100319975126383169nnreal ) ) ) ).
% prob_space.distributed_imp_emeasure_nonzero
thf(fact_1019_prob__space_Odistributed__imp__emeasure__nonzero,axiom,
! [M2: sigma_measure_a,MX: sigma_measure_o,X2: a > $o,Px: $o > extend8495563244428889912nnreal] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( probab177417448539121064ed_a_o @ M2 @ MX @ X2 @ Px )
=> ( ( sigma_emeasure_o @ MX
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( sigma_space_o @ MX ) )
& ( ( Px @ X )
!= zero_z7100319975126383169nnreal ) ) ) )
!= zero_z7100319975126383169nnreal ) ) ) ).
% prob_space.distributed_imp_emeasure_nonzero
thf(fact_1020_prob__space_Odistributed__imp__emeasure__nonzero,axiom,
! [M2: sigma_measure_a,MX: sigma_measure_nat,X2: a > nat,Px: nat > extend8495563244428889912nnreal] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( probab2712490043479616960_a_nat @ M2 @ MX @ X2 @ Px )
=> ( ( sigma_emeasure_nat @ MX
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( sigma_space_nat @ MX ) )
& ( ( Px @ X )
!= zero_z7100319975126383169nnreal ) ) ) )
!= zero_z7100319975126383169nnreal ) ) ) ).
% prob_space.distributed_imp_emeasure_nonzero
thf(fact_1021_prob__space_Odistributed__imp__emeasure__nonzero,axiom,
! [M2: sigma_measure_a,MX: sigma_measure_a,X2: a > a,Px: a > extend8495563244428889912nnreal] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( probab8876900357271971086ed_a_a @ M2 @ MX @ X2 @ Px )
=> ( ( sigma_emeasure_a @ MX
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ MX ) )
& ( ( Px @ X )
!= zero_z7100319975126383169nnreal ) ) ) )
!= zero_z7100319975126383169nnreal ) ) ) ).
% prob_space.distributed_imp_emeasure_nonzero
thf(fact_1022_finite__has__minimal2,axiom,
! [A2: set_a_real,A: a > real] :
( ( finite_finite_a_real @ A2 )
=> ( ( member_a_real @ A @ A2 )
=> ? [X3: a > real] :
( ( member_a_real @ X3 @ A2 )
& ( ord_less_eq_a_real @ X3 @ A )
& ! [Xa: a > real] :
( ( member_a_real @ Xa @ A2 )
=> ( ( ord_less_eq_a_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1023_finite__has__minimal2,axiom,
! [A2: set_a_o,A: a > $o] :
( ( finite_finite_a_o @ A2 )
=> ( ( member_a_o @ A @ A2 )
=> ? [X3: a > $o] :
( ( member_a_o @ X3 @ A2 )
& ( ord_less_eq_a_o @ X3 @ A )
& ! [Xa: a > $o] :
( ( member_a_o @ Xa @ A2 )
=> ( ( ord_less_eq_a_o @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1024_finite__has__minimal2,axiom,
! [A2: set_o,A: $o] :
( ( finite_finite_o @ A2 )
=> ( ( member_o @ A @ A2 )
=> ? [X3: $o] :
( ( member_o @ X3 @ A2 )
& ( ord_less_eq_o @ X3 @ A )
& ! [Xa: $o] :
( ( member_o @ Xa @ A2 )
=> ( ( ord_less_eq_o @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1025_finite__has__minimal2,axiom,
! [A2: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal] :
( ( finite3782138982310603983nnreal @ A2 )
=> ( ( member7908768830364227535nnreal @ A @ A2 )
=> ? [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A2 )
& ( ord_le3935885782089961368nnreal @ X3 @ A )
& ! [Xa: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ Xa @ A2 )
=> ( ( ord_le3935885782089961368nnreal @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1026_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1027_finite__has__minimal2,axiom,
! [A2: set_set_set_a,A: set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( member_set_set_a @ A @ A2 )
=> ? [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A2 )
& ( ord_le3724670747650509150_set_a @ X3 @ A )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1028_finite__has__minimal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ( ord_less_eq_real @ X3 @ A )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1029_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1030_finite__has__maximal2,axiom,
! [A2: set_a_real,A: a > real] :
( ( finite_finite_a_real @ A2 )
=> ( ( member_a_real @ A @ A2 )
=> ? [X3: a > real] :
( ( member_a_real @ X3 @ A2 )
& ( ord_less_eq_a_real @ A @ X3 )
& ! [Xa: a > real] :
( ( member_a_real @ Xa @ A2 )
=> ( ( ord_less_eq_a_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1031_finite__has__maximal2,axiom,
! [A2: set_a_o,A: a > $o] :
( ( finite_finite_a_o @ A2 )
=> ( ( member_a_o @ A @ A2 )
=> ? [X3: a > $o] :
( ( member_a_o @ X3 @ A2 )
& ( ord_less_eq_a_o @ A @ X3 )
& ! [Xa: a > $o] :
( ( member_a_o @ Xa @ A2 )
=> ( ( ord_less_eq_a_o @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1032_finite__has__maximal2,axiom,
! [A2: set_o,A: $o] :
( ( finite_finite_o @ A2 )
=> ( ( member_o @ A @ A2 )
=> ? [X3: $o] :
( ( member_o @ X3 @ A2 )
& ( ord_less_eq_o @ A @ X3 )
& ! [Xa: $o] :
( ( member_o @ Xa @ A2 )
=> ( ( ord_less_eq_o @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1033_finite__has__maximal2,axiom,
! [A2: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal] :
( ( finite3782138982310603983nnreal @ A2 )
=> ( ( member7908768830364227535nnreal @ A @ A2 )
=> ? [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A2 )
& ( ord_le3935885782089961368nnreal @ A @ X3 )
& ! [Xa: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ Xa @ A2 )
=> ( ( ord_le3935885782089961368nnreal @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1034_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1035_finite__has__maximal2,axiom,
! [A2: set_set_set_a,A: set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( member_set_set_a @ A @ A2 )
=> ? [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A2 )
& ( ord_le3724670747650509150_set_a @ A @ X3 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1036_finite__has__maximal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ( ord_less_eq_real @ A @ X3 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1037_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ A @ X3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1038_infinite__UNIV__char__0,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_char_0
thf(fact_1039_ex__new__if__finite,axiom,
! [A2: set_c_d] :
( ~ ( finite_finite_c_d @ top_top_set_c_d )
=> ( ( finite_finite_c_d @ A2 )
=> ? [A4: c > d] :
~ ( member_c_d @ A4 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_1040_ex__new__if__finite,axiom,
! [A2: set_a_real] :
( ~ ( finite_finite_a_real @ top_top_set_a_real )
=> ( ( finite_finite_a_real @ A2 )
=> ? [A4: a > real] :
~ ( member_a_real @ A4 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_1041_ex__new__if__finite,axiom,
! [A2: set_a_o] :
( ~ ( finite_finite_a_o @ top_top_set_a_o )
=> ( ( finite_finite_a_o @ A2 )
=> ? [A4: a > $o] :
~ ( member_a_o @ A4 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_1042_ex__new__if__finite,axiom,
! [A2: set_b] :
( ~ ( finite_finite_b @ top_top_set_b )
=> ( ( finite_finite_b @ A2 )
=> ? [A4: b] :
~ ( member_b @ A4 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_1043_ex__new__if__finite,axiom,
! [A2: set_a] :
( ~ ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_a @ A2 )
=> ? [A4: a] :
~ ( member_a @ A4 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_1044_ex__new__if__finite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ A2 )
=> ? [A4: nat] :
~ ( member_nat @ A4 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_1045_ex__new__if__finite,axiom,
! [A2: set_o] :
( ~ ( finite_finite_o @ top_top_set_o )
=> ( ( finite_finite_o @ A2 )
=> ? [A4: $o] :
~ ( member_o @ A4 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_1046_ex__new__if__finite,axiom,
! [A2: set_literal] :
( ~ ( finite5847741373460823677iteral @ top_top_set_literal )
=> ( ( finite5847741373460823677iteral @ A2 )
=> ? [A4: literal] :
~ ( member_literal @ A4 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_1047_ex__new__if__finite,axiom,
! [A2: set_Product_unit] :
( ~ ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite4290736615968046902t_unit @ A2 )
=> ? [A4: product_unit] :
~ ( member_Product_unit @ A4 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_1048_finite__class_Ofinite__UNIV,axiom,
finite_finite_o @ top_top_set_o ).
% finite_class.finite_UNIV
thf(fact_1049_finite__class_Ofinite__UNIV,axiom,
finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ).
% finite_class.finite_UNIV
thf(fact_1050_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_1051_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_o @ top_top_set_o )
=> ( finite5355008432043429460_nat_o @ top_to8070287629520841379_nat_o ) ) ) ).
% finite_Prod_UNIV
thf(fact_1052_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite5847741373460823677iteral @ top_top_set_literal )
=> ( finite211349803975347344iteral @ top_to6658620532179778271iteral ) ) ) ).
% finite_Prod_UNIV
thf(fact_1053_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( finite5113082511001691337t_unit @ top_to8544742955230171288t_unit ) ) ) ).
% finite_Prod_UNIV
thf(fact_1054_finite__Prod__UNIV,axiom,
( ( finite_finite_o @ top_top_set_o )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite1846473907987936430_o_nat @ top_to7022684507342537725_o_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_1055_finite__Prod__UNIV,axiom,
( ( finite_finite_o @ top_top_set_o )
=> ( ( finite_finite_o @ top_top_set_o )
=> ( finite6120865539452801872od_o_o @ top_to7721136755696657239od_o_o ) ) ) ).
% finite_Prod_UNIV
thf(fact_1056_finite__Prod__UNIV,axiom,
( ( finite_finite_o @ top_top_set_o )
=> ( ( finite5847741373460823677iteral @ top_top_set_literal )
=> ( finite8502321148819841420iteral @ top_to2491963433743229843iteral ) ) ) ).
% finite_Prod_UNIV
thf(fact_1057_finite__Prod__UNIV,axiom,
( ( finite_finite_o @ top_top_set_o )
=> ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( finite5959031561554722757t_unit @ top_to1629657009876642636t_unit ) ) ) ).
% finite_Prod_UNIV
thf(fact_1058_finite__Prod__UNIV,axiom,
( ( finite5847741373460823677iteral @ top_top_set_literal )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite4899330813501287658al_nat @ top_to4578518674692263481al_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_1059_finite__Prod__UNIV,axiom,
( ( finite5847741373460823677iteral @ top_top_set_literal )
=> ( ( finite_finite_o @ top_top_set_o )
=> ( finite6349991929781392404eral_o @ top_to5989182131927732763eral_o ) ) ) ).
% finite_Prod_UNIV
thf(fact_1060_finite__prod,axiom,
( ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_1061_finite__prod,axiom,
( ( finite5355008432043429460_nat_o @ top_to8070287629520841379_nat_o )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_o @ top_top_set_o ) ) ) ).
% finite_prod
thf(fact_1062_finite__prod,axiom,
( ( finite211349803975347344iteral @ top_to6658620532179778271iteral )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite5847741373460823677iteral @ top_top_set_literal ) ) ) ).
% finite_prod
thf(fact_1063_finite__prod,axiom,
( ( finite5113082511001691337t_unit @ top_to8544742955230171288t_unit )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_prod
thf(fact_1064_finite__prod,axiom,
( ( finite1846473907987936430_o_nat @ top_to7022684507342537725_o_nat )
= ( ( finite_finite_o @ top_top_set_o )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_1065_finite__prod,axiom,
( ( finite6120865539452801872od_o_o @ top_to7721136755696657239od_o_o )
= ( ( finite_finite_o @ top_top_set_o )
& ( finite_finite_o @ top_top_set_o ) ) ) ).
% finite_prod
thf(fact_1066_finite__prod,axiom,
( ( finite8502321148819841420iteral @ top_to2491963433743229843iteral )
= ( ( finite_finite_o @ top_top_set_o )
& ( finite5847741373460823677iteral @ top_top_set_literal ) ) ) ).
% finite_prod
thf(fact_1067_finite__prod,axiom,
( ( finite5959031561554722757t_unit @ top_to1629657009876642636t_unit )
= ( ( finite_finite_o @ top_top_set_o )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_prod
thf(fact_1068_finite__prod,axiom,
( ( finite4899330813501287658al_nat @ top_to4578518674692263481al_nat )
= ( ( finite5847741373460823677iteral @ top_top_set_literal )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_1069_finite__prod,axiom,
( ( finite6349991929781392404eral_o @ top_to5989182131927732763eral_o )
= ( ( finite5847741373460823677iteral @ top_top_set_literal )
& ( finite_finite_o @ top_top_set_o ) ) ) ).
% finite_prod
thf(fact_1070_Finite__Set_Ofinite__set,axiom,
( ( finite1152437895449049373et_nat @ top_top_set_set_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% Finite_Set.finite_set
thf(fact_1071_Finite__Set_Ofinite__set,axiom,
( ( finite_finite_set_o @ top_top_set_set_o )
= ( finite_finite_o @ top_top_set_o ) ) ).
% Finite_Set.finite_set
thf(fact_1072_Finite__Set_Ofinite__set,axiom,
( ( finite2869373537460367197iteral @ top_to5694933271948605156iteral )
= ( finite5847741373460823677iteral @ top_top_set_literal ) ) ).
% Finite_Set.finite_set
thf(fact_1073_Finite__Set_Ofinite__set,axiom,
( ( finite1772178364199683094t_unit @ top_to1767297665138865437t_unit )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).
% Finite_Set.finite_set
thf(fact_1074_rev__finite__subset,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_1075_rev__finite__subset,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_1076_rev__finite__subset,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_1077_infinite__super,axiom,
! [S2: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_1078_infinite__super,axiom,
! [S2: set_set_a,T: set_set_a] :
( ( ord_le3724670747650509150_set_a @ S2 @ T )
=> ( ~ ( finite_finite_set_a @ S2 )
=> ~ ( finite_finite_set_a @ T ) ) ) ).
% infinite_super
thf(fact_1079_infinite__super,axiom,
! [S2: set_a,T: set_a] :
( ( ord_less_eq_set_a @ S2 @ T )
=> ( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ T ) ) ) ).
% infinite_super
thf(fact_1080_finite__subset,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_1081_finite__subset,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( finite_finite_set_a @ B )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% finite_subset
thf(fact_1082_finite__subset,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_1083_infinite__imp__nonempty,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( S2 != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_1084_infinite__imp__nonempty,axiom,
! [S2: set_a] :
( ~ ( finite_finite_a @ S2 )
=> ( S2 != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_1085_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_1086_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_1087_prob__space_Oindep__events__finite__index__events,axiom,
! [M2: sigma_measure_a,F: set_a > set_a,I2: set_set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe451107616305015476_set_a @ M2 @ F @ I2 )
= ( ! [J2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ J2 @ I2 )
=> ( ( J2 != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ J2 )
=> ( indepe451107616305015476_set_a @ M2 @ F @ J2 ) ) ) ) ) ) ) ).
% prob_space.indep_events_finite_index_events
thf(fact_1088_prob__space_Oindep__events__finite__index__events,axiom,
! [M2: sigma_measure_a,F: a > set_a,I2: set_a] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe1948846426091821396ts_a_a @ M2 @ F @ I2 )
= ( ! [J2: set_a] :
( ( ord_less_eq_set_a @ J2 @ I2 )
=> ( ( J2 != bot_bot_set_a )
=> ( ( finite_finite_a @ J2 )
=> ( indepe1948846426091821396ts_a_a @ M2 @ F @ J2 ) ) ) ) ) ) ) ).
% prob_space.indep_events_finite_index_events
thf(fact_1089_prob__space_Oindep__events__finite__index__events,axiom,
! [M2: sigma_measure_a,F: nat > set_a,I2: set_nat] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe1551197314001032186_a_nat @ M2 @ F @ I2 )
= ( ! [J2: set_nat] :
( ( ord_less_eq_set_nat @ J2 @ I2 )
=> ( ( J2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ J2 )
=> ( indepe1551197314001032186_a_nat @ M2 @ F @ J2 ) ) ) ) ) ) ) ).
% prob_space.indep_events_finite_index_events
thf(fact_1090_prob__space_Oborel__0__1__law__AE,axiom,
! [M2: sigma_measure_c_d,P: nat > ( c > d ) > $o] :
( ( probab3693743499390067171ce_c_d @ M2 )
=> ( ( indepe4353200269427704460_d_nat @ M2
@ ^ [M4: nat] :
( collect_c_d
@ ^ [X: c > d] :
( ( member_c_d @ X @ ( sigma_space_c_d @ M2 ) )
& ( P @ M4 @ X ) ) )
@ top_top_set_nat )
=> ( ( eventually_c_d
@ ^ [X: c > d] :
~ ( finite_finite_nat
@ ( collect_nat
@ ^ [M4: nat] : ( P @ M4 @ X ) ) )
@ ( measur253975988404078935er_c_d @ M2 ) )
| ( eventually_c_d
@ ^ [X: c > d] :
( finite_finite_nat
@ ( collect_nat
@ ^ [M4: nat] : ( P @ M4 @ X ) ) )
@ ( measur253975988404078935er_c_d @ M2 ) ) ) ) ) ).
% prob_space.borel_0_1_law_AE
thf(fact_1091_prob__space_Oborel__0__1__law__AE,axiom,
! [M2: sigma_measure_a_real,P: nat > ( a > real ) > $o] :
( ( probab2024454272037758302a_real @ M2 )
=> ( ( indepe3916917635824975767al_nat @ M2
@ ^ [M4: nat] :
( collect_a_real
@ ^ [X: a > real] :
( ( member_a_real @ X @ ( sigma_space_a_real @ M2 ) )
& ( P @ M4 @ X ) ) )
@ top_top_set_nat )
=> ( ( eventually_a_real
@ ^ [X: a > real] :
~ ( finite_finite_nat
@ ( collect_nat
@ ^ [M4: nat] : ( P @ M4 @ X ) ) )
@ ( measur8946935934331175658a_real @ M2 ) )
| ( eventually_a_real
@ ^ [X: a > real] :
( finite_finite_nat
@ ( collect_nat
@ ^ [M4: nat] : ( P @ M4 @ X ) ) )
@ ( measur8946935934331175658a_real @ M2 ) ) ) ) ) ).
% prob_space.borel_0_1_law_AE
thf(fact_1092_prob__space_Oborel__0__1__law__AE,axiom,
! [M2: sigma_measure_a_o,P: nat > ( a > $o ) > $o] :
( ( probab7249042050958952188ce_a_o @ M2 )
=> ( ( indepe7519481913767141285_o_nat @ M2
@ ^ [M4: nat] :
( collect_a_o
@ ^ [X: a > $o] :
( ( member_a_o @ X @ ( sigma_space_a_o @ M2 ) )
& ( P @ M4 @ X ) ) )
@ top_top_set_nat )
=> ( ( eventually_a_o
@ ^ [X: a > $o] :
~ ( finite_finite_nat
@ ( collect_nat
@ ^ [M4: nat] : ( P @ M4 @ X ) ) )
@ ( measur1950989848205109360er_a_o @ M2 ) )
| ( eventually_a_o
@ ^ [X: a > $o] :
( finite_finite_nat
@ ( collect_nat
@ ^ [M4: nat] : ( P @ M4 @ X ) ) )
@ ( measur1950989848205109360er_a_o @ M2 ) ) ) ) ) ).
% prob_space.borel_0_1_law_AE
thf(fact_1093_prob__space_Oborel__0__1__law__AE,axiom,
! [M2: sigma_measure_b,P: nat > b > $o] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( indepe2786641642957426683_b_nat @ M2
@ ^ [M4: nat] :
( collect_b
@ ^ [X: b] :
( ( member_b @ X @ ( sigma_space_b @ M2 ) )
& ( P @ M4 @ X ) ) )
@ top_top_set_nat )
=> ( ( eventually_b
@ ^ [X: b] :
~ ( finite_finite_nat
@ ( collect_nat
@ ^ [M4: nat] : ( P @ M4 @ X ) ) )
@ ( measure_ae_filter_b @ M2 ) )
| ( eventually_b
@ ^ [X: b] :
( finite_finite_nat
@ ( collect_nat
@ ^ [M4: nat] : ( P @ M4 @ X ) ) )
@ ( measure_ae_filter_b @ M2 ) ) ) ) ) ).
% prob_space.borel_0_1_law_AE
thf(fact_1094_prob__space_Oborel__0__1__law__AE,axiom,
! [M2: sigma_measure_o,P: nat > $o > $o] :
( ( probab1190487603588612059pace_o @ M2 )
=> ( ( indepe8054989831391459860_o_nat @ M2
@ ^ [M4: nat] :
( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( sigma_space_o @ M2 ) )
& ( P @ M4 @ X ) ) )
@ top_top_set_nat )
=> ( ( eventually_o
@ ^ [X: $o] :
~ ( finite_finite_nat
@ ( collect_nat
@ ^ [M4: nat] : ( P @ M4 @ X ) ) )
@ ( measure_ae_filter_o @ M2 ) )
| ( eventually_o
@ ^ [X: $o] :
( finite_finite_nat
@ ( collect_nat
@ ^ [M4: nat] : ( P @ M4 @ X ) ) )
@ ( measure_ae_filter_o @ M2 ) ) ) ) ) ).
% prob_space.borel_0_1_law_AE
thf(fact_1095_prob__space_Oborel__0__1__law__AE,axiom,
! [M2: sigma_measure_nat,P: nat > nat > $o] :
( ( probab2904919403188438605ce_nat @ M2 )
=> ( ( indepe6019118560394178294at_nat @ M2
@ ^ [M4: nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( sigma_space_nat @ M2 ) )
& ( P @ M4 @ X ) ) )
@ top_top_set_nat )
=> ( ( eventually_nat
@ ^ [X: nat] :
~ ( finite_finite_nat
@ ( collect_nat
@ ^ [M4: nat] : ( P @ M4 @ X ) ) )
@ ( measur6539087422748349889er_nat @ M2 ) )
| ( eventually_nat
@ ^ [X: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [M4: nat] : ( P @ M4 @ X ) ) )
@ ( measur6539087422748349889er_nat @ M2 ) ) ) ) ) ).
% prob_space.borel_0_1_law_AE
thf(fact_1096_prob__space_Oborel__0__1__law__AE,axiom,
! [M2: sigma_measure_a,P: nat > a > $o] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( indepe1551197314001032186_a_nat @ M2
@ ^ [M4: nat] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ M2 ) )
& ( P @ M4 @ X ) ) )
@ top_top_set_nat )
=> ( ( eventually_a
@ ^ [X: a] :
~ ( finite_finite_nat
@ ( collect_nat
@ ^ [M4: nat] : ( P @ M4 @ X ) ) )
@ ( measure_ae_filter_a @ M2 ) )
| ( eventually_a
@ ^ [X: a] :
( finite_finite_nat
@ ( collect_nat
@ ^ [M4: nat] : ( P @ M4 @ X ) ) )
@ ( measure_ae_filter_a @ M2 ) ) ) ) ) ).
% prob_space.borel_0_1_law_AE
thf(fact_1097_prob__space_OAE__iff__emeasure__eq__1,axiom,
! [M2: sigma_measure_c_d,P: ( c > d ) > $o] :
( ( probab3693743499390067171ce_c_d @ M2 )
=> ( ( member_c_d_o @ P @ ( sigma_1714064210060623456_c_d_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( eventually_c_d @ P @ ( measur253975988404078935er_c_d @ M2 ) )
= ( ( sigma_emeasure_c_d @ M2
@ ( collect_c_d
@ ^ [X: c > d] :
( ( member_c_d @ X @ ( sigma_space_c_d @ M2 ) )
& ( P @ X ) ) ) )
= one_on2969667320475766781nnreal ) ) ) ) ).
% prob_space.AE_iff_emeasure_eq_1
thf(fact_1098_prob__space_OAE__iff__emeasure__eq__1,axiom,
! [M2: sigma_measure_a_real,P: ( a > real ) > $o] :
( ( probab2024454272037758302a_real @ M2 )
=> ( ( member_a_real_o @ P @ ( sigma_9085598459323199629real_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( eventually_a_real @ P @ ( measur8946935934331175658a_real @ M2 ) )
= ( ( sigma_5985106571655482610a_real @ M2
@ ( collect_a_real
@ ^ [X: a > real] :
( ( member_a_real @ X @ ( sigma_space_a_real @ M2 ) )
& ( P @ X ) ) ) )
= one_on2969667320475766781nnreal ) ) ) ) ).
% prob_space.AE_iff_emeasure_eq_1
thf(fact_1099_prob__space_OAE__iff__emeasure__eq__1,axiom,
! [M2: sigma_measure_a_o,P: ( a > $o ) > $o] :
( ( probab7249042050958952188ce_a_o @ M2 )
=> ( ( member_a_o_o @ P @ ( sigma_1195952539894209287_a_o_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( eventually_a_o @ P @ ( measur1950989848205109360er_a_o @ M2 ) )
= ( ( sigma_emeasure_a_o @ M2
@ ( collect_a_o
@ ^ [X: a > $o] :
( ( member_a_o @ X @ ( sigma_space_a_o @ M2 ) )
& ( P @ X ) ) ) )
= one_on2969667320475766781nnreal ) ) ) ) ).
% prob_space.AE_iff_emeasure_eq_1
thf(fact_1100_prob__space_OAE__iff__emeasure__eq__1,axiom,
! [M2: sigma_measure_b,P: b > $o] :
( ( probab7247484486040049090pace_b @ M2 )
=> ( ( member_b_o @ P @ ( sigma_measurable_b_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( eventually_b @ P @ ( measure_ae_filter_b @ M2 ) )
= ( ( sigma_emeasure_b @ M2
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ ( sigma_space_b @ M2 ) )
& ( P @ X ) ) ) )
= one_on2969667320475766781nnreal ) ) ) ) ).
% prob_space.AE_iff_emeasure_eq_1
thf(fact_1101_prob__space_OAE__iff__emeasure__eq__1,axiom,
! [M2: sigma_measure_o,P: $o > $o] :
( ( probab1190487603588612059pace_o @ M2 )
=> ( ( member_o_o @ P @ ( sigma_measurable_o_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( eventually_o @ P @ ( measure_ae_filter_o @ M2 ) )
= ( ( sigma_emeasure_o @ M2
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( sigma_space_o @ M2 ) )
& ( P @ X ) ) ) )
= one_on2969667320475766781nnreal ) ) ) ) ).
% prob_space.AE_iff_emeasure_eq_1
thf(fact_1102_prob__space_OAE__iff__emeasure__eq__1,axiom,
! [M2: sigma_measure_nat,P: nat > $o] :
( ( probab2904919403188438605ce_nat @ M2 )
=> ( ( member_nat_o @ P @ ( sigma_5101835498682829686_nat_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( eventually_nat @ P @ ( measur6539087422748349889er_nat @ M2 ) )
= ( ( sigma_emeasure_nat @ M2
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( sigma_space_nat @ M2 ) )
& ( P @ X ) ) ) )
= one_on2969667320475766781nnreal ) ) ) ) ).
% prob_space.AE_iff_emeasure_eq_1
thf(fact_1103_prob__space_OAE__iff__emeasure__eq__1,axiom,
! [M2: sigma_measure_a,P: a > $o] :
( ( probab7247484486040049089pace_a @ M2 )
=> ( ( member_a_o @ P @ ( sigma_measurable_a_o @ M2 @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( eventually_a @ P @ ( measure_ae_filter_a @ M2 ) )
= ( ( sigma_emeasure_a @ M2
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ M2 ) )
& ( P @ X ) ) ) )
= one_on2969667320475766781nnreal ) ) ) ) ).
% prob_space.AE_iff_emeasure_eq_1
thf(fact_1104_finite__has__maximal,axiom,
! [A2: set_Ex3793607809372303086nnreal] :
( ( finite3782138982310603983nnreal @ A2 )
=> ( ( A2 != bot_bo4854962954004695426nnreal )
=> ? [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A2 )
& ! [Xa: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ Xa @ A2 )
=> ( ( ord_le3935885782089961368nnreal @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1105_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1106_finite__has__maximal,axiom,
! [A2: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( A2 != bot_bo3380559777022489994_set_a )
=> ? [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A2 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1107_finite__has__maximal,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1108_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1109_finite__has__minimal,axiom,
! [A2: set_Ex3793607809372303086nnreal] :
( ( finite3782138982310603983nnreal @ A2 )
=> ( ( A2 != bot_bo4854962954004695426nnreal )
=> ? [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A2 )
& ! [Xa: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ Xa @ A2 )
=> ( ( ord_le3935885782089961368nnreal @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1110_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1111_finite__has__minimal,axiom,
! [A2: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( A2 != bot_bo3380559777022489994_set_a )
=> ? [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A2 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1112_finite__has__minimal,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1113_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1114_AE__I__eq__1,axiom,
! [P: a > $o] :
( ( ( sigma_emeasure_a @ m
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ X ) ) ) )
= one_on2969667320475766781nnreal )
=> ( ( member_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ X ) ) )
@ ( sigma_sets_a @ m ) )
=> ( eventually_a @ P @ ( measure_ae_filter_a @ m ) ) ) ) ).
% AE_I_eq_1
thf(fact_1115_indep__eventsI__indep__vars,axiom,
! [N: b > sigma_measure_b,X2: b > a > b,I2: set_b,P: b > b > $o] :
( ( indepe7639357355105118966_a_b_b @ m @ N @ X2 @ I2 )
=> ( ! [I3: b] :
( ( member_b @ I3 @ I2 )
=> ( member_set_b
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ ( sigma_space_b @ ( N @ I3 ) ) )
& ( P @ I3 @ X ) ) )
@ ( sigma_sets_b @ ( N @ I3 ) ) ) )
=> ( indepe1948846426091821397ts_a_b @ m
@ ^ [I4: b] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ I4 @ ( X2 @ I4 @ X ) ) ) )
@ I2 ) ) ) ).
% indep_eventsI_indep_vars
thf(fact_1116_indep__eventsI__indep__vars,axiom,
! [N: b > sigma_measure_o,X2: b > a > $o,I2: set_b,P: b > $o > $o] :
( ( indepe432941580615200527_a_b_o @ m @ N @ X2 @ I2 )
=> ( ! [I3: b] :
( ( member_b @ I3 @ I2 )
=> ( member_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( sigma_space_o @ ( N @ I3 ) ) )
& ( P @ I3 @ X ) ) )
@ ( sigma_sets_o @ ( N @ I3 ) ) ) )
=> ( indepe1948846426091821397ts_a_b @ m
@ ^ [I4: b] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ I4 @ ( X2 @ I4 @ X ) ) ) )
@ I2 ) ) ) ).
% indep_eventsI_indep_vars
thf(fact_1117_indep__eventsI__indep__vars,axiom,
! [N: a > sigma_measure_b,X2: a > a > b,I2: set_a,P: a > b > $o] :
( ( indepe1203440900223019191_a_a_b @ m @ N @ X2 @ I2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I2 )
=> ( member_set_b
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ ( sigma_space_b @ ( N @ I3 ) ) )
& ( P @ I3 @ X ) ) )
@ ( sigma_sets_b @ ( N @ I3 ) ) ) )
=> ( indepe1948846426091821396ts_a_a @ m
@ ^ [I4: a] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ I4 @ ( X2 @ I4 @ X ) ) ) )
@ I2 ) ) ) ).
% indep_eventsI_indep_vars
thf(fact_1118_indep__eventsI__indep__vars,axiom,
! [N: a > sigma_measure_o,X2: a > a > $o,I2: set_a,P: a > $o > $o] :
( ( indepe3332163980079594832_a_a_o @ m @ N @ X2 @ I2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I2 )
=> ( member_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( sigma_space_o @ ( N @ I3 ) ) )
& ( P @ I3 @ X ) ) )
@ ( sigma_sets_o @ ( N @ I3 ) ) ) )
=> ( indepe1948846426091821396ts_a_a @ m
@ ^ [I4: a] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ I4 @ ( X2 @ I4 @ X ) ) ) )
@ I2 ) ) ) ).
% indep_eventsI_indep_vars
thf(fact_1119_indep__eventsI__indep__vars,axiom,
! [N: $o > sigma_measure_b,X2: $o > a > b,I2: set_o,P: $o > b > $o] :
( ( indepe3252683823613847069_a_o_b @ m @ N @ X2 @ I2 )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( member_set_b
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ ( sigma_space_b @ ( N @ I3 ) ) )
& ( P @ I3 @ X ) ) )
@ ( sigma_sets_b @ ( N @ I3 ) ) ) )
=> ( indepe3695496658712714478ts_a_o @ m
@ ^ [I4: $o] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ I4 @ ( X2 @ I4 @ X ) ) ) )
@ I2 ) ) ) ).
% indep_eventsI_indep_vars
thf(fact_1120_indep__eventsI__indep__vars,axiom,
! [N: $o > sigma_measure_o,X2: $o > a > $o,I2: set_o,P: $o > $o > $o] :
( ( indepe9162428965118168502_a_o_o @ m @ N @ X2 @ I2 )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( member_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( sigma_space_o @ ( N @ I3 ) ) )
& ( P @ I3 @ X ) ) )
@ ( sigma_sets_o @ ( N @ I3 ) ) ) )
=> ( indepe3695496658712714478ts_a_o @ m
@ ^ [I4: $o] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ I4 @ ( X2 @ I4 @ X ) ) ) )
@ I2 ) ) ) ).
% indep_eventsI_indep_vars
thf(fact_1121_indep__eventsI__indep__vars,axiom,
! [N: b > sigma_measure_nat,X2: b > a > nat,I2: set_b,P: b > nat > $o] :
( ( indepe448710728707214361_b_nat @ m @ N @ X2 @ I2 )
=> ( ! [I3: b] :
( ( member_b @ I3 @ I2 )
=> ( member_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( sigma_space_nat @ ( N @ I3 ) ) )
& ( P @ I3 @ X ) ) )
@ ( sigma_sets_nat @ ( N @ I3 ) ) ) )
=> ( indepe1948846426091821397ts_a_b @ m
@ ^ [I4: b] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ I4 @ ( X2 @ I4 @ X ) ) ) )
@ I2 ) ) ) ).
% indep_eventsI_indep_vars
thf(fact_1122_indep__eventsI__indep__vars,axiom,
! [N: a > sigma_measure_nat,X2: a > a > nat,I2: set_a,P: a > nat > $o] :
( ( indepe8436638436605595672_a_nat @ m @ N @ X2 @ I2 )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I2 )
=> ( member_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( sigma_space_nat @ ( N @ I3 ) ) )
& ( P @ I3 @ X ) ) )
@ ( sigma_sets_nat @ ( N @ I3 ) ) ) )
=> ( indepe1948846426091821396ts_a_a @ m
@ ^ [I4: a] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ I4 @ ( X2 @ I4 @ X ) ) ) )
@ I2 ) ) ) ).
% indep_eventsI_indep_vars
thf(fact_1123_indep__eventsI__indep__vars,axiom,
! [N: $o > sigma_measure_nat,X2: $o > a > nat,I2: set_o,P: $o > nat > $o] :
( ( indepe3518881924824081714_o_nat @ m @ N @ X2 @ I2 )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I2 )
=> ( member_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( sigma_space_nat @ ( N @ I3 ) ) )
& ( P @ I3 @ X ) ) )
@ ( sigma_sets_nat @ ( N @ I3 ) ) ) )
=> ( indepe3695496658712714478ts_a_o @ m
@ ^ [I4: $o] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ I4 @ ( X2 @ I4 @ X ) ) ) )
@ I2 ) ) ) ).
% indep_eventsI_indep_vars
thf(fact_1124_indep__eventsI__indep__vars,axiom,
! [N: b > sigma_measure_a,X2: b > a > a,I2: set_b,P: b > a > $o] :
( ( indepe7639357355105118965_a_b_a @ m @ N @ X2 @ I2 )
=> ( ! [I3: b] :
( ( member_b @ I3 @ I2 )
=> ( member_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ ( N @ I3 ) ) )
& ( P @ I3 @ X ) ) )
@ ( sigma_sets_a @ ( N @ I3 ) ) ) )
=> ( indepe1948846426091821397ts_a_b @ m
@ ^ [I4: b] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ I4 @ ( X2 @ I4 @ X ) ) ) )
@ I2 ) ) ) ).
% indep_eventsI_indep_vars
thf(fact_1125_emeasure__eq__1__AE,axiom,
! [S2: set_a] :
( ( member_set_a @ S2 @ ( sigma_sets_a @ m ) )
=> ( ( eventually_a
@ ^ [X: a] : ( member_a @ X @ S2 )
@ ( measure_ae_filter_a @ m ) )
=> ( ( sigma_emeasure_a @ m @ S2 )
= one_on2969667320475766781nnreal ) ) ) ).
% emeasure_eq_1_AE
thf(fact_1126_finite__option__UNIV,axiom,
( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% finite_option_UNIV
thf(fact_1127_finite__option__UNIV,axiom,
( ( finite4093902646404507527tion_o @ top_top_set_option_o )
= ( finite_finite_o @ top_top_set_o ) ) ).
% finite_option_UNIV
thf(fact_1128_finite__option__UNIV,axiom,
( ( finite5071707688241699267iteral @ top_to8248435444729185354iteral )
= ( finite5847741373460823677iteral @ top_top_set_literal ) ) ).
% finite_option_UNIV
thf(fact_1129_finite__option__UNIV,axiom,
( ( finite1445617369574913404t_unit @ top_to2690860209552263555t_unit )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).
% finite_option_UNIV
thf(fact_1130_k__wise__indep__vars__subset,axiom,
! [K: nat,M: b > sigma_measure_d,X2: b > a > d,I2: set_b,J: set_b] :
( ( prob_k6574085460301583243_a_b_d @ m @ K @ M @ X2 @ I2 )
=> ( ( ord_less_eq_set_b @ J @ I2 )
=> ( ( finite_finite_b @ J )
=> ( ( ord_less_eq_nat @ ( finite_card_b @ J ) @ K )
=> ( indepe7639357355105118968_a_b_d @ m @ M @ X2 @ J ) ) ) ) ) ).
% k_wise_indep_vars_subset
thf(fact_1131_k__wise__indep__vars__subset,axiom,
! [K: nat,M: b > sigma_measure_c,X2: b > a > c,I2: set_b,J: set_b] :
( ( prob_k6574085460301583242_a_b_c @ m @ K @ M @ X2 @ I2 )
=> ( ( ord_less_eq_set_b @ J @ I2 )
=> ( ( finite_finite_b @ J )
=> ( ( ord_less_eq_nat @ ( finite_card_b @ J ) @ K )
=> ( indepe7639357355105118967_a_b_c @ m @ M @ X2 @ J ) ) ) ) ) ).
% k_wise_indep_vars_subset
thf(fact_1132_k__wise__indep__vars__def,axiom,
! [K: nat,M: b > sigma_measure_d,X2: b > a > d,I2: set_b] :
( ( prob_k6574085460301583243_a_b_d @ m @ K @ M @ X2 @ I2 )
= ( ! [J2: set_b] :
( ( ord_less_eq_set_b @ J2 @ I2 )
=> ( ( ord_less_eq_nat @ ( finite_card_b @ J2 ) @ K )
=> ( ( finite_finite_b @ J2 )
=> ( indepe7639357355105118968_a_b_d @ m @ M @ X2 @ J2 ) ) ) ) ) ) ).
% k_wise_indep_vars_def
thf(fact_1133_k__wise__indep__vars__def,axiom,
! [K: nat,M: b > sigma_measure_c,X2: b > a > c,I2: set_b] :
( ( prob_k6574085460301583242_a_b_c @ m @ K @ M @ X2 @ I2 )
= ( ! [J2: set_b] :
( ( ord_less_eq_set_b @ J2 @ I2 )
=> ( ( ord_less_eq_nat @ ( finite_card_b @ J2 ) @ K )
=> ( ( finite_finite_b @ J2 )
=> ( indepe7639357355105118967_a_b_c @ m @ M @ X2 @ J2 ) ) ) ) ) ) ).
% k_wise_indep_vars_def
thf(fact_1134_sets_Oempty__sets,axiom,
! [M2: sigma_measure_a] : ( member_set_a @ bot_bot_set_a @ ( sigma_sets_a @ M2 ) ) ).
% sets.empty_sets
thf(fact_1135_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_1136_card_Oempty,axiom,
( ( finite_card_literal @ bot_bot_set_literal )
= zero_zero_nat ) ).
% card.empty
thf(fact_1137_card_Oempty,axiom,
( ( finite410649719033368117t_unit @ bot_bo3957492148770167129t_unit )
= zero_zero_nat ) ).
% card.empty
thf(fact_1138_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_1139_sets_Otop,axiom,
! [M2: sigma_measure_a] : ( member_set_a @ ( sigma_space_a @ M2 ) @ ( sigma_sets_a @ M2 ) ) ).
% sets.top
thf(fact_1140_sets__completionI__sets,axiom,
! [A2: set_a,M2: sigma_measure_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ M2 ) )
=> ( member_set_a @ A2 @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M2 ) ) ) ) ).
% sets_completionI_sets
thf(fact_1141_sets__return,axiom,
! [M2: sigma_measure_a,X4: a] :
( ( sigma_sets_a @ ( giry_return_a @ M2 @ X4 ) )
= ( sigma_sets_a @ M2 ) ) ).
% sets_return
thf(fact_1142_card__0__eq,axiom,
! [A2: set_literal] :
( ( finite5847741373460823677iteral @ A2 )
=> ( ( ( finite_card_literal @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_literal ) ) ) ).
% card_0_eq
thf(fact_1143_card__0__eq,axiom,
! [A2: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A2 )
=> ( ( ( finite410649719033368117t_unit @ A2 )
= zero_zero_nat )
= ( A2 = bot_bo3957492148770167129t_unit ) ) ) ).
% card_0_eq
thf(fact_1144_card__0__eq,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_1145_card__0__eq,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ( finite_card_a @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_a ) ) ) ).
% card_0_eq
thf(fact_1146_tail__events__sets,axiom,
! [A2: nat > set_set_a] :
( ! [I3: nat] : ( ord_le3724670747650509150_set_a @ ( A2 @ I3 ) @ ( sigma_sets_a @ m ) )
=> ( ord_le3724670747650509150_set_a @ ( indepe7538416700049374166_a_nat @ m @ A2 ) @ ( sigma_sets_a @ m ) ) ) ).
% tail_events_sets
thf(fact_1147_cond__prob__eq__AE,axiom,
! [Q: a > $o,P: a > $o,P2: a > $o,Q2: a > $o] :
( ( eventually_a
@ ^ [X: a] :
( ( Q @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
@ ( measure_ae_filter_a @ m ) )
=> ( ( member_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ X ) ) )
@ ( sigma_sets_a @ m ) )
=> ( ( member_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P2 @ X ) ) )
@ ( sigma_sets_a @ m ) )
=> ( ( eventually_a
@ ^ [X: a] :
( ( Q @ X )
= ( Q2 @ X ) )
@ ( measure_ae_filter_a @ m ) )
=> ( ( member_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( Q @ X ) ) )
@ ( sigma_sets_a @ m ) )
=> ( ( member_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( Q2 @ X ) ) )
@ ( sigma_sets_a @ m ) )
=> ( ( probab5246806142355027601prob_a @ m @ P @ Q )
= ( probab5246806142355027601prob_a @ m @ P2 @ Q2 ) ) ) ) ) ) ) ) ).
% cond_prob_eq_AE
thf(fact_1148_sets__Ball,axiom,
! [I2: set_o,A2: $o > set_a,M2: $o > sigma_measure_a,I: $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ I2 )
=> ( member_set_a @ ( A2 @ X3 ) @ ( sigma_sets_a @ ( M2 @ X3 ) ) ) )
=> ( ( member_o @ I @ I2 )
=> ( member_set_a @ ( A2 @ I ) @ ( sigma_sets_a @ ( M2 @ I ) ) ) ) ) ).
% sets_Ball
thf(fact_1149_indep__setD__ev1,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( indepe2041756565122539606_set_a @ m @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( sigma_sets_a @ m ) ) ) ).
% indep_setD_ev1
thf(fact_1150_indep__setD__ev2,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( indepe2041756565122539606_set_a @ m @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ B @ ( sigma_sets_a @ m ) ) ) ).
% indep_setD_ev2
thf(fact_1151_prob__Collect__eq__0,axiom,
! [P: a > $o] :
( ( member_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ X ) ) )
@ ( sigma_sets_a @ m ) )
=> ( ( ( sigma_measure_a2 @ m
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ X ) ) ) )
= zero_zero_real )
= ( eventually_a
@ ^ [X: a] :
~ ( P @ X )
@ ( measure_ae_filter_a @ m ) ) ) ) ).
% prob_Collect_eq_0
thf(fact_1152_prob__eq__AE,axiom,
! [P: a > $o,Q: a > $o] :
( ( eventually_a
@ ^ [X: a] :
( ( P @ X )
= ( Q @ X ) )
@ ( measure_ae_filter_a @ m ) )
=> ( ( member_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ X ) ) )
@ ( sigma_sets_a @ m ) )
=> ( ( member_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( Q @ X ) ) )
@ ( sigma_sets_a @ m ) )
=> ( ( sigma_measure_a2 @ m
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ X ) ) ) )
= ( sigma_measure_a2 @ m
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( Q @ X ) ) ) ) ) ) ) ) ).
% prob_eq_AE
thf(fact_1153_bounded__measure,axiom,
! [A2: set_a] : ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ A2 ) @ ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) ) ) ).
% bounded_measure
thf(fact_1154_measure__increasing,axiom,
measur1776380161843274167a_real @ ( sigma_sets_a @ m ) @ ( sigma_measure_a2 @ m ) ).
% measure_increasing
thf(fact_1155_finite__measure__mono,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ A2 ) @ ( sigma_measure_a2 @ m @ B ) ) ) ) ).
% finite_measure_mono
thf(fact_1156_finite__measure__eq__AE,axiom,
! [A2: set_a,B: set_a] :
( ( eventually_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
= ( member_a @ X @ B ) )
@ ( measure_ae_filter_a @ m ) )
=> ( ( member_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ( sigma_measure_a2 @ m @ A2 )
= ( sigma_measure_a2 @ m @ B ) ) ) ) ) ).
% finite_measure_eq_AE
thf(fact_1157_finite__measure__mono__AE,axiom,
! [A2: set_a,B: set_a] :
( ( eventually_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
=> ( member_a @ X @ B ) )
@ ( measure_ae_filter_a @ m ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ A2 ) @ ( sigma_measure_a2 @ m @ B ) ) ) ) ).
% finite_measure_mono_AE
thf(fact_1158_prob__eq__0__AE,axiom,
! [P: a > $o] :
( ( eventually_a
@ ^ [X: a] :
~ ( P @ X )
@ ( measure_ae_filter_a @ m ) )
=> ( ( sigma_measure_a2 @ m
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ X ) ) ) )
= zero_zero_real ) ) ).
% prob_eq_0_AE
thf(fact_1159_prob__eq__0,axiom,
! [A2: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( ( sigma_measure_a2 @ m @ A2 )
= zero_zero_real )
= ( eventually_a
@ ^ [X: a] :
~ ( member_a @ X @ A2 )
@ ( measure_ae_filter_a @ m ) ) ) ) ).
% prob_eq_0
thf(fact_1160_measure__uniform__measure__eq__cond__prob,axiom,
! [P: a > $o,Q: a > $o] :
( ( member_a_o @ P @ ( sigma_measurable_a_o @ m @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( member_a_o @ Q @ ( sigma_measurable_a_o @ m @ ( sigma_count_space_o @ top_top_set_o ) ) )
=> ( ( sigma_measure_a2
@ ( nonneg6757527617543859701sure_a @ m
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( Q @ X ) ) ) )
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X
@ ( sigma_space_a
@ ( nonneg6757527617543859701sure_a @ m
@ ( collect_a
@ ^ [Y3: a] :
( ( member_a @ Y3 @ ( sigma_space_a @ m ) )
& ( Q @ Y3 ) ) ) ) ) )
& ( P @ X ) ) ) )
= ( probab5246806142355027601prob_a @ m @ P @ Q ) ) ) ) ).
% measure_uniform_measure_eq_cond_prob
thf(fact_1161_indep__set__def,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( indepe2041756565122539606_set_a @ m @ A2 @ B )
= ( indepe7780107833195774214ts_a_o @ m @ ( produc6113963288868236716_set_a @ A2 @ B ) @ top_top_set_o ) ) ).
% indep_set_def
thf(fact_1162_card__nat,axiom,
( ( finite_card_nat @ top_top_set_nat )
= zero_zero_nat ) ).
% card_nat
thf(fact_1163_AE__E__prob,axiom,
! [P: a > $o] :
( ( eventually_a @ P @ ( measure_ae_filter_a @ m ) )
=> ~ ! [S3: set_a] :
( ( ord_less_eq_set_a @ S3
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ X ) ) ) )
=> ( ( member_set_a @ S3 @ ( sigma_sets_a @ m ) )
=> ( ( sigma_measure_a2 @ m @ S3 )
!= one_one_real ) ) ) ) ).
% AE_E_prob
thf(fact_1164_prob__Collect__eq__1,axiom,
! [P: a > $o] :
( ( member_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ X ) ) )
@ ( sigma_sets_a @ m ) )
=> ( ( ( sigma_measure_a2 @ m
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ X ) ) ) )
= one_one_real )
= ( eventually_a @ P @ ( measure_ae_filter_a @ m ) ) ) ) ).
% prob_Collect_eq_1
thf(fact_1165_AE__in__set__eq__1,axiom,
! [A2: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( eventually_a
@ ^ [X: a] : ( member_a @ X @ A2 )
@ ( measure_ae_filter_a @ m ) )
= ( ( sigma_measure_a2 @ m @ A2 )
= one_one_real ) ) ) ).
% AE_in_set_eq_1
thf(fact_1166_prob__eq__1,axiom,
! [A2: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( ( sigma_measure_a2 @ m @ A2 )
= one_one_real )
= ( eventually_a
@ ^ [X: a] : ( member_a @ X @ A2 )
@ ( measure_ae_filter_a @ m ) ) ) ) ).
% prob_eq_1
thf(fact_1167_measure__ge__1__iff,axiom,
! [A2: set_a] :
( ( ord_less_eq_real @ one_one_real @ ( sigma_measure_a2 @ m @ A2 ) )
= ( ( sigma_measure_a2 @ m @ A2 )
= one_one_real ) ) ).
% measure_ge_1_iff
thf(fact_1168_prob__space,axiom,
( ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) )
= one_one_real ) ).
% prob_space
thf(fact_1169_AE__prob__1,axiom,
! [A2: set_a] :
( ( ( sigma_measure_a2 @ m @ A2 )
= one_one_real )
=> ( eventually_a
@ ^ [X: a] : ( member_a @ X @ A2 )
@ ( measure_ae_filter_a @ m ) ) ) ).
% AE_prob_1
thf(fact_1170_prob__le__1,axiom,
! [A2: set_a] : ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ A2 ) @ one_one_real ) ).
% prob_le_1
thf(fact_1171_subprob__measure__le__1,axiom,
! [X2: set_a] : ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ X2 ) @ one_one_real ) ).
% subprob_measure_le_1
thf(fact_1172_card__literal,axiom,
( ( finite_card_literal @ top_top_set_literal )
= zero_zero_nat ) ).
% card_literal
thf(fact_1173_kolmogorov__0__1__law,axiom,
! [A2: nat > set_set_a,X2: set_a] :
( ! [I3: nat] : ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ m ) @ ( A2 @ I3 ) )
=> ( ( indepe6267730027088848354_a_nat @ m @ A2 @ top_top_set_nat )
=> ( ( member_set_a @ X2 @ ( indepe7538416700049374166_a_nat @ m @ A2 ) )
=> ( ( ( sigma_measure_a2 @ m @ X2 )
= zero_zero_real )
| ( ( sigma_measure_a2 @ m @ X2 )
= one_one_real ) ) ) ) ) ).
% kolmogorov_0_1_law
thf(fact_1174_measure__exclude,axiom,
! [A2: set_a,B: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ( ( sigma_measure_a2 @ m @ A2 )
= ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) ) )
=> ( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
=> ( ( sigma_measure_a2 @ m @ B )
= zero_zero_real ) ) ) ) ) ).
% measure_exclude
thf(fact_1175_prob__neg,axiom,
! [P: a > $o] :
( ( member_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ X ) ) )
@ ( sigma_sets_a @ m ) )
=> ( ( sigma_measure_a2 @ m
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ~ ( P @ X ) ) ) )
= ( minus_minus_real @ one_one_real
@ ( sigma_measure_a2 @ m
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ X ) ) ) ) ) ) ) ).
% prob_neg
thf(fact_1176_measure__space__inter,axiom,
! [S4: set_a,T3: set_a] :
( ( member_set_a @ S4 @ ( 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 @ S4 @ T3 ) )
= ( sigma_measure_a2 @ m @ S4 ) ) ) ) ) ).
% measure_space_inter
thf(fact_1177_indep__setD,axiom,
! [A2: set_set_a,B: set_set_a,A: set_a,B3: set_a] :
( ( indepe2041756565122539606_set_a @ m @ A2 @ B )
=> ( ( member_set_a @ A @ A2 )
=> ( ( member_set_a @ B3 @ B )
=> ( ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ A @ B3 ) )
= ( times_times_real @ ( sigma_measure_a2 @ m @ A ) @ ( sigma_measure_a2 @ m @ B3 ) ) ) ) ) ) ).
% indep_setD
thf(fact_1178_sigma__algebra__tail__events,axiom,
! [A2: nat > set_set_a] :
( ! [I3: nat] : ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ m ) @ ( A2 @ I3 ) )
=> ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ m ) @ ( indepe7538416700049374166_a_nat @ m @ A2 ) ) ) ).
% sigma_algebra_tail_events
thf(fact_1179_indep__sets2__eq,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( indepe2041756565122539606_set_a @ m @ A2 @ B )
= ( ( ord_le3724670747650509150_set_a @ A2 @ ( sigma_sets_a @ m ) )
& ( ord_le3724670747650509150_set_a @ B @ ( sigma_sets_a @ m ) )
& ! [X: set_a] :
( ( member_set_a @ X @ A2 )
=> ! [Y3: set_a] :
( ( member_set_a @ Y3 @ B )
=> ( ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ X @ Y3 ) )
= ( times_times_real @ ( sigma_measure_a2 @ m @ X ) @ ( sigma_measure_a2 @ m @ Y3 ) ) ) ) ) ) ) ).
% indep_sets2_eq
thf(fact_1180_indep__setI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ! [A4: set_a,B4: set_a] :
( ( member_set_a @ A4 @ A2 )
=> ( ( 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 @ A2 @ B ) ) ) ) ).
% indep_setI
thf(fact_1181_prob__compl,axiom,
! [A2: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ ( sigma_space_a @ m ) @ A2 ) )
= ( minus_minus_real @ one_one_real @ ( sigma_measure_a2 @ m @ A2 ) ) ) ) ).
% prob_compl
thf(fact_1182_integral__bounded__eq__bound__then__AE,axiom,
! [F2: a > real,C: real] :
( ( eventually_a
@ ^ [X: a] : ( ord_less_eq_real @ ( F2 @ X ) @ C )
@ ( measure_ae_filter_a @ m ) )
=> ( ( bochne2139062162225249880a_real @ m @ F2 )
=> ( ( ( bochne378719280626478695a_real @ m @ F2 )
= ( times_times_real @ C @ ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) ) ) )
=> ( eventually_a
@ ^ [X: a] :
( ( F2 @ X )
= C )
@ ( measure_ae_filter_a @ m ) ) ) ) ) ).
% integral_bounded_eq_bound_then_AE
thf(fact_1183_finite__measure__Diff,axiom,
! [A2: set_a,B: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_minus_real @ ( sigma_measure_a2 @ m @ A2 ) @ ( sigma_measure_a2 @ m @ B ) ) ) ) ) ) ).
% finite_measure_Diff
thf(fact_1184_le0,axiom,
! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N4 ) ).
% le0
thf(fact_1185_diff__diff__cancel,axiom,
! [I: nat,N4: nat] :
( ( ord_less_eq_nat @ I @ N4 )
=> ( ( minus_minus_nat @ N4 @ ( minus_minus_nat @ N4 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1186_measure__eq__compl,axiom,
! [S4: set_a,T3: set_a] :
( ( member_set_a @ S4 @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ T3 @ ( sigma_sets_a @ m ) )
=> ( ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ ( sigma_space_a @ m ) @ S4 ) )
= ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ ( sigma_space_a @ m ) @ T3 ) ) )
=> ( ( sigma_measure_a2 @ m @ S4 )
= ( sigma_measure_a2 @ m @ T3 ) ) ) ) ) ).
% measure_eq_compl
thf(fact_1187_finite__measure__compl,axiom,
! [S2: set_a] :
( ( member_set_a @ S2 @ ( sigma_sets_a @ m ) )
=> ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ ( sigma_space_a @ m ) @ S2 ) )
= ( minus_minus_real @ ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) ) @ ( sigma_measure_a2 @ m @ S2 ) ) ) ) ).
% finite_measure_compl
thf(fact_1188_finite__measure__Diff_H,axiom,
! [A2: set_a,B: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_minus_real @ ( sigma_measure_a2 @ m @ A2 ) @ ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ) ) ).
% finite_measure_Diff'
thf(fact_1189_integral__le__const,axiom,
! [F2: a > real,C: real] :
( ( bochne2139062162225249880a_real @ m @ F2 )
=> ( ( eventually_a
@ ^ [X: a] : ( ord_less_eq_real @ ( F2 @ X ) @ C )
@ ( measure_ae_filter_a @ m ) )
=> ( ord_less_eq_real @ ( bochne378719280626478695a_real @ m @ F2 ) @ C ) ) ) ).
% integral_le_const
thf(fact_1190_integral__ge__const,axiom,
! [F2: a > real,C: real] :
( ( bochne2139062162225249880a_real @ m @ F2 )
=> ( ( eventually_a
@ ^ [X: a] : ( ord_less_eq_real @ C @ ( F2 @ X ) )
@ ( measure_ae_filter_a @ m ) )
=> ( ord_less_eq_real @ C @ ( bochne378719280626478695a_real @ m @ F2 ) ) ) ) ).
% integral_ge_const
thf(fact_1191_diff__is__0__eq_H,axiom,
! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ( minus_minus_nat @ M5 @ N4 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1192_diff__is__0__eq,axiom,
! [M5: nat,N4: nat] :
( ( ( minus_minus_nat @ M5 @ N4 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M5 @ N4 ) ) ).
% diff_is_0_eq
thf(fact_1193_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1194_le__cube,axiom,
! [M5: nat] : ( ord_less_eq_nat @ M5 @ ( times_times_nat @ M5 @ ( times_times_nat @ M5 @ M5 ) ) ) ).
% le_cube
thf(fact_1195_le__square,axiom,
! [M5: nat] : ( ord_less_eq_nat @ M5 @ ( times_times_nat @ M5 @ M5 ) ) ).
% le_square
thf(fact_1196_eq__diff__iff,axiom,
! [K: nat,M5: nat,N4: nat] :
( ( ord_less_eq_nat @ K @ M5 )
=> ( ( ord_less_eq_nat @ K @ N4 )
=> ( ( ( minus_minus_nat @ M5 @ K )
= ( minus_minus_nat @ N4 @ K ) )
= ( M5 = N4 ) ) ) ) ).
% eq_diff_iff
thf(fact_1197_le__diff__iff,axiom,
! [K: nat,M5: nat,N4: nat] :
( ( ord_less_eq_nat @ K @ M5 )
=> ( ( ord_less_eq_nat @ K @ N4 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M5 @ K ) @ ( minus_minus_nat @ N4 @ K ) )
= ( ord_less_eq_nat @ M5 @ N4 ) ) ) ) ).
% le_diff_iff
thf(fact_1198_Nat_Odiff__diff__eq,axiom,
! [K: nat,M5: nat,N4: nat] :
( ( ord_less_eq_nat @ K @ M5 )
=> ( ( ord_less_eq_nat @ K @ N4 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M5 @ K ) @ ( minus_minus_nat @ N4 @ K ) )
= ( minus_minus_nat @ M5 @ N4 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1199_diff__le__mono,axiom,
! [M5: nat,N4: nat,L2: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M5 @ L2 ) @ ( minus_minus_nat @ N4 @ L2 ) ) ) ).
% diff_le_mono
thf(fact_1200_diff__le__self,axiom,
! [M5: nat,N4: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M5 @ N4 ) @ M5 ) ).
% diff_le_self
thf(fact_1201_le__diff__iff_H,axiom,
! [A: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B3 ) )
= ( ord_less_eq_nat @ B3 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1202_mult__le__mono,axiom,
! [I: nat,J3: nat,K: nat,L2: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( ord_less_eq_nat @ K @ L2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J3 @ L2 ) ) ) ) ).
% mult_le_mono
thf(fact_1203_diff__le__mono2,axiom,
! [M5: nat,N4: nat,L2: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N4 ) @ ( minus_minus_nat @ L2 @ M5 ) ) ) ).
% diff_le_mono2
thf(fact_1204_mult__le__mono1,axiom,
! [I: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J3 @ K ) ) ) ).
% mult_le_mono1
thf(fact_1205_mult__le__mono2,axiom,
! [I: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J3 ) ) ) ).
% mult_le_mono2
thf(fact_1206_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1207_le__refl,axiom,
! [N4: nat] : ( ord_less_eq_nat @ N4 @ N4 ) ).
% le_refl
thf(fact_1208_le__trans,axiom,
! [I: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( ord_less_eq_nat @ J3 @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_1209_eq__imp__le,axiom,
! [M5: nat,N4: nat] :
( ( M5 = N4 )
=> ( ord_less_eq_nat @ M5 @ N4 ) ) ).
% eq_imp_le
thf(fact_1210_le__antisym,axiom,
! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ( ord_less_eq_nat @ N4 @ M5 )
=> ( M5 = N4 ) ) ) ).
% le_antisym
thf(fact_1211_nat__le__linear,axiom,
! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
| ( ord_less_eq_nat @ N4 @ M5 ) ) ).
% nat_le_linear
thf(fact_1212_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B3 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1213_less__eq__nat_Osimps_I1_J,axiom,
! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N4 ) ).
% less_eq_nat.simps(1)
thf(fact_1214_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1215_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1216_le__0__eq,axiom,
! [N4: nat] :
( ( ord_less_eq_nat @ N4 @ zero_zero_nat )
= ( N4 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1217_integral__less__AE,axiom,
! [X2: a > real,Y: a > real,A2: set_a] :
( ( bochne2139062162225249880a_real @ m @ X2 )
=> ( ( bochne2139062162225249880a_real @ m @ Y )
=> ( ( ( sigma_emeasure_a @ m @ A2 )
!= zero_z7100319975126383169nnreal )
=> ( ( member_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( eventually_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
=> ( ( X2 @ X )
!= ( Y @ X ) ) )
@ ( measure_ae_filter_a @ m ) )
=> ( ( eventually_a
@ ^ [X: a] : ( ord_less_eq_real @ ( X2 @ X ) @ ( Y @ X ) )
@ ( measure_ae_filter_a @ m ) )
=> ( ord_less_real @ ( bochne378719280626478695a_real @ m @ X2 ) @ ( bochne378719280626478695a_real @ m @ Y ) ) ) ) ) ) ) ) ).
% integral_less_AE
thf(fact_1218_integral__less__AE__space,axiom,
! [X2: a > real,Y: a > real] :
( ( bochne2139062162225249880a_real @ m @ X2 )
=> ( ( bochne2139062162225249880a_real @ m @ Y )
=> ( ( eventually_a
@ ^ [X: a] : ( ord_less_real @ ( X2 @ X ) @ ( Y @ X ) )
@ ( measure_ae_filter_a @ m ) )
=> ( ( ( sigma_emeasure_a @ m @ ( sigma_space_a @ m ) )
!= zero_z7100319975126383169nnreal )
=> ( ord_less_real @ ( bochne378719280626478695a_real @ m @ X2 ) @ ( bochne378719280626478695a_real @ m @ Y ) ) ) ) ) ) ).
% integral_less_AE_space
thf(fact_1219_expectation__less,axiom,
! [X2: a > real,B3: real] :
( ( bochne2139062162225249880a_real @ m @ X2 )
=> ( ( eventually_a
@ ^ [X: a] : ( ord_less_real @ ( X2 @ X ) @ B3 )
@ ( measure_ae_filter_a @ m ) )
=> ( ord_less_real @ ( bochne378719280626478695a_real @ m @ X2 ) @ B3 ) ) ) ).
% expectation_less
thf(fact_1220_expectation__greater,axiom,
! [X2: a > real,A: real] :
( ( bochne2139062162225249880a_real @ m @ X2 )
=> ( ( eventually_a
@ ^ [X: a] : ( ord_less_real @ A @ ( X2 @ X ) )
@ ( measure_ae_filter_a @ m ) )
=> ( ord_less_real @ A @ ( bochne378719280626478695a_real @ m @ X2 ) ) ) ) ).
% expectation_greater
thf(fact_1221_obtain__positive__integrable__function,axiom,
~ ! [F6: a > real] :
( ( member_a_real @ F6 @ ( sigma_9116425665531756122a_real @ m @ borel_5078946678739801102l_real ) )
=> ( ! [X6: a] : ( ord_less_real @ zero_zero_real @ ( F6 @ X6 ) )
=> ( ! [X6: a] : ( ord_less_eq_real @ ( F6 @ X6 ) @ one_one_real )
=> ~ ( bochne2139062162225249880a_real @ m @ F6 ) ) ) ) ).
% obtain_positive_integrable_function
thf(fact_1222_ennreal__top__mult__left,axiom,
! [X4: extend8495563244428889912nnreal] :
( ( X4 != zero_z7100319975126383169nnreal )
=> ( ( times_1893300245718287421nnreal @ X4 @ top_to1496364449551166952nnreal )
= top_to1496364449551166952nnreal ) ) ).
% ennreal_top_mult_left
thf(fact_1223_ennreal__top__mult__right,axiom,
! [X4: extend8495563244428889912nnreal] :
( ( X4 != zero_z7100319975126383169nnreal )
=> ( ( times_1893300245718287421nnreal @ top_to1496364449551166952nnreal @ X4 )
= top_to1496364449551166952nnreal ) ) ).
% ennreal_top_mult_right
thf(fact_1224_ennreal__top__minus,axiom,
! [X4: extend8495563244428889912nnreal] :
( ( minus_8429688780609304081nnreal @ top_to1496364449551166952nnreal @ X4 )
= top_to1496364449551166952nnreal ) ).
% ennreal_top_minus
thf(fact_1225_ennreal__minus__eq__top,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A @ B3 )
= top_to1496364449551166952nnreal )
= ( A = top_to1496364449551166952nnreal ) ) ).
% ennreal_minus_eq_top
thf(fact_1226_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_nat @ N3 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_1227_card__Collect__less__nat,axiom,
! [N4: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_nat @ I4 @ N4 ) ) )
= N4 ) ).
% card_Collect_less_nat
thf(fact_1228_ennreal__diff__self,axiom,
! [A: extend8495563244428889912nnreal] :
( ( A != top_to1496364449551166952nnreal )
=> ( ( minus_8429688780609304081nnreal @ A @ A )
= zero_z7100319975126383169nnreal ) ) ).
% ennreal_diff_self
thf(fact_1229_mult__le__cancel2,axiom,
! [M5: nat,K: nat,N4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M5 @ K ) @ ( times_times_nat @ N4 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M5 @ N4 ) ) ) ).
% mult_le_cancel2
thf(fact_1230_ennreal__mono__minus__cancel,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B3 ) @ ( minus_8429688780609304081nnreal @ A @ C ) )
=> ( ( ord_le7381754540660121996nnreal @ A @ top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ B3 @ A )
=> ( ( ord_le3935885782089961368nnreal @ C @ A )
=> ( ord_le3935885782089961368nnreal @ C @ B3 ) ) ) ) ) ).
% ennreal_mono_minus_cancel
thf(fact_1231_diff__less__top__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ ( minus_8429688780609304081nnreal @ A @ B3 ) @ top_to1496364449551166952nnreal )
= ( ord_le7381754540660121996nnreal @ A @ top_to1496364449551166952nnreal ) ) ).
% diff_less_top_ennreal
thf(fact_1232_ennreal__mult__strict__right__mono,axiom,
! [A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ C )
=> ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ B3 )
=> ( ( ord_le7381754540660121996nnreal @ B3 @ top_to1496364449551166952nnreal )
=> ( ord_le7381754540660121996nnreal @ ( times_1893300245718287421nnreal @ A @ B3 ) @ ( times_1893300245718287421nnreal @ C @ B3 ) ) ) ) ) ).
% ennreal_mult_strict_right_mono
thf(fact_1233_ennreal__mult__strict__left__mono,axiom,
! [A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ C )
=> ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ B3 )
=> ( ( ord_le7381754540660121996nnreal @ B3 @ top_to1496364449551166952nnreal )
=> ( ord_le7381754540660121996nnreal @ ( times_1893300245718287421nnreal @ B3 @ A ) @ ( times_1893300245718287421nnreal @ B3 @ C ) ) ) ) ) ).
% ennreal_mult_strict_left_mono
thf(fact_1234_diff__gt__0__iff__gt__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( minus_8429688780609304081nnreal @ A @ B3 ) )
= ( ( ( A = top_to1496364449551166952nnreal )
& ( B3 = top_to1496364449551166952nnreal ) )
| ( ord_le7381754540660121996nnreal @ B3 @ A ) ) ) ).
% diff_gt_0_iff_gt_ennreal
thf(fact_1235_ennreal__mult__less__top,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ ( times_1893300245718287421nnreal @ A @ B3 ) @ top_to1496364449551166952nnreal )
= ( ( A = zero_z7100319975126383169nnreal )
| ( B3 = zero_z7100319975126383169nnreal )
| ( ( ord_le7381754540660121996nnreal @ A @ top_to1496364449551166952nnreal )
& ( ord_le7381754540660121996nnreal @ B3 @ top_to1496364449551166952nnreal ) ) ) ) ).
% ennreal_mult_less_top
thf(fact_1236_ennreal__minus__pos__iff,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ( ord_le7381754540660121996nnreal @ A @ top_to1496364449551166952nnreal )
| ( ord_le7381754540660121996nnreal @ B3 @ top_to1496364449551166952nnreal ) )
=> ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( minus_8429688780609304081nnreal @ A @ B3 ) )
=> ( ord_le7381754540660121996nnreal @ B3 @ A ) ) ) ).
% ennreal_minus_pos_iff
thf(fact_1237_ennreal__between,axiom,
! [E: extend8495563244428889912nnreal,X4: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ E )
=> ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ X4 )
=> ( ( ord_le7381754540660121996nnreal @ X4 @ top_to1496364449551166952nnreal )
=> ( ord_le7381754540660121996nnreal @ ( minus_8429688780609304081nnreal @ X4 @ E ) @ X4 ) ) ) ) ).
% ennreal_between
thf(fact_1238_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I: nat,J3: nat] :
( ! [I3: nat,J4: nat] :
( ( ord_less_nat @ I3 @ J4 )
=> ( ord_less_nat @ ( F2 @ I3 ) @ ( F2 @ J4 ) ) )
=> ( ( ord_less_eq_nat @ I @ J3 )
=> ( ord_less_eq_nat @ ( F2 @ I ) @ ( F2 @ J3 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1239_le__neq__implies__less,axiom,
! [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
=> ( ( M5 != N4 )
=> ( ord_less_nat @ M5 @ N4 ) ) ) ).
% le_neq_implies_less
thf(fact_1240_less__or__eq__imp__le,axiom,
! [M5: nat,N4: nat] :
( ( ( ord_less_nat @ M5 @ N4 )
| ( M5 = N4 ) )
=> ( ord_less_eq_nat @ M5 @ N4 ) ) ).
% less_or_eq_imp_le
thf(fact_1241_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
| ( M4 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1242_less__imp__le__nat,axiom,
! [M5: nat,N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ( ord_less_eq_nat @ M5 @ N4 ) ) ).
% less_imp_le_nat
thf(fact_1243_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
& ( M4 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_1244_ennreal__zero__less__top,axiom,
ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ top_to1496364449551166952nnreal ).
% ennreal_zero_less_top
thf(fact_1245_ennreal__one__less__top,axiom,
ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ top_to1496364449551166952nnreal ).
% ennreal_one_less_top
thf(fact_1246_ex__least__nat__le,axiom,
! [P: nat > $o,N4: nat] :
( ( P @ N4 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N4 )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1247_diff__less__mono,axiom,
! [A: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B3 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1248_less__diff__iff,axiom,
! [K: nat,M5: nat,N4: nat] :
( ( ord_less_eq_nat @ K @ M5 )
=> ( ( ord_less_eq_nat @ K @ N4 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M5 @ K ) @ ( minus_minus_nat @ N4 @ K ) )
= ( ord_less_nat @ M5 @ N4 ) ) ) ) ).
% less_diff_iff
thf(fact_1249_diff__eq__0__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A @ top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ A @ B3 )
=> ( ( minus_8429688780609304081nnreal @ A @ B3 )
= zero_z7100319975126383169nnreal ) ) ) ).
% diff_eq_0_ennreal
thf(fact_1250_diff__eq__0__iff__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A @ B3 )
= zero_z7100319975126383169nnreal )
= ( ( ord_le7381754540660121996nnreal @ A @ top_to1496364449551166952nnreal )
& ( ord_le3935885782089961368nnreal @ A @ B3 ) ) ) ).
% diff_eq_0_iff_ennreal
thf(fact_1251_ennreal__approx__unit,axiom,
! [Z2: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ! [A4: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ A4 )
=> ( ( ord_le7381754540660121996nnreal @ A4 @ one_on2969667320475766781nnreal )
=> ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ A4 @ Z2 ) @ Y2 ) ) )
=> ( ord_le3935885782089961368nnreal @ Z2 @ Y2 ) ) ).
% ennreal_approx_unit
thf(fact_1252_bot__ennreal,axiom,
bot_bo841427958541957580nnreal = zero_z7100319975126383169nnreal ).
% bot_ennreal
thf(fact_1253_ennreal__minus__mono,axiom,
! [A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D2: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ C )
=> ( ( ord_le3935885782089961368nnreal @ D2 @ B3 )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B3 ) @ ( minus_8429688780609304081nnreal @ C @ D2 ) ) ) ) ).
% ennreal_minus_mono
thf(fact_1254_ennreal__mono__minus,axiom,
! [C: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ C @ B3 )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B3 ) @ ( minus_8429688780609304081nnreal @ A @ C ) ) ) ).
% ennreal_mono_minus
thf(fact_1255_diff__le__self__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B3 ) @ A ) ).
% diff_le_self_ennreal
thf(fact_1256_ennreal__diff__le__mono__left,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B3 )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ C ) @ B3 ) ) ).
% ennreal_diff_le_mono_left
thf(fact_1257_ennreal__right__diff__distrib,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( A != top_to1496364449551166952nnreal )
=> ( ( times_1893300245718287421nnreal @ A @ ( minus_8429688780609304081nnreal @ B3 @ C ) )
= ( minus_8429688780609304081nnreal @ ( times_1893300245718287421nnreal @ A @ B3 ) @ ( times_1893300245718287421nnreal @ A @ C ) ) ) ) ).
% ennreal_right_diff_distrib
thf(fact_1258_ennreal__minus__eq__0,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A @ B3 )
= zero_z7100319975126383169nnreal )
=> ( ord_le3935885782089961368nnreal @ A @ B3 ) ) ).
% ennreal_minus_eq_0
thf(fact_1259_minus__top__ennreal,axiom,
! [X4: extend8495563244428889912nnreal] :
( ( ( X4 = top_to1496364449551166952nnreal )
=> ( ( minus_8429688780609304081nnreal @ X4 @ top_to1496364449551166952nnreal )
= top_to1496364449551166952nnreal ) )
& ( ( X4 != top_to1496364449551166952nnreal )
=> ( ( minus_8429688780609304081nnreal @ X4 @ top_to1496364449551166952nnreal )
= zero_z7100319975126383169nnreal ) ) ) ).
% minus_top_ennreal
thf(fact_1260_ennreal__mult__cancel__left,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ( times_1893300245718287421nnreal @ A @ B3 )
= ( times_1893300245718287421nnreal @ A @ C ) )
= ( ( ( A = top_to1496364449551166952nnreal )
& ( B3 != zero_z7100319975126383169nnreal )
& ( C != zero_z7100319975126383169nnreal ) )
| ( A = zero_z7100319975126383169nnreal )
| ( B3 = C ) ) ) ).
% ennreal_mult_cancel_left
thf(fact_1261_ennreal__top__eq__mult__iff,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( top_to1496364449551166952nnreal
= ( times_1893300245718287421nnreal @ A @ B3 ) )
= ( ( ( A = top_to1496364449551166952nnreal )
& ( B3 != zero_z7100319975126383169nnreal ) )
| ( ( B3 = top_to1496364449551166952nnreal )
& ( A != zero_z7100319975126383169nnreal ) ) ) ) ).
% ennreal_top_eq_mult_iff
thf(fact_1262_ennreal__mult__eq__top__iff,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ( times_1893300245718287421nnreal @ A @ B3 )
= top_to1496364449551166952nnreal )
= ( ( ( A = top_to1496364449551166952nnreal )
& ( B3 != zero_z7100319975126383169nnreal ) )
| ( ( B3 = top_to1496364449551166952nnreal )
& ( A != zero_z7100319975126383169nnreal ) ) ) ) ).
% ennreal_mult_eq_top_iff
thf(fact_1263_ennreal__top__mult,axiom,
! [A: extend8495563244428889912nnreal] :
( ( ( A = zero_z7100319975126383169nnreal )
=> ( ( times_1893300245718287421nnreal @ top_to1496364449551166952nnreal @ A )
= zero_z7100319975126383169nnreal ) )
& ( ( A != zero_z7100319975126383169nnreal )
=> ( ( times_1893300245718287421nnreal @ top_to1496364449551166952nnreal @ A )
= top_to1496364449551166952nnreal ) ) ) ).
% ennreal_top_mult
thf(fact_1264_ennreal__mult__top,axiom,
! [A: extend8495563244428889912nnreal] :
( ( ( A = zero_z7100319975126383169nnreal )
=> ( ( times_1893300245718287421nnreal @ A @ top_to1496364449551166952nnreal )
= zero_z7100319975126383169nnreal ) )
& ( ( A != zero_z7100319975126383169nnreal )
=> ( ( times_1893300245718287421nnreal @ A @ top_to1496364449551166952nnreal )
= top_to1496364449551166952nnreal ) ) ) ).
% ennreal_mult_top
thf(fact_1265_ennreal__zero__neq__top,axiom,
zero_z7100319975126383169nnreal != top_to1496364449551166952nnreal ).
% ennreal_zero_neq_top
thf(fact_1266_ennreal__minus__cancel__iff,axiom,
! [A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A @ B3 )
= ( minus_8429688780609304081nnreal @ A @ C ) )
= ( ( B3 = C )
| ( ( ord_le3935885782089961368nnreal @ A @ B3 )
& ( ord_le3935885782089961368nnreal @ A @ C ) )
| ( A = top_to1496364449551166952nnreal ) ) ) ).
% ennreal_minus_cancel_iff
thf(fact_1267_ennreal__minus__cancel,axiom,
! [C: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( C != top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ A @ C )
=> ( ( ord_le3935885782089961368nnreal @ B3 @ C )
=> ( ( ( minus_8429688780609304081nnreal @ C @ A )
= ( minus_8429688780609304081nnreal @ C @ B3 ) )
=> ( A = B3 ) ) ) ) ) ).
% ennreal_minus_cancel
thf(fact_1268_neq__top__trans,axiom,
! [Y2: extend8495563244428889912nnreal,X4: extend8495563244428889912nnreal] :
( ( Y2 != top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ X4 @ Y2 )
=> ( X4 != top_to1496364449551166952nnreal ) ) ) ).
% neq_top_trans
thf(fact_1269_ennreal__top__neq__one,axiom,
top_to1496364449551166952nnreal != one_on2969667320475766781nnreal ).
% ennreal_top_neq_one
thf(fact_1270_ennreal__mult__le__mult__iff,axiom,
! [C: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( C != zero_z7100319975126383169nnreal )
=> ( ( C != top_to1496364449551166952nnreal )
=> ( ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ C @ A ) @ ( times_1893300245718287421nnreal @ C @ B3 ) )
= ( ord_le3935885782089961368nnreal @ A @ B3 ) ) ) ) ).
% ennreal_mult_le_mult_iff
thf(fact_1271_nat__mult__le__cancel__disj,axiom,
! [K: nat,M5: nat,N4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M5 ) @ ( times_times_nat @ K @ N4 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M5 @ N4 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1272_nat__mult__le__cancel1,axiom,
! [K: nat,M5: nat,N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M5 ) @ ( times_times_nat @ K @ N4 ) )
= ( ord_less_eq_nat @ M5 @ N4 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1273_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 ) ) ) )
=> ? [L3: ( nat > real ) > nat > nat] :
( ! [X6: nat > real,I5: nat] : ( ord_less_eq_nat @ ( L3 @ X6 @ I5 ) @ one_one_nat )
& ! [X6: nat > real,I5: nat] :
( ( ( P @ X6 )
& ( Q @ I5 )
& ( ( X6 @ I5 )
= zero_zero_real ) )
=> ( ( L3 @ X6 @ I5 )
= zero_zero_nat ) )
& ! [X6: nat > real,I5: nat] :
( ( ( P @ X6 )
& ( Q @ I5 )
& ( ( X6 @ I5 )
= one_one_real ) )
=> ( ( L3 @ X6 @ I5 )
= one_one_nat ) )
& ! [X6: nat > real,I5: nat] :
( ( ( P @ X6 )
& ( Q @ I5 )
& ( ( L3 @ X6 @ I5 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X6 @ I5 ) @ ( F2 @ X6 @ I5 ) ) )
& ! [X6: nat > real,I5: nat] :
( ( ( P @ X6 )
& ( Q @ I5 )
& ( ( L3 @ X6 @ I5 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F2 @ X6 @ I5 ) @ ( X6 @ I5 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1274_card__UNIV__unit,axiom,
( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
= one_one_nat ) ).
% card_UNIV_unit
thf(fact_1275_nn__integral__ge__const,axiom,
! [C: extend8495563244428889912nnreal,F2: a > extend8495563244428889912nnreal] :
( ( eventually_a
@ ^ [X: a] : ( ord_le3935885782089961368nnreal @ C @ ( F2 @ X ) )
@ ( measure_ae_filter_a @ m ) )
=> ( ord_le3935885782089961368nnreal @ C @ ( nonneg2725512125972007571gral_a @ m @ F2 ) ) ) ).
% nn_integral_ge_const
% Conjectures (1)
thf(conj_0,conjecture,
( prob_k6574085460301583243_a_b_d @ m @ k @ n
@ ^ [I4: b,X: a] : ( y @ I4 @ ( x @ I4 @ X ) )
@ i ) ).
%------------------------------------------------------------------------------