%------------------------------------------------------------------------------
% File     : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain   : Archive of Formal Proofs
% Problem  :
% Version  : Especial.
% English  :

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : Quasi_Borel_Spaces/0008_Probability_Space_QuasiBorel/prob_01018_039910__15390830_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1980 ( 518 unt; 699 typ;   0 def)
%            Number of atoms       : 3272 (1073 equ;   0 cnn)
%            Maximal formula atoms :    8 (   2 avg)
%            Number of connectives : 15687 (  86   ~;  17   |;  96   &;14072   @)
%                                         (   0 <=>;1416  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   21 (   8 avg)
%            Number of types       :  143 ( 142 usr)
%            Number of type conns  : 3665 (3665   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  560 ( 557 usr;  35 con; 0-6 aty)
%            Number of variables   : 4730 ( 652   ^;4054   !;  24   ?;4730   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:10:32.455
%------------------------------------------------------------------------------
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thf(ty_n_t__QuasiBorel__Oquasi____borel_It__Extended____Nonnegative____Real__Oennreal_J,type,
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thf(ty_n_t__QuasiBorel__Oquasi____borel_I_062_It__Real__Oreal_Mt__Complex__Ocomplex_J_J,type,
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thf(ty_n_t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J,type,
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thf(ty_n_t__Sigma____Algebra__Omeasure_It__Extended____Nonnegative____Real__Oennreal_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Extended____Nonnegative____Real__Oennreal_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Extended____Nonnegative____Real__Oennreal_Mtf__a_J_J,type,
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thf(ty_n_t__Sigma____Algebra__Omeasure_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
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thf(ty_n_t__Sigma____Algebra__Omeasure_It__QuasiBorel__Oquasi____borel_Itf__a_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
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thf(ty_n_t__QuasiBorel__Oquasi____borel_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_M_062_Itf__a_Mt__Real__Oreal_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_M_062_It__Real__Oreal_Mtf__a_J_J_J,type,
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thf(ty_n_t__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__a_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
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thf(ty_n_t__QuasiBorel__Oquasi____borel_I_062_Itf__a_Mt__Real__Oreal_J_J,type,
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thf(ty_n_t__QuasiBorel__Oquasi____borel_I_062_It__Real__Oreal_Mtf__a_J_J,type,
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thf(ty_n_t__Sigma____Algebra__Omeasure_I_062_It__Real__Oreal_Mtf__a_J_J,type,
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thf(ty_n_t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__a_M_062_It__Real__Oreal_Mtf__a_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Complex__Ocomplex_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Complex__Ocomplex_Mt__Real__Oreal_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
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thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Complex__Ocomplex_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
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thf(ty_n_t__QuasiBorel__Oquasi____borel_It__Complex__Ocomplex_J,type,
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thf(ty_n_t__Sigma____Algebra__Omeasure_It__Complex__Ocomplex_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Complex__Ocomplex_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Complex__Ocomplex_Mtf__a_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mtf__a_J,type,
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thf(ty_n_t__QuasiBorel__Oquasi____borel_It__Real__Oreal_J,type,
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thf(ty_n_t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
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thf(ty_n_t__Sigma____Algebra__Omeasure_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Real__Oreal_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J,type,
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thf(ty_n_t__Extended____Nonnegative____Real__Oennreal,type,
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thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
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thf(ty_n_t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
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thf(ty_n_t__Sigma____Algebra__Omeasure_Itf__a_J,type,
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thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
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thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
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thf(ty_n_t__Set__Oset_Itf__a_J,type,
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thf(ty_n_t__Complex__Ocomplex,type,
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thf(ty_n_t__Real__Oreal,type,
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (557)
thf(sy_c_Binary__Product__Measure_Opair__measure_001_062_It__Real__Oreal_Mtf__a_J_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
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thf(sy_c_Binary__Product__Measure_Opair__measure_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
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thf(sy_c_Binary__Product__Measure_Opair__measure_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal,type,
    binary3098606844978005306nnreal: sigma_7234349610311085201nnreal > sigma_7234349610311085201nnreal > sigma_8863766382501558222nnreal ).

thf(sy_c_Binary__Product__Measure_Opair__measure_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal,type,
    binary3818639336118950830l_real: sigma_7234349610311085201nnreal > sigma_measure_real > sigma_4689706013645835074l_real ).

thf(sy_c_Binary__Product__Measure_Opair__measure_001t__Product____Type__Oprod_I_062_It__Real__Oreal_Mtf__a_J_Mt__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
    binary2346186313421752393orel_a: sigma_8775847253591143008e_real > sigma_4063782130865963553orel_a > sigma_6107753212011660581orel_a ).

thf(sy_c_Binary__Product__Measure_Opair__measure_001t__QuasiBorel__Oquasi____borel_Itf__a_J_001t__Product____Type__Oprod_I_062_It__Real__Oreal_Mtf__a_J_Mt__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J,type,
    binary125940435690417031e_real: sigma_4063782130865963553orel_a > sigma_8775847253591143008e_real > sigma_1472180638263711203e_real ).

thf(sy_c_Binary__Product__Measure_Opair__measure_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal,type,
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thf(sy_c_Binary__Product__Measure_Opair__measure_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Binary__Product__Measure_Opair__measure_001t__Real__Oreal_001tf__a,type,
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thf(sy_c_Binary__Product__Measure_Opair__measure_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J_001_062_It__Real__Oreal_Mtf__a_J,type,
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thf(sy_c_Binary__Product__Measure_Opair__measure_001tf__a_001t__Complex__Ocomplex,type,
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thf(sy_c_Binary__Product__Measure_Opair__measure_001tf__a_001t__Extended____Nonnegative____Real__Oennreal,type,
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thf(sy_c_Binary__Product__Measure_Opair__measure_001tf__a_001t__Real__Oreal,type,
    binary932748531126180194a_real: sigma_measure_a > sigma_measure_real > sigma_2262136186458356274a_real ).

thf(sy_c_Binary__Product__Measure_Opair__measure_001tf__a_001tf__a,type,
    binary867438762418767560re_a_a: sigma_measure_a > sigma_measure_a > sigma_5711748576726957348od_a_a ).

thf(sy_c_Bochner__Integration_Ohas__bochner__integral_001t__Real__Oreal_001t__Real__Oreal,type,
    bochne663871741685100524l_real: sigma_measure_real > ( real > real ) > real > $o ).

thf(sy_c_Bochner__Integration_Ointegrable_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    bochne7086349386406454702x_real: sigma_3077487657436305159omplex > ( complex > real ) > $o ).

thf(sy_c_Bochner__Integration_Ointegrable_001t__Extended____Nonnegative____Real__Oennreal_001t__Complex__Ocomplex,type,
    bochne6309023331997297978omplex: sigma_7234349610311085201nnreal > ( extend8495563244428889912nnreal > complex ) > $o ).

thf(sy_c_Bochner__Integration_Ointegrable_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal,type,
    bochne9025062821074728248l_real: sigma_7234349610311085201nnreal > ( extend8495563244428889912nnreal > real ) > $o ).

thf(sy_c_Bochner__Integration_Ointegrable_001t__Real__Oreal_001t__Complex__Ocomplex,type,
    bochne7032760885902134062omplex: sigma_measure_real > ( real > complex ) > $o ).

thf(sy_c_Bochner__Integration_Ointegrable_001t__Real__Oreal_001t__Real__Oreal,type,
    bochne3340023020068487468l_real: sigma_measure_real > ( real > real ) > $o ).

thf(sy_c_Bochner__Integration_Ointegrable_001tf__a_001t__Complex__Ocomplex,type,
    bochne1348834467089073754omplex: sigma_measure_a > ( a > complex ) > $o ).

thf(sy_c_Bochner__Integration_Ointegrable_001tf__a_001t__Real__Oreal,type,
    bochne2139062162225249880a_real: sigma_measure_a > ( a > real ) > $o ).

thf(sy_c_Bochner__Integration_Olebesgue__integral_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    bochne8919328671811780063x_real: sigma_3077487657436305159omplex > ( complex > real ) > real ).

thf(sy_c_Bochner__Integration_Olebesgue__integral_001t__Extended____Nonnegative____Real__Oennreal_001t__Complex__Ocomplex,type,
    bochne7234769336060495339omplex: sigma_7234349610311085201nnreal > ( extend8495563244428889912nnreal > complex ) > complex ).

thf(sy_c_Bochner__Integration_Olebesgue__integral_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal,type,
    bochne2458729288719820649l_real: sigma_7234349610311085201nnreal > ( extend8495563244428889912nnreal > real ) > real ).

thf(sy_c_Bochner__Integration_Olebesgue__integral_001t__Real__Oreal_001t__Complex__Ocomplex,type,
    bochne8865740171307459423omplex: sigma_measure_real > ( real > complex ) > complex ).

thf(sy_c_Bochner__Integration_Olebesgue__integral_001t__Real__Oreal_001t__Real__Oreal,type,
    bochne3715101410578510557l_real: sigma_measure_real > ( real > real ) > real ).

thf(sy_c_Bochner__Integration_Olebesgue__integral_001tf__a_001t__Complex__Ocomplex,type,
    bochne4904656926214500329omplex: sigma_measure_a > ( a > complex ) > complex ).

thf(sy_c_Bochner__Integration_Olebesgue__integral_001tf__a_001t__Real__Oreal,type,
    bochne378719280626478695a_real: sigma_measure_a > ( a > real ) > real ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Complex__Ocomplex,type,
    borel_1392132677378845456omplex: sigma_3077487657436305159omplex ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Extended____Nonnegative____Real__Oennreal,type,
    borel_6524799422816628122nnreal: sigma_7234349610311085201nnreal ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Nat__Onat,type,
    borel_8449730974584783410el_nat: sigma_measure_nat ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Product____Type__Oprod_It__Extended____Nonnegative____Real__Oennreal_Mt__Complex__Ocomplex_J,type,
    borel_6932847338593847653omplex: sigma_4266603432115041092omplex ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Product____Type__Oprod_It__Extended____Nonnegative____Real__Oennreal_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
    borel_3951438148318096111nnreal: sigma_8863766382501558222nnreal ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Product____Type__Oprod_It__Extended____Nonnegative____Real__Oennreal_Mt__Real__Oreal_J,type,
    borel_4928740784729289315l_real: sigma_4689706013645835074l_real ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
    borel_4974289084073311971nnreal: sigma_4359661369728391106nnreal ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    borel_9155112475215227991l_real: sigma_2308072346491277622l_real ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Real__Oreal,type,
    borel_5078946678739801102l_real: sigma_measure_real ).

thf(sy_c_Characteristic__Functions_Ochar,type,
    characteristic_char: sigma_measure_real > real > complex ).

thf(sy_c_Convolution_Oconvolution_001t__Real__Oreal,type,
    convolution_real: sigma_measure_real > sigma_measure_real > sigma_measure_real ).

thf(sy_c_Distribution__Functions_Ocdf,type,
    distribution_cdf: sigma_measure_real > real > real ).

thf(sy_c_Distribution__Functions_Ofinite__borel__measure,type,
    distri7943378551711771532easure: sigma_measure_real > $o ).

thf(sy_c_Distribution__Functions_Oreal__distribution,type,
    distri2809703520229113005bution: sigma_measure_real > $o ).

thf(sy_c_Distributions_Oerlang__density,type,
    erlang_density: nat > real > real > real ).

thf(sy_c_Distributions_Onormal__density,type,
    normal_density: real > real > real > real ).

thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity_001t__Extended____Nonnegative____Real__Oennreal,type,
    extend2057119558705770725nnreal: extend8495563244428889912nnreal ).

thf(sy_c_Extended__Nonnegative__Real_Oenn2real,type,
    extend1669699412028896998n2real: extend8495563244428889912nnreal > real ).

thf(sy_c_Extended__Nonnegative__Real_Oennreal,type,
    extend7643940197134561352nnreal: real > extend8495563244428889912nnreal ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
    semiri2265585572941072030t_real: nat > real ).

thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    comp_c2117349707075585901x_real: ( complex > complex ) > ( real > complex ) > real > complex ).

thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Real__Oreal_001t__Real__Oreal,type,
    comp_c3333796836230738283l_real: ( complex > real ) > ( real > complex ) > real > real ).

thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001tf__a_001t__Real__Oreal,type,
    comp_complex_a_real: ( complex > a ) > ( real > complex ) > real > a ).

thf(sy_c_Fun_Ocomp_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal_001t__Real__Oreal,type,
    comp_E3822617923592311797l_real: ( extend8495563244428889912nnreal > real ) > ( real > extend8495563244428889912nnreal ) > real > real ).

thf(sy_c_Fun_Ocomp_001t__Extended____Nonnegative____Real__Oennreal_001tf__a_001t__Real__Oreal,type,
    comp_E4829442781247313743a_real: ( extend8495563244428889912nnreal > a ) > ( real > extend8495563244428889912nnreal ) > real > a ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    comp_r1968866223832618731x_real: ( real > complex ) > ( real > real ) > real > complex ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal,type,
    comp_r6279034453215524981l_real: ( real > extend8495563244428889912nnreal ) > ( real > real ) > real > extend8495563244428889912nnreal ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal_001tf__a,type,
    comp_r7806941060661185781real_a: ( real > extend8495563244428889912nnreal ) > ( a > real ) > a > extend8495563244428889912nnreal ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal,type,
    comp_real_real_real: ( real > real ) > ( real > real ) > real > real ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001tf__a,type,
    comp_real_real_a: ( real > real ) > ( a > real ) > a > real ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001tf__a_001t__Real__Oreal,type,
    comp_real_a_real: ( real > a ) > ( real > real ) > real > a ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001tf__a_001tf__a,type,
    comp_real_a_a: ( real > a ) > ( a > real ) > a > a ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    comp_a_complex_real: ( a > complex ) > ( real > a ) > real > complex ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal,type,
    comp_a8249376463644563905l_real: ( a > extend8495563244428889912nnreal ) > ( real > a ) > real > extend8495563244428889912nnreal ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Extended____Nonnegative____Real__Oennreal_001tf__a,type,
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thf(sy_c_Fun_Ocomp_001tf__a_001t__Real__Oreal_001t__Real__Oreal,type,
    comp_a_real_real: ( a > real ) > ( real > a ) > real > real ).

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

thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001t__Real__Oreal,type,
    comp_a_a_real: ( a > a ) > ( real > a ) > real > a ).

thf(sy_c_Giry__Monad_Osubprob__space_001t__Real__Oreal,type,
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thf(sy_c_Giry__Monad_Osubprob__space_001tf__a,type,
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    indepe4370139003212481807real_a: sigma_measure_a > ( ( real > real ) > sigma_measure_a ) > ( ( real > real ) > a > a ) > set_real_real > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001_062_It__Real__Oreal_Mtf__a_J_001t__Extended____Nonnegative____Real__Oennreal,type,
    indepe4152162118790851851nnreal: sigma_measure_a > ( ( real > a ) > sigma_7234349610311085201nnreal ) > ( ( real > a ) > a > extend8495563244428889912nnreal ) > set_real_a > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001_062_It__Real__Oreal_Mtf__a_J_001t__Real__Oreal,type,
    indepe8900428919897944831a_real: sigma_measure_a > ( ( real > a ) > sigma_measure_real ) > ( ( real > a ) > a > real ) > set_real_a > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001_062_It__Real__Oreal_Mtf__a_J_001tf__a,type,
    indepe4717112320896891883al_a_a: sigma_measure_a > ( ( real > a ) > sigma_measure_a ) > ( ( real > a ) > a > a ) > set_real_a > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001_062_Itf__a_Mt__Extended____Nonnegative____Real__Oennreal_J_001t__Extended____Nonnegative____Real__Oennreal,type,
    indepe8333295984245922857nnreal: sigma_measure_a > ( ( a > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal ) > ( ( a > extend8495563244428889912nnreal ) > a > extend8495563244428889912nnreal ) > set_a_7161065143582548615nnreal > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001_062_Itf__a_Mt__Extended____Nonnegative____Real__Oennreal_J_001t__Real__Oreal,type,
    indepe4651158806865961757l_real: sigma_measure_a > ( ( a > extend8495563244428889912nnreal ) > sigma_measure_real ) > ( ( a > extend8495563244428889912nnreal ) > a > real ) > set_a_7161065143582548615nnreal > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001_062_Itf__a_Mt__Extended____Nonnegative____Real__Oennreal_J_001tf__a,type,
    indepe6798734769030654029real_a: sigma_measure_a > ( ( a > extend8495563244428889912nnreal ) > sigma_measure_a ) > ( ( a > extend8495563244428889912nnreal ) > a > a ) > set_a_7161065143582548615nnreal > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001_062_Itf__a_Mt__Real__Oreal_J_001t__Extended____Nonnegative____Real__Oennreal,type,
    indepe308904231703567133nnreal: sigma_measure_a > ( ( a > real ) > sigma_7234349610311085201nnreal ) > ( ( a > real ) > a > extend8495563244428889912nnreal ) > set_a_real > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001_062_Itf__a_Mt__Real__Oreal_J_001t__Real__Oreal,type,
    indepe3165004708401367057l_real: sigma_measure_a > ( ( a > real ) > sigma_measure_real ) > ( ( a > real ) > a > real ) > set_a_real > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001_062_Itf__a_Mt__Real__Oreal_J_001tf__a,type,
    indepe7925344331933621849real_a: sigma_measure_a > ( ( a > real ) > sigma_measure_a ) > ( ( a > real ) > a > a ) > set_a_real > $o ).

thf(sy_c_Information_OKL__divergence_001t__Product____Type__Oprod_I_062_It__Real__Oreal_Mtf__a_J_Mt__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J,type,
    kL_div4552242606482481901e_real: real > sigma_8775847253591143008e_real > sigma_8775847253591143008e_real > real ).

thf(sy_c_Information_OKL__divergence_001t__Product____Type__Oprod_It__QuasiBorel__Oquasi____borel_Itf__a_J_Mt__Product____Type__Oprod_I_062_It__Real__Oreal_Mtf__a_J_Mt__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J_J,type,
    kL_div3327646984204289008e_real: real > sigma_1472180638263711203e_real > sigma_1472180638263711203e_real > real ).

thf(sy_c_Information_OKL__divergence_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    kL_div4114197932038040771l_real: real > sigma_2308072346491277622l_real > sigma_2308072346491277622l_real > real ).

thf(sy_c_Information_OKL__divergence_001t__Product____Type__Oprod_It__Real__Oreal_Mtf__a_J,type,
    kL_div2056522339270997053real_a: real > sigma_4670575602351775008real_a > sigma_4670575602351775008real_a > real ).

thf(sy_c_Information_OKL__divergence_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
    kL_div1539255837003659855a_real: real > sigma_2262136186458356274a_real > sigma_2262136186458356274a_real > real ).

thf(sy_c_Information_OKL__divergence_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    kL_div3267156980076932017od_a_a: real > sigma_5711748576726957348od_a_a > sigma_5711748576726957348od_a_a > real ).

thf(sy_c_Information_OKL__divergence_001t__Real__Oreal,type,
    kL_divergence_real: real > sigma_measure_real > sigma_measure_real > real ).

thf(sy_c_Information_OKL__divergence_001tf__a,type,
    kL_divergence_a: real > sigma_measure_a > sigma_measure_a > real ).

thf(sy_c_Information_Oprob__space_Oentropy_001t__Real__Oreal_001t__Real__Oreal,type,
    prob_e6953316728393294858l_real: sigma_measure_real > real > sigma_measure_real > ( real > real ) > real ).

thf(sy_c_Information_Oprob__space_Oentropy_001t__Real__Oreal_001tf__a,type,
    prob_entropy_real_a: sigma_measure_real > real > sigma_measure_a > ( real > a ) > real ).

thf(sy_c_Information_Oprob__space_Oentropy_001tf__a_001t__Real__Oreal,type,
    prob_entropy_a_real: sigma_measure_a > real > sigma_measure_real > ( a > real ) > real ).

thf(sy_c_Information_Oprob__space_Omutual__information_001t__Real__Oreal_001_062_It__Real__Oreal_Mtf__a_J_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
    prob_m9196104408708822272e_real: sigma_measure_real > real > sigma_measure_real_a > sigma_8927737637348964610e_real > ( real > real > a ) > ( real > sigma_measure_real ) > real ).

thf(sy_c_Information_Oprob__space_Omutual__information_001t__Real__Oreal_001t__QuasiBorel__Oquasi____borel_Itf__a_J_001t__Product____Type__Oprod_I_062_It__Real__Oreal_Mtf__a_J_Mt__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J,type,
    prob_m4228609518817447427e_real: sigma_measure_real > real > sigma_4063782130865963553orel_a > sigma_8775847253591143008e_real > ( real > quasi_borel_a ) > ( real > produc725540845905733987e_real ) > real ).

thf(sy_c_Information_Oprob__space_Omutual__information_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal,type,
    prob_m4172219917653797150l_real: sigma_measure_real > real > sigma_measure_real > sigma_measure_real > ( real > real ) > ( real > real ) > real ).

thf(sy_c_Information_Oprob__space_Omutual__information_001t__Real__Oreal_001t__Real__Oreal_001tf__a,type,
    prob_m6654432919154233356real_a: sigma_measure_real > real > sigma_measure_real > sigma_measure_a > ( real > real ) > ( real > a ) > real ).

thf(sy_c_Information_Oprob__space_Omutual__information_001t__Real__Oreal_001tf__a_001t__Real__Oreal,type,
    prob_m6024414151681283558a_real: sigma_measure_real > real > sigma_measure_a > sigma_measure_real > ( real > a ) > ( real > real ) > real ).

thf(sy_c_Information_Oprob__space_Omutual__information_001t__Real__Oreal_001tf__a_001tf__a,type,
    prob_m1941895425998922052al_a_a: sigma_measure_real > real > sigma_measure_a > sigma_measure_a > ( real > a ) > ( real > a ) > real ).

thf(sy_c_Information_Oprob__space_Omutual__information_001tf__a_001_062_It__Real__Oreal_Mtf__a_J_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
    prob_m7437826066704243362e_real: sigma_measure_a > real > sigma_measure_real_a > sigma_8927737637348964610e_real > ( a > real > a ) > ( a > sigma_measure_real ) > real ).

thf(sy_c_Information_Oprob__space_Omutual__information_001tf__a_001t__QuasiBorel__Oquasi____borel_Itf__a_J_001t__Product____Type__Oprod_I_062_It__Real__Oreal_Mtf__a_J_Mt__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J,type,
    prob_m4532829576834203813e_real: sigma_measure_a > real > sigma_4063782130865963553orel_a > sigma_8775847253591143008e_real > ( a > quasi_borel_a ) > ( a > produc725540845905733987e_real ) > real ).

thf(sy_c_Information_Oprob__space_Omutual__information_001tf__a_001t__Real__Oreal_001t__Real__Oreal,type,
    prob_m7207053172173760192l_real: sigma_measure_a > real > sigma_measure_real > sigma_measure_real > ( a > real ) > ( a > real ) > real ).

thf(sy_c_Lebesgue__Measure_Olborel_001t__Real__Oreal,type,
    lebesgue_lborel_real: sigma_measure_real ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Omeasure__to__qbs_001t__Complex__Ocomplex,type,
    measur1074055046195851610omplex: sigma_3077487657436305159omplex > quasi_borel_complex ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Omeasure__to__qbs_001t__Extended____Nonnegative____Real__Oennreal,type,
    measur2642298986910087140nnreal: sigma_7234349610311085201nnreal > quasi_9015997321629101608nnreal ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Omeasure__to__qbs_001t__Real__Oreal,type,
    measur6875533127466166616s_real: sigma_measure_real > quasi_borel_real ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Omeasure__to__qbs_001tf__a,type,
    measur6507891955840068946_qbs_a: sigma_measure_a > quasi_borel_a ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Oqbs__to__measure_001t__Complex__Ocomplex,type,
    measur3826415497239753490omplex: quasi_borel_complex > sigma_3077487657436305159omplex ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Oqbs__to__measure_001t__Extended____Nonnegative____Real__Oennreal,type,
    measur7384687747506661788nnreal: quasi_9015997321629101608nnreal > sigma_7234349610311085201nnreal ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Oqbs__to__measure_001t__Real__Oreal,type,
    measur1733462625046462224e_real: quasi_borel_real > sigma_measure_real ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Oqbs__to__measure_001tf__a,type,
    measur7857763439677503898sure_a: quasi_borel_a > sigma_measure_a ).

thf(sy_c_Measure__Space_Odistr_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    measur1675386140983903700x_real: sigma_3077487657436305159omplex > sigma_measure_real > ( complex > real ) > sigma_measure_real ).

thf(sy_c_Measure__Space_Odistr_001t__Complex__Ocomplex_001tf__a,type,
    measur724108212368259542plex_a: sigma_3077487657436305159omplex > sigma_measure_a > ( complex > a ) > sigma_measure_a ).

thf(sy_c_Measure__Space_Odistr_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal,type,
    measur6862244029252366686l_real: sigma_7234349610311085201nnreal > sigma_measure_real > ( extend8495563244428889912nnreal > real ) > sigma_measure_real ).

thf(sy_c_Measure__Space_Odistr_001t__Extended____Nonnegative____Real__Oennreal_001tf__a,type,
    measur7655964997769656268real_a: sigma_7234349610311085201nnreal > sigma_measure_a > ( extend8495563244428889912nnreal > a ) > sigma_measure_a ).

thf(sy_c_Measure__Space_Odistr_001t__Real__Oreal_001_062_It__Real__Oreal_Mtf__a_J,type,
    measur7864027549924149603real_a: sigma_measure_real > sigma_measure_real_a > ( real > real > a ) > sigma_measure_real_a ).

thf(sy_c_Measure__Space_Odistr_001t__Real__Oreal_001t__Complex__Ocomplex,type,
    measur1621797640479583060omplex: sigma_measure_real > sigma_3077487657436305159omplex > ( real > complex ) > sigma_3077487657436305159omplex ).

thf(sy_c_Measure__Space_Odistr_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal,type,
    measur8829990298702910942nnreal: sigma_measure_real > sigma_7234349610311085201nnreal > ( real > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal ).

thf(sy_c_Measure__Space_Odistr_001t__Real__Oreal_001t__Product____Type__Oprod_I_062_It__Real__Oreal_Mtf__a_J_Mt__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J,type,
    measur8637847926015211837e_real: sigma_measure_real > sigma_8775847253591143008e_real > ( real > produc725540845905733987e_real ) > sigma_8775847253591143008e_real ).

thf(sy_c_Measure__Space_Odistr_001t__Real__Oreal_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
    measur4452220837507949463omplex: sigma_measure_real > sigma_1667918933661321146omplex > ( real > produc4411394909380815293omplex ) > sigma_1667918933661321146omplex ).

thf(sy_c_Measure__Space_Odistr_001t__Real__Oreal_001t__Product____Type__Oprod_It__Extended____Nonnegative____Real__Oennreal_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
    measur4012415197360569771nnreal: sigma_measure_real > sigma_8863766382501558222nnreal > ( real > produc7414223468410354641nnreal ) > sigma_8863766382501558222nnreal ).

thf(sy_c_Measure__Space_Odistr_001t__Real__Oreal_001t__Product____Type__Oprod_It__QuasiBorel__Oquasi____borel_Itf__a_J_Mt__Product____Type__Oprod_I_062_It__Real__Oreal_Mtf__a_J_Mt__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J_J,type,
    measur2398198314208846400e_real: sigma_measure_real > sigma_1472180638263711203e_real > ( real > produc6543235832880896358e_real ) > sigma_1472180638263711203e_real ).

thf(sy_c_Measure__Space_Odistr_001t__Real__Oreal_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    measur6481026558495277843l_real: sigma_measure_real > sigma_2308072346491277622l_real > ( real > produc2422161461964618553l_real ) > sigma_2308072346491277622l_real ).

thf(sy_c_Measure__Space_Odistr_001t__Real__Oreal_001t__Product____Type__Oprod_It__Real__Oreal_Mtf__a_J,type,
    measur8388293263560174061real_a: sigma_measure_real > sigma_4670575602351775008real_a > ( real > product_prod_real_a ) > sigma_4670575602351775008real_a ).

thf(sy_c_Measure__Space_Odistr_001t__Real__Oreal_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
    measur7871026761292836863a_real: sigma_measure_real > sigma_2262136186458356274a_real > ( real > product_prod_a_real ) > sigma_2262136186458356274a_real ).

thf(sy_c_Measure__Space_Odistr_001t__Real__Oreal_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    measur2513335786126797313od_a_a: sigma_measure_real > sigma_5711748576726957348od_a_a > ( real > product_prod_a_a ) > sigma_5711748576726957348od_a_a ).

thf(sy_c_Measure__Space_Odistr_001t__Real__Oreal_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
    measur7149860273772831102orel_a: sigma_measure_real > sigma_4063782130865963553orel_a > ( real > quasi_borel_a ) > sigma_4063782130865963553orel_a ).

thf(sy_c_Measure__Space_Odistr_001t__Real__Oreal_001t__Real__Oreal,type,
    measur2993149975067245138l_real: sigma_measure_real > sigma_measure_real > ( real > real ) > sigma_measure_real ).

thf(sy_c_Measure__Space_Odistr_001t__Real__Oreal_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
    measur2366643943792126175e_real: sigma_measure_real > sigma_8927737637348964610e_real > ( real > sigma_measure_real ) > sigma_8927737637348964610e_real ).

thf(sy_c_Measure__Space_Odistr_001t__Real__Oreal_001tf__a,type,
    measure_distr_real_a: sigma_measure_real > sigma_measure_a > ( real > a ) > sigma_measure_a ).

thf(sy_c_Measure__Space_Odistr_001tf__a_001_062_It__Real__Oreal_Mtf__a_J,type,
    measur7323644686031903747real_a: sigma_measure_a > sigma_measure_real_a > ( a > real > a ) > sigma_measure_real_a ).

thf(sy_c_Measure__Space_Odistr_001tf__a_001t__Extended____Nonnegative____Real__Oennreal,type,
    measur4839436603801885502nnreal: sigma_measure_a > sigma_7234349610311085201nnreal > ( a > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal ).

thf(sy_c_Measure__Space_Odistr_001tf__a_001t__Product____Type__Oprod_I_062_It__Real__Oreal_Mtf__a_J_Mt__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J,type,
    measur5549411481742515165e_real: sigma_measure_a > sigma_8775847253591143008e_real > ( a > produc725540845905733987e_real ) > sigma_8775847253591143008e_real ).

thf(sy_c_Measure__Space_Odistr_001tf__a_001t__Product____Type__Oprod_It__Extended____Nonnegative____Real__Oennreal_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
    measur6341207572317192267nnreal: sigma_measure_a > sigma_8863766382501558222nnreal > ( a > produc7414223468410354641nnreal ) > sigma_8863766382501558222nnreal ).

thf(sy_c_Measure__Space_Odistr_001tf__a_001t__Product____Type__Oprod_It__QuasiBorel__Oquasi____borel_Itf__a_J_Mt__Product____Type__Oprod_I_062_It__Real__Oreal_Mtf__a_J_Mt__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J_J,type,
    measur2719181256529368288e_real: sigma_measure_a > sigma_1472180638263711203e_real > ( a > produc6543235832880896358e_real ) > sigma_1472180638263711203e_real ).

thf(sy_c_Measure__Space_Odistr_001tf__a_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    measur8266400719524636083l_real: sigma_measure_a > sigma_2308072346491277622l_real > ( a > produc2422161461964618553l_real ) > sigma_2308072346491277622l_real ).

thf(sy_c_Measure__Space_Odistr_001tf__a_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
    measur5725630100919690270orel_a: sigma_measure_a > sigma_4063782130865963553orel_a > ( a > quasi_borel_a ) > sigma_4063782130865963553orel_a ).

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_It__Real__Oreal_J,type,
    measur6656879888321211263e_real: sigma_measure_a > sigma_8927737637348964610e_real > ( a > sigma_measure_real ) > sigma_8927737637348964610e_real ).

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_Osigma__finite__measure_001t__Real__Oreal,type,
    measur487378040549452491e_real: sigma_measure_real > $o ).

thf(sy_c_Nat_OSuc,type,
    suc: nat > nat ).

thf(sy_c_Nonnegative__Lebesgue__Integration_Onn__integral_001_062_It__Real__Oreal_Mtf__a_J,type,
    nonneg43860225155639326real_a: sigma_measure_real_a > ( ( real > a ) > extend8495563244428889912nnreal ) > extend8495563244428889912nnreal ).

thf(sy_c_Nonnegative__Lebesgue__Integration_Onn__integral_001t__Complex__Ocomplex,type,
    nonneg6050707109158959065omplex: sigma_3077487657436305159omplex > ( complex > extend8495563244428889912nnreal ) > extend8495563244428889912nnreal ).

thf(sy_c_Nonnegative__Lebesgue__Integration_Onn__integral_001t__Extended____Nonnegative____Real__Oennreal,type,
    nonneg5898919932907209827nnreal: sigma_7234349610311085201nnreal > ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > extend8495563244428889912nnreal ).

thf(sy_c_Nonnegative__Lebesgue__Integration_Onn__integral_001t__Product____Type__Oprod_I_062_It__Real__Oreal_Mtf__a_J_Mt__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J,type,
    nonneg1471867029375019384e_real: sigma_8775847253591143008e_real > ( produc725540845905733987e_real > extend8495563244428889912nnreal ) > extend8495563244428889912nnreal ).

thf(sy_c_Nonnegative__Lebesgue__Integration_Onn__integral_001t__Product____Type__Oprod_It__QuasiBorel__Oquasi____borel_Itf__a_J_Mt__Product____Type__Oprod_I_062_It__Real__Oreal_Mtf__a_J_Mt__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J_J,type,
    nonneg8793934387659790843e_real: sigma_1472180638263711203e_real > ( produc6543235832880896358e_real > extend8495563244428889912nnreal ) > extend8495563244428889912nnreal ).

thf(sy_c_Nonnegative__Lebesgue__Integration_Onn__integral_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    nonneg1896927508495185742l_real: sigma_2308072346491277622l_real > ( produc2422161461964618553l_real > extend8495563244428889912nnreal ) > extend8495563244428889912nnreal ).

thf(sy_c_Nonnegative__Lebesgue__Integration_Onn__integral_001t__Product____Type__Oprod_It__Real__Oreal_Mtf__a_J,type,
    nonneg4568142736171598066real_a: sigma_4670575602351775008real_a > ( product_prod_real_a > extend8495563244428889912nnreal ) > extend8495563244428889912nnreal ).

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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Real__Oreal,type,
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    member_a: a > set_a > $o ).

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    alpha: real > a ).

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    mu: sigma_measure_real ).

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    f: a > real ).

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% Relevant facts (1277)
thf(fact_0_H_I2_J,axiom,
    probab7089802345832933103able_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) @ fa ).

% H(2)
thf(fact_1__092_060open_062integrable_A_092_060mu_062_A_If_A_092_060circ_062_A_092_060alpha_062_J_A_092_060Longrightarrow_062_Aqp_Oexpectation_A_If_A_092_060circ_062_A_092_060alpha_062_J_A_061_Aenn2real_A_I_092_060integral_062_092_060_094sup_062_L_Ax_O_Aennreal_A_I_If_A_092_060circ_062_A_092_060alpha_062_J_Ax_J_A_092_060partial_062_092_060mu_062_J_A_N_Aenn2real_A_I_092_060integral_062_092_060_094sup_062_L_Ax_O_Aennreal_A_I_N_A_If_A_092_060circ_062_A_092_060alpha_062_J_Ax_J_A_092_060partial_062_092_060mu_062_J_092_060close_062,axiom,
    ( ( bochne3340023020068487468l_real @ mu @ ( comp_a_real_real @ fa @ alpha ) )
   => ( ( bochne3715101410578510557l_real @ mu @ ( comp_a_real_real @ fa @ alpha ) )
      = ( minus_minus_real
        @ ( extend1669699412028896998n2real
          @ ( nonneg2667834350952324695l_real @ mu
            @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( comp_a_real_real @ fa @ alpha @ X ) ) ) )
        @ ( extend1669699412028896998n2real
          @ ( nonneg2667834350952324695l_real @ mu
            @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( uminus_uminus_real @ ( comp_a_real_real @ fa @ alpha @ X ) ) ) ) ) ) ) ) ).

% \<open>integrable \<mu> (f \<circ> \<alpha>) \<Longrightarrow> qp.expectation (f \<circ> \<alpha>) = enn2real (\<integral>\<^sup>+ x. ennreal ((f \<circ> \<alpha>) x) \<partial>\<mu>) - enn2real (\<integral>\<^sup>+ x. ennreal (- (f \<circ> \<alpha>) x) \<partial>\<mu>)\<close>
thf(fact_2_qp_Oin__Mx__axioms,axiom,
    probab9007417770424356215n_Mx_a @ x @ alpha ).

% qp.in_Mx_axioms
thf(fact_3_qp_Oqbs__prob__eq__refl,axiom,
    probab7355678800483015056b_eq_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ).

% qp.qbs_prob_eq_refl
thf(fact_4_qp_Oqbs__prob__eq2__refl,axiom,
    probab3918592701117320376_eq2_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ).

% qp.qbs_prob_eq2_refl
thf(fact_5_qp_Oqbs__prob__eq4__refl,axiom,
    probab7567053574026744118_eq4_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ).

% qp.qbs_prob_eq4_refl
thf(fact_6_qp_Oqbs__prob__eq3__refl,axiom,
    probab1131137119144644343_eq3_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ).

% qp.qbs_prob_eq3_refl
thf(fact_7_qp_Ointegrable__const,axiom,
    ! [A: real] :
      ( bochne3340023020068487468l_real @ mu
      @ ^ [X: real] : A ) ).

% qp.integrable_const
thf(fact_8_uminus__Pair,axiom,
    ! [A: real,B: produc2422161461964618553l_real] :
      ( ( uminus2261717358882270539l_real @ ( produc2480879129069238452l_real @ A @ B ) )
      = ( produc2480879129069238452l_real @ ( uminus_uminus_real @ A ) @ ( uminus2141826702334040752l_real @ B ) ) ) ).

% uminus_Pair
thf(fact_9_uminus__Pair,axiom,
    ! [A: produc2422161461964618553l_real,B: real] :
      ( ( uminus9098069779951357405l_real @ ( produc2485514626390451598l_real @ A @ B ) )
      = ( produc2485514626390451598l_real @ ( uminus2141826702334040752l_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).

% uminus_Pair
thf(fact_10_uminus__Pair,axiom,
    ! [A: produc2422161461964618553l_real,B: produc2422161461964618553l_real] :
      ( ( uminus3489145517733909598l_real @ ( produc4667175579286338263l_real @ A @ B ) )
      = ( produc4667175579286338263l_real @ ( uminus2141826702334040752l_real @ A ) @ ( uminus2141826702334040752l_real @ B ) ) ) ).

% uminus_Pair
thf(fact_11_uminus__Pair,axiom,
    ! [A: real,B: real] :
      ( ( uminus2141826702334040752l_real @ ( produc4511245868158468465l_real @ A @ B ) )
      = ( produc4511245868158468465l_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).

% uminus_Pair
thf(fact_12_minus__diff__eq,axiom,
    ! [A: produc2422161461964618553l_real,B: produc2422161461964618553l_real] :
      ( ( uminus2141826702334040752l_real @ ( minus_885040589139849760l_real @ A @ B ) )
      = ( minus_885040589139849760l_real @ B @ A ) ) ).

% minus_diff_eq
thf(fact_13_minus__diff__eq,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
      = ( minus_minus_real @ B @ A ) ) ).

% minus_diff_eq
thf(fact_14_diff__Pair,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( minus_885040589139849760l_real @ ( produc4511245868158468465l_real @ A @ B ) @ ( produc4511245868158468465l_real @ C @ D ) )
      = ( produc4511245868158468465l_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ).

% diff_Pair
thf(fact_15_diff__Pair,axiom,
    ! [A: real,B: extend8495563244428889912nnreal,C: real,D: extend8495563244428889912nnreal] :
      ( ( minus_7390125332462997804nnreal @ ( produc4778015194254607485nnreal @ A @ B ) @ ( produc4778015194254607485nnreal @ C @ D ) )
      = ( produc4778015194254607485nnreal @ ( minus_minus_real @ A @ C ) @ ( minus_8429688780609304081nnreal @ B @ D ) ) ) ).

% diff_Pair
thf(fact_16_diff__Pair,axiom,
    ! [A: real,B: nat,C: real,D: nat] :
      ( ( minus_1582581163013509572al_nat @ ( produc3181502643871035669al_nat @ A @ B ) @ ( produc3181502643871035669al_nat @ C @ D ) )
      = ( produc3181502643871035669al_nat @ ( minus_minus_real @ A @ C ) @ ( minus_minus_nat @ B @ D ) ) ) ).

% diff_Pair
thf(fact_17_diff__Pair,axiom,
    ! [A: extend8495563244428889912nnreal,B: real,C: extend8495563244428889912nnreal,D: real] :
      ( ( minus_7344577033118975148l_real @ ( produc2810268924804063229l_real @ A @ B ) @ ( produc2810268924804063229l_real @ C @ D ) )
      = ( produc2810268924804063229l_real @ ( minus_8429688780609304081nnreal @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ).

% diff_Pair
thf(fact_18_diff__Pair,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D: extend8495563244428889912nnreal] :
      ( ( minus_2620848512045058488nnreal @ ( produc344325839068023049nnreal @ A @ B ) @ ( produc344325839068023049nnreal @ C @ D ) )
      = ( produc344325839068023049nnreal @ ( minus_8429688780609304081nnreal @ A @ C ) @ ( minus_8429688780609304081nnreal @ B @ D ) ) ) ).

% diff_Pair
thf(fact_19_diff__Pair,axiom,
    ! [A: extend8495563244428889912nnreal,B: nat,C: extend8495563244428889912nnreal,D: nat] :
      ( ( minus_8921941125199129168al_nat @ ( produc625717604924970401al_nat @ A @ B ) @ ( produc625717604924970401al_nat @ C @ D ) )
      = ( produc625717604924970401al_nat @ ( minus_8429688780609304081nnreal @ A @ C ) @ ( minus_minus_nat @ B @ D ) ) ) ).

% diff_Pair
thf(fact_20_diff__Pair,axiom,
    ! [A: nat,B: real,C: nat,D: real] :
      ( ( minus_5557628854490389828t_real @ ( produc7837566107596912789t_real @ A @ B ) @ ( produc7837566107596912789t_real @ C @ D ) )
      = ( produc7837566107596912789t_real @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ).

% diff_Pair
thf(fact_21_diff__Pair,axiom,
    ! [A: nat,B: extend8495563244428889912nnreal,C: nat,D: extend8495563244428889912nnreal] :
      ( ( minus_1545026942176751184nnreal @ ( produc5075389201112886689nnreal @ A @ B ) @ ( produc5075389201112886689nnreal @ C @ D ) )
      = ( produc5075389201112886689nnreal @ ( minus_minus_nat @ A @ C ) @ ( minus_8429688780609304081nnreal @ B @ D ) ) ) ).

% diff_Pair
thf(fact_22_diff__Pair,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( minus_4365393887724441320at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
      = ( product_Pair_nat_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ D ) ) ) ).

% diff_Pair
thf(fact_23_qp_Oqbs__prob__axioms,axiom,
    probab701741629625904796prob_a @ x @ alpha @ mu ).

% qp.qbs_prob_axioms
thf(fact_24_qp_Ofinite__borel__measure__axioms,axiom,
    distri7943378551711771532easure @ mu ).

% qp.finite_borel_measure_axioms
thf(fact_25_verit__minus__simplify_I4_J,axiom,
    ! [B: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
      = B ) ).

% verit_minus_simplify(4)
thf(fact_26_verit__minus__simplify_I4_J,axiom,
    ! [B: produc2422161461964618553l_real] :
      ( ( uminus2141826702334040752l_real @ ( uminus2141826702334040752l_real @ B ) )
      = B ) ).

% verit_minus_simplify(4)
thf(fact_27_add_Oinverse__inverse,axiom,
    ! [A: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_28_add_Oinverse__inverse,axiom,
    ! [A: produc2422161461964618553l_real] :
      ( ( uminus2141826702334040752l_real @ ( uminus2141826702334040752l_real @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_29_neg__equal__iff__equal,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = ( uminus_uminus_real @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_30_neg__equal__iff__equal,axiom,
    ! [A: produc2422161461964618553l_real,B: produc2422161461964618553l_real] :
      ( ( ( uminus2141826702334040752l_real @ A )
        = ( uminus2141826702334040752l_real @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_31_qbs__prob_Oaxioms_I1_J,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu )
     => ( probab9007417770424356215n_Mx_a @ X2 @ Alpha ) ) ).

% qbs_prob.axioms(1)
thf(fact_32_qbs__prob__eq__equiv12,axiom,
    probab7355678800483015056b_eq_a = probab3918592701117320376_eq2_a ).

% qbs_prob_eq_equiv12
thf(fact_33_qbs__prob__eq__equiv14,axiom,
    probab7355678800483015056b_eq_a = probab7567053574026744118_eq4_a ).

% qbs_prob_eq_equiv14
thf(fact_34_qbs__prob__eq__equiv23,axiom,
    probab3918592701117320376_eq2_a = probab1131137119144644343_eq3_a ).

% qbs_prob_eq_equiv23
thf(fact_35_qbs__prob__eq__equiv24,axiom,
    probab3918592701117320376_eq2_a = probab7567053574026744118_eq4_a ).

% qbs_prob_eq_equiv24
thf(fact_36_qbs__prob__eq__equiv31,axiom,
    probab7355678800483015056b_eq_a = probab1131137119144644343_eq3_a ).

% qbs_prob_eq_equiv31
thf(fact_37_qbs__prob__eq__equiv34,axiom,
    probab1131137119144644343_eq3_a = probab7567053574026744118_eq4_a ).

% qbs_prob_eq_equiv34
thf(fact_38_qbs__prob__eq__1__implies__2,axiom,
    ! [P1: produc6543235832880896358e_real,P2: produc6543235832880896358e_real] :
      ( ( probab7355678800483015056b_eq_a @ P1 @ P2 )
     => ( probab3918592701117320376_eq2_a @ P1 @ P2 ) ) ).

% qbs_prob_eq_1_implies_2
thf(fact_39_qbs__prob__eq__1__implies__4,axiom,
    ! [P1: produc6543235832880896358e_real,P2: produc6543235832880896358e_real] :
      ( ( probab7355678800483015056b_eq_a @ P1 @ P2 )
     => ( probab7567053574026744118_eq4_a @ P1 @ P2 ) ) ).

% qbs_prob_eq_1_implies_4
thf(fact_40_qbs__prob__eq__2__implies__3,axiom,
    ! [P1: produc6543235832880896358e_real,P2: produc6543235832880896358e_real] :
      ( ( probab3918592701117320376_eq2_a @ P1 @ P2 )
     => ( probab1131137119144644343_eq3_a @ P1 @ P2 ) ) ).

% qbs_prob_eq_2_implies_3
thf(fact_41_qbs__prob__eq__3__implies__1,axiom,
    ! [P1: produc6543235832880896358e_real,P2: produc6543235832880896358e_real] :
      ( ( probab1131137119144644343_eq3_a @ P1 @ P2 )
     => ( probab7355678800483015056b_eq_a @ P1 @ P2 ) ) ).

% qbs_prob_eq_3_implies_1
thf(fact_42_qbs__prob__eq__4__implies__3,axiom,
    ! [P1: produc6543235832880896358e_real,P2: produc6543235832880896358e_real] :
      ( ( probab7567053574026744118_eq4_a @ P1 @ P2 )
     => ( probab1131137119144644343_eq3_a @ P1 @ P2 ) ) ).

% qbs_prob_eq_4_implies_3
thf(fact_43_qbs__prob_Oqbs__prob__eq__refl,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu )
     => ( probab7355678800483015056b_eq_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) ) ).

% qbs_prob.qbs_prob_eq_refl
thf(fact_44_qbs__prob__eq__dest_I1_J,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab7355678800483015056b_eq_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) )
     => ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu ) ) ).

% qbs_prob_eq_dest(1)
thf(fact_45_qbs__prob__eq__dest_I2_J,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab7355678800483015056b_eq_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) )
     => ( probab701741629625904796prob_a @ Y @ Beta @ Nu ) ) ).

% qbs_prob_eq_dest(2)
thf(fact_46_qbs__prob_Oqbs__prob__eq3__refl,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu )
     => ( probab1131137119144644343_eq3_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) ) ).

% qbs_prob.qbs_prob_eq3_refl
thf(fact_47_qbs__prob__eq3__dest_I1_J,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab1131137119144644343_eq3_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) )
     => ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu ) ) ).

% qbs_prob_eq3_dest(1)
thf(fact_48_qbs__prob__eq3__dest_I2_J,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab1131137119144644343_eq3_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) )
     => ( probab701741629625904796prob_a @ Y @ Beta @ Nu ) ) ).

% qbs_prob_eq3_dest(2)
thf(fact_49_qbs__prob_Oqbs__prob__eq4__refl,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu )
     => ( probab7567053574026744118_eq4_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) ) ).

% qbs_prob.qbs_prob_eq4_refl
thf(fact_50_qbs__prob_Oqbs__prob__eq2__refl,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu )
     => ( probab3918592701117320376_eq2_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) ) ).

% qbs_prob.qbs_prob_eq2_refl
thf(fact_51_qbs__prob__eq4__dest_I1_J,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab7567053574026744118_eq4_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) )
     => ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu ) ) ).

% qbs_prob_eq4_dest(1)
thf(fact_52_qbs__prob__eq4__dest_I2_J,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab7567053574026744118_eq4_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) )
     => ( probab701741629625904796prob_a @ Y @ Beta @ Nu ) ) ).

% qbs_prob_eq4_dest(2)
thf(fact_53_qbs__prob__eq2__dest_I1_J,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab3918592701117320376_eq2_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) )
     => ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu ) ) ).

% qbs_prob_eq2_dest(1)
thf(fact_54_qbs__prob__eq2__dest_I2_J,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab3918592701117320376_eq2_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) )
     => ( probab701741629625904796prob_a @ Y @ Beta @ Nu ) ) ).

% qbs_prob_eq2_dest(2)
thf(fact_55_qbs__prob__eq__dest_I3_J,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab7355678800483015056b_eq_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) )
     => ( Y = X2 ) ) ).

% qbs_prob_eq_dest(3)
thf(fact_56_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: real,C: real,B: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
      = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_57_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_58_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_59_minus__equation__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = B )
      = ( ( uminus_uminus_real @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_60_minus__equation__iff,axiom,
    ! [A: produc2422161461964618553l_real,B: produc2422161461964618553l_real] :
      ( ( ( uminus2141826702334040752l_real @ A )
        = B )
      = ( ( uminus2141826702334040752l_real @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_61_equation__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( uminus_uminus_real @ B ) )
      = ( B
        = ( uminus_uminus_real @ A ) ) ) ).

% equation_minus_iff
thf(fact_62_equation__minus__iff,axiom,
    ! [A: produc2422161461964618553l_real,B: produc2422161461964618553l_real] :
      ( ( A
        = ( uminus2141826702334040752l_real @ B ) )
      = ( B
        = ( uminus2141826702334040752l_real @ A ) ) ) ).

% equation_minus_iff
thf(fact_63_mem__Collect__eq,axiom,
    ! [A: real > complex,P: ( real > complex ) > $o] :
      ( ( member_real_complex @ A @ ( collect_real_complex @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_64_mem__Collect__eq,axiom,
    ! [A: real > real,P: ( real > real ) > $o] :
      ( ( member_real_real @ A @ ( collect_real_real @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_65_mem__Collect__eq,axiom,
    ! [A: real > a,P: ( real > a ) > $o] :
      ( ( member_real_a @ A @ ( collect_real_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_66_mem__Collect__eq,axiom,
    ! [A: a > extend8495563244428889912nnreal,P: ( a > extend8495563244428889912nnreal ) > $o] :
      ( ( member298456594901751504nnreal @ A @ ( collec5472405872578835474nnreal @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_67_mem__Collect__eq,axiom,
    ! [A: a > real,P: ( a > real ) > $o] :
      ( ( member_a_real @ A @ ( collect_a_real @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_68_Collect__mem__eq,axiom,
    ! [A2: set_real_complex] :
      ( ( collect_real_complex
        @ ^ [X: real > complex] : ( member_real_complex @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_69_Collect__mem__eq,axiom,
    ! [A2: set_real_real] :
      ( ( collect_real_real
        @ ^ [X: real > real] : ( member_real_real @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_70_Collect__mem__eq,axiom,
    ! [A2: set_real_a] :
      ( ( collect_real_a
        @ ^ [X: real > a] : ( member_real_a @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_71_Collect__mem__eq,axiom,
    ! [A2: set_a_7161065143582548615nnreal] :
      ( ( collec5472405872578835474nnreal
        @ ^ [X: a > extend8495563244428889912nnreal] : ( member298456594901751504nnreal @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_72_Collect__mem__eq,axiom,
    ! [A2: set_a_real] :
      ( ( collect_a_real
        @ ^ [X: a > real] : ( member_a_real @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_73_verit__negate__coefficient_I3_J,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
     => ( ( uminus_uminus_real @ A )
        = ( uminus_uminus_real @ B ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_74_verit__negate__coefficient_I3_J,axiom,
    ! [A: produc2422161461964618553l_real,B: produc2422161461964618553l_real] :
      ( ( A = B )
     => ( ( uminus2141826702334040752l_real @ A )
        = ( uminus2141826702334040752l_real @ B ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_75_qbs__prob__eq3__dest_I3_J,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab1131137119144644343_eq3_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) )
     => ( Y = X2 ) ) ).

% qbs_prob_eq3_dest(3)
thf(fact_76_qbs__prob__eq4__dest_I3_J,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab7567053574026744118_eq4_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) )
     => ( Y = X2 ) ) ).

% qbs_prob_eq4_dest(3)
thf(fact_77_qbs__prob__eq2__dest_I3_J,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab3918592701117320376_eq2_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) )
     => ( Y = X2 ) ) ).

% qbs_prob_eq2_dest(3)
thf(fact_78_minus__diff__commute,axiom,
    ! [B: real,A: real] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
      = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_79_minus__diff__commute,axiom,
    ! [B: produc2422161461964618553l_real,A: produc2422161461964618553l_real] :
      ( ( minus_885040589139849760l_real @ ( uminus2141826702334040752l_real @ B ) @ A )
      = ( minus_885040589139849760l_real @ ( uminus2141826702334040752l_real @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_80_Bochner__Integration_Ointegral__diff,axiom,
    ! [M: sigma_measure_real,F: real > real,G: real > real] :
      ( ( bochne3340023020068487468l_real @ M @ F )
     => ( ( bochne3340023020068487468l_real @ M @ G )
       => ( ( bochne3715101410578510557l_real @ M
            @ ^ [X: real] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) ) )
          = ( minus_minus_real @ ( bochne3715101410578510557l_real @ M @ F ) @ ( bochne3715101410578510557l_real @ M @ G ) ) ) ) ) ).

% Bochner_Integration.integral_diff
thf(fact_81_Bochner__Integration_Ointegral__diff,axiom,
    ! [M: sigma_measure_a,F: a > real,G: a > real] :
      ( ( bochne2139062162225249880a_real @ M @ F )
     => ( ( bochne2139062162225249880a_real @ M @ G )
       => ( ( bochne378719280626478695a_real @ M
            @ ^ [X: a] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) ) )
          = ( minus_minus_real @ ( bochne378719280626478695a_real @ M @ F ) @ ( bochne378719280626478695a_real @ M @ G ) ) ) ) ) ).

% Bochner_Integration.integral_diff
thf(fact_82_real__lebesgue__integral__def,axiom,
    ! [M: sigma_measure_real,F: real > real] :
      ( ( bochne3340023020068487468l_real @ M @ F )
     => ( ( bochne3715101410578510557l_real @ M @ F )
        = ( minus_minus_real
          @ ( extend1669699412028896998n2real
            @ ( nonneg2667834350952324695l_real @ M
              @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( F @ X ) ) ) )
          @ ( extend1669699412028896998n2real
            @ ( nonneg2667834350952324695l_real @ M
              @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( uminus_uminus_real @ ( F @ X ) ) ) ) ) ) ) ) ).

% real_lebesgue_integral_def
thf(fact_83_real__lebesgue__integral__def,axiom,
    ! [M: sigma_measure_a,F: a > real] :
      ( ( bochne2139062162225249880a_real @ M @ F )
     => ( ( bochne378719280626478695a_real @ M @ F )
        = ( minus_minus_real
          @ ( extend1669699412028896998n2real
            @ ( nonneg2725512125972007571gral_a @ M
              @ ^ [X: a] : ( extend7643940197134561352nnreal @ ( F @ X ) ) ) )
          @ ( extend1669699412028896998n2real
            @ ( nonneg2725512125972007571gral_a @ M
              @ ^ [X: a] : ( extend7643940197134561352nnreal @ ( uminus_uminus_real @ ( F @ X ) ) ) ) ) ) ) ) ).

% real_lebesgue_integral_def
thf(fact_84_integral__minus__iff,axiom,
    ! [M: sigma_measure_real,F: real > real] :
      ( ( bochne3340023020068487468l_real @ M
        @ ^ [X: real] : ( uminus_uminus_real @ ( F @ X ) ) )
      = ( bochne3340023020068487468l_real @ M @ F ) ) ).

% integral_minus_iff
thf(fact_85_integral__minus__iff,axiom,
    ! [M: sigma_measure_a,F: a > real] :
      ( ( bochne2139062162225249880a_real @ M
        @ ^ [X: a] : ( uminus_uminus_real @ ( F @ X ) ) )
      = ( bochne2139062162225249880a_real @ M @ F ) ) ).

% integral_minus_iff
thf(fact_86_integrable__minus,axiom,
    ! [M: sigma_measure_real,F: real > real] :
      ( ( bochne3340023020068487468l_real @ M @ F )
     => ( bochne3340023020068487468l_real @ M
        @ ^ [X: real] : ( uminus_uminus_real @ ( F @ X ) ) ) ) ).

% integrable_minus
thf(fact_87_integrable__minus,axiom,
    ! [M: sigma_measure_a,F: a > real] :
      ( ( bochne2139062162225249880a_real @ M @ F )
     => ( bochne2139062162225249880a_real @ M
        @ ^ [X: a] : ( uminus_uminus_real @ ( F @ X ) ) ) ) ).

% integrable_minus
thf(fact_88_integral__minus,axiom,
    ! [M: sigma_measure_real,F: real > real] :
      ( ( bochne3715101410578510557l_real @ M
        @ ^ [X: real] : ( uminus_uminus_real @ ( F @ X ) ) )
      = ( uminus_uminus_real @ ( bochne3715101410578510557l_real @ M @ F ) ) ) ).

% integral_minus
thf(fact_89_integral__minus,axiom,
    ! [M: sigma_measure_a,F: a > real] :
      ( ( bochne378719280626478695a_real @ M
        @ ^ [X: a] : ( uminus_uminus_real @ ( F @ X ) ) )
      = ( uminus_uminus_real @ ( bochne378719280626478695a_real @ M @ F ) ) ) ).

% integral_minus
thf(fact_90_Bochner__Integration_Ointegrable__diff,axiom,
    ! [M: sigma_measure_real,F: real > real,G: real > real] :
      ( ( bochne3340023020068487468l_real @ M @ F )
     => ( ( bochne3340023020068487468l_real @ M @ G )
       => ( bochne3340023020068487468l_real @ M
          @ ^ [X: real] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).

% Bochner_Integration.integrable_diff
thf(fact_91_Bochner__Integration_Ointegrable__diff,axiom,
    ! [M: sigma_measure_a,F: a > real,G: a > real] :
      ( ( bochne2139062162225249880a_real @ M @ F )
     => ( ( bochne2139062162225249880a_real @ M @ G )
       => ( bochne2139062162225249880a_real @ M
          @ ^ [X: a] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).

% Bochner_Integration.integrable_diff
thf(fact_92_K__record__comp,axiom,
    ! [C: real,F: real > a] :
      ( ( comp_a_real_real
        @ ^ [X: a] : C
        @ F )
      = ( ^ [X: real] : C ) ) ).

% K_record_comp
thf(fact_93_compose__const_I2_J,axiom,
    ! [A: real,G: real > a] :
      ( ( comp_a_real_real
        @ ^ [X: a] : A
        @ G )
      = ( ^ [X: real] : A ) ) ).

% compose_const(2)
thf(fact_94_comp__apply,axiom,
    ( comp_a_real_real
    = ( ^ [F2: a > real,G2: real > a,X: real] : ( F2 @ ( G2 @ X ) ) ) ) ).

% comp_apply
thf(fact_95_diff__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ A @ C ) )
     => ( B = C ) ) ).

% diff_left_imp_eq
thf(fact_96_comp__eq__dest__lhs,axiom,
    ! [A: a > real,B: real > a,C: real > real,V: real] :
      ( ( ( comp_a_real_real @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_97_comp__eq__elim,axiom,
    ! [A: a > real,B: real > a,C: a > real,D: real > a] :
      ( ( ( comp_a_real_real @ A @ B )
        = ( comp_a_real_real @ C @ D ) )
     => ! [V2: real] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_98_comp__eq__dest,axiom,
    ! [A: a > real,B: real > a,C: a > real,D: real > a,V: real] :
      ( ( ( comp_a_real_real @ A @ B )
        = ( comp_a_real_real @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_99_comp__assoc,axiom,
    ! [F: a > real,G: real > a,H: real > real] :
      ( ( comp_real_real_real @ ( comp_a_real_real @ F @ G ) @ H )
      = ( comp_a_real_real @ F @ ( comp_real_a_real @ G @ H ) ) ) ).

% comp_assoc
thf(fact_100_comp__assoc,axiom,
    ! [F: real > real,G: a > real,H: real > a] :
      ( ( comp_a_real_real @ ( comp_real_real_a @ F @ G ) @ H )
      = ( comp_real_real_real @ F @ ( comp_a_real_real @ G @ H ) ) ) ).

% comp_assoc
thf(fact_101_comp__assoc,axiom,
    ! [F: a > real,G: a > a,H: real > a] :
      ( ( comp_a_real_real @ ( comp_a_real_a @ F @ G ) @ H )
      = ( comp_a_real_real @ F @ ( comp_a_a_real @ G @ H ) ) ) ).

% comp_assoc
thf(fact_102_comp__def,axiom,
    ( comp_a_real_real
    = ( ^ [F2: a > real,G2: real > a,X: real] : ( F2 @ ( G2 @ X ) ) ) ) ).

% comp_def
thf(fact_103_compose__const_I1_J,axiom,
    ! [F: a > real,A: a] :
      ( ( comp_a_real_real @ F
        @ ^ [X: real] : A )
      = ( ^ [X: real] : ( F @ A ) ) ) ).

% compose_const(1)
thf(fact_104_integral__eq__cases,axiom,
    ! [M: sigma_measure_real,F: real > real,N: sigma_measure_real,G: real > real] :
      ( ( ( bochne3340023020068487468l_real @ M @ F )
        = ( bochne3340023020068487468l_real @ N @ G ) )
     => ( ( ( bochne3340023020068487468l_real @ M @ F )
         => ( ( bochne3340023020068487468l_real @ N @ G )
           => ( ( bochne3715101410578510557l_real @ M @ F )
              = ( bochne3715101410578510557l_real @ N @ G ) ) ) )
       => ( ( bochne3715101410578510557l_real @ M @ F )
          = ( bochne3715101410578510557l_real @ N @ G ) ) ) ) ).

% integral_eq_cases
thf(fact_105_integral__eq__cases,axiom,
    ! [M: sigma_measure_real,F: real > real,N: sigma_measure_a,G: a > real] :
      ( ( ( bochne3340023020068487468l_real @ M @ F )
        = ( bochne2139062162225249880a_real @ N @ G ) )
     => ( ( ( bochne3340023020068487468l_real @ M @ F )
         => ( ( bochne2139062162225249880a_real @ N @ G )
           => ( ( bochne3715101410578510557l_real @ M @ F )
              = ( bochne378719280626478695a_real @ N @ G ) ) ) )
       => ( ( bochne3715101410578510557l_real @ M @ F )
          = ( bochne378719280626478695a_real @ N @ G ) ) ) ) ).

% integral_eq_cases
thf(fact_106_integral__eq__cases,axiom,
    ! [M: sigma_measure_a,F: a > real,N: sigma_measure_real,G: real > real] :
      ( ( ( bochne2139062162225249880a_real @ M @ F )
        = ( bochne3340023020068487468l_real @ N @ G ) )
     => ( ( ( bochne2139062162225249880a_real @ M @ F )
         => ( ( bochne3340023020068487468l_real @ N @ G )
           => ( ( bochne378719280626478695a_real @ M @ F )
              = ( bochne3715101410578510557l_real @ N @ G ) ) ) )
       => ( ( bochne378719280626478695a_real @ M @ F )
          = ( bochne3715101410578510557l_real @ N @ G ) ) ) ) ).

% integral_eq_cases
thf(fact_107_integral__eq__cases,axiom,
    ! [M: sigma_measure_a,F: a > real,N: sigma_measure_a,G: a > real] :
      ( ( ( bochne2139062162225249880a_real @ M @ F )
        = ( bochne2139062162225249880a_real @ N @ G ) )
     => ( ( ( bochne2139062162225249880a_real @ M @ F )
         => ( ( bochne2139062162225249880a_real @ N @ G )
           => ( ( bochne378719280626478695a_real @ M @ F )
              = ( bochne378719280626478695a_real @ N @ G ) ) ) )
       => ( ( bochne378719280626478695a_real @ M @ F )
          = ( bochne378719280626478695a_real @ N @ G ) ) ) ) ).

% integral_eq_cases
thf(fact_108_prod_Oinject,axiom,
    ! [X1: quasi_borel_a,X22: produc725540845905733987e_real,Y1: quasi_borel_a,Y2: produc725540845905733987e_real] :
      ( ( ( produc4145838808978236886e_real @ X1 @ X22 )
        = ( produc4145838808978236886e_real @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_109_prod_Oinject,axiom,
    ! [X1: real > a,X22: sigma_measure_real,Y1: real > a,Y2: sigma_measure_real] :
      ( ( ( produc623176010801490259e_real @ X1 @ X22 )
        = ( produc623176010801490259e_real @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_110_prod_Oinject,axiom,
    ! [X1: real,X22: real,Y1: real,Y2: real] :
      ( ( ( produc4511245868158468465l_real @ X1 @ X22 )
        = ( produc4511245868158468465l_real @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_111_old_Oprod_Oinject,axiom,
    ! [A: quasi_borel_a,B: produc725540845905733987e_real,A3: quasi_borel_a,B2: produc725540845905733987e_real] :
      ( ( ( produc4145838808978236886e_real @ A @ B )
        = ( produc4145838808978236886e_real @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_112_old_Oprod_Oinject,axiom,
    ! [A: real > a,B: sigma_measure_real,A3: real > a,B2: sigma_measure_real] :
      ( ( ( produc623176010801490259e_real @ A @ B )
        = ( produc623176010801490259e_real @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_113_old_Oprod_Oinject,axiom,
    ! [A: real,B: real,A3: real,B2: real] :
      ( ( ( produc4511245868158468465l_real @ A @ B )
        = ( produc4511245868158468465l_real @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_114_qp_Oif__in__Rep_I2_J,axiom,
    ! [X3: quasi_borel_a,Alpha2: real > a,Mu2: sigma_measure_real] :
      ( ( member6844354795726785935e_real @ ( produc4145838808978236886e_real @ X3 @ ( produc623176010801490259e_real @ Alpha2 @ Mu2 ) ) @ ( probab8639044586466322086pace_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) ) )
     => ( probab701741629625904796prob_a @ X3 @ Alpha2 @ Mu2 ) ) ).

% qp.if_in_Rep(2)
thf(fact_115_qp_Oif__in__Rep_I3_J,axiom,
    ! [X3: quasi_borel_a,Alpha2: real > a,Mu2: sigma_measure_real] :
      ( ( member6844354795726785935e_real @ ( produc4145838808978236886e_real @ X3 @ ( produc623176010801490259e_real @ Alpha2 @ Mu2 ) ) @ ( probab8639044586466322086pace_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) ) )
     => ( probab7355678800483015056b_eq_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) @ ( produc4145838808978236886e_real @ X3 @ ( produc623176010801490259e_real @ Alpha2 @ Mu2 ) ) ) ) ).

% qp.if_in_Rep(3)
thf(fact_116_qp_Oin__Mx,axiom,
    member_real_a @ alpha @ ( qbs_Mx_a @ x ) ).

% qp.in_Mx
thf(fact_117_prod__cases3,axiom,
    ! [Y3: produc6543235832880896358e_real] :
      ~ ! [A4: quasi_borel_a,B3: real > a,C2: sigma_measure_real] :
          ( Y3
         != ( produc4145838808978236886e_real @ A4 @ ( produc623176010801490259e_real @ B3 @ C2 ) ) ) ).

% prod_cases3
thf(fact_118_qp_Oin__Rep,axiom,
    member6844354795726785935e_real @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) @ ( probab8639044586466322086pace_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) ) ).

% qp.in_Rep
thf(fact_119_qp_Oif__in__Rep_I1_J,axiom,
    ! [X3: quasi_borel_a,Alpha2: real > a,Mu2: sigma_measure_real] :
      ( ( member6844354795726785935e_real @ ( produc4145838808978236886e_real @ X3 @ ( produc623176010801490259e_real @ Alpha2 @ Mu2 ) ) @ ( probab8639044586466322086pace_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) ) )
     => ( X3 = x ) ) ).

% qp.if_in_Rep(1)
thf(fact_120_Rep__qbs__prob__space__inject,axiom,
    ! [X4: probab4737552673497767871pace_a,Y3: probab4737552673497767871pace_a] :
      ( ( ( probab8639044586466322086pace_a @ X4 )
        = ( probab8639044586466322086pace_a @ Y3 ) )
      = ( X4 = Y3 ) ) ).

% Rep_qbs_prob_space_inject
thf(fact_121_qbs__prob__space_Oabs__induct,axiom,
    ! [P: probab4737552673497767871pace_a > $o,X4: probab4737552673497767871pace_a] :
      ( ! [Y4: produc6543235832880896358e_real] :
          ( ( probab7355678800483015056b_eq_a @ Y4 @ Y4 )
         => ( P @ ( probab8173042092732894328pace_a @ Y4 ) ) )
     => ( P @ X4 ) ) ).

% qbs_prob_space.abs_induct
thf(fact_122_in__Mx_Oin__Mx,axiom,
    ! [X2: quasi_borel_complex,Alpha: real > complex] :
      ( ( probab8203172577112083445omplex @ X2 @ Alpha )
     => ( member_real_complex @ Alpha @ ( qbs_Mx_complex @ X2 ) ) ) ).

% in_Mx.in_Mx
thf(fact_123_in__Mx_Oin__Mx,axiom,
    ! [X2: quasi_borel_real,Alpha: real > real] :
      ( ( probab6852221862700129395x_real @ X2 @ Alpha )
     => ( member_real_real @ Alpha @ ( qbs_Mx_real @ X2 ) ) ) ).

% in_Mx.in_Mx
thf(fact_124_in__Mx_Oin__Mx,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a] :
      ( ( probab9007417770424356215n_Mx_a @ X2 @ Alpha )
     => ( member_real_a @ Alpha @ ( qbs_Mx_a @ X2 ) ) ) ).

% in_Mx.in_Mx
thf(fact_125_in__Mx_Ointro,axiom,
    ! [Alpha: real > complex,X2: quasi_borel_complex] :
      ( ( member_real_complex @ Alpha @ ( qbs_Mx_complex @ X2 ) )
     => ( probab8203172577112083445omplex @ X2 @ Alpha ) ) ).

% in_Mx.intro
thf(fact_126_in__Mx_Ointro,axiom,
    ! [Alpha: real > real,X2: quasi_borel_real] :
      ( ( member_real_real @ Alpha @ ( qbs_Mx_real @ X2 ) )
     => ( probab6852221862700129395x_real @ X2 @ Alpha ) ) ).

% in_Mx.intro
thf(fact_127_in__Mx_Ointro,axiom,
    ! [Alpha: real > a,X2: quasi_borel_a] :
      ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X2 ) )
     => ( probab9007417770424356215n_Mx_a @ X2 @ Alpha ) ) ).

% in_Mx.intro
thf(fact_128_in__Mx__def,axiom,
    ( probab8203172577112083445omplex
    = ( ^ [X5: quasi_borel_complex,Alpha3: real > complex] : ( member_real_complex @ Alpha3 @ ( qbs_Mx_complex @ X5 ) ) ) ) ).

% in_Mx_def
thf(fact_129_in__Mx__def,axiom,
    ( probab6852221862700129395x_real
    = ( ^ [X5: quasi_borel_real,Alpha3: real > real] : ( member_real_real @ Alpha3 @ ( qbs_Mx_real @ X5 ) ) ) ) ).

% in_Mx_def
thf(fact_130_in__Mx__def,axiom,
    ( probab9007417770424356215n_Mx_a
    = ( ^ [X5: quasi_borel_a,Alpha3: real > a] : ( member_real_a @ Alpha3 @ ( qbs_Mx_a @ X5 ) ) ) ) ).

% in_Mx_def
thf(fact_131_qbs__prob__space_Orep__prop,axiom,
    ! [Y3: probab4737552673497767871pace_a] :
    ? [X6: produc6543235832880896358e_real] :
      ( ( probab7355678800483015056b_eq_a @ X6 @ X6 )
      & ( ( probab8639044586466322086pace_a @ Y3 )
        = ( collec2245114308608258001e_real @ ( probab7355678800483015056b_eq_a @ X6 ) ) ) ) ).

% qbs_prob_space.rep_prop
thf(fact_132_qbs__prob_Oin__Rep,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu )
     => ( member6844354795726785935e_real @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( probab8639044586466322086pace_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) ) ) ) ).

% qbs_prob.in_Rep
thf(fact_133_qbs__prob_Oif__in__Rep_I1_J,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,X3: quasi_borel_a,Alpha2: real > a,Mu2: sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu )
     => ( ( member6844354795726785935e_real @ ( produc4145838808978236886e_real @ X3 @ ( produc623176010801490259e_real @ Alpha2 @ Mu2 ) ) @ ( probab8639044586466322086pace_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) ) )
       => ( X3 = X2 ) ) ) ).

% qbs_prob.if_in_Rep(1)
thf(fact_134_qbs__prob_Oif__in__Rep_I2_J,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,X3: quasi_borel_a,Alpha2: real > a,Mu2: sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu )
     => ( ( member6844354795726785935e_real @ ( produc4145838808978236886e_real @ X3 @ ( produc623176010801490259e_real @ Alpha2 @ Mu2 ) ) @ ( probab8639044586466322086pace_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) ) )
       => ( probab701741629625904796prob_a @ X3 @ Alpha2 @ Mu2 ) ) ) ).

% qbs_prob.if_in_Rep(2)
thf(fact_135_qbs__prob_Oif__in__Rep_I3_J,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,X3: quasi_borel_a,Alpha2: real > a,Mu2: sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu )
     => ( ( member6844354795726785935e_real @ ( produc4145838808978236886e_real @ X3 @ ( produc623176010801490259e_real @ Alpha2 @ Mu2 ) ) @ ( probab8639044586466322086pace_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) ) )
       => ( probab7355678800483015056b_eq_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ X3 @ ( produc623176010801490259e_real @ Alpha2 @ Mu2 ) ) ) ) ) ).

% qbs_prob.if_in_Rep(3)
thf(fact_136_Pair__inject,axiom,
    ! [A: quasi_borel_a,B: produc725540845905733987e_real,A3: quasi_borel_a,B2: produc725540845905733987e_real] :
      ( ( ( produc4145838808978236886e_real @ A @ B )
        = ( produc4145838808978236886e_real @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_137_Pair__inject,axiom,
    ! [A: real > a,B: sigma_measure_real,A3: real > a,B2: sigma_measure_real] :
      ( ( ( produc623176010801490259e_real @ A @ B )
        = ( produc623176010801490259e_real @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_138_Pair__inject,axiom,
    ! [A: real,B: real,A3: real,B2: real] :
      ( ( ( produc4511245868158468465l_real @ A @ B )
        = ( produc4511245868158468465l_real @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_139_prod__cases,axiom,
    ! [P: produc6543235832880896358e_real > $o,P3: produc6543235832880896358e_real] :
      ( ! [A4: quasi_borel_a,B3: produc725540845905733987e_real] : ( P @ ( produc4145838808978236886e_real @ A4 @ B3 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_140_prod__cases,axiom,
    ! [P: produc725540845905733987e_real > $o,P3: produc725540845905733987e_real] :
      ( ! [A4: real > a,B3: sigma_measure_real] : ( P @ ( produc623176010801490259e_real @ A4 @ B3 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_141_prod__cases,axiom,
    ! [P: produc2422161461964618553l_real > $o,P3: produc2422161461964618553l_real] :
      ( ! [A4: real,B3: real] : ( P @ ( produc4511245868158468465l_real @ A4 @ B3 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_142_surj__pair,axiom,
    ! [P3: produc6543235832880896358e_real] :
    ? [X6: quasi_borel_a,Y4: produc725540845905733987e_real] :
      ( P3
      = ( produc4145838808978236886e_real @ X6 @ Y4 ) ) ).

% surj_pair
thf(fact_143_surj__pair,axiom,
    ! [P3: produc725540845905733987e_real] :
    ? [X6: real > a,Y4: sigma_measure_real] :
      ( P3
      = ( produc623176010801490259e_real @ X6 @ Y4 ) ) ).

% surj_pair
thf(fact_144_surj__pair,axiom,
    ! [P3: produc2422161461964618553l_real] :
    ? [X6: real,Y4: real] :
      ( P3
      = ( produc4511245868158468465l_real @ X6 @ Y4 ) ) ).

% surj_pair
thf(fact_145_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc6543235832880896358e_real] :
      ~ ! [A4: quasi_borel_a,B3: produc725540845905733987e_real] :
          ( Y3
         != ( produc4145838808978236886e_real @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_146_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc725540845905733987e_real] :
      ~ ! [A4: real > a,B3: sigma_measure_real] :
          ( Y3
         != ( produc623176010801490259e_real @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_147_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc2422161461964618553l_real] :
      ~ ! [A4: real,B3: real] :
          ( Y3
         != ( produc4511245868158468465l_real @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_148_qbs__prob__space__eq,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab7355678800483015056b_eq_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) )
     => ( ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) )
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) ) ) ) ).

% qbs_prob_space_eq
thf(fact_149_qbs__prob__space__induct_H,axiom,
    ! [S: probab4737552673497767871pace_a,P: probab4737552673497767871pace_a > $o] :
      ( ! [X7: quasi_borel_a,Alpha4: real > a,Mu3: sigma_measure_real] :
          ( ( probab701741629625904796prob_a @ X7 @ Alpha4 @ Mu3 )
         => ( ( S
              = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X7 @ ( produc623176010801490259e_real @ Alpha4 @ Mu3 ) ) ) )
           => ( P @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X7 @ ( produc623176010801490259e_real @ Alpha4 @ Mu3 ) ) ) ) ) )
     => ( P @ S ) ) ).

% qbs_prob_space_induct'
thf(fact_150_qbs__prob__space__induct,axiom,
    ! [P: probab4737552673497767871pace_a > $o,S: probab4737552673497767871pace_a] :
      ( ! [X7: quasi_borel_a,Alpha4: real > a,Mu3: sigma_measure_real] :
          ( ( probab701741629625904796prob_a @ X7 @ Alpha4 @ Mu3 )
         => ( P @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X7 @ ( produc623176010801490259e_real @ Alpha4 @ Mu3 ) ) ) ) )
     => ( P @ S ) ) ).

% qbs_prob_space_induct
thf(fact_151_rep__qbs__prob__space,axiom,
    ! [P3: probab4737552673497767871pace_a] :
    ? [X7: quasi_borel_a,Alpha4: real > a,Mu3: sigma_measure_real] :
      ( ( P3
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X7 @ ( produc623176010801490259e_real @ Alpha4 @ Mu3 ) ) ) )
      & ( probab701741629625904796prob_a @ X7 @ Alpha4 @ Mu3 ) ) ).

% rep_qbs_prob_space
thf(fact_152_prod__induct3,axiom,
    ! [P: produc6543235832880896358e_real > $o,X4: produc6543235832880896358e_real] :
      ( ! [A4: quasi_borel_a,B3: real > a,C2: sigma_measure_real] : ( P @ ( produc4145838808978236886e_real @ A4 @ ( produc623176010801490259e_real @ B3 @ C2 ) ) )
     => ( P @ X4 ) ) ).

% prod_induct3
thf(fact_153_qp_Oin__Rep__induct,axiom,
    ! [P: produc6543235832880896358e_real > $o] :
      ( ! [Y5: quasi_borel_a,Beta2: real > a,Nu2: sigma_measure_real] :
          ( ( member6844354795726785935e_real @ ( produc4145838808978236886e_real @ Y5 @ ( produc623176010801490259e_real @ Beta2 @ Nu2 ) ) @ ( probab8639044586466322086pace_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) ) )
         => ( P @ ( produc4145838808978236886e_real @ Y5 @ ( produc623176010801490259e_real @ Beta2 @ Nu2 ) ) ) )
     => ( P @ ( probab221732815614317479pace_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) ) ) ) ).

% qp.in_Rep_induct
thf(fact_154_qp_Oqbs__prob__space__qbs__computation,axiom,
    ( ( probab1293289258141559360_qbs_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) )
    = x ) ).

% qp.qbs_prob_space_qbs_computation
thf(fact_155_qbs__prob_Oin__Rep__induct,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,P: produc6543235832880896358e_real > $o] :
      ( ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu )
     => ( ! [Y5: quasi_borel_a,Beta2: real > a,Nu2: sigma_measure_real] :
            ( ( member6844354795726785935e_real @ ( produc4145838808978236886e_real @ Y5 @ ( produc623176010801490259e_real @ Beta2 @ Nu2 ) ) @ ( probab8639044586466322086pace_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) ) )
           => ( P @ ( produc4145838808978236886e_real @ Y5 @ ( produc623176010801490259e_real @ Beta2 @ Nu2 ) ) ) )
       => ( P @ ( probab221732815614317479pace_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) ) ) ) ) ).

% qbs_prob.in_Rep_induct
thf(fact_156_minus__diff__minus,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
      = ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_157_minus__diff__minus,axiom,
    ! [A: produc2422161461964618553l_real,B: produc2422161461964618553l_real] :
      ( ( minus_885040589139849760l_real @ ( uminus2141826702334040752l_real @ A ) @ ( uminus2141826702334040752l_real @ B ) )
      = ( uminus2141826702334040752l_real @ ( minus_885040589139849760l_real @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_158_pair__qbs__prob_Oqbs__prob__space__eq__inverse_I2_J,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab5677843605262999830prob_a @ X2 @ Alpha @ Mu @ Y @ Beta @ Nu )
     => ( ( ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) )
          = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) ) )
       => ( probab3918592701117320376_eq2_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) ) ) ) ).

% pair_qbs_prob.qbs_prob_space_eq_inverse(2)
thf(fact_159_pair__qbs__prob_Oqbs__prob__space__eq__inverse_I4_J,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab5677843605262999830prob_a @ X2 @ Alpha @ Mu @ Y @ Beta @ Nu )
     => ( ( ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) )
          = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) ) )
       => ( probab7567053574026744118_eq4_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) ) ) ) ).

% pair_qbs_prob.qbs_prob_space_eq_inverse(4)
thf(fact_160_pair__qbs__prob_Oqbs__prob__space__eq__inverse_I3_J,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab5677843605262999830prob_a @ X2 @ Alpha @ Mu @ Y @ Beta @ Nu )
     => ( ( ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) )
          = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) ) )
       => ( probab1131137119144644343_eq3_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) ) ) ) ).

% pair_qbs_prob.qbs_prob_space_eq_inverse(3)
thf(fact_161_rep__qbs__prob__space_H,axiom,
    ! [S: probab4737552673497767871pace_a,X2: quasi_borel_a] :
      ( ( ( probab1293289258141559360_qbs_a @ S )
        = X2 )
     => ? [Alpha4: real > a,Mu3: sigma_measure_real] :
          ( ( S
            = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha4 @ Mu3 ) ) ) )
          & ( probab701741629625904796prob_a @ X2 @ Alpha4 @ Mu3 ) ) ) ).

% rep_qbs_prob_space'
thf(fact_162_qbs__prob_Oqbs__prob__space__qbs__computation,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu )
     => ( ( probab1293289258141559360_qbs_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) )
        = X2 ) ) ).

% qbs_prob.qbs_prob_space_qbs_computation
thf(fact_163_pair__qbs__prob_Osame__spaces,axiom,
    ! [X2: quasi_borel_complex,Alpha: real > complex,Mu: sigma_measure_real,Y: quasi_borel_complex,Beta: real > complex,Nu: sigma_measure_real] :
      ( ( probab2894914386679012502omplex @ X2 @ Alpha @ Mu @ Y @ Beta @ Nu )
     => ( ( Y = X2 )
       => ( member_real_complex @ Beta @ ( qbs_Mx_complex @ X2 ) ) ) ) ).

% pair_qbs_prob.same_spaces
thf(fact_164_pair__qbs__prob_Osame__spaces,axiom,
    ! [X2: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,Y: quasi_borel_real,Beta: real > real,Nu: sigma_measure_real] :
      ( ( probab6500215489368174228b_real @ X2 @ Alpha @ Mu @ Y @ Beta @ Nu )
     => ( ( Y = X2 )
       => ( member_real_real @ Beta @ ( qbs_Mx_real @ X2 ) ) ) ) ).

% pair_qbs_prob.same_spaces
thf(fact_165_pair__qbs__prob_Osame__spaces,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab5677843605262999830prob_a @ X2 @ Alpha @ Mu @ Y @ Beta @ Nu )
     => ( ( Y = X2 )
       => ( member_real_a @ Beta @ ( qbs_Mx_a @ X2 ) ) ) ) ).

% pair_qbs_prob.same_spaces
thf(fact_166_pair__qbs__prob__def,axiom,
    ( probab5677843605262999830prob_a
    = ( ^ [X5: quasi_borel_a,Alpha3: real > a,Mu4: sigma_measure_real,Y6: quasi_borel_a,Beta3: real > a,Nu3: sigma_measure_real] :
          ( ( probab701741629625904796prob_a @ X5 @ Alpha3 @ Mu4 )
          & ( probab701741629625904796prob_a @ Y6 @ Beta3 @ Nu3 ) ) ) ) ).

% pair_qbs_prob_def
thf(fact_167_pair__qbs__prob_Ointro,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu )
     => ( ( probab701741629625904796prob_a @ Y @ Beta @ Nu )
       => ( probab5677843605262999830prob_a @ X2 @ Alpha @ Mu @ Y @ Beta @ Nu ) ) ) ).

% pair_qbs_prob.intro
thf(fact_168_pair__qbs__prob_Oaxioms_I1_J,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab5677843605262999830prob_a @ X2 @ Alpha @ Mu @ Y @ Beta @ Nu )
     => ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu ) ) ).

% pair_qbs_prob.axioms(1)
thf(fact_169_pair__qbs__prob_Oaxioms_I2_J,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab5677843605262999830prob_a @ X2 @ Alpha @ Mu @ Y @ Beta @ Nu )
     => ( probab701741629625904796prob_a @ Y @ Beta @ Nu ) ) ).

% pair_qbs_prob.axioms(2)
thf(fact_170_pair__qbs__prob_Oqbs__prob__space__eq__inverse_I1_J,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab5677843605262999830prob_a @ X2 @ Alpha @ Mu @ Y @ Beta @ Nu )
     => ( ( ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) )
          = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) ) )
       => ( probab7355678800483015056b_eq_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) ) ) ) ).

% pair_qbs_prob.qbs_prob_space_eq_inverse(1)
thf(fact_171_rep__qbs__prob__space__def,axiom,
    ( probab221732815614317479pace_a
    = ( quot_r3987361144946391578e_real @ probab8639044586466322086pace_a ) ) ).

% rep_qbs_prob_space_def
thf(fact_172_qbs__prob__ennintegral_Oabs__eq,axiom,
    ! [X4: produc6543235832880896358e_real] :
      ( ( probab7355678800483015056b_eq_a @ X4 @ X4 )
     => ( ( probab3721531081081959085gral_a @ ( probab8173042092732894328pace_a @ X4 ) )
        = ( probab4322474783390693535gral_a @ X4 ) ) ) ).

% qbs_prob_ennintegral.abs_eq
thf(fact_173_qp_Osubprob__space__axioms,axiom,
    giry_s8208748868292234104e_real @ mu ).

% qp.subprob_space_axioms
thf(fact_174_qbs__prob__integral_Oabs__eq,axiom,
    ! [X4: produc6543235832880896358e_real] :
      ( ( probab7355678800483015056b_eq_a @ X4 @ X4 )
     => ( ( probab2419480525258570000gral_a @ ( probab8173042092732894328pace_a @ X4 ) )
        = ( probab5242164193669365150gral_a @ X4 ) ) ) ).

% qbs_prob_integral.abs_eq
thf(fact_175_pred__equals__eq2,axiom,
    ! [R: set_Pr7780167738718111686e_real,S2: set_Pr7780167738718111686e_real] :
      ( ( ( ^ [X: quasi_borel_a,Y7: produc725540845905733987e_real] : ( member6844354795726785935e_real @ ( produc4145838808978236886e_real @ X @ Y7 ) @ R ) )
        = ( ^ [X: quasi_borel_a,Y7: produc725540845905733987e_real] : ( member6844354795726785935e_real @ ( produc4145838808978236886e_real @ X @ Y7 ) @ S2 ) ) )
      = ( R = S2 ) ) ).

% pred_equals_eq2
thf(fact_176_pred__equals__eq2,axiom,
    ! [R: set_Pr4989138886603757763e_real,S2: set_Pr4989138886603757763e_real] :
      ( ( ( ^ [X: real > a,Y7: sigma_measure_real] : ( member5162606148374117132e_real @ ( produc623176010801490259e_real @ X @ Y7 ) @ R ) )
        = ( ^ [X: real > a,Y7: sigma_measure_real] : ( member5162606148374117132e_real @ ( produc623176010801490259e_real @ X @ Y7 ) @ S2 ) ) )
      = ( R = S2 ) ) ).

% pred_equals_eq2
thf(fact_177_pred__equals__eq2,axiom,
    ! [R: set_Pr6218003697084177305l_real,S2: set_Pr6218003697084177305l_real] :
      ( ( ( ^ [X: real,Y7: real] : ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ X @ Y7 ) @ R ) )
        = ( ^ [X: real,Y7: real] : ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ X @ Y7 ) @ S2 ) ) )
      = ( R = S2 ) ) ).

% pred_equals_eq2
thf(fact_178_qp_Oreal__distribution__axioms,axiom,
    distri2809703520229113005bution @ mu ).

% qp.real_distribution_axioms
thf(fact_179_integrableD_I3_J,axiom,
    ! [M: sigma_measure_real,F: real > real] :
      ( ( bochne3340023020068487468l_real @ M @ F )
     => ( ( nonneg2667834350952324695l_real @ M
          @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( uminus_uminus_real @ ( F @ X ) ) ) )
       != extend2057119558705770725nnreal ) ) ).

% integrableD(3)
thf(fact_180_integrableD_I3_J,axiom,
    ! [M: sigma_measure_a,F: a > real] :
      ( ( bochne2139062162225249880a_real @ M @ F )
     => ( ( nonneg2725512125972007571gral_a @ M
          @ ^ [X: a] : ( extend7643940197134561352nnreal @ ( uminus_uminus_real @ ( F @ X ) ) ) )
       != extend2057119558705770725nnreal ) ) ).

% integrableD(3)
thf(fact_181_qbs__prob__integral__const,axiom,
    ! [S: probab4737552673497767871pace_a,C: real] :
      ( ( probab2419480525258570000gral_a @ S
        @ ^ [X: a] : C )
      = C ) ).

% qbs_prob_integral_const
thf(fact_182_qbs__prob__ennintegral__const,axiom,
    ! [S: probab4737552673497767871pace_a,C: extend8495563244428889912nnreal] :
      ( ( probab3721531081081959085gral_a @ S
        @ ^ [X: a] : C )
      = C ) ).

% qbs_prob_ennintegral_const
thf(fact_183_qbs__prob_Oaxioms_I2_J,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu )
     => ( distri2809703520229113005bution @ Mu ) ) ).

% qbs_prob.axioms(2)
thf(fact_184_qbs__prob_Ointro,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real] :
      ( ( probab9007417770424356215n_Mx_a @ X2 @ Alpha )
     => ( ( distri2809703520229113005bution @ Mu )
       => ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu ) ) ) ).

% qbs_prob.intro
thf(fact_185_qbs__prob__def,axiom,
    ( probab701741629625904796prob_a
    = ( ^ [X5: quasi_borel_a,Alpha3: real > a,Mu4: sigma_measure_real] :
          ( ( probab9007417770424356215n_Mx_a @ X5 @ Alpha3 )
          & ( distri2809703520229113005bution @ Mu4 ) ) ) ) ).

% qbs_prob_def
thf(fact_186_fun_Omap__comp,axiom,
    ! [G: real > real,F: a > real,V: real > a] :
      ( ( comp_real_real_real @ G @ ( comp_a_real_real @ F @ V ) )
      = ( comp_a_real_real @ ( comp_real_real_a @ G @ F ) @ V ) ) ).

% fun.map_comp
thf(fact_187_fun_Omap__comp,axiom,
    ! [G: a > real,F: real > a,V: real > real] :
      ( ( comp_a_real_real @ G @ ( comp_real_a_real @ F @ V ) )
      = ( comp_real_real_real @ ( comp_a_real_real @ G @ F ) @ V ) ) ).

% fun.map_comp
thf(fact_188_fun_Omap__comp,axiom,
    ! [G: a > real,F: a > a,V: real > a] :
      ( ( comp_a_real_real @ G @ ( comp_a_a_real @ F @ V ) )
      = ( comp_a_real_real @ ( comp_a_real_a @ G @ F ) @ V ) ) ).

% fun.map_comp
thf(fact_189_integrableD_I2_J,axiom,
    ! [M: sigma_measure_real,F: real > real] :
      ( ( bochne3340023020068487468l_real @ M @ F )
     => ( ( nonneg2667834350952324695l_real @ M
          @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( F @ X ) ) )
       != extend2057119558705770725nnreal ) ) ).

% integrableD(2)
thf(fact_190_integrableD_I2_J,axiom,
    ! [M: sigma_measure_a,F: a > real] :
      ( ( bochne2139062162225249880a_real @ M @ F )
     => ( ( nonneg2725512125972007571gral_a @ M
          @ ^ [X: a] : ( extend7643940197134561352nnreal @ ( F @ X ) ) )
       != extend2057119558705770725nnreal ) ) ).

% integrableD(2)
thf(fact_191_qbs__prob__integral_Orep__eq,axiom,
    ( probab2419480525258570000gral_a
    = ( ^ [X: probab4737552673497767871pace_a] : ( probab5242164193669365150gral_a @ ( probab221732815614317479pace_a @ X ) ) ) ) ).

% qbs_prob_integral.rep_eq
thf(fact_192_qbs__prob__ennintegral_Orep__eq,axiom,
    ( probab3721531081081959085gral_a
    = ( ^ [X: probab4737552673497767871pace_a] : ( probab4322474783390693535gral_a @ ( probab221732815614317479pace_a @ X ) ) ) ) ).

% qbs_prob_ennintegral.rep_eq
thf(fact_193_real__distribution_Ofinite__borel__measure__M,axiom,
    ! [M: sigma_measure_real] :
      ( ( distri2809703520229113005bution @ M )
     => ( distri7943378551711771532easure @ M ) ) ).

% real_distribution.finite_borel_measure_M
thf(fact_194_qp_Oqbs__prob__integral__def2,axiom,
    ! [F: a > real] :
      ( ( probab2419480525258570000gral_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) @ F )
      = ( bochne378719280626478695a_real @ ( measure_distr_real_a @ mu @ ( measur7857763439677503898sure_a @ x ) @ alpha ) @ F ) ) ).

% qp.qbs_prob_integral_def2
thf(fact_195_cr__qbs__prob__space__def,axiom,
    ( probab4109582360957019945pace_a
    = ( ^ [X: produc6543235832880896358e_real,Y7: probab4737552673497767871pace_a] :
          ( ( probab7355678800483015056b_eq_a @ X @ X )
          & ( ( probab8173042092732894328pace_a @ X )
            = Y7 ) ) ) ) ).

% cr_qbs_prob_space_def
thf(fact_196_qp_Oqbs__prob__ennintegral__def,axiom,
    ! [F: a > extend8495563244428889912nnreal] :
      ( ( member298456594901751504nnreal @ F @ ( qbs_mo1434458643421888574nnreal @ x @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) ) )
     => ( ( probab3721531081081959085gral_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) @ F )
        = ( nonneg2667834350952324695l_real @ mu
          @ ^ [X: real] : ( F @ ( alpha @ X ) ) ) ) ) ).

% qp.qbs_prob_ennintegral_def
thf(fact_197_qp_Oqbs__prob__integral__def,axiom,
    ! [F: a > real] :
      ( ( member_a_real @ F @ ( qbs_morphism_a_real @ x @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) ) )
     => ( ( probab2419480525258570000gral_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) @ F )
        = ( bochne3715101410578510557l_real @ mu
          @ ^ [X: real] : ( F @ ( alpha @ X ) ) ) ) ) ).

% qp.qbs_prob_integral_def
thf(fact_198_qbs__integrable_Oabs__eq,axiom,
    ! [X4: produc6543235832880896358e_real] :
      ( ( probab7355678800483015056b_eq_a @ X4 @ X4 )
     => ( ( probab7312716125271441302able_a @ ( probab8173042092732894328pace_a @ X4 ) )
        = ( probab7089802345832933103able_a @ X4 ) ) ) ).

% qbs_integrable.abs_eq
thf(fact_199_integral__of__real,axiom,
    ! [M: sigma_measure_real,F: real > real] :
      ( ( bochne3340023020068487468l_real @ M @ F )
     => ( ( bochne3715101410578510557l_real @ M
          @ ^ [X: real] : ( real_V1803761363581548252l_real @ ( F @ X ) ) )
        = ( real_V1803761363581548252l_real @ ( bochne3715101410578510557l_real @ M @ F ) ) ) ) ).

% integral_of_real
thf(fact_200_integral__of__real,axiom,
    ! [M: sigma_measure_a,F: a > real] :
      ( ( bochne2139062162225249880a_real @ M @ F )
     => ( ( bochne378719280626478695a_real @ M
          @ ^ [X: a] : ( real_V1803761363581548252l_real @ ( F @ X ) ) )
        = ( real_V1803761363581548252l_real @ ( bochne378719280626478695a_real @ M @ F ) ) ) ) ).

% integral_of_real
thf(fact_201_qp_Oprob__space__axioms,axiom,
    probab535871623910865577e_real @ mu ).

% qp.prob_space_axioms
thf(fact_202_integral__complex__of__real,axiom,
    ! [M: sigma_measure_real,F: real > real] :
      ( ( bochne8865740171307459423omplex @ M
        @ ^ [X: real] : ( real_V4546457046886955230omplex @ ( F @ X ) ) )
      = ( real_V4546457046886955230omplex @ ( bochne3715101410578510557l_real @ M @ F ) ) ) ).

% integral_complex_of_real
thf(fact_203_integral__complex__of__real,axiom,
    ! [M: sigma_measure_a,F: a > real] :
      ( ( bochne4904656926214500329omplex @ M
        @ ^ [X: a] : ( real_V4546457046886955230omplex @ ( F @ X ) ) )
      = ( real_V4546457046886955230omplex @ ( bochne378719280626478695a_real @ M @ F ) ) ) ).

% integral_complex_of_real
thf(fact_204_qp_Oqbs__integrable__iff__integrable__distr,axiom,
    ! [F: a > real] :
      ( ( probab7312716125271441302able_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) @ F )
      = ( bochne2139062162225249880a_real @ ( measure_distr_real_a @ mu @ ( measur7857763439677503898sure_a @ x ) @ alpha ) @ F ) ) ).

% qp.qbs_integrable_iff_integrable_distr
thf(fact_205_integrable__of__real,axiom,
    ! [M: sigma_measure_real,F: real > real] :
      ( ( bochne3340023020068487468l_real @ M @ F )
     => ( bochne3340023020068487468l_real @ M
        @ ^ [X: real] : ( real_V1803761363581548252l_real @ ( F @ X ) ) ) ) ).

% integrable_of_real
thf(fact_206_integrable__of__real,axiom,
    ! [M: sigma_measure_a,F: a > real] :
      ( ( bochne2139062162225249880a_real @ M @ F )
     => ( bochne2139062162225249880a_real @ M
        @ ^ [X: a] : ( real_V1803761363581548252l_real @ ( F @ X ) ) ) ) ).

% integrable_of_real
thf(fact_207_qp_Oqbs__integrable__def,axiom,
    ! [F: a > real] :
      ( ( probab7312716125271441302able_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) @ F )
      = ( ( member_a_real @ F @ ( qbs_morphism_a_real @ x @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) ) )
        & ( bochne3340023020068487468l_real @ mu @ ( comp_a_real_real @ F @ alpha ) ) ) ) ).

% qp.qbs_integrable_def
thf(fact_208_qp_Oqbs__prob__ennintegral__def2,axiom,
    ! [F: a > extend8495563244428889912nnreal] :
      ( ( member298456594901751504nnreal @ F @ ( qbs_mo1434458643421888574nnreal @ x @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) ) )
     => ( ( probab3721531081081959085gral_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) @ F )
        = ( nonneg2725512125972007571gral_a @ ( measure_distr_real_a @ mu @ ( measur7857763439677503898sure_a @ x ) @ alpha ) @ F ) ) ) ).

% qp.qbs_prob_ennintegral_def2
thf(fact_209_qbs__integrable__morphism,axiom,
    ! [S: probab8009751763329705409e_real,X2: quasi_borel_real,F: real > real] :
      ( ( ( probab8185819741702177770s_real @ S )
        = X2 )
     => ( ( probab3847667120374951956e_real @ S @ F )
       => ( member_real_real @ F @ ( qbs_mo5229770564518008146l_real @ X2 @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) ) ) ) ) ).

% qbs_integrable_morphism
thf(fact_210_qbs__integrable__morphism,axiom,
    ! [S: probab4737552673497767871pace_a,X2: quasi_borel_a,F: a > real] :
      ( ( ( probab1293289258141559360_qbs_a @ S )
        = X2 )
     => ( ( probab7312716125271441302able_a @ S @ F )
       => ( member_a_real @ F @ ( qbs_morphism_a_real @ X2 @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) ) ) ) ) ).

% qbs_integrable_morphism
thf(fact_211_qbs__integrable__const,axiom,
    ! [S: probab4737552673497767871pace_a,C: real] :
      ( probab7312716125271441302able_a @ S
      @ ^ [X: a] : C ) ).

% qbs_integrable_const
thf(fact_212_real__distribution_Oaxioms_I1_J,axiom,
    ! [M: sigma_measure_real] :
      ( ( distri2809703520229113005bution @ M )
     => ( probab535871623910865577e_real @ M ) ) ).

% real_distribution.axioms(1)
thf(fact_213_qbs__prob_Oqbs__prob__ennintegral__def2,axiom,
    ! [X2: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,F: real > extend8495563244428889912nnreal] :
      ( ( probab3605210969150000782b_real @ X2 @ Alpha @ Mu )
     => ( ( member2919562650594848410nnreal @ F @ ( qbs_mo1317719164804411614nnreal @ X2 @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) ) )
       => ( ( probab7585390126108274877l_real @ ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X2 @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) ) @ F )
          = ( nonneg2667834350952324695l_real @ ( measur2993149975067245138l_real @ Mu @ ( measur1733462625046462224e_real @ X2 ) @ Alpha ) @ F ) ) ) ) ).

% qbs_prob.qbs_prob_ennintegral_def2
thf(fact_214_qbs__prob_Oqbs__prob__ennintegral__def2,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,F: a > extend8495563244428889912nnreal] :
      ( ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu )
     => ( ( member298456594901751504nnreal @ F @ ( qbs_mo1434458643421888574nnreal @ X2 @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) ) )
       => ( ( probab3721531081081959085gral_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) @ F )
          = ( nonneg2725512125972007571gral_a @ ( measure_distr_real_a @ Mu @ ( measur7857763439677503898sure_a @ X2 ) @ Alpha ) @ F ) ) ) ) ).

% qbs_prob.qbs_prob_ennintegral_def2
thf(fact_215_qbs__prob__eq__dest_I4_J,axiom,
    ! [X2: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,Y: quasi_borel_real,Beta: real > real,Nu: sigma_measure_real] :
      ( ( probab176830992722561178q_real @ ( produc4368523205619139320e_real @ X2 @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) @ ( produc4368523205619139320e_real @ Y @ ( produc1722724976708544245e_real @ Beta @ Nu ) ) )
     => ( ( measur2993149975067245138l_real @ Mu @ ( measur1733462625046462224e_real @ X2 ) @ Alpha )
        = ( measur2993149975067245138l_real @ Nu @ ( measur1733462625046462224e_real @ X2 ) @ Beta ) ) ) ).

% qbs_prob_eq_dest(4)
thf(fact_216_qbs__prob__eq__dest_I4_J,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab7355678800483015056b_eq_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) )
     => ( ( measure_distr_real_a @ Mu @ ( measur7857763439677503898sure_a @ X2 ) @ Alpha )
        = ( measure_distr_real_a @ Nu @ ( measur7857763439677503898sure_a @ X2 ) @ Beta ) ) ) ).

% qbs_prob_eq_dest(4)
thf(fact_217_qbs__prob_Oqbs__integrable__def,axiom,
    ! [X2: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,F: real > real] :
      ( ( probab3605210969150000782b_real @ X2 @ Alpha @ Mu )
     => ( ( probab3847667120374951956e_real @ ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X2 @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) ) @ F )
        = ( ( member_real_real @ F @ ( qbs_mo5229770564518008146l_real @ X2 @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) ) )
          & ( bochne3340023020068487468l_real @ Mu @ ( comp_real_real_real @ F @ Alpha ) ) ) ) ) ).

% qbs_prob.qbs_integrable_def
thf(fact_218_qbs__prob_Oqbs__integrable__def,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,F: a > real] :
      ( ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu )
     => ( ( probab7312716125271441302able_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) @ F )
        = ( ( member_a_real @ F @ ( qbs_morphism_a_real @ X2 @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) ) )
          & ( bochne3340023020068487468l_real @ Mu @ ( comp_a_real_real @ F @ Alpha ) ) ) ) ) ).

% qbs_prob.qbs_integrable_def
thf(fact_219_qbs__integrable__diff,axiom,
    ! [S: probab4737552673497767871pace_a,F: a > real,G: a > real] :
      ( ( probab7312716125271441302able_a @ S @ F )
     => ( ( probab7312716125271441302able_a @ S @ G )
       => ( probab7312716125271441302able_a @ S
          @ ^ [X: a] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).

% qbs_integrable_diff
thf(fact_220_qbs__prob_Oqbs__integrable__iff__integrable__distr,axiom,
    ! [X2: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,F: real > real] :
      ( ( probab3605210969150000782b_real @ X2 @ Alpha @ Mu )
     => ( ( probab3847667120374951956e_real @ ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X2 @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) ) @ F )
        = ( bochne3340023020068487468l_real @ ( measur2993149975067245138l_real @ Mu @ ( measur1733462625046462224e_real @ X2 ) @ Alpha ) @ F ) ) ) ).

% qbs_prob.qbs_integrable_iff_integrable_distr
thf(fact_221_qbs__prob_Oqbs__integrable__iff__integrable__distr,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,F: a > real] :
      ( ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu )
     => ( ( probab7312716125271441302able_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) @ F )
        = ( bochne2139062162225249880a_real @ ( measure_distr_real_a @ Mu @ ( measur7857763439677503898sure_a @ X2 ) @ Alpha ) @ F ) ) ) ).

% qbs_prob.qbs_integrable_iff_integrable_distr
thf(fact_222_pair__qbs__prob_Oqbs__prob__space__eq,axiom,
    ! [X2: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,Y: quasi_borel_real,Beta: real > real,Nu: sigma_measure_real] :
      ( ( probab6500215489368174228b_real @ X2 @ Alpha @ Mu @ Y @ Beta @ Nu )
     => ( ( Y = X2 )
       => ( ( ( measur2993149975067245138l_real @ Mu @ ( measur1733462625046462224e_real @ X2 ) @ Alpha )
            = ( measur2993149975067245138l_real @ Nu @ ( measur1733462625046462224e_real @ X2 ) @ Beta ) )
         => ( ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X2 @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) )
            = ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ Y @ ( produc1722724976708544245e_real @ Beta @ Nu ) ) ) ) ) ) ) ).

% pair_qbs_prob.qbs_prob_space_eq
thf(fact_223_pair__qbs__prob_Oqbs__prob__space__eq,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab5677843605262999830prob_a @ X2 @ Alpha @ Mu @ Y @ Beta @ Nu )
     => ( ( Y = X2 )
       => ( ( ( measure_distr_real_a @ Mu @ ( measur7857763439677503898sure_a @ X2 ) @ Alpha )
            = ( measure_distr_real_a @ Nu @ ( measur7857763439677503898sure_a @ X2 ) @ Beta ) )
         => ( ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) )
            = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) ) ) ) ) ) ).

% pair_qbs_prob.qbs_prob_space_eq
thf(fact_224_pair__qbs__prob_Oqbs__prob__eq__intro,axiom,
    ! [X2: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,Y: quasi_borel_real,Beta: real > real,Nu: sigma_measure_real] :
      ( ( probab6500215489368174228b_real @ X2 @ Alpha @ Mu @ Y @ Beta @ Nu )
     => ( ( X2 = Y )
       => ( ( ( measur2993149975067245138l_real @ Mu @ ( measur1733462625046462224e_real @ X2 ) @ Alpha )
            = ( measur2993149975067245138l_real @ Nu @ ( measur1733462625046462224e_real @ X2 ) @ Beta ) )
         => ( probab176830992722561178q_real @ ( produc4368523205619139320e_real @ X2 @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) @ ( produc4368523205619139320e_real @ Y @ ( produc1722724976708544245e_real @ Beta @ Nu ) ) ) ) ) ) ).

% pair_qbs_prob.qbs_prob_eq_intro
thf(fact_225_pair__qbs__prob_Oqbs__prob__eq__intro,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab5677843605262999830prob_a @ X2 @ Alpha @ Mu @ Y @ Beta @ Nu )
     => ( ( X2 = Y )
       => ( ( ( measure_distr_real_a @ Mu @ ( measur7857763439677503898sure_a @ X2 ) @ Alpha )
            = ( measure_distr_real_a @ Nu @ ( measur7857763439677503898sure_a @ X2 ) @ Beta ) )
         => ( probab7355678800483015056b_eq_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) ) ) ) ) ).

% pair_qbs_prob.qbs_prob_eq_intro
thf(fact_226_qbs__prob_Oqbs__prob__integral__def,axiom,
    ! [X2: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,F: real > real] :
      ( ( probab3605210969150000782b_real @ X2 @ Alpha @ Mu )
     => ( ( member_real_real @ F @ ( qbs_mo5229770564518008146l_real @ X2 @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) ) )
       => ( ( probab4207012259563505946l_real @ ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X2 @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) ) @ F )
          = ( bochne3715101410578510557l_real @ Mu
            @ ^ [X: real] : ( F @ ( Alpha @ X ) ) ) ) ) ) ).

% qbs_prob.qbs_prob_integral_def
thf(fact_227_qbs__prob_Oqbs__prob__integral__def,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,F: a > real] :
      ( ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu )
     => ( ( member_a_real @ F @ ( qbs_morphism_a_real @ X2 @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) ) )
       => ( ( probab2419480525258570000gral_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) @ F )
          = ( bochne3715101410578510557l_real @ Mu
            @ ^ [X: real] : ( F @ ( Alpha @ X ) ) ) ) ) ) ).

% qbs_prob.qbs_prob_integral_def
thf(fact_228_qbs__prob_Oqbs__prob__ennintegral__def,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,F: a > extend8495563244428889912nnreal] :
      ( ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu )
     => ( ( member298456594901751504nnreal @ F @ ( qbs_mo1434458643421888574nnreal @ X2 @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) ) )
       => ( ( probab3721531081081959085gral_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) @ F )
          = ( nonneg2667834350952324695l_real @ Mu
            @ ^ [X: real] : ( F @ ( Alpha @ X ) ) ) ) ) ) ).

% qbs_prob.qbs_prob_ennintegral_def
thf(fact_229_qbs__prob__integral__diff,axiom,
    ! [S: probab4737552673497767871pace_a,F: a > real,G: a > real] :
      ( ( probab7312716125271441302able_a @ S @ F )
     => ( ( probab7312716125271441302able_a @ S @ G )
       => ( ( probab2419480525258570000gral_a @ S
            @ ^ [X: a] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) ) )
          = ( minus_minus_real @ ( probab2419480525258570000gral_a @ S @ F ) @ ( probab2419480525258570000gral_a @ S @ G ) ) ) ) ) ).

% qbs_prob_integral_diff
thf(fact_230_qbs__integrable_Orep__eq,axiom,
    ( probab7312716125271441302able_a
    = ( ^ [X: probab4737552673497767871pace_a] : ( probab7089802345832933103able_a @ ( probab221732815614317479pace_a @ X ) ) ) ) ).

% qbs_integrable.rep_eq
thf(fact_231_qbs__prob_Oqbs__prob__integral__def2,axiom,
    ! [X2: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,F: real > real] :
      ( ( probab3605210969150000782b_real @ X2 @ Alpha @ Mu )
     => ( ( probab4207012259563505946l_real @ ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X2 @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) ) @ F )
        = ( bochne3715101410578510557l_real @ ( measur2993149975067245138l_real @ Mu @ ( measur1733462625046462224e_real @ X2 ) @ Alpha ) @ F ) ) ) ).

% qbs_prob.qbs_prob_integral_def2
thf(fact_232_qbs__prob_Oqbs__prob__integral__def2,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,F: a > real] :
      ( ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu )
     => ( ( probab2419480525258570000gral_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) @ F )
        = ( bochne378719280626478695a_real @ ( measure_distr_real_a @ Mu @ ( measur7857763439677503898sure_a @ X2 ) @ Alpha ) @ F ) ) ) ).

% qbs_prob.qbs_prob_integral_def2
thf(fact_233_qp_Oqbs__prob__ennintegral__not__morphism,axiom,
    ! [F: a > extend8495563244428889912nnreal] :
      ( ~ ( member298456594901751504nnreal @ F @ ( qbs_mo1434458643421888574nnreal @ x @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) ) )
     => ( ( probab3721531081081959085gral_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) @ F )
        = zero_z7100319975126383169nnreal ) ) ).

% qp.qbs_prob_ennintegral_not_morphism
thf(fact_234_qp_Oqbs__integrable__iff__integrable,axiom,
    ! [F: a > real] :
      ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ ( measur7857763439677503898sure_a @ x ) @ borel_5078946678739801102l_real ) )
     => ( ( probab7312716125271441302able_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) @ F )
        = ( bochne3340023020068487468l_real @ mu
          @ ^ [X: real] : ( F @ ( alpha @ X ) ) ) ) ) ).

% qp.qbs_integrable_iff_integrable
thf(fact_235_qp_Oqbs__integrable__measurable,axiom,
    ! [F: a > real] :
      ( ( probab7312716125271441302able_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) @ F )
     => ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ ( measur7857763439677503898sure_a @ x ) @ borel_5078946678739801102l_real ) ) ) ).

% qp.qbs_integrable_measurable
thf(fact_236_of__real__minus,axiom,
    ! [X4: real] :
      ( ( real_V1803761363581548252l_real @ ( uminus_uminus_real @ X4 ) )
      = ( uminus_uminus_real @ ( real_V1803761363581548252l_real @ X4 ) ) ) ).

% of_real_minus
thf(fact_237_minus__of__real__eq__of__real__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ( uminus_uminus_real @ ( real_V1803761363581548252l_real @ X4 ) )
        = ( real_V1803761363581548252l_real @ Y3 ) )
      = ( ( uminus_uminus_real @ X4 )
        = Y3 ) ) ).

% minus_of_real_eq_of_real_iff
thf(fact_238_of__real__eq__minus__of__real__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ( real_V1803761363581548252l_real @ X4 )
        = ( uminus_uminus_real @ ( real_V1803761363581548252l_real @ Y3 ) ) )
      = ( X4
        = ( uminus_uminus_real @ Y3 ) ) ) ).

% of_real_eq_minus_of_real_iff
thf(fact_239_of__real__diff,axiom,
    ! [X4: real,Y3: real] :
      ( ( real_V1803761363581548252l_real @ ( minus_minus_real @ X4 @ Y3 ) )
      = ( minus_minus_real @ ( real_V1803761363581548252l_real @ X4 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ).

% of_real_diff
thf(fact_240_qp_Oqbs__prob__measure__computation,axiom,
    ( ( probab7100426894406488384sure_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) )
    = ( measure_distr_real_a @ mu @ ( measur7857763439677503898sure_a @ x ) @ alpha ) ) ).

% qp.qbs_prob_measure_computation
thf(fact_241_distr__id,axiom,
    ! [N: sigma_measure_real] :
      ( ( measur2993149975067245138l_real @ N @ N
        @ ^ [X: real] : X )
      = N ) ).

% distr_id
thf(fact_242_assms,axiom,
    probab7312716125271441302able_a @ s @ f ).

% assms
thf(fact_243_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_244_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ A )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_245_diff__zero,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_zero
thf(fact_246_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_247_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_248_diff__0__right,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_0_right
thf(fact_249_diff__self,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% diff_self
thf(fact_250_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( ( uminus_uminus_real @ A )
        = A )
      = ( A = zero_zero_real ) ) ).

% neg_equal_zero
thf(fact_251_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A
        = ( uminus_uminus_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% equal_neg_zero
thf(fact_252_neg__equal__0__iff__equal,axiom,
    ! [A: real] :
      ( ( ( uminus_uminus_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% neg_equal_0_iff_equal
thf(fact_253_neg__equal__0__iff__equal,axiom,
    ! [A: produc2422161461964618553l_real] :
      ( ( ( uminus2141826702334040752l_real @ A )
        = zero_z1365759597461889520l_real )
      = ( A = zero_z1365759597461889520l_real ) ) ).

% neg_equal_0_iff_equal
thf(fact_254_neg__0__equal__iff__equal,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( uminus_uminus_real @ A ) )
      = ( zero_zero_real = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_255_neg__0__equal__iff__equal,axiom,
    ! [A: produc2422161461964618553l_real] :
      ( ( zero_z1365759597461889520l_real
        = ( uminus2141826702334040752l_real @ A ) )
      = ( zero_z1365759597461889520l_real = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_256_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_real @ zero_zero_real )
    = zero_zero_real ) ).

% add.inverse_neutral
thf(fact_257_add_Oinverse__neutral,axiom,
    ( ( uminus2141826702334040752l_real @ zero_z1365759597461889520l_real )
    = zero_z1365759597461889520l_real ) ).

% add.inverse_neutral
thf(fact_258_integral__zero,axiom,
    ! [M: sigma_measure_real] :
      ( ( bochne3715101410578510557l_real @ M
        @ ^ [X: real] : zero_zero_real )
      = zero_zero_real ) ).

% integral_zero
thf(fact_259_integral__zero,axiom,
    ! [M: sigma_measure_a] :
      ( ( bochne378719280626478695a_real @ M
        @ ^ [X: a] : zero_zero_real )
      = zero_zero_real ) ).

% integral_zero
thf(fact_260_integrable__zero,axiom,
    ! [M: sigma_measure_real] :
      ( bochne3340023020068487468l_real @ M
      @ ^ [X: real] : zero_zero_real ) ).

% integrable_zero
thf(fact_261_integrable__zero,axiom,
    ! [M: sigma_measure_a] :
      ( bochne2139062162225249880a_real @ M
      @ ^ [X: a] : zero_zero_real ) ).

% integrable_zero
thf(fact_262_qbs__prob__measure__prob__space_Ointegrable__const,axiom,
    ! [S: probab8009751763329705409e_real,A: real] :
      ( bochne3340023020068487468l_real @ ( probab4733579253584633066e_real @ S )
      @ ^ [X: real] : A ) ).

% qbs_prob_measure_prob_space.integrable_const
thf(fact_263_qbs__prob__measure__prob__space_Ointegrable__const,axiom,
    ! [S: probab4737552673497767871pace_a,A: real] :
      ( bochne2139062162225249880a_real @ ( probab7100426894406488384sure_a @ S )
      @ ^ [X: a] : A ) ).

% qbs_prob_measure_prob_space.integrable_const
thf(fact_264_diff__0,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ zero_zero_real @ A )
      = ( uminus_uminus_real @ A ) ) ).

% diff_0
thf(fact_265_diff__0,axiom,
    ! [A: produc2422161461964618553l_real] :
      ( ( minus_885040589139849760l_real @ zero_z1365759597461889520l_real @ A )
      = ( uminus2141826702334040752l_real @ A ) ) ).

% diff_0
thf(fact_266_verit__minus__simplify_I3_J,axiom,
    ! [B: real] :
      ( ( minus_minus_real @ zero_zero_real @ B )
      = ( uminus_uminus_real @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_267_verit__minus__simplify_I3_J,axiom,
    ! [B: produc2422161461964618553l_real] :
      ( ( minus_885040589139849760l_real @ zero_z1365759597461889520l_real @ B )
      = ( uminus2141826702334040752l_real @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_268_qbs__prob__measure__prob__space_Oreal__distribution__distr,axiom,
    ! [X2: real > real,S: probab8009751763329705409e_real] :
      ( ( member_real_real @ X2 @ ( sigma_5267869275261027754l_real @ ( probab4733579253584633066e_real @ S ) @ borel_5078946678739801102l_real ) )
     => ( distri2809703520229113005bution @ ( measur2993149975067245138l_real @ ( probab4733579253584633066e_real @ S ) @ borel_5078946678739801102l_real @ X2 ) ) ) ).

% qbs_prob_measure_prob_space.real_distribution_distr
thf(fact_269_qbs__prob__measure__prob__space_Oreal__distribution__distr,axiom,
    ! [X2: a > real,S: probab4737552673497767871pace_a] :
      ( ( member_a_real @ X2 @ ( sigma_9116425665531756122a_real @ ( probab7100426894406488384sure_a @ S ) @ borel_5078946678739801102l_real ) )
     => ( distri2809703520229113005bution @ ( measure_distr_a_real @ ( probab7100426894406488384sure_a @ S ) @ borel_5078946678739801102l_real @ X2 ) ) ) ).

% qbs_prob_measure_prob_space.real_distribution_distr
thf(fact_270_zero__prod__def,axiom,
    ( zero_z7224313980826913160nnreal
    = ( produc344325839068023049nnreal @ zero_z7100319975126383169nnreal @ zero_z7100319975126383169nnreal ) ) ).

% zero_prod_def
thf(fact_271_zero__prod__def,axiom,
    ( zero_z310983848714116220l_real
    = ( produc2810268924804063229l_real @ zero_z7100319975126383169nnreal @ zero_zero_real ) ) ).

% zero_prod_def
thf(fact_272_zero__prod__def,axiom,
    ( zero_z4496459946615171616al_nat
    = ( produc625717604924970401al_nat @ zero_z7100319975126383169nnreal @ zero_zero_nat ) ) ).

% zero_prod_def
thf(fact_273_zero__prod__def,axiom,
    ( zero_z356532148058138876nnreal
    = ( produc4778015194254607485nnreal @ zero_zero_real @ zero_z7100319975126383169nnreal ) ) ).

% zero_prod_def
thf(fact_274_zero__prod__def,axiom,
    ( zero_z1365759597461889520l_real
    = ( produc4511245868158468465l_real @ zero_zero_real @ zero_zero_real ) ) ).

% zero_prod_def
thf(fact_275_zero__prod__def,axiom,
    ( zero_z5987101913011988884al_nat
    = ( produc3181502643871035669al_nat @ zero_zero_real @ zero_zero_nat ) ) ).

% zero_prod_def
thf(fact_276_zero__prod__def,axiom,
    ( zero_z6342917800447569440nnreal
    = ( produc5075389201112886689nnreal @ zero_zero_nat @ zero_z7100319975126383169nnreal ) ) ).

% zero_prod_def
thf(fact_277_zero__prod__def,axiom,
    ( zero_z738777567634093332t_real
    = ( produc7837566107596912789t_real @ zero_zero_nat @ zero_zero_real ) ) ).

% zero_prod_def
thf(fact_278_zero__prod__def,axiom,
    ( zero_z3979849011205770936at_nat
    = ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ).

% zero_prod_def
thf(fact_279_zero__reorient,axiom,
    ! [X4: extend8495563244428889912nnreal] :
      ( ( zero_z7100319975126383169nnreal = X4 )
      = ( X4 = zero_z7100319975126383169nnreal ) ) ).

% zero_reorient
thf(fact_280_zero__reorient,axiom,
    ! [X4: real] :
      ( ( zero_zero_real = X4 )
      = ( X4 = zero_zero_real ) ) ).

% zero_reorient
thf(fact_281_zero__reorient,axiom,
    ! [X4: nat] :
      ( ( zero_zero_nat = X4 )
      = ( X4 = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_282_qbs__prob__measure__prob__space_Oprob__space__distr,axiom,
    ! [F: real > a,S: probab8009751763329705409e_real,M2: sigma_measure_a] :
      ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ ( probab4733579253584633066e_real @ S ) @ M2 ) )
     => ( probab7247484486040049089pace_a @ ( measure_distr_real_a @ ( probab4733579253584633066e_real @ S ) @ M2 @ F ) ) ) ).

% qbs_prob_measure_prob_space.prob_space_distr
thf(fact_283_qbs__prob__measure__prob__space_Oprob__space__distr,axiom,
    ! [F: real > extend8495563244428889912nnreal,S: probab8009751763329705409e_real,M2: sigma_7234349610311085201nnreal] :
      ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ ( probab4733579253584633066e_real @ S ) @ M2 ) )
     => ( probab6612481188548237749nnreal @ ( measur8829990298702910942nnreal @ ( probab4733579253584633066e_real @ S ) @ M2 @ F ) ) ) ).

% qbs_prob_measure_prob_space.prob_space_distr
thf(fact_284_qbs__prob__measure__prob__space_Oprob__space__distr,axiom,
    ! [F: real > complex,S: probab8009751763329705409e_real,M2: sigma_3077487657436305159omplex] :
      ( ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ ( probab4733579253584633066e_real @ S ) @ M2 ) )
     => ( probab6149883331606624555omplex @ ( measur1621797640479583060omplex @ ( probab4733579253584633066e_real @ S ) @ M2 @ F ) ) ) ).

% qbs_prob_measure_prob_space.prob_space_distr
thf(fact_285_qbs__prob__measure__prob__space_Oprob__space__distr,axiom,
    ! [F: real > real,S: probab8009751763329705409e_real,M2: sigma_measure_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ ( probab4733579253584633066e_real @ S ) @ M2 ) )
     => ( probab535871623910865577e_real @ ( measur2993149975067245138l_real @ ( probab4733579253584633066e_real @ S ) @ M2 @ F ) ) ) ).

% qbs_prob_measure_prob_space.prob_space_distr
thf(fact_286_qbs__prob__measure__prob__space_Oprob__space__distr,axiom,
    ! [F: a > extend8495563244428889912nnreal,S: probab4737552673497767871pace_a,M2: sigma_7234349610311085201nnreal] :
      ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ ( probab7100426894406488384sure_a @ S ) @ M2 ) )
     => ( probab6612481188548237749nnreal @ ( measur4839436603801885502nnreal @ ( probab7100426894406488384sure_a @ S ) @ M2 @ F ) ) ) ).

% qbs_prob_measure_prob_space.prob_space_distr
thf(fact_287_qbs__prob__measure__prob__space_Oprob__space__distr,axiom,
    ! [F: a > real,S: probab4737552673497767871pace_a,M2: sigma_measure_real] :
      ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ ( probab7100426894406488384sure_a @ S ) @ M2 ) )
     => ( probab535871623910865577e_real @ ( measure_distr_a_real @ ( probab7100426894406488384sure_a @ S ) @ M2 @ F ) ) ) ).

% qbs_prob_measure_prob_space.prob_space_distr
thf(fact_288_qbs__prob__measure__prob__space_Oprob__space__axioms,axiom,
    ! [S: probab8009751763329705409e_real] : ( probab535871623910865577e_real @ ( probab4733579253584633066e_real @ S ) ) ).

% qbs_prob_measure_prob_space.prob_space_axioms
thf(fact_289_qbs__prob__measure__prob__space_Oprob__space__axioms,axiom,
    ! [S: probab4737552673497767871pace_a] : ( probab7247484486040049089pace_a @ ( probab7100426894406488384sure_a @ S ) ) ).

% qbs_prob_measure_prob_space.prob_space_axioms
thf(fact_290_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y8: real,Z: real] : ( Y8 = Z ) )
    = ( ^ [A5: real,B4: real] :
          ( ( minus_minus_real @ A5 @ B4 )
          = zero_zero_real ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_291_qbs__prob__measure__prob__space_Osubprob__space__axioms,axiom,
    ! [S: probab8009751763329705409e_real] : ( giry_s8208748868292234104e_real @ ( probab4733579253584633066e_real @ S ) ) ).

% qbs_prob_measure_prob_space.subprob_space_axioms
thf(fact_292_qbs__prob__measure__prob__space_Osubprob__space__axioms,axiom,
    ! [S: probab4737552673497767871pace_a] : ( giry_subprob_space_a @ ( probab7100426894406488384sure_a @ S ) ) ).

% qbs_prob_measure_prob_space.subprob_space_axioms
thf(fact_293_distr__distr,axiom,
    ! [G: a > extend8495563244428889912nnreal,N: sigma_measure_a,L: sigma_7234349610311085201nnreal,F: real > a,M: sigma_measure_real] :
      ( ( member298456594901751504nnreal @ G @ ( sigma_214952329563889126nnreal @ N @ L ) )
     => ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ N ) )
       => ( ( measur4839436603801885502nnreal @ ( measure_distr_real_a @ M @ N @ F ) @ L @ G )
          = ( measur8829990298702910942nnreal @ M @ L @ ( comp_a8249376463644563905l_real @ G @ F ) ) ) ) ) ).

% distr_distr
thf(fact_294_distr__distr,axiom,
    ! [G: a > a,N: sigma_measure_a,L: sigma_measure_a,F: real > a,M: sigma_measure_real] :
      ( ( member_a_a @ G @ ( sigma_measurable_a_a @ N @ L ) )
     => ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ N ) )
       => ( ( measure_distr_a_a @ ( measure_distr_real_a @ M @ N @ F ) @ L @ G )
          = ( measure_distr_real_a @ M @ L @ ( comp_a_a_real @ G @ F ) ) ) ) ) ).

% distr_distr
thf(fact_295_distr__distr,axiom,
    ! [G: real > a,N: sigma_measure_real,L: sigma_measure_a,F: a > real,M: sigma_measure_a] :
      ( ( member_real_a @ G @ ( sigma_523072396149930112real_a @ N @ L ) )
     => ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ N ) )
       => ( ( measure_distr_real_a @ ( measure_distr_a_real @ M @ N @ F ) @ L @ G )
          = ( measure_distr_a_a @ M @ L @ ( comp_real_a_a @ G @ F ) ) ) ) ) ).

% distr_distr
thf(fact_296_distr__distr,axiom,
    ! [G: real > a,N: sigma_measure_real,L: sigma_measure_a,F: real > real,M: sigma_measure_real] :
      ( ( member_real_a @ G @ ( sigma_523072396149930112real_a @ N @ L ) )
     => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) )
       => ( ( measure_distr_real_a @ ( measur2993149975067245138l_real @ M @ N @ F ) @ L @ G )
          = ( measure_distr_real_a @ M @ L @ ( comp_real_a_real @ G @ F ) ) ) ) ) ).

% distr_distr
thf(fact_297_distr__distr,axiom,
    ! [G: extend8495563244428889912nnreal > a,N: sigma_7234349610311085201nnreal,L: sigma_measure_a,F: real > extend8495563244428889912nnreal,M: sigma_measure_real] :
      ( ( member4924430693770431270real_a @ G @ ( sigma_3031480723531659892real_a @ N @ L ) )
     => ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ N ) )
       => ( ( measur7655964997769656268real_a @ ( measur8829990298702910942nnreal @ M @ N @ F ) @ L @ G )
          = ( measure_distr_real_a @ M @ L @ ( comp_E4829442781247313743a_real @ G @ F ) ) ) ) ) ).

% distr_distr
thf(fact_298_distr__distr,axiom,
    ! [G: extend8495563244428889912nnreal > real,N: sigma_7234349610311085201nnreal,L: sigma_measure_real,F: real > extend8495563244428889912nnreal,M: sigma_measure_real] :
      ( ( member2874014351250825754l_real @ G @ ( sigma_7049758200512112822l_real @ N @ L ) )
     => ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ N ) )
       => ( ( measur6862244029252366686l_real @ ( measur8829990298702910942nnreal @ M @ N @ F ) @ L @ G )
          = ( measur2993149975067245138l_real @ M @ L @ ( comp_E3822617923592311797l_real @ G @ F ) ) ) ) ) ).

% distr_distr
thf(fact_299_distr__distr,axiom,
    ! [G: complex > a,N: sigma_3077487657436305159omplex,L: sigma_measure_a,F: real > complex,M: sigma_measure_real] :
      ( ( member_complex_a @ G @ ( sigma_6699518285080112254plex_a @ N @ L ) )
     => ( ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ M @ N ) )
       => ( ( measur724108212368259542plex_a @ ( measur1621797640479583060omplex @ M @ N @ F ) @ L @ G )
          = ( measure_distr_real_a @ M @ L @ ( comp_complex_a_real @ G @ F ) ) ) ) ) ).

% distr_distr
thf(fact_300_distr__distr,axiom,
    ! [G: complex > real,N: sigma_3077487657436305159omplex,L: sigma_measure_real,F: real > complex,M: sigma_measure_real] :
      ( ( member_complex_real @ G @ ( sigma_9165504702370893100x_real @ N @ L ) )
     => ( ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ M @ N ) )
       => ( ( measur1675386140983903700x_real @ ( measur1621797640479583060omplex @ M @ N @ F ) @ L @ G )
          = ( measur2993149975067245138l_real @ M @ L @ ( comp_c3333796836230738283l_real @ G @ F ) ) ) ) ) ).

% distr_distr
thf(fact_301_distr__distr,axiom,
    ! [G: a > real,N: sigma_measure_a,L: sigma_measure_real,F: real > a,M: sigma_measure_real] :
      ( ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ N @ L ) )
     => ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ N ) )
       => ( ( measure_distr_a_real @ ( measure_distr_real_a @ M @ N @ F ) @ L @ G )
          = ( measur2993149975067245138l_real @ M @ L @ ( comp_a_real_real @ G @ F ) ) ) ) ) ).

% distr_distr
thf(fact_302_distr__distr,axiom,
    ! [G: real > real,N: sigma_measure_real,L: sigma_measure_real,F: a > real,M: sigma_measure_a] :
      ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ N @ L ) )
     => ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ N ) )
       => ( ( measur2993149975067245138l_real @ ( measure_distr_a_real @ M @ N @ F ) @ L @ G )
          = ( measure_distr_a_real @ M @ L @ ( comp_real_real_a @ G @ F ) ) ) ) ) ).

% distr_distr
thf(fact_303_prob__space_Oreal__distribution__distr,axiom,
    ! [M: sigma_measure_a,X2: a > real] :
      ( ( probab7247484486040049089pace_a @ M )
     => ( ( member_a_real @ X2 @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
       => ( distri2809703520229113005bution @ ( measure_distr_a_real @ M @ borel_5078946678739801102l_real @ X2 ) ) ) ) ).

% prob_space.real_distribution_distr
thf(fact_304_prob__space_Oreal__distribution__distr,axiom,
    ! [M: sigma_measure_real,X2: real > real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( member_real_real @ X2 @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( distri2809703520229113005bution @ ( measur2993149975067245138l_real @ M @ borel_5078946678739801102l_real @ X2 ) ) ) ) ).

% prob_space.real_distribution_distr
thf(fact_305_borel__measurable__integrable,axiom,
    ! [M: sigma_measure_real,F: real > real] :
      ( ( bochne3340023020068487468l_real @ M @ F )
     => ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_integrable
thf(fact_306_borel__measurable__integrable,axiom,
    ! [M: sigma_measure_a,F: a > real] :
      ( ( bochne2139062162225249880a_real @ M @ F )
     => ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_integrable
thf(fact_307_borel__measurable__integrable,axiom,
    ! [M: sigma_measure_real,F: real > complex] :
      ( ( bochne7032760885902134062omplex @ M @ F )
     => ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ M @ borel_1392132677378845456omplex ) ) ) ).

% borel_measurable_integrable
thf(fact_308_integrableD_I1_J,axiom,
    ! [M: sigma_measure_real,F: real > real] :
      ( ( bochne3340023020068487468l_real @ M @ F )
     => ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% integrableD(1)
thf(fact_309_integrableD_I1_J,axiom,
    ! [M: sigma_measure_a,F: a > real] :
      ( ( bochne2139062162225249880a_real @ M @ F )
     => ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% integrableD(1)
thf(fact_310_qbs__integrable__iff__integrable,axiom,
    ( probab3847667120374951956e_real
    = ( ^ [S3: probab8009751763329705409e_real] : ( bochne3340023020068487468l_real @ ( probab4733579253584633066e_real @ S3 ) ) ) ) ).

% qbs_integrable_iff_integrable
thf(fact_311_qbs__integrable__iff__integrable,axiom,
    ( probab7312716125271441302able_a
    = ( ^ [S3: probab4737552673497767871pace_a] : ( bochne2139062162225249880a_real @ ( probab7100426894406488384sure_a @ S3 ) ) ) ) ).

% qbs_integrable_iff_integrable
thf(fact_312_qbs__integrable__if__integrable,axiom,
    ! [S: probab8009751763329705409e_real,F: real > real] :
      ( ( bochne3340023020068487468l_real @ ( probab4733579253584633066e_real @ S ) @ F )
     => ( probab3847667120374951956e_real @ S @ F ) ) ).

% qbs_integrable_if_integrable
thf(fact_313_qbs__integrable__if__integrable,axiom,
    ! [S: probab4737552673497767871pace_a,F: a > real] :
      ( ( bochne2139062162225249880a_real @ ( probab7100426894406488384sure_a @ S ) @ F )
     => ( probab7312716125271441302able_a @ S @ F ) ) ).

% qbs_integrable_if_integrable
thf(fact_314_integrable__if__qbs__integrable,axiom,
    ! [S: probab8009751763329705409e_real,F: real > real] :
      ( ( probab3847667120374951956e_real @ S @ F )
     => ( bochne3340023020068487468l_real @ ( probab4733579253584633066e_real @ S ) @ F ) ) ).

% integrable_if_qbs_integrable
thf(fact_315_integrable__if__qbs__integrable,axiom,
    ! [S: probab4737552673497767871pace_a,F: a > real] :
      ( ( probab7312716125271441302able_a @ S @ F )
     => ( bochne2139062162225249880a_real @ ( probab7100426894406488384sure_a @ S ) @ F ) ) ).

% integrable_if_qbs_integrable
thf(fact_316_not__integrable__integral__eq,axiom,
    ! [M: sigma_measure_real,F: real > real] :
      ( ~ ( bochne3340023020068487468l_real @ M @ F )
     => ( ( bochne3715101410578510557l_real @ M @ F )
        = zero_zero_real ) ) ).

% not_integrable_integral_eq
thf(fact_317_not__integrable__integral__eq,axiom,
    ! [M: sigma_measure_a,F: a > real] :
      ( ~ ( bochne2139062162225249880a_real @ M @ F )
     => ( ( bochne378719280626478695a_real @ M @ F )
        = zero_zero_real ) ) ).

% not_integrable_integral_eq
thf(fact_318_qbs__prob__integral__def2,axiom,
    ( probab4207012259563505946l_real
    = ( ^ [S3: probab8009751763329705409e_real] : ( bochne3715101410578510557l_real @ ( probab4733579253584633066e_real @ S3 ) ) ) ) ).

% qbs_prob_integral_def2
thf(fact_319_qbs__prob__integral__def2,axiom,
    ( probab2419480525258570000gral_a
    = ( ^ [S3: probab4737552673497767871pace_a] : ( bochne378719280626478695a_real @ ( probab7100426894406488384sure_a @ S3 ) ) ) ) ).

% qbs_prob_integral_def2
thf(fact_320_borel__measurable__integrable_H,axiom,
    ! [M: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > real,G: a > extend8495563244428889912nnreal,N: sigma_measure_a] :
      ( ( bochne9025062821074728248l_real @ M @ F )
     => ( ( member298456594901751504nnreal @ G @ ( sigma_214952329563889126nnreal @ N @ M ) )
       => ( member_a_real
          @ ^ [X: a] : ( F @ ( G @ X ) )
          @ ( sigma_9116425665531756122a_real @ N @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_integrable'
thf(fact_321_borel__measurable__integrable_H,axiom,
    ! [M: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > real,G: real > extend8495563244428889912nnreal,N: sigma_measure_real] :
      ( ( bochne9025062821074728248l_real @ M @ F )
     => ( ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ N @ M ) )
       => ( member_real_real
          @ ^ [X: real] : ( F @ ( G @ X ) )
          @ ( sigma_5267869275261027754l_real @ N @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_integrable'
thf(fact_322_borel__measurable__integrable_H,axiom,
    ! [M: sigma_3077487657436305159omplex,F: complex > real,G: real > complex,N: sigma_measure_real] :
      ( ( bochne7086349386406454702x_real @ M @ F )
     => ( ( member_real_complex @ G @ ( sigma_9111916201866572460omplex @ N @ M ) )
       => ( member_real_real
          @ ^ [X: real] : ( F @ ( G @ X ) )
          @ ( sigma_5267869275261027754l_real @ N @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_integrable'
thf(fact_323_borel__measurable__integrable_H,axiom,
    ! [M: sigma_measure_real,F: real > real,G: a > real,N: sigma_measure_a] :
      ( ( bochne3340023020068487468l_real @ M @ F )
     => ( ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ N @ M ) )
       => ( member_a_real
          @ ^ [X: a] : ( F @ ( G @ X ) )
          @ ( sigma_9116425665531756122a_real @ N @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_integrable'
thf(fact_324_borel__measurable__integrable_H,axiom,
    ! [M: sigma_measure_real,F: real > real,G: real > real,N: sigma_measure_real] :
      ( ( bochne3340023020068487468l_real @ M @ F )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ N @ M ) )
       => ( member_real_real
          @ ^ [X: real] : ( F @ ( G @ X ) )
          @ ( sigma_5267869275261027754l_real @ N @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_integrable'
thf(fact_325_borel__measurable__integrable_H,axiom,
    ! [M: sigma_measure_a,F: a > real,G: a > a,N: sigma_measure_a] :
      ( ( bochne2139062162225249880a_real @ M @ F )
     => ( ( member_a_a @ G @ ( sigma_measurable_a_a @ N @ M ) )
       => ( member_a_real
          @ ^ [X: a] : ( F @ ( G @ X ) )
          @ ( sigma_9116425665531756122a_real @ N @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_integrable'
thf(fact_326_borel__measurable__integrable_H,axiom,
    ! [M: sigma_measure_a,F: a > real,G: real > a,N: sigma_measure_real] :
      ( ( bochne2139062162225249880a_real @ M @ F )
     => ( ( member_real_a @ G @ ( sigma_523072396149930112real_a @ N @ M ) )
       => ( member_real_real
          @ ^ [X: real] : ( F @ ( G @ X ) )
          @ ( sigma_5267869275261027754l_real @ N @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_integrable'
thf(fact_327_borel__measurable__integrable_H,axiom,
    ! [M: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > complex,G: a > extend8495563244428889912nnreal,N: sigma_measure_a] :
      ( ( bochne6309023331997297978omplex @ M @ F )
     => ( ( member298456594901751504nnreal @ G @ ( sigma_214952329563889126nnreal @ N @ M ) )
       => ( member_a_complex
          @ ^ [X: a] : ( F @ ( G @ X ) )
          @ ( sigma_852363994732143452omplex @ N @ borel_1392132677378845456omplex ) ) ) ) ).

% borel_measurable_integrable'
thf(fact_328_borel__measurable__integrable_H,axiom,
    ! [M: sigma_measure_a,F: a > complex,G: real > a,N: sigma_measure_real] :
      ( ( bochne1348834467089073754omplex @ M @ F )
     => ( ( member_real_a @ G @ ( sigma_523072396149930112real_a @ N @ M ) )
       => ( member_real_complex
          @ ^ [X: real] : ( F @ ( G @ X ) )
          @ ( sigma_9111916201866572460omplex @ N @ borel_1392132677378845456omplex ) ) ) ) ).

% borel_measurable_integrable'
thf(fact_329_borel__measurable__integrable_H,axiom,
    ! [M: sigma_measure_real,F: real > complex,G: a > real,N: sigma_measure_a] :
      ( ( bochne7032760885902134062omplex @ M @ F )
     => ( ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ N @ M ) )
       => ( member_a_complex
          @ ^ [X: a] : ( F @ ( G @ X ) )
          @ ( sigma_852363994732143452omplex @ N @ borel_1392132677378845456omplex ) ) ) ) ).

% borel_measurable_integrable'
thf(fact_330_integrable__distr,axiom,
    ! [T: real > extend8495563244428889912nnreal,M: sigma_measure_real,M2: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > real] :
      ( ( member2919562650594848410nnreal @ T @ ( sigma_9017504469962657078nnreal @ M @ M2 ) )
     => ( ( bochne9025062821074728248l_real @ ( measur8829990298702910942nnreal @ M @ M2 @ T ) @ F )
       => ( bochne3340023020068487468l_real @ M
          @ ^ [X: real] : ( F @ ( T @ X ) ) ) ) ) ).

% integrable_distr
thf(fact_331_integrable__distr,axiom,
    ! [T: real > complex,M: sigma_measure_real,M2: sigma_3077487657436305159omplex,F: complex > real] :
      ( ( member_real_complex @ T @ ( sigma_9111916201866572460omplex @ M @ M2 ) )
     => ( ( bochne7086349386406454702x_real @ ( measur1621797640479583060omplex @ M @ M2 @ T ) @ F )
       => ( bochne3340023020068487468l_real @ M
          @ ^ [X: real] : ( F @ ( T @ X ) ) ) ) ) ).

% integrable_distr
thf(fact_332_integrable__distr,axiom,
    ! [T: a > extend8495563244428889912nnreal,M: sigma_measure_a,M2: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > real] :
      ( ( member298456594901751504nnreal @ T @ ( sigma_214952329563889126nnreal @ M @ M2 ) )
     => ( ( bochne9025062821074728248l_real @ ( measur4839436603801885502nnreal @ M @ M2 @ T ) @ F )
       => ( bochne2139062162225249880a_real @ M
          @ ^ [X: a] : ( F @ ( T @ X ) ) ) ) ) ).

% integrable_distr
thf(fact_333_integrable__distr,axiom,
    ! [T: real > real,M: sigma_measure_real,M2: sigma_measure_real,F: real > real] :
      ( ( member_real_real @ T @ ( sigma_5267869275261027754l_real @ M @ M2 ) )
     => ( ( bochne3340023020068487468l_real @ ( measur2993149975067245138l_real @ M @ M2 @ T ) @ F )
       => ( bochne3340023020068487468l_real @ M
          @ ^ [X: real] : ( F @ ( T @ X ) ) ) ) ) ).

% integrable_distr
thf(fact_334_integrable__distr,axiom,
    ! [T: a > real,M: sigma_measure_a,M2: sigma_measure_real,F: real > real] :
      ( ( member_a_real @ T @ ( sigma_9116425665531756122a_real @ M @ M2 ) )
     => ( ( bochne3340023020068487468l_real @ ( measure_distr_a_real @ M @ M2 @ T ) @ F )
       => ( bochne2139062162225249880a_real @ M
          @ ^ [X: a] : ( F @ ( T @ X ) ) ) ) ) ).

% integrable_distr
thf(fact_335_integrable__distr,axiom,
    ! [T: real > a,M: sigma_measure_real,M2: sigma_measure_a,F: a > real] :
      ( ( member_real_a @ T @ ( sigma_523072396149930112real_a @ M @ M2 ) )
     => ( ( bochne2139062162225249880a_real @ ( measure_distr_real_a @ M @ M2 @ T ) @ F )
       => ( bochne3340023020068487468l_real @ M
          @ ^ [X: real] : ( F @ ( T @ X ) ) ) ) ) ).

% integrable_distr
thf(fact_336_integrable__distr,axiom,
    ! [T: a > a,M: sigma_measure_a,M2: sigma_measure_a,F: a > real] :
      ( ( member_a_a @ T @ ( sigma_measurable_a_a @ M @ M2 ) )
     => ( ( bochne2139062162225249880a_real @ ( measure_distr_a_a @ M @ M2 @ T ) @ F )
       => ( bochne2139062162225249880a_real @ M
          @ ^ [X: a] : ( F @ ( T @ X ) ) ) ) ) ).

% integrable_distr
thf(fact_337_integral__distr,axiom,
    ! [G: real > extend8495563244428889912nnreal,M: sigma_measure_real,N: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > real] :
      ( ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ M @ N ) )
     => ( ( member2874014351250825754l_real @ F @ ( sigma_7049758200512112822l_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne2458729288719820649l_real @ ( measur8829990298702910942nnreal @ M @ N @ G ) @ F )
          = ( bochne3715101410578510557l_real @ M
            @ ^ [X: real] : ( F @ ( G @ X ) ) ) ) ) ) ).

% integral_distr
thf(fact_338_integral__distr,axiom,
    ! [G: real > complex,M: sigma_measure_real,N: sigma_3077487657436305159omplex,F: complex > real] :
      ( ( member_real_complex @ G @ ( sigma_9111916201866572460omplex @ M @ N ) )
     => ( ( member_complex_real @ F @ ( sigma_9165504702370893100x_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne8919328671811780063x_real @ ( measur1621797640479583060omplex @ M @ N @ G ) @ F )
          = ( bochne3715101410578510557l_real @ M
            @ ^ [X: real] : ( F @ ( G @ X ) ) ) ) ) ) ).

% integral_distr
thf(fact_339_integral__distr,axiom,
    ! [G: a > extend8495563244428889912nnreal,M: sigma_measure_a,N: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > real] :
      ( ( member298456594901751504nnreal @ G @ ( sigma_214952329563889126nnreal @ M @ N ) )
     => ( ( member2874014351250825754l_real @ F @ ( sigma_7049758200512112822l_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne2458729288719820649l_real @ ( measur4839436603801885502nnreal @ M @ N @ G ) @ F )
          = ( bochne378719280626478695a_real @ M
            @ ^ [X: a] : ( F @ ( G @ X ) ) ) ) ) ) ).

% integral_distr
thf(fact_340_integral__distr,axiom,
    ! [G: real > real,M: sigma_measure_real,N: sigma_measure_real,F: real > real] :
      ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ N ) )
     => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne3715101410578510557l_real @ ( measur2993149975067245138l_real @ M @ N @ G ) @ F )
          = ( bochne3715101410578510557l_real @ M
            @ ^ [X: real] : ( F @ ( G @ X ) ) ) ) ) ) ).

% integral_distr
thf(fact_341_integral__distr,axiom,
    ! [G: a > real,M: sigma_measure_a,N: sigma_measure_real,F: real > real] :
      ( ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ M @ N ) )
     => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne3715101410578510557l_real @ ( measure_distr_a_real @ M @ N @ G ) @ F )
          = ( bochne378719280626478695a_real @ M
            @ ^ [X: a] : ( F @ ( G @ X ) ) ) ) ) ) ).

% integral_distr
thf(fact_342_integral__distr,axiom,
    ! [G: real > a,M: sigma_measure_real,N: sigma_measure_a,F: a > real] :
      ( ( member_real_a @ G @ ( sigma_523072396149930112real_a @ M @ N ) )
     => ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne378719280626478695a_real @ ( measure_distr_real_a @ M @ N @ G ) @ F )
          = ( bochne3715101410578510557l_real @ M
            @ ^ [X: real] : ( F @ ( G @ X ) ) ) ) ) ) ).

% integral_distr
thf(fact_343_integral__distr,axiom,
    ! [G: a > a,M: sigma_measure_a,N: sigma_measure_a,F: a > real] :
      ( ( member_a_a @ G @ ( sigma_measurable_a_a @ M @ N ) )
     => ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne378719280626478695a_real @ ( measure_distr_a_a @ M @ N @ G ) @ F )
          = ( bochne378719280626478695a_real @ M
            @ ^ [X: a] : ( F @ ( G @ X ) ) ) ) ) ) ).

% integral_distr
thf(fact_344_integral__distr,axiom,
    ! [G: a > extend8495563244428889912nnreal,M: sigma_measure_a,N: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > complex] :
      ( ( member298456594901751504nnreal @ G @ ( sigma_214952329563889126nnreal @ M @ N ) )
     => ( ( member808392060238902940omplex @ F @ ( sigma_3243507219526583224omplex @ N @ borel_1392132677378845456omplex ) )
       => ( ( bochne7234769336060495339omplex @ ( measur4839436603801885502nnreal @ M @ N @ G ) @ F )
          = ( bochne4904656926214500329omplex @ M
            @ ^ [X: a] : ( F @ ( G @ X ) ) ) ) ) ) ).

% integral_distr
thf(fact_345_integral__distr,axiom,
    ! [G: real > a,M: sigma_measure_real,N: sigma_measure_a,F: a > complex] :
      ( ( member_real_a @ G @ ( sigma_523072396149930112real_a @ M @ N ) )
     => ( ( member_a_complex @ F @ ( sigma_852363994732143452omplex @ N @ borel_1392132677378845456omplex ) )
       => ( ( bochne4904656926214500329omplex @ ( measure_distr_real_a @ M @ N @ G ) @ F )
          = ( bochne8865740171307459423omplex @ M
            @ ^ [X: real] : ( F @ ( G @ X ) ) ) ) ) ) ).

% integral_distr
thf(fact_346_integral__distr,axiom,
    ! [G: a > real,M: sigma_measure_a,N: sigma_measure_real,F: real > complex] :
      ( ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ M @ N ) )
     => ( ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ N @ borel_1392132677378845456omplex ) )
       => ( ( bochne8865740171307459423omplex @ ( measure_distr_a_real @ M @ N @ G ) @ F )
          = ( bochne4904656926214500329omplex @ M
            @ ^ [X: a] : ( F @ ( G @ X ) ) ) ) ) ) ).

% integral_distr
thf(fact_347_integrable__distr__eq,axiom,
    ! [G: real > extend8495563244428889912nnreal,M: sigma_measure_real,N: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > real] :
      ( ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ M @ N ) )
     => ( ( member2874014351250825754l_real @ F @ ( sigma_7049758200512112822l_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne9025062821074728248l_real @ ( measur8829990298702910942nnreal @ M @ N @ G ) @ F )
          = ( bochne3340023020068487468l_real @ M
            @ ^ [X: real] : ( F @ ( G @ X ) ) ) ) ) ) ).

% integrable_distr_eq
thf(fact_348_integrable__distr__eq,axiom,
    ! [G: real > complex,M: sigma_measure_real,N: sigma_3077487657436305159omplex,F: complex > real] :
      ( ( member_real_complex @ G @ ( sigma_9111916201866572460omplex @ M @ N ) )
     => ( ( member_complex_real @ F @ ( sigma_9165504702370893100x_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne7086349386406454702x_real @ ( measur1621797640479583060omplex @ M @ N @ G ) @ F )
          = ( bochne3340023020068487468l_real @ M
            @ ^ [X: real] : ( F @ ( G @ X ) ) ) ) ) ) ).

% integrable_distr_eq
thf(fact_349_integrable__distr__eq,axiom,
    ! [G: a > extend8495563244428889912nnreal,M: sigma_measure_a,N: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > real] :
      ( ( member298456594901751504nnreal @ G @ ( sigma_214952329563889126nnreal @ M @ N ) )
     => ( ( member2874014351250825754l_real @ F @ ( sigma_7049758200512112822l_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne9025062821074728248l_real @ ( measur4839436603801885502nnreal @ M @ N @ G ) @ F )
          = ( bochne2139062162225249880a_real @ M
            @ ^ [X: a] : ( F @ ( G @ X ) ) ) ) ) ) ).

% integrable_distr_eq
thf(fact_350_integrable__distr__eq,axiom,
    ! [G: real > real,M: sigma_measure_real,N: sigma_measure_real,F: real > real] :
      ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ N ) )
     => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne3340023020068487468l_real @ ( measur2993149975067245138l_real @ M @ N @ G ) @ F )
          = ( bochne3340023020068487468l_real @ M
            @ ^ [X: real] : ( F @ ( G @ X ) ) ) ) ) ) ).

% integrable_distr_eq
thf(fact_351_integrable__distr__eq,axiom,
    ! [G: a > real,M: sigma_measure_a,N: sigma_measure_real,F: real > real] :
      ( ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ M @ N ) )
     => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne3340023020068487468l_real @ ( measure_distr_a_real @ M @ N @ G ) @ F )
          = ( bochne2139062162225249880a_real @ M
            @ ^ [X: a] : ( F @ ( G @ X ) ) ) ) ) ) ).

% integrable_distr_eq
thf(fact_352_integrable__distr__eq,axiom,
    ! [G: real > a,M: sigma_measure_real,N: sigma_measure_a,F: a > real] :
      ( ( member_real_a @ G @ ( sigma_523072396149930112real_a @ M @ N ) )
     => ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne2139062162225249880a_real @ ( measure_distr_real_a @ M @ N @ G ) @ F )
          = ( bochne3340023020068487468l_real @ M
            @ ^ [X: real] : ( F @ ( G @ X ) ) ) ) ) ) ).

% integrable_distr_eq
thf(fact_353_integrable__distr__eq,axiom,
    ! [G: a > a,M: sigma_measure_a,N: sigma_measure_a,F: a > real] :
      ( ( member_a_a @ G @ ( sigma_measurable_a_a @ M @ N ) )
     => ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne2139062162225249880a_real @ ( measure_distr_a_a @ M @ N @ G ) @ F )
          = ( bochne2139062162225249880a_real @ M
            @ ^ [X: a] : ( F @ ( G @ X ) ) ) ) ) ) ).

% integrable_distr_eq
thf(fact_354_integrable__distr__eq,axiom,
    ! [G: a > extend8495563244428889912nnreal,M: sigma_measure_a,N: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > complex] :
      ( ( member298456594901751504nnreal @ G @ ( sigma_214952329563889126nnreal @ M @ N ) )
     => ( ( member808392060238902940omplex @ F @ ( sigma_3243507219526583224omplex @ N @ borel_1392132677378845456omplex ) )
       => ( ( bochne6309023331997297978omplex @ ( measur4839436603801885502nnreal @ M @ N @ G ) @ F )
          = ( bochne1348834467089073754omplex @ M
            @ ^ [X: a] : ( F @ ( G @ X ) ) ) ) ) ) ).

% integrable_distr_eq
thf(fact_355_integrable__distr__eq,axiom,
    ! [G: real > a,M: sigma_measure_real,N: sigma_measure_a,F: a > complex] :
      ( ( member_real_a @ G @ ( sigma_523072396149930112real_a @ M @ N ) )
     => ( ( member_a_complex @ F @ ( sigma_852363994732143452omplex @ N @ borel_1392132677378845456omplex ) )
       => ( ( bochne1348834467089073754omplex @ ( measure_distr_real_a @ M @ N @ G ) @ F )
          = ( bochne7032760885902134062omplex @ M
            @ ^ [X: real] : ( F @ ( G @ X ) ) ) ) ) ) ).

% integrable_distr_eq
thf(fact_356_integrable__distr__eq,axiom,
    ! [G: a > real,M: sigma_measure_a,N: sigma_measure_real,F: real > complex] :
      ( ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ M @ N ) )
     => ( ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ N @ borel_1392132677378845456omplex ) )
       => ( ( bochne7032760885902134062omplex @ ( measure_distr_a_real @ M @ N @ G ) @ F )
          = ( bochne1348834467089073754omplex @ M
            @ ^ [X: a] : ( F @ ( G @ X ) ) ) ) ) ) ).

% integrable_distr_eq
thf(fact_357_real__integrable__def,axiom,
    ( bochne3340023020068487468l_real
    = ( ^ [M3: sigma_measure_real,F2: real > real] :
          ( ( member_real_real @ F2 @ ( sigma_5267869275261027754l_real @ M3 @ borel_5078946678739801102l_real ) )
          & ( ( nonneg2667834350952324695l_real @ M3
              @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( F2 @ X ) ) )
           != extend2057119558705770725nnreal )
          & ( ( nonneg2667834350952324695l_real @ M3
              @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( uminus_uminus_real @ ( F2 @ X ) ) ) )
           != extend2057119558705770725nnreal ) ) ) ) ).

% real_integrable_def
thf(fact_358_real__integrable__def,axiom,
    ( bochne2139062162225249880a_real
    = ( ^ [M3: sigma_measure_a,F2: a > real] :
          ( ( member_a_real @ F2 @ ( sigma_9116425665531756122a_real @ M3 @ borel_5078946678739801102l_real ) )
          & ( ( nonneg2725512125972007571gral_a @ M3
              @ ^ [X: a] : ( extend7643940197134561352nnreal @ ( F2 @ X ) ) )
           != extend2057119558705770725nnreal )
          & ( ( nonneg2725512125972007571gral_a @ M3
              @ ^ [X: a] : ( extend7643940197134561352nnreal @ ( uminus_uminus_real @ ( F2 @ X ) ) ) )
           != extend2057119558705770725nnreal ) ) ) ) ).

% real_integrable_def
thf(fact_359_qbs__prob_Oqbs__prob__measure__computation,axiom,
    ! [X2: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real] :
      ( ( probab3605210969150000782b_real @ X2 @ Alpha @ Mu )
     => ( ( probab4733579253584633066e_real @ ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X2 @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) ) )
        = ( measur2993149975067245138l_real @ Mu @ ( measur1733462625046462224e_real @ X2 ) @ Alpha ) ) ) ).

% qbs_prob.qbs_prob_measure_computation
thf(fact_360_qbs__prob_Oqbs__prob__measure__computation,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu )
     => ( ( probab7100426894406488384sure_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) )
        = ( measure_distr_real_a @ Mu @ ( measur7857763439677503898sure_a @ X2 ) @ Alpha ) ) ) ).

% qbs_prob.qbs_prob_measure_computation
thf(fact_361_qbs__prob__eq2__dest_I4_J,axiom,
    ! [X2: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,Y: quasi_borel_real,Beta: real > real,Nu: sigma_measure_real,F: real > real] :
      ( ( probab19465756045782014582_real @ ( produc4368523205619139320e_real @ X2 @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) @ ( produc4368523205619139320e_real @ Y @ ( produc1722724976708544245e_real @ Beta @ Nu ) ) )
     => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ ( measur1733462625046462224e_real @ X2 ) @ borel_5078946678739801102l_real ) )
       => ( ( bochne3715101410578510557l_real @ Mu
            @ ^ [X: real] : ( F @ ( Alpha @ X ) ) )
          = ( bochne3715101410578510557l_real @ Nu
            @ ^ [X: real] : ( F @ ( Beta @ X ) ) ) ) ) ) ).

% qbs_prob_eq2_dest(4)
thf(fact_362_qbs__prob__eq2__dest_I4_J,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real,F: a > real] :
      ( ( probab3918592701117320376_eq2_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) )
     => ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ ( measur7857763439677503898sure_a @ X2 ) @ borel_5078946678739801102l_real ) )
       => ( ( bochne3715101410578510557l_real @ Mu
            @ ^ [X: real] : ( F @ ( Alpha @ X ) ) )
          = ( bochne3715101410578510557l_real @ Nu
            @ ^ [X: real] : ( F @ ( Beta @ X ) ) ) ) ) ) ).

% qbs_prob_eq2_dest(4)
thf(fact_363_qbs__prob__ennintegral__def2,axiom,
    ! [S: probab8009751763329705409e_real,X2: quasi_borel_real,F: real > extend8495563244428889912nnreal] :
      ( ( ( probab8185819741702177770s_real @ S )
        = X2 )
     => ( ( member2919562650594848410nnreal @ F @ ( qbs_mo1317719164804411614nnreal @ X2 @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) ) )
       => ( ( probab7585390126108274877l_real @ S @ F )
          = ( nonneg2667834350952324695l_real @ ( probab4733579253584633066e_real @ S ) @ F ) ) ) ) ).

% qbs_prob_ennintegral_def2
thf(fact_364_qbs__prob__ennintegral__def2,axiom,
    ! [S: probab4737552673497767871pace_a,X2: quasi_borel_a,F: a > extend8495563244428889912nnreal] :
      ( ( ( probab1293289258141559360_qbs_a @ S )
        = X2 )
     => ( ( member298456594901751504nnreal @ F @ ( qbs_mo1434458643421888574nnreal @ X2 @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) ) )
       => ( ( probab3721531081081959085gral_a @ S @ F )
          = ( nonneg2725512125972007571gral_a @ ( probab7100426894406488384sure_a @ S ) @ F ) ) ) ) ).

% qbs_prob_ennintegral_def2
thf(fact_365_qbs__prob_Oqbs__integrable__measurable,axiom,
    ! [X2: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,F: real > real] :
      ( ( probab3605210969150000782b_real @ X2 @ Alpha @ Mu )
     => ( ( probab3847667120374951956e_real @ ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X2 @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) ) @ F )
       => ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ ( measur1733462625046462224e_real @ X2 ) @ borel_5078946678739801102l_real ) ) ) ) ).

% qbs_prob.qbs_integrable_measurable
thf(fact_366_qbs__prob_Oqbs__integrable__measurable,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,F: a > real] :
      ( ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu )
     => ( ( probab7312716125271441302able_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) @ F )
       => ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ ( measur7857763439677503898sure_a @ X2 ) @ borel_5078946678739801102l_real ) ) ) ) ).

% qbs_prob.qbs_integrable_measurable
thf(fact_367_pair__qbs__prob_Oqbs__prob__space__eq2,axiom,
    ! [X2: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,Y: quasi_borel_real,Beta: real > real,Nu: sigma_measure_real] :
      ( ( probab6500215489368174228b_real @ X2 @ Alpha @ Mu @ Y @ Beta @ Nu )
     => ( ( Y = X2 )
       => ( ! [F3: real > real] :
              ( ( member_real_real @ F3 @ ( sigma_5267869275261027754l_real @ ( measur1733462625046462224e_real @ X2 ) @ borel_5078946678739801102l_real ) )
             => ( ( bochne3715101410578510557l_real @ Mu
                  @ ^ [X: real] : ( F3 @ ( Alpha @ X ) ) )
                = ( bochne3715101410578510557l_real @ Nu
                  @ ^ [X: real] : ( F3 @ ( Beta @ X ) ) ) ) )
         => ( ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X2 @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) )
            = ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ Y @ ( produc1722724976708544245e_real @ Beta @ Nu ) ) ) ) ) ) ) ).

% pair_qbs_prob.qbs_prob_space_eq2
thf(fact_368_pair__qbs__prob_Oqbs__prob__space__eq2,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab5677843605262999830prob_a @ X2 @ Alpha @ Mu @ Y @ Beta @ Nu )
     => ( ( Y = X2 )
       => ( ! [F3: a > real] :
              ( ( member_a_real @ F3 @ ( sigma_9116425665531756122a_real @ ( measur7857763439677503898sure_a @ X2 ) @ borel_5078946678739801102l_real ) )
             => ( ( bochne3715101410578510557l_real @ Mu
                  @ ^ [X: real] : ( F3 @ ( Alpha @ X ) ) )
                = ( bochne3715101410578510557l_real @ Nu
                  @ ^ [X: real] : ( F3 @ ( Beta @ X ) ) ) ) )
         => ( ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) )
            = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) ) ) ) ) ) ).

% pair_qbs_prob.qbs_prob_space_eq2
thf(fact_369_pair__qbs__prob_Oqbs__prob__eq2__intro,axiom,
    ! [X2: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,Y: quasi_borel_real,Beta: real > real,Nu: sigma_measure_real] :
      ( ( probab6500215489368174228b_real @ X2 @ Alpha @ Mu @ Y @ Beta @ Nu )
     => ( ( X2 = Y )
       => ( ! [F3: real > real] :
              ( ( member_real_real @ F3 @ ( sigma_5267869275261027754l_real @ ( measur1733462625046462224e_real @ X2 ) @ borel_5078946678739801102l_real ) )
             => ( ( bochne3715101410578510557l_real @ Mu
                  @ ^ [X: real] : ( F3 @ ( Alpha @ X ) ) )
                = ( bochne3715101410578510557l_real @ Nu
                  @ ^ [X: real] : ( F3 @ ( Beta @ X ) ) ) ) )
         => ( probab19465756045782014582_real @ ( produc4368523205619139320e_real @ X2 @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) @ ( produc4368523205619139320e_real @ Y @ ( produc1722724976708544245e_real @ Beta @ Nu ) ) ) ) ) ) ).

% pair_qbs_prob.qbs_prob_eq2_intro
thf(fact_370_pair__qbs__prob_Oqbs__prob__eq2__intro,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab5677843605262999830prob_a @ X2 @ Alpha @ Mu @ Y @ Beta @ Nu )
     => ( ( X2 = Y )
       => ( ! [F3: a > real] :
              ( ( member_a_real @ F3 @ ( sigma_9116425665531756122a_real @ ( measur7857763439677503898sure_a @ X2 ) @ borel_5078946678739801102l_real ) )
             => ( ( bochne3715101410578510557l_real @ Mu
                  @ ^ [X: real] : ( F3 @ ( Alpha @ X ) ) )
                = ( bochne3715101410578510557l_real @ Nu
                  @ ^ [X: real] : ( F3 @ ( Beta @ X ) ) ) ) )
         => ( probab3918592701117320376_eq2_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) ) ) ) ) ).

% pair_qbs_prob.qbs_prob_eq2_intro
thf(fact_371_qbs__prob_Oqbs__integrable__iff__integrable,axiom,
    ! [X2: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,F: real > real] :
      ( ( probab3605210969150000782b_real @ X2 @ Alpha @ Mu )
     => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ ( measur1733462625046462224e_real @ X2 ) @ borel_5078946678739801102l_real ) )
       => ( ( probab3847667120374951956e_real @ ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X2 @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) ) @ F )
          = ( bochne3340023020068487468l_real @ Mu
            @ ^ [X: real] : ( F @ ( Alpha @ X ) ) ) ) ) ) ).

% qbs_prob.qbs_integrable_iff_integrable
thf(fact_372_qbs__prob_Oqbs__integrable__iff__integrable,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,F: a > real] :
      ( ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu )
     => ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ ( measur7857763439677503898sure_a @ X2 ) @ borel_5078946678739801102l_real ) )
       => ( ( probab7312716125271441302able_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) @ F )
          = ( bochne3340023020068487468l_real @ Mu
            @ ^ [X: real] : ( F @ ( Alpha @ X ) ) ) ) ) ) ).

% qbs_prob.qbs_integrable_iff_integrable
thf(fact_373_qbs__prob_Oqbs__prob__ennintegral__not__morphism,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,F: a > extend8495563244428889912nnreal] :
      ( ( probab701741629625904796prob_a @ X2 @ Alpha @ Mu )
     => ( ~ ( member298456594901751504nnreal @ F @ ( qbs_mo1434458643421888574nnreal @ X2 @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) ) )
       => ( ( probab3721531081081959085gral_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) @ F )
          = zero_z7100319975126383169nnreal ) ) ) ).

% qbs_prob.qbs_prob_ennintegral_not_morphism
thf(fact_374_borel__measurable__uminus__eq,axiom,
    ! [F: a > real,M: sigma_measure_a] :
      ( ( member_a_real
        @ ^ [X: a] : ( uminus_uminus_real @ ( F @ X ) )
        @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
      = ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_uminus_eq
thf(fact_375_borel__measurable__uminus__eq,axiom,
    ! [F: real > real,M: sigma_measure_real] :
      ( ( member_real_real
        @ ^ [X: real] : ( uminus_uminus_real @ ( F @ X ) )
        @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
      = ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_uminus_eq
thf(fact_376_borel__measurable__uminus__eq,axiom,
    ! [F: real > complex,M: sigma_measure_real] :
      ( ( member_real_complex
        @ ^ [X: real] : ( uminus1482373934393186551omplex @ ( F @ X ) )
        @ ( sigma_9111916201866572460omplex @ M @ borel_1392132677378845456omplex ) )
      = ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ M @ borel_1392132677378845456omplex ) ) ) ).

% borel_measurable_uminus_eq
thf(fact_377_nn__integral__distr,axiom,
    ! [T: real > extend8495563244428889912nnreal,M: sigma_measure_real,M2: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
      ( ( member2919562650594848410nnreal @ T @ ( sigma_9017504469962657078nnreal @ M @ M2 ) )
     => ( ( member8329810500450651686nnreal @ F @ ( sigma_7926153774531450434nnreal @ ( measur8829990298702910942nnreal @ M @ M2 @ T ) @ borel_6524799422816628122nnreal ) )
       => ( ( nonneg5898919932907209827nnreal @ ( measur8829990298702910942nnreal @ M @ M2 @ T ) @ F )
          = ( nonneg2667834350952324695l_real @ M
            @ ^ [X: real] : ( F @ ( T @ X ) ) ) ) ) ) ).

% nn_integral_distr
thf(fact_378_nn__integral__distr,axiom,
    ! [T: real > complex,M: sigma_measure_real,M2: sigma_3077487657436305159omplex,F: complex > extend8495563244428889912nnreal] :
      ( ( member_real_complex @ T @ ( sigma_9111916201866572460omplex @ M @ M2 ) )
     => ( ( member1891431242017442716nnreal @ F @ ( sigma_4389633852101207480nnreal @ ( measur1621797640479583060omplex @ M @ M2 @ T ) @ borel_6524799422816628122nnreal ) )
       => ( ( nonneg6050707109158959065omplex @ ( measur1621797640479583060omplex @ M @ M2 @ T ) @ F )
          = ( nonneg2667834350952324695l_real @ M
            @ ^ [X: real] : ( F @ ( T @ X ) ) ) ) ) ) ).

% nn_integral_distr
thf(fact_379_nn__integral__distr,axiom,
    ! [T: a > extend8495563244428889912nnreal,M: sigma_measure_a,M2: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
      ( ( member298456594901751504nnreal @ T @ ( sigma_214952329563889126nnreal @ M @ M2 ) )
     => ( ( member8329810500450651686nnreal @ F @ ( sigma_7926153774531450434nnreal @ ( measur4839436603801885502nnreal @ M @ M2 @ T ) @ borel_6524799422816628122nnreal ) )
       => ( ( nonneg5898919932907209827nnreal @ ( measur4839436603801885502nnreal @ M @ M2 @ T ) @ F )
          = ( nonneg2725512125972007571gral_a @ M
            @ ^ [X: a] : ( F @ ( T @ X ) ) ) ) ) ) ).

% nn_integral_distr
thf(fact_380_nn__integral__distr,axiom,
    ! [T: real > real,M: sigma_measure_real,M2: sigma_measure_real,F: real > extend8495563244428889912nnreal] :
      ( ( member_real_real @ T @ ( sigma_5267869275261027754l_real @ M @ M2 ) )
     => ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ ( measur2993149975067245138l_real @ M @ M2 @ T ) @ borel_6524799422816628122nnreal ) )
       => ( ( nonneg2667834350952324695l_real @ ( measur2993149975067245138l_real @ M @ M2 @ T ) @ F )
          = ( nonneg2667834350952324695l_real @ M
            @ ^ [X: real] : ( F @ ( T @ X ) ) ) ) ) ) ).

% nn_integral_distr
thf(fact_381_nn__integral__distr,axiom,
    ! [T: a > real,M: sigma_measure_a,M2: sigma_measure_real,F: real > extend8495563244428889912nnreal] :
      ( ( member_a_real @ T @ ( sigma_9116425665531756122a_real @ M @ M2 ) )
     => ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ ( measure_distr_a_real @ M @ M2 @ T ) @ borel_6524799422816628122nnreal ) )
       => ( ( nonneg2667834350952324695l_real @ ( measure_distr_a_real @ M @ M2 @ T ) @ F )
          = ( nonneg2725512125972007571gral_a @ M
            @ ^ [X: a] : ( F @ ( T @ X ) ) ) ) ) ) ).

% nn_integral_distr
thf(fact_382_nn__integral__distr,axiom,
    ! [T: real > a,M: sigma_measure_real,M2: sigma_measure_a,F: a > extend8495563244428889912nnreal] :
      ( ( member_real_a @ T @ ( sigma_523072396149930112real_a @ M @ M2 ) )
     => ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ ( measure_distr_real_a @ M @ M2 @ T ) @ borel_6524799422816628122nnreal ) )
       => ( ( nonneg2725512125972007571gral_a @ ( measure_distr_real_a @ M @ M2 @ T ) @ F )
          = ( nonneg2667834350952324695l_real @ M
            @ ^ [X: real] : ( F @ ( T @ X ) ) ) ) ) ) ).

% nn_integral_distr
thf(fact_383_nn__integral__distr,axiom,
    ! [T: a > a,M: sigma_measure_a,M2: sigma_measure_a,F: a > extend8495563244428889912nnreal] :
      ( ( member_a_a @ T @ ( sigma_measurable_a_a @ M @ M2 ) )
     => ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ ( measure_distr_a_a @ M @ M2 @ T ) @ borel_6524799422816628122nnreal ) )
       => ( ( nonneg2725512125972007571gral_a @ ( measure_distr_a_a @ M @ M2 @ T ) @ F )
          = ( nonneg2725512125972007571gral_a @ M
            @ ^ [X: a] : ( F @ ( T @ X ) ) ) ) ) ) ).

% nn_integral_distr
thf(fact_384_borel__measurable__uminus,axiom,
    ! [G: a > real,M: sigma_measure_a] :
      ( ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
     => ( member_a_real
        @ ^ [X: a] : ( uminus_uminus_real @ ( G @ X ) )
        @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_uminus
thf(fact_385_borel__measurable__uminus,axiom,
    ! [G: real > real,M: sigma_measure_real] :
      ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( member_real_real
        @ ^ [X: real] : ( uminus_uminus_real @ ( G @ X ) )
        @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_uminus
thf(fact_386_borel__measurable__uminus,axiom,
    ! [G: real > complex,M: sigma_measure_real] :
      ( ( member_real_complex @ G @ ( sigma_9111916201866572460omplex @ M @ borel_1392132677378845456omplex ) )
     => ( member_real_complex
        @ ^ [X: real] : ( uminus1482373934393186551omplex @ ( G @ X ) )
        @ ( sigma_9111916201866572460omplex @ M @ borel_1392132677378845456omplex ) ) ) ).

% borel_measurable_uminus
thf(fact_387_borel__measurable__Pair,axiom,
    ! [F: a > extend8495563244428889912nnreal,M: sigma_measure_a,G: a > extend8495563244428889912nnreal] :
      ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( ( member298456594901751504nnreal @ G @ ( sigma_214952329563889126nnreal @ M @ borel_6524799422816628122nnreal ) )
       => ( member5741711457236458191nnreal
          @ ^ [X: a] : ( produc344325839068023049nnreal @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_6220444669619866531nnreal @ M @ borel_3951438148318096111nnreal ) ) ) ) ).

% borel_measurable_Pair
thf(fact_388_borel__measurable__Pair,axiom,
    ! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,G: real > extend8495563244428889912nnreal] :
      ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) )
       => ( member7391843071750696581nnreal
          @ ^ [X: real] : ( produc344325839068023049nnreal @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_2101749304130238547nnreal @ M @ borel_3951438148318096111nnreal ) ) ) ) ).

% borel_measurable_Pair
thf(fact_389_borel__measurable__Pair,axiom,
    ! [F: a > extend8495563244428889912nnreal,M: sigma_measure_a,G: a > real] :
      ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
       => ( member5439804117191083459l_real
          @ ^ [X: a] : ( produc2810268924804063229l_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_7586584134059694359l_real @ M @ borel_4928740784729289315l_real ) ) ) ) ).

% borel_measurable_Pair
thf(fact_390_borel__measurable__Pair,axiom,
    ! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,G: real > real] :
      ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member4593507348387352185l_real
          @ ^ [X: real] : ( produc2810268924804063229l_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_1034331983380590023l_real @ M @ borel_4928740784729289315l_real ) ) ) ) ).

% borel_measurable_Pair
thf(fact_391_borel__measurable__Pair,axiom,
    ! [F: a > extend8495563244428889912nnreal,M: sigma_measure_a,G: a > complex] :
      ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( ( member_a_complex @ G @ ( sigma_852363994732143452omplex @ M @ borel_1392132677378845456omplex ) )
       => ( member520227739041225797omplex
          @ ^ [X: a] : ( produc1203308874072432767omplex @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_342526099188998681omplex @ M @ borel_6932847338593847653omplex ) ) ) ) ).

% borel_measurable_Pair
thf(fact_392_borel__measurable__Pair,axiom,
    ! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,G: real > complex] :
      ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( ( member_real_complex @ G @ ( sigma_9111916201866572460omplex @ M @ borel_1392132677378845456omplex ) )
       => ( member9010308190037439483omplex
          @ ^ [X: real] : ( produc1203308874072432767omplex @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_1610262524020092617omplex @ M @ borel_6932847338593847653omplex ) ) ) ) ).

% borel_measurable_Pair
thf(fact_393_borel__measurable__Pair,axiom,
    ! [F: a > real,M: sigma_measure_a,G: a > extend8495563244428889912nnreal] :
      ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member298456594901751504nnreal @ G @ ( sigma_214952329563889126nnreal @ M @ borel_6524799422816628122nnreal ) )
       => ( member5109759473273639491nnreal
          @ ^ [X: a] : ( produc4778015194254607485nnreal @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_7632132433403717015nnreal @ M @ borel_4974289084073311971nnreal ) ) ) ) ).

% borel_measurable_Pair
thf(fact_394_borel__measurable__Pair,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > extend8495563244428889912nnreal] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) )
       => ( member4263462704469908217nnreal
          @ ^ [X: real] : ( produc4778015194254607485nnreal @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_1079880282724612679nnreal @ M @ borel_4974289084073311971nnreal ) ) ) ) ).

% borel_measurable_Pair
thf(fact_395_borel__measurable__Pair,axiom,
    ! [F: a > real,M: sigma_measure_a,G: a > real] :
      ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
       => ( member2229928074028245815l_real
          @ ^ [X: a] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_414277600898586891l_real @ M @ borel_9155112475215227991l_real ) ) ) ) ).

% borel_measurable_Pair
thf(fact_396_borel__measurable__Pair,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member9086635009091248365l_real
          @ ^ [X: real] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_7998147297565726139l_real @ M @ borel_9155112475215227991l_real ) ) ) ) ).

% borel_measurable_Pair
thf(fact_397_borel__measurable__diff,axiom,
    ! [F: a > real,M: sigma_measure_a,G: a > real] :
      ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_a_real
          @ ^ [X: a] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_diff
thf(fact_398_borel__measurable__diff,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_real_real
          @ ^ [X: real] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_diff
thf(fact_399_borel__measurable__diff,axiom,
    ! [F: real > complex,M: sigma_measure_real,G: real > complex] :
      ( ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ M @ borel_1392132677378845456omplex ) )
     => ( ( member_real_complex @ G @ ( sigma_9111916201866572460omplex @ M @ borel_1392132677378845456omplex ) )
       => ( member_real_complex
          @ ^ [X: real] : ( minus_minus_complex @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_9111916201866572460omplex @ M @ borel_1392132677378845456omplex ) ) ) ) ).

% borel_measurable_diff
thf(fact_400_qbs__morphismE_I3_J,axiom,
    ! [F: complex > complex,X2: quasi_borel_complex,Y: quasi_borel_complex,Alpha: real > complex] :
      ( ( member5128974058612258834omplex @ F @ ( qbs_mo9200510921189519062omplex @ X2 @ Y ) )
     => ( ( member_real_complex @ Alpha @ ( qbs_Mx_complex @ X2 ) )
       => ( member_real_complex @ ( comp_c2117349707075585901x_real @ F @ Alpha ) @ ( qbs_Mx_complex @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_401_qbs__morphismE_I3_J,axiom,
    ! [F: complex > real,X2: quasi_borel_complex,Y: quasi_borel_real,Alpha: real > complex] :
      ( ( member_complex_real @ F @ ( qbs_mo6120686211186450644x_real @ X2 @ Y ) )
     => ( ( member_real_complex @ Alpha @ ( qbs_Mx_complex @ X2 ) )
       => ( member_real_real @ ( comp_c3333796836230738283l_real @ F @ Alpha ) @ ( qbs_Mx_real @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_402_qbs__morphismE_I3_J,axiom,
    ! [F: real > complex,X2: quasi_borel_real,Y: quasi_borel_complex,Alpha: real > real] :
      ( ( member_real_complex @ F @ ( qbs_mo6067097710682130004omplex @ X2 @ Y ) )
     => ( ( member_real_real @ Alpha @ ( qbs_Mx_real @ X2 ) )
       => ( member_real_complex @ ( comp_r1968866223832618731x_real @ F @ Alpha ) @ ( qbs_Mx_complex @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_403_qbs__morphismE_I3_J,axiom,
    ! [F: real > real,X2: quasi_borel_real,Y: quasi_borel_real,Alpha: real > real] :
      ( ( member_real_real @ F @ ( qbs_mo5229770564518008146l_real @ X2 @ Y ) )
     => ( ( member_real_real @ Alpha @ ( qbs_Mx_real @ X2 ) )
       => ( member_real_real @ ( comp_real_real_real @ F @ Alpha ) @ ( qbs_Mx_real @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_404_qbs__morphismE_I3_J,axiom,
    ! [F: complex > a,X2: quasi_borel_complex,Y: quasi_borel_a,Alpha: real > complex] :
      ( ( member_complex_a @ F @ ( qbs_mo6245657829219851990plex_a @ X2 @ Y ) )
     => ( ( member_real_complex @ Alpha @ ( qbs_Mx_complex @ X2 ) )
       => ( member_real_a @ ( comp_complex_a_real @ F @ Alpha ) @ ( qbs_Mx_a @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_405_qbs__morphismE_I3_J,axiom,
    ! [F: real > a,X2: quasi_borel_real,Y: quasi_borel_a,Alpha: real > real] :
      ( ( member_real_a @ F @ ( qbs_morphism_real_a @ X2 @ Y ) )
     => ( ( member_real_real @ Alpha @ ( qbs_Mx_real @ X2 ) )
       => ( member_real_a @ ( comp_real_a_real @ F @ Alpha ) @ ( qbs_Mx_a @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_406_qbs__morphismE_I3_J,axiom,
    ! [F: a > complex,X2: quasi_borel_a,Y: quasi_borel_complex,Alpha: real > a] :
      ( ( member_a_complex @ F @ ( qbs_mo398503538871883188omplex @ X2 @ Y ) )
     => ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X2 ) )
       => ( member_real_complex @ ( comp_a_complex_real @ F @ Alpha ) @ ( qbs_Mx_complex @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_407_qbs__morphismE_I3_J,axiom,
    ! [F: a > a,X2: quasi_borel_a,Y: quasi_borel_a,Alpha: real > a] :
      ( ( member_a_a @ F @ ( qbs_morphism_a_a @ X2 @ Y ) )
     => ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X2 ) )
       => ( member_real_a @ ( comp_a_a_real @ F @ Alpha ) @ ( qbs_Mx_a @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_408_qbs__morphismE_I3_J,axiom,
    ! [F: a > extend8495563244428889912nnreal,X2: quasi_borel_a,Y: quasi_9015997321629101608nnreal,Alpha: real > a] :
      ( ( member298456594901751504nnreal @ F @ ( qbs_mo1434458643421888574nnreal @ X2 @ Y ) )
     => ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X2 ) )
       => ( member2919562650594848410nnreal @ ( comp_a8249376463644563905l_real @ F @ Alpha ) @ ( qbs_Mx6523938229262583809nnreal @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_409_qbs__morphismE_I3_J,axiom,
    ! [F: a > real,X2: quasi_borel_a,Y: quasi_borel_real,Alpha: real > a] :
      ( ( member_a_real @ F @ ( qbs_morphism_a_real @ X2 @ Y ) )
     => ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X2 ) )
       => ( member_real_real @ ( comp_a_real_real @ F @ Alpha ) @ ( qbs_Mx_real @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_410_qbs__morphismI,axiom,
    ! [X2: quasi_borel_complex,F: complex > complex,Y: quasi_borel_complex] :
      ( ! [Alpha4: real > complex] :
          ( ( member_real_complex @ Alpha4 @ ( qbs_Mx_complex @ X2 ) )
         => ( member_real_complex @ ( comp_c2117349707075585901x_real @ F @ Alpha4 ) @ ( qbs_Mx_complex @ Y ) ) )
     => ( member5128974058612258834omplex @ F @ ( qbs_mo9200510921189519062omplex @ X2 @ Y ) ) ) ).

% qbs_morphismI
thf(fact_411_qbs__morphismI,axiom,
    ! [X2: quasi_borel_complex,F: complex > real,Y: quasi_borel_real] :
      ( ! [Alpha4: real > complex] :
          ( ( member_real_complex @ Alpha4 @ ( qbs_Mx_complex @ X2 ) )
         => ( member_real_real @ ( comp_c3333796836230738283l_real @ F @ Alpha4 ) @ ( qbs_Mx_real @ Y ) ) )
     => ( member_complex_real @ F @ ( qbs_mo6120686211186450644x_real @ X2 @ Y ) ) ) ).

% qbs_morphismI
thf(fact_412_qbs__morphismI,axiom,
    ! [X2: quasi_borel_real,F: real > complex,Y: quasi_borel_complex] :
      ( ! [Alpha4: real > real] :
          ( ( member_real_real @ Alpha4 @ ( qbs_Mx_real @ X2 ) )
         => ( member_real_complex @ ( comp_r1968866223832618731x_real @ F @ Alpha4 ) @ ( qbs_Mx_complex @ Y ) ) )
     => ( member_real_complex @ F @ ( qbs_mo6067097710682130004omplex @ X2 @ Y ) ) ) ).

% qbs_morphismI
thf(fact_413_qbs__morphismI,axiom,
    ! [X2: quasi_borel_real,F: real > real,Y: quasi_borel_real] :
      ( ! [Alpha4: real > real] :
          ( ( member_real_real @ Alpha4 @ ( qbs_Mx_real @ X2 ) )
         => ( member_real_real @ ( comp_real_real_real @ F @ Alpha4 ) @ ( qbs_Mx_real @ Y ) ) )
     => ( member_real_real @ F @ ( qbs_mo5229770564518008146l_real @ X2 @ Y ) ) ) ).

% qbs_morphismI
thf(fact_414_qbs__morphismI,axiom,
    ! [X2: quasi_borel_complex,F: complex > a,Y: quasi_borel_a] :
      ( ! [Alpha4: real > complex] :
          ( ( member_real_complex @ Alpha4 @ ( qbs_Mx_complex @ X2 ) )
         => ( member_real_a @ ( comp_complex_a_real @ F @ Alpha4 ) @ ( qbs_Mx_a @ Y ) ) )
     => ( member_complex_a @ F @ ( qbs_mo6245657829219851990plex_a @ X2 @ Y ) ) ) ).

% qbs_morphismI
thf(fact_415_qbs__morphismI,axiom,
    ! [X2: quasi_borel_real,F: real > a,Y: quasi_borel_a] :
      ( ! [Alpha4: real > real] :
          ( ( member_real_real @ Alpha4 @ ( qbs_Mx_real @ X2 ) )
         => ( member_real_a @ ( comp_real_a_real @ F @ Alpha4 ) @ ( qbs_Mx_a @ Y ) ) )
     => ( member_real_a @ F @ ( qbs_morphism_real_a @ X2 @ Y ) ) ) ).

% qbs_morphismI
thf(fact_416_qbs__morphismI,axiom,
    ! [X2: quasi_borel_a,F: a > complex,Y: quasi_borel_complex] :
      ( ! [Alpha4: real > a] :
          ( ( member_real_a @ Alpha4 @ ( qbs_Mx_a @ X2 ) )
         => ( member_real_complex @ ( comp_a_complex_real @ F @ Alpha4 ) @ ( qbs_Mx_complex @ Y ) ) )
     => ( member_a_complex @ F @ ( qbs_mo398503538871883188omplex @ X2 @ Y ) ) ) ).

% qbs_morphismI
thf(fact_417_qbs__morphismI,axiom,
    ! [X2: quasi_borel_a,F: a > a,Y: quasi_borel_a] :
      ( ! [Alpha4: real > a] :
          ( ( member_real_a @ Alpha4 @ ( qbs_Mx_a @ X2 ) )
         => ( member_real_a @ ( comp_a_a_real @ F @ Alpha4 ) @ ( qbs_Mx_a @ Y ) ) )
     => ( member_a_a @ F @ ( qbs_morphism_a_a @ X2 @ Y ) ) ) ).

% qbs_morphismI
thf(fact_418_qbs__morphismI,axiom,
    ! [X2: quasi_borel_a,F: a > extend8495563244428889912nnreal,Y: quasi_9015997321629101608nnreal] :
      ( ! [Alpha4: real > a] :
          ( ( member_real_a @ Alpha4 @ ( qbs_Mx_a @ X2 ) )
         => ( member2919562650594848410nnreal @ ( comp_a8249376463644563905l_real @ F @ Alpha4 ) @ ( qbs_Mx6523938229262583809nnreal @ Y ) ) )
     => ( member298456594901751504nnreal @ F @ ( qbs_mo1434458643421888574nnreal @ X2 @ Y ) ) ) ).

% qbs_morphismI
thf(fact_419_qbs__morphismI,axiom,
    ! [X2: quasi_borel_a,F: a > real,Y: quasi_borel_real] :
      ( ! [Alpha4: real > a] :
          ( ( member_real_a @ Alpha4 @ ( qbs_Mx_a @ X2 ) )
         => ( member_real_real @ ( comp_a_real_real @ F @ Alpha4 ) @ ( qbs_Mx_real @ Y ) ) )
     => ( member_a_real @ F @ ( qbs_morphism_a_real @ X2 @ Y ) ) ) ).

% qbs_morphismI
thf(fact_420_qp_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_a,X12: real > a,M22: sigma_measure_a,X23: real > a,Y12: a > extend8495563244428889912nnreal,N1: sigma_7234349610311085201nnreal,Y22: a > extend8495563244428889912nnreal,N2: sigma_7234349610311085201nnreal] :
      ( ( indepe365082296117321348real_a @ mu @ M1 @ X12 @ M22 @ X23 )
     => ( ( member298456594901751504nnreal @ Y12 @ ( sigma_214952329563889126nnreal @ M1 @ N1 ) )
       => ( ( member298456594901751504nnreal @ Y22 @ ( sigma_214952329563889126nnreal @ M22 @ N2 ) )
         => ( indepe6767359503340752434nnreal @ mu @ N1 @ ( comp_a8249376463644563905l_real @ Y12 @ X12 ) @ N2 @ ( comp_a8249376463644563905l_real @ Y22 @ X23 ) ) ) ) ) ).

% qp.indep_var_compose
thf(fact_421_qp_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_a,X12: real > a,M22: sigma_measure_a,X23: real > a,Y12: a > real,N1: sigma_measure_real,Y22: a > real,N2: sigma_measure_real] :
      ( ( indepe365082296117321348real_a @ mu @ M1 @ X12 @ M22 @ X23 )
     => ( ( member_a_real @ Y12 @ ( sigma_9116425665531756122a_real @ M1 @ N1 ) )
       => ( ( member_a_real @ Y22 @ ( sigma_9116425665531756122a_real @ M22 @ N2 ) )
         => ( indepe3760321310464026790l_real @ mu @ N1 @ ( comp_a_real_real @ Y12 @ X12 ) @ N2 @ ( comp_a_real_real @ Y22 @ X23 ) ) ) ) ) ).

% qp.indep_var_compose
thf(fact_422_qp_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_real,X12: real > real,M22: sigma_measure_real,X23: real > real,Y12: real > a,N1: sigma_measure_a,Y22: real > a,N2: sigma_measure_a] :
      ( ( indepe3760321310464026790l_real @ mu @ M1 @ X12 @ M22 @ X23 )
     => ( ( member_real_a @ Y12 @ ( sigma_523072396149930112real_a @ M1 @ N1 ) )
       => ( ( member_real_a @ Y22 @ ( sigma_523072396149930112real_a @ M22 @ N2 ) )
         => ( indepe365082296117321348real_a @ mu @ N1 @ ( comp_real_a_real @ Y12 @ X12 ) @ N2 @ ( comp_real_a_real @ Y22 @ X23 ) ) ) ) ) ).

% qp.indep_var_compose
thf(fact_423_qp_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_real,X12: real > real,M22: sigma_measure_real,X23: real > real,Y12: real > extend8495563244428889912nnreal,N1: sigma_7234349610311085201nnreal,Y22: real > extend8495563244428889912nnreal,N2: sigma_7234349610311085201nnreal] :
      ( ( indepe3760321310464026790l_real @ mu @ M1 @ X12 @ M22 @ X23 )
     => ( ( member2919562650594848410nnreal @ Y12 @ ( sigma_9017504469962657078nnreal @ M1 @ N1 ) )
       => ( ( member2919562650594848410nnreal @ Y22 @ ( sigma_9017504469962657078nnreal @ M22 @ N2 ) )
         => ( indepe6767359503340752434nnreal @ mu @ N1 @ ( comp_r6279034453215524981l_real @ Y12 @ X12 ) @ N2 @ ( comp_r6279034453215524981l_real @ Y22 @ X23 ) ) ) ) ) ).

% qp.indep_var_compose
thf(fact_424_qp_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_real,X12: real > real,M22: sigma_measure_real,X23: real > real,Y12: real > complex,N1: sigma_3077487657436305159omplex,Y22: real > complex,N2: sigma_3077487657436305159omplex] :
      ( ( indepe3760321310464026790l_real @ mu @ M1 @ X12 @ M22 @ X23 )
     => ( ( member_real_complex @ Y12 @ ( sigma_9111916201866572460omplex @ M1 @ N1 ) )
       => ( ( member_real_complex @ Y22 @ ( sigma_9111916201866572460omplex @ M22 @ N2 ) )
         => ( indepe1954327081502071720omplex @ mu @ N1 @ ( comp_r1968866223832618731x_real @ Y12 @ X12 ) @ N2 @ ( comp_r1968866223832618731x_real @ Y22 @ X23 ) ) ) ) ) ).

% qp.indep_var_compose
thf(fact_425_qp_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_real,X12: real > real,M22: sigma_measure_real,X23: real > real,Y12: real > real,N1: sigma_measure_real,Y22: real > real,N2: sigma_measure_real] :
      ( ( indepe3760321310464026790l_real @ mu @ M1 @ X12 @ M22 @ X23 )
     => ( ( member_real_real @ Y12 @ ( sigma_5267869275261027754l_real @ M1 @ N1 ) )
       => ( ( member_real_real @ Y22 @ ( sigma_5267869275261027754l_real @ M22 @ N2 ) )
         => ( indepe3760321310464026790l_real @ mu @ N1 @ ( comp_real_real_real @ Y12 @ X12 ) @ N2 @ ( comp_real_real_real @ Y22 @ X23 ) ) ) ) ) ).

% qp.indep_var_compose
thf(fact_426_qp_Oindep__var__rv1,axiom,
    ! [S2: sigma_measure_a,X2: real > a,T: sigma_measure_a,Y: real > a] :
      ( ( indepe365082296117321348real_a @ mu @ S2 @ X2 @ T @ Y )
     => ( member_real_a @ X2 @ ( sigma_523072396149930112real_a @ mu @ S2 ) ) ) ).

% qp.indep_var_rv1
thf(fact_427_qp_Oindep__var__rv1,axiom,
    ! [S2: sigma_7234349610311085201nnreal,X2: real > extend8495563244428889912nnreal,T: sigma_7234349610311085201nnreal,Y: real > extend8495563244428889912nnreal] :
      ( ( indepe6767359503340752434nnreal @ mu @ S2 @ X2 @ T @ Y )
     => ( member2919562650594848410nnreal @ X2 @ ( sigma_9017504469962657078nnreal @ mu @ S2 ) ) ) ).

% qp.indep_var_rv1
thf(fact_428_qp_Oindep__var__rv1,axiom,
    ! [S2: sigma_3077487657436305159omplex,X2: real > complex,T: sigma_3077487657436305159omplex,Y: real > complex] :
      ( ( indepe1954327081502071720omplex @ mu @ S2 @ X2 @ T @ Y )
     => ( member_real_complex @ X2 @ ( sigma_9111916201866572460omplex @ mu @ S2 ) ) ) ).

% qp.indep_var_rv1
thf(fact_429_qp_Oindep__var__rv1,axiom,
    ! [S2: sigma_measure_real,X2: real > real,T: sigma_measure_real,Y: real > real] :
      ( ( indepe3760321310464026790l_real @ mu @ S2 @ X2 @ T @ Y )
     => ( member_real_real @ X2 @ ( sigma_5267869275261027754l_real @ mu @ S2 ) ) ) ).

% qp.indep_var_rv1
thf(fact_430_qp_Oindep__var__rv2,axiom,
    ! [S2: sigma_measure_a,X2: real > a,T: sigma_measure_a,Y: real > a] :
      ( ( indepe365082296117321348real_a @ mu @ S2 @ X2 @ T @ Y )
     => ( member_real_a @ Y @ ( sigma_523072396149930112real_a @ mu @ T ) ) ) ).

% qp.indep_var_rv2
thf(fact_431_qp_Oindep__var__rv2,axiom,
    ! [S2: sigma_7234349610311085201nnreal,X2: real > extend8495563244428889912nnreal,T: sigma_7234349610311085201nnreal,Y: real > extend8495563244428889912nnreal] :
      ( ( indepe6767359503340752434nnreal @ mu @ S2 @ X2 @ T @ Y )
     => ( member2919562650594848410nnreal @ Y @ ( sigma_9017504469962657078nnreal @ mu @ T ) ) ) ).

% qp.indep_var_rv2
thf(fact_432_qp_Oindep__var__rv2,axiom,
    ! [S2: sigma_3077487657436305159omplex,X2: real > complex,T: sigma_3077487657436305159omplex,Y: real > complex] :
      ( ( indepe1954327081502071720omplex @ mu @ S2 @ X2 @ T @ Y )
     => ( member_real_complex @ Y @ ( sigma_9111916201866572460omplex @ mu @ T ) ) ) ).

% qp.indep_var_rv2
thf(fact_433_qp_Oindep__var__rv2,axiom,
    ! [S2: sigma_measure_real,X2: real > real,T: sigma_measure_real,Y: real > real] :
      ( ( indepe3760321310464026790l_real @ mu @ S2 @ X2 @ T @ Y )
     => ( member_real_real @ Y @ ( sigma_5267869275261027754l_real @ mu @ T ) ) ) ).

% qp.indep_var_rv2
thf(fact_434_qp_Omeasurable__finite__borel,axiom,
    ! [F: real > extend8495563244428889912nnreal] :
      ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ borel_6524799422816628122nnreal ) )
     => ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ mu @ borel_6524799422816628122nnreal ) ) ) ).

% qp.measurable_finite_borel
thf(fact_435_qp_Omeasurable__finite__borel,axiom,
    ! [F: real > real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
     => ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ mu @ borel_5078946678739801102l_real ) ) ) ).

% qp.measurable_finite_borel
thf(fact_436_qp_Omeasurable__finite__borel,axiom,
    ! [F: real > complex] :
      ( ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ borel_5078946678739801102l_real @ borel_1392132677378845456omplex ) )
     => ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ mu @ borel_1392132677378845456omplex ) ) ) ).

% qp.measurable_finite_borel
thf(fact_437_qp_Oprob__space__distr,axiom,
    ! [F: real > a,M2: sigma_measure_a] :
      ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ mu @ M2 ) )
     => ( probab7247484486040049089pace_a @ ( measure_distr_real_a @ mu @ M2 @ F ) ) ) ).

% qp.prob_space_distr
thf(fact_438_qp_Oprob__space__distr,axiom,
    ! [F: real > extend8495563244428889912nnreal,M2: sigma_7234349610311085201nnreal] :
      ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ mu @ M2 ) )
     => ( probab6612481188548237749nnreal @ ( measur8829990298702910942nnreal @ mu @ M2 @ F ) ) ) ).

% qp.prob_space_distr
thf(fact_439_qp_Oprob__space__distr,axiom,
    ! [F: real > complex,M2: sigma_3077487657436305159omplex] :
      ( ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ mu @ M2 ) )
     => ( probab6149883331606624555omplex @ ( measur1621797640479583060omplex @ mu @ M2 @ F ) ) ) ).

% qp.prob_space_distr
thf(fact_440_qp_Oprob__space__distr,axiom,
    ! [F: real > real,M2: sigma_measure_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ mu @ M2 ) )
     => ( probab535871623910865577e_real @ ( measur2993149975067245138l_real @ mu @ M2 @ F ) ) ) ).

% qp.prob_space_distr
thf(fact_441_qp_Oreal__distribution__distr,axiom,
    ! [X2: real > real] :
      ( ( member_real_real @ X2 @ ( sigma_5267869275261027754l_real @ mu @ borel_5078946678739801102l_real ) )
     => ( distri2809703520229113005bution @ ( measur2993149975067245138l_real @ mu @ borel_5078946678739801102l_real @ X2 ) ) ) ).

% qp.real_distribution_distr
thf(fact_442_qbs__closed3__dest_H,axiom,
    ! [P: real > nat,Fi: nat > real > complex,X2: quasi_borel_complex] :
      ( ( member_real_nat @ P @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) )
     => ( ! [I: nat] : ( member_real_complex @ ( Fi @ I ) @ ( qbs_Mx_complex @ X2 ) )
       => ( member_real_complex
          @ ^ [R2: real] : ( Fi @ ( P @ R2 ) @ R2 )
          @ ( qbs_Mx_complex @ X2 ) ) ) ) ).

% qbs_closed3_dest'
thf(fact_443_qbs__closed3__dest_H,axiom,
    ! [P: real > nat,Fi: nat > real > real,X2: quasi_borel_real] :
      ( ( member_real_nat @ P @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) )
     => ( ! [I: nat] : ( member_real_real @ ( Fi @ I ) @ ( qbs_Mx_real @ X2 ) )
       => ( member_real_real
          @ ^ [R2: real] : ( Fi @ ( P @ R2 ) @ R2 )
          @ ( qbs_Mx_real @ X2 ) ) ) ) ).

% qbs_closed3_dest'
thf(fact_444_qbs__closed3__dest_H,axiom,
    ! [P: real > nat,Fi: nat > real > a,X2: quasi_borel_a] :
      ( ( member_real_nat @ P @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) )
     => ( ! [I: nat] : ( member_real_a @ ( Fi @ I ) @ ( qbs_Mx_a @ X2 ) )
       => ( member_real_a
          @ ^ [R2: real] : ( Fi @ ( P @ R2 ) @ R2 )
          @ ( qbs_Mx_a @ X2 ) ) ) ) ).

% qbs_closed3_dest'
thf(fact_445_borel__measurable__minus__ennreal,axiom,
    ! [F: a > extend8495563244428889912nnreal,M: sigma_measure_a,G: a > extend8495563244428889912nnreal] :
      ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( ( member298456594901751504nnreal @ G @ ( sigma_214952329563889126nnreal @ M @ borel_6524799422816628122nnreal ) )
       => ( member298456594901751504nnreal
          @ ^ [X: a] : ( minus_8429688780609304081nnreal @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_214952329563889126nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ) ).

% borel_measurable_minus_ennreal
thf(fact_446_borel__measurable__minus__ennreal,axiom,
    ! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,G: real > extend8495563244428889912nnreal] :
      ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) )
       => ( member2919562650594848410nnreal
          @ ^ [X: real] : ( minus_8429688780609304081nnreal @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ) ).

% borel_measurable_minus_ennreal
thf(fact_447_qbs__closed1__dest,axiom,
    ! [Alpha: real > complex,X2: quasi_borel_complex,F: real > real] :
      ( ( member_real_complex @ Alpha @ ( qbs_Mx_complex @ X2 ) )
     => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
       => ( member_real_complex @ ( comp_r1968866223832618731x_real @ Alpha @ F ) @ ( qbs_Mx_complex @ X2 ) ) ) ) ).

% qbs_closed1_dest
thf(fact_448_qbs__closed1__dest,axiom,
    ! [Alpha: real > real,X2: quasi_borel_real,F: real > real] :
      ( ( member_real_real @ Alpha @ ( qbs_Mx_real @ X2 ) )
     => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
       => ( member_real_real @ ( comp_real_real_real @ Alpha @ F ) @ ( qbs_Mx_real @ X2 ) ) ) ) ).

% qbs_closed1_dest
thf(fact_449_qbs__closed1__dest,axiom,
    ! [Alpha: real > a,X2: quasi_borel_a,F: real > real] :
      ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X2 ) )
     => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
       => ( member_real_a @ ( comp_real_a_real @ Alpha @ F ) @ ( qbs_Mx_a @ X2 ) ) ) ) ).

% qbs_closed1_dest
thf(fact_450_measurable__ennreal,axiom,
    member2919562650594848410nnreal @ extend7643940197134561352nnreal @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ borel_6524799422816628122nnreal ) ).

% measurable_ennreal
thf(fact_451_qbs__prob__measure__prob__space_Oindep__var__rv2,axiom,
    ! [S: probab8009751763329705409e_real,S2: sigma_measure_a,X2: real > a,T: sigma_measure_a,Y: real > a] :
      ( ( indepe365082296117321348real_a @ ( probab4733579253584633066e_real @ S ) @ S2 @ X2 @ T @ Y )
     => ( member_real_a @ Y @ ( sigma_523072396149930112real_a @ ( probab4733579253584633066e_real @ S ) @ T ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_rv2
thf(fact_452_qbs__prob__measure__prob__space_Oindep__var__rv2,axiom,
    ! [S: probab8009751763329705409e_real,S2: sigma_7234349610311085201nnreal,X2: real > extend8495563244428889912nnreal,T: sigma_7234349610311085201nnreal,Y: real > extend8495563244428889912nnreal] :
      ( ( indepe6767359503340752434nnreal @ ( probab4733579253584633066e_real @ S ) @ S2 @ X2 @ T @ Y )
     => ( member2919562650594848410nnreal @ Y @ ( sigma_9017504469962657078nnreal @ ( probab4733579253584633066e_real @ S ) @ T ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_rv2
thf(fact_453_qbs__prob__measure__prob__space_Oindep__var__rv2,axiom,
    ! [S: probab8009751763329705409e_real,S2: sigma_3077487657436305159omplex,X2: real > complex,T: sigma_3077487657436305159omplex,Y: real > complex] :
      ( ( indepe1954327081502071720omplex @ ( probab4733579253584633066e_real @ S ) @ S2 @ X2 @ T @ Y )
     => ( member_real_complex @ Y @ ( sigma_9111916201866572460omplex @ ( probab4733579253584633066e_real @ S ) @ T ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_rv2
thf(fact_454_qbs__prob__measure__prob__space_Oindep__var__rv2,axiom,
    ! [S: probab8009751763329705409e_real,S2: sigma_measure_real,X2: real > real,T: sigma_measure_real,Y: real > real] :
      ( ( indepe3760321310464026790l_real @ ( probab4733579253584633066e_real @ S ) @ S2 @ X2 @ T @ Y )
     => ( member_real_real @ Y @ ( sigma_5267869275261027754l_real @ ( probab4733579253584633066e_real @ S ) @ T ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_rv2
thf(fact_455_qbs__prob__measure__prob__space_Oindep__var__rv2,axiom,
    ! [S: probab4737552673497767871pace_a,S2: sigma_7234349610311085201nnreal,X2: a > extend8495563244428889912nnreal,T: sigma_7234349610311085201nnreal,Y: a > extend8495563244428889912nnreal] :
      ( ( indepe3534117692041274858nnreal @ ( probab7100426894406488384sure_a @ S ) @ S2 @ X2 @ T @ Y )
     => ( member298456594901751504nnreal @ Y @ ( sigma_214952329563889126nnreal @ ( probab7100426894406488384sure_a @ S ) @ T ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_rv2
thf(fact_456_qbs__prob__measure__prob__space_Oindep__var__rv2,axiom,
    ! [S: probab4737552673497767871pace_a,S2: sigma_measure_real,X2: a > real,T: sigma_measure_real,Y: a > real] :
      ( ( indepe8958435565499147358a_real @ ( probab7100426894406488384sure_a @ S ) @ S2 @ X2 @ T @ Y )
     => ( member_a_real @ Y @ ( sigma_9116425665531756122a_real @ ( probab7100426894406488384sure_a @ S ) @ T ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_rv2
thf(fact_457_qbs__prob__measure__prob__space_Oindep__var__rv1,axiom,
    ! [S: probab8009751763329705409e_real,S2: sigma_measure_a,X2: real > a,T: sigma_measure_a,Y: real > a] :
      ( ( indepe365082296117321348real_a @ ( probab4733579253584633066e_real @ S ) @ S2 @ X2 @ T @ Y )
     => ( member_real_a @ X2 @ ( sigma_523072396149930112real_a @ ( probab4733579253584633066e_real @ S ) @ S2 ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_rv1
thf(fact_458_qbs__prob__measure__prob__space_Oindep__var__rv1,axiom,
    ! [S: probab8009751763329705409e_real,S2: sigma_7234349610311085201nnreal,X2: real > extend8495563244428889912nnreal,T: sigma_7234349610311085201nnreal,Y: real > extend8495563244428889912nnreal] :
      ( ( indepe6767359503340752434nnreal @ ( probab4733579253584633066e_real @ S ) @ S2 @ X2 @ T @ Y )
     => ( member2919562650594848410nnreal @ X2 @ ( sigma_9017504469962657078nnreal @ ( probab4733579253584633066e_real @ S ) @ S2 ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_rv1
thf(fact_459_qbs__prob__measure__prob__space_Oindep__var__rv1,axiom,
    ! [S: probab8009751763329705409e_real,S2: sigma_3077487657436305159omplex,X2: real > complex,T: sigma_3077487657436305159omplex,Y: real > complex] :
      ( ( indepe1954327081502071720omplex @ ( probab4733579253584633066e_real @ S ) @ S2 @ X2 @ T @ Y )
     => ( member_real_complex @ X2 @ ( sigma_9111916201866572460omplex @ ( probab4733579253584633066e_real @ S ) @ S2 ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_rv1
thf(fact_460_qbs__prob__measure__prob__space_Oindep__var__rv1,axiom,
    ! [S: probab8009751763329705409e_real,S2: sigma_measure_real,X2: real > real,T: sigma_measure_real,Y: real > real] :
      ( ( indepe3760321310464026790l_real @ ( probab4733579253584633066e_real @ S ) @ S2 @ X2 @ T @ Y )
     => ( member_real_real @ X2 @ ( sigma_5267869275261027754l_real @ ( probab4733579253584633066e_real @ S ) @ S2 ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_rv1
thf(fact_461_qbs__prob__measure__prob__space_Oindep__var__rv1,axiom,
    ! [S: probab4737552673497767871pace_a,S2: sigma_7234349610311085201nnreal,X2: a > extend8495563244428889912nnreal,T: sigma_7234349610311085201nnreal,Y: a > extend8495563244428889912nnreal] :
      ( ( indepe3534117692041274858nnreal @ ( probab7100426894406488384sure_a @ S ) @ S2 @ X2 @ T @ Y )
     => ( member298456594901751504nnreal @ X2 @ ( sigma_214952329563889126nnreal @ ( probab7100426894406488384sure_a @ S ) @ S2 ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_rv1
thf(fact_462_qbs__prob__measure__prob__space_Oindep__var__rv1,axiom,
    ! [S: probab4737552673497767871pace_a,S2: sigma_measure_real,X2: a > real,T: sigma_measure_real,Y: a > real] :
      ( ( indepe8958435565499147358a_real @ ( probab7100426894406488384sure_a @ S ) @ S2 @ X2 @ T @ Y )
     => ( member_a_real @ X2 @ ( sigma_9116425665531756122a_real @ ( probab7100426894406488384sure_a @ S ) @ S2 ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_rv1
thf(fact_463_real__distribution_Omeasurable__finite__borel,axiom,
    ! [M: sigma_measure_real,F: real > extend8495563244428889912nnreal] :
      ( ( distri2809703520229113005bution @ M )
     => ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ borel_6524799422816628122nnreal ) )
       => ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ) ).

% real_distribution.measurable_finite_borel
thf(fact_464_real__distribution_Omeasurable__finite__borel,axiom,
    ! [M: sigma_measure_real,F: real > real] :
      ( ( distri2809703520229113005bution @ M )
     => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
       => ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).

% real_distribution.measurable_finite_borel
thf(fact_465_real__distribution_Omeasurable__finite__borel,axiom,
    ! [M: sigma_measure_real,F: real > complex] :
      ( ( distri2809703520229113005bution @ M )
     => ( ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ borel_5078946678739801102l_real @ borel_1392132677378845456omplex ) )
       => ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ M @ borel_1392132677378845456omplex ) ) ) ) ).

% real_distribution.measurable_finite_borel
thf(fact_466_borel__measurable__enn2real,axiom,
    ! [F: a > extend8495563244428889912nnreal,M: sigma_measure_a] :
      ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( member_a_real
        @ ^ [X: a] : ( extend1669699412028896998n2real @ ( F @ X ) )
        @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_enn2real
thf(fact_467_borel__measurable__enn2real,axiom,
    ! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real] :
      ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( member_real_real
        @ ^ [X: real] : ( extend1669699412028896998n2real @ ( F @ X ) )
        @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_enn2real
thf(fact_468_qbs__prob__measure__prob__space_Oindep__var__compose,axiom,
    ! [S: probab8009751763329705409e_real,M1: sigma_measure_a,X12: real > a,M22: sigma_measure_a,X23: real > a,Y12: a > real,N1: sigma_measure_real,Y22: a > real,N2: sigma_measure_real] :
      ( ( indepe365082296117321348real_a @ ( probab4733579253584633066e_real @ S ) @ M1 @ X12 @ M22 @ X23 )
     => ( ( member_a_real @ Y12 @ ( sigma_9116425665531756122a_real @ M1 @ N1 ) )
       => ( ( member_a_real @ Y22 @ ( sigma_9116425665531756122a_real @ M22 @ N2 ) )
         => ( indepe3760321310464026790l_real @ ( probab4733579253584633066e_real @ S ) @ N1 @ ( comp_a_real_real @ Y12 @ X12 ) @ N2 @ ( comp_a_real_real @ Y22 @ X23 ) ) ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_compose
thf(fact_469_qbs__prob__measure__prob__space_Oindep__var__compose,axiom,
    ! [S: probab8009751763329705409e_real,M1: sigma_measure_real,X12: real > real,M22: sigma_measure_real,X23: real > real,Y12: real > a,N1: sigma_measure_a,Y22: real > a,N2: sigma_measure_a] :
      ( ( indepe3760321310464026790l_real @ ( probab4733579253584633066e_real @ S ) @ M1 @ X12 @ M22 @ X23 )
     => ( ( member_real_a @ Y12 @ ( sigma_523072396149930112real_a @ M1 @ N1 ) )
       => ( ( member_real_a @ Y22 @ ( sigma_523072396149930112real_a @ M22 @ N2 ) )
         => ( indepe365082296117321348real_a @ ( probab4733579253584633066e_real @ S ) @ N1 @ ( comp_real_a_real @ Y12 @ X12 ) @ N2 @ ( comp_real_a_real @ Y22 @ X23 ) ) ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_compose
thf(fact_470_qbs__prob__measure__prob__space_Oindep__var__compose,axiom,
    ! [S: probab8009751763329705409e_real,M1: sigma_measure_real,X12: real > real,M22: sigma_measure_real,X23: real > real,Y12: real > extend8495563244428889912nnreal,N1: sigma_7234349610311085201nnreal,Y22: real > extend8495563244428889912nnreal,N2: sigma_7234349610311085201nnreal] :
      ( ( indepe3760321310464026790l_real @ ( probab4733579253584633066e_real @ S ) @ M1 @ X12 @ M22 @ X23 )
     => ( ( member2919562650594848410nnreal @ Y12 @ ( sigma_9017504469962657078nnreal @ M1 @ N1 ) )
       => ( ( member2919562650594848410nnreal @ Y22 @ ( sigma_9017504469962657078nnreal @ M22 @ N2 ) )
         => ( indepe6767359503340752434nnreal @ ( probab4733579253584633066e_real @ S ) @ N1 @ ( comp_r6279034453215524981l_real @ Y12 @ X12 ) @ N2 @ ( comp_r6279034453215524981l_real @ Y22 @ X23 ) ) ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_compose
thf(fact_471_qbs__prob__measure__prob__space_Oindep__var__compose,axiom,
    ! [S: probab8009751763329705409e_real,M1: sigma_measure_real,X12: real > real,M22: sigma_measure_real,X23: real > real,Y12: real > complex,N1: sigma_3077487657436305159omplex,Y22: real > complex,N2: sigma_3077487657436305159omplex] :
      ( ( indepe3760321310464026790l_real @ ( probab4733579253584633066e_real @ S ) @ M1 @ X12 @ M22 @ X23 )
     => ( ( member_real_complex @ Y12 @ ( sigma_9111916201866572460omplex @ M1 @ N1 ) )
       => ( ( member_real_complex @ Y22 @ ( sigma_9111916201866572460omplex @ M22 @ N2 ) )
         => ( indepe1954327081502071720omplex @ ( probab4733579253584633066e_real @ S ) @ N1 @ ( comp_r1968866223832618731x_real @ Y12 @ X12 ) @ N2 @ ( comp_r1968866223832618731x_real @ Y22 @ X23 ) ) ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_compose
thf(fact_472_qbs__prob__measure__prob__space_Oindep__var__compose,axiom,
    ! [S: probab8009751763329705409e_real,M1: sigma_measure_real,X12: real > real,M22: sigma_measure_real,X23: real > real,Y12: real > real,N1: sigma_measure_real,Y22: real > real,N2: sigma_measure_real] :
      ( ( indepe3760321310464026790l_real @ ( probab4733579253584633066e_real @ S ) @ M1 @ X12 @ M22 @ X23 )
     => ( ( member_real_real @ Y12 @ ( sigma_5267869275261027754l_real @ M1 @ N1 ) )
       => ( ( member_real_real @ Y22 @ ( sigma_5267869275261027754l_real @ M22 @ N2 ) )
         => ( indepe3760321310464026790l_real @ ( probab4733579253584633066e_real @ S ) @ N1 @ ( comp_real_real_real @ Y12 @ X12 ) @ N2 @ ( comp_real_real_real @ Y22 @ X23 ) ) ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_compose
thf(fact_473_qbs__prob__measure__prob__space_Oindep__var__compose,axiom,
    ! [S: probab4737552673497767871pace_a,M1: sigma_measure_real,X12: a > real,M22: sigma_measure_real,X23: a > real,Y12: real > a,N1: sigma_measure_a,Y22: real > a,N2: sigma_measure_a] :
      ( ( indepe8958435565499147358a_real @ ( probab7100426894406488384sure_a @ S ) @ M1 @ X12 @ M22 @ X23 )
     => ( ( member_real_a @ Y12 @ ( sigma_523072396149930112real_a @ M1 @ N1 ) )
       => ( ( member_real_a @ Y22 @ ( sigma_523072396149930112real_a @ M22 @ N2 ) )
         => ( indepe2440653194691626188ar_a_a @ ( probab7100426894406488384sure_a @ S ) @ N1 @ ( comp_real_a_a @ Y12 @ X12 ) @ N2 @ ( comp_real_a_a @ Y22 @ X23 ) ) ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_compose
thf(fact_474_qbs__prob__measure__prob__space_Oindep__var__compose,axiom,
    ! [S: probab4737552673497767871pace_a,M1: sigma_measure_a,X12: a > a,M22: sigma_measure_a,X23: a > a,Y12: a > extend8495563244428889912nnreal,N1: sigma_7234349610311085201nnreal,Y22: a > extend8495563244428889912nnreal,N2: sigma_7234349610311085201nnreal] :
      ( ( indepe2440653194691626188ar_a_a @ ( probab7100426894406488384sure_a @ S ) @ M1 @ X12 @ M22 @ X23 )
     => ( ( member298456594901751504nnreal @ Y12 @ ( sigma_214952329563889126nnreal @ M1 @ N1 ) )
       => ( ( member298456594901751504nnreal @ Y22 @ ( sigma_214952329563889126nnreal @ M22 @ N2 ) )
         => ( indepe3534117692041274858nnreal @ ( probab7100426894406488384sure_a @ S ) @ N1 @ ( comp_a6042866249568583849real_a @ Y12 @ X12 ) @ N2 @ ( comp_a6042866249568583849real_a @ Y22 @ X23 ) ) ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_compose
thf(fact_475_qbs__prob__measure__prob__space_Oindep__var__compose,axiom,
    ! [S: probab4737552673497767871pace_a,M1: sigma_measure_a,X12: a > a,M22: sigma_measure_a,X23: a > a,Y12: a > real,N1: sigma_measure_real,Y22: a > real,N2: sigma_measure_real] :
      ( ( indepe2440653194691626188ar_a_a @ ( probab7100426894406488384sure_a @ S ) @ M1 @ X12 @ M22 @ X23 )
     => ( ( member_a_real @ Y12 @ ( sigma_9116425665531756122a_real @ M1 @ N1 ) )
       => ( ( member_a_real @ Y22 @ ( sigma_9116425665531756122a_real @ M22 @ N2 ) )
         => ( indepe8958435565499147358a_real @ ( probab7100426894406488384sure_a @ S ) @ N1 @ ( comp_a_real_a @ Y12 @ X12 ) @ N2 @ ( comp_a_real_a @ Y22 @ X23 ) ) ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_compose
thf(fact_476_qbs__prob__measure__prob__space_Oindep__var__compose,axiom,
    ! [S: probab4737552673497767871pace_a,M1: sigma_measure_real,X12: a > real,M22: sigma_measure_real,X23: a > real,Y12: real > real,N1: sigma_measure_real,Y22: real > real,N2: sigma_measure_real] :
      ( ( indepe8958435565499147358a_real @ ( probab7100426894406488384sure_a @ S ) @ M1 @ X12 @ M22 @ X23 )
     => ( ( member_real_real @ Y12 @ ( sigma_5267869275261027754l_real @ M1 @ N1 ) )
       => ( ( member_real_real @ Y22 @ ( sigma_5267869275261027754l_real @ M22 @ N2 ) )
         => ( indepe8958435565499147358a_real @ ( probab7100426894406488384sure_a @ S ) @ N1 @ ( comp_real_real_a @ Y12 @ X12 ) @ N2 @ ( comp_real_real_a @ Y22 @ X23 ) ) ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_compose
thf(fact_477_qbs__prob__measure__prob__space_Oindep__var__compose,axiom,
    ! [S: probab4737552673497767871pace_a,M1: sigma_measure_real,X12: a > real,M22: sigma_measure_real,X23: a > real,Y12: real > extend8495563244428889912nnreal,N1: sigma_7234349610311085201nnreal,Y22: real > extend8495563244428889912nnreal,N2: sigma_7234349610311085201nnreal] :
      ( ( indepe8958435565499147358a_real @ ( probab7100426894406488384sure_a @ S ) @ M1 @ X12 @ M22 @ X23 )
     => ( ( member2919562650594848410nnreal @ Y12 @ ( sigma_9017504469962657078nnreal @ M1 @ N1 ) )
       => ( ( member2919562650594848410nnreal @ Y22 @ ( sigma_9017504469962657078nnreal @ M22 @ N2 ) )
         => ( indepe3534117692041274858nnreal @ ( probab7100426894406488384sure_a @ S ) @ N1 @ ( comp_r7806941060661185781real_a @ Y12 @ X12 ) @ N2 @ ( comp_r7806941060661185781real_a @ Y22 @ X23 ) ) ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_compose
thf(fact_478_qbs__eqI,axiom,
    ! [X2: quasi_borel_a,Y: quasi_borel_a] :
      ( ( ( qbs_Mx_a @ X2 )
        = ( qbs_Mx_a @ Y ) )
     => ( X2 = Y ) ) ).

% qbs_eqI
thf(fact_479_qbs__morphism__ident_H,axiom,
    ! [X2: quasi_borel_real] :
      ( member_real_real
      @ ^ [X: real] : X
      @ ( qbs_mo5229770564518008146l_real @ X2 @ X2 ) ) ).

% qbs_morphism_ident'
thf(fact_480_qbs__morphism__comp,axiom,
    ! [F: real > complex,X2: quasi_borel_real,Y: quasi_borel_complex,G: complex > complex,Z2: quasi_borel_complex] :
      ( ( member_real_complex @ F @ ( qbs_mo6067097710682130004omplex @ X2 @ Y ) )
     => ( ( member5128974058612258834omplex @ G @ ( qbs_mo9200510921189519062omplex @ Y @ Z2 ) )
       => ( member_real_complex @ ( comp_c2117349707075585901x_real @ G @ F ) @ ( qbs_mo6067097710682130004omplex @ X2 @ Z2 ) ) ) ) ).

% qbs_morphism_comp
thf(fact_481_qbs__morphism__comp,axiom,
    ! [F: real > complex,X2: quasi_borel_real,Y: quasi_borel_complex,G: complex > real,Z2: quasi_borel_real] :
      ( ( member_real_complex @ F @ ( qbs_mo6067097710682130004omplex @ X2 @ Y ) )
     => ( ( member_complex_real @ G @ ( qbs_mo6120686211186450644x_real @ Y @ Z2 ) )
       => ( member_real_real @ ( comp_c3333796836230738283l_real @ G @ F ) @ ( qbs_mo5229770564518008146l_real @ X2 @ Z2 ) ) ) ) ).

% qbs_morphism_comp
thf(fact_482_qbs__morphism__comp,axiom,
    ! [F: real > complex,X2: quasi_borel_real,Y: quasi_borel_complex,G: complex > a,Z2: quasi_borel_a] :
      ( ( member_real_complex @ F @ ( qbs_mo6067097710682130004omplex @ X2 @ Y ) )
     => ( ( member_complex_a @ G @ ( qbs_mo6245657829219851990plex_a @ Y @ Z2 ) )
       => ( member_real_a @ ( comp_complex_a_real @ G @ F ) @ ( qbs_morphism_real_a @ X2 @ Z2 ) ) ) ) ).

% qbs_morphism_comp
thf(fact_483_qbs__morphism__comp,axiom,
    ! [F: real > real,X2: quasi_borel_real,Y: quasi_borel_real,G: real > complex,Z2: quasi_borel_complex] :
      ( ( member_real_real @ F @ ( qbs_mo5229770564518008146l_real @ X2 @ Y ) )
     => ( ( member_real_complex @ G @ ( qbs_mo6067097710682130004omplex @ Y @ Z2 ) )
       => ( member_real_complex @ ( comp_r1968866223832618731x_real @ G @ F ) @ ( qbs_mo6067097710682130004omplex @ X2 @ Z2 ) ) ) ) ).

% qbs_morphism_comp
thf(fact_484_qbs__morphism__comp,axiom,
    ! [F: real > real,X2: quasi_borel_real,Y: quasi_borel_real,G: real > real,Z2: quasi_borel_real] :
      ( ( member_real_real @ F @ ( qbs_mo5229770564518008146l_real @ X2 @ Y ) )
     => ( ( member_real_real @ G @ ( qbs_mo5229770564518008146l_real @ Y @ Z2 ) )
       => ( member_real_real @ ( comp_real_real_real @ G @ F ) @ ( qbs_mo5229770564518008146l_real @ X2 @ Z2 ) ) ) ) ).

% qbs_morphism_comp
thf(fact_485_qbs__morphism__comp,axiom,
    ! [F: real > real,X2: quasi_borel_real,Y: quasi_borel_real,G: real > a,Z2: quasi_borel_a] :
      ( ( member_real_real @ F @ ( qbs_mo5229770564518008146l_real @ X2 @ Y ) )
     => ( ( member_real_a @ G @ ( qbs_morphism_real_a @ Y @ Z2 ) )
       => ( member_real_a @ ( comp_real_a_real @ G @ F ) @ ( qbs_morphism_real_a @ X2 @ Z2 ) ) ) ) ).

% qbs_morphism_comp
thf(fact_486_qbs__morphism__comp,axiom,
    ! [F: real > a,X2: quasi_borel_real,Y: quasi_borel_a,G: a > complex,Z2: quasi_borel_complex] :
      ( ( member_real_a @ F @ ( qbs_morphism_real_a @ X2 @ Y ) )
     => ( ( member_a_complex @ G @ ( qbs_mo398503538871883188omplex @ Y @ Z2 ) )
       => ( member_real_complex @ ( comp_a_complex_real @ G @ F ) @ ( qbs_mo6067097710682130004omplex @ X2 @ Z2 ) ) ) ) ).

% qbs_morphism_comp
thf(fact_487_qbs__morphism__comp,axiom,
    ! [F: real > a,X2: quasi_borel_real,Y: quasi_borel_a,G: a > a,Z2: quasi_borel_a] :
      ( ( member_real_a @ F @ ( qbs_morphism_real_a @ X2 @ Y ) )
     => ( ( member_a_a @ G @ ( qbs_morphism_a_a @ Y @ Z2 ) )
       => ( member_real_a @ ( comp_a_a_real @ G @ F ) @ ( qbs_morphism_real_a @ X2 @ Z2 ) ) ) ) ).

% qbs_morphism_comp
thf(fact_488_qbs__morphism__comp,axiom,
    ! [F: real > a,X2: quasi_borel_real,Y: quasi_borel_a,G: a > extend8495563244428889912nnreal,Z2: quasi_9015997321629101608nnreal] :
      ( ( member_real_a @ F @ ( qbs_morphism_real_a @ X2 @ Y ) )
     => ( ( member298456594901751504nnreal @ G @ ( qbs_mo1434458643421888574nnreal @ Y @ Z2 ) )
       => ( member2919562650594848410nnreal @ ( comp_a8249376463644563905l_real @ G @ F ) @ ( qbs_mo1317719164804411614nnreal @ X2 @ Z2 ) ) ) ) ).

% qbs_morphism_comp
thf(fact_489_qbs__morphism__comp,axiom,
    ! [F: a > a,X2: quasi_borel_a,Y: quasi_borel_a,G: a > extend8495563244428889912nnreal,Z2: quasi_9015997321629101608nnreal] :
      ( ( member_a_a @ F @ ( qbs_morphism_a_a @ X2 @ Y ) )
     => ( ( member298456594901751504nnreal @ G @ ( qbs_mo1434458643421888574nnreal @ Y @ Z2 ) )
       => ( member298456594901751504nnreal @ ( comp_a6042866249568583849real_a @ G @ F ) @ ( qbs_mo1434458643421888574nnreal @ X2 @ Z2 ) ) ) ) ).

% qbs_morphism_comp
thf(fact_490_qbs__prob__eq4__dest_I4_J,axiom,
    ! [X2: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,Y: quasi_borel_real,Beta: real > real,Nu: sigma_measure_real,F: real > extend8495563244428889912nnreal] :
      ( ( probab39156407980996870284_real @ ( produc4368523205619139320e_real @ X2 @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) @ ( produc4368523205619139320e_real @ Y @ ( produc1722724976708544245e_real @ Beta @ Nu ) ) )
     => ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ ( measur1733462625046462224e_real @ X2 ) @ borel_6524799422816628122nnreal ) )
       => ( ( nonneg2667834350952324695l_real @ Mu
            @ ^ [X: real] : ( F @ ( Alpha @ X ) ) )
          = ( nonneg2667834350952324695l_real @ Nu
            @ ^ [X: real] : ( F @ ( Beta @ X ) ) ) ) ) ) ).

% qbs_prob_eq4_dest(4)
thf(fact_491_qbs__prob__eq4__dest_I4_J,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real,F: a > extend8495563244428889912nnreal] :
      ( ( probab7567053574026744118_eq4_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) )
     => ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ ( measur7857763439677503898sure_a @ X2 ) @ borel_6524799422816628122nnreal ) )
       => ( ( nonneg2667834350952324695l_real @ Mu
            @ ^ [X: real] : ( F @ ( Alpha @ X ) ) )
          = ( nonneg2667834350952324695l_real @ Nu
            @ ^ [X: real] : ( F @ ( Beta @ X ) ) ) ) ) ) ).

% qbs_prob_eq4_dest(4)
thf(fact_492_borel__measurable__const,axiom,
    ! [C: extend8495563244428889912nnreal,M: sigma_measure_a] :
      ( member298456594901751504nnreal
      @ ^ [X: a] : C
      @ ( sigma_214952329563889126nnreal @ M @ borel_6524799422816628122nnreal ) ) ).

% borel_measurable_const
thf(fact_493_borel__measurable__const,axiom,
    ! [C: extend8495563244428889912nnreal,M: sigma_measure_real] :
      ( member2919562650594848410nnreal
      @ ^ [X: real] : C
      @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) ) ).

% borel_measurable_const
thf(fact_494_borel__measurable__const,axiom,
    ! [C: real,M: sigma_measure_a] :
      ( member_a_real
      @ ^ [X: a] : C
      @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) ).

% borel_measurable_const
thf(fact_495_borel__measurable__const,axiom,
    ! [C: real,M: sigma_measure_real] :
      ( member_real_real
      @ ^ [X: real] : C
      @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ).

% borel_measurable_const
thf(fact_496_borel__measurable__const,axiom,
    ! [C: complex,M: sigma_measure_real] :
      ( member_real_complex
      @ ^ [X: real] : C
      @ ( sigma_9111916201866572460omplex @ M @ borel_1392132677378845456omplex ) ) ).

% borel_measurable_const
thf(fact_497_pair__qbs__prob_Oqbs__prob__space__eq4,axiom,
    ! [X2: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,Y: quasi_borel_real,Beta: real > real,Nu: sigma_measure_real] :
      ( ( probab6500215489368174228b_real @ X2 @ Alpha @ Mu @ Y @ Beta @ Nu )
     => ( ( Y = X2 )
       => ( ! [F3: real > extend8495563244428889912nnreal] :
              ( ( member2919562650594848410nnreal @ F3 @ ( sigma_9017504469962657078nnreal @ ( measur1733462625046462224e_real @ X2 ) @ borel_6524799422816628122nnreal ) )
             => ( ( nonneg2667834350952324695l_real @ Mu
                  @ ^ [X: real] : ( F3 @ ( Alpha @ X ) ) )
                = ( nonneg2667834350952324695l_real @ Nu
                  @ ^ [X: real] : ( F3 @ ( Beta @ X ) ) ) ) )
         => ( ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X2 @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) )
            = ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ Y @ ( produc1722724976708544245e_real @ Beta @ Nu ) ) ) ) ) ) ) ).

% pair_qbs_prob.qbs_prob_space_eq4
thf(fact_498_pair__qbs__prob_Oqbs__prob__space__eq4,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab5677843605262999830prob_a @ X2 @ Alpha @ Mu @ Y @ Beta @ Nu )
     => ( ( Y = X2 )
       => ( ! [F3: a > extend8495563244428889912nnreal] :
              ( ( member298456594901751504nnreal @ F3 @ ( sigma_214952329563889126nnreal @ ( measur7857763439677503898sure_a @ X2 ) @ borel_6524799422816628122nnreal ) )
             => ( ( nonneg2667834350952324695l_real @ Mu
                  @ ^ [X: real] : ( F3 @ ( Alpha @ X ) ) )
                = ( nonneg2667834350952324695l_real @ Nu
                  @ ^ [X: real] : ( F3 @ ( Beta @ X ) ) ) ) )
         => ( ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) )
            = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) ) ) ) ) ) ).

% pair_qbs_prob.qbs_prob_space_eq4
thf(fact_499_pair__qbs__prob_Oqbs__prob__eq4__intro,axiom,
    ! [X2: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,Y: quasi_borel_real,Beta: real > real,Nu: sigma_measure_real] :
      ( ( probab6500215489368174228b_real @ X2 @ Alpha @ Mu @ Y @ Beta @ Nu )
     => ( ( X2 = Y )
       => ( ! [F3: real > extend8495563244428889912nnreal] :
              ( ( member2919562650594848410nnreal @ F3 @ ( sigma_9017504469962657078nnreal @ ( measur1733462625046462224e_real @ X2 ) @ borel_6524799422816628122nnreal ) )
             => ( ( nonneg2667834350952324695l_real @ Mu
                  @ ^ [X: real] : ( F3 @ ( Alpha @ X ) ) )
                = ( nonneg2667834350952324695l_real @ Nu
                  @ ^ [X: real] : ( F3 @ ( Beta @ X ) ) ) ) )
         => ( probab39156407980996870284_real @ ( produc4368523205619139320e_real @ X2 @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) @ ( produc4368523205619139320e_real @ Y @ ( produc1722724976708544245e_real @ Beta @ Nu ) ) ) ) ) ) ).

% pair_qbs_prob.qbs_prob_eq4_intro
thf(fact_500_pair__qbs__prob_Oqbs__prob__eq4__intro,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab5677843605262999830prob_a @ X2 @ Alpha @ Mu @ Y @ Beta @ Nu )
     => ( ( X2 = Y )
       => ( ! [F3: a > extend8495563244428889912nnreal] :
              ( ( member298456594901751504nnreal @ F3 @ ( sigma_214952329563889126nnreal @ ( measur7857763439677503898sure_a @ X2 ) @ borel_6524799422816628122nnreal ) )
             => ( ( nonneg2667834350952324695l_real @ Mu
                  @ ^ [X: real] : ( F3 @ ( Alpha @ X ) ) )
                = ( nonneg2667834350952324695l_real @ Nu
                  @ ^ [X: real] : ( F3 @ ( Beta @ X ) ) ) ) )
         => ( probab7567053574026744118_eq4_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) ) ) ) ) ).

% pair_qbs_prob.qbs_prob_eq4_intro
thf(fact_501_qbs__Mx__R,axiom,
    ! [X2: sigma_measure_a] :
      ( ( qbs_Mx_a @ ( measur6507891955840068946_qbs_a @ X2 ) )
      = ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ X2 ) ) ).

% qbs_Mx_R
thf(fact_502_qbs__Mx__R,axiom,
    ! [X2: sigma_3077487657436305159omplex] :
      ( ( qbs_Mx_complex @ ( measur1074055046195851610omplex @ X2 ) )
      = ( sigma_9111916201866572460omplex @ borel_5078946678739801102l_real @ X2 ) ) ).

% qbs_Mx_R
thf(fact_503_qbs__Mx__R,axiom,
    ! [X2: sigma_7234349610311085201nnreal] :
      ( ( qbs_Mx6523938229262583809nnreal @ ( measur2642298986910087140nnreal @ X2 ) )
      = ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ X2 ) ) ).

% qbs_Mx_R
thf(fact_504_qbs__Mx__R,axiom,
    ! [X2: sigma_measure_real] :
      ( ( qbs_Mx_real @ ( measur6875533127466166616s_real @ X2 ) )
      = ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ X2 ) ) ).

% qbs_Mx_R
thf(fact_505_enn2real__0,axiom,
    ( ( extend1669699412028896998n2real @ zero_z7100319975126383169nnreal )
    = zero_zero_real ) ).

% enn2real_0
thf(fact_506_qp_Oindep__var__lebesgue__integral,axiom,
    ! [X12: real > real,X23: real > real] :
      ( ( indepe3760321310464026790l_real @ mu @ borel_5078946678739801102l_real @ X12 @ borel_5078946678739801102l_real @ X23 )
     => ( ( bochne3340023020068487468l_real @ mu @ X12 )
       => ( ( bochne3340023020068487468l_real @ mu @ X23 )
         => ( ( bochne3715101410578510557l_real @ mu
              @ ^ [Omega: real] : ( times_times_real @ ( X12 @ Omega ) @ ( X23 @ Omega ) ) )
            = ( times_times_real @ ( bochne3715101410578510557l_real @ mu @ X12 ) @ ( bochne3715101410578510557l_real @ mu @ X23 ) ) ) ) ) ) ).

% qp.indep_var_lebesgue_integral
thf(fact_507_qp_Oindep__var__lebesgue__integral,axiom,
    ! [X12: real > complex,X23: real > complex] :
      ( ( indepe1954327081502071720omplex @ mu @ borel_1392132677378845456omplex @ X12 @ borel_1392132677378845456omplex @ X23 )
     => ( ( bochne7032760885902134062omplex @ mu @ X12 )
       => ( ( bochne7032760885902134062omplex @ mu @ X23 )
         => ( ( bochne8865740171307459423omplex @ mu
              @ ^ [Omega: real] : ( times_times_complex @ ( X12 @ Omega ) @ ( X23 @ Omega ) ) )
            = ( times_times_complex @ ( bochne8865740171307459423omplex @ mu @ X12 ) @ ( bochne8865740171307459423omplex @ mu @ X23 ) ) ) ) ) ) ).

% qp.indep_var_lebesgue_integral
thf(fact_508_ennreal__0,axiom,
    ( ( extend7643940197134561352nnreal @ zero_zero_real )
    = zero_z7100319975126383169nnreal ) ).

% ennreal_0
thf(fact_509_qp_Oindep__var__integrable,axiom,
    ! [X12: real > real,X23: real > real] :
      ( ( indepe3760321310464026790l_real @ mu @ borel_5078946678739801102l_real @ X12 @ borel_5078946678739801102l_real @ X23 )
     => ( ( bochne3340023020068487468l_real @ mu @ X12 )
       => ( ( bochne3340023020068487468l_real @ mu @ X23 )
         => ( bochne3340023020068487468l_real @ mu
            @ ^ [Omega: real] : ( times_times_real @ ( X12 @ Omega ) @ ( X23 @ Omega ) ) ) ) ) ) ).

% qp.indep_var_integrable
thf(fact_510_qp_Oindep__var__integrable,axiom,
    ! [X12: real > complex,X23: real > complex] :
      ( ( indepe1954327081502071720omplex @ mu @ borel_1392132677378845456omplex @ X12 @ borel_1392132677378845456omplex @ X23 )
     => ( ( bochne7032760885902134062omplex @ mu @ X12 )
       => ( ( bochne7032760885902134062omplex @ mu @ X23 )
         => ( bochne7032760885902134062omplex @ mu
            @ ^ [Omega: real] : ( times_times_complex @ ( X12 @ Omega ) @ ( X23 @ Omega ) ) ) ) ) ) ).

% qp.indep_var_integrable
thf(fact_511_prob__space_Oindep__var__compose,axiom,
    ! [M: sigma_measure_real,M1: sigma_measure_a,X12: real > a,M22: sigma_measure_a,X23: real > a,Y12: a > extend8495563244428889912nnreal,N1: sigma_7234349610311085201nnreal,Y22: a > extend8495563244428889912nnreal,N2: sigma_7234349610311085201nnreal] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe365082296117321348real_a @ M @ M1 @ X12 @ M22 @ X23 )
       => ( ( member298456594901751504nnreal @ Y12 @ ( sigma_214952329563889126nnreal @ M1 @ N1 ) )
         => ( ( member298456594901751504nnreal @ Y22 @ ( sigma_214952329563889126nnreal @ M22 @ N2 ) )
           => ( indepe6767359503340752434nnreal @ M @ N1 @ ( comp_a8249376463644563905l_real @ Y12 @ X12 ) @ N2 @ ( comp_a8249376463644563905l_real @ Y22 @ X23 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_512_prob__space_Oindep__var__compose,axiom,
    ! [M: sigma_measure_real,M1: sigma_measure_a,X12: real > a,M22: sigma_measure_a,X23: real > a,Y12: a > real,N1: sigma_measure_real,Y22: a > real,N2: sigma_measure_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe365082296117321348real_a @ M @ M1 @ X12 @ M22 @ X23 )
       => ( ( member_a_real @ Y12 @ ( sigma_9116425665531756122a_real @ M1 @ N1 ) )
         => ( ( member_a_real @ Y22 @ ( sigma_9116425665531756122a_real @ M22 @ N2 ) )
           => ( indepe3760321310464026790l_real @ M @ N1 @ ( comp_a_real_real @ Y12 @ X12 ) @ N2 @ ( comp_a_real_real @ Y22 @ X23 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_513_prob__space_Oindep__var__compose,axiom,
    ! [M: sigma_measure_real,M1: sigma_measure_real,X12: real > real,M22: sigma_measure_real,X23: real > real,Y12: real > a,N1: sigma_measure_a,Y22: real > a,N2: sigma_measure_a] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe3760321310464026790l_real @ M @ M1 @ X12 @ M22 @ X23 )
       => ( ( member_real_a @ Y12 @ ( sigma_523072396149930112real_a @ M1 @ N1 ) )
         => ( ( member_real_a @ Y22 @ ( sigma_523072396149930112real_a @ M22 @ N2 ) )
           => ( indepe365082296117321348real_a @ M @ N1 @ ( comp_real_a_real @ Y12 @ X12 ) @ N2 @ ( comp_real_a_real @ Y22 @ X23 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_514_prob__space_Oindep__var__compose,axiom,
    ! [M: sigma_measure_real,M1: sigma_measure_real,X12: real > real,M22: sigma_measure_real,X23: real > real,Y12: real > extend8495563244428889912nnreal,N1: sigma_7234349610311085201nnreal,Y22: real > extend8495563244428889912nnreal,N2: sigma_7234349610311085201nnreal] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe3760321310464026790l_real @ M @ M1 @ X12 @ M22 @ X23 )
       => ( ( member2919562650594848410nnreal @ Y12 @ ( sigma_9017504469962657078nnreal @ M1 @ N1 ) )
         => ( ( member2919562650594848410nnreal @ Y22 @ ( sigma_9017504469962657078nnreal @ M22 @ N2 ) )
           => ( indepe6767359503340752434nnreal @ M @ N1 @ ( comp_r6279034453215524981l_real @ Y12 @ X12 ) @ N2 @ ( comp_r6279034453215524981l_real @ Y22 @ X23 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_515_prob__space_Oindep__var__compose,axiom,
    ! [M: sigma_measure_real,M1: sigma_measure_real,X12: real > real,M22: sigma_measure_real,X23: real > real,Y12: real > complex,N1: sigma_3077487657436305159omplex,Y22: real > complex,N2: sigma_3077487657436305159omplex] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe3760321310464026790l_real @ M @ M1 @ X12 @ M22 @ X23 )
       => ( ( member_real_complex @ Y12 @ ( sigma_9111916201866572460omplex @ M1 @ N1 ) )
         => ( ( member_real_complex @ Y22 @ ( sigma_9111916201866572460omplex @ M22 @ N2 ) )
           => ( indepe1954327081502071720omplex @ M @ N1 @ ( comp_r1968866223832618731x_real @ Y12 @ X12 ) @ N2 @ ( comp_r1968866223832618731x_real @ Y22 @ X23 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_516_prob__space_Oindep__var__compose,axiom,
    ! [M: sigma_measure_real,M1: sigma_measure_real,X12: real > real,M22: sigma_measure_real,X23: real > real,Y12: real > real,N1: sigma_measure_real,Y22: real > real,N2: sigma_measure_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe3760321310464026790l_real @ M @ M1 @ X12 @ M22 @ X23 )
       => ( ( member_real_real @ Y12 @ ( sigma_5267869275261027754l_real @ M1 @ N1 ) )
         => ( ( member_real_real @ Y22 @ ( sigma_5267869275261027754l_real @ M22 @ N2 ) )
           => ( indepe3760321310464026790l_real @ M @ N1 @ ( comp_real_real_real @ Y12 @ X12 ) @ N2 @ ( comp_real_real_real @ Y22 @ X23 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_517_qp_Oindep__var__distribution__eq,axiom,
    ! [S2: sigma_measure_a,X2: real > a,T: sigma_measure_a,Y: real > a] :
      ( ( indepe365082296117321348real_a @ mu @ S2 @ X2 @ T @ Y )
      = ( ( member_real_a @ X2 @ ( sigma_523072396149930112real_a @ mu @ S2 ) )
        & ( member_real_a @ Y @ ( sigma_523072396149930112real_a @ mu @ T ) )
        & ( ( binary867438762418767560re_a_a @ ( measure_distr_real_a @ mu @ S2 @ X2 ) @ ( measure_distr_real_a @ mu @ T @ Y ) )
          = ( measur2513335786126797313od_a_a @ mu @ ( binary867438762418767560re_a_a @ S2 @ T )
            @ ^ [X: real] : ( product_Pair_a_a @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ) ).

% qp.indep_var_distribution_eq
thf(fact_518_qp_Oindep__var__distribution__eq,axiom,
    ! [S2: sigma_7234349610311085201nnreal,X2: real > extend8495563244428889912nnreal,T: sigma_7234349610311085201nnreal,Y: real > extend8495563244428889912nnreal] :
      ( ( indepe6767359503340752434nnreal @ mu @ S2 @ X2 @ T @ Y )
      = ( ( member2919562650594848410nnreal @ X2 @ ( sigma_9017504469962657078nnreal @ mu @ S2 ) )
        & ( member2919562650594848410nnreal @ Y @ ( sigma_9017504469962657078nnreal @ mu @ T ) )
        & ( ( binary3098606844978005306nnreal @ ( measur8829990298702910942nnreal @ mu @ S2 @ X2 ) @ ( measur8829990298702910942nnreal @ mu @ T @ Y ) )
          = ( measur4012415197360569771nnreal @ mu @ ( binary3098606844978005306nnreal @ S2 @ T )
            @ ^ [X: real] : ( produc344325839068023049nnreal @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ) ).

% qp.indep_var_distribution_eq
thf(fact_519_qp_Oindep__var__distribution__eq,axiom,
    ! [S2: sigma_3077487657436305159omplex,X2: real > complex,T: sigma_3077487657436305159omplex,Y: real > complex] :
      ( ( indepe1954327081502071720omplex @ mu @ S2 @ X2 @ T @ Y )
      = ( ( member_real_complex @ X2 @ ( sigma_9111916201866572460omplex @ mu @ S2 ) )
        & ( member_real_complex @ Y @ ( sigma_9111916201866572460omplex @ mu @ T ) )
        & ( ( binary5145385660880348710omplex @ ( measur1621797640479583060omplex @ mu @ S2 @ X2 ) @ ( measur1621797640479583060omplex @ mu @ T @ Y ) )
          = ( measur4452220837507949463omplex @ mu @ ( binary5145385660880348710omplex @ S2 @ T )
            @ ^ [X: real] : ( produc101793102246108661omplex @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ) ).

% qp.indep_var_distribution_eq
thf(fact_520_qp_Oindep__var__distribution__eq,axiom,
    ! [S2: sigma_measure_real,X2: real > real,T: sigma_measure_real,Y: real > real] :
      ( ( indepe3760321310464026790l_real @ mu @ S2 @ X2 @ T @ Y )
      = ( ( member_real_real @ X2 @ ( sigma_5267869275261027754l_real @ mu @ S2 ) )
        & ( member_real_real @ Y @ ( sigma_5267869275261027754l_real @ mu @ T ) )
        & ( ( binary6478037234023840930l_real @ ( measur2993149975067245138l_real @ mu @ S2 @ X2 ) @ ( measur2993149975067245138l_real @ mu @ T @ Y ) )
          = ( measur6481026558495277843l_real @ mu @ ( binary6478037234023840930l_real @ S2 @ T )
            @ ^ [X: real] : ( produc4511245868158468465l_real @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ) ).

% qp.indep_var_distribution_eq
thf(fact_521_qp_Onn__integral__fst,axiom,
    ! [F: produc2422161461964618553l_real > extend8495563244428889912nnreal,M1: sigma_measure_real] :
      ( ( member2245694452317284363nnreal @ F @ ( sigma_219902274609641377nnreal @ ( binary6478037234023840930l_real @ M1 @ mu ) @ borel_6524799422816628122nnreal ) )
     => ( ( nonneg2667834350952324695l_real @ M1
          @ ^ [X: real] :
              ( nonneg2667834350952324695l_real @ mu
              @ ^ [Y7: real] : ( F @ ( produc4511245868158468465l_real @ X @ Y7 ) ) ) )
        = ( nonneg1896927508495185742l_real @ ( binary6478037234023840930l_real @ M1 @ mu ) @ F ) ) ) ).

% qp.nn_integral_fst
thf(fact_522_qp_Onn__integral__fst,axiom,
    ! [F: product_prod_a_real > extend8495563244428889912nnreal,M1: sigma_measure_a] :
      ( ( member4437793228276457543nnreal @ F @ ( sigma_8117956432444187619nnreal @ ( binary932748531126180194a_real @ M1 @ mu ) @ borel_6524799422816628122nnreal ) )
     => ( ( nonneg2725512125972007571gral_a @ M1
          @ ^ [X: a] :
              ( nonneg2667834350952324695l_real @ mu
              @ ^ [Y7: real] : ( F @ ( product_Pair_a_real @ X @ Y7 ) ) ) )
        = ( nonneg4050876233904260868a_real @ ( binary932748531126180194a_real @ M1 @ mu ) @ F ) ) ) ).

% qp.nn_integral_fst
thf(fact_523_qp_Oborel__measurable__nn__integral__fst,axiom,
    ! [F: product_prod_a_real > extend8495563244428889912nnreal,M1: sigma_measure_a] :
      ( ( member4437793228276457543nnreal @ F @ ( sigma_8117956432444187619nnreal @ ( binary932748531126180194a_real @ M1 @ mu ) @ borel_6524799422816628122nnreal ) )
     => ( member298456594901751504nnreal
        @ ^ [X: a] :
            ( nonneg2667834350952324695l_real @ mu
            @ ^ [Y7: real] : ( F @ ( product_Pair_a_real @ X @ Y7 ) ) )
        @ ( sigma_214952329563889126nnreal @ M1 @ borel_6524799422816628122nnreal ) ) ) ).

% qp.borel_measurable_nn_integral_fst
thf(fact_524_qp_Oborel__measurable__nn__integral__fst,axiom,
    ! [F: produc2422161461964618553l_real > extend8495563244428889912nnreal,M1: sigma_measure_real] :
      ( ( member2245694452317284363nnreal @ F @ ( sigma_219902274609641377nnreal @ ( binary6478037234023840930l_real @ M1 @ mu ) @ borel_6524799422816628122nnreal ) )
     => ( member2919562650594848410nnreal
        @ ^ [X: real] :
            ( nonneg2667834350952324695l_real @ mu
            @ ^ [Y7: real] : ( F @ ( produc4511245868158468465l_real @ X @ Y7 ) ) )
        @ ( sigma_9017504469962657078nnreal @ M1 @ borel_6524799422816628122nnreal ) ) ) ).

% qp.borel_measurable_nn_integral_fst
thf(fact_525_integral__mult__left__zero,axiom,
    ! [M: sigma_measure_real,F: real > real,C: real] :
      ( ( bochne3715101410578510557l_real @ M
        @ ^ [X: real] : ( times_times_real @ ( F @ X ) @ C ) )
      = ( times_times_real @ ( bochne3715101410578510557l_real @ M @ F ) @ C ) ) ).

% integral_mult_left_zero
thf(fact_526_integral__mult__left__zero,axiom,
    ! [M: sigma_measure_a,F: a > real,C: real] :
      ( ( bochne378719280626478695a_real @ M
        @ ^ [X: a] : ( times_times_real @ ( F @ X ) @ C ) )
      = ( times_times_real @ ( bochne378719280626478695a_real @ M @ F ) @ C ) ) ).

% integral_mult_left_zero
thf(fact_527_integral__mult__right__zero,axiom,
    ! [M: sigma_measure_real,C: real,F: real > real] :
      ( ( bochne3715101410578510557l_real @ M
        @ ^ [X: real] : ( times_times_real @ C @ ( F @ X ) ) )
      = ( times_times_real @ C @ ( bochne3715101410578510557l_real @ M @ F ) ) ) ).

% integral_mult_right_zero
thf(fact_528_integral__mult__right__zero,axiom,
    ! [M: sigma_measure_a,C: real,F: a > real] :
      ( ( bochne378719280626478695a_real @ M
        @ ^ [X: a] : ( times_times_real @ C @ ( F @ X ) ) )
      = ( times_times_real @ C @ ( bochne378719280626478695a_real @ M @ F ) ) ) ).

% integral_mult_right_zero
thf(fact_529_integrable__mult__left__iff,axiom,
    ! [M: sigma_measure_real,C: real,F: real > real] :
      ( ( bochne3340023020068487468l_real @ M
        @ ^ [X: real] : ( times_times_real @ C @ ( F @ X ) ) )
      = ( ( C = zero_zero_real )
        | ( bochne3340023020068487468l_real @ M @ F ) ) ) ).

% integrable_mult_left_iff
thf(fact_530_integrable__mult__left__iff,axiom,
    ! [M: sigma_measure_a,C: real,F: a > real] :
      ( ( bochne2139062162225249880a_real @ M
        @ ^ [X: a] : ( times_times_real @ C @ ( F @ X ) ) )
      = ( ( C = zero_zero_real )
        | ( bochne2139062162225249880a_real @ M @ F ) ) ) ).

% integrable_mult_left_iff
thf(fact_531_integrable__mult__right__iff,axiom,
    ! [M: sigma_measure_real,F: real > real,C: real] :
      ( ( bochne3340023020068487468l_real @ M
        @ ^ [X: real] : ( times_times_real @ ( F @ X ) @ C ) )
      = ( ( C = zero_zero_real )
        | ( bochne3340023020068487468l_real @ M @ F ) ) ) ).

% integrable_mult_right_iff
thf(fact_532_integrable__mult__right__iff,axiom,
    ! [M: sigma_measure_a,F: a > real,C: real] :
      ( ( bochne2139062162225249880a_real @ M
        @ ^ [X: a] : ( times_times_real @ ( F @ X ) @ C ) )
      = ( ( C = zero_zero_real )
        | ( bochne2139062162225249880a_real @ M @ F ) ) ) ).

% integrable_mult_right_iff
thf(fact_533_qbs__integrable__mult__iff,axiom,
    ! [S: probab4737552673497767871pace_a,C: real,F: a > real] :
      ( ( probab7312716125271441302able_a @ S
        @ ^ [X: a] : ( times_times_real @ C @ ( F @ X ) ) )
      = ( ( C = zero_zero_real )
        | ( probab7312716125271441302able_a @ S @ F ) ) ) ).

% qbs_integrable_mult_iff
thf(fact_534_integrable__mult__left,axiom,
    ! [C: real,M: sigma_measure_real,F: real > real] :
      ( ( ( C != zero_zero_real )
       => ( bochne3340023020068487468l_real @ M @ F ) )
     => ( bochne3340023020068487468l_real @ M
        @ ^ [X: real] : ( times_times_real @ ( F @ X ) @ C ) ) ) ).

% integrable_mult_left
thf(fact_535_integrable__mult__left,axiom,
    ! [C: real,M: sigma_measure_a,F: a > real] :
      ( ( ( C != zero_zero_real )
       => ( bochne2139062162225249880a_real @ M @ F ) )
     => ( bochne2139062162225249880a_real @ M
        @ ^ [X: a] : ( times_times_real @ ( F @ X ) @ C ) ) ) ).

% integrable_mult_left
thf(fact_536_integrable__mult__right,axiom,
    ! [C: real,M: sigma_measure_real,F: real > real] :
      ( ( ( C != zero_zero_real )
       => ( bochne3340023020068487468l_real @ M @ F ) )
     => ( bochne3340023020068487468l_real @ M
        @ ^ [X: real] : ( times_times_real @ C @ ( F @ X ) ) ) ) ).

% integrable_mult_right
thf(fact_537_integrable__mult__right,axiom,
    ! [C: real,M: sigma_measure_a,F: a > real] :
      ( ( ( C != zero_zero_real )
       => ( bochne2139062162225249880a_real @ M @ F ) )
     => ( bochne2139062162225249880a_real @ M
        @ ^ [X: a] : ( times_times_real @ C @ ( F @ X ) ) ) ) ).

% integrable_mult_right
thf(fact_538_Bochner__Integration_Ointegral__mult__left,axiom,
    ! [C: real,M: sigma_measure_real,F: real > real] :
      ( ( ( C != zero_zero_real )
       => ( bochne3340023020068487468l_real @ M @ F ) )
     => ( ( bochne3715101410578510557l_real @ M
          @ ^ [X: real] : ( times_times_real @ ( F @ X ) @ C ) )
        = ( times_times_real @ ( bochne3715101410578510557l_real @ M @ F ) @ C ) ) ) ).

% Bochner_Integration.integral_mult_left
thf(fact_539_Bochner__Integration_Ointegral__mult__left,axiom,
    ! [C: real,M: sigma_measure_a,F: a > real] :
      ( ( ( C != zero_zero_real )
       => ( bochne2139062162225249880a_real @ M @ F ) )
     => ( ( bochne378719280626478695a_real @ M
          @ ^ [X: a] : ( times_times_real @ ( F @ X ) @ C ) )
        = ( times_times_real @ ( bochne378719280626478695a_real @ M @ F ) @ C ) ) ) ).

% Bochner_Integration.integral_mult_left
thf(fact_540_Bochner__Integration_Ointegral__mult__right,axiom,
    ! [C: real,M: sigma_measure_real,F: real > real] :
      ( ( ( C != zero_zero_real )
       => ( bochne3340023020068487468l_real @ M @ F ) )
     => ( ( bochne3715101410578510557l_real @ M
          @ ^ [X: real] : ( times_times_real @ C @ ( F @ X ) ) )
        = ( times_times_real @ C @ ( bochne3715101410578510557l_real @ M @ F ) ) ) ) ).

% Bochner_Integration.integral_mult_right
thf(fact_541_Bochner__Integration_Ointegral__mult__right,axiom,
    ! [C: real,M: sigma_measure_a,F: a > real] :
      ( ( ( C != zero_zero_real )
       => ( bochne2139062162225249880a_real @ M @ F ) )
     => ( ( bochne378719280626478695a_real @ M
          @ ^ [X: a] : ( times_times_real @ C @ ( F @ X ) ) )
        = ( times_times_real @ C @ ( bochne378719280626478695a_real @ M @ F ) ) ) ) ).

% Bochner_Integration.integral_mult_right
thf(fact_542_enn2real__mult,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( extend1669699412028896998n2real @ ( times_1893300245718287421nnreal @ A @ B ) )
      = ( times_times_real @ ( extend1669699412028896998n2real @ A ) @ ( extend1669699412028896998n2real @ B ) ) ) ).

% enn2real_mult
thf(fact_543_mult_Oleft__commute,axiom,
    ! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( times_1893300245718287421nnreal @ B @ ( times_1893300245718287421nnreal @ A @ C ) )
      = ( times_1893300245718287421nnreal @ A @ ( times_1893300245718287421nnreal @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_544_mult_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] :
      ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_545_mult_Oleft__commute,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( times_times_complex @ B @ ( times_times_complex @ A @ C ) )
      = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_546_mult_Ocommute,axiom,
    ( times_1893300245718287421nnreal
    = ( ^ [A5: extend8495563244428889912nnreal,B4: extend8495563244428889912nnreal] : ( times_1893300245718287421nnreal @ B4 @ A5 ) ) ) ).

% mult.commute
thf(fact_547_mult_Ocommute,axiom,
    ( times_times_real
    = ( ^ [A5: real,B4: real] : ( times_times_real @ B4 @ A5 ) ) ) ).

% mult.commute
thf(fact_548_mult_Ocommute,axiom,
    ( times_times_complex
    = ( ^ [A5: complex,B4: complex] : ( times_times_complex @ B4 @ A5 ) ) ) ).

% mult.commute
thf(fact_549_mult_Oassoc,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( times_1893300245718287421nnreal @ ( times_1893300245718287421nnreal @ A @ B ) @ C )
      = ( times_1893300245718287421nnreal @ A @ ( times_1893300245718287421nnreal @ B @ C ) ) ) ).

% mult.assoc
thf(fact_550_mult_Oassoc,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% mult.assoc
thf(fact_551_mult_Oassoc,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
      = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).

% mult.assoc
thf(fact_552_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( times_1893300245718287421nnreal @ ( times_1893300245718287421nnreal @ A @ B ) @ C )
      = ( times_1893300245718287421nnreal @ A @ ( times_1893300245718287421nnreal @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_553_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_554_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
      = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_555_qbs__integrable__mult,axiom,
    ! [S: probab4737552673497767871pace_a,F: a > real,C: real] :
      ( ( probab7312716125271441302able_a @ S @ F )
     => ( probab7312716125271441302able_a @ S
        @ ^ [X: a] : ( times_times_real @ C @ ( F @ X ) ) ) ) ).

% qbs_integrable_mult
thf(fact_556_qbs__prob__integral__cmult,axiom,
    ! [S: probab4737552673497767871pace_a,C: real,F: a > real] :
      ( ( probab2419480525258570000gral_a @ S
        @ ^ [X: a] : ( times_times_real @ C @ ( F @ X ) ) )
      = ( times_times_real @ C @ ( probab2419480525258570000gral_a @ S @ F ) ) ) ).

% qbs_prob_integral_cmult
thf(fact_557_borel__measurable__times,axiom,
    ! [F: a > real,M: sigma_measure_a,G: a > real] :
      ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_a_real
          @ ^ [X: a] : ( times_times_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_times
thf(fact_558_borel__measurable__times,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_real_real
          @ ^ [X: real] : ( times_times_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_times
thf(fact_559_borel__measurable__times,axiom,
    ! [F: real > complex,M: sigma_measure_real,G: real > complex] :
      ( ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ M @ borel_1392132677378845456omplex ) )
     => ( ( member_real_complex @ G @ ( sigma_9111916201866572460omplex @ M @ borel_1392132677378845456omplex ) )
       => ( member_real_complex
          @ ^ [X: real] : ( times_times_complex @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_9111916201866572460omplex @ M @ borel_1392132677378845456omplex ) ) ) ) ).

% borel_measurable_times
thf(fact_560_borel__measurable__times__ennreal,axiom,
    ! [F: a > extend8495563244428889912nnreal,M: sigma_measure_a,G: a > extend8495563244428889912nnreal] :
      ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( ( member298456594901751504nnreal @ G @ ( sigma_214952329563889126nnreal @ M @ borel_6524799422816628122nnreal ) )
       => ( member298456594901751504nnreal
          @ ^ [X: a] : ( times_1893300245718287421nnreal @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_214952329563889126nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ) ).

% borel_measurable_times_ennreal
thf(fact_561_borel__measurable__times__ennreal,axiom,
    ! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,G: real > extend8495563244428889912nnreal] :
      ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) )
       => ( member2919562650594848410nnreal
          @ ^ [X: real] : ( times_1893300245718287421nnreal @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ) ).

% borel_measurable_times_ennreal
thf(fact_562_qbs__prob__ennintegral__cmult__noninfty,axiom,
    ! [C: extend8495563244428889912nnreal,S: probab4737552673497767871pace_a,F: a > extend8495563244428889912nnreal] :
      ( ( C != extend2057119558705770725nnreal )
     => ( ( probab3721531081081959085gral_a @ S
          @ ^ [X: a] : ( times_1893300245718287421nnreal @ C @ ( F @ X ) ) )
        = ( times_1893300245718287421nnreal @ C @ ( probab3721531081081959085gral_a @ S @ F ) ) ) ) ).

% qbs_prob_ennintegral_cmult_noninfty
thf(fact_563_ennreal__cong,axiom,
    ! [X4: real,Y3: real] :
      ( ( X4 = Y3 )
     => ( ( extend7643940197134561352nnreal @ X4 )
        = ( extend7643940197134561352nnreal @ Y3 ) ) ) ).

% ennreal_cong
thf(fact_564_prob__space_Oindep__var__integrable,axiom,
    ! [M: sigma_measure_a,X12: a > real,X23: a > real] :
      ( ( probab7247484486040049089pace_a @ M )
     => ( ( indepe8958435565499147358a_real @ M @ borel_5078946678739801102l_real @ X12 @ borel_5078946678739801102l_real @ X23 )
       => ( ( bochne2139062162225249880a_real @ M @ X12 )
         => ( ( bochne2139062162225249880a_real @ M @ X23 )
           => ( bochne2139062162225249880a_real @ M
              @ ^ [Omega: a] : ( times_times_real @ ( X12 @ Omega ) @ ( X23 @ Omega ) ) ) ) ) ) ) ).

% prob_space.indep_var_integrable
thf(fact_565_prob__space_Oindep__var__integrable,axiom,
    ! [M: sigma_measure_real,X12: real > real,X23: real > real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe3760321310464026790l_real @ M @ borel_5078946678739801102l_real @ X12 @ borel_5078946678739801102l_real @ X23 )
       => ( ( bochne3340023020068487468l_real @ M @ X12 )
         => ( ( bochne3340023020068487468l_real @ M @ X23 )
           => ( bochne3340023020068487468l_real @ M
              @ ^ [Omega: real] : ( times_times_real @ ( X12 @ Omega ) @ ( X23 @ Omega ) ) ) ) ) ) ) ).

% prob_space.indep_var_integrable
thf(fact_566_prob__space_Oindep__var__integrable,axiom,
    ! [M: sigma_measure_real,X12: real > complex,X23: real > complex] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe1954327081502071720omplex @ M @ borel_1392132677378845456omplex @ X12 @ borel_1392132677378845456omplex @ X23 )
       => ( ( bochne7032760885902134062omplex @ M @ X12 )
         => ( ( bochne7032760885902134062omplex @ M @ X23 )
           => ( bochne7032760885902134062omplex @ M
              @ ^ [Omega: real] : ( times_times_complex @ ( X12 @ Omega ) @ ( X23 @ Omega ) ) ) ) ) ) ) ).

% prob_space.indep_var_integrable
thf(fact_567_nn__integral__cmult,axiom,
    ! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,C: extend8495563244428889912nnreal] :
      ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( ( nonneg2667834350952324695l_real @ M
          @ ^ [X: real] : ( times_1893300245718287421nnreal @ C @ ( F @ X ) ) )
        = ( times_1893300245718287421nnreal @ C @ ( nonneg2667834350952324695l_real @ M @ F ) ) ) ) ).

% nn_integral_cmult
thf(fact_568_nn__integral__cmult,axiom,
    ! [F: a > extend8495563244428889912nnreal,M: sigma_measure_a,C: extend8495563244428889912nnreal] :
      ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( ( nonneg2725512125972007571gral_a @ M
          @ ^ [X: a] : ( times_1893300245718287421nnreal @ C @ ( F @ X ) ) )
        = ( times_1893300245718287421nnreal @ C @ ( nonneg2725512125972007571gral_a @ M @ F ) ) ) ) ).

% nn_integral_cmult
thf(fact_569_nn__integral__multc,axiom,
    ! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,C: extend8495563244428889912nnreal] :
      ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( ( nonneg2667834350952324695l_real @ M
          @ ^ [X: real] : ( times_1893300245718287421nnreal @ ( F @ X ) @ C ) )
        = ( times_1893300245718287421nnreal @ ( nonneg2667834350952324695l_real @ M @ F ) @ C ) ) ) ).

% nn_integral_multc
thf(fact_570_nn__integral__multc,axiom,
    ! [F: a > extend8495563244428889912nnreal,M: sigma_measure_a,C: extend8495563244428889912nnreal] :
      ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( ( nonneg2725512125972007571gral_a @ M
          @ ^ [X: a] : ( times_1893300245718287421nnreal @ ( F @ X ) @ C ) )
        = ( times_1893300245718287421nnreal @ ( nonneg2725512125972007571gral_a @ M @ F ) @ C ) ) ) ).

% nn_integral_multc
thf(fact_571_prob__space_Oindep__var__distribution__eq,axiom,
    ! [M: sigma_measure_a,S2: sigma_7234349610311085201nnreal,X2: a > extend8495563244428889912nnreal,T: sigma_7234349610311085201nnreal,Y: a > extend8495563244428889912nnreal] :
      ( ( probab7247484486040049089pace_a @ M )
     => ( ( indepe3534117692041274858nnreal @ M @ S2 @ X2 @ T @ Y )
        = ( ( member298456594901751504nnreal @ X2 @ ( sigma_214952329563889126nnreal @ M @ S2 ) )
          & ( member298456594901751504nnreal @ Y @ ( sigma_214952329563889126nnreal @ M @ T ) )
          & ( ( binary3098606844978005306nnreal @ ( measur4839436603801885502nnreal @ M @ S2 @ X2 ) @ ( measur4839436603801885502nnreal @ M @ T @ Y ) )
            = ( measur6341207572317192267nnreal @ M @ ( binary3098606844978005306nnreal @ S2 @ T )
              @ ^ [X: a] : ( produc344325839068023049nnreal @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ) ) ).

% prob_space.indep_var_distribution_eq
thf(fact_572_prob__space_Oindep__var__distribution__eq,axiom,
    ! [M: sigma_measure_a,S2: sigma_measure_real,X2: a > real,T: sigma_measure_real,Y: a > real] :
      ( ( probab7247484486040049089pace_a @ M )
     => ( ( indepe8958435565499147358a_real @ M @ S2 @ X2 @ T @ Y )
        = ( ( member_a_real @ X2 @ ( sigma_9116425665531756122a_real @ M @ S2 ) )
          & ( member_a_real @ Y @ ( sigma_9116425665531756122a_real @ M @ T ) )
          & ( ( binary6478037234023840930l_real @ ( measure_distr_a_real @ M @ S2 @ X2 ) @ ( measure_distr_a_real @ M @ T @ Y ) )
            = ( measur8266400719524636083l_real @ M @ ( binary6478037234023840930l_real @ S2 @ T )
              @ ^ [X: a] : ( produc4511245868158468465l_real @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ) ) ).

% prob_space.indep_var_distribution_eq
thf(fact_573_prob__space_Oindep__var__distribution__eq,axiom,
    ! [M: sigma_measure_real,S2: sigma_measure_a,X2: real > a,T: sigma_measure_a,Y: real > a] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe365082296117321348real_a @ M @ S2 @ X2 @ T @ Y )
        = ( ( member_real_a @ X2 @ ( sigma_523072396149930112real_a @ M @ S2 ) )
          & ( member_real_a @ Y @ ( sigma_523072396149930112real_a @ M @ T ) )
          & ( ( binary867438762418767560re_a_a @ ( measure_distr_real_a @ M @ S2 @ X2 ) @ ( measure_distr_real_a @ M @ T @ Y ) )
            = ( measur2513335786126797313od_a_a @ M @ ( binary867438762418767560re_a_a @ S2 @ T )
              @ ^ [X: real] : ( product_Pair_a_a @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ) ) ).

% prob_space.indep_var_distribution_eq
thf(fact_574_prob__space_Oindep__var__distribution__eq,axiom,
    ! [M: sigma_measure_real,S2: sigma_7234349610311085201nnreal,X2: real > extend8495563244428889912nnreal,T: sigma_7234349610311085201nnreal,Y: real > extend8495563244428889912nnreal] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe6767359503340752434nnreal @ M @ S2 @ X2 @ T @ Y )
        = ( ( member2919562650594848410nnreal @ X2 @ ( sigma_9017504469962657078nnreal @ M @ S2 ) )
          & ( member2919562650594848410nnreal @ Y @ ( sigma_9017504469962657078nnreal @ M @ T ) )
          & ( ( binary3098606844978005306nnreal @ ( measur8829990298702910942nnreal @ M @ S2 @ X2 ) @ ( measur8829990298702910942nnreal @ M @ T @ Y ) )
            = ( measur4012415197360569771nnreal @ M @ ( binary3098606844978005306nnreal @ S2 @ T )
              @ ^ [X: real] : ( produc344325839068023049nnreal @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ) ) ).

% prob_space.indep_var_distribution_eq
thf(fact_575_prob__space_Oindep__var__distribution__eq,axiom,
    ! [M: sigma_measure_real,S2: sigma_3077487657436305159omplex,X2: real > complex,T: sigma_3077487657436305159omplex,Y: real > complex] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe1954327081502071720omplex @ M @ S2 @ X2 @ T @ Y )
        = ( ( member_real_complex @ X2 @ ( sigma_9111916201866572460omplex @ M @ S2 ) )
          & ( member_real_complex @ Y @ ( sigma_9111916201866572460omplex @ M @ T ) )
          & ( ( binary5145385660880348710omplex @ ( measur1621797640479583060omplex @ M @ S2 @ X2 ) @ ( measur1621797640479583060omplex @ M @ T @ Y ) )
            = ( measur4452220837507949463omplex @ M @ ( binary5145385660880348710omplex @ S2 @ T )
              @ ^ [X: real] : ( produc101793102246108661omplex @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ) ) ).

% prob_space.indep_var_distribution_eq
thf(fact_576_prob__space_Oindep__var__distribution__eq,axiom,
    ! [M: sigma_measure_real,S2: sigma_measure_real,X2: real > real,T: sigma_measure_real,Y: real > real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe3760321310464026790l_real @ M @ S2 @ X2 @ T @ Y )
        = ( ( member_real_real @ X2 @ ( sigma_5267869275261027754l_real @ M @ S2 ) )
          & ( member_real_real @ Y @ ( sigma_5267869275261027754l_real @ M @ T ) )
          & ( ( binary6478037234023840930l_real @ ( measur2993149975067245138l_real @ M @ S2 @ X2 ) @ ( measur2993149975067245138l_real @ M @ T @ Y ) )
            = ( measur6481026558495277843l_real @ M @ ( binary6478037234023840930l_real @ S2 @ T )
              @ ^ [X: real] : ( produc4511245868158468465l_real @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ) ) ).

% prob_space.indep_var_distribution_eq
thf(fact_577_prob__space_Oindep__var__lebesgue__integral,axiom,
    ! [M: sigma_measure_a,X12: a > real,X23: a > real] :
      ( ( probab7247484486040049089pace_a @ M )
     => ( ( indepe8958435565499147358a_real @ M @ borel_5078946678739801102l_real @ X12 @ borel_5078946678739801102l_real @ X23 )
       => ( ( bochne2139062162225249880a_real @ M @ X12 )
         => ( ( bochne2139062162225249880a_real @ M @ X23 )
           => ( ( bochne378719280626478695a_real @ M
                @ ^ [Omega: a] : ( times_times_real @ ( X12 @ Omega ) @ ( X23 @ Omega ) ) )
              = ( times_times_real @ ( bochne378719280626478695a_real @ M @ X12 ) @ ( bochne378719280626478695a_real @ M @ X23 ) ) ) ) ) ) ) ).

% prob_space.indep_var_lebesgue_integral
thf(fact_578_prob__space_Oindep__var__lebesgue__integral,axiom,
    ! [M: sigma_measure_real,X12: real > real,X23: real > real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe3760321310464026790l_real @ M @ borel_5078946678739801102l_real @ X12 @ borel_5078946678739801102l_real @ X23 )
       => ( ( bochne3340023020068487468l_real @ M @ X12 )
         => ( ( bochne3340023020068487468l_real @ M @ X23 )
           => ( ( bochne3715101410578510557l_real @ M
                @ ^ [Omega: real] : ( times_times_real @ ( X12 @ Omega ) @ ( X23 @ Omega ) ) )
              = ( times_times_real @ ( bochne3715101410578510557l_real @ M @ X12 ) @ ( bochne3715101410578510557l_real @ M @ X23 ) ) ) ) ) ) ) ).

% prob_space.indep_var_lebesgue_integral
thf(fact_579_prob__space_Oindep__var__lebesgue__integral,axiom,
    ! [M: sigma_measure_real,X12: real > complex,X23: real > complex] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe1954327081502071720omplex @ M @ borel_1392132677378845456omplex @ X12 @ borel_1392132677378845456omplex @ X23 )
       => ( ( bochne7032760885902134062omplex @ M @ X12 )
         => ( ( bochne7032760885902134062omplex @ M @ X23 )
           => ( ( bochne8865740171307459423omplex @ M
                @ ^ [Omega: real] : ( times_times_complex @ ( X12 @ Omega ) @ ( X23 @ Omega ) ) )
              = ( times_times_complex @ ( bochne8865740171307459423omplex @ M @ X12 ) @ ( bochne8865740171307459423omplex @ M @ X23 ) ) ) ) ) ) ) ).

% prob_space.indep_var_lebesgue_integral
thf(fact_580_qbs__prob__measure__prob__space_Onn__integral__fst,axiom,
    ! [F: produc6543235832880896358e_real > extend8495563244428889912nnreal,M1: sigma_4063782130865963553orel_a,S: probab1516826487093506724e_real] :
      ( ( member7006052219480459064nnreal @ F @ ( sigma_7937771391406329678nnreal @ ( binary125940435690417031e_real @ M1 @ ( probab2194649109939266725e_real @ S ) ) @ borel_6524799422816628122nnreal ) )
     => ( ( nonneg7149001894848470201orel_a @ M1
          @ ^ [X: quasi_borel_a] :
              ( nonneg1471867029375019384e_real @ ( probab2194649109939266725e_real @ S )
              @ ^ [Y7: produc725540845905733987e_real] : ( F @ ( produc4145838808978236886e_real @ X @ Y7 ) ) ) )
        = ( nonneg8793934387659790843e_real @ ( binary125940435690417031e_real @ M1 @ ( probab2194649109939266725e_real @ S ) ) @ F ) ) ) ).

% qbs_prob_measure_prob_space.nn_integral_fst
thf(fact_581_qbs__prob__measure__prob__space_Onn__integral__fst,axiom,
    ! [F: produc725540845905733987e_real > extend8495563244428889912nnreal,M1: sigma_measure_real_a,S: probab8231748846206645574e_real] :
      ( ( member3165316040026113589nnreal @ F @ ( sigma_969860569848805835nnreal @ ( binary2119006201073916036e_real @ M1 @ ( probab673905877088250951e_real @ S ) ) @ borel_6524799422816628122nnreal ) )
     => ( ( nonneg43860225155639326real_a @ M1
          @ ^ [X: real > a] :
              ( nonneg768625725202329114e_real @ ( probab673905877088250951e_real @ S )
              @ ^ [Y7: sigma_measure_real] : ( F @ ( produc623176010801490259e_real @ X @ Y7 ) ) ) )
        = ( nonneg1471867029375019384e_real @ ( binary2119006201073916036e_real @ M1 @ ( probab673905877088250951e_real @ S ) ) @ F ) ) ) ).

% qbs_prob_measure_prob_space.nn_integral_fst
thf(fact_582_qbs__prob__measure__prob__space_Onn__integral__fst,axiom,
    ! [F: produc2422161461964618553l_real > extend8495563244428889912nnreal,M1: sigma_measure_real,S: probab8009751763329705409e_real] :
      ( ( member2245694452317284363nnreal @ F @ ( sigma_219902274609641377nnreal @ ( binary6478037234023840930l_real @ M1 @ ( probab4733579253584633066e_real @ S ) ) @ borel_6524799422816628122nnreal ) )
     => ( ( nonneg2667834350952324695l_real @ M1
          @ ^ [X: real] :
              ( nonneg2667834350952324695l_real @ ( probab4733579253584633066e_real @ S )
              @ ^ [Y7: real] : ( F @ ( produc4511245868158468465l_real @ X @ Y7 ) ) ) )
        = ( nonneg1896927508495185742l_real @ ( binary6478037234023840930l_real @ M1 @ ( probab4733579253584633066e_real @ S ) ) @ F ) ) ) ).

% qbs_prob_measure_prob_space.nn_integral_fst
thf(fact_583_qbs__prob__measure__prob__space_Onn__integral__fst,axiom,
    ! [F: product_prod_a_real > extend8495563244428889912nnreal,M1: sigma_measure_a,S: probab8009751763329705409e_real] :
      ( ( member4437793228276457543nnreal @ F @ ( sigma_8117956432444187619nnreal @ ( binary932748531126180194a_real @ M1 @ ( probab4733579253584633066e_real @ S ) ) @ borel_6524799422816628122nnreal ) )
     => ( ( nonneg2725512125972007571gral_a @ M1
          @ ^ [X: a] :
              ( nonneg2667834350952324695l_real @ ( probab4733579253584633066e_real @ S )
              @ ^ [Y7: real] : ( F @ ( product_Pair_a_real @ X @ Y7 ) ) ) )
        = ( nonneg4050876233904260868a_real @ ( binary932748531126180194a_real @ M1 @ ( probab4733579253584633066e_real @ S ) ) @ F ) ) ) ).

% qbs_prob_measure_prob_space.nn_integral_fst
thf(fact_584_qbs__prob__measure__prob__space_Onn__integral__fst,axiom,
    ! [F: product_prod_real_a > extend8495563244428889912nnreal,M1: sigma_measure_real,S: probab4737552673497767871pace_a] :
      ( ( member8281051115363742261nnreal @ F @ ( sigma_2737842282676696529nnreal @ ( binary1562767298599129992real_a @ M1 @ ( probab7100426894406488384sure_a @ S ) ) @ borel_6524799422816628122nnreal ) )
     => ( ( nonneg2667834350952324695l_real @ M1
          @ ^ [X: real] :
              ( nonneg2725512125972007571gral_a @ ( probab7100426894406488384sure_a @ S )
              @ ^ [Y7: a] : ( F @ ( product_Pair_real_a @ X @ Y7 ) ) ) )
        = ( nonneg4568142736171598066real_a @ ( binary1562767298599129992real_a @ M1 @ ( probab7100426894406488384sure_a @ S ) ) @ F ) ) ) ).

% qbs_prob_measure_prob_space.nn_integral_fst
thf(fact_585_qbs__prob__measure__prob__space_Onn__integral__fst,axiom,
    ! [F: product_prod_a_a > extend8495563244428889912nnreal,M1: sigma_measure_a,S: probab4737552673497767871pace_a] :
      ( ( member3238353849244381945nnreal @ F @ ( sigma_88170358281049359nnreal @ ( binary867438762418767560re_a_a @ M1 @ ( probab7100426894406488384sure_a @ S ) ) @ borel_6524799422816628122nnreal ) )
     => ( ( nonneg2725512125972007571gral_a @ M1
          @ ^ [X: a] :
              ( nonneg2725512125972007571gral_a @ ( probab7100426894406488384sure_a @ S )
              @ ^ [Y7: a] : ( F @ ( product_Pair_a_a @ X @ Y7 ) ) ) )
        = ( nonneg5307290267605202876od_a_a @ ( binary867438762418767560re_a_a @ M1 @ ( probab7100426894406488384sure_a @ S ) ) @ F ) ) ) ).

% qbs_prob_measure_prob_space.nn_integral_fst
thf(fact_586_qbs__prob__measure__prob__space_Oindep__var__integrable,axiom,
    ! [S: probab8009751763329705409e_real,X12: real > real,X23: real > real] :
      ( ( indepe3760321310464026790l_real @ ( probab4733579253584633066e_real @ S ) @ borel_5078946678739801102l_real @ X12 @ borel_5078946678739801102l_real @ X23 )
     => ( ( bochne3340023020068487468l_real @ ( probab4733579253584633066e_real @ S ) @ X12 )
       => ( ( bochne3340023020068487468l_real @ ( probab4733579253584633066e_real @ S ) @ X23 )
         => ( bochne3340023020068487468l_real @ ( probab4733579253584633066e_real @ S )
            @ ^ [Omega: real] : ( times_times_real @ ( X12 @ Omega ) @ ( X23 @ Omega ) ) ) ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_integrable
thf(fact_587_qbs__prob__measure__prob__space_Oindep__var__integrable,axiom,
    ! [S: probab4737552673497767871pace_a,X12: a > real,X23: a > real] :
      ( ( indepe8958435565499147358a_real @ ( probab7100426894406488384sure_a @ S ) @ borel_5078946678739801102l_real @ X12 @ borel_5078946678739801102l_real @ X23 )
     => ( ( bochne2139062162225249880a_real @ ( probab7100426894406488384sure_a @ S ) @ X12 )
       => ( ( bochne2139062162225249880a_real @ ( probab7100426894406488384sure_a @ S ) @ X23 )
         => ( bochne2139062162225249880a_real @ ( probab7100426894406488384sure_a @ S )
            @ ^ [Omega: a] : ( times_times_real @ ( X12 @ Omega ) @ ( X23 @ Omega ) ) ) ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_integrable
thf(fact_588_qbs__prob__measure__prob__space_Oindep__var__integrable,axiom,
    ! [S: probab4737552673497767871pace_a,X12: a > complex,X23: a > complex] :
      ( ( indepe3790908202538861408omplex @ ( probab7100426894406488384sure_a @ S ) @ borel_1392132677378845456omplex @ X12 @ borel_1392132677378845456omplex @ X23 )
     => ( ( bochne1348834467089073754omplex @ ( probab7100426894406488384sure_a @ S ) @ X12 )
       => ( ( bochne1348834467089073754omplex @ ( probab7100426894406488384sure_a @ S ) @ X23 )
         => ( bochne1348834467089073754omplex @ ( probab7100426894406488384sure_a @ S )
            @ ^ [Omega: a] : ( times_times_complex @ ( X12 @ Omega ) @ ( X23 @ Omega ) ) ) ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_integrable
thf(fact_589_qbs__prob__measure__prob__space_Oborel__measurable__nn__integral__fst,axiom,
    ! [F: produc6543235832880896358e_real > extend8495563244428889912nnreal,M1: sigma_4063782130865963553orel_a,S: probab1516826487093506724e_real] :
      ( ( member7006052219480459064nnreal @ F @ ( sigma_7937771391406329678nnreal @ ( binary125940435690417031e_real @ M1 @ ( probab2194649109939266725e_real @ S ) ) @ borel_6524799422816628122nnreal ) )
     => ( member2953391730036472438nnreal
        @ ^ [X: quasi_borel_a] :
            ( nonneg1471867029375019384e_real @ ( probab2194649109939266725e_real @ S )
            @ ^ [Y7: produc725540845905733987e_real] : ( F @ ( produc4145838808978236886e_real @ X @ Y7 ) ) )
        @ ( sigma_6209045778286148364nnreal @ M1 @ borel_6524799422816628122nnreal ) ) ) ).

% qbs_prob_measure_prob_space.borel_measurable_nn_integral_fst
thf(fact_590_qbs__prob__measure__prob__space_Oborel__measurable__nn__integral__fst,axiom,
    ! [F: produc725540845905733987e_real > extend8495563244428889912nnreal,M1: sigma_measure_real_a,S: probab8231748846206645574e_real] :
      ( ( member3165316040026113589nnreal @ F @ ( sigma_969860569848805835nnreal @ ( binary2119006201073916036e_real @ M1 @ ( probab673905877088250951e_real @ S ) ) @ borel_6524799422816628122nnreal ) )
     => ( member1591818685568139867nnreal
        @ ^ [X: real > a] :
            ( nonneg768625725202329114e_real @ ( probab673905877088250951e_real @ S )
            @ ^ [Y7: sigma_measure_real] : ( F @ ( produc623176010801490259e_real @ X @ Y7 ) ) )
        @ ( sigma_4817184057256041329nnreal @ M1 @ borel_6524799422816628122nnreal ) ) ) ).

% qbs_prob_measure_prob_space.borel_measurable_nn_integral_fst
thf(fact_591_qbs__prob__measure__prob__space_Oborel__measurable__nn__integral__fst,axiom,
    ! [F: product_prod_a_real > extend8495563244428889912nnreal,M1: sigma_measure_a,S: probab8009751763329705409e_real] :
      ( ( member4437793228276457543nnreal @ F @ ( sigma_8117956432444187619nnreal @ ( binary932748531126180194a_real @ M1 @ ( probab4733579253584633066e_real @ S ) ) @ borel_6524799422816628122nnreal ) )
     => ( member298456594901751504nnreal
        @ ^ [X: a] :
            ( nonneg2667834350952324695l_real @ ( probab4733579253584633066e_real @ S )
            @ ^ [Y7: real] : ( F @ ( product_Pair_a_real @ X @ Y7 ) ) )
        @ ( sigma_214952329563889126nnreal @ M1 @ borel_6524799422816628122nnreal ) ) ) ).

% qbs_prob_measure_prob_space.borel_measurable_nn_integral_fst
thf(fact_592_qbs__prob__measure__prob__space_Oborel__measurable__nn__integral__fst,axiom,
    ! [F: produc2422161461964618553l_real > extend8495563244428889912nnreal,M1: sigma_measure_real,S: probab8009751763329705409e_real] :
      ( ( member2245694452317284363nnreal @ F @ ( sigma_219902274609641377nnreal @ ( binary6478037234023840930l_real @ M1 @ ( probab4733579253584633066e_real @ S ) ) @ borel_6524799422816628122nnreal ) )
     => ( member2919562650594848410nnreal
        @ ^ [X: real] :
            ( nonneg2667834350952324695l_real @ ( probab4733579253584633066e_real @ S )
            @ ^ [Y7: real] : ( F @ ( produc4511245868158468465l_real @ X @ Y7 ) ) )
        @ ( sigma_9017504469962657078nnreal @ M1 @ borel_6524799422816628122nnreal ) ) ) ).

% qbs_prob_measure_prob_space.borel_measurable_nn_integral_fst
thf(fact_593_qbs__prob__measure__prob__space_Oborel__measurable__nn__integral__fst,axiom,
    ! [F: product_prod_a_a > extend8495563244428889912nnreal,M1: sigma_measure_a,S: probab4737552673497767871pace_a] :
      ( ( member3238353849244381945nnreal @ F @ ( sigma_88170358281049359nnreal @ ( binary867438762418767560re_a_a @ M1 @ ( probab7100426894406488384sure_a @ S ) ) @ borel_6524799422816628122nnreal ) )
     => ( member298456594901751504nnreal
        @ ^ [X: a] :
            ( nonneg2725512125972007571gral_a @ ( probab7100426894406488384sure_a @ S )
            @ ^ [Y7: a] : ( F @ ( product_Pair_a_a @ X @ Y7 ) ) )
        @ ( sigma_214952329563889126nnreal @ M1 @ borel_6524799422816628122nnreal ) ) ) ).

% qbs_prob_measure_prob_space.borel_measurable_nn_integral_fst
thf(fact_594_qbs__prob__measure__prob__space_Oborel__measurable__nn__integral__fst,axiom,
    ! [F: product_prod_real_a > extend8495563244428889912nnreal,M1: sigma_measure_real,S: probab4737552673497767871pace_a] :
      ( ( member8281051115363742261nnreal @ F @ ( sigma_2737842282676696529nnreal @ ( binary1562767298599129992real_a @ M1 @ ( probab7100426894406488384sure_a @ S ) ) @ borel_6524799422816628122nnreal ) )
     => ( member2919562650594848410nnreal
        @ ^ [X: real] :
            ( nonneg2725512125972007571gral_a @ ( probab7100426894406488384sure_a @ S )
            @ ^ [Y7: a] : ( F @ ( product_Pair_real_a @ X @ Y7 ) ) )
        @ ( sigma_9017504469962657078nnreal @ M1 @ borel_6524799422816628122nnreal ) ) ) ).

% qbs_prob_measure_prob_space.borel_measurable_nn_integral_fst
thf(fact_595_qbs__prob__measure__prob__space_Oindep__var__distribution__eq,axiom,
    ! [S: probab8009751763329705409e_real,S2: sigma_measure_a,X2: real > a,T: sigma_measure_a,Y: real > a] :
      ( ( indepe365082296117321348real_a @ ( probab4733579253584633066e_real @ S ) @ S2 @ X2 @ T @ Y )
      = ( ( member_real_a @ X2 @ ( sigma_523072396149930112real_a @ ( probab4733579253584633066e_real @ S ) @ S2 ) )
        & ( member_real_a @ Y @ ( sigma_523072396149930112real_a @ ( probab4733579253584633066e_real @ S ) @ T ) )
        & ( ( binary867438762418767560re_a_a @ ( measure_distr_real_a @ ( probab4733579253584633066e_real @ S ) @ S2 @ X2 ) @ ( measure_distr_real_a @ ( probab4733579253584633066e_real @ S ) @ T @ Y ) )
          = ( measur2513335786126797313od_a_a @ ( probab4733579253584633066e_real @ S ) @ ( binary867438762418767560re_a_a @ S2 @ T )
            @ ^ [X: real] : ( product_Pair_a_a @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_distribution_eq
thf(fact_596_qbs__prob__measure__prob__space_Oindep__var__distribution__eq,axiom,
    ! [S: probab8009751763329705409e_real,S2: sigma_7234349610311085201nnreal,X2: real > extend8495563244428889912nnreal,T: sigma_7234349610311085201nnreal,Y: real > extend8495563244428889912nnreal] :
      ( ( indepe6767359503340752434nnreal @ ( probab4733579253584633066e_real @ S ) @ S2 @ X2 @ T @ Y )
      = ( ( member2919562650594848410nnreal @ X2 @ ( sigma_9017504469962657078nnreal @ ( probab4733579253584633066e_real @ S ) @ S2 ) )
        & ( member2919562650594848410nnreal @ Y @ ( sigma_9017504469962657078nnreal @ ( probab4733579253584633066e_real @ S ) @ T ) )
        & ( ( binary3098606844978005306nnreal @ ( measur8829990298702910942nnreal @ ( probab4733579253584633066e_real @ S ) @ S2 @ X2 ) @ ( measur8829990298702910942nnreal @ ( probab4733579253584633066e_real @ S ) @ T @ Y ) )
          = ( measur4012415197360569771nnreal @ ( probab4733579253584633066e_real @ S ) @ ( binary3098606844978005306nnreal @ S2 @ T )
            @ ^ [X: real] : ( produc344325839068023049nnreal @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_distribution_eq
thf(fact_597_qbs__prob__measure__prob__space_Oindep__var__distribution__eq,axiom,
    ! [S: probab8009751763329705409e_real,S2: sigma_3077487657436305159omplex,X2: real > complex,T: sigma_3077487657436305159omplex,Y: real > complex] :
      ( ( indepe1954327081502071720omplex @ ( probab4733579253584633066e_real @ S ) @ S2 @ X2 @ T @ Y )
      = ( ( member_real_complex @ X2 @ ( sigma_9111916201866572460omplex @ ( probab4733579253584633066e_real @ S ) @ S2 ) )
        & ( member_real_complex @ Y @ ( sigma_9111916201866572460omplex @ ( probab4733579253584633066e_real @ S ) @ T ) )
        & ( ( binary5145385660880348710omplex @ ( measur1621797640479583060omplex @ ( probab4733579253584633066e_real @ S ) @ S2 @ X2 ) @ ( measur1621797640479583060omplex @ ( probab4733579253584633066e_real @ S ) @ T @ Y ) )
          = ( measur4452220837507949463omplex @ ( probab4733579253584633066e_real @ S ) @ ( binary5145385660880348710omplex @ S2 @ T )
            @ ^ [X: real] : ( produc101793102246108661omplex @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_distribution_eq
thf(fact_598_qbs__prob__measure__prob__space_Oindep__var__distribution__eq,axiom,
    ! [S: probab8009751763329705409e_real,S2: sigma_measure_real,X2: real > real,T: sigma_measure_real,Y: real > real] :
      ( ( indepe3760321310464026790l_real @ ( probab4733579253584633066e_real @ S ) @ S2 @ X2 @ T @ Y )
      = ( ( member_real_real @ X2 @ ( sigma_5267869275261027754l_real @ ( probab4733579253584633066e_real @ S ) @ S2 ) )
        & ( member_real_real @ Y @ ( sigma_5267869275261027754l_real @ ( probab4733579253584633066e_real @ S ) @ T ) )
        & ( ( binary6478037234023840930l_real @ ( measur2993149975067245138l_real @ ( probab4733579253584633066e_real @ S ) @ S2 @ X2 ) @ ( measur2993149975067245138l_real @ ( probab4733579253584633066e_real @ S ) @ T @ Y ) )
          = ( measur6481026558495277843l_real @ ( probab4733579253584633066e_real @ S ) @ ( binary6478037234023840930l_real @ S2 @ T )
            @ ^ [X: real] : ( produc4511245868158468465l_real @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_distribution_eq
thf(fact_599_qbs__prob__measure__prob__space_Oindep__var__distribution__eq,axiom,
    ! [S: probab4737552673497767871pace_a,S2: sigma_7234349610311085201nnreal,X2: a > extend8495563244428889912nnreal,T: sigma_7234349610311085201nnreal,Y: a > extend8495563244428889912nnreal] :
      ( ( indepe3534117692041274858nnreal @ ( probab7100426894406488384sure_a @ S ) @ S2 @ X2 @ T @ Y )
      = ( ( member298456594901751504nnreal @ X2 @ ( sigma_214952329563889126nnreal @ ( probab7100426894406488384sure_a @ S ) @ S2 ) )
        & ( member298456594901751504nnreal @ Y @ ( sigma_214952329563889126nnreal @ ( probab7100426894406488384sure_a @ S ) @ T ) )
        & ( ( binary3098606844978005306nnreal @ ( measur4839436603801885502nnreal @ ( probab7100426894406488384sure_a @ S ) @ S2 @ X2 ) @ ( measur4839436603801885502nnreal @ ( probab7100426894406488384sure_a @ S ) @ T @ Y ) )
          = ( measur6341207572317192267nnreal @ ( probab7100426894406488384sure_a @ S ) @ ( binary3098606844978005306nnreal @ S2 @ T )
            @ ^ [X: a] : ( produc344325839068023049nnreal @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_distribution_eq
thf(fact_600_qbs__prob__measure__prob__space_Oindep__var__distribution__eq,axiom,
    ! [S: probab4737552673497767871pace_a,S2: sigma_measure_real,X2: a > real,T: sigma_measure_real,Y: a > real] :
      ( ( indepe8958435565499147358a_real @ ( probab7100426894406488384sure_a @ S ) @ S2 @ X2 @ T @ Y )
      = ( ( member_a_real @ X2 @ ( sigma_9116425665531756122a_real @ ( probab7100426894406488384sure_a @ S ) @ S2 ) )
        & ( member_a_real @ Y @ ( sigma_9116425665531756122a_real @ ( probab7100426894406488384sure_a @ S ) @ T ) )
        & ( ( binary6478037234023840930l_real @ ( measure_distr_a_real @ ( probab7100426894406488384sure_a @ S ) @ S2 @ X2 ) @ ( measure_distr_a_real @ ( probab7100426894406488384sure_a @ S ) @ T @ Y ) )
          = ( measur8266400719524636083l_real @ ( probab7100426894406488384sure_a @ S ) @ ( binary6478037234023840930l_real @ S2 @ T )
            @ ^ [X: a] : ( produc4511245868158468465l_real @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_distribution_eq
thf(fact_601_qbs__prob__measure__prob__space_Oindep__var__lebesgue__integral,axiom,
    ! [S: probab8009751763329705409e_real,X12: real > real,X23: real > real] :
      ( ( indepe3760321310464026790l_real @ ( probab4733579253584633066e_real @ S ) @ borel_5078946678739801102l_real @ X12 @ borel_5078946678739801102l_real @ X23 )
     => ( ( bochne3340023020068487468l_real @ ( probab4733579253584633066e_real @ S ) @ X12 )
       => ( ( bochne3340023020068487468l_real @ ( probab4733579253584633066e_real @ S ) @ X23 )
         => ( ( bochne3715101410578510557l_real @ ( probab4733579253584633066e_real @ S )
              @ ^ [Omega: real] : ( times_times_real @ ( X12 @ Omega ) @ ( X23 @ Omega ) ) )
            = ( times_times_real @ ( bochne3715101410578510557l_real @ ( probab4733579253584633066e_real @ S ) @ X12 ) @ ( bochne3715101410578510557l_real @ ( probab4733579253584633066e_real @ S ) @ X23 ) ) ) ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_lebesgue_integral
thf(fact_602_qbs__prob__measure__prob__space_Oindep__var__lebesgue__integral,axiom,
    ! [S: probab4737552673497767871pace_a,X12: a > real,X23: a > real] :
      ( ( indepe8958435565499147358a_real @ ( probab7100426894406488384sure_a @ S ) @ borel_5078946678739801102l_real @ X12 @ borel_5078946678739801102l_real @ X23 )
     => ( ( bochne2139062162225249880a_real @ ( probab7100426894406488384sure_a @ S ) @ X12 )
       => ( ( bochne2139062162225249880a_real @ ( probab7100426894406488384sure_a @ S ) @ X23 )
         => ( ( bochne378719280626478695a_real @ ( probab7100426894406488384sure_a @ S )
              @ ^ [Omega: a] : ( times_times_real @ ( X12 @ Omega ) @ ( X23 @ Omega ) ) )
            = ( times_times_real @ ( bochne378719280626478695a_real @ ( probab7100426894406488384sure_a @ S ) @ X12 ) @ ( bochne378719280626478695a_real @ ( probab7100426894406488384sure_a @ S ) @ X23 ) ) ) ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_lebesgue_integral
thf(fact_603_qbs__prob__measure__prob__space_Oindep__var__lebesgue__integral,axiom,
    ! [S: probab4737552673497767871pace_a,X12: a > complex,X23: a > complex] :
      ( ( indepe3790908202538861408omplex @ ( probab7100426894406488384sure_a @ S ) @ borel_1392132677378845456omplex @ X12 @ borel_1392132677378845456omplex @ X23 )
     => ( ( bochne1348834467089073754omplex @ ( probab7100426894406488384sure_a @ S ) @ X12 )
       => ( ( bochne1348834467089073754omplex @ ( probab7100426894406488384sure_a @ S ) @ X23 )
         => ( ( bochne4904656926214500329omplex @ ( probab7100426894406488384sure_a @ S )
              @ ^ [Omega: a] : ( times_times_complex @ ( X12 @ Omega ) @ ( X23 @ Omega ) ) )
            = ( times_times_complex @ ( bochne4904656926214500329omplex @ ( probab7100426894406488384sure_a @ S ) @ X12 ) @ ( bochne4904656926214500329omplex @ ( probab7100426894406488384sure_a @ S ) @ X23 ) ) ) ) ) ) ).

% qbs_prob_measure_prob_space.indep_var_lebesgue_integral
thf(fact_604_qbs__prob__ennintegral__cmult,axiom,
    ! [S: probab4737552673497767871pace_a,X2: quasi_borel_a,F: a > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ( probab1293289258141559360_qbs_a @ S )
        = X2 )
     => ( ( member298456594901751504nnreal @ F @ ( qbs_mo1434458643421888574nnreal @ X2 @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) ) )
       => ( ( probab3721531081081959085gral_a @ S
            @ ^ [X: a] : ( times_1893300245718287421nnreal @ C @ ( F @ X ) ) )
          = ( times_1893300245718287421nnreal @ C @ ( probab3721531081081959085gral_a @ S @ F ) ) ) ) ) ).

% qbs_prob_ennintegral_cmult
thf(fact_605_qbs__Mx__are__measurable,axiom,
    ! [Alpha: real > real,X2: quasi_borel_real] :
      ( ( member_real_real @ Alpha @ ( qbs_Mx_real @ X2 ) )
     => ( member_real_real @ Alpha @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ ( measur1733462625046462224e_real @ X2 ) ) ) ) ).

% qbs_Mx_are_measurable
thf(fact_606_qbs__Mx__are__measurable,axiom,
    ! [Alpha: real > extend8495563244428889912nnreal,X2: quasi_9015997321629101608nnreal] :
      ( ( member2919562650594848410nnreal @ Alpha @ ( qbs_Mx6523938229262583809nnreal @ X2 ) )
     => ( member2919562650594848410nnreal @ Alpha @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ ( measur7384687747506661788nnreal @ X2 ) ) ) ) ).

% qbs_Mx_are_measurable
thf(fact_607_qbs__Mx__are__measurable,axiom,
    ! [Alpha: real > complex,X2: quasi_borel_complex] :
      ( ( member_real_complex @ Alpha @ ( qbs_Mx_complex @ X2 ) )
     => ( member_real_complex @ Alpha @ ( sigma_9111916201866572460omplex @ borel_5078946678739801102l_real @ ( measur3826415497239753490omplex @ X2 ) ) ) ) ).

% qbs_Mx_are_measurable
thf(fact_608_qbs__Mx__are__measurable,axiom,
    ! [Alpha: real > a,X2: quasi_borel_a] :
      ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X2 ) )
     => ( member_real_a @ Alpha @ ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ ( measur7857763439677503898sure_a @ X2 ) ) ) ) ).

% qbs_Mx_are_measurable
thf(fact_609_qbs__Mx__is__morphisms,axiom,
    ( qbs_Mx_a
    = ( qbs_morphism_real_a @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) ) ) ).

% qbs_Mx_is_morphisms
thf(fact_610_mult__minus__left,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_611_mult__minus__left,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
      = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_612_minus__mult__minus,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
      = ( times_times_complex @ A @ B ) ) ).

% minus_mult_minus
thf(fact_613_minus__mult__minus,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
      = ( times_times_real @ A @ B ) ) ).

% minus_mult_minus
thf(fact_614_mult__minus__right,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_615_mult__minus__right,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
      = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_616_vector__space__over__itself_Oscale__minus__left,axiom,
    ! [A: complex,X4: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ X4 )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ X4 ) ) ) ).

% vector_space_over_itself.scale_minus_left
thf(fact_617_vector__space__over__itself_Oscale__minus__left,axiom,
    ! [A: real,X4: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ X4 )
      = ( uminus_uminus_real @ ( times_times_real @ A @ X4 ) ) ) ).

% vector_space_over_itself.scale_minus_left
thf(fact_618_vector__space__over__itself_Oscale__minus__right,axiom,
    ! [A: complex,X4: complex] :
      ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ X4 ) )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ X4 ) ) ) ).

% vector_space_over_itself.scale_minus_right
thf(fact_619_vector__space__over__itself_Oscale__minus__right,axiom,
    ! [A: real,X4: real] :
      ( ( times_times_real @ A @ ( uminus_uminus_real @ X4 ) )
      = ( uminus_uminus_real @ ( times_times_real @ A @ X4 ) ) ) ).

% vector_space_over_itself.scale_minus_right
thf(fact_620_qp_OKL__same__eq__0,axiom,
    ! [B: real] :
      ( ( kL_divergence_real @ B @ mu @ mu )
      = zero_zero_real ) ).

% qp.KL_same_eq_0
thf(fact_621_measurable__Pair,axiom,
    ! [F: real > a,M: sigma_measure_real,M1: sigma_measure_a,G: real > a,M22: sigma_measure_a] :
      ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ M1 ) )
     => ( ( member_real_a @ G @ ( sigma_523072396149930112real_a @ M @ M22 ) )
       => ( member4570177857406309467od_a_a
          @ ^ [X: real] : ( product_Pair_a_a @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_5261090278765293737od_a_a @ M @ ( binary867438762418767560re_a_a @ M1 @ M22 ) ) ) ) ) ).

% measurable_Pair
thf(fact_622_measurable__Pair,axiom,
    ! [F: a > extend8495563244428889912nnreal,M: sigma_measure_a,M1: sigma_7234349610311085201nnreal,G: a > extend8495563244428889912nnreal,M22: sigma_7234349610311085201nnreal] :
      ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ M1 ) )
     => ( ( member298456594901751504nnreal @ G @ ( sigma_214952329563889126nnreal @ M @ M22 ) )
       => ( member5741711457236458191nnreal
          @ ^ [X: a] : ( produc344325839068023049nnreal @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_6220444669619866531nnreal @ M @ ( binary3098606844978005306nnreal @ M1 @ M22 ) ) ) ) ) ).

% measurable_Pair
thf(fact_623_measurable__Pair,axiom,
    ! [F: a > extend8495563244428889912nnreal,M: sigma_measure_a,M1: sigma_7234349610311085201nnreal,G: a > real,M22: sigma_measure_real] :
      ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ M1 ) )
     => ( ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ M @ M22 ) )
       => ( member5439804117191083459l_real
          @ ^ [X: a] : ( produc2810268924804063229l_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_7586584134059694359l_real @ M @ ( binary3818639336118950830l_real @ M1 @ M22 ) ) ) ) ) ).

% measurable_Pair
thf(fact_624_measurable__Pair,axiom,
    ! [F: real > a,M: sigma_measure_real,M1: sigma_measure_a,G: real > real,M22: sigma_measure_real] :
      ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ M1 ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ M22 ) )
       => ( member7821820149384076987a_real
          @ ^ [X: real] : ( product_Pair_a_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_18903642666787671a_real @ M @ ( binary932748531126180194a_real @ M1 @ M22 ) ) ) ) ) ).

% measurable_Pair
thf(fact_625_measurable__Pair,axiom,
    ! [F: real > a,M: sigma_measure_real,M1: sigma_measure_a,G: real > extend8495563244428889912nnreal,M22: sigma_7234349610311085201nnreal] :
      ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ M1 ) )
     => ( ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ M @ M22 ) )
       => ( member5478118246346387399nnreal
          @ ^ [X: real] : ( produc6870484446332933855nnreal @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_6229563495366030819nnreal @ M @ ( binary7199832230051445998nnreal @ M1 @ M22 ) ) ) ) ) ).

% measurable_Pair
thf(fact_626_measurable__Pair,axiom,
    ! [F: real > a,M: sigma_measure_real,M1: sigma_measure_a,G: real > complex,M22: sigma_3077487657436305159omplex] :
      ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ M1 ) )
     => ( ( member_real_complex @ G @ ( sigma_9111916201866572460omplex @ M @ M22 ) )
       => ( member5493493090628174141omplex
          @ ^ [X: real] : ( produc2214049761573155413omplex @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_8901046131853741145omplex @ M @ ( binary5206975929401438820omplex @ M1 @ M22 ) ) ) ) ) ).

% measurable_Pair
thf(fact_627_measurable__Pair,axiom,
    ! [F: a > real,M: sigma_measure_a,M1: sigma_measure_real,G: a > extend8495563244428889912nnreal,M22: sigma_7234349610311085201nnreal] :
      ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ M1 ) )
     => ( ( member298456594901751504nnreal @ G @ ( sigma_214952329563889126nnreal @ M @ M22 ) )
       => ( member5109759473273639491nnreal
          @ ^ [X: a] : ( produc4778015194254607485nnreal @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_7632132433403717015nnreal @ M @ ( binary5786385605569495086nnreal @ M1 @ M22 ) ) ) ) ) ).

% measurable_Pair
thf(fact_628_measurable__Pair,axiom,
    ! [F: a > real,M: sigma_measure_a,M1: sigma_measure_real,G: a > real,M22: sigma_measure_real] :
      ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ M1 ) )
     => ( ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ M @ M22 ) )
       => ( member2229928074028245815l_real
          @ ^ [X: a] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_414277600898586891l_real @ M @ ( binary6478037234023840930l_real @ M1 @ M22 ) ) ) ) ) ).

% measurable_Pair
thf(fact_629_measurable__Pair,axiom,
    ! [F: real > real,M: sigma_measure_real,M1: sigma_measure_real,G: real > a,M22: sigma_measure_a] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ M1 ) )
     => ( ( member_real_a @ G @ ( sigma_523072396149930112real_a @ M @ M22 ) )
       => ( member1006887528422719913real_a
          @ ^ [X: real] : ( product_Pair_real_a @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_536170144934124869real_a @ M @ ( binary1562767298599129992real_a @ M1 @ M22 ) ) ) ) ) ).

% measurable_Pair
thf(fact_630_measurable__Pair,axiom,
    ! [F: real > real,M: sigma_measure_real,M1: sigma_measure_real,G: real > real,M22: sigma_measure_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ M1 ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ M22 ) )
       => ( member9086635009091248365l_real
          @ ^ [X: real] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_7998147297565726139l_real @ M @ ( binary6478037234023840930l_real @ M1 @ M22 ) ) ) ) ) ).

% measurable_Pair
thf(fact_631_qbs__prob__measure__prob__space_OKL__same__eq__0,axiom,
    ! [B: real,S: probab8009751763329705409e_real] :
      ( ( kL_divergence_real @ B @ ( probab4733579253584633066e_real @ S ) @ ( probab4733579253584633066e_real @ S ) )
      = zero_zero_real ) ).

% qbs_prob_measure_prob_space.KL_same_eq_0
thf(fact_632_qbs__prob__measure__prob__space_OKL__same__eq__0,axiom,
    ! [B: real,S: probab4737552673497767871pace_a] :
      ( ( kL_divergence_a @ B @ ( probab7100426894406488384sure_a @ S ) @ ( probab7100426894406488384sure_a @ S ) )
      = zero_zero_real ) ).

% qbs_prob_measure_prob_space.KL_same_eq_0
thf(fact_633_vector__space__over__itself_Oscale__right__diff__distrib,axiom,
    ! [A: real,X4: real,Y3: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ X4 @ Y3 ) )
      = ( minus_minus_real @ ( times_times_real @ A @ X4 ) @ ( times_times_real @ A @ Y3 ) ) ) ).

% vector_space_over_itself.scale_right_diff_distrib
thf(fact_634_vector__space__over__itself_Oscale__right__diff__distrib,axiom,
    ! [A: complex,X4: complex,Y3: complex] :
      ( ( times_times_complex @ A @ ( minus_minus_complex @ X4 @ Y3 ) )
      = ( minus_minus_complex @ ( times_times_complex @ A @ X4 ) @ ( times_times_complex @ A @ Y3 ) ) ) ).

% vector_space_over_itself.scale_right_diff_distrib
thf(fact_635_vector__space__over__itself_Oscale__left__diff__distrib,axiom,
    ! [A: real,B: real,X4: real] :
      ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ X4 )
      = ( minus_minus_real @ ( times_times_real @ A @ X4 ) @ ( times_times_real @ B @ X4 ) ) ) ).

% vector_space_over_itself.scale_left_diff_distrib
thf(fact_636_vector__space__over__itself_Oscale__left__diff__distrib,axiom,
    ! [A: complex,B: complex,X4: complex] :
      ( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ X4 )
      = ( minus_minus_complex @ ( times_times_complex @ A @ X4 ) @ ( times_times_complex @ B @ X4 ) ) ) ).

% vector_space_over_itself.scale_left_diff_distrib
thf(fact_637_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
      = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_638_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_639_right__diff__distrib_H,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ A @ ( minus_minus_complex @ B @ C ) )
      = ( minus_minus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_640_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
      = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_641_left__diff__distrib_H,axiom,
    ! [B: real,C: real,A: real] :
      ( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
      = ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_642_left__diff__distrib_H,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( times_times_complex @ ( minus_minus_complex @ B @ C ) @ A )
      = ( minus_minus_complex @ ( times_times_complex @ B @ A ) @ ( times_times_complex @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_643_right__diff__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_644_right__diff__distrib,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ A @ ( minus_minus_complex @ B @ C ) )
      = ( minus_minus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_645_left__diff__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_646_left__diff__distrib,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ C )
      = ( minus_minus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_647_minus__mult__commute,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
      = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).

% minus_mult_commute
thf(fact_648_minus__mult__commute,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
      = ( times_times_real @ A @ ( uminus_uminus_real @ B ) ) ) ).

% minus_mult_commute
thf(fact_649_square__eq__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ A )
        = ( times_times_complex @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus1482373934393186551omplex @ B ) ) ) ) ).

% square_eq_iff
thf(fact_650_square__eq__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ A )
        = ( times_times_real @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_real @ B ) ) ) ) ).

% square_eq_iff
thf(fact_651_lambda__zero,axiom,
    ( ( ^ [H2: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_652_lambda__zero,axiom,
    ( ( ^ [H2: extend8495563244428889912nnreal] : zero_z7100319975126383169nnreal )
    = ( times_1893300245718287421nnreal @ zero_z7100319975126383169nnreal ) ) ).

% lambda_zero
thf(fact_653_lambda__zero,axiom,
    ( ( ^ [H2: real] : zero_zero_real )
    = ( times_times_real @ zero_zero_real ) ) ).

% lambda_zero
thf(fact_654_lambda__zero,axiom,
    ( ( ^ [H2: complex] : zero_zero_complex )
    = ( times_times_complex @ zero_zero_complex ) ) ).

% lambda_zero
thf(fact_655_measurable__Pair2__compose,axiom,
    ! [F: extend8495563244428889912nnreal > real,G: extend8495563244428889912nnreal > real,M: sigma_7234349610311085201nnreal,M1: sigma_measure_real,M22: sigma_measure_real,H: a > extend8495563244428889912nnreal,N: sigma_measure_a] :
      ( ( member7354208599470296673l_real
        @ ^ [X: extend8495563244428889912nnreal] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_1014563338549229999l_real @ M @ ( binary6478037234023840930l_real @ M1 @ M22 ) ) )
     => ( ( member298456594901751504nnreal @ H @ ( sigma_214952329563889126nnreal @ N @ M ) )
       => ( member_a_real
          @ ^ [X: a] : ( G @ ( H @ X ) )
          @ ( sigma_9116425665531756122a_real @ N @ M22 ) ) ) ) ).

% measurable_Pair2_compose
thf(fact_656_measurable__Pair2__compose,axiom,
    ! [F: a > real,G: a > real,M: sigma_measure_a,M1: sigma_measure_real,M22: sigma_measure_real,H: real > a,N: sigma_measure_real] :
      ( ( member2229928074028245815l_real
        @ ^ [X: a] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_414277600898586891l_real @ M @ ( binary6478037234023840930l_real @ M1 @ M22 ) ) )
     => ( ( member_real_a @ H @ ( sigma_523072396149930112real_a @ N @ M ) )
       => ( member_real_real
          @ ^ [X: real] : ( G @ ( H @ X ) )
          @ ( sigma_5267869275261027754l_real @ N @ M22 ) ) ) ) ).

% measurable_Pair2_compose
thf(fact_657_measurable__Pair2__compose,axiom,
    ! [F: real > real,G: real > real,M: sigma_measure_real,M1: sigma_measure_real,M22: sigma_measure_real,H: a > real,N: sigma_measure_a] :
      ( ( member9086635009091248365l_real
        @ ^ [X: real] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_7998147297565726139l_real @ M @ ( binary6478037234023840930l_real @ M1 @ M22 ) ) )
     => ( ( member_a_real @ H @ ( sigma_9116425665531756122a_real @ N @ M ) )
       => ( member_a_real
          @ ^ [X: a] : ( G @ ( H @ X ) )
          @ ( sigma_9116425665531756122a_real @ N @ M22 ) ) ) ) ).

% measurable_Pair2_compose
thf(fact_658_measurable__Pair2__compose,axiom,
    ! [F: real > real,G: real > real,M: sigma_measure_real,M1: sigma_measure_real,M22: sigma_measure_real,H: real > real,N: sigma_measure_real] :
      ( ( member9086635009091248365l_real
        @ ^ [X: real] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_7998147297565726139l_real @ M @ ( binary6478037234023840930l_real @ M1 @ M22 ) ) )
     => ( ( member_real_real @ H @ ( sigma_5267869275261027754l_real @ N @ M ) )
       => ( member_real_real
          @ ^ [X: real] : ( G @ ( H @ X ) )
          @ ( sigma_5267869275261027754l_real @ N @ M22 ) ) ) ) ).

% measurable_Pair2_compose
thf(fact_659_measurable__Pair2__compose,axiom,
    ! [F: extend8495563244428889912nnreal > real,G: extend8495563244428889912nnreal > real,M: sigma_7234349610311085201nnreal,M1: sigma_measure_real,M22: sigma_measure_real,H: real > extend8495563244428889912nnreal,N: sigma_measure_real] :
      ( ( member7354208599470296673l_real
        @ ^ [X: extend8495563244428889912nnreal] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_1014563338549229999l_real @ M @ ( binary6478037234023840930l_real @ M1 @ M22 ) ) )
     => ( ( member2919562650594848410nnreal @ H @ ( sigma_9017504469962657078nnreal @ N @ M ) )
       => ( member_real_real
          @ ^ [X: real] : ( G @ ( H @ X ) )
          @ ( sigma_5267869275261027754l_real @ N @ M22 ) ) ) ) ).

% measurable_Pair2_compose
thf(fact_660_measurable__Pair2__compose,axiom,
    ! [F: complex > real,G: complex > real,M: sigma_3077487657436305159omplex,M1: sigma_measure_real,M22: sigma_measure_real,H: real > complex,N: sigma_measure_real] :
      ( ( member7250751282632616811l_real
        @ ^ [X: complex] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_1557885496330159289l_real @ M @ ( binary6478037234023840930l_real @ M1 @ M22 ) ) )
     => ( ( member_real_complex @ H @ ( sigma_9111916201866572460omplex @ N @ M ) )
       => ( member_real_real
          @ ^ [X: real] : ( G @ ( H @ X ) )
          @ ( sigma_5267869275261027754l_real @ N @ M22 ) ) ) ) ).

% measurable_Pair2_compose
thf(fact_661_measurable__Pair2__compose,axiom,
    ! [F: a > real > a,G: a > sigma_measure_real,M: sigma_measure_a,M1: sigma_measure_real_a,M22: sigma_8927737637348964610e_real,H: real > a,N: sigma_measure_real] :
      ( ( member7666477768501999713e_real
        @ ^ [X: a] : ( produc623176010801490259e_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_6134520262524792629e_real @ M @ ( binary2119006201073916036e_real @ M1 @ M22 ) ) )
     => ( ( member_real_a @ H @ ( sigma_523072396149930112real_a @ N @ M ) )
       => ( member2630560753458908601e_real
          @ ^ [X: real] : ( G @ ( H @ X ) )
          @ ( sigma_5928869325259027335e_real @ N @ M22 ) ) ) ) ).

% measurable_Pair2_compose
thf(fact_662_measurable__Pair2__compose,axiom,
    ! [F: extend8495563244428889912nnreal > real > a,G: extend8495563244428889912nnreal > sigma_measure_real,M: sigma_7234349610311085201nnreal,M1: sigma_measure_real_a,M22: sigma_8927737637348964610e_real,H: a > extend8495563244428889912nnreal,N: sigma_measure_a] :
      ( ( member3748378782928841611e_real
        @ ^ [X: extend8495563244428889912nnreal] : ( produc623176010801490259e_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_8669861389021650905e_real @ M @ ( binary2119006201073916036e_real @ M1 @ M22 ) ) )
     => ( ( member298456594901751504nnreal @ H @ ( sigma_214952329563889126nnreal @ N @ M ) )
       => ( member997712892182982147e_real
          @ ^ [X: a] : ( G @ ( H @ X ) )
          @ ( sigma_3032266283304642263e_real @ N @ M22 ) ) ) ) ).

% measurable_Pair2_compose
thf(fact_663_measurable__Pair2__compose,axiom,
    ! [F: real > real > a,G: real > sigma_measure_real,M: sigma_measure_real,M1: sigma_measure_real_a,M22: sigma_8927737637348964610e_real,H: a > real,N: sigma_measure_a] :
      ( ( member4796208372699065879e_real
        @ ^ [X: real] : ( produc623176010801490259e_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_4238319853704539109e_real @ M @ ( binary2119006201073916036e_real @ M1 @ M22 ) ) )
     => ( ( member_a_real @ H @ ( sigma_9116425665531756122a_real @ N @ M ) )
       => ( member997712892182982147e_real
          @ ^ [X: a] : ( G @ ( H @ X ) )
          @ ( sigma_3032266283304642263e_real @ N @ M22 ) ) ) ) ).

% measurable_Pair2_compose
thf(fact_664_measurable__Pair2__compose,axiom,
    ! [F: real > real > a,G: real > sigma_measure_real,M: sigma_measure_real,M1: sigma_measure_real_a,M22: sigma_8927737637348964610e_real,H: real > real,N: sigma_measure_real] :
      ( ( member4796208372699065879e_real
        @ ^ [X: real] : ( produc623176010801490259e_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_4238319853704539109e_real @ M @ ( binary2119006201073916036e_real @ M1 @ M22 ) ) )
     => ( ( member_real_real @ H @ ( sigma_5267869275261027754l_real @ N @ M ) )
       => ( member2630560753458908601e_real
          @ ^ [X: real] : ( G @ ( H @ X ) )
          @ ( sigma_5928869325259027335e_real @ N @ M22 ) ) ) ) ).

% measurable_Pair2_compose
thf(fact_665_measurable__Pair1__compose,axiom,
    ! [F: extend8495563244428889912nnreal > real,G: extend8495563244428889912nnreal > real,M: sigma_7234349610311085201nnreal,M1: sigma_measure_real,M22: sigma_measure_real,H: a > extend8495563244428889912nnreal,N: sigma_measure_a] :
      ( ( member7354208599470296673l_real
        @ ^ [X: extend8495563244428889912nnreal] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_1014563338549229999l_real @ M @ ( binary6478037234023840930l_real @ M1 @ M22 ) ) )
     => ( ( member298456594901751504nnreal @ H @ ( sigma_214952329563889126nnreal @ N @ M ) )
       => ( member_a_real
          @ ^ [X: a] : ( F @ ( H @ X ) )
          @ ( sigma_9116425665531756122a_real @ N @ M1 ) ) ) ) ).

% measurable_Pair1_compose
thf(fact_666_measurable__Pair1__compose,axiom,
    ! [F: a > real,G: a > real,M: sigma_measure_a,M1: sigma_measure_real,M22: sigma_measure_real,H: real > a,N: sigma_measure_real] :
      ( ( member2229928074028245815l_real
        @ ^ [X: a] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_414277600898586891l_real @ M @ ( binary6478037234023840930l_real @ M1 @ M22 ) ) )
     => ( ( member_real_a @ H @ ( sigma_523072396149930112real_a @ N @ M ) )
       => ( member_real_real
          @ ^ [X: real] : ( F @ ( H @ X ) )
          @ ( sigma_5267869275261027754l_real @ N @ M1 ) ) ) ) ).

% measurable_Pair1_compose
thf(fact_667_measurable__Pair1__compose,axiom,
    ! [F: real > real,G: real > real,M: sigma_measure_real,M1: sigma_measure_real,M22: sigma_measure_real,H: a > real,N: sigma_measure_a] :
      ( ( member9086635009091248365l_real
        @ ^ [X: real] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_7998147297565726139l_real @ M @ ( binary6478037234023840930l_real @ M1 @ M22 ) ) )
     => ( ( member_a_real @ H @ ( sigma_9116425665531756122a_real @ N @ M ) )
       => ( member_a_real
          @ ^ [X: a] : ( F @ ( H @ X ) )
          @ ( sigma_9116425665531756122a_real @ N @ M1 ) ) ) ) ).

% measurable_Pair1_compose
thf(fact_668_measurable__Pair1__compose,axiom,
    ! [F: real > real,G: real > real,M: sigma_measure_real,M1: sigma_measure_real,M22: sigma_measure_real,H: real > real,N: sigma_measure_real] :
      ( ( member9086635009091248365l_real
        @ ^ [X: real] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_7998147297565726139l_real @ M @ ( binary6478037234023840930l_real @ M1 @ M22 ) ) )
     => ( ( member_real_real @ H @ ( sigma_5267869275261027754l_real @ N @ M ) )
       => ( member_real_real
          @ ^ [X: real] : ( F @ ( H @ X ) )
          @ ( sigma_5267869275261027754l_real @ N @ M1 ) ) ) ) ).

% measurable_Pair1_compose
thf(fact_669_measurable__Pair1__compose,axiom,
    ! [F: extend8495563244428889912nnreal > real,G: extend8495563244428889912nnreal > real,M: sigma_7234349610311085201nnreal,M1: sigma_measure_real,M22: sigma_measure_real,H: real > extend8495563244428889912nnreal,N: sigma_measure_real] :
      ( ( member7354208599470296673l_real
        @ ^ [X: extend8495563244428889912nnreal] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_1014563338549229999l_real @ M @ ( binary6478037234023840930l_real @ M1 @ M22 ) ) )
     => ( ( member2919562650594848410nnreal @ H @ ( sigma_9017504469962657078nnreal @ N @ M ) )
       => ( member_real_real
          @ ^ [X: real] : ( F @ ( H @ X ) )
          @ ( sigma_5267869275261027754l_real @ N @ M1 ) ) ) ) ).

% measurable_Pair1_compose
thf(fact_670_measurable__Pair1__compose,axiom,
    ! [F: complex > real,G: complex > real,M: sigma_3077487657436305159omplex,M1: sigma_measure_real,M22: sigma_measure_real,H: real > complex,N: sigma_measure_real] :
      ( ( member7250751282632616811l_real
        @ ^ [X: complex] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_1557885496330159289l_real @ M @ ( binary6478037234023840930l_real @ M1 @ M22 ) ) )
     => ( ( member_real_complex @ H @ ( sigma_9111916201866572460omplex @ N @ M ) )
       => ( member_real_real
          @ ^ [X: real] : ( F @ ( H @ X ) )
          @ ( sigma_5267869275261027754l_real @ N @ M1 ) ) ) ) ).

% measurable_Pair1_compose
thf(fact_671_measurable__Pair1__compose,axiom,
    ! [F: a > real > a,G: a > sigma_measure_real,M: sigma_measure_a,M1: sigma_measure_real_a,M22: sigma_8927737637348964610e_real,H: real > a,N: sigma_measure_real] :
      ( ( member7666477768501999713e_real
        @ ^ [X: a] : ( produc623176010801490259e_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_6134520262524792629e_real @ M @ ( binary2119006201073916036e_real @ M1 @ M22 ) ) )
     => ( ( member_real_a @ H @ ( sigma_523072396149930112real_a @ N @ M ) )
       => ( member_real_real_a
          @ ^ [X: real] : ( F @ ( H @ X ) )
          @ ( sigma_5735160441797593099real_a @ N @ M1 ) ) ) ) ).

% measurable_Pair1_compose
thf(fact_672_measurable__Pair1__compose,axiom,
    ! [F: extend8495563244428889912nnreal > real > a,G: extend8495563244428889912nnreal > sigma_measure_real,M: sigma_7234349610311085201nnreal,M1: sigma_measure_real_a,M22: sigma_8927737637348964610e_real,H: a > extend8495563244428889912nnreal,N: sigma_measure_a] :
      ( ( member3748378782928841611e_real
        @ ^ [X: extend8495563244428889912nnreal] : ( produc623176010801490259e_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_8669861389021650905e_real @ M @ ( binary2119006201073916036e_real @ M1 @ M22 ) ) )
     => ( ( member298456594901751504nnreal @ H @ ( sigma_214952329563889126nnreal @ N @ M ) )
       => ( member_a_real_a
          @ ^ [X: a] : ( F @ ( H @ X ) )
          @ ( sigma_5590391210564117339real_a @ N @ M1 ) ) ) ) ).

% measurable_Pair1_compose
thf(fact_673_measurable__Pair1__compose,axiom,
    ! [F: real > real > a,G: real > sigma_measure_real,M: sigma_measure_real,M1: sigma_measure_real_a,M22: sigma_8927737637348964610e_real,H: a > real,N: sigma_measure_a] :
      ( ( member4796208372699065879e_real
        @ ^ [X: real] : ( produc623176010801490259e_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_4238319853704539109e_real @ M @ ( binary2119006201073916036e_real @ M1 @ M22 ) ) )
     => ( ( member_a_real @ H @ ( sigma_9116425665531756122a_real @ N @ M ) )
       => ( member_a_real_a
          @ ^ [X: a] : ( F @ ( H @ X ) )
          @ ( sigma_5590391210564117339real_a @ N @ M1 ) ) ) ) ).

% measurable_Pair1_compose
thf(fact_674_measurable__Pair1__compose,axiom,
    ! [F: real > real > a,G: real > sigma_measure_real,M: sigma_measure_real,M1: sigma_measure_real_a,M22: sigma_8927737637348964610e_real,H: real > real,N: sigma_measure_real] :
      ( ( member4796208372699065879e_real
        @ ^ [X: real] : ( produc623176010801490259e_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_4238319853704539109e_real @ M @ ( binary2119006201073916036e_real @ M1 @ M22 ) ) )
     => ( ( member_real_real @ H @ ( sigma_5267869275261027754l_real @ N @ M ) )
       => ( member_real_real_a
          @ ^ [X: real] : ( F @ ( H @ X ) )
          @ ( sigma_5735160441797593099real_a @ N @ M1 ) ) ) ) ).

% measurable_Pair1_compose
thf(fact_675_qp_Oentropy__def,axiom,
    ! [B: real,S2: sigma_measure_a,X2: real > a] :
      ( ( prob_entropy_real_a @ mu @ B @ S2 @ X2 )
      = ( uminus_uminus_real @ ( kL_divergence_a @ B @ S2 @ ( measure_distr_real_a @ mu @ S2 @ X2 ) ) ) ) ).

% qp.entropy_def
thf(fact_676_qp_Oentropy__def,axiom,
    ! [B: real,S2: sigma_measure_real,X2: real > real] :
      ( ( prob_e6953316728393294858l_real @ mu @ B @ S2 @ X2 )
      = ( uminus_uminus_real @ ( kL_divergence_real @ B @ S2 @ ( measur2993149975067245138l_real @ mu @ S2 @ X2 ) ) ) ) ).

% qp.entropy_def
thf(fact_677_qp_Oborel__measurable__nn__integral,axiom,
    ! [F: a > real > extend8495563244428889912nnreal,N: sigma_measure_a] :
      ( ( member4437793228276457543nnreal @ ( produc5555093792979918904nnreal @ F ) @ ( sigma_8117956432444187619nnreal @ ( binary932748531126180194a_real @ N @ mu ) @ borel_6524799422816628122nnreal ) )
     => ( member298456594901751504nnreal
        @ ^ [X: a] : ( nonneg2667834350952324695l_real @ mu @ ( F @ X ) )
        @ ( sigma_214952329563889126nnreal @ N @ borel_6524799422816628122nnreal ) ) ) ).

% qp.borel_measurable_nn_integral
thf(fact_678_qp_Oborel__measurable__nn__integral,axiom,
    ! [F: real > real > extend8495563244428889912nnreal,N: sigma_measure_real] :
      ( ( member2245694452317284363nnreal @ ( produc4590977785667036862nnreal @ F ) @ ( sigma_219902274609641377nnreal @ ( binary6478037234023840930l_real @ N @ mu ) @ borel_6524799422816628122nnreal ) )
     => ( member2919562650594848410nnreal
        @ ^ [X: real] : ( nonneg2667834350952324695l_real @ mu @ ( F @ X ) )
        @ ( sigma_9017504469962657078nnreal @ N @ borel_6524799422816628122nnreal ) ) ) ).

% qp.borel_measurable_nn_integral
thf(fact_679_qp_Oborel__measurable__lebesgue__integral,axiom,
    ! [F: a > real > real,N: sigma_measure_a] :
      ( ( member2203687483360836539l_real @ ( produc2387036547305310124l_real @ F ) @ ( sigma_2779692123338079703l_real @ ( binary932748531126180194a_real @ N @ mu ) @ borel_5078946678739801102l_real ) )
     => ( member_a_real
        @ ^ [X: a] : ( bochne3715101410578510557l_real @ mu @ ( F @ X ) )
        @ ( sigma_9116425665531756122a_real @ N @ borel_5078946678739801102l_real ) ) ) ).

% qp.borel_measurable_lebesgue_integral
thf(fact_680_qp_Oborel__measurable__lebesgue__integral,axiom,
    ! [F: real > real > real,N: sigma_measure_real] :
      ( ( member6699615393305559423l_real @ ( produc313441363659479858l_real @ F ) @ ( sigma_8002782794886939285l_real @ ( binary6478037234023840930l_real @ N @ mu ) @ borel_5078946678739801102l_real ) )
     => ( member_real_real
        @ ^ [X: real] : ( bochne3715101410578510557l_real @ mu @ ( F @ X ) )
        @ ( sigma_5267869275261027754l_real @ N @ borel_5078946678739801102l_real ) ) ) ).

% qp.borel_measurable_lebesgue_integral
thf(fact_681_qp_Oborel__measurable__lebesgue__integral,axiom,
    ! [F: real > real > complex,N: sigma_measure_real] :
      ( ( member6750940717423799937omplex @ ( produc7870980171412995124omplex @ F ) @ ( sigma_5308840563538681623omplex @ ( binary6478037234023840930l_real @ N @ mu ) @ borel_1392132677378845456omplex ) )
     => ( member_real_complex
        @ ^ [X: real] : ( bochne8865740171307459423omplex @ mu @ ( F @ X ) )
        @ ( sigma_9111916201866572460omplex @ N @ borel_1392132677378845456omplex ) ) ) ).

% qp.borel_measurable_lebesgue_integral
thf(fact_682_integrableE,axiom,
    ! [M: sigma_measure_real,F: real > real] :
      ( ( bochne3340023020068487468l_real @ M @ F )
     => ~ ! [R3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ R3 )
           => ! [Q: real] :
                ( ( ord_less_eq_real @ zero_zero_real @ Q )
               => ( ( ( nonneg2667834350952324695l_real @ M
                      @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( F @ X ) ) )
                    = ( extend7643940197134561352nnreal @ R3 ) )
                 => ( ( ( nonneg2667834350952324695l_real @ M
                        @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( uminus_uminus_real @ ( F @ X ) ) ) )
                      = ( extend7643940197134561352nnreal @ Q ) )
                   => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
                     => ( ( bochne3715101410578510557l_real @ M @ F )
                       != ( minus_minus_real @ R3 @ Q ) ) ) ) ) ) ) ) ).

% integrableE
thf(fact_683_integrableE,axiom,
    ! [M: sigma_measure_a,F: a > real] :
      ( ( bochne2139062162225249880a_real @ M @ F )
     => ~ ! [R3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ R3 )
           => ! [Q: real] :
                ( ( ord_less_eq_real @ zero_zero_real @ Q )
               => ( ( ( nonneg2725512125972007571gral_a @ M
                      @ ^ [X: a] : ( extend7643940197134561352nnreal @ ( F @ X ) ) )
                    = ( extend7643940197134561352nnreal @ R3 ) )
                 => ( ( ( nonneg2725512125972007571gral_a @ M
                        @ ^ [X: a] : ( extend7643940197134561352nnreal @ ( uminus_uminus_real @ ( F @ X ) ) ) )
                      = ( extend7643940197134561352nnreal @ Q ) )
                   => ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
                     => ( ( bochne378719280626478695a_real @ M @ F )
                       != ( minus_minus_real @ R3 @ Q ) ) ) ) ) ) ) ) ).

% integrableE
thf(fact_684_qp_Oindep__vars__compose,axiom,
    ! [M2: ( real > complex ) > sigma_measure_real,X2: ( real > complex ) > real > real,I2: set_real_complex,Y: ( real > complex ) > real > a,N: ( real > complex ) > sigma_measure_a] :
      ( ( indepe3215717721046027291x_real @ mu @ M2 @ X2 @ I2 )
     => ( ! [I: real > complex] :
            ( ( member_real_complex @ I @ I2 )
           => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe1200421579086570447plex_a @ mu @ N
          @ ^ [I3: real > complex] : ( comp_real_a_real @ ( Y @ I3 ) @ ( X2 @ I3 ) )
          @ I2 ) ) ) ).

% qp.indep_vars_compose
thf(fact_685_qp_Oindep__vars__compose,axiom,
    ! [M2: ( real > complex ) > sigma_measure_a,X2: ( real > complex ) > real > a,I2: set_real_complex,Y: ( real > complex ) > a > extend8495563244428889912nnreal,N: ( real > complex ) > sigma_7234349610311085201nnreal] :
      ( ( indepe1200421579086570447plex_a @ mu @ M2 @ X2 @ I2 )
     => ( ! [I: real > complex] :
            ( ( member_real_complex @ I @ I2 )
           => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe6000904806191903783nnreal @ mu @ N
          @ ^ [I3: real > complex] : ( comp_a8249376463644563905l_real @ ( Y @ I3 ) @ ( X2 @ I3 ) )
          @ I2 ) ) ) ).

% qp.indep_vars_compose
thf(fact_686_qp_Oindep__vars__compose,axiom,
    ! [M2: ( real > real ) > sigma_measure_real,X2: ( real > real ) > real > real,I2: set_real_real,Y: ( real > real ) > real > a,N: ( real > real ) > sigma_measure_a] :
      ( ( indepe9089129998381042585l_real @ mu @ M2 @ X2 @ I2 )
     => ( ! [I: real > real] :
            ( ( member_real_real @ I @ I2 )
           => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe1927354855876929745real_a @ mu @ N
          @ ^ [I3: real > real] : ( comp_real_a_real @ ( Y @ I3 ) @ ( X2 @ I3 ) )
          @ I2 ) ) ) ).

% qp.indep_vars_compose
thf(fact_687_qp_Oindep__vars__compose,axiom,
    ! [M2: ( real > real ) > sigma_measure_a,X2: ( real > real ) > real > a,I2: set_real_real,Y: ( real > real ) > a > extend8495563244428889912nnreal,N: ( real > real ) > sigma_7234349610311085201nnreal] :
      ( ( indepe1927354855876929745real_a @ mu @ M2 @ X2 @ I2 )
     => ( ! [I: real > real] :
            ( ( member_real_real @ I @ I2 )
           => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe501623583335441061nnreal @ mu @ N
          @ ^ [I3: real > real] : ( comp_a8249376463644563905l_real @ ( Y @ I3 ) @ ( X2 @ I3 ) )
          @ I2 ) ) ) ).

% qp.indep_vars_compose
thf(fact_688_qp_Oindep__vars__compose,axiom,
    ! [M2: ( real > a ) > sigma_measure_real,X2: ( real > a ) > real > real,I2: set_real_a,Y: ( real > a ) > real > a,N: ( real > a ) > sigma_measure_a] :
      ( ( indepe6457644772562392769a_real @ mu @ M2 @ X2 @ I2 )
     => ( ! [I: real > a] :
            ( ( member_real_a @ I @ I2 )
           => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe357751042618000297al_a_a @ mu @ N
          @ ^ [I3: real > a] : ( comp_real_a_real @ ( Y @ I3 ) @ ( X2 @ I3 ) )
          @ I2 ) ) ) ).

% qp.indep_vars_compose
thf(fact_689_qp_Oindep__vars__compose,axiom,
    ! [M2: ( real > a ) > sigma_measure_a,X2: ( real > a ) > real > a,I2: set_real_a,Y: ( real > a ) > a > extend8495563244428889912nnreal,N: ( real > a ) > sigma_7234349610311085201nnreal] :
      ( ( indepe357751042618000297al_a_a @ mu @ M2 @ X2 @ I2 )
     => ( ! [I: real > a] :
            ( ( member_real_a @ I @ I2 )
           => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe1181869626592248269nnreal @ mu @ N
          @ ^ [I3: real > a] : ( comp_a8249376463644563905l_real @ ( Y @ I3 ) @ ( X2 @ I3 ) )
          @ I2 ) ) ) ).

% qp.indep_vars_compose
thf(fact_690_qp_Oindep__vars__compose,axiom,
    ! [M2: ( a > extend8495563244428889912nnreal ) > sigma_measure_real,X2: ( a > extend8495563244428889912nnreal ) > real > real,I2: set_a_7161065143582548615nnreal,Y: ( a > extend8495563244428889912nnreal ) > real > a,N: ( a > extend8495563244428889912nnreal ) > sigma_measure_a] :
      ( ( indepe1680866314667358175l_real @ mu @ M2 @ X2 @ I2 )
     => ( ! [I: a > extend8495563244428889912nnreal] :
            ( ( member298456594901751504nnreal @ I @ I2 )
           => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe8228798008660262667real_a @ mu @ N
          @ ^ [I3: a > extend8495563244428889912nnreal] : ( comp_real_a_real @ ( Y @ I3 ) @ ( X2 @ I3 ) )
          @ I2 ) ) ) ).

% qp.indep_vars_compose
thf(fact_691_qp_Oindep__vars__compose,axiom,
    ! [M2: ( a > extend8495563244428889912nnreal ) > sigma_measure_a,X2: ( a > extend8495563244428889912nnreal ) > real > a,I2: set_a_7161065143582548615nnreal,Y: ( a > extend8495563244428889912nnreal ) > a > extend8495563244428889912nnreal,N: ( a > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal] :
      ( ( indepe8228798008660262667real_a @ mu @ M2 @ X2 @ I2 )
     => ( ! [I: a > extend8495563244428889912nnreal] :
            ( ( member298456594901751504nnreal @ I @ I2 )
           => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe4617722330935435755nnreal @ mu @ N
          @ ^ [I3: a > extend8495563244428889912nnreal] : ( comp_a8249376463644563905l_real @ ( Y @ I3 ) @ ( X2 @ I3 ) )
          @ I2 ) ) ) ).

% qp.indep_vars_compose
thf(fact_692_qp_Oindep__vars__compose,axiom,
    ! [M2: ( a > real ) > sigma_measure_real,X2: ( a > real ) > real > real,I2: set_a_real,Y: ( a > real ) > real > a,N: ( a > real ) > sigma_measure_a] :
      ( ( indepe722220561065814995l_real @ mu @ M2 @ X2 @ I2 )
     => ( ! [I: a > real] :
            ( ( member_a_real @ I @ I2 )
           => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe3565983053654730263real_a @ mu @ N
          @ ^ [I3: a > real] : ( comp_real_a_real @ ( Y @ I3 ) @ ( X2 @ I3 ) )
          @ I2 ) ) ) ).

% qp.indep_vars_compose
thf(fact_693_qp_Oindep__vars__compose,axiom,
    ! [M2: ( a > real ) > sigma_measure_a,X2: ( a > real ) > real > a,I2: set_a_real,Y: ( a > real ) > a > extend8495563244428889912nnreal,N: ( a > real ) > sigma_7234349610311085201nnreal] :
      ( ( indepe3565983053654730263real_a @ mu @ M2 @ X2 @ I2 )
     => ( ! [I: a > real] :
            ( ( member_a_real @ I @ I2 )
           => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe6561983776359739359nnreal @ mu @ N
          @ ^ [I3: a > real] : ( comp_a8249376463644563905l_real @ ( Y @ I3 ) @ ( X2 @ I3 ) )
          @ I2 ) ) ) ).

% qp.indep_vars_compose
thf(fact_694_qp_Ochar__measurable,axiom,
    member_real_complex @ ( characteristic_char @ mu ) @ ( sigma_9111916201866572460omplex @ borel_5078946678739801102l_real @ borel_1392132677378845456omplex ) ).

% qp.char_measurable
thf(fact_695_qp_Oindep__vars__compose2,axiom,
    ! [M2: ( real > complex ) > sigma_measure_real,X2: ( real > complex ) > real > real,I2: set_real_complex,Y: ( real > complex ) > real > a,N: ( real > complex ) > sigma_measure_a] :
      ( ( indepe3215717721046027291x_real @ mu @ M2 @ X2 @ I2 )
     => ( ! [I: real > complex] :
            ( ( member_real_complex @ I @ I2 )
           => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe1200421579086570447plex_a @ mu @ N
          @ ^ [I3: real > complex,X: real] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
          @ I2 ) ) ) ).

% qp.indep_vars_compose2
thf(fact_696_qp_Oindep__vars__compose2,axiom,
    ! [M2: ( real > complex ) > sigma_measure_a,X2: ( real > complex ) > real > a,I2: set_real_complex,Y: ( real > complex ) > a > extend8495563244428889912nnreal,N: ( real > complex ) > sigma_7234349610311085201nnreal] :
      ( ( indepe1200421579086570447plex_a @ mu @ M2 @ X2 @ I2 )
     => ( ! [I: real > complex] :
            ( ( member_real_complex @ I @ I2 )
           => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe6000904806191903783nnreal @ mu @ N
          @ ^ [I3: real > complex,X: real] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
          @ I2 ) ) ) ).

% qp.indep_vars_compose2
thf(fact_697_qp_Oindep__vars__compose2,axiom,
    ! [M2: ( real > real ) > sigma_measure_real,X2: ( real > real ) > real > real,I2: set_real_real,Y: ( real > real ) > real > a,N: ( real > real ) > sigma_measure_a] :
      ( ( indepe9089129998381042585l_real @ mu @ M2 @ X2 @ I2 )
     => ( ! [I: real > real] :
            ( ( member_real_real @ I @ I2 )
           => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe1927354855876929745real_a @ mu @ N
          @ ^ [I3: real > real,X: real] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
          @ I2 ) ) ) ).

% qp.indep_vars_compose2
thf(fact_698_qp_Oindep__vars__compose2,axiom,
    ! [M2: ( real > real ) > sigma_measure_a,X2: ( real > real ) > real > a,I2: set_real_real,Y: ( real > real ) > a > extend8495563244428889912nnreal,N: ( real > real ) > sigma_7234349610311085201nnreal] :
      ( ( indepe1927354855876929745real_a @ mu @ M2 @ X2 @ I2 )
     => ( ! [I: real > real] :
            ( ( member_real_real @ I @ I2 )
           => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe501623583335441061nnreal @ mu @ N
          @ ^ [I3: real > real,X: real] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
          @ I2 ) ) ) ).

% qp.indep_vars_compose2
thf(fact_699_qp_Oindep__vars__compose2,axiom,
    ! [M2: ( real > a ) > sigma_measure_real,X2: ( real > a ) > real > real,I2: set_real_a,Y: ( real > a ) > real > a,N: ( real > a ) > sigma_measure_a] :
      ( ( indepe6457644772562392769a_real @ mu @ M2 @ X2 @ I2 )
     => ( ! [I: real > a] :
            ( ( member_real_a @ I @ I2 )
           => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe357751042618000297al_a_a @ mu @ N
          @ ^ [I3: real > a,X: real] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
          @ I2 ) ) ) ).

% qp.indep_vars_compose2
thf(fact_700_qp_Oindep__vars__compose2,axiom,
    ! [M2: ( real > a ) > sigma_measure_a,X2: ( real > a ) > real > a,I2: set_real_a,Y: ( real > a ) > a > extend8495563244428889912nnreal,N: ( real > a ) > sigma_7234349610311085201nnreal] :
      ( ( indepe357751042618000297al_a_a @ mu @ M2 @ X2 @ I2 )
     => ( ! [I: real > a] :
            ( ( member_real_a @ I @ I2 )
           => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe1181869626592248269nnreal @ mu @ N
          @ ^ [I3: real > a,X: real] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
          @ I2 ) ) ) ).

% qp.indep_vars_compose2
thf(fact_701_qp_Oindep__vars__compose2,axiom,
    ! [M2: ( a > extend8495563244428889912nnreal ) > sigma_measure_real,X2: ( a > extend8495563244428889912nnreal ) > real > real,I2: set_a_7161065143582548615nnreal,Y: ( a > extend8495563244428889912nnreal ) > real > a,N: ( a > extend8495563244428889912nnreal ) > sigma_measure_a] :
      ( ( indepe1680866314667358175l_real @ mu @ M2 @ X2 @ I2 )
     => ( ! [I: a > extend8495563244428889912nnreal] :
            ( ( member298456594901751504nnreal @ I @ I2 )
           => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe8228798008660262667real_a @ mu @ N
          @ ^ [I3: a > extend8495563244428889912nnreal,X: real] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
          @ I2 ) ) ) ).

% qp.indep_vars_compose2
thf(fact_702_qp_Oindep__vars__compose2,axiom,
    ! [M2: ( a > extend8495563244428889912nnreal ) > sigma_measure_a,X2: ( a > extend8495563244428889912nnreal ) > real > a,I2: set_a_7161065143582548615nnreal,Y: ( a > extend8495563244428889912nnreal ) > a > extend8495563244428889912nnreal,N: ( a > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal] :
      ( ( indepe8228798008660262667real_a @ mu @ M2 @ X2 @ I2 )
     => ( ! [I: a > extend8495563244428889912nnreal] :
            ( ( member298456594901751504nnreal @ I @ I2 )
           => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe4617722330935435755nnreal @ mu @ N
          @ ^ [I3: a > extend8495563244428889912nnreal,X: real] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
          @ I2 ) ) ) ).

% qp.indep_vars_compose2
thf(fact_703_qp_Oindep__vars__compose2,axiom,
    ! [M2: ( a > real ) > sigma_measure_real,X2: ( a > real ) > real > real,I2: set_a_real,Y: ( a > real ) > real > a,N: ( a > real ) > sigma_measure_a] :
      ( ( indepe722220561065814995l_real @ mu @ M2 @ X2 @ I2 )
     => ( ! [I: a > real] :
            ( ( member_a_real @ I @ I2 )
           => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe3565983053654730263real_a @ mu @ N
          @ ^ [I3: a > real,X: real] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
          @ I2 ) ) ) ).

% qp.indep_vars_compose2
thf(fact_704_qp_Oindep__vars__compose2,axiom,
    ! [M2: ( a > real ) > sigma_measure_a,X2: ( a > real ) > real > a,I2: set_a_real,Y: ( a > real ) > a > extend8495563244428889912nnreal,N: ( a > real ) > sigma_7234349610311085201nnreal] :
      ( ( indepe3565983053654730263real_a @ mu @ M2 @ X2 @ I2 )
     => ( ! [I: a > real] :
            ( ( member_a_real @ I @ I2 )
           => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe6561983776359739359nnreal @ mu @ N
          @ ^ [I3: a > real,X: real] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
          @ I2 ) ) ) ).

% qp.indep_vars_compose2
thf(fact_705_le__zero__eq,axiom,
    ! [N3: nat] :
      ( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
      = ( N3 = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_706_le__zero__eq,axiom,
    ! [N3: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ N3 @ zero_z7100319975126383169nnreal )
      = ( N3 = zero_z7100319975126383169nnreal ) ) ).

% le_zero_eq
thf(fact_707_neg__le__iff__le,axiom,
    ! [B: produc2422161461964618553l_real,A: produc2422161461964618553l_real] :
      ( ( ord_le1075799226346578649l_real @ ( uminus2141826702334040752l_real @ B ) @ ( uminus2141826702334040752l_real @ A ) )
      = ( ord_le1075799226346578649l_real @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_708_neg__le__iff__le,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_709_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
      = ( ord_less_eq_real @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_710_neg__less__eq__nonneg,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_711_less__eq__neg__nonpos,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% less_eq_neg_nonpos
thf(fact_712_neg__le__0__iff__le,axiom,
    ! [A: produc2422161461964618553l_real] :
      ( ( ord_le1075799226346578649l_real @ ( uminus2141826702334040752l_real @ A ) @ zero_z1365759597461889520l_real )
      = ( ord_le1075799226346578649l_real @ zero_z1365759597461889520l_real @ A ) ) ).

% neg_le_0_iff_le
thf(fact_713_neg__le__0__iff__le,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% neg_le_0_iff_le
thf(fact_714_neg__0__le__iff__le,axiom,
    ! [A: produc2422161461964618553l_real] :
      ( ( ord_le1075799226346578649l_real @ zero_z1365759597461889520l_real @ ( uminus2141826702334040752l_real @ A ) )
      = ( ord_le1075799226346578649l_real @ A @ zero_z1365759597461889520l_real ) ) ).

% neg_0_le_iff_le
thf(fact_715_neg__0__le__iff__le,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% neg_0_le_iff_le
thf(fact_716_ennreal__inj,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ( ( extend7643940197134561352nnreal @ A )
            = ( extend7643940197134561352nnreal @ B ) )
          = ( A = B ) ) ) ) ).

% ennreal_inj
thf(fact_717_ennreal__eq__zero__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ( extend7643940197134561352nnreal @ X4 )
          = zero_z7100319975126383169nnreal )
        = ( X4 = zero_zero_real ) ) ) ).

% ennreal_eq_zero_iff
thf(fact_718_enn2real__ennreal,axiom,
    ! [R4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R4 )
     => ( ( extend1669699412028896998n2real @ ( extend7643940197134561352nnreal @ R4 ) )
        = R4 ) ) ).

% enn2real_ennreal
thf(fact_719_complete__real,axiom,
    ! [S2: set_real] :
      ( ? [X8: real] : ( member_real @ X8 @ S2 )
     => ( ? [Z3: real] :
          ! [X6: real] :
            ( ( member_real @ X6 @ S2 )
           => ( ord_less_eq_real @ X6 @ Z3 ) )
       => ? [Y4: real] :
            ( ! [X8: real] :
                ( ( member_real @ X8 @ S2 )
               => ( ord_less_eq_real @ X8 @ Y4 ) )
            & ! [Z3: real] :
                ( ! [X6: real] :
                    ( ( member_real @ X6 @ S2 )
                   => ( ord_less_eq_real @ X6 @ Z3 ) )
               => ( ord_less_eq_real @ Y4 @ Z3 ) ) ) ) ) ).

% complete_real
thf(fact_720_verit__la__disequality,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
      | ~ ( ord_less_eq_real @ A @ B )
      | ~ ( ord_less_eq_real @ B @ A ) ) ).

% verit_la_disequality
thf(fact_721_verit__la__disequality,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( A = B )
      | ~ ( ord_le3935885782089961368nnreal @ A @ B )
      | ~ ( ord_le3935885782089961368nnreal @ B @ A ) ) ).

% verit_la_disequality
thf(fact_722_verit__comp__simplify1_I2_J,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_723_verit__comp__simplify1_I2_J,axiom,
    ! [A: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_724_zero__le,axiom,
    ! [X4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X4 ) ).

% zero_le
thf(fact_725_zero__le,axiom,
    ! [X4: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ X4 ) ).

% zero_le
thf(fact_726_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_eq_real @ A @ B )
        = ( ord_less_eq_real @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_727_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_728_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_729_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ D @ C )
       => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_730_le__imp__neg__le,axiom,
    ! [A: produc2422161461964618553l_real,B: produc2422161461964618553l_real] :
      ( ( ord_le1075799226346578649l_real @ A @ B )
     => ( ord_le1075799226346578649l_real @ ( uminus2141826702334040752l_real @ B ) @ ( uminus2141826702334040752l_real @ A ) ) ) ).

% le_imp_neg_le
thf(fact_731_le__imp__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% le_imp_neg_le
thf(fact_732_minus__le__iff,axiom,
    ! [A: produc2422161461964618553l_real,B: produc2422161461964618553l_real] :
      ( ( ord_le1075799226346578649l_real @ ( uminus2141826702334040752l_real @ A ) @ B )
      = ( ord_le1075799226346578649l_real @ ( uminus2141826702334040752l_real @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_733_minus__le__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_734_le__minus__iff,axiom,
    ! [A: produc2422161461964618553l_real,B: produc2422161461964618553l_real] :
      ( ( ord_le1075799226346578649l_real @ A @ ( uminus2141826702334040752l_real @ B ) )
      = ( ord_le1075799226346578649l_real @ B @ ( uminus2141826702334040752l_real @ A ) ) ) ).

% le_minus_iff
thf(fact_735_le__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
      = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).

% le_minus_iff
thf(fact_736_le__iff__diff__le__0,axiom,
    ( ord_less_eq_real
    = ( ^ [A5: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A5 @ B4 ) @ zero_zero_real ) ) ) ).

% le_iff_diff_le_0
thf(fact_737_real__minus__mult__self__le,axiom,
    ! [U: real,X4: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X4 @ X4 ) ) ).

% real_minus_mult_self_le
thf(fact_738_enn2real__nonneg,axiom,
    ! [X4: extend8495563244428889912nnreal] : ( ord_less_eq_real @ zero_zero_real @ ( extend1669699412028896998n2real @ X4 ) ) ).

% enn2real_nonneg
thf(fact_739_qbs__prob__measure__prob__space_Oindep__vars__compose2,axiom,
    ! [S: probab4737552673497767871pace_a,M2: ( real > complex ) > sigma_measure_real,X2: ( real > complex ) > a > real,I2: set_real_complex,Y: ( real > complex ) > real > a,N: ( real > complex ) > sigma_measure_a] :
      ( ( indepe2122272008937502813x_real @ ( probab7100426894406488384sure_a @ S ) @ M2 @ X2 @ I2 )
     => ( ! [I: real > complex] :
            ( ( member_real_complex @ I @ I2 )
           => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe7369748381470229261plex_a @ ( probab7100426894406488384sure_a @ S ) @ N
          @ ^ [I3: real > complex,X: a] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
          @ I2 ) ) ) ).

% qbs_prob_measure_prob_space.indep_vars_compose2
thf(fact_740_qbs__prob__measure__prob__space_Oindep__vars__compose2,axiom,
    ! [S: probab4737552673497767871pace_a,M2: ( real > complex ) > sigma_measure_a,X2: ( real > complex ) > a > a,I2: set_real_complex,Y: ( real > complex ) > a > extend8495563244428889912nnreal,N: ( real > complex ) > sigma_7234349610311085201nnreal] :
      ( ( indepe7369748381470229261plex_a @ ( probab7100426894406488384sure_a @ S ) @ M2 @ X2 @ I2 )
     => ( ! [I: real > complex] :
            ( ( member_real_complex @ I @ I2 )
           => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe2856984694482014569nnreal @ ( probab7100426894406488384sure_a @ S ) @ N
          @ ^ [I3: real > complex,X: a] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
          @ I2 ) ) ) ).

% qbs_prob_measure_prob_space.indep_vars_compose2
thf(fact_741_qbs__prob__measure__prob__space_Oindep__vars__compose2,axiom,
    ! [S: probab4737552673497767871pace_a,M2: ( real > real ) > sigma_measure_real,X2: ( real > real ) > a > real,I2: set_real_real,Y: ( real > real ) > real > a,N: ( real > real ) > sigma_measure_a] :
      ( ( indepe1495116825794919131l_real @ ( probab7100426894406488384sure_a @ S ) @ M2 @ X2 @ I2 )
     => ( ! [I: real > real] :
            ( ( member_real_real @ I @ I2 )
           => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe4370139003212481807real_a @ ( probab7100426894406488384sure_a @ S ) @ N
          @ ^ [I3: real > real,X: a] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
          @ I2 ) ) ) ).

% qbs_prob_measure_prob_space.indep_vars_compose2
thf(fact_742_qbs__prob__measure__prob__space_Oindep__vars__compose2,axiom,
    ! [S: probab4737552673497767871pace_a,M2: ( real > real ) > sigma_measure_a,X2: ( real > real ) > a > a,I2: set_real_real,Y: ( real > real ) > a > extend8495563244428889912nnreal,N: ( real > real ) > sigma_7234349610311085201nnreal] :
      ( ( indepe4370139003212481807real_a @ ( probab7100426894406488384sure_a @ S ) @ M2 @ X2 @ I2 )
     => ( ! [I: real > real] :
            ( ( member_real_real @ I @ I2 )
           => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe4467962090945823463nnreal @ ( probab7100426894406488384sure_a @ S ) @ N
          @ ^ [I3: real > real,X: a] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
          @ I2 ) ) ) ).

% qbs_prob_measure_prob_space.indep_vars_compose2
thf(fact_743_qbs__prob__measure__prob__space_Oindep__vars__compose2,axiom,
    ! [S: probab4737552673497767871pace_a,M2: ( real > a ) > sigma_measure_real,X2: ( real > a ) > a > real,I2: set_real_a,Y: ( real > a ) > real > a,N: ( real > a ) > sigma_measure_a] :
      ( ( indepe8900428919897944831a_real @ ( probab7100426894406488384sure_a @ S ) @ M2 @ X2 @ I2 )
     => ( ! [I: real > a] :
            ( ( member_real_a @ I @ I2 )
           => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe4717112320896891883al_a_a @ ( probab7100426894406488384sure_a @ S ) @ N
          @ ^ [I3: real > a,X: a] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
          @ I2 ) ) ) ).

% qbs_prob_measure_prob_space.indep_vars_compose2
thf(fact_744_qbs__prob__measure__prob__space_Oindep__vars__compose2,axiom,
    ! [S: probab4737552673497767871pace_a,M2: ( real > a ) > sigma_measure_a,X2: ( real > a ) > a > a,I2: set_real_a,Y: ( real > a ) > a > extend8495563244428889912nnreal,N: ( real > a ) > sigma_7234349610311085201nnreal] :
      ( ( indepe4717112320896891883al_a_a @ ( probab7100426894406488384sure_a @ S ) @ M2 @ X2 @ I2 )
     => ( ! [I: real > a] :
            ( ( member_real_a @ I @ I2 )
           => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe4152162118790851851nnreal @ ( probab7100426894406488384sure_a @ S ) @ N
          @ ^ [I3: real > a,X: a] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
          @ I2 ) ) ) ).

% qbs_prob_measure_prob_space.indep_vars_compose2
thf(fact_745_qbs__prob__measure__prob__space_Oindep__vars__compose2,axiom,
    ! [S: probab4737552673497767871pace_a,M2: ( a > extend8495563244428889912nnreal ) > sigma_measure_real,X2: ( a > extend8495563244428889912nnreal ) > a > real,I2: set_a_7161065143582548615nnreal,Y: ( a > extend8495563244428889912nnreal ) > real > a,N: ( a > extend8495563244428889912nnreal ) > sigma_measure_a] :
      ( ( indepe4651158806865961757l_real @ ( probab7100426894406488384sure_a @ S ) @ M2 @ X2 @ I2 )
     => ( ! [I: a > extend8495563244428889912nnreal] :
            ( ( member298456594901751504nnreal @ I @ I2 )
           => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe6798734769030654029real_a @ ( probab7100426894406488384sure_a @ S ) @ N
          @ ^ [I3: a > extend8495563244428889912nnreal,X: a] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
          @ I2 ) ) ) ).

% qbs_prob_measure_prob_space.indep_vars_compose2
thf(fact_746_qbs__prob__measure__prob__space_Oindep__vars__compose2,axiom,
    ! [S: probab4737552673497767871pace_a,M2: ( a > extend8495563244428889912nnreal ) > sigma_measure_a,X2: ( a > extend8495563244428889912nnreal ) > a > a,I2: set_a_7161065143582548615nnreal,Y: ( a > extend8495563244428889912nnreal ) > a > extend8495563244428889912nnreal,N: ( a > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal] :
      ( ( indepe6798734769030654029real_a @ ( probab7100426894406488384sure_a @ S ) @ M2 @ X2 @ I2 )
     => ( ! [I: a > extend8495563244428889912nnreal] :
            ( ( member298456594901751504nnreal @ I @ I2 )
           => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe8333295984245922857nnreal @ ( probab7100426894406488384sure_a @ S ) @ N
          @ ^ [I3: a > extend8495563244428889912nnreal,X: a] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
          @ I2 ) ) ) ).

% qbs_prob_measure_prob_space.indep_vars_compose2
thf(fact_747_qbs__prob__measure__prob__space_Oindep__vars__compose2,axiom,
    ! [S: probab4737552673497767871pace_a,M2: ( a > real ) > sigma_measure_real,X2: ( a > real ) > a > real,I2: set_a_real,Y: ( a > real ) > real > a,N: ( a > real ) > sigma_measure_a] :
      ( ( indepe3165004708401367057l_real @ ( probab7100426894406488384sure_a @ S ) @ M2 @ X2 @ I2 )
     => ( ! [I: a > real] :
            ( ( member_a_real @ I @ I2 )
           => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe7925344331933621849real_a @ ( probab7100426894406488384sure_a @ S ) @ N
          @ ^ [I3: a > real,X: a] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
          @ I2 ) ) ) ).

% qbs_prob_measure_prob_space.indep_vars_compose2
thf(fact_748_qbs__prob__measure__prob__space_Oindep__vars__compose2,axiom,
    ! [S: probab4737552673497767871pace_a,M2: ( a > real ) > sigma_measure_a,X2: ( a > real ) > a > a,I2: set_a_real,Y: ( a > real ) > a > extend8495563244428889912nnreal,N: ( a > real ) > sigma_7234349610311085201nnreal] :
      ( ( indepe7925344331933621849real_a @ ( probab7100426894406488384sure_a @ S ) @ M2 @ X2 @ I2 )
     => ( ! [I: a > real] :
            ( ( member_a_real @ I @ I2 )
           => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe308904231703567133nnreal @ ( probab7100426894406488384sure_a @ S ) @ N
          @ ^ [I3: a > real,X: a] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
          @ I2 ) ) ) ).

% qbs_prob_measure_prob_space.indep_vars_compose2
thf(fact_749_prob__space_Oindep__vars__compose2,axiom,
    ! [M: sigma_measure_real,M2: ( real > complex ) > sigma_measure_real,X2: ( real > complex ) > real > real,I2: set_real_complex,Y: ( real > complex ) > real > a,N: ( real > complex ) > sigma_measure_a] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe3215717721046027291x_real @ M @ M2 @ X2 @ I2 )
       => ( ! [I: real > complex] :
              ( ( member_real_complex @ I @ I2 )
             => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
         => ( indepe1200421579086570447plex_a @ M @ N
            @ ^ [I3: real > complex,X: real] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
            @ I2 ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_750_prob__space_Oindep__vars__compose2,axiom,
    ! [M: sigma_measure_real,M2: ( real > complex ) > sigma_measure_a,X2: ( real > complex ) > real > a,I2: set_real_complex,Y: ( real > complex ) > a > extend8495563244428889912nnreal,N: ( real > complex ) > sigma_7234349610311085201nnreal] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe1200421579086570447plex_a @ M @ M2 @ X2 @ I2 )
       => ( ! [I: real > complex] :
              ( ( member_real_complex @ I @ I2 )
             => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
         => ( indepe6000904806191903783nnreal @ M @ N
            @ ^ [I3: real > complex,X: real] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
            @ I2 ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_751_prob__space_Oindep__vars__compose2,axiom,
    ! [M: sigma_measure_real,M2: ( real > real ) > sigma_measure_real,X2: ( real > real ) > real > real,I2: set_real_real,Y: ( real > real ) > real > a,N: ( real > real ) > sigma_measure_a] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe9089129998381042585l_real @ M @ M2 @ X2 @ I2 )
       => ( ! [I: real > real] :
              ( ( member_real_real @ I @ I2 )
             => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
         => ( indepe1927354855876929745real_a @ M @ N
            @ ^ [I3: real > real,X: real] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
            @ I2 ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_752_prob__space_Oindep__vars__compose2,axiom,
    ! [M: sigma_measure_real,M2: ( real > real ) > sigma_measure_a,X2: ( real > real ) > real > a,I2: set_real_real,Y: ( real > real ) > a > extend8495563244428889912nnreal,N: ( real > real ) > sigma_7234349610311085201nnreal] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe1927354855876929745real_a @ M @ M2 @ X2 @ I2 )
       => ( ! [I: real > real] :
              ( ( member_real_real @ I @ I2 )
             => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
         => ( indepe501623583335441061nnreal @ M @ N
            @ ^ [I3: real > real,X: real] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
            @ I2 ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_753_prob__space_Oindep__vars__compose2,axiom,
    ! [M: sigma_measure_real,M2: ( real > a ) > sigma_measure_real,X2: ( real > a ) > real > real,I2: set_real_a,Y: ( real > a ) > real > a,N: ( real > a ) > sigma_measure_a] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe6457644772562392769a_real @ M @ M2 @ X2 @ I2 )
       => ( ! [I: real > a] :
              ( ( member_real_a @ I @ I2 )
             => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
         => ( indepe357751042618000297al_a_a @ M @ N
            @ ^ [I3: real > a,X: real] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
            @ I2 ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_754_prob__space_Oindep__vars__compose2,axiom,
    ! [M: sigma_measure_real,M2: ( real > a ) > sigma_measure_a,X2: ( real > a ) > real > a,I2: set_real_a,Y: ( real > a ) > a > extend8495563244428889912nnreal,N: ( real > a ) > sigma_7234349610311085201nnreal] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe357751042618000297al_a_a @ M @ M2 @ X2 @ I2 )
       => ( ! [I: real > a] :
              ( ( member_real_a @ I @ I2 )
             => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
         => ( indepe1181869626592248269nnreal @ M @ N
            @ ^ [I3: real > a,X: real] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
            @ I2 ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_755_prob__space_Oindep__vars__compose2,axiom,
    ! [M: sigma_measure_real,M2: ( a > extend8495563244428889912nnreal ) > sigma_measure_real,X2: ( a > extend8495563244428889912nnreal ) > real > real,I2: set_a_7161065143582548615nnreal,Y: ( a > extend8495563244428889912nnreal ) > real > a,N: ( a > extend8495563244428889912nnreal ) > sigma_measure_a] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe1680866314667358175l_real @ M @ M2 @ X2 @ I2 )
       => ( ! [I: a > extend8495563244428889912nnreal] :
              ( ( member298456594901751504nnreal @ I @ I2 )
             => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
         => ( indepe8228798008660262667real_a @ M @ N
            @ ^ [I3: a > extend8495563244428889912nnreal,X: real] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
            @ I2 ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_756_prob__space_Oindep__vars__compose2,axiom,
    ! [M: sigma_measure_real,M2: ( a > extend8495563244428889912nnreal ) > sigma_measure_a,X2: ( a > extend8495563244428889912nnreal ) > real > a,I2: set_a_7161065143582548615nnreal,Y: ( a > extend8495563244428889912nnreal ) > a > extend8495563244428889912nnreal,N: ( a > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe8228798008660262667real_a @ M @ M2 @ X2 @ I2 )
       => ( ! [I: a > extend8495563244428889912nnreal] :
              ( ( member298456594901751504nnreal @ I @ I2 )
             => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
         => ( indepe4617722330935435755nnreal @ M @ N
            @ ^ [I3: a > extend8495563244428889912nnreal,X: real] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
            @ I2 ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_757_prob__space_Oindep__vars__compose2,axiom,
    ! [M: sigma_measure_real,M2: ( a > real ) > sigma_measure_real,X2: ( a > real ) > real > real,I2: set_a_real,Y: ( a > real ) > real > a,N: ( a > real ) > sigma_measure_a] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe722220561065814995l_real @ M @ M2 @ X2 @ I2 )
       => ( ! [I: a > real] :
              ( ( member_a_real @ I @ I2 )
             => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
         => ( indepe3565983053654730263real_a @ M @ N
            @ ^ [I3: a > real,X: real] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
            @ I2 ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_758_prob__space_Oindep__vars__compose2,axiom,
    ! [M: sigma_measure_real,M2: ( a > real ) > sigma_measure_a,X2: ( a > real ) > real > a,I2: set_a_real,Y: ( a > real ) > a > extend8495563244428889912nnreal,N: ( a > real ) > sigma_7234349610311085201nnreal] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe3565983053654730263real_a @ M @ M2 @ X2 @ I2 )
       => ( ! [I: a > real] :
              ( ( member_a_real @ I @ I2 )
             => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
         => ( indepe6561983776359739359nnreal @ M @ N
            @ ^ [I3: a > real,X: real] : ( Y @ I3 @ ( X2 @ I3 @ X ) )
            @ I2 ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_759_measurable__Pair__compose__split,axiom,
    ! [F: a > a > a,M1: sigma_measure_a,M22: sigma_measure_a,N: sigma_measure_a,G: real > a,M: sigma_measure_real,H: real > a] :
      ( ( member1716570166360300819_a_a_a @ ( produc8815886927560695506_a_a_a @ F ) @ ( sigma_7861201367640403175_a_a_a @ ( binary867438762418767560re_a_a @ M1 @ M22 ) @ N ) )
     => ( ( member_real_a @ G @ ( sigma_523072396149930112real_a @ M @ M1 ) )
       => ( ( member_real_a @ H @ ( sigma_523072396149930112real_a @ M @ M22 ) )
         => ( member_real_a
            @ ^ [X: real] : ( F @ ( G @ X ) @ ( H @ X ) )
            @ ( sigma_523072396149930112real_a @ M @ N ) ) ) ) ) ).

% measurable_Pair_compose_split
thf(fact_760_measurable__Pair__compose__split,axiom,
    ! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal,M1: sigma_7234349610311085201nnreal,M22: sigma_7234349610311085201nnreal,N: sigma_7234349610311085201nnreal,G: a > extend8495563244428889912nnreal,M: sigma_measure_a,H: a > extend8495563244428889912nnreal] :
      ( ( member7009949782701513379nnreal @ ( produc8664085547722392150nnreal @ F ) @ ( sigma_4182148583319689017nnreal @ ( binary3098606844978005306nnreal @ M1 @ M22 ) @ N ) )
     => ( ( member298456594901751504nnreal @ G @ ( sigma_214952329563889126nnreal @ M @ M1 ) )
       => ( ( member298456594901751504nnreal @ H @ ( sigma_214952329563889126nnreal @ M @ M22 ) )
         => ( member298456594901751504nnreal
            @ ^ [X: a] : ( F @ ( G @ X ) @ ( H @ X ) )
            @ ( sigma_214952329563889126nnreal @ M @ N ) ) ) ) ) ).

% measurable_Pair_compose_split
thf(fact_761_measurable__Pair__compose__split,axiom,
    ! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal > real,M1: sigma_7234349610311085201nnreal,M22: sigma_7234349610311085201nnreal,N: sigma_measure_real,G: a > extend8495563244428889912nnreal,M: sigma_measure_a,H: a > extend8495563244428889912nnreal] :
      ( ( member8604482116299040791l_real @ ( produc959290528197307082l_real @ F ) @ ( sigma_944610991279855149l_real @ ( binary3098606844978005306nnreal @ M1 @ M22 ) @ N ) )
     => ( ( member298456594901751504nnreal @ G @ ( sigma_214952329563889126nnreal @ M @ M1 ) )
       => ( ( member298456594901751504nnreal @ H @ ( sigma_214952329563889126nnreal @ M @ M22 ) )
         => ( member_a_real
            @ ^ [X: a] : ( F @ ( G @ X ) @ ( H @ X ) )
            @ ( sigma_9116425665531756122a_real @ M @ N ) ) ) ) ) ).

% measurable_Pair_compose_split
thf(fact_762_measurable__Pair__compose__split,axiom,
    ! [F: a > a > real,M1: sigma_measure_a,M22: sigma_measure_a,N: sigma_measure_real,G: real > a,M: sigma_measure_real,H: real > a] :
      ( ( member4720799738798892397a_real @ ( produc603112735511730136a_real @ F ) @ ( sigma_1577484901308741891a_real @ ( binary867438762418767560re_a_a @ M1 @ M22 ) @ N ) )
     => ( ( member_real_a @ G @ ( sigma_523072396149930112real_a @ M @ M1 ) )
       => ( ( member_real_a @ H @ ( sigma_523072396149930112real_a @ M @ M22 ) )
         => ( member_real_real
            @ ^ [X: real] : ( F @ ( G @ X ) @ ( H @ X ) )
            @ ( sigma_5267869275261027754l_real @ M @ N ) ) ) ) ) ).

% measurable_Pair_compose_split
thf(fact_763_measurable__Pair__compose__split,axiom,
    ! [F: a > a > extend8495563244428889912nnreal,M1: sigma_measure_a,M22: sigma_measure_a,N: sigma_7234349610311085201nnreal,G: real > a,M: sigma_measure_real,H: real > a] :
      ( ( member3238353849244381945nnreal @ ( produc1994235879742596708nnreal @ F ) @ ( sigma_88170358281049359nnreal @ ( binary867438762418767560re_a_a @ M1 @ M22 ) @ N ) )
     => ( ( member_real_a @ G @ ( sigma_523072396149930112real_a @ M @ M1 ) )
       => ( ( member_real_a @ H @ ( sigma_523072396149930112real_a @ M @ M22 ) )
         => ( member2919562650594848410nnreal
            @ ^ [X: real] : ( F @ ( G @ X ) @ ( H @ X ) )
            @ ( sigma_9017504469962657078nnreal @ M @ N ) ) ) ) ) ).

% measurable_Pair_compose_split
thf(fact_764_measurable__Pair__compose__split,axiom,
    ! [F: a > a > complex,M1: sigma_measure_a,M22: sigma_measure_a,N: sigma_3077487657436305159omplex,G: real > a,M: sigma_measure_real,H: real > a] :
      ( ( member5834619665284812143omplex @ ( produc791145715742941146omplex @ F ) @ ( sigma_3522052638166126725omplex @ ( binary867438762418767560re_a_a @ M1 @ M22 ) @ N ) )
     => ( ( member_real_a @ G @ ( sigma_523072396149930112real_a @ M @ M1 ) )
       => ( ( member_real_a @ H @ ( sigma_523072396149930112real_a @ M @ M22 ) )
         => ( member_real_complex
            @ ^ [X: real] : ( F @ ( G @ X ) @ ( H @ X ) )
            @ ( sigma_9111916201866572460omplex @ M @ N ) ) ) ) ) ).

% measurable_Pair_compose_split
thf(fact_765_measurable__Pair__compose__split,axiom,
    ! [F: extend8495563244428889912nnreal > real > extend8495563244428889912nnreal,M1: sigma_7234349610311085201nnreal,M22: sigma_measure_real,N: sigma_7234349610311085201nnreal,G: a > extend8495563244428889912nnreal,M: sigma_measure_a,H: a > real] :
      ( ( member2340029571030124567nnreal @ ( produc2927036797647851338nnreal @ F ) @ ( sigma_5825728452972236333nnreal @ ( binary3818639336118950830l_real @ M1 @ M22 ) @ N ) )
     => ( ( member298456594901751504nnreal @ G @ ( sigma_214952329563889126nnreal @ M @ M1 ) )
       => ( ( member_a_real @ H @ ( sigma_9116425665531756122a_real @ M @ M22 ) )
         => ( member298456594901751504nnreal
            @ ^ [X: a] : ( F @ ( G @ X ) @ ( H @ X ) )
            @ ( sigma_214952329563889126nnreal @ M @ N ) ) ) ) ) ).

% measurable_Pair_compose_split
thf(fact_766_measurable__Pair__compose__split,axiom,
    ! [F: extend8495563244428889912nnreal > real > real,M1: sigma_7234349610311085201nnreal,M22: sigma_measure_real,N: sigma_measure_real,G: a > extend8495563244428889912nnreal,M: sigma_measure_a,H: a > real] :
      ( ( member5510275943784364939l_real @ ( produc166814986593279422l_real @ F ) @ ( sigma_1562285795970200353l_real @ ( binary3818639336118950830l_real @ M1 @ M22 ) @ N ) )
     => ( ( member298456594901751504nnreal @ G @ ( sigma_214952329563889126nnreal @ M @ M1 ) )
       => ( ( member_a_real @ H @ ( sigma_9116425665531756122a_real @ M @ M22 ) )
         => ( member_a_real
            @ ^ [X: a] : ( F @ ( G @ X ) @ ( H @ X ) )
            @ ( sigma_9116425665531756122a_real @ M @ N ) ) ) ) ) ).

% measurable_Pair_compose_split
thf(fact_767_measurable__Pair__compose__split,axiom,
    ! [F: a > real > a,M1: sigma_measure_a,M22: sigma_measure_real,N: sigma_measure_a,G: real > a,M: sigma_measure_real,H: real > real] :
      ( ( member77757556907816773real_a @ ( produc1233131502984679934real_a @ F ) @ ( sigma_6270567021478054675real_a @ ( binary932748531126180194a_real @ M1 @ M22 ) @ N ) )
     => ( ( member_real_a @ G @ ( sigma_523072396149930112real_a @ M @ M1 ) )
       => ( ( member_real_real @ H @ ( sigma_5267869275261027754l_real @ M @ M22 ) )
         => ( member_real_a
            @ ^ [X: real] : ( F @ ( G @ X ) @ ( H @ X ) )
            @ ( sigma_523072396149930112real_a @ M @ N ) ) ) ) ) ).

% measurable_Pair_compose_split
thf(fact_768_measurable__Pair__compose__split,axiom,
    ! [F: a > real > real,M1: sigma_measure_a,M22: sigma_measure_real,N: sigma_measure_real,G: real > a,M: sigma_measure_real,H: real > real] :
      ( ( member2203687483360836539l_real @ ( produc2387036547305310124l_real @ F ) @ ( sigma_2779692123338079703l_real @ ( binary932748531126180194a_real @ M1 @ M22 ) @ N ) )
     => ( ( member_real_a @ G @ ( sigma_523072396149930112real_a @ M @ M1 ) )
       => ( ( member_real_real @ H @ ( sigma_5267869275261027754l_real @ M @ M22 ) )
         => ( member_real_real
            @ ^ [X: real] : ( F @ ( G @ X ) @ ( H @ X ) )
            @ ( sigma_5267869275261027754l_real @ M @ N ) ) ) ) ) ).

% measurable_Pair_compose_split
thf(fact_769_ennreal__neg,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( extend7643940197134561352nnreal @ X4 )
        = zero_z7100319975126383169nnreal ) ) ).

% ennreal_neg
thf(fact_770_ennreal__eq__0__iff,axiom,
    ! [X4: real] :
      ( ( ( extend7643940197134561352nnreal @ X4 )
        = zero_z7100319975126383169nnreal )
      = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).

% ennreal_eq_0_iff
thf(fact_771_mult__right__ennreal__cancel,axiom,
    ! [A: extend8495563244428889912nnreal,C: real,B: extend8495563244428889912nnreal] :
      ( ( ( times_1893300245718287421nnreal @ A @ ( extend7643940197134561352nnreal @ C ) )
        = ( times_1893300245718287421nnreal @ B @ ( extend7643940197134561352nnreal @ C ) ) )
      = ( ( A = B )
        | ( ord_less_eq_real @ C @ zero_zero_real ) ) ) ).

% mult_right_ennreal_cancel
thf(fact_772_qbs__prob__measure__prob__space_Oindep__vars__compose,axiom,
    ! [S: probab8009751763329705409e_real,M2: ( real > complex ) > sigma_measure_a,X2: ( real > complex ) > real > a,I2: set_real_complex,Y: ( real > complex ) > a > real,N: ( real > complex ) > sigma_measure_real] :
      ( ( indepe1200421579086570447plex_a @ ( probab4733579253584633066e_real @ S ) @ M2 @ X2 @ I2 )
     => ( ! [I: real > complex] :
            ( ( member_real_complex @ I @ I2 )
           => ( member_a_real @ ( Y @ I ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe3215717721046027291x_real @ ( probab4733579253584633066e_real @ S ) @ N
          @ ^ [I3: real > complex] : ( comp_a_real_real @ ( Y @ I3 ) @ ( X2 @ I3 ) )
          @ I2 ) ) ) ).

% qbs_prob_measure_prob_space.indep_vars_compose
thf(fact_773_qbs__prob__measure__prob__space_Oindep__vars__compose,axiom,
    ! [S: probab8009751763329705409e_real,M2: ( real > real ) > sigma_measure_a,X2: ( real > real ) > real > a,I2: set_real_real,Y: ( real > real ) > a > real,N: ( real > real ) > sigma_measure_real] :
      ( ( indepe1927354855876929745real_a @ ( probab4733579253584633066e_real @ S ) @ M2 @ X2 @ I2 )
     => ( ! [I: real > real] :
            ( ( member_real_real @ I @ I2 )
           => ( member_a_real @ ( Y @ I ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe9089129998381042585l_real @ ( probab4733579253584633066e_real @ S ) @ N
          @ ^ [I3: real > real] : ( comp_a_real_real @ ( Y @ I3 ) @ ( X2 @ I3 ) )
          @ I2 ) ) ) ).

% qbs_prob_measure_prob_space.indep_vars_compose
thf(fact_774_qbs__prob__measure__prob__space_Oindep__vars__compose,axiom,
    ! [S: probab8009751763329705409e_real,M2: ( real > a ) > sigma_measure_a,X2: ( real > a ) > real > a,I2: set_real_a,Y: ( real > a ) > a > real,N: ( real > a ) > sigma_measure_real] :
      ( ( indepe357751042618000297al_a_a @ ( probab4733579253584633066e_real @ S ) @ M2 @ X2 @ I2 )
     => ( ! [I: real > a] :
            ( ( member_real_a @ I @ I2 )
           => ( member_a_real @ ( Y @ I ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe6457644772562392769a_real @ ( probab4733579253584633066e_real @ S ) @ N
          @ ^ [I3: real > a] : ( comp_a_real_real @ ( Y @ I3 ) @ ( X2 @ I3 ) )
          @ I2 ) ) ) ).

% qbs_prob_measure_prob_space.indep_vars_compose
thf(fact_775_qbs__prob__measure__prob__space_Oindep__vars__compose,axiom,
    ! [S: probab8009751763329705409e_real,M2: ( a > extend8495563244428889912nnreal ) > sigma_measure_a,X2: ( a > extend8495563244428889912nnreal ) > real > a,I2: set_a_7161065143582548615nnreal,Y: ( a > extend8495563244428889912nnreal ) > a > real,N: ( a > extend8495563244428889912nnreal ) > sigma_measure_real] :
      ( ( indepe8228798008660262667real_a @ ( probab4733579253584633066e_real @ S ) @ M2 @ X2 @ I2 )
     => ( ! [I: a > extend8495563244428889912nnreal] :
            ( ( member298456594901751504nnreal @ I @ I2 )
           => ( member_a_real @ ( Y @ I ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe1680866314667358175l_real @ ( probab4733579253584633066e_real @ S ) @ N
          @ ^ [I3: a > extend8495563244428889912nnreal] : ( comp_a_real_real @ ( Y @ I3 ) @ ( X2 @ I3 ) )
          @ I2 ) ) ) ).

% qbs_prob_measure_prob_space.indep_vars_compose
thf(fact_776_qbs__prob__measure__prob__space_Oindep__vars__compose,axiom,
    ! [S: probab8009751763329705409e_real,M2: ( a > real ) > sigma_measure_a,X2: ( a > real ) > real > a,I2: set_a_real,Y: ( a > real ) > a > real,N: ( a > real ) > sigma_measure_real] :
      ( ( indepe3565983053654730263real_a @ ( probab4733579253584633066e_real @ S ) @ M2 @ X2 @ I2 )
     => ( ! [I: a > real] :
            ( ( member_a_real @ I @ I2 )
           => ( member_a_real @ ( Y @ I ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe722220561065814995l_real @ ( probab4733579253584633066e_real @ S ) @ N
          @ ^ [I3: a > real] : ( comp_a_real_real @ ( Y @ I3 ) @ ( X2 @ I3 ) )
          @ I2 ) ) ) ).

% qbs_prob_measure_prob_space.indep_vars_compose
thf(fact_777_qbs__prob__measure__prob__space_Oindep__vars__compose,axiom,
    ! [S: probab4737552673497767871pace_a,M2: ( real > complex ) > sigma_measure_real,X2: ( real > complex ) > a > real,I2: set_real_complex,Y: ( real > complex ) > real > a,N: ( real > complex ) > sigma_measure_a] :
      ( ( indepe2122272008937502813x_real @ ( probab7100426894406488384sure_a @ S ) @ M2 @ X2 @ I2 )
     => ( ! [I: real > complex] :
            ( ( member_real_complex @ I @ I2 )
           => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe7369748381470229261plex_a @ ( probab7100426894406488384sure_a @ S ) @ N
          @ ^ [I3: real > complex] : ( comp_real_a_a @ ( Y @ I3 ) @ ( X2 @ I3 ) )
          @ I2 ) ) ) ).

% qbs_prob_measure_prob_space.indep_vars_compose
thf(fact_778_qbs__prob__measure__prob__space_Oindep__vars__compose,axiom,
    ! [S: probab4737552673497767871pace_a,M2: ( real > complex ) > sigma_measure_a,X2: ( real > complex ) > a > a,I2: set_real_complex,Y: ( real > complex ) > a > extend8495563244428889912nnreal,N: ( real > complex ) > sigma_7234349610311085201nnreal] :
      ( ( indepe7369748381470229261plex_a @ ( probab7100426894406488384sure_a @ S ) @ M2 @ X2 @ I2 )
     => ( ! [I: real > complex] :
            ( ( member_real_complex @ I @ I2 )
           => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe2856984694482014569nnreal @ ( probab7100426894406488384sure_a @ S ) @ N
          @ ^ [I3: real > complex] : ( comp_a6042866249568583849real_a @ ( Y @ I3 ) @ ( X2 @ I3 ) )
          @ I2 ) ) ) ).

% qbs_prob_measure_prob_space.indep_vars_compose
thf(fact_779_qbs__prob__measure__prob__space_Oindep__vars__compose,axiom,
    ! [S: probab4737552673497767871pace_a,M2: ( real > real ) > sigma_measure_real,X2: ( real > real ) > a > real,I2: set_real_real,Y: ( real > real ) > real > a,N: ( real > real ) > sigma_measure_a] :
      ( ( indepe1495116825794919131l_real @ ( probab7100426894406488384sure_a @ S ) @ M2 @ X2 @ I2 )
     => ( ! [I: real > real] :
            ( ( member_real_real @ I @ I2 )
           => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe4370139003212481807real_a @ ( probab7100426894406488384sure_a @ S ) @ N
          @ ^ [I3: real > real] : ( comp_real_a_a @ ( Y @ I3 ) @ ( X2 @ I3 ) )
          @ I2 ) ) ) ).

% qbs_prob_measure_prob_space.indep_vars_compose
thf(fact_780_qbs__prob__measure__prob__space_Oindep__vars__compose,axiom,
    ! [S: probab4737552673497767871pace_a,M2: ( real > real ) > sigma_measure_a,X2: ( real > real ) > a > a,I2: set_real_real,Y: ( real > real ) > a > extend8495563244428889912nnreal,N: ( real > real ) > sigma_7234349610311085201nnreal] :
      ( ( indepe4370139003212481807real_a @ ( probab7100426894406488384sure_a @ S ) @ M2 @ X2 @ I2 )
     => ( ! [I: real > real] :
            ( ( member_real_real @ I @ I2 )
           => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe4467962090945823463nnreal @ ( probab7100426894406488384sure_a @ S ) @ N
          @ ^ [I3: real > real] : ( comp_a6042866249568583849real_a @ ( Y @ I3 ) @ ( X2 @ I3 ) )
          @ I2 ) ) ) ).

% qbs_prob_measure_prob_space.indep_vars_compose
thf(fact_781_qbs__prob__measure__prob__space_Oindep__vars__compose,axiom,
    ! [S: probab4737552673497767871pace_a,M2: ( real > a ) > sigma_measure_real,X2: ( real > a ) > a > real,I2: set_real_a,Y: ( real > a ) > real > a,N: ( real > a ) > sigma_measure_a] :
      ( ( indepe8900428919897944831a_real @ ( probab7100426894406488384sure_a @ S ) @ M2 @ X2 @ I2 )
     => ( ! [I: real > a] :
            ( ( member_real_a @ I @ I2 )
           => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
       => ( indepe4717112320896891883al_a_a @ ( probab7100426894406488384sure_a @ S ) @ N
          @ ^ [I3: real > a] : ( comp_real_a_a @ ( Y @ I3 ) @ ( X2 @ I3 ) )
          @ I2 ) ) ) ).

% qbs_prob_measure_prob_space.indep_vars_compose
thf(fact_782_prob__space_Oindep__vars__compose,axiom,
    ! [M: sigma_measure_real,M2: ( real > complex ) > sigma_measure_real,X2: ( real > complex ) > real > real,I2: set_real_complex,Y: ( real > complex ) > real > a,N: ( real > complex ) > sigma_measure_a] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe3215717721046027291x_real @ M @ M2 @ X2 @ I2 )
       => ( ! [I: real > complex] :
              ( ( member_real_complex @ I @ I2 )
             => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
         => ( indepe1200421579086570447plex_a @ M @ N
            @ ^ [I3: real > complex] : ( comp_real_a_real @ ( Y @ I3 ) @ ( X2 @ I3 ) )
            @ I2 ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_783_prob__space_Oindep__vars__compose,axiom,
    ! [M: sigma_measure_real,M2: ( real > complex ) > sigma_measure_a,X2: ( real > complex ) > real > a,I2: set_real_complex,Y: ( real > complex ) > a > extend8495563244428889912nnreal,N: ( real > complex ) > sigma_7234349610311085201nnreal] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe1200421579086570447plex_a @ M @ M2 @ X2 @ I2 )
       => ( ! [I: real > complex] :
              ( ( member_real_complex @ I @ I2 )
             => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
         => ( indepe6000904806191903783nnreal @ M @ N
            @ ^ [I3: real > complex] : ( comp_a8249376463644563905l_real @ ( Y @ I3 ) @ ( X2 @ I3 ) )
            @ I2 ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_784_prob__space_Oindep__vars__compose,axiom,
    ! [M: sigma_measure_real,M2: ( real > real ) > sigma_measure_real,X2: ( real > real ) > real > real,I2: set_real_real,Y: ( real > real ) > real > a,N: ( real > real ) > sigma_measure_a] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe9089129998381042585l_real @ M @ M2 @ X2 @ I2 )
       => ( ! [I: real > real] :
              ( ( member_real_real @ I @ I2 )
             => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
         => ( indepe1927354855876929745real_a @ M @ N
            @ ^ [I3: real > real] : ( comp_real_a_real @ ( Y @ I3 ) @ ( X2 @ I3 ) )
            @ I2 ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_785_prob__space_Oindep__vars__compose,axiom,
    ! [M: sigma_measure_real,M2: ( real > real ) > sigma_measure_a,X2: ( real > real ) > real > a,I2: set_real_real,Y: ( real > real ) > a > extend8495563244428889912nnreal,N: ( real > real ) > sigma_7234349610311085201nnreal] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe1927354855876929745real_a @ M @ M2 @ X2 @ I2 )
       => ( ! [I: real > real] :
              ( ( member_real_real @ I @ I2 )
             => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
         => ( indepe501623583335441061nnreal @ M @ N
            @ ^ [I3: real > real] : ( comp_a8249376463644563905l_real @ ( Y @ I3 ) @ ( X2 @ I3 ) )
            @ I2 ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_786_prob__space_Oindep__vars__compose,axiom,
    ! [M: sigma_measure_real,M2: ( real > a ) > sigma_measure_real,X2: ( real > a ) > real > real,I2: set_real_a,Y: ( real > a ) > real > a,N: ( real > a ) > sigma_measure_a] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe6457644772562392769a_real @ M @ M2 @ X2 @ I2 )
       => ( ! [I: real > a] :
              ( ( member_real_a @ I @ I2 )
             => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
         => ( indepe357751042618000297al_a_a @ M @ N
            @ ^ [I3: real > a] : ( comp_real_a_real @ ( Y @ I3 ) @ ( X2 @ I3 ) )
            @ I2 ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_787_prob__space_Oindep__vars__compose,axiom,
    ! [M: sigma_measure_real,M2: ( real > a ) > sigma_measure_a,X2: ( real > a ) > real > a,I2: set_real_a,Y: ( real > a ) > a > extend8495563244428889912nnreal,N: ( real > a ) > sigma_7234349610311085201nnreal] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe357751042618000297al_a_a @ M @ M2 @ X2 @ I2 )
       => ( ! [I: real > a] :
              ( ( member_real_a @ I @ I2 )
             => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
         => ( indepe1181869626592248269nnreal @ M @ N
            @ ^ [I3: real > a] : ( comp_a8249376463644563905l_real @ ( Y @ I3 ) @ ( X2 @ I3 ) )
            @ I2 ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_788_prob__space_Oindep__vars__compose,axiom,
    ! [M: sigma_measure_real,M2: ( a > extend8495563244428889912nnreal ) > sigma_measure_real,X2: ( a > extend8495563244428889912nnreal ) > real > real,I2: set_a_7161065143582548615nnreal,Y: ( a > extend8495563244428889912nnreal ) > real > a,N: ( a > extend8495563244428889912nnreal ) > sigma_measure_a] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe1680866314667358175l_real @ M @ M2 @ X2 @ I2 )
       => ( ! [I: a > extend8495563244428889912nnreal] :
              ( ( member298456594901751504nnreal @ I @ I2 )
             => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
         => ( indepe8228798008660262667real_a @ M @ N
            @ ^ [I3: a > extend8495563244428889912nnreal] : ( comp_real_a_real @ ( Y @ I3 ) @ ( X2 @ I3 ) )
            @ I2 ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_789_prob__space_Oindep__vars__compose,axiom,
    ! [M: sigma_measure_real,M2: ( a > extend8495563244428889912nnreal ) > sigma_measure_a,X2: ( a > extend8495563244428889912nnreal ) > real > a,I2: set_a_7161065143582548615nnreal,Y: ( a > extend8495563244428889912nnreal ) > a > extend8495563244428889912nnreal,N: ( a > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe8228798008660262667real_a @ M @ M2 @ X2 @ I2 )
       => ( ! [I: a > extend8495563244428889912nnreal] :
              ( ( member298456594901751504nnreal @ I @ I2 )
             => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
         => ( indepe4617722330935435755nnreal @ M @ N
            @ ^ [I3: a > extend8495563244428889912nnreal] : ( comp_a8249376463644563905l_real @ ( Y @ I3 ) @ ( X2 @ I3 ) )
            @ I2 ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_790_prob__space_Oindep__vars__compose,axiom,
    ! [M: sigma_measure_real,M2: ( a > real ) > sigma_measure_real,X2: ( a > real ) > real > real,I2: set_a_real,Y: ( a > real ) > real > a,N: ( a > real ) > sigma_measure_a] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe722220561065814995l_real @ M @ M2 @ X2 @ I2 )
       => ( ! [I: a > real] :
              ( ( member_a_real @ I @ I2 )
             => ( member_real_a @ ( Y @ I ) @ ( sigma_523072396149930112real_a @ ( M2 @ I ) @ ( N @ I ) ) ) )
         => ( indepe3565983053654730263real_a @ M @ N
            @ ^ [I3: a > real] : ( comp_real_a_real @ ( Y @ I3 ) @ ( X2 @ I3 ) )
            @ I2 ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_791_prob__space_Oindep__vars__compose,axiom,
    ! [M: sigma_measure_real,M2: ( a > real ) > sigma_measure_a,X2: ( a > real ) > real > a,I2: set_a_real,Y: ( a > real ) > a > extend8495563244428889912nnreal,N: ( a > real ) > sigma_7234349610311085201nnreal] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe3565983053654730263real_a @ M @ M2 @ X2 @ I2 )
       => ( ! [I: a > real] :
              ( ( member_a_real @ I @ I2 )
             => ( member298456594901751504nnreal @ ( Y @ I ) @ ( sigma_214952329563889126nnreal @ ( M2 @ I ) @ ( N @ I ) ) ) )
         => ( indepe6561983776359739359nnreal @ M @ N
            @ ^ [I3: a > real] : ( comp_a8249376463644563905l_real @ ( Y @ I3 ) @ ( X2 @ I3 ) )
            @ I2 ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_792_ennreal__mult,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ( extend7643940197134561352nnreal @ ( times_times_real @ A @ B ) )
          = ( times_1893300245718287421nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B ) ) ) ) ) ).

% ennreal_mult
thf(fact_793_ennreal__mult_H,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( extend7643940197134561352nnreal @ ( times_times_real @ A @ B ) )
        = ( times_1893300245718287421nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B ) ) ) ) ).

% ennreal_mult'
thf(fact_794_ennreal__mult_H_H,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ B )
     => ( ( extend7643940197134561352nnreal @ ( times_times_real @ A @ B ) )
        = ( times_1893300245718287421nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B ) ) ) ) ).

% ennreal_mult''
thf(fact_795_ennreal__minus__if,axiom,
    ! [A: real,B: real] :
      ( ( minus_8429688780609304081nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B ) )
      = ( extend7643940197134561352nnreal @ ( if_real @ ( ord_less_eq_real @ zero_zero_real @ B ) @ ( if_real @ ( ord_less_eq_real @ B @ A ) @ ( minus_minus_real @ A @ B ) @ zero_zero_real ) @ A ) ) ) ).

% ennreal_minus_if
thf(fact_796_ennreal__minus,axiom,
    ! [Q2: real,R4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Q2 )
     => ( ( minus_8429688780609304081nnreal @ ( extend7643940197134561352nnreal @ R4 ) @ ( extend7643940197134561352nnreal @ Q2 ) )
        = ( extend7643940197134561352nnreal @ ( minus_minus_real @ R4 @ Q2 ) ) ) ) ).

% ennreal_minus
thf(fact_797_qbs__prob__measure__prob__space_Oentropy__def,axiom,
    ! [S: probab8009751763329705409e_real,B: real,S2: sigma_measure_a,X2: real > a] :
      ( ( prob_entropy_real_a @ ( probab4733579253584633066e_real @ S ) @ B @ S2 @ X2 )
      = ( uminus_uminus_real @ ( kL_divergence_a @ B @ S2 @ ( measure_distr_real_a @ ( probab4733579253584633066e_real @ S ) @ S2 @ X2 ) ) ) ) ).

% qbs_prob_measure_prob_space.entropy_def
thf(fact_798_qbs__prob__measure__prob__space_Oentropy__def,axiom,
    ! [S: probab8009751763329705409e_real,B: real,S2: sigma_measure_real,X2: real > real] :
      ( ( prob_e6953316728393294858l_real @ ( probab4733579253584633066e_real @ S ) @ B @ S2 @ X2 )
      = ( uminus_uminus_real @ ( kL_divergence_real @ B @ S2 @ ( measur2993149975067245138l_real @ ( probab4733579253584633066e_real @ S ) @ S2 @ X2 ) ) ) ) ).

% qbs_prob_measure_prob_space.entropy_def
thf(fact_799_qbs__prob__measure__prob__space_Oentropy__def,axiom,
    ! [S: probab4737552673497767871pace_a,B: real,S2: sigma_measure_real,X2: a > real] :
      ( ( prob_entropy_a_real @ ( probab7100426894406488384sure_a @ S ) @ B @ S2 @ X2 )
      = ( uminus_uminus_real @ ( kL_divergence_real @ B @ S2 @ ( measure_distr_a_real @ ( probab7100426894406488384sure_a @ S ) @ S2 @ X2 ) ) ) ) ).

% qbs_prob_measure_prob_space.entropy_def
thf(fact_800_qbs__prob__measure__prob__space_Oborel__measurable__lebesgue__integral,axiom,
    ! [F: a > real > real,N: sigma_measure_a,S: probab8009751763329705409e_real] :
      ( ( member2203687483360836539l_real @ ( produc2387036547305310124l_real @ F ) @ ( sigma_2779692123338079703l_real @ ( binary932748531126180194a_real @ N @ ( probab4733579253584633066e_real @ S ) ) @ borel_5078946678739801102l_real ) )
     => ( member_a_real
        @ ^ [X: a] : ( bochne3715101410578510557l_real @ ( probab4733579253584633066e_real @ S ) @ ( F @ X ) )
        @ ( sigma_9116425665531756122a_real @ N @ borel_5078946678739801102l_real ) ) ) ).

% qbs_prob_measure_prob_space.borel_measurable_lebesgue_integral
thf(fact_801_qbs__prob__measure__prob__space_Oborel__measurable__lebesgue__integral,axiom,
    ! [F: real > real > real,N: sigma_measure_real,S: probab8009751763329705409e_real] :
      ( ( member6699615393305559423l_real @ ( produc313441363659479858l_real @ F ) @ ( sigma_8002782794886939285l_real @ ( binary6478037234023840930l_real @ N @ ( probab4733579253584633066e_real @ S ) ) @ borel_5078946678739801102l_real ) )
     => ( member_real_real
        @ ^ [X: real] : ( bochne3715101410578510557l_real @ ( probab4733579253584633066e_real @ S ) @ ( F @ X ) )
        @ ( sigma_5267869275261027754l_real @ N @ borel_5078946678739801102l_real ) ) ) ).

% qbs_prob_measure_prob_space.borel_measurable_lebesgue_integral
thf(fact_802_qbs__prob__measure__prob__space_Oborel__measurable__lebesgue__integral,axiom,
    ! [F: a > a > real,N: sigma_measure_a,S: probab4737552673497767871pace_a] :
      ( ( member4720799738798892397a_real @ ( produc603112735511730136a_real @ F ) @ ( sigma_1577484901308741891a_real @ ( binary867438762418767560re_a_a @ N @ ( probab7100426894406488384sure_a @ S ) ) @ borel_5078946678739801102l_real ) )
     => ( member_a_real
        @ ^ [X: a] : ( bochne378719280626478695a_real @ ( probab7100426894406488384sure_a @ S ) @ ( F @ X ) )
        @ ( sigma_9116425665531756122a_real @ N @ borel_5078946678739801102l_real ) ) ) ).

% qbs_prob_measure_prob_space.borel_measurable_lebesgue_integral
thf(fact_803_qbs__prob__measure__prob__space_Oborel__measurable__lebesgue__integral,axiom,
    ! [F: real > a > real,N: sigma_measure_real,S: probab4737552673497767871pace_a] :
      ( ( member7939111694857414313a_real @ ( produc1204397526812833490a_real @ F ) @ ( sigma_8515116334834657477a_real @ ( binary1562767298599129992real_a @ N @ ( probab7100426894406488384sure_a @ S ) ) @ borel_5078946678739801102l_real ) )
     => ( member_real_real
        @ ^ [X: real] : ( bochne378719280626478695a_real @ ( probab7100426894406488384sure_a @ S ) @ ( F @ X ) )
        @ ( sigma_5267869275261027754l_real @ N @ borel_5078946678739801102l_real ) ) ) ).

% qbs_prob_measure_prob_space.borel_measurable_lebesgue_integral
thf(fact_804_qbs__prob__measure__prob__space_Oborel__measurable__lebesgue__integral,axiom,
    ! [F: real > a > complex,N: sigma_measure_real,S: probab4737552673497767871pace_a] :
      ( ( member115948512955364523omplex @ ( produc8077785557113432020omplex @ F ) @ ( sigma_6949951708218146119omplex @ ( binary1562767298599129992real_a @ N @ ( probab7100426894406488384sure_a @ S ) ) @ borel_1392132677378845456omplex ) )
     => ( member_real_complex
        @ ^ [X: real] : ( bochne4904656926214500329omplex @ ( probab7100426894406488384sure_a @ S ) @ ( F @ X ) )
        @ ( sigma_9111916201866572460omplex @ N @ borel_1392132677378845456omplex ) ) ) ).

% qbs_prob_measure_prob_space.borel_measurable_lebesgue_integral
thf(fact_805_qbs__prob__measure__prob__space_Oborel__measurable__nn__integral,axiom,
    ! [F: a > real > extend8495563244428889912nnreal,N: sigma_measure_a,S: probab8009751763329705409e_real] :
      ( ( member4437793228276457543nnreal @ ( produc5555093792979918904nnreal @ F ) @ ( sigma_8117956432444187619nnreal @ ( binary932748531126180194a_real @ N @ ( probab4733579253584633066e_real @ S ) ) @ borel_6524799422816628122nnreal ) )
     => ( member298456594901751504nnreal
        @ ^ [X: a] : ( nonneg2667834350952324695l_real @ ( probab4733579253584633066e_real @ S ) @ ( F @ X ) )
        @ ( sigma_214952329563889126nnreal @ N @ borel_6524799422816628122nnreal ) ) ) ).

% qbs_prob_measure_prob_space.borel_measurable_nn_integral
thf(fact_806_qbs__prob__measure__prob__space_Oborel__measurable__nn__integral,axiom,
    ! [F: real > real > extend8495563244428889912nnreal,N: sigma_measure_real,S: probab8009751763329705409e_real] :
      ( ( member2245694452317284363nnreal @ ( produc4590977785667036862nnreal @ F ) @ ( sigma_219902274609641377nnreal @ ( binary6478037234023840930l_real @ N @ ( probab4733579253584633066e_real @ S ) ) @ borel_6524799422816628122nnreal ) )
     => ( member2919562650594848410nnreal
        @ ^ [X: real] : ( nonneg2667834350952324695l_real @ ( probab4733579253584633066e_real @ S ) @ ( F @ X ) )
        @ ( sigma_9017504469962657078nnreal @ N @ borel_6524799422816628122nnreal ) ) ) ).

% qbs_prob_measure_prob_space.borel_measurable_nn_integral
thf(fact_807_qbs__prob__measure__prob__space_Oborel__measurable__nn__integral,axiom,
    ! [F: a > a > extend8495563244428889912nnreal,N: sigma_measure_a,S: probab4737552673497767871pace_a] :
      ( ( member3238353849244381945nnreal @ ( produc1994235879742596708nnreal @ F ) @ ( sigma_88170358281049359nnreal @ ( binary867438762418767560re_a_a @ N @ ( probab7100426894406488384sure_a @ S ) ) @ borel_6524799422816628122nnreal ) )
     => ( member298456594901751504nnreal
        @ ^ [X: a] : ( nonneg2725512125972007571gral_a @ ( probab7100426894406488384sure_a @ S ) @ ( F @ X ) )
        @ ( sigma_214952329563889126nnreal @ N @ borel_6524799422816628122nnreal ) ) ) ).

% qbs_prob_measure_prob_space.borel_measurable_nn_integral
thf(fact_808_qbs__prob__measure__prob__space_Oborel__measurable__nn__integral,axiom,
    ! [F: real > a > extend8495563244428889912nnreal,N: sigma_measure_real,S: probab4737552673497767871pace_a] :
      ( ( member8281051115363742261nnreal @ ( produc328383726578225758nnreal @ F ) @ ( sigma_2737842282676696529nnreal @ ( binary1562767298599129992real_a @ N @ ( probab7100426894406488384sure_a @ S ) ) @ borel_6524799422816628122nnreal ) )
     => ( member2919562650594848410nnreal
        @ ^ [X: real] : ( nonneg2725512125972007571gral_a @ ( probab7100426894406488384sure_a @ S ) @ ( F @ X ) )
        @ ( sigma_9017504469962657078nnreal @ N @ borel_6524799422816628122nnreal ) ) ) ).

% qbs_prob_measure_prob_space.borel_measurable_nn_integral
thf(fact_809_qp_Omutual__information__def,axiom,
    ! [B: real,S2: sigma_4063782130865963553orel_a,T: sigma_8775847253591143008e_real,X2: real > quasi_borel_a,Y: real > produc725540845905733987e_real] :
      ( ( prob_m4228609518817447427e_real @ mu @ B @ S2 @ T @ X2 @ Y )
      = ( kL_div3327646984204289008e_real @ B @ ( binary125940435690417031e_real @ ( measur7149860273772831102orel_a @ mu @ S2 @ X2 ) @ ( measur8637847926015211837e_real @ mu @ T @ Y ) )
        @ ( measur2398198314208846400e_real @ mu @ ( binary125940435690417031e_real @ S2 @ T )
          @ ^ [X: real] : ( produc4145838808978236886e_real @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ).

% qp.mutual_information_def
thf(fact_810_qp_Omutual__information__def,axiom,
    ! [B: real,S2: sigma_measure_real_a,T: sigma_8927737637348964610e_real,X2: real > real > a,Y: real > sigma_measure_real] :
      ( ( prob_m9196104408708822272e_real @ mu @ B @ S2 @ T @ X2 @ Y )
      = ( kL_div4552242606482481901e_real @ B @ ( binary2119006201073916036e_real @ ( measur7864027549924149603real_a @ mu @ S2 @ X2 ) @ ( measur2366643943792126175e_real @ mu @ T @ Y ) )
        @ ( measur8637847926015211837e_real @ mu @ ( binary2119006201073916036e_real @ S2 @ T )
          @ ^ [X: real] : ( produc623176010801490259e_real @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ).

% qp.mutual_information_def
thf(fact_811_qp_Omutual__information__def,axiom,
    ! [B: real,S2: sigma_measure_a,T: sigma_measure_a,X2: real > a,Y: real > a] :
      ( ( prob_m1941895425998922052al_a_a @ mu @ B @ S2 @ T @ X2 @ Y )
      = ( kL_div3267156980076932017od_a_a @ B @ ( binary867438762418767560re_a_a @ ( measure_distr_real_a @ mu @ S2 @ X2 ) @ ( measure_distr_real_a @ mu @ T @ Y ) )
        @ ( measur2513335786126797313od_a_a @ mu @ ( binary867438762418767560re_a_a @ S2 @ T )
          @ ^ [X: real] : ( product_Pair_a_a @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ).

% qp.mutual_information_def
thf(fact_812_qp_Omutual__information__def,axiom,
    ! [B: real,S2: sigma_measure_a,T: sigma_measure_real,X2: real > a,Y: real > real] :
      ( ( prob_m6024414151681283558a_real @ mu @ B @ S2 @ T @ X2 @ Y )
      = ( kL_div1539255837003659855a_real @ B @ ( binary932748531126180194a_real @ ( measure_distr_real_a @ mu @ S2 @ X2 ) @ ( measur2993149975067245138l_real @ mu @ T @ Y ) )
        @ ( measur7871026761292836863a_real @ mu @ ( binary932748531126180194a_real @ S2 @ T )
          @ ^ [X: real] : ( product_Pair_a_real @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ).

% qp.mutual_information_def
thf(fact_813_qp_Omutual__information__def,axiom,
    ! [B: real,S2: sigma_measure_real,T: sigma_measure_a,X2: real > real,Y: real > a] :
      ( ( prob_m6654432919154233356real_a @ mu @ B @ S2 @ T @ X2 @ Y )
      = ( kL_div2056522339270997053real_a @ B @ ( binary1562767298599129992real_a @ ( measur2993149975067245138l_real @ mu @ S2 @ X2 ) @ ( measure_distr_real_a @ mu @ T @ Y ) )
        @ ( measur8388293263560174061real_a @ mu @ ( binary1562767298599129992real_a @ S2 @ T )
          @ ^ [X: real] : ( product_Pair_real_a @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ).

% qp.mutual_information_def
thf(fact_814_qp_Omutual__information__def,axiom,
    ! [B: real,S2: sigma_measure_real,T: sigma_measure_real,X2: real > real,Y: real > real] :
      ( ( prob_m4172219917653797150l_real @ mu @ B @ S2 @ T @ X2 @ Y )
      = ( kL_div4114197932038040771l_real @ B @ ( binary6478037234023840930l_real @ ( measur2993149975067245138l_real @ mu @ S2 @ X2 ) @ ( measur2993149975067245138l_real @ mu @ T @ Y ) )
        @ ( measur6481026558495277843l_real @ mu @ ( binary6478037234023840930l_real @ S2 @ T )
          @ ^ [X: real] : ( produc4511245868158468465l_real @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ).

% qp.mutual_information_def
thf(fact_815_real__distribution_Ochar__measurable,axiom,
    ! [M: sigma_measure_real] :
      ( ( distri2809703520229113005bution @ M )
     => ( member_real_complex @ ( characteristic_char @ M ) @ ( sigma_9111916201866572460omplex @ borel_5078946678739801102l_real @ borel_1392132677378845456omplex ) ) ) ).

% real_distribution.char_measurable
thf(fact_816_prob__space_Oentropy__def,axiom,
    ! [M: sigma_measure_real,B: real,S2: sigma_measure_a,X2: real > a] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( prob_entropy_real_a @ M @ B @ S2 @ X2 )
        = ( uminus_uminus_real @ ( kL_divergence_a @ B @ S2 @ ( measure_distr_real_a @ M @ S2 @ X2 ) ) ) ) ) ).

% prob_space.entropy_def
thf(fact_817_prob__space_Oentropy__def,axiom,
    ! [M: sigma_measure_real,B: real,S2: sigma_measure_real,X2: real > real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( prob_e6953316728393294858l_real @ M @ B @ S2 @ X2 )
        = ( uminus_uminus_real @ ( kL_divergence_real @ B @ S2 @ ( measur2993149975067245138l_real @ M @ S2 @ X2 ) ) ) ) ) ).

% prob_space.entropy_def
thf(fact_818_Pair__le,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_le1075799226346578649l_real @ ( produc4511245868158468465l_real @ A @ B ) @ ( produc4511245868158468465l_real @ C @ D ) )
      = ( ( ord_less_eq_real @ A @ C )
        & ( ord_less_eq_real @ B @ D ) ) ) ).

% Pair_le
thf(fact_819_Pair__le,axiom,
    ! [A: real,B: extend8495563244428889912nnreal,C: real,D: extend8495563244428889912nnreal] :
      ( ( ord_le4096773168995780197nnreal @ ( produc4778015194254607485nnreal @ A @ B ) @ ( produc4778015194254607485nnreal @ C @ D ) )
      = ( ( ord_less_eq_real @ A @ C )
        & ( ord_le3935885782089961368nnreal @ B @ D ) ) ) ).

% Pair_le
thf(fact_820_Pair__le,axiom,
    ! [A: extend8495563244428889912nnreal,B: real,C: extend8495563244428889912nnreal,D: real] :
      ( ( ord_le4051224869651757541l_real @ ( produc2810268924804063229l_real @ A @ B ) @ ( produc2810268924804063229l_real @ C @ D ) )
      = ( ( ord_le3935885782089961368nnreal @ A @ C )
        & ( ord_less_eq_real @ B @ D ) ) ) ).

% Pair_le
thf(fact_821_Pair__le,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D: extend8495563244428889912nnreal] :
      ( ( ord_le1399272598019556209nnreal @ ( produc344325839068023049nnreal @ A @ B ) @ ( produc344325839068023049nnreal @ C @ D ) )
      = ( ( ord_le3935885782089961368nnreal @ A @ C )
        & ( ord_le3935885782089961368nnreal @ B @ D ) ) ) ).

% Pair_le
thf(fact_822_qp_Ochar__distr__add,axiom,
    ! [X12: real > real,X23: real > real,T2: real] :
      ( ( indepe3760321310464026790l_real @ mu @ borel_5078946678739801102l_real @ X12 @ borel_5078946678739801102l_real @ X23 )
     => ( ( characteristic_char
          @ ( measur2993149975067245138l_real @ mu @ borel_5078946678739801102l_real
            @ ^ [Omega: real] : ( plus_plus_real @ ( X12 @ Omega ) @ ( X23 @ Omega ) ) )
          @ T2 )
        = ( times_times_complex @ ( characteristic_char @ ( measur2993149975067245138l_real @ mu @ borel_5078946678739801102l_real @ X12 ) @ T2 ) @ ( characteristic_char @ ( measur2993149975067245138l_real @ mu @ borel_5078946678739801102l_real @ X23 ) @ T2 ) ) ) ) ).

% qp.char_distr_add
thf(fact_823_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_824_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_825_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_826_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_827_mem__case__prodI,axiom,
    ! [Z4: real > complex,C: real > real > set_real_complex,A: real,B: real] :
      ( ( member_real_complex @ Z4 @ ( C @ A @ B ) )
     => ( member_real_complex @ Z4 @ ( produc4115511989314032949omplex @ C @ ( produc4511245868158468465l_real @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_828_mem__case__prodI,axiom,
    ! [Z4: real > real,C: real > real > set_real_real,A: real,B: real] :
      ( ( member_real_real @ Z4 @ ( C @ A @ B ) )
     => ( member_real_real @ Z4 @ ( produc8597459571820422835l_real @ C @ ( produc4511245868158468465l_real @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_829_mem__case__prodI,axiom,
    ! [Z4: real > a,C: real > real > set_real_a,A: real,B: real] :
      ( ( member_real_a @ Z4 @ ( C @ A @ B ) )
     => ( member_real_a @ Z4 @ ( produc7433429769795935843real_a @ C @ ( produc4511245868158468465l_real @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_830_mem__case__prodI,axiom,
    ! [Z4: a > extend8495563244428889912nnreal,C: real > real > set_a_7161065143582548615nnreal,A: real,B: real] :
      ( ( member298456594901751504nnreal @ Z4 @ ( C @ A @ B ) )
     => ( member298456594901751504nnreal @ Z4 @ ( produc2007226135796332929nnreal @ C @ ( produc4511245868158468465l_real @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_831_mem__case__prodI,axiom,
    ! [Z4: a > real,C: real > real > set_a_real,A: real,B: real] :
      ( ( member_a_real @ Z4 @ ( C @ A @ B ) )
     => ( member_a_real @ Z4 @ ( produc5024990353902517109a_real @ C @ ( produc4511245868158468465l_real @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_832_mem__case__prodI,axiom,
    ! [Z4: real > complex,C: ( real > a ) > sigma_measure_real > set_real_complex,A: real > a,B: sigma_measure_real] :
      ( ( member_real_complex @ Z4 @ ( C @ A @ B ) )
     => ( member_real_complex @ Z4 @ ( produc9096342308652084503omplex @ C @ ( produc623176010801490259e_real @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_833_mem__case__prodI,axiom,
    ! [Z4: real > real,C: ( real > a ) > sigma_measure_real > set_real_real,A: real > a,B: sigma_measure_real] :
      ( ( member_real_real @ Z4 @ ( C @ A @ B ) )
     => ( member_real_real @ Z4 @ ( produc4072331520297120149l_real @ C @ ( produc623176010801490259e_real @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_834_mem__case__prodI,axiom,
    ! [Z4: real > a,C: ( real > a ) > sigma_measure_real > set_real_a,A: real > a,B: sigma_measure_real] :
      ( ( member_real_a @ Z4 @ ( C @ A @ B ) )
     => ( member_real_a @ Z4 @ ( produc4872297359009494785real_a @ C @ ( produc623176010801490259e_real @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_835_mem__case__prodI,axiom,
    ! [Z4: a > extend8495563244428889912nnreal,C: ( real > a ) > sigma_measure_real > set_a_7161065143582548615nnreal,A: real > a,B: sigma_measure_real] :
      ( ( member298456594901751504nnreal @ Z4 @ ( C @ A @ B ) )
     => ( member298456594901751504nnreal @ Z4 @ ( produc6281549229861490975nnreal @ C @ ( produc623176010801490259e_real @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_836_mem__case__prodI,axiom,
    ! [Z4: a > real,C: ( real > a ) > sigma_measure_real > set_a_real,A: real > a,B: sigma_measure_real] :
      ( ( member_a_real @ Z4 @ ( C @ A @ B ) )
     => ( member_a_real @ Z4 @ ( produc2463857943116076051a_real @ C @ ( produc623176010801490259e_real @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_837_mem__case__prodI2,axiom,
    ! [P3: produc2422161461964618553l_real,Z4: real > complex,C: real > real > set_real_complex] :
      ( ! [A4: real,B3: real] :
          ( ( P3
            = ( produc4511245868158468465l_real @ A4 @ B3 ) )
         => ( member_real_complex @ Z4 @ ( C @ A4 @ B3 ) ) )
     => ( member_real_complex @ Z4 @ ( produc4115511989314032949omplex @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_838_mem__case__prodI2,axiom,
    ! [P3: produc2422161461964618553l_real,Z4: real > real,C: real > real > set_real_real] :
      ( ! [A4: real,B3: real] :
          ( ( P3
            = ( produc4511245868158468465l_real @ A4 @ B3 ) )
         => ( member_real_real @ Z4 @ ( C @ A4 @ B3 ) ) )
     => ( member_real_real @ Z4 @ ( produc8597459571820422835l_real @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_839_mem__case__prodI2,axiom,
    ! [P3: produc2422161461964618553l_real,Z4: real > a,C: real > real > set_real_a] :
      ( ! [A4: real,B3: real] :
          ( ( P3
            = ( produc4511245868158468465l_real @ A4 @ B3 ) )
         => ( member_real_a @ Z4 @ ( C @ A4 @ B3 ) ) )
     => ( member_real_a @ Z4 @ ( produc7433429769795935843real_a @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_840_mem__case__prodI2,axiom,
    ! [P3: produc2422161461964618553l_real,Z4: a > extend8495563244428889912nnreal,C: real > real > set_a_7161065143582548615nnreal] :
      ( ! [A4: real,B3: real] :
          ( ( P3
            = ( produc4511245868158468465l_real @ A4 @ B3 ) )
         => ( member298456594901751504nnreal @ Z4 @ ( C @ A4 @ B3 ) ) )
     => ( member298456594901751504nnreal @ Z4 @ ( produc2007226135796332929nnreal @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_841_mem__case__prodI2,axiom,
    ! [P3: produc2422161461964618553l_real,Z4: a > real,C: real > real > set_a_real] :
      ( ! [A4: real,B3: real] :
          ( ( P3
            = ( produc4511245868158468465l_real @ A4 @ B3 ) )
         => ( member_a_real @ Z4 @ ( C @ A4 @ B3 ) ) )
     => ( member_a_real @ Z4 @ ( produc5024990353902517109a_real @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_842_mem__case__prodI2,axiom,
    ! [P3: produc725540845905733987e_real,Z4: real > complex,C: ( real > a ) > sigma_measure_real > set_real_complex] :
      ( ! [A4: real > a,B3: sigma_measure_real] :
          ( ( P3
            = ( produc623176010801490259e_real @ A4 @ B3 ) )
         => ( member_real_complex @ Z4 @ ( C @ A4 @ B3 ) ) )
     => ( member_real_complex @ Z4 @ ( produc9096342308652084503omplex @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_843_mem__case__prodI2,axiom,
    ! [P3: produc725540845905733987e_real,Z4: real > real,C: ( real > a ) > sigma_measure_real > set_real_real] :
      ( ! [A4: real > a,B3: sigma_measure_real] :
          ( ( P3
            = ( produc623176010801490259e_real @ A4 @ B3 ) )
         => ( member_real_real @ Z4 @ ( C @ A4 @ B3 ) ) )
     => ( member_real_real @ Z4 @ ( produc4072331520297120149l_real @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_844_mem__case__prodI2,axiom,
    ! [P3: produc725540845905733987e_real,Z4: real > a,C: ( real > a ) > sigma_measure_real > set_real_a] :
      ( ! [A4: real > a,B3: sigma_measure_real] :
          ( ( P3
            = ( produc623176010801490259e_real @ A4 @ B3 ) )
         => ( member_real_a @ Z4 @ ( C @ A4 @ B3 ) ) )
     => ( member_real_a @ Z4 @ ( produc4872297359009494785real_a @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_845_mem__case__prodI2,axiom,
    ! [P3: produc725540845905733987e_real,Z4: a > extend8495563244428889912nnreal,C: ( real > a ) > sigma_measure_real > set_a_7161065143582548615nnreal] :
      ( ! [A4: real > a,B3: sigma_measure_real] :
          ( ( P3
            = ( produc623176010801490259e_real @ A4 @ B3 ) )
         => ( member298456594901751504nnreal @ Z4 @ ( C @ A4 @ B3 ) ) )
     => ( member298456594901751504nnreal @ Z4 @ ( produc6281549229861490975nnreal @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_846_mem__case__prodI2,axiom,
    ! [P3: produc725540845905733987e_real,Z4: a > real,C: ( real > a ) > sigma_measure_real > set_a_real] :
      ( ! [A4: real > a,B3: sigma_measure_real] :
          ( ( P3
            = ( produc623176010801490259e_real @ A4 @ B3 ) )
         => ( member_a_real @ Z4 @ ( C @ A4 @ B3 ) ) )
     => ( member_a_real @ Z4 @ ( produc2463857943116076051a_real @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_847_case__prodI,axiom,
    ! [F: quasi_borel_a > produc725540845905733987e_real > $o,A: quasi_borel_a,B: produc725540845905733987e_real] :
      ( ( F @ A @ B )
     => ( produc7993906655439511789real_o @ F @ ( produc4145838808978236886e_real @ A @ B ) ) ) ).

% case_prodI
thf(fact_848_case__prodI,axiom,
    ! [F: ( real > a ) > sigma_measure_real > $o,A: real > a,B: sigma_measure_real] :
      ( ( F @ A @ B )
     => ( produc5798473187818486320real_o @ F @ ( produc623176010801490259e_real @ A @ B ) ) ) ).

% case_prodI
thf(fact_849_case__prodI,axiom,
    ! [F: real > real > $o,A: real,B: real] :
      ( ( F @ A @ B )
     => ( produc5414030515140494994real_o @ F @ ( produc4511245868158468465l_real @ A @ B ) ) ) ).

% case_prodI
thf(fact_850_case__prodI2,axiom,
    ! [P3: produc6543235832880896358e_real,C: quasi_borel_a > produc725540845905733987e_real > $o] :
      ( ! [A4: quasi_borel_a,B3: produc725540845905733987e_real] :
          ( ( P3
            = ( produc4145838808978236886e_real @ A4 @ B3 ) )
         => ( C @ A4 @ B3 ) )
     => ( produc7993906655439511789real_o @ C @ P3 ) ) ).

% case_prodI2
thf(fact_851_case__prodI2,axiom,
    ! [P3: produc725540845905733987e_real,C: ( real > a ) > sigma_measure_real > $o] :
      ( ! [A4: real > a,B3: sigma_measure_real] :
          ( ( P3
            = ( produc623176010801490259e_real @ A4 @ B3 ) )
         => ( C @ A4 @ B3 ) )
     => ( produc5798473187818486320real_o @ C @ P3 ) ) ).

% case_prodI2
thf(fact_852_case__prodI2,axiom,
    ! [P3: produc2422161461964618553l_real,C: real > real > $o] :
      ( ! [A4: real,B3: real] :
          ( ( P3
            = ( produc4511245868158468465l_real @ A4 @ B3 ) )
         => ( C @ A4 @ B3 ) )
     => ( produc5414030515140494994real_o @ C @ P3 ) ) ).

% case_prodI2
thf(fact_853_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_854_add__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_855_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_856_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_857_add_Oright__neutral,axiom,
    ! [A: extend8495563244428889912nnreal] :
      ( ( plus_p1859984266308609217nnreal @ A @ zero_z7100319975126383169nnreal )
      = A ) ).

% add.right_neutral
thf(fact_858_add_Oright__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.right_neutral
thf(fact_859_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_860_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A @ A ) )
      = ( A = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_861_add__cancel__left__left,axiom,
    ! [B: real,A: real] :
      ( ( ( plus_plus_real @ B @ A )
        = A )
      = ( B = zero_zero_real ) ) ).

% add_cancel_left_left
thf(fact_862_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_863_add__cancel__left__right,axiom,
    ! [A: real,B: real] :
      ( ( ( plus_plus_real @ A @ B )
        = A )
      = ( B = zero_zero_real ) ) ).

% add_cancel_left_right
thf(fact_864_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_865_add__cancel__right__left,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( plus_plus_real @ B @ A ) )
      = ( B = zero_zero_real ) ) ).

% add_cancel_right_left
thf(fact_866_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ A ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_867_add__cancel__right__right,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( plus_plus_real @ A @ B ) )
      = ( B = zero_zero_real ) ) ).

% add_cancel_right_right
thf(fact_868_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ A @ B ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_869_add__eq__0__iff__both__eq__0,axiom,
    ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
      ( ( ( plus_p1859984266308609217nnreal @ X4 @ Y3 )
        = zero_z7100319975126383169nnreal )
      = ( ( X4 = zero_z7100319975126383169nnreal )
        & ( Y3 = zero_z7100319975126383169nnreal ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_870_add__eq__0__iff__both__eq__0,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ( plus_plus_nat @ X4 @ Y3 )
        = zero_zero_nat )
      = ( ( X4 = zero_zero_nat )
        & ( Y3 = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_871_zero__eq__add__iff__both__eq__0,axiom,
    ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
      ( ( zero_z7100319975126383169nnreal
        = ( plus_p1859984266308609217nnreal @ X4 @ Y3 ) )
      = ( ( X4 = zero_z7100319975126383169nnreal )
        & ( Y3 = zero_z7100319975126383169nnreal ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_872_zero__eq__add__iff__both__eq__0,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X4 @ Y3 ) )
      = ( ( X4 = zero_zero_nat )
        & ( Y3 = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_873_add__0,axiom,
    ! [A: extend8495563244428889912nnreal] :
      ( ( plus_p1859984266308609217nnreal @ zero_z7100319975126383169nnreal @ A )
      = A ) ).

% add_0
thf(fact_874_add__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add_0
thf(fact_875_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_876_add__diff__cancel,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_877_diff__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_878_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_879_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_880_add__diff__cancel__left_H,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_881_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_882_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_883_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_884_add__diff__cancel__right_H,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_885_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_886_add__Pair,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( plus_p1196244663705802608l_real @ ( produc4511245868158468465l_real @ A @ B ) @ ( produc4511245868158468465l_real @ C @ D ) )
      = ( produc4511245868158468465l_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).

% add_Pair
thf(fact_887_add__Pair,axiom,
    ! [A: real,B: extend8495563244428889912nnreal,C: real,D: extend8495563244428889912nnreal] :
      ( ( plus_p6793480957787774588nnreal @ ( produc4778015194254607485nnreal @ A @ B ) @ ( produc4778015194254607485nnreal @ C @ D ) )
      = ( produc4778015194254607485nnreal @ ( plus_plus_real @ A @ C ) @ ( plus_p1859984266308609217nnreal @ B @ D ) ) ) ).

% add_Pair
thf(fact_888_add__Pair,axiom,
    ! [A: real,B: nat,C: real,D: nat] :
      ( ( plus_p4925795495032332052al_nat @ ( produc3181502643871035669al_nat @ A @ B ) @ ( produc3181502643871035669al_nat @ C @ D ) )
      = ( produc3181502643871035669al_nat @ ( plus_plus_real @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ).

% add_Pair
thf(fact_889_add__Pair,axiom,
    ! [A: extend8495563244428889912nnreal,B: real,C: extend8495563244428889912nnreal,D: real] :
      ( ( plus_p6747932658443751932l_real @ ( produc2810268924804063229l_real @ A @ B ) @ ( produc2810268924804063229l_real @ C @ D ) )
      = ( produc2810268924804063229l_real @ ( plus_p1859984266308609217nnreal @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).

% add_Pair
thf(fact_890_add__Pair,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D: extend8495563244428889912nnreal] :
      ( ( plus_p3686502382754676488nnreal @ ( produc344325839068023049nnreal @ A @ B ) @ ( produc344325839068023049nnreal @ C @ D ) )
      = ( produc344325839068023049nnreal @ ( plus_p1859984266308609217nnreal @ A @ C ) @ ( plus_p1859984266308609217nnreal @ B @ D ) ) ) ).

% add_Pair
thf(fact_891_add__Pair,axiom,
    ! [A: extend8495563244428889912nnreal,B: nat,C: extend8495563244428889912nnreal,D: nat] :
      ( ( plus_p2027530238859583392al_nat @ ( produc625717604924970401al_nat @ A @ B ) @ ( produc625717604924970401al_nat @ C @ D ) )
      = ( produc625717604924970401al_nat @ ( plus_p1859984266308609217nnreal @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ).

% add_Pair
thf(fact_892_add__Pair,axiom,
    ! [A: nat,B: real,C: nat,D: real] :
      ( ( plus_p8900843186509212308t_real @ ( produc7837566107596912789t_real @ A @ B ) @ ( produc7837566107596912789t_real @ C @ D ) )
      = ( produc7837566107596912789t_real @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).

% add_Pair
thf(fact_893_add__Pair,axiom,
    ! [A: nat,B: extend8495563244428889912nnreal,C: nat,D: extend8495563244428889912nnreal] :
      ( ( plus_p3873988092691981216nnreal @ ( produc5075389201112886689nnreal @ A @ B ) @ ( produc5075389201112886689nnreal @ C @ D ) )
      = ( produc5075389201112886689nnreal @ ( plus_plus_nat @ A @ C ) @ ( plus_p1859984266308609217nnreal @ B @ D ) ) ) ).

% add_Pair
thf(fact_894_add__Pair,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( plus_p9057090461656269880at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
      = ( product_Pair_nat_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ).

% add_Pair
thf(fact_895_add__minus__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_896_add__minus__cancel,axiom,
    ! [A: produc2422161461964618553l_real,B: produc2422161461964618553l_real] :
      ( ( plus_p1196244663705802608l_real @ A @ ( plus_p1196244663705802608l_real @ ( uminus2141826702334040752l_real @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_897_minus__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_898_minus__add__cancel,axiom,
    ! [A: produc2422161461964618553l_real,B: produc2422161461964618553l_real] :
      ( ( plus_p1196244663705802608l_real @ ( uminus2141826702334040752l_real @ A ) @ ( plus_p1196244663705802608l_real @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_899_minus__add__distrib,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).

% minus_add_distrib
thf(fact_900_minus__add__distrib,axiom,
    ! [A: produc2422161461964618553l_real,B: produc2422161461964618553l_real] :
      ( ( uminus2141826702334040752l_real @ ( plus_p1196244663705802608l_real @ A @ B ) )
      = ( plus_p1196244663705802608l_real @ ( uminus2141826702334040752l_real @ A ) @ ( uminus2141826702334040752l_real @ B ) ) ) ).

% minus_add_distrib
thf(fact_901_Bochner__Integration_Ointegrable__add,axiom,
    ! [M: sigma_measure_real,F: real > real,G: real > real] :
      ( ( bochne3340023020068487468l_real @ M @ F )
     => ( ( bochne3340023020068487468l_real @ M @ G )
       => ( bochne3340023020068487468l_real @ M
          @ ^ [X: real] : ( plus_plus_real @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).

% Bochner_Integration.integrable_add
thf(fact_902_Bochner__Integration_Ointegrable__add,axiom,
    ! [M: sigma_measure_a,F: a > real,G: a > real] :
      ( ( bochne2139062162225249880a_real @ M @ F )
     => ( ( bochne2139062162225249880a_real @ M @ G )
       => ( bochne2139062162225249880a_real @ M
          @ ^ [X: a] : ( plus_plus_real @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).

% Bochner_Integration.integrable_add
thf(fact_903_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_904_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_905_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel2
thf(fact_906_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).

% le_add_same_cancel2
thf(fact_907_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel1
thf(fact_908_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).

% le_add_same_cancel1
thf(fact_909_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_910_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% add_le_same_cancel2
thf(fact_911_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_912_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% add_le_same_cancel1
thf(fact_913_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_914_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_915_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_916_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_917_diff__add__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_918_add_Oright__inverse,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
      = zero_zero_real ) ).

% add.right_inverse
thf(fact_919_add_Oright__inverse,axiom,
    ! [A: produc2422161461964618553l_real] :
      ( ( plus_p1196244663705802608l_real @ A @ ( uminus2141826702334040752l_real @ A ) )
      = zero_z1365759597461889520l_real ) ).

% add.right_inverse
thf(fact_920_ab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_left_minus
thf(fact_921_ab__left__minus,axiom,
    ! [A: produc2422161461964618553l_real] :
      ( ( plus_p1196244663705802608l_real @ ( uminus2141826702334040752l_real @ A ) @ A )
      = zero_z1365759597461889520l_real ) ).

% ab_left_minus
thf(fact_922_diff__minus__eq__add,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
      = ( plus_plus_real @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_923_diff__minus__eq__add,axiom,
    ! [A: produc2422161461964618553l_real,B: produc2422161461964618553l_real] :
      ( ( minus_885040589139849760l_real @ A @ ( uminus2141826702334040752l_real @ B ) )
      = ( plus_p1196244663705802608l_real @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_924_uminus__add__conv__diff,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
      = ( minus_minus_real @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_925_uminus__add__conv__diff,axiom,
    ! [A: produc2422161461964618553l_real,B: produc2422161461964618553l_real] :
      ( ( plus_p1196244663705802608l_real @ ( uminus2141826702334040752l_real @ A ) @ B )
      = ( minus_885040589139849760l_real @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_926_ennreal__le__iff,axiom,
    ! [Y3: real,X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ( ( ord_le3935885782089961368nnreal @ ( extend7643940197134561352nnreal @ X4 ) @ ( extend7643940197134561352nnreal @ Y3 ) )
        = ( ord_less_eq_real @ X4 @ Y3 ) ) ) ).

% ennreal_le_iff
thf(fact_927_real__add__minus__iff,axiom,
    ! [X4: real,A: real] :
      ( ( ( plus_plus_real @ X4 @ ( uminus_uminus_real @ A ) )
        = zero_zero_real )
      = ( X4 = A ) ) ).

% real_add_minus_iff
thf(fact_928_Bochner__Integration_Ointegral__add,axiom,
    ! [M: sigma_measure_real,F: real > real,G: real > real] :
      ( ( bochne3340023020068487468l_real @ M @ F )
     => ( ( bochne3340023020068487468l_real @ M @ G )
       => ( ( bochne3715101410578510557l_real @ M
            @ ^ [X: real] : ( plus_plus_real @ ( F @ X ) @ ( G @ X ) ) )
          = ( plus_plus_real @ ( bochne3715101410578510557l_real @ M @ F ) @ ( bochne3715101410578510557l_real @ M @ G ) ) ) ) ) ).

% Bochner_Integration.integral_add
thf(fact_929_Bochner__Integration_Ointegral__add,axiom,
    ! [M: sigma_measure_a,F: a > real,G: a > real] :
      ( ( bochne2139062162225249880a_real @ M @ F )
     => ( ( bochne2139062162225249880a_real @ M @ G )
       => ( ( bochne378719280626478695a_real @ M
            @ ^ [X: a] : ( plus_plus_real @ ( F @ X ) @ ( G @ X ) ) )
          = ( plus_plus_real @ ( bochne378719280626478695a_real @ M @ F ) @ ( bochne378719280626478695a_real @ M @ G ) ) ) ) ) ).

% Bochner_Integration.integral_add
thf(fact_930_pred__subset__eq2,axiom,
    ! [R: set_Pr7780167738718111686e_real,S2: set_Pr7780167738718111686e_real] :
      ( ( ord_le3153059659696763535real_o
        @ ^ [X: quasi_borel_a,Y7: produc725540845905733987e_real] : ( member6844354795726785935e_real @ ( produc4145838808978236886e_real @ X @ Y7 ) @ R )
        @ ^ [X: quasi_borel_a,Y7: produc725540845905733987e_real] : ( member6844354795726785935e_real @ ( produc4145838808978236886e_real @ X @ Y7 ) @ S2 ) )
      = ( ord_le1172168791859281766e_real @ R @ S2 ) ) ).

% pred_subset_eq2
thf(fact_931_pred__subset__eq2,axiom,
    ! [R: set_Pr4989138886603757763e_real,S2: set_Pr4989138886603757763e_real] :
      ( ( ord_le2360185657988600274real_o
        @ ^ [X: real > a,Y7: sigma_measure_real] : ( member5162606148374117132e_real @ ( produc623176010801490259e_real @ X @ Y7 ) @ R )
        @ ^ [X: real > a,Y7: sigma_measure_real] : ( member5162606148374117132e_real @ ( produc623176010801490259e_real @ X @ Y7 ) @ S2 ) )
      = ( ord_le3323511981236633699e_real @ R @ S2 ) ) ).

% pred_subset_eq2
thf(fact_932_pred__subset__eq2,axiom,
    ! [R: set_Pr6218003697084177305l_real,S2: set_Pr6218003697084177305l_real] :
      ( ( ord_le1079842393864317646real_o
        @ ^ [X: real,Y7: real] : ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ X @ Y7 ) @ R )
        @ ^ [X: real,Y7: real] : ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ X @ Y7 ) @ S2 ) )
      = ( ord_le64383758879589177l_real @ R @ S2 ) ) ).

% pred_subset_eq2
thf(fact_933_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_934_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( plus_p1859984266308609217nnreal @ ( plus_p1859984266308609217nnreal @ A @ B ) @ C )
      = ( plus_p1859984266308609217nnreal @ A @ ( plus_p1859984266308609217nnreal @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_935_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_936_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I4: real,J: real,K: real,L2: real] :
      ( ( ( I4 = J )
        & ( K = L2 ) )
     => ( ( plus_plus_real @ I4 @ K )
        = ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_937_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I4: extend8495563244428889912nnreal,J: extend8495563244428889912nnreal,K: extend8495563244428889912nnreal,L2: extend8495563244428889912nnreal] :
      ( ( ( I4 = J )
        & ( K = L2 ) )
     => ( ( plus_p1859984266308609217nnreal @ I4 @ K )
        = ( plus_p1859984266308609217nnreal @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_938_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I4: nat,J: nat,K: nat,L2: nat] :
      ( ( ( I4 = J )
        & ( K = L2 ) )
     => ( ( plus_plus_nat @ I4 @ K )
        = ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_939_group__cancel_Oadd1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2
        = ( plus_plus_real @ K @ A ) )
     => ( ( plus_plus_real @ A2 @ B )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_940_group__cancel_Oadd1,axiom,
    ! [A2: extend8495563244428889912nnreal,K: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( A2
        = ( plus_p1859984266308609217nnreal @ K @ A ) )
     => ( ( plus_p1859984266308609217nnreal @ A2 @ B )
        = ( plus_p1859984266308609217nnreal @ K @ ( plus_p1859984266308609217nnreal @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_941_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2
        = ( plus_plus_nat @ K @ A ) )
     => ( ( plus_plus_nat @ A2 @ B )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_942_group__cancel_Oadd2,axiom,
    ! [B5: real,K: real,B: real,A: real] :
      ( ( B5
        = ( plus_plus_real @ K @ B ) )
     => ( ( plus_plus_real @ A @ B5 )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_943_group__cancel_Oadd2,axiom,
    ! [B5: extend8495563244428889912nnreal,K: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
      ( ( B5
        = ( plus_p1859984266308609217nnreal @ K @ B ) )
     => ( ( plus_p1859984266308609217nnreal @ A @ B5 )
        = ( plus_p1859984266308609217nnreal @ K @ ( plus_p1859984266308609217nnreal @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_944_group__cancel_Oadd2,axiom,
    ! [B5: nat,K: nat,B: nat,A: nat] :
      ( ( B5
        = ( plus_plus_nat @ K @ B ) )
     => ( ( plus_plus_nat @ A @ B5 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_945_add_Oassoc,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% add.assoc
thf(fact_946_add_Oassoc,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( plus_p1859984266308609217nnreal @ ( plus_p1859984266308609217nnreal @ A @ B ) @ C )
      = ( plus_p1859984266308609217nnreal @ A @ ( plus_p1859984266308609217nnreal @ B @ C ) ) ) ).

% add.assoc
thf(fact_947_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.assoc
thf(fact_948_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_949_add_Oright__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_950_add_Ocommute,axiom,
    ( plus_plus_real
    = ( ^ [A5: real,B4: real] : ( plus_plus_real @ B4 @ A5 ) ) ) ).

% add.commute
thf(fact_951_add_Ocommute,axiom,
    ( plus_p1859984266308609217nnreal
    = ( ^ [A5: extend8495563244428889912nnreal,B4: extend8495563244428889912nnreal] : ( plus_p1859984266308609217nnreal @ B4 @ A5 ) ) ) ).

% add.commute
thf(fact_952_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).

% add.commute
thf(fact_953_add_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] :
      ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% add.left_commute
thf(fact_954_add_Oleft__commute,axiom,
    ! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( plus_p1859984266308609217nnreal @ B @ ( plus_p1859984266308609217nnreal @ A @ C ) )
      = ( plus_p1859984266308609217nnreal @ A @ ( plus_p1859984266308609217nnreal @ B @ C ) ) ) ).

% add.left_commute
thf(fact_955_add_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.left_commute
thf(fact_956_add__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_957_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_958_add__right__imp__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_959_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_960_mem__case__prodE,axiom,
    ! [Z4: real > complex,C: real > real > set_real_complex,P3: produc2422161461964618553l_real] :
      ( ( member_real_complex @ Z4 @ ( produc4115511989314032949omplex @ C @ P3 ) )
     => ~ ! [X6: real,Y4: real] :
            ( ( P3
              = ( produc4511245868158468465l_real @ X6 @ Y4 ) )
           => ~ ( member_real_complex @ Z4 @ ( C @ X6 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_961_mem__case__prodE,axiom,
    ! [Z4: real > real,C: real > real > set_real_real,P3: produc2422161461964618553l_real] :
      ( ( member_real_real @ Z4 @ ( produc8597459571820422835l_real @ C @ P3 ) )
     => ~ ! [X6: real,Y4: real] :
            ( ( P3
              = ( produc4511245868158468465l_real @ X6 @ Y4 ) )
           => ~ ( member_real_real @ Z4 @ ( C @ X6 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_962_mem__case__prodE,axiom,
    ! [Z4: real > a,C: real > real > set_real_a,P3: produc2422161461964618553l_real] :
      ( ( member_real_a @ Z4 @ ( produc7433429769795935843real_a @ C @ P3 ) )
     => ~ ! [X6: real,Y4: real] :
            ( ( P3
              = ( produc4511245868158468465l_real @ X6 @ Y4 ) )
           => ~ ( member_real_a @ Z4 @ ( C @ X6 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_963_mem__case__prodE,axiom,
    ! [Z4: a > extend8495563244428889912nnreal,C: real > real > set_a_7161065143582548615nnreal,P3: produc2422161461964618553l_real] :
      ( ( member298456594901751504nnreal @ Z4 @ ( produc2007226135796332929nnreal @ C @ P3 ) )
     => ~ ! [X6: real,Y4: real] :
            ( ( P3
              = ( produc4511245868158468465l_real @ X6 @ Y4 ) )
           => ~ ( member298456594901751504nnreal @ Z4 @ ( C @ X6 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_964_mem__case__prodE,axiom,
    ! [Z4: a > real,C: real > real > set_a_real,P3: produc2422161461964618553l_real] :
      ( ( member_a_real @ Z4 @ ( produc5024990353902517109a_real @ C @ P3 ) )
     => ~ ! [X6: real,Y4: real] :
            ( ( P3
              = ( produc4511245868158468465l_real @ X6 @ Y4 ) )
           => ~ ( member_a_real @ Z4 @ ( C @ X6 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_965_mem__case__prodE,axiom,
    ! [Z4: real > complex,C: ( real > a ) > sigma_measure_real > set_real_complex,P3: produc725540845905733987e_real] :
      ( ( member_real_complex @ Z4 @ ( produc9096342308652084503omplex @ C @ P3 ) )
     => ~ ! [X6: real > a,Y4: sigma_measure_real] :
            ( ( P3
              = ( produc623176010801490259e_real @ X6 @ Y4 ) )
           => ~ ( member_real_complex @ Z4 @ ( C @ X6 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_966_mem__case__prodE,axiom,
    ! [Z4: real > real,C: ( real > a ) > sigma_measure_real > set_real_real,P3: produc725540845905733987e_real] :
      ( ( member_real_real @ Z4 @ ( produc4072331520297120149l_real @ C @ P3 ) )
     => ~ ! [X6: real > a,Y4: sigma_measure_real] :
            ( ( P3
              = ( produc623176010801490259e_real @ X6 @ Y4 ) )
           => ~ ( member_real_real @ Z4 @ ( C @ X6 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_967_mem__case__prodE,axiom,
    ! [Z4: real > a,C: ( real > a ) > sigma_measure_real > set_real_a,P3: produc725540845905733987e_real] :
      ( ( member_real_a @ Z4 @ ( produc4872297359009494785real_a @ C @ P3 ) )
     => ~ ! [X6: real > a,Y4: sigma_measure_real] :
            ( ( P3
              = ( produc623176010801490259e_real @ X6 @ Y4 ) )
           => ~ ( member_real_a @ Z4 @ ( C @ X6 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_968_mem__case__prodE,axiom,
    ! [Z4: a > extend8495563244428889912nnreal,C: ( real > a ) > sigma_measure_real > set_a_7161065143582548615nnreal,P3: produc725540845905733987e_real] :
      ( ( member298456594901751504nnreal @ Z4 @ ( produc6281549229861490975nnreal @ C @ P3 ) )
     => ~ ! [X6: real > a,Y4: sigma_measure_real] :
            ( ( P3
              = ( produc623176010801490259e_real @ X6 @ Y4 ) )
           => ~ ( member298456594901751504nnreal @ Z4 @ ( C @ X6 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_969_mem__case__prodE,axiom,
    ! [Z4: a > real,C: ( real > a ) > sigma_measure_real > set_a_real,P3: produc725540845905733987e_real] :
      ( ( member_a_real @ Z4 @ ( produc2463857943116076051a_real @ C @ P3 ) )
     => ~ ! [X6: real > a,Y4: sigma_measure_real] :
            ( ( P3
              = ( produc623176010801490259e_real @ X6 @ Y4 ) )
           => ~ ( member_a_real @ Z4 @ ( C @ X6 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_970_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I4: nat,J: nat,K: nat,L2: nat] :
      ( ( ( ord_less_eq_nat @ I4 @ J )
        & ( K = L2 ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_971_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I4: real,J: real,K: real,L2: real] :
      ( ( ( ord_less_eq_real @ I4 @ J )
        & ( K = L2 ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I4 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_972_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I4: extend8495563244428889912nnreal,J: extend8495563244428889912nnreal,K: extend8495563244428889912nnreal,L2: extend8495563244428889912nnreal] :
      ( ( ( ord_le3935885782089961368nnreal @ I4 @ J )
        & ( K = L2 ) )
     => ( ord_le3935885782089961368nnreal @ ( plus_p1859984266308609217nnreal @ I4 @ K ) @ ( plus_p1859984266308609217nnreal @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_973_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I4: nat,J: nat,K: nat,L2: nat] :
      ( ( ( I4 = J )
        & ( ord_less_eq_nat @ K @ L2 ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_974_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I4: real,J: real,K: real,L2: real] :
      ( ( ( I4 = J )
        & ( ord_less_eq_real @ K @ L2 ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I4 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_975_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I4: extend8495563244428889912nnreal,J: extend8495563244428889912nnreal,K: extend8495563244428889912nnreal,L2: extend8495563244428889912nnreal] :
      ( ( ( I4 = J )
        & ( ord_le3935885782089961368nnreal @ K @ L2 ) )
     => ( ord_le3935885782089961368nnreal @ ( plus_p1859984266308609217nnreal @ I4 @ K ) @ ( plus_p1859984266308609217nnreal @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_976_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I4: nat,J: nat,K: nat,L2: nat] :
      ( ( ( ord_less_eq_nat @ I4 @ J )
        & ( ord_less_eq_nat @ K @ L2 ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_977_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I4: real,J: real,K: real,L2: real] :
      ( ( ( ord_less_eq_real @ I4 @ J )
        & ( ord_less_eq_real @ K @ L2 ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I4 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_978_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I4: extend8495563244428889912nnreal,J: extend8495563244428889912nnreal,K: extend8495563244428889912nnreal,L2: extend8495563244428889912nnreal] :
      ( ( ( ord_le3935885782089961368nnreal @ I4 @ J )
        & ( ord_le3935885782089961368nnreal @ K @ L2 ) )
     => ( ord_le3935885782089961368nnreal @ ( plus_p1859984266308609217nnreal @ I4 @ K ) @ ( plus_p1859984266308609217nnreal @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_979_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_980_add__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_mono
thf(fact_981_add__mono,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A @ B )
     => ( ( ord_le3935885782089961368nnreal @ C @ D )
       => ( ord_le3935885782089961368nnreal @ ( plus_p1859984266308609217nnreal @ A @ C ) @ ( plus_p1859984266308609217nnreal @ B @ D ) ) ) ) ).

% add_mono
thf(fact_982_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_983_add__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).

% add_left_mono
thf(fact_984_add__left__mono,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A @ B )
     => ( ord_le3935885782089961368nnreal @ ( plus_p1859984266308609217nnreal @ C @ A ) @ ( plus_p1859984266308609217nnreal @ C @ B ) ) ) ).

% add_left_mono
thf(fact_985_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C2: nat] :
            ( B
           != ( plus_plus_nat @ A @ C2 ) ) ) ).

% less_eqE
thf(fact_986_less__eqE,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A @ B )
     => ~ ! [C2: extend8495563244428889912nnreal] :
            ( B
           != ( plus_p1859984266308609217nnreal @ A @ C2 ) ) ) ).

% less_eqE
thf(fact_987_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_988_add__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).

% add_right_mono
thf(fact_989_add__right__mono,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A @ B )
     => ( ord_le3935885782089961368nnreal @ ( plus_p1859984266308609217nnreal @ A @ C ) @ ( plus_p1859984266308609217nnreal @ B @ C ) ) ) ).

% add_right_mono
thf(fact_990_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] :
        ? [C3: nat] :
          ( B4
          = ( plus_plus_nat @ A5 @ C3 ) ) ) ) ).

% le_iff_add
thf(fact_991_le__iff__add,axiom,
    ( ord_le3935885782089961368nnreal
    = ( ^ [A5: extend8495563244428889912nnreal,B4: extend8495563244428889912nnreal] :
        ? [C3: extend8495563244428889912nnreal] :
          ( B4
          = ( plus_p1859984266308609217nnreal @ A5 @ C3 ) ) ) ) ).

% le_iff_add
thf(fact_992_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_993_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
     => ( ord_less_eq_real @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_994_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_995_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
     => ( ord_less_eq_real @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_996_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: extend8495563244428889912nnreal] :
      ( ( plus_p1859984266308609217nnreal @ zero_z7100319975126383169nnreal @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_997_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_998_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_999_add_Ocomm__neutral,axiom,
    ! [A: extend8495563244428889912nnreal] :
      ( ( plus_p1859984266308609217nnreal @ A @ zero_z7100319975126383169nnreal )
      = A ) ).

% add.comm_neutral
thf(fact_1000_add_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.comm_neutral
thf(fact_1001_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_1002_add_Ogroup__left__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_1003_verit__sum__simplify,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% verit_sum_simplify
thf(fact_1004_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_1005_add__diff__add,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
      = ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).

% add_diff_add
thf(fact_1006_group__cancel_Osub1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2
        = ( plus_plus_real @ K @ A ) )
     => ( ( minus_minus_real @ A2 @ B )
        = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_1007_diff__eq__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( minus_minus_real @ A @ B )
        = C )
      = ( A
        = ( plus_plus_real @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_1008_eq__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( A
        = ( minus_minus_real @ C @ B ) )
      = ( ( plus_plus_real @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_1009_add__diff__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_1010_diff__diff__eq2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_1011_diff__add__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_1012_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_1013_add__implies__diff,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ( plus_plus_real @ C @ B )
        = A )
     => ( C
        = ( minus_minus_real @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_1014_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ( plus_plus_nat @ C @ B )
        = A )
     => ( C
        = ( minus_minus_nat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_1015_diff__diff__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_1016_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_1017_group__cancel_Oneg1,axiom,
    ! [A2: real,K: real,A: real] :
      ( ( A2
        = ( plus_plus_real @ K @ A ) )
     => ( ( uminus_uminus_real @ A2 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_1018_group__cancel_Oneg1,axiom,
    ! [A2: produc2422161461964618553l_real,K: produc2422161461964618553l_real,A: produc2422161461964618553l_real] :
      ( ( A2
        = ( plus_p1196244663705802608l_real @ K @ A ) )
     => ( ( uminus2141826702334040752l_real @ A2 )
        = ( plus_p1196244663705802608l_real @ ( uminus2141826702334040752l_real @ K ) @ ( uminus2141826702334040752l_real @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_1019_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_1020_add_Oinverse__distrib__swap,axiom,
    ! [A: produc2422161461964618553l_real,B: produc2422161461964618553l_real] :
      ( ( uminus2141826702334040752l_real @ ( plus_p1196244663705802608l_real @ A @ B ) )
      = ( plus_p1196244663705802608l_real @ ( uminus2141826702334040752l_real @ B ) @ ( uminus2141826702334040752l_real @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_1021_subrelI,axiom,
    ! [R4: set_Pr7780167738718111686e_real,S: set_Pr7780167738718111686e_real] :
      ( ! [X6: quasi_borel_a,Y4: produc725540845905733987e_real] :
          ( ( member6844354795726785935e_real @ ( produc4145838808978236886e_real @ X6 @ Y4 ) @ R4 )
         => ( member6844354795726785935e_real @ ( produc4145838808978236886e_real @ X6 @ Y4 ) @ S ) )
     => ( ord_le1172168791859281766e_real @ R4 @ S ) ) ).

% subrelI
thf(fact_1022_subrelI,axiom,
    ! [R4: set_Pr4989138886603757763e_real,S: set_Pr4989138886603757763e_real] :
      ( ! [X6: real > a,Y4: sigma_measure_real] :
          ( ( member5162606148374117132e_real @ ( produc623176010801490259e_real @ X6 @ Y4 ) @ R4 )
         => ( member5162606148374117132e_real @ ( produc623176010801490259e_real @ X6 @ Y4 ) @ S ) )
     => ( ord_le3323511981236633699e_real @ R4 @ S ) ) ).

% subrelI
thf(fact_1023_subrelI,axiom,
    ! [R4: set_Pr6218003697084177305l_real,S: set_Pr6218003697084177305l_real] :
      ( ! [X6: real,Y4: real] :
          ( ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ X6 @ Y4 ) @ R4 )
         => ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ X6 @ Y4 ) @ S ) )
     => ( ord_le64383758879589177l_real @ R4 @ S ) ) ).

% subrelI
thf(fact_1024_case__prodD,axiom,
    ! [F: quasi_borel_a > produc725540845905733987e_real > $o,A: quasi_borel_a,B: produc725540845905733987e_real] :
      ( ( produc7993906655439511789real_o @ F @ ( produc4145838808978236886e_real @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_1025_case__prodD,axiom,
    ! [F: ( real > a ) > sigma_measure_real > $o,A: real > a,B: sigma_measure_real] :
      ( ( produc5798473187818486320real_o @ F @ ( produc623176010801490259e_real @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_1026_case__prodD,axiom,
    ! [F: real > real > $o,A: real,B: real] :
      ( ( produc5414030515140494994real_o @ F @ ( produc4511245868158468465l_real @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_1027_case__prodE,axiom,
    ! [C: quasi_borel_a > produc725540845905733987e_real > $o,P3: produc6543235832880896358e_real] :
      ( ( produc7993906655439511789real_o @ C @ P3 )
     => ~ ! [X6: quasi_borel_a,Y4: produc725540845905733987e_real] :
            ( ( P3
              = ( produc4145838808978236886e_real @ X6 @ Y4 ) )
           => ~ ( C @ X6 @ Y4 ) ) ) ).

% case_prodE
thf(fact_1028_case__prodE,axiom,
    ! [C: ( real > a ) > sigma_measure_real > $o,P3: produc725540845905733987e_real] :
      ( ( produc5798473187818486320real_o @ C @ P3 )
     => ~ ! [X6: real > a,Y4: sigma_measure_real] :
            ( ( P3
              = ( produc623176010801490259e_real @ X6 @ Y4 ) )
           => ~ ( C @ X6 @ Y4 ) ) ) ).

% case_prodE
thf(fact_1029_case__prodE,axiom,
    ! [C: real > real > $o,P3: produc2422161461964618553l_real] :
      ( ( produc5414030515140494994real_o @ C @ P3 )
     => ~ ! [X6: real,Y4: real] :
            ( ( P3
              = ( produc4511245868158468465l_real @ X6 @ Y4 ) )
           => ~ ( C @ X6 @ Y4 ) ) ) ).

% case_prodE
thf(fact_1030_qbs__integrable__add,axiom,
    ! [S: probab4737552673497767871pace_a,F: a > real,G: a > real] :
      ( ( probab7312716125271441302able_a @ S @ F )
     => ( ( probab7312716125271441302able_a @ S @ G )
       => ( probab7312716125271441302able_a @ S
          @ ^ [X: a] : ( plus_plus_real @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).

% qbs_integrable_add
thf(fact_1031_add__nonpos__eq__0__iff,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X4 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X4 @ Y3 )
            = zero_zero_nat )
          = ( ( X4 = zero_zero_nat )
            & ( Y3 = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1032_add__nonpos__eq__0__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
       => ( ( ( plus_plus_real @ X4 @ Y3 )
            = zero_zero_real )
          = ( ( X4 = zero_zero_real )
            & ( Y3 = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1033_add__nonpos__eq__0__iff,axiom,
    ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ X4 @ zero_z7100319975126383169nnreal )
     => ( ( ord_le3935885782089961368nnreal @ Y3 @ zero_z7100319975126383169nnreal )
       => ( ( ( plus_p1859984266308609217nnreal @ X4 @ Y3 )
            = zero_z7100319975126383169nnreal )
          = ( ( X4 = zero_z7100319975126383169nnreal )
            & ( Y3 = zero_z7100319975126383169nnreal ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1034_add__nonneg__eq__0__iff,axiom,
    ! [X4: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
       => ( ( ( plus_plus_nat @ X4 @ Y3 )
            = zero_zero_nat )
          = ( ( X4 = zero_zero_nat )
            & ( Y3 = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1035_add__nonneg__eq__0__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ( plus_plus_real @ X4 @ Y3 )
            = zero_zero_real )
          = ( ( X4 = zero_zero_real )
            & ( Y3 = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1036_add__nonneg__eq__0__iff,axiom,
    ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ X4 )
     => ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ Y3 )
       => ( ( ( plus_p1859984266308609217nnreal @ X4 @ Y3 )
            = zero_z7100319975126383169nnreal )
          = ( ( X4 = zero_z7100319975126383169nnreal )
            & ( Y3 = zero_z7100319975126383169nnreal ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1037_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_1038_add__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_nonpos_nonpos
thf(fact_1039_add__nonpos__nonpos,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A @ zero_z7100319975126383169nnreal )
     => ( ( ord_le3935885782089961368nnreal @ B @ zero_z7100319975126383169nnreal )
       => ( ord_le3935885782089961368nnreal @ ( plus_p1859984266308609217nnreal @ A @ B ) @ zero_z7100319975126383169nnreal ) ) ) ).

% add_nonpos_nonpos
thf(fact_1040_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_1041_add__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_1042_add__nonneg__nonneg,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ A )
     => ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ B )
       => ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( plus_p1859984266308609217nnreal @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_1043_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_1044_add__increasing2,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ B @ A )
       => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_1045_add__increasing2,axiom,
    ! [C: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ C )
     => ( ( ord_le3935885782089961368nnreal @ B @ A )
       => ( ord_le3935885782089961368nnreal @ B @ ( plus_p1859984266308609217nnreal @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_1046_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_1047_add__decreasing2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_1048_add__decreasing2,axiom,
    ! [C: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ C @ zero_z7100319975126383169nnreal )
     => ( ( ord_le3935885782089961368nnreal @ A @ B )
       => ( ord_le3935885782089961368nnreal @ ( plus_p1859984266308609217nnreal @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_1049_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_1050_add__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_1051_add__increasing,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ A )
     => ( ( ord_le3935885782089961368nnreal @ B @ C )
       => ( ord_le3935885782089961368nnreal @ B @ ( plus_p1859984266308609217nnreal @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_1052_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_1053_add__decreasing,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ C @ B )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_1054_add__decreasing,axiom,
    ! [A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A @ zero_z7100319975126383169nnreal )
     => ( ( ord_le3935885782089961368nnreal @ C @ B )
       => ( ord_le3935885782089961368nnreal @ ( plus_p1859984266308609217nnreal @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_1055_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ( ( minus_minus_nat @ B @ A )
            = C )
          = ( B
            = ( plus_plus_nat @ C @ A ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1056_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
        = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1057_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1058_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
        = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1059_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
        = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1060_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
        = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1061_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1062_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1063_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% le_add_diff
thf(fact_1064_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
        = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1065_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_1066_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_1067_add__le__add__imp__diff__le,axiom,
    ! [I4: nat,K: nat,N3: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ N3 )
     => ( ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ J @ K ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ N3 )
         => ( ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ J @ K ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N3 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1068_add__le__add__imp__diff__le,axiom,
    ! [I4: real,K: real,N3: real,J: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I4 @ K ) @ N3 )
     => ( ( ord_less_eq_real @ N3 @ ( plus_plus_real @ J @ K ) )
       => ( ( ord_less_eq_real @ ( plus_plus_real @ I4 @ K ) @ N3 )
         => ( ( ord_less_eq_real @ N3 @ ( plus_plus_real @ J @ K ) )
           => ( ord_less_eq_real @ ( minus_minus_real @ N3 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1069_add__le__imp__le__diff,axiom,
    ! [I4: nat,K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ N3 )
     => ( ord_less_eq_nat @ I4 @ ( minus_minus_nat @ N3 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_1070_add__le__imp__le__diff,axiom,
    ! [I4: real,K: real,N3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I4 @ K ) @ N3 )
     => ( ord_less_eq_real @ I4 @ ( minus_minus_real @ N3 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_1071_add__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( plus_plus_real @ A @ B )
        = zero_zero_real )
      = ( B
        = ( uminus_uminus_real @ A ) ) ) ).

% add_eq_0_iff
thf(fact_1072_add__eq__0__iff,axiom,
    ! [A: produc2422161461964618553l_real,B: produc2422161461964618553l_real] :
      ( ( ( plus_p1196244663705802608l_real @ A @ B )
        = zero_z1365759597461889520l_real )
      = ( B
        = ( uminus2141826702334040752l_real @ A ) ) ) ).

% add_eq_0_iff
thf(fact_1073_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_group_add_class.ab_left_minus
thf(fact_1074_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: produc2422161461964618553l_real] :
      ( ( plus_p1196244663705802608l_real @ ( uminus2141826702334040752l_real @ A ) @ A )
      = zero_z1365759597461889520l_real ) ).

% ab_group_add_class.ab_left_minus
thf(fact_1075_add_Oinverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( ( plus_plus_real @ A @ B )
        = zero_zero_real )
     => ( ( uminus_uminus_real @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_1076_add_Oinverse__unique,axiom,
    ! [A: produc2422161461964618553l_real,B: produc2422161461964618553l_real] :
      ( ( ( plus_p1196244663705802608l_real @ A @ B )
        = zero_z1365759597461889520l_real )
     => ( ( uminus2141826702334040752l_real @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_1077_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( uminus_uminus_real @ B ) )
      = ( ( plus_plus_real @ A @ B )
        = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_1078_eq__neg__iff__add__eq__0,axiom,
    ! [A: produc2422161461964618553l_real,B: produc2422161461964618553l_real] :
      ( ( A
        = ( uminus2141826702334040752l_real @ B ) )
      = ( ( plus_p1196244663705802608l_real @ A @ B )
        = zero_z1365759597461889520l_real ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_1079_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = B )
      = ( ( plus_plus_real @ A @ B )
        = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_1080_neg__eq__iff__add__eq__0,axiom,
    ! [A: produc2422161461964618553l_real,B: produc2422161461964618553l_real] :
      ( ( ( uminus2141826702334040752l_real @ A )
        = B )
      = ( ( plus_p1196244663705802608l_real @ A @ B )
        = zero_z1365759597461889520l_real ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_1081_square__diff__square__factored,axiom,
    ! [X4: real,Y3: real] :
      ( ( minus_minus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y3 @ Y3 ) )
      = ( times_times_real @ ( plus_plus_real @ X4 @ Y3 ) @ ( minus_minus_real @ X4 @ Y3 ) ) ) ).

% square_diff_square_factored
thf(fact_1082_square__diff__square__factored,axiom,
    ! [X4: complex,Y3: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X4 @ X4 ) @ ( times_times_complex @ Y3 @ Y3 ) )
      = ( times_times_complex @ ( plus_plus_complex @ X4 @ Y3 ) @ ( minus_minus_complex @ X4 @ Y3 ) ) ) ).

% square_diff_square_factored
thf(fact_1083_eq__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( C
        = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_1084_eq__add__iff2,axiom,
    ! [A: complex,E: complex,C: complex,B: complex,D: complex] :
      ( ( ( plus_plus_complex @ ( times_times_complex @ A @ E ) @ C )
        = ( plus_plus_complex @ ( times_times_complex @ B @ E ) @ D ) )
      = ( C
        = ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_1085_eq__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_1086_eq__add__iff1,axiom,
    ! [A: complex,E: complex,C: complex,B: complex,D: complex] :
      ( ( ( plus_plus_complex @ ( times_times_complex @ A @ E ) @ C )
        = ( plus_plus_complex @ ( times_times_complex @ B @ E ) @ D ) )
      = ( ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_1087_mult__diff__mult,axiom,
    ! [X4: real,Y3: real,A: real,B: real] :
      ( ( minus_minus_real @ ( times_times_real @ X4 @ Y3 ) @ ( times_times_real @ A @ B ) )
      = ( plus_plus_real @ ( times_times_real @ X4 @ ( minus_minus_real @ Y3 @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X4 @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_1088_mult__diff__mult,axiom,
    ! [X4: complex,Y3: complex,A: complex,B: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X4 @ Y3 ) @ ( times_times_complex @ A @ B ) )
      = ( plus_plus_complex @ ( times_times_complex @ X4 @ ( minus_minus_complex @ Y3 @ B ) ) @ ( times_times_complex @ ( minus_minus_complex @ X4 @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_1089_group__cancel_Osub2,axiom,
    ! [B5: real,K: real,B: real,A: real] :
      ( ( B5
        = ( plus_plus_real @ K @ B ) )
     => ( ( minus_minus_real @ A @ B5 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_1090_group__cancel_Osub2,axiom,
    ! [B5: produc2422161461964618553l_real,K: produc2422161461964618553l_real,B: produc2422161461964618553l_real,A: produc2422161461964618553l_real] :
      ( ( B5
        = ( plus_p1196244663705802608l_real @ K @ B ) )
     => ( ( minus_885040589139849760l_real @ A @ B5 )
        = ( plus_p1196244663705802608l_real @ ( uminus2141826702334040752l_real @ K ) @ ( minus_885040589139849760l_real @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_1091_diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A5: real,B4: real] : ( plus_plus_real @ A5 @ ( uminus_uminus_real @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_1092_diff__conv__add__uminus,axiom,
    ( minus_885040589139849760l_real
    = ( ^ [A5: produc2422161461964618553l_real,B4: produc2422161461964618553l_real] : ( plus_p1196244663705802608l_real @ A5 @ ( uminus2141826702334040752l_real @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_1093_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A5: real,B4: real] : ( plus_plus_real @ A5 @ ( uminus_uminus_real @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1094_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_885040589139849760l_real
    = ( ^ [A5: produc2422161461964618553l_real,B4: produc2422161461964618553l_real] : ( plus_p1196244663705802608l_real @ A5 @ ( uminus2141826702334040752l_real @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1095_ennreal__leI,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ X4 @ Y3 )
     => ( ord_le3935885782089961368nnreal @ ( extend7643940197134561352nnreal @ X4 ) @ ( extend7643940197134561352nnreal @ Y3 ) ) ) ).

% ennreal_leI
thf(fact_1096_minus__real__def,axiom,
    ( minus_minus_real
    = ( ^ [X: real,Y7: real] : ( plus_plus_real @ X @ ( uminus_uminus_real @ Y7 ) ) ) ) ).

% minus_real_def
thf(fact_1097_measurable__pair__swap_H,axiom,
    ! [M1: sigma_8775847253591143008e_real,M22: sigma_4063782130865963553orel_a] :
      ( member3320098977044907578e_real
      @ ( produc7940720747052681977e_real
        @ ^ [X: produc725540845905733987e_real,Y7: quasi_borel_a] : ( produc4145838808978236886e_real @ Y7 @ X ) )
      @ ( sigma_2231423344388583182e_real @ ( binary2346186313421752393orel_a @ M1 @ M22 ) @ ( binary125940435690417031e_real @ M22 @ M1 ) ) ) ).

% measurable_pair_swap'
thf(fact_1098_measurable__pair__swap_H,axiom,
    ! [M1: sigma_8927737637348964610e_real,M22: sigma_measure_real_a] :
      ( member2218098744970106036e_real
      @ ( produc5059234195088981235e_real
        @ ^ [X: sigma_measure_real,Y7: real > a] : ( produc623176010801490259e_real @ Y7 @ X ) )
      @ ( sigma_9176702948427284232e_real @ ( binary9009915518707683724real_a @ M1 @ M22 ) @ ( binary2119006201073916036e_real @ M22 @ M1 ) ) ) ).

% measurable_pair_swap'
thf(fact_1099_measurable__pair__swap_H,axiom,
    ! [M1: sigma_measure_real,M22: sigma_measure_real] :
      ( member1176752664076369724l_real
      @ ( produc3818700948422621747l_real
        @ ^ [X: real,Y7: real] : ( produc4511245868158468465l_real @ Y7 @ X ) )
      @ ( sigma_484132685431196816l_real @ ( binary6478037234023840930l_real @ M1 @ M22 ) @ ( binary6478037234023840930l_real @ M22 @ M1 ) ) ) ).

% measurable_pair_swap'
thf(fact_1100_qbs__prob__eq__def,axiom,
    ( probab176830992722561178q_real
    = ( ^ [P12: produc8908379489774204224e_real,P22: produc8908379489774204224e_real] :
          ( produc5925373591672580427real_o
          @ ^ [Qbs1: quasi_borel_real] :
              ( produc7147675743916775414real_o
              @ ^ [A1: real > real,M12: sigma_measure_real] :
                  ( produc5925373591672580427real_o
                  @ ^ [Qbs2: quasi_borel_real] :
                      ( produc7147675743916775414real_o
                      @ ^ [A22: real > real,M23: sigma_measure_real] :
                          ( ( probab3605210969150000782b_real @ Qbs1 @ A1 @ M12 )
                          & ( probab3605210969150000782b_real @ Qbs2 @ A22 @ M23 )
                          & ( Qbs1 = Qbs2 )
                          & ( ( measur2993149975067245138l_real @ M12 @ ( measur1733462625046462224e_real @ Qbs1 ) @ A1 )
                            = ( measur2993149975067245138l_real @ M23 @ ( measur1733462625046462224e_real @ Qbs2 ) @ A22 ) ) ) )
                  @ P22 ) )
          @ P12 ) ) ) ).

% qbs_prob_eq_def
thf(fact_1101_qbs__prob__eq__def,axiom,
    ( probab7355678800483015056b_eq_a
    = ( ^ [P12: produc6543235832880896358e_real,P22: produc6543235832880896358e_real] :
          ( produc7993906655439511789real_o
          @ ^ [Qbs1: quasi_borel_a] :
              ( produc5798473187818486320real_o
              @ ^ [A1: real > a,M12: sigma_measure_real] :
                  ( produc7993906655439511789real_o
                  @ ^ [Qbs2: quasi_borel_a] :
                      ( produc5798473187818486320real_o
                      @ ^ [A22: real > a,M23: sigma_measure_real] :
                          ( ( probab701741629625904796prob_a @ Qbs1 @ A1 @ M12 )
                          & ( probab701741629625904796prob_a @ Qbs2 @ A22 @ M23 )
                          & ( Qbs1 = Qbs2 )
                          & ( ( measure_distr_real_a @ M12 @ ( measur7857763439677503898sure_a @ Qbs1 ) @ A1 )
                            = ( measure_distr_real_a @ M23 @ ( measur7857763439677503898sure_a @ Qbs2 ) @ A22 ) ) ) )
                  @ P22 ) )
          @ P12 ) ) ) ).

% qbs_prob_eq_def
thf(fact_1102_diff__diff__ennreal,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A @ B )
     => ( ( B != extend2057119558705770725nnreal )
       => ( ( minus_8429688780609304081nnreal @ B @ ( minus_8429688780609304081nnreal @ B @ A ) )
          = A ) ) ) ).

% diff_diff_ennreal
thf(fact_1103_borel__measurable__add,axiom,
    ! [F: a > extend8495563244428889912nnreal,M: sigma_measure_a,G: a > extend8495563244428889912nnreal] :
      ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( ( member298456594901751504nnreal @ G @ ( sigma_214952329563889126nnreal @ M @ borel_6524799422816628122nnreal ) )
       => ( member298456594901751504nnreal
          @ ^ [X: a] : ( plus_p1859984266308609217nnreal @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_214952329563889126nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ) ).

% borel_measurable_add
thf(fact_1104_borel__measurable__add,axiom,
    ! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,G: real > extend8495563244428889912nnreal] :
      ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) )
       => ( member2919562650594848410nnreal
          @ ^ [X: real] : ( plus_p1859984266308609217nnreal @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ) ).

% borel_measurable_add
thf(fact_1105_borel__measurable__add,axiom,
    ! [F: a > real,M: sigma_measure_a,G: a > real] :
      ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_a_real
          @ ^ [X: a] : ( plus_plus_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_add
thf(fact_1106_borel__measurable__add,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_real_real
          @ ^ [X: real] : ( plus_plus_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_add
thf(fact_1107_borel__measurable__add,axiom,
    ! [F: real > complex,M: sigma_measure_real,G: real > complex] :
      ( ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ M @ borel_1392132677378845456omplex ) )
     => ( ( member_real_complex @ G @ ( sigma_9111916201866572460omplex @ M @ borel_1392132677378845456omplex ) )
       => ( member_real_complex
          @ ^ [X: real] : ( plus_plus_complex @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_9111916201866572460omplex @ M @ borel_1392132677378845456omplex ) ) ) ) ).

% borel_measurable_add
thf(fact_1108_borel__measurable__const__add,axiom,
    ! [F: a > real,M: sigma_measure_a,A: real] :
      ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
     => ( member_a_real
        @ ^ [X: a] : ( plus_plus_real @ A @ ( F @ X ) )
        @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_const_add
thf(fact_1109_borel__measurable__const__add,axiom,
    ! [F: real > real,M: sigma_measure_real,A: real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( member_real_real
        @ ^ [X: real] : ( plus_plus_real @ A @ ( F @ X ) )
        @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_const_add
thf(fact_1110_borel__measurable__const__add,axiom,
    ! [F: real > complex,M: sigma_measure_real,A: complex] :
      ( ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ M @ borel_1392132677378845456omplex ) )
     => ( member_real_complex
        @ ^ [X: real] : ( plus_plus_complex @ A @ ( F @ X ) )
        @ ( sigma_9111916201866572460omplex @ M @ borel_1392132677378845456omplex ) ) ) ).

% borel_measurable_const_add
thf(fact_1111_qbs__prob__integral__add,axiom,
    ! [S: probab4737552673497767871pace_a,F: a > real,G: a > real] :
      ( ( probab7312716125271441302able_a @ S @ F )
     => ( ( probab7312716125271441302able_a @ S @ G )
       => ( ( probab2419480525258570000gral_a @ S
            @ ^ [X: a] : ( plus_plus_real @ ( F @ X ) @ ( G @ X ) ) )
          = ( plus_plus_real @ ( probab2419480525258570000gral_a @ S @ F ) @ ( probab2419480525258570000gral_a @ S @ G ) ) ) ) ) ).

% qbs_prob_integral_add
thf(fact_1112_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: complex,E: complex,C: complex,B: complex,D: complex] :
      ( ( ord_less_eq_complex @ ( plus_plus_complex @ ( times_times_complex @ A @ E ) @ C ) @ ( plus_plus_complex @ ( times_times_complex @ B @ E ) @ D ) )
      = ( ord_less_eq_complex @ ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ E ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_1113_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_1114_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: complex,E: complex,C: complex,B: complex,D: complex] :
      ( ( ord_less_eq_complex @ ( plus_plus_complex @ ( times_times_complex @ A @ E ) @ C ) @ ( plus_plus_complex @ ( times_times_complex @ B @ E ) @ D ) )
      = ( ord_less_eq_complex @ C @ ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B @ A ) @ E ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_1115_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_1116_real__0__le__add__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X4 @ Y3 ) )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ X4 ) @ Y3 ) ) ).

% real_0_le_add_iff
thf(fact_1117_real__add__le__0__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ X4 @ Y3 ) @ zero_zero_real )
      = ( ord_less_eq_real @ Y3 @ ( uminus_uminus_real @ X4 ) ) ) ).

% real_add_le_0_iff
thf(fact_1118_ennreal__le__iff2,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_le3935885782089961368nnreal @ ( extend7643940197134561352nnreal @ X4 ) @ ( extend7643940197134561352nnreal @ Y3 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
          & ( ord_less_eq_real @ X4 @ Y3 ) )
        | ( ( ord_less_eq_real @ X4 @ zero_zero_real )
          & ( ord_less_eq_real @ Y3 @ zero_zero_real ) ) ) ) ).

% ennreal_le_iff2
thf(fact_1119_le__ennreal__iff,axiom,
    ! [R4: real,X4: extend8495563244428889912nnreal] :
      ( ( ord_less_eq_real @ zero_zero_real @ R4 )
     => ( ( ord_le3935885782089961368nnreal @ X4 @ ( extend7643940197134561352nnreal @ R4 ) )
        = ( ? [Q3: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ Q3 )
              & ( X4
                = ( extend7643940197134561352nnreal @ Q3 ) )
              & ( ord_less_eq_real @ Q3 @ R4 ) ) ) ) ) ).

% le_ennreal_iff
thf(fact_1120_prob__space_Ochar__distr__add,axiom,
    ! [M: sigma_measure_real,X12: real > real,X23: real > real,T2: real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe3760321310464026790l_real @ M @ borel_5078946678739801102l_real @ X12 @ borel_5078946678739801102l_real @ X23 )
       => ( ( characteristic_char
            @ ( measur2993149975067245138l_real @ M @ borel_5078946678739801102l_real
              @ ^ [Omega: real] : ( plus_plus_real @ ( X12 @ Omega ) @ ( X23 @ Omega ) ) )
            @ T2 )
          = ( times_times_complex @ ( characteristic_char @ ( measur2993149975067245138l_real @ M @ borel_5078946678739801102l_real @ X12 ) @ T2 ) @ ( characteristic_char @ ( measur2993149975067245138l_real @ M @ borel_5078946678739801102l_real @ X23 ) @ T2 ) ) ) ) ) ).

% prob_space.char_distr_add
thf(fact_1121_enn2real__leI,axiom,
    ! [B5: real,X4: extend8495563244428889912nnreal] :
      ( ( ord_less_eq_real @ zero_zero_real @ B5 )
     => ( ( ord_le3935885782089961368nnreal @ X4 @ ( extend7643940197134561352nnreal @ B5 ) )
       => ( ord_less_eq_real @ ( extend1669699412028896998n2real @ X4 ) @ B5 ) ) ) ).

% enn2real_leI
thf(fact_1122_qbs__Mx__subset__of__measurable,axiom,
    ! [X2: quasi_borel_real] : ( ord_le4198349162570665613l_real @ ( qbs_Mx_real @ X2 ) @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ ( measur1733462625046462224e_real @ X2 ) ) ) ).

% qbs_Mx_subset_of_measurable
thf(fact_1123_qbs__Mx__subset__of__measurable,axiom,
    ! [X2: quasi_9015997321629101608nnreal] : ( ord_le2462468573666744473nnreal @ ( qbs_Mx6523938229262583809nnreal @ X2 ) @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ ( measur7384687747506661788nnreal @ X2 ) ) ) ).

% qbs_Mx_subset_of_measurable
thf(fact_1124_qbs__Mx__subset__of__measurable,axiom,
    ! [X2: quasi_borel_complex] : ( ord_le2047140485929309711omplex @ ( qbs_Mx_complex @ X2 ) @ ( sigma_9111916201866572460omplex @ borel_5078946678739801102l_real @ ( measur3826415497239753490omplex @ X2 ) ) ) ).

% qbs_Mx_subset_of_measurable
thf(fact_1125_qbs__Mx__subset__of__measurable,axiom,
    ! [X2: quasi_borel_a] : ( ord_le5743406823621094409real_a @ ( qbs_Mx_a @ X2 ) @ ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ ( measur7857763439677503898sure_a @ X2 ) ) ) ).

% qbs_Mx_subset_of_measurable
thf(fact_1126_qbs__prob__measure__prob__space_Ochar__distr__add,axiom,
    ! [S: probab8009751763329705409e_real,X12: real > real,X23: real > real,T2: real] :
      ( ( indepe3760321310464026790l_real @ ( probab4733579253584633066e_real @ S ) @ borel_5078946678739801102l_real @ X12 @ borel_5078946678739801102l_real @ X23 )
     => ( ( characteristic_char
          @ ( measur2993149975067245138l_real @ ( probab4733579253584633066e_real @ S ) @ borel_5078946678739801102l_real
            @ ^ [Omega: real] : ( plus_plus_real @ ( X12 @ Omega ) @ ( X23 @ Omega ) ) )
          @ T2 )
        = ( times_times_complex @ ( characteristic_char @ ( measur2993149975067245138l_real @ ( probab4733579253584633066e_real @ S ) @ borel_5078946678739801102l_real @ X12 ) @ T2 ) @ ( characteristic_char @ ( measur2993149975067245138l_real @ ( probab4733579253584633066e_real @ S ) @ borel_5078946678739801102l_real @ X23 ) @ T2 ) ) ) ) ).

% qbs_prob_measure_prob_space.char_distr_add
thf(fact_1127_qbs__prob__measure__prob__space_Ochar__distr__add,axiom,
    ! [S: probab4737552673497767871pace_a,X12: a > real,X23: a > real,T2: real] :
      ( ( indepe8958435565499147358a_real @ ( probab7100426894406488384sure_a @ S ) @ borel_5078946678739801102l_real @ X12 @ borel_5078946678739801102l_real @ X23 )
     => ( ( characteristic_char
          @ ( measure_distr_a_real @ ( probab7100426894406488384sure_a @ S ) @ borel_5078946678739801102l_real
            @ ^ [Omega: a] : ( plus_plus_real @ ( X12 @ Omega ) @ ( X23 @ Omega ) ) )
          @ T2 )
        = ( times_times_complex @ ( characteristic_char @ ( measure_distr_a_real @ ( probab7100426894406488384sure_a @ S ) @ borel_5078946678739801102l_real @ X12 ) @ T2 ) @ ( characteristic_char @ ( measure_distr_a_real @ ( probab7100426894406488384sure_a @ S ) @ borel_5078946678739801102l_real @ X23 ) @ T2 ) ) ) ) ).

% qbs_prob_measure_prob_space.char_distr_add
thf(fact_1128_prob__space_Omutual__information__def,axiom,
    ! [M: sigma_measure_real,B: real,S2: sigma_4063782130865963553orel_a,T: sigma_8775847253591143008e_real,X2: real > quasi_borel_a,Y: real > produc725540845905733987e_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( prob_m4228609518817447427e_real @ M @ B @ S2 @ T @ X2 @ Y )
        = ( kL_div3327646984204289008e_real @ B @ ( binary125940435690417031e_real @ ( measur7149860273772831102orel_a @ M @ S2 @ X2 ) @ ( measur8637847926015211837e_real @ M @ T @ Y ) )
          @ ( measur2398198314208846400e_real @ M @ ( binary125940435690417031e_real @ S2 @ T )
            @ ^ [X: real] : ( produc4145838808978236886e_real @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ) ).

% prob_space.mutual_information_def
thf(fact_1129_prob__space_Omutual__information__def,axiom,
    ! [M: sigma_measure_real,B: real,S2: sigma_measure_real_a,T: sigma_8927737637348964610e_real,X2: real > real > a,Y: real > sigma_measure_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( prob_m9196104408708822272e_real @ M @ B @ S2 @ T @ X2 @ Y )
        = ( kL_div4552242606482481901e_real @ B @ ( binary2119006201073916036e_real @ ( measur7864027549924149603real_a @ M @ S2 @ X2 ) @ ( measur2366643943792126175e_real @ M @ T @ Y ) )
          @ ( measur8637847926015211837e_real @ M @ ( binary2119006201073916036e_real @ S2 @ T )
            @ ^ [X: real] : ( produc623176010801490259e_real @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ) ).

% prob_space.mutual_information_def
thf(fact_1130_prob__space_Omutual__information__def,axiom,
    ! [M: sigma_measure_real,B: real,S2: sigma_measure_a,T: sigma_measure_a,X2: real > a,Y: real > a] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( prob_m1941895425998922052al_a_a @ M @ B @ S2 @ T @ X2 @ Y )
        = ( kL_div3267156980076932017od_a_a @ B @ ( binary867438762418767560re_a_a @ ( measure_distr_real_a @ M @ S2 @ X2 ) @ ( measure_distr_real_a @ M @ T @ Y ) )
          @ ( measur2513335786126797313od_a_a @ M @ ( binary867438762418767560re_a_a @ S2 @ T )
            @ ^ [X: real] : ( product_Pair_a_a @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ) ).

% prob_space.mutual_information_def
thf(fact_1131_prob__space_Omutual__information__def,axiom,
    ! [M: sigma_measure_real,B: real,S2: sigma_measure_a,T: sigma_measure_real,X2: real > a,Y: real > real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( prob_m6024414151681283558a_real @ M @ B @ S2 @ T @ X2 @ Y )
        = ( kL_div1539255837003659855a_real @ B @ ( binary932748531126180194a_real @ ( measure_distr_real_a @ M @ S2 @ X2 ) @ ( measur2993149975067245138l_real @ M @ T @ Y ) )
          @ ( measur7871026761292836863a_real @ M @ ( binary932748531126180194a_real @ S2 @ T )
            @ ^ [X: real] : ( product_Pair_a_real @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ) ).

% prob_space.mutual_information_def
thf(fact_1132_prob__space_Omutual__information__def,axiom,
    ! [M: sigma_measure_real,B: real,S2: sigma_measure_real,T: sigma_measure_a,X2: real > real,Y: real > a] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( prob_m6654432919154233356real_a @ M @ B @ S2 @ T @ X2 @ Y )
        = ( kL_div2056522339270997053real_a @ B @ ( binary1562767298599129992real_a @ ( measur2993149975067245138l_real @ M @ S2 @ X2 ) @ ( measure_distr_real_a @ M @ T @ Y ) )
          @ ( measur8388293263560174061real_a @ M @ ( binary1562767298599129992real_a @ S2 @ T )
            @ ^ [X: real] : ( product_Pair_real_a @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ) ).

% prob_space.mutual_information_def
thf(fact_1133_prob__space_Omutual__information__def,axiom,
    ! [M: sigma_measure_real,B: real,S2: sigma_measure_real,T: sigma_measure_real,X2: real > real,Y: real > real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( prob_m4172219917653797150l_real @ M @ B @ S2 @ T @ X2 @ Y )
        = ( kL_div4114197932038040771l_real @ B @ ( binary6478037234023840930l_real @ ( measur2993149975067245138l_real @ M @ S2 @ X2 ) @ ( measur2993149975067245138l_real @ M @ T @ Y ) )
          @ ( measur6481026558495277843l_real @ M @ ( binary6478037234023840930l_real @ S2 @ T )
            @ ^ [X: real] : ( produc4511245868158468465l_real @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ) ).

% prob_space.mutual_information_def
thf(fact_1134_integral__real__bounded,axiom,
    ! [R4: real,M: sigma_measure_real,F: real > real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R4 )
     => ( ( ord_le3935885782089961368nnreal
          @ ( nonneg2667834350952324695l_real @ M
            @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( F @ X ) ) )
          @ ( extend7643940197134561352nnreal @ R4 ) )
       => ( ord_less_eq_real @ ( bochne3715101410578510557l_real @ M @ F ) @ R4 ) ) ) ).

% integral_real_bounded
thf(fact_1135_integral__real__bounded,axiom,
    ! [R4: real,M: sigma_measure_a,F: a > real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R4 )
     => ( ( ord_le3935885782089961368nnreal
          @ ( nonneg2725512125972007571gral_a @ M
            @ ^ [X: a] : ( extend7643940197134561352nnreal @ ( F @ X ) ) )
          @ ( extend7643940197134561352nnreal @ R4 ) )
       => ( ord_less_eq_real @ ( bochne378719280626478695a_real @ M @ F ) @ R4 ) ) ) ).

% integral_real_bounded
thf(fact_1136_Pair__mono,axiom,
    ! [X4: real,X9: real,Y3: real,Y9: real] :
      ( ( ord_less_eq_real @ X4 @ X9 )
     => ( ( ord_less_eq_real @ Y3 @ Y9 )
       => ( ord_le1075799226346578649l_real @ ( produc4511245868158468465l_real @ X4 @ Y3 ) @ ( produc4511245868158468465l_real @ X9 @ Y9 ) ) ) ) ).

% Pair_mono
thf(fact_1137_Pair__mono,axiom,
    ! [X4: real,X9: real,Y3: extend8495563244428889912nnreal,Y9: extend8495563244428889912nnreal] :
      ( ( ord_less_eq_real @ X4 @ X9 )
     => ( ( ord_le3935885782089961368nnreal @ Y3 @ Y9 )
       => ( ord_le4096773168995780197nnreal @ ( produc4778015194254607485nnreal @ X4 @ Y3 ) @ ( produc4778015194254607485nnreal @ X9 @ Y9 ) ) ) ) ).

% Pair_mono
thf(fact_1138_Pair__mono,axiom,
    ! [X4: extend8495563244428889912nnreal,X9: extend8495563244428889912nnreal,Y3: real,Y9: real] :
      ( ( ord_le3935885782089961368nnreal @ X4 @ X9 )
     => ( ( ord_less_eq_real @ Y3 @ Y9 )
       => ( ord_le4051224869651757541l_real @ ( produc2810268924804063229l_real @ X4 @ Y3 ) @ ( produc2810268924804063229l_real @ X9 @ Y9 ) ) ) ) ).

% Pair_mono
thf(fact_1139_Pair__mono,axiom,
    ! [X4: extend8495563244428889912nnreal,X9: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal,Y9: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ X4 @ X9 )
     => ( ( ord_le3935885782089961368nnreal @ Y3 @ Y9 )
       => ( ord_le1399272598019556209nnreal @ ( produc344325839068023049nnreal @ X4 @ Y3 ) @ ( produc344325839068023049nnreal @ X9 @ Y9 ) ) ) ) ).

% Pair_mono
thf(fact_1140_qbs__prob__measure__prob__space_Omutual__information__def,axiom,
    ! [S: probab8009751763329705409e_real,B: real,S2: sigma_measure_a,T: sigma_measure_a,X2: real > a,Y: real > a] :
      ( ( prob_m1941895425998922052al_a_a @ ( probab4733579253584633066e_real @ S ) @ B @ S2 @ T @ X2 @ Y )
      = ( kL_div3267156980076932017od_a_a @ B @ ( binary867438762418767560re_a_a @ ( measure_distr_real_a @ ( probab4733579253584633066e_real @ S ) @ S2 @ X2 ) @ ( measure_distr_real_a @ ( probab4733579253584633066e_real @ S ) @ T @ Y ) )
        @ ( measur2513335786126797313od_a_a @ ( probab4733579253584633066e_real @ S ) @ ( binary867438762418767560re_a_a @ S2 @ T )
          @ ^ [X: real] : ( product_Pair_a_a @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ).

% qbs_prob_measure_prob_space.mutual_information_def
thf(fact_1141_qbs__prob__measure__prob__space_Omutual__information__def,axiom,
    ! [S: probab8009751763329705409e_real,B: real,S2: sigma_measure_a,T: sigma_measure_real,X2: real > a,Y: real > real] :
      ( ( prob_m6024414151681283558a_real @ ( probab4733579253584633066e_real @ S ) @ B @ S2 @ T @ X2 @ Y )
      = ( kL_div1539255837003659855a_real @ B @ ( binary932748531126180194a_real @ ( measure_distr_real_a @ ( probab4733579253584633066e_real @ S ) @ S2 @ X2 ) @ ( measur2993149975067245138l_real @ ( probab4733579253584633066e_real @ S ) @ T @ Y ) )
        @ ( measur7871026761292836863a_real @ ( probab4733579253584633066e_real @ S ) @ ( binary932748531126180194a_real @ S2 @ T )
          @ ^ [X: real] : ( product_Pair_a_real @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ).

% qbs_prob_measure_prob_space.mutual_information_def
thf(fact_1142_qbs__prob__measure__prob__space_Omutual__information__def,axiom,
    ! [S: probab8009751763329705409e_real,B: real,S2: sigma_measure_real,T: sigma_measure_a,X2: real > real,Y: real > a] :
      ( ( prob_m6654432919154233356real_a @ ( probab4733579253584633066e_real @ S ) @ B @ S2 @ T @ X2 @ Y )
      = ( kL_div2056522339270997053real_a @ B @ ( binary1562767298599129992real_a @ ( measur2993149975067245138l_real @ ( probab4733579253584633066e_real @ S ) @ S2 @ X2 ) @ ( measure_distr_real_a @ ( probab4733579253584633066e_real @ S ) @ T @ Y ) )
        @ ( measur8388293263560174061real_a @ ( probab4733579253584633066e_real @ S ) @ ( binary1562767298599129992real_a @ S2 @ T )
          @ ^ [X: real] : ( product_Pair_real_a @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ).

% qbs_prob_measure_prob_space.mutual_information_def
thf(fact_1143_qbs__prob__measure__prob__space_Omutual__information__def,axiom,
    ! [S: probab8009751763329705409e_real,B: real,S2: sigma_measure_real,T: sigma_measure_real,X2: real > real,Y: real > real] :
      ( ( prob_m4172219917653797150l_real @ ( probab4733579253584633066e_real @ S ) @ B @ S2 @ T @ X2 @ Y )
      = ( kL_div4114197932038040771l_real @ B @ ( binary6478037234023840930l_real @ ( measur2993149975067245138l_real @ ( probab4733579253584633066e_real @ S ) @ S2 @ X2 ) @ ( measur2993149975067245138l_real @ ( probab4733579253584633066e_real @ S ) @ T @ Y ) )
        @ ( measur6481026558495277843l_real @ ( probab4733579253584633066e_real @ S ) @ ( binary6478037234023840930l_real @ S2 @ T )
          @ ^ [X: real] : ( produc4511245868158468465l_real @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ).

% qbs_prob_measure_prob_space.mutual_information_def
thf(fact_1144_qbs__prob__measure__prob__space_Omutual__information__def,axiom,
    ! [S: probab4737552673497767871pace_a,B: real,S2: sigma_4063782130865963553orel_a,T: sigma_8775847253591143008e_real,X2: a > quasi_borel_a,Y: a > produc725540845905733987e_real] :
      ( ( prob_m4532829576834203813e_real @ ( probab7100426894406488384sure_a @ S ) @ B @ S2 @ T @ X2 @ Y )
      = ( kL_div3327646984204289008e_real @ B @ ( binary125940435690417031e_real @ ( measur5725630100919690270orel_a @ ( probab7100426894406488384sure_a @ S ) @ S2 @ X2 ) @ ( measur5549411481742515165e_real @ ( probab7100426894406488384sure_a @ S ) @ T @ Y ) )
        @ ( measur2719181256529368288e_real @ ( probab7100426894406488384sure_a @ S ) @ ( binary125940435690417031e_real @ S2 @ T )
          @ ^ [X: a] : ( produc4145838808978236886e_real @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ).

% qbs_prob_measure_prob_space.mutual_information_def
thf(fact_1145_qbs__prob__measure__prob__space_Omutual__information__def,axiom,
    ! [S: probab4737552673497767871pace_a,B: real,S2: sigma_measure_real_a,T: sigma_8927737637348964610e_real,X2: a > real > a,Y: a > sigma_measure_real] :
      ( ( prob_m7437826066704243362e_real @ ( probab7100426894406488384sure_a @ S ) @ B @ S2 @ T @ X2 @ Y )
      = ( kL_div4552242606482481901e_real @ B @ ( binary2119006201073916036e_real @ ( measur7323644686031903747real_a @ ( probab7100426894406488384sure_a @ S ) @ S2 @ X2 ) @ ( measur6656879888321211263e_real @ ( probab7100426894406488384sure_a @ S ) @ T @ Y ) )
        @ ( measur5549411481742515165e_real @ ( probab7100426894406488384sure_a @ S ) @ ( binary2119006201073916036e_real @ S2 @ T )
          @ ^ [X: a] : ( produc623176010801490259e_real @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ).

% qbs_prob_measure_prob_space.mutual_information_def
thf(fact_1146_qbs__prob__measure__prob__space_Omutual__information__def,axiom,
    ! [S: probab4737552673497767871pace_a,B: real,S2: sigma_measure_real,T: sigma_measure_real,X2: a > real,Y: a > real] :
      ( ( prob_m7207053172173760192l_real @ ( probab7100426894406488384sure_a @ S ) @ B @ S2 @ T @ X2 @ Y )
      = ( kL_div4114197932038040771l_real @ B @ ( binary6478037234023840930l_real @ ( measure_distr_a_real @ ( probab7100426894406488384sure_a @ S ) @ S2 @ X2 ) @ ( measure_distr_a_real @ ( probab7100426894406488384sure_a @ S ) @ T @ Y ) )
        @ ( measur8266400719524636083l_real @ ( probab7100426894406488384sure_a @ S ) @ ( binary6478037234023840930l_real @ S2 @ T )
          @ ^ [X: a] : ( produc4511245868158468465l_real @ ( X2 @ X ) @ ( Y @ X ) ) ) ) ) ).

% qbs_prob_measure_prob_space.mutual_information_def
thf(fact_1147_qp_Osum__indep__random__variable,axiom,
    ! [X2: real > real,Y: real > real] :
      ( ( indepe3760321310464026790l_real @ mu @ borel_5078946678739801102l_real @ X2 @ borel_5078946678739801102l_real @ Y )
     => ( ( member_real_real @ X2 @ ( sigma_5267869275261027754l_real @ mu @ borel_5078946678739801102l_real ) )
       => ( ( member_real_real @ Y @ ( sigma_5267869275261027754l_real @ mu @ borel_5078946678739801102l_real ) )
         => ( ( measur2993149975067245138l_real @ mu @ borel_5078946678739801102l_real
              @ ^ [X: real] : ( plus_plus_real @ ( X2 @ X ) @ ( Y @ X ) ) )
            = ( convolution_real @ ( measur2993149975067245138l_real @ mu @ borel_5078946678739801102l_real @ X2 ) @ ( measur2993149975067245138l_real @ mu @ borel_5078946678739801102l_real @ Y ) ) ) ) ) ) ).

% qp.sum_indep_random_variable
thf(fact_1148_qp_Ochar__zero,axiom,
    ( ( characteristic_char @ mu @ zero_zero_real )
    = one_one_complex ) ).

% qp.char_zero
thf(fact_1149_qp_Oemeasure__real,axiom,
    ! [A2: set_real] :
    ? [R3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R3 )
      & ( ( sigma_emeasure_real @ mu @ A2 )
        = ( extend7643940197134561352nnreal @ R3 ) ) ) ).

% qp.emeasure_real
thf(fact_1150_pair__qbs__prob_Oqbs__prob__eq3__intro,axiom,
    ! [X2: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,Y: quasi_borel_real,Beta: real > real,Nu: sigma_measure_real] :
      ( ( probab6500215489368174228b_real @ X2 @ Alpha @ Mu @ Y @ Beta @ Nu )
     => ( ( X2 = Y )
       => ( ! [F3: real > real] :
              ( ( member_real_real @ F3 @ ( sigma_5267869275261027754l_real @ ( measur1733462625046462224e_real @ X2 ) @ borel_5078946678739801102l_real ) )
             => ( ! [X8: real] :
                    ( ( member_real @ X8 @ ( qbs_space_real @ X2 ) )
                   => ( ord_less_eq_real @ zero_zero_real @ ( F3 @ X8 ) ) )
               => ( ( bochne3715101410578510557l_real @ Mu
                    @ ^ [X: real] : ( F3 @ ( Alpha @ X ) ) )
                  = ( bochne3715101410578510557l_real @ Nu
                    @ ^ [X: real] : ( F3 @ ( Beta @ X ) ) ) ) ) )
         => ( probab75427942197663321473_real @ ( produc4368523205619139320e_real @ X2 @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) @ ( produc4368523205619139320e_real @ Y @ ( produc1722724976708544245e_real @ Beta @ Nu ) ) ) ) ) ) ).

% pair_qbs_prob.qbs_prob_eq3_intro
thf(fact_1151_pair__qbs__prob_Oqbs__prob__eq3__intro,axiom,
    ! [X2: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta: real > a,Nu: sigma_measure_real] :
      ( ( probab5677843605262999830prob_a @ X2 @ Alpha @ Mu @ Y @ Beta @ Nu )
     => ( ( X2 = Y )
       => ( ! [F3: a > real] :
              ( ( member_a_real @ F3 @ ( sigma_9116425665531756122a_real @ ( measur7857763439677503898sure_a @ X2 ) @ borel_5078946678739801102l_real ) )
             => ( ! [X8: a] :
                    ( ( member_a @ X8 @ ( qbs_space_a @ X2 ) )
                   => ( ord_less_eq_real @ zero_zero_real @ ( F3 @ X8 ) ) )
               => ( ( bochne3715101410578510557l_real @ Mu
                    @ ^ [X: real] : ( F3 @ ( Alpha @ X ) ) )
                  = ( bochne3715101410578510557l_real @ Nu
                    @ ^ [X: real] : ( F3 @ ( Beta @ X ) ) ) ) ) )
         => ( probab1131137119144644343_eq3_a @ ( produc4145838808978236886e_real @ X2 @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta @ Nu ) ) ) ) ) ) ).

% pair_qbs_prob.qbs_prob_eq3_intro
thf(fact_1152_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_1153_mult__1,axiom,
    ! [A: extend8495563244428889912nnreal] :
      ( ( times_1893300245718287421nnreal @ one_on2969667320475766781nnreal @ A )
      = A ) ).

% mult_1
thf(fact_1154_mult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% mult_1
thf(fact_1155_mult__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ one_one_complex @ A )
      = A ) ).

% mult_1
thf(fact_1156_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_1157_mult_Oright__neutral,axiom,
    ! [A: extend8495563244428889912nnreal] :
      ( ( times_1893300245718287421nnreal @ A @ one_on2969667320475766781nnreal )
      = A ) ).

% mult.right_neutral
thf(fact_1158_mult_Oright__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.right_neutral
thf(fact_1159_mult_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.right_neutral
thf(fact_1160_ennreal__plus,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ( extend7643940197134561352nnreal @ ( plus_plus_real @ A @ B ) )
          = ( plus_p1859984266308609217nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B ) ) ) ) ) ).

% ennreal_plus
thf(fact_1161_pred__subset__eq,axiom,
    ! [R: set_real_complex,S2: set_real_complex] :
      ( ( ord_le8896463590272329014plex_o
        @ ^ [X: real > complex] : ( member_real_complex @ X @ R )
        @ ^ [X: real > complex] : ( member_real_complex @ X @ S2 ) )
      = ( ord_le2047140485929309711omplex @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_1162_pred__subset__eq,axiom,
    ! [R: set_real_real,S2: set_real_real] :
      ( ( ord_le5273791883478943800real_o
        @ ^ [X: real > real] : ( member_real_real @ X @ R )
        @ ^ [X: real > real] : ( member_real_real @ X @ S2 ) )
      = ( ord_le4198349162570665613l_real @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_1163_pred__subset__eq,axiom,
    ! [R: set_real_a,S2: set_real_a] :
      ( ( ord_less_eq_real_a_o
        @ ^ [X: real > a] : ( member_real_a @ X @ R )
        @ ^ [X: real > a] : ( member_real_a @ X @ S2 ) )
      = ( ord_le5743406823621094409real_a @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_1164_pred__subset__eq,axiom,
    ! [R: set_a_7161065143582548615nnreal,S2: set_a_7161065143582548615nnreal] :
      ( ( ord_le1809911388877515638real_o
        @ ^ [X: a > extend8495563244428889912nnreal] : ( member298456594901751504nnreal @ X @ R )
        @ ^ [X: a > extend8495563244428889912nnreal] : ( member298456594901751504nnreal @ X @ S2 ) )
      = ( ord_le1007445205377960487nnreal @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_1165_pred__subset__eq,axiom,
    ! [R: set_a_real,S2: set_a_real] :
      ( ( ord_less_eq_a_real_o
        @ ^ [X: a > real] : ( member_a_real @ X @ R )
        @ ^ [X: a > real] : ( member_a_real @ X @ S2 ) )
      = ( ord_le3334967407727675675a_real @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_1166_one__reorient,axiom,
    ! [X4: complex] :
      ( ( one_one_complex = X4 )
      = ( X4 = one_one_complex ) ) ).

% one_reorient
thf(fact_1167_one__reorient,axiom,
    ! [X4: extend8495563244428889912nnreal] :
      ( ( one_on2969667320475766781nnreal = X4 )
      = ( X4 = one_on2969667320475766781nnreal ) ) ).

% one_reorient
thf(fact_1168_one__reorient,axiom,
    ! [X4: real] :
      ( ( one_one_real = X4 )
      = ( X4 = one_one_real ) ) ).

% one_reorient
thf(fact_1169_one__reorient,axiom,
    ! [X4: nat] :
      ( ( one_one_nat = X4 )
      = ( X4 = one_one_nat ) ) ).

% one_reorient
thf(fact_1170_qbs__space__eq__Mx,axiom,
    ! [X2: quasi_borel_a,Y: quasi_borel_a] :
      ( ( ( qbs_Mx_a @ X2 )
        = ( qbs_Mx_a @ Y ) )
     => ( ( qbs_space_a @ X2 )
        = ( qbs_space_a @ Y ) ) ) ).

% qbs_space_eq_Mx
thf(fact_1171_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > real > complex,X2: quasi_4275199384652633321omplex,R4: real] :
      ( ( member7402136750473155931omplex @ Alpha @ ( qbs_Mx_real_complex @ X2 ) )
     => ( member_real_complex @ ( Alpha @ R4 ) @ ( qbs_sp1878115117008156099omplex @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_1172_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > real > real,X2: quasi_1840791737016710247l_real,R4: real] :
      ( ( member8878224140454985689l_real @ Alpha @ ( qbs_Mx_real_real @ X2 ) )
     => ( member_real_real @ ( Alpha @ R4 ) @ ( qbs_space_real_real @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_1173_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > real > a,X2: quasi_borel_real_a,R4: real] :
      ( ( member_real_real_a @ Alpha @ ( qbs_Mx_real_a @ X2 ) )
     => ( member_real_a @ ( Alpha @ R4 ) @ ( qbs_space_real_a @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_1174_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > a > extend8495563244428889912nnreal,X2: quasi_6419473174764657869nnreal,R4: real] :
      ( ( member806324418734843867nnreal @ Alpha @ ( qbs_Mx4536111526588809758nnreal @ X2 ) )
     => ( member298456594901751504nnreal @ ( Alpha @ R4 ) @ ( qbs_sp2608499494640836445nnreal @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_1175_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > a > real,X2: quasi_borel_a_real,R4: real] :
      ( ( member_real_a_real @ Alpha @ ( qbs_Mx_a_real @ X2 ) )
     => ( member_a_real @ ( Alpha @ R4 ) @ ( qbs_space_a_real @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_1176_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > complex,X2: quasi_borel_complex,R4: real] :
      ( ( member_real_complex @ Alpha @ ( qbs_Mx_complex @ X2 ) )
     => ( member_complex @ ( Alpha @ R4 ) @ ( qbs_space_complex @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_1177_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > real,X2: quasi_borel_real,R4: real] :
      ( ( member_real_real @ Alpha @ ( qbs_Mx_real @ X2 ) )
     => ( member_real @ ( Alpha @ R4 ) @ ( qbs_space_real @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_1178_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > a,X2: quasi_borel_a,R4: real] :
      ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X2 ) )
     => ( member_a @ ( Alpha @ R4 ) @ ( qbs_space_a @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_1179_less__eq__quasi__borel_Ointros_I2_J,axiom,
    ! [X2: quasi_borel_a,Y: quasi_borel_a] :
      ( ( ( qbs_space_a @ X2 )
        = ( qbs_space_a @ Y ) )
     => ( ( ord_le5743406823621094409real_a @ ( qbs_Mx_a @ Y ) @ ( qbs_Mx_a @ X2 ) )
       => ( ord_le1843388692487544644orel_a @ X2 @ Y ) ) ) ).

% less_eq_quasi_borel.intros(2)
thf(fact_1180_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_1181_mult_Ocomm__neutral,axiom,
    ! [A: extend8495563244428889912nnreal] :
      ( ( times_1893300245718287421nnreal @ A @ one_on2969667320475766781nnreal )
      = A ) ).

% mult.comm_neutral
thf(fact_1182_mult_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.comm_neutral
thf(fact_1183_mult_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.comm_neutral
thf(fact_1184_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1185_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: extend8495563244428889912nnreal] :
      ( ( times_1893300245718287421nnreal @ one_on2969667320475766781nnreal @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1186_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1187_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ one_one_complex @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1188_ennreal__add__left__cancel,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ( plus_p1859984266308609217nnreal @ A @ B )
        = ( plus_p1859984266308609217nnreal @ A @ C ) )
      = ( ( A = extend2057119558705770725nnreal )
        | ( B = C ) ) ) ).

% ennreal_add_left_cancel
thf(fact_1189_ennreal__add__left__cancel__le,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ ( plus_p1859984266308609217nnreal @ A @ B ) @ ( plus_p1859984266308609217nnreal @ A @ C ) )
      = ( ( A = extend2057119558705770725nnreal )
        | ( ord_le3935885782089961368nnreal @ B @ C ) ) ) ).

% ennreal_add_left_cancel_le
thf(fact_1190_qbs__morphism__const,axiom,
    ! [Y3: complex,Y: quasi_borel_complex,X2: quasi_borel_real] :
      ( ( member_complex @ Y3 @ ( qbs_space_complex @ Y ) )
     => ( member_real_complex
        @ ^ [Uu: real] : Y3
        @ ( qbs_mo6067097710682130004omplex @ X2 @ Y ) ) ) ).

% qbs_morphism_const
thf(fact_1191_qbs__morphism__const,axiom,
    ! [Y3: real,Y: quasi_borel_real,X2: quasi_borel_real] :
      ( ( member_real @ Y3 @ ( qbs_space_real @ Y ) )
     => ( member_real_real
        @ ^ [Uu: real] : Y3
        @ ( qbs_mo5229770564518008146l_real @ X2 @ Y ) ) ) ).

% qbs_morphism_const
thf(fact_1192_qbs__morphism__const,axiom,
    ! [Y3: a,Y: quasi_borel_a,X2: quasi_borel_real] :
      ( ( member_a @ Y3 @ ( qbs_space_a @ Y ) )
     => ( member_real_a
        @ ^ [Uu: real] : Y3
        @ ( qbs_morphism_real_a @ X2 @ Y ) ) ) ).

% qbs_morphism_const
thf(fact_1193_qbs__morphism__const,axiom,
    ! [Y3: extend8495563244428889912nnreal,Y: quasi_9015997321629101608nnreal,X2: quasi_borel_a] :
      ( ( member7908768830364227535nnreal @ Y3 @ ( qbs_sp175953267596557954nnreal @ Y ) )
     => ( member298456594901751504nnreal
        @ ^ [Uu: a] : Y3
        @ ( qbs_mo1434458643421888574nnreal @ X2 @ Y ) ) ) ).

% qbs_morphism_const
thf(fact_1194_qbs__morphism__const,axiom,
    ! [Y3: real,Y: quasi_borel_real,X2: quasi_borel_a] :
      ( ( member_real @ Y3 @ ( qbs_space_real @ Y ) )
     => ( member_a_real
        @ ^ [Uu: a] : Y3
        @ ( qbs_morphism_a_real @ X2 @ Y ) ) ) ).

% qbs_morphism_const
thf(fact_1195_qbs__closed2__dest,axiom,
    ! [X4: a,X2: quasi_borel_a] :
      ( ( member_a @ X4 @ ( qbs_space_a @ X2 ) )
     => ( member_real_a
        @ ^ [R2: real] : X4
        @ ( qbs_Mx_a @ X2 ) ) ) ).

% qbs_closed2_dest
thf(fact_1196_ennreal__add__diff__cancel,axiom,
    ! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
      ( ( B != extend2057119558705770725nnreal )
     => ( ( minus_8429688780609304081nnreal @ ( plus_p1859984266308609217nnreal @ A @ B ) @ B )
        = A ) ) ).

% ennreal_add_diff_cancel
thf(fact_1197_real__distribution_Ochar__zero,axiom,
    ! [M: sigma_measure_real] :
      ( ( distri2809703520229113005bution @ M )
     => ( ( characteristic_char @ M @ zero_zero_real )
        = one_one_complex ) ) ).

% real_distribution.char_zero
thf(fact_1198_ennreal__plus__if,axiom,
    ! [A: real,B: real] :
      ( ( plus_p1859984266308609217nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B ) )
      = ( extend7643940197134561352nnreal @ ( if_real @ ( ord_less_eq_real @ zero_zero_real @ A ) @ ( if_real @ ( ord_less_eq_real @ zero_zero_real @ B ) @ ( plus_plus_real @ A @ B ) @ A ) @ B ) ) ) ).

% ennreal_plus_if
thf(fact_1199_qp_Oemeasure__ge__1__iff,axiom,
    ! [A2: set_real] :
      ( ( ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ ( sigma_emeasure_real @ mu @ A2 ) )
      = ( ( sigma_emeasure_real @ mu @ A2 )
        = one_on2969667320475766781nnreal ) ) ).

% qp.emeasure_ge_1_iff
thf(fact_1200_qp_Oemeasure__le__1,axiom,
    ! [S2: set_real] : ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_real @ mu @ S2 ) @ one_on2969667320475766781nnreal ) ).

% qp.emeasure_le_1
thf(fact_1201_qp_Osubprob__emeasure__le__1,axiom,
    ! [X2: set_real] : ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_real @ mu @ X2 ) @ one_on2969667320475766781nnreal ) ).

% qp.subprob_emeasure_le_1
thf(fact_1202_ennreal__eq__1,axiom,
    ! [X4: real] :
      ( ( ( extend7643940197134561352nnreal @ X4 )
        = one_on2969667320475766781nnreal )
      = ( X4 = one_one_real ) ) ).

% ennreal_eq_1
thf(fact_1203_ennreal__1,axiom,
    ( ( extend7643940197134561352nnreal @ one_one_real )
    = one_on2969667320475766781nnreal ) ).

% ennreal_1
thf(fact_1204_enn2real__1,axiom,
    ( ( extend1669699412028896998n2real @ one_on2969667320475766781nnreal )
    = one_one_real ) ).

% enn2real_1
thf(fact_1205_ennreal__le__1,axiom,
    ! [X4: real] :
      ( ( ord_le3935885782089961368nnreal @ ( extend7643940197134561352nnreal @ X4 ) @ one_on2969667320475766781nnreal )
      = ( ord_less_eq_real @ X4 @ one_one_real ) ) ).

% ennreal_le_1
thf(fact_1206_ennreal__ge__1,axiom,
    ! [X4: real] :
      ( ( ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ ( extend7643940197134561352nnreal @ X4 ) )
      = ( ord_less_eq_real @ one_one_real @ X4 ) ) ).

% ennreal_ge_1
thf(fact_1207_qp_Oobtain__positive__integrable__function,axiom,
    ~ ! [F3: real > real] :
        ( ( member_real_real @ F3 @ ( sigma_5267869275261027754l_real @ mu @ borel_5078946678739801102l_real ) )
       => ( ! [X8: real] : ( ord_less_real @ zero_zero_real @ ( F3 @ X8 ) )
         => ( ! [X8: real] : ( ord_less_eq_real @ ( F3 @ X8 ) @ one_one_real )
           => ~ ( bochne3340023020068487468l_real @ mu @ F3 ) ) ) ) ).

% qp.obtain_positive_integrable_function
thf(fact_1208_enn2real__eq__posreal__iff,axiom,
    ! [C: real,X4: extend8495563244428889912nnreal] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ( extend1669699412028896998n2real @ X4 )
          = C )
        = ( X4
          = ( extend7643940197134561352nnreal @ C ) ) ) ) ).

% enn2real_eq_posreal_iff
thf(fact_1209_qp_Ointegrable__cts__step,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( bochne3340023020068487468l_real @ mu @ ( weak_cts_step @ A @ B ) ) ) ).

% qp.integrable_cts_step
thf(fact_1210_less__eq__real__def,axiom,
    ( ord_less_eq_real
    = ( ^ [X: real,Y7: real] :
          ( ( ord_less_real @ X @ Y7 )
          | ( X = Y7 ) ) ) ) ).

% less_eq_real_def
thf(fact_1211_real__add__less__0__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X4 @ Y3 ) @ zero_zero_real )
      = ( ord_less_real @ Y3 @ ( uminus_uminus_real @ X4 ) ) ) ).

% real_add_less_0_iff
thf(fact_1212_real__0__less__add__iff,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X4 @ Y3 ) )
      = ( ord_less_real @ ( uminus_uminus_real @ X4 ) @ Y3 ) ) ).

% real_0_less_add_iff
thf(fact_1213_segment__bound__lemma,axiom,
    ! [B5: real,X4: real,Y3: real,U: real] :
      ( ( ord_less_eq_real @ B5 @ X4 )
     => ( ( ord_less_eq_real @ B5 @ Y3 )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ U @ one_one_real )
           => ( ord_less_eq_real @ B5 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ one_one_real @ U ) @ X4 ) @ ( times_times_real @ U @ Y3 ) ) ) ) ) ) ) ).

% segment_bound_lemma
thf(fact_1214_ennreal__less__zero__iff,axiom,
    ! [X4: real] :
      ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( extend7643940197134561352nnreal @ X4 ) )
      = ( ord_less_real @ zero_zero_real @ X4 ) ) ).

% ennreal_less_zero_iff
thf(fact_1215_one__less__ennreal,axiom,
    ! [X4: real] :
      ( ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ ( extend7643940197134561352nnreal @ X4 ) )
      = ( ord_less_real @ one_one_real @ X4 ) ) ).

% one_less_ennreal
thf(fact_1216_ennreal__less__one__iff,axiom,
    ! [X4: real] :
      ( ( ord_le7381754540660121996nnreal @ ( extend7643940197134561352nnreal @ X4 ) @ one_on2969667320475766781nnreal )
      = ( ord_less_real @ X4 @ one_one_real ) ) ).

% ennreal_less_one_iff
thf(fact_1217_ennreal__add__left__cancel__less,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le7381754540660121996nnreal @ ( plus_p1859984266308609217nnreal @ A @ B ) @ ( plus_p1859984266308609217nnreal @ A @ C ) )
      = ( ( A != extend2057119558705770725nnreal )
        & ( ord_le7381754540660121996nnreal @ B @ C ) ) ) ).

% ennreal_add_left_cancel_less
thf(fact_1218_ennreal__lessI,axiom,
    ! [Q2: real,R4: real] :
      ( ( ord_less_real @ zero_zero_real @ Q2 )
     => ( ( ord_less_real @ R4 @ Q2 )
       => ( ord_le7381754540660121996nnreal @ ( extend7643940197134561352nnreal @ R4 ) @ ( extend7643940197134561352nnreal @ Q2 ) ) ) ) ).

% ennreal_lessI
thf(fact_1219_add__mono__ennreal,axiom,
    ! [X4: extend8495563244428889912nnreal,Y3: real,X9: extend8495563244428889912nnreal,Y9: real] :
      ( ( ord_le7381754540660121996nnreal @ X4 @ ( extend7643940197134561352nnreal @ Y3 ) )
     => ( ( ord_le7381754540660121996nnreal @ X9 @ ( extend7643940197134561352nnreal @ Y9 ) )
       => ( ord_le7381754540660121996nnreal @ ( plus_p1859984266308609217nnreal @ X4 @ X9 ) @ ( extend7643940197134561352nnreal @ ( plus_plus_real @ Y3 @ Y9 ) ) ) ) ) ).

% add_mono_ennreal
thf(fact_1220_ennreal__less__iff,axiom,
    ! [R4: real,Q2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R4 )
     => ( ( ord_le7381754540660121996nnreal @ ( extend7643940197134561352nnreal @ R4 ) @ ( extend7643940197134561352nnreal @ Q2 ) )
        = ( ord_less_real @ R4 @ Q2 ) ) ) ).

% ennreal_less_iff
thf(fact_1221_lborel__distr__uminus,axiom,
    ( ( measur2993149975067245138l_real @ lebesgue_lborel_real @ borel_5078946678739801102l_real @ uminus_uminus_real )
    = lebesgue_lborel_real ) ).

% lborel_distr_uminus
thf(fact_1222_qp_OEx__finite__integrable__function,axiom,
    ? [X6: real > extend8495563244428889912nnreal] :
      ( ( member2919562650594848410nnreal @ X6 @ ( sigma_9017504469962657078nnreal @ mu @ borel_6524799422816628122nnreal ) )
      & ( ( nonneg2667834350952324695l_real @ mu @ X6 )
       != extend2057119558705770725nnreal )
      & ! [Xa: real] :
          ( ( member_real @ Xa @ ( sigma_space_real @ mu ) )
         => ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( X6 @ Xa ) )
            & ( ord_le7381754540660121996nnreal @ ( X6 @ Xa ) @ extend2057119558705770725nnreal ) ) ) ) ).

% qp.Ex_finite_integrable_function
thf(fact_1223_qp_Odistributed__convolution,axiom,
    ! [X2: real > real,Y: real > real,F: real > extend8495563244428889912nnreal,G: real > extend8495563244428889912nnreal] :
      ( ( indepe3760321310464026790l_real @ mu @ borel_5078946678739801102l_real @ X2 @ borel_5078946678739801102l_real @ Y )
     => ( ( probab1340766270110547944l_real @ mu @ lebesgue_lborel_real @ X2 @ F )
       => ( ( probab1340766270110547944l_real @ mu @ lebesgue_lborel_real @ Y @ G )
         => ( probab1340766270110547944l_real @ mu @ lebesgue_lborel_real
            @ ^ [X: real] : ( plus_plus_real @ ( X2 @ X ) @ ( Y @ X ) )
            @ ^ [X: real] :
                ( nonneg2667834350952324695l_real @ lebesgue_lborel_real
                @ ^ [Y7: real] : ( times_1893300245718287421nnreal @ ( F @ ( minus_minus_real @ X @ Y7 ) ) @ ( G @ Y7 ) ) ) ) ) ) ) ).

% qp.distributed_convolution
thf(fact_1224_qp_Ocdf__cts__step_I1_J,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ X4 @ Y3 )
     => ( ord_less_eq_real @ ( distribution_cdf @ mu @ X4 ) @ ( bochne3715101410578510557l_real @ mu @ ( weak_cts_step @ X4 @ Y3 ) ) ) ) ).

% qp.cdf_cts_step(1)
thf(fact_1225_qp_Ocdf__nondecreasing,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ X4 @ Y3 )
     => ( ord_less_eq_real @ ( distribution_cdf @ mu @ X4 ) @ ( distribution_cdf @ mu @ Y3 ) ) ) ).

% qp.cdf_nondecreasing
thf(fact_1226_qp_Ocdf__nonneg,axiom,
    ! [X4: real] : ( ord_less_eq_real @ zero_zero_real @ ( distribution_cdf @ mu @ X4 ) ) ).

% qp.cdf_nonneg
thf(fact_1227_qp_Ocdf__bounded__prob,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( distribution_cdf @ mu @ X4 ) @ one_one_real ) ).

% qp.cdf_bounded_prob
thf(fact_1228_qp_Oemeasure__space__1,axiom,
    ( ( sigma_emeasure_real @ mu @ ( sigma_space_real @ mu ) )
    = one_on2969667320475766781nnreal ) ).

% qp.emeasure_space_1
thf(fact_1229_qp_Oemeasure__space__le__1,axiom,
    ord_le3935885782089961368nnreal @ ( sigma_emeasure_real @ mu @ ( sigma_space_real @ mu ) ) @ one_on2969667320475766781nnreal ).

% qp.emeasure_space_le_1
thf(fact_1230_qp_Ocdf__cts__step_I2_J,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ X4 @ Y3 )
     => ( ord_less_eq_real @ ( bochne3715101410578510557l_real @ mu @ ( weak_cts_step @ X4 @ Y3 ) ) @ ( distribution_cdf @ mu @ Y3 ) ) ) ).

% qp.cdf_cts_step(2)
thf(fact_1231_cdf__unique_H,axiom,
    ! [M1: sigma_measure_real,M22: sigma_measure_real] :
      ( ( distri7943378551711771532easure @ M1 )
     => ( ( distri7943378551711771532easure @ M22 )
       => ( ( ( distribution_cdf @ M1 )
            = ( distribution_cdf @ M22 ) )
         => ( M1 = M22 ) ) ) ) ).

% cdf_unique'
thf(fact_1232_cdf__unique,axiom,
    ! [M1: sigma_measure_real,M22: sigma_measure_real] :
      ( ( distri2809703520229113005bution @ M1 )
     => ( ( distri2809703520229113005bution @ M22 )
       => ( ( ( distribution_cdf @ M1 )
            = ( distribution_cdf @ M22 ) )
         => ( M1 = M22 ) ) ) ) ).

% cdf_unique
thf(fact_1233_finite__borel__measure_Ocdf__nondecreasing,axiom,
    ! [M: sigma_measure_real,X4: real,Y3: real] :
      ( ( distri7943378551711771532easure @ M )
     => ( ( ord_less_eq_real @ X4 @ Y3 )
       => ( ord_less_eq_real @ ( distribution_cdf @ M @ X4 ) @ ( distribution_cdf @ M @ Y3 ) ) ) ) ).

% finite_borel_measure.cdf_nondecreasing
thf(fact_1234_real__distribution_Ocdf__bounded__prob,axiom,
    ! [M: sigma_measure_real,X4: real] :
      ( ( distri2809703520229113005bution @ M )
     => ( ord_less_eq_real @ ( distribution_cdf @ M @ X4 ) @ one_one_real ) ) ).

% real_distribution.cdf_bounded_prob
thf(fact_1235_finite__borel__measure_Ocdf__nonneg,axiom,
    ! [M: sigma_measure_real,X4: real] :
      ( ( distri7943378551711771532easure @ M )
     => ( ord_less_eq_real @ zero_zero_real @ ( distribution_cdf @ M @ X4 ) ) ) ).

% finite_borel_measure.cdf_nonneg
thf(fact_1236_real__distribution_Ocdf__cts__step_I1_J,axiom,
    ! [M: sigma_measure_real,X4: real,Y3: real] :
      ( ( distri2809703520229113005bution @ M )
     => ( ( ord_less_real @ X4 @ Y3 )
       => ( ord_less_eq_real @ ( distribution_cdf @ M @ X4 ) @ ( bochne3715101410578510557l_real @ M @ ( weak_cts_step @ X4 @ Y3 ) ) ) ) ) ).

% real_distribution.cdf_cts_step(1)
thf(fact_1237_real__distribution_Ocdf__cts__step_I2_J,axiom,
    ! [M: sigma_measure_real,X4: real,Y3: real] :
      ( ( distri2809703520229113005bution @ M )
     => ( ( ord_less_real @ X4 @ Y3 )
       => ( ord_less_eq_real @ ( bochne3715101410578510557l_real @ M @ ( weak_cts_step @ X4 @ Y3 ) ) @ ( distribution_cdf @ M @ Y3 ) ) ) ) ).

% real_distribution.cdf_cts_step(2)
thf(fact_1238_real__distribution_Ointegrable__cts__step,axiom,
    ! [M: sigma_measure_real,A: real,B: real] :
      ( ( distri2809703520229113005bution @ M )
     => ( ( ord_less_real @ A @ B )
       => ( bochne3340023020068487468l_real @ M @ ( weak_cts_step @ A @ B ) ) ) ) ).

% real_distribution.integrable_cts_step
thf(fact_1239_qp_Osigma__finite__measure__axioms,axiom,
    measur487378040549452491e_real @ mu ).

% qp.sigma_finite_measure_axioms
thf(fact_1240_qp_Osubprob__not__empty,axiom,
    ( ( sigma_space_real @ mu )
   != bot_bot_set_real ) ).

% qp.subprob_not_empty
thf(fact_1241_enn2real__bot,axiom,
    ( ( extend1669699412028896998n2real @ bot_bo841427958541957580nnreal )
    = zero_zero_real ) ).

% enn2real_bot
thf(fact_1242_qp_Ostandard__normal__distributed__expectation,axiom,
    ! [X2: real > real] :
      ( ( probab1340766270110547944l_real @ mu @ lebesgue_lborel_real @ X2
        @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( normal_density @ zero_zero_real @ one_one_real @ X ) ) )
     => ( ( bochne3715101410578510557l_real @ mu @ X2 )
        = zero_zero_real ) ) ).

% qp.standard_normal_distributed_expectation
thf(fact_1243_qp_Onormal__distributed__expectation,axiom,
    ! [Sigma: real,X2: real > real,Mu: real] :
      ( ( ord_less_real @ zero_zero_real @ Sigma )
     => ( ( probab1340766270110547944l_real @ mu @ lebesgue_lborel_real @ X2
          @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( normal_density @ Mu @ Sigma @ X ) ) )
       => ( ( bochne3715101410578510557l_real @ mu @ X2 )
          = Mu ) ) ) ).

% qp.normal_distributed_expectation
thf(fact_1244_ennreal__lt__0,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ zero_zero_real )
     => ( ( extend7643940197134561352nnreal @ X4 )
        = zero_z7100319975126383169nnreal ) ) ).

% ennreal_lt_0
thf(fact_1245_integral__normal__density,axiom,
    ! [Sigma: real,Mu: real] :
      ( ( ord_less_real @ zero_zero_real @ Sigma )
     => ( ( bochne3715101410578510557l_real @ lebesgue_lborel_real @ ( normal_density @ Mu @ Sigma ) )
        = one_one_real ) ) ).

% integral_normal_density
thf(fact_1246_integrable__normal__density,axiom,
    ! [Sigma: real,Mu: real] :
      ( ( ord_less_real @ zero_zero_real @ Sigma )
     => ( bochne3340023020068487468l_real @ lebesgue_lborel_real @ ( normal_density @ Mu @ Sigma ) ) ) ).

% integrable_normal_density
thf(fact_1247_integral__normal__moment__nz__1,axiom,
    ! [Sigma: real,Mu: real] :
      ( ( ord_less_real @ zero_zero_real @ Sigma )
     => ( ( bochne3715101410578510557l_real @ lebesgue_lborel_real
          @ ^ [X: real] : ( times_times_real @ ( normal_density @ Mu @ Sigma @ X ) @ X ) )
        = Mu ) ) ).

% integral_normal_moment_nz_1
thf(fact_1248_integrable__normal__moment__nz__1,axiom,
    ! [Sigma: real,Mu: real] :
      ( ( ord_less_real @ zero_zero_real @ Sigma )
     => ( bochne3340023020068487468l_real @ lebesgue_lborel_real
        @ ^ [X: real] : ( times_times_real @ ( normal_density @ Mu @ Sigma @ X ) @ X ) ) ) ).

% integrable_normal_moment_nz_1
thf(fact_1249_qp_Onormal__standard__normal__convert,axiom,
    ! [Sigma: real,X2: real > real,Mu: real] :
      ( ( ord_less_real @ zero_zero_real @ Sigma )
     => ( ( probab1340766270110547944l_real @ mu @ lebesgue_lborel_real @ X2
          @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( normal_density @ Mu @ Sigma @ X ) ) )
        = ( probab1340766270110547944l_real @ mu @ lebesgue_lborel_real
          @ ^ [X: real] : ( divide_divide_real @ ( minus_minus_real @ ( X2 @ X ) @ Mu ) @ Sigma )
          @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( normal_density @ zero_zero_real @ one_one_real @ X ) ) ) ) ) ).

% qp.normal_standard_normal_convert
thf(fact_1250_qp_Onormal__density__affine,axiom,
    ! [X2: real > real,Mu: real,Sigma: real,Alpha: real,Beta: real] :
      ( ( probab1340766270110547944l_real @ mu @ lebesgue_lborel_real @ X2
        @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( normal_density @ Mu @ Sigma @ X ) ) )
     => ( ( ord_less_real @ zero_zero_real @ Sigma )
       => ( ( Alpha != zero_zero_real )
         => ( probab1340766270110547944l_real @ mu @ lebesgue_lborel_real
            @ ^ [X: real] : ( plus_plus_real @ Beta @ ( times_times_real @ Alpha @ ( X2 @ X ) ) )
            @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( normal_density @ ( plus_plus_real @ Beta @ ( times_times_real @ Alpha @ Mu ) ) @ ( times_times_real @ ( abs_abs_real @ Alpha ) @ Sigma ) @ X ) ) ) ) ) ) ).

% qp.normal_density_affine
thf(fact_1251_qp_Odistributed__affineI,axiom,
    ! [X2: real > real,T2: real,C: real,F: real > extend8495563244428889912nnreal] :
      ( ( probab1340766270110547944l_real @ mu @ lebesgue_lborel_real
        @ ^ [X: real] : ( divide_divide_real @ ( minus_minus_real @ ( X2 @ X ) @ T2 ) @ C )
        @ ^ [X: real] : ( times_1893300245718287421nnreal @ ( extend7643940197134561352nnreal @ ( abs_abs_real @ C ) ) @ ( F @ ( plus_plus_real @ ( times_times_real @ X @ C ) @ T2 ) ) ) )
     => ( ( C != zero_zero_real )
       => ( probab1340766270110547944l_real @ mu @ lebesgue_lborel_real @ X2 @ F ) ) ) ).

% qp.distributed_affineI
thf(fact_1252_abs__real__def,axiom,
    ( abs_abs_real
    = ( ^ [A5: real] : ( if_real @ ( ord_less_real @ A5 @ zero_zero_real ) @ ( uminus_uminus_real @ A5 ) @ A5 ) ) ) ).

% abs_real_def
thf(fact_1253_cts__step__def,axiom,
    ( weak_cts_step
    = ( ^ [A5: real,B4: real,X: real] : ( if_real @ ( ord_less_eq_real @ X @ A5 ) @ one_one_real @ ( if_real @ ( ord_less_eq_real @ B4 @ X ) @ zero_zero_real @ ( divide_divide_real @ ( minus_minus_real @ B4 @ X ) @ ( minus_minus_real @ B4 @ A5 ) ) ) ) ) ) ).

% cts_step_def
thf(fact_1254_nn__integral__real__affine,axiom,
    ! [F: real > extend8495563244428889912nnreal,C: real,T2: real] :
      ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ borel_6524799422816628122nnreal ) )
     => ( ( C != zero_zero_real )
       => ( ( nonneg2667834350952324695l_real @ lebesgue_lborel_real @ F )
          = ( times_1893300245718287421nnreal @ ( extend7643940197134561352nnreal @ ( abs_abs_real @ C ) )
            @ ( nonneg2667834350952324695l_real @ lebesgue_lborel_real
              @ ^ [X: real] : ( F @ ( plus_plus_real @ T2 @ ( times_times_real @ C @ X ) ) ) ) ) ) ) ) ).

% nn_integral_real_affine
thf(fact_1255_qp_Odistributed__affine,axiom,
    ! [X2: real > real,F: real > extend8495563244428889912nnreal,C: real,T2: real] :
      ( ( probab1340766270110547944l_real @ mu @ lebesgue_lborel_real @ X2 @ F )
     => ( ( C != zero_zero_real )
       => ( probab1340766270110547944l_real @ mu @ lebesgue_lborel_real
          @ ^ [X: real] : ( plus_plus_real @ T2 @ ( times_times_real @ C @ ( X2 @ X ) ) )
          @ ^ [X: real] : ( divide4826598186094686858nnreal @ ( F @ ( divide_divide_real @ ( minus_minus_real @ X @ T2 ) @ C ) ) @ ( extend7643940197134561352nnreal @ ( abs_abs_real @ C ) ) ) ) ) ) ).

% qp.distributed_affine
thf(fact_1256_divide__mult__eq,axiom,
    ! [A: extend8495563244428889912nnreal,X4: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( A != zero_z7100319975126383169nnreal )
     => ( ( A != extend2057119558705770725nnreal )
       => ( ( divide4826598186094686858nnreal @ ( times_1893300245718287421nnreal @ X4 @ A ) @ ( times_1893300245718287421nnreal @ B @ A ) )
          = ( divide4826598186094686858nnreal @ X4 @ B ) ) ) ) ).

% divide_mult_eq
thf(fact_1257_divide__ennreal,axiom,
    ! [R4: real,Q2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R4 )
     => ( ( ord_less_real @ zero_zero_real @ Q2 )
       => ( ( divide4826598186094686858nnreal @ ( extend7643940197134561352nnreal @ R4 ) @ ( extend7643940197134561352nnreal @ Q2 ) )
          = ( extend7643940197134561352nnreal @ ( divide_divide_real @ R4 @ Q2 ) ) ) ) ) ).

% divide_ennreal
thf(fact_1258_qp_Oexponential__distributed__expectation,axiom,
    ! [L2: real,X2: real > real] :
      ( ( ord_less_real @ zero_zero_real @ L2 )
     => ( ( probab1340766270110547944l_real @ mu @ lebesgue_lborel_real @ X2
          @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( erlang_density @ zero_zero_nat @ L2 @ X ) ) )
       => ( ( bochne3715101410578510557l_real @ mu @ X2 )
          = ( divide_divide_real @ one_one_real @ L2 ) ) ) ) ).

% qp.exponential_distributed_expectation
thf(fact_1259_qp_Oerlang__distributed__mult__const,axiom,
    ! [X2: real > real,K: nat,L2: real,Alpha: real] :
      ( ( probab1340766270110547944l_real @ mu @ lebesgue_lborel_real @ X2
        @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( erlang_density @ K @ L2 @ X ) ) )
     => ( ( ord_less_real @ zero_zero_real @ Alpha )
       => ( ( ord_less_real @ zero_zero_real @ L2 )
         => ( probab1340766270110547944l_real @ mu @ lebesgue_lborel_real
            @ ^ [X: real] : ( times_times_real @ Alpha @ ( X2 @ X ) )
            @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( erlang_density @ K @ ( divide_divide_real @ L2 @ Alpha ) @ X ) ) ) ) ) ) ).

% qp.erlang_distributed_mult_const
thf(fact_1260_qp_Osum__indep__erlang,axiom,
    ! [X2: real > real,Y: real > real,L2: real,K_1: nat,K_2: nat] :
      ( ( indepe3760321310464026790l_real @ mu @ borel_5078946678739801102l_real @ X2 @ borel_5078946678739801102l_real @ Y )
     => ( ( ord_less_real @ zero_zero_real @ L2 )
       => ( ( probab1340766270110547944l_real @ mu @ lebesgue_lborel_real @ X2
            @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( erlang_density @ K_1 @ L2 @ X ) ) )
         => ( ( probab1340766270110547944l_real @ mu @ lebesgue_lborel_real @ Y
              @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( erlang_density @ K_2 @ L2 @ X ) ) )
           => ( probab1340766270110547944l_real @ mu @ lebesgue_lborel_real
              @ ^ [X: real] : ( plus_plus_real @ ( X2 @ X ) @ ( Y @ X ) )
              @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( erlang_density @ ( minus_minus_nat @ ( plus_plus_nat @ ( suc @ K_1 ) @ ( suc @ K_2 ) ) @ one_one_nat ) @ L2 @ X ) ) ) ) ) ) ) ).

% qp.sum_indep_erlang
thf(fact_1261_qp_Oexponential__distributed__min,axiom,
    ! [L2: real,U: real,X2: real > real,Y: real > real] :
      ( ( ord_less_real @ zero_zero_real @ L2 )
     => ( ( ord_less_real @ zero_zero_real @ U )
       => ( ( probab1340766270110547944l_real @ mu @ lebesgue_lborel_real @ X2
            @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( erlang_density @ zero_zero_nat @ L2 @ X ) ) )
         => ( ( probab1340766270110547944l_real @ mu @ lebesgue_lborel_real @ Y
              @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( erlang_density @ zero_zero_nat @ U @ X ) ) )
           => ( ( indepe3760321310464026790l_real @ mu @ borel_5078946678739801102l_real @ X2 @ borel_5078946678739801102l_real @ Y )
             => ( probab1340766270110547944l_real @ mu @ lebesgue_lborel_real
                @ ^ [X: real] : ( ord_min_real @ ( X2 @ X ) @ ( Y @ X ) )
                @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( erlang_density @ zero_zero_nat @ ( plus_plus_real @ L2 @ U ) @ X ) ) ) ) ) ) ) ) ).

% qp.exponential_distributed_min
thf(fact_1262_convolution__erlang__density,axiom,
    ! [L2: real,K_1: nat,K_2: nat] :
      ( ( ord_less_real @ zero_zero_real @ L2 )
     => ( ( ^ [X: real] :
              ( nonneg2667834350952324695l_real @ lebesgue_lborel_real
              @ ^ [Y7: real] : ( times_1893300245718287421nnreal @ ( extend7643940197134561352nnreal @ ( erlang_density @ K_1 @ L2 @ ( minus_minus_real @ X @ Y7 ) ) ) @ ( extend7643940197134561352nnreal @ ( erlang_density @ K_2 @ L2 @ Y7 ) ) ) ) )
        = ( ^ [X: real] : ( extend7643940197134561352nnreal @ ( erlang_density @ ( minus_minus_nat @ ( plus_plus_nat @ ( suc @ K_1 ) @ ( suc @ K_2 ) ) @ one_one_nat ) @ L2 @ X ) ) ) ) ) ).

% convolution_erlang_density
thf(fact_1263_qp_Oerlang__ith__moment__integrable,axiom,
    ! [L2: real,X2: real > real,K: nat,I4: nat] :
      ( ( ord_less_real @ zero_zero_real @ L2 )
     => ( ( probab1340766270110547944l_real @ mu @ lebesgue_lborel_real @ X2
          @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( erlang_density @ K @ L2 @ X ) ) )
       => ( bochne3340023020068487468l_real @ mu
          @ ^ [X: real] : ( power_power_real @ ( X2 @ X ) @ I4 ) ) ) ) ).

% qp.erlang_ith_moment_integrable
thf(fact_1264_real__arch__pow,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ X4 )
     => ? [N4: nat] : ( ord_less_real @ Y3 @ ( power_power_real @ X4 @ N4 ) ) ) ).

% real_arch_pow
thf(fact_1265_real__arch__pow__inv,axiom,
    ! [Y3: real,X4: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ? [N4: nat] : ( ord_less_real @ ( power_power_real @ X4 @ N4 ) @ Y3 ) ) ) ).

% real_arch_pow_inv
thf(fact_1266_min__ennreal,axiom,
    ! [X4: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ord_mi739588054667363089nnreal @ ( extend7643940197134561352nnreal @ X4 ) @ ( extend7643940197134561352nnreal @ Y3 ) )
          = ( extend7643940197134561352nnreal @ ( ord_min_real @ X4 @ Y3 ) ) ) ) ) ).

% min_ennreal
thf(fact_1267_integrable__std__normal__moment,axiom,
    ! [K: nat] :
      ( bochne3340023020068487468l_real @ lebesgue_lborel_real
      @ ^ [X: real] : ( times_times_real @ ( normal_density @ zero_zero_real @ one_one_real @ X ) @ ( power_power_real @ X @ K ) ) ) ).

% integrable_std_normal_moment
thf(fact_1268_integrable__normal__moment,axiom,
    ! [Sigma: real,Mu: real,K: nat] :
      ( ( ord_less_real @ zero_zero_real @ Sigma )
     => ( bochne3340023020068487468l_real @ lebesgue_lborel_real
        @ ^ [X: real] : ( times_times_real @ ( normal_density @ Mu @ Sigma @ X ) @ ( power_power_real @ ( minus_minus_real @ X @ Mu ) @ K ) ) ) ) ).

% integrable_normal_moment
thf(fact_1269_integrable__std__normal__moment__abs,axiom,
    ! [K: nat] :
      ( bochne3340023020068487468l_real @ lebesgue_lborel_real
      @ ^ [X: real] : ( times_times_real @ ( normal_density @ zero_zero_real @ one_one_real @ X ) @ ( power_power_real @ ( abs_abs_real @ X ) @ K ) ) ) ).

% integrable_std_normal_moment_abs
thf(fact_1270_integrable__normal__moment__abs,axiom,
    ! [Sigma: real,Mu: real,K: nat] :
      ( ( ord_less_real @ zero_zero_real @ Sigma )
     => ( bochne3340023020068487468l_real @ lebesgue_lborel_real
        @ ^ [X: real] : ( times_times_real @ ( normal_density @ Mu @ Sigma @ X ) @ ( power_power_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Mu ) ) @ K ) ) ) ) ).

% integrable_normal_moment_abs
thf(fact_1271_ennreal__power,axiom,
    ! [R4: real,N3: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ R4 )
     => ( ( power_6007165696250533058nnreal @ ( extend7643940197134561352nnreal @ R4 ) @ N3 )
        = ( extend7643940197134561352nnreal @ ( power_power_real @ R4 @ N3 ) ) ) ) ).

% ennreal_power
thf(fact_1272_qp_Oerlang__ith__moment,axiom,
    ! [L2: real,X2: real > real,K: nat,I4: nat] :
      ( ( ord_less_real @ zero_zero_real @ L2 )
     => ( ( probab1340766270110547944l_real @ mu @ lebesgue_lborel_real @ X2
          @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( erlang_density @ K @ L2 @ X ) ) )
       => ( ( bochne3715101410578510557l_real @ mu
            @ ^ [X: real] : ( power_power_real @ ( X2 @ X ) @ I4 ) )
          = ( divide_divide_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ I4 ) ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( power_power_real @ L2 @ I4 ) ) ) ) ) ) ).

% qp.erlang_ith_moment
thf(fact_1273_square__continuous,axiom,
    ! [E: real,X4: real] :
      ( ( ord_less_real @ zero_zero_real @ E )
     => ? [D2: real] :
          ( ( ord_less_real @ zero_zero_real @ D2 )
          & ! [Y10: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ Y10 @ X4 ) ) @ D2 )
             => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( times_times_real @ Y10 @ Y10 ) @ ( times_times_real @ X4 @ X4 ) ) ) @ E ) ) ) ) ).

% square_continuous
thf(fact_1274_nn__integral__erlang__ith__moment,axiom,
    ! [L2: real,K: nat,I4: nat] :
      ( ( ord_less_real @ zero_zero_real @ L2 )
     => ( ( nonneg2667834350952324695l_real @ lebesgue_lborel_real
          @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( times_times_real @ ( erlang_density @ K @ L2 @ X ) @ ( power_power_real @ X @ I4 ) ) ) )
        = ( extend7643940197134561352nnreal @ ( divide_divide_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ I4 ) ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( power_power_real @ L2 @ I4 ) ) ) ) ) ) ).

% nn_integral_erlang_ith_moment
thf(fact_1275_qp_Ohas__bochner__integral__erlang__ith__moment,axiom,
    ! [L2: real,X2: real > real,K: nat,I4: nat] :
      ( ( ord_less_real @ zero_zero_real @ L2 )
     => ( ( probab1340766270110547944l_real @ mu @ lebesgue_lborel_real @ X2
          @ ^ [X: real] : ( extend7643940197134561352nnreal @ ( erlang_density @ K @ L2 @ X ) ) )
       => ( bochne663871741685100524l_real @ mu
          @ ^ [X: real] : ( power_power_real @ ( X2 @ X ) @ I4 )
          @ ( divide_divide_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ I4 ) ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( power_power_real @ L2 @ I4 ) ) ) ) ) ) ).

% qp.has_bochner_integral_erlang_ith_moment
thf(fact_1276_lemma__interval,axiom,
    ! [A: real,X4: real,B: real] :
      ( ( ord_less_real @ A @ X4 )
     => ( ( ord_less_real @ X4 @ B )
       => ? [D2: real] :
            ( ( ord_less_real @ zero_zero_real @ D2 )
            & ! [Y10: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y10 ) ) @ D2 )
               => ( ( ord_less_eq_real @ A @ Y10 )
                  & ( ord_less_eq_real @ Y10 @ B ) ) ) ) ) ) ).

% lemma_interval

% Helper facts (3)
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X4: real,Y3: real] :
      ( ( if_real @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X4: real,Y3: real] :
      ( ( if_real @ $true @ X4 @ Y3 )
      = X4 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( probab5242164193669365150gral_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) @ fa )
    = ( minus_minus_real
      @ ( extend1669699412028896998n2real
        @ ( probab4322474783390693535gral_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) )
          @ ^ [X: a] : ( extend7643940197134561352nnreal @ ( fa @ X ) ) ) )
      @ ( extend1669699412028896998n2real
        @ ( probab4322474783390693535gral_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) )
          @ ^ [X: a] : ( extend7643940197134561352nnreal @ ( uminus_uminus_real @ ( fa @ X ) ) ) ) ) ) ) ).

%------------------------------------------------------------------------------