TPTP Problem File: SLH0198^1.p

%------------------------------------------------------------------------------
% 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_00080_002889__18475494_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1989 ( 246 unt; 710 typ;   0 def)
%            Number of atoms       : 4261 ( 918 equ;   0 cnn)
%            Maximal formula atoms :    7 (   3 avg)
%            Number of connectives : 16964 ( 305   ~;   8   |; 128   &;14082   @)
%                                         (   0 <=>;2441  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   20 (   9 avg)
%            Number of types       :  123 ( 122 usr)
%            Number of type conns  : 3736 (3736   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  591 ( 588 usr;  37 con; 0-5 aty)
%            Number of variables   : 4829 ( 462   ^;4299   !;  68   ?;4829   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:44:34.815
%------------------------------------------------------------------------------
% Could-be-implicit typings (122)
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    comple3428971583294703880tion_a: sigma_measure_a > sigma_measure_a ).

thf(sy_c_Complete__Measure_Ocompletion_001tf__b,type,
    comple3428971583294703881tion_b: sigma_measure_b > sigma_measure_b ).

thf(sy_c_Countable__Set_Ocountable_001tf__a,type,
    counta4098120917673242425able_a: set_a > $o ).

thf(sy_c_Countable__Set_Ocountable_001tf__b,type,
    counta4098120917673242426able_b: set_b > $o ).

thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001tf__a_001tf__a,type,
    prob_k138169005419483465_a_a_a: sigma_measure_a > nat > ( a > sigma_measure_a ) > ( a > a > a ) > set_a > $o ).

thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001tf__a_001tf__b,type,
    prob_k138169005419483466_a_a_b: sigma_measure_a > nat > ( a > sigma_measure_b ) > ( a > a > b ) > set_a > $o ).

thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001tf__a_001tf__c,type,
    prob_k138169005419483467_a_a_c: sigma_measure_a > nat > ( a > sigma_measure_c ) > ( a > a > c ) > set_a > $o ).

thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001tf__b_001tf__a,type,
    prob_k6574085460301583240_a_b_a: sigma_measure_a > nat > ( b > sigma_measure_a ) > ( b > a > a ) > set_b > $o ).

thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001tf__b_001tf__b,type,
    prob_k6574085460301583241_a_b_b: sigma_measure_a > nat > ( b > sigma_measure_b ) > ( b > a > b ) > set_b > $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__c_001tf__a,type,
    prob_k3786629878328907207_a_c_a: sigma_measure_a > nat > ( c > sigma_measure_a ) > ( c > a > a ) > set_c > $o ).

thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001tf__c_001tf__b,type,
    prob_k3786629878328907208_a_c_b: sigma_measure_a > nat > ( c > sigma_measure_b ) > ( c > a > b ) > set_c > $o ).

thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001tf__c_001tf__c,type,
    prob_k3786629878328907209_a_c_c: sigma_measure_a > nat > ( c > sigma_measure_c ) > ( c > a > c ) > set_c > $o ).

thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__b_001tf__a_001tf__a,type,
    prob_k5734387620607614154_b_a_a: sigma_measure_b > nat > ( a > sigma_measure_a ) > ( a > b > a ) > set_a > $o ).

thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__b_001tf__a_001tf__b,type,
    prob_k5734387620607614155_b_a_b: sigma_measure_b > nat > ( a > sigma_measure_b ) > ( a > b > b ) > set_a > $o ).

thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__b_001tf__a_001tf__c,type,
    prob_k5734387620607614156_b_a_c: sigma_measure_b > nat > ( a > sigma_measure_c ) > ( a > b > c ) > set_a > $o ).

thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__b_001tf__b_001tf__a,type,
    prob_k2946932038634938121_b_b_a: sigma_measure_b > nat > ( b > sigma_measure_a ) > ( b > b > a ) > set_b > $o ).

thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__b_001tf__b_001tf__b,type,
    prob_k2946932038634938122_b_b_b: sigma_measure_b > nat > ( b > sigma_measure_b ) > ( b > b > b ) > set_b > $o ).

thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__b_001tf__b_001tf__c,type,
    prob_k2946932038634938123_b_b_c: sigma_measure_b > nat > ( b > sigma_measure_c ) > ( b > b > c ) > set_b > $o ).

thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__b_001tf__c_001tf__a,type,
    prob_k159476456662262088_b_c_a: sigma_measure_b > nat > ( c > sigma_measure_a ) > ( c > b > a ) > set_c > $o ).

thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__b_001tf__c_001tf__b,type,
    prob_k159476456662262089_b_c_b: sigma_measure_b > nat > ( c > sigma_measure_b ) > ( c > b > b ) > set_c > $o ).

thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__b_001tf__c_001tf__c,type,
    prob_k159476456662262090_b_c_c: sigma_measure_b > nat > ( c > sigma_measure_c ) > ( c > b > c ) > set_c > $o ).

thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Ounit,type,
    finite410649719033368117t_unit: set_Product_unit > nat ).

thf(sy_c_Finite__Set_Ofinite_001_062_Itf__a_M_062_Itf__a_Mt__Real__Oreal_J_J,type,
    finite2647881322281930521a_real: set_a_a_real > $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_Eo,type,
    finite_finite_o: set_o > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
    finite_finite_nat: set_nat > $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__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__Set__Oset_Itf__b_J_J,type,
    finite7280322010187229928_set_b: set_set_set_b > $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__Set__Oset_Itf__b_J,type,
    finite_finite_set_b: set_set_b > $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__c,type,
    finite_finite_c: set_c > $o ).

thf(sy_c_Fun_Ocomp_001_062_Itf__a_Mt__Real__Oreal_J_001t__Sigma____Algebra__Omeasure_I_062_Itf__a_Mt__Real__Oreal_J_J_001tf__a,type,
    comp_a1543360357282447724real_a: ( ( a > real ) > sigma_measure_a_real ) > ( a > a > real ) > a > sigma_measure_a_real ).

thf(sy_c_Fun_Ocomp_001_062_Itf__a_Mtf__b_J_001t__Sigma____Algebra__Omeasure_I_062_Itf__a_Mtf__b_J_J_001tf__a,type,
    comp_a7517041177194281544_a_b_a: ( ( a > b ) > sigma_measure_a_b ) > ( a > a > b ) > a > sigma_measure_a_b ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J_001tf__a,type,
    comp_r70652391186890568real_a: ( real > sigma_measure_real ) > ( a > real ) > a > sigma_measure_real ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J_001tf__b,type,
    comp_r70652391186890569real_b: ( real > sigma_measure_real ) > ( b > real ) > b > sigma_measure_real ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001tf__b_001tf__a,type,
    comp_real_b_a: ( real > b ) > ( a > real ) > a > b ).

thf(sy_c_Fun_Ocomp_001tf__a_001_062_Itf__a_Mt__Real__Oreal_J_001tf__a,type,
    comp_a_a_real_a: ( a > a > real ) > ( a > a ) > a > a > real ).

thf(sy_c_Fun_Ocomp_001tf__a_001_062_Itf__a_Mt__Real__Oreal_J_001tf__b,type,
    comp_a_a_real_b: ( a > a > real ) > ( b > a ) > b > a > real ).

thf(sy_c_Fun_Ocomp_001tf__a_001_062_Itf__a_Mtf__b_J_001tf__a,type,
    comp_a_a_b_a: ( a > a > b ) > ( a > a ) > a > a > b ).

thf(sy_c_Fun_Ocomp_001tf__a_001_062_Itf__a_Mtf__b_J_001tf__b,type,
    comp_a_a_b_b: ( a > a > b ) > ( b > a ) > b > a > b ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Real__Oreal_001tf__a,type,
    comp_a_real_a: ( a > real ) > ( a > a ) > a > real ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Real__Oreal_001tf__b,type,
    comp_a_real_b: ( a > real ) > ( b > a ) > b > real ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Sigma____Algebra__Omeasure_Itf__a_J_001tf__a,type,
    comp_a5399727294799983398re_a_a: ( a > sigma_measure_a ) > ( a > a ) > a > sigma_measure_a ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Sigma____Algebra__Omeasure_Itf__a_J_001tf__b,type,
    comp_a5399727294799983399re_a_b: ( a > sigma_measure_a ) > ( b > a ) > b > sigma_measure_a ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001tf__a,type,
    comp_a_a_a: ( a > a ) > ( a > a ) > a > a ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001tf__b,type,
    comp_a_a_b: ( a > a ) > ( b > a ) > b > a ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__b_001tf__a,type,
    comp_a_b_a: ( a > b ) > ( a > a ) > a > b ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__b_001tf__b,type,
    comp_a_b_b: ( a > b ) > ( b > a ) > b > b ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__c_001tf__a,type,
    comp_a_c_a: ( a > c ) > ( a > a ) > a > c ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__c_001tf__b,type,
    comp_a_c_b: ( a > c ) > ( b > a ) > b > c ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__d_001tf__b,type,
    comp_a_d_b: ( a > d ) > ( b > a ) > b > d ).

thf(sy_c_Fun_Ocomp_001tf__b_001t__Real__Oreal_001tf__a,type,
    comp_b_real_a: ( b > real ) > ( a > b ) > a > real ).

thf(sy_c_Fun_Ocomp_001tf__b_001t__Real__Oreal_001tf__b,type,
    comp_b_real_b: ( b > real ) > ( b > b ) > b > real ).

thf(sy_c_Fun_Ocomp_001tf__b_001t__Sigma____Algebra__Omeasure_Itf__b_J_001tf__a,type,
    comp_b1755296839527801508re_b_a: ( b > sigma_measure_b ) > ( a > b ) > a > sigma_measure_b ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__a_001tf__a,type,
    comp_b_a_a: ( b > a ) > ( a > b ) > a > a ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__a_001tf__b,type,
    comp_b_a_b: ( b > a ) > ( b > b ) > b > a ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__b_001tf__a,type,
    comp_b_b_a: ( b > b ) > ( a > b ) > a > b ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__d_001tf__a,type,
    comp_b_d_a: ( b > d ) > ( a > b ) > a > d ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__d_001tf__b,type,
    comp_b_d_b: ( b > d ) > ( b > b ) > b > d ).

thf(sy_c_Fun_Ocomp_001tf__c_001t__Real__Oreal_001tf__a,type,
    comp_c_real_a: ( c > real ) > ( a > c ) > a > real ).

thf(sy_c_Fun_Ocomp_001tf__c_001t__Sigma____Algebra__Omeasure_Itf__c_J_001tf__a,type,
    comp_c7334238421110395426re_c_a: ( c > sigma_measure_c ) > ( a > c ) > a > sigma_measure_c ).

thf(sy_c_Fun_Ocomp_001tf__c_001tf__a_001tf__a,type,
    comp_c_a_a: ( c > a ) > ( a > c ) > a > a ).

thf(sy_c_Fun_Ocomp_001tf__c_001tf__b_001tf__a,type,
    comp_c_b_a: ( c > b ) > ( a > c ) > a > b ).

thf(sy_c_Fun_Ocomp_001tf__c_001tf__c_001tf__a,type,
    comp_c_c_a: ( c > c ) > ( a > c ) > a > c ).

thf(sy_c_Fun_Ocomp_001tf__d_001t__Sigma____Algebra__Omeasure_Itf__d_J_001tf__b,type,
    comp_d3689807965838213537re_d_b: ( d > sigma_measure_d ) > ( b > d ) > b > sigma_measure_d ).

thf(sy_c_FuncSet_OPi_001tf__a_001_062_Itf__a_Mt__Real__Oreal_J,type,
    pi_a_a_real: set_a > ( a > set_a_real ) > set_a_a_real ).

thf(sy_c_FuncSet_OPi_001tf__a_001_062_Itf__a_Mtf__b_J,type,
    pi_a_a_b: set_a > ( a > set_a_b ) > set_a_a_b ).

thf(sy_c_FuncSet_OPi_001tf__a_001t__Real__Oreal,type,
    pi_a_real: set_a > ( a > set_real ) > set_a_real ).

thf(sy_c_FuncSet_OPi_001tf__a_001tf__a,type,
    pi_a_a: set_a > ( a > set_a ) > set_a_a ).

thf(sy_c_FuncSet_OPi_001tf__a_001tf__b,type,
    pi_a_b: set_a > ( a > set_b ) > set_a_b ).

thf(sy_c_FuncSet_OPi_001tf__a_001tf__c,type,
    pi_a_c: set_a > ( a > set_c ) > set_a_c ).

thf(sy_c_FuncSet_OPi_001tf__b_001t__Set__Oset_Itf__b_J,type,
    pi_b_set_b: set_b > ( b > set_set_b ) > set_b_set_b ).

thf(sy_c_FuncSet_OPi_001tf__b_001tf__a,type,
    pi_b_a: set_b > ( b > set_a ) > set_b_a ).

thf(sy_c_FuncSet_OPi_001tf__b_001tf__b,type,
    pi_b_b: set_b > ( b > set_b ) > set_b_b ).

thf(sy_c_FuncSet_OPi_001tf__b_001tf__c,type,
    pi_b_c: set_b > ( b > set_c ) > set_b_c ).

thf(sy_c_FuncSet_OPi_001tf__c_001t__Set__Oset_Itf__b_J,type,
    pi_c_set_b: set_c > ( c > set_set_b ) > set_c_set_b ).

thf(sy_c_FuncSet_OPi_001tf__c_001tf__a,type,
    pi_c_a: set_c > ( c > set_a ) > set_c_a ).

thf(sy_c_FuncSet_OPi_001tf__c_001tf__b,type,
    pi_c_b: set_c > ( c > set_b ) > set_c_b ).

thf(sy_c_FuncSet_OPi_001tf__c_001tf__c,type,
    pi_c_c: set_c > ( c > set_c ) > set_c_c ).

thf(sy_c_Giry__Monad_Obind_001_Eo_001_Eo,type,
    giry_bind_o_o: sigma_measure_o > ( $o > sigma_measure_o ) > sigma_measure_o ).

thf(sy_c_Giry__Monad_Obind_001_Eo_001t__Real__Oreal,type,
    giry_bind_o_real: sigma_measure_o > ( $o > sigma_measure_real ) > sigma_measure_real ).

thf(sy_c_Giry__Monad_Obind_001_Eo_001tf__a,type,
    giry_bind_o_a: sigma_measure_o > ( $o > sigma_measure_a ) > sigma_measure_a ).

thf(sy_c_Giry__Monad_Obind_001_Eo_001tf__b,type,
    giry_bind_o_b: sigma_measure_o > ( $o > sigma_measure_b ) > sigma_measure_b ).

thf(sy_c_Giry__Monad_Obind_001_Eo_001tf__d,type,
    giry_bind_o_d: sigma_measure_o > ( $o > sigma_measure_d ) > sigma_measure_d ).

thf(sy_c_Giry__Monad_Obind_001t__Real__Oreal_001tf__b,type,
    giry_bind_real_b: sigma_measure_real > ( real > sigma_measure_b ) > sigma_measure_b ).

thf(sy_c_Giry__Monad_Obind_001t__Sigma____Algebra__Omeasure_Itf__a_J_001tf__a,type,
    giry_b1894039083993026922re_a_a: sigma_2662035542720850516sure_a > ( sigma_measure_a > sigma_measure_a ) > sigma_measure_a ).

thf(sy_c_Giry__Monad_Obind_001t__Sigma____Algebra__Omeasure_Itf__a_J_001tf__b,type,
    giry_b1894039083993026923re_a_b: sigma_2662035542720850516sure_a > ( sigma_measure_a > sigma_measure_b ) > sigma_measure_b ).

thf(sy_c_Giry__Monad_Obind_001t__Sigma____Algebra__Omeasure_Itf__b_J_001tf__a,type,
    giry_b1258390083388040233re_b_a: sigma_2733069582767196501sure_b > ( sigma_measure_b > sigma_measure_a ) > sigma_measure_a ).

thf(sy_c_Giry__Monad_Obind_001t__Sigma____Algebra__Omeasure_Itf__b_J_001tf__b,type,
    giry_b1258390083388040234re_b_b: sigma_2733069582767196501sure_b > ( sigma_measure_b > sigma_measure_b ) > sigma_measure_b ).

thf(sy_c_Giry__Monad_Obind_001t__Stream__Ostream_Itf__a_J_001tf__a,type,
    giry_bind_stream_a_a: sigma_6602114698961867009ream_a > ( stream_a > sigma_measure_a ) > sigma_measure_a ).

thf(sy_c_Giry__Monad_Obind_001t__Stream__Ostream_Itf__a_J_001tf__b,type,
    giry_bind_stream_a_b: sigma_6602114698961867009ream_a > ( stream_a > sigma_measure_b ) > sigma_measure_b ).

thf(sy_c_Giry__Monad_Obind_001t__Stream__Ostream_Itf__b_J_001tf__a,type,
    giry_bind_stream_b_a: sigma_6673148739008212994ream_b > ( stream_b > sigma_measure_a ) > sigma_measure_a ).

thf(sy_c_Giry__Monad_Obind_001t__Stream__Ostream_Itf__b_J_001tf__b,type,
    giry_bind_stream_b_b: sigma_6673148739008212994ream_b > ( stream_b > sigma_measure_b ) > sigma_measure_b ).

thf(sy_c_Giry__Monad_Obind_001tf__a_001_062_Itf__a_Mt__Real__Oreal_J,type,
    giry_bind_a_a_real: sigma_measure_a > ( a > sigma_measure_a_real ) > sigma_measure_a_real ).

thf(sy_c_Giry__Monad_Obind_001tf__a_001_062_Itf__a_Mtf__b_J,type,
    giry_bind_a_a_b: sigma_measure_a > ( a > sigma_measure_a_b ) > sigma_measure_a_b ).

thf(sy_c_Giry__Monad_Obind_001tf__a_001_Eo,type,
    giry_bind_a_o: sigma_measure_a > ( a > sigma_measure_o ) > sigma_measure_o ).

thf(sy_c_Giry__Monad_Obind_001tf__a_001t__Real__Oreal,type,
    giry_bind_a_real: sigma_measure_a > ( a > sigma_measure_real ) > sigma_measure_real ).

thf(sy_c_Giry__Monad_Obind_001tf__a_001t__Sigma____Algebra__Omeasure_Itf__a_J,type,
    giry_b5915752770402095394sure_a: sigma_measure_a > ( a > sigma_2662035542720850516sure_a ) > sigma_2662035542720850516sure_a ).

thf(sy_c_Giry__Monad_Obind_001tf__a_001t__Sigma____Algebra__Omeasure_Itf__b_J,type,
    giry_b5915752774705324195sure_b: sigma_measure_a > ( a > sigma_2733069582767196501sure_b ) > sigma_2733069582767196501sure_b ).

thf(sy_c_Giry__Monad_Obind_001tf__a_001tf__a,type,
    giry_bind_a_a: sigma_measure_a > ( a > sigma_measure_a ) > sigma_measure_a ).

thf(sy_c_Giry__Monad_Obind_001tf__a_001tf__b,type,
    giry_bind_a_b: sigma_measure_a > ( a > sigma_measure_b ) > sigma_measure_b ).

thf(sy_c_Giry__Monad_Obind_001tf__a_001tf__c,type,
    giry_bind_a_c: sigma_measure_a > ( a > sigma_measure_c ) > sigma_measure_c ).

thf(sy_c_Giry__Monad_Obind_001tf__a_001tf__d,type,
    giry_bind_a_d: sigma_measure_a > ( a > sigma_measure_d ) > sigma_measure_d ).

thf(sy_c_Giry__Monad_Obind_001tf__b_001_Eo,type,
    giry_bind_b_o: sigma_measure_b > ( b > sigma_measure_o ) > sigma_measure_o ).

thf(sy_c_Giry__Monad_Obind_001tf__b_001t__Real__Oreal,type,
    giry_bind_b_real: sigma_measure_b > ( b > sigma_measure_real ) > sigma_measure_real ).

thf(sy_c_Giry__Monad_Obind_001tf__b_001t__Sigma____Algebra__Omeasure_Itf__a_J,type,
    giry_b2825695392208082595sure_a: sigma_measure_b > ( b > sigma_2662035542720850516sure_a ) > sigma_2662035542720850516sure_a ).

thf(sy_c_Giry__Monad_Obind_001tf__b_001t__Sigma____Algebra__Omeasure_Itf__b_J,type,
    giry_b2825695396511311396sure_b: sigma_measure_b > ( b > sigma_2733069582767196501sure_b ) > sigma_2733069582767196501sure_b ).

thf(sy_c_Giry__Monad_Obind_001tf__b_001tf__a,type,
    giry_bind_b_a: sigma_measure_b > ( b > sigma_measure_a ) > sigma_measure_a ).

thf(sy_c_Giry__Monad_Obind_001tf__b_001tf__b,type,
    giry_bind_b_b: sigma_measure_b > ( b > sigma_measure_b ) > sigma_measure_b ).

thf(sy_c_Giry__Monad_Obind_001tf__b_001tf__d,type,
    giry_bind_b_d: sigma_measure_b > ( b > sigma_measure_d ) > sigma_measure_d ).

thf(sy_c_Giry__Monad_Obind_001tf__c_001tf__a,type,
    giry_bind_c_a: sigma_measure_c > ( c > sigma_measure_a ) > sigma_measure_a ).

thf(sy_c_Giry__Monad_Obind_001tf__c_001tf__b,type,
    giry_bind_c_b: sigma_measure_c > ( c > sigma_measure_b ) > sigma_measure_b ).

thf(sy_c_Giry__Monad_Obind_001tf__d_001tf__b,type,
    giry_bind_d_b: sigma_measure_d > ( d > sigma_measure_b ) > sigma_measure_b ).

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__a_Mtf__a_J,type,
    giry_return_a_a: sigma_measure_a_a > ( a > a ) > sigma_measure_a_a ).

thf(sy_c_Giry__Monad_Oreturn_001_062_Itf__a_Mtf__b_J,type,
    giry_return_a_b: sigma_measure_a_b > ( a > b ) > sigma_measure_a_b ).

thf(sy_c_Giry__Monad_Oreturn_001_062_Itf__a_Mtf__c_J,type,
    giry_return_a_c: sigma_measure_a_c > ( a > c ) > sigma_measure_a_c ).

thf(sy_c_Giry__Monad_Oreturn_001_062_Itf__b_Mtf__a_J,type,
    giry_return_b_a: sigma_measure_b_a > ( b > a ) > sigma_measure_b_a ).

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__Real__Oreal,type,
    giry_return_real: sigma_measure_real > real > sigma_measure_real ).

thf(sy_c_Giry__Monad_Oreturn_001t__Sigma____Algebra__Omeasure_Itf__a_J,type,
    giry_r3784365184592742sure_a: sigma_2662035542720850516sure_a > sigma_measure_a > sigma_2662035542720850516sure_a ).

thf(sy_c_Giry__Monad_Oreturn_001t__Sigma____Algebra__Omeasure_Itf__b_J,type,
    giry_r3784369487821543sure_b: sigma_2733069582767196501sure_b > sigma_measure_b > sigma_2733069582767196501sure_b ).

thf(sy_c_Giry__Monad_Oreturn_001t__Stream__Ostream_Itf__a_J,type,
    giry_return_stream_a: sigma_6602114698961867009ream_a > stream_a > sigma_6602114698961867009ream_a ).

thf(sy_c_Giry__Monad_Oreturn_001t__Stream__Ostream_Itf__b_J,type,
    giry_return_stream_b: sigma_6673148739008212994ream_b > stream_b > sigma_6673148739008212994ream_b ).

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__algebra_001_062_Itf__a_Mt__Real__Oreal_J,type,
    giry_s7054498699981855597a_real: sigma_measure_a_real > sigma_1818457195113549553a_real ).

thf(sy_c_Giry__Monad_Osubprob__algebra_001_062_Itf__a_Mtf__b_J,type,
    giry_s459149637493896852ra_a_b: sigma_measure_a_b > sigma_7611786102693776956re_a_b ).

thf(sy_c_Giry__Monad_Osubprob__algebra_001_Eo,type,
    giry_s3549050072915289962ebra_o: sigma_measure_o > sigma_8601284680732870894sure_o ).

thf(sy_c_Giry__Monad_Osubprob__algebra_001t__Real__Oreal,type,
    giry_s5092570657895779418a_real: sigma_measure_real > sigma_8927737637348964610e_real ).

thf(sy_c_Giry__Monad_Osubprob__algebra_001t__Sigma____Algebra__Omeasure_Itf__a_J,type,
    giry_s4089474288785316649sure_a: sigma_2662035542720850516sure_a > sigma_5260909708685749457sure_a ).

thf(sy_c_Giry__Monad_Osubprob__algebra_001t__Sigma____Algebra__Omeasure_Itf__b_J,type,
    giry_s4089474293088545450sure_b: sigma_2733069582767196501sure_b > sigma_2739725190396713810sure_b ).

thf(sy_c_Giry__Monad_Osubprob__algebra_001t__Stream__Ostream_Itf__a_J,type,
    giry_s8751189751103438934ream_a: sigma_6602114698961867009ream_a > sigma_779791719752606170ream_a ).

thf(sy_c_Giry__Monad_Osubprob__algebra_001t__Stream__Ostream_Itf__b_J,type,
    giry_s8751189755406667735ream_b: sigma_6673148739008212994ream_b > sigma_7481979238318346331ream_b ).

thf(sy_c_Giry__Monad_Osubprob__algebra_001tf__a,type,
    giry_s2448339315546285520ebra_a: sigma_measure_a > sigma_2662035542720850516sure_a ).

thf(sy_c_Giry__Monad_Osubprob__algebra_001tf__b,type,
    giry_s2448339315546285521ebra_b: sigma_measure_b > sigma_2733069582767196501sure_b ).

thf(sy_c_Giry__Monad_Osubprob__algebra_001tf__c,type,
    giry_s2448339315546285522ebra_c: sigma_measure_c > sigma_2804103622813542486sure_c ).

thf(sy_c_Giry__Monad_Osubprob__algebra_001tf__d,type,
    giry_s2448339315546285523ebra_d: sigma_measure_d > sigma_2875137662859888471sure_d ).

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__a_Mtf__a_J,type,
    giry_s8818617160626362673ce_a_a: sigma_measure_a_a > $o ).

thf(sy_c_Giry__Monad_Osubprob__space_001_062_Itf__a_Mtf__b_J,type,
    giry_s8818617164929591474ce_a_b: sigma_measure_a_b > $o ).

thf(sy_c_Giry__Monad_Osubprob__space_001_062_Itf__a_Mtf__c_J,type,
    giry_s8818617169232820275ce_a_c: sigma_measure_a_c > $o ).

thf(sy_c_Giry__Monad_Osubprob__space_001_062_Itf__b_Mtf__a_J,type,
    giry_s6031161578653686640ce_b_a: sigma_measure_b_a > $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__Real__Oreal,type,
    giry_s8208748868292234104e_real: sigma_measure_real > $o ).

thf(sy_c_Giry__Monad_Osubprob__space_001t__Sigma____Algebra__Omeasure_Itf__a_J,type,
    giry_s2296666607715182151sure_a: sigma_2662035542720850516sure_a > $o ).

thf(sy_c_Giry__Monad_Osubprob__space_001t__Sigma____Algebra__Omeasure_Itf__b_J,type,
    giry_s2296666612018410952sure_b: sigma_2733069582767196501sure_b > $o ).

thf(sy_c_Giry__Monad_Osubprob__space_001t__Stream__Ostream_Itf__a_J,type,
    giry_s6069289509726577336ream_a: sigma_6602114698961867009ream_a > $o ).

thf(sy_c_Giry__Monad_Osubprob__space_001t__Stream__Ostream_Itf__b_J,type,
    giry_s6069289514029806137ream_b: sigma_6673148739008212994ream_b > $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__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_Ominus__class_Ominus_001t__Set__Oset_Itf__b_J,type,
    minus_minus_set_b: set_b > set_b > set_b ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
    one_one_real: real ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
    plus_plus_real: real > real > real ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
    times_times_real: real > real > real ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
    zero_zero_real: real ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001tf__a_001t__Real__Oreal,type,
    groups2740460157737275248a_real: ( a > real ) > set_a > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001tf__b_001t__Real__Oreal,type,
    groups8336678772925405937b_real: ( b > real ) > set_b > real ).

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_001t__Set__Oset_Itf__b_J,type,
    indepe451107620608244277_set_b: sigma_measure_a > ( set_b > set_a ) > set_set_b > $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__a_001tf__c,type,
    indepe1948846426091821398ts_a_c: sigma_measure_a > ( c > set_a ) > set_c > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__events_001tf__b_001_Eo,type,
    indepe796274259248320173ts_b_o: sigma_measure_b > ( $o > set_b ) > set_o > $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__events_001tf__b_001t__Set__Oset_Itf__a_J,type,
    indepe1795572653114503411_set_a: sigma_measure_b > ( set_a > set_b ) > set_set_a > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__events_001tf__b_001t__Set__Oset_Itf__b_J,type,
    indepe1795572657417732212_set_b: sigma_measure_b > ( set_b > set_b ) > set_set_b > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__events_001tf__b_001tf__a,type,
    indepe8384762880973921171ts_b_a: sigma_measure_b > ( a > set_b ) > set_a > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__events_001tf__b_001tf__b,type,
    indepe8384762880973921172ts_b_b: sigma_measure_b > ( b > set_b ) > set_b > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__events_001tf__b_001tf__c,type,
    indepe8384762880973921173ts_b_c: sigma_measure_b > ( c > set_b ) > set_c > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__set_001tf__a,type,
    indepe2041756565122539606_set_a: sigma_measure_a > set_set_a > set_set_a > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__set_001tf__b,type,
    indepe2041756565122539607_set_b: sigma_measure_b > set_set_b > set_set_b > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001_062_Itf__a_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__a_Mtf__a_J,type,
    indepe1219004911012930679_a_a_a: sigma_measure_a > ( ( a > a ) > set_set_a ) > set_a_a > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001_062_Itf__a_Mtf__b_J,type,
    indepe1219004915316159480_a_a_b: sigma_measure_a > ( ( a > b ) > set_set_a ) > set_a_b > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001_062_Itf__a_Mtf__c_J,type,
    indepe1219004919619388281_a_a_c: sigma_measure_a > ( ( a > c ) > set_set_a ) > set_a_c > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001_062_Itf__b_Mtf__a_J,type,
    indepe7654921365895030454_a_b_a: sigma_measure_a > ( ( b > a ) > set_set_a ) > set_b_a > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001_Eo,type,
    indepe7780107833195774214ts_a_o: sigma_measure_a > ( $o > set_set_a ) > set_o > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001t__Nat__Onat,type,
    indepe6267730027088848354_a_nat: sigma_measure_a > ( nat > set_set_a ) > set_nat > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001t__Set__Oset_Itf__a_J,type,
    indepe4967106450811773644_set_a: sigma_measure_a > ( set_a > set_set_a ) > set_set_a > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001t__Set__Oset_Itf__b_J,type,
    indepe4967106455115002445_set_b: sigma_measure_a > ( set_b > set_set_a ) > set_set_b > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001tf__a,type,
    indepe8927441866673418604ts_a_a: sigma_measure_a > ( a > set_set_a ) > set_a > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001tf__b,type,
    indepe8927441866673418605ts_a_b: sigma_measure_a > ( b > set_set_a ) > set_b > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001tf__c,type,
    indepe8927441866673418606ts_a_c: sigma_measure_a > ( c > set_set_a ) > set_c > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001_062_Itf__a_Mt__Real__Oreal_J,type,
    indepe8950816178784143176a_real: sigma_measure_b > ( ( a > real ) > set_set_b ) > set_a_real > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001_062_Itf__a_Mtf__a_J,type,
    indepe7072862680909553272_b_a_a: sigma_measure_b > ( ( a > a ) > set_set_b ) > set_a_a > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001_062_Itf__a_Mtf__b_J,type,
    indepe7072862685212782073_b_a_b: sigma_measure_b > ( ( a > b ) > set_set_b ) > set_a_b > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001_062_Itf__a_Mtf__c_J,type,
    indepe7072862689516010874_b_a_c: sigma_measure_b > ( ( a > c ) > set_set_b ) > set_a_c > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001_062_Itf__b_Mtf__a_J,type,
    indepe4285407098936877239_b_b_a: sigma_measure_b > ( ( b > a ) > set_set_b ) > set_b_a > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001_Eo,type,
    indepe4880885433731379909ts_b_o: sigma_measure_b > ( $o > set_set_b ) > set_o > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001t__Nat__Onat,type,
    indepe7503174356045242851_b_nat: sigma_measure_b > ( nat > set_set_b ) > set_nat > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001t__Set__Oset_Itf__a_J,type,
    indepe6311571487621261579_set_a: sigma_measure_b > ( set_a > set_set_b ) > set_set_a > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001t__Set__Oset_Itf__b_J,type,
    indepe6311571491924490380_set_b: sigma_measure_b > ( set_b > set_set_b ) > set_set_b > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001tf__a,type,
    indepe6139986284700742571ts_b_a: sigma_measure_b > ( a > set_set_b ) > set_a > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001tf__b,type,
    indepe6139986284700742572ts_b_b: sigma_measure_b > ( b > set_set_b ) > set_b > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001tf__c,type,
    indepe6139986284700742573ts_b_c: sigma_measure_b > ( c > set_set_b ) > set_c > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001tf__a_001_062_Itf__a_Mt__Real__Oreal_J,type,
    indepe7138907624787055209a_real: sigma_measure_a > sigma_measure_a_real > ( a > a > real ) > sigma_measure_a_real > ( a > a > real ) > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001tf__a_001_062_Itf__a_Mtf__b_J,type,
    indepe6909679525661198488_a_a_b: sigma_measure_a > sigma_measure_a_b > ( a > a > b ) > sigma_measure_a_b > ( a > a > b ) > $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__b,type,
    indepe2440653194691626189ar_a_b: sigma_measure_a > sigma_measure_b > ( a > b ) > sigma_measure_b > ( a > b ) > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001tf__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__b_001_062_Itf__a_Mt__Real__Oreal_J,type,
    indepe2116752563100621480a_real: sigma_measure_b > sigma_measure_a_real > ( b > a > real ) > sigma_measure_a_real > ( b > a > real ) > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001tf__b_001_062_Itf__a_Mtf__b_J,type,
    indepe3540165258703045273_b_a_b: sigma_measure_b > sigma_measure_a_b > ( b > a > b ) > sigma_measure_a_b > ( b > a > b ) > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001tf__b_001t__Real__Oreal,type,
    indepe5331282143832502239b_real: sigma_measure_b > sigma_measure_real > ( b > real ) > sigma_measure_real > ( b > real ) > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001tf__b_001tf__a,type,
    indepe8876569649573725963ar_b_a: sigma_measure_b > sigma_measure_a > ( b > a ) > sigma_measure_a > ( b > a ) > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001tf__b_001tf__b,type,
    indepe8876569649573725964ar_b_b: sigma_measure_b > sigma_measure_b > ( b > b ) > sigma_measure_b > ( b > b ) > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001tf__b_001tf__c,type,
    indepe8876569649573725965ar_b_c: sigma_measure_b > sigma_measure_c > ( b > c ) > sigma_measure_c > ( b > c ) > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001tf__b_001tf__d,type,
    indepe8876569649573725966ar_b_d: sigma_measure_b > sigma_measure_d > ( b > d ) > sigma_measure_d > ( b > d ) > $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__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__c_001tf__a,type,
    indepe4851901773132442932_a_c_a: sigma_measure_a > ( c > sigma_measure_a ) > ( c > a > a ) > set_c > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001tf__c_001tf__b,type,
    indepe4851901773132442933_a_c_b: sigma_measure_a > ( c > sigma_measure_b ) > ( c > a > b ) > set_c > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001tf__c_001tf__c,type,
    indepe4851901773132442934_a_c_c: sigma_measure_a > ( c > sigma_measure_c ) > ( c > a > c ) > set_c > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001tf__c_001tf__d,type,
    indepe4851901773132442935_a_c_d: sigma_measure_a > ( c > sigma_measure_d ) > ( c > a > d ) > set_c > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__a_001t__Nat__Onat,type,
    indepe557731436560307799_a_nat: sigma_measure_b > ( a > sigma_measure_nat ) > ( a > b > nat ) > set_a > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__a_001tf__a,type,
    indepe6799659515411149879_b_a_a: sigma_measure_b > ( a > sigma_measure_a ) > ( a > b > a ) > set_a > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__a_001tf__b,type,
    indepe6799659515411149880_b_a_b: sigma_measure_b > ( a > sigma_measure_b ) > ( a > b > b ) > set_a > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__a_001tf__c,type,
    indepe6799659515411149881_b_a_c: sigma_measure_b > ( a > sigma_measure_c ) > ( a > b > c ) > set_a > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__b_001t__Nat__Onat,type,
    indepe1793175765516702296_b_nat: sigma_measure_b > ( b > sigma_measure_nat ) > ( b > b > nat ) > set_b > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__b_001tf__a,type,
    indepe4012203933438473846_b_b_a: sigma_measure_b > ( b > sigma_measure_a ) > ( b > b > a ) > set_b > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__b_001tf__b,type,
    indepe4012203933438473847_b_b_b: sigma_measure_b > ( b > sigma_measure_b ) > ( b > b > b ) > set_b > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__b_001tf__c,type,
    indepe4012203933438473848_b_b_c: sigma_measure_b > ( b > sigma_measure_c ) > ( b > b > c ) > set_b > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__c_001t__Nat__Onat,type,
    indepe3028620094473096793_c_nat: sigma_measure_b > ( c > sigma_measure_nat ) > ( c > b > nat ) > set_c > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__c_001tf__a,type,
    indepe1224748351465797813_b_c_a: sigma_measure_b > ( c > sigma_measure_a ) > ( c > b > a ) > set_c > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__c_001tf__b,type,
    indepe1224748351465797814_b_c_b: sigma_measure_b > ( c > sigma_measure_b ) > ( c > b > b ) > set_c > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__c_001tf__c,type,
    indepe1224748351465797815_b_c_c: sigma_measure_b > ( c > sigma_measure_c ) > ( c > b > c ) > set_c > $o ).

thf(sy_c_Independent__Family_Oprob__space_Otail__events_001tf__a_001t__Nat__Onat,type,
    indepe7538416700049374166_a_nat: sigma_measure_a > ( nat > set_set_a ) > set_set_a ).

thf(sy_c_Independent__Family_Oprob__space_Otail__events_001tf__b_001t__Nat__Onat,type,
    indepe8773861029005768663_b_nat: sigma_measure_b > ( nat > set_set_b ) > set_set_b ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
    inf_inf_set_a: set_a > set_a > set_a ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__b_J,type,
    inf_inf_set_b: set_b > set_b > set_b ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
    sup_sup_set_a: set_a > set_a > set_a ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__b_J,type,
    sup_sup_set_b: set_b > set_b > set_b ).

thf(sy_c_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_001_Eo_001tf__c,type,
    measure_distr_o_c: sigma_measure_o > sigma_measure_c > ( $o > c ) > sigma_measure_c ).

thf(sy_c_Measure__Space_Odistr_001t__Real__Oreal_001tf__b,type,
    measure_distr_real_b: sigma_measure_real > sigma_measure_b > ( real > b ) > sigma_measure_b ).

thf(sy_c_Measure__Space_Odistr_001t__Stream__Ostream_Itf__a_J_001tf__a,type,
    measur8481048967365464818am_a_a: sigma_6602114698961867009ream_a > sigma_measure_a > ( stream_a > a ) > sigma_measure_a ).

thf(sy_c_Measure__Space_Odistr_001t__Stream__Ostream_Itf__b_J_001tf__a,type,
    measur7845399966760478129am_b_a: sigma_6673148739008212994ream_b > sigma_measure_a > ( stream_b > a ) > sigma_measure_a ).

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__a_Mtf__a_J,type,
    measure_distr_a_a_a: sigma_measure_a > sigma_measure_a_a > ( a > a > a ) > sigma_measure_a_a ).

thf(sy_c_Measure__Space_Odistr_001tf__a_001_062_Itf__a_Mtf__b_J,type,
    measure_distr_a_a_b: sigma_measure_a > sigma_measure_a_b > ( a > a > b ) > sigma_measure_a_b ).

thf(sy_c_Measure__Space_Odistr_001tf__a_001_062_Itf__a_Mtf__c_J,type,
    measure_distr_a_a_c: sigma_measure_a > sigma_measure_a_c > ( a > a > c ) > sigma_measure_a_c ).

thf(sy_c_Measure__Space_Odistr_001tf__a_001_062_Itf__b_Mtf__a_J,type,
    measure_distr_a_b_a: sigma_measure_a > sigma_measure_b_a > ( a > b > a ) > sigma_measure_b_a ).

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__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__Sigma____Algebra__Omeasure_Itf__a_J,type,
    measur2827329750764064385sure_a: sigma_measure_a > sigma_2662035542720850516sure_a > ( a > sigma_measure_a ) > sigma_2662035542720850516sure_a ).

thf(sy_c_Measure__Space_Odistr_001tf__a_001t__Sigma____Algebra__Omeasure_Itf__b_J,type,
    measur2827329755067293186sure_b: sigma_measure_a > sigma_2733069582767196501sure_b > ( a > sigma_measure_b ) > sigma_2733069582767196501sure_b ).

thf(sy_c_Measure__Space_Odistr_001tf__a_001t__Stream__Ostream_Itf__a_J,type,
    measur8618403470260820734ream_a: sigma_measure_a > sigma_6602114698961867009ream_a > ( a > stream_a ) > sigma_6602114698961867009ream_a ).

thf(sy_c_Measure__Space_Odistr_001tf__a_001t__Stream__Ostream_Itf__b_J,type,
    measur8618403474564049535ream_b: sigma_measure_a > sigma_6673148739008212994ream_b > ( a > stream_b ) > sigma_6673148739008212994ream_b ).

thf(sy_c_Measure__Space_Odistr_001tf__a_001tf__a,type,
    measure_distr_a_a: sigma_measure_a > sigma_measure_a > ( a > a ) > sigma_measure_a ).

thf(sy_c_Measure__Space_Odistr_001tf__a_001tf__b,type,
    measure_distr_a_b: sigma_measure_a > sigma_measure_b > ( a > b ) > sigma_measure_b ).

thf(sy_c_Measure__Space_Odistr_001tf__a_001tf__c,type,
    measure_distr_a_c: sigma_measure_a > sigma_measure_c > ( a > c ) > sigma_measure_c ).

thf(sy_c_Measure__Space_Odistr_001tf__b_001_062_Itf__a_Mt__Real__Oreal_J,type,
    measur1784223122078132820a_real: sigma_measure_b > sigma_measure_a_real > ( b > a > real ) > sigma_measure_a_real ).

thf(sy_c_Measure__Space_Odistr_001tf__b_001_062_Itf__a_Mtf__a_J,type,
    measure_distr_b_a_a: sigma_measure_b > sigma_measure_a_a > ( b > a > a ) > sigma_measure_a_a ).

thf(sy_c_Measure__Space_Odistr_001tf__b_001_062_Itf__a_Mtf__b_J,type,
    measure_distr_b_a_b: sigma_measure_b > sigma_measure_a_b > ( b > a > b ) > sigma_measure_a_b ).

thf(sy_c_Measure__Space_Odistr_001tf__b_001_062_Itf__a_Mtf__c_J,type,
    measure_distr_b_a_c: sigma_measure_b > sigma_measure_a_c > ( b > a > c ) > sigma_measure_a_c ).

thf(sy_c_Measure__Space_Odistr_001tf__b_001_062_Itf__b_Mtf__a_J,type,
    measure_distr_b_b_a: sigma_measure_b > sigma_measure_b_a > ( b > b > a ) > sigma_measure_b_a ).

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__Real__Oreal,type,
    measure_distr_b_real: sigma_measure_b > sigma_measure_real > ( b > real ) > sigma_measure_real ).

thf(sy_c_Measure__Space_Odistr_001tf__b_001t__Sigma____Algebra__Omeasure_Itf__a_J,type,
    measur8960644409424827394sure_a: sigma_measure_b > sigma_2662035542720850516sure_a > ( b > sigma_measure_a ) > sigma_2662035542720850516sure_a ).

thf(sy_c_Measure__Space_Odistr_001tf__b_001t__Sigma____Algebra__Omeasure_Itf__b_J,type,
    measur8960644413728056195sure_b: sigma_measure_b > sigma_2733069582767196501sure_b > ( b > sigma_measure_b ) > sigma_2733069582767196501sure_b ).

thf(sy_c_Measure__Space_Odistr_001tf__b_001t__Stream__Ostream_Itf__a_J,type,
    measur5532647228270826173ream_a: sigma_measure_b > sigma_6602114698961867009ream_a > ( b > stream_a ) > sigma_6602114698961867009ream_a ).

thf(sy_c_Measure__Space_Odistr_001tf__b_001t__Stream__Ostream_Itf__b_J,type,
    measur5532647232574054974ream_b: sigma_measure_b > sigma_6673148739008212994ream_b > ( b > stream_b ) > sigma_6673148739008212994ream_b ).

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__b_001tf__c,type,
    measure_distr_b_c: sigma_measure_b > sigma_measure_c > ( b > c ) > sigma_measure_c ).

thf(sy_c_Measure__Space_Odistr_001tf__b_001tf__d,type,
    measure_distr_b_d: sigma_measure_b > sigma_measure_d > ( b > d ) > sigma_measure_d ).

thf(sy_c_Measure__Space_Odistr_001tf__c_001tf__b,type,
    measure_distr_c_b: sigma_measure_c > sigma_measure_b > ( c > b ) > sigma_measure_b ).

thf(sy_c_Measure__Space_Ofinite__measure_001_062_Itf__a_Mt__Real__Oreal_J,type,
    measur5960726210453282175a_real: sigma_measure_a_real > $o ).

thf(sy_c_Measure__Space_Ofinite__measure_001_062_Itf__a_Mtf__b_J,type,
    measur8914409478018001922re_a_b: sigma_measure_a_b > $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_001t__Stream__Ostream_Itf__a_J,type,
    measur2119334572981594984ream_a: sigma_6602114698961867009ream_a > $o ).

thf(sy_c_Measure__Space_Ofinite__measure_001t__Stream__Ostream_Itf__b_J,type,
    measur2119334577284823785ream_b: sigma_6673148739008212994ream_b > $o ).

thf(sy_c_Measure__Space_Ofinite__measure_001tf__a,type,
    measur930452917991658466sure_a: sigma_measure_a > $o ).

thf(sy_c_Measure__Space_Ofinite__measure_001tf__b,type,
    measur930452917991658467sure_b: sigma_measure_b > $o ).

thf(sy_c_Measure__Space_Ofinite__measure_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_001tf__a_001t__Real__Oreal,type,
    measur1776380161843274167a_real: set_set_a > ( set_a > real ) > $o ).

thf(sy_c_Measure__Space_Oincreasing_001tf__b_001t__Real__Oreal,type,
    measur7372598777031404856b_real: set_set_b > ( set_b > real ) > $o ).

thf(sy_c_Measure__Space_Osigma__finite__measure_001_062_Itf__a_Mt__Real__Oreal_J,type,
    measur5819825744149550268a_real: sigma_measure_a_real > $o ).

thf(sy_c_Measure__Space_Osigma__finite__measure_001_062_Itf__a_Mtf__b_J,type,
    measur5182267031777123205re_a_b: sigma_measure_a_b > $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_001t__Stream__Ostream_Itf__a_J,type,
    measur5334621511647899941ream_a: sigma_6602114698961867009ream_a > $o ).

thf(sy_c_Measure__Space_Osigma__finite__measure_001t__Stream__Ostream_Itf__b_J,type,
    measur5334621515951128742ream_b: sigma_6673148739008212994ream_b > $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__b,type,
    measur4308613598931908896sure_b: sigma_measure_b > $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_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_Itf__a_M_062_Itf__a_Mt__Real__Oreal_J_J_J,type,
    bot_bot_set_a_a_real: set_a_a_real ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_Itf__a_M_062_Itf__a_Mtf__b_J_J_J,type,
    bot_bot_set_a_a_b: set_a_a_b ).

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__a_Mtf__a_J_J,type,
    bot_bot_set_a_a: set_a_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
    bot_bot_set_a_b: set_a_b ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_Itf__a_Mtf__c_J_J,type,
    bot_bot_set_a_c: set_a_c ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_Itf__b_Mtf__a_J_J,type,
    bot_bot_set_b_a: set_b_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
    bot_bot_set_o: set_o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
    bot_bot_set_real: set_real ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_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_It__Set__Oset_Itf__b_J_J_J,type,
    bot_bo859375258733454347_set_b: set_set_set_b ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    bot_bot_set_set_a: set_set_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
    bot_bot_set_set_b: set_set_b ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Sigma____Algebra__Omeasure_I_Eo_J_J,type,
    bot_bo7838039659004643295sure_o: set_Sigma_measure_o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Sigma____Algebra__Omeasure_It__Sigma____Algebra__Omeasure_Itf__a_J_J_J,type,
    bot_bo3906570255302654624sure_a: set_Si8678843990275272244sure_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Sigma____Algebra__Omeasure_It__Sigma____Algebra__Omeasure_Itf__b_J_J_J,type,
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    member_b_d: ( b > d ) > set_b_d > $o ).

thf(sy_c_member_001_062_Itf__c_Mt__Real__Oreal_J,type,
    member_c_real: ( c > real ) > set_c_real > $o ).

thf(sy_c_member_001_062_Itf__c_Mt__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J,type,
    member4203522019703367553e_real: ( c > sigma_measure_real ) > set_c_122911134582766496e_real > $o ).

thf(sy_c_member_001_062_Itf__c_Mt__Sigma____Algebra__Omeasure_It__Sigma____Algebra__Omeasure_Itf__a_J_J_J,type,
    member6376300970257654096sure_a: ( c > sigma_2662035542720850516sure_a ) > set_c_649586175287248367sure_a > $o ).

thf(sy_c_member_001_062_Itf__c_Mt__Sigma____Algebra__Omeasure_It__Sigma____Algebra__Omeasure_Itf__b_J_J_J,type,
    member3855116451968618449sure_b: ( c > sigma_2733069582767196501sure_b ) > set_c_490803655962395632sure_b > $o ).

thf(sy_c_member_001_062_Itf__c_Mt__Sigma____Algebra__Omeasure_Itf__a_J_J,type,
    member4590190692143509013sure_a: ( c > sigma_measure_a ) > set_c_6209469361167710540sure_a > $o ).

thf(sy_c_member_001_062_Itf__c_Mt__Sigma____Algebra__Omeasure_Itf__b_J_J,type,
    member4661224732189854998sure_b: ( c > sigma_measure_b ) > set_c_3688284842878674893sure_b > $o ).

thf(sy_c_member_001_062_Itf__c_Mt__Sigma____Algebra__Omeasure_Itf__d_J_J,type,
    member4803292812282546968sure_d: ( c > sigma_measure_d ) > set_c_7869287843155379407sure_d > $o ).

thf(sy_c_member_001_062_Itf__c_Mtf__a_J,type,
    member_c_a: ( c > a ) > set_c_a > $o ).

thf(sy_c_member_001_062_Itf__c_Mtf__b_J,type,
    member_c_b: ( c > b ) > set_c_b > $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__d_Mt__Sigma____Algebra__Omeasure_Itf__b_J_J,type,
    member1571167353995842199sure_b: ( d > sigma_measure_b ) > set_d_5950004124339453134sure_b > $o ).

thf(sy_c_member_001_Eo,type,
    member_o: $o > set_o > $o ).

thf(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > set_nat > $o ).

thf(sy_c_member_001t__Real__Oreal,type,
    member_real: real > set_real > $o ).

thf(sy_c_member_001t__Set__Oset_It__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_It__Set__Oset_Itf__b_J_J,type,
    member_set_set_b: set_set_b > set_set_set_b > $o ).

thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
    member_set_a: set_a > set_set_a > $o ).

thf(sy_c_member_001t__Set__Oset_Itf__b_J,type,
    member_set_b: set_b > set_set_b > $o ).

thf(sy_c_member_001t__Set__Oset_Itf__c_J,type,
    member_set_c: set_c > set_set_c > $o ).

thf(sy_c_member_001t__Set__Oset_Itf__d_J,type,
    member_set_d: set_d > set_set_d > $o ).

thf(sy_c_member_001t__Sigma____Algebra__Omeasure_It__Stream__Ostream_Itf__a_J_J,type,
    member3412284305619680664ream_a: sigma_6602114698961867009ream_a > set_Si4836858405773242423ream_a > $o ).

thf(sy_c_member_001t__Sigma____Algebra__Omeasure_It__Stream__Ostream_Itf__b_J_J,type,
    member3483318345666026649ream_b: sigma_6673148739008212994ream_b > set_Si2315673887484206776ream_b > $o ).

thf(sy_c_member_001t__Sigma____Algebra__Omeasure_Itf__a_J,type,
    member3534519376729797778sure_a: sigma_measure_a > set_Sigma_measure_a > $o ).

thf(sy_c_member_001t__Sigma____Algebra__Omeasure_Itf__b_J,type,
    member3534519381033026579sure_b: sigma_measure_b > set_Sigma_measure_b > $o ).

thf(sy_c_member_001t__Stream__Ostream_Itf__a_J,type,
    member_stream_a: stream_a > set_stream_a > $o ).

thf(sy_c_member_001t__Stream__Ostream_Itf__b_J,type,
    member_stream_b: stream_b > set_stream_b > $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_c ).

thf(sy_v_M,type,
    m: sigma_measure_a ).

thf(sy_v_M_H,type,
    m2: c > sigma_measure_d ).

thf(sy_v_N,type,
    n: sigma_measure_b ).

thf(sy_v_X_H,type,
    x: c > b > d ).

thf(sy_v_f,type,
    f: a > b ).

% Relevant facts (1278)
thf(fact_0_prob__space__axioms,axiom,
    probab7247484486040049089pace_a @ m ).

% prob_space_axioms
thf(fact_1_assms_I1_J,axiom,
    member_a_b @ f @ ( sigma_measurable_a_b @ m @ n ) ).

% assms(1)
thf(fact_2_Pi__I,axiom,
    ! [A: set_c,F: c > c,B: c > set_c] :
      ( ! [X: c] :
          ( ( member_c @ X @ A )
         => ( member_c @ ( F @ X ) @ ( B @ X ) ) )
     => ( member_c_c @ F @ ( pi_c_c @ A @ B ) ) ) ).

% Pi_I
thf(fact_3_Pi__I,axiom,
    ! [A: set_c,F: c > b,B: c > set_b] :
      ( ! [X: c] :
          ( ( member_c @ X @ A )
         => ( member_b @ ( F @ X ) @ ( B @ X ) ) )
     => ( member_c_b @ F @ ( pi_c_b @ A @ B ) ) ) ).

% Pi_I
thf(fact_4_Pi__I,axiom,
    ! [A: set_c,F: c > a,B: c > set_a] :
      ( ! [X: c] :
          ( ( member_c @ X @ A )
         => ( member_a @ ( F @ X ) @ ( B @ X ) ) )
     => ( member_c_a @ F @ ( pi_c_a @ A @ B ) ) ) ).

% Pi_I
thf(fact_5_Pi__I,axiom,
    ! [A: set_b,F: b > c,B: b > set_c] :
      ( ! [X: b] :
          ( ( member_b @ X @ A )
         => ( member_c @ ( F @ X ) @ ( B @ X ) ) )
     => ( member_b_c @ F @ ( pi_b_c @ A @ B ) ) ) ).

% Pi_I
thf(fact_6_Pi__I,axiom,
    ! [A: set_b,F: b > b,B: b > set_b] :
      ( ! [X: b] :
          ( ( member_b @ X @ A )
         => ( member_b @ ( F @ X ) @ ( B @ X ) ) )
     => ( member_b_b @ F @ ( pi_b_b @ A @ B ) ) ) ).

% Pi_I
thf(fact_7_Pi__I,axiom,
    ! [A: set_b,F: b > a,B: b > set_a] :
      ( ! [X: b] :
          ( ( member_b @ X @ A )
         => ( member_a @ ( F @ X ) @ ( B @ X ) ) )
     => ( member_b_a @ F @ ( pi_b_a @ A @ B ) ) ) ).

% Pi_I
thf(fact_8_Pi__I,axiom,
    ! [A: set_a,F: a > real,B: a > set_real] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( member_real @ ( F @ X ) @ ( B @ X ) ) )
     => ( member_a_real @ F @ ( pi_a_real @ A @ B ) ) ) ).

% Pi_I
thf(fact_9_Pi__I,axiom,
    ! [A: set_a,F: a > c,B: a > set_c] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( member_c @ ( F @ X ) @ ( B @ X ) ) )
     => ( member_a_c @ F @ ( pi_a_c @ A @ B ) ) ) ).

% Pi_I
thf(fact_10_Pi__I,axiom,
    ! [A: set_a,F: a > b,B: a > set_b] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( member_b @ ( F @ X ) @ ( B @ X ) ) )
     => ( member_a_b @ F @ ( pi_a_b @ A @ B ) ) ) ).

% Pi_I
thf(fact_11_Pi__I,axiom,
    ! [A: set_a,F: a > a,B: a > set_a] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( member_a @ ( F @ X ) @ ( B @ X ) ) )
     => ( member_a_a @ F @ ( pi_a_a @ A @ B ) ) ) ).

% Pi_I
thf(fact_12_subprob__not__empty,axiom,
    ( ( sigma_space_a @ m )
   != bot_bot_set_a ) ).

% subprob_not_empty
thf(fact_13_subprob__space__axioms,axiom,
    giry_subprob_space_a @ m ).

% subprob_space_axioms
thf(fact_14_funcsetI,axiom,
    ! [A: set_c,F: c > c,B: set_c] :
      ( ! [X: c] :
          ( ( member_c @ X @ A )
         => ( member_c @ ( F @ X ) @ B ) )
     => ( member_c_c @ F
        @ ( pi_c_c @ A
          @ ^ [Uu: c] : B ) ) ) ).

% funcsetI
thf(fact_15_funcsetI,axiom,
    ! [A: set_c,F: c > b,B: set_b] :
      ( ! [X: c] :
          ( ( member_c @ X @ A )
         => ( member_b @ ( F @ X ) @ B ) )
     => ( member_c_b @ F
        @ ( pi_c_b @ A
          @ ^ [Uu: c] : B ) ) ) ).

% funcsetI
thf(fact_16_funcsetI,axiom,
    ! [A: set_c,F: c > a,B: set_a] :
      ( ! [X: c] :
          ( ( member_c @ X @ A )
         => ( member_a @ ( F @ X ) @ B ) )
     => ( member_c_a @ F
        @ ( pi_c_a @ A
          @ ^ [Uu: c] : B ) ) ) ).

% funcsetI
thf(fact_17_funcsetI,axiom,
    ! [A: set_b,F: b > c,B: set_c] :
      ( ! [X: b] :
          ( ( member_b @ X @ A )
         => ( member_c @ ( F @ X ) @ B ) )
     => ( member_b_c @ F
        @ ( pi_b_c @ A
          @ ^ [Uu: b] : B ) ) ) ).

% funcsetI
thf(fact_18_funcsetI,axiom,
    ! [A: set_b,F: b > b,B: set_b] :
      ( ! [X: b] :
          ( ( member_b @ X @ A )
         => ( member_b @ ( F @ X ) @ B ) )
     => ( member_b_b @ F
        @ ( pi_b_b @ A
          @ ^ [Uu: b] : B ) ) ) ).

% funcsetI
thf(fact_19_funcsetI,axiom,
    ! [A: set_b,F: b > a,B: set_a] :
      ( ! [X: b] :
          ( ( member_b @ X @ A )
         => ( member_a @ ( F @ X ) @ B ) )
     => ( member_b_a @ F
        @ ( pi_b_a @ A
          @ ^ [Uu: b] : B ) ) ) ).

% funcsetI
thf(fact_20_funcsetI,axiom,
    ! [A: set_a,F: a > real,B: set_real] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( member_real @ ( F @ X ) @ B ) )
     => ( member_a_real @ F
        @ ( pi_a_real @ A
          @ ^ [Uu: a] : B ) ) ) ).

% funcsetI
thf(fact_21_funcsetI,axiom,
    ! [A: set_a,F: a > c,B: set_c] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( member_c @ ( F @ X ) @ B ) )
     => ( member_a_c @ F
        @ ( pi_a_c @ A
          @ ^ [Uu: a] : B ) ) ) ).

% funcsetI
thf(fact_22_funcsetI,axiom,
    ! [A: set_a,F: a > b,B: set_b] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( member_b @ ( F @ X ) @ B ) )
     => ( member_a_b @ F
        @ ( pi_a_b @ A
          @ ^ [Uu: a] : B ) ) ) ).

% funcsetI
thf(fact_23_funcsetI,axiom,
    ! [A: set_a,F: a > a,B: set_a] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( member_a @ ( F @ X ) @ B ) )
     => ( member_a_a @ F
        @ ( pi_a_a @ A
          @ ^ [Uu: a] : B ) ) ) ).

% funcsetI
thf(fact_24_funcset__id,axiom,
    ! [A: set_a] :
      ( member_a_a
      @ ^ [X2: a] : X2
      @ ( pi_a_a @ A
        @ ^ [Uu: a] : A ) ) ).

% funcset_id
thf(fact_25_funcset__mem,axiom,
    ! [F: c > c,A: set_c,B: set_c,X3: c] :
      ( ( member_c_c @ F
        @ ( pi_c_c @ A
          @ ^ [Uu: c] : B ) )
     => ( ( member_c @ X3 @ A )
       => ( member_c @ ( F @ X3 ) @ B ) ) ) ).

% funcset_mem
thf(fact_26_funcset__mem,axiom,
    ! [F: c > b,A: set_c,B: set_b,X3: c] :
      ( ( member_c_b @ F
        @ ( pi_c_b @ A
          @ ^ [Uu: c] : B ) )
     => ( ( member_c @ X3 @ A )
       => ( member_b @ ( F @ X3 ) @ B ) ) ) ).

% funcset_mem
thf(fact_27_funcset__mem,axiom,
    ! [F: c > a,A: set_c,B: set_a,X3: c] :
      ( ( member_c_a @ F
        @ ( pi_c_a @ A
          @ ^ [Uu: c] : B ) )
     => ( ( member_c @ X3 @ A )
       => ( member_a @ ( F @ X3 ) @ B ) ) ) ).

% funcset_mem
thf(fact_28_funcset__mem,axiom,
    ! [F: b > c,A: set_b,B: set_c,X3: b] :
      ( ( member_b_c @ F
        @ ( pi_b_c @ A
          @ ^ [Uu: b] : B ) )
     => ( ( member_b @ X3 @ A )
       => ( member_c @ ( F @ X3 ) @ B ) ) ) ).

% funcset_mem
thf(fact_29_funcset__mem,axiom,
    ! [F: b > b,A: set_b,B: set_b,X3: b] :
      ( ( member_b_b @ F
        @ ( pi_b_b @ A
          @ ^ [Uu: b] : B ) )
     => ( ( member_b @ X3 @ A )
       => ( member_b @ ( F @ X3 ) @ B ) ) ) ).

% funcset_mem
thf(fact_30_funcset__mem,axiom,
    ! [F: b > a,A: set_b,B: set_a,X3: b] :
      ( ( member_b_a @ F
        @ ( pi_b_a @ A
          @ ^ [Uu: b] : B ) )
     => ( ( member_b @ X3 @ A )
       => ( member_a @ ( F @ X3 ) @ B ) ) ) ).

% funcset_mem
thf(fact_31_funcset__mem,axiom,
    ! [F: a > c,A: set_a,B: set_c,X3: a] :
      ( ( member_a_c @ F
        @ ( pi_a_c @ A
          @ ^ [Uu: a] : B ) )
     => ( ( member_a @ X3 @ A )
       => ( member_c @ ( F @ X3 ) @ B ) ) ) ).

% funcset_mem
thf(fact_32_funcset__mem,axiom,
    ! [F: a > a,A: set_a,B: set_a,X3: a] :
      ( ( member_a_a @ F
        @ ( pi_a_a @ A
          @ ^ [Uu: a] : B ) )
     => ( ( member_a @ X3 @ A )
       => ( member_a @ ( F @ X3 ) @ B ) ) ) ).

% funcset_mem
thf(fact_33_funcset__mem,axiom,
    ! [F: a > real,A: set_a,B: set_real,X3: a] :
      ( ( member_a_real @ F
        @ ( pi_a_real @ A
          @ ^ [Uu: a] : B ) )
     => ( ( member_a @ X3 @ A )
       => ( member_real @ ( F @ X3 ) @ B ) ) ) ).

% funcset_mem
thf(fact_34_funcset__mem,axiom,
    ! [F: a > b,A: set_a,B: set_b,X3: a] :
      ( ( member_a_b @ F
        @ ( pi_a_b @ A
          @ ^ [Uu: a] : B ) )
     => ( ( member_a @ X3 @ A )
       => ( member_b @ ( F @ X3 ) @ B ) ) ) ).

% funcset_mem
thf(fact_35_indep__vars__cong,axiom,
    ! [I: set_c,J: set_c,X4: c > a > d,Y: c > a > d,M: c > sigma_measure_d,N: c > sigma_measure_d] :
      ( ( I = J )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( ( X4 @ I2 )
              = ( Y @ I2 ) ) )
       => ( ! [I2: c] :
              ( ( member_c @ I2 @ I )
             => ( ( M @ I2 )
                = ( N @ I2 ) ) )
         => ( ( indepe4851901773132442935_a_c_d @ m @ M @ X4 @ I )
            = ( indepe4851901773132442935_a_c_d @ m @ N @ Y @ J ) ) ) ) ) ).

% indep_vars_cong
thf(fact_36_D_Osubprob__not__empty,axiom,
    ( ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) )
   != bot_bot_set_b ) ).

% D.subprob_not_empty
thf(fact_37_indep__sets__cong,axiom,
    ! [I: set_c,J: set_c,F2: c > set_set_a,G: c > set_set_a] :
      ( ( I = J )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe8927441866673418606ts_a_c @ m @ F2 @ I )
          = ( indepe8927441866673418606ts_a_c @ m @ G @ J ) ) ) ) ).

% indep_sets_cong
thf(fact_38_indep__sets__cong,axiom,
    ! [I: set_b,J: set_b,F2: b > set_set_a,G: b > set_set_a] :
      ( ( I = J )
     => ( ! [I2: b] :
            ( ( member_b @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe8927441866673418605ts_a_b @ m @ F2 @ I )
          = ( indepe8927441866673418605ts_a_b @ m @ G @ J ) ) ) ) ).

% indep_sets_cong
thf(fact_39_indep__sets__cong,axiom,
    ! [I: set_a,J: set_a,F2: a > set_set_a,G: a > set_set_a] :
      ( ( I = J )
     => ( ! [I2: a] :
            ( ( member_a @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe8927441866673418604ts_a_a @ m @ F2 @ I )
          = ( indepe8927441866673418604ts_a_a @ m @ G @ J ) ) ) ) ).

% indep_sets_cong
thf(fact_40_indep__sets__cong,axiom,
    ! [I: set_nat,J: set_nat,F2: nat > set_set_a,G: nat > set_set_a] :
      ( ( I = J )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe6267730027088848354_a_nat @ m @ F2 @ I )
          = ( indepe6267730027088848354_a_nat @ m @ G @ J ) ) ) ) ).

% indep_sets_cong
thf(fact_41_indep__sets__cong,axiom,
    ! [I: set_o,J: set_o,F2: $o > set_set_a,G: $o > set_set_a] :
      ( ( I = J )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe7780107833195774214ts_a_o @ m @ F2 @ I )
          = ( indepe7780107833195774214ts_a_o @ m @ G @ J ) ) ) ) ).

% indep_sets_cong
thf(fact_42_indep__sets__cong,axiom,
    ! [I: set_a_real,J: set_a_real,F2: ( a > real ) > set_set_a,G: ( a > real ) > set_set_a] :
      ( ( I = J )
     => ( ! [I2: a > real] :
            ( ( member_a_real @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe4749599203615801097a_real @ m @ F2 @ I )
          = ( indepe4749599203615801097a_real @ m @ G @ J ) ) ) ) ).

% indep_sets_cong
thf(fact_43_indep__sets__cong,axiom,
    ! [I: set_a_b,J: set_a_b,F2: ( a > b ) > set_set_a,G: ( a > b ) > set_set_a] :
      ( ( I = J )
     => ( ! [I2: a > b] :
            ( ( member_a_b @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe1219004915316159480_a_a_b @ m @ F2 @ I )
          = ( indepe1219004915316159480_a_a_b @ m @ G @ J ) ) ) ) ).

% indep_sets_cong
thf(fact_44_indep__sets__cong,axiom,
    ! [I: set_b_a,J: set_b_a,F2: ( b > a ) > set_set_a,G: ( b > a ) > set_set_a] :
      ( ( I = J )
     => ( ! [I2: b > a] :
            ( ( member_b_a @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe7654921365895030454_a_b_a @ m @ F2 @ I )
          = ( indepe7654921365895030454_a_b_a @ m @ G @ J ) ) ) ) ).

% indep_sets_cong
thf(fact_45_indep__sets__cong,axiom,
    ! [I: set_a_c,J: set_a_c,F2: ( a > c ) > set_set_a,G: ( a > c ) > set_set_a] :
      ( ( I = J )
     => ( ! [I2: a > c] :
            ( ( member_a_c @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe1219004919619388281_a_a_c @ m @ F2 @ I )
          = ( indepe1219004919619388281_a_a_c @ m @ G @ J ) ) ) ) ).

% indep_sets_cong
thf(fact_46_indep__sets__cong,axiom,
    ! [I: set_a_a,J: set_a_a,F2: ( a > a ) > set_set_a,G: ( a > a ) > set_set_a] :
      ( ( I = J )
     => ( ! [I2: a > a] :
            ( ( member_a_a @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe1219004911012930679_a_a_a @ m @ F2 @ I )
          = ( indepe1219004911012930679_a_a_a @ m @ G @ J ) ) ) ) ).

% indep_sets_cong
thf(fact_47_sigma__finite__measure__axioms,axiom,
    measur4308613598931908895sure_a @ m ).

% sigma_finite_measure_axioms
thf(fact_48_finite__measure__axioms,axiom,
    measur930452917991658466sure_a @ m ).

% finite_measure_axioms
thf(fact_49_PiE,axiom,
    ! [F: c > c,A: set_c,B: c > set_c,X3: c] :
      ( ( member_c_c @ F @ ( pi_c_c @ A @ B ) )
     => ( ~ ( member_c @ ( F @ X3 ) @ ( B @ X3 ) )
       => ~ ( member_c @ X3 @ A ) ) ) ).

% PiE
thf(fact_50_PiE,axiom,
    ! [F: b > c,A: set_b,B: b > set_c,X3: b] :
      ( ( member_b_c @ F @ ( pi_b_c @ A @ B ) )
     => ( ~ ( member_c @ ( F @ X3 ) @ ( B @ X3 ) )
       => ~ ( member_b @ X3 @ A ) ) ) ).

% PiE
thf(fact_51_PiE,axiom,
    ! [F: c > b,A: set_c,B: c > set_b,X3: c] :
      ( ( member_c_b @ F @ ( pi_c_b @ A @ B ) )
     => ( ~ ( member_b @ ( F @ X3 ) @ ( B @ X3 ) )
       => ~ ( member_c @ X3 @ A ) ) ) ).

% PiE
thf(fact_52_PiE,axiom,
    ! [F: b > b,A: set_b,B: b > set_b,X3: b] :
      ( ( member_b_b @ F @ ( pi_b_b @ A @ B ) )
     => ( ~ ( member_b @ ( F @ X3 ) @ ( B @ X3 ) )
       => ~ ( member_b @ X3 @ A ) ) ) ).

% PiE
thf(fact_53_PiE,axiom,
    ! [F: c > a,A: set_c,B: c > set_a,X3: c] :
      ( ( member_c_a @ F @ ( pi_c_a @ A @ B ) )
     => ( ~ ( member_a @ ( F @ X3 ) @ ( B @ X3 ) )
       => ~ ( member_c @ X3 @ A ) ) ) ).

% PiE
thf(fact_54_PiE,axiom,
    ! [F: a > real,A: set_a,B: a > set_real,X3: a] :
      ( ( member_a_real @ F @ ( pi_a_real @ A @ B ) )
     => ( ~ ( member_real @ ( F @ X3 ) @ ( B @ X3 ) )
       => ~ ( member_a @ X3 @ A ) ) ) ).

% PiE
thf(fact_55_PiE,axiom,
    ! [F: a > b,A: set_a,B: a > set_b,X3: a] :
      ( ( member_a_b @ F @ ( pi_a_b @ A @ B ) )
     => ( ~ ( member_b @ ( F @ X3 ) @ ( B @ X3 ) )
       => ~ ( member_a @ X3 @ A ) ) ) ).

% PiE
thf(fact_56_PiE,axiom,
    ! [F: b > a,A: set_b,B: b > set_a,X3: b] :
      ( ( member_b_a @ F @ ( pi_b_a @ A @ B ) )
     => ( ~ ( member_a @ ( F @ X3 ) @ ( B @ X3 ) )
       => ~ ( member_b @ X3 @ A ) ) ) ).

% PiE
thf(fact_57_PiE,axiom,
    ! [F: a > c,A: set_a,B: a > set_c,X3: a] :
      ( ( member_a_c @ F @ ( pi_a_c @ A @ B ) )
     => ( ~ ( member_c @ ( F @ X3 ) @ ( B @ X3 ) )
       => ~ ( member_a @ X3 @ A ) ) ) ).

% PiE
thf(fact_58_PiE,axiom,
    ! [F: a > a,A: set_a,B: a > set_a,X3: a] :
      ( ( member_a_a @ F @ ( pi_a_a @ A @ B ) )
     => ( ~ ( member_a @ ( F @ X3 ) @ ( B @ X3 ) )
       => ~ ( member_a @ X3 @ A ) ) ) ).

% PiE
thf(fact_59_indep__vars__compose2,axiom,
    ! [M: c > sigma_measure_b,X4: c > a > b,I: set_c,Y: c > b > a,N2: c > sigma_measure_a] :
      ( ( indepe4851901773132442933_a_c_b @ m @ M @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe4851901773132442932_a_c_a @ m @ N2
          @ ^ [I3: c,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% indep_vars_compose2
thf(fact_60_indep__vars__compose2,axiom,
    ! [M: c > sigma_measure_a,X4: c > a > a,I: set_c,Y: c > a > c,N2: c > sigma_measure_c] :
      ( ( indepe4851901773132442932_a_c_a @ m @ M @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe4851901773132442934_a_c_c @ m @ N2
          @ ^ [I3: c,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% indep_vars_compose2
thf(fact_61_indep__vars__compose2,axiom,
    ! [M: c > sigma_measure_a,X4: c > a > a,I: set_c,Y: c > a > a,N2: c > sigma_measure_a] :
      ( ( indepe4851901773132442932_a_c_a @ m @ M @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe4851901773132442932_a_c_a @ m @ N2
          @ ^ [I3: c,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% indep_vars_compose2
thf(fact_62_indep__vars__compose2,axiom,
    ! [M: b > sigma_measure_b,X4: b > a > b,I: set_b,Y: b > b > a,N2: b > sigma_measure_a] :
      ( ( indepe7639357355105118966_a_b_b @ m @ M @ X4 @ I )
     => ( ! [I2: b] :
            ( ( member_b @ I2 @ I )
           => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe7639357355105118965_a_b_a @ m @ N2
          @ ^ [I3: b,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% indep_vars_compose2
thf(fact_63_indep__vars__compose2,axiom,
    ! [M: b > sigma_measure_a,X4: b > a > a,I: set_b,Y: b > a > c,N2: b > sigma_measure_c] :
      ( ( indepe7639357355105118965_a_b_a @ m @ M @ X4 @ I )
     => ( ! [I2: b] :
            ( ( member_b @ I2 @ I )
           => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe7639357355105118967_a_b_c @ m @ N2
          @ ^ [I3: b,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% indep_vars_compose2
thf(fact_64_indep__vars__compose2,axiom,
    ! [M: b > sigma_measure_a,X4: b > a > a,I: set_b,Y: b > a > a,N2: b > sigma_measure_a] :
      ( ( indepe7639357355105118965_a_b_a @ m @ M @ X4 @ I )
     => ( ! [I2: b] :
            ( ( member_b @ I2 @ I )
           => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe7639357355105118965_a_b_a @ m @ N2
          @ ^ [I3: b,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% indep_vars_compose2
thf(fact_65_indep__vars__compose2,axiom,
    ! [M: a > sigma_measure_b,X4: a > a > b,I: set_a,Y: a > b > a,N2: a > sigma_measure_a] :
      ( ( indepe1203440900223019191_a_a_b @ m @ M @ X4 @ I )
     => ( ! [I2: a] :
            ( ( member_a @ I2 @ I )
           => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe1203440900223019190_a_a_a @ m @ N2
          @ ^ [I3: a,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% indep_vars_compose2
thf(fact_66_indep__vars__compose2,axiom,
    ! [M: a > sigma_measure_a,X4: a > a > a,I: set_a,Y: a > a > c,N2: a > sigma_measure_c] :
      ( ( indepe1203440900223019190_a_a_a @ m @ M @ X4 @ I )
     => ( ! [I2: a] :
            ( ( member_a @ I2 @ I )
           => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe1203440900223019192_a_a_c @ m @ N2
          @ ^ [I3: a,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% indep_vars_compose2
thf(fact_67_indep__vars__compose2,axiom,
    ! [M: a > sigma_measure_a,X4: a > a > a,I: set_a,Y: a > a > a,N2: a > sigma_measure_a] :
      ( ( indepe1203440900223019190_a_a_a @ m @ M @ X4 @ I )
     => ( ! [I2: a] :
            ( ( member_a @ I2 @ I )
           => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe1203440900223019190_a_a_a @ m @ N2
          @ ^ [I3: a,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% indep_vars_compose2
thf(fact_68_indep__vars__compose2,axiom,
    ! [M: c > sigma_measure_a,X4: c > a > a,I: set_c,Y: c > a > b,N2: c > sigma_measure_b] :
      ( ( indepe4851901773132442932_a_c_a @ m @ M @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_a_b @ ( Y @ I2 ) @ ( sigma_measurable_a_b @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe4851901773132442933_a_c_b @ m @ N2
          @ ^ [I3: c,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% indep_vars_compose2
thf(fact_69_prob__space__distr,axiom,
    ! [F: a > a > real,M: sigma_measure_a_real] :
      ( ( member_a_a_real @ F @ ( sigma_5073124708296780141a_real @ m @ M ) )
     => ( probab2024454272037758302a_real @ ( measur6806378183764566549a_real @ m @ M @ F ) ) ) ).

% prob_space_distr
thf(fact_70_prob__space__distr,axiom,
    ! [F: a > a > b,M: sigma_measure_a_b] :
      ( ( member_a_a_b @ F @ ( sigma_3971135317854480532_a_a_b @ m @ M ) )
     => ( probab45282617874185827ce_a_b @ ( measure_distr_a_a_b @ m @ M @ F ) ) ) ).

% prob_space_distr
thf(fact_71_prob__space__distr,axiom,
    ! [F: a > c,M: sigma_measure_c] :
      ( ( member_a_c @ F @ ( sigma_measurable_a_c @ m @ M ) )
     => ( probab7247484486040049091pace_c @ ( measure_distr_a_c @ m @ M @ F ) ) ) ).

% prob_space_distr
thf(fact_72_prob__space__distr,axiom,
    ! [F: a > real,M: sigma_measure_real] :
      ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ m @ M ) )
     => ( probab535871623910865577e_real @ ( measure_distr_a_real @ m @ M @ F ) ) ) ).

% prob_space_distr
thf(fact_73_prob__space__distr,axiom,
    ! [F: a > a,M: sigma_measure_a] :
      ( ( member_a_a @ F @ ( sigma_measurable_a_a @ m @ M ) )
     => ( probab7247484486040049089pace_a @ ( measure_distr_a_a @ m @ M @ F ) ) ) ).

% prob_space_distr
thf(fact_74_prob__space__distr,axiom,
    ! [F: a > b,M: sigma_measure_b] :
      ( ( member_a_b @ F @ ( sigma_measurable_a_b @ m @ M ) )
     => ( probab7247484486040049090pace_b @ ( measure_distr_a_b @ m @ M @ F ) ) ) ).

% prob_space_distr
thf(fact_75_prob__space__distr,axiom,
    ! [F: a > stream_b,M: sigma_6673148739008212994ream_b] :
      ( ( member_a_stream_b @ F @ ( sigma_6363237497282895319ream_b @ m @ M ) )
     => ( probab6181332906713481800ream_b @ ( measur8618403474564049535ream_b @ m @ M @ F ) ) ) ).

% prob_space_distr
thf(fact_76_prob__space__distr,axiom,
    ! [F: a > stream_a,M: sigma_6602114698961867009ream_a] :
      ( ( member_a_stream_a @ F @ ( sigma_6363237492979666518ream_a @ m @ M ) )
     => ( probab6181332902410252999ream_a @ ( measur8618403470260820734ream_a @ m @ M @ F ) ) ) ).

% prob_space_distr
thf(fact_77_finite__measure__distr,axiom,
    ! [F: a > a > real,M: sigma_measure_a_real] :
      ( ( member_a_a_real @ F @ ( sigma_5073124708296780141a_real @ m @ M ) )
     => ( measur5960726210453282175a_real @ ( measur6806378183764566549a_real @ m @ M @ F ) ) ) ).

% finite_measure_distr
thf(fact_78_finite__measure__distr,axiom,
    ! [F: a > a > b,M: sigma_measure_a_b] :
      ( ( member_a_a_b @ F @ ( sigma_3971135317854480532_a_a_b @ m @ M ) )
     => ( measur8914409478018001922re_a_b @ ( measure_distr_a_a_b @ m @ M @ F ) ) ) ).

% finite_measure_distr
thf(fact_79_finite__measure__distr,axiom,
    ! [F: a > c,M: sigma_measure_c] :
      ( ( member_a_c @ F @ ( sigma_measurable_a_c @ m @ M ) )
     => ( measur930452917991658468sure_c @ ( measure_distr_a_c @ m @ M @ F ) ) ) ).

% finite_measure_distr
thf(fact_80_finite__measure__distr,axiom,
    ! [F: a > real,M: sigma_measure_real] :
      ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ m @ M ) )
     => ( measur3606880022600206024e_real @ ( measure_distr_a_real @ m @ M @ F ) ) ) ).

% finite_measure_distr
thf(fact_81_finite__measure__distr,axiom,
    ! [F: a > a,M: sigma_measure_a] :
      ( ( member_a_a @ F @ ( sigma_measurable_a_a @ m @ M ) )
     => ( measur930452917991658466sure_a @ ( measure_distr_a_a @ m @ M @ F ) ) ) ).

% finite_measure_distr
thf(fact_82_finite__measure__distr,axiom,
    ! [F: a > b,M: sigma_measure_b] :
      ( ( member_a_b @ F @ ( sigma_measurable_a_b @ m @ M ) )
     => ( measur930452917991658467sure_b @ ( measure_distr_a_b @ m @ M @ F ) ) ) ).

% finite_measure_distr
thf(fact_83_subprob__space__distr,axiom,
    ! [F: a > a > real,M: sigma_measure_a_real] :
      ( ( member_a_a_real @ F @ ( sigma_5073124708296780141a_real @ m @ M ) )
     => ( ( ( sigma_space_a_real @ M )
         != bot_bot_set_a_real )
       => ( giry_s8245238245569539279a_real @ ( measur6806378183764566549a_real @ m @ M @ F ) ) ) ) ).

% subprob_space_distr
thf(fact_84_subprob__space__distr,axiom,
    ! [F: a > a > b,M: sigma_measure_a_b] :
      ( ( member_a_a_b @ F @ ( sigma_3971135317854480532_a_a_b @ m @ M ) )
     => ( ( ( sigma_space_a_b @ M )
         != bot_bot_set_a_b )
       => ( giry_s8818617164929591474ce_a_b @ ( measure_distr_a_a_b @ m @ M @ F ) ) ) ) ).

% subprob_space_distr
thf(fact_85_subprob__space__distr,axiom,
    ! [F: a > c,M: sigma_measure_c] :
      ( ( member_a_c @ F @ ( sigma_measurable_a_c @ m @ M ) )
     => ( ( ( sigma_space_c @ M )
         != bot_bot_set_c )
       => ( giry_subprob_space_c @ ( measure_distr_a_c @ m @ M @ F ) ) ) ) ).

% subprob_space_distr
thf(fact_86_subprob__space__distr,axiom,
    ! [F: a > sigma_measure_b,M: sigma_2733069582767196501sure_b] :
      ( ( member1617967451723104788sure_b @ F @ ( sigma_698462646940736682sure_b @ m @ M ) )
     => ( ( ( sigma_6149183809546982772sure_b @ M )
         != bot_bo5242124273094418374sure_b )
       => ( giry_s2296666612018410952sure_b @ ( measur2827329755067293186sure_b @ m @ M @ F ) ) ) ) ).

% subprob_space_distr
thf(fact_87_subprob__space__distr,axiom,
    ! [F: a > sigma_measure_a,M: sigma_2662035542720850516sure_a] :
      ( ( member1546933411676758803sure_a @ F @ ( sigma_698462642637507881sure_a @ m @ M ) )
     => ( ( ( sigma_6149183805243753971sure_a @ M )
         != bot_bo5171090233048072389sure_a )
       => ( giry_s2296666607715182151sure_a @ ( measur2827329750764064385sure_a @ m @ M @ F ) ) ) ) ).

% subprob_space_distr
thf(fact_88_subprob__space__distr,axiom,
    ! [F: a > $o,M: sigma_measure_o] :
      ( ( member_a_o @ F @ ( sigma_measurable_a_o @ m @ M ) )
     => ( ( ( sigma_space_o @ M )
         != bot_bot_set_o )
       => ( giry_subprob_space_o @ ( measure_distr_a_o @ m @ M @ F ) ) ) ) ).

% subprob_space_distr
thf(fact_89_subprob__space__distr,axiom,
    ! [F: a > a,M: sigma_measure_a] :
      ( ( member_a_a @ F @ ( sigma_measurable_a_a @ m @ M ) )
     => ( ( ( sigma_space_a @ M )
         != bot_bot_set_a )
       => ( giry_subprob_space_a @ ( measure_distr_a_a @ m @ M @ F ) ) ) ) ).

% subprob_space_distr
thf(fact_90_subprob__space__distr,axiom,
    ! [F: a > b,M: sigma_measure_b] :
      ( ( member_a_b @ F @ ( sigma_measurable_a_b @ m @ M ) )
     => ( ( ( sigma_space_b @ M )
         != bot_bot_set_b )
       => ( giry_subprob_space_b @ ( measure_distr_a_b @ m @ M @ F ) ) ) ) ).

% subprob_space_distr
thf(fact_91_subprob__space__distr,axiom,
    ! [F: a > real,M: sigma_measure_real] :
      ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ m @ M ) )
     => ( ( ( sigma_space_real @ M )
         != bot_bot_set_real )
       => ( giry_s8208748868292234104e_real @ ( measure_distr_a_real @ m @ M @ F ) ) ) ) ).

% subprob_space_distr
thf(fact_92_Pi__eq__empty,axiom,
    ! [A: set_a,B: a > set_real] :
      ( ( ( pi_a_real @ A @ B )
        = bot_bot_set_a_real )
      = ( ? [X2: a] :
            ( ( member_a @ X2 @ A )
            & ( ( B @ X2 )
              = bot_bot_set_real ) ) ) ) ).

% Pi_eq_empty
thf(fact_93_Pi__eq__empty,axiom,
    ! [A: set_a,B: a > set_a_real] :
      ( ( ( pi_a_a_real @ A @ B )
        = bot_bot_set_a_a_real )
      = ( ? [X2: a] :
            ( ( member_a @ X2 @ A )
            & ( ( B @ X2 )
              = bot_bot_set_a_real ) ) ) ) ).

% Pi_eq_empty
thf(fact_94_Pi__eq__empty,axiom,
    ! [A: set_a,B: a > set_a_b] :
      ( ( ( pi_a_a_b @ A @ B )
        = bot_bot_set_a_a_b )
      = ( ? [X2: a] :
            ( ( member_a @ X2 @ A )
            & ( ( B @ X2 )
              = bot_bot_set_a_b ) ) ) ) ).

% Pi_eq_empty
thf(fact_95_Pi__eq__empty,axiom,
    ! [A: set_a,B: a > set_c] :
      ( ( ( pi_a_c @ A @ B )
        = bot_bot_set_a_c )
      = ( ? [X2: a] :
            ( ( member_a @ X2 @ A )
            & ( ( B @ X2 )
              = bot_bot_set_c ) ) ) ) ).

% Pi_eq_empty
thf(fact_96_Pi__eq__empty,axiom,
    ! [A: set_a,B: a > set_a] :
      ( ( ( pi_a_a @ A @ B )
        = bot_bot_set_a_a )
      = ( ? [X2: a] :
            ( ( member_a @ X2 @ A )
            & ( ( B @ X2 )
              = bot_bot_set_a ) ) ) ) ).

% Pi_eq_empty
thf(fact_97_Pi__eq__empty,axiom,
    ! [A: set_a,B: a > set_b] :
      ( ( ( pi_a_b @ A @ B )
        = bot_bot_set_a_b )
      = ( ? [X2: a] :
            ( ( member_a @ X2 @ A )
            & ( ( B @ X2 )
              = bot_bot_set_b ) ) ) ) ).

% Pi_eq_empty
thf(fact_98_indep__sets__Dynkin,axiom,
    ! [F2: nat > set_set_a,I: set_nat] :
      ( ( indepe6267730027088848354_a_nat @ m @ F2 @ I )
     => ( indepe6267730027088848354_a_nat @ m
        @ ^ [I3: nat] : ( sigma_Dynkin_a @ ( sigma_space_a @ m ) @ ( F2 @ I3 ) )
        @ I ) ) ).

% indep_sets_Dynkin
thf(fact_99_indep__sets__Dynkin,axiom,
    ! [F2: $o > set_set_a,I: set_o] :
      ( ( indepe7780107833195774214ts_a_o @ m @ F2 @ I )
     => ( indepe7780107833195774214ts_a_o @ m
        @ ^ [I3: $o] : ( sigma_Dynkin_a @ ( sigma_space_a @ m ) @ ( F2 @ I3 ) )
        @ I ) ) ).

% indep_sets_Dynkin
thf(fact_100_Pi__cong,axiom,
    ! [A: set_b,F: b > a,G2: b > a,B: b > set_a] :
      ( ! [W: b] :
          ( ( member_b @ W @ A )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_b_a @ F @ ( pi_b_a @ A @ B ) )
        = ( member_b_a @ G2 @ ( pi_b_a @ A @ B ) ) ) ) ).

% Pi_cong
thf(fact_101_Pi__cong,axiom,
    ! [A: set_a,F: a > real,G2: a > real,B: a > set_real] :
      ( ! [W: a] :
          ( ( member_a @ W @ A )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_a_real @ F @ ( pi_a_real @ A @ B ) )
        = ( member_a_real @ G2 @ ( pi_a_real @ A @ B ) ) ) ) ).

% Pi_cong
thf(fact_102_Pi__cong,axiom,
    ! [A: set_a,F: a > b,G2: a > b,B: a > set_b] :
      ( ! [W: a] :
          ( ( member_a @ W @ A )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_a_b @ F @ ( pi_a_b @ A @ B ) )
        = ( member_a_b @ G2 @ ( pi_a_b @ A @ B ) ) ) ) ).

% Pi_cong
thf(fact_103_Pi__cong,axiom,
    ! [A: set_a,F: a > a > real,G2: a > a > real,B: a > set_a_real] :
      ( ! [W: a] :
          ( ( member_a @ W @ A )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_a_a_real @ F @ ( pi_a_a_real @ A @ B ) )
        = ( member_a_a_real @ G2 @ ( pi_a_a_real @ A @ B ) ) ) ) ).

% Pi_cong
thf(fact_104_Pi__cong,axiom,
    ! [A: set_a,F: a > a > b,G2: a > a > b,B: a > set_a_b] :
      ( ! [W: a] :
          ( ( member_a @ W @ A )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_a_a_b @ F @ ( pi_a_a_b @ A @ B ) )
        = ( member_a_a_b @ G2 @ ( pi_a_a_b @ A @ B ) ) ) ) ).

% Pi_cong
thf(fact_105_Pi__cong,axiom,
    ! [A: set_a,F: a > c,G2: a > c,B: a > set_c] :
      ( ! [W: a] :
          ( ( member_a @ W @ A )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_a_c @ F @ ( pi_a_c @ A @ B ) )
        = ( member_a_c @ G2 @ ( pi_a_c @ A @ B ) ) ) ) ).

% Pi_cong
thf(fact_106_Pi__cong,axiom,
    ! [A: set_a,F: a > a,G2: a > a,B: a > set_a] :
      ( ! [W: a] :
          ( ( member_a @ W @ A )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_a_a @ F @ ( pi_a_a @ A @ B ) )
        = ( member_a_a @ G2 @ ( pi_a_a @ A @ B ) ) ) ) ).

% Pi_cong
thf(fact_107_Pi__mem,axiom,
    ! [F: c > c,A: set_c,B: c > set_c,X3: c] :
      ( ( member_c_c @ F @ ( pi_c_c @ A @ B ) )
     => ( ( member_c @ X3 @ A )
       => ( member_c @ ( F @ X3 ) @ ( B @ X3 ) ) ) ) ).

% Pi_mem
thf(fact_108_Pi__mem,axiom,
    ! [F: c > b,A: set_c,B: c > set_b,X3: c] :
      ( ( member_c_b @ F @ ( pi_c_b @ A @ B ) )
     => ( ( member_c @ X3 @ A )
       => ( member_b @ ( F @ X3 ) @ ( B @ X3 ) ) ) ) ).

% Pi_mem
thf(fact_109_Pi__mem,axiom,
    ! [F: c > a,A: set_c,B: c > set_a,X3: c] :
      ( ( member_c_a @ F @ ( pi_c_a @ A @ B ) )
     => ( ( member_c @ X3 @ A )
       => ( member_a @ ( F @ X3 ) @ ( B @ X3 ) ) ) ) ).

% Pi_mem
thf(fact_110_Pi__mem,axiom,
    ! [F: b > c,A: set_b,B: b > set_c,X3: b] :
      ( ( member_b_c @ F @ ( pi_b_c @ A @ B ) )
     => ( ( member_b @ X3 @ A )
       => ( member_c @ ( F @ X3 ) @ ( B @ X3 ) ) ) ) ).

% Pi_mem
thf(fact_111_Pi__mem,axiom,
    ! [F: b > b,A: set_b,B: b > set_b,X3: b] :
      ( ( member_b_b @ F @ ( pi_b_b @ A @ B ) )
     => ( ( member_b @ X3 @ A )
       => ( member_b @ ( F @ X3 ) @ ( B @ X3 ) ) ) ) ).

% Pi_mem
thf(fact_112_Pi__mem,axiom,
    ! [F: a > real,A: set_a,B: a > set_real,X3: a] :
      ( ( member_a_real @ F @ ( pi_a_real @ A @ B ) )
     => ( ( member_a @ X3 @ A )
       => ( member_real @ ( F @ X3 ) @ ( B @ X3 ) ) ) ) ).

% Pi_mem
thf(fact_113_Pi__mem,axiom,
    ! [F: a > b,A: set_a,B: a > set_b,X3: a] :
      ( ( member_a_b @ F @ ( pi_a_b @ A @ B ) )
     => ( ( member_a @ X3 @ A )
       => ( member_b @ ( F @ X3 ) @ ( B @ X3 ) ) ) ) ).

% Pi_mem
thf(fact_114_Pi__mem,axiom,
    ! [F: b > a,A: set_b,B: b > set_a,X3: b] :
      ( ( member_b_a @ F @ ( pi_b_a @ A @ B ) )
     => ( ( member_b @ X3 @ A )
       => ( member_a @ ( F @ X3 ) @ ( B @ X3 ) ) ) ) ).

% Pi_mem
thf(fact_115_Pi__mem,axiom,
    ! [F: a > c,A: set_a,B: a > set_c,X3: a] :
      ( ( member_a_c @ F @ ( pi_a_c @ A @ B ) )
     => ( ( member_a @ X3 @ A )
       => ( member_c @ ( F @ X3 ) @ ( B @ X3 ) ) ) ) ).

% Pi_mem
thf(fact_116_Pi__mem,axiom,
    ! [F: a > a,A: set_a,B: a > set_a,X3: a] :
      ( ( member_a_a @ F @ ( pi_a_a @ A @ B ) )
     => ( ( member_a @ X3 @ A )
       => ( member_a @ ( F @ X3 ) @ ( B @ X3 ) ) ) ) ).

% Pi_mem
thf(fact_117_Pi__iff,axiom,
    ! [F: a > real,I: set_a,X4: a > set_real] :
      ( ( member_a_real @ F @ ( pi_a_real @ I @ X4 ) )
      = ( ! [X2: a] :
            ( ( member_a @ X2 @ I )
           => ( member_real @ ( F @ X2 ) @ ( X4 @ X2 ) ) ) ) ) ).

% Pi_iff
thf(fact_118_Pi__iff,axiom,
    ! [F: a > b,I: set_a,X4: a > set_b] :
      ( ( member_a_b @ F @ ( pi_a_b @ I @ X4 ) )
      = ( ! [X2: a] :
            ( ( member_a @ X2 @ I )
           => ( member_b @ ( F @ X2 ) @ ( X4 @ X2 ) ) ) ) ) ).

% Pi_iff
thf(fact_119_Pi__iff,axiom,
    ! [F: b > a,I: set_b,X4: b > set_a] :
      ( ( member_b_a @ F @ ( pi_b_a @ I @ X4 ) )
      = ( ! [X2: b] :
            ( ( member_b @ X2 @ I )
           => ( member_a @ ( F @ X2 ) @ ( X4 @ X2 ) ) ) ) ) ).

% Pi_iff
thf(fact_120_Pi__iff,axiom,
    ! [F: a > a > real,I: set_a,X4: a > set_a_real] :
      ( ( member_a_a_real @ F @ ( pi_a_a_real @ I @ X4 ) )
      = ( ! [X2: a] :
            ( ( member_a @ X2 @ I )
           => ( member_a_real @ ( F @ X2 ) @ ( X4 @ X2 ) ) ) ) ) ).

% Pi_iff
thf(fact_121_Pi__iff,axiom,
    ! [F: a > a > b,I: set_a,X4: a > set_a_b] :
      ( ( member_a_a_b @ F @ ( pi_a_a_b @ I @ X4 ) )
      = ( ! [X2: a] :
            ( ( member_a @ X2 @ I )
           => ( member_a_b @ ( F @ X2 ) @ ( X4 @ X2 ) ) ) ) ) ).

% Pi_iff
thf(fact_122_Pi__iff,axiom,
    ! [F: a > c,I: set_a,X4: a > set_c] :
      ( ( member_a_c @ F @ ( pi_a_c @ I @ X4 ) )
      = ( ! [X2: a] :
            ( ( member_a @ X2 @ I )
           => ( member_c @ ( F @ X2 ) @ ( X4 @ X2 ) ) ) ) ) ).

% Pi_iff
thf(fact_123_Pi__iff,axiom,
    ! [F: a > a,I: set_a,X4: a > set_a] :
      ( ( member_a_a @ F @ ( pi_a_a @ I @ X4 ) )
      = ( ! [X2: a] :
            ( ( member_a @ X2 @ I )
           => ( member_a @ ( F @ X2 ) @ ( X4 @ X2 ) ) ) ) ) ).

% Pi_iff
thf(fact_124_Pi__I_H,axiom,
    ! [A: set_c,F: c > c,B: c > set_c] :
      ( ! [X: c] :
          ( ( member_c @ X @ A )
         => ( member_c @ ( F @ X ) @ ( B @ X ) ) )
     => ( member_c_c @ F @ ( pi_c_c @ A @ B ) ) ) ).

% Pi_I'
thf(fact_125_Pi__I_H,axiom,
    ! [A: set_c,F: c > b,B: c > set_b] :
      ( ! [X: c] :
          ( ( member_c @ X @ A )
         => ( member_b @ ( F @ X ) @ ( B @ X ) ) )
     => ( member_c_b @ F @ ( pi_c_b @ A @ B ) ) ) ).

% Pi_I'
thf(fact_126_Pi__I_H,axiom,
    ! [A: set_c,F: c > a,B: c > set_a] :
      ( ! [X: c] :
          ( ( member_c @ X @ A )
         => ( member_a @ ( F @ X ) @ ( B @ X ) ) )
     => ( member_c_a @ F @ ( pi_c_a @ A @ B ) ) ) ).

% Pi_I'
thf(fact_127_Pi__I_H,axiom,
    ! [A: set_b,F: b > c,B: b > set_c] :
      ( ! [X: b] :
          ( ( member_b @ X @ A )
         => ( member_c @ ( F @ X ) @ ( B @ X ) ) )
     => ( member_b_c @ F @ ( pi_b_c @ A @ B ) ) ) ).

% Pi_I'
thf(fact_128_Pi__I_H,axiom,
    ! [A: set_b,F: b > b,B: b > set_b] :
      ( ! [X: b] :
          ( ( member_b @ X @ A )
         => ( member_b @ ( F @ X ) @ ( B @ X ) ) )
     => ( member_b_b @ F @ ( pi_b_b @ A @ B ) ) ) ).

% Pi_I'
thf(fact_129_Pi__I_H,axiom,
    ! [A: set_b,F: b > a,B: b > set_a] :
      ( ! [X: b] :
          ( ( member_b @ X @ A )
         => ( member_a @ ( F @ X ) @ ( B @ X ) ) )
     => ( member_b_a @ F @ ( pi_b_a @ A @ B ) ) ) ).

% Pi_I'
thf(fact_130_Pi__I_H,axiom,
    ! [A: set_a,F: a > real,B: a > set_real] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( member_real @ ( F @ X ) @ ( B @ X ) ) )
     => ( member_a_real @ F @ ( pi_a_real @ A @ B ) ) ) ).

% Pi_I'
thf(fact_131_Pi__I_H,axiom,
    ! [A: set_a,F: a > c,B: a > set_c] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( member_c @ ( F @ X ) @ ( B @ X ) ) )
     => ( member_a_c @ F @ ( pi_a_c @ A @ B ) ) ) ).

% Pi_I'
thf(fact_132_Pi__I_H,axiom,
    ! [A: set_a,F: a > b,B: a > set_b] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( member_b @ ( F @ X ) @ ( B @ X ) ) )
     => ( member_a_b @ F @ ( pi_a_b @ A @ B ) ) ) ).

% Pi_I'
thf(fact_133_Pi__I_H,axiom,
    ! [A: set_a,F: a > a,B: a > set_a] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( member_a @ ( F @ X ) @ ( B @ X ) ) )
     => ( member_a_a @ F @ ( pi_a_a @ A @ B ) ) ) ).

% Pi_I'
thf(fact_134_D_Ok__wise__indep__vars__compose,axiom,
    ! [K: nat,M: c > sigma_measure_b,X4: c > b > b,I: set_c,Y: c > b > a,N2: c > sigma_measure_a] :
      ( ( prob_k159476456662262089_b_c_b @ ( measure_distr_a_b @ m @ n @ f ) @ K @ M @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( prob_k159476456662262088_b_c_a @ ( measure_distr_a_b @ m @ n @ f ) @ K @ N2
          @ ^ [I3: c,X2: b] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% D.k_wise_indep_vars_compose
thf(fact_135_D_Ok__wise__indep__vars__compose,axiom,
    ! [K: nat,M: c > sigma_measure_a,X4: c > b > a,I: set_c,Y: c > a > c,N2: c > sigma_measure_c] :
      ( ( prob_k159476456662262088_b_c_a @ ( measure_distr_a_b @ m @ n @ f ) @ K @ M @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( prob_k159476456662262090_b_c_c @ ( measure_distr_a_b @ m @ n @ f ) @ K @ N2
          @ ^ [I3: c,X2: b] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% D.k_wise_indep_vars_compose
thf(fact_136_D_Ok__wise__indep__vars__compose,axiom,
    ! [K: nat,M: c > sigma_measure_a,X4: c > b > a,I: set_c,Y: c > a > a,N2: c > sigma_measure_a] :
      ( ( prob_k159476456662262088_b_c_a @ ( measure_distr_a_b @ m @ n @ f ) @ K @ M @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( prob_k159476456662262088_b_c_a @ ( measure_distr_a_b @ m @ n @ f ) @ K @ N2
          @ ^ [I3: c,X2: b] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% D.k_wise_indep_vars_compose
thf(fact_137_D_Ok__wise__indep__vars__compose,axiom,
    ! [K: nat,M: b > sigma_measure_b,X4: b > b > b,I: set_b,Y: b > b > a,N2: b > sigma_measure_a] :
      ( ( prob_k2946932038634938122_b_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ K @ M @ X4 @ I )
     => ( ! [I2: b] :
            ( ( member_b @ I2 @ I )
           => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( prob_k2946932038634938121_b_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ K @ N2
          @ ^ [I3: b,X2: b] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% D.k_wise_indep_vars_compose
thf(fact_138_D_Ok__wise__indep__vars__compose,axiom,
    ! [K: nat,M: b > sigma_measure_a,X4: b > b > a,I: set_b,Y: b > a > c,N2: b > sigma_measure_c] :
      ( ( prob_k2946932038634938121_b_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ K @ M @ X4 @ I )
     => ( ! [I2: b] :
            ( ( member_b @ I2 @ I )
           => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( prob_k2946932038634938123_b_b_c @ ( measure_distr_a_b @ m @ n @ f ) @ K @ N2
          @ ^ [I3: b,X2: b] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% D.k_wise_indep_vars_compose
thf(fact_139_D_Ok__wise__indep__vars__compose,axiom,
    ! [K: nat,M: b > sigma_measure_a,X4: b > b > a,I: set_b,Y: b > a > a,N2: b > sigma_measure_a] :
      ( ( prob_k2946932038634938121_b_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ K @ M @ X4 @ I )
     => ( ! [I2: b] :
            ( ( member_b @ I2 @ I )
           => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( prob_k2946932038634938121_b_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ K @ N2
          @ ^ [I3: b,X2: b] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% D.k_wise_indep_vars_compose
thf(fact_140_D_Ok__wise__indep__vars__compose,axiom,
    ! [K: nat,M: a > sigma_measure_b,X4: a > b > b,I: set_a,Y: a > b > a,N2: a > sigma_measure_a] :
      ( ( prob_k5734387620607614155_b_a_b @ ( measure_distr_a_b @ m @ n @ f ) @ K @ M @ X4 @ I )
     => ( ! [I2: a] :
            ( ( member_a @ I2 @ I )
           => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( prob_k5734387620607614154_b_a_a @ ( measure_distr_a_b @ m @ n @ f ) @ K @ N2
          @ ^ [I3: a,X2: b] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% D.k_wise_indep_vars_compose
thf(fact_141_D_Ok__wise__indep__vars__compose,axiom,
    ! [K: nat,M: a > sigma_measure_a,X4: a > b > a,I: set_a,Y: a > a > c,N2: a > sigma_measure_c] :
      ( ( prob_k5734387620607614154_b_a_a @ ( measure_distr_a_b @ m @ n @ f ) @ K @ M @ X4 @ I )
     => ( ! [I2: a] :
            ( ( member_a @ I2 @ I )
           => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( prob_k5734387620607614156_b_a_c @ ( measure_distr_a_b @ m @ n @ f ) @ K @ N2
          @ ^ [I3: a,X2: b] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% D.k_wise_indep_vars_compose
thf(fact_142_D_Ok__wise__indep__vars__compose,axiom,
    ! [K: nat,M: a > sigma_measure_a,X4: a > b > a,I: set_a,Y: a > a > a,N2: a > sigma_measure_a] :
      ( ( prob_k5734387620607614154_b_a_a @ ( measure_distr_a_b @ m @ n @ f ) @ K @ M @ X4 @ I )
     => ( ! [I2: a] :
            ( ( member_a @ I2 @ I )
           => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( prob_k5734387620607614154_b_a_a @ ( measure_distr_a_b @ m @ n @ f ) @ K @ N2
          @ ^ [I3: a,X2: b] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% D.k_wise_indep_vars_compose
thf(fact_143_D_Ok__wise__indep__vars__compose,axiom,
    ! [K: nat,M: c > sigma_measure_a,X4: c > b > a,I: set_c,Y: c > a > b,N2: c > sigma_measure_b] :
      ( ( prob_k159476456662262088_b_c_a @ ( measure_distr_a_b @ m @ n @ f ) @ K @ M @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_a_b @ ( Y @ I2 ) @ ( sigma_measurable_a_b @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( prob_k159476456662262089_b_c_b @ ( measure_distr_a_b @ m @ n @ f ) @ K @ N2
          @ ^ [I3: c,X2: b] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% D.k_wise_indep_vars_compose
thf(fact_144_k__wise__indep__vars__compose,axiom,
    ! [K: nat,M: c > sigma_measure_b,X4: c > a > b,I: set_c,Y: c > b > a,N2: c > sigma_measure_a] :
      ( ( prob_k3786629878328907208_a_c_b @ m @ K @ M @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( prob_k3786629878328907207_a_c_a @ m @ K @ N2
          @ ^ [I3: c,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% k_wise_indep_vars_compose
thf(fact_145_k__wise__indep__vars__compose,axiom,
    ! [K: nat,M: c > sigma_measure_a,X4: c > a > a,I: set_c,Y: c > a > c,N2: c > sigma_measure_c] :
      ( ( prob_k3786629878328907207_a_c_a @ m @ K @ M @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( prob_k3786629878328907209_a_c_c @ m @ K @ N2
          @ ^ [I3: c,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% k_wise_indep_vars_compose
thf(fact_146_k__wise__indep__vars__compose,axiom,
    ! [K: nat,M: c > sigma_measure_a,X4: c > a > a,I: set_c,Y: c > a > a,N2: c > sigma_measure_a] :
      ( ( prob_k3786629878328907207_a_c_a @ m @ K @ M @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( prob_k3786629878328907207_a_c_a @ m @ K @ N2
          @ ^ [I3: c,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% k_wise_indep_vars_compose
thf(fact_147_k__wise__indep__vars__compose,axiom,
    ! [K: nat,M: b > sigma_measure_b,X4: b > a > b,I: set_b,Y: b > b > a,N2: b > sigma_measure_a] :
      ( ( prob_k6574085460301583241_a_b_b @ m @ K @ M @ X4 @ I )
     => ( ! [I2: b] :
            ( ( member_b @ I2 @ I )
           => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( prob_k6574085460301583240_a_b_a @ m @ K @ N2
          @ ^ [I3: b,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% k_wise_indep_vars_compose
thf(fact_148_k__wise__indep__vars__compose,axiom,
    ! [K: nat,M: b > sigma_measure_a,X4: b > a > a,I: set_b,Y: b > a > c,N2: b > sigma_measure_c] :
      ( ( prob_k6574085460301583240_a_b_a @ m @ K @ M @ X4 @ I )
     => ( ! [I2: b] :
            ( ( member_b @ I2 @ I )
           => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( prob_k6574085460301583242_a_b_c @ m @ K @ N2
          @ ^ [I3: b,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% k_wise_indep_vars_compose
thf(fact_149_k__wise__indep__vars__compose,axiom,
    ! [K: nat,M: b > sigma_measure_a,X4: b > a > a,I: set_b,Y: b > a > a,N2: b > sigma_measure_a] :
      ( ( prob_k6574085460301583240_a_b_a @ m @ K @ M @ X4 @ I )
     => ( ! [I2: b] :
            ( ( member_b @ I2 @ I )
           => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( prob_k6574085460301583240_a_b_a @ m @ K @ N2
          @ ^ [I3: b,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% k_wise_indep_vars_compose
thf(fact_150_k__wise__indep__vars__compose,axiom,
    ! [K: nat,M: a > sigma_measure_b,X4: a > a > b,I: set_a,Y: a > b > a,N2: a > sigma_measure_a] :
      ( ( prob_k138169005419483466_a_a_b @ m @ K @ M @ X4 @ I )
     => ( ! [I2: a] :
            ( ( member_a @ I2 @ I )
           => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( prob_k138169005419483465_a_a_a @ m @ K @ N2
          @ ^ [I3: a,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% k_wise_indep_vars_compose
thf(fact_151_k__wise__indep__vars__compose,axiom,
    ! [K: nat,M: a > sigma_measure_a,X4: a > a > a,I: set_a,Y: a > a > c,N2: a > sigma_measure_c] :
      ( ( prob_k138169005419483465_a_a_a @ m @ K @ M @ X4 @ I )
     => ( ! [I2: a] :
            ( ( member_a @ I2 @ I )
           => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( prob_k138169005419483467_a_a_c @ m @ K @ N2
          @ ^ [I3: a,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% k_wise_indep_vars_compose
thf(fact_152_k__wise__indep__vars__compose,axiom,
    ! [K: nat,M: a > sigma_measure_a,X4: a > a > a,I: set_a,Y: a > a > a,N2: a > sigma_measure_a] :
      ( ( prob_k138169005419483465_a_a_a @ m @ K @ M @ X4 @ I )
     => ( ! [I2: a] :
            ( ( member_a @ I2 @ I )
           => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( prob_k138169005419483465_a_a_a @ m @ K @ N2
          @ ^ [I3: a,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% k_wise_indep_vars_compose
thf(fact_153_k__wise__indep__vars__compose,axiom,
    ! [K: nat,M: c > sigma_measure_a,X4: c > a > a,I: set_c,Y: c > a > b,N2: c > sigma_measure_b] :
      ( ( prob_k3786629878328907207_a_c_a @ m @ K @ M @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_a_b @ ( Y @ I2 ) @ ( sigma_measurable_a_b @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( prob_k3786629878328907208_a_c_b @ m @ K @ N2
          @ ^ [I3: c,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% k_wise_indep_vars_compose
thf(fact_154_D_Oindep__vars__compose2,axiom,
    ! [M: c > sigma_measure_b,X4: c > b > b,I: set_c,Y: c > b > a,N2: c > sigma_measure_a] :
      ( ( indepe1224748351465797814_b_c_b @ ( measure_distr_a_b @ m @ n @ f ) @ M @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe1224748351465797813_b_c_a @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [I3: c,X2: b] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% D.indep_vars_compose2
thf(fact_155_D_Oindep__vars__compose2,axiom,
    ! [M: c > sigma_measure_a,X4: c > b > a,I: set_c,Y: c > a > c,N2: c > sigma_measure_c] :
      ( ( indepe1224748351465797813_b_c_a @ ( measure_distr_a_b @ m @ n @ f ) @ M @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe1224748351465797815_b_c_c @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [I3: c,X2: b] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% D.indep_vars_compose2
thf(fact_156_D_Oindep__vars__compose2,axiom,
    ! [M: c > sigma_measure_a,X4: c > b > a,I: set_c,Y: c > a > a,N2: c > sigma_measure_a] :
      ( ( indepe1224748351465797813_b_c_a @ ( measure_distr_a_b @ m @ n @ f ) @ M @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe1224748351465797813_b_c_a @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [I3: c,X2: b] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% D.indep_vars_compose2
thf(fact_157_D_Oindep__vars__compose2,axiom,
    ! [M: b > sigma_measure_b,X4: b > b > b,I: set_b,Y: b > b > a,N2: b > sigma_measure_a] :
      ( ( indepe4012203933438473847_b_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ M @ X4 @ I )
     => ( ! [I2: b] :
            ( ( member_b @ I2 @ I )
           => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe4012203933438473846_b_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [I3: b,X2: b] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% D.indep_vars_compose2
thf(fact_158_D_Oindep__vars__compose2,axiom,
    ! [M: b > sigma_measure_a,X4: b > b > a,I: set_b,Y: b > a > c,N2: b > sigma_measure_c] :
      ( ( indepe4012203933438473846_b_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ M @ X4 @ I )
     => ( ! [I2: b] :
            ( ( member_b @ I2 @ I )
           => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe4012203933438473848_b_b_c @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [I3: b,X2: b] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% D.indep_vars_compose2
thf(fact_159_D_Oindep__vars__compose2,axiom,
    ! [M: b > sigma_measure_a,X4: b > b > a,I: set_b,Y: b > a > a,N2: b > sigma_measure_a] :
      ( ( indepe4012203933438473846_b_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ M @ X4 @ I )
     => ( ! [I2: b] :
            ( ( member_b @ I2 @ I )
           => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe4012203933438473846_b_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [I3: b,X2: b] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% D.indep_vars_compose2
thf(fact_160_D_Oindep__vars__compose2,axiom,
    ! [M: a > sigma_measure_b,X4: a > b > b,I: set_a,Y: a > b > a,N2: a > sigma_measure_a] :
      ( ( indepe6799659515411149880_b_a_b @ ( measure_distr_a_b @ m @ n @ f ) @ M @ X4 @ I )
     => ( ! [I2: a] :
            ( ( member_a @ I2 @ I )
           => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe6799659515411149879_b_a_a @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [I3: a,X2: b] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% D.indep_vars_compose2
thf(fact_161_D_Oindep__vars__compose2,axiom,
    ! [M: a > sigma_measure_a,X4: a > b > a,I: set_a,Y: a > a > c,N2: a > sigma_measure_c] :
      ( ( indepe6799659515411149879_b_a_a @ ( measure_distr_a_b @ m @ n @ f ) @ M @ X4 @ I )
     => ( ! [I2: a] :
            ( ( member_a @ I2 @ I )
           => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe6799659515411149881_b_a_c @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [I3: a,X2: b] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% D.indep_vars_compose2
thf(fact_162_D_Oindep__vars__compose2,axiom,
    ! [M: a > sigma_measure_a,X4: a > b > a,I: set_a,Y: a > a > a,N2: a > sigma_measure_a] :
      ( ( indepe6799659515411149879_b_a_a @ ( measure_distr_a_b @ m @ n @ f ) @ M @ X4 @ I )
     => ( ! [I2: a] :
            ( ( member_a @ I2 @ I )
           => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe6799659515411149879_b_a_a @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [I3: a,X2: b] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% D.indep_vars_compose2
thf(fact_163_D_Oindep__vars__compose2,axiom,
    ! [M: c > sigma_measure_a,X4: c > b > a,I: set_c,Y: c > a > b,N2: c > sigma_measure_b] :
      ( ( indepe1224748351465797813_b_c_a @ ( measure_distr_a_b @ m @ n @ f ) @ M @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_a_b @ ( Y @ I2 ) @ ( sigma_measurable_a_b @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe1224748351465797814_b_c_b @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [I3: c,X2: b] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
          @ I ) ) ) ).

% D.indep_vars_compose2
thf(fact_164_subprob__space_Osubprob__space__distr,axiom,
    ! [M2: sigma_measure_a,F: a > c,M: sigma_measure_c] :
      ( ( giry_subprob_space_a @ M2 )
     => ( ( member_a_c @ F @ ( sigma_measurable_a_c @ M2 @ M ) )
       => ( ( ( sigma_space_c @ M )
           != bot_bot_set_c )
         => ( giry_subprob_space_c @ ( measure_distr_a_c @ M2 @ M @ F ) ) ) ) ) ).

% subprob_space.subprob_space_distr
thf(fact_165_subprob__space_Osubprob__space__distr,axiom,
    ! [M2: sigma_measure_a,F: a > $o,M: sigma_measure_o] :
      ( ( giry_subprob_space_a @ M2 )
     => ( ( member_a_o @ F @ ( sigma_measurable_a_o @ M2 @ M ) )
       => ( ( ( sigma_space_o @ M )
           != bot_bot_set_o )
         => ( giry_subprob_space_o @ ( measure_distr_a_o @ M2 @ M @ F ) ) ) ) ) ).

% subprob_space.subprob_space_distr
thf(fact_166_subprob__space_Osubprob__space__distr,axiom,
    ! [M2: sigma_measure_a,F: a > a,M: sigma_measure_a] :
      ( ( giry_subprob_space_a @ M2 )
     => ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ M ) )
       => ( ( ( sigma_space_a @ M )
           != bot_bot_set_a )
         => ( giry_subprob_space_a @ ( measure_distr_a_a @ M2 @ M @ F ) ) ) ) ) ).

% subprob_space.subprob_space_distr
thf(fact_167_subprob__space_Osubprob__space__distr,axiom,
    ! [M2: sigma_measure_b,F: b > $o,M: sigma_measure_o] :
      ( ( giry_subprob_space_b @ M2 )
     => ( ( member_b_o @ F @ ( sigma_measurable_b_o @ M2 @ M ) )
       => ( ( ( sigma_space_o @ M )
           != bot_bot_set_o )
         => ( giry_subprob_space_o @ ( measure_distr_b_o @ M2 @ M @ F ) ) ) ) ) ).

% subprob_space.subprob_space_distr
thf(fact_168_subprob__space_Osubprob__space__distr,axiom,
    ! [M2: sigma_measure_b,F: b > a,M: sigma_measure_a] :
      ( ( giry_subprob_space_b @ M2 )
     => ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ M ) )
       => ( ( ( sigma_space_a @ M )
           != bot_bot_set_a )
         => ( giry_subprob_space_a @ ( measure_distr_b_a @ M2 @ M @ F ) ) ) ) ) ).

% subprob_space.subprob_space_distr
thf(fact_169_subprob__space_Osubprob__space__distr,axiom,
    ! [M2: sigma_measure_b,F: b > b,M: sigma_measure_b] :
      ( ( giry_subprob_space_b @ M2 )
     => ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M2 @ M ) )
       => ( ( ( sigma_space_b @ M )
           != bot_bot_set_b )
         => ( giry_subprob_space_b @ ( measure_distr_b_b @ M2 @ M @ F ) ) ) ) ) ).

% subprob_space.subprob_space_distr
thf(fact_170_subprob__space_Osubprob__space__distr,axiom,
    ! [M2: sigma_measure_a,F: a > b,M: sigma_measure_b] :
      ( ( giry_subprob_space_a @ M2 )
     => ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ M ) )
       => ( ( ( sigma_space_b @ M )
           != bot_bot_set_b )
         => ( giry_subprob_space_b @ ( measure_distr_a_b @ M2 @ M @ F ) ) ) ) ) ).

% subprob_space.subprob_space_distr
thf(fact_171_subprob__space_Osubprob__space__distr,axiom,
    ! [M2: sigma_measure_b,F: b > d,M: sigma_measure_d] :
      ( ( giry_subprob_space_b @ M2 )
     => ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M2 @ M ) )
       => ( ( ( sigma_space_d @ M )
           != bot_bot_set_d )
         => ( giry_subprob_space_d @ ( measure_distr_b_d @ M2 @ M @ F ) ) ) ) ) ).

% subprob_space.subprob_space_distr
thf(fact_172_subprob__space_Osubprob__space__distr,axiom,
    ! [M2: sigma_measure_b,F: b > real,M: sigma_measure_real] :
      ( ( giry_subprob_space_b @ M2 )
     => ( ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ M2 @ M ) )
       => ( ( ( sigma_space_real @ M )
           != bot_bot_set_real )
         => ( giry_s8208748868292234104e_real @ ( measure_distr_b_real @ M2 @ M @ F ) ) ) ) ) ).

% subprob_space.subprob_space_distr
thf(fact_173_subprob__space_Osubprob__space__distr,axiom,
    ! [M2: sigma_measure_a,F: a > real,M: sigma_measure_real] :
      ( ( giry_subprob_space_a @ M2 )
     => ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M2 @ M ) )
       => ( ( ( sigma_space_real @ M )
           != bot_bot_set_real )
         => ( giry_s8208748868292234104e_real @ ( measure_distr_a_real @ M2 @ M @ F ) ) ) ) ) ).

% subprob_space.subprob_space_distr
thf(fact_174_space__distr,axiom,
    ! [M2: sigma_measure_a,N2: sigma_measure_b,F: a > b] :
      ( ( sigma_space_b @ ( measure_distr_a_b @ M2 @ N2 @ F ) )
      = ( sigma_space_b @ N2 ) ) ).

% space_distr
thf(fact_175_measurable__distr__eq1,axiom,
    ! [Mf: sigma_measure_a,Nf: sigma_measure_b,F: a > b,Mf2: sigma_measure_d] :
      ( ( sigma_measurable_b_d @ ( measure_distr_a_b @ Mf @ Nf @ F ) @ Mf2 )
      = ( sigma_measurable_b_d @ Nf @ Mf2 ) ) ).

% measurable_distr_eq1
thf(fact_176_measurable__distr__eq1,axiom,
    ! [Mf: sigma_measure_a,Nf: sigma_measure_b,F: a > b,Mf2: sigma_measure_real] :
      ( ( sigma_5489272243865111003b_real @ ( measure_distr_a_b @ Mf @ Nf @ F ) @ Mf2 )
      = ( sigma_5489272243865111003b_real @ Nf @ Mf2 ) ) ).

% measurable_distr_eq1
thf(fact_177_measurable__distr__eq2,axiom,
    ! [Mg: sigma_measure_a,Mg2: sigma_measure_a,Ng: sigma_measure_b,G2: a > b] :
      ( ( sigma_measurable_a_b @ Mg @ ( measure_distr_a_b @ Mg2 @ Ng @ G2 ) )
      = ( sigma_measurable_a_b @ Mg @ Ng ) ) ).

% measurable_distr_eq2
thf(fact_178_D_Osubprob__space__axioms,axiom,
    giry_subprob_space_b @ ( measure_distr_a_b @ m @ n @ f ) ).

% D.subprob_space_axioms
thf(fact_179_D_Oindep__sets__cong,axiom,
    ! [I: set_c,J: set_c,F2: c > set_set_b,G: c > set_set_b] :
      ( ( I = J )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe6139986284700742573ts_b_c @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
          = ( indepe6139986284700742573ts_b_c @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).

% D.indep_sets_cong
thf(fact_180_D_Oindep__sets__cong,axiom,
    ! [I: set_b,J: set_b,F2: b > set_set_b,G: b > set_set_b] :
      ( ( I = J )
     => ( ! [I2: b] :
            ( ( member_b @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe6139986284700742572ts_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
          = ( indepe6139986284700742572ts_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).

% D.indep_sets_cong
thf(fact_181_D_Oindep__sets__cong,axiom,
    ! [I: set_a,J: set_a,F2: a > set_set_b,G: a > set_set_b] :
      ( ( I = J )
     => ( ! [I2: a] :
            ( ( member_a @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe6139986284700742571ts_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
          = ( indepe6139986284700742571ts_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).

% D.indep_sets_cong
thf(fact_182_D_Oindep__sets__cong,axiom,
    ! [I: set_nat,J: set_nat,F2: nat > set_set_b,G: nat > set_set_b] :
      ( ( I = J )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
          = ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).

% D.indep_sets_cong
thf(fact_183_D_Oindep__sets__cong,axiom,
    ! [I: set_o,J: set_o,F2: $o > set_set_b,G: $o > set_set_b] :
      ( ( I = J )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
          = ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).

% D.indep_sets_cong
thf(fact_184_D_Oindep__sets__cong,axiom,
    ! [I: set_a_real,J: set_a_real,F2: ( a > real ) > set_set_b,G: ( a > real ) > set_set_b] :
      ( ( I = J )
     => ( ! [I2: a > real] :
            ( ( member_a_real @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe8950816178784143176a_real @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
          = ( indepe8950816178784143176a_real @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).

% D.indep_sets_cong
thf(fact_185_D_Oindep__sets__cong,axiom,
    ! [I: set_a_b,J: set_a_b,F2: ( a > b ) > set_set_b,G: ( a > b ) > set_set_b] :
      ( ( I = J )
     => ( ! [I2: a > b] :
            ( ( member_a_b @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe7072862685212782073_b_a_b @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
          = ( indepe7072862685212782073_b_a_b @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).

% D.indep_sets_cong
thf(fact_186_D_Oindep__sets__cong,axiom,
    ! [I: set_b_a,J: set_b_a,F2: ( b > a ) > set_set_b,G: ( b > a ) > set_set_b] :
      ( ( I = J )
     => ( ! [I2: b > a] :
            ( ( member_b_a @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe4285407098936877239_b_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
          = ( indepe4285407098936877239_b_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).

% D.indep_sets_cong
thf(fact_187_D_Oindep__sets__cong,axiom,
    ! [I: set_a_c,J: set_a_c,F2: ( a > c ) > set_set_b,G: ( a > c ) > set_set_b] :
      ( ( I = J )
     => ( ! [I2: a > c] :
            ( ( member_a_c @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe7072862689516010874_b_a_c @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
          = ( indepe7072862689516010874_b_a_c @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).

% D.indep_sets_cong
thf(fact_188_D_Oindep__sets__cong,axiom,
    ! [I: set_a_a,J: set_a_a,F2: ( a > a ) > set_set_b,G: ( a > a ) > set_set_b] :
      ( ( I = J )
     => ( ! [I2: a > a] :
            ( ( member_a_a @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe7072862680909553272_b_a_a @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
          = ( indepe7072862680909553272_b_a_a @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).

% D.indep_sets_cong
thf(fact_189_indep__var__rv2,axiom,
    ! [S: sigma_measure_a_real,X4: a > a > real,T: sigma_measure_a_real,Y: a > a > real] :
      ( ( indepe7138907624787055209a_real @ m @ S @ X4 @ T @ Y )
     => ( member_a_a_real @ Y @ ( sigma_5073124708296780141a_real @ m @ T ) ) ) ).

% indep_var_rv2
thf(fact_190_indep__var__rv2,axiom,
    ! [S: sigma_measure_a_b,X4: a > a > b,T: sigma_measure_a_b,Y: a > a > b] :
      ( ( indepe6909679525661198488_a_a_b @ m @ S @ X4 @ T @ Y )
     => ( member_a_a_b @ Y @ ( sigma_3971135317854480532_a_a_b @ m @ T ) ) ) ).

% indep_var_rv2
thf(fact_191_indep__var__rv2,axiom,
    ! [S: sigma_measure_c,X4: a > c,T: sigma_measure_c,Y: a > c] :
      ( ( indepe2440653194691626190ar_a_c @ m @ S @ X4 @ T @ Y )
     => ( member_a_c @ Y @ ( sigma_measurable_a_c @ m @ T ) ) ) ).

% indep_var_rv2
thf(fact_192_indep__var__rv2,axiom,
    ! [S: sigma_measure_a,X4: a > a,T: sigma_measure_a,Y: a > a] :
      ( ( indepe2440653194691626188ar_a_a @ m @ S @ X4 @ T @ Y )
     => ( member_a_a @ Y @ ( sigma_measurable_a_a @ m @ T ) ) ) ).

% indep_var_rv2
thf(fact_193_indep__var__rv2,axiom,
    ! [S: sigma_measure_b,X4: a > b,T: sigma_measure_b,Y: a > b] :
      ( ( indepe2440653194691626189ar_a_b @ m @ S @ X4 @ T @ Y )
     => ( member_a_b @ Y @ ( sigma_measurable_a_b @ m @ T ) ) ) ).

% indep_var_rv2
thf(fact_194_indep__var__rv2,axiom,
    ! [S: sigma_measure_real,X4: a > real,T: sigma_measure_real,Y: a > real] :
      ( ( indepe8958435565499147358a_real @ m @ S @ X4 @ T @ Y )
     => ( member_a_real @ Y @ ( sigma_9116425665531756122a_real @ m @ T ) ) ) ).

% indep_var_rv2
thf(fact_195_indep__var__rv1,axiom,
    ! [S: sigma_measure_a_real,X4: a > a > real,T: sigma_measure_a_real,Y: a > a > real] :
      ( ( indepe7138907624787055209a_real @ m @ S @ X4 @ T @ Y )
     => ( member_a_a_real @ X4 @ ( sigma_5073124708296780141a_real @ m @ S ) ) ) ).

% indep_var_rv1
thf(fact_196_indep__var__rv1,axiom,
    ! [S: sigma_measure_a_b,X4: a > a > b,T: sigma_measure_a_b,Y: a > a > b] :
      ( ( indepe6909679525661198488_a_a_b @ m @ S @ X4 @ T @ Y )
     => ( member_a_a_b @ X4 @ ( sigma_3971135317854480532_a_a_b @ m @ S ) ) ) ).

% indep_var_rv1
thf(fact_197_indep__var__rv1,axiom,
    ! [S: sigma_measure_c,X4: a > c,T: sigma_measure_c,Y: a > c] :
      ( ( indepe2440653194691626190ar_a_c @ m @ S @ X4 @ T @ Y )
     => ( member_a_c @ X4 @ ( sigma_measurable_a_c @ m @ S ) ) ) ).

% indep_var_rv1
thf(fact_198_indep__var__rv1,axiom,
    ! [S: sigma_measure_a,X4: a > a,T: sigma_measure_a,Y: a > a] :
      ( ( indepe2440653194691626188ar_a_a @ m @ S @ X4 @ T @ Y )
     => ( member_a_a @ X4 @ ( sigma_measurable_a_a @ m @ S ) ) ) ).

% indep_var_rv1
thf(fact_199_indep__var__rv1,axiom,
    ! [S: sigma_measure_b,X4: a > b,T: sigma_measure_b,Y: a > b] :
      ( ( indepe2440653194691626189ar_a_b @ m @ S @ X4 @ T @ Y )
     => ( member_a_b @ X4 @ ( sigma_measurable_a_b @ m @ S ) ) ) ).

% indep_var_rv1
thf(fact_200_indep__var__rv1,axiom,
    ! [S: sigma_measure_real,X4: a > real,T: sigma_measure_real,Y: a > real] :
      ( ( indepe8958435565499147358a_real @ m @ S @ X4 @ T @ Y )
     => ( member_a_real @ X4 @ ( sigma_9116425665531756122a_real @ m @ S ) ) ) ).

% indep_var_rv1
thf(fact_201_D_Ofinite__measure__axioms,axiom,
    measur930452917991658467sure_b @ ( measure_distr_a_b @ m @ n @ f ) ).

% D.finite_measure_axioms
thf(fact_202_D_Oprob__space__axioms,axiom,
    probab7247484486040049090pace_b @ ( measure_distr_a_b @ m @ n @ f ) ).

% D.prob_space_axioms
thf(fact_203_D_Oprob__space__distr,axiom,
    ! [F: b > d,M: sigma_measure_d] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ M ) )
     => ( probab7247484486040049092pace_d @ ( measure_distr_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ M @ F ) ) ) ).

% D.prob_space_distr
thf(fact_204_D_Oprob__space__distr,axiom,
    ! [F: b > real,M: sigma_measure_real] :
      ( ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ ( measure_distr_a_b @ m @ n @ f ) @ M ) )
     => ( probab535871623910865577e_real @ ( measure_distr_b_real @ ( measure_distr_a_b @ m @ n @ f ) @ M @ F ) ) ) ).

% D.prob_space_distr
thf(fact_205_D_Oprob__space__distr,axiom,
    ! [F: b > a,M: sigma_measure_a] :
      ( ( member_b_a @ F @ ( sigma_measurable_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ M ) )
     => ( probab7247484486040049089pace_a @ ( measure_distr_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ M @ F ) ) ) ).

% D.prob_space_distr
thf(fact_206_D_Oprob__space__distr,axiom,
    ! [F: b > b,M: sigma_measure_b] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ M ) )
     => ( probab7247484486040049090pace_b @ ( measure_distr_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ M @ F ) ) ) ).

% D.prob_space_distr
thf(fact_207_D_Oprob__space__distr,axiom,
    ! [F: b > stream_b,M: sigma_6673148739008212994ream_b] :
      ( ( member_b_stream_b @ F @ ( sigma_3277481255292900758ream_b @ ( measure_distr_a_b @ m @ n @ f ) @ M ) )
     => ( probab6181332906713481800ream_b @ ( measur5532647232574054974ream_b @ ( measure_distr_a_b @ m @ n @ f ) @ M @ F ) ) ) ).

% D.prob_space_distr
thf(fact_208_D_Oprob__space__distr,axiom,
    ! [F: b > stream_a,M: sigma_6602114698961867009ream_a] :
      ( ( member_b_stream_a @ F @ ( sigma_3277481250989671957ream_a @ ( measure_distr_a_b @ m @ n @ f ) @ M ) )
     => ( probab6181332902410252999ream_a @ ( measur5532647228270826173ream_a @ ( measure_distr_a_b @ m @ n @ f ) @ M @ F ) ) ) ).

% D.prob_space_distr
thf(fact_209_D_Ofinite__measure__distr,axiom,
    ! [F: b > d,M: sigma_measure_d] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ M ) )
     => ( measur930452917991658469sure_d @ ( measure_distr_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ M @ F ) ) ) ).

% D.finite_measure_distr
thf(fact_210_D_Ofinite__measure__distr,axiom,
    ! [F: b > real,M: sigma_measure_real] :
      ( ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ ( measure_distr_a_b @ m @ n @ f ) @ M ) )
     => ( measur3606880022600206024e_real @ ( measure_distr_b_real @ ( measure_distr_a_b @ m @ n @ f ) @ M @ F ) ) ) ).

% D.finite_measure_distr
thf(fact_211_D_Ofinite__measure__distr,axiom,
    ! [F: b > a,M: sigma_measure_a] :
      ( ( member_b_a @ F @ ( sigma_measurable_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ M ) )
     => ( measur930452917991658466sure_a @ ( measure_distr_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ M @ F ) ) ) ).

% D.finite_measure_distr
thf(fact_212_D_Ofinite__measure__distr,axiom,
    ! [F: b > b,M: sigma_measure_b] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ M ) )
     => ( measur930452917991658467sure_b @ ( measure_distr_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ M @ F ) ) ) ).

% D.finite_measure_distr
thf(fact_213_D_Osubprob__space__distr,axiom,
    ! [F: b > sigma_measure_b,M: sigma_2733069582767196501sure_b] :
      ( ( member7751282110383867797sure_b @ F @ ( sigma_6831777305601499691sure_b @ ( measure_distr_a_b @ m @ n @ f ) @ M ) )
     => ( ( ( sigma_6149183809546982772sure_b @ M )
         != bot_bo5242124273094418374sure_b )
       => ( giry_s2296666612018410952sure_b @ ( measur8960644413728056195sure_b @ ( measure_distr_a_b @ m @ n @ f ) @ M @ F ) ) ) ) ).

% D.subprob_space_distr
thf(fact_214_D_Osubprob__space__distr,axiom,
    ! [F: b > sigma_measure_a,M: sigma_2662035542720850516sure_a] :
      ( ( member7680248070337521812sure_a @ F @ ( sigma_6831777301298270890sure_a @ ( measure_distr_a_b @ m @ n @ f ) @ M ) )
     => ( ( ( sigma_6149183805243753971sure_a @ M )
         != bot_bo5171090233048072389sure_a )
       => ( giry_s2296666607715182151sure_a @ ( measur8960644409424827394sure_a @ ( measure_distr_a_b @ m @ n @ f ) @ M @ F ) ) ) ) ).

% D.subprob_space_distr
thf(fact_215_D_Osubprob__space__distr,axiom,
    ! [F: b > $o,M: sigma_measure_o] :
      ( ( member_b_o @ F @ ( sigma_measurable_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ M ) )
     => ( ( ( sigma_space_o @ M )
         != bot_bot_set_o )
       => ( giry_subprob_space_o @ ( measure_distr_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ M @ F ) ) ) ) ).

% D.subprob_space_distr
thf(fact_216_D_Osubprob__space__distr,axiom,
    ! [F: b > a,M: sigma_measure_a] :
      ( ( member_b_a @ F @ ( sigma_measurable_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ M ) )
     => ( ( ( sigma_space_a @ M )
         != bot_bot_set_a )
       => ( giry_subprob_space_a @ ( measure_distr_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ M @ F ) ) ) ) ).

% D.subprob_space_distr
thf(fact_217_D_Osubprob__space__distr,axiom,
    ! [F: b > b,M: sigma_measure_b] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ M ) )
     => ( ( ( sigma_space_b @ M )
         != bot_bot_set_b )
       => ( giry_subprob_space_b @ ( measure_distr_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ M @ F ) ) ) ) ).

% D.subprob_space_distr
thf(fact_218_D_Osubprob__space__distr,axiom,
    ! [F: b > d,M: sigma_measure_d] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ M ) )
     => ( ( ( sigma_space_d @ M )
         != bot_bot_set_d )
       => ( giry_subprob_space_d @ ( measure_distr_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ M @ F ) ) ) ) ).

% D.subprob_space_distr
thf(fact_219_D_Osubprob__space__distr,axiom,
    ! [F: b > real,M: sigma_measure_real] :
      ( ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ ( measure_distr_a_b @ m @ n @ f ) @ M ) )
     => ( ( ( sigma_space_real @ M )
         != bot_bot_set_real )
       => ( giry_s8208748868292234104e_real @ ( measure_distr_b_real @ ( measure_distr_a_b @ m @ n @ f ) @ M @ F ) ) ) ) ).

% D.subprob_space_distr
thf(fact_220_D_Oindep__sets__Dynkin,axiom,
    ! [F2: nat > set_set_b,I: set_nat] :
      ( ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
     => ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f )
        @ ^ [I3: nat] : ( sigma_Dynkin_b @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ ( F2 @ I3 ) )
        @ I ) ) ).

% D.indep_sets_Dynkin
thf(fact_221_D_Oindep__sets__Dynkin,axiom,
    ! [F2: $o > set_set_b,I: set_o] :
      ( ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
     => ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f )
        @ ^ [I3: $o] : ( sigma_Dynkin_b @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ ( F2 @ I3 ) )
        @ I ) ) ).

% D.indep_sets_Dynkin
thf(fact_222_prob__space_Ok__wise__indep__vars__compose,axiom,
    ! [M2: sigma_measure_a,K: nat,M: c > sigma_measure_b,X4: c > a > b,I: set_c,Y: c > b > a,N2: c > sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( prob_k3786629878328907208_a_c_b @ M2 @ K @ M @ X4 @ I )
       => ( ! [I2: c] :
              ( ( member_c @ I2 @ I )
             => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( prob_k3786629878328907207_a_c_a @ M2 @ K @ N2
            @ ^ [I3: c,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.k_wise_indep_vars_compose
thf(fact_223_prob__space_Ok__wise__indep__vars__compose,axiom,
    ! [M2: sigma_measure_a,K: nat,M: c > sigma_measure_a,X4: c > a > a,I: set_c,Y: c > a > c,N2: c > sigma_measure_c] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( prob_k3786629878328907207_a_c_a @ M2 @ K @ M @ X4 @ I )
       => ( ! [I2: c] :
              ( ( member_c @ I2 @ I )
             => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( prob_k3786629878328907209_a_c_c @ M2 @ K @ N2
            @ ^ [I3: c,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.k_wise_indep_vars_compose
thf(fact_224_prob__space_Ok__wise__indep__vars__compose,axiom,
    ! [M2: sigma_measure_a,K: nat,M: c > sigma_measure_a,X4: c > a > a,I: set_c,Y: c > a > a,N2: c > sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( prob_k3786629878328907207_a_c_a @ M2 @ K @ M @ X4 @ I )
       => ( ! [I2: c] :
              ( ( member_c @ I2 @ I )
             => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( prob_k3786629878328907207_a_c_a @ M2 @ K @ N2
            @ ^ [I3: c,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.k_wise_indep_vars_compose
thf(fact_225_prob__space_Ok__wise__indep__vars__compose,axiom,
    ! [M2: sigma_measure_a,K: nat,M: b > sigma_measure_b,X4: b > a > b,I: set_b,Y: b > b > a,N2: b > sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( prob_k6574085460301583241_a_b_b @ M2 @ K @ M @ X4 @ I )
       => ( ! [I2: b] :
              ( ( member_b @ I2 @ I )
             => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( prob_k6574085460301583240_a_b_a @ M2 @ K @ N2
            @ ^ [I3: b,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.k_wise_indep_vars_compose
thf(fact_226_prob__space_Ok__wise__indep__vars__compose,axiom,
    ! [M2: sigma_measure_a,K: nat,M: b > sigma_measure_a,X4: b > a > a,I: set_b,Y: b > a > c,N2: b > sigma_measure_c] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( prob_k6574085460301583240_a_b_a @ M2 @ K @ M @ X4 @ I )
       => ( ! [I2: b] :
              ( ( member_b @ I2 @ I )
             => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( prob_k6574085460301583242_a_b_c @ M2 @ K @ N2
            @ ^ [I3: b,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.k_wise_indep_vars_compose
thf(fact_227_prob__space_Ok__wise__indep__vars__compose,axiom,
    ! [M2: sigma_measure_a,K: nat,M: b > sigma_measure_a,X4: b > a > a,I: set_b,Y: b > a > a,N2: b > sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( prob_k6574085460301583240_a_b_a @ M2 @ K @ M @ X4 @ I )
       => ( ! [I2: b] :
              ( ( member_b @ I2 @ I )
             => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( prob_k6574085460301583240_a_b_a @ M2 @ K @ N2
            @ ^ [I3: b,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.k_wise_indep_vars_compose
thf(fact_228_prob__space_Ok__wise__indep__vars__compose,axiom,
    ! [M2: sigma_measure_a,K: nat,M: a > sigma_measure_b,X4: a > a > b,I: set_a,Y: a > b > a,N2: a > sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( prob_k138169005419483466_a_a_b @ M2 @ K @ M @ X4 @ I )
       => ( ! [I2: a] :
              ( ( member_a @ I2 @ I )
             => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( prob_k138169005419483465_a_a_a @ M2 @ K @ N2
            @ ^ [I3: a,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.k_wise_indep_vars_compose
thf(fact_229_prob__space_Ok__wise__indep__vars__compose,axiom,
    ! [M2: sigma_measure_a,K: nat,M: a > sigma_measure_a,X4: a > a > a,I: set_a,Y: a > a > c,N2: a > sigma_measure_c] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( prob_k138169005419483465_a_a_a @ M2 @ K @ M @ X4 @ I )
       => ( ! [I2: a] :
              ( ( member_a @ I2 @ I )
             => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( prob_k138169005419483467_a_a_c @ M2 @ K @ N2
            @ ^ [I3: a,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.k_wise_indep_vars_compose
thf(fact_230_prob__space_Ok__wise__indep__vars__compose,axiom,
    ! [M2: sigma_measure_a,K: nat,M: a > sigma_measure_a,X4: a > a > a,I: set_a,Y: a > a > a,N2: a > sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( prob_k138169005419483465_a_a_a @ M2 @ K @ M @ X4 @ I )
       => ( ! [I2: a] :
              ( ( member_a @ I2 @ I )
             => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( prob_k138169005419483465_a_a_a @ M2 @ K @ N2
            @ ^ [I3: a,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.k_wise_indep_vars_compose
thf(fact_231_prob__space_Ok__wise__indep__vars__compose,axiom,
    ! [M2: sigma_measure_a,K: nat,M: c > sigma_measure_a,X4: c > a > a,I: set_c,Y: c > a > b,N2: c > sigma_measure_b] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( prob_k3786629878328907207_a_c_a @ M2 @ K @ M @ X4 @ I )
       => ( ! [I2: c] :
              ( ( member_c @ I2 @ I )
             => ( member_a_b @ ( Y @ I2 ) @ ( sigma_measurable_a_b @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( prob_k3786629878328907208_a_c_b @ M2 @ K @ N2
            @ ^ [I3: c,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.k_wise_indep_vars_compose
thf(fact_232_finite__measure_Oaxioms_I1_J,axiom,
    ! [M2: sigma_measure_a] :
      ( ( measur930452917991658466sure_a @ M2 )
     => ( measur4308613598931908895sure_a @ M2 ) ) ).

% finite_measure.axioms(1)
thf(fact_233_finite__measure_Oaxioms_I1_J,axiom,
    ! [M2: sigma_measure_b] :
      ( ( measur930452917991658467sure_b @ M2 )
     => ( measur4308613598931908896sure_b @ M2 ) ) ).

% finite_measure.axioms(1)
thf(fact_234_mem__Collect__eq,axiom,
    ! [A2: c,P: c > $o] :
      ( ( member_c @ A2 @ ( collect_c @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_235_mem__Collect__eq,axiom,
    ! [A2: nat,P: nat > $o] :
      ( ( member_nat @ A2 @ ( collect_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_236_mem__Collect__eq,axiom,
    ! [A2: b,P: b > $o] :
      ( ( member_b @ A2 @ ( collect_b @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_237_mem__Collect__eq,axiom,
    ! [A2: a,P: a > $o] :
      ( ( member_a @ A2 @ ( collect_a @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_238_mem__Collect__eq,axiom,
    ! [A2: a > real,P: ( a > real ) > $o] :
      ( ( member_a_real @ A2 @ ( collect_a_real @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_239_mem__Collect__eq,axiom,
    ! [A2: a > b,P: ( a > b ) > $o] :
      ( ( member_a_b @ A2 @ ( collect_a_b @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_240_mem__Collect__eq,axiom,
    ! [A2: b > a,P: ( b > a ) > $o] :
      ( ( member_b_a @ A2 @ ( collect_b_a @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_241_mem__Collect__eq,axiom,
    ! [A2: a > c,P: ( a > c ) > $o] :
      ( ( member_a_c @ A2 @ ( collect_a_c @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_242_mem__Collect__eq,axiom,
    ! [A2: a > a,P: ( a > a ) > $o] :
      ( ( member_a_a @ A2 @ ( collect_a_a @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_243_mem__Collect__eq,axiom,
    ! [A2: a > a > real,P: ( a > a > real ) > $o] :
      ( ( member_a_a_real @ A2 @ ( collect_a_a_real @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_244_Collect__mem__eq,axiom,
    ! [A: set_c] :
      ( ( collect_c
        @ ^ [X2: c] : ( member_c @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_245_Collect__mem__eq,axiom,
    ! [A: set_nat] :
      ( ( collect_nat
        @ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_246_Collect__mem__eq,axiom,
    ! [A: set_b] :
      ( ( collect_b
        @ ^ [X2: b] : ( member_b @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_247_Collect__mem__eq,axiom,
    ! [A: set_a] :
      ( ( collect_a
        @ ^ [X2: a] : ( member_a @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_248_Collect__mem__eq,axiom,
    ! [A: set_a_real] :
      ( ( collect_a_real
        @ ^ [X2: a > real] : ( member_a_real @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_249_Collect__mem__eq,axiom,
    ! [A: set_a_b] :
      ( ( collect_a_b
        @ ^ [X2: a > b] : ( member_a_b @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_250_Collect__mem__eq,axiom,
    ! [A: set_b_a] :
      ( ( collect_b_a
        @ ^ [X2: b > a] : ( member_b_a @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_251_Collect__mem__eq,axiom,
    ! [A: set_a_c] :
      ( ( collect_a_c
        @ ^ [X2: a > c] : ( member_a_c @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_252_Collect__mem__eq,axiom,
    ! [A: set_a_a] :
      ( ( collect_a_a
        @ ^ [X2: a > a] : ( member_a_a @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_253_Collect__mem__eq,axiom,
    ! [A: set_a_a_real] :
      ( ( collect_a_a_real
        @ ^ [X2: a > a > real] : ( member_a_a_real @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_254_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X: nat] :
          ( ( P @ X )
          = ( Q @ X ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_255_Collect__cong,axiom,
    ! [P: b > $o,Q: b > $o] :
      ( ! [X: b] :
          ( ( P @ X )
          = ( Q @ X ) )
     => ( ( collect_b @ P )
        = ( collect_b @ Q ) ) ) ).

% Collect_cong
thf(fact_256_Collect__cong,axiom,
    ! [P: a > $o,Q: a > $o] :
      ( ! [X: a] :
          ( ( P @ X )
          = ( Q @ X ) )
     => ( ( collect_a @ P )
        = ( collect_a @ Q ) ) ) ).

% Collect_cong
thf(fact_257_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_258_prob__space__imp__subprob__space,axiom,
    ! [M2: sigma_measure_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( giry_subprob_space_b @ M2 ) ) ).

% prob_space_imp_subprob_space
thf(fact_259_prob__space__imp__subprob__space,axiom,
    ! [M2: sigma_6673148739008212994ream_b] :
      ( ( probab6181332906713481800ream_b @ M2 )
     => ( giry_s6069289514029806137ream_b @ M2 ) ) ).

% prob_space_imp_subprob_space
thf(fact_260_prob__space__imp__subprob__space,axiom,
    ! [M2: sigma_6602114698961867009ream_a] :
      ( ( probab6181332902410252999ream_a @ M2 )
     => ( giry_s6069289509726577336ream_a @ M2 ) ) ).

% prob_space_imp_subprob_space
thf(fact_261_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_262_subprob__space__imp__sigma__finite,axiom,
    ! [M2: sigma_measure_b] :
      ( ( giry_subprob_space_b @ M2 )
     => ( measur4308613598931908896sure_b @ M2 ) ) ).

% subprob_space_imp_sigma_finite
thf(fact_263_subprob__space_Oaxioms_I1_J,axiom,
    ! [M2: sigma_measure_a] :
      ( ( giry_subprob_space_a @ M2 )
     => ( measur930452917991658466sure_a @ M2 ) ) ).

% subprob_space.axioms(1)
thf(fact_264_subprob__space_Oaxioms_I1_J,axiom,
    ! [M2: sigma_measure_b] :
      ( ( giry_subprob_space_b @ M2 )
     => ( measur930452917991658467sure_b @ M2 ) ) ).

% subprob_space.axioms(1)
thf(fact_265_finite__measure_Ofinite__measure__distr,axiom,
    ! [M2: sigma_measure_a,F: a > a > real,M: sigma_measure_a_real] :
      ( ( measur930452917991658466sure_a @ M2 )
     => ( ( member_a_a_real @ F @ ( sigma_5073124708296780141a_real @ M2 @ M ) )
       => ( measur5960726210453282175a_real @ ( measur6806378183764566549a_real @ M2 @ M @ F ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_266_finite__measure_Ofinite__measure__distr,axiom,
    ! [M2: sigma_measure_a,F: a > a > b,M: sigma_measure_a_b] :
      ( ( measur930452917991658466sure_a @ M2 )
     => ( ( member_a_a_b @ F @ ( sigma_3971135317854480532_a_a_b @ M2 @ M ) )
       => ( measur8914409478018001922re_a_b @ ( measure_distr_a_a_b @ M2 @ M @ F ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_267_finite__measure_Ofinite__measure__distr,axiom,
    ! [M2: sigma_measure_a,F: a > c,M: sigma_measure_c] :
      ( ( measur930452917991658466sure_a @ M2 )
     => ( ( member_a_c @ F @ ( sigma_measurable_a_c @ M2 @ M ) )
       => ( measur930452917991658468sure_c @ ( measure_distr_a_c @ M2 @ M @ F ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_268_finite__measure_Ofinite__measure__distr,axiom,
    ! [M2: sigma_measure_a,F: a > real,M: sigma_measure_real] :
      ( ( measur930452917991658466sure_a @ M2 )
     => ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M2 @ M ) )
       => ( measur3606880022600206024e_real @ ( measure_distr_a_real @ M2 @ M @ F ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_269_finite__measure_Ofinite__measure__distr,axiom,
    ! [M2: sigma_measure_a,F: a > a,M: sigma_measure_a] :
      ( ( measur930452917991658466sure_a @ M2 )
     => ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ M ) )
       => ( measur930452917991658466sure_a @ ( measure_distr_a_a @ M2 @ M @ F ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_270_finite__measure_Ofinite__measure__distr,axiom,
    ! [M2: sigma_measure_a,F: a > b,M: sigma_measure_b] :
      ( ( measur930452917991658466sure_a @ M2 )
     => ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ M ) )
       => ( measur930452917991658467sure_b @ ( measure_distr_a_b @ M2 @ M @ F ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_271_finite__measure_Ofinite__measure__distr,axiom,
    ! [M2: sigma_measure_b,F: b > d,M: sigma_measure_d] :
      ( ( measur930452917991658467sure_b @ M2 )
     => ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M2 @ M ) )
       => ( measur930452917991658469sure_d @ ( measure_distr_b_d @ M2 @ M @ F ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_272_finite__measure_Ofinite__measure__distr,axiom,
    ! [M2: sigma_measure_b,F: b > real,M: sigma_measure_real] :
      ( ( measur930452917991658467sure_b @ M2 )
     => ( ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ M2 @ M ) )
       => ( measur3606880022600206024e_real @ ( measure_distr_b_real @ M2 @ M @ F ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_273_finite__measure_Ofinite__measure__distr,axiom,
    ! [M2: sigma_measure_b,F: b > a,M: sigma_measure_a] :
      ( ( measur930452917991658467sure_b @ M2 )
     => ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ M ) )
       => ( measur930452917991658466sure_a @ ( measure_distr_b_a @ M2 @ M @ F ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_274_finite__measure_Ofinite__measure__distr,axiom,
    ! [M2: sigma_measure_b,F: b > b,M: sigma_measure_b] :
      ( ( measur930452917991658467sure_b @ M2 )
     => ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M2 @ M ) )
       => ( measur930452917991658467sure_b @ ( measure_distr_b_b @ M2 @ M @ F ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_275_subprob__space_Osubprob__not__empty,axiom,
    ! [M2: sigma_2733069582767196501sure_b] :
      ( ( giry_s2296666612018410952sure_b @ M2 )
     => ( ( sigma_6149183809546982772sure_b @ M2 )
       != bot_bo5242124273094418374sure_b ) ) ).

% subprob_space.subprob_not_empty
thf(fact_276_subprob__space_Osubprob__not__empty,axiom,
    ! [M2: sigma_2662035542720850516sure_a] :
      ( ( giry_s2296666607715182151sure_a @ M2 )
     => ( ( sigma_6149183805243753971sure_a @ M2 )
       != bot_bo5171090233048072389sure_a ) ) ).

% subprob_space.subprob_not_empty
thf(fact_277_subprob__space_Osubprob__not__empty,axiom,
    ! [M2: sigma_measure_o] :
      ( ( giry_subprob_space_o @ M2 )
     => ( ( sigma_space_o @ M2 )
       != bot_bot_set_o ) ) ).

% subprob_space.subprob_not_empty
thf(fact_278_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_279_subprob__space_Osubprob__not__empty,axiom,
    ! [M2: sigma_measure_b] :
      ( ( giry_subprob_space_b @ M2 )
     => ( ( sigma_space_b @ M2 )
       != bot_bot_set_b ) ) ).

% subprob_space.subprob_not_empty
thf(fact_280_D_Oindep__var__rv1,axiom,
    ! [S: sigma_measure_a,X4: b > a,T: sigma_measure_a,Y: b > a] :
      ( ( indepe8876569649573725963ar_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ S @ X4 @ T @ Y )
     => ( member_b_a @ X4 @ ( sigma_measurable_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ S ) ) ) ).

% D.indep_var_rv1
thf(fact_281_D_Oindep__var__rv1,axiom,
    ! [S: sigma_measure_d,X4: b > d,T: sigma_measure_d,Y: b > d] :
      ( ( indepe8876569649573725966ar_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ S @ X4 @ T @ Y )
     => ( member_b_d @ X4 @ ( sigma_measurable_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ S ) ) ) ).

% D.indep_var_rv1
thf(fact_282_D_Oindep__var__rv1,axiom,
    ! [S: sigma_measure_real,X4: b > real,T: sigma_measure_real,Y: b > real] :
      ( ( indepe5331282143832502239b_real @ ( measure_distr_a_b @ m @ n @ f ) @ S @ X4 @ T @ Y )
     => ( member_b_real @ X4 @ ( sigma_5489272243865111003b_real @ ( measure_distr_a_b @ m @ n @ f ) @ S ) ) ) ).

% D.indep_var_rv1
thf(fact_283_D_Oindep__var__rv2,axiom,
    ! [S: sigma_measure_a,X4: b > a,T: sigma_measure_a,Y: b > a] :
      ( ( indepe8876569649573725963ar_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ S @ X4 @ T @ Y )
     => ( member_b_a @ Y @ ( sigma_measurable_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ T ) ) ) ).

% D.indep_var_rv2
thf(fact_284_D_Oindep__var__rv2,axiom,
    ! [S: sigma_measure_d,X4: b > d,T: sigma_measure_d,Y: b > d] :
      ( ( indepe8876569649573725966ar_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ S @ X4 @ T @ Y )
     => ( member_b_d @ Y @ ( sigma_measurable_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ T ) ) ) ).

% D.indep_var_rv2
thf(fact_285_D_Oindep__var__rv2,axiom,
    ! [S: sigma_measure_real,X4: b > real,T: sigma_measure_real,Y: b > real] :
      ( ( indepe5331282143832502239b_real @ ( measure_distr_a_b @ m @ n @ f ) @ S @ X4 @ T @ Y )
     => ( member_b_real @ Y @ ( sigma_5489272243865111003b_real @ ( measure_distr_a_b @ m @ n @ f ) @ T ) ) ) ).

% D.indep_var_rv2
thf(fact_286_D_Oindep__sets__mono__sets,axiom,
    ! [F2: c > set_set_b,I: set_c,G: c > set_set_b] :
      ( ( indepe6139986284700742573ts_b_c @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe6139986284700742573ts_b_c @ ( measure_distr_a_b @ m @ n @ f ) @ G @ I ) ) ) ).

% D.indep_sets_mono_sets
thf(fact_287_D_Oindep__sets__mono__sets,axiom,
    ! [F2: b > set_set_b,I: set_b,G: b > set_set_b] :
      ( ( indepe6139986284700742572ts_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
     => ( ! [I2: b] :
            ( ( member_b @ I2 @ I )
           => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe6139986284700742572ts_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ G @ I ) ) ) ).

% D.indep_sets_mono_sets
thf(fact_288_D_Oindep__sets__mono__sets,axiom,
    ! [F2: a > set_set_b,I: set_a,G: a > set_set_b] :
      ( ( indepe6139986284700742571ts_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
     => ( ! [I2: a] :
            ( ( member_a @ I2 @ I )
           => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe6139986284700742571ts_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ G @ I ) ) ) ).

% D.indep_sets_mono_sets
thf(fact_289_D_Oindep__sets__mono__sets,axiom,
    ! [F2: nat > set_set_b,I: set_nat,G: nat > set_set_b] :
      ( ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I )
           => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ G @ I ) ) ) ).

% D.indep_sets_mono_sets
thf(fact_290_D_Oindep__sets__mono__sets,axiom,
    ! [F2: $o > set_set_b,I: set_o,G: $o > set_set_b] :
      ( ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ G @ I ) ) ) ).

% D.indep_sets_mono_sets
thf(fact_291_D_Oindep__sets__mono__sets,axiom,
    ! [F2: ( a > real ) > set_set_b,I: set_a_real,G: ( a > real ) > set_set_b] :
      ( ( indepe8950816178784143176a_real @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
     => ( ! [I2: a > real] :
            ( ( member_a_real @ I2 @ I )
           => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe8950816178784143176a_real @ ( measure_distr_a_b @ m @ n @ f ) @ G @ I ) ) ) ).

% D.indep_sets_mono_sets
thf(fact_292_D_Oindep__sets__mono__sets,axiom,
    ! [F2: ( a > b ) > set_set_b,I: set_a_b,G: ( a > b ) > set_set_b] :
      ( ( indepe7072862685212782073_b_a_b @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
     => ( ! [I2: a > b] :
            ( ( member_a_b @ I2 @ I )
           => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe7072862685212782073_b_a_b @ ( measure_distr_a_b @ m @ n @ f ) @ G @ I ) ) ) ).

% D.indep_sets_mono_sets
thf(fact_293_D_Oindep__sets__mono__sets,axiom,
    ! [F2: ( b > a ) > set_set_b,I: set_b_a,G: ( b > a ) > set_set_b] :
      ( ( indepe4285407098936877239_b_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
     => ( ! [I2: b > a] :
            ( ( member_b_a @ I2 @ I )
           => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe4285407098936877239_b_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ G @ I ) ) ) ).

% D.indep_sets_mono_sets
thf(fact_294_D_Oindep__sets__mono__sets,axiom,
    ! [F2: ( a > c ) > set_set_b,I: set_a_c,G: ( a > c ) > set_set_b] :
      ( ( indepe7072862689516010874_b_a_c @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
     => ( ! [I2: a > c] :
            ( ( member_a_c @ I2 @ I )
           => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe7072862689516010874_b_a_c @ ( measure_distr_a_b @ m @ n @ f ) @ G @ I ) ) ) ).

% D.indep_sets_mono_sets
thf(fact_295_D_Oindep__sets__mono__sets,axiom,
    ! [F2: ( a > a ) > set_set_b,I: set_a_a,G: ( a > a ) > set_set_b] :
      ( ( indepe7072862680909553272_b_a_a @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
     => ( ! [I2: a > a] :
            ( ( member_a_a @ I2 @ I )
           => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe7072862680909553272_b_a_a @ ( measure_distr_a_b @ m @ n @ f ) @ G @ I ) ) ) ).

% D.indep_sets_mono_sets
thf(fact_296_D_Oindep__sets__mono__index,axiom,
    ! [J: set_set_b,I: set_set_b,F2: set_b > set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ J @ I )
     => ( ( indepe6311571491924490380_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
       => ( indepe6311571491924490380_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ J ) ) ) ).

% D.indep_sets_mono_index
thf(fact_297_D_Oindep__sets__mono__index,axiom,
    ! [J: set_set_a,I: set_set_a,F2: set_a > set_set_b] :
      ( ( ord_le3724670747650509150_set_a @ J @ I )
     => ( ( indepe6311571487621261579_set_a @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
       => ( indepe6311571487621261579_set_a @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ J ) ) ) ).

% D.indep_sets_mono_index
thf(fact_298_D_Oindep__sets__mono__index,axiom,
    ! [J: set_b,I: set_b,F2: b > set_set_b] :
      ( ( ord_less_eq_set_b @ J @ I )
     => ( ( indepe6139986284700742572ts_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
       => ( indepe6139986284700742572ts_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ J ) ) ) ).

% D.indep_sets_mono_index
thf(fact_299_D_Oindep__sets__mono__index,axiom,
    ! [J: set_a,I: set_a,F2: a > set_set_b] :
      ( ( ord_less_eq_set_a @ J @ I )
     => ( ( indepe6139986284700742571ts_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
       => ( indepe6139986284700742571ts_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ J ) ) ) ).

% D.indep_sets_mono_index
thf(fact_300_D_Oindep__sets__mono__index,axiom,
    ! [J: set_nat,I: set_nat,F2: nat > set_set_b] :
      ( ( ord_less_eq_set_nat @ J @ I )
     => ( ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
       => ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ J ) ) ) ).

% D.indep_sets_mono_index
thf(fact_301_D_Oindep__sets__mono__index,axiom,
    ! [J: set_o,I: set_o,F2: $o > set_set_b] :
      ( ( ord_less_eq_set_o @ J @ I )
     => ( ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
       => ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ J ) ) ) ).

% D.indep_sets_mono_index
thf(fact_302_D_Oindep__vars__compose,axiom,
    ! [M: c > sigma_measure_b,X4: c > b > b,I: set_c,Y: c > b > a,N2: c > sigma_measure_a] :
      ( ( indepe1224748351465797814_b_c_b @ ( measure_distr_a_b @ m @ n @ f ) @ M @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe1224748351465797813_b_c_a @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [I3: c] : ( comp_b_a_b @ ( Y @ I3 ) @ ( X4 @ I3 ) )
          @ I ) ) ) ).

% D.indep_vars_compose
thf(fact_303_D_Oindep__vars__compose,axiom,
    ! [M: c > sigma_measure_a,X4: c > b > a,I: set_c,Y: c > a > c,N2: c > sigma_measure_c] :
      ( ( indepe1224748351465797813_b_c_a @ ( measure_distr_a_b @ m @ n @ f ) @ M @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe1224748351465797815_b_c_c @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [I3: c] : ( comp_a_c_b @ ( Y @ I3 ) @ ( X4 @ I3 ) )
          @ I ) ) ) ).

% D.indep_vars_compose
thf(fact_304_D_Oindep__vars__compose,axiom,
    ! [M: c > sigma_measure_a,X4: c > b > a,I: set_c,Y: c > a > a,N2: c > sigma_measure_a] :
      ( ( indepe1224748351465797813_b_c_a @ ( measure_distr_a_b @ m @ n @ f ) @ M @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe1224748351465797813_b_c_a @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [I3: c] : ( comp_a_a_b @ ( Y @ I3 ) @ ( X4 @ I3 ) )
          @ I ) ) ) ).

% D.indep_vars_compose
thf(fact_305_D_Oindep__vars__compose,axiom,
    ! [M: b > sigma_measure_b,X4: b > b > b,I: set_b,Y: b > b > a,N2: b > sigma_measure_a] :
      ( ( indepe4012203933438473847_b_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ M @ X4 @ I )
     => ( ! [I2: b] :
            ( ( member_b @ I2 @ I )
           => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe4012203933438473846_b_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [I3: b] : ( comp_b_a_b @ ( Y @ I3 ) @ ( X4 @ I3 ) )
          @ I ) ) ) ).

% D.indep_vars_compose
thf(fact_306_D_Oindep__vars__compose,axiom,
    ! [M: b > sigma_measure_a,X4: b > b > a,I: set_b,Y: b > a > c,N2: b > sigma_measure_c] :
      ( ( indepe4012203933438473846_b_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ M @ X4 @ I )
     => ( ! [I2: b] :
            ( ( member_b @ I2 @ I )
           => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe4012203933438473848_b_b_c @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [I3: b] : ( comp_a_c_b @ ( Y @ I3 ) @ ( X4 @ I3 ) )
          @ I ) ) ) ).

% D.indep_vars_compose
thf(fact_307_D_Oindep__vars__compose,axiom,
    ! [M: b > sigma_measure_a,X4: b > b > a,I: set_b,Y: b > a > a,N2: b > sigma_measure_a] :
      ( ( indepe4012203933438473846_b_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ M @ X4 @ I )
     => ( ! [I2: b] :
            ( ( member_b @ I2 @ I )
           => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe4012203933438473846_b_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [I3: b] : ( comp_a_a_b @ ( Y @ I3 ) @ ( X4 @ I3 ) )
          @ I ) ) ) ).

% D.indep_vars_compose
thf(fact_308_D_Oindep__vars__compose,axiom,
    ! [M: a > sigma_measure_b,X4: a > b > b,I: set_a,Y: a > b > a,N2: a > sigma_measure_a] :
      ( ( indepe6799659515411149880_b_a_b @ ( measure_distr_a_b @ m @ n @ f ) @ M @ X4 @ I )
     => ( ! [I2: a] :
            ( ( member_a @ I2 @ I )
           => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe6799659515411149879_b_a_a @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [I3: a] : ( comp_b_a_b @ ( Y @ I3 ) @ ( X4 @ I3 ) )
          @ I ) ) ) ).

% D.indep_vars_compose
thf(fact_309_D_Oindep__vars__compose,axiom,
    ! [M: a > sigma_measure_a,X4: a > b > a,I: set_a,Y: a > a > c,N2: a > sigma_measure_c] :
      ( ( indepe6799659515411149879_b_a_a @ ( measure_distr_a_b @ m @ n @ f ) @ M @ X4 @ I )
     => ( ! [I2: a] :
            ( ( member_a @ I2 @ I )
           => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe6799659515411149881_b_a_c @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [I3: a] : ( comp_a_c_b @ ( Y @ I3 ) @ ( X4 @ I3 ) )
          @ I ) ) ) ).

% D.indep_vars_compose
thf(fact_310_D_Oindep__vars__compose,axiom,
    ! [M: a > sigma_measure_a,X4: a > b > a,I: set_a,Y: a > a > a,N2: a > sigma_measure_a] :
      ( ( indepe6799659515411149879_b_a_a @ ( measure_distr_a_b @ m @ n @ f ) @ M @ X4 @ I )
     => ( ! [I2: a] :
            ( ( member_a @ I2 @ I )
           => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe6799659515411149879_b_a_a @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [I3: a] : ( comp_a_a_b @ ( Y @ I3 ) @ ( X4 @ I3 ) )
          @ I ) ) ) ).

% D.indep_vars_compose
thf(fact_311_D_Oindep__vars__compose,axiom,
    ! [M: c > sigma_measure_a,X4: c > b > a,I: set_c,Y: c > a > b,N2: c > sigma_measure_b] :
      ( ( indepe1224748351465797813_b_c_a @ ( measure_distr_a_b @ m @ n @ f ) @ M @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_a_b @ ( Y @ I2 ) @ ( sigma_measurable_a_b @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe1224748351465797814_b_c_b @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [I3: c] : ( comp_a_b_b @ ( Y @ I3 ) @ ( X4 @ I3 ) )
          @ I ) ) ) ).

% D.indep_vars_compose
thf(fact_312_prob__space_Oindep__sets__Dynkin,axiom,
    ! [M2: sigma_measure_b,F2: nat > set_set_b,I: set_nat] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe7503174356045242851_b_nat @ M2 @ F2 @ I )
       => ( indepe7503174356045242851_b_nat @ M2
          @ ^ [I3: nat] : ( sigma_Dynkin_b @ ( sigma_space_b @ M2 ) @ ( F2 @ I3 ) )
          @ I ) ) ) ).

% prob_space.indep_sets_Dynkin
thf(fact_313_prob__space_Oindep__sets__Dynkin,axiom,
    ! [M2: sigma_measure_a,F2: nat > set_set_a,I: set_nat] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe6267730027088848354_a_nat @ M2 @ F2 @ I )
       => ( indepe6267730027088848354_a_nat @ M2
          @ ^ [I3: nat] : ( sigma_Dynkin_a @ ( sigma_space_a @ M2 ) @ ( F2 @ I3 ) )
          @ I ) ) ) ).

% prob_space.indep_sets_Dynkin
thf(fact_314_prob__space_Oindep__sets__Dynkin,axiom,
    ! [M2: sigma_measure_a,F2: $o > set_set_a,I: set_o] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe7780107833195774214ts_a_o @ M2 @ F2 @ I )
       => ( indepe7780107833195774214ts_a_o @ M2
          @ ^ [I3: $o] : ( sigma_Dynkin_a @ ( sigma_space_a @ M2 ) @ ( F2 @ I3 ) )
          @ I ) ) ) ).

% prob_space.indep_sets_Dynkin
thf(fact_315_prob__space_Oindep__sets__Dynkin,axiom,
    ! [M2: sigma_measure_b,F2: $o > set_set_b,I: set_o] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe4880885433731379909ts_b_o @ M2 @ F2 @ I )
       => ( indepe4880885433731379909ts_b_o @ M2
          @ ^ [I3: $o] : ( sigma_Dynkin_b @ ( sigma_space_b @ M2 ) @ ( F2 @ I3 ) )
          @ I ) ) ) ).

% prob_space.indep_sets_Dynkin
thf(fact_316_D_Oprob__space__completion,axiom,
    probab7247484486040049090pace_b @ ( comple3428971583294703881tion_b @ ( measure_distr_a_b @ m @ n @ f ) ) ).

% D.prob_space_completion
thf(fact_317_prob__space_Oindep__vars__compose2,axiom,
    ! [M2: sigma_measure_a,M: c > sigma_measure_b,X4: c > a > b,I: set_c,Y: c > b > a,N2: c > sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe4851901773132442933_a_c_b @ M2 @ M @ X4 @ I )
       => ( ! [I2: c] :
              ( ( member_c @ I2 @ I )
             => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( indepe4851901773132442932_a_c_a @ M2 @ N2
            @ ^ [I3: c,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_318_prob__space_Oindep__vars__compose2,axiom,
    ! [M2: sigma_measure_a,M: c > sigma_measure_a,X4: c > a > a,I: set_c,Y: c > a > c,N2: c > sigma_measure_c] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe4851901773132442932_a_c_a @ M2 @ M @ X4 @ I )
       => ( ! [I2: c] :
              ( ( member_c @ I2 @ I )
             => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( indepe4851901773132442934_a_c_c @ M2 @ N2
            @ ^ [I3: c,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_319_prob__space_Oindep__vars__compose2,axiom,
    ! [M2: sigma_measure_a,M: c > sigma_measure_a,X4: c > a > a,I: set_c,Y: c > a > a,N2: c > sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe4851901773132442932_a_c_a @ M2 @ M @ X4 @ I )
       => ( ! [I2: c] :
              ( ( member_c @ I2 @ I )
             => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( indepe4851901773132442932_a_c_a @ M2 @ N2
            @ ^ [I3: c,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_320_prob__space_Oindep__vars__compose2,axiom,
    ! [M2: sigma_measure_a,M: b > sigma_measure_b,X4: b > a > b,I: set_b,Y: b > b > a,N2: b > sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe7639357355105118966_a_b_b @ M2 @ M @ X4 @ I )
       => ( ! [I2: b] :
              ( ( member_b @ I2 @ I )
             => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( indepe7639357355105118965_a_b_a @ M2 @ N2
            @ ^ [I3: b,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_321_prob__space_Oindep__vars__compose2,axiom,
    ! [M2: sigma_measure_a,M: b > sigma_measure_a,X4: b > a > a,I: set_b,Y: b > a > c,N2: b > sigma_measure_c] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe7639357355105118965_a_b_a @ M2 @ M @ X4 @ I )
       => ( ! [I2: b] :
              ( ( member_b @ I2 @ I )
             => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( indepe7639357355105118967_a_b_c @ M2 @ N2
            @ ^ [I3: b,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_322_prob__space_Oindep__vars__compose2,axiom,
    ! [M2: sigma_measure_a,M: b > sigma_measure_a,X4: b > a > a,I: set_b,Y: b > a > a,N2: b > sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe7639357355105118965_a_b_a @ M2 @ M @ X4 @ I )
       => ( ! [I2: b] :
              ( ( member_b @ I2 @ I )
             => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( indepe7639357355105118965_a_b_a @ M2 @ N2
            @ ^ [I3: b,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_323_prob__space_Oindep__vars__compose2,axiom,
    ! [M2: sigma_measure_a,M: a > sigma_measure_b,X4: a > a > b,I: set_a,Y: a > b > a,N2: a > sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe1203440900223019191_a_a_b @ M2 @ M @ X4 @ I )
       => ( ! [I2: a] :
              ( ( member_a @ I2 @ I )
             => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( indepe1203440900223019190_a_a_a @ M2 @ N2
            @ ^ [I3: a,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_324_prob__space_Oindep__vars__compose2,axiom,
    ! [M2: sigma_measure_a,M: a > sigma_measure_a,X4: a > a > a,I: set_a,Y: a > a > c,N2: a > sigma_measure_c] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe1203440900223019190_a_a_a @ M2 @ M @ X4 @ I )
       => ( ! [I2: a] :
              ( ( member_a @ I2 @ I )
             => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( indepe1203440900223019192_a_a_c @ M2 @ N2
            @ ^ [I3: a,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_325_prob__space_Oindep__vars__compose2,axiom,
    ! [M2: sigma_measure_a,M: a > sigma_measure_a,X4: a > a > a,I: set_a,Y: a > a > a,N2: a > sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe1203440900223019190_a_a_a @ M2 @ M @ X4 @ I )
       => ( ! [I2: a] :
              ( ( member_a @ I2 @ I )
             => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( indepe1203440900223019190_a_a_a @ M2 @ N2
            @ ^ [I3: a,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_326_prob__space_Oindep__vars__compose2,axiom,
    ! [M2: sigma_measure_a,M: c > sigma_measure_a,X4: c > a > a,I: set_c,Y: c > a > b,N2: c > sigma_measure_b] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe4851901773132442932_a_c_a @ M2 @ M @ X4 @ I )
       => ( ! [I2: c] :
              ( ( member_c @ I2 @ I )
             => ( member_a_b @ ( Y @ I2 ) @ ( sigma_measurable_a_b @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( indepe4851901773132442933_a_c_b @ M2 @ N2
            @ ^ [I3: c,X2: a] : ( Y @ I3 @ ( X4 @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_327_D_Osigma__finite__measure__axioms,axiom,
    measur4308613598931908896sure_b @ ( measure_distr_a_b @ m @ n @ f ) ).

% D.sigma_finite_measure_axioms
thf(fact_328_indep__sets__mono__sets,axiom,
    ! [F2: c > set_set_a,I: set_c,G: c > set_set_a] :
      ( ( indepe8927441866673418606ts_a_c @ m @ F2 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe8927441866673418606ts_a_c @ m @ G @ I ) ) ) ).

% indep_sets_mono_sets
thf(fact_329_indep__sets__mono__sets,axiom,
    ! [F2: b > set_set_a,I: set_b,G: b > set_set_a] :
      ( ( indepe8927441866673418605ts_a_b @ m @ F2 @ I )
     => ( ! [I2: b] :
            ( ( member_b @ I2 @ I )
           => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe8927441866673418605ts_a_b @ m @ G @ I ) ) ) ).

% indep_sets_mono_sets
thf(fact_330_indep__sets__mono__sets,axiom,
    ! [F2: a > set_set_a,I: set_a,G: a > set_set_a] :
      ( ( indepe8927441866673418604ts_a_a @ m @ F2 @ I )
     => ( ! [I2: a] :
            ( ( member_a @ I2 @ I )
           => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe8927441866673418604ts_a_a @ m @ G @ I ) ) ) ).

% indep_sets_mono_sets
thf(fact_331_indep__sets__mono__sets,axiom,
    ! [F2: nat > set_set_a,I: set_nat,G: nat > set_set_a] :
      ( ( indepe6267730027088848354_a_nat @ m @ F2 @ I )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I )
           => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe6267730027088848354_a_nat @ m @ G @ I ) ) ) ).

% indep_sets_mono_sets
thf(fact_332_indep__sets__mono__sets,axiom,
    ! [F2: $o > set_set_a,I: set_o,G: $o > set_set_a] :
      ( ( indepe7780107833195774214ts_a_o @ m @ F2 @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe7780107833195774214ts_a_o @ m @ G @ I ) ) ) ).

% indep_sets_mono_sets
thf(fact_333_indep__sets__mono__sets,axiom,
    ! [F2: ( a > real ) > set_set_a,I: set_a_real,G: ( a > real ) > set_set_a] :
      ( ( indepe4749599203615801097a_real @ m @ F2 @ I )
     => ( ! [I2: a > real] :
            ( ( member_a_real @ I2 @ I )
           => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe4749599203615801097a_real @ m @ G @ I ) ) ) ).

% indep_sets_mono_sets
thf(fact_334_indep__sets__mono__sets,axiom,
    ! [F2: ( a > b ) > set_set_a,I: set_a_b,G: ( a > b ) > set_set_a] :
      ( ( indepe1219004915316159480_a_a_b @ m @ F2 @ I )
     => ( ! [I2: a > b] :
            ( ( member_a_b @ I2 @ I )
           => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe1219004915316159480_a_a_b @ m @ G @ I ) ) ) ).

% indep_sets_mono_sets
thf(fact_335_indep__sets__mono__sets,axiom,
    ! [F2: ( b > a ) > set_set_a,I: set_b_a,G: ( b > a ) > set_set_a] :
      ( ( indepe7654921365895030454_a_b_a @ m @ F2 @ I )
     => ( ! [I2: b > a] :
            ( ( member_b_a @ I2 @ I )
           => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe7654921365895030454_a_b_a @ m @ G @ I ) ) ) ).

% indep_sets_mono_sets
thf(fact_336_indep__sets__mono__sets,axiom,
    ! [F2: ( a > c ) > set_set_a,I: set_a_c,G: ( a > c ) > set_set_a] :
      ( ( indepe1219004919619388281_a_a_c @ m @ F2 @ I )
     => ( ! [I2: a > c] :
            ( ( member_a_c @ I2 @ I )
           => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe1219004919619388281_a_a_c @ m @ G @ I ) ) ) ).

% indep_sets_mono_sets
thf(fact_337_indep__sets__mono__sets,axiom,
    ! [F2: ( a > a ) > set_set_a,I: set_a_a,G: ( a > a ) > set_set_a] :
      ( ( indepe1219004911012930679_a_a_a @ m @ F2 @ I )
     => ( ! [I2: a > a] :
            ( ( member_a_a @ I2 @ I )
           => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe1219004911012930679_a_a_a @ m @ G @ I ) ) ) ).

% indep_sets_mono_sets
thf(fact_338_indep__vars__subset,axiom,
    ! [M: c > sigma_measure_d,X4: c > a > d,I: set_c,J: set_c] :
      ( ( indepe4851901773132442935_a_c_d @ m @ M @ X4 @ I )
     => ( ( ord_less_eq_set_c @ J @ I )
       => ( indepe4851901773132442935_a_c_d @ m @ M @ X4 @ J ) ) ) ).

% indep_vars_subset
thf(fact_339_indep__sets__mono,axiom,
    ! [F2: c > set_set_a,I: set_c,J: set_c,G: c > set_set_a] :
      ( ( indepe8927441866673418606ts_a_c @ m @ F2 @ I )
     => ( ( ord_less_eq_set_c @ J @ I )
       => ( ! [I2: c] :
              ( ( member_c @ I2 @ J )
             => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe8927441866673418606ts_a_c @ m @ G @ J ) ) ) ) ).

% indep_sets_mono
thf(fact_340_indep__sets__mono,axiom,
    ! [F2: b > set_set_a,I: set_b,J: set_b,G: b > set_set_a] :
      ( ( indepe8927441866673418605ts_a_b @ m @ F2 @ I )
     => ( ( ord_less_eq_set_b @ J @ I )
       => ( ! [I2: b] :
              ( ( member_b @ I2 @ J )
             => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe8927441866673418605ts_a_b @ m @ G @ J ) ) ) ) ).

% indep_sets_mono
thf(fact_341_indep__sets__mono,axiom,
    ! [F2: a > set_set_a,I: set_a,J: set_a,G: a > set_set_a] :
      ( ( indepe8927441866673418604ts_a_a @ m @ F2 @ I )
     => ( ( ord_less_eq_set_a @ J @ I )
       => ( ! [I2: a] :
              ( ( member_a @ I2 @ J )
             => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe8927441866673418604ts_a_a @ m @ G @ J ) ) ) ) ).

% indep_sets_mono
thf(fact_342_indep__sets__mono,axiom,
    ! [F2: nat > set_set_a,I: set_nat,J: set_nat,G: nat > set_set_a] :
      ( ( indepe6267730027088848354_a_nat @ m @ F2 @ I )
     => ( ( ord_less_eq_set_nat @ J @ I )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ J )
             => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe6267730027088848354_a_nat @ m @ G @ J ) ) ) ) ).

% indep_sets_mono
thf(fact_343_indep__sets__mono,axiom,
    ! [F2: $o > set_set_a,I: set_o,J: set_o,G: $o > set_set_a] :
      ( ( indepe7780107833195774214ts_a_o @ m @ F2 @ I )
     => ( ( ord_less_eq_set_o @ J @ I )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ J )
             => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe7780107833195774214ts_a_o @ m @ G @ J ) ) ) ) ).

% indep_sets_mono
thf(fact_344_indep__sets__mono,axiom,
    ! [F2: set_b > set_set_a,I: set_set_b,J: set_set_b,G: set_b > set_set_a] :
      ( ( indepe4967106455115002445_set_b @ m @ F2 @ I )
     => ( ( ord_le3795704787696855135_set_b @ J @ I )
       => ( ! [I2: set_b] :
              ( ( member_set_b @ I2 @ J )
             => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe4967106455115002445_set_b @ m @ G @ J ) ) ) ) ).

% indep_sets_mono
thf(fact_345_indep__sets__mono,axiom,
    ! [F2: set_a > set_set_a,I: set_set_a,J: set_set_a,G: set_a > set_set_a] :
      ( ( indepe4967106450811773644_set_a @ m @ F2 @ I )
     => ( ( ord_le3724670747650509150_set_a @ J @ I )
       => ( ! [I2: set_a] :
              ( ( member_set_a @ I2 @ J )
             => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe4967106450811773644_set_a @ m @ G @ J ) ) ) ) ).

% indep_sets_mono
thf(fact_346_indep__sets__mono,axiom,
    ! [F2: ( a > real ) > set_set_a,I: set_a_real,J: set_a_real,G: ( a > real ) > set_set_a] :
      ( ( indepe4749599203615801097a_real @ m @ F2 @ I )
     => ( ( ord_le3334967407727675675a_real @ J @ I )
       => ( ! [I2: a > real] :
              ( ( member_a_real @ I2 @ J )
             => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe4749599203615801097a_real @ m @ G @ J ) ) ) ) ).

% indep_sets_mono
thf(fact_347_indep__sets__mono,axiom,
    ! [F2: ( a > b ) > set_set_a,I: set_a_b,J: set_a_b,G: ( a > b ) > set_set_a] :
      ( ( indepe1219004915316159480_a_a_b @ m @ F2 @ I )
     => ( ( ord_less_eq_set_a_b @ J @ I )
       => ( ! [I2: a > b] :
              ( ( member_a_b @ I2 @ J )
             => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe1219004915316159480_a_a_b @ m @ G @ J ) ) ) ) ).

% indep_sets_mono
thf(fact_348_indep__sets__mono,axiom,
    ! [F2: ( b > a ) > set_set_a,I: set_b_a,J: set_b_a,G: ( b > a ) > set_set_a] :
      ( ( indepe7654921365895030454_a_b_a @ m @ F2 @ I )
     => ( ( ord_less_eq_set_b_a @ J @ I )
       => ( ! [I2: b > a] :
              ( ( member_b_a @ I2 @ J )
             => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe7654921365895030454_a_b_a @ m @ G @ J ) ) ) ) ).

% indep_sets_mono
thf(fact_349_indep__sets__mono__index,axiom,
    ! [J: set_set_b,I: set_set_b,F2: set_b > set_set_a] :
      ( ( ord_le3795704787696855135_set_b @ J @ I )
     => ( ( indepe4967106455115002445_set_b @ m @ F2 @ I )
       => ( indepe4967106455115002445_set_b @ m @ F2 @ J ) ) ) ).

% indep_sets_mono_index
thf(fact_350_indep__sets__mono__index,axiom,
    ! [J: set_set_a,I: set_set_a,F2: set_a > set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ J @ I )
     => ( ( indepe4967106450811773644_set_a @ m @ F2 @ I )
       => ( indepe4967106450811773644_set_a @ m @ F2 @ J ) ) ) ).

% indep_sets_mono_index
thf(fact_351_indep__sets__mono__index,axiom,
    ! [J: set_b,I: set_b,F2: b > set_set_a] :
      ( ( ord_less_eq_set_b @ J @ I )
     => ( ( indepe8927441866673418605ts_a_b @ m @ F2 @ I )
       => ( indepe8927441866673418605ts_a_b @ m @ F2 @ J ) ) ) ).

% indep_sets_mono_index
thf(fact_352_indep__sets__mono__index,axiom,
    ! [J: set_a,I: set_a,F2: a > set_set_a] :
      ( ( ord_less_eq_set_a @ J @ I )
     => ( ( indepe8927441866673418604ts_a_a @ m @ F2 @ I )
       => ( indepe8927441866673418604ts_a_a @ m @ F2 @ J ) ) ) ).

% indep_sets_mono_index
thf(fact_353_indep__sets__mono__index,axiom,
    ! [J: set_nat,I: set_nat,F2: nat > set_set_a] :
      ( ( ord_less_eq_set_nat @ J @ I )
     => ( ( indepe6267730027088848354_a_nat @ m @ F2 @ I )
       => ( indepe6267730027088848354_a_nat @ m @ F2 @ J ) ) ) ).

% indep_sets_mono_index
thf(fact_354_indep__sets__mono__index,axiom,
    ! [J: set_o,I: set_o,F2: $o > set_set_a] :
      ( ( ord_less_eq_set_o @ J @ I )
     => ( ( indepe7780107833195774214ts_a_o @ m @ F2 @ I )
       => ( indepe7780107833195774214ts_a_o @ m @ F2 @ J ) ) ) ).

% indep_sets_mono_index
thf(fact_355_D_Oindep__sets__mono,axiom,
    ! [F2: c > set_set_b,I: set_c,J: set_c,G: c > set_set_b] :
      ( ( indepe6139986284700742573ts_b_c @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
     => ( ( ord_less_eq_set_c @ J @ I )
       => ( ! [I2: c] :
              ( ( member_c @ I2 @ J )
             => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe6139986284700742573ts_b_c @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).

% D.indep_sets_mono
thf(fact_356_D_Oindep__sets__mono,axiom,
    ! [F2: b > set_set_b,I: set_b,J: set_b,G: b > set_set_b] :
      ( ( indepe6139986284700742572ts_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
     => ( ( ord_less_eq_set_b @ J @ I )
       => ( ! [I2: b] :
              ( ( member_b @ I2 @ J )
             => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe6139986284700742572ts_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).

% D.indep_sets_mono
thf(fact_357_D_Oindep__sets__mono,axiom,
    ! [F2: a > set_set_b,I: set_a,J: set_a,G: a > set_set_b] :
      ( ( indepe6139986284700742571ts_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
     => ( ( ord_less_eq_set_a @ J @ I )
       => ( ! [I2: a] :
              ( ( member_a @ I2 @ J )
             => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe6139986284700742571ts_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).

% D.indep_sets_mono
thf(fact_358_D_Oindep__sets__mono,axiom,
    ! [F2: nat > set_set_b,I: set_nat,J: set_nat,G: nat > set_set_b] :
      ( ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
     => ( ( ord_less_eq_set_nat @ J @ I )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ J )
             => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).

% D.indep_sets_mono
thf(fact_359_D_Oindep__sets__mono,axiom,
    ! [F2: $o > set_set_b,I: set_o,J: set_o,G: $o > set_set_b] :
      ( ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
     => ( ( ord_less_eq_set_o @ J @ I )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ J )
             => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).

% D.indep_sets_mono
thf(fact_360_D_Oindep__sets__mono,axiom,
    ! [F2: set_b > set_set_b,I: set_set_b,J: set_set_b,G: set_b > set_set_b] :
      ( ( indepe6311571491924490380_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
     => ( ( ord_le3795704787696855135_set_b @ J @ I )
       => ( ! [I2: set_b] :
              ( ( member_set_b @ I2 @ J )
             => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe6311571491924490380_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).

% D.indep_sets_mono
thf(fact_361_D_Oindep__sets__mono,axiom,
    ! [F2: set_a > set_set_b,I: set_set_a,J: set_set_a,G: set_a > set_set_b] :
      ( ( indepe6311571487621261579_set_a @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
     => ( ( ord_le3724670747650509150_set_a @ J @ I )
       => ( ! [I2: set_a] :
              ( ( member_set_a @ I2 @ J )
             => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe6311571487621261579_set_a @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).

% D.indep_sets_mono
thf(fact_362_D_Oindep__sets__mono,axiom,
    ! [F2: ( a > real ) > set_set_b,I: set_a_real,J: set_a_real,G: ( a > real ) > set_set_b] :
      ( ( indepe8950816178784143176a_real @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
     => ( ( ord_le3334967407727675675a_real @ J @ I )
       => ( ! [I2: a > real] :
              ( ( member_a_real @ I2 @ J )
             => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe8950816178784143176a_real @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).

% D.indep_sets_mono
thf(fact_363_D_Oindep__sets__mono,axiom,
    ! [F2: ( a > b ) > set_set_b,I: set_a_b,J: set_a_b,G: ( a > b ) > set_set_b] :
      ( ( indepe7072862685212782073_b_a_b @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
     => ( ( ord_less_eq_set_a_b @ J @ I )
       => ( ! [I2: a > b] :
              ( ( member_a_b @ I2 @ J )
             => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe7072862685212782073_b_a_b @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).

% D.indep_sets_mono
thf(fact_364_D_Oindep__sets__mono,axiom,
    ! [F2: ( b > a ) > set_set_b,I: set_b_a,J: set_b_a,G: ( b > a ) > set_set_b] :
      ( ( indepe4285407098936877239_b_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
     => ( ( ord_less_eq_set_b_a @ J @ I )
       => ( ! [I2: b > a] :
              ( ( member_b_a @ I2 @ J )
             => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe4285407098936877239_b_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).

% D.indep_sets_mono
thf(fact_365_D_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_b,X1: b > b,M22: sigma_measure_b,X22: b > b,Y1: b > a,N1: sigma_measure_a,Y2: b > a,N22: sigma_measure_a] :
      ( ( indepe8876569649573725964ar_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ M1 @ X1 @ M22 @ X22 )
     => ( ( member_b_a @ Y1 @ ( sigma_measurable_b_a @ M1 @ N1 ) )
       => ( ( member_b_a @ Y2 @ ( sigma_measurable_b_a @ M22 @ N22 ) )
         => ( indepe8876569649573725963ar_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ N1 @ ( comp_b_a_b @ Y1 @ X1 ) @ N22 @ ( comp_b_a_b @ Y2 @ X22 ) ) ) ) ) ).

% D.indep_var_compose
thf(fact_366_D_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_a,X1: b > a,M22: sigma_measure_a,X22: b > a,Y1: a > a > real,N1: sigma_measure_a_real,Y2: a > a > real,N22: sigma_measure_a_real] :
      ( ( indepe8876569649573725963ar_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ M1 @ X1 @ M22 @ X22 )
     => ( ( member_a_a_real @ Y1 @ ( sigma_5073124708296780141a_real @ M1 @ N1 ) )
       => ( ( member_a_a_real @ Y2 @ ( sigma_5073124708296780141a_real @ M22 @ N22 ) )
         => ( indepe2116752563100621480a_real @ ( measure_distr_a_b @ m @ n @ f ) @ N1 @ ( comp_a_a_real_b @ Y1 @ X1 ) @ N22 @ ( comp_a_a_real_b @ Y2 @ X22 ) ) ) ) ) ).

% D.indep_var_compose
thf(fact_367_D_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_a,X1: b > a,M22: sigma_measure_a,X22: b > a,Y1: a > a > b,N1: sigma_measure_a_b,Y2: a > a > b,N22: sigma_measure_a_b] :
      ( ( indepe8876569649573725963ar_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ M1 @ X1 @ M22 @ X22 )
     => ( ( member_a_a_b @ Y1 @ ( sigma_3971135317854480532_a_a_b @ M1 @ N1 ) )
       => ( ( member_a_a_b @ Y2 @ ( sigma_3971135317854480532_a_a_b @ M22 @ N22 ) )
         => ( indepe3540165258703045273_b_a_b @ ( measure_distr_a_b @ m @ n @ f ) @ N1 @ ( comp_a_a_b_b @ Y1 @ X1 ) @ N22 @ ( comp_a_a_b_b @ Y2 @ X22 ) ) ) ) ) ).

% D.indep_var_compose
thf(fact_368_D_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_a,X1: b > a,M22: sigma_measure_a,X22: b > a,Y1: a > c,N1: sigma_measure_c,Y2: a > c,N22: sigma_measure_c] :
      ( ( indepe8876569649573725963ar_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ M1 @ X1 @ M22 @ X22 )
     => ( ( member_a_c @ Y1 @ ( sigma_measurable_a_c @ M1 @ N1 ) )
       => ( ( member_a_c @ Y2 @ ( sigma_measurable_a_c @ M22 @ N22 ) )
         => ( indepe8876569649573725965ar_b_c @ ( measure_distr_a_b @ m @ n @ f ) @ N1 @ ( comp_a_c_b @ Y1 @ X1 ) @ N22 @ ( comp_a_c_b @ Y2 @ X22 ) ) ) ) ) ).

% D.indep_var_compose
thf(fact_369_D_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_a,X1: b > a,M22: sigma_measure_a,X22: b > a,Y1: a > a,N1: sigma_measure_a,Y2: a > a,N22: sigma_measure_a] :
      ( ( indepe8876569649573725963ar_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ M1 @ X1 @ M22 @ X22 )
     => ( ( member_a_a @ Y1 @ ( sigma_measurable_a_a @ M1 @ N1 ) )
       => ( ( member_a_a @ Y2 @ ( sigma_measurable_a_a @ M22 @ N22 ) )
         => ( indepe8876569649573725963ar_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ N1 @ ( comp_a_a_b @ Y1 @ X1 ) @ N22 @ ( comp_a_a_b @ Y2 @ X22 ) ) ) ) ) ).

% D.indep_var_compose
thf(fact_370_D_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_a,X1: b > a,M22: sigma_measure_a,X22: b > a,Y1: a > b,N1: sigma_measure_b,Y2: a > b,N22: sigma_measure_b] :
      ( ( indepe8876569649573725963ar_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ M1 @ X1 @ M22 @ X22 )
     => ( ( member_a_b @ Y1 @ ( sigma_measurable_a_b @ M1 @ N1 ) )
       => ( ( member_a_b @ Y2 @ ( sigma_measurable_a_b @ M22 @ N22 ) )
         => ( indepe8876569649573725964ar_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ N1 @ ( comp_a_b_b @ Y1 @ X1 ) @ N22 @ ( comp_a_b_b @ Y2 @ X22 ) ) ) ) ) ).

% D.indep_var_compose
thf(fact_371_D_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_b,X1: b > b,M22: sigma_measure_b,X22: b > b,Y1: b > d,N1: sigma_measure_d,Y2: b > d,N22: sigma_measure_d] :
      ( ( indepe8876569649573725964ar_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ M1 @ X1 @ M22 @ X22 )
     => ( ( member_b_d @ Y1 @ ( sigma_measurable_b_d @ M1 @ N1 ) )
       => ( ( member_b_d @ Y2 @ ( sigma_measurable_b_d @ M22 @ N22 ) )
         => ( indepe8876569649573725966ar_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ N1 @ ( comp_b_d_b @ Y1 @ X1 ) @ N22 @ ( comp_b_d_b @ Y2 @ X22 ) ) ) ) ) ).

% D.indep_var_compose
thf(fact_372_D_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_b,X1: b > b,M22: sigma_measure_b,X22: b > b,Y1: b > real,N1: sigma_measure_real,Y2: b > real,N22: sigma_measure_real] :
      ( ( indepe8876569649573725964ar_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ M1 @ X1 @ M22 @ X22 )
     => ( ( member_b_real @ Y1 @ ( sigma_5489272243865111003b_real @ M1 @ N1 ) )
       => ( ( member_b_real @ Y2 @ ( sigma_5489272243865111003b_real @ M22 @ N22 ) )
         => ( indepe5331282143832502239b_real @ ( measure_distr_a_b @ m @ n @ f ) @ N1 @ ( comp_b_real_b @ Y1 @ X1 ) @ N22 @ ( comp_b_real_b @ Y2 @ X22 ) ) ) ) ) ).

% D.indep_var_compose
thf(fact_373_D_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_a,X1: b > a,M22: sigma_measure_a,X22: b > a,Y1: a > real,N1: sigma_measure_real,Y2: a > real,N22: sigma_measure_real] :
      ( ( indepe8876569649573725963ar_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ M1 @ X1 @ M22 @ X22 )
     => ( ( member_a_real @ Y1 @ ( sigma_9116425665531756122a_real @ M1 @ N1 ) )
       => ( ( member_a_real @ Y2 @ ( sigma_9116425665531756122a_real @ M22 @ N22 ) )
         => ( indepe5331282143832502239b_real @ ( measure_distr_a_b @ m @ n @ f ) @ N1 @ ( comp_a_real_b @ Y1 @ X1 ) @ N22 @ ( comp_a_real_b @ Y2 @ X22 ) ) ) ) ) ).

% D.indep_var_compose
thf(fact_374_function__factors__right,axiom,
    ! [G2: b > d,F: a > d] :
      ( ( ! [X2: a] :
          ? [Y3: b] :
            ( ( G2 @ Y3 )
            = ( F @ X2 ) ) )
      = ( ? [H: a > b] :
            ( F
            = ( comp_b_d_a @ G2 @ H ) ) ) ) ).

% function_factors_right
thf(fact_375_function__factors__left,axiom,
    ! [G2: a > b,F: a > d] :
      ( ( ! [X2: a,Y3: a] :
            ( ( ( G2 @ X2 )
              = ( G2 @ Y3 ) )
           => ( ( F @ X2 )
              = ( F @ Y3 ) ) ) )
      = ( ? [H: b > d] :
            ( F
            = ( comp_b_d_a @ H @ G2 ) ) ) ) ).

% function_factors_left
thf(fact_376_measurable__comp,axiom,
    ! [F: b > b,M2: sigma_measure_b,N2: sigma_measure_b,G2: b > a,L: sigma_measure_a] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M2 @ N2 ) )
     => ( ( member_b_a @ G2 @ ( sigma_measurable_b_a @ N2 @ L ) )
       => ( member_b_a @ ( comp_b_a_b @ G2 @ F ) @ ( sigma_measurable_b_a @ M2 @ L ) ) ) ) ).

% measurable_comp
thf(fact_377_measurable__comp,axiom,
    ! [F: b > a,M2: sigma_measure_b,N2: sigma_measure_a,G2: a > c,L: sigma_measure_c] :
      ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ N2 ) )
     => ( ( member_a_c @ G2 @ ( sigma_measurable_a_c @ N2 @ L ) )
       => ( member_b_c @ ( comp_a_c_b @ G2 @ F ) @ ( sigma_measurable_b_c @ M2 @ L ) ) ) ) ).

% measurable_comp
thf(fact_378_measurable__comp,axiom,
    ! [F: b > a,M2: sigma_measure_b,N2: sigma_measure_a,G2: a > a,L: sigma_measure_a] :
      ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ N2 ) )
     => ( ( member_a_a @ G2 @ ( sigma_measurable_a_a @ N2 @ L ) )
       => ( member_b_a @ ( comp_a_a_b @ G2 @ F ) @ ( sigma_measurable_b_a @ M2 @ L ) ) ) ) ).

% measurable_comp
thf(fact_379_measurable__comp,axiom,
    ! [F: a > c,M2: sigma_measure_a,N2: sigma_measure_c,G2: c > c,L: sigma_measure_c] :
      ( ( member_a_c @ F @ ( sigma_measurable_a_c @ M2 @ N2 ) )
     => ( ( member_c_c @ G2 @ ( sigma_measurable_c_c @ N2 @ L ) )
       => ( member_a_c @ ( comp_c_c_a @ G2 @ F ) @ ( sigma_measurable_a_c @ M2 @ L ) ) ) ) ).

% measurable_comp
thf(fact_380_measurable__comp,axiom,
    ! [F: a > c,M2: sigma_measure_a,N2: sigma_measure_c,G2: c > a,L: sigma_measure_a] :
      ( ( member_a_c @ F @ ( sigma_measurable_a_c @ M2 @ N2 ) )
     => ( ( member_c_a @ G2 @ ( sigma_measurable_c_a @ N2 @ L ) )
       => ( member_a_a @ ( comp_c_a_a @ G2 @ F ) @ ( sigma_measurable_a_a @ M2 @ L ) ) ) ) ).

% measurable_comp
thf(fact_381_measurable__comp,axiom,
    ! [F: a > a,M2: sigma_measure_a,N2: sigma_measure_a,G2: a > c,L: sigma_measure_c] :
      ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N2 ) )
     => ( ( member_a_c @ G2 @ ( sigma_measurable_a_c @ N2 @ L ) )
       => ( member_a_c @ ( comp_a_c_a @ G2 @ F ) @ ( sigma_measurable_a_c @ M2 @ L ) ) ) ) ).

% measurable_comp
thf(fact_382_measurable__comp,axiom,
    ! [F: a > a,M2: sigma_measure_a,N2: sigma_measure_a,G2: a > a,L: sigma_measure_a] :
      ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N2 ) )
     => ( ( member_a_a @ G2 @ ( sigma_measurable_a_a @ N2 @ L ) )
       => ( member_a_a @ ( comp_a_a_a @ G2 @ F ) @ ( sigma_measurable_a_a @ M2 @ L ) ) ) ) ).

% measurable_comp
thf(fact_383_measurable__comp,axiom,
    ! [F: a > c,M2: sigma_measure_a,N2: sigma_measure_c,G2: c > b,L: sigma_measure_b] :
      ( ( member_a_c @ F @ ( sigma_measurable_a_c @ M2 @ N2 ) )
     => ( ( member_c_b @ G2 @ ( sigma_measurable_c_b @ N2 @ L ) )
       => ( member_a_b @ ( comp_c_b_a @ G2 @ F ) @ ( sigma_measurable_a_b @ M2 @ L ) ) ) ) ).

% measurable_comp
thf(fact_384_measurable__comp,axiom,
    ! [F: b > a,M2: sigma_measure_b,N2: sigma_measure_a,G2: a > d,L: sigma_measure_d] :
      ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ N2 ) )
     => ( ( member_a_d @ G2 @ ( sigma_measurable_a_d @ N2 @ L ) )
       => ( member_b_d @ ( comp_a_d_b @ G2 @ F ) @ ( sigma_measurable_b_d @ M2 @ L ) ) ) ) ).

% measurable_comp
thf(fact_385_measurable__comp,axiom,
    ! [F: a > c,M2: sigma_measure_a,N2: sigma_measure_c,G2: c > real,L: sigma_measure_real] :
      ( ( member_a_c @ F @ ( sigma_measurable_a_c @ M2 @ N2 ) )
     => ( ( member_c_real @ G2 @ ( sigma_1862118822198465884c_real @ N2 @ L ) )
       => ( member_a_real @ ( comp_c_real_a @ G2 @ F ) @ ( sigma_9116425665531756122a_real @ M2 @ L ) ) ) ) ).

% measurable_comp
thf(fact_386_Pi__mono,axiom,
    ! [A: set_a,B: a > set_real,C: a > set_real] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( ord_less_eq_set_real @ ( B @ X ) @ ( C @ X ) ) )
     => ( ord_le3334967407727675675a_real @ ( pi_a_real @ A @ B ) @ ( pi_a_real @ A @ C ) ) ) ).

% Pi_mono
thf(fact_387_Pi__mono,axiom,
    ! [A: set_a,B: a > set_c,C: a > set_c] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( ord_less_eq_set_c @ ( B @ X ) @ ( C @ X ) ) )
     => ( ord_less_eq_set_a_c @ ( pi_a_c @ A @ B ) @ ( pi_a_c @ A @ C ) ) ) ).

% Pi_mono
thf(fact_388_Pi__mono,axiom,
    ! [A: set_c,B: c > set_b,C: c > set_b] :
      ( ! [X: c] :
          ( ( member_c @ X @ A )
         => ( ord_less_eq_set_b @ ( B @ X ) @ ( C @ X ) ) )
     => ( ord_less_eq_set_c_b @ ( pi_c_b @ A @ B ) @ ( pi_c_b @ A @ C ) ) ) ).

% Pi_mono
thf(fact_389_Pi__mono,axiom,
    ! [A: set_b,B: b > set_b,C: b > set_b] :
      ( ! [X: b] :
          ( ( member_b @ X @ A )
         => ( ord_less_eq_set_b @ ( B @ X ) @ ( C @ X ) ) )
     => ( ord_less_eq_set_b_b @ ( pi_b_b @ A @ B ) @ ( pi_b_b @ A @ C ) ) ) ).

% Pi_mono
thf(fact_390_Pi__mono,axiom,
    ! [A: set_a,B: a > set_b,C: a > set_b] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( ord_less_eq_set_b @ ( B @ X ) @ ( C @ X ) ) )
     => ( ord_less_eq_set_a_b @ ( pi_a_b @ A @ B ) @ ( pi_a_b @ A @ C ) ) ) ).

% Pi_mono
thf(fact_391_Pi__mono,axiom,
    ! [A: set_c,B: c > set_a,C: c > set_a] :
      ( ! [X: c] :
          ( ( member_c @ X @ A )
         => ( ord_less_eq_set_a @ ( B @ X ) @ ( C @ X ) ) )
     => ( ord_less_eq_set_c_a @ ( pi_c_a @ A @ B ) @ ( pi_c_a @ A @ C ) ) ) ).

% Pi_mono
thf(fact_392_Pi__mono,axiom,
    ! [A: set_b,B: b > set_a,C: b > set_a] :
      ( ! [X: b] :
          ( ( member_b @ X @ A )
         => ( ord_less_eq_set_a @ ( B @ X ) @ ( C @ X ) ) )
     => ( ord_less_eq_set_b_a @ ( pi_b_a @ A @ B ) @ ( pi_b_a @ A @ C ) ) ) ).

% Pi_mono
thf(fact_393_Pi__mono,axiom,
    ! [A: set_a,B: a > set_a,C: a > set_a] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( ord_less_eq_set_a @ ( B @ X ) @ ( C @ X ) ) )
     => ( ord_less_eq_set_a_a @ ( pi_a_a @ A @ B ) @ ( pi_a_a @ A @ C ) ) ) ).

% Pi_mono
thf(fact_394_Pi__mono,axiom,
    ! [A: set_c,B: c > set_set_b,C: c > set_set_b] :
      ( ! [X: c] :
          ( ( member_c @ X @ A )
         => ( ord_le3795704787696855135_set_b @ ( B @ X ) @ ( C @ X ) ) )
     => ( ord_le213790835969874778_set_b @ ( pi_c_set_b @ A @ B ) @ ( pi_c_set_b @ A @ C ) ) ) ).

% Pi_mono
thf(fact_395_Pi__mono,axiom,
    ! [A: set_b,B: b > set_set_b,C: b > set_set_b] :
      ( ! [X: b] :
          ( ( member_b @ X @ A )
         => ( ord_le3795704787696855135_set_b @ ( B @ X ) @ ( C @ X ) ) )
     => ( ord_le7949200835529852059_set_b @ ( pi_b_set_b @ A @ B ) @ ( pi_b_set_b @ A @ C ) ) ) ).

% Pi_mono
thf(fact_396_Pi__anti__mono,axiom,
    ! [A3: set_a,A: set_a,B: a > set_b] :
      ( ( ord_less_eq_set_a @ A3 @ A )
     => ( ord_less_eq_set_a_b @ ( pi_a_b @ A @ B ) @ ( pi_a_b @ A3 @ B ) ) ) ).

% Pi_anti_mono
thf(fact_397_Pi__anti__mono,axiom,
    ! [A3: set_a,A: set_a,B: a > set_real] :
      ( ( ord_less_eq_set_a @ A3 @ A )
     => ( ord_le3334967407727675675a_real @ ( pi_a_real @ A @ B ) @ ( pi_a_real @ A3 @ B ) ) ) ).

% Pi_anti_mono
thf(fact_398_Pi__anti__mono,axiom,
    ! [A3: set_a,A: set_a,B: a > set_a_real] :
      ( ( ord_less_eq_set_a @ A3 @ A )
     => ( ord_le8977394816241323160a_real @ ( pi_a_a_real @ A @ B ) @ ( pi_a_a_real @ A3 @ B ) ) ) ).

% Pi_anti_mono
thf(fact_399_Pi__anti__mono,axiom,
    ! [A3: set_a,A: set_a,B: a > set_a_b] :
      ( ( ord_less_eq_set_a @ A3 @ A )
     => ( ord_le7397243668584048853_a_a_b @ ( pi_a_a_b @ A @ B ) @ ( pi_a_a_b @ A3 @ B ) ) ) ).

% Pi_anti_mono
thf(fact_400_Pi__anti__mono,axiom,
    ! [A3: set_a,A: set_a,B: a > set_c] :
      ( ( ord_less_eq_set_a @ A3 @ A )
     => ( ord_less_eq_set_a_c @ ( pi_a_c @ A @ B ) @ ( pi_a_c @ A3 @ B ) ) ) ).

% Pi_anti_mono
thf(fact_401_Pi__anti__mono,axiom,
    ! [A3: set_a,A: set_a,B: a > set_a] :
      ( ( ord_less_eq_set_a @ A3 @ A )
     => ( ord_less_eq_set_a_a @ ( pi_a_a @ A @ B ) @ ( pi_a_a @ A3 @ B ) ) ) ).

% Pi_anti_mono
thf(fact_402_prob__space_Oindep__sets__mono,axiom,
    ! [M2: sigma_measure_a,F2: c > set_set_a,I: set_c,J: set_c,G: c > set_set_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe8927441866673418606ts_a_c @ M2 @ F2 @ I )
       => ( ( ord_less_eq_set_c @ J @ I )
         => ( ! [I2: c] :
                ( ( member_c @ I2 @ J )
               => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
           => ( indepe8927441866673418606ts_a_c @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_403_prob__space_Oindep__sets__mono,axiom,
    ! [M2: sigma_measure_a,F2: b > set_set_a,I: set_b,J: set_b,G: b > set_set_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe8927441866673418605ts_a_b @ M2 @ F2 @ I )
       => ( ( ord_less_eq_set_b @ J @ I )
         => ( ! [I2: b] :
                ( ( member_b @ I2 @ J )
               => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
           => ( indepe8927441866673418605ts_a_b @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_404_prob__space_Oindep__sets__mono,axiom,
    ! [M2: sigma_measure_a,F2: a > set_set_a,I: set_a,J: set_a,G: a > set_set_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe8927441866673418604ts_a_a @ M2 @ F2 @ I )
       => ( ( ord_less_eq_set_a @ J @ I )
         => ( ! [I2: a] :
                ( ( member_a @ I2 @ J )
               => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
           => ( indepe8927441866673418604ts_a_a @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_405_prob__space_Oindep__sets__mono,axiom,
    ! [M2: sigma_measure_b,F2: c > set_set_b,I: set_c,J: set_c,G: c > set_set_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe6139986284700742573ts_b_c @ M2 @ F2 @ I )
       => ( ( ord_less_eq_set_c @ J @ I )
         => ( ! [I2: c] :
                ( ( member_c @ I2 @ J )
               => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
           => ( indepe6139986284700742573ts_b_c @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_406_prob__space_Oindep__sets__mono,axiom,
    ! [M2: sigma_measure_b,F2: b > set_set_b,I: set_b,J: set_b,G: b > set_set_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe6139986284700742572ts_b_b @ M2 @ F2 @ I )
       => ( ( ord_less_eq_set_b @ J @ I )
         => ( ! [I2: b] :
                ( ( member_b @ I2 @ J )
               => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
           => ( indepe6139986284700742572ts_b_b @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_407_prob__space_Oindep__sets__mono,axiom,
    ! [M2: sigma_measure_b,F2: a > set_set_b,I: set_a,J: set_a,G: a > set_set_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe6139986284700742571ts_b_a @ M2 @ F2 @ I )
       => ( ( ord_less_eq_set_a @ J @ I )
         => ( ! [I2: a] :
                ( ( member_a @ I2 @ J )
               => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
           => ( indepe6139986284700742571ts_b_a @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_408_prob__space_Oindep__sets__mono,axiom,
    ! [M2: sigma_measure_b,F2: nat > set_set_b,I: set_nat,J: set_nat,G: nat > set_set_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe7503174356045242851_b_nat @ M2 @ F2 @ I )
       => ( ( ord_less_eq_set_nat @ J @ I )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ J )
               => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
           => ( indepe7503174356045242851_b_nat @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_409_prob__space_Oindep__sets__mono,axiom,
    ! [M2: sigma_measure_a,F2: nat > set_set_a,I: set_nat,J: set_nat,G: nat > set_set_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe6267730027088848354_a_nat @ M2 @ F2 @ I )
       => ( ( ord_less_eq_set_nat @ J @ I )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ J )
               => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
           => ( indepe6267730027088848354_a_nat @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_410_prob__space_Oindep__sets__mono,axiom,
    ! [M2: sigma_measure_a,F2: $o > set_set_a,I: set_o,J: set_o,G: $o > set_set_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe7780107833195774214ts_a_o @ M2 @ F2 @ I )
       => ( ( ord_less_eq_set_o @ J @ I )
         => ( ! [I2: $o] :
                ( ( member_o @ I2 @ J )
               => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
           => ( indepe7780107833195774214ts_a_o @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_411_prob__space_Oindep__sets__mono,axiom,
    ! [M2: sigma_measure_b,F2: $o > set_set_b,I: set_o,J: set_o,G: $o > set_set_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe4880885433731379909ts_b_o @ M2 @ F2 @ I )
       => ( ( ord_less_eq_set_o @ J @ I )
         => ( ! [I2: $o] :
                ( ( member_o @ I2 @ J )
               => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
           => ( indepe4880885433731379909ts_b_o @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_412_prob__space_Oindep__sets__mono__index,axiom,
    ! [M2: sigma_measure_a,J: set_b,I: set_b,F2: b > set_set_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( ord_less_eq_set_b @ J @ I )
       => ( ( indepe8927441866673418605ts_a_b @ M2 @ F2 @ I )
         => ( indepe8927441866673418605ts_a_b @ M2 @ F2 @ J ) ) ) ) ).

% prob_space.indep_sets_mono_index
thf(fact_413_prob__space_Oindep__sets__mono__index,axiom,
    ! [M2: sigma_measure_a,J: set_a,I: set_a,F2: a > set_set_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( ord_less_eq_set_a @ J @ I )
       => ( ( indepe8927441866673418604ts_a_a @ M2 @ F2 @ I )
         => ( indepe8927441866673418604ts_a_a @ M2 @ F2 @ J ) ) ) ) ).

% prob_space.indep_sets_mono_index
thf(fact_414_prob__space_Oindep__sets__mono__index,axiom,
    ! [M2: sigma_measure_b,J: set_b,I: set_b,F2: b > set_set_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( ord_less_eq_set_b @ J @ I )
       => ( ( indepe6139986284700742572ts_b_b @ M2 @ F2 @ I )
         => ( indepe6139986284700742572ts_b_b @ M2 @ F2 @ J ) ) ) ) ).

% prob_space.indep_sets_mono_index
thf(fact_415_prob__space_Oindep__sets__mono__index,axiom,
    ! [M2: sigma_measure_b,J: set_a,I: set_a,F2: a > set_set_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( ord_less_eq_set_a @ J @ I )
       => ( ( indepe6139986284700742571ts_b_a @ M2 @ F2 @ I )
         => ( indepe6139986284700742571ts_b_a @ M2 @ F2 @ J ) ) ) ) ).

% prob_space.indep_sets_mono_index
thf(fact_416_prob__space_Oindep__sets__mono__index,axiom,
    ! [M2: sigma_measure_b,J: set_nat,I: set_nat,F2: nat > set_set_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( ord_less_eq_set_nat @ J @ I )
       => ( ( indepe7503174356045242851_b_nat @ M2 @ F2 @ I )
         => ( indepe7503174356045242851_b_nat @ M2 @ F2 @ J ) ) ) ) ).

% prob_space.indep_sets_mono_index
thf(fact_417_prob__space_Oindep__sets__mono__index,axiom,
    ! [M2: sigma_measure_a,J: set_nat,I: set_nat,F2: nat > set_set_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( ord_less_eq_set_nat @ J @ I )
       => ( ( indepe6267730027088848354_a_nat @ M2 @ F2 @ I )
         => ( indepe6267730027088848354_a_nat @ M2 @ F2 @ J ) ) ) ) ).

% prob_space.indep_sets_mono_index
thf(fact_418_prob__space_Oindep__sets__mono__index,axiom,
    ! [M2: sigma_measure_a,J: set_o,I: set_o,F2: $o > set_set_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( ord_less_eq_set_o @ J @ I )
       => ( ( indepe7780107833195774214ts_a_o @ M2 @ F2 @ I )
         => ( indepe7780107833195774214ts_a_o @ M2 @ F2 @ J ) ) ) ) ).

% prob_space.indep_sets_mono_index
thf(fact_419_prob__space_Oindep__sets__mono__index,axiom,
    ! [M2: sigma_measure_b,J: set_o,I: set_o,F2: $o > set_set_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( ord_less_eq_set_o @ J @ I )
       => ( ( indepe4880885433731379909ts_b_o @ M2 @ F2 @ I )
         => ( indepe4880885433731379909ts_b_o @ M2 @ F2 @ J ) ) ) ) ).

% prob_space.indep_sets_mono_index
thf(fact_420_prob__space_Oindep__sets__mono__index,axiom,
    ! [M2: sigma_measure_a,J: set_set_b,I: set_set_b,F2: set_b > set_set_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( ord_le3795704787696855135_set_b @ J @ I )
       => ( ( indepe4967106455115002445_set_b @ M2 @ F2 @ I )
         => ( indepe4967106455115002445_set_b @ M2 @ F2 @ J ) ) ) ) ).

% prob_space.indep_sets_mono_index
thf(fact_421_prob__space_Oindep__sets__mono__index,axiom,
    ! [M2: sigma_measure_a,J: set_set_a,I: set_set_a,F2: set_a > set_set_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( ord_le3724670747650509150_set_a @ J @ I )
       => ( ( indepe4967106450811773644_set_a @ M2 @ F2 @ I )
         => ( indepe4967106450811773644_set_a @ M2 @ F2 @ J ) ) ) ) ).

% prob_space.indep_sets_mono_index
thf(fact_422_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M2: sigma_measure_a,F2: c > set_set_a,I: set_c,G: c > set_set_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe8927441866673418606ts_a_c @ M2 @ F2 @ I )
       => ( ! [I2: c] :
              ( ( member_c @ I2 @ I )
             => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe8927441866673418606ts_a_c @ M2 @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_423_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M2: sigma_measure_a,F2: b > set_set_a,I: set_b,G: b > set_set_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe8927441866673418605ts_a_b @ M2 @ F2 @ I )
       => ( ! [I2: b] :
              ( ( member_b @ I2 @ I )
             => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe8927441866673418605ts_a_b @ M2 @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_424_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M2: sigma_measure_a,F2: a > set_set_a,I: set_a,G: a > set_set_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe8927441866673418604ts_a_a @ M2 @ F2 @ I )
       => ( ! [I2: a] :
              ( ( member_a @ I2 @ I )
             => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe8927441866673418604ts_a_a @ M2 @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_425_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M2: sigma_measure_b,F2: c > set_set_b,I: set_c,G: c > set_set_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe6139986284700742573ts_b_c @ M2 @ F2 @ I )
       => ( ! [I2: c] :
              ( ( member_c @ I2 @ I )
             => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe6139986284700742573ts_b_c @ M2 @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_426_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M2: sigma_measure_b,F2: b > set_set_b,I: set_b,G: b > set_set_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe6139986284700742572ts_b_b @ M2 @ F2 @ I )
       => ( ! [I2: b] :
              ( ( member_b @ I2 @ I )
             => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe6139986284700742572ts_b_b @ M2 @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_427_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M2: sigma_measure_b,F2: a > set_set_b,I: set_a,G: a > set_set_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe6139986284700742571ts_b_a @ M2 @ F2 @ I )
       => ( ! [I2: a] :
              ( ( member_a @ I2 @ I )
             => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe6139986284700742571ts_b_a @ M2 @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_428_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M2: sigma_measure_b,F2: nat > set_set_b,I: set_nat,G: nat > set_set_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe7503174356045242851_b_nat @ M2 @ F2 @ I )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I )
             => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe7503174356045242851_b_nat @ M2 @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_429_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M2: sigma_measure_a,F2: nat > set_set_a,I: set_nat,G: nat > set_set_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe6267730027088848354_a_nat @ M2 @ F2 @ I )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I )
             => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe6267730027088848354_a_nat @ M2 @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_430_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M2: sigma_measure_a,F2: $o > set_set_a,I: set_o,G: $o > set_set_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe7780107833195774214ts_a_o @ M2 @ F2 @ I )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I )
             => ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe7780107833195774214ts_a_o @ M2 @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_431_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M2: sigma_measure_b,F2: $o > set_set_b,I: set_o,G: $o > set_set_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe4880885433731379909ts_b_o @ M2 @ F2 @ I )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I )
             => ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe4880885433731379909ts_b_o @ M2 @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_432_prob__space_Oindep__vars__subset,axiom,
    ! [M2: sigma_measure_a,M: c > sigma_measure_d,X4: c > a > d,I: set_c,J: set_c] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe4851901773132442935_a_c_d @ M2 @ M @ X4 @ I )
       => ( ( ord_less_eq_set_c @ J @ I )
         => ( indepe4851901773132442935_a_c_d @ M2 @ M @ X4 @ J ) ) ) ) ).

% prob_space.indep_vars_subset
thf(fact_433_distr__distr,axiom,
    ! [G2: a > c,N2: sigma_measure_a,L: sigma_measure_c,F: b > a,M2: sigma_measure_b] :
      ( ( member_a_c @ G2 @ ( sigma_measurable_a_c @ N2 @ L ) )
     => ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ N2 ) )
       => ( ( measure_distr_a_c @ ( measure_distr_b_a @ M2 @ N2 @ F ) @ L @ G2 )
          = ( measure_distr_b_c @ M2 @ L @ ( comp_a_c_b @ G2 @ F ) ) ) ) ) ).

% distr_distr
thf(fact_434_distr__distr,axiom,
    ! [G2: a > c,N2: sigma_measure_a,L: sigma_measure_c,F: a > a,M2: sigma_measure_a] :
      ( ( member_a_c @ G2 @ ( sigma_measurable_a_c @ N2 @ L ) )
     => ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N2 ) )
       => ( ( measure_distr_a_c @ ( measure_distr_a_a @ M2 @ N2 @ F ) @ L @ G2 )
          = ( measure_distr_a_c @ M2 @ L @ ( comp_a_c_a @ G2 @ F ) ) ) ) ) ).

% distr_distr
thf(fact_435_distr__distr,axiom,
    ! [G2: a > a,N2: sigma_measure_a,L: sigma_measure_a,F: b > a,M2: sigma_measure_b] :
      ( ( member_a_a @ G2 @ ( sigma_measurable_a_a @ N2 @ L ) )
     => ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ N2 ) )
       => ( ( measure_distr_a_a @ ( measure_distr_b_a @ M2 @ N2 @ F ) @ L @ G2 )
          = ( measure_distr_b_a @ M2 @ L @ ( comp_a_a_b @ G2 @ F ) ) ) ) ) ).

% distr_distr
thf(fact_436_distr__distr,axiom,
    ! [G2: a > a,N2: sigma_measure_a,L: sigma_measure_a,F: a > a,M2: sigma_measure_a] :
      ( ( member_a_a @ G2 @ ( sigma_measurable_a_a @ N2 @ L ) )
     => ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N2 ) )
       => ( ( measure_distr_a_a @ ( measure_distr_a_a @ M2 @ N2 @ F ) @ L @ G2 )
          = ( measure_distr_a_a @ M2 @ L @ ( comp_a_a_a @ G2 @ F ) ) ) ) ) ).

% distr_distr
thf(fact_437_distr__distr,axiom,
    ! [G2: c > b,N2: sigma_measure_c,L: sigma_measure_b,F: a > c,M2: sigma_measure_a] :
      ( ( member_c_b @ G2 @ ( sigma_measurable_c_b @ N2 @ L ) )
     => ( ( member_a_c @ F @ ( sigma_measurable_a_c @ M2 @ N2 ) )
       => ( ( measure_distr_c_b @ ( measure_distr_a_c @ M2 @ N2 @ F ) @ L @ G2 )
          = ( measure_distr_a_b @ M2 @ L @ ( comp_c_b_a @ G2 @ F ) ) ) ) ) ).

% distr_distr
thf(fact_438_distr__distr,axiom,
    ! [G2: b > a,N2: sigma_measure_b,L: sigma_measure_a,F: a > b,M2: sigma_measure_a] :
      ( ( member_b_a @ G2 @ ( sigma_measurable_b_a @ N2 @ L ) )
     => ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ N2 ) )
       => ( ( measure_distr_b_a @ ( measure_distr_a_b @ M2 @ N2 @ F ) @ L @ G2 )
          = ( measure_distr_a_a @ M2 @ L @ ( comp_b_a_a @ G2 @ F ) ) ) ) ) ).

% distr_distr
thf(fact_439_distr__distr,axiom,
    ! [G2: b > b,N2: sigma_measure_b,L: sigma_measure_b,F: a > b,M2: sigma_measure_a] :
      ( ( member_b_b @ G2 @ ( sigma_measurable_b_b @ N2 @ L ) )
     => ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ N2 ) )
       => ( ( measure_distr_b_b @ ( measure_distr_a_b @ M2 @ N2 @ F ) @ L @ G2 )
          = ( measure_distr_a_b @ M2 @ L @ ( comp_b_b_a @ G2 @ F ) ) ) ) ) ).

% distr_distr
thf(fact_440_distr__distr,axiom,
    ! [G2: real > b,N2: sigma_measure_real,L: sigma_measure_b,F: a > real,M2: sigma_measure_a] :
      ( ( member_real_b @ G2 @ ( sigma_523072396149930113real_b @ N2 @ L ) )
     => ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M2 @ N2 ) )
       => ( ( measure_distr_real_b @ ( measure_distr_a_real @ M2 @ N2 @ F ) @ L @ G2 )
          = ( measure_distr_a_b @ M2 @ L @ ( comp_real_b_a @ G2 @ F ) ) ) ) ) ).

% distr_distr
thf(fact_441_distr__distr,axiom,
    ! [G2: a > b,N2: sigma_measure_a,L: sigma_measure_b,F: b > a,M2: sigma_measure_b] :
      ( ( member_a_b @ G2 @ ( sigma_measurable_a_b @ N2 @ L ) )
     => ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ N2 ) )
       => ( ( measure_distr_a_b @ ( measure_distr_b_a @ M2 @ N2 @ F ) @ L @ G2 )
          = ( measure_distr_b_b @ M2 @ L @ ( comp_a_b_b @ G2 @ F ) ) ) ) ) ).

% distr_distr
thf(fact_442_distr__distr,axiom,
    ! [G2: a > b,N2: sigma_measure_a,L: sigma_measure_b,F: a > a,M2: sigma_measure_a] :
      ( ( member_a_b @ G2 @ ( sigma_measurable_a_b @ N2 @ L ) )
     => ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N2 ) )
       => ( ( measure_distr_a_b @ ( measure_distr_a_a @ M2 @ N2 @ F ) @ L @ G2 )
          = ( measure_distr_a_b @ M2 @ L @ ( comp_a_b_a @ G2 @ F ) ) ) ) ) ).

% distr_distr
thf(fact_443_prob__space_Oindep__var__compose,axiom,
    ! [M2: sigma_measure_a,M1: sigma_measure_b,X1: a > b,M22: sigma_measure_b,X22: a > b,Y1: b > a,N1: sigma_measure_a,Y2: b > a,N22: sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe2440653194691626189ar_a_b @ M2 @ M1 @ X1 @ M22 @ X22 )
       => ( ( member_b_a @ Y1 @ ( sigma_measurable_b_a @ M1 @ N1 ) )
         => ( ( member_b_a @ Y2 @ ( sigma_measurable_b_a @ M22 @ N22 ) )
           => ( indepe2440653194691626188ar_a_a @ M2 @ N1 @ ( comp_b_a_a @ Y1 @ X1 ) @ N22 @ ( comp_b_a_a @ Y2 @ X22 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_444_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 > c,N1: sigma_measure_c,Y2: a > c,N22: sigma_measure_c] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe2440653194691626188ar_a_a @ M2 @ M1 @ X1 @ M22 @ X22 )
       => ( ( member_a_c @ Y1 @ ( sigma_measurable_a_c @ M1 @ N1 ) )
         => ( ( member_a_c @ Y2 @ ( sigma_measurable_a_c @ M22 @ N22 ) )
           => ( indepe2440653194691626190ar_a_c @ M2 @ N1 @ ( comp_a_c_a @ Y1 @ X1 ) @ N22 @ ( comp_a_c_a @ Y2 @ X22 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_445_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 > a,N1: sigma_measure_a,Y2: a > a,N22: sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe2440653194691626188ar_a_a @ M2 @ M1 @ X1 @ M22 @ X22 )
       => ( ( member_a_a @ Y1 @ ( sigma_measurable_a_a @ M1 @ N1 ) )
         => ( ( member_a_a @ Y2 @ ( sigma_measurable_a_a @ M22 @ N22 ) )
           => ( indepe2440653194691626188ar_a_a @ M2 @ N1 @ ( comp_a_a_a @ Y1 @ X1 ) @ N22 @ ( comp_a_a_a @ Y2 @ X22 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_446_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 > b,N1: sigma_measure_b,Y2: a > b,N22: sigma_measure_b] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe2440653194691626188ar_a_a @ M2 @ M1 @ X1 @ M22 @ X22 )
       => ( ( member_a_b @ Y1 @ ( sigma_measurable_a_b @ M1 @ N1 ) )
         => ( ( member_a_b @ Y2 @ ( sigma_measurable_a_b @ M22 @ N22 ) )
           => ( indepe2440653194691626189ar_a_b @ M2 @ N1 @ ( comp_a_b_a @ Y1 @ X1 ) @ N22 @ ( comp_a_b_a @ Y2 @ X22 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_447_prob__space_Oindep__var__compose,axiom,
    ! [M2: sigma_measure_a,M1: sigma_measure_b,X1: a > b,M22: sigma_measure_b,X22: a > b,Y1: b > d,N1: sigma_measure_d,Y2: b > d,N22: sigma_measure_d] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe2440653194691626189ar_a_b @ M2 @ M1 @ X1 @ M22 @ X22 )
       => ( ( member_b_d @ Y1 @ ( sigma_measurable_b_d @ M1 @ N1 ) )
         => ( ( member_b_d @ Y2 @ ( sigma_measurable_b_d @ M22 @ N22 ) )
           => ( indepe2440653194691626191ar_a_d @ M2 @ N1 @ ( comp_b_d_a @ Y1 @ X1 ) @ N22 @ ( comp_b_d_a @ Y2 @ X22 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_448_prob__space_Oindep__var__compose,axiom,
    ! [M2: sigma_measure_a,M1: sigma_measure_b,X1: a > b,M22: sigma_measure_b,X22: a > b,Y1: b > real,N1: sigma_measure_real,Y2: b > real,N22: sigma_measure_real] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe2440653194691626189ar_a_b @ M2 @ M1 @ X1 @ M22 @ X22 )
       => ( ( member_b_real @ Y1 @ ( sigma_5489272243865111003b_real @ M1 @ N1 ) )
         => ( ( member_b_real @ Y2 @ ( sigma_5489272243865111003b_real @ M22 @ N22 ) )
           => ( indepe8958435565499147358a_real @ M2 @ N1 @ ( comp_b_real_a @ Y1 @ X1 ) @ N22 @ ( comp_b_real_a @ Y2 @ X22 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_449_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,Y2: 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 @ Y2 @ ( sigma_9116425665531756122a_real @ M22 @ N22 ) )
           => ( indepe8958435565499147358a_real @ M2 @ N1 @ ( comp_a_real_a @ Y1 @ X1 ) @ N22 @ ( comp_a_real_a @ Y2 @ X22 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_450_prob__space_Oindep__var__compose,axiom,
    ! [M2: sigma_measure_b,M1: sigma_measure_b,X1: b > b,M22: sigma_measure_b,X22: b > b,Y1: b > a,N1: sigma_measure_a,Y2: b > a,N22: sigma_measure_a] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe8876569649573725964ar_b_b @ M2 @ M1 @ X1 @ M22 @ X22 )
       => ( ( member_b_a @ Y1 @ ( sigma_measurable_b_a @ M1 @ N1 ) )
         => ( ( member_b_a @ Y2 @ ( sigma_measurable_b_a @ M22 @ N22 ) )
           => ( indepe8876569649573725963ar_b_a @ M2 @ N1 @ ( comp_b_a_b @ Y1 @ X1 ) @ N22 @ ( comp_b_a_b @ Y2 @ X22 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_451_prob__space_Oindep__var__compose,axiom,
    ! [M2: sigma_measure_b,M1: sigma_measure_a,X1: b > a,M22: sigma_measure_a,X22: b > a,Y1: a > c,N1: sigma_measure_c,Y2: a > c,N22: sigma_measure_c] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe8876569649573725963ar_b_a @ M2 @ M1 @ X1 @ M22 @ X22 )
       => ( ( member_a_c @ Y1 @ ( sigma_measurable_a_c @ M1 @ N1 ) )
         => ( ( member_a_c @ Y2 @ ( sigma_measurable_a_c @ M22 @ N22 ) )
           => ( indepe8876569649573725965ar_b_c @ M2 @ N1 @ ( comp_a_c_b @ Y1 @ X1 ) @ N22 @ ( comp_a_c_b @ Y2 @ X22 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_452_prob__space_Oindep__var__compose,axiom,
    ! [M2: sigma_measure_b,M1: sigma_measure_a,X1: b > a,M22: sigma_measure_a,X22: b > a,Y1: a > a,N1: sigma_measure_a,Y2: a > a,N22: sigma_measure_a] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe8876569649573725963ar_b_a @ M2 @ M1 @ X1 @ M22 @ X22 )
       => ( ( member_a_a @ Y1 @ ( sigma_measurable_a_a @ M1 @ N1 ) )
         => ( ( member_a_a @ Y2 @ ( sigma_measurable_a_a @ M22 @ N22 ) )
           => ( indepe8876569649573725963ar_b_a @ M2 @ N1 @ ( comp_a_a_b @ Y1 @ X1 ) @ N22 @ ( comp_a_a_b @ Y2 @ X22 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_453_prob__space_Oindep__vars__compose,axiom,
    ! [M2: sigma_measure_a,M: c > sigma_measure_b,X4: c > a > b,I: set_c,Y: c > b > a,N2: c > sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe4851901773132442933_a_c_b @ M2 @ M @ X4 @ I )
       => ( ! [I2: c] :
              ( ( member_c @ I2 @ I )
             => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( indepe4851901773132442932_a_c_a @ M2 @ N2
            @ ^ [I3: c] : ( comp_b_a_a @ ( Y @ I3 ) @ ( X4 @ I3 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_454_prob__space_Oindep__vars__compose,axiom,
    ! [M2: sigma_measure_a,M: c > sigma_measure_a,X4: c > a > a,I: set_c,Y: c > a > c,N2: c > sigma_measure_c] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe4851901773132442932_a_c_a @ M2 @ M @ X4 @ I )
       => ( ! [I2: c] :
              ( ( member_c @ I2 @ I )
             => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( indepe4851901773132442934_a_c_c @ M2 @ N2
            @ ^ [I3: c] : ( comp_a_c_a @ ( Y @ I3 ) @ ( X4 @ I3 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_455_prob__space_Oindep__vars__compose,axiom,
    ! [M2: sigma_measure_a,M: c > sigma_measure_a,X4: c > a > a,I: set_c,Y: c > a > a,N2: c > sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe4851901773132442932_a_c_a @ M2 @ M @ X4 @ I )
       => ( ! [I2: c] :
              ( ( member_c @ I2 @ I )
             => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( indepe4851901773132442932_a_c_a @ M2 @ N2
            @ ^ [I3: c] : ( comp_a_a_a @ ( Y @ I3 ) @ ( X4 @ I3 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_456_prob__space_Oindep__vars__compose,axiom,
    ! [M2: sigma_measure_a,M: b > sigma_measure_b,X4: b > a > b,I: set_b,Y: b > b > a,N2: b > sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe7639357355105118966_a_b_b @ M2 @ M @ X4 @ I )
       => ( ! [I2: b] :
              ( ( member_b @ I2 @ I )
             => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( indepe7639357355105118965_a_b_a @ M2 @ N2
            @ ^ [I3: b] : ( comp_b_a_a @ ( Y @ I3 ) @ ( X4 @ I3 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_457_prob__space_Oindep__vars__compose,axiom,
    ! [M2: sigma_measure_a,M: b > sigma_measure_a,X4: b > a > a,I: set_b,Y: b > a > c,N2: b > sigma_measure_c] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe7639357355105118965_a_b_a @ M2 @ M @ X4 @ I )
       => ( ! [I2: b] :
              ( ( member_b @ I2 @ I )
             => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( indepe7639357355105118967_a_b_c @ M2 @ N2
            @ ^ [I3: b] : ( comp_a_c_a @ ( Y @ I3 ) @ ( X4 @ I3 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_458_prob__space_Oindep__vars__compose,axiom,
    ! [M2: sigma_measure_a,M: b > sigma_measure_a,X4: b > a > a,I: set_b,Y: b > a > a,N2: b > sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe7639357355105118965_a_b_a @ M2 @ M @ X4 @ I )
       => ( ! [I2: b] :
              ( ( member_b @ I2 @ I )
             => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( indepe7639357355105118965_a_b_a @ M2 @ N2
            @ ^ [I3: b] : ( comp_a_a_a @ ( Y @ I3 ) @ ( X4 @ I3 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_459_prob__space_Oindep__vars__compose,axiom,
    ! [M2: sigma_measure_a,M: a > sigma_measure_b,X4: a > a > b,I: set_a,Y: a > b > a,N2: a > sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe1203440900223019191_a_a_b @ M2 @ M @ X4 @ I )
       => ( ! [I2: a] :
              ( ( member_a @ I2 @ I )
             => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( indepe1203440900223019190_a_a_a @ M2 @ N2
            @ ^ [I3: a] : ( comp_b_a_a @ ( Y @ I3 ) @ ( X4 @ I3 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_460_prob__space_Oindep__vars__compose,axiom,
    ! [M2: sigma_measure_a,M: a > sigma_measure_a,X4: a > a > a,I: set_a,Y: a > a > c,N2: a > sigma_measure_c] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe1203440900223019190_a_a_a @ M2 @ M @ X4 @ I )
       => ( ! [I2: a] :
              ( ( member_a @ I2 @ I )
             => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( indepe1203440900223019192_a_a_c @ M2 @ N2
            @ ^ [I3: a] : ( comp_a_c_a @ ( Y @ I3 ) @ ( X4 @ I3 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_461_prob__space_Oindep__vars__compose,axiom,
    ! [M2: sigma_measure_a,M: a > sigma_measure_a,X4: a > a > a,I: set_a,Y: a > a > a,N2: a > sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe1203440900223019190_a_a_a @ M2 @ M @ X4 @ I )
       => ( ! [I2: a] :
              ( ( member_a @ I2 @ I )
             => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( indepe1203440900223019190_a_a_a @ M2 @ N2
            @ ^ [I3: a] : ( comp_a_a_a @ ( Y @ I3 ) @ ( X4 @ I3 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_462_prob__space_Oindep__vars__compose,axiom,
    ! [M2: sigma_measure_a,M: c > sigma_measure_a,X4: c > a > a,I: set_c,Y: c > a > b,N2: c > sigma_measure_b] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe4851901773132442932_a_c_a @ M2 @ M @ X4 @ I )
       => ( ! [I2: c] :
              ( ( member_c @ I2 @ I )
             => ( member_a_b @ ( Y @ I2 ) @ ( sigma_measurable_a_b @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
         => ( indepe4851901773132442933_a_c_b @ M2 @ N2
            @ ^ [I3: c] : ( comp_a_b_a @ ( Y @ I3 ) @ ( X4 @ I3 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_463_prob__space_Oindep__sets_Ocong,axiom,
    indepe7503174356045242851_b_nat = indepe7503174356045242851_b_nat ).

% prob_space.indep_sets.cong
thf(fact_464_prob__space_Oindep__sets_Ocong,axiom,
    indepe6267730027088848354_a_nat = indepe6267730027088848354_a_nat ).

% prob_space.indep_sets.cong
thf(fact_465_prob__space_Oindep__sets_Ocong,axiom,
    indepe7780107833195774214ts_a_o = indepe7780107833195774214ts_a_o ).

% prob_space.indep_sets.cong
thf(fact_466_prob__space_Oindep__sets_Ocong,axiom,
    indepe4880885433731379909ts_b_o = indepe4880885433731379909ts_b_o ).

% prob_space.indep_sets.cong
thf(fact_467_prob__space_Oindep__vars_Ocong,axiom,
    indepe4851901773132442935_a_c_d = indepe4851901773132442935_a_c_d ).

% prob_space.indep_vars.cong
thf(fact_468_measurable__id,axiom,
    ! [M2: sigma_measure_a] :
      ( member_a_a
      @ ^ [X2: a] : X2
      @ ( sigma_measurable_a_a @ M2 @ M2 ) ) ).

% measurable_id
thf(fact_469_measurable__compose,axiom,
    ! [F: b > b,M2: sigma_measure_b,N2: sigma_measure_b,G2: b > a,L: sigma_measure_a] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M2 @ N2 ) )
     => ( ( member_b_a @ G2 @ ( sigma_measurable_b_a @ N2 @ L ) )
       => ( member_b_a
          @ ^ [X2: b] : ( G2 @ ( F @ X2 ) )
          @ ( sigma_measurable_b_a @ M2 @ L ) ) ) ) ).

% measurable_compose
thf(fact_470_measurable__compose,axiom,
    ! [F: b > a,M2: sigma_measure_b,N2: sigma_measure_a,G2: a > c,L: sigma_measure_c] :
      ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ N2 ) )
     => ( ( member_a_c @ G2 @ ( sigma_measurable_a_c @ N2 @ L ) )
       => ( member_b_c
          @ ^ [X2: b] : ( G2 @ ( F @ X2 ) )
          @ ( sigma_measurable_b_c @ M2 @ L ) ) ) ) ).

% measurable_compose
thf(fact_471_measurable__compose,axiom,
    ! [F: b > a,M2: sigma_measure_b,N2: sigma_measure_a,G2: a > a,L: sigma_measure_a] :
      ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ N2 ) )
     => ( ( member_a_a @ G2 @ ( sigma_measurable_a_a @ N2 @ L ) )
       => ( member_b_a
          @ ^ [X2: b] : ( G2 @ ( F @ X2 ) )
          @ ( sigma_measurable_b_a @ M2 @ L ) ) ) ) ).

% measurable_compose
thf(fact_472_measurable__compose,axiom,
    ! [F: a > c,M2: sigma_measure_a,N2: sigma_measure_c,G2: c > c,L: sigma_measure_c] :
      ( ( member_a_c @ F @ ( sigma_measurable_a_c @ M2 @ N2 ) )
     => ( ( member_c_c @ G2 @ ( sigma_measurable_c_c @ N2 @ L ) )
       => ( member_a_c
          @ ^ [X2: a] : ( G2 @ ( F @ X2 ) )
          @ ( sigma_measurable_a_c @ M2 @ L ) ) ) ) ).

% measurable_compose
thf(fact_473_measurable__compose,axiom,
    ! [F: a > c,M2: sigma_measure_a,N2: sigma_measure_c,G2: c > a,L: sigma_measure_a] :
      ( ( member_a_c @ F @ ( sigma_measurable_a_c @ M2 @ N2 ) )
     => ( ( member_c_a @ G2 @ ( sigma_measurable_c_a @ N2 @ L ) )
       => ( member_a_a
          @ ^ [X2: a] : ( G2 @ ( F @ X2 ) )
          @ ( sigma_measurable_a_a @ M2 @ L ) ) ) ) ).

% measurable_compose
thf(fact_474_measurable__compose,axiom,
    ! [F: a > a,M2: sigma_measure_a,N2: sigma_measure_a,G2: a > c,L: sigma_measure_c] :
      ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N2 ) )
     => ( ( member_a_c @ G2 @ ( sigma_measurable_a_c @ N2 @ L ) )
       => ( member_a_c
          @ ^ [X2: a] : ( G2 @ ( F @ X2 ) )
          @ ( sigma_measurable_a_c @ M2 @ L ) ) ) ) ).

% measurable_compose
thf(fact_475_measurable__compose,axiom,
    ! [F: a > a,M2: sigma_measure_a,N2: sigma_measure_a,G2: a > a,L: sigma_measure_a] :
      ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N2 ) )
     => ( ( member_a_a @ G2 @ ( sigma_measurable_a_a @ N2 @ L ) )
       => ( member_a_a
          @ ^ [X2: a] : ( G2 @ ( F @ X2 ) )
          @ ( sigma_measurable_a_a @ M2 @ L ) ) ) ) ).

% measurable_compose
thf(fact_476_measurable__compose,axiom,
    ! [F: a > c,M2: sigma_measure_a,N2: sigma_measure_c,G2: c > b,L: sigma_measure_b] :
      ( ( member_a_c @ F @ ( sigma_measurable_a_c @ M2 @ N2 ) )
     => ( ( member_c_b @ G2 @ ( sigma_measurable_c_b @ N2 @ L ) )
       => ( member_a_b
          @ ^ [X2: a] : ( G2 @ ( F @ X2 ) )
          @ ( sigma_measurable_a_b @ M2 @ L ) ) ) ) ).

% measurable_compose
thf(fact_477_measurable__compose,axiom,
    ! [F: b > a,M2: sigma_measure_b,N2: sigma_measure_a,G2: a > d,L: sigma_measure_d] :
      ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ N2 ) )
     => ( ( member_a_d @ G2 @ ( sigma_measurable_a_d @ N2 @ L ) )
       => ( member_b_d
          @ ^ [X2: b] : ( G2 @ ( F @ X2 ) )
          @ ( sigma_measurable_b_d @ M2 @ L ) ) ) ) ).

% measurable_compose
thf(fact_478_measurable__compose,axiom,
    ! [F: a > c,M2: sigma_measure_a,N2: sigma_measure_c,G2: c > real,L: sigma_measure_real] :
      ( ( member_a_c @ F @ ( sigma_measurable_a_c @ M2 @ N2 ) )
     => ( ( member_c_real @ G2 @ ( sigma_1862118822198465884c_real @ N2 @ L ) )
       => ( member_a_real
          @ ^ [X2: a] : ( G2 @ ( F @ X2 ) )
          @ ( sigma_9116425665531756122a_real @ M2 @ L ) ) ) ) ).

% measurable_compose
thf(fact_479_measurable__compose__rev,axiom,
    ! [F: c > c,L: sigma_measure_c,N2: sigma_measure_c,G2: a > c,M2: sigma_measure_a] :
      ( ( member_c_c @ F @ ( sigma_measurable_c_c @ L @ N2 ) )
     => ( ( member_a_c @ G2 @ ( sigma_measurable_a_c @ M2 @ L ) )
       => ( member_a_c
          @ ^ [X2: a] : ( F @ ( G2 @ X2 ) )
          @ ( sigma_measurable_a_c @ M2 @ N2 ) ) ) ) ).

% measurable_compose_rev
thf(fact_480_measurable__compose__rev,axiom,
    ! [F: c > a,L: sigma_measure_c,N2: sigma_measure_a,G2: a > c,M2: sigma_measure_a] :
      ( ( member_c_a @ F @ ( sigma_measurable_c_a @ L @ N2 ) )
     => ( ( member_a_c @ G2 @ ( sigma_measurable_a_c @ M2 @ L ) )
       => ( member_a_a
          @ ^ [X2: a] : ( F @ ( G2 @ X2 ) )
          @ ( sigma_measurable_a_a @ M2 @ N2 ) ) ) ) ).

% measurable_compose_rev
thf(fact_481_measurable__compose__rev,axiom,
    ! [F: b > a,L: sigma_measure_b,N2: sigma_measure_a,G2: b > b,M2: sigma_measure_b] :
      ( ( member_b_a @ F @ ( sigma_measurable_b_a @ L @ N2 ) )
     => ( ( member_b_b @ G2 @ ( sigma_measurable_b_b @ M2 @ L ) )
       => ( member_b_a
          @ ^ [X2: b] : ( F @ ( G2 @ X2 ) )
          @ ( sigma_measurable_b_a @ M2 @ N2 ) ) ) ) ).

% measurable_compose_rev
thf(fact_482_measurable__compose__rev,axiom,
    ! [F: a > c,L: sigma_measure_a,N2: sigma_measure_c,G2: b > a,M2: sigma_measure_b] :
      ( ( member_a_c @ F @ ( sigma_measurable_a_c @ L @ N2 ) )
     => ( ( member_b_a @ G2 @ ( sigma_measurable_b_a @ M2 @ L ) )
       => ( member_b_c
          @ ^ [X2: b] : ( F @ ( G2 @ X2 ) )
          @ ( sigma_measurable_b_c @ M2 @ N2 ) ) ) ) ).

% measurable_compose_rev
thf(fact_483_measurable__compose__rev,axiom,
    ! [F: a > c,L: sigma_measure_a,N2: sigma_measure_c,G2: a > a,M2: sigma_measure_a] :
      ( ( member_a_c @ F @ ( sigma_measurable_a_c @ L @ N2 ) )
     => ( ( member_a_a @ G2 @ ( sigma_measurable_a_a @ M2 @ L ) )
       => ( member_a_c
          @ ^ [X2: a] : ( F @ ( G2 @ X2 ) )
          @ ( sigma_measurable_a_c @ M2 @ N2 ) ) ) ) ).

% measurable_compose_rev
thf(fact_484_measurable__compose__rev,axiom,
    ! [F: a > a,L: sigma_measure_a,N2: sigma_measure_a,G2: b > a,M2: sigma_measure_b] :
      ( ( member_a_a @ F @ ( sigma_measurable_a_a @ L @ N2 ) )
     => ( ( member_b_a @ G2 @ ( sigma_measurable_b_a @ M2 @ L ) )
       => ( member_b_a
          @ ^ [X2: b] : ( F @ ( G2 @ X2 ) )
          @ ( sigma_measurable_b_a @ M2 @ N2 ) ) ) ) ).

% measurable_compose_rev
thf(fact_485_measurable__compose__rev,axiom,
    ! [F: a > a,L: sigma_measure_a,N2: sigma_measure_a,G2: a > a,M2: sigma_measure_a] :
      ( ( member_a_a @ F @ ( sigma_measurable_a_a @ L @ N2 ) )
     => ( ( member_a_a @ G2 @ ( sigma_measurable_a_a @ M2 @ L ) )
       => ( member_a_a
          @ ^ [X2: a] : ( F @ ( G2 @ X2 ) )
          @ ( sigma_measurable_a_a @ M2 @ N2 ) ) ) ) ).

% measurable_compose_rev
thf(fact_486_measurable__compose__rev,axiom,
    ! [F: c > b,L: sigma_measure_c,N2: sigma_measure_b,G2: a > c,M2: sigma_measure_a] :
      ( ( member_c_b @ F @ ( sigma_measurable_c_b @ L @ N2 ) )
     => ( ( member_a_c @ G2 @ ( sigma_measurable_a_c @ M2 @ L ) )
       => ( member_a_b
          @ ^ [X2: a] : ( F @ ( G2 @ X2 ) )
          @ ( sigma_measurable_a_b @ M2 @ N2 ) ) ) ) ).

% measurable_compose_rev
thf(fact_487_measurable__compose__rev,axiom,
    ! [F: a > d,L: sigma_measure_a,N2: sigma_measure_d,G2: b > a,M2: sigma_measure_b] :
      ( ( member_a_d @ F @ ( sigma_measurable_a_d @ L @ N2 ) )
     => ( ( member_b_a @ G2 @ ( sigma_measurable_b_a @ M2 @ L ) )
       => ( member_b_d
          @ ^ [X2: b] : ( F @ ( G2 @ X2 ) )
          @ ( sigma_measurable_b_d @ M2 @ N2 ) ) ) ) ).

% measurable_compose_rev
thf(fact_488_measurable__compose__rev,axiom,
    ! [F: c > real,L: sigma_measure_c,N2: sigma_measure_real,G2: a > c,M2: sigma_measure_a] :
      ( ( member_c_real @ F @ ( sigma_1862118822198465884c_real @ L @ N2 ) )
     => ( ( member_a_c @ G2 @ ( sigma_measurable_a_c @ M2 @ L ) )
       => ( member_a_real
          @ ^ [X2: a] : ( F @ ( G2 @ X2 ) )
          @ ( sigma_9116425665531756122a_real @ M2 @ N2 ) ) ) ) ).

% measurable_compose_rev
thf(fact_489_measurable__cong__simp,axiom,
    ! [M2: sigma_measure_a,N2: sigma_measure_a,M: sigma_measure_a_real,N: sigma_measure_a_real,F: a > a > real,G2: a > a > real] :
      ( ( M2 = N2 )
     => ( ( M = N )
       => ( ! [W: a] :
              ( ( member_a @ W @ ( sigma_space_a @ M2 ) )
             => ( ( F @ W )
                = ( G2 @ W ) ) )
         => ( ( member_a_a_real @ F @ ( sigma_5073124708296780141a_real @ M2 @ M ) )
            = ( member_a_a_real @ G2 @ ( sigma_5073124708296780141a_real @ N2 @ N ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_490_measurable__cong__simp,axiom,
    ! [M2: sigma_measure_a,N2: sigma_measure_a,M: sigma_measure_a_b,N: sigma_measure_a_b,F: a > a > b,G2: a > a > b] :
      ( ( M2 = N2 )
     => ( ( M = N )
       => ( ! [W: a] :
              ( ( member_a @ W @ ( sigma_space_a @ M2 ) )
             => ( ( F @ W )
                = ( G2 @ W ) ) )
         => ( ( member_a_a_b @ F @ ( sigma_3971135317854480532_a_a_b @ M2 @ M ) )
            = ( member_a_a_b @ G2 @ ( sigma_3971135317854480532_a_a_b @ N2 @ N ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_491_measurable__cong__simp,axiom,
    ! [M2: sigma_measure_a,N2: sigma_measure_a,M: sigma_measure_c,N: sigma_measure_c,F: a > c,G2: a > c] :
      ( ( M2 = N2 )
     => ( ( M = N )
       => ( ! [W: a] :
              ( ( member_a @ W @ ( sigma_space_a @ M2 ) )
             => ( ( F @ W )
                = ( G2 @ W ) ) )
         => ( ( member_a_c @ F @ ( sigma_measurable_a_c @ M2 @ M ) )
            = ( member_a_c @ G2 @ ( sigma_measurable_a_c @ N2 @ N ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_492_measurable__cong__simp,axiom,
    ! [M2: sigma_measure_a,N2: sigma_measure_a,M: sigma_measure_a,N: sigma_measure_a,F: a > a,G2: a > a] :
      ( ( M2 = N2 )
     => ( ( M = N )
       => ( ! [W: a] :
              ( ( member_a @ W @ ( sigma_space_a @ M2 ) )
             => ( ( F @ W )
                = ( G2 @ W ) ) )
         => ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ M ) )
            = ( member_a_a @ G2 @ ( sigma_measurable_a_a @ N2 @ N ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_493_measurable__cong__simp,axiom,
    ! [M2: sigma_measure_b,N2: sigma_measure_b,M: sigma_measure_a,N: sigma_measure_a,F: b > a,G2: b > a] :
      ( ( M2 = N2 )
     => ( ( M = N )
       => ( ! [W: b] :
              ( ( member_b @ W @ ( sigma_space_b @ M2 ) )
             => ( ( F @ W )
                = ( G2 @ W ) ) )
         => ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ M ) )
            = ( member_b_a @ G2 @ ( sigma_measurable_b_a @ N2 @ N ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_494_measurable__cong__simp,axiom,
    ! [M2: sigma_measure_a,N2: sigma_measure_a,M: sigma_measure_b,N: sigma_measure_b,F: a > b,G2: a > b] :
      ( ( M2 = N2 )
     => ( ( M = N )
       => ( ! [W: a] :
              ( ( member_a @ W @ ( sigma_space_a @ M2 ) )
             => ( ( F @ W )
                = ( G2 @ W ) ) )
         => ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ M ) )
            = ( member_a_b @ G2 @ ( sigma_measurable_a_b @ N2 @ N ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_495_measurable__cong__simp,axiom,
    ! [M2: sigma_measure_b,N2: sigma_measure_b,M: sigma_measure_d,N: sigma_measure_d,F: b > d,G2: b > d] :
      ( ( M2 = N2 )
     => ( ( M = N )
       => ( ! [W: b] :
              ( ( member_b @ W @ ( sigma_space_b @ M2 ) )
             => ( ( F @ W )
                = ( G2 @ W ) ) )
         => ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M2 @ M ) )
            = ( member_b_d @ G2 @ ( sigma_measurable_b_d @ N2 @ N ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_496_measurable__cong__simp,axiom,
    ! [M2: sigma_measure_b,N2: sigma_measure_b,M: sigma_measure_real,N: sigma_measure_real,F: b > real,G2: b > real] :
      ( ( M2 = N2 )
     => ( ( M = N )
       => ( ! [W: b] :
              ( ( member_b @ W @ ( sigma_space_b @ M2 ) )
             => ( ( F @ W )
                = ( G2 @ W ) ) )
         => ( ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ M2 @ M ) )
            = ( member_b_real @ G2 @ ( sigma_5489272243865111003b_real @ N2 @ N ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_497_measurable__cong__simp,axiom,
    ! [M2: sigma_measure_a,N2: sigma_measure_a,M: sigma_measure_real,N: sigma_measure_real,F: a > real,G2: a > real] :
      ( ( M2 = N2 )
     => ( ( M = N )
       => ( ! [W: a] :
              ( ( member_a @ W @ ( sigma_space_a @ M2 ) )
             => ( ( F @ W )
                = ( G2 @ W ) ) )
         => ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M2 @ M ) )
            = ( member_a_real @ G2 @ ( sigma_9116425665531756122a_real @ N2 @ N ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_498_measurable__space,axiom,
    ! [F: c > c,M2: sigma_measure_c,A: sigma_measure_c,X3: c] :
      ( ( member_c_c @ F @ ( sigma_measurable_c_c @ M2 @ A ) )
     => ( ( member_c @ X3 @ ( sigma_space_c @ M2 ) )
       => ( member_c @ ( F @ X3 ) @ ( sigma_space_c @ A ) ) ) ) ).

% measurable_space
thf(fact_499_measurable__space,axiom,
    ! [F: c > a,M2: sigma_measure_c,A: sigma_measure_a,X3: c] :
      ( ( member_c_a @ F @ ( sigma_measurable_c_a @ M2 @ A ) )
     => ( ( member_c @ X3 @ ( sigma_space_c @ M2 ) )
       => ( member_a @ ( F @ X3 ) @ ( sigma_space_a @ A ) ) ) ) ).

% measurable_space
thf(fact_500_measurable__space,axiom,
    ! [F: c > b,M2: sigma_measure_c,A: sigma_measure_b,X3: c] :
      ( ( member_c_b @ F @ ( sigma_measurable_c_b @ M2 @ A ) )
     => ( ( member_c @ X3 @ ( sigma_space_c @ M2 ) )
       => ( member_b @ ( F @ X3 ) @ ( sigma_space_b @ A ) ) ) ) ).

% measurable_space
thf(fact_501_measurable__space,axiom,
    ! [F: a > c,M2: sigma_measure_a,A: sigma_measure_c,X3: a] :
      ( ( member_a_c @ F @ ( sigma_measurable_a_c @ M2 @ A ) )
     => ( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
       => ( member_c @ ( F @ X3 ) @ ( sigma_space_c @ A ) ) ) ) ).

% measurable_space
thf(fact_502_measurable__space,axiom,
    ! [F: a > a,M2: sigma_measure_a,A: sigma_measure_a,X3: a] :
      ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ A ) )
     => ( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
       => ( member_a @ ( F @ X3 ) @ ( sigma_space_a @ A ) ) ) ) ).

% measurable_space
thf(fact_503_measurable__space,axiom,
    ! [F: b > c,M2: sigma_measure_b,A: sigma_measure_c,X3: b] :
      ( ( member_b_c @ F @ ( sigma_measurable_b_c @ M2 @ A ) )
     => ( ( member_b @ X3 @ ( sigma_space_b @ M2 ) )
       => ( member_c @ ( F @ X3 ) @ ( sigma_space_c @ A ) ) ) ) ).

% measurable_space
thf(fact_504_measurable__space,axiom,
    ! [F: b > a,M2: sigma_measure_b,A: sigma_measure_a,X3: b] :
      ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ A ) )
     => ( ( member_b @ X3 @ ( sigma_space_b @ M2 ) )
       => ( member_a @ ( F @ X3 ) @ ( sigma_space_a @ A ) ) ) ) ).

% measurable_space
thf(fact_505_measurable__space,axiom,
    ! [F: b > b,M2: sigma_measure_b,A: sigma_measure_b,X3: b] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M2 @ A ) )
     => ( ( member_b @ X3 @ ( sigma_space_b @ M2 ) )
       => ( member_b @ ( F @ X3 ) @ ( sigma_space_b @ A ) ) ) ) ).

% measurable_space
thf(fact_506_measurable__space,axiom,
    ! [F: a > b,M2: sigma_measure_a,A: sigma_measure_b,X3: a] :
      ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ A ) )
     => ( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
       => ( member_b @ ( F @ X3 ) @ ( sigma_space_b @ A ) ) ) ) ).

% measurable_space
thf(fact_507_measurable__space,axiom,
    ! [F: b > d,M2: sigma_measure_b,A: sigma_measure_d,X3: b] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M2 @ A ) )
     => ( ( member_b @ X3 @ ( sigma_space_b @ M2 ) )
       => ( member_d @ ( F @ X3 ) @ ( sigma_space_d @ A ) ) ) ) ).

% measurable_space
thf(fact_508_measurable__cong,axiom,
    ! [M2: sigma_measure_a,F: a > a > real,G2: a > a > real,M: sigma_measure_a_real] :
      ( ! [W: a] :
          ( ( member_a @ W @ ( sigma_space_a @ M2 ) )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_a_a_real @ F @ ( sigma_5073124708296780141a_real @ M2 @ M ) )
        = ( member_a_a_real @ G2 @ ( sigma_5073124708296780141a_real @ M2 @ M ) ) ) ) ).

% measurable_cong
thf(fact_509_measurable__cong,axiom,
    ! [M2: sigma_measure_a,F: a > a > b,G2: a > a > b,M: sigma_measure_a_b] :
      ( ! [W: a] :
          ( ( member_a @ W @ ( sigma_space_a @ M2 ) )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_a_a_b @ F @ ( sigma_3971135317854480532_a_a_b @ M2 @ M ) )
        = ( member_a_a_b @ G2 @ ( sigma_3971135317854480532_a_a_b @ M2 @ M ) ) ) ) ).

% measurable_cong
thf(fact_510_measurable__cong,axiom,
    ! [M2: sigma_measure_a,F: a > c,G2: a > c,M: sigma_measure_c] :
      ( ! [W: a] :
          ( ( member_a @ W @ ( sigma_space_a @ M2 ) )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_a_c @ F @ ( sigma_measurable_a_c @ M2 @ M ) )
        = ( member_a_c @ G2 @ ( sigma_measurable_a_c @ M2 @ M ) ) ) ) ).

% measurable_cong
thf(fact_511_measurable__cong,axiom,
    ! [M2: sigma_measure_a,F: a > a,G2: a > a,M: sigma_measure_a] :
      ( ! [W: a] :
          ( ( member_a @ W @ ( sigma_space_a @ M2 ) )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ M ) )
        = ( member_a_a @ G2 @ ( sigma_measurable_a_a @ M2 @ M ) ) ) ) ).

% measurable_cong
thf(fact_512_measurable__cong,axiom,
    ! [M2: sigma_measure_b,F: b > a,G2: b > a,M: sigma_measure_a] :
      ( ! [W: b] :
          ( ( member_b @ W @ ( sigma_space_b @ M2 ) )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ M ) )
        = ( member_b_a @ G2 @ ( sigma_measurable_b_a @ M2 @ M ) ) ) ) ).

% measurable_cong
thf(fact_513_measurable__cong,axiom,
    ! [M2: sigma_measure_a,F: a > b,G2: a > b,M: sigma_measure_b] :
      ( ! [W: a] :
          ( ( member_a @ W @ ( sigma_space_a @ M2 ) )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ M ) )
        = ( member_a_b @ G2 @ ( sigma_measurable_a_b @ M2 @ M ) ) ) ) ).

% measurable_cong
thf(fact_514_measurable__cong,axiom,
    ! [M2: sigma_measure_b,F: b > d,G2: b > d,M: sigma_measure_d] :
      ( ! [W: b] :
          ( ( member_b @ W @ ( sigma_space_b @ M2 ) )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M2 @ M ) )
        = ( member_b_d @ G2 @ ( sigma_measurable_b_d @ M2 @ M ) ) ) ) ).

% measurable_cong
thf(fact_515_measurable__cong,axiom,
    ! [M2: sigma_measure_b,F: b > real,G2: b > real,M: sigma_measure_real] :
      ( ! [W: b] :
          ( ( member_b @ W @ ( sigma_space_b @ M2 ) )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ M2 @ M ) )
        = ( member_b_real @ G2 @ ( sigma_5489272243865111003b_real @ M2 @ M ) ) ) ) ).

% measurable_cong
thf(fact_516_measurable__cong,axiom,
    ! [M2: sigma_measure_a,F: a > real,G2: a > real,M: sigma_measure_real] :
      ( ! [W: a] :
          ( ( member_a @ W @ ( sigma_space_a @ M2 ) )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M2 @ M ) )
        = ( member_a_real @ G2 @ ( sigma_9116425665531756122a_real @ M2 @ M ) ) ) ) ).

% measurable_cong
thf(fact_517_prob__space_Oindep__sets__cong,axiom,
    ! [M2: sigma_measure_a,I: set_c,J: set_c,F2: c > set_set_a,G: c > set_set_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( I = J )
       => ( ! [I2: c] :
              ( ( member_c @ I2 @ I )
             => ( ( F2 @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe8927441866673418606ts_a_c @ M2 @ F2 @ I )
            = ( indepe8927441866673418606ts_a_c @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_518_prob__space_Oindep__sets__cong,axiom,
    ! [M2: sigma_measure_a,I: set_b,J: set_b,F2: b > set_set_a,G: b > set_set_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( I = J )
       => ( ! [I2: b] :
              ( ( member_b @ I2 @ I )
             => ( ( F2 @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe8927441866673418605ts_a_b @ M2 @ F2 @ I )
            = ( indepe8927441866673418605ts_a_b @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_519_prob__space_Oindep__sets__cong,axiom,
    ! [M2: sigma_measure_a,I: set_a,J: set_a,F2: a > set_set_a,G: a > set_set_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( I = J )
       => ( ! [I2: a] :
              ( ( member_a @ I2 @ I )
             => ( ( F2 @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe8927441866673418604ts_a_a @ M2 @ F2 @ I )
            = ( indepe8927441866673418604ts_a_a @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_520_prob__space_Oindep__sets__cong,axiom,
    ! [M2: sigma_measure_b,I: set_c,J: set_c,F2: c > set_set_b,G: c > set_set_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( I = J )
       => ( ! [I2: c] :
              ( ( member_c @ I2 @ I )
             => ( ( F2 @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe6139986284700742573ts_b_c @ M2 @ F2 @ I )
            = ( indepe6139986284700742573ts_b_c @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_521_prob__space_Oindep__sets__cong,axiom,
    ! [M2: sigma_measure_b,I: set_b,J: set_b,F2: b > set_set_b,G: b > set_set_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( I = J )
       => ( ! [I2: b] :
              ( ( member_b @ I2 @ I )
             => ( ( F2 @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe6139986284700742572ts_b_b @ M2 @ F2 @ I )
            = ( indepe6139986284700742572ts_b_b @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_522_prob__space_Oindep__sets__cong,axiom,
    ! [M2: sigma_measure_b,I: set_a,J: set_a,F2: a > set_set_b,G: a > set_set_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( I = J )
       => ( ! [I2: a] :
              ( ( member_a @ I2 @ I )
             => ( ( F2 @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe6139986284700742571ts_b_a @ M2 @ F2 @ I )
            = ( indepe6139986284700742571ts_b_a @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_523_prob__space_Oindep__sets__cong,axiom,
    ! [M2: sigma_measure_b,I: set_nat,J: set_nat,F2: nat > set_set_b,G: nat > set_set_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( I = J )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I )
             => ( ( F2 @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe7503174356045242851_b_nat @ M2 @ F2 @ I )
            = ( indepe7503174356045242851_b_nat @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_524_prob__space_Oindep__sets__cong,axiom,
    ! [M2: sigma_measure_a,I: set_nat,J: set_nat,F2: nat > set_set_a,G: nat > set_set_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( I = J )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I )
             => ( ( F2 @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe6267730027088848354_a_nat @ M2 @ F2 @ I )
            = ( indepe6267730027088848354_a_nat @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_525_prob__space_Oindep__sets__cong,axiom,
    ! [M2: sigma_measure_a,I: set_o,J: set_o,F2: $o > set_set_a,G: $o > set_set_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( I = J )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I )
             => ( ( F2 @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe7780107833195774214ts_a_o @ M2 @ F2 @ I )
            = ( indepe7780107833195774214ts_a_o @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_526_prob__space_Oindep__sets__cong,axiom,
    ! [M2: sigma_measure_b,I: set_o,J: set_o,F2: $o > set_set_b,G: $o > set_set_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( I = J )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I )
             => ( ( F2 @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe4880885433731379909ts_b_o @ M2 @ F2 @ I )
            = ( indepe4880885433731379909ts_b_o @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_527_prob__space_Oindep__vars__cong,axiom,
    ! [M2: sigma_measure_a,I: set_c,J: set_c,X4: c > a > d,Y: c > a > d,M: c > sigma_measure_d,N: c > sigma_measure_d] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( I = J )
       => ( ! [I2: c] :
              ( ( member_c @ I2 @ I )
             => ( ( X4 @ I2 )
                = ( Y @ I2 ) ) )
         => ( ! [I2: c] :
                ( ( member_c @ I2 @ I )
               => ( ( M @ I2 )
                  = ( N @ I2 ) ) )
           => ( ( indepe4851901773132442935_a_c_d @ M2 @ M @ X4 @ I )
              = ( indepe4851901773132442935_a_c_d @ M2 @ N @ Y @ J ) ) ) ) ) ) ).

% prob_space.indep_vars_cong
thf(fact_528_measurable__const,axiom,
    ! [C2: a > real,M: sigma_measure_a_real,M2: sigma_measure_a] :
      ( ( member_a_real @ C2 @ ( sigma_space_a_real @ M ) )
     => ( member_a_a_real
        @ ^ [X2: a] : C2
        @ ( sigma_5073124708296780141a_real @ M2 @ M ) ) ) ).

% measurable_const
thf(fact_529_measurable__const,axiom,
    ! [C2: a > b,M: sigma_measure_a_b,M2: sigma_measure_a] :
      ( ( member_a_b @ C2 @ ( sigma_space_a_b @ M ) )
     => ( member_a_a_b
        @ ^ [X2: a] : C2
        @ ( sigma_3971135317854480532_a_a_b @ M2 @ M ) ) ) ).

% measurable_const
thf(fact_530_measurable__const,axiom,
    ! [C2: c,M: sigma_measure_c,M2: sigma_measure_a] :
      ( ( member_c @ C2 @ ( sigma_space_c @ M ) )
     => ( member_a_c
        @ ^ [X2: a] : C2
        @ ( sigma_measurable_a_c @ M2 @ M ) ) ) ).

% measurable_const
thf(fact_531_measurable__const,axiom,
    ! [C2: a,M: sigma_measure_a,M2: sigma_measure_b] :
      ( ( member_a @ C2 @ ( sigma_space_a @ M ) )
     => ( member_b_a
        @ ^ [X2: b] : C2
        @ ( sigma_measurable_b_a @ M2 @ M ) ) ) ).

% measurable_const
thf(fact_532_measurable__const,axiom,
    ! [C2: a,M: sigma_measure_a,M2: sigma_measure_a] :
      ( ( member_a @ C2 @ ( sigma_space_a @ M ) )
     => ( member_a_a
        @ ^ [X2: a] : C2
        @ ( sigma_measurable_a_a @ M2 @ M ) ) ) ).

% measurable_const
thf(fact_533_measurable__const,axiom,
    ! [C2: b,M: sigma_measure_b,M2: sigma_measure_a] :
      ( ( member_b @ C2 @ ( sigma_space_b @ M ) )
     => ( member_a_b
        @ ^ [X2: a] : C2
        @ ( sigma_measurable_a_b @ M2 @ M ) ) ) ).

% measurable_const
thf(fact_534_measurable__const,axiom,
    ! [C2: d,M: sigma_measure_d,M2: sigma_measure_b] :
      ( ( member_d @ C2 @ ( sigma_space_d @ M ) )
     => ( member_b_d
        @ ^ [X2: b] : C2
        @ ( sigma_measurable_b_d @ M2 @ M ) ) ) ).

% measurable_const
thf(fact_535_measurable__const,axiom,
    ! [C2: real,M: sigma_measure_real,M2: sigma_measure_b] :
      ( ( member_real @ C2 @ ( sigma_space_real @ M ) )
     => ( member_b_real
        @ ^ [X2: b] : C2
        @ ( sigma_5489272243865111003b_real @ M2 @ M ) ) ) ).

% measurable_const
thf(fact_536_measurable__const,axiom,
    ! [C2: real,M: sigma_measure_real,M2: sigma_measure_a] :
      ( ( member_real @ C2 @ ( sigma_space_real @ M ) )
     => ( member_a_real
        @ ^ [X2: a] : C2
        @ ( sigma_9116425665531756122a_real @ M2 @ M ) ) ) ).

% measurable_const
thf(fact_537_measurable__empty__iff,axiom,
    ! [N2: sigma_measure_c,F: a > c,M2: sigma_measure_a] :
      ( ( ( sigma_space_c @ N2 )
        = bot_bot_set_c )
     => ( ( member_a_c @ F @ ( sigma_measurable_a_c @ M2 @ N2 ) )
        = ( ( sigma_space_a @ M2 )
          = bot_bot_set_a ) ) ) ).

% measurable_empty_iff
thf(fact_538_measurable__empty__iff,axiom,
    ! [N2: sigma_measure_a,F: a > a,M2: sigma_measure_a] :
      ( ( ( sigma_space_a @ N2 )
        = bot_bot_set_a )
     => ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N2 ) )
        = ( ( sigma_space_a @ M2 )
          = bot_bot_set_a ) ) ) ).

% measurable_empty_iff
thf(fact_539_measurable__empty__iff,axiom,
    ! [N2: sigma_measure_a,F: b > a,M2: sigma_measure_b] :
      ( ( ( sigma_space_a @ N2 )
        = bot_bot_set_a )
     => ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ N2 ) )
        = ( ( sigma_space_b @ M2 )
          = bot_bot_set_b ) ) ) ).

% measurable_empty_iff
thf(fact_540_measurable__empty__iff,axiom,
    ! [N2: sigma_measure_a,F: $o > a,M2: sigma_measure_o] :
      ( ( ( sigma_space_a @ N2 )
        = bot_bot_set_a )
     => ( ( member_o_a @ F @ ( sigma_measurable_o_a @ M2 @ N2 ) )
        = ( ( sigma_space_o @ M2 )
          = bot_bot_set_o ) ) ) ).

% measurable_empty_iff
thf(fact_541_measurable__empty__iff,axiom,
    ! [N2: sigma_measure_b,F: b > b,M2: sigma_measure_b] :
      ( ( ( sigma_space_b @ N2 )
        = bot_bot_set_b )
     => ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M2 @ N2 ) )
        = ( ( sigma_space_b @ M2 )
          = bot_bot_set_b ) ) ) ).

% measurable_empty_iff
thf(fact_542_measurable__empty__iff,axiom,
    ! [N2: sigma_measure_b,F: $o > b,M2: sigma_measure_o] :
      ( ( ( sigma_space_b @ N2 )
        = bot_bot_set_b )
     => ( ( member_o_b @ F @ ( sigma_measurable_o_b @ M2 @ N2 ) )
        = ( ( sigma_space_o @ M2 )
          = bot_bot_set_o ) ) ) ).

% measurable_empty_iff
thf(fact_543_measurable__empty__iff,axiom,
    ! [N2: sigma_measure_o,F: a > $o,M2: sigma_measure_a] :
      ( ( ( sigma_space_o @ N2 )
        = bot_bot_set_o )
     => ( ( member_a_o @ F @ ( sigma_measurable_a_o @ M2 @ N2 ) )
        = ( ( sigma_space_a @ M2 )
          = bot_bot_set_a ) ) ) ).

% measurable_empty_iff
thf(fact_544_measurable__empty__iff,axiom,
    ! [N2: sigma_measure_o,F: b > $o,M2: sigma_measure_b] :
      ( ( ( sigma_space_o @ N2 )
        = bot_bot_set_o )
     => ( ( member_b_o @ F @ ( sigma_measurable_b_o @ M2 @ N2 ) )
        = ( ( sigma_space_b @ M2 )
          = bot_bot_set_b ) ) ) ).

% measurable_empty_iff
thf(fact_545_measurable__empty__iff,axiom,
    ! [N2: sigma_measure_o,F: $o > $o,M2: sigma_measure_o] :
      ( ( ( sigma_space_o @ N2 )
        = bot_bot_set_o )
     => ( ( member_o_o @ F @ ( sigma_measurable_o_o @ M2 @ N2 ) )
        = ( ( sigma_space_o @ M2 )
          = bot_bot_set_o ) ) ) ).

% measurable_empty_iff
thf(fact_546_measurable__empty__iff,axiom,
    ! [N2: sigma_measure_b,F: a > b,M2: sigma_measure_a] :
      ( ( ( sigma_space_b @ N2 )
        = bot_bot_set_b )
     => ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ N2 ) )
        = ( ( sigma_space_a @ M2 )
          = bot_bot_set_a ) ) ) ).

% measurable_empty_iff
thf(fact_547_prob__space_Oindep__var__rv1,axiom,
    ! [M2: sigma_measure_a,S: sigma_measure_a_real,X4: a > a > real,T: sigma_measure_a_real,Y: a > a > real] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe7138907624787055209a_real @ M2 @ S @ X4 @ T @ Y )
       => ( member_a_a_real @ X4 @ ( sigma_5073124708296780141a_real @ M2 @ S ) ) ) ) ).

% prob_space.indep_var_rv1
thf(fact_548_prob__space_Oindep__var__rv1,axiom,
    ! [M2: sigma_measure_a,S: sigma_measure_a_b,X4: a > a > b,T: sigma_measure_a_b,Y: a > a > b] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe6909679525661198488_a_a_b @ M2 @ S @ X4 @ T @ Y )
       => ( member_a_a_b @ X4 @ ( sigma_3971135317854480532_a_a_b @ M2 @ S ) ) ) ) ).

% prob_space.indep_var_rv1
thf(fact_549_prob__space_Oindep__var__rv1,axiom,
    ! [M2: sigma_measure_a,S: sigma_measure_c,X4: a > c,T: sigma_measure_c,Y: a > c] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe2440653194691626190ar_a_c @ M2 @ S @ X4 @ T @ Y )
       => ( member_a_c @ X4 @ ( sigma_measurable_a_c @ M2 @ S ) ) ) ) ).

% prob_space.indep_var_rv1
thf(fact_550_prob__space_Oindep__var__rv1,axiom,
    ! [M2: sigma_measure_a,S: sigma_measure_a,X4: a > a,T: sigma_measure_a,Y: a > a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe2440653194691626188ar_a_a @ M2 @ S @ X4 @ T @ Y )
       => ( member_a_a @ X4 @ ( sigma_measurable_a_a @ M2 @ S ) ) ) ) ).

% prob_space.indep_var_rv1
thf(fact_551_prob__space_Oindep__var__rv1,axiom,
    ! [M2: sigma_measure_a,S: sigma_measure_b,X4: a > b,T: sigma_measure_b,Y: a > b] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe2440653194691626189ar_a_b @ M2 @ S @ X4 @ T @ Y )
       => ( member_a_b @ X4 @ ( sigma_measurable_a_b @ M2 @ S ) ) ) ) ).

% prob_space.indep_var_rv1
thf(fact_552_prob__space_Oindep__var__rv1,axiom,
    ! [M2: sigma_measure_a,S: sigma_measure_real,X4: a > real,T: sigma_measure_real,Y: a > real] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe8958435565499147358a_real @ M2 @ S @ X4 @ T @ Y )
       => ( member_a_real @ X4 @ ( sigma_9116425665531756122a_real @ M2 @ S ) ) ) ) ).

% prob_space.indep_var_rv1
thf(fact_553_prob__space_Oindep__var__rv1,axiom,
    ! [M2: sigma_measure_b,S: sigma_measure_a,X4: b > a,T: sigma_measure_a,Y: b > a] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe8876569649573725963ar_b_a @ M2 @ S @ X4 @ T @ Y )
       => ( member_b_a @ X4 @ ( sigma_measurable_b_a @ M2 @ S ) ) ) ) ).

% prob_space.indep_var_rv1
thf(fact_554_prob__space_Oindep__var__rv1,axiom,
    ! [M2: sigma_measure_b,S: sigma_measure_d,X4: b > d,T: sigma_measure_d,Y: b > d] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe8876569649573725966ar_b_d @ M2 @ S @ X4 @ T @ Y )
       => ( member_b_d @ X4 @ ( sigma_measurable_b_d @ M2 @ S ) ) ) ) ).

% prob_space.indep_var_rv1
thf(fact_555_prob__space_Oindep__var__rv1,axiom,
    ! [M2: sigma_measure_b,S: sigma_measure_real,X4: b > real,T: sigma_measure_real,Y: b > real] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe5331282143832502239b_real @ M2 @ S @ X4 @ T @ Y )
       => ( member_b_real @ X4 @ ( sigma_5489272243865111003b_real @ M2 @ S ) ) ) ) ).

% prob_space.indep_var_rv1
thf(fact_556_prob__space_Oindep__var__rv2,axiom,
    ! [M2: sigma_measure_a,S: sigma_measure_a_real,X4: a > a > real,T: sigma_measure_a_real,Y: a > a > real] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe7138907624787055209a_real @ M2 @ S @ X4 @ T @ Y )
       => ( member_a_a_real @ Y @ ( sigma_5073124708296780141a_real @ M2 @ T ) ) ) ) ).

% prob_space.indep_var_rv2
thf(fact_557_prob__space_Oindep__var__rv2,axiom,
    ! [M2: sigma_measure_a,S: sigma_measure_a_b,X4: a > a > b,T: sigma_measure_a_b,Y: a > a > b] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe6909679525661198488_a_a_b @ M2 @ S @ X4 @ T @ Y )
       => ( member_a_a_b @ Y @ ( sigma_3971135317854480532_a_a_b @ M2 @ T ) ) ) ) ).

% prob_space.indep_var_rv2
thf(fact_558_prob__space_Oindep__var__rv2,axiom,
    ! [M2: sigma_measure_a,S: sigma_measure_c,X4: a > c,T: sigma_measure_c,Y: a > c] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe2440653194691626190ar_a_c @ M2 @ S @ X4 @ T @ Y )
       => ( member_a_c @ Y @ ( sigma_measurable_a_c @ M2 @ T ) ) ) ) ).

% prob_space.indep_var_rv2
thf(fact_559_prob__space_Oindep__var__rv2,axiom,
    ! [M2: sigma_measure_a,S: sigma_measure_a,X4: a > a,T: sigma_measure_a,Y: a > a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe2440653194691626188ar_a_a @ M2 @ S @ X4 @ T @ Y )
       => ( member_a_a @ Y @ ( sigma_measurable_a_a @ M2 @ T ) ) ) ) ).

% prob_space.indep_var_rv2
thf(fact_560_prob__space_Oindep__var__rv2,axiom,
    ! [M2: sigma_measure_a,S: sigma_measure_b,X4: a > b,T: sigma_measure_b,Y: a > b] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe2440653194691626189ar_a_b @ M2 @ S @ X4 @ T @ Y )
       => ( member_a_b @ Y @ ( sigma_measurable_a_b @ M2 @ T ) ) ) ) ).

% prob_space.indep_var_rv2
thf(fact_561_prob__space_Oindep__var__rv2,axiom,
    ! [M2: sigma_measure_a,S: sigma_measure_real,X4: a > real,T: sigma_measure_real,Y: a > real] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe8958435565499147358a_real @ M2 @ S @ X4 @ T @ Y )
       => ( member_a_real @ Y @ ( sigma_9116425665531756122a_real @ M2 @ T ) ) ) ) ).

% prob_space.indep_var_rv2
thf(fact_562_prob__space_Oindep__var__rv2,axiom,
    ! [M2: sigma_measure_b,S: sigma_measure_a,X4: b > a,T: sigma_measure_a,Y: b > a] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe8876569649573725963ar_b_a @ M2 @ S @ X4 @ T @ Y )
       => ( member_b_a @ Y @ ( sigma_measurable_b_a @ M2 @ T ) ) ) ) ).

% prob_space.indep_var_rv2
thf(fact_563_prob__space_Oindep__var__rv2,axiom,
    ! [M2: sigma_measure_b,S: sigma_measure_d,X4: b > d,T: sigma_measure_d,Y: b > d] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe8876569649573725966ar_b_d @ M2 @ S @ X4 @ T @ Y )
       => ( member_b_d @ Y @ ( sigma_measurable_b_d @ M2 @ T ) ) ) ) ).

% prob_space.indep_var_rv2
thf(fact_564_prob__space_Oindep__var__rv2,axiom,
    ! [M2: sigma_measure_b,S: sigma_measure_real,X4: b > real,T: sigma_measure_real,Y: b > real] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe5331282143832502239b_real @ M2 @ S @ X4 @ T @ Y )
       => ( member_b_real @ Y @ ( sigma_5489272243865111003b_real @ M2 @ T ) ) ) ) ).

% prob_space.indep_var_rv2
thf(fact_565_D_Oindep__sets__finite__index__sets,axiom,
    ! [F2: set_b > set_set_b,I: set_set_b] :
      ( ( indepe6311571491924490380_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
      = ( ! [J2: set_set_b] :
            ( ( ord_le3795704787696855135_set_b @ J2 @ I )
           => ( ( J2 != bot_bot_set_set_b )
             => ( ( finite_finite_set_b @ J2 )
               => ( indepe6311571491924490380_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ J2 ) ) ) ) ) ) ).

% D.indep_sets_finite_index_sets
thf(fact_566_D_Oindep__sets__finite__index__sets,axiom,
    ! [F2: set_a > set_set_b,I: set_set_a] :
      ( ( indepe6311571487621261579_set_a @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
      = ( ! [J2: set_set_a] :
            ( ( ord_le3724670747650509150_set_a @ J2 @ I )
           => ( ( J2 != bot_bot_set_set_a )
             => ( ( finite_finite_set_a @ J2 )
               => ( indepe6311571487621261579_set_a @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ J2 ) ) ) ) ) ) ).

% D.indep_sets_finite_index_sets
thf(fact_567_D_Oindep__sets__finite__index__sets,axiom,
    ! [F2: b > set_set_b,I: set_b] :
      ( ( indepe6139986284700742572ts_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
      = ( ! [J2: set_b] :
            ( ( ord_less_eq_set_b @ J2 @ I )
           => ( ( J2 != bot_bot_set_b )
             => ( ( finite_finite_b @ J2 )
               => ( indepe6139986284700742572ts_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ J2 ) ) ) ) ) ) ).

% D.indep_sets_finite_index_sets
thf(fact_568_D_Oindep__sets__finite__index__sets,axiom,
    ! [F2: a > set_set_b,I: set_a] :
      ( ( indepe6139986284700742571ts_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
      = ( ! [J2: set_a] :
            ( ( ord_less_eq_set_a @ J2 @ I )
           => ( ( J2 != bot_bot_set_a )
             => ( ( finite_finite_a @ J2 )
               => ( indepe6139986284700742571ts_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ J2 ) ) ) ) ) ) ).

% D.indep_sets_finite_index_sets
thf(fact_569_D_Oindep__sets__finite__index__sets,axiom,
    ! [F2: nat > set_set_b,I: set_nat] :
      ( ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
      = ( ! [J2: set_nat] :
            ( ( ord_less_eq_set_nat @ J2 @ I )
           => ( ( J2 != bot_bot_set_nat )
             => ( ( finite_finite_nat @ J2 )
               => ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ J2 ) ) ) ) ) ) ).

% D.indep_sets_finite_index_sets
thf(fact_570_D_Oindep__sets__finite__index__sets,axiom,
    ! [F2: $o > set_set_b,I: set_o] :
      ( ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
      = ( ! [J2: set_o] :
            ( ( ord_less_eq_set_o @ J2 @ I )
           => ( ( J2 != bot_bot_set_o )
             => ( ( finite_finite_o @ J2 )
               => ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ J2 ) ) ) ) ) ) ).

% D.indep_sets_finite_index_sets
thf(fact_571_space__completion,axiom,
    ! [M2: sigma_2733069582767196501sure_b] :
      ( ( sigma_6149183809546982772sure_b @ ( comple9004992072017958258sure_b @ M2 ) )
      = ( sigma_6149183809546982772sure_b @ M2 ) ) ).

% space_completion
thf(fact_572_space__completion,axiom,
    ! [M2: sigma_2662035542720850516sure_a] :
      ( ( sigma_6149183805243753971sure_a @ ( comple9004992067714729457sure_a @ M2 ) )
      = ( sigma_6149183805243753971sure_a @ M2 ) ) ).

% space_completion
thf(fact_573_space__completion,axiom,
    ! [M2: sigma_measure_b] :
      ( ( sigma_space_b @ ( comple3428971583294703881tion_b @ M2 ) )
      = ( sigma_space_b @ M2 ) ) ).

% space_completion
thf(fact_574_space__completion,axiom,
    ! [M2: sigma_measure_a] :
      ( ( sigma_space_a @ ( comple3428971583294703880tion_a @ M2 ) )
      = ( sigma_space_a @ M2 ) ) ).

% space_completion
thf(fact_575_subset__empty,axiom,
    ! [A: set_o] :
      ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
      = ( A = bot_bot_set_o ) ) ).

% subset_empty
thf(fact_576_subset__empty,axiom,
    ! [A: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A @ bot_bot_set_set_b )
      = ( A = bot_bot_set_set_b ) ) ).

% subset_empty
thf(fact_577_subset__empty,axiom,
    ! [A: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
      = ( A = bot_bot_set_set_a ) ) ).

% subset_empty
thf(fact_578_subset__empty,axiom,
    ! [A: set_b] :
      ( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
      = ( A = bot_bot_set_b ) ) ).

% subset_empty
thf(fact_579_subset__empty,axiom,
    ! [A: set_a] :
      ( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
      = ( A = bot_bot_set_a ) ) ).

% subset_empty
thf(fact_580_empty__subsetI,axiom,
    ! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).

% empty_subsetI
thf(fact_581_empty__subsetI,axiom,
    ! [A: set_set_b] : ( ord_le3795704787696855135_set_b @ bot_bot_set_set_b @ A ) ).

% empty_subsetI
thf(fact_582_empty__subsetI,axiom,
    ! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).

% empty_subsetI
thf(fact_583_empty__subsetI,axiom,
    ! [A: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A ) ).

% empty_subsetI
thf(fact_584_empty__subsetI,axiom,
    ! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).

% empty_subsetI
thf(fact_585_indep__sets__finite__index__sets,axiom,
    ! [F2: set_b > set_set_a,I: set_set_b] :
      ( ( indepe4967106455115002445_set_b @ m @ F2 @ I )
      = ( ! [J2: set_set_b] :
            ( ( ord_le3795704787696855135_set_b @ J2 @ I )
           => ( ( J2 != bot_bot_set_set_b )
             => ( ( finite_finite_set_b @ J2 )
               => ( indepe4967106455115002445_set_b @ m @ F2 @ J2 ) ) ) ) ) ) ).

% indep_sets_finite_index_sets
thf(fact_586_indep__sets__finite__index__sets,axiom,
    ! [F2: set_a > set_set_a,I: set_set_a] :
      ( ( indepe4967106450811773644_set_a @ m @ F2 @ I )
      = ( ! [J2: set_set_a] :
            ( ( ord_le3724670747650509150_set_a @ J2 @ I )
           => ( ( J2 != bot_bot_set_set_a )
             => ( ( finite_finite_set_a @ J2 )
               => ( indepe4967106450811773644_set_a @ m @ F2 @ J2 ) ) ) ) ) ) ).

% indep_sets_finite_index_sets
thf(fact_587_indep__sets__finite__index__sets,axiom,
    ! [F2: b > set_set_a,I: set_b] :
      ( ( indepe8927441866673418605ts_a_b @ m @ F2 @ I )
      = ( ! [J2: set_b] :
            ( ( ord_less_eq_set_b @ J2 @ I )
           => ( ( J2 != bot_bot_set_b )
             => ( ( finite_finite_b @ J2 )
               => ( indepe8927441866673418605ts_a_b @ m @ F2 @ J2 ) ) ) ) ) ) ).

% indep_sets_finite_index_sets
thf(fact_588_indep__sets__finite__index__sets,axiom,
    ! [F2: a > set_set_a,I: set_a] :
      ( ( indepe8927441866673418604ts_a_a @ m @ F2 @ I )
      = ( ! [J2: set_a] :
            ( ( ord_less_eq_set_a @ J2 @ I )
           => ( ( J2 != bot_bot_set_a )
             => ( ( finite_finite_a @ J2 )
               => ( indepe8927441866673418604ts_a_a @ m @ F2 @ J2 ) ) ) ) ) ) ).

% indep_sets_finite_index_sets
thf(fact_589_indep__sets__finite__index__sets,axiom,
    ! [F2: nat > set_set_a,I: set_nat] :
      ( ( indepe6267730027088848354_a_nat @ m @ F2 @ I )
      = ( ! [J2: set_nat] :
            ( ( ord_less_eq_set_nat @ J2 @ I )
           => ( ( J2 != bot_bot_set_nat )
             => ( ( finite_finite_nat @ J2 )
               => ( indepe6267730027088848354_a_nat @ m @ F2 @ J2 ) ) ) ) ) ) ).

% indep_sets_finite_index_sets
thf(fact_590_indep__sets__finite__index__sets,axiom,
    ! [F2: $o > set_set_a,I: set_o] :
      ( ( indepe7780107833195774214ts_a_o @ m @ F2 @ I )
      = ( ! [J2: set_o] :
            ( ( ord_less_eq_set_o @ J2 @ I )
           => ( ( J2 != bot_bot_set_o )
             => ( ( finite_finite_o @ J2 )
               => ( indepe7780107833195774214ts_a_o @ m @ F2 @ J2 ) ) ) ) ) ) ).

% indep_sets_finite_index_sets
thf(fact_591_compose__const_I2_J,axiom,
    ! [A2: d,G2: a > b] :
      ( ( comp_b_d_a
        @ ^ [X2: b] : A2
        @ G2 )
      = ( ^ [X2: a] : A2 ) ) ).

% compose_const(2)
thf(fact_592_K__record__comp,axiom,
    ! [C2: d,F: a > b] :
      ( ( comp_b_d_a
        @ ^ [X2: b] : C2
        @ F )
      = ( ^ [X2: a] : C2 ) ) ).

% K_record_comp
thf(fact_593_D_OM__in__subprob,axiom,
    member3534519381033026579sure_b @ ( measure_distr_a_b @ m @ n @ f ) @ ( sigma_6149183809546982772sure_b @ ( giry_s2448339315546285521ebra_b @ ( measure_distr_a_b @ m @ n @ f ) ) ) ).

% D.M_in_subprob
thf(fact_594_D_Oprob__space__stream__space,axiom,
    probab6181332906713481800ream_b @ ( stream2930637714550841298pace_b @ ( measure_distr_a_b @ m @ n @ f ) ) ).

% D.prob_space_stream_space
thf(fact_595_D_Odistr__const,axiom,
    ! [C2: c,N2: sigma_measure_c] :
      ( ( member_c @ C2 @ ( sigma_space_c @ N2 ) )
     => ( ( measure_distr_b_c @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [X2: b] : C2 )
        = ( giry_return_c @ N2 @ C2 ) ) ) ).

% D.distr_const
thf(fact_596_D_Odistr__const,axiom,
    ! [C2: a,N2: sigma_measure_a] :
      ( ( member_a @ C2 @ ( sigma_space_a @ N2 ) )
     => ( ( measure_distr_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [X2: b] : C2 )
        = ( giry_return_a @ N2 @ C2 ) ) ) ).

% D.distr_const
thf(fact_597_D_Odistr__const,axiom,
    ! [C2: b,N2: sigma_measure_b] :
      ( ( member_b @ C2 @ ( sigma_space_b @ N2 ) )
     => ( ( measure_distr_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [X2: b] : C2 )
        = ( giry_return_b @ N2 @ C2 ) ) ) ).

% D.distr_const
thf(fact_598_D_Odistr__const,axiom,
    ! [C2: sigma_measure_b,N2: sigma_2733069582767196501sure_b] :
      ( ( member3534519381033026579sure_b @ C2 @ ( sigma_6149183809546982772sure_b @ N2 ) )
     => ( ( measur8960644413728056195sure_b @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [X2: b] : C2 )
        = ( giry_r3784369487821543sure_b @ N2 @ C2 ) ) ) ).

% D.distr_const
thf(fact_599_D_Odistr__const,axiom,
    ! [C2: sigma_measure_a,N2: sigma_2662035542720850516sure_a] :
      ( ( member3534519376729797778sure_a @ C2 @ ( sigma_6149183805243753971sure_a @ N2 ) )
     => ( ( measur8960644409424827394sure_a @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [X2: b] : C2 )
        = ( giry_r3784365184592742sure_a @ N2 @ C2 ) ) ) ).

% D.distr_const
thf(fact_600_D_Odistr__const,axiom,
    ! [C2: a > real,N2: sigma_measure_a_real] :
      ( ( member_a_real @ C2 @ ( sigma_space_a_real @ N2 ) )
     => ( ( measur1784223122078132820a_real @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [X2: b] : C2 )
        = ( giry_return_a_real @ N2 @ C2 ) ) ) ).

% D.distr_const
thf(fact_601_D_Odistr__const,axiom,
    ! [C2: a > b,N2: sigma_measure_a_b] :
      ( ( member_a_b @ C2 @ ( sigma_space_a_b @ N2 ) )
     => ( ( measure_distr_b_a_b @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [X2: b] : C2 )
        = ( giry_return_a_b @ N2 @ C2 ) ) ) ).

% D.distr_const
thf(fact_602_D_Odistr__const,axiom,
    ! [C2: b > a,N2: sigma_measure_b_a] :
      ( ( member_b_a @ C2 @ ( sigma_space_b_a @ N2 ) )
     => ( ( measure_distr_b_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [X2: b] : C2 )
        = ( giry_return_b_a @ N2 @ C2 ) ) ) ).

% D.distr_const
thf(fact_603_D_Odistr__const,axiom,
    ! [C2: a > c,N2: sigma_measure_a_c] :
      ( ( member_a_c @ C2 @ ( sigma_space_a_c @ N2 ) )
     => ( ( measure_distr_b_a_c @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [X2: b] : C2 )
        = ( giry_return_a_c @ N2 @ C2 ) ) ) ).

% D.distr_const
thf(fact_604_D_Odistr__const,axiom,
    ! [C2: a > a,N2: sigma_measure_a_a] :
      ( ( member_a_a @ C2 @ ( sigma_space_a_a @ N2 ) )
     => ( ( measure_distr_b_a_a @ ( measure_distr_a_b @ m @ n @ f ) @ N2
          @ ^ [X2: b] : C2 )
        = ( giry_return_a_a @ N2 @ C2 ) ) ) ).

% D.distr_const
thf(fact_605_prob__space__completion,axiom,
    probab7247484486040049089pace_a @ ( comple3428971583294703880tion_a @ m ) ).

% prob_space_completion
thf(fact_606_indep__var__compose,axiom,
    ! [M1: sigma_measure_b,X1: a > b,M22: sigma_measure_b,X22: a > b,Y1: b > a,N1: sigma_measure_a,Y2: b > a,N22: sigma_measure_a] :
      ( ( indepe2440653194691626189ar_a_b @ m @ M1 @ X1 @ M22 @ X22 )
     => ( ( member_b_a @ Y1 @ ( sigma_measurable_b_a @ M1 @ N1 ) )
       => ( ( member_b_a @ Y2 @ ( sigma_measurable_b_a @ M22 @ N22 ) )
         => ( indepe2440653194691626188ar_a_a @ m @ N1 @ ( comp_b_a_a @ Y1 @ X1 ) @ N22 @ ( comp_b_a_a @ Y2 @ X22 ) ) ) ) ) ).

% indep_var_compose
thf(fact_607_indep__var__compose,axiom,
    ! [M1: sigma_measure_a,X1: a > a,M22: sigma_measure_a,X22: a > a,Y1: a > a > real,N1: sigma_measure_a_real,Y2: a > a > real,N22: sigma_measure_a_real] :
      ( ( indepe2440653194691626188ar_a_a @ m @ M1 @ X1 @ M22 @ X22 )
     => ( ( member_a_a_real @ Y1 @ ( sigma_5073124708296780141a_real @ M1 @ N1 ) )
       => ( ( member_a_a_real @ Y2 @ ( sigma_5073124708296780141a_real @ M22 @ N22 ) )
         => ( indepe7138907624787055209a_real @ m @ N1 @ ( comp_a_a_real_a @ Y1 @ X1 ) @ N22 @ ( comp_a_a_real_a @ Y2 @ X22 ) ) ) ) ) ).

% indep_var_compose
thf(fact_608_indep__var__compose,axiom,
    ! [M1: sigma_measure_a,X1: a > a,M22: sigma_measure_a,X22: a > a,Y1: a > a > b,N1: sigma_measure_a_b,Y2: a > a > b,N22: sigma_measure_a_b] :
      ( ( indepe2440653194691626188ar_a_a @ m @ M1 @ X1 @ M22 @ X22 )
     => ( ( member_a_a_b @ Y1 @ ( sigma_3971135317854480532_a_a_b @ M1 @ N1 ) )
       => ( ( member_a_a_b @ Y2 @ ( sigma_3971135317854480532_a_a_b @ M22 @ N22 ) )
         => ( indepe6909679525661198488_a_a_b @ m @ N1 @ ( comp_a_a_b_a @ Y1 @ X1 ) @ N22 @ ( comp_a_a_b_a @ Y2 @ X22 ) ) ) ) ) ).

% indep_var_compose
thf(fact_609_indep__var__compose,axiom,
    ! [M1: sigma_measure_a,X1: a > a,M22: sigma_measure_a,X22: a > a,Y1: a > c,N1: sigma_measure_c,Y2: a > c,N22: sigma_measure_c] :
      ( ( indepe2440653194691626188ar_a_a @ m @ M1 @ X1 @ M22 @ X22 )
     => ( ( member_a_c @ Y1 @ ( sigma_measurable_a_c @ M1 @ N1 ) )
       => ( ( member_a_c @ Y2 @ ( sigma_measurable_a_c @ M22 @ N22 ) )
         => ( indepe2440653194691626190ar_a_c @ m @ N1 @ ( comp_a_c_a @ Y1 @ X1 ) @ N22 @ ( comp_a_c_a @ Y2 @ X22 ) ) ) ) ) ).

% indep_var_compose
thf(fact_610_indep__var__compose,axiom,
    ! [M1: sigma_measure_a,X1: a > a,M22: sigma_measure_a,X22: a > a,Y1: a > a,N1: sigma_measure_a,Y2: a > a,N22: sigma_measure_a] :
      ( ( indepe2440653194691626188ar_a_a @ m @ M1 @ X1 @ M22 @ X22 )
     => ( ( member_a_a @ Y1 @ ( sigma_measurable_a_a @ M1 @ N1 ) )
       => ( ( member_a_a @ Y2 @ ( sigma_measurable_a_a @ M22 @ N22 ) )
         => ( indepe2440653194691626188ar_a_a @ m @ N1 @ ( comp_a_a_a @ Y1 @ X1 ) @ N22 @ ( comp_a_a_a @ Y2 @ X22 ) ) ) ) ) ).

% indep_var_compose
thf(fact_611_indep__var__compose,axiom,
    ! [M1: sigma_measure_a,X1: a > a,M22: sigma_measure_a,X22: a > a,Y1: a > b,N1: sigma_measure_b,Y2: a > b,N22: sigma_measure_b] :
      ( ( indepe2440653194691626188ar_a_a @ m @ M1 @ X1 @ M22 @ X22 )
     => ( ( member_a_b @ Y1 @ ( sigma_measurable_a_b @ M1 @ N1 ) )
       => ( ( member_a_b @ Y2 @ ( sigma_measurable_a_b @ M22 @ N22 ) )
         => ( indepe2440653194691626189ar_a_b @ m @ N1 @ ( comp_a_b_a @ Y1 @ X1 ) @ N22 @ ( comp_a_b_a @ Y2 @ X22 ) ) ) ) ) ).

% indep_var_compose
thf(fact_612_indep__var__compose,axiom,
    ! [M1: sigma_measure_b,X1: a > b,M22: sigma_measure_b,X22: a > b,Y1: b > d,N1: sigma_measure_d,Y2: b > d,N22: sigma_measure_d] :
      ( ( indepe2440653194691626189ar_a_b @ m @ M1 @ X1 @ M22 @ X22 )
     => ( ( member_b_d @ Y1 @ ( sigma_measurable_b_d @ M1 @ N1 ) )
       => ( ( member_b_d @ Y2 @ ( sigma_measurable_b_d @ M22 @ N22 ) )
         => ( indepe2440653194691626191ar_a_d @ m @ N1 @ ( comp_b_d_a @ Y1 @ X1 ) @ N22 @ ( comp_b_d_a @ Y2 @ X22 ) ) ) ) ) ).

% indep_var_compose
thf(fact_613_indep__var__compose,axiom,
    ! [M1: sigma_measure_b,X1: a > b,M22: sigma_measure_b,X22: a > b,Y1: b > real,N1: sigma_measure_real,Y2: b > real,N22: sigma_measure_real] :
      ( ( indepe2440653194691626189ar_a_b @ m @ M1 @ X1 @ M22 @ X22 )
     => ( ( member_b_real @ Y1 @ ( sigma_5489272243865111003b_real @ M1 @ N1 ) )
       => ( ( member_b_real @ Y2 @ ( sigma_5489272243865111003b_real @ M22 @ N22 ) )
         => ( indepe8958435565499147358a_real @ m @ N1 @ ( comp_b_real_a @ Y1 @ X1 ) @ N22 @ ( comp_b_real_a @ Y2 @ X22 ) ) ) ) ) ).

% indep_var_compose
thf(fact_614_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,Y2: 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 @ Y2 @ ( sigma_9116425665531756122a_real @ M22 @ N22 ) )
         => ( indepe8958435565499147358a_real @ m @ N1 @ ( comp_a_real_a @ Y1 @ X1 ) @ N22 @ ( comp_a_real_a @ Y2 @ X22 ) ) ) ) ) ).

% indep_var_compose
thf(fact_615_indep__vars__compose,axiom,
    ! [M: c > sigma_measure_b,X4: c > a > b,I: set_c,Y: c > b > a,N2: c > sigma_measure_a] :
      ( ( indepe4851901773132442933_a_c_b @ m @ M @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe4851901773132442932_a_c_a @ m @ N2
          @ ^ [I3: c] : ( comp_b_a_a @ ( Y @ I3 ) @ ( X4 @ I3 ) )
          @ I ) ) ) ).

% indep_vars_compose
thf(fact_616_indep__vars__compose,axiom,
    ! [M: c > sigma_measure_a,X4: c > a > a,I: set_c,Y: c > a > c,N2: c > sigma_measure_c] :
      ( ( indepe4851901773132442932_a_c_a @ m @ M @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe4851901773132442934_a_c_c @ m @ N2
          @ ^ [I3: c] : ( comp_a_c_a @ ( Y @ I3 ) @ ( X4 @ I3 ) )
          @ I ) ) ) ).

% indep_vars_compose
thf(fact_617_indep__vars__compose,axiom,
    ! [M: c > sigma_measure_a,X4: c > a > a,I: set_c,Y: c > a > a,N2: c > sigma_measure_a] :
      ( ( indepe4851901773132442932_a_c_a @ m @ M @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe4851901773132442932_a_c_a @ m @ N2
          @ ^ [I3: c] : ( comp_a_a_a @ ( Y @ I3 ) @ ( X4 @ I3 ) )
          @ I ) ) ) ).

% indep_vars_compose
thf(fact_618_indep__vars__compose,axiom,
    ! [M: b > sigma_measure_b,X4: b > a > b,I: set_b,Y: b > b > a,N2: b > sigma_measure_a] :
      ( ( indepe7639357355105118966_a_b_b @ m @ M @ X4 @ I )
     => ( ! [I2: b] :
            ( ( member_b @ I2 @ I )
           => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe7639357355105118965_a_b_a @ m @ N2
          @ ^ [I3: b] : ( comp_b_a_a @ ( Y @ I3 ) @ ( X4 @ I3 ) )
          @ I ) ) ) ).

% indep_vars_compose
thf(fact_619_indep__vars__compose,axiom,
    ! [M: b > sigma_measure_a,X4: b > a > a,I: set_b,Y: b > a > c,N2: b > sigma_measure_c] :
      ( ( indepe7639357355105118965_a_b_a @ m @ M @ X4 @ I )
     => ( ! [I2: b] :
            ( ( member_b @ I2 @ I )
           => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe7639357355105118967_a_b_c @ m @ N2
          @ ^ [I3: b] : ( comp_a_c_a @ ( Y @ I3 ) @ ( X4 @ I3 ) )
          @ I ) ) ) ).

% indep_vars_compose
thf(fact_620_indep__vars__compose,axiom,
    ! [M: b > sigma_measure_a,X4: b > a > a,I: set_b,Y: b > a > a,N2: b > sigma_measure_a] :
      ( ( indepe7639357355105118965_a_b_a @ m @ M @ X4 @ I )
     => ( ! [I2: b] :
            ( ( member_b @ I2 @ I )
           => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe7639357355105118965_a_b_a @ m @ N2
          @ ^ [I3: b] : ( comp_a_a_a @ ( Y @ I3 ) @ ( X4 @ I3 ) )
          @ I ) ) ) ).

% indep_vars_compose
thf(fact_621_indep__vars__compose,axiom,
    ! [M: a > sigma_measure_b,X4: a > a > b,I: set_a,Y: a > b > a,N2: a > sigma_measure_a] :
      ( ( indepe1203440900223019191_a_a_b @ m @ M @ X4 @ I )
     => ( ! [I2: a] :
            ( ( member_a @ I2 @ I )
           => ( member_b_a @ ( Y @ I2 ) @ ( sigma_measurable_b_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe1203440900223019190_a_a_a @ m @ N2
          @ ^ [I3: a] : ( comp_b_a_a @ ( Y @ I3 ) @ ( X4 @ I3 ) )
          @ I ) ) ) ).

% indep_vars_compose
thf(fact_622_indep__vars__compose,axiom,
    ! [M: a > sigma_measure_a,X4: a > a > a,I: set_a,Y: a > a > c,N2: a > sigma_measure_c] :
      ( ( indepe1203440900223019190_a_a_a @ m @ M @ X4 @ I )
     => ( ! [I2: a] :
            ( ( member_a @ I2 @ I )
           => ( member_a_c @ ( Y @ I2 ) @ ( sigma_measurable_a_c @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe1203440900223019192_a_a_c @ m @ N2
          @ ^ [I3: a] : ( comp_a_c_a @ ( Y @ I3 ) @ ( X4 @ I3 ) )
          @ I ) ) ) ).

% indep_vars_compose
thf(fact_623_indep__vars__compose,axiom,
    ! [M: a > sigma_measure_a,X4: a > a > a,I: set_a,Y: a > a > a,N2: a > sigma_measure_a] :
      ( ( indepe1203440900223019190_a_a_a @ m @ M @ X4 @ I )
     => ( ! [I2: a] :
            ( ( member_a @ I2 @ I )
           => ( member_a_a @ ( Y @ I2 ) @ ( sigma_measurable_a_a @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe1203440900223019190_a_a_a @ m @ N2
          @ ^ [I3: a] : ( comp_a_a_a @ ( Y @ I3 ) @ ( X4 @ I3 ) )
          @ I ) ) ) ).

% indep_vars_compose
thf(fact_624_indep__vars__compose,axiom,
    ! [M: c > sigma_measure_a,X4: c > a > a,I: set_c,Y: c > a > b,N2: c > sigma_measure_b] :
      ( ( indepe4851901773132442932_a_c_a @ m @ M @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_a_b @ ( Y @ I2 ) @ ( sigma_measurable_a_b @ ( M @ I2 ) @ ( N2 @ I2 ) ) ) )
       => ( indepe4851901773132442933_a_c_b @ m @ N2
          @ ^ [I3: c] : ( comp_a_b_a @ ( Y @ I3 ) @ ( X4 @ I3 ) )
          @ I ) ) ) ).

% indep_vars_compose
thf(fact_625_subset__antisym,axiom,
    ! [A: set_set_b,B: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A @ B )
     => ( ( ord_le3795704787696855135_set_b @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_626_subset__antisym,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B )
     => ( ( ord_le3724670747650509150_set_a @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_627_subset__antisym,axiom,
    ! [A: set_b,B: set_b] :
      ( ( ord_less_eq_set_b @ A @ B )
     => ( ( ord_less_eq_set_b @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_628_subset__antisym,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( ord_less_eq_set_a @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_629_subsetI,axiom,
    ! [A: set_c,B: set_c] :
      ( ! [X: c] :
          ( ( member_c @ X @ A )
         => ( member_c @ X @ B ) )
     => ( ord_less_eq_set_c @ A @ B ) ) ).

% subsetI
thf(fact_630_subsetI,axiom,
    ! [A: set_b,B: set_b] :
      ( ! [X: b] :
          ( ( member_b @ X @ A )
         => ( member_b @ X @ B ) )
     => ( ord_less_eq_set_b @ A @ B ) ) ).

% subsetI
thf(fact_631_subsetI,axiom,
    ! [A: set_a,B: set_a] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( member_a @ X @ B ) )
     => ( ord_less_eq_set_a @ A @ B ) ) ).

% subsetI
thf(fact_632_subsetI,axiom,
    ! [A: set_set_b,B: set_set_b] :
      ( ! [X: set_b] :
          ( ( member_set_b @ X @ A )
         => ( member_set_b @ X @ B ) )
     => ( ord_le3795704787696855135_set_b @ A @ B ) ) ).

% subsetI
thf(fact_633_subsetI,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ! [X: set_a] :
          ( ( member_set_a @ X @ A )
         => ( member_set_a @ X @ B ) )
     => ( ord_le3724670747650509150_set_a @ A @ B ) ) ).

% subsetI
thf(fact_634_subsetI,axiom,
    ! [A: set_a_real,B: set_a_real] :
      ( ! [X: a > real] :
          ( ( member_a_real @ X @ A )
         => ( member_a_real @ X @ B ) )
     => ( ord_le3334967407727675675a_real @ A @ B ) ) ).

% subsetI
thf(fact_635_subsetI,axiom,
    ! [A: set_a_b,B: set_a_b] :
      ( ! [X: a > b] :
          ( ( member_a_b @ X @ A )
         => ( member_a_b @ X @ B ) )
     => ( ord_less_eq_set_a_b @ A @ B ) ) ).

% subsetI
thf(fact_636_subsetI,axiom,
    ! [A: set_b_a,B: set_b_a] :
      ( ! [X: b > a] :
          ( ( member_b_a @ X @ A )
         => ( member_b_a @ X @ B ) )
     => ( ord_less_eq_set_b_a @ A @ B ) ) ).

% subsetI
thf(fact_637_subsetI,axiom,
    ! [A: set_a_c,B: set_a_c] :
      ( ! [X: a > c] :
          ( ( member_a_c @ X @ A )
         => ( member_a_c @ X @ B ) )
     => ( ord_less_eq_set_a_c @ A @ B ) ) ).

% subsetI
thf(fact_638_subsetI,axiom,
    ! [A: set_a_a,B: set_a_a] :
      ( ! [X: a > a] :
          ( ( member_a_a @ X @ A )
         => ( member_a_a @ X @ B ) )
     => ( ord_less_eq_set_a_a @ A @ B ) ) ).

% subsetI
thf(fact_639_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X2: nat] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_640_empty__Collect__eq,axiom,
    ! [P: a > $o] :
      ( ( bot_bot_set_a
        = ( collect_a @ P ) )
      = ( ! [X2: a] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_641_empty__Collect__eq,axiom,
    ! [P: b > $o] :
      ( ( bot_bot_set_b
        = ( collect_b @ P ) )
      = ( ! [X2: b] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_642_empty__Collect__eq,axiom,
    ! [P: $o > $o] :
      ( ( bot_bot_set_o
        = ( collect_o @ P ) )
      = ( ! [X2: $o] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_643_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X2: nat] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_644_Collect__empty__eq,axiom,
    ! [P: a > $o] :
      ( ( ( collect_a @ P )
        = bot_bot_set_a )
      = ( ! [X2: a] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_645_Collect__empty__eq,axiom,
    ! [P: b > $o] :
      ( ( ( collect_b @ P )
        = bot_bot_set_b )
      = ( ! [X2: b] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_646_Collect__empty__eq,axiom,
    ! [P: $o > $o] :
      ( ( ( collect_o @ P )
        = bot_bot_set_o )
      = ( ! [X2: $o] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_647_all__not__in__conv,axiom,
    ! [A: set_c] :
      ( ( ! [X2: c] :
            ~ ( member_c @ X2 @ A ) )
      = ( A = bot_bot_set_c ) ) ).

% all_not_in_conv
thf(fact_648_all__not__in__conv,axiom,
    ! [A: set_a] :
      ( ( ! [X2: a] :
            ~ ( member_a @ X2 @ A ) )
      = ( A = bot_bot_set_a ) ) ).

% all_not_in_conv
thf(fact_649_all__not__in__conv,axiom,
    ! [A: set_b] :
      ( ( ! [X2: b] :
            ~ ( member_b @ X2 @ A ) )
      = ( A = bot_bot_set_b ) ) ).

% all_not_in_conv
thf(fact_650_all__not__in__conv,axiom,
    ! [A: set_o] :
      ( ( ! [X2: $o] :
            ~ ( member_o @ X2 @ A ) )
      = ( A = bot_bot_set_o ) ) ).

% all_not_in_conv
thf(fact_651_all__not__in__conv,axiom,
    ! [A: set_a_real] :
      ( ( ! [X2: a > real] :
            ~ ( member_a_real @ X2 @ A ) )
      = ( A = bot_bot_set_a_real ) ) ).

% all_not_in_conv
thf(fact_652_all__not__in__conv,axiom,
    ! [A: set_a_b] :
      ( ( ! [X2: a > b] :
            ~ ( member_a_b @ X2 @ A ) )
      = ( A = bot_bot_set_a_b ) ) ).

% all_not_in_conv
thf(fact_653_all__not__in__conv,axiom,
    ! [A: set_b_a] :
      ( ( ! [X2: b > a] :
            ~ ( member_b_a @ X2 @ A ) )
      = ( A = bot_bot_set_b_a ) ) ).

% all_not_in_conv
thf(fact_654_all__not__in__conv,axiom,
    ! [A: set_a_c] :
      ( ( ! [X2: a > c] :
            ~ ( member_a_c @ X2 @ A ) )
      = ( A = bot_bot_set_a_c ) ) ).

% all_not_in_conv
thf(fact_655_all__not__in__conv,axiom,
    ! [A: set_a_a] :
      ( ( ! [X2: a > a] :
            ~ ( member_a_a @ X2 @ A ) )
      = ( A = bot_bot_set_a_a ) ) ).

% all_not_in_conv
thf(fact_656_all__not__in__conv,axiom,
    ! [A: set_a_a_real] :
      ( ( ! [X2: a > a > real] :
            ~ ( member_a_a_real @ X2 @ A ) )
      = ( A = bot_bot_set_a_a_real ) ) ).

% all_not_in_conv
thf(fact_657_empty__iff,axiom,
    ! [C2: c] :
      ~ ( member_c @ C2 @ bot_bot_set_c ) ).

% empty_iff
thf(fact_658_empty__iff,axiom,
    ! [C2: a] :
      ~ ( member_a @ C2 @ bot_bot_set_a ) ).

% empty_iff
thf(fact_659_empty__iff,axiom,
    ! [C2: b] :
      ~ ( member_b @ C2 @ bot_bot_set_b ) ).

% empty_iff
thf(fact_660_empty__iff,axiom,
    ! [C2: $o] :
      ~ ( member_o @ C2 @ bot_bot_set_o ) ).

% empty_iff
thf(fact_661_empty__iff,axiom,
    ! [C2: a > real] :
      ~ ( member_a_real @ C2 @ bot_bot_set_a_real ) ).

% empty_iff
thf(fact_662_empty__iff,axiom,
    ! [C2: a > b] :
      ~ ( member_a_b @ C2 @ bot_bot_set_a_b ) ).

% empty_iff
thf(fact_663_empty__iff,axiom,
    ! [C2: b > a] :
      ~ ( member_b_a @ C2 @ bot_bot_set_b_a ) ).

% empty_iff
thf(fact_664_empty__iff,axiom,
    ! [C2: a > c] :
      ~ ( member_a_c @ C2 @ bot_bot_set_a_c ) ).

% empty_iff
thf(fact_665_empty__iff,axiom,
    ! [C2: a > a] :
      ~ ( member_a_a @ C2 @ bot_bot_set_a_a ) ).

% empty_iff
thf(fact_666_empty__iff,axiom,
    ! [C2: a > a > real] :
      ~ ( member_a_a_real @ C2 @ bot_bot_set_a_a_real ) ).

% empty_iff
thf(fact_667_measurable__return2,axiom,
    ! [L: sigma_measure_a,N2: sigma_measure_b,X3: b] :
      ( ( sigma_measurable_a_b @ L @ ( giry_return_b @ N2 @ X3 ) )
      = ( sigma_measurable_a_b @ L @ N2 ) ) ).

% measurable_return2
thf(fact_668_measurable__return2,axiom,
    ! [L: sigma_measure_b,N2: sigma_measure_d,X3: d] :
      ( ( sigma_measurable_b_d @ L @ ( giry_return_d @ N2 @ X3 ) )
      = ( sigma_measurable_b_d @ L @ N2 ) ) ).

% measurable_return2
thf(fact_669_measurable__return2,axiom,
    ! [L: sigma_measure_b,N2: sigma_measure_real,X3: real] :
      ( ( sigma_5489272243865111003b_real @ L @ ( giry_return_real @ N2 @ X3 ) )
      = ( sigma_5489272243865111003b_real @ L @ N2 ) ) ).

% measurable_return2
thf(fact_670_measurable__return2,axiom,
    ! [L: sigma_measure_a,N2: sigma_measure_real,X3: real] :
      ( ( sigma_9116425665531756122a_real @ L @ ( giry_return_real @ N2 @ X3 ) )
      = ( sigma_9116425665531756122a_real @ L @ N2 ) ) ).

% measurable_return2
thf(fact_671_measurable__return1,axiom,
    ! [N2: sigma_measure_a,X3: a,L: sigma_measure_b] :
      ( ( sigma_measurable_a_b @ ( giry_return_a @ N2 @ X3 ) @ L )
      = ( sigma_measurable_a_b @ N2 @ L ) ) ).

% measurable_return1
thf(fact_672_measurable__return1,axiom,
    ! [N2: sigma_measure_b,X3: b,L: sigma_measure_d] :
      ( ( sigma_measurable_b_d @ ( giry_return_b @ N2 @ X3 ) @ L )
      = ( sigma_measurable_b_d @ N2 @ L ) ) ).

% measurable_return1
thf(fact_673_measurable__return1,axiom,
    ! [N2: sigma_measure_b,X3: b,L: sigma_measure_real] :
      ( ( sigma_5489272243865111003b_real @ ( giry_return_b @ N2 @ X3 ) @ L )
      = ( sigma_5489272243865111003b_real @ N2 @ L ) ) ).

% measurable_return1
thf(fact_674_measurable__return1,axiom,
    ! [N2: sigma_measure_a,X3: a,L: sigma_measure_real] :
      ( ( sigma_9116425665531756122a_real @ ( giry_return_a @ N2 @ X3 ) @ L )
      = ( sigma_9116425665531756122a_real @ N2 @ L ) ) ).

% measurable_return1
thf(fact_675_space__return,axiom,
    ! [M2: sigma_measure_a,X3: a] :
      ( ( sigma_space_a @ ( giry_return_a @ M2 @ X3 ) )
      = ( sigma_space_a @ M2 ) ) ).

% space_return
thf(fact_676_space__return,axiom,
    ! [M2: sigma_measure_b,X3: b] :
      ( ( sigma_space_b @ ( giry_return_b @ M2 @ X3 ) )
      = ( sigma_space_b @ M2 ) ) ).

% space_return
thf(fact_677_space__return,axiom,
    ! [M2: sigma_2733069582767196501sure_b,X3: sigma_measure_b] :
      ( ( sigma_6149183809546982772sure_b @ ( giry_r3784369487821543sure_b @ M2 @ X3 ) )
      = ( sigma_6149183809546982772sure_b @ M2 ) ) ).

% space_return
thf(fact_678_space__return,axiom,
    ! [M2: sigma_2662035542720850516sure_a,X3: sigma_measure_a] :
      ( ( sigma_6149183805243753971sure_a @ ( giry_r3784365184592742sure_a @ M2 @ X3 ) )
      = ( sigma_6149183805243753971sure_a @ M2 ) ) ).

% space_return
thf(fact_679_distr__const,axiom,
    ! [C2: c,N2: sigma_measure_c] :
      ( ( member_c @ C2 @ ( sigma_space_c @ N2 ) )
     => ( ( measure_distr_a_c @ m @ N2
          @ ^ [X2: a] : C2 )
        = ( giry_return_c @ N2 @ C2 ) ) ) ).

% distr_const
thf(fact_680_distr__const,axiom,
    ! [C2: a,N2: sigma_measure_a] :
      ( ( member_a @ C2 @ ( sigma_space_a @ N2 ) )
     => ( ( measure_distr_a_a @ m @ N2
          @ ^ [X2: a] : C2 )
        = ( giry_return_a @ N2 @ C2 ) ) ) ).

% distr_const
thf(fact_681_distr__const,axiom,
    ! [C2: b,N2: sigma_measure_b] :
      ( ( member_b @ C2 @ ( sigma_space_b @ N2 ) )
     => ( ( measure_distr_a_b @ m @ N2
          @ ^ [X2: a] : C2 )
        = ( giry_return_b @ N2 @ C2 ) ) ) ).

% distr_const
thf(fact_682_distr__const,axiom,
    ! [C2: sigma_measure_b,N2: sigma_2733069582767196501sure_b] :
      ( ( member3534519381033026579sure_b @ C2 @ ( sigma_6149183809546982772sure_b @ N2 ) )
     => ( ( measur2827329755067293186sure_b @ m @ N2
          @ ^ [X2: a] : C2 )
        = ( giry_r3784369487821543sure_b @ N2 @ C2 ) ) ) ).

% distr_const
thf(fact_683_distr__const,axiom,
    ! [C2: sigma_measure_a,N2: sigma_2662035542720850516sure_a] :
      ( ( member3534519376729797778sure_a @ C2 @ ( sigma_6149183805243753971sure_a @ N2 ) )
     => ( ( measur2827329750764064385sure_a @ m @ N2
          @ ^ [X2: a] : C2 )
        = ( giry_r3784365184592742sure_a @ N2 @ C2 ) ) ) ).

% distr_const
thf(fact_684_distr__const,axiom,
    ! [C2: a > real,N2: sigma_measure_a_real] :
      ( ( member_a_real @ C2 @ ( sigma_space_a_real @ N2 ) )
     => ( ( measur6806378183764566549a_real @ m @ N2
          @ ^ [X2: a] : C2 )
        = ( giry_return_a_real @ N2 @ C2 ) ) ) ).

% distr_const
thf(fact_685_distr__const,axiom,
    ! [C2: a > b,N2: sigma_measure_a_b] :
      ( ( member_a_b @ C2 @ ( sigma_space_a_b @ N2 ) )
     => ( ( measure_distr_a_a_b @ m @ N2
          @ ^ [X2: a] : C2 )
        = ( giry_return_a_b @ N2 @ C2 ) ) ) ).

% distr_const
thf(fact_686_distr__const,axiom,
    ! [C2: b > a,N2: sigma_measure_b_a] :
      ( ( member_b_a @ C2 @ ( sigma_space_b_a @ N2 ) )
     => ( ( measure_distr_a_b_a @ m @ N2
          @ ^ [X2: a] : C2 )
        = ( giry_return_b_a @ N2 @ C2 ) ) ) ).

% distr_const
thf(fact_687_distr__const,axiom,
    ! [C2: a > c,N2: sigma_measure_a_c] :
      ( ( member_a_c @ C2 @ ( sigma_space_a_c @ N2 ) )
     => ( ( measure_distr_a_a_c @ m @ N2
          @ ^ [X2: a] : C2 )
        = ( giry_return_a_c @ N2 @ C2 ) ) ) ).

% distr_const
thf(fact_688_distr__const,axiom,
    ! [C2: a > a,N2: sigma_measure_a_a] :
      ( ( member_a_a @ C2 @ ( sigma_space_a_a @ N2 ) )
     => ( ( measure_distr_a_a_a @ m @ N2
          @ ^ [X2: a] : C2 )
        = ( giry_return_a_a @ N2 @ C2 ) ) ) ).

% distr_const
thf(fact_689_return__measurable,axiom,
    ! [N2: sigma_measure_b] : ( member7751282110383867797sure_b @ ( giry_return_b @ N2 ) @ ( sigma_6831777305601499691sure_b @ N2 @ ( giry_s2448339315546285521ebra_b @ N2 ) ) ) ).

% return_measurable
thf(fact_690_return__measurable,axiom,
    ! [N2: sigma_measure_a] : ( member1546933411676758803sure_a @ ( giry_return_a @ N2 ) @ ( sigma_698462642637507881sure_a @ N2 @ ( giry_s2448339315546285520ebra_a @ N2 ) ) ) ).

% return_measurable
thf(fact_691_subprob__measurableD_I1_J,axiom,
    ! [N2: c > sigma_measure_b,M2: sigma_measure_c,S: sigma_measure_b,X3: c] :
      ( ( member4661224732189854998sure_b @ N2 @ ( sigma_3741719927407486892sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ S ) ) )
     => ( ( member_c @ X3 @ ( sigma_space_c @ M2 ) )
       => ( ( sigma_space_b @ ( N2 @ X3 ) )
          = ( sigma_space_b @ S ) ) ) ) ).

% subprob_measurableD(1)
thf(fact_692_subprob__measurableD_I1_J,axiom,
    ! [N2: a > sigma_measure_b,M2: sigma_measure_a,S: sigma_measure_b,X3: a] :
      ( ( member1617967451723104788sure_b @ N2 @ ( sigma_698462646940736682sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ S ) ) )
     => ( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
       => ( ( sigma_space_b @ ( N2 @ X3 ) )
          = ( sigma_space_b @ S ) ) ) ) ).

% subprob_measurableD(1)
thf(fact_693_subprob__measurableD_I1_J,axiom,
    ! [N2: b > sigma_measure_b,M2: sigma_measure_b,S: sigma_measure_b,X3: b] :
      ( ( member7751282110383867797sure_b @ N2 @ ( sigma_6831777305601499691sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ S ) ) )
     => ( ( member_b @ X3 @ ( sigma_space_b @ M2 ) )
       => ( ( sigma_space_b @ ( N2 @ X3 ) )
          = ( sigma_space_b @ S ) ) ) ) ).

% subprob_measurableD(1)
thf(fact_694_subprob__measurableD_I1_J,axiom,
    ! [N2: c > sigma_measure_a,M2: sigma_measure_c,S: sigma_measure_a,X3: c] :
      ( ( member4590190692143509013sure_a @ N2 @ ( sigma_3741719923104258091sure_a @ M2 @ ( giry_s2448339315546285520ebra_a @ S ) ) )
     => ( ( member_c @ X3 @ ( sigma_space_c @ M2 ) )
       => ( ( sigma_space_a @ ( N2 @ X3 ) )
          = ( sigma_space_a @ S ) ) ) ) ).

% subprob_measurableD(1)
thf(fact_695_subprob__measurableD_I1_J,axiom,
    ! [N2: a > sigma_measure_a,M2: sigma_measure_a,S: sigma_measure_a,X3: a] :
      ( ( member1546933411676758803sure_a @ N2 @ ( sigma_698462642637507881sure_a @ M2 @ ( giry_s2448339315546285520ebra_a @ S ) ) )
     => ( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
       => ( ( sigma_space_a @ ( N2 @ X3 ) )
          = ( sigma_space_a @ S ) ) ) ) ).

% subprob_measurableD(1)
thf(fact_696_subprob__measurableD_I1_J,axiom,
    ! [N2: b > sigma_measure_a,M2: sigma_measure_b,S: sigma_measure_a,X3: b] :
      ( ( member7680248070337521812sure_a @ N2 @ ( sigma_6831777301298270890sure_a @ M2 @ ( giry_s2448339315546285520ebra_a @ S ) ) )
     => ( ( member_b @ X3 @ ( sigma_space_b @ M2 ) )
       => ( ( sigma_space_a @ ( N2 @ X3 ) )
          = ( sigma_space_a @ S ) ) ) ) ).

% subprob_measurableD(1)
thf(fact_697_subprob__measurableD_I1_J,axiom,
    ! [N2: c > sigma_2733069582767196501sure_b,M2: sigma_measure_c,S: sigma_2733069582767196501sure_b,X3: c] :
      ( ( member3855116451968618449sure_b @ N2 @ ( sigma_7656852915902150309sure_b @ M2 @ ( giry_s4089474293088545450sure_b @ S ) ) )
     => ( ( member_c @ X3 @ ( sigma_space_c @ M2 ) )
       => ( ( sigma_6149183809546982772sure_b @ ( N2 @ X3 ) )
          = ( sigma_6149183809546982772sure_b @ S ) ) ) ) ).

% subprob_measurableD(1)
thf(fact_698_subprob__measurableD_I1_J,axiom,
    ! [N2: c > sigma_2662035542720850516sure_a,M2: sigma_measure_c,S: sigma_2662035542720850516sure_a,X3: c] :
      ( ( member6376300970257654096sure_a @ N2 @ ( sigma_7585818875855804324sure_a @ M2 @ ( giry_s4089474288785316649sure_a @ S ) ) )
     => ( ( member_c @ X3 @ ( sigma_space_c @ M2 ) )
       => ( ( sigma_6149183805243753971sure_a @ ( N2 @ X3 ) )
          = ( sigma_6149183805243753971sure_a @ S ) ) ) ) ).

% subprob_measurableD(1)
thf(fact_699_subprob__measurableD_I1_J,axiom,
    ! [N2: a > sigma_2733069582767196501sure_b,M2: sigma_measure_a,S: sigma_2733069582767196501sure_b,X3: a] :
      ( ( member1486180851757042259sure_b @ N2 @ ( sigma_5287917315690574119sure_b @ M2 @ ( giry_s4089474293088545450sure_b @ S ) ) )
     => ( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
       => ( ( sigma_6149183809546982772sure_b @ ( N2 @ X3 ) )
          = ( sigma_6149183809546982772sure_b @ S ) ) ) ) ).

% subprob_measurableD(1)
thf(fact_700_subprob__measurableD_I1_J,axiom,
    ! [N2: a > sigma_2662035542720850516sure_a,M2: sigma_measure_a,S: sigma_2662035542720850516sure_a,X3: a] :
      ( ( member4007365370046077906sure_a @ N2 @ ( sigma_5216883275644228134sure_a @ M2 @ ( giry_s4089474288785316649sure_a @ S ) ) )
     => ( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
       => ( ( sigma_6149183805243753971sure_a @ ( N2 @ X3 ) )
          = ( sigma_6149183805243753971sure_a @ S ) ) ) ) ).

% subprob_measurableD(1)
thf(fact_701_prob__space_OM__in__subprob,axiom,
    ! [M2: sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( member3534519376729797778sure_a @ M2 @ ( sigma_6149183805243753971sure_a @ ( giry_s2448339315546285520ebra_a @ M2 ) ) ) ) ).

% prob_space.M_in_subprob
thf(fact_702_prob__space_OM__in__subprob,axiom,
    ! [M2: sigma_measure_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( member3534519381033026579sure_b @ M2 @ ( sigma_6149183809546982772sure_b @ ( giry_s2448339315546285521ebra_b @ M2 ) ) ) ) ).

% prob_space.M_in_subprob
thf(fact_703_prob__space_OM__in__subprob,axiom,
    ! [M2: sigma_6673148739008212994ream_b] :
      ( ( probab6181332906713481800ream_b @ M2 )
     => ( member3483318345666026649ream_b @ M2 @ ( sigma_5419305464508041338ream_b @ ( giry_s8751189755406667735ream_b @ M2 ) ) ) ) ).

% prob_space.M_in_subprob
thf(fact_704_prob__space_OM__in__subprob,axiom,
    ! [M2: sigma_6602114698961867009ream_a] :
      ( ( probab6181332902410252999ream_a @ M2 )
     => ( member3412284305619680664ream_a @ M2 @ ( sigma_5348271424461695353ream_a @ ( giry_s8751189751103438934ream_a @ M2 ) ) ) ) ).

% prob_space.M_in_subprob
thf(fact_705_space__subprob__algebra__empty__iff,axiom,
    ! [N2: sigma_2733069582767196501sure_b] :
      ( ( ( sigma_6796319551096797917sure_b @ ( giry_s4089474293088545450sure_b @ N2 ) )
        = bot_bo1385385737013618977sure_b )
      = ( ( sigma_6149183809546982772sure_b @ N2 )
        = bot_bo5242124273094418374sure_b ) ) ).

% space_subprob_algebra_empty_iff
thf(fact_706_space__subprob__algebra__empty__iff,axiom,
    ! [N2: sigma_2662035542720850516sure_a] :
      ( ( ( sigma_6725285511050451932sure_a @ ( giry_s4089474288785316649sure_a @ N2 ) )
        = bot_bo3906570255302654624sure_a )
      = ( ( sigma_6149183805243753971sure_a @ N2 )
        = bot_bo5171090233048072389sure_a ) ) ).

% space_subprob_algebra_empty_iff
thf(fact_707_space__subprob__algebra__empty__iff,axiom,
    ! [N2: sigma_measure_o] :
      ( ( ( sigma_8736833305763468941sure_o @ ( giry_s3549050072915289962ebra_o @ N2 ) )
        = bot_bo7838039659004643295sure_o )
      = ( ( sigma_space_o @ N2 )
        = bot_bot_set_o ) ) ).

% space_subprob_algebra_empty_iff
thf(fact_708_space__subprob__algebra__empty__iff,axiom,
    ! [N2: sigma_measure_b] :
      ( ( ( sigma_6149183809546982772sure_b @ ( giry_s2448339315546285521ebra_b @ N2 ) )
        = bot_bo5242124273094418374sure_b )
      = ( ( sigma_space_b @ N2 )
        = bot_bot_set_b ) ) ).

% space_subprob_algebra_empty_iff
thf(fact_709_space__subprob__algebra__empty__iff,axiom,
    ! [N2: sigma_measure_a] :
      ( ( ( sigma_6149183805243753971sure_a @ ( giry_s2448339315546285520ebra_a @ N2 ) )
        = bot_bo5171090233048072389sure_a )
      = ( ( sigma_space_a @ N2 )
        = bot_bot_set_a ) ) ).

% space_subprob_algebra_empty_iff
thf(fact_710_subprob__measurableD_I3_J,axiom,
    ! [N2: c > sigma_measure_b,M2: sigma_measure_c,S: sigma_measure_b,X3: c,K2: sigma_measure_d] :
      ( ( member4661224732189854998sure_b @ N2 @ ( sigma_3741719927407486892sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ S ) ) )
     => ( ( member_c @ X3 @ ( sigma_space_c @ M2 ) )
       => ( ( sigma_measurable_b_d @ ( N2 @ X3 ) @ K2 )
          = ( sigma_measurable_b_d @ S @ K2 ) ) ) ) ).

% subprob_measurableD(3)
thf(fact_711_subprob__measurableD_I3_J,axiom,
    ! [N2: a > sigma_measure_b,M2: sigma_measure_a,S: sigma_measure_b,X3: a,K2: sigma_measure_d] :
      ( ( member1617967451723104788sure_b @ N2 @ ( sigma_698462646940736682sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ S ) ) )
     => ( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
       => ( ( sigma_measurable_b_d @ ( N2 @ X3 ) @ K2 )
          = ( sigma_measurable_b_d @ S @ K2 ) ) ) ) ).

% subprob_measurableD(3)
thf(fact_712_subprob__measurableD_I3_J,axiom,
    ! [N2: b > sigma_measure_b,M2: sigma_measure_b,S: sigma_measure_b,X3: b,K2: sigma_measure_d] :
      ( ( member7751282110383867797sure_b @ N2 @ ( sigma_6831777305601499691sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ S ) ) )
     => ( ( member_b @ X3 @ ( sigma_space_b @ M2 ) )
       => ( ( sigma_measurable_b_d @ ( N2 @ X3 ) @ K2 )
          = ( sigma_measurable_b_d @ S @ K2 ) ) ) ) ).

% subprob_measurableD(3)
thf(fact_713_subprob__measurableD_I3_J,axiom,
    ! [N2: c > sigma_measure_b,M2: sigma_measure_c,S: sigma_measure_b,X3: c,K2: sigma_measure_real] :
      ( ( member4661224732189854998sure_b @ N2 @ ( sigma_3741719927407486892sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ S ) ) )
     => ( ( member_c @ X3 @ ( sigma_space_c @ M2 ) )
       => ( ( sigma_5489272243865111003b_real @ ( N2 @ X3 ) @ K2 )
          = ( sigma_5489272243865111003b_real @ S @ K2 ) ) ) ) ).

% subprob_measurableD(3)
thf(fact_714_subprob__measurableD_I3_J,axiom,
    ! [N2: a > sigma_measure_b,M2: sigma_measure_a,S: sigma_measure_b,X3: a,K2: sigma_measure_real] :
      ( ( member1617967451723104788sure_b @ N2 @ ( sigma_698462646940736682sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ S ) ) )
     => ( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
       => ( ( sigma_5489272243865111003b_real @ ( N2 @ X3 ) @ K2 )
          = ( sigma_5489272243865111003b_real @ S @ K2 ) ) ) ) ).

% subprob_measurableD(3)
thf(fact_715_subprob__measurableD_I3_J,axiom,
    ! [N2: b > sigma_measure_b,M2: sigma_measure_b,S: sigma_measure_b,X3: b,K2: sigma_measure_real] :
      ( ( member7751282110383867797sure_b @ N2 @ ( sigma_6831777305601499691sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ S ) ) )
     => ( ( member_b @ X3 @ ( sigma_space_b @ M2 ) )
       => ( ( sigma_5489272243865111003b_real @ ( N2 @ X3 ) @ K2 )
          = ( sigma_5489272243865111003b_real @ S @ K2 ) ) ) ) ).

% subprob_measurableD(3)
thf(fact_716_subprob__measurableD_I3_J,axiom,
    ! [N2: c > sigma_measure_a,M2: sigma_measure_c,S: sigma_measure_a,X3: c,K2: sigma_measure_b] :
      ( ( member4590190692143509013sure_a @ N2 @ ( sigma_3741719923104258091sure_a @ M2 @ ( giry_s2448339315546285520ebra_a @ S ) ) )
     => ( ( member_c @ X3 @ ( sigma_space_c @ M2 ) )
       => ( ( sigma_measurable_a_b @ ( N2 @ X3 ) @ K2 )
          = ( sigma_measurable_a_b @ S @ K2 ) ) ) ) ).

% subprob_measurableD(3)
thf(fact_717_subprob__measurableD_I3_J,axiom,
    ! [N2: a > sigma_measure_a,M2: sigma_measure_a,S: sigma_measure_a,X3: a,K2: sigma_measure_b] :
      ( ( member1546933411676758803sure_a @ N2 @ ( sigma_698462642637507881sure_a @ M2 @ ( giry_s2448339315546285520ebra_a @ S ) ) )
     => ( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
       => ( ( sigma_measurable_a_b @ ( N2 @ X3 ) @ K2 )
          = ( sigma_measurable_a_b @ S @ K2 ) ) ) ) ).

% subprob_measurableD(3)
thf(fact_718_subprob__measurableD_I3_J,axiom,
    ! [N2: b > sigma_measure_a,M2: sigma_measure_b,S: sigma_measure_a,X3: b,K2: sigma_measure_b] :
      ( ( member7680248070337521812sure_a @ N2 @ ( sigma_6831777301298270890sure_a @ M2 @ ( giry_s2448339315546285520ebra_a @ S ) ) )
     => ( ( member_b @ X3 @ ( sigma_space_b @ M2 ) )
       => ( ( sigma_measurable_a_b @ ( N2 @ X3 ) @ K2 )
          = ( sigma_measurable_a_b @ S @ K2 ) ) ) ) ).

% subprob_measurableD(3)
thf(fact_719_subprob__measurableD_I3_J,axiom,
    ! [N2: c > sigma_measure_a,M2: sigma_measure_c,S: sigma_measure_a,X3: c,K2: sigma_measure_real] :
      ( ( member4590190692143509013sure_a @ N2 @ ( sigma_3741719923104258091sure_a @ M2 @ ( giry_s2448339315546285520ebra_a @ S ) ) )
     => ( ( member_c @ X3 @ ( sigma_space_c @ M2 ) )
       => ( ( sigma_9116425665531756122a_real @ ( N2 @ X3 ) @ K2 )
          = ( sigma_9116425665531756122a_real @ S @ K2 ) ) ) ) ).

% subprob_measurableD(3)
thf(fact_720_subprob__measurableD_I4_J,axiom,
    ! [N2: c > sigma_measure_d,M2: sigma_measure_c,S: sigma_measure_d,X3: c,K2: sigma_measure_b] :
      ( ( member4803292812282546968sure_d @ N2 @ ( sigma_3741719936013944494sure_d @ M2 @ ( giry_s2448339315546285523ebra_d @ S ) ) )
     => ( ( member_c @ X3 @ ( sigma_space_c @ M2 ) )
       => ( ( sigma_measurable_b_d @ K2 @ ( N2 @ X3 ) )
          = ( sigma_measurable_b_d @ K2 @ S ) ) ) ) ).

% subprob_measurableD(4)
thf(fact_721_subprob__measurableD_I4_J,axiom,
    ! [N2: a > sigma_measure_d,M2: sigma_measure_a,S: sigma_measure_d,X3: a,K2: sigma_measure_b] :
      ( ( member1760035531815796758sure_d @ N2 @ ( sigma_698462655547194284sure_d @ M2 @ ( giry_s2448339315546285523ebra_d @ S ) ) )
     => ( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
       => ( ( sigma_measurable_b_d @ K2 @ ( N2 @ X3 ) )
          = ( sigma_measurable_b_d @ K2 @ S ) ) ) ) ).

% subprob_measurableD(4)
thf(fact_722_subprob__measurableD_I4_J,axiom,
    ! [N2: b > sigma_measure_d,M2: sigma_measure_b,S: sigma_measure_d,X3: b,K2: sigma_measure_b] :
      ( ( member7893350190476559767sure_d @ N2 @ ( sigma_6831777314207957293sure_d @ M2 @ ( giry_s2448339315546285523ebra_d @ S ) ) )
     => ( ( member_b @ X3 @ ( sigma_space_b @ M2 ) )
       => ( ( sigma_measurable_b_d @ K2 @ ( N2 @ X3 ) )
          = ( sigma_measurable_b_d @ K2 @ S ) ) ) ) ).

% subprob_measurableD(4)
thf(fact_723_subprob__measurableD_I4_J,axiom,
    ! [N2: c > sigma_measure_real,M2: sigma_measure_c,S: sigma_measure_real,X3: c,K2: sigma_measure_b] :
      ( ( member4203522019703367553e_real @ N2 @ ( sigma_6238075410825027669e_real @ M2 @ ( giry_s5092570657895779418a_real @ S ) ) )
     => ( ( member_c @ X3 @ ( sigma_space_c @ M2 ) )
       => ( ( sigma_5489272243865111003b_real @ K2 @ ( N2 @ X3 ) )
          = ( sigma_5489272243865111003b_real @ K2 @ S ) ) ) ) ).

% subprob_measurableD(4)
thf(fact_724_subprob__measurableD_I4_J,axiom,
    ! [N2: a > sigma_measure_real,M2: sigma_measure_a,S: sigma_measure_real,X3: a,K2: sigma_measure_b] :
      ( ( member997712892182982147e_real @ N2 @ ( sigma_3032266283304642263e_real @ M2 @ ( giry_s5092570657895779418a_real @ S ) ) )
     => ( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
       => ( ( sigma_5489272243865111003b_real @ K2 @ ( N2 @ X3 ) )
          = ( sigma_5489272243865111003b_real @ K2 @ S ) ) ) ) ).

% subprob_measurableD(4)
thf(fact_725_subprob__measurableD_I4_J,axiom,
    ! [N2: b > sigma_measure_real,M2: sigma_measure_b,S: sigma_measure_real,X3: b,K2: sigma_measure_b] :
      ( ( member7212303474370562754e_real @ N2 @ ( sigma_23484828637447062e_real @ M2 @ ( giry_s5092570657895779418a_real @ S ) ) )
     => ( ( member_b @ X3 @ ( sigma_space_b @ M2 ) )
       => ( ( sigma_5489272243865111003b_real @ K2 @ ( N2 @ X3 ) )
          = ( sigma_5489272243865111003b_real @ K2 @ S ) ) ) ) ).

% subprob_measurableD(4)
thf(fact_726_subprob__measurableD_I4_J,axiom,
    ! [N2: c > sigma_measure_real,M2: sigma_measure_c,S: sigma_measure_real,X3: c,K2: sigma_measure_a] :
      ( ( member4203522019703367553e_real @ N2 @ ( sigma_6238075410825027669e_real @ M2 @ ( giry_s5092570657895779418a_real @ S ) ) )
     => ( ( member_c @ X3 @ ( sigma_space_c @ M2 ) )
       => ( ( sigma_9116425665531756122a_real @ K2 @ ( N2 @ X3 ) )
          = ( sigma_9116425665531756122a_real @ K2 @ S ) ) ) ) ).

% subprob_measurableD(4)
thf(fact_727_subprob__measurableD_I4_J,axiom,
    ! [N2: a > sigma_measure_real,M2: sigma_measure_a,S: sigma_measure_real,X3: a,K2: sigma_measure_a] :
      ( ( member997712892182982147e_real @ N2 @ ( sigma_3032266283304642263e_real @ M2 @ ( giry_s5092570657895779418a_real @ S ) ) )
     => ( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
       => ( ( sigma_9116425665531756122a_real @ K2 @ ( N2 @ X3 ) )
          = ( sigma_9116425665531756122a_real @ K2 @ S ) ) ) ) ).

% subprob_measurableD(4)
thf(fact_728_subprob__measurableD_I4_J,axiom,
    ! [N2: b > sigma_measure_real,M2: sigma_measure_b,S: sigma_measure_real,X3: b,K2: sigma_measure_a] :
      ( ( member7212303474370562754e_real @ N2 @ ( sigma_23484828637447062e_real @ M2 @ ( giry_s5092570657895779418a_real @ S ) ) )
     => ( ( member_b @ X3 @ ( sigma_space_b @ M2 ) )
       => ( ( sigma_9116425665531756122a_real @ K2 @ ( N2 @ X3 ) )
          = ( sigma_9116425665531756122a_real @ K2 @ S ) ) ) ) ).

% subprob_measurableD(4)
thf(fact_729_subprob__measurableD_I4_J,axiom,
    ! [N2: c > sigma_measure_b,M2: sigma_measure_c,S: sigma_measure_b,X3: c,K2: sigma_measure_a] :
      ( ( member4661224732189854998sure_b @ N2 @ ( sigma_3741719927407486892sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ S ) ) )
     => ( ( member_c @ X3 @ ( sigma_space_c @ M2 ) )
       => ( ( sigma_measurable_a_b @ K2 @ ( N2 @ X3 ) )
          = ( sigma_measurable_a_b @ K2 @ S ) ) ) ) ).

% subprob_measurableD(4)
thf(fact_730_subprob__space__kernel,axiom,
    ! [K2: c > sigma_measure_b,M2: sigma_measure_c,N2: sigma_measure_b,A2: c] :
      ( ( member4661224732189854998sure_b @ K2 @ ( sigma_3741719927407486892sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ N2 ) ) )
     => ( ( member_c @ A2 @ ( sigma_space_c @ M2 ) )
       => ( giry_subprob_space_b @ ( K2 @ A2 ) ) ) ) ).

% subprob_space_kernel
thf(fact_731_subprob__space__kernel,axiom,
    ! [K2: a > sigma_measure_b,M2: sigma_measure_a,N2: sigma_measure_b,A2: a] :
      ( ( member1617967451723104788sure_b @ K2 @ ( sigma_698462646940736682sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ N2 ) ) )
     => ( ( member_a @ A2 @ ( sigma_space_a @ M2 ) )
       => ( giry_subprob_space_b @ ( K2 @ A2 ) ) ) ) ).

% subprob_space_kernel
thf(fact_732_subprob__space__kernel,axiom,
    ! [K2: b > sigma_measure_b,M2: sigma_measure_b,N2: sigma_measure_b,A2: b] :
      ( ( member7751282110383867797sure_b @ K2 @ ( sigma_6831777305601499691sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ N2 ) ) )
     => ( ( member_b @ A2 @ ( sigma_space_b @ M2 ) )
       => ( giry_subprob_space_b @ ( K2 @ A2 ) ) ) ) ).

% subprob_space_kernel
thf(fact_733_subprob__space__kernel,axiom,
    ! [K2: c > sigma_measure_a,M2: sigma_measure_c,N2: sigma_measure_a,A2: c] :
      ( ( member4590190692143509013sure_a @ K2 @ ( sigma_3741719923104258091sure_a @ M2 @ ( giry_s2448339315546285520ebra_a @ N2 ) ) )
     => ( ( member_c @ A2 @ ( sigma_space_c @ M2 ) )
       => ( giry_subprob_space_a @ ( K2 @ A2 ) ) ) ) ).

% subprob_space_kernel
thf(fact_734_subprob__space__kernel,axiom,
    ! [K2: a > sigma_measure_a,M2: sigma_measure_a,N2: sigma_measure_a,A2: a] :
      ( ( member1546933411676758803sure_a @ K2 @ ( sigma_698462642637507881sure_a @ M2 @ ( giry_s2448339315546285520ebra_a @ N2 ) ) )
     => ( ( member_a @ A2 @ ( sigma_space_a @ M2 ) )
       => ( giry_subprob_space_a @ ( K2 @ A2 ) ) ) ) ).

% subprob_space_kernel
thf(fact_735_subprob__space__kernel,axiom,
    ! [K2: b > sigma_measure_a,M2: sigma_measure_b,N2: sigma_measure_a,A2: b] :
      ( ( member7680248070337521812sure_a @ K2 @ ( sigma_6831777301298270890sure_a @ M2 @ ( giry_s2448339315546285520ebra_a @ N2 ) ) )
     => ( ( member_b @ A2 @ ( sigma_space_b @ M2 ) )
       => ( giry_subprob_space_a @ ( K2 @ A2 ) ) ) ) ).

% subprob_space_kernel
thf(fact_736_subprob__space__kernel,axiom,
    ! [K2: sigma_measure_b > sigma_measure_b,M2: sigma_2733069582767196501sure_b,N2: sigma_measure_b,A2: sigma_measure_b] :
      ( ( member2971852239989997358sure_b @ K2 @ ( sigma_1495569021742745674sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ N2 ) ) )
     => ( ( member3534519381033026579sure_b @ A2 @ ( sigma_6149183809546982772sure_b @ M2 ) )
       => ( giry_subprob_space_b @ ( K2 @ A2 ) ) ) ) ).

% subprob_space_kernel
thf(fact_737_subprob__space__kernel,axiom,
    ! [K2: sigma_measure_a > sigma_measure_b,M2: sigma_2662035542720850516sure_a,N2: sigma_measure_b,A2: sigma_measure_a] :
      ( ( member710132958529219117sure_b @ K2 @ ( sigma_8457221777136743241sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ N2 ) ) )
     => ( ( member3534519376729797778sure_a @ A2 @ ( sigma_6149183805243753971sure_a @ M2 ) )
       => ( giry_subprob_space_b @ ( K2 @ A2 ) ) ) ) ).

% subprob_space_kernel
thf(fact_738_subprob__space__kernel,axiom,
    ! [K2: sigma_measure_b > sigma_measure_a,M2: sigma_2733069582767196501sure_b,N2: sigma_measure_a,A2: sigma_measure_b] :
      ( ( member2900818199943651373sure_a @ K2 @ ( sigma_1495569017439516873sure_a @ M2 @ ( giry_s2448339315546285520ebra_a @ N2 ) ) )
     => ( ( member3534519381033026579sure_b @ A2 @ ( sigma_6149183809546982772sure_b @ M2 ) )
       => ( giry_subprob_space_a @ ( K2 @ A2 ) ) ) ) ).

% subprob_space_kernel
thf(fact_739_subprob__space__kernel,axiom,
    ! [K2: sigma_measure_a > sigma_measure_a,M2: sigma_2662035542720850516sure_a,N2: sigma_measure_a,A2: sigma_measure_a] :
      ( ( member639098918482873132sure_a @ K2 @ ( sigma_8457221772833514440sure_a @ M2 @ ( giry_s2448339315546285520ebra_a @ N2 ) ) )
     => ( ( member3534519376729797778sure_a @ A2 @ ( sigma_6149183805243753971sure_a @ M2 ) )
       => ( giry_subprob_space_a @ ( K2 @ A2 ) ) ) ) ).

% subprob_space_kernel
thf(fact_740_measurable__distr,axiom,
    ! [F: b > d,M2: sigma_measure_b,N2: sigma_measure_d] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M2 @ N2 ) )
     => ( member3113920320082689328sure_d
        @ ^ [M3: sigma_measure_b] : ( measure_distr_b_d @ M3 @ N2 @ F )
        @ ( sigma_1495569030349203276sure_d @ ( giry_s2448339315546285521ebra_b @ M2 ) @ ( giry_s2448339315546285523ebra_d @ N2 ) ) ) ) ).

% measurable_distr
thf(fact_741_measurable__distr,axiom,
    ! [F: b > real,M2: sigma_measure_b,N2: sigma_measure_real] :
      ( ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ M2 @ N2 ) )
     => ( member504981398075599721e_real
        @ ^ [M3: sigma_measure_b] : ( measure_distr_b_real @ M3 @ N2 @ F )
        @ ( sigma_2902586055234986807e_real @ ( giry_s2448339315546285521ebra_b @ M2 ) @ ( giry_s5092570657895779418a_real @ N2 ) ) ) ) ).

% measurable_distr
thf(fact_742_measurable__distr,axiom,
    ! [F: b > b,M2: sigma_measure_b,N2: sigma_measure_b] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M2 @ N2 ) )
     => ( member2971852239989997358sure_b
        @ ^ [M3: sigma_measure_b] : ( measure_distr_b_b @ M3 @ N2 @ F )
        @ ( sigma_1495569021742745674sure_b @ ( giry_s2448339315546285521ebra_b @ M2 ) @ ( giry_s2448339315546285521ebra_b @ N2 ) ) ) ) ).

% measurable_distr
thf(fact_743_measurable__distr,axiom,
    ! [F: b > a,M2: sigma_measure_b,N2: sigma_measure_a] :
      ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ N2 ) )
     => ( member2900818199943651373sure_a
        @ ^ [M3: sigma_measure_b] : ( measure_distr_b_a @ M3 @ N2 @ F )
        @ ( sigma_1495569017439516873sure_a @ ( giry_s2448339315546285521ebra_b @ M2 ) @ ( giry_s2448339315546285520ebra_a @ N2 ) ) ) ) ).

% measurable_distr
thf(fact_744_measurable__distr,axiom,
    ! [F: a > a > real,M2: sigma_measure_a,N2: sigma_measure_a_real] :
      ( ( member_a_a_real @ F @ ( sigma_5073124708296780141a_real @ M2 @ N2 ) )
     => ( member9062029537003626441a_real
        @ ^ [M3: sigma_measure_a] : ( measur6806378183764566549a_real @ M3 @ N2 @ F )
        @ ( sigma_7630439489522207973a_real @ ( giry_s2448339315546285520ebra_a @ M2 ) @ ( giry_s7054498699981855597a_real @ N2 ) ) ) ) ).

% measurable_distr
thf(fact_745_measurable__distr,axiom,
    ! [F: a > a > b,M2: sigma_measure_a,N2: sigma_measure_a_b] :
      ( ( member_a_a_b @ F @ ( sigma_3971135317854480532_a_a_b @ M2 @ N2 ) )
     => ( member5860586603511027556re_a_b
        @ ^ [M3: sigma_measure_a] : ( measure_distr_a_a_b @ M3 @ N2 @ F )
        @ ( sigma_4154302075293779634re_a_b @ ( giry_s2448339315546285520ebra_a @ M2 ) @ ( giry_s459149637493896852ra_a_b @ N2 ) ) ) ) ).

% measurable_distr
thf(fact_746_measurable__distr,axiom,
    ! [F: a > c,M2: sigma_measure_a,N2: sigma_measure_c] :
      ( ( member_a_c @ F @ ( sigma_measurable_a_c @ M2 @ N2 ) )
     => ( member781166998575565102sure_c
        @ ^ [M3: sigma_measure_a] : ( measure_distr_a_c @ M3 @ N2 @ F )
        @ ( sigma_8457221781439972042sure_c @ ( giry_s2448339315546285520ebra_a @ M2 ) @ ( giry_s2448339315546285522ebra_c @ N2 ) ) ) ) ).

% measurable_distr
thf(fact_747_measurable__distr,axiom,
    ! [F: a > real,M2: sigma_measure_a,N2: sigma_measure_real] :
      ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M2 @ N2 ) )
     => ( member5873258777160121898e_real
        @ ^ [M3: sigma_measure_a] : ( measure_distr_a_real @ M3 @ N2 @ F )
        @ ( sigma_8270863434319508984e_real @ ( giry_s2448339315546285520ebra_a @ M2 ) @ ( giry_s5092570657895779418a_real @ N2 ) ) ) ) ).

% measurable_distr
thf(fact_748_measurable__distr,axiom,
    ! [F: a > b,M2: sigma_measure_a,N2: sigma_measure_b] :
      ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ N2 ) )
     => ( member710132958529219117sure_b
        @ ^ [M3: sigma_measure_a] : ( measure_distr_a_b @ M3 @ N2 @ F )
        @ ( sigma_8457221777136743241sure_b @ ( giry_s2448339315546285520ebra_a @ M2 ) @ ( giry_s2448339315546285521ebra_b @ N2 ) ) ) ) ).

% measurable_distr
thf(fact_749_measurable__distr,axiom,
    ! [F: a > a,M2: sigma_measure_a,N2: sigma_measure_a] :
      ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N2 ) )
     => ( member639098918482873132sure_a
        @ ^ [M3: sigma_measure_a] : ( measure_distr_a_a @ M3 @ N2 @ F )
        @ ( sigma_8457221772833514440sure_a @ ( giry_s2448339315546285520ebra_a @ M2 ) @ ( giry_s2448339315546285520ebra_a @ N2 ) ) ) ) ).

% measurable_distr
thf(fact_750_prob__space__return,axiom,
    ! [X3: c,M2: sigma_measure_c] :
      ( ( member_c @ X3 @ ( sigma_space_c @ M2 ) )
     => ( probab7247484486040049091pace_c @ ( giry_return_c @ M2 @ X3 ) ) ) ).

% prob_space_return
thf(fact_751_prob__space__return,axiom,
    ! [X3: a,M2: sigma_measure_a] :
      ( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
     => ( probab7247484486040049089pace_a @ ( giry_return_a @ M2 @ X3 ) ) ) ).

% prob_space_return
thf(fact_752_prob__space__return,axiom,
    ! [X3: b,M2: sigma_measure_b] :
      ( ( member_b @ X3 @ ( sigma_space_b @ M2 ) )
     => ( probab7247484486040049090pace_b @ ( giry_return_b @ M2 @ X3 ) ) ) ).

% prob_space_return
thf(fact_753_prob__space__return,axiom,
    ! [X3: sigma_measure_b,M2: sigma_2733069582767196501sure_b] :
      ( ( member3534519381033026579sure_b @ X3 @ ( sigma_6149183809546982772sure_b @ M2 ) )
     => ( probab6953849309755211513sure_b @ ( giry_r3784369487821543sure_b @ M2 @ X3 ) ) ) ).

% prob_space_return
thf(fact_754_prob__space__return,axiom,
    ! [X3: sigma_measure_a,M2: sigma_2662035542720850516sure_a] :
      ( ( member3534519376729797778sure_a @ X3 @ ( sigma_6149183805243753971sure_a @ M2 ) )
     => ( probab6953849305451982712sure_a @ ( giry_r3784365184592742sure_a @ M2 @ X3 ) ) ) ).

% prob_space_return
thf(fact_755_prob__space__return,axiom,
    ! [X3: stream_b,M2: sigma_6673148739008212994ream_b] :
      ( ( member_stream_b @ X3 @ ( sigma_space_stream_b @ M2 ) )
     => ( probab6181332906713481800ream_b @ ( giry_return_stream_b @ M2 @ X3 ) ) ) ).

% prob_space_return
thf(fact_756_prob__space__return,axiom,
    ! [X3: stream_a,M2: sigma_6602114698961867009ream_a] :
      ( ( member_stream_a @ X3 @ ( sigma_space_stream_a @ M2 ) )
     => ( probab6181332902410252999ream_a @ ( giry_return_stream_a @ M2 @ X3 ) ) ) ).

% prob_space_return
thf(fact_757_prob__space__return,axiom,
    ! [X3: a > real,M2: sigma_measure_a_real] :
      ( ( member_a_real @ X3 @ ( sigma_space_a_real @ M2 ) )
     => ( probab2024454272037758302a_real @ ( giry_return_a_real @ M2 @ X3 ) ) ) ).

% prob_space_return
thf(fact_758_prob__space__return,axiom,
    ! [X3: a > b,M2: sigma_measure_a_b] :
      ( ( member_a_b @ X3 @ ( sigma_space_a_b @ M2 ) )
     => ( probab45282617874185827ce_a_b @ ( giry_return_a_b @ M2 @ X3 ) ) ) ).

% prob_space_return
thf(fact_759_prob__space__return,axiom,
    ! [X3: b > a,M2: sigma_measure_b_a] :
      ( ( member_b_a @ X3 @ ( sigma_space_b_a @ M2 ) )
     => ( probab6481199068453056801ce_b_a @ ( giry_return_b_a @ M2 @ X3 ) ) ) ).

% prob_space_return
thf(fact_760_subprob__space__return,axiom,
    ! [X3: c,M2: sigma_measure_c] :
      ( ( member_c @ X3 @ ( sigma_space_c @ M2 ) )
     => ( giry_subprob_space_c @ ( giry_return_c @ M2 @ X3 ) ) ) ).

% subprob_space_return
thf(fact_761_subprob__space__return,axiom,
    ! [X3: a,M2: sigma_measure_a] :
      ( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
     => ( giry_subprob_space_a @ ( giry_return_a @ M2 @ X3 ) ) ) ).

% subprob_space_return
thf(fact_762_subprob__space__return,axiom,
    ! [X3: b,M2: sigma_measure_b] :
      ( ( member_b @ X3 @ ( sigma_space_b @ M2 ) )
     => ( giry_subprob_space_b @ ( giry_return_b @ M2 @ X3 ) ) ) ).

% subprob_space_return
thf(fact_763_subprob__space__return,axiom,
    ! [X3: sigma_measure_b,M2: sigma_2733069582767196501sure_b] :
      ( ( member3534519381033026579sure_b @ X3 @ ( sigma_6149183809546982772sure_b @ M2 ) )
     => ( giry_s2296666612018410952sure_b @ ( giry_r3784369487821543sure_b @ M2 @ X3 ) ) ) ).

% subprob_space_return
thf(fact_764_subprob__space__return,axiom,
    ! [X3: sigma_measure_a,M2: sigma_2662035542720850516sure_a] :
      ( ( member3534519376729797778sure_a @ X3 @ ( sigma_6149183805243753971sure_a @ M2 ) )
     => ( giry_s2296666607715182151sure_a @ ( giry_r3784365184592742sure_a @ M2 @ X3 ) ) ) ).

% subprob_space_return
thf(fact_765_subprob__space__return,axiom,
    ! [X3: a > real,M2: sigma_measure_a_real] :
      ( ( member_a_real @ X3 @ ( sigma_space_a_real @ M2 ) )
     => ( giry_s8245238245569539279a_real @ ( giry_return_a_real @ M2 @ X3 ) ) ) ).

% subprob_space_return
thf(fact_766_subprob__space__return,axiom,
    ! [X3: a > b,M2: sigma_measure_a_b] :
      ( ( member_a_b @ X3 @ ( sigma_space_a_b @ M2 ) )
     => ( giry_s8818617164929591474ce_a_b @ ( giry_return_a_b @ M2 @ X3 ) ) ) ).

% subprob_space_return
thf(fact_767_subprob__space__return,axiom,
    ! [X3: b > a,M2: sigma_measure_b_a] :
      ( ( member_b_a @ X3 @ ( sigma_space_b_a @ M2 ) )
     => ( giry_s6031161578653686640ce_b_a @ ( giry_return_b_a @ M2 @ X3 ) ) ) ).

% subprob_space_return
thf(fact_768_subprob__space__return,axiom,
    ! [X3: a > c,M2: sigma_measure_a_c] :
      ( ( member_a_c @ X3 @ ( sigma_space_a_c @ M2 ) )
     => ( giry_s8818617169232820275ce_a_c @ ( giry_return_a_c @ M2 @ X3 ) ) ) ).

% subprob_space_return
thf(fact_769_subprob__space__return,axiom,
    ! [X3: a > a,M2: sigma_measure_a_a] :
      ( ( member_a_a @ X3 @ ( sigma_space_a_a @ M2 ) )
     => ( giry_s8818617160626362673ce_a_a @ ( giry_return_a_a @ M2 @ X3 ) ) ) ).

% subprob_space_return
thf(fact_770_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X2: nat] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_771_Collect__mono__iff,axiom,
    ! [P: set_b > $o,Q: set_b > $o] :
      ( ( ord_le3795704787696855135_set_b @ ( collect_set_b @ P ) @ ( collect_set_b @ Q ) )
      = ( ! [X2: set_b] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_772_Collect__mono__iff,axiom,
    ! [P: set_a > $o,Q: set_a > $o] :
      ( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
      = ( ! [X2: set_a] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_773_Collect__mono__iff,axiom,
    ! [P: b > $o,Q: b > $o] :
      ( ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) )
      = ( ! [X2: b] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_774_Collect__mono__iff,axiom,
    ! [P: a > $o,Q: a > $o] :
      ( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
      = ( ! [X2: a] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_775_set__eq__subset,axiom,
    ( ( ^ [Y4: set_set_b,Z: set_set_b] : ( Y4 = Z ) )
    = ( ^ [A4: set_set_b,B2: set_set_b] :
          ( ( ord_le3795704787696855135_set_b @ A4 @ B2 )
          & ( ord_le3795704787696855135_set_b @ B2 @ A4 ) ) ) ) ).

% set_eq_subset
thf(fact_776_set__eq__subset,axiom,
    ( ( ^ [Y4: set_set_a,Z: set_set_a] : ( Y4 = Z ) )
    = ( ^ [A4: set_set_a,B2: set_set_a] :
          ( ( ord_le3724670747650509150_set_a @ A4 @ B2 )
          & ( ord_le3724670747650509150_set_a @ B2 @ A4 ) ) ) ) ).

% set_eq_subset
thf(fact_777_set__eq__subset,axiom,
    ( ( ^ [Y4: set_b,Z: set_b] : ( Y4 = Z ) )
    = ( ^ [A4: set_b,B2: set_b] :
          ( ( ord_less_eq_set_b @ A4 @ B2 )
          & ( ord_less_eq_set_b @ B2 @ A4 ) ) ) ) ).

% set_eq_subset
thf(fact_778_set__eq__subset,axiom,
    ( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
    = ( ^ [A4: set_a,B2: set_a] :
          ( ( ord_less_eq_set_a @ A4 @ B2 )
          & ( ord_less_eq_set_a @ B2 @ A4 ) ) ) ) ).

% set_eq_subset
thf(fact_779_subset__trans,axiom,
    ! [A: set_set_b,B: set_set_b,C: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A @ B )
     => ( ( ord_le3795704787696855135_set_b @ B @ C )
       => ( ord_le3795704787696855135_set_b @ A @ C ) ) ) ).

% subset_trans
thf(fact_780_subset__trans,axiom,
    ! [A: set_set_a,B: set_set_a,C: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B )
     => ( ( ord_le3724670747650509150_set_a @ B @ C )
       => ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).

% subset_trans
thf(fact_781_subset__trans,axiom,
    ! [A: set_b,B: set_b,C: set_b] :
      ( ( ord_less_eq_set_b @ A @ B )
     => ( ( ord_less_eq_set_b @ B @ C )
       => ( ord_less_eq_set_b @ A @ C ) ) ) ).

% subset_trans
thf(fact_782_subset__trans,axiom,
    ! [A: set_a,B: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( ord_less_eq_set_a @ B @ C )
       => ( ord_less_eq_set_a @ A @ C ) ) ) ).

% subset_trans
thf(fact_783_Collect__mono,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X: nat] :
          ( ( P @ X )
         => ( Q @ X ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_784_Collect__mono,axiom,
    ! [P: set_b > $o,Q: set_b > $o] :
      ( ! [X: set_b] :
          ( ( P @ X )
         => ( Q @ X ) )
     => ( ord_le3795704787696855135_set_b @ ( collect_set_b @ P ) @ ( collect_set_b @ Q ) ) ) ).

% Collect_mono
thf(fact_785_Collect__mono,axiom,
    ! [P: set_a > $o,Q: set_a > $o] :
      ( ! [X: set_a] :
          ( ( P @ X )
         => ( Q @ X ) )
     => ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).

% Collect_mono
thf(fact_786_Collect__mono,axiom,
    ! [P: b > $o,Q: b > $o] :
      ( ! [X: b] :
          ( ( P @ X )
         => ( Q @ X ) )
     => ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).

% Collect_mono
thf(fact_787_Collect__mono,axiom,
    ! [P: a > $o,Q: a > $o] :
      ( ! [X: a] :
          ( ( P @ X )
         => ( Q @ X ) )
     => ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).

% Collect_mono
thf(fact_788_subset__refl,axiom,
    ! [A: set_set_b] : ( ord_le3795704787696855135_set_b @ A @ A ) ).

% subset_refl
thf(fact_789_subset__refl,axiom,
    ! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).

% subset_refl
thf(fact_790_subset__refl,axiom,
    ! [A: set_b] : ( ord_less_eq_set_b @ A @ A ) ).

% subset_refl
thf(fact_791_subset__refl,axiom,
    ! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).

% subset_refl
thf(fact_792_subset__iff,axiom,
    ( ord_less_eq_set_c
    = ( ^ [A4: set_c,B2: set_c] :
        ! [T2: c] :
          ( ( member_c @ T2 @ A4 )
         => ( member_c @ T2 @ B2 ) ) ) ) ).

% subset_iff
thf(fact_793_subset__iff,axiom,
    ( ord_less_eq_set_b
    = ( ^ [A4: set_b,B2: set_b] :
        ! [T2: b] :
          ( ( member_b @ T2 @ A4 )
         => ( member_b @ T2 @ B2 ) ) ) ) ).

% subset_iff
thf(fact_794_subset__iff,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A4: set_a,B2: set_a] :
        ! [T2: a] :
          ( ( member_a @ T2 @ A4 )
         => ( member_a @ T2 @ B2 ) ) ) ) ).

% subset_iff
thf(fact_795_subset__iff,axiom,
    ( ord_le3795704787696855135_set_b
    = ( ^ [A4: set_set_b,B2: set_set_b] :
        ! [T2: set_b] :
          ( ( member_set_b @ T2 @ A4 )
         => ( member_set_b @ T2 @ B2 ) ) ) ) ).

% subset_iff
thf(fact_796_subset__iff,axiom,
    ( ord_le3724670747650509150_set_a
    = ( ^ [A4: set_set_a,B2: set_set_a] :
        ! [T2: set_a] :
          ( ( member_set_a @ T2 @ A4 )
         => ( member_set_a @ T2 @ B2 ) ) ) ) ).

% subset_iff
thf(fact_797_subset__iff,axiom,
    ( ord_le3334967407727675675a_real
    = ( ^ [A4: set_a_real,B2: set_a_real] :
        ! [T2: a > real] :
          ( ( member_a_real @ T2 @ A4 )
         => ( member_a_real @ T2 @ B2 ) ) ) ) ).

% subset_iff
thf(fact_798_subset__iff,axiom,
    ( ord_less_eq_set_a_b
    = ( ^ [A4: set_a_b,B2: set_a_b] :
        ! [T2: a > b] :
          ( ( member_a_b @ T2 @ A4 )
         => ( member_a_b @ T2 @ B2 ) ) ) ) ).

% subset_iff
thf(fact_799_subset__iff,axiom,
    ( ord_less_eq_set_b_a
    = ( ^ [A4: set_b_a,B2: set_b_a] :
        ! [T2: b > a] :
          ( ( member_b_a @ T2 @ A4 )
         => ( member_b_a @ T2 @ B2 ) ) ) ) ).

% subset_iff
thf(fact_800_subset__iff,axiom,
    ( ord_less_eq_set_a_c
    = ( ^ [A4: set_a_c,B2: set_a_c] :
        ! [T2: a > c] :
          ( ( member_a_c @ T2 @ A4 )
         => ( member_a_c @ T2 @ B2 ) ) ) ) ).

% subset_iff
thf(fact_801_subset__iff,axiom,
    ( ord_less_eq_set_a_a
    = ( ^ [A4: set_a_a,B2: set_a_a] :
        ! [T2: a > a] :
          ( ( member_a_a @ T2 @ A4 )
         => ( member_a_a @ T2 @ B2 ) ) ) ) ).

% subset_iff
thf(fact_802_equalityD2,axiom,
    ! [A: set_set_b,B: set_set_b] :
      ( ( A = B )
     => ( ord_le3795704787696855135_set_b @ B @ A ) ) ).

% equalityD2
thf(fact_803_equalityD2,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ( A = B )
     => ( ord_le3724670747650509150_set_a @ B @ A ) ) ).

% equalityD2
thf(fact_804_equalityD2,axiom,
    ! [A: set_b,B: set_b] :
      ( ( A = B )
     => ( ord_less_eq_set_b @ B @ A ) ) ).

% equalityD2
thf(fact_805_equalityD2,axiom,
    ! [A: set_a,B: set_a] :
      ( ( A = B )
     => ( ord_less_eq_set_a @ B @ A ) ) ).

% equalityD2
thf(fact_806_equalityD1,axiom,
    ! [A: set_set_b,B: set_set_b] :
      ( ( A = B )
     => ( ord_le3795704787696855135_set_b @ A @ B ) ) ).

% equalityD1
thf(fact_807_equalityD1,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ( A = B )
     => ( ord_le3724670747650509150_set_a @ A @ B ) ) ).

% equalityD1
thf(fact_808_equalityD1,axiom,
    ! [A: set_b,B: set_b] :
      ( ( A = B )
     => ( ord_less_eq_set_b @ A @ B ) ) ).

% equalityD1
thf(fact_809_equalityD1,axiom,
    ! [A: set_a,B: set_a] :
      ( ( A = B )
     => ( ord_less_eq_set_a @ A @ B ) ) ).

% equalityD1
thf(fact_810_subset__eq,axiom,
    ( ord_less_eq_set_c
    = ( ^ [A4: set_c,B2: set_c] :
        ! [X2: c] :
          ( ( member_c @ X2 @ A4 )
         => ( member_c @ X2 @ B2 ) ) ) ) ).

% subset_eq
thf(fact_811_subset__eq,axiom,
    ( ord_less_eq_set_b
    = ( ^ [A4: set_b,B2: set_b] :
        ! [X2: b] :
          ( ( member_b @ X2 @ A4 )
         => ( member_b @ X2 @ B2 ) ) ) ) ).

% subset_eq
thf(fact_812_subset__eq,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A4: set_a,B2: set_a] :
        ! [X2: a] :
          ( ( member_a @ X2 @ A4 )
         => ( member_a @ X2 @ B2 ) ) ) ) ).

% subset_eq
thf(fact_813_subset__eq,axiom,
    ( ord_le3795704787696855135_set_b
    = ( ^ [A4: set_set_b,B2: set_set_b] :
        ! [X2: set_b] :
          ( ( member_set_b @ X2 @ A4 )
         => ( member_set_b @ X2 @ B2 ) ) ) ) ).

% subset_eq
thf(fact_814_subset__eq,axiom,
    ( ord_le3724670747650509150_set_a
    = ( ^ [A4: set_set_a,B2: set_set_a] :
        ! [X2: set_a] :
          ( ( member_set_a @ X2 @ A4 )
         => ( member_set_a @ X2 @ B2 ) ) ) ) ).

% subset_eq
thf(fact_815_subset__eq,axiom,
    ( ord_le3334967407727675675a_real
    = ( ^ [A4: set_a_real,B2: set_a_real] :
        ! [X2: a > real] :
          ( ( member_a_real @ X2 @ A4 )
         => ( member_a_real @ X2 @ B2 ) ) ) ) ).

% subset_eq
thf(fact_816_subset__eq,axiom,
    ( ord_less_eq_set_a_b
    = ( ^ [A4: set_a_b,B2: set_a_b] :
        ! [X2: a > b] :
          ( ( member_a_b @ X2 @ A4 )
         => ( member_a_b @ X2 @ B2 ) ) ) ) ).

% subset_eq
thf(fact_817_subset__eq,axiom,
    ( ord_less_eq_set_b_a
    = ( ^ [A4: set_b_a,B2: set_b_a] :
        ! [X2: b > a] :
          ( ( member_b_a @ X2 @ A4 )
         => ( member_b_a @ X2 @ B2 ) ) ) ) ).

% subset_eq
thf(fact_818_subset__eq,axiom,
    ( ord_less_eq_set_a_c
    = ( ^ [A4: set_a_c,B2: set_a_c] :
        ! [X2: a > c] :
          ( ( member_a_c @ X2 @ A4 )
         => ( member_a_c @ X2 @ B2 ) ) ) ) ).

% subset_eq
thf(fact_819_subset__eq,axiom,
    ( ord_less_eq_set_a_a
    = ( ^ [A4: set_a_a,B2: set_a_a] :
        ! [X2: a > a] :
          ( ( member_a_a @ X2 @ A4 )
         => ( member_a_a @ X2 @ B2 ) ) ) ) ).

% subset_eq
thf(fact_820_equalityE,axiom,
    ! [A: set_set_b,B: set_set_b] :
      ( ( A = B )
     => ~ ( ( ord_le3795704787696855135_set_b @ A @ B )
         => ~ ( ord_le3795704787696855135_set_b @ B @ A ) ) ) ).

% equalityE
thf(fact_821_equalityE,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ( A = B )
     => ~ ( ( ord_le3724670747650509150_set_a @ A @ B )
         => ~ ( ord_le3724670747650509150_set_a @ B @ A ) ) ) ).

% equalityE
thf(fact_822_equalityE,axiom,
    ! [A: set_b,B: set_b] :
      ( ( A = B )
     => ~ ( ( ord_less_eq_set_b @ A @ B )
         => ~ ( ord_less_eq_set_b @ B @ A ) ) ) ).

% equalityE
thf(fact_823_equalityE,axiom,
    ! [A: set_a,B: set_a] :
      ( ( A = B )
     => ~ ( ( ord_less_eq_set_a @ A @ B )
         => ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).

% equalityE
thf(fact_824_subsetD,axiom,
    ! [A: set_c,B: set_c,C2: c] :
      ( ( ord_less_eq_set_c @ A @ B )
     => ( ( member_c @ C2 @ A )
       => ( member_c @ C2 @ B ) ) ) ).

% subsetD
thf(fact_825_subsetD,axiom,
    ! [A: set_b,B: set_b,C2: b] :
      ( ( ord_less_eq_set_b @ A @ B )
     => ( ( member_b @ C2 @ A )
       => ( member_b @ C2 @ B ) ) ) ).

% subsetD
thf(fact_826_subsetD,axiom,
    ! [A: set_a,B: set_a,C2: a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( member_a @ C2 @ A )
       => ( member_a @ C2 @ B ) ) ) ).

% subsetD
thf(fact_827_subsetD,axiom,
    ! [A: set_set_b,B: set_set_b,C2: set_b] :
      ( ( ord_le3795704787696855135_set_b @ A @ B )
     => ( ( member_set_b @ C2 @ A )
       => ( member_set_b @ C2 @ B ) ) ) ).

% subsetD
thf(fact_828_subsetD,axiom,
    ! [A: set_set_a,B: set_set_a,C2: set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B )
     => ( ( member_set_a @ C2 @ A )
       => ( member_set_a @ C2 @ B ) ) ) ).

% subsetD
thf(fact_829_subsetD,axiom,
    ! [A: set_a_real,B: set_a_real,C2: a > real] :
      ( ( ord_le3334967407727675675a_real @ A @ B )
     => ( ( member_a_real @ C2 @ A )
       => ( member_a_real @ C2 @ B ) ) ) ).

% subsetD
thf(fact_830_subsetD,axiom,
    ! [A: set_a_b,B: set_a_b,C2: a > b] :
      ( ( ord_less_eq_set_a_b @ A @ B )
     => ( ( member_a_b @ C2 @ A )
       => ( member_a_b @ C2 @ B ) ) ) ).

% subsetD
thf(fact_831_subsetD,axiom,
    ! [A: set_b_a,B: set_b_a,C2: b > a] :
      ( ( ord_less_eq_set_b_a @ A @ B )
     => ( ( member_b_a @ C2 @ A )
       => ( member_b_a @ C2 @ B ) ) ) ).

% subsetD
thf(fact_832_subsetD,axiom,
    ! [A: set_a_c,B: set_a_c,C2: a > c] :
      ( ( ord_less_eq_set_a_c @ A @ B )
     => ( ( member_a_c @ C2 @ A )
       => ( member_a_c @ C2 @ B ) ) ) ).

% subsetD
thf(fact_833_subsetD,axiom,
    ! [A: set_a_a,B: set_a_a,C2: a > a] :
      ( ( ord_less_eq_set_a_a @ A @ B )
     => ( ( member_a_a @ C2 @ A )
       => ( member_a_a @ C2 @ B ) ) ) ).

% subsetD
thf(fact_834_in__mono,axiom,
    ! [A: set_c,B: set_c,X3: c] :
      ( ( ord_less_eq_set_c @ A @ B )
     => ( ( member_c @ X3 @ A )
       => ( member_c @ X3 @ B ) ) ) ).

% in_mono
thf(fact_835_in__mono,axiom,
    ! [A: set_b,B: set_b,X3: b] :
      ( ( ord_less_eq_set_b @ A @ B )
     => ( ( member_b @ X3 @ A )
       => ( member_b @ X3 @ B ) ) ) ).

% in_mono
thf(fact_836_in__mono,axiom,
    ! [A: set_a,B: set_a,X3: a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( member_a @ X3 @ A )
       => ( member_a @ X3 @ B ) ) ) ).

% in_mono
thf(fact_837_in__mono,axiom,
    ! [A: set_set_b,B: set_set_b,X3: set_b] :
      ( ( ord_le3795704787696855135_set_b @ A @ B )
     => ( ( member_set_b @ X3 @ A )
       => ( member_set_b @ X3 @ B ) ) ) ).

% in_mono
thf(fact_838_in__mono,axiom,
    ! [A: set_set_a,B: set_set_a,X3: set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B )
     => ( ( member_set_a @ X3 @ A )
       => ( member_set_a @ X3 @ B ) ) ) ).

% in_mono
thf(fact_839_in__mono,axiom,
    ! [A: set_a_real,B: set_a_real,X3: a > real] :
      ( ( ord_le3334967407727675675a_real @ A @ B )
     => ( ( member_a_real @ X3 @ A )
       => ( member_a_real @ X3 @ B ) ) ) ).

% in_mono
thf(fact_840_in__mono,axiom,
    ! [A: set_a_b,B: set_a_b,X3: a > b] :
      ( ( ord_less_eq_set_a_b @ A @ B )
     => ( ( member_a_b @ X3 @ A )
       => ( member_a_b @ X3 @ B ) ) ) ).

% in_mono
thf(fact_841_in__mono,axiom,
    ! [A: set_b_a,B: set_b_a,X3: b > a] :
      ( ( ord_less_eq_set_b_a @ A @ B )
     => ( ( member_b_a @ X3 @ A )
       => ( member_b_a @ X3 @ B ) ) ) ).

% in_mono
thf(fact_842_in__mono,axiom,
    ! [A: set_a_c,B: set_a_c,X3: a > c] :
      ( ( ord_less_eq_set_a_c @ A @ B )
     => ( ( member_a_c @ X3 @ A )
       => ( member_a_c @ X3 @ B ) ) ) ).

% in_mono
thf(fact_843_in__mono,axiom,
    ! [A: set_a_a,B: set_a_a,X3: a > a] :
      ( ( ord_less_eq_set_a_a @ A @ B )
     => ( ( member_a_a @ X3 @ A )
       => ( member_a_a @ X3 @ B ) ) ) ).

% in_mono
thf(fact_844_ex__in__conv,axiom,
    ! [A: set_c] :
      ( ( ? [X2: c] : ( member_c @ X2 @ A ) )
      = ( A != bot_bot_set_c ) ) ).

% ex_in_conv
thf(fact_845_ex__in__conv,axiom,
    ! [A: set_a] :
      ( ( ? [X2: a] : ( member_a @ X2 @ A ) )
      = ( A != bot_bot_set_a ) ) ).

% ex_in_conv
thf(fact_846_ex__in__conv,axiom,
    ! [A: set_b] :
      ( ( ? [X2: b] : ( member_b @ X2 @ A ) )
      = ( A != bot_bot_set_b ) ) ).

% ex_in_conv
thf(fact_847_ex__in__conv,axiom,
    ! [A: set_o] :
      ( ( ? [X2: $o] : ( member_o @ X2 @ A ) )
      = ( A != bot_bot_set_o ) ) ).

% ex_in_conv
thf(fact_848_ex__in__conv,axiom,
    ! [A: set_a_real] :
      ( ( ? [X2: a > real] : ( member_a_real @ X2 @ A ) )
      = ( A != bot_bot_set_a_real ) ) ).

% ex_in_conv
thf(fact_849_ex__in__conv,axiom,
    ! [A: set_a_b] :
      ( ( ? [X2: a > b] : ( member_a_b @ X2 @ A ) )
      = ( A != bot_bot_set_a_b ) ) ).

% ex_in_conv
thf(fact_850_ex__in__conv,axiom,
    ! [A: set_b_a] :
      ( ( ? [X2: b > a] : ( member_b_a @ X2 @ A ) )
      = ( A != bot_bot_set_b_a ) ) ).

% ex_in_conv
thf(fact_851_ex__in__conv,axiom,
    ! [A: set_a_c] :
      ( ( ? [X2: a > c] : ( member_a_c @ X2 @ A ) )
      = ( A != bot_bot_set_a_c ) ) ).

% ex_in_conv
thf(fact_852_ex__in__conv,axiom,
    ! [A: set_a_a] :
      ( ( ? [X2: a > a] : ( member_a_a @ X2 @ A ) )
      = ( A != bot_bot_set_a_a ) ) ).

% ex_in_conv
thf(fact_853_ex__in__conv,axiom,
    ! [A: set_a_a_real] :
      ( ( ? [X2: a > a > real] : ( member_a_a_real @ X2 @ A ) )
      = ( A != bot_bot_set_a_a_real ) ) ).

% ex_in_conv
thf(fact_854_equals0I,axiom,
    ! [A: set_c] :
      ( ! [Y5: c] :
          ~ ( member_c @ Y5 @ A )
     => ( A = bot_bot_set_c ) ) ).

% equals0I
thf(fact_855_equals0I,axiom,
    ! [A: set_a] :
      ( ! [Y5: a] :
          ~ ( member_a @ Y5 @ A )
     => ( A = bot_bot_set_a ) ) ).

% equals0I
thf(fact_856_equals0I,axiom,
    ! [A: set_b] :
      ( ! [Y5: b] :
          ~ ( member_b @ Y5 @ A )
     => ( A = bot_bot_set_b ) ) ).

% equals0I
thf(fact_857_equals0I,axiom,
    ! [A: set_o] :
      ( ! [Y5: $o] :
          ~ ( member_o @ Y5 @ A )
     => ( A = bot_bot_set_o ) ) ).

% equals0I
thf(fact_858_equals0I,axiom,
    ! [A: set_a_real] :
      ( ! [Y5: a > real] :
          ~ ( member_a_real @ Y5 @ A )
     => ( A = bot_bot_set_a_real ) ) ).

% equals0I
thf(fact_859_equals0I,axiom,
    ! [A: set_a_b] :
      ( ! [Y5: a > b] :
          ~ ( member_a_b @ Y5 @ A )
     => ( A = bot_bot_set_a_b ) ) ).

% equals0I
thf(fact_860_equals0I,axiom,
    ! [A: set_b_a] :
      ( ! [Y5: b > a] :
          ~ ( member_b_a @ Y5 @ A )
     => ( A = bot_bot_set_b_a ) ) ).

% equals0I
thf(fact_861_equals0I,axiom,
    ! [A: set_a_c] :
      ( ! [Y5: a > c] :
          ~ ( member_a_c @ Y5 @ A )
     => ( A = bot_bot_set_a_c ) ) ).

% equals0I
thf(fact_862_equals0I,axiom,
    ! [A: set_a_a] :
      ( ! [Y5: a > a] :
          ~ ( member_a_a @ Y5 @ A )
     => ( A = bot_bot_set_a_a ) ) ).

% equals0I
thf(fact_863_equals0I,axiom,
    ! [A: set_a_a_real] :
      ( ! [Y5: a > a > real] :
          ~ ( member_a_a_real @ Y5 @ A )
     => ( A = bot_bot_set_a_a_real ) ) ).

% equals0I
thf(fact_864_equals0D,axiom,
    ! [A: set_c,A2: c] :
      ( ( A = bot_bot_set_c )
     => ~ ( member_c @ A2 @ A ) ) ).

% equals0D
thf(fact_865_equals0D,axiom,
    ! [A: set_a,A2: a] :
      ( ( A = bot_bot_set_a )
     => ~ ( member_a @ A2 @ A ) ) ).

% equals0D
thf(fact_866_equals0D,axiom,
    ! [A: set_b,A2: b] :
      ( ( A = bot_bot_set_b )
     => ~ ( member_b @ A2 @ A ) ) ).

% equals0D
thf(fact_867_equals0D,axiom,
    ! [A: set_o,A2: $o] :
      ( ( A = bot_bot_set_o )
     => ~ ( member_o @ A2 @ A ) ) ).

% equals0D
thf(fact_868_equals0D,axiom,
    ! [A: set_a_real,A2: a > real] :
      ( ( A = bot_bot_set_a_real )
     => ~ ( member_a_real @ A2 @ A ) ) ).

% equals0D
thf(fact_869_equals0D,axiom,
    ! [A: set_a_b,A2: a > b] :
      ( ( A = bot_bot_set_a_b )
     => ~ ( member_a_b @ A2 @ A ) ) ).

% equals0D
thf(fact_870_equals0D,axiom,
    ! [A: set_b_a,A2: b > a] :
      ( ( A = bot_bot_set_b_a )
     => ~ ( member_b_a @ A2 @ A ) ) ).

% equals0D
thf(fact_871_equals0D,axiom,
    ! [A: set_a_c,A2: a > c] :
      ( ( A = bot_bot_set_a_c )
     => ~ ( member_a_c @ A2 @ A ) ) ).

% equals0D
thf(fact_872_equals0D,axiom,
    ! [A: set_a_a,A2: a > a] :
      ( ( A = bot_bot_set_a_a )
     => ~ ( member_a_a @ A2 @ A ) ) ).

% equals0D
thf(fact_873_equals0D,axiom,
    ! [A: set_a_a_real,A2: a > a > real] :
      ( ( A = bot_bot_set_a_a_real )
     => ~ ( member_a_a_real @ A2 @ A ) ) ).

% equals0D
thf(fact_874_emptyE,axiom,
    ! [A2: c] :
      ~ ( member_c @ A2 @ bot_bot_set_c ) ).

% emptyE
thf(fact_875_emptyE,axiom,
    ! [A2: a] :
      ~ ( member_a @ A2 @ bot_bot_set_a ) ).

% emptyE
thf(fact_876_emptyE,axiom,
    ! [A2: b] :
      ~ ( member_b @ A2 @ bot_bot_set_b ) ).

% emptyE
thf(fact_877_emptyE,axiom,
    ! [A2: $o] :
      ~ ( member_o @ A2 @ bot_bot_set_o ) ).

% emptyE
thf(fact_878_emptyE,axiom,
    ! [A2: a > real] :
      ~ ( member_a_real @ A2 @ bot_bot_set_a_real ) ).

% emptyE
thf(fact_879_emptyE,axiom,
    ! [A2: a > b] :
      ~ ( member_a_b @ A2 @ bot_bot_set_a_b ) ).

% emptyE
thf(fact_880_emptyE,axiom,
    ! [A2: b > a] :
      ~ ( member_b_a @ A2 @ bot_bot_set_b_a ) ).

% emptyE
thf(fact_881_emptyE,axiom,
    ! [A2: a > c] :
      ~ ( member_a_c @ A2 @ bot_bot_set_a_c ) ).

% emptyE
thf(fact_882_emptyE,axiom,
    ! [A2: a > a] :
      ~ ( member_a_a @ A2 @ bot_bot_set_a_a ) ).

% emptyE
thf(fact_883_emptyE,axiom,
    ! [A2: a > a > real] :
      ~ ( member_a_a_real @ A2 @ bot_bot_set_a_a_real ) ).

% emptyE
thf(fact_884_distr__return,axiom,
    ! [F: a > a > real,M2: sigma_measure_a,N2: sigma_measure_a_real,X3: a] :
      ( ( member_a_a_real @ F @ ( sigma_5073124708296780141a_real @ M2 @ N2 ) )
     => ( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
       => ( ( measur6806378183764566549a_real @ ( giry_return_a @ M2 @ X3 ) @ N2 @ F )
          = ( giry_return_a_real @ N2 @ ( F @ X3 ) ) ) ) ) ).

% distr_return
thf(fact_885_distr__return,axiom,
    ! [F: a > a > b,M2: sigma_measure_a,N2: sigma_measure_a_b,X3: a] :
      ( ( member_a_a_b @ F @ ( sigma_3971135317854480532_a_a_b @ M2 @ N2 ) )
     => ( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
       => ( ( measure_distr_a_a_b @ ( giry_return_a @ M2 @ X3 ) @ N2 @ F )
          = ( giry_return_a_b @ N2 @ ( F @ X3 ) ) ) ) ) ).

% distr_return
thf(fact_886_distr__return,axiom,
    ! [F: a > c,M2: sigma_measure_a,N2: sigma_measure_c,X3: a] :
      ( ( member_a_c @ F @ ( sigma_measurable_a_c @ M2 @ N2 ) )
     => ( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
       => ( ( measure_distr_a_c @ ( giry_return_a @ M2 @ X3 ) @ N2 @ F )
          = ( giry_return_c @ N2 @ ( F @ X3 ) ) ) ) ) ).

% distr_return
thf(fact_887_distr__return,axiom,
    ! [F: a > a,M2: sigma_measure_a,N2: sigma_measure_a,X3: a] :
      ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N2 ) )
     => ( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
       => ( ( measure_distr_a_a @ ( giry_return_a @ M2 @ X3 ) @ N2 @ F )
          = ( giry_return_a @ N2 @ ( F @ X3 ) ) ) ) ) ).

% distr_return
thf(fact_888_distr__return,axiom,
    ! [F: b > a,M2: sigma_measure_b,N2: sigma_measure_a,X3: b] :
      ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ N2 ) )
     => ( ( member_b @ X3 @ ( sigma_space_b @ M2 ) )
       => ( ( measure_distr_b_a @ ( giry_return_b @ M2 @ X3 ) @ N2 @ F )
          = ( giry_return_a @ N2 @ ( F @ X3 ) ) ) ) ) ).

% distr_return
thf(fact_889_distr__return,axiom,
    ! [F: a > b,M2: sigma_measure_a,N2: sigma_measure_b,X3: a] :
      ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ N2 ) )
     => ( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
       => ( ( measure_distr_a_b @ ( giry_return_a @ M2 @ X3 ) @ N2 @ F )
          = ( giry_return_b @ N2 @ ( F @ X3 ) ) ) ) ) ).

% distr_return
thf(fact_890_distr__return,axiom,
    ! [F: b > d,M2: sigma_measure_b,N2: sigma_measure_d,X3: b] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M2 @ N2 ) )
     => ( ( member_b @ X3 @ ( sigma_space_b @ M2 ) )
       => ( ( measure_distr_b_d @ ( giry_return_b @ M2 @ X3 ) @ N2 @ F )
          = ( giry_return_d @ N2 @ ( F @ X3 ) ) ) ) ) ).

% distr_return
thf(fact_891_distr__return,axiom,
    ! [F: b > real,M2: sigma_measure_b,N2: sigma_measure_real,X3: b] :
      ( ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ M2 @ N2 ) )
     => ( ( member_b @ X3 @ ( sigma_space_b @ M2 ) )
       => ( ( measure_distr_b_real @ ( giry_return_b @ M2 @ X3 ) @ N2 @ F )
          = ( giry_return_real @ N2 @ ( F @ X3 ) ) ) ) ) ).

% distr_return
thf(fact_892_distr__return,axiom,
    ! [F: a > real,M2: sigma_measure_a,N2: sigma_measure_real,X3: a] :
      ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M2 @ N2 ) )
     => ( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
       => ( ( measure_distr_a_real @ ( giry_return_a @ M2 @ X3 ) @ N2 @ F )
          = ( giry_return_real @ N2 @ ( F @ X3 ) ) ) ) ) ).

% distr_return
thf(fact_893_subprob__space__return__ne,axiom,
    ! [M2: sigma_2733069582767196501sure_b,X3: sigma_measure_b] :
      ( ( ( sigma_6149183809546982772sure_b @ M2 )
       != bot_bo5242124273094418374sure_b )
     => ( giry_s2296666612018410952sure_b @ ( giry_r3784369487821543sure_b @ M2 @ X3 ) ) ) ).

% subprob_space_return_ne
thf(fact_894_subprob__space__return__ne,axiom,
    ! [M2: sigma_2662035542720850516sure_a,X3: sigma_measure_a] :
      ( ( ( sigma_6149183805243753971sure_a @ M2 )
       != bot_bo5171090233048072389sure_a )
     => ( giry_s2296666607715182151sure_a @ ( giry_r3784365184592742sure_a @ M2 @ X3 ) ) ) ).

% subprob_space_return_ne
thf(fact_895_subprob__space__return__ne,axiom,
    ! [M2: sigma_measure_o,X3: $o] :
      ( ( ( sigma_space_o @ M2 )
       != bot_bot_set_o )
     => ( giry_subprob_space_o @ ( giry_return_o @ M2 @ X3 ) ) ) ).

% subprob_space_return_ne
thf(fact_896_subprob__space__return__ne,axiom,
    ! [M2: sigma_measure_a,X3: a] :
      ( ( ( sigma_space_a @ M2 )
       != bot_bot_set_a )
     => ( giry_subprob_space_a @ ( giry_return_a @ M2 @ X3 ) ) ) ).

% subprob_space_return_ne
thf(fact_897_subprob__space__return__ne,axiom,
    ! [M2: sigma_measure_b,X3: b] :
      ( ( ( sigma_space_b @ M2 )
       != bot_bot_set_b )
     => ( giry_subprob_space_b @ ( giry_return_b @ M2 @ X3 ) ) ) ).

% subprob_space_return_ne
thf(fact_898_prob__space_Odistr__const,axiom,
    ! [M2: sigma_measure_a,C2: c,N2: sigma_measure_c] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( member_c @ C2 @ ( sigma_space_c @ N2 ) )
       => ( ( measure_distr_a_c @ M2 @ N2
            @ ^ [X2: a] : C2 )
          = ( giry_return_c @ N2 @ C2 ) ) ) ) ).

% prob_space.distr_const
thf(fact_899_prob__space_Odistr__const,axiom,
    ! [M2: sigma_measure_a,C2: a,N2: sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( member_a @ C2 @ ( sigma_space_a @ N2 ) )
       => ( ( measure_distr_a_a @ M2 @ N2
            @ ^ [X2: a] : C2 )
          = ( giry_return_a @ N2 @ C2 ) ) ) ) ).

% prob_space.distr_const
thf(fact_900_prob__space_Odistr__const,axiom,
    ! [M2: sigma_measure_a,C2: b,N2: sigma_measure_b] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( member_b @ C2 @ ( sigma_space_b @ N2 ) )
       => ( ( measure_distr_a_b @ M2 @ N2
            @ ^ [X2: a] : C2 )
          = ( giry_return_b @ N2 @ C2 ) ) ) ) ).

% prob_space.distr_const
thf(fact_901_prob__space_Odistr__const,axiom,
    ! [M2: sigma_measure_b,C2: c,N2: sigma_measure_c] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( member_c @ C2 @ ( sigma_space_c @ N2 ) )
       => ( ( measure_distr_b_c @ M2 @ N2
            @ ^ [X2: b] : C2 )
          = ( giry_return_c @ N2 @ C2 ) ) ) ) ).

% prob_space.distr_const
thf(fact_902_prob__space_Odistr__const,axiom,
    ! [M2: sigma_measure_b,C2: a,N2: sigma_measure_a] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( member_a @ C2 @ ( sigma_space_a @ N2 ) )
       => ( ( measure_distr_b_a @ M2 @ N2
            @ ^ [X2: b] : C2 )
          = ( giry_return_a @ N2 @ C2 ) ) ) ) ).

% prob_space.distr_const
thf(fact_903_prob__space_Odistr__const,axiom,
    ! [M2: sigma_measure_b,C2: b,N2: sigma_measure_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( member_b @ C2 @ ( sigma_space_b @ N2 ) )
       => ( ( measure_distr_b_b @ M2 @ N2
            @ ^ [X2: b] : C2 )
          = ( giry_return_b @ N2 @ C2 ) ) ) ) ).

% prob_space.distr_const
thf(fact_904_prob__space_Odistr__const,axiom,
    ! [M2: sigma_measure_a,C2: sigma_measure_b,N2: sigma_2733069582767196501sure_b] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( member3534519381033026579sure_b @ C2 @ ( sigma_6149183809546982772sure_b @ N2 ) )
       => ( ( measur2827329755067293186sure_b @ M2 @ N2
            @ ^ [X2: a] : C2 )
          = ( giry_r3784369487821543sure_b @ N2 @ C2 ) ) ) ) ).

% prob_space.distr_const
thf(fact_905_prob__space_Odistr__const,axiom,
    ! [M2: sigma_measure_a,C2: sigma_measure_a,N2: sigma_2662035542720850516sure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( member3534519376729797778sure_a @ C2 @ ( sigma_6149183805243753971sure_a @ N2 ) )
       => ( ( measur2827329750764064385sure_a @ M2 @ N2
            @ ^ [X2: a] : C2 )
          = ( giry_r3784365184592742sure_a @ N2 @ C2 ) ) ) ) ).

% prob_space.distr_const
thf(fact_906_prob__space_Odistr__const,axiom,
    ! [M2: sigma_measure_b,C2: sigma_measure_b,N2: sigma_2733069582767196501sure_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( member3534519381033026579sure_b @ C2 @ ( sigma_6149183809546982772sure_b @ N2 ) )
       => ( ( measur8960644413728056195sure_b @ M2 @ N2
            @ ^ [X2: b] : C2 )
          = ( giry_r3784369487821543sure_b @ N2 @ C2 ) ) ) ) ).

% prob_space.distr_const
thf(fact_907_prob__space_Odistr__const,axiom,
    ! [M2: sigma_measure_b,C2: sigma_measure_a,N2: sigma_2662035542720850516sure_a] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( member3534519376729797778sure_a @ C2 @ ( sigma_6149183805243753971sure_a @ N2 ) )
       => ( ( measur8960644409424827394sure_a @ M2 @ N2
            @ ^ [X2: b] : C2 )
          = ( giry_r3784365184592742sure_a @ N2 @ C2 ) ) ) ) ).

% prob_space.distr_const
thf(fact_908_prob__space_Oindep__sets__finite__index__sets,axiom,
    ! [M2: sigma_measure_a,F2: b > set_set_a,I: set_b] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe8927441866673418605ts_a_b @ M2 @ F2 @ I )
        = ( ! [J2: set_b] :
              ( ( ord_less_eq_set_b @ J2 @ I )
             => ( ( J2 != bot_bot_set_b )
               => ( ( finite_finite_b @ J2 )
                 => ( indepe8927441866673418605ts_a_b @ M2 @ F2 @ J2 ) ) ) ) ) ) ) ).

% prob_space.indep_sets_finite_index_sets
thf(fact_909_prob__space_Oindep__sets__finite__index__sets,axiom,
    ! [M2: sigma_measure_a,F2: a > set_set_a,I: set_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe8927441866673418604ts_a_a @ M2 @ F2 @ I )
        = ( ! [J2: set_a] :
              ( ( ord_less_eq_set_a @ J2 @ I )
             => ( ( J2 != bot_bot_set_a )
               => ( ( finite_finite_a @ J2 )
                 => ( indepe8927441866673418604ts_a_a @ M2 @ F2 @ J2 ) ) ) ) ) ) ) ).

% prob_space.indep_sets_finite_index_sets
thf(fact_910_prob__space_Oindep__sets__finite__index__sets,axiom,
    ! [M2: sigma_measure_b,F2: b > set_set_b,I: set_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe6139986284700742572ts_b_b @ M2 @ F2 @ I )
        = ( ! [J2: set_b] :
              ( ( ord_less_eq_set_b @ J2 @ I )
             => ( ( J2 != bot_bot_set_b )
               => ( ( finite_finite_b @ J2 )
                 => ( indepe6139986284700742572ts_b_b @ M2 @ F2 @ J2 ) ) ) ) ) ) ) ).

% prob_space.indep_sets_finite_index_sets
thf(fact_911_prob__space_Oindep__sets__finite__index__sets,axiom,
    ! [M2: sigma_measure_b,F2: a > set_set_b,I: set_a] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe6139986284700742571ts_b_a @ M2 @ F2 @ I )
        = ( ! [J2: set_a] :
              ( ( ord_less_eq_set_a @ J2 @ I )
             => ( ( J2 != bot_bot_set_a )
               => ( ( finite_finite_a @ J2 )
                 => ( indepe6139986284700742571ts_b_a @ M2 @ F2 @ J2 ) ) ) ) ) ) ) ).

% prob_space.indep_sets_finite_index_sets
thf(fact_912_prob__space_Oindep__sets__finite__index__sets,axiom,
    ! [M2: sigma_measure_b,F2: nat > set_set_b,I: set_nat] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe7503174356045242851_b_nat @ M2 @ F2 @ I )
        = ( ! [J2: set_nat] :
              ( ( ord_less_eq_set_nat @ J2 @ I )
             => ( ( J2 != bot_bot_set_nat )
               => ( ( finite_finite_nat @ J2 )
                 => ( indepe7503174356045242851_b_nat @ M2 @ F2 @ J2 ) ) ) ) ) ) ) ).

% prob_space.indep_sets_finite_index_sets
thf(fact_913_prob__space_Oindep__sets__finite__index__sets,axiom,
    ! [M2: sigma_measure_a,F2: nat > set_set_a,I: set_nat] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe6267730027088848354_a_nat @ M2 @ F2 @ I )
        = ( ! [J2: set_nat] :
              ( ( ord_less_eq_set_nat @ J2 @ I )
             => ( ( J2 != bot_bot_set_nat )
               => ( ( finite_finite_nat @ J2 )
                 => ( indepe6267730027088848354_a_nat @ M2 @ F2 @ J2 ) ) ) ) ) ) ) ).

% prob_space.indep_sets_finite_index_sets
thf(fact_914_prob__space_Oindep__sets__finite__index__sets,axiom,
    ! [M2: sigma_measure_a,F2: $o > set_set_a,I: set_o] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe7780107833195774214ts_a_o @ M2 @ F2 @ I )
        = ( ! [J2: set_o] :
              ( ( ord_less_eq_set_o @ J2 @ I )
             => ( ( J2 != bot_bot_set_o )
               => ( ( finite_finite_o @ J2 )
                 => ( indepe7780107833195774214ts_a_o @ M2 @ F2 @ J2 ) ) ) ) ) ) ) ).

% prob_space.indep_sets_finite_index_sets
thf(fact_915_prob__space_Oindep__sets__finite__index__sets,axiom,
    ! [M2: sigma_measure_b,F2: $o > set_set_b,I: set_o] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe4880885433731379909ts_b_o @ M2 @ F2 @ I )
        = ( ! [J2: set_o] :
              ( ( ord_less_eq_set_o @ J2 @ I )
             => ( ( J2 != bot_bot_set_o )
               => ( ( finite_finite_o @ J2 )
                 => ( indepe4880885433731379909ts_b_o @ M2 @ F2 @ J2 ) ) ) ) ) ) ) ).

% prob_space.indep_sets_finite_index_sets
thf(fact_916_prob__space_Oindep__sets__finite__index__sets,axiom,
    ! [M2: sigma_measure_a,F2: set_b > set_set_a,I: set_set_b] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe4967106455115002445_set_b @ M2 @ F2 @ I )
        = ( ! [J2: set_set_b] :
              ( ( ord_le3795704787696855135_set_b @ J2 @ I )
             => ( ( J2 != bot_bot_set_set_b )
               => ( ( finite_finite_set_b @ J2 )
                 => ( indepe4967106455115002445_set_b @ M2 @ F2 @ J2 ) ) ) ) ) ) ) ).

% prob_space.indep_sets_finite_index_sets
thf(fact_917_prob__space_Oindep__sets__finite__index__sets,axiom,
    ! [M2: sigma_measure_a,F2: set_a > set_set_a,I: set_set_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe4967106450811773644_set_a @ M2 @ F2 @ I )
        = ( ! [J2: set_set_a] :
              ( ( ord_le3724670747650509150_set_a @ J2 @ I )
             => ( ( J2 != bot_bot_set_set_a )
               => ( ( finite_finite_set_a @ J2 )
                 => ( indepe4967106450811773644_set_a @ M2 @ F2 @ J2 ) ) ) ) ) ) ) ).

% prob_space.indep_sets_finite_index_sets
thf(fact_918_Collect__subset,axiom,
    ! [A: set_c,P: c > $o] :
      ( ord_less_eq_set_c
      @ ( collect_c
        @ ^ [X2: c] :
            ( ( member_c @ X2 @ A )
            & ( P @ X2 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_919_Collect__subset,axiom,
    ! [A: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X2: nat] :
            ( ( member_nat @ X2 @ A )
            & ( P @ X2 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_920_Collect__subset,axiom,
    ! [A: set_b,P: b > $o] :
      ( ord_less_eq_set_b
      @ ( collect_b
        @ ^ [X2: b] :
            ( ( member_b @ X2 @ A )
            & ( P @ X2 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_921_Collect__subset,axiom,
    ! [A: set_a,P: a > $o] :
      ( ord_less_eq_set_a
      @ ( collect_a
        @ ^ [X2: a] :
            ( ( member_a @ X2 @ A )
            & ( P @ X2 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_922_Collect__subset,axiom,
    ! [A: set_set_b,P: set_b > $o] :
      ( ord_le3795704787696855135_set_b
      @ ( collect_set_b
        @ ^ [X2: set_b] :
            ( ( member_set_b @ X2 @ A )
            & ( P @ X2 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_923_Collect__subset,axiom,
    ! [A: set_set_a,P: set_a > $o] :
      ( ord_le3724670747650509150_set_a
      @ ( collect_set_a
        @ ^ [X2: set_a] :
            ( ( member_set_a @ X2 @ A )
            & ( P @ X2 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_924_Collect__subset,axiom,
    ! [A: set_a_real,P: ( a > real ) > $o] :
      ( ord_le3334967407727675675a_real
      @ ( collect_a_real
        @ ^ [X2: a > real] :
            ( ( member_a_real @ X2 @ A )
            & ( P @ X2 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_925_Collect__subset,axiom,
    ! [A: set_a_b,P: ( a > b ) > $o] :
      ( ord_less_eq_set_a_b
      @ ( collect_a_b
        @ ^ [X2: a > b] :
            ( ( member_a_b @ X2 @ A )
            & ( P @ X2 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_926_Collect__subset,axiom,
    ! [A: set_b_a,P: ( b > a ) > $o] :
      ( ord_less_eq_set_b_a
      @ ( collect_b_a
        @ ^ [X2: b > a] :
            ( ( member_b_a @ X2 @ A )
            & ( P @ X2 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_927_Collect__subset,axiom,
    ! [A: set_a_c,P: ( a > c ) > $o] :
      ( ord_less_eq_set_a_c
      @ ( collect_a_c
        @ ^ [X2: a > c] :
            ( ( member_a_c @ X2 @ A )
            & ( P @ X2 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_928_empty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X2: nat] : $false ) ) ).

% empty_def
thf(fact_929_empty__def,axiom,
    ( bot_bot_set_a
    = ( collect_a
      @ ^ [X2: a] : $false ) ) ).

% empty_def
thf(fact_930_empty__def,axiom,
    ( bot_bot_set_b
    = ( collect_b
      @ ^ [X2: b] : $false ) ) ).

% empty_def
thf(fact_931_empty__def,axiom,
    ( bot_bot_set_o
    = ( collect_o
      @ ^ [X2: $o] : $false ) ) ).

% empty_def
thf(fact_932_compose__const_I1_J,axiom,
    ! [F: b > d,A2: b] :
      ( ( comp_b_d_a @ F
        @ ^ [X2: a] : A2 )
      = ( ^ [X2: a] : ( F @ A2 ) ) ) ).

% compose_const(1)
thf(fact_933_measurable__completion,axiom,
    ! [F: b > a,M2: sigma_measure_b,N2: sigma_measure_a] :
      ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ N2 ) )
     => ( member_b_a @ F @ ( sigma_measurable_b_a @ ( comple3428971583294703881tion_b @ M2 ) @ N2 ) ) ) ).

% measurable_completion
thf(fact_934_measurable__completion,axiom,
    ! [F: b > d,M2: sigma_measure_b,N2: sigma_measure_d] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M2 @ N2 ) )
     => ( member_b_d @ F @ ( sigma_measurable_b_d @ ( comple3428971583294703881tion_b @ M2 ) @ N2 ) ) ) ).

% measurable_completion
thf(fact_935_measurable__completion,axiom,
    ! [F: b > real,M2: sigma_measure_b,N2: sigma_measure_real] :
      ( ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ M2 @ N2 ) )
     => ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ ( comple3428971583294703881tion_b @ M2 ) @ N2 ) ) ) ).

% measurable_completion
thf(fact_936_measurable__completion,axiom,
    ! [F: a > a > real,M2: sigma_measure_a,N2: sigma_measure_a_real] :
      ( ( member_a_a_real @ F @ ( sigma_5073124708296780141a_real @ M2 @ N2 ) )
     => ( member_a_a_real @ F @ ( sigma_5073124708296780141a_real @ ( comple3428971583294703880tion_a @ M2 ) @ N2 ) ) ) ).

% measurable_completion
thf(fact_937_measurable__completion,axiom,
    ! [F: a > a > b,M2: sigma_measure_a,N2: sigma_measure_a_b] :
      ( ( member_a_a_b @ F @ ( sigma_3971135317854480532_a_a_b @ M2 @ N2 ) )
     => ( member_a_a_b @ F @ ( sigma_3971135317854480532_a_a_b @ ( comple3428971583294703880tion_a @ M2 ) @ N2 ) ) ) ).

% measurable_completion
thf(fact_938_measurable__completion,axiom,
    ! [F: a > c,M2: sigma_measure_a,N2: sigma_measure_c] :
      ( ( member_a_c @ F @ ( sigma_measurable_a_c @ M2 @ N2 ) )
     => ( member_a_c @ F @ ( sigma_measurable_a_c @ ( comple3428971583294703880tion_a @ M2 ) @ N2 ) ) ) ).

% measurable_completion
thf(fact_939_measurable__completion,axiom,
    ! [F: a > a,M2: sigma_measure_a,N2: sigma_measure_a] :
      ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N2 ) )
     => ( member_a_a @ F @ ( sigma_measurable_a_a @ ( comple3428971583294703880tion_a @ M2 ) @ N2 ) ) ) ).

% measurable_completion
thf(fact_940_measurable__completion,axiom,
    ! [F: a > b,M2: sigma_measure_a,N2: sigma_measure_b] :
      ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ N2 ) )
     => ( member_a_b @ F @ ( sigma_measurable_a_b @ ( comple3428971583294703880tion_a @ M2 ) @ N2 ) ) ) ).

% measurable_completion
thf(fact_941_measurable__completion,axiom,
    ! [F: a > real,M2: sigma_measure_a,N2: sigma_measure_real] :
      ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M2 @ N2 ) )
     => ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ ( comple3428971583294703880tion_a @ M2 ) @ N2 ) ) ) ).

% measurable_completion
thf(fact_942_distr__completion,axiom,
    ! [X4: b > a,M2: sigma_measure_b,N2: sigma_measure_a] :
      ( ( member_b_a @ X4 @ ( sigma_measurable_b_a @ M2 @ N2 ) )
     => ( ( measure_distr_b_a @ ( comple3428971583294703881tion_b @ M2 ) @ N2 @ X4 )
        = ( measure_distr_b_a @ M2 @ N2 @ X4 ) ) ) ).

% distr_completion
thf(fact_943_distr__completion,axiom,
    ! [X4: b > d,M2: sigma_measure_b,N2: sigma_measure_d] :
      ( ( member_b_d @ X4 @ ( sigma_measurable_b_d @ M2 @ N2 ) )
     => ( ( measure_distr_b_d @ ( comple3428971583294703881tion_b @ M2 ) @ N2 @ X4 )
        = ( measure_distr_b_d @ M2 @ N2 @ X4 ) ) ) ).

% distr_completion
thf(fact_944_distr__completion,axiom,
    ! [X4: b > real,M2: sigma_measure_b,N2: sigma_measure_real] :
      ( ( member_b_real @ X4 @ ( sigma_5489272243865111003b_real @ M2 @ N2 ) )
     => ( ( measure_distr_b_real @ ( comple3428971583294703881tion_b @ M2 ) @ N2 @ X4 )
        = ( measure_distr_b_real @ M2 @ N2 @ X4 ) ) ) ).

% distr_completion
thf(fact_945_distr__completion,axiom,
    ! [X4: a > a > real,M2: sigma_measure_a,N2: sigma_measure_a_real] :
      ( ( member_a_a_real @ X4 @ ( sigma_5073124708296780141a_real @ M2 @ N2 ) )
     => ( ( measur6806378183764566549a_real @ ( comple3428971583294703880tion_a @ M2 ) @ N2 @ X4 )
        = ( measur6806378183764566549a_real @ M2 @ N2 @ X4 ) ) ) ).

% distr_completion
thf(fact_946_distr__completion,axiom,
    ! [X4: a > a > b,M2: sigma_measure_a,N2: sigma_measure_a_b] :
      ( ( member_a_a_b @ X4 @ ( sigma_3971135317854480532_a_a_b @ M2 @ N2 ) )
     => ( ( measure_distr_a_a_b @ ( comple3428971583294703880tion_a @ M2 ) @ N2 @ X4 )
        = ( measure_distr_a_a_b @ M2 @ N2 @ X4 ) ) ) ).

% distr_completion
thf(fact_947_distr__completion,axiom,
    ! [X4: a > c,M2: sigma_measure_a,N2: sigma_measure_c] :
      ( ( member_a_c @ X4 @ ( sigma_measurable_a_c @ M2 @ N2 ) )
     => ( ( measure_distr_a_c @ ( comple3428971583294703880tion_a @ M2 ) @ N2 @ X4 )
        = ( measure_distr_a_c @ M2 @ N2 @ X4 ) ) ) ).

% distr_completion
thf(fact_948_distr__completion,axiom,
    ! [X4: a > a,M2: sigma_measure_a,N2: sigma_measure_a] :
      ( ( member_a_a @ X4 @ ( sigma_measurable_a_a @ M2 @ N2 ) )
     => ( ( measure_distr_a_a @ ( comple3428971583294703880tion_a @ M2 ) @ N2 @ X4 )
        = ( measure_distr_a_a @ M2 @ N2 @ X4 ) ) ) ).

% distr_completion
thf(fact_949_distr__completion,axiom,
    ! [X4: a > b,M2: sigma_measure_a,N2: sigma_measure_b] :
      ( ( member_a_b @ X4 @ ( sigma_measurable_a_b @ M2 @ N2 ) )
     => ( ( measure_distr_a_b @ ( comple3428971583294703880tion_a @ M2 ) @ N2 @ X4 )
        = ( measure_distr_a_b @ M2 @ N2 @ X4 ) ) ) ).

% distr_completion
thf(fact_950_distr__completion,axiom,
    ! [X4: a > real,M2: sigma_measure_a,N2: sigma_measure_real] :
      ( ( member_a_real @ X4 @ ( sigma_9116425665531756122a_real @ M2 @ N2 ) )
     => ( ( measure_distr_a_real @ ( comple3428971583294703880tion_a @ M2 ) @ N2 @ X4 )
        = ( measure_distr_a_real @ M2 @ N2 @ X4 ) ) ) ).

% distr_completion
thf(fact_951_assms_I3_J,axiom,
    ( indepe4851901773132442935_a_c_d @ m @ m2
    @ ^ [I3: c] : ( comp_b_d_a @ ( x @ I3 ) @ f )
    @ i ) ).

% assms(3)
thf(fact_952_D_Oindep__events__finite__index__events,axiom,
    ! [F2: nat > set_b,I: set_nat] :
      ( ( indepe2786641642957426683_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
      = ( ! [J2: set_nat] :
            ( ( ord_less_eq_set_nat @ J2 @ I )
           => ( ( J2 != bot_bot_set_nat )
             => ( ( finite_finite_nat @ J2 )
               => ( indepe2786641642957426683_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ J2 ) ) ) ) ) ) ).

% D.indep_events_finite_index_events
thf(fact_953_D_Oindep__events__finite__index__events,axiom,
    ! [F2: $o > set_b,I: set_o] :
      ( ( indepe796274259248320173ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
      = ( ! [J2: set_o] :
            ( ( ord_less_eq_set_o @ J2 @ I )
           => ( ( J2 != bot_bot_set_o )
             => ( ( finite_finite_o @ J2 )
               => ( indepe796274259248320173ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ J2 ) ) ) ) ) ) ).

% D.indep_events_finite_index_events
thf(fact_954_D_Oindep__events__finite__index__events,axiom,
    ! [F2: set_b > set_b,I: set_set_b] :
      ( ( indepe1795572657417732212_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
      = ( ! [J2: set_set_b] :
            ( ( ord_le3795704787696855135_set_b @ J2 @ I )
           => ( ( J2 != bot_bot_set_set_b )
             => ( ( finite_finite_set_b @ J2 )
               => ( indepe1795572657417732212_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ J2 ) ) ) ) ) ) ).

% D.indep_events_finite_index_events
thf(fact_955_D_Oindep__events__finite__index__events,axiom,
    ! [F2: set_a > set_b,I: set_set_a] :
      ( ( indepe1795572653114503411_set_a @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
      = ( ! [J2: set_set_a] :
            ( ( ord_le3724670747650509150_set_a @ J2 @ I )
           => ( ( J2 != bot_bot_set_set_a )
             => ( ( finite_finite_set_a @ J2 )
               => ( indepe1795572653114503411_set_a @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ J2 ) ) ) ) ) ) ).

% D.indep_events_finite_index_events
thf(fact_956_D_Oindep__events__finite__index__events,axiom,
    ! [F2: b > set_b,I: set_b] :
      ( ( indepe8384762880973921172ts_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
      = ( ! [J2: set_b] :
            ( ( ord_less_eq_set_b @ J2 @ I )
           => ( ( J2 != bot_bot_set_b )
             => ( ( finite_finite_b @ J2 )
               => ( indepe8384762880973921172ts_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ J2 ) ) ) ) ) ) ).

% D.indep_events_finite_index_events
thf(fact_957_D_Oindep__events__finite__index__events,axiom,
    ! [F2: a > set_b,I: set_a] :
      ( ( indepe8384762880973921171ts_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ I )
      = ( ! [J2: set_a] :
            ( ( ord_less_eq_set_a @ J2 @ I )
           => ( ( J2 != bot_bot_set_a )
             => ( ( finite_finite_a @ J2 )
               => ( indepe8384762880973921171ts_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ F2 @ J2 ) ) ) ) ) ) ).

% D.indep_events_finite_index_events
thf(fact_958_indep__events__finite__index__events,axiom,
    ! [F2: nat > set_a,I: set_nat] :
      ( ( indepe1551197314001032186_a_nat @ m @ F2 @ I )
      = ( ! [J2: set_nat] :
            ( ( ord_less_eq_set_nat @ J2 @ I )
           => ( ( J2 != bot_bot_set_nat )
             => ( ( finite_finite_nat @ J2 )
               => ( indepe1551197314001032186_a_nat @ m @ F2 @ J2 ) ) ) ) ) ) ).

% indep_events_finite_index_events
thf(fact_959_indep__events__finite__index__events,axiom,
    ! [F2: $o > set_a,I: set_o] :
      ( ( indepe3695496658712714478ts_a_o @ m @ F2 @ I )
      = ( ! [J2: set_o] :
            ( ( ord_less_eq_set_o @ J2 @ I )
           => ( ( J2 != bot_bot_set_o )
             => ( ( finite_finite_o @ J2 )
               => ( indepe3695496658712714478ts_a_o @ m @ F2 @ J2 ) ) ) ) ) ) ).

% indep_events_finite_index_events
thf(fact_960_indep__events__finite__index__events,axiom,
    ! [F2: set_b > set_a,I: set_set_b] :
      ( ( indepe451107620608244277_set_b @ m @ F2 @ I )
      = ( ! [J2: set_set_b] :
            ( ( ord_le3795704787696855135_set_b @ J2 @ I )
           => ( ( J2 != bot_bot_set_set_b )
             => ( ( finite_finite_set_b @ J2 )
               => ( indepe451107620608244277_set_b @ m @ F2 @ J2 ) ) ) ) ) ) ).

% indep_events_finite_index_events
thf(fact_961_indep__events__finite__index__events,axiom,
    ! [F2: set_a > set_a,I: set_set_a] :
      ( ( indepe451107616305015476_set_a @ m @ F2 @ I )
      = ( ! [J2: set_set_a] :
            ( ( ord_le3724670747650509150_set_a @ J2 @ I )
           => ( ( J2 != bot_bot_set_set_a )
             => ( ( finite_finite_set_a @ J2 )
               => ( indepe451107616305015476_set_a @ m @ F2 @ J2 ) ) ) ) ) ) ).

% indep_events_finite_index_events
thf(fact_962_indep__events__finite__index__events,axiom,
    ! [F2: b > set_a,I: set_b] :
      ( ( indepe1948846426091821397ts_a_b @ m @ F2 @ I )
      = ( ! [J2: set_b] :
            ( ( ord_less_eq_set_b @ J2 @ I )
           => ( ( J2 != bot_bot_set_b )
             => ( ( finite_finite_b @ J2 )
               => ( indepe1948846426091821397ts_a_b @ m @ F2 @ J2 ) ) ) ) ) ) ).

% indep_events_finite_index_events
thf(fact_963_indep__events__finite__index__events,axiom,
    ! [F2: a > set_a,I: set_a] :
      ( ( indepe1948846426091821396ts_a_a @ m @ F2 @ I )
      = ( ! [J2: set_a] :
            ( ( ord_less_eq_set_a @ J2 @ I )
           => ( ( J2 != bot_bot_set_a )
             => ( ( finite_finite_a @ J2 )
               => ( indepe1948846426091821396ts_a_a @ m @ F2 @ J2 ) ) ) ) ) ) ).

% indep_events_finite_index_events
thf(fact_964_finite__Collect__subsets,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B2: set_nat] : ( ord_less_eq_set_nat @ B2 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_965_finite__Collect__subsets,axiom,
    ! [A: set_set_b] :
      ( ( finite_finite_set_b @ A )
     => ( finite7280322010187229928_set_b
        @ ( collect_set_set_b
          @ ^ [B2: set_set_b] : ( ord_le3795704787696855135_set_b @ B2 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_966_finite__Collect__subsets,axiom,
    ! [A: set_set_a] :
      ( ( finite_finite_set_a @ A )
     => ( finite7209287970140883943_set_a
        @ ( collect_set_set_a
          @ ^ [B2: set_set_a] : ( ord_le3724670747650509150_set_a @ B2 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_967_finite__Collect__subsets,axiom,
    ! [A: set_b] :
      ( ( finite_finite_b @ A )
     => ( finite_finite_set_b
        @ ( collect_set_b
          @ ^ [B2: set_b] : ( ord_less_eq_set_b @ B2 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_968_finite__Collect__subsets,axiom,
    ! [A: set_a] :
      ( ( finite_finite_a @ A )
     => ( finite_finite_set_a
        @ ( collect_set_a
          @ ^ [B2: set_a] : ( ord_less_eq_set_a @ B2 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_969_D_Obind__in__space,axiom,
    ! [A: b > sigma_measure_b,N2: sigma_measure_b] :
      ( ( member7751282110383867797sure_b @ A @ ( sigma_6831777305601499691sure_b @ ( measure_distr_a_b @ m @ n @ f ) @ ( giry_s2448339315546285521ebra_b @ N2 ) ) )
     => ( member3534519381033026579sure_b @ ( giry_bind_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ A ) @ ( sigma_6149183809546982772sure_b @ ( giry_s2448339315546285521ebra_b @ N2 ) ) ) ) ).

% D.bind_in_space
thf(fact_970_D_Obind__in__space,axiom,
    ! [A: b > sigma_measure_a,N2: sigma_measure_a] :
      ( ( member7680248070337521812sure_a @ A @ ( sigma_6831777301298270890sure_a @ ( measure_distr_a_b @ m @ n @ f ) @ ( giry_s2448339315546285520ebra_a @ N2 ) ) )
     => ( member3534519376729797778sure_a @ ( giry_bind_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ A ) @ ( sigma_6149183805243753971sure_a @ ( giry_s2448339315546285520ebra_a @ N2 ) ) ) ) ).

% D.bind_in_space
thf(fact_971_prob__space_Oprob__space__stream__space,axiom,
    ! [M2: sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( probab6181332902410252999ream_a @ ( stream2930637714550841297pace_a @ M2 ) ) ) ).

% prob_space.prob_space_stream_space
thf(fact_972_prob__space_Oprob__space__stream__space,axiom,
    ! [M2: sigma_measure_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( probab6181332906713481800ream_b @ ( stream2930637714550841298pace_b @ M2 ) ) ) ).

% prob_space.prob_space_stream_space
thf(fact_973_prob__space_Oprob__space__stream__space,axiom,
    ! [M2: sigma_6673148739008212994ream_b] :
      ( ( probab6181332906713481800ream_b @ M2 )
     => ( probab3987223172309149518ream_b @ ( stream5384723080203383896ream_b @ M2 ) ) ) ).

% prob_space.prob_space_stream_space
thf(fact_974_prob__space_Oprob__space__stream__space,axiom,
    ! [M2: sigma_6602114698961867009ream_a] :
      ( ( probab6181332902410252999ream_a @ M2 )
     => ( probab3916189132262803533ream_a @ ( stream5384723075900155095ream_a @ M2 ) ) ) ).

% prob_space.prob_space_stream_space
thf(fact_975_finite__Collect__disjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X2: nat] :
              ( ( P @ X2 )
              | ( Q @ X2 ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P ) )
        & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_976_finite__Collect__disjI,axiom,
    ! [P: b > $o,Q: b > $o] :
      ( ( finite_finite_b
        @ ( collect_b
          @ ^ [X2: b] :
              ( ( P @ X2 )
              | ( Q @ X2 ) ) ) )
      = ( ( finite_finite_b @ ( collect_b @ P ) )
        & ( finite_finite_b @ ( collect_b @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_977_finite__Collect__disjI,axiom,
    ! [P: a > $o,Q: a > $o] :
      ( ( finite_finite_a
        @ ( collect_a
          @ ^ [X2: a] :
              ( ( P @ X2 )
              | ( Q @ X2 ) ) ) )
      = ( ( finite_finite_a @ ( collect_a @ P ) )
        & ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_978_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
          @ ^ [X2: nat] :
              ( ( P @ X2 )
              & ( Q @ X2 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_979_finite__Collect__conjI,axiom,
    ! [P: b > $o,Q: b > $o] :
      ( ( ( finite_finite_b @ ( collect_b @ P ) )
        | ( finite_finite_b @ ( collect_b @ Q ) ) )
     => ( finite_finite_b
        @ ( collect_b
          @ ^ [X2: b] :
              ( ( P @ X2 )
              & ( Q @ X2 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_980_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
          @ ^ [X2: a] :
              ( ( P @ X2 )
              & ( Q @ X2 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_981_sigma__finite__measure__distr,axiom,
    ! [M2: sigma_measure_a,N2: sigma_measure_a_real,F: a > a > real] :
      ( ( measur5819825744149550268a_real @ ( measur6806378183764566549a_real @ M2 @ N2 @ F ) )
     => ( ( member_a_a_real @ F @ ( sigma_5073124708296780141a_real @ M2 @ N2 ) )
       => ( measur4308613598931908895sure_a @ M2 ) ) ) ).

% sigma_finite_measure_distr
thf(fact_982_sigma__finite__measure__distr,axiom,
    ! [M2: sigma_measure_a,N2: sigma_measure_a_b,F: a > a > b] :
      ( ( measur5182267031777123205re_a_b @ ( measure_distr_a_a_b @ M2 @ N2 @ F ) )
     => ( ( member_a_a_b @ F @ ( sigma_3971135317854480532_a_a_b @ M2 @ N2 ) )
       => ( measur4308613598931908895sure_a @ M2 ) ) ) ).

% sigma_finite_measure_distr
thf(fact_983_sigma__finite__measure__distr,axiom,
    ! [M2: sigma_measure_a,N2: sigma_measure_c,F: a > c] :
      ( ( measur4308613598931908897sure_c @ ( measure_distr_a_c @ M2 @ N2 @ F ) )
     => ( ( member_a_c @ F @ ( sigma_measurable_a_c @ M2 @ N2 ) )
       => ( measur4308613598931908895sure_a @ M2 ) ) ) ).

% sigma_finite_measure_distr
thf(fact_984_sigma__finite__measure__distr,axiom,
    ! [M2: sigma_measure_a,N2: sigma_measure_real,F: a > real] :
      ( ( measur487378040549452491e_real @ ( measure_distr_a_real @ M2 @ N2 @ F ) )
     => ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M2 @ N2 ) )
       => ( measur4308613598931908895sure_a @ M2 ) ) ) ).

% sigma_finite_measure_distr
thf(fact_985_sigma__finite__measure__distr,axiom,
    ! [M2: sigma_measure_b,N2: sigma_measure_d,F: b > d] :
      ( ( measur4308613598931908898sure_d @ ( measure_distr_b_d @ M2 @ N2 @ F ) )
     => ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M2 @ N2 ) )
       => ( measur4308613598931908896sure_b @ M2 ) ) ) ).

% sigma_finite_measure_distr
thf(fact_986_sigma__finite__measure__distr,axiom,
    ! [M2: sigma_measure_b,N2: sigma_measure_real,F: b > real] :
      ( ( measur487378040549452491e_real @ ( measure_distr_b_real @ M2 @ N2 @ F ) )
     => ( ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ M2 @ N2 ) )
       => ( measur4308613598931908896sure_b @ M2 ) ) ) ).

% sigma_finite_measure_distr
thf(fact_987_sigma__finite__measure__distr,axiom,
    ! [M2: sigma_measure_a,N2: sigma_measure_a,F: a > a] :
      ( ( measur4308613598931908895sure_a @ ( measure_distr_a_a @ M2 @ N2 @ F ) )
     => ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N2 ) )
       => ( measur4308613598931908895sure_a @ M2 ) ) ) ).

% sigma_finite_measure_distr
thf(fact_988_sigma__finite__measure__distr,axiom,
    ! [M2: sigma_measure_b,N2: sigma_measure_a,F: b > a] :
      ( ( measur4308613598931908895sure_a @ ( measure_distr_b_a @ M2 @ N2 @ F ) )
     => ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ N2 ) )
       => ( measur4308613598931908896sure_b @ M2 ) ) ) ).

% sigma_finite_measure_distr
thf(fact_989_sigma__finite__measure__distr,axiom,
    ! [M2: sigma_measure_a,N2: sigma_measure_b,F: a > b] :
      ( ( measur4308613598931908896sure_b @ ( measure_distr_a_b @ M2 @ N2 @ F ) )
     => ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ N2 ) )
       => ( measur4308613598931908895sure_a @ M2 ) ) ) ).

% sigma_finite_measure_distr
thf(fact_990_sigma__finite__measure__distr,axiom,
    ! [M2: sigma_measure_b,N2: sigma_measure_b,F: b > b] :
      ( ( measur4308613598931908896sure_b @ ( measure_distr_b_b @ M2 @ N2 @ F ) )
     => ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M2 @ N2 ) )
       => ( measur4308613598931908896sure_b @ M2 ) ) ) ).

% sigma_finite_measure_distr
thf(fact_991_prob__space__distrD,axiom,
    ! [F: a > c,M2: sigma_measure_a,N2: sigma_measure_c] :
      ( ( member_a_c @ F @ ( sigma_measurable_a_c @ M2 @ N2 ) )
     => ( ( probab7247484486040049091pace_c @ ( measure_distr_a_c @ M2 @ N2 @ F ) )
       => ( probab7247484486040049089pace_a @ M2 ) ) ) ).

% prob_space_distrD
thf(fact_992_prob__space__distrD,axiom,
    ! [F: a > real,M2: sigma_measure_a,N2: sigma_measure_real] :
      ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M2 @ N2 ) )
     => ( ( probab535871623910865577e_real @ ( measure_distr_a_real @ M2 @ N2 @ F ) )
       => ( probab7247484486040049089pace_a @ M2 ) ) ) ).

% prob_space_distrD
thf(fact_993_prob__space__distrD,axiom,
    ! [F: b > d,M2: sigma_measure_b,N2: sigma_measure_d] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M2 @ N2 ) )
     => ( ( probab7247484486040049092pace_d @ ( measure_distr_b_d @ M2 @ N2 @ F ) )
       => ( probab7247484486040049090pace_b @ M2 ) ) ) ).

% prob_space_distrD
thf(fact_994_prob__space__distrD,axiom,
    ! [F: b > real,M2: sigma_measure_b,N2: sigma_measure_real] :
      ( ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ M2 @ N2 ) )
     => ( ( probab535871623910865577e_real @ ( measure_distr_b_real @ M2 @ N2 @ F ) )
       => ( probab7247484486040049090pace_b @ M2 ) ) ) ).

% prob_space_distrD
thf(fact_995_prob__space__distrD,axiom,
    ! [F: a > a,M2: sigma_measure_a,N2: sigma_measure_a] :
      ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N2 ) )
     => ( ( probab7247484486040049089pace_a @ ( measure_distr_a_a @ M2 @ N2 @ F ) )
       => ( probab7247484486040049089pace_a @ M2 ) ) ) ).

% prob_space_distrD
thf(fact_996_prob__space__distrD,axiom,
    ! [F: b > a,M2: sigma_measure_b,N2: sigma_measure_a] :
      ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ N2 ) )
     => ( ( probab7247484486040049089pace_a @ ( measure_distr_b_a @ M2 @ N2 @ F ) )
       => ( probab7247484486040049090pace_b @ M2 ) ) ) ).

% prob_space_distrD
thf(fact_997_prob__space__distrD,axiom,
    ! [F: a > b,M2: sigma_measure_a,N2: sigma_measure_b] :
      ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ N2 ) )
     => ( ( probab7247484486040049090pace_b @ ( measure_distr_a_b @ M2 @ N2 @ F ) )
       => ( probab7247484486040049089pace_a @ M2 ) ) ) ).

% prob_space_distrD
thf(fact_998_prob__space__distrD,axiom,
    ! [F: b > b,M2: sigma_measure_b,N2: sigma_measure_b] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M2 @ N2 ) )
     => ( ( probab7247484486040049090pace_b @ ( measure_distr_b_b @ M2 @ N2 @ F ) )
       => ( probab7247484486040049090pace_b @ M2 ) ) ) ).

% prob_space_distrD
thf(fact_999_prob__space__distrD,axiom,
    ! [F: stream_b > a,M2: sigma_6673148739008212994ream_b,N2: sigma_measure_a] :
      ( ( member_stream_b_a @ F @ ( sigma_5590233989479323913am_b_a @ M2 @ N2 ) )
     => ( ( probab7247484486040049089pace_a @ ( measur7845399966760478129am_b_a @ M2 @ N2 @ F ) )
       => ( probab6181332906713481800ream_b @ M2 ) ) ) ).

% prob_space_distrD
thf(fact_1000_prob__space__distrD,axiom,
    ! [F: stream_a > a,M2: sigma_6602114698961867009ream_a,N2: sigma_measure_a] :
      ( ( member_stream_a_a @ F @ ( sigma_6225882990084310602am_a_a @ M2 @ N2 ) )
     => ( ( probab7247484486040049089pace_a @ ( measur8481048967365464818am_a_a @ M2 @ N2 @ F ) )
       => ( probab6181332902410252999ream_a @ M2 ) ) ) ).

% prob_space_distrD
thf(fact_1001_prob__space__stream__space,axiom,
    probab6181332902410252999ream_a @ ( stream2930637714550841297pace_a @ m ) ).

% prob_space_stream_space
thf(fact_1002_M__in__subprob,axiom,
    member3534519376729797778sure_a @ m @ ( sigma_6149183805243753971sure_a @ ( giry_s2448339315546285520ebra_a @ m ) ) ).

% M_in_subprob
thf(fact_1003_assms_I2_J,axiom,
    ! [I4: c] :
      ( ( member_c @ I4 @ i )
     => ( member_b_d @ ( x @ I4 ) @ ( sigma_measurable_b_d @ n @ ( m2 @ I4 ) ) ) ) ).

% assms(2)
thf(fact_1004_space__bind__empty,axiom,
    ! [M2: sigma_measure_a,F: a > sigma_measure_a] :
      ( ( ( sigma_space_a @ M2 )
        = bot_bot_set_a )
     => ( ( sigma_space_a @ ( giry_bind_a_a @ M2 @ F ) )
        = bot_bot_set_a ) ) ).

% space_bind_empty
thf(fact_1005_space__bind__empty,axiom,
    ! [M2: sigma_measure_a,F: a > sigma_measure_b] :
      ( ( ( sigma_space_a @ M2 )
        = bot_bot_set_a )
     => ( ( sigma_space_b @ ( giry_bind_a_b @ M2 @ F ) )
        = bot_bot_set_b ) ) ).

% space_bind_empty
thf(fact_1006_space__bind__empty,axiom,
    ! [M2: sigma_measure_a,F: a > sigma_measure_o] :
      ( ( ( sigma_space_a @ M2 )
        = bot_bot_set_a )
     => ( ( sigma_space_o @ ( giry_bind_a_o @ M2 @ F ) )
        = bot_bot_set_o ) ) ).

% space_bind_empty
thf(fact_1007_space__bind__empty,axiom,
    ! [M2: sigma_measure_b,F: b > sigma_measure_a] :
      ( ( ( sigma_space_b @ M2 )
        = bot_bot_set_b )
     => ( ( sigma_space_a @ ( giry_bind_b_a @ M2 @ F ) )
        = bot_bot_set_a ) ) ).

% space_bind_empty
thf(fact_1008_space__bind__empty,axiom,
    ! [M2: sigma_measure_b,F: b > sigma_measure_b] :
      ( ( ( sigma_space_b @ M2 )
        = bot_bot_set_b )
     => ( ( sigma_space_b @ ( giry_bind_b_b @ M2 @ F ) )
        = bot_bot_set_b ) ) ).

% space_bind_empty
thf(fact_1009_space__bind__empty,axiom,
    ! [M2: sigma_measure_b,F: b > sigma_measure_o] :
      ( ( ( sigma_space_b @ M2 )
        = bot_bot_set_b )
     => ( ( sigma_space_o @ ( giry_bind_b_o @ M2 @ F ) )
        = bot_bot_set_o ) ) ).

% space_bind_empty
thf(fact_1010_space__bind__empty,axiom,
    ! [M2: sigma_measure_o,F: $o > sigma_measure_a] :
      ( ( ( sigma_space_o @ M2 )
        = bot_bot_set_o )
     => ( ( sigma_space_a @ ( giry_bind_o_a @ M2 @ F ) )
        = bot_bot_set_a ) ) ).

% space_bind_empty
thf(fact_1011_space__bind__empty,axiom,
    ! [M2: sigma_measure_o,F: $o > sigma_measure_b] :
      ( ( ( sigma_space_o @ M2 )
        = bot_bot_set_o )
     => ( ( sigma_space_b @ ( giry_bind_o_b @ M2 @ F ) )
        = bot_bot_set_b ) ) ).

% space_bind_empty
thf(fact_1012_space__bind__empty,axiom,
    ! [M2: sigma_measure_o,F: $o > sigma_measure_o] :
      ( ( ( sigma_space_o @ M2 )
        = bot_bot_set_o )
     => ( ( sigma_space_o @ ( giry_bind_o_o @ M2 @ F ) )
        = bot_bot_set_o ) ) ).

% space_bind_empty
thf(fact_1013_space__bind__empty,axiom,
    ! [M2: sigma_2733069582767196501sure_b,F: sigma_measure_b > sigma_measure_a] :
      ( ( ( sigma_6149183809546982772sure_b @ M2 )
        = bot_bo5242124273094418374sure_b )
     => ( ( sigma_space_a @ ( giry_b1258390083388040233re_b_a @ M2 @ F ) )
        = bot_bot_set_a ) ) ).

% space_bind_empty
thf(fact_1014_subprob__space__bind,axiom,
    ! [M2: sigma_measure_a,F: a > sigma_measure_b,N2: sigma_measure_b] :
      ( ( giry_subprob_space_a @ M2 )
     => ( ( member1617967451723104788sure_b @ F @ ( sigma_698462646940736682sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ N2 ) ) )
       => ( giry_subprob_space_b @ ( giry_bind_a_b @ M2 @ F ) ) ) ) ).

% subprob_space_bind
thf(fact_1015_subprob__space__bind,axiom,
    ! [M2: sigma_measure_b,F: b > sigma_measure_b,N2: sigma_measure_b] :
      ( ( giry_subprob_space_b @ M2 )
     => ( ( member7751282110383867797sure_b @ F @ ( sigma_6831777305601499691sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ N2 ) ) )
       => ( giry_subprob_space_b @ ( giry_bind_b_b @ M2 @ F ) ) ) ) ).

% subprob_space_bind
thf(fact_1016_subprob__space__bind,axiom,
    ! [M2: sigma_measure_a,F: a > sigma_measure_a,N2: sigma_measure_a] :
      ( ( giry_subprob_space_a @ M2 )
     => ( ( member1546933411676758803sure_a @ F @ ( sigma_698462642637507881sure_a @ M2 @ ( giry_s2448339315546285520ebra_a @ N2 ) ) )
       => ( giry_subprob_space_a @ ( giry_bind_a_a @ M2 @ F ) ) ) ) ).

% subprob_space_bind
thf(fact_1017_subprob__space__bind,axiom,
    ! [M2: sigma_measure_b,F: b > sigma_measure_a,N2: sigma_measure_a] :
      ( ( giry_subprob_space_b @ M2 )
     => ( ( member7680248070337521812sure_a @ F @ ( sigma_6831777301298270890sure_a @ M2 @ ( giry_s2448339315546285520ebra_a @ N2 ) ) )
       => ( giry_subprob_space_a @ ( giry_bind_b_a @ M2 @ F ) ) ) ) ).

% subprob_space_bind
thf(fact_1018_bind__const_H,axiom,
    ! [M2: sigma_measure_a,N2: sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( giry_subprob_space_a @ N2 )
       => ( ( giry_bind_a_a @ M2
            @ ^ [X2: a] : N2 )
          = N2 ) ) ) ).

% bind_const'
thf(fact_1019_bind__const_H,axiom,
    ! [M2: sigma_measure_a,N2: sigma_measure_b] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( giry_subprob_space_b @ N2 )
       => ( ( giry_bind_a_b @ M2
            @ ^ [X2: a] : N2 )
          = N2 ) ) ) ).

% bind_const'
thf(fact_1020_bind__const_H,axiom,
    ! [M2: sigma_measure_b,N2: sigma_measure_a] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( giry_subprob_space_a @ N2 )
       => ( ( giry_bind_b_a @ M2
            @ ^ [X2: b] : N2 )
          = N2 ) ) ) ).

% bind_const'
thf(fact_1021_bind__const_H,axiom,
    ! [M2: sigma_measure_b,N2: sigma_measure_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( giry_subprob_space_b @ N2 )
       => ( ( giry_bind_b_b @ M2
            @ ^ [X2: b] : N2 )
          = N2 ) ) ) ).

% bind_const'
thf(fact_1022_bind__const_H,axiom,
    ! [M2: sigma_6673148739008212994ream_b,N2: sigma_measure_a] :
      ( ( probab6181332906713481800ream_b @ M2 )
     => ( ( giry_subprob_space_a @ N2 )
       => ( ( giry_bind_stream_b_a @ M2
            @ ^ [X2: stream_b] : N2 )
          = N2 ) ) ) ).

% bind_const'
thf(fact_1023_bind__const_H,axiom,
    ! [M2: sigma_6673148739008212994ream_b,N2: sigma_measure_b] :
      ( ( probab6181332906713481800ream_b @ M2 )
     => ( ( giry_subprob_space_b @ N2 )
       => ( ( giry_bind_stream_b_b @ M2
            @ ^ [X2: stream_b] : N2 )
          = N2 ) ) ) ).

% bind_const'
thf(fact_1024_bind__const_H,axiom,
    ! [M2: sigma_6602114698961867009ream_a,N2: sigma_measure_a] :
      ( ( probab6181332902410252999ream_a @ M2 )
     => ( ( giry_subprob_space_a @ N2 )
       => ( ( giry_bind_stream_a_a @ M2
            @ ^ [X2: stream_a] : N2 )
          = N2 ) ) ) ).

% bind_const'
thf(fact_1025_bind__const_H,axiom,
    ! [M2: sigma_6602114698961867009ream_a,N2: sigma_measure_b] :
      ( ( probab6181332902410252999ream_a @ M2 )
     => ( ( giry_subprob_space_b @ N2 )
       => ( ( giry_bind_stream_a_b @ M2
            @ ^ [X2: stream_a] : N2 )
          = N2 ) ) ) ).

% bind_const'
thf(fact_1026_space__bind__measurable,axiom,
    ! [F: a > sigma_measure_b,M2: sigma_measure_a,B: sigma_measure_b] :
      ( ( member1617967451723104788sure_b @ F @ ( sigma_698462646940736682sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ B ) ) )
     => ( ( ( sigma_space_a @ M2 )
         != bot_bot_set_a )
       => ( ( sigma_space_b @ ( giry_bind_a_b @ M2 @ F ) )
          = ( sigma_space_b @ B ) ) ) ) ).

% space_bind_measurable
thf(fact_1027_space__bind__measurable,axiom,
    ! [F: b > sigma_measure_b,M2: sigma_measure_b,B: sigma_measure_b] :
      ( ( member7751282110383867797sure_b @ F @ ( sigma_6831777305601499691sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ B ) ) )
     => ( ( ( sigma_space_b @ M2 )
         != bot_bot_set_b )
       => ( ( sigma_space_b @ ( giry_bind_b_b @ M2 @ F ) )
          = ( sigma_space_b @ B ) ) ) ) ).

% space_bind_measurable
thf(fact_1028_space__bind__measurable,axiom,
    ! [F: $o > sigma_measure_b,M2: sigma_measure_o,B: sigma_measure_b] :
      ( ( member5309494624352977582sure_b @ F @ ( sigma_4308157146278001604sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ B ) ) )
     => ( ( ( sigma_space_o @ M2 )
         != bot_bot_set_o )
       => ( ( sigma_space_b @ ( giry_bind_o_b @ M2 @ F ) )
          = ( sigma_space_b @ B ) ) ) ) ).

% space_bind_measurable
thf(fact_1029_space__bind__measurable,axiom,
    ! [F: a > sigma_measure_a,M2: sigma_measure_a,B: sigma_measure_a] :
      ( ( member1546933411676758803sure_a @ F @ ( sigma_698462642637507881sure_a @ M2 @ ( giry_s2448339315546285520ebra_a @ B ) ) )
     => ( ( ( sigma_space_a @ M2 )
         != bot_bot_set_a )
       => ( ( sigma_space_a @ ( giry_bind_a_a @ M2 @ F ) )
          = ( sigma_space_a @ B ) ) ) ) ).

% space_bind_measurable
thf(fact_1030_space__bind__measurable,axiom,
    ! [F: b > sigma_measure_a,M2: sigma_measure_b,B: sigma_measure_a] :
      ( ( member7680248070337521812sure_a @ F @ ( sigma_6831777301298270890sure_a @ M2 @ ( giry_s2448339315546285520ebra_a @ B ) ) )
     => ( ( ( sigma_space_b @ M2 )
         != bot_bot_set_b )
       => ( ( sigma_space_a @ ( giry_bind_b_a @ M2 @ F ) )
          = ( sigma_space_a @ B ) ) ) ) ).

% space_bind_measurable
thf(fact_1031_space__bind__measurable,axiom,
    ! [F: $o > sigma_measure_a,M2: sigma_measure_o,B: sigma_measure_a] :
      ( ( member5238460584306631597sure_a @ F @ ( sigma_4308157141974772803sure_a @ M2 @ ( giry_s2448339315546285520ebra_a @ B ) ) )
     => ( ( ( sigma_space_o @ M2 )
         != bot_bot_set_o )
       => ( ( sigma_space_a @ ( giry_bind_o_a @ M2 @ F ) )
          = ( sigma_space_a @ B ) ) ) ) ).

% space_bind_measurable
thf(fact_1032_space__bind__measurable,axiom,
    ! [F: a > sigma_2733069582767196501sure_b,M2: sigma_measure_a,B: sigma_2733069582767196501sure_b] :
      ( ( member1486180851757042259sure_b @ F @ ( sigma_5287917315690574119sure_b @ M2 @ ( giry_s4089474293088545450sure_b @ B ) ) )
     => ( ( ( sigma_space_a @ M2 )
         != bot_bot_set_a )
       => ( ( sigma_6149183809546982772sure_b @ ( giry_b5915752774705324195sure_b @ M2 @ F ) )
          = ( sigma_6149183809546982772sure_b @ B ) ) ) ) ).

% space_bind_measurable
thf(fact_1033_space__bind__measurable,axiom,
    ! [F: a > sigma_2662035542720850516sure_a,M2: sigma_measure_a,B: sigma_2662035542720850516sure_a] :
      ( ( member4007365370046077906sure_a @ F @ ( sigma_5216883275644228134sure_a @ M2 @ ( giry_s4089474288785316649sure_a @ B ) ) )
     => ( ( ( sigma_space_a @ M2 )
         != bot_bot_set_a )
       => ( ( sigma_6149183805243753971sure_a @ ( giry_b5915752770402095394sure_a @ M2 @ F ) )
          = ( sigma_6149183805243753971sure_a @ B ) ) ) ) ).

% space_bind_measurable
thf(fact_1034_space__bind__measurable,axiom,
    ! [F: b > sigma_2733069582767196501sure_b,M2: sigma_measure_b,B: sigma_2733069582767196501sure_b] :
      ( ( member7282334670290218258sure_b @ F @ ( sigma_1860699097368974310sure_b @ M2 @ ( giry_s4089474293088545450sure_b @ B ) ) )
     => ( ( ( sigma_space_b @ M2 )
         != bot_bot_set_b )
       => ( ( sigma_6149183809546982772sure_b @ ( giry_b2825695396511311396sure_b @ M2 @ F ) )
          = ( sigma_6149183809546982772sure_b @ B ) ) ) ) ).

% space_bind_measurable
thf(fact_1035_space__bind__measurable,axiom,
    ! [F: b > sigma_2662035542720850516sure_a,M2: sigma_measure_b,B: sigma_2662035542720850516sure_a] :
      ( ( member580147151724478097sure_a @ F @ ( sigma_1789665057322628325sure_a @ M2 @ ( giry_s4089474288785316649sure_a @ B ) ) )
     => ( ( ( sigma_space_b @ M2 )
         != bot_bot_set_b )
       => ( ( sigma_6149183805243753971sure_a @ ( giry_b2825695392208082595sure_a @ M2 @ F ) )
          = ( sigma_6149183805243753971sure_a @ B ) ) ) ) ).

% space_bind_measurable
thf(fact_1036_bind__return,axiom,
    ! [F: c > sigma_measure_b,M2: sigma_measure_c,N2: sigma_measure_b,X3: c] :
      ( ( member4661224732189854998sure_b @ F @ ( sigma_3741719927407486892sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ N2 ) ) )
     => ( ( member_c @ X3 @ ( sigma_space_c @ M2 ) )
       => ( ( giry_bind_c_b @ ( giry_return_c @ M2 @ X3 ) @ F )
          = ( F @ X3 ) ) ) ) ).

% bind_return
thf(fact_1037_bind__return,axiom,
    ! [F: a > sigma_measure_b,M2: sigma_measure_a,N2: sigma_measure_b,X3: a] :
      ( ( member1617967451723104788sure_b @ F @ ( sigma_698462646940736682sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ N2 ) ) )
     => ( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
       => ( ( giry_bind_a_b @ ( giry_return_a @ M2 @ X3 ) @ F )
          = ( F @ X3 ) ) ) ) ).

% bind_return
thf(fact_1038_bind__return,axiom,
    ! [F: b > sigma_measure_b,M2: sigma_measure_b,N2: sigma_measure_b,X3: b] :
      ( ( member7751282110383867797sure_b @ F @ ( sigma_6831777305601499691sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ N2 ) ) )
     => ( ( member_b @ X3 @ ( sigma_space_b @ M2 ) )
       => ( ( giry_bind_b_b @ ( giry_return_b @ M2 @ X3 ) @ F )
          = ( F @ X3 ) ) ) ) ).

% bind_return
thf(fact_1039_bind__return,axiom,
    ! [F: c > sigma_measure_a,M2: sigma_measure_c,N2: sigma_measure_a,X3: c] :
      ( ( member4590190692143509013sure_a @ F @ ( sigma_3741719923104258091sure_a @ M2 @ ( giry_s2448339315546285520ebra_a @ N2 ) ) )
     => ( ( member_c @ X3 @ ( sigma_space_c @ M2 ) )
       => ( ( giry_bind_c_a @ ( giry_return_c @ M2 @ X3 ) @ F )
          = ( F @ X3 ) ) ) ) ).

% bind_return
thf(fact_1040_bind__return,axiom,
    ! [F: a > sigma_measure_a,M2: sigma_measure_a,N2: sigma_measure_a,X3: a] :
      ( ( member1546933411676758803sure_a @ F @ ( sigma_698462642637507881sure_a @ M2 @ ( giry_s2448339315546285520ebra_a @ N2 ) ) )
     => ( ( member_a @ X3 @ ( sigma_space_a @ M2 ) )
       => ( ( giry_bind_a_a @ ( giry_return_a @ M2 @ X3 ) @ F )
          = ( F @ X3 ) ) ) ) ).

% bind_return
thf(fact_1041_bind__return,axiom,
    ! [F: b > sigma_measure_a,M2: sigma_measure_b,N2: sigma_measure_a,X3: b] :
      ( ( member7680248070337521812sure_a @ F @ ( sigma_6831777301298270890sure_a @ M2 @ ( giry_s2448339315546285520ebra_a @ N2 ) ) )
     => ( ( member_b @ X3 @ ( sigma_space_b @ M2 ) )
       => ( ( giry_bind_b_a @ ( giry_return_b @ M2 @ X3 ) @ F )
          = ( F @ X3 ) ) ) ) ).

% bind_return
thf(fact_1042_bind__return,axiom,
    ! [F: sigma_measure_b > sigma_measure_b,M2: sigma_2733069582767196501sure_b,N2: sigma_measure_b,X3: sigma_measure_b] :
      ( ( member2971852239989997358sure_b @ F @ ( sigma_1495569021742745674sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ N2 ) ) )
     => ( ( member3534519381033026579sure_b @ X3 @ ( sigma_6149183809546982772sure_b @ M2 ) )
       => ( ( giry_b1258390083388040234re_b_b @ ( giry_r3784369487821543sure_b @ M2 @ X3 ) @ F )
          = ( F @ X3 ) ) ) ) ).

% bind_return
thf(fact_1043_bind__return,axiom,
    ! [F: sigma_measure_a > sigma_measure_b,M2: sigma_2662035542720850516sure_a,N2: sigma_measure_b,X3: sigma_measure_a] :
      ( ( member710132958529219117sure_b @ F @ ( sigma_8457221777136743241sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ N2 ) ) )
     => ( ( member3534519376729797778sure_a @ X3 @ ( sigma_6149183805243753971sure_a @ M2 ) )
       => ( ( giry_b1894039083993026923re_a_b @ ( giry_r3784365184592742sure_a @ M2 @ X3 ) @ F )
          = ( F @ X3 ) ) ) ) ).

% bind_return
thf(fact_1044_bind__return,axiom,
    ! [F: sigma_measure_b > sigma_measure_a,M2: sigma_2733069582767196501sure_b,N2: sigma_measure_a,X3: sigma_measure_b] :
      ( ( member2900818199943651373sure_a @ F @ ( sigma_1495569017439516873sure_a @ M2 @ ( giry_s2448339315546285520ebra_a @ N2 ) ) )
     => ( ( member3534519381033026579sure_b @ X3 @ ( sigma_6149183809546982772sure_b @ M2 ) )
       => ( ( giry_b1258390083388040233re_b_a @ ( giry_r3784369487821543sure_b @ M2 @ X3 ) @ F )
          = ( F @ X3 ) ) ) ) ).

% bind_return
thf(fact_1045_bind__return,axiom,
    ! [F: sigma_measure_a > sigma_measure_a,M2: sigma_2662035542720850516sure_a,N2: sigma_measure_a,X3: sigma_measure_a] :
      ( ( member639098918482873132sure_a @ F @ ( sigma_8457221772833514440sure_a @ M2 @ ( giry_s2448339315546285520ebra_a @ N2 ) ) )
     => ( ( member3534519376729797778sure_a @ X3 @ ( sigma_6149183805243753971sure_a @ M2 ) )
       => ( ( giry_b1894039083993026922re_a_a @ ( giry_r3784365184592742sure_a @ M2 @ X3 ) @ F )
          = ( F @ X3 ) ) ) ) ).

% bind_return
thf(fact_1046_subprob__space_Obind__in__space,axiom,
    ! [M2: sigma_measure_a,A: a > sigma_measure_b,N2: sigma_measure_b] :
      ( ( giry_subprob_space_a @ M2 )
     => ( ( member1617967451723104788sure_b @ A @ ( sigma_698462646940736682sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ N2 ) ) )
       => ( member3534519381033026579sure_b @ ( giry_bind_a_b @ M2 @ A ) @ ( sigma_6149183809546982772sure_b @ ( giry_s2448339315546285521ebra_b @ N2 ) ) ) ) ) ).

% subprob_space.bind_in_space
thf(fact_1047_subprob__space_Obind__in__space,axiom,
    ! [M2: sigma_measure_b,A: b > sigma_measure_b,N2: sigma_measure_b] :
      ( ( giry_subprob_space_b @ M2 )
     => ( ( member7751282110383867797sure_b @ A @ ( sigma_6831777305601499691sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ N2 ) ) )
       => ( member3534519381033026579sure_b @ ( giry_bind_b_b @ M2 @ A ) @ ( sigma_6149183809546982772sure_b @ ( giry_s2448339315546285521ebra_b @ N2 ) ) ) ) ) ).

% subprob_space.bind_in_space
thf(fact_1048_subprob__space_Obind__in__space,axiom,
    ! [M2: sigma_measure_a,A: a > sigma_measure_a,N2: sigma_measure_a] :
      ( ( giry_subprob_space_a @ M2 )
     => ( ( member1546933411676758803sure_a @ A @ ( sigma_698462642637507881sure_a @ M2 @ ( giry_s2448339315546285520ebra_a @ N2 ) ) )
       => ( member3534519376729797778sure_a @ ( giry_bind_a_a @ M2 @ A ) @ ( sigma_6149183805243753971sure_a @ ( giry_s2448339315546285520ebra_a @ N2 ) ) ) ) ) ).

% subprob_space.bind_in_space
thf(fact_1049_subprob__space_Obind__in__space,axiom,
    ! [M2: sigma_measure_b,A: b > sigma_measure_a,N2: sigma_measure_a] :
      ( ( giry_subprob_space_b @ M2 )
     => ( ( member7680248070337521812sure_a @ A @ ( sigma_6831777301298270890sure_a @ M2 @ ( giry_s2448339315546285520ebra_a @ N2 ) ) )
       => ( member3534519376729797778sure_a @ ( giry_bind_b_a @ M2 @ A ) @ ( sigma_6149183805243753971sure_a @ ( giry_s2448339315546285520ebra_a @ N2 ) ) ) ) ) ).

% subprob_space.bind_in_space
thf(fact_1050_not__finite__existsD,axiom,
    ! [P: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
     => ? [X_1: nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_1051_not__finite__existsD,axiom,
    ! [P: b > $o] :
      ( ~ ( finite_finite_b @ ( collect_b @ P ) )
     => ? [X_1: b] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_1052_not__finite__existsD,axiom,
    ! [P: a > $o] :
      ( ~ ( finite_finite_a @ ( collect_a @ P ) )
     => ? [X_1: a] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_1053_pigeonhole__infinite__rel,axiom,
    ! [A: set_c,B: set_nat,R: c > nat > $o] :
      ( ~ ( finite_finite_c @ A )
     => ( ( finite_finite_nat @ B )
       => ( ! [X: c] :
              ( ( member_c @ X @ A )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B )
                  & ( R @ X @ Xa ) ) )
         => ? [X: nat] :
              ( ( member_nat @ X @ B )
              & ~ ( finite_finite_c
                  @ ( collect_c
                    @ ^ [A5: c] :
                        ( ( member_c @ A5 @ A )
                        & ( R @ A5 @ X ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1054_pigeonhole__infinite__rel,axiom,
    ! [A: set_c,B: set_b,R: c > b > $o] :
      ( ~ ( finite_finite_c @ A )
     => ( ( finite_finite_b @ B )
       => ( ! [X: c] :
              ( ( member_c @ X @ A )
             => ? [Xa: b] :
                  ( ( member_b @ Xa @ B )
                  & ( R @ X @ Xa ) ) )
         => ? [X: b] :
              ( ( member_b @ X @ B )
              & ~ ( finite_finite_c
                  @ ( collect_c
                    @ ^ [A5: c] :
                        ( ( member_c @ A5 @ A )
                        & ( R @ A5 @ X ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1055_pigeonhole__infinite__rel,axiom,
    ! [A: set_c,B: set_a,R: c > a > $o] :
      ( ~ ( finite_finite_c @ A )
     => ( ( finite_finite_a @ B )
       => ( ! [X: c] :
              ( ( member_c @ X @ A )
             => ? [Xa: a] :
                  ( ( member_a @ Xa @ B )
                  & ( R @ X @ Xa ) ) )
         => ? [X: a] :
              ( ( member_a @ X @ B )
              & ~ ( finite_finite_c
                  @ ( collect_c
                    @ ^ [A5: c] :
                        ( ( member_c @ A5 @ A )
                        & ( R @ A5 @ X ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1056_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat,B: set_nat,R: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ B )
       => ( ! [X: nat] :
              ( ( member_nat @ X @ A )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B )
                  & ( R @ X @ Xa ) ) )
         => ? [X: nat] :
              ( ( member_nat @ X @ B )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A5: nat] :
                        ( ( member_nat @ A5 @ A )
                        & ( R @ A5 @ X ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1057_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat,B: set_b,R: nat > b > $o] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite_finite_b @ B )
       => ( ! [X: nat] :
              ( ( member_nat @ X @ A )
             => ? [Xa: b] :
                  ( ( member_b @ Xa @ B )
                  & ( R @ X @ Xa ) ) )
         => ? [X: b] :
              ( ( member_b @ X @ B )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A5: nat] :
                        ( ( member_nat @ A5 @ A )
                        & ( R @ A5 @ X ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1058_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat,B: set_a,R: nat > a > $o] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite_finite_a @ B )
       => ( ! [X: nat] :
              ( ( member_nat @ X @ A )
             => ? [Xa: a] :
                  ( ( member_a @ Xa @ B )
                  & ( R @ X @ Xa ) ) )
         => ? [X: a] :
              ( ( member_a @ X @ B )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A5: nat] :
                        ( ( member_nat @ A5 @ A )
                        & ( R @ A5 @ X ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1059_pigeonhole__infinite__rel,axiom,
    ! [A: set_b,B: set_nat,R: b > nat > $o] :
      ( ~ ( finite_finite_b @ A )
     => ( ( finite_finite_nat @ B )
       => ( ! [X: b] :
              ( ( member_b @ X @ A )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B )
                  & ( R @ X @ Xa ) ) )
         => ? [X: nat] :
              ( ( member_nat @ X @ B )
              & ~ ( finite_finite_b
                  @ ( collect_b
                    @ ^ [A5: b] :
                        ( ( member_b @ A5 @ A )
                        & ( R @ A5 @ X ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1060_pigeonhole__infinite__rel,axiom,
    ! [A: set_b,B: set_b,R: b > b > $o] :
      ( ~ ( finite_finite_b @ A )
     => ( ( finite_finite_b @ B )
       => ( ! [X: b] :
              ( ( member_b @ X @ A )
             => ? [Xa: b] :
                  ( ( member_b @ Xa @ B )
                  & ( R @ X @ Xa ) ) )
         => ? [X: b] :
              ( ( member_b @ X @ B )
              & ~ ( finite_finite_b
                  @ ( collect_b
                    @ ^ [A5: b] :
                        ( ( member_b @ A5 @ A )
                        & ( R @ A5 @ X ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1061_pigeonhole__infinite__rel,axiom,
    ! [A: set_b,B: set_a,R: b > a > $o] :
      ( ~ ( finite_finite_b @ A )
     => ( ( finite_finite_a @ B )
       => ( ! [X: b] :
              ( ( member_b @ X @ A )
             => ? [Xa: a] :
                  ( ( member_a @ Xa @ B )
                  & ( R @ X @ Xa ) ) )
         => ? [X: a] :
              ( ( member_a @ X @ B )
              & ~ ( finite_finite_b
                  @ ( collect_b
                    @ ^ [A5: b] :
                        ( ( member_b @ A5 @ A )
                        & ( R @ A5 @ X ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1062_pigeonhole__infinite__rel,axiom,
    ! [A: set_a,B: set_nat,R: a > nat > $o] :
      ( ~ ( finite_finite_a @ A )
     => ( ( finite_finite_nat @ B )
       => ( ! [X: a] :
              ( ( member_a @ X @ A )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B )
                  & ( R @ X @ Xa ) ) )
         => ? [X: nat] :
              ( ( member_nat @ X @ B )
              & ~ ( finite_finite_a
                  @ ( collect_a
                    @ ^ [A5: a] :
                        ( ( member_a @ A5 @ A )
                        & ( R @ A5 @ X ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1063_bind__return__distr,axiom,
    ! [M2: sigma_measure_a,F: a > a > real,N2: sigma_measure_a_real] :
      ( ( ( sigma_space_a @ M2 )
       != bot_bot_set_a )
     => ( ( member_a_a_real @ F @ ( sigma_5073124708296780141a_real @ M2 @ N2 ) )
       => ( ( giry_bind_a_a_real @ M2 @ ( comp_a1543360357282447724real_a @ ( giry_return_a_real @ N2 ) @ F ) )
          = ( measur6806378183764566549a_real @ M2 @ N2 @ F ) ) ) ) ).

% bind_return_distr
thf(fact_1064_bind__return__distr,axiom,
    ! [M2: sigma_measure_a,F: a > a > b,N2: sigma_measure_a_b] :
      ( ( ( sigma_space_a @ M2 )
       != bot_bot_set_a )
     => ( ( member_a_a_b @ F @ ( sigma_3971135317854480532_a_a_b @ M2 @ N2 ) )
       => ( ( giry_bind_a_a_b @ M2 @ ( comp_a7517041177194281544_a_b_a @ ( giry_return_a_b @ N2 ) @ F ) )
          = ( measure_distr_a_a_b @ M2 @ N2 @ F ) ) ) ) ).

% bind_return_distr
thf(fact_1065_bind__return__distr,axiom,
    ! [M2: sigma_measure_a,F: a > c,N2: sigma_measure_c] :
      ( ( ( sigma_space_a @ M2 )
       != bot_bot_set_a )
     => ( ( member_a_c @ F @ ( sigma_measurable_a_c @ M2 @ N2 ) )
       => ( ( giry_bind_a_c @ M2 @ ( comp_c7334238421110395426re_c_a @ ( giry_return_c @ N2 ) @ F ) )
          = ( measure_distr_a_c @ M2 @ N2 @ F ) ) ) ) ).

% bind_return_distr
thf(fact_1066_bind__return__distr,axiom,
    ! [M2: sigma_measure_a,F: a > a,N2: sigma_measure_a] :
      ( ( ( sigma_space_a @ M2 )
       != bot_bot_set_a )
     => ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N2 ) )
       => ( ( giry_bind_a_a @ M2 @ ( comp_a5399727294799983398re_a_a @ ( giry_return_a @ N2 ) @ F ) )
          = ( measure_distr_a_a @ M2 @ N2 @ F ) ) ) ) ).

% bind_return_distr
thf(fact_1067_bind__return__distr,axiom,
    ! [M2: sigma_measure_b,F: b > a,N2: sigma_measure_a] :
      ( ( ( sigma_space_b @ M2 )
       != bot_bot_set_b )
     => ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ N2 ) )
       => ( ( giry_bind_b_a @ M2 @ ( comp_a5399727294799983399re_a_b @ ( giry_return_a @ N2 ) @ F ) )
          = ( measure_distr_b_a @ M2 @ N2 @ F ) ) ) ) ).

% bind_return_distr
thf(fact_1068_bind__return__distr,axiom,
    ! [M2: sigma_measure_a,F: a > b,N2: sigma_measure_b] :
      ( ( ( sigma_space_a @ M2 )
       != bot_bot_set_a )
     => ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ N2 ) )
       => ( ( giry_bind_a_b @ M2 @ ( comp_b1755296839527801508re_b_a @ ( giry_return_b @ N2 ) @ F ) )
          = ( measure_distr_a_b @ M2 @ N2 @ F ) ) ) ) ).

% bind_return_distr
thf(fact_1069_bind__return__distr,axiom,
    ! [M2: sigma_measure_b,F: b > d,N2: sigma_measure_d] :
      ( ( ( sigma_space_b @ M2 )
       != bot_bot_set_b )
     => ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M2 @ N2 ) )
       => ( ( giry_bind_b_d @ M2 @ ( comp_d3689807965838213537re_d_b @ ( giry_return_d @ N2 ) @ F ) )
          = ( measure_distr_b_d @ M2 @ N2 @ F ) ) ) ) ).

% bind_return_distr
thf(fact_1070_bind__return__distr,axiom,
    ! [M2: sigma_measure_b,F: b > real,N2: sigma_measure_real] :
      ( ( ( sigma_space_b @ M2 )
       != bot_bot_set_b )
     => ( ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ M2 @ N2 ) )
       => ( ( giry_bind_b_real @ M2 @ ( comp_r70652391186890569real_b @ ( giry_return_real @ N2 ) @ F ) )
          = ( measure_distr_b_real @ M2 @ N2 @ F ) ) ) ) ).

% bind_return_distr
thf(fact_1071_bind__return__distr,axiom,
    ! [M2: sigma_measure_a,F: a > real,N2: sigma_measure_real] :
      ( ( ( sigma_space_a @ M2 )
       != bot_bot_set_a )
     => ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M2 @ N2 ) )
       => ( ( giry_bind_a_real @ M2 @ ( comp_r70652391186890568real_a @ ( giry_return_real @ N2 ) @ F ) )
          = ( measure_distr_a_real @ M2 @ N2 @ F ) ) ) ) ).

% bind_return_distr
thf(fact_1072_bind__distr,axiom,
    ! [F: a > c,M2: sigma_measure_a,X4: sigma_measure_c,N2: c > sigma_measure_b,K2: sigma_measure_b] :
      ( ( member_a_c @ F @ ( sigma_measurable_a_c @ M2 @ X4 ) )
     => ( ( member4661224732189854998sure_b @ N2 @ ( sigma_3741719927407486892sure_b @ X4 @ ( giry_s2448339315546285521ebra_b @ K2 ) ) )
       => ( ( ( sigma_space_a @ M2 )
           != bot_bot_set_a )
         => ( ( giry_bind_c_b @ ( measure_distr_a_c @ M2 @ X4 @ F ) @ N2 )
            = ( giry_bind_a_b @ M2
              @ ^ [X2: a] : ( N2 @ ( F @ X2 ) ) ) ) ) ) ) ).

% bind_distr
thf(fact_1073_bind__distr,axiom,
    ! [F: a > a,M2: sigma_measure_a,X4: sigma_measure_a,N2: a > sigma_measure_b,K2: sigma_measure_b] :
      ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ X4 ) )
     => ( ( member1617967451723104788sure_b @ N2 @ ( sigma_698462646940736682sure_b @ X4 @ ( giry_s2448339315546285521ebra_b @ K2 ) ) )
       => ( ( ( sigma_space_a @ M2 )
           != bot_bot_set_a )
         => ( ( giry_bind_a_b @ ( measure_distr_a_a @ M2 @ X4 @ F ) @ N2 )
            = ( giry_bind_a_b @ M2
              @ ^ [X2: a] : ( N2 @ ( F @ X2 ) ) ) ) ) ) ) ).

% bind_distr
thf(fact_1074_bind__distr,axiom,
    ! [F: b > a,M2: sigma_measure_b,X4: sigma_measure_a,N2: a > sigma_measure_b,K2: sigma_measure_b] :
      ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ X4 ) )
     => ( ( member1617967451723104788sure_b @ N2 @ ( sigma_698462646940736682sure_b @ X4 @ ( giry_s2448339315546285521ebra_b @ K2 ) ) )
       => ( ( ( sigma_space_b @ M2 )
           != bot_bot_set_b )
         => ( ( giry_bind_a_b @ ( measure_distr_b_a @ M2 @ X4 @ F ) @ N2 )
            = ( giry_bind_b_b @ M2
              @ ^ [X2: b] : ( N2 @ ( F @ X2 ) ) ) ) ) ) ) ).

% bind_distr
thf(fact_1075_bind__distr,axiom,
    ! [F: a > b,M2: sigma_measure_a,X4: sigma_measure_b,N2: b > sigma_measure_b,K2: sigma_measure_b] :
      ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ X4 ) )
     => ( ( member7751282110383867797sure_b @ N2 @ ( sigma_6831777305601499691sure_b @ X4 @ ( giry_s2448339315546285521ebra_b @ K2 ) ) )
       => ( ( ( sigma_space_a @ M2 )
           != bot_bot_set_a )
         => ( ( giry_bind_b_b @ ( measure_distr_a_b @ M2 @ X4 @ F ) @ N2 )
            = ( giry_bind_a_b @ M2
              @ ^ [X2: a] : ( N2 @ ( F @ X2 ) ) ) ) ) ) ) ).

% bind_distr
thf(fact_1076_bind__distr,axiom,
    ! [F: b > d,M2: sigma_measure_b,X4: sigma_measure_d,N2: d > sigma_measure_b,K2: sigma_measure_b] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M2 @ X4 ) )
     => ( ( member1571167353995842199sure_b @ N2 @ ( sigma_651662549213474093sure_b @ X4 @ ( giry_s2448339315546285521ebra_b @ K2 ) ) )
       => ( ( ( sigma_space_b @ M2 )
           != bot_bot_set_b )
         => ( ( giry_bind_d_b @ ( measure_distr_b_d @ M2 @ X4 @ F ) @ N2 )
            = ( giry_bind_b_b @ M2
              @ ^ [X2: b] : ( N2 @ ( F @ X2 ) ) ) ) ) ) ) ).

% bind_distr
thf(fact_1077_bind__distr,axiom,
    ! [F: b > real,M2: sigma_measure_b,X4: sigma_measure_real,N2: real > sigma_measure_b,K2: sigma_measure_b] :
      ( ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ M2 @ X4 ) )
     => ( ( member7352859679829567710sure_b @ N2 @ ( sigma_7716911761598366714sure_b @ X4 @ ( giry_s2448339315546285521ebra_b @ K2 ) ) )
       => ( ( ( sigma_space_b @ M2 )
           != bot_bot_set_b )
         => ( ( giry_bind_real_b @ ( measure_distr_b_real @ M2 @ X4 @ F ) @ N2 )
            = ( giry_bind_b_b @ M2
              @ ^ [X2: b] : ( N2 @ ( F @ X2 ) ) ) ) ) ) ) ).

% bind_distr
thf(fact_1078_bind__distr,axiom,
    ! [F: a > real,M2: sigma_measure_a,X4: sigma_measure_real,N2: real > sigma_measure_b,K2: sigma_measure_b] :
      ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M2 @ X4 ) )
     => ( ( member7352859679829567710sure_b @ N2 @ ( sigma_7716911761598366714sure_b @ X4 @ ( giry_s2448339315546285521ebra_b @ K2 ) ) )
       => ( ( ( sigma_space_a @ M2 )
           != bot_bot_set_a )
         => ( ( giry_bind_real_b @ ( measure_distr_a_real @ M2 @ X4 @ F ) @ N2 )
            = ( giry_bind_a_b @ M2
              @ ^ [X2: a] : ( N2 @ ( F @ X2 ) ) ) ) ) ) ) ).

% bind_distr
thf(fact_1079_bind__distr,axiom,
    ! [F: a > c,M2: sigma_measure_a,X4: sigma_measure_c,N2: c > sigma_measure_a,K2: sigma_measure_a] :
      ( ( member_a_c @ F @ ( sigma_measurable_a_c @ M2 @ X4 ) )
     => ( ( member4590190692143509013sure_a @ N2 @ ( sigma_3741719923104258091sure_a @ X4 @ ( giry_s2448339315546285520ebra_a @ K2 ) ) )
       => ( ( ( sigma_space_a @ M2 )
           != bot_bot_set_a )
         => ( ( giry_bind_c_a @ ( measure_distr_a_c @ M2 @ X4 @ F ) @ N2 )
            = ( giry_bind_a_a @ M2
              @ ^ [X2: a] : ( N2 @ ( F @ X2 ) ) ) ) ) ) ) ).

% bind_distr
thf(fact_1080_bind__distr,axiom,
    ! [F: a > a,M2: sigma_measure_a,X4: sigma_measure_a,N2: a > sigma_measure_a,K2: sigma_measure_a] :
      ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ X4 ) )
     => ( ( member1546933411676758803sure_a @ N2 @ ( sigma_698462642637507881sure_a @ X4 @ ( giry_s2448339315546285520ebra_a @ K2 ) ) )
       => ( ( ( sigma_space_a @ M2 )
           != bot_bot_set_a )
         => ( ( giry_bind_a_a @ ( measure_distr_a_a @ M2 @ X4 @ F ) @ N2 )
            = ( giry_bind_a_a @ M2
              @ ^ [X2: a] : ( N2 @ ( F @ X2 ) ) ) ) ) ) ) ).

% bind_distr
thf(fact_1081_bind__distr,axiom,
    ! [F: b > a,M2: sigma_measure_b,X4: sigma_measure_a,N2: a > sigma_measure_a,K2: sigma_measure_a] :
      ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ X4 ) )
     => ( ( member1546933411676758803sure_a @ N2 @ ( sigma_698462642637507881sure_a @ X4 @ ( giry_s2448339315546285520ebra_a @ K2 ) ) )
       => ( ( ( sigma_space_b @ M2 )
           != bot_bot_set_b )
         => ( ( giry_bind_a_a @ ( measure_distr_b_a @ M2 @ X4 @ F ) @ N2 )
            = ( giry_bind_b_a @ M2
              @ ^ [X2: b] : ( N2 @ ( F @ X2 ) ) ) ) ) ) ) ).

% bind_distr
thf(fact_1082_distr__bind,axiom,
    ! [N2: a > sigma_measure_b,M2: sigma_measure_a,K2: sigma_measure_b,F: b > a,R: sigma_measure_a] :
      ( ( member1617967451723104788sure_b @ N2 @ ( sigma_698462646940736682sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ K2 ) ) )
     => ( ( ( sigma_space_a @ M2 )
         != bot_bot_set_a )
       => ( ( member_b_a @ F @ ( sigma_measurable_b_a @ K2 @ R ) )
         => ( ( measure_distr_b_a @ ( giry_bind_a_b @ M2 @ N2 ) @ R @ F )
            = ( giry_bind_a_a @ M2
              @ ^ [X2: a] : ( measure_distr_b_a @ ( N2 @ X2 ) @ R @ F ) ) ) ) ) ) ).

% distr_bind
thf(fact_1083_distr__bind,axiom,
    ! [N2: b > sigma_measure_b,M2: sigma_measure_b,K2: sigma_measure_b,F: b > a,R: sigma_measure_a] :
      ( ( member7751282110383867797sure_b @ N2 @ ( sigma_6831777305601499691sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ K2 ) ) )
     => ( ( ( sigma_space_b @ M2 )
         != bot_bot_set_b )
       => ( ( member_b_a @ F @ ( sigma_measurable_b_a @ K2 @ R ) )
         => ( ( measure_distr_b_a @ ( giry_bind_b_b @ M2 @ N2 ) @ R @ F )
            = ( giry_bind_b_a @ M2
              @ ^ [X2: b] : ( measure_distr_b_a @ ( N2 @ X2 ) @ R @ F ) ) ) ) ) ) ).

% distr_bind
thf(fact_1084_distr__bind,axiom,
    ! [N2: $o > sigma_measure_b,M2: sigma_measure_o,K2: sigma_measure_b,F: b > a,R: sigma_measure_a] :
      ( ( member5309494624352977582sure_b @ N2 @ ( sigma_4308157146278001604sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ K2 ) ) )
     => ( ( ( sigma_space_o @ M2 )
         != bot_bot_set_o )
       => ( ( member_b_a @ F @ ( sigma_measurable_b_a @ K2 @ R ) )
         => ( ( measure_distr_b_a @ ( giry_bind_o_b @ M2 @ N2 ) @ R @ F )
            = ( giry_bind_o_a @ M2
              @ ^ [X2: $o] : ( measure_distr_b_a @ ( N2 @ X2 ) @ R @ F ) ) ) ) ) ) ).

% distr_bind
thf(fact_1085_distr__bind,axiom,
    ! [N2: a > sigma_measure_b,M2: sigma_measure_a,K2: sigma_measure_b,F: b > d,R: sigma_measure_d] :
      ( ( member1617967451723104788sure_b @ N2 @ ( sigma_698462646940736682sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ K2 ) ) )
     => ( ( ( sigma_space_a @ M2 )
         != bot_bot_set_a )
       => ( ( member_b_d @ F @ ( sigma_measurable_b_d @ K2 @ R ) )
         => ( ( measure_distr_b_d @ ( giry_bind_a_b @ M2 @ N2 ) @ R @ F )
            = ( giry_bind_a_d @ M2
              @ ^ [X2: a] : ( measure_distr_b_d @ ( N2 @ X2 ) @ R @ F ) ) ) ) ) ) ).

% distr_bind
thf(fact_1086_distr__bind,axiom,
    ! [N2: b > sigma_measure_b,M2: sigma_measure_b,K2: sigma_measure_b,F: b > d,R: sigma_measure_d] :
      ( ( member7751282110383867797sure_b @ N2 @ ( sigma_6831777305601499691sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ K2 ) ) )
     => ( ( ( sigma_space_b @ M2 )
         != bot_bot_set_b )
       => ( ( member_b_d @ F @ ( sigma_measurable_b_d @ K2 @ R ) )
         => ( ( measure_distr_b_d @ ( giry_bind_b_b @ M2 @ N2 ) @ R @ F )
            = ( giry_bind_b_d @ M2
              @ ^ [X2: b] : ( measure_distr_b_d @ ( N2 @ X2 ) @ R @ F ) ) ) ) ) ) ).

% distr_bind
thf(fact_1087_distr__bind,axiom,
    ! [N2: $o > sigma_measure_b,M2: sigma_measure_o,K2: sigma_measure_b,F: b > d,R: sigma_measure_d] :
      ( ( member5309494624352977582sure_b @ N2 @ ( sigma_4308157146278001604sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ K2 ) ) )
     => ( ( ( sigma_space_o @ M2 )
         != bot_bot_set_o )
       => ( ( member_b_d @ F @ ( sigma_measurable_b_d @ K2 @ R ) )
         => ( ( measure_distr_b_d @ ( giry_bind_o_b @ M2 @ N2 ) @ R @ F )
            = ( giry_bind_o_d @ M2
              @ ^ [X2: $o] : ( measure_distr_b_d @ ( N2 @ X2 ) @ R @ F ) ) ) ) ) ) ).

% distr_bind
thf(fact_1088_distr__bind,axiom,
    ! [N2: a > sigma_measure_b,M2: sigma_measure_a,K2: sigma_measure_b,F: b > real,R: sigma_measure_real] :
      ( ( member1617967451723104788sure_b @ N2 @ ( sigma_698462646940736682sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ K2 ) ) )
     => ( ( ( sigma_space_a @ M2 )
         != bot_bot_set_a )
       => ( ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ K2 @ R ) )
         => ( ( measure_distr_b_real @ ( giry_bind_a_b @ M2 @ N2 ) @ R @ F )
            = ( giry_bind_a_real @ M2
              @ ^ [X2: a] : ( measure_distr_b_real @ ( N2 @ X2 ) @ R @ F ) ) ) ) ) ) ).

% distr_bind
thf(fact_1089_distr__bind,axiom,
    ! [N2: b > sigma_measure_b,M2: sigma_measure_b,K2: sigma_measure_b,F: b > real,R: sigma_measure_real] :
      ( ( member7751282110383867797sure_b @ N2 @ ( sigma_6831777305601499691sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ K2 ) ) )
     => ( ( ( sigma_space_b @ M2 )
         != bot_bot_set_b )
       => ( ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ K2 @ R ) )
         => ( ( measure_distr_b_real @ ( giry_bind_b_b @ M2 @ N2 ) @ R @ F )
            = ( giry_bind_b_real @ M2
              @ ^ [X2: b] : ( measure_distr_b_real @ ( N2 @ X2 ) @ R @ F ) ) ) ) ) ) ).

% distr_bind
thf(fact_1090_distr__bind,axiom,
    ! [N2: $o > sigma_measure_b,M2: sigma_measure_o,K2: sigma_measure_b,F: b > real,R: sigma_measure_real] :
      ( ( member5309494624352977582sure_b @ N2 @ ( sigma_4308157146278001604sure_b @ M2 @ ( giry_s2448339315546285521ebra_b @ K2 ) ) )
     => ( ( ( sigma_space_o @ M2 )
         != bot_bot_set_o )
       => ( ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ K2 @ R ) )
         => ( ( measure_distr_b_real @ ( giry_bind_o_b @ M2 @ N2 ) @ R @ F )
            = ( giry_bind_o_real @ M2
              @ ^ [X2: $o] : ( measure_distr_b_real @ ( N2 @ X2 ) @ R @ F ) ) ) ) ) ) ).

% distr_bind
thf(fact_1091_distr__bind,axiom,
    ! [N2: a > sigma_measure_a,M2: sigma_measure_a,K2: sigma_measure_a,F: a > c,R: sigma_measure_c] :
      ( ( member1546933411676758803sure_a @ N2 @ ( sigma_698462642637507881sure_a @ M2 @ ( giry_s2448339315546285520ebra_a @ K2 ) ) )
     => ( ( ( sigma_space_a @ M2 )
         != bot_bot_set_a )
       => ( ( member_a_c @ F @ ( sigma_measurable_a_c @ K2 @ R ) )
         => ( ( measure_distr_a_c @ ( giry_bind_a_a @ M2 @ N2 ) @ R @ F )
            = ( giry_bind_a_c @ M2
              @ ^ [X2: a] : ( measure_distr_a_c @ ( N2 @ X2 ) @ R @ F ) ) ) ) ) ) ).

% distr_bind
thf(fact_1092_bind__distr__return,axiom,
    ! [F: a > a,M2: sigma_measure_a,N2: sigma_measure_a,G2: a > c,L: sigma_measure_c] :
      ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N2 ) )
     => ( ( member_a_c @ G2 @ ( sigma_measurable_a_c @ N2 @ L ) )
       => ( ( ( sigma_space_a @ M2 )
           != bot_bot_set_a )
         => ( ( giry_bind_a_c @ ( measure_distr_a_a @ M2 @ N2 @ F )
              @ ^ [X2: a] : ( giry_return_c @ L @ ( G2 @ X2 ) ) )
            = ( measure_distr_a_c @ M2 @ L
              @ ^ [X2: a] : ( G2 @ ( F @ X2 ) ) ) ) ) ) ) ).

% bind_distr_return
thf(fact_1093_bind__distr__return,axiom,
    ! [F: a > a,M2: sigma_measure_a,N2: sigma_measure_a,G2: a > a,L: sigma_measure_a] :
      ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N2 ) )
     => ( ( member_a_a @ G2 @ ( sigma_measurable_a_a @ N2 @ L ) )
       => ( ( ( sigma_space_a @ M2 )
           != bot_bot_set_a )
         => ( ( giry_bind_a_a @ ( measure_distr_a_a @ M2 @ N2 @ F )
              @ ^ [X2: a] : ( giry_return_a @ L @ ( G2 @ X2 ) ) )
            = ( measure_distr_a_a @ M2 @ L
              @ ^ [X2: a] : ( G2 @ ( F @ X2 ) ) ) ) ) ) ) ).

% bind_distr_return
thf(fact_1094_bind__distr__return,axiom,
    ! [F: a > c,M2: sigma_measure_a,N2: sigma_measure_c,G2: c > b,L: sigma_measure_b] :
      ( ( member_a_c @ F @ ( sigma_measurable_a_c @ M2 @ N2 ) )
     => ( ( member_c_b @ G2 @ ( sigma_measurable_c_b @ N2 @ L ) )
       => ( ( ( sigma_space_a @ M2 )
           != bot_bot_set_a )
         => ( ( giry_bind_c_b @ ( measure_distr_a_c @ M2 @ N2 @ F )
              @ ^ [X2: c] : ( giry_return_b @ L @ ( G2 @ X2 ) ) )
            = ( measure_distr_a_b @ M2 @ L
              @ ^ [X2: a] : ( G2 @ ( F @ X2 ) ) ) ) ) ) ) ).

% bind_distr_return
thf(fact_1095_bind__distr__return,axiom,
    ! [F: b > b,M2: sigma_measure_b,N2: sigma_measure_b,G2: b > a,L: sigma_measure_a] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M2 @ N2 ) )
     => ( ( member_b_a @ G2 @ ( sigma_measurable_b_a @ N2 @ L ) )
       => ( ( ( sigma_space_b @ M2 )
           != bot_bot_set_b )
         => ( ( giry_bind_b_a @ ( measure_distr_b_b @ M2 @ N2 @ F )
              @ ^ [X2: b] : ( giry_return_a @ L @ ( G2 @ X2 ) ) )
            = ( measure_distr_b_a @ M2 @ L
              @ ^ [X2: b] : ( G2 @ ( F @ X2 ) ) ) ) ) ) ) ).

% bind_distr_return
thf(fact_1096_bind__distr__return,axiom,
    ! [F: b > a,M2: sigma_measure_b,N2: sigma_measure_a,G2: a > c,L: sigma_measure_c] :
      ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ N2 ) )
     => ( ( member_a_c @ G2 @ ( sigma_measurable_a_c @ N2 @ L ) )
       => ( ( ( sigma_space_b @ M2 )
           != bot_bot_set_b )
         => ( ( giry_bind_a_c @ ( measure_distr_b_a @ M2 @ N2 @ F )
              @ ^ [X2: a] : ( giry_return_c @ L @ ( G2 @ X2 ) ) )
            = ( measure_distr_b_c @ M2 @ L
              @ ^ [X2: b] : ( G2 @ ( F @ X2 ) ) ) ) ) ) ) ).

% bind_distr_return
thf(fact_1097_bind__distr__return,axiom,
    ! [F: b > a,M2: sigma_measure_b,N2: sigma_measure_a,G2: a > a,L: sigma_measure_a] :
      ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ N2 ) )
     => ( ( member_a_a @ G2 @ ( sigma_measurable_a_a @ N2 @ L ) )
       => ( ( ( sigma_space_b @ M2 )
           != bot_bot_set_b )
         => ( ( giry_bind_a_a @ ( measure_distr_b_a @ M2 @ N2 @ F )
              @ ^ [X2: a] : ( giry_return_a @ L @ ( G2 @ X2 ) ) )
            = ( measure_distr_b_a @ M2 @ L
              @ ^ [X2: b] : ( G2 @ ( F @ X2 ) ) ) ) ) ) ) ).

% bind_distr_return
thf(fact_1098_bind__distr__return,axiom,
    ! [F: $o > b,M2: sigma_measure_o,N2: sigma_measure_b,G2: b > a,L: sigma_measure_a] :
      ( ( member_o_b @ F @ ( sigma_measurable_o_b @ M2 @ N2 ) )
     => ( ( member_b_a @ G2 @ ( sigma_measurable_b_a @ N2 @ L ) )
       => ( ( ( sigma_space_o @ M2 )
           != bot_bot_set_o )
         => ( ( giry_bind_b_a @ ( measure_distr_o_b @ M2 @ N2 @ F )
              @ ^ [X2: b] : ( giry_return_a @ L @ ( G2 @ X2 ) ) )
            = ( measure_distr_o_a @ M2 @ L
              @ ^ [X2: $o] : ( G2 @ ( F @ X2 ) ) ) ) ) ) ) ).

% bind_distr_return
thf(fact_1099_bind__distr__return,axiom,
    ! [F: $o > a,M2: sigma_measure_o,N2: sigma_measure_a,G2: a > c,L: sigma_measure_c] :
      ( ( member_o_a @ F @ ( sigma_measurable_o_a @ M2 @ N2 ) )
     => ( ( member_a_c @ G2 @ ( sigma_measurable_a_c @ N2 @ L ) )
       => ( ( ( sigma_space_o @ M2 )
           != bot_bot_set_o )
         => ( ( giry_bind_a_c @ ( measure_distr_o_a @ M2 @ N2 @ F )
              @ ^ [X2: a] : ( giry_return_c @ L @ ( G2 @ X2 ) ) )
            = ( measure_distr_o_c @ M2 @ L
              @ ^ [X2: $o] : ( G2 @ ( F @ X2 ) ) ) ) ) ) ) ).

% bind_distr_return
thf(fact_1100_bind__distr__return,axiom,
    ! [F: $o > a,M2: sigma_measure_o,N2: sigma_measure_a,G2: a > a,L: sigma_measure_a] :
      ( ( member_o_a @ F @ ( sigma_measurable_o_a @ M2 @ N2 ) )
     => ( ( member_a_a @ G2 @ ( sigma_measurable_a_a @ N2 @ L ) )
       => ( ( ( sigma_space_o @ M2 )
           != bot_bot_set_o )
         => ( ( giry_bind_a_a @ ( measure_distr_o_a @ M2 @ N2 @ F )
              @ ^ [X2: a] : ( giry_return_a @ L @ ( G2 @ X2 ) ) )
            = ( measure_distr_o_a @ M2 @ L
              @ ^ [X2: $o] : ( G2 @ ( F @ X2 ) ) ) ) ) ) ) ).

% bind_distr_return
thf(fact_1101_bind__distr__return,axiom,
    ! [F: a > a,M2: sigma_measure_a,N2: sigma_measure_a,G2: a > b,L: sigma_measure_b] :
      ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N2 ) )
     => ( ( member_a_b @ G2 @ ( sigma_measurable_a_b @ N2 @ L ) )
       => ( ( ( sigma_space_a @ M2 )
           != bot_bot_set_a )
         => ( ( giry_bind_a_b @ ( measure_distr_a_a @ M2 @ N2 @ F )
              @ ^ [X2: a] : ( giry_return_b @ L @ ( G2 @ X2 ) ) )
            = ( measure_distr_a_b @ M2 @ L
              @ ^ [X2: a] : ( G2 @ ( F @ X2 ) ) ) ) ) ) ) ).

% bind_distr_return
thf(fact_1102_bind__return__distr_H,axiom,
    ! [M2: sigma_measure_a,F: a > a > real,N2: sigma_measure_a_real] :
      ( ( ( sigma_space_a @ M2 )
       != bot_bot_set_a )
     => ( ( member_a_a_real @ F @ ( sigma_5073124708296780141a_real @ M2 @ N2 ) )
       => ( ( giry_bind_a_a_real @ M2
            @ ^ [X2: a] : ( giry_return_a_real @ N2 @ ( F @ X2 ) ) )
          = ( measur6806378183764566549a_real @ M2 @ N2 @ F ) ) ) ) ).

% bind_return_distr'
thf(fact_1103_bind__return__distr_H,axiom,
    ! [M2: sigma_measure_a,F: a > a > b,N2: sigma_measure_a_b] :
      ( ( ( sigma_space_a @ M2 )
       != bot_bot_set_a )
     => ( ( member_a_a_b @ F @ ( sigma_3971135317854480532_a_a_b @ M2 @ N2 ) )
       => ( ( giry_bind_a_a_b @ M2
            @ ^ [X2: a] : ( giry_return_a_b @ N2 @ ( F @ X2 ) ) )
          = ( measure_distr_a_a_b @ M2 @ N2 @ F ) ) ) ) ).

% bind_return_distr'
thf(fact_1104_bind__return__distr_H,axiom,
    ! [M2: sigma_measure_a,F: a > c,N2: sigma_measure_c] :
      ( ( ( sigma_space_a @ M2 )
       != bot_bot_set_a )
     => ( ( member_a_c @ F @ ( sigma_measurable_a_c @ M2 @ N2 ) )
       => ( ( giry_bind_a_c @ M2
            @ ^ [X2: a] : ( giry_return_c @ N2 @ ( F @ X2 ) ) )
          = ( measure_distr_a_c @ M2 @ N2 @ F ) ) ) ) ).

% bind_return_distr'
thf(fact_1105_bind__return__distr_H,axiom,
    ! [M2: sigma_measure_a,F: a > a,N2: sigma_measure_a] :
      ( ( ( sigma_space_a @ M2 )
       != bot_bot_set_a )
     => ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ N2 ) )
       => ( ( giry_bind_a_a @ M2
            @ ^ [X2: a] : ( giry_return_a @ N2 @ ( F @ X2 ) ) )
          = ( measure_distr_a_a @ M2 @ N2 @ F ) ) ) ) ).

% bind_return_distr'
thf(fact_1106_bind__return__distr_H,axiom,
    ! [M2: sigma_measure_b,F: b > a,N2: sigma_measure_a] :
      ( ( ( sigma_space_b @ M2 )
       != bot_bot_set_b )
     => ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ N2 ) )
       => ( ( giry_bind_b_a @ M2
            @ ^ [X2: b] : ( giry_return_a @ N2 @ ( F @ X2 ) ) )
          = ( measure_distr_b_a @ M2 @ N2 @ F ) ) ) ) ).

% bind_return_distr'
thf(fact_1107_bind__return__distr_H,axiom,
    ! [M2: sigma_measure_a,F: a > b,N2: sigma_measure_b] :
      ( ( ( sigma_space_a @ M2 )
       != bot_bot_set_a )
     => ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ N2 ) )
       => ( ( giry_bind_a_b @ M2
            @ ^ [X2: a] : ( giry_return_b @ N2 @ ( F @ X2 ) ) )
          = ( measure_distr_a_b @ M2 @ N2 @ F ) ) ) ) ).

% bind_return_distr'
thf(fact_1108_bind__return__distr_H,axiom,
    ! [M2: sigma_measure_b,F: b > d,N2: sigma_measure_d] :
      ( ( ( sigma_space_b @ M2 )
       != bot_bot_set_b )
     => ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M2 @ N2 ) )
       => ( ( giry_bind_b_d @ M2
            @ ^ [X2: b] : ( giry_return_d @ N2 @ ( F @ X2 ) ) )
          = ( measure_distr_b_d @ M2 @ N2 @ F ) ) ) ) ).

% bind_return_distr'
thf(fact_1109_bind__return__distr_H,axiom,
    ! [M2: sigma_measure_b,F: b > real,N2: sigma_measure_real] :
      ( ( ( sigma_space_b @ M2 )
       != bot_bot_set_b )
     => ( ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ M2 @ N2 ) )
       => ( ( giry_bind_b_real @ M2
            @ ^ [X2: b] : ( giry_return_real @ N2 @ ( F @ X2 ) ) )
          = ( measure_distr_b_real @ M2 @ N2 @ F ) ) ) ) ).

% bind_return_distr'
thf(fact_1110_bind__return__distr_H,axiom,
    ! [M2: sigma_measure_a,F: a > real,N2: sigma_measure_real] :
      ( ( ( sigma_space_a @ M2 )
       != bot_bot_set_a )
     => ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M2 @ N2 ) )
       => ( ( giry_bind_a_real @ M2
            @ ^ [X2: a] : ( giry_return_real @ N2 @ ( F @ X2 ) ) )
          = ( measure_distr_a_real @ M2 @ N2 @ F ) ) ) ) ).

% bind_return_distr'
thf(fact_1111_finite__has__maximal2,axiom,
    ! [A: set_a_real,A2: a > real] :
      ( ( finite_finite_a_real @ A )
     => ( ( member_a_real @ A2 @ A )
       => ? [X: a > real] :
            ( ( member_a_real @ X @ A )
            & ( ord_less_eq_a_real @ A2 @ X )
            & ! [Xa: a > real] :
                ( ( member_a_real @ Xa @ A )
               => ( ( ord_less_eq_a_real @ X @ Xa )
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1112_finite__has__maximal2,axiom,
    ! [A: set_a_a_real,A2: a > a > real] :
      ( ( finite2647881322281930521a_real @ A )
     => ( ( member_a_a_real @ A2 @ A )
       => ? [X: a > a > real] :
            ( ( member_a_a_real @ X @ A )
            & ( ord_less_eq_a_a_real @ A2 @ X )
            & ! [Xa: a > a > real] :
                ( ( member_a_a_real @ Xa @ A )
               => ( ( ord_less_eq_a_a_real @ X @ Xa )
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1113_finite__has__maximal2,axiom,
    ! [A: set_set_set_b,A2: set_set_b] :
      ( ( finite7280322010187229928_set_b @ A )
     => ( ( member_set_set_b @ A2 @ A )
       => ? [X: set_set_b] :
            ( ( member_set_set_b @ X @ A )
            & ( ord_le3795704787696855135_set_b @ A2 @ X )
            & ! [Xa: set_set_b] :
                ( ( member_set_set_b @ Xa @ A )
               => ( ( ord_le3795704787696855135_set_b @ X @ Xa )
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1114_finite__has__maximal2,axiom,
    ! [A: set_set_set_a,A2: set_set_a] :
      ( ( finite7209287970140883943_set_a @ A )
     => ( ( member_set_set_a @ A2 @ A )
       => ? [X: set_set_a] :
            ( ( member_set_set_a @ X @ A )
            & ( ord_le3724670747650509150_set_a @ A2 @ X )
            & ! [Xa: set_set_a] :
                ( ( member_set_set_a @ Xa @ A )
               => ( ( ord_le3724670747650509150_set_a @ X @ Xa )
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1115_finite__has__maximal2,axiom,
    ! [A: set_set_b,A2: set_b] :
      ( ( finite_finite_set_b @ A )
     => ( ( member_set_b @ A2 @ A )
       => ? [X: set_b] :
            ( ( member_set_b @ X @ A )
            & ( ord_less_eq_set_b @ A2 @ X )
            & ! [Xa: set_b] :
                ( ( member_set_b @ Xa @ A )
               => ( ( ord_less_eq_set_b @ X @ Xa )
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1116_finite__has__maximal2,axiom,
    ! [A: set_set_a,A2: set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( member_set_a @ A2 @ A )
       => ? [X: set_a] :
            ( ( member_set_a @ X @ A )
            & ( ord_less_eq_set_a @ A2 @ X )
            & ! [Xa: set_a] :
                ( ( member_set_a @ Xa @ A )
               => ( ( ord_less_eq_set_a @ X @ Xa )
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1117_finite__has__maximal2,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A2 @ A )
       => ? [X: nat] :
            ( ( member_nat @ X @ A )
            & ( ord_less_eq_nat @ A2 @ X )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ X @ Xa )
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1118_finite__has__minimal2,axiom,
    ! [A: set_a_real,A2: a > real] :
      ( ( finite_finite_a_real @ A )
     => ( ( member_a_real @ A2 @ A )
       => ? [X: a > real] :
            ( ( member_a_real @ X @ A )
            & ( ord_less_eq_a_real @ X @ A2 )
            & ! [Xa: a > real] :
                ( ( member_a_real @ Xa @ A )
               => ( ( ord_less_eq_a_real @ Xa @ X )
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1119_finite__has__minimal2,axiom,
    ! [A: set_a_a_real,A2: a > a > real] :
      ( ( finite2647881322281930521a_real @ A )
     => ( ( member_a_a_real @ A2 @ A )
       => ? [X: a > a > real] :
            ( ( member_a_a_real @ X @ A )
            & ( ord_less_eq_a_a_real @ X @ A2 )
            & ! [Xa: a > a > real] :
                ( ( member_a_a_real @ Xa @ A )
               => ( ( ord_less_eq_a_a_real @ Xa @ X )
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1120_finite__has__minimal2,axiom,
    ! [A: set_set_set_b,A2: set_set_b] :
      ( ( finite7280322010187229928_set_b @ A )
     => ( ( member_set_set_b @ A2 @ A )
       => ? [X: set_set_b] :
            ( ( member_set_set_b @ X @ A )
            & ( ord_le3795704787696855135_set_b @ X @ A2 )
            & ! [Xa: set_set_b] :
                ( ( member_set_set_b @ Xa @ A )
               => ( ( ord_le3795704787696855135_set_b @ Xa @ X )
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1121_finite__has__minimal2,axiom,
    ! [A: set_set_set_a,A2: set_set_a] :
      ( ( finite7209287970140883943_set_a @ A )
     => ( ( member_set_set_a @ A2 @ A )
       => ? [X: set_set_a] :
            ( ( member_set_set_a @ X @ A )
            & ( ord_le3724670747650509150_set_a @ X @ A2 )
            & ! [Xa: set_set_a] :
                ( ( member_set_set_a @ Xa @ A )
               => ( ( ord_le3724670747650509150_set_a @ Xa @ X )
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1122_finite__has__minimal2,axiom,
    ! [A: set_set_b,A2: set_b] :
      ( ( finite_finite_set_b @ A )
     => ( ( member_set_b @ A2 @ A )
       => ? [X: set_b] :
            ( ( member_set_b @ X @ A )
            & ( ord_less_eq_set_b @ X @ A2 )
            & ! [Xa: set_b] :
                ( ( member_set_b @ Xa @ A )
               => ( ( ord_less_eq_set_b @ Xa @ X )
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1123_finite__has__minimal2,axiom,
    ! [A: set_set_a,A2: set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( member_set_a @ A2 @ A )
       => ? [X: set_a] :
            ( ( member_set_a @ X @ A )
            & ( ord_less_eq_set_a @ X @ A2 )
            & ! [Xa: set_a] :
                ( ( member_set_a @ Xa @ A )
               => ( ( ord_less_eq_set_a @ Xa @ X )
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1124_finite__has__minimal2,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A2 @ A )
       => ? [X: nat] :
            ( ( member_nat @ X @ A )
            & ( ord_less_eq_nat @ X @ A2 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ Xa @ X )
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1125_finite__subset,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( finite_finite_nat @ B )
       => ( finite_finite_nat @ A ) ) ) ).

% finite_subset
thf(fact_1126_finite__subset,axiom,
    ! [A: set_set_b,B: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A @ B )
     => ( ( finite_finite_set_b @ B )
       => ( finite_finite_set_b @ A ) ) ) ).

% finite_subset
thf(fact_1127_finite__subset,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B )
     => ( ( finite_finite_set_a @ B )
       => ( finite_finite_set_a @ A ) ) ) ).

% finite_subset
thf(fact_1128_finite__subset,axiom,
    ! [A: set_b,B: set_b] :
      ( ( ord_less_eq_set_b @ A @ B )
     => ( ( finite_finite_b @ B )
       => ( finite_finite_b @ A ) ) ) ).

% finite_subset
thf(fact_1129_finite__subset,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( finite_finite_a @ B )
       => ( finite_finite_a @ A ) ) ) ).

% finite_subset
thf(fact_1130_infinite__super,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( ord_less_eq_set_nat @ S @ T )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ T ) ) ) ).

% infinite_super
thf(fact_1131_infinite__super,axiom,
    ! [S: set_set_b,T: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ S @ T )
     => ( ~ ( finite_finite_set_b @ S )
       => ~ ( finite_finite_set_b @ T ) ) ) ).

% infinite_super
thf(fact_1132_infinite__super,axiom,
    ! [S: set_set_a,T: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ S @ T )
     => ( ~ ( finite_finite_set_a @ S )
       => ~ ( finite_finite_set_a @ T ) ) ) ).

% infinite_super
thf(fact_1133_infinite__super,axiom,
    ! [S: set_b,T: set_b] :
      ( ( ord_less_eq_set_b @ S @ T )
     => ( ~ ( finite_finite_b @ S )
       => ~ ( finite_finite_b @ T ) ) ) ).

% infinite_super
thf(fact_1134_infinite__super,axiom,
    ! [S: set_a,T: set_a] :
      ( ( ord_less_eq_set_a @ S @ T )
     => ( ~ ( finite_finite_a @ S )
       => ~ ( finite_finite_a @ T ) ) ) ).

% infinite_super
thf(fact_1135_rev__finite__subset,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( finite_finite_nat @ A ) ) ) ).

% rev_finite_subset
thf(fact_1136_rev__finite__subset,axiom,
    ! [B: set_set_b,A: set_set_b] :
      ( ( finite_finite_set_b @ B )
     => ( ( ord_le3795704787696855135_set_b @ A @ B )
       => ( finite_finite_set_b @ A ) ) ) ).

% rev_finite_subset
thf(fact_1137_rev__finite__subset,axiom,
    ! [B: set_set_a,A: set_set_a] :
      ( ( finite_finite_set_a @ B )
     => ( ( ord_le3724670747650509150_set_a @ A @ B )
       => ( finite_finite_set_a @ A ) ) ) ).

% rev_finite_subset
thf(fact_1138_rev__finite__subset,axiom,
    ! [B: set_b,A: set_b] :
      ( ( finite_finite_b @ B )
     => ( ( ord_less_eq_set_b @ A @ B )
       => ( finite_finite_b @ A ) ) ) ).

% rev_finite_subset
thf(fact_1139_rev__finite__subset,axiom,
    ! [B: set_a,A: set_a] :
      ( ( finite_finite_a @ B )
     => ( ( ord_less_eq_set_a @ A @ B )
       => ( finite_finite_a @ A ) ) ) ).

% rev_finite_subset
thf(fact_1140_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_1141_finite_OemptyI,axiom,
    finite_finite_a @ bot_bot_set_a ).

% finite.emptyI
thf(fact_1142_finite_OemptyI,axiom,
    finite_finite_b @ bot_bot_set_b ).

% finite.emptyI
thf(fact_1143_finite_OemptyI,axiom,
    finite_finite_o @ bot_bot_set_o ).

% finite.emptyI
thf(fact_1144_infinite__imp__nonempty,axiom,
    ! [S: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( S != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_1145_infinite__imp__nonempty,axiom,
    ! [S: set_a] :
      ( ~ ( finite_finite_a @ S )
     => ( S != bot_bot_set_a ) ) ).

% infinite_imp_nonempty
thf(fact_1146_infinite__imp__nonempty,axiom,
    ! [S: set_b] :
      ( ~ ( finite_finite_b @ S )
     => ( S != bot_bot_set_b ) ) ).

% infinite_imp_nonempty
thf(fact_1147_infinite__imp__nonempty,axiom,
    ! [S: set_o] :
      ( ~ ( finite_finite_o @ S )
     => ( S != bot_bot_set_o ) ) ).

% infinite_imp_nonempty
thf(fact_1148_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_1149_prob__space_Oprob__space__completion,axiom,
    ! [M2: sigma_measure_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( probab7247484486040049090pace_b @ ( comple3428971583294703881tion_b @ M2 ) ) ) ).

% prob_space.prob_space_completion
thf(fact_1150_prob__space_Oprob__space__completion,axiom,
    ! [M2: sigma_6673148739008212994ream_b] :
      ( ( probab6181332906713481800ream_b @ M2 )
     => ( probab6181332906713481800ream_b @ ( comple9045182969540656911ream_b @ M2 ) ) ) ).

% prob_space.prob_space_completion
thf(fact_1151_prob__space_Oprob__space__completion,axiom,
    ! [M2: sigma_6602114698961867009ream_a] :
      ( ( probab6181332902410252999ream_a @ M2 )
     => ( probab6181332902410252999ream_a @ ( comple9045182965237428110ream_a @ M2 ) ) ) ).

% prob_space.prob_space_completion
thf(fact_1152_prob__space__imp__sigma__finite,axiom,
    ! [M2: sigma_6673148739008212994ream_b] :
      ( ( probab6181332906713481800ream_b @ M2 )
     => ( measur5334621515951128742ream_b @ M2 ) ) ).

% prob_space_imp_sigma_finite
thf(fact_1153_prob__space__imp__sigma__finite,axiom,
    ! [M2: sigma_6602114698961867009ream_a] :
      ( ( probab6181332902410252999ream_a @ M2 )
     => ( measur5334621511647899941ream_a @ M2 ) ) ).

% prob_space_imp_sigma_finite
thf(fact_1154_prob__space__imp__sigma__finite,axiom,
    ! [M2: sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( measur4308613598931908895sure_a @ M2 ) ) ).

% prob_space_imp_sigma_finite
thf(fact_1155_prob__space__imp__sigma__finite,axiom,
    ! [M2: sigma_measure_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( measur4308613598931908896sure_b @ M2 ) ) ).

% prob_space_imp_sigma_finite
thf(fact_1156_prob__space_Oindep__events__finite__index__events,axiom,
    ! [M2: sigma_measure_a,F2: nat > set_a,I: set_nat] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe1551197314001032186_a_nat @ M2 @ F2 @ I )
        = ( ! [J2: set_nat] :
              ( ( ord_less_eq_set_nat @ J2 @ I )
             => ( ( J2 != bot_bot_set_nat )
               => ( ( finite_finite_nat @ J2 )
                 => ( indepe1551197314001032186_a_nat @ M2 @ F2 @ J2 ) ) ) ) ) ) ) ).

% prob_space.indep_events_finite_index_events
thf(fact_1157_prob__space_Oindep__events__finite__index__events,axiom,
    ! [M2: sigma_measure_a,F2: $o > set_a,I: set_o] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe3695496658712714478ts_a_o @ M2 @ F2 @ I )
        = ( ! [J2: set_o] :
              ( ( ord_less_eq_set_o @ J2 @ I )
             => ( ( J2 != bot_bot_set_o )
               => ( ( finite_finite_o @ J2 )
                 => ( indepe3695496658712714478ts_a_o @ M2 @ F2 @ J2 ) ) ) ) ) ) ) ).

% prob_space.indep_events_finite_index_events
thf(fact_1158_prob__space_Oindep__events__finite__index__events,axiom,
    ! [M2: sigma_measure_a,F2: b > set_a,I: set_b] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe1948846426091821397ts_a_b @ M2 @ F2 @ I )
        = ( ! [J2: set_b] :
              ( ( ord_less_eq_set_b @ J2 @ I )
             => ( ( J2 != bot_bot_set_b )
               => ( ( finite_finite_b @ J2 )
                 => ( indepe1948846426091821397ts_a_b @ M2 @ F2 @ J2 ) ) ) ) ) ) ) ).

% prob_space.indep_events_finite_index_events
thf(fact_1159_prob__space_Oindep__events__finite__index__events,axiom,
    ! [M2: sigma_measure_a,F2: a > set_a,I: set_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe1948846426091821396ts_a_a @ M2 @ F2 @ I )
        = ( ! [J2: set_a] :
              ( ( ord_less_eq_set_a @ J2 @ I )
             => ( ( J2 != bot_bot_set_a )
               => ( ( finite_finite_a @ J2 )
                 => ( indepe1948846426091821396ts_a_a @ M2 @ F2 @ J2 ) ) ) ) ) ) ) ).

% prob_space.indep_events_finite_index_events
thf(fact_1160_prob__space_Oindep__events__finite__index__events,axiom,
    ! [M2: sigma_measure_b,F2: nat > set_b,I: set_nat] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe2786641642957426683_b_nat @ M2 @ F2 @ I )
        = ( ! [J2: set_nat] :
              ( ( ord_less_eq_set_nat @ J2 @ I )
             => ( ( J2 != bot_bot_set_nat )
               => ( ( finite_finite_nat @ J2 )
                 => ( indepe2786641642957426683_b_nat @ M2 @ F2 @ J2 ) ) ) ) ) ) ) ).

% prob_space.indep_events_finite_index_events
thf(fact_1161_prob__space_Oindep__events__finite__index__events,axiom,
    ! [M2: sigma_measure_b,F2: $o > set_b,I: set_o] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe796274259248320173ts_b_o @ M2 @ F2 @ I )
        = ( ! [J2: set_o] :
              ( ( ord_less_eq_set_o @ J2 @ I )
             => ( ( J2 != bot_bot_set_o )
               => ( ( finite_finite_o @ J2 )
                 => ( indepe796274259248320173ts_b_o @ M2 @ F2 @ J2 ) ) ) ) ) ) ) ).

% prob_space.indep_events_finite_index_events
thf(fact_1162_prob__space_Oindep__events__finite__index__events,axiom,
    ! [M2: sigma_measure_b,F2: b > set_b,I: set_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe8384762880973921172ts_b_b @ M2 @ F2 @ I )
        = ( ! [J2: set_b] :
              ( ( ord_less_eq_set_b @ J2 @ I )
             => ( ( J2 != bot_bot_set_b )
               => ( ( finite_finite_b @ J2 )
                 => ( indepe8384762880973921172ts_b_b @ M2 @ F2 @ J2 ) ) ) ) ) ) ) ).

% prob_space.indep_events_finite_index_events
thf(fact_1163_prob__space_Oindep__events__finite__index__events,axiom,
    ! [M2: sigma_measure_b,F2: a > set_b,I: set_a] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( indepe8384762880973921171ts_b_a @ M2 @ F2 @ I )
        = ( ! [J2: set_a] :
              ( ( ord_less_eq_set_a @ J2 @ I )
             => ( ( J2 != bot_bot_set_a )
               => ( ( finite_finite_a @ J2 )
                 => ( indepe8384762880973921171ts_b_a @ M2 @ F2 @ J2 ) ) ) ) ) ) ) ).

% prob_space.indep_events_finite_index_events
thf(fact_1164_prob__space_Oindep__events__finite__index__events,axiom,
    ! [M2: sigma_measure_a,F2: set_b > set_a,I: set_set_b] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe451107620608244277_set_b @ M2 @ F2 @ I )
        = ( ! [J2: set_set_b] :
              ( ( ord_le3795704787696855135_set_b @ J2 @ I )
             => ( ( J2 != bot_bot_set_set_b )
               => ( ( finite_finite_set_b @ J2 )
                 => ( indepe451107620608244277_set_b @ M2 @ F2 @ J2 ) ) ) ) ) ) ) ).

% prob_space.indep_events_finite_index_events
thf(fact_1165_prob__space_Oindep__events__finite__index__events,axiom,
    ! [M2: sigma_measure_a,F2: set_a > set_a,I: set_set_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( indepe451107616305015476_set_a @ M2 @ F2 @ I )
        = ( ! [J2: set_set_a] :
              ( ( ord_le3724670747650509150_set_a @ J2 @ I )
             => ( ( J2 != bot_bot_set_set_a )
               => ( ( finite_finite_set_a @ J2 )
                 => ( indepe451107616305015476_set_a @ M2 @ F2 @ J2 ) ) ) ) ) ) ) ).

% prob_space.indep_events_finite_index_events
thf(fact_1166_prob__space_Ofinite__measure,axiom,
    ! [M2: sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( measur930452917991658466sure_a @ M2 ) ) ).

% prob_space.finite_measure
thf(fact_1167_prob__space_Ofinite__measure,axiom,
    ! [M2: sigma_measure_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( measur930452917991658467sure_b @ M2 ) ) ).

% prob_space.finite_measure
thf(fact_1168_prob__space_Ofinite__measure,axiom,
    ! [M2: sigma_6673148739008212994ream_b] :
      ( ( probab6181332906713481800ream_b @ M2 )
     => ( measur2119334577284823785ream_b @ M2 ) ) ).

% prob_space.finite_measure
thf(fact_1169_prob__space_Ofinite__measure,axiom,
    ! [M2: sigma_6602114698961867009ream_a] :
      ( ( probab6181332902410252999ream_a @ M2 )
     => ( measur2119334572981594984ream_a @ M2 ) ) ).

% prob_space.finite_measure
thf(fact_1170_finite__has__maximal,axiom,
    ! [A: set_o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ? [X: $o] :
            ( ( member_o @ X @ A )
            & ! [Xa: $o] :
                ( ( member_o @ Xa @ A )
               => ( ( ord_less_eq_o @ X @ Xa )
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1171_finite__has__maximal,axiom,
    ! [A: set_set_set_b] :
      ( ( finite7280322010187229928_set_b @ A )
     => ( ( A != bot_bo859375258733454347_set_b )
       => ? [X: set_set_b] :
            ( ( member_set_set_b @ X @ A )
            & ! [Xa: set_set_b] :
                ( ( member_set_set_b @ Xa @ A )
               => ( ( ord_le3795704787696855135_set_b @ X @ Xa )
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1172_finite__has__maximal,axiom,
    ! [A: set_set_set_a] :
      ( ( finite7209287970140883943_set_a @ A )
     => ( ( A != bot_bo3380559777022489994_set_a )
       => ? [X: set_set_a] :
            ( ( member_set_set_a @ X @ A )
            & ! [Xa: set_set_a] :
                ( ( member_set_set_a @ Xa @ A )
               => ( ( ord_le3724670747650509150_set_a @ X @ Xa )
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1173_finite__has__maximal,axiom,
    ! [A: set_set_b] :
      ( ( finite_finite_set_b @ A )
     => ( ( A != bot_bot_set_set_b )
       => ? [X: set_b] :
            ( ( member_set_b @ X @ A )
            & ! [Xa: set_b] :
                ( ( member_set_b @ Xa @ A )
               => ( ( ord_less_eq_set_b @ X @ Xa )
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1174_finite__has__maximal,axiom,
    ! [A: set_set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( A != bot_bot_set_set_a )
       => ? [X: set_a] :
            ( ( member_set_a @ X @ A )
            & ! [Xa: set_a] :
                ( ( member_set_a @ Xa @ A )
               => ( ( ord_less_eq_set_a @ X @ Xa )
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1175_finite__has__maximal,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ? [X: nat] :
            ( ( member_nat @ X @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ X @ Xa )
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1176_finite__has__minimal,axiom,
    ! [A: set_o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ? [X: $o] :
            ( ( member_o @ X @ A )
            & ! [Xa: $o] :
                ( ( member_o @ Xa @ A )
               => ( ( ord_less_eq_o @ Xa @ X )
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1177_finite__has__minimal,axiom,
    ! [A: set_set_set_b] :
      ( ( finite7280322010187229928_set_b @ A )
     => ( ( A != bot_bo859375258733454347_set_b )
       => ? [X: set_set_b] :
            ( ( member_set_set_b @ X @ A )
            & ! [Xa: set_set_b] :
                ( ( member_set_set_b @ Xa @ A )
               => ( ( ord_le3795704787696855135_set_b @ Xa @ X )
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1178_finite__has__minimal,axiom,
    ! [A: set_set_set_a] :
      ( ( finite7209287970140883943_set_a @ A )
     => ( ( A != bot_bo3380559777022489994_set_a )
       => ? [X: set_set_a] :
            ( ( member_set_set_a @ X @ A )
            & ! [Xa: set_set_a] :
                ( ( member_set_set_a @ Xa @ A )
               => ( ( ord_le3724670747650509150_set_a @ Xa @ X )
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1179_finite__has__minimal,axiom,
    ! [A: set_set_b] :
      ( ( finite_finite_set_b @ A )
     => ( ( A != bot_bot_set_set_b )
       => ? [X: set_b] :
            ( ( member_set_b @ X @ A )
            & ! [Xa: set_b] :
                ( ( member_set_b @ Xa @ A )
               => ( ( ord_less_eq_set_b @ Xa @ X )
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1180_finite__has__minimal,axiom,
    ! [A: set_set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( A != bot_bot_set_set_a )
       => ? [X: set_a] :
            ( ( member_set_a @ X @ A )
            & ! [Xa: set_a] :
                ( ( member_set_a @ Xa @ A )
               => ( ( ord_less_eq_set_a @ Xa @ X )
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1181_finite__has__minimal,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ? [X: nat] :
            ( ( member_nat @ X @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ Xa @ X )
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1182_prob__space_Onot__empty,axiom,
    ! [M2: sigma_2733069582767196501sure_b] :
      ( ( probab6953849309755211513sure_b @ M2 )
     => ( ( sigma_6149183809546982772sure_b @ M2 )
       != bot_bo5242124273094418374sure_b ) ) ).

% prob_space.not_empty
thf(fact_1183_prob__space_Onot__empty,axiom,
    ! [M2: sigma_2662035542720850516sure_a] :
      ( ( probab6953849305451982712sure_a @ M2 )
     => ( ( sigma_6149183805243753971sure_a @ M2 )
       != bot_bo5171090233048072389sure_a ) ) ).

% prob_space.not_empty
thf(fact_1184_prob__space_Onot__empty,axiom,
    ! [M2: sigma_measure_o] :
      ( ( probab1190487603588612059pace_o @ M2 )
     => ( ( sigma_space_o @ M2 )
       != bot_bot_set_o ) ) ).

% prob_space.not_empty
thf(fact_1185_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_1186_prob__space_Onot__empty,axiom,
    ! [M2: sigma_measure_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( sigma_space_b @ M2 )
       != bot_bot_set_b ) ) ).

% prob_space.not_empty
thf(fact_1187_prob__space_Onot__empty,axiom,
    ! [M2: sigma_6673148739008212994ream_b] :
      ( ( probab6181332906713481800ream_b @ M2 )
     => ( ( sigma_space_stream_b @ M2 )
       != bot_bot_set_stream_b ) ) ).

% prob_space.not_empty
thf(fact_1188_prob__space_Onot__empty,axiom,
    ! [M2: sigma_6602114698961867009ream_a] :
      ( ( probab6181332902410252999ream_a @ M2 )
     => ( ( sigma_space_stream_a @ M2 )
       != bot_bot_set_stream_a ) ) ).

% prob_space.not_empty
thf(fact_1189_prob__space_Oprob__space__distr,axiom,
    ! [M2: sigma_measure_a,F: a > c,M: sigma_measure_c] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( member_a_c @ F @ ( sigma_measurable_a_c @ M2 @ M ) )
       => ( probab7247484486040049091pace_c @ ( measure_distr_a_c @ M2 @ M @ F ) ) ) ) ).

% prob_space.prob_space_distr
thf(fact_1190_prob__space_Oprob__space__distr,axiom,
    ! [M2: sigma_measure_a,F: a > real,M: sigma_measure_real] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M2 @ M ) )
       => ( probab535871623910865577e_real @ ( measure_distr_a_real @ M2 @ M @ F ) ) ) ) ).

% prob_space.prob_space_distr
thf(fact_1191_prob__space_Oprob__space__distr,axiom,
    ! [M2: sigma_measure_a,F: a > a,M: sigma_measure_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M2 @ M ) )
       => ( probab7247484486040049089pace_a @ ( measure_distr_a_a @ M2 @ M @ F ) ) ) ) ).

% prob_space.prob_space_distr
thf(fact_1192_prob__space_Oprob__space__distr,axiom,
    ! [M2: sigma_measure_a,F: a > b,M: sigma_measure_b] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M2 @ M ) )
       => ( probab7247484486040049090pace_b @ ( measure_distr_a_b @ M2 @ M @ F ) ) ) ) ).

% prob_space.prob_space_distr
thf(fact_1193_prob__space_Oprob__space__distr,axiom,
    ! [M2: sigma_measure_b,F: b > d,M: sigma_measure_d] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M2 @ M ) )
       => ( probab7247484486040049092pace_d @ ( measure_distr_b_d @ M2 @ M @ F ) ) ) ) ).

% prob_space.prob_space_distr
thf(fact_1194_prob__space_Oprob__space__distr,axiom,
    ! [M2: sigma_measure_b,F: b > real,M: sigma_measure_real] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ M2 @ M ) )
       => ( probab535871623910865577e_real @ ( measure_distr_b_real @ M2 @ M @ F ) ) ) ) ).

% prob_space.prob_space_distr
thf(fact_1195_prob__space_Oprob__space__distr,axiom,
    ! [M2: sigma_measure_b,F: b > a,M: sigma_measure_a] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M2 @ M ) )
       => ( probab7247484486040049089pace_a @ ( measure_distr_b_a @ M2 @ M @ F ) ) ) ) ).

% prob_space.prob_space_distr
thf(fact_1196_prob__space_Oprob__space__distr,axiom,
    ! [M2: sigma_measure_b,F: b > b,M: sigma_measure_b] :
      ( ( probab7247484486040049090pace_b @ M2 )
     => ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M2 @ M ) )
       => ( probab7247484486040049090pace_b @ ( measure_distr_b_b @ M2 @ M @ F ) ) ) ) ).

% prob_space.prob_space_distr
thf(fact_1197_prob__space_Oprob__space__distr,axiom,
    ! [M2: sigma_measure_a,F: a > stream_b,M: sigma_6673148739008212994ream_b] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( member_a_stream_b @ F @ ( sigma_6363237497282895319ream_b @ M2 @ M ) )
       => ( probab6181332906713481800ream_b @ ( measur8618403474564049535ream_b @ M2 @ M @ F ) ) ) ) ).

% prob_space.prob_space_distr
thf(fact_1198_prob__space_Oprob__space__distr,axiom,
    ! [M2: sigma_measure_a,F: a > stream_a,M: sigma_6602114698961867009ream_a] :
      ( ( probab7247484486040049089pace_a @ M2 )
     => ( ( member_a_stream_a @ F @ ( sigma_6363237492979666518ream_a @ M2 @ M ) )
       => ( probab6181332902410252999ream_a @ ( measur8618403470260820734ream_a @ M2 @ M @ F ) ) ) ) ).

% prob_space.prob_space_distr
thf(fact_1199_D_Oindep__eventsI__indep__vars,axiom,
    ! [N2: c > sigma_measure_c,X4: c > b > c,I: set_c,P: c > c > $o] :
      ( ( indepe1224748351465797815_b_c_c @ ( measure_distr_a_b @ m @ n @ f ) @ N2 @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_set_c
              @ ( collect_c
                @ ^ [X2: c] :
                    ( ( member_c @ X2 @ ( sigma_space_c @ ( N2 @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_c @ ( N2 @ I2 ) ) ) )
       => ( indepe8384762880973921173ts_b_c @ ( measure_distr_a_b @ m @ n @ f )
          @ ^ [I3: c] :
              ( collect_b
              @ ^ [X2: b] :
                  ( ( member_b @ X2 @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
                  & ( P @ I3 @ ( X4 @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% D.indep_eventsI_indep_vars
thf(fact_1200_D_Oindep__eventsI__indep__vars,axiom,
    ! [N2: b > sigma_measure_c,X4: b > b > c,I: set_b,P: b > c > $o] :
      ( ( indepe4012203933438473848_b_b_c @ ( measure_distr_a_b @ m @ n @ f ) @ N2 @ X4 @ I )
     => ( ! [I2: b] :
            ( ( member_b @ I2 @ I )
           => ( member_set_c
              @ ( collect_c
                @ ^ [X2: c] :
                    ( ( member_c @ X2 @ ( sigma_space_c @ ( N2 @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_c @ ( N2 @ I2 ) ) ) )
       => ( indepe8384762880973921172ts_b_b @ ( measure_distr_a_b @ m @ n @ f )
          @ ^ [I3: b] :
              ( collect_b
              @ ^ [X2: b] :
                  ( ( member_b @ X2 @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
                  & ( P @ I3 @ ( X4 @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% D.indep_eventsI_indep_vars
thf(fact_1201_D_Oindep__eventsI__indep__vars,axiom,
    ! [N2: a > sigma_measure_c,X4: a > b > c,I: set_a,P: a > c > $o] :
      ( ( indepe6799659515411149881_b_a_c @ ( measure_distr_a_b @ m @ n @ f ) @ N2 @ X4 @ I )
     => ( ! [I2: a] :
            ( ( member_a @ I2 @ I )
           => ( member_set_c
              @ ( collect_c
                @ ^ [X2: c] :
                    ( ( member_c @ X2 @ ( sigma_space_c @ ( N2 @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_c @ ( N2 @ I2 ) ) ) )
       => ( indepe8384762880973921171ts_b_a @ ( measure_distr_a_b @ m @ n @ f )
          @ ^ [I3: a] :
              ( collect_b
              @ ^ [X2: b] :
                  ( ( member_b @ X2 @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
                  & ( P @ I3 @ ( X4 @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% D.indep_eventsI_indep_vars
thf(fact_1202_D_Oindep__eventsI__indep__vars,axiom,
    ! [N2: c > sigma_measure_nat,X4: c > b > nat,I: set_c,P: c > nat > $o] :
      ( ( indepe3028620094473096793_c_nat @ ( measure_distr_a_b @ m @ n @ f ) @ N2 @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_set_nat
              @ ( collect_nat
                @ ^ [X2: nat] :
                    ( ( member_nat @ X2 @ ( sigma_space_nat @ ( N2 @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_nat @ ( N2 @ I2 ) ) ) )
       => ( indepe8384762880973921173ts_b_c @ ( measure_distr_a_b @ m @ n @ f )
          @ ^ [I3: c] :
              ( collect_b
              @ ^ [X2: b] :
                  ( ( member_b @ X2 @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
                  & ( P @ I3 @ ( X4 @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% D.indep_eventsI_indep_vars
thf(fact_1203_D_Oindep__eventsI__indep__vars,axiom,
    ! [N2: b > sigma_measure_nat,X4: b > b > nat,I: set_b,P: b > nat > $o] :
      ( ( indepe1793175765516702296_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ N2 @ X4 @ I )
     => ( ! [I2: b] :
            ( ( member_b @ I2 @ I )
           => ( member_set_nat
              @ ( collect_nat
                @ ^ [X2: nat] :
                    ( ( member_nat @ X2 @ ( sigma_space_nat @ ( N2 @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_nat @ ( N2 @ I2 ) ) ) )
       => ( indepe8384762880973921172ts_b_b @ ( measure_distr_a_b @ m @ n @ f )
          @ ^ [I3: b] :
              ( collect_b
              @ ^ [X2: b] :
                  ( ( member_b @ X2 @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
                  & ( P @ I3 @ ( X4 @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% D.indep_eventsI_indep_vars
thf(fact_1204_D_Oindep__eventsI__indep__vars,axiom,
    ! [N2: a > sigma_measure_nat,X4: a > b > nat,I: set_a,P: a > nat > $o] :
      ( ( indepe557731436560307799_a_nat @ ( measure_distr_a_b @ m @ n @ f ) @ N2 @ X4 @ I )
     => ( ! [I2: a] :
            ( ( member_a @ I2 @ I )
           => ( member_set_nat
              @ ( collect_nat
                @ ^ [X2: nat] :
                    ( ( member_nat @ X2 @ ( sigma_space_nat @ ( N2 @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_nat @ ( N2 @ I2 ) ) ) )
       => ( indepe8384762880973921171ts_b_a @ ( measure_distr_a_b @ m @ n @ f )
          @ ^ [I3: a] :
              ( collect_b
              @ ^ [X2: b] :
                  ( ( member_b @ X2 @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
                  & ( P @ I3 @ ( X4 @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% D.indep_eventsI_indep_vars
thf(fact_1205_D_Oindep__eventsI__indep__vars,axiom,
    ! [N2: c > sigma_measure_a,X4: c > b > a,I: set_c,P: c > a > $o] :
      ( ( indepe1224748351465797813_b_c_a @ ( measure_distr_a_b @ m @ n @ f ) @ N2 @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_set_a
              @ ( collect_a
                @ ^ [X2: a] :
                    ( ( member_a @ X2 @ ( sigma_space_a @ ( N2 @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_a @ ( N2 @ I2 ) ) ) )
       => ( indepe8384762880973921173ts_b_c @ ( measure_distr_a_b @ m @ n @ f )
          @ ^ [I3: c] :
              ( collect_b
              @ ^ [X2: b] :
                  ( ( member_b @ X2 @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
                  & ( P @ I3 @ ( X4 @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% D.indep_eventsI_indep_vars
thf(fact_1206_D_Oindep__eventsI__indep__vars,axiom,
    ! [N2: b > sigma_measure_a,X4: b > b > a,I: set_b,P: b > a > $o] :
      ( ( indepe4012203933438473846_b_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ N2 @ X4 @ I )
     => ( ! [I2: b] :
            ( ( member_b @ I2 @ I )
           => ( member_set_a
              @ ( collect_a
                @ ^ [X2: a] :
                    ( ( member_a @ X2 @ ( sigma_space_a @ ( N2 @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_a @ ( N2 @ I2 ) ) ) )
       => ( indepe8384762880973921172ts_b_b @ ( measure_distr_a_b @ m @ n @ f )
          @ ^ [I3: b] :
              ( collect_b
              @ ^ [X2: b] :
                  ( ( member_b @ X2 @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
                  & ( P @ I3 @ ( X4 @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% D.indep_eventsI_indep_vars
thf(fact_1207_D_Oindep__eventsI__indep__vars,axiom,
    ! [N2: a > sigma_measure_a,X4: a > b > a,I: set_a,P: a > a > $o] :
      ( ( indepe6799659515411149879_b_a_a @ ( measure_distr_a_b @ m @ n @ f ) @ N2 @ X4 @ I )
     => ( ! [I2: a] :
            ( ( member_a @ I2 @ I )
           => ( member_set_a
              @ ( collect_a
                @ ^ [X2: a] :
                    ( ( member_a @ X2 @ ( sigma_space_a @ ( N2 @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_a @ ( N2 @ I2 ) ) ) )
       => ( indepe8384762880973921171ts_b_a @ ( measure_distr_a_b @ m @ n @ f )
          @ ^ [I3: a] :
              ( collect_b
              @ ^ [X2: b] :
                  ( ( member_b @ X2 @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
                  & ( P @ I3 @ ( X4 @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% D.indep_eventsI_indep_vars
thf(fact_1208_D_Oindep__eventsI__indep__vars,axiom,
    ! [N2: c > sigma_measure_b,X4: c > b > b,I: set_c,P: c > b > $o] :
      ( ( indepe1224748351465797814_b_c_b @ ( measure_distr_a_b @ m @ n @ f ) @ N2 @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_set_b
              @ ( collect_b
                @ ^ [X2: b] :
                    ( ( member_b @ X2 @ ( sigma_space_b @ ( N2 @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_b @ ( N2 @ I2 ) ) ) )
       => ( indepe8384762880973921173ts_b_c @ ( measure_distr_a_b @ m @ n @ f )
          @ ^ [I3: c] :
              ( collect_b
              @ ^ [X2: b] :
                  ( ( member_b @ X2 @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
                  & ( P @ I3 @ ( X4 @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% D.indep_eventsI_indep_vars
thf(fact_1209_D_Oindep__events__def__alt,axiom,
    ! [A: nat > set_b,I: set_nat] :
      ( ( indepe2786641642957426683_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ A @ I )
      = ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f )
        @ ^ [I3: nat] : ( insert_set_b @ ( A @ I3 ) @ bot_bot_set_set_b )
        @ I ) ) ).

% D.indep_events_def_alt
thf(fact_1210_D_Oindep__events__def__alt,axiom,
    ! [A: $o > set_b,I: set_o] :
      ( ( indepe796274259248320173ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ A @ I )
      = ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f )
        @ ^ [I3: $o] : ( insert_set_b @ ( A @ I3 ) @ bot_bot_set_set_b )
        @ I ) ) ).

% D.indep_events_def_alt
thf(fact_1211_indep__eventsI__indep__vars,axiom,
    ! [N2: c > sigma_measure_d,X4: c > a > d,I: set_c,P: c > d > $o] :
      ( ( indepe4851901773132442935_a_c_d @ m @ N2 @ X4 @ I )
     => ( ! [I2: c] :
            ( ( member_c @ I2 @ I )
           => ( member_set_d
              @ ( collect_d
                @ ^ [X2: d] :
                    ( ( member_d @ X2 @ ( sigma_space_d @ ( N2 @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_d @ ( N2 @ I2 ) ) ) )
       => ( indepe1948846426091821398ts_a_c @ m
          @ ^ [I3: c] :
              ( collect_a
              @ ^ [X2: a] :
                  ( ( member_a @ X2 @ ( sigma_space_a @ m ) )
                  & ( P @ I3 @ ( X4 @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% indep_eventsI_indep_vars
thf(fact_1212_tail__events__sets,axiom,
    ! [A: nat > set_set_a] :
      ( ! [I2: nat] : ( ord_le3724670747650509150_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ m ) )
     => ( ord_le3724670747650509150_set_a @ ( indepe7538416700049374166_a_nat @ m @ A ) @ ( sigma_sets_a @ m ) ) ) ).

% tail_events_sets
thf(fact_1213_D_Otail__events__sets,axiom,
    ! [A: nat > set_set_b] :
      ( ! [I2: nat] : ( ord_le3795704787696855135_set_b @ ( A @ I2 ) @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
     => ( ord_le3795704787696855135_set_b @ ( indepe8773861029005768663_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ A ) @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) ) ) ).

% D.tail_events_sets
thf(fact_1214_D_Oindep__setD__ev1,axiom,
    ! [A: set_set_b,B: set_set_b] :
      ( ( indepe2041756565122539607_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ A @ B )
     => ( ord_le3795704787696855135_set_b @ A @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) ) ) ).

% D.indep_setD_ev1
thf(fact_1215_D_Oindep__setD__ev2,axiom,
    ! [A: set_set_b,B: set_set_b] :
      ( ( indepe2041756565122539607_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ A @ B )
     => ( ord_le3795704787696855135_set_b @ B @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) ) ) ).

% D.indep_setD_ev2
thf(fact_1216_indep__setD__ev1,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ( indepe2041756565122539606_set_a @ m @ A @ B )
     => ( ord_le3724670747650509150_set_a @ A @ ( sigma_sets_a @ m ) ) ) ).

% indep_setD_ev1
thf(fact_1217_indep__setD__ev2,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ( indepe2041756565122539606_set_a @ m @ A @ B )
     => ( ord_le3724670747650509150_set_a @ B @ ( sigma_sets_a @ m ) ) ) ).

% indep_setD_ev2
thf(fact_1218_D_Obounded__measure,axiom,
    ! [A: set_b] : ( ord_less_eq_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A ) @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) ) ) ).

% D.bounded_measure
thf(fact_1219_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_1220_D_Ofinite__measure__mono,axiom,
    ! [A: set_b,B: set_b] :
      ( ( ord_less_eq_set_b @ A @ B )
     => ( ( member_set_b @ B @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
       => ( ord_less_eq_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A ) @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ B ) ) ) ) ).

% D.finite_measure_mono
thf(fact_1221_D_Omeasure__increasing,axiom,
    measur7372598777031404856b_real @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) ) ).

% D.measure_increasing
thf(fact_1222_D_Oprob__space,axiom,
    ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
    = one_one_real ) ).

% D.prob_space
thf(fact_1223_D_Omeasure__ge__1__iff,axiom,
    ! [A: set_b] :
      ( ( ord_less_eq_real @ one_one_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A ) )
      = ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A )
        = one_one_real ) ) ).

% D.measure_ge_1_iff
thf(fact_1224_measure__ge__1__iff,axiom,
    ! [A: set_a] :
      ( ( ord_less_eq_real @ one_one_real @ ( sigma_measure_a2 @ m @ A ) )
      = ( ( sigma_measure_a2 @ m @ A )
        = one_one_real ) ) ).

% measure_ge_1_iff
thf(fact_1225_bounded__measure,axiom,
    ! [A: set_a] : ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ A ) @ ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) ) ) ).

% bounded_measure
thf(fact_1226_finite__measure__mono,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
       => ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ A ) @ ( sigma_measure_a2 @ m @ B ) ) ) ) ).

% finite_measure_mono
thf(fact_1227_prob__space,axiom,
    ( ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) )
    = one_one_real ) ).

% prob_space
thf(fact_1228_measure__increasing,axiom,
    measur1776380161843274167a_real @ ( sigma_sets_a @ m ) @ ( sigma_measure_a2 @ m ) ).

% measure_increasing
thf(fact_1229_subprob__measure__le__1,axiom,
    ! [X4: set_a] : ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ X4 ) @ one_one_real ) ).

% subprob_measure_le_1
thf(fact_1230_prob__le__1,axiom,
    ! [A: set_a] : ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ A ) @ one_one_real ) ).

% prob_le_1
thf(fact_1231_D_Oprob__le__1,axiom,
    ! [A: set_b] : ( ord_less_eq_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A ) @ one_one_real ) ).

% D.prob_le_1
thf(fact_1232_D_Osubprob__measure__le__1,axiom,
    ! [X4: set_b] : ( ord_less_eq_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ X4 ) @ one_one_real ) ).

% D.subprob_measure_le_1
thf(fact_1233_D_Oindep__setI,axiom,
    ! [A: set_set_b,B: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
     => ( ( ord_le3795704787696855135_set_b @ B @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
       => ( ! [A6: set_b,B3: set_b] :
              ( ( member_set_b @ A6 @ A )
             => ( ( member_set_b @ B3 @ B )
               => ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( inf_inf_set_b @ A6 @ B3 ) )
                  = ( times_times_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A6 ) @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ B3 ) ) ) ) )
         => ( indepe2041756565122539607_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ A @ B ) ) ) ) ).

% D.indep_setI
thf(fact_1234_D_Oindep__sets2__eq,axiom,
    ! [A: set_set_b,B: set_set_b] :
      ( ( indepe2041756565122539607_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ A @ B )
      = ( ( ord_le3795704787696855135_set_b @ A @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
        & ( ord_le3795704787696855135_set_b @ B @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
        & ! [X2: set_b] :
            ( ( member_set_b @ X2 @ A )
           => ! [Y3: set_b] :
                ( ( member_set_b @ Y3 @ B )
               => ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( inf_inf_set_b @ X2 @ Y3 ) )
                  = ( times_times_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ X2 ) @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ Y3 ) ) ) ) ) ) ) ).

% D.indep_sets2_eq
thf(fact_1235_D_Oprob__neg,axiom,
    ! [P: b > $o] :
      ( ( member_set_b
        @ ( collect_b
          @ ^ [X2: b] :
              ( ( member_b @ X2 @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
     => ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f )
          @ ( collect_b
            @ ^ [X2: b] :
                ( ( member_b @ X2 @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
                & ~ ( P @ X2 ) ) ) )
        = ( minus_minus_real @ one_one_real
          @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f )
            @ ( collect_b
              @ ^ [X2: b] :
                  ( ( member_b @ X2 @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
                  & ( P @ X2 ) ) ) ) ) ) ) ).

% D.prob_neg
thf(fact_1236_D_Oindep__setD,axiom,
    ! [A: set_set_b,B: set_set_b,A2: set_b,B4: set_b] :
      ( ( indepe2041756565122539607_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ A @ B )
     => ( ( member_set_b @ A2 @ A )
       => ( ( member_set_b @ B4 @ B )
         => ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( inf_inf_set_b @ A2 @ B4 ) )
            = ( times_times_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A2 ) @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ B4 ) ) ) ) ) ) ).

% D.indep_setD
thf(fact_1237_D_Omeasure__eq__compl,axiom,
    ! [S2: set_b,T3: set_b] :
      ( ( member_set_b @ S2 @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
     => ( ( member_set_b @ T3 @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
       => ( ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( minus_minus_set_b @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ S2 ) )
            = ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( minus_minus_set_b @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ T3 ) ) )
         => ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ S2 )
            = ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ T3 ) ) ) ) ) ).

% D.measure_eq_compl
thf(fact_1238_prob__neg,axiom,
    ! [P: a > $o] :
      ( ( member_set_a
        @ ( collect_a
          @ ^ [X2: a] :
              ( ( member_a @ X2 @ ( sigma_space_a @ m ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_a @ m ) )
     => ( ( sigma_measure_a2 @ m
          @ ( collect_a
            @ ^ [X2: a] :
                ( ( member_a @ X2 @ ( sigma_space_a @ m ) )
                & ~ ( P @ X2 ) ) ) )
        = ( minus_minus_real @ one_one_real
          @ ( sigma_measure_a2 @ m
            @ ( collect_a
              @ ^ [X2: a] :
                  ( ( member_a @ X2 @ ( sigma_space_a @ m ) )
                  & ( P @ X2 ) ) ) ) ) ) ) ).

% prob_neg
thf(fact_1239_D_Omeasure__space__inter,axiom,
    ! [S2: set_b,T3: set_b] :
      ( ( member_set_b @ S2 @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
     => ( ( member_set_b @ T3 @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
       => ( ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ T3 )
            = ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) ) )
         => ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( inf_inf_set_b @ S2 @ T3 ) )
            = ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ S2 ) ) ) ) ) ).

% D.measure_space_inter
thf(fact_1240_D_Ofinite__measure__compl,axiom,
    ! [S: set_b] :
      ( ( member_set_b @ S @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
     => ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( minus_minus_set_b @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ S ) )
        = ( minus_minus_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) ) @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ S ) ) ) ) ).

% D.finite_measure_compl
thf(fact_1241_D_Ofinite__measure__Diff_H,axiom,
    ! [A: set_b,B: set_b] :
      ( ( member_set_b @ A @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
     => ( ( member_set_b @ B @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
       => ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( minus_minus_set_b @ A @ B ) )
          = ( minus_minus_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A ) @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( inf_inf_set_b @ A @ B ) ) ) ) ) ) ).

% D.finite_measure_Diff'
thf(fact_1242_D_Ofinite__measure__Diff,axiom,
    ! [A: set_b,B: set_b] :
      ( ( member_set_b @ A @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
     => ( ( member_set_b @ B @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
       => ( ( ord_less_eq_set_b @ B @ A )
         => ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( minus_minus_set_b @ A @ B ) )
            = ( minus_minus_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A ) @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ B ) ) ) ) ) ) ).

% D.finite_measure_Diff
thf(fact_1243_D_Oprob__compl,axiom,
    ! [A: set_b] :
      ( ( member_set_b @ A @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
     => ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( minus_minus_set_b @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ A ) )
        = ( minus_minus_real @ one_one_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A ) ) ) ) ).

% D.prob_compl
thf(fact_1244_D_Omeasure__exclude,axiom,
    ! [A: set_b,B: set_b] :
      ( ( member_set_b @ A @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
     => ( ( member_set_b @ B @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
       => ( ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A )
            = ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) ) )
         => ( ( ( inf_inf_set_b @ A @ B )
              = bot_bot_set_b )
           => ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ B )
              = zero_zero_real ) ) ) ) ) ).

% D.measure_exclude
thf(fact_1245_D_Osigma__algebra__tail__events,axiom,
    ! [A: nat > set_set_b] :
      ( ! [I2: nat] : ( sigma_4968961713055010668ebra_b @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ ( A @ I2 ) )
     => ( sigma_4968961713055010668ebra_b @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ ( indepe8773861029005768663_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ A ) ) ) ).

% D.sigma_algebra_tail_events
thf(fact_1246_D_Ofinite__measure__eq__sum__singleton,axiom,
    ! [S: set_b] :
      ( ( finite_finite_b @ S )
     => ( ! [X: b] :
            ( ( member_b @ X @ S )
           => ( member_set_b @ ( insert_b @ X @ bot_bot_set_b ) @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) ) )
       => ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ S )
          = ( groups8336678772925405937b_real
            @ ^ [X2: b] : ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( insert_b @ X2 @ bot_bot_set_b ) )
            @ S ) ) ) ) ).

% D.finite_measure_eq_sum_singleton
thf(fact_1247_finite__measure__Diff_H,axiom,
    ! [A: set_a,B: set_a] :
      ( ( member_set_a @ A @ ( sigma_sets_a @ m ) )
     => ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
       => ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ A @ B ) )
          = ( minus_minus_real @ ( sigma_measure_a2 @ m @ A ) @ ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ A @ B ) ) ) ) ) ) ).

% finite_measure_Diff'
thf(fact_1248_measure__space__inter,axiom,
    ! [S2: set_a,T3: set_a] :
      ( ( member_set_a @ S2 @ ( sigma_sets_a @ m ) )
     => ( ( member_set_a @ T3 @ ( sigma_sets_a @ m ) )
       => ( ( ( sigma_measure_a2 @ m @ T3 )
            = ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) ) )
         => ( ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ S2 @ T3 ) )
            = ( sigma_measure_a2 @ m @ S2 ) ) ) ) ) ).

% measure_space_inter
thf(fact_1249_measure__eq__compl,axiom,
    ! [S2: set_a,T3: set_a] :
      ( ( member_set_a @ S2 @ ( sigma_sets_a @ m ) )
     => ( ( member_set_a @ T3 @ ( sigma_sets_a @ m ) )
       => ( ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ ( sigma_space_a @ m ) @ S2 ) )
            = ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ ( sigma_space_a @ m ) @ T3 ) ) )
         => ( ( sigma_measure_a2 @ m @ S2 )
            = ( sigma_measure_a2 @ m @ T3 ) ) ) ) ) ).

% measure_eq_compl
thf(fact_1250_indep__setD,axiom,
    ! [A: set_set_a,B: set_set_a,A2: set_a,B4: set_a] :
      ( ( indepe2041756565122539606_set_a @ m @ A @ B )
     => ( ( member_set_a @ A2 @ A )
       => ( ( member_set_a @ B4 @ B )
         => ( ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ A2 @ B4 ) )
            = ( times_times_real @ ( sigma_measure_a2 @ m @ A2 ) @ ( sigma_measure_a2 @ m @ B4 ) ) ) ) ) ) ).

% indep_setD
thf(fact_1251_finite__measure__compl,axiom,
    ! [S: set_a] :
      ( ( member_set_a @ S @ ( sigma_sets_a @ m ) )
     => ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ ( sigma_space_a @ m ) @ S ) )
        = ( minus_minus_real @ ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) ) @ ( sigma_measure_a2 @ m @ S ) ) ) ) ).

% finite_measure_compl
thf(fact_1252_finite__measure__Diff,axiom,
    ! [A: set_a,B: set_a] :
      ( ( member_set_a @ A @ ( sigma_sets_a @ m ) )
     => ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
       => ( ( ord_less_eq_set_a @ B @ A )
         => ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ A @ B ) )
            = ( minus_minus_real @ ( sigma_measure_a2 @ m @ A ) @ ( sigma_measure_a2 @ m @ B ) ) ) ) ) ) ).

% finite_measure_Diff
thf(fact_1253_prob__compl,axiom,
    ! [A: set_a] :
      ( ( member_set_a @ A @ ( sigma_sets_a @ m ) )
     => ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ ( sigma_space_a @ m ) @ A ) )
        = ( minus_minus_real @ one_one_real @ ( sigma_measure_a2 @ m @ A ) ) ) ) ).

% prob_compl
thf(fact_1254_measure__exclude,axiom,
    ! [A: set_a,B: set_a] :
      ( ( member_set_a @ A @ ( sigma_sets_a @ m ) )
     => ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
       => ( ( ( sigma_measure_a2 @ m @ A )
            = ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) ) )
         => ( ( ( inf_inf_set_a @ A @ B )
              = bot_bot_set_a )
           => ( ( sigma_measure_a2 @ m @ B )
              = zero_zero_real ) ) ) ) ) ).

% measure_exclude
thf(fact_1255_indep__sets2__eq,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ( indepe2041756565122539606_set_a @ m @ A @ B )
      = ( ( ord_le3724670747650509150_set_a @ A @ ( sigma_sets_a @ m ) )
        & ( ord_le3724670747650509150_set_a @ B @ ( sigma_sets_a @ m ) )
        & ! [X2: set_a] :
            ( ( member_set_a @ X2 @ A )
           => ! [Y3: set_a] :
                ( ( member_set_a @ Y3 @ B )
               => ( ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ X2 @ Y3 ) )
                  = ( times_times_real @ ( sigma_measure_a2 @ m @ X2 ) @ ( sigma_measure_a2 @ m @ Y3 ) ) ) ) ) ) ) ).

% indep_sets2_eq
thf(fact_1256_indep__setI,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ ( sigma_sets_a @ m ) )
     => ( ( ord_le3724670747650509150_set_a @ B @ ( sigma_sets_a @ m ) )
       => ( ! [A6: set_a,B3: set_a] :
              ( ( member_set_a @ A6 @ A )
             => ( ( member_set_a @ B3 @ B )
               => ( ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ A6 @ B3 ) )
                  = ( times_times_real @ ( sigma_measure_a2 @ m @ A6 ) @ ( sigma_measure_a2 @ m @ B3 ) ) ) ) )
         => ( indepe2041756565122539606_set_a @ m @ A @ B ) ) ) ) ).

% indep_setI
thf(fact_1257_D_Okolmogorov__0__1__law,axiom,
    ! [A: nat > set_set_b,X4: set_b] :
      ( ! [I2: nat] : ( sigma_4968961713055010668ebra_b @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ ( A @ I2 ) )
     => ( ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ A @ top_top_set_nat )
       => ( ( member_set_b @ X4 @ ( indepe8773861029005768663_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ A ) )
         => ( ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ X4 )
              = zero_zero_real )
            | ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ X4 )
              = one_one_real ) ) ) ) ) ).

% D.kolmogorov_0_1_law
thf(fact_1258_sigma__algebra__tail__events,axiom,
    ! [A: nat > set_set_a] :
      ( ! [I2: nat] : ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ m ) @ ( A @ I2 ) )
     => ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ m ) @ ( indepe7538416700049374166_a_nat @ m @ A ) ) ) ).

% sigma_algebra_tail_events
thf(fact_1259_finite__measure__eq__sum__singleton,axiom,
    ! [S: set_a] :
      ( ( finite_finite_a @ S )
     => ( ! [X: a] :
            ( ( member_a @ X @ S )
           => ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ ( sigma_sets_a @ m ) ) )
       => ( ( sigma_measure_a2 @ m @ S )
          = ( groups2740460157737275248a_real
            @ ^ [X2: a] : ( sigma_measure_a2 @ m @ ( insert_a @ X2 @ bot_bot_set_a ) )
            @ S ) ) ) ) ).

% finite_measure_eq_sum_singleton
thf(fact_1260_kolmogorov__0__1__law,axiom,
    ! [A: nat > set_set_a,X4: set_a] :
      ( ! [I2: nat] : ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ m ) @ ( A @ I2 ) )
     => ( ( indepe6267730027088848354_a_nat @ m @ A @ top_top_set_nat )
       => ( ( member_set_a @ X4 @ ( indepe7538416700049374166_a_nat @ m @ A ) )
         => ( ( ( sigma_measure_a2 @ m @ X4 )
              = zero_zero_real )
            | ( ( sigma_measure_a2 @ m @ X4 )
              = one_one_real ) ) ) ) ) ).

% kolmogorov_0_1_law
thf(fact_1261_infinite__UNIV__nat,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% infinite_UNIV_nat
thf(fact_1262_D_Ocountable__support,axiom,
    ( counta4098120917673242426able_b
    @ ( collect_b
      @ ^ [X2: b] :
          ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( insert_b @ X2 @ bot_bot_set_b ) )
         != zero_zero_real ) ) ) ).

% D.countable_support
thf(fact_1263_indep__set__def,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ( indepe2041756565122539606_set_a @ m @ A @ B )
      = ( indepe7780107833195774214ts_a_o @ m @ ( produc6113963288868236716_set_a @ A @ B ) @ top_top_set_o ) ) ).

% indep_set_def
thf(fact_1264_D_Oindep__set__def,axiom,
    ! [A: set_set_b,B: set_set_b] :
      ( ( indepe2041756565122539607_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ A @ B )
      = ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ ( produc6184997328914582701_set_b @ A @ B ) @ top_top_set_o ) ) ).

% D.indep_set_def
thf(fact_1265_D_Omeasure__zero__union,axiom,
    ! [S2: set_b,T3: set_b] :
      ( ( member_set_b @ S2 @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
     => ( ( member_set_b @ T3 @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
       => ( ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ T3 )
            = zero_zero_real )
         => ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( sup_sup_set_b @ S2 @ T3 ) )
            = ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ S2 ) ) ) ) ) ).

% D.measure_zero_union
thf(fact_1266_card__UNIV__unit,axiom,
    ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
    = one_one_nat ) ).

% card_UNIV_unit
thf(fact_1267_countable__support,axiom,
    ( counta4098120917673242425able_a
    @ ( collect_a
      @ ^ [X2: a] :
          ( ( sigma_measure_a2 @ m @ ( insert_a @ X2 @ bot_bot_set_a ) )
         != zero_zero_real ) ) ) ).

% countable_support
thf(fact_1268_UNIV__bool,axiom,
    ( top_top_set_o
    = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).

% UNIV_bool
thf(fact_1269_D_Ofinite__measure__Union,axiom,
    ! [A: set_b,B: set_b] :
      ( ( member_set_b @ A @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
     => ( ( member_set_b @ B @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
       => ( ( ( inf_inf_set_b @ A @ B )
            = bot_bot_set_b )
         => ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( sup_sup_set_b @ A @ B ) )
            = ( plus_plus_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A ) @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ B ) ) ) ) ) ) ).

% D.finite_measure_Union
thf(fact_1270_D_Ofinite__measure__Union_H,axiom,
    ! [A: set_b,B: set_b] :
      ( ( member_set_b @ A @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
     => ( ( member_set_b @ B @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
       => ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( sup_sup_set_b @ A @ B ) )
          = ( plus_plus_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A ) @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( minus_minus_set_b @ B @ A ) ) ) ) ) ) ).

% D.finite_measure_Union'
thf(fact_1271_D_Ofinite__measure__subadditive,axiom,
    ! [A: set_b,B: set_b] :
      ( ( member_set_b @ A @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
     => ( ( member_set_b @ B @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
       => ( ord_less_eq_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( sup_sup_set_b @ A @ B ) ) @ ( plus_plus_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A ) @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ B ) ) ) ) ) ).

% D.finite_measure_subadditive
thf(fact_1272_measure__zero__union,axiom,
    ! [S2: set_a,T3: set_a] :
      ( ( member_set_a @ S2 @ ( sigma_sets_a @ m ) )
     => ( ( member_set_a @ T3 @ ( sigma_sets_a @ m ) )
       => ( ( ( sigma_measure_a2 @ m @ T3 )
            = zero_zero_real )
         => ( ( sigma_measure_a2 @ m @ ( sup_sup_set_a @ S2 @ T3 ) )
            = ( sigma_measure_a2 @ m @ S2 ) ) ) ) ) ).

% measure_zero_union
thf(fact_1273_finite__measure__subadditive,axiom,
    ! [A: set_a,B: set_a] :
      ( ( member_set_a @ A @ ( sigma_sets_a @ m ) )
     => ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
       => ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ ( sup_sup_set_a @ A @ B ) ) @ ( plus_plus_real @ ( sigma_measure_a2 @ m @ A ) @ ( sigma_measure_a2 @ m @ B ) ) ) ) ) ).

% finite_measure_subadditive
thf(fact_1274_finite__measure__Union_H,axiom,
    ! [A: set_a,B: set_a] :
      ( ( member_set_a @ A @ ( sigma_sets_a @ m ) )
     => ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
       => ( ( sigma_measure_a2 @ m @ ( sup_sup_set_a @ A @ B ) )
          = ( plus_plus_real @ ( sigma_measure_a2 @ m @ A ) @ ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ B @ A ) ) ) ) ) ) ).

% finite_measure_Union'
thf(fact_1275_finite__measure__Union,axiom,
    ! [A: set_a,B: set_a] :
      ( ( member_set_a @ A @ ( sigma_sets_a @ m ) )
     => ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
       => ( ( ( inf_inf_set_a @ A @ B )
            = bot_bot_set_a )
         => ( ( sigma_measure_a2 @ m @ ( sup_sup_set_a @ A @ B ) )
            = ( plus_plus_real @ ( sigma_measure_a2 @ m @ A ) @ ( sigma_measure_a2 @ m @ B ) ) ) ) ) ) ).

% finite_measure_Union
thf(fact_1276_D_Oobtain__positive__integrable__function,axiom,
    ~ ! [F3: b > real] :
        ( ( member_b_real @ F3 @ ( sigma_5489272243865111003b_real @ ( measure_distr_a_b @ m @ n @ f ) @ borel_5078946678739801102l_real ) )
       => ( ! [X5: b] : ( ord_less_real @ zero_zero_real @ ( F3 @ X5 ) )
         => ( ! [X5: b] : ( ord_less_eq_real @ ( F3 @ X5 ) @ one_one_real )
           => ~ ( bochne7735280777413380569b_real @ ( measure_distr_a_b @ m @ n @ f ) @ F3 ) ) ) ) ).

% D.obtain_positive_integrable_function
thf(fact_1277_obtain__positive__integrable__function,axiom,
    ~ ! [F3: a > real] :
        ( ( member_a_real @ F3 @ ( sigma_9116425665531756122a_real @ m @ borel_5078946678739801102l_real ) )
       => ( ! [X5: a] : ( ord_less_real @ zero_zero_real @ ( F3 @ X5 ) )
         => ( ! [X5: a] : ( ord_less_eq_real @ ( F3 @ X5 ) @ one_one_real )
           => ~ ( bochne2139062162225249880a_real @ m @ F3 ) ) ) ) ).

% obtain_positive_integrable_function

% Conjectures (1)
thf(conj_0,conjecture,
    ( member_a_b @ f
    @ ( pi_a_b @ ( sigma_space_a @ m )
      @ ^ [Uu: a] : ( sigma_space_b @ n ) ) ) ).

%------------------------------------------------------------------------------