%------------------------------------------------------------------------------
% 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/0009_Monad_QuasiBorel/prob_01368_066499__15461278_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2304 ( 333 unt;1024 typ;   0 def)
%            Number of atoms       : 3851 (1003 equ;   1 cnn)
%            Maximal formula atoms :   12 (   3 avg)
%            Number of connectives : 15350 ( 110   ~;   2   |; 112   &;13018   @)
%                                         (   0 <=>;2108  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   19 (   8 avg)
%            Number of types       :  235 ( 234 usr)
%            Number of type conns  : 3652 (3652   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  792 ( 790 usr;  38 con; 0-5 aty)
%            Number of variables   : 4381 ( 231   ^;4066   !;  84   ?;4381   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:11:12.284
%------------------------------------------------------------------------------
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    set_real_complex: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    produc2422161461964618553l_real: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Complex__Ocomplex_J_J,type,
    set_nat_complex: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
    produc7716430852924023517t_real: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    set_real_real: $tType ).

thf(ty_n_t__QuasiBorel__Oquasi____borel_It__Complex__Ocomplex_J,type,
    quasi_borel_complex: $tType ).

thf(ty_n_t__Sigma____Algebra__Omeasure_It__Complex__Ocomplex_J,type,
    sigma_3077487657436305159omplex: $tType ).

thf(ty_n_t__Set__Oset_It__Sigma____Algebra__Omeasure_I_Eo_J_J,type,
    set_Sigma_measure_o: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    set_real_nat: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    set_nat_real: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    set_nat_nat: $tType ).

thf(ty_n_t__Sigma____Algebra__Omeasure_I_062_I_Eo_M_Eo_J_J,type,
    sigma_measure_o_o: $tType ).

thf(ty_n_t__Set__Oset_I_062_I_Eo_Mt__Complex__Ocomplex_J_J,type,
    set_o_complex: $tType ).

thf(ty_n_t__QuasiBorel__Oquasi____borel_It__Real__Oreal_J,type,
    quasi_borel_real: $tType ).

thf(ty_n_t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
    sigma_measure_real: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
    set_set_real: $tType ).

thf(ty_n_t__QuasiBorel__Oquasi____borel_It__Nat__Onat_J,type,
    quasi_borel_nat: $tType ).

thf(ty_n_t__Sigma____Algebra__Omeasure_It__Nat__Onat_J,type,
    sigma_measure_nat: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Real__Oreal_J_J,type,
    set_a_real: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J,type,
    set_real_b: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J,type,
    set_real_a: $tType ).

thf(ty_n_t__Extended____Nonnegative____Real__Oennreal,type,
    extend8495563244428889912nnreal: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_M_Eo_J_J,type,
    set_real_o: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
    set_nat_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_I_Eo_Mt__Real__Oreal_J_J,type,
    set_o_real: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
    set_nat_o: $tType ).

thf(ty_n_t__Set__Oset_I_062_I_Eo_Mt__Nat__Onat_J_J,type,
    set_o_nat: $tType ).

thf(ty_n_t__QuasiBorel__Oquasi____borel_Itf__b_J,type,
    quasi_borel_b: $tType ).

thf(ty_n_t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
    quasi_borel_a: $tType ).

thf(ty_n_t__Sigma____Algebra__Omeasure_Itf__b_J,type,
    sigma_measure_b: $tType ).

thf(ty_n_t__Sigma____Algebra__Omeasure_Itf__a_J,type,
    sigma_measure_a: $tType ).

thf(ty_n_t__QuasiBorel__Oquasi____borel_I_Eo_J,type,
    quasi_borel_o: $tType ).

thf(ty_n_t__Sigma____Algebra__Omeasure_I_Eo_J,type,
    sigma_measure_o: $tType ).

thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
    set_complex: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__b_Mtf__b_J_J,type,
    set_b_b: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__b_Mtf__a_J_J,type,
    set_b_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
    set_a_b: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
    set_a_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
    set_a_o: $tType ).

thf(ty_n_t__Set__Oset_I_062_I_Eo_Mtf__a_J_J,type,
    set_o_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
    set_o_o: $tType ).

thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
    set_real: $tType ).

thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
    set_nat: $tType ).

thf(ty_n_t__Set__Oset_Itf__b_J,type,
    set_b: $tType ).

thf(ty_n_t__Set__Oset_Itf__a_J,type,
    set_a: $tType ).

thf(ty_n_t__Complex__Ocomplex,type,
    complex: $tType ).

thf(ty_n_t__Set__Oset_I_Eo_J,type,
    set_o: $tType ).

thf(ty_n_t__Real__Oreal,type,
    real: $tType ).

thf(ty_n_t__Nat__Onat,type,
    nat: $tType ).

thf(ty_n_tf__b,type,
    b: $tType ).

thf(ty_n_tf__a,type,
    a: $tType ).

% Explicit typings (790)
thf(sy_c_Binary__Product__Measure_Opair__measure_001t__Nat__Onat_001t__Real__Oreal,type,
    binary2880700947547503686t_real: sigma_measure_nat > sigma_measure_real > sigma_5310753476256395226t_real ).

thf(sy_c_Binary__Product__Measure_Opair__measure_001t__Real__Oreal_001t__Real__Oreal,type,
    binary6478037234023840930l_real: sigma_measure_real > sigma_measure_real > sigma_2308072346491277622l_real ).

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

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

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

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001_Eo,type,
    borel_5500255247093592246orel_o: sigma_measure_o ).

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__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_Complete__Measure_Ocompletion_001_Eo,type,
    comple48332195503990434tion_o: sigma_measure_o > sigma_measure_o ).

thf(sy_c_Complete__Measure_Ocompletion_001t__Nat__Onat,type,
    comple4529072586887470918on_nat: sigma_measure_nat > sigma_measure_nat ).

thf(sy_c_Complete__Measure_Ocompletion_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
    comple520438860280874727t_real: sigma_5310753476256395226t_real > sigma_5310753476256395226t_real ).

thf(sy_c_Complete__Measure_Ocompletion_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    comple4976096958823823939l_real: sigma_2308072346491277622l_real > sigma_2308072346491277622l_real ).

thf(sy_c_Complete__Measure_Ocompletion_001t__Real__Oreal,type,
    comple3506806835435775778n_real: sigma_measure_real > sigma_measure_real ).

thf(sy_c_Complete__Measure_Ocompletion_001tf__a,type,
    comple3428971583294703880tion_a: sigma_measure_a > sigma_measure_a ).

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

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_Extended__Nat_Oinfinity__class_Oinfinity_001t__Extended____Nonnegative____Real__Oennreal,type,
    extend2057119558705770725nnreal: extend8495563244428889912nnreal ).

thf(sy_c_Finite__Set_Ofinite_001t__Extended____Nonnegative____Real__Oennreal,type,
    finite3782138982310603983nnreal: set_Ex3793607809372303086nnreal > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
    finite_finite_real: set_real > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Real__Oreal_J,type,
    finite9007344921179782393t_real: set_set_real > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
    finite4560333769392153135t_real: set_set_set_real > $o ).

thf(sy_c_Fun_Ocomp_001_Eo_001_Eo_001_Eo,type,
    comp_o_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).

thf(sy_c_Fun_Ocomp_001_Eo_001_Eo_001t__Real__Oreal,type,
    comp_o_o_real: ( $o > $o ) > ( real > $o ) > real > $o ).

thf(sy_c_Fun_Ocomp_001_Eo_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    comp_o_complex_real: ( $o > complex ) > ( real > $o ) > real > complex ).

thf(sy_c_Fun_Ocomp_001_Eo_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal,type,
    comp_o592059427571696603l_real: ( $o > extend8495563244428889912nnreal ) > ( real > $o ) > real > extend8495563244428889912nnreal ).

thf(sy_c_Fun_Ocomp_001_Eo_001t__Nat__Onat_001t__Real__Oreal,type,
    comp_o_nat_real: ( $o > nat ) > ( real > $o ) > real > nat ).

thf(sy_c_Fun_Ocomp_001_Eo_001t__Real__Oreal_001_Eo,type,
    comp_o_real_o: ( $o > real ) > ( $o > $o ) > $o > real ).

thf(sy_c_Fun_Ocomp_001_Eo_001t__Real__Oreal_001t__Real__Oreal,type,
    comp_o_real_real: ( $o > real ) > ( real > $o ) > real > real ).

thf(sy_c_Fun_Ocomp_001_Eo_001tf__a_001t__Real__Oreal,type,
    comp_o_a_real: ( $o > a ) > ( real > $o ) > real > a ).

thf(sy_c_Fun_Ocomp_001t__Extended____Nonnegative____Real__Oennreal_001_Eo_001t__Real__Oreal,type,
    comp_E6134263049385049321o_real: ( extend8495563244428889912nnreal > $o ) > ( real > extend8495563244428889912nnreal ) > real > $o ).

thf(sy_c_Fun_Ocomp_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal_001tf__a,type,
    comp_E1477048112619722217real_a: ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > ( a > extend8495563244428889912nnreal ) > a > extend8495563244428889912nnreal ).

thf(sy_c_Fun_Ocomp_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal_001tf__b,type,
    comp_E1477048112619722218real_b: ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > ( b > extend8495563244428889912nnreal ) > b > extend8495563244428889912nnreal ).

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_001t__Real__Oreal_001tf__a,type,
    comp_E5459461548720263541real_a: ( extend8495563244428889912nnreal > real ) > ( a > extend8495563244428889912nnreal ) > a > real ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001_Eo_001t__Real__Oreal,type,
    comp_nat_o_real: ( nat > $o ) > ( real > nat ) > real > $o ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    comp_n4215249288434654095x_real: ( nat > complex ) > ( real > nat ) > real > complex ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal,type,
    comp_n3455504981484479769l_real: ( nat > extend8495563244428889912nnreal ) > ( real > nat ) > real > extend8495563244428889912nnreal ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
    comp_nat_nat_real: ( nat > nat ) > ( real > nat ) > real > nat ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Nat__Onat,type,
    comp_nat_real_nat: ( nat > real ) > ( nat > nat ) > nat > real ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Real__Oreal,type,
    comp_nat_real_real: ( nat > real ) > ( real > nat ) > real > real ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001tf__a_001t__Real__Oreal,type,
    comp_nat_a_real: ( nat > a ) > ( real > nat ) > real > a ).

thf(sy_c_Fun_Ocomp_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_I_Eo_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_I_Eo_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_I_Eo_J,type,
    comp_P8454002975481202678pace_o: ( probab1241297377463522905pace_o > probab1241297377463522905pace_o ) > ( probab1241297377463522905pace_o > probab1241297377463522905pace_o ) > probab1241297377463522905pace_o > probab1241297377463522905pace_o ).

thf(sy_c_Fun_Ocomp_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_I_Eo_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_I_Eo_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J,type,
    comp_P6982291091032798180e_real: ( probab1241297377463522905pace_o > probab1241297377463522905pace_o ) > ( probab8009751763329705409e_real > probab1241297377463522905pace_o ) > probab8009751763329705409e_real > probab1241297377463522905pace_o ).

thf(sy_c_Fun_Ocomp_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_I_Eo_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_I_Eo_J,type,
    comp_P4701177297104927972pace_o: ( probab8009751763329705409e_real > probab1241297377463522905pace_o ) > ( probab1241297377463522905pace_o > probab8009751763329705409e_real ) > probab1241297377463522905pace_o > probab1241297377463522905pace_o ).

thf(sy_c_Fun_Ocomp_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_I_Eo_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J,type,
    comp_P1838334351674764662e_real: ( probab8009751763329705409e_real > probab1241297377463522905pace_o ) > ( probab8009751763329705409e_real > probab8009751763329705409e_real ) > probab8009751763329705409e_real > probab1241297377463522905pace_o ).

thf(sy_c_Fun_Ocomp_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Complex__Ocomplex_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J,type,
    comp_P6469183723381889322e_real: ( probab8009751763329705409e_real > probab3778977310982246339omplex ) > ( probab8009751763329705409e_real > probab8009751763329705409e_real ) > probab8009751763329705409e_real > probab3778977310982246339omplex ).

thf(sy_c_Fun_Ocomp_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Extended____Nonnegative____Real__Oennreal_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J,type,
    comp_P3464484273722719156e_real: ( probab8009751763329705409e_real > probab1359147627358338381nnreal ) > ( probab8009751763329705409e_real > probab8009751763329705409e_real ) > probab8009751763329705409e_real > probab1359147627358338381nnreal ).

thf(sy_c_Fun_Ocomp_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Nat__Onat_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Nat__Onat_J,type,
    comp_P655014663824291312ce_nat: ( probab8009751763329705409e_real > probab8370124847414953445ce_nat ) > ( probab8370124847414953445ce_nat > probab8009751763329705409e_real ) > probab8370124847414953445ce_nat > probab8370124847414953445ce_nat ).

thf(sy_c_Fun_Ocomp_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Nat__Onat_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J,type,
    comp_P5219563683584835660e_real: ( probab8009751763329705409e_real > probab8370124847414953445ce_nat ) > ( probab8009751763329705409e_real > probab8009751763329705409e_real ) > probab8009751763329705409e_real > probab8370124847414953445ce_nat ).

thf(sy_c_Fun_Ocomp_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J,type,
    comp_P6332119684870606760e_real: ( probab8009751763329705409e_real > probab8009751763329705409e_real ) > ( probab8009751763329705409e_real > probab8009751763329705409e_real ) > probab8009751763329705409e_real > probab8009751763329705409e_real ).

thf(sy_c_Fun_Ocomp_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__a_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J,type,
    comp_P2634432118360280224e_real: ( probab4737552673497767871pace_a > probab8009751763329705409e_real ) > ( probab8009751763329705409e_real > probab4737552673497767871pace_a ) > probab8009751763329705409e_real > probab8009751763329705409e_real ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001_Eo_001_Eo,type,
    comp_real_o_o: ( real > $o ) > ( $o > real ) > $o > $o ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001_Eo_001t__Real__Oreal,type,
    comp_real_o_real: ( real > $o ) > ( real > real ) > real > $o ).

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__Extended____Nonnegative____Real__Oennreal,type,
    comp_r6281409797179841921nnreal: ( real > extend8495563244428889912nnreal ) > ( extend8495563244428889912nnreal > real ) > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).

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__Nat__Onat_001t__Nat__Onat,type,
    comp_real_nat_nat: ( real > nat ) > ( nat > real ) > nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Nat__Onat_001t__Real__Oreal,type,
    comp_real_nat_real: ( real > nat ) > ( real > real ) > real > nat ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
    comp_r4825950202331181903t_real: ( real > produc7716430852924023517t_real ) > ( produc7716430852924023517t_real > real ) > produc7716430852924023517t_real > produc7716430852924023517t_real ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    comp_r97482993418849103l_real: ( real > produc2422161461964618553l_real ) > ( produc2422161461964618553l_real > real ) > produc2422161461964618553l_real > produc2422161461964618553l_real ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
    comp_real_real_o: ( real > real ) > ( $o > real ) > $o > real ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Nat__Onat,type,
    comp_real_real_nat: ( real > real ) > ( nat > real ) > nat > real ).

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_001_Eo,type,
    comp_real_a_o: ( real > a ) > ( $o > real ) > $o > a ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001tf__a_001t__Nat__Onat,type,
    comp_real_a_nat: ( real > a ) > ( nat > real ) > nat > a ).

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_001t__Real__Oreal_001tf__b_001t__Real__Oreal,type,
    comp_real_b_real: ( real > b ) > ( real > real ) > real > b ).

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,
    comp_a6042866249568583849real_a: ( a > extend8495563244428889912nnreal ) > ( a > a ) > a > extend8495563244428889912nnreal ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Extended____Nonnegative____Real__Oennreal_001tf__b,type,
    comp_a6042866249568583850real_b: ( a > extend8495563244428889912nnreal ) > ( b > a ) > b > extend8495563244428889912nnreal ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_I_Eo_J_001t__Real__Oreal,type,
    comp_a281010391317482986o_real: ( a > probab1241297377463522905pace_o ) > ( real > a ) > real > probab1241297377463522905pace_o ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Complex__Ocomplex_J_001t__Real__Oreal,type,
    comp_a2485845550147043532x_real: ( a > probab3778977310982246339omplex ) > ( real > a ) > real > probab3778977310982246339omplex ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Extended____Nonnegative____Real__Oennreal_J_001t__Real__Oreal,type,
    comp_a4151604154359478614l_real: ( a > probab1359147627358338381nnreal ) > ( real > a ) > real > probab1359147627358338381nnreal ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Nat__Onat_J_001t__Real__Oreal,type,
    comp_a1663454009834812910t_real: ( a > probab8370124847414953445ce_nat ) > ( real > a ) > real > probab8370124847414953445ce_nat ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J_001t__Real__Oreal,type,
    comp_a677806100398348746l_real: ( a > probab8009751763329705409e_real ) > ( real > a ) > real > probab8009751763329705409e_real ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__a_J_001t__Real__Oreal,type,
    comp_a423948744115472976a_real: ( a > probab4737552673497767871pace_a ) > ( real > a ) > real > probab4737552673497767871pace_a ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__b_J_001t__Real__Oreal,type,
    comp_a6868616473110226257b_real: ( a > probab4737552677800996672pace_b ) > ( real > a ) > real > probab4737552677800996672pace_b ).

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_Fun_Ocomp_001tf__a_001tf__b_001t__Real__Oreal,type,
    comp_a_b_real: ( a > b ) > ( real > a ) > real > b ).

thf(sy_c_Fun_Ocomp_001tf__b_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal,type,
    comp_b5186278242990201154l_real: ( b > extend8495563244428889912nnreal ) > ( real > b ) > real > extend8495563244428889912nnreal ).

thf(sy_c_Fun_Ocomp_001tf__b_001t__Extended____Nonnegative____Real__Oennreal_001tf__a,type,
    comp_b2719268283915226856real_a: ( b > extend8495563244428889912nnreal ) > ( a > b ) > a > extend8495563244428889912nnreal ).

thf(sy_c_Fun_Ocomp_001tf__b_001t__Extended____Nonnegative____Real__Oennreal_001tf__b,type,
    comp_b2719268283915226857real_b: ( b > extend8495563244428889912nnreal ) > ( b > b ) > b > extend8495563244428889912nnreal ).

thf(sy_c_Fun_Ocomp_001tf__b_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_I_Eo_J_001t__Real__Oreal,type,
    comp_b3676460685241377833o_real: ( b > probab1241297377463522905pace_o ) > ( real > b ) > real > probab1241297377463522905pace_o ).

thf(sy_c_Fun_Ocomp_001tf__b_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Complex__Ocomplex_J_001t__Real__Oreal,type,
    comp_b2988533236785854029x_real: ( b > probab3778977310982246339omplex ) > ( real > b ) > real > probab3778977310982246339omplex ).

thf(sy_c_Fun_Ocomp_001tf__b_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Extended____Nonnegative____Real__Oennreal_J_001t__Real__Oreal,type,
    comp_b7132781594232482007l_real: ( b > probab1359147627358338381nnreal ) > ( real > b ) > real > probab1359147627358338381nnreal ).

thf(sy_c_Fun_Ocomp_001tf__b_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Nat__Onat_J_001t__Real__Oreal,type,
    comp_b384277876254322031t_real: ( b > probab8370124847414953445ce_nat ) > ( real > b ) > real > probab8370124847414953445ce_nat ).

thf(sy_c_Fun_Ocomp_001tf__b_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J_001t__Real__Oreal,type,
    comp_b7676164988068126923l_real: ( b > probab8009751763329705409e_real ) > ( real > b ) > real > probab8009751763329705409e_real ).

thf(sy_c_Fun_Ocomp_001tf__b_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__a_J_001t__Real__Oreal,type,
    comp_b7704093361825181967a_real: ( b > probab4737552673497767871pace_a ) > ( real > b ) > real > probab4737552673497767871pace_a ).

thf(sy_c_Fun_Ocomp_001tf__b_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__b_J_001t__Real__Oreal,type,
    comp_b4925389053965159440b_real: ( b > probab4737552677800996672pace_b ) > ( real > b ) > real > probab4737552677800996672pace_b ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__a_001t__Real__Oreal,type,
    comp_b_a_real: ( b > a ) > ( real > b ) > real > a ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__b_001t__Real__Oreal,type,
    comp_b_b_real: ( b > b ) > ( real > b ) > real > b ).

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

thf(sy_c_Giry__Monad_Obind_001_Eo_001t__Nat__Onat,type,
    giry_bind_o_nat: sigma_measure_o > ( $o > sigma_measure_nat ) > sigma_measure_nat ).

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

thf(sy_c_Giry__Monad_Obind_001t__Nat__Onat_001_Eo,type,
    giry_bind_nat_o: sigma_measure_nat > ( nat > sigma_measure_o ) > sigma_measure_o ).

thf(sy_c_Giry__Monad_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
    giry_bind_nat_nat: sigma_measure_nat > ( nat > sigma_measure_nat ) > sigma_measure_nat ).

thf(sy_c_Giry__Monad_Obind_001t__Nat__Onat_001t__Real__Oreal,type,
    giry_bind_nat_real: sigma_measure_nat > ( nat > sigma_measure_real ) > sigma_measure_real ).

thf(sy_c_Giry__Monad_Obind_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_001_Eo,type,
    giry_b1272544242581952950real_o: sigma_2308072346491277622l_real > ( produc2422161461964618553l_real > sigma_measure_o ) > sigma_measure_o ).

thf(sy_c_Giry__Monad_Obind_001t__Real__Oreal_001_Eo,type,
    giry_bind_real_o: sigma_measure_real > ( real > sigma_measure_o ) > sigma_measure_o ).

thf(sy_c_Giry__Monad_Obind_001t__Real__Oreal_001t__Nat__Onat,type,
    giry_bind_real_nat: sigma_measure_real > ( real > sigma_measure_nat ) > sigma_measure_nat ).

thf(sy_c_Giry__Monad_Obind_001t__Real__Oreal_001t__Real__Oreal,type,
    giry_bind_real_real: sigma_measure_real > ( real > sigma_measure_real ) > sigma_measure_real ).

thf(sy_c_Giry__Monad_Oprob__algebra_001t__Real__Oreal,type,
    giry_p6081475675320601670a_real: sigma_measure_real > sigma_8927737637348964610e_real ).

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

thf(sy_c_Giry__Monad_Osubprob__algebra_001t__Complex__Ocomplex,type,
    giry_s6563750351226335580omplex: sigma_3077487657436305159omplex > sigma_4317789280431751428omplex ).

thf(sy_c_Giry__Monad_Osubprob__algebra_001t__Extended____Nonnegative____Real__Oennreal,type,
    giry_s4961427825854884838nnreal: sigma_7234349610311085201nnreal > sigma_4746704660585072014nnreal ).

thf(sy_c_Giry__Monad_Osubprob__algebra_001t__Nat__Onat,type,
    giry_s8280036963460128894ra_nat: sigma_measure_nat > sigma_7132509806855041574re_nat ).

thf(sy_c_Giry__Monad_Osubprob__algebra_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
    giry_s6733618172312976303t_real: sigma_5310753476256395226t_real > sigma_6565123957185080115t_real ).

thf(sy_c_Giry__Monad_Osubprob__algebra_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    giry_s9089345087316773643l_real: sigma_2308072346491277622l_real > sigma_5254896642142037903l_real ).

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

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

thf(sy_c_Giry__Monad_Osubprob__space_001_Eo,type,
    giry_subprob_space_o: sigma_measure_o > $o ).

thf(sy_c_Giry__Monad_Osubprob__space_001t__Complex__Ocomplex,type,
    giry_s5699845841807254394omplex: sigma_3077487657436305159omplex > $o ).

thf(sy_c_Giry__Monad_Osubprob__space_001t__Extended____Nonnegative____Real__Oennreal,type,
    giry_s9043694198952438276nnreal: sigma_7234349610311085201nnreal > $o ).

thf(sy_c_Giry__Monad_Osubprob__space_001t__Nat__Onat,type,
    giry_s459323515522551452ce_nat: sigma_measure_nat > $o ).

thf(sy_c_Giry__Monad_Osubprob__space_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
    giry_s632251592358338321t_real: sigma_5310753476256395226t_real > $o ).

thf(sy_c_Giry__Monad_Osubprob__space_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    giry_s5019895290865573229l_real: sigma_2308072346491277622l_real > $o ).

thf(sy_c_Giry__Monad_Osubprob__space_001t__Real__Oreal,type,
    giry_s8208748868292234104e_real: sigma_measure_real > $o ).

thf(sy_c_Giry__Monad_Osubprob__space_001tf__a,type,
    giry_subprob_space_a: sigma_measure_a > $o ).

thf(sy_c_Giry__Monad_Osubprob__space_001tf__b,type,
    giry_subprob_space_b: sigma_measure_b > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nonnegative____Real__Oennreal,type,
    minus_8429688780609304081nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
    minus_minus_real: real > real > real ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
    minus_minus_set_real: set_real > set_real > set_real ).

thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
    one_one_complex: complex ).

thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
    one_one_real: real ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nonnegative____Real__Oennreal,type,
    plus_p1859984266308609217nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).

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

thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
    times_times_complex: complex > complex > complex ).

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

thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nonnegative____Real__Oennreal,type,
    zero_z7100319975126383169nnreal: extend8495563244428889912nnreal ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
    zero_zero_real: real ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__events_001t__Real__Oreal_001t__Real__Oreal,type,
    indepe570525225537247534l_real: sigma_measure_real > ( real > set_real ) > set_real > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__events_001t__Real__Oreal_001t__Set__Oset_It__Real__Oreal_J,type,
    indepe3416441470874335588t_real: sigma_measure_real > ( set_real > set_real ) > set_set_real > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__set_001t__Real__Oreal,type,
    indepe5067751462249938772t_real: sigma_measure_real > set_set_real > set_set_real > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001t__Real__Oreal_001_062_I_Eo_M_Eo_J,type,
    indepe6032362129116668415al_o_o: sigma_measure_real > ( ( $o > $o ) > set_set_real ) > set_o_o > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001t__Real__Oreal_001_062_It__Real__Oreal_M_Eo_J,type,
    indepe8014066527605497865real_o: sigma_measure_real > ( ( real > $o ) > set_set_real ) > set_real_o > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001t__Real__Oreal_001_062_It__Real__Oreal_Mt__Complex__Ocomplex_J,type,
    indepe6244138955099140627omplex: sigma_measure_real > ( ( real > complex ) > set_set_real ) > set_real_complex > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001t__Real__Oreal_001_062_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
    indepe4643477836443271069nnreal: sigma_measure_real > ( ( real > extend8495563244428889912nnreal ) > set_set_real ) > set_re5328672808648366137nnreal > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001t__Real__Oreal_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
    indepe7235330332140030005al_nat: sigma_measure_real > ( ( real > nat ) > set_set_real ) > set_real_nat > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001t__Real__Oreal_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
    indepe4905764430350991377l_real: sigma_measure_real > ( ( real > real ) > set_set_real ) > set_real_real > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001t__Real__Oreal_001_062_Itf__a_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
    indepe7934793495794061069nnreal: sigma_measure_real > ( ( a > extend8495563244428889912nnreal ) > set_set_real ) > set_a_7161065143582548615nnreal > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001t__Real__Oreal_001_062_Itf__a_Mt__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__b_J_J,type,
    indepe1711633972345727297pace_b: sigma_measure_real > ( ( a > probab4737552677800996672pace_b ) > set_set_real ) > set_a_3263942136899480541pace_b > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001t__Real__Oreal_001_062_Itf__a_Mt__Real__Oreal_J,type,
    indepe7634599452152747905a_real: sigma_measure_real > ( ( a > real ) > set_set_real ) > set_a_real > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001t__Real__Oreal_001_062_Itf__b_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
    indepe4831269413077975822nnreal: sigma_measure_real > ( ( b > extend8495563244428889912nnreal ) > set_set_real ) > set_b_6825823330181178888nnreal > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001t__Real__Oreal_001t__Real__Oreal,type,
    indepe8783372407961027910l_real: sigma_measure_real > ( real > set_set_real ) > set_real > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001t__Real__Oreal_001t__Set__Oset_It__Real__Oreal_J,type,
    indepe8752365572443096444t_real: sigma_measure_real > ( set_real > set_set_real ) > set_set_real > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001_Eo_001_Eo,type,
    indepe6952847619639754700ar_o_o: sigma_measure_o > sigma_measure_o > ( $o > $o ) > sigma_measure_o > ( $o > $o ) > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001_Eo_001t__Real__Oreal,type,
    indepe8209737011595649400o_real: sigma_measure_o > sigma_measure_real > ( $o > real ) > sigma_measure_real > ( $o > real ) > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001t__Nat__Onat_001t__Nat__Onat,type,
    indepe2246810989395310190at_nat: sigma_measure_nat > sigma_measure_nat > ( nat > nat ) > sigma_measure_nat > ( nat > nat ) > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001t__Nat__Onat_001t__Real__Oreal,type,
    indepe8057041690633792074t_real: sigma_measure_nat > sigma_measure_real > ( nat > real ) > sigma_measure_real > ( nat > real ) > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001t__Real__Oreal_001_Eo,type,
    indepe495429349782043166real_o: sigma_measure_real > sigma_measure_o > ( real > $o ) > sigma_measure_o > ( real > $o ) > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001t__Real__Oreal_001t__Complex__Ocomplex,type,
    indepe1954327081502071720omplex: sigma_measure_real > sigma_3077487657436305159omplex > ( real > complex ) > sigma_3077487657436305159omplex > ( real > complex ) > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal,type,
    indepe6767359503340752434nnreal: sigma_measure_real > sigma_7234349610311085201nnreal > ( real > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal > ( real > extend8495563244428889912nnreal ) > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001t__Real__Oreal_001t__Nat__Onat,type,
    indepe3400978226907914954al_nat: sigma_measure_real > sigma_measure_nat > ( real > nat ) > sigma_measure_nat > ( real > nat ) > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001t__Real__Oreal_001t__Real__Oreal,type,
    indepe3760321310464026790l_real: sigma_measure_real > sigma_measure_real > ( real > real ) > sigma_measure_real > ( real > real ) > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001t__Real__Oreal_001tf__a,type,
    indepe365082296117321348real_a: sigma_measure_real > sigma_measure_a > ( real > a ) > sigma_measure_a > ( real > a ) > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001t__Real__Oreal_001tf__b,type,
    indepe365082296117321349real_b: sigma_measure_real > sigma_measure_b > ( real > b ) > sigma_measure_b > ( real > b ) > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001tf__a_001t__Extended____Nonnegative____Real__Oennreal,type,
    indepe3534117692041274858nnreal: sigma_measure_a > sigma_7234349610311085201nnreal > ( a > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal > ( a > extend8495563244428889912nnreal ) > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001tf__a_001t__Real__Oreal,type,
    indepe8958435565499147358a_real: sigma_measure_a > sigma_measure_real > ( a > real ) > sigma_measure_real > ( a > real ) > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001tf__b_001t__Extended____Nonnegative____Real__Oennreal,type,
    indepe430593609325189611nnreal: sigma_measure_b > sigma_7234349610311085201nnreal > ( b > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal > ( b > extend8495563244428889912nnreal ) > $o ).

thf(sy_c_Independent__Family_Oprob__space_Otail__events_001t__Real__Oreal_001t__Nat__Onat,type,
    indepe6490915699233070710al_nat: sigma_measure_real > ( nat > set_set_real ) > set_set_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_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Real__Oreal_J,type,
    inf_inf_set_real: set_real > set_real > set_real ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Real__Oreal_J,type,
    sup_sup_set_real: set_real > set_real > set_real ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Omeasure__to__qbs_001_Eo,type,
    measur2705496967258476524_qbs_o: sigma_measure_o > quasi_borel_o ).

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__Nat__Onat,type,
    measur4416158800429964412bs_nat: sigma_measure_nat > quasi_borel_nat ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Omeasure__to__qbs_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
    measur2011440102552004913t_real: sigma_5310753476256395226t_real > quasi_7702169472194803971t_real ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Omeasure__to__qbs_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    measur3029786804310284173l_real: sigma_2308072346491277622l_real > quasi_5476411728266286559l_real ).

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_Omeasure__to__qbs_001tf__b,type,
    measur6507891955840068947_qbs_b: sigma_measure_b > quasi_borel_b ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Oqbs__to__measure_001_Eo,type,
    measur2926627334652526644sure_o: quasi_borel_o > sigma_measure_o ).

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__Nat__Onat,type,
    measur7418878410283781684re_nat: quasi_borel_nat > sigma_measure_nat ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Oqbs__to__measure_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
    measur6583539744506187129t_real: quasi_7702169472194803971t_real > sigma_5310753476256395226t_real ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Oqbs__to__measure_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    measur6658399249024522197l_real: quasi_5476411728266286559l_real > sigma_2308072346491277622l_real ).

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__QuasiBorel__Adjunction_Oqbs__to__measure_001tf__b,type,
    measur7857763439677503899sure_b: quasi_borel_b > sigma_measure_b ).

thf(sy_c_Measure__Space_Odistr_001_Eo_001_Eo,type,
    measure_distr_o_o: sigma_measure_o > sigma_measure_o > ( $o > $o ) > sigma_measure_o ).

thf(sy_c_Measure__Space_Odistr_001_Eo_001t__Real__Oreal,type,
    measure_distr_o_real: sigma_measure_o > sigma_measure_real > ( $o > real ) > sigma_measure_real ).

thf(sy_c_Measure__Space_Odistr_001_Eo_001tf__a,type,
    measure_distr_o_a: sigma_measure_o > sigma_measure_a > ( $o > a ) > sigma_measure_a ).

thf(sy_c_Measure__Space_Odistr_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__Nat__Onat_001t__Real__Oreal,type,
    measur4910586471993217526t_real: sigma_measure_nat > sigma_measure_real > ( nat > real ) > sigma_measure_real ).

thf(sy_c_Measure__Space_Odistr_001t__Nat__Onat_001tf__a,type,
    measure_distr_nat_a: sigma_measure_nat > sigma_measure_a > ( nat > a ) > sigma_measure_a ).

thf(sy_c_Measure__Space_Odistr_001t__Real__Oreal_001_Eo,type,
    measure_distr_real_o: sigma_measure_real > sigma_measure_o > ( real > $o ) > sigma_measure_o ).

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__Nat__Onat,type,
    measur254523008267340406al_nat: sigma_measure_real > sigma_measure_nat > ( real > nat ) > sigma_measure_nat ).

thf(sy_c_Measure__Space_Odistr_001t__Real__Oreal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
    measur3655679046452803511t_real: sigma_measure_real > sigma_5310753476256395226t_real > ( real > produc7716430852924023517t_real ) > sigma_5310753476256395226t_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__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_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_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__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_001tf__a,type,
    measure_distr_a_a: sigma_measure_a > sigma_measure_a > ( a > a ) > sigma_measure_a ).

thf(sy_c_Measure__Space_Odistr_001tf__b_001t__Extended____Nonnegative____Real__Oennreal,type,
    measur1735912521085800255nnreal: sigma_measure_b > sigma_7234349610311085201nnreal > ( b > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal ).

thf(sy_c_Measure__Space_Ofinite__measure_001_Eo,type,
    measur2447921437955784316sure_o: sigma_measure_o > $o ).

thf(sy_c_Measure__Space_Ofinite__measure_001t__Complex__Ocomplex,type,
    measur5795638154895664842omplex: sigma_3077487657436305159omplex > $o ).

thf(sy_c_Measure__Space_Ofinite__measure_001t__Extended____Nonnegative____Real__Oennreal,type,
    measur8478876643349974356nnreal: sigma_7234349610311085201nnreal > $o ).

thf(sy_c_Measure__Space_Ofinite__measure_001t__Nat__Onat,type,
    measur8338831127414845932re_nat: sigma_measure_nat > $o ).

thf(sy_c_Measure__Space_Ofinite__measure_001t__Real__Oreal,type,
    measur3606880022600206024e_real: sigma_measure_real > $o ).

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

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

thf(sy_c_Measure__Space_Oincreasing_001t__Real__Oreal_001t__Real__Oreal,type,
    measur4480787322886042509l_real: set_set_real > ( set_real > real ) > $o ).

thf(sy_c_Measure__Space_Osigma__finite__measure_001_Eo,type,
    measur1827666076404920889sure_o: sigma_measure_o > $o ).

thf(sy_c_Measure__Space_Osigma__finite__measure_001t__Complex__Ocomplex,type,
    measur2063495708654786125omplex: sigma_3077487657436305159omplex > $o ).

thf(sy_c_Measure__Space_Osigma__finite__measure_001t__Extended____Nonnegative____Real__Oennreal,type,
    measur6426964080664357591nnreal: sigma_7234349610311085201nnreal > $o ).

thf(sy_c_Measure__Space_Osigma__finite__measure_001t__Nat__Onat,type,
    measur8258956421386577775re_nat: sigma_measure_nat > $o ).

thf(sy_c_Measure__Space_Osigma__finite__measure_001t__Real__Oreal,type,
    measur487378040549452491e_real: sigma_measure_real > $o ).

thf(sy_c_Measure__Space_Osigma__finite__measure_001tf__a,type,
    measur4308613598931908895sure_a: sigma_measure_a > $o ).

thf(sy_c_Measure__Space_Osigma__finite__measure_001tf__b,type,
    measur4308613598931908896sure_b: sigma_measure_b > $o ).

thf(sy_c_Monad__QuasiBorel_Oin__MPx_001_Eo,type,
    monad_in_MPx_o: quasi_borel_o > ( real > probab1241297377463522905pace_o ) > $o ).

thf(sy_c_Monad__QuasiBorel_Oin__MPx_001t__Complex__Ocomplex,type,
    monad_in_MPx_complex: quasi_borel_complex > ( real > probab3778977310982246339omplex ) > $o ).

thf(sy_c_Monad__QuasiBorel_Oin__MPx_001t__Extended____Nonnegative____Real__Oennreal,type,
    monad_915879624692006445nnreal: quasi_9015997321629101608nnreal > ( real > probab1359147627358338381nnreal ) > $o ).

thf(sy_c_Monad__QuasiBorel_Oin__MPx_001t__Nat__Onat,type,
    monad_in_MPx_nat: quasi_borel_nat > ( real > probab8370124847414953445ce_nat ) > $o ).

thf(sy_c_Monad__QuasiBorel_Oin__MPx_001t__Real__Oreal,type,
    monad_in_MPx_real: quasi_borel_real > ( real > probab8009751763329705409e_real ) > $o ).

thf(sy_c_Monad__QuasiBorel_Oin__MPx_001tf__a,type,
    monad_in_MPx_a: quasi_borel_a > ( real > probab4737552673497767871pace_a ) > $o ).

thf(sy_c_Monad__QuasiBorel_Oin__MPx_001tf__b,type,
    monad_in_MPx_b: quasi_borel_b > ( real > probab4737552677800996672pace_b ) > $o ).

thf(sy_c_Monad__QuasiBorel_OmonadP__qbs_001_Eo,type,
    monad_monadP_qbs_o: quasi_borel_o > quasi_5673379896094141183pace_o ).

thf(sy_c_Monad__QuasiBorel_OmonadP__qbs_001t__Complex__Ocomplex,type,
    monad_3228211519047171924omplex: quasi_borel_complex > quasi_8687197021405177651omplex ).

thf(sy_c_Monad__QuasiBorel_OmonadP__qbs_001t__Extended____Nonnegative____Real__Oennreal,type,
    monad_8737719049617959390nnreal: quasi_9015997321629101608nnreal > quasi_2877558397483028925nnreal ).

thf(sy_c_Monad__QuasiBorel_OmonadP__qbs_001t__Nat__Onat,type,
    monad_monadP_qbs_nat: quasi_borel_nat > quasi_4026513206518478421ce_nat ).

thf(sy_c_Monad__QuasiBorel_OmonadP__qbs_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__b_J,type,
    monad_1549008831671149082pace_b: quasi_3431323906171122406pace_b > quasi_5637586813637594663pace_b ).

thf(sy_c_Monad__QuasiBorel_OmonadP__qbs_001t__Real__Oreal,type,
    monad_2887651017592114770s_real: quasi_borel_real > quasi_4837629875193058737e_real ).

thf(sy_c_Monad__QuasiBorel_OmonadP__qbs_001tf__a,type,
    monad_monadP_qbs_a: quasi_borel_a > quasi_3360289866124776421pace_a ).

thf(sy_c_Monad__QuasiBorel_OmonadP__qbs_001tf__b,type,
    monad_monadP_qbs_b: quasi_borel_b > quasi_3431323906171122406pace_b ).

thf(sy_c_Monad__QuasiBorel_OmonadP__qbs__MPx_001_Eo,type,
    monad_7120508265431748167_MPx_o: quasi_borel_o > set_re8832080236242745156pace_o ).

thf(sy_c_Monad__QuasiBorel_OmonadP__qbs__MPx_001t__Complex__Ocomplex,type,
    monad_1302594500181521855omplex: quasi_borel_complex > set_re8901213273154387908omplex ).

thf(sy_c_Monad__QuasiBorel_OmonadP__qbs__MPx_001t__Extended____Nonnegative____Real__Oennreal,type,
    monad_5049177628250775113nnreal: quasi_9015997321629101608nnreal > set_re7593075551311995214nnreal ).

thf(sy_c_Monad__QuasiBorel_OmonadP__qbs__MPx_001t__Nat__Onat,type,
    monad_6889595236306528993Px_nat: quasi_borel_nat > set_re2274660837269498342ce_nat ).

thf(sy_c_Monad__QuasiBorel_OmonadP__qbs__MPx_001t__Real__Oreal,type,
    monad_5590613879340176701x_real: quasi_borel_real > set_re1721576651363272770e_real ).

thf(sy_c_Monad__QuasiBorel_OmonadP__qbs__MPx_001tf__a,type,
    monad_4223963853064302125_MPx_a: quasi_borel_a > set_re6101349637198077994pace_a ).

thf(sy_c_Monad__QuasiBorel_OmonadP__qbs__MPx_001tf__b,type,
    monad_4223963853064302126_MPx_b: quasi_borel_b > set_re3580165118909042347pace_b ).

thf(sy_c_Monad__QuasiBorel_OmonadP__qbs__Pf_001_Eo_001_Eo,type,
    monad_3447778872307538587Pf_o_o: quasi_borel_o > quasi_borel_o > ( $o > $o ) > probab1241297377463522905pace_o > probab1241297377463522905pace_o ).

thf(sy_c_Monad__QuasiBorel_OmonadP__qbs__Pf_001_Eo_001t__Real__Oreal,type,
    monad_5553685203771961321o_real: quasi_borel_o > quasi_borel_real > ( $o > real ) > probab1241297377463522905pace_o > probab8009751763329705409e_real ).

thf(sy_c_Monad__QuasiBorel_OmonadP__qbs__Pf_001t__Nat__Onat_001t__Nat__Onat,type,
    monad_2263650980725743037at_nat: quasi_borel_nat > quasi_borel_nat > ( nat > nat ) > probab8370124847414953445ce_nat > probab8370124847414953445ce_nat ).

thf(sy_c_Monad__QuasiBorel_OmonadP__qbs__Pf_001t__Nat__Onat_001t__Real__Oreal,type,
    monad_6765637433395911193t_real: quasi_borel_nat > quasi_borel_real > ( nat > real ) > probab8370124847414953445ce_nat > probab8009751763329705409e_real ).

thf(sy_c_Monad__QuasiBorel_OmonadP__qbs__Pf_001t__Real__Oreal_001_Eo,type,
    monad_7062749578813130895real_o: quasi_borel_real > quasi_borel_o > ( real > $o ) > probab8009751763329705409e_real > probab1241297377463522905pace_o ).

thf(sy_c_Monad__QuasiBorel_OmonadP__qbs__Pf_001t__Real__Oreal_001t__Complex__Ocomplex,type,
    monad_4232241275811048823omplex: quasi_borel_real > quasi_borel_complex > ( real > complex ) > probab8009751763329705409e_real > probab3778977310982246339omplex ).

thf(sy_c_Monad__QuasiBorel_OmonadP__qbs__Pf_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal,type,
    monad_6772047417230001665nnreal: quasi_borel_real > quasi_9015997321629101608nnreal > ( real > extend8495563244428889912nnreal ) > probab8009751763329705409e_real > probab1359147627358338381nnreal ).

thf(sy_c_Monad__QuasiBorel_OmonadP__qbs__Pf_001t__Real__Oreal_001t__Nat__Onat,type,
    monad_2109573969670034073al_nat: quasi_borel_real > quasi_borel_nat > ( real > nat ) > probab8009751763329705409e_real > probab8370124847414953445ce_nat ).

thf(sy_c_Monad__QuasiBorel_OmonadP__qbs__Pf_001t__Real__Oreal_001t__Real__Oreal,type,
    monad_4235422856176591093l_real: quasi_borel_real > quasi_borel_real > ( real > real ) > probab8009751763329705409e_real > probab8009751763329705409e_real ).

thf(sy_c_Monad__QuasiBorel_OmonadP__qbs__Pf_001t__Real__Oreal_001tf__a,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Real__Oreal_Mt__Probability____Space____QuasiBorel__Oqbs____prob____space_I_Eo_J_J_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
    produc982178768907016120e_real: ( real > probab1241297377463522905pace_o ) > sigma_measure_real > produc4079060523694651464e_real ).

thf(sy_c_Product__Type_OPair_001_062_It__Real__Oreal_Mt__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Complex__Ocomplex_J_J_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
    produc4083023864984044190e_real: ( real > probab3778977310982246339omplex ) > sigma_measure_real > produc4770755024348780e_real ).

thf(sy_c_Product__Type_OPair_001_062_It__Real__Oreal_Mt__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Extended____Nonnegative____Real__Oennreal_J_J_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
    produc3526280104234912148e_real: ( real > probab1359147627358338381nnreal ) > sigma_measure_real > produc453011888977720162e_real ).

thf(sy_c_Product__Type_OPair_001_062_It__Real__Oreal_Mt__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Nat__Onat_J_J_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
    produc6214735565764730620e_real: ( real > probab8370124847414953445ce_nat ) > sigma_measure_real > produc7313919164992340170e_real ).

thf(sy_c_Product__Type_OPair_001_062_It__Real__Oreal_Mt__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J_J_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
    produc1095797480378445344e_real: ( real > probab8009751763329705409e_real ) > sigma_measure_real > produc7188788373697618030e_real ).

thf(sy_c_Product__Type_OPair_001_062_It__Real__Oreal_Mt__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__a_J_J_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
    produc2709302780738849234e_real: ( real > probab4737552673497767871pace_a ) > sigma_measure_real > produc2433583426017294818e_real ).

thf(sy_c_Product__Type_OPair_001_062_It__Real__Oreal_Mt__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__b_J_J_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
    produc2065420038842681233e_real: ( real > probab4737552677800996672pace_b ) > sigma_measure_real > produc1789700684121126817e_real ).

thf(sy_c_Product__Type_OPair_001_062_It__Real__Oreal_Mt__Real__Oreal_J_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
    produc1722724976708544245e_real: ( real > real ) > sigma_measure_real > produc7602496232715749699e_real ).

thf(sy_c_Product__Type_OPair_001_062_It__Real__Oreal_Mtf__a_J_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
    produc623176010801490259e_real: ( real > a ) > sigma_measure_real > produc725540845905733987e_real ).

thf(sy_c_Product__Type_OPair_001_062_It__Real__Oreal_Mtf__b_J_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
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    produc344325839068023049nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > produc7414223468410354641nnreal ).

thf(sy_c_Product__Type_OPair_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal,type,
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thf(sy_c_Product__Type_OPair_001t__Extended____Nonnegative____Real__Oennreal_001t__Set__Oset_It__Real__Oreal_J,type,
    produc3019872141242647603t_real: extend8495563244428889912nnreal > set_real > produc4039787570298959739t_real ).

thf(sy_c_Product__Type_OPair_001t__QuasiBorel__Oquasi____borel_I_Eo_J_001t__Product____Type__Oprod_I_062_It__Real__Oreal_M_Eo_J_Mt__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J,type,
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thf(sy_c_Product__Type_OPair_001t__QuasiBorel__Oquasi____borel_It__Complex__Ocomplex_J_001t__Product____Type__Oprod_I_062_It__Real__Oreal_Mt__Complex__Ocomplex_J_Mt__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J,type,
    produc7812373664507111416e_real: quasi_borel_complex > produc7114270865491226433e_real > produc3405552003670437440e_real ).

thf(sy_c_Product__Type_OPair_001t__QuasiBorel__Oquasi____borel_It__Extended____Nonnegative____Real__Oennreal_J_001t__Product____Type__Oprod_I_062_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J_Mt__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J,type,
    produc3311748387682954872e_real: quasi_9015997321629101608nnreal > produc1621850594562464567e_real > produc8502452040433673664e_real ).

thf(sy_c_Product__Type_OPair_001t__QuasiBorel__Oquasi____borel_It__Nat__Onat_J_001t__Product____Type__Oprod_I_062_It__Real__Oreal_Mt__Nat__Onat_J_Mt__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J,type,
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thf(sy_c_Product__Type_OPair_001t__QuasiBorel__Oquasi____borel_It__Probability____Space____QuasiBorel__Oqbs____prob____space_I_Eo_J_J_001t__Product____Type__Oprod_I_062_It__Real__Oreal_Mt__Probability____Space____QuasiBorel__Oqbs____prob____space_I_Eo_J_J_Mt__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J,type,
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thf(sy_c_Product__Type_OPair_001t__QuasiBorel__Oquasi____borel_It__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Complex__Ocomplex_J_J_001t__Product____Type__Oprod_I_062_It__Real__Oreal_Mt__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Complex__Ocomplex_J_J_Mt__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J,type,
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thf(sy_c_Product__Type_OPair_001t__QuasiBorel__Oquasi____borel_It__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Nat__Onat_J_J_001t__Product____Type__Oprod_I_062_It__Real__Oreal_Mt__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Nat__Onat_J_J_Mt__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J,type,
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thf(sy_c_Product__Type_OPair_001t__QuasiBorel__Oquasi____borel_It__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J_J_001t__Product____Type__Oprod_I_062_It__Real__Oreal_Mt__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J_J_Mt__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J,type,
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thf(sy_c_Product__Type_OPair_001t__QuasiBorel__Oquasi____borel_It__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__a_J_J_001t__Product____Type__Oprod_I_062_It__Real__Oreal_Mt__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__a_J_J_Mt__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J,type,
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thf(sy_c_Product__Type_OPair_001t__QuasiBorel__Oquasi____borel_It__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__b_J_J_001t__Product____Type__Oprod_I_062_It__Real__Oreal_Mt__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__b_J_J_Mt__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J,type,
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thf(sy_c_Product__Type_OPair_001t__QuasiBorel__Oquasi____borel_It__Real__Oreal_J_001t__Product____Type__Oprod_I_062_It__Real__Oreal_Mt__Real__Oreal_J_Mt__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J,type,
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thf(sy_c_Product__Type_OPair_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,
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thf(sy_c_Product__Type_OPair_001t__QuasiBorel__Oquasi____borel_Itf__b_J_001t__Product____Type__Oprod_I_062_It__Real__Oreal_Mtf__b_J_Mt__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J,type,
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thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Complex__Ocomplex_J_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Complex__Ocomplex_J_J,type,
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thf(sy_c_QuasiBorel_Oqbs__Mx_001_Eo,type,
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thf(sy_c_QuasiBorel_Oqbs__Mx_001t__Complex__Ocomplex,type,
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thf(sy_c_QuasiBorel_Oqbs__Mx_001tf__a,type,
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thf(sy_c_QuasiBorel_Oqbs__Mx_001tf__b,type,
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    qbs_morphism_nat_nat: quasi_borel_nat > quasi_borel_nat > set_nat_nat ).

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    qbs_mo2896996399151425555t_real: quasi_borel_nat > quasi_7702169472194803971t_real > set_na4391433010951109924t_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Nat__Onat_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    qbs_mo6520021129653552495l_real: quasi_borel_nat > quasi_5476411728266286559l_real > set_na5185791421324906752l_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Nat__Onat_001t__Real__Oreal,type,
    qbs_mo2000642995705457910t_real: quasi_borel_nat > quasi_borel_real > set_nat_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_I_Eo_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_I_Eo_J,type,
    qbs_mo5587578735296274488pace_o: quasi_5673379896094141183pace_o > quasi_5673379896094141183pace_o > set_Pr5902255448072599515pace_o ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__b_J_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__b_J,type,
    qbs_mo3188246409730626167pace_b: quasi_5637586813637594663pace_b > quasi_3431323906171122406pace_b > set_Pr9001386418803518490pace_b ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_I_Eo_J,type,
    qbs_mo6162687862215152862pace_o: quasi_4837629875193058737e_real > quasi_5673379896094141183pace_o > set_Pr5085879303345694575pace_o ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Complex__Ocomplex_J,type,
    qbs_mo3866313822246798974omplex: quasi_4837629875193058737e_real > quasi_8687197021405177651omplex > set_Pr1361619117094143577omplex ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Extended____Nonnegative____Real__Oennreal_J,type,
    qbs_mo2680574766477480456nnreal: quasi_4837629875193058737e_real > quasi_2877558397483028925nnreal > set_Pr3430538925378988515nnreal ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Nat__Onat_J,type,
    qbs_mo3330508391926297248ce_nat: quasi_4837629875193058737e_real > quasi_4026513206518478421ce_nat > set_Pr1990545267392273019ce_nat ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J,type,
    qbs_mo7502123496470045948e_real: quasi_4837629875193058737e_real > quasi_4837629875193058737e_real > set_Pr7678246286638956119e_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__a_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Extended____Nonnegative____Real__Oennreal_J,type,
    qbs_mo6857830617226182420nnreal: quasi_3360289866124776421pace_a > quasi_2877558397483028925nnreal > set_Pr8491145367753866205nnreal ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__a_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J,type,
    qbs_mo3705727718672329736e_real: quasi_3360289866124776421pace_a > quasi_4837629875193058737e_real > set_Pr1457397031307248465e_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__b_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Extended____Nonnegative____Real__Oennreal_J,type,
    qbs_mo3432782386221347477nnreal: quasi_3431323906171122406pace_b > quasi_2877558397483028925nnreal > set_Pr3292545666760900318nnreal ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__b_J_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__b_J,type,
    qbs_mo5665329929810820088pace_b: quasi_3431323906171122406pace_b > quasi_3431323906171122406pace_b > set_Pr4612125890992911643pace_b ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
    qbs_mo1022283140530407736t_real: quasi_7702169472194803971t_real > quasi_7702169472194803971t_real > set_Pr1824005347406644699t_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J_001t__Real__Oreal,type,
    qbs_mo8506632985670901265l_real: quasi_7702169472194803971t_real > quasi_borel_real > set_Pr618161673012039770l_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Real__Oreal_001_Eo,type,
    qbs_morphism_real_o: quasi_borel_real > quasi_borel_o > set_real_o ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Real__Oreal_001t__Complex__Ocomplex,type,
    qbs_mo6067097710682130004omplex: quasi_borel_real > quasi_borel_complex > set_real_complex ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal,type,
    qbs_mo1317719164804411614nnreal: quasi_borel_real > quasi_9015997321629101608nnreal > set_re5328672808648366137nnreal ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Real__Oreal_001t__Nat__Onat,type,
    qbs_mo6567951568834356598al_nat: quasi_borel_real > quasi_borel_nat > set_real_nat ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Real__Oreal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
    qbs_mo3601394221918002359t_real: quasi_borel_real > quasi_7702169472194803971t_real > set_re7361189984880485448t_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Real__Oreal_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    qbs_mo6845038431372961811l_real: quasi_borel_real > quasi_5476411728266286559l_real > set_re3939601646622484004l_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Real__Oreal_001t__Real__Oreal,type,
    qbs_mo5229770564518008146l_real: quasi_borel_real > quasi_borel_real > set_real_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Real__Oreal_001tf__a,type,
    qbs_morphism_real_a: quasi_borel_real > quasi_borel_a > set_real_a ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Real__Oreal_001tf__b,type,
    qbs_morphism_real_b: quasi_borel_real > quasi_borel_b > set_real_b ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__a_001_Eo,type,
    qbs_morphism_a_o: quasi_borel_a > quasi_borel_o > set_a_o ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__a_001t__Extended____Nonnegative____Real__Oennreal,type,
    qbs_mo1434458643421888574nnreal: quasi_borel_a > quasi_9015997321629101608nnreal > set_a_7161065143582548615nnreal ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__a_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_I_Eo_J,type,
    qbs_mo1614836618362447827pace_o: quasi_borel_a > quasi_5673379896094141183pace_o > set_a_2487995229096676726pace_o ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__a_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Complex__Ocomplex_J,type,
    qbs_mo1042063135579555017omplex: quasi_borel_a > quasi_8687197021405177651omplex > set_a_6212107248488283666omplex ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__a_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Extended____Nonnegative____Real__Oennreal_J,type,
    qbs_mo990775079526714963nnreal: quasi_borel_a > quasi_2877558397483028925nnreal > set_a_7827055764996705180nnreal ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__a_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Nat__Onat_J,type,
    qbs_mo7775059402295510763ce_nat: quasi_borel_a > quasi_4026513206518478421ce_nat > set_a_1288009948629238836ce_nat ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__a_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J,type,
    qbs_mo1035972366611710919e_real: quasi_borel_a > quasi_4837629875193058737e_real > set_a_741245887377775504e_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__a_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__a_J,type,
    qbs_mo6277074938832049209pace_a: quasi_borel_a > quasi_3360289866124776421pace_a > set_a_5785126655188516188pace_a ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__a_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__b_J,type,
    qbs_mo6277074943135278010pace_b: quasi_borel_a > quasi_3431323906171122406pace_b > set_a_3263942136899480541pace_b ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__a_001t__Real__Oreal,type,
    qbs_morphism_a_real: quasi_borel_a > quasi_borel_real > set_a_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__a_001tf__a,type,
    qbs_morphism_a_a: quasi_borel_a > quasi_borel_a > set_a_a ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__a_001tf__b,type,
    qbs_morphism_a_b: quasi_borel_a > quasi_borel_b > set_a_b ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__b_001t__Extended____Nonnegative____Real__Oennreal,type,
    qbs_mo7554306597560579135nnreal: quasi_borel_b > quasi_9015997321629101608nnreal > set_b_6825823330181178888nnreal ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__b_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_I_Eo_J,type,
    qbs_mo7566660958362347922pace_o: quasi_borel_b > quasi_5673379896094141183pace_o > set_b_5938708947977461685pace_o ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__b_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Complex__Ocomplex_J,type,
    qbs_mo8077574125540273866omplex: quasi_borel_b > quasi_8687197021405177651omplex > set_b_9144620356085364627omplex ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__b_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Extended____Nonnegative____Real__Oennreal_J,type,
    qbs_mo7003404251872637524nnreal: quasi_borel_b > quasi_2877558397483028925nnreal > set_b_4402007533991870237nnreal ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__b_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Nat__Onat_J,type,
    qbs_mo4747267137793946348ce_nat: quasi_borel_b > quasi_4026513206518478421ce_nat > set_b_1159273042777767861ce_nat ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__b_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J,type,
    qbs_mo907235460760239944e_real: quasi_borel_b > quasi_4837629875193058737e_real > set_b_2833242828649588369e_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__b_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__a_J,type,
    qbs_mo504416620858058360pace_a: quasi_borel_b > quasi_3360289866124776421pace_a > set_b_2357908436866916379pace_a ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__b_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__b_J,type,
    qbs_mo504416625161287161pace_b: quasi_borel_b > quasi_3431323906171122406pace_b > set_b_9060095955432656540pace_b ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__b_001tf__a,type,
    qbs_morphism_b_a: quasi_borel_b > quasi_borel_a > set_b_a ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__b_001tf__b,type,
    qbs_morphism_b_b: quasi_borel_b > quasi_borel_b > set_b_b ).

thf(sy_c_QuasiBorel_Oquasi__borel_ORep__quasi__borel_001_Eo,type,
    quasi_3788804942992426063orel_o: quasi_borel_o > produc679738297434340018real_o ).

thf(sy_c_QuasiBorel_Oquasi__borel_ORep__quasi__borel_001t__Complex__Ocomplex,type,
    quasi_5592172293381431735omplex: quasi_borel_complex > produc6484079244375271668omplex ).

thf(sy_c_QuasiBorel_Oquasi__borel_ORep__quasi__borel_001t__Extended____Nonnegative____Real__Oennreal,type,
    quasi_5951729137868634689nnreal: quasi_9015997321629101608nnreal > produc3431711466136494088nnreal ).

thf(sy_c_QuasiBorel_Oquasi__borel_ORep__quasi__borel_001t__Nat__Onat,type,
    quasi_2638801612224710361el_nat: quasi_borel_nat > produc3490415624965685560al_nat ).

thf(sy_c_QuasiBorel_Oquasi__borel_ORep__quasi__borel_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__a_J,type,
    quasi_159326145307389174pace_a: quasi_3360289866124776421pace_a > produc5176701162915795698pace_a ).

thf(sy_c_QuasiBorel_Oquasi__borel_ORep__quasi__borel_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__b_J,type,
    quasi_159326149610617975pace_b: quasi_3431323906171122406pace_b > produc7033823874502446642pace_b ).

thf(sy_c_QuasiBorel_Oquasi__borel_ORep__quasi__borel_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
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thf(sy_c_QuasiBorel_Oquasi__borel_ORep__quasi__borel_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
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thf(sy_c_QuasiBorel_Oquasi__borel_ORep__quasi__borel_001t__Real__Oreal,type,
    quasi_4307383193256703285l_real: quasi_borel_real > produc1975794761236386672l_real ).

thf(sy_c_Set_OCollect_001_062_I_Eo_M_Eo_J,type,
    collect_o_o: ( ( $o > $o ) > $o ) > set_o_o ).

thf(sy_c_Set_OCollect_001_062_It__Real__Oreal_M_Eo_J,type,
    collect_real_o: ( ( real > $o ) > $o ) > set_real_o ).

thf(sy_c_Set_OCollect_001_062_It__Real__Oreal_Mt__Complex__Ocomplex_J,type,
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thf(sy_c_Set_OCollect_001_062_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
    collec9130413544115709400nnreal: ( ( real > extend8495563244428889912nnreal ) > $o ) > set_re5328672808648366137nnreal ).

thf(sy_c_Set_OCollect_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
    collect_real_nat: ( ( real > nat ) > $o ) > set_real_nat ).

thf(sy_c_Set_OCollect_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
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thf(sy_c_Set_OCollect_001_062_Itf__a_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
    collec5472405872578835474nnreal: ( ( a > extend8495563244428889912nnreal ) > $o ) > set_a_7161065143582548615nnreal ).

thf(sy_c_Set_OCollect_001_062_Itf__a_Mt__Real__Oreal_J,type,
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thf(sy_c_Set_OCollect_001_062_Itf__b_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
    collec2368881789862750227nnreal: ( ( b > extend8495563244428889912nnreal ) > $o ) > set_b_6825823330181178888nnreal ).

thf(sy_c_Set_OCollect_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,
    collec2245114308608258001e_real: ( produc6543235832880896358e_real > $o ) > set_Pr7780167738718111686e_real ).

thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
    collect_real: ( real > $o ) > set_real ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Real__Oreal_J,type,
    collect_set_real: ( set_real > $o ) > set_set_real ).

thf(sy_c_Sigma__Algebra_Omeasurable_001_062_I_Eo_M_Eo_J_001t__Sigma____Algebra__Omeasure_I_Eo_J,type,
    sigma_6141153081871920972sure_o: sigma_measure_o_o > sigma_8601284680732870894sure_o > set_o_8108268051690476111sure_o ).

thf(sy_c_Sigma__Algebra_Omeasurable_001_062_I_Eo_M_Eo_J_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
    sigma_9205866881352898830e_real: sigma_measure_o_o > sigma_8927737637348964610e_real > set_o_802602961120082807e_real ).

thf(sy_c_Sigma__Algebra_Omeasurable_001_062_It__Real__Oreal_M_Eo_J_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
    sigma_2943736883462281714e_real: sigma_measure_real_o > sigma_8927737637348964610e_real > set_re8722879854129957181e_real ).

thf(sy_c_Sigma__Algebra_Omeasurable_001_062_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
    sigma_2864275607600219376e_real: sigma_5394977995791401948nnreal > sigma_8927737637348964610e_real > set_re6081088085470630745e_real ).

thf(sy_c_Sigma__Algebra_Omeasurable_001_062_It__Real__Oreal_Mt__Nat__Onat_J_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
    sigma_4915400539034359384e_real: sigma_6586288717683155060al_nat > sigma_8927737637348964610e_real > set_re2278671248297355457e_real ).

thf(sy_c_Sigma__Algebra_Omeasurable_001_062_Itf__a_Mt__Extended____Nonnegative____Real__Oennreal_J_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
    sigma_4648059183711430126e_real: sigma_3251133792989648932nnreal > sigma_8927737637348964610e_real > set_a_9220888951947300665e_real ).

thf(sy_c_Sigma__Algebra_Omeasurable_001_062_Itf__a_Mt__Real__Oreal_J_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
    sigma_5461457353648996602e_real: sigma_measure_a_real > sigma_8927737637348964610e_real > set_a_2480322683688530885e_real ).

thf(sy_c_Sigma__Algebra_Omeasurable_001_062_Itf__b_Mt__Extended____Nonnegative____Real__Oennreal_J_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
    sigma_6211663741840559789e_real: sigma_2915891979588279205nnreal > sigma_8927737637348964610e_real > set_b_1222426755260090488e_real ).

thf(sy_c_Sigma__Algebra_Omeasurable_001_Eo_001_Eo,type,
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thf(sy_c_Sigma__Algebra_Omeasurable_001_Eo_001t__Extended____Nonnegative____Real__Oennreal,type,
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thf(sy_c_Sigma__Algebra_Omeasurable_001_Eo_001t__Nat__Onat,type,
    sigma_1999164137574644376_o_nat: sigma_measure_o > sigma_measure_nat > set_o_nat ).

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    member_nat: nat > set_nat > $o ).

thf(sy_c_member_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_I_Eo_J,type,
    member8440415008478396930pace_o: probab1241297377463522905pace_o > set_Pr6115912531763461305pace_o > $o ).

thf(sy_c_member_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Nat__Onat_J,type,
    member1134292470272169596ce_nat: probab8370124847414953445ce_nat > set_Pr2653088245073381147ce_nat > $o ).

thf(sy_c_member_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_It__Real__Oreal_J,type,
    member7522313270622477016e_real: probab8009751763329705409e_real > set_Pr7030517981848544119e_real > $o ).

thf(sy_c_member_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__a_J,type,
    member6485455074645559016pace_a: probab4737552673497767871pace_a > set_Pr8456636223741192351pace_a > $o ).

thf(sy_c_member_001t__Probability____Space____QuasiBorel__Oqbs____prob____space_Itf__b_J,type,
    member6485455078948787817pace_b: probab4737552677800996672pace_b > set_Pr8527670263787538336pace_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
    member557208447399453958t_real: produc7716430852924023517t_real > set_Pr320017278500174781t_real > $o ).

thf(sy_c_member_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,
    member6844354795726785935e_real: produc6543235832880896358e_real > set_Pr7780167738718111686e_real > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__QuasiBorel__Oquasi____borel_Itf__b_J_Mt__Product____Type__Oprod_I_062_It__Real__Oreal_Mtf__b_J_Mt__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J_J,type,
    member379764716372043661e_real: produc78645753526154084e_real > set_Pr6060481869236011716e_real > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    member7849222048561428706l_real: produc2422161461964618553l_real > set_Pr6218003697084177305l_real > $o ).

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

thf(sy_c_member_001t__Set__Oset_It__Real__Oreal_J,type,
    member_set_real: set_real > set_set_real > $o ).

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
    member_set_set_real: set_set_real > set_set_set_real > $o ).

thf(sy_c_member_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
    member4553183543495551918e_real: sigma_measure_real > set_Si6059263944882162789e_real > $o ).

thf(sy_c_member_001tf__a,type,
    member_a: a > set_a > $o ).

thf(sy_c_member_001tf__b,type,
    member_b: b > set_b > $o ).

thf(sy_v_X,type,
    x: quasi_borel_a ).

thf(sy_v_Y,type,
    y: quasi_borel_b ).

thf(sy_v__092_060alpha_062____,type,
    alpha: real > a ).

thf(sy_v__092_060beta_062____,type,
    beta: real > b ).

thf(sy_v__092_060mu_062____,type,
    mu: sigma_measure_real ).

thf(sy_v_f,type,
    f: a > probab4737552677800996672pace_b ).

thf(sy_v_g,type,
    g: b > extend8495563244428889912nnreal ).

thf(sy_v_h____,type,
    h: real > sigma_measure_real ).

thf(sy_v_s,type,
    s: probab4737552673497767871pace_a ).

% Relevant facts (1279)
thf(fact_0_hb_I2_J,axiom,
    member2630560753458908601e_real @ h @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) ).

% hb(2)
thf(fact_1_hb_I1_J,axiom,
    member_real_b @ beta @ ( qbs_Mx_b @ y ) ).

% hb(1)
thf(fact_2_hb_I3_J,axiom,
    ( ( comp_a6868616473110226257b_real @ f @ alpha )
    = ( ^ [R: real] : ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ y @ ( produc4478270668571743890e_real @ beta @ ( h @ R ) ) ) ) ) ) ).

% hb(3)
thf(fact_3_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_4_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_5_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_6_qp_Omeasurable__finite__borel,axiom,
    ! [F: real > nat] :
      ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) )
     => ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ mu @ borel_8449730974584783410el_nat ) ) ) ).

% qp.measurable_finite_borel
thf(fact_7_qp_Omeasurable__finite__borel,axiom,
    ! [F: real > $o] :
      ( ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ borel_5078946678739801102l_real @ borel_5500255247093592246orel_o ) )
     => ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ mu @ borel_5500255247093592246orel_o ) ) ) ).

% qp.measurable_finite_borel
thf(fact_8_qp_Ofinite__borel__measure__axioms,axiom,
    distri7943378551711771532easure @ mu ).

% qp.finite_borel_measure_axioms
thf(fact_9_qp_Osubprob__space__axioms,axiom,
    giry_s8208748868292234104e_real @ mu ).

% qp.subprob_space_axioms
thf(fact_10_qp_Oreal__distribution__axioms,axiom,
    distri2809703520229113005bution @ mu ).

% qp.real_distribution_axioms
thf(fact_11_qp_Oprob__space__axioms,axiom,
    probab535871623910865577e_real @ mu ).

% qp.prob_space_axioms
thf(fact_12_qp_Osigma__finite__measure__axioms,axiom,
    measur487378040549452491e_real @ mu ).

% qp.sigma_finite_measure_axioms
thf(fact_13_qp_Ofinite__measure__axioms,axiom,
    measur3606880022600206024e_real @ mu ).

% qp.finite_measure_axioms
thf(fact_14_qp_Oindep__sets__cong,axiom,
    ! [I: set_o_o,J: set_o_o,F2: ( $o > $o ) > set_set_real,G: ( $o > $o ) > set_set_real] :
      ( ( I = J )
     => ( ! [I2: $o > $o] :
            ( ( member_o_o @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe6032362129116668415al_o_o @ mu @ F2 @ I )
          = ( indepe6032362129116668415al_o_o @ mu @ G @ J ) ) ) ) ).

% qp.indep_sets_cong
thf(fact_15_qp_Oindep__sets__cong,axiom,
    ! [I: set_b_6825823330181178888nnreal,J: set_b_6825823330181178888nnreal,F2: ( b > extend8495563244428889912nnreal ) > set_set_real,G: ( b > extend8495563244428889912nnreal ) > set_set_real] :
      ( ( I = J )
     => ( ! [I2: b > extend8495563244428889912nnreal] :
            ( ( member6418304549040442065nnreal @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe4831269413077975822nnreal @ mu @ F2 @ I )
          = ( indepe4831269413077975822nnreal @ mu @ G @ J ) ) ) ) ).

% qp.indep_sets_cong
thf(fact_16_qp_Oindep__sets__cong,axiom,
    ! [I: set_a_3263942136899480541pace_b,J: set_a_3263942136899480541pace_b,F2: ( a > probab4737552677800996672pace_b ) > set_set_real,G: ( a > probab4737552677800996672pace_b ) > set_set_real] :
      ( ( I = J )
     => ( ! [I2: a > probab4737552677800996672pace_b] :
            ( ( member7340901614391157822pace_b @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe1711633972345727297pace_b @ mu @ F2 @ I )
          = ( indepe1711633972345727297pace_b @ mu @ G @ J ) ) ) ) ).

% qp.indep_sets_cong
thf(fact_17_qp_Oindep__sets__cong,axiom,
    ! [I: set_a_7161065143582548615nnreal,J: set_a_7161065143582548615nnreal,F2: ( a > extend8495563244428889912nnreal ) > set_set_real,G: ( a > extend8495563244428889912nnreal ) > set_set_real] :
      ( ( I = J )
     => ( ! [I2: a > extend8495563244428889912nnreal] :
            ( ( member298456594901751504nnreal @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe7934793495794061069nnreal @ mu @ F2 @ I )
          = ( indepe7934793495794061069nnreal @ mu @ G @ J ) ) ) ) ).

% qp.indep_sets_cong
thf(fact_18_qp_Oindep__sets__cong,axiom,
    ! [I: set_a_real,J: set_a_real,F2: ( a > real ) > set_set_real,G: ( a > real ) > set_set_real] :
      ( ( I = J )
     => ( ! [I2: a > real] :
            ( ( member_a_real @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe7634599452152747905a_real @ mu @ F2 @ I )
          = ( indepe7634599452152747905a_real @ mu @ G @ J ) ) ) ) ).

% qp.indep_sets_cong
thf(fact_19_qp_Oindep__sets__cong,axiom,
    ! [I: set_real_o,J: set_real_o,F2: ( real > $o ) > set_set_real,G: ( real > $o ) > set_set_real] :
      ( ( I = J )
     => ( ! [I2: real > $o] :
            ( ( member_real_o @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe8014066527605497865real_o @ mu @ F2 @ I )
          = ( indepe8014066527605497865real_o @ mu @ G @ J ) ) ) ) ).

% qp.indep_sets_cong
thf(fact_20_qp_Oindep__sets__cong,axiom,
    ! [I: set_real_nat,J: set_real_nat,F2: ( real > nat ) > set_set_real,G: ( real > nat ) > set_set_real] :
      ( ( I = J )
     => ( ! [I2: real > nat] :
            ( ( member_real_nat @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe7235330332140030005al_nat @ mu @ F2 @ I )
          = ( indepe7235330332140030005al_nat @ mu @ G @ J ) ) ) ) ).

% qp.indep_sets_cong
thf(fact_21_qp_Oindep__sets__cong,axiom,
    ! [I: set_re5328672808648366137nnreal,J: set_re5328672808648366137nnreal,F2: ( real > extend8495563244428889912nnreal ) > set_set_real,G: ( real > extend8495563244428889912nnreal ) > set_set_real] :
      ( ( I = J )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe4643477836443271069nnreal @ mu @ F2 @ I )
          = ( indepe4643477836443271069nnreal @ mu @ G @ J ) ) ) ) ).

% qp.indep_sets_cong
thf(fact_22_qp_Oindep__sets__cong,axiom,
    ! [I: set_real_complex,J: set_real_complex,F2: ( real > complex ) > set_set_real,G: ( real > complex ) > set_set_real] :
      ( ( I = J )
     => ( ! [I2: real > complex] :
            ( ( member_real_complex @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe6244138955099140627omplex @ mu @ F2 @ I )
          = ( indepe6244138955099140627omplex @ mu @ G @ J ) ) ) ) ).

% qp.indep_sets_cong
thf(fact_23_qp_Oindep__sets__cong,axiom,
    ! [I: set_real_real,J: set_real_real,F2: ( real > real ) > set_set_real,G: ( real > real ) > set_set_real] :
      ( ( I = J )
     => ( ! [I2: real > real] :
            ( ( member_real_real @ I2 @ I )
           => ( ( F2 @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe4905764430350991377l_real @ mu @ F2 @ I )
          = ( indepe4905764430350991377l_real @ mu @ G @ J ) ) ) ) ).

% qp.indep_sets_cong
thf(fact_24_qp_Oindep__var__rv1,axiom,
    ! [S: sigma_measure_o,X: real > $o,T: sigma_measure_o,Y: real > $o] :
      ( ( indepe495429349782043166real_o @ mu @ S @ X @ T @ Y )
     => ( member_real_o @ X @ ( sigma_3939073009482781210real_o @ mu @ S ) ) ) ).

% qp.indep_var_rv1
thf(fact_25_qp_Oindep__var__rv1,axiom,
    ! [S: sigma_measure_nat,X: real > nat,T: sigma_measure_nat,Y: real > nat] :
      ( ( indepe3400978226907914954al_nat @ mu @ S @ X @ T @ Y )
     => ( member_real_nat @ X @ ( sigma_6315060578831106510al_nat @ mu @ S ) ) ) ).

% qp.indep_var_rv1
thf(fact_26_qp_Oindep__var__rv1,axiom,
    ! [S: sigma_7234349610311085201nnreal,X: real > extend8495563244428889912nnreal,T: sigma_7234349610311085201nnreal,Y: real > extend8495563244428889912nnreal] :
      ( ( indepe6767359503340752434nnreal @ mu @ S @ X @ T @ Y )
     => ( member2919562650594848410nnreal @ X @ ( sigma_9017504469962657078nnreal @ mu @ S ) ) ) ).

% qp.indep_var_rv1
thf(fact_27_qp_Oindep__var__rv1,axiom,
    ! [S: sigma_3077487657436305159omplex,X: real > complex,T: sigma_3077487657436305159omplex,Y: real > complex] :
      ( ( indepe1954327081502071720omplex @ mu @ S @ X @ T @ Y )
     => ( member_real_complex @ X @ ( sigma_9111916201866572460omplex @ mu @ S ) ) ) ).

% qp.indep_var_rv1
thf(fact_28_qp_Oindep__var__rv1,axiom,
    ! [S: sigma_measure_real,X: real > real,T: sigma_measure_real,Y: real > real] :
      ( ( indepe3760321310464026790l_real @ mu @ S @ X @ T @ Y )
     => ( member_real_real @ X @ ( sigma_5267869275261027754l_real @ mu @ S ) ) ) ).

% qp.indep_var_rv1
thf(fact_29_qp_Oindep__var__rv2,axiom,
    ! [S: sigma_measure_o,X: real > $o,T: sigma_measure_o,Y: real > $o] :
      ( ( indepe495429349782043166real_o @ mu @ S @ X @ T @ Y )
     => ( member_real_o @ Y @ ( sigma_3939073009482781210real_o @ mu @ T ) ) ) ).

% qp.indep_var_rv2
thf(fact_30_qp_Oindep__var__rv2,axiom,
    ! [S: sigma_measure_nat,X: real > nat,T: sigma_measure_nat,Y: real > nat] :
      ( ( indepe3400978226907914954al_nat @ mu @ S @ X @ T @ Y )
     => ( member_real_nat @ Y @ ( sigma_6315060578831106510al_nat @ mu @ T ) ) ) ).

% qp.indep_var_rv2
thf(fact_31_qp_Oindep__var__rv2,axiom,
    ! [S: sigma_7234349610311085201nnreal,X: real > extend8495563244428889912nnreal,T: sigma_7234349610311085201nnreal,Y: real > extend8495563244428889912nnreal] :
      ( ( indepe6767359503340752434nnreal @ mu @ S @ X @ T @ Y )
     => ( member2919562650594848410nnreal @ Y @ ( sigma_9017504469962657078nnreal @ mu @ T ) ) ) ).

% qp.indep_var_rv2
thf(fact_32_qp_Oindep__var__rv2,axiom,
    ! [S: sigma_3077487657436305159omplex,X: real > complex,T: sigma_3077487657436305159omplex,Y: real > complex] :
      ( ( indepe1954327081502071720omplex @ mu @ S @ X @ T @ Y )
     => ( member_real_complex @ Y @ ( sigma_9111916201866572460omplex @ mu @ T ) ) ) ).

% qp.indep_var_rv2
thf(fact_33_qp_Oindep__var__rv2,axiom,
    ! [S: sigma_measure_real,X: real > real,T: sigma_measure_real,Y: real > real] :
      ( ( indepe3760321310464026790l_real @ mu @ S @ X @ T @ Y )
     => ( member_real_real @ Y @ ( sigma_5267869275261027754l_real @ mu @ T ) ) ) ).

% qp.indep_var_rv2
thf(fact_34_qp_Om__in__space__prob__algebra,axiom,
    member4553183543495551918e_real @ mu @ ( sigma_2594925453452915853e_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) ).

% qp.m_in_space_prob_algebra
thf(fact_35_qp_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_o,X1: real > $o,M2: sigma_measure_o,X2: real > $o,Y1: $o > $o,N1: sigma_measure_o,Y2: $o > $o,N2: sigma_measure_o] :
      ( ( indepe495429349782043166real_o @ mu @ M1 @ X1 @ M2 @ X2 )
     => ( ( member_o_o @ Y1 @ ( sigma_measurable_o_o @ M1 @ N1 ) )
       => ( ( member_o_o @ Y2 @ ( sigma_measurable_o_o @ M2 @ N2 ) )
         => ( indepe495429349782043166real_o @ mu @ N1 @ ( comp_o_o_real @ Y1 @ X1 ) @ N2 @ ( comp_o_o_real @ Y2 @ X2 ) ) ) ) ) ).

% qp.indep_var_compose
thf(fact_36_qp_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_b,X1: real > b,M2: sigma_measure_b,X2: real > b,Y1: b > extend8495563244428889912nnreal,N1: sigma_7234349610311085201nnreal,Y2: b > extend8495563244428889912nnreal,N2: sigma_7234349610311085201nnreal] :
      ( ( indepe365082296117321349real_b @ mu @ M1 @ X1 @ M2 @ X2 )
     => ( ( member6418304549040442065nnreal @ Y1 @ ( sigma_6334800283702579687nnreal @ M1 @ N1 ) )
       => ( ( member6418304549040442065nnreal @ Y2 @ ( sigma_6334800283702579687nnreal @ M2 @ N2 ) )
         => ( indepe6767359503340752434nnreal @ mu @ N1 @ ( comp_b5186278242990201154l_real @ Y1 @ X1 ) @ N2 @ ( comp_b5186278242990201154l_real @ Y2 @ X2 ) ) ) ) ) ).

% qp.indep_var_compose
thf(fact_37_qp_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_a,X1: real > a,M2: sigma_measure_a,X2: real > a,Y1: a > extend8495563244428889912nnreal,N1: sigma_7234349610311085201nnreal,Y2: a > extend8495563244428889912nnreal,N2: sigma_7234349610311085201nnreal] :
      ( ( indepe365082296117321348real_a @ mu @ M1 @ X1 @ M2 @ X2 )
     => ( ( member298456594901751504nnreal @ Y1 @ ( sigma_214952329563889126nnreal @ M1 @ N1 ) )
       => ( ( member298456594901751504nnreal @ Y2 @ ( sigma_214952329563889126nnreal @ M2 @ N2 ) )
         => ( indepe6767359503340752434nnreal @ mu @ N1 @ ( comp_a8249376463644563905l_real @ Y1 @ X1 ) @ N2 @ ( comp_a8249376463644563905l_real @ Y2 @ X2 ) ) ) ) ) ).

% qp.indep_var_compose
thf(fact_38_qp_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_o,X1: real > $o,M2: sigma_measure_o,X2: real > $o,Y1: $o > real,N1: sigma_measure_real,Y2: $o > real,N2: sigma_measure_real] :
      ( ( indepe495429349782043166real_o @ mu @ M1 @ X1 @ M2 @ X2 )
     => ( ( member_o_real @ Y1 @ ( sigma_2430008634441611636o_real @ M1 @ N1 ) )
       => ( ( member_o_real @ Y2 @ ( sigma_2430008634441611636o_real @ M2 @ N2 ) )
         => ( indepe3760321310464026790l_real @ mu @ N1 @ ( comp_o_real_real @ Y1 @ X1 ) @ N2 @ ( comp_o_real_real @ Y2 @ X2 ) ) ) ) ) ).

% qp.indep_var_compose
thf(fact_39_qp_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_nat,X1: real > nat,M2: sigma_measure_nat,X2: real > nat,Y1: nat > real,N1: sigma_measure_real,Y2: nat > real,N2: sigma_measure_real] :
      ( ( indepe3400978226907914954al_nat @ mu @ M1 @ X1 @ M2 @ X2 )
     => ( ( member_nat_real @ Y1 @ ( sigma_1747752005702207822t_real @ M1 @ N1 ) )
       => ( ( member_nat_real @ Y2 @ ( sigma_1747752005702207822t_real @ M2 @ N2 ) )
         => ( indepe3760321310464026790l_real @ mu @ N1 @ ( comp_nat_real_real @ Y1 @ X1 ) @ N2 @ ( comp_nat_real_real @ Y2 @ X2 ) ) ) ) ) ).

% qp.indep_var_compose
thf(fact_40_qp_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_a,X1: real > a,M2: sigma_measure_a,X2: real > a,Y1: a > real,N1: sigma_measure_real,Y2: a > real,N2: sigma_measure_real] :
      ( ( indepe365082296117321348real_a @ mu @ M1 @ X1 @ M2 @ X2 )
     => ( ( member_a_real @ Y1 @ ( sigma_9116425665531756122a_real @ M1 @ N1 ) )
       => ( ( member_a_real @ Y2 @ ( sigma_9116425665531756122a_real @ M2 @ N2 ) )
         => ( indepe3760321310464026790l_real @ mu @ N1 @ ( comp_a_real_real @ Y1 @ X1 ) @ N2 @ ( comp_a_real_real @ Y2 @ X2 ) ) ) ) ) ).

% qp.indep_var_compose
thf(fact_41_qp_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_real,X1: real > real,M2: sigma_measure_real,X2: real > real,Y1: real > $o,N1: sigma_measure_o,Y2: real > $o,N2: sigma_measure_o] :
      ( ( indepe3760321310464026790l_real @ mu @ M1 @ X1 @ M2 @ X2 )
     => ( ( member_real_o @ Y1 @ ( sigma_3939073009482781210real_o @ M1 @ N1 ) )
       => ( ( member_real_o @ Y2 @ ( sigma_3939073009482781210real_o @ M2 @ N2 ) )
         => ( indepe495429349782043166real_o @ mu @ N1 @ ( comp_real_o_real @ Y1 @ X1 ) @ N2 @ ( comp_real_o_real @ Y2 @ X2 ) ) ) ) ) ).

% qp.indep_var_compose
thf(fact_42_qp_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_real,X1: real > real,M2: sigma_measure_real,X2: real > real,Y1: real > nat,N1: sigma_measure_nat,Y2: real > nat,N2: sigma_measure_nat] :
      ( ( indepe3760321310464026790l_real @ mu @ M1 @ X1 @ M2 @ X2 )
     => ( ( member_real_nat @ Y1 @ ( sigma_6315060578831106510al_nat @ M1 @ N1 ) )
       => ( ( member_real_nat @ Y2 @ ( sigma_6315060578831106510al_nat @ M2 @ N2 ) )
         => ( indepe3400978226907914954al_nat @ mu @ N1 @ ( comp_real_nat_real @ Y1 @ X1 ) @ N2 @ ( comp_real_nat_real @ Y2 @ X2 ) ) ) ) ) ).

% qp.indep_var_compose
thf(fact_43_qp_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_real,X1: real > real,M2: sigma_measure_real,X2: real > real,Y1: real > extend8495563244428889912nnreal,N1: sigma_7234349610311085201nnreal,Y2: real > extend8495563244428889912nnreal,N2: sigma_7234349610311085201nnreal] :
      ( ( indepe3760321310464026790l_real @ mu @ M1 @ X1 @ M2 @ X2 )
     => ( ( member2919562650594848410nnreal @ Y1 @ ( sigma_9017504469962657078nnreal @ M1 @ N1 ) )
       => ( ( member2919562650594848410nnreal @ Y2 @ ( sigma_9017504469962657078nnreal @ M2 @ N2 ) )
         => ( indepe6767359503340752434nnreal @ mu @ N1 @ ( comp_r6279034453215524981l_real @ Y1 @ X1 ) @ N2 @ ( comp_r6279034453215524981l_real @ Y2 @ X2 ) ) ) ) ) ).

% qp.indep_var_compose
thf(fact_44_qp_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_real,X1: real > real,M2: sigma_measure_real,X2: real > real,Y1: real > complex,N1: sigma_3077487657436305159omplex,Y2: real > complex,N2: sigma_3077487657436305159omplex] :
      ( ( indepe3760321310464026790l_real @ mu @ M1 @ X1 @ M2 @ X2 )
     => ( ( member_real_complex @ Y1 @ ( sigma_9111916201866572460omplex @ M1 @ N1 ) )
       => ( ( member_real_complex @ Y2 @ ( sigma_9111916201866572460omplex @ M2 @ N2 ) )
         => ( indepe1954327081502071720omplex @ mu @ N1 @ ( comp_r1968866223832618731x_real @ Y1 @ X1 ) @ N2 @ ( comp_r1968866223832618731x_real @ Y2 @ X2 ) ) ) ) ) ).

% qp.indep_var_compose
thf(fact_45__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062_092_060beta_062_Ah_O_A_092_060lbrakk_062_092_060beta_062_A_092_060in_062_Aqbs__Mx_AY_059_Ah_A_092_060in_062_Areal__borel_A_092_060rightarrow_062_092_060_094sub_062M_Aprob__algebra_Areal__borel_059_Af_A_092_060circ_062_A_092_060alpha_062_A_061_A_I_092_060lambda_062r_O_Aqbs__prob__space_A_IY_M_A_092_060beta_062_M_Ah_Ar_J_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Beta: real > b] :
        ( ( member_real_b @ Beta @ ( qbs_Mx_b @ y ) )
       => ! [H: real > sigma_measure_real] :
            ( ( member2630560753458908601e_real @ H @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( comp_a6868616473110226257b_real @ f @ alpha )
             != ( ^ [R: real] : ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ y @ ( produc4478270668571743890e_real @ Beta @ ( H @ R ) ) ) ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>\<beta> h. \<lbrakk>\<beta> \<in> qbs_Mx Y; h \<in> real_borel \<rightarrow>\<^sub>M prob_algebra real_borel; f \<circ> \<alpha> = (\<lambda>r. qbs_prob_space (Y, \<beta>, h r))\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_46_qbs__prob__MPx,axiom,
    ! [Alpha: real > $o,X: quasi_borel_o,G2: real > sigma_measure_real,R2: real] :
      ( ( member_real_o @ Alpha @ ( qbs_Mx_o @ X ) )
     => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
       => ( probab163731277575722550prob_o @ X @ Alpha @ ( G2 @ R2 ) ) ) ) ).

% qbs_prob_MPx
thf(fact_47_qbs__prob__MPx,axiom,
    ! [Alpha: real > nat,X: quasi_borel_nat,G2: real > sigma_measure_real,R2: real] :
      ( ( member_real_nat @ Alpha @ ( qbs_Mx_nat @ X ) )
     => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
       => ( probab2851505236026752178ob_nat @ X @ Alpha @ ( G2 @ R2 ) ) ) ) ).

% qbs_prob_MPx
thf(fact_48_qbs__prob__MPx,axiom,
    ! [Alpha: real > extend8495563244428889912nnreal,X: quasi_9015997321629101608nnreal,G2: real > sigma_measure_real,R2: real] :
      ( ( member2919562650594848410nnreal @ Alpha @ ( qbs_Mx6523938229262583809nnreal @ X ) )
     => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
       => ( probab8888997264033409050nnreal @ X @ Alpha @ ( G2 @ R2 ) ) ) ) ).

% qbs_prob_MPx
thf(fact_49_qbs__prob__MPx,axiom,
    ! [Alpha: real > complex,X: quasi_borel_complex,G2: real > sigma_measure_real,R2: real] :
      ( ( member_real_complex @ Alpha @ ( qbs_Mx_complex @ X ) )
     => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
       => ( probab963564174529062288omplex @ X @ Alpha @ ( G2 @ R2 ) ) ) ) ).

% qbs_prob_MPx
thf(fact_50_qbs__prob__MPx,axiom,
    ! [Alpha: real > real,X: quasi_borel_real,G2: real > sigma_measure_real,R2: real] :
      ( ( member_real_real @ Alpha @ ( qbs_Mx_real @ X ) )
     => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
       => ( probab3605210969150000782b_real @ X @ Alpha @ ( G2 @ R2 ) ) ) ) ).

% qbs_prob_MPx
thf(fact_51_qbs__prob__MPx,axiom,
    ! [Alpha: real > b,X: quasi_borel_b,G2: real > sigma_measure_real,R2: real] :
      ( ( member_real_b @ Alpha @ ( qbs_Mx_b @ X ) )
     => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
       => ( probab701741629625904797prob_b @ X @ Alpha @ ( G2 @ R2 ) ) ) ) ).

% qbs_prob_MPx
thf(fact_52_qbs__prob__MPx,axiom,
    ! [Alpha: real > a,X: quasi_borel_a,G2: real > sigma_measure_real,R2: real] :
      ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X ) )
     => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
       => ( probab701741629625904796prob_a @ X @ Alpha @ ( G2 @ R2 ) ) ) ) ).

% qbs_prob_MPx
thf(fact_53_prob__space__bind_H,axiom,
    ! [A: sigma_measure_real,M: sigma_measure_real,B: real > sigma_measure_real,N: sigma_measure_real] :
      ( ( member4553183543495551918e_real @ A @ ( sigma_2594925453452915853e_real @ ( giry_p6081475675320601670a_real @ M ) ) )
     => ( ( member2630560753458908601e_real @ B @ ( sigma_5928869325259027335e_real @ M @ ( giry_p6081475675320601670a_real @ N ) ) )
       => ( probab535871623910865577e_real @ ( giry_bind_real_real @ A @ B ) ) ) ) ).

% prob_space_bind'
thf(fact_54_prob__algebra__real__prob__measure,axiom,
    ! [P: sigma_measure_real] :
      ( ( member4553183543495551918e_real @ P @ ( sigma_2594925453452915853e_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
      = ( distri2809703520229113005bution @ P ) ) ).

% prob_algebra_real_prob_measure
thf(fact_55_qbs__prob_Om__in__space__prob__algebra,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( member4553183543495551918e_real @ Mu @ ( sigma_2594925453452915853e_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) ) ) ).

% qbs_prob.m_in_space_prob_algebra
thf(fact_56_qbs__prob_Om__in__space__prob__algebra,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X @ Alpha @ Mu )
     => ( member4553183543495551918e_real @ Mu @ ( sigma_2594925453452915853e_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) ) ) ).

% qbs_prob.m_in_space_prob_algebra
thf(fact_57_in__MPx_Oex,axiom,
    ! [X: quasi_borel_b,Beta2: real > probab4737552677800996672pace_b] :
      ( ( monad_in_MPx_b @ X @ Beta2 )
     => ? [X3: real > b] :
          ( ( member_real_b @ X3 @ ( qbs_Mx_b @ X ) )
          & ? [Xa: real > sigma_measure_real] :
              ( ( member2630560753458908601e_real @ Xa @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
              & ! [R3: real] :
                  ( ( Beta2 @ R3 )
                  = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ X3 @ ( Xa @ R3 ) ) ) ) ) ) ) ) ).

% in_MPx.ex
thf(fact_58_in__MPx_Oex,axiom,
    ! [X: quasi_borel_a,Beta2: real > probab4737552673497767871pace_a] :
      ( ( monad_in_MPx_a @ X @ Beta2 )
     => ? [X3: real > a] :
          ( ( member_real_a @ X3 @ ( qbs_Mx_a @ X ) )
          & ? [Xa: real > sigma_measure_real] :
              ( ( member2630560753458908601e_real @ Xa @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
              & ! [R3: real] :
                  ( ( Beta2 @ R3 )
                  = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ X3 @ ( Xa @ R3 ) ) ) ) ) ) ) ) ).

% in_MPx.ex
thf(fact_59_in__MPx_Ointro,axiom,
    ! [X: quasi_borel_b,Beta2: real > probab4737552677800996672pace_b] :
      ( ? [X4: real > b] :
          ( ( member_real_b @ X4 @ ( qbs_Mx_b @ X ) )
          & ? [Xa2: real > sigma_measure_real] :
              ( ( member2630560753458908601e_real @ Xa2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
              & ! [R4: real] :
                  ( ( Beta2 @ R4 )
                  = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ X4 @ ( Xa2 @ R4 ) ) ) ) ) ) )
     => ( monad_in_MPx_b @ X @ Beta2 ) ) ).

% in_MPx.intro
thf(fact_60_in__MPx_Ointro,axiom,
    ! [X: quasi_borel_a,Beta2: real > probab4737552673497767871pace_a] :
      ( ? [X4: real > a] :
          ( ( member_real_a @ X4 @ ( qbs_Mx_a @ X ) )
          & ? [Xa2: real > sigma_measure_real] :
              ( ( member2630560753458908601e_real @ Xa2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
              & ! [R4: real] :
                  ( ( Beta2 @ R4 )
                  = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ X4 @ ( Xa2 @ R4 ) ) ) ) ) ) )
     => ( monad_in_MPx_a @ X @ Beta2 ) ) ).

% in_MPx.intro
thf(fact_61_in__MPx_Orep__inMPx,axiom,
    ! [X: quasi_borel_o,Beta2: real > probab1241297377463522905pace_o] :
      ( ( monad_in_MPx_o @ X @ Beta2 )
     => ? [Alpha2: real > $o,G3: real > sigma_measure_real] :
          ( ( member_real_o @ Alpha2 @ ( qbs_Mx_o @ X ) )
          & ( member2630560753458908601e_real @ G3 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
          & ( Beta2
            = ( ^ [R: real] : ( probab414947219978445330pace_o @ ( produc2942336022369489698e_real @ X @ ( produc787877863769208121e_real @ Alpha2 @ ( G3 @ R ) ) ) ) ) ) ) ) ).

% in_MPx.rep_inMPx
thf(fact_62_in__MPx_Orep__inMPx,axiom,
    ! [X: quasi_borel_nat,Beta2: real > probab8370124847414953445ce_nat] :
      ( ( monad_in_MPx_nat @ X @ Beta2 )
     => ? [Alpha2: real > nat,G3: real > sigma_measure_real] :
          ( ( member_real_nat @ Alpha2 @ ( qbs_Mx_nat @ X ) )
          & ( member2630560753458908601e_real @ G3 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
          & ( Beta2
            = ( ^ [R: real] : ( probab4893816680317050838ce_nat @ ( produc2796221405228754040e_real @ X @ ( produc6760937697707383505e_real @ Alpha2 @ ( G3 @ R ) ) ) ) ) ) ) ) ).

% in_MPx.rep_inMPx
thf(fact_63_in__MPx_Orep__inMPx,axiom,
    ! [X: quasi_9015997321629101608nnreal,Beta2: real > probab1359147627358338381nnreal] :
      ( ( monad_915879624692006445nnreal @ X @ Beta2 )
     => ? [Alpha2: real > extend8495563244428889912nnreal,G3: real > sigma_measure_real] :
          ( ( member2919562650594848410nnreal @ Alpha2 @ ( qbs_Mx6523938229262583809nnreal @ X ) )
          & ( member2630560753458908601e_real @ G3 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
          & ( Beta2
            = ( ^ [R: real] : ( probab2095897665094527806nnreal @ ( produc3311748387682954872e_real @ X @ ( produc8591767778467529065e_real @ Alpha2 @ ( G3 @ R ) ) ) ) ) ) ) ) ).

% in_MPx.rep_inMPx
thf(fact_64_in__MPx_Orep__inMPx,axiom,
    ! [X: quasi_borel_complex,Beta2: real > probab3778977310982246339omplex] :
      ( ( monad_in_MPx_complex @ X @ Beta2 )
     => ? [Alpha2: real > complex,G3: real > sigma_measure_real] :
          ( ( member_real_complex @ Alpha2 @ ( qbs_Mx_complex @ X ) )
          & ( member2630560753458908601e_real @ G3 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
          & ( Beta2
            = ( ^ [R: real] : ( probab2835974231434789044omplex @ ( produc7812373664507111416e_real @ X @ ( produc8166640644904819571e_real @ Alpha2 @ ( G3 @ R ) ) ) ) ) ) ) ) ).

% in_MPx.rep_inMPx
thf(fact_65_in__MPx_Orep__inMPx,axiom,
    ! [X: quasi_borel_real,Beta2: real > probab8009751763329705409e_real] :
      ( ( monad_in_MPx_real @ X @ Beta2 )
     => ? [Alpha2: real > real,G3: real > sigma_measure_real] :
          ( ( member_real_real @ Alpha2 @ ( qbs_Mx_real @ X ) )
          & ( member2630560753458908601e_real @ G3 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
          & ( Beta2
            = ( ^ [R: real] : ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X @ ( produc1722724976708544245e_real @ Alpha2 @ ( G3 @ R ) ) ) ) ) ) ) ) ).

% in_MPx.rep_inMPx
thf(fact_66_in__MPx_Orep__inMPx,axiom,
    ! [X: quasi_borel_b,Beta2: real > probab4737552677800996672pace_b] :
      ( ( monad_in_MPx_b @ X @ Beta2 )
     => ? [Alpha2: real > b,G3: real > sigma_measure_real] :
          ( ( member_real_b @ Alpha2 @ ( qbs_Mx_b @ X ) )
          & ( member2630560753458908601e_real @ G3 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
          & ( Beta2
            = ( ^ [R: real] : ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha2 @ ( G3 @ R ) ) ) ) ) ) ) ) ).

% in_MPx.rep_inMPx
thf(fact_67_in__MPx_Orep__inMPx,axiom,
    ! [X: quasi_borel_a,Beta2: real > probab4737552673497767871pace_a] :
      ( ( monad_in_MPx_a @ X @ Beta2 )
     => ? [Alpha2: real > a,G3: real > sigma_measure_real] :
          ( ( member_real_a @ Alpha2 @ ( qbs_Mx_a @ X ) )
          & ( member2630560753458908601e_real @ G3 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
          & ( Beta2
            = ( ^ [R: real] : ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha2 @ ( G3 @ R ) ) ) ) ) ) ) ) ).

% in_MPx.rep_inMPx
thf(fact_68_in__MPx__def,axiom,
    ( monad_in_MPx_b
    = ( ^ [X5: quasi_borel_b,Beta3: real > probab4737552677800996672pace_b] :
        ? [X6: real > b] :
          ( ( member_real_b @ X6 @ ( qbs_Mx_b @ X5 ) )
          & ? [Y3: real > sigma_measure_real] :
              ( ( member2630560753458908601e_real @ Y3 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
              & ! [R: real] :
                  ( ( Beta3 @ R )
                  = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X5 @ ( produc4478270668571743890e_real @ X6 @ ( Y3 @ R ) ) ) ) ) ) ) ) ) ).

% in_MPx_def
thf(fact_69_in__MPx__def,axiom,
    ( monad_in_MPx_a
    = ( ^ [X5: quasi_borel_a,Beta3: real > probab4737552673497767871pace_a] :
        ? [X6: real > a] :
          ( ( member_real_a @ X6 @ ( qbs_Mx_a @ X5 ) )
          & ? [Y3: real > sigma_measure_real] :
              ( ( member2630560753458908601e_real @ Y3 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
              & ! [R: real] :
                  ( ( Beta3 @ R )
                  = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X5 @ ( produc623176010801490259e_real @ X6 @ ( Y3 @ R ) ) ) ) ) ) ) ) ) ).

% in_MPx_def
thf(fact_70_qbs__prob__space__induct_H,axiom,
    ! [S2: probab4737552677800996672pace_b,P2: probab4737552677800996672pace_b > $o] :
      ( ! [X7: quasi_borel_b,Alpha2: real > b,Mu2: sigma_measure_real] :
          ( ( probab701741629625904797prob_b @ X7 @ Alpha2 @ Mu2 )
         => ( ( S2
              = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X7 @ ( produc4478270668571743890e_real @ Alpha2 @ Mu2 ) ) ) )
           => ( P2 @ ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X7 @ ( produc4478270668571743890e_real @ Alpha2 @ Mu2 ) ) ) ) ) )
     => ( P2 @ S2 ) ) ).

% qbs_prob_space_induct'
thf(fact_71_qbs__prob__space__induct_H,axiom,
    ! [S2: probab4737552673497767871pace_a,P2: probab4737552673497767871pace_a > $o] :
      ( ! [X7: quasi_borel_a,Alpha2: real > a,Mu2: sigma_measure_real] :
          ( ( probab701741629625904796prob_a @ X7 @ Alpha2 @ Mu2 )
         => ( ( S2
              = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X7 @ ( produc623176010801490259e_real @ Alpha2 @ Mu2 ) ) ) )
           => ( P2 @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X7 @ ( produc623176010801490259e_real @ Alpha2 @ Mu2 ) ) ) ) ) )
     => ( P2 @ S2 ) ) ).

% qbs_prob_space_induct'
thf(fact_72_qbs__prob__space__induct,axiom,
    ! [P2: probab4737552677800996672pace_b > $o,S2: probab4737552677800996672pace_b] :
      ( ! [X7: quasi_borel_b,Alpha2: real > b,Mu2: sigma_measure_real] :
          ( ( probab701741629625904797prob_b @ X7 @ Alpha2 @ Mu2 )
         => ( P2 @ ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X7 @ ( produc4478270668571743890e_real @ Alpha2 @ Mu2 ) ) ) ) )
     => ( P2 @ S2 ) ) ).

% qbs_prob_space_induct
thf(fact_73_qbs__prob__space__induct,axiom,
    ! [P2: probab4737552673497767871pace_a > $o,S2: probab4737552673497767871pace_a] :
      ( ! [X7: quasi_borel_a,Alpha2: real > a,Mu2: sigma_measure_real] :
          ( ( probab701741629625904796prob_a @ X7 @ Alpha2 @ Mu2 )
         => ( P2 @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X7 @ ( produc623176010801490259e_real @ Alpha2 @ Mu2 ) ) ) ) )
     => ( P2 @ S2 ) ) ).

% qbs_prob_space_induct
thf(fact_74_rep__qbs__prob__space,axiom,
    ! [P: probab4737552677800996672pace_b] :
    ? [X7: quasi_borel_b,Alpha2: real > b,Mu2: sigma_measure_real] :
      ( ( P
        = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X7 @ ( produc4478270668571743890e_real @ Alpha2 @ Mu2 ) ) ) )
      & ( probab701741629625904797prob_b @ X7 @ Alpha2 @ Mu2 ) ) ).

% rep_qbs_prob_space
thf(fact_75_rep__qbs__prob__space,axiom,
    ! [P: probab4737552673497767871pace_a] :
    ? [X7: quasi_borel_a,Alpha2: real > a,Mu2: sigma_measure_real] :
      ( ( P
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X7 @ ( produc623176010801490259e_real @ Alpha2 @ Mu2 ) ) ) )
      & ( probab701741629625904796prob_a @ X7 @ Alpha2 @ Mu2 ) ) ).

% rep_qbs_prob_space
thf(fact_76_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_77_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_78_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_79_real__distribution_Omeasurable__finite__borel,axiom,
    ! [M: sigma_measure_real,F: real > nat] :
      ( ( distri2809703520229113005bution @ M )
     => ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) )
       => ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ borel_8449730974584783410el_nat ) ) ) ) ).

% real_distribution.measurable_finite_borel
thf(fact_80_real__distribution_Omeasurable__finite__borel,axiom,
    ! [M: sigma_measure_real,F: real > $o] :
      ( ( distri2809703520229113005bution @ M )
     => ( ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ borel_5078946678739801102l_real @ borel_5500255247093592246orel_o ) )
       => ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ M @ borel_5500255247093592246orel_o ) ) ) ) ).

% real_distribution.measurable_finite_borel
thf(fact_81_prob__space_Oindep__var__compose,axiom,
    ! [M: sigma_measure_o,M1: sigma_measure_real,X1: $o > real,M2: sigma_measure_real,X2: $o > real,Y1: real > $o,N1: sigma_measure_o,Y2: real > $o,N2: sigma_measure_o] :
      ( ( probab1190487603588612059pace_o @ M )
     => ( ( indepe8209737011595649400o_real @ M @ M1 @ X1 @ M2 @ X2 )
       => ( ( member_real_o @ Y1 @ ( sigma_3939073009482781210real_o @ M1 @ N1 ) )
         => ( ( member_real_o @ Y2 @ ( sigma_3939073009482781210real_o @ M2 @ N2 ) )
           => ( indepe6952847619639754700ar_o_o @ M @ N1 @ ( comp_real_o_o @ Y1 @ X1 ) @ N2 @ ( comp_real_o_o @ Y2 @ X2 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_82_prob__space_Oindep__var__compose,axiom,
    ! [M: sigma_measure_nat,M1: sigma_measure_real,X1: nat > real,M2: sigma_measure_real,X2: nat > real,Y1: real > nat,N1: sigma_measure_nat,Y2: real > nat,N2: sigma_measure_nat] :
      ( ( probab2904919403188438605ce_nat @ M )
     => ( ( indepe8057041690633792074t_real @ M @ M1 @ X1 @ M2 @ X2 )
       => ( ( member_real_nat @ Y1 @ ( sigma_6315060578831106510al_nat @ M1 @ N1 ) )
         => ( ( member_real_nat @ Y2 @ ( sigma_6315060578831106510al_nat @ M2 @ N2 ) )
           => ( indepe2246810989395310190at_nat @ M @ N1 @ ( comp_real_nat_nat @ Y1 @ X1 ) @ N2 @ ( comp_real_nat_nat @ Y2 @ X2 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_83_prob__space_Oindep__var__compose,axiom,
    ! [M: sigma_measure_real,M1: sigma_measure_o,X1: real > $o,M2: sigma_measure_o,X2: real > $o,Y1: $o > $o,N1: sigma_measure_o,Y2: $o > $o,N2: sigma_measure_o] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe495429349782043166real_o @ M @ M1 @ X1 @ M2 @ X2 )
       => ( ( member_o_o @ Y1 @ ( sigma_measurable_o_o @ M1 @ N1 ) )
         => ( ( member_o_o @ Y2 @ ( sigma_measurable_o_o @ M2 @ N2 ) )
           => ( indepe495429349782043166real_o @ M @ N1 @ ( comp_o_o_real @ Y1 @ X1 ) @ N2 @ ( comp_o_o_real @ Y2 @ X2 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_84_prob__space_Oindep__var__compose,axiom,
    ! [M: sigma_measure_real,M1: sigma_measure_b,X1: real > b,M2: sigma_measure_b,X2: real > b,Y1: b > extend8495563244428889912nnreal,N1: sigma_7234349610311085201nnreal,Y2: b > extend8495563244428889912nnreal,N2: sigma_7234349610311085201nnreal] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe365082296117321349real_b @ M @ M1 @ X1 @ M2 @ X2 )
       => ( ( member6418304549040442065nnreal @ Y1 @ ( sigma_6334800283702579687nnreal @ M1 @ N1 ) )
         => ( ( member6418304549040442065nnreal @ Y2 @ ( sigma_6334800283702579687nnreal @ M2 @ N2 ) )
           => ( indepe6767359503340752434nnreal @ M @ N1 @ ( comp_b5186278242990201154l_real @ Y1 @ X1 ) @ N2 @ ( comp_b5186278242990201154l_real @ Y2 @ X2 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_85_prob__space_Oindep__var__compose,axiom,
    ! [M: sigma_measure_real,M1: sigma_measure_a,X1: real > a,M2: sigma_measure_a,X2: real > a,Y1: a > extend8495563244428889912nnreal,N1: sigma_7234349610311085201nnreal,Y2: a > extend8495563244428889912nnreal,N2: sigma_7234349610311085201nnreal] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe365082296117321348real_a @ M @ M1 @ X1 @ M2 @ X2 )
       => ( ( member298456594901751504nnreal @ Y1 @ ( sigma_214952329563889126nnreal @ M1 @ N1 ) )
         => ( ( member298456594901751504nnreal @ Y2 @ ( sigma_214952329563889126nnreal @ M2 @ N2 ) )
           => ( indepe6767359503340752434nnreal @ M @ N1 @ ( comp_a8249376463644563905l_real @ Y1 @ X1 ) @ N2 @ ( comp_a8249376463644563905l_real @ Y2 @ X2 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_86_prob__space_Oindep__var__compose,axiom,
    ! [M: sigma_measure_real,M1: sigma_measure_o,X1: real > $o,M2: sigma_measure_o,X2: real > $o,Y1: $o > real,N1: sigma_measure_real,Y2: $o > real,N2: sigma_measure_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe495429349782043166real_o @ M @ M1 @ X1 @ M2 @ X2 )
       => ( ( member_o_real @ Y1 @ ( sigma_2430008634441611636o_real @ M1 @ N1 ) )
         => ( ( member_o_real @ Y2 @ ( sigma_2430008634441611636o_real @ M2 @ N2 ) )
           => ( indepe3760321310464026790l_real @ M @ N1 @ ( comp_o_real_real @ Y1 @ X1 ) @ N2 @ ( comp_o_real_real @ Y2 @ X2 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_87_prob__space_Oindep__var__compose,axiom,
    ! [M: sigma_measure_real,M1: sigma_measure_nat,X1: real > nat,M2: sigma_measure_nat,X2: real > nat,Y1: nat > real,N1: sigma_measure_real,Y2: nat > real,N2: sigma_measure_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe3400978226907914954al_nat @ M @ M1 @ X1 @ M2 @ X2 )
       => ( ( member_nat_real @ Y1 @ ( sigma_1747752005702207822t_real @ M1 @ N1 ) )
         => ( ( member_nat_real @ Y2 @ ( sigma_1747752005702207822t_real @ M2 @ N2 ) )
           => ( indepe3760321310464026790l_real @ M @ N1 @ ( comp_nat_real_real @ Y1 @ X1 ) @ N2 @ ( comp_nat_real_real @ Y2 @ X2 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_88_prob__space_Oindep__var__compose,axiom,
    ! [M: sigma_measure_real,M1: sigma_measure_a,X1: real > a,M2: sigma_measure_a,X2: real > a,Y1: a > real,N1: sigma_measure_real,Y2: a > real,N2: sigma_measure_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe365082296117321348real_a @ M @ M1 @ X1 @ M2 @ X2 )
       => ( ( member_a_real @ Y1 @ ( sigma_9116425665531756122a_real @ M1 @ N1 ) )
         => ( ( member_a_real @ Y2 @ ( sigma_9116425665531756122a_real @ M2 @ N2 ) )
           => ( indepe3760321310464026790l_real @ M @ N1 @ ( comp_a_real_real @ Y1 @ X1 ) @ N2 @ ( comp_a_real_real @ Y2 @ X2 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_89_prob__space_Oindep__var__compose,axiom,
    ! [M: sigma_measure_real,M1: sigma_measure_real,X1: real > real,M2: sigma_measure_real,X2: real > real,Y1: real > $o,N1: sigma_measure_o,Y2: real > $o,N2: sigma_measure_o] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe3760321310464026790l_real @ M @ M1 @ X1 @ M2 @ X2 )
       => ( ( member_real_o @ Y1 @ ( sigma_3939073009482781210real_o @ M1 @ N1 ) )
         => ( ( member_real_o @ Y2 @ ( sigma_3939073009482781210real_o @ M2 @ N2 ) )
           => ( indepe495429349782043166real_o @ M @ N1 @ ( comp_real_o_real @ Y1 @ X1 ) @ N2 @ ( comp_real_o_real @ Y2 @ X2 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_90_prob__space_Oindep__var__compose,axiom,
    ! [M: sigma_measure_real,M1: sigma_measure_real,X1: real > real,M2: sigma_measure_real,X2: real > real,Y1: real > nat,N1: sigma_measure_nat,Y2: real > nat,N2: sigma_measure_nat] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe3760321310464026790l_real @ M @ M1 @ X1 @ M2 @ X2 )
       => ( ( member_real_nat @ Y1 @ ( sigma_6315060578831106510al_nat @ M1 @ N1 ) )
         => ( ( member_real_nat @ Y2 @ ( sigma_6315060578831106510al_nat @ M2 @ N2 ) )
           => ( indepe3400978226907914954al_nat @ M @ N1 @ ( comp_real_nat_real @ Y1 @ X1 ) @ N2 @ ( comp_real_nat_real @ Y2 @ X2 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_91_prob__space_Oindep__var_Ocong,axiom,
    indepe3760321310464026790l_real = indepe3760321310464026790l_real ).

% prob_space.indep_var.cong
thf(fact_92_bind__cong,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_real,F: real > sigma_measure_real,G2: real > sigma_measure_real] :
      ( ( M = N )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( sigma_space_real @ M ) )
           => ( ( F @ X3 )
              = ( G2 @ X3 ) ) )
       => ( ( giry_bind_real_real @ M @ F )
          = ( giry_bind_real_real @ N @ G2 ) ) ) ) ).

% bind_cong
thf(fact_93_bind__cong__All,axiom,
    ! [M: sigma_measure_real,F: real > sigma_measure_real,G2: real > sigma_measure_real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( sigma_space_real @ M ) )
         => ( ( F @ X3 )
            = ( G2 @ X3 ) ) )
     => ( ( giry_bind_real_real @ M @ F )
        = ( giry_bind_real_real @ M @ G2 ) ) ) ).

% bind_cong_All
thf(fact_94_prob__space_Oindep__sets__cong,axiom,
    ! [M: sigma_measure_real,I: set_o_o,J: set_o_o,F2: ( $o > $o ) > set_set_real,G: ( $o > $o ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( I = J )
       => ( ! [I2: $o > $o] :
              ( ( member_o_o @ I2 @ I )
             => ( ( F2 @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe6032362129116668415al_o_o @ M @ F2 @ I )
            = ( indepe6032362129116668415al_o_o @ M @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_95_prob__space_Oindep__sets__cong,axiom,
    ! [M: sigma_measure_real,I: set_b_6825823330181178888nnreal,J: set_b_6825823330181178888nnreal,F2: ( b > extend8495563244428889912nnreal ) > set_set_real,G: ( b > extend8495563244428889912nnreal ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( I = J )
       => ( ! [I2: b > extend8495563244428889912nnreal] :
              ( ( member6418304549040442065nnreal @ I2 @ I )
             => ( ( F2 @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe4831269413077975822nnreal @ M @ F2 @ I )
            = ( indepe4831269413077975822nnreal @ M @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_96_prob__space_Oindep__sets__cong,axiom,
    ! [M: sigma_measure_real,I: set_a_3263942136899480541pace_b,J: set_a_3263942136899480541pace_b,F2: ( a > probab4737552677800996672pace_b ) > set_set_real,G: ( a > probab4737552677800996672pace_b ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( I = J )
       => ( ! [I2: a > probab4737552677800996672pace_b] :
              ( ( member7340901614391157822pace_b @ I2 @ I )
             => ( ( F2 @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe1711633972345727297pace_b @ M @ F2 @ I )
            = ( indepe1711633972345727297pace_b @ M @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_97_prob__space_Oindep__sets__cong,axiom,
    ! [M: sigma_measure_real,I: set_a_7161065143582548615nnreal,J: set_a_7161065143582548615nnreal,F2: ( a > extend8495563244428889912nnreal ) > set_set_real,G: ( a > extend8495563244428889912nnreal ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( I = J )
       => ( ! [I2: a > extend8495563244428889912nnreal] :
              ( ( member298456594901751504nnreal @ I2 @ I )
             => ( ( F2 @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe7934793495794061069nnreal @ M @ F2 @ I )
            = ( indepe7934793495794061069nnreal @ M @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_98_prob__space_Oindep__sets__cong,axiom,
    ! [M: sigma_measure_real,I: set_a_real,J: set_a_real,F2: ( a > real ) > set_set_real,G: ( a > real ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( I = J )
       => ( ! [I2: a > real] :
              ( ( member_a_real @ I2 @ I )
             => ( ( F2 @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe7634599452152747905a_real @ M @ F2 @ I )
            = ( indepe7634599452152747905a_real @ M @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_99_prob__space_Oindep__sets__cong,axiom,
    ! [M: sigma_measure_real,I: set_real_o,J: set_real_o,F2: ( real > $o ) > set_set_real,G: ( real > $o ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( I = J )
       => ( ! [I2: real > $o] :
              ( ( member_real_o @ I2 @ I )
             => ( ( F2 @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe8014066527605497865real_o @ M @ F2 @ I )
            = ( indepe8014066527605497865real_o @ M @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_100_prob__space_Oindep__sets__cong,axiom,
    ! [M: sigma_measure_real,I: set_real_nat,J: set_real_nat,F2: ( real > nat ) > set_set_real,G: ( real > nat ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( I = J )
       => ( ! [I2: real > nat] :
              ( ( member_real_nat @ I2 @ I )
             => ( ( F2 @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe7235330332140030005al_nat @ M @ F2 @ I )
            = ( indepe7235330332140030005al_nat @ M @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_101_prob__space_Oindep__sets__cong,axiom,
    ! [M: sigma_measure_real,I: set_re5328672808648366137nnreal,J: set_re5328672808648366137nnreal,F2: ( real > extend8495563244428889912nnreal ) > set_set_real,G: ( real > extend8495563244428889912nnreal ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( I = J )
       => ( ! [I2: real > extend8495563244428889912nnreal] :
              ( ( member2919562650594848410nnreal @ I2 @ I )
             => ( ( F2 @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe4643477836443271069nnreal @ M @ F2 @ I )
            = ( indepe4643477836443271069nnreal @ M @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_102_prob__space_Oindep__sets__cong,axiom,
    ! [M: sigma_measure_real,I: set_real_complex,J: set_real_complex,F2: ( real > complex ) > set_set_real,G: ( real > complex ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( I = J )
       => ( ! [I2: real > complex] :
              ( ( member_real_complex @ I2 @ I )
             => ( ( F2 @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe6244138955099140627omplex @ M @ F2 @ I )
            = ( indepe6244138955099140627omplex @ M @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_103_prob__space_Oindep__sets__cong,axiom,
    ! [M: sigma_measure_real,I: set_real_real,J: set_real_real,F2: ( real > real ) > set_set_real,G: ( real > real ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( I = J )
       => ( ! [I2: real > real] :
              ( ( member_real_real @ I2 @ I )
             => ( ( F2 @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe4905764430350991377l_real @ M @ F2 @ I )
            = ( indepe4905764430350991377l_real @ M @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_104_prob__space__imp__subprob__space,axiom,
    ! [M: sigma_measure_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( giry_s8208748868292234104e_real @ M ) ) ).

% prob_space_imp_subprob_space
thf(fact_105_subprob__space__imp__sigma__finite,axiom,
    ! [M: sigma_measure_real] :
      ( ( giry_s8208748868292234104e_real @ M )
     => ( measur487378040549452491e_real @ M ) ) ).

% subprob_space_imp_sigma_finite
thf(fact_106_subprob__space_Oaxioms_I1_J,axiom,
    ! [M: sigma_measure_real] :
      ( ( giry_s8208748868292234104e_real @ M )
     => ( measur3606880022600206024e_real @ M ) ) ).

% subprob_space.axioms(1)
thf(fact_107_qbs__prob_Oaxioms_I2_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( distri2809703520229113005bution @ Mu ) ) ).

% qbs_prob.axioms(2)
thf(fact_108_qbs__prob_Oaxioms_I2_J,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X @ Alpha @ Mu )
     => ( distri2809703520229113005bution @ Mu ) ) ).

% qbs_prob.axioms(2)
thf(fact_109_real__distribution_Ofinite__borel__measure__M,axiom,
    ! [M: sigma_measure_real] :
      ( ( distri2809703520229113005bution @ M )
     => ( distri7943378551711771532easure @ M ) ) ).

% real_distribution.finite_borel_measure_M
thf(fact_110_real__distribution_Oaxioms_I1_J,axiom,
    ! [M: sigma_measure_real] :
      ( ( distri2809703520229113005bution @ M )
     => ( probab535871623910865577e_real @ M ) ) ).

% real_distribution.axioms(1)
thf(fact_111_finite__borel__measure_Oaxioms_I1_J,axiom,
    ! [M: sigma_measure_real] :
      ( ( distri7943378551711771532easure @ M )
     => ( measur3606880022600206024e_real @ M ) ) ).

% finite_borel_measure.axioms(1)
thf(fact_112_prob__space_Oindep__var__rv2,axiom,
    ! [M: sigma_measure_o,S: sigma_measure_o,X: $o > $o,T: sigma_measure_o,Y: $o > $o] :
      ( ( probab1190487603588612059pace_o @ M )
     => ( ( indepe6952847619639754700ar_o_o @ M @ S @ X @ T @ Y )
       => ( member_o_o @ Y @ ( sigma_measurable_o_o @ M @ T ) ) ) ) ).

% prob_space.indep_var_rv2
thf(fact_113_prob__space_Oindep__var__rv2,axiom,
    ! [M: sigma_measure_b,S: sigma_7234349610311085201nnreal,X: b > extend8495563244428889912nnreal,T: sigma_7234349610311085201nnreal,Y: b > extend8495563244428889912nnreal] :
      ( ( probab7247484486040049090pace_b @ M )
     => ( ( indepe430593609325189611nnreal @ M @ S @ X @ T @ Y )
       => ( member6418304549040442065nnreal @ Y @ ( sigma_6334800283702579687nnreal @ M @ T ) ) ) ) ).

% prob_space.indep_var_rv2
thf(fact_114_prob__space_Oindep__var__rv2,axiom,
    ! [M: sigma_measure_a,S: sigma_7234349610311085201nnreal,X: a > extend8495563244428889912nnreal,T: sigma_7234349610311085201nnreal,Y: a > extend8495563244428889912nnreal] :
      ( ( probab7247484486040049089pace_a @ M )
     => ( ( indepe3534117692041274858nnreal @ M @ S @ X @ T @ Y )
       => ( member298456594901751504nnreal @ Y @ ( sigma_214952329563889126nnreal @ M @ T ) ) ) ) ).

% prob_space.indep_var_rv2
thf(fact_115_prob__space_Oindep__var__rv2,axiom,
    ! [M: sigma_measure_o,S: sigma_measure_real,X: $o > real,T: sigma_measure_real,Y: $o > real] :
      ( ( probab1190487603588612059pace_o @ M )
     => ( ( indepe8209737011595649400o_real @ M @ S @ X @ T @ Y )
       => ( member_o_real @ Y @ ( sigma_2430008634441611636o_real @ M @ T ) ) ) ) ).

% prob_space.indep_var_rv2
thf(fact_116_prob__space_Oindep__var__rv2,axiom,
    ! [M: sigma_measure_nat,S: sigma_measure_real,X: nat > real,T: sigma_measure_real,Y: nat > real] :
      ( ( probab2904919403188438605ce_nat @ M )
     => ( ( indepe8057041690633792074t_real @ M @ S @ X @ T @ Y )
       => ( member_nat_real @ Y @ ( sigma_1747752005702207822t_real @ M @ T ) ) ) ) ).

% prob_space.indep_var_rv2
thf(fact_117_prob__space_Oindep__var__rv2,axiom,
    ! [M: sigma_measure_a,S: sigma_measure_real,X: a > real,T: sigma_measure_real,Y: a > real] :
      ( ( probab7247484486040049089pace_a @ M )
     => ( ( indepe8958435565499147358a_real @ M @ S @ X @ T @ Y )
       => ( member_a_real @ Y @ ( sigma_9116425665531756122a_real @ M @ T ) ) ) ) ).

% prob_space.indep_var_rv2
thf(fact_118_prob__space_Oindep__var__rv2,axiom,
    ! [M: sigma_measure_real,S: sigma_measure_o,X: real > $o,T: sigma_measure_o,Y: real > $o] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe495429349782043166real_o @ M @ S @ X @ T @ Y )
       => ( member_real_o @ Y @ ( sigma_3939073009482781210real_o @ M @ T ) ) ) ) ).

% prob_space.indep_var_rv2
thf(fact_119_prob__space_Oindep__var__rv2,axiom,
    ! [M: sigma_measure_real,S: sigma_measure_nat,X: real > nat,T: sigma_measure_nat,Y: real > nat] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe3400978226907914954al_nat @ M @ S @ X @ T @ Y )
       => ( member_real_nat @ Y @ ( sigma_6315060578831106510al_nat @ M @ T ) ) ) ) ).

% prob_space.indep_var_rv2
thf(fact_120_prob__space_Oindep__var__rv2,axiom,
    ! [M: sigma_measure_real,S: sigma_7234349610311085201nnreal,X: real > extend8495563244428889912nnreal,T: sigma_7234349610311085201nnreal,Y: real > extend8495563244428889912nnreal] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe6767359503340752434nnreal @ M @ S @ X @ T @ Y )
       => ( member2919562650594848410nnreal @ Y @ ( sigma_9017504469962657078nnreal @ M @ T ) ) ) ) ).

% prob_space.indep_var_rv2
thf(fact_121_prob__space_Oindep__var__rv2,axiom,
    ! [M: sigma_measure_real,S: sigma_3077487657436305159omplex,X: real > complex,T: sigma_3077487657436305159omplex,Y: real > complex] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe1954327081502071720omplex @ M @ S @ X @ T @ Y )
       => ( member_real_complex @ Y @ ( sigma_9111916201866572460omplex @ M @ T ) ) ) ) ).

% prob_space.indep_var_rv2
thf(fact_122_mem__Collect__eq,axiom,
    ! [A2: real,P2: real > $o] :
      ( ( member_real @ A2 @ ( collect_real @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_123_mem__Collect__eq,axiom,
    ! [A2: $o > $o,P2: ( $o > $o ) > $o] :
      ( ( member_o_o @ A2 @ ( collect_o_o @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_124_mem__Collect__eq,axiom,
    ! [A2: b > extend8495563244428889912nnreal,P2: ( b > extend8495563244428889912nnreal ) > $o] :
      ( ( member6418304549040442065nnreal @ A2 @ ( collec2368881789862750227nnreal @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_125_mem__Collect__eq,axiom,
    ! [A2: a > extend8495563244428889912nnreal,P2: ( a > extend8495563244428889912nnreal ) > $o] :
      ( ( member298456594901751504nnreal @ A2 @ ( collec5472405872578835474nnreal @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_126_mem__Collect__eq,axiom,
    ! [A2: a > real,P2: ( a > real ) > $o] :
      ( ( member_a_real @ A2 @ ( collect_a_real @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_127_mem__Collect__eq,axiom,
    ! [A2: real > $o,P2: ( real > $o ) > $o] :
      ( ( member_real_o @ A2 @ ( collect_real_o @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_128_mem__Collect__eq,axiom,
    ! [A2: real > nat,P2: ( real > nat ) > $o] :
      ( ( member_real_nat @ A2 @ ( collect_real_nat @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_129_mem__Collect__eq,axiom,
    ! [A2: real > extend8495563244428889912nnreal,P2: ( real > extend8495563244428889912nnreal ) > $o] :
      ( ( member2919562650594848410nnreal @ A2 @ ( collec9130413544115709400nnreal @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_130_mem__Collect__eq,axiom,
    ! [A2: real > complex,P2: ( real > complex ) > $o] :
      ( ( member_real_complex @ A2 @ ( collect_real_complex @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_131_mem__Collect__eq,axiom,
    ! [A2: real > real,P2: ( real > real ) > $o] :
      ( ( member_real_real @ A2 @ ( collect_real_real @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_132_Collect__mem__eq,axiom,
    ! [A: set_real] :
      ( ( collect_real
        @ ^ [X6: real] : ( member_real @ X6 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_133_Collect__mem__eq,axiom,
    ! [A: set_o_o] :
      ( ( collect_o_o
        @ ^ [X6: $o > $o] : ( member_o_o @ X6 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_134_Collect__mem__eq,axiom,
    ! [A: set_b_6825823330181178888nnreal] :
      ( ( collec2368881789862750227nnreal
        @ ^ [X6: b > extend8495563244428889912nnreal] : ( member6418304549040442065nnreal @ X6 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_135_Collect__mem__eq,axiom,
    ! [A: set_a_7161065143582548615nnreal] :
      ( ( collec5472405872578835474nnreal
        @ ^ [X6: a > extend8495563244428889912nnreal] : ( member298456594901751504nnreal @ X6 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_136_Collect__mem__eq,axiom,
    ! [A: set_a_real] :
      ( ( collect_a_real
        @ ^ [X6: a > real] : ( member_a_real @ X6 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_137_Collect__mem__eq,axiom,
    ! [A: set_real_o] :
      ( ( collect_real_o
        @ ^ [X6: real > $o] : ( member_real_o @ X6 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_138_Collect__mem__eq,axiom,
    ! [A: set_real_nat] :
      ( ( collect_real_nat
        @ ^ [X6: real > nat] : ( member_real_nat @ X6 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_139_Collect__mem__eq,axiom,
    ! [A: set_re5328672808648366137nnreal] :
      ( ( collec9130413544115709400nnreal
        @ ^ [X6: real > extend8495563244428889912nnreal] : ( member2919562650594848410nnreal @ X6 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_140_Collect__mem__eq,axiom,
    ! [A: set_real_complex] :
      ( ( collect_real_complex
        @ ^ [X6: real > complex] : ( member_real_complex @ X6 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_141_Collect__mem__eq,axiom,
    ! [A: set_real_real] :
      ( ( collect_real_real
        @ ^ [X6: real > real] : ( member_real_real @ X6 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_142_Collect__cong,axiom,
    ! [P2: real > $o,Q: real > $o] :
      ( ! [X3: real] :
          ( ( P2 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_real @ P2 )
        = ( collect_real @ Q ) ) ) ).

% Collect_cong
thf(fact_143_prob__space_Oindep__var__rv1,axiom,
    ! [M: sigma_measure_o,S: sigma_measure_o,X: $o > $o,T: sigma_measure_o,Y: $o > $o] :
      ( ( probab1190487603588612059pace_o @ M )
     => ( ( indepe6952847619639754700ar_o_o @ M @ S @ X @ T @ Y )
       => ( member_o_o @ X @ ( sigma_measurable_o_o @ M @ S ) ) ) ) ).

% prob_space.indep_var_rv1
thf(fact_144_prob__space_Oindep__var__rv1,axiom,
    ! [M: sigma_measure_b,S: sigma_7234349610311085201nnreal,X: b > extend8495563244428889912nnreal,T: sigma_7234349610311085201nnreal,Y: b > extend8495563244428889912nnreal] :
      ( ( probab7247484486040049090pace_b @ M )
     => ( ( indepe430593609325189611nnreal @ M @ S @ X @ T @ Y )
       => ( member6418304549040442065nnreal @ X @ ( sigma_6334800283702579687nnreal @ M @ S ) ) ) ) ).

% prob_space.indep_var_rv1
thf(fact_145_prob__space_Oindep__var__rv1,axiom,
    ! [M: sigma_measure_a,S: sigma_7234349610311085201nnreal,X: a > extend8495563244428889912nnreal,T: sigma_7234349610311085201nnreal,Y: a > extend8495563244428889912nnreal] :
      ( ( probab7247484486040049089pace_a @ M )
     => ( ( indepe3534117692041274858nnreal @ M @ S @ X @ T @ Y )
       => ( member298456594901751504nnreal @ X @ ( sigma_214952329563889126nnreal @ M @ S ) ) ) ) ).

% prob_space.indep_var_rv1
thf(fact_146_prob__space_Oindep__var__rv1,axiom,
    ! [M: sigma_measure_o,S: sigma_measure_real,X: $o > real,T: sigma_measure_real,Y: $o > real] :
      ( ( probab1190487603588612059pace_o @ M )
     => ( ( indepe8209737011595649400o_real @ M @ S @ X @ T @ Y )
       => ( member_o_real @ X @ ( sigma_2430008634441611636o_real @ M @ S ) ) ) ) ).

% prob_space.indep_var_rv1
thf(fact_147_prob__space_Oindep__var__rv1,axiom,
    ! [M: sigma_measure_nat,S: sigma_measure_real,X: nat > real,T: sigma_measure_real,Y: nat > real] :
      ( ( probab2904919403188438605ce_nat @ M )
     => ( ( indepe8057041690633792074t_real @ M @ S @ X @ T @ Y )
       => ( member_nat_real @ X @ ( sigma_1747752005702207822t_real @ M @ S ) ) ) ) ).

% prob_space.indep_var_rv1
thf(fact_148_prob__space_Oindep__var__rv1,axiom,
    ! [M: sigma_measure_a,S: sigma_measure_real,X: a > real,T: sigma_measure_real,Y: a > real] :
      ( ( probab7247484486040049089pace_a @ M )
     => ( ( indepe8958435565499147358a_real @ M @ S @ X @ T @ Y )
       => ( member_a_real @ X @ ( sigma_9116425665531756122a_real @ M @ S ) ) ) ) ).

% prob_space.indep_var_rv1
thf(fact_149_prob__space_Oindep__var__rv1,axiom,
    ! [M: sigma_measure_real,S: sigma_measure_o,X: real > $o,T: sigma_measure_o,Y: real > $o] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe495429349782043166real_o @ M @ S @ X @ T @ Y )
       => ( member_real_o @ X @ ( sigma_3939073009482781210real_o @ M @ S ) ) ) ) ).

% prob_space.indep_var_rv1
thf(fact_150_prob__space_Oindep__var__rv1,axiom,
    ! [M: sigma_measure_real,S: sigma_measure_nat,X: real > nat,T: sigma_measure_nat,Y: real > nat] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe3400978226907914954al_nat @ M @ S @ X @ T @ Y )
       => ( member_real_nat @ X @ ( sigma_6315060578831106510al_nat @ M @ S ) ) ) ) ).

% prob_space.indep_var_rv1
thf(fact_151_prob__space_Oindep__var__rv1,axiom,
    ! [M: sigma_measure_real,S: sigma_7234349610311085201nnreal,X: real > extend8495563244428889912nnreal,T: sigma_7234349610311085201nnreal,Y: real > extend8495563244428889912nnreal] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe6767359503340752434nnreal @ M @ S @ X @ T @ Y )
       => ( member2919562650594848410nnreal @ X @ ( sigma_9017504469962657078nnreal @ M @ S ) ) ) ) ).

% prob_space.indep_var_rv1
thf(fact_152_prob__space_Oindep__var__rv1,axiom,
    ! [M: sigma_measure_real,S: sigma_3077487657436305159omplex,X: real > complex,T: sigma_3077487657436305159omplex,Y: real > complex] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe1954327081502071720omplex @ M @ S @ X @ T @ Y )
       => ( member_real_complex @ X @ ( sigma_9111916201866572460omplex @ M @ S ) ) ) ) ).

% prob_space.indep_var_rv1
thf(fact_153_rep__monadP__qbs__MPx,axiom,
    ! [Beta2: real > probab1241297377463522905pace_o,X: quasi_borel_o] :
      ( ( member150931338705623565pace_o @ Beta2 @ ( monad_7120508265431748167_MPx_o @ X ) )
     => ? [Alpha2: real > $o,G3: real > sigma_measure_real] :
          ( ( member_real_o @ Alpha2 @ ( qbs_Mx_o @ X ) )
          & ( member2630560753458908601e_real @ G3 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
          & ( Beta2
            = ( ^ [R: real] : ( probab414947219978445330pace_o @ ( produc2942336022369489698e_real @ X @ ( produc787877863769208121e_real @ Alpha2 @ ( G3 @ R ) ) ) ) ) ) ) ) ).

% rep_monadP_qbs_MPx
thf(fact_154_rep__monadP__qbs__MPx,axiom,
    ! [Beta2: real > probab8370124847414953445ce_nat,X: quasi_borel_nat] :
      ( ( member5760356831232730823ce_nat @ Beta2 @ ( monad_6889595236306528993Px_nat @ X ) )
     => ? [Alpha2: real > nat,G3: real > sigma_measure_real] :
          ( ( member_real_nat @ Alpha2 @ ( qbs_Mx_nat @ X ) )
          & ( member2630560753458908601e_real @ G3 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
          & ( Beta2
            = ( ^ [R: real] : ( probab4893816680317050838ce_nat @ ( produc2796221405228754040e_real @ X @ ( produc6760937697707383505e_real @ Alpha2 @ ( G3 @ R ) ) ) ) ) ) ) ) ).

% rep_monadP_qbs_MPx
thf(fact_155_rep__monadP__qbs__MPx,axiom,
    ! [Beta2: real > probab1359147627358338381nnreal,X: quasi_9015997321629101608nnreal] :
      ( ( member1466800385206301231nnreal @ Beta2 @ ( monad_5049177628250775113nnreal @ X ) )
     => ? [Alpha2: real > extend8495563244428889912nnreal,G3: real > sigma_measure_real] :
          ( ( member2919562650594848410nnreal @ Alpha2 @ ( qbs_Mx6523938229262583809nnreal @ X ) )
          & ( member2630560753458908601e_real @ G3 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
          & ( Beta2
            = ( ^ [R: real] : ( probab2095897665094527806nnreal @ ( produc3311748387682954872e_real @ X @ ( produc8591767778467529065e_real @ Alpha2 @ ( G3 @ R ) ) ) ) ) ) ) ) ).

% rep_monadP_qbs_MPx
thf(fact_156_rep__monadP__qbs__MPx,axiom,
    ! [Beta2: real > probab3778977310982246339omplex,X: quasi_borel_complex] :
      ( ( member22330080113046437omplex @ Beta2 @ ( monad_1302594500181521855omplex @ X ) )
     => ? [Alpha2: real > complex,G3: real > sigma_measure_real] :
          ( ( member_real_complex @ Alpha2 @ ( qbs_Mx_complex @ X ) )
          & ( member2630560753458908601e_real @ G3 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
          & ( Beta2
            = ( ^ [R: real] : ( probab2835974231434789044omplex @ ( produc7812373664507111416e_real @ X @ ( produc8166640644904819571e_real @ Alpha2 @ ( G3 @ R ) ) ) ) ) ) ) ) ).

% rep_monadP_qbs_MPx
thf(fact_157_rep__monadP__qbs__MPx,axiom,
    ! [Beta2: real > probab8009751763329705409e_real,X: quasi_borel_real] :
      ( ( member895572740875562275e_real @ Beta2 @ ( monad_5590613879340176701x_real @ X ) )
     => ? [Alpha2: real > real,G3: real > sigma_measure_real] :
          ( ( member_real_real @ Alpha2 @ ( qbs_Mx_real @ X ) )
          & ( member2630560753458908601e_real @ G3 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
          & ( Beta2
            = ( ^ [R: real] : ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X @ ( produc1722724976708544245e_real @ Alpha2 @ ( G3 @ R ) ) ) ) ) ) ) ) ).

% rep_monadP_qbs_MPx
thf(fact_158_rep__monadP__qbs__MPx,axiom,
    ! [Beta2: real > probab4737552677800996672pace_b,X: quasi_borel_b] :
      ( ( member5892728515905534708pace_b @ Beta2 @ ( monad_4223963853064302126_MPx_b @ X ) )
     => ? [Alpha2: real > b,G3: real > sigma_measure_real] :
          ( ( member_real_b @ Alpha2 @ ( qbs_Mx_b @ X ) )
          & ( member2630560753458908601e_real @ G3 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
          & ( Beta2
            = ( ^ [R: real] : ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha2 @ ( G3 @ R ) ) ) ) ) ) ) ) ).

% rep_monadP_qbs_MPx
thf(fact_159_rep__monadP__qbs__MPx,axiom,
    ! [Beta2: real > probab4737552673497767871pace_a,X: quasi_borel_a] :
      ( ( member5821694475859188723pace_a @ Beta2 @ ( monad_4223963853064302125_MPx_a @ X ) )
     => ? [Alpha2: real > a,G3: real > sigma_measure_real] :
          ( ( member_real_a @ Alpha2 @ ( qbs_Mx_a @ X ) )
          & ( member2630560753458908601e_real @ G3 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
          & ( Beta2
            = ( ^ [R: real] : ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha2 @ ( G3 @ R ) ) ) ) ) ) ) ) ).

% rep_monadP_qbs_MPx
thf(fact_160_qp_Oqbs__prob__axioms,axiom,
    probab701741629625904796prob_a @ x @ alpha @ mu ).

% qp.qbs_prob_axioms
thf(fact_161_qp_Obind__in__space,axiom,
    ! [A: real > sigma_measure_real,N: sigma_measure_real] :
      ( ( member2630560753458908601e_real @ A @ ( sigma_5928869325259027335e_real @ mu @ ( giry_s5092570657895779418a_real @ N ) ) )
     => ( member4553183543495551918e_real @ ( giry_bind_real_real @ mu @ A ) @ ( sigma_2594925453452915853e_real @ ( giry_s5092570657895779418a_real @ N ) ) ) ) ).

% qp.bind_in_space
thf(fact_162_qbs__join__computation_I1_J,axiom,
    ! [X: quasi_borel_o,Beta2: real > probab1241297377463522905pace_o,Mu: sigma_measure_real,Ssx: probab691856902995763354pace_o,Alpha: real > $o,G2: real > sigma_measure_real] :
      ( ( probab3620450583171887351pace_o @ ( monad_monadP_qbs_o @ X ) @ Beta2 @ Mu )
     => ( ( Ssx
          = ( probab1393150374121012435pace_o @ ( produc1093366075506243232e_real @ ( monad_monadP_qbs_o @ X ) @ ( produc982178768907016120e_real @ Beta2 @ Mu ) ) ) )
       => ( ( member_real_o @ Alpha @ ( qbs_Mx_o @ X ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( Beta2
                = ( ^ [R: real] : ( probab414947219978445330pace_o @ ( produc2942336022369489698e_real @ X @ ( produc787877863769208121e_real @ Alpha @ ( G2 @ R ) ) ) ) ) )
             => ( probab163731277575722550prob_o @ X @ Alpha @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ).

% qbs_join_computation(1)
thf(fact_163_qbs__join__computation_I1_J,axiom,
    ! [X: quasi_borel_nat,Beta2: real > probab8370124847414953445ce_nat,Mu: sigma_measure_real,Ssx: probab3005566686192770426ce_nat,Alpha: real > nat,G2: real > sigma_measure_real] :
      ( ( probab2468401811917278919ce_nat @ ( monad_monadP_qbs_nat @ X ) @ Beta2 @ Mu )
     => ( ( Ssx
          = ( probab1586022766323937771ce_nat @ ( produc5084857187822765240e_real @ ( monad_monadP_qbs_nat @ X ) @ ( produc6214735565764730620e_real @ Beta2 @ Mu ) ) ) )
       => ( ( member_real_nat @ Alpha @ ( qbs_Mx_nat @ X ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( Beta2
                = ( ^ [R: real] : ( probab4893816680317050838ce_nat @ ( produc2796221405228754040e_real @ X @ ( produc6760937697707383505e_real @ Alpha @ ( G2 @ R ) ) ) ) ) )
             => ( probab2851505236026752178ob_nat @ X @ Alpha @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ).

% qbs_join_computation(1)
thf(fact_164_qbs__join__computation_I1_J,axiom,
    ! [X: quasi_9015997321629101608nnreal,Beta2: real > probab1359147627358338381nnreal,Mu: sigma_measure_real,Ssx: probab3223874146922238178nnreal,Alpha: real > extend8495563244428889912nnreal,G2: real > sigma_measure_real] :
      ( ( probab8011796075923174959nnreal @ ( monad_8737719049617959390nnreal @ X ) @ Beta2 @ Mu )
     => ( ( Ssx
          = ( probab2630004677939362131nnreal @ ( produc9122815929407448760e_real @ ( monad_8737719049617959390nnreal @ X ) @ ( produc3526280104234912148e_real @ Beta2 @ Mu ) ) ) )
       => ( ( member2919562650594848410nnreal @ Alpha @ ( qbs_Mx6523938229262583809nnreal @ X ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( Beta2
                = ( ^ [R: real] : ( probab2095897665094527806nnreal @ ( produc3311748387682954872e_real @ X @ ( produc8591767778467529065e_real @ Alpha @ ( G2 @ R ) ) ) ) ) )
             => ( probab8888997264033409050nnreal @ X @ Alpha @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ).

% qbs_join_computation(1)
thf(fact_165_qbs__join__computation_I1_J,axiom,
    ! [X: quasi_borel_complex,Beta2: real > probab3778977310982246339omplex,Mu: sigma_measure_real,Ssx: probab3364748380892496984omplex,Alpha: real > complex,G2: real > sigma_measure_real] :
      ( ( probab6260968705227822245omplex @ ( monad_3228211519047171924omplex @ X ) @ Beta2 @ Mu )
     => ( ( Ssx
          = ( probab8929889938064994249omplex @ ( produc4831291127269080632e_real @ ( monad_3228211519047171924omplex @ X ) @ ( produc4083023864984044190e_real @ Beta2 @ Mu ) ) ) )
       => ( ( member_real_complex @ Alpha @ ( qbs_Mx_complex @ X ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( Beta2
                = ( ^ [R: real] : ( probab2835974231434789044omplex @ ( produc7812373664507111416e_real @ X @ ( produc8166640644904819571e_real @ Alpha @ ( G2 @ R ) ) ) ) ) )
             => ( probab963564174529062288omplex @ X @ Alpha @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ).

% qbs_join_computation(1)
thf(fact_166_qbs__join__computation_I1_J,axiom,
    ! [X: quasi_borel_real,Beta2: real > probab8009751763329705409e_real,Mu: sigma_measure_real,Ssx: probab3093221366759455318e_real,Alpha: real > real,G2: real > sigma_measure_real] :
      ( ( probab8234793495611891107e_real @ ( monad_2887651017592114770s_real @ X ) @ Beta2 @ Mu )
     => ( ( Ssx
          = ( probab1999026297785200327e_real @ ( produc2581243556061882552e_real @ ( monad_2887651017592114770s_real @ X ) @ ( produc1095797480378445344e_real @ Beta2 @ Mu ) ) ) )
       => ( ( member_real_real @ Alpha @ ( qbs_Mx_real @ X ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( Beta2
                = ( ^ [R: real] : ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X @ ( produc1722724976708544245e_real @ Alpha @ ( G2 @ R ) ) ) ) ) )
             => ( probab3605210969150000782b_real @ X @ Alpha @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ).

% qbs_join_computation(1)
thf(fact_167_qbs__join__computation_I1_J,axiom,
    ! [X: quasi_borel_b,Beta2: real > probab4737552677800996672pace_b,Mu: sigma_measure_real,Ssx: probab8624551251471354625pace_b,Alpha: real > b,G2: real > sigma_measure_real] :
      ( ( probab2008080876895902430pace_b @ ( monad_monadP_qbs_b @ X ) @ Beta2 @ Mu )
     => ( ( Ssx
          = ( probab2758740287680477370pace_b @ ( produc4607355092896878674e_real @ ( monad_monadP_qbs_b @ X ) @ ( produc2065420038842681233e_real @ Beta2 @ Mu ) ) ) )
       => ( ( member_real_b @ Alpha @ ( qbs_Mx_b @ X ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( Beta2
                = ( ^ [R: real] : ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ ( G2 @ R ) ) ) ) ) )
             => ( probab701741629625904797prob_b @ X @ Alpha @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ).

% qbs_join_computation(1)
thf(fact_168_qbs__join__computation_I1_J,axiom,
    ! [X: quasi_borel_a,Beta2: real > probab4737552673497767871pace_a,Mu: sigma_measure_real,Ssx: probab8553517211425008640pace_a,Alpha: real > a,G2: real > sigma_measure_real] :
      ( ( probab2008080872592673629pace_a @ ( monad_monadP_qbs_a @ X ) @ Beta2 @ Mu )
     => ( ( Ssx
          = ( probab2758740283377248569pace_a @ ( produc6726600922647680468e_real @ ( monad_monadP_qbs_a @ X ) @ ( produc2709302780738849234e_real @ Beta2 @ Mu ) ) ) )
       => ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( Beta2
                = ( ^ [R: real] : ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ ( G2 @ R ) ) ) ) ) )
             => ( probab701741629625904796prob_a @ X @ Alpha @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ).

% qbs_join_computation(1)
thf(fact_169_qp_Ochar__measurable,axiom,
    member_real_complex @ ( characteristic_char @ mu ) @ ( sigma_9111916201866572460omplex @ borel_5078946678739801102l_real @ borel_1392132677378845456omplex ) ).

% qp.char_measurable
thf(fact_170_qp_Oindep__sets__mono__sets,axiom,
    ! [F2: ( $o > $o ) > set_set_real,I: set_o_o,G: ( $o > $o ) > set_set_real] :
      ( ( indepe6032362129116668415al_o_o @ mu @ F2 @ I )
     => ( ! [I2: $o > $o] :
            ( ( member_o_o @ I2 @ I )
           => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe6032362129116668415al_o_o @ mu @ G @ I ) ) ) ).

% qp.indep_sets_mono_sets
thf(fact_171_qp_Oindep__sets__mono__sets,axiom,
    ! [F2: ( b > extend8495563244428889912nnreal ) > set_set_real,I: set_b_6825823330181178888nnreal,G: ( b > extend8495563244428889912nnreal ) > set_set_real] :
      ( ( indepe4831269413077975822nnreal @ mu @ F2 @ I )
     => ( ! [I2: b > extend8495563244428889912nnreal] :
            ( ( member6418304549040442065nnreal @ I2 @ I )
           => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe4831269413077975822nnreal @ mu @ G @ I ) ) ) ).

% qp.indep_sets_mono_sets
thf(fact_172_qp_Oindep__sets__mono__sets,axiom,
    ! [F2: ( a > probab4737552677800996672pace_b ) > set_set_real,I: set_a_3263942136899480541pace_b,G: ( a > probab4737552677800996672pace_b ) > set_set_real] :
      ( ( indepe1711633972345727297pace_b @ mu @ F2 @ I )
     => ( ! [I2: a > probab4737552677800996672pace_b] :
            ( ( member7340901614391157822pace_b @ I2 @ I )
           => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe1711633972345727297pace_b @ mu @ G @ I ) ) ) ).

% qp.indep_sets_mono_sets
thf(fact_173_qp_Oindep__sets__mono__sets,axiom,
    ! [F2: ( a > extend8495563244428889912nnreal ) > set_set_real,I: set_a_7161065143582548615nnreal,G: ( a > extend8495563244428889912nnreal ) > set_set_real] :
      ( ( indepe7934793495794061069nnreal @ mu @ F2 @ I )
     => ( ! [I2: a > extend8495563244428889912nnreal] :
            ( ( member298456594901751504nnreal @ I2 @ I )
           => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe7934793495794061069nnreal @ mu @ G @ I ) ) ) ).

% qp.indep_sets_mono_sets
thf(fact_174_qp_Oindep__sets__mono__sets,axiom,
    ! [F2: ( a > real ) > set_set_real,I: set_a_real,G: ( a > real ) > set_set_real] :
      ( ( indepe7634599452152747905a_real @ mu @ F2 @ I )
     => ( ! [I2: a > real] :
            ( ( member_a_real @ I2 @ I )
           => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe7634599452152747905a_real @ mu @ G @ I ) ) ) ).

% qp.indep_sets_mono_sets
thf(fact_175_qp_Oindep__sets__mono__sets,axiom,
    ! [F2: ( real > $o ) > set_set_real,I: set_real_o,G: ( real > $o ) > set_set_real] :
      ( ( indepe8014066527605497865real_o @ mu @ F2 @ I )
     => ( ! [I2: real > $o] :
            ( ( member_real_o @ I2 @ I )
           => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe8014066527605497865real_o @ mu @ G @ I ) ) ) ).

% qp.indep_sets_mono_sets
thf(fact_176_qp_Oindep__sets__mono__sets,axiom,
    ! [F2: ( real > nat ) > set_set_real,I: set_real_nat,G: ( real > nat ) > set_set_real] :
      ( ( indepe7235330332140030005al_nat @ mu @ F2 @ I )
     => ( ! [I2: real > nat] :
            ( ( member_real_nat @ I2 @ I )
           => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe7235330332140030005al_nat @ mu @ G @ I ) ) ) ).

% qp.indep_sets_mono_sets
thf(fact_177_qp_Oindep__sets__mono__sets,axiom,
    ! [F2: ( real > extend8495563244428889912nnreal ) > set_set_real,I: set_re5328672808648366137nnreal,G: ( real > extend8495563244428889912nnreal ) > set_set_real] :
      ( ( indepe4643477836443271069nnreal @ mu @ F2 @ I )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe4643477836443271069nnreal @ mu @ G @ I ) ) ) ).

% qp.indep_sets_mono_sets
thf(fact_178_qp_Oindep__sets__mono__sets,axiom,
    ! [F2: ( real > complex ) > set_set_real,I: set_real_complex,G: ( real > complex ) > set_set_real] :
      ( ( indepe6244138955099140627omplex @ mu @ F2 @ I )
     => ( ! [I2: real > complex] :
            ( ( member_real_complex @ I2 @ I )
           => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe6244138955099140627omplex @ mu @ G @ I ) ) ) ).

% qp.indep_sets_mono_sets
thf(fact_179_qp_Oindep__sets__mono__sets,axiom,
    ! [F2: ( real > real ) > set_set_real,I: set_real_real,G: ( real > real ) > set_set_real] :
      ( ( indepe4905764430350991377l_real @ mu @ F2 @ I )
     => ( ! [I2: real > real] :
            ( ( member_real_real @ I2 @ I )
           => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
       => ( indepe4905764430350991377l_real @ mu @ G @ I ) ) ) ).

% qp.indep_sets_mono_sets
thf(fact_180_qp_Oindep__sets__mono__index,axiom,
    ! [J: set_set_real,I: set_set_real,F2: set_real > set_set_real] :
      ( ( ord_le3558479182127378552t_real @ J @ I )
     => ( ( indepe8752365572443096444t_real @ mu @ F2 @ I )
       => ( indepe8752365572443096444t_real @ mu @ F2 @ J ) ) ) ).

% qp.indep_sets_mono_index
thf(fact_181_qp_Oindep__sets__mono__index,axiom,
    ! [J: set_real,I: set_real,F2: real > set_set_real] :
      ( ( ord_less_eq_set_real @ J @ I )
     => ( ( indepe8783372407961027910l_real @ mu @ F2 @ I )
       => ( indepe8783372407961027910l_real @ mu @ F2 @ J ) ) ) ).

% qp.indep_sets_mono_index
thf(fact_182_comp__apply,axiom,
    ( comp_real_real_real
    = ( ^ [F3: real > real,G4: real > real,X6: real] : ( F3 @ ( G4 @ X6 ) ) ) ) ).

% comp_apply
thf(fact_183_comp__apply,axiom,
    ( comp_real_o_o
    = ( ^ [F3: real > $o,G4: $o > real,X6: $o] : ( F3 @ ( G4 @ X6 ) ) ) ) ).

% comp_apply
thf(fact_184_comp__apply,axiom,
    ( comp_real_nat_nat
    = ( ^ [F3: real > nat,G4: nat > real,X6: nat] : ( F3 @ ( G4 @ X6 ) ) ) ) ).

% comp_apply
thf(fact_185_comp__apply,axiom,
    ( comp_a6868616473110226257b_real
    = ( ^ [F3: a > probab4737552677800996672pace_b,G4: real > a,X6: real] : ( F3 @ ( G4 @ X6 ) ) ) ) ).

% comp_apply
thf(fact_186_comp__apply,axiom,
    ( comp_a_real_real
    = ( ^ [F3: a > real,G4: real > a,X6: real] : ( F3 @ ( G4 @ X6 ) ) ) ) ).

% comp_apply
thf(fact_187_old_Oprod_Oinject,axiom,
    ! [A2: quasi_borel_b,B2: produc4580635503675987618e_real,A3: quasi_borel_b,B3: produc4580635503675987618e_real] :
      ( ( ( produc2180226129289916244e_real @ A2 @ B2 )
        = ( produc2180226129289916244e_real @ A3 @ B3 ) )
      = ( ( A2 = A3 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_188_old_Oprod_Oinject,axiom,
    ! [A2: real > b,B2: sigma_measure_real,A3: real > b,B3: sigma_measure_real] :
      ( ( ( produc4478270668571743890e_real @ A2 @ B2 )
        = ( produc4478270668571743890e_real @ A3 @ B3 ) )
      = ( ( A2 = A3 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_189_old_Oprod_Oinject,axiom,
    ! [A2: quasi_borel_a,B2: produc725540845905733987e_real,A3: quasi_borel_a,B3: produc725540845905733987e_real] :
      ( ( ( produc4145838808978236886e_real @ A2 @ B2 )
        = ( produc4145838808978236886e_real @ A3 @ B3 ) )
      = ( ( A2 = A3 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_190_old_Oprod_Oinject,axiom,
    ! [A2: real > a,B2: sigma_measure_real,A3: real > a,B3: sigma_measure_real] :
      ( ( ( produc623176010801490259e_real @ A2 @ B2 )
        = ( produc623176010801490259e_real @ A3 @ B3 ) )
      = ( ( A2 = A3 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_191_prod_Oinject,axiom,
    ! [X12: quasi_borel_b,X22: produc4580635503675987618e_real,Y12: quasi_borel_b,Y22: produc4580635503675987618e_real] :
      ( ( ( produc2180226129289916244e_real @ X12 @ X22 )
        = ( produc2180226129289916244e_real @ Y12 @ Y22 ) )
      = ( ( X12 = Y12 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_192_prod_Oinject,axiom,
    ! [X12: real > b,X22: sigma_measure_real,Y12: real > b,Y22: sigma_measure_real] :
      ( ( ( produc4478270668571743890e_real @ X12 @ X22 )
        = ( produc4478270668571743890e_real @ Y12 @ Y22 ) )
      = ( ( X12 = Y12 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_193_prod_Oinject,axiom,
    ! [X12: quasi_borel_a,X22: produc725540845905733987e_real,Y12: quasi_borel_a,Y22: produc725540845905733987e_real] :
      ( ( ( produc4145838808978236886e_real @ X12 @ X22 )
        = ( produc4145838808978236886e_real @ Y12 @ Y22 ) )
      = ( ( X12 = Y12 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_194_prod_Oinject,axiom,
    ! [X12: real > a,X22: sigma_measure_real,Y12: real > a,Y22: sigma_measure_real] :
      ( ( ( produc623176010801490259e_real @ X12 @ X22 )
        = ( produc623176010801490259e_real @ Y12 @ Y22 ) )
      = ( ( X12 = Y12 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_195_qp_OM__in__subprob,axiom,
    member4553183543495551918e_real @ mu @ ( sigma_2594925453452915853e_real @ ( giry_s5092570657895779418a_real @ mu ) ) ).

% qp.M_in_subprob
thf(fact_196_qp_Oin__Mx,axiom,
    member_real_a @ alpha @ ( qbs_Mx_a @ x ) ).

% qp.in_Mx
thf(fact_197_hs_I2_J,axiom,
    ( s
    = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) ) ).

% hs(2)
thf(fact_198_qp_Oindep__sets__mono,axiom,
    ! [F2: real > set_set_real,I: set_real,J: set_real,G: real > set_set_real] :
      ( ( indepe8783372407961027910l_real @ mu @ F2 @ I )
     => ( ( ord_less_eq_set_real @ J @ I )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ J )
             => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe8783372407961027910l_real @ mu @ G @ J ) ) ) ) ).

% qp.indep_sets_mono
thf(fact_199_qp_Oindep__sets__mono,axiom,
    ! [F2: set_real > set_set_real,I: set_set_real,J: set_set_real,G: set_real > set_set_real] :
      ( ( indepe8752365572443096444t_real @ mu @ F2 @ I )
     => ( ( ord_le3558479182127378552t_real @ J @ I )
       => ( ! [I2: set_real] :
              ( ( member_set_real @ I2 @ J )
             => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe8752365572443096444t_real @ mu @ G @ J ) ) ) ) ).

% qp.indep_sets_mono
thf(fact_200_qp_Oindep__sets__mono,axiom,
    ! [F2: ( $o > $o ) > set_set_real,I: set_o_o,J: set_o_o,G: ( $o > $o ) > set_set_real] :
      ( ( indepe6032362129116668415al_o_o @ mu @ F2 @ I )
     => ( ( ord_less_eq_set_o_o @ J @ I )
       => ( ! [I2: $o > $o] :
              ( ( member_o_o @ I2 @ J )
             => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe6032362129116668415al_o_o @ mu @ G @ J ) ) ) ) ).

% qp.indep_sets_mono
thf(fact_201_qp_Oindep__sets__mono,axiom,
    ! [F2: ( b > extend8495563244428889912nnreal ) > set_set_real,I: set_b_6825823330181178888nnreal,J: set_b_6825823330181178888nnreal,G: ( b > extend8495563244428889912nnreal ) > set_set_real] :
      ( ( indepe4831269413077975822nnreal @ mu @ F2 @ I )
     => ( ( ord_le672203391976590760nnreal @ J @ I )
       => ( ! [I2: b > extend8495563244428889912nnreal] :
              ( ( member6418304549040442065nnreal @ I2 @ J )
             => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe4831269413077975822nnreal @ mu @ G @ J ) ) ) ) ).

% qp.indep_sets_mono
thf(fact_202_qp_Oindep__sets__mono,axiom,
    ! [F2: ( a > extend8495563244428889912nnreal ) > set_set_real,I: set_a_7161065143582548615nnreal,J: set_a_7161065143582548615nnreal,G: ( a > extend8495563244428889912nnreal ) > set_set_real] :
      ( ( indepe7934793495794061069nnreal @ mu @ F2 @ I )
     => ( ( ord_le1007445205377960487nnreal @ J @ I )
       => ( ! [I2: a > extend8495563244428889912nnreal] :
              ( ( member298456594901751504nnreal @ I2 @ J )
             => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe7934793495794061069nnreal @ mu @ G @ J ) ) ) ) ).

% qp.indep_sets_mono
thf(fact_203_qp_Oindep__sets__mono,axiom,
    ! [F2: ( a > real ) > set_set_real,I: set_a_real,J: set_a_real,G: ( a > real ) > set_set_real] :
      ( ( indepe7634599452152747905a_real @ mu @ F2 @ I )
     => ( ( ord_le3334967407727675675a_real @ J @ I )
       => ( ! [I2: a > real] :
              ( ( member_a_real @ I2 @ J )
             => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe7634599452152747905a_real @ mu @ G @ J ) ) ) ) ).

% qp.indep_sets_mono
thf(fact_204_qp_Oindep__sets__mono,axiom,
    ! [F2: ( real > $o ) > set_set_real,I: set_real_o,J: set_real_o,G: ( real > $o ) > set_set_real] :
      ( ( indepe8014066527605497865real_o @ mu @ F2 @ I )
     => ( ( ord_le1615110227528160547real_o @ J @ I )
       => ( ! [I2: real > $o] :
              ( ( member_real_o @ I2 @ J )
             => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe8014066527605497865real_o @ mu @ G @ J ) ) ) ) ).

% qp.indep_sets_mono
thf(fact_205_qp_Oindep__sets__mono,axiom,
    ! [F2: ( real > nat ) > set_set_real,I: set_real_nat,J: set_real_nat,G: ( real > nat ) > set_set_real] :
      ( ( indepe7235330332140030005al_nat @ mu @ F2 @ I )
     => ( ( ord_le6098800555920186673al_nat @ J @ I )
       => ( ! [I2: real > nat] :
              ( ( member_real_nat @ I2 @ J )
             => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe7235330332140030005al_nat @ mu @ G @ J ) ) ) ) ).

% qp.indep_sets_mono
thf(fact_206_qp_Oindep__sets__mono,axiom,
    ! [F2: ( real > extend8495563244428889912nnreal ) > set_set_real,I: set_re5328672808648366137nnreal,J: set_re5328672808648366137nnreal,G: ( real > extend8495563244428889912nnreal ) > set_set_real] :
      ( ( indepe4643477836443271069nnreal @ mu @ F2 @ I )
     => ( ( ord_le2462468573666744473nnreal @ J @ I )
       => ( ! [I2: real > extend8495563244428889912nnreal] :
              ( ( member2919562650594848410nnreal @ I2 @ J )
             => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe4643477836443271069nnreal @ mu @ G @ J ) ) ) ) ).

% qp.indep_sets_mono
thf(fact_207_qp_Oindep__sets__mono,axiom,
    ! [F2: ( real > complex ) > set_set_real,I: set_real_complex,J: set_real_complex,G: ( real > complex ) > set_set_real] :
      ( ( indepe6244138955099140627omplex @ mu @ F2 @ I )
     => ( ( ord_le2047140485929309711omplex @ J @ I )
       => ( ! [I2: real > complex] :
              ( ( member_real_complex @ I2 @ J )
             => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe6244138955099140627omplex @ mu @ G @ J ) ) ) ) ).

% qp.indep_sets_mono
thf(fact_208_monadP__qbs__Mx,axiom,
    ! [X: quasi_borel_b] :
      ( ( qbs_Mx4221877620443553463pace_b @ ( monad_monadP_qbs_b @ X ) )
      = ( monad_4223963853064302126_MPx_b @ X ) ) ).

% monadP_qbs_Mx
thf(fact_209__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062_092_060alpha_062_A_092_060mu_062_O_A_092_060lbrakk_062qbs__prob_AX_A_092_060alpha_062_A_092_060mu_062_059_As_A_061_Aqbs__prob__space_A_IX_M_A_092_060alpha_062_M_A_092_060mu_062_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Alpha2: real > a,Mu2: sigma_measure_real] :
        ( ( probab701741629625904796prob_a @ x @ Alpha2 @ Mu2 )
       => ( s
         != ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ Alpha2 @ Mu2 ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>\<alpha> \<mu>. \<lbrakk>qbs_prob X \<alpha> \<mu>; s = qbs_prob_space (X, \<alpha>, \<mu>)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_210_qp_Oin__Mx__axioms,axiom,
    probab9007417770424356215n_Mx_a @ x @ alpha ).

% qp.in_Mx_axioms
thf(fact_211_assms_I2_J,axiom,
    member7340901614391157822pace_b @ f @ ( qbs_mo6277074943135278010pace_b @ x @ ( monad_monadP_qbs_b @ y ) ) ).

% assms(2)
thf(fact_212_prob__space_Oindep__sets__mono,axiom,
    ! [M: sigma_measure_real,F2: real > set_set_real,I: set_real,J: set_real,G: real > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe8783372407961027910l_real @ M @ F2 @ I )
       => ( ( ord_less_eq_set_real @ J @ I )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ J )
               => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
           => ( indepe8783372407961027910l_real @ M @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_213_prob__space_Oindep__sets__mono,axiom,
    ! [M: sigma_measure_real,F2: set_real > set_set_real,I: set_set_real,J: set_set_real,G: set_real > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe8752365572443096444t_real @ M @ F2 @ I )
       => ( ( ord_le3558479182127378552t_real @ J @ I )
         => ( ! [I2: set_real] :
                ( ( member_set_real @ I2 @ J )
               => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
           => ( indepe8752365572443096444t_real @ M @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_214_prob__space_Oindep__sets__mono,axiom,
    ! [M: sigma_measure_real,F2: ( $o > $o ) > set_set_real,I: set_o_o,J: set_o_o,G: ( $o > $o ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe6032362129116668415al_o_o @ M @ F2 @ I )
       => ( ( ord_less_eq_set_o_o @ J @ I )
         => ( ! [I2: $o > $o] :
                ( ( member_o_o @ I2 @ J )
               => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
           => ( indepe6032362129116668415al_o_o @ M @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_215_prob__space_Oindep__sets__mono,axiom,
    ! [M: sigma_measure_real,F2: ( b > extend8495563244428889912nnreal ) > set_set_real,I: set_b_6825823330181178888nnreal,J: set_b_6825823330181178888nnreal,G: ( b > extend8495563244428889912nnreal ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe4831269413077975822nnreal @ M @ F2 @ I )
       => ( ( ord_le672203391976590760nnreal @ J @ I )
         => ( ! [I2: b > extend8495563244428889912nnreal] :
                ( ( member6418304549040442065nnreal @ I2 @ J )
               => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
           => ( indepe4831269413077975822nnreal @ M @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_216_prob__space_Oindep__sets__mono,axiom,
    ! [M: sigma_measure_real,F2: ( a > extend8495563244428889912nnreal ) > set_set_real,I: set_a_7161065143582548615nnreal,J: set_a_7161065143582548615nnreal,G: ( a > extend8495563244428889912nnreal ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe7934793495794061069nnreal @ M @ F2 @ I )
       => ( ( ord_le1007445205377960487nnreal @ J @ I )
         => ( ! [I2: a > extend8495563244428889912nnreal] :
                ( ( member298456594901751504nnreal @ I2 @ J )
               => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
           => ( indepe7934793495794061069nnreal @ M @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_217_prob__space_Oindep__sets__mono,axiom,
    ! [M: sigma_measure_real,F2: ( a > real ) > set_set_real,I: set_a_real,J: set_a_real,G: ( a > real ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe7634599452152747905a_real @ M @ F2 @ I )
       => ( ( ord_le3334967407727675675a_real @ J @ I )
         => ( ! [I2: a > real] :
                ( ( member_a_real @ I2 @ J )
               => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
           => ( indepe7634599452152747905a_real @ M @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_218_prob__space_Oindep__sets__mono,axiom,
    ! [M: sigma_measure_real,F2: ( real > $o ) > set_set_real,I: set_real_o,J: set_real_o,G: ( real > $o ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe8014066527605497865real_o @ M @ F2 @ I )
       => ( ( ord_le1615110227528160547real_o @ J @ I )
         => ( ! [I2: real > $o] :
                ( ( member_real_o @ I2 @ J )
               => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
           => ( indepe8014066527605497865real_o @ M @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_219_prob__space_Oindep__sets__mono,axiom,
    ! [M: sigma_measure_real,F2: ( real > nat ) > set_set_real,I: set_real_nat,J: set_real_nat,G: ( real > nat ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe7235330332140030005al_nat @ M @ F2 @ I )
       => ( ( ord_le6098800555920186673al_nat @ J @ I )
         => ( ! [I2: real > nat] :
                ( ( member_real_nat @ I2 @ J )
               => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
           => ( indepe7235330332140030005al_nat @ M @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_220_prob__space_Oindep__sets__mono,axiom,
    ! [M: sigma_measure_real,F2: ( real > extend8495563244428889912nnreal ) > set_set_real,I: set_re5328672808648366137nnreal,J: set_re5328672808648366137nnreal,G: ( real > extend8495563244428889912nnreal ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe4643477836443271069nnreal @ M @ F2 @ I )
       => ( ( ord_le2462468573666744473nnreal @ J @ I )
         => ( ! [I2: real > extend8495563244428889912nnreal] :
                ( ( member2919562650594848410nnreal @ I2 @ J )
               => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
           => ( indepe4643477836443271069nnreal @ M @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_221_prob__space_Oindep__sets__mono,axiom,
    ! [M: sigma_measure_real,F2: ( real > complex ) > set_set_real,I: set_real_complex,J: set_real_complex,G: ( real > complex ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe6244138955099140627omplex @ M @ F2 @ I )
       => ( ( ord_le2047140485929309711omplex @ J @ I )
         => ( ! [I2: real > complex] :
                ( ( member_real_complex @ I2 @ J )
               => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
           => ( indepe6244138955099140627omplex @ M @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_222_subprob__measurableD_I1_J,axiom,
    ! [N: real > sigma_measure_o,M: sigma_measure_real,S: sigma_measure_o,X8: real] :
      ( ( member1287260281327066359sure_o @ N @ ( sigma_8971795097834940435sure_o @ M @ ( giry_s3549050072915289962ebra_o @ S ) ) )
     => ( ( member_real @ X8 @ ( sigma_space_real @ M ) )
       => ( ( sigma_space_o @ ( N @ X8 ) )
          = ( sigma_space_o @ S ) ) ) ) ).

% subprob_measurableD(1)
thf(fact_223_subprob__measurableD_I1_J,axiom,
    ! [N: real > sigma_measure_nat,M: sigma_measure_real,S: sigma_measure_nat,X8: real] :
      ( ( member8263806380797784669re_nat @ N @ ( sigma_854259722376812203re_nat @ M @ ( giry_s8280036963460128894ra_nat @ S ) ) )
     => ( ( member_real @ X8 @ ( sigma_space_real @ M ) )
       => ( ( sigma_space_nat @ ( N @ X8 ) )
          = ( sigma_space_nat @ S ) ) ) ) ).

% subprob_measurableD(1)
thf(fact_224_subprob__measurableD_I1_J,axiom,
    ! [N: $o > sigma_measure_o,M: sigma_measure_o,S: sigma_measure_o,X8: $o] :
      ( ( member4999217005381492423sure_o @ N @ ( sigma_2256073753033958621sure_o @ M @ ( giry_s3549050072915289962ebra_o @ S ) ) )
     => ( ( member_o @ X8 @ ( sigma_space_o @ M ) )
       => ( ( sigma_space_o @ ( N @ X8 ) )
          = ( sigma_space_o @ S ) ) ) ) ).

% subprob_measurableD(1)
thf(fact_225_subprob__measurableD_I1_J,axiom,
    ! [N: $o > sigma_measure_nat,M: sigma_measure_o,S: sigma_measure_nat,X8: $o] :
      ( ( member275192172136629389re_nat @ N @ ( sigma_2980789436339249249re_nat @ M @ ( giry_s8280036963460128894ra_nat @ S ) ) )
     => ( ( member_o @ X8 @ ( sigma_space_o @ M ) )
       => ( ( sigma_space_nat @ ( N @ X8 ) )
          = ( sigma_space_nat @ S ) ) ) ) ).

% subprob_measurableD(1)
thf(fact_226_subprob__measurableD_I1_J,axiom,
    ! [N: nat > sigma_measure_o,M: sigma_measure_nat,S: sigma_measure_o,X8: nat] :
      ( ( member3907955456775592219sure_o @ N @ ( sigma_3614372446148830647sure_o @ M @ ( giry_s3549050072915289962ebra_o @ S ) ) )
     => ( ( member_nat @ X8 @ ( sigma_space_nat @ M ) )
       => ( ( sigma_space_o @ ( N @ X8 ) )
          = ( sigma_space_o @ S ) ) ) ) ).

% subprob_measurableD(1)
thf(fact_227_subprob__measurableD_I1_J,axiom,
    ! [N: nat > sigma_measure_nat,M: sigma_measure_nat,S: sigma_measure_nat,X8: nat] :
      ( ( member8953738614771599161re_nat @ N @ ( sigma_2960785956597205511re_nat @ M @ ( giry_s8280036963460128894ra_nat @ S ) ) )
     => ( ( member_nat @ X8 @ ( sigma_space_nat @ M ) )
       => ( ( sigma_space_nat @ ( N @ X8 ) )
          = ( sigma_space_nat @ S ) ) ) ) ).

% subprob_measurableD(1)
thf(fact_228_subprob__measurableD_I1_J,axiom,
    ! [N: real > sigma_measure_real,M: sigma_measure_real,S: sigma_measure_real,X8: real] :
      ( ( member2630560753458908601e_real @ N @ ( sigma_5928869325259027335e_real @ M @ ( giry_s5092570657895779418a_real @ S ) ) )
     => ( ( member_real @ X8 @ ( sigma_space_real @ M ) )
       => ( ( sigma_space_real @ ( N @ X8 ) )
          = ( sigma_space_real @ S ) ) ) ) ).

% subprob_measurableD(1)
thf(fact_229_subprob__measurableD_I1_J,axiom,
    ! [N: $o > sigma_measure_real,M: sigma_measure_o,S: sigma_measure_real,X8: $o] :
      ( ( member1425558209591478249e_real @ N @ ( sigma_4942308948664123965e_real @ M @ ( giry_s5092570657895779418a_real @ S ) ) )
     => ( ( member_o @ X8 @ ( sigma_space_o @ M ) )
       => ( ( sigma_space_real @ ( N @ X8 ) )
          = ( sigma_space_real @ S ) ) ) ) ).

% subprob_measurableD(1)
thf(fact_230_subprob__measurableD_I1_J,axiom,
    ! [N: nat > sigma_measure_real,M: sigma_measure_nat,S: sigma_measure_real,X8: nat] :
      ( ( member5430058146565829781e_real @ N @ ( sigma_76919172735023331e_real @ M @ ( giry_s5092570657895779418a_real @ S ) ) )
     => ( ( member_nat @ X8 @ ( sigma_space_nat @ M ) )
       => ( ( sigma_space_real @ ( N @ X8 ) )
          = ( sigma_space_real @ S ) ) ) ) ).

% subprob_measurableD(1)
thf(fact_231_subprob__measurableD_I1_J,axiom,
    ! [N: ( $o > $o ) > sigma_measure_o,M: sigma_measure_o_o,S: sigma_measure_o,X8: $o > $o] :
      ( ( member9162187650169808304sure_o @ N @ ( sigma_6141153081871920972sure_o @ M @ ( giry_s3549050072915289962ebra_o @ S ) ) )
     => ( ( member_o_o @ X8 @ ( sigma_space_o_o @ M ) )
       => ( ( sigma_space_o @ ( N @ X8 ) )
          = ( sigma_space_o @ S ) ) ) ) ).

% subprob_measurableD(1)
thf(fact_232_prob__space_OM__in__subprob,axiom,
    ! [M: sigma_measure_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( member4553183543495551918e_real @ M @ ( sigma_2594925453452915853e_real @ ( giry_s5092570657895779418a_real @ M ) ) ) ) ).

% prob_space.M_in_subprob
thf(fact_233_prob__space_Oindep__sets__mono__index,axiom,
    ! [M: sigma_measure_real,J: set_set_real,I: set_set_real,F2: set_real > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( ord_le3558479182127378552t_real @ J @ I )
       => ( ( indepe8752365572443096444t_real @ M @ F2 @ I )
         => ( indepe8752365572443096444t_real @ M @ F2 @ J ) ) ) ) ).

% prob_space.indep_sets_mono_index
thf(fact_234_prob__space_Oindep__sets__mono__index,axiom,
    ! [M: sigma_measure_real,J: set_real,I: set_real,F2: real > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( ord_less_eq_set_real @ J @ I )
       => ( ( indepe8783372407961027910l_real @ M @ F2 @ I )
         => ( indepe8783372407961027910l_real @ M @ F2 @ J ) ) ) ) ).

% prob_space.indep_sets_mono_index
thf(fact_235_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M: sigma_measure_real,F2: ( $o > $o ) > set_set_real,I: set_o_o,G: ( $o > $o ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe6032362129116668415al_o_o @ M @ F2 @ I )
       => ( ! [I2: $o > $o] :
              ( ( member_o_o @ I2 @ I )
             => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe6032362129116668415al_o_o @ M @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_236_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M: sigma_measure_real,F2: ( b > extend8495563244428889912nnreal ) > set_set_real,I: set_b_6825823330181178888nnreal,G: ( b > extend8495563244428889912nnreal ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe4831269413077975822nnreal @ M @ F2 @ I )
       => ( ! [I2: b > extend8495563244428889912nnreal] :
              ( ( member6418304549040442065nnreal @ I2 @ I )
             => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe4831269413077975822nnreal @ M @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_237_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M: sigma_measure_real,F2: ( a > probab4737552677800996672pace_b ) > set_set_real,I: set_a_3263942136899480541pace_b,G: ( a > probab4737552677800996672pace_b ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe1711633972345727297pace_b @ M @ F2 @ I )
       => ( ! [I2: a > probab4737552677800996672pace_b] :
              ( ( member7340901614391157822pace_b @ I2 @ I )
             => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe1711633972345727297pace_b @ M @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_238_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M: sigma_measure_real,F2: ( a > extend8495563244428889912nnreal ) > set_set_real,I: set_a_7161065143582548615nnreal,G: ( a > extend8495563244428889912nnreal ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe7934793495794061069nnreal @ M @ F2 @ I )
       => ( ! [I2: a > extend8495563244428889912nnreal] :
              ( ( member298456594901751504nnreal @ I2 @ I )
             => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe7934793495794061069nnreal @ M @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_239_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M: sigma_measure_real,F2: ( a > real ) > set_set_real,I: set_a_real,G: ( a > real ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe7634599452152747905a_real @ M @ F2 @ I )
       => ( ! [I2: a > real] :
              ( ( member_a_real @ I2 @ I )
             => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe7634599452152747905a_real @ M @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_240_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M: sigma_measure_real,F2: ( real > $o ) > set_set_real,I: set_real_o,G: ( real > $o ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe8014066527605497865real_o @ M @ F2 @ I )
       => ( ! [I2: real > $o] :
              ( ( member_real_o @ I2 @ I )
             => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe8014066527605497865real_o @ M @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_241_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M: sigma_measure_real,F2: ( real > nat ) > set_set_real,I: set_real_nat,G: ( real > nat ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe7235330332140030005al_nat @ M @ F2 @ I )
       => ( ! [I2: real > nat] :
              ( ( member_real_nat @ I2 @ I )
             => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe7235330332140030005al_nat @ M @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_242_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M: sigma_measure_real,F2: ( real > extend8495563244428889912nnreal ) > set_set_real,I: set_re5328672808648366137nnreal,G: ( real > extend8495563244428889912nnreal ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe4643477836443271069nnreal @ M @ F2 @ I )
       => ( ! [I2: real > extend8495563244428889912nnreal] :
              ( ( member2919562650594848410nnreal @ I2 @ I )
             => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe4643477836443271069nnreal @ M @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_243_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M: sigma_measure_real,F2: ( real > complex ) > set_set_real,I: set_real_complex,G: ( real > complex ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe6244138955099140627omplex @ M @ F2 @ I )
       => ( ! [I2: real > complex] :
              ( ( member_real_complex @ I2 @ I )
             => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe6244138955099140627omplex @ M @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_244_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M: sigma_measure_real,F2: ( real > real ) > set_set_real,I: set_real_real,G: ( real > real ) > set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe4905764430350991377l_real @ M @ F2 @ I )
       => ( ! [I2: real > real] :
              ( ( member_real_real @ I2 @ I )
             => ( ord_le3558479182127378552t_real @ ( G @ I2 ) @ ( F2 @ I2 ) ) )
         => ( indepe4905764430350991377l_real @ M @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_245_subprob__measurableD_I4_J,axiom,
    ! [N: real > sigma_measure_o,M: sigma_measure_real,S: sigma_measure_o,X8: real,K: sigma_measure_real] :
      ( ( member1287260281327066359sure_o @ N @ ( sigma_8971795097834940435sure_o @ M @ ( giry_s3549050072915289962ebra_o @ S ) ) )
     => ( ( member_real @ X8 @ ( sigma_space_real @ M ) )
       => ( ( sigma_3939073009482781210real_o @ K @ ( N @ X8 ) )
          = ( sigma_3939073009482781210real_o @ K @ S ) ) ) ) ).

% subprob_measurableD(4)
thf(fact_246_subprob__measurableD_I4_J,axiom,
    ! [N: $o > sigma_measure_o,M: sigma_measure_o,S: sigma_measure_o,X8: $o,K: sigma_measure_real] :
      ( ( member4999217005381492423sure_o @ N @ ( sigma_2256073753033958621sure_o @ M @ ( giry_s3549050072915289962ebra_o @ S ) ) )
     => ( ( member_o @ X8 @ ( sigma_space_o @ M ) )
       => ( ( sigma_3939073009482781210real_o @ K @ ( N @ X8 ) )
          = ( sigma_3939073009482781210real_o @ K @ S ) ) ) ) ).

% subprob_measurableD(4)
thf(fact_247_subprob__measurableD_I4_J,axiom,
    ! [N: nat > sigma_measure_o,M: sigma_measure_nat,S: sigma_measure_o,X8: nat,K: sigma_measure_real] :
      ( ( member3907955456775592219sure_o @ N @ ( sigma_3614372446148830647sure_o @ M @ ( giry_s3549050072915289962ebra_o @ S ) ) )
     => ( ( member_nat @ X8 @ ( sigma_space_nat @ M ) )
       => ( ( sigma_3939073009482781210real_o @ K @ ( N @ X8 ) )
          = ( sigma_3939073009482781210real_o @ K @ S ) ) ) ) ).

% subprob_measurableD(4)
thf(fact_248_subprob__measurableD_I4_J,axiom,
    ! [N: real > sigma_measure_nat,M: sigma_measure_real,S: sigma_measure_nat,X8: real,K: sigma_measure_real] :
      ( ( member8263806380797784669re_nat @ N @ ( sigma_854259722376812203re_nat @ M @ ( giry_s8280036963460128894ra_nat @ S ) ) )
     => ( ( member_real @ X8 @ ( sigma_space_real @ M ) )
       => ( ( sigma_6315060578831106510al_nat @ K @ ( N @ X8 ) )
          = ( sigma_6315060578831106510al_nat @ K @ S ) ) ) ) ).

% subprob_measurableD(4)
thf(fact_249_subprob__measurableD_I4_J,axiom,
    ! [N: $o > sigma_measure_nat,M: sigma_measure_o,S: sigma_measure_nat,X8: $o,K: sigma_measure_real] :
      ( ( member275192172136629389re_nat @ N @ ( sigma_2980789436339249249re_nat @ M @ ( giry_s8280036963460128894ra_nat @ S ) ) )
     => ( ( member_o @ X8 @ ( sigma_space_o @ M ) )
       => ( ( sigma_6315060578831106510al_nat @ K @ ( N @ X8 ) )
          = ( sigma_6315060578831106510al_nat @ K @ S ) ) ) ) ).

% subprob_measurableD(4)
thf(fact_250_subprob__measurableD_I4_J,axiom,
    ! [N: nat > sigma_measure_nat,M: sigma_measure_nat,S: sigma_measure_nat,X8: nat,K: sigma_measure_real] :
      ( ( member8953738614771599161re_nat @ N @ ( sigma_2960785956597205511re_nat @ M @ ( giry_s8280036963460128894ra_nat @ S ) ) )
     => ( ( member_nat @ X8 @ ( sigma_space_nat @ M ) )
       => ( ( sigma_6315060578831106510al_nat @ K @ ( N @ X8 ) )
          = ( sigma_6315060578831106510al_nat @ K @ S ) ) ) ) ).

% subprob_measurableD(4)
thf(fact_251_subprob__measurableD_I4_J,axiom,
    ! [N: real > sigma_7234349610311085201nnreal,M: sigma_measure_real,S: sigma_7234349610311085201nnreal,X8: real,K: sigma_measure_real] :
      ( ( member6511058623620042181nnreal @ N @ ( sigma_5916024246782451219nnreal @ M @ ( giry_s4961427825854884838nnreal @ S ) ) )
     => ( ( member_real @ X8 @ ( sigma_space_real @ M ) )
       => ( ( sigma_9017504469962657078nnreal @ K @ ( N @ X8 ) )
          = ( sigma_9017504469962657078nnreal @ K @ S ) ) ) ) ).

% subprob_measurableD(4)
thf(fact_252_subprob__measurableD_I4_J,axiom,
    ! [N: $o > sigma_7234349610311085201nnreal,M: sigma_measure_o,S: sigma_7234349610311085201nnreal,X8: $o,K: sigma_measure_real] :
      ( ( member7435840198782050293nnreal @ N @ ( sigma_2165380899935909833nnreal @ M @ ( giry_s4961427825854884838nnreal @ S ) ) )
     => ( ( member_o @ X8 @ ( sigma_space_o @ M ) )
       => ( ( sigma_9017504469962657078nnreal @ K @ ( N @ X8 ) )
          = ( sigma_9017504469962657078nnreal @ K @ S ) ) ) ) ).

% subprob_measurableD(4)
thf(fact_253_subprob__measurableD_I4_J,axiom,
    ! [N: nat > sigma_7234349610311085201nnreal,M: sigma_measure_nat,S: sigma_7234349610311085201nnreal,X8: nat,K: sigma_measure_real] :
      ( ( member749279831001449121nnreal @ N @ ( sigma_4547119271277850991nnreal @ M @ ( giry_s4961427825854884838nnreal @ S ) ) )
     => ( ( member_nat @ X8 @ ( sigma_space_nat @ M ) )
       => ( ( sigma_9017504469962657078nnreal @ K @ ( N @ X8 ) )
          = ( sigma_9017504469962657078nnreal @ K @ S ) ) ) ) ).

% subprob_measurableD(4)
thf(fact_254_subprob__measurableD_I4_J,axiom,
    ! [N: real > sigma_3077487657436305159omplex,M: sigma_measure_real,S: sigma_3077487657436305159omplex,X8: real,K: sigma_measure_real] :
      ( ( member2519266086233363259omplex @ N @ ( sigma_6659613801874150537omplex @ M @ ( giry_s6563750351226335580omplex @ S ) ) )
     => ( ( member_real @ X8 @ ( sigma_space_real @ M ) )
       => ( ( sigma_9111916201866572460omplex @ K @ ( N @ X8 ) )
          = ( sigma_9111916201866572460omplex @ K @ S ) ) ) ) ).

% subprob_measurableD(4)
thf(fact_255_subprob__measurableD_I3_J,axiom,
    ! [N: real > sigma_measure_o,M: sigma_measure_real,S: sigma_measure_o,X8: real,K: sigma_measure_real] :
      ( ( member1287260281327066359sure_o @ N @ ( sigma_8971795097834940435sure_o @ M @ ( giry_s3549050072915289962ebra_o @ S ) ) )
     => ( ( member_real @ X8 @ ( sigma_space_real @ M ) )
       => ( ( sigma_2430008634441611636o_real @ ( N @ X8 ) @ K )
          = ( sigma_2430008634441611636o_real @ S @ K ) ) ) ) ).

% subprob_measurableD(3)
thf(fact_256_subprob__measurableD_I3_J,axiom,
    ! [N: $o > sigma_measure_o,M: sigma_measure_o,S: sigma_measure_o,X8: $o,K: sigma_measure_real] :
      ( ( member4999217005381492423sure_o @ N @ ( sigma_2256073753033958621sure_o @ M @ ( giry_s3549050072915289962ebra_o @ S ) ) )
     => ( ( member_o @ X8 @ ( sigma_space_o @ M ) )
       => ( ( sigma_2430008634441611636o_real @ ( N @ X8 ) @ K )
          = ( sigma_2430008634441611636o_real @ S @ K ) ) ) ) ).

% subprob_measurableD(3)
thf(fact_257_subprob__measurableD_I3_J,axiom,
    ! [N: nat > sigma_measure_o,M: sigma_measure_nat,S: sigma_measure_o,X8: nat,K: sigma_measure_real] :
      ( ( member3907955456775592219sure_o @ N @ ( sigma_3614372446148830647sure_o @ M @ ( giry_s3549050072915289962ebra_o @ S ) ) )
     => ( ( member_nat @ X8 @ ( sigma_space_nat @ M ) )
       => ( ( sigma_2430008634441611636o_real @ ( N @ X8 ) @ K )
          = ( sigma_2430008634441611636o_real @ S @ K ) ) ) ) ).

% subprob_measurableD(3)
thf(fact_258_subprob__measurableD_I3_J,axiom,
    ! [N: real > sigma_measure_nat,M: sigma_measure_real,S: sigma_measure_nat,X8: real,K: sigma_measure_real] :
      ( ( member8263806380797784669re_nat @ N @ ( sigma_854259722376812203re_nat @ M @ ( giry_s8280036963460128894ra_nat @ S ) ) )
     => ( ( member_real @ X8 @ ( sigma_space_real @ M ) )
       => ( ( sigma_1747752005702207822t_real @ ( N @ X8 ) @ K )
          = ( sigma_1747752005702207822t_real @ S @ K ) ) ) ) ).

% subprob_measurableD(3)
thf(fact_259_subprob__measurableD_I3_J,axiom,
    ! [N: $o > sigma_measure_nat,M: sigma_measure_o,S: sigma_measure_nat,X8: $o,K: sigma_measure_real] :
      ( ( member275192172136629389re_nat @ N @ ( sigma_2980789436339249249re_nat @ M @ ( giry_s8280036963460128894ra_nat @ S ) ) )
     => ( ( member_o @ X8 @ ( sigma_space_o @ M ) )
       => ( ( sigma_1747752005702207822t_real @ ( N @ X8 ) @ K )
          = ( sigma_1747752005702207822t_real @ S @ K ) ) ) ) ).

% subprob_measurableD(3)
thf(fact_260_subprob__measurableD_I3_J,axiom,
    ! [N: nat > sigma_measure_nat,M: sigma_measure_nat,S: sigma_measure_nat,X8: nat,K: sigma_measure_real] :
      ( ( member8953738614771599161re_nat @ N @ ( sigma_2960785956597205511re_nat @ M @ ( giry_s8280036963460128894ra_nat @ S ) ) )
     => ( ( member_nat @ X8 @ ( sigma_space_nat @ M ) )
       => ( ( sigma_1747752005702207822t_real @ ( N @ X8 ) @ K )
          = ( sigma_1747752005702207822t_real @ S @ K ) ) ) ) ).

% subprob_measurableD(3)
thf(fact_261_subprob__measurableD_I3_J,axiom,
    ! [N: real > sigma_measure_a,M: sigma_measure_real,S: sigma_measure_a,X8: real,K: sigma_measure_real] :
      ( ( member7281825639783221725sure_a @ N @ ( sigma_7716911757295137913sure_a @ M @ ( giry_s2448339315546285520ebra_a @ S ) ) )
     => ( ( member_real @ X8 @ ( sigma_space_real @ M ) )
       => ( ( sigma_9116425665531756122a_real @ ( N @ X8 ) @ K )
          = ( sigma_9116425665531756122a_real @ S @ K ) ) ) ) ).

% subprob_measurableD(3)
thf(fact_262_subprob__measurableD_I3_J,axiom,
    ! [N: $o > sigma_measure_a,M: sigma_measure_o,S: sigma_measure_a,X8: $o,K: sigma_measure_real] :
      ( ( member5238460584306631597sure_a @ N @ ( sigma_4308157141974772803sure_a @ M @ ( giry_s2448339315546285520ebra_a @ S ) ) )
     => ( ( member_o @ X8 @ ( sigma_space_o @ M ) )
       => ( ( sigma_9116425665531756122a_real @ ( N @ X8 ) @ K )
          = ( sigma_9116425665531756122a_real @ S @ K ) ) ) ) ).

% subprob_measurableD(3)
thf(fact_263_subprob__measurableD_I3_J,axiom,
    ! [N: nat > sigma_measure_a,M: sigma_measure_nat,S: sigma_measure_a,X8: nat,K: sigma_measure_real] :
      ( ( member5772124096947614721sure_a @ N @ ( sigma_7689407562410516509sure_a @ M @ ( giry_s2448339315546285520ebra_a @ S ) ) )
     => ( ( member_nat @ X8 @ ( sigma_space_nat @ M ) )
       => ( ( sigma_9116425665531756122a_real @ ( N @ X8 ) @ K )
          = ( sigma_9116425665531756122a_real @ S @ K ) ) ) ) ).

% subprob_measurableD(3)
thf(fact_264_subprob__measurableD_I3_J,axiom,
    ! [N: real > sigma_measure_real,M: sigma_measure_real,S: sigma_measure_real,X8: real,K: sigma_measure_o] :
      ( ( member2630560753458908601e_real @ N @ ( sigma_5928869325259027335e_real @ M @ ( giry_s5092570657895779418a_real @ S ) ) )
     => ( ( member_real @ X8 @ ( sigma_space_real @ M ) )
       => ( ( sigma_3939073009482781210real_o @ ( N @ X8 ) @ K )
          = ( sigma_3939073009482781210real_o @ S @ K ) ) ) ) ).

% subprob_measurableD(3)
thf(fact_265_subprob__space__kernel,axiom,
    ! [K: real > sigma_measure_real,M: sigma_measure_real,N: sigma_measure_real,A2: real] :
      ( ( member2630560753458908601e_real @ K @ ( sigma_5928869325259027335e_real @ M @ ( giry_s5092570657895779418a_real @ N ) ) )
     => ( ( member_real @ A2 @ ( sigma_space_real @ M ) )
       => ( giry_s8208748868292234104e_real @ ( K @ A2 ) ) ) ) ).

% subprob_space_kernel
thf(fact_266_subprob__space__kernel,axiom,
    ! [K: $o > sigma_measure_real,M: sigma_measure_o,N: sigma_measure_real,A2: $o] :
      ( ( member1425558209591478249e_real @ K @ ( sigma_4942308948664123965e_real @ M @ ( giry_s5092570657895779418a_real @ N ) ) )
     => ( ( member_o @ A2 @ ( sigma_space_o @ M ) )
       => ( giry_s8208748868292234104e_real @ ( K @ A2 ) ) ) ) ).

% subprob_space_kernel
thf(fact_267_subprob__space__kernel,axiom,
    ! [K: nat > sigma_measure_real,M: sigma_measure_nat,N: sigma_measure_real,A2: nat] :
      ( ( member5430058146565829781e_real @ K @ ( sigma_76919172735023331e_real @ M @ ( giry_s5092570657895779418a_real @ N ) ) )
     => ( ( member_nat @ A2 @ ( sigma_space_nat @ M ) )
       => ( giry_s8208748868292234104e_real @ ( K @ A2 ) ) ) ) ).

% subprob_space_kernel
thf(fact_268_subprob__space__kernel,axiom,
    ! [K: ( $o > $o ) > sigma_measure_real,M: sigma_measure_o_o,N: sigma_measure_real,A2: $o > $o] :
      ( ( member3097274092748018112e_real @ K @ ( sigma_9205866881352898830e_real @ M @ ( giry_s5092570657895779418a_real @ N ) ) )
     => ( ( member_o_o @ A2 @ ( sigma_space_o_o @ M ) )
       => ( giry_s8208748868292234104e_real @ ( K @ A2 ) ) ) ) ).

% subprob_space_kernel
thf(fact_269_subprob__space__kernel,axiom,
    ! [K: ( b > extend8495563244428889912nnreal ) > sigma_measure_real,M: sigma_2915891979588279205nnreal,N: sigma_measure_real,A2: b > extend8495563244428889912nnreal] :
      ( ( member7169416981491269209e_real @ K @ ( sigma_6211663741840559789e_real @ M @ ( giry_s5092570657895779418a_real @ N ) ) )
     => ( ( member6418304549040442065nnreal @ A2 @ ( sigma_1088381056518117040nnreal @ M ) )
       => ( giry_s8208748868292234104e_real @ ( K @ A2 ) ) ) ) ).

% subprob_space_kernel
thf(fact_270_subprob__space__kernel,axiom,
    ! [K: ( a > extend8495563244428889912nnreal ) > sigma_measure_real,M: sigma_3251133792989648932nnreal,N: sigma_measure_real,A2: a > extend8495563244428889912nnreal] :
      ( ( member5605812423362139546e_real @ K @ ( sigma_4648059183711430126e_real @ M @ ( giry_s5092570657895779418a_real @ N ) ) )
     => ( ( member298456594901751504nnreal @ A2 @ ( sigma_4191905139234202287nnreal @ M ) )
       => ( giry_s8208748868292234104e_real @ ( K @ A2 ) ) ) ) ).

% subprob_space_kernel
thf(fact_271_subprob__space__kernel,axiom,
    ! [K: ( a > real ) > sigma_measure_real,M: sigma_measure_a_real,N: sigma_measure_real,A2: a > real] :
      ( ( member5267239262543387174e_real @ K @ ( sigma_5461457353648996602e_real @ M @ ( giry_s5092570657895779418a_real @ N ) ) )
     => ( ( member_a_real @ A2 @ ( sigma_space_a_real @ M ) )
       => ( giry_s8208748868292234104e_real @ ( K @ A2 ) ) ) ) ).

% subprob_space_kernel
thf(fact_272_subprob__space__kernel,axiom,
    ! [K: ( real > $o ) > sigma_measure_real,M: sigma_measure_real_o,N: sigma_measure_real,A2: real > $o] :
      ( ( member3725182231601361310e_real @ K @ ( sigma_2943736883462281714e_real @ M @ ( giry_s5092570657895779418a_real @ N ) ) )
     => ( ( member_real_o @ A2 @ ( sigma_space_real_o @ M ) )
       => ( giry_s8208748868292234104e_real @ ( K @ A2 ) ) ) ) ).

% subprob_space_kernel
thf(fact_273_subprob__space__kernel,axiom,
    ! [K: ( real > nat ) > sigma_measure_real,M: sigma_6586288717683155060al_nat,N: sigma_measure_real,A2: real > nat] :
      ( ( member5335330379516739594e_real @ K @ ( sigma_4915400539034359384e_real @ M @ ( giry_s5092570657895779418a_real @ N ) ) )
     => ( ( member_real_nat @ A2 @ ( sigma_space_real_nat @ M ) )
       => ( giry_s8208748868292234104e_real @ ( K @ A2 ) ) ) ) ).

% subprob_space_kernel
thf(fact_274_subprob__space__kernel,axiom,
    ! [K: ( real > extend8495563244428889912nnreal ) > sigma_measure_real,M: sigma_5394977995791401948nnreal,N: sigma_measure_real,A2: real > extend8495563244428889912nnreal] :
      ( ( member6221223627903047330e_real @ K @ ( sigma_2864275607600219376e_real @ M @ ( giry_s5092570657895779418a_real @ N ) ) )
     => ( ( member2919562650594848410nnreal @ A2 @ ( sigma_2369682286586992763nnreal @ M ) )
       => ( giry_s8208748868292234104e_real @ ( K @ A2 ) ) ) ) ).

% subprob_space_kernel
thf(fact_275_subprob__space__bind,axiom,
    ! [M: sigma_measure_real,F: real > sigma_measure_real,N: sigma_measure_real] :
      ( ( giry_s8208748868292234104e_real @ M )
     => ( ( member2630560753458908601e_real @ F @ ( sigma_5928869325259027335e_real @ M @ ( giry_s5092570657895779418a_real @ N ) ) )
       => ( giry_s8208748868292234104e_real @ ( giry_bind_real_real @ M @ F ) ) ) ) ).

% subprob_space_bind
thf(fact_276_old_Oprod_Oexhaust,axiom,
    ! [Y4: produc78645753526154084e_real] :
      ~ ! [A4: quasi_borel_b,B4: produc4580635503675987618e_real] :
          ( Y4
         != ( produc2180226129289916244e_real @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_277_old_Oprod_Oexhaust,axiom,
    ! [Y4: produc4580635503675987618e_real] :
      ~ ! [A4: real > b,B4: sigma_measure_real] :
          ( Y4
         != ( produc4478270668571743890e_real @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_278_old_Oprod_Oexhaust,axiom,
    ! [Y4: produc6543235832880896358e_real] :
      ~ ! [A4: quasi_borel_a,B4: produc725540845905733987e_real] :
          ( Y4
         != ( produc4145838808978236886e_real @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_279_old_Oprod_Oexhaust,axiom,
    ! [Y4: produc725540845905733987e_real] :
      ~ ! [A4: real > a,B4: sigma_measure_real] :
          ( Y4
         != ( produc623176010801490259e_real @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_280_surj__pair,axiom,
    ! [P: produc78645753526154084e_real] :
    ? [X3: quasi_borel_b,Y5: produc4580635503675987618e_real] :
      ( P
      = ( produc2180226129289916244e_real @ X3 @ Y5 ) ) ).

% surj_pair
thf(fact_281_surj__pair,axiom,
    ! [P: produc4580635503675987618e_real] :
    ? [X3: real > b,Y5: sigma_measure_real] :
      ( P
      = ( produc4478270668571743890e_real @ X3 @ Y5 ) ) ).

% surj_pair
thf(fact_282_surj__pair,axiom,
    ! [P: produc6543235832880896358e_real] :
    ? [X3: quasi_borel_a,Y5: produc725540845905733987e_real] :
      ( P
      = ( produc4145838808978236886e_real @ X3 @ Y5 ) ) ).

% surj_pair
thf(fact_283_surj__pair,axiom,
    ! [P: produc725540845905733987e_real] :
    ? [X3: real > a,Y5: sigma_measure_real] :
      ( P
      = ( produc623176010801490259e_real @ X3 @ Y5 ) ) ).

% surj_pair
thf(fact_284_prod__cases,axiom,
    ! [P2: produc78645753526154084e_real > $o,P: produc78645753526154084e_real] :
      ( ! [A4: quasi_borel_b,B4: produc4580635503675987618e_real] : ( P2 @ ( produc2180226129289916244e_real @ A4 @ B4 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_285_prod__cases,axiom,
    ! [P2: produc4580635503675987618e_real > $o,P: produc4580635503675987618e_real] :
      ( ! [A4: real > b,B4: sigma_measure_real] : ( P2 @ ( produc4478270668571743890e_real @ A4 @ B4 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_286_prod__cases,axiom,
    ! [P2: produc6543235832880896358e_real > $o,P: produc6543235832880896358e_real] :
      ( ! [A4: quasi_borel_a,B4: produc725540845905733987e_real] : ( P2 @ ( produc4145838808978236886e_real @ A4 @ B4 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_287_prod__cases,axiom,
    ! [P2: produc725540845905733987e_real > $o,P: produc725540845905733987e_real] :
      ( ! [A4: real > a,B4: sigma_measure_real] : ( P2 @ ( produc623176010801490259e_real @ A4 @ B4 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_288_Pair__inject,axiom,
    ! [A2: quasi_borel_b,B2: produc4580635503675987618e_real,A3: quasi_borel_b,B3: produc4580635503675987618e_real] :
      ( ( ( produc2180226129289916244e_real @ A2 @ B2 )
        = ( produc2180226129289916244e_real @ A3 @ B3 ) )
     => ~ ( ( A2 = A3 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_289_Pair__inject,axiom,
    ! [A2: real > b,B2: sigma_measure_real,A3: real > b,B3: sigma_measure_real] :
      ( ( ( produc4478270668571743890e_real @ A2 @ B2 )
        = ( produc4478270668571743890e_real @ A3 @ B3 ) )
     => ~ ( ( A2 = A3 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_290_Pair__inject,axiom,
    ! [A2: quasi_borel_a,B2: produc725540845905733987e_real,A3: quasi_borel_a,B3: produc725540845905733987e_real] :
      ( ( ( produc4145838808978236886e_real @ A2 @ B2 )
        = ( produc4145838808978236886e_real @ A3 @ B3 ) )
     => ~ ( ( A2 = A3 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_291_Pair__inject,axiom,
    ! [A2: real > a,B2: sigma_measure_real,A3: real > a,B3: sigma_measure_real] :
      ( ( ( produc623176010801490259e_real @ A2 @ B2 )
        = ( produc623176010801490259e_real @ A3 @ B3 ) )
     => ~ ( ( A2 = A3 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_292_comp__def,axiom,
    ( comp_real_real_real
    = ( ^ [F3: real > real,G4: real > real,X6: real] : ( F3 @ ( G4 @ X6 ) ) ) ) ).

% comp_def
thf(fact_293_comp__def,axiom,
    ( comp_real_o_o
    = ( ^ [F3: real > $o,G4: $o > real,X6: $o] : ( F3 @ ( G4 @ X6 ) ) ) ) ).

% comp_def
thf(fact_294_comp__def,axiom,
    ( comp_real_nat_nat
    = ( ^ [F3: real > nat,G4: nat > real,X6: nat] : ( F3 @ ( G4 @ X6 ) ) ) ) ).

% comp_def
thf(fact_295_comp__def,axiom,
    ( comp_a6868616473110226257b_real
    = ( ^ [F3: a > probab4737552677800996672pace_b,G4: real > a,X6: real] : ( F3 @ ( G4 @ X6 ) ) ) ) ).

% comp_def
thf(fact_296_comp__def,axiom,
    ( comp_a_real_real
    = ( ^ [F3: a > real,G4: real > a,X6: real] : ( F3 @ ( G4 @ X6 ) ) ) ) ).

% comp_def
thf(fact_297_comp__assoc,axiom,
    ! [F: real > $o,G2: $o > real,H2: $o > $o] :
      ( ( comp_o_o_o @ ( comp_real_o_o @ F @ G2 ) @ H2 )
      = ( comp_real_o_o @ F @ ( comp_o_real_o @ G2 @ H2 ) ) ) ).

% comp_assoc
thf(fact_298_comp__assoc,axiom,
    ! [F: real > nat,G2: nat > real,H2: nat > nat] :
      ( ( comp_nat_nat_nat @ ( comp_real_nat_nat @ F @ G2 ) @ H2 )
      = ( comp_real_nat_nat @ F @ ( comp_nat_real_nat @ G2 @ H2 ) ) ) ).

% comp_assoc
thf(fact_299_comp__assoc,axiom,
    ! [F: real > real,G2: real > real,H2: real > real] :
      ( ( comp_real_real_real @ ( comp_real_real_real @ F @ G2 ) @ H2 )
      = ( comp_real_real_real @ F @ ( comp_real_real_real @ G2 @ H2 ) ) ) ).

% comp_assoc
thf(fact_300_comp__assoc,axiom,
    ! [F: a > real,G2: real > a,H2: real > real] :
      ( ( comp_real_real_real @ ( comp_a_real_real @ F @ G2 ) @ H2 )
      = ( comp_a_real_real @ F @ ( comp_real_a_real @ G2 @ H2 ) ) ) ).

% comp_assoc
thf(fact_301_comp__assoc,axiom,
    ! [F: $o > $o,G2: real > $o,H2: $o > real] :
      ( ( comp_real_o_o @ ( comp_o_o_real @ F @ G2 ) @ H2 )
      = ( comp_o_o_o @ F @ ( comp_real_o_o @ G2 @ H2 ) ) ) ).

% comp_assoc
thf(fact_302_comp__assoc,axiom,
    ! [F: real > $o,G2: real > real,H2: $o > real] :
      ( ( comp_real_o_o @ ( comp_real_o_real @ F @ G2 ) @ H2 )
      = ( comp_real_o_o @ F @ ( comp_real_real_o @ G2 @ H2 ) ) ) ).

% comp_assoc
thf(fact_303_comp__assoc,axiom,
    ! [F: nat > nat,G2: real > nat,H2: nat > real] :
      ( ( comp_real_nat_nat @ ( comp_nat_nat_real @ F @ G2 ) @ H2 )
      = ( comp_nat_nat_nat @ F @ ( comp_real_nat_nat @ G2 @ H2 ) ) ) ).

% comp_assoc
thf(fact_304_comp__assoc,axiom,
    ! [F: real > nat,G2: real > real,H2: nat > real] :
      ( ( comp_real_nat_nat @ ( comp_real_nat_real @ F @ G2 ) @ H2 )
      = ( comp_real_nat_nat @ F @ ( comp_real_real_nat @ G2 @ H2 ) ) ) ).

% comp_assoc
thf(fact_305_comp__assoc,axiom,
    ! [F: real > real,G2: a > real,H2: real > a] :
      ( ( comp_a_real_real @ ( comp_real_real_a @ F @ G2 ) @ H2 )
      = ( comp_real_real_real @ F @ ( comp_a_real_real @ G2 @ H2 ) ) ) ).

% comp_assoc
thf(fact_306_comp__assoc,axiom,
    ! [F: a > real,G2: a > a,H2: real > a] :
      ( ( comp_a_real_real @ ( comp_a_real_a @ F @ G2 ) @ H2 )
      = ( comp_a_real_real @ F @ ( comp_a_a_real @ G2 @ H2 ) ) ) ).

% comp_assoc
thf(fact_307_comp__eq__dest,axiom,
    ! [A2: real > real,B2: real > real,C: real > real,D: real > real,V: real] :
      ( ( ( comp_real_real_real @ A2 @ B2 )
        = ( comp_real_real_real @ C @ D ) )
     => ( ( A2 @ ( B2 @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_308_comp__eq__dest,axiom,
    ! [A2: real > real,B2: real > real,C: a > real,D: real > a,V: real] :
      ( ( ( comp_real_real_real @ A2 @ B2 )
        = ( comp_a_real_real @ C @ D ) )
     => ( ( A2 @ ( B2 @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_309_comp__eq__dest,axiom,
    ! [A2: real > $o,B2: $o > real,C: real > $o,D: $o > real,V: $o] :
      ( ( ( comp_real_o_o @ A2 @ B2 )
        = ( comp_real_o_o @ C @ D ) )
     => ( ( A2 @ ( B2 @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_310_comp__eq__dest,axiom,
    ! [A2: real > nat,B2: nat > real,C: real > nat,D: nat > real,V: nat] :
      ( ( ( comp_real_nat_nat @ A2 @ B2 )
        = ( comp_real_nat_nat @ C @ D ) )
     => ( ( A2 @ ( B2 @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_311_comp__eq__dest,axiom,
    ! [A2: a > probab4737552677800996672pace_b,B2: real > a,C: a > probab4737552677800996672pace_b,D: real > a,V: real] :
      ( ( ( comp_a6868616473110226257b_real @ A2 @ B2 )
        = ( comp_a6868616473110226257b_real @ C @ D ) )
     => ( ( A2 @ ( B2 @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_312_comp__eq__dest,axiom,
    ! [A2: a > real,B2: real > a,C: real > real,D: real > real,V: real] :
      ( ( ( comp_a_real_real @ A2 @ B2 )
        = ( comp_real_real_real @ C @ D ) )
     => ( ( A2 @ ( B2 @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_313_comp__eq__dest,axiom,
    ! [A2: a > real,B2: real > a,C: a > real,D: real > a,V: real] :
      ( ( ( comp_a_real_real @ A2 @ B2 )
        = ( comp_a_real_real @ C @ D ) )
     => ( ( A2 @ ( B2 @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_314_comp__eq__elim,axiom,
    ! [A2: real > real,B2: real > real,C: real > real,D: real > real] :
      ( ( ( comp_real_real_real @ A2 @ B2 )
        = ( comp_real_real_real @ C @ D ) )
     => ! [V2: real] :
          ( ( A2 @ ( B2 @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_315_comp__eq__elim,axiom,
    ! [A2: real > real,B2: real > real,C: a > real,D: real > a] :
      ( ( ( comp_real_real_real @ A2 @ B2 )
        = ( comp_a_real_real @ C @ D ) )
     => ! [V2: real] :
          ( ( A2 @ ( B2 @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_316_comp__eq__elim,axiom,
    ! [A2: real > $o,B2: $o > real,C: real > $o,D: $o > real] :
      ( ( ( comp_real_o_o @ A2 @ B2 )
        = ( comp_real_o_o @ C @ D ) )
     => ! [V2: $o] :
          ( ( A2 @ ( B2 @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_317_comp__eq__elim,axiom,
    ! [A2: real > nat,B2: nat > real,C: real > nat,D: nat > real] :
      ( ( ( comp_real_nat_nat @ A2 @ B2 )
        = ( comp_real_nat_nat @ C @ D ) )
     => ! [V2: nat] :
          ( ( A2 @ ( B2 @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_318_comp__eq__elim,axiom,
    ! [A2: a > probab4737552677800996672pace_b,B2: real > a,C: a > probab4737552677800996672pace_b,D: real > a] :
      ( ( ( comp_a6868616473110226257b_real @ A2 @ B2 )
        = ( comp_a6868616473110226257b_real @ C @ D ) )
     => ! [V2: real] :
          ( ( A2 @ ( B2 @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_319_comp__eq__elim,axiom,
    ! [A2: a > real,B2: real > a,C: real > real,D: real > real] :
      ( ( ( comp_a_real_real @ A2 @ B2 )
        = ( comp_real_real_real @ C @ D ) )
     => ! [V2: real] :
          ( ( A2 @ ( B2 @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_320_comp__eq__elim,axiom,
    ! [A2: a > real,B2: real > a,C: a > real,D: real > a] :
      ( ( ( comp_a_real_real @ A2 @ B2 )
        = ( comp_a_real_real @ C @ D ) )
     => ! [V2: real] :
          ( ( A2 @ ( B2 @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_321_comp__eq__dest__lhs,axiom,
    ! [A2: real > real,B2: real > real,C: real > real,V: real] :
      ( ( ( comp_real_real_real @ A2 @ B2 )
        = C )
     => ( ( A2 @ ( B2 @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_322_comp__eq__dest__lhs,axiom,
    ! [A2: real > $o,B2: $o > real,C: $o > $o,V: $o] :
      ( ( ( comp_real_o_o @ A2 @ B2 )
        = C )
     => ( ( A2 @ ( B2 @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_323_comp__eq__dest__lhs,axiom,
    ! [A2: real > nat,B2: nat > real,C: nat > nat,V: nat] :
      ( ( ( comp_real_nat_nat @ A2 @ B2 )
        = C )
     => ( ( A2 @ ( B2 @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_324_comp__eq__dest__lhs,axiom,
    ! [A2: a > probab4737552677800996672pace_b,B2: real > a,C: real > probab4737552677800996672pace_b,V: real] :
      ( ( ( comp_a6868616473110226257b_real @ A2 @ B2 )
        = C )
     => ( ( A2 @ ( B2 @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_325_comp__eq__dest__lhs,axiom,
    ! [A2: a > real,B2: real > a,C: real > real,V: real] :
      ( ( ( comp_a_real_real @ A2 @ B2 )
        = C )
     => ( ( A2 @ ( B2 @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_326_measurable__prob__algebraI,axiom,
    ! [N: sigma_measure_real,F: real > sigma_measure_real,M: sigma_measure_real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( sigma_space_real @ N ) )
         => ( probab535871623910865577e_real @ ( F @ X3 ) ) )
     => ( ( member2630560753458908601e_real @ F @ ( sigma_5928869325259027335e_real @ N @ ( giry_s5092570657895779418a_real @ M ) ) )
       => ( member2630560753458908601e_real @ F @ ( sigma_5928869325259027335e_real @ N @ ( giry_p6081475675320601670a_real @ M ) ) ) ) ) ).

% measurable_prob_algebraI
thf(fact_327_measurable__prob__algebraI,axiom,
    ! [N: sigma_measure_o,F: $o > sigma_measure_real,M: sigma_measure_real] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( sigma_space_o @ N ) )
         => ( probab535871623910865577e_real @ ( F @ X3 ) ) )
     => ( ( member1425558209591478249e_real @ F @ ( sigma_4942308948664123965e_real @ N @ ( giry_s5092570657895779418a_real @ M ) ) )
       => ( member1425558209591478249e_real @ F @ ( sigma_4942308948664123965e_real @ N @ ( giry_p6081475675320601670a_real @ M ) ) ) ) ) ).

% measurable_prob_algebraI
thf(fact_328_measurable__prob__algebraI,axiom,
    ! [N: sigma_measure_nat,F: nat > sigma_measure_real,M: sigma_measure_real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( sigma_space_nat @ N ) )
         => ( probab535871623910865577e_real @ ( F @ X3 ) ) )
     => ( ( member5430058146565829781e_real @ F @ ( sigma_76919172735023331e_real @ N @ ( giry_s5092570657895779418a_real @ M ) ) )
       => ( member5430058146565829781e_real @ F @ ( sigma_76919172735023331e_real @ N @ ( giry_p6081475675320601670a_real @ M ) ) ) ) ) ).

% measurable_prob_algebraI
thf(fact_329_measurable__prob__algebraI,axiom,
    ! [N: sigma_measure_o_o,F: ( $o > $o ) > sigma_measure_real,M: sigma_measure_real] :
      ( ! [X3: $o > $o] :
          ( ( member_o_o @ X3 @ ( sigma_space_o_o @ N ) )
         => ( probab535871623910865577e_real @ ( F @ X3 ) ) )
     => ( ( member3097274092748018112e_real @ F @ ( sigma_9205866881352898830e_real @ N @ ( giry_s5092570657895779418a_real @ M ) ) )
       => ( member3097274092748018112e_real @ F @ ( sigma_9205866881352898830e_real @ N @ ( giry_p6081475675320601670a_real @ M ) ) ) ) ) ).

% measurable_prob_algebraI
thf(fact_330_measurable__prob__algebraI,axiom,
    ! [N: sigma_2915891979588279205nnreal,F: ( b > extend8495563244428889912nnreal ) > sigma_measure_real,M: sigma_measure_real] :
      ( ! [X3: b > extend8495563244428889912nnreal] :
          ( ( member6418304549040442065nnreal @ X3 @ ( sigma_1088381056518117040nnreal @ N ) )
         => ( probab535871623910865577e_real @ ( F @ X3 ) ) )
     => ( ( member7169416981491269209e_real @ F @ ( sigma_6211663741840559789e_real @ N @ ( giry_s5092570657895779418a_real @ M ) ) )
       => ( member7169416981491269209e_real @ F @ ( sigma_6211663741840559789e_real @ N @ ( giry_p6081475675320601670a_real @ M ) ) ) ) ) ).

% measurable_prob_algebraI
thf(fact_331_measurable__prob__algebraI,axiom,
    ! [N: sigma_3251133792989648932nnreal,F: ( a > extend8495563244428889912nnreal ) > sigma_measure_real,M: sigma_measure_real] :
      ( ! [X3: a > extend8495563244428889912nnreal] :
          ( ( member298456594901751504nnreal @ X3 @ ( sigma_4191905139234202287nnreal @ N ) )
         => ( probab535871623910865577e_real @ ( F @ X3 ) ) )
     => ( ( member5605812423362139546e_real @ F @ ( sigma_4648059183711430126e_real @ N @ ( giry_s5092570657895779418a_real @ M ) ) )
       => ( member5605812423362139546e_real @ F @ ( sigma_4648059183711430126e_real @ N @ ( giry_p6081475675320601670a_real @ M ) ) ) ) ) ).

% measurable_prob_algebraI
thf(fact_332_measurable__prob__algebraI,axiom,
    ! [N: sigma_measure_a_real,F: ( a > real ) > sigma_measure_real,M: sigma_measure_real] :
      ( ! [X3: a > real] :
          ( ( member_a_real @ X3 @ ( sigma_space_a_real @ N ) )
         => ( probab535871623910865577e_real @ ( F @ X3 ) ) )
     => ( ( member5267239262543387174e_real @ F @ ( sigma_5461457353648996602e_real @ N @ ( giry_s5092570657895779418a_real @ M ) ) )
       => ( member5267239262543387174e_real @ F @ ( sigma_5461457353648996602e_real @ N @ ( giry_p6081475675320601670a_real @ M ) ) ) ) ) ).

% measurable_prob_algebraI
thf(fact_333_measurable__prob__algebraI,axiom,
    ! [N: sigma_measure_real_o,F: ( real > $o ) > sigma_measure_real,M: sigma_measure_real] :
      ( ! [X3: real > $o] :
          ( ( member_real_o @ X3 @ ( sigma_space_real_o @ N ) )
         => ( probab535871623910865577e_real @ ( F @ X3 ) ) )
     => ( ( member3725182231601361310e_real @ F @ ( sigma_2943736883462281714e_real @ N @ ( giry_s5092570657895779418a_real @ M ) ) )
       => ( member3725182231601361310e_real @ F @ ( sigma_2943736883462281714e_real @ N @ ( giry_p6081475675320601670a_real @ M ) ) ) ) ) ).

% measurable_prob_algebraI
thf(fact_334_measurable__prob__algebraI,axiom,
    ! [N: sigma_6586288717683155060al_nat,F: ( real > nat ) > sigma_measure_real,M: sigma_measure_real] :
      ( ! [X3: real > nat] :
          ( ( member_real_nat @ X3 @ ( sigma_space_real_nat @ N ) )
         => ( probab535871623910865577e_real @ ( F @ X3 ) ) )
     => ( ( member5335330379516739594e_real @ F @ ( sigma_4915400539034359384e_real @ N @ ( giry_s5092570657895779418a_real @ M ) ) )
       => ( member5335330379516739594e_real @ F @ ( sigma_4915400539034359384e_real @ N @ ( giry_p6081475675320601670a_real @ M ) ) ) ) ) ).

% measurable_prob_algebraI
thf(fact_335_measurable__prob__algebraI,axiom,
    ! [N: sigma_5394977995791401948nnreal,F: ( real > extend8495563244428889912nnreal ) > sigma_measure_real,M: sigma_measure_real] :
      ( ! [X3: real > extend8495563244428889912nnreal] :
          ( ( member2919562650594848410nnreal @ X3 @ ( sigma_2369682286586992763nnreal @ N ) )
         => ( probab535871623910865577e_real @ ( F @ X3 ) ) )
     => ( ( member6221223627903047330e_real @ F @ ( sigma_2864275607600219376e_real @ N @ ( giry_s5092570657895779418a_real @ M ) ) )
       => ( member6221223627903047330e_real @ F @ ( sigma_2864275607600219376e_real @ N @ ( giry_p6081475675320601670a_real @ M ) ) ) ) ) ).

% measurable_prob_algebraI
thf(fact_336_subprob__space_Obind__in__space,axiom,
    ! [M: sigma_measure_real,A: real > sigma_measure_real,N: sigma_measure_real] :
      ( ( giry_s8208748868292234104e_real @ M )
     => ( ( member2630560753458908601e_real @ A @ ( sigma_5928869325259027335e_real @ M @ ( giry_s5092570657895779418a_real @ N ) ) )
       => ( member4553183543495551918e_real @ ( giry_bind_real_real @ M @ A ) @ ( sigma_2594925453452915853e_real @ ( giry_s5092570657895779418a_real @ N ) ) ) ) ) ).

% subprob_space.bind_in_space
thf(fact_337_prod__cases3,axiom,
    ! [Y4: produc78645753526154084e_real] :
      ~ ! [A4: quasi_borel_b,B4: real > b,C2: sigma_measure_real] :
          ( Y4
         != ( produc2180226129289916244e_real @ A4 @ ( produc4478270668571743890e_real @ B4 @ C2 ) ) ) ).

% prod_cases3
thf(fact_338_prod__cases3,axiom,
    ! [Y4: produc6543235832880896358e_real] :
      ~ ! [A4: quasi_borel_a,B4: real > a,C2: sigma_measure_real] :
          ( Y4
         != ( produc4145838808978236886e_real @ A4 @ ( produc623176010801490259e_real @ B4 @ C2 ) ) ) ).

% prod_cases3
thf(fact_339_prod__induct3,axiom,
    ! [P2: produc78645753526154084e_real > $o,X8: produc78645753526154084e_real] :
      ( ! [A4: quasi_borel_b,B4: real > b,C2: sigma_measure_real] : ( P2 @ ( produc2180226129289916244e_real @ A4 @ ( produc4478270668571743890e_real @ B4 @ C2 ) ) )
     => ( P2 @ X8 ) ) ).

% prod_induct3
thf(fact_340_prod__induct3,axiom,
    ! [P2: produc6543235832880896358e_real > $o,X8: produc6543235832880896358e_real] :
      ( ! [A4: quasi_borel_a,B4: real > a,C2: sigma_measure_real] : ( P2 @ ( produc4145838808978236886e_real @ A4 @ ( produc623176010801490259e_real @ B4 @ C2 ) ) )
     => ( P2 @ X8 ) ) ).

% prod_induct3
thf(fact_341_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_342_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_343_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_344_qbs__join__computation_I2_J,axiom,
    ! [X: quasi_borel_o,Beta2: real > probab1241297377463522905pace_o,Mu: sigma_measure_real,Ssx: probab691856902995763354pace_o,Alpha: real > $o,G2: real > sigma_measure_real] :
      ( ( probab3620450583171887351pace_o @ ( monad_monadP_qbs_o @ X ) @ Beta2 @ Mu )
     => ( ( Ssx
          = ( probab1393150374121012435pace_o @ ( produc1093366075506243232e_real @ ( monad_monadP_qbs_o @ X ) @ ( produc982178768907016120e_real @ Beta2 @ Mu ) ) ) )
       => ( ( member_real_o @ Alpha @ ( qbs_Mx_o @ X ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( Beta2
                = ( ^ [R: real] : ( probab414947219978445330pace_o @ ( produc2942336022369489698e_real @ X @ ( produc787877863769208121e_real @ Alpha @ ( G2 @ R ) ) ) ) ) )
             => ( ( monad_qbs_join_o @ Ssx )
                = ( probab414947219978445330pace_o @ ( produc2942336022369489698e_real @ X @ ( produc787877863769208121e_real @ Alpha @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ) ) ).

% qbs_join_computation(2)
thf(fact_345_qbs__join__computation_I2_J,axiom,
    ! [X: quasi_borel_nat,Beta2: real > probab8370124847414953445ce_nat,Mu: sigma_measure_real,Ssx: probab3005566686192770426ce_nat,Alpha: real > nat,G2: real > sigma_measure_real] :
      ( ( probab2468401811917278919ce_nat @ ( monad_monadP_qbs_nat @ X ) @ Beta2 @ Mu )
     => ( ( Ssx
          = ( probab1586022766323937771ce_nat @ ( produc5084857187822765240e_real @ ( monad_monadP_qbs_nat @ X ) @ ( produc6214735565764730620e_real @ Beta2 @ Mu ) ) ) )
       => ( ( member_real_nat @ Alpha @ ( qbs_Mx_nat @ X ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( Beta2
                = ( ^ [R: real] : ( probab4893816680317050838ce_nat @ ( produc2796221405228754040e_real @ X @ ( produc6760937697707383505e_real @ Alpha @ ( G2 @ R ) ) ) ) ) )
             => ( ( monad_qbs_join_nat @ Ssx )
                = ( probab4893816680317050838ce_nat @ ( produc2796221405228754040e_real @ X @ ( produc6760937697707383505e_real @ Alpha @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ) ) ).

% qbs_join_computation(2)
thf(fact_346_qbs__join__computation_I2_J,axiom,
    ! [X: quasi_9015997321629101608nnreal,Beta2: real > probab1359147627358338381nnreal,Mu: sigma_measure_real,Ssx: probab3223874146922238178nnreal,Alpha: real > extend8495563244428889912nnreal,G2: real > sigma_measure_real] :
      ( ( probab8011796075923174959nnreal @ ( monad_8737719049617959390nnreal @ X ) @ Beta2 @ Mu )
     => ( ( Ssx
          = ( probab2630004677939362131nnreal @ ( produc9122815929407448760e_real @ ( monad_8737719049617959390nnreal @ X ) @ ( produc3526280104234912148e_real @ Beta2 @ Mu ) ) ) )
       => ( ( member2919562650594848410nnreal @ Alpha @ ( qbs_Mx6523938229262583809nnreal @ X ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( Beta2
                = ( ^ [R: real] : ( probab2095897665094527806nnreal @ ( produc3311748387682954872e_real @ X @ ( produc8591767778467529065e_real @ Alpha @ ( G2 @ R ) ) ) ) ) )
             => ( ( monad_1133032965969113481nnreal @ Ssx )
                = ( probab2095897665094527806nnreal @ ( produc3311748387682954872e_real @ X @ ( produc8591767778467529065e_real @ Alpha @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ) ) ).

% qbs_join_computation(2)
thf(fact_347_qbs__join__computation_I2_J,axiom,
    ! [X: quasi_borel_complex,Beta2: real > probab3778977310982246339omplex,Mu: sigma_measure_real,Ssx: probab3364748380892496984omplex,Alpha: real > complex,G2: real > sigma_measure_real] :
      ( ( probab6260968705227822245omplex @ ( monad_3228211519047171924omplex @ X ) @ Beta2 @ Mu )
     => ( ( Ssx
          = ( probab8929889938064994249omplex @ ( produc4831291127269080632e_real @ ( monad_3228211519047171924omplex @ X ) @ ( produc4083023864984044190e_real @ Beta2 @ Mu ) ) ) )
       => ( ( member_real_complex @ Alpha @ ( qbs_Mx_complex @ X ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( Beta2
                = ( ^ [R: real] : ( probab2835974231434789044omplex @ ( produc7812373664507111416e_real @ X @ ( produc8166640644904819571e_real @ Alpha @ ( G2 @ R ) ) ) ) ) )
             => ( ( monad_519594680078223615omplex @ Ssx )
                = ( probab2835974231434789044omplex @ ( produc7812373664507111416e_real @ X @ ( produc8166640644904819571e_real @ Alpha @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ) ) ).

% qbs_join_computation(2)
thf(fact_348_qbs__join__computation_I2_J,axiom,
    ! [X: quasi_borel_real,Beta2: real > probab8009751763329705409e_real,Mu: sigma_measure_real,Ssx: probab3093221366759455318e_real,Alpha: real > real,G2: real > sigma_measure_real] :
      ( ( probab8234793495611891107e_real @ ( monad_2887651017592114770s_real @ X ) @ Beta2 @ Mu )
     => ( ( Ssx
          = ( probab1999026297785200327e_real @ ( produc2581243556061882552e_real @ ( monad_2887651017592114770s_real @ X ) @ ( produc1095797480378445344e_real @ Beta2 @ Mu ) ) ) )
       => ( ( member_real_real @ Alpha @ ( qbs_Mx_real @ X ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( Beta2
                = ( ^ [R: real] : ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X @ ( produc1722724976708544245e_real @ Alpha @ ( G2 @ R ) ) ) ) ) )
             => ( ( monad_qbs_join_real @ Ssx )
                = ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X @ ( produc1722724976708544245e_real @ Alpha @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ) ) ).

% qbs_join_computation(2)
thf(fact_349_qbs__join__computation_I2_J,axiom,
    ! [X: quasi_borel_b,Beta2: real > probab4737552677800996672pace_b,Mu: sigma_measure_real,Ssx: probab8624551251471354625pace_b,Alpha: real > b,G2: real > sigma_measure_real] :
      ( ( probab2008080876895902430pace_b @ ( monad_monadP_qbs_b @ X ) @ Beta2 @ Mu )
     => ( ( Ssx
          = ( probab2758740287680477370pace_b @ ( produc4607355092896878674e_real @ ( monad_monadP_qbs_b @ X ) @ ( produc2065420038842681233e_real @ Beta2 @ Mu ) ) ) )
       => ( ( member_real_b @ Alpha @ ( qbs_Mx_b @ X ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( Beta2
                = ( ^ [R: real] : ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ ( G2 @ R ) ) ) ) ) )
             => ( ( monad_qbs_join_b @ Ssx )
                = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ) ) ).

% qbs_join_computation(2)
thf(fact_350_qbs__join__computation_I2_J,axiom,
    ! [X: quasi_borel_a,Beta2: real > probab4737552673497767871pace_a,Mu: sigma_measure_real,Ssx: probab8553517211425008640pace_a,Alpha: real > a,G2: real > sigma_measure_real] :
      ( ( probab2008080872592673629pace_a @ ( monad_monadP_qbs_a @ X ) @ Beta2 @ Mu )
     => ( ( Ssx
          = ( probab2758740283377248569pace_a @ ( produc6726600922647680468e_real @ ( monad_monadP_qbs_a @ X ) @ ( produc2709302780738849234e_real @ Beta2 @ Mu ) ) ) )
       => ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( Beta2
                = ( ^ [R: real] : ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ ( G2 @ R ) ) ) ) ) )
             => ( ( monad_qbs_join_a @ Ssx )
                = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ) ) ).

% qbs_join_computation(2)
thf(fact_351_qp_Oqbs__bind__computation_I1_J,axiom,
    ! [S2: probab4737552673497767871pace_a,F: a > probab1241297377463522905pace_o,Y: quasi_borel_o,Beta2: real > $o,G2: real > sigma_measure_real] :
      ( ( S2
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) )
     => ( ( member7312870768605366359pace_o @ F @ ( qbs_mo1614836618362447827pace_o @ x @ ( monad_monadP_qbs_o @ Y ) ) )
       => ( ( member_real_o @ Beta2 @ ( qbs_Mx_o @ Y ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( ( comp_a281010391317482986o_real @ F @ alpha )
                = ( ^ [R: real] : ( probab414947219978445330pace_o @ ( produc2942336022369489698e_real @ Y @ ( produc787877863769208121e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
             => ( probab163731277575722550prob_o @ Y @ Beta2 @ ( giry_bind_real_real @ mu @ G2 ) ) ) ) ) ) ) ).

% qp.qbs_bind_computation(1)
thf(fact_352_qp_Oqbs__bind__computation_I1_J,axiom,
    ! [S2: probab4737552673497767871pace_a,F: a > probab8370124847414953445ce_nat,Y: quasi_borel_nat,Beta2: real > nat,G2: real > sigma_measure_real] :
      ( ( S2
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) )
     => ( ( member6439975221678120701ce_nat @ F @ ( qbs_mo7775059402295510763ce_nat @ x @ ( monad_monadP_qbs_nat @ Y ) ) )
       => ( ( member_real_nat @ Beta2 @ ( qbs_Mx_nat @ Y ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( ( comp_a1663454009834812910t_real @ F @ alpha )
                = ( ^ [R: real] : ( probab4893816680317050838ce_nat @ ( produc2796221405228754040e_real @ Y @ ( produc6760937697707383505e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
             => ( probab2851505236026752178ob_nat @ Y @ Beta2 @ ( giry_bind_real_real @ mu @ G2 ) ) ) ) ) ) ) ).

% qp.qbs_bind_computation(1)
thf(fact_353_qp_Oqbs__bind__computation_I1_J,axiom,
    ! [S2: probab4737552673497767871pace_a,F: a > probab1359147627358338381nnreal,Y: quasi_9015997321629101608nnreal,Beta2: real > extend8495563244428889912nnreal,G2: real > sigma_measure_real] :
      ( ( S2
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) )
     => ( ( member2352135308469533285nnreal @ F @ ( qbs_mo990775079526714963nnreal @ x @ ( monad_8737719049617959390nnreal @ Y ) ) )
       => ( ( member2919562650594848410nnreal @ Beta2 @ ( qbs_Mx6523938229262583809nnreal @ Y ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( ( comp_a4151604154359478614l_real @ F @ alpha )
                = ( ^ [R: real] : ( probab2095897665094527806nnreal @ ( produc3311748387682954872e_real @ Y @ ( produc8591767778467529065e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
             => ( probab8888997264033409050nnreal @ Y @ Beta2 @ ( giry_bind_real_real @ mu @ G2 ) ) ) ) ) ) ) ).

% qp.qbs_bind_computation(1)
thf(fact_354_qp_Oqbs__bind__computation_I1_J,axiom,
    ! [S2: probab4737552673497767871pace_a,F: a > probab3778977310982246339omplex,Y: quasi_borel_complex,Beta2: real > complex,G2: real > sigma_measure_real] :
      ( ( S2
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) )
     => ( ( member2695794483689275355omplex @ F @ ( qbs_mo1042063135579555017omplex @ x @ ( monad_3228211519047171924omplex @ Y ) ) )
       => ( ( member_real_complex @ Beta2 @ ( qbs_Mx_complex @ Y ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( ( comp_a2485845550147043532x_real @ F @ alpha )
                = ( ^ [R: real] : ( probab2835974231434789044omplex @ ( produc7812373664507111416e_real @ Y @ ( produc8166640644904819571e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
             => ( probab963564174529062288omplex @ Y @ Beta2 @ ( giry_bind_real_real @ mu @ G2 ) ) ) ) ) ) ) ).

% qp.qbs_bind_computation(1)
thf(fact_355_qp_Oqbs__bind__computation_I1_J,axiom,
    ! [S2: probab4737552673497767871pace_a,F: a > probab8009751763329705409e_real,Y: quasi_borel_real,Beta2: real > real,G2: real > sigma_measure_real] :
      ( ( S2
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) )
     => ( ( member4522402906629368409e_real @ F @ ( qbs_mo1035972366611710919e_real @ x @ ( monad_2887651017592114770s_real @ Y ) ) )
       => ( ( member_real_real @ Beta2 @ ( qbs_Mx_real @ Y ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( ( comp_a677806100398348746l_real @ F @ alpha )
                = ( ^ [R: real] : ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ Y @ ( produc1722724976708544245e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
             => ( probab3605210969150000782b_real @ Y @ Beta2 @ ( giry_bind_real_real @ mu @ G2 ) ) ) ) ) ) ) ).

% qp.qbs_bind_computation(1)
thf(fact_356_qp_Oqbs__bind__computation_I1_J,axiom,
    ! [S2: probab4737552673497767871pace_a,F: a > probab4737552677800996672pace_b,Y: quasi_borel_b,Beta2: real > b,G2: real > sigma_measure_real] :
      ( ( S2
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) )
     => ( ( member7340901614391157822pace_b @ F @ ( qbs_mo6277074943135278010pace_b @ x @ ( monad_monadP_qbs_b @ Y ) ) )
       => ( ( member_real_b @ Beta2 @ ( qbs_Mx_b @ Y ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( ( comp_a6868616473110226257b_real @ F @ alpha )
                = ( ^ [R: real] : ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ Y @ ( produc4478270668571743890e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
             => ( probab701741629625904797prob_b @ Y @ Beta2 @ ( giry_bind_real_real @ mu @ G2 ) ) ) ) ) ) ) ).

% qp.qbs_bind_computation(1)
thf(fact_357_qp_Oqbs__bind__computation_I1_J,axiom,
    ! [S2: probab4737552673497767871pace_a,F: a > probab4737552673497767871pace_a,Y: quasi_borel_a,Beta2: real > a,G2: real > sigma_measure_real] :
      ( ( S2
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) )
     => ( ( member7269867574344811837pace_a @ F @ ( qbs_mo6277074938832049209pace_a @ x @ ( monad_monadP_qbs_a @ Y ) ) )
       => ( ( member_real_a @ Beta2 @ ( qbs_Mx_a @ Y ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( ( comp_a423948744115472976a_real @ F @ alpha )
                = ( ^ [R: real] : ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
             => ( probab701741629625904796prob_a @ Y @ Beta2 @ ( giry_bind_real_real @ mu @ G2 ) ) ) ) ) ) ) ).

% qp.qbs_bind_computation(1)
thf(fact_358_qp_Oif__in__Rep_I2_J,axiom,
    ! [X9: quasi_borel_a,Alpha3: real > a,Mu3: sigma_measure_real] :
      ( ( member6844354795726785935e_real @ ( produc4145838808978236886e_real @ X9 @ ( produc623176010801490259e_real @ Alpha3 @ Mu3 ) ) @ ( probab8639044586466322086pace_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) ) )
     => ( probab701741629625904796prob_a @ X9 @ Alpha3 @ Mu3 ) ) ).

% qp.if_in_Rep(2)
thf(fact_359_qp_Oif__in__Rep_I1_J,axiom,
    ! [X9: quasi_borel_a,Alpha3: real > a,Mu3: sigma_measure_real] :
      ( ( member6844354795726785935e_real @ ( produc4145838808978236886e_real @ X9 @ ( produc623176010801490259e_real @ Alpha3 @ Mu3 ) ) @ ( probab8639044586466322086pace_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) ) )
     => ( X9 = x ) ) ).

% qp.if_in_Rep(1)
thf(fact_360_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_361_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_362_qp_OmonadP__qbs__Pf__computation_I1_J,axiom,
    ! [S2: probab4737552673497767871pace_a,F: a > probab4737552677800996672pace_b,Y: quasi_3431323906171122406pace_b] :
      ( ( S2
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) )
     => ( ( member7340901614391157822pace_b @ F @ ( qbs_mo6277074943135278010pace_b @ x @ Y ) )
       => ( probab2008080876895902430pace_b @ Y @ ( comp_a6868616473110226257b_real @ F @ alpha ) @ mu ) ) ) ).

% qp.monadP_qbs_Pf_computation(1)
thf(fact_363_qp_OmonadP__qbs__Pf__computation_I1_J,axiom,
    ! [S2: probab4737552673497767871pace_a,F: a > extend8495563244428889912nnreal,Y: quasi_9015997321629101608nnreal] :
      ( ( S2
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) )
     => ( ( member298456594901751504nnreal @ F @ ( qbs_mo1434458643421888574nnreal @ x @ Y ) )
       => ( probab8888997264033409050nnreal @ Y @ ( comp_a8249376463644563905l_real @ F @ alpha ) @ mu ) ) ) ).

% qp.monadP_qbs_Pf_computation(1)
thf(fact_364_qp_OmonadP__qbs__Pf__computation_I1_J,axiom,
    ! [S2: probab4737552673497767871pace_a,F: a > real,Y: quasi_borel_real] :
      ( ( S2
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) )
     => ( ( member_a_real @ F @ ( qbs_morphism_a_real @ x @ Y ) )
       => ( probab3605210969150000782b_real @ Y @ ( comp_a_real_real @ F @ alpha ) @ mu ) ) ) ).

% qp.monadP_qbs_Pf_computation(1)
thf(fact_365_qp_OmonadP__qbs__Pf__computation_I1_J,axiom,
    ! [S2: probab4737552673497767871pace_a,F: a > b,Y: quasi_borel_b] :
      ( ( S2
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) )
     => ( ( member_a_b @ F @ ( qbs_morphism_a_b @ x @ Y ) )
       => ( probab701741629625904797prob_b @ Y @ ( comp_a_b_real @ F @ alpha ) @ mu ) ) ) ).

% qp.monadP_qbs_Pf_computation(1)
thf(fact_366_qp_OmonadP__qbs__Pf__computation_I1_J,axiom,
    ! [S2: probab4737552673497767871pace_a,F: a > a,Y: quasi_borel_a] :
      ( ( S2
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) )
     => ( ( member_a_a @ F @ ( qbs_morphism_a_a @ x @ Y ) )
       => ( probab701741629625904796prob_a @ Y @ ( comp_a_a_real @ F @ alpha ) @ mu ) ) ) ).

% qp.monadP_qbs_Pf_computation(1)
thf(fact_367_qp_Oqbs__prob__space__in__Px,axiom,
    member6485455074645559016pace_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) @ ( monad_3932415646498543856s_Px_a @ x ) ).

% qp.qbs_prob_space_in_Px
thf(fact_368_assms_I1_J,axiom,
    member6485455074645559016pace_a @ s @ ( monad_3932415646498543856s_Px_a @ x ) ).

% assms(1)
thf(fact_369_qp_OmonadP__qbs__Pf__computation_I2_J,axiom,
    ! [S2: probab4737552673497767871pace_a,F: a > probab4737552677800996672pace_b,Y: quasi_3431323906171122406pace_b] :
      ( ( S2
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) )
     => ( ( member7340901614391157822pace_b @ F @ ( qbs_mo6277074943135278010pace_b @ x @ Y ) )
       => ( ( monad_6220640177326107869pace_b @ x @ Y @ F @ S2 )
          = ( probab2758740287680477370pace_b @ ( produc4607355092896878674e_real @ Y @ ( produc2065420038842681233e_real @ ( comp_a6868616473110226257b_real @ F @ alpha ) @ mu ) ) ) ) ) ) ).

% qp.monadP_qbs_Pf_computation(2)
thf(fact_370_qp_OmonadP__qbs__Pf__computation_I2_J,axiom,
    ! [S2: probab4737552673497767871pace_a,F: a > extend8495563244428889912nnreal,Y: quasi_9015997321629101608nnreal] :
      ( ( S2
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) )
     => ( ( member298456594901751504nnreal @ F @ ( qbs_mo1434458643421888574nnreal @ x @ Y ) )
       => ( ( monad_4746400152670514267nnreal @ x @ Y @ F @ S2 )
          = ( probab2095897665094527806nnreal @ ( produc3311748387682954872e_real @ Y @ ( produc8591767778467529065e_real @ ( comp_a8249376463644563905l_real @ F @ alpha ) @ mu ) ) ) ) ) ) ).

% qp.monadP_qbs_Pf_computation(2)
thf(fact_371_qp_OmonadP__qbs__Pf__computation_I2_J,axiom,
    ! [S2: probab4737552673497767871pace_a,F: a > real,Y: quasi_borel_real] :
      ( ( S2
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) )
     => ( ( member_a_real @ F @ ( qbs_morphism_a_real @ x @ Y ) )
       => ( ( monad_6933770117061659215a_real @ x @ Y @ F @ S2 )
          = ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ Y @ ( produc1722724976708544245e_real @ ( comp_a_real_real @ F @ alpha ) @ mu ) ) ) ) ) ) ).

% qp.monadP_qbs_Pf_computation(2)
thf(fact_372_qp_OmonadP__qbs__Pf__computation_I2_J,axiom,
    ! [S2: probab4737552673497767871pace_a,F: a > b,Y: quasi_borel_b] :
      ( ( S2
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) )
     => ( ( member_a_b @ F @ ( qbs_morphism_a_b @ x @ Y ) )
       => ( ( monad_399656224284192924Pf_a_b @ x @ Y @ F @ S2 )
          = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ Y @ ( produc4478270668571743890e_real @ ( comp_a_b_real @ F @ alpha ) @ mu ) ) ) ) ) ) ).

% qp.monadP_qbs_Pf_computation(2)
thf(fact_373_qp_OmonadP__qbs__Pf__computation_I2_J,axiom,
    ! [S2: probab4737552673497767871pace_a,F: a > a,Y: quasi_borel_a] :
      ( ( S2
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) )
     => ( ( member_a_a @ F @ ( qbs_morphism_a_a @ x @ Y ) )
       => ( ( monad_399656224284192923Pf_a_a @ x @ Y @ F @ S2 )
          = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ ( comp_a_a_real @ F @ alpha ) @ mu ) ) ) ) ) ) ).

% qp.monadP_qbs_Pf_computation(2)
thf(fact_374_Rep__qbs__prob__space__inject,axiom,
    ! [X8: probab4737552673497767871pace_a,Y4: probab4737552673497767871pace_a] :
      ( ( ( probab8639044586466322086pace_a @ X8 )
        = ( probab8639044586466322086pace_a @ Y4 ) )
      = ( X8 = Y4 ) ) ).

% Rep_qbs_prob_space_inject
thf(fact_375_qbs__prob__eq__equiv23,axiom,
    probab3918592701117320376_eq2_a = probab1131137119144644343_eq3_a ).

% qbs_prob_eq_equiv23
thf(fact_376_qbs__prob__eq__equiv24,axiom,
    probab3918592701117320376_eq2_a = probab7567053574026744118_eq4_a ).

% qbs_prob_eq_equiv24
thf(fact_377_qbs__prob__eq__equiv34,axiom,
    probab1131137119144644343_eq3_a = probab7567053574026744118_eq4_a ).

% qbs_prob_eq_equiv34
thf(fact_378_qbs__prob__eq__2__implies__3,axiom,
    ! [P1: produc6543235832880896358e_real,P22: produc6543235832880896358e_real] :
      ( ( probab3918592701117320376_eq2_a @ P1 @ P22 )
     => ( probab1131137119144644343_eq3_a @ P1 @ P22 ) ) ).

% qbs_prob_eq_2_implies_3
thf(fact_379_qbs__prob__eq__4__implies__3,axiom,
    ! [P1: produc6543235832880896358e_real,P22: produc6543235832880896358e_real] :
      ( ( probab7567053574026744118_eq4_a @ P1 @ P22 )
     => ( probab1131137119144644343_eq3_a @ P1 @ P22 ) ) ).

% qbs_prob_eq_4_implies_3
thf(fact_380_in__Mx__def,axiom,
    ( probab5158868910350447249n_Mx_o
    = ( ^ [X5: quasi_borel_o,Alpha4: real > $o] : ( member_real_o @ Alpha4 @ ( qbs_Mx_o @ X5 ) ) ) ) ).

% in_Mx_def
thf(fact_381_in__Mx__def,axiom,
    ( probab3648538964331556119Mx_nat
    = ( ^ [X5: quasi_borel_nat,Alpha4: real > nat] : ( member_real_nat @ Alpha4 @ ( qbs_Mx_nat @ X5 ) ) ) ) ).

% in_Mx_def
thf(fact_382_in__Mx__def,axiom,
    ( probab1290144374689509503nnreal
    = ( ^ [X5: quasi_9015997321629101608nnreal,Alpha4: real > extend8495563244428889912nnreal] : ( member2919562650594848410nnreal @ Alpha4 @ ( qbs_Mx6523938229262583809nnreal @ X5 ) ) ) ) ).

% in_Mx_def
thf(fact_383_in__Mx__def,axiom,
    ( probab8203172577112083445omplex
    = ( ^ [X5: quasi_borel_complex,Alpha4: real > complex] : ( member_real_complex @ Alpha4 @ ( qbs_Mx_complex @ X5 ) ) ) ) ).

% in_Mx_def
thf(fact_384_in__Mx__def,axiom,
    ( probab6852221862700129395x_real
    = ( ^ [X5: quasi_borel_real,Alpha4: real > real] : ( member_real_real @ Alpha4 @ ( qbs_Mx_real @ X5 ) ) ) ) ).

% in_Mx_def
thf(fact_385_in__Mx__def,axiom,
    ( probab9007417770424356216n_Mx_b
    = ( ^ [X5: quasi_borel_b,Alpha4: real > b] : ( member_real_b @ Alpha4 @ ( qbs_Mx_b @ X5 ) ) ) ) ).

% in_Mx_def
thf(fact_386_in__Mx__def,axiom,
    ( probab9007417770424356215n_Mx_a
    = ( ^ [X5: quasi_borel_a,Alpha4: real > a] : ( member_real_a @ Alpha4 @ ( qbs_Mx_a @ X5 ) ) ) ) ).

% in_Mx_def
thf(fact_387_in__Mx_Ointro,axiom,
    ! [Alpha: real > $o,X: quasi_borel_o] :
      ( ( member_real_o @ Alpha @ ( qbs_Mx_o @ X ) )
     => ( probab5158868910350447249n_Mx_o @ X @ Alpha ) ) ).

% in_Mx.intro
thf(fact_388_in__Mx_Ointro,axiom,
    ! [Alpha: real > nat,X: quasi_borel_nat] :
      ( ( member_real_nat @ Alpha @ ( qbs_Mx_nat @ X ) )
     => ( probab3648538964331556119Mx_nat @ X @ Alpha ) ) ).

% in_Mx.intro
thf(fact_389_in__Mx_Ointro,axiom,
    ! [Alpha: real > extend8495563244428889912nnreal,X: quasi_9015997321629101608nnreal] :
      ( ( member2919562650594848410nnreal @ Alpha @ ( qbs_Mx6523938229262583809nnreal @ X ) )
     => ( probab1290144374689509503nnreal @ X @ Alpha ) ) ).

% in_Mx.intro
thf(fact_390_in__Mx_Ointro,axiom,
    ! [Alpha: real > complex,X: quasi_borel_complex] :
      ( ( member_real_complex @ Alpha @ ( qbs_Mx_complex @ X ) )
     => ( probab8203172577112083445omplex @ X @ Alpha ) ) ).

% in_Mx.intro
thf(fact_391_in__Mx_Ointro,axiom,
    ! [Alpha: real > real,X: quasi_borel_real] :
      ( ( member_real_real @ Alpha @ ( qbs_Mx_real @ X ) )
     => ( probab6852221862700129395x_real @ X @ Alpha ) ) ).

% in_Mx.intro
thf(fact_392_in__Mx_Ointro,axiom,
    ! [Alpha: real > b,X: quasi_borel_b] :
      ( ( member_real_b @ Alpha @ ( qbs_Mx_b @ X ) )
     => ( probab9007417770424356216n_Mx_b @ X @ Alpha ) ) ).

% in_Mx.intro
thf(fact_393_in__Mx_Ointro,axiom,
    ! [Alpha: real > a,X: quasi_borel_a] :
      ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X ) )
     => ( probab9007417770424356215n_Mx_a @ X @ Alpha ) ) ).

% in_Mx.intro
thf(fact_394_in__Mx_Oin__Mx,axiom,
    ! [X: quasi_borel_o,Alpha: real > $o] :
      ( ( probab5158868910350447249n_Mx_o @ X @ Alpha )
     => ( member_real_o @ Alpha @ ( qbs_Mx_o @ X ) ) ) ).

% in_Mx.in_Mx
thf(fact_395_in__Mx_Oin__Mx,axiom,
    ! [X: quasi_borel_nat,Alpha: real > nat] :
      ( ( probab3648538964331556119Mx_nat @ X @ Alpha )
     => ( member_real_nat @ Alpha @ ( qbs_Mx_nat @ X ) ) ) ).

% in_Mx.in_Mx
thf(fact_396_in__Mx_Oin__Mx,axiom,
    ! [X: quasi_9015997321629101608nnreal,Alpha: real > extend8495563244428889912nnreal] :
      ( ( probab1290144374689509503nnreal @ X @ Alpha )
     => ( member2919562650594848410nnreal @ Alpha @ ( qbs_Mx6523938229262583809nnreal @ X ) ) ) ).

% in_Mx.in_Mx
thf(fact_397_in__Mx_Oin__Mx,axiom,
    ! [X: quasi_borel_complex,Alpha: real > complex] :
      ( ( probab8203172577112083445omplex @ X @ Alpha )
     => ( member_real_complex @ Alpha @ ( qbs_Mx_complex @ X ) ) ) ).

% in_Mx.in_Mx
thf(fact_398_in__Mx_Oin__Mx,axiom,
    ! [X: quasi_borel_real,Alpha: real > real] :
      ( ( probab6852221862700129395x_real @ X @ Alpha )
     => ( member_real_real @ Alpha @ ( qbs_Mx_real @ X ) ) ) ).

% in_Mx.in_Mx
thf(fact_399_in__Mx_Oin__Mx,axiom,
    ! [X: quasi_borel_b,Alpha: real > b] :
      ( ( probab9007417770424356216n_Mx_b @ X @ Alpha )
     => ( member_real_b @ Alpha @ ( qbs_Mx_b @ X ) ) ) ).

% in_Mx.in_Mx
thf(fact_400_in__Mx_Oin__Mx,axiom,
    ! [X: quasi_borel_a,Alpha: real > a] :
      ( ( probab9007417770424356215n_Mx_a @ X @ Alpha )
     => ( member_real_a @ Alpha @ ( qbs_Mx_a @ X ) ) ) ).

% in_Mx.in_Mx
thf(fact_401_qbs__prob_Oaxioms_I1_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( probab9007417770424356216n_Mx_b @ X @ Alpha ) ) ).

% qbs_prob.axioms(1)
thf(fact_402_qbs__prob_Oaxioms_I1_J,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X @ Alpha @ Mu )
     => ( probab9007417770424356215n_Mx_a @ X @ Alpha ) ) ).

% qbs_prob.axioms(1)
thf(fact_403_qbs__prob__eq3__dest_I3_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,Y: quasi_borel_b,Beta2: real > b,Nu: sigma_measure_real] :
      ( ( probab1131137119144644344_eq3_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) @ ( produc2180226129289916244e_real @ Y @ ( produc4478270668571743890e_real @ Beta2 @ Nu ) ) )
     => ( Y = X ) ) ).

% qbs_prob_eq3_dest(3)
thf(fact_404_qbs__prob__eq3__dest_I3_J,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta2: real > a,Nu: sigma_measure_real] :
      ( ( probab1131137119144644343_eq3_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta2 @ Nu ) ) )
     => ( Y = X ) ) ).

% qbs_prob_eq3_dest(3)
thf(fact_405_qbs__prob__eq4__dest_I3_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,Y: quasi_borel_b,Beta2: real > b,Nu: sigma_measure_real] :
      ( ( probab7567053574026744119_eq4_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) @ ( produc2180226129289916244e_real @ Y @ ( produc4478270668571743890e_real @ Beta2 @ Nu ) ) )
     => ( Y = X ) ) ).

% qbs_prob_eq4_dest(3)
thf(fact_406_qbs__prob__eq4__dest_I3_J,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta2: real > a,Nu: sigma_measure_real] :
      ( ( probab7567053574026744118_eq4_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta2 @ Nu ) ) )
     => ( Y = X ) ) ).

% qbs_prob_eq4_dest(3)
thf(fact_407_qbs__prob__eq2__dest_I3_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,Y: quasi_borel_b,Beta2: real > b,Nu: sigma_measure_real] :
      ( ( probab3918592701117320377_eq2_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) @ ( produc2180226129289916244e_real @ Y @ ( produc4478270668571743890e_real @ Beta2 @ Nu ) ) )
     => ( Y = X ) ) ).

% qbs_prob_eq2_dest(3)
thf(fact_408_qbs__prob__eq2__dest_I3_J,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta2: real > a,Nu: sigma_measure_real] :
      ( ( probab3918592701117320376_eq2_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta2 @ Nu ) ) )
     => ( Y = X ) ) ).

% qbs_prob_eq2_dest(3)
thf(fact_409_qbs__prob__eq3__dest_I2_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,Y: quasi_borel_b,Beta2: real > b,Nu: sigma_measure_real] :
      ( ( probab1131137119144644344_eq3_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) @ ( produc2180226129289916244e_real @ Y @ ( produc4478270668571743890e_real @ Beta2 @ Nu ) ) )
     => ( probab701741629625904797prob_b @ Y @ Beta2 @ Nu ) ) ).

% qbs_prob_eq3_dest(2)
thf(fact_410_qbs__prob__eq3__dest_I2_J,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta2: real > a,Nu: sigma_measure_real] :
      ( ( probab1131137119144644343_eq3_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta2 @ Nu ) ) )
     => ( probab701741629625904796prob_a @ Y @ Beta2 @ Nu ) ) ).

% qbs_prob_eq3_dest(2)
thf(fact_411_qbs__prob__eq3__dest_I1_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,Y: quasi_borel_b,Beta2: real > b,Nu: sigma_measure_real] :
      ( ( probab1131137119144644344_eq3_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) @ ( produc2180226129289916244e_real @ Y @ ( produc4478270668571743890e_real @ Beta2 @ Nu ) ) )
     => ( probab701741629625904797prob_b @ X @ Alpha @ Mu ) ) ).

% qbs_prob_eq3_dest(1)
thf(fact_412_qbs__prob__eq3__dest_I1_J,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta2: real > a,Nu: sigma_measure_real] :
      ( ( probab1131137119144644343_eq3_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta2 @ Nu ) ) )
     => ( probab701741629625904796prob_a @ X @ Alpha @ Mu ) ) ).

% qbs_prob_eq3_dest(1)
thf(fact_413_qbs__prob_Oqbs__prob__eq3__refl,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( probab1131137119144644344_eq3_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) ) ).

% qbs_prob.qbs_prob_eq3_refl
thf(fact_414_qbs__prob_Oqbs__prob__eq3__refl,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X @ Alpha @ Mu )
     => ( probab1131137119144644343_eq3_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) ) ).

% qbs_prob.qbs_prob_eq3_refl
thf(fact_415_qbs__prob_Ointro,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real] :
      ( ( probab9007417770424356216n_Mx_b @ X @ Alpha )
     => ( ( distri2809703520229113005bution @ Mu )
       => ( probab701741629625904797prob_b @ X @ Alpha @ Mu ) ) ) ).

% qbs_prob.intro
thf(fact_416_qbs__prob_Ointro,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real] :
      ( ( probab9007417770424356215n_Mx_a @ X @ Alpha )
     => ( ( distri2809703520229113005bution @ Mu )
       => ( probab701741629625904796prob_a @ X @ Alpha @ Mu ) ) ) ).

% qbs_prob.intro
thf(fact_417_qbs__prob__def,axiom,
    ( probab701741629625904797prob_b
    = ( ^ [X5: quasi_borel_b,Alpha4: real > b,Mu4: sigma_measure_real] :
          ( ( probab9007417770424356216n_Mx_b @ X5 @ Alpha4 )
          & ( distri2809703520229113005bution @ Mu4 ) ) ) ) ).

% qbs_prob_def
thf(fact_418_qbs__prob__def,axiom,
    ( probab701741629625904796prob_a
    = ( ^ [X5: quasi_borel_a,Alpha4: real > a,Mu4: sigma_measure_real] :
          ( ( probab9007417770424356215n_Mx_a @ X5 @ Alpha4 )
          & ( distri2809703520229113005bution @ Mu4 ) ) ) ) ).

% qbs_prob_def
thf(fact_419_qbs__prob__eq2__dest_I2_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,Y: quasi_borel_b,Beta2: real > b,Nu: sigma_measure_real] :
      ( ( probab3918592701117320377_eq2_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) @ ( produc2180226129289916244e_real @ Y @ ( produc4478270668571743890e_real @ Beta2 @ Nu ) ) )
     => ( probab701741629625904797prob_b @ Y @ Beta2 @ Nu ) ) ).

% qbs_prob_eq2_dest(2)
thf(fact_420_qbs__prob__eq2__dest_I2_J,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta2: real > a,Nu: sigma_measure_real] :
      ( ( probab3918592701117320376_eq2_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta2 @ Nu ) ) )
     => ( probab701741629625904796prob_a @ Y @ Beta2 @ Nu ) ) ).

% qbs_prob_eq2_dest(2)
thf(fact_421_qbs__prob__eq2__dest_I1_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,Y: quasi_borel_b,Beta2: real > b,Nu: sigma_measure_real] :
      ( ( probab3918592701117320377_eq2_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) @ ( produc2180226129289916244e_real @ Y @ ( produc4478270668571743890e_real @ Beta2 @ Nu ) ) )
     => ( probab701741629625904797prob_b @ X @ Alpha @ Mu ) ) ).

% qbs_prob_eq2_dest(1)
thf(fact_422_qbs__prob__eq2__dest_I1_J,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta2: real > a,Nu: sigma_measure_real] :
      ( ( probab3918592701117320376_eq2_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta2 @ Nu ) ) )
     => ( probab701741629625904796prob_a @ X @ Alpha @ Mu ) ) ).

% qbs_prob_eq2_dest(1)
thf(fact_423_qbs__prob__eq4__dest_I2_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,Y: quasi_borel_b,Beta2: real > b,Nu: sigma_measure_real] :
      ( ( probab7567053574026744119_eq4_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) @ ( produc2180226129289916244e_real @ Y @ ( produc4478270668571743890e_real @ Beta2 @ Nu ) ) )
     => ( probab701741629625904797prob_b @ Y @ Beta2 @ Nu ) ) ).

% qbs_prob_eq4_dest(2)
thf(fact_424_qbs__prob__eq4__dest_I2_J,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta2: real > a,Nu: sigma_measure_real] :
      ( ( probab7567053574026744118_eq4_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta2 @ Nu ) ) )
     => ( probab701741629625904796prob_a @ Y @ Beta2 @ Nu ) ) ).

% qbs_prob_eq4_dest(2)
thf(fact_425_qbs__prob__eq4__dest_I1_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,Y: quasi_borel_b,Beta2: real > b,Nu: sigma_measure_real] :
      ( ( probab7567053574026744119_eq4_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) @ ( produc2180226129289916244e_real @ Y @ ( produc4478270668571743890e_real @ Beta2 @ Nu ) ) )
     => ( probab701741629625904797prob_b @ X @ Alpha @ Mu ) ) ).

% qbs_prob_eq4_dest(1)
thf(fact_426_qbs__prob__eq4__dest_I1_J,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta2: real > a,Nu: sigma_measure_real] :
      ( ( probab7567053574026744118_eq4_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta2 @ Nu ) ) )
     => ( probab701741629625904796prob_a @ X @ Alpha @ Mu ) ) ).

% qbs_prob_eq4_dest(1)
thf(fact_427_qbs__prob_Oqbs__prob__eq2__refl,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( probab3918592701117320377_eq2_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) ) ).

% qbs_prob.qbs_prob_eq2_refl
thf(fact_428_qbs__prob_Oqbs__prob__eq2__refl,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X @ Alpha @ Mu )
     => ( probab3918592701117320376_eq2_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) ) ).

% qbs_prob.qbs_prob_eq2_refl
thf(fact_429_qbs__prob_Oqbs__prob__eq4__refl,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( probab7567053574026744119_eq4_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) ) ).

% qbs_prob.qbs_prob_eq4_refl
thf(fact_430_qbs__prob_Oqbs__prob__eq4__refl,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X @ Alpha @ Mu )
     => ( probab7567053574026744118_eq4_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) ) ).

% qbs_prob.qbs_prob_eq4_refl
thf(fact_431_qbs__prob_Oqbs__prob__space__in__Px,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( member6485455078948787817pace_b @ ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) @ ( monad_3932415646498543857s_Px_b @ X ) ) ) ).

% qbs_prob.qbs_prob_space_in_Px
thf(fact_432_qbs__prob_Oqbs__prob__space__in__Px,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X @ Alpha @ Mu )
     => ( member6485455074645559016pace_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) @ ( monad_3932415646498543856s_Px_a @ X ) ) ) ).

% qbs_prob.qbs_prob_space_in_Px
thf(fact_433_rep__monadP__qbs__Px,axiom,
    ! [S2: probab4737552677800996672pace_b,X: quasi_borel_b] :
      ( ( member6485455078948787817pace_b @ S2 @ ( monad_3932415646498543857s_Px_b @ X ) )
     => ? [Alpha2: real > b,Mu2: sigma_measure_real] :
          ( ( S2
            = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha2 @ Mu2 ) ) ) )
          & ( probab701741629625904797prob_b @ X @ Alpha2 @ Mu2 ) ) ) ).

% rep_monadP_qbs_Px
thf(fact_434_rep__monadP__qbs__Px,axiom,
    ! [S2: probab4737552673497767871pace_a,X: quasi_borel_a] :
      ( ( member6485455074645559016pace_a @ S2 @ ( monad_3932415646498543856s_Px_a @ X ) )
     => ? [Alpha2: real > a,Mu2: sigma_measure_real] :
          ( ( S2
            = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha2 @ Mu2 ) ) ) )
          & ( probab701741629625904796prob_a @ X @ Alpha2 @ Mu2 ) ) ) ).

% rep_monadP_qbs_Px
thf(fact_435_qbs__prob_OmonadP__qbs__Pf__computation_I1_J,axiom,
    ! [X: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,S2: probab8009751763329705409e_real,F: real > $o,Y: quasi_borel_o] :
      ( ( probab3605210969150000782b_real @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) ) )
       => ( ( member_real_o @ F @ ( qbs_morphism_real_o @ X @ Y ) )
         => ( probab163731277575722550prob_o @ Y @ ( comp_real_o_real @ F @ Alpha ) @ Mu ) ) ) ) ).

% qbs_prob.monadP_qbs_Pf_computation(1)
thf(fact_436_qbs__prob_OmonadP__qbs__Pf__computation_I1_J,axiom,
    ! [X: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,S2: probab8009751763329705409e_real,F: real > nat,Y: quasi_borel_nat] :
      ( ( probab3605210969150000782b_real @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) ) )
       => ( ( member_real_nat @ F @ ( qbs_mo6567951568834356598al_nat @ X @ Y ) )
         => ( probab2851505236026752178ob_nat @ Y @ ( comp_real_nat_real @ F @ Alpha ) @ Mu ) ) ) ) ).

% qbs_prob.monadP_qbs_Pf_computation(1)
thf(fact_437_qbs__prob_OmonadP__qbs__Pf__computation_I1_J,axiom,
    ! [X: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,S2: probab8009751763329705409e_real,F: real > extend8495563244428889912nnreal,Y: quasi_9015997321629101608nnreal] :
      ( ( probab3605210969150000782b_real @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) ) )
       => ( ( member2919562650594848410nnreal @ F @ ( qbs_mo1317719164804411614nnreal @ X @ Y ) )
         => ( probab8888997264033409050nnreal @ Y @ ( comp_r6279034453215524981l_real @ F @ Alpha ) @ Mu ) ) ) ) ).

% qbs_prob.monadP_qbs_Pf_computation(1)
thf(fact_438_qbs__prob_OmonadP__qbs__Pf__computation_I1_J,axiom,
    ! [X: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,S2: probab8009751763329705409e_real,F: real > complex,Y: quasi_borel_complex] :
      ( ( probab3605210969150000782b_real @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) ) )
       => ( ( member_real_complex @ F @ ( qbs_mo6067097710682130004omplex @ X @ Y ) )
         => ( probab963564174529062288omplex @ Y @ ( comp_r1968866223832618731x_real @ F @ Alpha ) @ Mu ) ) ) ) ).

% qbs_prob.monadP_qbs_Pf_computation(1)
thf(fact_439_qbs__prob_OmonadP__qbs__Pf__computation_I1_J,axiom,
    ! [X: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,S2: probab8009751763329705409e_real,F: real > real,Y: quasi_borel_real] :
      ( ( probab3605210969150000782b_real @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) ) )
       => ( ( member_real_real @ F @ ( qbs_mo5229770564518008146l_real @ X @ Y ) )
         => ( probab3605210969150000782b_real @ Y @ ( comp_real_real_real @ F @ Alpha ) @ Mu ) ) ) ) ).

% qbs_prob.monadP_qbs_Pf_computation(1)
thf(fact_440_qbs__prob_OmonadP__qbs__Pf__computation_I1_J,axiom,
    ! [X: quasi_borel_o,Alpha: real > $o,Mu: sigma_measure_real,S2: probab1241297377463522905pace_o,F: $o > $o,Y: quasi_borel_o] :
      ( ( probab163731277575722550prob_o @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab414947219978445330pace_o @ ( produc2942336022369489698e_real @ X @ ( produc787877863769208121e_real @ Alpha @ Mu ) ) ) )
       => ( ( member_o_o @ F @ ( qbs_morphism_o_o @ X @ Y ) )
         => ( probab163731277575722550prob_o @ Y @ ( comp_o_o_real @ F @ Alpha ) @ Mu ) ) ) ) ).

% qbs_prob.monadP_qbs_Pf_computation(1)
thf(fact_441_qbs__prob_OmonadP__qbs__Pf__computation_I1_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,S2: probab4737552677800996672pace_b,F: b > extend8495563244428889912nnreal,Y: quasi_9015997321629101608nnreal] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) )
       => ( ( member6418304549040442065nnreal @ F @ ( qbs_mo7554306597560579135nnreal @ X @ Y ) )
         => ( probab8888997264033409050nnreal @ Y @ ( comp_b5186278242990201154l_real @ F @ Alpha ) @ Mu ) ) ) ) ).

% qbs_prob.monadP_qbs_Pf_computation(1)
thf(fact_442_qbs__prob_OmonadP__qbs__Pf__computation_I1_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,S2: probab4737552677800996672pace_b,F: b > b,Y: quasi_borel_b] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) )
       => ( ( member_b_b @ F @ ( qbs_morphism_b_b @ X @ Y ) )
         => ( probab701741629625904797prob_b @ Y @ ( comp_b_b_real @ F @ Alpha ) @ Mu ) ) ) ) ).

% qbs_prob.monadP_qbs_Pf_computation(1)
thf(fact_443_qbs__prob_OmonadP__qbs__Pf__computation_I1_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,S2: probab4737552677800996672pace_b,F: b > a,Y: quasi_borel_a] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) )
       => ( ( member_b_a @ F @ ( qbs_morphism_b_a @ X @ Y ) )
         => ( probab701741629625904796prob_a @ Y @ ( comp_b_a_real @ F @ Alpha ) @ Mu ) ) ) ) ).

% qbs_prob.monadP_qbs_Pf_computation(1)
thf(fact_444_qbs__prob_OmonadP__qbs__Pf__computation_I1_J,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,S2: probab4737552673497767871pace_a,F: a > extend8495563244428889912nnreal,Y: quasi_9015997321629101608nnreal] :
      ( ( probab701741629625904796prob_a @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) )
       => ( ( member298456594901751504nnreal @ F @ ( qbs_mo1434458643421888574nnreal @ X @ Y ) )
         => ( probab8888997264033409050nnreal @ Y @ ( comp_a8249376463644563905l_real @ F @ Alpha ) @ Mu ) ) ) ) ).

% qbs_prob.monadP_qbs_Pf_computation(1)
thf(fact_445_qbs__prob_Oif__in__Rep_I2_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,X9: quasi_borel_b,Alpha3: real > b,Mu3: sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( ( member379764716372043661e_real @ ( produc2180226129289916244e_real @ X9 @ ( produc4478270668571743890e_real @ Alpha3 @ Mu3 ) ) @ ( probab8639044586466322087pace_b @ ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) ) )
       => ( probab701741629625904797prob_b @ X9 @ Alpha3 @ Mu3 ) ) ) ).

% qbs_prob.if_in_Rep(2)
thf(fact_446_qbs__prob_Oif__in__Rep_I2_J,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,X9: quasi_borel_a,Alpha3: real > a,Mu3: sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X @ Alpha @ Mu )
     => ( ( member6844354795726785935e_real @ ( produc4145838808978236886e_real @ X9 @ ( produc623176010801490259e_real @ Alpha3 @ Mu3 ) ) @ ( probab8639044586466322086pace_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) ) )
       => ( probab701741629625904796prob_a @ X9 @ Alpha3 @ Mu3 ) ) ) ).

% qbs_prob.if_in_Rep(2)
thf(fact_447_qbs__prob_Oif__in__Rep_I1_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,X9: quasi_borel_b,Alpha3: real > b,Mu3: sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( ( member379764716372043661e_real @ ( produc2180226129289916244e_real @ X9 @ ( produc4478270668571743890e_real @ Alpha3 @ Mu3 ) ) @ ( probab8639044586466322087pace_b @ ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) ) )
       => ( X9 = X ) ) ) ).

% qbs_prob.if_in_Rep(1)
thf(fact_448_qbs__prob_Oif__in__Rep_I1_J,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,X9: quasi_borel_a,Alpha3: real > a,Mu3: sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X @ Alpha @ Mu )
     => ( ( member6844354795726785935e_real @ ( produc4145838808978236886e_real @ X9 @ ( produc623176010801490259e_real @ Alpha3 @ Mu3 ) ) @ ( probab8639044586466322086pace_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) ) )
       => ( X9 = X ) ) ) ).

% qbs_prob.if_in_Rep(1)
thf(fact_449_qbs__prob_Oin__Rep,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( member379764716372043661e_real @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) @ ( probab8639044586466322087pace_b @ ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) ) ) ) ).

% qbs_prob.in_Rep
thf(fact_450_qbs__prob_Oin__Rep,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X @ Alpha @ Mu )
     => ( member6844354795726785935e_real @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( probab8639044586466322086pace_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) ) ) ) ).

% qbs_prob.in_Rep
thf(fact_451_qbs__prob_Oqbs__bind__computation_I1_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,S2: probab4737552677800996672pace_b,F: b > probab1241297377463522905pace_o,Y: quasi_borel_o,Beta2: real > $o,G2: real > sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) )
       => ( ( member4041323071750490646pace_o @ F @ ( qbs_mo7566660958362347922pace_o @ X @ ( monad_monadP_qbs_o @ Y ) ) )
         => ( ( member_real_o @ Beta2 @ ( qbs_Mx_o @ Y ) )
           => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
             => ( ( ( comp_b3676460685241377833o_real @ F @ Alpha )
                  = ( ^ [R: real] : ( probab414947219978445330pace_o @ ( produc2942336022369489698e_real @ Y @ ( produc787877863769208121e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
               => ( probab163731277575722550prob_o @ Y @ Beta2 @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ).

% qbs_prob.qbs_bind_computation(1)
thf(fact_452_qbs__prob_Oqbs__bind__computation_I1_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,S2: probab4737552677800996672pace_b,F: b > probab8370124847414953445ce_nat,Y: quasi_borel_nat,Beta2: real > nat,G2: real > sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) )
       => ( ( member3412182957176556286ce_nat @ F @ ( qbs_mo4747267137793946348ce_nat @ X @ ( monad_monadP_qbs_nat @ Y ) ) )
         => ( ( member_real_nat @ Beta2 @ ( qbs_Mx_nat @ Y ) )
           => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
             => ( ( ( comp_b384277876254322031t_real @ F @ Alpha )
                  = ( ^ [R: real] : ( probab4893816680317050838ce_nat @ ( produc2796221405228754040e_real @ Y @ ( produc6760937697707383505e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
               => ( probab2851505236026752178ob_nat @ Y @ Beta2 @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ).

% qbs_prob.qbs_bind_computation(1)
thf(fact_453_qbs__prob_Oqbs__bind__computation_I1_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,S2: probab4737552677800996672pace_b,F: b > probab1359147627358338381nnreal,Y: quasi_9015997321629101608nnreal,Beta2: real > extend8495563244428889912nnreal,G2: real > sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) )
       => ( ( member8364764480815455846nnreal @ F @ ( qbs_mo7003404251872637524nnreal @ X @ ( monad_8737719049617959390nnreal @ Y ) ) )
         => ( ( member2919562650594848410nnreal @ Beta2 @ ( qbs_Mx6523938229262583809nnreal @ Y ) )
           => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
             => ( ( ( comp_b7132781594232482007l_real @ F @ Alpha )
                  = ( ^ [R: real] : ( probab2095897665094527806nnreal @ ( produc3311748387682954872e_real @ Y @ ( produc8591767778467529065e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
               => ( probab8888997264033409050nnreal @ Y @ Beta2 @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ).

% qbs_prob.qbs_bind_computation(1)
thf(fact_454_qbs__prob_Oqbs__bind__computation_I1_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,S2: probab4737552677800996672pace_b,F: b > probab3778977310982246339omplex,Y: quasi_borel_complex,Beta2: real > complex,G2: real > sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) )
       => ( ( member507933436795218396omplex @ F @ ( qbs_mo8077574125540273866omplex @ X @ ( monad_3228211519047171924omplex @ Y ) ) )
         => ( ( member_real_complex @ Beta2 @ ( qbs_Mx_complex @ Y ) )
           => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
             => ( ( ( comp_b2988533236785854029x_real @ F @ Alpha )
                  = ( ^ [R: real] : ( probab2835974231434789044omplex @ ( produc7812373664507111416e_real @ Y @ ( produc8166640644904819571e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
               => ( probab963564174529062288omplex @ Y @ Beta2 @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ).

% qbs_prob.qbs_bind_computation(1)
thf(fact_455_qbs__prob_Oqbs__bind__computation_I1_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,S2: probab4737552677800996672pace_b,F: b > probab8009751763329705409e_real,Y: quasi_borel_real,Beta2: real > real,G2: real > sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) )
       => ( ( member4393666000777897434e_real @ F @ ( qbs_mo907235460760239944e_real @ X @ ( monad_2887651017592114770s_real @ Y ) ) )
         => ( ( member_real_real @ Beta2 @ ( qbs_Mx_real @ Y ) )
           => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
             => ( ( ( comp_b7676164988068126923l_real @ F @ Alpha )
                  = ( ^ [R: real] : ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ Y @ ( produc1722724976708544245e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
               => ( probab3605210969150000782b_real @ Y @ Beta2 @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ).

% qbs_prob.qbs_bind_computation(1)
thf(fact_456_qbs__prob_Oqbs__bind__computation_I1_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,S2: probab4737552677800996672pace_b,F: b > probab4737552677800996672pace_b,Y: quasi_borel_b,Beta2: real > b,G2: real > sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) )
       => ( ( member1568243296417166973pace_b @ F @ ( qbs_mo504416625161287161pace_b @ X @ ( monad_monadP_qbs_b @ Y ) ) )
         => ( ( member_real_b @ Beta2 @ ( qbs_Mx_b @ Y ) )
           => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
             => ( ( ( comp_b4925389053965159440b_real @ F @ Alpha )
                  = ( ^ [R: real] : ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ Y @ ( produc4478270668571743890e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
               => ( probab701741629625904797prob_b @ Y @ Beta2 @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ).

% qbs_prob.qbs_bind_computation(1)
thf(fact_457_qbs__prob_Oqbs__bind__computation_I1_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,S2: probab4737552677800996672pace_b,F: b > probab4737552673497767871pace_a,Y: quasi_borel_a,Beta2: real > a,G2: real > sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) )
       => ( ( member1497209256370820988pace_a @ F @ ( qbs_mo504416620858058360pace_a @ X @ ( monad_monadP_qbs_a @ Y ) ) )
         => ( ( member_real_a @ Beta2 @ ( qbs_Mx_a @ Y ) )
           => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
             => ( ( ( comp_b7704093361825181967a_real @ F @ Alpha )
                  = ( ^ [R: real] : ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
               => ( probab701741629625904796prob_a @ Y @ Beta2 @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ).

% qbs_prob.qbs_bind_computation(1)
thf(fact_458_qbs__prob_Oqbs__bind__computation_I1_J,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,S2: probab4737552673497767871pace_a,F: a > probab1241297377463522905pace_o,Y: quasi_borel_o,Beta2: real > $o,G2: real > sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) )
       => ( ( member7312870768605366359pace_o @ F @ ( qbs_mo1614836618362447827pace_o @ X @ ( monad_monadP_qbs_o @ Y ) ) )
         => ( ( member_real_o @ Beta2 @ ( qbs_Mx_o @ Y ) )
           => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
             => ( ( ( comp_a281010391317482986o_real @ F @ Alpha )
                  = ( ^ [R: real] : ( probab414947219978445330pace_o @ ( produc2942336022369489698e_real @ Y @ ( produc787877863769208121e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
               => ( probab163731277575722550prob_o @ Y @ Beta2 @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ).

% qbs_prob.qbs_bind_computation(1)
thf(fact_459_qbs__prob_Oqbs__bind__computation_I1_J,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,S2: probab4737552673497767871pace_a,F: a > probab8370124847414953445ce_nat,Y: quasi_borel_nat,Beta2: real > nat,G2: real > sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) )
       => ( ( member6439975221678120701ce_nat @ F @ ( qbs_mo7775059402295510763ce_nat @ X @ ( monad_monadP_qbs_nat @ Y ) ) )
         => ( ( member_real_nat @ Beta2 @ ( qbs_Mx_nat @ Y ) )
           => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
             => ( ( ( comp_a1663454009834812910t_real @ F @ Alpha )
                  = ( ^ [R: real] : ( probab4893816680317050838ce_nat @ ( produc2796221405228754040e_real @ Y @ ( produc6760937697707383505e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
               => ( probab2851505236026752178ob_nat @ Y @ Beta2 @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ).

% qbs_prob.qbs_bind_computation(1)
thf(fact_460_qbs__prob_Oqbs__bind__computation_I1_J,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,S2: probab4737552673497767871pace_a,F: a > probab1359147627358338381nnreal,Y: quasi_9015997321629101608nnreal,Beta2: real > extend8495563244428889912nnreal,G2: real > sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) )
       => ( ( member2352135308469533285nnreal @ F @ ( qbs_mo990775079526714963nnreal @ X @ ( monad_8737719049617959390nnreal @ Y ) ) )
         => ( ( member2919562650594848410nnreal @ Beta2 @ ( qbs_Mx6523938229262583809nnreal @ Y ) )
           => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
             => ( ( ( comp_a4151604154359478614l_real @ F @ Alpha )
                  = ( ^ [R: real] : ( probab2095897665094527806nnreal @ ( produc3311748387682954872e_real @ Y @ ( produc8591767778467529065e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
               => ( probab8888997264033409050nnreal @ Y @ Beta2 @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ).

% qbs_prob.qbs_bind_computation(1)
thf(fact_461_qp_Oin__Rep__induct,axiom,
    ! [P2: produc6543235832880896358e_real > $o] :
      ( ! [Y6: quasi_borel_a,Beta: real > a,Nu2: sigma_measure_real] :
          ( ( member6844354795726785935e_real @ ( produc4145838808978236886e_real @ Y6 @ ( produc623176010801490259e_real @ Beta @ Nu2 ) ) @ ( probab8639044586466322086pace_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) ) )
         => ( P2 @ ( produc4145838808978236886e_real @ Y6 @ ( produc623176010801490259e_real @ Beta @ Nu2 ) ) ) )
     => ( P2 @ ( probab221732815614317479pace_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) ) ) ) ).

% qp.in_Rep_induct
thf(fact_462_qp_Oqbs__bind__computation_I2_J,axiom,
    ! [S2: probab4737552673497767871pace_a,F: a > probab1241297377463522905pace_o,Y: quasi_borel_o,Beta2: real > $o,G2: real > sigma_measure_real] :
      ( ( S2
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) )
     => ( ( member7312870768605366359pace_o @ F @ ( qbs_mo1614836618362447827pace_o @ x @ ( monad_monadP_qbs_o @ Y ) ) )
       => ( ( member_real_o @ Beta2 @ ( qbs_Mx_o @ Y ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( ( comp_a281010391317482986o_real @ F @ alpha )
                = ( ^ [R: real] : ( probab414947219978445330pace_o @ ( produc2942336022369489698e_real @ Y @ ( produc787877863769208121e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
             => ( ( monad_qbs_bind_a_o @ S2 @ F )
                = ( probab414947219978445330pace_o @ ( produc2942336022369489698e_real @ Y @ ( produc787877863769208121e_real @ Beta2 @ ( giry_bind_real_real @ mu @ G2 ) ) ) ) ) ) ) ) ) ) ).

% qp.qbs_bind_computation(2)
thf(fact_463_qp_Oqbs__bind__computation_I2_J,axiom,
    ! [S2: probab4737552673497767871pace_a,F: a > probab8370124847414953445ce_nat,Y: quasi_borel_nat,Beta2: real > nat,G2: real > sigma_measure_real] :
      ( ( S2
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) )
     => ( ( member6439975221678120701ce_nat @ F @ ( qbs_mo7775059402295510763ce_nat @ x @ ( monad_monadP_qbs_nat @ Y ) ) )
       => ( ( member_real_nat @ Beta2 @ ( qbs_Mx_nat @ Y ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( ( comp_a1663454009834812910t_real @ F @ alpha )
                = ( ^ [R: real] : ( probab4893816680317050838ce_nat @ ( produc2796221405228754040e_real @ Y @ ( produc6760937697707383505e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
             => ( ( monad_qbs_bind_a_nat @ S2 @ F )
                = ( probab4893816680317050838ce_nat @ ( produc2796221405228754040e_real @ Y @ ( produc6760937697707383505e_real @ Beta2 @ ( giry_bind_real_real @ mu @ G2 ) ) ) ) ) ) ) ) ) ) ).

% qp.qbs_bind_computation(2)
thf(fact_464_qp_Oqbs__bind__computation_I2_J,axiom,
    ! [S2: probab4737552673497767871pace_a,F: a > probab1359147627358338381nnreal,Y: quasi_9015997321629101608nnreal,Beta2: real > extend8495563244428889912nnreal,G2: real > sigma_measure_real] :
      ( ( S2
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) )
     => ( ( member2352135308469533285nnreal @ F @ ( qbs_mo990775079526714963nnreal @ x @ ( monad_8737719049617959390nnreal @ Y ) ) )
       => ( ( member2919562650594848410nnreal @ Beta2 @ ( qbs_Mx6523938229262583809nnreal @ Y ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( ( comp_a4151604154359478614l_real @ F @ alpha )
                = ( ^ [R: real] : ( probab2095897665094527806nnreal @ ( produc3311748387682954872e_real @ Y @ ( produc8591767778467529065e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
             => ( ( monad_5632564317481720659nnreal @ S2 @ F )
                = ( probab2095897665094527806nnreal @ ( produc3311748387682954872e_real @ Y @ ( produc8591767778467529065e_real @ Beta2 @ ( giry_bind_real_real @ mu @ G2 ) ) ) ) ) ) ) ) ) ) ).

% qp.qbs_bind_computation(2)
thf(fact_465_qp_Oqbs__bind__computation_I2_J,axiom,
    ! [S2: probab4737552673497767871pace_a,F: a > probab3778977310982246339omplex,Y: quasi_borel_complex,Beta2: real > complex,G2: real > sigma_measure_real] :
      ( ( S2
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) )
     => ( ( member2695794483689275355omplex @ F @ ( qbs_mo1042063135579555017omplex @ x @ ( monad_3228211519047171924omplex @ Y ) ) )
       => ( ( member_real_complex @ Beta2 @ ( qbs_Mx_complex @ Y ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( ( comp_a2485845550147043532x_real @ F @ alpha )
                = ( ^ [R: real] : ( probab2835974231434789044omplex @ ( produc7812373664507111416e_real @ Y @ ( produc8166640644904819571e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
             => ( ( monad_5863511652491161289omplex @ S2 @ F )
                = ( probab2835974231434789044omplex @ ( produc7812373664507111416e_real @ Y @ ( produc8166640644904819571e_real @ Beta2 @ ( giry_bind_real_real @ mu @ G2 ) ) ) ) ) ) ) ) ) ) ).

% qp.qbs_bind_computation(2)
thf(fact_466_qp_Oqbs__bind__computation_I2_J,axiom,
    ! [S2: probab4737552673497767871pace_a,F: a > probab8009751763329705409e_real,Y: quasi_borel_real,Beta2: real > real,G2: real > sigma_measure_real] :
      ( ( S2
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) )
     => ( ( member4522402906629368409e_real @ F @ ( qbs_mo1035972366611710919e_real @ x @ ( monad_2887651017592114770s_real @ Y ) ) )
       => ( ( member_real_real @ Beta2 @ ( qbs_Mx_real @ Y ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( ( comp_a677806100398348746l_real @ F @ alpha )
                = ( ^ [R: real] : ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ Y @ ( produc1722724976708544245e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
             => ( ( monad_6610920709787368775a_real @ S2 @ F )
                = ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ Y @ ( produc1722724976708544245e_real @ Beta2 @ ( giry_bind_real_real @ mu @ G2 ) ) ) ) ) ) ) ) ) ) ).

% qp.qbs_bind_computation(2)
thf(fact_467_qp_Oqbs__bind__computation_I2_J,axiom,
    ! [S2: probab4737552673497767871pace_a,F: a > probab4737552677800996672pace_b,Y: quasi_borel_b,Beta2: real > b,G2: real > sigma_measure_real] :
      ( ( S2
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) )
     => ( ( member7340901614391157822pace_b @ F @ ( qbs_mo6277074943135278010pace_b @ x @ ( monad_monadP_qbs_b @ Y ) ) )
       => ( ( member_real_b @ Beta2 @ ( qbs_Mx_b @ Y ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( ( comp_a6868616473110226257b_real @ F @ alpha )
                = ( ^ [R: real] : ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ Y @ ( produc4478270668571743890e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
             => ( ( monad_qbs_bind_a_b @ S2 @ F )
                = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ Y @ ( produc4478270668571743890e_real @ Beta2 @ ( giry_bind_real_real @ mu @ G2 ) ) ) ) ) ) ) ) ) ) ).

% qp.qbs_bind_computation(2)
thf(fact_468_qp_Oqbs__bind__computation_I2_J,axiom,
    ! [S2: probab4737552673497767871pace_a,F: a > probab4737552673497767871pace_a,Y: quasi_borel_a,Beta2: real > a,G2: real > sigma_measure_real] :
      ( ( S2
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) )
     => ( ( member7269867574344811837pace_a @ F @ ( qbs_mo6277074938832049209pace_a @ x @ ( monad_monadP_qbs_a @ Y ) ) )
       => ( ( member_real_a @ Beta2 @ ( qbs_Mx_a @ Y ) )
         => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
           => ( ( ( comp_a423948744115472976a_real @ F @ alpha )
                = ( ^ [R: real] : ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
             => ( ( monad_qbs_bind_a_a @ S2 @ F )
                = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta2 @ ( giry_bind_real_real @ mu @ G2 ) ) ) ) ) ) ) ) ) ) ).

% qp.qbs_bind_computation(2)
thf(fact_469_qp_Oif__in__Rep_I3_J,axiom,
    ! [X9: quasi_borel_a,Alpha3: real > a,Mu3: sigma_measure_real] :
      ( ( member6844354795726785935e_real @ ( produc4145838808978236886e_real @ X9 @ ( produc623176010801490259e_real @ Alpha3 @ Mu3 ) ) @ ( probab8639044586466322086pace_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) ) ) )
     => ( probab7355678800483015056b_eq_a @ ( produc4145838808978236886e_real @ x @ ( produc623176010801490259e_real @ alpha @ mu ) ) @ ( produc4145838808978236886e_real @ X9 @ ( produc623176010801490259e_real @ Alpha3 @ Mu3 ) ) ) ) ).

% qp.if_in_Rep(3)
thf(fact_470_qbs__prob_Oqbs__bind__computation_I2_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,S2: probab4737552677800996672pace_b,F: b > probab1241297377463522905pace_o,Y: quasi_borel_o,Beta2: real > $o,G2: real > sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) )
       => ( ( member4041323071750490646pace_o @ F @ ( qbs_mo7566660958362347922pace_o @ X @ ( monad_monadP_qbs_o @ Y ) ) )
         => ( ( member_real_o @ Beta2 @ ( qbs_Mx_o @ Y ) )
           => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
             => ( ( ( comp_b3676460685241377833o_real @ F @ Alpha )
                  = ( ^ [R: real] : ( probab414947219978445330pace_o @ ( produc2942336022369489698e_real @ Y @ ( produc787877863769208121e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
               => ( ( monad_qbs_bind_b_o @ S2 @ F )
                  = ( probab414947219978445330pace_o @ ( produc2942336022369489698e_real @ Y @ ( produc787877863769208121e_real @ Beta2 @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ) ) ) ).

% qbs_prob.qbs_bind_computation(2)
thf(fact_471_qbs__prob_Oqbs__bind__computation_I2_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,S2: probab4737552677800996672pace_b,F: b > probab8370124847414953445ce_nat,Y: quasi_borel_nat,Beta2: real > nat,G2: real > sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) )
       => ( ( member3412182957176556286ce_nat @ F @ ( qbs_mo4747267137793946348ce_nat @ X @ ( monad_monadP_qbs_nat @ Y ) ) )
         => ( ( member_real_nat @ Beta2 @ ( qbs_Mx_nat @ Y ) )
           => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
             => ( ( ( comp_b384277876254322031t_real @ F @ Alpha )
                  = ( ^ [R: real] : ( probab4893816680317050838ce_nat @ ( produc2796221405228754040e_real @ Y @ ( produc6760937697707383505e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
               => ( ( monad_qbs_bind_b_nat @ S2 @ F )
                  = ( probab4893816680317050838ce_nat @ ( produc2796221405228754040e_real @ Y @ ( produc6760937697707383505e_real @ Beta2 @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ) ) ) ).

% qbs_prob.qbs_bind_computation(2)
thf(fact_472_qbs__prob_Oqbs__bind__computation_I2_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,S2: probab4737552677800996672pace_b,F: b > probab1359147627358338381nnreal,Y: quasi_9015997321629101608nnreal,Beta2: real > extend8495563244428889912nnreal,G2: real > sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) )
       => ( ( member8364764480815455846nnreal @ F @ ( qbs_mo7003404251872637524nnreal @ X @ ( monad_8737719049617959390nnreal @ Y ) ) )
         => ( ( member2919562650594848410nnreal @ Beta2 @ ( qbs_Mx6523938229262583809nnreal @ Y ) )
           => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
             => ( ( ( comp_b7132781594232482007l_real @ F @ Alpha )
                  = ( ^ [R: real] : ( probab2095897665094527806nnreal @ ( produc3311748387682954872e_real @ Y @ ( produc8591767778467529065e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
               => ( ( monad_2529040234765635412nnreal @ S2 @ F )
                  = ( probab2095897665094527806nnreal @ ( produc3311748387682954872e_real @ Y @ ( produc8591767778467529065e_real @ Beta2 @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ) ) ) ).

% qbs_prob.qbs_bind_computation(2)
thf(fact_473_qbs__prob_Oqbs__bind__computation_I2_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,S2: probab4737552677800996672pace_b,F: b > probab3778977310982246339omplex,Y: quasi_borel_complex,Beta2: real > complex,G2: real > sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) )
       => ( ( member507933436795218396omplex @ F @ ( qbs_mo8077574125540273866omplex @ X @ ( monad_3228211519047171924omplex @ Y ) ) )
         => ( ( member_real_complex @ Beta2 @ ( qbs_Mx_complex @ Y ) )
           => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
             => ( ( ( comp_b2988533236785854029x_real @ F @ Alpha )
                  = ( ^ [R: real] : ( probab2835974231434789044omplex @ ( produc7812373664507111416e_real @ Y @ ( produc8166640644904819571e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
               => ( ( monad_2493997385533008074omplex @ S2 @ F )
                  = ( probab2835974231434789044omplex @ ( produc7812373664507111416e_real @ Y @ ( produc8166640644904819571e_real @ Beta2 @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ) ) ) ).

% qbs_prob.qbs_bind_computation(2)
thf(fact_474_qbs__prob_Oqbs__bind__computation_I2_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,S2: probab4737552677800996672pace_b,F: b > probab8009751763329705409e_real,Y: quasi_borel_real,Beta2: real > real,G2: real > sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) )
       => ( ( member4393666000777897434e_real @ F @ ( qbs_mo907235460760239944e_real @ X @ ( monad_2887651017592114770s_real @ Y ) ) )
         => ( ( member_real_real @ Beta2 @ ( qbs_Mx_real @ Y ) )
           => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
             => ( ( ( comp_b7676164988068126923l_real @ F @ Alpha )
                  = ( ^ [R: real] : ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ Y @ ( produc1722724976708544245e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
               => ( ( monad_2983767288120723656b_real @ S2 @ F )
                  = ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ Y @ ( produc1722724976708544245e_real @ Beta2 @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ) ) ) ).

% qbs_prob.qbs_bind_computation(2)
thf(fact_475_qbs__prob_Oqbs__bind__computation_I2_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,S2: probab4737552677800996672pace_b,F: b > probab4737552677800996672pace_b,Y: quasi_borel_b,Beta2: real > b,G2: real > sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) )
       => ( ( member1568243296417166973pace_b @ F @ ( qbs_mo504416625161287161pace_b @ X @ ( monad_monadP_qbs_b @ Y ) ) )
         => ( ( member_real_b @ Beta2 @ ( qbs_Mx_b @ Y ) )
           => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
             => ( ( ( comp_b4925389053965159440b_real @ F @ Alpha )
                  = ( ^ [R: real] : ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ Y @ ( produc4478270668571743890e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
               => ( ( monad_qbs_bind_b_b @ S2 @ F )
                  = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ Y @ ( produc4478270668571743890e_real @ Beta2 @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ) ) ) ).

% qbs_prob.qbs_bind_computation(2)
thf(fact_476_qbs__prob_Oqbs__bind__computation_I2_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,S2: probab4737552677800996672pace_b,F: b > probab4737552673497767871pace_a,Y: quasi_borel_a,Beta2: real > a,G2: real > sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) )
       => ( ( member1497209256370820988pace_a @ F @ ( qbs_mo504416620858058360pace_a @ X @ ( monad_monadP_qbs_a @ Y ) ) )
         => ( ( member_real_a @ Beta2 @ ( qbs_Mx_a @ Y ) )
           => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
             => ( ( ( comp_b7704093361825181967a_real @ F @ Alpha )
                  = ( ^ [R: real] : ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
               => ( ( monad_qbs_bind_b_a @ S2 @ F )
                  = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta2 @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ) ) ) ).

% qbs_prob.qbs_bind_computation(2)
thf(fact_477_qbs__prob_Oqbs__bind__computation_I2_J,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,S2: probab4737552673497767871pace_a,F: a > probab1241297377463522905pace_o,Y: quasi_borel_o,Beta2: real > $o,G2: real > sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) )
       => ( ( member7312870768605366359pace_o @ F @ ( qbs_mo1614836618362447827pace_o @ X @ ( monad_monadP_qbs_o @ Y ) ) )
         => ( ( member_real_o @ Beta2 @ ( qbs_Mx_o @ Y ) )
           => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
             => ( ( ( comp_a281010391317482986o_real @ F @ Alpha )
                  = ( ^ [R: real] : ( probab414947219978445330pace_o @ ( produc2942336022369489698e_real @ Y @ ( produc787877863769208121e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
               => ( ( monad_qbs_bind_a_o @ S2 @ F )
                  = ( probab414947219978445330pace_o @ ( produc2942336022369489698e_real @ Y @ ( produc787877863769208121e_real @ Beta2 @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ) ) ) ).

% qbs_prob.qbs_bind_computation(2)
thf(fact_478_qbs__prob_Oqbs__bind__computation_I2_J,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,S2: probab4737552673497767871pace_a,F: a > probab8370124847414953445ce_nat,Y: quasi_borel_nat,Beta2: real > nat,G2: real > sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) )
       => ( ( member6439975221678120701ce_nat @ F @ ( qbs_mo7775059402295510763ce_nat @ X @ ( monad_monadP_qbs_nat @ Y ) ) )
         => ( ( member_real_nat @ Beta2 @ ( qbs_Mx_nat @ Y ) )
           => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
             => ( ( ( comp_a1663454009834812910t_real @ F @ Alpha )
                  = ( ^ [R: real] : ( probab4893816680317050838ce_nat @ ( produc2796221405228754040e_real @ Y @ ( produc6760937697707383505e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
               => ( ( monad_qbs_bind_a_nat @ S2 @ F )
                  = ( probab4893816680317050838ce_nat @ ( produc2796221405228754040e_real @ Y @ ( produc6760937697707383505e_real @ Beta2 @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ) ) ) ).

% qbs_prob.qbs_bind_computation(2)
thf(fact_479_qbs__prob_Oqbs__bind__computation_I2_J,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,S2: probab4737552673497767871pace_a,F: a > probab1359147627358338381nnreal,Y: quasi_9015997321629101608nnreal,Beta2: real > extend8495563244428889912nnreal,G2: real > sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) )
       => ( ( member2352135308469533285nnreal @ F @ ( qbs_mo990775079526714963nnreal @ X @ ( monad_8737719049617959390nnreal @ Y ) ) )
         => ( ( member2919562650594848410nnreal @ Beta2 @ ( qbs_Mx6523938229262583809nnreal @ Y ) )
           => ( ( member2630560753458908601e_real @ G2 @ ( sigma_5928869325259027335e_real @ borel_5078946678739801102l_real @ ( giry_p6081475675320601670a_real @ borel_5078946678739801102l_real ) ) )
             => ( ( ( comp_a4151604154359478614l_real @ F @ Alpha )
                  = ( ^ [R: real] : ( probab2095897665094527806nnreal @ ( produc3311748387682954872e_real @ Y @ ( produc8591767778467529065e_real @ Beta2 @ ( G2 @ R ) ) ) ) ) )
               => ( ( monad_5632564317481720659nnreal @ S2 @ F )
                  = ( probab2095897665094527806nnreal @ ( produc3311748387682954872e_real @ Y @ ( produc8591767778467529065e_real @ Beta2 @ ( giry_bind_real_real @ Mu @ G2 ) ) ) ) ) ) ) ) ) ) ) ).

% qbs_prob.qbs_bind_computation(2)
thf(fact_480_Pair__le,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( ( ord_le1075799226346578649l_real @ ( produc4511245868158468465l_real @ A2 @ B2 ) @ ( produc4511245868158468465l_real @ C @ D ) )
      = ( ( ord_less_eq_real @ A2 @ C )
        & ( ord_less_eq_real @ B2 @ D ) ) ) ).

% Pair_le
thf(fact_481_Pair__le,axiom,
    ! [A2: real,B2: extend8495563244428889912nnreal,C: real,D: extend8495563244428889912nnreal] :
      ( ( ord_le4096773168995780197nnreal @ ( produc4778015194254607485nnreal @ A2 @ B2 ) @ ( produc4778015194254607485nnreal @ C @ D ) )
      = ( ( ord_less_eq_real @ A2 @ C )
        & ( ord_le3935885782089961368nnreal @ B2 @ D ) ) ) ).

% Pair_le
thf(fact_482_Pair__le,axiom,
    ! [A2: extend8495563244428889912nnreal,B2: real,C: extend8495563244428889912nnreal,D: real] :
      ( ( ord_le4051224869651757541l_real @ ( produc2810268924804063229l_real @ A2 @ B2 ) @ ( produc2810268924804063229l_real @ C @ D ) )
      = ( ( ord_le3935885782089961368nnreal @ A2 @ C )
        & ( ord_less_eq_real @ B2 @ D ) ) ) ).

% Pair_le
thf(fact_483_Pair__le,axiom,
    ! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D: extend8495563244428889912nnreal] :
      ( ( ord_le1399272598019556209nnreal @ ( produc344325839068023049nnreal @ A2 @ B2 ) @ ( produc344325839068023049nnreal @ C @ D ) )
      = ( ( ord_le3935885782089961368nnreal @ A2 @ C )
        & ( ord_le3935885782089961368nnreal @ B2 @ D ) ) ) ).

% Pair_le
thf(fact_484_Pair__le,axiom,
    ! [A2: real,B2: set_real,C: real,D: set_real] :
      ( ( ord_le4945634015433394191t_real @ ( produc3307717890874201511t_real @ A2 @ B2 ) @ ( produc3307717890874201511t_real @ C @ D ) )
      = ( ( ord_less_eq_real @ A2 @ C )
        & ( ord_less_eq_set_real @ B2 @ D ) ) ) ).

% Pair_le
thf(fact_485_Pair__le,axiom,
    ! [A2: set_real,B2: real,C: set_real,D: real] :
      ( ( ord_le5399745455957431311l_real @ ( produc6322126818539730599l_real @ A2 @ B2 ) @ ( produc6322126818539730599l_real @ C @ D ) )
      = ( ( ord_less_eq_set_real @ A2 @ C )
        & ( ord_less_eq_real @ B2 @ D ) ) ) ).

% Pair_le
thf(fact_486_Pair__le,axiom,
    ! [A2: set_real,B2: extend8495563244428889912nnreal,C: set_real,D: extend8495563244428889912nnreal] :
      ( ( ord_le4558722300798010011nnreal @ ( produc8660806124923995315nnreal @ A2 @ B2 ) @ ( produc8660806124923995315nnreal @ C @ D ) )
      = ( ( ord_less_eq_set_real @ A2 @ C )
        & ( ord_le3935885782089961368nnreal @ B2 @ D ) ) ) ).

% Pair_le
thf(fact_487_Pair__le,axiom,
    ! [A2: extend8495563244428889912nnreal,B2: set_real,C: extend8495563244428889912nnreal,D: set_real] :
      ( ( ord_le3400283774364675867t_real @ ( produc3019872141242647603t_real @ A2 @ B2 ) @ ( produc3019872141242647603t_real @ C @ D ) )
      = ( ( ord_le3935885782089961368nnreal @ A2 @ C )
        & ( ord_less_eq_set_real @ B2 @ D ) ) ) ).

% Pair_le
thf(fact_488_Pair__le,axiom,
    ! [A2: real,B2: set_set_real,C: real,D: set_set_real] :
      ( ( ord_le6676645311053956677t_real @ ( produc7184536090204209885t_real @ A2 @ B2 ) @ ( produc7184536090204209885t_real @ C @ D ) )
      = ( ( ord_less_eq_real @ A2 @ C )
        & ( ord_le3558479182127378552t_real @ B2 @ D ) ) ) ).

% Pair_le
thf(fact_489_Pair__le,axiom,
    ! [A2: set_set_real,B2: real,C: set_set_real,D: real] :
      ( ( ord_le4824006318743526981l_real @ ( produc8911628951482418397l_real @ A2 @ B2 ) @ ( produc8911628951482418397l_real @ C @ D ) )
      = ( ( ord_le3558479182127378552t_real @ A2 @ C )
        & ( ord_less_eq_real @ B2 @ D ) ) ) ).

% Pair_le
thf(fact_490_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_491_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_492_qbs__prob_OmonadP__qbs__Pf__computation_I2_J,axiom,
    ! [X: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,S2: probab8009751763329705409e_real,F: real > $o,Y: quasi_borel_o] :
      ( ( probab3605210969150000782b_real @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) ) )
       => ( ( member_real_o @ F @ ( qbs_morphism_real_o @ X @ Y ) )
         => ( ( monad_7062749578813130895real_o @ X @ Y @ F @ S2 )
            = ( probab414947219978445330pace_o @ ( produc2942336022369489698e_real @ Y @ ( produc787877863769208121e_real @ ( comp_real_o_real @ F @ Alpha ) @ Mu ) ) ) ) ) ) ) ).

% qbs_prob.monadP_qbs_Pf_computation(2)
thf(fact_493_qbs__prob_OmonadP__qbs__Pf__computation_I2_J,axiom,
    ! [X: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,S2: probab8009751763329705409e_real,F: real > nat,Y: quasi_borel_nat] :
      ( ( probab3605210969150000782b_real @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) ) )
       => ( ( member_real_nat @ F @ ( qbs_mo6567951568834356598al_nat @ X @ Y ) )
         => ( ( monad_2109573969670034073al_nat @ X @ Y @ F @ S2 )
            = ( probab4893816680317050838ce_nat @ ( produc2796221405228754040e_real @ Y @ ( produc6760937697707383505e_real @ ( comp_real_nat_real @ F @ Alpha ) @ Mu ) ) ) ) ) ) ) ).

% qbs_prob.monadP_qbs_Pf_computation(2)
thf(fact_494_qbs__prob_OmonadP__qbs__Pf__computation_I2_J,axiom,
    ! [X: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,S2: probab8009751763329705409e_real,F: real > extend8495563244428889912nnreal,Y: quasi_9015997321629101608nnreal] :
      ( ( probab3605210969150000782b_real @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) ) )
       => ( ( member2919562650594848410nnreal @ F @ ( qbs_mo1317719164804411614nnreal @ X @ Y ) )
         => ( ( monad_6772047417230001665nnreal @ X @ Y @ F @ S2 )
            = ( probab2095897665094527806nnreal @ ( produc3311748387682954872e_real @ Y @ ( produc8591767778467529065e_real @ ( comp_r6279034453215524981l_real @ F @ Alpha ) @ Mu ) ) ) ) ) ) ) ).

% qbs_prob.monadP_qbs_Pf_computation(2)
thf(fact_495_qbs__prob_OmonadP__qbs__Pf__computation_I2_J,axiom,
    ! [X: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,S2: probab8009751763329705409e_real,F: real > complex,Y: quasi_borel_complex] :
      ( ( probab3605210969150000782b_real @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) ) )
       => ( ( member_real_complex @ F @ ( qbs_mo6067097710682130004omplex @ X @ Y ) )
         => ( ( monad_4232241275811048823omplex @ X @ Y @ F @ S2 )
            = ( probab2835974231434789044omplex @ ( produc7812373664507111416e_real @ Y @ ( produc8166640644904819571e_real @ ( comp_r1968866223832618731x_real @ F @ Alpha ) @ Mu ) ) ) ) ) ) ) ).

% qbs_prob.monadP_qbs_Pf_computation(2)
thf(fact_496_qbs__prob_OmonadP__qbs__Pf__computation_I2_J,axiom,
    ! [X: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,S2: probab8009751763329705409e_real,F: real > real,Y: quasi_borel_real] :
      ( ( probab3605210969150000782b_real @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) ) )
       => ( ( member_real_real @ F @ ( qbs_mo5229770564518008146l_real @ X @ Y ) )
         => ( ( monad_4235422856176591093l_real @ X @ Y @ F @ S2 )
            = ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ Y @ ( produc1722724976708544245e_real @ ( comp_real_real_real @ F @ Alpha ) @ Mu ) ) ) ) ) ) ) ).

% qbs_prob.monadP_qbs_Pf_computation(2)
thf(fact_497_qbs__prob_OmonadP__qbs__Pf__computation_I2_J,axiom,
    ! [X: quasi_borel_o,Alpha: real > $o,Mu: sigma_measure_real,S2: probab1241297377463522905pace_o,F: $o > $o,Y: quasi_borel_o] :
      ( ( probab163731277575722550prob_o @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab414947219978445330pace_o @ ( produc2942336022369489698e_real @ X @ ( produc787877863769208121e_real @ Alpha @ Mu ) ) ) )
       => ( ( member_o_o @ F @ ( qbs_morphism_o_o @ X @ Y ) )
         => ( ( monad_3447778872307538587Pf_o_o @ X @ Y @ F @ S2 )
            = ( probab414947219978445330pace_o @ ( produc2942336022369489698e_real @ Y @ ( produc787877863769208121e_real @ ( comp_o_o_real @ F @ Alpha ) @ Mu ) ) ) ) ) ) ) ).

% qbs_prob.monadP_qbs_Pf_computation(2)
thf(fact_498_qbs__prob_OmonadP__qbs__Pf__computation_I2_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,S2: probab4737552677800996672pace_b,F: b > extend8495563244428889912nnreal,Y: quasi_9015997321629101608nnreal] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) )
       => ( ( member6418304549040442065nnreal @ F @ ( qbs_mo7554306597560579135nnreal @ X @ Y ) )
         => ( ( monad_1642876069954429020nnreal @ X @ Y @ F @ S2 )
            = ( probab2095897665094527806nnreal @ ( produc3311748387682954872e_real @ Y @ ( produc8591767778467529065e_real @ ( comp_b5186278242990201154l_real @ F @ Alpha ) @ Mu ) ) ) ) ) ) ) ).

% qbs_prob.monadP_qbs_Pf_computation(2)
thf(fact_499_qbs__prob_OmonadP__qbs__Pf__computation_I2_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,S2: probab4737552677800996672pace_b,F: b > b,Y: quasi_borel_b] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) )
       => ( ( member_b_b @ F @ ( qbs_morphism_b_b @ X @ Y ) )
         => ( ( monad_6835572679166292699Pf_b_b @ X @ Y @ F @ S2 )
            = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ Y @ ( produc4478270668571743890e_real @ ( comp_b_b_real @ F @ Alpha ) @ Mu ) ) ) ) ) ) ) ).

% qbs_prob.monadP_qbs_Pf_computation(2)
thf(fact_500_qbs__prob_OmonadP__qbs__Pf__computation_I2_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,S2: probab4737552677800996672pace_b,F: b > a,Y: quasi_borel_a] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) )
       => ( ( member_b_a @ F @ ( qbs_morphism_b_a @ X @ Y ) )
         => ( ( monad_6835572679166292698Pf_b_a @ X @ Y @ F @ S2 )
            = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ ( comp_b_a_real @ F @ Alpha ) @ Mu ) ) ) ) ) ) ) ).

% qbs_prob.monadP_qbs_Pf_computation(2)
thf(fact_501_qbs__prob_OmonadP__qbs__Pf__computation_I2_J,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,S2: probab4737552673497767871pace_a,F: a > extend8495563244428889912nnreal,Y: quasi_9015997321629101608nnreal] :
      ( ( probab701741629625904796prob_a @ X @ Alpha @ Mu )
     => ( ( S2
          = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) )
       => ( ( member298456594901751504nnreal @ F @ ( qbs_mo1434458643421888574nnreal @ X @ Y ) )
         => ( ( monad_4746400152670514267nnreal @ X @ Y @ F @ S2 )
            = ( probab2095897665094527806nnreal @ ( produc3311748387682954872e_real @ Y @ ( produc8591767778467529065e_real @ ( comp_a8249376463644563905l_real @ F @ Alpha ) @ Mu ) ) ) ) ) ) ) ).

% qbs_prob.monadP_qbs_Pf_computation(2)
thf(fact_502_qbs__prob__space_Oabs__induct,axiom,
    ! [P2: probab4737552677800996672pace_b > $o,X8: probab4737552677800996672pace_b] :
      ( ! [Y5: produc78645753526154084e_real] :
          ( ( probab7355678800483015057b_eq_b @ Y5 @ Y5 )
         => ( P2 @ ( probab8173042092732894329pace_b @ Y5 ) ) )
     => ( P2 @ X8 ) ) ).

% qbs_prob_space.abs_induct
thf(fact_503_qbs__prob__space_Oabs__induct,axiom,
    ! [P2: probab4737552673497767871pace_a > $o,X8: probab4737552673497767871pace_a] :
      ( ! [Y5: produc6543235832880896358e_real] :
          ( ( probab7355678800483015056b_eq_a @ Y5 @ Y5 )
         => ( P2 @ ( probab8173042092732894328pace_a @ Y5 ) ) )
     => ( P2 @ X8 ) ) ).

% qbs_prob_space.abs_induct
thf(fact_504_qbs__prob__space_Orep__prop,axiom,
    ! [Y4: probab4737552673497767871pace_a] :
    ? [X3: produc6543235832880896358e_real] :
      ( ( probab7355678800483015056b_eq_a @ X3 @ X3 )
      & ( ( probab8639044586466322086pace_a @ Y4 )
        = ( collec2245114308608258001e_real @ ( probab7355678800483015056b_eq_a @ X3 ) ) ) ) ).

% qbs_prob_space.rep_prop
thf(fact_505_qbs__prob__eq__3__implies__1,axiom,
    ! [P1: produc6543235832880896358e_real,P22: produc6543235832880896358e_real] :
      ( ( probab1131137119144644343_eq3_a @ P1 @ P22 )
     => ( probab7355678800483015056b_eq_a @ P1 @ P22 ) ) ).

% qbs_prob_eq_3_implies_1
thf(fact_506_qbs__prob__eq__equiv31,axiom,
    probab7355678800483015056b_eq_a = probab1131137119144644343_eq3_a ).

% qbs_prob_eq_equiv31
thf(fact_507_qbs__prob__eq__1__implies__4,axiom,
    ! [P1: produc6543235832880896358e_real,P22: produc6543235832880896358e_real] :
      ( ( probab7355678800483015056b_eq_a @ P1 @ P22 )
     => ( probab7567053574026744118_eq4_a @ P1 @ P22 ) ) ).

% qbs_prob_eq_1_implies_4
thf(fact_508_qbs__prob__eq__1__implies__2,axiom,
    ! [P1: produc6543235832880896358e_real,P22: produc6543235832880896358e_real] :
      ( ( probab7355678800483015056b_eq_a @ P1 @ P22 )
     => ( probab3918592701117320376_eq2_a @ P1 @ P22 ) ) ).

% qbs_prob_eq_1_implies_2
thf(fact_509_qbs__prob__eq__equiv14,axiom,
    probab7355678800483015056b_eq_a = probab7567053574026744118_eq4_a ).

% qbs_prob_eq_equiv14
thf(fact_510_qbs__prob__eq__equiv12,axiom,
    probab7355678800483015056b_eq_a = probab3918592701117320376_eq2_a ).

% qbs_prob_eq_equiv12
thf(fact_511_qbs__of__bind,axiom,
    ! [S2: probab4737552673497767871pace_a,X: quasi_borel_a,F: a > probab4737552677800996672pace_b,Y: quasi_borel_b] :
      ( ( member6485455074645559016pace_a @ S2 @ ( monad_3932415646498543856s_Px_a @ X ) )
     => ( ( member7340901614391157822pace_b @ F @ ( qbs_mo6277074943135278010pace_b @ X @ ( monad_monadP_qbs_b @ Y ) ) )
       => ( ( probab1293289258141559361_qbs_b @ ( monad_qbs_bind_a_b @ S2 @ F ) )
          = Y ) ) ) ).

% qbs_of_bind
thf(fact_512_qbs__of__bind,axiom,
    ! [S2: probab4737552673497767871pace_a,X: quasi_borel_a,F: a > probab4737552673497767871pace_a,Y: quasi_borel_a] :
      ( ( member6485455074645559016pace_a @ S2 @ ( monad_3932415646498543856s_Px_a @ X ) )
     => ( ( member7269867574344811837pace_a @ F @ ( qbs_mo6277074938832049209pace_a @ X @ ( monad_monadP_qbs_a @ Y ) ) )
       => ( ( probab1293289258141559360_qbs_a @ ( monad_qbs_bind_a_a @ S2 @ F ) )
          = Y ) ) ) ).

% qbs_of_bind
thf(fact_513_qbs__prob__eq__dest_I3_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,Y: quasi_borel_b,Beta2: real > b,Nu: sigma_measure_real] :
      ( ( probab7355678800483015057b_eq_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) @ ( produc2180226129289916244e_real @ Y @ ( produc4478270668571743890e_real @ Beta2 @ Nu ) ) )
     => ( Y = X ) ) ).

% qbs_prob_eq_dest(3)
thf(fact_514_qbs__prob__eq__dest_I3_J,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta2: real > a,Nu: sigma_measure_real] :
      ( ( probab7355678800483015056b_eq_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta2 @ Nu ) ) )
     => ( Y = X ) ) ).

% qbs_prob_eq_dest(3)
thf(fact_515_qbs__join__morphism,axiom,
    ! [X: quasi_borel_b] : ( member7687570473057579771pace_b @ monad_qbs_join_b @ ( qbs_mo3188246409730626167pace_b @ ( monad_1549008831671149082pace_b @ ( monad_monadP_qbs_b @ X ) ) @ ( monad_monadP_qbs_b @ X ) ) ) ).

% qbs_join_morphism
thf(fact_516_qbs__prob__space__eq,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,Y: quasi_borel_b,Beta2: real > b,Nu: sigma_measure_real] :
      ( ( probab7355678800483015057b_eq_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) @ ( produc2180226129289916244e_real @ Y @ ( produc4478270668571743890e_real @ Beta2 @ Nu ) ) )
     => ( ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) )
        = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ Y @ ( produc4478270668571743890e_real @ Beta2 @ Nu ) ) ) ) ) ).

% qbs_prob_space_eq
thf(fact_517_qbs__prob__space__eq,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta2: real > a,Nu: sigma_measure_real] :
      ( ( probab7355678800483015056b_eq_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta2 @ Nu ) ) )
     => ( ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) )
        = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta2 @ Nu ) ) ) ) ) ).

% qbs_prob_space_eq
thf(fact_518_qbs__prob_Oqbs__prob__eq__refl,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( probab7355678800483015057b_eq_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) ) ).

% qbs_prob.qbs_prob_eq_refl
thf(fact_519_qbs__prob_Oqbs__prob__eq__refl,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X @ Alpha @ Mu )
     => ( probab7355678800483015056b_eq_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) ) ).

% qbs_prob.qbs_prob_eq_refl
thf(fact_520_qbs__prob__eq__dest_I1_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,Y: quasi_borel_b,Beta2: real > b,Nu: sigma_measure_real] :
      ( ( probab7355678800483015057b_eq_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) @ ( produc2180226129289916244e_real @ Y @ ( produc4478270668571743890e_real @ Beta2 @ Nu ) ) )
     => ( probab701741629625904797prob_b @ X @ Alpha @ Mu ) ) ).

% qbs_prob_eq_dest(1)
thf(fact_521_qbs__prob__eq__dest_I1_J,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta2: real > a,Nu: sigma_measure_real] :
      ( ( probab7355678800483015056b_eq_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta2 @ Nu ) ) )
     => ( probab701741629625904796prob_a @ X @ Alpha @ Mu ) ) ).

% qbs_prob_eq_dest(1)
thf(fact_522_qbs__prob__eq__dest_I2_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,Y: quasi_borel_b,Beta2: real > b,Nu: sigma_measure_real] :
      ( ( probab7355678800483015057b_eq_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) @ ( produc2180226129289916244e_real @ Y @ ( produc4478270668571743890e_real @ Beta2 @ Nu ) ) )
     => ( probab701741629625904797prob_b @ Y @ Beta2 @ Nu ) ) ).

% qbs_prob_eq_dest(2)
thf(fact_523_qbs__prob__eq__dest_I2_J,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,Y: quasi_borel_a,Beta2: real > a,Nu: sigma_measure_real] :
      ( ( probab7355678800483015056b_eq_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ Y @ ( produc623176010801490259e_real @ Beta2 @ Nu ) ) )
     => ( probab701741629625904796prob_a @ Y @ Beta2 @ Nu ) ) ).

% qbs_prob_eq_dest(2)
thf(fact_524_monadP__qbs__Pf__morphism,axiom,
    ! [F: real > $o,X: quasi_borel_real,Y: quasi_borel_o] :
      ( ( member_real_o @ F @ ( qbs_morphism_real_o @ X @ Y ) )
     => ( member303193890323951544pace_o @ ( monad_7062749578813130895real_o @ X @ Y @ F ) @ ( qbs_mo6162687862215152862pace_o @ ( monad_2887651017592114770s_real @ X ) @ ( monad_monadP_qbs_o @ Y ) ) ) ) ).

% monadP_qbs_Pf_morphism
thf(fact_525_monadP__qbs__Pf__morphism,axiom,
    ! [F: real > nat,X: quasi_borel_real,Y: quasi_borel_nat] :
      ( ( member_real_nat @ F @ ( qbs_mo6567951568834356598al_nat @ X @ Y ) )
     => ( member6245316948443533788ce_nat @ ( monad_2109573969670034073al_nat @ X @ Y @ F ) @ ( qbs_mo3330508391926297248ce_nat @ ( monad_2887651017592114770s_real @ X ) @ ( monad_monadP_qbs_nat @ Y ) ) ) ) ).

% monadP_qbs_Pf_morphism
thf(fact_526_monadP__qbs__Pf__morphism,axiom,
    ! [F: real > extend8495563244428889912nnreal,X: quasi_borel_real,Y: quasi_9015997321629101608nnreal] :
      ( ( member2919562650594848410nnreal @ F @ ( qbs_mo1317719164804411614nnreal @ X @ Y ) )
     => ( member6561125330500187460nnreal @ ( monad_6772047417230001665nnreal @ X @ Y @ F ) @ ( qbs_mo2680574766477480456nnreal @ ( monad_2887651017592114770s_real @ X ) @ ( monad_8737719049617959390nnreal @ Y ) ) ) ) ).

% monadP_qbs_Pf_morphism
thf(fact_527_monadP__qbs__Pf__morphism,axiom,
    ! [F: real > complex,X: quasi_borel_real,Y: quasi_borel_complex] :
      ( ( member_real_complex @ F @ ( qbs_mo6067097710682130004omplex @ X @ Y ) )
     => ( member2120525692860708026omplex @ ( monad_4232241275811048823omplex @ X @ Y @ F ) @ ( qbs_mo3866313822246798974omplex @ ( monad_2887651017592114770s_real @ X ) @ ( monad_3228211519047171924omplex @ Y ) ) ) ) ).

% monadP_qbs_Pf_morphism
thf(fact_528_monadP__qbs__Pf__morphism,axiom,
    ! [F: real > real,X: quasi_borel_real,Y: quasi_borel_real] :
      ( ( member_real_real @ F @ ( qbs_mo5229770564518008146l_real @ X @ Y ) )
     => ( member2008481591236484536e_real @ ( monad_4235422856176591093l_real @ X @ Y @ F ) @ ( qbs_mo7502123496470045948e_real @ ( monad_2887651017592114770s_real @ X ) @ ( monad_2887651017592114770s_real @ Y ) ) ) ) ).

% monadP_qbs_Pf_morphism
thf(fact_529_monadP__qbs__Pf__morphism,axiom,
    ! [F: a > extend8495563244428889912nnreal,X: quasi_borel_a,Y: quasi_9015997321629101608nnreal] :
      ( ( member298456594901751504nnreal @ F @ ( qbs_mo1434458643421888574nnreal @ X @ Y ) )
     => ( member4985931665141988134nnreal @ ( monad_4746400152670514267nnreal @ X @ Y @ F ) @ ( qbs_mo6857830617226182420nnreal @ ( monad_monadP_qbs_a @ X ) @ ( monad_8737719049617959390nnreal @ Y ) ) ) ) ).

% monadP_qbs_Pf_morphism
thf(fact_530_monadP__qbs__Pf__morphism,axiom,
    ! [F: a > real,X: quasi_borel_a,Y: quasi_borel_real] :
      ( ( member_a_real @ F @ ( qbs_morphism_a_real @ X @ Y ) )
     => ( member7559386242808357530e_real @ ( monad_6933770117061659215a_real @ X @ Y @ F ) @ ( qbs_mo3705727718672329736e_real @ ( monad_monadP_qbs_a @ X ) @ ( monad_2887651017592114770s_real @ Y ) ) ) ) ).

% monadP_qbs_Pf_morphism
thf(fact_531_monadP__qbs__Pf__morphism,axiom,
    ! [F: $o > $o,X: quasi_borel_o,Y: quasi_borel_o] :
      ( ( member_o_o @ F @ ( qbs_morphism_o_o @ X @ Y ) )
     => ( member8288871389756152636pace_o @ ( monad_3447778872307538587Pf_o_o @ X @ Y @ F ) @ ( qbs_mo5587578735296274488pace_o @ ( monad_monadP_qbs_o @ X ) @ ( monad_monadP_qbs_o @ Y ) ) ) ) ).

% monadP_qbs_Pf_morphism
thf(fact_532_monadP__qbs__Pf__morphism,axiom,
    ! [F: b > extend8495563244428889912nnreal,X: quasi_borel_b,Y: quasi_9015997321629101608nnreal] :
      ( ( member6418304549040442065nnreal @ F @ ( qbs_mo7554306597560579135nnreal @ X @ Y ) )
     => ( member1560883434137153191nnreal @ ( monad_1642876069954429020nnreal @ X @ Y @ F ) @ ( qbs_mo3432782386221347477nnreal @ ( monad_monadP_qbs_b @ X ) @ ( monad_8737719049617959390nnreal @ Y ) ) ) ) ).

% monadP_qbs_Pf_morphism
thf(fact_533_monadP__qbs__Pf__morphism,axiom,
    ! [F: b > b,X: quasi_borel_b,Y: quasi_borel_b] :
      ( ( member_b_b @ F @ ( qbs_morphism_b_b @ X @ Y ) )
     => ( member3240531344000085116pace_b @ ( monad_6835572679166292699Pf_b_b @ X @ Y @ F ) @ ( qbs_mo5665329929810820088pace_b @ ( monad_monadP_qbs_b @ X ) @ ( monad_monadP_qbs_b @ Y ) ) ) ) ).

% monadP_qbs_Pf_morphism
thf(fact_534_qbs__prob_Oif__in__Rep_I3_J,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,X9: quasi_borel_b,Alpha3: real > b,Mu3: sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( ( member379764716372043661e_real @ ( produc2180226129289916244e_real @ X9 @ ( produc4478270668571743890e_real @ Alpha3 @ Mu3 ) ) @ ( probab8639044586466322087pace_b @ ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) ) )
       => ( probab7355678800483015057b_eq_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) @ ( produc2180226129289916244e_real @ X9 @ ( produc4478270668571743890e_real @ Alpha3 @ Mu3 ) ) ) ) ) ).

% qbs_prob.if_in_Rep(3)
thf(fact_535_qbs__prob_Oif__in__Rep_I3_J,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,X9: quasi_borel_a,Alpha3: real > a,Mu3: sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X @ Alpha @ Mu )
     => ( ( member6844354795726785935e_real @ ( produc4145838808978236886e_real @ X9 @ ( produc623176010801490259e_real @ Alpha3 @ Mu3 ) ) @ ( probab8639044586466322086pace_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) ) )
       => ( probab7355678800483015056b_eq_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) @ ( produc4145838808978236886e_real @ X9 @ ( produc623176010801490259e_real @ Alpha3 @ Mu3 ) ) ) ) ) ).

% qbs_prob.if_in_Rep(3)
thf(fact_536_rep__qbs__prob__space_H,axiom,
    ! [S2: probab4737552677800996672pace_b,X: quasi_borel_b] :
      ( ( ( probab1293289258141559361_qbs_b @ S2 )
        = X )
     => ? [Alpha2: real > b,Mu2: sigma_measure_real] :
          ( ( S2
            = ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha2 @ Mu2 ) ) ) )
          & ( probab701741629625904797prob_b @ X @ Alpha2 @ Mu2 ) ) ) ).

% rep_qbs_prob_space'
thf(fact_537_rep__qbs__prob__space_H,axiom,
    ! [S2: probab4737552673497767871pace_a,X: quasi_borel_a] :
      ( ( ( probab1293289258141559360_qbs_a @ S2 )
        = X )
     => ? [Alpha2: real > a,Mu2: sigma_measure_real] :
          ( ( S2
            = ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha2 @ Mu2 ) ) ) )
          & ( probab701741629625904796prob_a @ X @ Alpha2 @ Mu2 ) ) ) ).

% rep_qbs_prob_space'
thf(fact_538_qbs__prob_Oqbs__prob__space__qbs__computation,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( ( probab1293289258141559361_qbs_b @ ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) )
        = X ) ) ).

% qbs_prob.qbs_prob_space_qbs_computation
thf(fact_539_qbs__prob_Oqbs__prob__space__qbs__computation,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real] :
      ( ( probab701741629625904796prob_a @ X @ Alpha @ Mu )
     => ( ( probab1293289258141559360_qbs_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) )
        = X ) ) ).

% qbs_prob.qbs_prob_space_qbs_computation
thf(fact_540_monadP__qbs__Pf__comp,axiom,
    ! [S2: probab1241297377463522905pace_o,X: quasi_borel_o,F: $o > real,Y: quasi_borel_real,G2: real > $o,Z: quasi_borel_o] :
      ( ( member8440415008478396930pace_o @ S2 @ ( monad_8932450573223570058s_Px_o @ X ) )
     => ( ( member_o_real @ F @ ( qbs_morphism_o_real @ X @ Y ) )
       => ( ( member_real_o @ G2 @ ( qbs_morphism_real_o @ Y @ Z ) )
         => ( ( comp_P4701177297104927972pace_o @ ( monad_7062749578813130895real_o @ Y @ Z @ G2 ) @ ( monad_5553685203771961321o_real @ X @ Y @ F ) @ S2 )
            = ( monad_3447778872307538587Pf_o_o @ X @ Z @ ( comp_real_o_o @ G2 @ F ) @ S2 ) ) ) ) ) ).

% monadP_qbs_Pf_comp
thf(fact_541_monadP__qbs__Pf__comp,axiom,
    ! [S2: probab8370124847414953445ce_nat,X: quasi_borel_nat,F: nat > real,Y: quasi_borel_real,G2: real > nat,Z: quasi_borel_nat] :
      ( ( member1134292470272169596ce_nat @ S2 @ ( monad_5642707204117833054Px_nat @ X ) )
     => ( ( member_nat_real @ F @ ( qbs_mo2000642995705457910t_real @ X @ Y ) )
       => ( ( member_real_nat @ G2 @ ( qbs_mo6567951568834356598al_nat @ Y @ Z ) )
         => ( ( comp_P655014663824291312ce_nat @ ( monad_2109573969670034073al_nat @ Y @ Z @ G2 ) @ ( monad_6765637433395911193t_real @ X @ Y @ F ) @ S2 )
            = ( monad_2263650980725743037at_nat @ X @ Z @ ( comp_real_nat_nat @ G2 @ F ) @ S2 ) ) ) ) ) ).

% monadP_qbs_Pf_comp
thf(fact_542_monadP__qbs__Pf__comp,axiom,
    ! [S2: probab8009751763329705409e_real,X: quasi_borel_real,F: real > real,Y: quasi_borel_real,G2: real > $o,Z: quasi_borel_o] :
      ( ( member7522313270622477016e_real @ S2 @ ( monad_1231911713131340090x_real @ X ) )
     => ( ( member_real_real @ F @ ( qbs_mo5229770564518008146l_real @ X @ Y ) )
       => ( ( member_real_o @ G2 @ ( qbs_morphism_real_o @ Y @ Z ) )
         => ( ( comp_P1838334351674764662e_real @ ( monad_7062749578813130895real_o @ Y @ Z @ G2 ) @ ( monad_4235422856176591093l_real @ X @ Y @ F ) @ S2 )
            = ( monad_7062749578813130895real_o @ X @ Z @ ( comp_real_o_real @ G2 @ F ) @ S2 ) ) ) ) ) ).

% monadP_qbs_Pf_comp
thf(fact_543_monadP__qbs__Pf__comp,axiom,
    ! [S2: probab8009751763329705409e_real,X: quasi_borel_real,F: real > real,Y: quasi_borel_real,G2: real > nat,Z: quasi_borel_nat] :
      ( ( member7522313270622477016e_real @ S2 @ ( monad_1231911713131340090x_real @ X ) )
     => ( ( member_real_real @ F @ ( qbs_mo5229770564518008146l_real @ X @ Y ) )
       => ( ( member_real_nat @ G2 @ ( qbs_mo6567951568834356598al_nat @ Y @ Z ) )
         => ( ( comp_P5219563683584835660e_real @ ( monad_2109573969670034073al_nat @ Y @ Z @ G2 ) @ ( monad_4235422856176591093l_real @ X @ Y @ F ) @ S2 )
            = ( monad_2109573969670034073al_nat @ X @ Z @ ( comp_real_nat_real @ G2 @ F ) @ S2 ) ) ) ) ) ).

% monadP_qbs_Pf_comp
thf(fact_544_monadP__qbs__Pf__comp,axiom,
    ! [S2: probab8009751763329705409e_real,X: quasi_borel_real,F: real > real,Y: quasi_borel_real,G2: real > extend8495563244428889912nnreal,Z: quasi_9015997321629101608nnreal] :
      ( ( member7522313270622477016e_real @ S2 @ ( monad_1231911713131340090x_real @ X ) )
     => ( ( member_real_real @ F @ ( qbs_mo5229770564518008146l_real @ X @ Y ) )
       => ( ( member2919562650594848410nnreal @ G2 @ ( qbs_mo1317719164804411614nnreal @ Y @ Z ) )
         => ( ( comp_P3464484273722719156e_real @ ( monad_6772047417230001665nnreal @ Y @ Z @ G2 ) @ ( monad_4235422856176591093l_real @ X @ Y @ F ) @ S2 )
            = ( monad_6772047417230001665nnreal @ X @ Z @ ( comp_r6279034453215524981l_real @ G2 @ F ) @ S2 ) ) ) ) ) ).

% monadP_qbs_Pf_comp
thf(fact_545_monadP__qbs__Pf__comp,axiom,
    ! [S2: probab8009751763329705409e_real,X: quasi_borel_real,F: real > real,Y: quasi_borel_real,G2: real > complex,Z: quasi_borel_complex] :
      ( ( member7522313270622477016e_real @ S2 @ ( monad_1231911713131340090x_real @ X ) )
     => ( ( member_real_real @ F @ ( qbs_mo5229770564518008146l_real @ X @ Y ) )
       => ( ( member_real_complex @ G2 @ ( qbs_mo6067097710682130004omplex @ Y @ Z ) )
         => ( ( comp_P6469183723381889322e_real @ ( monad_4232241275811048823omplex @ Y @ Z @ G2 ) @ ( monad_4235422856176591093l_real @ X @ Y @ F ) @ S2 )
            = ( monad_4232241275811048823omplex @ X @ Z @ ( comp_r1968866223832618731x_real @ G2 @ F ) @ S2 ) ) ) ) ) ).

% monadP_qbs_Pf_comp
thf(fact_546_monadP__qbs__Pf__comp,axiom,
    ! [S2: probab8009751763329705409e_real,X: quasi_borel_real,F: real > real,Y: quasi_borel_real,G2: real > real,Z: quasi_borel_real] :
      ( ( member7522313270622477016e_real @ S2 @ ( monad_1231911713131340090x_real @ X ) )
     => ( ( member_real_real @ F @ ( qbs_mo5229770564518008146l_real @ X @ Y ) )
       => ( ( member_real_real @ G2 @ ( qbs_mo5229770564518008146l_real @ Y @ Z ) )
         => ( ( comp_P6332119684870606760e_real @ ( monad_4235422856176591093l_real @ Y @ Z @ G2 ) @ ( monad_4235422856176591093l_real @ X @ Y @ F ) @ S2 )
            = ( monad_4235422856176591093l_real @ X @ Z @ ( comp_real_real_real @ G2 @ F ) @ S2 ) ) ) ) ) ).

% monadP_qbs_Pf_comp
thf(fact_547_monadP__qbs__Pf__comp,axiom,
    ! [S2: probab8009751763329705409e_real,X: quasi_borel_real,F: real > a,Y: quasi_borel_a,G2: a > real,Z: quasi_borel_real] :
      ( ( member7522313270622477016e_real @ S2 @ ( monad_1231911713131340090x_real @ X ) )
     => ( ( member_real_a @ F @ ( qbs_morphism_real_a @ X @ Y ) )
       => ( ( member_a_real @ G2 @ ( qbs_morphism_a_real @ Y @ Z ) )
         => ( ( comp_P2634432118360280224e_real @ ( monad_6933770117061659215a_real @ Y @ Z @ G2 ) @ ( monad_7563788884534609013real_a @ X @ Y @ F ) @ S2 )
            = ( monad_4235422856176591093l_real @ X @ Z @ ( comp_a_real_real @ G2 @ F ) @ S2 ) ) ) ) ) ).

% monadP_qbs_Pf_comp
thf(fact_548_monadP__qbs__Pf__comp,axiom,
    ! [S2: probab8009751763329705409e_real,X: quasi_borel_real,F: real > $o,Y: quasi_borel_o,G2: $o > $o,Z: quasi_borel_o] :
      ( ( member7522313270622477016e_real @ S2 @ ( monad_1231911713131340090x_real @ X ) )
     => ( ( member_real_o @ F @ ( qbs_morphism_real_o @ X @ Y ) )
       => ( ( member_o_o @ G2 @ ( qbs_morphism_o_o @ Y @ Z ) )
         => ( ( comp_P6982291091032798180e_real @ ( monad_3447778872307538587Pf_o_o @ Y @ Z @ G2 ) @ ( monad_7062749578813130895real_o @ X @ Y @ F ) @ S2 )
            = ( monad_7062749578813130895real_o @ X @ Z @ ( comp_o_o_real @ G2 @ F ) @ S2 ) ) ) ) ) ).

% monadP_qbs_Pf_comp
thf(fact_549_monadP__qbs__Pf__comp,axiom,
    ! [S2: probab1241297377463522905pace_o,X: quasi_borel_o,F: $o > $o,Y: quasi_borel_o,G2: $o > $o,Z: quasi_borel_o] :
      ( ( member8440415008478396930pace_o @ S2 @ ( monad_8932450573223570058s_Px_o @ X ) )
     => ( ( member_o_o @ F @ ( qbs_morphism_o_o @ X @ Y ) )
       => ( ( member_o_o @ G2 @ ( qbs_morphism_o_o @ Y @ Z ) )
         => ( ( comp_P8454002975481202678pace_o @ ( monad_3447778872307538587Pf_o_o @ Y @ Z @ G2 ) @ ( monad_3447778872307538587Pf_o_o @ X @ Y @ F ) @ S2 )
            = ( monad_3447778872307538587Pf_o_o @ X @ Z @ ( comp_o_o_o @ G2 @ F ) @ S2 ) ) ) ) ) ).

% monadP_qbs_Pf_comp
thf(fact_550_Pair__mono,axiom,
    ! [X8: real,X10: real,Y4: real,Y7: real] :
      ( ( ord_less_eq_real @ X8 @ X10 )
     => ( ( ord_less_eq_real @ Y4 @ Y7 )
       => ( ord_le1075799226346578649l_real @ ( produc4511245868158468465l_real @ X8 @ Y4 ) @ ( produc4511245868158468465l_real @ X10 @ Y7 ) ) ) ) ).

% Pair_mono
thf(fact_551_Pair__mono,axiom,
    ! [X8: real,X10: real,Y4: extend8495563244428889912nnreal,Y7: extend8495563244428889912nnreal] :
      ( ( ord_less_eq_real @ X8 @ X10 )
     => ( ( ord_le3935885782089961368nnreal @ Y4 @ Y7 )
       => ( ord_le4096773168995780197nnreal @ ( produc4778015194254607485nnreal @ X8 @ Y4 ) @ ( produc4778015194254607485nnreal @ X10 @ Y7 ) ) ) ) ).

% Pair_mono
thf(fact_552_Pair__mono,axiom,
    ! [X8: extend8495563244428889912nnreal,X10: extend8495563244428889912nnreal,Y4: real,Y7: real] :
      ( ( ord_le3935885782089961368nnreal @ X8 @ X10 )
     => ( ( ord_less_eq_real @ Y4 @ Y7 )
       => ( ord_le4051224869651757541l_real @ ( produc2810268924804063229l_real @ X8 @ Y4 ) @ ( produc2810268924804063229l_real @ X10 @ Y7 ) ) ) ) ).

% Pair_mono
thf(fact_553_Pair__mono,axiom,
    ! [X8: extend8495563244428889912nnreal,X10: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal,Y7: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ X8 @ X10 )
     => ( ( ord_le3935885782089961368nnreal @ Y4 @ Y7 )
       => ( ord_le1399272598019556209nnreal @ ( produc344325839068023049nnreal @ X8 @ Y4 ) @ ( produc344325839068023049nnreal @ X10 @ Y7 ) ) ) ) ).

% Pair_mono
thf(fact_554_Pair__mono,axiom,
    ! [X8: real,X10: real,Y4: set_real,Y7: set_real] :
      ( ( ord_less_eq_real @ X8 @ X10 )
     => ( ( ord_less_eq_set_real @ Y4 @ Y7 )
       => ( ord_le4945634015433394191t_real @ ( produc3307717890874201511t_real @ X8 @ Y4 ) @ ( produc3307717890874201511t_real @ X10 @ Y7 ) ) ) ) ).

% Pair_mono
thf(fact_555_Pair__mono,axiom,
    ! [X8: set_real,X10: set_real,Y4: real,Y7: real] :
      ( ( ord_less_eq_set_real @ X8 @ X10 )
     => ( ( ord_less_eq_real @ Y4 @ Y7 )
       => ( ord_le5399745455957431311l_real @ ( produc6322126818539730599l_real @ X8 @ Y4 ) @ ( produc6322126818539730599l_real @ X10 @ Y7 ) ) ) ) ).

% Pair_mono
thf(fact_556_Pair__mono,axiom,
    ! [X8: set_real,X10: set_real,Y4: extend8495563244428889912nnreal,Y7: extend8495563244428889912nnreal] :
      ( ( ord_less_eq_set_real @ X8 @ X10 )
     => ( ( ord_le3935885782089961368nnreal @ Y4 @ Y7 )
       => ( ord_le4558722300798010011nnreal @ ( produc8660806124923995315nnreal @ X8 @ Y4 ) @ ( produc8660806124923995315nnreal @ X10 @ Y7 ) ) ) ) ).

% Pair_mono
thf(fact_557_Pair__mono,axiom,
    ! [X8: extend8495563244428889912nnreal,X10: extend8495563244428889912nnreal,Y4: set_real,Y7: set_real] :
      ( ( ord_le3935885782089961368nnreal @ X8 @ X10 )
     => ( ( ord_less_eq_set_real @ Y4 @ Y7 )
       => ( ord_le3400283774364675867t_real @ ( produc3019872141242647603t_real @ X8 @ Y4 ) @ ( produc3019872141242647603t_real @ X10 @ Y7 ) ) ) ) ).

% Pair_mono
thf(fact_558_Pair__mono,axiom,
    ! [X8: real,X10: real,Y4: set_set_real,Y7: set_set_real] :
      ( ( ord_less_eq_real @ X8 @ X10 )
     => ( ( ord_le3558479182127378552t_real @ Y4 @ Y7 )
       => ( ord_le6676645311053956677t_real @ ( produc7184536090204209885t_real @ X8 @ Y4 ) @ ( produc7184536090204209885t_real @ X10 @ Y7 ) ) ) ) ).

% Pair_mono
thf(fact_559_Pair__mono,axiom,
    ! [X8: set_set_real,X10: set_set_real,Y4: real,Y7: real] :
      ( ( ord_le3558479182127378552t_real @ X8 @ X10 )
     => ( ( ord_less_eq_real @ Y4 @ Y7 )
       => ( ord_le4824006318743526981l_real @ ( produc8911628951482418397l_real @ X8 @ Y4 ) @ ( produc8911628951482418397l_real @ X10 @ Y7 ) ) ) ) ).

% Pair_mono
thf(fact_560_qbs__prob_Oin__Rep__induct,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,P2: produc78645753526154084e_real > $o] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( ! [Y6: quasi_borel_b,Beta: real > b,Nu2: sigma_measure_real] :
            ( ( member379764716372043661e_real @ ( produc2180226129289916244e_real @ Y6 @ ( produc4478270668571743890e_real @ Beta @ Nu2 ) ) @ ( probab8639044586466322087pace_b @ ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) ) )
           => ( P2 @ ( produc2180226129289916244e_real @ Y6 @ ( produc4478270668571743890e_real @ Beta @ Nu2 ) ) ) )
       => ( P2 @ ( probab221732815614317480pace_b @ ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) ) ) ) ) ).

% qbs_prob.in_Rep_induct
thf(fact_561_qbs__prob_Oin__Rep__induct,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,P2: produc6543235832880896358e_real > $o] :
      ( ( probab701741629625904796prob_a @ X @ Alpha @ Mu )
     => ( ! [Y6: quasi_borel_a,Beta: real > a,Nu2: sigma_measure_real] :
            ( ( member6844354795726785935e_real @ ( produc4145838808978236886e_real @ Y6 @ ( produc623176010801490259e_real @ Beta @ Nu2 ) ) @ ( probab8639044586466322086pace_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) ) )
           => ( P2 @ ( produc4145838808978236886e_real @ Y6 @ ( produc623176010801490259e_real @ Beta @ Nu2 ) ) ) )
       => ( P2 @ ( probab221732815614317479pace_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) ) ) ) ) ).

% qbs_prob.in_Rep_induct
thf(fact_562_assms_I3_J,axiom,
    member6418304549040442065nnreal @ g @ ( qbs_mo7554306597560579135nnreal @ y @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) ) ).

% assms(3)
thf(fact_563_qbs__morphismE_I3_J,axiom,
    ! [F: $o > nat,X: quasi_borel_o,Y: quasi_borel_nat,Alpha: real > $o] :
      ( ( member_o_nat @ F @ ( qbs_morphism_o_nat @ X @ Y ) )
     => ( ( member_real_o @ Alpha @ ( qbs_Mx_o @ X ) )
       => ( member_real_nat @ ( comp_o_nat_real @ F @ Alpha ) @ ( qbs_Mx_nat @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_564_qbs__morphismE_I3_J,axiom,
    ! [F: $o > extend8495563244428889912nnreal,X: quasi_borel_o,Y: quasi_9015997321629101608nnreal,Alpha: real > $o] :
      ( ( member5265953103328284778nnreal @ F @ ( qbs_mo8025287178234086488nnreal @ X @ Y ) )
     => ( ( member_real_o @ Alpha @ ( qbs_Mx_o @ X ) )
       => ( member2919562650594848410nnreal @ ( comp_o592059427571696603l_real @ F @ Alpha ) @ ( qbs_Mx6523938229262583809nnreal @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_565_qbs__morphismE_I3_J,axiom,
    ! [F: $o > complex,X: quasi_borel_o,Y: quasi_borel_complex,Alpha: real > $o] :
      ( ( member_o_complex @ F @ ( qbs_mo5261837971759248846omplex @ X @ Y ) )
     => ( ( member_real_o @ Alpha @ ( qbs_Mx_o @ X ) )
       => ( member_real_complex @ ( comp_o_complex_real @ F @ Alpha ) @ ( qbs_Mx_complex @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_566_qbs__morphismE_I3_J,axiom,
    ! [F: $o > real,X: quasi_borel_o,Y: quasi_borel_real,Alpha: real > $o] :
      ( ( member_o_real @ F @ ( qbs_morphism_o_real @ X @ Y ) )
     => ( ( member_real_o @ Alpha @ ( qbs_Mx_o @ X ) )
       => ( member_real_real @ ( comp_o_real_real @ F @ Alpha ) @ ( qbs_Mx_real @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_567_qbs__morphismE_I3_J,axiom,
    ! [F: nat > $o,X: quasi_borel_nat,Y: quasi_borel_o,Alpha: real > nat] :
      ( ( member_nat_o @ F @ ( qbs_morphism_nat_o @ X @ Y ) )
     => ( ( member_real_nat @ Alpha @ ( qbs_Mx_nat @ X ) )
       => ( member_real_o @ ( comp_nat_o_real @ F @ Alpha ) @ ( qbs_Mx_o @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_568_qbs__morphismE_I3_J,axiom,
    ! [F: nat > nat,X: quasi_borel_nat,Y: quasi_borel_nat,Alpha: real > nat] :
      ( ( member_nat_nat @ F @ ( qbs_morphism_nat_nat @ X @ Y ) )
     => ( ( member_real_nat @ Alpha @ ( qbs_Mx_nat @ X ) )
       => ( member_real_nat @ ( comp_nat_nat_real @ F @ Alpha ) @ ( qbs_Mx_nat @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_569_qbs__morphismE_I3_J,axiom,
    ! [F: nat > extend8495563244428889912nnreal,X: quasi_borel_nat,Y: quasi_9015997321629101608nnreal,Alpha: real > nat] :
      ( ( member8283130129095025342nnreal @ F @ ( qbs_mo4632421585747671682nnreal @ X @ Y ) )
     => ( ( member_real_nat @ Alpha @ ( qbs_Mx_nat @ X ) )
       => ( member2919562650594848410nnreal @ ( comp_n3455504981484479769l_real @ F @ Alpha ) @ ( qbs_Mx6523938229262583809nnreal @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_570_qbs__morphismE_I3_J,axiom,
    ! [F: nat > complex,X: quasi_borel_nat,Y: quasi_borel_complex,Alpha: real > nat] :
      ( ( member_nat_complex @ F @ ( qbs_mo6368047727621400568omplex @ X @ Y ) )
     => ( ( member_real_nat @ Alpha @ ( qbs_Mx_nat @ X ) )
       => ( member_real_complex @ ( comp_n4215249288434654095x_real @ F @ Alpha ) @ ( qbs_Mx_complex @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_571_qbs__morphismE_I3_J,axiom,
    ! [F: nat > real,X: quasi_borel_nat,Y: quasi_borel_real,Alpha: real > nat] :
      ( ( member_nat_real @ F @ ( qbs_mo2000642995705457910t_real @ X @ Y ) )
     => ( ( member_real_nat @ Alpha @ ( qbs_Mx_nat @ X ) )
       => ( member_real_real @ ( comp_nat_real_real @ F @ Alpha ) @ ( qbs_Mx_real @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_572_qbs__morphismE_I3_J,axiom,
    ! [F: extend8495563244428889912nnreal > $o,X: quasi_9015997321629101608nnreal,Y: quasi_borel_o,Alpha: real > extend8495563244428889912nnreal] :
      ( ( member8095236870201115968real_o @ F @ ( qbs_mo7845494039804091494real_o @ X @ Y ) )
     => ( ( member2919562650594848410nnreal @ Alpha @ ( qbs_Mx6523938229262583809nnreal @ X ) )
       => ( member_real_o @ ( comp_E6134263049385049321o_real @ F @ Alpha ) @ ( qbs_Mx_o @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_573_qbs__morphismI,axiom,
    ! [X: quasi_borel_o,F: $o > nat,Y: quasi_borel_nat] :
      ( ! [Alpha2: real > $o] :
          ( ( member_real_o @ Alpha2 @ ( qbs_Mx_o @ X ) )
         => ( member_real_nat @ ( comp_o_nat_real @ F @ Alpha2 ) @ ( qbs_Mx_nat @ Y ) ) )
     => ( member_o_nat @ F @ ( qbs_morphism_o_nat @ X @ Y ) ) ) ).

% qbs_morphismI
thf(fact_574_qbs__morphismI,axiom,
    ! [X: quasi_borel_o,F: $o > extend8495563244428889912nnreal,Y: quasi_9015997321629101608nnreal] :
      ( ! [Alpha2: real > $o] :
          ( ( member_real_o @ Alpha2 @ ( qbs_Mx_o @ X ) )
         => ( member2919562650594848410nnreal @ ( comp_o592059427571696603l_real @ F @ Alpha2 ) @ ( qbs_Mx6523938229262583809nnreal @ Y ) ) )
     => ( member5265953103328284778nnreal @ F @ ( qbs_mo8025287178234086488nnreal @ X @ Y ) ) ) ).

% qbs_morphismI
thf(fact_575_qbs__morphismI,axiom,
    ! [X: quasi_borel_o,F: $o > complex,Y: quasi_borel_complex] :
      ( ! [Alpha2: real > $o] :
          ( ( member_real_o @ Alpha2 @ ( qbs_Mx_o @ X ) )
         => ( member_real_complex @ ( comp_o_complex_real @ F @ Alpha2 ) @ ( qbs_Mx_complex @ Y ) ) )
     => ( member_o_complex @ F @ ( qbs_mo5261837971759248846omplex @ X @ Y ) ) ) ).

% qbs_morphismI
thf(fact_576_qbs__morphismI,axiom,
    ! [X: quasi_borel_o,F: $o > real,Y: quasi_borel_real] :
      ( ! [Alpha2: real > $o] :
          ( ( member_real_o @ Alpha2 @ ( qbs_Mx_o @ X ) )
         => ( member_real_real @ ( comp_o_real_real @ F @ Alpha2 ) @ ( qbs_Mx_real @ Y ) ) )
     => ( member_o_real @ F @ ( qbs_morphism_o_real @ X @ Y ) ) ) ).

% qbs_morphismI
thf(fact_577_qbs__morphismI,axiom,
    ! [X: quasi_borel_nat,F: nat > $o,Y: quasi_borel_o] :
      ( ! [Alpha2: real > nat] :
          ( ( member_real_nat @ Alpha2 @ ( qbs_Mx_nat @ X ) )
         => ( member_real_o @ ( comp_nat_o_real @ F @ Alpha2 ) @ ( qbs_Mx_o @ Y ) ) )
     => ( member_nat_o @ F @ ( qbs_morphism_nat_o @ X @ Y ) ) ) ).

% qbs_morphismI
thf(fact_578_qbs__morphismI,axiom,
    ! [X: quasi_borel_nat,F: nat > nat,Y: quasi_borel_nat] :
      ( ! [Alpha2: real > nat] :
          ( ( member_real_nat @ Alpha2 @ ( qbs_Mx_nat @ X ) )
         => ( member_real_nat @ ( comp_nat_nat_real @ F @ Alpha2 ) @ ( qbs_Mx_nat @ Y ) ) )
     => ( member_nat_nat @ F @ ( qbs_morphism_nat_nat @ X @ Y ) ) ) ).

% qbs_morphismI
thf(fact_579_qbs__morphismI,axiom,
    ! [X: quasi_borel_nat,F: nat > extend8495563244428889912nnreal,Y: quasi_9015997321629101608nnreal] :
      ( ! [Alpha2: real > nat] :
          ( ( member_real_nat @ Alpha2 @ ( qbs_Mx_nat @ X ) )
         => ( member2919562650594848410nnreal @ ( comp_n3455504981484479769l_real @ F @ Alpha2 ) @ ( qbs_Mx6523938229262583809nnreal @ Y ) ) )
     => ( member8283130129095025342nnreal @ F @ ( qbs_mo4632421585747671682nnreal @ X @ Y ) ) ) ).

% qbs_morphismI
thf(fact_580_qbs__morphismI,axiom,
    ! [X: quasi_borel_nat,F: nat > complex,Y: quasi_borel_complex] :
      ( ! [Alpha2: real > nat] :
          ( ( member_real_nat @ Alpha2 @ ( qbs_Mx_nat @ X ) )
         => ( member_real_complex @ ( comp_n4215249288434654095x_real @ F @ Alpha2 ) @ ( qbs_Mx_complex @ Y ) ) )
     => ( member_nat_complex @ F @ ( qbs_mo6368047727621400568omplex @ X @ Y ) ) ) ).

% qbs_morphismI
thf(fact_581_qbs__morphismI,axiom,
    ! [X: quasi_borel_nat,F: nat > real,Y: quasi_borel_real] :
      ( ! [Alpha2: real > nat] :
          ( ( member_real_nat @ Alpha2 @ ( qbs_Mx_nat @ X ) )
         => ( member_real_real @ ( comp_nat_real_real @ F @ Alpha2 ) @ ( qbs_Mx_real @ Y ) ) )
     => ( member_nat_real @ F @ ( qbs_mo2000642995705457910t_real @ X @ Y ) ) ) ).

% qbs_morphismI
thf(fact_582_qbs__morphismI,axiom,
    ! [X: quasi_9015997321629101608nnreal,F: extend8495563244428889912nnreal > $o,Y: quasi_borel_o] :
      ( ! [Alpha2: real > extend8495563244428889912nnreal] :
          ( ( member2919562650594848410nnreal @ Alpha2 @ ( qbs_Mx6523938229262583809nnreal @ X ) )
         => ( member_real_o @ ( comp_E6134263049385049321o_real @ F @ Alpha2 ) @ ( qbs_Mx_o @ Y ) ) )
     => ( member8095236870201115968real_o @ F @ ( qbs_mo7845494039804091494real_o @ X @ Y ) ) ) ).

% qbs_morphismI
thf(fact_583_subsetI,axiom,
    ! [A: set_real,B: set_real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A )
         => ( member_real @ X3 @ B ) )
     => ( ord_less_eq_set_real @ A @ B ) ) ).

% subsetI
thf(fact_584_subsetI,axiom,
    ! [A: set_set_real,B: set_set_real] :
      ( ! [X3: set_real] :
          ( ( member_set_real @ X3 @ A )
         => ( member_set_real @ X3 @ B ) )
     => ( ord_le3558479182127378552t_real @ A @ B ) ) ).

% subsetI
thf(fact_585_subsetI,axiom,
    ! [A: set_o_o,B: set_o_o] :
      ( ! [X3: $o > $o] :
          ( ( member_o_o @ X3 @ A )
         => ( member_o_o @ X3 @ B ) )
     => ( ord_less_eq_set_o_o @ A @ B ) ) ).

% subsetI
thf(fact_586_subsetI,axiom,
    ! [A: set_b_6825823330181178888nnreal,B: set_b_6825823330181178888nnreal] :
      ( ! [X3: b > extend8495563244428889912nnreal] :
          ( ( member6418304549040442065nnreal @ X3 @ A )
         => ( member6418304549040442065nnreal @ X3 @ B ) )
     => ( ord_le672203391976590760nnreal @ A @ B ) ) ).

% subsetI
thf(fact_587_subsetI,axiom,
    ! [A: set_a_7161065143582548615nnreal,B: set_a_7161065143582548615nnreal] :
      ( ! [X3: a > extend8495563244428889912nnreal] :
          ( ( member298456594901751504nnreal @ X3 @ A )
         => ( member298456594901751504nnreal @ X3 @ B ) )
     => ( ord_le1007445205377960487nnreal @ A @ B ) ) ).

% subsetI
thf(fact_588_subsetI,axiom,
    ! [A: set_a_real,B: set_a_real] :
      ( ! [X3: a > real] :
          ( ( member_a_real @ X3 @ A )
         => ( member_a_real @ X3 @ B ) )
     => ( ord_le3334967407727675675a_real @ A @ B ) ) ).

% subsetI
thf(fact_589_subsetI,axiom,
    ! [A: set_real_o,B: set_real_o] :
      ( ! [X3: real > $o] :
          ( ( member_real_o @ X3 @ A )
         => ( member_real_o @ X3 @ B ) )
     => ( ord_le1615110227528160547real_o @ A @ B ) ) ).

% subsetI
thf(fact_590_subsetI,axiom,
    ! [A: set_real_nat,B: set_real_nat] :
      ( ! [X3: real > nat] :
          ( ( member_real_nat @ X3 @ A )
         => ( member_real_nat @ X3 @ B ) )
     => ( ord_le6098800555920186673al_nat @ A @ B ) ) ).

% subsetI
thf(fact_591_subsetI,axiom,
    ! [A: set_re5328672808648366137nnreal,B: set_re5328672808648366137nnreal] :
      ( ! [X3: real > extend8495563244428889912nnreal] :
          ( ( member2919562650594848410nnreal @ X3 @ A )
         => ( member2919562650594848410nnreal @ X3 @ B ) )
     => ( ord_le2462468573666744473nnreal @ A @ B ) ) ).

% subsetI
thf(fact_592_subsetI,axiom,
    ! [A: set_real_complex,B: set_real_complex] :
      ( ! [X3: real > complex] :
          ( ( member_real_complex @ X3 @ A )
         => ( member_real_complex @ X3 @ B ) )
     => ( ord_le2047140485929309711omplex @ A @ B ) ) ).

% subsetI
thf(fact_593_subset__antisym,axiom,
    ! [A: set_set_real,B: set_set_real] :
      ( ( ord_le3558479182127378552t_real @ A @ B )
     => ( ( ord_le3558479182127378552t_real @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_594_subset__antisym,axiom,
    ! [A: set_real,B: set_real] :
      ( ( ord_less_eq_set_real @ A @ B )
     => ( ( ord_less_eq_set_real @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_595_order__refl,axiom,
    ! [X8: real] : ( ord_less_eq_real @ X8 @ X8 ) ).

% order_refl
thf(fact_596_order__refl,axiom,
    ! [X8: set_set_real] : ( ord_le3558479182127378552t_real @ X8 @ X8 ) ).

% order_refl
thf(fact_597_order__refl,axiom,
    ! [X8: set_real] : ( ord_less_eq_set_real @ X8 @ X8 ) ).

% order_refl
thf(fact_598_order__refl,axiom,
    ! [X8: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ X8 @ X8 ) ).

% order_refl
thf(fact_599_dual__order_Orefl,axiom,
    ! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_600_dual__order_Orefl,axiom,
    ! [A2: set_set_real] : ( ord_le3558479182127378552t_real @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_601_dual__order_Orefl,axiom,
    ! [A2: set_real] : ( ord_less_eq_set_real @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_602_dual__order_Orefl,axiom,
    ! [A2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_603_Levy__uniqueness,axiom,
    ! [M1: sigma_measure_real,M2: sigma_measure_real] :
      ( ( distri2809703520229113005bution @ M1 )
     => ( ( distri2809703520229113005bution @ M2 )
       => ( ( ( characteristic_char @ M1 )
            = ( characteristic_char @ M2 ) )
         => ( M1 = M2 ) ) ) ) ).

% Levy_uniqueness
thf(fact_604_order__antisym__conv,axiom,
    ! [Y4: real,X8: real] :
      ( ( ord_less_eq_real @ Y4 @ X8 )
     => ( ( ord_less_eq_real @ X8 @ Y4 )
        = ( X8 = Y4 ) ) ) ).

% order_antisym_conv
thf(fact_605_order__antisym__conv,axiom,
    ! [Y4: set_set_real,X8: set_set_real] :
      ( ( ord_le3558479182127378552t_real @ Y4 @ X8 )
     => ( ( ord_le3558479182127378552t_real @ X8 @ Y4 )
        = ( X8 = Y4 ) ) ) ).

% order_antisym_conv
thf(fact_606_order__antisym__conv,axiom,
    ! [Y4: set_real,X8: set_real] :
      ( ( ord_less_eq_set_real @ Y4 @ X8 )
     => ( ( ord_less_eq_set_real @ X8 @ Y4 )
        = ( X8 = Y4 ) ) ) ).

% order_antisym_conv
thf(fact_607_order__antisym__conv,axiom,
    ! [Y4: extend8495563244428889912nnreal,X8: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ Y4 @ X8 )
     => ( ( ord_le3935885782089961368nnreal @ X8 @ Y4 )
        = ( X8 = Y4 ) ) ) ).

% order_antisym_conv
thf(fact_608_linorder__le__cases,axiom,
    ! [X8: real,Y4: real] :
      ( ~ ( ord_less_eq_real @ X8 @ Y4 )
     => ( ord_less_eq_real @ Y4 @ X8 ) ) ).

% linorder_le_cases
thf(fact_609_linorder__le__cases,axiom,
    ! [X8: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
      ( ~ ( ord_le3935885782089961368nnreal @ X8 @ Y4 )
     => ( ord_le3935885782089961368nnreal @ Y4 @ X8 ) ) ).

% linorder_le_cases
thf(fact_610_ord__le__eq__subst,axiom,
    ! [A2: real,B2: real,F: real > real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_eq_real @ X3 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_611_ord__le__eq__subst,axiom,
    ! [A2: real,B2: real,F: real > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_eq_real @ X3 @ Y5 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_le3935885782089961368nnreal @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_612_ord__le__eq__subst,axiom,
    ! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > real,C: real] :
      ( ( ord_le3935885782089961368nnreal @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_613_ord__le__eq__subst,axiom,
    ! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_le3935885782089961368nnreal @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_614_ord__le__eq__subst,axiom,
    ! [A2: real,B2: real,F: real > set_real,C: set_real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_eq_real @ X3 @ Y5 )
             => ( ord_less_eq_set_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_set_real @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_615_ord__le__eq__subst,axiom,
    ! [A2: set_real,B2: set_real,F: set_real > real,C: real] :
      ( ( ord_less_eq_set_real @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: set_real,Y5: set_real] :
              ( ( ord_less_eq_set_real @ X3 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_616_ord__le__eq__subst,axiom,
    ! [A2: set_real,B2: set_real,F: set_real > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_less_eq_set_real @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: set_real,Y5: set_real] :
              ( ( ord_less_eq_set_real @ X3 @ Y5 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_le3935885782089961368nnreal @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_617_ord__le__eq__subst,axiom,
    ! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > set_real,C: set_real] :
      ( ( ord_le3935885782089961368nnreal @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
             => ( ord_less_eq_set_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_set_real @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_618_ord__le__eq__subst,axiom,
    ! [A2: real,B2: real,F: real > set_set_real,C: set_set_real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_eq_real @ X3 @ Y5 )
             => ( ord_le3558479182127378552t_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_le3558479182127378552t_real @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_619_ord__le__eq__subst,axiom,
    ! [A2: set_set_real,B2: set_set_real,F: set_set_real > real,C: real] :
      ( ( ord_le3558479182127378552t_real @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: set_set_real,Y5: set_set_real] :
              ( ( ord_le3558479182127378552t_real @ X3 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_620_ord__eq__le__subst,axiom,
    ! [A2: real,F: real > real,B2: real,C: real] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_eq_real @ X3 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_621_ord__eq__le__subst,axiom,
    ! [A2: extend8495563244428889912nnreal,F: real > extend8495563244428889912nnreal,B2: real,C: real] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_eq_real @ X3 @ Y5 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_le3935885782089961368nnreal @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_622_ord__eq__le__subst,axiom,
    ! [A2: real,F: extend8495563244428889912nnreal > real,B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le3935885782089961368nnreal @ B2 @ C )
       => ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_623_ord__eq__le__subst,axiom,
    ! [A2: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le3935885782089961368nnreal @ B2 @ C )
       => ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_le3935885782089961368nnreal @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_624_ord__eq__le__subst,axiom,
    ! [A2: set_real,F: real > set_real,B2: real,C: real] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_eq_real @ X3 @ Y5 )
             => ( ord_less_eq_set_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_set_real @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_625_ord__eq__le__subst,axiom,
    ! [A2: real,F: set_real > real,B2: set_real,C: set_real] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_set_real @ B2 @ C )
       => ( ! [X3: set_real,Y5: set_real] :
              ( ( ord_less_eq_set_real @ X3 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_626_ord__eq__le__subst,axiom,
    ! [A2: extend8495563244428889912nnreal,F: set_real > extend8495563244428889912nnreal,B2: set_real,C: set_real] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_set_real @ B2 @ C )
       => ( ! [X3: set_real,Y5: set_real] :
              ( ( ord_less_eq_set_real @ X3 @ Y5 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_le3935885782089961368nnreal @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_627_ord__eq__le__subst,axiom,
    ! [A2: set_real,F: extend8495563244428889912nnreal > set_real,B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le3935885782089961368nnreal @ B2 @ C )
       => ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
             => ( ord_less_eq_set_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_set_real @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_628_ord__eq__le__subst,axiom,
    ! [A2: set_set_real,F: real > set_set_real,B2: real,C: real] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_eq_real @ X3 @ Y5 )
             => ( ord_le3558479182127378552t_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_le3558479182127378552t_real @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_629_ord__eq__le__subst,axiom,
    ! [A2: real,F: set_set_real > real,B2: set_set_real,C: set_set_real] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le3558479182127378552t_real @ B2 @ C )
       => ( ! [X3: set_set_real,Y5: set_set_real] :
              ( ( ord_le3558479182127378552t_real @ X3 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_630_linorder__linear,axiom,
    ! [X8: real,Y4: real] :
      ( ( ord_less_eq_real @ X8 @ Y4 )
      | ( ord_less_eq_real @ Y4 @ X8 ) ) ).

% linorder_linear
thf(fact_631_linorder__linear,axiom,
    ! [X8: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ X8 @ Y4 )
      | ( ord_le3935885782089961368nnreal @ Y4 @ X8 ) ) ).

% linorder_linear
thf(fact_632_order__eq__refl,axiom,
    ! [X8: real,Y4: real] :
      ( ( X8 = Y4 )
     => ( ord_less_eq_real @ X8 @ Y4 ) ) ).

% order_eq_refl
thf(fact_633_order__eq__refl,axiom,
    ! [X8: set_set_real,Y4: set_set_real] :
      ( ( X8 = Y4 )
     => ( ord_le3558479182127378552t_real @ X8 @ Y4 ) ) ).

% order_eq_refl
thf(fact_634_order__eq__refl,axiom,
    ! [X8: set_real,Y4: set_real] :
      ( ( X8 = Y4 )
     => ( ord_less_eq_set_real @ X8 @ Y4 ) ) ).

% order_eq_refl
thf(fact_635_order__eq__refl,axiom,
    ! [X8: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
      ( ( X8 = Y4 )
     => ( ord_le3935885782089961368nnreal @ X8 @ Y4 ) ) ).

% order_eq_refl
thf(fact_636_order__subst2,axiom,
    ! [A2: real,B2: real,F: real > real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_eq_real @ X3 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_637_order__subst2,axiom,
    ! [A2: real,B2: real,F: real > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_le3935885782089961368nnreal @ ( F @ B2 ) @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_eq_real @ X3 @ Y5 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_le3935885782089961368nnreal @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_638_order__subst2,axiom,
    ! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > real,C: real] :
      ( ( ord_le3935885782089961368nnreal @ A2 @ B2 )
     => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
       => ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_639_order__subst2,axiom,
    ! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A2 @ B2 )
     => ( ( ord_le3935885782089961368nnreal @ ( F @ B2 ) @ C )
       => ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_le3935885782089961368nnreal @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_640_order__subst2,axiom,
    ! [A2: real,B2: real,F: real > set_real,C: set_real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_set_real @ ( F @ B2 ) @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_eq_real @ X3 @ Y5 )
             => ( ord_less_eq_set_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_set_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_641_order__subst2,axiom,
    ! [A2: set_real,B2: set_real,F: set_real > real,C: real] :
      ( ( ord_less_eq_set_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
       => ( ! [X3: set_real,Y5: set_real] :
              ( ( ord_less_eq_set_real @ X3 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_642_order__subst2,axiom,
    ! [A2: set_real,B2: set_real,F: set_real > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_less_eq_set_real @ A2 @ B2 )
     => ( ( ord_le3935885782089961368nnreal @ ( F @ B2 ) @ C )
       => ( ! [X3: set_real,Y5: set_real] :
              ( ( ord_less_eq_set_real @ X3 @ Y5 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_le3935885782089961368nnreal @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_643_order__subst2,axiom,
    ! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > set_real,C: set_real] :
      ( ( ord_le3935885782089961368nnreal @ A2 @ B2 )
     => ( ( ord_less_eq_set_real @ ( F @ B2 ) @ C )
       => ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
             => ( ord_less_eq_set_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_set_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_644_order__subst2,axiom,
    ! [A2: real,B2: real,F: real > set_set_real,C: set_set_real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_le3558479182127378552t_real @ ( F @ B2 ) @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_eq_real @ X3 @ Y5 )
             => ( ord_le3558479182127378552t_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_le3558479182127378552t_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_645_order__subst2,axiom,
    ! [A2: set_set_real,B2: set_set_real,F: set_set_real > real,C: real] :
      ( ( ord_le3558479182127378552t_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
       => ( ! [X3: set_set_real,Y5: set_set_real] :
              ( ( ord_le3558479182127378552t_real @ X3 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_646_order__subst1,axiom,
    ! [A2: real,F: real > real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_eq_real @ X3 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_647_order__subst1,axiom,
    ! [A2: real,F: extend8495563244428889912nnreal > real,B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
     => ( ( ord_le3935885782089961368nnreal @ B2 @ C )
       => ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_648_order__subst1,axiom,
    ! [A2: extend8495563244428889912nnreal,F: real > extend8495563244428889912nnreal,B2: real,C: real] :
      ( ( ord_le3935885782089961368nnreal @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_eq_real @ X3 @ Y5 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_le3935885782089961368nnreal @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_649_order__subst1,axiom,
    ! [A2: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A2 @ ( F @ B2 ) )
     => ( ( ord_le3935885782089961368nnreal @ B2 @ C )
       => ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_le3935885782089961368nnreal @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_650_order__subst1,axiom,
    ! [A2: real,F: set_real > real,B2: set_real,C: set_real] :
      ( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_real @ B2 @ C )
       => ( ! [X3: set_real,Y5: set_real] :
              ( ( ord_less_eq_set_real @ X3 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_651_order__subst1,axiom,
    ! [A2: set_real,F: real > set_real,B2: real,C: real] :
      ( ( ord_less_eq_set_real @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_eq_real @ X3 @ Y5 )
             => ( ord_less_eq_set_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_set_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_652_order__subst1,axiom,
    ! [A2: set_real,F: extend8495563244428889912nnreal > set_real,B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_less_eq_set_real @ A2 @ ( F @ B2 ) )
     => ( ( ord_le3935885782089961368nnreal @ B2 @ C )
       => ( ! [X3: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X3 @ Y5 )
             => ( ord_less_eq_set_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_set_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_653_order__subst1,axiom,
    ! [A2: extend8495563244428889912nnreal,F: set_real > extend8495563244428889912nnreal,B2: set_real,C: set_real] :
      ( ( ord_le3935885782089961368nnreal @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_real @ B2 @ C )
       => ( ! [X3: set_real,Y5: set_real] :
              ( ( ord_less_eq_set_real @ X3 @ Y5 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_le3935885782089961368nnreal @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_654_order__subst1,axiom,
    ! [A2: real,F: set_set_real > real,B2: set_set_real,C: set_set_real] :
      ( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
     => ( ( ord_le3558479182127378552t_real @ B2 @ C )
       => ( ! [X3: set_set_real,Y5: set_set_real] :
              ( ( ord_le3558479182127378552t_real @ X3 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_655_order__subst1,axiom,
    ! [A2: set_set_real,F: real > set_set_real,B2: real,C: real] :
      ( ( ord_le3558479182127378552t_real @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_eq_real @ X3 @ Y5 )
             => ( ord_le3558479182127378552t_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_le3558479182127378552t_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_656_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y8: real,Z2: real] : ( Y8 = Z2 ) )
    = ( ^ [A5: real,B5: real] :
          ( ( ord_less_eq_real @ A5 @ B5 )
          & ( ord_less_eq_real @ B5 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_657_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y8: set_set_real,Z2: set_set_real] : ( Y8 = Z2 ) )
    = ( ^ [A5: set_set_real,B5: set_set_real] :
          ( ( ord_le3558479182127378552t_real @ A5 @ B5 )
          & ( ord_le3558479182127378552t_real @ B5 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_658_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y8: set_real,Z2: set_real] : ( Y8 = Z2 ) )
    = ( ^ [A5: set_real,B5: set_real] :
          ( ( ord_less_eq_set_real @ A5 @ B5 )
          & ( ord_less_eq_set_real @ B5 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_659_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y8: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] : ( Y8 = Z2 ) )
    = ( ^ [A5: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
          ( ( ord_le3935885782089961368nnreal @ A5 @ B5 )
          & ( ord_le3935885782089961368nnreal @ B5 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_660_antisym,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_661_antisym,axiom,
    ! [A2: set_set_real,B2: set_set_real] :
      ( ( ord_le3558479182127378552t_real @ A2 @ B2 )
     => ( ( ord_le3558479182127378552t_real @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_662_antisym,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ B2 )
     => ( ( ord_less_eq_set_real @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_663_antisym,axiom,
    ! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A2 @ B2 )
     => ( ( ord_le3935885782089961368nnreal @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_664_dual__order_Otrans,axiom,
    ! [B2: real,A2: real,C: real] :
      ( ( ord_less_eq_real @ B2 @ A2 )
     => ( ( ord_less_eq_real @ C @ B2 )
       => ( ord_less_eq_real @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_665_dual__order_Otrans,axiom,
    ! [B2: set_set_real,A2: set_set_real,C: set_set_real] :
      ( ( ord_le3558479182127378552t_real @ B2 @ A2 )
     => ( ( ord_le3558479182127378552t_real @ C @ B2 )
       => ( ord_le3558479182127378552t_real @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_666_dual__order_Otrans,axiom,
    ! [B2: set_real,A2: set_real,C: set_real] :
      ( ( ord_less_eq_set_real @ B2 @ A2 )
     => ( ( ord_less_eq_set_real @ C @ B2 )
       => ( ord_less_eq_set_real @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_667_dual__order_Otrans,axiom,
    ! [B2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ B2 @ A2 )
     => ( ( ord_le3935885782089961368nnreal @ C @ B2 )
       => ( ord_le3935885782089961368nnreal @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_668_dual__order_Oantisym,axiom,
    ! [B2: real,A2: real] :
      ( ( ord_less_eq_real @ B2 @ A2 )
     => ( ( ord_less_eq_real @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_669_dual__order_Oantisym,axiom,
    ! [B2: set_set_real,A2: set_set_real] :
      ( ( ord_le3558479182127378552t_real @ B2 @ A2 )
     => ( ( ord_le3558479182127378552t_real @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_670_dual__order_Oantisym,axiom,
    ! [B2: set_real,A2: set_real] :
      ( ( ord_less_eq_set_real @ B2 @ A2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_671_dual__order_Oantisym,axiom,
    ! [B2: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ B2 @ A2 )
     => ( ( ord_le3935885782089961368nnreal @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_672_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y8: real,Z2: real] : ( Y8 = Z2 ) )
    = ( ^ [A5: real,B5: real] :
          ( ( ord_less_eq_real @ B5 @ A5 )
          & ( ord_less_eq_real @ A5 @ B5 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_673_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y8: set_set_real,Z2: set_set_real] : ( Y8 = Z2 ) )
    = ( ^ [A5: set_set_real,B5: set_set_real] :
          ( ( ord_le3558479182127378552t_real @ B5 @ A5 )
          & ( ord_le3558479182127378552t_real @ A5 @ B5 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_674_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y8: set_real,Z2: set_real] : ( Y8 = Z2 ) )
    = ( ^ [A5: set_real,B5: set_real] :
          ( ( ord_less_eq_set_real @ B5 @ A5 )
          & ( ord_less_eq_set_real @ A5 @ B5 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_675_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y8: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] : ( Y8 = Z2 ) )
    = ( ^ [A5: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
          ( ( ord_le3935885782089961368nnreal @ B5 @ A5 )
          & ( ord_le3935885782089961368nnreal @ A5 @ B5 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_676_linorder__wlog,axiom,
    ! [P2: real > real > $o,A2: real,B2: real] :
      ( ! [A4: real,B4: real] :
          ( ( ord_less_eq_real @ A4 @ B4 )
         => ( P2 @ A4 @ B4 ) )
     => ( ! [A4: real,B4: real] :
            ( ( P2 @ B4 @ A4 )
           => ( P2 @ A4 @ B4 ) )
       => ( P2 @ A2 @ B2 ) ) ) ).

% linorder_wlog
thf(fact_677_linorder__wlog,axiom,
    ! [P2: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o,A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
      ( ! [A4: extend8495563244428889912nnreal,B4: extend8495563244428889912nnreal] :
          ( ( ord_le3935885782089961368nnreal @ A4 @ B4 )
         => ( P2 @ A4 @ B4 ) )
     => ( ! [A4: extend8495563244428889912nnreal,B4: extend8495563244428889912nnreal] :
            ( ( P2 @ B4 @ A4 )
           => ( P2 @ A4 @ B4 ) )
       => ( P2 @ A2 @ B2 ) ) ) ).

% linorder_wlog
thf(fact_678_order__trans,axiom,
    ! [X8: real,Y4: real,Z3: real] :
      ( ( ord_less_eq_real @ X8 @ Y4 )
     => ( ( ord_less_eq_real @ Y4 @ Z3 )
       => ( ord_less_eq_real @ X8 @ Z3 ) ) ) ).

% order_trans
thf(fact_679_order__trans,axiom,
    ! [X8: set_set_real,Y4: set_set_real,Z3: set_set_real] :
      ( ( ord_le3558479182127378552t_real @ X8 @ Y4 )
     => ( ( ord_le3558479182127378552t_real @ Y4 @ Z3 )
       => ( ord_le3558479182127378552t_real @ X8 @ Z3 ) ) ) ).

% order_trans
thf(fact_680_order__trans,axiom,
    ! [X8: set_real,Y4: set_real,Z3: set_real] :
      ( ( ord_less_eq_set_real @ X8 @ Y4 )
     => ( ( ord_less_eq_set_real @ Y4 @ Z3 )
       => ( ord_less_eq_set_real @ X8 @ Z3 ) ) ) ).

% order_trans
thf(fact_681_order__trans,axiom,
    ! [X8: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ X8 @ Y4 )
     => ( ( ord_le3935885782089961368nnreal @ Y4 @ Z3 )
       => ( ord_le3935885782089961368nnreal @ X8 @ Z3 ) ) ) ).

% order_trans
thf(fact_682_order_Otrans,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ord_less_eq_real @ A2 @ C ) ) ) ).

% order.trans
thf(fact_683_order_Otrans,axiom,
    ! [A2: set_set_real,B2: set_set_real,C: set_set_real] :
      ( ( ord_le3558479182127378552t_real @ A2 @ B2 )
     => ( ( ord_le3558479182127378552t_real @ B2 @ C )
       => ( ord_le3558479182127378552t_real @ A2 @ C ) ) ) ).

% order.trans
thf(fact_684_order_Otrans,axiom,
    ! [A2: set_real,B2: set_real,C: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ B2 )
     => ( ( ord_less_eq_set_real @ B2 @ C )
       => ( ord_less_eq_set_real @ A2 @ C ) ) ) ).

% order.trans
thf(fact_685_order_Otrans,axiom,
    ! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A2 @ B2 )
     => ( ( ord_le3935885782089961368nnreal @ B2 @ C )
       => ( ord_le3935885782089961368nnreal @ A2 @ C ) ) ) ).

% order.trans
thf(fact_686_order__antisym,axiom,
    ! [X8: real,Y4: real] :
      ( ( ord_less_eq_real @ X8 @ Y4 )
     => ( ( ord_less_eq_real @ Y4 @ X8 )
       => ( X8 = Y4 ) ) ) ).

% order_antisym
thf(fact_687_order__antisym,axiom,
    ! [X8: set_set_real,Y4: set_set_real] :
      ( ( ord_le3558479182127378552t_real @ X8 @ Y4 )
     => ( ( ord_le3558479182127378552t_real @ Y4 @ X8 )
       => ( X8 = Y4 ) ) ) ).

% order_antisym
thf(fact_688_order__antisym,axiom,
    ! [X8: set_real,Y4: set_real] :
      ( ( ord_less_eq_set_real @ X8 @ Y4 )
     => ( ( ord_less_eq_set_real @ Y4 @ X8 )
       => ( X8 = Y4 ) ) ) ).

% order_antisym
thf(fact_689_order__antisym,axiom,
    ! [X8: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ X8 @ Y4 )
     => ( ( ord_le3935885782089961368nnreal @ Y4 @ X8 )
       => ( X8 = Y4 ) ) ) ).

% order_antisym
thf(fact_690_ord__le__eq__trans,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_real @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_691_ord__le__eq__trans,axiom,
    ! [A2: set_set_real,B2: set_set_real,C: set_set_real] :
      ( ( ord_le3558479182127378552t_real @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_le3558479182127378552t_real @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_692_ord__le__eq__trans,axiom,
    ! [A2: set_real,B2: set_real,C: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_set_real @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_693_ord__le__eq__trans,axiom,
    ! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_le3935885782089961368nnreal @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_694_ord__eq__le__trans,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ord_less_eq_real @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_695_ord__eq__le__trans,axiom,
    ! [A2: set_set_real,B2: set_set_real,C: set_set_real] :
      ( ( A2 = B2 )
     => ( ( ord_le3558479182127378552t_real @ B2 @ C )
       => ( ord_le3558479182127378552t_real @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_696_ord__eq__le__trans,axiom,
    ! [A2: set_real,B2: set_real,C: set_real] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_set_real @ B2 @ C )
       => ( ord_less_eq_set_real @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_697_ord__eq__le__trans,axiom,
    ! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( A2 = B2 )
     => ( ( ord_le3935885782089961368nnreal @ B2 @ C )
       => ( ord_le3935885782089961368nnreal @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_698_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y8: real,Z2: real] : ( Y8 = Z2 ) )
    = ( ^ [X6: real,Y3: real] :
          ( ( ord_less_eq_real @ X6 @ Y3 )
          & ( ord_less_eq_real @ Y3 @ X6 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_699_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y8: set_set_real,Z2: set_set_real] : ( Y8 = Z2 ) )
    = ( ^ [X6: set_set_real,Y3: set_set_real] :
          ( ( ord_le3558479182127378552t_real @ X6 @ Y3 )
          & ( ord_le3558479182127378552t_real @ Y3 @ X6 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_700_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y8: set_real,Z2: set_real] : ( Y8 = Z2 ) )
    = ( ^ [X6: set_real,Y3: set_real] :
          ( ( ord_less_eq_set_real @ X6 @ Y3 )
          & ( ord_less_eq_set_real @ Y3 @ X6 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_701_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y8: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] : ( Y8 = Z2 ) )
    = ( ^ [X6: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
          ( ( ord_le3935885782089961368nnreal @ X6 @ Y3 )
          & ( ord_le3935885782089961368nnreal @ Y3 @ X6 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_702_le__cases3,axiom,
    ! [X8: real,Y4: real,Z3: real] :
      ( ( ( ord_less_eq_real @ X8 @ Y4 )
       => ~ ( ord_less_eq_real @ Y4 @ Z3 ) )
     => ( ( ( ord_less_eq_real @ Y4 @ X8 )
         => ~ ( ord_less_eq_real @ X8 @ Z3 ) )
       => ( ( ( ord_less_eq_real @ X8 @ Z3 )
           => ~ ( ord_less_eq_real @ Z3 @ Y4 ) )
         => ( ( ( ord_less_eq_real @ Z3 @ Y4 )
             => ~ ( ord_less_eq_real @ Y4 @ X8 ) )
           => ( ( ( ord_less_eq_real @ Y4 @ Z3 )
               => ~ ( ord_less_eq_real @ Z3 @ X8 ) )
             => ~ ( ( ord_less_eq_real @ Z3 @ X8 )
                 => ~ ( ord_less_eq_real @ X8 @ Y4 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_703_le__cases3,axiom,
    ! [X8: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] :
      ( ( ( ord_le3935885782089961368nnreal @ X8 @ Y4 )
       => ~ ( ord_le3935885782089961368nnreal @ Y4 @ Z3 ) )
     => ( ( ( ord_le3935885782089961368nnreal @ Y4 @ X8 )
         => ~ ( ord_le3935885782089961368nnreal @ X8 @ Z3 ) )
       => ( ( ( ord_le3935885782089961368nnreal @ X8 @ Z3 )
           => ~ ( ord_le3935885782089961368nnreal @ Z3 @ Y4 ) )
         => ( ( ( ord_le3935885782089961368nnreal @ Z3 @ Y4 )
             => ~ ( ord_le3935885782089961368nnreal @ Y4 @ X8 ) )
           => ( ( ( ord_le3935885782089961368nnreal @ Y4 @ Z3 )
               => ~ ( ord_le3935885782089961368nnreal @ Z3 @ X8 ) )
             => ~ ( ( ord_le3935885782089961368nnreal @ Z3 @ X8 )
                 => ~ ( ord_le3935885782089961368nnreal @ X8 @ Y4 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_704_nle__le,axiom,
    ! [A2: real,B2: real] :
      ( ( ~ ( ord_less_eq_real @ A2 @ B2 ) )
      = ( ( ord_less_eq_real @ B2 @ A2 )
        & ( B2 != A2 ) ) ) ).

% nle_le
thf(fact_705_nle__le,axiom,
    ! [A2: extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal] :
      ( ( ~ ( ord_le3935885782089961368nnreal @ A2 @ B2 ) )
      = ( ( ord_le3935885782089961368nnreal @ B2 @ A2 )
        & ( B2 != A2 ) ) ) ).

% nle_le
thf(fact_706_Collect__mono__iff,axiom,
    ! [P2: set_real > $o,Q: set_real > $o] :
      ( ( ord_le3558479182127378552t_real @ ( collect_set_real @ P2 ) @ ( collect_set_real @ Q ) )
      = ( ! [X6: set_real] :
            ( ( P2 @ X6 )
           => ( Q @ X6 ) ) ) ) ).

% Collect_mono_iff
thf(fact_707_Collect__mono__iff,axiom,
    ! [P2: real > $o,Q: real > $o] :
      ( ( ord_less_eq_set_real @ ( collect_real @ P2 ) @ ( collect_real @ Q ) )
      = ( ! [X6: real] :
            ( ( P2 @ X6 )
           => ( Q @ X6 ) ) ) ) ).

% Collect_mono_iff
thf(fact_708_set__eq__subset,axiom,
    ( ( ^ [Y8: set_set_real,Z2: set_set_real] : ( Y8 = Z2 ) )
    = ( ^ [A6: set_set_real,B6: set_set_real] :
          ( ( ord_le3558479182127378552t_real @ A6 @ B6 )
          & ( ord_le3558479182127378552t_real @ B6 @ A6 ) ) ) ) ).

% set_eq_subset
thf(fact_709_set__eq__subset,axiom,
    ( ( ^ [Y8: set_real,Z2: set_real] : ( Y8 = Z2 ) )
    = ( ^ [A6: set_real,B6: set_real] :
          ( ( ord_less_eq_set_real @ A6 @ B6 )
          & ( ord_less_eq_set_real @ B6 @ A6 ) ) ) ) ).

% set_eq_subset
thf(fact_710_subset__trans,axiom,
    ! [A: set_set_real,B: set_set_real,C3: set_set_real] :
      ( ( ord_le3558479182127378552t_real @ A @ B )
     => ( ( ord_le3558479182127378552t_real @ B @ C3 )
       => ( ord_le3558479182127378552t_real @ A @ C3 ) ) ) ).

% subset_trans
thf(fact_711_subset__trans,axiom,
    ! [A: set_real,B: set_real,C3: set_real] :
      ( ( ord_less_eq_set_real @ A @ B )
     => ( ( ord_less_eq_set_real @ B @ C3 )
       => ( ord_less_eq_set_real @ A @ C3 ) ) ) ).

% subset_trans
thf(fact_712_Collect__mono,axiom,
    ! [P2: set_real > $o,Q: set_real > $o] :
      ( ! [X3: set_real] :
          ( ( P2 @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le3558479182127378552t_real @ ( collect_set_real @ P2 ) @ ( collect_set_real @ Q ) ) ) ).

% Collect_mono
thf(fact_713_Collect__mono,axiom,
    ! [P2: real > $o,Q: real > $o] :
      ( ! [X3: real] :
          ( ( P2 @ X3 )
         => ( Q @ X3 ) )
     => ( ord_less_eq_set_real @ ( collect_real @ P2 ) @ ( collect_real @ Q ) ) ) ).

% Collect_mono
thf(fact_714_subset__refl,axiom,
    ! [A: set_set_real] : ( ord_le3558479182127378552t_real @ A @ A ) ).

% subset_refl
thf(fact_715_subset__refl,axiom,
    ! [A: set_real] : ( ord_less_eq_set_real @ A @ A ) ).

% subset_refl
thf(fact_716_subset__iff,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B6: set_real] :
        ! [T2: real] :
          ( ( member_real @ T2 @ A6 )
         => ( member_real @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_717_subset__iff,axiom,
    ( ord_le3558479182127378552t_real
    = ( ^ [A6: set_set_real,B6: set_set_real] :
        ! [T2: set_real] :
          ( ( member_set_real @ T2 @ A6 )
         => ( member_set_real @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_718_subset__iff,axiom,
    ( ord_less_eq_set_o_o
    = ( ^ [A6: set_o_o,B6: set_o_o] :
        ! [T2: $o > $o] :
          ( ( member_o_o @ T2 @ A6 )
         => ( member_o_o @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_719_subset__iff,axiom,
    ( ord_le672203391976590760nnreal
    = ( ^ [A6: set_b_6825823330181178888nnreal,B6: set_b_6825823330181178888nnreal] :
        ! [T2: b > extend8495563244428889912nnreal] :
          ( ( member6418304549040442065nnreal @ T2 @ A6 )
         => ( member6418304549040442065nnreal @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_720_subset__iff,axiom,
    ( ord_le1007445205377960487nnreal
    = ( ^ [A6: set_a_7161065143582548615nnreal,B6: set_a_7161065143582548615nnreal] :
        ! [T2: a > extend8495563244428889912nnreal] :
          ( ( member298456594901751504nnreal @ T2 @ A6 )
         => ( member298456594901751504nnreal @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_721_subset__iff,axiom,
    ( ord_le3334967407727675675a_real
    = ( ^ [A6: set_a_real,B6: set_a_real] :
        ! [T2: a > real] :
          ( ( member_a_real @ T2 @ A6 )
         => ( member_a_real @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_722_subset__iff,axiom,
    ( ord_le1615110227528160547real_o
    = ( ^ [A6: set_real_o,B6: set_real_o] :
        ! [T2: real > $o] :
          ( ( member_real_o @ T2 @ A6 )
         => ( member_real_o @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_723_subset__iff,axiom,
    ( ord_le6098800555920186673al_nat
    = ( ^ [A6: set_real_nat,B6: set_real_nat] :
        ! [T2: real > nat] :
          ( ( member_real_nat @ T2 @ A6 )
         => ( member_real_nat @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_724_subset__iff,axiom,
    ( ord_le2462468573666744473nnreal
    = ( ^ [A6: set_re5328672808648366137nnreal,B6: set_re5328672808648366137nnreal] :
        ! [T2: real > extend8495563244428889912nnreal] :
          ( ( member2919562650594848410nnreal @ T2 @ A6 )
         => ( member2919562650594848410nnreal @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_725_subset__iff,axiom,
    ( ord_le2047140485929309711omplex
    = ( ^ [A6: set_real_complex,B6: set_real_complex] :
        ! [T2: real > complex] :
          ( ( member_real_complex @ T2 @ A6 )
         => ( member_real_complex @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_726_equalityD2,axiom,
    ! [A: set_set_real,B: set_set_real] :
      ( ( A = B )
     => ( ord_le3558479182127378552t_real @ B @ A ) ) ).

% equalityD2
thf(fact_727_equalityD2,axiom,
    ! [A: set_real,B: set_real] :
      ( ( A = B )
     => ( ord_less_eq_set_real @ B @ A ) ) ).

% equalityD2
thf(fact_728_equalityD1,axiom,
    ! [A: set_set_real,B: set_set_real] :
      ( ( A = B )
     => ( ord_le3558479182127378552t_real @ A @ B ) ) ).

% equalityD1
thf(fact_729_equalityD1,axiom,
    ! [A: set_real,B: set_real] :
      ( ( A = B )
     => ( ord_less_eq_set_real @ A @ B ) ) ).

% equalityD1
thf(fact_730_subset__eq,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B6: set_real] :
        ! [X6: real] :
          ( ( member_real @ X6 @ A6 )
         => ( member_real @ X6 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_731_subset__eq,axiom,
    ( ord_le3558479182127378552t_real
    = ( ^ [A6: set_set_real,B6: set_set_real] :
        ! [X6: set_real] :
          ( ( member_set_real @ X6 @ A6 )
         => ( member_set_real @ X6 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_732_subset__eq,axiom,
    ( ord_less_eq_set_o_o
    = ( ^ [A6: set_o_o,B6: set_o_o] :
        ! [X6: $o > $o] :
          ( ( member_o_o @ X6 @ A6 )
         => ( member_o_o @ X6 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_733_subset__eq,axiom,
    ( ord_le672203391976590760nnreal
    = ( ^ [A6: set_b_6825823330181178888nnreal,B6: set_b_6825823330181178888nnreal] :
        ! [X6: b > extend8495563244428889912nnreal] :
          ( ( member6418304549040442065nnreal @ X6 @ A6 )
         => ( member6418304549040442065nnreal @ X6 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_734_subset__eq,axiom,
    ( ord_le1007445205377960487nnreal
    = ( ^ [A6: set_a_7161065143582548615nnreal,B6: set_a_7161065143582548615nnreal] :
        ! [X6: a > extend8495563244428889912nnreal] :
          ( ( member298456594901751504nnreal @ X6 @ A6 )
         => ( member298456594901751504nnreal @ X6 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_735_subset__eq,axiom,
    ( ord_le3334967407727675675a_real
    = ( ^ [A6: set_a_real,B6: set_a_real] :
        ! [X6: a > real] :
          ( ( member_a_real @ X6 @ A6 )
         => ( member_a_real @ X6 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_736_subset__eq,axiom,
    ( ord_le1615110227528160547real_o
    = ( ^ [A6: set_real_o,B6: set_real_o] :
        ! [X6: real > $o] :
          ( ( member_real_o @ X6 @ A6 )
         => ( member_real_o @ X6 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_737_subset__eq,axiom,
    ( ord_le6098800555920186673al_nat
    = ( ^ [A6: set_real_nat,B6: set_real_nat] :
        ! [X6: real > nat] :
          ( ( member_real_nat @ X6 @ A6 )
         => ( member_real_nat @ X6 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_738_subset__eq,axiom,
    ( ord_le2462468573666744473nnreal
    = ( ^ [A6: set_re5328672808648366137nnreal,B6: set_re5328672808648366137nnreal] :
        ! [X6: real > extend8495563244428889912nnreal] :
          ( ( member2919562650594848410nnreal @ X6 @ A6 )
         => ( member2919562650594848410nnreal @ X6 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_739_subset__eq,axiom,
    ( ord_le2047140485929309711omplex
    = ( ^ [A6: set_real_complex,B6: set_real_complex] :
        ! [X6: real > complex] :
          ( ( member_real_complex @ X6 @ A6 )
         => ( member_real_complex @ X6 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_740_equalityE,axiom,
    ! [A: set_set_real,B: set_set_real] :
      ( ( A = B )
     => ~ ( ( ord_le3558479182127378552t_real @ A @ B )
         => ~ ( ord_le3558479182127378552t_real @ B @ A ) ) ) ).

% equalityE
thf(fact_741_equalityE,axiom,
    ! [A: set_real,B: set_real] :
      ( ( A = B )
     => ~ ( ( ord_less_eq_set_real @ A @ B )
         => ~ ( ord_less_eq_set_real @ B @ A ) ) ) ).

% equalityE
thf(fact_742_subsetD,axiom,
    ! [A: set_real,B: set_real,C: real] :
      ( ( ord_less_eq_set_real @ A @ B )
     => ( ( member_real @ C @ A )
       => ( member_real @ C @ B ) ) ) ).

% subsetD
thf(fact_743_subsetD,axiom,
    ! [A: set_set_real,B: set_set_real,C: set_real] :
      ( ( ord_le3558479182127378552t_real @ A @ B )
     => ( ( member_set_real @ C @ A )
       => ( member_set_real @ C @ B ) ) ) ).

% subsetD
thf(fact_744_subsetD,axiom,
    ! [A: set_o_o,B: set_o_o,C: $o > $o] :
      ( ( ord_less_eq_set_o_o @ A @ B )
     => ( ( member_o_o @ C @ A )
       => ( member_o_o @ C @ B ) ) ) ).

% subsetD
thf(fact_745_subsetD,axiom,
    ! [A: set_b_6825823330181178888nnreal,B: set_b_6825823330181178888nnreal,C: b > extend8495563244428889912nnreal] :
      ( ( ord_le672203391976590760nnreal @ A @ B )
     => ( ( member6418304549040442065nnreal @ C @ A )
       => ( member6418304549040442065nnreal @ C @ B ) ) ) ).

% subsetD
thf(fact_746_subsetD,axiom,
    ! [A: set_a_7161065143582548615nnreal,B: set_a_7161065143582548615nnreal,C: a > extend8495563244428889912nnreal] :
      ( ( ord_le1007445205377960487nnreal @ A @ B )
     => ( ( member298456594901751504nnreal @ C @ A )
       => ( member298456594901751504nnreal @ C @ B ) ) ) ).

% subsetD
thf(fact_747_subsetD,axiom,
    ! [A: set_a_real,B: set_a_real,C: a > real] :
      ( ( ord_le3334967407727675675a_real @ A @ B )
     => ( ( member_a_real @ C @ A )
       => ( member_a_real @ C @ B ) ) ) ).

% subsetD
thf(fact_748_subsetD,axiom,
    ! [A: set_real_o,B: set_real_o,C: real > $o] :
      ( ( ord_le1615110227528160547real_o @ A @ B )
     => ( ( member_real_o @ C @ A )
       => ( member_real_o @ C @ B ) ) ) ).

% subsetD
thf(fact_749_subsetD,axiom,
    ! [A: set_real_nat,B: set_real_nat,C: real > nat] :
      ( ( ord_le6098800555920186673al_nat @ A @ B )
     => ( ( member_real_nat @ C @ A )
       => ( member_real_nat @ C @ B ) ) ) ).

% subsetD
thf(fact_750_subsetD,axiom,
    ! [A: set_re5328672808648366137nnreal,B: set_re5328672808648366137nnreal,C: real > extend8495563244428889912nnreal] :
      ( ( ord_le2462468573666744473nnreal @ A @ B )
     => ( ( member2919562650594848410nnreal @ C @ A )
       => ( member2919562650594848410nnreal @ C @ B ) ) ) ).

% subsetD
thf(fact_751_subsetD,axiom,
    ! [A: set_real_complex,B: set_real_complex,C: real > complex] :
      ( ( ord_le2047140485929309711omplex @ A @ B )
     => ( ( member_real_complex @ C @ A )
       => ( member_real_complex @ C @ B ) ) ) ).

% subsetD
thf(fact_752_in__mono,axiom,
    ! [A: set_real,B: set_real,X8: real] :
      ( ( ord_less_eq_set_real @ A @ B )
     => ( ( member_real @ X8 @ A )
       => ( member_real @ X8 @ B ) ) ) ).

% in_mono
thf(fact_753_in__mono,axiom,
    ! [A: set_set_real,B: set_set_real,X8: set_real] :
      ( ( ord_le3558479182127378552t_real @ A @ B )
     => ( ( member_set_real @ X8 @ A )
       => ( member_set_real @ X8 @ B ) ) ) ).

% in_mono
thf(fact_754_in__mono,axiom,
    ! [A: set_o_o,B: set_o_o,X8: $o > $o] :
      ( ( ord_less_eq_set_o_o @ A @ B )
     => ( ( member_o_o @ X8 @ A )
       => ( member_o_o @ X8 @ B ) ) ) ).

% in_mono
thf(fact_755_in__mono,axiom,
    ! [A: set_b_6825823330181178888nnreal,B: set_b_6825823330181178888nnreal,X8: b > extend8495563244428889912nnreal] :
      ( ( ord_le672203391976590760nnreal @ A @ B )
     => ( ( member6418304549040442065nnreal @ X8 @ A )
       => ( member6418304549040442065nnreal @ X8 @ B ) ) ) ).

% in_mono
thf(fact_756_in__mono,axiom,
    ! [A: set_a_7161065143582548615nnreal,B: set_a_7161065143582548615nnreal,X8: a > extend8495563244428889912nnreal] :
      ( ( ord_le1007445205377960487nnreal @ A @ B )
     => ( ( member298456594901751504nnreal @ X8 @ A )
       => ( member298456594901751504nnreal @ X8 @ B ) ) ) ).

% in_mono
thf(fact_757_in__mono,axiom,
    ! [A: set_a_real,B: set_a_real,X8: a > real] :
      ( ( ord_le3334967407727675675a_real @ A @ B )
     => ( ( member_a_real @ X8 @ A )
       => ( member_a_real @ X8 @ B ) ) ) ).

% in_mono
thf(fact_758_in__mono,axiom,
    ! [A: set_real_o,B: set_real_o,X8: real > $o] :
      ( ( ord_le1615110227528160547real_o @ A @ B )
     => ( ( member_real_o @ X8 @ A )
       => ( member_real_o @ X8 @ B ) ) ) ).

% in_mono
thf(fact_759_in__mono,axiom,
    ! [A: set_real_nat,B: set_real_nat,X8: real > nat] :
      ( ( ord_le6098800555920186673al_nat @ A @ B )
     => ( ( member_real_nat @ X8 @ A )
       => ( member_real_nat @ X8 @ B ) ) ) ).

% in_mono
thf(fact_760_in__mono,axiom,
    ! [A: set_re5328672808648366137nnreal,B: set_re5328672808648366137nnreal,X8: real > extend8495563244428889912nnreal] :
      ( ( ord_le2462468573666744473nnreal @ A @ B )
     => ( ( member2919562650594848410nnreal @ X8 @ A )
       => ( member2919562650594848410nnreal @ X8 @ B ) ) ) ).

% in_mono
thf(fact_761_in__mono,axiom,
    ! [A: set_real_complex,B: set_real_complex,X8: real > complex] :
      ( ( ord_le2047140485929309711omplex @ A @ B )
     => ( ( member_real_complex @ X8 @ A )
       => ( member_real_complex @ X8 @ B ) ) ) ).

% in_mono
thf(fact_762_qbs__eqI,axiom,
    ! [X: quasi_borel_b,Y: quasi_borel_b] :
      ( ( ( qbs_Mx_b @ X )
        = ( qbs_Mx_b @ Y ) )
     => ( X = Y ) ) ).

% qbs_eqI
thf(fact_763_qbs__eqI,axiom,
    ! [X: quasi_borel_a,Y: quasi_borel_a] :
      ( ( ( qbs_Mx_a @ X )
        = ( qbs_Mx_a @ Y ) )
     => ( X = Y ) ) ).

% qbs_eqI
thf(fact_764_qbs__morphism__comp,axiom,
    ! [F: nat > real,X: quasi_borel_nat,Y: quasi_borel_real,G2: real > nat,Z: quasi_borel_nat] :
      ( ( member_nat_real @ F @ ( qbs_mo2000642995705457910t_real @ X @ Y ) )
     => ( ( member_real_nat @ G2 @ ( qbs_mo6567951568834356598al_nat @ Y @ Z ) )
       => ( member_nat_nat @ ( comp_real_nat_nat @ G2 @ F ) @ ( qbs_morphism_nat_nat @ X @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_765_qbs__morphism__comp,axiom,
    ! [F: real > $o,X: quasi_borel_real,Y: quasi_borel_o,G2: $o > nat,Z: quasi_borel_nat] :
      ( ( member_real_o @ F @ ( qbs_morphism_real_o @ X @ Y ) )
     => ( ( member_o_nat @ G2 @ ( qbs_morphism_o_nat @ Y @ Z ) )
       => ( member_real_nat @ ( comp_o_nat_real @ G2 @ F ) @ ( qbs_mo6567951568834356598al_nat @ X @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_766_qbs__morphism__comp,axiom,
    ! [F: real > $o,X: quasi_borel_real,Y: quasi_borel_o,G2: $o > extend8495563244428889912nnreal,Z: quasi_9015997321629101608nnreal] :
      ( ( member_real_o @ F @ ( qbs_morphism_real_o @ X @ Y ) )
     => ( ( member5265953103328284778nnreal @ G2 @ ( qbs_mo8025287178234086488nnreal @ Y @ Z ) )
       => ( member2919562650594848410nnreal @ ( comp_o592059427571696603l_real @ G2 @ F ) @ ( qbs_mo1317719164804411614nnreal @ X @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_767_qbs__morphism__comp,axiom,
    ! [F: real > $o,X: quasi_borel_real,Y: quasi_borel_o,G2: $o > complex,Z: quasi_borel_complex] :
      ( ( member_real_o @ F @ ( qbs_morphism_real_o @ X @ Y ) )
     => ( ( member_o_complex @ G2 @ ( qbs_mo5261837971759248846omplex @ Y @ Z ) )
       => ( member_real_complex @ ( comp_o_complex_real @ G2 @ F ) @ ( qbs_mo6067097710682130004omplex @ X @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_768_qbs__morphism__comp,axiom,
    ! [F: real > $o,X: quasi_borel_real,Y: quasi_borel_o,G2: $o > real,Z: quasi_borel_real] :
      ( ( member_real_o @ F @ ( qbs_morphism_real_o @ X @ Y ) )
     => ( ( member_o_real @ G2 @ ( qbs_morphism_o_real @ Y @ Z ) )
       => ( member_real_real @ ( comp_o_real_real @ G2 @ F ) @ ( qbs_mo5229770564518008146l_real @ X @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_769_qbs__morphism__comp,axiom,
    ! [F: real > nat,X: quasi_borel_real,Y: quasi_borel_nat,G2: nat > $o,Z: quasi_borel_o] :
      ( ( member_real_nat @ F @ ( qbs_mo6567951568834356598al_nat @ X @ Y ) )
     => ( ( member_nat_o @ G2 @ ( qbs_morphism_nat_o @ Y @ Z ) )
       => ( member_real_o @ ( comp_nat_o_real @ G2 @ F ) @ ( qbs_morphism_real_o @ X @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_770_qbs__morphism__comp,axiom,
    ! [F: real > nat,X: quasi_borel_real,Y: quasi_borel_nat,G2: nat > nat,Z: quasi_borel_nat] :
      ( ( member_real_nat @ F @ ( qbs_mo6567951568834356598al_nat @ X @ Y ) )
     => ( ( member_nat_nat @ G2 @ ( qbs_morphism_nat_nat @ Y @ Z ) )
       => ( member_real_nat @ ( comp_nat_nat_real @ G2 @ F ) @ ( qbs_mo6567951568834356598al_nat @ X @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_771_qbs__morphism__comp,axiom,
    ! [F: real > nat,X: quasi_borel_real,Y: quasi_borel_nat,G2: nat > extend8495563244428889912nnreal,Z: quasi_9015997321629101608nnreal] :
      ( ( member_real_nat @ F @ ( qbs_mo6567951568834356598al_nat @ X @ Y ) )
     => ( ( member8283130129095025342nnreal @ G2 @ ( qbs_mo4632421585747671682nnreal @ Y @ Z ) )
       => ( member2919562650594848410nnreal @ ( comp_n3455504981484479769l_real @ G2 @ F ) @ ( qbs_mo1317719164804411614nnreal @ X @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_772_qbs__morphism__comp,axiom,
    ! [F: real > nat,X: quasi_borel_real,Y: quasi_borel_nat,G2: nat > complex,Z: quasi_borel_complex] :
      ( ( member_real_nat @ F @ ( qbs_mo6567951568834356598al_nat @ X @ Y ) )
     => ( ( member_nat_complex @ G2 @ ( qbs_mo6368047727621400568omplex @ Y @ Z ) )
       => ( member_real_complex @ ( comp_n4215249288434654095x_real @ G2 @ F ) @ ( qbs_mo6067097710682130004omplex @ X @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_773_qbs__morphism__comp,axiom,
    ! [F: real > nat,X: quasi_borel_real,Y: quasi_borel_nat,G2: nat > real,Z: quasi_borel_real] :
      ( ( member_real_nat @ F @ ( qbs_mo6567951568834356598al_nat @ X @ Y ) )
     => ( ( member_nat_real @ G2 @ ( qbs_mo2000642995705457910t_real @ Y @ Z ) )
       => ( member_real_real @ ( comp_nat_real_real @ G2 @ F ) @ ( qbs_mo5229770564518008146l_real @ X @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_774_qbs__closed1__dest,axiom,
    ! [Alpha: real > $o,X: quasi_borel_o,F: real > real] :
      ( ( member_real_o @ Alpha @ ( qbs_Mx_o @ X ) )
     => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
       => ( member_real_o @ ( comp_real_o_real @ Alpha @ F ) @ ( qbs_Mx_o @ X ) ) ) ) ).

% qbs_closed1_dest
thf(fact_775_qbs__closed1__dest,axiom,
    ! [Alpha: real > nat,X: quasi_borel_nat,F: real > real] :
      ( ( member_real_nat @ Alpha @ ( qbs_Mx_nat @ X ) )
     => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
       => ( member_real_nat @ ( comp_real_nat_real @ Alpha @ F ) @ ( qbs_Mx_nat @ X ) ) ) ) ).

% qbs_closed1_dest
thf(fact_776_qbs__closed1__dest,axiom,
    ! [Alpha: real > extend8495563244428889912nnreal,X: quasi_9015997321629101608nnreal,F: real > real] :
      ( ( member2919562650594848410nnreal @ Alpha @ ( qbs_Mx6523938229262583809nnreal @ X ) )
     => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
       => ( member2919562650594848410nnreal @ ( comp_r6279034453215524981l_real @ Alpha @ F ) @ ( qbs_Mx6523938229262583809nnreal @ X ) ) ) ) ).

% qbs_closed1_dest
thf(fact_777_qbs__closed1__dest,axiom,
    ! [Alpha: real > complex,X: quasi_borel_complex,F: real > real] :
      ( ( member_real_complex @ Alpha @ ( qbs_Mx_complex @ X ) )
     => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
       => ( member_real_complex @ ( comp_r1968866223832618731x_real @ Alpha @ F ) @ ( qbs_Mx_complex @ X ) ) ) ) ).

% qbs_closed1_dest
thf(fact_778_qbs__closed1__dest,axiom,
    ! [Alpha: real > real,X: quasi_borel_real,F: real > real] :
      ( ( member_real_real @ Alpha @ ( qbs_Mx_real @ X ) )
     => ( ( 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 @ X ) ) ) ) ).

% qbs_closed1_dest
thf(fact_779_qbs__closed1__dest,axiom,
    ! [Alpha: real > b,X: quasi_borel_b,F: real > real] :
      ( ( member_real_b @ Alpha @ ( qbs_Mx_b @ X ) )
     => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
       => ( member_real_b @ ( comp_real_b_real @ Alpha @ F ) @ ( qbs_Mx_b @ X ) ) ) ) ).

% qbs_closed1_dest
thf(fact_780_qbs__closed1__dest,axiom,
    ! [Alpha: real > a,X: quasi_borel_a,F: real > real] :
      ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X ) )
     => ( ( 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 @ X ) ) ) ) ).

% qbs_closed1_dest
thf(fact_781_qbs__Mx__R,axiom,
    ! [X: sigma_measure_b] :
      ( ( qbs_Mx_b @ ( measur6507891955840068947_qbs_b @ X ) )
      = ( sigma_523072396149930113real_b @ borel_5078946678739801102l_real @ X ) ) ).

% qbs_Mx_R
thf(fact_782_qbs__Mx__R,axiom,
    ! [X: sigma_measure_a] :
      ( ( qbs_Mx_a @ ( measur6507891955840068946_qbs_a @ X ) )
      = ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ X ) ) ).

% qbs_Mx_R
thf(fact_783_qbs__Mx__R,axiom,
    ! [X: sigma_measure_nat] :
      ( ( qbs_Mx_nat @ ( measur4416158800429964412bs_nat @ X ) )
      = ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ X ) ) ).

% qbs_Mx_R
thf(fact_784_qbs__Mx__R,axiom,
    ! [X: sigma_3077487657436305159omplex] :
      ( ( qbs_Mx_complex @ ( measur1074055046195851610omplex @ X ) )
      = ( sigma_9111916201866572460omplex @ borel_5078946678739801102l_real @ X ) ) ).

% qbs_Mx_R
thf(fact_785_qbs__Mx__R,axiom,
    ! [X: sigma_7234349610311085201nnreal] :
      ( ( qbs_Mx6523938229262583809nnreal @ ( measur2642298986910087140nnreal @ X ) )
      = ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ X ) ) ).

% qbs_Mx_R
thf(fact_786_qbs__Mx__R,axiom,
    ! [X: sigma_measure_real] :
      ( ( qbs_Mx_real @ ( measur6875533127466166616s_real @ X ) )
      = ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ X ) ) ).

% qbs_Mx_R
thf(fact_787_qbs__Mx__R,axiom,
    ! [X: sigma_measure_o] :
      ( ( qbs_Mx_o @ ( measur2705496967258476524_qbs_o @ X ) )
      = ( sigma_3939073009482781210real_o @ borel_5078946678739801102l_real @ X ) ) ).

% qbs_Mx_R
thf(fact_788_qbs__Mx__R,axiom,
    ! [X: sigma_2308072346491277622l_real] :
      ( ( qbs_Mx2863504943393711152l_real @ ( measur3029786804310284173l_real @ X ) )
      = ( sigma_7998147297565726139l_real @ borel_5078946678739801102l_real @ X ) ) ).

% qbs_Mx_R
thf(fact_789_qbs__Mx__R,axiom,
    ! [X: sigma_5310753476256395226t_real] :
      ( ( qbs_Mx6884344600386557780t_real @ ( measur2011440102552004913t_real @ X ) )
      = ( sigma_2975298441655967327t_real @ borel_5078946678739801102l_real @ X ) ) ).

% qbs_Mx_R
thf(fact_790_ennreal_Oqbs__morphism__measurable__intro,axiom,
    ! [K2: extend8495563244428889912nnreal > extend8495563244428889912nnreal,Y: sigma_7234349610311085201nnreal] :
      ( ( member8329810500450651686nnreal @ K2 @ ( sigma_7926153774531450434nnreal @ borel_6524799422816628122nnreal @ Y ) )
     => ( member8329810500450651686nnreal @ K2 @ ( qbs_mo660571752308592106nnreal @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur2642298986910087140nnreal @ Y ) ) ) ) ).

% ennreal.qbs_morphism_measurable_intro
thf(fact_791_ennreal_Oqbs__morphism__measurable__intro,axiom,
    ! [K2: extend8495563244428889912nnreal > real,Y: sigma_measure_real] :
      ( ( member2874014351250825754l_real @ K2 @ ( sigma_7049758200512112822l_real @ borel_6524799422816628122nnreal @ Y ) )
     => ( member2874014351250825754l_real @ K2 @ ( qbs_mo8573344932208643166l_real @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur6875533127466166616s_real @ Y ) ) ) ) ).

% ennreal.qbs_morphism_measurable_intro
thf(fact_792_ennreal_Oqbs__morphism__measurable__intro,axiom,
    ! [K2: extend8495563244428889912nnreal > $o,Y: sigma_measure_o] :
      ( ( member8095236870201115968real_o @ K2 @ ( sigma_6279906219187228174real_o @ borel_6524799422816628122nnreal @ Y ) )
     => ( member8095236870201115968real_o @ K2 @ ( qbs_mo7845494039804091494real_o @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur2705496967258476524_qbs_o @ Y ) ) ) ) ).

% ennreal.qbs_morphism_measurable_intro
thf(fact_793_ennreal_Oqbs__morphism__measurable__intro,axiom,
    ! [K2: extend8495563244428889912nnreal > produc2422161461964618553l_real,Y: sigma_2308072346491277622l_real] :
      ( ( member7354208599470296673l_real @ K2 @ ( sigma_1014563338549229999l_real @ borel_6524799422816628122nnreal @ Y ) )
     => ( member7354208599470296673l_real @ K2 @ ( qbs_mo8045053474288007175l_real @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur3029786804310284173l_real @ Y ) ) ) ) ).

% ennreal.qbs_morphism_measurable_intro
thf(fact_794_ennreal_Oqbs__morphism__measurable__intro,axiom,
    ! [K2: extend8495563244428889912nnreal > produc7716430852924023517t_real,Y: sigma_5310753476256395226t_real] :
      ( ( member2197300837325509253t_real @ K2 @ ( sigma_2672489485171276883t_real @ borel_6524799422816628122nnreal @ Y ) )
     => ( member2197300837325509253t_real @ K2 @ ( qbs_mo3970737951806219947t_real @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur2011440102552004913t_real @ Y ) ) ) ) ).

% ennreal.qbs_morphism_measurable_intro
thf(fact_795_real_Oqbs__morphism__measurable__intro,axiom,
    ! [K2: real > nat,Y: sigma_measure_nat] :
      ( ( member_real_nat @ K2 @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ Y ) )
     => ( member_real_nat @ K2 @ ( qbs_mo6567951568834356598al_nat @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur4416158800429964412bs_nat @ Y ) ) ) ) ).

% real.qbs_morphism_measurable_intro
thf(fact_796_real_Oqbs__morphism__measurable__intro,axiom,
    ! [K2: real > complex,Y: sigma_3077487657436305159omplex] :
      ( ( member_real_complex @ K2 @ ( sigma_9111916201866572460omplex @ borel_5078946678739801102l_real @ Y ) )
     => ( member_real_complex @ K2 @ ( qbs_mo6067097710682130004omplex @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur1074055046195851610omplex @ Y ) ) ) ) ).

% real.qbs_morphism_measurable_intro
thf(fact_797_real_Oqbs__morphism__measurable__intro,axiom,
    ! [K2: real > extend8495563244428889912nnreal,Y: sigma_7234349610311085201nnreal] :
      ( ( member2919562650594848410nnreal @ K2 @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ Y ) )
     => ( member2919562650594848410nnreal @ K2 @ ( qbs_mo1317719164804411614nnreal @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur2642298986910087140nnreal @ Y ) ) ) ) ).

% real.qbs_morphism_measurable_intro
thf(fact_798_real_Oqbs__morphism__measurable__intro,axiom,
    ! [K2: real > real,Y: sigma_measure_real] :
      ( ( member_real_real @ K2 @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ Y ) )
     => ( member_real_real @ K2 @ ( qbs_mo5229770564518008146l_real @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur6875533127466166616s_real @ Y ) ) ) ) ).

% real.qbs_morphism_measurable_intro
thf(fact_799_real_Oqbs__morphism__measurable__intro,axiom,
    ! [K2: real > $o,Y: sigma_measure_o] :
      ( ( member_real_o @ K2 @ ( sigma_3939073009482781210real_o @ borel_5078946678739801102l_real @ Y ) )
     => ( member_real_o @ K2 @ ( qbs_morphism_real_o @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur2705496967258476524_qbs_o @ Y ) ) ) ) ).

% real.qbs_morphism_measurable_intro
thf(fact_800_real_Oqbs__morphism__measurable__intro,axiom,
    ! [K2: real > produc2422161461964618553l_real,Y: sigma_2308072346491277622l_real] :
      ( ( member9086635009091248365l_real @ K2 @ ( sigma_7998147297565726139l_real @ borel_5078946678739801102l_real @ Y ) )
     => ( member9086635009091248365l_real @ K2 @ ( qbs_mo6845038431372961811l_real @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur3029786804310284173l_real @ Y ) ) ) ) ).

% real.qbs_morphism_measurable_intro
thf(fact_801_real_Oqbs__morphism__measurable__intro,axiom,
    ! [K2: real > produc7716430852924023517t_real,Y: sigma_5310753476256395226t_real] :
      ( ( member969486235890772753t_real @ K2 @ ( sigma_2975298441655967327t_real @ borel_5078946678739801102l_real @ Y ) )
     => ( member969486235890772753t_real @ K2 @ ( qbs_mo3601394221918002359t_real @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur2011440102552004913t_real @ Y ) ) ) ) ).

% real.qbs_morphism_measurable_intro
thf(fact_802_ennreal_Oqbs__morphism__dest,axiom,
    ! [K2: extend8495563244428889912nnreal > extend8495563244428889912nnreal,Y: sigma_7234349610311085201nnreal] :
      ( ( member8329810500450651686nnreal @ K2 @ ( qbs_mo660571752308592106nnreal @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur2642298986910087140nnreal @ Y ) ) )
     => ( member8329810500450651686nnreal @ K2 @ ( sigma_7926153774531450434nnreal @ borel_6524799422816628122nnreal @ Y ) ) ) ).

% ennreal.qbs_morphism_dest
thf(fact_803_ennreal_Oqbs__morphism__dest,axiom,
    ! [K2: extend8495563244428889912nnreal > real,Y: sigma_measure_real] :
      ( ( member2874014351250825754l_real @ K2 @ ( qbs_mo8573344932208643166l_real @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur6875533127466166616s_real @ Y ) ) )
     => ( member2874014351250825754l_real @ K2 @ ( sigma_7049758200512112822l_real @ borel_6524799422816628122nnreal @ Y ) ) ) ).

% ennreal.qbs_morphism_dest
thf(fact_804_ennreal_Oqbs__morphism__dest,axiom,
    ! [K2: extend8495563244428889912nnreal > $o,Y: sigma_measure_o] :
      ( ( member8095236870201115968real_o @ K2 @ ( qbs_mo7845494039804091494real_o @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur2705496967258476524_qbs_o @ Y ) ) )
     => ( member8095236870201115968real_o @ K2 @ ( sigma_6279906219187228174real_o @ borel_6524799422816628122nnreal @ Y ) ) ) ).

% ennreal.qbs_morphism_dest
thf(fact_805_ennreal_Oqbs__morphism__dest,axiom,
    ! [K2: extend8495563244428889912nnreal > produc2422161461964618553l_real,Y: sigma_2308072346491277622l_real] :
      ( ( member7354208599470296673l_real @ K2 @ ( qbs_mo8045053474288007175l_real @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur3029786804310284173l_real @ Y ) ) )
     => ( member7354208599470296673l_real @ K2 @ ( sigma_1014563338549229999l_real @ borel_6524799422816628122nnreal @ Y ) ) ) ).

% ennreal.qbs_morphism_dest
thf(fact_806_ennreal_Oqbs__morphism__dest,axiom,
    ! [K2: extend8495563244428889912nnreal > produc7716430852924023517t_real,Y: sigma_5310753476256395226t_real] :
      ( ( member2197300837325509253t_real @ K2 @ ( qbs_mo3970737951806219947t_real @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur2011440102552004913t_real @ Y ) ) )
     => ( member2197300837325509253t_real @ K2 @ ( sigma_2672489485171276883t_real @ borel_6524799422816628122nnreal @ Y ) ) ) ).

% ennreal.qbs_morphism_dest
thf(fact_807_ennreal_Ostandard__borel__r__full__faithful,axiom,
    ! [Y: sigma_7234349610311085201nnreal] :
      ( ( sigma_7926153774531450434nnreal @ borel_6524799422816628122nnreal @ Y )
      = ( qbs_mo660571752308592106nnreal @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur2642298986910087140nnreal @ Y ) ) ) ).

% ennreal.standard_borel_r_full_faithful
thf(fact_808_ennreal_Ostandard__borel__r__full__faithful,axiom,
    ! [Y: sigma_measure_real] :
      ( ( sigma_7049758200512112822l_real @ borel_6524799422816628122nnreal @ Y )
      = ( qbs_mo8573344932208643166l_real @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur6875533127466166616s_real @ Y ) ) ) ).

% ennreal.standard_borel_r_full_faithful
thf(fact_809_ennreal_Ostandard__borel__r__full__faithful,axiom,
    ! [Y: sigma_measure_o] :
      ( ( sigma_6279906219187228174real_o @ borel_6524799422816628122nnreal @ Y )
      = ( qbs_mo7845494039804091494real_o @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur2705496967258476524_qbs_o @ Y ) ) ) ).

% ennreal.standard_borel_r_full_faithful
thf(fact_810_ennreal_Ostandard__borel__r__full__faithful,axiom,
    ! [Y: sigma_2308072346491277622l_real] :
      ( ( sigma_1014563338549229999l_real @ borel_6524799422816628122nnreal @ Y )
      = ( qbs_mo8045053474288007175l_real @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur3029786804310284173l_real @ Y ) ) ) ).

% ennreal.standard_borel_r_full_faithful
thf(fact_811_ennreal_Ostandard__borel__r__full__faithful,axiom,
    ! [Y: sigma_5310753476256395226t_real] :
      ( ( sigma_2672489485171276883t_real @ borel_6524799422816628122nnreal @ Y )
      = ( qbs_mo3970737951806219947t_real @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur2011440102552004913t_real @ Y ) ) ) ).

% ennreal.standard_borel_r_full_faithful
thf(fact_812_real_Ostandard__borel__r__full__faithful,axiom,
    ! [Y: sigma_measure_nat] :
      ( ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ Y )
      = ( qbs_mo6567951568834356598al_nat @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur4416158800429964412bs_nat @ Y ) ) ) ).

% real.standard_borel_r_full_faithful
thf(fact_813_real_Ostandard__borel__r__full__faithful,axiom,
    ! [Y: sigma_3077487657436305159omplex] :
      ( ( sigma_9111916201866572460omplex @ borel_5078946678739801102l_real @ Y )
      = ( qbs_mo6067097710682130004omplex @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur1074055046195851610omplex @ Y ) ) ) ).

% real.standard_borel_r_full_faithful
thf(fact_814_real_Ostandard__borel__r__full__faithful,axiom,
    ! [Y: sigma_7234349610311085201nnreal] :
      ( ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ Y )
      = ( qbs_mo1317719164804411614nnreal @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur2642298986910087140nnreal @ Y ) ) ) ).

% real.standard_borel_r_full_faithful
thf(fact_815_real_Ostandard__borel__r__full__faithful,axiom,
    ! [Y: sigma_measure_real] :
      ( ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ Y )
      = ( qbs_mo5229770564518008146l_real @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur6875533127466166616s_real @ Y ) ) ) ).

% real.standard_borel_r_full_faithful
thf(fact_816_real_Ostandard__borel__r__full__faithful,axiom,
    ! [Y: sigma_measure_o] :
      ( ( sigma_3939073009482781210real_o @ borel_5078946678739801102l_real @ Y )
      = ( qbs_morphism_real_o @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur2705496967258476524_qbs_o @ Y ) ) ) ).

% real.standard_borel_r_full_faithful
thf(fact_817_real_Ostandard__borel__r__full__faithful,axiom,
    ! [Y: sigma_2308072346491277622l_real] :
      ( ( sigma_7998147297565726139l_real @ borel_5078946678739801102l_real @ Y )
      = ( qbs_mo6845038431372961811l_real @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur3029786804310284173l_real @ Y ) ) ) ).

% real.standard_borel_r_full_faithful
thf(fact_818_real_Ostandard__borel__r__full__faithful,axiom,
    ! [Y: sigma_5310753476256395226t_real] :
      ( ( sigma_2975298441655967327t_real @ borel_5078946678739801102l_real @ Y )
      = ( qbs_mo3601394221918002359t_real @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur2011440102552004913t_real @ Y ) ) ) ).

% real.standard_borel_r_full_faithful
thf(fact_819_real_Oqbs__morphism__dest,axiom,
    ! [K2: real > nat,Y: sigma_measure_nat] :
      ( ( member_real_nat @ K2 @ ( qbs_mo6567951568834356598al_nat @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur4416158800429964412bs_nat @ Y ) ) )
     => ( member_real_nat @ K2 @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ Y ) ) ) ).

% real.qbs_morphism_dest
thf(fact_820_real_Oqbs__morphism__dest,axiom,
    ! [K2: real > complex,Y: sigma_3077487657436305159omplex] :
      ( ( member_real_complex @ K2 @ ( qbs_mo6067097710682130004omplex @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur1074055046195851610omplex @ Y ) ) )
     => ( member_real_complex @ K2 @ ( sigma_9111916201866572460omplex @ borel_5078946678739801102l_real @ Y ) ) ) ).

% real.qbs_morphism_dest
thf(fact_821_real_Oqbs__morphism__dest,axiom,
    ! [K2: real > extend8495563244428889912nnreal,Y: sigma_7234349610311085201nnreal] :
      ( ( member2919562650594848410nnreal @ K2 @ ( qbs_mo1317719164804411614nnreal @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur2642298986910087140nnreal @ Y ) ) )
     => ( member2919562650594848410nnreal @ K2 @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ Y ) ) ) ).

% real.qbs_morphism_dest
thf(fact_822_real_Oqbs__morphism__dest,axiom,
    ! [K2: real > real,Y: sigma_measure_real] :
      ( ( member_real_real @ K2 @ ( qbs_mo5229770564518008146l_real @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur6875533127466166616s_real @ Y ) ) )
     => ( member_real_real @ K2 @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ Y ) ) ) ).

% real.qbs_morphism_dest
thf(fact_823_real_Oqbs__morphism__dest,axiom,
    ! [K2: real > $o,Y: sigma_measure_o] :
      ( ( member_real_o @ K2 @ ( qbs_morphism_real_o @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur2705496967258476524_qbs_o @ Y ) ) )
     => ( member_real_o @ K2 @ ( sigma_3939073009482781210real_o @ borel_5078946678739801102l_real @ Y ) ) ) ).

% real.qbs_morphism_dest
thf(fact_824_real_Oqbs__morphism__dest,axiom,
    ! [K2: real > produc2422161461964618553l_real,Y: sigma_2308072346491277622l_real] :
      ( ( member9086635009091248365l_real @ K2 @ ( qbs_mo6845038431372961811l_real @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur3029786804310284173l_real @ Y ) ) )
     => ( member9086635009091248365l_real @ K2 @ ( sigma_7998147297565726139l_real @ borel_5078946678739801102l_real @ Y ) ) ) ).

% real.qbs_morphism_dest
thf(fact_825_real_Oqbs__morphism__dest,axiom,
    ! [K2: real > produc7716430852924023517t_real,Y: sigma_5310753476256395226t_real] :
      ( ( member969486235890772753t_real @ K2 @ ( qbs_mo3601394221918002359t_real @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur2011440102552004913t_real @ Y ) ) )
     => ( member969486235890772753t_real @ K2 @ ( sigma_2975298441655967327t_real @ borel_5078946678739801102l_real @ Y ) ) ) ).

% real.qbs_morphism_dest
thf(fact_826_r__preserves__morphisms,axiom,
    ! [X: sigma_measure_b,Y: sigma_7234349610311085201nnreal] : ( ord_le672203391976590760nnreal @ ( sigma_6334800283702579687nnreal @ X @ Y ) @ ( qbs_mo7554306597560579135nnreal @ ( measur6507891955840068947_qbs_b @ X ) @ ( measur2642298986910087140nnreal @ Y ) ) ) ).

% r_preserves_morphisms
thf(fact_827_r__preserves__morphisms,axiom,
    ! [X: sigma_measure_a,Y: sigma_7234349610311085201nnreal] : ( ord_le1007445205377960487nnreal @ ( sigma_214952329563889126nnreal @ X @ Y ) @ ( qbs_mo1434458643421888574nnreal @ ( measur6507891955840068946_qbs_a @ X ) @ ( measur2642298986910087140nnreal @ Y ) ) ) ).

% r_preserves_morphisms
thf(fact_828_r__preserves__morphisms,axiom,
    ! [X: sigma_measure_nat,Y: sigma_measure_real] : ( ord_le2908806416726583473t_real @ ( sigma_1747752005702207822t_real @ X @ Y ) @ ( qbs_mo2000642995705457910t_real @ ( measur4416158800429964412bs_nat @ X ) @ ( measur6875533127466166616s_real @ Y ) ) ) ).

% r_preserves_morphisms
thf(fact_829_r__preserves__morphisms,axiom,
    ! [X: sigma_measure_a,Y: sigma_measure_real] : ( ord_le3334967407727675675a_real @ ( sigma_9116425665531756122a_real @ X @ Y ) @ ( qbs_morphism_a_real @ ( measur6507891955840068946_qbs_a @ X ) @ ( measur6875533127466166616s_real @ Y ) ) ) ).

% r_preserves_morphisms
thf(fact_830_r__preserves__morphisms,axiom,
    ! [X: sigma_7234349610311085201nnreal,Y: sigma_7234349610311085201nnreal] : ( ord_le2847260637007690789nnreal @ ( sigma_7926153774531450434nnreal @ X @ Y ) @ ( qbs_mo660571752308592106nnreal @ ( measur2642298986910087140nnreal @ X ) @ ( measur2642298986910087140nnreal @ Y ) ) ) ).

% r_preserves_morphisms
thf(fact_831_r__preserves__morphisms,axiom,
    ! [X: sigma_7234349610311085201nnreal,Y: sigma_measure_real] : ( ord_le2792513217584188441l_real @ ( sigma_7049758200512112822l_real @ X @ Y ) @ ( qbs_mo8573344932208643166l_real @ ( measur2642298986910087140nnreal @ X ) @ ( measur6875533127466166616s_real @ Y ) ) ) ).

% r_preserves_morphisms
thf(fact_832_r__preserves__morphisms,axiom,
    ! [X: sigma_7234349610311085201nnreal,Y: sigma_measure_o] : ( ord_le5553135326598321815real_o @ ( sigma_6279906219187228174real_o @ X @ Y ) @ ( qbs_mo7845494039804091494real_o @ ( measur2642298986910087140nnreal @ X ) @ ( measur2705496967258476524_qbs_o @ Y ) ) ) ).

% r_preserves_morphisms
thf(fact_833_r__preserves__morphisms,axiom,
    ! [X: sigma_measure_real,Y: sigma_measure_nat] : ( ord_le6098800555920186673al_nat @ ( sigma_6315060578831106510al_nat @ X @ Y ) @ ( qbs_mo6567951568834356598al_nat @ ( measur6875533127466166616s_real @ X ) @ ( measur4416158800429964412bs_nat @ Y ) ) ) ).

% r_preserves_morphisms
thf(fact_834_r__preserves__morphisms,axiom,
    ! [X: sigma_measure_real,Y: sigma_3077487657436305159omplex] : ( ord_le2047140485929309711omplex @ ( sigma_9111916201866572460omplex @ X @ Y ) @ ( qbs_mo6067097710682130004omplex @ ( measur6875533127466166616s_real @ X ) @ ( measur1074055046195851610omplex @ Y ) ) ) ).

% r_preserves_morphisms
thf(fact_835_r__preserves__morphisms,axiom,
    ! [X: sigma_measure_real,Y: sigma_7234349610311085201nnreal] : ( ord_le2462468573666744473nnreal @ ( sigma_9017504469962657078nnreal @ X @ Y ) @ ( qbs_mo1317719164804411614nnreal @ ( measur6875533127466166616s_real @ X ) @ ( measur2642298986910087140nnreal @ Y ) ) ) ).

% r_preserves_morphisms
thf(fact_836_bool_Oqbs__morphism__measurable__intro,axiom,
    ! [K2: $o > extend8495563244428889912nnreal,Y: sigma_7234349610311085201nnreal] :
      ( ( member5265953103328284778nnreal @ K2 @ ( sigma_6459699357617223168nnreal @ borel_5500255247093592246orel_o @ Y ) )
     => ( member5265953103328284778nnreal @ K2 @ ( qbs_mo8025287178234086488nnreal @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur2642298986910087140nnreal @ Y ) ) ) ) ).

% bool.qbs_morphism_measurable_intro
thf(fact_837_bool_Oqbs__morphism__measurable__intro,axiom,
    ! [K2: $o > real,Y: sigma_measure_real] :
      ( ( member_o_real @ K2 @ ( sigma_2430008634441611636o_real @ borel_5500255247093592246orel_o @ Y ) )
     => ( member_o_real @ K2 @ ( qbs_morphism_o_real @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur6875533127466166616s_real @ Y ) ) ) ) ).

% bool.qbs_morphism_measurable_intro
thf(fact_838_bool_Oqbs__morphism__measurable__intro,axiom,
    ! [K2: $o > $o,Y: sigma_measure_o] :
      ( ( member_o_o @ K2 @ ( sigma_measurable_o_o @ borel_5500255247093592246orel_o @ Y ) )
     => ( member_o_o @ K2 @ ( qbs_morphism_o_o @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur2705496967258476524_qbs_o @ Y ) ) ) ) ).

% bool.qbs_morphism_measurable_intro
thf(fact_839_bool_Oqbs__morphism__measurable__intro,axiom,
    ! [K2: $o > produc2422161461964618553l_real,Y: sigma_2308072346491277622l_real] :
      ( ( member8602590400058854173l_real @ K2 @ ( sigma_2489502367751881329l_real @ borel_5500255247093592246orel_o @ Y ) )
     => ( member8602590400058854173l_real @ K2 @ ( qbs_mo6076292073755557401l_real @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur3029786804310284173l_real @ Y ) ) ) ) ).

% bool.qbs_morphism_measurable_intro
thf(fact_840_bool_Oqbs__morphism__measurable__intro,axiom,
    ! [K2: $o > produc7716430852924023517t_real,Y: sigma_5310753476256395226t_real] :
      ( ( member8739982677887872321t_real @ K2 @ ( sigma_680772226823374357t_real @ borel_5500255247093592246orel_o @ Y ) )
     => ( member8739982677887872321t_real @ K2 @ ( qbs_mo928686258813670845t_real @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur2011440102552004913t_real @ Y ) ) ) ) ).

% bool.qbs_morphism_measurable_intro
thf(fact_841_nat_Oqbs__morphism__measurable__intro,axiom,
    ! [K2: nat > extend8495563244428889912nnreal,Y: sigma_7234349610311085201nnreal] :
      ( ( member8283130129095025342nnreal @ K2 @ ( sigma_6306161311797543642nnreal @ borel_8449730974584783410el_nat @ Y ) )
     => ( member8283130129095025342nnreal @ K2 @ ( qbs_mo4632421585747671682nnreal @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur2642298986910087140nnreal @ Y ) ) ) ) ).

% nat.qbs_morphism_measurable_intro
thf(fact_842_nat_Oqbs__morphism__measurable__intro,axiom,
    ! [K2: nat > real,Y: sigma_measure_real] :
      ( ( member_nat_real @ K2 @ ( sigma_1747752005702207822t_real @ borel_8449730974584783410el_nat @ Y ) )
     => ( member_nat_real @ K2 @ ( qbs_mo2000642995705457910t_real @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur6875533127466166616s_real @ Y ) ) ) ) ).

% nat.qbs_morphism_measurable_intro
thf(fact_843_nat_Oqbs__morphism__measurable__intro,axiom,
    ! [K2: nat > $o,Y: sigma_measure_o] :
      ( ( member_nat_o @ K2 @ ( sigma_5101835498682829686_nat_o @ borel_8449730974584783410el_nat @ Y ) )
     => ( member_nat_o @ K2 @ ( qbs_morphism_nat_o @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur2705496967258476524_qbs_o @ Y ) ) ) ) ).

% nat.qbs_morphism_measurable_intro
thf(fact_844_nat_Oqbs__morphism__measurable__intro,axiom,
    ! [K2: nat > produc2422161461964618553l_real,Y: sigma_2308072346491277622l_real] :
      ( ( member4944736856719700937l_real @ K2 @ ( sigma_5893925349391517463l_real @ borel_8449730974584783410el_nat @ Y ) )
     => ( member4944736856719700937l_real @ K2 @ ( qbs_mo6520021129653552495l_real @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur3029786804310284173l_real @ Y ) ) ) ) ).

% nat.qbs_morphism_measurable_intro
thf(fact_845_nat_Oqbs__morphism__measurable__intro,axiom,
    ! [K2: nat > produc7716430852924023517t_real,Y: sigma_5310753476256395226t_real] :
      ( ( member4237364305948650477t_real @ K2 @ ( sigma_8750250338757262779t_real @ borel_8449730974584783410el_nat @ Y ) )
     => ( member4237364305948650477t_real @ K2 @ ( qbs_mo2896996399151425555t_real @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur2011440102552004913t_real @ Y ) ) ) ) ).

% nat.qbs_morphism_measurable_intro
thf(fact_846_qbs__Mx__is__morphisms,axiom,
    ( qbs_Mx_b
    = ( qbs_morphism_real_b @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) ) ) ).

% qbs_Mx_is_morphisms
thf(fact_847_qbs__Mx__is__morphisms,axiom,
    ( qbs_Mx_a
    = ( qbs_morphism_real_a @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) ) ) ).

% qbs_Mx_is_morphisms
thf(fact_848_bool_Ostandard__borel__r__full__faithful,axiom,
    ! [Y: sigma_7234349610311085201nnreal] :
      ( ( sigma_6459699357617223168nnreal @ borel_5500255247093592246orel_o @ Y )
      = ( qbs_mo8025287178234086488nnreal @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur2642298986910087140nnreal @ Y ) ) ) ).

% bool.standard_borel_r_full_faithful
thf(fact_849_bool_Ostandard__borel__r__full__faithful,axiom,
    ! [Y: sigma_measure_real] :
      ( ( sigma_2430008634441611636o_real @ borel_5500255247093592246orel_o @ Y )
      = ( qbs_morphism_o_real @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur6875533127466166616s_real @ Y ) ) ) ).

% bool.standard_borel_r_full_faithful
thf(fact_850_bool_Ostandard__borel__r__full__faithful,axiom,
    ! [Y: sigma_measure_o] :
      ( ( sigma_measurable_o_o @ borel_5500255247093592246orel_o @ Y )
      = ( qbs_morphism_o_o @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur2705496967258476524_qbs_o @ Y ) ) ) ).

% bool.standard_borel_r_full_faithful
thf(fact_851_bool_Ostandard__borel__r__full__faithful,axiom,
    ! [Y: sigma_2308072346491277622l_real] :
      ( ( sigma_2489502367751881329l_real @ borel_5500255247093592246orel_o @ Y )
      = ( qbs_mo6076292073755557401l_real @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur3029786804310284173l_real @ Y ) ) ) ).

% bool.standard_borel_r_full_faithful
thf(fact_852_bool_Ostandard__borel__r__full__faithful,axiom,
    ! [Y: sigma_5310753476256395226t_real] :
      ( ( sigma_680772226823374357t_real @ borel_5500255247093592246orel_o @ Y )
      = ( qbs_mo928686258813670845t_real @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur2011440102552004913t_real @ Y ) ) ) ).

% bool.standard_borel_r_full_faithful
thf(fact_853_nat_Ostandard__borel__r__full__faithful,axiom,
    ! [Y: sigma_7234349610311085201nnreal] :
      ( ( sigma_6306161311797543642nnreal @ borel_8449730974584783410el_nat @ Y )
      = ( qbs_mo4632421585747671682nnreal @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur2642298986910087140nnreal @ Y ) ) ) ).

% nat.standard_borel_r_full_faithful
thf(fact_854_nat_Ostandard__borel__r__full__faithful,axiom,
    ! [Y: sigma_measure_real] :
      ( ( sigma_1747752005702207822t_real @ borel_8449730974584783410el_nat @ Y )
      = ( qbs_mo2000642995705457910t_real @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur6875533127466166616s_real @ Y ) ) ) ).

% nat.standard_borel_r_full_faithful
thf(fact_855_nat_Ostandard__borel__r__full__faithful,axiom,
    ! [Y: sigma_measure_o] :
      ( ( sigma_5101835498682829686_nat_o @ borel_8449730974584783410el_nat @ Y )
      = ( qbs_morphism_nat_o @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur2705496967258476524_qbs_o @ Y ) ) ) ).

% nat.standard_borel_r_full_faithful
thf(fact_856_nat_Ostandard__borel__r__full__faithful,axiom,
    ! [Y: sigma_2308072346491277622l_real] :
      ( ( sigma_5893925349391517463l_real @ borel_8449730974584783410el_nat @ Y )
      = ( qbs_mo6520021129653552495l_real @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur3029786804310284173l_real @ Y ) ) ) ).

% nat.standard_borel_r_full_faithful
thf(fact_857_nat_Ostandard__borel__r__full__faithful,axiom,
    ! [Y: sigma_5310753476256395226t_real] :
      ( ( sigma_8750250338757262779t_real @ borel_8449730974584783410el_nat @ Y )
      = ( qbs_mo2896996399151425555t_real @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur2011440102552004913t_real @ Y ) ) ) ).

% nat.standard_borel_r_full_faithful
thf(fact_858_bool_Oqbs__morphism__dest,axiom,
    ! [K2: $o > extend8495563244428889912nnreal,Y: sigma_7234349610311085201nnreal] :
      ( ( member5265953103328284778nnreal @ K2 @ ( qbs_mo8025287178234086488nnreal @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur2642298986910087140nnreal @ Y ) ) )
     => ( member5265953103328284778nnreal @ K2 @ ( sigma_6459699357617223168nnreal @ borel_5500255247093592246orel_o @ Y ) ) ) ).

% bool.qbs_morphism_dest
thf(fact_859_bool_Oqbs__morphism__dest,axiom,
    ! [K2: $o > real,Y: sigma_measure_real] :
      ( ( member_o_real @ K2 @ ( qbs_morphism_o_real @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur6875533127466166616s_real @ Y ) ) )
     => ( member_o_real @ K2 @ ( sigma_2430008634441611636o_real @ borel_5500255247093592246orel_o @ Y ) ) ) ).

% bool.qbs_morphism_dest
thf(fact_860_bool_Oqbs__morphism__dest,axiom,
    ! [K2: $o > $o,Y: sigma_measure_o] :
      ( ( member_o_o @ K2 @ ( qbs_morphism_o_o @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur2705496967258476524_qbs_o @ Y ) ) )
     => ( member_o_o @ K2 @ ( sigma_measurable_o_o @ borel_5500255247093592246orel_o @ Y ) ) ) ).

% bool.qbs_morphism_dest
thf(fact_861_bool_Oqbs__morphism__dest,axiom,
    ! [K2: $o > produc2422161461964618553l_real,Y: sigma_2308072346491277622l_real] :
      ( ( member8602590400058854173l_real @ K2 @ ( qbs_mo6076292073755557401l_real @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur3029786804310284173l_real @ Y ) ) )
     => ( member8602590400058854173l_real @ K2 @ ( sigma_2489502367751881329l_real @ borel_5500255247093592246orel_o @ Y ) ) ) ).

% bool.qbs_morphism_dest
thf(fact_862_bool_Oqbs__morphism__dest,axiom,
    ! [K2: $o > produc7716430852924023517t_real,Y: sigma_5310753476256395226t_real] :
      ( ( member8739982677887872321t_real @ K2 @ ( qbs_mo928686258813670845t_real @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur2011440102552004913t_real @ Y ) ) )
     => ( member8739982677887872321t_real @ K2 @ ( sigma_680772226823374357t_real @ borel_5500255247093592246orel_o @ Y ) ) ) ).

% bool.qbs_morphism_dest
thf(fact_863_nat_Oqbs__morphism__dest,axiom,
    ! [K2: nat > extend8495563244428889912nnreal,Y: sigma_7234349610311085201nnreal] :
      ( ( member8283130129095025342nnreal @ K2 @ ( qbs_mo4632421585747671682nnreal @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur2642298986910087140nnreal @ Y ) ) )
     => ( member8283130129095025342nnreal @ K2 @ ( sigma_6306161311797543642nnreal @ borel_8449730974584783410el_nat @ Y ) ) ) ).

% nat.qbs_morphism_dest
thf(fact_864_nat_Oqbs__morphism__dest,axiom,
    ! [K2: nat > real,Y: sigma_measure_real] :
      ( ( member_nat_real @ K2 @ ( qbs_mo2000642995705457910t_real @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur6875533127466166616s_real @ Y ) ) )
     => ( member_nat_real @ K2 @ ( sigma_1747752005702207822t_real @ borel_8449730974584783410el_nat @ Y ) ) ) ).

% nat.qbs_morphism_dest
thf(fact_865_nat_Oqbs__morphism__dest,axiom,
    ! [K2: nat > $o,Y: sigma_measure_o] :
      ( ( member_nat_o @ K2 @ ( qbs_morphism_nat_o @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur2705496967258476524_qbs_o @ Y ) ) )
     => ( member_nat_o @ K2 @ ( sigma_5101835498682829686_nat_o @ borel_8449730974584783410el_nat @ Y ) ) ) ).

% nat.qbs_morphism_dest
thf(fact_866_nat_Oqbs__morphism__dest,axiom,
    ! [K2: nat > produc2422161461964618553l_real,Y: sigma_2308072346491277622l_real] :
      ( ( member4944736856719700937l_real @ K2 @ ( qbs_mo6520021129653552495l_real @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur3029786804310284173l_real @ Y ) ) )
     => ( member4944736856719700937l_real @ K2 @ ( sigma_5893925349391517463l_real @ borel_8449730974584783410el_nat @ Y ) ) ) ).

% nat.qbs_morphism_dest
thf(fact_867_nat_Oqbs__morphism__dest,axiom,
    ! [K2: nat > produc7716430852924023517t_real,Y: sigma_5310753476256395226t_real] :
      ( ( member4237364305948650477t_real @ K2 @ ( qbs_mo2896996399151425555t_real @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur2011440102552004913t_real @ Y ) ) )
     => ( member4237364305948650477t_real @ K2 @ ( sigma_8750250338757262779t_real @ borel_8449730974584783410el_nat @ Y ) ) ) ).

% nat.qbs_morphism_dest
thf(fact_868_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_869_R__correct,axiom,
    ! [X: sigma_measure_nat] :
      ( ( quasi_2638801612224710361el_nat @ ( measur4416158800429964412bs_nat @ X ) )
      = ( produc3843710909396041968al_nat @ ( sigma_space_nat @ X ) @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ X ) ) ) ).

% R_correct
thf(fact_870_R__correct,axiom,
    ! [X: sigma_3077487657436305159omplex] :
      ( ( quasi_5592172293381431735omplex @ ( measur1074055046195851610omplex @ X ) )
      = ( produc5917795231952384940omplex @ ( sigma_space_complex @ X ) @ ( sigma_9111916201866572460omplex @ borel_5078946678739801102l_real @ X ) ) ) ).

% R_correct
thf(fact_871_R__correct,axiom,
    ! [X: sigma_7234349610311085201nnreal] :
      ( ( quasi_5951729137868634689nnreal @ ( measur2642298986910087140nnreal @ X ) )
      = ( produc5490069393840030656nnreal @ ( sigma_3147302497200244656nnreal @ X ) @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ X ) ) ) ).

% R_correct
thf(fact_872_R__correct,axiom,
    ! [X: sigma_measure_real] :
      ( ( quasi_4307383193256703285l_real @ ( measur6875533127466166616s_real @ X ) )
      = ( produc3470952364605845544l_real @ ( sigma_space_real @ X ) @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ X ) ) ) ).

% R_correct
thf(fact_873_R__correct,axiom,
    ! [X: sigma_measure_o] :
      ( ( quasi_3788804942992426063orel_o @ ( measur2705496967258476524_qbs_o @ X ) )
      = ( produc2133903995702845986real_o @ ( sigma_space_o @ X ) @ ( sigma_3939073009482781210real_o @ borel_5078946678739801102l_real @ X ) ) ) ).

% R_correct
thf(fact_874_R__correct,axiom,
    ! [X: sigma_2308072346491277622l_real] :
      ( ( quasi_2376713246294942192l_real @ ( measur3029786804310284173l_real @ X ) )
      = ( produc5420172574571962722l_real @ ( sigma_2519298556039103681l_real @ X ) @ ( sigma_7998147297565726139l_real @ borel_5078946678739801102l_real @ X ) ) ) ).

% R_correct
thf(fact_875_R__correct,axiom,
    ! [X: sigma_5310753476256395226t_real] :
      ( ( quasi_1341009766961362196t_real @ ( measur2011440102552004913t_real @ X ) )
      = ( produc2405820699097220962t_real @ ( sigma_2177939267068080229t_real @ X ) @ ( sigma_2975298441655967327t_real @ borel_5078946678739801102l_real @ X ) ) ) ).

% R_correct
thf(fact_876_qp_Oprob__space__completion,axiom,
    probab535871623910865577e_real @ ( comple3506806835435775778n_real @ mu ) ).

% qp.prob_space_completion
thf(fact_877_qp_Ofinite__measure__distr,axiom,
    ! [F: real > a,M3: sigma_measure_a] :
      ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ mu @ M3 ) )
     => ( measur930452917991658466sure_a @ ( measure_distr_real_a @ mu @ M3 @ F ) ) ) ).

% qp.finite_measure_distr
thf(fact_878_qp_Ofinite__measure__distr,axiom,
    ! [F: real > $o,M3: sigma_measure_o] :
      ( ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ mu @ M3 ) )
     => ( measur2447921437955784316sure_o @ ( measure_distr_real_o @ mu @ M3 @ F ) ) ) ).

% qp.finite_measure_distr
thf(fact_879_qp_Ofinite__measure__distr,axiom,
    ! [F: real > nat,M3: sigma_measure_nat] :
      ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ mu @ M3 ) )
     => ( measur8338831127414845932re_nat @ ( measur254523008267340406al_nat @ mu @ M3 @ F ) ) ) ).

% qp.finite_measure_distr
thf(fact_880_qp_Ofinite__measure__distr,axiom,
    ! [F: real > extend8495563244428889912nnreal,M3: sigma_7234349610311085201nnreal] :
      ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ mu @ M3 ) )
     => ( measur8478876643349974356nnreal @ ( measur8829990298702910942nnreal @ mu @ M3 @ F ) ) ) ).

% qp.finite_measure_distr
thf(fact_881_qp_Ofinite__measure__distr,axiom,
    ! [F: real > complex,M3: sigma_3077487657436305159omplex] :
      ( ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ mu @ M3 ) )
     => ( measur5795638154895664842omplex @ ( measur1621797640479583060omplex @ mu @ M3 @ F ) ) ) ).

% qp.finite_measure_distr
thf(fact_882_qp_Ofinite__measure__distr,axiom,
    ! [F: real > real,M3: sigma_measure_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ mu @ M3 ) )
     => ( measur3606880022600206024e_real @ ( measur2993149975067245138l_real @ mu @ M3 @ F ) ) ) ).

% qp.finite_measure_distr
thf(fact_883_qp_Oprob__space__distr,axiom,
    ! [F: real > a,M3: sigma_measure_a] :
      ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ mu @ M3 ) )
     => ( probab7247484486040049089pace_a @ ( measure_distr_real_a @ mu @ M3 @ F ) ) ) ).

% qp.prob_space_distr
thf(fact_884_qp_Oprob__space__distr,axiom,
    ! [F: real > $o,M3: sigma_measure_o] :
      ( ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ mu @ M3 ) )
     => ( probab1190487603588612059pace_o @ ( measure_distr_real_o @ mu @ M3 @ F ) ) ) ).

% qp.prob_space_distr
thf(fact_885_qp_Oprob__space__distr,axiom,
    ! [F: real > nat,M3: sigma_measure_nat] :
      ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ mu @ M3 ) )
     => ( probab2904919403188438605ce_nat @ ( measur254523008267340406al_nat @ mu @ M3 @ F ) ) ) ).

% qp.prob_space_distr
thf(fact_886_qp_Oprob__space__distr,axiom,
    ! [F: real > extend8495563244428889912nnreal,M3: sigma_7234349610311085201nnreal] :
      ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ mu @ M3 ) )
     => ( probab6612481188548237749nnreal @ ( measur8829990298702910942nnreal @ mu @ M3 @ F ) ) ) ).

% qp.prob_space_distr
thf(fact_887_qp_Oprob__space__distr,axiom,
    ! [F: real > complex,M3: sigma_3077487657436305159omplex] :
      ( ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ mu @ M3 ) )
     => ( probab6149883331606624555omplex @ ( measur1621797640479583060omplex @ mu @ M3 @ F ) ) ) ).

% qp.prob_space_distr
thf(fact_888_qp_Oprob__space__distr,axiom,
    ! [F: real > real,M3: sigma_measure_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ mu @ M3 ) )
     => ( probab535871623910865577e_real @ ( measur2993149975067245138l_real @ mu @ M3 @ F ) ) ) ).

% qp.prob_space_distr
thf(fact_889_finite__measure_Osigma__finite__measure,axiom,
    ! [M: sigma_measure_real] :
      ( ( measur3606880022600206024e_real @ M )
     => ( measur487378040549452491e_real @ M ) ) ).

% finite_measure.sigma_finite_measure
thf(fact_890_real_Oexist__fg,axiom,
    ? [X3: real > real] :
      ( ( member_real_real @ X3 @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
      & ? [Xa: real > real] :
          ( ( member_real_real @ Xa @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
          & ! [Xb: real] :
              ( ( member_real @ Xb @ ( sigma_space_real @ borel_5078946678739801102l_real ) )
             => ( ( comp_real_real_real @ Xa @ X3 @ Xb )
                = Xb ) ) ) ) ).

% real.exist_fg
thf(fact_891_qp_Oreal__distribution__distr,axiom,
    ! [X: real > real] :
      ( ( member_real_real @ X @ ( sigma_5267869275261027754l_real @ mu @ borel_5078946678739801102l_real ) )
     => ( distri2809703520229113005bution @ ( measur2993149975067245138l_real @ mu @ borel_5078946678739801102l_real @ X ) ) ) ).

% qp.real_distribution_distr
thf(fact_892_nat_Oexist__fg,axiom,
    ? [X3: nat > real] :
      ( ( member_nat_real @ X3 @ ( sigma_1747752005702207822t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) )
      & ? [Xa: real > nat] :
          ( ( member_real_nat @ Xa @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) )
          & ! [Xb: nat] :
              ( ( member_nat @ Xb @ ( sigma_space_nat @ borel_8449730974584783410el_nat ) )
             => ( ( comp_real_nat_nat @ Xa @ X3 @ Xb )
                = Xb ) ) ) ) ).

% nat.exist_fg
thf(fact_893_bool_Oexist__fg,axiom,
    ? [X3: $o > real] :
      ( ( member_o_real @ X3 @ ( sigma_2430008634441611636o_real @ borel_5500255247093592246orel_o @ borel_5078946678739801102l_real ) )
      & ? [Xa: real > $o] :
          ( ( member_real_o @ Xa @ ( sigma_3939073009482781210real_o @ borel_5078946678739801102l_real @ borel_5500255247093592246orel_o ) )
          & ! [Xb: $o] :
              ( ( member_o @ Xb @ ( sigma_space_o @ borel_5500255247093592246orel_o ) )
             => ( ( comp_real_o_o @ Xa @ X3 @ Xb )
                = Xb ) ) ) ) ).

% bool.exist_fg
thf(fact_894_qbs__prob_Oqbs__prob__ennintegral__not__morphism,axiom,
    ! [X: quasi_borel_real,Alpha: real > real,Mu: sigma_measure_real,F: real > extend8495563244428889912nnreal] :
      ( ( probab3605210969150000782b_real @ X @ Alpha @ Mu )
     => ( ~ ( member2919562650594848410nnreal @ F @ ( qbs_mo1317719164804411614nnreal @ X @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) ) )
       => ( ( probab7585390126108274877l_real @ ( probab8451368711090282418e_real @ ( produc4368523205619139320e_real @ X @ ( produc1722724976708544245e_real @ Alpha @ Mu ) ) ) @ F )
          = zero_z7100319975126383169nnreal ) ) ) ).

% qbs_prob.qbs_prob_ennintegral_not_morphism
thf(fact_895_qbs__prob_Oqbs__prob__ennintegral__not__morphism,axiom,
    ! [X: quasi_borel_b,Alpha: real > b,Mu: sigma_measure_real,F: b > extend8495563244428889912nnreal] :
      ( ( probab701741629625904797prob_b @ X @ Alpha @ Mu )
     => ( ~ ( member6418304549040442065nnreal @ F @ ( qbs_mo7554306597560579135nnreal @ X @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) ) )
       => ( ( probab3721531081081959086gral_b @ ( probab8173042092732894329pace_b @ ( produc2180226129289916244e_real @ X @ ( produc4478270668571743890e_real @ Alpha @ Mu ) ) ) @ F )
          = zero_z7100319975126383169nnreal ) ) ) ).

% qbs_prob.qbs_prob_ennintegral_not_morphism
thf(fact_896_qbs__prob_Oqbs__prob__ennintegral__not__morphism,axiom,
    ! [X: quasi_borel_a,Alpha: real > a,Mu: sigma_measure_real,F: a > extend8495563244428889912nnreal] :
      ( ( probab701741629625904796prob_a @ X @ Alpha @ Mu )
     => ( ~ ( member298456594901751504nnreal @ F @ ( qbs_mo1434458643421888574nnreal @ X @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) ) )
       => ( ( probab3721531081081959085gral_a @ ( probab8173042092732894328pace_a @ ( produc4145838808978236886e_real @ X @ ( produc623176010801490259e_real @ Alpha @ Mu ) ) ) @ F )
          = zero_z7100319975126383169nnreal ) ) ) ).

% qbs_prob.qbs_prob_ennintegral_not_morphism
thf(fact_897_monadP__qbs__correct,axiom,
    ! [X: quasi_borel_b] :
      ( ( quasi_159326149610617975pace_b @ ( monad_monadP_qbs_b @ X ) )
      = ( produc5532427567629089954pace_b @ ( monad_3932415646498543857s_Px_b @ X ) @ ( monad_4223963853064302126_MPx_b @ X ) ) ) ).

% monadP_qbs_correct
thf(fact_898_monadP__qbs__correct,axiom,
    ! [X: quasi_borel_a] :
      ( ( quasi_159326145307389174pace_a @ ( monad_monadP_qbs_a @ X ) )
      = ( produc6037706855006621922pace_a @ ( monad_3932415646498543856s_Px_a @ X ) @ ( monad_4223963853064302125_MPx_a @ X ) ) ) ).

% monadP_qbs_correct
thf(fact_899_ennreal_Oexist__fg,axiom,
    ? [X3: extend8495563244428889912nnreal > real] :
      ( ( member2874014351250825754l_real @ X3 @ ( sigma_7049758200512112822l_real @ borel_6524799422816628122nnreal @ borel_5078946678739801102l_real ) )
      & ? [Xa: real > extend8495563244428889912nnreal] :
          ( ( member2919562650594848410nnreal @ Xa @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ borel_6524799422816628122nnreal ) )
          & ! [Xb: extend8495563244428889912nnreal] :
              ( ( member7908768830364227535nnreal @ Xb @ ( sigma_3147302497200244656nnreal @ borel_6524799422816628122nnreal ) )
             => ( ( comp_r6281409797179841921nnreal @ Xa @ X3 @ Xb )
                = Xb ) ) ) ) ).

% ennreal.exist_fg
thf(fact_900_qp_Osubprob__space__distr,axiom,
    ! [F: real > a,M3: sigma_measure_a] :
      ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ mu @ M3 ) )
     => ( ( ( sigma_space_a @ M3 )
         != bot_bot_set_a )
       => ( giry_subprob_space_a @ ( measure_distr_real_a @ mu @ M3 @ F ) ) ) ) ).

% qp.subprob_space_distr
thf(fact_901_qp_Osubprob__space__distr,axiom,
    ! [F: real > produc2422161461964618553l_real,M3: sigma_2308072346491277622l_real] :
      ( ( member9086635009091248365l_real @ F @ ( sigma_7998147297565726139l_real @ mu @ M3 ) )
     => ( ( ( sigma_2519298556039103681l_real @ M3 )
         != bot_bo3948376660626123781l_real )
       => ( giry_s5019895290865573229l_real @ ( measur6481026558495277843l_real @ mu @ M3 @ F ) ) ) ) ).

% qp.subprob_space_distr
thf(fact_902_qp_Osubprob__space__distr,axiom,
    ! [F: real > produc7716430852924023517t_real,M3: sigma_5310753476256395226t_real] :
      ( ( member969486235890772753t_real @ F @ ( sigma_2975298441655967327t_real @ mu @ M3 ) )
     => ( ( ( sigma_2177939267068080229t_real @ M3 )
         != bot_bo4585460402199822889t_real )
       => ( giry_s632251592358338321t_real @ ( measur3655679046452803511t_real @ mu @ M3 @ F ) ) ) ) ).

% qp.subprob_space_distr
thf(fact_903_qp_Osubprob__space__distr,axiom,
    ! [F: real > $o,M3: sigma_measure_o] :
      ( ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ mu @ M3 ) )
     => ( ( ( sigma_space_o @ M3 )
         != bot_bot_set_o )
       => ( giry_subprob_space_o @ ( measure_distr_real_o @ mu @ M3 @ F ) ) ) ) ).

% qp.subprob_space_distr
thf(fact_904_qp_Osubprob__space__distr,axiom,
    ! [F: real > nat,M3: sigma_measure_nat] :
      ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ mu @ M3 ) )
     => ( ( ( sigma_space_nat @ M3 )
         != bot_bot_set_nat )
       => ( giry_s459323515522551452ce_nat @ ( measur254523008267340406al_nat @ mu @ M3 @ F ) ) ) ) ).

% qp.subprob_space_distr
thf(fact_905_qp_Osubprob__space__distr,axiom,
    ! [F: real > extend8495563244428889912nnreal,M3: sigma_7234349610311085201nnreal] :
      ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ mu @ M3 ) )
     => ( ( ( sigma_3147302497200244656nnreal @ M3 )
         != bot_bo4854962954004695426nnreal )
       => ( giry_s9043694198952438276nnreal @ ( measur8829990298702910942nnreal @ mu @ M3 @ F ) ) ) ) ).

% qp.subprob_space_distr
thf(fact_906_qp_Osubprob__space__distr,axiom,
    ! [F: real > complex,M3: sigma_3077487657436305159omplex] :
      ( ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ mu @ M3 ) )
     => ( ( ( sigma_space_complex @ M3 )
         != bot_bot_set_complex )
       => ( giry_s5699845841807254394omplex @ ( measur1621797640479583060omplex @ mu @ M3 @ F ) ) ) ) ).

% qp.subprob_space_distr
thf(fact_907_qp_Osubprob__space__distr,axiom,
    ! [F: real > real,M3: sigma_measure_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ mu @ M3 ) )
     => ( ( ( sigma_space_real @ M3 )
         != bot_bot_set_real )
       => ( giry_s8208748868292234104e_real @ ( measur2993149975067245138l_real @ mu @ M3 @ F ) ) ) ) ).

% qp.subprob_space_distr
thf(fact_908_space__distr,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_a,F: real > a] :
      ( ( sigma_space_a @ ( measure_distr_real_a @ M @ N @ F ) )
      = ( sigma_space_a @ N ) ) ).

% space_distr
thf(fact_909_space__distr,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_real,F: real > real] :
      ( ( sigma_space_real @ ( measur2993149975067245138l_real @ M @ N @ F ) )
      = ( sigma_space_real @ N ) ) ).

% space_distr
thf(fact_910_measurable__distr__eq2,axiom,
    ! [Mg: sigma_measure_o,Mg2: sigma_measure_real,Ng: sigma_measure_real,G2: real > real] :
      ( ( sigma_2430008634441611636o_real @ Mg @ ( measur2993149975067245138l_real @ Mg2 @ Ng @ G2 ) )
      = ( sigma_2430008634441611636o_real @ Mg @ Ng ) ) ).

% measurable_distr_eq2
thf(fact_911_measurable__distr__eq2,axiom,
    ! [Mg: sigma_measure_nat,Mg2: sigma_measure_real,Ng: sigma_measure_real,G2: real > real] :
      ( ( sigma_1747752005702207822t_real @ Mg @ ( measur2993149975067245138l_real @ Mg2 @ Ng @ G2 ) )
      = ( sigma_1747752005702207822t_real @ Mg @ Ng ) ) ).

% measurable_distr_eq2
thf(fact_912_measurable__distr__eq2,axiom,
    ! [Mg: sigma_measure_a,Mg2: sigma_measure_real,Ng: sigma_measure_real,G2: real > real] :
      ( ( sigma_9116425665531756122a_real @ Mg @ ( measur2993149975067245138l_real @ Mg2 @ Ng @ G2 ) )
      = ( sigma_9116425665531756122a_real @ Mg @ Ng ) ) ).

% measurable_distr_eq2
thf(fact_913_measurable__distr__eq2,axiom,
    ! [Mg: sigma_measure_real,Mg2: sigma_measure_real,Ng: sigma_measure_real,G2: real > real] :
      ( ( sigma_5267869275261027754l_real @ Mg @ ( measur2993149975067245138l_real @ Mg2 @ Ng @ G2 ) )
      = ( sigma_5267869275261027754l_real @ Mg @ Ng ) ) ).

% measurable_distr_eq2
thf(fact_914_measurable__distr__eq1,axiom,
    ! [Mf: sigma_measure_real,Nf: sigma_measure_real,F: real > real,Mf2: sigma_measure_o] :
      ( ( sigma_3939073009482781210real_o @ ( measur2993149975067245138l_real @ Mf @ Nf @ F ) @ Mf2 )
      = ( sigma_3939073009482781210real_o @ Nf @ Mf2 ) ) ).

% measurable_distr_eq1
thf(fact_915_measurable__distr__eq1,axiom,
    ! [Mf: sigma_measure_real,Nf: sigma_measure_real,F: real > real,Mf2: sigma_measure_nat] :
      ( ( sigma_6315060578831106510al_nat @ ( measur2993149975067245138l_real @ Mf @ Nf @ F ) @ Mf2 )
      = ( sigma_6315060578831106510al_nat @ Nf @ Mf2 ) ) ).

% measurable_distr_eq1
thf(fact_916_measurable__distr__eq1,axiom,
    ! [Mf: sigma_measure_real,Nf: sigma_measure_a,F: real > a,Mf2: sigma_measure_real] :
      ( ( sigma_9116425665531756122a_real @ ( measure_distr_real_a @ Mf @ Nf @ F ) @ Mf2 )
      = ( sigma_9116425665531756122a_real @ Nf @ Mf2 ) ) ).

% measurable_distr_eq1
thf(fact_917_measurable__distr__eq1,axiom,
    ! [Mf: sigma_measure_real,Nf: sigma_measure_real,F: real > real,Mf2: sigma_7234349610311085201nnreal] :
      ( ( sigma_9017504469962657078nnreal @ ( measur2993149975067245138l_real @ Mf @ Nf @ F ) @ Mf2 )
      = ( sigma_9017504469962657078nnreal @ Nf @ Mf2 ) ) ).

% measurable_distr_eq1
thf(fact_918_measurable__distr__eq1,axiom,
    ! [Mf: sigma_measure_real,Nf: sigma_measure_real,F: real > real,Mf2: sigma_3077487657436305159omplex] :
      ( ( sigma_9111916201866572460omplex @ ( measur2993149975067245138l_real @ Mf @ Nf @ F ) @ Mf2 )
      = ( sigma_9111916201866572460omplex @ Nf @ Mf2 ) ) ).

% measurable_distr_eq1
thf(fact_919_measurable__distr__eq1,axiom,
    ! [Mf: sigma_measure_real,Nf: sigma_measure_real,F: real > real,Mf2: sigma_measure_real] :
      ( ( sigma_5267869275261027754l_real @ ( measur2993149975067245138l_real @ Mf @ Nf @ F ) @ Mf2 )
      = ( sigma_5267869275261027754l_real @ Nf @ Mf2 ) ) ).

% measurable_distr_eq1
thf(fact_920_space__completion,axiom,
    ! [M: sigma_2308072346491277622l_real] :
      ( ( sigma_2519298556039103681l_real @ ( comple4976096958823823939l_real @ M ) )
      = ( sigma_2519298556039103681l_real @ M ) ) ).

% space_completion
thf(fact_921_space__completion,axiom,
    ! [M: sigma_5310753476256395226t_real] :
      ( ( sigma_2177939267068080229t_real @ ( comple520438860280874727t_real @ M ) )
      = ( sigma_2177939267068080229t_real @ M ) ) ).

% space_completion
thf(fact_922_space__completion,axiom,
    ! [M: sigma_measure_o] :
      ( ( sigma_space_o @ ( comple48332195503990434tion_o @ M ) )
      = ( sigma_space_o @ M ) ) ).

% space_completion
thf(fact_923_space__completion,axiom,
    ! [M: sigma_measure_nat] :
      ( ( sigma_space_nat @ ( comple4529072586887470918on_nat @ M ) )
      = ( sigma_space_nat @ M ) ) ).

% space_completion
thf(fact_924_space__completion,axiom,
    ! [M: sigma_measure_real] :
      ( ( sigma_space_real @ ( comple3506806835435775778n_real @ M ) )
      = ( sigma_space_real @ M ) ) ).

% space_completion
thf(fact_925_le__zero__eq,axiom,
    ! [N3: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ N3 @ zero_z7100319975126383169nnreal )
      = ( N3 = zero_z7100319975126383169nnreal ) ) ).

% le_zero_eq
thf(fact_926_finite__measure_Ofinite__measure__distr,axiom,
    ! [M: sigma_measure_o,F: $o > $o,M3: sigma_measure_o] :
      ( ( measur2447921437955784316sure_o @ M )
     => ( ( member_o_o @ F @ ( sigma_measurable_o_o @ M @ M3 ) )
       => ( measur2447921437955784316sure_o @ ( measure_distr_o_o @ M @ M3 @ F ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_927_finite__measure_Ofinite__measure__distr,axiom,
    ! [M: sigma_measure_b,F: b > extend8495563244428889912nnreal,M3: sigma_7234349610311085201nnreal] :
      ( ( measur930452917991658467sure_b @ M )
     => ( ( member6418304549040442065nnreal @ F @ ( sigma_6334800283702579687nnreal @ M @ M3 ) )
       => ( measur8478876643349974356nnreal @ ( measur1735912521085800255nnreal @ M @ M3 @ F ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_928_finite__measure_Ofinite__measure__distr,axiom,
    ! [M: sigma_measure_a,F: a > extend8495563244428889912nnreal,M3: sigma_7234349610311085201nnreal] :
      ( ( measur930452917991658466sure_a @ M )
     => ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ M3 ) )
       => ( measur8478876643349974356nnreal @ ( measur4839436603801885502nnreal @ M @ M3 @ F ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_929_finite__measure_Ofinite__measure__distr,axiom,
    ! [M: sigma_measure_o,F: $o > real,M3: sigma_measure_real] :
      ( ( measur2447921437955784316sure_o @ M )
     => ( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ M3 ) )
       => ( measur3606880022600206024e_real @ ( measure_distr_o_real @ M @ M3 @ F ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_930_finite__measure_Ofinite__measure__distr,axiom,
    ! [M: sigma_measure_nat,F: nat > real,M3: sigma_measure_real] :
      ( ( measur8338831127414845932re_nat @ M )
     => ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ M3 ) )
       => ( measur3606880022600206024e_real @ ( measur4910586471993217526t_real @ M @ M3 @ F ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_931_finite__measure_Ofinite__measure__distr,axiom,
    ! [M: sigma_measure_a,F: a > real,M3: sigma_measure_real] :
      ( ( measur930452917991658466sure_a @ M )
     => ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ M3 ) )
       => ( measur3606880022600206024e_real @ ( measure_distr_a_real @ M @ M3 @ F ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_932_finite__measure_Ofinite__measure__distr,axiom,
    ! [M: sigma_measure_real,F: real > a,M3: sigma_measure_a] :
      ( ( measur3606880022600206024e_real @ M )
     => ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ M3 ) )
       => ( measur930452917991658466sure_a @ ( measure_distr_real_a @ M @ M3 @ F ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_933_finite__measure_Ofinite__measure__distr,axiom,
    ! [M: sigma_measure_real,F: real > $o,M3: sigma_measure_o] :
      ( ( measur3606880022600206024e_real @ M )
     => ( ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ M @ M3 ) )
       => ( measur2447921437955784316sure_o @ ( measure_distr_real_o @ M @ M3 @ F ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_934_finite__measure_Ofinite__measure__distr,axiom,
    ! [M: sigma_measure_real,F: real > nat,M3: sigma_measure_nat] :
      ( ( measur3606880022600206024e_real @ M )
     => ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ M3 ) )
       => ( measur8338831127414845932re_nat @ ( measur254523008267340406al_nat @ M @ M3 @ F ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_935_finite__measure_Ofinite__measure__distr,axiom,
    ! [M: sigma_measure_real,F: real > extend8495563244428889912nnreal,M3: sigma_7234349610311085201nnreal] :
      ( ( measur3606880022600206024e_real @ M )
     => ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ M3 ) )
       => ( measur8478876643349974356nnreal @ ( measur8829990298702910942nnreal @ M @ M3 @ F ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_936_qp_Osubprob__not__empty,axiom,
    ( ( sigma_space_real @ mu )
   != bot_bot_set_real ) ).

% qp.subprob_not_empty
thf(fact_937_subset__empty,axiom,
    ! [A: set_set_real] :
      ( ( ord_le3558479182127378552t_real @ A @ bot_bot_set_set_real )
      = ( A = bot_bot_set_set_real ) ) ).

% subset_empty
thf(fact_938_subset__empty,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
      = ( A = bot_bot_set_real ) ) ).

% subset_empty
thf(fact_939_empty__subsetI,axiom,
    ! [A: set_set_real] : ( ord_le3558479182127378552t_real @ bot_bot_set_set_real @ A ) ).

% empty_subsetI
thf(fact_940_empty__subsetI,axiom,
    ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).

% empty_subsetI
thf(fact_941_bot__prod__def,axiom,
    ( bot_bo987450378295171601t_real
    = ( produc6377492392003646173t_real @ bot_bot_set_real @ bot_bot_set_real ) ) ).

% bot_prod_def
thf(fact_942_bot_Oextremum,axiom,
    ! [A2: set_set_real] : ( ord_le3558479182127378552t_real @ bot_bot_set_set_real @ A2 ) ).

% bot.extremum
thf(fact_943_bot_Oextremum,axiom,
    ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).

% bot.extremum
thf(fact_944_bot_Oextremum,axiom,
    ! [A2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ bot_bo841427958541957580nnreal @ A2 ) ).

% bot.extremum
thf(fact_945_bot_Oextremum__unique,axiom,
    ! [A2: set_set_real] :
      ( ( ord_le3558479182127378552t_real @ A2 @ bot_bot_set_set_real )
      = ( A2 = bot_bot_set_set_real ) ) ).

% bot.extremum_unique
thf(fact_946_bot_Oextremum__unique,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
      = ( A2 = bot_bot_set_real ) ) ).

% bot.extremum_unique
thf(fact_947_bot_Oextremum__unique,axiom,
    ! [A2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A2 @ bot_bo841427958541957580nnreal )
      = ( A2 = bot_bo841427958541957580nnreal ) ) ).

% bot.extremum_unique
thf(fact_948_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_set_real] :
      ( ( ord_le3558479182127378552t_real @ A2 @ bot_bot_set_set_real )
     => ( A2 = bot_bot_set_set_real ) ) ).

% bot.extremum_uniqueI
thf(fact_949_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
     => ( A2 = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
thf(fact_950_bot_Oextremum__uniqueI,axiom,
    ! [A2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A2 @ bot_bo841427958541957580nnreal )
     => ( A2 = bot_bo841427958541957580nnreal ) ) ).

% bot.extremum_uniqueI
thf(fact_951_space__subprob__algebra__empty__iff,axiom,
    ! [N: sigma_2308072346491277622l_real] :
      ( ( ( sigma_7301438298888579502l_real @ ( giry_s9089345087316773643l_real @ N ) )
        = bot_bo1104398002864633984l_real )
      = ( ( sigma_2519298556039103681l_real @ N )
        = bot_bo3948376660626123781l_real ) ) ).

% space_subprob_algebra_empty_iff
thf(fact_952_space__subprob__algebra__empty__iff,axiom,
    ! [N: sigma_5310753476256395226t_real] :
      ( ( ( sigma_8570436206843495506t_real @ ( giry_s6733618172312976303t_real @ N ) )
        = bot_bo6147983440946193572t_real )
      = ( ( sigma_2177939267068080229t_real @ N )
        = bot_bo4585460402199822889t_real ) ) ).

% space_subprob_algebra_empty_iff
thf(fact_953_space__subprob__algebra__empty__iff,axiom,
    ! [N: sigma_measure_o] :
      ( ( ( sigma_8736833305763468941sure_o @ ( giry_s3549050072915289962ebra_o @ N ) )
        = bot_bo7838039659004643295sure_o )
      = ( ( sigma_space_o @ N )
        = bot_bot_set_o ) ) ).

% space_subprob_algebra_empty_iff
thf(fact_954_space__subprob__algebra__empty__iff,axiom,
    ! [N: sigma_measure_nat] :
      ( ( ( sigma_6480959204629101233re_nat @ ( giry_s8280036963460128894ra_nat @ N ) )
        = bot_bo8872222457363190133re_nat )
      = ( ( sigma_space_nat @ N )
        = bot_bot_set_nat ) ) ).

% space_subprob_algebra_empty_iff
thf(fact_955_space__subprob__algebra__empty__iff,axiom,
    ! [N: sigma_measure_real] :
      ( ( ( sigma_2594925453452915853e_real @ ( giry_s5092570657895779418a_real @ N ) )
        = bot_bo5686449298802467025e_real )
      = ( ( sigma_space_real @ N )
        = bot_bot_set_real ) ) ).

% space_subprob_algebra_empty_iff
thf(fact_956_space__bind__empty,axiom,
    ! [M: sigma_measure_o,F: $o > sigma_measure_o] :
      ( ( ( sigma_space_o @ M )
        = bot_bot_set_o )
     => ( ( sigma_space_o @ ( giry_bind_o_o @ M @ F ) )
        = bot_bot_set_o ) ) ).

% space_bind_empty
thf(fact_957_space__bind__empty,axiom,
    ! [M: sigma_measure_o,F: $o > sigma_measure_nat] :
      ( ( ( sigma_space_o @ M )
        = bot_bot_set_o )
     => ( ( sigma_space_nat @ ( giry_bind_o_nat @ M @ F ) )
        = bot_bot_set_nat ) ) ).

% space_bind_empty
thf(fact_958_space__bind__empty,axiom,
    ! [M: sigma_measure_nat,F: nat > sigma_measure_o] :
      ( ( ( sigma_space_nat @ M )
        = bot_bot_set_nat )
     => ( ( sigma_space_o @ ( giry_bind_nat_o @ M @ F ) )
        = bot_bot_set_o ) ) ).

% space_bind_empty
thf(fact_959_space__bind__empty,axiom,
    ! [M: sigma_measure_nat,F: nat > sigma_measure_nat] :
      ( ( ( sigma_space_nat @ M )
        = bot_bot_set_nat )
     => ( ( sigma_space_nat @ ( giry_bind_nat_nat @ M @ F ) )
        = bot_bot_set_nat ) ) ).

% space_bind_empty
thf(fact_960_space__bind__empty,axiom,
    ! [M: sigma_measure_o,F: $o > sigma_measure_real] :
      ( ( ( sigma_space_o @ M )
        = bot_bot_set_o )
     => ( ( sigma_space_real @ ( giry_bind_o_real @ M @ F ) )
        = bot_bot_set_real ) ) ).

% space_bind_empty
thf(fact_961_space__bind__empty,axiom,
    ! [M: sigma_measure_nat,F: nat > sigma_measure_real] :
      ( ( ( sigma_space_nat @ M )
        = bot_bot_set_nat )
     => ( ( sigma_space_real @ ( giry_bind_nat_real @ M @ F ) )
        = bot_bot_set_real ) ) ).

% space_bind_empty
thf(fact_962_space__bind__empty,axiom,
    ! [M: sigma_measure_real,F: real > sigma_measure_o] :
      ( ( ( sigma_space_real @ M )
        = bot_bot_set_real )
     => ( ( sigma_space_o @ ( giry_bind_real_o @ M @ F ) )
        = bot_bot_set_o ) ) ).

% space_bind_empty
thf(fact_963_space__bind__empty,axiom,
    ! [M: sigma_measure_real,F: real > sigma_measure_nat] :
      ( ( ( sigma_space_real @ M )
        = bot_bot_set_real )
     => ( ( sigma_space_nat @ ( giry_bind_real_nat @ M @ F ) )
        = bot_bot_set_nat ) ) ).

% space_bind_empty
thf(fact_964_space__bind__empty,axiom,
    ! [M: sigma_measure_real,F: real > sigma_measure_real] :
      ( ( ( sigma_space_real @ M )
        = bot_bot_set_real )
     => ( ( sigma_space_real @ ( giry_bind_real_real @ M @ F ) )
        = bot_bot_set_real ) ) ).

% space_bind_empty
thf(fact_965_space__bind__empty,axiom,
    ! [M: sigma_2308072346491277622l_real,F: produc2422161461964618553l_real > sigma_measure_o] :
      ( ( ( sigma_2519298556039103681l_real @ M )
        = bot_bo3948376660626123781l_real )
     => ( ( sigma_space_o @ ( giry_b1272544242581952950real_o @ M @ F ) )
        = bot_bot_set_o ) ) ).

% space_bind_empty
thf(fact_966_subprob__space_Osubprob__not__empty,axiom,
    ! [M: sigma_2308072346491277622l_real] :
      ( ( giry_s5019895290865573229l_real @ M )
     => ( ( sigma_2519298556039103681l_real @ M )
       != bot_bo3948376660626123781l_real ) ) ).

% subprob_space.subprob_not_empty
thf(fact_967_subprob__space_Osubprob__not__empty,axiom,
    ! [M: sigma_5310753476256395226t_real] :
      ( ( giry_s632251592358338321t_real @ M )
     => ( ( sigma_2177939267068080229t_real @ M )
       != bot_bo4585460402199822889t_real ) ) ).

% subprob_space.subprob_not_empty
thf(fact_968_subprob__space_Osubprob__not__empty,axiom,
    ! [M: sigma_measure_o] :
      ( ( giry_subprob_space_o @ M )
     => ( ( sigma_space_o @ M )
       != bot_bot_set_o ) ) ).

% subprob_space.subprob_not_empty
thf(fact_969_subprob__space_Osubprob__not__empty,axiom,
    ! [M: sigma_measure_nat] :
      ( ( giry_s459323515522551452ce_nat @ M )
     => ( ( sigma_space_nat @ M )
       != bot_bot_set_nat ) ) ).

% subprob_space.subprob_not_empty
thf(fact_970_subprob__space_Osubprob__not__empty,axiom,
    ! [M: sigma_measure_real] :
      ( ( giry_s8208748868292234104e_real @ M )
     => ( ( sigma_space_real @ M )
       != bot_bot_set_real ) ) ).

% subprob_space.subprob_not_empty
thf(fact_971_prob__space_Oreal__distribution__distr,axiom,
    ! [M: sigma_measure_o,X: $o > real] :
      ( ( probab1190487603588612059pace_o @ M )
     => ( ( member_o_real @ X @ ( sigma_2430008634441611636o_real @ M @ borel_5078946678739801102l_real ) )
       => ( distri2809703520229113005bution @ ( measure_distr_o_real @ M @ borel_5078946678739801102l_real @ X ) ) ) ) ).

% prob_space.real_distribution_distr
thf(fact_972_prob__space_Oreal__distribution__distr,axiom,
    ! [M: sigma_measure_nat,X: nat > real] :
      ( ( probab2904919403188438605ce_nat @ M )
     => ( ( member_nat_real @ X @ ( sigma_1747752005702207822t_real @ M @ borel_5078946678739801102l_real ) )
       => ( distri2809703520229113005bution @ ( measur4910586471993217526t_real @ M @ borel_5078946678739801102l_real @ X ) ) ) ) ).

% prob_space.real_distribution_distr
thf(fact_973_prob__space_Oreal__distribution__distr,axiom,
    ! [M: sigma_measure_a,X: a > real] :
      ( ( probab7247484486040049089pace_a @ M )
     => ( ( member_a_real @ X @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
       => ( distri2809703520229113005bution @ ( measure_distr_a_real @ M @ borel_5078946678739801102l_real @ X ) ) ) ) ).

% prob_space.real_distribution_distr
thf(fact_974_prob__space_Oreal__distribution__distr,axiom,
    ! [M: sigma_measure_real,X: real > real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( member_real_real @ X @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( distri2809703520229113005bution @ ( measur2993149975067245138l_real @ M @ borel_5078946678739801102l_real @ X ) ) ) ) ).

% prob_space.real_distribution_distr
thf(fact_975_subprob__space_Osubprob__space__distr,axiom,
    ! [M: sigma_measure_b,F: b > extend8495563244428889912nnreal,M3: sigma_7234349610311085201nnreal] :
      ( ( giry_subprob_space_b @ M )
     => ( ( member6418304549040442065nnreal @ F @ ( sigma_6334800283702579687nnreal @ M @ M3 ) )
       => ( ( ( sigma_3147302497200244656nnreal @ M3 )
           != bot_bo4854962954004695426nnreal )
         => ( giry_s9043694198952438276nnreal @ ( measur1735912521085800255nnreal @ M @ M3 @ F ) ) ) ) ) ).

% subprob_space.subprob_space_distr
thf(fact_976_subprob__space_Osubprob__space__distr,axiom,
    ! [M: sigma_measure_a,F: a > extend8495563244428889912nnreal,M3: sigma_7234349610311085201nnreal] :
      ( ( giry_subprob_space_a @ M )
     => ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ M3 ) )
       => ( ( ( sigma_3147302497200244656nnreal @ M3 )
           != bot_bo4854962954004695426nnreal )
         => ( giry_s9043694198952438276nnreal @ ( measur4839436603801885502nnreal @ M @ M3 @ F ) ) ) ) ) ).

% subprob_space.subprob_space_distr
thf(fact_977_subprob__space_Osubprob__space__distr,axiom,
    ! [M: sigma_measure_o,F: $o > $o,M3: sigma_measure_o] :
      ( ( giry_subprob_space_o @ M )
     => ( ( member_o_o @ F @ ( sigma_measurable_o_o @ M @ M3 ) )
       => ( ( ( sigma_space_o @ M3 )
           != bot_bot_set_o )
         => ( giry_subprob_space_o @ ( measure_distr_o_o @ M @ M3 @ F ) ) ) ) ) ).

% subprob_space.subprob_space_distr
thf(fact_978_subprob__space_Osubprob__space__distr,axiom,
    ! [M: sigma_measure_real,F: real > a,M3: sigma_measure_a] :
      ( ( giry_s8208748868292234104e_real @ M )
     => ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ M3 ) )
       => ( ( ( sigma_space_a @ M3 )
           != bot_bot_set_a )
         => ( giry_subprob_space_a @ ( measure_distr_real_a @ M @ M3 @ F ) ) ) ) ) ).

% subprob_space.subprob_space_distr
thf(fact_979_subprob__space_Osubprob__space__distr,axiom,
    ! [M: sigma_measure_real,F: real > $o,M3: sigma_measure_o] :
      ( ( giry_s8208748868292234104e_real @ M )
     => ( ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ M @ M3 ) )
       => ( ( ( sigma_space_o @ M3 )
           != bot_bot_set_o )
         => ( giry_subprob_space_o @ ( measure_distr_real_o @ M @ M3 @ F ) ) ) ) ) ).

% subprob_space.subprob_space_distr
thf(fact_980_subprob__space_Osubprob__space__distr,axiom,
    ! [M: sigma_measure_real,F: real > nat,M3: sigma_measure_nat] :
      ( ( giry_s8208748868292234104e_real @ M )
     => ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ M3 ) )
       => ( ( ( sigma_space_nat @ M3 )
           != bot_bot_set_nat )
         => ( giry_s459323515522551452ce_nat @ ( measur254523008267340406al_nat @ M @ M3 @ F ) ) ) ) ) ).

% subprob_space.subprob_space_distr
thf(fact_981_subprob__space_Osubprob__space__distr,axiom,
    ! [M: sigma_measure_o,F: $o > real,M3: sigma_measure_real] :
      ( ( giry_subprob_space_o @ M )
     => ( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ M3 ) )
       => ( ( ( sigma_space_real @ M3 )
           != bot_bot_set_real )
         => ( giry_s8208748868292234104e_real @ ( measure_distr_o_real @ M @ M3 @ F ) ) ) ) ) ).

% subprob_space.subprob_space_distr
thf(fact_982_subprob__space_Osubprob__space__distr,axiom,
    ! [M: sigma_measure_nat,F: nat > real,M3: sigma_measure_real] :
      ( ( giry_s459323515522551452ce_nat @ M )
     => ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ M3 ) )
       => ( ( ( sigma_space_real @ M3 )
           != bot_bot_set_real )
         => ( giry_s8208748868292234104e_real @ ( measur4910586471993217526t_real @ M @ M3 @ F ) ) ) ) ) ).

% subprob_space.subprob_space_distr
thf(fact_983_subprob__space_Osubprob__space__distr,axiom,
    ! [M: sigma_measure_a,F: a > real,M3: sigma_measure_real] :
      ( ( giry_subprob_space_a @ M )
     => ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ M3 ) )
       => ( ( ( sigma_space_real @ M3 )
           != bot_bot_set_real )
         => ( giry_s8208748868292234104e_real @ ( measure_distr_a_real @ M @ M3 @ F ) ) ) ) ) ).

% subprob_space.subprob_space_distr
thf(fact_984_subprob__space_Osubprob__space__distr,axiom,
    ! [M: sigma_measure_real,F: real > extend8495563244428889912nnreal,M3: sigma_7234349610311085201nnreal] :
      ( ( giry_s8208748868292234104e_real @ M )
     => ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ M3 ) )
       => ( ( ( sigma_3147302497200244656nnreal @ M3 )
           != bot_bo4854962954004695426nnreal )
         => ( giry_s9043694198952438276nnreal @ ( measur8829990298702910942nnreal @ M @ M3 @ F ) ) ) ) ) ).

% subprob_space.subprob_space_distr
thf(fact_985_space__bind__measurable,axiom,
    ! [F: $o > sigma_measure_o,M: sigma_measure_o,B: sigma_measure_o] :
      ( ( member4999217005381492423sure_o @ F @ ( sigma_2256073753033958621sure_o @ M @ ( giry_s3549050072915289962ebra_o @ B ) ) )
     => ( ( ( sigma_space_o @ M )
         != bot_bot_set_o )
       => ( ( sigma_space_o @ ( giry_bind_o_o @ M @ F ) )
          = ( sigma_space_o @ B ) ) ) ) ).

% space_bind_measurable
thf(fact_986_space__bind__measurable,axiom,
    ! [F: $o > sigma_measure_nat,M: sigma_measure_o,B: sigma_measure_nat] :
      ( ( member275192172136629389re_nat @ F @ ( sigma_2980789436339249249re_nat @ M @ ( giry_s8280036963460128894ra_nat @ B ) ) )
     => ( ( ( sigma_space_o @ M )
         != bot_bot_set_o )
       => ( ( sigma_space_nat @ ( giry_bind_o_nat @ M @ F ) )
          = ( sigma_space_nat @ B ) ) ) ) ).

% space_bind_measurable
thf(fact_987_space__bind__measurable,axiom,
    ! [F: nat > sigma_measure_o,M: sigma_measure_nat,B: sigma_measure_o] :
      ( ( member3907955456775592219sure_o @ F @ ( sigma_3614372446148830647sure_o @ M @ ( giry_s3549050072915289962ebra_o @ B ) ) )
     => ( ( ( sigma_space_nat @ M )
         != bot_bot_set_nat )
       => ( ( sigma_space_o @ ( giry_bind_nat_o @ M @ F ) )
          = ( sigma_space_o @ B ) ) ) ) ).

% space_bind_measurable
thf(fact_988_space__bind__measurable,axiom,
    ! [F: nat > sigma_measure_nat,M: sigma_measure_nat,B: sigma_measure_nat] :
      ( ( member8953738614771599161re_nat @ F @ ( sigma_2960785956597205511re_nat @ M @ ( giry_s8280036963460128894ra_nat @ B ) ) )
     => ( ( ( sigma_space_nat @ M )
         != bot_bot_set_nat )
       => ( ( sigma_space_nat @ ( giry_bind_nat_nat @ M @ F ) )
          = ( sigma_space_nat @ B ) ) ) ) ).

% space_bind_measurable
thf(fact_989_space__bind__measurable,axiom,
    ! [F: real > sigma_measure_o,M: sigma_measure_real,B: sigma_measure_o] :
      ( ( member1287260281327066359sure_o @ F @ ( sigma_8971795097834940435sure_o @ M @ ( giry_s3549050072915289962ebra_o @ B ) ) )
     => ( ( ( sigma_space_real @ M )
         != bot_bot_set_real )
       => ( ( sigma_space_o @ ( giry_bind_real_o @ M @ F ) )
          = ( sigma_space_o @ B ) ) ) ) ).

% space_bind_measurable
thf(fact_990_space__bind__measurable,axiom,
    ! [F: real > sigma_measure_nat,M: sigma_measure_real,B: sigma_measure_nat] :
      ( ( member8263806380797784669re_nat @ F @ ( sigma_854259722376812203re_nat @ M @ ( giry_s8280036963460128894ra_nat @ B ) ) )
     => ( ( ( sigma_space_real @ M )
         != bot_bot_set_real )
       => ( ( sigma_space_nat @ ( giry_bind_real_nat @ M @ F ) )
          = ( sigma_space_nat @ B ) ) ) ) ).

% space_bind_measurable
thf(fact_991_space__bind__measurable,axiom,
    ! [F: $o > sigma_measure_real,M: sigma_measure_o,B: sigma_measure_real] :
      ( ( member1425558209591478249e_real @ F @ ( sigma_4942308948664123965e_real @ M @ ( giry_s5092570657895779418a_real @ B ) ) )
     => ( ( ( sigma_space_o @ M )
         != bot_bot_set_o )
       => ( ( sigma_space_real @ ( giry_bind_o_real @ M @ F ) )
          = ( sigma_space_real @ B ) ) ) ) ).

% space_bind_measurable
thf(fact_992_space__bind__measurable,axiom,
    ! [F: nat > sigma_measure_real,M: sigma_measure_nat,B: sigma_measure_real] :
      ( ( member5430058146565829781e_real @ F @ ( sigma_76919172735023331e_real @ M @ ( giry_s5092570657895779418a_real @ B ) ) )
     => ( ( ( sigma_space_nat @ M )
         != bot_bot_set_nat )
       => ( ( sigma_space_real @ ( giry_bind_nat_real @ M @ F ) )
          = ( sigma_space_real @ B ) ) ) ) ).

% space_bind_measurable
thf(fact_993_space__bind__measurable,axiom,
    ! [F: real > sigma_measure_real,M: sigma_measure_real,B: sigma_measure_real] :
      ( ( member2630560753458908601e_real @ F @ ( sigma_5928869325259027335e_real @ M @ ( giry_s5092570657895779418a_real @ B ) ) )
     => ( ( ( sigma_space_real @ M )
         != bot_bot_set_real )
       => ( ( sigma_space_real @ ( giry_bind_real_real @ M @ F ) )
          = ( sigma_space_real @ B ) ) ) ) ).

% space_bind_measurable
thf(fact_994_space__bind__measurable,axiom,
    ! [F: produc2422161461964618553l_real > sigma_measure_o,M: sigma_2308072346491277622l_real,B: sigma_measure_o] :
      ( ( member3321517115727214696sure_o @ F @ ( sigma_3382329464100874366sure_o @ M @ ( giry_s3549050072915289962ebra_o @ B ) ) )
     => ( ( ( sigma_2519298556039103681l_real @ M )
         != bot_bo3948376660626123781l_real )
       => ( ( sigma_space_o @ ( giry_b1272544242581952950real_o @ M @ F ) )
          = ( sigma_space_o @ B ) ) ) ) ).

% space_bind_measurable
thf(fact_995_zero__le,axiom,
    ! [X8: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ X8 ) ).

% zero_le
thf(fact_996_le__measureD1,axiom,
    ! [A: sigma_2308072346491277622l_real,B: sigma_2308072346491277622l_real] :
      ( ( ord_le7533194067943772566l_real @ A @ B )
     => ( ord_le64383758879589177l_real @ ( sigma_2519298556039103681l_real @ A ) @ ( sigma_2519298556039103681l_real @ B ) ) ) ).

% le_measureD1
thf(fact_997_le__measureD1,axiom,
    ! [A: sigma_5310753476256395226t_real,B: sigma_5310753476256395226t_real] :
      ( ( ord_le8248399539298082874t_real @ A @ B )
     => ( ord_le5802650104174031709t_real @ ( sigma_2177939267068080229t_real @ A ) @ ( sigma_2177939267068080229t_real @ B ) ) ) ).

% le_measureD1
thf(fact_998_le__measureD1,axiom,
    ! [A: sigma_measure_o,B: sigma_measure_o] :
      ( ( ord_le478349814012620405sure_o @ A @ B )
     => ( ord_less_eq_set_o @ ( sigma_space_o @ A ) @ ( sigma_space_o @ B ) ) ) ).

% le_measureD1
thf(fact_999_le__measureD1,axiom,
    ! [A: sigma_measure_nat,B: sigma_measure_nat] :
      ( ( ord_le2862109966718184649re_nat @ A @ B )
     => ( ord_less_eq_set_nat @ ( sigma_space_nat @ A ) @ ( sigma_space_nat @ B ) ) ) ).

% le_measureD1
thf(fact_1000_le__measureD1,axiom,
    ! [A: sigma_3733394171116455995t_real,B: sigma_3733394171116455995t_real] :
      ( ( ord_le4015499323295518683t_real @ A @ B )
     => ( ord_le3558479182127378552t_real @ ( sigma_space_set_real @ A ) @ ( sigma_space_set_real @ B ) ) ) ).

% le_measureD1
thf(fact_1001_le__measureD1,axiom,
    ! [A: sigma_measure_real,B: sigma_measure_real] :
      ( ( ord_le487379304121309861e_real @ A @ B )
     => ( ord_less_eq_set_real @ ( sigma_space_real @ A ) @ ( sigma_space_real @ B ) ) ) ).

% le_measureD1
thf(fact_1002_measurable__completion,axiom,
    ! [F: $o > $o,M: sigma_measure_o,N: sigma_measure_o] :
      ( ( member_o_o @ F @ ( sigma_measurable_o_o @ M @ N ) )
     => ( member_o_o @ F @ ( sigma_measurable_o_o @ ( comple48332195503990434tion_o @ M ) @ N ) ) ) ).

% measurable_completion
thf(fact_1003_measurable__completion,axiom,
    ! [F: b > extend8495563244428889912nnreal,M: sigma_measure_b,N: sigma_7234349610311085201nnreal] :
      ( ( member6418304549040442065nnreal @ F @ ( sigma_6334800283702579687nnreal @ M @ N ) )
     => ( member6418304549040442065nnreal @ F @ ( sigma_6334800283702579687nnreal @ ( comple3428971583294703881tion_b @ M ) @ N ) ) ) ).

% measurable_completion
thf(fact_1004_measurable__completion,axiom,
    ! [F: a > extend8495563244428889912nnreal,M: sigma_measure_a,N: sigma_7234349610311085201nnreal] :
      ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ N ) )
     => ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ ( comple3428971583294703880tion_a @ M ) @ N ) ) ) ).

% measurable_completion
thf(fact_1005_measurable__completion,axiom,
    ! [F: $o > real,M: sigma_measure_o,N: sigma_measure_real] :
      ( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ N ) )
     => ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ ( comple48332195503990434tion_o @ M ) @ N ) ) ) ).

% measurable_completion
thf(fact_1006_measurable__completion,axiom,
    ! [F: nat > real,M: sigma_measure_nat,N: sigma_measure_real] :
      ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ N ) )
     => ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ ( comple4529072586887470918on_nat @ M ) @ N ) ) ) ).

% measurable_completion
thf(fact_1007_measurable__completion,axiom,
    ! [F: a > real,M: sigma_measure_a,N: sigma_measure_real] :
      ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ N ) )
     => ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ ( comple3428971583294703880tion_a @ M ) @ N ) ) ) ).

% measurable_completion
thf(fact_1008_measurable__completion,axiom,
    ! [F: real > $o,M: sigma_measure_real,N: sigma_measure_o] :
      ( ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ M @ N ) )
     => ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ ( comple3506806835435775778n_real @ M ) @ N ) ) ) ).

% measurable_completion
thf(fact_1009_measurable__completion,axiom,
    ! [F: real > nat,M: sigma_measure_real,N: sigma_measure_nat] :
      ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ N ) )
     => ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ ( comple3506806835435775778n_real @ M ) @ N ) ) ) ).

% measurable_completion
thf(fact_1010_measurable__completion,axiom,
    ! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,N: sigma_7234349610311085201nnreal] :
      ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ N ) )
     => ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ ( comple3506806835435775778n_real @ M ) @ N ) ) ) ).

% measurable_completion
thf(fact_1011_measurable__completion,axiom,
    ! [F: real > complex,M: sigma_measure_real,N: sigma_3077487657436305159omplex] :
      ( ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ M @ N ) )
     => ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ ( comple3506806835435775778n_real @ M ) @ N ) ) ) ).

% measurable_completion
thf(fact_1012_distr__distr,axiom,
    ! [G2: $o > $o,N: sigma_measure_o,L: sigma_measure_o,F: $o > $o,M: sigma_measure_o] :
      ( ( member_o_o @ G2 @ ( sigma_measurable_o_o @ N @ L ) )
     => ( ( member_o_o @ F @ ( sigma_measurable_o_o @ M @ N ) )
       => ( ( measure_distr_o_o @ ( measure_distr_o_o @ M @ N @ F ) @ L @ G2 )
          = ( measure_distr_o_o @ M @ L @ ( comp_o_o_o @ G2 @ F ) ) ) ) ) ).

% distr_distr
thf(fact_1013_distr__distr,axiom,
    ! [G2: a > extend8495563244428889912nnreal,N: sigma_measure_a,L: sigma_7234349610311085201nnreal,F: real > a,M: sigma_measure_real] :
      ( ( member298456594901751504nnreal @ G2 @ ( sigma_214952329563889126nnreal @ N @ L ) )
     => ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ N ) )
       => ( ( measur4839436603801885502nnreal @ ( measure_distr_real_a @ M @ N @ F ) @ L @ G2 )
          = ( measur8829990298702910942nnreal @ M @ L @ ( comp_a8249376463644563905l_real @ G2 @ F ) ) ) ) ) ).

% distr_distr
thf(fact_1014_distr__distr,axiom,
    ! [G2: a > a,N: sigma_measure_a,L: sigma_measure_a,F: real > a,M: sigma_measure_real] :
      ( ( member_a_a @ G2 @ ( 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 @ G2 )
          = ( measure_distr_real_a @ M @ L @ ( comp_a_a_real @ G2 @ F ) ) ) ) ) ).

% distr_distr
thf(fact_1015_distr__distr,axiom,
    ! [G2: $o > $o,N: sigma_measure_o,L: sigma_measure_o,F: real > $o,M: sigma_measure_real] :
      ( ( member_o_o @ G2 @ ( sigma_measurable_o_o @ N @ L ) )
     => ( ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ M @ N ) )
       => ( ( measure_distr_o_o @ ( measure_distr_real_o @ M @ N @ F ) @ L @ G2 )
          = ( measure_distr_real_o @ M @ L @ ( comp_o_o_real @ G2 @ F ) ) ) ) ) ).

% distr_distr
thf(fact_1016_distr__distr,axiom,
    ! [G2: $o > a,N: sigma_measure_o,L: sigma_measure_a,F: real > $o,M: sigma_measure_real] :
      ( ( member_o_a @ G2 @ ( sigma_measurable_o_a @ N @ L ) )
     => ( ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ M @ N ) )
       => ( ( measure_distr_o_a @ ( measure_distr_real_o @ M @ N @ F ) @ L @ G2 )
          = ( measure_distr_real_a @ M @ L @ ( comp_o_a_real @ G2 @ F ) ) ) ) ) ).

% distr_distr
thf(fact_1017_distr__distr,axiom,
    ! [G2: nat > a,N: sigma_measure_nat,L: sigma_measure_a,F: real > nat,M: sigma_measure_real] :
      ( ( member_nat_a @ G2 @ ( sigma_4105081583803843548_nat_a @ N @ L ) )
     => ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ N ) )
       => ( ( measure_distr_nat_a @ ( measur254523008267340406al_nat @ M @ N @ F ) @ L @ G2 )
          = ( measure_distr_real_a @ M @ L @ ( comp_nat_a_real @ G2 @ F ) ) ) ) ) ).

% distr_distr
thf(fact_1018_distr__distr,axiom,
    ! [G2: real > a,N: sigma_measure_real,L: sigma_measure_a,F: $o > real,M: sigma_measure_o] :
      ( ( member_real_a @ G2 @ ( sigma_523072396149930112real_a @ N @ L ) )
     => ( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ N ) )
       => ( ( measure_distr_real_a @ ( measure_distr_o_real @ M @ N @ F ) @ L @ G2 )
          = ( measure_distr_o_a @ M @ L @ ( comp_real_a_o @ G2 @ F ) ) ) ) ) ).

% distr_distr
thf(fact_1019_distr__distr,axiom,
    ! [G2: real > a,N: sigma_measure_real,L: sigma_measure_a,F: nat > real,M: sigma_measure_nat] :
      ( ( member_real_a @ G2 @ ( sigma_523072396149930112real_a @ N @ L ) )
     => ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ N ) )
       => ( ( measure_distr_real_a @ ( measur4910586471993217526t_real @ M @ N @ F ) @ L @ G2 )
          = ( measure_distr_nat_a @ M @ L @ ( comp_real_a_nat @ G2 @ F ) ) ) ) ) ).

% distr_distr
thf(fact_1020_distr__distr,axiom,
    ! [G2: real > a,N: sigma_measure_real,L: sigma_measure_a,F: a > real,M: sigma_measure_a] :
      ( ( member_real_a @ G2 @ ( 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 @ G2 )
          = ( measure_distr_a_a @ M @ L @ ( comp_real_a_a @ G2 @ F ) ) ) ) ) ).

% distr_distr
thf(fact_1021_distr__distr,axiom,
    ! [G2: extend8495563244428889912nnreal > real,N: sigma_7234349610311085201nnreal,L: sigma_measure_real,F: real > extend8495563244428889912nnreal,M: sigma_measure_real] :
      ( ( member2874014351250825754l_real @ G2 @ ( sigma_7049758200512112822l_real @ N @ L ) )
     => ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ N ) )
       => ( ( measur6862244029252366686l_real @ ( measur8829990298702910942nnreal @ M @ N @ F ) @ L @ G2 )
          = ( measur2993149975067245138l_real @ M @ L @ ( comp_E3822617923592311797l_real @ G2 @ F ) ) ) ) ) ).

% distr_distr
thf(fact_1022_distr__completion,axiom,
    ! [X: $o > $o,M: sigma_measure_o,N: sigma_measure_o] :
      ( ( member_o_o @ X @ ( sigma_measurable_o_o @ M @ N ) )
     => ( ( measure_distr_o_o @ ( comple48332195503990434tion_o @ M ) @ N @ X )
        = ( measure_distr_o_o @ M @ N @ X ) ) ) ).

% distr_completion
thf(fact_1023_distr__completion,axiom,
    ! [X: b > extend8495563244428889912nnreal,M: sigma_measure_b,N: sigma_7234349610311085201nnreal] :
      ( ( member6418304549040442065nnreal @ X @ ( sigma_6334800283702579687nnreal @ M @ N ) )
     => ( ( measur1735912521085800255nnreal @ ( comple3428971583294703881tion_b @ M ) @ N @ X )
        = ( measur1735912521085800255nnreal @ M @ N @ X ) ) ) ).

% distr_completion
thf(fact_1024_distr__completion,axiom,
    ! [X: a > extend8495563244428889912nnreal,M: sigma_measure_a,N: sigma_7234349610311085201nnreal] :
      ( ( member298456594901751504nnreal @ X @ ( sigma_214952329563889126nnreal @ M @ N ) )
     => ( ( measur4839436603801885502nnreal @ ( comple3428971583294703880tion_a @ M ) @ N @ X )
        = ( measur4839436603801885502nnreal @ M @ N @ X ) ) ) ).

% distr_completion
thf(fact_1025_distr__completion,axiom,
    ! [X: $o > real,M: sigma_measure_o,N: sigma_measure_real] :
      ( ( member_o_real @ X @ ( sigma_2430008634441611636o_real @ M @ N ) )
     => ( ( measure_distr_o_real @ ( comple48332195503990434tion_o @ M ) @ N @ X )
        = ( measure_distr_o_real @ M @ N @ X ) ) ) ).

% distr_completion
thf(fact_1026_distr__completion,axiom,
    ! [X: nat > real,M: sigma_measure_nat,N: sigma_measure_real] :
      ( ( member_nat_real @ X @ ( sigma_1747752005702207822t_real @ M @ N ) )
     => ( ( measur4910586471993217526t_real @ ( comple4529072586887470918on_nat @ M ) @ N @ X )
        = ( measur4910586471993217526t_real @ M @ N @ X ) ) ) ).

% distr_completion
thf(fact_1027_distr__completion,axiom,
    ! [X: a > real,M: sigma_measure_a,N: sigma_measure_real] :
      ( ( member_a_real @ X @ ( sigma_9116425665531756122a_real @ M @ N ) )
     => ( ( measure_distr_a_real @ ( comple3428971583294703880tion_a @ M ) @ N @ X )
        = ( measure_distr_a_real @ M @ N @ X ) ) ) ).

% distr_completion
thf(fact_1028_distr__completion,axiom,
    ! [X: real > a,M: sigma_measure_real,N: sigma_measure_a] :
      ( ( member_real_a @ X @ ( sigma_523072396149930112real_a @ M @ N ) )
     => ( ( measure_distr_real_a @ ( comple3506806835435775778n_real @ M ) @ N @ X )
        = ( measure_distr_real_a @ M @ N @ X ) ) ) ).

% distr_completion
thf(fact_1029_distr__completion,axiom,
    ! [X: real > $o,M: sigma_measure_real,N: sigma_measure_o] :
      ( ( member_real_o @ X @ ( sigma_3939073009482781210real_o @ M @ N ) )
     => ( ( measure_distr_real_o @ ( comple3506806835435775778n_real @ M ) @ N @ X )
        = ( measure_distr_real_o @ M @ N @ X ) ) ) ).

% distr_completion
thf(fact_1030_distr__completion,axiom,
    ! [X: real > nat,M: sigma_measure_real,N: sigma_measure_nat] :
      ( ( member_real_nat @ X @ ( sigma_6315060578831106510al_nat @ M @ N ) )
     => ( ( measur254523008267340406al_nat @ ( comple3506806835435775778n_real @ M ) @ N @ X )
        = ( measur254523008267340406al_nat @ M @ N @ X ) ) ) ).

% distr_completion
thf(fact_1031_distr__completion,axiom,
    ! [X: real > extend8495563244428889912nnreal,M: sigma_measure_real,N: sigma_7234349610311085201nnreal] :
      ( ( member2919562650594848410nnreal @ X @ ( sigma_9017504469962657078nnreal @ M @ N ) )
     => ( ( measur8829990298702910942nnreal @ ( comple3506806835435775778n_real @ M ) @ N @ X )
        = ( measur8829990298702910942nnreal @ M @ N @ X ) ) ) ).

% distr_completion
thf(fact_1032_qp_Oindep__sets__finite__index__sets,axiom,
    ! [F2: set_real > set_set_real,I: set_set_real] :
      ( ( indepe8752365572443096444t_real @ mu @ F2 @ I )
      = ( ! [J2: set_set_real] :
            ( ( ord_le3558479182127378552t_real @ J2 @ I )
           => ( ( J2 != bot_bot_set_set_real )
             => ( ( finite9007344921179782393t_real @ J2 )
               => ( indepe8752365572443096444t_real @ mu @ F2 @ J2 ) ) ) ) ) ) ).

% qp.indep_sets_finite_index_sets
thf(fact_1033_qp_Oindep__sets__finite__index__sets,axiom,
    ! [F2: real > set_set_real,I: set_real] :
      ( ( indepe8783372407961027910l_real @ mu @ F2 @ I )
      = ( ! [J2: set_real] :
            ( ( ord_less_eq_set_real @ J2 @ I )
           => ( ( J2 != bot_bot_set_real )
             => ( ( finite_finite_real @ J2 )
               => ( indepe8783372407961027910l_real @ mu @ F2 @ J2 ) ) ) ) ) ) ).

% qp.indep_sets_finite_index_sets
thf(fact_1034_qp_OKL__same__eq__0,axiom,
    ! [B2: real] :
      ( ( kL_divergence_real @ B2 @ mu @ mu )
      = zero_zero_real ) ).

% qp.KL_same_eq_0
thf(fact_1035_sigma__finite__measure__distr,axiom,
    ! [M: sigma_measure_o,N: sigma_measure_o,F: $o > $o] :
      ( ( measur1827666076404920889sure_o @ ( measure_distr_o_o @ M @ N @ F ) )
     => ( ( member_o_o @ F @ ( sigma_measurable_o_o @ M @ N ) )
       => ( measur1827666076404920889sure_o @ M ) ) ) ).

% sigma_finite_measure_distr
thf(fact_1036_sigma__finite__measure__distr,axiom,
    ! [M: sigma_measure_b,N: sigma_7234349610311085201nnreal,F: b > extend8495563244428889912nnreal] :
      ( ( measur6426964080664357591nnreal @ ( measur1735912521085800255nnreal @ M @ N @ F ) )
     => ( ( member6418304549040442065nnreal @ F @ ( sigma_6334800283702579687nnreal @ M @ N ) )
       => ( measur4308613598931908896sure_b @ M ) ) ) ).

% sigma_finite_measure_distr
thf(fact_1037_sigma__finite__measure__distr,axiom,
    ! [M: sigma_measure_a,N: sigma_7234349610311085201nnreal,F: a > extend8495563244428889912nnreal] :
      ( ( measur6426964080664357591nnreal @ ( measur4839436603801885502nnreal @ M @ N @ F ) )
     => ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ N ) )
       => ( measur4308613598931908895sure_a @ M ) ) ) ).

% sigma_finite_measure_distr
thf(fact_1038_sigma__finite__measure__distr,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_a,F: real > a] :
      ( ( measur4308613598931908895sure_a @ ( measure_distr_real_a @ M @ N @ F ) )
     => ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ N ) )
       => ( measur487378040549452491e_real @ M ) ) ) ).

% sigma_finite_measure_distr
thf(fact_1039_sigma__finite__measure__distr,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_o,F: real > $o] :
      ( ( measur1827666076404920889sure_o @ ( measure_distr_real_o @ M @ N @ F ) )
     => ( ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ M @ N ) )
       => ( measur487378040549452491e_real @ M ) ) ) ).

% sigma_finite_measure_distr
thf(fact_1040_sigma__finite__measure__distr,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_nat,F: real > nat] :
      ( ( measur8258956421386577775re_nat @ ( measur254523008267340406al_nat @ M @ N @ F ) )
     => ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ N ) )
       => ( measur487378040549452491e_real @ M ) ) ) ).

% sigma_finite_measure_distr
thf(fact_1041_sigma__finite__measure__distr,axiom,
    ! [M: sigma_measure_real,N: sigma_7234349610311085201nnreal,F: real > extend8495563244428889912nnreal] :
      ( ( measur6426964080664357591nnreal @ ( measur8829990298702910942nnreal @ M @ N @ F ) )
     => ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ N ) )
       => ( measur487378040549452491e_real @ M ) ) ) ).

% sigma_finite_measure_distr
thf(fact_1042_sigma__finite__measure__distr,axiom,
    ! [M: sigma_measure_real,N: sigma_3077487657436305159omplex,F: real > complex] :
      ( ( measur2063495708654786125omplex @ ( measur1621797640479583060omplex @ M @ N @ F ) )
     => ( ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ M @ N ) )
       => ( measur487378040549452491e_real @ M ) ) ) ).

% sigma_finite_measure_distr
thf(fact_1043_sigma__finite__measure__distr,axiom,
    ! [M: sigma_measure_o,N: sigma_measure_real,F: $o > real] :
      ( ( measur487378040549452491e_real @ ( measure_distr_o_real @ M @ N @ F ) )
     => ( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ N ) )
       => ( measur1827666076404920889sure_o @ M ) ) ) ).

% sigma_finite_measure_distr
thf(fact_1044_sigma__finite__measure__distr,axiom,
    ! [M: sigma_measure_nat,N: sigma_measure_real,F: nat > real] :
      ( ( measur487378040549452491e_real @ ( measur4910586471993217526t_real @ M @ N @ F ) )
     => ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ N ) )
       => ( measur8258956421386577775re_nat @ M ) ) ) ).

% sigma_finite_measure_distr
thf(fact_1045_prob__space_Oprob__space__distr,axiom,
    ! [M: sigma_measure_o,F: $o > $o,M3: sigma_measure_o] :
      ( ( probab1190487603588612059pace_o @ M )
     => ( ( member_o_o @ F @ ( sigma_measurable_o_o @ M @ M3 ) )
       => ( probab1190487603588612059pace_o @ ( measure_distr_o_o @ M @ M3 @ F ) ) ) ) ).

% prob_space.prob_space_distr
thf(fact_1046_prob__space_Oprob__space__distr,axiom,
    ! [M: sigma_measure_b,F: b > extend8495563244428889912nnreal,M3: sigma_7234349610311085201nnreal] :
      ( ( probab7247484486040049090pace_b @ M )
     => ( ( member6418304549040442065nnreal @ F @ ( sigma_6334800283702579687nnreal @ M @ M3 ) )
       => ( probab6612481188548237749nnreal @ ( measur1735912521085800255nnreal @ M @ M3 @ F ) ) ) ) ).

% prob_space.prob_space_distr
thf(fact_1047_prob__space_Oprob__space__distr,axiom,
    ! [M: sigma_measure_a,F: a > extend8495563244428889912nnreal,M3: sigma_7234349610311085201nnreal] :
      ( ( probab7247484486040049089pace_a @ M )
     => ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ M3 ) )
       => ( probab6612481188548237749nnreal @ ( measur4839436603801885502nnreal @ M @ M3 @ F ) ) ) ) ).

% prob_space.prob_space_distr
thf(fact_1048_prob__space_Oprob__space__distr,axiom,
    ! [M: sigma_measure_o,F: $o > real,M3: sigma_measure_real] :
      ( ( probab1190487603588612059pace_o @ M )
     => ( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ M3 ) )
       => ( probab535871623910865577e_real @ ( measure_distr_o_real @ M @ M3 @ F ) ) ) ) ).

% prob_space.prob_space_distr
thf(fact_1049_prob__space_Oprob__space__distr,axiom,
    ! [M: sigma_measure_nat,F: nat > real,M3: sigma_measure_real] :
      ( ( probab2904919403188438605ce_nat @ M )
     => ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ M3 ) )
       => ( probab535871623910865577e_real @ ( measur4910586471993217526t_real @ M @ M3 @ F ) ) ) ) ).

% prob_space.prob_space_distr
thf(fact_1050_prob__space_Oprob__space__distr,axiom,
    ! [M: sigma_measure_a,F: a > real,M3: sigma_measure_real] :
      ( ( probab7247484486040049089pace_a @ M )
     => ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ M3 ) )
       => ( probab535871623910865577e_real @ ( measure_distr_a_real @ M @ M3 @ F ) ) ) ) ).

% prob_space.prob_space_distr
thf(fact_1051_prob__space_Oprob__space__distr,axiom,
    ! [M: sigma_measure_real,F: real > a,M3: sigma_measure_a] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ M3 ) )
       => ( probab7247484486040049089pace_a @ ( measure_distr_real_a @ M @ M3 @ F ) ) ) ) ).

% prob_space.prob_space_distr
thf(fact_1052_prob__space_Oprob__space__distr,axiom,
    ! [M: sigma_measure_real,F: real > $o,M3: sigma_measure_o] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ M @ M3 ) )
       => ( probab1190487603588612059pace_o @ ( measure_distr_real_o @ M @ M3 @ F ) ) ) ) ).

% prob_space.prob_space_distr
thf(fact_1053_prob__space_Oprob__space__distr,axiom,
    ! [M: sigma_measure_real,F: real > nat,M3: sigma_measure_nat] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ M3 ) )
       => ( probab2904919403188438605ce_nat @ ( measur254523008267340406al_nat @ M @ M3 @ F ) ) ) ) ).

% prob_space.prob_space_distr
thf(fact_1054_prob__space_Oprob__space__distr,axiom,
    ! [M: sigma_measure_real,F: real > extend8495563244428889912nnreal,M3: sigma_7234349610311085201nnreal] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ M3 ) )
       => ( probab6612481188548237749nnreal @ ( measur8829990298702910942nnreal @ M @ M3 @ F ) ) ) ) ).

% prob_space.prob_space_distr
thf(fact_1055_prob__space__distrD,axiom,
    ! [F: $o > $o,M: sigma_measure_o,N: sigma_measure_o] :
      ( ( member_o_o @ F @ ( sigma_measurable_o_o @ M @ N ) )
     => ( ( probab1190487603588612059pace_o @ ( measure_distr_o_o @ M @ N @ F ) )
       => ( probab1190487603588612059pace_o @ M ) ) ) ).

% prob_space_distrD
thf(fact_1056_prob__space__distrD,axiom,
    ! [F: b > extend8495563244428889912nnreal,M: sigma_measure_b,N: sigma_7234349610311085201nnreal] :
      ( ( member6418304549040442065nnreal @ F @ ( sigma_6334800283702579687nnreal @ M @ N ) )
     => ( ( probab6612481188548237749nnreal @ ( measur1735912521085800255nnreal @ M @ N @ F ) )
       => ( probab7247484486040049090pace_b @ M ) ) ) ).

% prob_space_distrD
thf(fact_1057_prob__space__distrD,axiom,
    ! [F: a > extend8495563244428889912nnreal,M: sigma_measure_a,N: sigma_7234349610311085201nnreal] :
      ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ N ) )
     => ( ( probab6612481188548237749nnreal @ ( measur4839436603801885502nnreal @ M @ N @ F ) )
       => ( probab7247484486040049089pace_a @ M ) ) ) ).

% prob_space_distrD
thf(fact_1058_prob__space__distrD,axiom,
    ! [F: real > a,M: sigma_measure_real,N: sigma_measure_a] :
      ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ N ) )
     => ( ( probab7247484486040049089pace_a @ ( measure_distr_real_a @ M @ N @ F ) )
       => ( probab535871623910865577e_real @ M ) ) ) ).

% prob_space_distrD
thf(fact_1059_prob__space__distrD,axiom,
    ! [F: real > $o,M: sigma_measure_real,N: sigma_measure_o] :
      ( ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ M @ N ) )
     => ( ( probab1190487603588612059pace_o @ ( measure_distr_real_o @ M @ N @ F ) )
       => ( probab535871623910865577e_real @ M ) ) ) ).

% prob_space_distrD
thf(fact_1060_prob__space__distrD,axiom,
    ! [F: real > nat,M: sigma_measure_real,N: sigma_measure_nat] :
      ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ N ) )
     => ( ( probab2904919403188438605ce_nat @ ( measur254523008267340406al_nat @ M @ N @ F ) )
       => ( probab535871623910865577e_real @ M ) ) ) ).

% prob_space_distrD
thf(fact_1061_prob__space__distrD,axiom,
    ! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,N: sigma_7234349610311085201nnreal] :
      ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ N ) )
     => ( ( probab6612481188548237749nnreal @ ( measur8829990298702910942nnreal @ M @ N @ F ) )
       => ( probab535871623910865577e_real @ M ) ) ) ).

% prob_space_distrD
thf(fact_1062_prob__space__distrD,axiom,
    ! [F: real > complex,M: sigma_measure_real,N: sigma_3077487657436305159omplex] :
      ( ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ M @ N ) )
     => ( ( probab6149883331606624555omplex @ ( measur1621797640479583060omplex @ M @ N @ F ) )
       => ( probab535871623910865577e_real @ M ) ) ) ).

% prob_space_distrD
thf(fact_1063_prob__space__distrD,axiom,
    ! [F: $o > real,M: sigma_measure_o,N: sigma_measure_real] :
      ( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ N ) )
     => ( ( probab535871623910865577e_real @ ( measure_distr_o_real @ M @ N @ F ) )
       => ( probab1190487603588612059pace_o @ M ) ) ) ).

% prob_space_distrD
thf(fact_1064_prob__space__distrD,axiom,
    ! [F: nat > real,M: sigma_measure_nat,N: sigma_measure_real] :
      ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ N ) )
     => ( ( probab535871623910865577e_real @ ( measur4910586471993217526t_real @ M @ N @ F ) )
       => ( probab2904919403188438605ce_nat @ M ) ) ) ).

% prob_space_distrD
thf(fact_1065_prob__space_Onot__empty,axiom,
    ! [M: sigma_2308072346491277622l_real] :
      ( ( probab381214941161092476l_real @ M )
     => ( ( sigma_2519298556039103681l_real @ M )
       != bot_bo3948376660626123781l_real ) ) ).

% prob_space.not_empty
thf(fact_1066_prob__space_Onot__empty,axiom,
    ! [M: sigma_5310753476256395226t_real] :
      ( ( probab2963434746955566752t_real @ M )
     => ( ( sigma_2177939267068080229t_real @ M )
       != bot_bo4585460402199822889t_real ) ) ).

% prob_space.not_empty
thf(fact_1067_prob__space_Onot__empty,axiom,
    ! [M: sigma_measure_o] :
      ( ( probab1190487603588612059pace_o @ M )
     => ( ( sigma_space_o @ M )
       != bot_bot_set_o ) ) ).

% prob_space.not_empty
thf(fact_1068_prob__space_Onot__empty,axiom,
    ! [M: sigma_measure_nat] :
      ( ( probab2904919403188438605ce_nat @ M )
     => ( ( sigma_space_nat @ M )
       != bot_bot_set_nat ) ) ).

% prob_space.not_empty
thf(fact_1069_prob__space_Onot__empty,axiom,
    ! [M: sigma_measure_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( sigma_space_real @ M )
       != bot_bot_set_real ) ) ).

% prob_space.not_empty
thf(fact_1070_measurable__empty__iff,axiom,
    ! [N: sigma_7234349610311085201nnreal,F: b > extend8495563244428889912nnreal,M: sigma_measure_b] :
      ( ( ( sigma_3147302497200244656nnreal @ N )
        = bot_bo4854962954004695426nnreal )
     => ( ( member6418304549040442065nnreal @ F @ ( sigma_6334800283702579687nnreal @ M @ N ) )
        = ( ( sigma_space_b @ M )
          = bot_bot_set_b ) ) ) ).

% measurable_empty_iff
thf(fact_1071_measurable__empty__iff,axiom,
    ! [N: sigma_7234349610311085201nnreal,F: a > extend8495563244428889912nnreal,M: sigma_measure_a] :
      ( ( ( sigma_3147302497200244656nnreal @ N )
        = bot_bo4854962954004695426nnreal )
     => ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ N ) )
        = ( ( sigma_space_a @ M )
          = bot_bot_set_a ) ) ) ).

% measurable_empty_iff
thf(fact_1072_measurable__empty__iff,axiom,
    ! [N: sigma_measure_o,F: $o > $o,M: sigma_measure_o] :
      ( ( ( sigma_space_o @ N )
        = bot_bot_set_o )
     => ( ( member_o_o @ F @ ( sigma_measurable_o_o @ M @ N ) )
        = ( ( sigma_space_o @ M )
          = bot_bot_set_o ) ) ) ).

% measurable_empty_iff
thf(fact_1073_measurable__empty__iff,axiom,
    ! [N: sigma_measure_o,F: nat > $o,M: sigma_measure_nat] :
      ( ( ( sigma_space_o @ N )
        = bot_bot_set_o )
     => ( ( member_nat_o @ F @ ( sigma_5101835498682829686_nat_o @ M @ N ) )
        = ( ( sigma_space_nat @ M )
          = bot_bot_set_nat ) ) ) ).

% measurable_empty_iff
thf(fact_1074_measurable__empty__iff,axiom,
    ! [N: sigma_measure_nat,F: $o > nat,M: sigma_measure_o] :
      ( ( ( sigma_space_nat @ N )
        = bot_bot_set_nat )
     => ( ( member_o_nat @ F @ ( sigma_1999164137574644376_o_nat @ M @ N ) )
        = ( ( sigma_space_o @ M )
          = bot_bot_set_o ) ) ) ).

% measurable_empty_iff
thf(fact_1075_measurable__empty__iff,axiom,
    ! [N: sigma_measure_nat,F: nat > nat,M: sigma_measure_nat] :
      ( ( ( sigma_space_nat @ N )
        = bot_bot_set_nat )
     => ( ( member_nat_nat @ F @ ( sigma_4350458207664084850at_nat @ M @ N ) )
        = ( ( sigma_space_nat @ M )
          = bot_bot_set_nat ) ) ) ).

% measurable_empty_iff
thf(fact_1076_measurable__empty__iff,axiom,
    ! [N: sigma_measure_o,F: real > $o,M: sigma_measure_real] :
      ( ( ( sigma_space_o @ N )
        = bot_bot_set_o )
     => ( ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ M @ N ) )
        = ( ( sigma_space_real @ M )
          = bot_bot_set_real ) ) ) ).

% measurable_empty_iff
thf(fact_1077_measurable__empty__iff,axiom,
    ! [N: sigma_measure_nat,F: real > nat,M: sigma_measure_real] :
      ( ( ( sigma_space_nat @ N )
        = bot_bot_set_nat )
     => ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ N ) )
        = ( ( sigma_space_real @ M )
          = bot_bot_set_real ) ) ) ).

% measurable_empty_iff
thf(fact_1078_measurable__empty__iff,axiom,
    ! [N: sigma_measure_real,F: $o > real,M: sigma_measure_o] :
      ( ( ( sigma_space_real @ N )
        = bot_bot_set_real )
     => ( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ N ) )
        = ( ( sigma_space_o @ M )
          = bot_bot_set_o ) ) ) ).

% measurable_empty_iff
thf(fact_1079_measurable__empty__iff,axiom,
    ! [N: sigma_measure_real,F: nat > real,M: sigma_measure_nat] :
      ( ( ( sigma_space_real @ N )
        = bot_bot_set_real )
     => ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ N ) )
        = ( ( sigma_space_nat @ M )
          = bot_bot_set_nat ) ) ) ).

% measurable_empty_iff
thf(fact_1080_space__bot,axiom,
    ( ( sigma_2519298556039103681l_real @ bot_bo1645959323927710666l_real )
    = bot_bo3948376660626123781l_real ) ).

% space_bot
thf(fact_1081_space__bot,axiom,
    ( ( sigma_2177939267068080229t_real @ bot_bo4334674888575956078t_real )
    = bot_bo4585460402199822889t_real ) ).

% space_bot
thf(fact_1082_space__bot,axiom,
    ( ( sigma_space_o @ bot_bo5758314138661044393sure_o )
    = bot_bot_set_o ) ).

% space_bot
thf(fact_1083_space__bot,axiom,
    ( ( sigma_space_nat @ bot_bo6718502177978453909re_nat )
    = bot_bot_set_nat ) ).

% space_bot
thf(fact_1084_space__bot,axiom,
    ( ( sigma_space_real @ bot_bo5982154664989874033e_real )
    = bot_bot_set_real ) ).

% space_bot
thf(fact_1085_space__empty__eq__bot,axiom,
    ! [A2: sigma_2308072346491277622l_real] :
      ( ( ( sigma_2519298556039103681l_real @ A2 )
        = bot_bo3948376660626123781l_real )
      = ( A2 = bot_bo1645959323927710666l_real ) ) ).

% space_empty_eq_bot
thf(fact_1086_space__empty__eq__bot,axiom,
    ! [A2: sigma_5310753476256395226t_real] :
      ( ( ( sigma_2177939267068080229t_real @ A2 )
        = bot_bo4585460402199822889t_real )
      = ( A2 = bot_bo4334674888575956078t_real ) ) ).

% space_empty_eq_bot
thf(fact_1087_space__empty__eq__bot,axiom,
    ! [A2: sigma_measure_o] :
      ( ( ( sigma_space_o @ A2 )
        = bot_bot_set_o )
      = ( A2 = bot_bo5758314138661044393sure_o ) ) ).

% space_empty_eq_bot
thf(fact_1088_space__empty__eq__bot,axiom,
    ! [A2: sigma_measure_nat] :
      ( ( ( sigma_space_nat @ A2 )
        = bot_bot_set_nat )
      = ( A2 = bot_bo6718502177978453909re_nat ) ) ).

% space_empty_eq_bot
thf(fact_1089_space__empty__eq__bot,axiom,
    ! [A2: sigma_measure_real] :
      ( ( ( sigma_space_real @ A2 )
        = bot_bot_set_real )
      = ( A2 = bot_bo5982154664989874033e_real ) ) ).

% space_empty_eq_bot
thf(fact_1090_prob__space_Oindep__sets__finite__index__sets,axiom,
    ! [M: sigma_measure_real,F2: set_real > set_set_real,I: set_set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe8752365572443096444t_real @ M @ F2 @ I )
        = ( ! [J2: set_set_real] :
              ( ( ord_le3558479182127378552t_real @ J2 @ I )
             => ( ( J2 != bot_bot_set_set_real )
               => ( ( finite9007344921179782393t_real @ J2 )
                 => ( indepe8752365572443096444t_real @ M @ F2 @ J2 ) ) ) ) ) ) ) ).

% prob_space.indep_sets_finite_index_sets
thf(fact_1091_prob__space_Oindep__sets__finite__index__sets,axiom,
    ! [M: sigma_measure_real,F2: real > set_set_real,I: set_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( ( indepe8783372407961027910l_real @ M @ F2 @ I )
        = ( ! [J2: set_real] :
              ( ( ord_less_eq_set_real @ J2 @ I )
             => ( ( J2 != bot_bot_set_real )
               => ( ( finite_finite_real @ J2 )
                 => ( indepe8783372407961027910l_real @ M @ F2 @ J2 ) ) ) ) ) ) ) ).

% prob_space.indep_sets_finite_index_sets
thf(fact_1092_measurable__cong__simp,axiom,
    ! [M: sigma_measure_b,N: sigma_measure_b,M3: sigma_7234349610311085201nnreal,N4: sigma_7234349610311085201nnreal,F: b > extend8495563244428889912nnreal,G2: b > extend8495563244428889912nnreal] :
      ( ( M = N )
     => ( ( M3 = N4 )
       => ( ! [W: b] :
              ( ( member_b @ W @ ( sigma_space_b @ M ) )
             => ( ( F @ W )
                = ( G2 @ W ) ) )
         => ( ( member6418304549040442065nnreal @ F @ ( sigma_6334800283702579687nnreal @ M @ M3 ) )
            = ( member6418304549040442065nnreal @ G2 @ ( sigma_6334800283702579687nnreal @ N @ N4 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_1093_measurable__cong__simp,axiom,
    ! [M: sigma_measure_a,N: sigma_measure_a,M3: sigma_7234349610311085201nnreal,N4: sigma_7234349610311085201nnreal,F: a > extend8495563244428889912nnreal,G2: a > extend8495563244428889912nnreal] :
      ( ( M = N )
     => ( ( M3 = N4 )
       => ( ! [W: a] :
              ( ( member_a @ W @ ( sigma_space_a @ M ) )
             => ( ( F @ W )
                = ( G2 @ W ) ) )
         => ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ M3 ) )
            = ( member298456594901751504nnreal @ G2 @ ( sigma_214952329563889126nnreal @ N @ N4 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_1094_measurable__cong__simp,axiom,
    ! [M: sigma_measure_o,N: sigma_measure_o,M3: sigma_measure_o,N4: sigma_measure_o,F: $o > $o,G2: $o > $o] :
      ( ( M = N )
     => ( ( M3 = N4 )
       => ( ! [W: $o] :
              ( ( member_o @ W @ ( sigma_space_o @ M ) )
             => ( ( F @ W )
                = ( G2 @ W ) ) )
         => ( ( member_o_o @ F @ ( sigma_measurable_o_o @ M @ M3 ) )
            = ( member_o_o @ G2 @ ( sigma_measurable_o_o @ N @ N4 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_1095_measurable__cong__simp,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_real,M3: sigma_measure_o,N4: sigma_measure_o,F: real > $o,G2: real > $o] :
      ( ( M = N )
     => ( ( M3 = N4 )
       => ( ! [W: real] :
              ( ( member_real @ W @ ( sigma_space_real @ M ) )
             => ( ( F @ W )
                = ( G2 @ W ) ) )
         => ( ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ M @ M3 ) )
            = ( member_real_o @ G2 @ ( sigma_3939073009482781210real_o @ N @ N4 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_1096_measurable__cong__simp,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_real,M3: sigma_measure_nat,N4: sigma_measure_nat,F: real > nat,G2: real > nat] :
      ( ( M = N )
     => ( ( M3 = N4 )
       => ( ! [W: real] :
              ( ( member_real @ W @ ( sigma_space_real @ M ) )
             => ( ( F @ W )
                = ( G2 @ W ) ) )
         => ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ M3 ) )
            = ( member_real_nat @ G2 @ ( sigma_6315060578831106510al_nat @ N @ N4 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_1097_measurable__cong__simp,axiom,
    ! [M: sigma_measure_o,N: sigma_measure_o,M3: sigma_measure_real,N4: sigma_measure_real,F: $o > real,G2: $o > real] :
      ( ( M = N )
     => ( ( M3 = N4 )
       => ( ! [W: $o] :
              ( ( member_o @ W @ ( sigma_space_o @ M ) )
             => ( ( F @ W )
                = ( G2 @ W ) ) )
         => ( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ M3 ) )
            = ( member_o_real @ G2 @ ( sigma_2430008634441611636o_real @ N @ N4 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_1098_measurable__cong__simp,axiom,
    ! [M: sigma_measure_nat,N: sigma_measure_nat,M3: sigma_measure_real,N4: sigma_measure_real,F: nat > real,G2: nat > real] :
      ( ( M = N )
     => ( ( M3 = N4 )
       => ( ! [W: nat] :
              ( ( member_nat @ W @ ( sigma_space_nat @ M ) )
             => ( ( F @ W )
                = ( G2 @ W ) ) )
         => ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ M3 ) )
            = ( member_nat_real @ G2 @ ( sigma_1747752005702207822t_real @ N @ N4 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_1099_measurable__cong__simp,axiom,
    ! [M: sigma_measure_a,N: sigma_measure_a,M3: sigma_measure_real,N4: sigma_measure_real,F: a > real,G2: a > real] :
      ( ( M = N )
     => ( ( M3 = N4 )
       => ( ! [W: a] :
              ( ( member_a @ W @ ( sigma_space_a @ M ) )
             => ( ( F @ W )
                = ( G2 @ W ) ) )
         => ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ M3 ) )
            = ( member_a_real @ G2 @ ( sigma_9116425665531756122a_real @ N @ N4 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_1100_measurable__cong__simp,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_real,M3: sigma_7234349610311085201nnreal,N4: sigma_7234349610311085201nnreal,F: real > extend8495563244428889912nnreal,G2: real > extend8495563244428889912nnreal] :
      ( ( M = N )
     => ( ( M3 = N4 )
       => ( ! [W: real] :
              ( ( member_real @ W @ ( sigma_space_real @ M ) )
             => ( ( F @ W )
                = ( G2 @ W ) ) )
         => ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ M3 ) )
            = ( member2919562650594848410nnreal @ G2 @ ( sigma_9017504469962657078nnreal @ N @ N4 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_1101_measurable__cong__simp,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_real,M3: sigma_3077487657436305159omplex,N4: sigma_3077487657436305159omplex,F: real > complex,G2: real > complex] :
      ( ( M = N )
     => ( ( M3 = N4 )
       => ( ! [W: real] :
              ( ( member_real @ W @ ( sigma_space_real @ M ) )
             => ( ( F @ W )
                = ( G2 @ W ) ) )
         => ( ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ M @ M3 ) )
            = ( member_real_complex @ G2 @ ( sigma_9111916201866572460omplex @ N @ N4 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_1102_measurable__space,axiom,
    ! [F: b > extend8495563244428889912nnreal,M: sigma_measure_b,A: sigma_7234349610311085201nnreal,X8: b] :
      ( ( member6418304549040442065nnreal @ F @ ( sigma_6334800283702579687nnreal @ M @ A ) )
     => ( ( member_b @ X8 @ ( sigma_space_b @ M ) )
       => ( member7908768830364227535nnreal @ ( F @ X8 ) @ ( sigma_3147302497200244656nnreal @ A ) ) ) ) ).

% measurable_space
thf(fact_1103_measurable__space,axiom,
    ! [F: a > extend8495563244428889912nnreal,M: sigma_measure_a,A: sigma_7234349610311085201nnreal,X8: a] :
      ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ A ) )
     => ( ( member_a @ X8 @ ( sigma_space_a @ M ) )
       => ( member7908768830364227535nnreal @ ( F @ X8 ) @ ( sigma_3147302497200244656nnreal @ A ) ) ) ) ).

% measurable_space
thf(fact_1104_measurable__space,axiom,
    ! [F: $o > $o,M: sigma_measure_o,A: sigma_measure_o,X8: $o] :
      ( ( member_o_o @ F @ ( sigma_measurable_o_o @ M @ A ) )
     => ( ( member_o @ X8 @ ( sigma_space_o @ M ) )
       => ( member_o @ ( F @ X8 ) @ ( sigma_space_o @ A ) ) ) ) ).

% measurable_space
thf(fact_1105_measurable__space,axiom,
    ! [F: $o > nat,M: sigma_measure_o,A: sigma_measure_nat,X8: $o] :
      ( ( member_o_nat @ F @ ( sigma_1999164137574644376_o_nat @ M @ A ) )
     => ( ( member_o @ X8 @ ( sigma_space_o @ M ) )
       => ( member_nat @ ( F @ X8 ) @ ( sigma_space_nat @ A ) ) ) ) ).

% measurable_space
thf(fact_1106_measurable__space,axiom,
    ! [F: nat > $o,M: sigma_measure_nat,A: sigma_measure_o,X8: nat] :
      ( ( member_nat_o @ F @ ( sigma_5101835498682829686_nat_o @ M @ A ) )
     => ( ( member_nat @ X8 @ ( sigma_space_nat @ M ) )
       => ( member_o @ ( F @ X8 ) @ ( sigma_space_o @ A ) ) ) ) ).

% measurable_space
thf(fact_1107_measurable__space,axiom,
    ! [F: nat > nat,M: sigma_measure_nat,A: sigma_measure_nat,X8: nat] :
      ( ( member_nat_nat @ F @ ( sigma_4350458207664084850at_nat @ M @ A ) )
     => ( ( member_nat @ X8 @ ( sigma_space_nat @ M ) )
       => ( member_nat @ ( F @ X8 ) @ ( sigma_space_nat @ A ) ) ) ) ).

% measurable_space
thf(fact_1108_measurable__space,axiom,
    ! [F: real > $o,M: sigma_measure_real,A: sigma_measure_o,X8: real] :
      ( ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ M @ A ) )
     => ( ( member_real @ X8 @ ( sigma_space_real @ M ) )
       => ( member_o @ ( F @ X8 ) @ ( sigma_space_o @ A ) ) ) ) ).

% measurable_space
thf(fact_1109_measurable__space,axiom,
    ! [F: real > nat,M: sigma_measure_real,A: sigma_measure_nat,X8: real] :
      ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ A ) )
     => ( ( member_real @ X8 @ ( sigma_space_real @ M ) )
       => ( member_nat @ ( F @ X8 ) @ ( sigma_space_nat @ A ) ) ) ) ).

% measurable_space
thf(fact_1110_measurable__space,axiom,
    ! [F: $o > real,M: sigma_measure_o,A: sigma_measure_real,X8: $o] :
      ( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ A ) )
     => ( ( member_o @ X8 @ ( sigma_space_o @ M ) )
       => ( member_real @ ( F @ X8 ) @ ( sigma_space_real @ A ) ) ) ) ).

% measurable_space
thf(fact_1111_measurable__space,axiom,
    ! [F: nat > real,M: sigma_measure_nat,A: sigma_measure_real,X8: nat] :
      ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ A ) )
     => ( ( member_nat @ X8 @ ( sigma_space_nat @ M ) )
       => ( member_real @ ( F @ X8 ) @ ( sigma_space_real @ A ) ) ) ) ).

% measurable_space
thf(fact_1112_measurable__cong,axiom,
    ! [M: sigma_measure_b,F: b > extend8495563244428889912nnreal,G2: b > extend8495563244428889912nnreal,M3: sigma_7234349610311085201nnreal] :
      ( ! [W: b] :
          ( ( member_b @ W @ ( sigma_space_b @ M ) )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member6418304549040442065nnreal @ F @ ( sigma_6334800283702579687nnreal @ M @ M3 ) )
        = ( member6418304549040442065nnreal @ G2 @ ( sigma_6334800283702579687nnreal @ M @ M3 ) ) ) ) ).

% measurable_cong
thf(fact_1113_measurable__cong,axiom,
    ! [M: sigma_measure_a,F: a > extend8495563244428889912nnreal,G2: a > extend8495563244428889912nnreal,M3: sigma_7234349610311085201nnreal] :
      ( ! [W: a] :
          ( ( member_a @ W @ ( sigma_space_a @ M ) )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ M3 ) )
        = ( member298456594901751504nnreal @ G2 @ ( sigma_214952329563889126nnreal @ M @ M3 ) ) ) ) ).

% measurable_cong
thf(fact_1114_measurable__cong,axiom,
    ! [M: sigma_measure_o,F: $o > $o,G2: $o > $o,M3: sigma_measure_o] :
      ( ! [W: $o] :
          ( ( member_o @ W @ ( sigma_space_o @ M ) )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_o_o @ F @ ( sigma_measurable_o_o @ M @ M3 ) )
        = ( member_o_o @ G2 @ ( sigma_measurable_o_o @ M @ M3 ) ) ) ) ).

% measurable_cong
thf(fact_1115_measurable__cong,axiom,
    ! [M: sigma_measure_real,F: real > $o,G2: real > $o,M3: sigma_measure_o] :
      ( ! [W: real] :
          ( ( member_real @ W @ ( sigma_space_real @ M ) )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ M @ M3 ) )
        = ( member_real_o @ G2 @ ( sigma_3939073009482781210real_o @ M @ M3 ) ) ) ) ).

% measurable_cong
thf(fact_1116_measurable__cong,axiom,
    ! [M: sigma_measure_real,F: real > nat,G2: real > nat,M3: sigma_measure_nat] :
      ( ! [W: real] :
          ( ( member_real @ W @ ( sigma_space_real @ M ) )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ M3 ) )
        = ( member_real_nat @ G2 @ ( sigma_6315060578831106510al_nat @ M @ M3 ) ) ) ) ).

% measurable_cong
thf(fact_1117_measurable__cong,axiom,
    ! [M: sigma_measure_o,F: $o > real,G2: $o > real,M3: sigma_measure_real] :
      ( ! [W: $o] :
          ( ( member_o @ W @ ( sigma_space_o @ M ) )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ M3 ) )
        = ( member_o_real @ G2 @ ( sigma_2430008634441611636o_real @ M @ M3 ) ) ) ) ).

% measurable_cong
thf(fact_1118_measurable__cong,axiom,
    ! [M: sigma_measure_nat,F: nat > real,G2: nat > real,M3: sigma_measure_real] :
      ( ! [W: nat] :
          ( ( member_nat @ W @ ( sigma_space_nat @ M ) )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ M3 ) )
        = ( member_nat_real @ G2 @ ( sigma_1747752005702207822t_real @ M @ M3 ) ) ) ) ).

% measurable_cong
thf(fact_1119_measurable__cong,axiom,
    ! [M: sigma_measure_a,F: a > real,G2: a > real,M3: sigma_measure_real] :
      ( ! [W: a] :
          ( ( member_a @ W @ ( sigma_space_a @ M ) )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ M3 ) )
        = ( member_a_real @ G2 @ ( sigma_9116425665531756122a_real @ M @ M3 ) ) ) ) ).

% measurable_cong
thf(fact_1120_measurable__cong,axiom,
    ! [M: sigma_measure_real,F: real > extend8495563244428889912nnreal,G2: real > extend8495563244428889912nnreal,M3: sigma_7234349610311085201nnreal] :
      ( ! [W: real] :
          ( ( member_real @ W @ ( sigma_space_real @ M ) )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ M3 ) )
        = ( member2919562650594848410nnreal @ G2 @ ( sigma_9017504469962657078nnreal @ M @ M3 ) ) ) ) ).

% measurable_cong
thf(fact_1121_measurable__cong,axiom,
    ! [M: sigma_measure_real,F: real > complex,G2: real > complex,M3: sigma_3077487657436305159omplex] :
      ( ! [W: real] :
          ( ( member_real @ W @ ( sigma_space_real @ M ) )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ M @ M3 ) )
        = ( member_real_complex @ G2 @ ( sigma_9111916201866572460omplex @ M @ M3 ) ) ) ) ).

% measurable_cong
thf(fact_1122_measurable__comp,axiom,
    ! [F: b > b,M: sigma_measure_b,N: sigma_measure_b,G2: b > extend8495563244428889912nnreal,L: sigma_7234349610311085201nnreal] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ N ) )
     => ( ( member6418304549040442065nnreal @ G2 @ ( sigma_6334800283702579687nnreal @ N @ L ) )
       => ( member6418304549040442065nnreal @ ( comp_b2719268283915226857real_b @ G2 @ F ) @ ( sigma_6334800283702579687nnreal @ M @ L ) ) ) ) ).

% measurable_comp
thf(fact_1123_measurable__comp,axiom,
    ! [F: a > b,M: sigma_measure_a,N: sigma_measure_b,G2: b > extend8495563244428889912nnreal,L: sigma_7234349610311085201nnreal] :
      ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M @ N ) )
     => ( ( member6418304549040442065nnreal @ G2 @ ( sigma_6334800283702579687nnreal @ N @ L ) )
       => ( member298456594901751504nnreal @ ( comp_b2719268283915226856real_a @ G2 @ F ) @ ( sigma_214952329563889126nnreal @ M @ L ) ) ) ) ).

% measurable_comp
thf(fact_1124_measurable__comp,axiom,
    ! [F: b > a,M: sigma_measure_b,N: sigma_measure_a,G2: a > extend8495563244428889912nnreal,L: sigma_7234349610311085201nnreal] :
      ( ( member_b_a @ F @ ( sigma_measurable_b_a @ M @ N ) )
     => ( ( member298456594901751504nnreal @ G2 @ ( sigma_214952329563889126nnreal @ N @ L ) )
       => ( member6418304549040442065nnreal @ ( comp_a6042866249568583850real_b @ G2 @ F ) @ ( sigma_6334800283702579687nnreal @ M @ L ) ) ) ) ).

% measurable_comp
thf(fact_1125_measurable__comp,axiom,
    ! [F: a > a,M: sigma_measure_a,N: sigma_measure_a,G2: a > extend8495563244428889912nnreal,L: sigma_7234349610311085201nnreal] :
      ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ N ) )
     => ( ( member298456594901751504nnreal @ G2 @ ( sigma_214952329563889126nnreal @ N @ L ) )
       => ( member298456594901751504nnreal @ ( comp_a6042866249568583849real_a @ G2 @ F ) @ ( sigma_214952329563889126nnreal @ M @ L ) ) ) ) ).

% measurable_comp
thf(fact_1126_measurable__comp,axiom,
    ! [F: $o > $o,M: sigma_measure_o,N: sigma_measure_o,G2: $o > $o,L: sigma_measure_o] :
      ( ( member_o_o @ F @ ( sigma_measurable_o_o @ M @ N ) )
     => ( ( member_o_o @ G2 @ ( sigma_measurable_o_o @ N @ L ) )
       => ( member_o_o @ ( comp_o_o_o @ G2 @ F ) @ ( sigma_measurable_o_o @ M @ L ) ) ) ) ).

% measurable_comp
thf(fact_1127_measurable__comp,axiom,
    ! [F: b > extend8495563244428889912nnreal,M: sigma_measure_b,N: sigma_7234349610311085201nnreal,G2: extend8495563244428889912nnreal > extend8495563244428889912nnreal,L: sigma_7234349610311085201nnreal] :
      ( ( member6418304549040442065nnreal @ F @ ( sigma_6334800283702579687nnreal @ M @ N ) )
     => ( ( member8329810500450651686nnreal @ G2 @ ( sigma_7926153774531450434nnreal @ N @ L ) )
       => ( member6418304549040442065nnreal @ ( comp_E1477048112619722218real_b @ G2 @ F ) @ ( sigma_6334800283702579687nnreal @ M @ L ) ) ) ) ).

% measurable_comp
thf(fact_1128_measurable__comp,axiom,
    ! [F: a > extend8495563244428889912nnreal,M: sigma_measure_a,N: sigma_7234349610311085201nnreal,G2: extend8495563244428889912nnreal > extend8495563244428889912nnreal,L: sigma_7234349610311085201nnreal] :
      ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ N ) )
     => ( ( member8329810500450651686nnreal @ G2 @ ( sigma_7926153774531450434nnreal @ N @ L ) )
       => ( member298456594901751504nnreal @ ( comp_E1477048112619722217real_a @ G2 @ F ) @ ( sigma_214952329563889126nnreal @ M @ L ) ) ) ) ).

% measurable_comp
thf(fact_1129_measurable__comp,axiom,
    ! [F: a > extend8495563244428889912nnreal,M: sigma_measure_a,N: sigma_7234349610311085201nnreal,G2: extend8495563244428889912nnreal > real,L: sigma_measure_real] :
      ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ N ) )
     => ( ( member2874014351250825754l_real @ G2 @ ( sigma_7049758200512112822l_real @ N @ L ) )
       => ( member_a_real @ ( comp_E5459461548720263541real_a @ G2 @ F ) @ ( sigma_9116425665531756122a_real @ M @ L ) ) ) ) ).

% measurable_comp
thf(fact_1130_measurable__comp,axiom,
    ! [F: real > b,M: sigma_measure_real,N: sigma_measure_b,G2: b > extend8495563244428889912nnreal,L: sigma_7234349610311085201nnreal] :
      ( ( member_real_b @ F @ ( sigma_523072396149930113real_b @ M @ N ) )
     => ( ( member6418304549040442065nnreal @ G2 @ ( sigma_6334800283702579687nnreal @ N @ L ) )
       => ( member2919562650594848410nnreal @ ( comp_b5186278242990201154l_real @ G2 @ F ) @ ( sigma_9017504469962657078nnreal @ M @ L ) ) ) ) ).

% measurable_comp
thf(fact_1131_measurable__comp,axiom,
    ! [F: real > a,M: sigma_measure_real,N: sigma_measure_a,G2: a > extend8495563244428889912nnreal,L: sigma_7234349610311085201nnreal] :
      ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ N ) )
     => ( ( member298456594901751504nnreal @ G2 @ ( sigma_214952329563889126nnreal @ N @ L ) )
       => ( member2919562650594848410nnreal @ ( comp_a8249376463644563905l_real @ G2 @ F ) @ ( sigma_9017504469962657078nnreal @ M @ L ) ) ) ) ).

% measurable_comp
thf(fact_1132_prob__space_Oprob__space__completion,axiom,
    ! [M: sigma_measure_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( probab535871623910865577e_real @ ( comple3506806835435775778n_real @ M ) ) ) ).

% prob_space.prob_space_completion
thf(fact_1133_prob__space__imp__sigma__finite,axiom,
    ! [M: sigma_measure_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( measur487378040549452491e_real @ M ) ) ).

% prob_space_imp_sigma_finite
thf(fact_1134_prob__space_Ofinite__measure,axiom,
    ! [M: sigma_measure_real] :
      ( ( probab535871623910865577e_real @ M )
     => ( measur3606880022600206024e_real @ M ) ) ).

% prob_space.finite_measure
thf(fact_1135_qp_Oindep__events__finite__index__events,axiom,
    ! [F2: set_real > set_real,I: set_set_real] :
      ( ( indepe3416441470874335588t_real @ mu @ F2 @ I )
      = ( ! [J2: set_set_real] :
            ( ( ord_le3558479182127378552t_real @ J2 @ I )
           => ( ( J2 != bot_bot_set_set_real )
             => ( ( finite9007344921179782393t_real @ J2 )
               => ( indepe3416441470874335588t_real @ mu @ F2 @ J2 ) ) ) ) ) ) ).

% qp.indep_events_finite_index_events
thf(fact_1136_qp_Oindep__events__finite__index__events,axiom,
    ! [F2: real > set_real,I: set_real] :
      ( ( indepe570525225537247534l_real @ mu @ F2 @ I )
      = ( ! [J2: set_real] :
            ( ( ord_less_eq_set_real @ J2 @ I )
           => ( ( J2 != bot_bot_set_real )
             => ( ( finite_finite_real @ J2 )
               => ( indepe570525225537247534l_real @ mu @ F2 @ J2 ) ) ) ) ) ) ).

% qp.indep_events_finite_index_events
thf(fact_1137_qp_Ochar__zero,axiom,
    ( ( characteristic_char @ mu @ zero_zero_real )
    = one_one_complex ) ).

% qp.char_zero
thf(fact_1138_sigma__finite__measure_OKL__same__eq__0,axiom,
    ! [M: sigma_measure_real,B2: real] :
      ( ( measur487378040549452491e_real @ M )
     => ( ( kL_divergence_real @ B2 @ M @ M )
        = zero_zero_real ) ) ).

% sigma_finite_measure.KL_same_eq_0
thf(fact_1139_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_1140_finite__has__minimal,axiom,
    ! [A: set_real] :
      ( ( finite_finite_real @ A )
     => ( ( A != bot_bot_set_real )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A )
            & ! [Xa2: real] :
                ( ( member_real @ Xa2 @ A )
               => ( ( ord_less_eq_real @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1141_finite__has__minimal,axiom,
    ! [A: set_set_set_real] :
      ( ( finite4560333769392153135t_real @ A )
     => ( ( A != bot_bo3378928929837779682t_real )
       => ? [X3: set_set_real] :
            ( ( member_set_set_real @ X3 @ A )
            & ! [Xa2: set_set_real] :
                ( ( member_set_set_real @ Xa2 @ A )
               => ( ( ord_le3558479182127378552t_real @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1142_finite__has__minimal,axiom,
    ! [A: set_set_real] :
      ( ( finite9007344921179782393t_real @ A )
     => ( ( A != bot_bot_set_set_real )
       => ? [X3: set_real] :
            ( ( member_set_real @ X3 @ A )
            & ! [Xa2: set_real] :
                ( ( member_set_real @ Xa2 @ A )
               => ( ( ord_less_eq_set_real @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1143_finite__has__minimal,axiom,
    ! [A: set_Ex3793607809372303086nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ? [X3: extend8495563244428889912nnreal] :
            ( ( member7908768830364227535nnreal @ X3 @ A )
            & ! [Xa2: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ Xa2 @ A )
               => ( ( ord_le3935885782089961368nnreal @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1144_finite__has__maximal,axiom,
    ! [A: set_real] :
      ( ( finite_finite_real @ A )
     => ( ( A != bot_bot_set_real )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A )
            & ! [Xa2: real] :
                ( ( member_real @ Xa2 @ A )
               => ( ( ord_less_eq_real @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1145_finite__has__maximal,axiom,
    ! [A: set_set_set_real] :
      ( ( finite4560333769392153135t_real @ A )
     => ( ( A != bot_bo3378928929837779682t_real )
       => ? [X3: set_set_real] :
            ( ( member_set_set_real @ X3 @ A )
            & ! [Xa2: set_set_real] :
                ( ( member_set_set_real @ Xa2 @ A )
               => ( ( ord_le3558479182127378552t_real @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1146_finite__has__maximal,axiom,
    ! [A: set_set_real] :
      ( ( finite9007344921179782393t_real @ A )
     => ( ( A != bot_bot_set_set_real )
       => ? [X3: set_real] :
            ( ( member_set_real @ X3 @ A )
            & ! [Xa2: set_real] :
                ( ( member_set_real @ Xa2 @ A )
               => ( ( ord_less_eq_set_real @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1147_finite__has__maximal,axiom,
    ! [A: set_Ex3793607809372303086nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ? [X3: extend8495563244428889912nnreal] :
            ( ( member7908768830364227535nnreal @ X3 @ A )
            & ! [Xa2: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ Xa2 @ A )
               => ( ( ord_le3935885782089961368nnreal @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1148_qbs__morphism__measurable__intro,axiom,
    ! [F: b > extend8495563244428889912nnreal,X: quasi_borel_b,Y: sigma_7234349610311085201nnreal] :
      ( ( member6418304549040442065nnreal @ F @ ( sigma_6334800283702579687nnreal @ ( measur7857763439677503899sure_b @ X ) @ Y ) )
     => ( member6418304549040442065nnreal @ F @ ( qbs_mo7554306597560579135nnreal @ X @ ( measur2642298986910087140nnreal @ Y ) ) ) ) ).

% qbs_morphism_measurable_intro
thf(fact_1149_qbs__morphism__measurable__intro,axiom,
    ! [F: $o > real,X: quasi_borel_o,Y: sigma_measure_real] :
      ( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ ( measur2926627334652526644sure_o @ X ) @ Y ) )
     => ( member_o_real @ F @ ( qbs_morphism_o_real @ X @ ( measur6875533127466166616s_real @ Y ) ) ) ) ).

% qbs_morphism_measurable_intro
thf(fact_1150_qbs__morphism__measurable__intro,axiom,
    ! [F: nat > real,X: quasi_borel_nat,Y: sigma_measure_real] :
      ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ ( measur7418878410283781684re_nat @ X ) @ Y ) )
     => ( member_nat_real @ F @ ( qbs_mo2000642995705457910t_real @ X @ ( measur6875533127466166616s_real @ Y ) ) ) ) ).

% qbs_morphism_measurable_intro
thf(fact_1151_qbs__morphism__measurable__intro,axiom,
    ! [F: $o > $o,X: quasi_borel_o,Y: sigma_measure_o] :
      ( ( member_o_o @ F @ ( sigma_measurable_o_o @ ( measur2926627334652526644sure_o @ X ) @ Y ) )
     => ( member_o_o @ F @ ( qbs_morphism_o_o @ X @ ( measur2705496967258476524_qbs_o @ Y ) ) ) ) ).

% qbs_morphism_measurable_intro
thf(fact_1152_qbs__morphism__measurable__intro,axiom,
    ! [F: a > extend8495563244428889912nnreal,X: quasi_borel_a,Y: sigma_7234349610311085201nnreal] :
      ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ ( measur7857763439677503898sure_a @ X ) @ Y ) )
     => ( member298456594901751504nnreal @ F @ ( qbs_mo1434458643421888574nnreal @ X @ ( measur2642298986910087140nnreal @ Y ) ) ) ) ).

% qbs_morphism_measurable_intro
thf(fact_1153_qbs__morphism__measurable__intro,axiom,
    ! [F: a > real,X: quasi_borel_a,Y: sigma_measure_real] :
      ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ ( measur7857763439677503898sure_a @ X ) @ Y ) )
     => ( member_a_real @ F @ ( qbs_morphism_a_real @ X @ ( measur6875533127466166616s_real @ Y ) ) ) ) ).

% qbs_morphism_measurable_intro
thf(fact_1154_qbs__morphism__measurable__intro,axiom,
    ! [F: a > $o,X: quasi_borel_a,Y: sigma_measure_o] :
      ( ( member_a_o @ F @ ( sigma_measurable_a_o @ ( measur7857763439677503898sure_a @ X ) @ Y ) )
     => ( member_a_o @ F @ ( qbs_morphism_a_o @ X @ ( measur2705496967258476524_qbs_o @ Y ) ) ) ) ).

% qbs_morphism_measurable_intro
thf(fact_1155_qbs__morphism__measurable__intro,axiom,
    ! [F: real > nat,X: quasi_borel_real,Y: sigma_measure_nat] :
      ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ ( measur1733462625046462224e_real @ X ) @ Y ) )
     => ( member_real_nat @ F @ ( qbs_mo6567951568834356598al_nat @ X @ ( measur4416158800429964412bs_nat @ Y ) ) ) ) ).

% qbs_morphism_measurable_intro
thf(fact_1156_qbs__morphism__measurable__intro,axiom,
    ! [F: real > complex,X: quasi_borel_real,Y: sigma_3077487657436305159omplex] :
      ( ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ ( measur1733462625046462224e_real @ X ) @ Y ) )
     => ( member_real_complex @ F @ ( qbs_mo6067097710682130004omplex @ X @ ( measur1074055046195851610omplex @ Y ) ) ) ) ).

% qbs_morphism_measurable_intro
thf(fact_1157_qbs__morphism__measurable__intro,axiom,
    ! [F: real > extend8495563244428889912nnreal,X: quasi_borel_real,Y: sigma_7234349610311085201nnreal] :
      ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ ( measur1733462625046462224e_real @ X ) @ Y ) )
     => ( member2919562650594848410nnreal @ F @ ( qbs_mo1317719164804411614nnreal @ X @ ( measur2642298986910087140nnreal @ Y ) ) ) ) ).

% qbs_morphism_measurable_intro
thf(fact_1158_rl__order__adjunction,axiom,
    ! [X: sigma_7234349610311085201nnreal,Y: quasi_9015997321629101608nnreal] :
      ( ( ord_le1854472233513649201nnreal @ X @ ( measur7384687747506661788nnreal @ Y ) )
      = ( ord_le8324202960911679112nnreal @ ( measur2642298986910087140nnreal @ X ) @ Y ) ) ).

% rl_order_adjunction
thf(fact_1159_rl__order__adjunction,axiom,
    ! [X: sigma_measure_o,Y: quasi_borel_o] :
      ( ( ord_le478349814012620405sure_o @ X @ ( measur2926627334652526644sure_o @ Y ) )
      = ( ord_le6535282547291637854orel_o @ ( measur2705496967258476524_qbs_o @ X ) @ Y ) ) ).

% rl_order_adjunction
thf(fact_1160_rl__order__adjunction,axiom,
    ! [X: sigma_measure_a,Y: quasi_borel_a] :
      ( ( ord_le254669795585780187sure_a @ X @ ( measur7857763439677503898sure_a @ Y ) )
      = ( ord_le1843388692487544644orel_a @ ( measur6507891955840068946_qbs_a @ X ) @ Y ) ) ).

% rl_order_adjunction
thf(fact_1161_rl__order__adjunction,axiom,
    ! [X: sigma_2308072346491277622l_real,Y: quasi_5476411728266286559l_real] :
      ( ( ord_le7533194067943772566l_real @ X @ ( measur6658399249024522197l_real @ Y ) )
      = ( ord_le61211656448765311l_real @ ( measur3029786804310284173l_real @ X ) @ Y ) ) ).

% rl_order_adjunction
thf(fact_1162_rl__order__adjunction,axiom,
    ! [X: sigma_5310753476256395226t_real,Y: quasi_7702169472194803971t_real] :
      ( ( ord_le8248399539298082874t_real @ X @ ( measur6583539744506187129t_real @ Y ) )
      = ( ord_le1210920900177092771t_real @ ( measur2011440102552004913t_real @ X ) @ Y ) ) ).

% rl_order_adjunction
thf(fact_1163_rl__order__adjunction,axiom,
    ! [X: sigma_measure_real,Y: quasi_borel_real] :
      ( ( ord_le487379304121309861e_real @ X @ ( measur1733462625046462224e_real @ Y ) )
      = ( ord_le1181865976659726716l_real @ ( measur6875533127466166616s_real @ X ) @ Y ) ) ).

% rl_order_adjunction
thf(fact_1164_lr__adjunction__correspondence,axiom,
    ! [X: quasi_borel_b,Y: sigma_7234349610311085201nnreal] :
      ( ( qbs_mo7554306597560579135nnreal @ X @ ( measur2642298986910087140nnreal @ Y ) )
      = ( sigma_6334800283702579687nnreal @ ( measur7857763439677503899sure_b @ X ) @ Y ) ) ).

% lr_adjunction_correspondence
thf(fact_1165_lr__adjunction__correspondence,axiom,
    ! [X: quasi_borel_o,Y: sigma_measure_real] :
      ( ( qbs_morphism_o_real @ X @ ( measur6875533127466166616s_real @ Y ) )
      = ( sigma_2430008634441611636o_real @ ( measur2926627334652526644sure_o @ X ) @ Y ) ) ).

% lr_adjunction_correspondence
thf(fact_1166_lr__adjunction__correspondence,axiom,
    ! [X: quasi_borel_nat,Y: sigma_measure_real] :
      ( ( qbs_mo2000642995705457910t_real @ X @ ( measur6875533127466166616s_real @ Y ) )
      = ( sigma_1747752005702207822t_real @ ( measur7418878410283781684re_nat @ X ) @ Y ) ) ).

% lr_adjunction_correspondence
thf(fact_1167_lr__adjunction__correspondence,axiom,
    ! [X: quasi_borel_o,Y: sigma_measure_o] :
      ( ( qbs_morphism_o_o @ X @ ( measur2705496967258476524_qbs_o @ Y ) )
      = ( sigma_measurable_o_o @ ( measur2926627334652526644sure_o @ X ) @ Y ) ) ).

% lr_adjunction_correspondence
thf(fact_1168_lr__adjunction__correspondence,axiom,
    ! [X: quasi_borel_a,Y: sigma_7234349610311085201nnreal] :
      ( ( qbs_mo1434458643421888574nnreal @ X @ ( measur2642298986910087140nnreal @ Y ) )
      = ( sigma_214952329563889126nnreal @ ( measur7857763439677503898sure_a @ X ) @ Y ) ) ).

% lr_adjunction_correspondence
thf(fact_1169_lr__adjunction__correspondence,axiom,
    ! [X: quasi_borel_a,Y: sigma_measure_real] :
      ( ( qbs_morphism_a_real @ X @ ( measur6875533127466166616s_real @ Y ) )
      = ( sigma_9116425665531756122a_real @ ( measur7857763439677503898sure_a @ X ) @ Y ) ) ).

% lr_adjunction_correspondence
thf(fact_1170_lr__adjunction__correspondence,axiom,
    ! [X: quasi_borel_a,Y: sigma_measure_o] :
      ( ( qbs_morphism_a_o @ X @ ( measur2705496967258476524_qbs_o @ Y ) )
      = ( sigma_measurable_a_o @ ( measur7857763439677503898sure_a @ X ) @ Y ) ) ).

% lr_adjunction_correspondence
thf(fact_1171_lr__adjunction__correspondence,axiom,
    ! [X: quasi_borel_real,Y: sigma_measure_nat] :
      ( ( qbs_mo6567951568834356598al_nat @ X @ ( measur4416158800429964412bs_nat @ Y ) )
      = ( sigma_6315060578831106510al_nat @ ( measur1733462625046462224e_real @ X ) @ Y ) ) ).

% lr_adjunction_correspondence
thf(fact_1172_lr__adjunction__correspondence,axiom,
    ! [X: quasi_borel_real,Y: sigma_3077487657436305159omplex] :
      ( ( qbs_mo6067097710682130004omplex @ X @ ( measur1074055046195851610omplex @ Y ) )
      = ( sigma_9111916201866572460omplex @ ( measur1733462625046462224e_real @ X ) @ Y ) ) ).

% lr_adjunction_correspondence
thf(fact_1173_lr__adjunction__correspondence,axiom,
    ! [X: quasi_borel_real,Y: sigma_7234349610311085201nnreal] :
      ( ( qbs_mo1317719164804411614nnreal @ X @ ( measur2642298986910087140nnreal @ Y ) )
      = ( sigma_9017504469962657078nnreal @ ( measur1733462625046462224e_real @ X ) @ Y ) ) ).

% lr_adjunction_correspondence
thf(fact_1174_qbs__morphism__dest,axiom,
    ! [F: b > extend8495563244428889912nnreal,X: quasi_borel_b,Y: sigma_7234349610311085201nnreal] :
      ( ( member6418304549040442065nnreal @ F @ ( qbs_mo7554306597560579135nnreal @ X @ ( measur2642298986910087140nnreal @ Y ) ) )
     => ( member6418304549040442065nnreal @ F @ ( sigma_6334800283702579687nnreal @ ( measur7857763439677503899sure_b @ X ) @ Y ) ) ) ).

% qbs_morphism_dest
thf(fact_1175_qbs__morphism__dest,axiom,
    ! [F: $o > real,X: quasi_borel_o,Y: sigma_measure_real] :
      ( ( member_o_real @ F @ ( qbs_morphism_o_real @ X @ ( measur6875533127466166616s_real @ Y ) ) )
     => ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ ( measur2926627334652526644sure_o @ X ) @ Y ) ) ) ).

% qbs_morphism_dest
thf(fact_1176_qbs__morphism__dest,axiom,
    ! [F: nat > real,X: quasi_borel_nat,Y: sigma_measure_real] :
      ( ( member_nat_real @ F @ ( qbs_mo2000642995705457910t_real @ X @ ( measur6875533127466166616s_real @ Y ) ) )
     => ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ ( measur7418878410283781684re_nat @ X ) @ Y ) ) ) ).

% qbs_morphism_dest
thf(fact_1177_qbs__morphism__dest,axiom,
    ! [F: $o > $o,X: quasi_borel_o,Y: sigma_measure_o] :
      ( ( member_o_o @ F @ ( qbs_morphism_o_o @ X @ ( measur2705496967258476524_qbs_o @ Y ) ) )
     => ( member_o_o @ F @ ( sigma_measurable_o_o @ ( measur2926627334652526644sure_o @ X ) @ Y ) ) ) ).

% qbs_morphism_dest
thf(fact_1178_qbs__morphism__dest,axiom,
    ! [F: a > extend8495563244428889912nnreal,X: quasi_borel_a,Y: sigma_7234349610311085201nnreal] :
      ( ( member298456594901751504nnreal @ F @ ( qbs_mo1434458643421888574nnreal @ X @ ( measur2642298986910087140nnreal @ Y ) ) )
     => ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ ( measur7857763439677503898sure_a @ X ) @ Y ) ) ) ).

% qbs_morphism_dest
thf(fact_1179_qbs__morphism__dest,axiom,
    ! [F: a > real,X: quasi_borel_a,Y: sigma_measure_real] :
      ( ( member_a_real @ F @ ( qbs_morphism_a_real @ X @ ( measur6875533127466166616s_real @ Y ) ) )
     => ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ ( measur7857763439677503898sure_a @ X ) @ Y ) ) ) ).

% qbs_morphism_dest
thf(fact_1180_qbs__morphism__dest,axiom,
    ! [F: a > $o,X: quasi_borel_a,Y: sigma_measure_o] :
      ( ( member_a_o @ F @ ( qbs_morphism_a_o @ X @ ( measur2705496967258476524_qbs_o @ Y ) ) )
     => ( member_a_o @ F @ ( sigma_measurable_a_o @ ( measur7857763439677503898sure_a @ X ) @ Y ) ) ) ).

% qbs_morphism_dest
thf(fact_1181_qbs__morphism__dest,axiom,
    ! [F: real > nat,X: quasi_borel_real,Y: sigma_measure_nat] :
      ( ( member_real_nat @ F @ ( qbs_mo6567951568834356598al_nat @ X @ ( measur4416158800429964412bs_nat @ Y ) ) )
     => ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ ( measur1733462625046462224e_real @ X ) @ Y ) ) ) ).

% qbs_morphism_dest
thf(fact_1182_qbs__morphism__dest,axiom,
    ! [F: real > complex,X: quasi_borel_real,Y: sigma_3077487657436305159omplex] :
      ( ( member_real_complex @ F @ ( qbs_mo6067097710682130004omplex @ X @ ( measur1074055046195851610omplex @ Y ) ) )
     => ( member_real_complex @ F @ ( sigma_9111916201866572460omplex @ ( measur1733462625046462224e_real @ X ) @ Y ) ) ) ).

% qbs_morphism_dest
thf(fact_1183_qbs__morphism__dest,axiom,
    ! [F: real > extend8495563244428889912nnreal,X: quasi_borel_real,Y: sigma_7234349610311085201nnreal] :
      ( ( member2919562650594848410nnreal @ F @ ( qbs_mo1317719164804411614nnreal @ X @ ( measur2642298986910087140nnreal @ Y ) ) )
     => ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ ( measur1733462625046462224e_real @ X ) @ Y ) ) ) ).

% qbs_morphism_dest
thf(fact_1184_qbs__Mx__subset__of__measurable,axiom,
    ! [X: quasi_borel_b] : ( ord_le5814440863667440394real_b @ ( qbs_Mx_b @ X ) @ ( sigma_523072396149930113real_b @ borel_5078946678739801102l_real @ ( measur7857763439677503899sure_b @ X ) ) ) ).

% qbs_Mx_subset_of_measurable
thf(fact_1185_qbs__Mx__subset__of__measurable,axiom,
    ! [X: quasi_borel_o] : ( ord_le1615110227528160547real_o @ ( qbs_Mx_o @ X ) @ ( sigma_3939073009482781210real_o @ borel_5078946678739801102l_real @ ( measur2926627334652526644sure_o @ X ) ) ) ).

% qbs_Mx_subset_of_measurable
thf(fact_1186_qbs__Mx__subset__of__measurable,axiom,
    ! [X: quasi_borel_nat] : ( ord_le6098800555920186673al_nat @ ( qbs_Mx_nat @ X ) @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ ( measur7418878410283781684re_nat @ X ) ) ) ).

% qbs_Mx_subset_of_measurable
thf(fact_1187_qbs__Mx__subset__of__measurable,axiom,
    ! [X: quasi_9015997321629101608nnreal] : ( ord_le2462468573666744473nnreal @ ( qbs_Mx6523938229262583809nnreal @ X ) @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ ( measur7384687747506661788nnreal @ X ) ) ) ).

% qbs_Mx_subset_of_measurable
thf(fact_1188_qbs__Mx__subset__of__measurable,axiom,
    ! [X: quasi_borel_complex] : ( ord_le2047140485929309711omplex @ ( qbs_Mx_complex @ X ) @ ( sigma_9111916201866572460omplex @ borel_5078946678739801102l_real @ ( measur3826415497239753490omplex @ X ) ) ) ).

% qbs_Mx_subset_of_measurable
thf(fact_1189_qbs__Mx__subset__of__measurable,axiom,
    ! [X: quasi_borel_a] : ( ord_le5743406823621094409real_a @ ( qbs_Mx_a @ X ) @ ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ ( measur7857763439677503898sure_a @ X ) ) ) ).

% qbs_Mx_subset_of_measurable
thf(fact_1190_qbs__Mx__subset__of__measurable,axiom,
    ! [X: quasi_5476411728266286559l_real] : ( ord_le3446602116853026244l_real @ ( qbs_Mx2863504943393711152l_real @ X ) @ ( sigma_7998147297565726139l_real @ borel_5078946678739801102l_real @ ( measur6658399249024522197l_real @ X ) ) ) ).

% qbs_Mx_subset_of_measurable
thf(fact_1191_qbs__Mx__subset__of__measurable,axiom,
    ! [X: quasi_7702169472194803971t_real] : ( ord_le3197061595610012648t_real @ ( qbs_Mx6884344600386557780t_real @ X ) @ ( sigma_2975298441655967327t_real @ borel_5078946678739801102l_real @ ( measur6583539744506187129t_real @ X ) ) ) ).

% qbs_Mx_subset_of_measurable
thf(fact_1192_qbs__Mx__subset__of__measurable,axiom,
    ! [X: quasi_borel_real] : ( ord_le4198349162570665613l_real @ ( qbs_Mx_real @ X ) @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ ( measur1733462625046462224e_real @ X ) ) ) ).

% qbs_Mx_subset_of_measurable
thf(fact_1193_qbs__Mx__are__measurable,axiom,
    ! [Alpha: real > b,X: quasi_borel_b] :
      ( ( member_real_b @ Alpha @ ( qbs_Mx_b @ X ) )
     => ( member_real_b @ Alpha @ ( sigma_523072396149930113real_b @ borel_5078946678739801102l_real @ ( measur7857763439677503899sure_b @ X ) ) ) ) ).

% qbs_Mx_are_measurable
thf(fact_1194_qbs__Mx__are__measurable,axiom,
    ! [Alpha: real > $o,X: quasi_borel_o] :
      ( ( member_real_o @ Alpha @ ( qbs_Mx_o @ X ) )
     => ( member_real_o @ Alpha @ ( sigma_3939073009482781210real_o @ borel_5078946678739801102l_real @ ( measur2926627334652526644sure_o @ X ) ) ) ) ).

% qbs_Mx_are_measurable
thf(fact_1195_qbs__Mx__are__measurable,axiom,
    ! [Alpha: real > nat,X: quasi_borel_nat] :
      ( ( member_real_nat @ Alpha @ ( qbs_Mx_nat @ X ) )
     => ( member_real_nat @ Alpha @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ ( measur7418878410283781684re_nat @ X ) ) ) ) ).

% qbs_Mx_are_measurable
thf(fact_1196_qbs__Mx__are__measurable,axiom,
    ! [Alpha: real > extend8495563244428889912nnreal,X: quasi_9015997321629101608nnreal] :
      ( ( member2919562650594848410nnreal @ Alpha @ ( qbs_Mx6523938229262583809nnreal @ X ) )
     => ( member2919562650594848410nnreal @ Alpha @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ ( measur7384687747506661788nnreal @ X ) ) ) ) ).

% qbs_Mx_are_measurable
thf(fact_1197_qbs__Mx__are__measurable,axiom,
    ! [Alpha: real > complex,X: quasi_borel_complex] :
      ( ( member_real_complex @ Alpha @ ( qbs_Mx_complex @ X ) )
     => ( member_real_complex @ Alpha @ ( sigma_9111916201866572460omplex @ borel_5078946678739801102l_real @ ( measur3826415497239753490omplex @ X ) ) ) ) ).

% qbs_Mx_are_measurable
thf(fact_1198_qbs__Mx__are__measurable,axiom,
    ! [Alpha: real > a,X: quasi_borel_a] :
      ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X ) )
     => ( member_real_a @ Alpha @ ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ ( measur7857763439677503898sure_a @ X ) ) ) ) ).

% qbs_Mx_are_measurable
thf(fact_1199_qbs__Mx__are__measurable,axiom,
    ! [Alpha: real > produc2422161461964618553l_real,X: quasi_5476411728266286559l_real] :
      ( ( member9086635009091248365l_real @ Alpha @ ( qbs_Mx2863504943393711152l_real @ X ) )
     => ( member9086635009091248365l_real @ Alpha @ ( sigma_7998147297565726139l_real @ borel_5078946678739801102l_real @ ( measur6658399249024522197l_real @ X ) ) ) ) ).

% qbs_Mx_are_measurable
thf(fact_1200_qbs__Mx__are__measurable,axiom,
    ! [Alpha: real > produc7716430852924023517t_real,X: quasi_7702169472194803971t_real] :
      ( ( member969486235890772753t_real @ Alpha @ ( qbs_Mx6884344600386557780t_real @ X ) )
     => ( member969486235890772753t_real @ Alpha @ ( sigma_2975298441655967327t_real @ borel_5078946678739801102l_real @ ( measur6583539744506187129t_real @ X ) ) ) ) ).

% qbs_Mx_are_measurable
thf(fact_1201_qbs__Mx__are__measurable,axiom,
    ! [Alpha: real > real,X: quasi_borel_real] :
      ( ( member_real_real @ Alpha @ ( qbs_Mx_real @ X ) )
     => ( member_real_real @ Alpha @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ ( measur1733462625046462224e_real @ X ) ) ) ) ).

% qbs_Mx_are_measurable
thf(fact_1202_l__preserves__morphisms,axiom,
    ! [X: quasi_7702169472194803971t_real,Y: quasi_7702169472194803971t_real] : ( ord_le8153730267393820219t_real @ ( qbs_mo1022283140530407736t_real @ X @ Y ) @ ( sigma_6707289319217010064t_real @ ( measur6583539744506187129t_real @ X ) @ ( measur6583539744506187129t_real @ Y ) ) ) ).

% l_preserves_morphisms
thf(fact_1203_l__preserves__morphisms,axiom,
    ! [X: quasi_7702169472194803971t_real,Y: quasi_borel_real] : ( ord_le5677405320596342778l_real @ ( qbs_mo8506632985670901265l_real @ X @ Y ) @ ( sigma_7880537205408866233l_real @ ( measur6583539744506187129t_real @ X ) @ ( measur1733462625046462224e_real @ Y ) ) ) ).

% l_preserves_morphisms
thf(fact_1204_l__preserves__morphisms,axiom,
    ! [X: quasi_borel_real,Y: quasi_borel_o] : ( ord_le1615110227528160547real_o @ ( qbs_morphism_real_o @ X @ Y ) @ ( sigma_3939073009482781210real_o @ ( measur1733462625046462224e_real @ X ) @ ( measur2926627334652526644sure_o @ Y ) ) ) ).

% l_preserves_morphisms
thf(fact_1205_l__preserves__morphisms,axiom,
    ! [X: quasi_borel_real,Y: quasi_borel_nat] : ( ord_le6098800555920186673al_nat @ ( qbs_mo6567951568834356598al_nat @ X @ Y ) @ ( sigma_6315060578831106510al_nat @ ( measur1733462625046462224e_real @ X ) @ ( measur7418878410283781684re_nat @ Y ) ) ) ).

% l_preserves_morphisms
thf(fact_1206_l__preserves__morphisms,axiom,
    ! [X: quasi_borel_real,Y: quasi_9015997321629101608nnreal] : ( ord_le2462468573666744473nnreal @ ( qbs_mo1317719164804411614nnreal @ X @ Y ) @ ( sigma_9017504469962657078nnreal @ ( measur1733462625046462224e_real @ X ) @ ( measur7384687747506661788nnreal @ Y ) ) ) ).

% l_preserves_morphisms
thf(fact_1207_l__preserves__morphisms,axiom,
    ! [X: quasi_borel_real,Y: quasi_borel_complex] : ( ord_le2047140485929309711omplex @ ( qbs_mo6067097710682130004omplex @ X @ Y ) @ ( sigma_9111916201866572460omplex @ ( measur1733462625046462224e_real @ X ) @ ( measur3826415497239753490omplex @ Y ) ) ) ).

% l_preserves_morphisms
thf(fact_1208_l__preserves__morphisms,axiom,
    ! [X: quasi_borel_real,Y: quasi_borel_a] : ( ord_le5743406823621094409real_a @ ( qbs_morphism_real_a @ X @ Y ) @ ( sigma_523072396149930112real_a @ ( measur1733462625046462224e_real @ X ) @ ( measur7857763439677503898sure_a @ Y ) ) ) ).

% l_preserves_morphisms
thf(fact_1209_l__preserves__morphisms,axiom,
    ! [X: quasi_borel_real,Y: quasi_5476411728266286559l_real] : ( ord_le3446602116853026244l_real @ ( qbs_mo6845038431372961811l_real @ X @ Y ) @ ( sigma_7998147297565726139l_real @ ( measur1733462625046462224e_real @ X ) @ ( measur6658399249024522197l_real @ Y ) ) ) ).

% l_preserves_morphisms
thf(fact_1210_l__preserves__morphisms,axiom,
    ! [X: quasi_borel_real,Y: quasi_7702169472194803971t_real] : ( ord_le3197061595610012648t_real @ ( qbs_mo3601394221918002359t_real @ X @ Y ) @ ( sigma_2975298441655967327t_real @ ( measur1733462625046462224e_real @ X ) @ ( measur6583539744506187129t_real @ Y ) ) ) ).

% l_preserves_morphisms
thf(fact_1211_l__preserves__morphisms,axiom,
    ! [X: quasi_borel_real,Y: quasi_borel_real] : ( ord_le4198349162570665613l_real @ ( qbs_mo5229770564518008146l_real @ X @ Y ) @ ( sigma_5267869275261027754l_real @ ( measur1733462625046462224e_real @ X ) @ ( measur1733462625046462224e_real @ Y ) ) ) ).

% l_preserves_morphisms
thf(fact_1212_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_1213_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_1214_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_1215_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_1216_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_1217_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
          @ ^ [X6: real] : ( F @ ( alpha @ X6 ) ) ) ) ) ).

% qp.qbs_prob_ennintegral_def
thf(fact_1218_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_1219_not__qbs__morphism,axiom,
    member_o_o @ (~) @ ( qbs_morphism_o_o @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) ) ).

% not_qbs_morphism
thf(fact_1220_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
          @ ^ [X6: real] : ( F @ ( alpha @ X6 ) ) ) ) ) ).

% qp.qbs_integrable_iff_integrable
thf(fact_1221_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
          @ ^ [X6: real] : ( F @ ( alpha @ X6 ) ) ) ) ) ).

% qp.qbs_prob_integral_def
thf(fact_1222_qp_Oobtain__positive__integrable__function,axiom,
    ~ ! [F4: real > real] :
        ( ( member_real_real @ F4 @ ( sigma_5267869275261027754l_real @ mu @ borel_5078946678739801102l_real ) )
       => ( ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( F4 @ X4 ) )
         => ( ! [X4: real] : ( ord_less_eq_real @ ( F4 @ X4 ) @ one_one_real )
           => ~ ( bochne3340023020068487468l_real @ mu @ F4 ) ) ) ) ).

% qp.obtain_positive_integrable_function
thf(fact_1223_real__real_Ostandard__borel__lr__sets__ident,axiom,
    ( ( sigma_1543982426361076760l_real @ ( measur6658399249024522197l_real @ ( measur3029786804310284173l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) ) )
    = ( sigma_1543982426361076760l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) ) ).

% real_real.standard_borel_lr_sets_ident
thf(fact_1224_nat__real_Ostandard__borel__lr__sets__ident,axiom,
    ( ( sigma_2363344875859395388t_real @ ( measur6583539744506187129t_real @ ( measur2011440102552004913t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) ) ) )
    = ( sigma_2363344875859395388t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) ) ) ).

% nat_real.standard_borel_lr_sets_ident
thf(fact_1225_real_Ostandard__borel__lr__sets__ident,axiom,
    ( ( sigma_sets_real @ ( measur1733462625046462224e_real @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) ) )
    = ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ).

% real.standard_borel_lr_sets_ident
thf(fact_1226_qp_Oevents__eq__borel,axiom,
    ( ( sigma_sets_real @ mu )
    = ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ).

% qp.events_eq_borel
thf(fact_1227_qp_Osets__M,axiom,
    ! [A2: set_real] :
      ( ( member_set_real @ A2 @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
     => ( member_set_real @ A2 @ ( sigma_sets_real @ mu ) ) ) ).

% qp.sets_M
thf(fact_1228_qp_Ointegrable__cts__step,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( bochne3340023020068487468l_real @ mu @ ( weak_cts_step @ A2 @ B2 ) ) ) ).

% qp.integrable_cts_step
thf(fact_1229_real__real_Oexist__fg,axiom,
    ? [X3: produc2422161461964618553l_real > real] :
      ( ( member6699615393305559423l_real @ X3 @ ( sigma_8002782794886939285l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) @ borel_5078946678739801102l_real ) )
      & ? [Xa: real > produc2422161461964618553l_real] :
          ( ( member9086635009091248365l_real @ Xa @ ( sigma_7998147297565726139l_real @ borel_5078946678739801102l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) )
          & ! [Xb: produc2422161461964618553l_real] :
              ( ( member7849222048561428706l_real @ Xb @ ( sigma_2519298556039103681l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) )
             => ( ( comp_r97482993418849103l_real @ Xa @ X3 @ Xb )
                = Xb ) ) ) ) ).

% real_real.exist_fg
thf(fact_1230_nat__real_Oexist__fg,axiom,
    ? [X3: produc7716430852924023517t_real > real] :
      ( ( member4539693599630147619l_real @ X3 @ ( sigma_7880537205408866233l_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) @ borel_5078946678739801102l_real ) )
      & ? [Xa: real > produc7716430852924023517t_real] :
          ( ( member969486235890772753t_real @ Xa @ ( sigma_2975298441655967327t_real @ borel_5078946678739801102l_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) ) )
          & ! [Xb: produc7716430852924023517t_real] :
              ( ( member557208447399453958t_real @ Xb @ ( sigma_2177939267068080229t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) ) )
             => ( ( comp_r4825950202331181903t_real @ Xa @ X3 @ Xb )
                = Xb ) ) ) ) ).

% nat_real.exist_fg
thf(fact_1231_real__distribution_Oevents__eq__borel,axiom,
    ! [M: sigma_measure_real] :
      ( ( distri2809703520229113005bution @ M )
     => ( ( sigma_sets_real @ M )
        = ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ).

% real_distribution.events_eq_borel
thf(fact_1232_finite__borel__measure_OM__is__borel,axiom,
    ! [M: sigma_measure_real] :
      ( ( distri7943378551711771532easure @ M )
     => ( ( sigma_sets_real @ M )
        = ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ).

% finite_borel_measure.M_is_borel
thf(fact_1233_finite__borel__measure_Osets__M,axiom,
    ! [M: sigma_measure_real,A2: set_real] :
      ( ( distri7943378551711771532easure @ M )
     => ( ( member_set_real @ A2 @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
       => ( member_set_real @ A2 @ ( sigma_sets_real @ M ) ) ) ) ).

% finite_borel_measure.sets_M
thf(fact_1234_qp_Otail__events__sets,axiom,
    ! [A: nat > set_set_real] :
      ( ! [I2: nat] : ( ord_le3558479182127378552t_real @ ( A @ I2 ) @ ( sigma_sets_real @ mu ) )
     => ( ord_le3558479182127378552t_real @ ( indepe6490915699233070710al_nat @ mu @ A ) @ ( sigma_sets_real @ mu ) ) ) ).

% qp.tail_events_sets
thf(fact_1235_qp_Omeasure__ge__1__iff,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_real @ one_one_real @ ( sigma_measure_real2 @ mu @ A ) )
      = ( ( sigma_measure_real2 @ mu @ A )
        = one_one_real ) ) ).

% qp.measure_ge_1_iff
thf(fact_1236_qp_Ofinite__measure__mono,axiom,
    ! [A: set_real,B: set_real] :
      ( ( ord_less_eq_set_real @ A @ B )
     => ( ( member_set_real @ B @ ( sigma_sets_real @ mu ) )
       => ( ord_less_eq_real @ ( sigma_measure_real2 @ mu @ A ) @ ( sigma_measure_real2 @ mu @ B ) ) ) ) ).

% qp.finite_measure_mono
thf(fact_1237_qp_Obounded__measure,axiom,
    ! [A: set_real] : ( ord_less_eq_real @ ( sigma_measure_real2 @ mu @ A ) @ ( sigma_measure_real2 @ mu @ ( sigma_space_real @ mu ) ) ) ).

% qp.bounded_measure
thf(fact_1238_qp_Oprob__space,axiom,
    ( ( sigma_measure_real2 @ mu @ ( sigma_space_real @ mu ) )
    = one_one_real ) ).

% qp.prob_space
thf(fact_1239_qp_Oprob__le__1,axiom,
    ! [A: set_real] : ( ord_less_eq_real @ ( sigma_measure_real2 @ mu @ A ) @ one_one_real ) ).

% qp.prob_le_1
thf(fact_1240_qp_Osubprob__measure__le__1,axiom,
    ! [X: set_real] : ( ord_less_eq_real @ ( sigma_measure_real2 @ mu @ X ) @ one_one_real ) ).

% qp.subprob_measure_le_1
thf(fact_1241_qp_Omeasure__increasing,axiom,
    measur4480787322886042509l_real @ ( sigma_sets_real @ mu ) @ ( sigma_measure_real2 @ mu ) ).

% qp.measure_increasing
thf(fact_1242_qp_Oindep__setD__ev2,axiom,
    ! [A: set_set_real,B: set_set_real] :
      ( ( indepe5067751462249938772t_real @ mu @ A @ B )
     => ( ord_le3558479182127378552t_real @ B @ ( sigma_sets_real @ mu ) ) ) ).

% qp.indep_setD_ev2
thf(fact_1243_qp_Oindep__setD__ev1,axiom,
    ! [A: set_set_real,B: set_set_real] :
      ( ( indepe5067751462249938772t_real @ mu @ A @ B )
     => ( ord_le3558479182127378552t_real @ A @ ( sigma_sets_real @ mu ) ) ) ).

% qp.indep_setD_ev1
thf(fact_1244_qp_OEx__finite__integrable__function,axiom,
    ? [X3: real > extend8495563244428889912nnreal] :
      ( ( member2919562650594848410nnreal @ X3 @ ( sigma_9017504469962657078nnreal @ mu @ borel_6524799422816628122nnreal ) )
      & ( ( nonneg2667834350952324695l_real @ mu @ X3 )
       != extend2057119558705770725nnreal )
      & ! [Xa2: real] :
          ( ( member_real @ Xa2 @ ( sigma_space_real @ mu ) )
         => ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( X3 @ Xa2 ) )
            & ( ord_le7381754540660121996nnreal @ ( X3 @ Xa2 ) @ extend2057119558705770725nnreal ) ) ) ) ).

% qp.Ex_finite_integrable_function
thf(fact_1245_qp_Omeasure__exclude,axiom,
    ! [A: set_real,B: set_real] :
      ( ( member_set_real @ A @ ( sigma_sets_real @ mu ) )
     => ( ( member_set_real @ B @ ( sigma_sets_real @ mu ) )
       => ( ( ( sigma_measure_real2 @ mu @ A )
            = ( sigma_measure_real2 @ mu @ ( sigma_space_real @ mu ) ) )
         => ( ( ( inf_inf_set_real @ A @ B )
              = bot_bot_set_real )
           => ( ( sigma_measure_real2 @ mu @ B )
              = zero_zero_real ) ) ) ) ) ).

% qp.measure_exclude
thf(fact_1246_qp_Oindep__sets2__eq,axiom,
    ! [A: set_set_real,B: set_set_real] :
      ( ( indepe5067751462249938772t_real @ mu @ A @ B )
      = ( ( ord_le3558479182127378552t_real @ A @ ( sigma_sets_real @ mu ) )
        & ( ord_le3558479182127378552t_real @ B @ ( sigma_sets_real @ mu ) )
        & ! [X6: set_real] :
            ( ( member_set_real @ X6 @ A )
           => ! [Y3: set_real] :
                ( ( member_set_real @ Y3 @ B )
               => ( ( sigma_measure_real2 @ mu @ ( inf_inf_set_real @ X6 @ Y3 ) )
                  = ( times_times_real @ ( sigma_measure_real2 @ mu @ X6 ) @ ( sigma_measure_real2 @ mu @ Y3 ) ) ) ) ) ) ) ).

% qp.indep_sets2_eq
thf(fact_1247_qp_Omeasure__space__inter,axiom,
    ! [S2: set_real,T3: set_real] :
      ( ( member_set_real @ S2 @ ( sigma_sets_real @ mu ) )
     => ( ( member_set_real @ T3 @ ( sigma_sets_real @ mu ) )
       => ( ( ( sigma_measure_real2 @ mu @ T3 )
            = ( sigma_measure_real2 @ mu @ ( sigma_space_real @ mu ) ) )
         => ( ( sigma_measure_real2 @ mu @ ( inf_inf_set_real @ S2 @ T3 ) )
            = ( sigma_measure_real2 @ mu @ S2 ) ) ) ) ) ).

% qp.measure_space_inter
thf(fact_1248_qp_Oindep__setD,axiom,
    ! [A: set_set_real,B: set_set_real,A2: set_real,B2: set_real] :
      ( ( indepe5067751462249938772t_real @ mu @ A @ B )
     => ( ( member_set_real @ A2 @ A )
       => ( ( member_set_real @ B2 @ B )
         => ( ( sigma_measure_real2 @ mu @ ( inf_inf_set_real @ A2 @ B2 ) )
            = ( times_times_real @ ( sigma_measure_real2 @ mu @ A2 ) @ ( sigma_measure_real2 @ mu @ B2 ) ) ) ) ) ) ).

% qp.indep_setD
thf(fact_1249_qp_Oindep__setI,axiom,
    ! [A: set_set_real,B: set_set_real] :
      ( ( ord_le3558479182127378552t_real @ A @ ( sigma_sets_real @ mu ) )
     => ( ( ord_le3558479182127378552t_real @ B @ ( sigma_sets_real @ mu ) )
       => ( ! [A4: set_real,B4: set_real] :
              ( ( member_set_real @ A4 @ A )
             => ( ( member_set_real @ B4 @ B )
               => ( ( sigma_measure_real2 @ mu @ ( inf_inf_set_real @ A4 @ B4 ) )
                  = ( times_times_real @ ( sigma_measure_real2 @ mu @ A4 ) @ ( sigma_measure_real2 @ mu @ B4 ) ) ) ) )
         => ( indepe5067751462249938772t_real @ mu @ A @ B ) ) ) ) ).

% qp.indep_setI
thf(fact_1250_qp_Ocdf__cts__step_I2_J,axiom,
    ! [X8: real,Y4: real] :
      ( ( ord_less_real @ X8 @ Y4 )
     => ( ord_less_eq_real @ ( bochne3715101410578510557l_real @ mu @ ( weak_cts_step @ X8 @ Y4 ) ) @ ( distribution_cdf @ mu @ Y4 ) ) ) ).

% qp.cdf_cts_step(2)
thf(fact_1251_qp_Ocdf__cts__step_I1_J,axiom,
    ! [X8: real,Y4: real] :
      ( ( ord_less_real @ X8 @ Y4 )
     => ( ord_less_eq_real @ ( distribution_cdf @ mu @ X8 ) @ ( bochne3715101410578510557l_real @ mu @ ( weak_cts_step @ X8 @ Y4 ) ) ) ) ).

% qp.cdf_cts_step(1)
thf(fact_1252_qp_Oprob__neg,axiom,
    ! [P2: real > $o] :
      ( ( member_set_real
        @ ( collect_real
          @ ^ [X6: real] :
              ( ( member_real @ X6 @ ( sigma_space_real @ mu ) )
              & ( P2 @ X6 ) ) )
        @ ( sigma_sets_real @ mu ) )
     => ( ( sigma_measure_real2 @ mu
          @ ( collect_real
            @ ^ [X6: real] :
                ( ( member_real @ X6 @ ( sigma_space_real @ mu ) )
                & ~ ( P2 @ X6 ) ) ) )
        = ( minus_minus_real @ one_one_real
          @ ( sigma_measure_real2 @ mu
            @ ( collect_real
              @ ^ [X6: real] :
                  ( ( member_real @ X6 @ ( sigma_space_real @ mu ) )
                  & ( P2 @ X6 ) ) ) ) ) ) ) ).

% qp.prob_neg
thf(fact_1253_qp_Ocdf__nondecreasing,axiom,
    ! [X8: real,Y4: real] :
      ( ( ord_less_eq_real @ X8 @ Y4 )
     => ( ord_less_eq_real @ ( distribution_cdf @ mu @ X8 ) @ ( distribution_cdf @ mu @ Y4 ) ) ) ).

% qp.cdf_nondecreasing
thf(fact_1254_qp_Ocdf__nonneg,axiom,
    ! [X8: real] : ( ord_less_eq_real @ zero_zero_real @ ( distribution_cdf @ mu @ X8 ) ) ).

% qp.cdf_nonneg
thf(fact_1255_qp_Ocdf__bounded__prob,axiom,
    ! [X8: real] : ( ord_less_eq_real @ ( distribution_cdf @ mu @ X8 ) @ one_one_real ) ).

% qp.cdf_bounded_prob
thf(fact_1256_qp_Ocdf__bounded,axiom,
    ! [X8: real] : ( ord_less_eq_real @ ( distribution_cdf @ mu @ X8 ) @ ( sigma_measure_real2 @ mu @ ( sigma_space_real @ mu ) ) ) ).

% qp.cdf_bounded
thf(fact_1257_cdf__unique_H,axiom,
    ! [M1: sigma_measure_real,M2: sigma_measure_real] :
      ( ( distri7943378551711771532easure @ M1 )
     => ( ( distri7943378551711771532easure @ M2 )
       => ( ( ( distribution_cdf @ M1 )
            = ( distribution_cdf @ M2 ) )
         => ( M1 = M2 ) ) ) ) ).

% cdf_unique'
thf(fact_1258_cdf__unique,axiom,
    ! [M1: sigma_measure_real,M2: sigma_measure_real] :
      ( ( distri2809703520229113005bution @ M1 )
     => ( ( distri2809703520229113005bution @ M2 )
       => ( ( ( distribution_cdf @ M1 )
            = ( distribution_cdf @ M2 ) )
         => ( M1 = M2 ) ) ) ) ).

% cdf_unique
thf(fact_1259_finite__borel__measure_Ocdf__nondecreasing,axiom,
    ! [M: sigma_measure_real,X8: real,Y4: real] :
      ( ( distri7943378551711771532easure @ M )
     => ( ( ord_less_eq_real @ X8 @ Y4 )
       => ( ord_less_eq_real @ ( distribution_cdf @ M @ X8 ) @ ( distribution_cdf @ M @ Y4 ) ) ) ) ).

% finite_borel_measure.cdf_nondecreasing
thf(fact_1260_real__distribution_Ocdf__bounded__prob,axiom,
    ! [M: sigma_measure_real,X8: real] :
      ( ( distri2809703520229113005bution @ M )
     => ( ord_less_eq_real @ ( distribution_cdf @ M @ X8 ) @ one_one_real ) ) ).

% real_distribution.cdf_bounded_prob
thf(fact_1261_finite__borel__measure_Ocdf__nonneg,axiom,
    ! [M: sigma_measure_real,X8: real] :
      ( ( distri7943378551711771532easure @ M )
     => ( ord_less_eq_real @ zero_zero_real @ ( distribution_cdf @ M @ X8 ) ) ) ).

% finite_borel_measure.cdf_nonneg
thf(fact_1262_finite__borel__measure_Ocdf__bounded,axiom,
    ! [M: sigma_measure_real,X8: real] :
      ( ( distri7943378551711771532easure @ M )
     => ( ord_less_eq_real @ ( distribution_cdf @ M @ X8 ) @ ( sigma_measure_real2 @ M @ ( sigma_space_real @ M ) ) ) ) ).

% finite_borel_measure.cdf_bounded
thf(fact_1263_real__distribution_Ocdf__cts__step_I2_J,axiom,
    ! [M: sigma_measure_real,X8: real,Y4: real] :
      ( ( distri2809703520229113005bution @ M )
     => ( ( ord_less_real @ X8 @ Y4 )
       => ( ord_less_eq_real @ ( bochne3715101410578510557l_real @ M @ ( weak_cts_step @ X8 @ Y4 ) ) @ ( distribution_cdf @ M @ Y4 ) ) ) ) ).

% real_distribution.cdf_cts_step(2)
thf(fact_1264_real__distribution_Ocdf__cts__step_I1_J,axiom,
    ! [M: sigma_measure_real,X8: real,Y4: real] :
      ( ( distri2809703520229113005bution @ M )
     => ( ( ord_less_real @ X8 @ Y4 )
       => ( ord_less_eq_real @ ( distribution_cdf @ M @ X8 ) @ ( bochne3715101410578510557l_real @ M @ ( weak_cts_step @ X8 @ Y4 ) ) ) ) ) ).

% real_distribution.cdf_cts_step(1)
thf(fact_1265_qp_Omeasure__eq__compl,axiom,
    ! [S2: set_real,T3: set_real] :
      ( ( member_set_real @ S2 @ ( sigma_sets_real @ mu ) )
     => ( ( member_set_real @ T3 @ ( sigma_sets_real @ mu ) )
       => ( ( ( sigma_measure_real2 @ mu @ ( minus_minus_set_real @ ( sigma_space_real @ mu ) @ S2 ) )
            = ( sigma_measure_real2 @ mu @ ( minus_minus_set_real @ ( sigma_space_real @ mu ) @ T3 ) ) )
         => ( ( sigma_measure_real2 @ mu @ S2 )
            = ( sigma_measure_real2 @ mu @ T3 ) ) ) ) ) ).

% qp.measure_eq_compl
thf(fact_1266_qp_Ofinite__measure__compl,axiom,
    ! [S: set_real] :
      ( ( member_set_real @ S @ ( sigma_sets_real @ mu ) )
     => ( ( sigma_measure_real2 @ mu @ ( minus_minus_set_real @ ( sigma_space_real @ mu ) @ S ) )
        = ( minus_minus_real @ ( sigma_measure_real2 @ mu @ ( sigma_space_real @ mu ) ) @ ( sigma_measure_real2 @ mu @ S ) ) ) ) ).

% qp.finite_measure_compl
thf(fact_1267_qp_Ofinite__measure__Diff,axiom,
    ! [A: set_real,B: set_real] :
      ( ( member_set_real @ A @ ( sigma_sets_real @ mu ) )
     => ( ( member_set_real @ B @ ( sigma_sets_real @ mu ) )
       => ( ( ord_less_eq_set_real @ B @ A )
         => ( ( sigma_measure_real2 @ mu @ ( minus_minus_set_real @ A @ B ) )
            = ( minus_minus_real @ ( sigma_measure_real2 @ mu @ A ) @ ( sigma_measure_real2 @ mu @ B ) ) ) ) ) ) ).

% qp.finite_measure_Diff
thf(fact_1268_qp_Ofinite__measure__Diff_H,axiom,
    ! [A: set_real,B: set_real] :
      ( ( member_set_real @ A @ ( sigma_sets_real @ mu ) )
     => ( ( member_set_real @ B @ ( sigma_sets_real @ mu ) )
       => ( ( sigma_measure_real2 @ mu @ ( minus_minus_set_real @ A @ B ) )
          = ( minus_minus_real @ ( sigma_measure_real2 @ mu @ A ) @ ( sigma_measure_real2 @ mu @ ( inf_inf_set_real @ A @ B ) ) ) ) ) ) ).

% qp.finite_measure_Diff'
thf(fact_1269_qp_Oprob__compl,axiom,
    ! [A: set_real] :
      ( ( member_set_real @ A @ ( sigma_sets_real @ mu ) )
     => ( ( sigma_measure_real2 @ mu @ ( minus_minus_set_real @ ( sigma_space_real @ mu ) @ A ) )
        = ( minus_minus_real @ one_one_real @ ( sigma_measure_real2 @ mu @ A ) ) ) ) ).

% qp.prob_compl
thf(fact_1270_real__distribution_Ointegrable__cts__step,axiom,
    ! [M: sigma_measure_real,A2: real,B2: real] :
      ( ( distri2809703520229113005bution @ M )
     => ( ( ord_less_real @ A2 @ B2 )
       => ( bochne3340023020068487468l_real @ M @ ( weak_cts_step @ A2 @ B2 ) ) ) ) ).

% real_distribution.integrable_cts_step
thf(fact_1271_qp_Ochar__distr__add,axiom,
    ! [X1: real > real,X2: real > real,T3: real] :
      ( ( indepe3760321310464026790l_real @ mu @ borel_5078946678739801102l_real @ X1 @ borel_5078946678739801102l_real @ X2 )
     => ( ( characteristic_char
          @ ( measur2993149975067245138l_real @ mu @ borel_5078946678739801102l_real
            @ ^ [Omega: real] : ( plus_plus_real @ ( X1 @ Omega ) @ ( X2 @ Omega ) ) )
          @ T3 )
        = ( times_times_complex @ ( characteristic_char @ ( measur2993149975067245138l_real @ mu @ borel_5078946678739801102l_real @ X1 ) @ T3 ) @ ( characteristic_char @ ( measur2993149975067245138l_real @ mu @ borel_5078946678739801102l_real @ X2 ) @ T3 ) ) ) ) ).

% qp.char_distr_add
thf(fact_1272_segment__bound__lemma,axiom,
    ! [B: real,X8: real,Y4: real,U: real] :
      ( ( ord_less_eq_real @ B @ X8 )
     => ( ( ord_less_eq_real @ B @ Y4 )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ U @ one_one_real )
           => ( ord_less_eq_real @ B @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ one_one_real @ U ) @ X8 ) @ ( times_times_real @ U @ Y4 ) ) ) ) ) ) ) ).

% segment_bound_lemma
thf(fact_1273_sum__le__prod1,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ A2 @ one_one_real )
     => ( ( ord_less_eq_real @ B2 @ one_one_real )
       => ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B2 ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ A2 @ B2 ) ) ) ) ) ).

% sum_le_prod1
thf(fact_1274_add__diff__eq__iff__ennreal,axiom,
    ! [X8: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
      ( ( ( plus_p1859984266308609217nnreal @ X8 @ ( minus_8429688780609304081nnreal @ Y4 @ X8 ) )
        = Y4 )
      = ( ord_le3935885782089961368nnreal @ X8 @ Y4 ) ) ).

% add_diff_eq_iff_ennreal
thf(fact_1275_qp_Ofinite__measure__Union,axiom,
    ! [A: set_real,B: set_real] :
      ( ( member_set_real @ A @ ( sigma_sets_real @ mu ) )
     => ( ( member_set_real @ B @ ( sigma_sets_real @ mu ) )
       => ( ( ( inf_inf_set_real @ A @ B )
            = bot_bot_set_real )
         => ( ( sigma_measure_real2 @ mu @ ( sup_sup_set_real @ A @ B ) )
            = ( plus_plus_real @ ( sigma_measure_real2 @ mu @ A ) @ ( sigma_measure_real2 @ mu @ B ) ) ) ) ) ) ).

% qp.finite_measure_Union
thf(fact_1276_qp_Omeasure__zero__union,axiom,
    ! [S2: set_real,T3: set_real] :
      ( ( member_set_real @ S2 @ ( sigma_sets_real @ mu ) )
     => ( ( member_set_real @ T3 @ ( sigma_sets_real @ mu ) )
       => ( ( ( sigma_measure_real2 @ mu @ T3 )
            = zero_zero_real )
         => ( ( sigma_measure_real2 @ mu @ ( sup_sup_set_real @ S2 @ T3 ) )
            = ( sigma_measure_real2 @ mu @ S2 ) ) ) ) ) ).

% qp.measure_zero_union
thf(fact_1277_qp_Ofinite__measure__subadditive,axiom,
    ! [A: set_real,B: set_real] :
      ( ( member_set_real @ A @ ( sigma_sets_real @ mu ) )
     => ( ( member_set_real @ B @ ( sigma_sets_real @ mu ) )
       => ( ord_less_eq_real @ ( sigma_measure_real2 @ mu @ ( sup_sup_set_real @ A @ B ) ) @ ( plus_plus_real @ ( sigma_measure_real2 @ mu @ A ) @ ( sigma_measure_real2 @ mu @ B ) ) ) ) ) ).

% qp.finite_measure_subadditive
thf(fact_1278_qp_Ofinite__measure__Union_H,axiom,
    ! [A: set_real,B: set_real] :
      ( ( member_set_real @ A @ ( sigma_sets_real @ mu ) )
     => ( ( member_set_real @ B @ ( sigma_sets_real @ mu ) )
       => ( ( sigma_measure_real2 @ mu @ ( sup_sup_set_real @ A @ B ) )
          = ( plus_plus_real @ ( sigma_measure_real2 @ mu @ A ) @ ( sigma_measure_real2 @ mu @ ( minus_minus_set_real @ B @ A ) ) ) ) ) ) ).

% qp.finite_measure_Union'

% Conjectures (1)
thf(conj_0,conjecture,
    probab701741629625904797prob_b @ y @ beta @ ( giry_bind_real_real @ mu @ h ) ).

%------------------------------------------------------------------------------