TPTP Problem File: SLH0986^1.p

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

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : Frequency_Moments/0087_Frequency_Moment_k/prob_00662_028451__20038090_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1977 ( 440 unt; 749 typ;   0 def)
%            Number of atoms       : 3988 ( 880 equ;   0 cnn)
%            Maximal formula atoms :   36 (   3 avg)
%            Number of connectives : 18279 ( 207   ~;  24   |; 121   &;15853   @)
%                                         (   0 <=>;2074  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   22 (   9 avg)
%            Number of types       :  153 ( 152 usr)
%            Number of type conns  : 4490 (4490   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  600 ( 597 usr;  67 con; 0-6 aty)
%            Number of variables   : 4323 ( 549   ^;3757   !;  17   ?;4323   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 12:20:10.047
%------------------------------------------------------------------------------
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    borel_5572891200955083722nnreal: sigma_3116700849277497537nnreal ).

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    borel_4974289084073311971nnreal: sigma_4359661369728391106nnreal ).

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    prob_k5128454954192185136l_real: sigma_9047027012034273406at_nat > nat > ( ( product_prod_nat_nat > real ) > sigma_measure_real ) > ( ( product_prod_nat_nat > real ) > ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_Pr1002607673312053630t_real > $o ).

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    prob_k1742953041526194391nnreal: sigma_9047027012034273406at_nat > nat > ( ( real > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal ) > ( ( real > extend8495563244428889912nnreal ) > ( product_prod_nat_nat > product_prod_nat_nat ) > extend8495563244428889912nnreal ) > set_re5328672808648366137nnreal > $o ).

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    prob_k2910374860881282210at_nat: sigma_9047027012034273406at_nat > nat > ( ( real > extend8495563244428889912nnreal ) > sigma_5515648953823433982at_nat ) > ( ( real > extend8495563244428889912nnreal ) > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat ) > set_re5328672808648366137nnreal > $o ).

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    prob_k515836242556158155l_real: sigma_9047027012034273406at_nat > nat > ( ( real > extend8495563244428889912nnreal ) > sigma_measure_real ) > ( ( real > extend8495563244428889912nnreal ) > ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_re5328672808648366137nnreal > $o ).

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    prob_k1493464604931706806nnreal: sigma_9047027012034273406at_nat > nat > ( $o > sigma_7234349610311085201nnreal ) > ( $o > ( product_prod_nat_nat > product_prod_nat_nat ) > extend8495563244428889912nnreal ) > set_o > $o ).

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    prob_k658605421894945539at_nat: sigma_9047027012034273406at_nat > nat > ( $o > sigma_5515648953823433982at_nat ) > ( $o > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat ) > set_o > $o ).

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    prob_k6935007176262895658o_real: sigma_9047027012034273406at_nat > nat > ( $o > sigma_measure_real ) > ( $o > ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_o > $o ).

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    prob_k3726877565640988844nnreal: sigma_9047027012034273406at_nat > nat > ( rat > sigma_7234349610311085201nnreal ) > ( rat > ( product_prod_nat_nat > product_prod_nat_nat ) > extend8495563244428889912nnreal ) > set_rat > $o ).

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    prob_k8272357602513536653at_nat: sigma_9047027012034273406at_nat > nat > ( rat > sigma_5515648953823433982at_nat ) > ( rat > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat ) > set_rat > $o ).

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    prob_k4779580098842486304t_real: sigma_9047027012034273406at_nat > nat > ( rat > sigma_measure_real ) > ( rat > ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_rat > $o ).

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    prob_k2282667604904944778nnreal: sigma_5515648953823433982at_nat > nat > ( ( product_prod_nat_nat > real ) > sigma_7234349610311085201nnreal ) > ( ( product_prod_nat_nat > real ) > product_prod_nat_nat > extend8495563244428889912nnreal ) > set_Pr1002607673312053630t_real > $o ).

thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Real__Oreal_J_001t__Real__Oreal,type,
    prob_k5633034945198330622l_real: sigma_5515648953823433982at_nat > nat > ( ( product_prod_nat_nat > real ) > sigma_measure_real ) > ( ( product_prod_nat_nat > real ) > product_prod_nat_nat > real ) > set_Pr1002607673312053630t_real > $o ).

thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J_001t__Extended____Nonnegative____Real__Oennreal,type,
    prob_k1953979174828017545nnreal: sigma_5515648953823433982at_nat > nat > ( ( real > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal ) > ( ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > extend8495563244428889912nnreal ) > set_re5328672808648366137nnreal > $o ).

thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    prob_k8442391182763575152at_nat: sigma_5515648953823433982at_nat > nat > ( ( real > extend8495563244428889912nnreal ) > sigma_5515648953823433982at_nat ) > ( ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > product_prod_nat_nat ) > set_re5328672808648366137nnreal > $o ).

thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J_001t__Real__Oreal,type,
    prob_k5108616785771355773l_real: sigma_5515648953823433982at_nat > nat > ( ( real > extend8495563244428889912nnreal ) > sigma_measure_real ) > ( ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > real ) > set_re5328672808648366137nnreal > $o ).

thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001t__Extended____Nonnegative____Real__Oennreal,type,
    prob_k147916592727809156nnreal: sigma_5515648953823433982at_nat > nat > ( $o > sigma_7234349610311085201nnreal ) > ( $o > product_prod_nat_nat > extend8495563244428889912nnreal ) > set_o > $o ).

thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    prob_k129107890564256949at_nat: sigma_5515648953823433982at_nat > nat > ( $o > sigma_5515648953823433982at_nat ) > ( $o > product_prod_nat_nat > product_prod_nat_nat ) > set_o > $o ).

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    prob_k1749327348295208283at_nat: sigma_5515648953823433982at_nat > nat > ( rat > sigma_5515648953823433982at_nat ) > ( rat > product_prod_nat_nat > product_prod_nat_nat ) > set_rat > $o ).

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    indepe2783168947688421667nnreal: sigma_9047027012034273406at_nat > ( $o > sigma_7234349610311085201nnreal ) > ( $o > ( product_prod_nat_nat > product_prod_nat_nat ) > extend8495563244428889912nnreal ) > set_o > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_Eo_001t__Nat__Onat,type,
    indepe2989309374999565755_o_nat: sigma_9047027012034273406at_nat > ( $o > sigma_measure_nat ) > ( $o > ( product_prod_nat_nat > product_prod_nat_nat ) > nat ) > set_o > $o ).

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thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Rat__Orat_001_Eo,type,
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    indepe4547145070500471935nnreal: sigma_9047027012034273406at_nat > ( rat > sigma_7234349610311085201nnreal ) > ( rat > ( product_prod_nat_nat > product_prod_nat_nat ) > extend8495563244428889912nnreal ) > set_rat > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Rat__Orat_001t__Nat__Onat,type,
    indepe494537661987070231at_nat: sigma_9047027012034273406at_nat > ( rat > sigma_measure_nat ) > ( rat > ( product_prod_nat_nat > product_prod_nat_nat ) > nat ) > set_rat > $o ).

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thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Rat__Orat_001t__Rat__Orat,type,
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thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Rat__Orat_001t__Real__Oreal,type,
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    indepe3718687268563708247l_real: sigma_5515648953823433982at_nat > ( ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > sigma_measure_real ) > ( ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > product_prod_nat_nat > real ) > set_Pr947837736998463782t_real > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Real__Oreal,type,
    indepe2220390954403586572t_real: sigma_5515648953823433982at_nat > ( ( product_prod_nat_nat > product_prod_nat_nat ) > sigma_measure_real ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > real ) > set_Pr2458342521480944603at_nat > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Real__Oreal_J_001t__Extended____Nonnegative____Real__Oennreal,type,
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thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Real__Oreal_J_001t__Real__Oreal,type,
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thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J_001_Eo,type,
    indepe1951750190343126618real_o: sigma_5515648953823433982at_nat > ( ( real > extend8495563244428889912nnreal ) > sigma_measure_o ) > ( ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > $o ) > set_re5328672808648366137nnreal > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J_001t__Extended____Nonnegative____Real__Oennreal,type,
    indepe3243683517584732406nnreal: sigma_5515648953823433982at_nat > ( ( real > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal ) > ( ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > extend8495563244428889912nnreal ) > set_re5328672808648366137nnreal > $o ).

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    indepe3593890525012728259at_nat: sigma_5515648953823433982at_nat > ( ( real > extend8495563244428889912nnreal ) > sigma_5515648953823433982at_nat ) > ( ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > product_prod_nat_nat ) > set_re5328672808648366137nnreal > $o ).

thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J_001t__Rat__Orat,type,
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thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J_001t__Real__Oreal,type,
    indepe579324506763905386l_real: sigma_5515648953823433982at_nat > ( ( real > extend8495563244428889912nnreal ) > sigma_measure_real ) > ( ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > real ) > set_re5328672808648366137nnreal > $o ).

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thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Real__Oreal,type,
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    indepe7529601277151292294nnreal: sigma_measure_real > ( rat > sigma_7234349610311085201nnreal ) > ( rat > real > extend8495563244428889912nnreal ) > set_rat > $o ).

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    indepe3710239703223808239at_nat: sigma_9047027012034273406at_nat > ( nat > set_se1666487788256820497at_nat ) > set_se1666487788256820497at_nat ).

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    indepe3837050773481735769at_nat: sigma_5515648953823433982at_nat > ( nat > set_se7855581050983116737at_nat ) > set_se7855581050983116737at_nat ).

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    kL_div5695791497786497691at_nat: real > sigma_9047027012034273406at_nat > sigma_9047027012034273406at_nat > real ).

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    prob_m3062007823079647199at_nat: sigma_9047027012034273406at_nat > real > sigma_measure_nat > sigma_measure_nat > ( ( product_prod_nat_nat > product_prod_nat_nat ) > nat ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > nat ) > real ).

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    lp_spa812643495979991928at_nat: extend8495563244428889912nnreal > sigma_9047027012034273406at_nat > functi5165143733218492903t_real ).

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thf(sy_c_member_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
    member_real_real: ( real > real ) > set_real_real > $o ).

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

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_member_001t__Rat__Orat,type,
    member_rat: rat > set_rat > $o ).

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

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    member3752267775663534962at_nat: set_Pr2458342521480944603at_nat > set_se1666487788256820497at_nat > $o ).

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    member524040414084610768nnreal: set_re5328672808648366137nnreal > set_se2490721793304844655nnreal > $o ).

thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
    member_set_o: set_o > set_set_o > $o ).

thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: set_nat > set_set_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member2643936169264416010at_nat: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Rat__Orat_J,type,
    member_set_rat: set_rat > set_set_rat > $o ).

thf(sy_v__092_060delta_062,type,
    delta: rat ).

thf(sy_v__092_060epsilon_062,type,
    epsilon: rat ).

thf(sy_v_as,type,
    as: list_nat ).

thf(sy_v_i_092_060_094sub_0621____,type,
    i_1: nat ).

thf(sy_v_i_092_060_094sub_0622____,type,
    i_2: nat ).

thf(sy_v_k,type,
    k: nat ).

thf(sy_v_n,type,
    n: nat ).

% Relevant facts (1222)
thf(fact_0_False,axiom,
    as != nil_nat ).

% False
thf(fact_1_s1__nonzero,axiom,
    ord_less_nat @ zero_zero_nat @ ( frequency_Moment_s_1 @ delta @ n @ k ) ).

% s1_nonzero
thf(fact_2__092_060Omega_062_092_060_094sub_0622_Osubprob__space__axioms,axiom,
    giry_s2972128636810349169at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ).

% \<Omega>\<^sub>2.subprob_space_axioms
thf(fact_3__092_060Omega_062_092_060_094sub_0622_Oindep__sets__cong,axiom,
    ! [I: set_rat,J: set_rat,F: rat > set_se1666487788256820497at_nat,G: rat > set_se1666487788256820497at_nat] :
      ( ( I = J )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( ( F @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe2998401279100655083at_rat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
          = ( indepe2998401279100655083at_rat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ G @ J ) ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_cong
thf(fact_4__092_060Omega_062_092_060_094sub_0622_Oindep__sets__cong,axiom,
    ! [I: set_Pr2458342521480944603at_nat,J: set_Pr2458342521480944603at_nat,F: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se1666487788256820497at_nat,G: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se1666487788256820497at_nat] :
      ( ( I = J )
     => ( ! [I2: product_prod_nat_nat > product_prod_nat_nat] :
            ( ( member8885076297122219836at_nat @ I2 @ I )
           => ( ( F @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe7028276433782144696at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
          = ( indepe7028276433782144696at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ G @ J ) ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_cong
thf(fact_5__092_060Omega_062_092_060_094sub_0622_Oindep__sets__cong,axiom,
    ! [I: set_Pr1002607673312053630t_real,J: set_Pr1002607673312053630t_real,F: ( product_prod_nat_nat > real ) > set_se1666487788256820497at_nat,G: ( product_prod_nat_nat > real ) > set_se1666487788256820497at_nat] :
      ( ( I = J )
     => ( ! [I2: product_prod_nat_nat > real] :
            ( ( member4564283293661824327t_real @ I2 @ I )
           => ( ( F @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe4000825420875437771t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
          = ( indepe4000825420875437771t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ G @ J ) ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_cong
thf(fact_6__092_060Omega_062_092_060_094sub_0622_Oindep__sets__cong,axiom,
    ! [I: set_re5328672808648366137nnreal,J: set_re5328672808648366137nnreal,F: ( real > extend8495563244428889912nnreal ) > set_se1666487788256820497at_nat,G: ( real > extend8495563244428889912nnreal ) > set_se1666487788256820497at_nat] :
      ( ( I = J )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( ( F @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe6665278795436257814nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
          = ( indepe6665278795436257814nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ G @ J ) ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_cong
thf(fact_7__092_060Omega_062_092_060_094sub_0622_Oindep__sets__cong,axiom,
    ! [I: set_Pr947837736998463782t_real,J: set_Pr947837736998463782t_real,F: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_se1666487788256820497at_nat,G: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_se1666487788256820497at_nat] :
      ( ( I = J )
     => ( ! [I2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
            ( ( member8159409068225774087t_real @ I2 @ I )
           => ( ( F @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe5134244768989041027t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
          = ( indepe5134244768989041027t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ G @ J ) ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_cong
thf(fact_8__092_060Omega_062_092_060_094sub_0622_Oindep__sets__cong,axiom,
    ! [I: set_o,J: set_o,F: $o > set_se1666487788256820497at_nat,G: $o > set_se1666487788256820497at_nat] :
      ( ( I = J )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( ( F @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe9198310886692500677_nat_o @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
          = ( indepe9198310886692500677_nat_o @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ G @ J ) ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_cong
thf(fact_9__092_060Omega_062_092_060_094sub_0622_Oprob__space__axioms,axiom,
    probab8562894880268318498at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ).

% \<Omega>\<^sub>2.prob_space_axioms
thf(fact_10__092_060Omega_062_092_060_094sub_0622_Osigma__finite__measure,axiom,
    measur1801591924826899012at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ).

% \<Omega>\<^sub>2.sigma_finite_measure
thf(fact_11__092_060Omega_062_092_060_094sub_0622_Ofinite__measure,axiom,
    measur5156515106886843841at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ).

% \<Omega>\<^sub>2.finite_measure
thf(fact_12_s2__nonzero,axiom,
    ord_less_nat @ zero_zero_nat @ ( frequency_Moment_s_2 @ epsilon ) ).

% s2_nonzero
thf(fact_13_n__nonzero,axiom,
    ord_less_nat @ zero_zero_nat @ n ).

% n_nonzero
thf(fact_14__092_060Omega_062_092_060_094sub_0622_Oindep__sets__mono__index,axiom,
    ! [J: set_o,I: set_o,F: $o > set_se1666487788256820497at_nat] :
      ( ( ord_less_eq_set_o @ J @ I )
     => ( ( indepe9198310886692500677_nat_o @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
       => ( indepe9198310886692500677_nat_o @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ J ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_mono_index
thf(fact_15__092_060Omega_062_092_060_094sub_0622_Oindep__sets__mono__index,axiom,
    ! [J: set_Pr947837736998463782t_real,I: set_Pr947837736998463782t_real,F: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_se1666487788256820497at_nat] :
      ( ( ord_le8622349663015078278t_real @ J @ I )
     => ( ( indepe5134244768989041027t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
       => ( indepe5134244768989041027t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ J ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_mono_index
thf(fact_16__092_060Omega_062_092_060_094sub_0622_Oindep__sets__mono__index,axiom,
    ! [J: set_re5328672808648366137nnreal,I: set_re5328672808648366137nnreal,F: ( real > extend8495563244428889912nnreal ) > set_se1666487788256820497at_nat] :
      ( ( ord_le2462468573666744473nnreal @ J @ I )
     => ( ( indepe6665278795436257814nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
       => ( indepe6665278795436257814nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ J ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_mono_index
thf(fact_17__092_060Omega_062_092_060_094sub_0622_Oindep__sets__mono__index,axiom,
    ! [J: set_Pr1002607673312053630t_real,I: set_Pr1002607673312053630t_real,F: ( product_prod_nat_nat > real ) > set_se1666487788256820497at_nat] :
      ( ( ord_le1994730768851292446t_real @ J @ I )
     => ( ( indepe4000825420875437771t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
       => ( indepe4000825420875437771t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ J ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_mono_index
thf(fact_18__092_060Omega_062_092_060_094sub_0622_Oindep__sets__mono__index,axiom,
    ! [J: set_Pr2458342521480944603at_nat,I: set_Pr2458342521480944603at_nat,F: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se1666487788256820497at_nat] :
      ( ( ord_le3857079194666040379at_nat @ J @ I )
     => ( ( indepe7028276433782144696at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
       => ( indepe7028276433782144696at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ J ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_mono_index
thf(fact_19__092_060Omega_062_092_060_094sub_0622_Oindep__sets__mono__index,axiom,
    ! [J: set_rat,I: set_rat,F: rat > set_se1666487788256820497at_nat] :
      ( ( ord_less_eq_set_rat @ J @ I )
     => ( ( indepe2998401279100655083at_rat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
       => ( indepe2998401279100655083at_rat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ J ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_mono_index
thf(fact_20_zero__less__of__rat__iff,axiom,
    ! [R: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( field_2639924705303425560at_rat @ R ) )
      = ( ord_less_rat @ zero_zero_rat @ R ) ) ).

% zero_less_of_rat_iff
thf(fact_21_zero__less__of__rat__iff,axiom,
    ! [R: rat] :
      ( ( ord_less_real @ zero_zero_real @ ( field_7254667332652039916t_real @ R ) )
      = ( ord_less_rat @ zero_zero_rat @ R ) ) ).

% zero_less_of_rat_iff
thf(fact_22_of__rat__less__0__iff,axiom,
    ! [R: rat] :
      ( ( ord_less_rat @ ( field_2639924705303425560at_rat @ R ) @ zero_zero_rat )
      = ( ord_less_rat @ R @ zero_zero_rat ) ) ).

% of_rat_less_0_iff
thf(fact_23_of__rat__less__0__iff,axiom,
    ! [R: rat] :
      ( ( ord_less_real @ ( field_7254667332652039916t_real @ R ) @ zero_zero_real )
      = ( ord_less_rat @ R @ zero_zero_rat ) ) ).

% of_rat_less_0_iff
thf(fact_24__092_060Omega_062_092_060_094sub_0622_Oindep__vars__compose2,axiom,
    ! [M: rat > sigma_measure_real,X: rat > ( product_prod_nat_nat > product_prod_nat_nat ) > real,I: set_rat,Y: rat > real > extend8495563244428889912nnreal,N: rat > sigma_7234349610311085201nnreal] :
      ( ( indepe4371592291047002227t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ X @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe4547145070500471935nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N
          @ ^ [I3: rat,X2: product_prod_nat_nat > product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_compose2
thf(fact_25__092_060Omega_062_092_060_094sub_0622_Oindep__vars__compose2,axiom,
    ! [M: $o > sigma_measure_real,X: $o > ( product_prod_nat_nat > product_prod_nat_nat ) > real,I: set_o,Y: $o > real > extend8495563244428889912nnreal,N: $o > sigma_7234349610311085201nnreal] :
      ( ( indepe2405714897255445271o_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ X @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe2783168947688421667nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N
          @ ^ [I3: $o,X2: product_prod_nat_nat > product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_compose2
thf(fact_26__092_060Omega_062_092_060_094sub_0622_Oindep__vars__compose2,axiom,
    ! [M: rat > sigma_5515648953823433982at_nat,X: rat > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,I: set_rat,Y: rat > product_prod_nat_nat > real,N: rat > sigma_measure_real] :
      ( ( indepe6783812011079266042at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ X @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe4371592291047002227t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N
          @ ^ [I3: rat,X2: product_prod_nat_nat > product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_compose2
thf(fact_27__092_060Omega_062_092_060_094sub_0622_Oindep__vars__compose2,axiom,
    ! [M: $o > sigma_5515648953823433982at_nat,X: $o > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,I: set_o,Y: $o > product_prod_nat_nat > real,N: $o > sigma_measure_real] :
      ( ( indepe5033476800998874454at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ X @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe2405714897255445271o_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N
          @ ^ [I3: $o,X2: product_prod_nat_nat > product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_compose2
thf(fact_28__092_060Omega_062_092_060_094sub_0622_Oindep__vars__compose2,axiom,
    ! [M: ( real > extend8495563244428889912nnreal ) > sigma_measure_real,X: ( real > extend8495563244428889912nnreal ) > ( product_prod_nat_nat > product_prod_nat_nat ) > real,I: set_re5328672808648366137nnreal,Y: ( real > extend8495563244428889912nnreal ) > real > extend8495563244428889912nnreal,N: ( real > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal] :
      ( ( indepe3660082575425806622l_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ X @ I )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe675575072008979114nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N
          @ ^ [I3: real > extend8495563244428889912nnreal,X2: product_prod_nat_nat > product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_compose2
thf(fact_29__092_060Omega_062_092_060_094sub_0622_Oindep__vars__compose2,axiom,
    ! [M: rat > sigma_5515648953823433982at_nat,X: rat > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,I: set_rat,Y: rat > product_prod_nat_nat > product_prod_nat_nat,N: rat > sigma_5515648953823433982at_nat] :
      ( ( indepe6783812011079266042at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ X @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member8885076297122219836at_nat @ ( Y @ I2 ) @ ( sigma_3682087626466204304at_nat @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe6783812011079266042at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N
          @ ^ [I3: rat,X2: product_prod_nat_nat > product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_compose2
thf(fact_30__092_060Omega_062_092_060_094sub_0622_Oindep__vars__compose2,axiom,
    ! [M: $o > sigma_5515648953823433982at_nat,X: $o > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,I: set_o,Y: $o > product_prod_nat_nat > product_prod_nat_nat,N: $o > sigma_5515648953823433982at_nat] :
      ( ( indepe5033476800998874454at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ X @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member8885076297122219836at_nat @ ( Y @ I2 ) @ ( sigma_3682087626466204304at_nat @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe5033476800998874454at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N
          @ ^ [I3: $o,X2: product_prod_nat_nat > product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_compose2
thf(fact_31__092_060Omega_062_092_060_094sub_0622_Oindep__vars__compose2,axiom,
    ! [M: ( real > extend8495563244428889912nnreal ) > sigma_5515648953823433982at_nat,X: ( real > extend8495563244428889912nnreal ) > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,I: set_re5328672808648366137nnreal,Y: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > real,N: ( real > extend8495563244428889912nnreal ) > sigma_measure_real] :
      ( ( indepe1677428707908761359at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ X @ I )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe3660082575425806622l_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N
          @ ^ [I3: real > extend8495563244428889912nnreal,X2: product_prod_nat_nat > product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_compose2
thf(fact_32__092_060Omega_062_092_060_094sub_0622_Oindep__vars__compose2,axiom,
    ! [M: ( product_prod_nat_nat > real ) > sigma_measure_real,X: ( product_prod_nat_nat > real ) > ( product_prod_nat_nat > product_prod_nat_nat ) > real,I: set_Pr1002607673312053630t_real,Y: ( product_prod_nat_nat > real ) > real > extend8495563244428889912nnreal,N: ( product_prod_nat_nat > real ) > sigma_7234349610311085201nnreal] :
      ( ( indepe1199366807463074589l_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ X @ I )
     => ( ! [I2: product_prod_nat_nat > real] :
            ( ( member4564283293661824327t_real @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe4741077166904906793nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N
          @ ^ [I3: product_prod_nat_nat > real,X2: product_prod_nat_nat > product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_compose2
thf(fact_33__092_060Omega_062_092_060_094sub_0622_Oindep__vars__compose2,axiom,
    ! [M: ( real > extend8495563244428889912nnreal ) > sigma_5515648953823433982at_nat,X: ( real > extend8495563244428889912nnreal ) > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,I: set_re5328672808648366137nnreal,Y: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > product_prod_nat_nat,N: ( real > extend8495563244428889912nnreal ) > sigma_5515648953823433982at_nat] :
      ( ( indepe1677428707908761359at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ X @ I )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member8885076297122219836at_nat @ ( Y @ I2 ) @ ( sigma_3682087626466204304at_nat @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe1677428707908761359at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N
          @ ^ [I3: real > extend8495563244428889912nnreal,X2: product_prod_nat_nat > product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_compose2
thf(fact_34__092_060Omega_062_092_060_094sub_0622_Oindep__sets__mono__sets,axiom,
    ! [F: $o > set_se1666487788256820497at_nat,I: set_o,G: $o > set_se1666487788256820497at_nat] :
      ( ( indepe9198310886692500677_nat_o @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( ord_le8269821659064819057at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
       => ( indepe9198310886692500677_nat_o @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ G @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_mono_sets
thf(fact_35__092_060Omega_062_092_060_094sub_0622_Oindep__sets__mono__sets,axiom,
    ! [F: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_se1666487788256820497at_nat,I: set_Pr947837736998463782t_real,G: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_se1666487788256820497at_nat] :
      ( ( indepe5134244768989041027t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
     => ( ! [I2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
            ( ( member8159409068225774087t_real @ I2 @ I )
           => ( ord_le8269821659064819057at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
       => ( indepe5134244768989041027t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ G @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_mono_sets
thf(fact_36__092_060Omega_062_092_060_094sub_0622_Oindep__sets__mono__sets,axiom,
    ! [F: ( real > extend8495563244428889912nnreal ) > set_se1666487788256820497at_nat,I: set_re5328672808648366137nnreal,G: ( real > extend8495563244428889912nnreal ) > set_se1666487788256820497at_nat] :
      ( ( indepe6665278795436257814nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( ord_le8269821659064819057at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
       => ( indepe6665278795436257814nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ G @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_mono_sets
thf(fact_37__092_060Omega_062_092_060_094sub_0622_Oindep__sets__mono__sets,axiom,
    ! [F: ( product_prod_nat_nat > real ) > set_se1666487788256820497at_nat,I: set_Pr1002607673312053630t_real,G: ( product_prod_nat_nat > real ) > set_se1666487788256820497at_nat] :
      ( ( indepe4000825420875437771t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
     => ( ! [I2: product_prod_nat_nat > real] :
            ( ( member4564283293661824327t_real @ I2 @ I )
           => ( ord_le8269821659064819057at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
       => ( indepe4000825420875437771t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ G @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_mono_sets
thf(fact_38__092_060Omega_062_092_060_094sub_0622_Oindep__sets__mono__sets,axiom,
    ! [F: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se1666487788256820497at_nat,I: set_Pr2458342521480944603at_nat,G: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se1666487788256820497at_nat] :
      ( ( indepe7028276433782144696at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
     => ( ! [I2: product_prod_nat_nat > product_prod_nat_nat] :
            ( ( member8885076297122219836at_nat @ I2 @ I )
           => ( ord_le8269821659064819057at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
       => ( indepe7028276433782144696at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ G @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_mono_sets
thf(fact_39__092_060Omega_062_092_060_094sub_0622_Oindep__sets__mono__sets,axiom,
    ! [F: rat > set_se1666487788256820497at_nat,I: set_rat,G: rat > set_se1666487788256820497at_nat] :
      ( ( indepe2998401279100655083at_rat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( ord_le8269821659064819057at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
       => ( indepe2998401279100655083at_rat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ G @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_mono_sets
thf(fact_40__092_060Omega_062_092_060_094sub_0622_Oprob__space__completion,axiom,
    probab8562894880268318498at_nat @ ( comple4843269853529550107at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) ).

% \<Omega>\<^sub>2.prob_space_completion
thf(fact_41_integral__zero,axiom,
    ! [M2: sigma_9047027012034273406at_nat] :
      ( ( bochne5509773249985062230t_real @ M2
        @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : zero_zero_real )
      = zero_zero_real ) ).

% integral_zero
thf(fact_42_integral__zero,axiom,
    ! [M2: sigma_5515648953823433982at_nat] :
      ( ( bochne6384019433803981034t_real @ M2
        @ ^ [X2: product_prod_nat_nat] : zero_zero_real )
      = zero_zero_real ) ).

% integral_zero
thf(fact_43_integrable__2,axiom,
    ! [G2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( as != nil_nat )
     => ( bochne7117808529828525605t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ G2 ) ) ).

% integrable_2
thf(fact_44_of__rat__0,axiom,
    ( ( field_2639924705303425560at_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% of_rat_0
thf(fact_45_of__rat__0,axiom,
    ( ( field_7254667332652039916t_real @ zero_zero_rat )
    = zero_zero_real ) ).

% of_rat_0
thf(fact_46_of__rat__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( field_2639924705303425560at_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% of_rat_eq_0_iff
thf(fact_47_of__rat__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( field_7254667332652039916t_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_rat ) ) ).

% of_rat_eq_0_iff
thf(fact_48__092_060delta_062__range,axiom,
    ord_less_rat @ zero_zero_rat @ delta ).

% \<delta>_range
thf(fact_49_of__rat__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( field_7254667332652039916t_real @ A )
        = ( field_7254667332652039916t_real @ B ) )
      = ( A = B ) ) ).

% of_rat_eq_iff
thf(fact_50_fk__nonzero,axiom,
    ord_less_rat @ zero_zero_rat @ ( frequency_F_nat @ k @ as ) ).

% fk_nonzero
thf(fact_51_integrable__zero,axiom,
    ! [M2: sigma_9047027012034273406at_nat] :
      ( bochne7117808529828525605t_real @ M2
      @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : zero_zero_real ) ).

% integrable_zero
thf(fact_52_integrable__zero,axiom,
    ! [M2: sigma_5515648953823433982at_nat] :
      ( bochne2596016609597520987t_real @ M2
      @ ^ [X2: product_prod_nat_nat] : zero_zero_real ) ).

% integrable_zero
thf(fact_53__092_060Omega_062_092_060_094sub_0622_Oindep__sets__mono,axiom,
    ! [F: $o > set_se1666487788256820497at_nat,I: set_o,J: set_o,G: $o > set_se1666487788256820497at_nat] :
      ( ( indepe9198310886692500677_nat_o @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
     => ( ( ord_less_eq_set_o @ J @ I )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ J )
             => ( ord_le8269821659064819057at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
         => ( indepe9198310886692500677_nat_o @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ G @ J ) ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_mono
thf(fact_54__092_060Omega_062_092_060_094sub_0622_Oindep__sets__mono,axiom,
    ! [F: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_se1666487788256820497at_nat,I: set_Pr947837736998463782t_real,J: set_Pr947837736998463782t_real,G: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_se1666487788256820497at_nat] :
      ( ( indepe5134244768989041027t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
     => ( ( ord_le8622349663015078278t_real @ J @ I )
       => ( ! [I2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
              ( ( member8159409068225774087t_real @ I2 @ J )
             => ( ord_le8269821659064819057at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
         => ( indepe5134244768989041027t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ G @ J ) ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_mono
thf(fact_55__092_060Omega_062_092_060_094sub_0622_Oindep__sets__mono,axiom,
    ! [F: ( real > extend8495563244428889912nnreal ) > set_se1666487788256820497at_nat,I: set_re5328672808648366137nnreal,J: set_re5328672808648366137nnreal,G: ( real > extend8495563244428889912nnreal ) > set_se1666487788256820497at_nat] :
      ( ( indepe6665278795436257814nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
     => ( ( ord_le2462468573666744473nnreal @ J @ I )
       => ( ! [I2: real > extend8495563244428889912nnreal] :
              ( ( member2919562650594848410nnreal @ I2 @ J )
             => ( ord_le8269821659064819057at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
         => ( indepe6665278795436257814nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ G @ J ) ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_mono
thf(fact_56__092_060Omega_062_092_060_094sub_0622_Oindep__sets__mono,axiom,
    ! [F: ( product_prod_nat_nat > real ) > set_se1666487788256820497at_nat,I: set_Pr1002607673312053630t_real,J: set_Pr1002607673312053630t_real,G: ( product_prod_nat_nat > real ) > set_se1666487788256820497at_nat] :
      ( ( indepe4000825420875437771t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
     => ( ( ord_le1994730768851292446t_real @ J @ I )
       => ( ! [I2: product_prod_nat_nat > real] :
              ( ( member4564283293661824327t_real @ I2 @ J )
             => ( ord_le8269821659064819057at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
         => ( indepe4000825420875437771t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ G @ J ) ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_mono
thf(fact_57__092_060Omega_062_092_060_094sub_0622_Oindep__sets__mono,axiom,
    ! [F: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se1666487788256820497at_nat,I: set_Pr2458342521480944603at_nat,J: set_Pr2458342521480944603at_nat,G: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se1666487788256820497at_nat] :
      ( ( indepe7028276433782144696at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
     => ( ( ord_le3857079194666040379at_nat @ J @ I )
       => ( ! [I2: product_prod_nat_nat > product_prod_nat_nat] :
              ( ( member8885076297122219836at_nat @ I2 @ J )
             => ( ord_le8269821659064819057at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
         => ( indepe7028276433782144696at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ G @ J ) ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_mono
thf(fact_58__092_060Omega_062_092_060_094sub_0622_Oindep__sets__mono,axiom,
    ! [F: rat > set_se1666487788256820497at_nat,I: set_rat,J: set_rat,G: rat > set_se1666487788256820497at_nat] :
      ( ( indepe2998401279100655083at_rat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
     => ( ( ord_less_eq_set_rat @ J @ I )
       => ( ! [I2: rat] :
              ( ( member_rat @ I2 @ J )
             => ( ord_le8269821659064819057at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
         => ( indepe2998401279100655083at_rat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ G @ J ) ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_mono
thf(fact_59_zero__eq__of__rat__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( field_2639924705303425560at_rat @ A ) )
      = ( zero_zero_rat = A ) ) ).

% zero_eq_of_rat_iff
thf(fact_60_zero__eq__of__rat__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_real
        = ( field_7254667332652039916t_real @ A ) )
      = ( zero_zero_rat = A ) ) ).

% zero_eq_of_rat_iff
thf(fact_61_zero__le__of__rat__iff,axiom,
    ! [R: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( field_2639924705303425560at_rat @ R ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ R ) ) ).

% zero_le_of_rat_iff
thf(fact_62_zero__le__of__rat__iff,axiom,
    ! [R: rat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( field_7254667332652039916t_real @ R ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ R ) ) ).

% zero_le_of_rat_iff
thf(fact_63_of__rat__le__0__iff,axiom,
    ! [R: rat] :
      ( ( ord_less_eq_rat @ ( field_2639924705303425560at_rat @ R ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ R @ zero_zero_rat ) ) ).

% of_rat_le_0_iff
thf(fact_64_of__rat__le__0__iff,axiom,
    ! [R: rat] :
      ( ( ord_less_eq_real @ ( field_7254667332652039916t_real @ R ) @ zero_zero_real )
      = ( ord_less_eq_rat @ R @ zero_zero_rat ) ) ).

% of_rat_le_0_iff
thf(fact_65__092_060Omega_062_092_060_094sub_0622_Ointegrable__const,axiom,
    ! [A: real] :
      ( bochne7117808529828525605t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
      @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : A ) ).

% \<Omega>\<^sub>2.integrable_const
thf(fact_66_of__rat__less__eq,axiom,
    ! [R: rat,S: rat] :
      ( ( ord_less_eq_rat @ ( field_2639924705303425560at_rat @ R ) @ ( field_2639924705303425560at_rat @ S ) )
      = ( ord_less_eq_rat @ R @ S ) ) ).

% of_rat_less_eq
thf(fact_67_of__rat__less__eq,axiom,
    ! [R: rat,S: rat] :
      ( ( ord_less_eq_real @ ( field_7254667332652039916t_real @ R ) @ ( field_7254667332652039916t_real @ S ) )
      = ( ord_less_eq_rat @ R @ S ) ) ).

% of_rat_less_eq
thf(fact_68_integral__eq__cases,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,N: sigma_9047027012034273406at_nat,G2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( ( bochne7117808529828525605t_real @ M2 @ F2 )
        = ( bochne7117808529828525605t_real @ N @ G2 ) )
     => ( ( ( bochne7117808529828525605t_real @ M2 @ F2 )
         => ( ( bochne7117808529828525605t_real @ N @ G2 )
           => ( ( bochne5509773249985062230t_real @ M2 @ F2 )
              = ( bochne5509773249985062230t_real @ N @ G2 ) ) ) )
       => ( ( bochne5509773249985062230t_real @ M2 @ F2 )
          = ( bochne5509773249985062230t_real @ N @ G2 ) ) ) ) ).

% integral_eq_cases
thf(fact_69_integral__eq__cases,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,N: sigma_5515648953823433982at_nat,G2: product_prod_nat_nat > real] :
      ( ( ( bochne7117808529828525605t_real @ M2 @ F2 )
        = ( bochne2596016609597520987t_real @ N @ G2 ) )
     => ( ( ( bochne7117808529828525605t_real @ M2 @ F2 )
         => ( ( bochne2596016609597520987t_real @ N @ G2 )
           => ( ( bochne5509773249985062230t_real @ M2 @ F2 )
              = ( bochne6384019433803981034t_real @ N @ G2 ) ) ) )
       => ( ( bochne5509773249985062230t_real @ M2 @ F2 )
          = ( bochne6384019433803981034t_real @ N @ G2 ) ) ) ) ).

% integral_eq_cases
thf(fact_70_integral__eq__cases,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real,N: sigma_9047027012034273406at_nat,G2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( ( bochne2596016609597520987t_real @ M2 @ F2 )
        = ( bochne7117808529828525605t_real @ N @ G2 ) )
     => ( ( ( bochne2596016609597520987t_real @ M2 @ F2 )
         => ( ( bochne7117808529828525605t_real @ N @ G2 )
           => ( ( bochne6384019433803981034t_real @ M2 @ F2 )
              = ( bochne5509773249985062230t_real @ N @ G2 ) ) ) )
       => ( ( bochne6384019433803981034t_real @ M2 @ F2 )
          = ( bochne5509773249985062230t_real @ N @ G2 ) ) ) ) ).

% integral_eq_cases
thf(fact_71_integral__eq__cases,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real,N: sigma_5515648953823433982at_nat,G2: product_prod_nat_nat > real] :
      ( ( ( bochne2596016609597520987t_real @ M2 @ F2 )
        = ( bochne2596016609597520987t_real @ N @ G2 ) )
     => ( ( ( bochne2596016609597520987t_real @ M2 @ F2 )
         => ( ( bochne2596016609597520987t_real @ N @ G2 )
           => ( ( bochne6384019433803981034t_real @ M2 @ F2 )
              = ( bochne6384019433803981034t_real @ N @ G2 ) ) ) )
       => ( ( bochne6384019433803981034t_real @ M2 @ F2 )
          = ( bochne6384019433803981034t_real @ N @ G2 ) ) ) ) ).

% integral_eq_cases
thf(fact_72_finite__measure_Ointegrable__const,axiom,
    ! [M2: sigma_9047027012034273406at_nat,A: real] :
      ( ( measur5156515106886843841at_nat @ M2 )
     => ( bochne7117808529828525605t_real @ M2
        @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : A ) ) ).

% finite_measure.integrable_const
thf(fact_73_finite__measure_Ointegrable__const,axiom,
    ! [M2: sigma_5515648953823433982at_nat,A: real] :
      ( ( measur7911732083032432613at_nat @ M2 )
     => ( bochne2596016609597520987t_real @ M2
        @ ^ [X2: product_prod_nat_nat] : A ) ) ).

% finite_measure.integrable_const
thf(fact_74_not__integrable__integral__eq,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ~ ( bochne7117808529828525605t_real @ M2 @ F2 )
     => ( ( bochne5509773249985062230t_real @ M2 @ F2 )
        = zero_zero_real ) ) ).

% not_integrable_integral_eq
thf(fact_75_not__integrable__integral__eq,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real] :
      ( ~ ( bochne2596016609597520987t_real @ M2 @ F2 )
     => ( ( bochne6384019433803981034t_real @ M2 @ F2 )
        = zero_zero_real ) ) ).

% not_integrable_integral_eq
thf(fact_76_of__rat__less,axiom,
    ! [R: rat,S: rat] :
      ( ( ord_less_rat @ ( field_2639924705303425560at_rat @ R ) @ ( field_2639924705303425560at_rat @ S ) )
      = ( ord_less_rat @ R @ S ) ) ).

% of_rat_less
thf(fact_77_of__rat__less,axiom,
    ! [R: rat,S: rat] :
      ( ( ord_less_real @ ( field_7254667332652039916t_real @ R ) @ ( field_7254667332652039916t_real @ S ) )
      = ( ord_less_rat @ R @ S ) ) ).

% of_rat_less
thf(fact_78__092_060Omega_062_092_060_094sub_0622_Ok__wise__indep__vars__compose,axiom,
    ! [K: nat,M: rat > sigma_measure_real,X: rat > ( product_prod_nat_nat > product_prod_nat_nat ) > real,I: set_rat,Y: rat > real > extend8495563244428889912nnreal,N: rat > sigma_7234349610311085201nnreal] :
      ( ( prob_k4779580098842486304t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ K @ M @ X @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( prob_k3726877565640988844nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ K @ N
          @ ^ [I3: rat,X2: product_prod_nat_nat > product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.k_wise_indep_vars_compose
thf(fact_79__092_060Omega_062_092_060_094sub_0622_Ok__wise__indep__vars__compose,axiom,
    ! [K: nat,M: $o > sigma_measure_real,X: $o > ( product_prod_nat_nat > product_prod_nat_nat ) > real,I: set_o,Y: $o > real > extend8495563244428889912nnreal,N: $o > sigma_7234349610311085201nnreal] :
      ( ( prob_k6935007176262895658o_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ K @ M @ X @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( prob_k1493464604931706806nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ K @ N
          @ ^ [I3: $o,X2: product_prod_nat_nat > product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.k_wise_indep_vars_compose
thf(fact_80__092_060Omega_062_092_060_094sub_0622_Ok__wise__indep__vars__compose,axiom,
    ! [K: nat,M: rat > sigma_5515648953823433982at_nat,X: rat > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,I: set_rat,Y: rat > product_prod_nat_nat > real,N: rat > sigma_measure_real] :
      ( ( prob_k8272357602513536653at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ K @ M @ X @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( prob_k4779580098842486304t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ K @ N
          @ ^ [I3: rat,X2: product_prod_nat_nat > product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.k_wise_indep_vars_compose
thf(fact_81__092_060Omega_062_092_060_094sub_0622_Ok__wise__indep__vars__compose,axiom,
    ! [K: nat,M: $o > sigma_5515648953823433982at_nat,X: $o > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,I: set_o,Y: $o > product_prod_nat_nat > real,N: $o > sigma_measure_real] :
      ( ( prob_k658605421894945539at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ K @ M @ X @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( prob_k6935007176262895658o_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ K @ N
          @ ^ [I3: $o,X2: product_prod_nat_nat > product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.k_wise_indep_vars_compose
thf(fact_82__092_060Omega_062_092_060_094sub_0622_Ok__wise__indep__vars__compose,axiom,
    ! [K: nat,M: ( real > extend8495563244428889912nnreal ) > sigma_measure_real,X: ( real > extend8495563244428889912nnreal ) > ( product_prod_nat_nat > product_prod_nat_nat ) > real,I: set_re5328672808648366137nnreal,Y: ( real > extend8495563244428889912nnreal ) > real > extend8495563244428889912nnreal,N: ( real > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal] :
      ( ( prob_k515836242556158155l_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ K @ M @ X @ I )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( prob_k1742953041526194391nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ K @ N
          @ ^ [I3: real > extend8495563244428889912nnreal,X2: product_prod_nat_nat > product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.k_wise_indep_vars_compose
thf(fact_83__092_060Omega_062_092_060_094sub_0622_Ok__wise__indep__vars__compose,axiom,
    ! [K: nat,M: rat > sigma_5515648953823433982at_nat,X: rat > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,I: set_rat,Y: rat > product_prod_nat_nat > product_prod_nat_nat,N: rat > sigma_5515648953823433982at_nat] :
      ( ( prob_k8272357602513536653at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ K @ M @ X @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member8885076297122219836at_nat @ ( Y @ I2 ) @ ( sigma_3682087626466204304at_nat @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( prob_k8272357602513536653at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ K @ N
          @ ^ [I3: rat,X2: product_prod_nat_nat > product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.k_wise_indep_vars_compose
thf(fact_84__092_060Omega_062_092_060_094sub_0622_Ok__wise__indep__vars__compose,axiom,
    ! [K: nat,M: $o > sigma_5515648953823433982at_nat,X: $o > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,I: set_o,Y: $o > product_prod_nat_nat > product_prod_nat_nat,N: $o > sigma_5515648953823433982at_nat] :
      ( ( prob_k658605421894945539at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ K @ M @ X @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member8885076297122219836at_nat @ ( Y @ I2 ) @ ( sigma_3682087626466204304at_nat @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( prob_k658605421894945539at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ K @ N
          @ ^ [I3: $o,X2: product_prod_nat_nat > product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.k_wise_indep_vars_compose
thf(fact_85__092_060Omega_062_092_060_094sub_0622_Ok__wise__indep__vars__compose,axiom,
    ! [K: nat,M: ( real > extend8495563244428889912nnreal ) > sigma_5515648953823433982at_nat,X: ( real > extend8495563244428889912nnreal ) > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,I: set_re5328672808648366137nnreal,Y: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > real,N: ( real > extend8495563244428889912nnreal ) > sigma_measure_real] :
      ( ( prob_k2910374860881282210at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ K @ M @ X @ I )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( prob_k515836242556158155l_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ K @ N
          @ ^ [I3: real > extend8495563244428889912nnreal,X2: product_prod_nat_nat > product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.k_wise_indep_vars_compose
thf(fact_86__092_060Omega_062_092_060_094sub_0622_Ok__wise__indep__vars__compose,axiom,
    ! [K: nat,M: ( product_prod_nat_nat > real ) > sigma_measure_real,X: ( product_prod_nat_nat > real ) > ( product_prod_nat_nat > product_prod_nat_nat ) > real,I: set_Pr1002607673312053630t_real,Y: ( product_prod_nat_nat > real ) > real > extend8495563244428889912nnreal,N: ( product_prod_nat_nat > real ) > sigma_7234349610311085201nnreal] :
      ( ( prob_k5128454954192185136l_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ K @ M @ X @ I )
     => ( ! [I2: product_prod_nat_nat > real] :
            ( ( member4564283293661824327t_real @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( prob_k5974023319877427644nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ K @ N
          @ ^ [I3: product_prod_nat_nat > real,X2: product_prod_nat_nat > product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.k_wise_indep_vars_compose
thf(fact_87__092_060Omega_062_092_060_094sub_0622_Ok__wise__indep__vars__compose,axiom,
    ! [K: nat,M: ( real > extend8495563244428889912nnreal ) > sigma_5515648953823433982at_nat,X: ( real > extend8495563244428889912nnreal ) > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,I: set_re5328672808648366137nnreal,Y: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > product_prod_nat_nat,N: ( real > extend8495563244428889912nnreal ) > sigma_5515648953823433982at_nat] :
      ( ( prob_k2910374860881282210at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ K @ M @ X @ I )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member8885076297122219836at_nat @ ( Y @ I2 ) @ ( sigma_3682087626466204304at_nat @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( prob_k2910374860881282210at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ K @ N
          @ ^ [I3: real > extend8495563244428889912nnreal,X2: product_prod_nat_nat > product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.k_wise_indep_vars_compose
thf(fact_88__092_060Omega_062_092_060_094sub_0622_OKL__same__eq__0,axiom,
    ! [B: real] :
      ( ( kL_div5695791497786497691at_nat @ B @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) )
      = zero_zero_real ) ).

% \<Omega>\<^sub>2.KL_same_eq_0
thf(fact_89__092_060Omega_062_092_060_094sub_0622_Oindep__var__rv1,axiom,
    ! [S2: sigma_measure_real,X: ( product_prod_nat_nat > product_prod_nat_nat ) > real,T: sigma_measure_real,Y: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( indepe1938972608428940447t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ S2 @ X @ T @ Y )
     => ( member8159409068225774087t_real @ X @ ( sigma_5064276549609164707t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ S2 ) ) ) ).

% \<Omega>\<^sub>2.indep_var_rv1
thf(fact_90__092_060Omega_062_092_060_094sub_0622_Oindep__var__rv2,axiom,
    ! [S2: sigma_measure_real,X: ( product_prod_nat_nat > product_prod_nat_nat ) > real,T: sigma_measure_real,Y: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( indepe1938972608428940447t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ S2 @ X @ T @ Y )
     => ( member8159409068225774087t_real @ Y @ ( sigma_5064276549609164707t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ T ) ) ) ).

% \<Omega>\<^sub>2.indep_var_rv2
thf(fact_91_result__exp__1,axiom,
    ( ( as != nil_nat )
   => ( ( bochne6384019433803981034t_real @ ( freque5010624893710627907mega_1 @ as ) @ ( freque135395702774845718t_real @ k @ as ) )
      = ( field_7254667332652039916t_real @ ( frequency_F_nat @ k @ as ) ) ) ) ).

% result_exp_1
thf(fact_92_mem__Collect__eq,axiom,
    ! [A: rat,P: rat > $o] :
      ( ( member_rat @ A @ ( collect_rat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_93_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat > real,P: ( product_prod_nat_nat > real ) > $o] :
      ( ( member4564283293661824327t_real @ A @ ( collec1908255412270979209t_real @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_94_mem__Collect__eq,axiom,
    ! [A: real > extend8495563244428889912nnreal,P: ( real > extend8495563244428889912nnreal ) > $o] :
      ( ( member2919562650594848410nnreal @ A @ ( collec9130413544115709400nnreal @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_95_mem__Collect__eq,axiom,
    ! [A: ( product_prod_nat_nat > product_prod_nat_nat ) > real,P: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > $o] :
      ( ( member8159409068225774087t_real @ A @ ( collec4597233195267612485t_real @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_96_mem__Collect__eq,axiom,
    ! [A: $o,P: $o > $o] :
      ( ( member_o @ A @ ( collect_o @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_97_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_98_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat > product_prod_nat_nat,P: ( product_prod_nat_nat > product_prod_nat_nat ) > $o] :
      ( ( member8885076297122219836at_nat @ A @ ( collec248804655031457914at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_99_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_100_Collect__mem__eq,axiom,
    ! [A2: set_rat] :
      ( ( collect_rat
        @ ^ [X2: rat] : ( member_rat @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_101_Collect__mem__eq,axiom,
    ! [A2: set_Pr1002607673312053630t_real] :
      ( ( collec1908255412270979209t_real
        @ ^ [X2: product_prod_nat_nat > real] : ( member4564283293661824327t_real @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_102_Collect__mem__eq,axiom,
    ! [A2: set_re5328672808648366137nnreal] :
      ( ( collec9130413544115709400nnreal
        @ ^ [X2: real > extend8495563244428889912nnreal] : ( member2919562650594848410nnreal @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_103_Collect__mem__eq,axiom,
    ! [A2: set_Pr947837736998463782t_real] :
      ( ( collec4597233195267612485t_real
        @ ^ [X2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] : ( member8159409068225774087t_real @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_104_Collect__mem__eq,axiom,
    ! [A2: set_o] :
      ( ( collect_o
        @ ^ [X2: $o] : ( member_o @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_105_Collect__mem__eq,axiom,
    ! [A2: set_nat] :
      ( ( collect_nat
        @ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_106_Collect__mem__eq,axiom,
    ! [A2: set_Pr2458342521480944603at_nat] :
      ( ( collec248804655031457914at_nat
        @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( member8885076297122219836at_nat @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_107_Collect__mem__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_108_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_109_Collect__cong,axiom,
    ! [P: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,Q: ( product_prod_nat_nat > product_prod_nat_nat ) > $o] :
      ( ! [X3: product_prod_nat_nat > product_prod_nat_nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec248804655031457914at_nat @ P )
        = ( collec248804655031457914at_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_110_Collect__cong,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec3392354462482085612at_nat @ P )
        = ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_111__092_060Omega_062_092_060_094sub_0622_Oindep__vars__compose,axiom,
    ! [M: rat > sigma_measure_real,X: rat > ( product_prod_nat_nat > product_prod_nat_nat ) > real,I: set_rat,Y: rat > real > extend8495563244428889912nnreal,N: rat > sigma_7234349610311085201nnreal] :
      ( ( indepe4371592291047002227t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ X @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe4547145070500471935nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N
          @ ^ [I3: rat] : ( comp_r2164139168250337006at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_compose
thf(fact_112__092_060Omega_062_092_060_094sub_0622_Oindep__vars__compose,axiom,
    ! [M: $o > sigma_measure_real,X: $o > ( product_prod_nat_nat > product_prod_nat_nat ) > real,I: set_o,Y: $o > real > extend8495563244428889912nnreal,N: $o > sigma_7234349610311085201nnreal] :
      ( ( indepe2405714897255445271o_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ X @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe2783168947688421667nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N
          @ ^ [I3: $o] : ( comp_r2164139168250337006at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_compose
thf(fact_113__092_060Omega_062_092_060_094sub_0622_Oindep__vars__compose,axiom,
    ! [M: rat > sigma_5515648953823433982at_nat,X: rat > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,I: set_rat,Y: rat > product_prod_nat_nat > real,N: rat > sigma_measure_real] :
      ( ( indepe6783812011079266042at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ X @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe4371592291047002227t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N
          @ ^ [I3: rat] : ( comp_P2183065948301422769at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_compose
thf(fact_114__092_060Omega_062_092_060_094sub_0622_Oindep__vars__compose,axiom,
    ! [M: $o > sigma_5515648953823433982at_nat,X: $o > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,I: set_o,Y: $o > product_prod_nat_nat > real,N: $o > sigma_measure_real] :
      ( ( indepe5033476800998874454at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ X @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe2405714897255445271o_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N
          @ ^ [I3: $o] : ( comp_P2183065948301422769at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_compose
thf(fact_115__092_060Omega_062_092_060_094sub_0622_Oindep__vars__compose,axiom,
    ! [M: ( real > extend8495563244428889912nnreal ) > sigma_measure_real,X: ( real > extend8495563244428889912nnreal ) > ( product_prod_nat_nat > product_prod_nat_nat ) > real,I: set_re5328672808648366137nnreal,Y: ( real > extend8495563244428889912nnreal ) > real > extend8495563244428889912nnreal,N: ( real > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal] :
      ( ( indepe3660082575425806622l_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ X @ I )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe675575072008979114nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N
          @ ^ [I3: real > extend8495563244428889912nnreal] : ( comp_r2164139168250337006at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_compose
thf(fact_116__092_060Omega_062_092_060_094sub_0622_Oindep__vars__compose,axiom,
    ! [M: rat > sigma_5515648953823433982at_nat,X: rat > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,I: set_rat,Y: rat > product_prod_nat_nat > product_prod_nat_nat,N: rat > sigma_5515648953823433982at_nat] :
      ( ( indepe6783812011079266042at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ X @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member8885076297122219836at_nat @ ( Y @ I2 ) @ ( sigma_3682087626466204304at_nat @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe6783812011079266042at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N
          @ ^ [I3: rat] : ( comp_P7516068803530355784at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_compose
thf(fact_117__092_060Omega_062_092_060_094sub_0622_Oindep__vars__compose,axiom,
    ! [M: $o > sigma_5515648953823433982at_nat,X: $o > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,I: set_o,Y: $o > product_prod_nat_nat > product_prod_nat_nat,N: $o > sigma_5515648953823433982at_nat] :
      ( ( indepe5033476800998874454at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ X @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member8885076297122219836at_nat @ ( Y @ I2 ) @ ( sigma_3682087626466204304at_nat @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe5033476800998874454at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N
          @ ^ [I3: $o] : ( comp_P7516068803530355784at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_compose
thf(fact_118__092_060Omega_062_092_060_094sub_0622_Oindep__vars__compose,axiom,
    ! [M: ( real > extend8495563244428889912nnreal ) > sigma_5515648953823433982at_nat,X: ( real > extend8495563244428889912nnreal ) > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,I: set_re5328672808648366137nnreal,Y: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > real,N: ( real > extend8495563244428889912nnreal ) > sigma_measure_real] :
      ( ( indepe1677428707908761359at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ X @ I )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe3660082575425806622l_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N
          @ ^ [I3: real > extend8495563244428889912nnreal] : ( comp_P2183065948301422769at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_compose
thf(fact_119__092_060Omega_062_092_060_094sub_0622_Oindep__vars__compose,axiom,
    ! [M: ( product_prod_nat_nat > real ) > sigma_measure_real,X: ( product_prod_nat_nat > real ) > ( product_prod_nat_nat > product_prod_nat_nat ) > real,I: set_Pr1002607673312053630t_real,Y: ( product_prod_nat_nat > real ) > real > extend8495563244428889912nnreal,N: ( product_prod_nat_nat > real ) > sigma_7234349610311085201nnreal] :
      ( ( indepe1199366807463074589l_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ X @ I )
     => ( ! [I2: product_prod_nat_nat > real] :
            ( ( member4564283293661824327t_real @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe4741077166904906793nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N
          @ ^ [I3: product_prod_nat_nat > real] : ( comp_r2164139168250337006at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_compose
thf(fact_120__092_060Omega_062_092_060_094sub_0622_Oindep__vars__compose,axiom,
    ! [M: ( real > extend8495563244428889912nnreal ) > sigma_5515648953823433982at_nat,X: ( real > extend8495563244428889912nnreal ) > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,I: set_re5328672808648366137nnreal,Y: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > product_prod_nat_nat,N: ( real > extend8495563244428889912nnreal ) > sigma_5515648953823433982at_nat] :
      ( ( indepe1677428707908761359at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ X @ I )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member8885076297122219836at_nat @ ( Y @ I2 ) @ ( sigma_3682087626466204304at_nat @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe1677428707908761359at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N
          @ ^ [I3: real > extend8495563244428889912nnreal] : ( comp_P7516068803530355784at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_compose
thf(fact_121_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_122_neq0__conv,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% neq0_conv
thf(fact_123_less__nat__zero__code,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_124__092_060Omega_062_092_060_094sub_0622_Ofinite__measure__distr,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,M: sigma_measure_real] :
      ( ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M ) )
     => ( measur3606880022600206024e_real @ ( measur6433597746558672971t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ F2 ) ) ) ).

% \<Omega>\<^sub>2.finite_measure_distr
thf(fact_125__092_060Omega_062_092_060_094sub_0622_Ofinite__measure__distr,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat,M: sigma_9047027012034273406at_nat] :
      ( ( member2646312932426349184at_nat @ F2 @ ( sigma_1496667973943888668at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M ) )
     => ( measur5156515106886843841at_nat @ ( measur7731843055261497284at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ F2 ) ) ) ).

% \<Omega>\<^sub>2.finite_measure_distr
thf(fact_126__092_060Omega_062_092_060_094sub_0622_Ofinite__measure__distr,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,M: sigma_5515648953823433982at_nat] :
      ( ( member7151659165045815932at_nat @ F2 @ ( sigma_3543877168342358986at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M ) )
     => ( measur7911732083032432613at_nat @ ( measur4728020197420827938at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ F2 ) ) ) ).

% \<Omega>\<^sub>2.finite_measure_distr
thf(fact_127__092_060Omega_062_092_060_094sub_0622_Oprob__space__distr,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,M: sigma_measure_real] :
      ( ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M ) )
     => ( probab535871623910865577e_real @ ( measur6433597746558672971t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ F2 ) ) ) ).

% \<Omega>\<^sub>2.prob_space_distr
thf(fact_128__092_060Omega_062_092_060_094sub_0622_Oprob__space__distr,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat,M: sigma_9047027012034273406at_nat] :
      ( ( member2646312932426349184at_nat @ F2 @ ( sigma_1496667973943888668at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M ) )
     => ( probab8562894880268318498at_nat @ ( measur7731843055261497284at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ F2 ) ) ) ).

% \<Omega>\<^sub>2.prob_space_distr
thf(fact_129__092_060Omega_062_092_060_094sub_0622_Oprob__space__distr,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,M: sigma_5515648953823433982at_nat] :
      ( ( member7151659165045815932at_nat @ F2 @ ( sigma_3543877168342358986at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M ) )
     => ( probab2019946041432190532at_nat @ ( measur4728020197420827938at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ F2 ) ) ) ).

% \<Omega>\<^sub>2.prob_space_distr
thf(fact_130_finite__measure__axioms,axiom,
    measur7911732083032432613at_nat @ ( freque5010624893710627907mega_1 @ as ) ).

% finite_measure_axioms
thf(fact_131_prob__space__axioms,axiom,
    probab2019946041432190532at_nat @ ( freque5010624893710627907mega_1 @ as ) ).

% prob_space_axioms
thf(fact_132_integrable__1,axiom,
    ! [F2: product_prod_nat_nat > real] :
      ( ( as != nil_nat )
     => ( bochne2596016609597520987t_real @ ( freque5010624893710627907mega_1 @ as ) @ F2 ) ) ).

% integrable_1
thf(fact_133_k__ge__1,axiom,
    ord_less_eq_nat @ one_one_nat @ k ).

% k_ge_1
thf(fact_134_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_135_le0,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% le0
thf(fact_136_local_Ointegrable__const,axiom,
    ! [A: real] :
      ( bochne2596016609597520987t_real @ ( freque5010624893710627907mega_1 @ as )
      @ ^ [X2: product_prod_nat_nat] : A ) ).

% local.integrable_const
thf(fact_137__092_060Omega_062_092_060_094sub_0622_Oindep__var__compose,axiom,
    ! [M1: sigma_5515648953823433982at_nat,X1: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,M22: sigma_5515648953823433982at_nat,X22: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,Y1: product_prod_nat_nat > product_prod_nat_nat,N1: sigma_5515648953823433982at_nat,Y2: product_prod_nat_nat > product_prod_nat_nat,N22: sigma_5515648953823433982at_nat] :
      ( ( indepe6532217695432402382at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M1 @ X1 @ M22 @ X22 )
     => ( ( member8885076297122219836at_nat @ Y1 @ ( sigma_3682087626466204304at_nat @ M1 @ N1 ) )
       => ( ( member8885076297122219836at_nat @ Y2 @ ( sigma_3682087626466204304at_nat @ M22 @ N22 ) )
         => ( indepe6532217695432402382at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N1 @ ( comp_P7516068803530355784at_nat @ Y1 @ X1 ) @ N22 @ ( comp_P7516068803530355784at_nat @ Y2 @ X22 ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_var_compose
thf(fact_138__092_060Omega_062_092_060_094sub_0622_Oindep__var__compose,axiom,
    ! [M1: sigma_9047027012034273406at_nat,X1: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat,M22: sigma_9047027012034273406at_nat,X22: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat,Y1: ( product_prod_nat_nat > product_prod_nat_nat ) > real,N1: sigma_measure_real,Y2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,N22: sigma_measure_real] :
      ( ( indepe4309971216286138904at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M1 @ X1 @ M22 @ X22 )
     => ( ( member8159409068225774087t_real @ Y1 @ ( sigma_5064276549609164707t_real @ M1 @ N1 ) )
       => ( ( member8159409068225774087t_real @ Y2 @ ( sigma_5064276549609164707t_real @ M22 @ N22 ) )
         => ( indepe1938972608428940447t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N1 @ ( comp_P137399349585858651at_nat @ Y1 @ X1 ) @ N22 @ ( comp_P137399349585858651at_nat @ Y2 @ X22 ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_var_compose
thf(fact_139__092_060Omega_062_092_060_094sub_0622_Oindep__var__compose,axiom,
    ! [M1: sigma_5515648953823433982at_nat,X1: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,M22: sigma_5515648953823433982at_nat,X22: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,Y1: product_prod_nat_nat > real,N1: sigma_measure_real,Y2: product_prod_nat_nat > real,N22: sigma_measure_real] :
      ( ( indepe6532217695432402382at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M1 @ X1 @ M22 @ X22 )
     => ( ( member4564283293661824327t_real @ Y1 @ ( sigma_1188828689629184861t_real @ M1 @ N1 ) )
       => ( ( member4564283293661824327t_real @ Y2 @ ( sigma_1188828689629184861t_real @ M22 @ N22 ) )
         => ( indepe1938972608428940447t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N1 @ ( comp_P2183065948301422769at_nat @ Y1 @ X1 ) @ N22 @ ( comp_P2183065948301422769at_nat @ Y2 @ X22 ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_var_compose
thf(fact_140__092_060Omega_062_092_060_094sub_0622_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_real,X1: ( product_prod_nat_nat > product_prod_nat_nat ) > real,M22: sigma_measure_real,X22: ( product_prod_nat_nat > product_prod_nat_nat ) > real,Y1: real > extend8495563244428889912nnreal,N1: sigma_7234349610311085201nnreal,Y2: real > extend8495563244428889912nnreal,N22: sigma_7234349610311085201nnreal] :
      ( ( indepe1938972608428940447t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M1 @ X1 @ M22 @ X22 )
     => ( ( member2919562650594848410nnreal @ Y1 @ ( sigma_9017504469962657078nnreal @ M1 @ N1 ) )
       => ( ( member2919562650594848410nnreal @ Y2 @ ( sigma_9017504469962657078nnreal @ M22 @ N22 ) )
         => ( indepe1784549585806403243nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N1 @ ( comp_r2164139168250337006at_nat @ Y1 @ X1 ) @ N22 @ ( comp_r2164139168250337006at_nat @ Y2 @ X22 ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_var_compose
thf(fact_141__092_060Omega_062_092_060_094sub_0622_Oindep__var__compose,axiom,
    ! [M1: sigma_measure_real,X1: ( product_prod_nat_nat > product_prod_nat_nat ) > real,M22: sigma_measure_real,X22: ( product_prod_nat_nat > product_prod_nat_nat ) > real,Y1: real > real,N1: sigma_measure_real,Y2: real > real,N22: sigma_measure_real] :
      ( ( indepe1938972608428940447t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M1 @ X1 @ M22 @ X22 )
     => ( ( member_real_real @ Y1 @ ( sigma_5267869275261027754l_real @ M1 @ N1 ) )
       => ( ( member_real_real @ Y2 @ ( sigma_5267869275261027754l_real @ M22 @ N22 ) )
         => ( indepe1938972608428940447t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N1 @ ( comp_r4584311727289514594at_nat @ Y1 @ X1 ) @ N22 @ ( comp_r4584311727289514594at_nat @ Y2 @ X22 ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_var_compose
thf(fact_142__092_060Omega_062_092_060_094sub_0622_Oindep__vars__distr,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,N: sigma_measure_real,I: set_rat,X4: rat > real > extend8495563244428889912nnreal,M: rat > sigma_7234349610311085201nnreal] :
      ( ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N ) )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( X4 @ I2 ) @ ( sigma_9017504469962657078nnreal @ N @ ( M @ I2 ) ) ) )
       => ( ( indepe4547145070500471935nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M
            @ ^ [I3: rat] : ( comp_r2164139168250337006at_nat @ ( X4 @ I3 ) @ F2 )
            @ I )
         => ( indepe7529601277151292294nnreal @ ( measur6433597746558672971t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_distr
thf(fact_143__092_060Omega_062_092_060_094sub_0622_Oindep__vars__distr,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,N: sigma_measure_real,I: set_o,X4: $o > real > extend8495563244428889912nnreal,M: $o > sigma_7234349610311085201nnreal] :
      ( ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N ) )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( X4 @ I2 ) @ ( sigma_9017504469962657078nnreal @ N @ ( M @ I2 ) ) ) )
       => ( ( indepe2783168947688421667nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M
            @ ^ [I3: $o] : ( comp_r2164139168250337006at_nat @ ( X4 @ I3 ) @ F2 )
            @ I )
         => ( indepe1757858165614604892nnreal @ ( measur6433597746558672971t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_distr
thf(fact_144__092_060Omega_062_092_060_094sub_0622_Oindep__vars__distr,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,N: sigma_5515648953823433982at_nat,I: set_rat,X4: rat > product_prod_nat_nat > real,M: rat > sigma_measure_real] :
      ( ( member7151659165045815932at_nat @ F2 @ ( sigma_3543877168342358986at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N ) )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member4564283293661824327t_real @ ( X4 @ I2 ) @ ( sigma_1188828689629184861t_real @ N @ ( M @ I2 ) ) ) )
       => ( ( indepe4371592291047002227t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M
            @ ^ [I3: rat] : ( comp_P2183065948301422769at_nat @ ( X4 @ I3 ) @ F2 )
            @ I )
         => ( indepe5387908695526973631t_real @ ( measur4728020197420827938at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_distr
thf(fact_145__092_060Omega_062_092_060_094sub_0622_Oindep__vars__distr,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,N: sigma_5515648953823433982at_nat,I: set_o,X4: $o > product_prod_nat_nat > real,M: $o > sigma_measure_real] :
      ( ( member7151659165045815932at_nat @ F2 @ ( sigma_3543877168342358986at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N ) )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member4564283293661824327t_real @ ( X4 @ I2 ) @ ( sigma_1188828689629184861t_real @ N @ ( M @ I2 ) ) ) )
       => ( ( indepe2405714897255445271o_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M
            @ ^ [I3: $o] : ( comp_P2183065948301422769at_nat @ ( X4 @ I3 ) @ F2 )
            @ I )
         => ( indepe6599791682495051723o_real @ ( measur4728020197420827938at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_distr
thf(fact_146__092_060Omega_062_092_060_094sub_0622_Oindep__vars__distr,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,N: sigma_measure_real,I: set_re5328672808648366137nnreal,X4: ( real > extend8495563244428889912nnreal ) > real > extend8495563244428889912nnreal,M: ( real > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal] :
      ( ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N ) )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( X4 @ I2 ) @ ( sigma_9017504469962657078nnreal @ N @ ( M @ I2 ) ) ) )
       => ( ( indepe675575072008979114nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M
            @ ^ [I3: real > extend8495563244428889912nnreal] : ( comp_r2164139168250337006at_nat @ ( X4 @ I3 ) @ F2 )
            @ I )
         => ( indepe8147369424511330225nnreal @ ( measur6433597746558672971t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_distr
thf(fact_147__092_060Omega_062_092_060_094sub_0622_Oindep__vars__distr,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,N: sigma_5515648953823433982at_nat,I: set_rat,X4: rat > product_prod_nat_nat > product_prod_nat_nat,M: rat > sigma_5515648953823433982at_nat] :
      ( ( member7151659165045815932at_nat @ F2 @ ( sigma_3543877168342358986at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N ) )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member8885076297122219836at_nat @ ( X4 @ I2 ) @ ( sigma_3682087626466204304at_nat @ N @ ( M @ I2 ) ) ) )
       => ( ( indepe6783812011079266042at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M
            @ ^ [I3: rat] : ( comp_P7516068803530355784at_nat @ ( X4 @ I3 ) @ F2 )
            @ I )
         => ( indepe2321685950664944046at_nat @ ( measur4728020197420827938at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_distr
thf(fact_148__092_060Omega_062_092_060_094sub_0622_Oindep__vars__distr,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,N: sigma_5515648953823433982at_nat,I: set_o,X4: $o > product_prod_nat_nat > product_prod_nat_nat,M: $o > sigma_5515648953823433982at_nat] :
      ( ( member7151659165045815932at_nat @ F2 @ ( sigma_3543877168342358986at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N ) )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member8885076297122219836at_nat @ ( X4 @ I2 ) @ ( sigma_3682087626466204304at_nat @ N @ ( M @ I2 ) ) ) )
       => ( ( indepe5033476800998874454at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M
            @ ^ [I3: $o] : ( comp_P7516068803530355784at_nat @ ( X4 @ I3 ) @ F2 )
            @ I )
         => ( indepe504693989766925730at_nat @ ( measur4728020197420827938at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_distr
thf(fact_149__092_060Omega_062_092_060_094sub_0622_Oindep__vars__distr,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,N: sigma_5515648953823433982at_nat,I: set_re5328672808648366137nnreal,X4: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > real,M: ( real > extend8495563244428889912nnreal ) > sigma_measure_real] :
      ( ( member7151659165045815932at_nat @ F2 @ ( sigma_3543877168342358986at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N ) )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member4564283293661824327t_real @ ( X4 @ I2 ) @ ( sigma_1188828689629184861t_real @ N @ ( M @ I2 ) ) ) )
       => ( ( indepe3660082575425806622l_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M
            @ ^ [I3: real > extend8495563244428889912nnreal] : ( comp_P2183065948301422769at_nat @ ( X4 @ I3 ) @ F2 )
            @ I )
         => ( indepe579324506763905386l_real @ ( measur4728020197420827938at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_distr
thf(fact_150__092_060Omega_062_092_060_094sub_0622_Oindep__vars__distr,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,N: sigma_measure_real,I: set_Pr1002607673312053630t_real,X4: ( product_prod_nat_nat > real ) > real > extend8495563244428889912nnreal,M: ( product_prod_nat_nat > real ) > sigma_7234349610311085201nnreal] :
      ( ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N ) )
     => ( ! [I2: product_prod_nat_nat > real] :
            ( ( member4564283293661824327t_real @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( X4 @ I2 ) @ ( sigma_9017504469962657078nnreal @ N @ ( M @ I2 ) ) ) )
       => ( ( indepe4741077166904906793nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M
            @ ^ [I3: product_prod_nat_nat > real] : ( comp_r2164139168250337006at_nat @ ( X4 @ I3 ) @ F2 )
            @ I )
         => ( indepe3984226279986541154nnreal @ ( measur6433597746558672971t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_distr
thf(fact_151__092_060Omega_062_092_060_094sub_0622_Oindep__vars__distr,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,N: sigma_5515648953823433982at_nat,I: set_re5328672808648366137nnreal,X4: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > product_prod_nat_nat,M: ( real > extend8495563244428889912nnreal ) > sigma_5515648953823433982at_nat] :
      ( ( member7151659165045815932at_nat @ F2 @ ( sigma_3543877168342358986at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N ) )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member8885076297122219836at_nat @ ( X4 @ I2 ) @ ( sigma_3682087626466204304at_nat @ N @ ( M @ I2 ) ) ) )
       => ( ( indepe1677428707908761359at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M
            @ ^ [I3: real > extend8495563244428889912nnreal] : ( comp_P7516068803530355784at_nat @ ( X4 @ I3 ) @ F2 )
            @ I )
         => ( indepe3593890525012728259at_nat @ ( measur4728020197420827938at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_distr
thf(fact_152_less__eq__nat_Osimps_I1_J,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% less_eq_nat.simps(1)
thf(fact_153_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_154_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_155_le__0__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_156_less__mono__imp__le__mono,axiom,
    ! [F2: nat > nat,I4: nat,J2: nat] :
      ( ! [I2: nat,J3: nat] :
          ( ( ord_less_nat @ I2 @ J3 )
         => ( ord_less_nat @ ( F2 @ I2 ) @ ( F2 @ J3 ) ) )
     => ( ( ord_less_eq_nat @ I4 @ J2 )
       => ( ord_less_eq_nat @ ( F2 @ I4 ) @ ( F2 @ J2 ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_157_le__neq__implies__less,axiom,
    ! [M3: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M3 @ N2 )
     => ( ( M3 != N2 )
       => ( ord_less_nat @ M3 @ N2 ) ) ) ).

% le_neq_implies_less
thf(fact_158_less__or__eq__imp__le,axiom,
    ! [M3: nat,N2: nat] :
      ( ( ( ord_less_nat @ M3 @ N2 )
        | ( M3 = N2 ) )
     => ( ord_less_eq_nat @ M3 @ N2 ) ) ).

% less_or_eq_imp_le
thf(fact_159_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M4: nat,N3: nat] :
          ( ( ord_less_nat @ M4 @ N3 )
          | ( M4 = N3 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_160_less__imp__le__nat,axiom,
    ! [M3: nat,N2: nat] :
      ( ( ord_less_nat @ M3 @ N2 )
     => ( ord_less_eq_nat @ M3 @ N2 ) ) ).

% less_imp_le_nat
thf(fact_161_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
          & ( M4 != N3 ) ) ) ) ).

% nat_less_le
thf(fact_162_less__eq__rat__def,axiom,
    ( ord_less_eq_rat
    = ( ^ [X2: rat,Y3: rat] :
          ( ( ord_less_rat @ X2 @ Y3 )
          | ( X2 = Y3 ) ) ) ) ).

% less_eq_rat_def
thf(fact_163_ex__least__nat__le,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N2 )
            & ! [I5: nat] :
                ( ( ord_less_nat @ I5 @ K2 )
               => ~ ( P @ I5 ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_164_integrable__distr,axiom,
    ! [T: ( product_prod_nat_nat > product_prod_nat_nat ) > real,M2: sigma_9047027012034273406at_nat,M: sigma_measure_real,F2: real > real] :
      ( ( member8159409068225774087t_real @ T @ ( sigma_5064276549609164707t_real @ M2 @ M ) )
     => ( ( bochne3340023020068487468l_real @ ( measur6433597746558672971t_real @ M2 @ M @ T ) @ F2 )
       => ( bochne7117808529828525605t_real @ M2
          @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( F2 @ ( T @ X2 ) ) ) ) ) ).

% integrable_distr
thf(fact_165_integrable__distr,axiom,
    ! [T: product_prod_nat_nat > real,M2: sigma_5515648953823433982at_nat,M: sigma_measure_real,F2: real > real] :
      ( ( member4564283293661824327t_real @ T @ ( sigma_1188828689629184861t_real @ M2 @ M ) )
     => ( ( bochne3340023020068487468l_real @ ( measur4284412874431678645t_real @ M2 @ M @ T ) @ F2 )
       => ( bochne2596016609597520987t_real @ M2
          @ ^ [X2: product_prod_nat_nat] : ( F2 @ ( T @ X2 ) ) ) ) ) ).

% integrable_distr
thf(fact_166_integrable__distr,axiom,
    ! [T: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat,M2: sigma_9047027012034273406at_nat,M: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( member2646312932426349184at_nat @ T @ ( sigma_1496667973943888668at_nat @ M2 @ M ) )
     => ( ( bochne7117808529828525605t_real @ ( measur7731843055261497284at_nat @ M2 @ M @ T ) @ F2 )
       => ( bochne7117808529828525605t_real @ M2
          @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( F2 @ ( T @ X2 ) ) ) ) ) ).

% integrable_distr
thf(fact_167_integrable__distr,axiom,
    ! [T: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,M2: sigma_5515648953823433982at_nat,M: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( member8598435083969174976at_nat @ T @ ( sigma_7146669294626004438at_nat @ M2 @ M ) )
     => ( ( bochne7117808529828525605t_real @ ( measur8330812323704473390at_nat @ M2 @ M @ T ) @ F2 )
       => ( bochne2596016609597520987t_real @ M2
          @ ^ [X2: product_prod_nat_nat] : ( F2 @ ( T @ X2 ) ) ) ) ) ).

% integrable_distr
thf(fact_168_integrable__distr,axiom,
    ! [T: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,M2: sigma_9047027012034273406at_nat,M: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real] :
      ( ( member7151659165045815932at_nat @ T @ ( sigma_3543877168342358986at_nat @ M2 @ M ) )
     => ( ( bochne2596016609597520987t_real @ ( measur4728020197420827938at_nat @ M2 @ M @ T ) @ F2 )
       => ( bochne7117808529828525605t_real @ M2
          @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( F2 @ ( T @ X2 ) ) ) ) ) ).

% integrable_distr
thf(fact_169_integrable__distr,axiom,
    ! [T: product_prod_nat_nat > product_prod_nat_nat,M2: sigma_5515648953823433982at_nat,M: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real] :
      ( ( member8885076297122219836at_nat @ T @ ( sigma_3682087626466204304at_nat @ M2 @ M ) )
     => ( ( bochne2596016609597520987t_real @ ( measur6503400250167368504at_nat @ M2 @ M @ T ) @ F2 )
       => ( bochne2596016609597520987t_real @ M2
          @ ^ [X2: product_prod_nat_nat] : ( F2 @ ( T @ X2 ) ) ) ) ) ).

% integrable_distr
thf(fact_170_linorder__neqE__nat,axiom,
    ! [X5: nat,Y4: nat] :
      ( ( X5 != Y4 )
     => ( ~ ( ord_less_nat @ X5 @ Y4 )
       => ( ord_less_nat @ Y4 @ X5 ) ) ) ).

% linorder_neqE_nat
thf(fact_171_infinite__descent,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N4: nat] :
          ( ~ ( P @ N4 )
         => ? [M5: nat] :
              ( ( ord_less_nat @ M5 @ N4 )
              & ~ ( P @ M5 ) ) )
     => ( P @ N2 ) ) ).

% infinite_descent
thf(fact_172_nat__less__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N4: nat] :
          ( ! [M5: nat] :
              ( ( ord_less_nat @ M5 @ N4 )
             => ( P @ M5 ) )
         => ( P @ N4 ) )
     => ( P @ N2 ) ) ).

% nat_less_induct
thf(fact_173_less__irrefl__nat,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_irrefl_nat
thf(fact_174_less__not__refl3,axiom,
    ! [S: nat,T2: nat] :
      ( ( ord_less_nat @ S @ T2 )
     => ( S != T2 ) ) ).

% less_not_refl3
thf(fact_175_less__not__refl2,axiom,
    ! [N2: nat,M3: nat] :
      ( ( ord_less_nat @ N2 @ M3 )
     => ( M3 != N2 ) ) ).

% less_not_refl2
thf(fact_176_less__not__refl,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_not_refl
thf(fact_177_nat__neq__iff,axiom,
    ! [M3: nat,N2: nat] :
      ( ( M3 != N2 )
      = ( ( ord_less_nat @ M3 @ N2 )
        | ( ord_less_nat @ N2 @ M3 ) ) ) ).

% nat_neq_iff
thf(fact_178_infinite__descent0,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N4: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N4 )
           => ( ~ ( P @ N4 )
             => ? [M5: nat] :
                  ( ( ord_less_nat @ M5 @ N4 )
                  & ~ ( P @ M5 ) ) ) )
       => ( P @ N2 ) ) ) ).

% infinite_descent0
thf(fact_179_gr__implies__not0,axiom,
    ! [M3: nat,N2: nat] :
      ( ( ord_less_nat @ M3 @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_180_less__zeroE,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_zeroE
thf(fact_181_not__less0,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less0
thf(fact_182_not__gr0,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr0
thf(fact_183_gr0I,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr0I
thf(fact_184_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_185_k__wise__indep__vars__compose,axiom,
    ! [K: nat,M: rat > sigma_measure_real,X: rat > product_prod_nat_nat > real,I: set_rat,Y: rat > real > extend8495563244428889912nnreal,N: rat > sigma_7234349610311085201nnreal] :
      ( ( prob_k9046839836125593554t_real @ ( freque5010624893710627907mega_1 @ as ) @ K @ M @ X @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( prob_k9060532812646980446nnreal @ ( freque5010624893710627907mega_1 @ as ) @ K @ N
          @ ^ [I3: rat,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% k_wise_indep_vars_compose
thf(fact_186_k__wise__indep__vars__compose,axiom,
    ! [K: nat,M: $o > sigma_measure_real,X: $o > product_prod_nat_nat > real,I: set_o,Y: $o > real > extend8495563244428889912nnreal,N: $o > sigma_7234349610311085201nnreal] :
      ( ( prob_k5626656578853715448o_real @ ( freque5010624893710627907mega_1 @ as ) @ K @ M @ X @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( prob_k147916592727809156nnreal @ ( freque5010624893710627907mega_1 @ as ) @ K @ N
          @ ^ [I3: $o,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% k_wise_indep_vars_compose
thf(fact_187_k__wise__indep__vars__compose,axiom,
    ! [K: nat,M: rat > sigma_5515648953823433982at_nat,X: rat > product_prod_nat_nat > product_prod_nat_nat,I: set_rat,Y: rat > product_prod_nat_nat > real,N: rat > sigma_measure_real] :
      ( ( prob_k1749327348295208283at_nat @ ( freque5010624893710627907mega_1 @ as ) @ K @ M @ X @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( prob_k9046839836125593554t_real @ ( freque5010624893710627907mega_1 @ as ) @ K @ N
          @ ^ [I3: rat,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% k_wise_indep_vars_compose
thf(fact_188_k__wise__indep__vars__compose,axiom,
    ! [K: nat,M: $o > sigma_5515648953823433982at_nat,X: $o > product_prod_nat_nat > product_prod_nat_nat,I: set_o,Y: $o > product_prod_nat_nat > real,N: $o > sigma_measure_real] :
      ( ( prob_k129107890564256949at_nat @ ( freque5010624893710627907mega_1 @ as ) @ K @ M @ X @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( prob_k5626656578853715448o_real @ ( freque5010624893710627907mega_1 @ as ) @ K @ N
          @ ^ [I3: $o,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% k_wise_indep_vars_compose
thf(fact_189_k__wise__indep__vars__compose,axiom,
    ! [K: nat,M: ( real > extend8495563244428889912nnreal ) > sigma_measure_real,X: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > real,I: set_re5328672808648366137nnreal,Y: ( real > extend8495563244428889912nnreal ) > real > extend8495563244428889912nnreal,N: ( real > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal] :
      ( ( prob_k5108616785771355773l_real @ ( freque5010624893710627907mega_1 @ as ) @ K @ M @ X @ I )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( prob_k1953979174828017545nnreal @ ( freque5010624893710627907mega_1 @ as ) @ K @ N
          @ ^ [I3: real > extend8495563244428889912nnreal,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% k_wise_indep_vars_compose
thf(fact_190_k__wise__indep__vars__compose,axiom,
    ! [K: nat,M: rat > sigma_5515648953823433982at_nat,X: rat > product_prod_nat_nat > product_prod_nat_nat,I: set_rat,Y: rat > product_prod_nat_nat > product_prod_nat_nat,N: rat > sigma_5515648953823433982at_nat] :
      ( ( prob_k1749327348295208283at_nat @ ( freque5010624893710627907mega_1 @ as ) @ K @ M @ X @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member8885076297122219836at_nat @ ( Y @ I2 ) @ ( sigma_3682087626466204304at_nat @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( prob_k1749327348295208283at_nat @ ( freque5010624893710627907mega_1 @ as ) @ K @ N
          @ ^ [I3: rat,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% k_wise_indep_vars_compose
thf(fact_191_k__wise__indep__vars__compose,axiom,
    ! [K: nat,M: $o > sigma_5515648953823433982at_nat,X: $o > product_prod_nat_nat > product_prod_nat_nat,I: set_o,Y: $o > product_prod_nat_nat > product_prod_nat_nat,N: $o > sigma_5515648953823433982at_nat] :
      ( ( prob_k129107890564256949at_nat @ ( freque5010624893710627907mega_1 @ as ) @ K @ M @ X @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member8885076297122219836at_nat @ ( Y @ I2 ) @ ( sigma_3682087626466204304at_nat @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( prob_k129107890564256949at_nat @ ( freque5010624893710627907mega_1 @ as ) @ K @ N
          @ ^ [I3: $o,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% k_wise_indep_vars_compose
thf(fact_192_k__wise__indep__vars__compose,axiom,
    ! [K: nat,M: ( real > extend8495563244428889912nnreal ) > sigma_5515648953823433982at_nat,X: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > product_prod_nat_nat,I: set_re5328672808648366137nnreal,Y: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > real,N: ( real > extend8495563244428889912nnreal ) > sigma_measure_real] :
      ( ( prob_k8442391182763575152at_nat @ ( freque5010624893710627907mega_1 @ as ) @ K @ M @ X @ I )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( prob_k5108616785771355773l_real @ ( freque5010624893710627907mega_1 @ as ) @ K @ N
          @ ^ [I3: real > extend8495563244428889912nnreal,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% k_wise_indep_vars_compose
thf(fact_193_k__wise__indep__vars__compose,axiom,
    ! [K: nat,M: ( product_prod_nat_nat > real ) > sigma_measure_real,X: ( product_prod_nat_nat > real ) > product_prod_nat_nat > real,I: set_Pr1002607673312053630t_real,Y: ( product_prod_nat_nat > real ) > real > extend8495563244428889912nnreal,N: ( product_prod_nat_nat > real ) > sigma_7234349610311085201nnreal] :
      ( ( prob_k5633034945198330622l_real @ ( freque5010624893710627907mega_1 @ as ) @ K @ M @ X @ I )
     => ( ! [I2: product_prod_nat_nat > real] :
            ( ( member4564283293661824327t_real @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( prob_k2282667604904944778nnreal @ ( freque5010624893710627907mega_1 @ as ) @ K @ N
          @ ^ [I3: product_prod_nat_nat > real,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% k_wise_indep_vars_compose
thf(fact_194_k__wise__indep__vars__compose,axiom,
    ! [K: nat,M: ( real > extend8495563244428889912nnreal ) > sigma_5515648953823433982at_nat,X: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > product_prod_nat_nat,I: set_re5328672808648366137nnreal,Y: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > product_prod_nat_nat,N: ( real > extend8495563244428889912nnreal ) > sigma_5515648953823433982at_nat] :
      ( ( prob_k8442391182763575152at_nat @ ( freque5010624893710627907mega_1 @ as ) @ K @ M @ X @ I )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member8885076297122219836at_nat @ ( Y @ I2 ) @ ( sigma_3682087626466204304at_nat @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( prob_k8442391182763575152at_nat @ ( freque5010624893710627907mega_1 @ as ) @ K @ N
          @ ^ [I3: real > extend8495563244428889912nnreal,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% k_wise_indep_vars_compose
thf(fact_195__092_060Omega_062_092_060_094sub_0622_Oindep__var__distribution__eq,axiom,
    ! [S2: sigma_measure_nat,X: ( product_prod_nat_nat > product_prod_nat_nat ) > nat,T: sigma_measure_nat,Y: ( product_prod_nat_nat > product_prod_nat_nat ) > nat] :
      ( ( indepe8633027689432630083at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ S2 @ X @ T @ Y )
      = ( ( member3219910559795078443at_nat @ X @ ( sigma_4532147426969876551at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ S2 ) )
        & ( member3219910559795078443at_nat @ Y @ ( sigma_4532147426969876551at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ T ) )
        & ( ( binary6458575275919672938at_nat @ ( measur6421871825836039407at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ S2 @ X ) @ ( measur6421871825836039407at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ T @ Y ) )
          = ( measur4728020197420827938at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ ( binary6458575275919672938at_nat @ S2 @ T )
            @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( product_Pair_nat_nat @ ( X @ X2 ) @ ( Y @ X2 ) ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_var_distribution_eq
thf(fact_196__092_060Omega_062_092_060_094sub_0622_Oindep__var__distribution__eq,axiom,
    ! [S2: sigma_measure_real,X: ( product_prod_nat_nat > product_prod_nat_nat ) > real,T: sigma_measure_real,Y: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( indepe1938972608428940447t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ S2 @ X @ T @ Y )
      = ( ( member8159409068225774087t_real @ X @ ( sigma_5064276549609164707t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ S2 ) )
        & ( member8159409068225774087t_real @ Y @ ( sigma_5064276549609164707t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ T ) )
        & ( ( binary6478037234023840930l_real @ ( measur6433597746558672971t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ S2 @ X ) @ ( measur6433597746558672971t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ T @ Y ) )
          = ( measur3108995680308112986l_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ ( binary6478037234023840930l_real @ S2 @ T )
            @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( produc4511245868158468465l_real @ ( X @ X2 ) @ ( Y @ X2 ) ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_var_distribution_eq
thf(fact_197_prob__space_Oindep__vars__distr,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real,N: sigma_measure_real,I: set_rat,X4: rat > real > extend8495563244428889912nnreal,M: rat > sigma_7234349610311085201nnreal] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ M2 @ N ) )
       => ( ! [I2: rat] :
              ( ( member_rat @ I2 @ I )
             => ( member2919562650594848410nnreal @ ( X4 @ I2 ) @ ( sigma_9017504469962657078nnreal @ N @ ( M @ I2 ) ) ) )
         => ( ( indepe317362771116669643nnreal @ M2 @ M
              @ ^ [I3: rat] : ( comp_r868043356317784952at_nat @ ( X4 @ I3 ) @ F2 )
              @ I )
           => ( indepe7529601277151292294nnreal @ ( measur4284412874431678645t_real @ M2 @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ) ).

% prob_space.indep_vars_distr
thf(fact_198_prob__space_Oindep__vars__distr,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real,N: sigma_measure_real,I: set_o,X4: $o > real > extend8495563244428889912nnreal,M: $o > sigma_7234349610311085201nnreal] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ M2 @ N ) )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I )
             => ( member2919562650594848410nnreal @ ( X4 @ I2 ) @ ( sigma_9017504469962657078nnreal @ N @ ( M @ I2 ) ) ) )
         => ( ( indepe7306336255242631127nnreal @ M2 @ M
              @ ^ [I3: $o] : ( comp_r868043356317784952at_nat @ ( X4 @ I3 ) @ F2 )
              @ I )
           => ( indepe1757858165614604892nnreal @ ( measur4284412874431678645t_real @ M2 @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ) ).

% prob_space.indep_vars_distr
thf(fact_199_prob__space_Oindep__vars__distr,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > product_prod_nat_nat,N: sigma_5515648953823433982at_nat,I: set_rat,X4: rat > product_prod_nat_nat > real,M: rat > sigma_measure_real] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( member8885076297122219836at_nat @ F2 @ ( sigma_3682087626466204304at_nat @ M2 @ N ) )
       => ( ! [I2: rat] :
              ( ( member_rat @ I2 @ I )
             => ( member4564283293661824327t_real @ ( X4 @ I2 ) @ ( sigma_1188828689629184861t_real @ N @ ( M @ I2 ) ) ) )
         => ( ( indepe5387908695526973631t_real @ M2 @ M
              @ ^ [I3: rat] : ( comp_P6083776177191121141at_nat @ ( X4 @ I3 ) @ F2 )
              @ I )
           => ( indepe5387908695526973631t_real @ ( measur6503400250167368504at_nat @ M2 @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ) ).

% prob_space.indep_vars_distr
thf(fact_200_prob__space_Oindep__vars__distr,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > product_prod_nat_nat,N: sigma_5515648953823433982at_nat,I: set_o,X4: $o > product_prod_nat_nat > real,M: $o > sigma_measure_real] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( member8885076297122219836at_nat @ F2 @ ( sigma_3682087626466204304at_nat @ M2 @ N ) )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I )
             => ( member4564283293661824327t_real @ ( X4 @ I2 ) @ ( sigma_1188828689629184861t_real @ N @ ( M @ I2 ) ) ) )
         => ( ( indepe6599791682495051723o_real @ M2 @ M
              @ ^ [I3: $o] : ( comp_P6083776177191121141at_nat @ ( X4 @ I3 ) @ F2 )
              @ I )
           => ( indepe6599791682495051723o_real @ ( measur6503400250167368504at_nat @ M2 @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ) ).

% prob_space.indep_vars_distr
thf(fact_201_prob__space_Oindep__vars__distr,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real,N: sigma_measure_real,I: set_re5328672808648366137nnreal,X4: ( real > extend8495563244428889912nnreal ) > real > extend8495563244428889912nnreal,M: ( real > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ M2 @ N ) )
       => ( ! [I2: real > extend8495563244428889912nnreal] :
              ( ( member2919562650594848410nnreal @ I2 @ I )
             => ( member2919562650594848410nnreal @ ( X4 @ I2 ) @ ( sigma_9017504469962657078nnreal @ N @ ( M @ I2 ) ) ) )
         => ( ( indepe3243683517584732406nnreal @ M2 @ M
              @ ^ [I3: real > extend8495563244428889912nnreal] : ( comp_r868043356317784952at_nat @ ( X4 @ I3 ) @ F2 )
              @ I )
           => ( indepe8147369424511330225nnreal @ ( measur4284412874431678645t_real @ M2 @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ) ).

% prob_space.indep_vars_distr
thf(fact_202_prob__space_Oindep__vars__distr,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,N: sigma_measure_real,I: set_rat,X4: rat > real > extend8495563244428889912nnreal,M: rat > sigma_7234349610311085201nnreal] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ M2 @ N ) )
       => ( ! [I2: rat] :
              ( ( member_rat @ I2 @ I )
             => ( member2919562650594848410nnreal @ ( X4 @ I2 ) @ ( sigma_9017504469962657078nnreal @ N @ ( M @ I2 ) ) ) )
         => ( ( indepe4547145070500471935nnreal @ M2 @ M
              @ ^ [I3: rat] : ( comp_r2164139168250337006at_nat @ ( X4 @ I3 ) @ F2 )
              @ I )
           => ( indepe7529601277151292294nnreal @ ( measur6433597746558672971t_real @ M2 @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ) ).

% prob_space.indep_vars_distr
thf(fact_203_prob__space_Oindep__vars__distr,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,N: sigma_measure_real,I: set_o,X4: $o > real > extend8495563244428889912nnreal,M: $o > sigma_7234349610311085201nnreal] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ M2 @ N ) )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I )
             => ( member2919562650594848410nnreal @ ( X4 @ I2 ) @ ( sigma_9017504469962657078nnreal @ N @ ( M @ I2 ) ) ) )
         => ( ( indepe2783168947688421667nnreal @ M2 @ M
              @ ^ [I3: $o] : ( comp_r2164139168250337006at_nat @ ( X4 @ I3 ) @ F2 )
              @ I )
           => ( indepe1757858165614604892nnreal @ ( measur6433597746558672971t_real @ M2 @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ) ).

% prob_space.indep_vars_distr
thf(fact_204_prob__space_Oindep__vars__distr,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > product_prod_nat_nat,N: sigma_5515648953823433982at_nat,I: set_rat,X4: rat > product_prod_nat_nat > product_prod_nat_nat,M: rat > sigma_5515648953823433982at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( member8885076297122219836at_nat @ F2 @ ( sigma_3682087626466204304at_nat @ M2 @ N ) )
       => ( ! [I2: rat] :
              ( ( member_rat @ I2 @ I )
             => ( member8885076297122219836at_nat @ ( X4 @ I2 ) @ ( sigma_3682087626466204304at_nat @ N @ ( M @ I2 ) ) ) )
         => ( ( indepe2321685950664944046at_nat @ M2 @ M
              @ ^ [I3: rat] : ( comp_P2240441846945064862at_nat @ ( X4 @ I3 ) @ F2 )
              @ I )
           => ( indepe2321685950664944046at_nat @ ( measur6503400250167368504at_nat @ M2 @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ) ).

% prob_space.indep_vars_distr
thf(fact_205_prob__space_Oindep__vars__distr,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > product_prod_nat_nat,N: sigma_5515648953823433982at_nat,I: set_o,X4: $o > product_prod_nat_nat > product_prod_nat_nat,M: $o > sigma_5515648953823433982at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( member8885076297122219836at_nat @ F2 @ ( sigma_3682087626466204304at_nat @ M2 @ N ) )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I )
             => ( member8885076297122219836at_nat @ ( X4 @ I2 ) @ ( sigma_3682087626466204304at_nat @ N @ ( M @ I2 ) ) ) )
         => ( ( indepe504693989766925730at_nat @ M2 @ M
              @ ^ [I3: $o] : ( comp_P2240441846945064862at_nat @ ( X4 @ I3 ) @ F2 )
              @ I )
           => ( indepe504693989766925730at_nat @ ( measur6503400250167368504at_nat @ M2 @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ) ).

% prob_space.indep_vars_distr
thf(fact_206_prob__space_Oindep__vars__distr,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > product_prod_nat_nat,N: sigma_5515648953823433982at_nat,I: set_re5328672808648366137nnreal,X4: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > real,M: ( real > extend8495563244428889912nnreal ) > sigma_measure_real] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( member8885076297122219836at_nat @ F2 @ ( sigma_3682087626466204304at_nat @ M2 @ N ) )
       => ( ! [I2: real > extend8495563244428889912nnreal] :
              ( ( member2919562650594848410nnreal @ I2 @ I )
             => ( member4564283293661824327t_real @ ( X4 @ I2 ) @ ( sigma_1188828689629184861t_real @ N @ ( M @ I2 ) ) ) )
         => ( ( indepe579324506763905386l_real @ M2 @ M
              @ ^ [I3: real > extend8495563244428889912nnreal] : ( comp_P6083776177191121141at_nat @ ( X4 @ I3 ) @ F2 )
              @ I )
           => ( indepe579324506763905386l_real @ ( measur6503400250167368504at_nat @ M2 @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ) ).

% prob_space.indep_vars_distr
thf(fact_207_KL__same__eq__0,axiom,
    ! [B: real] :
      ( ( kL_div6543510193410477195at_nat @ B @ ( freque5010624893710627907mega_1 @ as ) @ ( freque5010624893710627907mega_1 @ as ) )
      = zero_zero_real ) ).

% KL_same_eq_0
thf(fact_208_Pair__le,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_le4635361258860028081at_rat @ ( product_Pair_rat_rat @ A @ B ) @ ( product_Pair_rat_rat @ C @ D ) )
      = ( ( ord_less_eq_rat @ A @ C )
        & ( ord_less_eq_rat @ B @ D ) ) ) ).

% Pair_le
thf(fact_209_Pair__le,axiom,
    ! [A: rat,B: nat,C: rat,D: nat] :
      ( ( ord_le8143589817253424809at_nat @ ( product_Pair_rat_nat @ A @ B ) @ ( product_Pair_rat_nat @ C @ D ) )
      = ( ( ord_less_eq_rat @ A @ C )
        & ( ord_less_eq_nat @ B @ D ) ) ) ).

% Pair_le
thf(fact_210_Pair__le,axiom,
    ! [A: nat,B: rat,C: nat,D: rat] :
      ( ( ord_le4951915902794893993at_rat @ ( product_Pair_nat_rat @ A @ B ) @ ( product_Pair_nat_rat @ C @ D ) )
      = ( ( ord_less_eq_nat @ A @ C )
        & ( ord_less_eq_rat @ B @ D ) ) ) ).

% Pair_le
thf(fact_211_Pair__le,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
      = ( ( ord_less_eq_nat @ A @ C )
        & ( ord_less_eq_nat @ B @ D ) ) ) ).

% Pair_le
thf(fact_212_Pair__le,axiom,
    ! [A: set_Pr1002607673312053630t_real,B: rat,C: set_Pr1002607673312053630t_real,D: rat] :
      ( ( ord_le2259096304072862181al_rat @ ( produc370051823116057599al_rat @ A @ B ) @ ( produc370051823116057599al_rat @ C @ D ) )
      = ( ( ord_le1994730768851292446t_real @ A @ C )
        & ( ord_less_eq_rat @ B @ D ) ) ) ).

% Pair_le
thf(fact_213_Pair__le,axiom,
    ! [A: set_Pr1002607673312053630t_real,B: nat,C: set_Pr1002607673312053630t_real,D: nat] :
      ( ( ord_le5767324862466258909al_nat @ ( produc1005181883202553335al_nat @ A @ B ) @ ( produc1005181883202553335al_nat @ C @ D ) )
      = ( ( ord_le1994730768851292446t_real @ A @ C )
        & ( ord_less_eq_nat @ B @ D ) ) ) ).

% Pair_le
thf(fact_214_Pair__le,axiom,
    ! [A: rat,B: set_Pr1002607673312053630t_real,C: rat,D: set_Pr1002607673312053630t_real] :
      ( ( ord_le6637847787101155707t_real @ ( produc8291022738106172877t_real @ A @ B ) @ ( produc8291022738106172877t_real @ C @ D ) )
      = ( ( ord_less_eq_rat @ A @ C )
        & ( ord_le1994730768851292446t_real @ B @ D ) ) ) ).

% Pair_le
thf(fact_215_Pair__le,axiom,
    ! [A: nat,B: set_Pr1002607673312053630t_real,C: nat,D: set_Pr1002607673312053630t_real] :
      ( ( ord_le1943484209260552579t_real @ ( produc3596659160265569749t_real @ A @ B ) @ ( produc3596659160265569749t_real @ C @ D ) )
      = ( ( ord_less_eq_nat @ A @ C )
        & ( ord_le1994730768851292446t_real @ B @ D ) ) ) ).

% Pair_le
thf(fact_216_Pair__le,axiom,
    ! [A: set_Pr2458342521480944603at_nat,B: rat,C: set_Pr2458342521480944603at_nat,D: rat] :
      ( ( ord_le7912795254584737716at_rat @ ( produc5584217423264826444at_rat @ A @ B ) @ ( produc5584217423264826444at_rat @ C @ D ) )
      = ( ( ord_le3857079194666040379at_nat @ A @ C )
        & ( ord_less_eq_rat @ B @ D ) ) ) ).

% Pair_le
thf(fact_217_Pair__le,axiom,
    ! [A: set_Pr2458342521480944603at_nat,B: nat,C: set_Pr2458342521480944603at_nat,D: nat] :
      ( ( ord_le2197651776123358636at_nat @ ( produc6219347483351322180at_nat @ A @ B ) @ ( produc6219347483351322180at_nat @ C @ D ) )
      = ( ( ord_le3857079194666040379at_nat @ A @ C )
        & ( ord_less_eq_nat @ B @ D ) ) ) ).

% Pair_le
thf(fact_218_not__gr__zero,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_219_not__gr__zero,axiom,
    ! [N2: extend8495563244428889912nnreal] :
      ( ( ~ ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ N2 ) )
      = ( N2 = zero_z7100319975126383169nnreal ) ) ).

% not_gr_zero
thf(fact_220__092_060Omega_062_092_060_094sub_0622_OLp__subset__Lq_I2_J,axiom,
    ! [P2: extend8495563244428889912nnreal,Q2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ P2 @ Q2 )
     => ( ord_le1044648919696853383t_real @ ( lp_spa812643495979991928at_nat @ Q2 @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) @ ( lp_spa812643495979991928at_nat @ P2 @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) ) ) ).

% \<Omega>\<^sub>2.Lp_subset_Lq(2)
thf(fact_221_zero__order_I2_J,axiom,
    ! [N2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ N2 @ zero_z7100319975126383169nnreal )
      = ( N2 = zero_z7100319975126383169nnreal ) ) ).

% zero_order(2)
thf(fact_222_zero__order_I2_J,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% zero_order(2)
thf(fact_223_prob__space__distr,axiom,
    ! [F2: product_prod_nat_nat > real,M: sigma_measure_real] :
      ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ ( freque5010624893710627907mega_1 @ as ) @ M ) )
     => ( probab535871623910865577e_real @ ( measur4284412874431678645t_real @ ( freque5010624893710627907mega_1 @ as ) @ M @ F2 ) ) ) ).

% prob_space_distr
thf(fact_224_prob__space__distr,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,M: sigma_9047027012034273406at_nat] :
      ( ( member8598435083969174976at_nat @ F2 @ ( sigma_7146669294626004438at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M ) )
     => ( probab8562894880268318498at_nat @ ( measur8330812323704473390at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M @ F2 ) ) ) ).

% prob_space_distr
thf(fact_225_prob__space__distr,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,M: sigma_5515648953823433982at_nat] :
      ( ( member8885076297122219836at_nat @ F2 @ ( sigma_3682087626466204304at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M ) )
     => ( probab2019946041432190532at_nat @ ( measur6503400250167368504at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M @ F2 ) ) ) ).

% prob_space_distr
thf(fact_226_finite__measure__distr,axiom,
    ! [F2: product_prod_nat_nat > real,M: sigma_measure_real] :
      ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ ( freque5010624893710627907mega_1 @ as ) @ M ) )
     => ( measur3606880022600206024e_real @ ( measur4284412874431678645t_real @ ( freque5010624893710627907mega_1 @ as ) @ M @ F2 ) ) ) ).

% finite_measure_distr
thf(fact_227_finite__measure__distr,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,M: sigma_9047027012034273406at_nat] :
      ( ( member8598435083969174976at_nat @ F2 @ ( sigma_7146669294626004438at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M ) )
     => ( measur5156515106886843841at_nat @ ( measur8330812323704473390at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M @ F2 ) ) ) ).

% finite_measure_distr
thf(fact_228_finite__measure__distr,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,M: sigma_5515648953823433982at_nat] :
      ( ( member8885076297122219836at_nat @ F2 @ ( sigma_3682087626466204304at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M ) )
     => ( measur7911732083032432613at_nat @ ( measur6503400250167368504at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M @ F2 ) ) ) ).

% finite_measure_distr
thf(fact_229_indep__vars__compose,axiom,
    ! [M: rat > sigma_measure_real,X: rat > product_prod_nat_nat > real,I: set_rat,Y: rat > real > extend8495563244428889912nnreal,N: rat > sigma_7234349610311085201nnreal] :
      ( ( indepe5387908695526973631t_real @ ( freque5010624893710627907mega_1 @ as ) @ M @ X @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe317362771116669643nnreal @ ( freque5010624893710627907mega_1 @ as ) @ N
          @ ^ [I3: rat] : ( comp_r868043356317784952at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
          @ I ) ) ) ).

% indep_vars_compose
thf(fact_230_indep__vars__compose,axiom,
    ! [M: $o > sigma_measure_real,X: $o > product_prod_nat_nat > real,I: set_o,Y: $o > real > extend8495563244428889912nnreal,N: $o > sigma_7234349610311085201nnreal] :
      ( ( indepe6599791682495051723o_real @ ( freque5010624893710627907mega_1 @ as ) @ M @ X @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe7306336255242631127nnreal @ ( freque5010624893710627907mega_1 @ as ) @ N
          @ ^ [I3: $o] : ( comp_r868043356317784952at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
          @ I ) ) ) ).

% indep_vars_compose
thf(fact_231_indep__vars__compose,axiom,
    ! [M: rat > sigma_5515648953823433982at_nat,X: rat > product_prod_nat_nat > product_prod_nat_nat,I: set_rat,Y: rat > product_prod_nat_nat > real,N: rat > sigma_measure_real] :
      ( ( indepe2321685950664944046at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M @ X @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe5387908695526973631t_real @ ( freque5010624893710627907mega_1 @ as ) @ N
          @ ^ [I3: rat] : ( comp_P6083776177191121141at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
          @ I ) ) ) ).

% indep_vars_compose
thf(fact_232_indep__vars__compose,axiom,
    ! [M: $o > sigma_5515648953823433982at_nat,X: $o > product_prod_nat_nat > product_prod_nat_nat,I: set_o,Y: $o > product_prod_nat_nat > real,N: $o > sigma_measure_real] :
      ( ( indepe504693989766925730at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M @ X @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe6599791682495051723o_real @ ( freque5010624893710627907mega_1 @ as ) @ N
          @ ^ [I3: $o] : ( comp_P6083776177191121141at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
          @ I ) ) ) ).

% indep_vars_compose
thf(fact_233_indep__vars__compose,axiom,
    ! [M: ( real > extend8495563244428889912nnreal ) > sigma_measure_real,X: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > real,I: set_re5328672808648366137nnreal,Y: ( real > extend8495563244428889912nnreal ) > real > extend8495563244428889912nnreal,N: ( real > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal] :
      ( ( indepe579324506763905386l_real @ ( freque5010624893710627907mega_1 @ as ) @ M @ X @ I )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe3243683517584732406nnreal @ ( freque5010624893710627907mega_1 @ as ) @ N
          @ ^ [I3: real > extend8495563244428889912nnreal] : ( comp_r868043356317784952at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
          @ I ) ) ) ).

% indep_vars_compose
thf(fact_234_indep__vars__compose,axiom,
    ! [M: rat > sigma_5515648953823433982at_nat,X: rat > product_prod_nat_nat > product_prod_nat_nat,I: set_rat,Y: rat > product_prod_nat_nat > product_prod_nat_nat,N: rat > sigma_5515648953823433982at_nat] :
      ( ( indepe2321685950664944046at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M @ X @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member8885076297122219836at_nat @ ( Y @ I2 ) @ ( sigma_3682087626466204304at_nat @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe2321685950664944046at_nat @ ( freque5010624893710627907mega_1 @ as ) @ N
          @ ^ [I3: rat] : ( comp_P2240441846945064862at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
          @ I ) ) ) ).

% indep_vars_compose
thf(fact_235_indep__vars__compose,axiom,
    ! [M: $o > sigma_5515648953823433982at_nat,X: $o > product_prod_nat_nat > product_prod_nat_nat,I: set_o,Y: $o > product_prod_nat_nat > product_prod_nat_nat,N: $o > sigma_5515648953823433982at_nat] :
      ( ( indepe504693989766925730at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M @ X @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member8885076297122219836at_nat @ ( Y @ I2 ) @ ( sigma_3682087626466204304at_nat @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe504693989766925730at_nat @ ( freque5010624893710627907mega_1 @ as ) @ N
          @ ^ [I3: $o] : ( comp_P2240441846945064862at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
          @ I ) ) ) ).

% indep_vars_compose
thf(fact_236_indep__vars__compose,axiom,
    ! [M: ( real > extend8495563244428889912nnreal ) > sigma_5515648953823433982at_nat,X: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > product_prod_nat_nat,I: set_re5328672808648366137nnreal,Y: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > real,N: ( real > extend8495563244428889912nnreal ) > sigma_measure_real] :
      ( ( indepe3593890525012728259at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M @ X @ I )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe579324506763905386l_real @ ( freque5010624893710627907mega_1 @ as ) @ N
          @ ^ [I3: real > extend8495563244428889912nnreal] : ( comp_P6083776177191121141at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
          @ I ) ) ) ).

% indep_vars_compose
thf(fact_237_indep__vars__compose,axiom,
    ! [M: ( product_prod_nat_nat > real ) > sigma_measure_real,X: ( product_prod_nat_nat > real ) > product_prod_nat_nat > real,I: set_Pr1002607673312053630t_real,Y: ( product_prod_nat_nat > real ) > real > extend8495563244428889912nnreal,N: ( product_prod_nat_nat > real ) > sigma_7234349610311085201nnreal] :
      ( ( indepe3389242201104679889l_real @ ( freque5010624893710627907mega_1 @ as ) @ M @ X @ I )
     => ( ! [I2: product_prod_nat_nat > real] :
            ( ( member4564283293661824327t_real @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe6657538984008873693nnreal @ ( freque5010624893710627907mega_1 @ as ) @ N
          @ ^ [I3: product_prod_nat_nat > real] : ( comp_r868043356317784952at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
          @ I ) ) ) ).

% indep_vars_compose
thf(fact_238_indep__vars__compose,axiom,
    ! [M: ( real > extend8495563244428889912nnreal ) > sigma_5515648953823433982at_nat,X: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > product_prod_nat_nat,I: set_re5328672808648366137nnreal,Y: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > product_prod_nat_nat,N: ( real > extend8495563244428889912nnreal ) > sigma_5515648953823433982at_nat] :
      ( ( indepe3593890525012728259at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M @ X @ I )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member8885076297122219836at_nat @ ( Y @ I2 ) @ ( sigma_3682087626466204304at_nat @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe3593890525012728259at_nat @ ( freque5010624893710627907mega_1 @ as ) @ N
          @ ^ [I3: real > extend8495563244428889912nnreal] : ( comp_P2240441846945064862at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
          @ I ) ) ) ).

% indep_vars_compose
thf(fact_239_indep__vars__compose2,axiom,
    ! [M: rat > sigma_measure_real,X: rat > product_prod_nat_nat > real,I: set_rat,Y: rat > real > extend8495563244428889912nnreal,N: rat > sigma_7234349610311085201nnreal] :
      ( ( indepe5387908695526973631t_real @ ( freque5010624893710627907mega_1 @ as ) @ M @ X @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe317362771116669643nnreal @ ( freque5010624893710627907mega_1 @ as ) @ N
          @ ^ [I3: rat,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% indep_vars_compose2
thf(fact_240_indep__vars__compose2,axiom,
    ! [M: $o > sigma_measure_real,X: $o > product_prod_nat_nat > real,I: set_o,Y: $o > real > extend8495563244428889912nnreal,N: $o > sigma_7234349610311085201nnreal] :
      ( ( indepe6599791682495051723o_real @ ( freque5010624893710627907mega_1 @ as ) @ M @ X @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe7306336255242631127nnreal @ ( freque5010624893710627907mega_1 @ as ) @ N
          @ ^ [I3: $o,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% indep_vars_compose2
thf(fact_241_indep__vars__compose2,axiom,
    ! [M: rat > sigma_5515648953823433982at_nat,X: rat > product_prod_nat_nat > product_prod_nat_nat,I: set_rat,Y: rat > product_prod_nat_nat > real,N: rat > sigma_measure_real] :
      ( ( indepe2321685950664944046at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M @ X @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe5387908695526973631t_real @ ( freque5010624893710627907mega_1 @ as ) @ N
          @ ^ [I3: rat,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% indep_vars_compose2
thf(fact_242_indep__vars__compose2,axiom,
    ! [M: $o > sigma_5515648953823433982at_nat,X: $o > product_prod_nat_nat > product_prod_nat_nat,I: set_o,Y: $o > product_prod_nat_nat > real,N: $o > sigma_measure_real] :
      ( ( indepe504693989766925730at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M @ X @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe6599791682495051723o_real @ ( freque5010624893710627907mega_1 @ as ) @ N
          @ ^ [I3: $o,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% indep_vars_compose2
thf(fact_243_indep__vars__compose2,axiom,
    ! [M: ( real > extend8495563244428889912nnreal ) > sigma_measure_real,X: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > real,I: set_re5328672808648366137nnreal,Y: ( real > extend8495563244428889912nnreal ) > real > extend8495563244428889912nnreal,N: ( real > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal] :
      ( ( indepe579324506763905386l_real @ ( freque5010624893710627907mega_1 @ as ) @ M @ X @ I )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe3243683517584732406nnreal @ ( freque5010624893710627907mega_1 @ as ) @ N
          @ ^ [I3: real > extend8495563244428889912nnreal,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% indep_vars_compose2
thf(fact_244_indep__vars__compose2,axiom,
    ! [M: rat > sigma_5515648953823433982at_nat,X: rat > product_prod_nat_nat > product_prod_nat_nat,I: set_rat,Y: rat > product_prod_nat_nat > product_prod_nat_nat,N: rat > sigma_5515648953823433982at_nat] :
      ( ( indepe2321685950664944046at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M @ X @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member8885076297122219836at_nat @ ( Y @ I2 ) @ ( sigma_3682087626466204304at_nat @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe2321685950664944046at_nat @ ( freque5010624893710627907mega_1 @ as ) @ N
          @ ^ [I3: rat,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% indep_vars_compose2
thf(fact_245_indep__vars__compose2,axiom,
    ! [M: $o > sigma_5515648953823433982at_nat,X: $o > product_prod_nat_nat > product_prod_nat_nat,I: set_o,Y: $o > product_prod_nat_nat > product_prod_nat_nat,N: $o > sigma_5515648953823433982at_nat] :
      ( ( indepe504693989766925730at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M @ X @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member8885076297122219836at_nat @ ( Y @ I2 ) @ ( sigma_3682087626466204304at_nat @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe504693989766925730at_nat @ ( freque5010624893710627907mega_1 @ as ) @ N
          @ ^ [I3: $o,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% indep_vars_compose2
thf(fact_246_indep__vars__compose2,axiom,
    ! [M: ( real > extend8495563244428889912nnreal ) > sigma_5515648953823433982at_nat,X: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > product_prod_nat_nat,I: set_re5328672808648366137nnreal,Y: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > real,N: ( real > extend8495563244428889912nnreal ) > sigma_measure_real] :
      ( ( indepe3593890525012728259at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M @ X @ I )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe579324506763905386l_real @ ( freque5010624893710627907mega_1 @ as ) @ N
          @ ^ [I3: real > extend8495563244428889912nnreal,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% indep_vars_compose2
thf(fact_247_indep__vars__compose2,axiom,
    ! [M: ( product_prod_nat_nat > real ) > sigma_measure_real,X: ( product_prod_nat_nat > real ) > product_prod_nat_nat > real,I: set_Pr1002607673312053630t_real,Y: ( product_prod_nat_nat > real ) > real > extend8495563244428889912nnreal,N: ( product_prod_nat_nat > real ) > sigma_7234349610311085201nnreal] :
      ( ( indepe3389242201104679889l_real @ ( freque5010624893710627907mega_1 @ as ) @ M @ X @ I )
     => ( ! [I2: product_prod_nat_nat > real] :
            ( ( member4564283293661824327t_real @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe6657538984008873693nnreal @ ( freque5010624893710627907mega_1 @ as ) @ N
          @ ^ [I3: product_prod_nat_nat > real,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% indep_vars_compose2
thf(fact_248_indep__vars__compose2,axiom,
    ! [M: ( real > extend8495563244428889912nnreal ) > sigma_5515648953823433982at_nat,X: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > product_prod_nat_nat,I: set_re5328672808648366137nnreal,Y: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > product_prod_nat_nat,N: ( real > extend8495563244428889912nnreal ) > sigma_5515648953823433982at_nat] :
      ( ( indepe3593890525012728259at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M @ X @ I )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member8885076297122219836at_nat @ ( Y @ I2 ) @ ( sigma_3682087626466204304at_nat @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
       => ( indepe3593890525012728259at_nat @ ( freque5010624893710627907mega_1 @ as ) @ N
          @ ^ [I3: real > extend8495563244428889912nnreal,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
          @ I ) ) ) ).

% indep_vars_compose2
thf(fact_249_indep__vars__distr,axiom,
    ! [F2: product_prod_nat_nat > real,N: sigma_measure_real,I: set_rat,X4: rat > real > extend8495563244428889912nnreal,M: rat > sigma_7234349610311085201nnreal] :
      ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ ( freque5010624893710627907mega_1 @ as ) @ N ) )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( X4 @ I2 ) @ ( sigma_9017504469962657078nnreal @ N @ ( M @ I2 ) ) ) )
       => ( ( indepe317362771116669643nnreal @ ( freque5010624893710627907mega_1 @ as ) @ M
            @ ^ [I3: rat] : ( comp_r868043356317784952at_nat @ ( X4 @ I3 ) @ F2 )
            @ I )
         => ( indepe7529601277151292294nnreal @ ( measur4284412874431678645t_real @ ( freque5010624893710627907mega_1 @ as ) @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ).

% indep_vars_distr
thf(fact_250_indep__vars__distr,axiom,
    ! [F2: product_prod_nat_nat > real,N: sigma_measure_real,I: set_o,X4: $o > real > extend8495563244428889912nnreal,M: $o > sigma_7234349610311085201nnreal] :
      ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ ( freque5010624893710627907mega_1 @ as ) @ N ) )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( X4 @ I2 ) @ ( sigma_9017504469962657078nnreal @ N @ ( M @ I2 ) ) ) )
       => ( ( indepe7306336255242631127nnreal @ ( freque5010624893710627907mega_1 @ as ) @ M
            @ ^ [I3: $o] : ( comp_r868043356317784952at_nat @ ( X4 @ I3 ) @ F2 )
            @ I )
         => ( indepe1757858165614604892nnreal @ ( measur4284412874431678645t_real @ ( freque5010624893710627907mega_1 @ as ) @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ).

% indep_vars_distr
thf(fact_251_indep__vars__distr,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,N: sigma_5515648953823433982at_nat,I: set_rat,X4: rat > product_prod_nat_nat > real,M: rat > sigma_measure_real] :
      ( ( member8885076297122219836at_nat @ F2 @ ( sigma_3682087626466204304at_nat @ ( freque5010624893710627907mega_1 @ as ) @ N ) )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member4564283293661824327t_real @ ( X4 @ I2 ) @ ( sigma_1188828689629184861t_real @ N @ ( M @ I2 ) ) ) )
       => ( ( indepe5387908695526973631t_real @ ( freque5010624893710627907mega_1 @ as ) @ M
            @ ^ [I3: rat] : ( comp_P6083776177191121141at_nat @ ( X4 @ I3 ) @ F2 )
            @ I )
         => ( indepe5387908695526973631t_real @ ( measur6503400250167368504at_nat @ ( freque5010624893710627907mega_1 @ as ) @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ).

% indep_vars_distr
thf(fact_252_indep__vars__distr,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,N: sigma_5515648953823433982at_nat,I: set_o,X4: $o > product_prod_nat_nat > real,M: $o > sigma_measure_real] :
      ( ( member8885076297122219836at_nat @ F2 @ ( sigma_3682087626466204304at_nat @ ( freque5010624893710627907mega_1 @ as ) @ N ) )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member4564283293661824327t_real @ ( X4 @ I2 ) @ ( sigma_1188828689629184861t_real @ N @ ( M @ I2 ) ) ) )
       => ( ( indepe6599791682495051723o_real @ ( freque5010624893710627907mega_1 @ as ) @ M
            @ ^ [I3: $o] : ( comp_P6083776177191121141at_nat @ ( X4 @ I3 ) @ F2 )
            @ I )
         => ( indepe6599791682495051723o_real @ ( measur6503400250167368504at_nat @ ( freque5010624893710627907mega_1 @ as ) @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ).

% indep_vars_distr
thf(fact_253_indep__vars__distr,axiom,
    ! [F2: product_prod_nat_nat > real,N: sigma_measure_real,I: set_re5328672808648366137nnreal,X4: ( real > extend8495563244428889912nnreal ) > real > extend8495563244428889912nnreal,M: ( real > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal] :
      ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ ( freque5010624893710627907mega_1 @ as ) @ N ) )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( X4 @ I2 ) @ ( sigma_9017504469962657078nnreal @ N @ ( M @ I2 ) ) ) )
       => ( ( indepe3243683517584732406nnreal @ ( freque5010624893710627907mega_1 @ as ) @ M
            @ ^ [I3: real > extend8495563244428889912nnreal] : ( comp_r868043356317784952at_nat @ ( X4 @ I3 ) @ F2 )
            @ I )
         => ( indepe8147369424511330225nnreal @ ( measur4284412874431678645t_real @ ( freque5010624893710627907mega_1 @ as ) @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ).

% indep_vars_distr
thf(fact_254_indep__vars__distr,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,N: sigma_5515648953823433982at_nat,I: set_rat,X4: rat > product_prod_nat_nat > product_prod_nat_nat,M: rat > sigma_5515648953823433982at_nat] :
      ( ( member8885076297122219836at_nat @ F2 @ ( sigma_3682087626466204304at_nat @ ( freque5010624893710627907mega_1 @ as ) @ N ) )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member8885076297122219836at_nat @ ( X4 @ I2 ) @ ( sigma_3682087626466204304at_nat @ N @ ( M @ I2 ) ) ) )
       => ( ( indepe2321685950664944046at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M
            @ ^ [I3: rat] : ( comp_P2240441846945064862at_nat @ ( X4 @ I3 ) @ F2 )
            @ I )
         => ( indepe2321685950664944046at_nat @ ( measur6503400250167368504at_nat @ ( freque5010624893710627907mega_1 @ as ) @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ).

% indep_vars_distr
thf(fact_255_indep__vars__distr,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,N: sigma_5515648953823433982at_nat,I: set_o,X4: $o > product_prod_nat_nat > product_prod_nat_nat,M: $o > sigma_5515648953823433982at_nat] :
      ( ( member8885076297122219836at_nat @ F2 @ ( sigma_3682087626466204304at_nat @ ( freque5010624893710627907mega_1 @ as ) @ N ) )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member8885076297122219836at_nat @ ( X4 @ I2 ) @ ( sigma_3682087626466204304at_nat @ N @ ( M @ I2 ) ) ) )
       => ( ( indepe504693989766925730at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M
            @ ^ [I3: $o] : ( comp_P2240441846945064862at_nat @ ( X4 @ I3 ) @ F2 )
            @ I )
         => ( indepe504693989766925730at_nat @ ( measur6503400250167368504at_nat @ ( freque5010624893710627907mega_1 @ as ) @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ).

% indep_vars_distr
thf(fact_256_indep__vars__distr,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,N: sigma_5515648953823433982at_nat,I: set_re5328672808648366137nnreal,X4: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > real,M: ( real > extend8495563244428889912nnreal ) > sigma_measure_real] :
      ( ( member8885076297122219836at_nat @ F2 @ ( sigma_3682087626466204304at_nat @ ( freque5010624893710627907mega_1 @ as ) @ N ) )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member4564283293661824327t_real @ ( X4 @ I2 ) @ ( sigma_1188828689629184861t_real @ N @ ( M @ I2 ) ) ) )
       => ( ( indepe579324506763905386l_real @ ( freque5010624893710627907mega_1 @ as ) @ M
            @ ^ [I3: real > extend8495563244428889912nnreal] : ( comp_P6083776177191121141at_nat @ ( X4 @ I3 ) @ F2 )
            @ I )
         => ( indepe579324506763905386l_real @ ( measur6503400250167368504at_nat @ ( freque5010624893710627907mega_1 @ as ) @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ).

% indep_vars_distr
thf(fact_257_indep__vars__distr,axiom,
    ! [F2: product_prod_nat_nat > real,N: sigma_measure_real,I: set_Pr1002607673312053630t_real,X4: ( product_prod_nat_nat > real ) > real > extend8495563244428889912nnreal,M: ( product_prod_nat_nat > real ) > sigma_7234349610311085201nnreal] :
      ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ ( freque5010624893710627907mega_1 @ as ) @ N ) )
     => ( ! [I2: product_prod_nat_nat > real] :
            ( ( member4564283293661824327t_real @ I2 @ I )
           => ( member2919562650594848410nnreal @ ( X4 @ I2 ) @ ( sigma_9017504469962657078nnreal @ N @ ( M @ I2 ) ) ) )
       => ( ( indepe6657538984008873693nnreal @ ( freque5010624893710627907mega_1 @ as ) @ M
            @ ^ [I3: product_prod_nat_nat > real] : ( comp_r868043356317784952at_nat @ ( X4 @ I3 ) @ F2 )
            @ I )
         => ( indepe3984226279986541154nnreal @ ( measur4284412874431678645t_real @ ( freque5010624893710627907mega_1 @ as ) @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ).

% indep_vars_distr
thf(fact_258_indep__vars__distr,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,N: sigma_5515648953823433982at_nat,I: set_re5328672808648366137nnreal,X4: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > product_prod_nat_nat,M: ( real > extend8495563244428889912nnreal ) > sigma_5515648953823433982at_nat] :
      ( ( member8885076297122219836at_nat @ F2 @ ( sigma_3682087626466204304at_nat @ ( freque5010624893710627907mega_1 @ as ) @ N ) )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member8885076297122219836at_nat @ ( X4 @ I2 ) @ ( sigma_3682087626466204304at_nat @ N @ ( M @ I2 ) ) ) )
       => ( ( indepe3593890525012728259at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M
            @ ^ [I3: real > extend8495563244428889912nnreal] : ( comp_P2240441846945064862at_nat @ ( X4 @ I3 ) @ F2 )
            @ I )
         => ( indepe3593890525012728259at_nat @ ( measur6503400250167368504at_nat @ ( freque5010624893710627907mega_1 @ as ) @ N @ F2 ) @ M @ X4 @ I ) ) ) ) ).

% indep_vars_distr
thf(fact_259_one__eq__of__rat__iff,axiom,
    ! [A: rat] :
      ( ( one_one_rat
        = ( field_2639924705303425560at_rat @ A ) )
      = ( one_one_rat = A ) ) ).

% one_eq_of_rat_iff
thf(fact_260_one__eq__of__rat__iff,axiom,
    ! [A: rat] :
      ( ( one_one_real
        = ( field_7254667332652039916t_real @ A ) )
      = ( one_one_rat = A ) ) ).

% one_eq_of_rat_iff
thf(fact_261_of__rat__eq__1__iff,axiom,
    ! [A: rat] :
      ( ( ( field_2639924705303425560at_rat @ A )
        = one_one_rat )
      = ( A = one_one_rat ) ) ).

% of_rat_eq_1_iff
thf(fact_262_of__rat__eq__1__iff,axiom,
    ! [A: rat] :
      ( ( ( field_7254667332652039916t_real @ A )
        = one_one_real )
      = ( A = one_one_rat ) ) ).

% of_rat_eq_1_iff
thf(fact_263_of__rat__1,axiom,
    ( ( field_2639924705303425560at_rat @ one_one_rat )
    = one_one_rat ) ).

% of_rat_1
thf(fact_264_of__rat__1,axiom,
    ( ( field_7254667332652039916t_real @ one_one_rat )
    = one_one_real ) ).

% of_rat_1
thf(fact_265_less__one,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ one_one_nat )
      = ( N2 = zero_zero_nat ) ) ).

% less_one
thf(fact_266_of__rat__less__1__iff,axiom,
    ! [R: rat] :
      ( ( ord_less_rat @ ( field_2639924705303425560at_rat @ R ) @ one_one_rat )
      = ( ord_less_rat @ R @ one_one_rat ) ) ).

% of_rat_less_1_iff
thf(fact_267_of__rat__less__1__iff,axiom,
    ! [R: rat] :
      ( ( ord_less_real @ ( field_7254667332652039916t_real @ R ) @ one_one_real )
      = ( ord_less_rat @ R @ one_one_rat ) ) ).

% of_rat_less_1_iff
thf(fact_268_one__less__of__rat__iff,axiom,
    ! [R: rat] :
      ( ( ord_less_rat @ one_one_rat @ ( field_2639924705303425560at_rat @ R ) )
      = ( ord_less_rat @ one_one_rat @ R ) ) ).

% one_less_of_rat_iff
thf(fact_269_one__less__of__rat__iff,axiom,
    ! [R: rat] :
      ( ( ord_less_real @ one_one_real @ ( field_7254667332652039916t_real @ R ) )
      = ( ord_less_rat @ one_one_rat @ R ) ) ).

% one_less_of_rat_iff
thf(fact_270_one__le__of__rat__iff,axiom,
    ! [R: rat] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( field_2639924705303425560at_rat @ R ) )
      = ( ord_less_eq_rat @ one_one_rat @ R ) ) ).

% one_le_of_rat_iff
thf(fact_271_one__le__of__rat__iff,axiom,
    ! [R: rat] :
      ( ( ord_less_eq_real @ one_one_real @ ( field_7254667332652039916t_real @ R ) )
      = ( ord_less_eq_rat @ one_one_rat @ R ) ) ).

% one_le_of_rat_iff
thf(fact_272_of__rat__le__1__iff,axiom,
    ! [R: rat] :
      ( ( ord_less_eq_rat @ ( field_2639924705303425560at_rat @ R ) @ one_one_rat )
      = ( ord_less_eq_rat @ R @ one_one_rat ) ) ).

% of_rat_le_1_iff
thf(fact_273_of__rat__le__1__iff,axiom,
    ! [R: rat] :
      ( ( ord_less_eq_real @ ( field_7254667332652039916t_real @ R ) @ one_one_real )
      = ( ord_less_eq_rat @ R @ one_one_rat ) ) ).

% of_rat_le_1_iff
thf(fact_274_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y5: nat] :
            ( ( P @ Y5 )
           => ( ord_less_eq_nat @ Y5 @ B ) )
       => ? [X3: nat] :
            ( ( P @ X3 )
            & ! [Y6: nat] :
                ( ( P @ Y6 )
               => ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_275_nat__le__linear,axiom,
    ! [M3: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M3 @ N2 )
      | ( ord_less_eq_nat @ N2 @ M3 ) ) ).

% nat_le_linear
thf(fact_276_le__antisym,axiom,
    ! [M3: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M3 @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ M3 )
       => ( M3 = N2 ) ) ) ).

% le_antisym
thf(fact_277_eq__imp__le,axiom,
    ! [M3: nat,N2: nat] :
      ( ( M3 = N2 )
     => ( ord_less_eq_nat @ M3 @ N2 ) ) ).

% eq_imp_le
thf(fact_278_le__trans,axiom,
    ! [I4: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I4 @ J2 )
     => ( ( ord_less_eq_nat @ J2 @ K )
       => ( ord_less_eq_nat @ I4 @ K ) ) ) ).

% le_trans
thf(fact_279_le__refl,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).

% le_refl
thf(fact_280_one__reorient,axiom,
    ! [X5: nat] :
      ( ( one_one_nat = X5 )
      = ( X5 = one_one_nat ) ) ).

% one_reorient
thf(fact_281_one__reorient,axiom,
    ! [X5: rat] :
      ( ( one_one_rat = X5 )
      = ( X5 = one_one_rat ) ) ).

% one_reorient
thf(fact_282_one__reorient,axiom,
    ! [X5: real] :
      ( ( one_one_real = X5 )
      = ( X5 = one_one_real ) ) ).

% one_reorient
thf(fact_283_one__reorient,axiom,
    ! [X5: extend8495563244428889912nnreal] :
      ( ( one_on2969667320475766781nnreal = X5 )
      = ( X5 = one_on2969667320475766781nnreal ) ) ).

% one_reorient
thf(fact_284_fk__update__2_Ocases,axiom,
    ! [X5: produc8642769642335960151at_nat] :
      ~ ! [A3: nat,M6: nat,X3: nat,L: nat] :
          ( X5
         != ( produc6385450045882626063at_nat @ A3 @ ( produc487386426758144856at_nat @ M6 @ ( product_Pair_nat_nat @ X3 @ L ) ) ) ) ).

% fk_update_2.cases
thf(fact_285_zero__reorient,axiom,
    ! [X5: nat] :
      ( ( zero_zero_nat = X5 )
      = ( X5 = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_286_zero__reorient,axiom,
    ! [X5: rat] :
      ( ( zero_zero_rat = X5 )
      = ( X5 = zero_zero_rat ) ) ).

% zero_reorient
thf(fact_287_zero__reorient,axiom,
    ! [X5: real] :
      ( ( zero_zero_real = X5 )
      = ( X5 = zero_zero_real ) ) ).

% zero_reorient
thf(fact_288_zero__reorient,axiom,
    ! [X5: extend8495563244428889912nnreal] :
      ( ( zero_z7100319975126383169nnreal = X5 )
      = ( X5 = zero_z7100319975126383169nnreal ) ) ).

% zero_reorient
thf(fact_289_zero__le,axiom,
    ! [X5: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ X5 ) ).

% zero_le
thf(fact_290_zero__le,axiom,
    ! [X5: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X5 ) ).

% zero_le
thf(fact_291_zero__order_I4_J,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
      = ( N2 != zero_zero_nat ) ) ).

% zero_order(4)
thf(fact_292_zero__order_I4_J,axiom,
    ! [N2: extend8495563244428889912nnreal] :
      ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ N2 )
      = ( N2 != zero_z7100319975126383169nnreal ) ) ).

% zero_order(4)
thf(fact_293_zero__order_I3_J,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% zero_order(3)
thf(fact_294_zero__order_I3_J,axiom,
    ! [N2: extend8495563244428889912nnreal] :
      ~ ( ord_le7381754540660121996nnreal @ N2 @ zero_z7100319975126383169nnreal ) ).

% zero_order(3)
thf(fact_295_gr__zeroI,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr_zeroI
thf(fact_296_gr__zeroI,axiom,
    ! [N2: extend8495563244428889912nnreal] :
      ( ( N2 != zero_z7100319975126383169nnreal )
     => ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ N2 ) ) ).

% gr_zeroI
thf(fact_297_gr__implies__not__zero,axiom,
    ! [M3: nat,N2: nat] :
      ( ( ord_less_nat @ M3 @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_298_gr__implies__not__zero,axiom,
    ! [M3: extend8495563244428889912nnreal,N2: extend8495563244428889912nnreal] :
      ( ( ord_le7381754540660121996nnreal @ M3 @ N2 )
     => ( N2 != zero_z7100319975126383169nnreal ) ) ).

% gr_implies_not_zero
thf(fact_299_Pair__mono,axiom,
    ! [X5: rat,X6: rat,Y4: rat,Y7: rat] :
      ( ( ord_less_eq_rat @ X5 @ X6 )
     => ( ( ord_less_eq_rat @ Y4 @ Y7 )
       => ( ord_le4635361258860028081at_rat @ ( product_Pair_rat_rat @ X5 @ Y4 ) @ ( product_Pair_rat_rat @ X6 @ Y7 ) ) ) ) ).

% Pair_mono
thf(fact_300_Pair__mono,axiom,
    ! [X5: rat,X6: rat,Y4: nat,Y7: nat] :
      ( ( ord_less_eq_rat @ X5 @ X6 )
     => ( ( ord_less_eq_nat @ Y4 @ Y7 )
       => ( ord_le8143589817253424809at_nat @ ( product_Pair_rat_nat @ X5 @ Y4 ) @ ( product_Pair_rat_nat @ X6 @ Y7 ) ) ) ) ).

% Pair_mono
thf(fact_301_Pair__mono,axiom,
    ! [X5: nat,X6: nat,Y4: rat,Y7: rat] :
      ( ( ord_less_eq_nat @ X5 @ X6 )
     => ( ( ord_less_eq_rat @ Y4 @ Y7 )
       => ( ord_le4951915902794893993at_rat @ ( product_Pair_nat_rat @ X5 @ Y4 ) @ ( product_Pair_nat_rat @ X6 @ Y7 ) ) ) ) ).

% Pair_mono
thf(fact_302_Pair__mono,axiom,
    ! [X5: nat,X6: nat,Y4: nat,Y7: nat] :
      ( ( ord_less_eq_nat @ X5 @ X6 )
     => ( ( ord_less_eq_nat @ Y4 @ Y7 )
       => ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ ( product_Pair_nat_nat @ X6 @ Y7 ) ) ) ) ).

% Pair_mono
thf(fact_303_Pair__mono,axiom,
    ! [X5: set_Pr1002607673312053630t_real,X6: set_Pr1002607673312053630t_real,Y4: rat,Y7: rat] :
      ( ( ord_le1994730768851292446t_real @ X5 @ X6 )
     => ( ( ord_less_eq_rat @ Y4 @ Y7 )
       => ( ord_le2259096304072862181al_rat @ ( produc370051823116057599al_rat @ X5 @ Y4 ) @ ( produc370051823116057599al_rat @ X6 @ Y7 ) ) ) ) ).

% Pair_mono
thf(fact_304_Pair__mono,axiom,
    ! [X5: set_Pr1002607673312053630t_real,X6: set_Pr1002607673312053630t_real,Y4: nat,Y7: nat] :
      ( ( ord_le1994730768851292446t_real @ X5 @ X6 )
     => ( ( ord_less_eq_nat @ Y4 @ Y7 )
       => ( ord_le5767324862466258909al_nat @ ( produc1005181883202553335al_nat @ X5 @ Y4 ) @ ( produc1005181883202553335al_nat @ X6 @ Y7 ) ) ) ) ).

% Pair_mono
thf(fact_305_Pair__mono,axiom,
    ! [X5: rat,X6: rat,Y4: set_Pr1002607673312053630t_real,Y7: set_Pr1002607673312053630t_real] :
      ( ( ord_less_eq_rat @ X5 @ X6 )
     => ( ( ord_le1994730768851292446t_real @ Y4 @ Y7 )
       => ( ord_le6637847787101155707t_real @ ( produc8291022738106172877t_real @ X5 @ Y4 ) @ ( produc8291022738106172877t_real @ X6 @ Y7 ) ) ) ) ).

% Pair_mono
thf(fact_306_Pair__mono,axiom,
    ! [X5: nat,X6: nat,Y4: set_Pr1002607673312053630t_real,Y7: set_Pr1002607673312053630t_real] :
      ( ( ord_less_eq_nat @ X5 @ X6 )
     => ( ( ord_le1994730768851292446t_real @ Y4 @ Y7 )
       => ( ord_le1943484209260552579t_real @ ( produc3596659160265569749t_real @ X5 @ Y4 ) @ ( produc3596659160265569749t_real @ X6 @ Y7 ) ) ) ) ).

% Pair_mono
thf(fact_307_Pair__mono,axiom,
    ! [X5: set_Pr2458342521480944603at_nat,X6: set_Pr2458342521480944603at_nat,Y4: rat,Y7: rat] :
      ( ( ord_le3857079194666040379at_nat @ X5 @ X6 )
     => ( ( ord_less_eq_rat @ Y4 @ Y7 )
       => ( ord_le7912795254584737716at_rat @ ( produc5584217423264826444at_rat @ X5 @ Y4 ) @ ( produc5584217423264826444at_rat @ X6 @ Y7 ) ) ) ) ).

% Pair_mono
thf(fact_308_Pair__mono,axiom,
    ! [X5: set_Pr2458342521480944603at_nat,X6: set_Pr2458342521480944603at_nat,Y4: nat,Y7: nat] :
      ( ( ord_le3857079194666040379at_nat @ X5 @ X6 )
     => ( ( ord_less_eq_nat @ Y4 @ Y7 )
       => ( ord_le2197651776123358636at_nat @ ( produc6219347483351322180at_nat @ X5 @ Y4 ) @ ( produc6219347483351322180at_nat @ X6 @ Y7 ) ) ) ) ).

% Pair_mono
thf(fact_309_finite__measure_Oaxioms_I1_J,axiom,
    ! [M2: sigma_9047027012034273406at_nat] :
      ( ( measur5156515106886843841at_nat @ M2 )
     => ( measur1801591924826899012at_nat @ M2 ) ) ).

% finite_measure.axioms(1)
thf(fact_310_finite__measure_Oaxioms_I1_J,axiom,
    ! [M2: sigma_5515648953823433982at_nat] :
      ( ( measur7911732083032432613at_nat @ M2 )
     => ( measur1424910518806321826at_nat @ M2 ) ) ).

% finite_measure.axioms(1)
thf(fact_311_distr__distr,axiom,
    ! [G2: product_prod_nat_nat > product_prod_nat_nat,N: sigma_5515648953823433982at_nat,L2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > product_prod_nat_nat,M2: sigma_5515648953823433982at_nat] :
      ( ( member8885076297122219836at_nat @ G2 @ ( sigma_3682087626466204304at_nat @ N @ L2 ) )
     => ( ( member8885076297122219836at_nat @ F2 @ ( sigma_3682087626466204304at_nat @ M2 @ N ) )
       => ( ( measur6503400250167368504at_nat @ ( measur6503400250167368504at_nat @ M2 @ N @ F2 ) @ L2 @ G2 )
          = ( measur6503400250167368504at_nat @ M2 @ L2 @ ( comp_P2240441846945064862at_nat @ G2 @ F2 ) ) ) ) ) ).

% distr_distr
thf(fact_312_distr__distr,axiom,
    ! [G2: product_prod_nat_nat > real,N: sigma_5515648953823433982at_nat,L2: sigma_measure_real,F2: product_prod_nat_nat > product_prod_nat_nat,M2: sigma_5515648953823433982at_nat] :
      ( ( member4564283293661824327t_real @ G2 @ ( sigma_1188828689629184861t_real @ N @ L2 ) )
     => ( ( member8885076297122219836at_nat @ F2 @ ( sigma_3682087626466204304at_nat @ M2 @ N ) )
       => ( ( measur4284412874431678645t_real @ ( measur6503400250167368504at_nat @ M2 @ N @ F2 ) @ L2 @ G2 )
          = ( measur4284412874431678645t_real @ M2 @ L2 @ ( comp_P6083776177191121141at_nat @ G2 @ F2 ) ) ) ) ) ).

% distr_distr
thf(fact_313_distr__distr,axiom,
    ! [G2: real > extend8495563244428889912nnreal,N: sigma_measure_real,L2: sigma_7234349610311085201nnreal,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,M2: sigma_9047027012034273406at_nat] :
      ( ( member2919562650594848410nnreal @ G2 @ ( sigma_9017504469962657078nnreal @ N @ L2 ) )
     => ( ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ M2 @ N ) )
       => ( ( measur8829990298702910942nnreal @ ( measur6433597746558672971t_real @ M2 @ N @ F2 ) @ L2 @ G2 )
          = ( measur4206041593328951383nnreal @ M2 @ L2 @ ( comp_r2164139168250337006at_nat @ G2 @ F2 ) ) ) ) ) ).

% distr_distr
thf(fact_314_distr__distr,axiom,
    ! [G2: real > extend8495563244428889912nnreal,N: sigma_measure_real,L2: sigma_7234349610311085201nnreal,F2: product_prod_nat_nat > real,M2: sigma_5515648953823433982at_nat] :
      ( ( member2919562650594848410nnreal @ G2 @ ( sigma_9017504469962657078nnreal @ N @ L2 ) )
     => ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ M2 @ N ) )
       => ( ( measur8829990298702910942nnreal @ ( measur4284412874431678645t_real @ M2 @ N @ F2 ) @ L2 @ G2 )
          = ( measur803599879579684289nnreal @ M2 @ L2 @ ( comp_r868043356317784952at_nat @ G2 @ F2 ) ) ) ) ) ).

% distr_distr
thf(fact_315_finite__measure_Ofinite__measure__distr,axiom,
    ! [M2: sigma_measure_real,F2: real > extend8495563244428889912nnreal,M: sigma_7234349610311085201nnreal] :
      ( ( measur3606880022600206024e_real @ M2 )
     => ( ( member2919562650594848410nnreal @ F2 @ ( sigma_9017504469962657078nnreal @ M2 @ M ) )
       => ( measur8478876643349974356nnreal @ ( measur8829990298702910942nnreal @ M2 @ M @ F2 ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_316_finite__measure_Ofinite__measure__distr,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,M: sigma_measure_real] :
      ( ( measur5156515106886843841at_nat @ M2 )
     => ( ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ M2 @ M ) )
       => ( measur3606880022600206024e_real @ ( measur6433597746558672971t_real @ M2 @ M @ F2 ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_317_finite__measure_Ofinite__measure__distr,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat,M: sigma_9047027012034273406at_nat] :
      ( ( measur5156515106886843841at_nat @ M2 )
     => ( ( member2646312932426349184at_nat @ F2 @ ( sigma_1496667973943888668at_nat @ M2 @ M ) )
       => ( measur5156515106886843841at_nat @ ( measur7731843055261497284at_nat @ M2 @ M @ F2 ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_318_finite__measure_Ofinite__measure__distr,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,M: sigma_5515648953823433982at_nat] :
      ( ( measur5156515106886843841at_nat @ M2 )
     => ( ( member7151659165045815932at_nat @ F2 @ ( sigma_3543877168342358986at_nat @ M2 @ M ) )
       => ( measur7911732083032432613at_nat @ ( measur4728020197420827938at_nat @ M2 @ M @ F2 ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_319_finite__measure_Ofinite__measure__distr,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real,M: sigma_measure_real] :
      ( ( measur7911732083032432613at_nat @ M2 )
     => ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ M2 @ M ) )
       => ( measur3606880022600206024e_real @ ( measur4284412874431678645t_real @ M2 @ M @ F2 ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_320_finite__measure_Ofinite__measure__distr,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,M: sigma_9047027012034273406at_nat] :
      ( ( measur7911732083032432613at_nat @ M2 )
     => ( ( member8598435083969174976at_nat @ F2 @ ( sigma_7146669294626004438at_nat @ M2 @ M ) )
       => ( measur5156515106886843841at_nat @ ( measur8330812323704473390at_nat @ M2 @ M @ F2 ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_321_finite__measure_Ofinite__measure__distr,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > product_prod_nat_nat,M: sigma_5515648953823433982at_nat] :
      ( ( measur7911732083032432613at_nat @ M2 )
     => ( ( member8885076297122219836at_nat @ F2 @ ( sigma_3682087626466204304at_nat @ M2 @ M ) )
       => ( measur7911732083032432613at_nat @ ( measur6503400250167368504at_nat @ M2 @ M @ F2 ) ) ) ) ).

% finite_measure.finite_measure_distr
thf(fact_322_prob__space_Ok__wise__indep__vars__compose,axiom,
    ! [M2: sigma_5515648953823433982at_nat,K: nat,M: rat > sigma_measure_real,X: rat > product_prod_nat_nat > real,I: set_rat,Y: rat > real > extend8495563244428889912nnreal,N: rat > sigma_7234349610311085201nnreal] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( prob_k9046839836125593554t_real @ M2 @ K @ M @ X @ I )
       => ( ! [I2: rat] :
              ( ( member_rat @ I2 @ I )
             => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( prob_k9060532812646980446nnreal @ M2 @ K @ N
            @ ^ [I3: rat,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.k_wise_indep_vars_compose
thf(fact_323_prob__space_Ok__wise__indep__vars__compose,axiom,
    ! [M2: sigma_5515648953823433982at_nat,K: nat,M: $o > sigma_measure_real,X: $o > product_prod_nat_nat > real,I: set_o,Y: $o > real > extend8495563244428889912nnreal,N: $o > sigma_7234349610311085201nnreal] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( prob_k5626656578853715448o_real @ M2 @ K @ M @ X @ I )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I )
             => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( prob_k147916592727809156nnreal @ M2 @ K @ N
            @ ^ [I3: $o,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.k_wise_indep_vars_compose
thf(fact_324_prob__space_Ok__wise__indep__vars__compose,axiom,
    ! [M2: sigma_5515648953823433982at_nat,K: nat,M: rat > sigma_5515648953823433982at_nat,X: rat > product_prod_nat_nat > product_prod_nat_nat,I: set_rat,Y: rat > product_prod_nat_nat > real,N: rat > sigma_measure_real] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( prob_k1749327348295208283at_nat @ M2 @ K @ M @ X @ I )
       => ( ! [I2: rat] :
              ( ( member_rat @ I2 @ I )
             => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( prob_k9046839836125593554t_real @ M2 @ K @ N
            @ ^ [I3: rat,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.k_wise_indep_vars_compose
thf(fact_325_prob__space_Ok__wise__indep__vars__compose,axiom,
    ! [M2: sigma_5515648953823433982at_nat,K: nat,M: $o > sigma_5515648953823433982at_nat,X: $o > product_prod_nat_nat > product_prod_nat_nat,I: set_o,Y: $o > product_prod_nat_nat > real,N: $o > sigma_measure_real] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( prob_k129107890564256949at_nat @ M2 @ K @ M @ X @ I )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I )
             => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( prob_k5626656578853715448o_real @ M2 @ K @ N
            @ ^ [I3: $o,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.k_wise_indep_vars_compose
thf(fact_326_prob__space_Ok__wise__indep__vars__compose,axiom,
    ! [M2: sigma_5515648953823433982at_nat,K: nat,M: ( real > extend8495563244428889912nnreal ) > sigma_measure_real,X: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > real,I: set_re5328672808648366137nnreal,Y: ( real > extend8495563244428889912nnreal ) > real > extend8495563244428889912nnreal,N: ( real > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( prob_k5108616785771355773l_real @ M2 @ K @ M @ X @ I )
       => ( ! [I2: real > extend8495563244428889912nnreal] :
              ( ( member2919562650594848410nnreal @ I2 @ I )
             => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( prob_k1953979174828017545nnreal @ M2 @ K @ N
            @ ^ [I3: real > extend8495563244428889912nnreal,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.k_wise_indep_vars_compose
thf(fact_327_prob__space_Ok__wise__indep__vars__compose,axiom,
    ! [M2: sigma_9047027012034273406at_nat,K: nat,M: rat > sigma_measure_real,X: rat > ( product_prod_nat_nat > product_prod_nat_nat ) > real,I: set_rat,Y: rat > real > extend8495563244428889912nnreal,N: rat > sigma_7234349610311085201nnreal] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( prob_k4779580098842486304t_real @ M2 @ K @ M @ X @ I )
       => ( ! [I2: rat] :
              ( ( member_rat @ I2 @ I )
             => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( prob_k3726877565640988844nnreal @ M2 @ K @ N
            @ ^ [I3: rat,X2: product_prod_nat_nat > product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.k_wise_indep_vars_compose
thf(fact_328_prob__space_Ok__wise__indep__vars__compose,axiom,
    ! [M2: sigma_9047027012034273406at_nat,K: nat,M: $o > sigma_measure_real,X: $o > ( product_prod_nat_nat > product_prod_nat_nat ) > real,I: set_o,Y: $o > real > extend8495563244428889912nnreal,N: $o > sigma_7234349610311085201nnreal] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( prob_k6935007176262895658o_real @ M2 @ K @ M @ X @ I )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I )
             => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( prob_k1493464604931706806nnreal @ M2 @ K @ N
            @ ^ [I3: $o,X2: product_prod_nat_nat > product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.k_wise_indep_vars_compose
thf(fact_329_prob__space_Ok__wise__indep__vars__compose,axiom,
    ! [M2: sigma_5515648953823433982at_nat,K: nat,M: rat > sigma_5515648953823433982at_nat,X: rat > product_prod_nat_nat > product_prod_nat_nat,I: set_rat,Y: rat > product_prod_nat_nat > product_prod_nat_nat,N: rat > sigma_5515648953823433982at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( prob_k1749327348295208283at_nat @ M2 @ K @ M @ X @ I )
       => ( ! [I2: rat] :
              ( ( member_rat @ I2 @ I )
             => ( member8885076297122219836at_nat @ ( Y @ I2 ) @ ( sigma_3682087626466204304at_nat @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( prob_k1749327348295208283at_nat @ M2 @ K @ N
            @ ^ [I3: rat,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.k_wise_indep_vars_compose
thf(fact_330_prob__space_Ok__wise__indep__vars__compose,axiom,
    ! [M2: sigma_5515648953823433982at_nat,K: nat,M: $o > sigma_5515648953823433982at_nat,X: $o > product_prod_nat_nat > product_prod_nat_nat,I: set_o,Y: $o > product_prod_nat_nat > product_prod_nat_nat,N: $o > sigma_5515648953823433982at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( prob_k129107890564256949at_nat @ M2 @ K @ M @ X @ I )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I )
             => ( member8885076297122219836at_nat @ ( Y @ I2 ) @ ( sigma_3682087626466204304at_nat @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( prob_k129107890564256949at_nat @ M2 @ K @ N
            @ ^ [I3: $o,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.k_wise_indep_vars_compose
thf(fact_331_prob__space_Ok__wise__indep__vars__compose,axiom,
    ! [M2: sigma_5515648953823433982at_nat,K: nat,M: ( real > extend8495563244428889912nnreal ) > sigma_5515648953823433982at_nat,X: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > product_prod_nat_nat,I: set_re5328672808648366137nnreal,Y: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > real,N: ( real > extend8495563244428889912nnreal ) > sigma_measure_real] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( prob_k8442391182763575152at_nat @ M2 @ K @ M @ X @ I )
       => ( ! [I2: real > extend8495563244428889912nnreal] :
              ( ( member2919562650594848410nnreal @ I2 @ I )
             => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( prob_k5108616785771355773l_real @ M2 @ K @ N
            @ ^ [I3: real > extend8495563244428889912nnreal,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.k_wise_indep_vars_compose
thf(fact_332__092_060Omega_062_092_060_094sub_0622_Omutual__information__def,axiom,
    ! [B: real,S2: sigma_measure_nat,T: sigma_measure_nat,X: ( product_prod_nat_nat > product_prod_nat_nat ) > nat,Y: ( product_prod_nat_nat > product_prod_nat_nat ) > nat] :
      ( ( prob_m3062007823079647199at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ B @ S2 @ T @ X @ Y )
      = ( kL_div6543510193410477195at_nat @ B @ ( binary6458575275919672938at_nat @ ( measur6421871825836039407at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ S2 @ X ) @ ( measur6421871825836039407at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ T @ Y ) )
        @ ( measur4728020197420827938at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ ( binary6458575275919672938at_nat @ S2 @ T )
          @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( product_Pair_nat_nat @ ( X @ X2 ) @ ( Y @ X2 ) ) ) ) ) ).

% \<Omega>\<^sub>2.mutual_information_def
thf(fact_333_mutual__information__def,axiom,
    ! [B: real,S2: sigma_measure_nat,T: sigma_measure_nat,X: product_prod_nat_nat > nat,Y: product_prod_nat_nat > nat] :
      ( ( prob_m7791635597327167627at_nat @ ( freque5010624893710627907mega_1 @ as ) @ B @ S2 @ T @ X @ Y )
      = ( kL_div6543510193410477195at_nat @ B @ ( binary6458575275919672938at_nat @ ( measur843052931239516761at_nat @ ( freque5010624893710627907mega_1 @ as ) @ S2 @ X ) @ ( measur843052931239516761at_nat @ ( freque5010624893710627907mega_1 @ as ) @ T @ Y ) )
        @ ( measur6503400250167368504at_nat @ ( freque5010624893710627907mega_1 @ as ) @ ( binary6458575275919672938at_nat @ S2 @ T )
          @ ^ [X2: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( X @ X2 ) @ ( Y @ X2 ) ) ) ) ) ).

% mutual_information_def
thf(fact_334_prob__space_Oindep__var__distribution__eq,axiom,
    ! [M2: sigma_measure_real,S2: sigma_7234349610311085201nnreal,X: real > extend8495563244428889912nnreal,T: sigma_7234349610311085201nnreal,Y: real > extend8495563244428889912nnreal] :
      ( ( probab535871623910865577e_real @ M2 )
     => ( ( indepe6767359503340752434nnreal @ M2 @ S2 @ X @ T @ Y )
        = ( ( member2919562650594848410nnreal @ X @ ( sigma_9017504469962657078nnreal @ M2 @ S2 ) )
          & ( member2919562650594848410nnreal @ Y @ ( sigma_9017504469962657078nnreal @ M2 @ T ) )
          & ( ( binary3098606844978005306nnreal @ ( measur8829990298702910942nnreal @ M2 @ S2 @ X ) @ ( measur8829990298702910942nnreal @ M2 @ T @ Y ) )
            = ( measur4012415197360569771nnreal @ M2 @ ( binary3098606844978005306nnreal @ S2 @ T )
              @ ^ [X2: real] : ( produc344325839068023049nnreal @ ( X @ X2 ) @ ( Y @ X2 ) ) ) ) ) ) ) ).

% prob_space.indep_var_distribution_eq
thf(fact_335_prob__space_Oindep__var__distribution__eq,axiom,
    ! [M2: sigma_9047027012034273406at_nat,S2: sigma_measure_nat,X: ( product_prod_nat_nat > product_prod_nat_nat ) > nat,T: sigma_measure_nat,Y: ( product_prod_nat_nat > product_prod_nat_nat ) > nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe8633027689432630083at_nat @ M2 @ S2 @ X @ T @ Y )
        = ( ( member3219910559795078443at_nat @ X @ ( sigma_4532147426969876551at_nat @ M2 @ S2 ) )
          & ( member3219910559795078443at_nat @ Y @ ( sigma_4532147426969876551at_nat @ M2 @ T ) )
          & ( ( binary6458575275919672938at_nat @ ( measur6421871825836039407at_nat @ M2 @ S2 @ X ) @ ( measur6421871825836039407at_nat @ M2 @ T @ Y ) )
            = ( measur4728020197420827938at_nat @ M2 @ ( binary6458575275919672938at_nat @ S2 @ T )
              @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( product_Pair_nat_nat @ ( X @ X2 ) @ ( Y @ X2 ) ) ) ) ) ) ) ).

% prob_space.indep_var_distribution_eq
thf(fact_336_prob__space_Oindep__var__distribution__eq,axiom,
    ! [M2: sigma_5515648953823433982at_nat,S2: sigma_5515648953823433982at_nat,X: product_prod_nat_nat > product_prod_nat_nat,T: sigma_5515648953823433982at_nat,Y: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe9184017870484869004at_nat @ M2 @ S2 @ X @ T @ Y )
        = ( ( member8885076297122219836at_nat @ X @ ( sigma_3682087626466204304at_nat @ M2 @ S2 ) )
          & ( member8885076297122219836at_nat @ Y @ ( sigma_3682087626466204304at_nat @ M2 @ T ) )
          & ( ( binary8350690942515918216at_nat @ ( measur6503400250167368504at_nat @ M2 @ S2 @ X ) @ ( measur6503400250167368504at_nat @ M2 @ T @ Y ) )
            = ( measur1639216089608557598at_nat @ M2 @ ( binary8350690942515918216at_nat @ S2 @ T )
              @ ^ [X2: product_prod_nat_nat] : ( produc6161850002892822231at_nat @ ( X @ X2 ) @ ( Y @ X2 ) ) ) ) ) ) ) ).

% prob_space.indep_var_distribution_eq
thf(fact_337_prob__space_Oindep__var__distribution__eq,axiom,
    ! [M2: sigma_5515648953823433982at_nat,S2: sigma_measure_nat,X: product_prod_nat_nat > nat,T: sigma_measure_nat,Y: product_prod_nat_nat > nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe8085942211766624005at_nat @ M2 @ S2 @ X @ T @ Y )
        = ( ( member7868840539957676139at_nat @ X @ ( sigma_8719606837941807361at_nat @ M2 @ S2 ) )
          & ( member7868840539957676139at_nat @ Y @ ( sigma_8719606837941807361at_nat @ M2 @ T ) )
          & ( ( binary6458575275919672938at_nat @ ( measur843052931239516761at_nat @ M2 @ S2 @ X ) @ ( measur843052931239516761at_nat @ M2 @ T @ Y ) )
            = ( measur6503400250167368504at_nat @ M2 @ ( binary6458575275919672938at_nat @ S2 @ T )
              @ ^ [X2: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( X @ X2 ) @ ( Y @ X2 ) ) ) ) ) ) ) ).

% prob_space.indep_var_distribution_eq
thf(fact_338_prob__space_Oindep__var__distribution__eq,axiom,
    ! [M2: sigma_9047027012034273406at_nat,S2: sigma_measure_real,X: ( product_prod_nat_nat > product_prod_nat_nat ) > real,T: sigma_measure_real,Y: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe1938972608428940447t_real @ M2 @ S2 @ X @ T @ Y )
        = ( ( member8159409068225774087t_real @ X @ ( sigma_5064276549609164707t_real @ M2 @ S2 ) )
          & ( member8159409068225774087t_real @ Y @ ( sigma_5064276549609164707t_real @ M2 @ T ) )
          & ( ( binary6478037234023840930l_real @ ( measur6433597746558672971t_real @ M2 @ S2 @ X ) @ ( measur6433597746558672971t_real @ M2 @ T @ Y ) )
            = ( measur3108995680308112986l_real @ M2 @ ( binary6478037234023840930l_real @ S2 @ T )
              @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( produc4511245868158468465l_real @ ( X @ X2 ) @ ( Y @ X2 ) ) ) ) ) ) ) ).

% prob_space.indep_var_distribution_eq
thf(fact_339_prob__space_Oindep__var__distribution__eq,axiom,
    ! [M2: sigma_5515648953823433982at_nat,S2: sigma_measure_real,X: product_prod_nat_nat > real,T: sigma_measure_real,Y: product_prod_nat_nat > real] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe1845147739501508449t_real @ M2 @ S2 @ X @ T @ Y )
        = ( ( member4564283293661824327t_real @ X @ ( sigma_1188828689629184861t_real @ M2 @ S2 ) )
          & ( member4564283293661824327t_real @ Y @ ( sigma_1188828689629184861t_real @ M2 @ T ) )
          & ( ( binary6478037234023840930l_real @ ( measur4284412874431678645t_real @ M2 @ S2 @ X ) @ ( measur4284412874431678645t_real @ M2 @ T @ Y ) )
            = ( measur6140588988970102384l_real @ M2 @ ( binary6478037234023840930l_real @ S2 @ T )
              @ ^ [X2: product_prod_nat_nat] : ( produc4511245868158468465l_real @ ( X @ X2 ) @ ( Y @ X2 ) ) ) ) ) ) ) ).

% prob_space.indep_var_distribution_eq
thf(fact_340__092_060Omega_062_092_060_094sub_0622_OLp__subset__Lq_I1_J,axiom,
    ! [P2: extend8495563244428889912nnreal,Q2: extend8495563244428889912nnreal,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( ord_le3935885782089961368nnreal @ P2 @ Q2 )
     => ( ord_le3935885782089961368nnreal @ ( functi3128325879070182998t_real @ ( lp_spa812643495979991928at_nat @ P2 @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) @ F2 ) @ ( functi3128325879070182998t_real @ ( lp_spa812643495979991928at_nat @ Q2 @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) @ F2 ) ) ) ).

% \<Omega>\<^sub>2.Lp_subset_Lq(1)
thf(fact_341__092_060Omega_062_092_060_094sub_0622_OLp__subset__Lq_I3_J,axiom,
    ! [P2: extend8495563244428889912nnreal,Q2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ P2 @ Q2 )
     => ( ord_le8622349663015078278t_real @ ( functi8590574888837585149t_real @ ( lp_spa812643495979991928at_nat @ Q2 @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) ) @ ( functi8590574888837585149t_real @ ( lp_spa812643495979991928at_nat @ P2 @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) ) ) ) ).

% \<Omega>\<^sub>2.Lp_subset_Lq(3)
thf(fact_342_indep__var__distribution__eq,axiom,
    ! [S2: sigma_5515648953823433982at_nat,X: product_prod_nat_nat > product_prod_nat_nat,T: sigma_5515648953823433982at_nat,Y: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( indepe9184017870484869004at_nat @ ( freque5010624893710627907mega_1 @ as ) @ S2 @ X @ T @ Y )
      = ( ( member8885076297122219836at_nat @ X @ ( sigma_3682087626466204304at_nat @ ( freque5010624893710627907mega_1 @ as ) @ S2 ) )
        & ( member8885076297122219836at_nat @ Y @ ( sigma_3682087626466204304at_nat @ ( freque5010624893710627907mega_1 @ as ) @ T ) )
        & ( ( binary8350690942515918216at_nat @ ( measur6503400250167368504at_nat @ ( freque5010624893710627907mega_1 @ as ) @ S2 @ X ) @ ( measur6503400250167368504at_nat @ ( freque5010624893710627907mega_1 @ as ) @ T @ Y ) )
          = ( measur1639216089608557598at_nat @ ( freque5010624893710627907mega_1 @ as ) @ ( binary8350690942515918216at_nat @ S2 @ T )
            @ ^ [X2: product_prod_nat_nat] : ( produc6161850002892822231at_nat @ ( X @ X2 ) @ ( Y @ X2 ) ) ) ) ) ) ).

% indep_var_distribution_eq
thf(fact_343_indep__var__distribution__eq,axiom,
    ! [S2: sigma_measure_nat,X: product_prod_nat_nat > nat,T: sigma_measure_nat,Y: product_prod_nat_nat > nat] :
      ( ( indepe8085942211766624005at_nat @ ( freque5010624893710627907mega_1 @ as ) @ S2 @ X @ T @ Y )
      = ( ( member7868840539957676139at_nat @ X @ ( sigma_8719606837941807361at_nat @ ( freque5010624893710627907mega_1 @ as ) @ S2 ) )
        & ( member7868840539957676139at_nat @ Y @ ( sigma_8719606837941807361at_nat @ ( freque5010624893710627907mega_1 @ as ) @ T ) )
        & ( ( binary6458575275919672938at_nat @ ( measur843052931239516761at_nat @ ( freque5010624893710627907mega_1 @ as ) @ S2 @ X ) @ ( measur843052931239516761at_nat @ ( freque5010624893710627907mega_1 @ as ) @ T @ Y ) )
          = ( measur6503400250167368504at_nat @ ( freque5010624893710627907mega_1 @ as ) @ ( binary6458575275919672938at_nat @ S2 @ T )
            @ ^ [X2: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( X @ X2 ) @ ( Y @ X2 ) ) ) ) ) ) ).

% indep_var_distribution_eq
thf(fact_344_indep__var__distribution__eq,axiom,
    ! [S2: sigma_measure_real,X: product_prod_nat_nat > real,T: sigma_measure_real,Y: product_prod_nat_nat > real] :
      ( ( indepe1845147739501508449t_real @ ( freque5010624893710627907mega_1 @ as ) @ S2 @ X @ T @ Y )
      = ( ( member4564283293661824327t_real @ X @ ( sigma_1188828689629184861t_real @ ( freque5010624893710627907mega_1 @ as ) @ S2 ) )
        & ( member4564283293661824327t_real @ Y @ ( sigma_1188828689629184861t_real @ ( freque5010624893710627907mega_1 @ as ) @ T ) )
        & ( ( binary6478037234023840930l_real @ ( measur4284412874431678645t_real @ ( freque5010624893710627907mega_1 @ as ) @ S2 @ X ) @ ( measur4284412874431678645t_real @ ( freque5010624893710627907mega_1 @ as ) @ T @ Y ) )
          = ( measur6140588988970102384l_real @ ( freque5010624893710627907mega_1 @ as ) @ ( binary6478037234023840930l_real @ S2 @ T )
            @ ^ [X2: product_prod_nat_nat] : ( produc4511245868158468465l_real @ ( X @ X2 ) @ ( Y @ X2 ) ) ) ) ) ) ).

% indep_var_distribution_eq
thf(fact_345_prob__space_OLp__subset__Lq_I2_J,axiom,
    ! [M2: sigma_9047027012034273406at_nat,P2: extend8495563244428889912nnreal,Q2: extend8495563244428889912nnreal] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( ord_le3935885782089961368nnreal @ P2 @ Q2 )
       => ( ord_le1044648919696853383t_real @ ( lp_spa812643495979991928at_nat @ Q2 @ M2 ) @ ( lp_spa812643495979991928at_nat @ P2 @ M2 ) ) ) ) ).

% prob_space.Lp_subset_Lq(2)
thf(fact_346_prob__space_OLp__subset__Lq_I2_J,axiom,
    ! [M2: sigma_5515648953823433982at_nat,P2: extend8495563244428889912nnreal,Q2: extend8495563244428889912nnreal] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( ord_le3935885782089961368nnreal @ P2 @ Q2 )
       => ( ord_le6049248562894827741t_real @ ( lp_spa8733284142207339630at_nat @ Q2 @ M2 ) @ ( lp_spa8733284142207339630at_nat @ P2 @ M2 ) ) ) ) ).

% prob_space.Lp_subset_Lq(2)
thf(fact_347_F__gr__0,axiom,
    ! [As: list_nat,K: nat] :
      ( ( As != nil_nat )
     => ( ord_less_rat @ zero_zero_rat @ ( frequency_F_nat @ K @ As ) ) ) ).

% F_gr_0
thf(fact_348_sigma__finite__measure__axioms,axiom,
    measur1424910518806321826at_nat @ ( freque5010624893710627907mega_1 @ as ) ).

% sigma_finite_measure_axioms
thf(fact_349_Lp__subset__Lq_I2_J,axiom,
    ! [P2: extend8495563244428889912nnreal,Q2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ P2 @ Q2 )
     => ( ord_le6049248562894827741t_real @ ( lp_spa8733284142207339630at_nat @ Q2 @ ( freque5010624893710627907mega_1 @ as ) ) @ ( lp_spa8733284142207339630at_nat @ P2 @ ( freque5010624893710627907mega_1 @ as ) ) ) ) ).

% Lp_subset_Lq(2)
thf(fact_350_indep__var__rv2,axiom,
    ! [S2: sigma_5515648953823433982at_nat,X: product_prod_nat_nat > product_prod_nat_nat,T: sigma_5515648953823433982at_nat,Y: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( indepe9184017870484869004at_nat @ ( freque5010624893710627907mega_1 @ as ) @ S2 @ X @ T @ Y )
     => ( member8885076297122219836at_nat @ Y @ ( sigma_3682087626466204304at_nat @ ( freque5010624893710627907mega_1 @ as ) @ T ) ) ) ).

% indep_var_rv2
thf(fact_351_indep__var__rv2,axiom,
    ! [S2: sigma_measure_real,X: product_prod_nat_nat > real,T: sigma_measure_real,Y: product_prod_nat_nat > real] :
      ( ( indepe1845147739501508449t_real @ ( freque5010624893710627907mega_1 @ as ) @ S2 @ X @ T @ Y )
     => ( member4564283293661824327t_real @ Y @ ( sigma_1188828689629184861t_real @ ( freque5010624893710627907mega_1 @ as ) @ T ) ) ) ).

% indep_var_rv2
thf(fact_352_indep__var__rv1,axiom,
    ! [S2: sigma_5515648953823433982at_nat,X: product_prod_nat_nat > product_prod_nat_nat,T: sigma_5515648953823433982at_nat,Y: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( indepe9184017870484869004at_nat @ ( freque5010624893710627907mega_1 @ as ) @ S2 @ X @ T @ Y )
     => ( member8885076297122219836at_nat @ X @ ( sigma_3682087626466204304at_nat @ ( freque5010624893710627907mega_1 @ as ) @ S2 ) ) ) ).

% indep_var_rv1
thf(fact_353_indep__var__rv1,axiom,
    ! [S2: sigma_measure_real,X: product_prod_nat_nat > real,T: sigma_measure_real,Y: product_prod_nat_nat > real] :
      ( ( indepe1845147739501508449t_real @ ( freque5010624893710627907mega_1 @ as ) @ S2 @ X @ T @ Y )
     => ( member4564283293661824327t_real @ X @ ( sigma_1188828689629184861t_real @ ( freque5010624893710627907mega_1 @ as ) @ S2 ) ) ) ).

% indep_var_rv1
thf(fact_354_indep__var__compose,axiom,
    ! [M1: sigma_5515648953823433982at_nat,X1: product_prod_nat_nat > product_prod_nat_nat,M22: sigma_5515648953823433982at_nat,X22: product_prod_nat_nat > product_prod_nat_nat,Y1: product_prod_nat_nat > product_prod_nat_nat,N1: sigma_5515648953823433982at_nat,Y2: product_prod_nat_nat > product_prod_nat_nat,N22: sigma_5515648953823433982at_nat] :
      ( ( indepe9184017870484869004at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M1 @ X1 @ M22 @ X22 )
     => ( ( member8885076297122219836at_nat @ Y1 @ ( sigma_3682087626466204304at_nat @ M1 @ N1 ) )
       => ( ( member8885076297122219836at_nat @ Y2 @ ( sigma_3682087626466204304at_nat @ M22 @ N22 ) )
         => ( indepe9184017870484869004at_nat @ ( freque5010624893710627907mega_1 @ as ) @ N1 @ ( comp_P2240441846945064862at_nat @ Y1 @ X1 ) @ N22 @ ( comp_P2240441846945064862at_nat @ Y2 @ X22 ) ) ) ) ) ).

% indep_var_compose
thf(fact_355_indep__var__compose,axiom,
    ! [M1: sigma_9047027012034273406at_nat,X1: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,M22: sigma_9047027012034273406at_nat,X22: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Y1: ( product_prod_nat_nat > product_prod_nat_nat ) > real,N1: sigma_measure_real,Y2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,N22: sigma_measure_real] :
      ( ( indepe911637784861272026at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M1 @ X1 @ M22 @ X22 )
     => ( ( member8159409068225774087t_real @ Y1 @ ( sigma_5064276549609164707t_real @ M1 @ N1 ) )
       => ( ( member8159409068225774087t_real @ Y2 @ ( sigma_5064276549609164707t_real @ M22 @ N22 ) )
         => ( indepe1845147739501508449t_real @ ( freque5010624893710627907mega_1 @ as ) @ N1 @ ( comp_P3287744429501015755at_nat @ Y1 @ X1 ) @ N22 @ ( comp_P3287744429501015755at_nat @ Y2 @ X22 ) ) ) ) ) ).

% indep_var_compose
thf(fact_356_indep__var__compose,axiom,
    ! [M1: sigma_5515648953823433982at_nat,X1: product_prod_nat_nat > product_prod_nat_nat,M22: sigma_5515648953823433982at_nat,X22: product_prod_nat_nat > product_prod_nat_nat,Y1: product_prod_nat_nat > real,N1: sigma_measure_real,Y2: product_prod_nat_nat > real,N22: sigma_measure_real] :
      ( ( indepe9184017870484869004at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M1 @ X1 @ M22 @ X22 )
     => ( ( member4564283293661824327t_real @ Y1 @ ( sigma_1188828689629184861t_real @ M1 @ N1 ) )
       => ( ( member4564283293661824327t_real @ Y2 @ ( sigma_1188828689629184861t_real @ M22 @ N22 ) )
         => ( indepe1845147739501508449t_real @ ( freque5010624893710627907mega_1 @ as ) @ N1 @ ( comp_P6083776177191121141at_nat @ Y1 @ X1 ) @ N22 @ ( comp_P6083776177191121141at_nat @ Y2 @ X22 ) ) ) ) ) ).

% indep_var_compose
thf(fact_357_indep__var__compose,axiom,
    ! [M1: sigma_measure_real,X1: product_prod_nat_nat > real,M22: sigma_measure_real,X22: product_prod_nat_nat > real,Y1: real > extend8495563244428889912nnreal,N1: sigma_7234349610311085201nnreal,Y2: real > extend8495563244428889912nnreal,N22: sigma_7234349610311085201nnreal] :
      ( ( indepe1845147739501508449t_real @ ( freque5010624893710627907mega_1 @ as ) @ M1 @ X1 @ M22 @ X22 )
     => ( ( member2919562650594848410nnreal @ Y1 @ ( sigma_9017504469962657078nnreal @ M1 @ N1 ) )
       => ( ( member2919562650594848410nnreal @ Y2 @ ( sigma_9017504469962657078nnreal @ M22 @ N22 ) )
         => ( indepe8239858677677355629nnreal @ ( freque5010624893710627907mega_1 @ as ) @ N1 @ ( comp_r868043356317784952at_nat @ Y1 @ X1 ) @ N22 @ ( comp_r868043356317784952at_nat @ Y2 @ X22 ) ) ) ) ) ).

% indep_var_compose
thf(fact_358_indep__var__compose,axiom,
    ! [M1: sigma_measure_real,X1: product_prod_nat_nat > real,M22: sigma_measure_real,X22: product_prod_nat_nat > real,Y1: real > real,N1: sigma_measure_real,Y2: real > real,N22: sigma_measure_real] :
      ( ( indepe1845147739501508449t_real @ ( freque5010624893710627907mega_1 @ as ) @ M1 @ X1 @ M22 @ X22 )
     => ( ( member_real_real @ Y1 @ ( sigma_5267869275261027754l_real @ M1 @ N1 ) )
       => ( ( member_real_real @ Y2 @ ( sigma_5267869275261027754l_real @ M22 @ N22 ) )
         => ( indepe1845147739501508449t_real @ ( freque5010624893710627907mega_1 @ as ) @ N1 @ ( comp_r3517367148340353796at_nat @ Y1 @ X1 ) @ N22 @ ( comp_r3517367148340353796at_nat @ Y2 @ X22 ) ) ) ) ) ).

% indep_var_compose
thf(fact_359__092_060epsilon_062__range,axiom,
    member_rat @ epsilon @ ( set_or5199638295745620268an_rat @ zero_zero_rat @ one_one_rat ) ).

% \<epsilon>_range
thf(fact_360_L1__D_I2_J,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,M2: sigma_9047027012034273406at_nat] :
      ( ( member8159409068225774087t_real @ F2 @ ( functi8590574888837585149t_real @ ( lp_spa812643495979991928at_nat @ one_on2969667320475766781nnreal @ M2 ) ) )
     => ( bochne7117808529828525605t_real @ M2 @ F2 ) ) ).

% L1_D(2)
thf(fact_361_L1__D_I2_J,axiom,
    ! [F2: product_prod_nat_nat > real,M2: sigma_5515648953823433982at_nat] :
      ( ( member4564283293661824327t_real @ F2 @ ( functi6492146676480127185t_real @ ( lp_spa8733284142207339630at_nat @ one_on2969667320475766781nnreal @ M2 ) ) )
     => ( bochne2596016609597520987t_real @ M2 @ F2 ) ) ).

% L1_D(2)
thf(fact_362_L1__I_I1_J,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( bochne7117808529828525605t_real @ M2 @ F2 )
     => ( member8159409068225774087t_real @ F2 @ ( functi8590574888837585149t_real @ ( lp_spa812643495979991928at_nat @ one_on2969667320475766781nnreal @ M2 ) ) ) ) ).

% L1_I(1)
thf(fact_363_L1__I_I1_J,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real] :
      ( ( bochne2596016609597520987t_real @ M2 @ F2 )
     => ( member4564283293661824327t_real @ F2 @ ( functi6492146676480127185t_real @ ( lp_spa8733284142207339630at_nat @ one_on2969667320475766781nnreal @ M2 ) ) ) ) ).

% L1_I(1)
thf(fact_364_L1__space,axiom,
    ! [M2: sigma_9047027012034273406at_nat] :
      ( ( functi8590574888837585149t_real @ ( lp_spa812643495979991928at_nat @ one_on2969667320475766781nnreal @ M2 ) )
      = ( collec4597233195267612485t_real @ ( bochne7117808529828525605t_real @ M2 ) ) ) ).

% L1_space
thf(fact_365_L1__space,axiom,
    ! [M2: sigma_5515648953823433982at_nat] :
      ( ( functi6492146676480127185t_real @ ( lp_spa8733284142207339630at_nat @ one_on2969667320475766781nnreal @ M2 ) )
      = ( collec1908255412270979209t_real @ ( bochne2596016609597520987t_real @ M2 ) ) ) ).

% L1_space
thf(fact_366_prob__space_OLp__subset__Lq_I3_J,axiom,
    ! [M2: sigma_9047027012034273406at_nat,P2: extend8495563244428889912nnreal,Q2: extend8495563244428889912nnreal] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( ord_le3935885782089961368nnreal @ P2 @ Q2 )
       => ( ord_le8622349663015078278t_real @ ( functi8590574888837585149t_real @ ( lp_spa812643495979991928at_nat @ Q2 @ M2 ) ) @ ( functi8590574888837585149t_real @ ( lp_spa812643495979991928at_nat @ P2 @ M2 ) ) ) ) ) ).

% prob_space.Lp_subset_Lq(3)
thf(fact_367_prob__space_OLp__subset__Lq_I3_J,axiom,
    ! [M2: sigma_5515648953823433982at_nat,P2: extend8495563244428889912nnreal,Q2: extend8495563244428889912nnreal] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( ord_le3935885782089961368nnreal @ P2 @ Q2 )
       => ( ord_le1994730768851292446t_real @ ( functi6492146676480127185t_real @ ( lp_spa8733284142207339630at_nat @ Q2 @ M2 ) ) @ ( functi6492146676480127185t_real @ ( lp_spa8733284142207339630at_nat @ P2 @ M2 ) ) ) ) ) ).

% prob_space.Lp_subset_Lq(3)
thf(fact_368_prob__space_OLp__subset__Lq_I1_J,axiom,
    ! [M2: sigma_9047027012034273406at_nat,P2: extend8495563244428889912nnreal,Q2: extend8495563244428889912nnreal,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( ord_le3935885782089961368nnreal @ P2 @ Q2 )
       => ( ord_le3935885782089961368nnreal @ ( functi3128325879070182998t_real @ ( lp_spa812643495979991928at_nat @ P2 @ M2 ) @ F2 ) @ ( functi3128325879070182998t_real @ ( lp_spa812643495979991928at_nat @ Q2 @ M2 ) @ F2 ) ) ) ) ).

% prob_space.Lp_subset_Lq(1)
thf(fact_369_prob__space_OLp__subset__Lq_I1_J,axiom,
    ! [M2: sigma_5515648953823433982at_nat,P2: extend8495563244428889912nnreal,Q2: extend8495563244428889912nnreal,F2: product_prod_nat_nat > real] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( ord_le3935885782089961368nnreal @ P2 @ Q2 )
       => ( ord_le3935885782089961368nnreal @ ( functi5516212835536045368t_real @ ( lp_spa8733284142207339630at_nat @ P2 @ M2 ) @ F2 ) @ ( functi5516212835536045368t_real @ ( lp_spa8733284142207339630at_nat @ Q2 @ M2 ) @ F2 ) ) ) ) ).

% prob_space.Lp_subset_Lq(1)
thf(fact_370_prob__space_Oindep__sets_Ocong,axiom,
    indepe9198310886692500677_nat_o = indepe9198310886692500677_nat_o ).

% prob_space.indep_sets.cong
thf(fact_371_prob__space_Oindep__sets_Ocong,axiom,
    indepe619418257930843587_nat_o = indepe619418257930843587_nat_o ).

% prob_space.indep_sets.cong
thf(fact_372_prob__space_Oindep__sets_Ocong,axiom,
    indepe5134244768989041027t_real = indepe5134244768989041027t_real ).

% prob_space.indep_sets.cong
thf(fact_373_prob__space_Oindep__sets_Ocong,axiom,
    indepe6665278795436257814nnreal = indepe6665278795436257814nnreal ).

% prob_space.indep_sets.cong
thf(fact_374_prob__space_Oindep__sets_Ocong,axiom,
    indepe4000825420875437771t_real = indepe4000825420875437771t_real ).

% prob_space.indep_sets.cong
thf(fact_375_prob__space_Oindep__sets_Ocong,axiom,
    indepe7028276433782144696at_nat = indepe7028276433782144696at_nat ).

% prob_space.indep_sets.cong
thf(fact_376_prob__space_Oindep__sets_Ocong,axiom,
    indepe2998401279100655083at_rat = indepe2998401279100655083at_rat ).

% prob_space.indep_sets.cong
thf(fact_377_prob__space_Oindep__var_Ocong,axiom,
    indepe1938972608428940447t_real = indepe1938972608428940447t_real ).

% prob_space.indep_var.cong
thf(fact_378_prob__space_Oindep__var_Ocong,axiom,
    indepe1845147739501508449t_real = indepe1845147739501508449t_real ).

% prob_space.indep_var.cong
thf(fact_379_prob__space_Oindep__sets__cong,axiom,
    ! [M2: sigma_5515648953823433982at_nat,I: set_rat,J: set_rat,F: rat > set_se7855581050983116737at_nat,G: rat > set_se7855581050983116737at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( I = J )
       => ( ! [I2: rat] :
              ( ( member_rat @ I2 @ I )
             => ( ( F @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe4562553564394819693at_rat @ M2 @ F @ I )
            = ( indepe4562553564394819693at_rat @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_380_prob__space_Oindep__sets__cong,axiom,
    ! [M2: sigma_5515648953823433982at_nat,I: set_o,J: set_o,F: $o > set_se7855581050983116737at_nat,G: $o > set_se7855581050983116737at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( I = J )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I )
             => ( ( F @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe619418257930843587_nat_o @ M2 @ F @ I )
            = ( indepe619418257930843587_nat_o @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_381_prob__space_Oindep__sets__cong,axiom,
    ! [M2: sigma_5515648953823433982at_nat,I: set_re5328672808648366137nnreal,J: set_re5328672808648366137nnreal,F: ( real > extend8495563244428889912nnreal ) > set_se7855581050983116737at_nat,G: ( real > extend8495563244428889912nnreal ) > set_se7855581050983116737at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( I = J )
       => ( ! [I2: real > extend8495563244428889912nnreal] :
              ( ( member2919562650594848410nnreal @ I2 @ I )
             => ( ( F @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe334513521632759960nnreal @ M2 @ F @ I )
            = ( indepe334513521632759960nnreal @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_382_prob__space_Oindep__sets__cong,axiom,
    ! [M2: sigma_5515648953823433982at_nat,I: set_Pr1002607673312053630t_real,J: set_Pr1002607673312053630t_real,F: ( product_prod_nat_nat > real ) > set_se7855581050983116737at_nat,G: ( product_prod_nat_nat > real ) > set_se7855581050983116737at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( I = J )
       => ( ! [I2: product_prod_nat_nat > real] :
              ( ( member4564283293661824327t_real @ I2 @ I )
             => ( ( F @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe3818579677853040073t_real @ M2 @ F @ I )
            = ( indepe3818579677853040073t_real @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_383_prob__space_Oindep__sets__cong,axiom,
    ! [M2: sigma_9047027012034273406at_nat,I: set_o,J: set_o,F: $o > set_se1666487788256820497at_nat,G: $o > set_se1666487788256820497at_nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( I = J )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I )
             => ( ( F @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe9198310886692500677_nat_o @ M2 @ F @ I )
            = ( indepe9198310886692500677_nat_o @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_384_prob__space_Oindep__sets__cong,axiom,
    ! [M2: sigma_9047027012034273406at_nat,I: set_rat,J: set_rat,F: rat > set_se1666487788256820497at_nat,G: rat > set_se1666487788256820497at_nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( I = J )
       => ( ! [I2: rat] :
              ( ( member_rat @ I2 @ I )
             => ( ( F @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe2998401279100655083at_rat @ M2 @ F @ I )
            = ( indepe2998401279100655083at_rat @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_385_prob__space_Oindep__sets__cong,axiom,
    ! [M2: sigma_5515648953823433982at_nat,I: set_Pr2458342521480944603at_nat,J: set_Pr2458342521480944603at_nat,F: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se7855581050983116737at_nat,G: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se7855581050983116737at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( I = J )
       => ( ! [I2: product_prod_nat_nat > product_prod_nat_nat] :
              ( ( member8885076297122219836at_nat @ I2 @ I )
             => ( ( F @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe6545576254369104698at_nat @ M2 @ F @ I )
            = ( indepe6545576254369104698at_nat @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_386_prob__space_Oindep__sets__cong,axiom,
    ! [M2: sigma_9047027012034273406at_nat,I: set_re5328672808648366137nnreal,J: set_re5328672808648366137nnreal,F: ( real > extend8495563244428889912nnreal ) > set_se1666487788256820497at_nat,G: ( real > extend8495563244428889912nnreal ) > set_se1666487788256820497at_nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( I = J )
       => ( ! [I2: real > extend8495563244428889912nnreal] :
              ( ( member2919562650594848410nnreal @ I2 @ I )
             => ( ( F @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe6665278795436257814nnreal @ M2 @ F @ I )
            = ( indepe6665278795436257814nnreal @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_387_prob__space_Oindep__sets__cong,axiom,
    ! [M2: sigma_5515648953823433982at_nat,I: set_Pr947837736998463782t_real,J: set_Pr947837736998463782t_real,F: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_se7855581050983116737at_nat,G: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_se7855581050983116737at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( I = J )
       => ( ! [I2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
              ( ( member8159409068225774087t_real @ I2 @ I )
             => ( ( F @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe2662975882062776837t_real @ M2 @ F @ I )
            = ( indepe2662975882062776837t_real @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_388_prob__space_Oindep__sets__cong,axiom,
    ! [M2: sigma_9047027012034273406at_nat,I: set_Pr1002607673312053630t_real,J: set_Pr1002607673312053630t_real,F: ( product_prod_nat_nat > real ) > set_se1666487788256820497at_nat,G: ( product_prod_nat_nat > real ) > set_se1666487788256820497at_nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( I = J )
       => ( ! [I2: product_prod_nat_nat > real] :
              ( ( member4564283293661824327t_real @ I2 @ I )
             => ( ( F @ I2 )
                = ( G @ I2 ) ) )
         => ( ( indepe4000825420875437771t_real @ M2 @ F @ I )
            = ( indepe4000825420875437771t_real @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_cong
thf(fact_389_prob__space_Oindep__sets__mono__index,axiom,
    ! [M2: sigma_5515648953823433982at_nat,J: set_Pr2458342521480944603at_nat,I: set_Pr2458342521480944603at_nat,F: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se7855581050983116737at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( ord_le3857079194666040379at_nat @ J @ I )
       => ( ( indepe6545576254369104698at_nat @ M2 @ F @ I )
         => ( indepe6545576254369104698at_nat @ M2 @ F @ J ) ) ) ) ).

% prob_space.indep_sets_mono_index
thf(fact_390_prob__space_Oindep__sets__mono__index,axiom,
    ! [M2: sigma_5515648953823433982at_nat,J: set_Pr1002607673312053630t_real,I: set_Pr1002607673312053630t_real,F: ( product_prod_nat_nat > real ) > set_se7855581050983116737at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( ord_le1994730768851292446t_real @ J @ I )
       => ( ( indepe3818579677853040073t_real @ M2 @ F @ I )
         => ( indepe3818579677853040073t_real @ M2 @ F @ J ) ) ) ) ).

% prob_space.indep_sets_mono_index
thf(fact_391_prob__space_Oindep__sets__mono__index,axiom,
    ! [M2: sigma_5515648953823433982at_nat,J: set_Pr947837736998463782t_real,I: set_Pr947837736998463782t_real,F: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_se7855581050983116737at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( ord_le8622349663015078278t_real @ J @ I )
       => ( ( indepe2662975882062776837t_real @ M2 @ F @ I )
         => ( indepe2662975882062776837t_real @ M2 @ F @ J ) ) ) ) ).

% prob_space.indep_sets_mono_index
thf(fact_392_prob__space_Oindep__sets__mono__index,axiom,
    ! [M2: sigma_9047027012034273406at_nat,J: set_o,I: set_o,F: $o > set_se1666487788256820497at_nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( ord_less_eq_set_o @ J @ I )
       => ( ( indepe9198310886692500677_nat_o @ M2 @ F @ I )
         => ( indepe9198310886692500677_nat_o @ M2 @ F @ J ) ) ) ) ).

% prob_space.indep_sets_mono_index
thf(fact_393_prob__space_Oindep__sets__mono__index,axiom,
    ! [M2: sigma_5515648953823433982at_nat,J: set_o,I: set_o,F: $o > set_se7855581050983116737at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( ord_less_eq_set_o @ J @ I )
       => ( ( indepe619418257930843587_nat_o @ M2 @ F @ I )
         => ( indepe619418257930843587_nat_o @ M2 @ F @ J ) ) ) ) ).

% prob_space.indep_sets_mono_index
thf(fact_394_prob__space_Oindep__sets__mono__index,axiom,
    ! [M2: sigma_9047027012034273406at_nat,J: set_Pr947837736998463782t_real,I: set_Pr947837736998463782t_real,F: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_se1666487788256820497at_nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( ord_le8622349663015078278t_real @ J @ I )
       => ( ( indepe5134244768989041027t_real @ M2 @ F @ I )
         => ( indepe5134244768989041027t_real @ M2 @ F @ J ) ) ) ) ).

% prob_space.indep_sets_mono_index
thf(fact_395_prob__space_Oindep__sets__mono__index,axiom,
    ! [M2: sigma_9047027012034273406at_nat,J: set_re5328672808648366137nnreal,I: set_re5328672808648366137nnreal,F: ( real > extend8495563244428889912nnreal ) > set_se1666487788256820497at_nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( ord_le2462468573666744473nnreal @ J @ I )
       => ( ( indepe6665278795436257814nnreal @ M2 @ F @ I )
         => ( indepe6665278795436257814nnreal @ M2 @ F @ J ) ) ) ) ).

% prob_space.indep_sets_mono_index
thf(fact_396_prob__space_Oindep__sets__mono__index,axiom,
    ! [M2: sigma_9047027012034273406at_nat,J: set_Pr1002607673312053630t_real,I: set_Pr1002607673312053630t_real,F: ( product_prod_nat_nat > real ) > set_se1666487788256820497at_nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( ord_le1994730768851292446t_real @ J @ I )
       => ( ( indepe4000825420875437771t_real @ M2 @ F @ I )
         => ( indepe4000825420875437771t_real @ M2 @ F @ J ) ) ) ) ).

% prob_space.indep_sets_mono_index
thf(fact_397_prob__space_Oindep__sets__mono__index,axiom,
    ! [M2: sigma_9047027012034273406at_nat,J: set_Pr2458342521480944603at_nat,I: set_Pr2458342521480944603at_nat,F: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se1666487788256820497at_nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( ord_le3857079194666040379at_nat @ J @ I )
       => ( ( indepe7028276433782144696at_nat @ M2 @ F @ I )
         => ( indepe7028276433782144696at_nat @ M2 @ F @ J ) ) ) ) ).

% prob_space.indep_sets_mono_index
thf(fact_398_prob__space_Oindep__sets__mono__index,axiom,
    ! [M2: sigma_9047027012034273406at_nat,J: set_rat,I: set_rat,F: rat > set_se1666487788256820497at_nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( ord_less_eq_set_rat @ J @ I )
       => ( ( indepe2998401279100655083at_rat @ M2 @ F @ I )
         => ( indepe2998401279100655083at_rat @ M2 @ F @ J ) ) ) ) ).

% prob_space.indep_sets_mono_index
thf(fact_399_prob__space_Oindep__var__rv2,axiom,
    ! [M2: sigma_measure_real,S2: sigma_7234349610311085201nnreal,X: real > extend8495563244428889912nnreal,T: sigma_7234349610311085201nnreal,Y: real > extend8495563244428889912nnreal] :
      ( ( probab535871623910865577e_real @ M2 )
     => ( ( indepe6767359503340752434nnreal @ M2 @ S2 @ X @ T @ Y )
       => ( member2919562650594848410nnreal @ Y @ ( sigma_9017504469962657078nnreal @ M2 @ T ) ) ) ) ).

% prob_space.indep_var_rv2
thf(fact_400_prob__space_Oindep__var__rv2,axiom,
    ! [M2: sigma_5515648953823433982at_nat,S2: sigma_5515648953823433982at_nat,X: product_prod_nat_nat > product_prod_nat_nat,T: sigma_5515648953823433982at_nat,Y: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe9184017870484869004at_nat @ M2 @ S2 @ X @ T @ Y )
       => ( member8885076297122219836at_nat @ Y @ ( sigma_3682087626466204304at_nat @ M2 @ T ) ) ) ) ).

% prob_space.indep_var_rv2
thf(fact_401_prob__space_Oindep__var__rv2,axiom,
    ! [M2: sigma_9047027012034273406at_nat,S2: sigma_measure_real,X: ( product_prod_nat_nat > product_prod_nat_nat ) > real,T: sigma_measure_real,Y: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe1938972608428940447t_real @ M2 @ S2 @ X @ T @ Y )
       => ( member8159409068225774087t_real @ Y @ ( sigma_5064276549609164707t_real @ M2 @ T ) ) ) ) ).

% prob_space.indep_var_rv2
thf(fact_402_prob__space_Oindep__var__rv2,axiom,
    ! [M2: sigma_5515648953823433982at_nat,S2: sigma_measure_real,X: product_prod_nat_nat > real,T: sigma_measure_real,Y: product_prod_nat_nat > real] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe1845147739501508449t_real @ M2 @ S2 @ X @ T @ Y )
       => ( member4564283293661824327t_real @ Y @ ( sigma_1188828689629184861t_real @ M2 @ T ) ) ) ) ).

% prob_space.indep_var_rv2
thf(fact_403_prob__space_Oindep__var__rv1,axiom,
    ! [M2: sigma_measure_real,S2: sigma_7234349610311085201nnreal,X: real > extend8495563244428889912nnreal,T: sigma_7234349610311085201nnreal,Y: real > extend8495563244428889912nnreal] :
      ( ( probab535871623910865577e_real @ M2 )
     => ( ( indepe6767359503340752434nnreal @ M2 @ S2 @ X @ T @ Y )
       => ( member2919562650594848410nnreal @ X @ ( sigma_9017504469962657078nnreal @ M2 @ S2 ) ) ) ) ).

% prob_space.indep_var_rv1
thf(fact_404_prob__space_Oindep__var__rv1,axiom,
    ! [M2: sigma_5515648953823433982at_nat,S2: sigma_5515648953823433982at_nat,X: product_prod_nat_nat > product_prod_nat_nat,T: sigma_5515648953823433982at_nat,Y: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe9184017870484869004at_nat @ M2 @ S2 @ X @ T @ Y )
       => ( member8885076297122219836at_nat @ X @ ( sigma_3682087626466204304at_nat @ M2 @ S2 ) ) ) ) ).

% prob_space.indep_var_rv1
thf(fact_405_prob__space_Oindep__var__rv1,axiom,
    ! [M2: sigma_9047027012034273406at_nat,S2: sigma_measure_real,X: ( product_prod_nat_nat > product_prod_nat_nat ) > real,T: sigma_measure_real,Y: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe1938972608428940447t_real @ M2 @ S2 @ X @ T @ Y )
       => ( member8159409068225774087t_real @ X @ ( sigma_5064276549609164707t_real @ M2 @ S2 ) ) ) ) ).

% prob_space.indep_var_rv1
thf(fact_406_prob__space_Oindep__var__rv1,axiom,
    ! [M2: sigma_5515648953823433982at_nat,S2: sigma_measure_real,X: product_prod_nat_nat > real,T: sigma_measure_real,Y: product_prod_nat_nat > real] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe1845147739501508449t_real @ M2 @ S2 @ X @ T @ Y )
       => ( member4564283293661824327t_real @ X @ ( sigma_1188828689629184861t_real @ M2 @ S2 ) ) ) ) ).

% prob_space.indep_var_rv1
thf(fact_407_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F: rat > set_se7855581050983116737at_nat,I: set_rat,G: rat > set_se7855581050983116737at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe4562553564394819693at_rat @ M2 @ F @ I )
       => ( ! [I2: rat] :
              ( ( member_rat @ I2 @ I )
             => ( ord_le2077887516847798113at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
         => ( indepe4562553564394819693at_rat @ M2 @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_408_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F: $o > set_se7855581050983116737at_nat,I: set_o,G: $o > set_se7855581050983116737at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe619418257930843587_nat_o @ M2 @ F @ I )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I )
             => ( ord_le2077887516847798113at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
         => ( indepe619418257930843587_nat_o @ M2 @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_409_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F: ( real > extend8495563244428889912nnreal ) > set_se7855581050983116737at_nat,I: set_re5328672808648366137nnreal,G: ( real > extend8495563244428889912nnreal ) > set_se7855581050983116737at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe334513521632759960nnreal @ M2 @ F @ I )
       => ( ! [I2: real > extend8495563244428889912nnreal] :
              ( ( member2919562650594848410nnreal @ I2 @ I )
             => ( ord_le2077887516847798113at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
         => ( indepe334513521632759960nnreal @ M2 @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_410_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F: ( product_prod_nat_nat > real ) > set_se7855581050983116737at_nat,I: set_Pr1002607673312053630t_real,G: ( product_prod_nat_nat > real ) > set_se7855581050983116737at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe3818579677853040073t_real @ M2 @ F @ I )
       => ( ! [I2: product_prod_nat_nat > real] :
              ( ( member4564283293661824327t_real @ I2 @ I )
             => ( ord_le2077887516847798113at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
         => ( indepe3818579677853040073t_real @ M2 @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_411_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F: $o > set_se1666487788256820497at_nat,I: set_o,G: $o > set_se1666487788256820497at_nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe9198310886692500677_nat_o @ M2 @ F @ I )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I )
             => ( ord_le8269821659064819057at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
         => ( indepe9198310886692500677_nat_o @ M2 @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_412_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F: rat > set_se1666487788256820497at_nat,I: set_rat,G: rat > set_se1666487788256820497at_nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe2998401279100655083at_rat @ M2 @ F @ I )
       => ( ! [I2: rat] :
              ( ( member_rat @ I2 @ I )
             => ( ord_le8269821659064819057at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
         => ( indepe2998401279100655083at_rat @ M2 @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_413_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se7855581050983116737at_nat,I: set_Pr2458342521480944603at_nat,G: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se7855581050983116737at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe6545576254369104698at_nat @ M2 @ F @ I )
       => ( ! [I2: product_prod_nat_nat > product_prod_nat_nat] :
              ( ( member8885076297122219836at_nat @ I2 @ I )
             => ( ord_le2077887516847798113at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
         => ( indepe6545576254369104698at_nat @ M2 @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_414_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F: ( real > extend8495563244428889912nnreal ) > set_se1666487788256820497at_nat,I: set_re5328672808648366137nnreal,G: ( real > extend8495563244428889912nnreal ) > set_se1666487788256820497at_nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe6665278795436257814nnreal @ M2 @ F @ I )
       => ( ! [I2: real > extend8495563244428889912nnreal] :
              ( ( member2919562650594848410nnreal @ I2 @ I )
             => ( ord_le8269821659064819057at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
         => ( indepe6665278795436257814nnreal @ M2 @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_415_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_se7855581050983116737at_nat,I: set_Pr947837736998463782t_real,G: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_se7855581050983116737at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe2662975882062776837t_real @ M2 @ F @ I )
       => ( ! [I2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
              ( ( member8159409068225774087t_real @ I2 @ I )
             => ( ord_le2077887516847798113at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
         => ( indepe2662975882062776837t_real @ M2 @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_416_prob__space_Oindep__sets__mono__sets,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F: ( product_prod_nat_nat > real ) > set_se1666487788256820497at_nat,I: set_Pr1002607673312053630t_real,G: ( product_prod_nat_nat > real ) > set_se1666487788256820497at_nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe4000825420875437771t_real @ M2 @ F @ I )
       => ( ! [I2: product_prod_nat_nat > real] :
              ( ( member4564283293661824327t_real @ I2 @ I )
             => ( ord_le8269821659064819057at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
         => ( indepe4000825420875437771t_real @ M2 @ G @ I ) ) ) ) ).

% prob_space.indep_sets_mono_sets
thf(fact_417_prob__space_Oindep__vars__compose2,axiom,
    ! [M2: sigma_5515648953823433982at_nat,M: rat > sigma_measure_real,X: rat > product_prod_nat_nat > real,I: set_rat,Y: rat > real > extend8495563244428889912nnreal,N: rat > sigma_7234349610311085201nnreal] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe5387908695526973631t_real @ M2 @ M @ X @ I )
       => ( ! [I2: rat] :
              ( ( member_rat @ I2 @ I )
             => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( indepe317362771116669643nnreal @ M2 @ N
            @ ^ [I3: rat,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_418_prob__space_Oindep__vars__compose2,axiom,
    ! [M2: sigma_5515648953823433982at_nat,M: $o > sigma_measure_real,X: $o > product_prod_nat_nat > real,I: set_o,Y: $o > real > extend8495563244428889912nnreal,N: $o > sigma_7234349610311085201nnreal] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe6599791682495051723o_real @ M2 @ M @ X @ I )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I )
             => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( indepe7306336255242631127nnreal @ M2 @ N
            @ ^ [I3: $o,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_419_prob__space_Oindep__vars__compose2,axiom,
    ! [M2: sigma_5515648953823433982at_nat,M: rat > sigma_5515648953823433982at_nat,X: rat > product_prod_nat_nat > product_prod_nat_nat,I: set_rat,Y: rat > product_prod_nat_nat > real,N: rat > sigma_measure_real] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe2321685950664944046at_nat @ M2 @ M @ X @ I )
       => ( ! [I2: rat] :
              ( ( member_rat @ I2 @ I )
             => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( indepe5387908695526973631t_real @ M2 @ N
            @ ^ [I3: rat,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_420_prob__space_Oindep__vars__compose2,axiom,
    ! [M2: sigma_5515648953823433982at_nat,M: $o > sigma_5515648953823433982at_nat,X: $o > product_prod_nat_nat > product_prod_nat_nat,I: set_o,Y: $o > product_prod_nat_nat > real,N: $o > sigma_measure_real] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe504693989766925730at_nat @ M2 @ M @ X @ I )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I )
             => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( indepe6599791682495051723o_real @ M2 @ N
            @ ^ [I3: $o,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_421_prob__space_Oindep__vars__compose2,axiom,
    ! [M2: sigma_5515648953823433982at_nat,M: ( real > extend8495563244428889912nnreal ) > sigma_measure_real,X: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > real,I: set_re5328672808648366137nnreal,Y: ( real > extend8495563244428889912nnreal ) > real > extend8495563244428889912nnreal,N: ( real > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe579324506763905386l_real @ M2 @ M @ X @ I )
       => ( ! [I2: real > extend8495563244428889912nnreal] :
              ( ( member2919562650594848410nnreal @ I2 @ I )
             => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( indepe3243683517584732406nnreal @ M2 @ N
            @ ^ [I3: real > extend8495563244428889912nnreal,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_422_prob__space_Oindep__vars__compose2,axiom,
    ! [M2: sigma_9047027012034273406at_nat,M: rat > sigma_measure_real,X: rat > ( product_prod_nat_nat > product_prod_nat_nat ) > real,I: set_rat,Y: rat > real > extend8495563244428889912nnreal,N: rat > sigma_7234349610311085201nnreal] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe4371592291047002227t_real @ M2 @ M @ X @ I )
       => ( ! [I2: rat] :
              ( ( member_rat @ I2 @ I )
             => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( indepe4547145070500471935nnreal @ M2 @ N
            @ ^ [I3: rat,X2: product_prod_nat_nat > product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_423_prob__space_Oindep__vars__compose2,axiom,
    ! [M2: sigma_9047027012034273406at_nat,M: $o > sigma_measure_real,X: $o > ( product_prod_nat_nat > product_prod_nat_nat ) > real,I: set_o,Y: $o > real > extend8495563244428889912nnreal,N: $o > sigma_7234349610311085201nnreal] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe2405714897255445271o_real @ M2 @ M @ X @ I )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I )
             => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( indepe2783168947688421667nnreal @ M2 @ N
            @ ^ [I3: $o,X2: product_prod_nat_nat > product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_424_prob__space_Oindep__vars__compose2,axiom,
    ! [M2: sigma_5515648953823433982at_nat,M: rat > sigma_5515648953823433982at_nat,X: rat > product_prod_nat_nat > product_prod_nat_nat,I: set_rat,Y: rat > product_prod_nat_nat > product_prod_nat_nat,N: rat > sigma_5515648953823433982at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe2321685950664944046at_nat @ M2 @ M @ X @ I )
       => ( ! [I2: rat] :
              ( ( member_rat @ I2 @ I )
             => ( member8885076297122219836at_nat @ ( Y @ I2 ) @ ( sigma_3682087626466204304at_nat @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( indepe2321685950664944046at_nat @ M2 @ N
            @ ^ [I3: rat,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_425_prob__space_Oindep__vars__compose2,axiom,
    ! [M2: sigma_5515648953823433982at_nat,M: $o > sigma_5515648953823433982at_nat,X: $o > product_prod_nat_nat > product_prod_nat_nat,I: set_o,Y: $o > product_prod_nat_nat > product_prod_nat_nat,N: $o > sigma_5515648953823433982at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe504693989766925730at_nat @ M2 @ M @ X @ I )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I )
             => ( member8885076297122219836at_nat @ ( Y @ I2 ) @ ( sigma_3682087626466204304at_nat @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( indepe504693989766925730at_nat @ M2 @ N
            @ ^ [I3: $o,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_426_prob__space_Oindep__vars__compose2,axiom,
    ! [M2: sigma_5515648953823433982at_nat,M: ( real > extend8495563244428889912nnreal ) > sigma_5515648953823433982at_nat,X: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > product_prod_nat_nat,I: set_re5328672808648366137nnreal,Y: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > real,N: ( real > extend8495563244428889912nnreal ) > sigma_measure_real] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe3593890525012728259at_nat @ M2 @ M @ X @ I )
       => ( ! [I2: real > extend8495563244428889912nnreal] :
              ( ( member2919562650594848410nnreal @ I2 @ I )
             => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( indepe579324506763905386l_real @ M2 @ N
            @ ^ [I3: real > extend8495563244428889912nnreal,X2: product_prod_nat_nat] : ( Y @ I3 @ ( X @ I3 @ X2 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose2
thf(fact_427_F__ge__0,axiom,
    ! [K: nat,As: list_nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( frequency_F_nat @ K @ As ) ) ).

% F_ge_0
thf(fact_428_prob__space_Oindep__sets__mono,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F: rat > set_se7855581050983116737at_nat,I: set_rat,J: set_rat,G: rat > set_se7855581050983116737at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe4562553564394819693at_rat @ M2 @ F @ I )
       => ( ( ord_less_eq_set_rat @ J @ I )
         => ( ! [I2: rat] :
                ( ( member_rat @ I2 @ J )
               => ( ord_le2077887516847798113at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
           => ( indepe4562553564394819693at_rat @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_429_prob__space_Oindep__sets__mono,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F: $o > set_se7855581050983116737at_nat,I: set_o,J: set_o,G: $o > set_se7855581050983116737at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe619418257930843587_nat_o @ M2 @ F @ I )
       => ( ( ord_less_eq_set_o @ J @ I )
         => ( ! [I2: $o] :
                ( ( member_o @ I2 @ J )
               => ( ord_le2077887516847798113at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
           => ( indepe619418257930843587_nat_o @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_430_prob__space_Oindep__sets__mono,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F: ( real > extend8495563244428889912nnreal ) > set_se7855581050983116737at_nat,I: set_re5328672808648366137nnreal,J: set_re5328672808648366137nnreal,G: ( real > extend8495563244428889912nnreal ) > set_se7855581050983116737at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe334513521632759960nnreal @ M2 @ F @ I )
       => ( ( ord_le2462468573666744473nnreal @ J @ I )
         => ( ! [I2: real > extend8495563244428889912nnreal] :
                ( ( member2919562650594848410nnreal @ I2 @ J )
               => ( ord_le2077887516847798113at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
           => ( indepe334513521632759960nnreal @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_431_prob__space_Oindep__sets__mono,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F: ( product_prod_nat_nat > real ) > set_se7855581050983116737at_nat,I: set_Pr1002607673312053630t_real,J: set_Pr1002607673312053630t_real,G: ( product_prod_nat_nat > real ) > set_se7855581050983116737at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe3818579677853040073t_real @ M2 @ F @ I )
       => ( ( ord_le1994730768851292446t_real @ J @ I )
         => ( ! [I2: product_prod_nat_nat > real] :
                ( ( member4564283293661824327t_real @ I2 @ J )
               => ( ord_le2077887516847798113at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
           => ( indepe3818579677853040073t_real @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_432_prob__space_Oindep__sets__mono,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F: $o > set_se1666487788256820497at_nat,I: set_o,J: set_o,G: $o > set_se1666487788256820497at_nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe9198310886692500677_nat_o @ M2 @ F @ I )
       => ( ( ord_less_eq_set_o @ J @ I )
         => ( ! [I2: $o] :
                ( ( member_o @ I2 @ J )
               => ( ord_le8269821659064819057at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
           => ( indepe9198310886692500677_nat_o @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_433_prob__space_Oindep__sets__mono,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F: rat > set_se1666487788256820497at_nat,I: set_rat,J: set_rat,G: rat > set_se1666487788256820497at_nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe2998401279100655083at_rat @ M2 @ F @ I )
       => ( ( ord_less_eq_set_rat @ J @ I )
         => ( ! [I2: rat] :
                ( ( member_rat @ I2 @ J )
               => ( ord_le8269821659064819057at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
           => ( indepe2998401279100655083at_rat @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_434_prob__space_Oindep__sets__mono,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se7855581050983116737at_nat,I: set_Pr2458342521480944603at_nat,J: set_Pr2458342521480944603at_nat,G: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se7855581050983116737at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe6545576254369104698at_nat @ M2 @ F @ I )
       => ( ( ord_le3857079194666040379at_nat @ J @ I )
         => ( ! [I2: product_prod_nat_nat > product_prod_nat_nat] :
                ( ( member8885076297122219836at_nat @ I2 @ J )
               => ( ord_le2077887516847798113at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
           => ( indepe6545576254369104698at_nat @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_435_prob__space_Oindep__sets__mono,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F: ( real > extend8495563244428889912nnreal ) > set_se1666487788256820497at_nat,I: set_re5328672808648366137nnreal,J: set_re5328672808648366137nnreal,G: ( real > extend8495563244428889912nnreal ) > set_se1666487788256820497at_nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe6665278795436257814nnreal @ M2 @ F @ I )
       => ( ( ord_le2462468573666744473nnreal @ J @ I )
         => ( ! [I2: real > extend8495563244428889912nnreal] :
                ( ( member2919562650594848410nnreal @ I2 @ J )
               => ( ord_le8269821659064819057at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
           => ( indepe6665278795436257814nnreal @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_436_prob__space_Oindep__sets__mono,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_se7855581050983116737at_nat,I: set_Pr947837736998463782t_real,J: set_Pr947837736998463782t_real,G: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_se7855581050983116737at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe2662975882062776837t_real @ M2 @ F @ I )
       => ( ( ord_le8622349663015078278t_real @ J @ I )
         => ( ! [I2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
                ( ( member8159409068225774087t_real @ I2 @ J )
               => ( ord_le2077887516847798113at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
           => ( indepe2662975882062776837t_real @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_437_prob__space_Oindep__sets__mono,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F: ( product_prod_nat_nat > real ) > set_se1666487788256820497at_nat,I: set_Pr1002607673312053630t_real,J: set_Pr1002607673312053630t_real,G: ( product_prod_nat_nat > real ) > set_se1666487788256820497at_nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe4000825420875437771t_real @ M2 @ F @ I )
       => ( ( ord_le1994730768851292446t_real @ J @ I )
         => ( ! [I2: product_prod_nat_nat > real] :
                ( ( member4564283293661824327t_real @ I2 @ J )
               => ( ord_le8269821659064819057at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
           => ( indepe4000825420875437771t_real @ M2 @ G @ J ) ) ) ) ) ).

% prob_space.indep_sets_mono
thf(fact_438_prob__space_Oindep__var__compose,axiom,
    ! [M2: sigma_9047027012034273406at_nat,M1: sigma_5515648953823433982at_nat,X1: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,M22: sigma_5515648953823433982at_nat,X22: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,Y1: product_prod_nat_nat > product_prod_nat_nat,N1: sigma_5515648953823433982at_nat,Y2: product_prod_nat_nat > product_prod_nat_nat,N22: sigma_5515648953823433982at_nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe6532217695432402382at_nat @ M2 @ M1 @ X1 @ M22 @ X22 )
       => ( ( member8885076297122219836at_nat @ Y1 @ ( sigma_3682087626466204304at_nat @ M1 @ N1 ) )
         => ( ( member8885076297122219836at_nat @ Y2 @ ( sigma_3682087626466204304at_nat @ M22 @ N22 ) )
           => ( indepe6532217695432402382at_nat @ M2 @ N1 @ ( comp_P7516068803530355784at_nat @ Y1 @ X1 ) @ N22 @ ( comp_P7516068803530355784at_nat @ Y2 @ X22 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_439_prob__space_Oindep__var__compose,axiom,
    ! [M2: sigma_5515648953823433982at_nat,M1: sigma_5515648953823433982at_nat,X1: product_prod_nat_nat > product_prod_nat_nat,M22: sigma_5515648953823433982at_nat,X22: product_prod_nat_nat > product_prod_nat_nat,Y1: product_prod_nat_nat > product_prod_nat_nat,N1: sigma_5515648953823433982at_nat,Y2: product_prod_nat_nat > product_prod_nat_nat,N22: sigma_5515648953823433982at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe9184017870484869004at_nat @ M2 @ M1 @ X1 @ M22 @ X22 )
       => ( ( member8885076297122219836at_nat @ Y1 @ ( sigma_3682087626466204304at_nat @ M1 @ N1 ) )
         => ( ( member8885076297122219836at_nat @ Y2 @ ( sigma_3682087626466204304at_nat @ M22 @ N22 ) )
           => ( indepe9184017870484869004at_nat @ M2 @ N1 @ ( comp_P2240441846945064862at_nat @ Y1 @ X1 ) @ N22 @ ( comp_P2240441846945064862at_nat @ Y2 @ X22 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_440_prob__space_Oindep__var__compose,axiom,
    ! [M2: sigma_9047027012034273406at_nat,M1: sigma_9047027012034273406at_nat,X1: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat,M22: sigma_9047027012034273406at_nat,X22: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat,Y1: ( product_prod_nat_nat > product_prod_nat_nat ) > real,N1: sigma_measure_real,Y2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,N22: sigma_measure_real] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe4309971216286138904at_nat @ M2 @ M1 @ X1 @ M22 @ X22 )
       => ( ( member8159409068225774087t_real @ Y1 @ ( sigma_5064276549609164707t_real @ M1 @ N1 ) )
         => ( ( member8159409068225774087t_real @ Y2 @ ( sigma_5064276549609164707t_real @ M22 @ N22 ) )
           => ( indepe1938972608428940447t_real @ M2 @ N1 @ ( comp_P137399349585858651at_nat @ Y1 @ X1 ) @ N22 @ ( comp_P137399349585858651at_nat @ Y2 @ X22 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_441_prob__space_Oindep__var__compose,axiom,
    ! [M2: sigma_9047027012034273406at_nat,M1: sigma_5515648953823433982at_nat,X1: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,M22: sigma_5515648953823433982at_nat,X22: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,Y1: product_prod_nat_nat > real,N1: sigma_measure_real,Y2: product_prod_nat_nat > real,N22: sigma_measure_real] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe6532217695432402382at_nat @ M2 @ M1 @ X1 @ M22 @ X22 )
       => ( ( member4564283293661824327t_real @ Y1 @ ( sigma_1188828689629184861t_real @ M1 @ N1 ) )
         => ( ( member4564283293661824327t_real @ Y2 @ ( sigma_1188828689629184861t_real @ M22 @ N22 ) )
           => ( indepe1938972608428940447t_real @ M2 @ N1 @ ( comp_P2183065948301422769at_nat @ Y1 @ X1 ) @ N22 @ ( comp_P2183065948301422769at_nat @ Y2 @ X22 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_442_prob__space_Oindep__var__compose,axiom,
    ! [M2: sigma_5515648953823433982at_nat,M1: sigma_9047027012034273406at_nat,X1: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,M22: sigma_9047027012034273406at_nat,X22: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Y1: ( product_prod_nat_nat > product_prod_nat_nat ) > real,N1: sigma_measure_real,Y2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,N22: sigma_measure_real] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe911637784861272026at_nat @ M2 @ M1 @ X1 @ M22 @ X22 )
       => ( ( member8159409068225774087t_real @ Y1 @ ( sigma_5064276549609164707t_real @ M1 @ N1 ) )
         => ( ( member8159409068225774087t_real @ Y2 @ ( sigma_5064276549609164707t_real @ M22 @ N22 ) )
           => ( indepe1845147739501508449t_real @ M2 @ N1 @ ( comp_P3287744429501015755at_nat @ Y1 @ X1 ) @ N22 @ ( comp_P3287744429501015755at_nat @ Y2 @ X22 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_443_prob__space_Oindep__var__compose,axiom,
    ! [M2: sigma_5515648953823433982at_nat,M1: sigma_5515648953823433982at_nat,X1: product_prod_nat_nat > product_prod_nat_nat,M22: sigma_5515648953823433982at_nat,X22: product_prod_nat_nat > product_prod_nat_nat,Y1: product_prod_nat_nat > real,N1: sigma_measure_real,Y2: product_prod_nat_nat > real,N22: sigma_measure_real] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe9184017870484869004at_nat @ M2 @ M1 @ X1 @ M22 @ X22 )
       => ( ( member4564283293661824327t_real @ Y1 @ ( sigma_1188828689629184861t_real @ M1 @ N1 ) )
         => ( ( member4564283293661824327t_real @ Y2 @ ( sigma_1188828689629184861t_real @ M22 @ N22 ) )
           => ( indepe1845147739501508449t_real @ M2 @ N1 @ ( comp_P6083776177191121141at_nat @ Y1 @ X1 ) @ N22 @ ( comp_P6083776177191121141at_nat @ Y2 @ X22 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_444_prob__space_Oindep__var__compose,axiom,
    ! [M2: sigma_9047027012034273406at_nat,M1: sigma_measure_real,X1: ( product_prod_nat_nat > product_prod_nat_nat ) > real,M22: sigma_measure_real,X22: ( product_prod_nat_nat > product_prod_nat_nat ) > real,Y1: real > extend8495563244428889912nnreal,N1: sigma_7234349610311085201nnreal,Y2: real > extend8495563244428889912nnreal,N22: sigma_7234349610311085201nnreal] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe1938972608428940447t_real @ M2 @ M1 @ X1 @ M22 @ X22 )
       => ( ( member2919562650594848410nnreal @ Y1 @ ( sigma_9017504469962657078nnreal @ M1 @ N1 ) )
         => ( ( member2919562650594848410nnreal @ Y2 @ ( sigma_9017504469962657078nnreal @ M22 @ N22 ) )
           => ( indepe1784549585806403243nnreal @ M2 @ N1 @ ( comp_r2164139168250337006at_nat @ Y1 @ X1 ) @ N22 @ ( comp_r2164139168250337006at_nat @ Y2 @ X22 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_445_prob__space_Oindep__var__compose,axiom,
    ! [M2: sigma_9047027012034273406at_nat,M1: sigma_measure_real,X1: ( product_prod_nat_nat > product_prod_nat_nat ) > real,M22: sigma_measure_real,X22: ( product_prod_nat_nat > product_prod_nat_nat ) > real,Y1: real > real,N1: sigma_measure_real,Y2: real > real,N22: sigma_measure_real] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe1938972608428940447t_real @ M2 @ M1 @ X1 @ M22 @ X22 )
       => ( ( member_real_real @ Y1 @ ( sigma_5267869275261027754l_real @ M1 @ N1 ) )
         => ( ( member_real_real @ Y2 @ ( sigma_5267869275261027754l_real @ M22 @ N22 ) )
           => ( indepe1938972608428940447t_real @ M2 @ N1 @ ( comp_r4584311727289514594at_nat @ Y1 @ X1 ) @ N22 @ ( comp_r4584311727289514594at_nat @ Y2 @ X22 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_446_prob__space_Oindep__var__compose,axiom,
    ! [M2: sigma_5515648953823433982at_nat,M1: sigma_measure_real,X1: product_prod_nat_nat > real,M22: sigma_measure_real,X22: product_prod_nat_nat > real,Y1: real > extend8495563244428889912nnreal,N1: sigma_7234349610311085201nnreal,Y2: real > extend8495563244428889912nnreal,N22: sigma_7234349610311085201nnreal] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe1845147739501508449t_real @ M2 @ M1 @ X1 @ M22 @ X22 )
       => ( ( member2919562650594848410nnreal @ Y1 @ ( sigma_9017504469962657078nnreal @ M1 @ N1 ) )
         => ( ( member2919562650594848410nnreal @ Y2 @ ( sigma_9017504469962657078nnreal @ M22 @ N22 ) )
           => ( indepe8239858677677355629nnreal @ M2 @ N1 @ ( comp_r868043356317784952at_nat @ Y1 @ X1 ) @ N22 @ ( comp_r868043356317784952at_nat @ Y2 @ X22 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_447_prob__space_Oindep__var__compose,axiom,
    ! [M2: sigma_5515648953823433982at_nat,M1: sigma_measure_real,X1: product_prod_nat_nat > real,M22: sigma_measure_real,X22: product_prod_nat_nat > real,Y1: real > real,N1: sigma_measure_real,Y2: real > real,N22: sigma_measure_real] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe1845147739501508449t_real @ M2 @ M1 @ X1 @ M22 @ X22 )
       => ( ( member_real_real @ Y1 @ ( sigma_5267869275261027754l_real @ M1 @ N1 ) )
         => ( ( member_real_real @ Y2 @ ( sigma_5267869275261027754l_real @ M22 @ N22 ) )
           => ( indepe1845147739501508449t_real @ M2 @ N1 @ ( comp_r3517367148340353796at_nat @ Y1 @ X1 ) @ N22 @ ( comp_r3517367148340353796at_nat @ Y2 @ X22 ) ) ) ) ) ) ).

% prob_space.indep_var_compose
thf(fact_448_prob__space_Oindep__vars__compose,axiom,
    ! [M2: sigma_5515648953823433982at_nat,M: rat > sigma_measure_real,X: rat > product_prod_nat_nat > real,I: set_rat,Y: rat > real > extend8495563244428889912nnreal,N: rat > sigma_7234349610311085201nnreal] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe5387908695526973631t_real @ M2 @ M @ X @ I )
       => ( ! [I2: rat] :
              ( ( member_rat @ I2 @ I )
             => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( indepe317362771116669643nnreal @ M2 @ N
            @ ^ [I3: rat] : ( comp_r868043356317784952at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_449_prob__space_Oindep__vars__compose,axiom,
    ! [M2: sigma_5515648953823433982at_nat,M: $o > sigma_measure_real,X: $o > product_prod_nat_nat > real,I: set_o,Y: $o > real > extend8495563244428889912nnreal,N: $o > sigma_7234349610311085201nnreal] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe6599791682495051723o_real @ M2 @ M @ X @ I )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I )
             => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( indepe7306336255242631127nnreal @ M2 @ N
            @ ^ [I3: $o] : ( comp_r868043356317784952at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_450_prob__space_Oindep__vars__compose,axiom,
    ! [M2: sigma_5515648953823433982at_nat,M: rat > sigma_5515648953823433982at_nat,X: rat > product_prod_nat_nat > product_prod_nat_nat,I: set_rat,Y: rat > product_prod_nat_nat > real,N: rat > sigma_measure_real] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe2321685950664944046at_nat @ M2 @ M @ X @ I )
       => ( ! [I2: rat] :
              ( ( member_rat @ I2 @ I )
             => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( indepe5387908695526973631t_real @ M2 @ N
            @ ^ [I3: rat] : ( comp_P6083776177191121141at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_451_prob__space_Oindep__vars__compose,axiom,
    ! [M2: sigma_5515648953823433982at_nat,M: $o > sigma_5515648953823433982at_nat,X: $o > product_prod_nat_nat > product_prod_nat_nat,I: set_o,Y: $o > product_prod_nat_nat > real,N: $o > sigma_measure_real] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe504693989766925730at_nat @ M2 @ M @ X @ I )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I )
             => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( indepe6599791682495051723o_real @ M2 @ N
            @ ^ [I3: $o] : ( comp_P6083776177191121141at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_452_prob__space_Oindep__vars__compose,axiom,
    ! [M2: sigma_5515648953823433982at_nat,M: ( real > extend8495563244428889912nnreal ) > sigma_measure_real,X: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > real,I: set_re5328672808648366137nnreal,Y: ( real > extend8495563244428889912nnreal ) > real > extend8495563244428889912nnreal,N: ( real > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe579324506763905386l_real @ M2 @ M @ X @ I )
       => ( ! [I2: real > extend8495563244428889912nnreal] :
              ( ( member2919562650594848410nnreal @ I2 @ I )
             => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( indepe3243683517584732406nnreal @ M2 @ N
            @ ^ [I3: real > extend8495563244428889912nnreal] : ( comp_r868043356317784952at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_453_prob__space_Oindep__vars__compose,axiom,
    ! [M2: sigma_9047027012034273406at_nat,M: rat > sigma_measure_real,X: rat > ( product_prod_nat_nat > product_prod_nat_nat ) > real,I: set_rat,Y: rat > real > extend8495563244428889912nnreal,N: rat > sigma_7234349610311085201nnreal] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe4371592291047002227t_real @ M2 @ M @ X @ I )
       => ( ! [I2: rat] :
              ( ( member_rat @ I2 @ I )
             => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( indepe4547145070500471935nnreal @ M2 @ N
            @ ^ [I3: rat] : ( comp_r2164139168250337006at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_454_prob__space_Oindep__vars__compose,axiom,
    ! [M2: sigma_9047027012034273406at_nat,M: $o > sigma_measure_real,X: $o > ( product_prod_nat_nat > product_prod_nat_nat ) > real,I: set_o,Y: $o > real > extend8495563244428889912nnreal,N: $o > sigma_7234349610311085201nnreal] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe2405714897255445271o_real @ M2 @ M @ X @ I )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I )
             => ( member2919562650594848410nnreal @ ( Y @ I2 ) @ ( sigma_9017504469962657078nnreal @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( indepe2783168947688421667nnreal @ M2 @ N
            @ ^ [I3: $o] : ( comp_r2164139168250337006at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_455_prob__space_Oindep__vars__compose,axiom,
    ! [M2: sigma_5515648953823433982at_nat,M: rat > sigma_5515648953823433982at_nat,X: rat > product_prod_nat_nat > product_prod_nat_nat,I: set_rat,Y: rat > product_prod_nat_nat > product_prod_nat_nat,N: rat > sigma_5515648953823433982at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe2321685950664944046at_nat @ M2 @ M @ X @ I )
       => ( ! [I2: rat] :
              ( ( member_rat @ I2 @ I )
             => ( member8885076297122219836at_nat @ ( Y @ I2 ) @ ( sigma_3682087626466204304at_nat @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( indepe2321685950664944046at_nat @ M2 @ N
            @ ^ [I3: rat] : ( comp_P2240441846945064862at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_456_prob__space_Oindep__vars__compose,axiom,
    ! [M2: sigma_5515648953823433982at_nat,M: $o > sigma_5515648953823433982at_nat,X: $o > product_prod_nat_nat > product_prod_nat_nat,I: set_o,Y: $o > product_prod_nat_nat > product_prod_nat_nat,N: $o > sigma_5515648953823433982at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe504693989766925730at_nat @ M2 @ M @ X @ I )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I )
             => ( member8885076297122219836at_nat @ ( Y @ I2 ) @ ( sigma_3682087626466204304at_nat @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( indepe504693989766925730at_nat @ M2 @ N
            @ ^ [I3: $o] : ( comp_P2240441846945064862at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_457_prob__space_Oindep__vars__compose,axiom,
    ! [M2: sigma_5515648953823433982at_nat,M: ( real > extend8495563244428889912nnreal ) > sigma_5515648953823433982at_nat,X: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > product_prod_nat_nat,I: set_re5328672808648366137nnreal,Y: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > real,N: ( real > extend8495563244428889912nnreal ) > sigma_measure_real] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe3593890525012728259at_nat @ M2 @ M @ X @ I )
       => ( ! [I2: real > extend8495563244428889912nnreal] :
              ( ( member2919562650594848410nnreal @ I2 @ I )
             => ( member4564283293661824327t_real @ ( Y @ I2 ) @ ( sigma_1188828689629184861t_real @ ( M @ I2 ) @ ( N @ I2 ) ) ) )
         => ( indepe579324506763905386l_real @ M2 @ N
            @ ^ [I3: real > extend8495563244428889912nnreal] : ( comp_P6083776177191121141at_nat @ ( Y @ I3 ) @ ( X @ I3 ) )
            @ I ) ) ) ) ).

% prob_space.indep_vars_compose
thf(fact_458__092_060Omega_062_092_060_094sub_0622_Oobtain__positive__integrable__function,axiom,
    ~ ! [F3: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
        ( ( member8159409068225774087t_real @ F3 @ ( sigma_5064276549609164707t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ borel_5078946678739801102l_real ) )
       => ( ! [X7: product_prod_nat_nat > product_prod_nat_nat] : ( ord_less_real @ zero_zero_real @ ( F3 @ X7 ) )
         => ( ! [X7: product_prod_nat_nat > product_prod_nat_nat] : ( ord_less_eq_real @ ( F3 @ X7 ) @ one_one_real )
           => ~ ( bochne7117808529828525605t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F3 ) ) ) ) ).

% \<Omega>\<^sub>2.obtain_positive_integrable_function
thf(fact_459_prob__space_Omutual__information__def,axiom,
    ! [M2: sigma_9047027012034273406at_nat,B: real,S2: sigma_measure_nat,T: sigma_measure_nat,X: ( product_prod_nat_nat > product_prod_nat_nat ) > nat,Y: ( product_prod_nat_nat > product_prod_nat_nat ) > nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( prob_m3062007823079647199at_nat @ M2 @ B @ S2 @ T @ X @ Y )
        = ( kL_div6543510193410477195at_nat @ B @ ( binary6458575275919672938at_nat @ ( measur6421871825836039407at_nat @ M2 @ S2 @ X ) @ ( measur6421871825836039407at_nat @ M2 @ T @ Y ) )
          @ ( measur4728020197420827938at_nat @ M2 @ ( binary6458575275919672938at_nat @ S2 @ T )
            @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( product_Pair_nat_nat @ ( X @ X2 ) @ ( Y @ X2 ) ) ) ) ) ) ).

% prob_space.mutual_information_def
thf(fact_460_prob__space_Omutual__information__def,axiom,
    ! [M2: sigma_5515648953823433982at_nat,B: real,S2: sigma_measure_nat,T: sigma_measure_nat,X: product_prod_nat_nat > nat,Y: product_prod_nat_nat > nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( prob_m7791635597327167627at_nat @ M2 @ B @ S2 @ T @ X @ Y )
        = ( kL_div6543510193410477195at_nat @ B @ ( binary6458575275919672938at_nat @ ( measur843052931239516761at_nat @ M2 @ S2 @ X ) @ ( measur843052931239516761at_nat @ M2 @ T @ Y ) )
          @ ( measur6503400250167368504at_nat @ M2 @ ( binary6458575275919672938at_nat @ S2 @ T )
            @ ^ [X2: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( X @ X2 ) @ ( Y @ X2 ) ) ) ) ) ) ).

% prob_space.mutual_information_def
thf(fact_461__092_060Omega_062_092_060_094sub_0622_OLp__subset__Lq_I4_J,axiom,
    ! [P2: extend8495563244428889912nnreal,Q2: extend8495563244428889912nnreal,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( ord_le3935885782089961368nnreal @ P2 @ Q2 )
     => ( ( member8159409068225774087t_real @ F2 @ ( functi8590574888837585149t_real @ ( lp_spa812643495979991928at_nat @ Q2 @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) ) )
       => ( ord_less_eq_real @ ( functi7696537667341841001t_real @ ( lp_spa812643495979991928at_nat @ P2 @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) @ F2 ) @ ( functi7696537667341841001t_real @ ( lp_spa812643495979991928at_nat @ Q2 @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) @ F2 ) ) ) ) ).

% \<Omega>\<^sub>2.Lp_subset_Lq(4)
thf(fact_462__092_060Omega_062_092_060_094sub_0622_Osubprob__emeasure__le__1,axiom,
    ! [X: set_Pr2458342521480944603at_nat] : ( ord_le3935885782089961368nnreal @ ( sigma_3593834561172453006at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ X ) @ one_on2969667320475766781nnreal ) ).

% \<Omega>\<^sub>2.subprob_emeasure_le_1
thf(fact_463__092_060Omega_062_092_060_094sub_0622_Oemeasure__le__1,axiom,
    ! [S2: set_Pr2458342521480944603at_nat] : ( ord_le3935885782089961368nnreal @ ( sigma_3593834561172453006at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ S2 ) @ one_on2969667320475766781nnreal ) ).

% \<Omega>\<^sub>2.emeasure_le_1
thf(fact_464__092_060Omega_062_092_060_094sub_0622_Oemeasure__ge__1__iff,axiom,
    ! [A2: set_Pr2458342521480944603at_nat] :
      ( ( ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ ( sigma_3593834561172453006at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 ) )
      = ( ( sigma_3593834561172453006at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 )
        = one_on2969667320475766781nnreal ) ) ).

% \<Omega>\<^sub>2.emeasure_ge_1_iff
thf(fact_465_indep__sets__mono__sets,axiom,
    ! [F: rat > set_se7855581050983116737at_nat,I: set_rat,G: rat > set_se7855581050983116737at_nat] :
      ( ( indepe4562553564394819693at_rat @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( ord_le2077887516847798113at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
       => ( indepe4562553564394819693at_rat @ ( freque5010624893710627907mega_1 @ as ) @ G @ I ) ) ) ).

% indep_sets_mono_sets
thf(fact_466_indep__sets__mono__sets,axiom,
    ! [F: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se7855581050983116737at_nat,I: set_Pr2458342521480944603at_nat,G: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se7855581050983116737at_nat] :
      ( ( indepe6545576254369104698at_nat @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
     => ( ! [I2: product_prod_nat_nat > product_prod_nat_nat] :
            ( ( member8885076297122219836at_nat @ I2 @ I )
           => ( ord_le2077887516847798113at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
       => ( indepe6545576254369104698at_nat @ ( freque5010624893710627907mega_1 @ as ) @ G @ I ) ) ) ).

% indep_sets_mono_sets
thf(fact_467_indep__sets__mono__sets,axiom,
    ! [F: ( product_prod_nat_nat > real ) > set_se7855581050983116737at_nat,I: set_Pr1002607673312053630t_real,G: ( product_prod_nat_nat > real ) > set_se7855581050983116737at_nat] :
      ( ( indepe3818579677853040073t_real @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
     => ( ! [I2: product_prod_nat_nat > real] :
            ( ( member4564283293661824327t_real @ I2 @ I )
           => ( ord_le2077887516847798113at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
       => ( indepe3818579677853040073t_real @ ( freque5010624893710627907mega_1 @ as ) @ G @ I ) ) ) ).

% indep_sets_mono_sets
thf(fact_468_indep__sets__mono__sets,axiom,
    ! [F: ( real > extend8495563244428889912nnreal ) > set_se7855581050983116737at_nat,I: set_re5328672808648366137nnreal,G: ( real > extend8495563244428889912nnreal ) > set_se7855581050983116737at_nat] :
      ( ( indepe334513521632759960nnreal @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( ord_le2077887516847798113at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
       => ( indepe334513521632759960nnreal @ ( freque5010624893710627907mega_1 @ as ) @ G @ I ) ) ) ).

% indep_sets_mono_sets
thf(fact_469_indep__sets__mono__sets,axiom,
    ! [F: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_se7855581050983116737at_nat,I: set_Pr947837736998463782t_real,G: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_se7855581050983116737at_nat] :
      ( ( indepe2662975882062776837t_real @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
     => ( ! [I2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
            ( ( member8159409068225774087t_real @ I2 @ I )
           => ( ord_le2077887516847798113at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
       => ( indepe2662975882062776837t_real @ ( freque5010624893710627907mega_1 @ as ) @ G @ I ) ) ) ).

% indep_sets_mono_sets
thf(fact_470_indep__sets__mono__sets,axiom,
    ! [F: $o > set_se7855581050983116737at_nat,I: set_o,G: $o > set_se7855581050983116737at_nat] :
      ( ( indepe619418257930843587_nat_o @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( ord_le2077887516847798113at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
       => ( indepe619418257930843587_nat_o @ ( freque5010624893710627907mega_1 @ as ) @ G @ I ) ) ) ).

% indep_sets_mono_sets
thf(fact_471_indep__sets__cong,axiom,
    ! [I: set_rat,J: set_rat,F: rat > set_se7855581050983116737at_nat,G: rat > set_se7855581050983116737at_nat] :
      ( ( I = J )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( ( F @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe4562553564394819693at_rat @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
          = ( indepe4562553564394819693at_rat @ ( freque5010624893710627907mega_1 @ as ) @ G @ J ) ) ) ) ).

% indep_sets_cong
thf(fact_472_indep__sets__cong,axiom,
    ! [I: set_Pr2458342521480944603at_nat,J: set_Pr2458342521480944603at_nat,F: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se7855581050983116737at_nat,G: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se7855581050983116737at_nat] :
      ( ( I = J )
     => ( ! [I2: product_prod_nat_nat > product_prod_nat_nat] :
            ( ( member8885076297122219836at_nat @ I2 @ I )
           => ( ( F @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe6545576254369104698at_nat @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
          = ( indepe6545576254369104698at_nat @ ( freque5010624893710627907mega_1 @ as ) @ G @ J ) ) ) ) ).

% indep_sets_cong
thf(fact_473_indep__sets__cong,axiom,
    ! [I: set_Pr1002607673312053630t_real,J: set_Pr1002607673312053630t_real,F: ( product_prod_nat_nat > real ) > set_se7855581050983116737at_nat,G: ( product_prod_nat_nat > real ) > set_se7855581050983116737at_nat] :
      ( ( I = J )
     => ( ! [I2: product_prod_nat_nat > real] :
            ( ( member4564283293661824327t_real @ I2 @ I )
           => ( ( F @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe3818579677853040073t_real @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
          = ( indepe3818579677853040073t_real @ ( freque5010624893710627907mega_1 @ as ) @ G @ J ) ) ) ) ).

% indep_sets_cong
thf(fact_474_indep__sets__cong,axiom,
    ! [I: set_re5328672808648366137nnreal,J: set_re5328672808648366137nnreal,F: ( real > extend8495563244428889912nnreal ) > set_se7855581050983116737at_nat,G: ( real > extend8495563244428889912nnreal ) > set_se7855581050983116737at_nat] :
      ( ( I = J )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( ( F @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe334513521632759960nnreal @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
          = ( indepe334513521632759960nnreal @ ( freque5010624893710627907mega_1 @ as ) @ G @ J ) ) ) ) ).

% indep_sets_cong
thf(fact_475_indep__sets__cong,axiom,
    ! [I: set_Pr947837736998463782t_real,J: set_Pr947837736998463782t_real,F: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_se7855581050983116737at_nat,G: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_se7855581050983116737at_nat] :
      ( ( I = J )
     => ( ! [I2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
            ( ( member8159409068225774087t_real @ I2 @ I )
           => ( ( F @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe2662975882062776837t_real @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
          = ( indepe2662975882062776837t_real @ ( freque5010624893710627907mega_1 @ as ) @ G @ J ) ) ) ) ).

% indep_sets_cong
thf(fact_476_indep__sets__cong,axiom,
    ! [I: set_o,J: set_o,F: $o > set_se7855581050983116737at_nat,G: $o > set_se7855581050983116737at_nat] :
      ( ( I = J )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( ( F @ I2 )
              = ( G @ I2 ) ) )
       => ( ( indepe619418257930843587_nat_o @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
          = ( indepe619418257930843587_nat_o @ ( freque5010624893710627907mega_1 @ as ) @ G @ J ) ) ) ) ).

% indep_sets_cong
thf(fact_477_indep__sets__mono,axiom,
    ! [F: rat > set_se7855581050983116737at_nat,I: set_rat,J: set_rat,G: rat > set_se7855581050983116737at_nat] :
      ( ( indepe4562553564394819693at_rat @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
     => ( ( ord_less_eq_set_rat @ J @ I )
       => ( ! [I2: rat] :
              ( ( member_rat @ I2 @ J )
             => ( ord_le2077887516847798113at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
         => ( indepe4562553564394819693at_rat @ ( freque5010624893710627907mega_1 @ as ) @ G @ J ) ) ) ) ).

% indep_sets_mono
thf(fact_478_indep__sets__mono,axiom,
    ! [F: ( real > extend8495563244428889912nnreal ) > set_se7855581050983116737at_nat,I: set_re5328672808648366137nnreal,J: set_re5328672808648366137nnreal,G: ( real > extend8495563244428889912nnreal ) > set_se7855581050983116737at_nat] :
      ( ( indepe334513521632759960nnreal @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
     => ( ( ord_le2462468573666744473nnreal @ J @ I )
       => ( ! [I2: real > extend8495563244428889912nnreal] :
              ( ( member2919562650594848410nnreal @ I2 @ J )
             => ( ord_le2077887516847798113at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
         => ( indepe334513521632759960nnreal @ ( freque5010624893710627907mega_1 @ as ) @ G @ J ) ) ) ) ).

% indep_sets_mono
thf(fact_479_indep__sets__mono,axiom,
    ! [F: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se7855581050983116737at_nat,I: set_Pr2458342521480944603at_nat,J: set_Pr2458342521480944603at_nat,G: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se7855581050983116737at_nat] :
      ( ( indepe6545576254369104698at_nat @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
     => ( ( ord_le3857079194666040379at_nat @ J @ I )
       => ( ! [I2: product_prod_nat_nat > product_prod_nat_nat] :
              ( ( member8885076297122219836at_nat @ I2 @ J )
             => ( ord_le2077887516847798113at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
         => ( indepe6545576254369104698at_nat @ ( freque5010624893710627907mega_1 @ as ) @ G @ J ) ) ) ) ).

% indep_sets_mono
thf(fact_480_indep__sets__mono,axiom,
    ! [F: ( product_prod_nat_nat > real ) > set_se7855581050983116737at_nat,I: set_Pr1002607673312053630t_real,J: set_Pr1002607673312053630t_real,G: ( product_prod_nat_nat > real ) > set_se7855581050983116737at_nat] :
      ( ( indepe3818579677853040073t_real @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
     => ( ( ord_le1994730768851292446t_real @ J @ I )
       => ( ! [I2: product_prod_nat_nat > real] :
              ( ( member4564283293661824327t_real @ I2 @ J )
             => ( ord_le2077887516847798113at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
         => ( indepe3818579677853040073t_real @ ( freque5010624893710627907mega_1 @ as ) @ G @ J ) ) ) ) ).

% indep_sets_mono
thf(fact_481_indep__sets__mono,axiom,
    ! [F: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_se7855581050983116737at_nat,I: set_Pr947837736998463782t_real,J: set_Pr947837736998463782t_real,G: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_se7855581050983116737at_nat] :
      ( ( indepe2662975882062776837t_real @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
     => ( ( ord_le8622349663015078278t_real @ J @ I )
       => ( ! [I2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
              ( ( member8159409068225774087t_real @ I2 @ J )
             => ( ord_le2077887516847798113at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
         => ( indepe2662975882062776837t_real @ ( freque5010624893710627907mega_1 @ as ) @ G @ J ) ) ) ) ).

% indep_sets_mono
thf(fact_482_indep__sets__mono,axiom,
    ! [F: $o > set_se7855581050983116737at_nat,I: set_o,J: set_o,G: $o > set_se7855581050983116737at_nat] :
      ( ( indepe619418257930843587_nat_o @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
     => ( ( ord_less_eq_set_o @ J @ I )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ J )
             => ( ord_le2077887516847798113at_nat @ ( G @ I2 ) @ ( F @ I2 ) ) )
         => ( indepe619418257930843587_nat_o @ ( freque5010624893710627907mega_1 @ as ) @ G @ J ) ) ) ) ).

% indep_sets_mono
thf(fact_483_indep__sets__mono__index,axiom,
    ! [J: set_Pr2458342521480944603at_nat,I: set_Pr2458342521480944603at_nat,F: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se7855581050983116737at_nat] :
      ( ( ord_le3857079194666040379at_nat @ J @ I )
     => ( ( indepe6545576254369104698at_nat @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
       => ( indepe6545576254369104698at_nat @ ( freque5010624893710627907mega_1 @ as ) @ F @ J ) ) ) ).

% indep_sets_mono_index
thf(fact_484_indep__sets__mono__index,axiom,
    ! [J: set_Pr1002607673312053630t_real,I: set_Pr1002607673312053630t_real,F: ( product_prod_nat_nat > real ) > set_se7855581050983116737at_nat] :
      ( ( ord_le1994730768851292446t_real @ J @ I )
     => ( ( indepe3818579677853040073t_real @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
       => ( indepe3818579677853040073t_real @ ( freque5010624893710627907mega_1 @ as ) @ F @ J ) ) ) ).

% indep_sets_mono_index
thf(fact_485_indep__sets__mono__index,axiom,
    ! [J: set_Pr947837736998463782t_real,I: set_Pr947837736998463782t_real,F: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_se7855581050983116737at_nat] :
      ( ( ord_le8622349663015078278t_real @ J @ I )
     => ( ( indepe2662975882062776837t_real @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
       => ( indepe2662975882062776837t_real @ ( freque5010624893710627907mega_1 @ as ) @ F @ J ) ) ) ).

% indep_sets_mono_index
thf(fact_486_indep__sets__mono__index,axiom,
    ! [J: set_o,I: set_o,F: $o > set_se7855581050983116737at_nat] :
      ( ( ord_less_eq_set_o @ J @ I )
     => ( ( indepe619418257930843587_nat_o @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
       => ( indepe619418257930843587_nat_o @ ( freque5010624893710627907mega_1 @ as ) @ F @ J ) ) ) ).

% indep_sets_mono_index
thf(fact_487_Lp__subset__Lq_I3_J,axiom,
    ! [P2: extend8495563244428889912nnreal,Q2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ P2 @ Q2 )
     => ( ord_le1994730768851292446t_real @ ( functi6492146676480127185t_real @ ( lp_spa8733284142207339630at_nat @ Q2 @ ( freque5010624893710627907mega_1 @ as ) ) ) @ ( functi6492146676480127185t_real @ ( lp_spa8733284142207339630at_nat @ P2 @ ( freque5010624893710627907mega_1 @ as ) ) ) ) ) ).

% Lp_subset_Lq(3)
thf(fact_488_Lp__subset__Lq_I1_J,axiom,
    ! [P2: extend8495563244428889912nnreal,Q2: extend8495563244428889912nnreal,F2: product_prod_nat_nat > real] :
      ( ( ord_le3935885782089961368nnreal @ P2 @ Q2 )
     => ( ord_le3935885782089961368nnreal @ ( functi5516212835536045368t_real @ ( lp_spa8733284142207339630at_nat @ P2 @ ( freque5010624893710627907mega_1 @ as ) ) @ F2 ) @ ( functi5516212835536045368t_real @ ( lp_spa8733284142207339630at_nat @ Q2 @ ( freque5010624893710627907mega_1 @ as ) ) @ F2 ) ) ) ).

% Lp_subset_Lq(1)
thf(fact_489_obtain__positive__integrable__function,axiom,
    ~ ! [F3: product_prod_nat_nat > real] :
        ( ( member4564283293661824327t_real @ F3 @ ( sigma_1188828689629184861t_real @ ( freque5010624893710627907mega_1 @ as ) @ borel_5078946678739801102l_real ) )
       => ( ! [X7: product_prod_nat_nat] : ( ord_less_real @ zero_zero_real @ ( F3 @ X7 ) )
         => ( ! [X7: product_prod_nat_nat] : ( ord_less_eq_real @ ( F3 @ X7 ) @ one_one_real )
           => ~ ( bochne2596016609597520987t_real @ ( freque5010624893710627907mega_1 @ as ) @ F3 ) ) ) ) ).

% obtain_positive_integrable_function
thf(fact_490_L__zero__space_I1_J,axiom,
    ! [M2: sigma_9047027012034273406at_nat] :
      ( ( functi8590574888837585149t_real @ ( lp_spa812643495979991928at_nat @ zero_z7100319975126383169nnreal @ M2 ) )
      = ( sigma_5064276549609164707t_real @ M2 @ borel_5078946678739801102l_real ) ) ).

% L_zero_space(1)
thf(fact_491_L__zero__space_I1_J,axiom,
    ! [M2: sigma_5515648953823433982at_nat] :
      ( ( functi6492146676480127185t_real @ ( lp_spa8733284142207339630at_nat @ zero_z7100319975126383169nnreal @ M2 ) )
      = ( sigma_1188828689629184861t_real @ M2 @ borel_5078946678739801102l_real ) ) ).

% L_zero_space(1)
thf(fact_492_borel__prod,axiom,
    ( ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real )
    = borel_9155112475215227991l_real ) ).

% borel_prod
thf(fact_493_borel__prod,axiom,
    ( ( binary5786385605569495086nnreal @ borel_5078946678739801102l_real @ borel_6524799422816628122nnreal )
    = borel_4974289084073311971nnreal ) ).

% borel_prod
thf(fact_494_borel__prod,axiom,
    ( ( binary3818639336118950830l_real @ borel_6524799422816628122nnreal @ borel_5078946678739801102l_real )
    = borel_4928740784729289315l_real ) ).

% borel_prod
thf(fact_495_borel__prod,axiom,
    ( ( binary3098606844978005306nnreal @ borel_6524799422816628122nnreal @ borel_6524799422816628122nnreal )
    = borel_3951438148318096111nnreal ) ).

% borel_prod
thf(fact_496_integrableD_I1_J,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( bochne7117808529828525605t_real @ M2 @ F2 )
     => ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ M2 @ borel_5078946678739801102l_real ) ) ) ).

% integrableD(1)
thf(fact_497_integrableD_I1_J,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real] :
      ( ( bochne2596016609597520987t_real @ M2 @ F2 )
     => ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ M2 @ borel_5078946678739801102l_real ) ) ) ).

% integrableD(1)
thf(fact_498_borel__measurable__integrable,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( bochne7117808529828525605t_real @ M2 @ F2 )
     => ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ M2 @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_integrable
thf(fact_499_borel__measurable__integrable,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real] :
      ( ( bochne2596016609597520987t_real @ M2 @ F2 )
     => ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ M2 @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_integrable
thf(fact_500_L1__D_I1_J,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,M2: sigma_9047027012034273406at_nat] :
      ( ( member8159409068225774087t_real @ F2 @ ( functi8590574888837585149t_real @ ( lp_spa812643495979991928at_nat @ one_on2969667320475766781nnreal @ M2 ) ) )
     => ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ M2 @ borel_5078946678739801102l_real ) ) ) ).

% L1_D(1)
thf(fact_501_L1__D_I1_J,axiom,
    ! [F2: product_prod_nat_nat > real,M2: sigma_5515648953823433982at_nat] :
      ( ( member4564283293661824327t_real @ F2 @ ( functi6492146676480127185t_real @ ( lp_spa8733284142207339630at_nat @ one_on2969667320475766781nnreal @ M2 ) ) )
     => ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ M2 @ borel_5078946678739801102l_real ) ) ) ).

% L1_D(1)
thf(fact_502_Lp__measurable,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,P2: extend8495563244428889912nnreal,M2: sigma_9047027012034273406at_nat] :
      ( ( member8159409068225774087t_real @ F2 @ ( functi8590574888837585149t_real @ ( lp_spa812643495979991928at_nat @ P2 @ M2 ) ) )
     => ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ M2 @ borel_5078946678739801102l_real ) ) ) ).

% Lp_measurable
thf(fact_503_Lp__measurable,axiom,
    ! [F2: product_prod_nat_nat > real,P2: extend8495563244428889912nnreal,M2: sigma_5515648953823433982at_nat] :
      ( ( member4564283293661824327t_real @ F2 @ ( functi6492146676480127185t_real @ ( lp_spa8733284142207339630at_nat @ P2 @ M2 ) ) )
     => ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ M2 @ borel_5078946678739801102l_real ) ) ) ).

% Lp_measurable
thf(fact_504_borel__measurable__integrable_H,axiom,
    ! [M2: sigma_measure_real,F2: real > real,G2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,N: sigma_9047027012034273406at_nat] :
      ( ( bochne3340023020068487468l_real @ M2 @ F2 )
     => ( ( member8159409068225774087t_real @ G2 @ ( sigma_5064276549609164707t_real @ N @ M2 ) )
       => ( member8159409068225774087t_real
          @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( F2 @ ( G2 @ X2 ) )
          @ ( sigma_5064276549609164707t_real @ N @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_integrable'
thf(fact_505_borel__measurable__integrable_H,axiom,
    ! [M2: sigma_measure_real,F2: real > real,G2: product_prod_nat_nat > real,N: sigma_5515648953823433982at_nat] :
      ( ( bochne3340023020068487468l_real @ M2 @ F2 )
     => ( ( member4564283293661824327t_real @ G2 @ ( sigma_1188828689629184861t_real @ N @ M2 ) )
       => ( member4564283293661824327t_real
          @ ^ [X2: product_prod_nat_nat] : ( F2 @ ( G2 @ X2 ) )
          @ ( sigma_1188828689629184861t_real @ N @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_integrable'
thf(fact_506_borel__measurable__integrable_H,axiom,
    ! [M2: sigma_7234349610311085201nnreal,F2: extend8495563244428889912nnreal > real,G2: real > extend8495563244428889912nnreal,N: sigma_measure_real] :
      ( ( bochne9025062821074728248l_real @ M2 @ F2 )
     => ( ( member2919562650594848410nnreal @ G2 @ ( sigma_9017504469962657078nnreal @ N @ M2 ) )
       => ( member_real_real
          @ ^ [X2: real] : ( F2 @ ( G2 @ X2 ) )
          @ ( sigma_5267869275261027754l_real @ N @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_integrable'
thf(fact_507_borel__measurable__integrable_H,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,G2: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat,N: sigma_9047027012034273406at_nat] :
      ( ( bochne7117808529828525605t_real @ M2 @ F2 )
     => ( ( member2646312932426349184at_nat @ G2 @ ( sigma_1496667973943888668at_nat @ N @ M2 ) )
       => ( member8159409068225774087t_real
          @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( F2 @ ( G2 @ X2 ) )
          @ ( sigma_5064276549609164707t_real @ N @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_integrable'
thf(fact_508_borel__measurable__integrable_H,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,G2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,N: sigma_5515648953823433982at_nat] :
      ( ( bochne7117808529828525605t_real @ M2 @ F2 )
     => ( ( member8598435083969174976at_nat @ G2 @ ( sigma_7146669294626004438at_nat @ N @ M2 ) )
       => ( member4564283293661824327t_real
          @ ^ [X2: product_prod_nat_nat] : ( F2 @ ( G2 @ X2 ) )
          @ ( sigma_1188828689629184861t_real @ N @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_integrable'
thf(fact_509_borel__measurable__integrable_H,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real,G2: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,N: sigma_9047027012034273406at_nat] :
      ( ( bochne2596016609597520987t_real @ M2 @ F2 )
     => ( ( member7151659165045815932at_nat @ G2 @ ( sigma_3543877168342358986at_nat @ N @ M2 ) )
       => ( member8159409068225774087t_real
          @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( F2 @ ( G2 @ X2 ) )
          @ ( sigma_5064276549609164707t_real @ N @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_integrable'
thf(fact_510_borel__measurable__integrable_H,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real,G2: product_prod_nat_nat > product_prod_nat_nat,N: sigma_5515648953823433982at_nat] :
      ( ( bochne2596016609597520987t_real @ M2 @ F2 )
     => ( ( member8885076297122219836at_nat @ G2 @ ( sigma_3682087626466204304at_nat @ N @ M2 ) )
       => ( member4564283293661824327t_real
          @ ^ [X2: product_prod_nat_nat] : ( F2 @ ( G2 @ X2 ) )
          @ ( sigma_1188828689629184861t_real @ N @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_integrable'
thf(fact_511_integral__distr,axiom,
    ! [G2: real > extend8495563244428889912nnreal,M2: sigma_measure_real,N: sigma_7234349610311085201nnreal,F2: extend8495563244428889912nnreal > real] :
      ( ( member2919562650594848410nnreal @ G2 @ ( sigma_9017504469962657078nnreal @ M2 @ N ) )
     => ( ( member2874014351250825754l_real @ F2 @ ( sigma_7049758200512112822l_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne2458729288719820649l_real @ ( measur8829990298702910942nnreal @ M2 @ N @ G2 ) @ F2 )
          = ( bochne3715101410578510557l_real @ M2
            @ ^ [X2: real] : ( F2 @ ( G2 @ X2 ) ) ) ) ) ) ).

% integral_distr
thf(fact_512_integral__distr,axiom,
    ! [G2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,M2: sigma_9047027012034273406at_nat,N: sigma_measure_real,F2: real > real] :
      ( ( member8159409068225774087t_real @ G2 @ ( sigma_5064276549609164707t_real @ M2 @ N ) )
     => ( ( member_real_real @ F2 @ ( sigma_5267869275261027754l_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne3715101410578510557l_real @ ( measur6433597746558672971t_real @ M2 @ N @ G2 ) @ F2 )
          = ( bochne5509773249985062230t_real @ M2
            @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( F2 @ ( G2 @ X2 ) ) ) ) ) ) ).

% integral_distr
thf(fact_513_integral__distr,axiom,
    ! [G2: product_prod_nat_nat > real,M2: sigma_5515648953823433982at_nat,N: sigma_measure_real,F2: real > real] :
      ( ( member4564283293661824327t_real @ G2 @ ( sigma_1188828689629184861t_real @ M2 @ N ) )
     => ( ( member_real_real @ F2 @ ( sigma_5267869275261027754l_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne3715101410578510557l_real @ ( measur4284412874431678645t_real @ M2 @ N @ G2 ) @ F2 )
          = ( bochne6384019433803981034t_real @ M2
            @ ^ [X2: product_prod_nat_nat] : ( F2 @ ( G2 @ X2 ) ) ) ) ) ) ).

% integral_distr
thf(fact_514_integral__distr,axiom,
    ! [G2: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat,M2: sigma_9047027012034273406at_nat,N: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( member2646312932426349184at_nat @ G2 @ ( sigma_1496667973943888668at_nat @ M2 @ N ) )
     => ( ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne5509773249985062230t_real @ ( measur7731843055261497284at_nat @ M2 @ N @ G2 ) @ F2 )
          = ( bochne5509773249985062230t_real @ M2
            @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( F2 @ ( G2 @ X2 ) ) ) ) ) ) ).

% integral_distr
thf(fact_515_integral__distr,axiom,
    ! [G2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,M2: sigma_5515648953823433982at_nat,N: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( member8598435083969174976at_nat @ G2 @ ( sigma_7146669294626004438at_nat @ M2 @ N ) )
     => ( ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne5509773249985062230t_real @ ( measur8330812323704473390at_nat @ M2 @ N @ G2 ) @ F2 )
          = ( bochne6384019433803981034t_real @ M2
            @ ^ [X2: product_prod_nat_nat] : ( F2 @ ( G2 @ X2 ) ) ) ) ) ) ).

% integral_distr
thf(fact_516_integral__distr,axiom,
    ! [G2: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,M2: sigma_9047027012034273406at_nat,N: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real] :
      ( ( member7151659165045815932at_nat @ G2 @ ( sigma_3543877168342358986at_nat @ M2 @ N ) )
     => ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne6384019433803981034t_real @ ( measur4728020197420827938at_nat @ M2 @ N @ G2 ) @ F2 )
          = ( bochne5509773249985062230t_real @ M2
            @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( F2 @ ( G2 @ X2 ) ) ) ) ) ) ).

% integral_distr
thf(fact_517_integral__distr,axiom,
    ! [G2: product_prod_nat_nat > product_prod_nat_nat,M2: sigma_5515648953823433982at_nat,N: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real] :
      ( ( member8885076297122219836at_nat @ G2 @ ( sigma_3682087626466204304at_nat @ M2 @ N ) )
     => ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne6384019433803981034t_real @ ( measur6503400250167368504at_nat @ M2 @ N @ G2 ) @ F2 )
          = ( bochne6384019433803981034t_real @ M2
            @ ^ [X2: product_prod_nat_nat] : ( F2 @ ( G2 @ X2 ) ) ) ) ) ) ).

% integral_distr
thf(fact_518_integrable__distr__eq,axiom,
    ! [G2: real > extend8495563244428889912nnreal,M2: sigma_measure_real,N: sigma_7234349610311085201nnreal,F2: extend8495563244428889912nnreal > real] :
      ( ( member2919562650594848410nnreal @ G2 @ ( sigma_9017504469962657078nnreal @ M2 @ N ) )
     => ( ( member2874014351250825754l_real @ F2 @ ( sigma_7049758200512112822l_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne9025062821074728248l_real @ ( measur8829990298702910942nnreal @ M2 @ N @ G2 ) @ F2 )
          = ( bochne3340023020068487468l_real @ M2
            @ ^ [X2: real] : ( F2 @ ( G2 @ X2 ) ) ) ) ) ) ).

% integrable_distr_eq
thf(fact_519_integrable__distr__eq,axiom,
    ! [G2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,M2: sigma_9047027012034273406at_nat,N: sigma_measure_real,F2: real > real] :
      ( ( member8159409068225774087t_real @ G2 @ ( sigma_5064276549609164707t_real @ M2 @ N ) )
     => ( ( member_real_real @ F2 @ ( sigma_5267869275261027754l_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne3340023020068487468l_real @ ( measur6433597746558672971t_real @ M2 @ N @ G2 ) @ F2 )
          = ( bochne7117808529828525605t_real @ M2
            @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( F2 @ ( G2 @ X2 ) ) ) ) ) ) ).

% integrable_distr_eq
thf(fact_520_integrable__distr__eq,axiom,
    ! [G2: product_prod_nat_nat > real,M2: sigma_5515648953823433982at_nat,N: sigma_measure_real,F2: real > real] :
      ( ( member4564283293661824327t_real @ G2 @ ( sigma_1188828689629184861t_real @ M2 @ N ) )
     => ( ( member_real_real @ F2 @ ( sigma_5267869275261027754l_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne3340023020068487468l_real @ ( measur4284412874431678645t_real @ M2 @ N @ G2 ) @ F2 )
          = ( bochne2596016609597520987t_real @ M2
            @ ^ [X2: product_prod_nat_nat] : ( F2 @ ( G2 @ X2 ) ) ) ) ) ) ).

% integrable_distr_eq
thf(fact_521_integrable__distr__eq,axiom,
    ! [G2: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat,M2: sigma_9047027012034273406at_nat,N: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( member2646312932426349184at_nat @ G2 @ ( sigma_1496667973943888668at_nat @ M2 @ N ) )
     => ( ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne7117808529828525605t_real @ ( measur7731843055261497284at_nat @ M2 @ N @ G2 ) @ F2 )
          = ( bochne7117808529828525605t_real @ M2
            @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( F2 @ ( G2 @ X2 ) ) ) ) ) ) ).

% integrable_distr_eq
thf(fact_522_integrable__distr__eq,axiom,
    ! [G2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,M2: sigma_5515648953823433982at_nat,N: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( member8598435083969174976at_nat @ G2 @ ( sigma_7146669294626004438at_nat @ M2 @ N ) )
     => ( ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne7117808529828525605t_real @ ( measur8330812323704473390at_nat @ M2 @ N @ G2 ) @ F2 )
          = ( bochne2596016609597520987t_real @ M2
            @ ^ [X2: product_prod_nat_nat] : ( F2 @ ( G2 @ X2 ) ) ) ) ) ) ).

% integrable_distr_eq
thf(fact_523_integrable__distr__eq,axiom,
    ! [G2: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,M2: sigma_9047027012034273406at_nat,N: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real] :
      ( ( member7151659165045815932at_nat @ G2 @ ( sigma_3543877168342358986at_nat @ M2 @ N ) )
     => ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne2596016609597520987t_real @ ( measur4728020197420827938at_nat @ M2 @ N @ G2 ) @ F2 )
          = ( bochne7117808529828525605t_real @ M2
            @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( F2 @ ( G2 @ X2 ) ) ) ) ) ) ).

% integrable_distr_eq
thf(fact_524_integrable__distr__eq,axiom,
    ! [G2: product_prod_nat_nat > product_prod_nat_nat,M2: sigma_5515648953823433982at_nat,N: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real] :
      ( ( member8885076297122219836at_nat @ G2 @ ( sigma_3682087626466204304at_nat @ M2 @ N ) )
     => ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ N @ borel_5078946678739801102l_real ) )
       => ( ( bochne2596016609597520987t_real @ ( measur6503400250167368504at_nat @ M2 @ N @ G2 ) @ F2 )
          = ( bochne2596016609597520987t_real @ M2
            @ ^ [X2: product_prod_nat_nat] : ( F2 @ ( G2 @ X2 ) ) ) ) ) ) ).

% integrable_distr_eq
thf(fact_525_Lp__measurable__subset,axiom,
    ! [P2: extend8495563244428889912nnreal,M2: sigma_9047027012034273406at_nat] : ( ord_le8622349663015078278t_real @ ( functi8590574888837585149t_real @ ( lp_spa812643495979991928at_nat @ P2 @ M2 ) ) @ ( sigma_5064276549609164707t_real @ M2 @ borel_5078946678739801102l_real ) ) ).

% Lp_measurable_subset
thf(fact_526_Lp__measurable__subset,axiom,
    ! [P2: extend8495563244428889912nnreal,M2: sigma_5515648953823433982at_nat] : ( ord_le1994730768851292446t_real @ ( functi6492146676480127185t_real @ ( lp_spa8733284142207339630at_nat @ P2 @ M2 ) ) @ ( sigma_1188828689629184861t_real @ M2 @ borel_5078946678739801102l_real ) ) ).

% Lp_measurable_subset
thf(fact_527_prob__space_OLp__subset__Lq_I4_J,axiom,
    ! [M2: sigma_9047027012034273406at_nat,P2: extend8495563244428889912nnreal,Q2: extend8495563244428889912nnreal,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( ord_le3935885782089961368nnreal @ P2 @ Q2 )
       => ( ( member8159409068225774087t_real @ F2 @ ( functi8590574888837585149t_real @ ( lp_spa812643495979991928at_nat @ Q2 @ M2 ) ) )
         => ( ord_less_eq_real @ ( functi7696537667341841001t_real @ ( lp_spa812643495979991928at_nat @ P2 @ M2 ) @ F2 ) @ ( functi7696537667341841001t_real @ ( lp_spa812643495979991928at_nat @ Q2 @ M2 ) @ F2 ) ) ) ) ) ).

% prob_space.Lp_subset_Lq(4)
thf(fact_528_prob__space_OLp__subset__Lq_I4_J,axiom,
    ! [M2: sigma_5515648953823433982at_nat,P2: extend8495563244428889912nnreal,Q2: extend8495563244428889912nnreal,F2: product_prod_nat_nat > real] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( ord_le3935885782089961368nnreal @ P2 @ Q2 )
       => ( ( member4564283293661824327t_real @ F2 @ ( functi6492146676480127185t_real @ ( lp_spa8733284142207339630at_nat @ Q2 @ M2 ) ) )
         => ( ord_less_eq_real @ ( functi1179182286487682149t_real @ ( lp_spa8733284142207339630at_nat @ P2 @ M2 ) @ F2 ) @ ( functi1179182286487682149t_real @ ( lp_spa8733284142207339630at_nat @ Q2 @ M2 ) @ F2 ) ) ) ) ) ).

% prob_space.Lp_subset_Lq(4)
thf(fact_529_sigma__finite__pair__measure,axiom,
    ! [A2: sigma_measure_nat,B2: sigma_measure_nat] :
      ( ( measur8258956421386577775re_nat @ A2 )
     => ( ( measur8258956421386577775re_nat @ B2 )
       => ( measur1424910518806321826at_nat @ ( binary6458575275919672938at_nat @ A2 @ B2 ) ) ) ) ).

% sigma_finite_pair_measure
thf(fact_530_sigma__finite__pair__measure,axiom,
    ! [A2: sigma_9047027012034273406at_nat,B2: sigma_9047027012034273406at_nat] :
      ( ( measur1801591924826899012at_nat @ A2 )
     => ( ( measur1801591924826899012at_nat @ B2 )
       => ( measur1587578714913708492at_nat @ ( binary1747650524874713620at_nat @ A2 @ B2 ) ) ) ) ).

% sigma_finite_pair_measure
thf(fact_531_sigma__finite__pair__measure,axiom,
    ! [A2: sigma_9047027012034273406at_nat,B2: sigma_5515648953823433982at_nat] :
      ( ( measur1801591924826899012at_nat @ A2 )
     => ( ( measur1424910518806321826at_nat @ B2 )
       => ( measur4892767040697796656at_nat @ ( binary7478230798343962322at_nat @ A2 @ B2 ) ) ) ) ).

% sigma_finite_pair_measure
thf(fact_532_sigma__finite__pair__measure,axiom,
    ! [A2: sigma_5515648953823433982at_nat,B2: sigma_9047027012034273406at_nat] :
      ( ( measur1424910518806321826at_nat @ A2 )
     => ( ( measur1801591924826899012at_nat @ B2 )
       => ( measur6339542959621155700at_nat @ ( binary1857650887772831966at_nat @ A2 @ B2 ) ) ) ) ).

% sigma_finite_pair_measure
thf(fact_533_sigma__finite__pair__measure,axiom,
    ! [A2: sigma_5515648953823433982at_nat,B2: sigma_5515648953823433982at_nat] :
      ( ( measur1424910518806321826at_nat @ A2 )
     => ( ( measur1424910518806321826at_nat @ B2 )
       => ( measur3407119333870111880at_nat @ ( binary8350690942515918216at_nat @ A2 @ B2 ) ) ) ) ).

% sigma_finite_pair_measure
thf(fact_534_sigma__finite__measure__distr,axiom,
    ! [M2: sigma_measure_real,N: sigma_7234349610311085201nnreal,F2: real > extend8495563244428889912nnreal] :
      ( ( measur6426964080664357591nnreal @ ( measur8829990298702910942nnreal @ M2 @ N @ F2 ) )
     => ( ( member2919562650594848410nnreal @ F2 @ ( sigma_9017504469962657078nnreal @ M2 @ N ) )
       => ( measur487378040549452491e_real @ M2 ) ) ) ).

% sigma_finite_measure_distr
thf(fact_535_sigma__finite__measure__distr,axiom,
    ! [M2: sigma_9047027012034273406at_nat,N: sigma_measure_real,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( measur487378040549452491e_real @ ( measur6433597746558672971t_real @ M2 @ N @ F2 ) )
     => ( ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ M2 @ N ) )
       => ( measur1801591924826899012at_nat @ M2 ) ) ) ).

% sigma_finite_measure_distr
thf(fact_536_sigma__finite__measure__distr,axiom,
    ! [M2: sigma_5515648953823433982at_nat,N: sigma_measure_real,F2: product_prod_nat_nat > real] :
      ( ( measur487378040549452491e_real @ ( measur4284412874431678645t_real @ M2 @ N @ F2 ) )
     => ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ M2 @ N ) )
       => ( measur1424910518806321826at_nat @ M2 ) ) ) ).

% sigma_finite_measure_distr
thf(fact_537_sigma__finite__measure__distr,axiom,
    ! [M2: sigma_9047027012034273406at_nat,N: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( measur1801591924826899012at_nat @ ( measur7731843055261497284at_nat @ M2 @ N @ F2 ) )
     => ( ( member2646312932426349184at_nat @ F2 @ ( sigma_1496667973943888668at_nat @ M2 @ N ) )
       => ( measur1801591924826899012at_nat @ M2 ) ) ) ).

% sigma_finite_measure_distr
thf(fact_538_sigma__finite__measure__distr,axiom,
    ! [M2: sigma_5515648953823433982at_nat,N: sigma_9047027012034273406at_nat,F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( measur1801591924826899012at_nat @ ( measur8330812323704473390at_nat @ M2 @ N @ F2 ) )
     => ( ( member8598435083969174976at_nat @ F2 @ ( sigma_7146669294626004438at_nat @ M2 @ N ) )
       => ( measur1424910518806321826at_nat @ M2 ) ) ) ).

% sigma_finite_measure_distr
thf(fact_539_sigma__finite__measure__distr,axiom,
    ! [M2: sigma_9047027012034273406at_nat,N: sigma_5515648953823433982at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat] :
      ( ( measur1424910518806321826at_nat @ ( measur4728020197420827938at_nat @ M2 @ N @ F2 ) )
     => ( ( member7151659165045815932at_nat @ F2 @ ( sigma_3543877168342358986at_nat @ M2 @ N ) )
       => ( measur1801591924826899012at_nat @ M2 ) ) ) ).

% sigma_finite_measure_distr
thf(fact_540_sigma__finite__measure__distr,axiom,
    ! [M2: sigma_5515648953823433982at_nat,N: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( measur1424910518806321826at_nat @ ( measur6503400250167368504at_nat @ M2 @ N @ F2 ) )
     => ( ( member8885076297122219836at_nat @ F2 @ ( sigma_3682087626466204304at_nat @ M2 @ N ) )
       => ( measur1424910518806321826at_nat @ M2 ) ) ) ).

% sigma_finite_measure_distr
thf(fact_541_sigma__finite__measure_OKL__same__eq__0,axiom,
    ! [M2: sigma_9047027012034273406at_nat,B: real] :
      ( ( measur1801591924826899012at_nat @ M2 )
     => ( ( kL_div5695791497786497691at_nat @ B @ M2 @ M2 )
        = zero_zero_real ) ) ).

% sigma_finite_measure.KL_same_eq_0
thf(fact_542_sigma__finite__measure_OKL__same__eq__0,axiom,
    ! [M2: sigma_5515648953823433982at_nat,B: real] :
      ( ( measur1424910518806321826at_nat @ M2 )
     => ( ( kL_div6543510193410477195at_nat @ B @ M2 @ M2 )
        = zero_zero_real ) ) ).

% sigma_finite_measure.KL_same_eq_0
thf(fact_543_measurable__Pair2__compose,axiom,
    ! [F2: product_prod_nat_nat > nat,G2: product_prod_nat_nat > nat,M2: sigma_5515648953823433982at_nat,M1: sigma_measure_nat,M22: sigma_measure_nat,H: product_prod_nat_nat > product_prod_nat_nat,N: sigma_5515648953823433982at_nat] :
      ( ( member8885076297122219836at_nat
        @ ^ [X2: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
        @ ( sigma_3682087626466204304at_nat @ M2 @ ( binary6458575275919672938at_nat @ M1 @ M22 ) ) )
     => ( ( member8885076297122219836at_nat @ H @ ( sigma_3682087626466204304at_nat @ N @ M2 ) )
       => ( member7868840539957676139at_nat
          @ ^ [X2: product_prod_nat_nat] : ( G2 @ ( H @ X2 ) )
          @ ( sigma_8719606837941807361at_nat @ N @ M22 ) ) ) ) ).

% measurable_Pair2_compose
thf(fact_544_measurable__Pair2__compose,axiom,
    ! [F2: real > nat,G2: real > nat,M2: sigma_measure_real,M1: sigma_measure_nat,M22: sigma_measure_nat,H: ( product_prod_nat_nat > product_prod_nat_nat ) > real,N: sigma_9047027012034273406at_nat] :
      ( ( member278708040391457717at_nat
        @ ^ [X2: real] : ( product_Pair_nat_nat @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
        @ ( sigma_707772041119837571at_nat @ M2 @ ( binary6458575275919672938at_nat @ M1 @ M22 ) ) )
     => ( ( member8159409068225774087t_real @ H @ ( sigma_5064276549609164707t_real @ N @ M2 ) )
       => ( member3219910559795078443at_nat
          @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( G2 @ ( H @ X2 ) )
          @ ( sigma_4532147426969876551at_nat @ N @ M22 ) ) ) ) ).

% measurable_Pair2_compose
thf(fact_545_measurable__Pair2__compose,axiom,
    ! [F2: real > nat,G2: real > nat,M2: sigma_measure_real,M1: sigma_measure_nat,M22: sigma_measure_nat,H: product_prod_nat_nat > real,N: sigma_5515648953823433982at_nat] :
      ( ( member278708040391457717at_nat
        @ ^ [X2: real] : ( product_Pair_nat_nat @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
        @ ( sigma_707772041119837571at_nat @ M2 @ ( binary6458575275919672938at_nat @ M1 @ M22 ) ) )
     => ( ( member4564283293661824327t_real @ H @ ( sigma_1188828689629184861t_real @ N @ M2 ) )
       => ( member7868840539957676139at_nat
          @ ^ [X2: product_prod_nat_nat] : ( G2 @ ( H @ X2 ) )
          @ ( sigma_8719606837941807361at_nat @ N @ M22 ) ) ) ) ).

% measurable_Pair2_compose
thf(fact_546_measurable__Pair2__compose,axiom,
    ! [F2: extend8495563244428889912nnreal > nat,G2: extend8495563244428889912nnreal > nat,M2: sigma_7234349610311085201nnreal,M1: sigma_measure_nat,M22: sigma_measure_nat,H: real > extend8495563244428889912nnreal,N: sigma_measure_real] :
      ( ( member2900893821049216297at_nat
        @ ^ [X2: extend8495563244428889912nnreal] : ( product_Pair_nat_nat @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
        @ ( sigma_7317317259004200311at_nat @ M2 @ ( binary6458575275919672938at_nat @ M1 @ M22 ) ) )
     => ( ( member2919562650594848410nnreal @ H @ ( sigma_9017504469962657078nnreal @ N @ M2 ) )
       => ( member_real_nat
          @ ^ [X2: real] : ( G2 @ ( H @ X2 ) )
          @ ( sigma_6315060578831106510al_nat @ N @ M22 ) ) ) ) ).

% measurable_Pair2_compose
thf(fact_547_measurable__Pair1__compose,axiom,
    ! [F2: product_prod_nat_nat > nat,G2: product_prod_nat_nat > nat,M2: sigma_5515648953823433982at_nat,M1: sigma_measure_nat,M22: sigma_measure_nat,H: product_prod_nat_nat > product_prod_nat_nat,N: sigma_5515648953823433982at_nat] :
      ( ( member8885076297122219836at_nat
        @ ^ [X2: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
        @ ( sigma_3682087626466204304at_nat @ M2 @ ( binary6458575275919672938at_nat @ M1 @ M22 ) ) )
     => ( ( member8885076297122219836at_nat @ H @ ( sigma_3682087626466204304at_nat @ N @ M2 ) )
       => ( member7868840539957676139at_nat
          @ ^ [X2: product_prod_nat_nat] : ( F2 @ ( H @ X2 ) )
          @ ( sigma_8719606837941807361at_nat @ N @ M1 ) ) ) ) ).

% measurable_Pair1_compose
thf(fact_548_measurable__Pair1__compose,axiom,
    ! [F2: real > nat,G2: real > nat,M2: sigma_measure_real,M1: sigma_measure_nat,M22: sigma_measure_nat,H: ( product_prod_nat_nat > product_prod_nat_nat ) > real,N: sigma_9047027012034273406at_nat] :
      ( ( member278708040391457717at_nat
        @ ^ [X2: real] : ( product_Pair_nat_nat @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
        @ ( sigma_707772041119837571at_nat @ M2 @ ( binary6458575275919672938at_nat @ M1 @ M22 ) ) )
     => ( ( member8159409068225774087t_real @ H @ ( sigma_5064276549609164707t_real @ N @ M2 ) )
       => ( member3219910559795078443at_nat
          @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( F2 @ ( H @ X2 ) )
          @ ( sigma_4532147426969876551at_nat @ N @ M1 ) ) ) ) ).

% measurable_Pair1_compose
thf(fact_549_measurable__Pair1__compose,axiom,
    ! [F2: real > nat,G2: real > nat,M2: sigma_measure_real,M1: sigma_measure_nat,M22: sigma_measure_nat,H: product_prod_nat_nat > real,N: sigma_5515648953823433982at_nat] :
      ( ( member278708040391457717at_nat
        @ ^ [X2: real] : ( product_Pair_nat_nat @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
        @ ( sigma_707772041119837571at_nat @ M2 @ ( binary6458575275919672938at_nat @ M1 @ M22 ) ) )
     => ( ( member4564283293661824327t_real @ H @ ( sigma_1188828689629184861t_real @ N @ M2 ) )
       => ( member7868840539957676139at_nat
          @ ^ [X2: product_prod_nat_nat] : ( F2 @ ( H @ X2 ) )
          @ ( sigma_8719606837941807361at_nat @ N @ M1 ) ) ) ) ).

% measurable_Pair1_compose
thf(fact_550_measurable__Pair1__compose,axiom,
    ! [F2: extend8495563244428889912nnreal > nat,G2: extend8495563244428889912nnreal > nat,M2: sigma_7234349610311085201nnreal,M1: sigma_measure_nat,M22: sigma_measure_nat,H: real > extend8495563244428889912nnreal,N: sigma_measure_real] :
      ( ( member2900893821049216297at_nat
        @ ^ [X2: extend8495563244428889912nnreal] : ( product_Pair_nat_nat @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
        @ ( sigma_7317317259004200311at_nat @ M2 @ ( binary6458575275919672938at_nat @ M1 @ M22 ) ) )
     => ( ( member2919562650594848410nnreal @ H @ ( sigma_9017504469962657078nnreal @ N @ M2 ) )
       => ( member_real_nat
          @ ^ [X2: real] : ( F2 @ ( H @ X2 ) )
          @ ( sigma_6315060578831106510al_nat @ N @ M1 ) ) ) ) ).

% measurable_Pair1_compose
thf(fact_551_measurable__Pair,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,M2: sigma_5515648953823433982at_nat,M1: sigma_5515648953823433982at_nat,G2: product_prod_nat_nat > product_prod_nat_nat,M22: sigma_5515648953823433982at_nat] :
      ( ( member8885076297122219836at_nat @ F2 @ ( sigma_3682087626466204304at_nat @ M2 @ M1 ) )
     => ( ( member8885076297122219836at_nat @ G2 @ ( sigma_3682087626466204304at_nat @ M2 @ M22 ) )
       => ( member1255236750638408610at_nat
          @ ^ [X2: product_prod_nat_nat] : ( produc6161850002892822231at_nat @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
          @ ( sigma_4220525006649896822at_nat @ M2 @ ( binary8350690942515918216at_nat @ M1 @ M22 ) ) ) ) ) ).

% measurable_Pair
thf(fact_552_measurable__Pair,axiom,
    ! [F2: product_prod_nat_nat > nat,M2: sigma_5515648953823433982at_nat,M1: sigma_measure_nat,G2: product_prod_nat_nat > nat,M22: sigma_measure_nat] :
      ( ( member7868840539957676139at_nat @ F2 @ ( sigma_8719606837941807361at_nat @ M2 @ M1 ) )
     => ( ( member7868840539957676139at_nat @ G2 @ ( sigma_8719606837941807361at_nat @ M2 @ M22 ) )
       => ( member8885076297122219836at_nat
          @ ^ [X2: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
          @ ( sigma_3682087626466204304at_nat @ M2 @ ( binary6458575275919672938at_nat @ M1 @ M22 ) ) ) ) ) ).

% measurable_Pair
thf(fact_553_measurable__Pair,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,M2: sigma_5515648953823433982at_nat,M1: sigma_5515648953823433982at_nat,G2: product_prod_nat_nat > real,M22: sigma_measure_real] :
      ( ( member8885076297122219836at_nat @ F2 @ ( sigma_3682087626466204304at_nat @ M2 @ M1 ) )
     => ( ( member4564283293661824327t_real @ G2 @ ( sigma_1188828689629184861t_real @ M2 @ M22 ) )
       => ( member7102390617368680951t_real
          @ ^ [X2: product_prod_nat_nat] : ( produc904203886716845910t_real @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
          @ ( sigma_3852182464305326733t_real @ M2 @ ( binary2981866668218350181t_real @ M1 @ M22 ) ) ) ) ) ).

% measurable_Pair
thf(fact_554_measurable__Pair,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,M2: sigma_9047027012034273406at_nat,M1: sigma_measure_real,G2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,M22: sigma_measure_real] :
      ( ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ M2 @ M1 ) )
     => ( ( member8159409068225774087t_real @ G2 @ ( sigma_5064276549609164707t_real @ M2 @ M22 ) )
       => ( member1941033684658813108l_real
          @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( produc4511245868158468465l_real @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
          @ ( sigma_4131590825631767810l_real @ M2 @ ( binary6478037234023840930l_real @ M1 @ M22 ) ) ) ) ) ).

% measurable_Pair
thf(fact_555_measurable__Pair,axiom,
    ! [F2: product_prod_nat_nat > real,M2: sigma_5515648953823433982at_nat,M1: sigma_measure_real,G2: product_prod_nat_nat > product_prod_nat_nat,M22: sigma_5515648953823433982at_nat] :
      ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ M2 @ M1 ) )
     => ( ( member8885076297122219836at_nat @ G2 @ ( sigma_3682087626466204304at_nat @ M2 @ M22 ) )
       => ( member95027015144503653at_nat
          @ ^ [X2: product_prod_nat_nat] : ( produc423147238207498620at_nat @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
          @ ( sigma_8789979247889735931at_nat @ M2 @ ( binary2500810019709002891at_nat @ M1 @ M22 ) ) ) ) ) ).

% measurable_Pair
thf(fact_556_measurable__Pair,axiom,
    ! [F2: product_prod_nat_nat > real,M2: sigma_5515648953823433982at_nat,M1: sigma_measure_real,G2: product_prod_nat_nat > real,M22: sigma_measure_real] :
      ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ M2 @ M1 ) )
     => ( ( member4564283293661824327t_real @ G2 @ ( sigma_1188828689629184861t_real @ M2 @ M22 ) )
       => ( member2619278773872280948l_real
          @ ^ [X2: product_prod_nat_nat] : ( produc4511245868158468465l_real @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
          @ ( sigma_8891564814549521864l_real @ M2 @ ( binary6478037234023840930l_real @ M1 @ M22 ) ) ) ) ) ).

% measurable_Pair
thf(fact_557_measurable__Pair,axiom,
    ! [F2: real > extend8495563244428889912nnreal,M2: sigma_measure_real,M1: sigma_7234349610311085201nnreal,G2: real > extend8495563244428889912nnreal,M22: sigma_7234349610311085201nnreal] :
      ( ( member2919562650594848410nnreal @ F2 @ ( sigma_9017504469962657078nnreal @ M2 @ M1 ) )
     => ( ( member2919562650594848410nnreal @ G2 @ ( sigma_9017504469962657078nnreal @ M2 @ M22 ) )
       => ( member7391843071750696581nnreal
          @ ^ [X2: real] : ( produc344325839068023049nnreal @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
          @ ( sigma_2101749304130238547nnreal @ M2 @ ( binary3098606844978005306nnreal @ M1 @ M22 ) ) ) ) ) ).

% measurable_Pair
thf(fact_558_measurable__fst_H,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,M2: sigma_5515648953823433982at_nat,N: sigma_measure_nat,P: sigma_measure_nat] :
      ( ( member8885076297122219836at_nat @ F2 @ ( sigma_3682087626466204304at_nat @ M2 @ ( binary6458575275919672938at_nat @ N @ P ) ) )
     => ( member7868840539957676139at_nat
        @ ^ [X2: product_prod_nat_nat] : ( product_fst_nat_nat @ ( F2 @ X2 ) )
        @ ( sigma_8719606837941807361at_nat @ M2 @ N ) ) ) ).

% measurable_fst'
thf(fact_559_finite__measure__pair__measure,axiom,
    ! [M2: sigma_measure_nat,N: sigma_measure_nat] :
      ( ( measur8338831127414845932re_nat @ M2 )
     => ( ( measur8338831127414845932re_nat @ N )
       => ( measur7911732083032432613at_nat @ ( binary6458575275919672938at_nat @ N @ M2 ) ) ) ) ).

% finite_measure_pair_measure
thf(fact_560_finite__measure__pair__measure,axiom,
    ! [M2: sigma_9047027012034273406at_nat,N: sigma_9047027012034273406at_nat] :
      ( ( measur5156515106886843841at_nat @ M2 )
     => ( ( measur5156515106886843841at_nat @ N )
       => ( measur2265838613734309135at_nat @ ( binary1747650524874713620at_nat @ N @ M2 ) ) ) ) ).

% finite_measure_pair_measure
thf(fact_561_finite__measure__pair__measure,axiom,
    ! [M2: sigma_9047027012034273406at_nat,N: sigma_5515648953823433982at_nat] :
      ( ( measur5156515106886843841at_nat @ M2 )
     => ( ( measur7911732083032432613at_nat @ N )
       => ( measur6653163171691485041at_nat @ ( binary1857650887772831966at_nat @ N @ M2 ) ) ) ) ).

% finite_measure_pair_measure
thf(fact_562_finite__measure__pair__measure,axiom,
    ! [M2: sigma_5515648953823433982at_nat,N: sigma_9047027012034273406at_nat] :
      ( ( measur7911732083032432613at_nat @ M2 )
     => ( ( measur5156515106886843841at_nat @ N )
       => ( measur5206387252768125997at_nat @ ( binary7478230798343962322at_nat @ N @ M2 ) ) ) ) ).

% finite_measure_pair_measure
thf(fact_563_finite__measure__pair__measure,axiom,
    ! [M2: sigma_5515648953823433982at_nat,N: sigma_5515648953823433982at_nat] :
      ( ( measur7911732083032432613at_nat @ M2 )
     => ( ( measur7911732083032432613at_nat @ N )
       => ( measur5936237848286085451at_nat @ ( binary8350690942515918216at_nat @ N @ M2 ) ) ) ) ).

% finite_measure_pair_measure
thf(fact_564_measurable__fst_H_H,axiom,
    ! [F2: nat > product_prod_nat_nat,M2: sigma_measure_nat,N: sigma_5515648953823433982at_nat,P: sigma_measure_nat] :
      ( ( member6743353555990375057at_nat @ F2 @ ( sigma_2856282194129746655at_nat @ M2 @ N ) )
     => ( member8885076297122219836at_nat
        @ ^ [X2: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ X2 ) )
        @ ( sigma_3682087626466204304at_nat @ ( binary6458575275919672938at_nat @ M2 @ P ) @ N ) ) ) ).

% measurable_fst''
thf(fact_565_measurable__fst_H_H,axiom,
    ! [F2: nat > real,M2: sigma_measure_nat,N: sigma_measure_real,P: sigma_measure_nat] :
      ( ( member_nat_real @ F2 @ ( sigma_1747752005702207822t_real @ M2 @ N ) )
     => ( member4564283293661824327t_real
        @ ^ [X2: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ X2 ) )
        @ ( sigma_1188828689629184861t_real @ ( binary6458575275919672938at_nat @ M2 @ P ) @ N ) ) ) ).

% measurable_fst''
thf(fact_566__092_060Omega_062_092_060_094sub_0622_Ocovar__indep__eq__zero,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,G2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( bochne7117808529828525605t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F2 )
     => ( ( bochne7117808529828525605t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ G2 )
       => ( ( indepe1938972608428940447t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ borel_5078946678739801102l_real @ F2 @ borel_5078946678739801102l_real @ G2 )
         => ( ( probab3351693472195852697t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F2 @ G2 )
            = zero_zero_real ) ) ) ) ).

% \<Omega>\<^sub>2.covar_indep_eq_zero
thf(fact_567__092_060Omega_062_092_060_094sub_0622_Oindep__var__lebesgue__integral,axiom,
    ! [X1: ( product_prod_nat_nat > product_prod_nat_nat ) > real,X22: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( indepe1938972608428940447t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ borel_5078946678739801102l_real @ X1 @ borel_5078946678739801102l_real @ X22 )
     => ( ( bochne7117808529828525605t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ X1 )
       => ( ( bochne7117808529828525605t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ X22 )
         => ( ( bochne5509773249985062230t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
              @ ^ [Omega: product_prod_nat_nat > product_prod_nat_nat] : ( times_times_real @ ( X1 @ Omega ) @ ( X22 @ Omega ) ) )
            = ( times_times_real @ ( bochne5509773249985062230t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ X1 ) @ ( bochne5509773249985062230t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ X22 ) ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_var_lebesgue_integral
thf(fact_568_covar__indep__eq__zero,axiom,
    ! [F2: product_prod_nat_nat > real,G2: product_prod_nat_nat > real] :
      ( ( bochne2596016609597520987t_real @ ( freque5010624893710627907mega_1 @ as ) @ F2 )
     => ( ( bochne2596016609597520987t_real @ ( freque5010624893710627907mega_1 @ as ) @ G2 )
       => ( ( indepe1845147739501508449t_real @ ( freque5010624893710627907mega_1 @ as ) @ borel_5078946678739801102l_real @ F2 @ borel_5078946678739801102l_real @ G2 )
         => ( ( probab7722952755062086631t_real @ ( freque5010624893710627907mega_1 @ as ) @ F2 @ G2 )
            = zero_zero_real ) ) ) ) ).

% covar_indep_eq_zero
thf(fact_569__092_060Omega_062_092_060_094sub_0622_Oindep__var__integrable,axiom,
    ! [X1: ( product_prod_nat_nat > product_prod_nat_nat ) > real,X22: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( indepe1938972608428940447t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ borel_5078946678739801102l_real @ X1 @ borel_5078946678739801102l_real @ X22 )
     => ( ( bochne7117808529828525605t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ X1 )
       => ( ( bochne7117808529828525605t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ X22 )
         => ( bochne7117808529828525605t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
            @ ^ [Omega: product_prod_nat_nat > product_prod_nat_nat] : ( times_times_real @ ( X1 @ Omega ) @ ( X22 @ Omega ) ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_var_integrable
thf(fact_570__092_060Omega_062_092_060_094sub_0622_Oemeasure__real,axiom,
    ! [A2: set_Pr2458342521480944603at_nat] :
    ? [R2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R2 )
      & ( ( sigma_3593834561172453006at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 )
        = ( extend7643940197134561352nnreal @ R2 ) ) ) ).

% \<Omega>\<^sub>2.emeasure_real
thf(fact_571_sigma__finite__measure_Oobtain__positive__integrable__function,axiom,
    ! [M2: sigma_9047027012034273406at_nat] :
      ( ( measur1801591924826899012at_nat @ M2 )
     => ~ ! [F3: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
            ( ( member8159409068225774087t_real @ F3 @ ( sigma_5064276549609164707t_real @ M2 @ borel_5078946678739801102l_real ) )
           => ( ! [X7: product_prod_nat_nat > product_prod_nat_nat] : ( ord_less_real @ zero_zero_real @ ( F3 @ X7 ) )
             => ( ! [X7: product_prod_nat_nat > product_prod_nat_nat] : ( ord_less_eq_real @ ( F3 @ X7 ) @ one_one_real )
               => ~ ( bochne7117808529828525605t_real @ M2 @ F3 ) ) ) ) ) ).

% sigma_finite_measure.obtain_positive_integrable_function
thf(fact_572_sigma__finite__measure_Oobtain__positive__integrable__function,axiom,
    ! [M2: sigma_5515648953823433982at_nat] :
      ( ( measur1424910518806321826at_nat @ M2 )
     => ~ ! [F3: product_prod_nat_nat > real] :
            ( ( member4564283293661824327t_real @ F3 @ ( sigma_1188828689629184861t_real @ M2 @ borel_5078946678739801102l_real ) )
           => ( ! [X7: product_prod_nat_nat] : ( ord_less_real @ zero_zero_real @ ( F3 @ X7 ) )
             => ( ! [X7: product_prod_nat_nat] : ( ord_less_eq_real @ ( F3 @ X7 ) @ one_one_real )
               => ~ ( bochne2596016609597520987t_real @ M2 @ F3 ) ) ) ) ) ).

% sigma_finite_measure.obtain_positive_integrable_function
thf(fact_573__092_060Omega_062_092_060_094sub_0622_Oemeasure__space__le__1,axiom,
    ord_le3935885782089961368nnreal @ ( sigma_3593834561172453006at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) ) @ one_on2969667320475766781nnreal ).

% \<Omega>\<^sub>2.emeasure_space_le_1
thf(fact_574_Norm__zero,axiom,
    ! [N: functi5165143733218492903t_real] :
      ( ( functi7696537667341841001t_real @ N @ zero_z7307146680010048121t_real )
      = zero_zero_real ) ).

% Norm_zero
thf(fact_575_Norm__zero,axiom,
    ! [N: functi4297716468133224573t_real] :
      ( ( functi1179182286487682149t_real @ N @ zero_z5900165307960944725t_real )
      = zero_zero_real ) ).

% Norm_zero
thf(fact_576_Norm__zero,axiom,
    ! [N: functi4972691055096080419m_real] :
      ( ( functional_Norm_real @ N @ zero_zero_real )
      = zero_zero_real ) ).

% Norm_zero
thf(fact_577_mult__is__0,axiom,
    ! [M3: nat,N2: nat] :
      ( ( ( times_times_nat @ M3 @ N2 )
        = zero_zero_nat )
      = ( ( M3 = zero_zero_nat )
        | ( N2 = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_578_mult__0__right,axiom,
    ! [M3: nat] :
      ( ( times_times_nat @ M3 @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_579_mult__cancel1,axiom,
    ! [K: nat,M3: nat,N2: nat] :
      ( ( ( times_times_nat @ K @ M3 )
        = ( times_times_nat @ K @ N2 ) )
      = ( ( M3 = N2 )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_580_mult__cancel2,axiom,
    ! [M3: nat,K: nat,N2: nat] :
      ( ( ( times_times_nat @ M3 @ K )
        = ( times_times_nat @ N2 @ K ) )
      = ( ( M3 = N2 )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_581_nat__mult__eq__1__iff,axiom,
    ! [M3: nat,N2: nat] :
      ( ( ( times_times_nat @ M3 @ N2 )
        = one_one_nat )
      = ( ( M3 = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_582_nat__1__eq__mult__iff,axiom,
    ! [M3: nat,N2: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M3 @ N2 ) )
      = ( ( M3 = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_583_indep__var__integrable,axiom,
    ! [X1: product_prod_nat_nat > real,X22: product_prod_nat_nat > real] :
      ( ( indepe1845147739501508449t_real @ ( freque5010624893710627907mega_1 @ as ) @ borel_5078946678739801102l_real @ X1 @ borel_5078946678739801102l_real @ X22 )
     => ( ( bochne2596016609597520987t_real @ ( freque5010624893710627907mega_1 @ as ) @ X1 )
       => ( ( bochne2596016609597520987t_real @ ( freque5010624893710627907mega_1 @ as ) @ X22 )
         => ( bochne2596016609597520987t_real @ ( freque5010624893710627907mega_1 @ as )
            @ ^ [Omega: product_prod_nat_nat] : ( times_times_real @ ( X1 @ Omega ) @ ( X22 @ Omega ) ) ) ) ) ) ).

% indep_var_integrable
thf(fact_584_Lp__subset__Lq_I4_J,axiom,
    ! [P2: extend8495563244428889912nnreal,Q2: extend8495563244428889912nnreal,F2: product_prod_nat_nat > real] :
      ( ( ord_le3935885782089961368nnreal @ P2 @ Q2 )
     => ( ( member4564283293661824327t_real @ F2 @ ( functi6492146676480127185t_real @ ( lp_spa8733284142207339630at_nat @ Q2 @ ( freque5010624893710627907mega_1 @ as ) ) ) )
       => ( ord_less_eq_real @ ( functi1179182286487682149t_real @ ( lp_spa8733284142207339630at_nat @ P2 @ ( freque5010624893710627907mega_1 @ as ) ) @ F2 ) @ ( functi1179182286487682149t_real @ ( lp_spa8733284142207339630at_nat @ Q2 @ ( freque5010624893710627907mega_1 @ as ) ) @ F2 ) ) ) ) ).

% Lp_subset_Lq(4)
thf(fact_585_mult__1__left,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% mult_1_left
thf(fact_586_mult__1__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1_left
thf(fact_587_mult__1__left,axiom,
    ! [A: extend8495563244428889912nnreal] :
      ( ( times_1893300245718287421nnreal @ one_on2969667320475766781nnreal @ A )
      = A ) ).

% mult_1_left
thf(fact_588_mult__1__left,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% mult_1_left
thf(fact_589_mult__1__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult_1_right
thf(fact_590_mult__1__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult_1_right
thf(fact_591_mult__1__right,axiom,
    ! [A: extend8495563244428889912nnreal] :
      ( ( times_1893300245718287421nnreal @ A @ one_on2969667320475766781nnreal )
      = A ) ).

% mult_1_right
thf(fact_592_mult__1__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult_1_right
thf(fact_593_mult__less__cancel1,axiom,
    ! [K: nat,M3: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M3 @ N2 ) ) ) ).

% mult_less_cancel1
thf(fact_594_mult__less__cancel2,axiom,
    ! [M3: nat,K: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M3 @ K ) @ ( times_times_nat @ N2 @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M3 @ N2 ) ) ) ).

% mult_less_cancel2
thf(fact_595_nat__0__less__mult__iff,axiom,
    ! [M3: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M3 @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M3 )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% nat_0_less_mult_iff
thf(fact_596_indep__var__lebesgue__integral,axiom,
    ! [X1: product_prod_nat_nat > real,X22: product_prod_nat_nat > real] :
      ( ( indepe1845147739501508449t_real @ ( freque5010624893710627907mega_1 @ as ) @ borel_5078946678739801102l_real @ X1 @ borel_5078946678739801102l_real @ X22 )
     => ( ( bochne2596016609597520987t_real @ ( freque5010624893710627907mega_1 @ as ) @ X1 )
       => ( ( bochne2596016609597520987t_real @ ( freque5010624893710627907mega_1 @ as ) @ X22 )
         => ( ( bochne6384019433803981034t_real @ ( freque5010624893710627907mega_1 @ as )
              @ ^ [Omega: product_prod_nat_nat] : ( times_times_real @ ( X1 @ Omega ) @ ( X22 @ Omega ) ) )
            = ( times_times_real @ ( bochne6384019433803981034t_real @ ( freque5010624893710627907mega_1 @ as ) @ X1 ) @ ( bochne6384019433803981034t_real @ ( freque5010624893710627907mega_1 @ as ) @ X22 ) ) ) ) ) ) ).

% indep_var_lebesgue_integral
thf(fact_597_integral__mult__right__zero,axiom,
    ! [M2: sigma_9047027012034273406at_nat,C: real,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( bochne5509773249985062230t_real @ M2
        @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( times_times_real @ C @ ( F2 @ X2 ) ) )
      = ( times_times_real @ C @ ( bochne5509773249985062230t_real @ M2 @ F2 ) ) ) ).

% integral_mult_right_zero
thf(fact_598_integral__mult__right__zero,axiom,
    ! [M2: sigma_5515648953823433982at_nat,C: real,F2: product_prod_nat_nat > real] :
      ( ( bochne6384019433803981034t_real @ M2
        @ ^ [X2: product_prod_nat_nat] : ( times_times_real @ C @ ( F2 @ X2 ) ) )
      = ( times_times_real @ C @ ( bochne6384019433803981034t_real @ M2 @ F2 ) ) ) ).

% integral_mult_right_zero
thf(fact_599_integral__mult__left__zero,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,C: real] :
      ( ( bochne5509773249985062230t_real @ M2
        @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( times_times_real @ ( F2 @ X2 ) @ C ) )
      = ( times_times_real @ ( bochne5509773249985062230t_real @ M2 @ F2 ) @ C ) ) ).

% integral_mult_left_zero
thf(fact_600_integral__mult__left__zero,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real,C: real] :
      ( ( bochne6384019433803981034t_real @ M2
        @ ^ [X2: product_prod_nat_nat] : ( times_times_real @ ( F2 @ X2 ) @ C ) )
      = ( times_times_real @ ( bochne6384019433803981034t_real @ M2 @ F2 ) @ C ) ) ).

% integral_mult_left_zero
thf(fact_601_integrable__mult__right__iff,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,C: real] :
      ( ( bochne7117808529828525605t_real @ M2
        @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( times_times_real @ ( F2 @ X2 ) @ C ) )
      = ( ( C = zero_zero_real )
        | ( bochne7117808529828525605t_real @ M2 @ F2 ) ) ) ).

% integrable_mult_right_iff
thf(fact_602_integrable__mult__right__iff,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real,C: real] :
      ( ( bochne2596016609597520987t_real @ M2
        @ ^ [X2: product_prod_nat_nat] : ( times_times_real @ ( F2 @ X2 ) @ C ) )
      = ( ( C = zero_zero_real )
        | ( bochne2596016609597520987t_real @ M2 @ F2 ) ) ) ).

% integrable_mult_right_iff
thf(fact_603_integrable__mult__left__iff,axiom,
    ! [M2: sigma_9047027012034273406at_nat,C: real,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( bochne7117808529828525605t_real @ M2
        @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( times_times_real @ C @ ( F2 @ X2 ) ) )
      = ( ( C = zero_zero_real )
        | ( bochne7117808529828525605t_real @ M2 @ F2 ) ) ) ).

% integrable_mult_left_iff
thf(fact_604_integrable__mult__left__iff,axiom,
    ! [M2: sigma_5515648953823433982at_nat,C: real,F2: product_prod_nat_nat > real] :
      ( ( bochne2596016609597520987t_real @ M2
        @ ^ [X2: product_prod_nat_nat] : ( times_times_real @ C @ ( F2 @ X2 ) ) )
      = ( ( C = zero_zero_real )
        | ( bochne2596016609597520987t_real @ M2 @ F2 ) ) ) ).

% integrable_mult_left_iff
thf(fact_605__092_060Omega_062_092_060_094sub_0622_Oemeasure__space__1,axiom,
    ( ( sigma_3593834561172453006at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
    = one_on2969667320475766781nnreal ) ).

% \<Omega>\<^sub>2.emeasure_space_1
thf(fact_606_mult__le__cancel1,axiom,
    ! [K: nat,M3: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M3 @ N2 ) ) ) ).

% mult_le_cancel1
thf(fact_607_mult__le__cancel2,axiom,
    ! [M3: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M3 @ K ) @ ( times_times_nat @ N2 @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M3 @ N2 ) ) ) ).

% mult_le_cancel2
thf(fact_608_eNorm__zero,axiom,
    ! [N: functi4972691055096080419m_real] :
      ( ( functi690790994237603730m_real @ N @ zero_zero_real )
      = zero_z7100319975126383169nnreal ) ).

% eNorm_zero
thf(fact_609_eNorm__zero,axiom,
    ! [N: functi5165143733218492903t_real] :
      ( ( functi3128325879070182998t_real @ N @ zero_z7307146680010048121t_real )
      = zero_z7100319975126383169nnreal ) ).

% eNorm_zero
thf(fact_610_eNorm__zero,axiom,
    ! [N: functi4297716468133224573t_real] :
      ( ( functi5516212835536045368t_real @ N @ zero_z5900165307960944725t_real )
      = zero_z7100319975126383169nnreal ) ).

% eNorm_zero
thf(fact_611_integrable__mult__left,axiom,
    ! [C: real,M2: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( ( C != zero_zero_real )
       => ( bochne7117808529828525605t_real @ M2 @ F2 ) )
     => ( bochne7117808529828525605t_real @ M2
        @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( times_times_real @ ( F2 @ X2 ) @ C ) ) ) ).

% integrable_mult_left
thf(fact_612_integrable__mult__left,axiom,
    ! [C: real,M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real] :
      ( ( ( C != zero_zero_real )
       => ( bochne2596016609597520987t_real @ M2 @ F2 ) )
     => ( bochne2596016609597520987t_real @ M2
        @ ^ [X2: product_prod_nat_nat] : ( times_times_real @ ( F2 @ X2 ) @ C ) ) ) ).

% integrable_mult_left
thf(fact_613_integrable__mult__right,axiom,
    ! [C: real,M2: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( ( C != zero_zero_real )
       => ( bochne7117808529828525605t_real @ M2 @ F2 ) )
     => ( bochne7117808529828525605t_real @ M2
        @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( times_times_real @ C @ ( F2 @ X2 ) ) ) ) ).

% integrable_mult_right
thf(fact_614_integrable__mult__right,axiom,
    ! [C: real,M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real] :
      ( ( ( C != zero_zero_real )
       => ( bochne2596016609597520987t_real @ M2 @ F2 ) )
     => ( bochne2596016609597520987t_real @ M2
        @ ^ [X2: product_prod_nat_nat] : ( times_times_real @ C @ ( F2 @ X2 ) ) ) ) ).

% integrable_mult_right
thf(fact_615_Bochner__Integration_Ointegral__mult__left,axiom,
    ! [C: real,M2: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( ( C != zero_zero_real )
       => ( bochne7117808529828525605t_real @ M2 @ F2 ) )
     => ( ( bochne5509773249985062230t_real @ M2
          @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( times_times_real @ ( F2 @ X2 ) @ C ) )
        = ( times_times_real @ ( bochne5509773249985062230t_real @ M2 @ F2 ) @ C ) ) ) ).

% Bochner_Integration.integral_mult_left
thf(fact_616_Bochner__Integration_Ointegral__mult__left,axiom,
    ! [C: real,M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real] :
      ( ( ( C != zero_zero_real )
       => ( bochne2596016609597520987t_real @ M2 @ F2 ) )
     => ( ( bochne6384019433803981034t_real @ M2
          @ ^ [X2: product_prod_nat_nat] : ( times_times_real @ ( F2 @ X2 ) @ C ) )
        = ( times_times_real @ ( bochne6384019433803981034t_real @ M2 @ F2 ) @ C ) ) ) ).

% Bochner_Integration.integral_mult_left
thf(fact_617_Bochner__Integration_Ointegral__mult__right,axiom,
    ! [C: real,M2: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( ( C != zero_zero_real )
       => ( bochne7117808529828525605t_real @ M2 @ F2 ) )
     => ( ( bochne5509773249985062230t_real @ M2
          @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( times_times_real @ C @ ( F2 @ X2 ) ) )
        = ( times_times_real @ C @ ( bochne5509773249985062230t_real @ M2 @ F2 ) ) ) ) ).

% Bochner_Integration.integral_mult_right
thf(fact_618_Bochner__Integration_Ointegral__mult__right,axiom,
    ! [C: real,M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real] :
      ( ( ( C != zero_zero_real )
       => ( bochne2596016609597520987t_real @ M2 @ F2 ) )
     => ( ( bochne6384019433803981034t_real @ M2
          @ ^ [X2: product_prod_nat_nat] : ( times_times_real @ C @ ( F2 @ X2 ) ) )
        = ( times_times_real @ C @ ( bochne6384019433803981034t_real @ M2 @ F2 ) ) ) ) ).

% Bochner_Integration.integral_mult_right
thf(fact_619__092_060Omega_062_092_060_094sub_0622_Oindep__sets__Dynkin,axiom,
    ! [F: $o > set_se1666487788256820497at_nat,I: set_o] :
      ( ( indepe9198310886692500677_nat_o @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
     => ( indepe9198310886692500677_nat_o @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
        @ ^ [I3: $o] : ( sigma_3745205859527822540at_nat @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) @ ( F @ I3 ) )
        @ I ) ) ).

% \<Omega>\<^sub>2.indep_sets_Dynkin
thf(fact_620__092_060Omega_062_092_060_094sub_0622_Oindep__sets__Dynkin,axiom,
    ! [F: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_se1666487788256820497at_nat,I: set_Pr947837736998463782t_real] :
      ( ( indepe5134244768989041027t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
     => ( indepe5134244768989041027t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
        @ ^ [I3: ( product_prod_nat_nat > product_prod_nat_nat ) > real] : ( sigma_3745205859527822540at_nat @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) @ ( F @ I3 ) )
        @ I ) ) ).

% \<Omega>\<^sub>2.indep_sets_Dynkin
thf(fact_621__092_060Omega_062_092_060_094sub_0622_Oindep__sets__Dynkin,axiom,
    ! [F: ( real > extend8495563244428889912nnreal ) > set_se1666487788256820497at_nat,I: set_re5328672808648366137nnreal] :
      ( ( indepe6665278795436257814nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
     => ( indepe6665278795436257814nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
        @ ^ [I3: real > extend8495563244428889912nnreal] : ( sigma_3745205859527822540at_nat @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) @ ( F @ I3 ) )
        @ I ) ) ).

% \<Omega>\<^sub>2.indep_sets_Dynkin
thf(fact_622__092_060Omega_062_092_060_094sub_0622_Oindep__sets__Dynkin,axiom,
    ! [F: ( product_prod_nat_nat > real ) > set_se1666487788256820497at_nat,I: set_Pr1002607673312053630t_real] :
      ( ( indepe4000825420875437771t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
     => ( indepe4000825420875437771t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
        @ ^ [I3: product_prod_nat_nat > real] : ( sigma_3745205859527822540at_nat @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) @ ( F @ I3 ) )
        @ I ) ) ).

% \<Omega>\<^sub>2.indep_sets_Dynkin
thf(fact_623__092_060Omega_062_092_060_094sub_0622_Oindep__sets__Dynkin,axiom,
    ! [F: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se1666487788256820497at_nat,I: set_Pr2458342521480944603at_nat] :
      ( ( indepe7028276433782144696at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
     => ( indepe7028276433782144696at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
        @ ^ [I3: product_prod_nat_nat > product_prod_nat_nat] : ( sigma_3745205859527822540at_nat @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) @ ( F @ I3 ) )
        @ I ) ) ).

% \<Omega>\<^sub>2.indep_sets_Dynkin
thf(fact_624__092_060Omega_062_092_060_094sub_0622_Oindep__sets__Dynkin,axiom,
    ! [F: rat > set_se1666487788256820497at_nat,I: set_rat] :
      ( ( indepe2998401279100655083at_rat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
     => ( indepe2998401279100655083at_rat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
        @ ^ [I3: rat] : ( sigma_3745205859527822540at_nat @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) @ ( F @ I3 ) )
        @ I ) ) ).

% \<Omega>\<^sub>2.indep_sets_Dynkin
thf(fact_625_less__eq__measure_Ointros_I1_J,axiom,
    ! [M2: sigma_9047027012034273406at_nat,N: sigma_9047027012034273406at_nat] :
      ( ( ord_le2250674462368129583at_nat @ ( sigma_4468984517727773725at_nat @ M2 ) @ ( sigma_4468984517727773725at_nat @ N ) )
     => ( ord_le5477907261385988126at_nat @ M2 @ N ) ) ).

% less_eq_measure.intros(1)
thf(fact_626_less__eq__measure_Ointros_I1_J,axiom,
    ! [M2: sigma_5515648953823433982at_nat,N: sigma_5515648953823433982at_nat] :
      ( ( ord_le7866589430770878221at_nat @ ( sigma_4797346298676585097at_nat @ M2 ) @ ( sigma_4797346298676585097at_nat @ N ) )
     => ( ord_le1401241647504520542at_nat @ M2 @ N ) ) ).

% less_eq_measure.intros(1)
thf(fact_627_mult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_628_mult_Oleft__commute,axiom,
    ! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( times_1893300245718287421nnreal @ B @ ( times_1893300245718287421nnreal @ A @ C ) )
      = ( times_1893300245718287421nnreal @ A @ ( times_1893300245718287421nnreal @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_629_mult_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] :
      ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_630_mult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A4: nat,B3: nat] : ( times_times_nat @ B3 @ A4 ) ) ) ).

% mult.commute
thf(fact_631_mult_Ocommute,axiom,
    ( times_1893300245718287421nnreal
    = ( ^ [A4: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] : ( times_1893300245718287421nnreal @ B3 @ A4 ) ) ) ).

% mult.commute
thf(fact_632_mult_Ocommute,axiom,
    ( times_times_real
    = ( ^ [A4: real,B3: real] : ( times_times_real @ B3 @ A4 ) ) ) ).

% mult.commute
thf(fact_633_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_634_mult_Oassoc,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( times_1893300245718287421nnreal @ ( times_1893300245718287421nnreal @ A @ B ) @ C )
      = ( times_1893300245718287421nnreal @ A @ ( times_1893300245718287421nnreal @ B @ C ) ) ) ).

% mult.assoc
thf(fact_635_mult_Oassoc,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% mult.assoc
thf(fact_636_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_637_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( times_1893300245718287421nnreal @ ( times_1893300245718287421nnreal @ A @ B ) @ C )
      = ( times_1893300245718287421nnreal @ A @ ( times_1893300245718287421nnreal @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_638_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_639_of__rat__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( field_7254667332652039916t_real @ ( times_times_rat @ A @ B ) )
      = ( times_times_real @ ( field_7254667332652039916t_real @ A ) @ ( field_7254667332652039916t_real @ B ) ) ) ).

% of_rat_mult
thf(fact_640_le__measureD1,axiom,
    ! [A2: sigma_5515648953823433982at_nat,B2: sigma_5515648953823433982at_nat] :
      ( ( ord_le1401241647504520542at_nat @ A2 @ B2 )
     => ( ord_le3146513528884898305at_nat @ ( sigma_4797346298676585097at_nat @ A2 ) @ ( sigma_4797346298676585097at_nat @ B2 ) ) ) ).

% le_measureD1
thf(fact_641_le__measureD1,axiom,
    ! [A2: sigma_9047027012034273406at_nat,B2: sigma_9047027012034273406at_nat] :
      ( ( ord_le5477907261385988126at_nat @ A2 @ B2 )
     => ( ord_le3857079194666040379at_nat @ ( sigma_4468984517727773725at_nat @ A2 ) @ ( sigma_4468984517727773725at_nat @ B2 ) ) ) ).

% le_measureD1
thf(fact_642_le__measureD1,axiom,
    ! [A2: sigma_2424604449015074331t_real,B2: sigma_2424604449015074331t_real] :
      ( ( ord_le6138776903863807611t_real @ A2 @ B2 )
     => ( ord_le1994730768851292446t_real @ ( sigma_7766831664147846694t_real @ A2 ) @ ( sigma_7766831664147846694t_real @ B2 ) ) ) ).

% le_measureD1
thf(fact_643_le__measureD1,axiom,
    ! [A2: sigma_3568842061677812681t_real,B2: sigma_3568842061677812681t_real] :
      ( ( ord_le9078708787208573801t_real @ A2 @ B2 )
     => ( ord_le8622349663015078278t_real @ ( sigma_8760150869319784680t_real @ A2 ) @ ( sigma_8760150869319784680t_real @ B2 ) ) ) ).

% le_measureD1
thf(fact_644_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_645_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_646_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: extend8495563244428889912nnreal] :
      ( ( times_1893300245718287421nnreal @ one_on2969667320475766781nnreal @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_647_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_648_mult_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.comm_neutral
thf(fact_649_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_650_mult_Ocomm__neutral,axiom,
    ! [A: extend8495563244428889912nnreal] :
      ( ( times_1893300245718287421nnreal @ A @ one_on2969667320475766781nnreal )
      = A ) ).

% mult.comm_neutral
thf(fact_651_mult_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.comm_neutral
thf(fact_652_Bochner__Integration_Ointegral__cong,axiom,
    ! [M2: sigma_9047027012034273406at_nat,N: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,G2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( M2 = N )
     => ( ! [X3: product_prod_nat_nat > product_prod_nat_nat] :
            ( ( member8885076297122219836at_nat @ X3 @ ( sigma_4468984517727773725at_nat @ N ) )
           => ( ( F2 @ X3 )
              = ( G2 @ X3 ) ) )
       => ( ( bochne5509773249985062230t_real @ M2 @ F2 )
          = ( bochne5509773249985062230t_real @ N @ G2 ) ) ) ) ).

% Bochner_Integration.integral_cong
thf(fact_653_Bochner__Integration_Ointegral__cong,axiom,
    ! [M2: sigma_5515648953823433982at_nat,N: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real,G2: product_prod_nat_nat > real] :
      ( ( M2 = N )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ ( sigma_4797346298676585097at_nat @ N ) )
           => ( ( F2 @ X3 )
              = ( G2 @ X3 ) ) )
       => ( ( bochne6384019433803981034t_real @ M2 @ F2 )
          = ( bochne6384019433803981034t_real @ N @ G2 ) ) ) ) ).

% Bochner_Integration.integral_cong
thf(fact_654_times__nat_Osimps_I1_J,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% times_nat.simps(1)
thf(fact_655_Bochner__Integration_Ointegrable__cong,axiom,
    ! [M2: sigma_9047027012034273406at_nat,N: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,G2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( M2 = N )
     => ( ! [X3: product_prod_nat_nat > product_prod_nat_nat] :
            ( ( member8885076297122219836at_nat @ X3 @ ( sigma_4468984517727773725at_nat @ N ) )
           => ( ( F2 @ X3 )
              = ( G2 @ X3 ) ) )
       => ( ( bochne7117808529828525605t_real @ M2 @ F2 )
          = ( bochne7117808529828525605t_real @ N @ G2 ) ) ) ) ).

% Bochner_Integration.integrable_cong
thf(fact_656_Bochner__Integration_Ointegrable__cong,axiom,
    ! [M2: sigma_5515648953823433982at_nat,N: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real,G2: product_prod_nat_nat > real] :
      ( ( M2 = N )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ ( sigma_4797346298676585097at_nat @ N ) )
           => ( ( F2 @ X3 )
              = ( G2 @ X3 ) ) )
       => ( ( bochne2596016609597520987t_real @ M2 @ F2 )
          = ( bochne2596016609597520987t_real @ N @ G2 ) ) ) ) ).

% Bochner_Integration.integrable_cong
thf(fact_657_le__cube,axiom,
    ! [M3: nat] : ( ord_less_eq_nat @ M3 @ ( times_times_nat @ M3 @ ( times_times_nat @ M3 @ M3 ) ) ) ).

% le_cube
thf(fact_658_le__square,axiom,
    ! [M3: nat] : ( ord_less_eq_nat @ M3 @ ( times_times_nat @ M3 @ M3 ) ) ).

% le_square
thf(fact_659_mult__le__mono,axiom,
    ! [I4: nat,J2: nat,K: nat,L3: nat] :
      ( ( ord_less_eq_nat @ I4 @ J2 )
     => ( ( ord_less_eq_nat @ K @ L3 )
       => ( ord_less_eq_nat @ ( times_times_nat @ I4 @ K ) @ ( times_times_nat @ J2 @ L3 ) ) ) ) ).

% mult_le_mono
thf(fact_660_mult__le__mono1,axiom,
    ! [I4: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I4 @ J2 )
     => ( ord_less_eq_nat @ ( times_times_nat @ I4 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ).

% mult_le_mono1
thf(fact_661_mult__le__mono2,axiom,
    ! [I4: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I4 @ J2 )
     => ( ord_less_eq_nat @ ( times_times_nat @ K @ I4 ) @ ( times_times_nat @ K @ J2 ) ) ) ).

% mult_le_mono2
thf(fact_662_nat__mult__1__right,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ N2 @ one_one_nat )
      = N2 ) ).

% nat_mult_1_right
thf(fact_663_nat__mult__1,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ one_one_nat @ N2 )
      = N2 ) ).

% nat_mult_1
thf(fact_664_quasinorm__subsetI,axiom,
    ! [N1: functi5165143733218492903t_real,N22: functi5165143733218492903t_real,C2: real] :
      ( ! [F3: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
          ( ( member8159409068225774087t_real @ F3 @ ( functi8590574888837585149t_real @ N1 ) )
         => ( ord_le3935885782089961368nnreal @ ( functi3128325879070182998t_real @ N22 @ F3 ) @ ( times_1893300245718287421nnreal @ ( extend7643940197134561352nnreal @ C2 ) @ ( functi3128325879070182998t_real @ N1 @ F3 ) ) ) )
     => ( ord_le1044648919696853383t_real @ N1 @ N22 ) ) ).

% quasinorm_subsetI
thf(fact_665_quasinorm__subsetI,axiom,
    ! [N1: functi4297716468133224573t_real,N22: functi4297716468133224573t_real,C2: real] :
      ( ! [F3: product_prod_nat_nat > real] :
          ( ( member4564283293661824327t_real @ F3 @ ( functi6492146676480127185t_real @ N1 ) )
         => ( ord_le3935885782089961368nnreal @ ( functi5516212835536045368t_real @ N22 @ F3 ) @ ( times_1893300245718287421nnreal @ ( extend7643940197134561352nnreal @ C2 ) @ ( functi5516212835536045368t_real @ N1 @ F3 ) ) ) )
     => ( ord_le6049248562894827741t_real @ N1 @ N22 ) ) ).

% quasinorm_subsetI
thf(fact_666_less__eq__quasinorm__def,axiom,
    ( ord_le1044648919696853383t_real
    = ( ^ [N12: functi5165143733218492903t_real,N23: functi5165143733218492903t_real] :
        ? [C3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ C3 )
          & ! [F4: ( product_prod_nat_nat > product_prod_nat_nat ) > real] : ( ord_le3935885782089961368nnreal @ ( functi3128325879070182998t_real @ N23 @ F4 ) @ ( times_1893300245718287421nnreal @ ( extend7643940197134561352nnreal @ C3 ) @ ( functi3128325879070182998t_real @ N12 @ F4 ) ) ) ) ) ) ).

% less_eq_quasinorm_def
thf(fact_667_less__eq__quasinorm__def,axiom,
    ( ord_le6049248562894827741t_real
    = ( ^ [N12: functi4297716468133224573t_real,N23: functi4297716468133224573t_real] :
        ? [C3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ C3 )
          & ! [F4: product_prod_nat_nat > real] : ( ord_le3935885782089961368nnreal @ ( functi5516212835536045368t_real @ N23 @ F4 ) @ ( times_1893300245718287421nnreal @ ( extend7643940197134561352nnreal @ C3 ) @ ( functi5516212835536045368t_real @ N12 @ F4 ) ) ) ) ) ) ).

% less_eq_quasinorm_def
thf(fact_668_quasinorm__subsetD,axiom,
    ! [N1: functi5165143733218492903t_real,N22: functi5165143733218492903t_real] :
      ( ( ord_le1044648919696853383t_real @ N1 @ N22 )
     => ? [C4: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ C4 )
          & ! [F5: ( product_prod_nat_nat > product_prod_nat_nat ) > real] : ( ord_le3935885782089961368nnreal @ ( functi3128325879070182998t_real @ N22 @ F5 ) @ ( times_1893300245718287421nnreal @ ( extend7643940197134561352nnreal @ C4 ) @ ( functi3128325879070182998t_real @ N1 @ F5 ) ) ) ) ) ).

% quasinorm_subsetD
thf(fact_669_quasinorm__subsetD,axiom,
    ! [N1: functi4297716468133224573t_real,N22: functi4297716468133224573t_real] :
      ( ( ord_le6049248562894827741t_real @ N1 @ N22 )
     => ? [C4: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ C4 )
          & ! [F5: product_prod_nat_nat > real] : ( ord_le3935885782089961368nnreal @ ( functi5516212835536045368t_real @ N22 @ F5 ) @ ( times_1893300245718287421nnreal @ ( extend7643940197134561352nnreal @ C4 ) @ ( functi5516212835536045368t_real @ N1 @ F5 ) ) ) ) ) ).

% quasinorm_subsetD
thf(fact_670_measurable__Pair1_H,axiom,
    ! [X5: nat,M1: sigma_measure_nat,M22: sigma_measure_nat] :
      ( ( member_nat @ X5 @ ( sigma_space_nat @ M1 ) )
     => ( member6743353555990375057at_nat @ ( product_Pair_nat_nat @ X5 ) @ ( sigma_2856282194129746655at_nat @ M22 @ ( binary6458575275919672938at_nat @ M1 @ M22 ) ) ) ) ).

% measurable_Pair1'
thf(fact_671_mult__less__mono1,axiom,
    ! [I4: nat,J2: nat,K: nat] :
      ( ( ord_less_nat @ I4 @ J2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ I4 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ) ).

% mult_less_mono1
thf(fact_672_mult__less__mono2,axiom,
    ! [I4: nat,J2: nat,K: nat] :
      ( ( ord_less_nat @ I4 @ J2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ K @ I4 ) @ ( times_times_nat @ K @ J2 ) ) ) ) ).

% mult_less_mono2
thf(fact_673_mult__eq__self__implies__10,axiom,
    ! [M3: nat,N2: nat] :
      ( ( M3
        = ( times_times_nat @ M3 @ N2 ) )
     => ( ( N2 = one_one_nat )
        | ( M3 = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_674_emeasure__space,axiom,
    ! [M2: sigma_9047027012034273406at_nat,A2: set_Pr2458342521480944603at_nat] : ( ord_le3935885782089961368nnreal @ ( sigma_3593834561172453006at_nat @ M2 @ A2 ) @ ( sigma_3593834561172453006at_nat @ M2 @ ( sigma_4468984517727773725at_nat @ M2 ) ) ) ).

% emeasure_space
thf(fact_675_emeasure__space,axiom,
    ! [M2: sigma_5515648953823433982at_nat,A2: set_Pr1261947904930325089at_nat] : ( ord_le3935885782089961368nnreal @ ( sigma_411563132819280856at_nat @ M2 @ A2 ) @ ( sigma_411563132819280856at_nat @ M2 @ ( sigma_4797346298676585097at_nat @ M2 ) ) ) ).

% emeasure_space
thf(fact_676_eNorm__Norm,axiom,
    ! [X5: ( product_prod_nat_nat > product_prod_nat_nat ) > real,N: functi5165143733218492903t_real] :
      ( ( member8159409068225774087t_real @ X5 @ ( functi8590574888837585149t_real @ N ) )
     => ( ( functi3128325879070182998t_real @ N @ X5 )
        = ( extend7643940197134561352nnreal @ ( functi7696537667341841001t_real @ N @ X5 ) ) ) ) ).

% eNorm_Norm
thf(fact_677_eNorm__Norm,axiom,
    ! [X5: product_prod_nat_nat > real,N: functi4297716468133224573t_real] :
      ( ( member4564283293661824327t_real @ X5 @ ( functi6492146676480127185t_real @ N ) )
     => ( ( functi5516212835536045368t_real @ N @ X5 )
        = ( extend7643940197134561352nnreal @ ( functi1179182286487682149t_real @ N @ X5 ) ) ) ) ).

% eNorm_Norm
thf(fact_678_quasinorm__subset__Norm__eNorm,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,N1: functi5165143733218492903t_real,N22: functi5165143733218492903t_real,C2: real] :
      ( ( ( member8159409068225774087t_real @ F2 @ ( functi8590574888837585149t_real @ N1 ) )
       => ( ord_less_eq_real @ ( functi7696537667341841001t_real @ N22 @ F2 ) @ ( times_times_real @ C2 @ ( functi7696537667341841001t_real @ N1 @ F2 ) ) ) )
     => ( ( ord_le1044648919696853383t_real @ N1 @ N22 )
       => ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_le3935885782089961368nnreal @ ( functi3128325879070182998t_real @ N22 @ F2 ) @ ( times_1893300245718287421nnreal @ ( extend7643940197134561352nnreal @ C2 ) @ ( functi3128325879070182998t_real @ N1 @ F2 ) ) ) ) ) ) ).

% quasinorm_subset_Norm_eNorm
thf(fact_679_quasinorm__subset__Norm__eNorm,axiom,
    ! [F2: product_prod_nat_nat > real,N1: functi4297716468133224573t_real,N22: functi4297716468133224573t_real,C2: real] :
      ( ( ( member4564283293661824327t_real @ F2 @ ( functi6492146676480127185t_real @ N1 ) )
       => ( ord_less_eq_real @ ( functi1179182286487682149t_real @ N22 @ F2 ) @ ( times_times_real @ C2 @ ( functi1179182286487682149t_real @ N1 @ F2 ) ) ) )
     => ( ( ord_le6049248562894827741t_real @ N1 @ N22 )
       => ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_le3935885782089961368nnreal @ ( functi5516212835536045368t_real @ N22 @ F2 ) @ ( times_1893300245718287421nnreal @ ( extend7643940197134561352nnreal @ C2 ) @ ( functi5516212835536045368t_real @ N1 @ F2 ) ) ) ) ) ) ).

% quasinorm_subset_Norm_eNorm
thf(fact_680_measurable__Pair2_H,axiom,
    ! [Y4: nat,M22: sigma_measure_nat,M1: sigma_measure_nat] :
      ( ( member_nat @ Y4 @ ( sigma_space_nat @ M22 ) )
     => ( member6743353555990375057at_nat
        @ ^ [X2: nat] : ( product_Pair_nat_nat @ X2 @ Y4 )
        @ ( sigma_2856282194129746655at_nat @ M1 @ ( binary6458575275919672938at_nat @ M1 @ M22 ) ) ) ) ).

% measurable_Pair2'
thf(fact_681_Bochner__Integration_Ointegral__nonneg,axiom,
    ! [M2: sigma_measure_rat,F2: rat > real] :
      ( ! [X3: rat] :
          ( ( member_rat @ X3 @ ( sigma_space_rat @ M2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( bochne6128977971683193353t_real @ M2 @ F2 ) ) ) ).

% Bochner_Integration.integral_nonneg
thf(fact_682_Bochner__Integration_Ointegral__nonneg,axiom,
    ! [M2: sigma_2424604449015074331t_real,F2: ( product_prod_nat_nat > real ) > real] :
      ( ! [X3: product_prod_nat_nat > real] :
          ( ( member4564283293661824327t_real @ X3 @ ( sigma_7766831664147846694t_real @ M2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( bochne2904387722813762055l_real @ M2 @ F2 ) ) ) ).

% Bochner_Integration.integral_nonneg
thf(fact_683_Bochner__Integration_Ointegral__nonneg,axiom,
    ! [M2: sigma_5394977995791401948nnreal,F2: ( real > extend8495563244428889912nnreal ) > real] :
      ( ! [X3: real > extend8495563244428889912nnreal] :
          ( ( member2919562650594848410nnreal @ X3 @ ( sigma_2369682286586992763nnreal @ M2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( bochne1920604158554476212l_real @ M2 @ F2 ) ) ) ).

% Bochner_Integration.integral_nonneg
thf(fact_684_Bochner__Integration_Ointegral__nonneg,axiom,
    ! [M2: sigma_3568842061677812681t_real,F2: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > real] :
      ( ! [X3: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
          ( ( member8159409068225774087t_real @ X3 @ ( sigma_8760150869319784680t_real @ M2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( bochne7370261254949204129l_real @ M2 @ F2 ) ) ) ).

% Bochner_Integration.integral_nonneg
thf(fact_685_Bochner__Integration_Ointegral__nonneg,axiom,
    ! [M2: sigma_measure_o,F2: $o > real] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( sigma_space_o @ M2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( bochne8006308498830368769o_real @ M2 @ F2 ) ) ) ).

% Bochner_Integration.integral_nonneg
thf(fact_686_Bochner__Integration_Ointegral__nonneg,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ! [X3: product_prod_nat_nat > product_prod_nat_nat] :
          ( ( member8885076297122219836at_nat @ X3 @ ( sigma_4468984517727773725at_nat @ M2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( bochne5509773249985062230t_real @ M2 @ F2 ) ) ) ).

% Bochner_Integration.integral_nonneg
thf(fact_687_Bochner__Integration_Ointegral__nonneg,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ ( sigma_4797346298676585097at_nat @ M2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( bochne6384019433803981034t_real @ M2 @ F2 ) ) ) ).

% Bochner_Integration.integral_nonneg
thf(fact_688_integral__mono,axiom,
    ! [M2: sigma_measure_rat,F2: rat > real,G2: rat > real] :
      ( ( bochne6810293860105484248t_real @ M2 @ F2 )
     => ( ( bochne6810293860105484248t_real @ M2 @ G2 )
       => ( ! [X3: rat] :
              ( ( member_rat @ X3 @ ( sigma_space_rat @ M2 ) )
             => ( ord_less_eq_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) )
         => ( ord_less_eq_real @ ( bochne6128977971683193353t_real @ M2 @ F2 ) @ ( bochne6128977971683193353t_real @ M2 @ G2 ) ) ) ) ) ).

% integral_mono
thf(fact_689_integral__mono,axiom,
    ! [M2: sigma_2424604449015074331t_real,F2: ( product_prod_nat_nat > real ) > real,G2: ( product_prod_nat_nat > real ) > real] :
      ( ( bochne1222508287783722872l_real @ M2 @ F2 )
     => ( ( bochne1222508287783722872l_real @ M2 @ G2 )
       => ( ! [X3: product_prod_nat_nat > real] :
              ( ( member4564283293661824327t_real @ X3 @ ( sigma_7766831664147846694t_real @ M2 ) )
             => ( ord_less_eq_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) )
         => ( ord_less_eq_real @ ( bochne2904387722813762055l_real @ M2 @ F2 ) @ ( bochne2904387722813762055l_real @ M2 @ G2 ) ) ) ) ) ).

% integral_mono
thf(fact_690_integral__mono,axiom,
    ! [M2: sigma_5394977995791401948nnreal,F2: ( real > extend8495563244428889912nnreal ) > real,G2: ( real > extend8495563244428889912nnreal ) > real] :
      ( ( bochne4404133754790937219l_real @ M2 @ F2 )
     => ( ( bochne4404133754790937219l_real @ M2 @ G2 )
       => ( ! [X3: real > extend8495563244428889912nnreal] :
              ( ( member2919562650594848410nnreal @ X3 @ ( sigma_2369682286586992763nnreal @ M2 ) )
             => ( ord_less_eq_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) )
         => ( ord_less_eq_real @ ( bochne1920604158554476212l_real @ M2 @ F2 ) @ ( bochne1920604158554476212l_real @ M2 @ G2 ) ) ) ) ) ).

% integral_mono
thf(fact_691_integral__mono,axiom,
    ! [M2: sigma_3568842061677812681t_real,F2: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > real,G2: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > real] :
      ( ( bochne6626571995716118384l_real @ M2 @ F2 )
     => ( ( bochne6626571995716118384l_real @ M2 @ G2 )
       => ( ! [X3: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
              ( ( member8159409068225774087t_real @ X3 @ ( sigma_8760150869319784680t_real @ M2 ) )
             => ( ord_less_eq_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) )
         => ( ord_less_eq_real @ ( bochne7370261254949204129l_real @ M2 @ F2 ) @ ( bochne7370261254949204129l_real @ M2 @ G2 ) ) ) ) ) ).

% integral_mono
thf(fact_692_integral__mono,axiom,
    ! [M2: sigma_measure_o,F2: $o > real,G2: $o > real] :
      ( ( bochne661340805755426930o_real @ M2 @ F2 )
     => ( ( bochne661340805755426930o_real @ M2 @ G2 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( sigma_space_o @ M2 ) )
             => ( ord_less_eq_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) )
         => ( ord_less_eq_real @ ( bochne8006308498830368769o_real @ M2 @ F2 ) @ ( bochne8006308498830368769o_real @ M2 @ G2 ) ) ) ) ) ).

% integral_mono
thf(fact_693_integral__mono,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,G2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( bochne7117808529828525605t_real @ M2 @ F2 )
     => ( ( bochne7117808529828525605t_real @ M2 @ G2 )
       => ( ! [X3: product_prod_nat_nat > product_prod_nat_nat] :
              ( ( member8885076297122219836at_nat @ X3 @ ( sigma_4468984517727773725at_nat @ M2 ) )
             => ( ord_less_eq_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) )
         => ( ord_less_eq_real @ ( bochne5509773249985062230t_real @ M2 @ F2 ) @ ( bochne5509773249985062230t_real @ M2 @ G2 ) ) ) ) ) ).

% integral_mono
thf(fact_694_integral__mono,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real,G2: product_prod_nat_nat > real] :
      ( ( bochne2596016609597520987t_real @ M2 @ F2 )
     => ( ( bochne2596016609597520987t_real @ M2 @ G2 )
       => ( ! [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ ( sigma_4797346298676585097at_nat @ M2 ) )
             => ( ord_less_eq_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) )
         => ( ord_less_eq_real @ ( bochne6384019433803981034t_real @ M2 @ F2 ) @ ( bochne6384019433803981034t_real @ M2 @ G2 ) ) ) ) ) ).

% integral_mono
thf(fact_695_quasinorm__subsetI_H,axiom,
    ! [N1: functi5165143733218492903t_real,N22: functi5165143733218492903t_real,C2: real] :
      ( ! [F3: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
          ( ( member8159409068225774087t_real @ F3 @ ( functi8590574888837585149t_real @ N1 ) )
         => ( member8159409068225774087t_real @ F3 @ ( functi8590574888837585149t_real @ N22 ) ) )
     => ( ! [F3: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
            ( ( member8159409068225774087t_real @ F3 @ ( functi8590574888837585149t_real @ N1 ) )
           => ( ord_less_eq_real @ ( functi7696537667341841001t_real @ N22 @ F3 ) @ ( times_times_real @ C2 @ ( functi7696537667341841001t_real @ N1 @ F3 ) ) ) )
       => ( ord_le1044648919696853383t_real @ N1 @ N22 ) ) ) ).

% quasinorm_subsetI'
thf(fact_696_quasinorm__subsetI_H,axiom,
    ! [N1: functi4297716468133224573t_real,N22: functi4297716468133224573t_real,C2: real] :
      ( ! [F3: product_prod_nat_nat > real] :
          ( ( member4564283293661824327t_real @ F3 @ ( functi6492146676480127185t_real @ N1 ) )
         => ( member4564283293661824327t_real @ F3 @ ( functi6492146676480127185t_real @ N22 ) ) )
     => ( ! [F3: product_prod_nat_nat > real] :
            ( ( member4564283293661824327t_real @ F3 @ ( functi6492146676480127185t_real @ N1 ) )
           => ( ord_less_eq_real @ ( functi1179182286487682149t_real @ N22 @ F3 ) @ ( times_times_real @ C2 @ ( functi1179182286487682149t_real @ N1 @ F3 ) ) ) )
       => ( ord_le6049248562894827741t_real @ N1 @ N22 ) ) ) ).

% quasinorm_subsetI'
thf(fact_697_finite__measure_Oemeasure__real,axiom,
    ! [M2: sigma_9047027012034273406at_nat,A2: set_Pr2458342521480944603at_nat] :
      ( ( measur5156515106886843841at_nat @ M2 )
     => ? [R2: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ R2 )
          & ( ( sigma_3593834561172453006at_nat @ M2 @ A2 )
            = ( extend7643940197134561352nnreal @ R2 ) ) ) ) ).

% finite_measure.emeasure_real
thf(fact_698_finite__measure_Oemeasure__real,axiom,
    ! [M2: sigma_5515648953823433982at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( measur7911732083032432613at_nat @ M2 )
     => ? [R2: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ R2 )
          & ( ( sigma_411563132819280856at_nat @ M2 @ A2 )
            = ( extend7643940197134561352nnreal @ R2 ) ) ) ) ).

% finite_measure.emeasure_real
thf(fact_699_integral__mono_H,axiom,
    ! [M2: sigma_measure_rat,F2: rat > real,G2: rat > real] :
      ( ( bochne6810293860105484248t_real @ M2 @ F2 )
     => ( ! [X3: rat] :
            ( ( member_rat @ X3 @ ( sigma_space_rat @ M2 ) )
           => ( ord_less_eq_real @ ( G2 @ X3 ) @ ( F2 @ X3 ) ) )
       => ( ! [X3: rat] :
              ( ( member_rat @ X3 @ ( sigma_space_rat @ M2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
         => ( ord_less_eq_real @ ( bochne6128977971683193353t_real @ M2 @ G2 ) @ ( bochne6128977971683193353t_real @ M2 @ F2 ) ) ) ) ) ).

% integral_mono'
thf(fact_700_integral__mono_H,axiom,
    ! [M2: sigma_2424604449015074331t_real,F2: ( product_prod_nat_nat > real ) > real,G2: ( product_prod_nat_nat > real ) > real] :
      ( ( bochne1222508287783722872l_real @ M2 @ F2 )
     => ( ! [X3: product_prod_nat_nat > real] :
            ( ( member4564283293661824327t_real @ X3 @ ( sigma_7766831664147846694t_real @ M2 ) )
           => ( ord_less_eq_real @ ( G2 @ X3 ) @ ( F2 @ X3 ) ) )
       => ( ! [X3: product_prod_nat_nat > real] :
              ( ( member4564283293661824327t_real @ X3 @ ( sigma_7766831664147846694t_real @ M2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
         => ( ord_less_eq_real @ ( bochne2904387722813762055l_real @ M2 @ G2 ) @ ( bochne2904387722813762055l_real @ M2 @ F2 ) ) ) ) ) ).

% integral_mono'
thf(fact_701_integral__mono_H,axiom,
    ! [M2: sigma_5394977995791401948nnreal,F2: ( real > extend8495563244428889912nnreal ) > real,G2: ( real > extend8495563244428889912nnreal ) > real] :
      ( ( bochne4404133754790937219l_real @ M2 @ F2 )
     => ( ! [X3: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ X3 @ ( sigma_2369682286586992763nnreal @ M2 ) )
           => ( ord_less_eq_real @ ( G2 @ X3 ) @ ( F2 @ X3 ) ) )
       => ( ! [X3: real > extend8495563244428889912nnreal] :
              ( ( member2919562650594848410nnreal @ X3 @ ( sigma_2369682286586992763nnreal @ M2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
         => ( ord_less_eq_real @ ( bochne1920604158554476212l_real @ M2 @ G2 ) @ ( bochne1920604158554476212l_real @ M2 @ F2 ) ) ) ) ) ).

% integral_mono'
thf(fact_702_integral__mono_H,axiom,
    ! [M2: sigma_3568842061677812681t_real,F2: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > real,G2: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > real] :
      ( ( bochne6626571995716118384l_real @ M2 @ F2 )
     => ( ! [X3: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
            ( ( member8159409068225774087t_real @ X3 @ ( sigma_8760150869319784680t_real @ M2 ) )
           => ( ord_less_eq_real @ ( G2 @ X3 ) @ ( F2 @ X3 ) ) )
       => ( ! [X3: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
              ( ( member8159409068225774087t_real @ X3 @ ( sigma_8760150869319784680t_real @ M2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
         => ( ord_less_eq_real @ ( bochne7370261254949204129l_real @ M2 @ G2 ) @ ( bochne7370261254949204129l_real @ M2 @ F2 ) ) ) ) ) ).

% integral_mono'
thf(fact_703_integral__mono_H,axiom,
    ! [M2: sigma_measure_o,F2: $o > real,G2: $o > real] :
      ( ( bochne661340805755426930o_real @ M2 @ F2 )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ ( sigma_space_o @ M2 ) )
           => ( ord_less_eq_real @ ( G2 @ X3 ) @ ( F2 @ X3 ) ) )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ ( sigma_space_o @ M2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
         => ( ord_less_eq_real @ ( bochne8006308498830368769o_real @ M2 @ G2 ) @ ( bochne8006308498830368769o_real @ M2 @ F2 ) ) ) ) ) ).

% integral_mono'
thf(fact_704_integral__mono_H,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,G2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( bochne7117808529828525605t_real @ M2 @ F2 )
     => ( ! [X3: product_prod_nat_nat > product_prod_nat_nat] :
            ( ( member8885076297122219836at_nat @ X3 @ ( sigma_4468984517727773725at_nat @ M2 ) )
           => ( ord_less_eq_real @ ( G2 @ X3 ) @ ( F2 @ X3 ) ) )
       => ( ! [X3: product_prod_nat_nat > product_prod_nat_nat] :
              ( ( member8885076297122219836at_nat @ X3 @ ( sigma_4468984517727773725at_nat @ M2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
         => ( ord_less_eq_real @ ( bochne5509773249985062230t_real @ M2 @ G2 ) @ ( bochne5509773249985062230t_real @ M2 @ F2 ) ) ) ) ) ).

% integral_mono'
thf(fact_705_integral__mono_H,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real,G2: product_prod_nat_nat > real] :
      ( ( bochne2596016609597520987t_real @ M2 @ F2 )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ ( sigma_4797346298676585097at_nat @ M2 ) )
           => ( ord_less_eq_real @ ( G2 @ X3 ) @ ( F2 @ X3 ) ) )
       => ( ! [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ ( sigma_4797346298676585097at_nat @ M2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
         => ( ord_less_eq_real @ ( bochne6384019433803981034t_real @ M2 @ G2 ) @ ( bochne6384019433803981034t_real @ M2 @ F2 ) ) ) ) ) ).

% integral_mono'
thf(fact_706_measurable__Pair1,axiom,
    ! [F2: produc233154603053746533al_rat > extend8495563244428889912nnreal,M1: sigma_measure_real,M22: sigma_measure_rat,M2: sigma_7234349610311085201nnreal,Y4: rat] :
      ( ( member4283970898438528823nnreal @ F2 @ ( sigma_8016388916404342093nnreal @ ( binary6812879460589906638al_rat @ M1 @ M22 ) @ M2 ) )
     => ( ( member_rat @ Y4 @ ( sigma_space_rat @ M22 ) )
       => ( member2919562650594848410nnreal
          @ ^ [X2: real] : ( F2 @ ( produc2546372583784539933al_rat @ X2 @ Y4 ) )
          @ ( sigma_9017504469962657078nnreal @ M1 @ M2 ) ) ) ) ).

% measurable_Pair1
thf(fact_707_measurable__Pair1,axiom,
    ! [F2: product_prod_real_o > extend8495563244428889912nnreal,M1: sigma_measure_real,M22: sigma_measure_o,M2: sigma_7234349610311085201nnreal,Y4: $o] :
      ( ( member8834806718775753679nnreal @ F2 @ ( sigma_968900072706916587nnreal @ ( binary651916263276196130real_o @ M1 @ M22 ) @ M2 ) )
     => ( ( member_o @ Y4 @ ( sigma_space_o @ M22 ) )
       => ( member2919562650594848410nnreal
          @ ^ [X2: real] : ( F2 @ ( product_Pair_real_o @ X2 @ Y4 ) )
          @ ( sigma_9017504469962657078nnreal @ M1 @ M2 ) ) ) ) ).

% measurable_Pair1
thf(fact_708_measurable__Pair1,axiom,
    ! [F2: product_prod_nat_nat > real,M1: sigma_measure_nat,M22: sigma_measure_nat,M2: sigma_measure_real,Y4: nat] :
      ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ ( binary6458575275919672938at_nat @ M1 @ M22 ) @ M2 ) )
     => ( ( member_nat @ Y4 @ ( sigma_space_nat @ M22 ) )
       => ( member_nat_real
          @ ^ [X2: nat] : ( F2 @ ( product_Pair_nat_nat @ X2 @ Y4 ) )
          @ ( sigma_1747752005702207822t_real @ M1 @ M2 ) ) ) ) ).

% measurable_Pair1
thf(fact_709_measurable__Pair1,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,M1: sigma_measure_nat,M22: sigma_measure_nat,M2: sigma_5515648953823433982at_nat,Y4: nat] :
      ( ( member8885076297122219836at_nat @ F2 @ ( sigma_3682087626466204304at_nat @ ( binary6458575275919672938at_nat @ M1 @ M22 ) @ M2 ) )
     => ( ( member_nat @ Y4 @ ( sigma_space_nat @ M22 ) )
       => ( member6743353555990375057at_nat
          @ ^ [X2: nat] : ( F2 @ ( product_Pair_nat_nat @ X2 @ Y4 ) )
          @ ( sigma_2856282194129746655at_nat @ M1 @ M2 ) ) ) ) ).

% measurable_Pair1
thf(fact_710_measurable__Pair1,axiom,
    ! [F2: produc4865670479116712712at_rat > real,M1: sigma_5515648953823433982at_nat,M22: sigma_measure_rat,M2: sigma_measure_real,Y4: rat] :
      ( ( member3063989173860411626t_real @ F2 @ ( sigma_1716267658361341702t_real @ ( binary1275730340088535569at_rat @ M1 @ M22 ) @ M2 ) )
     => ( ( member_rat @ Y4 @ ( sigma_space_rat @ M22 ) )
       => ( member4564283293661824327t_real
          @ ^ [X2: product_prod_nat_nat] : ( F2 @ ( produc5715581010483709826at_rat @ X2 @ Y4 ) )
          @ ( sigma_1188828689629184861t_real @ M1 @ M2 ) ) ) ) ).

% measurable_Pair1
thf(fact_711_measurable__Pair1,axiom,
    ! [F2: produc652263959640188542_nat_o > real,M1: sigma_5515648953823433982at_nat,M22: sigma_measure_o,M2: sigma_measure_real,Y4: $o] :
      ( ( member8831524769400975172o_real @ F2 @ ( sigma_4883534621586810586o_real @ ( binary1863826757812796191_nat_o @ M1 @ M22 ) @ M2 ) )
     => ( ( member_o @ Y4 @ ( sigma_space_o @ M22 ) )
       => ( member4564283293661824327t_real
          @ ^ [X2: product_prod_nat_nat] : ( F2 @ ( produc855456346497908590_nat_o @ X2 @ Y4 ) )
          @ ( sigma_1188828689629184861t_real @ M1 @ M2 ) ) ) ) ).

% measurable_Pair1
thf(fact_712_measurable__Pair1,axiom,
    ! [F2: produc340901343777058320nnreal > extend8495563244428889912nnreal,M1: sigma_measure_real,M22: sigma_5394977995791401948nnreal,M2: sigma_7234349610311085201nnreal,Y4: real > extend8495563244428889912nnreal] :
      ( ( member958556886534734818nnreal @ F2 @ ( sigma_3700830109642529784nnreal @ ( binary7428713593190571257nnreal @ M1 @ M22 ) @ M2 ) )
     => ( ( member2919562650594848410nnreal @ Y4 @ ( sigma_2369682286586992763nnreal @ M22 ) )
       => ( member2919562650594848410nnreal
          @ ^ [X2: real] : ( F2 @ ( produc4340527341261162312nnreal @ X2 @ Y4 ) )
          @ ( sigma_9017504469962657078nnreal @ M1 @ M2 ) ) ) ) ).

% measurable_Pair1
thf(fact_713_measurable__Pair1,axiom,
    ! [F2: produc2006704064980238794at_nat > extend8495563244428889912nnreal,M1: sigma_measure_real,M22: sigma_5515648953823433982at_nat,M2: sigma_7234349610311085201nnreal,Y4: product_prod_nat_nat] :
      ( ( member7845912703658599608nnreal @ F2 @ ( sigma_1225396849305551572nnreal @ ( binary2500810019709002891at_nat @ M1 @ M22 ) @ M2 ) )
     => ( ( member8440522571783428010at_nat @ Y4 @ ( sigma_4797346298676585097at_nat @ M22 ) )
       => ( member2919562650594848410nnreal
          @ ^ [X2: real] : ( F2 @ ( produc423147238207498620at_nat @ X2 @ Y4 ) )
          @ ( sigma_9017504469962657078nnreal @ M1 @ M2 ) ) ) ) ).

% measurable_Pair1
thf(fact_714_measurable__Pair1,axiom,
    ! [F2: produc4865670479116712712at_rat > product_prod_nat_nat,M1: sigma_5515648953823433982at_nat,M22: sigma_measure_rat,M2: sigma_5515648953823433982at_nat,Y4: rat] :
      ( ( member9178413912459831897at_nat @ F2 @ ( sigma_2126037268676729639at_nat @ ( binary1275730340088535569at_rat @ M1 @ M22 ) @ M2 ) )
     => ( ( member_rat @ Y4 @ ( sigma_space_rat @ M22 ) )
       => ( member8885076297122219836at_nat
          @ ^ [X2: product_prod_nat_nat] : ( F2 @ ( produc5715581010483709826at_rat @ X2 @ Y4 ) )
          @ ( sigma_3682087626466204304at_nat @ M1 @ M2 ) ) ) ) ).

% measurable_Pair1
thf(fact_715_measurable__Pair1,axiom,
    ! [F2: produc652263959640188542_nat_o > product_prod_nat_nat,M1: sigma_5515648953823433982at_nat,M22: sigma_measure_o,M2: sigma_5515648953823433982at_nat,Y4: $o] :
      ( ( member5715356904927313535at_nat @ F2 @ ( sigma_977161685228763219at_nat @ ( binary1863826757812796191_nat_o @ M1 @ M22 ) @ M2 ) )
     => ( ( member_o @ Y4 @ ( sigma_space_o @ M22 ) )
       => ( member8885076297122219836at_nat
          @ ^ [X2: product_prod_nat_nat] : ( F2 @ ( produc855456346497908590_nat_o @ X2 @ Y4 ) )
          @ ( sigma_3682087626466204304at_nat @ M1 @ M2 ) ) ) ) ).

% measurable_Pair1
thf(fact_716_measurable__Pair2,axiom,
    ! [F2: produc3315746662415021285t_real > extend8495563244428889912nnreal,M1: sigma_measure_rat,M22: sigma_measure_real,M2: sigma_7234349610311085201nnreal,X5: rat] :
      ( ( member4168167158652291767nnreal @ F2 @ ( sigma_7900585176618105037nnreal @ ( binary7703388793893277262t_real @ M1 @ M22 ) @ M2 ) )
     => ( ( member_rat @ X5 @ ( sigma_space_rat @ M1 ) )
       => ( member2919562650594848410nnreal
          @ ^ [Y3: real] : ( F2 @ ( produc3436881917087910557t_real @ X5 @ Y3 ) )
          @ ( sigma_9017504469962657078nnreal @ M22 @ M2 ) ) ) ) ).

% measurable_Pair2
thf(fact_717_measurable__Pair2,axiom,
    ! [F2: product_prod_o_real > extend8495563244428889912nnreal,M1: sigma_measure_o,M22: sigma_measure_real,M2: sigma_7234349610311085201nnreal,X5: $o] :
      ( ( member2741464431984739937nnreal @ F2 @ ( sigma_4098929822770678653nnreal @ ( binary8366223925089802364o_real @ M1 @ M22 ) @ M2 ) )
     => ( ( member_o @ X5 @ ( sigma_space_o @ M1 ) )
       => ( member2919562650594848410nnreal
          @ ^ [Y3: real] : ( F2 @ ( product_Pair_o_real @ X5 @ Y3 ) )
          @ ( sigma_9017504469962657078nnreal @ M22 @ M2 ) ) ) ) ).

% measurable_Pair2
thf(fact_718_measurable__Pair2,axiom,
    ! [F2: product_prod_nat_nat > real,M1: sigma_measure_nat,M22: sigma_measure_nat,M2: sigma_measure_real,X5: nat] :
      ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ ( binary6458575275919672938at_nat @ M1 @ M22 ) @ M2 ) )
     => ( ( member_nat @ X5 @ ( sigma_space_nat @ M1 ) )
       => ( member_nat_real
          @ ^ [Y3: nat] : ( F2 @ ( product_Pair_nat_nat @ X5 @ Y3 ) )
          @ ( sigma_1747752005702207822t_real @ M22 @ M2 ) ) ) ) ).

% measurable_Pair2
thf(fact_719_measurable__Pair2,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,M1: sigma_measure_nat,M22: sigma_measure_nat,M2: sigma_5515648953823433982at_nat,X5: nat] :
      ( ( member8885076297122219836at_nat @ F2 @ ( sigma_3682087626466204304at_nat @ ( binary6458575275919672938at_nat @ M1 @ M22 ) @ M2 ) )
     => ( ( member_nat @ X5 @ ( sigma_space_nat @ M1 ) )
       => ( member6743353555990375057at_nat
          @ ^ [Y3: nat] : ( F2 @ ( product_Pair_nat_nat @ X5 @ Y3 ) )
          @ ( sigma_2856282194129746655at_nat @ M22 @ M2 ) ) ) ) ).

% measurable_Pair2
thf(fact_720_measurable__Pair2,axiom,
    ! [F2: produc1354869487384554270at_nat > real,M1: sigma_measure_rat,M22: sigma_5515648953823433982at_nat,M2: sigma_measure_real,X5: rat] :
      ( ( member2709382946904397312t_real @ F2 @ ( sigma_1361661431405327388t_real @ ( binary8600737263914268127at_nat @ M1 @ M22 ) @ M2 ) )
     => ( ( member_rat @ X5 @ ( sigma_space_rat @ M1 ) )
       => ( member4564283293661824327t_real
          @ ^ [Y3: product_prod_nat_nat] : ( F2 @ ( produc3817215897454666576at_nat @ X5 @ Y3 ) )
          @ ( sigma_1188828689629184861t_real @ M22 @ M2 ) ) ) ) ).

% measurable_Pair2
thf(fact_721_measurable__Pair2,axiom,
    ! [F2: produc5146536252030154256at_nat > real,M1: sigma_measure_o,M22: sigma_5515648953823433982at_nat,M2: sigma_measure_real,X5: $o] :
      ( ( member3854150731224862166t_real @ F2 @ ( sigma_9129532620265473388t_real @ ( binary2026124553305310001at_nat @ M1 @ M22 ) @ M2 ) )
     => ( ( member_o @ X5 @ ( sigma_space_o @ M1 ) )
       => ( member4564283293661824327t_real
          @ ^ [Y3: product_prod_nat_nat] : ( F2 @ ( produc1017754141990422400at_nat @ X5 @ Y3 ) )
          @ ( sigma_1188828689629184861t_real @ M22 @ M2 ) ) ) ) ).

% measurable_Pair2
thf(fact_722_measurable__Pair2,axiom,
    ! [F2: produc5979837638821899152l_real > extend8495563244428889912nnreal,M1: sigma_5394977995791401948nnreal,M22: sigma_measure_real,M2: sigma_7234349610311085201nnreal,X5: real > extend8495563244428889912nnreal] :
      ( ( member1302632477427941730nnreal @ F2 @ ( sigma_4044905700535736696nnreal @ ( binary398587456444785913l_real @ M1 @ M22 ) @ M2 ) )
     => ( ( member2919562650594848410nnreal @ X5 @ ( sigma_2369682286586992763nnreal @ M1 ) )
       => ( member2919562650594848410nnreal
          @ ^ [Y3: real] : ( F2 @ ( produc6533773241370152776l_real @ X5 @ Y3 ) )
          @ ( sigma_9017504469962657078nnreal @ M22 @ M2 ) ) ) ) ).

% measurable_Pair2
thf(fact_723_measurable__Pair2,axiom,
    ! [F2: produc6292279318250605404t_real > extend8495563244428889912nnreal,M1: sigma_5515648953823433982at_nat,M22: sigma_measure_real,M2: sigma_7234349610311085201nnreal,X5: product_prod_nat_nat] :
      ( ( member6532988213140674122nnreal @ F2 @ ( sigma_9135844395642401894nnreal @ ( binary2981866668218350181t_real @ M1 @ M22 ) @ M2 ) )
     => ( ( member8440522571783428010at_nat @ X5 @ ( sigma_4797346298676585097at_nat @ M1 ) )
       => ( member2919562650594848410nnreal
          @ ^ [Y3: real] : ( F2 @ ( produc904203886716845910t_real @ X5 @ Y3 ) )
          @ ( sigma_9017504469962657078nnreal @ M22 @ M2 ) ) ) ) ).

% measurable_Pair2
thf(fact_724_measurable__Pair2,axiom,
    ! [F2: produc1354869487384554270at_nat > product_prod_nat_nat,M1: sigma_measure_rat,M22: sigma_5515648953823433982at_nat,M2: sigma_5515648953823433982at_nat,X5: rat] :
      ( ( member7164277158852124355at_nat @ F2 @ ( sigma_111900515069022097at_nat @ ( binary8600737263914268127at_nat @ M1 @ M22 ) @ M2 ) )
     => ( ( member_rat @ X5 @ ( sigma_space_rat @ M1 ) )
       => ( member8885076297122219836at_nat
          @ ^ [Y3: product_prod_nat_nat] : ( F2 @ ( produc3817215897454666576at_nat @ X5 @ Y3 ) )
          @ ( sigma_3682087626466204304at_nat @ M22 @ M2 ) ) ) ) ).

% measurable_Pair2
thf(fact_725_measurable__Pair2,axiom,
    ! [F2: produc5146536252030154256at_nat > product_prod_nat_nat,M1: sigma_measure_o,M22: sigma_5515648953823433982at_nat,M2: sigma_5515648953823433982at_nat,X5: $o] :
      ( ( member1082810179976869997at_nat @ F2 @ ( sigma_5567986997133095489at_nat @ ( binary2026124553305310001at_nat @ M1 @ M22 ) @ M2 ) )
     => ( ( member_o @ X5 @ ( sigma_space_o @ M1 ) )
       => ( member8885076297122219836at_nat
          @ ^ [Y3: product_prod_nat_nat] : ( F2 @ ( produc1017754141990422400at_nat @ X5 @ Y3 ) )
          @ ( sigma_3682087626466204304at_nat @ M22 @ M2 ) ) ) ) ).

% measurable_Pair2
thf(fact_726_measurable__compose__Pair1,axiom,
    ! [X5: rat,M1: sigma_measure_rat,G2: produc3315746662415021285t_real > extend8495563244428889912nnreal,M22: sigma_measure_real,L2: sigma_7234349610311085201nnreal] :
      ( ( member_rat @ X5 @ ( sigma_space_rat @ M1 ) )
     => ( ( member4168167158652291767nnreal @ G2 @ ( sigma_7900585176618105037nnreal @ ( binary7703388793893277262t_real @ M1 @ M22 ) @ L2 ) )
       => ( member2919562650594848410nnreal
          @ ^ [Y3: real] : ( G2 @ ( produc3436881917087910557t_real @ X5 @ Y3 ) )
          @ ( sigma_9017504469962657078nnreal @ M22 @ L2 ) ) ) ) ).

% measurable_compose_Pair1
thf(fact_727_measurable__compose__Pair1,axiom,
    ! [X5: $o,M1: sigma_measure_o,G2: product_prod_o_real > extend8495563244428889912nnreal,M22: sigma_measure_real,L2: sigma_7234349610311085201nnreal] :
      ( ( member_o @ X5 @ ( sigma_space_o @ M1 ) )
     => ( ( member2741464431984739937nnreal @ G2 @ ( sigma_4098929822770678653nnreal @ ( binary8366223925089802364o_real @ M1 @ M22 ) @ L2 ) )
       => ( member2919562650594848410nnreal
          @ ^ [Y3: real] : ( G2 @ ( product_Pair_o_real @ X5 @ Y3 ) )
          @ ( sigma_9017504469962657078nnreal @ M22 @ L2 ) ) ) ) ).

% measurable_compose_Pair1
thf(fact_728_measurable__compose__Pair1,axiom,
    ! [X5: nat,M1: sigma_measure_nat,G2: product_prod_nat_nat > real,M22: sigma_measure_nat,L2: sigma_measure_real] :
      ( ( member_nat @ X5 @ ( sigma_space_nat @ M1 ) )
     => ( ( member4564283293661824327t_real @ G2 @ ( sigma_1188828689629184861t_real @ ( binary6458575275919672938at_nat @ M1 @ M22 ) @ L2 ) )
       => ( member_nat_real
          @ ^ [Y3: nat] : ( G2 @ ( product_Pair_nat_nat @ X5 @ Y3 ) )
          @ ( sigma_1747752005702207822t_real @ M22 @ L2 ) ) ) ) ).

% measurable_compose_Pair1
thf(fact_729_measurable__compose__Pair1,axiom,
    ! [X5: nat,M1: sigma_measure_nat,G2: product_prod_nat_nat > product_prod_nat_nat,M22: sigma_measure_nat,L2: sigma_5515648953823433982at_nat] :
      ( ( member_nat @ X5 @ ( sigma_space_nat @ M1 ) )
     => ( ( member8885076297122219836at_nat @ G2 @ ( sigma_3682087626466204304at_nat @ ( binary6458575275919672938at_nat @ M1 @ M22 ) @ L2 ) )
       => ( member6743353555990375057at_nat
          @ ^ [Y3: nat] : ( G2 @ ( product_Pair_nat_nat @ X5 @ Y3 ) )
          @ ( sigma_2856282194129746655at_nat @ M22 @ L2 ) ) ) ) ).

% measurable_compose_Pair1
thf(fact_730_measurable__compose__Pair1,axiom,
    ! [X5: rat,M1: sigma_measure_rat,G2: produc1354869487384554270at_nat > real,M22: sigma_5515648953823433982at_nat,L2: sigma_measure_real] :
      ( ( member_rat @ X5 @ ( sigma_space_rat @ M1 ) )
     => ( ( member2709382946904397312t_real @ G2 @ ( sigma_1361661431405327388t_real @ ( binary8600737263914268127at_nat @ M1 @ M22 ) @ L2 ) )
       => ( member4564283293661824327t_real
          @ ^ [Y3: product_prod_nat_nat] : ( G2 @ ( produc3817215897454666576at_nat @ X5 @ Y3 ) )
          @ ( sigma_1188828689629184861t_real @ M22 @ L2 ) ) ) ) ).

% measurable_compose_Pair1
thf(fact_731_measurable__compose__Pair1,axiom,
    ! [X5: $o,M1: sigma_measure_o,G2: produc5146536252030154256at_nat > real,M22: sigma_5515648953823433982at_nat,L2: sigma_measure_real] :
      ( ( member_o @ X5 @ ( sigma_space_o @ M1 ) )
     => ( ( member3854150731224862166t_real @ G2 @ ( sigma_9129532620265473388t_real @ ( binary2026124553305310001at_nat @ M1 @ M22 ) @ L2 ) )
       => ( member4564283293661824327t_real
          @ ^ [Y3: product_prod_nat_nat] : ( G2 @ ( produc1017754141990422400at_nat @ X5 @ Y3 ) )
          @ ( sigma_1188828689629184861t_real @ M22 @ L2 ) ) ) ) ).

% measurable_compose_Pair1
thf(fact_732_measurable__compose__Pair1,axiom,
    ! [X5: real > extend8495563244428889912nnreal,M1: sigma_5394977995791401948nnreal,G2: produc5979837638821899152l_real > extend8495563244428889912nnreal,M22: sigma_measure_real,L2: sigma_7234349610311085201nnreal] :
      ( ( member2919562650594848410nnreal @ X5 @ ( sigma_2369682286586992763nnreal @ M1 ) )
     => ( ( member1302632477427941730nnreal @ G2 @ ( sigma_4044905700535736696nnreal @ ( binary398587456444785913l_real @ M1 @ M22 ) @ L2 ) )
       => ( member2919562650594848410nnreal
          @ ^ [Y3: real] : ( G2 @ ( produc6533773241370152776l_real @ X5 @ Y3 ) )
          @ ( sigma_9017504469962657078nnreal @ M22 @ L2 ) ) ) ) ).

% measurable_compose_Pair1
thf(fact_733_measurable__compose__Pair1,axiom,
    ! [X5: product_prod_nat_nat,M1: sigma_5515648953823433982at_nat,G2: produc6292279318250605404t_real > extend8495563244428889912nnreal,M22: sigma_measure_real,L2: sigma_7234349610311085201nnreal] :
      ( ( member8440522571783428010at_nat @ X5 @ ( sigma_4797346298676585097at_nat @ M1 ) )
     => ( ( member6532988213140674122nnreal @ G2 @ ( sigma_9135844395642401894nnreal @ ( binary2981866668218350181t_real @ M1 @ M22 ) @ L2 ) )
       => ( member2919562650594848410nnreal
          @ ^ [Y3: real] : ( G2 @ ( produc904203886716845910t_real @ X5 @ Y3 ) )
          @ ( sigma_9017504469962657078nnreal @ M22 @ L2 ) ) ) ) ).

% measurable_compose_Pair1
thf(fact_734_measurable__compose__Pair1,axiom,
    ! [X5: rat,M1: sigma_measure_rat,G2: produc1354869487384554270at_nat > product_prod_nat_nat,M22: sigma_5515648953823433982at_nat,L2: sigma_5515648953823433982at_nat] :
      ( ( member_rat @ X5 @ ( sigma_space_rat @ M1 ) )
     => ( ( member7164277158852124355at_nat @ G2 @ ( sigma_111900515069022097at_nat @ ( binary8600737263914268127at_nat @ M1 @ M22 ) @ L2 ) )
       => ( member8885076297122219836at_nat
          @ ^ [Y3: product_prod_nat_nat] : ( G2 @ ( produc3817215897454666576at_nat @ X5 @ Y3 ) )
          @ ( sigma_3682087626466204304at_nat @ M22 @ L2 ) ) ) ) ).

% measurable_compose_Pair1
thf(fact_735_measurable__compose__Pair1,axiom,
    ! [X5: $o,M1: sigma_measure_o,G2: produc5146536252030154256at_nat > product_prod_nat_nat,M22: sigma_5515648953823433982at_nat,L2: sigma_5515648953823433982at_nat] :
      ( ( member_o @ X5 @ ( sigma_space_o @ M1 ) )
     => ( ( member1082810179976869997at_nat @ G2 @ ( sigma_5567986997133095489at_nat @ ( binary2026124553305310001at_nat @ M1 @ M22 ) @ L2 ) )
       => ( member8885076297122219836at_nat
          @ ^ [Y3: product_prod_nat_nat] : ( G2 @ ( produc1017754141990422400at_nat @ X5 @ Y3 ) )
          @ ( sigma_3682087626466204304at_nat @ M22 @ L2 ) ) ) ) ).

% measurable_compose_Pair1
thf(fact_736_prob__space_Oindep__var__integrable,axiom,
    ! [M2: sigma_9047027012034273406at_nat,X1: ( product_prod_nat_nat > product_prod_nat_nat ) > real,X22: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe1938972608428940447t_real @ M2 @ borel_5078946678739801102l_real @ X1 @ borel_5078946678739801102l_real @ X22 )
       => ( ( bochne7117808529828525605t_real @ M2 @ X1 )
         => ( ( bochne7117808529828525605t_real @ M2 @ X22 )
           => ( bochne7117808529828525605t_real @ M2
              @ ^ [Omega: product_prod_nat_nat > product_prod_nat_nat] : ( times_times_real @ ( X1 @ Omega ) @ ( X22 @ Omega ) ) ) ) ) ) ) ).

% prob_space.indep_var_integrable
thf(fact_737_prob__space_Oindep__var__integrable,axiom,
    ! [M2: sigma_5515648953823433982at_nat,X1: product_prod_nat_nat > real,X22: product_prod_nat_nat > real] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe1845147739501508449t_real @ M2 @ borel_5078946678739801102l_real @ X1 @ borel_5078946678739801102l_real @ X22 )
       => ( ( bochne2596016609597520987t_real @ M2 @ X1 )
         => ( ( bochne2596016609597520987t_real @ M2 @ X22 )
           => ( bochne2596016609597520987t_real @ M2
              @ ^ [Omega: product_prod_nat_nat] : ( times_times_real @ ( X1 @ Omega ) @ ( X22 @ Omega ) ) ) ) ) ) ) ).

% prob_space.indep_var_integrable
thf(fact_738_prob__space_Oindep__var__lebesgue__integral,axiom,
    ! [M2: sigma_9047027012034273406at_nat,X1: ( product_prod_nat_nat > product_prod_nat_nat ) > real,X22: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe1938972608428940447t_real @ M2 @ borel_5078946678739801102l_real @ X1 @ borel_5078946678739801102l_real @ X22 )
       => ( ( bochne7117808529828525605t_real @ M2 @ X1 )
         => ( ( bochne7117808529828525605t_real @ M2 @ X22 )
           => ( ( bochne5509773249985062230t_real @ M2
                @ ^ [Omega: product_prod_nat_nat > product_prod_nat_nat] : ( times_times_real @ ( X1 @ Omega ) @ ( X22 @ Omega ) ) )
              = ( times_times_real @ ( bochne5509773249985062230t_real @ M2 @ X1 ) @ ( bochne5509773249985062230t_real @ M2 @ X22 ) ) ) ) ) ) ) ).

% prob_space.indep_var_lebesgue_integral
thf(fact_739_prob__space_Oindep__var__lebesgue__integral,axiom,
    ! [M2: sigma_5515648953823433982at_nat,X1: product_prod_nat_nat > real,X22: product_prod_nat_nat > real] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe1845147739501508449t_real @ M2 @ borel_5078946678739801102l_real @ X1 @ borel_5078946678739801102l_real @ X22 )
       => ( ( bochne2596016609597520987t_real @ M2 @ X1 )
         => ( ( bochne2596016609597520987t_real @ M2 @ X22 )
           => ( ( bochne6384019433803981034t_real @ M2
                @ ^ [Omega: product_prod_nat_nat] : ( times_times_real @ ( X1 @ Omega ) @ ( X22 @ Omega ) ) )
              = ( times_times_real @ ( bochne6384019433803981034t_real @ M2 @ X1 ) @ ( bochne6384019433803981034t_real @ M2 @ X22 ) ) ) ) ) ) ) ).

% prob_space.indep_var_lebesgue_integral
thf(fact_740_Lp__D_I1_J,axiom,
    ! [P2: real,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,M2: sigma_9047027012034273406at_nat] :
      ( ( ord_less_real @ zero_zero_real @ P2 )
     => ( ( member8159409068225774087t_real @ F2 @ ( functi8590574888837585149t_real @ ( lp_spa812643495979991928at_nat @ ( extend7643940197134561352nnreal @ P2 ) @ M2 ) ) )
       => ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ M2 @ borel_5078946678739801102l_real ) ) ) ) ).

% Lp_D(1)
thf(fact_741_Lp__D_I1_J,axiom,
    ! [P2: real,F2: product_prod_nat_nat > real,M2: sigma_5515648953823433982at_nat] :
      ( ( ord_less_real @ zero_zero_real @ P2 )
     => ( ( member4564283293661824327t_real @ F2 @ ( functi6492146676480127185t_real @ ( lp_spa8733284142207339630at_nat @ ( extend7643940197134561352nnreal @ P2 ) @ M2 ) ) )
       => ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ M2 @ borel_5078946678739801102l_real ) ) ) ) ).

% Lp_D(1)
thf(fact_742_spaceN__contains__zero,axiom,
    ! [N: functi4972691055096080419m_real] : ( member_real @ zero_zero_real @ ( functi7867165335332326841N_real @ N ) ) ).

% spaceN_contains_zero
thf(fact_743_spaceN__contains__zero,axiom,
    ! [N: functi5165143733218492903t_real] : ( member8159409068225774087t_real @ zero_z7307146680010048121t_real @ ( functi8590574888837585149t_real @ N ) ) ).

% spaceN_contains_zero
thf(fact_744_spaceN__contains__zero,axiom,
    ! [N: functi4297716468133224573t_real] : ( member4564283293661824327t_real @ zero_z5900165307960944725t_real @ ( functi6492146676480127185t_real @ N ) ) ).

% spaceN_contains_zero
thf(fact_745_Norm__nonneg,axiom,
    ! [N: functi5165143733218492903t_real,X5: ( product_prod_nat_nat > product_prod_nat_nat ) > real] : ( ord_less_eq_real @ zero_zero_real @ ( functi7696537667341841001t_real @ N @ X5 ) ) ).

% Norm_nonneg
thf(fact_746_Norm__nonneg,axiom,
    ! [N: functi4297716468133224573t_real,X5: product_prod_nat_nat > real] : ( ord_less_eq_real @ zero_zero_real @ ( functi1179182286487682149t_real @ N @ X5 ) ) ).

% Norm_nonneg
thf(fact_747_eNorm__Norm_H,axiom,
    ! [X5: ( product_prod_nat_nat > product_prod_nat_nat ) > real,N: functi5165143733218492903t_real] :
      ( ~ ( member8159409068225774087t_real @ X5 @ ( functi8590574888837585149t_real @ N ) )
     => ( ( functi7696537667341841001t_real @ N @ X5 )
        = zero_zero_real ) ) ).

% eNorm_Norm'
thf(fact_748_eNorm__Norm_H,axiom,
    ! [X5: product_prod_nat_nat > real,N: functi4297716468133224573t_real] :
      ( ~ ( member4564283293661824327t_real @ X5 @ ( functi6492146676480127185t_real @ N ) )
     => ( ( functi1179182286487682149t_real @ N @ X5 )
        = zero_zero_real ) ) ).

% eNorm_Norm'
thf(fact_749_quasinorm__subset__space,axiom,
    ! [N1: functi5165143733218492903t_real,N22: functi5165143733218492903t_real] :
      ( ( ord_le1044648919696853383t_real @ N1 @ N22 )
     => ( ord_le8622349663015078278t_real @ ( functi8590574888837585149t_real @ N1 ) @ ( functi8590574888837585149t_real @ N22 ) ) ) ).

% quasinorm_subset_space
thf(fact_750_quasinorm__subset__space,axiom,
    ! [N1: functi4297716468133224573t_real,N22: functi4297716468133224573t_real] :
      ( ( ord_le6049248562894827741t_real @ N1 @ N22 )
     => ( ord_le1994730768851292446t_real @ ( functi6492146676480127185t_real @ N1 ) @ ( functi6492146676480127185t_real @ N22 ) ) ) ).

% quasinorm_subset_space
thf(fact_751_borel__measurable__ennreal__iff,axiom,
    ! [M2: sigma_measure_rat,F2: rat > real] :
      ( ! [X3: rat] :
          ( ( member_rat @ X3 @ ( sigma_space_rat @ M2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
     => ( ( member8415191394374006982nnreal
          @ ^ [X2: rat] : ( extend7643940197134561352nnreal @ ( F2 @ X2 ) )
          @ ( sigma_6438222577076525282nnreal @ M2 @ borel_6524799422816628122nnreal ) )
        = ( member_rat_real @ F2 @ ( sigma_6570439852047981398t_real @ M2 @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_ennreal_iff
thf(fact_752_borel__measurable__ennreal__iff,axiom,
    ! [M2: sigma_2424604449015074331t_real,F2: ( product_prod_nat_nat > real ) > real] :
      ( ! [X3: product_prod_nat_nat > real] :
          ( ( member4564283293661824327t_real @ X3 @ ( sigma_7766831664147846694t_real @ M2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
     => ( ( member2974749651384298352nnreal
          @ ^ [X2: product_prod_nat_nat > real] : ( extend7643940197134561352nnreal @ ( F2 @ X2 ) )
          @ ( sigma_2456675030176926214nnreal @ M2 @ borel_6524799422816628122nnreal ) )
        = ( member7144638408329579108l_real @ F2 @ ( sigma_8243241406658468474l_real @ M2 @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_ennreal_iff
thf(fact_753_borel__measurable__ennreal__iff,axiom,
    ! [M2: sigma_5394977995791401948nnreal,F2: ( real > extend8495563244428889912nnreal ) > real] :
      ( ! [X3: real > extend8495563244428889912nnreal] :
          ( ( member2919562650594848410nnreal @ X3 @ ( sigma_2369682286586992763nnreal @ M2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
     => ( ( member753266501825730289nnreal
          @ ^ [X2: real > extend8495563244428889912nnreal] : ( extend7643940197134561352nnreal @ ( F2 @ X2 ) )
          @ ( sigma_1327742956433181965nnreal @ M2 @ borel_6524799422816628122nnreal ) )
        = ( member6501617229604179301l_real @ F2 @ ( sigma_5459402728545780225l_real @ M2 @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_ennreal_iff
thf(fact_754_borel__measurable__ennreal__iff,axiom,
    ! [M2: sigma_3568842061677812681t_real,F2: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > real] :
      ( ! [X3: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
          ( ( member8159409068225774087t_real @ X3 @ ( sigma_8760150869319784680t_real @ M2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
     => ( ( member5002615375154731358nnreal
          @ ^ [X2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] : ( extend7643940197134561352nnreal @ ( F2 @ X2 ) )
          @ ( sigma_1072441452098868858nnreal @ M2 @ borel_6524799422816628122nnreal ) )
        = ( member4884756941283332434l_real @ F2 @ ( sigma_7001161438373432558l_real @ M2 @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_ennreal_iff
thf(fact_755_borel__measurable__ennreal__iff,axiom,
    ! [M2: sigma_measure_o,F2: $o > real] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( sigma_space_o @ M2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
     => ( ( member5265953103328284778nnreal
          @ ^ [X2: $o] : ( extend7643940197134561352nnreal @ ( F2 @ X2 ) )
          @ ( sigma_6459699357617223168nnreal @ M2 @ borel_6524799422816628122nnreal ) )
        = ( member_o_real @ F2 @ ( sigma_2430008634441611636o_real @ M2 @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_ennreal_iff
thf(fact_756_borel__measurable__ennreal__iff,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ! [X3: product_prod_nat_nat > product_prod_nat_nat] :
          ( ( member8885076297122219836at_nat @ X3 @ ( sigma_4468984517727773725at_nat @ M2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
     => ( ( member6378568689046301843nnreal
          @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( extend7643940197134561352nnreal @ ( F2 @ X2 ) )
          @ ( sigma_2988370261648260015nnreal @ M2 @ borel_6524799422816628122nnreal ) )
        = ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ M2 @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_ennreal_iff
thf(fact_757_borel__measurable__ennreal__iff,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ ( sigma_4797346298676585097at_nat @ M2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
     => ( ( member8497640837515055059nnreal
          @ ^ [X2: product_prod_nat_nat] : ( extend7643940197134561352nnreal @ ( F2 @ X2 ) )
          @ ( sigma_1808371603960765545nnreal @ M2 @ borel_6524799422816628122nnreal ) )
        = ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ M2 @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_ennreal_iff
thf(fact_758_borel__measurable__ennreal__iff,axiom,
    ! [M2: sigma_measure_real,F2: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( sigma_space_real @ M2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
     => ( ( member2919562650594848410nnreal
          @ ^ [X2: real] : ( extend7643940197134561352nnreal @ ( F2 @ X2 ) )
          @ ( sigma_9017504469962657078nnreal @ M2 @ borel_6524799422816628122nnreal ) )
        = ( member_real_real @ F2 @ ( sigma_5267869275261027754l_real @ M2 @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_ennreal_iff
thf(fact_759_ennreal__le__1,axiom,
    ! [X5: real] :
      ( ( ord_le3935885782089961368nnreal @ ( extend7643940197134561352nnreal @ X5 ) @ one_on2969667320475766781nnreal )
      = ( ord_less_eq_real @ X5 @ one_one_real ) ) ).

% ennreal_le_1
thf(fact_760_ennreal__ge__1,axiom,
    ! [X5: real] :
      ( ( ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ ( extend7643940197134561352nnreal @ X5 ) )
      = ( ord_less_eq_real @ one_one_real @ X5 ) ) ).

% ennreal_ge_1
thf(fact_761__092_060Omega_062_092_060_094sub_0622_Osubprob__space__distr,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat,M: sigma_9047027012034273406at_nat] :
      ( ( member2646312932426349184at_nat @ F2 @ ( sigma_1496667973943888668at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M ) )
     => ( ( ( sigma_4468984517727773725at_nat @ M )
         != bot_bo2540092574283496047at_nat )
       => ( giry_s2972128636810349169at_nat @ ( measur7731843055261497284at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ F2 ) ) ) ) ).

% \<Omega>\<^sub>2.subprob_space_distr
thf(fact_762__092_060Omega_062_092_060_094sub_0622_Osubprob__space__distr,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat,M: sigma_5515648953823433982at_nat] :
      ( ( member7151659165045815932at_nat @ F2 @ ( sigma_3543877168342358986at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M ) )
     => ( ( ( sigma_4797346298676585097at_nat @ M )
         != bot_bo2099793752762293965at_nat )
       => ( giry_s1779952876826699573at_nat @ ( measur4728020197420827938at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ F2 ) ) ) ) ).

% \<Omega>\<^sub>2.subprob_space_distr
thf(fact_763__092_060Omega_062_092_060_094sub_0622_Osubprob__space__distr,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,M: sigma_measure_real] :
      ( ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M ) )
     => ( ( ( sigma_space_real @ M )
         != bot_bot_set_real )
       => ( giry_s8208748868292234104e_real @ ( measur6433597746558672971t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ F2 ) ) ) ) ).

% \<Omega>\<^sub>2.subprob_space_distr
thf(fact_764_ennreal__less__one__iff,axiom,
    ! [X5: real] :
      ( ( ord_le7381754540660121996nnreal @ ( extend7643940197134561352nnreal @ X5 ) @ one_on2969667320475766781nnreal )
      = ( ord_less_real @ X5 @ one_one_real ) ) ).

% ennreal_less_one_iff
thf(fact_765_one__less__ennreal,axiom,
    ! [X5: real] :
      ( ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ ( extend7643940197134561352nnreal @ X5 ) )
      = ( ord_less_real @ one_one_real @ X5 ) ) ).

% one_less_ennreal
thf(fact_766_ennreal__less__zero__iff,axiom,
    ! [X5: real] :
      ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( extend7643940197134561352nnreal @ X5 ) )
      = ( ord_less_real @ zero_zero_real @ X5 ) ) ).

% ennreal_less_zero_iff
thf(fact_767_not__empty,axiom,
    ( ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) )
   != bot_bo2099793752762293965at_nat ) ).

% not_empty
thf(fact_768_subprob__space__axioms,axiom,
    giry_s1779952876826699573at_nat @ ( freque5010624893710627907mega_1 @ as ) ).

% subprob_space_axioms
thf(fact_769_emeasure__space__1,axiom,
    ( ( sigma_411563132819280856at_nat @ ( freque5010624893710627907mega_1 @ as ) @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
    = one_on2969667320475766781nnreal ) ).

% emeasure_space_1
thf(fact_770_emeasure__space__le__1,axiom,
    ord_le3935885782089961368nnreal @ ( sigma_411563132819280856at_nat @ ( freque5010624893710627907mega_1 @ as ) @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) ) @ one_on2969667320475766781nnreal ).

% emeasure_space_le_1
thf(fact_771_emeasure__ge__1__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ ( sigma_411563132819280856at_nat @ ( freque5010624893710627907mega_1 @ as ) @ A2 ) )
      = ( ( sigma_411563132819280856at_nat @ ( freque5010624893710627907mega_1 @ as ) @ A2 )
        = one_on2969667320475766781nnreal ) ) ).

% emeasure_ge_1_iff
thf(fact_772_emeasure__le__1,axiom,
    ! [S2: set_Pr1261947904930325089at_nat] : ( ord_le3935885782089961368nnreal @ ( sigma_411563132819280856at_nat @ ( freque5010624893710627907mega_1 @ as ) @ S2 ) @ one_on2969667320475766781nnreal ) ).

% emeasure_le_1
thf(fact_773_measure__le__1,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ( ord_le3935885782089961368nnreal @ ( sigma_411563132819280856at_nat @ ( freque5010624893710627907mega_1 @ as ) @ X ) @ one_on2969667320475766781nnreal ) ).

% measure_le_1
thf(fact_774_emeasure__real,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
    ? [R2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R2 )
      & ( ( sigma_411563132819280856at_nat @ ( freque5010624893710627907mega_1 @ as ) @ A2 )
        = ( extend7643940197134561352nnreal @ R2 ) ) ) ).

% emeasure_real
thf(fact_775_subprob__space__distr,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,M: sigma_9047027012034273406at_nat] :
      ( ( member8598435083969174976at_nat @ F2 @ ( sigma_7146669294626004438at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M ) )
     => ( ( ( sigma_4468984517727773725at_nat @ M )
         != bot_bo2540092574283496047at_nat )
       => ( giry_s2972128636810349169at_nat @ ( measur8330812323704473390at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M @ F2 ) ) ) ) ).

% subprob_space_distr
thf(fact_776_subprob__space__distr,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,M: sigma_5515648953823433982at_nat] :
      ( ( member8885076297122219836at_nat @ F2 @ ( sigma_3682087626466204304at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M ) )
     => ( ( ( sigma_4797346298676585097at_nat @ M )
         != bot_bo2099793752762293965at_nat )
       => ( giry_s1779952876826699573at_nat @ ( measur6503400250167368504at_nat @ ( freque5010624893710627907mega_1 @ as ) @ M @ F2 ) ) ) ) ).

% subprob_space_distr
thf(fact_777_subprob__space__distr,axiom,
    ! [F2: product_prod_nat_nat > real,M: sigma_measure_real] :
      ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ ( freque5010624893710627907mega_1 @ as ) @ M ) )
     => ( ( ( sigma_space_real @ M )
         != bot_bot_set_real )
       => ( giry_s8208748868292234104e_real @ ( measur4284412874431678645t_real @ ( freque5010624893710627907mega_1 @ as ) @ M @ F2 ) ) ) ) ).

% subprob_space_distr
thf(fact_778__092_060Omega_062_092_060_094sub_0622_Onot__empty,axiom,
    ( ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) )
   != bot_bo2540092574283496047at_nat ) ).

% \<Omega>\<^sub>2.not_empty
thf(fact_779_ennreal__inj,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ( ( extend7643940197134561352nnreal @ A )
            = ( extend7643940197134561352nnreal @ B ) )
          = ( A = B ) ) ) ) ).

% ennreal_inj
thf(fact_780_emeasure__empty,axiom,
    ! [M2: sigma_5515648953823433982at_nat] :
      ( ( sigma_411563132819280856at_nat @ M2 @ bot_bo2099793752762293965at_nat )
      = zero_z7100319975126383169nnreal ) ).

% emeasure_empty
thf(fact_781_emeasure__empty,axiom,
    ! [M2: sigma_9047027012034273406at_nat] :
      ( ( sigma_3593834561172453006at_nat @ M2 @ bot_bo2540092574283496047at_nat )
      = zero_z7100319975126383169nnreal ) ).

% emeasure_empty
thf(fact_782_ennreal__0,axiom,
    ( ( extend7643940197134561352nnreal @ zero_zero_real )
    = zero_z7100319975126383169nnreal ) ).

% ennreal_0
thf(fact_783_ennreal__eq__1,axiom,
    ! [X5: real] :
      ( ( ( extend7643940197134561352nnreal @ X5 )
        = one_on2969667320475766781nnreal )
      = ( X5 = one_one_real ) ) ).

% ennreal_eq_1
thf(fact_784_ennreal__1,axiom,
    ( ( extend7643940197134561352nnreal @ one_one_real )
    = one_on2969667320475766781nnreal ) ).

% ennreal_1
thf(fact_785_ennreal__eq__zero__iff,axiom,
    ! [X5: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X5 )
     => ( ( ( extend7643940197134561352nnreal @ X5 )
          = zero_z7100319975126383169nnreal )
        = ( X5 = zero_zero_real ) ) ) ).

% ennreal_eq_zero_iff
thf(fact_786_ennreal__le__iff,axiom,
    ! [Y4: real,X5: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
     => ( ( ord_le3935885782089961368nnreal @ ( extend7643940197134561352nnreal @ X5 ) @ ( extend7643940197134561352nnreal @ Y4 ) )
        = ( ord_less_eq_real @ X5 @ Y4 ) ) ) ).

% ennreal_le_iff
thf(fact_787_indep__sets__Dynkin,axiom,
    ! [F: $o > set_se7855581050983116737at_nat,I: set_o] :
      ( ( indepe619418257930843587_nat_o @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
     => ( indepe619418257930843587_nat_o @ ( freque5010624893710627907mega_1 @ as )
        @ ^ [I3: $o] : ( sigma_906991758010117658at_nat @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) @ ( F @ I3 ) )
        @ I ) ) ).

% indep_sets_Dynkin
thf(fact_788_borel__measurable__times__ennreal,axiom,
    ! [F2: real > extend8495563244428889912nnreal,M2: sigma_measure_real,G2: real > extend8495563244428889912nnreal] :
      ( ( member2919562650594848410nnreal @ F2 @ ( sigma_9017504469962657078nnreal @ M2 @ borel_6524799422816628122nnreal ) )
     => ( ( member2919562650594848410nnreal @ G2 @ ( sigma_9017504469962657078nnreal @ M2 @ borel_6524799422816628122nnreal ) )
       => ( member2919562650594848410nnreal
          @ ^ [X2: real] : ( times_1893300245718287421nnreal @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
          @ ( sigma_9017504469962657078nnreal @ M2 @ borel_6524799422816628122nnreal ) ) ) ) ).

% borel_measurable_times_ennreal
thf(fact_789_bot__prod__def,axiom,
    ( bot_bo4893629854819490515at_nat
    = ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) ) ).

% bot_prod_def
thf(fact_790_bot__prod__def,axiom,
    ( bot_bo4864135842559243903at_nat
    = ( produc5120233859229291749at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2540092574283496047at_nat ) ) ).

% bot_prod_def
thf(fact_791_bot__prod__def,axiom,
    ( bot_bo31374950876252916at_nat
    = ( produc3149970401713881818at_nat @ bot_bo2099793752762293965at_nat @ bot_bot_nat ) ) ).

% bot_prod_def
thf(fact_792_bot__prod__def,axiom,
    ( bot_bo2428222629654921807at_nat
    = ( produc2906465383384717037at_nat @ bot_bo2540092574283496047at_nat @ bot_bo2099793752762293965at_nat ) ) ).

% bot_prod_def
thf(fact_793_bot__prod__def,axiom,
    ( bot_bo2239160001760357891at_nat
    = ( produc6813982156787026511at_nat @ bot_bo2540092574283496047at_nat @ bot_bo2540092574283496047at_nat ) ) ).

% bot_prod_def
thf(fact_794_bot__prod__def,axiom,
    ( bot_bo4498404034370125432at_nat
    = ( produc6219347483351322180at_nat @ bot_bo2540092574283496047at_nat @ bot_bot_nat ) ) ).

% bot_prod_def
thf(fact_795_bot__prod__def,axiom,
    ( bot_bo2978901588506915098at_nat
    = ( produc5463602008962177208at_nat @ bot_bot_nat @ bot_bo2099793752762293965at_nat ) ) ).

% bot_prod_def
thf(fact_796_bot__prod__def,axiom,
    ( bot_bo722169136944652792at_nat
    = ( produc1587072818483735364at_nat @ bot_bot_nat @ bot_bo2540092574283496047at_nat ) ) ).

% bot_prod_def
thf(fact_797_bot__prod__def,axiom,
    ( bot_bo2769642828321324397at_nat
    = ( product_Pair_nat_nat @ bot_bot_nat @ bot_bot_nat ) ) ).

% bot_prod_def
thf(fact_798_Bochner__Integration_Ointegral__empty,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( ( sigma_4468984517727773725at_nat @ M2 )
        = bot_bo2540092574283496047at_nat )
     => ( ( bochne5509773249985062230t_real @ M2 @ F2 )
        = zero_zero_real ) ) ).

% Bochner_Integration.integral_empty
thf(fact_799_Bochner__Integration_Ointegral__empty,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real] :
      ( ( ( sigma_4797346298676585097at_nat @ M2 )
        = bot_bo2099793752762293965at_nat )
     => ( ( bochne6384019433803981034t_real @ M2 @ F2 )
        = zero_zero_real ) ) ).

% Bochner_Integration.integral_empty
thf(fact_800_ennreal__zero__less__mult__iff,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( times_1893300245718287421nnreal @ A @ B ) )
      = ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ A )
        & ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ B ) ) ) ).

% ennreal_zero_less_mult_iff
thf(fact_801_measurable__ennreal,axiom,
    member2919562650594848410nnreal @ extend7643940197134561352nnreal @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ borel_6524799422816628122nnreal ) ).

% measurable_ennreal
thf(fact_802_ennreal__zero__less__one,axiom,
    ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ one_on2969667320475766781nnreal ).

% ennreal_zero_less_one
thf(fact_803_prob__space_Oindep__sets__Dynkin,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F: $o > set_se1666487788256820497at_nat,I: set_o] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe9198310886692500677_nat_o @ M2 @ F @ I )
       => ( indepe9198310886692500677_nat_o @ M2
          @ ^ [I3: $o] : ( sigma_3745205859527822540at_nat @ ( sigma_4468984517727773725at_nat @ M2 ) @ ( F @ I3 ) )
          @ I ) ) ) ).

% prob_space.indep_sets_Dynkin
thf(fact_804_prob__space_Oindep__sets__Dynkin,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F: $o > set_se7855581050983116737at_nat,I: set_o] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe619418257930843587_nat_o @ M2 @ F @ I )
       => ( indepe619418257930843587_nat_o @ M2
          @ ^ [I3: $o] : ( sigma_906991758010117658at_nat @ ( sigma_4797346298676585097at_nat @ M2 ) @ ( F @ I3 ) )
          @ I ) ) ) ).

% prob_space.indep_sets_Dynkin
thf(fact_805_prob__space_Oindep__sets__Dynkin,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_se1666487788256820497at_nat,I: set_Pr947837736998463782t_real] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe5134244768989041027t_real @ M2 @ F @ I )
       => ( indepe5134244768989041027t_real @ M2
          @ ^ [I3: ( product_prod_nat_nat > product_prod_nat_nat ) > real] : ( sigma_3745205859527822540at_nat @ ( sigma_4468984517727773725at_nat @ M2 ) @ ( F @ I3 ) )
          @ I ) ) ) ).

% prob_space.indep_sets_Dynkin
thf(fact_806_prob__space_Oindep__sets__Dynkin,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F: ( real > extend8495563244428889912nnreal ) > set_se1666487788256820497at_nat,I: set_re5328672808648366137nnreal] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe6665278795436257814nnreal @ M2 @ F @ I )
       => ( indepe6665278795436257814nnreal @ M2
          @ ^ [I3: real > extend8495563244428889912nnreal] : ( sigma_3745205859527822540at_nat @ ( sigma_4468984517727773725at_nat @ M2 ) @ ( F @ I3 ) )
          @ I ) ) ) ).

% prob_space.indep_sets_Dynkin
thf(fact_807_prob__space_Oindep__sets__Dynkin,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F: ( product_prod_nat_nat > real ) > set_se1666487788256820497at_nat,I: set_Pr1002607673312053630t_real] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe4000825420875437771t_real @ M2 @ F @ I )
       => ( indepe4000825420875437771t_real @ M2
          @ ^ [I3: product_prod_nat_nat > real] : ( sigma_3745205859527822540at_nat @ ( sigma_4468984517727773725at_nat @ M2 ) @ ( F @ I3 ) )
          @ I ) ) ) ).

% prob_space.indep_sets_Dynkin
thf(fact_808_prob__space_Oindep__sets__Dynkin,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se1666487788256820497at_nat,I: set_Pr2458342521480944603at_nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe7028276433782144696at_nat @ M2 @ F @ I )
       => ( indepe7028276433782144696at_nat @ M2
          @ ^ [I3: product_prod_nat_nat > product_prod_nat_nat] : ( sigma_3745205859527822540at_nat @ ( sigma_4468984517727773725at_nat @ M2 ) @ ( F @ I3 ) )
          @ I ) ) ) ).

% prob_space.indep_sets_Dynkin
thf(fact_809_prob__space_Oindep__sets__Dynkin,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F: rat > set_se1666487788256820497at_nat,I: set_rat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe2998401279100655083at_rat @ M2 @ F @ I )
       => ( indepe2998401279100655083at_rat @ M2
          @ ^ [I3: rat] : ( sigma_3745205859527822540at_nat @ ( sigma_4468984517727773725at_nat @ M2 ) @ ( F @ I3 ) )
          @ I ) ) ) ).

% prob_space.indep_sets_Dynkin
thf(fact_810_ennreal__rat__dense,axiom,
    ! [X5: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
      ( ( ord_le7381754540660121996nnreal @ X5 @ Y4 )
     => ? [R2: rat] :
          ( ( ord_le7381754540660121996nnreal @ X5 @ ( extend7643940197134561352nnreal @ ( field_7254667332652039916t_real @ R2 ) ) )
          & ( ord_le7381754540660121996nnreal @ ( extend7643940197134561352nnreal @ ( field_7254667332652039916t_real @ R2 ) ) @ Y4 ) ) ) ).

% ennreal_rat_dense
thf(fact_811_borel__measurable__const,axiom,
    ! [C: product_prod_nat_nat,M2: sigma_5515648953823433982at_nat] :
      ( member8885076297122219836at_nat
      @ ^ [X2: product_prod_nat_nat] : C
      @ ( sigma_3682087626466204304at_nat @ M2 @ borel_2901874158916493343at_nat ) ) ).

% borel_measurable_const
thf(fact_812_borel__measurable__const,axiom,
    ! [C: real,M2: sigma_9047027012034273406at_nat] :
      ( member8159409068225774087t_real
      @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : C
      @ ( sigma_5064276549609164707t_real @ M2 @ borel_5078946678739801102l_real ) ) ).

% borel_measurable_const
thf(fact_813_borel__measurable__const,axiom,
    ! [C: real,M2: sigma_5515648953823433982at_nat] :
      ( member4564283293661824327t_real
      @ ^ [X2: product_prod_nat_nat] : C
      @ ( sigma_1188828689629184861t_real @ M2 @ borel_5078946678739801102l_real ) ) ).

% borel_measurable_const
thf(fact_814_borel__measurable__const,axiom,
    ! [C: extend8495563244428889912nnreal,M2: sigma_measure_real] :
      ( member2919562650594848410nnreal
      @ ^ [X2: real] : C
      @ ( sigma_9017504469962657078nnreal @ M2 @ borel_6524799422816628122nnreal ) ) ).

% borel_measurable_const
thf(fact_815_mult__right__ennreal__cancel,axiom,
    ! [A: extend8495563244428889912nnreal,C: real,B: extend8495563244428889912nnreal] :
      ( ( ( times_1893300245718287421nnreal @ A @ ( extend7643940197134561352nnreal @ C ) )
        = ( times_1893300245718287421nnreal @ B @ ( extend7643940197134561352nnreal @ C ) ) )
      = ( ( A = B )
        | ( ord_less_eq_real @ C @ zero_zero_real ) ) ) ).

% mult_right_ennreal_cancel
thf(fact_816_ennreal__mult_H_H,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ B )
     => ( ( extend7643940197134561352nnreal @ ( times_times_real @ A @ B ) )
        = ( times_1893300245718287421nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B ) ) ) ) ).

% ennreal_mult''
thf(fact_817_ennreal__mult_H,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( extend7643940197134561352nnreal @ ( times_times_real @ A @ B ) )
        = ( times_1893300245718287421nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B ) ) ) ) ).

% ennreal_mult'
thf(fact_818_ennreal__mult,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ( extend7643940197134561352nnreal @ ( times_times_real @ A @ B ) )
          = ( times_1893300245718287421nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B ) ) ) ) ) ).

% ennreal_mult
thf(fact_819_ennreal__lessI,axiom,
    ! [Q2: real,R: real] :
      ( ( ord_less_real @ zero_zero_real @ Q2 )
     => ( ( ord_less_real @ R @ Q2 )
       => ( ord_le7381754540660121996nnreal @ ( extend7643940197134561352nnreal @ R ) @ ( extend7643940197134561352nnreal @ Q2 ) ) ) ) ).

% ennreal_lessI
thf(fact_820_ennreal__leI,axiom,
    ! [X5: real,Y4: real] :
      ( ( ord_less_eq_real @ X5 @ Y4 )
     => ( ord_le3935885782089961368nnreal @ ( extend7643940197134561352nnreal @ X5 ) @ ( extend7643940197134561352nnreal @ Y4 ) ) ) ).

% ennreal_leI
thf(fact_821_ennreal__approx__unit,axiom,
    ! [Z: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
      ( ! [A3: extend8495563244428889912nnreal] :
          ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ A3 )
         => ( ( ord_le7381754540660121996nnreal @ A3 @ one_on2969667320475766781nnreal )
           => ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ A3 @ Z ) @ Y4 ) ) )
     => ( ord_le3935885782089961368nnreal @ Z @ Y4 ) ) ).

% ennreal_approx_unit
thf(fact_822_borel__measurable__times,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,M2: sigma_9047027012034273406at_nat,G2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ M2 @ borel_5078946678739801102l_real ) )
     => ( ( member8159409068225774087t_real @ G2 @ ( sigma_5064276549609164707t_real @ M2 @ borel_5078946678739801102l_real ) )
       => ( member8159409068225774087t_real
          @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( times_times_real @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
          @ ( sigma_5064276549609164707t_real @ M2 @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_times
thf(fact_823_borel__measurable__times,axiom,
    ! [F2: product_prod_nat_nat > real,M2: sigma_5515648953823433982at_nat,G2: product_prod_nat_nat > real] :
      ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ M2 @ borel_5078946678739801102l_real ) )
     => ( ( member4564283293661824327t_real @ G2 @ ( sigma_1188828689629184861t_real @ M2 @ borel_5078946678739801102l_real ) )
       => ( member4564283293661824327t_real
          @ ^ [X2: product_prod_nat_nat] : ( times_times_real @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
          @ ( sigma_1188828689629184861t_real @ M2 @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_times
thf(fact_824_mult__eq__1,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ A )
     => ( ( ord_le3935885782089961368nnreal @ A @ one_on2969667320475766781nnreal )
       => ( ( ord_le3935885782089961368nnreal @ B @ one_on2969667320475766781nnreal )
         => ( ( ( times_1893300245718287421nnreal @ A @ B )
              = one_on2969667320475766781nnreal )
            = ( ( A = one_on2969667320475766781nnreal )
              & ( B = one_on2969667320475766781nnreal ) ) ) ) ) ) ).

% mult_eq_1
thf(fact_825_mult__eq__1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ( ord_less_eq_real @ B @ one_one_real )
         => ( ( ( times_times_real @ A @ B )
              = one_one_real )
            = ( ( A = one_one_real )
              & ( B = one_one_real ) ) ) ) ) ) ).

% mult_eq_1
thf(fact_826_mult__eq__1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ( ord_less_eq_rat @ B @ one_one_rat )
         => ( ( ( times_times_rat @ A @ B )
              = one_one_rat )
            = ( ( A = one_one_rat )
              & ( B = one_one_rat ) ) ) ) ) ) ).

% mult_eq_1
thf(fact_827_mult__eq__1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ A @ one_one_nat )
       => ( ( ord_less_eq_nat @ B @ one_one_nat )
         => ( ( ( times_times_nat @ A @ B )
              = one_one_nat )
            = ( ( A = one_one_nat )
              & ( B = one_one_nat ) ) ) ) ) ) ).

% mult_eq_1
thf(fact_828_ennreal__less__iff,axiom,
    ! [R: real,Q2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R )
     => ( ( ord_le7381754540660121996nnreal @ ( extend7643940197134561352nnreal @ R ) @ ( extend7643940197134561352nnreal @ Q2 ) )
        = ( ord_less_real @ R @ Q2 ) ) ) ).

% ennreal_less_iff
thf(fact_829_ennreal__neg,axiom,
    ! [X5: real] :
      ( ( ord_less_eq_real @ X5 @ zero_zero_real )
     => ( ( extend7643940197134561352nnreal @ X5 )
        = zero_z7100319975126383169nnreal ) ) ).

% ennreal_neg
thf(fact_830_ennreal__eq__0__iff,axiom,
    ! [X5: real] :
      ( ( ( extend7643940197134561352nnreal @ X5 )
        = zero_z7100319975126383169nnreal )
      = ( ord_less_eq_real @ X5 @ zero_zero_real ) ) ).

% ennreal_eq_0_iff
thf(fact_831_ennreal__le__iff2,axiom,
    ! [X5: real,Y4: real] :
      ( ( ord_le3935885782089961368nnreal @ ( extend7643940197134561352nnreal @ X5 ) @ ( extend7643940197134561352nnreal @ Y4 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
          & ( ord_less_eq_real @ X5 @ Y4 ) )
        | ( ( ord_less_eq_real @ X5 @ zero_zero_real )
          & ( ord_less_eq_real @ Y4 @ zero_zero_real ) ) ) ) ).

% ennreal_le_iff2
thf(fact_832_le__ennreal__iff,axiom,
    ! [R: real,X5: extend8495563244428889912nnreal] :
      ( ( ord_less_eq_real @ zero_zero_real @ R )
     => ( ( ord_le3935885782089961368nnreal @ X5 @ ( extend7643940197134561352nnreal @ R ) )
        = ( ? [Q3: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ Q3 )
              & ( X5
                = ( extend7643940197134561352nnreal @ Q3 ) )
              & ( ord_less_eq_real @ Q3 @ R ) ) ) ) ) ).

% le_ennreal_iff
thf(fact_833_borel__measurable__Pair,axiom,
    ! [F2: real > real,M2: sigma_measure_real,G2: real > extend8495563244428889912nnreal] :
      ( ( member_real_real @ F2 @ ( sigma_5267869275261027754l_real @ M2 @ borel_5078946678739801102l_real ) )
     => ( ( member2919562650594848410nnreal @ G2 @ ( sigma_9017504469962657078nnreal @ M2 @ borel_6524799422816628122nnreal ) )
       => ( member4263462704469908217nnreal
          @ ^ [X2: real] : ( produc4778015194254607485nnreal @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
          @ ( sigma_1079880282724612679nnreal @ M2 @ borel_4974289084073311971nnreal ) ) ) ) ).

% borel_measurable_Pair
thf(fact_834_borel__measurable__Pair,axiom,
    ! [F2: real > extend8495563244428889912nnreal,M2: sigma_measure_real,G2: real > real] :
      ( ( member2919562650594848410nnreal @ F2 @ ( sigma_9017504469962657078nnreal @ M2 @ borel_6524799422816628122nnreal ) )
     => ( ( member_real_real @ G2 @ ( sigma_5267869275261027754l_real @ M2 @ borel_5078946678739801102l_real ) )
       => ( member4593507348387352185l_real
          @ ^ [X2: real] : ( produc2810268924804063229l_real @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
          @ ( sigma_1034331983380590023l_real @ M2 @ borel_4928740784729289315l_real ) ) ) ) ).

% borel_measurable_Pair
thf(fact_835_borel__measurable__Pair,axiom,
    ! [F2: real > extend8495563244428889912nnreal,M2: sigma_measure_real,G2: real > extend8495563244428889912nnreal] :
      ( ( member2919562650594848410nnreal @ F2 @ ( sigma_9017504469962657078nnreal @ M2 @ borel_6524799422816628122nnreal ) )
     => ( ( member2919562650594848410nnreal @ G2 @ ( sigma_9017504469962657078nnreal @ M2 @ borel_6524799422816628122nnreal ) )
       => ( member7391843071750696581nnreal
          @ ^ [X2: real] : ( produc344325839068023049nnreal @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
          @ ( sigma_2101749304130238547nnreal @ M2 @ borel_3951438148318096111nnreal ) ) ) ) ).

% borel_measurable_Pair
thf(fact_836_borel__measurable__Pair,axiom,
    ! [F2: product_prod_nat_nat > nat,M2: sigma_5515648953823433982at_nat,G2: product_prod_nat_nat > nat] :
      ( ( member7868840539957676139at_nat @ F2 @ ( sigma_8719606837941807361at_nat @ M2 @ borel_8449730974584783410el_nat ) )
     => ( ( member7868840539957676139at_nat @ G2 @ ( sigma_8719606837941807361at_nat @ M2 @ borel_8449730974584783410el_nat ) )
       => ( member8885076297122219836at_nat
          @ ^ [X2: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
          @ ( sigma_3682087626466204304at_nat @ M2 @ borel_2901874158916493343at_nat ) ) ) ) ).

% borel_measurable_Pair
thf(fact_837_borel__measurable__Pair,axiom,
    ! [F2: product_prod_nat_nat > real,M2: sigma_5515648953823433982at_nat,G2: product_prod_nat_nat > real] :
      ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ M2 @ borel_5078946678739801102l_real ) )
     => ( ( member4564283293661824327t_real @ G2 @ ( sigma_1188828689629184861t_real @ M2 @ borel_5078946678739801102l_real ) )
       => ( member2619278773872280948l_real
          @ ^ [X2: product_prod_nat_nat] : ( produc4511245868158468465l_real @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
          @ ( sigma_8891564814549521864l_real @ M2 @ borel_9155112475215227991l_real ) ) ) ) ).

% borel_measurable_Pair
thf(fact_838_borel__measurable__Pair,axiom,
    ! [F2: product_prod_nat_nat > real,M2: sigma_5515648953823433982at_nat,G2: product_prod_nat_nat > extend8495563244428889912nnreal] :
      ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ M2 @ borel_5078946678739801102l_real ) )
     => ( ( member8497640837515055059nnreal @ G2 @ ( sigma_1808371603960765545nnreal @ M2 @ borel_6524799422816628122nnreal ) )
       => ( member7022753855621176064nnreal
          @ ^ [X2: product_prod_nat_nat] : ( produc4778015194254607485nnreal @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
          @ ( sigma_8467819794085251284nnreal @ M2 @ borel_4974289084073311971nnreal ) ) ) ) ).

% borel_measurable_Pair
thf(fact_839_borel__measurable__Pair,axiom,
    ! [F2: product_prod_nat_nat > extend8495563244428889912nnreal,M2: sigma_5515648953823433982at_nat,G2: product_prod_nat_nat > real] :
      ( ( member8497640837515055059nnreal @ F2 @ ( sigma_1808371603960765545nnreal @ M2 @ borel_6524799422816628122nnreal ) )
     => ( ( member4564283293661824327t_real @ G2 @ ( sigma_1188828689629184861t_real @ M2 @ borel_5078946678739801102l_real ) )
       => ( member7352798499538620032l_real
          @ ^ [X2: product_prod_nat_nat] : ( produc2810268924804063229l_real @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
          @ ( sigma_8422271494741228628l_real @ M2 @ borel_4928740784729289315l_real ) ) ) ) ).

% borel_measurable_Pair
thf(fact_840_borel__measurable__Pair,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,M2: sigma_5515648953823433982at_nat,G2: product_prod_nat_nat > real] :
      ( ( member8885076297122219836at_nat @ F2 @ ( sigma_3682087626466204304at_nat @ M2 @ borel_2901874158916493343at_nat ) )
     => ( ( member4564283293661824327t_real @ G2 @ ( sigma_1188828689629184861t_real @ M2 @ borel_5078946678739801102l_real ) )
       => ( member7102390617368680951t_real
          @ ^ [X2: product_prod_nat_nat] : ( produc904203886716845910t_real @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
          @ ( sigma_3852182464305326733t_real @ M2 @ borel_5044688924370832830t_real ) ) ) ) ).

% borel_measurable_Pair
thf(fact_841_borel__measurable__Pair,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,M2: sigma_5515648953823433982at_nat,G2: product_prod_nat_nat > extend8495563244428889912nnreal] :
      ( ( member8885076297122219836at_nat @ F2 @ ( sigma_3682087626466204304at_nat @ M2 @ borel_2901874158916493343at_nat ) )
     => ( ( member8497640837515055059nnreal @ G2 @ ( sigma_1808371603960765545nnreal @ M2 @ borel_6524799422816628122nnreal ) )
       => ( member2392587564862597379nnreal
          @ ^ [X2: product_prod_nat_nat] : ( produc9096837261389135074nnreal @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
          @ ( sigma_4363137424377088025nnreal @ M2 @ borel_5572891200955083722nnreal ) ) ) ) ).

% borel_measurable_Pair
thf(fact_842_borel__measurable__Pair,axiom,
    ! [F2: product_prod_nat_nat > real,M2: sigma_5515648953823433982at_nat,G2: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ M2 @ borel_5078946678739801102l_real ) )
     => ( ( member8885076297122219836at_nat @ G2 @ ( sigma_3682087626466204304at_nat @ M2 @ borel_2901874158916493343at_nat ) )
       => ( member95027015144503653at_nat
          @ ^ [X2: product_prod_nat_nat] : ( produc423147238207498620at_nat @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
          @ ( sigma_8789979247889735931at_nat @ M2 @ borel_759113671100466220at_nat ) ) ) ) ).

% borel_measurable_Pair
thf(fact_843_non__empty__space,axiom,
    ( ( as != nil_nat )
   => ( ( frequency_Moment_M_1 @ as )
     != bot_bo2099793752762293965at_nat ) ) ).

% non_empty_space
thf(fact_844__092_060Omega_062_092_060_094sub_0622_Oindep__var__from__indep__vars,axiom,
    ! [I4: product_prod_nat_nat,J2: product_prod_nat_nat,M: sigma_measure_real,F2: product_prod_nat_nat > ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( I4 != J2 )
     => ( ( indepe4215498690062702080t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
          @ ^ [Uu: product_prod_nat_nat] : M
          @ F2
          @ ( insert8211810215607154385at_nat @ I4 @ ( insert8211810215607154385at_nat @ J2 @ bot_bo2099793752762293965at_nat ) ) )
       => ( indepe1938972608428940447t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ ( F2 @ I4 ) @ M @ ( F2 @ J2 ) ) ) ) ).

% \<Omega>\<^sub>2.indep_var_from_indep_vars
thf(fact_845__092_060Omega_062_092_060_094sub_0622_Oindep__var__from__indep__vars,axiom,
    ! [I4: product_prod_nat_nat > product_prod_nat_nat,J2: product_prod_nat_nat > product_prod_nat_nat,M: sigma_measure_real,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( I4 != J2 )
     => ( ( indepe5511084738810189248t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
          @ ^ [Uu: product_prod_nat_nat > product_prod_nat_nat] : M
          @ F2
          @ ( insert5766096353517822421at_nat @ I4 @ ( insert5766096353517822421at_nat @ J2 @ bot_bo2540092574283496047at_nat ) ) )
       => ( indepe1938972608428940447t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ M @ ( F2 @ I4 ) @ M @ ( F2 @ J2 ) ) ) ) ).

% \<Omega>\<^sub>2.indep_var_from_indep_vars
thf(fact_846__092_060Omega_062_092_060_094sub_0622_Ocovariance__def,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,G2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( probab3351693472195852697t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F2 @ G2 )
      = ( bochne5509773249985062230t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
        @ ^ [Omega: product_prod_nat_nat > product_prod_nat_nat] : ( times_times_real @ ( minus_minus_real @ ( F2 @ Omega ) @ ( bochne5509773249985062230t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F2 ) ) @ ( minus_minus_real @ ( G2 @ Omega ) @ ( bochne5509773249985062230t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ G2 ) ) ) ) ) ).

% \<Omega>\<^sub>2.covariance_def
thf(fact_847__092_060Omega_062_092_060_094sub_0622_Oindep__sets__finite__index__sets,axiom,
    ! [F: nat > set_se1666487788256820497at_nat,I: set_nat] :
      ( ( indepe3633531339187150819at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
      = ( ! [J4: set_nat] :
            ( ( ord_less_eq_set_nat @ J4 @ I )
           => ( ( J4 != bot_bot_set_nat )
             => ( ( finite_finite_nat @ J4 )
               => ( indepe3633531339187150819at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ J4 ) ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_finite_index_sets
thf(fact_848__092_060Omega_062_092_060_094sub_0622_Oindep__sets__finite__index__sets,axiom,
    ! [F: product_prod_nat_nat > set_se1666487788256820497at_nat,I: set_Pr1261947904930325089at_nat] :
      ( ( indepe2942784128085459246at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
      = ( ! [J4: set_Pr1261947904930325089at_nat] :
            ( ( ord_le3146513528884898305at_nat @ J4 @ I )
           => ( ( J4 != bot_bo2099793752762293965at_nat )
             => ( ( finite6177210948735845034at_nat @ J4 )
               => ( indepe2942784128085459246at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ J4 ) ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_finite_index_sets
thf(fact_849__092_060Omega_062_092_060_094sub_0622_Oindep__sets__finite__index__sets,axiom,
    ! [F: $o > set_se1666487788256820497at_nat,I: set_o] :
      ( ( indepe9198310886692500677_nat_o @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
      = ( ! [J4: set_o] :
            ( ( ord_less_eq_set_o @ J4 @ I )
           => ( ( J4 != bot_bot_set_o )
             => ( ( finite_finite_o @ J4 )
               => ( indepe9198310886692500677_nat_o @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ J4 ) ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_finite_index_sets
thf(fact_850__092_060Omega_062_092_060_094sub_0622_Oindep__sets__finite__index__sets,axiom,
    ! [F: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_se1666487788256820497at_nat,I: set_Pr947837736998463782t_real] :
      ( ( indepe5134244768989041027t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
      = ( ! [J4: set_Pr947837736998463782t_real] :
            ( ( ord_le8622349663015078278t_real @ J4 @ I )
           => ( ( J4 != bot_bo2850223541457833914t_real )
             => ( ( finite3361987442264708871t_real @ J4 )
               => ( indepe5134244768989041027t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ J4 ) ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_finite_index_sets
thf(fact_851__092_060Omega_062_092_060_094sub_0622_Oindep__sets__finite__index__sets,axiom,
    ! [F: ( real > extend8495563244428889912nnreal ) > set_se1666487788256820497at_nat,I: set_re5328672808648366137nnreal] :
      ( ( indepe6665278795436257814nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
      = ( ! [J4: set_re5328672808648366137nnreal] :
            ( ( ord_le2462468573666744473nnreal @ J4 @ I )
           => ( ( J4 != bot_bo6037503491064675021nnreal )
             => ( ( finite7684081742213514138nnreal @ J4 )
               => ( indepe6665278795436257814nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ J4 ) ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_finite_index_sets
thf(fact_852__092_060Omega_062_092_060_094sub_0622_Oindep__sets__finite__index__sets,axiom,
    ! [F: ( product_prod_nat_nat > real ) > set_se1666487788256820497at_nat,I: set_Pr1002607673312053630t_real] :
      ( ( indepe4000825420875437771t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
      = ( ! [J4: set_Pr1002607673312053630t_real] :
            ( ( ord_le1994730768851292446t_real @ J4 @ I )
           => ( ( J4 != bot_bo8258804065021230570t_real )
             => ( ( finite7134436996427083847t_real @ J4 )
               => ( indepe4000825420875437771t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ J4 ) ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_finite_index_sets
thf(fact_853__092_060Omega_062_092_060_094sub_0622_Oindep__sets__finite__index__sets,axiom,
    ! [F: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se1666487788256820497at_nat,I: set_Pr2458342521480944603at_nat] :
      ( ( indepe7028276433782144696at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
      = ( ! [J4: set_Pr2458342521480944603at_nat] :
            ( ( ord_le3857079194666040379at_nat @ J4 @ I )
           => ( ( J4 != bot_bo2540092574283496047at_nat )
             => ( ( finite1779803868683885628at_nat @ J4 )
               => ( indepe7028276433782144696at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ J4 ) ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_finite_index_sets
thf(fact_854__092_060Omega_062_092_060_094sub_0622_Oindep__sets__finite__index__sets,axiom,
    ! [F: rat > set_se1666487788256820497at_nat,I: set_rat] :
      ( ( indepe2998401279100655083at_rat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
      = ( ! [J4: set_rat] :
            ( ( ord_less_eq_set_rat @ J4 @ I )
           => ( ( J4 != bot_bot_set_rat )
             => ( ( finite_finite_rat @ J4 )
               => ( indepe2998401279100655083at_rat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ J4 ) ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_sets_finite_index_sets
thf(fact_855__092_060Omega_062_092_060_094sub_0622_Ofinite__emeasure__space,axiom,
    ( ( sigma_3593834561172453006at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
   != top_to1496364449551166952nnreal ) ).

% \<Omega>\<^sub>2.finite_emeasure_space
thf(fact_856_fin__space,axiom,
    ( ( as != nil_nat )
   => ( finite6177210948735845034at_nat @ ( frequency_Moment_M_1 @ as ) ) ) ).

% fin_space
thf(fact_857_diff__0__eq__0,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_858_diff__self__eq__0,axiom,
    ! [M3: nat] :
      ( ( minus_minus_nat @ M3 @ M3 )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_859_diff__diff__cancel,axiom,
    ! [I4: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I4 @ N2 )
     => ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I4 ) )
        = I4 ) ) ).

% diff_diff_cancel
thf(fact_860_finite__emeasure__space,axiom,
    ( ( sigma_411563132819280856at_nat @ ( freque5010624893710627907mega_1 @ as ) @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
   != top_to1496364449551166952nnreal ) ).

% finite_emeasure_space
thf(fact_861_indep__sets__finite__index__sets,axiom,
    ! [F: nat > set_se7855581050983116737at_nat,I: set_nat] :
      ( ( indepe5197683624481315429at_nat @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
      = ( ! [J4: set_nat] :
            ( ( ord_less_eq_set_nat @ J4 @ I )
           => ( ( J4 != bot_bot_set_nat )
             => ( ( finite_finite_nat @ J4 )
               => ( indepe5197683624481315429at_nat @ ( freque5010624893710627907mega_1 @ as ) @ F @ J4 ) ) ) ) ) ) ).

% indep_sets_finite_index_sets
thf(fact_862_indep__sets__finite__index__sets,axiom,
    ! [F: product_prod_nat_nat > set_se7855581050983116737at_nat,I: set_Pr1261947904930325089at_nat] :
      ( ( indepe2425276932038335532at_nat @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
      = ( ! [J4: set_Pr1261947904930325089at_nat] :
            ( ( ord_le3146513528884898305at_nat @ J4 @ I )
           => ( ( J4 != bot_bo2099793752762293965at_nat )
             => ( ( finite6177210948735845034at_nat @ J4 )
               => ( indepe2425276932038335532at_nat @ ( freque5010624893710627907mega_1 @ as ) @ F @ J4 ) ) ) ) ) ) ).

% indep_sets_finite_index_sets
thf(fact_863_indep__sets__finite__index__sets,axiom,
    ! [F: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se7855581050983116737at_nat,I: set_Pr2458342521480944603at_nat] :
      ( ( indepe6545576254369104698at_nat @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
      = ( ! [J4: set_Pr2458342521480944603at_nat] :
            ( ( ord_le3857079194666040379at_nat @ J4 @ I )
           => ( ( J4 != bot_bo2540092574283496047at_nat )
             => ( ( finite1779803868683885628at_nat @ J4 )
               => ( indepe6545576254369104698at_nat @ ( freque5010624893710627907mega_1 @ as ) @ F @ J4 ) ) ) ) ) ) ).

% indep_sets_finite_index_sets
thf(fact_864_indep__sets__finite__index__sets,axiom,
    ! [F: ( product_prod_nat_nat > real ) > set_se7855581050983116737at_nat,I: set_Pr1002607673312053630t_real] :
      ( ( indepe3818579677853040073t_real @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
      = ( ! [J4: set_Pr1002607673312053630t_real] :
            ( ( ord_le1994730768851292446t_real @ J4 @ I )
           => ( ( J4 != bot_bo8258804065021230570t_real )
             => ( ( finite7134436996427083847t_real @ J4 )
               => ( indepe3818579677853040073t_real @ ( freque5010624893710627907mega_1 @ as ) @ F @ J4 ) ) ) ) ) ) ).

% indep_sets_finite_index_sets
thf(fact_865_indep__sets__finite__index__sets,axiom,
    ! [F: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_se7855581050983116737at_nat,I: set_Pr947837736998463782t_real] :
      ( ( indepe2662975882062776837t_real @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
      = ( ! [J4: set_Pr947837736998463782t_real] :
            ( ( ord_le8622349663015078278t_real @ J4 @ I )
           => ( ( J4 != bot_bo2850223541457833914t_real )
             => ( ( finite3361987442264708871t_real @ J4 )
               => ( indepe2662975882062776837t_real @ ( freque5010624893710627907mega_1 @ as ) @ F @ J4 ) ) ) ) ) ) ).

% indep_sets_finite_index_sets
thf(fact_866_indep__sets__finite__index__sets,axiom,
    ! [F: $o > set_se7855581050983116737at_nat,I: set_o] :
      ( ( indepe619418257930843587_nat_o @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
      = ( ! [J4: set_o] :
            ( ( ord_less_eq_set_o @ J4 @ I )
           => ( ( J4 != bot_bot_set_o )
             => ( ( finite_finite_o @ J4 )
               => ( indepe619418257930843587_nat_o @ ( freque5010624893710627907mega_1 @ as ) @ F @ J4 ) ) ) ) ) ) ).

% indep_sets_finite_index_sets
thf(fact_867_indep__var__from__indep__vars,axiom,
    ! [I4: product_prod_nat_nat,J2: product_prod_nat_nat,M: sigma_measure_real,F2: product_prod_nat_nat > product_prod_nat_nat > real] :
      ( ( I4 != J2 )
     => ( ( indepe4688187045501433524t_real @ ( freque5010624893710627907mega_1 @ as )
          @ ^ [Uu: product_prod_nat_nat] : M
          @ F2
          @ ( insert8211810215607154385at_nat @ I4 @ ( insert8211810215607154385at_nat @ J2 @ bot_bo2099793752762293965at_nat ) ) )
       => ( indepe1845147739501508449t_real @ ( freque5010624893710627907mega_1 @ as ) @ M @ ( F2 @ I4 ) @ M @ ( F2 @ J2 ) ) ) ) ).

% indep_var_from_indep_vars
thf(fact_868_indep__var__from__indep__vars,axiom,
    ! [I4: product_prod_nat_nat > product_prod_nat_nat,J2: product_prod_nat_nat > product_prod_nat_nat,M: sigma_measure_real,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > real] :
      ( ( I4 != J2 )
     => ( ( indepe2220390954403586572t_real @ ( freque5010624893710627907mega_1 @ as )
          @ ^ [Uu: product_prod_nat_nat > product_prod_nat_nat] : M
          @ F2
          @ ( insert5766096353517822421at_nat @ I4 @ ( insert5766096353517822421at_nat @ J2 @ bot_bo2540092574283496047at_nat ) ) )
       => ( indepe1845147739501508449t_real @ ( freque5010624893710627907mega_1 @ as ) @ M @ ( F2 @ I4 ) @ M @ ( F2 @ J2 ) ) ) ) ).

% indep_var_from_indep_vars
thf(fact_869_covariance__def,axiom,
    ! [F2: product_prod_nat_nat > real,G2: product_prod_nat_nat > real] :
      ( ( probab7722952755062086631t_real @ ( freque5010624893710627907mega_1 @ as ) @ F2 @ G2 )
      = ( bochne6384019433803981034t_real @ ( freque5010624893710627907mega_1 @ as )
        @ ^ [Omega: product_prod_nat_nat] : ( times_times_real @ ( minus_minus_real @ ( F2 @ Omega ) @ ( bochne6384019433803981034t_real @ ( freque5010624893710627907mega_1 @ as ) @ F2 ) ) @ ( minus_minus_real @ ( G2 @ Omega ) @ ( bochne6384019433803981034t_real @ ( freque5010624893710627907mega_1 @ as ) @ G2 ) ) ) ) ) ).

% covariance_def
thf(fact_870_semiring__norm_I63_J,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% semiring_norm(63)
thf(fact_871_semiring__norm_I63_J,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% semiring_norm(63)
thf(fact_872_semiring__norm_I63_J,axiom,
    ! [A: extend8495563244428889912nnreal] :
      ( ( times_1893300245718287421nnreal @ zero_z7100319975126383169nnreal @ A )
      = zero_z7100319975126383169nnreal ) ).

% semiring_norm(63)
thf(fact_873_semiring__norm_I63_J,axiom,
    ! [A: real] :
      ( ( times_times_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% semiring_norm(63)
thf(fact_874_semiring__norm_I64_J,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% semiring_norm(64)
thf(fact_875_semiring__norm_I64_J,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% semiring_norm(64)
thf(fact_876_semiring__norm_I64_J,axiom,
    ! [A: extend8495563244428889912nnreal] :
      ( ( times_1893300245718287421nnreal @ A @ zero_z7100319975126383169nnreal )
      = zero_z7100319975126383169nnreal ) ).

% semiring_norm(64)
thf(fact_877_semiring__norm_I64_J,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% semiring_norm(64)
thf(fact_878_semiring__norm_I58_J,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% semiring_norm(58)
thf(fact_879_semiring__norm_I58_J,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% semiring_norm(58)
thf(fact_880_diff__self,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% diff_self
thf(fact_881_diff__self,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% diff_self
thf(fact_882_right__minus__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = zero_zero_rat )
      = ( A = B ) ) ).

% right_minus_eq
thf(fact_883_right__minus__eq,axiom,
    ! [A: real,B: real] :
      ( ( ( minus_minus_real @ A @ B )
        = zero_zero_real )
      = ( A = B ) ) ).

% right_minus_eq
thf(fact_884_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_885_diff__zero,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_zero
thf(fact_886_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_887_diff__zero,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_zero
thf(fact_888_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_889_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ A )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_890_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_891_space__bot,axiom,
    ( ( sigma_4797346298676585097at_nat @ bot_bo8824583409880654738at_nat )
    = bot_bo2099793752762293965at_nat ) ).

% space_bot
thf(fact_892_space__bot,axiom,
    ( ( sigma_4468984517727773725at_nat @ bot_bo7613792866757523178at_nat )
    = bot_bo2540092574283496047at_nat ) ).

% space_bot
thf(fact_893_zero__less__diff,axiom,
    ! [N2: nat,M3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M3 ) )
      = ( ord_less_nat @ M3 @ N2 ) ) ).

% zero_less_diff
thf(fact_894_diff__is__0__eq,axiom,
    ! [M3: nat,N2: nat] :
      ( ( ( minus_minus_nat @ M3 @ N2 )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M3 @ N2 ) ) ).

% diff_is_0_eq
thf(fact_895_diff__is__0__eq_H,axiom,
    ! [M3: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M3 @ N2 )
     => ( ( minus_minus_nat @ M3 @ N2 )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_896_emeasure__bot,axiom,
    ! [X: set_Pr2458342521480944603at_nat] :
      ( ( sigma_3593834561172453006at_nat @ bot_bo7613792866757523178at_nat @ X )
      = zero_z7100319975126383169nnreal ) ).

% emeasure_bot
thf(fact_897_emeasure__bot,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( sigma_411563132819280856at_nat @ bot_bo8824583409880654738at_nat @ X )
      = zero_z7100319975126383169nnreal ) ).

% emeasure_bot
thf(fact_898_Bochner__Integration_Ointegrable__diff,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,G2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( bochne7117808529828525605t_real @ M2 @ F2 )
     => ( ( bochne7117808529828525605t_real @ M2 @ G2 )
       => ( bochne7117808529828525605t_real @ M2
          @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( minus_minus_real @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).

% Bochner_Integration.integrable_diff
thf(fact_899_Bochner__Integration_Ointegrable__diff,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real,G2: product_prod_nat_nat > real] :
      ( ( bochne2596016609597520987t_real @ M2 @ F2 )
     => ( ( bochne2596016609597520987t_real @ M2 @ G2 )
       => ( bochne2596016609597520987t_real @ M2
          @ ^ [X2: product_prod_nat_nat] : ( minus_minus_real @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).

% Bochner_Integration.integrable_diff
thf(fact_900_diff__le__0__iff__le,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ zero_zero_real )
      = ( ord_less_eq_real @ A @ B ) ) ).

% diff_le_0_iff_le
thf(fact_901_diff__le__0__iff__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% diff_le_0_iff_le
thf(fact_902_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
      = ( ord_less_eq_real @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_903_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_eq_rat @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_904_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_rat @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_905_diff__gt__0__iff__gt,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
      = ( ord_less_real @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_906_diff__less__0__iff__less,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ B ) ) ).

% diff_less_0_iff_less
thf(fact_907_diff__less__0__iff__less,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ zero_zero_real )
      = ( ord_less_real @ A @ B ) ) ).

% diff_less_0_iff_less
thf(fact_908_Bochner__Integration_Ointegral__diff,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,G2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( bochne7117808529828525605t_real @ M2 @ F2 )
     => ( ( bochne7117808529828525605t_real @ M2 @ G2 )
       => ( ( bochne5509773249985062230t_real @ M2
            @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( minus_minus_real @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) )
          = ( minus_minus_real @ ( bochne5509773249985062230t_real @ M2 @ F2 ) @ ( bochne5509773249985062230t_real @ M2 @ G2 ) ) ) ) ) ).

% Bochner_Integration.integral_diff
thf(fact_909_Bochner__Integration_Ointegral__diff,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > real,G2: product_prod_nat_nat > real] :
      ( ( bochne2596016609597520987t_real @ M2 @ F2 )
     => ( ( bochne2596016609597520987t_real @ M2 @ G2 )
       => ( ( bochne6384019433803981034t_real @ M2
            @ ^ [X2: product_prod_nat_nat] : ( minus_minus_real @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) )
          = ( minus_minus_real @ ( bochne6384019433803981034t_real @ M2 @ F2 ) @ ( bochne6384019433803981034t_real @ M2 @ G2 ) ) ) ) ) ).

% Bochner_Integration.integral_diff
thf(fact_910_emeasure__subprob__space__less__top,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( sigma_411563132819280856at_nat @ ( freque5010624893710627907mega_1 @ as ) @ A2 )
     != top_to1496364449551166952nnreal ) ).

% emeasure_subprob_space_less_top
thf(fact_911__092_060Omega_062_092_060_094sub_0622_Oemeasure__subprob__space__less__top,axiom,
    ! [A2: set_Pr2458342521480944603at_nat] :
      ( ( sigma_3593834561172453006at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 )
     != top_to1496364449551166952nnreal ) ).

% \<Omega>\<^sub>2.emeasure_subprob_space_less_top
thf(fact_912_ennreal__minus__cancel__iff,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ( minus_8429688780609304081nnreal @ A @ B )
        = ( minus_8429688780609304081nnreal @ A @ C ) )
      = ( ( B = C )
        | ( ( ord_le3935885782089961368nnreal @ A @ B )
          & ( ord_le3935885782089961368nnreal @ A @ C ) )
        | ( A = top_to1496364449551166952nnreal ) ) ) ).

% ennreal_minus_cancel_iff
thf(fact_913_ennreal__minus__cancel,axiom,
    ! [C: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( C != top_to1496364449551166952nnreal )
     => ( ( ord_le3935885782089961368nnreal @ A @ C )
       => ( ( ord_le3935885782089961368nnreal @ B @ C )
         => ( ( ( minus_8429688780609304081nnreal @ C @ A )
              = ( minus_8429688780609304081nnreal @ C @ B ) )
           => ( A = B ) ) ) ) ) ).

% ennreal_minus_cancel
thf(fact_914_diff__less__top__ennreal,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le7381754540660121996nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ top_to1496364449551166952nnreal )
      = ( ord_le7381754540660121996nnreal @ A @ top_to1496364449551166952nnreal ) ) ).

% diff_less_top_ennreal
thf(fact_915_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_916_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: real,C: real,B: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
      = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_917_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_918_of__rat__diff,axiom,
    ! [A: rat,B: rat] :
      ( ( field_7254667332652039916t_real @ ( minus_minus_rat @ A @ B ) )
      = ( minus_minus_real @ ( field_7254667332652039916t_real @ A ) @ ( field_7254667332652039916t_real @ B ) ) ) ).

% of_rat_diff
thf(fact_919_Pair__top__top,axiom,
    ( ( produc344325839068023049nnreal @ top_to1496364449551166952nnreal @ top_to1496364449551166952nnreal )
    = top_to1606914339385671969nnreal ) ).

% Pair_top_top
thf(fact_920_Pair__top__top,axiom,
    ( ( produc1819051116115787367_set_o @ top_to1496364449551166952nnreal @ top_top_set_o )
    = top_to8481901701325284453_set_o ) ).

% Pair_top_top
thf(fact_921_Pair__top__top,axiom,
    ( ( produc2694532298418486364t_unit @ top_to1496364449551166952nnreal @ top_to1996260823553986621t_unit )
    = top_to5978268291654745946t_unit ) ).

% Pair_top_top
thf(fact_922_Pair__top__top,axiom,
    ( ( produc2045468279157342809nnreal @ top_top_set_o @ top_to1496364449551166952nnreal )
    = top_to4281738276809083791nnreal ) ).

% Pair_top_top
thf(fact_923_Pair__top__top,axiom,
    ( ( produc8758509063432514254nnreal @ top_to1996260823553986621t_unit @ top_to1496364449551166952nnreal )
    = top_to6254695844867088772nnreal ) ).

% Pair_top_top
thf(fact_924_Pair__top__top,axiom,
    ( ( produc5838405689764958487_set_o @ top_top_set_o @ top_top_set_o )
    = top_to8356300885640301175_set_o ) ).

% Pair_top_top
thf(fact_925_Pair__top__top,axiom,
    ( ( produc1413254743864018700t_unit @ top_top_set_o @ top_to1996260823553986621t_unit )
    = top_to7387087233619026540t_unit ) ).

% Pair_top_top
thf(fact_926_Pair__top__top,axiom,
    ( ( produc2330249535568903522_set_o @ top_to1996260823553986621t_unit @ top_top_set_o )
    = top_to6943113401079982402_set_o ) ).

% Pair_top_top
thf(fact_927_Pair__top__top,axiom,
    ( ( produc7269801343787132631t_unit @ top_to1996260823553986621t_unit @ top_to1996260823553986621t_unit )
    = top_to1373874035669595319t_unit ) ).

% Pair_top_top
thf(fact_928_Pair__top__top,axiom,
    ( ( produc5330676427504004780at_nat @ top_to1496364449551166952nnreal @ top_to7002412479815936651at_nat )
    = top_to4912648571690621508at_nat ) ).

% Pair_top_top
thf(fact_929_ennreal__between,axiom,
    ! [E: extend8495563244428889912nnreal,X5: extend8495563244428889912nnreal] :
      ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ E )
     => ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ X5 )
       => ( ( ord_le7381754540660121996nnreal @ X5 @ top_to1496364449551166952nnreal )
         => ( ord_le7381754540660121996nnreal @ ( minus_8429688780609304081nnreal @ X5 @ E ) @ X5 ) ) ) ) ).

% ennreal_between
thf(fact_930_ennreal__minus__pos__iff,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ( ord_le7381754540660121996nnreal @ A @ top_to1496364449551166952nnreal )
        | ( ord_le7381754540660121996nnreal @ B @ top_to1496364449551166952nnreal ) )
     => ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) )
       => ( ord_le7381754540660121996nnreal @ B @ A ) ) ) ).

% ennreal_minus_pos_iff
thf(fact_931_diff__gt__0__iff__gt__ennreal,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) )
      = ( ( ( A = top_to1496364449551166952nnreal )
          & ( B = top_to1496364449551166952nnreal ) )
        | ( ord_le7381754540660121996nnreal @ B @ A ) ) ) ).

% diff_gt_0_iff_gt_ennreal
thf(fact_932_ennreal__mono__minus__cancel,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ ( minus_8429688780609304081nnreal @ A @ C ) )
     => ( ( ord_le7381754540660121996nnreal @ A @ top_to1496364449551166952nnreal )
       => ( ( ord_le3935885782089961368nnreal @ B @ A )
         => ( ( ord_le3935885782089961368nnreal @ C @ A )
           => ( ord_le3935885782089961368nnreal @ C @ B ) ) ) ) ) ).

% ennreal_mono_minus_cancel
thf(fact_933_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_934_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ D @ C )
       => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_935_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ D @ C )
       => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_936_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_937_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_938_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_939_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_940_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_eq_real @ A @ B )
        = ( ord_less_eq_real @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_941_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_eq_rat @ A @ B )
        = ( ord_less_eq_rat @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_942_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_943_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_944_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_945_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_946_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_rat @ A @ B )
        = ( ord_less_rat @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_947_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_real @ A @ B )
        = ( ord_less_real @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_948_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ D @ C )
       => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_949_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ D @ C )
       => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_950_minus__nat_Osimps_I1_J,axiom,
    ! [M3: nat] :
      ( ( minus_minus_nat @ M3 @ zero_zero_nat )
      = M3 ) ).

% minus_nat.simps(1)
thf(fact_951_diffs0__imp__equal,axiom,
    ! [M3: nat,N2: nat] :
      ( ( ( minus_minus_nat @ M3 @ N2 )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N2 @ M3 )
          = zero_zero_nat )
       => ( M3 = N2 ) ) ) ).

% diffs0_imp_equal
thf(fact_952_less__imp__diff__less,axiom,
    ! [J2: nat,K: nat,N2: nat] :
      ( ( ord_less_nat @ J2 @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J2 @ N2 ) @ K ) ) ).

% less_imp_diff_less
thf(fact_953_diff__less__mono2,axiom,
    ! [M3: nat,N2: nat,L3: nat] :
      ( ( ord_less_nat @ M3 @ N2 )
     => ( ( ord_less_nat @ M3 @ L3 )
       => ( ord_less_nat @ ( minus_minus_nat @ L3 @ N2 ) @ ( minus_minus_nat @ L3 @ M3 ) ) ) ) ).

% diff_less_mono2
thf(fact_954_eq__diff__iff,axiom,
    ! [K: nat,M3: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M3 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ( minus_minus_nat @ M3 @ K )
            = ( minus_minus_nat @ N2 @ K ) )
          = ( M3 = N2 ) ) ) ) ).

% eq_diff_iff
thf(fact_955_le__diff__iff,axiom,
    ! [K: nat,M3: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M3 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( ord_less_eq_nat @ M3 @ N2 ) ) ) ) ).

% le_diff_iff
thf(fact_956_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M3: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M3 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( minus_minus_nat @ M3 @ N2 ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_957_diff__le__mono,axiom,
    ! [M3: nat,N2: nat,L3: nat] :
      ( ( ord_less_eq_nat @ M3 @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ L3 ) @ ( minus_minus_nat @ N2 @ L3 ) ) ) ).

% diff_le_mono
thf(fact_958_diff__le__self,axiom,
    ! [M3: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ N2 ) @ M3 ) ).

% diff_le_self
thf(fact_959_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_960_diff__le__mono2,axiom,
    ! [M3: nat,N2: nat,L3: nat] :
      ( ( ord_less_eq_nat @ M3 @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L3 @ N2 ) @ ( minus_minus_nat @ L3 @ M3 ) ) ) ).

% diff_le_mono2
thf(fact_961_nat__distrib_I4_J,axiom,
    ! [K: nat,M3: nat,N2: nat] :
      ( ( times_times_nat @ K @ ( minus_minus_nat @ M3 @ N2 ) )
      = ( minus_minus_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N2 ) ) ) ).

% nat_distrib(4)
thf(fact_962_nat__distrib_I3_J,axiom,
    ! [M3: nat,N2: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M3 @ N2 ) @ K )
      = ( minus_minus_nat @ ( times_times_nat @ M3 @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).

% nat_distrib(3)
thf(fact_963_neq__top__trans,axiom,
    ! [Y4: extend8495563244428889912nnreal,X5: extend8495563244428889912nnreal] :
      ( ( Y4 != top_to1496364449551166952nnreal )
     => ( ( ord_le3935885782089961368nnreal @ X5 @ Y4 )
       => ( X5 != top_to1496364449551166952nnreal ) ) ) ).

% neq_top_trans
thf(fact_964_ennreal__diff__le__mono__left,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A @ B )
     => ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ C ) @ B ) ) ).

% ennreal_diff_le_mono_left
thf(fact_965_diff__le__self__ennreal,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ A ) ).

% diff_le_self_ennreal
thf(fact_966_ennreal__mono__minus,axiom,
    ! [C: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ C @ B )
     => ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ ( minus_8429688780609304081nnreal @ A @ C ) ) ) ).

% ennreal_mono_minus
thf(fact_967_ennreal__minus__mono,axiom,
    ! [A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A @ C )
     => ( ( ord_le3935885782089961368nnreal @ D @ B )
       => ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ ( minus_8429688780609304081nnreal @ C @ D ) ) ) ) ).

% ennreal_minus_mono
thf(fact_968_ennreal__top__neq__one,axiom,
    top_to1496364449551166952nnreal != one_on2969667320475766781nnreal ).

% ennreal_top_neq_one
thf(fact_969_spaceN__diff,axiom,
    ! [X5: real,N: functi4972691055096080419m_real,Y4: real] :
      ( ( member_real @ X5 @ ( functi7867165335332326841N_real @ N ) )
     => ( ( member_real @ Y4 @ ( functi7867165335332326841N_real @ N ) )
       => ( member_real @ ( minus_minus_real @ X5 @ Y4 ) @ ( functi7867165335332326841N_real @ N ) ) ) ) ).

% spaceN_diff
thf(fact_970_spaceN__diff,axiom,
    ! [X5: ( product_prod_nat_nat > product_prod_nat_nat ) > real,N: functi5165143733218492903t_real,Y4: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( member8159409068225774087t_real @ X5 @ ( functi8590574888837585149t_real @ N ) )
     => ( ( member8159409068225774087t_real @ Y4 @ ( functi8590574888837585149t_real @ N ) )
       => ( member8159409068225774087t_real @ ( minus_4447934758107655753t_real @ X5 @ Y4 ) @ ( functi8590574888837585149t_real @ N ) ) ) ) ).

% spaceN_diff
thf(fact_971_spaceN__diff,axiom,
    ! [X5: product_prod_nat_nat > real,N: functi4297716468133224573t_real,Y4: product_prod_nat_nat > real] :
      ( ( member4564283293661824327t_real @ X5 @ ( functi6492146676480127185t_real @ N ) )
     => ( ( member4564283293661824327t_real @ Y4 @ ( functi6492146676480127185t_real @ N ) )
       => ( member4564283293661824327t_real @ ( minus_6672239299375745157t_real @ X5 @ Y4 ) @ ( functi6492146676480127185t_real @ N ) ) ) ) ).

% spaceN_diff
thf(fact_972_ennreal__minus,axiom,
    ! [Q2: real,R: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Q2 )
     => ( ( minus_8429688780609304081nnreal @ ( extend7643940197134561352nnreal @ R ) @ ( extend7643940197134561352nnreal @ Q2 ) )
        = ( extend7643940197134561352nnreal @ ( minus_minus_real @ R @ Q2 ) ) ) ) ).

% ennreal_minus
thf(fact_973_ennreal__minus__if,axiom,
    ! [A: real,B: real] :
      ( ( minus_8429688780609304081nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B ) )
      = ( extend7643940197134561352nnreal @ ( if_real @ ( ord_less_eq_real @ zero_zero_real @ B ) @ ( if_real @ ( ord_less_eq_real @ B @ A ) @ ( minus_minus_real @ A @ B ) @ zero_zero_real ) @ A ) ) ) ).

% ennreal_minus_if
thf(fact_974_diff__eq__0__iff__ennreal,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ( minus_8429688780609304081nnreal @ A @ B )
        = zero_z7100319975126383169nnreal )
      = ( ( ord_le7381754540660121996nnreal @ A @ top_to1496364449551166952nnreal )
        & ( ord_le3935885782089961368nnreal @ A @ B ) ) ) ).

% diff_eq_0_iff_ennreal
thf(fact_975_diff__eq__0__ennreal,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le7381754540660121996nnreal @ A @ top_to1496364449551166952nnreal )
     => ( ( ord_le3935885782089961368nnreal @ A @ B )
       => ( ( minus_8429688780609304081nnreal @ A @ B )
          = zero_z7100319975126383169nnreal ) ) ) ).

% diff_eq_0_ennreal
thf(fact_976_space__empty__eq__bot,axiom,
    ! [A: sigma_5515648953823433982at_nat] :
      ( ( ( sigma_4797346298676585097at_nat @ A )
        = bot_bo2099793752762293965at_nat )
      = ( A = bot_bo8824583409880654738at_nat ) ) ).

% space_empty_eq_bot
thf(fact_977_space__empty__eq__bot,axiom,
    ! [A: sigma_9047027012034273406at_nat] :
      ( ( ( sigma_4468984517727773725at_nat @ A )
        = bot_bo2540092574283496047at_nat )
      = ( A = bot_bo7613792866757523178at_nat ) ) ).

% space_empty_eq_bot
thf(fact_978_borel__measurable__diff_H,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,M2: sigma_9047027012034273406at_nat,G2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ M2 @ borel_5078946678739801102l_real ) )
     => ( ( member8159409068225774087t_real @ G2 @ ( sigma_5064276549609164707t_real @ M2 @ borel_5078946678739801102l_real ) )
       => ( member8159409068225774087t_real @ ( minus_4447934758107655753t_real @ F2 @ G2 ) @ ( sigma_5064276549609164707t_real @ M2 @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_diff'
thf(fact_979_borel__measurable__diff_H,axiom,
    ! [F2: product_prod_nat_nat > real,M2: sigma_5515648953823433982at_nat,G2: product_prod_nat_nat > real] :
      ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ M2 @ borel_5078946678739801102l_real ) )
     => ( ( member4564283293661824327t_real @ G2 @ ( sigma_1188828689629184861t_real @ M2 @ borel_5078946678739801102l_real ) )
       => ( member4564283293661824327t_real @ ( minus_6672239299375745157t_real @ F2 @ G2 ) @ ( sigma_1188828689629184861t_real @ M2 @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_diff'
thf(fact_980_diff__less,axiom,
    ! [N2: nat,M3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ M3 )
       => ( ord_less_nat @ ( minus_minus_nat @ M3 @ N2 ) @ M3 ) ) ) ).

% diff_less
thf(fact_981_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).

% diff_less_mono
thf(fact_982_less__diff__iff,axiom,
    ! [K: nat,M3: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M3 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( ord_less_nat @ M3 @ N2 ) ) ) ) ).

% less_diff_iff
thf(fact_983_ennreal__less__top,axiom,
    ! [X5: real] : ( ord_le7381754540660121996nnreal @ ( extend7643940197134561352nnreal @ X5 ) @ top_to1496364449551166952nnreal ) ).

% ennreal_less_top
thf(fact_984_ennreal__zero__less__top,axiom,
    ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ top_to1496364449551166952nnreal ).

% ennreal_zero_less_top
thf(fact_985_ennreal__minus__eq__0,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ( minus_8429688780609304081nnreal @ A @ B )
        = zero_z7100319975126383169nnreal )
     => ( ord_le3935885782089961368nnreal @ A @ B ) ) ).

% ennreal_minus_eq_0
thf(fact_986_diff__gr0__ennreal,axiom,
    ! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
      ( ( ord_le7381754540660121996nnreal @ B @ A )
     => ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) ) ) ).

% diff_gr0_ennreal
thf(fact_987_finite__measure_Oemeasure__finite,axiom,
    ! [M2: sigma_9047027012034273406at_nat,A2: set_Pr2458342521480944603at_nat] :
      ( ( measur5156515106886843841at_nat @ M2 )
     => ( ( sigma_3593834561172453006at_nat @ M2 @ A2 )
       != top_to1496364449551166952nnreal ) ) ).

% finite_measure.emeasure_finite
thf(fact_988_finite__measure_Oemeasure__finite,axiom,
    ! [M2: sigma_5515648953823433982at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( measur7911732083032432613at_nat @ M2 )
     => ( ( sigma_411563132819280856at_nat @ M2 @ A2 )
       != top_to1496364449551166952nnreal ) ) ).

% finite_measure.emeasure_finite
thf(fact_989_ennreal__one__less__top,axiom,
    ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ top_to1496364449551166952nnreal ).

% ennreal_one_less_top
thf(fact_990_borel__measurable__diff,axiom,
    ! [F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,M2: sigma_9047027012034273406at_nat,G2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ M2 @ borel_5078946678739801102l_real ) )
     => ( ( member8159409068225774087t_real @ G2 @ ( sigma_5064276549609164707t_real @ M2 @ borel_5078946678739801102l_real ) )
       => ( member8159409068225774087t_real
          @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( minus_minus_real @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
          @ ( sigma_5064276549609164707t_real @ M2 @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_diff
thf(fact_991_borel__measurable__diff,axiom,
    ! [F2: product_prod_nat_nat > real,M2: sigma_5515648953823433982at_nat,G2: product_prod_nat_nat > real] :
      ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ M2 @ borel_5078946678739801102l_real ) )
     => ( ( member4564283293661824327t_real @ G2 @ ( sigma_1188828689629184861t_real @ M2 @ borel_5078946678739801102l_real ) )
       => ( member4564283293661824327t_real
          @ ^ [X2: product_prod_nat_nat] : ( minus_minus_real @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
          @ ( sigma_1188828689629184861t_real @ M2 @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_diff
thf(fact_992_borel__measurable__minus__ennreal,axiom,
    ! [F2: real > extend8495563244428889912nnreal,M2: sigma_measure_real,G2: real > extend8495563244428889912nnreal] :
      ( ( member2919562650594848410nnreal @ F2 @ ( sigma_9017504469962657078nnreal @ M2 @ borel_6524799422816628122nnreal ) )
     => ( ( member2919562650594848410nnreal @ G2 @ ( sigma_9017504469962657078nnreal @ M2 @ borel_6524799422816628122nnreal ) )
       => ( member2919562650594848410nnreal
          @ ^ [X2: real] : ( minus_8429688780609304081nnreal @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
          @ ( sigma_9017504469962657078nnreal @ M2 @ borel_6524799422816628122nnreal ) ) ) ) ).

% borel_measurable_minus_ennreal
thf(fact_993_ennreal__cases,axiom,
    ! [X5: extend8495563244428889912nnreal] :
      ( ! [R2: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ R2 )
         => ( X5
           != ( extend7643940197134561352nnreal @ R2 ) ) )
     => ( X5 = top_to1496364449551166952nnreal ) ) ).

% ennreal_cases
thf(fact_994_ennreal2__cases,axiom,
    ! [X5: extend8495563244428889912nnreal,Xa: extend8495563244428889912nnreal] :
      ( ! [R2: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ R2 )
         => ( ( X5
              = ( extend7643940197134561352nnreal @ R2 ) )
           => ! [Ra: real] :
                ( ( ord_less_eq_real @ zero_zero_real @ Ra )
               => ( Xa
                 != ( extend7643940197134561352nnreal @ Ra ) ) ) ) )
     => ( ! [R2: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ R2 )
           => ( ( X5
                = ( extend7643940197134561352nnreal @ R2 ) )
             => ( Xa != top_to1496364449551166952nnreal ) ) )
       => ( ( ( X5 = top_to1496364449551166952nnreal )
           => ! [R2: real] :
                ( ( ord_less_eq_real @ zero_zero_real @ R2 )
               => ( Xa
                 != ( extend7643940197134561352nnreal @ R2 ) ) ) )
         => ~ ( ( X5 = top_to1496364449551166952nnreal )
             => ( Xa != top_to1496364449551166952nnreal ) ) ) ) ) ).

% ennreal2_cases
thf(fact_995_ennreal3__cases,axiom,
    ! [X5: extend8495563244428889912nnreal,Xa: extend8495563244428889912nnreal,Xaa: extend8495563244428889912nnreal] :
      ( ! [R2: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ R2 )
         => ( ( X5
              = ( extend7643940197134561352nnreal @ R2 ) )
           => ! [Ra: real] :
                ( ( ord_less_eq_real @ zero_zero_real @ Ra )
               => ( ( Xa
                    = ( extend7643940197134561352nnreal @ Ra ) )
                 => ! [Raa: real] :
                      ( ( ord_less_eq_real @ zero_zero_real @ Raa )
                     => ( Xaa
                       != ( extend7643940197134561352nnreal @ Raa ) ) ) ) ) ) )
     => ( ! [R2: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ R2 )
           => ( ( X5
                = ( extend7643940197134561352nnreal @ R2 ) )
             => ! [Ra: real] :
                  ( ( ord_less_eq_real @ zero_zero_real @ Ra )
                 => ( ( Xa
                      = ( extend7643940197134561352nnreal @ Ra ) )
                   => ( Xaa != top_to1496364449551166952nnreal ) ) ) ) )
       => ( ! [R2: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ R2 )
             => ( ( X5
                  = ( extend7643940197134561352nnreal @ R2 ) )
               => ( ( Xa = top_to1496364449551166952nnreal )
                 => ! [Ra: real] :
                      ( ( ord_less_eq_real @ zero_zero_real @ Ra )
                     => ( Xaa
                       != ( extend7643940197134561352nnreal @ Ra ) ) ) ) ) )
         => ( ! [R2: real] :
                ( ( ord_less_eq_real @ zero_zero_real @ R2 )
               => ( ( X5
                    = ( extend7643940197134561352nnreal @ R2 ) )
                 => ( ( Xa = top_to1496364449551166952nnreal )
                   => ( Xaa != top_to1496364449551166952nnreal ) ) ) )
           => ( ( ( X5 = top_to1496364449551166952nnreal )
               => ! [R2: real] :
                    ( ( ord_less_eq_real @ zero_zero_real @ R2 )
                   => ( ( Xa
                        = ( extend7643940197134561352nnreal @ R2 ) )
                     => ! [Ra: real] :
                          ( ( ord_less_eq_real @ zero_zero_real @ Ra )
                         => ( Xaa
                           != ( extend7643940197134561352nnreal @ Ra ) ) ) ) ) )
             => ( ( ( X5 = top_to1496364449551166952nnreal )
                 => ! [R2: real] :
                      ( ( ord_less_eq_real @ zero_zero_real @ R2 )
                     => ( ( Xa
                          = ( extend7643940197134561352nnreal @ R2 ) )
                       => ( Xaa != top_to1496364449551166952nnreal ) ) ) )
               => ( ( ( X5 = top_to1496364449551166952nnreal )
                   => ( ( Xa = top_to1496364449551166952nnreal )
                     => ! [R2: real] :
                          ( ( ord_less_eq_real @ zero_zero_real @ R2 )
                         => ( Xaa
                           != ( extend7643940197134561352nnreal @ R2 ) ) ) ) )
                 => ~ ( ( X5 = top_to1496364449551166952nnreal )
                     => ( ( Xa = top_to1496364449551166952nnreal )
                       => ( Xaa != top_to1496364449551166952nnreal ) ) ) ) ) ) ) ) ) ) ).

% ennreal3_cases
thf(fact_996_finite__measure_Ofinite__emeasure__space,axiom,
    ! [M2: sigma_9047027012034273406at_nat] :
      ( ( measur5156515106886843841at_nat @ M2 )
     => ( ( sigma_3593834561172453006at_nat @ M2 @ ( sigma_4468984517727773725at_nat @ M2 ) )
       != top_to1496364449551166952nnreal ) ) ).

% finite_measure.finite_emeasure_space
thf(fact_997_finite__measure_Ofinite__emeasure__space,axiom,
    ! [M2: sigma_5515648953823433982at_nat] :
      ( ( measur7911732083032432613at_nat @ M2 )
     => ( ( sigma_411563132819280856at_nat @ M2 @ ( sigma_4797346298676585097at_nat @ M2 ) )
       != top_to1496364449551166952nnreal ) ) ).

% finite_measure.finite_emeasure_space
thf(fact_998_ennreal__mult__le__mult__iff,axiom,
    ! [C: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( C != zero_z7100319975126383169nnreal )
     => ( ( C != top_to1496364449551166952nnreal )
       => ( ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ C @ A ) @ ( times_1893300245718287421nnreal @ C @ B ) )
          = ( ord_le3935885782089961368nnreal @ A @ B ) ) ) ) ).

% ennreal_mult_le_mult_iff
thf(fact_999_ennreal__mult__less__top,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le7381754540660121996nnreal @ ( times_1893300245718287421nnreal @ A @ B ) @ top_to1496364449551166952nnreal )
      = ( ( A = zero_z7100319975126383169nnreal )
        | ( B = zero_z7100319975126383169nnreal )
        | ( ( ord_le7381754540660121996nnreal @ A @ top_to1496364449551166952nnreal )
          & ( ord_le7381754540660121996nnreal @ B @ top_to1496364449551166952nnreal ) ) ) ) ).

% ennreal_mult_less_top
thf(fact_1000_ennreal__mult__strict__left__mono,axiom,
    ! [A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le7381754540660121996nnreal @ A @ C )
     => ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ B )
       => ( ( ord_le7381754540660121996nnreal @ B @ top_to1496364449551166952nnreal )
         => ( ord_le7381754540660121996nnreal @ ( times_1893300245718287421nnreal @ B @ A ) @ ( times_1893300245718287421nnreal @ B @ C ) ) ) ) ) ).

% ennreal_mult_strict_left_mono
thf(fact_1001_ennreal__mult__strict__right__mono,axiom,
    ! [A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le7381754540660121996nnreal @ A @ C )
     => ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ B )
       => ( ( ord_le7381754540660121996nnreal @ B @ top_to1496364449551166952nnreal )
         => ( ord_le7381754540660121996nnreal @ ( times_1893300245718287421nnreal @ A @ B ) @ ( times_1893300245718287421nnreal @ C @ B ) ) ) ) ) ).

% ennreal_mult_strict_right_mono
thf(fact_1002_less__top__ennreal,axiom,
    ! [X5: extend8495563244428889912nnreal] :
      ( ( ord_le7381754540660121996nnreal @ X5 @ top_to1496364449551166952nnreal )
      = ( ? [R3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ R3 )
            & ( X5
              = ( extend7643940197134561352nnreal @ R3 ) ) ) ) ) ).

% less_top_ennreal
thf(fact_1003_emeasure__single__in__space,axiom,
    ! [M2: sigma_measure_rat,X5: rat] :
      ( ( ( sigma_emeasure_rat @ M2 @ ( insert_rat @ X5 @ bot_bot_set_rat ) )
       != zero_z7100319975126383169nnreal )
     => ( member_rat @ X5 @ ( sigma_space_rat @ M2 ) ) ) ).

% emeasure_single_in_space
thf(fact_1004_emeasure__single__in__space,axiom,
    ! [M2: sigma_2424604449015074331t_real,X5: product_prod_nat_nat > real] :
      ( ( ( sigma_7320754503854894453t_real @ M2 @ ( insert2113160732575218286t_real @ X5 @ bot_bo8258804065021230570t_real ) )
       != zero_z7100319975126383169nnreal )
     => ( member4564283293661824327t_real @ X5 @ ( sigma_7766831664147846694t_real @ M2 ) ) ) ).

% emeasure_single_in_space
thf(fact_1005_emeasure__single__in__space,axiom,
    ! [M2: sigma_5394977995791401948nnreal,X5: real > extend8495563244428889912nnreal] :
      ( ( ( sigma_5209345488743548908nnreal @ M2 @ ( insert152533262698245683nnreal @ X5 @ bot_bo6037503491064675021nnreal ) )
       != zero_z7100319975126383169nnreal )
     => ( member2919562650594848410nnreal @ X5 @ ( sigma_2369682286586992763nnreal @ M2 ) ) ) ).

% emeasure_single_in_space
thf(fact_1006_emeasure__single__in__space,axiom,
    ! [M2: sigma_3568842061677812681t_real,X5: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( ( sigma_7977965363854542169t_real @ M2 @ ( insert5637722417281287840t_real @ X5 @ bot_bo2850223541457833914t_real ) )
       != zero_z7100319975126383169nnreal )
     => ( member8159409068225774087t_real @ X5 @ ( sigma_8760150869319784680t_real @ M2 ) ) ) ).

% emeasure_single_in_space
thf(fact_1007_emeasure__single__in__space,axiom,
    ! [M2: sigma_measure_o,X5: $o] :
      ( ( ( sigma_emeasure_o @ M2 @ ( insert_o @ X5 @ bot_bot_set_o ) )
       != zero_z7100319975126383169nnreal )
     => ( member_o @ X5 @ ( sigma_space_o @ M2 ) ) ) ).

% emeasure_single_in_space
thf(fact_1008_emeasure__single__in__space,axiom,
    ! [M2: sigma_5515648953823433982at_nat,X5: product_prod_nat_nat] :
      ( ( ( sigma_411563132819280856at_nat @ M2 @ ( insert8211810215607154385at_nat @ X5 @ bot_bo2099793752762293965at_nat ) )
       != zero_z7100319975126383169nnreal )
     => ( member8440522571783428010at_nat @ X5 @ ( sigma_4797346298676585097at_nat @ M2 ) ) ) ).

% emeasure_single_in_space
thf(fact_1009_emeasure__single__in__space,axiom,
    ! [M2: sigma_9047027012034273406at_nat,X5: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( ( sigma_3593834561172453006at_nat @ M2 @ ( insert5766096353517822421at_nat @ X5 @ bot_bo2540092574283496047at_nat ) )
       != zero_z7100319975126383169nnreal )
     => ( member8885076297122219836at_nat @ X5 @ ( sigma_4468984517727773725at_nat @ M2 ) ) ) ).

% emeasure_single_in_space
thf(fact_1010_prob__space_Oindep__sets__finite__index__sets,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F: nat > set_se7855581050983116737at_nat,I: set_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe5197683624481315429at_nat @ M2 @ F @ I )
        = ( ! [J4: set_nat] :
              ( ( ord_less_eq_set_nat @ J4 @ I )
             => ( ( J4 != bot_bot_set_nat )
               => ( ( finite_finite_nat @ J4 )
                 => ( indepe5197683624481315429at_nat @ M2 @ F @ J4 ) ) ) ) ) ) ) ).

% prob_space.indep_sets_finite_index_sets
thf(fact_1011_prob__space_Oindep__sets__finite__index__sets,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F: $o > set_se7855581050983116737at_nat,I: set_o] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe619418257930843587_nat_o @ M2 @ F @ I )
        = ( ! [J4: set_o] :
              ( ( ord_less_eq_set_o @ J4 @ I )
             => ( ( J4 != bot_bot_set_o )
               => ( ( finite_finite_o @ J4 )
                 => ( indepe619418257930843587_nat_o @ M2 @ F @ J4 ) ) ) ) ) ) ) ).

% prob_space.indep_sets_finite_index_sets
thf(fact_1012_prob__space_Oindep__sets__finite__index__sets,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F: product_prod_nat_nat > set_se7855581050983116737at_nat,I: set_Pr1261947904930325089at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe2425276932038335532at_nat @ M2 @ F @ I )
        = ( ! [J4: set_Pr1261947904930325089at_nat] :
              ( ( ord_le3146513528884898305at_nat @ J4 @ I )
             => ( ( J4 != bot_bo2099793752762293965at_nat )
               => ( ( finite6177210948735845034at_nat @ J4 )
                 => ( indepe2425276932038335532at_nat @ M2 @ F @ J4 ) ) ) ) ) ) ) ).

% prob_space.indep_sets_finite_index_sets
thf(fact_1013_prob__space_Oindep__sets__finite__index__sets,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F: nat > set_se1666487788256820497at_nat,I: set_nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe3633531339187150819at_nat @ M2 @ F @ I )
        = ( ! [J4: set_nat] :
              ( ( ord_less_eq_set_nat @ J4 @ I )
             => ( ( J4 != bot_bot_set_nat )
               => ( ( finite_finite_nat @ J4 )
                 => ( indepe3633531339187150819at_nat @ M2 @ F @ J4 ) ) ) ) ) ) ) ).

% prob_space.indep_sets_finite_index_sets
thf(fact_1014_prob__space_Oindep__sets__finite__index__sets,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F: ( product_prod_nat_nat > real ) > set_se7855581050983116737at_nat,I: set_Pr1002607673312053630t_real] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe3818579677853040073t_real @ M2 @ F @ I )
        = ( ! [J4: set_Pr1002607673312053630t_real] :
              ( ( ord_le1994730768851292446t_real @ J4 @ I )
             => ( ( J4 != bot_bo8258804065021230570t_real )
               => ( ( finite7134436996427083847t_real @ J4 )
                 => ( indepe3818579677853040073t_real @ M2 @ F @ J4 ) ) ) ) ) ) ) ).

% prob_space.indep_sets_finite_index_sets
thf(fact_1015_prob__space_Oindep__sets__finite__index__sets,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F: $o > set_se1666487788256820497at_nat,I: set_o] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe9198310886692500677_nat_o @ M2 @ F @ I )
        = ( ! [J4: set_o] :
              ( ( ord_less_eq_set_o @ J4 @ I )
             => ( ( J4 != bot_bot_set_o )
               => ( ( finite_finite_o @ J4 )
                 => ( indepe9198310886692500677_nat_o @ M2 @ F @ J4 ) ) ) ) ) ) ) ).

% prob_space.indep_sets_finite_index_sets
thf(fact_1016_prob__space_Oindep__sets__finite__index__sets,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F: rat > set_se1666487788256820497at_nat,I: set_rat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe2998401279100655083at_rat @ M2 @ F @ I )
        = ( ! [J4: set_rat] :
              ( ( ord_less_eq_set_rat @ J4 @ I )
             => ( ( J4 != bot_bot_set_rat )
               => ( ( finite_finite_rat @ J4 )
                 => ( indepe2998401279100655083at_rat @ M2 @ F @ J4 ) ) ) ) ) ) ) ).

% prob_space.indep_sets_finite_index_sets
thf(fact_1017_prob__space_Oindep__sets__finite__index__sets,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F: product_prod_nat_nat > set_se1666487788256820497at_nat,I: set_Pr1261947904930325089at_nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe2942784128085459246at_nat @ M2 @ F @ I )
        = ( ! [J4: set_Pr1261947904930325089at_nat] :
              ( ( ord_le3146513528884898305at_nat @ J4 @ I )
             => ( ( J4 != bot_bo2099793752762293965at_nat )
               => ( ( finite6177210948735845034at_nat @ J4 )
                 => ( indepe2942784128085459246at_nat @ M2 @ F @ J4 ) ) ) ) ) ) ) ).

% prob_space.indep_sets_finite_index_sets
thf(fact_1018_prob__space_Oindep__sets__finite__index__sets,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F: ( product_prod_nat_nat > product_prod_nat_nat ) > set_se7855581050983116737at_nat,I: set_Pr2458342521480944603at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe6545576254369104698at_nat @ M2 @ F @ I )
        = ( ! [J4: set_Pr2458342521480944603at_nat] :
              ( ( ord_le3857079194666040379at_nat @ J4 @ I )
             => ( ( J4 != bot_bo2540092574283496047at_nat )
               => ( ( finite1779803868683885628at_nat @ J4 )
                 => ( indepe6545576254369104698at_nat @ M2 @ F @ J4 ) ) ) ) ) ) ) ).

% prob_space.indep_sets_finite_index_sets
thf(fact_1019_prob__space_Oindep__sets__finite__index__sets,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F: ( real > extend8495563244428889912nnreal ) > set_se1666487788256820497at_nat,I: set_re5328672808648366137nnreal] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe6665278795436257814nnreal @ M2 @ F @ I )
        = ( ! [J4: set_re5328672808648366137nnreal] :
              ( ( ord_le2462468573666744473nnreal @ J4 @ I )
             => ( ( J4 != bot_bo6037503491064675021nnreal )
               => ( ( finite7684081742213514138nnreal @ J4 )
                 => ( indepe6665278795436257814nnreal @ M2 @ F @ J4 ) ) ) ) ) ) ) ).

% prob_space.indep_sets_finite_index_sets
thf(fact_1020_semiring__norm_I113_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% semiring_norm(113)
thf(fact_1021_semiring__norm_I113_J,axiom,
    ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ zero_z7100319975126383169nnreal ).

% semiring_norm(113)
thf(fact_1022_semiring__norm_I113_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).

% semiring_norm(113)
thf(fact_1023_semiring__norm_I113_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% semiring_norm(113)
thf(fact_1024_semiring__norm_I114_J,axiom,
    ord_less_eq_real @ one_one_real @ one_one_real ).

% semiring_norm(114)
thf(fact_1025_semiring__norm_I114_J,axiom,
    ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ one_on2969667320475766781nnreal ).

% semiring_norm(114)
thf(fact_1026_semiring__norm_I114_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% semiring_norm(114)
thf(fact_1027_semiring__norm_I114_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% semiring_norm(114)
thf(fact_1028_semiring__norm_I137_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% semiring_norm(137)
thf(fact_1029_semiring__norm_I137_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).

% semiring_norm(137)
thf(fact_1030_semiring__norm_I137_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).

% semiring_norm(137)
thf(fact_1031_semiring__norm_I137_J,axiom,
    ~ ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ zero_z7100319975126383169nnreal ) ).

% semiring_norm(137)
thf(fact_1032_semiring__norm_I160_J,axiom,
    one_one_nat != zero_zero_nat ).

% semiring_norm(160)
thf(fact_1033_semiring__norm_I160_J,axiom,
    one_one_rat != zero_zero_rat ).

% semiring_norm(160)
thf(fact_1034_semiring__norm_I160_J,axiom,
    one_one_real != zero_zero_real ).

% semiring_norm(160)
thf(fact_1035_semiring__norm_I160_J,axiom,
    one_on2969667320475766781nnreal != zero_z7100319975126383169nnreal ).

% semiring_norm(160)
thf(fact_1036_semiring__norm_I138_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% semiring_norm(138)
thf(fact_1037_semiring__norm_I138_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% semiring_norm(138)
thf(fact_1038_semiring__norm_I138_J,axiom,
    ~ ( ord_less_real @ one_one_real @ one_one_real ) ).

% semiring_norm(138)
thf(fact_1039_semiring__norm_I138_J,axiom,
    ~ ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ one_on2969667320475766781nnreal ) ).

% semiring_norm(138)
thf(fact_1040_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M3: nat,N2: nat] :
      ( ( ( times_times_nat @ K @ M3 )
        = ( times_times_nat @ K @ N2 ) )
      = ( ( K = zero_zero_nat )
        | ( M3 = N2 ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_1041_semiring__norm_I111_J,axiom,
    ord_less_eq_real @ zero_zero_real @ one_one_real ).

% semiring_norm(111)
thf(fact_1042_semiring__norm_I111_J,axiom,
    ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ one_on2969667320475766781nnreal ).

% semiring_norm(111)
thf(fact_1043_semiring__norm_I111_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).

% semiring_norm(111)
thf(fact_1044_semiring__norm_I111_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% semiring_norm(111)
thf(fact_1045_semiring__norm_I112_J,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% semiring_norm(112)
thf(fact_1046_semiring__norm_I112_J,axiom,
    ~ ( ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ zero_z7100319975126383169nnreal ) ).

% semiring_norm(112)
thf(fact_1047_semiring__norm_I112_J,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% semiring_norm(112)
thf(fact_1048_semiring__norm_I112_J,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% semiring_norm(112)
thf(fact_1049_semiring__norm_I136_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% semiring_norm(136)
thf(fact_1050_semiring__norm_I136_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% semiring_norm(136)
thf(fact_1051_semiring__norm_I136_J,axiom,
    ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).

% semiring_norm(136)
thf(fact_1052_semiring__norm_I136_J,axiom,
    ~ ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ zero_z7100319975126383169nnreal ) ).

% semiring_norm(136)
thf(fact_1053_semiring__norm_I135_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% semiring_norm(135)
thf(fact_1054_semiring__norm_I135_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% semiring_norm(135)
thf(fact_1055_semiring__norm_I135_J,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% semiring_norm(135)
thf(fact_1056_semiring__norm_I135_J,axiom,
    ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ one_on2969667320475766781nnreal ).

% semiring_norm(135)
thf(fact_1057_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M3: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ( times_times_nat @ K @ M3 )
          = ( times_times_nat @ K @ N2 ) )
        = ( M3 = N2 ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_1058_nat__mult__less__cancel1,axiom,
    ! [K: nat,M3: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N2 ) )
        = ( ord_less_nat @ M3 @ N2 ) ) ) ).

% nat_mult_less_cancel1
thf(fact_1059_nat__mult__le__cancel1,axiom,
    ! [K: nat,M3: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N2 ) )
        = ( ord_less_eq_nat @ M3 @ N2 ) ) ) ).

% nat_mult_le_cancel1
thf(fact_1060__092_060Omega_062_092_060_094sub_0622_Oindep__events__finite__index__events,axiom,
    ! [F: nat > set_Pr2458342521480944603at_nat,I: set_nat] :
      ( ( indepe7450211079141483467at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
      = ( ! [J4: set_nat] :
            ( ( ord_less_eq_set_nat @ J4 @ I )
           => ( ( J4 != bot_bot_set_nat )
             => ( ( finite_finite_nat @ J4 )
               => ( indepe7450211079141483467at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ J4 ) ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_events_finite_index_events
thf(fact_1061__092_060Omega_062_092_060_094sub_0622_Oindep__events__finite__index__events,axiom,
    ! [F: product_prod_nat_nat > set_Pr2458342521480944603at_nat,I: set_Pr1261947904930325089at_nat] :
      ( ( indepe3568070906090945350at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
      = ( ! [J4: set_Pr1261947904930325089at_nat] :
            ( ( ord_le3146513528884898305at_nat @ J4 @ I )
           => ( ( J4 != bot_bo2099793752762293965at_nat )
             => ( ( finite6177210948735845034at_nat @ J4 )
               => ( indepe3568070906090945350at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ J4 ) ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_events_finite_index_events
thf(fact_1062__092_060Omega_062_092_060_094sub_0622_Oindep__events__finite__index__events,axiom,
    ! [F: ( product_prod_nat_nat > product_prod_nat_nat ) > set_Pr2458342521480944603at_nat,I: set_Pr2458342521480944603at_nat] :
      ( ( indepe2449102103094332064at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
      = ( ! [J4: set_Pr2458342521480944603at_nat] :
            ( ( ord_le3857079194666040379at_nat @ J4 @ I )
           => ( ( J4 != bot_bo2540092574283496047at_nat )
             => ( ( finite1779803868683885628at_nat @ J4 )
               => ( indepe2449102103094332064at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ J4 ) ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_events_finite_index_events
thf(fact_1063__092_060Omega_062_092_060_094sub_0622_Oindep__events__finite__index__events,axiom,
    ! [F: ( product_prod_nat_nat > real ) > set_Pr2458342521480944603at_nat,I: set_Pr1002607673312053630t_real] :
      ( ( indepe5332547468274401507t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
      = ( ! [J4: set_Pr1002607673312053630t_real] :
            ( ( ord_le1994730768851292446t_real @ J4 @ I )
           => ( ( J4 != bot_bo8258804065021230570t_real )
             => ( ( finite7134436996427083847t_real @ J4 )
               => ( indepe5332547468274401507t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ J4 ) ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_events_finite_index_events
thf(fact_1064__092_060Omega_062_092_060_094sub_0622_Oindep__events__finite__index__events,axiom,
    ! [F: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_Pr2458342521480944603at_nat,I: set_Pr947837736998463782t_real] :
      ( ( indepe6336448564937256299t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ I )
      = ( ! [J4: set_Pr947837736998463782t_real] :
            ( ( ord_le8622349663015078278t_real @ J4 @ I )
           => ( ( J4 != bot_bo2850223541457833914t_real )
             => ( ( finite3361987442264708871t_real @ J4 )
               => ( indepe6336448564937256299t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F @ J4 ) ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_events_finite_index_events
thf(fact_1065__092_060Omega_062_092_060_094sub_0622_Oindep__events__def__alt,axiom,
    ! [A2: $o > set_Pr2458342521480944603at_nat,I: set_o] :
      ( ( indepe9001878988524906205_nat_o @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 @ I )
      = ( indepe9198310886692500677_nat_o @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
        @ ^ [I3: $o] : ( insert1813694899509750283at_nat @ ( A2 @ I3 ) @ bot_bo3048727414778170277at_nat )
        @ I ) ) ).

% \<Omega>\<^sub>2.indep_events_def_alt
thf(fact_1066__092_060Omega_062_092_060_094sub_0622_Oindep__events__def__alt,axiom,
    ! [A2: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_Pr2458342521480944603at_nat,I: set_Pr947837736998463782t_real] :
      ( ( indepe6336448564937256299t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 @ I )
      = ( indepe5134244768989041027t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
        @ ^ [I3: ( product_prod_nat_nat > product_prod_nat_nat ) > real] : ( insert1813694899509750283at_nat @ ( A2 @ I3 ) @ bot_bo3048727414778170277at_nat )
        @ I ) ) ).

% \<Omega>\<^sub>2.indep_events_def_alt
thf(fact_1067__092_060Omega_062_092_060_094sub_0622_Oindep__events__def__alt,axiom,
    ! [A2: ( real > extend8495563244428889912nnreal ) > set_Pr2458342521480944603at_nat,I: set_re5328672808648366137nnreal] :
      ( ( indepe8842583234502444542nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 @ I )
      = ( indepe6665278795436257814nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
        @ ^ [I3: real > extend8495563244428889912nnreal] : ( insert1813694899509750283at_nat @ ( A2 @ I3 ) @ bot_bo3048727414778170277at_nat )
        @ I ) ) ).

% \<Omega>\<^sub>2.indep_events_def_alt
thf(fact_1068__092_060Omega_062_092_060_094sub_0622_Oindep__events__def__alt,axiom,
    ! [A2: ( product_prod_nat_nat > real ) > set_Pr2458342521480944603at_nat,I: set_Pr1002607673312053630t_real] :
      ( ( indepe5332547468274401507t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 @ I )
      = ( indepe4000825420875437771t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
        @ ^ [I3: product_prod_nat_nat > real] : ( insert1813694899509750283at_nat @ ( A2 @ I3 ) @ bot_bo3048727414778170277at_nat )
        @ I ) ) ).

% \<Omega>\<^sub>2.indep_events_def_alt
thf(fact_1069__092_060Omega_062_092_060_094sub_0622_Oindep__events__def__alt,axiom,
    ! [A2: ( product_prod_nat_nat > product_prod_nat_nat ) > set_Pr2458342521480944603at_nat,I: set_Pr2458342521480944603at_nat] :
      ( ( indepe2449102103094332064at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 @ I )
      = ( indepe7028276433782144696at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
        @ ^ [I3: product_prod_nat_nat > product_prod_nat_nat] : ( insert1813694899509750283at_nat @ ( A2 @ I3 ) @ bot_bo3048727414778170277at_nat )
        @ I ) ) ).

% \<Omega>\<^sub>2.indep_events_def_alt
thf(fact_1070__092_060Omega_062_092_060_094sub_0622_Oindep__events__def__alt,axiom,
    ! [A2: rat > set_Pr2458342521480944603at_nat,I: set_rat] :
      ( ( indepe6815081019054987731at_rat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 @ I )
      = ( indepe2998401279100655083at_rat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
        @ ^ [I3: rat] : ( insert1813694899509750283at_nat @ ( A2 @ I3 ) @ bot_bo3048727414778170277at_nat )
        @ I ) ) ).

% \<Omega>\<^sub>2.indep_events_def_alt
thf(fact_1071_indep__events__finite__index__events,axiom,
    ! [F: nat > set_Pr1261947904930325089at_nat,I: set_nat] :
      ( ( indepe8857210806783466621at_nat @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
      = ( ! [J4: set_nat] :
            ( ( ord_less_eq_set_nat @ J4 @ I )
           => ( ( J4 != bot_bot_set_nat )
             => ( ( finite_finite_nat @ J4 )
               => ( indepe8857210806783466621at_nat @ ( freque5010624893710627907mega_1 @ as ) @ F @ J4 ) ) ) ) ) ) ).

% indep_events_finite_index_events
thf(fact_1072_indep__events__finite__index__events,axiom,
    ! [F: product_prod_nat_nat > set_Pr1261947904930325089at_nat,I: set_Pr1261947904930325089at_nat] :
      ( ( indepe3472028322304269332at_nat @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
      = ( ! [J4: set_Pr1261947904930325089at_nat] :
            ( ( ord_le3146513528884898305at_nat @ J4 @ I )
           => ( ( J4 != bot_bo2099793752762293965at_nat )
             => ( ( finite6177210948735845034at_nat @ J4 )
               => ( indepe3472028322304269332at_nat @ ( freque5010624893710627907mega_1 @ as ) @ F @ J4 ) ) ) ) ) ) ).

% indep_events_finite_index_events
thf(fact_1073_indep__events__finite__index__events,axiom,
    ! [F: ( product_prod_nat_nat > product_prod_nat_nat ) > set_Pr1261947904930325089at_nat,I: set_Pr2458342521480944603at_nat] :
      ( ( indepe7170863032374590802at_nat @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
      = ( ! [J4: set_Pr2458342521480944603at_nat] :
            ( ( ord_le3857079194666040379at_nat @ J4 @ I )
           => ( ( J4 != bot_bo2540092574283496047at_nat )
             => ( ( finite1779803868683885628at_nat @ J4 )
               => ( indepe7170863032374590802at_nat @ ( freque5010624893710627907mega_1 @ as ) @ F @ J4 ) ) ) ) ) ) ).

% indep_events_finite_index_events
thf(fact_1074_indep__events__finite__index__events,axiom,
    ! [F: ( product_prod_nat_nat > real ) > set_Pr1261947904930325089at_nat,I: set_Pr1002607673312053630t_real] :
      ( ( indepe4391344641829096881t_real @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
      = ( ! [J4: set_Pr1002607673312053630t_real] :
            ( ( ord_le1994730768851292446t_real @ J4 @ I )
           => ( ( J4 != bot_bo8258804065021230570t_real )
             => ( ( finite7134436996427083847t_real @ J4 )
               => ( indepe4391344641829096881t_real @ ( freque5010624893710627907mega_1 @ as ) @ F @ J4 ) ) ) ) ) ) ).

% indep_events_finite_index_events
thf(fact_1075_indep__events__finite__index__events,axiom,
    ! [F: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_Pr1261947904930325089at_nat,I: set_Pr947837736998463782t_real] :
      ( ( indepe3994697929461740573t_real @ ( freque5010624893710627907mega_1 @ as ) @ F @ I )
      = ( ! [J4: set_Pr947837736998463782t_real] :
            ( ( ord_le8622349663015078278t_real @ J4 @ I )
           => ( ( J4 != bot_bo2850223541457833914t_real )
             => ( ( finite3361987442264708871t_real @ J4 )
               => ( indepe3994697929461740573t_real @ ( freque5010624893710627907mega_1 @ as ) @ F @ J4 ) ) ) ) ) ) ).

% indep_events_finite_index_events
thf(fact_1076_indep__events__def__alt,axiom,
    ! [A2: $o > set_Pr1261947904930325089at_nat,I: set_o] :
      ( ( indepe8047530689963900331_nat_o @ ( freque5010624893710627907mega_1 @ as ) @ A2 @ I )
      = ( indepe619418257930843587_nat_o @ ( freque5010624893710627907mega_1 @ as )
        @ ^ [I3: $o] : ( insert9200635055090092081at_nat @ ( A2 @ I3 ) @ bot_bo3083307316010499117at_nat )
        @ I ) ) ).

% indep_events_def_alt
thf(fact_1077_space___092_060Omega_062_092_060_094sub_0622,axiom,
    ( ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) )
    = top_to7002412479815936651at_nat ) ).

% space_\<Omega>\<^sub>2
thf(fact_1078_diff__commute,axiom,
    ! [I4: nat,J2: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J2 ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I4 @ K ) @ J2 ) ) ).

% diff_commute
thf(fact_1079_prob__space_Oindep__events__def__alt,axiom,
    ! [M2: sigma_9047027012034273406at_nat,A2: $o > set_Pr2458342521480944603at_nat,I: set_o] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe9001878988524906205_nat_o @ M2 @ A2 @ I )
        = ( indepe9198310886692500677_nat_o @ M2
          @ ^ [I3: $o] : ( insert1813694899509750283at_nat @ ( A2 @ I3 ) @ bot_bo3048727414778170277at_nat )
          @ I ) ) ) ).

% prob_space.indep_events_def_alt
thf(fact_1080_prob__space_Oindep__events__def__alt,axiom,
    ! [M2: sigma_5515648953823433982at_nat,A2: $o > set_Pr1261947904930325089at_nat,I: set_o] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe8047530689963900331_nat_o @ M2 @ A2 @ I )
        = ( indepe619418257930843587_nat_o @ M2
          @ ^ [I3: $o] : ( insert9200635055090092081at_nat @ ( A2 @ I3 ) @ bot_bo3083307316010499117at_nat )
          @ I ) ) ) ).

% prob_space.indep_events_def_alt
thf(fact_1081_prob__space_Oindep__events__def__alt,axiom,
    ! [M2: sigma_9047027012034273406at_nat,A2: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_Pr2458342521480944603at_nat,I: set_Pr947837736998463782t_real] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe6336448564937256299t_real @ M2 @ A2 @ I )
        = ( indepe5134244768989041027t_real @ M2
          @ ^ [I3: ( product_prod_nat_nat > product_prod_nat_nat ) > real] : ( insert1813694899509750283at_nat @ ( A2 @ I3 ) @ bot_bo3048727414778170277at_nat )
          @ I ) ) ) ).

% prob_space.indep_events_def_alt
thf(fact_1082_prob__space_Oindep__events__def__alt,axiom,
    ! [M2: sigma_9047027012034273406at_nat,A2: ( real > extend8495563244428889912nnreal ) > set_Pr2458342521480944603at_nat,I: set_re5328672808648366137nnreal] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe8842583234502444542nnreal @ M2 @ A2 @ I )
        = ( indepe6665278795436257814nnreal @ M2
          @ ^ [I3: real > extend8495563244428889912nnreal] : ( insert1813694899509750283at_nat @ ( A2 @ I3 ) @ bot_bo3048727414778170277at_nat )
          @ I ) ) ) ).

% prob_space.indep_events_def_alt
thf(fact_1083_prob__space_Oindep__events__def__alt,axiom,
    ! [M2: sigma_9047027012034273406at_nat,A2: ( product_prod_nat_nat > real ) > set_Pr2458342521480944603at_nat,I: set_Pr1002607673312053630t_real] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe5332547468274401507t_real @ M2 @ A2 @ I )
        = ( indepe4000825420875437771t_real @ M2
          @ ^ [I3: product_prod_nat_nat > real] : ( insert1813694899509750283at_nat @ ( A2 @ I3 ) @ bot_bo3048727414778170277at_nat )
          @ I ) ) ) ).

% prob_space.indep_events_def_alt
thf(fact_1084_prob__space_Oindep__events__def__alt,axiom,
    ! [M2: sigma_9047027012034273406at_nat,A2: ( product_prod_nat_nat > product_prod_nat_nat ) > set_Pr2458342521480944603at_nat,I: set_Pr2458342521480944603at_nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe2449102103094332064at_nat @ M2 @ A2 @ I )
        = ( indepe7028276433782144696at_nat @ M2
          @ ^ [I3: product_prod_nat_nat > product_prod_nat_nat] : ( insert1813694899509750283at_nat @ ( A2 @ I3 ) @ bot_bo3048727414778170277at_nat )
          @ I ) ) ) ).

% prob_space.indep_events_def_alt
thf(fact_1085_prob__space_Oindep__events__def__alt,axiom,
    ! [M2: sigma_9047027012034273406at_nat,A2: rat > set_Pr2458342521480944603at_nat,I: set_rat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe6815081019054987731at_rat @ M2 @ A2 @ I )
        = ( indepe2998401279100655083at_rat @ M2
          @ ^ [I3: rat] : ( insert1813694899509750283at_nat @ ( A2 @ I3 ) @ bot_bo3048727414778170277at_nat )
          @ I ) ) ) ).

% prob_space.indep_events_def_alt
thf(fact_1086_prob__space_Oindep__events__finite__index__events,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F: nat > set_Pr2458342521480944603at_nat,I: set_nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe7450211079141483467at_nat @ M2 @ F @ I )
        = ( ! [J4: set_nat] :
              ( ( ord_less_eq_set_nat @ J4 @ I )
             => ( ( J4 != bot_bot_set_nat )
               => ( ( finite_finite_nat @ J4 )
                 => ( indepe7450211079141483467at_nat @ M2 @ F @ J4 ) ) ) ) ) ) ) ).

% prob_space.indep_events_finite_index_events
thf(fact_1087_prob__space_Oindep__events__finite__index__events,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F: product_prod_nat_nat > set_Pr2458342521480944603at_nat,I: set_Pr1261947904930325089at_nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe3568070906090945350at_nat @ M2 @ F @ I )
        = ( ! [J4: set_Pr1261947904930325089at_nat] :
              ( ( ord_le3146513528884898305at_nat @ J4 @ I )
             => ( ( J4 != bot_bo2099793752762293965at_nat )
               => ( ( finite6177210948735845034at_nat @ J4 )
                 => ( indepe3568070906090945350at_nat @ M2 @ F @ J4 ) ) ) ) ) ) ) ).

% prob_space.indep_events_finite_index_events
thf(fact_1088_prob__space_Oindep__events__finite__index__events,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F: ( product_prod_nat_nat > product_prod_nat_nat ) > set_Pr2458342521480944603at_nat,I: set_Pr2458342521480944603at_nat] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe2449102103094332064at_nat @ M2 @ F @ I )
        = ( ! [J4: set_Pr2458342521480944603at_nat] :
              ( ( ord_le3857079194666040379at_nat @ J4 @ I )
             => ( ( J4 != bot_bo2540092574283496047at_nat )
               => ( ( finite1779803868683885628at_nat @ J4 )
                 => ( indepe2449102103094332064at_nat @ M2 @ F @ J4 ) ) ) ) ) ) ) ).

% prob_space.indep_events_finite_index_events
thf(fact_1089_prob__space_Oindep__events__finite__index__events,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F: ( product_prod_nat_nat > real ) > set_Pr2458342521480944603at_nat,I: set_Pr1002607673312053630t_real] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe5332547468274401507t_real @ M2 @ F @ I )
        = ( ! [J4: set_Pr1002607673312053630t_real] :
              ( ( ord_le1994730768851292446t_real @ J4 @ I )
             => ( ( J4 != bot_bo8258804065021230570t_real )
               => ( ( finite7134436996427083847t_real @ J4 )
                 => ( indepe5332547468274401507t_real @ M2 @ F @ J4 ) ) ) ) ) ) ) ).

% prob_space.indep_events_finite_index_events
thf(fact_1090_prob__space_Oindep__events__finite__index__events,axiom,
    ! [M2: sigma_9047027012034273406at_nat,F: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_Pr2458342521480944603at_nat,I: set_Pr947837736998463782t_real] :
      ( ( probab8562894880268318498at_nat @ M2 )
     => ( ( indepe6336448564937256299t_real @ M2 @ F @ I )
        = ( ! [J4: set_Pr947837736998463782t_real] :
              ( ( ord_le8622349663015078278t_real @ J4 @ I )
             => ( ( J4 != bot_bo2850223541457833914t_real )
               => ( ( finite3361987442264708871t_real @ J4 )
                 => ( indepe6336448564937256299t_real @ M2 @ F @ J4 ) ) ) ) ) ) ) ).

% prob_space.indep_events_finite_index_events
thf(fact_1091_prob__space_Oindep__events__finite__index__events,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F: nat > set_Pr1261947904930325089at_nat,I: set_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe8857210806783466621at_nat @ M2 @ F @ I )
        = ( ! [J4: set_nat] :
              ( ( ord_less_eq_set_nat @ J4 @ I )
             => ( ( J4 != bot_bot_set_nat )
               => ( ( finite_finite_nat @ J4 )
                 => ( indepe8857210806783466621at_nat @ M2 @ F @ J4 ) ) ) ) ) ) ) ).

% prob_space.indep_events_finite_index_events
thf(fact_1092_prob__space_Oindep__events__finite__index__events,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F: product_prod_nat_nat > set_Pr1261947904930325089at_nat,I: set_Pr1261947904930325089at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe3472028322304269332at_nat @ M2 @ F @ I )
        = ( ! [J4: set_Pr1261947904930325089at_nat] :
              ( ( ord_le3146513528884898305at_nat @ J4 @ I )
             => ( ( J4 != bot_bo2099793752762293965at_nat )
               => ( ( finite6177210948735845034at_nat @ J4 )
                 => ( indepe3472028322304269332at_nat @ M2 @ F @ J4 ) ) ) ) ) ) ) ).

% prob_space.indep_events_finite_index_events
thf(fact_1093_prob__space_Oindep__events__finite__index__events,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F: ( product_prod_nat_nat > product_prod_nat_nat ) > set_Pr1261947904930325089at_nat,I: set_Pr2458342521480944603at_nat] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe7170863032374590802at_nat @ M2 @ F @ I )
        = ( ! [J4: set_Pr2458342521480944603at_nat] :
              ( ( ord_le3857079194666040379at_nat @ J4 @ I )
             => ( ( J4 != bot_bo2540092574283496047at_nat )
               => ( ( finite1779803868683885628at_nat @ J4 )
                 => ( indepe7170863032374590802at_nat @ M2 @ F @ J4 ) ) ) ) ) ) ) ).

% prob_space.indep_events_finite_index_events
thf(fact_1094_prob__space_Oindep__events__finite__index__events,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F: ( product_prod_nat_nat > real ) > set_Pr1261947904930325089at_nat,I: set_Pr1002607673312053630t_real] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe4391344641829096881t_real @ M2 @ F @ I )
        = ( ! [J4: set_Pr1002607673312053630t_real] :
              ( ( ord_le1994730768851292446t_real @ J4 @ I )
             => ( ( J4 != bot_bo8258804065021230570t_real )
               => ( ( finite7134436996427083847t_real @ J4 )
                 => ( indepe4391344641829096881t_real @ M2 @ F @ J4 ) ) ) ) ) ) ) ).

% prob_space.indep_events_finite_index_events
thf(fact_1095_prob__space_Oindep__events__finite__index__events,axiom,
    ! [M2: sigma_5515648953823433982at_nat,F: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > set_Pr1261947904930325089at_nat,I: set_Pr947837736998463782t_real] :
      ( ( probab2019946041432190532at_nat @ M2 )
     => ( ( indepe3994697929461740573t_real @ M2 @ F @ I )
        = ( ! [J4: set_Pr947837736998463782t_real] :
              ( ( ord_le8622349663015078278t_real @ J4 @ I )
             => ( ( J4 != bot_bo2850223541457833914t_real )
               => ( ( finite3361987442264708871t_real @ J4 )
                 => ( indepe3994697929461740573t_real @ M2 @ F @ J4 ) ) ) ) ) ) ) ).

% prob_space.indep_events_finite_index_events
thf(fact_1096__092_060Omega_062_092_060_094sub_0622_Oindep__eventsI__indep__vars,axiom,
    ! [N: rat > sigma_measure_rat,X: rat > ( product_prod_nat_nat > product_prod_nat_nat ) > rat,I: set_rat,P: rat > rat > $o] :
      ( ( indepe9082779638755350303at_rat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N @ X @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member_set_rat
              @ ( collect_rat
                @ ^ [X2: rat] :
                    ( ( member_rat @ X2 @ ( sigma_space_rat @ ( N @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_rat @ ( N @ I2 ) ) ) )
       => ( indepe6815081019054987731at_rat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
          @ ^ [I3: rat] :
              ( collec248804655031457914at_nat
              @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] :
                  ( ( member8885076297122219836at_nat @ X2 @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
                  & ( P @ I3 @ ( X @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_eventsI_indep_vars
thf(fact_1097__092_060Omega_062_092_060_094sub_0622_Oindep__eventsI__indep__vars,axiom,
    ! [N: rat > sigma_measure_o,X: rat > ( product_prod_nat_nat > product_prod_nat_nat ) > $o,I: set_rat,P: rat > $o > $o] :
      ( ( indepe2354647808791182993_rat_o @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N @ X @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member_set_o
              @ ( collect_o
                @ ^ [X2: $o] :
                    ( ( member_o @ X2 @ ( sigma_space_o @ ( N @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_o @ ( N @ I2 ) ) ) )
       => ( indepe6815081019054987731at_rat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
          @ ^ [I3: rat] :
              ( collec248804655031457914at_nat
              @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] :
                  ( ( member8885076297122219836at_nat @ X2 @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
                  & ( P @ I3 @ ( X @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_eventsI_indep_vars
thf(fact_1098__092_060Omega_062_092_060_094sub_0622_Oindep__eventsI__indep__vars,axiom,
    ! [N: $o > sigma_measure_rat,X: $o > ( product_prod_nat_nat > product_prod_nat_nat ) > rat,I: set_o,P: $o > rat > $o] :
      ( ( indepe2354179314913070019_o_rat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N @ X @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member_set_rat
              @ ( collect_rat
                @ ^ [X2: rat] :
                    ( ( member_rat @ X2 @ ( sigma_space_rat @ ( N @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_rat @ ( N @ I2 ) ) ) )
       => ( indepe9001878988524906205_nat_o @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
          @ ^ [I3: $o] :
              ( collec248804655031457914at_nat
              @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] :
                  ( ( member8885076297122219836at_nat @ X2 @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
                  & ( P @ I3 @ ( X @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_eventsI_indep_vars
thf(fact_1099__092_060Omega_062_092_060_094sub_0622_Oindep__eventsI__indep__vars,axiom,
    ! [N: $o > sigma_measure_o,X: $o > ( product_prod_nat_nat > product_prod_nat_nat ) > $o,I: set_o,P: $o > $o > $o] :
      ( ( indepe2527157288950651117at_o_o @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N @ X @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member_set_o
              @ ( collect_o
                @ ^ [X2: $o] :
                    ( ( member_o @ X2 @ ( sigma_space_o @ ( N @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_o @ ( N @ I2 ) ) ) )
       => ( indepe9001878988524906205_nat_o @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
          @ ^ [I3: $o] :
              ( collec248804655031457914at_nat
              @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] :
                  ( ( member8885076297122219836at_nat @ X2 @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
                  & ( P @ I3 @ ( X @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_eventsI_indep_vars
thf(fact_1100__092_060Omega_062_092_060_094sub_0622_Oindep__eventsI__indep__vars,axiom,
    ! [N: rat > sigma_measure_nat,X: rat > ( product_prod_nat_nat > product_prod_nat_nat ) > nat,I: set_rat,P: rat > nat > $o] :
      ( ( indepe494537661987070231at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N @ X @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member_set_nat
              @ ( collect_nat
                @ ^ [X2: nat] :
                    ( ( member_nat @ X2 @ ( sigma_space_nat @ ( N @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_nat @ ( N @ I2 ) ) ) )
       => ( indepe6815081019054987731at_rat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
          @ ^ [I3: rat] :
              ( collec248804655031457914at_nat
              @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] :
                  ( ( member8885076297122219836at_nat @ X2 @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
                  & ( P @ I3 @ ( X @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_eventsI_indep_vars
thf(fact_1101__092_060Omega_062_092_060_094sub_0622_Oindep__eventsI__indep__vars,axiom,
    ! [N: $o > sigma_measure_nat,X: $o > ( product_prod_nat_nat > product_prod_nat_nat ) > nat,I: set_o,P: $o > nat > $o] :
      ( ( indepe2989309374999565755_o_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N @ X @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member_set_nat
              @ ( collect_nat
                @ ^ [X2: nat] :
                    ( ( member_nat @ X2 @ ( sigma_space_nat @ ( N @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_nat @ ( N @ I2 ) ) ) )
       => ( indepe9001878988524906205_nat_o @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
          @ ^ [I3: $o] :
              ( collec248804655031457914at_nat
              @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] :
                  ( ( member8885076297122219836at_nat @ X2 @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
                  & ( P @ I3 @ ( X @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_eventsI_indep_vars
thf(fact_1102__092_060Omega_062_092_060_094sub_0622_Oindep__eventsI__indep__vars,axiom,
    ! [N: rat > sigma_5394977995791401948nnreal,X: rat > ( product_prod_nat_nat > product_prod_nat_nat ) > real > extend8495563244428889912nnreal,I: set_rat,P: rat > ( real > extend8495563244428889912nnreal ) > $o] :
      ( ( indepe8916110873896082250nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N @ X @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member524040414084610768nnreal
              @ ( collec9130413544115709400nnreal
                @ ^ [X2: real > extend8495563244428889912nnreal] :
                    ( ( member2919562650594848410nnreal @ X2 @ ( sigma_2369682286586992763nnreal @ ( N @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_3125015092713243876nnreal @ ( N @ I2 ) ) ) )
       => ( indepe6815081019054987731at_rat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
          @ ^ [I3: rat] :
              ( collec248804655031457914at_nat
              @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] :
                  ( ( member8885076297122219836at_nat @ X2 @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
                  & ( P @ I3 @ ( X @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_eventsI_indep_vars
thf(fact_1103__092_060Omega_062_092_060_094sub_0622_Oindep__eventsI__indep__vars,axiom,
    ! [N: ( real > extend8495563244428889912nnreal ) > sigma_measure_rat,X: ( real > extend8495563244428889912nnreal ) > ( product_prod_nat_nat > product_prod_nat_nat ) > rat,I: set_re5328672808648366137nnreal,P: ( real > extend8495563244428889912nnreal ) > rat > $o] :
      ( ( indepe3746081685005995338al_rat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N @ X @ I )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member_set_rat
              @ ( collect_rat
                @ ^ [X2: rat] :
                    ( ( member_rat @ X2 @ ( sigma_space_rat @ ( N @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_rat @ ( N @ I2 ) ) ) )
       => ( indepe8842583234502444542nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
          @ ^ [I3: real > extend8495563244428889912nnreal] :
              ( collec248804655031457914at_nat
              @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] :
                  ( ( member8885076297122219836at_nat @ X2 @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
                  & ( P @ I3 @ ( X @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_eventsI_indep_vars
thf(fact_1104__092_060Omega_062_092_060_094sub_0622_Oindep__eventsI__indep__vars,axiom,
    ! [N: ( real > extend8495563244428889912nnreal ) > sigma_measure_o,X: ( real > extend8495563244428889912nnreal ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o,I: set_re5328672808648366137nnreal,P: ( real > extend8495563244428889912nnreal ) > $o > $o] :
      ( ( indepe4299347172207196070real_o @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N @ X @ I )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member_set_o
              @ ( collect_o
                @ ^ [X2: $o] :
                    ( ( member_o @ X2 @ ( sigma_space_o @ ( N @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_o @ ( N @ I2 ) ) ) )
       => ( indepe8842583234502444542nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
          @ ^ [I3: real > extend8495563244428889912nnreal] :
              ( collec248804655031457914at_nat
              @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] :
                  ( ( member8885076297122219836at_nat @ X2 @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
                  & ( P @ I3 @ ( X @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_eventsI_indep_vars
thf(fact_1105__092_060Omega_062_092_060_094sub_0622_Oindep__eventsI__indep__vars,axiom,
    ! [N: $o > sigma_5394977995791401948nnreal,X: $o > ( product_prod_nat_nat > product_prod_nat_nat ) > real > extend8495563244428889912nnreal,I: set_o,P: $o > ( real > extend8495563244428889912nnreal ) > $o] :
      ( ( indepe1363157079359510510nnreal @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ N @ X @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member524040414084610768nnreal
              @ ( collec9130413544115709400nnreal
                @ ^ [X2: real > extend8495563244428889912nnreal] :
                    ( ( member2919562650594848410nnreal @ X2 @ ( sigma_2369682286586992763nnreal @ ( N @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_3125015092713243876nnreal @ ( N @ I2 ) ) ) )
       => ( indepe9001878988524906205_nat_o @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
          @ ^ [I3: $o] :
              ( collec248804655031457914at_nat
              @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] :
                  ( ( member8885076297122219836at_nat @ X2 @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
                  & ( P @ I3 @ ( X @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% \<Omega>\<^sub>2.indep_eventsI_indep_vars
thf(fact_1106__092_060Omega_062_092_060_094sub_0622_Oindep__vars__sum,axiom,
    ! [I: set_rat,I4: rat,X: rat > ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( finite_finite_rat @ I )
     => ( ~ ( member_rat @ I4 @ I )
       => ( ( indepe4371592291047002227t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
            @ ^ [Uu: rat] : borel_5078946678739801102l_real
            @ X
            @ ( insert_rat @ I4 @ I ) )
         => ( indepe1938972608428940447t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ borel_5078946678739801102l_real @ ( X @ I4 ) @ borel_5078946678739801102l_real
            @ ^ [Omega: product_prod_nat_nat > product_prod_nat_nat] :
                ( groups2190756095862149312t_real
                @ ^ [I3: rat] : ( X @ I3 @ Omega )
                @ I ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_sum
thf(fact_1107__092_060Omega_062_092_060_094sub_0622_Oindep__vars__sum,axiom,
    ! [I: set_Pr1002607673312053630t_real,I4: product_prod_nat_nat > real,X: ( product_prod_nat_nat > real ) > ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( finite7134436996427083847t_real @ I )
     => ( ~ ( member4564283293661824327t_real @ I4 @ I )
       => ( ( indepe1199366807463074589l_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
            @ ^ [Uu: product_prod_nat_nat > real] : borel_5078946678739801102l_real
            @ X
            @ ( insert2113160732575218286t_real @ I4 @ I ) )
         => ( indepe1938972608428940447t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ borel_5078946678739801102l_real @ ( X @ I4 ) @ borel_5078946678739801102l_real
            @ ^ [Omega: product_prod_nat_nat > product_prod_nat_nat] :
                ( groups8217084703900529808l_real
                @ ^ [I3: product_prod_nat_nat > real] : ( X @ I3 @ Omega )
                @ I ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_sum
thf(fact_1108__092_060Omega_062_092_060_094sub_0622_Oindep__vars__sum,axiom,
    ! [I: set_re5328672808648366137nnreal,I4: real > extend8495563244428889912nnreal,X: ( real > extend8495563244428889912nnreal ) > ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( finite7684081742213514138nnreal @ I )
     => ( ~ ( member2919562650594848410nnreal @ I4 @ I )
       => ( ( indepe3660082575425806622l_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
            @ ^ [Uu: real > extend8495563244428889912nnreal] : borel_5078946678739801102l_real
            @ X
            @ ( insert152533262698245683nnreal @ I4 @ I ) )
         => ( indepe1938972608428940447t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ borel_5078946678739801102l_real @ ( X @ I4 ) @ borel_5078946678739801102l_real
            @ ^ [Omega: product_prod_nat_nat > product_prod_nat_nat] :
                ( groups8532390058693651307l_real
                @ ^ [I3: real > extend8495563244428889912nnreal] : ( X @ I3 @ Omega )
                @ I ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_sum
thf(fact_1109__092_060Omega_062_092_060_094sub_0622_Oindep__vars__sum,axiom,
    ! [I: set_Pr947837736998463782t_real,I4: ( product_prod_nat_nat > product_prod_nat_nat ) > real,X: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( finite3361987442264708871t_real @ I )
     => ( ~ ( member8159409068225774087t_real @ I4 @ I )
       => ( ( indepe8350486816430810379l_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
            @ ^ [Uu: ( product_prod_nat_nat > product_prod_nat_nat ) > real] : borel_5078946678739801102l_real
            @ X
            @ ( insert5637722417281287840t_real @ I4 @ I ) )
         => ( indepe1938972608428940447t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ borel_5078946678739801102l_real @ ( X @ I4 ) @ borel_5078946678739801102l_real
            @ ^ [Omega: product_prod_nat_nat > product_prod_nat_nat] :
                ( groups5775555352902572120l_real
                @ ^ [I3: ( product_prod_nat_nat > product_prod_nat_nat ) > real] : ( X @ I3 @ Omega )
                @ I ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_sum
thf(fact_1110__092_060Omega_062_092_060_094sub_0622_Oindep__vars__sum,axiom,
    ! [I: set_o,I4: $o,X: $o > ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( finite_finite_o @ I )
     => ( ~ ( member_o @ I4 @ I )
       => ( ( indepe2405714897255445271o_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
            @ ^ [Uu: $o] : borel_5078946678739801102l_real
            @ X
            @ ( insert_o @ I4 @ I ) )
         => ( indepe1938972608428940447t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ borel_5078946678739801102l_real @ ( X @ I4 ) @ borel_5078946678739801102l_real
            @ ^ [Omega: product_prod_nat_nat > product_prod_nat_nat] :
                ( groups8691415230153176458o_real
                @ ^ [I3: $o] : ( X @ I3 @ Omega )
                @ I ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_sum
thf(fact_1111__092_060Omega_062_092_060_094sub_0622_Oindep__vars__sum,axiom,
    ! [I: set_nat,I4: nat,X: nat > ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( finite_finite_nat @ I )
     => ( ~ ( member_nat @ I4 @ I )
       => ( ( indepe8772276481556004459t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
            @ ^ [Uu: nat] : borel_5078946678739801102l_real
            @ X
            @ ( insert_nat @ I4 @ I ) )
         => ( indepe1938972608428940447t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ borel_5078946678739801102l_real @ ( X @ I4 ) @ borel_5078946678739801102l_real
            @ ^ [Omega: product_prod_nat_nat > product_prod_nat_nat] :
                ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( X @ I3 @ Omega )
                @ I ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_sum
thf(fact_1112__092_060Omega_062_092_060_094sub_0622_Oindep__vars__sum,axiom,
    ! [I: set_Pr2458342521480944603at_nat,I4: product_prod_nat_nat > product_prod_nat_nat,X: ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( finite1779803868683885628at_nat @ I )
     => ( ~ ( member8885076297122219836at_nat @ I4 @ I )
       => ( ( indepe5511084738810189248t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
            @ ^ [Uu: product_prod_nat_nat > product_prod_nat_nat] : borel_5078946678739801102l_real
            @ X
            @ ( insert5766096353517822421at_nat @ I4 @ I ) )
         => ( indepe1938972608428940447t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ borel_5078946678739801102l_real @ ( X @ I4 ) @ borel_5078946678739801102l_real
            @ ^ [Omega: product_prod_nat_nat > product_prod_nat_nat] :
                ( groups4036814449015215885t_real
                @ ^ [I3: product_prod_nat_nat > product_prod_nat_nat] : ( X @ I3 @ Omega )
                @ I ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_sum
thf(fact_1113__092_060Omega_062_092_060_094sub_0622_Oindep__vars__sum,axiom,
    ! [I: set_Pr1261947904930325089at_nat,I4: product_prod_nat_nat,X: product_prod_nat_nat > ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( finite6177210948735845034at_nat @ I )
     => ( ~ ( member8440522571783428010at_nat @ I4 @ I )
       => ( ( indepe4215498690062702080t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
            @ ^ [Uu: product_prod_nat_nat] : borel_5078946678739801102l_real
            @ X
            @ ( insert8211810215607154385at_nat @ I4 @ I ) )
         => ( indepe1938972608428940447t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ borel_5078946678739801102l_real @ ( X @ I4 ) @ borel_5078946678739801102l_real
            @ ^ [Omega: product_prod_nat_nat > product_prod_nat_nat] :
                ( groups4567486121110086003t_real
                @ ^ [I3: product_prod_nat_nat] : ( X @ I3 @ Omega )
                @ I ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_vars_sum
thf(fact_1114_indep__vars__sum,axiom,
    ! [I: set_rat,I4: rat,X: rat > product_prod_nat_nat > real] :
      ( ( finite_finite_rat @ I )
     => ( ~ ( member_rat @ I4 @ I )
       => ( ( indepe5387908695526973631t_real @ ( freque5010624893710627907mega_1 @ as )
            @ ^ [Uu: rat] : borel_5078946678739801102l_real
            @ X
            @ ( insert_rat @ I4 @ I ) )
         => ( indepe1845147739501508449t_real @ ( freque5010624893710627907mega_1 @ as ) @ borel_5078946678739801102l_real @ ( X @ I4 ) @ borel_5078946678739801102l_real
            @ ^ [Omega: product_prod_nat_nat] :
                ( groups2190756095862149312t_real
                @ ^ [I3: rat] : ( X @ I3 @ Omega )
                @ I ) ) ) ) ) ).

% indep_vars_sum
thf(fact_1115_indep__vars__sum,axiom,
    ! [I: set_Pr1002607673312053630t_real,I4: product_prod_nat_nat > real,X: ( product_prod_nat_nat > real ) > product_prod_nat_nat > real] :
      ( ( finite7134436996427083847t_real @ I )
     => ( ~ ( member4564283293661824327t_real @ I4 @ I )
       => ( ( indepe3389242201104679889l_real @ ( freque5010624893710627907mega_1 @ as )
            @ ^ [Uu: product_prod_nat_nat > real] : borel_5078946678739801102l_real
            @ X
            @ ( insert2113160732575218286t_real @ I4 @ I ) )
         => ( indepe1845147739501508449t_real @ ( freque5010624893710627907mega_1 @ as ) @ borel_5078946678739801102l_real @ ( X @ I4 ) @ borel_5078946678739801102l_real
            @ ^ [Omega: product_prod_nat_nat] :
                ( groups8217084703900529808l_real
                @ ^ [I3: product_prod_nat_nat > real] : ( X @ I3 @ Omega )
                @ I ) ) ) ) ) ).

% indep_vars_sum
thf(fact_1116_indep__vars__sum,axiom,
    ! [I: set_re5328672808648366137nnreal,I4: real > extend8495563244428889912nnreal,X: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > real] :
      ( ( finite7684081742213514138nnreal @ I )
     => ( ~ ( member2919562650594848410nnreal @ I4 @ I )
       => ( ( indepe579324506763905386l_real @ ( freque5010624893710627907mega_1 @ as )
            @ ^ [Uu: real > extend8495563244428889912nnreal] : borel_5078946678739801102l_real
            @ X
            @ ( insert152533262698245683nnreal @ I4 @ I ) )
         => ( indepe1845147739501508449t_real @ ( freque5010624893710627907mega_1 @ as ) @ borel_5078946678739801102l_real @ ( X @ I4 ) @ borel_5078946678739801102l_real
            @ ^ [Omega: product_prod_nat_nat] :
                ( groups8532390058693651307l_real
                @ ^ [I3: real > extend8495563244428889912nnreal] : ( X @ I3 @ Omega )
                @ I ) ) ) ) ) ).

% indep_vars_sum
thf(fact_1117_indep__vars__sum,axiom,
    ! [I: set_Pr947837736998463782t_real,I4: ( product_prod_nat_nat > product_prod_nat_nat ) > real,X: ( ( product_prod_nat_nat > product_prod_nat_nat ) > real ) > product_prod_nat_nat > real] :
      ( ( finite3361987442264708871t_real @ I )
     => ( ~ ( member8159409068225774087t_real @ I4 @ I )
       => ( ( indepe3718687268563708247l_real @ ( freque5010624893710627907mega_1 @ as )
            @ ^ [Uu: ( product_prod_nat_nat > product_prod_nat_nat ) > real] : borel_5078946678739801102l_real
            @ X
            @ ( insert5637722417281287840t_real @ I4 @ I ) )
         => ( indepe1845147739501508449t_real @ ( freque5010624893710627907mega_1 @ as ) @ borel_5078946678739801102l_real @ ( X @ I4 ) @ borel_5078946678739801102l_real
            @ ^ [Omega: product_prod_nat_nat] :
                ( groups5775555352902572120l_real
                @ ^ [I3: ( product_prod_nat_nat > product_prod_nat_nat ) > real] : ( X @ I3 @ Omega )
                @ I ) ) ) ) ) ).

% indep_vars_sum
thf(fact_1118_indep__vars__sum,axiom,
    ! [I: set_o,I4: $o,X: $o > product_prod_nat_nat > real] :
      ( ( finite_finite_o @ I )
     => ( ~ ( member_o @ I4 @ I )
       => ( ( indepe6599791682495051723o_real @ ( freque5010624893710627907mega_1 @ as )
            @ ^ [Uu: $o] : borel_5078946678739801102l_real
            @ X
            @ ( insert_o @ I4 @ I ) )
         => ( indepe1845147739501508449t_real @ ( freque5010624893710627907mega_1 @ as ) @ borel_5078946678739801102l_real @ ( X @ I4 ) @ borel_5078946678739801102l_real
            @ ^ [Omega: product_prod_nat_nat] :
                ( groups8691415230153176458o_real
                @ ^ [I3: $o] : ( X @ I3 @ Omega )
                @ I ) ) ) ) ) ).

% indep_vars_sum
thf(fact_1119_indep__vars__sum,axiom,
    ! [I: set_nat,I4: nat,X: nat > product_prod_nat_nat > real] :
      ( ( finite_finite_nat @ I )
     => ( ~ ( member_nat @ I4 @ I )
       => ( ( indepe565220849181200055t_real @ ( freque5010624893710627907mega_1 @ as )
            @ ^ [Uu: nat] : borel_5078946678739801102l_real
            @ X
            @ ( insert_nat @ I4 @ I ) )
         => ( indepe1845147739501508449t_real @ ( freque5010624893710627907mega_1 @ as ) @ borel_5078946678739801102l_real @ ( X @ I4 ) @ borel_5078946678739801102l_real
            @ ^ [Omega: product_prod_nat_nat] :
                ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( X @ I3 @ Omega )
                @ I ) ) ) ) ) ).

% indep_vars_sum
thf(fact_1120_indep__vars__sum,axiom,
    ! [I: set_Pr2458342521480944603at_nat,I4: product_prod_nat_nat > product_prod_nat_nat,X: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > real] :
      ( ( finite1779803868683885628at_nat @ I )
     => ( ~ ( member8885076297122219836at_nat @ I4 @ I )
       => ( ( indepe2220390954403586572t_real @ ( freque5010624893710627907mega_1 @ as )
            @ ^ [Uu: product_prod_nat_nat > product_prod_nat_nat] : borel_5078946678739801102l_real
            @ X
            @ ( insert5766096353517822421at_nat @ I4 @ I ) )
         => ( indepe1845147739501508449t_real @ ( freque5010624893710627907mega_1 @ as ) @ borel_5078946678739801102l_real @ ( X @ I4 ) @ borel_5078946678739801102l_real
            @ ^ [Omega: product_prod_nat_nat] :
                ( groups4036814449015215885t_real
                @ ^ [I3: product_prod_nat_nat > product_prod_nat_nat] : ( X @ I3 @ Omega )
                @ I ) ) ) ) ) ).

% indep_vars_sum
thf(fact_1121_indep__vars__sum,axiom,
    ! [I: set_Pr1261947904930325089at_nat,I4: product_prod_nat_nat,X: product_prod_nat_nat > product_prod_nat_nat > real] :
      ( ( finite6177210948735845034at_nat @ I )
     => ( ~ ( member8440522571783428010at_nat @ I4 @ I )
       => ( ( indepe4688187045501433524t_real @ ( freque5010624893710627907mega_1 @ as )
            @ ^ [Uu: product_prod_nat_nat] : borel_5078946678739801102l_real
            @ X
            @ ( insert8211810215607154385at_nat @ I4 @ I ) )
         => ( indepe1845147739501508449t_real @ ( freque5010624893710627907mega_1 @ as ) @ borel_5078946678739801102l_real @ ( X @ I4 ) @ borel_5078946678739801102l_real
            @ ^ [Omega: product_prod_nat_nat] :
                ( groups4567486121110086003t_real
                @ ^ [I3: product_prod_nat_nat] : ( X @ I3 @ Omega )
                @ I ) ) ) ) ) ).

% indep_vars_sum
thf(fact_1122_indep__eventsI__indep__vars,axiom,
    ! [N: rat > sigma_measure_rat,X: rat > product_prod_nat_nat > rat,I: set_rat,P: rat > rat > $o] :
      ( ( indepe1532306160154479979at_rat @ ( freque5010624893710627907mega_1 @ as ) @ N @ X @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member_set_rat
              @ ( collect_rat
                @ ^ [X2: rat] :
                    ( ( member_rat @ X2 @ ( sigma_space_rat @ ( N @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_rat @ ( N @ I2 ) ) ) )
       => ( indepe8222080746696970885at_rat @ ( freque5010624893710627907mega_1 @ as )
          @ ^ [I3: rat] :
              ( collec3392354462482085612at_nat
              @ ^ [X2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ X2 @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
                  & ( P @ I3 @ ( X @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% indep_eventsI_indep_vars
thf(fact_1123_indep__eventsI__indep__vars,axiom,
    ! [N: rat > sigma_measure_o,X: rat > product_prod_nat_nat > $o,I: set_rat,P: rat > $o > $o] :
      ( ( indepe5353759272196758853_rat_o @ ( freque5010624893710627907mega_1 @ as ) @ N @ X @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member_set_o
              @ ( collect_o
                @ ^ [X2: $o] :
                    ( ( member_o @ X2 @ ( sigma_space_o @ ( N @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_o @ ( N @ I2 ) ) ) )
       => ( indepe8222080746696970885at_rat @ ( freque5010624893710627907mega_1 @ as )
          @ ^ [I3: rat] :
              ( collec3392354462482085612at_nat
              @ ^ [X2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ X2 @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
                  & ( P @ I3 @ ( X @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% indep_eventsI_indep_vars
thf(fact_1124_indep__eventsI__indep__vars,axiom,
    ! [N: $o > sigma_measure_rat,X: $o > product_prod_nat_nat > rat,I: set_o,P: $o > rat > $o] :
      ( ( indepe5353290778318645879_o_rat @ ( freque5010624893710627907mega_1 @ as ) @ N @ X @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member_set_rat
              @ ( collect_rat
                @ ^ [X2: rat] :
                    ( ( member_rat @ X2 @ ( sigma_space_rat @ ( N @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_rat @ ( N @ I2 ) ) ) )
       => ( indepe8047530689963900331_nat_o @ ( freque5010624893710627907mega_1 @ as )
          @ ^ [I3: $o] :
              ( collec3392354462482085612at_nat
              @ ^ [X2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ X2 @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
                  & ( P @ I3 @ ( X @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% indep_eventsI_indep_vars
thf(fact_1125_indep__eventsI__indep__vars,axiom,
    ! [N: $o > sigma_measure_o,X: $o > product_prod_nat_nat > $o,I: set_o,P: $o > $o > $o] :
      ( ( indepe8934423792353453369at_o_o @ ( freque5010624893710627907mega_1 @ as ) @ N @ X @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member_set_o
              @ ( collect_o
                @ ^ [X2: $o] :
                    ( ( member_o @ X2 @ ( sigma_space_o @ ( N @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_o @ ( N @ I2 ) ) ) )
       => ( indepe8047530689963900331_nat_o @ ( freque5010624893710627907mega_1 @ as )
          @ ^ [I3: $o] :
              ( collec3392354462482085612at_nat
              @ ^ [X2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ X2 @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
                  & ( P @ I3 @ ( X @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% indep_eventsI_indep_vars
thf(fact_1126_indep__eventsI__indep__vars,axiom,
    ! [N: rat > sigma_measure_nat,X: rat > product_prod_nat_nat > nat,I: set_rat,P: rat > nat > $o] :
      ( ( indepe2167436220240975715at_nat @ ( freque5010624893710627907mega_1 @ as ) @ N @ X @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member_set_nat
              @ ( collect_nat
                @ ^ [X2: nat] :
                    ( ( member_nat @ X2 @ ( sigma_space_nat @ ( N @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_nat @ ( N @ I2 ) ) ) )
       => ( indepe8222080746696970885at_rat @ ( freque5010624893710627907mega_1 @ as )
          @ ^ [I3: rat] :
              ( collec3392354462482085612at_nat
              @ ^ [X2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ X2 @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
                  & ( P @ I3 @ ( X @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% indep_eventsI_indep_vars
thf(fact_1127_indep__eventsI__indep__vars,axiom,
    ! [N: $o > sigma_measure_nat,X: $o > product_prod_nat_nat > nat,I: set_o,P: $o > nat > $o] :
      ( ( indepe5988420838405141615_o_nat @ ( freque5010624893710627907mega_1 @ as ) @ N @ X @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member_set_nat
              @ ( collect_nat
                @ ^ [X2: nat] :
                    ( ( member_nat @ X2 @ ( sigma_space_nat @ ( N @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_nat @ ( N @ I2 ) ) ) )
       => ( indepe8047530689963900331_nat_o @ ( freque5010624893710627907mega_1 @ as )
          @ ^ [I3: $o] :
              ( collec3392354462482085612at_nat
              @ ^ [X2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ X2 @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
                  & ( P @ I3 @ ( X @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% indep_eventsI_indep_vars
thf(fact_1128_indep__eventsI__indep__vars,axiom,
    ! [N: rat > sigma_5394977995791401948nnreal,X: rat > product_prod_nat_nat > real > extend8495563244428889912nnreal,I: set_rat,P: rat > ( real > extend8495563244428889912nnreal ) > $o] :
      ( ( indepe429544034647304598nnreal @ ( freque5010624893710627907mega_1 @ as ) @ N @ X @ I )
     => ( ! [I2: rat] :
            ( ( member_rat @ I2 @ I )
           => ( member524040414084610768nnreal
              @ ( collec9130413544115709400nnreal
                @ ^ [X2: real > extend8495563244428889912nnreal] :
                    ( ( member2919562650594848410nnreal @ X2 @ ( sigma_2369682286586992763nnreal @ ( N @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_3125015092713243876nnreal @ ( N @ I2 ) ) ) )
       => ( indepe8222080746696970885at_rat @ ( freque5010624893710627907mega_1 @ as )
          @ ^ [I3: rat] :
              ( collec3392354462482085612at_nat
              @ ^ [X2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ X2 @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
                  & ( P @ I3 @ ( X @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% indep_eventsI_indep_vars
thf(fact_1129_indep__eventsI__indep__vars,axiom,
    ! [N: ( real > extend8495563244428889912nnreal ) > sigma_measure_rat,X: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > rat,I: set_re5328672808648366137nnreal,P: ( real > extend8495563244428889912nnreal ) > rat > $o] :
      ( ( indepe4482886882611993494al_rat @ ( freque5010624893710627907mega_1 @ as ) @ N @ X @ I )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member_set_rat
              @ ( collect_rat
                @ ^ [X2: rat] :
                    ( ( member_rat @ X2 @ ( sigma_space_rat @ ( N @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_rat @ ( N @ I2 ) ) ) )
       => ( indepe138081623465165488nnreal @ ( freque5010624893710627907mega_1 @ as )
          @ ^ [I3: real > extend8495563244428889912nnreal] :
              ( collec3392354462482085612at_nat
              @ ^ [X2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ X2 @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
                  & ( P @ I3 @ ( X @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% indep_eventsI_indep_vars
thf(fact_1130_indep__eventsI__indep__vars,axiom,
    ! [N: ( real > extend8495563244428889912nnreal ) > sigma_measure_o,X: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > $o,I: set_re5328672808648366137nnreal,P: ( real > extend8495563244428889912nnreal ) > $o > $o] :
      ( ( indepe1951750190343126618real_o @ ( freque5010624893710627907mega_1 @ as ) @ N @ X @ I )
     => ( ! [I2: real > extend8495563244428889912nnreal] :
            ( ( member2919562650594848410nnreal @ I2 @ I )
           => ( member_set_o
              @ ( collect_o
                @ ^ [X2: $o] :
                    ( ( member_o @ X2 @ ( sigma_space_o @ ( N @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_sets_o @ ( N @ I2 ) ) ) )
       => ( indepe138081623465165488nnreal @ ( freque5010624893710627907mega_1 @ as )
          @ ^ [I3: real > extend8495563244428889912nnreal] :
              ( collec3392354462482085612at_nat
              @ ^ [X2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ X2 @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
                  & ( P @ I3 @ ( X @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% indep_eventsI_indep_vars
thf(fact_1131_indep__eventsI__indep__vars,axiom,
    ! [N: $o > sigma_5394977995791401948nnreal,X: $o > product_prod_nat_nat > real > extend8495563244428889912nnreal,I: set_o,P: $o > ( real > extend8495563244428889912nnreal ) > $o] :
      ( ( indepe8238932134350216866nnreal @ ( freque5010624893710627907mega_1 @ as ) @ N @ X @ I )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I )
           => ( member524040414084610768nnreal
              @ ( collec9130413544115709400nnreal
                @ ^ [X2: real > extend8495563244428889912nnreal] :
                    ( ( member2919562650594848410nnreal @ X2 @ ( sigma_2369682286586992763nnreal @ ( N @ I2 ) ) )
                    & ( P @ I2 @ X2 ) ) )
              @ ( sigma_3125015092713243876nnreal @ ( N @ I2 ) ) ) )
       => ( indepe8047530689963900331_nat_o @ ( freque5010624893710627907mega_1 @ as )
          @ ^ [I3: $o] :
              ( collec3392354462482085612at_nat
              @ ^ [X2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ X2 @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
                  & ( P @ I3 @ ( X @ I3 @ X2 ) ) ) )
          @ I ) ) ) ).

% indep_eventsI_indep_vars
thf(fact_1132_Bochner__Integration_Ointegrable__sum,axiom,
    ! [I: set_rat,M2: sigma_5515648953823433982at_nat,F2: rat > product_prod_nat_nat > real] :
      ( ! [I2: rat] :
          ( ( member_rat @ I2 @ I )
         => ( bochne2596016609597520987t_real @ M2 @ ( F2 @ I2 ) ) )
     => ( bochne2596016609597520987t_real @ M2
        @ ^ [X2: product_prod_nat_nat] :
            ( groups2190756095862149312t_real
            @ ^ [I3: rat] : ( F2 @ I3 @ X2 )
            @ I ) ) ) ).

% Bochner_Integration.integrable_sum
thf(fact_1133_Bochner__Integration_Ointegrable__sum,axiom,
    ! [I: set_o,M2: sigma_5515648953823433982at_nat,F2: $o > product_prod_nat_nat > real] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ I )
         => ( bochne2596016609597520987t_real @ M2 @ ( F2 @ I2 ) ) )
     => ( bochne2596016609597520987t_real @ M2
        @ ^ [X2: product_prod_nat_nat] :
            ( groups8691415230153176458o_real
            @ ^ [I3: $o] : ( F2 @ I3 @ X2 )
            @ I ) ) ) ).

% Bochner_Integration.integrable_sum
thf(fact_1134_Bochner__Integration_Ointegrable__sum,axiom,
    ! [I: set_re5328672808648366137nnreal,M2: sigma_5515648953823433982at_nat,F2: ( real > extend8495563244428889912nnreal ) > product_prod_nat_nat > real] :
      ( ! [I2: real > extend8495563244428889912nnreal] :
          ( ( member2919562650594848410nnreal @ I2 @ I )
         => ( bochne2596016609597520987t_real @ M2 @ ( F2 @ I2 ) ) )
     => ( bochne2596016609597520987t_real @ M2
        @ ^ [X2: product_prod_nat_nat] :
            ( groups8532390058693651307l_real
            @ ^ [I3: real > extend8495563244428889912nnreal] : ( F2 @ I3 @ X2 )
            @ I ) ) ) ).

% Bochner_Integration.integrable_sum
thf(fact_1135_Bochner__Integration_Ointegrable__sum,axiom,
    ! [I: set_Pr1261947904930325089at_nat,M2: sigma_5515648953823433982at_nat,F2: product_prod_nat_nat > product_prod_nat_nat > real] :
      ( ! [I2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ I2 @ I )
         => ( bochne2596016609597520987t_real @ M2 @ ( F2 @ I2 ) ) )
     => ( bochne2596016609597520987t_real @ M2
        @ ^ [X2: product_prod_nat_nat] :
            ( groups4567486121110086003t_real
            @ ^ [I3: product_prod_nat_nat] : ( F2 @ I3 @ X2 )
            @ I ) ) ) ).

% Bochner_Integration.integrable_sum
thf(fact_1136_Bochner__Integration_Ointegrable__sum,axiom,
    ! [I: set_rat,M2: sigma_9047027012034273406at_nat,F2: rat > ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ! [I2: rat] :
          ( ( member_rat @ I2 @ I )
         => ( bochne7117808529828525605t_real @ M2 @ ( F2 @ I2 ) ) )
     => ( bochne7117808529828525605t_real @ M2
        @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] :
            ( groups2190756095862149312t_real
            @ ^ [I3: rat] : ( F2 @ I3 @ X2 )
            @ I ) ) ) ).

% Bochner_Integration.integrable_sum
thf(fact_1137_Bochner__Integration_Ointegrable__sum,axiom,
    ! [I: set_o,M2: sigma_9047027012034273406at_nat,F2: $o > ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ I )
         => ( bochne7117808529828525605t_real @ M2 @ ( F2 @ I2 ) ) )
     => ( bochne7117808529828525605t_real @ M2
        @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] :
            ( groups8691415230153176458o_real
            @ ^ [I3: $o] : ( F2 @ I3 @ X2 )
            @ I ) ) ) ).

% Bochner_Integration.integrable_sum
thf(fact_1138_Bochner__Integration_Ointegrable__sum,axiom,
    ! [I: set_Pr1002607673312053630t_real,M2: sigma_5515648953823433982at_nat,F2: ( product_prod_nat_nat > real ) > product_prod_nat_nat > real] :
      ( ! [I2: product_prod_nat_nat > real] :
          ( ( member4564283293661824327t_real @ I2 @ I )
         => ( bochne2596016609597520987t_real @ M2 @ ( F2 @ I2 ) ) )
     => ( bochne2596016609597520987t_real @ M2
        @ ^ [X2: product_prod_nat_nat] :
            ( groups8217084703900529808l_real
            @ ^ [I3: product_prod_nat_nat > real] : ( F2 @ I3 @ X2 )
            @ I ) ) ) ).

% Bochner_Integration.integrable_sum
thf(fact_1139_Bochner__Integration_Ointegrable__sum,axiom,
    ! [I: set_re5328672808648366137nnreal,M2: sigma_9047027012034273406at_nat,F2: ( real > extend8495563244428889912nnreal ) > ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ! [I2: real > extend8495563244428889912nnreal] :
          ( ( member2919562650594848410nnreal @ I2 @ I )
         => ( bochne7117808529828525605t_real @ M2 @ ( F2 @ I2 ) ) )
     => ( bochne7117808529828525605t_real @ M2
        @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] :
            ( groups8532390058693651307l_real
            @ ^ [I3: real > extend8495563244428889912nnreal] : ( F2 @ I3 @ X2 )
            @ I ) ) ) ).

% Bochner_Integration.integrable_sum
thf(fact_1140_Bochner__Integration_Ointegrable__sum,axiom,
    ! [I: set_Pr2458342521480944603at_nat,M2: sigma_5515648953823433982at_nat,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > real] :
      ( ! [I2: product_prod_nat_nat > product_prod_nat_nat] :
          ( ( member8885076297122219836at_nat @ I2 @ I )
         => ( bochne2596016609597520987t_real @ M2 @ ( F2 @ I2 ) ) )
     => ( bochne2596016609597520987t_real @ M2
        @ ^ [X2: product_prod_nat_nat] :
            ( groups4036814449015215885t_real
            @ ^ [I3: product_prod_nat_nat > product_prod_nat_nat] : ( F2 @ I3 @ X2 )
            @ I ) ) ) ).

% Bochner_Integration.integrable_sum
thf(fact_1141_Bochner__Integration_Ointegrable__sum,axiom,
    ! [I: set_Pr1261947904930325089at_nat,M2: sigma_9047027012034273406at_nat,F2: product_prod_nat_nat > ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ! [I2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ I2 @ I )
         => ( bochne7117808529828525605t_real @ M2 @ ( F2 @ I2 ) ) )
     => ( bochne7117808529828525605t_real @ M2
        @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] :
            ( groups4567486121110086003t_real
            @ ^ [I3: product_prod_nat_nat] : ( F2 @ I3 @ X2 )
            @ I ) ) ) ).

% Bochner_Integration.integrable_sum
thf(fact_1142_sets__bot,axiom,
    ( ( sigma_7545971424795947232at_nat @ bot_bo8824583409880654738at_nat )
    = ( insert9200635055090092081at_nat @ bot_bo2099793752762293965at_nat @ bot_bo3083307316010499117at_nat ) ) ).

% sets_bot
thf(fact_1143_sets__bot,axiom,
    ( ( sigma_1566336194983598982at_nat @ bot_bo7613792866757523178at_nat )
    = ( insert1813694899509750283at_nat @ bot_bo2540092574283496047at_nat @ bot_bo3048727414778170277at_nat ) ) ).

% sets_bot
thf(fact_1144_tail__events__sets,axiom,
    ! [A2: nat > set_se7855581050983116737at_nat] :
      ( ! [I2: nat] : ( ord_le2077887516847798113at_nat @ ( A2 @ I2 ) @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
     => ( ord_le2077887516847798113at_nat @ ( indepe3837050773481735769at_nat @ ( freque5010624893710627907mega_1 @ as ) @ A2 ) @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) ) ) ).

% tail_events_sets
thf(fact_1145__092_060Omega_062_092_060_094sub_0622_Otail__events__sets,axiom,
    ! [A2: nat > set_se1666487788256820497at_nat] :
      ( ! [I2: nat] : ( ord_le8269821659064819057at_nat @ ( A2 @ I2 ) @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
     => ( ord_le8269821659064819057at_nat @ ( indepe3710239703223808239at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 ) @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) ) ) ).

% \<Omega>\<^sub>2.tail_events_sets
thf(fact_1146__092_060Omega_062_092_060_094sub_0622_Oindep__setD__ev2,axiom,
    ! [A2: set_se1666487788256820497at_nat,B2: set_se1666487788256820497at_nat] :
      ( ( indepe866335810092677197at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 @ B2 )
     => ( ord_le8269821659064819057at_nat @ B2 @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) ) ) ).

% \<Omega>\<^sub>2.indep_setD_ev2
thf(fact_1147__092_060Omega_062_092_060_094sub_0622_Oindep__setD__ev1,axiom,
    ! [A2: set_se1666487788256820497at_nat,B2: set_se1666487788256820497at_nat] :
      ( ( indepe866335810092677197at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 @ B2 )
     => ( ord_le8269821659064819057at_nat @ A2 @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) ) ) ).

% \<Omega>\<^sub>2.indep_setD_ev1
thf(fact_1148__092_060Omega_062_092_060_094sub_0622_Oindep__set__def,axiom,
    ! [A2: set_se1666487788256820497at_nat,B2: set_se1666487788256820497at_nat] :
      ( ( indepe866335810092677197at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 @ B2 )
      = ( indepe9198310886692500677_nat_o @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ ( produc8027521732861500323at_nat @ A2 @ B2 ) @ top_top_set_o ) ) ).

% \<Omega>\<^sub>2.indep_set_def
thf(fact_1149_indep__setD__ev2,axiom,
    ! [A2: set_se7855581050983116737at_nat,B2: set_se7855581050983116737at_nat] :
      ( ( indepe617343343079735897at_nat @ ( freque5010624893710627907mega_1 @ as ) @ A2 @ B2 )
     => ( ord_le2077887516847798113at_nat @ B2 @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) ) ) ).

% indep_setD_ev2
thf(fact_1150_indep__setD__ev1,axiom,
    ! [A2: set_se7855581050983116737at_nat,B2: set_se7855581050983116737at_nat] :
      ( ( indepe617343343079735897at_nat @ ( freque5010624893710627907mega_1 @ as ) @ A2 @ B2 )
     => ( ord_le2077887516847798113at_nat @ A2 @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) ) ) ).

% indep_setD_ev1
thf(fact_1151_prob__space__completion,axiom,
    probab2019946041432190532at_nat @ ( comple3409133383357356555at_nat @ ( freque5010624893710627907mega_1 @ as ) ) ).

% prob_space_completion
thf(fact_1152_card__UNIV__unit,axiom,
    ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
    = one_one_nat ) ).

% card_UNIV_unit
thf(fact_1153_indep__set__def,axiom,
    ! [A2: set_se7855581050983116737at_nat,B2: set_se7855581050983116737at_nat] :
      ( ( indepe617343343079735897at_nat @ ( freque5010624893710627907mega_1 @ as ) @ A2 @ B2 )
      = ( indepe619418257930843587_nat_o @ ( freque5010624893710627907mega_1 @ as ) @ ( produc6107030024498867375at_nat @ A2 @ B2 ) @ top_top_set_o ) ) ).

% indep_set_def
thf(fact_1154_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_nat @ N3 @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_1155_card__Collect__less__nat,axiom,
    ! [N2: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N2 ) ) )
      = N2 ) ).

% card_Collect_less_nat
thf(fact_1156_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_1157__092_060Omega_062_092_060_094sub_0622_Oapprox__PInf__emeasure__with__finite,axiom,
    ! [W: set_Pr2458342521480944603at_nat,C2: real] :
      ( ( member3752267775663534962at_nat @ W @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
     => ( ( ( sigma_3593834561172453006at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ W )
          = extend2057119558705770725nnreal )
       => ~ ! [Z2: set_Pr2458342521480944603at_nat] :
              ( ( member3752267775663534962at_nat @ Z2 @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
             => ( ( ord_le3857079194666040379at_nat @ Z2 @ W )
               => ( ( ord_le7381754540660121996nnreal @ ( sigma_3593834561172453006at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ Z2 ) @ extend2057119558705770725nnreal )
                 => ~ ( ord_le7381754540660121996nnreal @ ( extend7643940197134561352nnreal @ C2 ) @ ( sigma_3593834561172453006at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ Z2 ) ) ) ) ) ) ) ).

% \<Omega>\<^sub>2.approx_PInf_emeasure_with_finite
thf(fact_1158__092_060Omega_062_092_060_094sub_0622_Oapprox__with__finite__emeasure,axiom,
    ! [W: set_Pr2458342521480944603at_nat,C2: extend8495563244428889912nnreal] :
      ( ( member3752267775663534962at_nat @ W @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
     => ( ( ord_le7381754540660121996nnreal @ C2 @ ( sigma_3593834561172453006at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ W ) )
       => ~ ! [Z2: set_Pr2458342521480944603at_nat] :
              ( ( member3752267775663534962at_nat @ Z2 @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
             => ( ( ord_le3857079194666040379at_nat @ Z2 @ W )
               => ( ( ord_le7381754540660121996nnreal @ ( sigma_3593834561172453006at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ Z2 ) @ extend2057119558705770725nnreal )
                 => ~ ( ord_le7381754540660121996nnreal @ C2 @ ( sigma_3593834561172453006at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ Z2 ) ) ) ) ) ) ) ).

% \<Omega>\<^sub>2.approx_with_finite_emeasure
thf(fact_1159_approx__with__finite__emeasure,axiom,
    ! [W: set_Pr1261947904930325089at_nat,C2: extend8495563244428889912nnreal] :
      ( ( member2643936169264416010at_nat @ W @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
     => ( ( ord_le7381754540660121996nnreal @ C2 @ ( sigma_411563132819280856at_nat @ ( freque5010624893710627907mega_1 @ as ) @ W ) )
       => ~ ! [Z2: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Z2 @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
             => ( ( ord_le3146513528884898305at_nat @ Z2 @ W )
               => ( ( ord_le7381754540660121996nnreal @ ( sigma_411563132819280856at_nat @ ( freque5010624893710627907mega_1 @ as ) @ Z2 ) @ extend2057119558705770725nnreal )
                 => ~ ( ord_le7381754540660121996nnreal @ C2 @ ( sigma_411563132819280856at_nat @ ( freque5010624893710627907mega_1 @ as ) @ Z2 ) ) ) ) ) ) ) ).

% approx_with_finite_emeasure
thf(fact_1160_approx__PInf__emeasure__with__finite,axiom,
    ! [W: set_Pr1261947904930325089at_nat,C2: real] :
      ( ( member2643936169264416010at_nat @ W @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
     => ( ( ( sigma_411563132819280856at_nat @ ( freque5010624893710627907mega_1 @ as ) @ W )
          = extend2057119558705770725nnreal )
       => ~ ! [Z2: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Z2 @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
             => ( ( ord_le3146513528884898305at_nat @ Z2 @ W )
               => ( ( ord_le7381754540660121996nnreal @ ( sigma_411563132819280856at_nat @ ( freque5010624893710627907mega_1 @ as ) @ Z2 ) @ extend2057119558705770725nnreal )
                 => ~ ( ord_le7381754540660121996nnreal @ ( extend7643940197134561352nnreal @ C2 ) @ ( sigma_411563132819280856at_nat @ ( freque5010624893710627907mega_1 @ as ) @ Z2 ) ) ) ) ) ) ) ).

% approx_PInf_emeasure_with_finite
thf(fact_1161__092_060Omega_062_092_060_094sub_0622_Omeasure__ge__1__iff,axiom,
    ! [A2: set_Pr2458342521480944603at_nat] :
      ( ( ord_less_eq_real @ one_one_real @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 ) )
      = ( ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 )
        = one_one_real ) ) ).

% \<Omega>\<^sub>2.measure_ge_1_iff
thf(fact_1162__092_060Omega_062_092_060_094sub_0622_Obounded__measure,axiom,
    ! [A2: set_Pr2458342521480944603at_nat] : ( ord_less_eq_real @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 ) @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) ) ) ).

% \<Omega>\<^sub>2.bounded_measure
thf(fact_1163__092_060Omega_062_092_060_094sub_0622_Oprob__space,axiom,
    ( ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
    = one_one_real ) ).

% \<Omega>\<^sub>2.prob_space
thf(fact_1164__092_060Omega_062_092_060_094sub_0622_Oemeasure__eq__measure,axiom,
    ! [A2: set_Pr2458342521480944603at_nat] :
      ( ( sigma_3593834561172453006at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 )
      = ( extend7643940197134561352nnreal @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 ) ) ) ).

% \<Omega>\<^sub>2.emeasure_eq_measure
thf(fact_1165__092_060Omega_062_092_060_094sub_0622_Omeasure__eq__compl,axiom,
    ! [S: set_Pr2458342521480944603at_nat,T2: set_Pr2458342521480944603at_nat] :
      ( ( member3752267775663534962at_nat @ S @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
     => ( ( member3752267775663534962at_nat @ T2 @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
       => ( ( ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ ( minus_9197513832320918452at_nat @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) @ S ) )
            = ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ ( minus_9197513832320918452at_nat @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) @ T2 ) ) )
         => ( ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ S )
            = ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ T2 ) ) ) ) ) ).

% \<Omega>\<^sub>2.measure_eq_compl
thf(fact_1166__092_060Omega_062_092_060_094sub_0622_Ofinite__measure__eq__sum__singleton,axiom,
    ! [S2: set_Pr2458342521480944603at_nat] :
      ( ( finite1779803868683885628at_nat @ S2 )
     => ( ! [X3: product_prod_nat_nat > product_prod_nat_nat] :
            ( ( member8885076297122219836at_nat @ X3 @ S2 )
           => ( member3752267775663534962at_nat @ ( insert5766096353517822421at_nat @ X3 @ bot_bo2540092574283496047at_nat ) @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) ) )
       => ( ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ S2 )
          = ( groups4036814449015215885t_real
            @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ ( insert5766096353517822421at_nat @ X2 @ bot_bo2540092574283496047at_nat ) )
            @ S2 ) ) ) ) ).

% \<Omega>\<^sub>2.finite_measure_eq_sum_singleton
thf(fact_1167__092_060Omega_062_092_060_094sub_0622_Ofinite__measure__compl,axiom,
    ! [S2: set_Pr2458342521480944603at_nat] :
      ( ( member3752267775663534962at_nat @ S2 @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
     => ( ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ ( minus_9197513832320918452at_nat @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) @ S2 ) )
        = ( minus_minus_real @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) ) @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ S2 ) ) ) ) ).

% \<Omega>\<^sub>2.finite_measure_compl
thf(fact_1168__092_060Omega_062_092_060_094sub_0622_Ofinite__measure__mono,axiom,
    ! [A2: set_Pr2458342521480944603at_nat,B2: set_Pr2458342521480944603at_nat] :
      ( ( ord_le3857079194666040379at_nat @ A2 @ B2 )
     => ( ( member3752267775663534962at_nat @ B2 @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
       => ( ord_less_eq_real @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 ) @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ B2 ) ) ) ) ).

% \<Omega>\<^sub>2.finite_measure_mono
thf(fact_1169__092_060Omega_062_092_060_094sub_0622_Ofinite__measure__Diff,axiom,
    ! [A2: set_Pr2458342521480944603at_nat,B2: set_Pr2458342521480944603at_nat] :
      ( ( member3752267775663534962at_nat @ A2 @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
     => ( ( member3752267775663534962at_nat @ B2 @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
       => ( ( ord_le3857079194666040379at_nat @ B2 @ A2 )
         => ( ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ ( minus_9197513832320918452at_nat @ A2 @ B2 ) )
            = ( minus_minus_real @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 ) @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ B2 ) ) ) ) ) ) ).

% \<Omega>\<^sub>2.finite_measure_Diff
thf(fact_1170__092_060Omega_062_092_060_094sub_0622_Oprob__compl,axiom,
    ! [A2: set_Pr2458342521480944603at_nat] :
      ( ( member3752267775663534962at_nat @ A2 @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
     => ( ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ ( minus_9197513832320918452at_nat @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) @ A2 ) )
        = ( minus_minus_real @ one_one_real @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 ) ) ) ) ).

% \<Omega>\<^sub>2.prob_compl
thf(fact_1171__092_060Omega_062_092_060_094sub_0622_Oprob__neg,axiom,
    ! [P: ( product_prod_nat_nat > product_prod_nat_nat ) > $o] :
      ( ( member3752267775663534962at_nat
        @ ( collec248804655031457914at_nat
          @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] :
              ( ( member8885076297122219836at_nat @ X2 @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
              & ( P @ X2 ) ) )
        @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
     => ( ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
          @ ( collec248804655031457914at_nat
            @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] :
                ( ( member8885076297122219836at_nat @ X2 @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
                & ~ ( P @ X2 ) ) ) )
        = ( minus_minus_real @ one_one_real
          @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
            @ ( collec248804655031457914at_nat
              @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] :
                  ( ( member8885076297122219836at_nat @ X2 @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
                  & ( P @ X2 ) ) ) ) ) ) ) ).

% \<Omega>\<^sub>2.prob_neg
thf(fact_1172__092_060Omega_062_092_060_094sub_0622_Osubprob__measure__le__1,axiom,
    ! [X: set_Pr2458342521480944603at_nat] : ( ord_less_eq_real @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ X ) @ one_one_real ) ).

% \<Omega>\<^sub>2.subprob_measure_le_1
thf(fact_1173__092_060Omega_062_092_060_094sub_0622_Oprob__le__1,axiom,
    ! [A2: set_Pr2458342521480944603at_nat] : ( ord_less_eq_real @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 ) @ one_one_real ) ).

% \<Omega>\<^sub>2.prob_le_1
thf(fact_1174__092_060Omega_062_092_060_094sub_0622_Omeasure__increasing,axiom,
    measur5766409560513867270t_real @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) ).

% \<Omega>\<^sub>2.measure_increasing
thf(fact_1175_diff__diff__ennreal,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A @ B )
     => ( ( B != extend2057119558705770725nnreal )
       => ( ( minus_8429688780609304081nnreal @ B @ ( minus_8429688780609304081nnreal @ B @ A ) )
          = A ) ) ) ).

% diff_diff_ennreal
thf(fact_1176_Lp__cases,axiom,
    ! [P2: extend8495563244428889912nnreal] :
      ( ! [P22: real] :
          ( ( P2
            = ( extend7643940197134561352nnreal @ P22 ) )
         => ( ( ord_less_real @ zero_zero_real @ P22 )
           => ~ ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ P2 ) ) )
     => ( ( P2 != zero_z7100319975126383169nnreal )
       => ( P2 = extend2057119558705770725nnreal ) ) ) ).

% Lp_cases
thf(fact_1177_Lp__cases__1__PInf,axiom,
    ! [P2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ P2 )
     => ( ! [P22: real] :
            ( ( P2
              = ( extend7643940197134561352nnreal @ P22 ) )
           => ( ( ord_less_real @ one_one_real @ P22 )
             => ~ ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ P2 ) ) )
       => ( ( P2 != one_on2969667320475766781nnreal )
         => ( P2 = extend2057119558705770725nnreal ) ) ) ) ).

% Lp_cases_1_PInf
thf(fact_1178__092_060Omega_062_092_060_094sub_0622_Orandom__variable__small__tails,axiom,
    ! [Alpha: real,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real,K3: real] :
      ( ( ord_less_real @ zero_zero_real @ Alpha )
     => ( ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ borel_5078946678739801102l_real ) )
       => ? [C4: real] :
            ( ( ord_less_real
              @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
                @ ( collec248804655031457914at_nat
                  @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] :
                      ( ( member8885076297122219836at_nat @ X2 @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
                      & ( ord_less_eq_real @ C4 @ ( abs_abs_real @ ( F2 @ X2 ) ) ) ) ) )
              @ Alpha )
            & ( ord_less_eq_real @ K3 @ C4 ) ) ) ) ).

% \<Omega>\<^sub>2.random_variable_small_tails
thf(fact_1179__092_060Omega_062_092_060_094sub_0622_Oindep__sets2__eq,axiom,
    ! [A2: set_se1666487788256820497at_nat,B2: set_se1666487788256820497at_nat] :
      ( ( indepe866335810092677197at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 @ B2 )
      = ( ( ord_le8269821659064819057at_nat @ A2 @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
        & ( ord_le8269821659064819057at_nat @ B2 @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
        & ! [X2: set_Pr2458342521480944603at_nat] :
            ( ( member3752267775663534962at_nat @ X2 @ A2 )
           => ! [Y3: set_Pr2458342521480944603at_nat] :
                ( ( member3752267775663534962at_nat @ Y3 @ B2 )
               => ( ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ ( inf_in8856262298555431021at_nat @ X2 @ Y3 ) )
                  = ( times_times_real @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ X2 ) @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ Y3 ) ) ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_sets2_eq
thf(fact_1180_measure__ge__1__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_eq_real @ one_one_real @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ A2 ) )
      = ( ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ A2 )
        = one_one_real ) ) ).

% measure_ge_1_iff
thf(fact_1181_bounded__measure,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : ( ord_less_eq_real @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ A2 ) @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) ) ) ).

% bounded_measure
thf(fact_1182_prob__space,axiom,
    ( ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
    = one_one_real ) ).

% prob_space
thf(fact_1183_emeasure__eq__measure,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( sigma_411563132819280856at_nat @ ( freque5010624893710627907mega_1 @ as ) @ A2 )
      = ( extend7643940197134561352nnreal @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ A2 ) ) ) ).

% emeasure_eq_measure
thf(fact_1184_measure__eq__compl,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
      ( ( member2643936169264416010at_nat @ S @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
     => ( ( member2643936169264416010at_nat @ T2 @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
       => ( ( ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ ( minus_1356011639430497352at_nat @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) @ S ) )
            = ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ ( minus_1356011639430497352at_nat @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) @ T2 ) ) )
         => ( ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ S )
            = ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ T2 ) ) ) ) ) ).

% measure_eq_compl
thf(fact_1185_finite__measure__compl,axiom,
    ! [S2: set_Pr1261947904930325089at_nat] :
      ( ( member2643936169264416010at_nat @ S2 @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
     => ( ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ ( minus_1356011639430497352at_nat @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) @ S2 ) )
        = ( minus_minus_real @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) ) @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ S2 ) ) ) ) ).

% finite_measure_compl
thf(fact_1186_finite__measure__mono,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
     => ( ( member2643936169264416010at_nat @ B2 @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
       => ( ord_less_eq_real @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ A2 ) @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ B2 ) ) ) ) ).

% finite_measure_mono
thf(fact_1187_finite__measure__Diff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( member2643936169264416010at_nat @ A2 @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
     => ( ( member2643936169264416010at_nat @ B2 @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
       => ( ( ord_le3146513528884898305at_nat @ B2 @ A2 )
         => ( ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
            = ( minus_minus_real @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ A2 ) @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ B2 ) ) ) ) ) ) ).

% finite_measure_Diff
thf(fact_1188_prob__compl,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( member2643936169264416010at_nat @ A2 @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
     => ( ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ ( minus_1356011639430497352at_nat @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) @ A2 ) )
        = ( minus_minus_real @ one_one_real @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ A2 ) ) ) ) ).

% prob_compl
thf(fact_1189_finite__measure__eq__sum__singleton,axiom,
    ! [S2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ S2 )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ S2 )
           => ( member2643936169264416010at_nat @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) ) )
       => ( ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ S2 )
          = ( groups4567486121110086003t_real
            @ ^ [X2: product_prod_nat_nat] : ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
            @ S2 ) ) ) ) ).

% finite_measure_eq_sum_singleton
thf(fact_1190_prob__neg,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( member2643936169264416010at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X2 @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
              & ( P @ X2 ) ) )
        @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
     => ( ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as )
          @ ( collec3392354462482085612at_nat
            @ ^ [X2: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X2 @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
                & ~ ( P @ X2 ) ) ) )
        = ( minus_minus_real @ one_one_real
          @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as )
            @ ( collec3392354462482085612at_nat
              @ ^ [X2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ X2 @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
                  & ( P @ X2 ) ) ) ) ) ) ) ).

% prob_neg
thf(fact_1191_random__variable__small__tails,axiom,
    ! [Alpha: real,F2: product_prod_nat_nat > real,K3: real] :
      ( ( ord_less_real @ zero_zero_real @ Alpha )
     => ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ ( freque5010624893710627907mega_1 @ as ) @ borel_5078946678739801102l_real ) )
       => ? [C4: real] :
            ( ( ord_less_real
              @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as )
                @ ( collec3392354462482085612at_nat
                  @ ^ [X2: product_prod_nat_nat] :
                      ( ( member8440522571783428010at_nat @ X2 @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
                      & ( ord_less_eq_real @ C4 @ ( abs_abs_real @ ( F2 @ X2 ) ) ) ) ) )
              @ Alpha )
            & ( ord_less_eq_real @ K3 @ C4 ) ) ) ) ).

% random_variable_small_tails
thf(fact_1192__092_060Omega_062_092_060_094sub_0622_Omeasure__space__inter,axiom,
    ! [S: set_Pr2458342521480944603at_nat,T2: set_Pr2458342521480944603at_nat] :
      ( ( member3752267775663534962at_nat @ S @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
     => ( ( member3752267775663534962at_nat @ T2 @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
       => ( ( ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ T2 )
            = ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) ) )
         => ( ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ ( inf_in8856262298555431021at_nat @ S @ T2 ) )
            = ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ S ) ) ) ) ) ).

% \<Omega>\<^sub>2.measure_space_inter
thf(fact_1193__092_060Omega_062_092_060_094sub_0622_Oindep__setD,axiom,
    ! [A2: set_se1666487788256820497at_nat,B2: set_se1666487788256820497at_nat,A: set_Pr2458342521480944603at_nat,B: set_Pr2458342521480944603at_nat] :
      ( ( indepe866335810092677197at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 @ B2 )
     => ( ( member3752267775663534962at_nat @ A @ A2 )
       => ( ( member3752267775663534962at_nat @ B @ B2 )
         => ( ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ ( inf_in8856262298555431021at_nat @ A @ B ) )
            = ( times_times_real @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A ) @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ B ) ) ) ) ) ) ).

% \<Omega>\<^sub>2.indep_setD
thf(fact_1194__092_060Omega_062_092_060_094sub_0622_Ofinite__measure__Diff_H,axiom,
    ! [A2: set_Pr2458342521480944603at_nat,B2: set_Pr2458342521480944603at_nat] :
      ( ( member3752267775663534962at_nat @ A2 @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
     => ( ( member3752267775663534962at_nat @ B2 @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
       => ( ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ ( minus_9197513832320918452at_nat @ A2 @ B2 ) )
          = ( minus_minus_real @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 ) @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ ( inf_in8856262298555431021at_nat @ A2 @ B2 ) ) ) ) ) ) ).

% \<Omega>\<^sub>2.finite_measure_Diff'
thf(fact_1195__092_060Omega_062_092_060_094sub_0622_Omeasure__exclude,axiom,
    ! [A2: set_Pr2458342521480944603at_nat,B2: set_Pr2458342521480944603at_nat] :
      ( ( member3752267775663534962at_nat @ A2 @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
     => ( ( member3752267775663534962at_nat @ B2 @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
       => ( ( ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 )
            = ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ ( sigma_4468984517727773725at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) ) )
         => ( ( ( inf_in8856262298555431021at_nat @ A2 @ B2 )
              = bot_bo2540092574283496047at_nat )
           => ( ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ B2 )
              = zero_zero_real ) ) ) ) ) ).

% \<Omega>\<^sub>2.measure_exclude
thf(fact_1196__092_060Omega_062_092_060_094sub_0622_Oindep__setI,axiom,
    ! [A2: set_se1666487788256820497at_nat,B2: set_se1666487788256820497at_nat] :
      ( ( ord_le8269821659064819057at_nat @ A2 @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
     => ( ( ord_le8269821659064819057at_nat @ B2 @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
       => ( ! [A3: set_Pr2458342521480944603at_nat,B4: set_Pr2458342521480944603at_nat] :
              ( ( member3752267775663534962at_nat @ A3 @ A2 )
             => ( ( member3752267775663534962at_nat @ B4 @ B2 )
               => ( ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ ( inf_in8856262298555431021at_nat @ A3 @ B4 ) )
                  = ( times_times_real @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A3 ) @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ B4 ) ) ) ) )
         => ( indepe866335810092677197at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 @ B2 ) ) ) ) ).

% \<Omega>\<^sub>2.indep_setI
thf(fact_1197_prob__le__1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : ( ord_less_eq_real @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ A2 ) @ one_one_real ) ).

% prob_le_1
thf(fact_1198_subprob__measure__le__1,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ( ord_less_eq_real @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ X ) @ one_one_real ) ).

% subprob_measure_le_1
thf(fact_1199_measure__increasing,axiom,
    measur4974978767913102394t_real @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) ) ).

% measure_increasing
thf(fact_1200_measure__space__inter,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
      ( ( member2643936169264416010at_nat @ S @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
     => ( ( member2643936169264416010at_nat @ T2 @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
       => ( ( ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ T2 )
            = ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) ) )
         => ( ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ ( inf_in2572325071724192079at_nat @ S @ T2 ) )
            = ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ S ) ) ) ) ) ).

% measure_space_inter
thf(fact_1201_indep__setD,axiom,
    ! [A2: set_se7855581050983116737at_nat,B2: set_se7855581050983116737at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( indepe617343343079735897at_nat @ ( freque5010624893710627907mega_1 @ as ) @ A2 @ B2 )
     => ( ( member2643936169264416010at_nat @ A @ A2 )
       => ( ( member2643936169264416010at_nat @ B @ B2 )
         => ( ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ ( inf_in2572325071724192079at_nat @ A @ B ) )
            = ( times_times_real @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ A ) @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ B ) ) ) ) ) ) ).

% indep_setD
thf(fact_1202_finite__measure__Diff_H,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( member2643936169264416010at_nat @ A2 @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
     => ( ( member2643936169264416010at_nat @ B2 @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
       => ( ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
          = ( minus_minus_real @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ A2 ) @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ) ) ) ) ).

% finite_measure_Diff'
thf(fact_1203_measure__exclude,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( member2643936169264416010at_nat @ A2 @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
     => ( ( member2643936169264416010at_nat @ B2 @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
       => ( ( ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ A2 )
            = ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ ( sigma_4797346298676585097at_nat @ ( freque5010624893710627907mega_1 @ as ) ) ) )
         => ( ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
              = bot_bo2099793752762293965at_nat )
           => ( ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ B2 )
              = zero_zero_real ) ) ) ) ) ).

% measure_exclude
thf(fact_1204_indep__sets2__eq,axiom,
    ! [A2: set_se7855581050983116737at_nat,B2: set_se7855581050983116737at_nat] :
      ( ( indepe617343343079735897at_nat @ ( freque5010624893710627907mega_1 @ as ) @ A2 @ B2 )
      = ( ( ord_le2077887516847798113at_nat @ A2 @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
        & ( ord_le2077887516847798113at_nat @ B2 @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
        & ! [X2: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X2 @ A2 )
           => ! [Y3: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ Y3 @ B2 )
               => ( ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ ( inf_in2572325071724192079at_nat @ X2 @ Y3 ) )
                  = ( times_times_real @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ X2 ) @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ Y3 ) ) ) ) ) ) ) ).

% indep_sets2_eq
thf(fact_1205_indep__setI,axiom,
    ! [A2: set_se7855581050983116737at_nat,B2: set_se7855581050983116737at_nat] :
      ( ( ord_le2077887516847798113at_nat @ A2 @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
     => ( ( ord_le2077887516847798113at_nat @ B2 @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
       => ( ! [A3: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ A3 @ A2 )
             => ( ( member2643936169264416010at_nat @ B4 @ B2 )
               => ( ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ ( inf_in2572325071724192079at_nat @ A3 @ B4 ) )
                  = ( times_times_real @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ A3 ) @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ B4 ) ) ) ) )
         => ( indepe617343343079735897at_nat @ ( freque5010624893710627907mega_1 @ as ) @ A2 @ B2 ) ) ) ) ).

% indep_setI
thf(fact_1206_ennreal__lt__0,axiom,
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ zero_zero_real )
     => ( ( extend7643940197134561352nnreal @ X5 )
        = zero_z7100319975126383169nnreal ) ) ).

% ennreal_lt_0
thf(fact_1207__092_060Omega_062_092_060_094sub_0622_Osum__measure__le__measure__inter,axiom,
    ! [A2: set_Pr2458342521480944603at_nat,B2: set_Pr2458342521480944603at_nat] :
      ( ( member3752267775663534962at_nat @ A2 @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
     => ( ( member3752267775663534962at_nat @ B2 @ ( sigma_1566336194983598982at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) ) )
       => ( ord_less_eq_real @ ( plus_plus_real @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ A2 ) @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ B2 ) ) @ ( plus_plus_real @ one_one_real @ ( sigma_8204706121814932565at_nat @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ ( inf_in8856262298555431021at_nat @ A2 @ B2 ) ) ) ) ) ) ).

% \<Omega>\<^sub>2.sum_measure_le_measure_inter
thf(fact_1208__092_060Omega_062_092_060_094sub_0622_Obound__L1__Lp_I2_J,axiom,
    ! [P2: real,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( ord_less_eq_real @ one_one_real @ P2 )
     => ( ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ borel_5078946678739801102l_real ) )
       => ( ( bochne7117808529828525605t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
            @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( powr_real @ ( abs_abs_real @ ( F2 @ X2 ) ) @ P2 ) )
         => ( ord_less_eq_real @ ( powr_real @ ( abs_abs_real @ ( bochne5509773249985062230t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F2 ) ) @ P2 )
            @ ( bochne5509773249985062230t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
              @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( powr_real @ ( abs_abs_real @ ( F2 @ X2 ) ) @ P2 ) ) ) ) ) ) ).

% \<Omega>\<^sub>2.bound_L1_Lp(2)
thf(fact_1209_sum__measure__le__measure__inter,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( member2643936169264416010at_nat @ A2 @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
     => ( ( member2643936169264416010at_nat @ B2 @ ( sigma_7545971424795947232at_nat @ ( freque5010624893710627907mega_1 @ as ) ) )
       => ( ord_less_eq_real @ ( plus_plus_real @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ A2 ) @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ B2 ) ) @ ( plus_plus_real @ one_one_real @ ( sigma_4260610412020759633at_nat @ ( freque5010624893710627907mega_1 @ as ) @ ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ) ) ) ) ).

% sum_measure_le_measure_inter
thf(fact_1210_bound__L1__Lp_I1_J,axiom,
    ! [P2: real,F2: product_prod_nat_nat > real] :
      ( ( ord_less_eq_real @ one_one_real @ P2 )
     => ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ ( freque5010624893710627907mega_1 @ as ) @ borel_5078946678739801102l_real ) )
       => ( ( bochne2596016609597520987t_real @ ( freque5010624893710627907mega_1 @ as )
            @ ^ [X2: product_prod_nat_nat] : ( powr_real @ ( abs_abs_real @ ( F2 @ X2 ) ) @ P2 ) )
         => ( bochne2596016609597520987t_real @ ( freque5010624893710627907mega_1 @ as ) @ F2 ) ) ) ) ).

% bound_L1_Lp(1)
thf(fact_1211_bound__L1__Lp_I2_J,axiom,
    ! [P2: real,F2: product_prod_nat_nat > real] :
      ( ( ord_less_eq_real @ one_one_real @ P2 )
     => ( ( member4564283293661824327t_real @ F2 @ ( sigma_1188828689629184861t_real @ ( freque5010624893710627907mega_1 @ as ) @ borel_5078946678739801102l_real ) )
       => ( ( bochne2596016609597520987t_real @ ( freque5010624893710627907mega_1 @ as )
            @ ^ [X2: product_prod_nat_nat] : ( powr_real @ ( abs_abs_real @ ( F2 @ X2 ) ) @ P2 ) )
         => ( ord_less_eq_real @ ( powr_real @ ( abs_abs_real @ ( bochne6384019433803981034t_real @ ( freque5010624893710627907mega_1 @ as ) @ F2 ) ) @ P2 )
            @ ( bochne6384019433803981034t_real @ ( freque5010624893710627907mega_1 @ as )
              @ ^ [X2: product_prod_nat_nat] : ( powr_real @ ( abs_abs_real @ ( F2 @ X2 ) ) @ P2 ) ) ) ) ) ) ).

% bound_L1_Lp(2)
thf(fact_1212__092_060Omega_062_092_060_094sub_0622_Obound__L1__Lp_I1_J,axiom,
    ! [P2: real,F2: ( product_prod_nat_nat > product_prod_nat_nat ) > real] :
      ( ( ord_less_eq_real @ one_one_real @ P2 )
     => ( ( member8159409068225774087t_real @ F2 @ ( sigma_5064276549609164707t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ borel_5078946678739801102l_real ) )
       => ( ( bochne7117808529828525605t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
            @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( powr_real @ ( abs_abs_real @ ( F2 @ X2 ) ) @ P2 ) )
         => ( bochne7117808529828525605t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as ) @ F2 ) ) ) ) ).

% \<Omega>\<^sub>2.bound_L1_Lp(1)
thf(fact_1213_add__is__0,axiom,
    ! [M3: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M3 @ N2 )
        = zero_zero_nat )
      = ( ( M3 = zero_zero_nat )
        & ( N2 = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_1214_Nat_Oadd__0__right,axiom,
    ! [M3: nat] :
      ( ( plus_plus_nat @ M3 @ zero_zero_nat )
      = M3 ) ).

% Nat.add_0_right
thf(fact_1215_nat__add__left__cancel__less,axiom,
    ! [K: nat,M3: nat,N2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_nat @ M3 @ N2 ) ) ).

% nat_add_left_cancel_less
thf(fact_1216_nat__add__left__cancel__le,axiom,
    ! [K: nat,M3: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_eq_nat @ M3 @ N2 ) ) ).

% nat_add_left_cancel_le
thf(fact_1217_diff__diff__left,axiom,
    ! [I4: nat,J2: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J2 ) @ K )
      = ( minus_minus_nat @ I4 @ ( plus_plus_nat @ J2 @ K ) ) ) ).

% diff_diff_left
thf(fact_1218_add__gr__0,axiom,
    ! [M3: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M3 @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M3 )
        | ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% add_gr_0
thf(fact_1219_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J2: nat,I4: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( minus_minus_nat @ I4 @ ( minus_minus_nat @ J2 @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I4 @ K ) @ J2 ) ) ) ).

% Nat.diff_diff_right
thf(fact_1220_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J2: nat,I4: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I4 )
        = ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I4 ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_1221_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J2: nat,I4: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( plus_plus_nat @ I4 @ ( minus_minus_nat @ J2 @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I4 @ J2 ) @ K ) ) ) ).

% Nat.add_diff_assoc

% Helper facts (3)
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X5: real,Y4: real] :
      ( ( if_real @ $false @ X5 @ Y4 )
      = Y4 ) ).

thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X5: real,Y4: real] :
      ( ( if_real @ $true @ X5 @ Y4 )
      = X5 ) ).

% Conjectures (3)
thf(conj_0,hypothesis,
    ord_less_nat @ i_1 @ ( frequency_Moment_s_1 @ delta @ n @ k ) ).

thf(conj_1,hypothesis,
    ord_less_nat @ i_2 @ ( frequency_Moment_s_2 @ epsilon ) ).

thf(conj_2,conjecture,
    ( ( bochne5509773249985062230t_real @ ( freque5010624893710627908mega_2 @ epsilon @ delta @ n @ k @ as )
      @ ^ [X2: product_prod_nat_nat > product_prod_nat_nat] : ( freque135395702774845718t_real @ k @ as @ ( X2 @ ( product_Pair_nat_nat @ i_1 @ i_2 ) ) ) )
    = ( field_7254667332652039916t_real @ ( frequency_F_nat @ k @ as ) ) ) ).

%------------------------------------------------------------------------------